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The most general isoplanatism theorem
Volume 3. number 1
OPTICS COMMIJNICATIONS
THE
MOST
GENERAL
ISOPLANATISM
THEOREM
W. T. WELFORD
Received
5 January 1971
The theorem gives the change in optical path aberration for infinitesimal rotations about any axes in object and image space of a chosen pencil in an optical system of no symmetry. It is applicable to an.” system governed by E’ermat’s principle, including holographic systems. It includes as special cases less general results such as T. Smith’s oatical cosine law and Conrad.vy’s formula for offense against the sine condition. I7
1. INTRODUCTION:
T. SMITH’S OPTICAL COSINE LAW AND OTHER RESULTS RELATING TO ISOPLANATISM
The optical cosine law of T. Smith [ 11, also discussed by G. C. Steward [2], states the conditions under which an infinitesimal translation of a pencil in object space produces an infinitesimal translation without change of shape of the corresponding pencil in image space. There are no restrictions as to symmetry in the optical system but on the other hand the restriction to infinitesimal lranslalions means that we cannot derive other more specialized results, such as the Staeble-Lihotzky condition (F. Staeble [3], E. Lihotzky [4]) and the Conrady offence against the sine condition [5]: this is because an infinitesimal translation is not the most general infinitesimal displacement of a pencil - an infinitesimal rota/ion is what is really needed. To put this into terms relating to conventional optical systems, the optical cosine law can only be applied to systems where the entrance pupil and the exit pupil are both at infinity or where the pencils are aberration-free. There is a further limitation which applies to some of the isoplanatism formulae mentioned: they are conditions for isoplanatism but they do not give explicitly the amount of the change in aberration for a small displacement if the condition is not fulfilled. Conrady’s formula does do this and it is consequently useful in practical optical design. It is perhaps worthwhile to underline this distinction by calling results of the first kind isoplanafism conditions and results of the second kind nonisoplrcnnlism formulae. In the present communication we derive a non-
isoplanatism formula for infinitesimal rotations about arbitrarily chosen axes and with no assump tions of symmetry in the optical system. The result then includes all previous formulae.
2. THE THEOREM The result to be proved can be stated as follows. Let there be an optical system with no postulated symmetry but with isotropic media; also let the object and image space media be homogeneous with refractive indices n and n’. Let p be a unit vector defining a certain axis of rotation in object space, let I- be a unit vector along a ray of a pencil from an object point 0. let S be the length of the perpendicular from p to r and let H be the angle between p and r. Similarly p’ defines another rotation axis in image space, r’ is along the same ray as r but in image space and S’ and 0’ are defined in relation to p’ and :’ in the same way. Then the necessary and sufficient condition that a small rotation E of the object pencil about p produces a small rotation t’ of the image pencil about p’ without any deformation of the image pencil is t’n’S’ sin 8’ = EnSsin8
+ C + 0 (rr2)
(1)
for all rays of the pencil, where C is a constant and 0 is a quantity of order ES. Furthermore, if this condition is not fulfilled the increment in optical path aberration in the pencil due to the rotation is dW = c’n’S’sint?
- EnSsin
+ C + 0(tr2).
(2)
1
Volume
3. number
3. PROOF
1
OPTIC’S
COMMUNICATIONS
OF THE THEOREM
In order to prove the theorem of section 2 we require an auxiliary theorem which expresses the effect of a small rotation of a pencil on a wavefront of the pencil. Let W(R,. e) be the point characteristic of the optical system; we fix R, as the position vector of the object point 0 of the pencil and R as the position vector of another point, in the image space or in any intermediate space which is home geneous. The equation of a wavefront 1: in this space is then
- ___
Is‘ig. 1. Rotation
of a Iwncil
;d~orit the ;utis
p.
where the gradient is taken with respect to h?. But VIV is simply ar. by the basic property of the characteristic function, so the rotated surface X1 has the equation
marking off along each ray the distance eS sin ii as in fig. I: the error is Cj(g2). The position vector R has disappeared from the final result for the path difference; this is at it should be since one would not expect the choice vf a particular wavefront of the pencil to be relevant. Also the position vector R. to the object point of the pencil plays no part in the proof: it was used in the symbol for the point characteristic simply in orderto make explicit the dependence of Won the coordinates of two points. The result applies to a rotation of a wavefront in a.ny homogeneous space of the system. We can now prove the theorem of section 2. Let X and C’ be wavefronts of a pencil in object and image space (fig. 21 and let IXbe a ray of this pencil in the two spaces. Let X become Cl after a rotation E about an axis p and let Xi be a wavefront of the corresponding pencil in image space. Let 0, shown in broken line, be a ray of the new pencil. Let N meet \‘ and X1 in P and Q and Y and l;i in P’ and Q’t: let h meet Z and 5 1 in PI and Q and X:’ and Ci in Pi and Q’. Let Xi be chosen so that
w(lp,.R,-EI?{p.~-~l.f;
IQQ’],
W(I?,,. /iI
= I,.
