Abstract

The eigenvalues for all the significant low-order resonant modes of an unstable optical resonator with circular mirrors are computed using an eigenvalue method called the Prony method. A general equivalence relation is also given, by means of which one can obtain the design parameters for a single-ended unstable resonator of the type usually employed in practical lasers, from the calculated or tabulated values for an equivalent symmetric or double-ended unstable resonator.

© 1970 Optical Society of America

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References

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  1. A. E. Siegman, Proc. IEEE 53, 277 (1965).
    [CrossRef]
  2. A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electronics QE-3, 156 (1967).
    [CrossRef]
  3. W. K. Kahn, Appl. Opt. 5, 407 (1966).
    [CrossRef] [PubMed]
  4. S. R. Barone, Appl. Opt. 6, 861 (1967).
    [CrossRef] [PubMed]
  5. L. Bergstein, Appl. Opt. 7, 495 (1968).
    [CrossRef] [PubMed]
  6. W. Streifer, IEEE J. Quantum Electronics QE-4, 229 (1968).
    [CrossRef]
  7. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2129 (1969).
    [CrossRef] [PubMed]
  8. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
    [CrossRef] [PubMed]
  9. D. C. Sinclair, T. H. E. Cottrell, Appl. Opt. 6, 845 (1967).
    [CrossRef] [PubMed]
  10. Yu A. Anenev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—Doklady 13, 351 (1968).
  11. W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electronics QE-5, 575 (1969).
    [CrossRef]
  12. Y. A. Ananev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—JETP 28, 69 (1969).
  13. A. G. Fox, T. Li, Bell Sys. Tech. J. 40, 453 (1961).
  14. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 131 (1969).
    [CrossRef] [PubMed]
  15. E. Bodewig, Matrix Calculus (North-Holland Publ. Co., Amsterdam, and Wiley, New York, 1956), esp. Part IV-A, Sec. 2.5.2, p. 256.
  16. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965), especially p. 369, and Chap. 9.
  17. M. Clint, A. Jennings, Computer J. 13, 76 (1970).
    [CrossRef]
  18. Gaspard Riche Prony, J. l’Ecole Polytechnique Paris, 2nd Cahier 1, 24 (December1795).
  19. L. Weiss, R. N. McDonough, SIAM Rev. 5, 145 (1963).
    [CrossRef]
  20. D. F. Tuttle, in Aspects of Network and System Theory, R. E. Kalman, N. DeClaris, Eds. (Holt, Rinehart and Winston, New York, 1971), Chap. 7.
  21. J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

1970 (1)

M. Clint, A. Jennings, Computer J. 13, 76 (1970).
[CrossRef]

1969 (5)

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electronics QE-5, 575 (1969).
[CrossRef]

Y. A. Ananev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—JETP 28, 69 (1969).

R. L. Sanderson, W. Streifer, Appl. Opt. 8, 131 (1969).
[CrossRef] [PubMed]

R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2129 (1969).
[CrossRef] [PubMed]

R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
[CrossRef] [PubMed]

1968 (3)

L. Bergstein, Appl. Opt. 7, 495 (1968).
[CrossRef] [PubMed]

W. Streifer, IEEE J. Quantum Electronics QE-4, 229 (1968).
[CrossRef]

Yu A. Anenev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—Doklady 13, 351 (1968).

1967 (3)

1966 (1)

1965 (1)

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

1964 (1)

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

1963 (1)

L. Weiss, R. N. McDonough, SIAM Rev. 5, 145 (1963).
[CrossRef]

1961 (1)

A. G. Fox, T. Li, Bell Sys. Tech. J. 40, 453 (1961).

Ananev, Y. A.

Y. A. Ananev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—JETP 28, 69 (1969).

Anenev, Yu A.

Yu A. Anenev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—Doklady 13, 351 (1968).

Arrathoon, R.

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electronics QE-3, 156 (1967).
[CrossRef]

Barone, S. R.

Bergstein, L.

Bodewig, E.

E. Bodewig, Matrix Calculus (North-Holland Publ. Co., Amsterdam, and Wiley, New York, 1956), esp. Part IV-A, Sec. 2.5.2, p. 256.

Clint, M.

