Abstract

The plasma electrons in pulsed far ir lasers constitute a temporary divergent refractive element. This lens element can be strong enough to prohibit stable laser modes and serves to explain the delay commonly reported between current and laser pulses. With electron recombination the cavity rapidly becomes stable, Q switching the laser pulse. The lens effect and Q-switching rate are calculated in terms of tube dimensions, laser frequency, electron density, and radial electron density distribution. Higher laser frequencies appear earlier when this effect is operative. A simple criterion is given to distinguish this delay effect from other sources of laser pulse delay. Experimental measurements are presented of cavity stability as a function of mirror curvature and electron density; wall reflections are shown to be important. The radial electron density distribution is compared with plasma theory results. Lens effects associated with the laser gain are shown to be negligible.

© 1970 Optical Society of America

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References

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  1. F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
    [CrossRef]
  2. S. Kon, M. Yamanaka, J. Yamamoto, H. Yoshinaga, Japan. J. Phys. 6, 612 (1967).
    [CrossRef]
  3. S. Kon, M. Otsuka, M. Yamanaka, H. Yoshinaga, Japan. J. Appl. Phys. 7, 434 (1968).
    [CrossRef]
  4. V. Sochor, E. Brannen, Appl. Phys. Lett. 10, 232 (1967).
    [CrossRef]
  5. H. Steffen, B. Keller, F. K. Kneubühl, Electron. Lett. 3, 562 (1967).
    [CrossRef]
  6. R. G. Jones, C. C. Bradley, J. Chamberlain, H. A. Gebbie, N. W. B. Stone, H. Sixsmith, Appl. Opt. 8, 701 (1969).
    [CrossRef] [PubMed]
  7. E. Brannen, V. Sochor, W. J. Sarjeant, H. R. Froelich, Proc. IEEE 55, 562 (1967).
  8. M. Yamanaka, S. Kon, J. Yamamoto, H. Yoshinaga, Japan. J. Appl. Phys. 7, 554 (1968).
    [CrossRef]
  9. H. Steffen, F. K. Kneubühl, IEEE J. Quant. Electron. QE-4, 992 (1968).
    [CrossRef]
  10. L. E. S. Mathias, A. Crocker, M. S. Wills, IEEE J. Quantum Electron. QE-4, 205 (1968).
    [CrossRef]
  11. R. Turner, T. O. Poehler, J. Appl. Phys. 39, 5726 (1968).
    [CrossRef]
  12. See, for example, R. Turner, A. K. Hochberg, T. O. Poehler, Appl. Phys. Lett. 12, 104 (1968), and P. G. Frayne, J. Phys. B, 2, 247 (1969).
    [CrossRef]
  13. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  14. M. Bertolotti, Nuovo Cimento 32, 1242 (1964); E. R. Caianiello, A. Turrin, Nuovo Cimento 10, 594 (1953).
    [CrossRef]
  15. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  16. V. Sochor, Czech. J. Phys. B18, 910 (1968).
    [CrossRef]
  17. J. P. Markiewicz, J. L. Emmett, Appl. Opt. 5, 1687 (1966).
    [CrossRef] [PubMed]
  18. E. D. Nelson, J. Y. Wong, Appl. Opt. 6, 1259 (1967).
    [CrossRef] [PubMed]
  19. See E. H. Putley, Appl. Opt. 4, 649 (1965).
    [CrossRef]
  20. D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).
  21. T. Li, H. Zucker, J. Opt. Soc. Amer. 57, 984 (1967).
    [CrossRef]
  22. A. E. Siegman, Proc. IEEE 53, 277 (1965); see also A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
    [CrossRef]
  23. A. L. Bloom, Gas Lasers (John Wiley & Sons, New York, 1968), Chap. 3.
  24. W. Schottky, Physik. Z. 25, 635 (1929); L. Tonks, I. Langmuir, Phys. Rev. 34, 876 (1929); G. Francis, Handbuch der Physik (Julius Springer-Verlag, Berlin, 1956), Vol. 22, p. 53.
    [CrossRef]
  25. S. A. Self, H. N. Ewald, Phys. Fluids 9, 2486 (1966).
    [CrossRef]
  26. H. Greenstein, Phys. Rev. 175, 438 (1968). Greenstein treats inhomogeneously-broadened lines and gain saturation in terms of phenomenological relaxation constants using a generalized Bloch formalism. The results given here for an unsaturated, homogeneously-broadened gain line are readily extended using the Greenstein development. This extension, however, would yield no stronger effects than the present analysis does.
    [CrossRef]
  27. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  28. L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
    [CrossRef]

