The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation

15 June 1998

Optics Communications 152 Ž1998. 108–118

Full length article

The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation P. Varga a , P. Torok ¨¨

b,1

a

b

Research Institute for Materials Science, P.O. Box 49, H-1525 Budapest, Hungary Department of Engineering Science, UniÕersity of Oxford, Parks Rd, Oxford OX1 3PJ, UK Received 27 August 1997; revised 12 January 1998; accepted 9 February 1998

Abstract We present an exact solution of Maxwell’s equations for a non-paraxial Gaussian wave. Paraxial vectorial and paraxial scalar approximations of the exact solution are also obtained. We examine in detail, via several numerical examples, the validity conditions for the paraxial scalar and paraxial vectorial approximations when compared to the exact solution. Our findings show that when the half width of the beam waist is greater than 10 l the paraxial scalar approximation yields accurate results. When, however, the half width of the beam waist is comparable to the wavelength the exact solution predicts significant deviations from the paraxial scalar approximation. These results might prove important in semiconductor laser research. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The Gaussian wave is commonly used in theoretical and experimental optics and its mathematical representation has successfully been applied by many workers. The fundamental problem with the most frequently used mathematical descriptions of the Gaussian beam is that they do not satisfy Maxwell’s equations, even though they are readily applicable to experimental problems. A need arises therefore to construct and numerically analyse a consistent solution of Maxwell’s equations that is applicable to Gaussian beams of high divergencerconvergence angles Ži.e. possessing an arbitrary beam waist. and which, on the paraxial approximation, reduces to the well-known scalar Gaussian wave. It is also essential to establish the conditions under which the exact solution of Maxwell’s equations can legitimately be approximated by the scalar Gaussian wave. A Gaussian wave is usually considered in practice to be linearly polarised. This implies that the electric and magnetic field components are mutually perpendicular to the

1

E-mail: [email protected]

corresponding propagation direction. It is known from experimental results that it is impossible to generate a perfectly unidirectional laser beam which means that the beam does not possess a single propagation direction. It follows that it is impossible to define the plane of polarisation for a Gaussian beam, which contradicts to the assumption of the Gaussian beam being plane polarised. It is even more apparent when plane waves are focused by high aperture lenses w1x. As shown by Richards and Wolf, in the image space of a high aperture lens the structure of polarisation becomes fairly complicated and thus the distribution cannot be represented by any single polarisation direction. In early theories, which are briefly discussed in Section 2, the Gaussian wave was derived as a solution of the homogeneous scalar wave ŽHelmholtz. equation. The resulting equations were, however, applied to the electric or the magnetic field. The wave equation can be derived from Maxwell’s equations for homogeneous space thus any solution of Maxwell’s equations for homogeneous space satisfies the wave equation. Conversely, Maxwell’s equations cannot be derived from the scalar or vectorial form of the wave equation, hence any particular solution of the wave equation does not necessarily satisfy Maxwell’s

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 9 2 - 3

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

equations. An example is the spherical wave, E s AexpwyiŽ v t y kr .xrr, with constant amplitude A. Its divergence does not vanish and, as pointed out by Luneburg w2x, the amplitude of an electric spherical wave must depend on the angular coordinates. Another example is a solution of the Fresnel-Kirchhoff integral, which is derived with the use of Green’s theorem assuming that both field and probe functions satisfy the Helmholtz equation w3x. It can, in general, be stated that field quantities obtained as solutions of the wave equation may not satisfy Maxwell’s equations. Most of the previous works, based on the scalar diffraction theory, employ the paraxial approximation, i.e. the Gaussian beam is considered to possess a mean propagation direction and therefore it can be described by the function AŽ x, y, z .expwyiŽ v t y kr .x, where the amplitude AŽ x, y, z . varies slowly compared to the exponential term. This approximation may be acceptable for long laser cavities but in optical systems when a Gaussian beam is focused by a high numerical aperture lens it becomes invalid. A similar situation may occur for semiconductor lasers. In this case, although waveguide properties may well play an important role, the small diameter of the active medium is likely to cause significant deviations, in terms of beam properties, from that predicted by the scalar paraxial theory. The vectorial nature of the Gaussian beams has not been observed in experimental linear optics and therefore there have been no experimental results reported on properties of laser beams different from that predicted by the scalar theory. In this paper we show that the waist of laser beams used in experimental optics is large enough to produce the traditional scalar Gaussian beam. Results of numerical computations confirm that the use of scalar theory in linear optics is justified up to beam waists 5–10 times the wavelength. When, however, small beam waists and near field distributions are considered, the difference between scalar and vectorial Gaussian waves becomes appreciable. The vectorial nature of Gaussian beams becomes important in nonlinear optics w4x. Self-focusing properties of intensive beams are severely affected by the structure of the field in the focal region of a lens. It was shown that the intensity distribution in the self-trapped beam exhibits several maxima along the propagation direction instead of exhibiting a single maximum as predicted by the scalar diffraction theory. The construction of this paper is as follows. In Section 2 a brief review is presented on the scalar and electromagnetic theories. In Section 3 the rigorous theory is presented that satisfies Maxwell’s equations. The derivation is based on a self-consistent solution of the electromagnetic boundary value problem and the use of the Hertz vector. In this way we avoid problems associated with the fact that the Huygens-Fresnel integral does not satisfy Maxwell’s equations. Section 4 describes the paraxial vectorial approxima-

