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On the asymptotic evaluation of diffraction integrals with a special view to the theory of defocusing and optical contrast
Physica
XVIII,
no 6-7
Juni-Juli
1952
ON- THE ASYMPTOTIC EVALUATION OF DIFFRACTION INTEGRALS WITH A SPECIAL VIEW TO THE THEORY OF DEFOCUSING AND OPTICAL CONTRAST by H. BREMMER N.V.
Philips’
Philips Research Gloeilampenfabrieken
Laboratories Eindhoven,
Nederland
synopsis An expansion is given of the function u(x, y. z) that satisfies the scalar wave equation in the half-space z > 0 and equals a given distribution U&Y, y) in the plane z = 0. If ue is zero beyond a closed contour L in z = 0, part (which vanishes u(x, y. z) splits into a so-called geometrical-optical outside the cylinder passing through L and having generating lines parallel to the z-asis) and a diffraction part (determined by the values of uc near L). Each part can be expanded into terms depending on U&Y, y) itself and on its iterative two-dimensional Laplace operators d%c = ($/a2 + $/@)” U&C, y). The expansions are in general asymptotic for small wave lengths; however, their exact validity can be proved for functions uo that are polynomials inside the contour L. The terms of the diffraction part consist of contour integrals along L, the corresponding terms for the geometrical-optical part do not depend on any integration. The first few terms of the latter part are essential for defocusing effects and can be connected with the brightness-contrast existing in the plane z = 0.
1. Introduction. One of the few boundary problems occurring in diffraction theory which can be solved rigorously concerns the determination of a function u that satisfies the scalar wave equation (Aa + k?)~ = 0 and equals a given distribution Z&Y, y) in some plane z = 0. The notation A, is used for the three-dimensional Laplace operator in order to distinguish it from the corresponding two-dimensional operator A, to be discussed later on. The problem in question is uniquely determined for the half space z > 0 provided the radiation condition is fulfilled there at infinity. This condition -
469 -
470
H.
BREMMER
amounts to a decrease of u proportional to eikR/R at great distances (R + co). Green’s method leads to the following solution:
u(P) =-;g
eWQp) dO,%@)
(Qp)
J
(2~ >
0)
(1)
P
in which do, is a surface element of the plane z = 0 and (QP) is the distance from P to the variable point Q. A detailed discussion of the rigorous conditions for the validity of (1) is given by L u n eb e r g l). The scalar treatment considered here is also applicable to the electromagnetic theory of optics because each vector component by itself does satisfy the above wave equation. However, the complete vectorial problem should also deal with the complicated connections between the various fieldcomponents. In practical problems the integration in (1) extends over a finite part only of z = 0 because zcOis always zero beyond some closed boundary curve L (depending on some aperture or other). The diffraction integral then is of a type which can be reduced to suitable approximations in the two following cases: 1) Distances great compared to the finite dimensions of the domain of integration E inside L. The exponent ik(QP) can be approximated there by the linear terms of its Taylor development with respect to xQ and yQ. The Fraunhofer approximation thus obtained shows the connection of the u-distribution at great distances with the Fourier transform of U&V, y); 2) Distances great compared to the wavelength. The saddle-point method for exponential integrals can be used here. In its evaluation the factor exp (ik(QP)) has to be considered together with other exponential factors occurring occasionally in zbO(Q) (such a factor may arise, e.g., from a primary wave describing the illumination of the plane z = 0 by a light source situated somewhere in z < 0). Neither of these methods is adapted in its usual form for an investigation of u near the plane z = 0. In the theory of lenses this domain is of interest with a view to the theory of defocusing. The present paper deals with an expansion which is suitable in that case. As in other diffraction theories, the result to be derived splits into two contributions : u) a part of geometrical-optical character, the main term of which can be interpreted as a shadow effect, b) a diffraction part (in a limited sense of the word) which only
ON
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47.1
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depends on the properties of zbOalong the boundary L of X. In accordance with the theory of V a n K a m p e n “) the first contribution can be expanded for small wavelengths into an asymptotic series containing integral powers of k-l, the second contribution into a similar series depending on odd-numbered powers of kd. The basic results of this paper will be derived provisionally in the two following sections. The details of the theory follow afterwards. 2. Formal derivation of a series for u(P) in the domain The wave equation can be written as if
z > 0.
a22qa2 + (k2 + A,)26 = 0, A, = azjax2 + a21ay2
represents the two-dimensional Laplace operator. Our boundary problem of finding a solution u in z > 0 that equals a given function z&r, y) in z = 0 can be solved symbolically by: ,21(x,y, z) = e”Z(k2+d2)tuo(x, y) = eikz(1+d2’k2)fq)(x, y).
(z > 0)
(2)
The sign of the exponent of (2) is in accordance with the radiation condition, as may be verified for the special case of u,, equal to a constant. We now apply the expansion e’k’(l-‘2/kz)t = (4 nkz)i X:=0 (H~~,,(kz)/n!)
(n2z/2k)“,
(3)
which is valid for 1A 1 < 1 k ) and which results from elementary properties of Bessel functions “). By substituting A, for - A2 we obtain from (2): u(x, y, z) = (&nkz)* IZ;& (Hill-)-&kz)/n!) (- z/2k)” A; q,(x, y).
(4)
Each term of this series has a definite meaning if we understand by A’%,, the application, n times in succession, of the two-dimensional Laplace operator A, to the function zc,,(x, y). For instance we have: 4:=(;+3=$+2&+$. The expansion (4) formally satisfies the wave equation and reduces to u,, for z = 0. The correctness of (4) is easily verified for those special functions u,,(x, y) for which the operator ‘A2 amounts to multiplication by a constant a. Examples of such functions are
472
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BREMMER
the Bessel functions l&a!) and Jo{e(- a)&} for a > 0 and u < 0 respectively, and also the exponential function exp (ax + /?y) if c? + ,92 = a. For functions of this type (2) represents the exact solution of the boundary problem if d, is simply replaced by a so as to obtain
In these cases the derivation of the series (4) reduces to an ordinary non-symbolical application of (3) for A2 = -a. For the functions under consideration we conclude that (4) is convergent if 1 a 1 < 1 K I2 but divergent if 1 a 1 > I k 1’. The exponentials exp (ax + py) constitute the components of the Fourier integral of u,&, y), each of them having a period 1 = 2nluk. The above condition of convergence thus amounts to 1 > l.A convergence of (4) may therefore be expected for structures u&z, y) without periods I < 1. The contributions for which 1 > 1 are in general unimportant in optical problems since they do not contribute to the diffraction pattern at great distances, as has been shown by B o o k e r and C 1 e m m o w “). Anyhow, (4) cannot represent u(P) exactly under all circumstances. Another disadvantage of (4) is that it does not give explicitly the typical edge effects arising from the vicinity of L for functions U&X, y) that are zero outside L. Nevertheless such edge effects are contained implicitly in (4), as can be shown in the following way. Let @(x, y) be a continuous function that is positive inside and negative outside L. The function representing ztOover the complete plane z = 0 then is given by u,,(x, y)U{@(x, y)} if U(t) marks Heaviside’s unit function (which is 1 for positive arguments and zero for negative arguments). The iterative Laplace operators of the product of 2~~and U(@) lead amongst others to delta functions and their derivatives according to the relation 6(t) = dU(t)/df. Each term containing such functions shows the argument t = @ and thus becomes singular at @ = 0, that is along the boundary L. The contributions containing these delta functions constitute formally the diffraction part of our solution. A more convenient representation of this diffraction part can be deduced with the aid of Green’s integral theorem and is indicated without proof in the next section. Its correctness is demonstrated in the sections 5 and 6 for functions u&, y) that are representable by polynomials in x and y inside L.
