Abstract

Laser resonators containing self-focusing elements are described using the matrix formalism appropriate for Gaussian beams. The laser rods are not approximated by traditional optical elements but treated as thick materials showing a radial quadratic dependence of their refractive index. Cavities incorporating an arbitrary number of lenses, mirrors, and crystals (including rods with spherical endfaces) are described with no restriction on the value of the pump power. Stability domains and beam size in the resonator and inside the rods are calculated. A few experimental tests with Nd:YAP lasers are presented.

© 1987 Optical Society of America

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References

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  1. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, New York, 1976).
  2. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, NJ, 1981).
  3. L. Tarassov, Physique des processus dans les générateurs de rayonnement optique cohérent (MIR, Moscow, 1981; traduction française, 1985).
  4. W. Koechner, “Absorbed Pump Power, Thermal Profile and Stresses in a cw Pumped Nd:YAG Laser Rod,” Appl. Opt. 9, 1429 (1970);“Thermal Lensing in a Nd:YAG Laser Rod,” 9, 2548 (1970).
    [CrossRef] [PubMed]
  5. J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
    [CrossRef]
  6. See, for example, D. Vivien, “Les matériaux pour lasers,” Rev. Phys. Appl. 21, 709 (1986) orJ. J. Aubert, C. Wyon, D. Vivien, A. M. Lejus, “Crystal Growth and Optical Characteristics of Lanthanide Hexaluminate La1−xNdxMgAl11O19,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1986), paper TUK35.
    [CrossRef]
  7. G. A. Massey, J. M. Yarborough, “High Average Power Operation and Nonlinear Optical Generation with the Nd:YAlO3 Laser,” Appl. Phys. Lett. 18, 576 (1971);A. A. Kaminski et al., “Anisotropy of Spectroscopic Characteristics in Triaxial YAlO3-Nd3+ Laser Crystals,” Phys. Status Solidi 51, 509 (1979).
    [CrossRef]
  8. E. Reed, “A Flashlamp-Pumped Q-Switched Cr:Nd:GSGG Laser,” IEEE J. Quantum Electron. QE-21, 1625 (1985).
    [CrossRef]
  9. D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).
  10. L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
    [CrossRef]
  11. T. Y. Fan, A. Cordova-Plaza, M. J. F. Digonnet, R. L. Byer, H. J. Shaw, “Nd MgO LiNbO3 Spectroscopy and Laser Devices,” J. Opt. Soc. Am. B 3, 140 (1986).
    [CrossRef]
  12. P. Moulton, Opt. News 9, 000 (1982);P. Hammerling, A. B. Budgor, A. Pinto, Tunable Solid-State Lasers (Springer-Verlag, Berlin, 1985).
  13. M. J. P. Payne, H. W. Evans, “Flashlamp-Pumped Lasing of Chromium: GSGG,” in Technical Digest, Topical Meeting on Tunable Solid-State Lasers (Optical Society of America, Washington, DC, 1985), paper FA4.
  14. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  15. V. Magni, “Resonators for Solid-State Lasers with Large-Volume Fundamental Mode and High Alignment Stability,” Appl. Opt. 25, 107 (1986).
    [CrossRef] [PubMed]
  16. G. Herziger, H. Weber, “Equivalent Optical Resonators,” Appl. Opt. 23, 1450 (1984).
    [CrossRef] [PubMed]
  17. H. P. Kortz, R. Iffländer, H. Weber, “Stability and Beam Divergence of Multimode Lasers with Internal Variable Lenses,” Appl. Opt. 20, 4124 (1981).
    [CrossRef] [PubMed]
  18. S. Reynaud, Ecole Normale Superieure, Paris; private communication.