109

tion of the rigorous solution. In Section 5 a detailed numerical analysis is presented and, in particular, the validity conditions of the paraxial approximation are discussed.

2. Historical review 2.1. Scalar theories 2.1.1. Paraxial theories There are two ways to formulate the problem of mathematical description of Gaussian waves. The first method is to find an equation for a standing wave inside a long laser cavity and then to describe the propagating wave emitted from the resonator by essentially the same functional form. The second, most frequently used, method is to approximate the scalar Helmholtz equation paraxially and then to obtain a solution, i.e. the Gaussian wave, from the approximated Helmholtz equation. Fox and Li w5x constructed, for the first time, a self-consistent solution for a laser resonator. They studied scalar diffraction in a Fabry-Perot interferometer used as a laser resonator. Their numerical model was obtained such that an initial wave propagated towards one of the mirrors and then it was reflected back and reached the other mirror. It was found, by computing the field on one of the mirrors, that after many round-trips the relative field distribution did not vary and thus a steady state was reached. This was regarded as the fundamental mode of the interferometer. A closed-form solution of the same problem was find by Boyd and Gordon w6x for an open confocal resonator. The authors assumed that a particular field distribution exists which is identical on both mirrors. The field on one mirror is reproduced by the identical field on the second mirror by diffraction. The problem was treated by the Fresnel-Kirchhoff integral in the paraxial approximation. Their treatment resulted in a Fredholm-type integral equation with its kernel being equal to the unknown distribution. The lowest order eigenfunction of the resonator was shown to be the Gaussian wave and the higher order modes were shown to be a multiplication of the Gaussian wave and the Hermite polynomials. Kogelnik and Li w7x found the same solution by employing the paraxial approximation of the homogeneous wave equation. The authors pointed out that any function that satisfies the wave equation can either be a field or a potential quantity. Because in this paper the results of Kogelnik and Li are directly compared to those obtained from the rigorous theory we give now a brief account of the work of Kogelnik and Li. A solution of the scalar wave equation Duy

1 E 2u c2 E t 2

s0

Ž1.

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

110

2

for light propagating in the qz direction

is

u s c Ž x , y, z . exp w yi Ž v t y k 0 z . x ,

Ž2.

where k 0 is the wave number in Õacuo, k 0 s vrc. If the paraxial condition E 2CrE z 2 < k 0 ECrE z holds then the fundamental Gaussian wave is given by

C Ž x , y, z ;w 0 . s

w0 w

½

exp yr 2

1 w2 Ž z .

y

ik 0 2 RŽ z .

5

y iF Ž z . ,

Ž3. 2

2

2

where r s x q y , and R s z Ž 1 q z 02rz 2 . ,

Ž4.

w 2 s w 02 Ž 1 q z 2rz 02 . ,

Ž5.

y1

F s tan

Ž zrz 0 . ,

z 0 s k 0 w 02r2,

Ž6. Ž7.

where w 0 denotes the beam waist and z 0 denotes the Rayleigh length. For a detailed discussion on the scalar Gaussian wave the reader is referred to the work of Saleh and Teich w8x. It was shown by Deschamps w9x that the above results can also be obtained by generalising the phase of a spherical wave which, in the paraxial approximation, is given by k 0 Ž z q r 2r2 z . , the generalised expression for the phase is k 0 z q r 2r2 q Ž z . , which leads to the Gaussian wave with q s z y iz 0 . 2.1.2. Non-paraxial theories Non-paraxial theories are usually based on the angular spectrum representation and considered as boundary value problems. The boundary function, in our case the Gaussian function, is determined at the plane where the boundary value is defined Ž z s 0.. A general optical beam was studied by Porras w10x. His formalism was not restricted to Gaussian beams. Beam characteristics, such as the total power and beam width defined as the second order moment of the transversal irradiance distribution, were introduced and their propagation was discussed. It is worth to note that the author found for a general beam that the widening of the beam width during propagation obeys a hyperbolic relation. This should be contrasted with Eq. Ž5. from which it follows that the hyperbolic widening is not an exclusive property of the Gaussian beams.

2

The authors of the present paper prefer to use a sign convention of expwy iŽ v t y kz .x instead of expw iŽ v t y kz .x, which is, sometimes, used by others Žsee, e.g., Ref. w7x..