(3)
where I, is the optical path length from 0 to C. Let RI be the position vector of any point A on the proposed axis of rotation p. If now the pencil is rotated through t as in the theorem. this is equivalent to a rigid rotation of the wavefront S and the position vectors e about p. Let the new position of ?I be YL*: we obtain Xl by replacing the vectors R by R- tp A (R - RJ ), assuming squares of E are negljgible. so that the equatjon of X:] is W(Ro.
@tp
A CR-& 1) = L* ChJ2)
(4 1
The posjtion vectors I? now refer to the surface Yl. Expanding by Taylor’s theorem we have ~(~,.~)-E(P~(IZ-~~).~~W)
. (5)
-= L+O&
= L+OCc?,
*
(6)
} denote the scalar where the brackets 1 triple product of the three vectors inside them. This quantity is equal in magnitude to the volume of the parallelepiped with sides p. R- R1 and r and since p and f are unit vectors this volume is the lengths of the perpendicular from p to f multiplied by the sine of the angle 13 between p and r. Eq. (6) can therefore be written w(Ro,
I?) - t>zSsin
ii
L+ 0((r2) :
-[PP’la r 0:
here the square brackets denote optical path lengths and the subscripts n and 0 denote the rays along which the optical paths are measured. But
(7)
this is the required result. The sign of B must be chosen correctly to agree with the sign of the intended rotation E. Eq. (7) has the interpretation that the surface 721 can be obtained from the wavefront 1 by * This
surface C 1 is not necessarily a physically pos sible wavefront. it is merely the surface S rotated through the angle E.
2
Fig.
2. The theorem.
Volume 3. number 1
and. since the angle between n and h is of order E in all spaces. Fermat’s principle yields [PIPi]/,
= [PP’],
+ O(2)
;
RQlr, = P’Qlc, + O(a2) I LPiQ’lD = [P’Qilll Combining we find [P’Qi]
-
+ G(o’)
these results
(10)
.
[PQ] = O(O~).
(11) eq.
[PQ]
(‘2)
= EnSsin
[P’QB]
= E’H’S’ sin 8’ + c
(14)
[P’Q’I]
= [P’QiI
+ [QbQi]
.
(13)
where C is a constant. This constant C arises because Xi is not necessarily the wavefront obtained by a rotation of X’, it is onlv parallel to this rotated wavefront. Thus the necessity of eq. (1) is proved. To prove sufficiency it is enough to note that if eq. (1) is fulfilled then [P’Q’l] must be given by an equation of the form of eq. (13): thus Xi must be parallel to a surface obtained by a rigid rotation of X’. i.e. it is a wavefront of a rotated pencil. The first part of the theorem is thus proved. If axes p and p’ exist which give isoplanatism in relation to a certain pencil of rays then the ratio E’ E of the rotations is fixed. its value can in principle be determined by a procedure analogous to paraxial raytracing and the tracing of close tangential or sagittal pencils, although for a system with no symmetry the details will be more complicated. To prove the second part of the theorem, suppose that eq. (1) does not hold and let C’l be a wavefront of the pencil corresponding to the incident rotated pencil. as in fig. 3. We have assigned a proposed axis p’ and we can rotate \“
Fig. 3. Change of optical path when the wavefront shape changes.
(15)
Thus from eq. (11) [Q’2Q’J
= [PQj
-
[P’Q;]
= E’IZ’S’ sin 8’ - EnSsinH + C .
If Cl is in fact a wavefront of a pencil which is obtained by a rigid rotation E’ of the original image space pencil about some axis p’ then by eq. (7) = E’u’S’ sin 8’ + C ,
about this axis through any angle t’ to give a surface Xi , this surface would have been a wavefront parallel to 21 if eq. (1) had been satisfied. Let the ray n meet Cb in Qi : then
and also lve have
with eqs. (8) and (9)
Since ?;, was obtained from X’ by a rotation, (7) gives
[P’Qi]
March 19’71
OPTICS COMMUNICATIONS
(16)
Since [QiQ’J gives the change in shape of the wavefront the second part of the theorem is proved. Clearly the theorem may if desired be applied to a subset only of the complete pencil of rays. as is usual in optical designing applications of the various isoplanatism conditions and formulae. The result then holds only for the rays which are tested. The constant C which occurs in the formulation of the theorem is not of great importance: it appears because in testing for isoplanatism of a complete pencil it does not signify which wavefront of the pencil is tested. The test has to be applied to at least two rays in order to use it and the constant vanishes in comparing these rays. In the special cases given in the next section a general ray of the pencil is always tested with respect to a ray which stands out as the principal ray in some sense and then the formulation involves the difference between eq. (2) for the principal ray and for the general ray. In deriving special cases it is sometimes useful to obtain the quantity S sin 0 from the scalar triple product {p.D,r) where D is taken as a vector from an)’ point on the axis p to anq’ point on the ray r. The conditions of the problem usually suggest a particular choice of D to simplify the calculation. This theorem has been derived from Fermat’s principle and so it applies to any image-forming system governed by Fermat’s principle. For example. the image formation of diffraction gratings it generally formulated by stating that the diffracted wavefront has increments of Nh in optical path per grating ruling, where N is an integer, and with this convention the theorem applies to gratings. Since holographic image formation can be formulated in terms of elementary superposed gratings the theorem also applies to holography.
