Abstract

In end-pumped Nd:YVO4 amplifiers, beam quality improvement is obtained both in theory and in experiments. A theoretical model of gain-guided laser amplifier is developed by comprehensively considering thermal effect, gain guiding and gain saturation effect. Several key parameters of the amplifier are discussed such as the input beam quality, the beam filling factor between input beam and pump beam, the ratio between input power and pump power, and the length of laser crystal. The theoretical results are confirmed by the experiments.

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References

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  1. F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17(19), 1352–1354 (1992).
    [CrossRef] [PubMed]
  2. A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
    [CrossRef]
  3. X. Yan, Q. Liu, D. Wang, and M. Gong, “Combined guiding effect in the end-pumped laser resonator,” Opt. Express 19(7), 6883–6902 (2011).
    [CrossRef] [PubMed]
  4. P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused-gain amplifier,” Phys. Rev. Lett. 70(11), 1607–1610 (1993).
    [CrossRef] [PubMed]
  5. X. Yan, Q. Liu, X. Fu, D. Wang, and M. Gong, “Gain guiding effect in end-pumped Nd:YVO4 MOPA lasers,” J. Opt. Soc. Am. B 27(6), 1286–1290 (2010).
    [CrossRef]
  6. I. H. Deutsch, J. C. Garrison, and E. M. Wright, “Excess noise in gain-guided amplifiers,” J. Opt. Soc. Am. B 8(6), 1244–1251 (1991).
    [CrossRef]
  7. A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
    [CrossRef]
  8. J. E. LaSala, D. A. Deacon, and J. M. Madey, “Optical guiding in a free-electron-laser oscillator,” Phys. Rev. Lett. 59(18), 2047–2050 (1987).
    [CrossRef] [PubMed]
  9. A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35(4), 675–681 (1999).
    [CrossRef]
  10. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39(3), 1253–1263 (1989).
    [CrossRef] [PubMed]
  11. B. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27(4), 1989–2002 (1983).
    [CrossRef]
  12. J. H. Marburger, “Self-focusing theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  13. J. Fleck, J. Morris, and E. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14(5), 353–363 (1978).
    [CrossRef]
  14. W. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5(7), 1412–1423 (1988).
    [CrossRef]
  15. M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. 47(13), 2317–2325 (2008).
    [CrossRef] [PubMed]
  16. W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer-Verlag Publication, 2006), Chap. 7.
  17. C. Serrat, M. Van Exter, N. Van Druten, and J. Woerdman, “Transverse mode formation in microlasers by combined gain-and index-guiding,” IEEE J. Quantum Electron. 35(9), 1314–1321 (1999).
    [CrossRef]
  18. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation, Fundamentals, Advanced Concepts and Applications, 2nd ed. (Springer-Verlag Publication, 2005), Chap. 24.
  19. A. Minassian, B. Thompson, G. Smith, and M. Damzen, “High-power scaling (>100 W) of a diode-pumped TEM00 Nd:GdVO4 laser system,” IEEE J. Sel. Top. Quantum Electron. 11(3), 621–625 (2005).
    [CrossRef]
  20. B. Schulz, M. Frede, R. Wilhelm, and D. Kracht, “High power end-pumped Nd:YVO4 amplifier,” in Conference on Advanced Solid-State Photonics, Technical Digest (CD) (Optical Society of America, 2006), paper WB15.

2011 (1)

2010 (1)

2008 (1)

2006 (1)

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

2005 (1)

A. Minassian, B. Thompson, G. Smith, and M. Damzen, “High-power scaling (>100 W) of a diode-pumped TEM00 Nd:GdVO4 laser system,” IEEE J. Sel. Top. Quantum Electron. 11(3), 621–625 (2005).
[CrossRef]

1999 (2)

C. Serrat, M. Van Exter, N. Van Druten, and J. Woerdman, “Transverse mode formation in microlasers by combined gain-and index-guiding,” IEEE J. Quantum Electron. 35(9), 1314–1321 (1999).
[CrossRef]

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35(4), 675–681 (1999).
[CrossRef]

1998 (1)

