- Email: [email protected]
III Principles and Design of Optical Arrays
E. WOLF, PROGRESS IN OPTICS XXV
0 ELSEVIER SCIENCE PUBLISHERS B.V., 1988
I11 PRINCIPLES AND DESXCN OF OPTICAL ARRAYS BY
WANG SHAOMIN Hangzhou University Hangzhou, China
L. RONCHI Istiiuto di Ricerca sulle Onde Elettromagnetiche del CNR Florence, Italy
CONTENTS $ 1 . INTRODUCTION
....................
PAGE
281
$ 2. THE 2 x 2 MATRIX TREATMENT OF AN ALIGNED OPTI284 CAL ELEMENT . . . . . . . . . . . . . . . . . . . . .
$ 3 . MATRIX TREATMENT FOR MISALIGNED ELEMENTS
. 305 Q 4. MATRIX TREATMENT OF ARRAYS . . . . . . . . . . . 311 Q 5. PSEUDO CONJUGATOR ARRAYS . . . . . . . . . . . 323 $ 6. ARRAYS WITH VANISHING DETERMINANT . . . . . . 341 $ 7 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . 346 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 347 347 REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
# 1. Introduction Arrays of optical elements can be found in nature, such as the compound eyes of insects, and in every day life, such as the comer-cube arrays on cars and bicycles or on road signs. In recent years optical arrays have attracted new interest, since they have been found not only to give rise to some unusual phenomena but also to have many potential applications in present and future industry and in scientific research. The most unusual new phenomena associated with arrays are the nonGaussian imaging property of GRIN (gradient index) rod arrays (REES[ 19821) and the pseudo phase conjugation phenomenon of comer-cube arrays (JACOBS [ 19821). A brief description of these phenomena follows, starting with the image formation property. First, it should be pointed out that the image given by an array is a synthesized image in the sense that it is formed by the superposition of the beams emerging from the various elements of the array. It is the best focus of the beam emerging from the array. As such, it depends not only on the optical properties of the elements of the array but also on their mutual positions, namely, on the shape or curvature of the array's input surface X and output surface Z'.Thus, in general, the image of a source given by an array does not coincide with the images given by the single elements. As is well known, a GRIN rod consists of a medium whose refractive index is a parabolic function of a cylindrical coordinate r = no(1
- @or2)
(1.1)
9
where no clearly is the value of n on the axis, r = 0. It is also well known (MARCUSE[ 19721) that the meridional rays in such a medium are sinusoids independentlyof their initial height and slope with half-periodicityp = n/&, (at least in the paraxial approximation). Consider then (REES [ 19821) an array constituted by a number of GRIN rods of the same length 1 and with the axes parallel to each other (fig. 1). If we choose I = p, any ray entering a rod at a distance r from its axis and with slope I' will arrive at the exit surface with opposite height and slope. Now assume a point source S, on the axis of the axial rod of the array (labeled 1 in fig. 1) and look for its image given by the 28 1
282
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
Fig. 1. Non-Gaussian imaging in of a GRIN rod array: The distance O,S, of the image from X’ is equal to the distance O,S, of the source from X, for any O,S,.
array. To this end, let us consider a single ray per rod - the ray impinging at the axial point of the rod: It will arrive at the axial point of the exit surface with opposite slope. Hence the emerging ray will cross the axis of the array at a point S2,such that I02S,I = IS,O,l for any S,O,. This means that, at least for a GRIN rod array of the type shown in fig. 1, the relation between conjugate planes is not the usual Gaussian law. Experimental tests of such a property have been carried out by KAWAZUand OGURA [ 19801. In considering pseudo-phase conjugation, there are a number of experiments which prove that corner-cube arrays may be used to approximate phase conKALININ jugation. For example, ORLOV, VIRNIK,VOROTILIN, GERASIMOV, and SAGALOVICH [ 19781 reported dynamic compensation of optical inhomogeneities in an optically pumped neodymium/glass rod by means of a retroreflecting array (fig. 2). When compared with the use of a plane mirror at the position of the array, the retroreflecting array dramatically reduces the beam divergence due to thermal distortion of the rod. On the other hand, BARRETTand JACOBS [ 19791 proved that the wavefront distortion caused by an inhomogeneous medium in an imaging system can be compensated if a corner-cube array, or a small-period retroarray, is used as shown in fig. 3. Indeed, MATHIEUand BELANGER[ 19801 described the use of small-period retroarrays as mirrors for laser resonators to compensate for distorting ele-
111, § 11
1NTRODUCTtON
283
Fig. 3. Point source imaging: (a) without any phase perturber; (b) with a phase perturber.
ments inside the laser. Two kinds of lasers were considered. One of them was aTEA (Transverse Excitation, Atmospheric Pressure) C0,laser ( A = 10.6 pm) with a resonator that included a 36% reflecting flat mirror, 1 m away from the retroarray. The burn spot of the retroarray could be tilted by as much as 25 degrees before laser action ceased. Various transparent phase perturbers could be inserted up to 15 cm from the retroarray without stopping laser action. The other laser resonator was a pulsed dye laser (rhodamine 6G), with a 2 cm2 lasing aperture in the same resonator as the TEA laser. The burn spot of this laser, operating in the visible spectrum, was even better than that of the infrared laser. In both cases the retroarrays were common plastic retroreflectors like those used on bicycles and highway signs. The periodicity of the arrays was about 2.5 mm. The significance of these applications of corner-cube arrays, and of some other arrays, is that no other combination of optical elements has yet been discovered that performs conjugation (or reversal) of a wavefront of arbitrary shape and aberration. On the other hand, there is no proof that such conjugators are impossible (O’MEARA[ 19821). Thus, corner-cube arrays have been called “pseudo phase conjugators”. Present industrial applications of arrays include a major use of GRIN rod arrays (fig. 1) in photocopying machines as compact unit magnification devices [ 19801). (MATSUSHITA and TOYAMA From the foregoing considerations it follows that arrays form a new area in optics: It is not trivial to analyze their behavior in terms of classical optics just because they do not follow the Gaussian law of conjugated points. Fortunately, however, the non-Gaussian imaging process may be easily treated (WANG SHAOMIN [ 19851)by recasting the classical matrix-optics methods (BROUWER, ONEILand WALTHER [ 19631, GERRARD and BURCH [ 19751) or by extending the ABCD matrix introduced by KOGELNIK [ 19651 (see also KOGELNIK and Lr [1966]) to describe Gaussian beam propagation and optical resonator properties.
284
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
[III, 0 2
In J 2 of this chapter we recall the essential features of the 2 x 2 ray transfer matrix method. In J 3 we derive the augmented 4 x 4 matrix introduced to describe the optical behavior of a misaligned optical element: This section is the basis of treating arrays, where the arrangement of the individual optical elements can be regarded as a “regular” misalignment. In J 4 we show how, because of the regularity of the misalignment of the elements forming the array, an array may be treated in terms of a 2 x 2 matrix (WANG SHAOMIN[ 1985]), which accounts for imaging properties, phase conjugation properties, and some other new properties of arrays. In 8 5 and 5 6 some examples are given of arrays that possess interesting properties. The problem of the quality of synthesized images is treated also by analyzing the so-called additional aberrations of the arrays, and interference fringe formation, with particular regard to the conditions under which they can be eliminated.
0 2. The 2 x 2 Matrix Treatment of an Aligned Optical Element 2.1. SIGN CONVENTIONS
Consider a centered optical element, namely an element with symmetry of revolution around an axis, and two reference planes, RP, and RP,, both normal to the symmetry axis (fig. 4). The symmetry axis will be taken as the z-axis, directed in the sense of propagation of radiation, conventionally from left to right. Such an element will be termed “aligned”. A ray impinging onto the optical system is completely identified by the position q of the point where it crosses RP, and by its “momentum” p (see, for example, DRAGT[ 19821). If RP, is referred to a Cartesian system of coordinate axes i , j , with origin 0, on the z-axis, one has
where x’ = dR/dZ, y’
=
djId2.
Fig. 4. Sign conventions, in the forward propagation,for the parameters of a ray at two reference planes and for the coordinates u and u of the axial points S, and S,.
111, B 21
2X 2
MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
285
For rays in a meridional plane, which are of interest for the paraxial approximation when aberrations are disregarded, one can use r in the place of x or y , and r’ in the place of x’ or y’. Figure 4 shows the sign convention adopted in this chapter for r and r’. From this .sign convention follows the sign convention for the distances of the axial points from the reference planes. If, as just stated, r; and r; in fig. 4 are positive, the distance u of S, from RP, and the distance v of S, from RP, are to be counted as positive, since we can write
r;
rl =, U
r ; = -r2. v
(2.2)
Accordingly, the points have positive distance from the reference plane if they are on the left of it; more precisely, if one proceeds along the z-axis in the sense of radiation, one fist encounters that point and then the reference plane. Thus the distance v from RP, of any point of the z-axis in the image space on the right of RP, is negative. When the optical system contains a reflector, the sign convention for r‘ changes, as shown in fig. 5. Correspondingly,even the sign of u changes so that points on the left of the reflector have a negative axial coordinate. It is useful to introduce the sign matrix S,defined as
s=(’0
- 10).
Some examples of the use of the matrix S are given in $2.2 and 5 2.3. The sign of the radius of curvature of a spherical surface will be considered positive if, proceedingin the positive direction of the z-axis,one first encounters the surface and then the center of curvature. Hence, for example, the radius R of the spherical interface of fig. 6b (5 2.2) will be considered as positive.
Fig. 5. Sign convention for r‘ in the case of a reflector.
286
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
2.2 DEFINITION OF THE RAY TRANSFER MATRIX
If a paraxial ray is passing through a linear optical system, there are linear relations between the parameters r, ,r; of the outgoing ray at the reference plane RP,, and those r , , r; of the incoming ray at RP,: r,
=
at,
+ br; ,
ri
=
cr,
+ dr; ,
(2.4)
where the coefficients a, b, c, and d depend on the features of the optical system included between RP,and RP2 and also on the positions of RP,and RP, with respect to the optical element. Equations (2.4) may be written in a matrix form
r,
=
yr,
9
where
The arrow under the matrix symbol indicates that it describes the propagation from left to right. The matrix y is called the “ray transfer matrix” and is analogous to the ABCD matrix introduced by KOGELNIK[ 19651 (see also ARNAUD[ 19761). It may be verified that
n, Det(rvl) = ad - bc = -,
(2.7)
n2
where (fig. 4) n, denotes the (constant) refractive index on the left of RP, (namely, that of the medium containing the incident rays) and n2 denotes the refractive index on the right of RP, . If r’ indicates a Cartesian component of the momentum p, instead of x’ or y’, the corresponding matrix h J would have unit determinant, or, in other words, it would be unimodular (GERRARD and BURCH [1975]). The matrix is unimodular in most common cases when n, = n2. Evaluating the determinant in any particular application and verifying that (2.7) is satisfied provide a useful check of the calculations. The matrix formalism is particularly useful when the optical system is formed by several elements linked up axially. If y(‘)indicates the ray transfer J of the whole matrix of the ith element starting from the left, the matrix N system is given by
111, P 21
2X 2
MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
287
[Note the order of the indices in eq. (2.8).] One has It can be noted that eq. (2.5) may be solved for rl in terms of r,, yielding
2, = y-'Jzy
(2.9)
where arrows have been added to emphasize the sense of propagation. For reciprocity of the optical path equation (2.9) may be used to study the ray paths in reverse propagation, by taking into account that, due to the sign conventions, the rays in reverse propagation have slopes - r; and - r; . More precisely, we have
c,=
s,'2,
57-2= SJ,
From eq. (2.9) we derive SJ, =
sy-'s-'sJ*
or also
(2.10) When Det(y) = 1, p reduces to the well-known expression (FOG [ 1982]), where only the main diagonal elements of matrix 5 are interchanged. Equation (2.10) is a generalized expression for the ray transfer matrix in reverse [ 1983b1) and is particularly useful in treating propagation (WANG SHAOMIN optical systems containing reflectors. 2.2 BASIC METHODS FOR DERIVING RAY TRANSFER MATRICES
There are three basic methods to derive the ray transfer matrix associated with an optical element. In the present section we start choosing RP, and RP2 to coincide with the first and the last surfaces, respectively, of the optical element. (1) One method is based on eqs. (2.4), according to which:
288
[III, 8 2
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
1-1
b-0
a*-1
b.0
C
=2/R
c
.o
dsl
d=-1
Fig. 6. Determination of the elements a, b, c, and d of the ray transfer matrices of the simplest optical elements:(a) plane-parallel layer of uniform medium; (b) spherical interface; (c) spherical mirror; (d) comer-cube reflector with reference planes at its vertex.
Equations (2.11) may be used to derive the ray transfer matrices of a plane parallel layer of thickness 1 in a uniform medium (fig. 6a) and the matrix of a spherical interface of radius R (fig. 6b refers to refraction; fig. 6c refers to reflection). It appears clearly from the figures that the ray transfer matrix 5 , = ,(1) for the Iayer is
91(1)=
(A :>
whereas the matrix
plane pardel layer
L)
Y 2for the spherical interface is
92(R n,,n 2 ) = ( I n2R
(n1 - n2) n1
refraction,
(2.12)
I K I 21
2 X 2 MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
reflection,
289
(2.13)
where the label I refers to reflection. It turns out that In other words the matrix valid in reflection may be obtained by starting with the matrix valid in refraction, replacing n2 by - n, and multiplying it on the left by the sign matrix S. Recall that, in figs. 6b and c, R must be counted as positive, according to the sign convention for radii of curvature. Among the optical elements the corner cube reflector is of particular interest. For it one finds
(
y lr = -
0
O) ,
-1
comer-cube reflector,
(2.14)
when RP, and RP, are chosen as shown in fig, 6d. (2) Some ray transfer matrices may be derived starting with hill and For example, by taking the limit R --t co in eqs. (2.13), one obtains the plane for refraction and reflection respecinterface matrices y,(n,,n,) and tively:
z2.
plane mirror.
(2.15)
Other matrices may be obtained by using known matrices, hence NJ 9, and y3,in connection with eq. (2.8). For example, the matrix bJ4(l, n,,n,) associated with a plane parallel slab of thickness 1 and refractive index n,, bordered on both sides by a medium with refractive index n,,may be obtained --t = y3(n2,nl). Hence, by putting $') = y 3 ( n , , n,),, Y(') = M,,(l) and MC3)
(2.16)
290
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, B 2
Again, the ray transfer matrix of a thick lens, limited by two spherical surfaces of radius R and R2, respectively, of thickness I, and of refractive index n (outer refractive index = 1 on both sides), may be obtained by combining three factors in eq. (2.8), namely, = h+l,(Rl, 1, n), kJ(*) = h$(I), 4 M(3)= N12(R2, n, 1). Hence, (2.17) with al=l--,
(n - 1)l
1
bl = - , n
nR 1
cl=(n-l)
1
1
1
(n - 1)l
R2
R,
nR1R2
,
(n - 1)l dl=l+---. nR2
(2.18)
The thin-lens matrix NJ6 may be obtained either by taking the limit I + 0 y5,or by putting only two factors in eq. (2.8), namely, 4 M(') = Y2(R1, 1, n) and kJ(2) = Y2(R2, n, 1). Hence,
in
(2.19) where (2.20) f denoting the usual focal length. Thin-lens matrices may be applied to find the telescope matrix kl, corresponding to fig. 7a or the matrix rv18 corresponding to the general telescopic
a)
b)
Fig. 7. (a) Telescopic scheme; (b) generalized telescopic scheme.
111.8 21
2 X 2 MATRIX TREATMENT OF
f2
5 7 = r
: 1.
--( A 6,
fi + f 2
-fi
+
1
3
29 1
ALIGNED OPTICAL ELEMENT
98=[
f2
fi+ fi + 6
-~ (f1+S)
fIf2
f2
(2.21)
The preceding matrices are collected in Tables 1 and 2. Recall that the ray transfer matrix of an optical system depends not only on the intrinsic properties of the system but also on the reference planes. Up to now the reference planes have been chosen at the input and output surfaces of the optical element. But we can apply the so-called reference plane moving technique, which is a simple application of eq. (2.8) to get different matrix forms or to find some new matrices. The technique is illustrated in fig. 8: The ray transfer matrix, relative to the new reference planes RP; and RP; ,can be given, according to eq. (2.8), by
9' = yl(b2)y y 1 ( 0 where b,, b, > 0. Hence,
(:
3
Fig. 8. Moved reference planes RP;and RP;.
TABLE1 Some 2 x 2 ray transfer matrices ,M (forward propagation). System
Schematic
Matrix
Expression(s) for matrix elements
RPd
Spherical interface
Plane-parallel slab 0 1
N W N
~~
Thin lens
w X h)
~
Telescope I
~~
Telescope 11
Lenslike medium
293
h)
W W
294
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
TABLE 2 Some 2 x 2 ray transfer matrices M (reflection).
System
Schematic
Matrix
Comer cube reflector I
Spherical reflector
Flat reflector
Comer cube reflector I1
Cat's eye reflector
=(
a
+ cb,
ab,
+ b + b,(cb, + d ) cb,
c
RP;GRPi
a)
(2.22)
+d
RPlIRPz b)
Fig, 9. (a) Comer-cube reflector with reference planes at its mouth; (b) cat's eye reflector.
III,8 21
2
2 MATRIX
X
TREATMENT OF ALIGNED OPTICAL ELEMENT
295
As an example of application of eq. (2.22), we can derive the matrix M ir of the comer cube reflector with reference planes at its mouth, as indicated + in fig. 9a, starting with matrix Ylr given by eq. (2.14), and noting that b , = b, = I :
If then we consider a possible refraction at the new reference planes and apply eq. (2.8), we obtain
Another example may be the cat’s eye (Fig. 9b), with reference planes at the anterior focal plane of the lens. Its matrix ,Ms,may be calculated as !$~r =
y2(f)
!$j(f)
!$,(f)y3r yl
and is given by (2.23)
In 3 5 the preceding technique will be applied to a particular problem: Given an optical element whose diagonal elements of the ray transfer matrix M, are equal, a = d, to find the moved reference planes RP;, RP;, with b, = b2 = uo, such that B = 0 and D = 1. (3) In some cases, when the optical system consists of a rod of inhomogeneous medium, ray transfer matrices may be obtained by solving the eikonal differential equation (BORN and WOLF[ 19801):
i(n2)
=
gradn,
where s denotes the arc of the ray trajectory measured from the origin. For example, in the case of a GRIN rod, when the radial distribution of the refractive index is given by eq. (l.l), the eikonal equation in the paraxial approximation (z s) becomes
-
d2r + 2/3,r ds2
-
=
0
(2.24)
296
PRlNCIPLES AND DESIGN OF OPTICAL ARRAYS
[IK 8 2
The solution of eq. (2.24) can be written in the form
(2.25) where r, denotes the value of r at the input plane z = 0 and r; denotes the slope of the ray at z = O + . Differentiating eq. (2.25) with respect to z yields rl =
- -A rl sin (f z ) + r; cos ('z)
.
P
Hence, by putting r2 = r(l), r; = r'(l), where I denotes the length of the rod, we obtain r2 = cos (al>r, r;
=
--sin PA
+
sin ($1) r; ,
(3 (3 -I
rl
+ cos
-I
r; .
The ray transfer matrix k19of the GRIN rod therefore may be written in the form
with U,
= cos (
l a ) ,
bo =
WJm J2Bo
CO
=
-2/3obo,
do = u O . (2.26)
Note that matrix lU19 describes the optical behavior of a GRIN rod bordered on both sides by a medium with refractive index no, since it does not account for any refraction at the input and output surfaces. Such refraction can be taken into account by applying eq. (2.8),with results depending on the shape of the input and output surfaces. For example, if the rod is limited by plane interfaces, one has
I K 8 21
2 X 2 MATRIX
291
TREATMENT OF ALIGNED OPTICAL ELEMENT
fore,
(2.27)
Note that if
fiO -= 0, the circular functions become hyperbolic functions.
2.4. IMAGE FORMING MATRIX
The "moved" matrix ,M' defined by eq. (2.22) has a very important application to the study of the image formation properties of an optical system. With reference to fig. 10, let
Y=
(; );
(2.28)
denote the ray transfer matrix of the image-formingblock, from its first surface RP, to its last surface RP2, including these two interfaces, and MI=(A
+
B
)
(2.29)
C D
the ray transfer matrix from RP; to RP;. The expressions for the matrix elements A, B, C, D of Y ' are given in eqs. (2.22) in terms of the elements RP; I I I I I I I I I
I
I
RP; I I I
I I I I
I I
I I
I
I
1 I
I I
I I
I I I I
-Z
I I
Fig. 10. Definition of symbols for the determinationof the elements A , B, C, and D of the matrix relative to two "moved" reference planes.
298
[III, 5 2
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
6 2 = - z , so that u and z denote, in value and sign, the coordinate of RP; with respect to RP, and the coordinate of RP; with respect to RP,, respectively. From eq. (2.22) we have
a, b, c, d and of 6 , and b, as well. Let us put 6 , = u and
A=a-cz,
B=au+b-z(cu+d),
C=c,
D=cu+d. (2.30)
It is useful for further applications to recall the meaning of each element of
M ’. Let us start with B. -+
From the fist eq. (2.4) applied to the pair RP;, RP;, we write r,
= Ar,
+ Br; .
