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beam, using the equations of Richards and Wolf.11. Their treatment applies ... suming a Gaussian waist. Simon et al.20 analyzed to the first order in f beams.
C. J. R. Sheppard and S. Saghafi

Vol. 16, No. 6 / June 1999 / J. Opt. Soc. Am. A

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Electromagnetic Gaussian beams beyond the paraxial approximation C. J. R. Sheppard* and S. Saghafi† Department of Physical Optics, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia Received July 31, 1998; revised manuscript received November 24, 1998; accepted December 2, 1998 The lowest-order Gaussian beam mode is considered in a high-aperture theory based on a new variation of the complex source point model. To avoid singularities, combinations of sources and sinks are assumed. The resultant beam is a rigorous solution of Maxwell’s equations for all space that reduces to the conventional Gaussian beam in the paraxial limit and that is physically realizable. The field in the region of the waist and far from the waist is explored. It is demonstrated that direct rigorous evaluation of the field is feasible. © 1999 Optical Society of America [S0740-3232(99)01205-3] OCIS codes: 050.1940, 140.3410, 250.5500.

1. INTRODUCTION There is continuing interest in extending the treatment of beam modes beyond the paraxial approximation, in particular for the analysis of emission from lasers and of the behavior of resonant cavities. Recently we presented a scalar treatment for beam modes beyond the paraxial approximation, based on the complex source point model.1 The amplitude in the focal region was explored, and conditions for oscillation of a resonant cavity were investigated. In this model, first introduced for scalar beams by Deschamps,2 a source is assumed to be placed at an imaginary distance along the z axis. The field of a complex source point is a rigorous solution of Maxwell’s equation. Deschamps pointed out that there is a singularity in the solution, which is thus nonphysical. This approach was generalized to higher-order modes in rectangular coordinates by Shin and Felsen,3 who showed the connection with Siegman’s expansion in terms of Hermite functions of complex argument.4 Couture and Be´langer5 suggested that this theory can be generalized to the spherical harmonics. Cullen and Yu6 extended the complex source point method for the lowest-order mode to the electromagnetic case. They showed that an almost plane-polarized beam is produced by crossed electric and magnetic dipoles positioned at an imaginary distance. They gave explicit expressions for the field in the focal region and developed an approximate theory containing higher-order terms in addition to those in the conventional Gaussian beam. It has been pointed out that the presence of these sources results in the occurrence of a singularity in the focal plane at a radius equal to the confocal parameter.2,7 This singularity gives rise to no problems as long as the confocal parameter is large and the field in the focal region only is considered. We introduced1 a modified theory in which the sources are accompanied by sinks. This can be recognized as a description of Huygens’s principle: The external focused radiation excites the source, which then reradiates. In this modified theory the sin0740-3232/99/061381-06$15.00

gularities are replaced by phase singularities, which are physically realizable.8 Many authors have considered high-aperture Gaussian beams, as we briefly review in chronological order. Van Nie9 first considered correction terms to the Gaussian beam for high apertures. He considered a beam with a transverse electric field and a Gaussian cross section. Yoshida and Asakura10 calculated the field in the focal region of an aplanatic lens illuminated by a Gaussian beam, using the equations of Richards and Wolf.11 Their treatment applies only to beams truncated by the lens aperture, which can extend over a hemisphere at most. Lax et al.12 derived corrections in a power series of f, where f5

w0 2z 0

5

1 kw 0

5

1

A2kz 0

,

(1)

k 5 2 p /l, w 0 is the beam-waist radius, and z 0 is the confocal parameter, such that z 0 5 kw 0 2 /2.

