Far-field radiation of planar Gaussiansources and comparison with solutionsbased on the parabolic approximation

Xiaodong Zeng, Changhong Liang, and Yuying An

Previous research on the Gaussian beam was restricted in scope because of the paraxial assumption.Based on a rigorous solution of the Helmholtz equation, the radiation properties of a Gaussian source arereexamined. The results are different from what previous paraxial theory predicted. It is shown thatthe commonly used concept of the Gaussian beam and the separability of the field distribution along thetwo transverse axes may be misleading in certain conditions. Furthermore the theoretical upper limiton the beam divergence angle is shown to be 65.5°. © 1997 Optical Society of America

Key words: Gaussian beam, light propagation, divergence angle.

1. Introduction

As a simple mathematical model, the Gaussian beamhas been widely used in optical theories and applica-tions. For example, most laser beams and the fielddistribution in single-mode optical fibers can be de-scribed approximately by Gaussian beams. Becauseof the importance of Gaussian beams in optics, par-ticularly in the area of lasers, a great deal has beenwritten about Gaussian beams and their propagation~see, for example, Refs. 1–5!. But, because of math-ematical difficulties, paraxial approximations wereused in most previous research. For most laserbeams the divergence angle is very small, approxi-mately several milliradians. In this case the fieldamplitude becomes insignificant in the off-axis re-gion, and the errors arising from the paraxial approx-imation can be neglected. Although paraxialtheories can describe experimental phenomena wellwhen the divergence angle is small, they cannot ex-plain the propagation features of a beam with alarger divergence angle. Hereafter a Gaussianbeam within the framework of paraxial theory iscalled the conventional Gaussian beam.We point out that when the beam emitted from a

Gaussian source has a large divergence angle, the

The authors are with the Department of Technical Physics, Xid-ian University, 710071 Xian, China.Received 8 April 1996; revised manuscript received 16 August

1996.0003-6935y97y102042-06$10.00y0© 1997 Optical Society of America

2042 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997

beam-propagation properties are significantly differ-ent from what paraxial theories predict. We refer tothe beam we discuss as the general Gaussian wave.An example of the difference in behavior between theconventional Gaussian beam and the general Gauss-ian wave occurs when the source distribution is sep-arable; i.e., it can be represented as a product of twofunctions. In this case the general Gaussian wave isno longer separable. This is inexplicable on the ba-sis of paraxial theories, which predict that propaga-tion does not alter the separable structure of thebeam. Another example is that the modulus of thegeneral Gaussian wave in the plane transverse tothe propagation direction will not have a truly Gauss-ian profile; but the amplitude of the conventionalGaussian beam is an accurate Gaussian function.For an accurate estimation of the field distribution,particularly in the off-axis region, it may be necessaryto take into account these differences. Inmany prac-tical applications the paraxial condition cannot besatisfied, such as holography, audiovisual devices,and optical sensing. In these fields, largernumerical-aperture lenses are usually used. To de-sign an optical system, one should have an accurateexpression for the field distribution of the beam.In this paper we present a rigorous far-field solu-

tion of the Helmholtz equation, using an appropriateboundary condition. Based on this rigorous solu-tion, the propagation behavior of the beam emittedfrom a plane Gaussian source is reexamined. Math-ematical relations for the intensity, phase distribu-tion, beam spot size, and beam-divergence angle arederived. These results are used to examine closely

the theoretical upper limit on the divergence angle ofthe general Gaussian wave and the errors of theparaxial approximation. Our results are differentfrom previous theories and are expected to provide abetter understanding of light propagation phenom-ena.

2. Solution of the Wave Equation

We handle the treatment of propagation of a wave bysolving the scalar wave equation. We limit our dis-cussion to monochromatic radiation, that is, radia-tion at a certain frequency. More general forms ofradiation can always be described by the superposi-tion of all its sinusoidal components. A three-dimensional wave expressed by c~r, t! 5u~r!exp~2ivt! is considered. The space-dependentpart satisfies the Helmholtz equation, i.e.,

¹2u~r! 1 k2u~r! 5 0, (1)

where k 5 2pyl is the propagation constant in vac-uum. A solution to Eq. ~1! can be expressed in termsof its boundary values6:

u~r! 5 212p *

s

u~r*!]G~r, r*!