The propagation of a beam with a Gaussian boundary value was studied by Agrawal and Pattanayak w11x. The authors constructed a highly convergent orthogonal expansion to represent the function expw ikz Ž1 y b 2 .1r2 x for b 4 0. From the numerical results it was shown that the paraxial theory fails for small initial beam half widths w 0 . For example, for kw 0 s 2 and z s 5rk the paraxial theory produces significant discrepancies, however for kw 0 s 4 and z s 10rk the difference is negligible. Recently Zeng et al. w12x considered the solution as a boundary value problem with a Gaussian amplitude distribution on the boundary. The initial wave was chosen to be scalar and it was assumed that this property is maintained during propagation. The Rayleigh-Sommerfeld diffraction integral was solved for the far field. No restriction was made for the initial beam parameter waist. The authors found that the beam parameters R and w differed from those of Eqs. Ž4. and Ž5., where the r s x 2 q y 2 q z 2 substitution is to be used for z 0 . The authors also stated that the beam divergence is limited to f 65.58 with the initial beam waist approaching zero.

(

2.2. Vectorial solutions 2.2.1. Paraxial theories The need for considering the Gaussian wave to be vectorial first arose in quantum electronics. Lax et al. w13x found a vectorial solution for the Gaussian wave having been dissatisfied with the non-Maxwellian character of the Gaussian wave. The medium filling the cavity was assumed to be amplifying. By looking for transÕersal and longitudinal components, a pair of coupled differential equations was constructed for the transversal and the longitudinal components of the field. The source term in the equations was proportional to the amplification of the active medium and to the transversal or longitudinal field, respectively, as it is expected from laser theory. The longitudinal component was shown to be proportional to the gradient of the transversal component. Pattanayak and Agrawall w14x attempted to find the form of a general optical beam and in particular of a paraxial beam. The authors considered the Whittaker potentials as the starting point of their formulation that satisfy the scalar Helmholtz equation. The field was produced as a superposition of plane electromagnetic waves with amplitudes expressed by the Whittaker potentials. It is clear, as stated in Section 1, that the superposition does not result in a unidirectional and polarised wave. It was shown that in the paraxial approximation the field possesses transversal and longitudinal components. Small beam parameters Ž kw 0 . were also considered and high order corrections have been obtained. Two principal modes were found in the paraxial approximation, an H and an E beam mode. The mean propagation direction of the E beam was found to coincide with the z axis and also E x s 0. The

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

other two components of the E beam was found not to vanish and hence the E field was not perpendicular to the mean propagation direction. A study on paraxial beams with general w15,16x and Gaussian w17,18x boundary value was reported. In these works it was emphasised that paraxial beams cannot, in principle, be plane polarised: ‘‘The polarization of the field cannot be chosen independently of spatial variation’’ w17x. Simon et al. w17x found an explicit solution to the paraxial Gaussian beam for which they showed that the field components E x and B y are identical to those predicted by the scalar wave theory of Kogelnik and Li w7x. They also found the other two transversal components to vanish. The longitudinal component of the electric and magnetic fields was found not to vanish and, in addition, having the form of E z f x expwyŽ x 2 q y 2 .rw 2 x and B z f y expwyŽ x 2 q y 2 .rw 2 x. In a later paper w18x, after performing more accurate calculations, the authors showed that E y and B x do not actually vanish and hence the field to be elliptically and longitudinally polarised. A different vectorial paraxial approximation was obtained recently by Varga and Torok ¨ ¨ w19x who assumed that the paraxial scalar solution is associated with the Hertz vector rather than the field. The solution of Varga and Torok, ¨ ¨ in this paper, is obtained as the approximation of the exact results.

2.2.2. Non-paraxial theories Similar to the problem of a scalar wave Porras w10x also studied beam characteristics of a vectorial beam. He showed that the hyperbolic relation of the second order moment remains valid for the vectorial case. The divergence of the beam was shown to remain finite even for w 0 ™ 0; with the highest beam divergence being 638. Cullen and Yu w20x gave a full electromagnetic solution for the Gaussian wave by generalising the Hertz dipole wave. The electromagnetic field of an electric dipole is given from a Hertz Õector by ZsI

exp Ž ikr . r

,

Ž8.

where I denotes the unit vector along the x axis. The field quantities can be obtained from the Hertz vector, as shown in Section 3 if, according to Deschamps w9x, the radial distance r is replaced by r s x 2 q y 2 q Ž z y iz 0 .

2 1r2

.

Ž9.