(2)

They went on to assume that the transverse component of the electric field is polarized in the x direction. This assumption is consistent with fields only of the magnetic type, i.e., with polarization similar to that of a magnetic dipole. Shin and Felsen3 considered complex multipole sources in the transverse plane. They showed, using a scalar treatment, that these sources give rise to fields that in the paraxial limit are identical to Siegman’s modes of complex argument. Davis13 assumed that the magnetic vector potential is plane polarized. For such waves the transverse component of the magnetic field is plane polarized, corresponding only to a beam of the electric type, i.e., with a polarization similar to that of an electric dipole. Agrawal and Pattanayak14 considered plane-polarized electric field with a Gaussian amplitude variation in the waist. They considered only the x component of the elec© 1999 Optical Society of America

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tric field. They obtained a solution for a half-space, which included evanescent waves, and went on to assume that these waves are negligible. They claimed that this assumption is good for f , 0.4 and that the results reduce to the paraxial case for f , 0.1. Cullen and Yu6 considered the complex source point method. They assumed a combination of crossed electric and magnetic dipoles to achieve symmetrical behavior of the electric and the magnetic fields. They gave an exact solution for the field but indicated that that solution was difficult to calculate. They therefore developed a solution to second order in f and investigated the shape of the phase fronts and resonant frequencies of a cavity. Pattanayak and Agrawal14 obtained a solution in terms of Whittaker potentials, which are the magnitudes of the Hertz potentials. They assumed that both Whittaker potentials are oriented in the x direction, which would permit solutions that are polarized in any direction. However, the polarization behaviors of these beams of electric and magnetic types are different, so the symmetry between electric and magnetic fields disappears. They went on to consider Whittaker potentials of Gaussian variation. Davis and Patsakos15 assumed that the magnetic vector potential is oriented along the z axis. By assuming that its magnitude obeys the paraxial Gaussian beam behavior they generated TM beam modes and, using duality, TE modes. In the lowest-order modes, these correspond to radial and circumferential electric fields, respectively. Such beam modes were described previously by Kogelnik and Li,16 who showed that in the paraxial approximation they can be generated from plane-polarized components. Couture and Be´langer5 considered a beam in which the magnetic vector potential is transverse and exhibits an axial variation identical to that for a paraxial Gaussian beam. It should be noted that this configuration results in an electric field on axis that is modification of the comparable field for a Gaussian beam. They showed that this beam corresponds to a complex source point of magnetic vector potential. The beams are of the electric type. They went on to propose that complex multipole sources could be associated with Laguerre–Gaussian beams of complex argument. Davis and Patsakos17 showed that assuming Whittaker potentials oriented in the z direction gives rise to TM and TE modes. Agrawal and Lax7 discussed how differences in the various beams result from different boundary conditions. They assumed that the transverse component of an electric field is oriented in a constant direction, which is equivalent to assuming beams only of the magnetic type. They continued to assume a Gaussian variation in the transverse electric field in the waist. This field necessarily has evanescent components, thus restricting the validity to a half-space. They claimed that their approach is superior to that of Couture and Be´langer5 because it avoids the nonphysical singularity in the complex source point model. Takenaka et al.18 and Zauderer19 derived corrections to higher-order beams by using a scalar approach and assuming a Gaussian waist.

C. J. R. Sheppard and S. Saghafi

Simon et al.20 analyzed to the first order in f beams with plane-polarized transverse components of electric and magnetic fields. In a later paper21 they extended this analysis to second order for beams with E y and B x components zero in the far field. Varga and To¨ro¨k22,23 analyzed a beam in which the electric Hertz vector is in the x direction and has a Gaussian variation in the waist or the far field. Their beam is thus of the electric type. These theories are based on a variety of assumptions, and, although they tend to ordinary Gaussian beams for low angles of convergence, they have a variety of behaviors for high angles of convergence. The important distinction of the complex source–sink solution developed in this paper is that it is a rigorous solution of Maxwell’s equations throughout all space. The theories that assume a Gaussian variation in amplitude in the waist7,14,18–23 are valid only for a half-space, and evanescent components exist in the waist region.