1969 (1)

1968 (9)

H. Greenstein, Phys. Rev. 175, 438 (1968). Greenstein treats inhomogeneously-broadened lines and gain saturation in terms of phenomenological relaxation constants using a generalized Bloch formalism. The results given here for an unsaturated, homogeneously-broadened gain line are readily extended using the Greenstein development. This extension, however, would yield no stronger effects than the present analysis does.
[CrossRef]

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

M. Yamanaka, S. Kon, J. Yamamoto, H. Yoshinaga, Japan. J. Appl. Phys. 7, 554 (1968).
[CrossRef]

H. Steffen, F. K. Kneubühl, IEEE J. Quant. Electron. QE-4, 992 (1968).
[CrossRef]

L. E. S. Mathias, A. Crocker, M. S. Wills, IEEE J. Quantum Electron. QE-4, 205 (1968).
[CrossRef]

R. Turner, T. O. Poehler, J. Appl. Phys. 39, 5726 (1968).
[CrossRef]

See, for example, R. Turner, A. K. Hochberg, T. O. Poehler, Appl. Phys. Lett. 12, 104 (1968), and P. G. Frayne, J. Phys. B, 2, 247 (1969).
[CrossRef]

S. Kon, M. Otsuka, M. Yamanaka, H. Yoshinaga, Japan. J. Appl. Phys. 7, 434 (1968).
[CrossRef]

V. Sochor, Czech. J. Phys. B18, 910 (1968).
[CrossRef]

1967 (7)

F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
[CrossRef]

S. Kon, M. Yamanaka, J. Yamamoto, H. Yoshinaga, Japan. J. Phys. 6, 612 (1967).
[CrossRef]

T. Li, H. Zucker, J. Opt. Soc. Amer. 57, 984 (1967).
[CrossRef]

V. Sochor, E. Brannen, Appl. Phys. Lett. 10, 232 (1967).
[CrossRef]

H. Steffen, B. Keller, F. K. Kneubühl, Electron. Lett. 3, 562 (1967).
[CrossRef]

E. Brannen, V. Sochor, W. J. Sarjeant, H. R. Froelich, Proc. IEEE 55, 562 (1967).

E. D. Nelson, J. Y. Wong, Appl. Opt. 6, 1259 (1967).
[CrossRef] [PubMed]

1966 (2)

1965 (5)

See E. H. Putley, Appl. Opt. 4, 649 (1965).
[CrossRef]

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).

A. E. Siegman, Proc. IEEE 53, 277 (1965); see also A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

1964 (1)

M. Bertolotti, Nuovo Cimento 32, 1242 (1964); E. R. Caianiello, A. Turrin, Nuovo Cimento 10, 594 (1953).
[CrossRef]

1961 (1)

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

1929 (1)

W. Schottky, Physik. Z. 25, 635 (1929); L. Tonks, I. Langmuir, Phys. Rev. 34, 876 (1929); G. Francis, Handbuch der Physik (Julius Springer-Verlag, Berlin, 1956), Vol. 22, p. 53.
[CrossRef]

Allen, C.

F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
[CrossRef]

Arams, F.

F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
[CrossRef]

Bertolotti, M.

M. Bertolotti, Nuovo Cimento 32, 1242 (1964); E. R. Caianiello, A. Turrin, Nuovo Cimento 10, 594 (1953).
[CrossRef]

Bloom, A. L.

A. L. Bloom, Gas Lasers (John Wiley & Sons, New York, 1968), Chap. 3.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Bradley, C. C.

Brannen, E.

V. Sochor, E. Brannen, Appl. Phys. Lett. 10, 232 (1967).
[CrossRef]

E. Brannen, V. Sochor, W. J. Sarjeant, H. R. Froelich, Proc. IEEE 55, 562 (1967).

Button, K.