ON
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3. The sp!ditting of u(P) into a geometrical-optical part and a diffraction part. Such a splitting is obtained by the following formal
procedure. We transform (1) with the aid of Sommerfeld’s for eiWP) /(QP) into the th ree-dimensional integral u(P) = &
dA
;leikz(l-Az/kz)h do,
s- 0
!...I
.?I
integral
2co(Q)JoV(QPo)),
in which P,(x, y, 0) is the orthogonal projection of P(x, y, z) on the plane z = 0. A substitution of the expansion (3) then leads to integrals the integrands of which depend amongst others on a factor ~2”JoMQPoH. Th is expression should be replaced by the Laplace operator terms now obtained,
(- ~,)“Jo(~(QPo)), referring to the coordinates viz.
of Q. The various
are further to be transformed by n successive applications i=o, 1, . . . . n - 1) of Green’s formula: &dOa +
s
. di zco(Q).d;+Jo(~(QPo))
=/IZdOp
.A;‘-‘-I
(for
Jo(;c(QPo)) . Ai+’ u,(Q) +
Lds[d'zuo(Q).g{~~-i-'~o(~(QPo))}-~~-i-'~o(~(QPo)).~{~'l,,(Q)H.
(6) Q
Q
In this identity ds is an element of the boundary curve L, an arbitrary point of which is marked Q (s increases when moving along L in a sense which is observed as counter-clockwise by an observer viewing from the space z > 0), whilst a/&, refers to a differentiation along the normal at Q in the outward direction. When evaluating (5) with the aid of (6) the surface integrals occurring in the right side of (6) lead to contributions proportional to the integrals , which
j;dl
A/i,
dO,.d;-‘-’
can also be written
Jo(A(QPo)) .dp’
vo(Q),
as follows in view of the symmetry
(QP,) with respect to Q and PO:
of
474
According
H.
to Neumann’s 2n
(
$+P
BREMMER
integral
a2 a-j-I
as )
theorem
this expression
equals
A;++’ uo(P) = 2nA;‘u,(P),
provided P, is situated inside L, whereas it does vanish for P, outside L. The surface integrals occurring in the right side of (6) thus result in a final contribution that differs from zero only in the cylindrical region @(x, y) > 0 the points P(x, y, z) of which show projections P&x, y, 0) inside L (see fig. 1). This contribution, to
be termed geometrical-ofitical @art or ugeo,,,, proves to be identical with (4) in @(x, y) > 0. This first contribution can therefore be represented throughout the half space z > 0 by
The region @(x, y) > 0 will be called the illuminated region because it constitutes the space that is lit through the area C if the latter is illuminated by a parallel beam of light arriving from the negative z-direction. The remaining contribution to G(P), to be termed diffraction arises from the contour integrals of (6) when this $art or udi/f9 formula is applied to the various terms (5) of u(P). The diffraction part thus appears as a sum over n and i of terms proportional to OmdI ;I [4$ uO(Q). ;
{A;-‘-‘J,(l(QP,))}
-
Q
-A
;‘+-I
J~(~(QP~. =JQ
4
~Q)I.
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A formal summation over n for fixed j now proves to be possible, again by using (4). The final result of the deduction briefly indicated here can be represented as follows: aLdiN= -
1
C”,--o -;,
2.75
ds .Ls
[d&,(Q)
$
-
@
K,$
U~~,(Q)>l~ (8) Q
In (9b) 2F0 refers to Pochhammer’s notation for hypergeometric series. A direct proof of the final representation of u(P) in z > 0 by the sum 4P) = %wn(~) + %ij,PL (10) will be performed in sections 5 and 6 for the case of polynomial functions U&X, y). We conclude this section by remarking that the equivalence of (9~) and (9b) can be shown by the substitutions u = 5+
W=oQ)2 + (C 7 d2>”
in the two consecutive terms of (9~) and by applying identity *) for the finite series of H$(k[): eikt (&nkQ6H/‘&(k<)
= ii2F,
j, 1 -
j;; &)
the following
.
(11)
4. The diffraction $art expressed as a surface integral. A splitting corresponding to (10) is well known in diffraction problems. The diffraction part always appears as a line integral along a boundary curve L. M a g g i “), R u b i n o w i c z S), and B o u w k a m p ‘) discussed a Kirchhoff’s integral which amounts to a special function u0 inside L orginating from a primary spherical wave. In our case, however, z.+,is supposed to be an arbitrary function of x and y. *)
see
w at
s 0 n,
l.c.,
page
198.
476
H.
BREMMER
An acoustical problem corresponding to a function 2t0 that is constant inside L has been treated by S c h o c h “). M a g g i has already pointed out that surface integrals like (1) can be transformed, with the aid of Stokes’s integral theorem, into a line integral, apart from a remainder term corresponding to our has been investigated thoroughly by ~‘geo,,r~This transformation K o t t 1 e r “). In its turn the line integral (8) for zbdi,,can be converted at once into a surface integral over the area C of z = 0 inside L by applying Green’s integral theorem for the two-dimensional case. The result reads: 1 1 ZL,i// = - Ci”==o~ zdO, {d$ u&Q) .A,Ki - Kj.A;+’ u,,(Q)}, (12) 2n 1. in which formula A, refers to differentiations with respect to the coordinates of Q. 5. The wave equation for the series of uug,,,, + udirl cut off after a finite number of terms. A substitution of (7), and (8) or (12), into (10) leads to an infinite series for u(P). In this section we consider the corresponding series consisting of a finite number of terms (N). By applying (12) for the diffraction part, this finite series proves to be:
Z&N(P) = uN,
geont(P)
+
Z‘N, cti//(P)
= “ Hl,'i, 7-
(kz) A;
-
dx,
+$+'u,(Q)l.
Y) +
(13)
Owing to the vanishing of Ki for z = 0 this series fulfils the boundary condition u(P) + uo(P) for .z + 0. Further, the corresponding infinite series (N + m) automatically reduces to a finite one if ztOis a polynomial in x and y of some degree m. In fact, A’II,, then vanishes for j > m/2 so that the infinite series may just as well be replaced by the finite series provided N exceeds the quantity $rn - 1. The correctness of (10) for polynomial functions z&r, y) is therefore shown by demonstrating that (13) satisfies the wave equation for N > am - 1. The proof to be given here will be based on an investigation of (A,, + k2)Ki(Q, P); the subscript P or Q in A, and A, henceforth indicates, when necessary, that the Laplace operator should refer to the coordinates of P or Q.
ON
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477.
The formula (9a) represents Kj as the effect of sources situated (for a fixed position of Q) at the points (x0, yQ, 5‘) on a line drawn through Q parallel to the z-axis. The wave function eikr/r describing a single point source at Y = 0, or x = y = z = 0 satisfies the differential equation : (A, + k2) (eiky/r) = -
4n 6(x) d(y) d(z).
(14) This can be verified by remarking that the right-hand side.does vanish everywhere except at the source, as it should; moreover, the integration over an infinitely small sphere around the source yields - 4n for the right-hand side, which is in accordance with the result obtained from an integration of the left-hand side with the aid of Gauss’s integral theorem. The extension of (14) to a superposition of sources such as represented by (9~) leads to: (A,, + WKj n%(-
qi+
=
1
-6(z+~)}. o=d5. [‘+‘H;.“,(KC){G(z-0 s The <-integration of the contribution depending on the factor 6(z - [) is performed by substituting 5 = z in the remaining part of the integrand in so far as 5‘ = z is inside the interval of integration. This happens to be for z > 0. On the other hand, the term S(z + [) results in a non-vanishing contribution only if z < 0. In our consideration referring to the space z > 0 we have only to account for the first d-function which yields: G--NY
(2k)i-”
-Ye)
(A,, + k2)Kj = {&(1)‘+‘/(2K)‘-‘}
S(X -
XQ) S(y -ye)
zi+*H;.“*(Kz).
(15)
Before applying this formula to ZL~,~~,,we transform the diffraction part of (13) into the following form taking adventage of the symmetry property AQKi = A,Ki:
We next derive with the aid of (15) : ((‘3
+
k2)
%~,di//
X ~ c do,
1
=
{Ah
4Q)
qi+l i!
