Abstract

In this paper we show that numerical and kernel expansion procedures for solving the laser mode problem do not differ in essence; both convert the integral equation into a matrix equation. Furthermore, the Fox and Li iterative method is shown to be a matrix diagonalization technique. A particular kernel expansion using Gaussian-Hermite functions is discussed, as are matrix diagonalization techniques. Numerical results are compared with other published values. We conclude that the optimum procedure is to use gaussian quadrature numerical integration to convert to a matrix equation and diagonalize the matrix with the computer program ALLMAT. This method is computationally simple and simultaneously determines many modes. Also, it is applicable to unstable and/or tilted mirror resonators with selectively coated reflectors.

© 1969 Optical Society of America

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References

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  1. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  2. P. F. Checcacci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
    [CrossRef]
  3. V. I. Krylov, Approximate Calculaton of Integrals (The Macmillan Company, New York, 1962).
  4. A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Inc., Englewood Cliffs, 1966).
  5. H. Schachter, L. Bergstein, in Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 173.
  6. L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
    [CrossRef]
  7. P. J. Eberlein, J. Soc. Ind. Appl. Math. 10, 74 (1962).
    [CrossRef]
  8. P. J. Eberlein, J. Boothroyd, Numerisch. Math. 11, 1 (1968).
    [CrossRef]
  9. J. G. P. Francis, Computer J. 4, 265, 332 (1961, 1962).
    [CrossRef]
  10. A. E. Siegman, R. Arrathoon, IEEE Trans. QE-3, 156 (1967).
  11. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965), Chap. 9.
  12. W. Streifer, J. Opt. Soc. Amer. 55, 868 (1965).
    [CrossRef]
  13. J. H. Wilkinson, Computer J. 4, 230 (1961a).
    [CrossRef]
  14. Ref. 11, p. 86.
  15. W. Streifer, H. Gamo in Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 351.
  16. G. N. Watson, J. London Math. Soc. 8, 194 (1933a).
    [CrossRef]
  17. D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).
  18. L. A. Vainshtein, Sov. Phys.–JETP 17, 714 (1963).
  19. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  20. G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

1968 (1)

P. J. Eberlein, J. Boothroyd, Numerisch. Math. 11, 1 (1968).
[CrossRef]

1967 (1)

A. E. Siegman, R. Arrathoon, IEEE Trans. QE-3, 156 (1967).

1966 (2)

P. F. Checcacci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

1965 (1)

W. Streifer, J. Opt. Soc. Amer. 55, 868 (1965).
[CrossRef]

1963 (1)

L. A. Vainshtein, Sov. Phys.–JETP 17, 714 (1963).

1962 (2)

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

P. J. Eberlein, J. Soc. Ind. Appl. Math. 10, 74 (1962).
[CrossRef]

1961 (4)

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

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

J. H. Wilkinson, Computer J. 4, 230 (1961a).
[CrossRef]

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

1933 (1)

G. N. Watson, J. London Math. Soc. 8, 194 (1933a).
[CrossRef]

Arrathoon, R.

A. E. Siegman, R. Arrathoon, IEEE Trans. QE-3, 156 (1967).

Bergstein, L.

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

H. Schachter, L. Bergstein, in Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 173.

Boothroyd, J.

P. J. Eberlein, J. Boothroyd, Numerisch. Math. 11, 1 (1968).
[CrossRef]

Boyd, G. D.

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

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

Checcacci, P. F.

P. F. Checcacci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

Consortini, A.

P. F. Checcacci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

Eberlein, P. J.

P. J. Eberlein, J. Boothroyd, Numerisch. Math. 11, 1 (1968).
[CrossRef]

P. J. Eberlein, J. Soc. Ind. Appl. Math. 10, 74 (1962).
[CrossRef]

Fox, A. G.

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

Francis, J. G. P.

J. G. P. Francis, Computer J. 4, 265, 332 (1961, 1962).
[CrossRef]

Gamo, H.

W. Streifer, H. Gamo in Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 351.

Gordon, J. P.

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

Kogelnik, H.

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Krylov, V. I.

V. I. Krylov, Approximate Calculaton of Integrals (The Macmillan Company, New York, 1962).

Li, T.

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

Marom, E.

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Schachter, H.

H. Schachter, L. Bergstein, in Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 173.

Scheggi, A.

P. F. Checcacci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

Secrest, D.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Inc., Englewood Cliffs, 1966).

Siegman, A. E.

A. E. Siegman, R. Arrathoon, IEEE Trans. QE-3, 156 (1967).

Slepian, D.

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Streifer, W.

W. Streifer, J. Opt. Soc. Amer. 55, 868 (1965).
[CrossRef]

W. Streifer, H. Gamo in Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 351.

Stroud, A. H.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Inc., Englewood Cliffs, 1966).

Vainshtein, L. A.

