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1196 J. Opt. Soc. Am. A/Vol. 6, No. 8/August 1989

Huygens-Fresnel-Kirchhoff wave-front diffractionformulation: spherical waves

Hal G. Kraus

Idaho National Engineering Laboratory, EG&G Idaho, Incorporated, P.O. Box 1625, Idaho Falls, Idaho 83415

Received October 20, 1988; accepted March 6, 1989The Huygens-Fresnel diffraction integral has been formulated for incident spherical waves with use of theKirchhoff obliquity factor and the wave front as the surface of integration instead of the aperture plane. Accuratenumerical integration calculations were used to investigate very-near-field (a few aperture diameters or less)diffraction for the well-established case of a circular aperture. It is shown that the classical aperture-planeformulation degenerates when the wave front, as truncated at the aperture, has any degree of curvature to it,whereas the wave-front formulation produces accurate results from up to one aperture diameter behind theaperture plane. It is also shown that the Huygens-Fresnel-Kirchhoff incident-plane-wave-aperture-plane-inte-gration and incident-spherical-wave-wave-front-integration formulations produce equally accurate results forapertures with exit f-numbers as small as 1.

INTRODUCTION

In theory, the classical Huygens-Fresnel principle must usethe wave front as a reference for calculations of propagationand diffraction interference." 2 However, in studies of dif-fracting apertures, practical application of this principle hasnearly always resulted in the use of the aperture plane as theplane of integration. Examples of this dominate the relatedjournal and textbook literature. 3 -" Strictly speaking, thisassumption is correct only if the incident wave is planar and -is propagating parallel to the optic axis. For a nonplanarincident wave propagating along the optic axis, assumingthat the wave front, at the aperture, lies nearly in the aper-ture plane is a good approximation for calculating diffrac-tion fields in most of the region beyond the aperture. In theKirchhoff and the Rayleigh-Sommerfeld theories of diffrac-tion, line or surface integration in the aperture plane isused.' 2 The aperture plane of integration is also used inFourier-transform and linear systems-convolution diffrac-tion theory.13"14 It is believed that this simplification is usedin part because of the complexity of the integrals involved inan integration across the wave front. However, the primaryreason for the use of this simplifying assumption is probablythe lack of the realization that it is being used. In contrast,examples of well-known simplifying assumptions includeapproximating the obliquity factor as equal to 1 and retain-ing the first two terms of the binomial expansion of thesquare-root term that defines the propagation distance fromthe integration plane to the observation plane (the well-known Fresnel approximation). The purpose of this re-search was twofold: (1) to investigate the limit of accurateprediction for near-field Fresnel diffraction by using Huy-gens-Fresnel diffraction theory in the strictest form (with noassumptions other than those concomitant with scalar wavetheory), including integrating across a circular-aperturetruncated spherical wave front, and (2) to investigate theimprovements realized in diffraction patterns calculated byusing this approach as compared with the aperture-plane-integration formulation. It is in this near-field region that

the wave-front curvature, in conjunction with the obliquityfactor, has the potential to improve the predictive capabilityof the theory.

There appears to be little published literature in this spe-cific area. In related literature, the classical Huygens-Fres-nel principle has been applied strictly in studies of Fresnelzones and Fresnel-zone plates or lenses in which the curva-ture of the spherical wave front is essential to the analysis. 2"15The error introduced by the Fresnel approximation has beeninvestigated for circular apertures131 6 as well as for rectan-gular apertures.' 7 Southwell' 6 claimed that the Fresnel ap-proximation begins to break down for spherical wave propa-gation for beams faster than f/12. Circular apertures werechosen for the investigation described in the present paperbecause the diffraction patterns for low Fresnel numbers arewell established by agreement between theoretical and ex-perimental results.9-11 This choice permitted the genera-tion of a set of reference solutions, which are described belowin the Investigation and Discussion section, that could beverified by the results of this previous research. These ref-erence solutions were used to detect degeneration in thediffraction patterns that is due to differing approximationsin the formulation, concentrating on those approximationsrelated to wave-front curvature. In the case of the Fresneldiffraction region, it is well known that the scalar Huygens-Fresnel theory produces inaccurate results as the aperture isapproached and/or as the aperture size approaches thewavelength of the light. However, these regions are seldomwell defined because of the analytical intractability of thefunctions that describe diffraction field solutions. Often,quantitative information about the applicability of the the-ory can be derived only through numerical calculations, as isdone in this paper.

FORMULATIONFigure 1 shows the geometric configuration for the usualHuygens-Fresnel-Kirchhoff diffraction integral formula-

0740-3232/89/081196-10$02.00 © 1989 Optical Society of America

Hal G. Kraus

Vol. 6, No. 8/August 1989/J. Opt. Soc. Am. A 1197

Ep(r', 0) = C El: f2t (1 + COS O')exp(ikR')sin dadq

(3)

Circular aperture of radius aiin x-y plane

Fig. 1. Geometry for Huygens-Fresnel-Kirchhoff aperture-planediffraction formulation for spherical waves.

tion for a spherical wave diffracted by a circular aperture,using aperture-plane integration. The formulation is wellestablished2 and is repeated here for comparison with thewave-front-integration formulation. The appropriate inte-gral for calculation of the wave amplitude at a point in theobservation plane is

27r Ja (1 + Cos O')exp[ik(R + R')]Ep(r', Of=C R rdrd/i,

(1)

in which

C = -(EO/2X)i,

k = 27r/X,R = (r2 + Z2)1/2,

R = (r,2 + r2 - 2rr' cos(k - 'p') + Z2]1/2,

cos O' = (-a2 + 2 + a2)/2bc,a = [rp 2 + rQ2 - 2 rprQf cos(¢,Q'- _ I- i 1/2

b = [rQ' 2+ r2 - 2 rrQ' cos(, - kQ') + z,2 1/2

= RI,

rQ' = r(z + z')/z.