4 dxCiN_o
- A,,
zi+B (2~)‘-4
- AF'
Hj!$(kZ)
u,(Q)} 8(x -x0)
X
S(y -
yo).
478
H.
BREMMER
The operator d,, can be shifted to the left of the integral signs. Further, the effect of the d-functions is zero unless P&x, y, 0) be situated in C, that is if @(x, y) > 0. Thus we get : (A,, + k2)u N,di,, = 4 +-c ipLo y’ x [A,, {U(Q) .A; u,(P)} -
&&
H;.ti,(kz)
U(G) . A;;’
q,(P)].
x (16)
We now proceed to the evaluation of (d, + k’)~~,~~~,,,. The differentiation with respect to z can be performed by applying the identity (k* + d*/d2) {z”+~ Hi,!! (kz)) = 2nk zfl-* H&(kz), which follows from elementary Hence, in view of the definition (k* ‘+ d*/d.+,,
properties of the Hankel functions. of uN, gco,,,according to (13))
geom=
11-k up) (+k)+ lqz, (I- l)” -I?- H$,,Jkz) (n -
= Moreover, ‘2
uN,
geonr =
U(@) (&nkz)* c;:;
(-
z/2k)”
n,
. A; u&> y) =
H~,‘~,(kz).A;‘+‘u,(x,y).
(17)
we have:
(&nkz)“X”-,(-~%)nH”~,(kz)A2{U(@).A&,(z,y)}.
An addition (‘3 +
1) ! (2k)l--1
k2)UN,geom
(+ nkz)* EfZo q?
(18)
of (17) and (18) leads to : =
H!,‘L,(kz) [A2 { U(G). A; zto} + U(Q) ($nkz)lqr
H;&(kz).A,N+‘u,.
The series occurring here proves to be identical apart from the sign. The only term remaining of (16) and (19) therefore yields:
(A, + k*)U, = u(G) (inkz)*
qf
which is the basic relation for the proof functions u0 provided N > &n - 1.
U(Q) . A;+’ uo] + (19)
with that of (16), after an adding
HgL(kz).
A;+’ u,,, (20)
of (13) for polynomial
ON
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479
6. The representation of the wave function in z > 0 by the series (13) for N -+ co. For polynomials uO(x, y) of degree m the right-hand
side of (20) vanishes if N > m/2 - 1 so that the wave equation (A3 + k2)u = 0 is satisfied in this case by the finite series (13) as well as by the infinite series which is identical with it. Further, this series reduces to U(@)u,(x, y) at z = 0 as it should do. Finally the radiation condition is fulfilled as may be seen by considering first the point at infinity in a direction differing from that of the z-axis. Such a point is situated beyond the illuminated region so that only the diffraction part has to be accounted for; the vanishing proportional to eikR/R here results from the form of the wave functions describing the sources occurring in (9a). The radiation condition also holds on the z-axis itself as follows by considering moreover the asymptotic approximations of the Hankel functions occurring in ZLOTY,,,. The validity of (10) has thus been proved completely for polynomial functions 2~~. The sum of (7) and (8) also represents the extension of the wave function to the space z < 0 provided z is replaced by 1 z 1 and the Hankel function Ho) of the first kind by that of the second kind Ht2’ throughout. This follo,ws from the fact that (k 1z I)“‘“HjFL!l_,(k ) z I) constitutes the analytical continuation for z < 0 of the function (kz)“+’ Hj,‘)+(kz) when arriving from z > 0 through the aperture C (n being an integer). We now pass to functions u0 differing from a polynomial so that (13) becomes an infinite series for N + co. Apparently the successive terms of zigeon,are of the orders of magnitude of k-” for k -+ 00. The n-th term of zcdi,,is even of a lower order, viz. k-“-“, as will be shown in the next section. The decrease of the orders of magnitude with respect to k-’ suggests that 24, is asymptotic for k + 00. This is also in accordance with (20), the right-hand side of which is of the order of kpN, so that (20) approaches to the wave equation (da + k2)2t, = 0 as k tends to infinity. A rigorous proof of the asymptotic character of the series U, (in Poincare’s sense) for k.+ co will not be given here. 7. The dependence of the diffraction term on odd powers of k-“. A substitution of (9b) into (8) yields the following explicit formula for the diffraction part of u(P) for finite N:
480
21l\',
H.
-
di//
=
Lx?; -(i/2k)’ 2n
I-0
BREMMER
ds [d’u,(Q)
j !
. -& - -& Q
{d’zco(Q))] x Q
iktr
dzb e--
X
ZL' -
[=(I&-(PQ)z)/2(rr--r)
[i+'
(PQ)2
2F
0
(21)
The hypergeometric function consists of terms proportional to the first i - 1 integral powers of k-‘. The dependence of the j-th term of zidi,, on k-’ (remembering that zbois independent of k) can therefore be indicated as follows :
in which f, is independent of k. The zb-integral is of a well-known type that can be expanded, by means of successive partial integrations, into the following series which in general is asymptotic for k --f co:
0 .s+l
du eik” f,(u) - eikcPQ) Czo A substitution following type
i-
s
&
fPQ)-
of this expression into (22) leads to a series of the (zLdi,,)i = Cz,
&J1
ds vi,,(P) Q)eiktPQ).
The exponential factor of this integral shows saddlepoints QS at those points of Q on L at which (PQ) passes through an extreme value as function of s. For k---f 00 the integral depends mainly on the contributions from the points near these saddle points. The theory concerning such points yields, e.g. in the case of a minimum of (PQ), a first-order approximation
/
23Ti 4 I W2(PQ)/ds2), 1 ’
ds pi,,(P, Q) eiktPQ) N vi,,(P, QJ eik (‘Qs)
whereas the remaining contributions prove to be proportional to k-“I-’ (m a positive integer) instead of k-*. The term (u~~,,)~thus depends asymptotically on terms proportional to odd powers of k-t beginning with k +--l. The complete diffraction part of u(P), viz. uudi,,= X~Eo (u~~,,)~therefore depends on contributions of the orders km&, k-‘/z, kd’z . . . This result is in accordance with van Kampen’s statement that diffraction integrals of any type consist asymptotically of terms
ON
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481.
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proportional to k”, k-‘, k-*. . . and others proportional to K-i, P’2, k---Q . . . . In our case the former terms are component parts of ~‘geotn 1 the latter of u,;,,. Moreover, ugco,,,depends, in the first instance, on the behaviour of u. near the projection PO of P on the plane of integration of (1). This position of PO corresponds to a twodimensional saddlepoint of the exponent ik(QP) of (1) since (QP) is minimum at PO as function of xa and yQ. In van Kampen’s nomenclature we have to do here with a critical point of the first kind.’ On the other hand the saddlepoints of zldiif, which lead to odd powers of k-l, are one-dimensional insofar as they are connected with an extreme value of (QP) as a function of the position of Q on a curve. This type of saddlepoint was named by V a n K a m p e n critical point of the second kind. The so-called critical points of the third kind are left out of consideration here; they concern contributions to u(P) arising from points near occasional discontinuities in the boundary curve L. The latter contributions, if present, are once again proportional to integer powers of k-l. 8. The special case of a constant function zto(x. y). In this case all the functions A”uo vanish for n > 0, so that both ZLOTY,,, and ,z+,, reduce to a single term. Remembering that fF$((kz)
we obtain
= (2/nkz)* eikr,
from (7) and (8) u(P) = u,U(@) eik’ + 2
According
to (9b), K. is given by:
. J
aK0
L ds r.
(23)
Q
ikrr
K. = z s- (PQ)
d%L e u* -
The integration symbol disappears aK,lav,, which proves to be:
in the differential
eW’Q) =0 -z---z
Substituting
ay,
’
into (23) yields: eW’Q)
u(P) = zc,U(@) eikz-
2
z s
the sense of increasing Physica
XVIII
coefficient
W'Q)
(PQ)* -7
avQ
z* .
