- Email: [email protected]
Polarization effects in interferograms of radial GRIN rods
1 March 2000
Optics Communications 175 Ž2000. 259–263 www.elsevier.comrlocateroptcom
Polarization effects in interferograms of radial GRIN rods ) Marcial Montoya-Hernandez , Daniel Malacara-Hernandez ´ ´ Centro de InÕestigaciones en Optica, A.C., Apdo. Postal 1-948, C.P. 37000, Loma de Bosque 115, Col. Lomas del Campestre, Leon, ´ Gto., Mexico Received 23 September 1999; received in revised form 13 December 1999; accepted 14 December 1999
Abstract Birefringence in rods with a gradient refractive index ŽGRIN rods. is caused by anisotropic stresses produced in the doping process used to introduce its inhomogeneity. This birefringence is studied using a modified Mach–Zehnder interferometer to observe the effect of a GRIN rod on circularly polarized light. Computer simulations agree roughly with measurements obtained using a GRIN rod of length 12 cm and diameter 0.267 cm. We establish the basis for separating the contributions of birefringence and wavefront aberrations from the observed fringe patterns. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.25.Hz; 42.25.Ja; 42.25.Kb; 42.25.Lc Keywords: Interferometry; GRIN rods; Polarization
1. Introduction It has been reported w1x that the nonuniformity in the doping used to produce a gradient index of refraction ŽGRIN. also produces anisotropic internal stresses. Due to these stresses the medium becomes birefringent. This anisotropy splits the incident wavefront into two orthogonally polarized wavefronts, one in the radial direction and the other in the tangential direction. The purpose of this paper is to describe the formation of an interferogram with spiral fringes using these birefringent gradient index
) Corresponding author. Tel.: q52-4773-1017; fax: q52-47175000; e-mail: [email protected]
rods. There have been efforts to evaluate the quality in the GRIN rods imaging w1x but they do not deal with the birefringence effect in the same manner. This paper establishes the basis to separate the birefringence from the gradient index of refraction. We should mention a related phenomenon, optical vortices, widely studied in some fields. An optical vortex is a screw-like wavefront like an helical staircase that has a phase singularity at its core. Its phase increases around its core from zero to 2 p n, where n is a real number often referred as the charge of the singularity. The similitude in mathematical expressions with those previously given in the literature w3–5x that describe these vortices makes them of interest to study from the point of view of this paper. These vortices have been observed in high power lasers w6,7x, in speckle patterns w8x and in optical
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 4 5 1 - X
260
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
Fig. 1. Set-up of the modified Mach–Zehnder interferometer.
fibers w2x. Also, vortices have been produced by holographic methods w7,9x and they have been researched as techniques for micro manipulation of atoms, molecules and organic cells. However, to our knowledge, there has not been enough research on the role of polarization in these phenomena, even when birefringence is clearly present in at least some laser cavities w10x and optical fibers w2x. In the same way it is possible to observe some polarization phenomena in speckle patterns w8x. 2. Experimental set-up Fig. 1 shows the diagram of a modified Mach– Zehnder interferometer. The illuminating light beam
is linearly polarized at 458 with respect to the horizontal plane. Here, BS1 is a polarizing cubic beam splitter and BS 2 is a nonpolarizing cube beam splitter. The net result is that QWR 1 and QWR 2 are illuminated with linearly polarized light. QWR 1 and QWR 2 are quarter wave retarder plates. M 1 and M 2 are mirrors. The elements SF1 and SF2 are spatial filters, the GRIN-rod is the optical system under test, and L 1 and L 2 are two identical lenses that collimate the wavefront at both the exit of the GRIN rod and the reference wavefront. The lens L 3 forms the image of the exit pupil of the GRIN rod on the CCD camera. The retarder plates change the polarization from linear to circularly polarized light. Oblique 458 incidence at mirrors M 1 and M 2 introduce a small difference in phase shift between the s and p components. In order to partially compensate this effect and to obtain circularly polarized light after reflection, the quarter wave plate retarder is slightly rotated. The set-up was used to produce the interferograms of Fig. 2, the first of which results when the polarization at the GRIN rod’s entrance pupil and the test beam at the CCD are both circular and in the same sense. In this case fringes look like any other interferogram. However, towards the edge the fringe visibility decreases due to a change in polarization caused by the birefringence of the GRIN rod. The second interferogram was obtained when the two
Fig. 2. Experimental interferograms with circularly polarized beams: Ža. in the same sense; and Žb. in the opposite sense.