(2.3 1)
It is evident that r2 is independent of r; if B = 0. In other words, if B = 0, all rays crossing RP; at r , will cross RP; at the same point r,, independently of their initial slope r; . Hence B=O
(2.32)
defines the plane RP; ,which contains the images of the points of RP; . In other words, if u is the value of z such that eq. (2.32) is satisfied, RP; is the plane image of RP;. From the second eq. (2.30) we find (2.33)
When B = 0, the matrix V I J’ is called the “image-forming matrix” or “imaging matrix“. Equation (2.33)is the generalized equation for conjugated points and is often referred to as the ABCD law for image formation. It should be noted that, in general, eq. (2.33) does not coincide with the usual Gaussian law of conjugated points, since u and u denote the distances from the fist and last surfaces of the imaging block, not from its principal planes. However, the ABCD law and the Gaussian law coincide when RP, and RP, coincide with the principal planes as, for example, in the case of thin lenses. When z # u, B (which is different from 0) is a length that is called the “effective thickness” of the optical system. From eq. (2.31) it follows also that i f B = 0, then A = r&,, independently of r; .Hence, in the image-forming matrices the first diagonal element represents the transverse magnification m,, relative to the pair of conjugated planes RP; and RP;:
mo 21
2 X 2 MATRIX
TREATMENT OF ALIGNED OPTICAL ELEMENT
(2.34)
B=O.
A=m2,
299
On account of eqs. (2.30), (2.33), and (2.7), we have
- bc cu + d
ad m 2 = a - c v = -=
n,/n2 cu + d '
-
(2.35)
Analogously, we h d that the second diagonal element D represents, by definition,the angular magnification m1of the optical system. In fact, ifwe write the second eq. (2.4) in the form appropriate to the pair RP;,RP;, namely, r;
=
Cr, t Dr;
and choose ri
=
(2.36)
0, we see that
D=';
r; Hence, m , = D = cu t d .
(2.37)
It follows, in particular, that n1 , mlm2 = (cu + d ) (a - c v ) = n2
which is equivalent to the well-known Lagrange formula. Consider finally C. From eq. (2.36) it follows that, for r; = 0, namely for incident rays parallel to the axis,
(2.38) Hence, C is related to the optical power of the optical system. For example, in the case of a positive thin lens (fig. 11) one has r,/r; = -f,f denoting the R P,Z
rl
R P2
-. 1 . . ~
1 I
R Pk
I
I
I
I r;S-r,
/f
I I I
I
Fig. 1 1 . A thin lens with reference plane RP; at the rear focus.
300
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 2
focal length, and therefore C = - 1M'. In the sequel it will be shown that C or c are in any case equal to the inverse of the rear focal length fi, with the sign changed. In conclusion, the imaging matrix Y ' can be written in the form (2.39)
As is well known (GERRARD and BURCH [ 1975]), with the help of the matrix
Y ' of eq. (2.29) it is possible to express all quantities of interest for describing
the paraxial behavior of the optical system in terms of the elements of the matrix drJ of eq. (2.28). For example, one can derive the coordinate u = s, of the anterior or front focal plane with respect to the fist surface of the optical system, namely, with respect to RP, ,by noting that for such a plane D = 0. In fact, when D = 0, eq. (2.36) yields r;
=
Cr, ,
which indicates that all rays crossing RP; at the same height I , emerge from RP; parallel to each other, since their slope r; does not depend on r; . Hence, u=s1=
d
(2.40)
- - *
C
Analogously, the rear focal plane may be found by requiring that a set of parallel rays incident onto RP; at their emergence pass through the same point r2 of RP; . This requires that r2 be independent of r , , hence A = 0. From the first eq. (2.30) it is seen that the rear focal plane has coordinate z = - s2 with s2 given by s2=--.
a
(2.41)
C
The principal points may be found by looking for that pair of conjugated planes for which the linear magnification is equal to 1, hence, by imposingB = 0 andA = 1. Their coordinatesfor RP, and RP2, respectively, are given by u = h , and z = - h2 with 1
- z = - (1 - a ) , C
principal planes. (2.42)
111, B 21
2 X 2 MATRIX
301
TREATMENT OF ALIGNED OPTICAL ELEMENT
Accordingly, the front and rear focal lengths f, and f2 are given by fI
- S 1
n 1 - h I -- - A n2 c
f -3
-h - - _ 1 , focal lengths. (2.43) C
From the second eq. (2.43)it appears that c, and also C in virtue of the third eq. (2.30), are in any case equal to - l / f 2 , as anticipated. The nodal planes are, by definition,that pair of conjugated planes for which the angular magnification is equal to 1 . Hence, by imposing B = 0, D = 1, we find that their coordinates are u = hi and z = - h i , with hi and h; given by
1 h;=u=-(l-d), C
h2'-- - z = -
2:(
-1-a
C
),
nodal planes.
2.5. APPLICATION OF MATRIX METHODS TO NON-GAUSSIAN IMAGING
Equation (2.33) has been derived with reference to point sources or to straight rays, but it may be used also in the case of non-Gaussian imaging, namely, to describe the propagation of a Gaussian beam through an optical instrument. This application derives from the fact that in eq. (2.33) u and u may be considered both as (axial) coordinates of conjugate points, and as radii of curvature of the incident and emerging wavefronts at the first and last surfaces RP, and RP, of the optical block, respectively (fig. 12). In other words we can say that eq. (2.33) holds not only when u specifies the center of an impinging spherical wavefront and u specifies that of the correspondingly emerging sphericalwavefront, but also when u and u represent the curvature radii p , and p2 of those two wavefronts. This may be directly proved by noting that p , = r,/ri and p2 = r2&, so that, by using eq. (2.4), we can write
r2 p2
ar, + br; - ar,/r; + b =--ap, + b = cr, + dr; cr,/r; + d c p , + d '
Fig. 12. Transformation of a spherical wave by an optical system.
(2.45)
302
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[I14 5 2
If, now, we recall that a Gaussian beam may be considered as a spherical wave with complex radius of curvature q = q(z), which follows the Kogelnik transformation (or ABCD law) in its propagation through an optical system (KOGELNIK [ 19651, ARNAUD[ 1976]), we see that eq. (2.45) holds also with complex p, and p2 and therefore eq. (2.33) holds with complex u and u. Another generalization of the matrix method is that the elements of a ray transfer matrix may take complex values in order to describe situations where there is a transverse distribution of loss or gain, in particular the effects of a mirror with a Gaussian profile of its reflectivity (YARIV[ 19761). Such generalized matrices are referred to as “equivalent” transfer matrices. Other--equivalent transfer matrices have been introduced for particular purposes. For example, sometimes one needs to know what is the physical meaning of an optical element known only through its ray transfer matrix, such as
Since an object at u is imaged at a position u given by (see eq. 2.33), v=-
+ b, c,u + d, a,u
which can also be written as
u(s t) U
+ 1’
-
then a transfer matrix a2b2c2d2given by (WANGSHAOMIN[ 19841) + Me
with
=(:: 2>=L2
(2.46)
+ d, a,u + b,
(2.47)
1
C]U
c2=---
1 u
0 1)
is equivalent to Y in the sense that an optical element described by Ye images an object at u at the same position as the element described by matrix M. This simple consideration indicates that any optical element may be +
2
111, § 21
X
2
MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
303
regarded as a thin lens with focal length f = - lie2, which depends on the position of the object. This result will be usefully applied in 5. It should be noted that the equivalent transfer matrices may be applied to beams, not to rays. 2.6. THE EIKONAL FUNCTION AND THE FRESNEL NUMBER OF A
CENTERED SYSTEM
The eikonal function of an optical system, which, as is well known, expresses every possible optical path from the input plane RP;to the output (or observation) plane RP; of the system, is useful to analyze both the diffraction effects qualitatively and the interference phenomena quantitatively. In this section we derive an expression for it, holding for a general centered optical system, in terms of the elements of the ray transfer matrix of the system. To this end it should be noted that the eikonal function in a centered system may be written as (COLLINS[ 19701)
(2.48) Here, Lo is the optical path length from RP; to RP; along the axis, rl and r, are the usual radial coordinates at RP; and RP;, and $, n and Y denote coefficients that depend on the properties of the optical system only. Since the momentum p of a ray is the radial gradient of the eikonal (ARNAUD [ 1976]), it follows from eqs. (2.1) that the slope of a ray in a medium can be evaluated as the radial gradient of the eikonal divided by the refractive index of the medium. For the sake of simplicity, let the refractive index be equal to 1 on both sides of the optical system. On account of the sign convention for ray slopes, we obtain
or also
r,
=
$ -r,
n
1 + -r;
n
,
r;
=
(y
- O ) r , + oYr ; .
(2.49)
By comparing eqs. (2.49) with (2.4), we find the relations between the coefficients of the eikonal function and the transfer matrix elements, namely, A B
$=-,
a = -1 , B
D Y=-. B
(2.50)
304
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
[IK5 2
Hence, the expression for the eikonal function L in terms of the ray transfer matrix elements is 1 L = Lo + -(Ar: - 2r1r2+ O r : ) . 2B
(2.5 1)
A fist application of eq. (2.51) is the evaluation of the Fresnel number of the input pupil of the optical system. It is well known that the Fresnel number of a circular aperture of radius a, illuminated by a plane wave, and seen at an observation point on the axis of the aperture at a distance 1, is given by
(2.52) where 3, denotes the wavelength. If a spherical wave with wavefront curvature l/p impinges onto the aperture, the Fresnel number becomes (CARTER [ 19821) (2.53) However, eqs. (2.52) and (2.53) hold only in the particular case,when a homogeneous space follows the aperture. In a more general case a series of optical elements may be present between the aperture and the observation point. In this case it is complicated to evaluate N by means of conventional optics. It becomes easy if use is made of eq. (2.51) by taking into account that the Fresnel number is given by the optical path difference between the wavelets arriving at the observation point from the edge of the aperture (r, = a) and from its center (rI = 0), divided by ;A. Since the observation point is on the axis of the instrument, one has to put in eq. (2.5 1) r, = 0. Hence, the Fresnel number due to the optical system included between RP; and RP; is simply given by
(2.54) In the case of an input phase difference caused by an incident spherical wave of curvature radius p, the complete form of the Fresnel number in terms of the transfer matrix elements is (FANDIANYUAN [ 19831) (2.55)
111, B 31
MATRIX TREATMENT FOR MISALIGNED ELEMENTS
305
It is easily verified that for a plane wave ( p = a)incident on the aperture and followed by free space [A = 1, B = I as given by eq. (2.12)], expression (2.55) reduces to eq. (2.52).
6 3. Matrix Treatment for Misaligned Elements 3.1. THE 4 x 4 RAY TRANSFER MATRIX
The 2 x 2 ray transfer matrices of the previous section hold for centered systems only. In practice, however, centered systems do not exist. An optical system has always some element slightly decentered or misaligned because of tolerances of manufacturing, mechanical and thermal instabilities, and other reasons. To treat paraxial ray propagation through a misaligned optical ele[ 19793). ment, one can have recourse to 4 x 4 matrices (WANG SHAOMIN Let us start with a misaligned optical element used in forward propagation. Let the element, specified by the matrix
Y
=(:
1)’
be included between the two planes RP; , RP; (fig. 13), with a symmetry axis z’ that makes the angle E’ with the reference z-axis. Assume ,M to include the effect of possible refractions at RP; and RP;. Let RP, and RP, be the two planes normal to the z-axis that are as close as possible to RP; and RP;, respectively. The distance I from RP, to RP, is h o s t equal to the distance from RP; to RP; . Finally, let us denote by E the distance from the z-axis to the point where the z’-axis crosses RP, (or RP;). In other words, E and E‘ are the “ray” parameters of the z‘-axis.
RS
RR2
Fig. 13. Misalignment diagram for a system in forward propagation.
306
PRINCIPLES AND DESlGN OF OPTICAL ARRAYS
[III, B 3
A paraxial ray w ill be specified by the parameters rl, r; at the input plane RP, and by r,, r; at the exit plane RP, .To establish the (linear) relations among r,, r; and rl, r ; , we may introduce the ray parameters TI, 7; at RP;, and T,, 7; at RP;. Clearly, we have
7, = a?,
+ b7; ,
7;
= c7,
+ d7; .
(3.2)
On the other hand, 7, and 7; are simply related to rl and r; (in the paraxial approximation) by 7; x r;
7, x rl - E ,
- E‘ ,
(3.3)
and r,, r; are related to 7,, 7; by
+ E + iEJ ,
r, x F,
r; x 7;+ 8’
.
(3.4)
By combining eqs. (3.2) to (3.4) and writing them in matrix form, we obtain
where u, 8, y, and 6 are called “misalignment matrix” elements, and they are given by a=l-a,
p=l-b,
6=1-d.
y=-c,
(3.6)
To suit the needs of matrix multiplication, eq. (3.5) can be written in the following form: r$4) = M(4)r(4), 4
(3.7)
1
where a b
CIE
BE’ (3.8)
0 0 0
1
This is a general form of 4 x 4 (or “augmented”) matrix that is very useful in treating misaligned optical elements or systems. In the case of reflectors one simply has to put
6 = - 1- d ,
for reflectors,
(3.9)
because in eqs. (3.2) and (3.4) the sign of both and r; must be changed according to the sign conventions recalled in 8 2.1.
307
MATRIX TREATMENT FOR MISALIGNED ELEMENTS
Fig. 14. Case of a system where the choice of the misaligned axis z' is arbitrary.
The augmented matrices as derived here only hold for rays and not for beams (WANG SHAOMINand WEBER[ 19821). It is worth noting that when the z'-axis of a misaligned optical element is not well defined, as occurs with spherical interfaces or mirrors, the misalignment parameters E and E' are not uniquely determined but can be arbitrarily chosen, at least within certain limits. For example, in the case of fig. 14, where R is positive, one can choose either E' = 0 and E = E ~ or, E = 0 and E' = E J R ,or other values of E and E', provided that E
-
R
+ E'
=
El const. = R
(3.10)
Depending on the choice of E and E', different augmented matrices are found to describe the same case; however, they yield the same results when applied to practical problems. A particular case of interest is when the misaligned optical element is a plane-parallel layer of uniform medium. In this case [see eq. (2.12)] a = 1, b = 1, c = 0,d = 1, and therefore a = 0, /3 = 0, y = 0 , b = 0, and the augmented matrix MI4)(l)becomes /1
1 0
o\ (3.11)
\o 0 0
I /
The above matrix allows us to write the ray transfer matrix y'(4) relating the ray parameters at two planes u and z by applying eq. (2.8): Hence, (3.12)
\ o o o
1
I
308
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 3
where A, B, C, and D are still given by eqs. (2.30) and
A=fl-iiz,
O+=CX-YZ,
yT=y,
&=a.
(3.13)
The quantities O+ , A , yT, and & are referred to as “misalignment matrix elements for the total optical system”. Constructing the 4 x 4 matrix of an optical element is simple if use is made of eqs. (3.6) and (3.9). For example, for a spherical mirror of radius R one has from the second eq. (2.13) a = 1, b = 0, c = 2/R, d = 1 (and 1 = 0), and therefore, ci = 0, /3 = 0, y = - 2/R, and 6 = - 2. Hence, Nl:4)(R, E, E ’ )
=
/I 0 2/R 1 0 0 \o 0
0 o \ - 2 ~ l R -28’ 1 0 0 1 /
(3.14)
It follows directly from expression eq. (3.8) for Yc4)that an aligned system specified by the 2 x 2 matrix 9 of eq. (3.1) may be described also in terms of the augmented matrix /a b
0 O \
(3.15)
\o
0 0 I /
since, for the aligned elements, E = E’ = 0. This observation allows one to apply the 4 x 4 matrix method to the propagation of a paraxial ray through an optical system containing aligned and misaligned elements. In particular, the 4 x 4 matrix associated with a layer of thickness I of uniform medium (without refraction at its ends) coincides with + Mi4)(l),indicating that a plane-parallel layer of uniform medium is insensitive to misalignments. Analogously, the 4 x 4 sign matrix S(4) is found to be /I
0 0
o\ (3.16)
\o
0 0
I /
As discussed in 3 2, when a paraxial ray through a centered optical system is reversed, a new ray transfer matrix is derived from the original one by interchanging the elements a and d and dividing by the determinant of the matrix (see eq. 2.10). When the paraxial ray is passing through a misaligned
111%I 31
MATRIX TREATMENT FOR MISALIGNED ELEMENTS
309
optical system in reverse propagation, a reversed 4 x 4 matrix also can be obtained as follows: M(4) = S(4)[M(4)]+ iS(4) t
([S(4)]-
1
= S(4)).
(3.17)
Hence,
PE’\
/ d b a’&
(3.18)
\ o o o
q ’ /
with a’ = by
- da ,
p‘ = q’
=
p’
=
b6 - d/3,
y’ = a y
-CM,
6’ = a6 - c/3, (3.19)
Det(bl),
which, on account of eqs. (3.6), becomes
(3.20)
\o
0 0
1
1
in the case Det( rvl) = 1. It should be noted that the use of these augmented matrices may be laborious, especiallywhen many 4 x 4 matrices must be multiplied by one another to apply eq. (2.8) to some practical system. However, the treatment may be greatly simplified by the use of ray transfer flow graphs (HENLEYand WILLIAMS [ 19731). (For a brief introduction to these flow graphs and some examples of their application to the optical problems treated in this chapter see WANG SHAOMIN and RONCHI [ 19861).
3.2. ElKONAL FUNCTION FOR MISALIGNED SYSTEMS
In the case of misaligned systems the general expression for the eikonal function (2.48) must be replaced by
L = L, + qr1 + $r2
+ &prf - Or,r2 + f Yrg,
(3.21)
where the linear terms account for the variations of the eikonal function due to misalignment.
310
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
By proceeding as in 5 2.5, we find
Hence, r 2 = -1[ $ r I + r ;
n
r ; = [ F - n ] , , + - - r!P; + - ++cp$ .
+p],
n
o
(3.22)
Equations (3.22) must be compared with those derivable by means of matrix given by eq. (3.12), namely,
M '(4) +
r,
=Ar, t
Br;
+ a,,.& + he',
r; = Cr,
+ D r ; + y+ + b e ' ,
(3.23)
where the coefficients are given by eqs. (2.30) and (3.13). Hence, we find
a = -1 ,
A
$=-, B
B
D Y=BY
(3.24)
and for coefficients of the linear terms in eq. (3.21) cp=
(w+ h E ' ) , B
+=
[(BYr - D%)& + (Bs, B
- %)&'I .
(3.25)
Substituting eqs. (3.24) and (3.25) into eq. (3.21), the eikonal function for misaligned systems can be completely written in terms of ray transfer matrix elements as follows: L
=
1 Lo + -[A< - 2r,r, + Dr; + 2(%e t &.&')r, 2B + 2 { ( B h - D%)E + ( B S , - D&.)&'}r2],
(3.26)
The expression (3.26) for L will be used in 4.4 to study the interference effects in the plane of the synthesized image of an optical array. Note that if no optical instrument is placed between RP; and RP;, one has Lo = 1, A = 1, B = I, D = 1, M = 0, /? = 0, y = 0, and S = 0; hence, cc, = 0, & = 0, = 0, and S, = 0, and eq. (3.26) reduces to
(3.27)
MATRIX TREATMENT OF ARRAYS
311
# 4. Matrix Treatment of Arrays 4.1. GENERAL CONSIDERATIONS
An array may be considered as an ensemble of regularly misaligned optical elements. As such, an array may be treated by utilizing the considerations of 3, holding for generally misaligned elements, with important simplifications due to the regularity of the misalignment. For each impinging beam the emerging beam is constituted by the ensemble of the beams (“individual” beams) emerging from the individual elements. As a first step, the study of the optical properties of an array may be carried out by considering a single ray impinging onto each element of the array, for example the ray through the axial point (“vertex”) of each element, and the corresponding emerging ray. The fact that each element has a finite-size input pupil, so that the various rays belonging to the beam impingingonto the element behave in a different manner, is taken into account as a second step by studying the so-called “residual aberrations,’. In any case the size of each element must be so small,and therefore the beam entering the element so thin, that the ray through the vertex gives a good indication of the behavior of the individual thin beam. 4.2. THE RAY TRANSFER MATRIX FOR AN ARRAY
Because of the regularity of the misalignment of the elements of an array, the position of the ith element may be easily identified by the position E, of its vertex (fig. 15), and the inclination .$ of its axis may be simply expressed in terms of
I
I I
I
I
/
!
-1 Fig, 15. An array of optical elements.
312
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
WI,8 4
the geometrical parameters of the array. An important role is obviously played by the shapes of the input and output surfaces of the array, defined as the two surfaces containing the input and output vertices of the elements. For example, in the case of fig. 15 the input surface of the array is a spherical surface of (positive) radius R, and the output surface is a concentric sphere with radius R - I, where I is the common length of the elements. From the figure it appears that we can write E; =
- Ei/R .
(4.1)
Consider, now, a ray impinging with slope r; on the vertex of the ith element. Assuming a reference plane RP, through that vertex, the ray is specified by rl , r ; , with
r,
(4.2)
= Ei.
Hence, we can write
or, in matrix form, 1
(:)
=(-:
0
o!