Volume 3. number 4.
SOME SPECIAL
OPTICS
1
COMMUXICATIONS
Mnrch 1971
CASES
We now derive several special already known, from the theorem
cases, some of section 2.
The o,!diccll eosinc Into sf T. Stnilh Let the axes p and p’ both move to infinity and the rotations E and E’ decrease in such a way that in the limit the wavefront 1; has an infinitesimal translation 6s and X’ an infinitesimal translation 6s’. Then eq. (1) states as a necessary and sufficient condition for stationarity of shapes of the pencils. 4.1.
n’6s’* r’ - n6.s.f
+C = O(4),
(l-7)
for all rays. which is T. Smith’s optical cosine law. Our theorem shows further that if this condition is not fulfilled the change in wavefront aberration is given by the left-hand-side of eq. (1’7). which was not pointed out by Smith.
Let OXKZZbe axes at the exit pupil of a symmetrical optical system with Oz along the optical axis and 0~) in the tangential section, as in fig. 4. Let I’ be the paraxial image distance and let 61’ be the longitudinal spherical aberration for a finite ray of the axial pencil with direction cosines (L’,M’,N’). There are similar definitions in object space. In order to derive the usual formula we take the unit vector p along the x axis. We thus take
p’ = (1.0.0) D’ = (0.0,1’-61’)
(18)
r’ = (L’.M’,N’) and the scalar triple product is easily found to be -M’(I’ -6Z’). with a corresponding value in object space. We take c = 17 1 where 77 is the (small) object displacement from the axis and E’ = 7’ I’. Then the change in aberration as the object point moves from the axis a distance 17 in the tangential section is given by Linear
coma
= -tf’~‘M’~
=
related to Conrady’s the sine condition:
formula
for offence
against
(20) This is actually the ratio of sagittal image heights formed by paraxial rays and rays at finite convergence angle U and it forms a convenient dimensionless measure of linear coma (Conrady [7]). If the bracket in eq. (19) is equated to zero or if we put OSC’ equal to zero we obtain in effect the Staeble-Lihotzky condition, given in several variations by Berek [8]. 4.3. A .fortnula J‘OVnoniso~lanalism of oblique pencils in 0 sytntrzelricrrl ofilical s_vsIetw Let O’P’ be the axis of a symmetrical optical system in image space, 0’ being the centre of the exit pupil and P’ the axial gaussian image point (see fig. 5). We take the usual axes OXJZ in the pupil and similarly oriented P’57 axes in the image plane. Let 17’ be the intersection height in the image plane of the principal ray of an oblique pencil, this off-axis image point being Pi. We derive from the general nonisoplanatism formula of section 2 an expression for the change in wavefront aberration of the pencil for a small change in object height dq. Let Oi be the tangential image on the principal ray of the point where the principal ray cuts the axis in object space. The desired infinitesimal movement of the wavefront moves P’1 along the 17 axis and 0; along the principal ray: let Q be a ?
- - ~ 1 (19)
where H is the Lagrange invariant, ?zrlq, and ~1 is the paraxial convergence angle (Welford [6]). The constant C of the theorem is clearly to be taken as zero since the principal ray coincides with the axis and its y-direction cosines are zero in both spaces. The quantity in eq. (19) is closely 4
Fig. 4. Axial isoplnnatism.
Fig. 5. Off-axis
isoplanatism for transverse ments.
displace-
Volume 3. number
OPTICS COMMUNICATIONS
1
March
1971
point in the y- z plane on the perpendicular to the principal ray and at height 77 above the i: axis: then the desired axis p’ must be through Q and parallel to the s axis. We define p similarly in object space but 01 coincides with the centre of the entrance pupil. Let OiP’ = !’ and let the vec“tar D’ be taken from Q to the point where the general ray meets the image plane. We take P’
= (l.O,O)
D’
= (6(‘,6q’,t’
,F)
r’
= (L’.M’,N’)
.
, UI
(21)
where (65’,dq’) are the transverse ray aberration components of the ray with direction cosines (L’, I’@, N) and barred quantities refer to the principal ray. It follows that
D’.
{p’.
r’}
= NT&,’ - &,‘t’
rsi’ ;
axi5
ci
Fig. 6. Off-axis isoplanatism ments (the generalized
(22),
but E’ =N’dn’ t’ and similarly for E so the change in wavefront aberration along the ray (L, M, N) is. if the entering pencil is aberration-free, -dg
1
-@‘NY 6~’ /‘) - nM
n’$+’
t
+ C.