L ds
s being defined
(pQ)*
V'Q) -7
av,
’
(24)
as in (6). 31
482
H.
BREMMER
This simple formula may be compared with a similar one derived by Rub i n o w i c z *) for a convergent wave passing through an aperture of arbitrary size. The specialization of R u b i n ow i c z’s formula to a parallel beam at right angle to the aperture leads to an expression containing a denominator (PQ) - z instead of (PQ)’ - 2. The reason for this discrepancy is that Rubinowicz’s wave function ,has not to satisfy the boundary condition u = u0 at z = 0 which is essential in our case. This very condition requires the occurrence of virtual sources, as expressed by (9a), at either side of 2 = 0. A saddle-point analysis similar to that applied by R u b i n ow i c z shows that the jump -u0 eikr, undergone by %gco,n = u0 eikzx U(Q) when passing across the cylindrical limit @(x, y) = 0 of the illuminated region from the inner to the outer side, is just compensated by an opposite jump of zbdi,,.The continuity of ug,,,, + zbdi,, at this limit is therefore guaranteed. 9. A further evaluation of the series for ugrom. In special circumstances we are only interested in the geometrical-optical part of u(P), for instance when investigating the effect of defocusing which depends on the change of the wave function in the vicinity of the image plane of an optical system. In this case we can identify ztOwith the wave function in the image plane itself (z = 0) whereas zbdi,, can be neglected if P is not near the edge of the image. In order to discuss ~&WI8in more detail for small z we transform (7) into the following series by expressing H,,-, in terms of elementary functions:
According to the remark made also to be valid for z < 0 (at because it is unchanged if z is that is if z and i are replaced by first few terms yields:
=u(Q) eiks Z‘gNU, c
*) l.c.
formula
(6) on p. 934.
in section 6 this expression proves least beyond the imaging system) replaced by 1z 1 and H(l) by H”‘, - z and - i. An evaluation of the
ON
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483 .
This series can be simplified considerably for kz> 1 (distances s great compared to the wavelength) in which case we may approximate each coefficient by its first term only. The series then obtained reads
or symbolically This expansion
: u(P) N U(Q) eikz.e(ir’Zk)d22~~. (27) follows straightforwardly from (2) by putting eikz( I +&x/k’W
-
eikz
e(iz/2k)d,
10. The interpretation of the various terms of the series for ~~geO,,,. The approximation given by the first term of (254, viz. eikkO(X, y), represents a plane wave eikZ the amplitude of which is identical at all points of a line parallel to the z-axis. The factor U&Z, y) would also occur if a wave eikZarriving from z < 0 were modified in the plane z = 0 by an object having a transparency varying from point to point according to q,(x, y). The case of real zfO(purely absorbing virtual object) thus suggests interpreting the first term of (23) as a “shadow term” describing the distribution of light in the shadow cast beyond z = 0 by this object (or by the optical image of such an object). The first correction to the shadow term, viz. eikz (iz/2k) d,zc.,(x, y),
can be connected with the “contrast” C of the distribution zl,(n-, y) in 2 = 0 if C is defined as follows. We compute the average ziOof 11~over a small circle (radius I) around the point under observation P(q,, yO), that is: 1 1 2rr de e ~60(xo + e cos v, y. + e sin d. Uo(xoJ ~0) = z o dp, s ..O /” The linear terms of the Taylor expansion of u. with respect to Q lead to zero terms in view of the p-integration. The/ second-order terms yield the contribution:
= (12/8n) A, uo(x, y).
484 Obviously,
H.
BREMMER
the next terms are of the orders of 14, l6 . . . so that 4’”
%;i$
= ( 1/8n2)d,z&,
y).
(28)
The operator Au, here proves to be a measure for the deviation of ZL~at P from its average over the surrounding vicinity. In the case of real positive u,,, this function zhOis connected directly with the brightness in the image plane which is proportional to 1 z+ 12. Now, the contrast of the brightness depends on the difference in brightness at neighbouring points. We might speak of positive contrast at a point P if the brightness there exceeds the average of its value over the immediate vicinity, so that U: > ~7;. In view of (26) and of our assumption of a real positive u,-function this inequality leads to a negative value of A,zc,. In the opposite case, viz. a spot which is dark relative to its environment (negative contrast), A2u0 proves to be positive. We might therefore define C = - A,u, as the “contrast” of zto and call the second term of (25) the contrast term. The factor 1182 of (28) is not essential; it merely depends on the shape of the area concerned with the averaging procedure. The connection of the optical contrast with the quantity - A,u, has already been pointed out in 1865 by M a c h lo) when investigating the physiological sensitivity of the human eye. The third term of (25), which is proportional to A&,, apparently depends on the contrast of A,u,, that is on a contrast of second order of z+,.* The complete series (25) thus represents the corrections to the simple shadow effect in terms of contrasts of increasing orders. 11. Final remarks concerning the expansion of the geometricaloptical part. The interpretation of the various terms of (25) is less simple in the general case of a complex function u0 which may describe both absorption effects and phase retardations caused by an object (or by its image due to an optical system) in the plane .z = 0. The contrast term, e.g., now depends on the distribution of the brightness as well as on that of the phase retardations at z = 0. An interesting limiting case, given by 1 u0 1 = 1 or zfO= ei@ (@ real), corresponds to the distribution at z = 0 caused by a purely transparent non-absorbing object (or by the ideal image of such an object). The evaluation of the brightness 1 U(X, y, z) I2 near r = 0 from (25) leads to an expansion starting with ( ZI I2 = 1 - (I z I/k) A,@ . . . . . (29)
ON
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ASYMPTOTIC
EVALUATION
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DIFFRACTION
INTEGRALS
445
This formula indicates how special features of an object at z = 0 which is invisible there (owing to the constant value of 1 zcOI2 = 1) become visible for z # 0, an explanation of which has been given by 2 e r n i k e n). The expression (29) thus explains quantitatively the improved visibility of transparent objects in coherent light by means of defocusing since it shows the variation of 1ZI l2 as a function of x and y (depending on A,@), such a variation not existing for adjustment in focus (1ZL1= 1 throughout). Moreover, (29) is in accordance with a special case of small phase retardations considered by Zemike *). It will be obvious that also other phenomena observed during defocusing depend on the value of 1u I2 which results from (25). The author is indebted to his colleague Dr. B o u w k a m p for helpful advice concerning the wording of this paper. Eindhoven, Received
13 February
1952.
21-3-52.
REFERENCES 1) Luneberg, R. I<., Mathematical Theory of Optics (Providence, R. J.: Brown University; 1944), page 351-359. 2) V a n K a m p e n, N. G., Physica 14 (1948-1949) 575. 3) W a t s o n, G. N., Theory of Bessel functions (Cambridge 1944), page 140. 4) Booker, H. G:and Clemmow, P. C., Proc. Inst. Elect. Engrs, pt. 111, 97 (1950) 11. 5) AI a g g i, G. A., Ann. di Matem. 16 (1888) 21. 6) Rubinowicz, A., Phys. Rev.54(1938)931. 7) Bouwkamp, C. J., Physica 7 (1940) 485. 8) S c h o c h, A., Ak. Z. 6 (1941) 318. 9) K o t t 1 e r, F., Ann. Physik (4) 70 (1923) 416. 10) 11 a c h, E., Wiener Ber. 52 (1865) 303; see also Wien-Harms Handbuch der Esperimentalphysik, tome 20 second part (Leipzig 1929), p. 176. 11) Z e r n i k e, F., Physica 9 (1942) 686.
____-*) l.c.,
page
696 formula
(11).
XVIII,
no 6-7
Juni-Juli
1952
ON- THE ASYMPTOTIC EVALUATION OF DIFFRACTION INTEGRALS WITH A SPECIAL VIEW TO THE THEORY OF DEFOCUSING AND OPTICAL CONTRAST by H. BREMMER N.V.