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
261
Fig. 3. Vector diagram that shows the phase of the radial and tangential electric field components.
beams are circularly polarized but in opposite senses. We will demonstrate that the spiral fringes’ visibility changes due to the birefringence in the GRIN rod, with zero birefringence along the optic axis and maximum birefringence at the edge. Notice that in this interferogram the interference does not occur in the center, as expected, since the test polarization does not change in this area and remains completely orthogonal to the reference beam. This is in contrast with the first interferogram where at the pupil center interference is complete. 3. Vectorial interpretation of interferograms We shall present a mathematical description for the case in which the polarization of the reference beam and that of the entrance pupil of the GRIN rod are circular. To represent the electric vector of a circularly polarized light beam at a point on the pupil of the GRIN rod, we use polar coordinates Ž r, u ., where u is the angle measured from the y axis as in Fig. 3Ža.. Circularly polarized light can be represented by two beams plane polarized in mutually perpendicular directions, with the same magnitude and with a phase difference of 908. If the phase of the tangential component is ahead of the radial component, we speak of right-handed circular polarization, otherwise, circulation is said to be left-handed. Also, the phase of both components change linearly with the angle u as shown in Fig. 3Ža., increasing for right-handed polarization and decreasing for left handed. The phases of the radial and tangential components are as shown in Fig. 3Ža., with the positive sign for right handed light and the negative sign otherwise. The phase is v t when u s 0.
Let us now consider a right-handed circularly polarized beam after it has passed through the GRIN rod. The gradient index of refraction introduces an isotropic wavefront distortion whose phase we represent by f Ž r, u .. This phase is added to both the radial and the tangential components of the electric vector, as illustrated in Fig. 3Žb.. However, as we pointed out earlier, in addition to the gradient index of refraction, an anisotropic birefringence is introduced. To take this anisotropy into consideration, an additional phase term d Ž r, u . is introduced in the expression for the tangential component. The interference between these two circularly polarized beams is now produced by the Mach–Zehnder interferometer shown in Fig. 1. Then, in the exiting light beam the radial component ER of the electric field is: ER s a exp i w v t " u x 4 q a exp i v t q u q f Ž r , u .
4,
Ž 1.
where a is the maximum amplitude of each component. The tangential component E T is: E T s a exp i v t " Ž u q pr2 .
vtqu
4 q a exp i qpr2q f Ž r , u . q d Ž r , u . 4 .
Ž 2.
Since these two components are orthogonal, the resultant irradiance in the interferogram is the sum of the irradiances of the radial and tangential components. The radial irradiance is: IR s ² ER2 : s a2 1 q cos u . u q f Ž r , u .
4
Ž 3.
and the tangential irradiance is: I T s ² ET2 : s a2 1 q cos u . u q pr2 . pr2 qf Ž r , u . q d Ž r , u .
4.
Ž 4.
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
262
Fig. 4. Computer simulated interferograms resembling the experimental results.
Hence, the total irradiance at the interference pattern is given by:
or
I s a2 2 q cos u . u q f Ž r , u . q cos u . u
I H s 2 a 2 1 q sin
qpr2. pr2 q f Ž r , u . q d Ž r , u .
4.
Ž 5.
When both beams have the same polarization sense the upper sign is used and Eq. Ž5. becomes: I 5 s a 2 2 q cos f Ž r , u . q cos f Ž r , u . qd Ž r , u .
Ž 6.
4
or
d Ž r ,u .
½
I 5 s 2 a2 1 q cos
q
d Ž r ,u . 2
2
5
cos f Ž r , u .
.
Ž 7.
Fig. 4Ža. shows a computer simulation made using Eq. Ž7., corresponding to the experimental interferogram shown in Fig. 2Ža.. A close resemblance is evident. When both beams are circularly polarized in opposite senses, the lower sign should be used and Eq. Ž5. becomes: I H s a 2 2 q cos 2 u q f Ž r , u . ycos 2 u q f Ž r , u . q d Ž r , u .
4
Ž 8.