Cullen and Yu used the generalised distance r and the method described in Section 3 to derive the electromagnetic Gaussian wave. Their rigorous solution is fairly difficult to analyse. It contains a spherical wave expŽ ik 0 r . that is multiplied by a sum whose terms are inversely proportional to the higher powers of the complex distance

111

r. Contrary to the Hertz dipole, the solution possesses no singularity which is the consequence of using the generalised distance given by Eq. Ž9.. It is important to note, as Cullen and Yu pointed out, that the electric and magnetic fields are asymmetrical and thus they cannot represent beam-waves. The paraxial approximation was also found by Cullen and Yu. The paraxially approximated vectorial wave was shown to possess three components even though the Hertz vector was oriented along the x axis. For large beam waists the mean component of the electric field was shown to coincide with the x axis, similar to that given by the paraxial scalar theory. There were, however, two other components found; one lateral component along the y axis and a longitudinal component along the z axis. The approximated results of Cullen and Yu agree with those, obtained in a different way, by Simon et al. w17x and Varga and Torok ¨ ¨ w19x. Davis and Patsakos w21,22x obtained the scalar wave solution from the vector potential. This approach automatically results in the vector components of the field from the derivation of the scalar Gaussian wave. By looking for a more accurate solution they did not neglect the term E 2CrE z 2 and obtained a series expansion Cs C q s 2C 2 q . . . , where C is defined by Eq. Ž3. and s, which is small for the usual lasers, is the ratio of the beam waist width w 0 and the Rayleigh length z 0 . Using this result Doicu and Wriedt w23x produced a plane wave spectrum of this corrected vector potential and have obtained expressions that are suitable for numerical purposes. As a conclusion, prior to the work of Varga and Torok ¨¨ w19x, only Cullen and Yu w20x presented an exact solution for the vectorial Gaussian wave whilst others used various approximations. In this work we present a different exact solution that is more suitable for numerical analysis than that of Cullen and Yu.

3. The electromagnetic Gaussian wave We now obtain a solution of the Maxwell equations. The medium of propagation is assumed to be homogeneous, charge and current free. The solution may be obtained in a straightforward manner. Once the boundary value is known the solution is given by the plane wave spectrum method. The only problem remaining is to define the boundary value. Now we assume that the result of Boyd and Gordon w6x, which is a self-consistent solution of the diffraction problem for an open resonator Ži.e. the eigenfunction of an open resonator. describes a physically existing wave. This, however, cannot possibly be an electromagnetic wave, because of its non vanishing divergence, but it may well be some other physical quantity which satisfies the Helmholtz equation. There are two such physical quantities: Ža. the Hertz vector and Žb. a set of electromagnetic vector and scalar potentials. Our solution

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

112

is obtained in terms for the Hertz vector. A second solution is presented in the Appendix which is based on the knowledge of the boundary values of E x and E y in the plane z s 0. The field strength vectors may be determined from the Hertz vector in the following manner. The Hertz vector Z satisfies the Helmholtz equation 2

DZy

´m E Z c

2

Et

2

s 0,

Ž 10.

vector may be represented by its angular spectrum representation 1

V Ž x , y, z . s

2

Esy

´m E Z c2 E t 2

q =Ž = P Z .

Ž 11.

´m Bs

==

c

EZ Et

.

Ž 12.

In homogeneous space another independent pair of field vectors can be obtained from Hertz vector:

em Dsy

c

==

EZ

where `

HHy` V Ž x , y,0. 2p

is the Fourier spectrum of the boundary function and Ž k 2x q k 2y q k 2z .1r2 s Ž em .1r2 k 0 . From Eqs. Ž11. and Ž12. the field strength vectors are given by 1 2p k

` 2

HHy` g Ž k

x ,k y

. F Ž k x ,k y .

=exp i Ž k x x q k y y q k z z . d k x d k y

em E 2 Z c2 E t 2

Bsy

'

i ´m 2p k

`

HHy` m Ž k

x ,k y

. F Ž k x ,k y .

=exp i Ž k x x q k y y q k z z . d k x d k y ,

1

V Ž x , y, z . exp Ž yi v t . ,

Ž 14.

Z y s Zz s 0 ,

m Ž k x ,k y . s k z J y k y K ,

Ž 20.

Ž 15.

where the coefficient 1rk 2 was introduced for convenience, k s ´m k 0 and V possesses field dimensions. For a general case Z y and Z z may be chosen different from zero and then the procedure, described below, can be applied to these components too. Let Z x be given by the boundary value at z s 0, Z x Ž x, y,0. s V Ž x, y,0.. This boundary value cannot be measured in a straightforward manner but may be deduced from measuring the field outside the laser cavity. The x component of the Hertz

Ž 21.

with I, J, K being the unit vectors of the Cartesian system. The Fourier amplitudes of the field vector and the wave vector k s k x I q k y J q k z K are orthogonal hence E and B are represented by their vector-wave spectrum. Eqs. Ž18. and Ž19. are the rigorous solutions of Maxwell’s equations when the boundary value of one of the components of the Hertz vector is given in the z s 0 plane. They can also be applied to problems different from that presented in this work. If the boundary value is defined by the Gaussian function

ž

V Ž x , y,0 . s exp y

'

Ž 19.

and q =Ž = P Z . .

