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Vectorial analytical description of propagation of a highly nonparaxial beam
1 February 2002
Optics Communications 202 (2002) 17–20 www.elsevier.com/locate/optcom
Vectorial analytical description of propagation of a highly nonparaxial beam Alessandro Ciattoni a,b,d,*, Bruno Crosignani c,d, Paolo Di Porto c,d a Dipartimento di Fisica, Universit a Roma Tre, I-00146 Rome, Italy Istituto Nazionale di Fisica della Materia Unit a di Roma 3, Rome, Italy c Dipartimento di Fisica, Universit a dell’Aquila, 67010 L’Aquila, Italy Istituto Nazionale di Fisica della Materia, Unit a di Roma ‘‘La Sapienza’’, 00185 Rome, Italy b
d
Received 5 September 2001; received in revised form 12 November 2001; accepted 22 November 2001
Abstract We present a formalism describing optical propagation in a homogeneous medium of a fully vectorial highly nonparaxial field, characterized by a waist smaller than the wavelength. The method allows us to derive an analytical expression for a field possessing an initial Gaussian transverse distribution of width w, in the extreme nonparaxial regime w < k, valid for propagation distances z J d, where d ¼ w2 =k is the diffraction length. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25.-p; 42.25.Fx Keywords: Nonparaxial propagation; Gaussian beams
Propagation optics mainly deals with the paraxial regime (that is f ¼ k=w 1, w being the transverse beam width) which is described by the so-called scalar parabolic equation, derived from Helmholtz’s equation by assuming the slowly varying approximation hypothesis. The analytical approaches, from the first [1] to the most recent ones [2,3], aimed at removing the limitations associated with the smallness of f and the scalar
*
Corresponding author. Fax: +39-655-79-078. E-mail addresses: [email protected], alessandro.ciattoni@fis.uniroma3.it (A. Ciattoni).
assumption are of practical relevance only when f < 1, and all of them basically deal with asymptotic expansions in the parameter f. On the other hand, the description of optical propagation in the nonparaxial regime, dealing with values of w such that f > 1, is becoming more and more important with the advent of new optical structures, like, e.g., microcavities and photonic band-gap crystals [4], possessing linear dimensions or spatial scales of variation comparable or even smaller than k. The description of highly nonparaxial fields is commonly achieved through an ab initio numerical solution of Maxwell’s equation and turns out to be time consuming and not particularly illuminating. Thus, it is natural
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 7 2 2 - 9
18
A. Ciattoni et al. / Optics Communications 202 (2002) 17–20
to look for the development of a propagation equation analogous to the one valid in the paraxial regime or, at least, for some exact analytical solutions which could enlighten the main features of nonparaxial propagation and serve as a test for the numerical results. Some analytical solutions available in the highly nonparaxial regime [5–10] have been derived by employing the complex source and sink-point method. This approach furnishes exact solutions, but it is not suitable to treat propagation of an arbitrary boundary field. Other approches to highly nonparaxial propagation are limited to particular boundary transverse fields [11]. In this paper, conversely, we develop a general method apt to describing propagation in the nonparaxial regime, which in particular enables us to find an analytical solution associated with a realistic Gaussian boundary condition, valid for any value of f > 1 and for propagation distance z J d ¼ w2 =k. In a homogeneous medium of refractive index n, the forward monochromatic propagating field Eðr? ; zÞ can be related to its distribution on the plane z ¼ 0 by means of the relations [12] Z 1 o expðikRÞ 2 0 0 d r? E? ðr? ; 0Þ E? ðr? ; zÞ ¼ ; 2p oz R Z 1 o expðikRÞ d2 r0? Ex ðr0? ; 0Þ Ez ðr? ; zÞ ¼ 2p ox R o expðikRÞ þ Ey ðr0? ; 0Þ ; ð1Þ oy R
where r? ¼ o=ox i þ o=oy j. Hereafter, we consider highly nonparaxial fields characterized by a boundary transverse component E? ðr0? ; 0Þ practically vanishing for jr0? j > k. In this case, Eq. (1) is particularly suitable in obtaining approximate expressions for the propagating vector field. As a matter of fact, the main contribution to the integral in Eq. (2) comes from the small area around r0? ¼ 0, whose diameter w we assume to be smaller than k ¼ 2p=k. This allows us to expand in Eq. (1) the nucleus expðikRÞ=R in the smallness parameter r?0 =k K w=k, obtaining, to the first significant order, the expression ( expðikRÞ 1 ikðxx0 þ yy 0 Þ ’ exp ikr R r r
) ik ðy 2 þ z2 Þx02 þ ðx2 þ z2 Þy 02 2xyx0 y 0 þ ; 2r3
where
with
r? ¼ xi þ yj;
r0?