When the object duces to
(23)
dW = -d<
= -dq
I’) -n(M-&j)
n’$$(A&&i%N’6~’ 1
t
Here dq’ dl) is simply the local magnification. The quantity dW of eq. (16) does not, of course, consist only of coma-like terms in this case: it is a measure of the total change in optical pat11 aberration relative to the path along the principal ray. If 6~’ = 0 and dW is set equal to zero in eq. (24) we obtain the “extended sine law”. a condition for perfect off-axis imagery deduced by T. Smith [l] from the optical cosine law. Axial shifts Consider a shift parallel to the axis of an offaxis object point, as in fig. 6: the appropriate axis then passes through the intersection of the perpendicular to 0’1 with the 17 axis, following the same reasoning as in section 4.3. We have
4.4.
p’
= (l.O,O)
D’
= (6C,bq’+t’,‘$?,O)
r’
= (L’,M’.N’) Thus
(25)
.
the change
in aberration
if’-N’+M’&f~‘P)
is -n(8-N)
. (26)
rr$(l-N’) 1
In order to remove the constant we can subtract from this the change in path along the principal ray, and we have finally for the change in aberration for a shift dn of the object point. (24) dW
point
for longitudinal displaceHerschel theorem). is on the axis
-n(l-N)
.
this
re
(27)
I
which is the generalized Herschel condition [9] valid in the presence of spherical aberration. In the special cases dealt with in this section there are local magnification factors such as dl)’ ‘dn: these are all special cases of the ratio E’ E which occurs in the general statement of the theorem in section 2 and their values are easily found in terms of the optics of the “paraxial” region around the principal ray.
5. SOME COMPARISONS References have been made to several papers in which particular isoplanatism theorems are The logical interdependence of these may given. be displayed in a diagram (fig. 7), in which the theorems are shown as blocks and an arrow indicates the direction of derivability. In naming the blocks no distinction has been drawn between the formulation as isoplanatism conditions or as nonisoplanatism formulae. Some variations may be mentioned. M.di Jorio [lo] gave a version for the tangential section only of the off-axis nonisoplanatism formula of section 4.3: his result has a slightly different coefficient for the transverse ray aberration component 6))’ and it seems the error occurred through an assumption in the rather complicated derivation that the extreme rays of the incoming 5
Volume
3. number
1 The
OPTICS present theorem
Off- axis isoplanatism
Fig. 7. I.ogic;il interdependence theorems.
COMMI’NICATIONS
,M/I:~ t.ch 1971
a treatment of off-axis isoplanatism and his term in the ray aberration again differs from that in eq. (24) for similar reasons: also in this treatment he neglected the effect of pupil aberration in shifting the axis of rotation of the principal ray in image space. The numerical difference between the formulae is probably small in many practical situations in lens optics but the traditional criterion of isopianatism. constancy of pencil shape. seems to be the only one which can be extended to the general non-symmetric system with arbitrarily large aberrations; e.g. holographic systems. Finally it may be noted that the quantity n.Ssinil which occurs in the statement of the basic theorem can be regarded as a measure of the relative skewness between the ray r and the rotation axis p. Then if p is taken as the axis of a symmetrical optical system the theorem reduces to a statement of the skew invariant theorem [14].
law (T.Smlthl
of the isoplanatism REFERENCES
original and displaced pencils are parallel. although the entrance pupil is not at infinity. H. H. Hopkins [ 111 gave a nonisoplanatism formula for axial image formation in which the term in the longitudinal ray aberration differed from that given by Conrady and others; in effect Hopkins gave N’61’ I’ instead of 61’ ‘I’ in our eq. (19) *. The difference seems to be due to an implied but unstated difference in the definition of isoplanatism: the usual definition is constancy of shape of the pencil but the Hopkins treatment in effect takes as the criterion constancy of wavefront aberration at a given point in the pupil: if the axial wavefront has aberrations this condition leads to a change in shape of the off-axis wavefront which is of the same order of magnitude as the field angle. The point was treated in detail by W. Weinstein [ 121. H. H. Hopkins [ 131 also gave : Hopkins cl:Limeti his formula was equivalent rndy’s hut this is clearly untrue.
6
to (‘on-
[l] ‘I’.Smith.
‘Trans. Opt. Sot. London 24 (1922-3) 31. [2] 6. C. Steward. The s,wnmetrical optical system (Cambridge Ilniv. Press. London. 1928). [3] I;. Staehle. Mtlnchener Sitzungsherichtc (1919) 163. 141 15. Lihotzky. \Viener Sitzungsberichte 128 (1919) 85. [S] A. IC. Conrady. Monthly Kotices Roy. Astron. Sot. 65 (1905) 501. IS] \\‘. T. \Velford. in: Handbuch der Physik. Vol. 29 (Springer. Heidelberg. 1967) p. 1. [7] A. E:. Conrady. Applied optics and optical design. part I (Oxford Univ. Press London. 1929). [RI M. Berek. Grundlngen der praktischen Optik (De C;ru> tcr Berlin. 1930). [9] J. F. LV. Herschel. Phil.‘I‘rans. Roy. Sot. 111 (1421) 222. IlO] M. til ,Jorio. :J. Opt. Sot. Am. 39 (1949) 305. [ll] ft. H. Hopl&ns. Proc. Phys. Sot. (I.ondon) 58 (1946)
92. [I21 \V. \Veinstein, Ph. U. Thesis, Irniversity of London (19jO). [lS] II. II. IIopkins. .Japan. J. Appl. Phps. 4(19%) Suppl. 1. p. 31. [14] \V. T. \Velford. Opt. Act:1 15 (196;s) 621.