2. LINEARLY POLARIZED (LP01) BEAM Cullen and Yu6 showed that an approximately planepolarized beam is produced by a combination of an electric dipole and a magnetic dipole, oriented along the x and y axes, respectively. We call this the LP01 beam, where 0 and 1 are the angular and the radial mode numbers, respectively. In the paraxial limit this beam tends to the usual TEM00 mode. It has been shown24 that the polarization in the far field for this mixed-dipole case is the same as that produced by focusing of a plane-polarized plane wave, as considered by Richards and Wolf.11 The amplitude distribution for a simple source–sink combination placed at the origin was given by Sheppard and Matthews.25 The field distribution for the mixed-dipole combination, placed at the origin, has been investigated by Sheppard and Larkin.26 The field distribution in the focal region for any axially symmetric beam can be expressed in terms of higher-order mixed multipole pairs.27 The electric field of an electric and magnetic dipole placed at the origin can be written in a compact form in terms of the functions f(kr) and g(kr), defined as f ~ kr ! 5 j 0 ~ kr ! 1 j 2 ~ kr ! 5 23 g ~ kr ! 5 j 0 ~ kr ! 2 5

1 2

F

cos kr ~ kr !

2

2

sin kr ~ kr ! 3

j 2 ~ kr !

F

G

,

G

cos kr 3 sin kr sin kr 1 , 2 2 2 kr ~ kr ! ~ kr ! 3

(3)

where j n is a spherical Bessel function of order n. Then the electric field at point x, y, z and spherical radius r, normalized to unity at the origin, is E5

H

g ~ kr ! 1 @ f ~ kr ! 2 g ~ kr !#

1 @ f ~ kr ! 2 g ~ kr !# 1

H

x2 r

2

1

i 2

J

f ~ kr ! kz i

xy r2

@ f ~ kr ! 2 g ~ kr !#

j

xz r

2

2

i 2

J

f ~ kr ! kx k,

(4)

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where k 5 2 p /l. In the complex source point model the dipoles are placed at the point z 5 iz 0 , so r should be replaced by R, given by R 5 @ x 2 1 y 2 1 ~ z 2 iz 0 ! 2 # 1/2,

(5)

and z by z 2 iz 0 . The electric field is thus expressed in closed form and can be simply calculated numerically. However, to our knowledge no plots of the behavior have been published either for the conventional dipole sources or for source– sink combinations. Figure 1 shows the variation in timeaveraged electric energy density WE 5

e 2 ^E & 2

(6)

(which might be termed intensity) along the x, y, and z axes, normalized to unity at the origin, for different values of kz 0 . For kz 0 5 0 the behavior degenerates to that previously published for the mixed-dipole wave.26 The intensity exhibits peak values displaced along the x axis as a result of the longitudinal field component: For values of kz 0 5 2 f 2 greater than approximately unity this behavior is absent. For small values of kz 0 the intensity exhibits some oscillations in the plane of the waist, but these are weak for kz 0 greater than ;2. The distribution

Fig. 2. Time-averaged electric energy density, normalized to unity at the focus, in the waist for the LP01 beam for kz 0 5 2.

Fig. 3. Transverse variation in time-averaged total energy density, normalized to unity at the focus, for different values of kz 0 .

becomes broader in all three directions (especially in z) as kz 0 becomes larger. As kz 0 becomes larger, the asymmetry in the waist decreases. The intensity in the plane of the waist for kz 0 5 2 is shown in Fig. 2. It is seen to be broader in the x direction, as is well known to occur for a highly focused plane-polarized beam.11 The distribution of the magnetic field is identical to that of the electric field but rotated through 90°.11 The time-averaged total energy density is axially symmetric and is given by WT 5

e e @ f 2 ~ kR ! 1 g 2 ~ kR !# 1 ~ kR ! 2 f 2 ~ kR ! , 2 8

(7)

which is shown for the waist, after normalizing to unity at the origin, in Fig. 3. The variation changes only slightly for values of kz 0 less than ;2. Along the z axis the electric field is E 5 @ g ~ kR ! 1 ikR f ~ kR ! /2# i,

(8)

so the phase is F 5 Arg$ g @ k ~ z 2 iz 0 !# 1 ik ~ z 2 iz 0 ! f @ k ~ z 2 iz 0 !# /2% , (9)

Fig. 1. Variation in time-averaged electric energy density, normalized to unity at the focus, for the LP01 beam along the three axes for different values of kz 0 . Dashed curves, (sin kr/kr)2.

which is shown in Fig. 4, with a phase kz suppressed, illustrating the Gouy phase anomaly. Figure 5 shows contours of constant intensity (timeaveraged electric energy density) in azimuthal planes for kz 0 5 2, showing that the beam has a smooth behavior for intensities greater than 0.1 times the peak.