0
0
¼ x i þ y j;
E? ¼ Ex i þ Ey j;
ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y 2 þ z2 ¼ r?2 þ z2 . To derive the conditions under which Eq. (4) holds true, it is sufficient to warrant that the first-order in the exponential is much smaller than the zeroth-order term kr. It is straightforwardly seen that this happens if w < k and z > d ¼ w2 =k. Inserting Eq. (4) in Eq. (2) we obtain for F? the expression F? ðr? ; zÞ ¼
G? ðr? ;zÞ ¼
1 expðikrÞ G? ðr? ; zÞ; 2p r Z
( d2 r0? exp
ð5Þ
ikðxx0 þyy 0 Þ r
k ¼ ðx=cÞn and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðx x0 Þ2 þ ðy y 0 Þ2 þ z2 :
) ik ðy 2 þz2 Þx02 þðx2 þz2 Þy 02 2xyx0 y 0 þ 2r3
If we introduce the potential F? defined by [13] Z 1 expðikRÞ d2 r0? E? ðr0? ; 0Þ ; ð2Þ F? ðr? ; zÞ ¼ 2p R
E? ðr0? ;0Þ:
it is immediate to check that the field can be expressed through the relations E? ðr? ; zÞ ¼
oF? ; oz
Ez ðr? ; zÞ ¼ r? F? ;
ð3Þ
ð6Þ
We wish to note that our approximation basically differs from the standard Fraunhofer approximation [14] even if the regions of validity z > w2 =k are the same. In fact, the Fraunhofer approximation is based on the asymptotic expansion in the smallness parameter jr0? r? j=z ’ w=z while our expansion deals with the case w=k < 1.
A. Ciattoni et al. / Optics Communications 202 (2002) 17–20
The potential F? in Eq. (5) is the product of an outgoing spherical wave and a slowly varying factor G? . In fact, for the extreme nonparaxial field (i.e., E? ðr? ; 0Þ ¼ E0 dðr? Þ, E0 being a fixed vector), G? ¼ E0 which suggests that, for highly nonparaxial fields, G? behaves as a slowly varying function of r, and the potential F? essentially behaves as a spherical wave. This remarkable feature underlines the central role played by spherical waves in describing highly nonparaxial fields. For the opposite case of a paraxial beam, the potential F? is the product of a plane wave expðikzÞ and of a slowly varying amplitude. Comparing the potentials in these two extreme situations, we can argue that the presence of the spherical wave expðikrÞ=r in Eq. (5) is related to the large divergence angle characterizing highly nonparaxial field propagation, as much as the presence of the plane wave expðikzÞ in the paraxial potential is related to the smallness of the divergence angle in paraxial propagation. These general considerations can be made explicit to the particular case of the boundary distribution r2 E? ðr? ; 0Þ ¼ E0 exp ?2 i: ð7Þ 2w By inserting Eq. (7) in Eq. (6), we obtain Z $ G? ðr? ; zÞ ¼ iE0 d2 r0? exp r0? ðA : r0? Þ þ b r0? ; ð8Þ $
where the matrix A is $
A¼ 1=2w2 ikðy 2 þ z2 Þ=2r3 ikxy=2r3 ; ikxy=2r3 1=2w2 ikðx2 þ z2 Þ=2r3 ð9Þ $
b ¼ ikr? =r and A : r0? indicates the standard rowby-column product. The integral appearing in Eq. (8) can be analytically evaluated [15], yielding $ $ 1 1 1=2 b ðA : bÞ exp G? ðr? ; zÞ ¼ iE0 p ½det A 4 ð10Þ or, after some algebra
G? ðr? ; zÞ ¼ iE0 2pw2 Q1=2 P 1=2 exp
19
k 2 w2 r?2 ; 2P r2 ð11Þ