OPTICS COMMIJNICATIONS
THE
MOST
GENERAL
ISOPLANATISM
THEOREM
W. T. WELFORD
Received
5 January 1971
The theorem gives the change in optical path aberration for infinitesimal rotations about any axes in object and image space of a chosen pencil in an optical system of no symmetry. It is applicable to an.” system governed by E’ermat’s principle, including holographic systems. It includes as special cases less general results such as T. Smith’s oatical cosine law and Conrad.vy’s formula for offense against the sine condition. I7
1. INTRODUCTION:
T. SMITH’S OPTICAL COSINE LAW AND OTHER RESULTS RELATING TO ISOPLANATISM
The optical cosine law of T. Smith [ 11, also discussed by G. C. Steward [2], states the conditions under which an infinitesimal translation of a pencil in object space produces an infinitesimal translation without change of shape of the corresponding pencil in image space. There are no restrictions as to symmetry in the optical system but on the other hand the restriction to infinitesimal lranslalions means that we cannot derive other more specialized results, such as the Staeble-Lihotzky condition (F. Staeble [3], E. Lihotzky [4]) and the Conrady offence against the sine condition [5]: this is because an infinitesimal translation is not the most general infinitesimal displacement of a pencil - an infinitesimal rota/ion is what is really needed. To put this into terms relating to conventional optical systems, the optical cosine law can only be applied to systems where the entrance pupil and the exit pupil are both at infinity or where the pencils are aberration-free. There is a further limitation which applies to some of the isoplanatism formulae mentioned: they are conditions for isoplanatism but they do not give explicitly the amount of the change in aberration for a small displacement if the condition is not fulfilled. Conrady’s formula does do this and it is consequently useful in practical optical design. It is perhaps worthwhile to underline this distinction by calling results of the first kind isoplanafism conditions and results of the second kind nonisoplrcnnlism formulae. In the present communication we derive a non-
isoplanatism formula for infinitesimal rotations about arbitrarily chosen axes and with no assump tions of symmetry in the optical system. The result then includes all previous formulae.
2. THE THEOREM The result to be proved can be stated as follows. Let there be an optical system with no postulated symmetry but with isotropic media; also let the object and image space media be homogeneous with refractive indices n and n’. Let p be a unit vector defining a certain axis of rotation in object space, let I- be a unit vector along a ray of a pencil from an object point 0. let S be the length of the perpendicular from p to r and let H be the angle between p and r. Similarly p’ defines another rotation axis in image space, r’ is along the same ray as r but in image space and S’ and 0’ are defined in relation to p’ and :’ in the same way. Then the necessary and sufficient condition that a small rotation E of the object pencil about p produces a small rotation t’ of the image pencil about p’ without any deformation of the image pencil is t’n’S’ sin 8’ = EnSsin8
+ C + 0 (rr2)
(1)
for all rays of the pencil, where C is a constant and 0 is a quantity of order ES. Furthermore, if this condition is not fulfilled the increment in optical path aberration in the pencil due to the rotation is dW = c’n’S’sint?
- EnSsin
+ C + 0(tr2).
(2)
1
Volume
3. number
3. PROOF
1
OPTIC’S
COMMUNICATIONS
OF THE THEOREM
In order to prove the theorem of section 2 we require an auxiliary theorem which expresses the effect of a small rotation of a pencil on a wavefront of the pencil. Let W(R,. e) be the point characteristic of the optical system; we fix R, as the position vector of the object point 0 of the pencil and R as the position vector of another point, in the image space or in any intermediate space which is home geneous. The equation of a wavefront 1: in this space is then
- ___
Is‘ig. 1. Rotation
of a Iwncil
;d~orit the ;utis
p.
where the gradient is taken with respect to h?. But VIV is simply ar. by the basic property of the characteristic function, so the rotated surface X1 has the equation
marking off along each ray the distance eS sin ii as in fig. I: the error is Cj(g2). The position vector R has disappeared from the final result for the path difference; this is at it should be since one would not expect the choice vf a particular wavefront of the pencil to be relevant. Also the position vector R. to the object point of the pencil plays no part in the proof: it was used in the symbol for the point characteristic simply in orderto make explicit the dependence of Won the coordinates of two points. The result applies to a rotation of a wavefront in a.ny homogeneous space of the system. We can now prove the theorem of section 2. Let X and C’ be wavefronts of a pencil in object and image space (fig. 21 and let IXbe a ray of this pencil in the two spaces. Let X become Cl after a rotation E about an axis p and let Xi be a wavefront of the corresponding pencil in image space. Let 0, shown in broken line, be a ray of the new pencil. Let N meet \‘ and X1 in P and Q and Y and l;i in P’ and Q’t: let h meet Z and 5 1 in PI and Q and X:’ and Ci in Pi and Q’. Let Xi be chosen so that
w(lp,.R,-EI?{p.~-~l.f;
IQQ’],
W(I?,,. /iI
= I,.