(a)
(4.3)
Introducing eq. (4.3) into eq. ( 3 . 9 , where LJ = kli (and therefore ai,pi, yi, S,) may be different for different array elements, yields
or also, r2 = rv$)r,
with
111.8 41
MATRIX TREATMENT OF ARRAYS
313
where
$) is the ray transfer matrix of the ith element of the array. If the elements of the array are equal to one another, WJi and also are independent of i: Nli = 2, ,M$ = Nla, and
WJ$
with
Being independent of i, the matrix ,Ma is the ray transfer matrix of each element and therefore also of the array (whence the label “a”). One passes from one element to another by varying r,. ,Ma may be used as any ray transfer matrix examined in Q 2.3. However, a peculiar feature of WJa is that 1 Det(hla) = d - - (@ - bb)
R
(4.8)
depends on R and is generally different from n,/nz.Accordingly, the arrays are called “nonlinear” optical systems. It could be noted that, in general, different values of i correspond to different input planes RP,,; however, in the paraxial approximation the input planes of the individual optical elements of the array almost coincide. The most important application of WJa is to establish the relations between the ray parameters at two (moved) planes RP;,RP; (fig. 10, Q 2.4). By proceeding as in Q 2.4, we find that the matrix NJL for the array is given bY
314
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
with A = 1 - - B+ - 62 R R’
c = --s
R,
B = ( 1 -:)u
+6+
(t
-d)z,
6U
D=--+d. R
(4.10)
Equations (4.10) are derived by substituting for a, b, c and d in eqs. (2.30) the expressions (4.7) for the elements a’, b‘, c’, and d’. The matrix NJ; describes the effect on the plane z of all elements of the array. It indicates that, besides the conventional individual images, there is a synthesized image (at the plane z for which B = 0) which possibly has some unique properties. All the first-order synthesized imaging properties are embedded in this matrix. is The physical meaning of the various elements of the 2 x 2 matrix the same as that described in 5 2.4 for the matrix b J ’. By denoting by Y the value of z for which B = 0, we obtain a conjugate distance equation for the synthesized image:
2:
I/=
(R - B)u/R + 6 (Rd - SU)/R
(4.11)
With the help of eqs. (3.6) it can be easily verified that
V=
+ b + (uR - B)u/R cu + d + ( Y R - 6)u/R
au
(4.12)
Evidently, in general, the image plane for the best focus does not coincide with the image plane of the individual element, which is given by the ABCD law (2.33). Therefore, the synthesized image is not a Gaussian image. When B = 0, the matrix bl1:is called “synthesized image-formingmatrix“. The geometrical-optics parameters of the array, namely, the angular and transverse magnificationsm, and m, ,the focal lengths fi and f i ,the distances s, and s2 of the foci from the input and output surfaces of the array, and the distancesh, ,h, ,h i , and hi of the principal planes and of the nodal planes from the same surfaces, may be written either in terms of the elements of the matrix NJa or in terms of the parameters of the individual elements of the array. In the former case one has simply to substitute a’, 6’, c’, and d‘ for a, 6, c, and d in eqs. (2.35), (2.37), and (2.40)-(2.44), V for u, and Det(NJ,) for
MATRIX TREATMENT OF ARRAYS
315
n, I n z , obtaining
m, = c’u + d ,
(4.13)
m, = a’ - c’V ,
(4.14)
31
=
32 =
hl
d --
c’
(4.15)
’
a’ --9
(4.16)
C’
Det(M,)
=
-
d
(4.17)
Y
C’
1 - a’
(4.18)
h2=-,
C’
(4.19) 1
f2=
-2’
(4.20) (4.21)
hi
Det(M,)
=
- a’
(4.22)
C’
In the latter case one has to express a‘, b’, c’, and d in eqs. (4.13)-(4.22) in terms of a, b, c, and d by means of eqs. (4.7), obtaining (4.23) (4.24)
dR 6
(4.25)
s1=-,
R - 8 -R - l + b s2=----
6
6
Y
(4.26)
316
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
/3d - b6 6
[III, 8 4
(4.27)
hl=-,
(4.28)
fl
=
d(R - 1 + b)
+b,
R
(4.30)
f2=b’
hi = h;
=
(4.29)
R(d - 1)
s
9
(R - I + b)(l - d ) - b. 6
(4.3 1) (4.32)
Note that, in general, fl # f2, which confirms the non-Gaussian character of array imaging. In all the preceding expressions 6 = 1 - d, except in the expression for reflectors, when 6 = - 1 - d, in agreement with eq. (3.9). Matrix lMa plays the same role and may be used in the same way as the matrix Y of eq. (2.6). Hence, a reversed propagation matrix pafor arrays may be deduced by applying eq. (2.10), and an augmented 4 x 4 matrix for arrays can be given analogous to eq. (3.8) to treat misaligned arrays:
(4.33)
-
5
y=-,
R
(4.34)
where the plus (minus) sign holds for transmission (reflection). As an example of application of the matrix we can show that some arrays, like the thin lens array, are misalignment-insensitive devices. From
111, I41
table 1 (0 2.3) one derives for the thin lens array b = 0, d = 1, and I RP, coincides with RP,), and therefore, from (4.34) a=o,
317
MATRIX TREATMENT OF ARRAYS
p=o,
y=o,
s=o,
=
0 (since (4.35)
which constitute the requirements for insensitivity to misalignment. The same property is presented also by corner-cube reflector arrays and by cat’s eye reflector arrays (table 2, 0 2.3) for which b = 0, t = 0, and d = - 1.
4.3. ADDITIONAL ABERRATIONS FOR ARRAYS
The treatment of the previous section describes the behavior of a single ray per element of the array, the bbprincipal”ray, that impinges onto the element at its input vertex. The other rays incident on the same element are expected to behave differently, even if only slightly, since the input pupil of the elements is assumed to be “small”. In other words, there are aberrations due to the finite (non-infinitesimal) size of the input pupil of each element of the array, which are denoted either as “additional” aberrations or “synthesized imaging’’ aberra[1984]), in order not to tions (WANG SHAOMINand ZHOU GUOSHENG confuse them with conventional aberrations, which are due to the h i t e size of the entire optical device. It should be noted that in an array the conventional aberrations are generally eliminated by choosing the position of each element of the array suitably, according to the well-known principle of Fresnel lenses and mirrors. To evaluate the additional aberrations of an element of the may, for example, the ith element, one has to consider two incident rays, the principal ray and a marginal one, starting from the same point source on the object plane, and to find the transverse coordinates r,, rz,o of the two points where the emerging rays cross the plane of the synthesized image z = V. Here, V is given by eq. (4.11) or (4.12). In the sequel to this section the label “0” refers to the principal ray, and the label i specifjiag the element of the array will be suppressed to simplify the writing. Clearly, the ith element of the array is misaligned, so that use will be made of the equations in 0 3, in particular of eqs. (3.12) and (3.13). By introducing WJ’(4) given by eq. (3.12) into the equation
ri4)(V ) = -+M ’C4)
r(P)(u),
where the coordinates u and V of the object and synthesized image are evident,
318
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
we obtain r*(V) = (a - cV)r,(u)+ [(au + b) - V(cu t d)]ri(u)
+ (a - yV)&+ (/I - SV)&’.
(4.36)
This equation holds for all rays incident on that element of the array which differ from one another only by r; . For the principal ray we have (see fig. 16)
and for the marginal rays U - r;,o f r;(u)N & f 0 - r,(u) -
U
U
3
where CT denotes the radius of the pupil of the element. Hence, the synthesized imaging aberration A = r2(V) - r2,0(V)is given by U
A=f-[autbU
(4.37)
V(CU+~)].
This is a general expression for additional aberrations of synthesized images. It appears from eq. (4.37) that the synthesized imaging aberrations are eliminated if the following condition is satisfied:
(4.38)
2
LU Fig. 16. Scheme for the evaluation of the additional aberrations.
111, § 41
319
MATRIX TREATMENT OF ARRAYS
hence, recalling eq. (2.33), if the synthesized image coincides with the image of the individual elements, Y = u. Introducing eq. (4.38) into eq. (4.12) yields (OlR - / ~ ) ( c + u d) = (YR- 6 ) ( +~ b ) ,
(4.39)
which says that with any array there is a position u of the object, that satisfying eq. (4.39), for which V = u and the synthesized image is free from aberrations. This value of u is given by u=
R(l - d) + Id - b(d + 6) c(R - 1 + b ) + a6
(4.40)
and corresponds to
The condition u = V is satisfied anyway, as follows from eq. (4.12), if (4.41) Equations (4.41) are called the abyb condition. Note that ifthey are satisfied the synthesized image is free from additional aberrations for any position u, rl of the object.
4.4. INTERFERENCE EFFECTS IN THE SYNTHESIZED IMAGE
By definition, the synthesized image given by an array is the superposition of many beams. The fact that, in general, the additional aberrations of a misaligned element are different from zero means that, even in the case of a point source, each individualemerging beam crosses the image plane in a region of nonvanishing area around the image point. Hence, when the array is illuminated by coherent radiation, interferencephenomena are expected to take place. In fact, interference fringes may be observed (fig. 17a) through a synthesized image, even if sometimes they are not observed (fig. 17b). The interference phenomena may be studied by means of the eikonal function L of misaligned elements of eq. (3.26). Let us fist consider a point source in front of an array simply constituted
320
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 4
Fig. 17. Examples ofimages given by an array illuminated by coherent tight: (a) With interference fringes; (b) without interference fringes.
by two elements (fig. 18) and two emerging rays, which arrive at the same point of a plane z (not necessarily the synthesized image plane). Let us denote by E > 0, e' = - e/R the misalignment parameters of the upper element, R being the radius of curvature of the input surface of the array. The parameters of the lower element will be - E, - 6'. According to eq. (3.26), the eikonal function along the lower ray will differ from that along the upper ray by the sign of the linear terms. Hence, there will be a difference of optical path along the two rays given by
It appears from eq. (4.42) that AL is a function of r, ;hence, one must expect
Fig. 18. A two-element array.
111, 41
MATRIX TREATMENT OF ARRAYS
32 1
to observe interference fringes in the plane of the synthesized image. The condition for the absence of fringes is (4.43) With the help of eqs. (2.30), (3.13), and (3.6) this equation reduces to eq. (4.39) for the elimination of additional aberrations. Hence, we conclude that fringes are eliminated, together with the additional aberrations, when the synthesized image coincides with the images of the individual elements. However, in most applications of arrays the synthesized image does not coincide with individual images. Thus, in general, interference fringes are expected to be observed through the synthesized images. For an extended source the treatment is a little more laborious, since the field (e.g., E ) at any point P2 of the z plane should be calculated by means of the so-called diffraction integral. For a single generally misaligned element we have W 2 )=
S { E ( P ’ ) K(P1, P 2 ) 8x1
Y
(4.44)
XI
where P I is a point in the object plane, the integration should be extended to the surface of the object, and K ( P l , P 2 ) is given in terms of the eikonal function L by K f P ,,P z ) =
i
-exp (ik,,L) , LB
(4.45)
where L is given by eq. (3.26) and ko is the free-space wavenumber. It can be [ 19701) noted that eq. (4.45) is a generalization of Collins’ formula (COLLINS and reduces to it when the optical instrument is composed of a single centered element, for which L is given by eq. (2.51). Note also that when no optical instrument is placed between the object and the observation plane, L assumes the expression in eq. (3.27) and eq. (4.44) reduces to the Huygens-Fresnel principle. When the array is composed of many elements, each specified by a couple of indices j and k, eq. (4.44) must be replaced by (4.46) where Kjkdepends on the indices j and k through the misalignment parameters
322
[III, § 4
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
of the (j, k)th element appearing in the associated expression Ljk of L. To perform the integrations, it is useful to pass to Cartesian coordinates by replacing the scalar parameters r, r’, E, and E‘ by vectors: ri = ( X i , Y i )
r2 = ( X 2 7 Y 2 )
9
8
9
= ( E x , Ey)
8‘
$1
= &(,:
and the products in the expression for L by scalar products. Thus we will write for the element of orders j and k Ljk(pl,
(4.47)
p2) = L ’ ( ~ Ip2) , + ex1 + f v l + gX2 + hy2 7
where L‘(P,,P z ) is given by L ’ ( P , , P 2 ) = L 0 + A $ - 2 r 1 - r 2 +D< =
Lo + A(x:
+ y:)
- 2(x1x2 + y I y 2 )+ D ( x + ~~
3)
(4.48)
and is independent of j and k, whereas
+ &EL),
e = 2(c+,
g
=
2(B7+ - DLI+)E,
f = ~ ( C G Z E , , + &$),
+ 2(B&
- D&)&:,
h = 2(B7+ - D%)E, + 2(B& - D & ) E ~ ,
(4.49)
depend on j and k. For the element of indices j and k we have E, =
2jax,
cU = 2kay,
I&
8’Y
- 2 j -0, . R a = -2k2, R =
j = f 1, ..., + _ M ,
k = f l , ...,& N ,
(4.50)
2ax and 2ay being the sizes of the pupil of the individual element in the x and y directions, respectively. By introducing eqs. (4.50) into eqs. (4.49) and recalling eqs. (2.30), (3.6), and (3.13), we obtain e = ejk = 4e0aJ, h
=
hik
=
f
= fik =
(:)
eo=-z y--
g = gjk = 4g0aJ, (4.5 1)
4goa,,k,
with
4eoayk,
P
+a---,
R
(4.52)
111, § 51
PSEUDO CONJUGATOR ARRAYS
323
Then eq. (4.47) becomes Ljk =
L'(pI, p2) + 4(e0xl + g 0 ~ 2 ) j g x+ 4(e0~,+ g0~2)kgy
(4.53)
with L' still given by eq. (4.48). The field distribution obtained by introducing eq. (4.53) for L into eq. (4.45) should now be summed over the indices j and k, according to eq. (4.46). The calculation of E(P2) is particularly easy when the source is a point source. The intensity distribution in the image plane has the same features as that produced by a rectangular diffraction grating formed by (2M + 1) x (2N + 1) equispaced elements (MA JIAN, Luo XUEMIN, WANG SHAOMIN, RISALITIand RONCHI[ 19861) and presents main maxima, separated by 1B Ax2 = -, x direction, 2gxgo
LB Ay2 = -,
y direction,
2 aygo
with a number of secondary maxima in between. The halfwidth of the main maxima is found to be ax2 =
Ax2
(2M + 1) ,
6y,
=
AY2 (2N + 1) *
~
It follows that the complete elimination of fringes requires go = 0, which again yields the condition (4.39), first found for the elimination of the additional aberrations. In such a case the fringes are infinitely large. Then one concludes, as in the case of a two-element array, that, in general, the interference fringes disappear when the synthesized image coincides with the individual images. This happens either for any position of the object, when the a/?$ condition is satisfied, or for a particular position of the object, given by eq. (4.40), when the a/?$ condition is not satisfied. Clearly the fringes disappear from the image even when 6x2 and Sy2are larger than the size of the beam emerging from the array at the best focus.
0 5. Pseudo Conjugator Arrays 5.1, PHASE-CONJUGATIONMIRRORS
As anticipated in the Introduction, some arrays behave like phase conjugators. This property is made possible by the fact that the determinant of the
324
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[IIL 8 5
ray transfer matrix rvla of an array may be different from nJnz (see eq. 4.8), which means that arrays do not behave like conventional (linear) optical systems. To analyze which arrays demonstrate the phase-conjugator property, let us briefly review, in terms of ray transfer matrices, the properties of the so-called phase-conjugation mirrors (PCM). It is well known (Au YEUNG, FEKETE,PEPPERand YARN [ 19791) that phase-conjugation mirrors, which are usually obtained by some nonlinear process such as degenerate four-wave mixing, can reflect an incident field so as to yield its conjugate replica, that is, to leave wavefront and amplitude distribution unchanged, whatever they are. In fig. 19 the reflection property of the PCM is compared with that of the flat reflector and with that of the cat’s eye reflector. It is clear that the PCM ray transfer matrix (Au YEUNG,FEKETE, PEPPERand YARIV[1979J) is
yl=(’0
”>.
-1
for which Det(lMI) = - 1 .
(5.2)
It is a general and characteristic property of phase conjugators to have the determinant equal to - 1. As a fist consequence of eq. (5.2), the matrix 9,for reverse propagation is equal to the matrix yIitself:
Fig. 19. Comparison of the properties of a flat reflector, a cat’s eye reflector, and a PCM.
111, I 51
325
PSEUDO CONJUGATOR ARRAYS
The matrix ulJ may be used to prove some properties of a PCM;for example, its ability to compensate for wavefront distortion when a wave passes through an inhomogeneous medium, is reflected by a reflector, and passes back through the same inhomogeneous medium (fig. 20). If a flat mirror is used as reflector, the whole ray transfer matrix is given by
2ac
ad+ be
If the reflector is a cat's eye reflector, we have
- (ad + bc)
("c :)=(%:) (-: -:) (: :)=(
-2ac
-(ad+bc)
On the other hand, for the PCM mirror we have
In other words, with the iirst two reflectors the parameters of the emerging beam depend on the medium crossed, whereas for the PCM the effect introduced by the medium after the first passage is compensated for at the second passage, so that the ABCD matrix does not depend on the characteristics of the medium. The same result is obtained if the inhomogeneous medium is misaligned, so that it is described by the 4 x 4 augmented matrix $4) given by eq. (3.8). The augmented matrix for the (aligned) PCM mirror is /l 0 0 o\
\o
0 0
l i
picture"
Fig. 20. Experimental apparatus to test the property of some array or PCM of compensatingfor the wavefront distortiondue to the inhomogeneitiesin a test tube with 0 , b, c, dray transfer matrix elements.
326
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
and we find that where ?1(4) is given by eq. (3.18). Another property of PCMs which is easily proved by the matrix algorithm is that they are insensitive to misalignment. This property may be easily proved (WANG SHAOMIN [ 1983a1) with the help of eqs. (3.6) and (3.9), by noting that a = 1 - a = 0 , 8 = I - b = 0 (since I = 0), y = - c = 0, and 6 = - 1 - d = 0. The features of a PCM may be revealed even by an equivalent matrix RAIIof the type (2.46). On account of expression (5.1) for the ray transfer , we find matrix 2
c2=--,
U
so that
Note that Det(Nl?,,)= 1. By recalling the expression (2.13) of the ray transfer matrix lW2,(R) of a spherical mirror with curvature radius R, we conclude that a PCM is equivalent to a spherical mirror whose curvature is identically equal to the curvature of the incident wavefront. This means that a phase conjugator behaves like a zoom lens, by keeping the image distance identically equal to the object distance:
v = -24.
(5.10)
Note that eq. (5.10) holds both in reflection and transmission, in spite of (or because of) the different sign conventions for the axial distances assumed for the two cases. Another interesting result derivable by comparing eq. (5.9) with eq. (2.39) is that both the transverse magnification m, and the angular magnification m ,of a PCM are unity. 5.2. PHASE-CONJUGATOR ARRAYS
There are several arrays that can perform phase conjugation. For example, consider an array of spherical mirrors (FREEMAN,FREIBERG and GARCIA
111, $ 51
PSEUDO CONJUGATOR ARRAYS
321
[ 19781) with
l=O,
R=-u,
(5.11)
where R now represents the radius of curvature of the input surface of the array. With the help of eq. (4.11) or (4.12) we may easily verify that eq. (5.10) is satisfied. Even an array formed by corner-cube reflectors (SATO,NAGURA,IKEDA [ 1982]), for which a = - 1, b = c = 0, and d = - 1 (see and HATSUZAWA eqs. 2.14), satisfies eq. (5. lo), provided that eqs. (5.11) are satisfied. However, in both cases the second eq. (5.11) indicates that, when u varies (namely, in the case of objects at different distances from the array), R must be varied if phase conjugation must be realized. To change R means to change the mutual positions of the elements of the array, which is generally obtained by means of a rather complicated mechanical structure including pistons, according to the phase-conjugation principle of the well-known coherent optical adaptive techniques (COAT). The complicated COAT structure may be avoided with plane arrays (R = a),provided that the optical elements composing them have d=-l.
b=O,
(5.12)
In this case eq. (4.11) yields directly V = u/d = - u. Clearly, arrays of this type offer many advantages over phase conjugators based on COAT: an unlimited dynamic range, much higher response speed, lower weight, lower cost, and greater simplicity. It is immediately verified that for R = 00 the ray transfer matrix Wla of eq. (4.6) becomes (5.13)
and the equivalent transfer matrix (2.46) becomes (5.14) c2 being given
c2
=
by
d 1 - -. u t b u
~
(5.15)
At the same time expression (4.11) for the coordinate of the synthesized
328
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
image takes the simpler form y=- u + b d
[III, c2 5
(5.16)
If, in addition to R = a,we choose optical elements with b = 0, d = - 1, takes the form of the matrix of PCM (see eq. 5.1), and c2 becomes - 2/24 as for PCM (see eq. 5.8). Clearly, arrays operating as phase conjugators offer significant potential advantages even with respect to PCM (or other nonlinear devices) in addition to lower weight, lower cost, and greater simplicity: Since they are composed of passive and linear elements, arbitrarily weak signals can be conjugated and the return frequency is always equal to the input frequency. However, it must be noted that the principles of wavefront replication are quite different between nonlinear processes and arrays. In the case of nonlinear processes a wavefront is replicated point by point, with continuity, whereas in the case of an array some elements of the wavefront are not replicated (fig. 21), and, moreover, the elements that are replicated are "reversed". Consequently, the phase conjugation performed by an array is neither complete nor perfect: This is the reason why phase-conjugation arrays are called "pseudo conjugators". Despite this imperfection, pseudo conjugator arrays have many practical applicationsbecause they are compact imaging devices with unit magnification, as follows by comparing the matrix Ye of eq. (5.14) with the general ray transfer matrix NJt' of eq. (2.39). Pseudo conjugator arrays may be designed to operate both in backward and/or in forward propagation. An example of an array operating both in forward and in backward propagation is the GRIN rod array described in 5 1 (see fig. 1). Even the cat's eye may be used to form arrays operating in both directions (see § 5.3). As a general method, given an optical (linear) element and the associated ray Pila
YI
de art fromS simuPtaneously
s*+":fsimultaneously
1
Fig. 21. Approximate replica of a wavefront by an array.