]nds, (2)

where G~r, r*! 5 exp~ikur 2 r*u!yur 2 r*u, s denotes theboundary surface, r* takes on the values of r on theboundary surface, and n is the local outward normalto this surface. Equation ~2! is the first Rayleigh–Sommerfeld integral formula and corresponds to Dir-ichlet boundary conditions. We define a Cartesiancoordinate system ~x, y, z! as shown in Fig. 1. Thesource ~i.e., the boundary surface! lies in the x–yplane and radiates in the half-space, z . 0. Ac-cording to the geometry, on the boundary source wehave

]G~r, r*!

]n5 2

]G~r, r*!

]z. (3)

Fig. 1. Coordinate system. The ~x, y, z! represents a far-fieldobservation point.

The partial derivative of G with respect to z can nowbe obtained as follows. It is easy to show that

]

]zur 2 r*u 5

]

]z@~x 2 x9!2 1 ~y 2 y9!2 1 z2#1y2

5z

ur 2 r*u, (4)

so that

]G~r, r*!

]z5

]

]z@exp~ikur 2 r*u!yur 2 r*u#

5z

ur 2 r*u3@ikur 2 r*u 2 1#exp~ikur 2 r*u!.

(5)

By substituting Eq. ~5! and the definition of theboundary surface into Eq. ~2!, we obtain

u~r! 512p *

2`

`

*2`

`

u~r*!z

ur 2 r*u3~ikur 2 r*u 2 1!

3 exp~ikur 2 r*u!dx9dy9, (6)

where r 5 xi 1 yj 1 zk and r* 5 x9i 1 y9j, the latterbeing the argument for the source field distribution.To this point we have made no approximations what-soever, so that Eq. ~6! is the rigorous solution of thewave equation @Eq. ~1!#.The so-called far field means here that the distance

from the source to the observation point is muchgreater than the wavelength as well as the size of thesource. Therefore the following approximations canbe made:

kur 2 r*u..1, (7)

1ur 2 r*u

<1uru. (8)

In the phase of the integrand, a Taylor series expan-sion permits us to introduce the following simplifica-tion:

ur 2 r*u < uru 2 ~¹uru! z r*

5 uru 21uru

r z r*. (9)

Substituting Eqs. ~7!–~9! into Eq. ~6! yields

u~x, y, z! 5izlr

exp~ikr!r *

2`

`

*2`

`

u~x9, y9!

3 expF2ikr

~xx9 1 yy9!Gdx9dy9,(10)

where r 5 ~x2 1 y2 1 z2!1y2. To obtain Eq. ~10! weused only the far-field conditions ~7!–~9!; thereforeEq. ~10! could be regarded as a rigorous solution of

1 April 1997 y Vol. 36, No. 10 y APPLIED OPTICS 2043

the wave equation ~1! for the far field. The far-fieldapproximations are reasonable for many practicalcases, because field measuring is usually attemptedat distances significantly greater than the wave-length or the source size.

3. Propagation of a General Gaussian Wave

Suppose the field distribution of the source is aGaussian function with a plane phase front, i.e.,

u~x9, y9! 5 A expS2x92 1 y92

v02 D , (11)

where the parameter v0 is the beam halfwidth at z 50. The latter is the distance from the peak of thefield distribution at which the function decays to 1yeof its maximum value. By substituting Eq. ~11! intoEq. ~10!, one can obtain

u~x, y, z! 5 Bzr2exp~ikr!expS2

1C2

x2 1 y2

r2 D ,(12)

where

B 5iApv0

2

l, (13)

C2 54

k2v02 . (14)

This is our basic result. According to Eq. ~12! thepropagation of a general Gaussian wave is uniquelydetermined once the source spot size v0 and thewavelength l are specified. The first exponentialfactor in Eq. ~12! describes the phase of a sphericalwave with a radius of curvature r centered at theorigin. The second exponential factor and the factorzyr2 in Eq. ~12! determine the far-field amplitude asa function of the coordinates and how this changes asthe wave propagates in the half-space, z . 0. Wecan see that Eq. ~12! is different from the conven-tional Gaussian beam4:

u~x, y, z! 5 Bexp~ikz!

zexpF2i

k~x2 1 y2!2z G

3 expS21C2

x2 1 y2

z2 D . (15)

In Eq. ~15! the far-field condition 4z2 .. k2v04 has

been invoked. Obviously Eq. ~15! can be regarded asthe special case of Eq. ~12! in the paraxial condition.If in the paraxial region we can take r ' z in theamplitude term and r ' z 1 ~x2 1 y2!y2z in the phaseterm in Eq. ~12!, Eq. ~12! changes to Eq. ~15!. FromEq. ~15! we can see that the expression of conven-tional Gaussian beam is a product of two functions,each dependent on only one of the two transversecoordinates ~x or y!. According to Eq. ~12!, the gen-eral Gaussian wave is not represented as a product oftwo separable functions, despite the fact that thesource function is assumed to be separable. The

2044 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997

properties of the general Gaussian wave are dis-cussed in Subsections 3.A–3.C.