In what follows the electric strength and magnetic induction vectors, Eqs. Ž11. and Ž12., shall be used. It can now be readily verified that the electromagnetic waves may only be plane waves if the Hertz vector is also a plane wave and hence the Gaussian beam cannot be plane polarised w14,17,20x . Eq. Ž10. represents three, mathematically independent solutions for Z x ,Z y and Z z , so it is sufficient to solve the problem for Z x k 0:

k2

Ž 18.

and

g Ž k x ,k y . s Ž k 2 y k 2x . I y k x k y J I k x k z K ,

Hsy

Ž 17.

where

and

Zx s

1

=exp yi Ž k x x q k y y . d x d y

Ž 13.

Et

. Ž 16.

Es

and

x ,k y

=exp i Ž k x x q k y y q k z z . d k x d k y ,

F Ž k x ,k y . s where ´ is the dielectric constant and m is the magnetic susceptibility. The electromagnetic field vectors are given by w24x

`

HHy` F Ž k 2p

x2qy2 w 02

/

.

Ž 22.

then, on substituting Eq. Ž22. into Eq. Ž17., and with the use of Eqs. Ž18. and Ž19., the field strength vectors are given by Es

w 02 4p k 2

`

HHy` g Ž k

x ,k y

. exp

y

w 02 4

Ž k 2x q k 2y .

=exp i Ž k x x q k y y q k z z . d k x d k y ,

Ž 23.

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

and

113

with

Bsy

iw 02

'´m

`

HHy` m Ž k

4p k

x ,k y

. exp

y

w 02 4

Ž k 2x q k 2y .

=exp i Ž k x x q k y y q k z z . d k x d k y ,

k y s k sin u X ,

x s r cos up ,

w 02 2

`

ž

H0 k exp

w 02 k 2

y

/

4

=exp iz Ž k 2 y k 2 .

1r2

J0 Ž rk . dk ,

Ey s

w 02

I0Ž e. q I2Ž e. cos Ž 2 up . ,

4k2 w 02 4k

I Ž e. sin 2 2

Ž 2 up . ,

Ez s y2 i

w 02 4k

I Ž e. cos up , 2 1

Ž 28 .

`

H0

ž

Ž 2 k 2k y k 3 . exp y

=exp iz Ž k 2 y k 2 . I1Ž e. s

`

H0 k

2

2 1r2

2

Ž k yk .

1r2

I2Ž e. s

`

H0 k

3

ž

exp y

4

=exp iz Ž k 2 y k 2 .

/

/

J0 Ž rk .

w 02 k 2 4

/

B x s 0, Bz s i

B ys y i

w 02

'´m

2k

'´m

2k

I1Ž m. sin up ,

ž

exp y

w 02 k 2 4

1r2

/

w 02 k 2 4

/

J0 Ž rk .

dk

J1Ž rk .

1r2

dk .

Ž 31.

We note that Eqs. Ž28. and Ž29. can be brought to a form, on confining the upper integral limit to k, that is formally identical to that obtained by Stamnes and Dhayalan w25x for the far-field distribution of a converging electric dipole wave.

4. Paraxial approximation In the following the approximation of the rigorous solution is obtained for large beam waists. It is shown that this approximation results in the usual expression of Eq. Ž3. for the Gaussian beam. The paraxial vector wave, given in Refs. w20,21x, can also be obtained on approximating Eqs. Ž23. and Ž24.. It is now assumed that k z s k y Ž k 2x q k 2y . r2 k

Ž 32.

2

l

H0 exp Žy

1 4

w 02 k 2 . J0 Ž rk .

=exp Ž yiz k 2r2 k . k d k .

dk ,

dk .

w 02 exp Ž ikz .

J1Ž rk .

Ž 33.

Let VG s VG,0 y VG,l where

J2 Ž rk .

1r2

2

=exp iz Ž k 2 y k 2 .

VG s

VG,0 s

2

`

H0 exp Žy

1 4

w 02 k 2 . J0 Ž rk .

w 02 exp Ž ikz . 2

`

Hl

exp Ž y 14 w 02 k 2 . J0 Ž rk .

=exp Ž yiz k 2r2 k . k d k .

Ž 30.

Ž 34 .

and VG,l s

I0Ž m. ,

w 02 exp Ž ikz .

=exp Ž yiz k 2r2 k . k d k ,

Ž 29.