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The beam is not strictly plane polarized; the field components are illustrated separately for the focal plane in Fig. 6. These values were plotted directly from Eq. (2), so the relative strengths of the cross components can be determined. It can be seen that the axial z component of field is approximately a factor of 4 smaller than the x component and that the y component is a further factor of 4 smaller. These cross components become smaller rapidly as kz 0 increases. The absolute value of the z component exhibits twofold rotational symmetry and is zero for x 5 0, and that of the y component exhibits fourfold symmetry and is zero for x 5 0 or y 5 0. At a general point in space, all three components are nonzero and complex. Thus we cannot define a single phase in a simple way. The symmetry properties are as given by Richards and Wolf.11 In the waist, R is purely

Fig. 4. Axial variation in phase for the LP01 beam for different values of kz 0 , showing the Gouy phase anomaly.

Fig. 6. Variation in the three components of the electric field strength in the waist for the LP01 beam for kz 0 5 2: (a) E x , (b) E y , (c) iE z .

Fig. 5. Contours of constant time-averaged electric energy density, normalized to unity at the focus, in meridional planes, for kz 0 5 2.

real for x 2 1 y 2 . z 0 2 and purely imaginary for x 2 1 y 2 , z 0 2 . Thus f(kR) and g(kR) are both real. The axial component of the electric field is thus in phase quadrature to the transverse component.

C. J. R. Sheppard and S. Saghafi

Vol. 16, No. 6 / June 1999 / J. Opt. Soc. Am. A

In the far field, the terms in Eq. (1) that decay most quickly can be neglected. The electric field of the outgoing component (i.e., from the source), calculated from Eqs. (3)–(5) by expanding the trigonometric functions of kR into complex exponentials and retaining only the positive exponents, is 3i E52 exp~ ikr ! exp~ kz 0 cos u !@~ 1 1 cos u 4kr 2 sin2 u cos2 f ! i 2 sin2 u sin f cos f j 2 sin u ~ 1 1 cos u ! cos f k# ,

3i E52 exp~ ikr ! exp~ kz 0 ! 4kr

3. SECOND-ORDER THEORY Cullen and Yu6 derived an approximate theory that is valid to the second order in f. In the theory presented in Section 2, the sink term is of strength exp(2f 2) and hence is negligible to second order in f. Thus the results presented by Cullen and Yu are valid. The electric field is

X H

Ex 5 1 2 f 2

3 exp@ 2kz 0 ~ 1 2 cos u !#~ 1 1 cos u ! .

(11)

The magnitude is axially symmetric; this is an important characteristic of the mixed-dipole solution. The normalized far-field intensity is shown as a polar plot in Fig. 7, where its amplitude is 1 1 cos u a~ u ! 5 exp@ 2kz 0 ~ 1 2 cos u !# 2

S

of the dipole combination. As kz 0 increases, the directionality increases as the beam approximates a paraxial Gaussian beam more closely. The angle u 0 , where the intensity in a transverse plane in the far field drops to one half of the intensity on the axis, is plotted in Fig. 8. Note that this quantity, discussed by Porras31 and by Zeng et al.,32 is somewhat misleading, because a spherically symmetric wave, as a result of the inverse square law, exhibits a value of u 0 5 45° but is, by symmetry, not directional. For the electromagnetic mixed-dipole wave the maximum value of u 0 is 37.8°.