L. A. Vainshtein, Sov. Phys.–JETP 17, 714 (1963).

Watson, G. N.

G. N. Watson, J. London Math. Soc. 8, 194 (1933a).
[CrossRef]

Wilkinson, J. H.

J. H. Wilkinson, Computer J. 4, 230 (1961a).
[CrossRef]

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

Bell Syst. Tech. J. (4)

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

D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

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

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Computer J. (2)

J. H. Wilkinson, Computer J. 4, 230 (1961a).
[CrossRef]

J. G. P. Francis, Computer J. 4, 265, 332 (1961, 1962).
[CrossRef]

IEEE Trans. (1)

A. E. Siegman, R. Arrathoon, IEEE Trans. QE-3, 156 (1967).

J. London Math. Soc. (1)

G. N. Watson, J. London Math. Soc. 8, 194 (1933a).
[CrossRef]

J. Opt. Soc. Amer. (2)

W. Streifer, J. Opt. Soc. Amer. 55, 868 (1965).
[CrossRef]

L. Bergstein, E. Marom, J. Opt. Soc. Amer. 56, 16 (1966).
[CrossRef]

J. Soc. Ind. Appl. Math. (1)

P. J. Eberlein, J. Soc. Ind. Appl. Math. 10, 74 (1962).
[CrossRef]

Numerisch. Math. (1)

P. J. Eberlein, J. Boothroyd, Numerisch. Math. 11, 1 (1968).
[CrossRef]

Proc. IEEE (1)

P. F. Checcacci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

Sov. Phys.–JETP (1)

L. A. Vainshtein, Sov. Phys.–JETP 17, 714 (1963).

Other (6)

Ref. 11, p. 86.

W. Streifer, H. Gamo in Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 351.

V. I. Krylov, Approximate Calculaton of Integrals (The Macmillan Company, New York, 1962).

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Inc., Englewood Cliffs, 1966).

H. Schachter, L. Bergstein, in Optical Masers, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 173.

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

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Figures (1)

Fig. 1
Fig. 1

The resonator geometry.

Tables (6)

Tables Icon

Table I Comparison of Percentage Power Loss for Confocal Resonator (ξ = 0) with c = 4.0

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Table II Comparison of Percentage Power Loss of Confocal Resonators (ξ = 0)

Tables Icon

Table III Comparison of Percentage Power Loss for Plane Parallel Resonator (ξ = 1)

Tables Icon

Table IV Comparison of Percentage Power Loss for Plane Parallel Resonators (ξ = 1)

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Table V Comparison of Percentage Power Loss for Plane Parallel Resonators (ξ = 1) with large c number

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Table VI Computation Times for Gaussian Quadrature Calculation of Plane Parallel Modes

Equations (22)

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

γ ɛ ( s ) = ( c 1 2 ) ( c 1 2 ) K ( s , t ) ɛ ( t ) d t ,
K ( s , t ) = [ i e - i k d 2 π ] 1 exp { - i [ ξ 2 ( s 2 + t 2 ) - s t ] } ,
power loss per transit = 1 - γ 2 .
γ ɛ ( s m ) = n = 1 N K ( s n , t n ) D n n ɛ ( t n ) ,
D n n = h ( 1 2 , 1 , 1 , , 1 , 1 2 , )             trapezoidal rule D n n = ( h / 3 ) ( 1 , 4 , 2 , 4 , , 2 , 4 , 1 )             Simpson rule
B ɛ = ɛ γ .
- a a f ( x ) d x n = 1 N D n n f ( x n ) ,
n = 1 N K ( s i , t n ) D n n ɛ ( t n ) ,
K ( s , t ) = n , m = 1 K n m ψ n ( s ) ϕ m ( t ) .
K ( s , t ) n , m = 1 N K n m ψ n ( s ) ϕ m ( t ) ,
ɛ ( t ) p = 1 N 1 ɛ p ϕ p ( t ) .
D m p = ( c 1 2 ) ( c 1 2 ) ϕ m ( t ) ϕ p ( t ) d t ,
ψ n ( s ) q = 1 N A q n ϕ q ( s ) ,
γ q = 1 N ɛ q ϕ q ( s ) = n , m , p , q = 1 N A q n K n m D m p ɛ p ϕ q ( s ) ,
AKD ɛ = ɛ γ .
K ( s , t ) n , m = 1 N K n m ϕ n ( s ) ϕ m ( t ) ,
v ( 0 ) = n = 0 N α n u n .
B u n = γ n u n .
γ n γ n + 1 ,             n = 1 , 2 , , N - 1.
v ( m ) = γ 1 m [ α 1 u , + n = 2 N α n ( γ n / γ 1 ) m u n ] ,
ɛ ( t ) = e - ( i ξ t 2 / 2 ) n = 1 N ɛ n e - ( t 2 / 2 ) H n ( t ) ,
K n m = ( i ) n 2 n n ! δ n m ,

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