Abstract

We consider a compact three-mirror cavity consisting of a flat output coupler, a curved folding mirror, and an active medium with one facet cut at the Brewster angle and the other facet coated for unit reflectivity. We examine the sensitivity to thermal lensing and to self-focusing in the active medium of the Gaussian beam that is circulating in that cavity. We use a simple thin-lens model; the astigmatism of the beam that is circulating in the cavity and the nonlinear coupling between the field distributions along the two orthogonal axes are taken into account. We find configurations in which beam ellipticity is compensated for at either end of the cavity in the presence of thermal lensing. We have derived an analytical criterion that predicts the sensitivity of the beam size to nonlinear lensing. The ability of the cavity to favor self-mode locking is found to be sensitive to the strength of thermal lensing. In the absence of thermal lensing, cavities operated as telescopic systems (C = 0) or self-imaging systems (B = 0) are most appropriate for achieving self-mode locking, with nonlinear mode selection accomplished through saturation of the spatially varying laser gain. We identify conditions for which self-mode locking can be produced by variable-reflectivity output couplers with either maximum or minimum reflectivity at the center of the coupler. We use our model to estimate the nonlinear gain produced in laser cavities equipped with such output couplers. We identify a cavity configuration for which nonlinear lensing can simultaneously produce mode locking and correction of beam ellipticity at the output coupler.

© 2000 Optical Society of America

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1998 (2)

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

X. G. Huang, M. R. Wang, “Analytical design for Kerr-lens mode locking of compact solid-state lasers,” Opt. Commun. 158, 322–330 (1998).
[CrossRef]

1997 (6)

1996 (2)

1995 (2)

1994 (8)

1993 (2)

1992 (6)

1991 (4)

1990 (1)

M. E. Innoncenzi, H. T. Yura, C. L. Fincher, R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Agnesi, A.

A. Agnesi, E. Piccinini, G. C. Reali, “Influence of thermal effects in Kerr-lens mode-locked femtosecond Cr4+:forsterite lasers,” Opt. Commun. 135, 77–82 (1997).
[CrossRef]

Agrawal, G. P.

Albers, P.

J. Frauchiger, P. Albers, H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped cw Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1055 (1992).
[CrossRef]

Alfano, R. R.

Anctil, G.

G. Anctil, “Régimes d’émission d’un laser à l’état solide muni d’une cavité en V,” master’s degree thesis (Université Laval, Cité Universitaire, Québec, (Canada, 1998).

Andreas, W. A.

M. Mehendale, T. R. Nelson, F. G. Omenetto, W. A. Andreas, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

Barthelemy, A.

Bouma, B. E.

Boyd, R. W.

Brabec, T.

Bridges, R. E.

Buck, P.

M. Sheik-Bahae, P. Li-Kam-Wa, P. Buck, “Optimization of Kerr-lens resonators,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 179.

Burns, D.

Cassanho, A.

Cerullo, G.

Chai, B. H. T.

Chilla, J. L. A.

Christov, I. P.

Conlon, P. J.

Cormier, J.-F.

J.-F. Cormier, M. Piché, F. Salin, “Suppression of beam breakup in self-mode-locked Ti:sapphire lasers,” Opt. Lett. 19, 1225–1227 (1994).
[CrossRef] [PubMed]

J.-F. Cormier, “Étude de la génération d’impulsions brèves dans les lasers à autosynchronisation modale,” doctoral dissertation (Université Laval, Cité Universitaire, Québec, Canada, 1996).

Couderc, V.

Curley, P. F.

De Silvestri, S.

Dymott, M. J. P.

Ferguson, A. I.

Fields, R. A.

M. E. Innoncenzi, H. T. Yura, C. L. Fincher, R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innoncenzi, H. T. Yura, C. L. Fincher, R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Frauchiger, J.

J. Frauchiger, P. Albers, H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped cw Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1055 (1992).
[CrossRef]

French, P. M. W.

