Solution and focus property of the nonparaxial vector beams in the parabolic coordinates

PENG Ji; CUI Zhifeng; QU Jun

doi:10.7510/jgjs.issn.1001-3806.2014.05.027

Volume 38 Issue 5

Oct. 2014

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Solution and focus property of the nonparaxial vector beams in the parabolic coordinates

1.
College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, China

Corresponding author: QU Jun, qujun70@mail.ahnu.edu.cn
Received Date: 2013-10-11
Accepted Date: 2013-11-13

Abstract

In order to solve the nonparaxial vector wave equation in the cylindrical coordinates and obtain electric field expression of the beams, based on the electric field along the azimuthal polarization under the axisymmetric circumstance, the vector wave equation under the nonparaxiality similar circumstances was transformed to the parabolic coordinates and was solved appropriately with the separation variables method. The corresponding numerical calculation was made. The results show that the new analytical solution of the nonparaxial vector wave equation is discussed to describe the propagation of a laser beam. The electric field of such a beam is found to be based on the solutions of the confluent hypergeometric function and the Meijer functions. The intensity distribution of beam is similar to the first-class zero-order Bessel beam mode. The intensity of the light beam near the optical axis is nearly infinite, and decays rapidly along the peripheral direction and decreases sharply along the radial direction in the focal plane. The acquired results are of certain significance for exploring the propagation properties of vector beams in case of nonparaxial approximation.
- laser optics,
- nonparaxial vector wave equation,
- coordinate transformation,
- confluent hypergeometric function,
- Meijer function