(10)

where u and f are the spherical polar and azimuthal coordinates, respectively. The polarization is as given previously for a mixed dipole wave.28,29 It is the same as that produced by focusing a plane-polarized wave,11,30 except that the polar variation in strength is different, the field magnitude being

D

u u 5 cos exp 22kz 0 sin2 . 2 2 2

1385

1

Ey 5

2 @ 1 1 ~ iz ! /z 0 #

2

@~ x/w 0 ! 2 1 ~ y/w 0 ! 2 # 2 @ 1 1 ~ iz ! /z 0 # 3

2 f 2 ~ x/w 0 !~ y/w 0 ! @ 1 1 ~ iz ! /z 0 # 2

5 ~ x/w 0 ! 2 1 3 ~ y/w 0 ! 2

JC

@ 1 1 ~ iz ! /z 0 # 2

c,

c,

2i f ~ x/w 0 ! Ez 5 2 c, @ 1 1 ~ iz ! /z 0 # (12)

Note that Eq. (12) is valid for u in the range 0 < u < p . The beam is directional even for kz 0 5 0, which is in contrast to the scalar case1 and results from the handedness

(13)

where c is the paraxial Gaussian beam solution, given by16

c5

H

J

1 ~ x/w 0 ! 2 1 ~ y/w 0 ! 2 exp 2 exp~ ikz ! . @ 1 1 ~ iz ! /z 0 # @ 1 1 ~ iz ! /z 0 # (14)

It can be seen that the waist is not circularly symmetric. The axial field component is of order f and the crosspolarization component of order f 2 .

4. DISCUSSION

Fig. 7. Radiation pattern showing the normalized far-field intensity for different values of kz 0 .

Fig. 8. Variation in the angle at which the far-field intensity drops to one half of the on-axis value for different values of kz 0 .

The field has been investigated for a beam that is a rigorous solution of Maxwell’s equations, is valid for all space, and avoids nonphysical singularities. It reduces to the usual Gaussian beam in the paraxial limit. It was demonstrated that it is straightforward to evaluate this solution directly. Singularities were avoided by introduction of dipole sinks. It might be argued that sinks of radiation are not physical, as they could be interpreted as violating Sommerfeld’s radiation condition. However, there is no reason why a converging spherical wave cannot be generated experimentally. The radiation condition is based on the argument that a converging wave is unlikely to occur naturally and spontaneously. Another factor is that the sinks are not real but fictitious and are balanced by fictitious sources. The fact that the solution does satisfy Maxwell’s equations and approximates observed behavior well is strong justification for the model.

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ACKNOWLEDGMENTS The authors acknowledge support from the Australian Research Council and from the Science Foundation for Physics within the University of Sydney.

15.

*Also at The Australian Key Centre for Microscopy and

17.

Microanalysis, University of Sydney, New South Wales 2009, Australia. † Present address, Department of Physics, Macquarie University, Sydney, New South Wales 2109, Australia.

18.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14.

C. J. R. Sheppard and S. Saghafi, ‘‘Beam modes beyond the paraxial approximation: a scalar treatment,’’ Phys. Rev. A 57, 2971–2979 (1998). G. A. Deschamps, ‘‘Gaussian beam as a bundle of complex rays,’’ Electron. Lett. 7, 684–685 (1971). S. Y. Shin and L. Felsen, ‘‘Gaussian beam modes by multipoles with complex source points,’’ J. Opt. Soc. Am. 67, 699–700 (1977). A. E. Siegman, ‘‘Hermite–Gaussian functions of complex arguments as optical-beam eigenfunctions,’’ J. Opt. Soc. Am. 63, 1093–1094 (1973). M. Couture and P.-A. Be´langer, ‘‘From Gaussian beam to complex-source-point spherical wave,’’ Phys. Rev. A 24, 355–359 (1981). A. L. Cullen and P. K. Yu, ‘‘Complex source-point theory of the electromagnetic open resonator,’’ Proc. R. Soc. London, Ser. A 366, 155–171 (1979). G. P. Agrawal and M. Lax, ‘‘Free-space wave propagation beyond the paraxial approximation,’’ Phys. Rev. A 27, 1693–1695 (1983). J. F. Nye and M. Berry, ‘‘Dislocations in wave trains,’’ Proc. R. Soc. London, Ser. A 336, 165–190 (1974). A. G. van Nie, ‘‘Rigorous calculation of the electromagnetic field of a wave beam,’’ Philips Res. Rep. 19, 378–394 (1964). A. Yoshida and T. Asakura, ‘‘Electromagnetic field near the focus of Gaussian beams,’’ Optik (Stuttgart) 41, 281–291 (1974). B. Richards and E. Wolf, ‘‘Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,’’ Proc. R. Soc. London, Ser. A 253, 358– 379 (1959). M. Lax, W. H. Louisel, and W. B. McKnight, ‘‘From Maxwell to paraxial wave optics,’’ Phys. Rev. A 11, 1365–1370 (1975). L. W. Davis, ‘‘Theory of electromagnetic beams,’’ Phys. Rev. A 19, 1177–1179 (1979). G. P. Agrawal and D. N. Pattanayck, ‘‘Gaussian beam