Philips’
Philips Research Gloeilampenfabrieken
Laboratories Eindhoven,
Nederland
synopsis An expansion is given of the function u(x, y. z) that satisfies the scalar wave equation in the half-space z > 0 and equals a given distribution U&Y, y) in the plane z = 0. If ue is zero beyond a closed contour L in z = 0, part (which vanishes u(x, y. z) splits into a so-called geometrical-optical outside the cylinder passing through L and having generating lines parallel to the z-asis) and a diffraction part (determined by the values of uc near L). Each part can be expanded into terms depending on U&Y, y) itself and on its iterative two-dimensional Laplace operators d%c = ($/a2 + $/@)” U&C, y). The expansions are in general asymptotic for small wave lengths; however, their exact validity can be proved for functions uo that are polynomials inside the contour L. The terms of the diffraction part consist of contour integrals along L, the corresponding terms for the geometrical-optical part do not depend on any integration. The first few terms of the latter part are essential for defocusing effects and can be connected with the brightness-contrast existing in the plane z = 0.
1. Introduction. One of the few boundary problems occurring in diffraction theory which can be solved rigorously concerns the determination of a function u that satisfies the scalar wave equation (Aa + k?)~ = 0 and equals a given distribution Z&Y, y) in some plane z = 0. The notation A, is used for the three-dimensional Laplace operator in order to distinguish it from the corresponding two-dimensional operator A, to be discussed later on. The problem in question is uniquely determined for the half space z > 0 provided the radiation condition is fulfilled there at infinity. This condition -
469 -
470
H.
BREMMER
amounts to a decrease of u proportional to eikR/R at great distances (R + co). Green’s method leads to the following solution:
u(P) =-;g
eWQp) dO,%@)
(Qp)
J
(2~ >
0)
(1)
P
in which do, is a surface element of the plane z = 0 and (QP) is the distance from P to the variable point Q. A detailed discussion of the rigorous conditions for the validity of (1) is given by L u n eb e r g l). The scalar treatment considered here is also applicable to the electromagnetic theory of optics because each vector component by itself does satisfy the above wave equation. However, the complete vectorial problem should also deal with the complicated connections between the various fieldcomponents. In practical problems the integration in (1) extends over a finite part only of z = 0 because zcOis always zero beyond some closed boundary curve L (depending on some aperture or other). The diffraction integral then is of a type which can be reduced to suitable approximations in the two following cases: 1) Distances great compared to the finite dimensions of the domain of integration E inside L. The exponent ik(QP) can be approximated there by the linear terms of its Taylor development with respect to xQ and yQ. The Fraunhofer approximation thus obtained shows the connection of the u-distribution at great distances with the Fourier transform of U&V, y); 2) Distances great compared to the wavelength. The saddle-point method for exponential integrals can be used here. In its evaluation the factor exp (ik(QP)) has to be considered together with other exponential factors occurring occasionally in zbO(Q) (such a factor may arise, e.g., from a primary wave describing the illumination of the plane z = 0 by a light source situated somewhere in z < 0). Neither of these methods is adapted in its usual form for an investigation of u near the plane z = 0. In the theory of lenses this domain is of interest with a view to the theory of defocusing. The present paper deals with an expansion which is suitable in that case. As in other diffraction theories, the result to be derived splits into two contributions : u) a part of geometrical-optical character, the main term of which can be interpreted as a shadow effect, b) a diffraction part (in a limited sense of the word) which only
ON
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47.1
INTEGRALS
depends on the properties of zbOalong the boundary L of X. In accordance with the theory of V a n K a m p e n “) the first contribution can be expanded for small wavelengths into an asymptotic series containing integral powers of k-l, the second contribution into a similar series depending on odd-numbered powers of kd. The basic results of this paper will be derived provisionally in the two following sections. The details of the theory follow afterwards. 2. Formal derivation of a series for u(P) in the domain The wave equation can be written as if
z > 0.
a22qa2 + (k2 + A,)26 = 0, A, = azjax2 + a21ay2
represents the two-dimensional Laplace operator. Our boundary problem of finding a solution u in z > 0 that equals a given function z&r, y) in z = 0 can be solved symbolically by: ,21(x,y, z) = e”Z(k2+d2)tuo(x, y) = eikz(1+d2’k2)fq)(x, y).
(z > 0)
(2)
The sign of the exponent of (2) is in accordance with the radiation condition, as may be verified for the special case of u,, equal to a constant. We now apply the expansion e’k’(l-‘2/kz)t = (4 nkz)i X:=0 (H~~,,(kz)/n!)
(n2z/2k)“,
(3)
which is valid for 1A 1 < 1 k ) and which results from elementary properties of Bessel functions “). By substituting A, for - A2 we obtain from (2): u(x, y, z) = (&nkz)* IZ;& (Hill-)-&kz)/n!) (- z/2k)” A; q,(x, y).
(4)
Each term of this series has a definite meaning if we understand by A’%,, the application, n times in succession, of the two-dimensional Laplace operator A, to the function zc,,(x, y). For instance we have: 4:=(;+3=$+2&+$. The expansion (4) formally satisfies the wave equation and reduces to u,, for z = 0. The correctness of (4) is easily verified for those special functions u,,(x, y) for which the operator ‘A2 amounts to multiplication by a constant a. Examples of such functions are
472
H.
BREMMER
the Bessel functions l&a!) and Jo{e(- a)&} for a > 0 and u < 0 respectively, and also the exponential function exp (ax + /?y) if c? + ,92 = a. For functions of this type (2) represents the exact solution of the boundary problem if d, is simply replaced by a so as to obtain
In these cases the derivation of the series (4) reduces to an ordinary non-symbolical application of (3) for A2 = -a. For the functions under consideration we conclude that (4) is convergent if 1 a 1 < 1 K I2 but divergent if 1 a 1 > I k 1’. The exponentials exp (ax + py) constitute the components of the Fourier integral of u,&, y), each of them having a period 1 = 2nluk. The above condition of convergence thus amounts to 1 > l.A convergence of (4) may therefore be expected for structures u&z, y) without periods I < 1. The contributions for which 1 > 1 are in general unimportant in optical problems since they do not contribute to the diffraction pattern at great distances, as has been shown by B o o k e r and C 1 e m m o w “). Anyhow, (4) cannot represent u(P) exactly under all circumstances. Another disadvantage of (4) is that it does not give explicitly the typical edge effects arising from the vicinity of L for functions U&X, y) that are zero outside L. Nevertheless such edge effects are contained implicitly in (4), as can be shown in the following way. Let @(x, y) be a continuous function that is positive inside and negative outside L. The function representing ztOover the complete plane z = 0 then is given by u,,(x, y)U{@(x, y)} if U(t) marks Heaviside’s unit function (which is 1 for positive arguments and zero for negative arguments). The iterative Laplace operators of the product of 2~~and U(@) lead amongst others to delta functions and their derivatives according to the relation 6(t) = dU(t)/df. Each term containing such functions shows the argument t = @ and thus becomes singular at @ = 0, that is along the boundary L. The contributions containing these delta functions constitute formally the diffraction part of our solution. A more convenient representation of this diffraction part can be deduced with the aid of Green’s integral theorem and is indicated without proof in the next section. Its correctness is demonstrated in the sections 5 and 6 for functions u&, y) that are representable by polynomials in x and y inside L.