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

P. J. Conlon, Y. P. Tong, P. M. W. French, J. R. Taylor, A. V. Shestakov, “Passive mode locking and dispersion measurement of a sub-100-fs Cr4+:YAG laser,” Opt. Lett. 19, 1468–1470 (1994).
[CrossRef] [PubMed]

Froehly, C.

Fujimoto, J. G.

Guy, O.

Haus, H. A.

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

Hermann, J.

Hopkins, J.-M.

Huang, C.-P.

Huang, F.

X. G. Huang, F. Huang, W.-K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

Huang, X. G.

X. G. Huang, M. R. Wang, “Analytical design for Kerr-lens mode locking of compact solid-state lasers,” Opt. Commun. 158, 322–330 (1998).
[CrossRef]

X. G. Huang, F. Huang, W.-K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

Innoncenzi, M. E.

M. E. Innoncenzi, H. T. Yura, C. L. Fincher, R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

Jenssen, H. P.

Jung, I. D.

Kalashnikov, V. L.

Kalosha, V. P.

Kapteyn, H. C.

Kärtner, F. X.

Kean, P. N.

Keller, U.

Kennedy, G. T.

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 4th ed. (Springer-Verlag, New York, 1996), Chap. 7.
[CrossRef]

Krausz, F.

Lee, W.-K.

X. G. Huang, F. Huang, W.-K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

LiKamWa, P.

Li-Kam-Wa, P.

M. Sheik-Bahae, P. Li-Kam-Wa, P. Buck, “Optimization of Kerr-lens resonators,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 179.

Louradour, F.

Loza-Alvarez, D.

Magni, V.

Malcolm, G. P. A.

Martinez, O. E.

Matuschek, N.

Mehendale, M.

M. Mehendale, T. R. Nelson, F. G. Omenetto, W. A. Andreas, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

Mellish, R.

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

Mikhailov, V. P.

Miller, A.

Monguzzi, A.

Morier-Genoud, F.

Murnane, M. M.

Nelson, T. R.

M. Mehendale, T. R. Nelson, F. G. Omenetto, W. A. Andreas, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

New, G. H. C.

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

New, S. C. W.

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

Omenetto, F. G.

M. Mehendale, T. R. Nelson, F. G. Omenetto, W. A. Andreas, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

Pallaro, L.

Petricevic, V.

Piccinini, E.

A. Agnesi, E. Piccinini, G. C. Reali, “Influence of thermal effects in Kerr-lens mode-locked femtosecond Cr4+:forsterite lasers,” Opt. Commun. 135, 77–82 (1997).
[CrossRef]

Piché, M.

Poloyko, I. G.

Ramaswamy-Paye, M.

Reali, G. C.

A. Agnesi, E. Piccinini, G. C. Reali, “Influence of thermal effects in Kerr-lens mode-locked femtosecond Cr4+:forsterite lasers,” Opt. Commun. 135, 77–82 (1997).
[CrossRef]

Ritsataki, A.

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

Salin, F.

Scheuer, V.

Seas, A.

Sheik-Bahae, M.

M. Sheik-Bahae, P. Li-Kam-Wa, P. Buck, “Optimization of Kerr-lens resonators,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 179.

Shestakov, A. V.

Sibbett, W.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Spence, D. E.

Spielman, Ch.

Spielmann, Ch.

Squier, J. A.

Sutter, D. H.

Szipöcs, R.

Taft, G.

Taylor, J. R.

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

P. J. Conlon, Y. P. Tong, P. M. W. French, J. R. Taylor, A. V. Shestakov, “Passive mode locking and dispersion measurement of a sub-100-fs Cr4+:YAG laser,” Opt. Lett. 19, 1468–1470 (1994).
[CrossRef] [PubMed]

Tilsch, M.

Tong, Y. P.

Tschudi, T.

Valentine, G. J.

Valster, A.

Van Stryland, E. W.

Wang, M. R.

X. G. Huang, M. R. Wang, “Analytical design for Kerr-lens mode locking of compact solid-state lasers,” Opt. Commun. 158, 322–330 (1998).
[CrossRef]

X. G. Huang, F. Huang, W.-K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

Weber, H. P.