F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
[CrossRef]

Casperson, L.

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Chamberlain, J.

Crocker, A.

L. E. S. Mathias, A. Crocker, M. S. Wills, IEEE J. Quantum Electron. QE-4, 205 (1968).
[CrossRef]

Emmett, J. L.

Ewald, H. N.

S. A. Self, H. N. Ewald, Phys. Fluids 9, 2486 (1966).
[CrossRef]

Froelich, H. R.

E. Brannen, V. Sochor, W. J. Sarjeant, H. R. Froelich, Proc. IEEE 55, 562 (1967).

Gebbie, H. A.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Greenstein, H.

H. Greenstein, Phys. Rev. 175, 438 (1968). Greenstein treats inhomogeneously-broadened lines and gain saturation in terms of phenomenological relaxation constants using a generalized Bloch formalism. The results given here for an unsaturated, homogeneously-broadened gain line are readily extended using the Greenstein development. This extension, however, would yield no stronger effects than the present analysis does.
[CrossRef]

Hochberg, A. K.

See, for example, R. Turner, A. K. Hochberg, T. O. Poehler, Appl. Phys. Lett. 12, 104 (1968), and P. G. Frayne, J. Phys. B, 2, 247 (1969).
[CrossRef]

Jones, R. G.

Keller, B.

H. Steffen, B. Keller, F. K. Kneubühl, Electron. Lett. 3, 562 (1967).
[CrossRef]

Kneubühl, F. K.

H. Steffen, F. K. Kneubühl, IEEE J. Quant. Electron. QE-4, 992 (1968).
[CrossRef]

H. Steffen, B. Keller, F. K. Kneubühl, Electron. Lett. 3, 562 (1967).
[CrossRef]

Kogelnik, H.

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Kon, S.

M. Yamanaka, S. Kon, J. Yamamoto, H. Yoshinaga, Japan. J. Appl. Phys. 7, 554 (1968).
[CrossRef]

S. Kon, M. Otsuka, M. Yamanaka, H. Yoshinaga, Japan. J. Appl. Phys. 7, 434 (1968).
[CrossRef]

S. Kon, M. Yamanaka, J. Yamamoto, H. Yoshinaga, Japan. J. Phys. 6, 612 (1967).
[CrossRef]

Li, T.

T. Li, H. Zucker, J. Opt. Soc. Amer. 57, 984 (1967).
[CrossRef]

Markiewicz, J. P.

Mathias, L. E. S.

L. E. S. Mathias, A. Crocker, M. S. Wills, IEEE J. Quantum Electron. QE-4, 205 (1968).
[CrossRef]

McCumber, D. E.

D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).

Nelson, E. D.

Otsuka, M.

S. Kon, M. Otsuka, M. Yamanaka, H. Yoshinaga, Japan. J. Appl. Phys. 7, 434 (1968).
[CrossRef]

Poehler, T. O.

R. Turner, T. O. Poehler, J. Appl. Phys. 39, 5726 (1968).
[CrossRef]

See, for example, R. Turner, A. K. Hochberg, T. O. Poehler, Appl. Phys. Lett. 12, 104 (1968), and P. G. Frayne, J. Phys. B, 2, 247 (1969).
[CrossRef]

Putley, E. H.

Rubin, L.

F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
[CrossRef]

Sarjeant, W. J.

E. Brannen, V. Sochor, W. J. Sarjeant, H. R. Froelich, Proc. IEEE 55, 562 (1967).

Schottky, W.

W. Schottky, Physik. Z. 25, 635 (1929); L. Tonks, I. Langmuir, Phys. Rev. 34, 876 (1929); G. Francis, Handbuch der Physik (Julius Springer-Verlag, Berlin, 1956), Vol. 22, p. 53.
[CrossRef]

Self, S. A.

S. A. Self, H. N. Ewald, Phys. Fluids 9, 2486 (1966).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Proc. IEEE 53, 277 (1965); see also A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

Sixsmith, H.

Sochor, V.