References

[1]	HAYAZAWA N, SAITO Y, KAWATA S. Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy[J]. Applied Physics Letters, 2004, 85(25): 6239-6241.
[2]	ZHANG Zh H, LI Ch G, LI J Zh, et al. Phase compensation in lensless Fourier transform digital holography [J]. Laser Technology, 2013, 37(5): 569-600(in Chinese).
[3]	ZHAN Q W. Trapping metallic Rayleigh particles with radial polarization[J]. Optics Express, 2004, 12(15): 3377-3382.
[4]	MASAKI M, TERUTAKE H, YASUHIRO T. Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam[J]. Applied Optics, 2009, 48(32): 6143-6151.
[5]	SILER M, JAKL P, BRZOBOHATY O, et al. Optical forces induced behavior of a particle in a non-diffracting vortex beam[J]. Optics Express, 2012, 20(22): 24304-24319.
[6]	ZHOU P, MA Y X, WANG X L, et al. Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere[J]. Applied Optics, 2009, 48(28): 5251-5258.
[7]	GE X L, FENG X X, FAN Ch Y. Progress of the study of phase discontinuity of laser propagation through atmosphere[J]. Laser Technology, 2012, 36(4): 485-489(in Chinese).
[8]	ZHOU P, LIU Z J, XU X J, et al. Propagation of phase-locked partially coherent flattened beam array in turbulent atmosphere[J]. Optics and Lasers in Engineering, 2009, 47(11): 1254-1258.
[9]	ZHU Zh W, SU Zh P. Spectral change of J0-correlated partially coherent flat-topped beam in turbulent atmosphere[J]. Laser Technology, 2012, 36(4): 532-535(in Chinese).
[10]	XU H F, CUI Zh F, QU J. Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence[J]. Optics Express, 2011, 19(22): 21163-21173.
[11]	WANG L, SHEN X J, ZHANG W A, et al. Analysis of spectral propagating properties of Gaussian beam[J]. Laser Technology, 2012, 36(5): 700-703(in Chinese).
[12]	XU H F, LUO H, CUI Zh F, et al. Polarization characteristics of partially coherent elegant Laguerre-Gaussian beams in non-Kolmogorov turbulence[J]. Optics and Lasers in Engineering, 2012, 50(5): 760-766.
[13]	WANG B, FEI J Ch, CUI Zh F, et al. Reserch of degree of polarization of PCELG beam propagating through a circular aperture[J].Laser Technology, 2013, 37(5): 672-678(in Chinese).
[14]	BARREIRO J T, WEI T Ch, KWIAT P G. Remote preparation of single-photon "Hybrid" entangled and vector-polarization states[J]. Physical Review Letters, 2010, 105(3): 030407.
[15]	LI X P, CAO Y Y, GU M. Superresolution-focal-volume induced 3.0Tbytes/disk capacity by focusing a radially polarized beam[J]. Optics Letters, 2011, 36(13): 2510-2512.
[16]	ZHAN Q W. Cylindrical vector beams: from mathematical concepts to applications[J]. Advances in Optics and Photonics, 2009, 1(1): 1-57.
[17]	MIGUEL A B, JULIO C G V, SABINO Ch C. Parabolic nondiffracting optical wave fields[J]. Optics Letters, 2004, 29(1): 44-46.
[18]	BORN M, WOLF E. Principles of optics [M]. 7th ed. Cambridgeshire,United Kingdom: Cambridge University Press, 1999: 11-19.
[19]	DURNIN J, MICELI J J, EBERLY J H. Diffraction-free beams[J]. Physical Review Letters, 1987, 58(15): 1499-1501.
[20]	KHONINA S N, KOTLYAR V V, SKIDANOV R V, et al. Rotation of microparticles with Bessel beams generated by diffractive elements[J]. Journal of Modern Optics, 2004, 51(14): 2167-2184.
[21]	GUTIERREZ-VEGA J C, ITURBE-CASTILLO M D, CHAVEZ-CERDA S. Alternative formulation for invariant optical fields: Mathieu beams[J]. Optics Letters, 2000, 25(20): 1493-1495.
[22]	KOGELNIK H, LI T. Laser beams and resonators[J]. Proceedings of the IEEE, 1966, 54(10): 1312-1329.
[23]	DUAN K L, LV B D. Application of the Wigner distribution function to complex-argument Hermite- and Laguerre-Gaussian beams beyond the paraxial approximation[J]. Optics & Laser Technology, 2007, 39(1): 110-115.
[24]	SESHADRI S R. Self-interaction and mutual interaction of complex-argument Laguerre-Gauss beams[J]. Optics Letters, 2006, 31(5): 619-621.
[25]	KOSTENBAUDER A, SUN Y, SIEGMAN A E. Eigenmode expansions using biorthogonal functions: complex-valued Hermite-Gaussians: reply to comment[J]. Journal of the Optical Society of America, 2006,A23(6):1528-1529.
[26]	HALL D G. Vector-beam solutions of Maxwell’s wave equation[J]. Optics Letters, 1996, 21(1): 9-11.
[27]	XIN J T, GAO Ch Q, LI Ch. Combination of Hermit-Gaussian beams to arbitery order vector beams[J]. Scientia Sinica Physica Mechanica & Astronomica, 2012, 42(10): 1017-1021(in Chinese).
[28]	LIM B C, PHUA P B, LAI W J, et al. Fast switchable electro-optic radial polarization retarder[J]. Optics Letters, 2008, 33(9): 950-952.
[29]	TIDWELL S C, DENNIS H F, WAYNE D K. Generating radially polarized beams interferometrically[J]. Applied Optics, 1990, 29(15): 2234-2239.
[30]	MAURER C, JESACHER A, FURHAPTER S, et al. Tailoring of arbitrary optical vector beams[J]. New Journal of Physics, 2007, 9(3): 78.
[31]	WANG X L, DING J P, NI W J, et al. Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement[J]. Optics Letters,2007,32(24):3549-3551.
[32]	KOTLYAR V V, SKIDANOV R V, KHONINA S N, et al. Hypergeometric modes[J]. Optics Letters, 2007, 32(7): 742-744.
[33]	KARIMI E,ZITO G, PICCIRILLO B, et al. Hypergeometric-Gaussian modes[J]. Optics Letters, 2007, 32(21): 3053-3055.
[34]	KOTIYAR V V, KOVALEV A A, SOIFER V A. Hankel-Bessel laser beams[J]. Journal of the Optical Society of America, 2012,A29(5):741-747.
[35]	ABRAMOWITZ M, STEGUN I. Handbook of mathematical functions[M]. 9th ed. New York,USA:Dover Publishing Inc, 1970: 504-510.