J. Frauchiger, P. Albers, H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped cw Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1055 (1992).
[CrossRef]

Wise, F. W.

Xu, L.

Yanovsky, V. P.

Yura, H. T.

M. E. Innoncenzi, H. T. Yura, C. L. Fincher, R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Zhang, G.

Zhou, J.

Appl. Phys. Lett. (1)

M. E. Innoncenzi, H. T. Yura, C. L. Fincher, R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. Frauchiger, P. Albers, H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped cw Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1055 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

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

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

Opt. Commun. (5)

M. Piché, “Beam reshaping and self-mode-locking in nonlinear laser resonators,” Opt. Commun. 86, 156–160 (1991).
[CrossRef]

A. Agnesi, E. Piccinini, G. C. Reali, “Influence of thermal effects in Kerr-lens mode-locked femtosecond Cr4+:forsterite lasers,” Opt. Commun. 135, 77–82 (1997).
[CrossRef]

M. Mehendale, T. R. Nelson, F. G. Omenetto, W. A. Andreas, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

X. G. Huang, M. R. Wang, “Analytical design for Kerr-lens mode locking of compact solid-state lasers,” Opt. Commun. 158, 322–330 (1998).
[CrossRef]

X. G. Huang, F. Huang, W.-K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

Opt. Lett. (21)

D. E. Spence, P. N. Kean, W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991).
[CrossRef] [PubMed]

G. P. A. Malcolm, A. I. Ferguson, “Self-mode locking of a diode-pumped Nd:YLF laser,” Opt. Lett. 16, 1967–1969 (1991).
[CrossRef] [PubMed]

A. Miller, P. LiKamWa, B. H. T. Chai, E. W. Van Stryland, “Generation of 150-fs tunable pulses in Cr:LiSrAlF6,” Opt. Lett. 17, 195–197 (1992).
[CrossRef] [PubMed]

A. Seas, V. Petricevic, R. R. Alfano, “Generation of sub-100-fs pulses from a cw mode-locked chromium-doped forsterite laser,” Opt. Lett. 17, 937–939 (1992).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielman, P. F. Curley, F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292–1294 (1992).
[CrossRef] [PubMed]

R. E. Bridges, R. W. Boyd, G. P. Agrawal, “Effect of beam ellipticity on self-mode locking in lasers,” Opt. Lett. 18, 2026–2028 (1993).
[CrossRef] [PubMed]

G. Cerullo, S. De Silvestri, V. Magni, L. Pallaro, “Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers,” Opt. Lett. 19, 807–809 (1994).
[CrossRef] [PubMed]

V. Couderc, O. Guy, A. Barthelemy, C. Froehly, F. Louradour, “Self-optimized resonator for optical pumping of solid-state lasers,” Opt. Lett. 19, 1134–1136 (1994).
[CrossRef] [PubMed]

J. Zhou, G. Taft, C.-P. Huang, M. M. Murnane, H. C. Kapteyn, I. P. Christov, “Pulse evolution in a broad-bandwidth Ti:sapphire laser,” Opt. Lett. 19, 1149–1151 (1994).
[CrossRef] [PubMed]

J.-F. Cormier, M. Piché, F. Salin, “Suppression of beam breakup in self-mode-locked Ti:sapphire lasers,” Opt. Lett. 19, 1225–1227 (1994).
[CrossRef] [PubMed]

P. J. Conlon, Y. P. Tong, P. M. W. French, J. R. Taylor, A. V. Shestakov, “Passive mode locking and dispersion measurement of a sub-100-fs Cr4+:YAG laser,” Opt. Lett. 19, 1468–1470 (1994).
[CrossRef] [PubMed]

M. Ramaswamy-Paye, J. G. Fujimoto, “Compact dispersion-compensating geometry for Kerr-lens mode-locked femtosecond lasers,” Opt. Lett. 19, 1756–1758 (1994).
[CrossRef] [PubMed]