V. Sochor, Czech. J. Phys. B18, 910 (1968).
[CrossRef]

E. Brannen, V. Sochor, W. J. Sarjeant, H. R. Froelich, Proc. IEEE 55, 562 (1967).

V. Sochor, E. Brannen, Appl. Phys. Lett. 10, 232 (1967).
[CrossRef]

Steffen, H.

H. Steffen, F. K. Kneubühl, IEEE J. Quant. Electron. QE-4, 992 (1968).
[CrossRef]

H. Steffen, B. Keller, F. K. Kneubühl, Electron. Lett. 3, 562 (1967).
[CrossRef]

Stone, N. W. B.

Turner, R.

R. Turner, T. O. Poehler, J. Appl. Phys. 39, 5726 (1968).
[CrossRef]

See, for example, R. Turner, A. K. Hochberg, T. O. Poehler, Appl. Phys. Lett. 12, 104 (1968), and P. G. Frayne, J. Phys. B, 2, 247 (1969).
[CrossRef]

Wang, M.

F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
[CrossRef]

Wills, M. S.

L. E. S. Mathias, A. Crocker, M. S. Wills, IEEE J. Quantum Electron. QE-4, 205 (1968).
[CrossRef]

Wong, J. Y.

Yamamoto, J.

M. Yamanaka, S. Kon, J. Yamamoto, H. Yoshinaga, Japan. J. Appl. Phys. 7, 554 (1968).
[CrossRef]

S. Kon, M. Yamanaka, J. Yamamoto, H. Yoshinaga, Japan. J. Phys. 6, 612 (1967).
[CrossRef]

Yamanaka, M.

S. Kon, M. Otsuka, M. Yamanaka, H. Yoshinaga, Japan. J. Appl. Phys. 7, 434 (1968).
[CrossRef]

M. Yamanaka, S. Kon, J. Yamamoto, H. Yoshinaga, Japan. J. Appl. Phys. 7, 554 (1968).
[CrossRef]

S. Kon, M. Yamanaka, J. Yamamoto, H. Yoshinaga, Japan. J. Phys. 6, 612 (1967).
[CrossRef]

Yariv, A.

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Yoshinaga, H.

S. Kon, M. Otsuka, M. Yamanaka, H. Yoshinaga, Japan. J. Appl. Phys. 7, 434 (1968).
[CrossRef]

M. Yamanaka, S. Kon, J. Yamamoto, H. Yoshinaga, Japan. J. Appl. Phys. 7, 554 (1968).
[CrossRef]

S. Kon, M. Yamanaka, J. Yamamoto, H. Yoshinaga, Japan. J. Phys. 6, 612 (1967).
[CrossRef]

Zucker, H.

T. Li, H. Zucker, J. Opt. Soc. Amer. 57, 984 (1967).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (3)

V. Sochor, E. Brannen, Appl. Phys. Lett. 10, 232 (1967).
[CrossRef]

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

See, for example, R. Turner, A. K. Hochberg, T. O. Poehler, Appl. Phys. Lett. 12, 104 (1968), and P. G. Frayne, J. Phys. B, 2, 247 (1969).
[CrossRef]

Bell Syst. Tech. J. (3)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).

Czech. J. Phys. (1)

V. Sochor, Czech. J. Phys. B18, 910 (1968).
[CrossRef]

Electron. Lett. (1)

H. Steffen, B. Keller, F. K. Kneubühl, Electron. Lett. 3, 562 (1967).
[CrossRef]

IEEE J. Quant. Electron. (1)

H. Steffen, F. K. Kneubühl, IEEE J. Quant. Electron. QE-4, 992 (1968).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. E. S. Mathias, A. Crocker, M. S. Wills, IEEE J. Quantum Electron. QE-4, 205 (1968).
[CrossRef]

J. Appl. Phys. (1)

R. Turner, T. O. Poehler, J. Appl. Phys. 39, 5726 (1968).
[CrossRef]

J. Opt. Soc. Amer. (1)

T. Li, H. Zucker, J. Opt. Soc. Amer. 57, 984 (1967).
[CrossRef]

Japan. J. Appl. Phys. (2)