E0 is the emission amplitude at point S, and X is the wave-length of light; the other parameters are as depicted in Fig. 1.In Eq. (1), as with all formulations described in this paper,the obliquity factor, -(i/2X)(1 + cos 0'), as derived from theFresnel-Kirchhoff diffraction formulation for sphericalwaves15 has been used. The associated equation for deter-mining the Fresnel number, Fr, can be derived easily byusing the geometry of Fig. 1 to define the optical path differ-ence between S to 0 to 0' and S to the aperture edge to 0', interms of multiples of X/2, to give

in which

C = -[Eop exp(ikp)/2X]i,

01 = sin'1(a/p),

R' = [p2 sin2 0 + rp 2 - 2pr' sin 0 cos(O - p)

+ (Z' + h)2 ]l/2 ,

h = p(l - cos 0),

Cos 0' = (-a 2 + b2 + c2)/2bc,

a = [r_ 2 + rQ/2 - 2rplrQl cos(op/ - (Q/)]1/2

b = [p2 sin2 0 + rQ2 -2rQ'p sin 0 cos(k - IQ')

+ (' + h)2 ]1/2 ,

c = R,

0Q = 0rQ = (p + z')tan 0,

and the other parameters are as defined in Fig. 2 or asdefined earlier in this section. Equation (3) was derived in amanner analogous to the derivation of Eq. (1) except thatthe wave front at the aperture and truncated by the apertureis the surface of integration. Note the simplification in theexponential term (see the definition of C) that results fromthis formulation, owing to the fact that the phase at the wavefront, which is also the surface of integration, is constant. Inthis case, the Fresnel-number equation was derived by usingthe Fresnel-zone-area approach. The surface area, Al, of theIth Fresnel zone on the surface of a spherical wave front canbe shown to be2

(4)

The total surface area, Al, of the portion of the sphericalwave truncated at the aperture is (see any standard mathe-matics book)

A = 27rph, (5)

in which

h=p [1- -a2) = p(1 - cos 0). (6)

Fr = (2/X)[(z 2 + a2)1/2 + (Z'2 + a2 )1/2 - (z + Z)].

Thus, as in the usual definition, Fr is a measure of themultiples of /2 or half-period zones resulting from thispath-length difference.

Illustrated in Fig. 2 is the geometric configuration used forthe Huygens-Fresnel-Kirchhoff spherical-wave-front for-mulation for a spherical wave diffracted by a circular aper-ture, resulting in the following equation for the wave ampli-tude at a point in the observation plane:

Spherical wave front truncated 81249by circular aperture of radius a

Fig. 2. Geometry for Huygens-Fresnel-Kirchhoff wave-front dif-fraction formulation for spherical waves.

(2)

Hal G. Kraus

Al = 7rpX Z' + X(21 - 1) -(P + ZI) 1 4


Using Eq. (6) in Eq. (5) and equating it to the sum of the Al'sfor m half-period zones gives the results

mA = > Al,

1=1

2[ ( a2)1/2] -rpx [

m

- (21-4 =

1=1

(7)

1)1.

(8)

By simplifying, rearranging, and noting that the summationterm is equal to m2

, Eq. (8) can be reduced to

4(p + z') ( + ') m =P\ p

2 J

m=Fr. (9)

If the first term of Eq. (9) is neglected because Xm/4 << z' and

the square-root term is approximated by using the first twoterms of a binomial expansion by assuming that a2

/p2

<< 1,this expression reduces to

(p + z/)a2 a-2 1 1 Apz' X P Z'J

(10)

Equation (10) describes the relationship that has been usedso widely for calculating Fresnel numbers for sphericalwaves impinging upon a circular aperture.2 It is generallyaccepted that this expression is valid until the Fresnel num-ber becomes large.2'2 0 However, results presented belowreveal that this expression can produce a large error in thenear field even for Fresnel numbers <10. Equation (10) canbe reduced to the case of an incident plane wave by letting papproach -, producing the well-known expression

Fr = a2 (11)

Table 1 summarizes the Fresnel-number equations, which

Table 1. Characteristic Circular-Aperture Fresnel-Number Equations of the Form Am2 + Bm + C for Plane-Wave and Spherical-Wave Huygens-Fresnel-Kirchhoff Diffraction Formulationsa

Formulation A B C

Exact aperture plane, plane waveb 1 A-- (a2)

4z' 4 z j+O Aperture plane, spherical wavec 1 4

Exact aperture plane, spherical wave [Eq. (2)] 0 A -4 12z'[(z 2 + a2)12 + (Z,2 + a2)12- (z + z')])

Exact wave front, spherical wave [Eq. (9)] 1 4z'-4 2p(p + z') 1 -(1 _ a2)1]}

0 m is the Fresnel number Fr.b This more general form may be derived easily by using optical-path-length differences, 2 and it has no restrictions. Equation (11) results from this characteristic

equation when m2 is small so that the first term can be neglected.I Derived using Eqs. (4) and (17), assuming that A, = 7ra

2.

Table 2. Huygens-Fresnel-Kirchhoff Aperture-Plane Diffraction Formulation: Plane-Wave Casesaz' As in Fig. 1 Determined from Eq. (10) z' As in Fig. 1 Determined from Eq. (2)

Aperture Normalized NormalizedFresnel Radius Intensity Exit Intensity Exit

Case Number Fr a (mm) z' (mm) I,, (r = 0) f'-Number z' (mm) In (r' = 0) f'-Number

1 2 0.00625 0.03086481 0.002060 2.469 0.03054854 0.000053 2.4442 2 0.01250 0.12345922 0.000135 4.938 0.12314336 0.000003 4.9263 2 0.02500 0.49383691 0.000008 9.877 0.49352262 0.000000 9.8704 2 0.05000 1.97534750 0.000000 19.753 1.97503969 0.000000 19.7505 4 0.00625 0.01543240 0.111700 1.235 0.01479963 0.000806 1.1846 4 0.01250 0.06172961 0.008052 2.469 0.06109690 0.000053 2.4447 4 0.02500 0.24691845 0.000525 4.938 0.24628600 0.000003 4.9268 4 0.05000 0.98767380 0.000033 9.877 0.98704240 0.000000 9.8709 6 0.00625 0.01028827 1.000000 0.823 0.00933907 0.003799 0.747

10 6 0.01250 0.04115308 0.086556 1.645 0.04020389 0.000262 1.60811 6 0.02500 0.16461230 0.005841 3.292 0.16366318 0.000017 3.27312 6 0.05000 0.65844922 0.000374 6.584 0.65750030 0.000001 6.57513 8 0.00625 0.00771620 1.000000 0.617 0.00045061 0.010927 0.51614 8 0.01250 0.03086481 0.483360 1.235 0.02959925 0.000807 1.18415 8 0.02500 0.12345922 0.032214 2.469 0.12219380 0.000053 2.44416 8 0.05000 0.49383688 0.002100 4.938 0.49257200 0.000003 4.92617 10 0.00625 0.00617296 1.000000 0.494 0.00459097 0.023927 0.36718 10 0.01250 0.02469185 1.000000 0.988 0.02310987 0.001909 0.92419 10 0.02500 0.09876740 0.121140 1.975 0.09718546 0.000128 1.94420 10 0.05000 0.39506953 0.007938 3.951 0.39348785 0.000008 3.935

a Other parameters: z = 1000.0 m, = 0.6328 X 10-6, r = 1.6a, Nb Nt = 16.