(3)
where I, is the optical path length from 0 to C. Let RI be the position vector of any point A on the proposed axis of rotation p. If now the pencil is rotated through t as in the theorem. this is equivalent to a rigid rotation of the wavefront S and the position vectors e about p. Let the new position of ?I be YL*: we obtain Xl by replacing the vectors R by R- tp A (R - RJ ), assuming squares of E are negljgible. so that the equatjon of X:] is W(Ro.
@tp
A CR-& 1) = L* ChJ2)
(4 1
The posjtion vectors I? now refer to the surface Yl. Expanding by Taylor’s theorem we have ~(~,.~)-E(P~(IZ-~~).~~W)
. (5)
-= L+O&
= L+OCc?,
*
(6)
} denote the scalar where the brackets 1 triple product of the three vectors inside them. This quantity is equal in magnitude to the volume of the parallelepiped with sides p. R- R1 and r and since p and f are unit vectors this volume is the lengths of the perpendicular from p to f multiplied by the sine of the angle 13 between p and r. Eq. (6) can therefore be written w(Ro,
I?) - t>zSsin
ii
L+ 0((r2) :
-[PP’la r 0:
here the square brackets denote optical path lengths and the subscripts n and 0 denote the rays along which the optical paths are measured. But
(7)
this is the required result. The sign of B must be chosen correctly to agree with the sign of the intended rotation E. Eq. (7) has the interpretation that the surface 721 can be obtained from the wavefront 1 by * This
surface C 1 is not necessarily a physically pos sible wavefront. it is merely the surface S rotated through the angle E.
2
Fig.
2. The theorem.
Volume 3. number 1
and. since the angle between n and h is of order E in all spaces. Fermat’s principle yields [PIPi]/,
= [PP’],
+ O(2)
;
RQlr, = P’Qlc, + O(a2) I LPiQ’lD = [P’Qilll Combining we find [P’Qi]
-
+ G(o’)
these results
(10)
.
[PQ] = O(O~).
(11) eq.
[PQ]
(‘2)
= EnSsin
[P’QB]
= E’H’S’ sin 8’ + c
(14)
[P’Q’I]
= [P’QiI
+ [QbQi]
.
(13)
where C is a constant. This constant C arises because Xi is not necessarily the wavefront obtained by a rotation of X’, it is onlv parallel to this rotated wavefront. Thus the necessity of eq. (1) is proved. To prove sufficiency it is enough to note that if eq. (1) is fulfilled then [P’Q’l] must be given by an equation of the form of eq. (13): thus Xi must be parallel to a surface obtained by a rigid rotation of X’. i.e. it is a wavefront of a rotated pencil. The first part of the theorem is thus proved. If axes p and p’ exist which give isoplanatism in relation to a certain pencil of rays then the ratio E’ E of the rotations is fixed. its value can in principle be determined by a procedure analogous to paraxial raytracing and the tracing of close tangential or sagittal pencils, although for a system with no symmetry the details will be more complicated. To prove the second part of the theorem, suppose that eq. (1) does not hold and let C’l be a wavefront of the pencil corresponding to the incident rotated pencil. as in fig. 3. We have assigned a proposed axis p’ and we can rotate \“
Fig. 3. Change of optical path when the wavefront shape changes.
(15)
Thus from eq. (11) [Q’2Q’J
= [PQj
-
[P’Q;]
= E’IZ’S’ sin 8’ - EnSsinH + C .
If Cl is in fact a wavefront of a pencil which is obtained by a rigid rotation E’ of the original image space pencil about some axis p’ then by eq. (7) = E’u’S’ sin 8’ + C ,
about this axis through any angle t’ to give a surface Xi , this surface would have been a wavefront parallel to 21 if eq. (1) had been satisfied. Let the ray n meet Cb in Qi : then
and also lve have
with eqs. (8) and (9)
Since ?;, was obtained from X’ by a rotation, (7) gives
[P’Qi]
March 19’71
OPTICS COMMUNICATIONS
(16)
Since [QiQ’J gives the change in shape of the wavefront the second part of the theorem is proved. Clearly the theorem may if desired be applied to a subset only of the complete pencil of rays. as is usual in optical designing applications of the various isoplanatism conditions and formulae. The result then holds only for the rays which are tested. The constant C which occurs in the formulation of the theorem is not of great importance: it appears because in testing for isoplanatism of a complete pencil it does not signify which wavefront of the pencil is tested. The test has to be applied to at least two rays in order to use it and the constant vanishes in comparing these rays. In the special cases given in the next section a general ray of the pencil is always tested with respect to a ray which stands out as the principal ray in some sense and then the formulation involves the difference between eq. (2) for the principal ray and for the general ray. In deriving special cases it is sometimes useful to obtain the quantity S sin 0 from the scalar triple product {p.D,r) where D is taken as a vector from an)’ point on the axis p to anq’ point on the ray r. The conditions of the problem usually suggest a particular choice of D to simplify the calculation. This theorem has been derived from Fermat’s principle and so it applies to any image-forming system governed by Fermat’s principle. For example. the image formation of diffraction gratings it generally formulated by stating that the diffracted wavefront has increments of Nh in optical path per grating ruling, where N is an integer, and with this convention the theorem applies to gratings. Since holographic image formation can be formulated in terms of elementary superposed gratings the theorem also applies to holography.