where QðrÞ ¼ ð1 ikw2 =rÞ and P ðr; zÞ ¼ ð1 ikw2 z2 =r3 Þ: By inserting Eq. (11) into Eq. (5) and the resulting potential in Eq. (3) we obtain for the electric field
1 z E? ðr? ; zÞ ¼ iE0 w2 ik r r Q1 z 1P 2 3 þ þ 2Q r2 2P z r i 2 2 2 h k w zr r þ 2 ?4 Q þ ð1 P Þ P r 2z expðikrÞ k 2 w2 r?2
Q1=2 P 1=2 exp ; r 2P r2
1 x Q1 x 2 þ ik Ez ðr? ; zÞ ¼ E0 w r r 2Q r2 2 2 P 1 3x k w x r?2 þ 2 ð3 P Þ 2P r2 2Pr2 Pr2 expðikrÞ k 2 w2 r?2
Q1=2 P 1=2 exp r 2P r2 ð12Þ which is the main result of this paper. Eq. (12) exhibit some relevant features which are worth enphasizing since they are peculiar of the highly nonparaxial regime. The asymptotic behavior of the field for r? ! 1 at any given plane z ¼ z0 is 2 w k 2 w2 þ ikr? ; ðikz0 Þ exp E? iE0 r? 2 2 2 w k w þ ikr? : ðikwÞ exp ð13Þ Ez E0 r? 2 Note that the transverse component E? vanishes as r?2 , while for the usual paraxial Gaussian beam it vanishes as exp½ðr? =wÞ2 =2. This is a quite general property of highly nonparaxial fields as it can be straightforwardly verified by starting from the general expression given in Eq. (5). In fact, G? becomes asymptotically constant as r? ! 1 (Oðr?1 Þ, see Eq. (6)) so that F? behaves, for
20
A. Ciattoni et al. / Optics Communications 202 (2002) 17–20
r? ! 1, as a spherical wave and, consequently, the electric field as in Eq. (13). For the sake of completeness, we show the level plots of the moduli of both the transverse and longitudinal components of the electric field (Fig. 1). They exhibit the main features of the solutions, and, in particular, they show, as expected, that longitudinal and transverse components are comparable.
References [1] M. Lax, W.H. Louisell, W. McKnight, Phys. Rev. A 11 (1975) 1365. [2] A. Yu Savchencko, B. Ya Zel’dovich, J. Opt. Soc. Am. B 13 (1996) 273. [3] A. Ciattoni, B. Crosignani, P. Di Porto, A. Yariv, J. Opt. Soc. Am. B 17 (2000) 809. [4] J.D. Joannopoulis, R.D. Meade, J.N. Winn, Photonic Crystals, Princeton University, Princeton, NJ, 1995. [5] G.A. Deschamps, Electron. Lett. 7 (1971) 684. [6] C.J.R. Sheppard, S. Saghafi, Phys. Rev. A 57 (1998) 2971. [7] C.J.R. Sheppard, S. Saghafi, J. Opt. Soc. Am. A 16 (1999) 1381. [8] C.J.R. Sheppard, S. Saghafi, Opt. Lett. 15 (1999) 1543. [9] Z. Bouchal, M. Olivik, J. Mod. Opt. 42 (1995) 1555. [10] A.L. Cullen, P.K. Yu, Proc. R. Soc. A 366 (1979) 155. [11] P. Varga, P. Torok, Opt. Commun. 152 (1998) 108. [12] See, e.g., R.K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1966, sec. 45.7. [13] We believe F? to deserve the name of potential since the electric field is invariant under the gauge transformation Fx ! Fx oy w , Fy ! Fy þ ox w, where wðx; yÞ is an arbitrary scalar function. [14] See, e.g., J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968, section 4-1. [15] See, e.g., P. Ramond, Field Theory: A Modern Primer, Addison-Wesley, UK, Inc., 1990, appendix A.
Fig. 1. Level plot of the normalized moduli jEx j=jE0 j (a1, a2, a3) and jEz j=jE0 j (b1, b2, b3) of a highly nonparaxial gaussian beam with w ¼ 0:1 lm on the planes z1 ¼ 2kw2 , z2 ¼ 2:5kw2 and z3 ¼ 3kw2 . The wavelength is k ¼ 0:5 lm.
