Abstract

We develop a round-trip matrix diagonalization method for quantitative description of selection of radially or azimuthally polarized beams by birefringence-induced bifocusing in a simple laser resonator. We employ different focusing between radially and tangentially polarized light in thermally stressed laser rods to obtain low-loss stable oscillation in a radially polarized Laguerre–Gaussian, LG(0,1)*, mode. We derive a free-space propagator for the radially and azimuthally polarized LG(0,1)* modes and explain basic principles of mode selection by use of a round-trip matrix diagonalization method. Within this method we calculate round-trip diffraction losses and intensity distributions for the lowest-loss transverse modes. We show that, for the considered laser configuration, the round-trip loss obtained for the radially polarized LG(0,1)* mode is significantly smaller than that of the azimuthally polarized mode. Our experimental results, obtained with a diode side-pumped Nd:YAG rod in a flat–convex resonator, confirm the theoretical predictions. We achieved a pure radially polarized LG(0,1)* beam with M2=2.5 and tens of watts of output power.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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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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    [CrossRef]
  25. W. Koechner, Solid State Laser Engineering, 5th ed. (Springer, 1999).
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    [CrossRef] [PubMed]
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    [CrossRef]
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  30. R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, "Transverse mode shaping and selection in laser resonators," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325-385.
    [CrossRef]
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    [CrossRef]

2006

2005

2004

2003

2002

2001

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

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

1999

Y. Liu, D. Cline, and P. He, "Vacuum laser acceleration using a radially polarized CO2 laser beam," Nucl. Instrum. Methods Phys. Res. A 424, 296-303 (1999).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

1998

1993

1990

1972

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE 60, 1107-1109 (1972).
[CrossRef]

D. Pohl, "Operation of a ruby laser in the purely transverse electric mode TE01," Appl. Phys. Lett. 20, 266-267 (1972).
[CrossRef]

1970

1968

L. M. Osterink and J. D. Foster, "Thermal effects and transverse mode control in a Nd:YAG laser," Appl. Phys. Lett. 12, 128-131 (1968).
[CrossRef]

1966

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

1965

T. Li, "Diffraction loss and selection of modes in maser resonators with circular mirrors," Bell Syst. Tech. J. 44, 917-932 (1965).

Ahmed, M. A.

T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and Th. Graf, "Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors," Laser Phys. Lett. 1, 234-236 (2004).
[CrossRef]

Armstrong, D. J.

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]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Bokor, N.

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1981).

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]

Cline, D.

Y. Liu, D. Cline, and P. He, "Vacuum laser acceleration using a radially polarized CO2 laser beam," Nucl. Instrum. Methods Phys. Res. A 424, 296-303 (1999).
[CrossRef]

Davidson, N.

N. Davidson and N. Bokor, "High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens," Opt. Lett. 29, 1318-1320 (2004).
[CrossRef] [PubMed]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, "Transverse mode shaping and selection in laser resonators," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325-385.
[CrossRef]

Deng, D.

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]

Ehrlichmann, D.

Ford, D. H.

Foster, J. D.

L. M. Osterink and J. D. Foster, "Thermal effects and transverse mode control in a Nd:YAG laser," Appl. Phys. Lett. 12, 128-131 (1968).
[CrossRef]

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, "Transverse mode shaping and selection in laser resonators," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325-385.
[CrossRef]

Glur, H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Graf, Th.

T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and Th. Graf, "Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors," Laser Phys. Lett. 1, 234-236 (2004).
[CrossRef]

Habich, U.

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, "Transverse mode shaping and selection in laser resonators," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325-385.
[CrossRef]

He, P.

Y. Liu, D. Cline, and P. He, "Vacuum laser acceleration using a radially polarized CO2 laser beam," Nucl. Instrum. Methods Phys. Res. A 424, 296-303 (1999).
[CrossRef]

Jackel, S.

Kimura, W. D.

Koechner, W.