½
d Ž r ,u . 2
=sin 2 u q f Ž r , u . q
d Ž r ,u . 2
5
.
Ž 9.
The computer simulation in Fig. 4Žb. shows the spiral fringes that correspond to the experimental results shown in Fig. 2Žb.. The angular dependance 2 u in Eq. Ž9. closely resembles the representation of a vortex but its physical nature could be different, as will be shown next. Eq. Ž7. represents an interferogram with fringes that correspond to the sum of the gradient index and the birefringence, so that a bright fringe occurs whenever
f Ž r ,u . q
d Ž r ,u . 2
s 2 mp
Ž 10 .
where m is an integer. However, the contrast is modulated by the birefringence, that is, by cosŽ d Ž r, u .r2., explaining the maximum contrast at the center of the interferogram. As shown in Fig. 4Ža. the first minimum in the contrast occurs for d Ž r, u . s p and there is a contrast inversion for d Ž r, u . s 2 p.
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
Eq. Ž9. represents an interferogram with spiral fringes, as in Fig. 4Žb.. Again, the fringes are determined by the gradient index and the birefringence, but with an angular dependence on u that produces the spirals. The contrast is again modulated by birefringence, in this case by sinŽ d Ž r, u .r2.. This produces a contrast minimum at the center of the pattern and the first maximum occurs at d Ž r, u . s p. It is interesting to note that if there is no birefringence Ž d Ž r, u . s 0. the first fringe pattern described by Eq. Ž7. becomes a normal two wave interferogram with a phase deformation f Ž r, u .. On the other hand, the second interferogram ŽEq. Ž9.. becomes a uniform field with constant irradiance or, equivalently, a spiral fringe pattern with zero contrast. 4. Conclusions A mathematical description of interferograms obtained with GRIN rods with circularly polarized light has been presented. This paper establishes a basis for further work to evaluate the birefringence effects in GRIN-rod imaging. The light beam being measured can be assumed to be formed by two wavefronts, one with a radial polarization and the other with a tangential polarization, as pointed out before. The fringe shapes in both interferograms are described by combining the information from the two orthogonally polarized wavefronts as in Eq. Ž10.. However, both interferograms are modulated by a function of the birefringence. By measuring this contrast variation, and taking into account other possible causes for contrast variations, the two wavefront shapes can be
263
separated. Work on this subject is being carried out and will be reported in a future paper. Acknowledgements The authors would like to thank to Dr. Orestes N. Stavroudis for useful comments and acknowledge the economic support in this research to the Consejo Nacional de Ciencia y Tecnologia ŽCONACyT. with research contract No. 28472-E. References w1x W.A. Wozniak, Birefringence of gradient-index lenses of SELFOC w type, Proc. SPIE 2169 Ž1994. 156–167. w2x G. Indebetouw, Optical vortices and their propagation, J. Mod. Opt. 40 Ž1993. 73–87. w3x M. Mansuripur, E.M. Wright, Linear optical vortices, Opt. Photon. News 10 Ž1999. 40–43. w4x D. Rozas, Z.S. Sacks, G.A. Swartzlander Jr., Experimental observation of fluidlike motion of optical vortices, Phys. Rev. Lett. 79 Ž1997. 3399–3402. w5x A.E. Siegman, Lasers, University Science Books, Sausalito, 1986, pp. 689–690. w6x J.M. Vaughan, D.V. Willetts, Temporal and interference fringe analysis of TEMU01 laser modes, J. Opt. Soc. Am. 73 Ž1983. 1018–1021. w7x N.R. Heckenberg, R. McDuff, C.P. Smith, H. RubinszteinDunlop, M.J. Wegener, Laser beams with phase singularities, Opt. Quantum Electron. 24 Ž1992. S951–S962. w8x N.B. Baranova, A.V. Mamaev, N.M. Pilipetsky, V.V. Shkunov, B.Ya. Zel’dovich, Wave-front dislocations: Topological limitations for adaptive systems with phase conjugation, J. Opt. Soc. Am. 73 Ž1983. 525–528. w9x A.G. Poleshchuk et al., Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure, Appl. Opt. 38 Ž1999. 1295–1301. w10x W. Koechner, Solid-State Laser Engineering, Springer- Verlag, Berlin, 1988, p. 378.