Optics Communications 202 (2002) 17–20 www.elsevier.com/locate/optcom
Vectorial analytical description of propagation of a highly nonparaxial beam Alessandro Ciattoni a,b,d,*, Bruno Crosignani c,d, Paolo Di Porto c,d a Dipartimento di Fisica, Universit a Roma Tre, I-00146 Rome, Italy Istituto Nazionale di Fisica della Materia Unit a di Roma 3, Rome, Italy c Dipartimento di Fisica, Universit a dell’Aquila, 67010 L’Aquila, Italy Istituto Nazionale di Fisica della Materia, Unit a di Roma ‘‘La Sapienza’’, 00185 Rome, Italy b
d
Received 5 September 2001; received in revised form 12 November 2001; accepted 22 November 2001
Abstract We present a formalism describing optical propagation in a homogeneous medium of a fully vectorial highly nonparaxial field, characterized by a waist smaller than the wavelength. The method allows us to derive an analytical expression for a field possessing an initial Gaussian transverse distribution of width w, in the extreme nonparaxial regime w < k, valid for propagation distances z J d, where d ¼ w2 =k is the diffraction length. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25.-p; 42.25.Fx Keywords: Nonparaxial propagation; Gaussian beams
Propagation optics mainly deals with the paraxial regime (that is f ¼ k=w 1, w being the transverse beam width) which is described by the so-called scalar parabolic equation, derived from Helmholtz’s equation by assuming the slowly varying approximation hypothesis. The analytical approaches, from the first [1] to the most recent ones [2,3], aimed at removing the limitations associated with the smallness of f and the scalar
*
Corresponding author. Fax: +39-655-79-078. E-mail addresses: [email protected], alessandro.ciattoni@fis.uniroma3.it (A. Ciattoni).
assumption are of practical relevance only when f < 1, and all of them basically deal with asymptotic expansions in the parameter f. On the other hand, the description of optical propagation in the nonparaxial regime, dealing with values of w such that f > 1, is becoming more and more important with the advent of new optical structures, like, e.g., microcavities and photonic band-gap crystals [4], possessing linear dimensions or spatial scales of variation comparable or even smaller than k. The description of highly nonparaxial fields is commonly achieved through an ab initio numerical solution of Maxwell’s equation and turns out to be time consuming and not particularly illuminating. Thus, it is natural
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 7 2 2 - 9
18
A. Ciattoni et al. / Optics Communications 202 (2002) 17–20
to look for the development of a propagation equation analogous to the one valid in the paraxial regime or, at least, for some exact analytical solutions which could enlighten the main features of nonparaxial propagation and serve as a test for the numerical results. Some analytical solutions available in the highly nonparaxial regime [5–10] have been derived by employing the complex source and sink-point method. This approach furnishes exact solutions, but it is not suitable to treat propagation of an arbitrary boundary field. Other approches to highly nonparaxial propagation are limited to particular boundary transverse fields [11]. In this paper, conversely, we develop a general method apt to describing propagation in the nonparaxial regime, which in particular enables us to find an analytical solution associated with a realistic Gaussian boundary condition, valid for any value of f > 1 and for propagation distance z J d ¼ w2 =k. In a homogeneous medium of refractive index n, the forward monochromatic propagating field Eðr? ; zÞ can be related to its distribution on the plane z ¼ 0 by means of the relations [12] Z 1 o expðikRÞ 2 0 0 d r? E? ðr? ; 0Þ E? ðr? ; zÞ ¼ ; 2p oz R Z 1 o expðikRÞ d2 r0? Ex ðr0? ; 0Þ Ez ðr? ; zÞ ¼ 2p ox R o expðikRÞ þ Ey ðr0? ; 0Þ ; ð1Þ oy R
where r? ¼ o=ox i þ o=oy j. Hereafter, we consider highly nonparaxial fields characterized by a boundary transverse component E? ðr0? ; 0Þ practically vanishing for jr0? j > k. In this case, Eq. (1) is particularly suitable in obtaining approximate expressions for the propagating vector field. As a matter of fact, the main contribution to the integral in Eq. (2) comes from the small area around r0? ¼ 0, whose diameter w we assume to be smaller than k ¼ 2p=k. This allows us to expand in Eq. (1) the nucleus expðikRÞ=R in the smallness parameter r?0 =k K w=k, obtaining, to the first significant order, the expression ( expðikRÞ 1 ikðxx0 þ yy 0 Þ ’ exp ikr R r r
) ik ðy 2 þ z2 Þx02 þ ðx2 þ z2 Þy 02 2xyx0 y 0 þ ; 2r3
where
with
r? ¼ xi þ yj;
r0?