W. Koechner, "Thermal lensing in a Nd:YAG rod," Appl. Opt. 9, 2548-2553 (1970).
[CrossRef] [PubMed]

W. Koechner, Solid State Laser Engineering, 5th ed. (Springer, 1999).

Kogelnik, H.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Kozawa, Y.

Leger, J.

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.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

T. Li, "Diffraction loss and selection of modes in maser resonators with circular mirrors," Bell Syst. Tech. J. 44, 917-932 (1965).

Liu, Y.

Y. Liu, D. Cline, and P. He, "Vacuum laser acceleration using a radially polarized CO2 laser beam," Nucl. Instrum. Methods Phys. Res. A 424, 296-303 (1999).
[CrossRef]

Matsumura, K.

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE 60, 1107-1109 (1972).
[CrossRef]

Meir, A.

Moser, T.

T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and Th. Graf, "Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors," Laser Phys. Lett. 1, 234-236 (2004).
[CrossRef]

Moshe, I.

Mushiake, Y.

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE 60, 1107-1109 (1972).
[CrossRef]

Nakajima, N.

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE 60, 1107-1109 (1972).
[CrossRef]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (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]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, "Transverse mode shaping and selection in laser resonators," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325-385.
[CrossRef]

Osterink, L. M.

L. M. Osterink and J. D. Foster, "Thermal effects and transverse mode control in a Nd:YAG laser," Appl. Phys. Lett. 12, 128-131 (1968).
[CrossRef]

Parriaux, O.

T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and Th. Graf, "Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors," Laser Phys. Lett. 1, 234-236 (2004).
[CrossRef]

Phillips, M. C.

Pigeon, F.

T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and Th. Graf, "Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors," Laser Phys. Lett. 1, 234-236 (2004).
[CrossRef]

Plum, H.-D.

Pohl, D.

D. Pohl, "Operation of a ruby laser in the purely transverse electric mode TE01," Appl. Phys. Lett. 20, 266-267 (1972).
[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]

Roth, M. S.

Sato, S.

Siegman, A. E.

A. E. Siegman, "New developments in laser resonators," Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Smith, A. V.

Tidwell, S. C.

Tovar, A. A.

Weber, H. P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1981).

Wyss, E.

T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and Th. Graf, "Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors," Laser Phys. Lett. 1, 234-236 (2004).
[CrossRef]

Wyss, E. W.

Yakunin, V. P.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

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]

Zhan, Q.

Appl. Opt.

Appl. Phys. Lett.

L. M. Osterink and J. D. Foster, "Thermal effects and transverse mode control in a Nd:YAG laser," Appl. Phys. Lett. 12, 128-131 (1968).
[CrossRef]

D. Pohl, "Operation of a ruby laser in the purely transverse electric mode TE01," Appl. Phys. Lett. 20, 266-267 (1972).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Bell Syst. Tech. J.

T. Li, "Diffraction loss and selection of modes in maser resonators with circular mirrors," Bell Syst. Tech. J. 44, 917-932 (1965).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. D

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

Laser Phys. Lett.

T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and Th. Graf, "Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors," Laser Phys. Lett. 1, 234-236 (2004).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A

Y. Liu, D. Cline, and P. He, "Vacuum laser acceleration using a radially polarized CO2 laser beam," Nucl. Instrum. Methods Phys. Res. A 424, 296-303 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

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

Proc. IEEE

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE 60, 1107-1109 (1972).
[CrossRef]

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Proc. SPIE

A. E. Siegman, "New developments in laser resonators," Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1981).

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, "Transverse mode shaping and selection in laser resonators," in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325-385.
[CrossRef]

W. Koechner, Solid State Laser Engineering, 5th ed. (Springer, 1999).

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Figures (8)

Fig. 1
Fig. 1

Basic resonator configuration for selection of a radially polarized mode. The principal planes in the rod are denoted by dotted lines. While the mode with radial polarization (dashed lines) is well focused through the aperture, the mode with azimuthal polarization (dotted lines) suffers high diffraction losses because of the aperture.