ON
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473
INTEGRALS
3. The sp!ditting of u(P) into a geometrical-optical part and a diffraction part. Such a splitting is obtained by the following formal
procedure. We transform (1) with the aid of Sommerfeld’s for eiWP) /(QP) into the th ree-dimensional integral u(P) = &
dA
;leikz(l-Az/kz)h do,
s- 0
!...I
.?I
integral
2co(Q)JoV(QPo)),
in which P,(x, y, 0) is the orthogonal projection of P(x, y, z) on the plane z = 0. A substitution of the expansion (3) then leads to integrals the integrands of which depend amongst others on a factor ~2”JoMQPoH. Th is expression should be replaced by the Laplace operator terms now obtained,
(- ~,)“Jo(~(QPo)), referring to the coordinates viz.
of Q. The various
are further to be transformed by n successive applications i=o, 1, . . . . n - 1) of Green’s formula: &dOa +
s
. di zco(Q).d;+Jo(~(QPo))
=/IZdOp
.A;‘-‘-I
(for
Jo(;c(QPo)) . Ai+’ u,(Q) +
Lds[d'zuo(Q).g{~~-i-'~o(~(QPo))}-~~-i-'~o(~(QPo)).~{~'l,,(Q)H.
(6) Q
Q
In this identity ds is an element of the boundary curve L, an arbitrary point of which is marked Q (s increases when moving along L in a sense which is observed as counter-clockwise by an observer viewing from the space z > 0), whilst a/&, refers to a differentiation along the normal at Q in the outward direction. When evaluating (5) with the aid of (6) the surface integrals occurring in the right side of (6) lead to contributions proportional to the integrals , which
j;dl
A/i,
dO,.d;-‘-’
can also be written
Jo(A(QPo)) .dp’
vo(Q),
as follows in view of the symmetry
(QP,) with respect to Q and PO:
of
474
According
H.
to Neumann’s 2n
(
$+P
BREMMER
integral
a2 a-j-I
as )
theorem
this expression
equals
A;++’ uo(P) = 2nA;‘u,(P),
provided P, is situated inside L, whereas it does vanish for P, outside L. The surface integrals occurring in the right side of (6) thus result in a final contribution that differs from zero only in the cylindrical region @(x, y) > 0 the points P(x, y, z) of which show projections P&x, y, 0) inside L (see fig. 1). This contribution, to
be termed geometrical-ofitical @art or ugeo,,,, proves to be identical with (4) in @(x, y) > 0. This first contribution can therefore be represented throughout the half space z > 0 by
The region @(x, y) > 0 will be called the illuminated region because it constitutes the space that is lit through the area C if the latter is illuminated by a parallel beam of light arriving from the negative z-direction. The remaining contribution to G(P), to be termed diffraction arises from the contour integrals of (6) when this $art or udi/f9 formula is applied to the various terms (5) of u(P). The diffraction part thus appears as a sum over n and i of terms proportional to OmdI ;I [4$ uO(Q). ;
{A;-‘-‘J,(l(QP,))}
-
Q
-A
;‘+-I
J~(~(QP~. =JQ
4
~Q)I.
ON
THE
ASYMPTOTIC
EVALUATION
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475
INTEGRALS
A formal summation over n for fixed j now proves to be possible, again by using (4). The final result of the deduction briefly indicated here can be represented as follows: aLdiN= -
1
C”,--o -;,
2.75
ds .Ls
[d&,(Q)
$
-
@
K,$
U~~,(Q)>l~ (8) Q
In (9b) 2F0 refers to Pochhammer’s notation for hypergeometric series. A direct proof of the final representation of u(P) in z > 0 by the sum 4P) = %wn(~) + %ij,PL (10) will be performed in sections 5 and 6 for the case of polynomial functions U&X, y). We conclude this section by remarking that the equivalence of (9~) and (9b) can be shown by the substitutions u = 5+
W=oQ)2 + (C 7 d2>”
in the two consecutive terms of (9~) and by applying identity *) for the finite series of H$(k[): eikt (&nkQ6H/‘&(k<)
= ii2F,
j, 1 -
j;; &)
the following
.
(11)
4. The diffraction $art expressed as a surface integral. A splitting corresponding to (10) is well known in diffraction problems. The diffraction part always appears as a line integral along a boundary curve L. M a g g i “), R u b i n o w i c z S), and B o u w k a m p ‘) discussed a Kirchhoff’s integral which amounts to a special function u0 inside L orginating from a primary spherical wave. In our case, however, z.+,is supposed to be an arbitrary function of x and y. *)
see
w at
s 0 n,
l.c.,
page
198.
476
H.
BREMMER
An acoustical problem corresponding to a function 2t0 that is constant inside L has been treated by S c h o c h “). M a g g i has already pointed out that surface integrals like (1) can be transformed, with the aid of Stokes’s integral theorem, into a line integral, apart from a remainder term corresponding to our has been investigated thoroughly by ~‘geo,,r~This transformation K o t t 1 e r “). In its turn the line integral (8) for zbdi,,can be converted at once into a surface integral over the area C of z = 0 inside L by applying Green’s integral theorem for the two-dimensional case. The result reads: 1 1 ZL,i// = - Ci”==o~ zdO, {d$ u&Q) .A,Ki - Kj.A;+’ u,,(Q)}, (12) 2n 1. in which formula A, refers to differentiations with respect to the coordinates of Q. 5. The wave equation for the series of uug,,,, + udirl cut off after a finite number of terms. A substitution of (7), and (8) or (12), into (10) leads to an infinite series for u(P). In this section we consider the corresponding series consisting of a finite number of terms (N). By applying (12) for the diffraction part, this finite series proves to be:
Z&N(P) = uN,
geont(P)
+
Z‘N, cti//(P)
= “ Hl,'i, 7-
(kz) A;
-
dx,
+$+'u,(Q)l.
Y) +
(13)
Owing to the vanishing of Ki for z = 0 this series fulfils the boundary condition u(P) + uo(P) for .z + 0. Further, the corresponding infinite series (N + m) automatically reduces to a finite one if ztOis a polynomial in x and y of some degree m. In fact, A’II,, then vanishes for j > m/2 so that the infinite series may just as well be replaced by the finite series provided N exceeds the quantity $rn - 1. The correctness of (10) for polynomial functions z&r, y) is therefore shown by demonstrating that (13) satisfies the wave equation for N > am - 1. The proof to be given here will be based on an investigation of (A,, + k2)Ki(Q, P); the subscript P or Q in A, and A, henceforth indicates, when necessary, that the Laplace operator should refer to the coordinates of P or Q.
ON
THE
ASYMPTOTIC
EVALUATION
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DIFFRACTION
INTEGRALS
477.
The formula (9a) represents Kj as the effect of sources situated (for a fixed position of Q) at the points (x0, yQ, 5‘) on a line drawn through Q parallel to the z-axis. The wave function eikr/r describing a single point source at Y = 0, or x = y = z = 0 satisfies the differential equation : (A, + k2) (eiky/r) = -
4n 6(x) d(y) d(z).
(14) This can be verified by remarking that the right-hand side.does vanish everywhere except at the source, as it should; moreover, the integration over an infinitely small sphere around the source yields - 4n for the right-hand side, which is in accordance with the result obtained from an integration of the left-hand side with the aid of Gauss’s integral theorem. The extension of (14) to a superposition of sources such as represented by (9~) leads to: (A,, + WKj n%(-
qi+
=
1
-6(z+~)}. o=d5. [‘+‘H;.“,(KC){G(z-0 s The <-integration of the contribution depending on the factor 6(z - [) is performed by substituting 5 = z in the remaining part of the integrand in so far as 5‘ = z is inside the interval of integration. This happens to be for z > 0. On the other hand, the term S(z + [) results in a non-vanishing contribution only if z < 0. In our consideration referring to the space z > 0 we have only to account for the first d-function which yields: G--NY
(2k)i-”
-Ye)
(A,, + k2)Kj = {&(1)‘+‘/(2K)‘-‘}
S(X -
XQ) S(y -ye)
zi+*H;.“*(Kz).
(15)
Before applying this formula to ZL~,~~,,we transform the diffraction part of (13) into the following form taking adventage of the symmetry property AQKi = A,Ki:
We next derive with the aid of (15) : ((‘3
+
k2)
%~,di//
X ~ c do,
1
=
{Ah
4Q)
qi+l i!