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

1993 (1)

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused-gain amplifier,” Phys. Rev. Lett. 70(11), 1607–1610 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

1989 (1)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39(3), 1253–1263 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

J. E. LaSala, D. A. Deacon, and J. M. Madey, “Optical guiding in a free-electron-laser oscillator,” Phys. Rev. Lett. 59(18), 2047–2050 (1987).
[CrossRef] [PubMed]

1983 (1)

B. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27(4), 1989–2002 (1983).
[CrossRef]

1978 (1)

J. Fleck, J. Morris, and E. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14(5), 353–363 (1978).
[CrossRef]

1975 (1)

J. H. Marburger, “Self-focusing theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Ballato, J.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

Bass, M.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

Battle, P. R.

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused-gain amplifier,” Phys. Rev. Lett. 70(11), 1607–1610 (1993).
[CrossRef] [PubMed]

Bliss, E.

J. Fleck, J. Morris, and E. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14(5), 353–363 (1978).
[CrossRef]

Carlsten, J. L.

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused-gain amplifier,” Phys. Rev. Lett. 70(11), 1607–1610 (1993).
[CrossRef] [PubMed]

Chen, Y.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

Conroy, R. S.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35(4), 675–681 (1999).
[CrossRef]

Damzen, M.

A. Minassian, B. Thompson, G. Smith, and M. Damzen, “High-power scaling (>100 W) of a diode-pumped TEM00 Nd:GdVO4 laser system,” IEEE J. Sel. Top. Quantum Electron. 11(3), 621–625 (2005).
[CrossRef]

Deacon, D. A.

J. E. LaSala, D. A. Deacon, and J. M. Madey, “Optical guiding in a free-electron-laser oscillator,” Phys. Rev. Lett. 59(18), 2047–2050 (1987).
[CrossRef] [PubMed]

Deutsch, I. H.

Fleck, J.

J. Fleck, J. Morris, and E. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14(5), 353–363 (1978).
[CrossRef]

Foy, P.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

French, P. M. W.

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

Friel, G. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35(4), 675–681 (1999).
[CrossRef]

Fu, X.

Garrison, J. C.

Gong, M.

Hawkins, W.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

Hyde, S. C. W.

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

Kemp, A. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35(4), 675–681 (1999).
[CrossRef]

LaSala, J. E.

J. E. LaSala, D. A. Deacon, and J. M. Madey, “Optical guiding in a free-electron-laser oscillator,” Phys. Rev. Lett. 59(18), 2047–2050 (1987).
[CrossRef] [PubMed]

Liu, Q.

Madey, J. M.

J. E. LaSala, D. A. Deacon, and J. M. Madey, “Optical guiding in a free-electron-laser oscillator,” Phys. Rev. Lett. 59(18), 2047–2050 (1987).
[CrossRef] [PubMed]

Marburger, J. H.

J. H. Marburger, “Self-focusing theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Melish, R.

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

Minassian, A.

A. Minassian, B. Thompson, G. Smith, and M. Damzen, “High-power scaling (>100 W) of a diode-pumped TEM00 Nd:GdVO4 laser system,” IEEE J. Sel. Top. Quantum Electron. 11(3), 621–625 (2005).
[CrossRef]

Morris, J.

J. Fleck, J. Morris, and E. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14(5), 353–363 (1978).
[CrossRef]

Mousave, L.

Nadgaran, H.

New, G.

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

Newstein, M.

B. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27(4), 1989–2002 (1983).
[CrossRef]

Perry, B.

B. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27(4), 1989–2002 (1983).
[CrossRef]

Rabinowitz, P.

B. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27(4), 1989–2002 (1983).
[CrossRef]

Richardson, M. C.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

Risk, W.

Ritsataki, A.

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

Sabaeian, M.

Salin, F.

Serrat, C.

C. Serrat, M. Van Exter, N. Van Druten, and J. Woerdman, “Transverse mode formation in microlasers by combined gain-and index-guiding,” IEEE J. Quantum Electron. 35(9), 1314–1321 (1999).
[CrossRef]

Siegman, A.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39(3), 1253–1263 (1989).
[CrossRef] [PubMed]

Sinclair, B. D.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35(4), 675–681 (1999).
[CrossRef]

Smith, G.