Hal G. Kraus


0.1 are rearranged in the form of characteristic equations, forthe various cases that are considered in this paper: (1) aplane incident upon a circular aperture, (2) a spherical waveincident upon a circular aperture (an approximate aperture-plane formulation is used, and it is assumed that the spheri-cal wave front is nearly coincident with the aperture plane),

0.01- (3) a spherical wave incident upon a circular aperture (theexact aperture-plane formulation is used), and (4) a spheri-cal wave incident upon a circular aperture (the exact wave-

Fr=2 front formulation is used).

Fr=0 . \ \ \^\\\ Fr=6 INVESTIGATION AND DISCUSSION

0.001 - \ \ \Fr=8 This research was conducted by making a few different setsFr 10 of calculations in the near Fresnel diffraction region for

investigating the effects of wave-front-related approxima-tions in the Huygens-Fresnel-Kirchhoff diffraction formu-

s \ \ \ \ \ lation. The helium-neon laser wavelength of 0.6328 Am was.N \ \ \ \ \used for all test calculations reported below. This wave-

0.0001 \ length was chosen because nearly all the experimental dataz -\ \ available for verification of circular-aperture Fresnel dif-

- h \ fraction patterns were obtained with a helium-neon laser_ Above this value, there is with this emission wavelength. 9"1 '2 0 These experimental

significant visual error in_ the diffraction profiles; \ data and supporting theoretical calculations9"10'11 2 0 were

below this level profiles are used to support the claims of relative performance of the0.00001 visually and numerically theoretical formulations and approximations examined be-

accurate low. This paper is concerned not with efficient computa-tional procedures but rather with proper theoretical formu-lation for diffraction field calculations. As such, Eqs. (1)and (3) were integrated numerically by using Gaussianquadrature with Nt terms in each of Nb radial regions in r

0.000001 I I I I (Fig. 1) or in angular regions in 0 (Fig. 2) by Nb angular0 2 4 6 8 10 regions in 0 (Figs. 1 and 2). Convergence was ensured by

f'JNo. successively increasing Nt and Nb until the observation

Fig. 3. Normalized intensity on optic axis in observation plane, plane intensities no longer changed. In all cases, a 16-termI,(r' = 0), versus f-number (exiting f-number) for even Fresnel quadrature with 16 X 16, or 256, subregions of integrationnumbers 2 through 10. was sufficient to achieve this convergence. The observa-

Table 3. Huygens-Fresnel-Kirchhoff Aperture-Plane Diffraction Formulation: Spherical-Wave Casesa

z' as in Fig. 1, Wave-Front NormalizedFresnel Aperture Determined from Protrusion Angle Entrance Exit Intensity

Case Number Fr Radius a (mm) Eq. (2) (mm) ha/a (mrad) f-Number f-Number I,, (r' = 0)

1 2 0.00625 0.03064494 0.31248006 800 2.452 0.000242 2 0.01250 0.12469001 0.62496048 400 4.988 0.003053 2 0.02500 0.51919051 1.25000390 200 10.384 0.046674 2 0.05000 2.46134142 2.50003130 100 24.613 0.683265 4 0.00625 0.01482443 0.31248006 800 1.186 0.000986 4 0.01250 0.06148414 0.62496048 400 2.459 0.003087 4 0.02500 0.25255250 1.25000390 200 5.051 0.047918 4 0.05000 1.09534339 2.50003130 100 10.953 0.809049 6 0.00625 0.00935064 0.31248006 800 0.748 0.00395

10 6 0.01250 0.04037784 0.62496048 400 1.615 0.0032711 6 0.02500 0.16643380 1.25000390 200 3.329 0.0482712 6 0.05000 0.70397371 2.50003130 100 7.040 0.8421113 8 0.00625 0.00645754 0.31248006 800 0.517 0.0110514 8 0.01250 0.02969867 0.62496048 400 1.188 0.0037815 8 0.02500 0.12375252 1.25000390 200 2.475 0.0486216 8 0.05000 0.51828805 2.50003130 100 5.183 0.8552117 10 0.00625 0.00459575 0.31248006 800 0.367 0.0239918 10 0.01250 0.02317488 0.62496048 400 0.927 0.0048519 10 0.02500 0.09818627 1.25000390 200 1.964 0.0487220 10 0.05000 0.40980004 2.50003130 100 4.098 0.87680

a Other parameters: z 0.01 m, A = 0.6328 X 10-6 m, r = 1.6a, Nb = Nt 16. From Fig. 1, z = z-ha, 01 = tan-1(a/z.), ha = p(l - cos 01), p (Z2 + a2)1/

2.

Hal G. Kraus


1.0 -

0.8

._

c 0.6

~0

'a 0-4 -C'0

z0.2

0.010.00

(0.00)

1.0

C.2

a)

.N

0z

19.94 39.87 59.81 79.75(16.00) (32.00) (48.00) (64.00)Observation plane radius, r' (m)

(a)

99.68(80.00)

the small laboratory laser and the rapid maturation of pho-todiode and charge-coupled-device technologies, detailedquantitative measurements are now possible.9 ""1,20"2 How-ever, the common use of lasers has also meant that a spatial-ly filtered collimated laser beam is normally used to producenearly planar incident waves, with a focusing lens oftenbeing used before or after the aperture to compress theFresnel diffraction region into a workable space in the lab-oratory.9 '., 20 For this reason, Fresnel diffraction fields pro-duced by a plane wave incident upon a circular aperture, fora range of Fr values, were used here as standard referencecases8 "10 "'1 that the incident-spherical-wave calculationscould be compared with for accuracy.

A set of 20 cases was used to establish the initial or basesolutions for a plane wave incident upon a circular aperture,using Eq. (1). These cases are summarized in Table 2. Theplane wave was simulated by setting z = 1000.0 m. Diffrac-

C

._

.E0Cco

0

z

0.00 16.00 32.00 48.00 64.00 80.00Observation plane radius, r' (jum)

(b)Fig. 4. Normalized intensity, I,,, versus observation-plane radius,r', for Fr = 2: (a) Table 4, case 4 (Table 2, case 4); (b) Table 3, case 4.

tion-plane light intensity was calculated and normalized byusing the relationship

In(r', 0') = (2)EpEp*/Imax, (12)

where Ep* is the complex conjugate of Ep and Imax is themaximum intensity, (1/2)EpEpmax*, calculated for any pointin the observation field. Since the field is axisymmetric, 201points from r' = 0.0 to r = rf = 1.6a and for bp' = 0 werecalculated as a representative cross section of the field inten-sity. All calculations were done with a dual-processor CrayXMP/24 computer. Solution times were 4-6 min per case.

