Abstract

The modes of an unstable aligned empty strip resonator are found using the kernel expansion technique with the linear prolate functions as a basis set. Calculations of the modes are used to examine the accuracy of the asymptotic methods at low equivalent Fresnel numbers and to demonstrate the orthogonality of the cavity modes.

© 1983 Optical Society of America

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References

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  1. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 454 (1961).
  2. W. P. Latham, G. C. Dente, Appl. Opt. 19, 1618 (1980).
    [CrossRef] [PubMed]
  3. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  4. P. Horwitz, J. Opt. Soc. Am. 63, 1528 (1973).
    [CrossRef]
  5. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 131 (1969).
    [CrossRef] [PubMed]
  6. L. Bergstein, E. Maron, J. Opt. Soc. Am. 56, 16 (1966).
    [CrossRef]
  7. L. Bergstein, H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).
    [CrossRef]
  8. L. Bergstein, Appl. Opt. 7, 495 (1968).
    [CrossRef] [PubMed]
  9. C. Y. She, H. Heffner, Appl. Opt. 3, 703 (1961).
    [CrossRef]
  10. R. Freiden, “Evaluation, Design, and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions,” in Progress in Optics, Vol. 9 (North-Holland, Amsterdam, 1971), Chap. 8.
    [CrossRef]
  11. D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 43, 43 (1961).
  12. D. Slepian, J. Math. Phys. Cambridge, Mass. 44, 99 (1965).
  13. W. Streifer, J. Opt. Soc. Am. 55, 868 (1965).
    [CrossRef]
  14. International Mathematical and Statistical Libraries, Inc. (1977).
  15. A. L. VanBuren, “A fortran Computer Program for Calculating the Linear Prolate Functions,” NRL Report 7944 (10May1976).
  16. C. Flammer, Spheroidal Wave Functions (Stanford U. P., Stanford, Calif., 1972).
  17. D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973).
    [CrossRef] [PubMed]
  18. A. H. Paxton, Mission Research Corp., Albuquerque, N.M.; private communication.
  19. E. Henderson, W. P. Latham, Laser Digest, AFWL TR 80-4, 21 (1980).

1980 (1)

1973 (2)

1970 (1)

1969 (1)

1968 (1)

1966 (1)

1965 (2)

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

D. Slepian, J. Math. Phys. Cambridge, Mass. 44, 99 (1965).

1964 (1)

1961 (3)

C. Y. She, H. Heffner, Appl. Opt. 3, 703 (1961).
[CrossRef]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 454 (1961).

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

Bergstein, L.

Chester, A. N.

Dente, G. C.

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. P., Stanford, Calif., 1972).

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 454 (1961).

Freiden, R.

R. Freiden, “Evaluation, Design, and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions,” in Progress in Optics, Vol. 9 (North-Holland, Amsterdam, 1971), Chap. 8.
[CrossRef]

Heffner, H.

Henderson, E.

E. Henderson, W. P. Latham, Laser Digest, AFWL TR 80-4, 21 (1980).

Horwitz, P.

Latham, W. P.

W. P. Latham, G. C. Dente, Appl. Opt. 19, 1618 (1980).
[CrossRef] [PubMed]

E. Henderson, W. P. Latham, Laser Digest, AFWL TR 80-4, 21 (1980).

Li, T.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 454 (1961).

Maron, E.

Miller, H. Y.

Paxton, A. H.

A. H. Paxton, Mission Research Corp., Albuquerque, N.M.; private communication.

Pollak, H. O.

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

Rensch, D. B.

Sanderson, R. L.

Schachter, H.

She, C. Y.

Siegman, A. E.

Slepian, D.

D. Slepian, J. Math. Phys. Cambridge, Mass. 44, 99 (1965).

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

Streifer, W.

VanBuren, A. L.

A. L. VanBuren, “A fortran Computer Program for Calculating the Linear Prolate Functions,” NRL Report 7944 (10May1976).

Appl. Opt. (6)

Bell Syst. Tech. J. (2)

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

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 454 (1961).

J. Math. Phys. Cambridge, Mass. (1)

D. Slepian, J. Math. Phys. Cambridge, Mass. 44, 99 (1965).

J. Opt. Soc. Am. (4)

Other (6)

International Mathematical and Statistical Libraries, Inc. (1977).

A. L. VanBuren, “A fortran Computer Program for Calculating the Linear Prolate Functions,” NRL Report 7944 (10May1976).

C. Flammer, Spheroidal Wave Functions (Stanford U. P., Stanford, Calif., 1972).

A. H. Paxton, Mission Research Corp., Albuquerque, N.M.; private communication.

E. Henderson, W. P. Latham, Laser Digest, AFWL TR 80-4, 21 (1980).

R. Freiden, “Evaluation, Design, and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions,” in Progress in Optics, Vol. 9 (North-Holland, Amsterdam, 1971), Chap. 8.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Geometry of the strip resonator.

Fig. 2
Fig. 2

Modes 0, 1, and 2: intensity F = 0.57 and M = 2.5.

Fig. 3
Fig. 3

Modes 0 and 1: phase F = 0.57 and M = 2.5.

Tables (4)

Tables Icon

Table I Magnitudes of the Mode Eigenvalues

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Table II Comparison of Magnitudes of Mode Eigenvalues Near Point of Maximum Mode Separation a

Tables Icon

Table III Comparison of Magnitudes of Mode Eigenvalues Near Mode Crossing a

Tables Icon

Table IV Numerical Verification of Mode Orthogonality

Equations (18)

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g l = 1 - ( L / R l ) ,
γ u ( x ) = i F - 1 1 d y exp { - i π F [ g ( x 2 + y 2 ) - 2 x y ] } u ( y ) ,
F = a 1 2 / ( 2 L λ g 2 ) ,
g = 2 g 1 g 2 - 1.
v ( x ) = j = 0 a j i - j / 2 η j - 3 / 4 ψ j ( x ) ,
u ( x ) = exp [ i π F 2 ( M + 1 M ) x 2 ] v ( x ) ,
i j 2 π η j c ψ j ( x ) = - 1 1 d y exp ( i c x y ) ψ j ( y ) ,
M = g + g 2 - 1 .
γ v ( x ) = i F exp [ - i π F ( M + 1 M ) x 2 ] - 1 1 d y exp ( i 2 π F x y ) v ( y ) .
- 1 1 d y ψ j ( y ) ψ k ( y ) = η j δ j k ,
exp ( i c x y ) = k = 0 i k 2 π c η k ψ k ( x ) ψ k ( y ) .
γ a j = k = 0 B j k a k ,
B j k = i ( j + k + 1 ) / 2 ( η j η k ) - 1 / 4 - 1 1 d y × exp [ - i π F ( M + 1 M ) y 2 ] ψ j ( y ) ψ k ( y ) .
E ( x ) = j = N + 1 a j i - 1 / 2 η j - 3 / 4 ψ j ( x )
( 1 - x 2 ) d 2 ψ j d x 2 - 2 x d ψ j d x + ( Γ j - c 2 x 2 ) ψ j = 0.
N 0 = M 2 a 1 2 λ L = 5.0
t = π M F 1
- 1 1 d x u n ( x ) u m ( x ) = δ n m .

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