1986 (4)

See, for example, D. Vivien, “Les matériaux pour lasers,” Rev. Phys. Appl. 21, 709 (1986) orJ. J. Aubert, C. Wyon, D. Vivien, A. M. Lejus, “Crystal Growth and Optical Characteristics of Lanthanide Hexaluminate La1−xNdxMgAl11O19,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1986), paper TUK35.
[CrossRef]

L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
[CrossRef]

T. Y. Fan, A. Cordova-Plaza, M. J. F. Digonnet, R. L. Byer, H. J. Shaw, “Nd MgO LiNbO3 Spectroscopy and Laser Devices,” J. Opt. Soc. Am. B 3, 140 (1986).
[CrossRef]

V. Magni, “Resonators for Solid-State Lasers with Large-Volume Fundamental Mode and High Alignment Stability,” Appl. Opt. 25, 107 (1986).
[CrossRef] [PubMed]

1985 (1)

E. Reed, “A Flashlamp-Pumped Q-Switched Cr:Nd:GSGG Laser,” IEEE J. Quantum Electron. QE-21, 1625 (1985).
[CrossRef]

1984 (2)

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

G. Herziger, H. Weber, “Equivalent Optical Resonators,” Appl. Opt. 23, 1450 (1984).
[CrossRef] [PubMed]

1982 (1)

P. Moulton, Opt. News 9, 000 (1982);P. Hammerling, A. B. Budgor, A. Pinto, Tunable Solid-State Lasers (Springer-Verlag, Berlin, 1985).

1981 (1)

1971 (1)

G. A. Massey, J. M. Yarborough, “High Average Power Operation and Nonlinear Optical Generation with the Nd:YAlO3 Laser,” Appl. Phys. Lett. 18, 576 (1971);A. A. Kaminski et al., “Anisotropy of Spectroscopic Characteristics in Triaxial YAlO3-Nd3+ Laser Crystals,” Phys. Status Solidi 51, 509 (1979).
[CrossRef]

1970 (2)

1966 (1)

Aubert, J. J.

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

Auzel, F.

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

Byer, R. L.

Collongues, R.

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

Cordova-Plaza, A.

Digonnet, M. J. F.

Evans, H. W.

M. J. P. Payne, H. W. Evans, “Flashlamp-Pumped Lasing of Chromium: GSGG,” in Technical Digest, Topical Meeting on Tunable Solid-State Lasers (Optical Society of America, Washington, DC, 1985), paper FA4.

Fan, T. Y.

Foster, J. D.

J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

Herziger, G.

Iffländer, R.

Koechner, W.

Kogelnik, H.

Kortz, H. P.

Leduc, M.

L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
[CrossRef]

Lejus, A. M.

L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
[CrossRef]

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

Li, T.

Magni, V.

Massey, G. A.

G. A. Massey, J. M. Yarborough, “High Average Power Operation and Nonlinear Optical Generation with the Nd:YAlO3 Laser,” Appl. Phys. Lett. 18, 576 (1971);A. A. Kaminski et al., “Anisotropy of Spectroscopic Characteristics in Triaxial YAlO3-Nd3+ Laser Crystals,” Phys. Status Solidi 51, 509 (1979).
[CrossRef]

Moncorgé, R.

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

Moulton, P.

P. Moulton, Opt. News 9, 000 (1982);P. Hammerling, A. B. Budgor, A. Pinto, Tunable Solid-State Lasers (Springer-Verlag, Berlin, 1985).

Osterink, L. M.

J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

Payne, M. J. P.

M. J. P. Payne, H. W. Evans, “Flashlamp-Pumped Lasing of Chromium: GSGG,” in Technical Digest, Topical Meeting on Tunable Solid-State Lasers (Optical Society of America, Washington, DC, 1985), paper FA4.

Reed, E.

E. Reed, “A Flashlamp-Pumped Q-Switched Cr:Nd:GSGG Laser,” IEEE J. Quantum Electron. QE-21, 1625 (1985).
[CrossRef]

Reynaud, S.

S. Reynaud, Ecole Normale Superieure, Paris; private communication.

Schearer, L. D.

L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
[CrossRef]

Shaw, H. J.

Tarassov, L.

L. Tarassov, Physique des processus dans les générateurs de rayonnement optique cohérent (MIR, Moscow, 1981; traduction française, 1985).

Thery, J.

L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
[CrossRef]

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, NJ, 1981).

Vivien, D.

L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
[CrossRef]

See, for example, D. Vivien, “Les matériaux pour lasers,” Rev. Phys. Appl. 21, 709 (1986) orJ. J. Aubert, C. Wyon, D. Vivien, A. M. Lejus, “Crystal Growth and Optical Characteristics of Lanthanide Hexaluminate La1−xNdxMgAl11O19,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1986), paper TUK35.
[CrossRef]

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

Weber, H.