I__&& _I
Examples of pseudo-conjugators arrays (forward propagation). System
Schematic
M=(-'
Cat's eye array
I
I
a
V
Expressions for matrix elements
Matricesa
0
M'=(-'
0
-
3 I
M IJI v
O),
-1
O) -1
Ray transfer matrix of a single element of the array, as defined by eq. (2.6) (M) and eq. (2.22) (M'),
System
Schematic
330
TABLE4 Examples of pseudo-conjugator arrays (reflection). Expressions for matrix elements
Matrices"
I
= 0
I
-
I
1
r 4 l \
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
- 0 I I 1 I
I
0 I
I
n
I
- 0
J II
a
-px
c (
i
B
IL
B
WI,8 5
Half-bead array
Concave-mirror array (radius R > 0)
Concave-mirror array (radius R > 0)
PSEUDO CONJUGATOR ARRAYS
GRIN-fiber array
Bead array
a
Ray transfer matrix of a single element of the array, with respect to reference planes RP (M) and RP’ (M’).
-
W
w
332
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
[III, 9 5
transfer matrix Y [with Det(9) = 11, one can investigate if it may be used to form pseudo conjugator arrays by examining the matrix 9’obtained by the plane-moving technique [see eq. (2.22) or (2.29)]: If a value u, of u may be found for which eqs. (5.12) are satisfied for z = - u,, that element may be used to form a pseudo conjugator array. By using eqs. (2.30), eqs. (5.12) can be written in the form UU,
+ b + U,(CU,t d) = 0 ,
CU, t
d=-1;
hence,
(a-l)u,=-b,
~uO=-l-d.
(5.17)
Since ad - bc = 1, eqs. (5.17) may be simultaneously satisfied only if
a=d.
(5.18)
This condition is a generalized condition for obtaining plane arrays as phase conjugators. From eqs. (5.17) one has
b l-a
l+d
uo = -= --
C
(5.19)
Tables 3 and 4 show some examples of pseudo conjugator arrays. From the standpoint of the additional aberrations it may be noted that for the plane arrays (R = co), for which V is given by eq. (5.16), eq. (4.37) becomes
A
=
d
k - ( l - d - CU). d
(5.20)
It follows that additional aberrations are eliminated if u=-
l-d C
(5.21)
namely, if the object is placed at the principal plane of the individual elements forming the array (see eq. 2.42). In the case of pseudo-conjugator plane arrays, for which d = - 1, eq. (5.20) becomes A
=
~ ( C-U2),
(5.22)
which indicates that when the object is at u = 2/c, pseudo conjugator arrays with c # 0 may behave better than arrays with c = 0.
111, J 51
PSEUDO CONJUGATOR ARRAYS
333
Let us now examine in detail some arrays with phase-conjugation properties that have been built, tested, and in some cases also applied. 5.3. CORNER-CUBE ARRAY
As anticipated in $! 1, corner-cube arrays have been used to approximate phase conjugation (JACOBS [ 19821) and to compensate for optical inhomogeneities in laser amplifiers (ORLOV,VIRNIK, VOROTILIN, GERASIMOV, KALININand SAGALOVICH [ 19781) and optical resonators (MATHIEUand BELANGER[1980]). In such applications the comer-cube arrays were the common plastic retroreflectors with about 2.5 mm pitch used MI bicycles and highway signs (fig. 22). As discussed in the previous section, the ray transfer matrix of this array has the form (5.1), in particular, c = 0. Hence, the additional aberration A becomes
A
=
+20,
(5.23)
which indicates that if the object is a point source, the image is not “a point” but has radius 20. This result has been checked by MA JIAN, TANGW u , ZHU FUXIANG, TONGDINWANG and WANG SHAOMIN [ 19851 (fig. 23), with the experiment drawn in fig. 24. Corner-cube arrays are typically backward systems.
Fig. 22. Corner-cube array with 2.5 mrn pitch.
334
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
5.4. CATS EYE ARRAY
The cat’s eye, which has the same ray transfer matrix as the corner-cube reflector (see table 2), may be used to form pseudo conjugator arrays, which operate also as forward systems (MA JIAN, TANGWu, ZHU FUXIANG,TONG DINWANG and WANGSHAOMIN[ 19851). Note that, as a backward system compensates for the inhomogeneities of a medium in front of it, just because the medium remains the same, analogously, for a forward system the inhomogeneous medium in front of it must be strictly related to that on the other side if compensation must be realized. If we write the ray transfer matrix of the system in fig. 25, we obtain
Hence, the compensating conditions require (5.25)
Fig. 23. Image of a point source given by a comer-cube array.
335
PSEUDO CONJUGATOR ARRAYS
Fig. 24. Experimental setup to measure the additional aberrations of a corner-cube reflector.
namely, recalling that a,d, - b,c, a2 = dl
,
b2 = b, ,
=
a2d2 - b,c, = 1,
c2 = c, ,
d2 = a,
.
(5.26)
5.5. ROOF-MIRROR ARRAY
The roof-mirror array is a backward system. If one combines a comer-cube array with a cat’s eye array, as shown in fig. 26, one obtains by suitably choosing the distances, the so-called roof-mirror array, with c # 0. Hence, as noted in § 5.2, there is a position of the object for which the synthesized image is free from additional aberration. The total transfer matrix 5 for an individual element of the arrays (fig. 26) can be written as (5.27)
with (5.28)
Fig. 25. A forward-propagation cat’s eye system with inhomogeneous media on both sides.
336
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
Fig. 26. A roof mirror.
Hence, 12
a, = 1 - -,
f
b, = I ,
hl2 + l2 - f’
(5.29) and a=
- (a,d, + 6 , ~, ~ ) 6
=
- 26,d,,
c=
- 2a,c,,
=
- (a,d, + 6
d
, ~. ~ )
Accordingly, since a = d. the systems of fig. 26 can be used to form arrays with phase-conjugation properties (see eq. 5.18). The planes RP; = RP; to which one must refer in order to satisfy the pseudo-conjugation conditions eq. (5.12) are located at a distance u, from RP, = RP,, given, according to eq. (5.19), by (5.30) where use has been made of the unimodular property of matrix ys. The additional aberration A given by eq. (5.20) takes the particular form
A
=
20 T - (U,C,U + 1) . d
Hence, for an object placed at a distance u=
-- 1 a,%
from RP;, A vanishes: The image of a point source is “a point”.
(5.31)
111, B 51
331
PSEUDO CONJUGATOR ARRAYS
For the preceding property, added to the fact that the magnification is 1, a device of this type may be used, as it has been used in Japan, in compact photocopying machines.
5.6. GRIN ROD ARRAYS, THICK-LENS ARRAYS A N D BEAD ARRAYS
Plane GRIN rod arrays are usually employed in compact unit-magnification and TOYAMA imaging devices in the photocopying industry (MATSUSHITA [1980]). Since they operate in air, the proper ray transfer matrix to be considered is rf (1, no, Po, I) as given by eq. (2.27), namely, (5.32) with a,, boy coy and do given by eqs. (2.26). With the diagonal elements of M ,o being equal, GRIN rods may be used to form a pseudo conjugator array + (see eq. 5.18). The reference plane RP; ,for which eqs. (5.12) are verified, is at a distance u, from the input surface of the rod, which is given, according to eq. (5.19), by u0 =
1 + do --*
con0
As a numerical example, one can take 1 = 6.8 mm, no = 1.7, and one finds u,=
(5.33)
& = 0.5 mm-I,
-0.15mm.
The negative sign of u, means that RP; is on the right of the input surface of the rod. An array employed in photocopying machines may be formed by approximately 400 rods, with radius = 0.4mm. Since co # 0, the additional aberrations may be completely corrected by putting the object at the distance u=-
2 con0
from RP;, which in the preceding numerical example results in a distance of about 9 mm.
338
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 5
An array of this type has been successfully tested (WANGSHAOMIN, ZHOU GUOSHENG, Wu MEIYING,PENG LIANHUIand TIANLIJUAN[ 19831). Pseudo phase-conjugator arrays may be formed also by thick lenses (WANG SHAOMIN, Z u JINMIN, WANG SHIAOJING, YIN CHENGRENGand ZHANG ZEXUN [1983]), and even by beads with the rear half-surface reflecting (HUANGWEISHAI, JIANGXIUMING,CHENYINGLI,ZHAOJIAJU, Z u JINMIN and WANGSHAOMIN [ 19831). Let us consider in some detail this last array, formed by rather unusual optical elements (fig. 27). of a single bead (fig. 28) may be found by The ray transfer matrix applying eq. (2.8). Let us first introduce the matrix y, describing the refraction at the input spherical surface and the transit through the bead:
where p > 0 denotes the radius of the bead, n is the refractive index of the material by which the bead is formed, ,Ml(2p) is given by of eq. (2.12), and y2(p,1, n) is given by the first of eqs . (2.13). For the elements a,, b,, c, , and d, of h $ we find
2-n , a, = n
b , = -2P , n
1-n
cs=-,
nP
1
d,=-, n
Fig. 27. A bead array with 0.07 m m pitch.
PSEUDO CONJUGATOR ARRAYS
ug=-
339
9
Fig. 28. A couple of beads composing an array.
and, obviously, Det(bl?,)= l/n. Then the ray transfer matrix vrJ of the bead, with respect to the reference plane RP, = RP, indicated in fig. 28, is obtained as
with ?Isgiven by eq. (2.10) and I&,.(- p ) given by the second eq. (2.13). The result of the matrix multiplication is a=---(n - 4)
n
c=-
2(n - 2 ) , nP
b=
--, 4P n
d = -(n. - 4) n
Hence, a = d: The beads with a reflecting half-surface may be used to form pseudo conjugator arrays. For them (fig. 28) ug
= -p.
The phase-conjugation properties of bead arrays, including the capability of compensating for inhomogeneities of a medium in front of them, have been checked with the experiment shown in fig. 20 (5 5.1). An example of the results is shown in fig. 29. It may be worth noting that, because of the smallness of the diameter of the beads, the main effect determining the quality of the synthesized image given
W
0 P
c (
Fig. 29. (a) The object imaged by the apparatus of fig. 20, with a flat mirror and without test tube; (b) with a flat mirror and with a test tube; (c) with a bead array and without the test tube; (d) with a bead array and with the test tube.
1 cn
111,s 61
ARRAYS WITH VANISHING DETERMINANT
34 1
Fig. 30. The image of a point source in coherent light, given by a bead array.
by a bead array is not the additional aberration, as in the case of comer-cube arrays, but diffraction. This may be proved by means of fig. 30, representing the image of a point source obtained with the apparatus shown in fig. 24 (Q 5.3), with the comer-cube array replaced by a bead array.
4 6. Arrays with Vanishing Determinant 6.1. THEORETICAL CONSIDERATIONS
All optical systems considered up to now have a ray transfer matrix with a detenninant equal to n,/n, (if they are linear) or equal to - 1 (if they are nonlinear systems or phase conjugator arrays). In Q 6 we now consider another type of array, whose ray transfer matrix blahas a determinant equal to 0. According to eq. (4.8), the determinant of the ray transfer matrix of an array is given by 1
Det(y,) = DET = d - - (@ - bS) ,
R
342
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 0 6
where R denotes the radius of the input surface of the may, b and d are two of the elements of the ray transfer matrix of the optical element forming the array, and (see eqs. 3.6) j= 1 - b and 6 = 1 - d. Recall that in the case of reflectors, 6 = - 1 - d. The requirement
(6.2)
Det(y,) = 0 is satisfied simply by choosing b R=j--6. d
(6.3)
Then the elements of matrix (4.7) to be a ’ = -~ 66
j d - b6
,
5, are simply found with the help of eqs.
b’= b ,
The first observation is that for these arrays the reversed matrix defined by eq. (2.10), does not exist because of eq. (6.2). If now we introduce eq. (6.3) into eq. (4.1 l), we obtain
pa,
b V=d’
from which follows the important result that the position of the synthesized image is independent of the position of the object. Moreover, by noting that eq. (6.3) can also be written as b R=lT-, d
where the lower sign holds for reflectors, eq. (6.5) can be rewritten in the form
V=T(R-I),
(6.6)
which indicates that the synthesized image forms in all cases at the common center of the input and output surfaces of the may. All sources in object space are imaged at the same plane: The depth of the field is infinite.
I U 8 61
ARRAYS WITH VANISHING DETERMINANT
343
Another important feature of this type of arrays is found ir we evaluate the transverse magnification m2 of the array by introducing eqs. (6.4) and (6.5) into eq. (4.14). We find
m,=O.
(6.7)
Consequently, arrays with DET = 0 are not imaging devices, since, because of eq. (6.7), the synthesized image of any object should be a ‘‘geometrical point”. This, of course, is not true because of the additional aberrations, by virtue of which the image of a point is a “spot”. By application of eq. (4.37) the additional aberration, namely the size of the spot, can be shown to be
Hence, the size of the synthesized image of any object depends only on the size of the individual optical element and on the matrix element d of the individual element forming the array. In comparison with the effects of the additional aberration, the diffraction effects of individual elements appear to be negligible. As a matter of fact, the Fresnel number of the input pupil of the individual element, as seen from the
Fig. 31. An array with DET = 0,formed by elements with a square pupil.
344
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
Fig. 32. An array with DET
=
0, formed by elements with a circular pupil.
synthesized image plane, is infinite, as follows from the application of eq. (2.55) and by noting that even the image of the input plane (u = 0) forms at V = b/d, where, therefore, B = 0. The fact that this type of array cannot be applied as an imaging system does not mean that these arrays do not have practical applications: They may fkd several other applications, such as exposure meters, shadowless lamps, solar furnaces, and others. Even arrays with DET close to 0 appear to have interesting properties in terms of applications, since their transverse magnification can be controlled by the value of R. For example, a GRIN rod array capable of reduction or enlargement has been designed both by means of conventional optics (REES and LAMA[ 19841) and by matrix methods (WANG SHAOMIN and RONCHI [ 19851).
Fig. 33. Experimental scheme used to test the arrays shown in figs. 31 and 32.
111, § 61
ARRAYS WITH VANISHING DETERMINANT
345
6.2. EXPERIMENTAL TESTS
Two kinds of arrays with DET = 0 have been built (WANG SHAOMINand ZHANGZEXUN[ 19851) and tested. One array was made ofglass elements,with a square pupil of side 12 mm, and d = - 1 (fig. 31). The other array was made of Plexiglas, with a circular pupil of diameter 1.75 mm and d = - 0.5 (fig. 32).
Fig. 34. Images obtained with the arrays shown in figs. 31 (left) and 32 (right), with DET = 0.
346
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
WI,§1
The experimental scheme is shown in fig. 33. The images are shown in fig. 34a and b. They are in complete agreement with theoretical expectations: The image is independent of the position of the source, and the image has the same shape as the pupil of the elements of the array, which is clearly due to the additional aberrations. As a last observation, we can note that the synthesized image in coherent light may present interference fringes (see Q 4.4), which, however, may disappear when the object is at the position u given by eq. (4.40). In the present case this becomes
Figure 17, where such an effect has already been shown, is, m fact, a picture photographed with an array with DET = 0.
7. Conclusion
Optical arrays constitute a class of optical instruments whose features cannot be studied by means of classical optics. This characteristic is probably because their structure interrupts the continuity or the analyticity of the medium where waves propagate. From this point of view they are similarto the so-called Fresnel lenses, whose incapability of being treated by the differential laws of ordinary optics was demonstrated several years ago (RONCHIand TORALDO DI FRANCIA[ 19521). The matrix technique has proved to be a powerful tool in investigating the paraxial behavior of both linear and nonlinear optical systems. It may be worth recalling that we define as linear those systems whose ray transfer matrices or or IM) have a determinant equal to nl/n2,n1 being the refractive index of the medium containing the incident rays and n2 that of the medium containing the emerging rays. The ray transfer matrices Nla of arrays (see eq. 4.6) may have a determinant different from n,/n,, even if, or despite the fact that, they are made of linear optical elements. This is so because the elements of the ray transfer matrix, and therefore its determinant, depend on the geometry of the array, in particular on the radius R of the input surface of the array. The possibility follows, therefore, of obtaining devices with nonlinear proper-
(v v’
1111
REFERENCES
347
ties without making recourse to complicated techniques like coherent optical adaptive techniques or degenerate four-wave mixing. This is the case for arrays with DET = - 1, described in # 5, which have phase conjugation properties, even if only approximated, and for arrays with DET = 0. These last arrays with DET = 0, described in !$ 6, seem to form a new type of optical instrument, with the unusual property of having infinite Eeld depth, and image size independent of the position of the object. Such arrays have only recently been discovered, so that one can expect to find other interesting properties or applications with further investigations. It is interesting to note that the compound eye of insects is similar to the preceding arrays, since its DET is close to 0. Acknowledgements One of the authors (Wang Shaomin) is grateful to ICTP (Trieste, Italy) and to NSFC (Beijing, China) for their support. References ARNAUD, J.A., 1976, Beam and Fiber Optics (Academic Press, New York). AU YEUNG,J., D. FEKETE, D.M. PEPPERand A. YARIV,1979. IEEE J. Quantum Electron. QE-15, 1180-1 188. BARRETT, H.H., and S.F. JACOBS.1979, Opt. Lett. 4, 190-192. BORN,M., and E. WOLF, 1980, Principles of Optics, 6th Ed. (Pergamon Press, Oxford). and A. WALTHER,1963, Appl. Opt. 2, 1239-1246. BROUWER, W., E.L. O”EIL CARTER,W.H., 1982, Appl. Opt. 21, 1989-1994. COLLINSJR, S.A., 1970, J. Opt. SOC.Am. 60, 1168-1177. DRAGT,A.J., 1982, J. Opt. SOC.Am. 72, 372-379. FAN,DIANYUAN, 1983, Acta Opt. Sinica 3, 319-325 (in Chinese). FOG,C., 1982, Appl. Opt. 21, 1530-1531. FREEMAN, R.H., R.J. FREIBERG and H.R. GARCIA, 1978, Opt. Lett. 2, 6143. GERRARD, A., and J.M. BURCH,1975, Introduction to Matrix Methods in Optics (Wiley, London). HENLEY,E.J., and R.A. WILLIAMS,1973, Graph Theory in Modern Engineering (Academic Press, New York). HUANG,WEISHAI, JIANGXIUMING, CHEN YINGLI, ZHAOJIMU, ZU JINMIN and WANG SHAOMIN, 1983, Chin. J. Lasers 16, 191-192 (in Chinese). JACOBS,S.F., 1982, Opt. Eng. 21, 281-283. KAWAZU, M., and Y.OGURA,1980, Appl. Opt. 19, 1105-1 112. KOGELNIK, H., 1965, Bell Sys. Techn. J. 4 , 4 5 5 4 9 4 . KOGELNIK, H., and T. LI, 1966, Appl. Opt. 5, 1550-1567. MA, JIAN,TANGwu, ZHU FUXIANG, TONGDINWANGand WANGSHAOMIN, 1985, Physics, pp. 491-492 (in Chinese).
348
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
1111
MA,JIAN,Luo XUEMIN,WANGSHAOMIN, R. RISALITIand L. RONCHI,1986, Nuovo Cimento D 8,91-104. MARCUSE,D., 1972, Light Transmission Optics (Van Nostrand Reinhold, New York). MATHIEU,P., and P.A. BELANGER, 1980, Appl. Opt. 19,2262-2264. MATSUSHITA, K., and M. TOYAMA, 1980, Appl. Opt. 19, 1070-1075. OMEARA,T.R., 1982, Opt. Eng. 21,271-280. ORLOV,V.K., YAZ. VIRNIK, S.P. VOROTILIN, V.B. GERASIMOV, Yu.A. KALININand A.YA. SAGALOVICH, 1978, Sov. J. Quantum Electron. 8, 799. REES,J.D., 1982, Appl. Opt. 21, 1009-1016. REES,J.D., and W. LAMA,1984, Appl. Opt. 23, 1715-1724. RONCHI,L., and G. TORALDODI FRANCIA, 1952, Atti Fond. G. Ronchi 7, 145-160. SATO,T., Y.NAGURA,0. IKEDAand T. HATSUZAWA, 1982, Appl. Opt. 21, 1778-1784. WANG,SHAOMIN,1979, J. Hangzhou Univ. 3,42-52 (in Chinese). WANG,SHAOMIN,1983a, J. Hangzhou Univ. 10,476-490 (in Chinese). WANG,SHAOMIN,1983b, Appl. Lasers 3(2), 13-16 (in Chinese). WANG,SHAOMIN, 1984, J. Hangzhou Univ. 11,79-91 (in Chinese). WANG,SHAOMIN, 1985, Opt. & Quantum Electron. 17, 1-14. WANG,SHAOMIN, and L. RONCHI,1985, GRIN imaging systems described by transfer matrices, in: Gradient Index Optical Imaging Systems, Proc. Topical Meeting (Palermo, Italy). WANG,SHAOMIN, and L. RONCHI, 1986, Atti Fond. G. Ronchi 41, 53-66. WANG,SHAOMIN, and H. WEBER,1982, Opt. Cornmun. 41, 360-362. WANG, SHAOMIN,and H. WEBER, 1985, Matrix Methods in Treating Phase Conjugate Phenomena, in: Trends in Quantum Optics, Proc. 2nd Int. Conf. (Bucharest, Romania). WANG,SHAOMIN, and ZHANGZEXUN,1985, J. Hangzhou Univ. 12, 70-73 (in Chinese). WANG,SHAOMIN, and ZHoU GUOSHENG,1984, Acta Opt. Sinica 4, 1119-1 123 (in Chinese). WANG,SHAOMIN, Zu JINMIN,WANGSHIAOJING, YIN CHENGRENG and ZHANGZEXUN,1983, Appl. Lasers 3(5), 27-28 (in Chinese). WANG,SHAOMIN, ZHOU GUOSHENG, WU MEIYING,PENGLIANHUIand TIANLIJUAN,1983, Acta Phys. Sinica 32, 1357-1360 (in Chinese). YARIV,A., 1976, Introduction to Optical Electronics (Holt, Rinehart & Winston).
