Abstract

This study reports the first systematic approach to the excitation of all high-order Hermite-Gaussian modes (HGMs) in end-pumped solid-state lasers. This study uses a metal-wire-inserted laser resonator accompanied with the “off axis pumping” approach. This study presents numerical analysis of the excitation of HGMs in end-pumped solid-state lasers and experimentally generated HGM patterns. This study also experimentally demonstrates the generation of an square vortex array laser beams by passing specific high-order HGMs (HGn,n + 1 or HGn + 1,n modes) through a Dove prism-embedded unbalanced Mach-Zehnder interferometer [Optics Express 16, 19934-19949]. The resulting square vortex array laser beams with embedded vortexes aligned in a square array can be applied to multi-spot dark optical traps in the future.

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References

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  4. M. P. Thirugnanasambandam, Y. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. 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).
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    [CrossRef]
  10. Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
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  14. M. Woerdemann, K. Berghoff, and C. Denz, “Dynamic multiple-beam counter-propagating optical traps using optical phase-conjugation,” Opt. Express 18(21), 22348–22357 (2010).
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  15. R. L. Eriksen, P. C. Mogensen, and J. Glückstad, “Multiple-beam optical tweezers generated by the generalized phase-contrast method,” Opt. Lett. 27(4), 267–269 (2002).
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  16. R. Eriksen, V. Daria, and J. Gluckstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10(14), 597–602 (2002).
    [PubMed]
  17. S.-C. Chu, C.-S. Yang, and K. Otsuka, “Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer,” Opt. Express 16(24), 19934–19949 (2008).
    [CrossRef] [PubMed]
  18. E. Santamato, A. Sasso, B. Piccirillo, and A. Vella, “Optical angular momentum transfer to transparent isotropic particles using laser beam carrying zero average angular momentum,” Opt. Express 10(17), 871–878 (2002).
    [PubMed]
  19. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
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    [CrossRef]
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    [CrossRef]
  30. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express 12(15), 3548–3553 (2004).
    [CrossRef] [PubMed]
  31. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), 105–107.

2010 (2)

M. P. Thirugnanasambandam, Y. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010).
[CrossRef]

M. Woerdemann, K. Berghoff, and C. Denz, “Dynamic multiple-beam counter-propagating optical traps using optical phase-conjugation,” Opt. Express 18(21), 22348–22357 (2010).
[CrossRef] [PubMed]

2008 (2)

2007 (2)

2004 (5)

2002 (3)

2001 (3)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Y. F. Chen, Y. P. Lan, and S. C. Wang, “Generation of Laguerre–Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72(2), 167–170 (2001).
[CrossRef]

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A 63(6), 063401 (2001).
[CrossRef]

1999 (2)

1998 (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998).
[CrossRef]

1997 (2)

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite–Gaussian Modes in Fiber-Coupled Laser-Diode End-Pumped Lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

1996 (1)

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28(3), 213–214 (1996).
[CrossRef]

1983 (1)

1979 (1)

1966 (1)

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998).
[CrossRef]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998).
[CrossRef]

Bandres, M. A.

Berghoff, K.

Bhowmik, A.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Chen, Y. F.

Y. F. Chen, Y. P. Lan, and S. C. Wang, “Generation of Laguerre–Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72(2), 167–170 (2001).
[CrossRef]

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite–Gaussian Modes in Fiber-Coupled Laser-Diode End-Pumped Lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

Chu, S.-C.

Daria, V.

Denz, C.

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998).
[CrossRef]

Endo, M.

Eriksen, R.

Eriksen, R. L.

Fujioka, T.

Gluckstad, J.

Glückstad, J.

Gutiérrez-Vega, J. C.

Hill III, W. T.

Huang, T. M.

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite–Gaussian Modes in Fiber-Coupled Laser-Diode End-Pumped Lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Jhe, W.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A 63(6), 063401 (2001).
[CrossRef]

Kamikariya, K.

Kao, C. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite–Gaussian Modes in Fiber-Coupled Laser-Diode End-Pumped Lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

Kawakami, M.

Kim, K.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A 63(6), 063401 (2001).
[CrossRef]

Kogelnik, H.