Yarborough, J. M.

G. A. Massey, J. M. Yarborough, “High Average Power Operation and Nonlinear Optical Generation with the Nd:YAlO3 Laser,” Appl. Phys. Lett. 18, 576 (1971);A. A. Kaminski et al., “Anisotropy of Spectroscopic Characteristics in Triaxial YAlO3-Nd3+ Laser Crystals,” Phys. Status Solidi 51, 509 (1979).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

G. A. Massey, J. M. Yarborough, “High Average Power Operation and Nonlinear Optical Generation with the Nd:YAlO3 Laser,” Appl. Phys. Lett. 18, 576 (1971);A. A. Kaminski et al., “Anisotropy of Spectroscopic Characteristics in Triaxial YAlO3-Nd3+ Laser Crystals,” Phys. Status Solidi 51, 509 (1979).
[CrossRef]

C. R. Acad. Sci. Paris (1)

D. Vivien, A. M. Lejus, J. Thery, R. Collongues, J. J. Aubert, R. Moncorgé, F. Auzel, “Observation de l'effet laser continu dans l'aluminate La0.3Nd0.1MgAl11O19 Monocristallin (LNA) elabori par la methode Czochralski,” C. R. Acad. Sci. Paris 298, 195 (1984).

IEEE J. Quantum Electron. (2)

L. D. Schearer, M. Leduc, D. Vivien, A. M. Lejus, J. Thery, “LNA: A New CW Nd Laser Tunable Around 1.05 and 1.08 μm,” IEEE J. Quantum Electron. QE-22, 713 (1986).
[CrossRef]

E. Reed, “A Flashlamp-Pumped Q-Switched Cr:Nd:GSGG Laser,” IEEE J. Quantum Electron. QE-21, 1625 (1985).
[CrossRef]

J. Appl. Phys. (1)

J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

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

Opt. News (1)

P. Moulton, Opt. News 9, 000 (1982);P. Hammerling, A. B. Budgor, A. Pinto, Tunable Solid-State Lasers (Springer-Verlag, Berlin, 1985).

Rev. Phys. Appl. (1)

See, for example, D. Vivien, “Les matériaux pour lasers,” Rev. Phys. Appl. 21, 709 (1986) orJ. J. Aubert, C. Wyon, D. Vivien, A. M. Lejus, “Crystal Growth and Optical Characteristics of Lanthanide Hexaluminate La1−xNdxMgAl11O19,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1986), paper TUK35.
[CrossRef]

Other (5)

W. Koechner, Solid-State Laser Engineering (Springer-Verlag, New York, 1976).

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, NJ, 1981).

L. Tarassov, Physique des processus dans les générateurs de rayonnement optique cohérent (MIR, Moscow, 1981; traduction française, 1985).

M. J. P. Payne, H. W. Evans, “Flashlamp-Pumped Lasing of Chromium: GSGG,” in Technical Digest, Topical Meeting on Tunable Solid-State Lasers (Optical Society of America, Washington, DC, 1985), paper FA4.

S. Reynaud, Ecole Normale Superieure, Paris; private communication.

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Figures (20)

Fig. 1
Fig. 1

Principal features of a Gaussian beam. w0 is the beam-waist size, R(z) is the radius of curvature of the wavefront at abscissa z, w(z) is the spot size at z. z0 is the confocal parameter.

Fig. 2
Fig. 2

Beam propagation through a laser crystal showing a quadratic radial gradient of refractive index: (a) the incident beam is divergent; (b) the incident beam has its beam waist located on the entrance surface of the rod.

Fig. 3
Fig. 3

Position d′ of the beam waist of the merging beam after crossing the focusing rod as a function of the absorbed pump power Pa in the rod (for the definition of P0, see text). The shape of the beam profile inside the rod is shown in the additional drawings corresponding in turn to positive and negative sections of the curve.

Fig. 4
Fig. 4

Example of the general laser resonators calculated in this paper; they may contain lenses, mirrors, and focusing rods (including rods with spherical endfaces).

Fig. 5
Fig. 5

Laser cavity corresponding to calculations shown in Figs. 511. The rod is 5 cm long, its refractive index n0 is 1.9; the end mirrors are identical and spherical with radius R; R′ is the curvature of the two end surfaces of the rod.