A. Beam Spot Size

From Eq. ~12! we can see that the amplitude distri-bution in the transverse direction is no longer aGaussian function; it depends on a product of twoterms, zyr2 and an exponential factor. We introducethe beam spot size parameter r0 by the definition thatit is equal to the distance ~in transverse direction! atwhich the field intensity decays to one-half of its max-imum value. By use of Eq. ~12! we can obtain theintensity distribution,

I~x, y, z! 5 uu~x, y, z!u2

5 uBu2z2

~r2 1 z2!2expS2D

r2

r2 1 z2D , (16)

where

D 5k2v0

2

2, r2 5 x2 1 y2.

Let I~r0! 5 I~r 5 0!y2; one can obtain

uBu2

2z25

uBu2z2

~r02 1 z2!2

expS2Dr0

2

r02 1 z2D

or

12

5 S1 2r0

2

r02 1 z2D

2

expS2Dr0

2

r02 1 z2D . (17)

It is impossible to solve Eq. ~17! to obtain an ex-plicit form r0 directly. However, it is not difficult toshow that within the interval @0, 1#, h 5 ~1 2 t!2

exp~2Dt! has a negative first derivative with respectto t, i.e., h is a decreasing function, and moreover wehave h~0! 5 1, h~1! 5 0. According to Rolle’s theo-rem we know that Eq. ~17! has a unique solution forr0

2y~r02 1 z2!; let it be t0, say,

r02

r02 1 z2

5 t0

or

r0 5 S t01 2 t0

D1y2

z. (18)

Equation ~18! shows that the beam spot size isdirectly proportional to z. The coefficient @t0y~1 2t0!#

1y2 depends on the ratio v0yl; it can be evaluatedby the numerical method.

B. Phase Distribution

The phase front of the general Gaussian wave is asphere around the origin with radius r. This issomewhat different from the conventional Gaussianbeam. In the case of the paraxial approximation wehave z2 .. r2; the phase function of the general

Gaussian wave can be expressed as

kr 5 kz~1 1 r2!1y2 < kz~1 1 r2y2z2!. (19)

This expression is just the same as the phase functionof the conventional Gaussian beam.

C. Beam-Divergence Angle

We define the full width at half-maximum intensityas

u 5 2 tan21Sr0z D 5 2 tan21S t0

1 2 t0D1y2

. (20)

Because t0 is related to v0yl, u is related to v0yl.Figure 2 shows the variation in the beam-divergenceangle u with v0yl. It indicates that the smallerv0yl, the larger the beam-divergence angle. How-ever, this curve is surprisingly different from theparaxial result,4

u 5~2 ln 2!1y2

p

l

v0, (21)

particularly for a small value of v0yl. As we de-scribe below, the divergence angle of a general Gauss-ianwave has an upper limit. This upper limit can befound with the help of Eq. ~17!. To this end, werewrite Eq. ~17! as

12

5 S 11 1 r0

2yz2D2

expS2Dr0

2yz2

1 1 r02yz2D . (22)

Obviously, we have

S 11 1 r0

2yz2D2

$12

(23)

Fig. 2. Variation of the beam-divergence angle u with v0yl. Thedashed curve corresponds to the conventional Gaussian beam.

or

r0z

# SÎ2 2 1D1y2

. (24)

Therefore the maximum divergence angle is

umax 5 2 tan21SÎ2 2 1D1y2

< 65.5°. (25)

This result shows that the far-field beam-divergenceangle of any Gaussian source cannot be larger than65.5°. This result cannot be predicted by paraxialoptics. From Eq. ~22! we can see that when D 3 0,i.e., the source size approaches zero, the divergenceangle takes on its maximum value. Note that thelimiting value cannot be realized in practice becauseit requires the size of the source to be zero; in this casethe far-field intensity is also zero unless the source isa d function. The above theoretical limit also ex-plains why the beam-divergence angle approaching65.5° has never been found in practice.