Before the emergence of photodetector and charge-cou-pled-device technology, diffraction fields were studied qual-itatively by using photographic methods. 8"9 The photo-graphs by Harris' 8 were made by using arc lamps as a lightsource, which would generate spherical waves. No quantita-tive diffraction field measurements that used sphericalwaves have been found. As a result of the proliferation of

a,

c

0C'a

0z

0.00 17.74 35.49 53.23 70.98(0.00) (16.00) (32.00) (48.00) (64.00)

Observation plane radius, r' (am)(a)

1.0

0.8

0.6

0.4

0.2

0.0I0.00 16.00 32.00 48.00 64.00

Observation plane radius, r' (m)(b)

88.72(80.00)

80.00

Fig. 5. Normalized intensity, I,,, versus observation-plane radius,r', for Fr = 4: (a) Table 4, case 8 (Table 2, case 8); (b) Table 3, case 8.

Hal G. Kraus


1.0

0.8

3 0.6

N*< 0.4_

z.0.2

0.00.00(0.00)

1.0

U)C

'a

G)N

0z

17.11 34.21 51.32 68.43(16.00) (32.00) (48.00) (64.00)Observation plane radius, r' (m)

(a)

85.53(80.00)

in the observation-plane field intensity plots, and calcula-tions show an accuracy of 3 to 5 decimal places. It can beseen that increasing Fr or decreasing the f-number, i.e., theaperture radius, eventually results in the failure of this Huy-gens-Fresnel-Kirchhoff diffraction theory to describe thediffraction field accurately. The second method was identi-cal to the first except that z' was calculated by using Eq. (2).Results analogous to those of the first method are reportedin the last three columns of Table 2. Notice the orders ofmagnitude of the improvement in the I(r' = 0) values,indicating that using Eq. (10) can result in considerableerror in this region, even when Fr is small. However, theprimary purpose of these data is to establish the nominalcases for well-known diffraction field solutions. Figures4(a), 5(a), 6(a), 7(a), and 8(a) show plots of the observationplane I,, versus r', with the r' coordinate values given inparentheses, for cases 4, 8, 12, 16, and 20 of Table 2. These

:A

Ea

0z

0.00 16.00 32.00 48.00 64.00 80.00Observation plane radius, r' (um)

(b)Fig. 6. Normalized intensity, I, versus observation-plane radius,r', for Fr = 6: (a) Table 4, case 12 (Table 2, case 12); (b) Table 3, case12.

tion fields for even Fresnel numbers from 2 to 10 were calcu-lated by two different methods. In the first method fixedvalues of Fr, a, z, and X were used, and z' was calculated fromEq. (10). Note that z' is a few to a few thousand times Xbehind the aperture and that the aperture diameter is withinthe range 20-160X so that these fields are in the very nearFresnel diffraction region and close to the limit of applicabil-ity for scalar diffraction theory. The fifth column of Table 2lists I at the center of the diffraction field. This parameteris one of the primary indicators that the diffraction theory isdegenerating, as this number should always be zero. EvenFresnel numbers were chosen for all the test cases so thatthis indicator could be used. The sixth column of Table 2 isthe exiting f-number defined as the ratio of z' to the aperturediameter. Figure 3 shows a plot of I(r' = 0) versus f-number summarizing these data. As indicated in this fig-ure, when I(r' = 0) < 0.001 no degeneration can be detected

0.00 16.79 33.58 50.38 67.17(0.00) (16.00) (32.00) (48.00) (64.00)

Observation plane radius, r' (m)(a)

1.0

0.8

C

a

a)N

0z

0.6

0.4

0.2

0.0L-0.00 16.00 32.00 48.00 64.00

Observation plane radius, r' (m)

83.96(80.00)

80.00

(b) 8-1255

Fig. 7. Normalized intensity, I, versus observation-plane radius,7r, for Fr = 8: (a) Table 4, case 16 (Table 2, case 16); (b) Table 3, case16.

Hal G. Kraus


1.0

0.8

-

c 0.6._

a)N,, 0.4

0Z0.2

0.000.00

(0.00)

_.F

C:._C~0N

0

16.60 33.19 49.79 66.39(16.00) (32.00) (48.00) (64.00)Observation plane radius, r' (nm)

(a)

.0! l I I0.00 16.00 32.00 48.00 64.00

Observation plane radius, r' (am)(b)

82.98(80.00)

80.00

Fig. 8. Normalized intensity, I,,, versus observation plane radius,r', for Fr = 10: (a) Table 4, case 20 (Table 2, case 20); (b) Table 3,case 20.

profiles reproduce accurately those profiles reported in theliterature. 9 -1120

The 20 cases described in Table 2 were recalculated for thecase of an incident spherical wave, with z = 0.01 m, asreported in Table 3. Equation (2) was used to calculate z'.The entrance f-number, defined as the ratio of z to theaperture diameter, and the f-number, are reported. Thereappears to be no correlation of the performance of the aper-ture-plane formulation with the f-number. However, theresults show severe degeneration in I,(r' = 0) for decreasingf-number. The parameter ha is the distance from the aper-ture plane to the intersection of the spherical wave front andthe optic axis at the point of wave truncation by the aperture[i.e., ha = p(l - cos 01)]. The quotient ha/a is equal to thetangent of the angle from the aperture edge to this intersec-tion point or nearly equal to the angle itself for these smallangles. This is a measure of wave-front protrusion through

the aperture plane. Therefore this quotient will be referredto as the wave-front protrusion angle. The aperture-planeformulation degenerates rapidly as this wave-front protru-sion increases and as the Fresnel number increases. Notethat increasing the Fresnel number also decreases z' for afixed aperture radius. This in effect increases the magni-tude and the variation of 0', the obliquity angle. Figures4(b), 5(b), 6(b), 7(b), and 8(b) are analogous to Figs. 4(a)5(a), 6(a), 7(a), and 8(a) but are for cases 4, 8, 12, 16, and 20 ofTable 3. These figures illustrate the severe degeneration inthe intensity profiles for the largest apertures (these havethe largest ha/a values).

If the degeneration in the diffraction intensity profiles isindeed due to the wave-front protrusion through the aper-ture, then the wave-front formulation should correct thiserror. In fact, the results obtained by using the wave-frontformulation should then closely reproduce those of column 8of Table 2 for the incident-plane-wave case in terms of In(r'= 0) values. Table 4, in which the results for these 20 casesas obtained by using the wave-front formulation are report-ed, shows that this is true. The diffraction field intensityprofiles of Figs. 4(a), 5(a), 6(a), 7(a), and 8(a) were repro-duced accurately as long as In(r' = 0) < 0.001, although thescale and the location of the diffraction pattern change onthe basis of the value of the parameter Sf as is defined andexplained below. In this case, the r' values of Figs. 4(a), 5(a),6(a), 7(a) and 8(a) are those without parentheses and shouldbe compared with the r' values of Figs. 4(b), 5(b), 6(b), 7(b),and 8(b) of the aperture-plane formulation. These scaledifferences are explained below.