0 ELSEVIER SCIENCE PUBLISHERS B.V., 1988
I11 PRINCIPLES AND DESXCN OF OPTICAL ARRAYS BY
WANG SHAOMIN Hangzhou University Hangzhou, China
L. RONCHI Istiiuto di Ricerca sulle Onde Elettromagnetiche del CNR Florence, Italy
CONTENTS $ 1 . INTRODUCTION
....................
PAGE
281
$ 2. THE 2 x 2 MATRIX TREATMENT OF AN ALIGNED OPTI284 CAL ELEMENT . . . . . . . . . . . . . . . . . . . . .
$ 3 . MATRIX TREATMENT FOR MISALIGNED ELEMENTS
. 305 Q 4. MATRIX TREATMENT OF ARRAYS . . . . . . . . . . . 311 Q 5. PSEUDO CONJUGATOR ARRAYS . . . . . . . . . . . 323 $ 6. ARRAYS WITH VANISHING DETERMINANT . . . . . . 341 $ 7 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . 346 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 347 347 REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
# 1. Introduction Arrays of optical elements can be found in nature, such as the compound eyes of insects, and in every day life, such as the comer-cube arrays on cars and bicycles or on road signs. In recent years optical arrays have attracted new interest, since they have been found not only to give rise to some unusual phenomena but also to have many potential applications in present and future industry and in scientific research. The most unusual new phenomena associated with arrays are the nonGaussian imaging property of GRIN (gradient index) rod arrays (REES[ 19821) and the pseudo phase conjugation phenomenon of comer-cube arrays (JACOBS [ 19821). A brief description of these phenomena follows, starting with the image formation property. First, it should be pointed out that the image given by an array is a synthesized image in the sense that it is formed by the superposition of the beams emerging from the various elements of the array. It is the best focus of the beam emerging from the array. As such, it depends not only on the optical properties of the elements of the array but also on their mutual positions, namely, on the shape or curvature of the array's input surface X and output surface Z'.Thus, in general, the image of a source given by an array does not coincide with the images given by the single elements. As is well known, a GRIN rod consists of a medium whose refractive index is a parabolic function of a cylindrical coordinate r = no(1
- @or2)
(1.1)
9
where no clearly is the value of n on the axis, r = 0. It is also well known (MARCUSE[ 19721) that the meridional rays in such a medium are sinusoids independentlyof their initial height and slope with half-periodicityp = n/&, (at least in the paraxial approximation). Consider then (REES [ 19821) an array constituted by a number of GRIN rods of the same length 1 and with the axes parallel to each other (fig. 1). If we choose I = p, any ray entering a rod at a distance r from its axis and with slope I' will arrive at the exit surface with opposite height and slope. Now assume a point source S, on the axis of the axial rod of the array (labeled 1 in fig. 1) and look for its image given by the 28 1
282
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
Fig. 1. Non-Gaussian imaging in of a GRIN rod array: The distance O,S, of the image from X’ is equal to the distance O,S, of the source from X, for any O,S,.
array. To this end, let us consider a single ray per rod - the ray impinging at the axial point of the rod: It will arrive at the axial point of the exit surface with opposite slope. Hence the emerging ray will cross the axis of the array at a point S2,such that I02S,I = IS,O,l for any S,O,. This means that, at least for a GRIN rod array of the type shown in fig. 1, the relation between conjugate planes is not the usual Gaussian law. Experimental tests of such a property have been carried out by KAWAZUand OGURA [ 19801. In considering pseudo-phase conjugation, there are a number of experiments which prove that corner-cube arrays may be used to approximate phase conKALININ jugation. For example, ORLOV, VIRNIK,VOROTILIN, GERASIMOV, and SAGALOVICH [ 19781 reported dynamic compensation of optical inhomogeneities in an optically pumped neodymium/glass rod by means of a retroreflecting array (fig. 2). When compared with the use of a plane mirror at the position of the array, the retroreflecting array dramatically reduces the beam divergence due to thermal distortion of the rod. On the other hand, BARRETTand JACOBS [ 19791 proved that the wavefront distortion caused by an inhomogeneous medium in an imaging system can be compensated if a corner-cube array, or a small-period retroarray, is used as shown in fig. 3. Indeed, MATHIEUand BELANGER[ 19801 described the use of small-period retroarrays as mirrors for laser resonators to compensate for distorting ele-
111, § 11
1NTRODUCTtON
283
Fig. 3. Point source imaging: (a) without any phase perturber; (b) with a phase perturber.
ments inside the laser. Two kinds of lasers were considered. One of them was aTEA (Transverse Excitation, Atmospheric Pressure) C0,laser ( A = 10.6 pm) with a resonator that included a 36% reflecting flat mirror, 1 m away from the retroarray. The burn spot of the retroarray could be tilted by as much as 25 degrees before laser action ceased. Various transparent phase perturbers could be inserted up to 15 cm from the retroarray without stopping laser action. The other laser resonator was a pulsed dye laser (rhodamine 6G), with a 2 cm2 lasing aperture in the same resonator as the TEA laser. The burn spot of this laser, operating in the visible spectrum, was even better than that of the infrared laser. In both cases the retroarrays were common plastic retroreflectors like those used on bicycles and highway signs. The periodicity of the arrays was about 2.5 mm. The significance of these applications of corner-cube arrays, and of some other arrays, is that no other combination of optical elements has yet been discovered that performs conjugation (or reversal) of a wavefront of arbitrary shape and aberration. On the other hand, there is no proof that such conjugators are impossible (O’MEARA[ 19821). Thus, corner-cube arrays have been called “pseudo phase conjugators”. Present industrial applications of arrays include a major use of GRIN rod arrays (fig. 1) in photocopying machines as compact unit magnification devices [ 19801). (MATSUSHITA and TOYAMA From the foregoing considerations it follows that arrays form a new area in optics: It is not trivial to analyze their behavior in terms of classical optics just because they do not follow the Gaussian law of conjugated points. Fortunately, however, the non-Gaussian imaging process may be easily treated (WANG SHAOMIN [ 19851)by recasting the classical matrix-optics methods (BROUWER, ONEILand WALTHER [ 19631, GERRARD and BURCH [ 19751) or by extending the ABCD matrix introduced by KOGELNIK [ 19651 (see also KOGELNIK and Lr [1966]) to describe Gaussian beam propagation and optical resonator properties.
284
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
[III, 0 2
In J 2 of this chapter we recall the essential features of the 2 x 2 ray transfer matrix method. In J 3 we derive the augmented 4 x 4 matrix introduced to describe the optical behavior of a misaligned optical element: This section is the basis of treating arrays, where the arrangement of the individual optical elements can be regarded as a “regular” misalignment. In J 4 we show how, because of the regularity of the misalignment of the elements forming the array, an array may be treated in terms of a 2 x 2 matrix (WANG SHAOMIN[ 1985]), which accounts for imaging properties, phase conjugation properties, and some other new properties of arrays. In 8 5 and 5 6 some examples are given of arrays that possess interesting properties. The problem of the quality of synthesized images is treated also by analyzing the so-called additional aberrations of the arrays, and interference fringe formation, with particular regard to the conditions under which they can be eliminated.
0 2. The 2 x 2 Matrix Treatment of an Aligned Optical Element 2.1. SIGN CONVENTIONS
Consider a centered optical element, namely an element with symmetry of revolution around an axis, and two reference planes, RP, and RP,, both normal to the symmetry axis (fig. 4). The symmetry axis will be taken as the z-axis, directed in the sense of propagation of radiation, conventionally from left to right. Such an element will be termed “aligned”. A ray impinging onto the optical system is completely identified by the position q of the point where it crosses RP, and by its “momentum” p (see, for example, DRAGT[ 19821). If RP, is referred to a Cartesian system of coordinate axes i , j , with origin 0, on the z-axis, one has
where x’ = dR/dZ, y’
=
djId2.
Fig. 4. Sign conventions, in the forward propagation,for the parameters of a ray at two reference planes and for the coordinates u and u of the axial points S, and S,.
111, B 21
2X 2
MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
285
For rays in a meridional plane, which are of interest for the paraxial approximation when aberrations are disregarded, one can use r in the place of x or y , and r’ in the place of x’ or y’. Figure 4 shows the sign convention adopted in this chapter for r and r’. From this .sign convention follows the sign convention for the distances of the axial points from the reference planes. If, as just stated, r; and r; in fig. 4 are positive, the distance u of S, from RP, and the distance v of S, from RP, are to be counted as positive, since we can write
r;
rl =, U
r ; = -r2. v
(2.2)
Accordingly, the points have positive distance from the reference plane if they are on the left of it; more precisely, if one proceeds along the z-axis in the sense of radiation, one fist encounters that point and then the reference plane. Thus the distance v from RP, of any point of the z-axis in the image space on the right of RP, is negative. When the optical system contains a reflector, the sign convention for r‘ changes, as shown in fig. 5. Correspondingly,even the sign of u changes so that points on the left of the reflector have a negative axial coordinate. It is useful to introduce the sign matrix S,defined as
s=(’0
- 10).
Some examples of the use of the matrix S are given in $2.2 and 5 2.3. The sign of the radius of curvature of a spherical surface will be considered positive if, proceedingin the positive direction of the z-axis,one first encounters the surface and then the center of curvature. Hence, for example, the radius R of the spherical interface of fig. 6b (5 2.2) will be considered as positive.
Fig. 5. Sign convention for r‘ in the case of a reflector.
286
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
2.2 DEFINITION OF THE RAY TRANSFER MATRIX
If a paraxial ray is passing through a linear optical system, there are linear relations between the parameters r, ,r; of the outgoing ray at the reference plane RP,, and those r , , r; of the incoming ray at RP,: r,
=
at,
+ br; ,
ri
=
cr,
+ dr; ,
(2.4)
where the coefficients a, b, c, and d depend on the features of the optical system included between RP,and RP2 and also on the positions of RP,and RP, with respect to the optical element. Equations (2.4) may be written in a matrix form
r,
=
yr,
9
where
The arrow under the matrix symbol indicates that it describes the propagation from left to right. The matrix y is called the “ray transfer matrix” and is analogous to the ABCD matrix introduced by KOGELNIK[ 19651 (see also ARNAUD[ 19761). It may be verified that
n, Det(rvl) = ad - bc = -,
(2.7)
n2
where (fig. 4) n, denotes the (constant) refractive index on the left of RP, (namely, that of the medium containing the incident rays) and n2 denotes the refractive index on the right of RP, . If r’ indicates a Cartesian component of the momentum p, instead of x’ or y’, the corresponding matrix h J would have unit determinant, or, in other words, it would be unimodular (GERRARD and BURCH [1975]). The matrix is unimodular in most common cases when n, = n2. Evaluating the determinant in any particular application and verifying that (2.7) is satisfied provide a useful check of the calculations. The matrix formalism is particularly useful when the optical system is formed by several elements linked up axially. If y(‘)indicates the ray transfer J of the whole matrix of the ith element starting from the left, the matrix N system is given by
111, P 21
2X 2
MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
287
[Note the order of the indices in eq. (2.8).] One has It can be noted that eq. (2.5) may be solved for rl in terms of r,, yielding
2, = y-'Jzy
(2.9)
where arrows have been added to emphasize the sense of propagation. For reciprocity of the optical path equation (2.9) may be used to study the ray paths in reverse propagation, by taking into account that, due to the sign conventions, the rays in reverse propagation have slopes - r; and - r; . More precisely, we have
c,=
s,'2,
57-2= SJ,
From eq. (2.9) we derive SJ, =
sy-'s-'sJ*
or also
(2.10) When Det(y) = 1, p reduces to the well-known expression (FOG [ 1982]), where only the main diagonal elements of matrix 5 are interchanged. Equation (2.10) is a generalized expression for the ray transfer matrix in reverse [ 1983b1) and is particularly useful in treating propagation (WANG SHAOMIN optical systems containing reflectors. 2.2 BASIC METHODS FOR DERIVING RAY TRANSFER MATRICES
There are three basic methods to derive the ray transfer matrix associated with an optical element. In the present section we start choosing RP, and RP2 to coincide with the first and the last surfaces, respectively, of the optical element. (1) One method is based on eqs. (2.4), according to which:
288
[III, 8 2
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
1-1
b-0
a*-1
b.0
C
=2/R
c
.o
dsl
d=-1
Fig. 6. Determination of the elements a, b, c, and d of the ray transfer matrices of the simplest optical elements:(a) plane-parallel layer of uniform medium; (b) spherical interface; (c) spherical mirror; (d) comer-cube reflector with reference planes at its vertex.
Equations (2.11) may be used to derive the ray transfer matrices of a plane parallel layer of thickness 1 in a uniform medium (fig. 6a) and the matrix of a spherical interface of radius R (fig. 6b refers to refraction; fig. 6c refers to reflection). It appears clearly from the figures that the ray transfer matrix 5 , = ,(1) for the Iayer is
91(1)=
(A :>
whereas the matrix
plane pardel layer
L)
Y 2for the spherical interface is
92(R n,,n 2 ) = ( I n2R
(n1 - n2) n1
refraction,
(2.12)
I K I 21
2 X 2 MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
reflection,
289
(2.13)
where the label I refers to reflection. It turns out that In other words the matrix valid in reflection may be obtained by starting with the matrix valid in refraction, replacing n2 by - n, and multiplying it on the left by the sign matrix S. Recall that, in figs. 6b and c, R must be counted as positive, according to the sign convention for radii of curvature. Among the optical elements the corner cube reflector is of particular interest. For it one finds
(
y lr = -
0
O) ,
-1
comer-cube reflector,
(2.14)
when RP, and RP, are chosen as shown in fig, 6d. (2) Some ray transfer matrices may be derived starting with hill and For example, by taking the limit R --t co in eqs. (2.13), one obtains the plane for refraction and reflection respecinterface matrices y,(n,,n,) and tively:
z2.
plane mirror.
(2.15)
Other matrices may be obtained by using known matrices, hence NJ 9, and y3,in connection with eq. (2.8). For example, the matrix bJ4(l, n,,n,) associated with a plane parallel slab of thickness 1 and refractive index n,, bordered on both sides by a medium with refractive index n,,may be obtained --t = y3(n2,nl). Hence, by putting $') = y 3 ( n , , n,),, Y(') = M,,(l) and MC3)
(2.16)
290
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, B 2
Again, the ray transfer matrix of a thick lens, limited by two spherical surfaces of radius R and R2, respectively, of thickness I, and of refractive index n (outer refractive index = 1 on both sides), may be obtained by combining three factors in eq. (2.8), namely, = h+l,(Rl, 1, n), kJ(*) = h$(I), 4 M(3)= N12(R2, n, 1). Hence, (2.17) with al=l--,
(n - 1)l
1
bl = - , n
nR 1
cl=(n-l)
1
1
1
(n - 1)l
R2
R,
nR1R2
,
(n - 1)l dl=l+---. nR2
(2.18)
The thin-lens matrix NJ6 may be obtained either by taking the limit I + 0 y5,or by putting only two factors in eq. (2.8), namely, 4 M(') = Y2(R1, 1, n) and kJ(2) = Y2(R2, n, 1). Hence,
in
(2.19) where (2.20) f denoting the usual focal length. Thin-lens matrices may be applied to find the telescope matrix kl, corresponding to fig. 7a or the matrix rv18 corresponding to the general telescopic
a)
b)
Fig. 7. (a) Telescopic scheme; (b) generalized telescopic scheme.
111.8 21
2 X 2 MATRIX TREATMENT OF
f2
5 7 = r
: 1.
--( A 6,
fi + f 2
-fi
+
1
3
29 1
ALIGNED OPTICAL ELEMENT
98=[
f2
fi+ fi + 6
-~ (f1+S)
fIf2
f2
(2.21)
The preceding matrices are collected in Tables 1 and 2. Recall that the ray transfer matrix of an optical system depends not only on the intrinsic properties of the system but also on the reference planes. Up to now the reference planes have been chosen at the input and output surfaces of the optical element. But we can apply the so-called reference plane moving technique, which is a simple application of eq. (2.8) to get different matrix forms or to find some new matrices. The technique is illustrated in fig. 8: The ray transfer matrix, relative to the new reference planes RP; and RP; ,can be given, according to eq. (2.8), by
9' = yl(b2)y y 1 ( 0 where b,, b, > 0. Hence,
(:
3
Fig. 8. Moved reference planes RP;and RP;.
TABLE1 Some 2 x 2 ray transfer matrices ,M (forward propagation). System
Schematic
Matrix
Expression(s) for matrix elements
RPd
Spherical interface
Plane-parallel slab 0 1
N W N
~~
Thin lens
w X h)
~
Telescope I
~~
Telescope 11
Lenslike medium
293
h)
W W
294
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
TABLE 2 Some 2 x 2 ray transfer matrices M (reflection).
System
Schematic
Matrix
Comer cube reflector I
Spherical reflector
Flat reflector
Comer cube reflector I1
Cat's eye reflector
=(
a
+ cb,
ab,
+ b + b,(cb, + d ) cb,
c
RP;GRPi
a)
(2.22)
+d
RPlIRPz b)
Fig, 9. (a) Comer-cube reflector with reference planes at its mouth; (b) cat's eye reflector.
III,8 21
2
2 MATRIX
X
TREATMENT OF ALIGNED OPTICAL ELEMENT
295
As an example of application of eq. (2.22), we can derive the matrix M ir of the comer cube reflector with reference planes at its mouth, as indicated + in fig. 9a, starting with matrix Ylr given by eq. (2.14), and noting that b , = b, = I :
If then we consider a possible refraction at the new reference planes and apply eq. (2.8), we obtain
Another example may be the cat’s eye (Fig. 9b), with reference planes at the anterior focal plane of the lens. Its matrix ,Ms,may be calculated as !$~r =
y2(f)
!$j(f)
!$,(f)y3r yl
and is given by (2.23)
In 3 5 the preceding technique will be applied to a particular problem: Given an optical element whose diagonal elements of the ray transfer matrix M, are equal, a = d, to find the moved reference planes RP;, RP;, with b, = b2 = uo, such that B = 0 and D = 1. (3) In some cases, when the optical system consists of a rod of inhomogeneous medium, ray transfer matrices may be obtained by solving the eikonal differential equation (BORN and WOLF[ 19801):
i(n2)
=
gradn,
where s denotes the arc of the ray trajectory measured from the origin. For example, in the case of a GRIN rod, when the radial distribution of the refractive index is given by eq. (l.l), the eikonal equation in the paraxial approximation (z s) becomes
-
d2r + 2/3,r ds2
-
=
0
(2.24)
296
PRlNCIPLES AND DESIGN OF OPTICAL ARRAYS
[IK 8 2
The solution of eq. (2.24) can be written in the form
(2.25) where r, denotes the value of r at the input plane z = 0 and r; denotes the slope of the ray at z = O + . Differentiating eq. (2.25) with respect to z yields rl =
- -A rl sin (f z ) + r; cos ('z)
.
P
Hence, by putting r2 = r(l), r; = r'(l), where I denotes the length of the rod, we obtain r2 = cos (al>r, r;
=
--sin PA
+
sin ($1) r; ,
(3 (3 -I
rl
+ cos
-I
r; .
The ray transfer matrix k19of the GRIN rod therefore may be written in the form
with U,
= cos (
l a ) ,
bo =
WJm J2Bo
CO
=
-2/3obo,
do = u O . (2.26)
Note that matrix lU19 describes the optical behavior of a GRIN rod bordered on both sides by a medium with refractive index no, since it does not account for any refraction at the input and output surfaces. Such refraction can be taken into account by applying eq. (2.8),with results depending on the shape of the input and output surfaces. For example, if the rod is limited by plane interfaces, one has
I K 8 21
2 X 2 MATRIX
291
TREATMENT OF ALIGNED OPTICAL ELEMENT
fore,
(2.27)
Note that if
fiO -= 0, the circular functions become hyperbolic functions.
2.4. IMAGE FORMING MATRIX
The "moved" matrix ,M' defined by eq. (2.22) has a very important application to the study of the image formation properties of an optical system. With reference to fig. 10, let
Y=
(; );
(2.28)
denote the ray transfer matrix of the image-formingblock, from its first surface RP, to its last surface RP2, including these two interfaces, and MI=(A
+
B
)
(2.29)
C D
the ray transfer matrix from RP; to RP;. The expressions for the matrix elements A, B, C, D of Y ' are given in eqs. (2.22) in terms of the elements RP; I I I I I I I I I
I
I
RP; I I I
I I I I
I I
I I
I
I
1 I
I I
I I
I I I I
-Z
I I
Fig. 10. Definition of symbols for the determinationof the elements A , B, C, and D of the matrix relative to two "moved" reference planes.
298
[III, 5 2
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
6 2 = - z , so that u and z denote, in value and sign, the coordinate of RP; with respect to RP, and the coordinate of RP; with respect to RP,, respectively. From eq. (2.22) we have
a, b, c, d and of 6 , and b, as well. Let us put 6 , = u and
A=a-cz,
B=au+b-z(cu+d),
C=c,
D=cu+d. (2.30)
It is useful for further applications to recall the meaning of each element of
M ’. Let us start with B. -+
From the fist eq. (2.4) applied to the pair RP;, RP;, we write r,
= Ar,
+ Br; .