Optics Communications 175 Ž2000. 259–263 www.elsevier.comrlocateroptcom
Polarization effects in interferograms of radial GRIN rods ) Marcial Montoya-Hernandez , Daniel Malacara-Hernandez ´ ´ Centro de InÕestigaciones en Optica, A.C., Apdo. Postal 1-948, C.P. 37000, Loma de Bosque 115, Col. Lomas del Campestre, Leon, ´ Gto., Mexico Received 23 September 1999; received in revised form 13 December 1999; accepted 14 December 1999
Abstract Birefringence in rods with a gradient refractive index ŽGRIN rods. is caused by anisotropic stresses produced in the doping process used to introduce its inhomogeneity. This birefringence is studied using a modified Mach–Zehnder interferometer to observe the effect of a GRIN rod on circularly polarized light. Computer simulations agree roughly with measurements obtained using a GRIN rod of length 12 cm and diameter 0.267 cm. We establish the basis for separating the contributions of birefringence and wavefront aberrations from the observed fringe patterns. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.25.Hz; 42.25.Ja; 42.25.Kb; 42.25.Lc Keywords: Interferometry; GRIN rods; Polarization
1. Introduction It has been reported w1x that the nonuniformity in the doping used to produce a gradient index of refraction ŽGRIN. also produces anisotropic internal stresses. Due to these stresses the medium becomes birefringent. This anisotropy splits the incident wavefront into two orthogonally polarized wavefronts, one in the radial direction and the other in the tangential direction. The purpose of this paper is to describe the formation of an interferogram with spiral fringes using these birefringent gradient index
) Corresponding author. Tel.: q52-4773-1017; fax: q52-47175000; e-mail: [email protected]
rods. There have been efforts to evaluate the quality in the GRIN rods imaging w1x but they do not deal with the birefringence effect in the same manner. This paper establishes the basis to separate the birefringence from the gradient index of refraction. We should mention a related phenomenon, optical vortices, widely studied in some fields. An optical vortex is a screw-like wavefront like an helical staircase that has a phase singularity at its core. Its phase increases around its core from zero to 2 p n, where n is a real number often referred as the charge of the singularity. The similitude in mathematical expressions with those previously given in the literature w3–5x that describe these vortices makes them of interest to study from the point of view of this paper. These vortices have been observed in high power lasers w6,7x, in speckle patterns w8x and in optical
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 4 5 1 - X
260
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
Fig. 1. Set-up of the modified Mach–Zehnder interferometer.
fibers w2x. Also, vortices have been produced by holographic methods w7,9x and they have been researched as techniques for micro manipulation of atoms, molecules and organic cells. However, to our knowledge, there has not been enough research on the role of polarization in these phenomena, even when birefringence is clearly present in at least some laser cavities w10x and optical fibers w2x. In the same way it is possible to observe some polarization phenomena in speckle patterns w8x. 2. Experimental set-up Fig. 1 shows the diagram of a modified Mach– Zehnder interferometer. The illuminating light beam
is linearly polarized at 458 with respect to the horizontal plane. Here, BS1 is a polarizing cubic beam splitter and BS 2 is a nonpolarizing cube beam splitter. The net result is that QWR 1 and QWR 2 are illuminated with linearly polarized light. QWR 1 and QWR 2 are quarter wave retarder plates. M 1 and M 2 are mirrors. The elements SF1 and SF2 are spatial filters, the GRIN-rod is the optical system under test, and L 1 and L 2 are two identical lenses that collimate the wavefront at both the exit of the GRIN rod and the reference wavefront. The lens L 3 forms the image of the exit pupil of the GRIN rod on the CCD camera. The retarder plates change the polarization from linear to circularly polarized light. Oblique 458 incidence at mirrors M 1 and M 2 introduce a small difference in phase shift between the s and p components. In order to partially compensate this effect and to obtain circularly polarized light after reflection, the quarter wave plate retarder is slightly rotated. The set-up was used to produce the interferograms of Fig. 2, the first of which results when the polarization at the GRIN rod’s entrance pupil and the test beam at the CCD are both circular and in the same sense. In this case fringes look like any other interferogram. However, towards the edge the fringe visibility decreases due to a change in polarization caused by the birefringence of the GRIN rod. The second interferogram was obtained when the two
Fig. 2. Experimental interferograms with circularly polarized beams: Ža. in the same sense; and Žb. in the opposite sense.