The parameter z' is defined as the distance from the opticaxis point of aperture-plane or wave-front integration to theaxis point in the observation plane (Figs. 1 and 2). In thecase of the wave-front formulation, the circular aperture is adistance ha farther away from the observation plane than forthe aperture-plane formulation. As the wave-front protru-sion through the aperture becomes significant for the wave-front formulation, the scale of the diffraction pattern in theobservation plane changes by the ratio of z' + ha for thewave-front formulation to z' calculated for an incident planewave. This scale factor is defined as

Sf = (z,' + ha)/ZPX (13)

in which z,' was calculated from Eq. (9) and zp' was calculat-ed from Eq. (11). Values of Sf are listed in Table 4 also.This parameter indicates simply that, when the distancefrom the aperture to the observation plane changes owing towave-front protrusion, the scale of the diffraction patternchanges. This parameter also provides an indication of theaccuracy limit of the theory, for as S becomes significantlyless than 1.0, the diffraction profiles become inaccurate.

Close examination of the wave-front formulation revealsthat it corrects for two errors that are present in the aper-ture-plane formulation. The first is the precise optical pathlength from the source point to an observation-plane pointthat is due to the Huygens-principle secondary sources' be-ing located on the wave front instead of in the apertureplane. Note, however, that the optical path length from thesource to the aperture edge to the center of the observationplane should be identical if the problem is defined equiva-lently for the two different formulations. In actuality, thesmall differences in z' values for the respective cases de-

Hal G. Kraus


Table 4. Huygens-Fresnel-Kirchhoff Wave-Front Diffraction Formulation: Spherical-Wave Casesa

z' As in Fig. 1, Wave-Front NormalizedFresnel Aperture Determined from Protrusion Angle Exit Scale Intensity

Case Number Fr Radius a (mm) Eq. (9) (mm) ha/a (mrad) f'-Number Factor Sf I, (r' = 0)

1 2 0.00625 0.03064299 0.31250004 2.451 0.99288 0.0000532 2 0.01250 0.12468219 0.62500025 4.987 1.00997 0.0000033 2 0.02500 0.51915933 1.25000200 10.383 1.05134 0.0000004 2 0.05000 2.46122394 2.50002000 24.612 1.24603 0.0000005 4 0.00625 0.01482248 0.31250004 1.186 0.96060 0.0008076 4 0.01250 0.06147633 0.62500025 2.459 0.99602 0.0000537 4 0.02500 0.25252127 1.25000200 5.050 1.02282 0.0000038 4 0.05000 1.09521988 2.50002000 10.952 1.10901 0.0000009 6 0.00625 0.00934869 0.31250004 0.748 0.90886 0.003806

10 6 0.01250 0.04037003 0.62500025 1.615 0.98116 0.00026311 6 0.02500 0.16640256 1.25000200 3.328 1.01107 0.00001712 6 0.05000 0.70384933 2.50002000 7.038 1.06914 0.00000113 8 0.00625 0.00645558 0.31250004 0.516 0.83688 0.01094714 8 0.01250 0.02969086 0.62500025 1.188 0.96222 0.00081115 8 0.02500 0.12372128 1.25000200 2.474 1.00238 0.00005316 8 0.05000 0.51816338 2.50002000 5.182 1.04951 0.00000317 10 0.00625 0.00459379 0.31250004 0.368 0.74449 0.02379218 10 0.01250 0.02316706 0.62500024 0.927 0.93856 0.00191919 10 0.02500 0.09815499 1.25000200 1.963 0.99412 0.00012820 10 0.05000 0.40967512 2.50002000 4.097 1.03729 0.000008

a Other parameters: p = 0.01 m, X = 0.6328 X 10-6 m, rf = 1.6aSf, Nb = Nt = 16. From Fig. 2, z = p - h = p cos 01, 01 = sin-1 (a/p), h = p(l - cos 01).

scribed in Tables 3 and 4 result from using z = 0.01 for theaperture-plane formulation versus p = 0.01 for the wave-front formulation. To be exactly equivalent, R at r = a ofthe aperture-plane formulation should have been set equalto p of the wave-front formulation. However, these differ-ences are very small, 0.00125%, referenced to either z or p,thus making the cases described in Tables 3 and 4 nearlyidentical. The second error is due to the change and varia-tion of the obliquity factor relative to either the apertureplane or the wave front; the latter approach has been shownto be the correct one. This also brings up the question ofwhether a more sophisticated form of the obliquity factorderived from a more detailed theory would extend this Huy-

35

30-

25

:Ec 20 Fr=10

1 , X Fr=8L2 15 ~~Fr=6

10 ~~~Fr=4

CD6)

6)

6.25 12.50 18.75 25.00 31.25 37.50 43.75 50.00Aperture radius, a (jim)

Fig. 9. Percent error in Fresnel number calculated by using z'values from Table 4 in Eq. (10) versus aperture radius for evenFresnel numbers from 2 to 10 with p = 0.01 m, X = 0.6328 m.

gens-Fresnel-Kirchhoff wave-front formulation even fur-ther. This possibility was not investigated in this research.

Finally, it is interesting to calculate the error introducedinto the determination of the true value of the Fresnel num-ber by using the correct values of z' from Table 4 in therelationship normally used for calculation of Fr [Eq. (10)].The results of such calculations are plotted in Fig. 9, whichshows that this error increases rapidly in the near-field dif-fraction region as the circular-aperture radius decreases fora fixed Fresnel number or as the true Fresnel number in-creases for a fixed aperture radius.

SUMMARY AND CONCLUSIONS

A comparison of two different Huygens-Fresnel-Kirchhoffdiffraction formulations for spherical waves incident upon acircular aperture was made, using the aperture plane as theintegration surface for one and using the wave front truncat-ed at the aperture as the integration surface for the other.The region of primary concern was the very near-field regionof diffraction beyond a circular aperture. It was shown thatthe wave-front formulation produces results for sphericalwaves incident upon a circular aperture that are comparablewith the results obtained for the aperture-plane formulationwhen the incident wave is planar. The aperture-plane for-mulation degenerates prematurely when spherical waves in-cident upon the aperture are considered with decreasingspherical-wave radius at the aperture, increasing apertureradius, and increasing Fresnel number. This degenerationwas found to be caused by two errors that become significantwhen there is significant protrusion of the aperture-truncat-ed spherical wave front beyond the aperture plane:

1. The first error is due to the optical path length's notfollowing the proper trajectory to the wave front and then tothe observation plane as required by Huygens's principle.

Hal G. Kraus


This error is most significant between the optic axis and theedge of the aperture. At the optic axis and at the edge of theaperture this error decreases to zero. For this treatment it isassumed that the problems are defined equivalently for thetwo formulations, with R at r = a of Fig. 2 equal to p of Fig. 2.

