rado (1977).16 C. Roddier and F. Roddier, "Influence of exposure time on spectral

properties of turbulence-degraded astronomical images," J. Opt.Soc. Am. 65, 664-667 (1975).

17C. Roddier, "Measurements of the atmospheric attenuation of thespectral components of astronomical images," J. Opt. Soc. Am. 66,478-482 (1976).

18D. P. Karo and A. M. Schneiderman, "Speckle interferometrylens-atmosphere MTF measurements," J. Opt. Soc. Am. 66,

1252-1256 (1976).

19A. M. Schneiderman and D. P. Karo, "Double Star Speckle Inter-ferometry Measurements of Atmospheric Non-Isoplanicity," AERLRR 441, May 1977. Also see A. M. Schneiderman and D. P. Karo,"Speckle interferometry measurements of atmospheric noniso-planicity using double stars," J. Opt. Soc. Am. 68, 338 (1978).

20The possibility (noted in Ref. 18) that approximations exist in thetheory (Ref. 2) make the predicted MTF level uncertain by as much

Wave propagation in optical systems with large aperturesFrank D. Feiock*

Rocketdyne Division, Rockwell International, 6633 Canoga Avenue, Canoga Park, California 91304(Received 12 September 1977)

The Kirchhoff-Huygens equation is used to investigate wave propagation in optical systems, withlarge propagation Fresnel numbers N, and aperture-to-length ratios (a /L) which are not small. Thelimit of applicability of the Fresnel approximation is analytically established for a thin rectangularaperture. It is shown that the error introduced by the Fresnel approximation to the Kirchhoff integralis comparable to the effects of diffraction, computed by the approximation, times the dimensionlessparameter 7rNF (a /2L )2

INTRODUCTION

The subject of wave propagation in optical systems withlarge apertures is of current practical interest, particularly inthe design of high-power laser systems. Depending onwavelength and propagation lengths, optical systems withlarge apertures will generally contain one or more segmentswith a large propagation Fresnel number.

Wave propagation at high Fresnel number has previouslyreceived considerable attention." 2 This work has addressedthe problem in the Fresnel approximation, which neglectshigher-order terms in the phase factor. This approach isjustifiable when the large Fresnel number results from theshort wavelength of the radiation, but may neglect importanteffects specifically related to the large aperture. The purposeof this paper is to examine the effects of large apertures onwave propagation with large Fresnel numbers and to assessthe validity of the Fresnel approximation.

REVIEW OF FRESNEL APPROXIMATION

To propagate the field we use the Kirchhoff integral for-mulation of Huygen's principle, which gives the field at a pointp in terms of the field on an aperture

up = (iIX) UA e-ikR (cosO/Ri) dA.

In Eq. (1)

k = 27r/X,

where X is the wavelength of the radiation,

R = [(x2 - x1)2 + (Y2 - yl) 2 + (Z2 - Zl)2]1/2

is the distance between the aperture surface element dA andthe point p, and 0 is the angle between R and the normal to

the surface of the aperture. We have used the Sommerfeldform for the obliquity factor cosO.

For simplicity, consider a rectangular aperture in the z1plane

-a <x,<a, (4)-b < Yi < b.

The field is to be computed at points p(x2 , Y2) in the Z2 planewith

L = 1z2 -zI, (5)

which is large compared to the dimensions of the aperture.Then the square root given in Eq. (3) can be expanded in apower series. The discussion is facilitated by introducingdimensionless variables (x, y) for the point p and (x', y') forpoints on the aperture using the relations

x 2=ax, x1=ax', (6)Y2 = by, Y, = by'.

To second order, the expansion gives

R =L {1+ [(a)2(x-X')2 + (b)2 (y - y)2 (

-1[(a) I(x -)2+(L)2 (y yt21 .(1)

At this point the usual procedure is to keep only the first-order terms in the expansion and neglect the slow variation

(2) of

cosO/R = L/R 2 (8)

in comparison with the rapid oscillation of the exponentialfactor in the integral. This leads to the Fresnel approxima-tion to the Kirchhoff integral

(3)

485 J. Opt. Soc. Am., Vol. 68, No. 4, April 1978 0030-3941/78/6804-0485$00.50 � 1978 Optical Society of America 485

.UP(X, y) = i(NXNY)1/ 2 fJ UA(x, Y')

X exp -iir[Nx(x-x') 2 + Ny(y - y') 2 ] dx' dy', (9)

where

Nx = a2 /XL, Ny = b2/XL (10)

are the Fresnel numbers of the aperture. We have droppedthe constant phase factor exp(-ikL).