M. J. P. Dymott, A. I. Ferguson, “Self-mode-locked diode-pumped Cr:LiSAF laser,” Opt. Lett. 19, 1988–1990 (1994).
[CrossRef] [PubMed]

V. P. Yanovsky, F. W. Wise, A. Cassanho, H. P. Jenssen, “Kerr-lens mode-locked diode-pumped Cr:LiSGAF laser,” Opt. Lett. 20, 1304–1306 (1995).
[CrossRef] [PubMed]

I. D. Jung, F. X. Kärtner, N. Matuschek, D. H. Sutter, F. Morier-Genoud, G. Zhang, U. Keller, V. Scheuer, M. Tilsch, T. Tschudi, “Self-starting 6.5-fs pulses from a Ti:sapphire laser,” Opt. Lett. 22, 1009–1011 (1997).
[CrossRef] [PubMed]

G. J. Valentine, J.-M. Hopkins, D. Loza-Alvarez, G. T. Kennedy, W. Sibbett, D. Burns, A. Valster, “Ultralow-pump-threshold, femtosecond Cr3+:LiSrAlF6 laser pumped by a single narrow-stripe AlGaInP laser diode,” Opt. Lett. 22, 1639–1641 (1997).
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Other (6)

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M. Sheik-Bahae, P. Li-Kam-Wa, P. Buck, “Optimization of Kerr-lens resonators,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 179.

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J.-F. Cormier, “Étude de la génération d’impulsions brèves dans les lasers à autosynchronisation modale,” doctoral dissertation (Université Laval, Cité Universitaire, Québec, Canada, 1996).

W. Koechner, Solid-State Laser Engineering, 4th ed. (Springer-Verlag, New York, 1996), Chap. 7.
[CrossRef]

G. Anctil, “Régimes d’émission d’un laser à l’état solide muni d’une cavité en V,” master’s degree thesis (Université Laval, Cité Universitaire, Québec, (Canada, 1998).

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of the V cavity.

Fig. 2
Fig. 2

Geometric stability diagrams along the tangential (continuous curves) and sagittal (dashed curves) directions for the bare V cavity under consideration. The following values of angle θ were used: (a) 5°, (b) 10°, (c) 20°. The common stability zones are indicated by the shaded areas.

Fig. 3
Fig. 3

Angles θ that yield the same beam waist at M3 along tangential and sagittal directions in a bare V cavity.

Fig. 4
Fig. 4

Zones of geometric stability with θ = 11° along the tangential (continuous curves) and the sagittal (dashed curves) directions in the presence of (a) a weak thermal lens with F th/R m = 0.5 and (b) a strong thermal lens with F th/R m = 0.1. The common stability zones are indicated by the shaded areas and the arrows.

Fig. 5
Fig. 5

Angles θ that produce the same beam waist at M3 along tangential and sagittal directions in the presence of (a) a weak thermal lens with F th/R m = 0.5 and (b) a strong thermal lens with F th/R m = 0.1.

Fig. 6
Fig. 6

Angles θ that produce the same beam waist at M1 in tangential and sagittal directions in the presence of (a) a weak thermal lens with F th/R m = 0.5 and (b) a strong thermal lens with F th/R m = 0.1.

Fig. 7
Fig. 7

Beam waist along the tangential (continuous curves) and the sagittal (dashed curves) directions in the presence of a weak thermal lens (F th/R m = 0.5) as a function of circulating power with L 1/R m = L 2/R m = 0.5 (a) at M3 with θ = 15.5° and (b) at M1 with θ = 2°.

Fig. 8
Fig. 8

Beam waist at M3 along the tangential (continuous curves) and the sagittal (dashed curves) directions in the presence of a strong thermal lens (F th/R m = 0.1) as a function of circulating power for (a) L 1/R m = 0.491, L 2/R m = 0.5, and θ = 15.5°; (b) L 1/R m = 0.6, L 2/R m = 2.7, and θ = 9.5°.

Fig. 9
Fig. 9

Spatial profile of the reflectivity of output coupler M3 with a valley at the center (dashed curve, with δ = 0.10) and a crest at the center (continuous curve, δ = -0.10).