16.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

propagation beyond the paraxial approximation,’’ J. Opt. Soc. Am. 69, 575–578 (1979). L. W. Davis and G. Patsakos, ‘‘TM and TE electromagnetic beams in free space,’’ Opt. Lett. 6, 22–23 (1981). H. Kogelnik and T. Li, ‘‘Laser beams and resonators,’’ Proc. IEEE 54, 1312–1329 (1966). L. W. Davis and G. Patsakos, ‘‘Comment on representation of vector electromagnetic beams,’’ Phys. Rev. A 26, 3702– 3703 (1982). T. Takenaka, M. Yokota, and O. Fukumitsu, ‘‘Propagation of light beams beyond the paraxial approximation,’’ J. Opt. Soc. Am. A 2, 826–829 (1985). E. Zauderer, ‘‘Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,’’ J. Opt. Soc. Am. A 3, 465–469 (1986). R. Simon, E. C. G. Sudarshan, and N. Mukunda, ‘‘Gaussian–Maxwell beams,’’ J. Opt. Soc. Am. A 3, 536–540 (1986). R. Simon, E. C. G. Sudarshan, and N. Mukunda, ‘‘Cross polarization in laser beams,’’ Appl. Opt. 26, 1589–1593 (1987). P. Varga and P. To¨ro¨k, ‘‘Exact and approximate solutions of Maxwell’s equations for a confocal cavity,’’ Opt. Lett. 21, 1523–1525 (1996). P. Varga and P. To¨ro¨k, ‘‘Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave equation,’’ Opt. Commun. 152, 1–3 (1998). C. J. R. Sheppard and T. Wilson, ‘‘Gaussian-beam theory of lenses with annular aperture,’’ IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978). C. J. R. Sheppard and H. J. Matthews, ‘‘Imaging in highaperture optical systems,’’ J. Opt. Soc. Am. A 4, 1354–1360 (1987). C. J. R. Sheppard and K. G. Larkin, ‘‘Optimal concentration of electromagnetic radiation,’’ J. Mod. Opt. 41, 1495–1505 (1994). C. J. R. Sheppard and P. To¨ro¨k, ‘‘Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,’’ J. Mod. Opt. 44, 803–818 (1997). J. J. Stamnes and V. Dhayalan, ‘‘Focusing of electric dipole waves,’’ Pure Appl. Phys. 5, 195–226 (1996). C. J. R. Sheppard and P. To¨ro¨k, ‘‘Electromagnetic field in the focal region of an electric dipole wave,’’ Optik (Stuttgart) 104, 175–177 (1997). C. J. R. Sheppard, ‘‘Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,’’ IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978). M. A. Porras, ‘‘The best optical beam beyond the paraxial approximation,’’ Opt. Commun. 111, 338–349 (1994). X. D. Zeng, C. H. Liang, and Y. Y. An, ‘‘Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,’’ Appl. Opt. 36, 2042–2047 (1997).