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
261
Fig. 3. Vector diagram that shows the phase of the radial and tangential electric field components.
beams are circularly polarized but in opposite senses. We will demonstrate that the spiral fringes’ visibility changes due to the birefringence in the GRIN rod, with zero birefringence along the optic axis and maximum birefringence at the edge. Notice that in this interferogram the interference does not occur in the center, as expected, since the test polarization does not change in this area and remains completely orthogonal to the reference beam. This is in contrast with the first interferogram where at the pupil center interference is complete. 3. Vectorial interpretation of interferograms We shall present a mathematical description for the case in which the polarization of the reference beam and that of the entrance pupil of the GRIN rod are circular. To represent the electric vector of a circularly polarized light beam at a point on the pupil of the GRIN rod, we use polar coordinates Ž r, u ., where u is the angle measured from the y axis as in Fig. 3Ža.. Circularly polarized light can be represented by two beams plane polarized in mutually perpendicular directions, with the same magnitude and with a phase difference of 908. If the phase of the tangential component is ahead of the radial component, we speak of right-handed circular polarization, otherwise, circulation is said to be left-handed. Also, the phase of both components change linearly with the angle u as shown in Fig. 3Ža., increasing for right-handed polarization and decreasing for left handed. The phases of the radial and tangential components are as shown in Fig. 3Ža., with the positive sign for right handed light and the negative sign otherwise. The phase is v t when u s 0.
Let us now consider a right-handed circularly polarized beam after it has passed through the GRIN rod. The gradient index of refraction introduces an isotropic wavefront distortion whose phase we represent by f Ž r, u .. This phase is added to both the radial and the tangential components of the electric vector, as illustrated in Fig. 3Žb.. However, as we pointed out earlier, in addition to the gradient index of refraction, an anisotropic birefringence is introduced. To take this anisotropy into consideration, an additional phase term d Ž r, u . is introduced in the expression for the tangential component. The interference between these two circularly polarized beams is now produced by the Mach–Zehnder interferometer shown in Fig. 1. Then, in the exiting light beam the radial component ER of the electric field is: ER s a exp i w v t " u x 4 q a exp i v t q u q f Ž r , u .
4,
Ž 1.
where a is the maximum amplitude of each component. The tangential component E T is: E T s a exp i v t " Ž u q pr2 .
vtqu
4 q a exp i qpr2q f Ž r , u . q d Ž r , u . 4 .
Ž 2.
Since these two components are orthogonal, the resultant irradiance in the interferogram is the sum of the irradiances of the radial and tangential components. The radial irradiance is: IR s ² ER2 : s a2 1 q cos u . u q f Ž r , u .
4
Ž 3.
and the tangential irradiance is: I T s ² ET2 : s a2 1 q cos u . u q pr2 . pr2 qf Ž r , u . q d Ž r , u .
4.
Ž 4.
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
262
Fig. 4. Computer simulated interferograms resembling the experimental results.
Hence, the total irradiance at the interference pattern is given by:
or
I s a2 2 q cos u . u q f Ž r , u . q cos u . u
I H s 2 a 2 1 q sin
qpr2. pr2 q f Ž r , u . q d Ž r , u .
4.
Ž 5.
When both beams have the same polarization sense the upper sign is used and Eq. Ž5. becomes: I 5 s a 2 2 q cos f Ž r , u . q cos f Ž r , u . qd Ž r , u .
Ž 6.
4
or
d Ž r ,u .
½
I 5 s 2 a2 1 q cos
q
d Ž r ,u . 2
2
5
cos f Ž r , u .
.
Ž 7.
Fig. 4Ža. shows a computer simulation made using Eq. Ž7., corresponding to the experimental interferogram shown in Fig. 2Ža.. A close resemblance is evident. When both beams are circularly polarized in opposite senses, the lower sign should be used and Eq. Ž5. becomes: I H s a 2 2 q cos 2 u q f Ž r , u . ycos 2 u q f Ž r , u . q d Ž r , u .
4
Ž 8.
½
d Ž r ,u . 2
=sin 2 u q f Ž r , u . q
d Ž r ,u . 2
5
.
Ž 9.