0
0
¼ x i þ y j;
E? ¼ Ex i þ Ey j;
ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y 2 þ z2 ¼ r?2 þ z2 . To derive the conditions under which Eq. (4) holds true, it is sufficient to warrant that the first-order in the exponential is much smaller than the zeroth-order term kr. It is straightforwardly seen that this happens if w < k and z > d ¼ w2 =k. Inserting Eq. (4) in Eq. (2) we obtain for F? the expression F? ðr? ; zÞ ¼
G? ðr? ;zÞ ¼
1 expðikrÞ G? ðr? ; zÞ; 2p r Z
( d2 r0? exp
ð5Þ
ikðxx0 þyy 0 Þ r
k ¼ ðx=cÞn and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðx x0 Þ2 þ ðy y 0 Þ2 þ z2 :
) ik ðy 2 þz2 Þx02 þðx2 þz2 Þy 02 2xyx0 y 0 þ 2r3
If we introduce the potential F? defined by [13] Z 1 expðikRÞ d2 r0? E? ðr0? ; 0Þ ; ð2Þ F? ðr? ; zÞ ¼ 2p R
E? ðr0? ;0Þ:
it is immediate to check that the field can be expressed through the relations E? ðr? ; zÞ ¼
oF? ; oz
Ez ðr? ; zÞ ¼ r? F? ;
ð3Þ
ð6Þ
We wish to note that our approximation basically differs from the standard Fraunhofer approximation [14] even if the regions of validity z > w2 =k are the same. In fact, the Fraunhofer approximation is based on the asymptotic expansion in the smallness parameter jr0? r? j=z ’ w=z while our expansion deals with the case w=k < 1.
A. Ciattoni et al. / Optics Communications 202 (2002) 17–20
The potential F? in Eq. (5) is the product of an outgoing spherical wave and a slowly varying factor G? . In fact, for the extreme nonparaxial field (i.e., E? ðr? ; 0Þ ¼ E0 dðr? Þ, E0 being a fixed vector), G? ¼ E0 which suggests that, for highly nonparaxial fields, G? behaves as a slowly varying function of r, and the potential F? essentially behaves as a spherical wave. This remarkable feature underlines the central role played by spherical waves in describing highly nonparaxial fields. For the opposite case of a paraxial beam, the potential F? is the product of a plane wave expðikzÞ and of a slowly varying amplitude. Comparing the potentials in these two extreme situations, we can argue that the presence of the spherical wave expðikrÞ=r in Eq. (5) is related to the large divergence angle characterizing highly nonparaxial field propagation, as much as the presence of the plane wave expðikzÞ in the paraxial potential is related to the smallness of the divergence angle in paraxial propagation. These general considerations can be made explicit to the particular case of the boundary distribution r2 E? ðr? ; 0Þ ¼ E0 exp ?2 i: ð7Þ 2w By inserting Eq. (7) in Eq. (6), we obtain Z $ G? ðr? ; zÞ ¼ iE0 d2 r0? exp r0? ðA : r0? Þ þ b r0? ; ð8Þ $
where the matrix A is $
A¼ 1=2w2 ikðy 2 þ z2 Þ=2r3 ikxy=2r3 ; ikxy=2r3 1=2w2 ikðx2 þ z2 Þ=2r3 ð9Þ $
b ¼ ikr? =r and A : r0? indicates the standard rowby-column product. The integral appearing in Eq. (8) can be analytically evaluated [15], yielding $ $ 1 1 1=2 b ðA : bÞ exp G? ðr? ; zÞ ¼ iE0 p ½det A 4 ð10Þ or, after some algebra
G? ðr? ; zÞ ¼ iE0 2pw2 Q1=2 P 1=2 exp
19
k 2 w2 r?2 ; 2P r2 ð11Þ