The Cartesian components of the magnetic field are given by w 02

H0 k

ž

exp y

dk ,

ž

=exp iz Ž k y k . w 02 k 2

4

exp y

2 1r2

2

w 02 k 2

1r2

from which the approximation of the Hertz vector component in Eq. Ž27. is obtained. Let l s Ž k 2x q k 2y .1r2 be the wavenumber for which Eq. Ž32. is satisfied, i.e. the sum of the high order terms in the power expansion is much less than 2p . Let w 0 be so large that expŽyw 02 l 2r4. becomes negligibly small, then

with I0Ž e. s

yk 2 .

Ž 27.

where J0 is the Bessel function, zero order, first kind. After substituting Eqs. Ž25. and Ž26. into Eqs. Ž20. and Ž23. and applying the transformation rules w1x of Bessel functions the Cartesian components of the electric field are given by Ex s

`

I1Ž m. s

Ž 26.

This will be applied to the x component of the Hertz vector to give

2

=exp iz Ž k 2 y k 2 .

Ž 25.

y s r sin up .

VG Ž x , y, z . s

H0 k Ž k

Ž 24.

In order to ease numerical computations, Eq. Ž23. is transformed by using k x s k cos u X ,

`

I0Ž m. s

Ž 35.

It can be readily shown that VG,0 reproduces the scalar Gaussian wave and the contribution of VG,l is negligibly

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

114

small. By applying the inverse transformation of Eqs. Ž25. and Ž26. to Eq. Ž34. we obtain VG,0 s

w 02

w 02

`

HHy` exp y 4 Ž k 4p

ž

=exp i k x x q k y y y

2 2 x qky

E s E0 q e ,

.

k 2x q k 2y 2k

z

/

B s B0 q b,

dkx dky.

w

½

exp yr 2

1 w2 Ž z .

y

ik 0 2 RŽ z .

'

5

2

`

Ž 37.

exs

ez s

x2

i

ž

y kqm x

qm

ž

qm2

/

x2qy2 2 qm2

E0 ,

VG,l F

2

Hl

exp Ž y

w 02 k 2

. k d k s exp

that is negligibly small as anticipated.

ž

y

E0 ,

/

ž

'

b y s ´m y

bx s 0,

1 4

xy qm2

Ž 42 .

and

exp Ž y 14 w 02 k 2 . J0 Ž rk . k d k

`

e ys y

y 1 E0 ,

and for J0 Ž x . F 1 w 02

Ž 41.

and with additional waves e and b

'

Hl

Ž 40 .

B0 s J ´m VG ,0 Ž x , y, z . exp w yi Ž v t y kz . x , y iF Ž z .

where quantities R, w and F are the same as in Eqs. Ž4. – Ž6. except here we used, instead of k 0 , k s em vrc. The integral Eq. Ž35. gives w 02

E0 s I VG ,0 Ž x , y, z . exp w yi Ž v t y kz . x , and

=exp Ž ikz . ,

VG,l F

Ž 39 .

with fundamental waves

Carrying out the integration results in w0

Ž 38.

and

Ž 36.

VG,0 s

As VG,0 is a slowly varying function of co-ordinates the electric and magnetic fields can be expressed as a sum of two vectors

w 02 l 2 4

/

,

'

bz s y ´m

y qm

E0 .

x2qy2

i q kqm

2 qm2

/

E0 ,

Ž 43.

Here qm s z y iz m0 . The electric and magnetic fields, represented by Eqs. Ž40. – Ž43., possess x, y and z compo-

Fig. 1. Figure showing the sum of evanescent and propagating fields Žand the evanescent wave – inset. as function of half width w 0 at the x s y s z s 0 position.

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

nents. The elliptical polarisation was first found by Simon et al. w18x and the longitudinal component was discovered by Lax et al. w13x. Eqs. Ž37. – Ž41. represent the paraxial TM Gaussian wave. The paraxial TE Gaussian wave can be obtained by interchanging the electric and magnetic field components. In practice additional waves are negligible with respect to the fundamental waves. Function
'

5. Numerical results

115

5.2. Comparison of the rigorous and approximate solutions A possible measure of inaccuracy of the paraxial scalar approximation is defined as the ratio of the time-averaged electric energy density, E P E ) , as computed from Eq. Ž28. and the time-averaged electric energy density of the paraxial scalar approximation ŽCase I.. The accuracy of the paraxial vectorial approximation, Eq. Ž42., as compared to the paraxial scalar approximation is defined as the ratio of the former and the latter approximations, respectively ŽCase II.. The inaccuracy of the scalar approximation is shown in Fig. 2Ža. for a half width of w 0 s l as a function of the normalised axial position zrz 0 . The continuous line corresponds to Case I whilst the dashed line corresponds to