2. The second error is due to the incorrect obliquityangle's being calculated relative to the surface of integration.Huygens's principle defines this surface of integration as thewave front. As long as the incident wave at the apertureshows little curvature or protrusion through the aperture,aperture-plane-integration results show no premature de-generation. However, when this curvature or protrusionbecomes significant, the proper surface of integration is thewave front at the aperture as truncated by the aperture.

It was found that wave-front integration should be usedwhen the wave-front protrusion angle, ha/a (see Table 4), islarger than 0.001 rad. Furthermore, it was found that thiswave-front formulation degenerates for incident sphericalwaves at the same point that the aperture-plane formulationdegenerates for incident planar waves, i.e., when the exitingf-number is <1.0 (see Tables 2 and 4).

It was determined that for even Fresnel numbers the fail-ure of the theoretical predictions to produce the properdiffraction patterns could in all cases be detected by check-ing the magnitude of the diffraction field intensity in thecenter of the observation plane, In(r' = 0), which should bezero. Values of this parameter of <0.001 indicated that theaccuracy of the calculated diffraction patterns was at least3-5 decimal places. This result suggests that, in general, theaccuracy of calculated diffraction patterns could be checkedby calculating I,(r' = 0) for the nearest even Fresnel number.

The specific relationships used for determining the Fres-nel number were also shown to be important in this near-field region. The usual relationship used for sphericalwaves as defined by Eq. (10) was shown to produce signifi-cant error in this region of diffraction, even for low Fresnelnumbers (<10). The exact relationships for determiningthe Fresnel number for the aperture-plane and wave-frontformulations were derived and are summarized in Table 1.

Research in which the Huygens-Fresnel-Kirchhoff wave-front diffraction integral has been formulated for the classi-cal (paraxial) 22 and exact23 scalar wave solutions for a Gauss-ian laser beam incident upon a circular aperture has alsobeen completed.2 4 This research addresses laser-beamwave-front-curvature effects for various distances from thelaser-beam waist and for various levels of beam truncationby a circular aperture by means of an analogous comparisonof aperture-plane and wave-front formulations. The condi-tions under which the wave-front formulation producesmore accurate results and thus should be used instead of theaperture-plane formulation have also been defined. Aswould be expected, these conditions differ somewhat fromthose for spherical waves. Current research by this authorincludes the addition of a focusing lens to these Huygens-Fresnel-Kirchhoff spherical-wave-front and Gaussian-la-ser-beam-wave-front formulations, which will be describedin papers submitted for publication in the future.

The above conclusions have more-extensive implicationsthan have been discussed to this point. Since the properintegration surface is the aperture-truncated wave front,Fourier-transform and linear-systems-convolution tech-

niques suffer from the same deficiences as long as the aper-ture plane is used as the plane of integration. This is alsotrue for the Kirchhoff and Rayleigh-Sommerfeld formula-tions, since integration is applied across the aperture planeas part of the surface integral.12 These formulations couldbe modified so as to use the circular-aperture-truncatedwave front as part of the surface of integration. Spherical-wave-front truncation before but close to a circular aperturewas formulated conceptually by means of the Fresnel-Kirchhoff diffraction formulation,' 5 but subsequent simpli-fications resulted in the use of the aperture plane as part ofthe surface of integration. This problem is exacerbatedfurther by the fact that wave-front curvature is eliminatedor altered in experimental measurements by means of spa-tial filtering and beam collimation or focusing lenses, as ismentioned above. These procedures eliminate or mask truewave-front-curvature effects. Furthermore, when noncir-cular apertures are considered or when the incident direc-tion of the wave is not along the optic axis (perpendicular toaperture plane), the impinging spherical or nonplanar waveis not truncated uniformly by the aperture boundary, leav-ing an undefinable surface of integration for all these meth-ods. Therefore one of the possible methods to follow thewave accurately in the region near the aperture is to solve thescalar wave equation and track the wave as it impinges,possibly nonuniformly, upon and propagates through theaperture. Numerical solutions to the scalar-wave equationhave in fact been used for many years for diffraction calcual-tions, but not in this specific context.2 023 This approachshould be superior to any of the methods based on the Huy-gens-Fresnel principle when applied to nonplanar wavefronts and arbitrarily shaped apertures. The only limita-tion of such a generalized scalar-wave-equation solution fortracking the actual wave front would be that of the scalar-wave theory in general, i.e., if the aperture dimensions aremuch larger than the wavelength of light being considered,so that aperture edge effects can be neglected, and the ab-sence of significant polarization effects. This approachshould therefore also be superior to any of the Fourier-transform and linear-systems-convolution techniques inwhich the aperture plane is used as the plane of integrationwhen nonplanar or off-optic-axis incident waves are beingconsidered. Future research will be focused on developing anumerical solution to the scalar-wave equation for such gen-eralized diffraction calculations.

ACKNOWLEDGMENTS

This research was funded under the Exploratory Researchand Development Program of EG&G Idaho, Incorporated,at the Idaho National Energy Laboratory under U.S. De-partment of Energy contract DE-AC07-76-ID01570.

REFERENCES

1. B. B. Baker and E. T. Copson, The Mathematical Theory ofHuygens's Principle, 2nd ed. (Oxford U. Press, Oxford, 1969).

2. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Pa., 1987).3. A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Phys-

ics (Academic, New York, 1964).4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,

New York, 1968).5. C. S. Williams and 0. A. Beckland, Optics: A Short Course for

Hal G. Kraus


Scientists and Engineers (Wiley-Interscience, New York,1972).

6. F. A. Jenkins and H. E. White, Fundamental Optics, 4th ed.(McGraw-Hill, New York, 1976).

7. A. Nussbaum and R. A. Phillips, Contemporary Optics for Sci-entists and Engineers (Prentice-Hall, Englewood Cliffs, N.J.,1976).

8. J. A. Hudson, "Fresnel-Kirchhoff diffraction in optical systems:an approximate computational algorithm," Appl. Opt. 23,2292-2295 (1984).

9. Y. P. Kathuria, "Fresnel and far-field diffraction due to anelliptical aperture," J. Opt. Soc. Am. A 2, 852-857 (1985).

10. Y. P. Kathuria, "Computer modelling of three-dimensionalFresnel-diffraction pattern at circular, rectangular and squareapertures," Opt. Appl. 14, 509-514 (1984).

11. D. S. Burch, "Fresnel diffraction by a circular aperture," Am. J.Phys. 53, 255-260 (1985).

12. E. Wolf and E. W. Marchand, "Comparison of the Kirchhoffand the Rayleigh-Sommerfeld theories of diffraction at anaperture," J. Opt. Soc. Am. 54, 587-594 (1964).