M. Clint, A. Jennings, Computer J. 13, 76 (1970).
[CrossRef]

Cottrell, T. H. E.

Fox, A. G.

A. G. Fox, T. Li, Bell Sys. Tech. J. 40, 453 (1961).

Gordon, J. P.

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

Jennings, A.

M. Clint, A. Jennings, Computer J. 13, 76 (1970).
[CrossRef]

Kahn, W. K.

Kogelnik, H.

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

Krupke, W. F.

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electronics QE-5, 575 (1969).
[CrossRef]

Li, T.

A. G. Fox, T. Li, Bell Sys. Tech. J. 40, 453 (1961).

McDonough, R. N.

L. Weiss, R. N. McDonough, SIAM Rev. 5, 145 (1963).
[CrossRef]

Prony, Gaspard Riche

Gaspard Riche Prony, J. l’Ecole Polytechnique Paris, 2nd Cahier 1, 24 (December1795).

Sanderson, R. L.

Sherstobitov, V. E.

Y. A. Ananev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—JETP 28, 69 (1969).

Yu A. Anenev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—Doklady 13, 351 (1968).

Siegman, A. E.

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electronics QE-3, 156 (1967).
[CrossRef]

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

Sinclair, D. C.

Sooy, W. R.

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electronics QE-5, 575 (1969).
[CrossRef]

Streifer, W.

Sventsitskaya, N. A.

Y. A. Ananev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—JETP 28, 69 (1969).

Yu A. Anenev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—Doklady 13, 351 (1968).

Tuttle, D. F.

D. F. Tuttle, in Aspects of Network and System Theory, R. E. Kalman, N. DeClaris, Eds. (Holt, Rinehart and Winston, New York, 1971), Chap. 7.

Weiss, L.

L. Weiss, R. N. McDonough, SIAM Rev. 5, 145 (1963).
[CrossRef]

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965), especially p. 369, and Chap. 9.

Appl. Opt. (7)

Bell Sys. Tech. J. (1)

A. G. Fox, T. Li, Bell Sys. Tech. J. 40, 453 (1961).

Bell Syst. Tech. J. (1)

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

Computer J. (1)

M. Clint, A. Jennings, Computer J. 13, 76 (1970).
[CrossRef]

IEEE J. Quantum Electronics (3)

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electronics QE-3, 156 (1967).
[CrossRef]

W. F. Krupke, W. R. Sooy, IEEE J. Quantum Electronics QE-5, 575 (1969).
[CrossRef]

W. Streifer, IEEE J. Quantum Electronics QE-4, 229 (1968).
[CrossRef]

Proc. IEEE (1)

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

SIAM Rev (1)

L. Weiss, R. N. McDonough, SIAM Rev. 5, 145 (1963).
[CrossRef]

Sov. Phys.—Doklady (1)

Yu A. Anenev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—Doklady 13, 351 (1968).

Sov. Phys.—JETP (1)

Y. A. Ananev, N. A. Sventsitskaya, V. E. Sherstobitov, Sov. Phys.—JETP 28, 69 (1969).

Other (4)

Gaspard Riche Prony, J. l’Ecole Polytechnique Paris, 2nd Cahier 1, 24 (December1795).

E. Bodewig, Matrix Calculus (North-Holland Publ. Co., Amsterdam, and Wiley, New York, 1956), esp. Part IV-A, Sec. 2.5.2, p. 256.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965), especially p. 369, and Chap. 9.

D. F. Tuttle, in Aspects of Network and System Theory, R. E. Kalman, N. DeClaris, Eds. (Holt, Rinehart and Winston, New York, 1971), Chap. 7.

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Figures (11)

Fig. 1
Fig. 1

Unstable optical resonator configurations: (a) symmetric or double-ended case; (b) single-ended case.

Fig. 2
Fig. 2

Eigenvalues of all significant l = 0 (axially symmetric) eigenmodes for the unstable resonator with circular mirrors, for the case g = 2.6 or M = 5.0, and for 0 ≤ Neq ≤ 28 (0 ≤ N ≲ 13).

Fig. 3
Fig. 3

Eigenvalue magnitudes for the l = 0 modes for g = 1.25 and M = 2.

Fig. 4
Fig. 4

Eigenvalue magnitudes for the l = 1 modes for g = 1.25 and M = 2.

