Abstract

This study reports a possible first systematic approach to the selective excitations of all Mathieu-Gauss modes (MGMs) in end-pumped solid-state lasers with a new kind of axicon-based stable laser resonator. The study classifies MGMs into two categories, and explores and verifies the approach to excite each MGM category using numerical simulations. Controlling both the “cavity mode gain” and the “cavity conical asymmetry” of the axicon-based stable laser resonator achieves the proposed selective MGM-excitation approach.

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
    [CrossRef]
  2. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
    [CrossRef]
  4. C. A. Dartora and H. E. Hernández-Figueroa, “Properties of a localized Mathieu pulse,” J. Opt. Soc. Am. A 21(4), 662–667 (2004).
    [CrossRef]
  5. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett. 31(2), 238–240 (2006).
    [CrossRef] [PubMed]
  6. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14(9), 4182–4187 (2006).
    [CrossRef] [PubMed]
  7. F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [CrossRef]
  8. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
    [CrossRef]
  9. V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
    [CrossRef]
  10. Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84(4), 423–430 (2006).
    [CrossRef]
  11. E. Mcleod, AndC. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
    [CrossRef] [PubMed]
  12. K. Wang, L. Zeng, and Ch. Yin, “Influence of the incident wave-front on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216(1-3), 99–103 (2003).
    [CrossRef]
  13. L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18(23), 24287–24292 (2010).
    [CrossRef] [PubMed]
  14. V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
    [CrossRef]
  15. M. B. Alvarez-Elizondo, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, “Generation of Mathieu-Gauss modes with an axicon-based laser resonator,” Opt. Express 16(23), 18770–18775 (2008).
    [CrossRef]
  16. K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6(9), 635–638 (2009).
    [CrossRef]
  17. The Language of Technical Computing, See http://www.mathworks.com/ .
  18. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel–Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A 20(11), 2113–2122 (2003).
    [CrossRef]
  19. S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 (2007).
    [CrossRef] [PubMed]
  20. M. Endo, M. Kawakami, K. Nanri, S. Takeda, and T. Fujioka, “Two-dimensional Simulation of an Unstable Resonator with a Stable Core,” Appl. Opt. 38(15), 3298–3307 (1999).
    [CrossRef]
  21. M. Endo, “Numerical simulation of an optical resonator for generation of a doughnut-like laser beam,” Opt. Express 12(9), 1959–1965 (2004).
    [CrossRef] [PubMed]
  22. A. Bhowmik, “Closed-cavity solutions with partially coherent fields in the space-frequency domain,” Appl. Opt. 22(21), 3338 (1983).
    [CrossRef] [PubMed]
  23. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004), Chap. 4.
  24. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004) pp.97–101.
  25. T. Ohtomo, K. Kamikariya, K. Otsuka, and S. -C. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15(17), 10705–10717 (2007).
    [CrossRef] [PubMed]

2010

2009

K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6(9), 635–638 (2009).
[CrossRef]

2008

E. Mcleod, AndC. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[CrossRef] [PubMed]

E. Mcleod, AndC. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[CrossRef] [PubMed]

M. B. Alvarez-Elizondo, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, “Generation of Mathieu-Gauss modes with an axicon-based laser resonator,” Opt. Express 16(23), 18770–18775 (2008).
[CrossRef]

2007

2006

2005

2004

2003

J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel–Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A 20(11), 2113–2122 (2003).
[CrossRef]

K. Wang, L. Zeng, and Ch. Yin, “Influence of the incident wave-front on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216(1-3), 99–103 (2003).
[CrossRef]

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

2000

1999

1987

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

1983

Alvarez-Elizondo, M. B.

Ambrosio, L. A.

Arnold, C. B.

E. Mcleod, AndC. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[CrossRef] [PubMed]

Bandres, M. A.

Bhowmik, A.

Bouchal, Z.

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
[CrossRef]

Celechovsky, R.

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
[CrossRef]

Chávez-Cerda, S.