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通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

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Solution and focus property of the nonparaxial vector beams in the parabolic coordinates

Corresponding author: QU Jun, qujun70@mail.ahnu.edu.cn

1. College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, China

Received Date: 2013-10-11

Accepted Date: 2013-11-13

Available Online: 2014-09-25

Abstract: In order to solve the nonparaxial vector wave equation in the cylindrical coordinates and obtain electric field expression of the beams, based on the electric field along the azimuthal polarization under the axisymmetric circumstance, the vector wave equation under the nonparaxiality similar circumstances was transformed to the parabolic coordinates and was solved appropriately with the separation variables method. The corresponding numerical calculation was made. The results show that the new analytical solution of the nonparaxial vector wave equation is discussed to describe the propagation of a laser beam. The electric field of such a beam is found to be based on the solutions of the confluent hypergeometric function and the Meijer functions. The intensity distribution of beam is similar to the first-class zero-order Bessel beam mode. The intensity of the light beam near the optical axis is nearly infinite, and decays rapidly along the peripheral direction and decreases sharply along the radial direction in the focal plane. The acquired results are of certain significance for exploring the propagation properties of vector beams in case of nonparaxial approximation.

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Reference (35)

[1]	HAYAZAWA N, SAITO Y, KAWATA S. Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy[J]. Applied Physics Letters, 2004, 85(25): 6239-6241.
[2]	ZHANG Zh H, LI Ch G, LI J Zh, et al. Phase compensation in lensless Fourier transform digital holography [J]. Laser Technology, 2013, 37(5): 569-600(in Chinese).
[3]	ZHAN Q W. Trapping metallic Rayleigh particles with radial polarization[J]. Optics Express, 2004, 12(15): 3377-3382.
[4]	MASAKI M, TERUTAKE H, YASUHIRO T. Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam[J]. Applied Optics, 2009, 48(32): 6143-6151.
[5]	SILER M, JAKL P, BRZOBOHATY O, et al. Optical forces induced behavior of a particle in a non-diffracting vortex beam[J]. Optics Express, 2012, 20(22): 24304-24319.
[6]	ZHOU P, MA Y X, WANG X L, et al. Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere[J]. Applied Optics, 2009, 48(28): 5251-5258.
[7]	GE X L, FENG X X, FAN Ch Y. Progress of the study of phase discontinuity of laser propagation through atmosphere[J]. Laser Technology, 2012, 36(4): 485-489(in Chinese).
[8]	ZHOU P, LIU Z J, XU X J, et al. Propagation of phase-locked partially coherent flattened beam array in turbulent atmosphere[J]. Optics and Lasers in Engineering, 2009, 47(11): 1254-1258.
[9]	ZHU Zh W, SU Zh P. Spectral change of J0-correlated partially coherent flat-topped beam in turbulent atmosphere[J]. Laser Technology, 2012, 36(4): 532-535(in Chinese).
[10]	XU H F, CUI Zh F, QU J. Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence[J]. Optics Express, 2011, 19(22): 21163-21173.
[11]	WANG L, SHEN X J, ZHANG W A, et al. Analysis of spectral propagating properties of Gaussian beam[J]. Laser Technology, 2012, 36(5): 700-703(in Chinese).
[12]	XU H F, LUO H, CUI Zh F, et al. Polarization characteristics of partially coherent elegant Laguerre-Gaussian beams in non-Kolmogorov turbulence[J]. Optics and Lasers in Engineering, 2012, 50(5): 760-766.