For the first-order approximation to the phase to be valid,it is sufficient for the additional phase factor resulting fromthe second-order terms to be small compared with one radian,which is equivalent to the requirement

max {Nx (a) 2 NY ( a N. (1)2 N ( b )21

It can be argued that this mathematical requirement is overlystringent. 3 Physically, the first-order approximation is usefulprovided the higher-order terms not change the value of thesuperposition integral. For values of the parameters whichfail to meet the strict inequality, the phase due to the first-order terms oscillates rapidly and cancels the effects due tothe higher-order terms. The metliod of stationary phaseprovides a mathematical realization of this argument. In thenext section we shall obtain a more general asymptotic ex-pansion of the integral, which can be used to evaluate thesignificance of the higher-order terms.

The Fresnel-Kirchhoff integral can also be derived directlyfrom the scalar Helmholtz equation for the field 41(x1 , yl,Z1) ( 2 02 02

X2+ k 2 ) p -. (11)

Write the solution in the form of a wave propagating along thez 1 axis

'(x1, yi, z1 ) = U(xi, Yi, z1) e-ikzl. (12)

The second partial derivative with respect to z , gives

d / 0 e-ikzl (-k 2 - 2ik d +d 2) U(xi, Yj,z1). (13)

Assuming U(x1, yi, z1) to be a slowly varying function of z1,the second derivative can be neglected in comparison with thefirst two terms. In this approximation, combining Eqs. (11),(12), and (13) gives

02 02 0( + -+ -2ik d ) U(xi 1,yi, Z) = 0. (14)

Equation (14) is the paraxial approximation to the scalarHelmholtz equation. By direct substitution, it can be shownthat Up (x, y) given by Eq. (9) is a solution of Eq. (14). Be-cause this derivation does not explicitly deal with the ex-pansion of R or the obliquity factor, it is sometimes arguedthat Eq. (9) has a validity beyond the approximations usedto derive it from the Kirchhoff integral. It should be noted,however, that an approximation was made to obtain Eq. (14)which is equivalent to the requirement that

a2 U a0 2 02« ( a2±+ - - 2ikyd) U1

everywhere in the propagation region.

486 J. Opt. Soc. Am., Vol. 68, No. 4, April 1978

(15)

It is difficult to establish the validity of this approximation,especially in the integral form of Eq. (9), which may accu-mulate a large number of small errors. The remainder of thisreport is concerned with assessing the validity of the Fresnelapproximation, Eq. (9), to the Kirchhoff integral given in Eq.(1).

HIGHER-ORDER TERMS

We wish to evaluate the effect on the Kirchhoff integral dueto higher-order terms in the expansion of R, given by Eq. (7).For simplicity we consider a thin rectangular slit for which

b <<a <L.

Then to first order in (b/L)2 and second order in (a/L)2, thephase factor is proportional to

A = L + Nx(x -x') 2[1 - a(x -x')2 ] + 2 NY(y -y) 2 ,X X 2 2

(16)

where

a = (a/2L) 2 . (17)

We also include the obliquity factor to first order in a

cosO/R = 1/1L[1 + 4a(x - x') 2]j. (18)

In this approximation, the Kirchhoff integral is given by

UP(X,y) = i(NXNY)1"2 f [UA(x',y')/1 + 4a(x-X)2]

X exp (- i7r[Nx(x - x')2[1 - a(x -X)2]

+ Ny(y - y') 21) dx' dy'. (19)

Even with the extra terms, the two-dimensional kernel for athin rectangular slit can be written in the form of a productof two one-dimensional kernels.