Fig. 6
Fig. 6

Resonator of Fig. 5 with R = 30 cm, Pa = 0.2P0 (Pa, absorbed pump power in the rod), R′ = ∞: (a) stability zones for variable position L1 and L2 of the end mirrors; (b) volume v of the Gaussian beam in the laser rod as a function of the end mirror position L for a symmetrical cavity (L1 = L2 = L); (b) corresponds to points on the diagonal of (a).

Fig. 7
Fig. 7

Same caption as for Fig. 6, except for the value of R′. Here R′ = 60 cm.

Fig. 8
Fig. 8

Same caption as for Fig. 6, except for the value of R′. Here R′ = 40 cm.

Fig. 9
Fig. 9

Resonator of Fig. 5 with R = 30 cm and L1 = L2 = L (symmetrical cavity). Stability zones showing L, position of end mirrors, as a function of Pa, the absorbed pump power in the rod: (a) R′ = ∞;(b) R′ = 10 cm.

Fig. 10
Fig. 10

Profile of the Gaussian beam in the resonators of Fig. 9 for L = 10 cm and Pa/P0 = 0.4. Curve A corresponds to point A in Fig. 9(a) (R′ = ∞) and curve B to point B in Fig. 9(b) (R′ = 10 cm).

Fig. 11
Fig. 11

Resonator of Fig. 5 with R = 10 cm, R′ = ∞: (a) Stability map showing L, position of end mirrors, as a function of Pa, the absorbed pump power in the rod, up to very high values of Pa. (b) Profile of the Gaussian beam in the resonators of Fig. 12(a) for L = 7.5 cm and Pa/P0 = 10 (curve V) and Pa/P0 = 40 (curve W), corresponding to points V and W in Fig. 12(a).

Fig. 12
Fig. 12

(a) Ring symmetrical resonator including two laser crystals; (b) equivalent linear half-cavity.

Fig. 13
Fig. 13

Resonator of Fig. 12b with f = R/2 = 10 cm, d = 10 cm, n0 = 1.85: (a) Stability map in the (L2,L3) plane. L1 = 20 cm, Pa/P0 = 0.32. Horizontal axis: L2 varies from 60 to 100 cm; vertical axis: L3 varies from 0 to 20 cm. (b) Profile of the Gaussian beam in the cavity corresponding to point A of (a): Pa/P0 = 0.32, L1 = 20 cm, L2 = 87 cm, L3 = 5.5 cm. (c) Pumped volume in the rod as a function of absorbed power Pa for the same parameters of cavity as for (b), except for Pa/P0 which varies between 0.27 and 0.35. Operation at Pa/P0 = 0.32 corresponds to point A.

Fig. 14
Fig. 14

Experimental results obtained with a Nd:YAP crystal of length d = 9 cm, diameter 4 mm, and refractive index n0 = 1.8. The crystal is pumped by two Kr+ lamps in Microcontrole YAG 904 cavity. Vertical axis: d/n0d′, where d′ is the distance between the beam waist and the crystal for an incident plane wave (d′ is defined in Fig. 2). Horizontal axis: electrical power supplied by the lamps.

Fig. 15
Fig. 15

Experimental results obtained with the Nd:YAP crystal studied in Fig. 14; R = 25 cm, d = 9 cm, n0 = 1.8: (a) laser resonator; (b) stability map in the (L1,Pa) plane for L2 = 45 cm; L1, distance between the spherical mirror and the crystal; Pa, absorbed pump power; (c) corresponding results in the (L2,Pa) plane for L1 = 55 cm.

Fig. 16
Fig. 16

For a resonator with a spherical end mirror of center C, located at M, the beam waist is located at O between C and M, and the confocal parameter z0 is equal to OH (see text).

Fig. 17
Fig. 17

Geometrical construction of the beam waist in a two-mirror resonator. Mirrors M1 and M2 are centered in C1 and C2, respectively. If the two circles shown in dashed lines intersect, the resonator is stable. O is the position of the beam waist, OH is equal to the confocal parameter z0.

Fig. 18
Fig. 18

Transformation of a Gaussian beam through a thin lens. O and O′ are the positions of the beam waists. The wavefronts at focal points F and F′ are centered at C and C′, respectively. C and O′ (and also O and C′) are conjugate points in geometrical optics.