Numerical results were computed on an IBM PC compatible 486DX2 66 MHz computer. Programs were written in FORTRAN77 and using the NAG ŽNumerical Algorithm Group, Oxford. numerical subroutine packages. Numerical data was plotted by using the Tecplot software. No post data processing was performed prior to plotting. The aim of this paper is to determine the limit within which the paraxial scalar theory gives a reasonably good approximation. This approximation shall be referred to as the ‘paraxial scalar approximation’ or PSA. In what follows we examine in detail how much the boundary value affects the electric field in the vicinity of the origin of the co-ordinate system.We also examine how the free parameters affect the propagation of the beam. In all numerical examples the wavelength of light, l, is taken to be unity and thus k s 2p . The free parameter is chosen to be the half width, w 0 , of the Gaussian function representing the boundary value for the Hertz vector. The scaling parameter for longitudinal directions is the wavelength or for longer distances the Rayleigh diffraction length z 0 s p w 02rl. The scaling parameter for the transversal directions is the actual width w Ž z . of the beam. 5.1. The field at z s 0 In Eq. Ž29. the wave number, k , is defined within the Ž0,`x range. For k ) k the resulting wave becomes irregular or evanescent and for 0 F k F k the resulting wave is regular or propagating. In Fig. 1 the amplitude of the propagating component of the electric field Epr , at x s y s z s 0 is shown as a function of the half width w 0 . This figure shows that even for w 0 s l the electric field is nearly unity in the vicinity of the origin of the co-ordinate system. The evanescent component of the electric field is shown in the inset which reveals that for half widths w 0 F l the evanescent waves have a profound effect on the electric field.

Fig. 2. Figures showing the inaccuracy of the paraxial scalar approximation when compared to the exact solution Žcontinuous line. and paraxial vectorial approximation Ždashed line.. Ža. and Žb. correspond to half widths of w 0 s l and w 0 s10 l, respectively.

116

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

Case II. As the figure shows the paraxial scalar approximation results in an f 90% accuracy in the waist Ž zrz 0 s 0. which slowly but gradually improves for zrz 0 ) 2.5. It can

thus be concluded that the paraxial vectorial approximation does not improve significantly the accuracy of computations compared to the exact method. Fig. 2Žb. shows the same functional dependence as Fig. 2Ža. but for an initial half width of w 0 s 10 l. The continuous line corresponds to Case I whilst the dashed line corresponds to Case II. As the figure shows the paraxial scalar approximation results in a more accurate computation for this case. The accuracy is f 0.1% for zrz 0 s 0 which becomes f 0.03% for zrz 0 s 300. The paraxial vectorial approximation, again, does not improve the accuracy of computations. We can thus conclude from the above figures that for small w 0 the paraxial vectorial approximation does not improve the accuracy of computations and for large w 0 the paraxial scalar approximation gives an accurate description of the distribution. The paraxial vectorial approximation therefore does not seem to give an appreciable improvement on the axial distribution compared to the paraxial scalar approximation. The inaccuracy of the scalar approximation along a principal transversal direction and at the wings of the Gaussian distribution is shown in Fig. 3Ža. –3Žc. as function of normalised x co-ordinate Ž xrx M . where x M is defined by:

C Ž x M ,0, z . C Ž 0,0, z .

2

s 0.01

for a particular z position. Figs. 3Ža. –3Žc. are plotted for half widths of w 0 s l, 10 l and 100 l, respectively, and individual curves correspond to z 0 s 0.5p , z 0 s p , z 0 s 2p and z 0 s 3p . For small w 0 the exact transversal distribution is essentially different from the paraxial scalar approximation. For large lateral co-ordinates the inaccuracy can be as high as several 100%. For w 0 s 10 l and 100 l the inaccuracy of the paraxial scalar approximation becomes sufficiently low, 10 and 5%, respectively.

6. Conclusion In this paper we have presented an exact solution for the Gaussian vectorial wave. Our solution satisfies Maxwell’s equations. This exact solution has been compared to results obtained previously by other workers, notably the paraxial scalar and paraxial vectorial approximations. Our results show that when the half width of the Gaussian beam Žat the waist. is of the order of f 10 l the

Fig. 3. Figures showing the inaccuracy of the paraxial scalar approximation when compared to the exact solution. Ža., Žb. and Žc. correspond to half widths of w 0 s l, 10 l and 100 l, respectively. Individual curves correspond to z 0 s 0.5p , z 0 sp , z 0 s 2p and z 0 s 3p . Note that in Žc. curves corresponding to z 0 s 0.5p and z 0 s 2p nearly overlap.

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

paraxial scalar approximation gives a result of reasonable accuracy compared to that given by the exact solution. For common gas and solid state lasers the resonator lengths are great enough so that the paraxial scalar approximation gives a good accuracy. In semiconductor lasers, however, the width of the active zone is comparable to the wavelength which indicates that there the paraxial scalar approximation does not give sufficiently accurate result. We note that for semiconductor lasers there are other phenomena to be considered Ži.e. the inhomogeneity of the active zone and the epitaxial layer.. These conclusions imply that the focusing of a Gaussian beam by a high aperture lens might not be treated in a rigorous manner by a scalar Gaussian apodisation function in the Richards and Wolf w1x integrals.