In order to obtain analytical solutions of Eqs. (9) and (19),we assume uniform irradiation of the aperture. For bothequations, the resulting field at the receiver point p (x, y) canbe written as a product

(20)U (x,y) = U(x)V(y),

where the function V(y) is given by the integral

V(y) = (iNy)112 J- eirNy(y-Y)2 dy'.

f-1 I

Thus the assessment of the influence of the extra term in thephase and retention of the obliquity factor ieduces to theevaluation of the integral

U(a, x) = (iN.)112

X f expf-iirN.,(x - X')2[1 - a(X - X') 2] dx'.f-1 1 + 4a(x - X)2

(22)

The integral in Eq. (22) can be put into more tractable formfor evaluation by the following changes of variables. Let sx' - x and split the resulting integral into two parts-1 - x< s < O and 0 < s < 1 - x. Change the sign of s in the firstpart and introduce the second change of variable t = irNxs 2.

Then the function U(a, x) can be written in the form

U(B, x) = (i/2)1121F[/, 7rNx(1 - x) 2] +P [,B, 7rNx(I X)2]j,

(23)

Frank D. Feiock 486

TABLE I. Propagation parameters for comiparison of solutions.

a =6.75 a = (a ) 1.11 X 10-3

L = 101.25 X =-= 7.86 X 10-74irL

X = 1 X 10- 3 wo = 7rN = 450 ir

Nx = - = 450 =r (= a N. =XL 0 2

where

d = a/irNx = X/4IrL,

and the function F(l3, w) is defined by

f w e -it(l-t) dtF(O, w) = I

Jo 1 + 403t (2i7rt)1 /2(25)

In the limit O0, the function F(l3, w) reduces to the Fresnelintegral

lim F(fl, w) = IF(W) = -it dt (26),6-0 fo V27t

and Eq. (23) becomes the standard solution of the Fresnel-Kirchhoff equation for a uniformly irradiated aperture.

RESULTS

The function F(fl, w), defined by Eq. (25), is evaluated inthe Appendix as an asymptotic series in

w = 7rN. (1 + X)2

and

B = 1 - 2#w > 1 - 2(a/L)2

(27)

(28)

for points I x I < 1. We wish to use the asymptotic expansionfor a discussion of the significance of the extra term in thephase factor and the retention of the obliquity factor in thecomputation of fields for large-aperture propagations.

To establish the validity of the asymptotic expansion, valuescomputed from combining Eqs. (23) and (All) are comparedwith direct integration of Eq. (22) for the propagation pa-rameters given in Table I. For this example the third-orderterm 2N. (a/2L)4 is completely negligible. The results for theamplitude and phase of the field are given in Table II for fivevalues of x.

The numerical values were obtained as follows: U(O.F.)(where O.F. means obliquity factor) was the result of directnumerical integration of Eq. (22) including both the extraphase term and the obliquity factor. U(No O.F.) was ob-tained by numerical integration of Eq. (22), keeping the extraterm in the phase but setting the obliquity factor to unity.The U. (O.F.) values were the result of evaluating the as-ymptotic expansion given in Eq. (All), including both thephase term A and the obliquity term y. U, (No O.F.) resultedfrom the same expansion with the obliquity factor set to unity,i.e., with oy = 0. Lastly, the Ua(F.A.) (where F.A. meansFresnel approximation) values were obtained both by usingnumerical values for the Fresnel integral4 in Eq. (23) and bydirect numerical integration of Eq. (22) with a = 0. Equalityof the two values for Ua (F.A.) was used to assure accuracy ofthe numerical integration procedure.

From a comparison of the values given in Table II for theamplitude and phase of the field, the following observations.can be made:

(i) The asymptotic expansion is in excellent agreement withthe direct numerical integration for values of x from 0 to8/9.

(ii) The effect of retaining the obliquity factor is identifi-able, but so small as to be negligible.

(iii) The Fresnel approximation is not in good agreementin amplitude or phase, with the exception of the on-axis am-plitude. The phase on axis has the right magnitude but thewrong sign.

(iv) Retention of the obliquity factor does not compensatefor the Fresnel approximation.

To obtain a qualitative understanding of these results, wecompute the field with and without the extra term in the phasefactor, using the leading terms of the expansion of F(f3, w) asgiven by Eqs. (All) and (A12), respectively. Substituting intoEq. (23) the Fresnel approximation, without the extra term,gives

Ui(x) = 1 + [(1 + i)/2]IH[7rNx(I - x) 2 ]

+ H[7rNx(1 + x)2]},

where

H(w) = i e-iw/(2rw)1 1 2.