Kubodera, K.

Kwon, N.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A 63(6), 063401 (2001).
[CrossRef]

Laabs, H.

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28(3), 213–214 (1996).
[CrossRef]

Lan, Y. P.

Y. F. Chen, Y. P. Lan, and S. C. Wang, “Generation of Laguerre–Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72(2), 167–170 (2001).
[CrossRef]

Li, T.

Lin, K. H.

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Milam, D.

Miyaji, G.

Miyanaga, N.

Miyazawa, S.

Mogensen, P. C.

Nakatsuka, M.

Nanri, K.

Ohtomo, T.

Otsuka, K.

Ozygus, B.

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28(3), 213–214 (1996).
[CrossRef]

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998).
[CrossRef]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Piccirillo, B.

Santamato, E.

Sasso, A.

Schwarz, U. T.

Senatsky, Y.

M. P. Thirugnanasambandam, Y. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010).
[CrossRef]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Song, Y.

Sueda, K.

Takeda, S.

Thirugnanasambandam, M. P.

M. P. Thirugnanasambandam, Y. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010).
[CrossRef]

Ueda, K.

M. P. Thirugnanasambandam, Y. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010).
[CrossRef]

Vella, A.

Wang, C. L.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite–Gaussian Modes in Fiber-Coupled Laser-Diode End-Pumped Lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

Wang, S. C.

Y. F. Chen, Y. P. Lan, and S. C. Wang, “Generation of Laguerre–Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72(2), 167–170 (2001).
[CrossRef]

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite–Gaussian Modes in Fiber-Coupled Laser-Diode End-Pumped Lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Woerdemann, M.

Xu, X.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A 63(6), 063401 (2001).
[CrossRef]

Yang, C.-S.

Appl. Opt. (5)

Appl. Phys. B (1)

Y. F. Chen, Y. P. Lan, and S. C. Wang, “Generation of Laguerre–Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72(2), 167–170 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite–Gaussian Modes in Fiber-Coupled Laser-Diode End-Pumped Lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

J. Mod. Opt. (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

Laser Phys. Lett. (1)

M. P. Thirugnanasambandam, Y. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010).
[CrossRef]

Opt. Commun. (1)

Y. F. Chen, T. M. Huang, K. H. Lin, C. F. Kao, C. L. Wang, and S. C. Wang, “Analysis of the effect of pump position on transverse modes in fiber-coupled laser-diode end pumped lasers,” Opt. Commun. 136(5-6), 399–404 (1997).
[CrossRef]

Opt. Express (8)

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]

K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express 12(15), 3548–3553 (2004).
[CrossRef] [PubMed]

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]

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]

M. Woerdemann, K. Berghoff, and C. Denz, “Dynamic multiple-beam counter-propagating optical traps using optical phase-conjugation,” Opt. Express 18(21), 22348–22357 (2010).
[CrossRef] [PubMed]

R. Eriksen, V. Daria, and J. Gluckstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10(14), 597–602 (2002).
[PubMed]

S.-C. Chu, C.-S. Yang, and K. Otsuka, “Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer,” Opt. Express 16(24), 19934–19949 (2008).
[CrossRef] [PubMed]

E. Santamato, A. Sasso, B. Piccirillo, and A. Vella, “Optical angular momentum transfer to transparent isotropic particles using laser beam carrying zero average angular momentum,” Opt. Express 10(17), 871–878 (2002).
[PubMed]

Opt. Laser Technol. (1)

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28(3), 213–214 (1996).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (1)

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A 63(6), 063401 (2001).
[CrossRef]

Science (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Other (5)

H. Kogelnik and W. W. Rigrod, “Visual display of isolated optical-resonator modes,” Proc. IRE 50, 220 (1962).

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

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), 105–107.

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 (4174 KB)     
» Media 2: MOV (1847 KB)     
» Media 3: MOV (5454 KB)     

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic diagram of opaque-wire-inserted half-symmetric laser resonator. (b) Relative transverse positions of opaque-wire and end pumping beam to the oscillation HGM.