Chu, S. -C.

Chu, S.-C.

K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6(9), 635–638 (2009).
[CrossRef]

S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 (2007).
[CrossRef] [PubMed]

Dartora, C. A.

Dholakia, K.

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14(9), 4182–4187 (2006).
[CrossRef] [PubMed]

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

Dultz, W.

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Egorov, A. A.

Endo, M.

Fujioka, T.

Garce´s-Cha´vez, V.

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Gutiérrez-Vega, J. C.

Hernández-Figueroa, H. E.

Hsiao, H.-Y.

K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6(9), 635–638 (2009).
[CrossRef]

Inoue, T.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84(4), 423–430 (2006).
[CrossRef]

Iturbe-Castillo, M. D.

Kamikariya, K.

Kartashov, Y. V.

Kawakami, M.

Kizuka, Y.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84(4), 423–430 (2006).
[CrossRef]

Kolka, Z.

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
[CrossRef]

Kollarova, V.

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
[CrossRef]

López-Mariscal, C.

Matsuoka, Y.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84(4), 423–430 (2006).
[CrossRef]

McGloin, D.

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

Mcleod, E.

E. Mcleod, AndC. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[CrossRef] [PubMed]

Medrik, T.

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
[CrossRef]

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Milne, G.

Nanri, K.

Ohtomo, T.

K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6(9), 635–638 (2009).
[CrossRef]

T. Ohtomo, K. Kamikariya, K. Otsuka, and S. -C. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15(17), 10705–10717 (2007).
[CrossRef] [PubMed]

Otsuka, K.

Padgett, M. J.

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Rodríguez-Masegosa, R.

Schmitzer, H.

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

Takeda, S.

Tokunaga, K.

K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6(9), 635–638 (2009).
[CrossRef]

Torner, L.

Vysloukh, V. A.

Wang, K.

K. Wang, L. Zeng, and Ch. Yin, “Influence of the incident wave-front on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216(1-3), 99–103 (2003).
[CrossRef]

Wilfert, O.

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
[CrossRef]

Yin, Ch.

K. Wang, L. Zeng, and Ch. Yin, “Influence of the incident wave-front on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216(1-3), 99–103 (2003).
[CrossRef]

Zeng, L.

K. Wang, L. Zeng, and Ch. Yin, “Influence of the incident wave-front on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216(1-3), 99–103 (2003).
[CrossRef]

Appl. Opt.

Appl. Phys., A Mater. Sci. Process.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84(4), 423–430 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Laser Phys. Lett.

K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6(9), 635–638 (2009).
[CrossRef]

Nat. Nanotechnol.

E. Mcleod, AndC. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[CrossRef] [PubMed]

Opt. Commun.

K. Wang, L. Zeng, and Ch. Yin, “Influence of the incident wave-front on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216(1-3), 99–103 (2003).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

V. Garce´s-Cha´vez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia,“Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[CrossRef]

Proc. SPIE

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C, 67361C-9 (2007).
[CrossRef]

Other

The Language of Technical Computing, See http://www.mathworks.com/ .

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004), Chap. 4.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004) pp.97–101.

Supplementary Material (3)

» Media 1: MOV (1322 KB)     
» Media 2: MOV (613 KB)     
» Media 3: MOV (998 KB)     

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Figures (14)

Fig. 1
Fig. 1

(a) Amplitude distributions of an m=2, q=5 even Mathieu-Gauss mode (MGM) at different z planes: z=0, z=0.6zmax and z=1.2zmax, (b) propagation of the amplitude patterns along the planes (x, z) and (y, z) in the range [-1.6 zmax, 1.6 zmax], (c) spectrum of an analytical m=2, q=5 MGM. The square window sizes of all figures in Fig. (a) are dimensions of 6w0 ×6w0 .

Fig. 2
Fig. 2

Simulation model of the proposed axicon-based stable laser resonator

Fig. 3
Fig. 3

Transverse amplitude distributions of Mathieu-Gauss modes (a) q=0 MGM, and (b) q=4.5 MGMs.