(29)

(30)

TABLE II. Amplitude and phase of fields.

x U(O.F.) U(No O.F.) Ux (O.F.) Ux (No. O.F.) U,,(F.A.)

0 0.989 47 0.989 42 0.989 47 0.989 43 0.989 43-0.010 718 -0.010 767 -0.010 702 -0.010 750 0.010 723

0.996 74 0.996 74 0.996 74 0.996 74 0.998 732/9 0.002 037 0.002 069 0.002 096 0.002 081 0.015 756

1.018 1 1.018 2 0.018 0 1.018 1 0.017 0-0.003 124 -0.003 125 -0.003 132 -0.003 131 -0.007 753

0.982 43 0.982 41 0.982 44 0.982 42 0.981 026/9 0.020 127 0.020 187 0.020 122 0.020 184 0.019 419

8/9 0.947 69 0.947 70 0.947 69 0.947 70 0.943 62-0.046 557 -0.046 626 -0.046 534 -0.046 599 -0.045 051

487 J. Opt. Soc. Am., Vol. 68, No. 4, April 1978 Frank D. Feiock 487

With the extra term, but neglecting the obliquity factor -y, wehave

U2(x) = 1 + [(1 + i)/2]IG[7rN (1 - x)2 1 + G[7rN (1 + x)2]},

(31)

where

G(w) = H(w) eiflw2 . (32)

For the point on axis, x = 0, the values of wo and /3w' for thenumerical example are given in Table I. For these values weobtain

U2(0) = 1 + [(1 + j)j2/(2,rw 0 )1/2]

= 0.98944 e-0 .0 10 723i,

and

U1(O) = 1 + [(1 + i)i/(2irwo)'1 2]

= 0.98944 e+0.010 723i

Using only the lowest-order terms gives numerical values ingood agreement with the more exact values given in Table II.The extra term in the expansion of the phase k.R results in achange of sign of the phase at the point on axis, x = 0.

In the asymptotic limit, both results limit to the geometricalsolution U = 1 for x < 1. In the Fresnel approximation theeffect due to diffraction is

DUi(x) = Ul(x) - 1

=[(1 + i)/2IH[brNx(1-x) 2 1 + H[7rN.(1 + x)2]}, (33)

where H(w) is given by Eq. (30). The error resulting from theFresnel approximation is given by

AU= U2(x) - U1(x)

[(1 + i)/21 E[7rNx(1 - x) 2] + E[7rNx(1 + x)2]}, (34)

where

E(W) = [eif/w2- 1]H(W) iflw 2H(w).

the phase factor for any desired set of propagation parameters.The error introduced by the Fresnel approximation is ap-proximately i7r(a/2L)2 Nx times the diffractive effects beingcomputed. If this parameter is not small, one may as well usegeometrical propagation as the usual diffraction theory basedon the Fresnel approximation.

ACKNOWLEDGMENTS

The author is grateful to C. E. Greninger for helpful tech-nical discussions and to V. L. Gamiz for help with the nu-merical evaluations.

APPENDIX

To determine the functional dependence of F(/3, w) weevaluate the integral in Eq. (25) as an asymptotic series in w.5,6

For this purpose, consider the contour integral

e =r e-ih(t) dtC-c 1 + yt (27rt)1/2 ' (Al)

where

h(t) = t - /t 2,

and the contour C = C1 + C 2 + C3 + C4 + C5 is shown in Fig.1. In Eq. (Al) we have written -y = 4/ to distinguish theobliquity factor dependence from the extra term in the phasefactor.

The segments of the contour C are summarized in Table III.In the limit 3 - 0, the integral along C 3 becomes the desiredintegral F(,, w) given by Eq. (25) and the contribution ofsegment C2 vanishes. On the range of integration on contourC5 , 0 < t < w, we have

(1 - 2,Bt)> 1 - 2,w = 1 - a(l L X)2

(A2)

(35)

Thus the error introduced in the computation of the field inthe Fresnel approximation is comparable to the effects ofdiffraction times the dimensionless parameter

0w2 = 7r(a/2L) 2Nx. (36)

If this parameter is not small, one may as well use the geo-metrical result

UW = 11lxi <1

01Ixl>0as the diffraction theory in the Fresnel approximation.