(2.3 1)
It is evident that r2 is independent of r; if B = 0. In other words, if B = 0, all rays crossing RP; at r , will cross RP; at the same point r,, independently of their initial slope r; . Hence B=O
(2.32)
defines the plane RP; ,which contains the images of the points of RP; . In other words, if u is the value of z such that eq. (2.32) is satisfied, RP; is the plane image of RP;. From the second eq. (2.30) we find (2.33)
When B = 0, the matrix V I J’ is called the “image-forming matrix” or “imaging matrix“. Equation (2.33)is the generalized equation for conjugated points and is often referred to as the ABCD law for image formation. It should be noted that, in general, eq. (2.33) does not coincide with the usual Gaussian law of conjugated points, since u and u denote the distances from the fist and last surfaces of the imaging block, not from its principal planes. However, the ABCD law and the Gaussian law coincide when RP, and RP, coincide with the principal planes as, for example, in the case of thin lenses. When z # u, B (which is different from 0) is a length that is called the “effective thickness” of the optical system. From eq. (2.31) it follows also that i f B = 0, then A = r&,, independently of r; .Hence, in the image-forming matrices the first diagonal element represents the transverse magnification m,, relative to the pair of conjugated planes RP; and RP;:
mo 21
2 X 2 MATRIX
TREATMENT OF ALIGNED OPTICAL ELEMENT
(2.34)
B=O.
A=m2,
299
On account of eqs. (2.30), (2.33), and (2.7), we have
- bc cu + d
ad m 2 = a - c v = -=
n,/n2 cu + d '
-
(2.35)
Analogously, we h d that the second diagonal element D represents, by definition,the angular magnification m1of the optical system. In fact, ifwe write the second eq. (2.4) in the form appropriate to the pair RP;,RP;, namely, r;
=
Cr, t Dr;
and choose ri
=
(2.36)
0, we see that
D=';
r; Hence, m , = D = cu t d .
(2.37)
It follows, in particular, that n1 , mlm2 = (cu + d ) (a - c v ) = n2
which is equivalent to the well-known Lagrange formula. Consider finally C. From eq. (2.36) it follows that, for r; = 0, namely for incident rays parallel to the axis,
(2.38) Hence, C is related to the optical power of the optical system. For example, in the case of a positive thin lens (fig. 11) one has r,/r; = -f,f denoting the R P,Z
rl
R P2
-. 1 . . ~
1 I
R Pk
I
I
I
I r;S-r,
/f
I I I
I
Fig. 1 1 . A thin lens with reference plane RP; at the rear focus.
300
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 2
focal length, and therefore C = - 1M'. In the sequel it will be shown that C or c are in any case equal to the inverse of the rear focal length fi, with the sign changed. In conclusion, the imaging matrix Y ' can be written in the form (2.39)
As is well known (GERRARD and BURCH [ 1975]), with the help of the matrix
Y ' of eq. (2.29) it is possible to express all quantities of interest for describing
the paraxial behavior of the optical system in terms of the elements of the matrix drJ of eq. (2.28). For example, one can derive the coordinate u = s, of the anterior or front focal plane with respect to the fist surface of the optical system, namely, with respect to RP, ,by noting that for such a plane D = 0. In fact, when D = 0, eq. (2.36) yields r;
=
Cr, ,
which indicates that all rays crossing RP; at the same height I , emerge from RP; parallel to each other, since their slope r; does not depend on r; . Hence, u=s1=
d
(2.40)
- - *
C
Analogously, the rear focal plane may be found by requiring that a set of parallel rays incident onto RP; at their emergence pass through the same point r2 of RP; . This requires that r2 be independent of r , , hence A = 0. From the first eq. (2.30) it is seen that the rear focal plane has coordinate z = - s2 with s2 given by s2=--.
a
(2.41)
C
The principal points may be found by looking for that pair of conjugated planes for which the linear magnification is equal to 1, hence, by imposingB = 0 andA = 1. Their coordinatesfor RP, and RP2, respectively, are given by u = h , and z = - h2 with 1
- z = - (1 - a ) , C
principal planes. (2.42)
111, B 21
2 X 2 MATRIX
301
TREATMENT OF ALIGNED OPTICAL ELEMENT
Accordingly, the front and rear focal lengths f, and f2 are given by fI
- S 1
n 1 - h I -- - A n2 c
f -3
-h - - _ 1 , focal lengths. (2.43) C
From the second eq. (2.43)it appears that c, and also C in virtue of the third eq. (2.30), are in any case equal to - l / f 2 , as anticipated. The nodal planes are, by definition,that pair of conjugated planes for which the angular magnification is equal to 1 . Hence, by imposing B = 0, D = 1, we find that their coordinates are u = hi and z = - h i , with hi and h; given by
1 h;=u=-(l-d), C
h2'-- - z = -
2:(
-1-a
C
),
nodal planes.
2.5. APPLICATION OF MATRIX METHODS TO NON-GAUSSIAN IMAGING
Equation (2.33) has been derived with reference to point sources or to straight rays, but it may be used also in the case of non-Gaussian imaging, namely, to describe the propagation of a Gaussian beam through an optical instrument. This application derives from the fact that in eq. (2.33) u and u may be considered both as (axial) coordinates of conjugate points, and as radii of curvature of the incident and emerging wavefronts at the first and last surfaces RP, and RP, of the optical block, respectively (fig. 12). In other words we can say that eq. (2.33) holds not only when u specifies the center of an impinging spherical wavefront and u specifies that of the correspondingly emerging sphericalwavefront, but also when u and u represent the curvature radii p , and p2 of those two wavefronts. This may be directly proved by noting that p , = r,/ri and p2 = r2&, so that, by using eq. (2.4), we can write
r2 p2
ar, + br; - ar,/r; + b =--ap, + b = cr, + dr; cr,/r; + d c p , + d '
Fig. 12. Transformation of a spherical wave by an optical system.
(2.45)
302
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[I14 5 2
If, now, we recall that a Gaussian beam may be considered as a spherical wave with complex radius of curvature q = q(z), which follows the Kogelnik transformation (or ABCD law) in its propagation through an optical system (KOGELNIK [ 19651, ARNAUD[ 1976]), we see that eq. (2.45) holds also with complex p, and p2 and therefore eq. (2.33) holds with complex u and u. Another generalization of the matrix method is that the elements of a ray transfer matrix may take complex values in order to describe situations where there is a transverse distribution of loss or gain, in particular the effects of a mirror with a Gaussian profile of its reflectivity (YARIV[ 19761). Such generalized matrices are referred to as “equivalent” transfer matrices. Other--equivalent transfer matrices have been introduced for particular purposes. For example, sometimes one needs to know what is the physical meaning of an optical element known only through its ray transfer matrix, such as
Since an object at u is imaged at a position u given by (see eq. 2.33), v=-
+ b, c,u + d, a,u
which can also be written as
u(s t) U
+ 1’
-
then a transfer matrix a2b2c2d2given by (WANGSHAOMIN[ 19841) + Me
with
=(:: 2>=L2
(2.46)
+ d, a,u + b,
(2.47)
1
C]U
c2=---
1 u
0 1)
is equivalent to Y in the sense that an optical element described by Ye images an object at u at the same position as the element described by matrix M. This simple consideration indicates that any optical element may be +
2
111, § 21
X
2
MATRIX TREATMENT OF ALIGNED OPTICAL ELEMENT
303
regarded as a thin lens with focal length f = - lie2, which depends on the position of the object. This result will be usefully applied in 5. It should be noted that the equivalent transfer matrices may be applied to beams, not to rays. 2.6. THE EIKONAL FUNCTION AND THE FRESNEL NUMBER OF A
CENTERED SYSTEM
The eikonal function of an optical system, which, as is well known, expresses every possible optical path from the input plane RP;to the output (or observation) plane RP; of the system, is useful to analyze both the diffraction effects qualitatively and the interference phenomena quantitatively. In this section we derive an expression for it, holding for a general centered optical system, in terms of the elements of the ray transfer matrix of the system. To this end it should be noted that the eikonal function in a centered system may be written as (COLLINS[ 19701)
(2.48) Here, Lo is the optical path length from RP; to RP; along the axis, rl and r, are the usual radial coordinates at RP; and RP;, and $, n and Y denote coefficients that depend on the properties of the optical system only. Since the momentum p of a ray is the radial gradient of the eikonal (ARNAUD [ 1976]), it follows from eqs. (2.1) that the slope of a ray in a medium can be evaluated as the radial gradient of the eikonal divided by the refractive index of the medium. For the sake of simplicity, let the refractive index be equal to 1 on both sides of the optical system. On account of the sign convention for ray slopes, we obtain
or also
r,
=
$ -r,
n
1 + -r;
n
,
r;
=
(y
- O ) r , + oYr ; .
(2.49)
By comparing eqs. (2.49) with (2.4), we find the relations between the coefficients of the eikonal function and the transfer matrix elements, namely, A B
$=-,
a = -1 , B
D Y=-. B
(2.50)
304
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
[IK5 2
Hence, the expression for the eikonal function L in terms of the ray transfer matrix elements is 1 L = Lo + -(Ar: - 2r1r2+ O r : ) . 2B
(2.5 1)
A fist application of eq. (2.51) is the evaluation of the Fresnel number of the input pupil of the optical system. It is well known that the Fresnel number of a circular aperture of radius a, illuminated by a plane wave, and seen at an observation point on the axis of the aperture at a distance 1, is given by
(2.52) where 3, denotes the wavelength. If a spherical wave with wavefront curvature l/p impinges onto the aperture, the Fresnel number becomes (CARTER [ 19821) (2.53) However, eqs. (2.52) and (2.53) hold only in the particular case,when a homogeneous space follows the aperture. In a more general case a series of optical elements may be present between the aperture and the observation point. In this case it is complicated to evaluate N by means of conventional optics. It becomes easy if use is made of eq. (2.51) by taking into account that the Fresnel number is given by the optical path difference between the wavelets arriving at the observation point from the edge of the aperture (r, = a) and from its center (rI = 0), divided by ;A. Since the observation point is on the axis of the instrument, one has to put in eq. (2.5 1) r, = 0. Hence, the Fresnel number due to the optical system included between RP; and RP; is simply given by
(2.54) In the case of an input phase difference caused by an incident spherical wave of curvature radius p, the complete form of the Fresnel number in terms of the transfer matrix elements is (FANDIANYUAN [ 19831) (2.55)
111, B 31
MATRIX TREATMENT FOR MISALIGNED ELEMENTS
305
It is easily verified that for a plane wave ( p = a)incident on the aperture and followed by free space [A = 1, B = I as given by eq. (2.12)], expression (2.55) reduces to eq. (2.52).
6 3. Matrix Treatment for Misaligned Elements 3.1. THE 4 x 4 RAY TRANSFER MATRIX
The 2 x 2 ray transfer matrices of the previous section hold for centered systems only. In practice, however, centered systems do not exist. An optical system has always some element slightly decentered or misaligned because of tolerances of manufacturing, mechanical and thermal instabilities, and other reasons. To treat paraxial ray propagation through a misaligned optical ele[ 19793). ment, one can have recourse to 4 x 4 matrices (WANG SHAOMIN Let us start with a misaligned optical element used in forward propagation. Let the element, specified by the matrix
Y
=(:
1)’
be included between the two planes RP; , RP; (fig. 13), with a symmetry axis z’ that makes the angle E’ with the reference z-axis. Assume ,M to include the effect of possible refractions at RP; and RP;. Let RP, and RP, be the two planes normal to the z-axis that are as close as possible to RP; and RP;, respectively. The distance I from RP, to RP, is h o s t equal to the distance from RP; to RP; . Finally, let us denote by E the distance from the z-axis to the point where the z’-axis crosses RP, (or RP;). In other words, E and E‘ are the “ray” parameters of the z‘-axis.
RS
RR2
Fig. 13. Misalignment diagram for a system in forward propagation.
306
PRINCIPLES AND DESlGN OF OPTICAL ARRAYS
[III, B 3
A paraxial ray w ill be specified by the parameters rl, r; at the input plane RP, and by r,, r; at the exit plane RP, .To establish the (linear) relations among r,, r; and rl, r ; , we may introduce the ray parameters TI, 7; at RP;, and T,, 7; at RP;. Clearly, we have
7, = a?,
+ b7; ,
7;
= c7,
+ d7; .
(3.2)
On the other hand, 7, and 7; are simply related to rl and r; (in the paraxial approximation) by 7; x r;
7, x rl - E ,
- E‘ ,
(3.3)
and r,, r; are related to 7,, 7; by
+ E + iEJ ,
r, x F,
r; x 7;+ 8’
.
(3.4)
By combining eqs. (3.2) to (3.4) and writing them in matrix form, we obtain
where u, 8, y, and 6 are called “misalignment matrix” elements, and they are given by a=l-a,
p=l-b,
6=1-d.
y=-c,
(3.6)
To suit the needs of matrix multiplication, eq. (3.5) can be written in the following form: r$4) = M(4)r(4), 4
(3.7)
1
where a b
CIE
BE’ (3.8)
0 0 0
1
This is a general form of 4 x 4 (or “augmented”) matrix that is very useful in treating misaligned optical elements or systems. In the case of reflectors one simply has to put
6 = - 1- d ,
for reflectors,
(3.9)
because in eqs. (3.2) and (3.4) the sign of both and r; must be changed according to the sign conventions recalled in 8 2.1.
307
MATRIX TREATMENT FOR MISALIGNED ELEMENTS
Fig. 14. Case of a system where the choice of the misaligned axis z' is arbitrary.
The augmented matrices as derived here only hold for rays and not for beams (WANG SHAOMINand WEBER[ 19821). It is worth noting that when the z'-axis of a misaligned optical element is not well defined, as occurs with spherical interfaces or mirrors, the misalignment parameters E and E' are not uniquely determined but can be arbitrarily chosen, at least within certain limits. For example, in the case of fig. 14, where R is positive, one can choose either E' = 0 and E = E ~ or, E = 0 and E' = E J R ,or other values of E and E', provided that E
-
R
+ E'
=
El const. = R
(3.10)
Depending on the choice of E and E', different augmented matrices are found to describe the same case; however, they yield the same results when applied to practical problems. A particular case of interest is when the misaligned optical element is a plane-parallel layer of uniform medium. In this case [see eq. (2.12)] a = 1, b = 1, c = 0,d = 1, and therefore a = 0, /3 = 0, y = 0 , b = 0, and the augmented matrix MI4)(l)becomes /1
1 0
o\ (3.11)
\o 0 0
I /
The above matrix allows us to write the ray transfer matrix y'(4) relating the ray parameters at two planes u and z by applying eq. (2.8): Hence, (3.12)
\ o o o
1
I
308
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 3
where A, B, C, and D are still given by eqs. (2.30) and
A=fl-iiz,
O+=CX-YZ,
yT=y,
&=a.
(3.13)
The quantities O+ , A , yT, and & are referred to as “misalignment matrix elements for the total optical system”. Constructing the 4 x 4 matrix of an optical element is simple if use is made of eqs. (3.6) and (3.9). For example, for a spherical mirror of radius R one has from the second eq. (2.13) a = 1, b = 0, c = 2/R, d = 1 (and 1 = 0), and therefore, ci = 0, /3 = 0, y = - 2/R, and 6 = - 2. Hence, Nl:4)(R, E, E ’ )
=
/I 0 2/R 1 0 0 \o 0
0 o \ - 2 ~ l R -28’ 1 0 0 1 /
(3.14)
It follows directly from expression eq. (3.8) for Yc4)that an aligned system specified by the 2 x 2 matrix 9 of eq. (3.1) may be described also in terms of the augmented matrix /a b
0 O \
(3.15)
\o
0 0 I /
since, for the aligned elements, E = E’ = 0. This observation allows one to apply the 4 x 4 matrix method to the propagation of a paraxial ray through an optical system containing aligned and misaligned elements. In particular, the 4 x 4 matrix associated with a layer of thickness I of uniform medium (without refraction at its ends) coincides with + Mi4)(l),indicating that a plane-parallel layer of uniform medium is insensitive to misalignments. Analogously, the 4 x 4 sign matrix S(4) is found to be /I
0 0
o\ (3.16)
\o
0 0
I /
As discussed in 3 2, when a paraxial ray through a centered optical system is reversed, a new ray transfer matrix is derived from the original one by interchanging the elements a and d and dividing by the determinant of the matrix (see eq. 2.10). When the paraxial ray is passing through a misaligned
111%I 31
MATRIX TREATMENT FOR MISALIGNED ELEMENTS
309
optical system in reverse propagation, a reversed 4 x 4 matrix also can be obtained as follows: M(4) = S(4)[M(4)]+ iS(4) t
([S(4)]-
1
= S(4)).
(3.17)
Hence,
PE’\
/ d b a’&
(3.18)
\ o o o
q ’ /
with a’ = by
- da ,
p‘ = q’
=
p’
=
b6 - d/3,
y’ = a y
-CM,
6’ = a6 - c/3, (3.19)
Det(bl),
which, on account of eqs. (3.6), becomes
(3.20)
\o
0 0
1
1
in the case Det( rvl) = 1. It should be noted that the use of these augmented matrices may be laborious, especiallywhen many 4 x 4 matrices must be multiplied by one another to apply eq. (2.8) to some practical system. However, the treatment may be greatly simplified by the use of ray transfer flow graphs (HENLEYand WILLIAMS [ 19731). (For a brief introduction to these flow graphs and some examples of their application to the optical problems treated in this chapter see WANG SHAOMIN and RONCHI [ 19861).
3.2. ElKONAL FUNCTION FOR MISALIGNED SYSTEMS
In the case of misaligned systems the general expression for the eikonal function (2.48) must be replaced by
L = L, + qr1 + $r2
+ &prf - Or,r2 + f Yrg,
(3.21)
where the linear terms account for the variations of the eikonal function due to misalignment.
310
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
By proceeding as in 5 2.5, we find
Hence, r 2 = -1[ $ r I + r ;
n
r ; = [ F - n ] , , + - - r!P; + - ++cp$ .
+p],
n
o
(3.22)
Equations (3.22) must be compared with those derivable by means of matrix given by eq. (3.12), namely,
M '(4) +
r,
=Ar, t
Br;
+ a,,.& + he',
r; = Cr,
+ D r ; + y+ + b e ' ,
(3.23)
where the coefficients are given by eqs. (2.30) and (3.13). Hence, we find
a = -1 ,
A
$=-, B
B
D Y=BY
(3.24)
and for coefficients of the linear terms in eq. (3.21) cp=
(w+ h E ' ) , B
+=
[(BYr - D%)& + (Bs, B
- %)&'I .
(3.25)
Substituting eqs. (3.24) and (3.25) into eq. (3.21), the eikonal function for misaligned systems can be completely written in terms of ray transfer matrix elements as follows: L
=
1 Lo + -[A< - 2r,r, + Dr; + 2(%e t &.&')r, 2B + 2 { ( B h - D%)E + ( B S , - D&.)&'}r2],
(3.26)
The expression (3.26) for L will be used in 4.4 to study the interference effects in the plane of the synthesized image of an optical array. Note that if no optical instrument is placed between RP; and RP;, one has Lo = 1, A = 1, B = I, D = 1, M = 0, /? = 0, y = 0, and S = 0; hence, cc, = 0, & = 0, = 0, and S, = 0, and eq. (3.26) reduces to
(3.27)
MATRIX TREATMENT OF ARRAYS
311
# 4. Matrix Treatment of Arrays 4.1. GENERAL CONSIDERATIONS
An array may be considered as an ensemble of regularly misaligned optical elements. As such, an array may be treated by utilizing the considerations of 3, holding for generally misaligned elements, with important simplifications due to the regularity of the misalignment. For each impinging beam the emerging beam is constituted by the ensemble of the beams (“individual” beams) emerging from the individual elements. As a first step, the study of the optical properties of an array may be carried out by considering a single ray impinging onto each element of the array, for example the ray through the axial point (“vertex”) of each element, and the corresponding emerging ray. The fact that each element has a finite-size input pupil, so that the various rays belonging to the beam impingingonto the element behave in a different manner, is taken into account as a second step by studying the so-called “residual aberrations,’. In any case the size of each element must be so small,and therefore the beam entering the element so thin, that the ray through the vertex gives a good indication of the behavior of the individual thin beam. 4.2. THE RAY TRANSFER MATRIX FOR AN ARRAY
Because of the regularity of the misalignment of the elements of an array, the position of the ith element may be easily identified by the position E, of its vertex (fig. 15), and the inclination .$ of its axis may be simply expressed in terms of
I
I I
I
I
/
!
-1 Fig, 15. An array of optical elements.
312
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
WI,8 4
the geometrical parameters of the array. An important role is obviously played by the shapes of the input and output surfaces of the array, defined as the two surfaces containing the input and output vertices of the elements. For example, in the case of fig. 15 the input surface of the array is a spherical surface of (positive) radius R, and the output surface is a concentric sphere with radius R - I, where I is the common length of the elements. From the figure it appears that we can write E; =
- Ei/R .
(4.1)
Consider, now, a ray impinging with slope r; on the vertex of the ith element. Assuming a reference plane RP, through that vertex, the ray is specified by rl , r ; , with
r,
(4.2)
= Ei.
Hence, we can write
or, in matrix form, 1
(:)
=(-:
0
o!
