Abstract

We propose an improved version of the earlier developed optical arrangement for generating inhomogeneously polarized laser light modes with the aid of a diffractive optical element (DOE) with carrier frequency. By eliminating lenses from the optical arrangement, we achieve the miniaturization, reduced light losses, a smaller number of parameters being matched, and a simpler system adjustment procedure. Note that all the capabilities of the previous version, namely, the universality and simple readjustment to different polarization types, are fully retained. The numerical modeling of the polarization mode combiner has made it possible to analyze its per formance and capabilities. In the experiments, the quality of the resulting beams is shown to be improved. For generating higher-order cylindrical beams, a lower-order mode at the output of the polarization mode combiner is additionally transformed with a DOE that operates in the zero diffraction order, introducing radial phase changes.

© 2011 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2011 (1)

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760(2011).
[Crossref]

2010 (1)

2009 (2)

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
[Crossref]

2008 (1)

2007 (3)

2006 (4)

2005 (4)

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005).
[Crossref] [PubMed]

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, “A new method for generating axially-symmetric and radially-polarized beams,” J. Phys. D 38, 827–832 (2005).
[Crossref]

N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

2004 (1)

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

2000 (3)

1999 (1)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

1998 (1)

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Diffraction optical elements matched to the Gauss-Laguerre modes,” Opt. Spectrosc. 85, 636–644 (1998).

1996 (1)

1990 (1)

1966 (1)

Ahmed, M. A.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Aït-Ameur, K.

Balalayev, S. A.

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Biener, G.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87(2000).
[Crossref] [PubMed]

Cottrell, D. M.

Davis, J. A.

de Saint Denis, R.

Dong, B. Z.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, “A new method for generating axially-symmetric and radially-polarized beams,” J. Phys. D 38, 827–832 (2005).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Ford, D. H.

Friberg, A. T.

Glur, H.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Graf, T.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Gu, B. Y.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, “A new method for generating axially-symmetric and radially-polarized beams,” J. Phys. D 38, 827–832 (2005).
[Crossref]

Gupta, D. N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Haist, T.

Hasman, E.

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Hernández, L.

Hierle, R.

Hirayama, T.

Juskaitis, R.

Kant, N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Karpeev, S. V.

Kazanskiy, N. L.

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760(2011).
[Crossref]

Khonina, S. N.

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760(2011).
[Crossref]

S. N. Khonina and S. V. Karpeev, “Grating-based optical scheme for the universal generation of inhomogeneously polarized laser beams,” Appl. Opt. 49, 1734–1738 (2010).
[Crossref] [PubMed]

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Diffraction optical elements matched to the Gauss-Laguerre modes,” Opt. Spectrosc. 85, 636–644 (1998).

Kim, D. E.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Kimura, W. D.

Kleiner, V.

Kogelnik, H.

Kohler, C.

Kotlyar, V. V.

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Diffraction optical elements matched to the Gauss-Laguerre modes,” Opt. Spectrosc. 85, 636–644 (1998).

Kozawa, Y.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Li, T.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966)

Massoumian, F.

McNamara, D. E.

Moser, T.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Nakamura, T.

Neil, M. A. A.

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Niu, C. H.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, “A new method for generating axially-symmetric and radially-polarized beams,” J. Phys. D 38, 827–832 (2005).
[Crossref]

Niv, A.

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Osten, W.

Paivanranta, B.

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

Parriaux, O.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Passilly, N.

Petrov, D.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

Pigeon, F.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Roch, J.-F.

Romano, V.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

Romero, J. A.

Sato, S.

Schwab, X.

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Skidanov, R. V.

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Diffraction optical elements matched to the Gauss-Laguerre modes,” Opt. Spectrosc. 85, 636–644 (1998).

Sonehara, T.

Suk, H.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Tan, B.

K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603–2607 (2006).
[Crossref]

Tidwell, S. C.

Treussart, F.

Turunen, J.

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

Venkatakrishnan, K.

K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603–2607 (2006).
[Crossref]

Volotovsky, S. G.

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760(2011).
[Crossref]

Volpe, G.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

Wilson, T.

Yirmiyahu, Y.

Yonezawa, K.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87(2000).
[Crossref] [PubMed]

Zhan, Q.

Zhang, Y.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, “A new method for generating axially-symmetric and radially-polarized beams,” J. Phys. D 38, 827–832 (2005).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (4)

Appl. Phys. B (1)

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80, 707–713 (2005).
[Crossref]

J. Micromech. Microeng. (1)

K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603–2607 (2006).
[Crossref]

J. Mod. Opt. (1)

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760(2011).
[Crossref]

J. Opt. A (1)

S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A 11, 065702–065708 (2009).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Phys. D (2)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, “A new method for generating axially-symmetric and radially-polarized beams,” J. Phys. D 38, 827–832 (2005).
[Crossref]

Opt. Commun. (1)

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Opt. Spectrosc. (1)

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “Diffraction optical elements matched to the Gauss-Laguerre modes,” Opt. Spectrosc. 85, 636–644 (1998).

Phys. Lett. A (1)

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1966)

V.A.Soifer, ed., Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

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Figures (9)

Fig. 1
Fig. 1

Arrangement for a simplified coherent superposition of two light fields with arbitrary complex coefficients by use of multiorder DOEs and a diffraction grating.

Fig. 2
Fig. 2

Generation of the HG modes ( 0 , 9 ) propagating at angles to the optical axis: (a) phase of a partially encoded DOE and (b) intensity distribution (negative) calculated at distance z = 500 mm from the DOE plane.

Fig. 3
Fig. 3

Modeling the scheme for the coherent superposition of two HG modes, ( 0 , 9 ) and ( 9 , 0 ) : (a) phases of two DOEs sharing the same substrate and (b) intensity distribution (negative), calculated at distance z = 250 mm from the DOE plane.