Fig. 10
Fig. 10

Nonlinear gain ΔG as a function of circulating power for (a) the cavity described in Fig. 8(a) with w m /R m = 0.010 and δ = 0.10 and (b) the cavity described in Fig. 8(b) with w m /R m = 0.0015 and δ = -0.10. In (b) the continuous and dashed curves represent ΔG obtained, respectively, with and without considering the reflectivity profile of M3 in the calculation of the beam waist.

Equations (52)

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s-1Lc/n0s0s,  10-2/Re1,  10-1/F1,
ε1=2L1cRe-1,  ε2=2L2Re-1,
A=D=2ε1ε2-1,
B=-s2Reε11-ε1ε2,
C=4ε2/s2Re.
πw12λ=s2Re|ε1|1-ε1ε21-1-2ε1ε221/2.
πw32λ=Re2ε2ε11-ε1ε21/2.
A=D=A-B/Fth,
C=-2AFth+BFth2+C,
-1<A-BFth<1.
ε1=2Fths2Re-1,  ε2=-11+2Fth/s2Re.
πw32λ=Re21-ε1ε2ε2+s2Re2Fth1-ε1ε2ε11-ε1s2Re2Fth1/2.
πw12λ=s2Re|ε1|1-ε1ε21-1-2ε1ε2-s2ε1ReFth1-ε1ε221/2.
αiwi6+βiwi3wj+γiwi2+δiwj2=0,
αi=1-Ai-BiFth2,
βi=16BiRmneffPcircπLcRmAi-BiFth,
γi=-λBiπ2,
δi=-8BiRmneffPcircπLcRm2.
Sp=1wpdwpdp,
S1p=-12|ε1|s2Re1-ε1ε21-2ε1ε2-ε1s21-ε1ε2Rep1-1-2ε1ε2-ε1s21-ε1ε2Rep2.
S1p±s2Re/8ε2,
S1p-18s2Reε2+s2Rep/2,
pcrit=1-2ε1ε2ε1s2Re1-ε1ε2,
S3p=s2Re8f1pf2p=s2Re81-ε1s2Rep2ε2+s2Rep1-ε1ε22.
Rx, y=1+δ-δ exp-2x2+y2/wm2,
Rx, y=1-δ exp-2x2+y2/wm2,
ΔG=Lpcw-LpNL,
Lpcw or NL=δ1+wtan2wm21+wsag2wm21/2+const.
1rddrr ddr Tz, r+1KT Qz, r=0,
Qz, r=Q0zexp-2r2/wp2,
Q0z=2αhπwp2 P0 exp-αz,
h=1-λpλ+1-ηλpλ,
Tz, 0-Tz, r=Q0zwp28KTE12r2wp2+ln2r2wp2+γ,
Tz, 0-Tz, r=-Q0zwp28KTn=11n1n!-2r2wp2n,
E1u=ue-ttdt,
Tz, 0-Tz, rQ0zr24KT.
ϕthr=dn/dTwp2λKT hP01-exp-αLcr2,
ϕthr=πr2/λFth,
Fth=πKTwp2PhdndT,
Ph=hP01-exp-αLc,
Eoutx, y=Einx, yexp-iϕNLx, y,
ϕNLx, y=2πλ Lcn2Imax exp-2x2wx2-2y2wy2,
ϕNLx, y-2πλ Lcn2Imaxbax2x2wx2+2y2aywy2.
Imax=2Pcircπwxwy.
Fx=πwx3wy8LcneffPcirc,
Fy=πwxwy38LcneffPcirc.
Rx, y=exp-2x2+y2/wm2,
10-1/F˜1,
1F˜=iλπwm2.
Ã=D˜=A-BF+Re21-ε1ε2ε2+Re2F1-ε1ε21F˜,
B˜=B-s2Re241-ε1ε221F˜,
C˜=C-2AF+BF2-s2ε2s2+Re2F1-ε1ε221F˜.

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