M. Yamanaka, S. Kon, J. Yamamoto, H. Yoshinaga, Japan. J. Appl. Phys. 7, 554 (1968).
[CrossRef]

S. Kon, M. Otsuka, M. Yamanaka, H. Yoshinaga, Japan. J. Appl. Phys. 7, 434 (1968).
[CrossRef]

Japan. J. Phys. (1)

S. Kon, M. Yamanaka, J. Yamamoto, H. Yoshinaga, Japan. J. Phys. 6, 612 (1967).
[CrossRef]

Nuovo Cimento (1)

M. Bertolotti, Nuovo Cimento 32, 1242 (1964); E. R. Caianiello, A. Turrin, Nuovo Cimento 10, 594 (1953).
[CrossRef]

Phys. Fluids (1)

S. A. Self, H. N. Ewald, Phys. Fluids 9, 2486 (1966).
[CrossRef]

Phys. Rev. (1)

H. Greenstein, Phys. Rev. 175, 438 (1968). Greenstein treats inhomogeneously-broadened lines and gain saturation in terms of phenomenological relaxation constants using a generalized Bloch formalism. The results given here for an unsaturated, homogeneously-broadened gain line are readily extended using the Greenstein development. This extension, however, would yield no stronger effects than the present analysis does.
[CrossRef]

Physik. Z. (1)

W. Schottky, Physik. Z. 25, 635 (1929); L. Tonks, I. Langmuir, Phys. Rev. 34, 876 (1929); G. Francis, Handbuch der Physik (Julius Springer-Verlag, Berlin, 1956), Vol. 22, p. 53.
[CrossRef]

Proc. IEEE (3)

F. Arams, C. Allen, M. Wang, K. Button, L. Rubin, Proc. IEEE 55, 420 (1967).
[CrossRef]

A. E. Siegman, Proc. IEEE 53, 277 (1965); see also A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

E. Brannen, V. Sochor, W. J. Sarjeant, H. R. Froelich, Proc. IEEE 55, 562 (1967).

Other (1)

A. L. Bloom, Gas Lasers (John Wiley & Sons, New York, 1968), Chap. 3.

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Figures (15)

Fig. 1
Fig. 1

Laser resonator stability analysis.

Fig. 2
Fig. 2

Cavity stability diagram including electron lens effect for the experimental laser.

Fig. 3
Fig. 3

Cavity stability diagram including electron lens effect for L/r0 = 60.

Fig. 4
Fig. 4

Cavity stability dynamics.

Fig. 5
Fig. 5

Experimental apparatus.

Fig. 6
Fig. 6

Microwave interferometer data. Upper traces: 1000-A peak current pulse. Lower traces: microwave signal as noted.

Fig. 7
Fig. 7

Radial electron distributions.

Fig. 8
Fig. 8

Microwave interferometer fringes.

Fig. 9
Fig. 9

Electron density decay.

Fig. 10
Fig. 10

Experimental results.

Fig. 11
Fig. 11

(a) Radial model comparison: β = 0.65. (b) Radial model comparison: β = 1.0. (c) Radial model comparison: β = 1.44.

Fig. 12
Fig. 12

Typical laser emission data. Upper trace: 2 μsec/cm, 500 A/cm current pulses. Lower traces: 2 μsec/cm, laser pulses as a function of cavity length near cavity resonance.

Fig. 13
Fig. 13

Dispersion associated with laser again.

Fig. 14
Fig. 14

Molecular refraction lens focal lengths.

Fig. 15
Fig. 15

Molecular gain guided beam radius.

Tables (1)

Tables Icon

Table I Quantities Plotted in Figs. 14 and 15

Equations (38)