Fig. 2
Fig. 2

Calculated round-trip loss of four lowest-order modes with radial (dots) and azimuthal (diamonds) polarization as a function of the intracavity aperture diameter for the resonator configuration in Fig. 1. The losses of the radially polarized modes are significantly lower than those of the azimuthally polarized modes.

Fig. 3
Fig. 3

Calculated intensity distributions of the lowest-order (a) LG ( 0 , 1 ) * , (b) LG ( 1 , 1 ) * , and (c) LG ( 2 , 1 ) * modes with radial (solid curve) and azimuthal (diamonds) polarizations for the resonator configuration in Fig. 1 with an aperture diameter of 1.2   mm . As expected, the modes with azimuthal polarization have a larger beam size. We determined that the calculated intensity distributions shown in Fig. 3 coincide with the cross sections of the ideal nondegenerate LG modes that are shown in the insets.

Fig. 4
Fig. 4

Basic resonator configuration for selection of an azimuthally polarized mode. While the mode with azimuthal polarization (dotted lines) is stable and well focused through the aperture, the mode with radial polarization (dashed lines) is unstable and suffers high diffraction losses.

Fig. 5
Fig. 5

Calculated round-trip loss of four lowest-order modes with radial (dots) and azimuthal (diamonds) polarization as a function of the intracavity aperture diameter for the resonator configuration in Fig. 4. The losses of the azimuthally polarized modes are significantly lower than those of the radially polarized modes.

Fig. 6
Fig. 6

Calculated intensity distributions of the lowest-order modes with radial (solid curve) and azimuthal (diamonds) polarizations for the resonator configuration of Fig. 4 with an aperture diameter of 1.2   mm . As expected, the mode with radial polarization is the mode of the unstable resonator.

Fig. 7
Fig. 7

Representative experimental results: (a), (b) near-field and far-field intensity distributions of the obtained radially polarized LG ( 0 , 1 ) * mode; (c)–(f) near-field distributions after the mode passes through a λ / 2 retardation plate together with a linear polarizer. The λ / 2 plate orientations are (c) 0 deg, (d) 45 deg, (e) 22.5 deg, (f) 22.5   deg ; (g), (h) distributions obtained after the mode passes through a horizontal λ / 4 plate, a λ / 2 plate at (g) 22.5 deg or (h) 22.5 deg and a linear polarizer.

Fig. 8
Fig. 8

(a) Absolute value of the deviation (in radians) of the azimuthal angle ψ = 1 / 2 arctan ( S 2 / S 1 ) from the radial direction in each transverse point and (b) the recorded intensity distribution. It is evident that the deviation is significant mainly in areas with nearly zero intensity.

Equations (120)

Equations on this page are rendered with MathJax. Learn more.