4 dxCiN_o
- A,,
zi+B (2~)‘-4
- AF'
Hj!$(kZ)
u,(Q)} 8(x -x0)
X
S(y -
yo).
478
H.
BREMMER
The operator d,, can be shifted to the left of the integral signs. Further, the effect of the d-functions is zero unless P&x, y, 0) be situated in C, that is if @(x, y) > 0. Thus we get : (A,, + k2)u N,di,, = 4 +-c ipLo y’ x [A,, {U(Q) .A; u,(P)} -
&&
H;.ti,(kz)
U(G) . A;;’
q,(P)].
x (16)
We now proceed to the evaluation of (d, + k’)~~,~~~,,,. The differentiation with respect to z can be performed by applying the identity (k* + d*/d2) {z”+~ Hi,!! (kz)) = 2nk zfl-* H&(kz), which follows from elementary Hence, in view of the definition (k* ‘+ d*/d.+,,
properties of the Hankel functions. of uN, gco,,,according to (13))
geom=
11-k up) (+k)+ lqz, (I- l)” -I?- H$,,Jkz) (n -
= Moreover, ‘2
uN,
geonr =
U(@) (&nkz)* c;:;
(-
z/2k)”
n,
. A; u&> y) =
H~,‘~,(kz).A;‘+‘u,(x,y).
(17)
we have:
(&nkz)“X”-,(-~%)nH”~,(kz)A2{U(@).A&,(z,y)}.
An addition (‘3 +
1) ! (2k)l--1
k2)UN,geom
(+ nkz)* EfZo q?
(18)
of (17) and (18) leads to : =
H!,‘L,(kz) [A2 { U(G). A; zto} + U(Q) ($nkz)lqr
H;&(kz).A,N+‘u,.
The series occurring here proves to be identical apart from the sign. The only term remaining of (16) and (19) therefore yields:
(A, + k*)U, = u(G) (inkz)*
qf
which is the basic relation for the proof functions u0 provided N > &n - 1.
U(Q) . A;+’ uo] + (19)
with that of (16), after an adding
HgL(kz).
A;+’ u,,, (20)
of (13) for polynomial
ON
THE
ASYMPTOTIC
EVALUATION
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INTEGRALS
479
6. The representation of the wave function in z > 0 by the series (13) for N -+ co. For polynomials uO(x, y) of degree m the right-hand
side of (20) vanishes if N > m/2 - 1 so that the wave equation (A3 + k2)u = 0 is satisfied in this case by the finite series (13) as well as by the infinite series which is identical with it. Further, this series reduces to U(@)u,(x, y) at z = 0 as it should do. Finally the radiation condition is fulfilled as may be seen by considering first the point at infinity in a direction differing from that of the z-axis. Such a point is situated beyond the illuminated region so that only the diffraction part has to be accounted for; the vanishing proportional to eikR/R here results from the form of the wave functions describing the sources occurring in (9a). The radiation condition also holds on the z-axis itself as follows by considering moreover the asymptotic approximations of the Hankel functions occurring in ZLOTY,,,. The validity of (10) has thus been proved completely for polynomial functions 2~~. The sum of (7) and (8) also represents the extension of the wave function to the space z < 0 provided z is replaced by 1 z 1 and the Hankel function Ho) of the first kind by that of the second kind Ht2’ throughout. This follo,ws from the fact that (k 1z I)“‘“HjFL!l_,(k ) z I) constitutes the analytical continuation for z < 0 of the function (kz)“+’ Hj,‘)+(kz) when arriving from z > 0 through the aperture C (n being an integer). We now pass to functions u0 differing from a polynomial so that (13) becomes an infinite series for N + co. Apparently the successive terms of zigeon,are of the orders of magnitude of k-” for k -+ 00. The n-th term of zcdi,,is even of a lower order, viz. k-“-“, as will be shown in the next section. The decrease of the orders of magnitude with respect to k-’ suggests that 24, is asymptotic for k + 00. This is also in accordance with (20), the right-hand side of which is of the order of kpN, so that (20) approaches to the wave equation (da + k2)2t, = 0 as k tends to infinity. A rigorous proof of the asymptotic character of the series U, (in Poincare’s sense) for k.+ co will not be given here. 7. The dependence of the diffraction term on odd powers of k-“. A substitution of (9b) into (8) yields the following explicit formula for the diffraction part of u(P) for finite N:
480
21l\',
H.
-
di//
=
Lx?; -(i/2k)’ 2n
I-0
BREMMER
ds [d’u,(Q)
j !
. -& - -& Q
{d’zco(Q))] x Q
iktr
dzb e--
X
ZL' -
[=(I&-(PQ)z)/2(rr--r)
[i+'
(PQ)2
2F
0
(21)
The hypergeometric function consists of terms proportional to the first i - 1 integral powers of k-‘. The dependence of the j-th term of zidi,, on k-’ (remembering that zbois independent of k) can therefore be indicated as follows :
in which f, is independent of k. The zb-integral is of a well-known type that can be expanded, by means of successive partial integrations, into the following series which in general is asymptotic for k --f co:
0 .s+l
du eik” f,(u) - eikcPQ) Czo A substitution following type
i-
s
&
fPQ)-
of this expression into (22) leads to a series of the (zLdi,,)i = Cz,
&J1
ds vi,,(P) Q)eiktPQ).
The exponential factor of this integral shows saddlepoints QS at those points of Q on L at which (PQ) passes through an extreme value as function of s. For k---f 00 the integral depends mainly on the contributions from the points near these saddle points. The theory concerning such points yields, e.g. in the case of a minimum of (PQ), a first-order approximation
/
23Ti 4 I W2(PQ)/ds2), 1 ’
ds pi,,(P, Q) eiktPQ) N vi,,(P, QJ eik (‘Qs)
whereas the remaining contributions prove to be proportional to k-“I-’ (m a positive integer) instead of k-*. The term (u~~,,)~thus depends asymptotically on terms proportional to odd powers of k-t beginning with k +--l. The complete diffraction part of u(P), viz. uudi,,= X~Eo (u~~,,)~therefore depends on contributions of the orders km&, k-‘/z, kd’z . . . This result is in accordance with van Kampen’s statement that diffraction integrals of any type consist asymptotically of terms
ON
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DIFFRACTION
481.
INTEGRALS
proportional to k”, k-‘, k-*. . . and others proportional to K-i, P’2, k---Q . . . . In our case the former terms are component parts of ~‘geotn 1 the latter of u,;,,. Moreover, ugco,,,depends, in the first instance, on the behaviour of u. near the projection PO of P on the plane of integration of (1). This position of PO corresponds to a twodimensional saddlepoint of the exponent ik(QP) of (1) since (QP) is minimum at PO as function of xa and yQ. In van Kampen’s nomenclature we have to do here with a critical point of the first kind.’ On the other hand the saddlepoints of zldiif, which lead to odd powers of k-l, are one-dimensional insofar as they are connected with an extreme value of (QP) as a function of the position of Q on a curve. This type of saddlepoint was named by V a n K a m p e n critical point of the second kind. The so-called critical points of the third kind are left out of consideration here; they concern contributions to u(P) arising from points near occasional discontinuities in the boundary curve L. The latter contributions, if present, are once again proportional to integer powers of k-l. 8. The special case of a constant function zto(x. y). In this case all the functions A”uo vanish for n > 0, so that both ZLOTY,,, and ,z+,, reduce to a single term. Remembering that fF$((kz)
we obtain
= (2/nkz)* eikr,
from (7) and (8) u(P) = u,U(@) eik’ + 2
According
to (9b), K. is given by:
. J
aK0
L ds r.