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ω 2 = ω p 2 + c 2 k 2 ,
n 2 = 1 N / N p , N p = 0 m ω 2 / e 2 .
N ( r ) = N 0 [ 1 β ( r / r 0 ) 2 ] .
n 1 1 2 N 0 N + 1 2 ( β N 0 r 0 2 N p ) r 2 , N 0 N p 1 , d n d r α 2 r , α 2 = β r 0 2 N 0 N p .
( d / d s ) [ n ( r ) ( d r / d s ) ] = grad n ( r )
d 2 r d z 2 = 1 n d n d r = α 2 r .
| r 2 r 2 | = | cosh α L 1 / α sinh α L α sinh α L cosh α L | | r 1 r 1 | = | a b c d | | r 1 r 1 | ,
0 ( cosh α L sinh α L α R 1 ) ( cosh α L sinh α L α R 2 ) = G 1 G 2 1
0 ( L R sinh ( m ) m cosh ( m ) ) 2 = G 2 1 , M = 1 ( α L ) 2 = r 0 2 L 2 N p N 0 = 1 m 2 .
G 2 = 0 = > L / R = m coth ( m ) ( confocal cavity ) , G 2 = 1 = > L / R = m coth ( m ) ± m / sinh ( m ) ( + concentric cavity plane parallel cavity ) .
F = r 0 2 / λ L 1
M = ( r 0 2 / L 2 ) ( N p / N 0 ) 1
N 0 / N p λ / L .
p = 1 λ [ 2 r 0 2 0 r 0 Δ n ( r ) d r ] , n 2 = 1 ( Δ N 0 / N p ) [ 1 β ( r / r 0 ) 2 ] .
0 r 0 [ 1 β ( r / r 0 ) 2 ] d r r 0 1 = 1 β 3 , p = 2 r 0 λ { 1 [ 1 Δ N 0 N p ( 1 β 3 ) ] 1 2 } .
1 + [ 1 ( g 1 g 2 ) 1 ] 1 2 1 [ 1 ( g 1 g 2 ) 1 ] 1 2 .
1 / q = ( 1 / R ) ( i λ / π w 2 ) .
q 2 = ( A q 1 + B ) / ( C q 1 + D ) ,
M = | A B C D | .
w 2 = 2 λ B / π [ 4 ( A + D ) 2 ] 1 2 .
N ( r ) = N 0 J 0 ( 2.4 r / r 0 ) .
J 0 ( x ) 1 ( x / 2 ) 2 , N ( r ) = N 0 [ 1 ( 2.4 2 r r 0 ) 2 ] = N 0 [ 1 1.44 ( r r 0 ) 2 ] ,
A = ν I / ( ν I + ν i n ) ,
n 1 1 2 ( N 0 / N p ) ,
n L = q λ vac 2 ( q c 2 ) ( 1 f ) ,
q = 2 n L / λ vac = 9563.
Δ q q = Δ n n = 1 2 Δ N 0 N p { 0 r 0 [ 1 β ( r r 0 ) 2 ] 2 π r d r / π r 0 2 } = 1 2 Δ N 0 N p ( 1 β 2 ) ;
Δ N 0 = 2 ( N p ) ( Δ q ) / 0.675 q = 3.0 × 10 12 .
n 2 = μ 1 + χ ( ω ) = 1 + χ ( ω ) + i χ ( ω ) .
χ ( ω ) = i χ 0 / ( 1 + i x ) = χ 0 ( x 1 + x 2 + i 1 + x 2 ) , x = ( ω ω 0 ) T 2 ,
χ ( r ) = χ ( ω ) [ 1 β ( r / r 0 ) 2 ] , n 1 + 1 2 χ ( ω ) [ 1 β ( r / r 0 ) 2 ] 1 1 2 χ ( ω ) β ( r 2 / r 0 2 ) ,
n 1 ± 1 2 ( β / r 0 2 ) ( χ 0 / 2 ) r 2 , + negative lens , x < 0. positive lens , x > 0.
α 2 = β r 0 2 χ 0 2 , d 2 r d z 2 ± α 2 r = 0.
f = 1 / α sin α L , positive lens , f = 1 / α sinh α L , negative lens .
χ 0 2 = λ 2 π α 0 = λ η 2 π 8.686 ,
n 2 = 1 α 2 r 2 , α 2 = i β χ 0 / r 0 2 ,
1 q m = i α = ( 1 i ) [ β χ 0 2 r 0 2 ] 1 2 = 1 R m i λ π w m 2
n 2 = 1 + α 2 r 2 , α 2 = β N 0 / r 0 2 N p , 1 q m = α = 1 R m 1 λ π w m 2

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