13. J. E. Harvey, "Fourier treatment of near-field scalar diffractiontheory," Am J. Phys. 47, 974-980 (1979).

14. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics(Wiley, New York, 1978).

15. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon,New York, 1980).

16. W. H. Southwell, "Validity of the Fresnel approximation in thenear field," J. Opt. Soc. Am. 71, 7-14 (1981).

17. F. D. Feiock, "Wave propagation in optical systems with largeapertures," J. Opt. Soc. Am. 68, 485-489 (1978).

18. F. S. Harris, Jr., "Light diffraction patterns," Appl. Opt. 3,909-913 (1964).

19. R. C. Smith and J. S. Marsh, "Diffraction patterns of simpleapertures," J. Opt. Soc. Am. 64, 798-803 (1974).

20. A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr.,"Fresnel diffraction effects in the design of high-power lasersystems," Appl. Phys. Lett. 23, 85-87 (1973).

21. J. T. Wesley and A. F. Behof, "Optical diffraction pattern mea-surements using a self-scanning photodiode array interfaced toa microcomputer," Am. J. Phys. 55, 835-844 (1987).

22. A. E. Siegman, Lasers (University Science Books, Mill Valley,Calif., 1986).

23. B. T. Landesman and H. H. Barrett, "Gaussian amplitude func-tions that are exact solutions to the scalar Helmholtz equation,"J. Opt. Soc. Am. A 5, 1610-1619 (1988).

24. H. G. Kraus, "Huygens-Fresnel-Kirchhoff wave-front diffrac-tion formulation: paraxial and exact Gaussian laser beams,"submitted to J. Opt. Soc. Am. A.

Hal G. Kraus

JOSA COMMUNICATIONSCommunications are short papers. Appropriate material for this section includes reports of incidental researchresults, comments on papers previously published, and short descriptions of theoretical and experimental tech-niques. Communications are handled much the same as regular papers. Galley proofs are provided.

Gaussian amplitude functions that are exact solutions to thescalar Helmholtz equation;

Geometrical representation of the fundamental mode of aGaussian beam in oblate spheroidal coordinates: comment

L. B. Felsen

Department of Electrical Engineering and Computer Science, Weber Research Institute, Polytechnic University,Farmingdale, New York 11735

Received January 30, 1989; accepted June 6, 1989

Some perspectives are provided on inferences relating to the connection between complex rays and Gaussian beamsin the subject papers [J. Opt. Soc. Am. A 5, 1610 (1988); 6, 5 (1989)].

In the subject publications,12 the authors discovered thatexact wave solutions that behave as Gaussian beams can berepresented naturally in spheroidal coordinates. Althoughthe authors make some reference to related earlier studies,they seem to be unaware of the extensive literature pertain-ing especially to the relation between Gaussian beams andcomplex rays. In the Introduction of Ref. 2, in the contextof a review of earlier studies, the following sentence appears:"Further work has since extended the representation of thefundamental mode of a Gaussian beam by complex rays, aconcept that Felsen vigorously disputed." The paper onwhich my "vigorous dispute" is based was published in1976,1 and, on rereading that paper, I am at a loss to detectthe negative view toward complex rays that is alleged. Inthat paper I attempted to place in perspective various tech-niques that had been investigated in connection with thedifficult subject of propagation and diffraction of generalhigh-frequency wave fields with complex phase (evanescentwaves). In fact, Sec. D of that paper contains what I regardas a strong advocacy of the complex-source-point techniquefor generating exact field solutions that behave as Gaussianbeams. The following statements were made in that paper,first for two-dimensional fields and then for three-dimen-sional fields:

1. The simple artifice of replacing the real source coordi-nates p' = (y', z') [or r' = (x', y', z')] of a line source (or pointsource) by the complex values Po' + ib, (or r + ib 2) generatesan exact beam solution of the wave equation with a waistcenter at Po (or r) and a beam axis along b (or b 2), whichbehaves in the vicinity of the positive z axis as a paraxialGaussian beam (note that the paraxial beam formulas areonly approximate solutions of the wave equation).

2. Because of statement 1, the complex-source-pointsubstitution applied to conventional exact or approximate

field solutions (Green's functions) in the presence of obsta-cles or other medium perturbations converts these propaga-tion and diffraction solutions for incident cylindrical orspherical wave fields into exact or approximate propagationand diffraction solutions for incident two-dimensional orthree-dimensional Gaussian beam fields.

3. By performing the complex-source-point substitutionon vector electric or magnetic current dipoles, one may gen-erate exact vector Gaussian beam solutions of the Maxwellfield equations. This general class of solutions includesnonconventional polarizations generated by various inclina-tions of the dipole axis with respect to the beam axis vectorb. In a later paper,2 it is shown how exact higher order beamsolutions can be generated from complex displacement ofmultipole sources.

Having strongly advocated, rather than disputed, thecomplex-source-point method for dealing with beam propa-gation and diffraction problems, I have also been concernedwith the interpretation of the results in real coordinatespace. Here, the constructs of evanescent wave trackingexplored in Ref. 3, which, for Gaussian beams, occur natural-ly in elliptic (or spheroidal) coordinates, become relevant.

Since neither Ref. 1 nor Ref. 2 lists subsequent studieswherein my co-workers and I "vigorously" used the com-plex-source-point and complex-ray techniques, and sincemy own orientation in this regard may be placed in questionby the above-cited statement from Ref. 2, I have taken theliberty of listing various publications in which I have partici-pated that pertain to this subject area.420

REFERENCES

1. B. T. Landesmann and H. H. Barrett, "Gaussian amplitudefunctions that are exact solutions to the scalar Helmholtz equa-tion," J. Opt. Soc. Am. A 5, 1610-1619 (1988).


JOSA Communications

2. B. T. Landesmann, "Geometrical representation of the funda-mental mode of a Gaussian beam in oblate spheroidal coordi-nates," J. Opt. Soc. Am. A 6, 5-17 (1989).

3. L. B. Felsen, "Evanescent waves," J. Opt. Soc. Am. 66, 751-760(1976).

4. S. Y. Shin and L. B. Felsen, "Gaussian beam modes by multi-poles with complex source points," J. Opt. Soc. Am. 67, 699-700(1977).

5. L. B. Felsen, "Complex-source-point solutions of the field equa-tions and their relation to the propagation and scattering ofGaussian beams," in Symposia Matematica, Istituto Nazion-ale di Alta Matematica (Academic, London, 1976), Vol. 18, pp.40-56.

6. S. Y. Shin and L. B. Felsen, "Lateral shifts of totally reflectedGaussian beams," Radio Sci. 12, 551-564 (1977).

7. S. Y. Shin and L. B. Felsen, "Multiply reflected Gaussian beamsin a circular cross section," IEEE Trans. Microwave TheoryTech. MTT-26, 845-851 (1978).