(a)
(4.3)
Introducing eq. (4.3) into eq. ( 3 . 9 , where LJ = kli (and therefore ai,pi, yi, S,) may be different for different array elements, yields
or also, r2 = rv$)r,
with
111.8 41
MATRIX TREATMENT OF ARRAYS
313
where
$) is the ray transfer matrix of the ith element of the array. If the elements of the array are equal to one another, WJi and also are independent of i: Nli = 2, ,M$ = Nla, and
WJ$
with
Being independent of i, the matrix ,Ma is the ray transfer matrix of each element and therefore also of the array (whence the label “a”). One passes from one element to another by varying r,. ,Ma may be used as any ray transfer matrix examined in Q 2.3. However, a peculiar feature of WJa is that 1 Det(hla) = d - - (@ - bb)
R
(4.8)
depends on R and is generally different from n,/nz.Accordingly, the arrays are called “nonlinear” optical systems. It could be noted that, in general, different values of i correspond to different input planes RP,,; however, in the paraxial approximation the input planes of the individual optical elements of the array almost coincide. The most important application of WJa is to establish the relations between the ray parameters at two (moved) planes RP;,RP; (fig. 10, Q 2.4). By proceeding as in Q 2.4, we find that the matrix NJL for the array is given bY
314
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
with A = 1 - - B+ - 62 R R’
c = --s
R,
B = ( 1 -:)u
+6+
(t
-d)z,
6U
D=--+d. R
(4.10)
Equations (4.10) are derived by substituting for a, b, c and d in eqs. (2.30) the expressions (4.7) for the elements a’, b‘, c’, and d’. The matrix NJ; describes the effect on the plane z of all elements of the array. It indicates that, besides the conventional individual images, there is a synthesized image (at the plane z for which B = 0) which possibly has some unique properties. All the first-order synthesized imaging properties are embedded in this matrix. is The physical meaning of the various elements of the 2 x 2 matrix the same as that described in 5 2.4 for the matrix b J ’. By denoting by Y the value of z for which B = 0, we obtain a conjugate distance equation for the synthesized image:
2:
I/=
(R - B)u/R + 6 (Rd - SU)/R
(4.11)
With the help of eqs. (3.6) it can be easily verified that
V=
+ b + (uR - B)u/R cu + d + ( Y R - 6)u/R
au
(4.12)
Evidently, in general, the image plane for the best focus does not coincide with the image plane of the individual element, which is given by the ABCD law (2.33). Therefore, the synthesized image is not a Gaussian image. When B = 0, the matrix bl1:is called “synthesized image-formingmatrix“. The geometrical-optics parameters of the array, namely, the angular and transverse magnificationsm, and m, ,the focal lengths fi and f i ,the distances s, and s2 of the foci from the input and output surfaces of the array, and the distancesh, ,h, ,h i , and hi of the principal planes and of the nodal planes from the same surfaces, may be written either in terms of the elements of the matrix NJa or in terms of the parameters of the individual elements of the array. In the former case one has simply to substitute a’, 6’, c’, and d‘ for a, 6, c, and d in eqs. (2.35), (2.37), and (2.40)-(2.44), V for u, and Det(NJ,) for
MATRIX TREATMENT OF ARRAYS
315
n, I n z , obtaining
m, = c’u + d ,
(4.13)
m, = a’ - c’V ,
(4.14)
31
=
32 =
hl
d --
c’
(4.15)
’
a’ --9
(4.16)
C’
Det(M,)
=
-
d
(4.17)
Y
C’
1 - a’
(4.18)
h2=-,
C’
(4.19) 1
f2=
-2’
(4.20) (4.21)
hi
Det(M,)
=
- a’
(4.22)
C’
In the latter case one has to express a‘, b’, c’, and d in eqs. (4.13)-(4.22) in terms of a, b, c, and d by means of eqs. (4.7), obtaining (4.23) (4.24)
dR 6
(4.25)
s1=-,
R - 8 -R - l + b s2=----
6
6
Y
(4.26)
316
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
/3d - b6 6
[III, 8 4
(4.27)
hl=-,
(4.28)
fl
=
d(R - 1 + b)
+b,
R
(4.30)
f2=b’
hi = h;
=
(4.29)
R(d - 1)
s
9
(R - I + b)(l - d ) - b. 6
(4.3 1) (4.32)
Note that, in general, fl # f2, which confirms the non-Gaussian character of array imaging. In all the preceding expressions 6 = 1 - d, except in the expression for reflectors, when 6 = - 1 - d, in agreement with eq. (3.9). Matrix lMa plays the same role and may be used in the same way as the matrix Y of eq. (2.6). Hence, a reversed propagation matrix pafor arrays may be deduced by applying eq. (2.10), and an augmented 4 x 4 matrix for arrays can be given analogous to eq. (3.8) to treat misaligned arrays:
(4.33)
-
5
y=-,
R
(4.34)
where the plus (minus) sign holds for transmission (reflection). As an example of application of the matrix we can show that some arrays, like the thin lens array, are misalignment-insensitive devices. From
111, I41
table 1 (0 2.3) one derives for the thin lens array b = 0, d = 1, and I RP, coincides with RP,), and therefore, from (4.34) a=o,
317
MATRIX TREATMENT OF ARRAYS
p=o,
y=o,
s=o,
=
0 (since (4.35)
which constitute the requirements for insensitivity to misalignment. The same property is presented also by corner-cube reflector arrays and by cat’s eye reflector arrays (table 2, 0 2.3) for which b = 0, t = 0, and d = - 1.
4.3. ADDITIONAL ABERRATIONS FOR ARRAYS
The treatment of the previous section describes the behavior of a single ray per element of the array, the bbprincipal”ray, that impinges onto the element at its input vertex. The other rays incident on the same element are expected to behave differently, even if only slightly, since the input pupil of the elements is assumed to be “small”. In other words, there are aberrations due to the finite (non-infinitesimal) size of the input pupil of each element of the array, which are denoted either as “additional” aberrations or “synthesized imaging’’ aberra[1984]), in order not to tions (WANG SHAOMINand ZHOU GUOSHENG confuse them with conventional aberrations, which are due to the h i t e size of the entire optical device. It should be noted that in an array the conventional aberrations are generally eliminated by choosing the position of each element of the array suitably, according to the well-known principle of Fresnel lenses and mirrors. To evaluate the additional aberrations of an element of the may, for example, the ith element, one has to consider two incident rays, the principal ray and a marginal one, starting from the same point source on the object plane, and to find the transverse coordinates r,, rz,o of the two points where the emerging rays cross the plane of the synthesized image z = V. Here, V is given by eq. (4.11) or (4.12). In the sequel to this section the label “0” refers to the principal ray, and the label i specifjiag the element of the array will be suppressed to simplify the writing. Clearly, the ith element of the array is misaligned, so that use will be made of the equations in 0 3, in particular of eqs. (3.12) and (3.13). By introducing WJ’(4) given by eq. (3.12) into the equation
ri4)(V ) = -+M ’C4)
r(P)(u),
where the coordinates u and V of the object and synthesized image are evident,
318
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
we obtain r*(V) = (a - cV)r,(u)+ [(au + b) - V(cu t d)]ri(u)
+ (a - yV)&+ (/I - SV)&’.
(4.36)
This equation holds for all rays incident on that element of the array which differ from one another only by r; . For the principal ray we have (see fig. 16)
and for the marginal rays U - r;,o f r;(u)N & f 0 - r,(u) -
U
U
3
where CT denotes the radius of the pupil of the element. Hence, the synthesized imaging aberration A = r2(V) - r2,0(V)is given by U
A=f-[autbU
(4.37)
V(CU+~)].
This is a general expression for additional aberrations of synthesized images. It appears from eq. (4.37) that the synthesized imaging aberrations are eliminated if the following condition is satisfied:
(4.38)
2
LU Fig. 16. Scheme for the evaluation of the additional aberrations.
111, § 41
319
MATRIX TREATMENT OF ARRAYS
hence, recalling eq. (2.33), if the synthesized image coincides with the image of the individual elements, Y = u. Introducing eq. (4.38) into eq. (4.12) yields (OlR - / ~ ) ( c + u d) = (YR- 6 ) ( +~ b ) ,
(4.39)
which says that with any array there is a position u of the object, that satisfying eq. (4.39), for which V = u and the synthesized image is free from aberrations. This value of u is given by u=
R(l - d) + Id - b(d + 6) c(R - 1 + b ) + a6
(4.40)
and corresponds to
The condition u = V is satisfied anyway, as follows from eq. (4.12), if (4.41) Equations (4.41) are called the abyb condition. Note that ifthey are satisfied the synthesized image is free from additional aberrations for any position u, rl of the object.
4.4. INTERFERENCE EFFECTS IN THE SYNTHESIZED IMAGE
By definition, the synthesized image given by an array is the superposition of many beams. The fact that, in general, the additional aberrations of a misaligned element are different from zero means that, even in the case of a point source, each individualemerging beam crosses the image plane in a region of nonvanishing area around the image point. Hence, when the array is illuminated by coherent radiation, interferencephenomena are expected to take place. In fact, interference fringes may be observed (fig. 17a) through a synthesized image, even if sometimes they are not observed (fig. 17b). The interference phenomena may be studied by means of the eikonal function L of misaligned elements of eq. (3.26). Let us fist consider a point source in front of an array simply constituted
320
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 4
Fig. 17. Examples ofimages given by an array illuminated by coherent tight: (a) With interference fringes; (b) without interference fringes.
by two elements (fig. 18) and two emerging rays, which arrive at the same point of a plane z (not necessarily the synthesized image plane). Let us denote by E > 0, e' = - e/R the misalignment parameters of the upper element, R being the radius of curvature of the input surface of the array. The parameters of the lower element will be - E, - 6'. According to eq. (3.26), the eikonal function along the lower ray will differ from that along the upper ray by the sign of the linear terms. Hence, there will be a difference of optical path along the two rays given by
It appears from eq. (4.42) that AL is a function of r, ;hence, one must expect
Fig. 18. A two-element array.
111, 41
MATRIX TREATMENT OF ARRAYS
32 1
to observe interference fringes in the plane of the synthesized image. The condition for the absence of fringes is (4.43) With the help of eqs. (2.30), (3.13), and (3.6) this equation reduces to eq. (4.39) for the elimination of additional aberrations. Hence, we conclude that fringes are eliminated, together with the additional aberrations, when the synthesized image coincides with the images of the individual elements. However, in most applications of arrays the synthesized image does not coincide with individual images. Thus, in general, interference fringes are expected to be observed through the synthesized images. For an extended source the treatment is a little more laborious, since the field (e.g., E ) at any point P2 of the z plane should be calculated by means of the so-called diffraction integral. For a single generally misaligned element we have W 2 )=
S { E ( P ’ ) K(P1, P 2 ) 8x1
Y
(4.44)
XI
where P I is a point in the object plane, the integration should be extended to the surface of the object, and K ( P l , P 2 ) is given in terms of the eikonal function L by K f P ,,P z ) =
i
-exp (ik,,L) , LB
(4.45)
where L is given by eq. (3.26) and ko is the free-space wavenumber. It can be [ 19701) noted that eq. (4.45) is a generalization of Collins’ formula (COLLINS and reduces to it when the optical instrument is composed of a single centered element, for which L is given by eq. (2.51). Note also that when no optical instrument is placed between the object and the observation plane, L assumes the expression in eq. (3.27) and eq. (4.44) reduces to the Huygens-Fresnel principle. When the array is composed of many elements, each specified by a couple of indices j and k, eq. (4.44) must be replaced by (4.46) where Kjkdepends on the indices j and k through the misalignment parameters
322
[III, § 4
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
of the (j, k)th element appearing in the associated expression Ljk of L. To perform the integrations, it is useful to pass to Cartesian coordinates by replacing the scalar parameters r, r’, E, and E‘ by vectors: ri = ( X i , Y i )
r2 = ( X 2 7 Y 2 )
9
8
9
= ( E x , Ey)
8‘
$1
= &(,:
and the products in the expression for L by scalar products. Thus we will write for the element of orders j and k Ljk(pl,
(4.47)
p2) = L ’ ( ~ Ip2) , + ex1 + f v l + gX2 + hy2 7
where L‘(P,,P z ) is given by L ’ ( P , , P 2 ) = L 0 + A $ - 2 r 1 - r 2 +D< =
Lo + A(x:
+ y:)
- 2(x1x2 + y I y 2 )+ D ( x + ~~
3)
(4.48)
and is independent of j and k, whereas
+ &EL),
e = 2(c+,
g
=
2(B7+ - DLI+)E,
f = ~ ( C G Z E , , + &$),
+ 2(B&
- D&)&:,
h = 2(B7+ - D%)E, + 2(B& - D & ) E ~ ,
(4.49)
depend on j and k. For the element of indices j and k we have E, =
2jax,
cU = 2kay,
I&
8’Y
- 2 j -0, . R a = -2k2, R =
j = f 1, ..., + _ M ,
k = f l , ...,& N ,
(4.50)
2ax and 2ay being the sizes of the pupil of the individual element in the x and y directions, respectively. By introducing eqs. (4.50) into eqs. (4.49) and recalling eqs. (2.30), (3.6), and (3.13), we obtain e = ejk = 4e0aJ, h
=
hik
=
f
= fik =
(:)
eo=-z y--
g = gjk = 4g0aJ, (4.5 1)
4goa,,k,
with
4eoayk,
P
+a---,
R
(4.52)
111, § 51
PSEUDO CONJUGATOR ARRAYS
323
Then eq. (4.47) becomes Ljk =
L'(pI, p2) + 4(e0xl + g 0 ~ 2 ) j g x+ 4(e0~,+ g0~2)kgy
(4.53)
with L' still given by eq. (4.48). The field distribution obtained by introducing eq. (4.53) for L into eq. (4.45) should now be summed over the indices j and k, according to eq. (4.46). The calculation of E(P2) is particularly easy when the source is a point source. The intensity distribution in the image plane has the same features as that produced by a rectangular diffraction grating formed by (2M + 1) x (2N + 1) equispaced elements (MA JIAN, Luo XUEMIN, WANG SHAOMIN, RISALITIand RONCHI[ 19861) and presents main maxima, separated by 1B Ax2 = -, x direction, 2gxgo
LB Ay2 = -,
y direction,
2 aygo
with a number of secondary maxima in between. The halfwidth of the main maxima is found to be ax2 =
Ax2
(2M + 1) ,
6y,
=
AY2 (2N + 1) *
~
It follows that the complete elimination of fringes requires go = 0, which again yields the condition (4.39), first found for the elimination of the additional aberrations. In such a case the fringes are infinitely large. Then one concludes, as in the case of a two-element array, that, in general, the interference fringes disappear when the synthesized image coincides with the individual images. This happens either for any position of the object, when the a/?$ condition is satisfied, or for a particular position of the object, given by eq. (4.40), when the a/?$ condition is not satisfied. Clearly the fringes disappear from the image even when 6x2 and Sy2are larger than the size of the beam emerging from the array at the best focus.
0 5. Pseudo Conjugator Arrays 5.1, PHASE-CONJUGATIONMIRRORS
As anticipated in the Introduction, some arrays behave like phase conjugators. This property is made possible by the fact that the determinant of the
324
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[IIL 8 5
ray transfer matrix rvla of an array may be different from nJnz (see eq. 4.8), which means that arrays do not behave like conventional (linear) optical systems. To analyze which arrays demonstrate the phase-conjugator property, let us briefly review, in terms of ray transfer matrices, the properties of the so-called phase-conjugation mirrors (PCM). It is well known (Au YEUNG, FEKETE,PEPPERand YARN [ 19791) that phase-conjugation mirrors, which are usually obtained by some nonlinear process such as degenerate four-wave mixing, can reflect an incident field so as to yield its conjugate replica, that is, to leave wavefront and amplitude distribution unchanged, whatever they are. In fig. 19 the reflection property of the PCM is compared with that of the flat reflector and with that of the cat’s eye reflector. It is clear that the PCM ray transfer matrix (Au YEUNG,FEKETE, PEPPERand YARIV[1979J) is
yl=(’0
”>.
-1
for which Det(lMI) = - 1 .
(5.2)
It is a general and characteristic property of phase conjugators to have the determinant equal to - 1. As a fist consequence of eq. (5.2), the matrix 9,for reverse propagation is equal to the matrix yIitself:
Fig. 19. Comparison of the properties of a flat reflector, a cat’s eye reflector, and a PCM.
111, I 51
325
PSEUDO CONJUGATOR ARRAYS
The matrix ulJ may be used to prove some properties of a PCM;for example, its ability to compensate for wavefront distortion when a wave passes through an inhomogeneous medium, is reflected by a reflector, and passes back through the same inhomogeneous medium (fig. 20). If a flat mirror is used as reflector, the whole ray transfer matrix is given by
2ac
ad+ be
If the reflector is a cat's eye reflector, we have
- (ad + bc)
("c :)=(%:) (-: -:) (: :)=(
-2ac
-(ad+bc)
On the other hand, for the PCM mirror we have
In other words, with the iirst two reflectors the parameters of the emerging beam depend on the medium crossed, whereas for the PCM the effect introduced by the medium after the first passage is compensated for at the second passage, so that the ABCD matrix does not depend on the characteristics of the medium. The same result is obtained if the inhomogeneous medium is misaligned, so that it is described by the 4 x 4 augmented matrix $4) given by eq. (3.8). The augmented matrix for the (aligned) PCM mirror is /l 0 0 o\
\o
0 0
l i
picture"
Fig. 20. Experimental apparatus to test the property of some array or PCM of compensatingfor the wavefront distortiondue to the inhomogeneitiesin a test tube with 0 , b, c, dray transfer matrix elements.
326
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
and we find that where ?1(4) is given by eq. (3.18). Another property of PCMs which is easily proved by the matrix algorithm is that they are insensitive to misalignment. This property may be easily proved (WANG SHAOMIN [ 1983a1) with the help of eqs. (3.6) and (3.9), by noting that a = 1 - a = 0 , 8 = I - b = 0 (since I = 0), y = - c = 0, and 6 = - 1 - d = 0. The features of a PCM may be revealed even by an equivalent matrix RAIIof the type (2.46). On account of expression (5.1) for the ray transfer , we find matrix 2
c2=--,
U
so that
Note that Det(Nl?,,)= 1. By recalling the expression (2.13) of the ray transfer matrix lW2,(R) of a spherical mirror with curvature radius R, we conclude that a PCM is equivalent to a spherical mirror whose curvature is identically equal to the curvature of the incident wavefront. This means that a phase conjugator behaves like a zoom lens, by keeping the image distance identically equal to the object distance:
v = -24.
(5.10)
Note that eq. (5.10) holds both in reflection and transmission, in spite of (or because of) the different sign conventions for the axial distances assumed for the two cases. Another interesting result derivable by comparing eq. (5.9) with eq. (2.39) is that both the transverse magnification m, and the angular magnification m ,of a PCM are unity. 5.2. PHASE-CONJUGATOR ARRAYS
There are several arrays that can perform phase conjugation. For example, consider an array of spherical mirrors (FREEMAN,FREIBERG and GARCIA
111, $ 51
PSEUDO CONJUGATOR ARRAYS
321
[ 19781) with
l=O,
R=-u,
(5.11)
where R now represents the radius of curvature of the input surface of the array. With the help of eq. (4.11) or (4.12) we may easily verify that eq. (5.10) is satisfied. Even an array formed by corner-cube reflectors (SATO,NAGURA,IKEDA [ 1982]), for which a = - 1, b = c = 0, and d = - 1 (see and HATSUZAWA eqs. 2.14), satisfies eq. (5. lo), provided that eqs. (5.11) are satisfied. However, in both cases the second eq. (5.11) indicates that, when u varies (namely, in the case of objects at different distances from the array), R must be varied if phase conjugation must be realized. To change R means to change the mutual positions of the elements of the array, which is generally obtained by means of a rather complicated mechanical structure including pistons, according to the phase-conjugation principle of the well-known coherent optical adaptive techniques (COAT). The complicated COAT structure may be avoided with plane arrays (R = a),provided that the optical elements composing them have d=-l.
b=O,
(5.12)
In this case eq. (4.11) yields directly V = u/d = - u. Clearly, arrays of this type offer many advantages over phase conjugators based on COAT: an unlimited dynamic range, much higher response speed, lower weight, lower cost, and greater simplicity. It is immediately verified that for R = 00 the ray transfer matrix Wla of eq. (4.6) becomes (5.13)
and the equivalent transfer matrix (2.46) becomes (5.14) c2 being given
c2
=
by
d 1 - -. u t b u
~
(5.15)
At the same time expression (4.11) for the coordinate of the synthesized
328
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
image takes the simpler form y=- u + b d
[III, c2 5
(5.16)
If, in addition to R = a,we choose optical elements with b = 0, d = - 1, takes the form of the matrix of PCM (see eq. 5.1), and c2 becomes - 2/24 as for PCM (see eq. 5.8). Clearly, arrays operating as phase conjugators offer significant potential advantages even with respect to PCM (or other nonlinear devices) in addition to lower weight, lower cost, and greater simplicity: Since they are composed of passive and linear elements, arbitrarily weak signals can be conjugated and the return frequency is always equal to the input frequency. However, it must be noted that the principles of wavefront replication are quite different between nonlinear processes and arrays. In the case of nonlinear processes a wavefront is replicated point by point, with continuity, whereas in the case of an array some elements of the wavefront are not replicated (fig. 21), and, moreover, the elements that are replicated are "reversed". Consequently, the phase conjugation performed by an array is neither complete nor perfect: This is the reason why phase-conjugation arrays are called "pseudo conjugators". Despite this imperfection, pseudo conjugator arrays have many practical applicationsbecause they are compact imaging devices with unit magnification, as follows by comparing the matrix Ye of eq. (5.14) with the general ray transfer matrix NJt' of eq. (2.39). Pseudo conjugator arrays may be designed to operate both in backward and/or in forward propagation. An example of an array operating both in forward and in backward propagation is the GRIN rod array described in 5 1 (see fig. 1). Even the cat's eye may be used to form arrays operating in both directions (see § 5.3). As a general method, given an optical (linear) element and the associated ray Pila
YI
de art fromS simuPtaneously
s*+":fsimultaneously
1
Fig. 21. Approximate replica of a wavefront by an array.