Fig. 19
Fig. 19

Cavity α, containing a thin lens of focal length f, is equivalent to cavity β, containing no lens. The beam waist is located at O (see text).

Fig. 20
Fig. 20

Stability zones of the resonator shown in Fig. 19(α). The figure is drawn for the following values of R1, the radius of the spherical mirror, and f, the focal length of the thin lens: R1 = 30 cm, f = 40 cm. The analytical functions corresponding to each curve of the stability map are: (α) L1 = R1 + ff2(fL2); (β) L1 = f; (γ) L1 = f + R1; (δ) L1 = f + f2/(L2f); () L1 = f + f2/(L2f) + R1.

Equations (45)

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u ( r , z ) = F ( z ) exp [ r 2 / w ( z ) 2 ] exp [ i n π r 2 / λ 0 R ( z ) ] ,
r 2 = x 2 + y 2 ,
1 / n q ( z ) = 1 / R ( z ) + i λ 0 / ( n π ) [ w ( z ) ] 2 ,
w ( z ) 2 = w 0 2 [ 1 + ( z / z 0 ) 2 ] ;
R ( z ) = z [ 1 + ( z 0 / z ) 2 ] ;
q ( z ) = z / n i z 0 / n ;
z 0 = n π w 0 2 / λ 0 .
q ( z ) = [ A q ( z ) + B ] / [ C q ( z ) + D ] .
M = ( A B C D ) ,
M = ( 1 d / n 0 1 ) ,
d = z z .
M = ( 1 0 V 1 )
n = n 0 η r 2 / 2
M = [ cos b ( d n 0 ) ( sin b b ) ( n 0 d ) b sin b cos b ] ,
b = d η / n 0 .
M = ( 1 d / n 0 η d 1 ) .
T ( r ) = T ( 0 ) α r 2
α = P a 4 π r 0 2 L κ ,
n ( r ) = n 0 α ( dn / dT ) r 2 ,
η = P a ( dn / dT ) 2 π r 0 2 L κ = P a P 0 L 2 ,
b = d [ P a ( d n / d T ) 2 π n 0 r 0 2 L κ ] 1 / 2 = d L P a P 0 ,
P 0 = 2 π n 0 r 0 2 κ ( d n / d T ) L .
l / q ( z ) = [ c q ( z ) + D ] / [ A q ( z ) + B ] ,
Im [ 1 / q ( z ) ] = λ 0 ( n 0 π ) [ ω ( z ) ] 2 = w 0 2 w ( z ) 2 z 0 .
w ( z ) = w 0 cos 2 b + [ β sin b + ( z / z 0 ) cos b ] 2 ,
β = ( d / n 0 b ) ( 1 / z 0 ) = ( L / n 0 z 0 ) P 0 P a .
w ( z ) = w 0 cos 2 b + β 2 sin 2 b ,
w ( z ) = w 0 cos b .
Re [ q ( z + L ) ] = z ( z + L ) = d .
d = ( L / n 0 ) P a P 0 + ( n 0 z 0 2 / L ) P 0 P a cotan b + [ ( n 0 z 0 / L ) P 0 P a ] 2 tan b .
d = L n 0 b tan b = L n 0 P a P 0 tan P a P 0 .
P a / P 0 = m π 2 ,
d L n 0 ( P a / P 0 ) ;
q = [ Aq + B ] / [ Cq + D ] ,
{ | A + D | < 2 , ( A D ) / C > 0 , C 0 .
q p = A D 2 C i [ [ A D ] / 2 C ] 2 B .
υ = z z + L π [ w ( z ) ] 2 dz ,
O H 2 ¯ = C O ¯ O M ¯ .
O M ¯ = z .
C O ¯ = R ( z ) z = ( z 0 ) 2 z .
O H 2 ¯ = ( z 0 ) 2 .
{ O < M 1 M 2 < R 1 . R 2 < M 1 M 2 < R 1 + R 2 .
C F ¯ O F ¯ = f 2 .
M 2 F ¯ = f 2 / M 2 F ¯ = f 2 / L 2 f ,
{ O < M 1 F < R 1 f 2 / ( L 2 f ) < M 1 F < [ f 2 / ( L 2 f ) ] + R 1 .

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