where QðrÞ ¼ ð1 ikw2 =rÞ and P ðr; zÞ ¼ ð1 ikw2 z2 =r3 Þ: By inserting Eq. (11) into Eq. (5) and the resulting potential in Eq. (3) we obtain for the electric field
1 z E? ðr? ; zÞ ¼ iE0 w2 ik r r Q1 z 1P 2 3 þ þ 2Q r2 2P z r i 2 2 2 h k w zr r þ 2 ?4 Q þ ð1 P Þ P r 2z expðikrÞ k 2 w2 r?2
Q1=2 P 1=2 exp ; r 2P r2
1 x Q1 x 2 þ ik Ez ðr? ; zÞ ¼ E0 w r r 2Q r2 2 2 P 1 3x k w x r?2 þ 2 ð3 P Þ 2P r2 2Pr2 Pr2 expðikrÞ k 2 w2 r?2
Q1=2 P 1=2 exp r 2P r2 ð12Þ which is the main result of this paper. Eq. (12) exhibit some relevant features which are worth enphasizing since they are peculiar of the highly nonparaxial regime. The asymptotic behavior of the field for r? ! 1 at any given plane z ¼ z0 is 2 w k 2 w2 þ ikr? ; ðikz0 Þ exp E? iE0 r? 2 2 2 w k w þ ikr? : ðikwÞ exp ð13Þ Ez E0 r? 2 Note that the transverse component E? vanishes as r?2 , while for the usual paraxial Gaussian beam it vanishes as exp½ðr? =wÞ2 =2. This is a quite general property of highly nonparaxial fields as it can be straightforwardly verified by starting from the general expression given in Eq. (5). In fact, G? becomes asymptotically constant as r? ! 1 (Oðr?1 Þ, see Eq. (6)) so that F? behaves, for
20
A. Ciattoni et al. / Optics Communications 202 (2002) 17–20
r? ! 1, as a spherical wave and, consequently, the electric field as in Eq. (13). For the sake of completeness, we show the level plots of the moduli of both the transverse and longitudinal components of the electric field (Fig. 1). They exhibit the main features of the solutions, and, in particular, they show, as expected, that longitudinal and transverse components are comparable.
References [1] M. Lax, W.H. Louisell, W. McKnight, Phys. Rev. A 11 (1975) 1365. [2] A. Yu Savchencko, B. Ya Zel’dovich, J. Opt. Soc. Am. B 13 (1996) 273. [3] A. Ciattoni, B. Crosignani, P. Di Porto, A. Yariv, J. Opt. Soc. Am. B 17 (2000) 809. [4] J.D. Joannopoulis, R.D. Meade, J.N. Winn, Photonic Crystals, Princeton University, Princeton, NJ, 1995. [5] G.A. Deschamps, Electron. Lett. 7 (1971) 684. [6] C.J.R. Sheppard, S. Saghafi, Phys. Rev. A 57 (1998) 2971. [7] C.J.R. Sheppard, S. Saghafi, J. Opt. Soc. Am. A 16 (1999) 1381. [8] C.J.R. Sheppard, S. Saghafi, Opt. Lett. 15 (1999) 1543. [9] Z. Bouchal, M. Olivik, J. Mod. Opt. 42 (1995) 1555. [10] A.L. Cullen, P.K. Yu, Proc. R. Soc. A 366 (1979) 155. [11] P. Varga, P. Torok, Opt. Commun. 152 (1998) 108. [12] See, e.g., R.K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1966, sec. 45.7. [13] We believe F? to deserve the name of potential since the electric field is invariant under the gauge transformation Fx ! Fx oy w , Fy ! Fy þ ox w, where wðx; yÞ is an arbitrary scalar function. [14] See, e.g., J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968, section 4-1. [15] See, e.g., P. Ramond, Field Theory: A Modern Primer, Addison-Wesley, UK, Inc., 1990, appendix A.
Fig. 1. Level plot of the normalized moduli jEx j=jE0 j (a1, a2, a3) and jEz j=jE0 j (b1, b2, b3) of a highly nonparaxial gaussian beam with w ¼ 0:1 lm on the planes z1 ¼ 2kw2 , z2 ¼ 2:5kw2 and z3 ¼ 3kw2 . The wavelength is k ¼ 0:5 lm.