Volume 3. number 4.
SOME SPECIAL
OPTICS
1
COMMUXICATIONS
Mnrch 1971
CASES
We now derive several special already known, from the theorem
cases, some of section 2.
The o,!diccll eosinc Into sf T. Stnilh Let the axes p and p’ both move to infinity and the rotations E and E’ decrease in such a way that in the limit the wavefront 1; has an infinitesimal translation 6s and X’ an infinitesimal translation 6s’. Then eq. (1) states as a necessary and sufficient condition for stationarity of shapes of the pencils. 4.1.
n’6s’* r’ - n6.s.f
+C = O(4),
(l-7)
for all rays. which is T. Smith’s optical cosine law. Our theorem shows further that if this condition is not fulfilled the change in wavefront aberration is given by the left-hand-side of eq. (1’7). which was not pointed out by Smith.
Let OXKZZbe axes at the exit pupil of a symmetrical optical system with Oz along the optical axis and 0~) in the tangential section, as in fig. 4. Let I’ be the paraxial image distance and let 61’ be the longitudinal spherical aberration for a finite ray of the axial pencil with direction cosines (L’,M’,N’). There are similar definitions in object space. In order to derive the usual formula we take the unit vector p along the x axis. We thus take
p’ = (1.0.0) D’ = (0.0,1’-61’)
(18)
r’ = (L’.M’,N’) and the scalar triple product is easily found to be -M’(I’ -6Z’). with a corresponding value in object space. We take c = 17 1 where 77 is the (small) object displacement from the axis and E’ = 7’ I’. Then the change in aberration as the object point moves from the axis a distance 17 in the tangential section is given by Linear
coma
= -tf’~‘M’~
=
related to Conrady’s the sine condition:
formula
for offence
against
(20) This is actually the ratio of sagittal image heights formed by paraxial rays and rays at finite convergence angle U and it forms a convenient dimensionless measure of linear coma (Conrady [7]). If the bracket in eq. (19) is equated to zero or if we put OSC’ equal to zero we obtain in effect the Staeble-Lihotzky condition, given in several variations by Berek [8]. 4.3. A .fortnula J‘OVnoniso~lanalism of oblique pencils in 0 sytntrzelricrrl ofilical s_vsIetw Let O’P’ be the axis of a symmetrical optical system in image space, 0’ being the centre of the exit pupil and P’ the axial gaussian image point (see fig. 5). We take the usual axes OXJZ in the pupil and similarly oriented P’57 axes in the image plane. Let 17’ be the intersection height in the image plane of the principal ray of an oblique pencil, this off-axis image point being Pi. We derive from the general nonisoplanatism formula of section 2 an expression for the change in wavefront aberration of the pencil for a small change in object height dq. Let Oi be the tangential image on the principal ray of the point where the principal ray cuts the axis in object space. The desired infinitesimal movement of the wavefront moves P’1 along the 17 axis and 0; along the principal ray: let Q be a ?
- - ~ 1 (19)
where H is the Lagrange invariant, ?zrlq, and ~1 is the paraxial convergence angle (Welford [6]). The constant C of the theorem is clearly to be taken as zero since the principal ray coincides with the axis and its y-direction cosines are zero in both spaces. The quantity in eq. (19) is closely 4
Fig. 4. Axial isoplnnatism.
Fig. 5. Off-axis
isoplanatism for transverse ments.
displace-
Volume 3. number
OPTICS COMMUNICATIONS
1
March
1971
point in the y- z plane on the perpendicular to the principal ray and at height 77 above the i: axis: then the desired axis p’ must be through Q and parallel to the s axis. We define p similarly in object space but 01 coincides with the centre of the entrance pupil. Let OiP’ = !’ and let the vec“tar D’ be taken from Q to the point where the general ray meets the image plane. We take P’
= (l.O,O)
D’
= (6(‘,6q’,t’
,F)
r’
= (L’.M’,N’)
.
, UI
(21)
where (65’,dq’) are the transverse ray aberration components of the ray with direction cosines (L’, I’@, N) and barred quantities refer to the principal ray. It follows that
D’.
{p’.
r’}
= NT&,’ - &,‘t’
rsi’ ;
axi5
ci
Fig. 6. Off-axis isoplanatism ments (the generalized
(22),
but E’ =N’dn’ t’ and similarly for E so the change in wavefront aberration along the ray (L, M, N) is. if the entering pencil is aberration-free, -dg
1
-@‘NY 6~’ /‘) - nM
n’$+’
t
+ C.