(23)
Q
ikrr
K. = z s- (PQ)
d%L e u* -
The integration symbol disappears aK,lav,, which proves to be:
in the differential
eW’Q) =0 -z---z
Substituting
ay,
’
into (23) yields: eW’Q)
u(P) = zc,U(@) eikz-
2
z s
the sense of increasing Physica
XVIII
coefficient
W'Q)
(PQ)* -7
avQ
z* .
L ds
s being defined
(pQ)*
V'Q) -7
av,
’
(24)
as in (6). 31
482
H.
BREMMER
This simple formula may be compared with a similar one derived by Rub i n o w i c z *) for a convergent wave passing through an aperture of arbitrary size. The specialization of R u b i n ow i c z’s formula to a parallel beam at right angle to the aperture leads to an expression containing a denominator (PQ) - z instead of (PQ)’ - 2. The reason for this discrepancy is that Rubinowicz’s wave function ,has not to satisfy the boundary condition u = u0 at z = 0 which is essential in our case. This very condition requires the occurrence of virtual sources, as expressed by (9a), at either side of 2 = 0. A saddle-point analysis similar to that applied by R u b i n ow i c z shows that the jump -u0 eikr, undergone by %gco,n = u0 eikzx U(Q) when passing across the cylindrical limit @(x, y) = 0 of the illuminated region from the inner to the outer side, is just compensated by an opposite jump of zbdi,,.The continuity of ug,,,, + zbdi,, at this limit is therefore guaranteed. 9. A further evaluation of the series for ugrom. In special circumstances we are only interested in the geometrical-optical part of u(P), for instance when investigating the effect of defocusing which depends on the change of the wave function in the vicinity of the image plane of an optical system. In this case we can identify ztOwith the wave function in the image plane itself (z = 0) whereas zbdi,, can be neglected if P is not near the edge of the image. In order to discuss ~&WI8in more detail for small z we transform (7) into the following series by expressing H,,-, in terms of elementary functions:
According to the remark made also to be valid for z < 0 (at because it is unchanged if z is that is if z and i are replaced by first few terms yields:
=u(Q) eiks Z‘gNU, c
*) l.c.
formula
(6) on p. 934.
in section 6 this expression proves least beyond the imaging system) replaced by 1z 1 and H(l) by H”‘, - z and - i. An evaluation of the
ON
THE
ASYMPTOTIC
EVALUATION
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INTEGRALS
483 .
This series can be simplified considerably for kz> 1 (distances s great compared to the wavelength) in which case we may approximate each coefficient by its first term only. The series then obtained reads
or symbolically This expansion
: u(P) N U(Q) eikz.e(ir’Zk)d22~~. (27) follows straightforwardly from (2) by putting eikz( I +&x/k’W
-
eikz
e(iz/2k)d,
10. The interpretation of the various terms of the series for ~~geO,,,. The approximation given by the first term of (254, viz. eikkO(X, y), represents a plane wave eikZ the amplitude of which is identical at all points of a line parallel to the z-axis. The factor U&Z, y) would also occur if a wave eikZarriving from z < 0 were modified in the plane z = 0 by an object having a transparency varying from point to point according to q,(x, y). The case of real zfO(purely absorbing virtual object) thus suggests interpreting the first term of (23) as a “shadow term” describing the distribution of light in the shadow cast beyond z = 0 by this object (or by the optical image of such an object). The first correction to the shadow term, viz. eikz (iz/2k) d,zc.,(x, y),
can be connected with the “contrast” C of the distribution zl,(n-, y) in 2 = 0 if C is defined as follows. We compute the average ziOof 11~over a small circle (radius I) around the point under observation P(q,, yO), that is: 1 1 2rr de e ~60(xo + e cos v, y. + e sin d. Uo(xoJ ~0) = z o dp, s ..O /” The linear terms of the Taylor expansion of u. with respect to Q lead to zero terms in view of the p-integration. The/ second-order terms yield the contribution:
= (12/8n) A, uo(x, y).
484 Obviously,
H.
BREMMER
the next terms are of the orders of 14, l6 . . . so that 4’”
%;i$
= ( 1/8n2)d,z&,
y).
(28)
The operator Au, here proves to be a measure for the deviation of ZL~at P from its average over the surrounding vicinity. In the case of real positive u,,, this function zhOis connected directly with the brightness in the image plane which is proportional to 1 z+ 12. Now, the contrast of the brightness depends on the difference in brightness at neighbouring points. We might speak of positive contrast at a point P if the brightness there exceeds the average of its value over the immediate vicinity, so that U: > ~7;. In view of (26) and of our assumption of a real positive u,-function this inequality leads to a negative value of A,zc,. In the opposite case, viz. a spot which is dark relative to its environment (negative contrast), A2u0 proves to be positive. We might therefore define C = - A,u, as the “contrast” of zto and call the second term of (25) the contrast term. The factor 1182 of (28) is not essential; it merely depends on the shape of the area concerned with the averaging procedure. The connection of the optical contrast with the quantity - A,u, has already been pointed out in 1865 by M a c h lo) when investigating the physiological sensitivity of the human eye. The third term of (25), which is proportional to A&,, apparently depends on the contrast of A,u,, that is on a contrast of second order of z+,.* The complete series (25) thus represents the corrections to the simple shadow effect in terms of contrasts of increasing orders. 11. Final remarks concerning the expansion of the geometricaloptical part. The interpretation of the various terms of (25) is less simple in the general case of a complex function u0 which may describe both absorption effects and phase retardations caused by an object (or by its image due to an optical system) in the plane .z = 0. The contrast term, e.g., now depends on the distribution of the brightness as well as on that of the phase retardations at z = 0. An interesting limiting case, given by 1 u0 1 = 1 or zfO= ei@ (@ real), corresponds to the distribution at z = 0 caused by a purely transparent non-absorbing object (or by the ideal image of such an object). The evaluation of the brightness 1 U(X, y, z) I2 near r = 0 from (25) leads to an expansion starting with ( ZI I2 = 1 - (I z I/k) A,@ . . . . . (29)
ON
THE
ASYMPTOTIC
EVALUATION
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DIFFRACTION
INTEGRALS
445
This formula indicates how special features of an object at z = 0 which is invisible there (owing to the constant value of 1 zcOI2 = 1) become visible for z # 0, an explanation of which has been given by 2 e r n i k e n). The expression (29) thus explains quantitatively the improved visibility of transparent objects in coherent light by means of defocusing since it shows the variation of 1ZI l2 as a function of x and y (depending on A,@), such a variation not existing for adjustment in focus (1ZL1= 1 throughout). Moreover, (29) is in accordance with a special case of small phase retardations considered by Zemike *). It will be obvious that also other phenomena observed during defocusing depend on the value of 1u I2 which results from (25). The author is indebted to his colleague Dr. B o u w k a m p for helpful advice concerning the wording of this paper. Eindhoven, Received
13 February
1952.
21-3-52.
REFERENCES 1) Luneberg, R. I<., Mathematical Theory of Optics (Providence, R. J.: Brown University; 1944), page 351-359. 2) V a n K a m p e n, N. G., Physica 14 (1948-1949) 575. 3) W a t s o n, G. N., Theory of Bessel functions (Cambridge 1944), page 140. 4) Booker, H. G:and Clemmow, P. C., Proc. Inst. Elect. Engrs, pt. 111, 97 (1950) 11. 5) AI a g g i, G. A., Ann. di Matem. 16 (1888) 21. 6) Rubinowicz, A., Phys. Rev.54(1938)931. 7) Bouwkamp, C. J., Physica 7 (1940) 485. 8) S c h o c h, A., Ak. Z. 6 (1941) 318. 9) K o t t 1 e r, F., Ann. Physik (4) 70 (1923) 416. 10) 11 a c h, E., Wiener Ber. 52 (1865) 303; see also Wien-Harms Handbuch der Esperimentalphysik, tome 20 second part (Leipzig 1929), p. 176. 11) Z e r n i k e, F., Physica 9 (1942) 686.
____-*) l.c.,
page
696 formula
(11).