4. Errors of Paraxial Approximation

According to Eqs. ~12! and ~15!, when the laser beamshave a very small divergence angle, the condition z2

.. x2 1 y2 is always satisfied. In this case the gen-eral Gaussian wave can be described approximatelyby a conventional Gaussian beam. However, whenthe beam has a larger divergence angle, the conven-tional Gaussian beam cannot be described well inpractice. We will discuss the errors owing to theparaxial approximation. Because the relative errorsare of most practical importance, in the following weconsider only the relative errors of the phase and theamplitude.The phase functions of the conventional and gen-

eral Gaussian wave are

wc 5 kz 1kr2

2z, (26)

wg 5 kzS1 1r2

z2D1y2

, (27)

respectively. The relative phase error is

ep 5 Uwg 2 wc

wgU 5 U1 2

1 1 r2y2z2

~1 1 r2yz2!1y2U5 U1 2

1 1 ~tan2 u!y2~1 1 tan2 u!1y2U , (28)

where tan u 5 ryz. Figure 3 shows the variation ofepwith u. It can be seen that the error may be largerin the off-axis region.Using Eqs. ~12! and ~15!, one can obtain the rela-

1 April 1997 y Vol. 36, No. 10 y APPLIED OPTICS 2045

tive amplitude error

ea 5 Uuugu 2 uucuuugu

U5 U1 2 S1 1

r2

z2DexpF21C2

r4

z2~r2 1 z2!GU5 U1 2 ~1 1 tan2 u!expS2

1C2

tan4 u

1 1 tan2 uDU . (29)

Figure 4 shows the relationship between ea and u.In Fig. 4 curves a, b, c, d, and e are obtained with v0yl5 0.25, 0.5, 1.0, 2.0, 4.0, respectively. It can be seenthat, in the paraxial region, i.e., x2 1 y2 ' 0, theamplitude error ea is approximately zero, and more-

Fig. 3. Relative phase errors between the general and conven-tional Gaussian beams with angle u.

Fig. 4. Relative amplitude errors between the general and con-ventional Gaussian beams with angle u. For curves a, b, c, d, e,the parameters are v0yl 5 0.25, 0.5, 1.0, 2.0, 4.0, respectively.

2046 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997

over each curve has two zeros. This can be ex-plained as follows. From point r 5 0 the amplitudeof the general Gaussian wave decreases somewhatmore rapidly with r than the conventional Gaussianbeam does. But, with further increasing values of r,the general Gaussian wave decreases less rapidlythan the conventional Gaussian beam does. Thenwe know that the amplitude curves of the two typesof wave have two intersection points. Therefore theerror curves have two zeros. From Fig. 4 we can seethat the amplitude errors may be much larger in theoff-axis region. When the beam has a larger diver-gence angle, this error cannot be neglected.

5. Conclusion

Based on a rigorous solution of the Helmholtz equa-tion, the far-field distribution and propagation prop-erties of a general Gaussian wave have beendiscussed. The results indicate that the radiationproperties of a Gaussian source are different from theconventional paraxial theory. The far-field distribu-tion is not a truly Gaussian function. This is inex-plicable on the basis of paraxial theory, which pointsout that propagation does not alter the Gaussian dis-tribution. Our results show that the separability ofthe beam does not occur, although the source distri-bution is separable, and the far field may have non-trivial errors resulting from a paraxial assumptionthat the optical field would be expressed as a productof two Gaussian functions, each dependent on onlyone of the two transverse coordinates. For an accu-rate estimation of the field distribution ~e.g., in pro-cesses requiring coherent light and uniform wavefronts, such as holography and laser Doppler an-emometry! it may be necessary to take into accountthe coupling between the field distribution along thetwo principal axes.The results also indicate that the limiting diver-

gence angle of the general Gaussian wave is 65.5°,which cannot be realized in practice because it re-quires the source distribution to be a d function.This knowledge cannot be predicted from previousparaxial theories.Finally, note that our results cannot be applied in

the near-field region, because the far-field conditionbreaks down in the near zone of the source, that is,close to z 5 0. The near-field properties are nottaken into account in this paper, because they lead tomore complicated mathematical forms. Neverthe-less the results pertaining to the propagation, theseparability, and the divergence angle remain un-changed. The general Gaussian wave could be amore reasonable model for describing the propaga-tion of the beam emitted from a Gaussian source.

This study was supported by the Chinese NationalNatural Sciences Foundation.

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through lenslike media including those with a loss and gainvariation,” Appl. Opt. 4, 1562–1569 ~1965!.

2. L. D. Dickson, “Characteristics of a propagating Gaussianbeam,” Appl. Opt. 9, 1854–1861 ~1970!.

3. A. Yariv, Introduction to Optical Electronics ~Holt, Rinehart &Winston, New York, 1971!, Chap. 3, p. 30.

4. D. Marcuse, Light Transmission Optics ~Van Nostrand Rein-

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