8. P. D. Einziger and L. B. Felsen, "Evanescent waves and complexrays," IEEE Trans. Antennas Propag. AP-30, 594-604 (1982).

9. J. V. Hasselmann and L. B. Felsen, "Asymptotic analysis ofparabolic reflector antennas," IEEE Trans. Antennas Propag.AP-30, 677-685 (1982).

10. G. Ghione, I. Montrosset, and L. B. Felsen, "Complex ray analy-sis of radiation from large apertures with tapered illumination,"IEEE Trans. Antennas and Propag. AP-30, 689-693 (1984).

11. L. B. Felsen, "Geometrical theory of diffraction, evanescentwaves, complex rays and Gaussian beams," Geophys. J. R. As-tron. Soc. 79, 77-88 (1984).

Vol. 6/No. 10/October 1989/J. Opt. Soc. Am. A 1641

12. X. J. Gao and L. B. Felsen, "Complex ray analysis of beamtransmission through two-dimensional radomes," IEEE Trans.Antennas Propag. AP-33, 963-975 (1985).

13. L. B. Felsen, "Novel ways for tracking rays," J. Opt. Soc. Am. A2, 954-963 (1985).

14. Y. Z. Ruan and L. B. Felsen, "Reflections and transmission ofbeams at a curved interface," J. Opt. Soc. Am. A 3, 566-579(1986).

15. I. T. Lu, L. B. Felsen, Y. Z. Ruan, and Z. L. Zhang, "Evaluationof beam fields reflected at a plane interface," IEEE Trans.Antennas Propag. AP-35, 809-817 (1987).

16. L., B. Felsen, "Real spectra, complex spectra, compact spectra,"J. Opt. Soc. Am. A 3,486-496 (1986).

17. P. D. Einziger, Y. Haramaty, and L. B. Felsen, "Complex raysfor radiation from discretized aperture distributions," IEEETrans. Antennas Propag. AP-35, 1031-1044 (1987).

18. E. Heyman and L. B. Felsen, "Propagating pulsed beam solu-tions by complex source parameter substitutions," IEEE Trans.Antennas Propag. AP-34, 1062-1065 (1986).

19. E. Heyman and L. B. Felsen, "Complex-source pulsed-beamfields," J. Opt. Soc. Am. A 6,806-817 (1989).

20. J. Maciel and L. B. Felsen, "Systematic study of fields due toextended apertures by Gaussian beam discretization," IEEETrans. Antennas Propag. 37, 884-892 (1989).

21. J. Maciel and L. B. Felsen, "Gaussian beam analysis of propaga-tion from an extended plane aperture through dielectric layers:I-plane layer; II-circular cylindrical layer," submitted to IEEETrans. Antennas Propag.

1642 J. Opt. Soc. Am. A/Vol. 6, No. 10/October 1989

Gaussian amplitude functions that are exact solutions to thescalar Helmholtz equation;

Geometrical representation of the fundamental mode of aGaussian beam in oblate spheroidal coordinates:

reply to comment

B. T. Landesman

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Received April 25,1989; accepted May 16, 1989

A reply is made to Felsen's comment [J. Opt. Soc. Am. A 6, 1640 (1989)], and the basis is given for the statement towhich he objected.

The statement to which Felsen objected1 in the Introductionof my paper 2 was made on the basis of the first paragraph inthe second column of p. 753 of his 1976 paper on evanescentwaves,3 and it referred only to the interpretation given tocomplex rays by some authors. Nowhere in either of mypapers 2 4 did I state or imply that Felsen disagrees with theconcept of complex-source-point theory, and I regret that heperceived that I had done so. Indeed, I am well aware ofFelsen's many contributions to this body of work; however, Idid not cite this earlier literature at length for two reasons.First, the amount of research into complex-source-point the-ory is quite extensive, and I did not believe that either paperwas the proper place for a literature survey. Second, I be-lieve that these new wave functions differ significantlyenough from complex-source-point wave functions that anexhaustive citation of research on the latter was not calledfor.

In my opinion, the central difficulty with complex-source-point wave functions rests with the radical in the argumentof the exponent. In order to retrieve phase and amplitudeinformation from this representation, the square-root termmust be evaluated in some fashion. The usual procedure isto use the binomial expansion and then to approximate thewave function expression by eliminating the higher-orderterms. This maneuver results in the limitation of the valid-ity of the wave function to the paraxial region. The othermethod of removing the radical consists of resolving thecomplex number into magnitude and phase terms. Unfor-tunately, this manipulation leads to an expression that hasno simple interpretation.

In contrast, the wave functions that Barrett and I intro-duced are not hampered by a radical in the argument of theexponent, and, because of this, the phase and the amplitudeare readily available and easily interpreted. Furthermore,this simple result is not limited to the paraxial region. In-deed, there are no approximations in this wave function at

all, whereas the paraxial approximation is almost unavoid-able in the complex-source-point case. Finally, although inRefs. 2 and 4 wave functions were presented that are solu-tions to the scalar wave equation, there is nothing implicit ineither the physics or the mathematics that prohibits expan-sion of these wave functions to solutions of the vector waveequation. This subject is left for future research.

We now have two wave functions, both of which are exactsolutions to the scalar Helmholtz equation, that can be usedto represent Gaussian amplitude functions. The complex-source-point wave function must sacrifice exactitude in fa-vor of an accessible interpretation. In contrast, the newwave functions possess a simple and powerful interpretationwhile remaining free of approximations. We can debate themeaning of "exact" ad infinitum, but let the resolution restwith physical interpretation. As Felsen pointed out, he hasbeen concerned with interpretation of results in real coordi-nate space, and this is precisely the thrust of Ref. 2. Ibelieve that the geometrical representation given in thatpaper provides a better understanding of the physical natureof Gaussian amplitude beams, one that does not rely on thenonintuitive concept of complex point sources or complexrays.

REFERENCES

1. L. B. Felsen, "Gaussian amplitude functions that are exact solu-tions to the scalar Helmholtz equation; Geometrical representa-tion of the fundamental mode of a Gaussian beam in oblatespheroidal coordinates: comment," J. Opt. Soc. Am. A 6, 1640(1989).

2. B. T. Landesman, "Geometrical representation of the fundamen-tal mode of a Gaussian beam in oblate spheroidal coordinates," J.Opt. Soc. Am. A 6, 5-17 (1989).

3. L. B. Felsen, "Evanescent waves," J. Opt. Soc. Am. 66, 751-760.4. B. T. Landesman and H. H. Barrett, "Gaussian amplitude func-

tions that are exact solutions to the scalar Helmholtz equation,"J. Opt. Soc. Am. A 5, 1610-1619 (1988).


JOSA Communications

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