Fig. 4
Fig. 4

Amplitudes of the components and the total intensity (negative) of the complex field distribution resulting from the polarization mode superposition of the HG modes with the indices ( n , 0 ) and ( 0 , n ) at n = 1 , 4, 9 in the focal plane of a lens with focus f = 100 mm . * indicates that the longitudinal component values are two orders smaller than those of the transverse component.

Fig. 5
Fig. 5

Appearance of the substrate with three pairs of DOEs to generate HG modes with the indices ( 0 , n ) and ( n , 0 ) at n = 1 , 4, 9.

Fig. 6
Fig. 6

Experimental results on the coherent superposition of modes with various coefficients by use of an optical setup in Fig. 1: (a) and (b) for the HG modes ( 0 , 4 ) and ( 4 , 0 ) , (c) and (d) for the HG modes ( 0 , 9 ) and ( 9 , 0 ) .

Fig. 7
Fig. 7

Intensity distribution at the optical setup output for the HG modes ( 0 , 1 ) and ( 1 , 0 ) : (a) experimental pattern and (b) a comparative axial profile—experimentally derived (thick curve) and theoretically estimated (thin curve).

Fig. 8
Fig. 8

(a) Appearance of a ring phase converter of lower-order modes into higher-order beams and its effect on the beam shown in Fig. 7 at distance (b)  L = 200 mm and (c)  L = 400 mm ; and (d) intensity in the focal plane of an additional lens.

Fig. 9
Fig. 9

Intensity distributions (negative) at the optical setup output for various analyzer positions (1, vertical position; 2, rotated to the right through 45 ° ; 3, horizontal position; 4, rotated to the left through 45 ° ) for a radially polarized LG beam ( 1 , 0 ) derived (a) by the superposition of linearly polarized HG modes ( 0 , 1 ) and ( 1 , 0 ) , (b) and (c) with the aid of a circular DOE found at the combiner output, and (d) in the focal plane of an additional lens.

Equations (19)

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f ( x , y ) = [ Ψ 1 ( x a , y ) + Ψ 2 ( x + a , y ) ] [ 1 + cos ( β x ) ] .
F ( u , v , z ) = f ( x , y ) exp ( i k 2 z x 2 ) exp ( i k z x u ) d x ,
x s = u ± z k β , x s = u ,
S 1 , 2 = Ψ 1 ( u ± λ z 0 2 π β a , v ) exp [ i k 2 z ( u ± λ z 0 2 π β ) 2 ] ,
S 3 = Ψ 1 ( u a , v ) exp [ i k 2 z u 2 ] ,
S 4 , 5 = Ψ 2 ( u ± λ z 0 2 π β + a , v ) exp [ i k 2 z ( u ± λ z 0 2 π β ) 2 ] ,
S 6 = Ψ 2 ( u + a , v ) exp [ i k 2 z u 2 ] .
F ( u , v , z 0 ) = { Ψ 1 ( u , v ) exp ( i β u ) + Ψ 2 ( u , v ) exp ( i β u ) } exp [ i k 2 z 0 ( u 2 + a 2 ) ] + { Ψ 1 ( u a , v ) + Ψ 2 ( u + a , v ) } exp [ i k 2 z 0 u 2 ] + { Ψ 1 ( u 2 a , v ) exp [ i k 2 z 0 ( u + a ) 2 ] + Ψ 2 ( u + 2 a , v ) exp [ i k 2 z 0 ( u a ) 2 ] } .
τ ( u ) = [ 1 + cos ( β u + b ) ] ,
G ( u , v , z 0 ) { Ψ 1 ( u , v ) exp ( i b ) + Ψ 2 ( u , v ) exp ( i b ) } + { Ψ 1 ( u , v ) exp ( i β u ) + Ψ 2 ( u , v ) exp ( i β u ) } + { Ψ 1 ( u , v ) exp ( i 2 β u ) exp ( i b ) + Ψ 2 ( u , v ) exp ( i 2 β u ) exp ( i b ) } .
G 0 ( u , v , z 0 ) Ψ 1 ( u , v ) + c Ψ 2 ( u , v ) ,
G 0 ( u , v , z 0 ) Ψ 1 ( u , v ) e x + c Ψ 2 ( u , v ) e y .
z 0 = a d λ .
HG 0 , 0 ( x , y ) = 1 σ exp [ ( x 2 + y 2 ) 2 σ 2 ] ,
S 0 ( x , y , c ) = HG 0 , 0 ( x , y ) [ e x + c e y ] = HG 0 , 0 ( x , y ) [ e r ( cos φ + c sin φ ) + e φ ( c cos φ sin φ ) ] ,
HG 1 , 0 ( x , y ) = x σ · HG 0 , 0 ( x , y ) , HG 0 , 1 ( x , y ) = y σ · HG 0 , 0 ( x , y ) .
S 1 x ( x , y , c ) = HG 1 , 0 ( x , y ) e x + c HG 0 , 1 ( x , y ) e y = HG 0 , 0 ( x , y ) [ x e x + c y e y ] = HG 0 , 0 ( x , y ) · r · [ e r ( cos 2 φ + c sin 2 φ ) + e φ sin φ cos φ ( c 1 ) ] ,
S 1 y ( x , y , c ) = HG 0 , 1 ( x , y ) e x + c HG 1 , 0 ( x , y ) e y = HG 0 , 0 ( x , y ) [ y e x + c x e y ] = HG 0 , 0 ( x , y ) · r · [ e r sin φ cos φ ( 1 + c ) + e φ ( c cos 2 φ sin 2 φ ) ] .
LG 0 , 1 ( r ) = r σ exp [ r 2 2 σ 2 ] e r ,

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