LG ( 0 , 1 ) *
LG ( 0 , 1 ) *
LG ( 0 , 1 ) *
LG ( 0 , 1 ) *
M 2 = 2.5
TEM 01
CO 2
LG ( 0 , 1 ) *
E p 1 r ( r , φ , 0 ) = E 0 ρ 1 / 2 L p 1 ( ρ ) exp ( ρ / 2 ) e ^ r = f p 1 ( r ) cos ( φ ) e ^ x + f p 1 ( r ) sin ( φ ) e ^ y ,
E p 1 a ( r , φ , 0 ) = E 0 ρ 1 / 2 L p 1 ( ρ ) exp ( ρ / 2 ) e ^ φ = f p 1 ( r ) sin ( φ ) e ^ x + f p 1 ( r ) cos ( φ ) e ^ y ,
ρ = 2 r 2 / w 2
E 0
L p 1
l = 1
f p 1 ( r )
e ^ r
e ^ x
e ^ y
E ( r 1 , φ 1 , z ) = i exp ( i k z ) λ z 0 0 2 π E p 1 ( r , φ , 0 ) × exp [ i k 2 z ( r 2 + r 1 2 2 r r 1 cos ( φ φ 1 ) ) ] × r d r d φ ,
k = 2 π / λ
r 1
φ 1
z = 0
E ( r 1 , φ 1 , z ) = i exp ( i k z ) λ z 0 f p 1 ( r ) exp [ i k 2 z ( r 2 + r 1 2 ) ] r × 0 2 π ( cos φ e ^ x + sin φ e ^ y ) × exp ( i k z r r 1 cos ( φ φ 1 ) ) d r d φ .
e ^ x
e ^ y
φ φ + φ 1
  e ^ x 0 2 π cos φ exp [ i k r r 1 z cos ( φ φ 1 ) ] d φ   = e ^ x 0 2 π cos ( φ + φ 1 ) exp [ i k r r 1 z cos ( φ ) ] d φ   = e ^ x 0 2 π { cos φ cos φ 1 sin φ sin φ 1 }     × exp ( i k r r 1 z cos ( φ ) ) d φ   = e ^ x 2 π i cos φ 1 J 1 ( k r r 1 z ) ,
J n ( ξ ) = 1 2 π i n 0 2 π exp ( i ξ cos φ ) cos ( n φ ) d φ
0 2 π exp ( i ξ cos φ ) sin ( n φ ) d φ = 0 .
2 π i J 1 ( k r r 1 z ) ( cos φ e ^ x + sin φ e ^ y ) = 2 π i J 1 ( k r r 1 z ) e ^ r ,
E ( r 1 , φ 1 , z ) = 2 π exp ( i k z ) λ z 0 f p 1 ( r ) × exp [ i k 2 z ( r 2 + r 1 2 ) ] J 1 ( k r r 1 z ) r d r e ^ r .
exp ( i l φ )
l = 1
LG ( 0 ,   1 ) *
LG ( 0 , 1 ) *
LG ( 0 , 1 ) *
E 0 f ( r ) e i φ ( e x + e y e i π / 2 ) + e i φ ( e x + e y e i π / 2 )   = E 0 f ( r ) { e x ( e i φ + e i φ ) + e y [ e i ( φ π / 2 ) + e i ( φ π / 2 ) ] }   = 2 E 0 f ( r ) ( e x cos φ + e y sin φ ) ,
f φ / f r = 1.2
f φ / f r
f φ / f r
R = 50   cm
3   kW
26 .5   cm
f r = 23.8   cm
f φ = 29.3   cm
K U n = γ n U n ,
U n
γ n
1 | γ n | 2
U n
γ n
U n
U n
K r = P 1 A P 2 T L r P 3 B M P 2 T L r P 3 A P 1 ,
K φ = P 1 A P 2 T L φ P 3 B M P 2 T L φ P 3 A P 1 ,
P 1
P 2
P 3
z 1
z 2
z 3
T L r
T L φ
P 2
P 3
z 1
z 4
z 4 z 1
z 1 ( 0.5   cm )
z 2 ( 50.75   cm )
z 3 ( 43.85   cm )
z 1
λ = 1064   nm
1500 × 1500
LG ( 0 , 1 ) *
0.4   to   0.8   mm
1 .2   mm
g 1
g 2
g 1 g 2 = 1.36
f r = 23.8 c m
g 1 g 2 = 0.31
f φ = 29.3 c m
L G ( 0 , 1 ) *
0.635   cm × 14.6   cm
0.8   mm
0.5   cm
M 2
M 2 = 2.52
30 35   W
λ / 2
λ / 2
22.5
λ / 4
λ / 2
22.5   deg
S 0
S 1
S 2
S 3
ψ = 1 / 2 arctan ( S 2 / S 1 )
LG ( 0 , 1 ) *
LG ( 0 , 1 ) *
LG ( 0 , 1 ) *
M 2 = 2.5
TE 01
LG ( 0 , 1 ) *
LG ( 1 , 1 ) *
LG ( 2 , 1 ) *
1.2   mm
1.2   mm
LG ( 0 , 1 ) *
λ / 2
λ / 2
22.5   deg
λ / 4
λ / 2
22.5 deg
ψ = 1 / 2 arctan ( S 2 / S 1 )

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