Fig. 5
Fig. 5

Eigenvalue magnitudes and phases for l = 0 and g = 1.1, M = 1.558.

Fig. 6
Fig. 6

Eigenvalue magnitudes and phases for l = 1 and g = 1.1, M = 1.558.

Fig. 7
Fig. 7

Eigenvalue magnitudes and phases for l = 0 and g = 1.05, M = 1.370.

Fig. 8
Fig. 8

Eigenvalue magnitudes and phases for l = 1 and g = 1.05, M = 1.370.

Fig. 9
Fig. 9

Eigenvalue magnitudes and phases for l = 0 and g = 1.01, M = 1.152.

Fig. 10
Fig. 10

Eigenvalue magnitudes and phases for l = 1 and g = 1.01, M = 1.152.

Fig. 11
Fig. 11

The power loss per bounce at the first dominant mode peak Neq = 0.5 plotted vs the magnification parameter [(M2 − 1)/M2]2. The loss predicted by the geometrical optics analysis is also shown for comparison.

Equations (16)

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Mu = λ u ,
v 0 = k 1 u 1 + k 2 u 2 + k 3 u 3 + v 1 = k 1 λ 1 u 1 + k 2 λ 2 u 2 + k 3 λ 3 u 3 + v m = k 1 λ m 1 u 1 + k 2 λ m 2 u 2 + k 3 λ m 3 u 3 +
u n · u m i ( u n ) i ( u m ) i = δ n m ,
F m v m · v m - m = k 1 2 λ 1 m + k 2 2 λ 2 m + k 3 2 λ 3 m +
λ N + Q 1 λ N - 1 + + Q N - 1 λ + Q N = 0.
F m + N + Q 1 F m + N - 1 + + Q N - 1 F m + 1 + Q N F m = 0
F N Q 1 + F N - 1 Q 2 +         + F 1 Q N = - F N + 1 F N + 1 Q 1 + F N Q 2 +         + F 2 Q N = - F N + 2 F 2 N - 1 Q 1 + F 2 N - 2 Q 2 + + F N Q N = - F 2 N
u n l ( r , θ ) = φ n l ( r a ) e - j l θ ,
γ φ ( x ) = j l + 1 c 0 1 y J l ( c x y ) exp [ - j ( c g / 2 ) ( x 2 + y 2 ) ] φ ( y ) d y ,
0 1 x φ n l ( x ) φ m l ( x ) d x = δ n m .
N e q = N ( g 2 - 1 ) 1 2 = N 2 ( M - 1 M ) .
loss = 1 - γ 2 frequency = ( c / 2 L ) [ q + ( ψ / π ) ] ,
γ 1 φ 1 ( x ) = j l + 1 ( k / L 1 ) 0 a 2 y J l ( k x y / L 1 ) exp [ - j ( k / 2 L ) × ( g 1 x 2 + g 2 y 2 ) ] φ 2 ( y ) d y γ 2 φ 2 ( y ) = j l + 1 ( k / L 1 ) 0 a 1 z J l ( k y z / L 1 ) exp [ - j ( k / 2 L 1 ) × ( g 2 y 2 + g 1 z 2 ) ] φ 1 ( z ) d z ,
γ 1 γ 2 φ ( x ) = j l + 1 ( c 1 / 2 g 2 ) 0 1 y J [ ( c 1 / 2 g 2 ) x y ] × exp [ - j ( c 1 / 4 g 2 ) ( 2 g 1 g 2 - 1 ) ( x 2 + y 2 ) ] φ 1 ( y ) d y ,
N = N 1 / 2 g 2 g = 1 - 2 g 1 g 2 γ = γ 1 γ 2 N e q = N 1 [ g 1 g 2 ( g 1 g 2 - 1 ) ] 1 2 .
γ = { γ 1 γ 2 ( g 1 g 2 > 1 , g 2 > 0 ) γ 1 * γ 2 * ( g 1 g 2 > 1 , g 2 < 0 ) ( - 1 ) l + 1 γ 1 * γ 2 * ( g 1 g 2 < 0 , g 2 > 0 ) ( - 1 ) l + 1 γ 1 γ 2 ( g 1 g 2 < 0 , g 2 < 0 ) .

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