A. Minassian, B. Thompson, G. Smith, and M. Damzen, “High-power scaling (>100 W) of a diode-pumped TEM00 Nd:GdVO4 laser system,” IEEE J. Sel. Top. Quantum Electron. 11(3), 621–625 (2005).
[CrossRef]

Squier, J.

Sudesh, V.

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

Taylor, J.

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

Thompson, B.

A. Minassian, B. Thompson, G. Smith, and M. Damzen, “High-power scaling (>100 W) of a diode-pumped TEM00 Nd:GdVO4 laser system,” IEEE J. Sel. Top. Quantum Electron. 11(3), 621–625 (2005).
[CrossRef]

Van Druten, N.

C. Serrat, M. Van Exter, N. Van Druten, and J. Woerdman, “Transverse mode formation in microlasers by combined gain-and index-guiding,” IEEE J. Quantum Electron. 35(9), 1314–1321 (1999).
[CrossRef]

Van Exter, M.

C. Serrat, M. Van Exter, N. Van Druten, and J. Woerdman, “Transverse mode formation in microlasers by combined gain-and index-guiding,” IEEE J. Quantum Electron. 35(9), 1314–1321 (1999).
[CrossRef]

Wang, D.

Wessel, J. G.

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused-gain amplifier,” Phys. Rev. Lett. 70(11), 1607–1610 (1993).
[CrossRef] [PubMed]

Woerdman, J.

C. Serrat, M. Van Exter, N. Van Druten, and J. Woerdman, “Transverse mode formation in microlasers by combined gain-and index-guiding,” IEEE J. Quantum Electron. 35(9), 1314–1321 (1999).
[CrossRef]

Wright, E. M.

Yan, X.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89(25), 251101 (2006).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. Serrat, M. Van Exter, N. Van Druten, and J. Woerdman, “Transverse mode formation in microlasers by combined gain-and index-guiding,” IEEE J. Quantum Electron. 35(9), 1314–1321 (1999).
[CrossRef]

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35(4), 675–681 (1999).
[CrossRef]

J. Fleck, J. Morris, and E. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14(5), 353–363 (1978).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

A. Ritsataki, G. New, R. Melish, S. C. W. Hyde, P. M. W. French, and J. Taylor, “Theoretical modeling of gain-guiding effects in experimental all-solid-state KLM lasers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 185–192 (1998).
[CrossRef]

A. Minassian, B. Thompson, G. Smith, and M. Damzen, “High-power scaling (>100 W) of a diode-pumped TEM00 Nd:GdVO4 laser system,” IEEE J. Sel. Top. Quantum Electron. 11(3), 621–625 (2005).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39(3), 1253–1263 (1989).
[CrossRef] [PubMed]

B. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27(4), 1989–2002 (1983).
[CrossRef]

Phys. Rev. Lett. (2)

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused-gain amplifier,” Phys. Rev. Lett. 70(11), 1607–1610 (1993).
[CrossRef] [PubMed]

J. E. LaSala, D. A. Deacon, and J. M. Madey, “Optical guiding in a free-electron-laser oscillator,” Phys. Rev. Lett. 59(18), 2047–2050 (1987).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

J. H. Marburger, “Self-focusing theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Other (3)

W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer-Verlag Publication, 2006), Chap. 7.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation, Fundamentals, Advanced Concepts and Applications, 2nd ed. (Springer-Verlag Publication, 2005), Chap. 24.

B. Schulz, M. Frede, R. Wilhelm, and D. Kracht, “High power end-pumped Nd:YVO4 amplifier,” in Conference on Advanced Solid-State Photonics, Technical Digest (CD) (Optical Society of America, 2006), paper WB15.

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Figures (11)

Fig. 1
Fig. 1

The scheme for the iterative procedure of the slice algorithm operating in solving the Maxwell's equations.

Fig. 2
Fig. 2

The configuration of a typical end-pumped laser amplifier.