Fig. 2
Fig. 2

Demonstration of resulting progress of stable intensity distribution of (a) HG2,2 mode, (b) HG3,1 mode and (c) HG4,5 mode. The most left-hand side images show simulated spontaneous emission patterns with partially coherent random fields.

Fig. 3
Fig. 3

Movie of resulting progress of stable amplitude distribution of (a) HG2,2 mode (Media 1), (b) HG3,1 mode (Media 2) and (c) HG4,5 mode (Media 3)

Fig. 4
Fig. 4

The resulting oscillation HGM patterns in the opaque-wire-inserted resonator from simulations. The numbers in the parentheses is HGM numbers, and the percentages shown in the figure are the mode purity of selective HGMs from resonator.

Fig. 5
Fig. 5

Experimental setup for generating high-order HGMs

Fig. 6
Fig. 6

High-order HGMs patterns produced by end-pumped SSLs. The straight lines in figures (a) and (b) describe the orientation of metal-wire for lasing HGMs. (a) The orientation of lasing high-order HGMs will be according to the orientation of inserted metal-wire. (b) HGm,3 modes (m = 1, 2, and 3). Fixing the metal-wire, changing the pumping beam position could control the lasing HGMs’ order m. (c) HGn,n modes (n = 1, 2, and 3). (d) HGn,n + 1 modes or HGn + 1,n modes (n = 1, 2, and 3).

Fig. 7
Fig. 7

Output lasing HGMs’ beam powers versus input pumping beam powers. The symbols “■”, “▲”, “□”, “△” indicate output/input powers of lasing HGMs in Fig. 6(a) to Fig. 6(d), respectively. Two numbers in the parentheses right to the symbol denotes the mode number of the lasing HGM, i.e. (nx, ny).

Fig. 8
Fig. 8

Output lasing HGn,0 modes’ beam powers versus input pumping beam powers (n = 1 to 4). The “solid” and “hollow” symbols indicate the output/input power of lasing HGMs from a laser resonator without and with metal-wire inserted, respectively.

Fig. 9
Fig. 9

(a) Vortex array beam generation from two Hermite-Gaussian modes superposition. (b) Schematic diagram of the Dove prism-embedded unbalanced Mach-Zehnder interferometer for vortex array laser beam generation.

Fig. 10
Fig. 10

(a) (b) Intensity of the incident HGn,n + 1 modes or HGn + 1,n modes (n = 1 and 2, respectively). (c) (d) Intensity distributions of square vortex array laser beams resulted by passing HGn,n + 1 modes or HGn + 1,n modes through the Dove-prism embedded Mach-Zehnder interferometer. The symbol z denotes the distance from the CCD to the output port of the interferometer, BS2.

Fig. 11
Fig. 11

(a) Analytical amplitude distribution of a square vortex array beam resulting from superposing HG10,11 mode and HG11,10 mode with π/2 phase difference . (b) (c) Enlarged center amplitude and phase distribution of square vortex array laser beams (d) The interferogram (a calculation of interference fringes of the vortex array laser beam with a tilted plane wave) Red and blue spots indicate the cross-section position of the vortices.

Equations (8)

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E q + 1 ( x , y ) ~ E q ( x , y ) ,
M P = | u , u H G | 2 = | u ( u H G ) * d A | 2 .
U V L = H G n , n + 1 ± i × H G n + 1 , n .
H G n x , n y ( x , y , z ) = ( 1 2 n x + n y 1 π n x ! n y ! ) 1 / 2 1 w z × H n x ( 2 x w z ) H n y ( 2 y w z ) exp [ r 2 w z 2 ] × exp i [ k z + k r 2 2 R z ( n x + n y + 1 ) ψ G ( z ) ] .
H n ( 2 x w z ) H n + 1 ( 2 y w z ) + i × H n + 1 ( 2 x w z ) H n ( 2 y w z ) = 0.
{ H n ( 2 x w z ) H n + 1 ( 2 y w z ) = 0 H n + 1 ( 2 x w z ) H n ( 2 y w z ) = 0 .
H n ( 2 x w z ) = H n ( 2 y w z ) = 0
H n + 1 ( 2 x w z ) = H n + 1 ( 2 y w z ) = 0.

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