[13]	WANG B, FEI J Ch, CUI Zh F, et al. Reserch of degree of polarization of PCELG beam propagating through a circular aperture[J].Laser Technology, 2013, 37(5): 672-678(in Chinese).
[14]	BARREIRO J T, WEI T Ch, KWIAT P G. Remote preparation of single-photon "Hybrid" entangled and vector-polarization states[J]. Physical Review Letters, 2010, 105(3): 030407.
[15]	LI X P, CAO Y Y, GU M. Superresolution-focal-volume induced 3.0Tbytes/disk capacity by focusing a radially polarized beam[J]. Optics Letters, 2011, 36(13): 2510-2512.
[16]	ZHAN Q W. Cylindrical vector beams: from mathematical concepts to applications[J]. Advances in Optics and Photonics, 2009, 1(1): 1-57.
[17]	MIGUEL A B, JULIO C G V, SABINO Ch C. Parabolic nondiffracting optical wave fields[J]. Optics Letters, 2004, 29(1): 44-46.
[18]	BORN M, WOLF E. Principles of optics [M]. 7th ed. Cambridgeshire,United Kingdom: Cambridge University Press, 1999: 11-19.
[19]	DURNIN J, MICELI J J, EBERLY J H. Diffraction-free beams[J]. Physical Review Letters, 1987, 58(15): 1499-1501.
[20]	KHONINA S N, KOTLYAR V V, SKIDANOV R V, et al. Rotation of microparticles with Bessel beams generated by diffractive elements[J]. Journal of Modern Optics, 2004, 51(14): 2167-2184.
[21]	GUTIERREZ-VEGA J C, ITURBE-CASTILLO M D, CHAVEZ-CERDA S. Alternative formulation for invariant optical fields: Mathieu beams[J]. Optics Letters, 2000, 25(20): 1493-1495.
[22]	KOGELNIK H, LI T. Laser beams and resonators[J]. Proceedings of the IEEE, 1966, 54(10): 1312-1329.
[23]	DUAN K L, LV B D. Application of the Wigner distribution function to complex-argument Hermite- and Laguerre-Gaussian beams beyond the paraxial approximation[J]. Optics & Laser Technology, 2007, 39(1): 110-115.
[24]	SESHADRI S R. Self-interaction and mutual interaction of complex-argument Laguerre-Gauss beams[J]. Optics Letters, 2006, 31(5): 619-621.
[25]	KOSTENBAUDER A, SUN Y, SIEGMAN A E. Eigenmode expansions using biorthogonal functions: complex-valued Hermite-Gaussians: reply to comment[J]. Journal of the Optical Society of America, 2006,A23(6):1528-1529.
[26]	HALL D G. Vector-beam solutions of Maxwell’s wave equation[J]. Optics Letters, 1996, 21(1): 9-11.
[27]	XIN J T, GAO Ch Q, LI Ch. Combination of Hermit-Gaussian beams to arbitery order vector beams[J]. Scientia Sinica Physica Mechanica & Astronomica, 2012, 42(10): 1017-1021(in Chinese).
[28]	LIM B C, PHUA P B, LAI W J, et al. Fast switchable electro-optic radial polarization retarder[J]. Optics Letters, 2008, 33(9): 950-952.
[29]	TIDWELL S C, DENNIS H F, WAYNE D K. Generating radially polarized beams interferometrically[J]. Applied Optics, 1990, 29(15): 2234-2239.
[30]	MAURER C, JESACHER A, FURHAPTER S, et al. Tailoring of arbitrary optical vector beams[J]. New Journal of Physics, 2007, 9(3): 78.
[31]	WANG X L, DING J P, NI W J, et al. Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement[J]. Optics Letters,2007,32(24):3549-3551.
[32]	KOTLYAR V V, SKIDANOV R V, KHONINA S N, et al. Hypergeometric modes[J]. Optics Letters, 2007, 32(7): 742-744.
[33]	KARIMI E,ZITO G, PICCIRILLO B, et al. Hypergeometric-Gaussian modes[J]. Optics Letters, 2007, 32(21): 3053-3055.
[34]	KOTIYAR V V, KOVALEV A A, SOIFER V A. Hankel-Bessel laser beams[J]. Journal of the Optical Society of America, 2012,A29(5):741-747.
[35]	ABRAMOWITZ M, STEGUN I. Handbook of mathematical functions[M]. 9th ed. New York,USA:Dover Publishing Inc, 1970: 504-510.

Solution and focus property of the nonparaxial vector beams in the parabolic coordinates

Abstract

References

Proportional views

通讯作者: 陈斌, bchen63@163.com

Proportional views