I__&& _I
Examples of pseudo-conjugators arrays (forward propagation). System
Schematic
M=(-'
Cat's eye array
I
I
a
V
Expressions for matrix elements
Matricesa
0
M'=(-'
0
-
3 I
M IJI v
O),
-1
O) -1
Ray transfer matrix of a single element of the array, as defined by eq. (2.6) (M) and eq. (2.22) (M'),
System
Schematic
330
TABLE4 Examples of pseudo-conjugator arrays (reflection). Expressions for matrix elements
Matrices"
I
= 0
I
-
I
1
r 4 l \
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
- 0 I I 1 I
I
0 I
I
n
I
- 0
J II
a
-px
c (
i
B
IL
B
WI,8 5
Half-bead array
Concave-mirror array (radius R > 0)
Concave-mirror array (radius R > 0)
PSEUDO CONJUGATOR ARRAYS
GRIN-fiber array
Bead array
a
Ray transfer matrix of a single element of the array, with respect to reference planes RP (M) and RP’ (M’).
-
W
w
332
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
[III, 9 5
transfer matrix Y [with Det(9) = 11, one can investigate if it may be used to form pseudo conjugator arrays by examining the matrix 9’obtained by the plane-moving technique [see eq. (2.22) or (2.29)]: If a value u, of u may be found for which eqs. (5.12) are satisfied for z = - u,, that element may be used to form a pseudo conjugator array. By using eqs. (2.30), eqs. (5.12) can be written in the form UU,
+ b + U,(CU,t d) = 0 ,
CU, t
d=-1;
hence,
(a-l)u,=-b,
~uO=-l-d.
(5.17)
Since ad - bc = 1, eqs. (5.17) may be simultaneously satisfied only if
a=d.
(5.18)
This condition is a generalized condition for obtaining plane arrays as phase conjugators. From eqs. (5.17) one has
b l-a
l+d
uo = -= --
C
(5.19)
Tables 3 and 4 show some examples of pseudo conjugator arrays. From the standpoint of the additional aberrations it may be noted that for the plane arrays (R = co), for which V is given by eq. (5.16), eq. (4.37) becomes
A
=
d
k - ( l - d - CU). d
(5.20)
It follows that additional aberrations are eliminated if u=-
l-d C
(5.21)
namely, if the object is placed at the principal plane of the individual elements forming the array (see eq. 2.42). In the case of pseudo-conjugator plane arrays, for which d = - 1, eq. (5.20) becomes A
=
~ ( C-U2),
(5.22)
which indicates that when the object is at u = 2/c, pseudo conjugator arrays with c # 0 may behave better than arrays with c = 0.
111, J 51
PSEUDO CONJUGATOR ARRAYS
333
Let us now examine in detail some arrays with phase-conjugation properties that have been built, tested, and in some cases also applied. 5.3. CORNER-CUBE ARRAY
As anticipated in $! 1, corner-cube arrays have been used to approximate phase conjugation (JACOBS [ 19821) and to compensate for optical inhomogeneities in laser amplifiers (ORLOV,VIRNIK, VOROTILIN, GERASIMOV, KALININand SAGALOVICH [ 19781) and optical resonators (MATHIEUand BELANGER[1980]). In such applications the comer-cube arrays were the common plastic retroreflectors with about 2.5 mm pitch used MI bicycles and highway signs (fig. 22). As discussed in the previous section, the ray transfer matrix of this array has the form (5.1), in particular, c = 0. Hence, the additional aberration A becomes
A
=
+20,
(5.23)
which indicates that if the object is a point source, the image is not “a point” but has radius 20. This result has been checked by MA JIAN, TANGW u , ZHU FUXIANG, TONGDINWANG and WANG SHAOMIN [ 19851 (fig. 23), with the experiment drawn in fig. 24. Corner-cube arrays are typically backward systems.
Fig. 22. Corner-cube array with 2.5 mrn pitch.
334
PRINCIPLES A N D DESIGN OF OPTICAL ARRAYS
5.4. CATS EYE ARRAY
The cat’s eye, which has the same ray transfer matrix as the corner-cube reflector (see table 2), may be used to form pseudo conjugator arrays, which operate also as forward systems (MA JIAN, TANGWu, ZHU FUXIANG,TONG DINWANG and WANGSHAOMIN[ 19851). Note that, as a backward system compensates for the inhomogeneities of a medium in front of it, just because the medium remains the same, analogously, for a forward system the inhomogeneous medium in front of it must be strictly related to that on the other side if compensation must be realized. If we write the ray transfer matrix of the system in fig. 25, we obtain
Hence, the compensating conditions require (5.25)
Fig. 23. Image of a point source given by a comer-cube array.
335
PSEUDO CONJUGATOR ARRAYS
Fig. 24. Experimental setup to measure the additional aberrations of a corner-cube reflector.
namely, recalling that a,d, - b,c, a2 = dl
,
b2 = b, ,
=
a2d2 - b,c, = 1,
c2 = c, ,
d2 = a,
.
(5.26)
5.5. ROOF-MIRROR ARRAY
The roof-mirror array is a backward system. If one combines a comer-cube array with a cat’s eye array, as shown in fig. 26, one obtains by suitably choosing the distances, the so-called roof-mirror array, with c # 0. Hence, as noted in § 5.2, there is a position of the object for which the synthesized image is free from additional aberration. The total transfer matrix 5 for an individual element of the arrays (fig. 26) can be written as (5.27)
with (5.28)
Fig. 25. A forward-propagation cat’s eye system with inhomogeneous media on both sides.
336
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
Fig. 26. A roof mirror.
Hence, 12
a, = 1 - -,
f
b, = I ,
hl2 + l2 - f’
(5.29) and a=
- (a,d, + 6 , ~, ~ ) 6
=
- 26,d,,
c=
- 2a,c,,
=
- (a,d, + 6
d
, ~. ~ )
Accordingly, since a = d. the systems of fig. 26 can be used to form arrays with phase-conjugation properties (see eq. 5.18). The planes RP; = RP; to which one must refer in order to satisfy the pseudo-conjugation conditions eq. (5.12) are located at a distance u, from RP, = RP,, given, according to eq. (5.19), by (5.30) where use has been made of the unimodular property of matrix ys. The additional aberration A given by eq. (5.20) takes the particular form
A
=
20 T - (U,C,U + 1) . d
Hence, for an object placed at a distance u=
-- 1 a,%
from RP;, A vanishes: The image of a point source is “a point”.
(5.31)
111, B 51
331
PSEUDO CONJUGATOR ARRAYS
For the preceding property, added to the fact that the magnification is 1, a device of this type may be used, as it has been used in Japan, in compact photocopying machines.
5.6. GRIN ROD ARRAYS, THICK-LENS ARRAYS A N D BEAD ARRAYS
Plane GRIN rod arrays are usually employed in compact unit-magnification and TOYAMA imaging devices in the photocopying industry (MATSUSHITA [1980]). Since they operate in air, the proper ray transfer matrix to be considered is rf (1, no, Po, I) as given by eq. (2.27), namely, (5.32) with a,, boy coy and do given by eqs. (2.26). With the diagonal elements of M ,o being equal, GRIN rods may be used to form a pseudo conjugator array + (see eq. 5.18). The reference plane RP; ,for which eqs. (5.12) are verified, is at a distance u, from the input surface of the rod, which is given, according to eq. (5.19), by u0 =
1 + do --*
con0
As a numerical example, one can take 1 = 6.8 mm, no = 1.7, and one finds u,=
(5.33)
& = 0.5 mm-I,
-0.15mm.
The negative sign of u, means that RP; is on the right of the input surface of the rod. An array employed in photocopying machines may be formed by approximately 400 rods, with radius = 0.4mm. Since co # 0, the additional aberrations may be completely corrected by putting the object at the distance u=-
2 con0
from RP;, which in the preceding numerical example results in a distance of about 9 mm.
338
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 8 5
An array of this type has been successfully tested (WANGSHAOMIN, ZHOU GUOSHENG, Wu MEIYING,PENG LIANHUIand TIANLIJUAN[ 19831). Pseudo phase-conjugator arrays may be formed also by thick lenses (WANG SHAOMIN, Z u JINMIN, WANG SHIAOJING, YIN CHENGRENGand ZHANG ZEXUN [1983]), and even by beads with the rear half-surface reflecting (HUANGWEISHAI, JIANGXIUMING,CHENYINGLI,ZHAOJIAJU, Z u JINMIN and WANGSHAOMIN [ 19831). Let us consider in some detail this last array, formed by rather unusual optical elements (fig. 27). of a single bead (fig. 28) may be found by The ray transfer matrix applying eq. (2.8). Let us first introduce the matrix y, describing the refraction at the input spherical surface and the transit through the bead:
where p > 0 denotes the radius of the bead, n is the refractive index of the material by which the bead is formed, ,Ml(2p) is given by of eq. (2.12), and y2(p,1, n) is given by the first of eqs . (2.13). For the elements a,, b,, c, , and d, of h $ we find
2-n , a, = n
b , = -2P , n
1-n
cs=-,
nP
1
d,=-, n
Fig. 27. A bead array with 0.07 m m pitch.
PSEUDO CONJUGATOR ARRAYS
ug=-
339
9
Fig. 28. A couple of beads composing an array.
and, obviously, Det(bl?,)= l/n. Then the ray transfer matrix vrJ of the bead, with respect to the reference plane RP, = RP, indicated in fig. 28, is obtained as
with ?Isgiven by eq. (2.10) and I&,.(- p ) given by the second eq. (2.13). The result of the matrix multiplication is a=---(n - 4)
n
c=-
2(n - 2 ) , nP
b=
--, 4P n
d = -(n. - 4) n
Hence, a = d: The beads with a reflecting half-surface may be used to form pseudo conjugator arrays. For them (fig. 28) ug
= -p.
The phase-conjugation properties of bead arrays, including the capability of compensating for inhomogeneities of a medium in front of them, have been checked with the experiment shown in fig. 20 (5 5.1). An example of the results is shown in fig. 29. It may be worth noting that, because of the smallness of the diameter of the beads, the main effect determining the quality of the synthesized image given
W
0 P
c (
Fig. 29. (a) The object imaged by the apparatus of fig. 20, with a flat mirror and without test tube; (b) with a flat mirror and with a test tube; (c) with a bead array and without the test tube; (d) with a bead array and with the test tube.
1 cn
111,s 61
ARRAYS WITH VANISHING DETERMINANT
34 1
Fig. 30. The image of a point source in coherent light, given by a bead array.
by a bead array is not the additional aberration, as in the case of comer-cube arrays, but diffraction. This may be proved by means of fig. 30, representing the image of a point source obtained with the apparatus shown in fig. 24 (Q 5.3), with the comer-cube array replaced by a bead array.
4 6. Arrays with Vanishing Determinant 6.1. THEORETICAL CONSIDERATIONS
All optical systems considered up to now have a ray transfer matrix with a detenninant equal to n,/n, (if they are linear) or equal to - 1 (if they are nonlinear systems or phase conjugator arrays). In Q 6 we now consider another type of array, whose ray transfer matrix blahas a determinant equal to 0. According to eq. (4.8), the determinant of the ray transfer matrix of an array is given by 1
Det(y,) = DET = d - - (@ - bS) ,
R
342
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
[III, 0 6
where R denotes the radius of the input surface of the may, b and d are two of the elements of the ray transfer matrix of the optical element forming the array, and (see eqs. 3.6) j= 1 - b and 6 = 1 - d. Recall that in the case of reflectors, 6 = - 1 - d. The requirement
(6.2)
Det(y,) = 0 is satisfied simply by choosing b R=j--6. d
(6.3)
Then the elements of matrix (4.7) to be a ’ = -~ 66
j d - b6
,
5, are simply found with the help of eqs.
b’= b ,
The first observation is that for these arrays the reversed matrix defined by eq. (2.10), does not exist because of eq. (6.2). If now we introduce eq. (6.3) into eq. (4.1 l), we obtain
pa,
b V=d’
from which follows the important result that the position of the synthesized image is independent of the position of the object. Moreover, by noting that eq. (6.3) can also be written as b R=lT-, d
where the lower sign holds for reflectors, eq. (6.5) can be rewritten in the form
V=T(R-I),
(6.6)
which indicates that the synthesized image forms in all cases at the common center of the input and output surfaces of the may. All sources in object space are imaged at the same plane: The depth of the field is infinite.
I U 8 61
ARRAYS WITH VANISHING DETERMINANT
343
Another important feature of this type of arrays is found ir we evaluate the transverse magnification m2 of the array by introducing eqs. (6.4) and (6.5) into eq. (4.14). We find
m,=O.
(6.7)
Consequently, arrays with DET = 0 are not imaging devices, since, because of eq. (6.7), the synthesized image of any object should be a ‘‘geometrical point”. This, of course, is not true because of the additional aberrations, by virtue of which the image of a point is a “spot”. By application of eq. (4.37) the additional aberration, namely the size of the spot, can be shown to be
Hence, the size of the synthesized image of any object depends only on the size of the individual optical element and on the matrix element d of the individual element forming the array. In comparison with the effects of the additional aberration, the diffraction effects of individual elements appear to be negligible. As a matter of fact, the Fresnel number of the input pupil of the individual element, as seen from the
Fig. 31. An array with DET = 0,formed by elements with a square pupil.
344
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
Fig. 32. An array with DET
=
0, formed by elements with a circular pupil.
synthesized image plane, is infinite, as follows from the application of eq. (2.55) and by noting that even the image of the input plane (u = 0) forms at V = b/d, where, therefore, B = 0. The fact that this type of array cannot be applied as an imaging system does not mean that these arrays do not have practical applications: They may fkd several other applications, such as exposure meters, shadowless lamps, solar furnaces, and others. Even arrays with DET close to 0 appear to have interesting properties in terms of applications, since their transverse magnification can be controlled by the value of R. For example, a GRIN rod array capable of reduction or enlargement has been designed both by means of conventional optics (REES and LAMA[ 19841) and by matrix methods (WANG SHAOMIN and RONCHI [ 19851).
Fig. 33. Experimental scheme used to test the arrays shown in figs. 31 and 32.
111, § 61
ARRAYS WITH VANISHING DETERMINANT
345
6.2. EXPERIMENTAL TESTS
Two kinds of arrays with DET = 0 have been built (WANG SHAOMINand ZHANGZEXUN[ 19851) and tested. One array was made ofglass elements,with a square pupil of side 12 mm, and d = - 1 (fig. 31). The other array was made of Plexiglas, with a circular pupil of diameter 1.75 mm and d = - 0.5 (fig. 32).
Fig. 34. Images obtained with the arrays shown in figs. 31 (left) and 32 (right), with DET = 0.
346
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
WI,§1
The experimental scheme is shown in fig. 33. The images are shown in fig. 34a and b. They are in complete agreement with theoretical expectations: The image is independent of the position of the source, and the image has the same shape as the pupil of the elements of the array, which is clearly due to the additional aberrations. As a last observation, we can note that the synthesized image in coherent light may present interference fringes (see Q 4.4), which, however, may disappear when the object is at the position u given by eq. (4.40). In the present case this becomes
Figure 17, where such an effect has already been shown, is, m fact, a picture photographed with an array with DET = 0.
7. Conclusion
Optical arrays constitute a class of optical instruments whose features cannot be studied by means of classical optics. This characteristic is probably because their structure interrupts the continuity or the analyticity of the medium where waves propagate. From this point of view they are similarto the so-called Fresnel lenses, whose incapability of being treated by the differential laws of ordinary optics was demonstrated several years ago (RONCHIand TORALDO DI FRANCIA[ 19521). The matrix technique has proved to be a powerful tool in investigating the paraxial behavior of both linear and nonlinear optical systems. It may be worth recalling that we define as linear those systems whose ray transfer matrices or or IM) have a determinant equal to nl/n2,n1 being the refractive index of the medium containing the incident rays and n2 that of the medium containing the emerging rays. The ray transfer matrices Nla of arrays (see eq. 4.6) may have a determinant different from n,/n,, even if, or despite the fact that, they are made of linear optical elements. This is so because the elements of the ray transfer matrix, and therefore its determinant, depend on the geometry of the array, in particular on the radius R of the input surface of the array. The possibility follows, therefore, of obtaining devices with nonlinear proper-
(v v’
1111
REFERENCES
347
ties without making recourse to complicated techniques like coherent optical adaptive techniques or degenerate four-wave mixing. This is the case for arrays with DET = - 1, described in # 5, which have phase conjugation properties, even if only approximated, and for arrays with DET = 0. These last arrays with DET = 0, described in !$ 6, seem to form a new type of optical instrument, with the unusual property of having infinite Eeld depth, and image size independent of the position of the object. Such arrays have only recently been discovered, so that one can expect to find other interesting properties or applications with further investigations. It is interesting to note that the compound eye of insects is similar to the preceding arrays, since its DET is close to 0. Acknowledgements One of the authors (Wang Shaomin) is grateful to ICTP (Trieste, Italy) and to NSFC (Beijing, China) for their support. References ARNAUD, J.A., 1976, Beam and Fiber Optics (Academic Press, New York). AU YEUNG,J., D. FEKETE, D.M. PEPPERand A. YARIV,1979. IEEE J. Quantum Electron. QE-15, 1180-1 188. BARRETT, H.H., and S.F. JACOBS.1979, Opt. Lett. 4, 190-192. BORN,M., and E. WOLF, 1980, Principles of Optics, 6th Ed. (Pergamon Press, Oxford). and A. WALTHER,1963, Appl. Opt. 2, 1239-1246. BROUWER, W., E.L. O”EIL CARTER,W.H., 1982, Appl. Opt. 21, 1989-1994. COLLINSJR, S.A., 1970, J. Opt. SOC.Am. 60, 1168-1177. DRAGT,A.J., 1982, J. Opt. SOC.Am. 72, 372-379. FAN,DIANYUAN, 1983, Acta Opt. Sinica 3, 319-325 (in Chinese). FOG,C., 1982, Appl. Opt. 21, 1530-1531. FREEMAN, R.H., R.J. FREIBERG and H.R. GARCIA, 1978, Opt. Lett. 2, 6143. GERRARD, A., and J.M. BURCH,1975, Introduction to Matrix Methods in Optics (Wiley, London). HENLEY,E.J., and R.A. WILLIAMS,1973, Graph Theory in Modern Engineering (Academic Press, New York). HUANG,WEISHAI, JIANGXIUMING, CHEN YINGLI, ZHAOJIMU, ZU JINMIN and WANG SHAOMIN, 1983, Chin. J. Lasers 16, 191-192 (in Chinese). JACOBS,S.F., 1982, Opt. Eng. 21, 281-283. KAWAZU, M., and Y.OGURA,1980, Appl. Opt. 19, 1105-1 112. KOGELNIK, H., 1965, Bell Sys. Techn. J. 4 , 4 5 5 4 9 4 . KOGELNIK, H., and T. LI, 1966, Appl. Opt. 5, 1550-1567. MA, JIAN,TANGwu, ZHU FUXIANG, TONGDINWANGand WANGSHAOMIN, 1985, Physics, pp. 491-492 (in Chinese).
348
PRINCIPLES AND DESIGN OF OPTICAL ARRAYS
1111
MA,JIAN,Luo XUEMIN,WANGSHAOMIN, R. RISALITIand L. RONCHI,1986, Nuovo Cimento D 8,91-104. MARCUSE,D., 1972, Light Transmission Optics (Van Nostrand Reinhold, New York). MATHIEU,P., and P.A. BELANGER, 1980, Appl. Opt. 19,2262-2264. MATSUSHITA, K., and M. TOYAMA, 1980, Appl. Opt. 19, 1070-1075. OMEARA,T.R., 1982, Opt. Eng. 21,271-280. ORLOV,V.K., YAZ. VIRNIK, S.P. VOROTILIN, V.B. GERASIMOV, Yu.A. KALININand A.YA. SAGALOVICH, 1978, Sov. J. Quantum Electron. 8, 799. REES,J.D., 1982, Appl. Opt. 21, 1009-1016. REES,J.D., and W. LAMA,1984, Appl. Opt. 23, 1715-1724. RONCHI,L., and G. TORALDODI FRANCIA, 1952, Atti Fond. G. Ronchi 7, 145-160. SATO,T., Y.NAGURA,0. IKEDAand T. HATSUZAWA, 1982, Appl. Opt. 21, 1778-1784. WANG,SHAOMIN,1979, J. Hangzhou Univ. 3,42-52 (in Chinese). WANG,SHAOMIN,1983a, J. Hangzhou Univ. 10,476-490 (in Chinese). WANG,SHAOMIN,1983b, Appl. Lasers 3(2), 13-16 (in Chinese). WANG,SHAOMIN, 1984, J. Hangzhou Univ. 11,79-91 (in Chinese). WANG,SHAOMIN, 1985, Opt. & Quantum Electron. 17, 1-14. WANG,SHAOMIN, and L. RONCHI,1985, GRIN imaging systems described by transfer matrices, in: Gradient Index Optical Imaging Systems, Proc. Topical Meeting (Palermo, Italy). WANG,SHAOMIN, and L. RONCHI, 1986, Atti Fond. G. Ronchi 41, 53-66. WANG,SHAOMIN, and H. WEBER,1982, Opt. Cornmun. 41, 360-362. WANG, SHAOMIN,and H. WEBER, 1985, Matrix Methods in Treating Phase Conjugate Phenomena, in: Trends in Quantum Optics, Proc. 2nd Int. Conf. (Bucharest, Romania). WANG,SHAOMIN, and ZHANGZEXUN,1985, J. Hangzhou Univ. 12, 70-73 (in Chinese). WANG,SHAOMIN, and ZHoU GUOSHENG,1984, Acta Opt. Sinica 4, 1119-1 123 (in Chinese). WANG,SHAOMIN, Zu JINMIN,WANGSHIAOJING, YIN CHENGRENG and ZHANGZEXUN,1983, Appl. Lasers 3(5), 27-28 (in Chinese). WANG,SHAOMIN, ZHOU GUOSHENG, WU MEIYING,PENGLIANHUIand TIANLIJUAN,1983, Acta Phys. Sinica 32, 1357-1360 (in Chinese). YARIV,A., 1976, Introduction to Optical Electronics (Holt, Rinehart & Winston).