Fig. 4
Fig. 4

Relative effective gain region at the laser crystal to excite q=0 MG modes. Red circles in Fig. (b) indicate the position of target spots of each MGM.

Fig. 5
Fig. 5

Resulting oscillation field patterns in the cavity from simulations with the effective gain region situated according to the parameters in Table 1.

Fig. 6
Fig. 6

Relative effective gain region at the laser crystal to excite q>0 MG modes. Red ellipses in Fig. (b) indicate the position of target spots of each MGM.

Fig. 7
Fig. 7

(a) Cross-section of the bi-conical lens on the y-z plane. (b) The profiles of the reflective asymmetric conical surface C1 on the plane (y,z) and plane (x,z).

Fig. 8
Fig. 8

Resulting oscillation field patterns in the cavity from simulations with the effective gain region situated according to parameters in Table 2.

Fig. 9
Fig. 9

Movie of resulting progress of stable amplitude distribution of (a) q=0, m=3 even MGM (Media 1), (b) q>0, m=3 even MGM (Media 2) and (c) q>0, m=3 odd MGM (Media 3).

Fig. 10
Fig. 10

Propagation of amplitude profile along plane (x, z) or plane (y, z) of (a) q=0, m=2 even MG modes, (b) q>0, m=3 even MG modes, and (c) q>0, m=3 odd MG modes. The symbol d denotes the MGM propagation distance from the conical surface C2 .

Fig. 11
Fig. 11

(a) Transverse amplitude of lasing MG modes from the proposed MGM excitation scheme from simulation, (b) The spectrum of lasing MGMs shown in figure (a), and (c) The corresponding analytical spectrum of each MGM in figure (a).

Fig. 12
Fig. 12

Amplitude distribution and spectrum of resulting lasing q>0 MG modes from the proposed asymmetrical axicon-based stable resonator of different cavity asymmetry parameters σ and the corresponding analytical results: (a) σ=0.99, (b) σ=0.98, and (c) σ=0.97.

Fig. 13
Fig. 13

Amplitude distributions and spectrums of resulting lasing high order MG modes from the proposed asymmetric axicon-based stable resonator (a) σ=1, m=5, q=0 MGM, (b) σ=0.98, m=5, q=4.5 even MGM, and (c) σ=0.98, m = 5, q=4.5 odd MGM.

Fig. 14
Fig. 14

Normalized intensity distributions along x-axis at laser crystal of analytical q=4.5 even MGMs of orders m=1 to m=8. The intensities are normalized by their target spot intensity.

Tables (2)

Tables Icon

Table 1 Estimated pumping beam off-axis distance x, y and effective gain region size at the crystal to q=0 MGMs excitation a

Tables Icon

Table 2 Estimated pumping beam off-axis distance x,y and effective gain region size at the crystal to q>0 MGMs excitation a

Equations (8)

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MG m e ( r ) = exp ( i k t 2 2 k z μ ) GB ( r ) Je m ( ξ , q ) ce m ( η , q )
MG m o ( r ) = exp ( i k t 2 2 k z μ ) GB ( r ) Jo m ( ξ , q ) co m ( η , q ) .
E j + 1 ( x , y ) ~ E j ( x , y ) ,
z r ( x , y ) = ρ × ( σ × x ) 2 + y 2 .
z t ( x , y ) = n + 1 n 1 × z i ( x , y ) + t ,
BG m ( r ) = exp ( i k t 2 2 k z μ ) GB ( r ) J m ( k t r μ ) exp ( i m φ ) ,
BG m e ( r ) = exp ( i k t 2 2 k z μ ) GB ( r ) J m ( k t r μ ) cos ( m ϕ )
BG m o ( r ) = exp ( i k t 2 2 k z μ ) GB ( r ) J m ( k t r μ ) sin ( m ϕ ) .

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