Fig. 3
Fig. 3

The power distributions and fitting curves of three composed input beams. (a) the Gaussian beam with spherical aberration (GA), (b) the hollow beam with spherical aberration (HA), (c) the quasi-Gaussian beam with spherical aberration (QGA).

Fig. 4
Fig. 4

The relationship between M2 factor of input beam and M2 factor of output beam.

Fig. 5
Fig. 5

The dependency of the M2 factor of output beam and output power on the pump radii in the situation of wl = 300, 400, 500μm respectively.

Fig. 6
Fig. 6

The dependency of the M2 factor of output beam on the input power in the situation of Ip = 25, 40W respectively.

Fig. 7
Fig. 7

The M2 factor of output beam and output power versus the length of crystal.

Fig. 8
Fig. 8

The beam profiles and beam quality factors before and after the amplification. (a), (c) are from the same signal beam but at different positions near the focal region; (b), (d) are from the corresponding output beam respectively.

Fig. 9
Fig. 9

The M2 factor of output beam with different input beam radius in experiments and in theory.

Fig. 10
Fig. 10

The M2 factor of output beam with the input beam power in experiment and theory for Ip = 25, 40W respectively.

Fig. 11
Fig. 11

The M2 factor of output beam with LD operating temperature in experiment.

Tables (1)

Tables Icon

Table 1 Parameters for the Theoretical Model of Gain-Guided Laser Amplifier

Equations (20)

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

2 ψ2ik ψ z =( ikg+2k k 0 Δn )ψ,
E=Re( ψ )exp[ i( ωtkz ) ].
ψ( x,y,z+Δz )=exp[ ( i 2k 2 + g 2 i k 0 Δn )Δz ]ψ( x,y,z ).
ψ( x,y,z+Δz )=exp( iΔz 4k 2 )exp[ ( g 2 i k 0 Δn )Δz ]exp( iΔz 4k 2 )ψ( x,y,z ).
ψ 1 ( x,y,z+ Δz /2 )=exp( iΔz 4k 2 )ψ( x,y,z ),
ψ 1 ( x,y,z+ Δz /2 )= F 1 { F[ ψ( x,y,z ) ]exp( ik( Δz /2 ) 1 λ l 2 f x 2 λ l 2 f y 2 ) },
ψ 2 ( x,y,z+ Δz /2 )=exp[ Δz( g 2 i k 0 Δn ) ] ψ 1 ( x,y,z+ Δz /2 ),
ψ( x,y,z+Δz )=exp( iΔz 4k 2 ) ψ 2 ( x,y,z+ Δz /2 ).
ψ( x,y,z+Δz )= F 1 { F[ ψ 2 ( x,y,z+ Δz /2 ) ]exp( ik( Δz /2 ). 1 λ l 2 f x 2 λ l 2 f y 2 ) }.
g 0 = α σ 21 τ f P pump h v p exp( αz ).
g( x,y,z )= 1 I l ( x,y,z ) d I l ( x,y,z ) dz = g 0 ( x,y,z ) 1+ I l ( x,y,z ) / I sat ,
I sat = h v l / σ 21 τ f .
K x 2 T( x,y,z ) x 2 + K y 2 T( x,y,z ) y 2 + K z 2 T( x,y,z ) z 2 =Q( x,y,z ),
Q( x,y,z )= Q 0 exp[ 2 ( x 2 + y 2 ) 2 / w p 4 ]exp( αz ),
Q 0 = η( I p I extract ) crystal exp[ 2 ( x 2 + y 2 ) 2 / w p 4 ]exp( αz ) dxdydz ,
Δn= dn dT ( T T 0 ).
M 2 = π λ AC B 2 4 ,
w x 2 = 4 + + x 2 E( x,y,z ) E * ( x,y,z ) dxdy + + E( x,y,z ) E * ( x,y,z ) dxdy ,
ψ( x,y,0 )=A( x,y,0 )exp[ iφ( x,y,0 ) ],
φ( x,y,0 )= c 0 ( x 2 + y 2 ) 2 + φ 0 ,

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