The authors would like to thank Professor Colin Sheppard of the University of Sydney, Australia for discussions. P. Torok ¨ ¨ is on leave from the Central Research Institute for Physics of the Hungarian Academy of Sciences ŽBudapest..

Appendix A One of our reviewers has pointed out that Eqs. Ž18. and Ž19. can be derived in a different way. The knowledge of the tangential component of the electric or magnetic field vector on the boundary is sufficient to construct the solution provided the space is charge and current free w26,27x. The way to obtain the solution is well known and described by Stamnes w28x. We now follow the derivation of Stamnes and Dhayalan w25x and Stamnes w28x. Let us assume that the tangential component of the electric field Et is known at z s 0. The Fourier transform of Et is given by E˜j Ž k t . s

1

`

2p

HHy` E Ž x , y,0. exp Žyik P r . d x d y, j

j s x , y.

t

t

Ž A.1 .

where k t s Ik x q Jk y and r t s Ix q Jy. In order to have a divergence free field we must set the third component of E˜ to satisfy E˜z Ž k x ,k y . s y

1 kz

k t P E˜ Ž k t . .

Ž A.2 .

The Fourier transform of the magnetic induction vector on the boundary is then given by B˜ Ž k t . s

1 k0

where k s k t q Kk z . The field for z G 0 is given by the angular spectrum representations ` 1 EŽ r. s E˜ Ž k t . 2p y`

HH

=exp i Ž k x x q k y y q k z z . d k x d k y , BŽ r. s

1

k = E˜ Ž k t . ,

Ž A.3 .

Ž A.4 .

`

HHy` B˜ Ž k . 2p t

=exp i Ž k x x q k y y q k z z . d k x d k y ,

Ž A.5 .

where r s r t q Kz. The tangential components of the field can be obtained from Eqs. Ž11. and Ž15.

ž

E x Ž x , y,0 . s 1 q E y Ž x , y,0 . s

Acknowledgements

117

1

1 E2 k2 E x2

/

V Ž x , y,0 . ,

E2

k 2 E xE y

V Ž x , y,0 . .

Ž A.6 .

Let V˜Ž k x ,k y . now denote F Ž k x ,k y . for the sake of consistency, then 1 E˜x Ž k x ,k y . s 2 Ž k 2 y k 2x . V˜ Ž k x ,k y . , k kxky E˜y Ž k x ,k y . s y 2 V˜ Ž k x ,k y . , Ž A.7 . k and the z component of the Fourier transform according to Eq. ŽA.2. is given by kxkz E˜z Ž k x ,k y . s y 2 V˜ Ž k x ,k y . . Ž A.8 . k Substituting Eqs. ŽA.7. and ŽA.8. into Eqs. ŽA.4. and ŽA.5. the result clearly identical to those of Eqs. Ž23. and Ž24.. The same reviewer also suggested that perhaps it would be possible to set up a boundary value simpler than Eq. ŽA.6.. In an attempt of doing so we now use the scalar and vector potentials to obtain a self-consistent solution of the Helmholtz equation. The half space z ) 0 is charge free hence the scalar potential F s const. Furthermore, the Lorentz condition is satisfied if the vector potential is divergence free, = P AŽ r,t . s 0. The field quantities can be derived from the vector potential by 1 EA Esy , B s = = A. Ž A.9 . c Et For a constant frequency v the electric field proportional to the vector potential. If we assume that the boundary value relates to the electric field by EŽ x, y,0. s W Ž x, y,0. and we apply Eqs. ŽA.1., ŽA.2., ŽA.3., ŽA.4. and ŽA.5. then the solution ` 1 E Ž x , y, z . s m Ž k x ,k y . W˜ Ž k x ,k y . y` k z

HH

=exp i Ž k x x q k y y q k z z . d k x d k y ,

Ž A.10 .

P. Varga, P. Torokr ¨ ¨ Optics Communications 152 (1998) 108–118

118

and B Ž x , y, z . s

1 k

`

1

HHy` k

g Ž k x ,k y . W˜ Ž k x ,k y .

z

=exp i Ž k x x q k y y q k z z . d k x d k y

Ž A.11 . is obtained. Here g and m are defined by Eqs. Ž20. and Ž21., respectively and W˜ Ž k x ,k y . is the Fourier transform of W Ž x, y,0.. It should be pointed out that the d k x d k yrk z remains small of the second order even if k z ™ 0. The electromagnetic vectors given by Eqs. ŽA.10. and ŽA.11. differ from those of Eqs. Ž23. and Ž24.. In the present case the beam is clearly more divergent for small beam waist w 0 but it is smoother in the vicinity of the plane z s 0. This, alternative, solution, just like Eqs. Ž23. and Ž24., does not represent a plane polarised wave.

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