When the object duces to
(23)
dW = -d<
= -dq
I’) -n(M-&j)
n’$$(A&&i%N’6~’ 1
t
Here dq’ dl) is simply the local magnification. The quantity dW of eq. (16) does not, of course, consist only of coma-like terms in this case: it is a measure of the total change in optical pat11 aberration relative to the path along the principal ray. If 6~’ = 0 and dW is set equal to zero in eq. (24) we obtain the “extended sine law”. a condition for perfect off-axis imagery deduced by T. Smith [l] from the optical cosine law. Axial shifts Consider a shift parallel to the axis of an offaxis object point, as in fig. 6: the appropriate axis then passes through the intersection of the perpendicular to 0’1 with the 17 axis, following the same reasoning as in section 4.3. We have
4.4.
p’
= (l.O,O)
D’
= (6C,bq’+t’,‘$?,O)
r’
= (L’,M’.N’) Thus
(25)
.
the change
in aberration
if’-N’+M’&f~‘P)
is -n(8-N)
. (26)
rr$(l-N’) 1
In order to remove the constant we can subtract from this the change in path along the principal ray, and we have finally for the change in aberration for a shift dn of the object point. (24) dW
point
for longitudinal displaceHerschel theorem). is on the axis
-n(l-N)
.
this
re
(27)
I
which is the generalized Herschel condition [9] valid in the presence of spherical aberration. In the special cases dealt with in this section there are local magnification factors such as dl)’ ‘dn: these are all special cases of the ratio E’ E which occurs in the general statement of the theorem in section 2 and their values are easily found in terms of the optics of the “paraxial” region around the principal ray.
5. SOME COMPARISONS References have been made to several papers in which particular isoplanatism theorems are The logical interdependence of these may given. be displayed in a diagram (fig. 7), in which the theorems are shown as blocks and an arrow indicates the direction of derivability. In naming the blocks no distinction has been drawn between the formulation as isoplanatism conditions or as nonisoplanatism formulae. Some variations may be mentioned. M.di Jorio [lo] gave a version for the tangential section only of the off-axis nonisoplanatism formula of section 4.3: his result has a slightly different coefficient for the transverse ray aberration component 6))’ and it seems the error occurred through an assumption in the rather complicated derivation that the extreme rays of the incoming 5
Volume
3. number
1 The
OPTICS present theorem
Off- axis isoplanatism
Fig. 7. I.ogic;il interdependence theorems.
COMMI’NICATIONS
,M/I:~ t.ch 1971
a treatment of off-axis isoplanatism and his term in the ray aberration again differs from that in eq. (24) for similar reasons: also in this treatment he neglected the effect of pupil aberration in shifting the axis of rotation of the principal ray in image space. The numerical difference between the formulae is probably small in many practical situations in lens optics but the traditional criterion of isopianatism. constancy of pencil shape. seems to be the only one which can be extended to the general non-symmetric system with arbitrarily large aberrations; e.g. holographic systems. Finally it may be noted that the quantity n.Ssinil which occurs in the statement of the basic theorem can be regarded as a measure of the relative skewness between the ray r and the rotation axis p. Then if p is taken as the axis of a symmetrical optical system the theorem reduces to a statement of the skew invariant theorem [14].
law (T.Smlthl
of the isoplanatism REFERENCES
original and displaced pencils are parallel. although the entrance pupil is not at infinity. H. H. Hopkins [ 111 gave a nonisoplanatism formula for axial image formation in which the term in the longitudinal ray aberration differed from that given by Conrady and others; in effect Hopkins gave N’61’ I’ instead of 61’ ‘I’ in our eq. (19) *. The difference seems to be due to an implied but unstated difference in the definition of isoplanatism: the usual definition is constancy of shape of the pencil but the Hopkins treatment in effect takes as the criterion constancy of wavefront aberration at a given point in the pupil: if the axial wavefront has aberrations this condition leads to a change in shape of the off-axis wavefront which is of the same order of magnitude as the field angle. The point was treated in detail by W. Weinstein [ 121. H. H. Hopkins [ 131 also gave : Hopkins cl:Limeti his formula was equivalent rndy’s hut this is clearly untrue.
6
to (‘on-
[l] ‘I’.Smith.
‘Trans. Opt. Sot. London 24 (1922-3) 31. [2] 6. C. Steward. The s,wnmetrical optical system (Cambridge Ilniv. Press. London. 1928). [3] I;. Staehle. Mtlnchener Sitzungsherichtc (1919) 163. 141 15. Lihotzky. \Viener Sitzungsberichte 128 (1919) 85. [S] A. IC. Conrady. Monthly Kotices Roy. Astron. Sot. 65 (1905) 501. IS] \\‘. T. \Velford. in: Handbuch der Physik. Vol. 29 (Springer. Heidelberg. 1967) p. 1. [7] A. E:. Conrady. Applied optics and optical design. part I (Oxford Univ. Press London. 1929). [RI M. Berek. Grundlngen der praktischen Optik (De C;ru> tcr Berlin. 1930). [9] J. F. LV. Herschel. Phil.‘I‘rans. Roy. Sot. 111 (1421) 222. IlO] M. til ,Jorio. :J. Opt. Sot. Am. 39 (1949) 305. [ll] ft. H. Hopl&ns. Proc. Phys. Sot. (I.ondon) 58 (1946)
92. [I21 \V. \Veinstein, Ph. U. Thesis, Irniversity of London (19jO). [lS] II. II. IIopkins. .Japan. J. Appl. Phps. 4(19%) Suppl. 1. p. 31. [14] \V. T. \Velford. Opt. Act:1 15 (196;s) 621.