CONCLUSIONS

We have obtained an asymptotic expansion for the de-pendence of a propagated field on the obliquity factor y andthe extra term / in the expansion of the phase factor kR in theKirchhoff integral. The validity of the expansion given in Eq.(All) has been established for a thin rectangular slit bycomparison with results obtained from direct numerical in-tegration of the Kirchhoff integral. The asymptotic expan-sion (All) can be used to evaluate the significance of theobliquity factor -y and the extra term /w 2 in the expansion of

i rCOMPLEX

t

FIG. 1. Contour for evaluation F(/, w).

488 J. Opt. Soc. Am., Vol. 68, No. 4, April 1978

,> 1-(a/L)2 >0.

Frank D. Meock 488

TABLE III. Summary of segments for contour integration.

C1: tc = -ir -ih = -r -i/3r2

dtC2 : t, = 6 eioedt - i4

ei/2 do

C3: t, = t -ih = _i(t - flt2)

C4: t, = W - ir -ih = -r(l - 20 - i[w(l - Ow) + Or2]C5: t, = t -iT -ih = -T(1 - 20t) - i[t(l - 00 + OT21

Thus in the limit T - the contribution along segment C5vanishes.

The integrand of Eq. (Al) has no poles inside the closedcontour and thus I, = 0. It follows that the desired finiteintegral can be written as the sum of two infinite integrals

F(flw) = (-i/22r)l/2f-I1 + I41,

LXe-r-iflr2 dr11 JO 1- iyr v;:

= e-Br-if3r 2 dr4 = e-iW(lW) Jo + -y(w-ir) (r + iw)1/2

where

B = 1- 2fw.

(A3)

(A4)

(A5)

(A6)

At this point no approximations have been made. It remainsto evaluate the two infinite integrals.

To evaluate the first integral, make the change of variabler = s2. Then

e-ii3s4ll = 2 J e-s2 (l ds. (A7)

We compute the integral as a power series in / and y << 1 byexpanding the bracket as a series in s2 and integrating termby term. To second order in / and y the result is

I, = v"7 {1 + (i/4)(2-y - 30) - (3 /3 2)(8'y2 - 20f/y + 3502)1.

(A8)

The second integral can be evaluated as an asymptotic seriesin

B = 1 - 2,w >_ 1-(a/L)2 (A9)

and

(A10)

For points near the axis and propagations with large Fresnelnumber Nx, we have I x I < 1 and w >> 1. In this region, theintegrand in Eq. (A5) can be written as the product of thefactor

e-Br/(iw)l/2 (l + yw)

and a power series in r. The integration can then be doneterm by term. After rearranging terms and combining theresults with Eq. (A8), the function F(fl, w) is given by

F(l, w)

- 1 (I + - (2y - 30) - 2 (8&y2 - 20/y + 35/2))

+ ie-iw(1-) r1 +i 1 + 3yw(2irw)1/2 (l + yw) LB B2 2w(l + -yw)

-+ 3 + 10yw + 15,y2w2

B3(2w) 2(l + yw3 /3(1 + 3&yw) .5 + 21yw + 35,y2w2 + 359y3w3

B4 W( + -yw) (2w)3(l + yw)3

(All)

In the limit 4/ = y - 0, B = 1 and Eq. (All) becomes the as-ymptotic expansion of the Fresnel integral

1F-( i ie=w 23 3 7 -[ -IF(W)= _+ Il- +

2 (2irw) 1/2 I (2w) 2 (2w) 4

(A12)

*Present address: MS L-476, Lawrence Livermore Laboratory, P.O.Box 808, Livermore, Calif. 94550.

'N. G. VanKampen, "An Asymptotic Treatment of DiffractionProblems," Physica XIV, 575-589 (1949).

2Paul Horwitz, "Asymptotic theory of unstable resonator modes," J.Opt. Soc. Am. 63, 1528-1543 (1973).

3Joseph W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968), p. 59.

4J. Boersma, "Computation of Fresnel Integrals," Math of Compu-tation 14, 380 (1960).

5A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956).6F. D. Feiock, "Finite-size effects in Rayleigh scattering," Phys. Rev.

169, 165-171 (1968).

489 J. Opt. Soc. Am., Vol. 68, No. 4, April 1978

W = '7rNX (I ± X)2.

Frank D. Feiock 489

+ i I - 3 - 5+ - . .) 1.(2w (2w)3