The computer simulation in Fig. 4Žb. shows the spiral fringes that correspond to the experimental results shown in Fig. 2Žb.. The angular dependance 2 u in Eq. Ž9. closely resembles the representation of a vortex but its physical nature could be different, as will be shown next. Eq. Ž7. represents an interferogram with fringes that correspond to the sum of the gradient index and the birefringence, so that a bright fringe occurs whenever
f Ž r ,u . q
d Ž r ,u . 2
s 2 mp
Ž 10 .
where m is an integer. However, the contrast is modulated by the birefringence, that is, by cosŽ d Ž r, u .r2., explaining the maximum contrast at the center of the interferogram. As shown in Fig. 4Ža. the first minimum in the contrast occurs for d Ž r, u . s p and there is a contrast inversion for d Ž r, u . s 2 p.
M. Montoya-Hernandez, D. Malacara-Hernandezr Optics Communications 175 (2000) 259–263 ´ ´
Eq. Ž9. represents an interferogram with spiral fringes, as in Fig. 4Žb.. Again, the fringes are determined by the gradient index and the birefringence, but with an angular dependence on u that produces the spirals. The contrast is again modulated by birefringence, in this case by sinŽ d Ž r, u .r2.. This produces a contrast minimum at the center of the pattern and the first maximum occurs at d Ž r, u . s p. It is interesting to note that if there is no birefringence Ž d Ž r, u . s 0. the first fringe pattern described by Eq. Ž7. becomes a normal two wave interferogram with a phase deformation f Ž r, u .. On the other hand, the second interferogram ŽEq. Ž9.. becomes a uniform field with constant irradiance or, equivalently, a spiral fringe pattern with zero contrast. 4. Conclusions A mathematical description of interferograms obtained with GRIN rods with circularly polarized light has been presented. This paper establishes a basis for further work to evaluate the birefringence effects in GRIN-rod imaging. The light beam being measured can be assumed to be formed by two wavefronts, one with a radial polarization and the other with a tangential polarization, as pointed out before. The fringe shapes in both interferograms are described by combining the information from the two orthogonally polarized wavefronts as in Eq. Ž10.. However, both interferograms are modulated by a function of the birefringence. By measuring this contrast variation, and taking into account other possible causes for contrast variations, the two wavefront shapes can be
263
separated. Work on this subject is being carried out and will be reported in a future paper. Acknowledgements The authors would like to thank to Dr. Orestes N. Stavroudis for useful comments and acknowledge the economic support in this research to the Consejo Nacional de Ciencia y Tecnologia ŽCONACyT. with research contract No. 28472-E. References w1x W.A. Wozniak, Birefringence of gradient-index lenses of SELFOC w type, Proc. SPIE 2169 Ž1994. 156–167. w2x G. Indebetouw, Optical vortices and their propagation, J. Mod. Opt. 40 Ž1993. 73–87. w3x M. Mansuripur, E.M. Wright, Linear optical vortices, Opt. Photon. News 10 Ž1999. 40–43. w4x D. Rozas, Z.S. Sacks, G.A. Swartzlander Jr., Experimental observation of fluidlike motion of optical vortices, Phys. Rev. Lett. 79 Ž1997. 3399–3402. w5x A.E. Siegman, Lasers, University Science Books, Sausalito, 1986, pp. 689–690. w6x J.M. Vaughan, D.V. Willetts, Temporal and interference fringe analysis of TEMU01 laser modes, J. Opt. Soc. Am. 73 Ž1983. 1018–1021. w7x N.R. Heckenberg, R. McDuff, C.P. Smith, H. RubinszteinDunlop, M.J. Wegener, Laser beams with phase singularities, Opt. Quantum Electron. 24 Ž1992. S951–S962. w8x N.B. Baranova, A.V. Mamaev, N.M. Pilipetsky, V.V. Shkunov, B.Ya. Zel’dovich, Wave-front dislocations: Topological limitations for adaptive systems with phase conjugation, J. Opt. Soc. Am. 73 Ž1983. 525–528. w9x A.G. Poleshchuk et al., Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure, Appl. Opt. 38 Ž1999. 1295–1301. w10x W. Koechner, Solid-State Laser Engineering, Springer- Verlag, Berlin, 1988, p. 378.