Abstract

Modes of resonators formed by paraxial elements of infinite extent and single internal aperture are shown to be the solutions of two coupled symmetric integral equations involving four parameters: two Fresnel numbers N1 and N2 and two geometrical factors A1 and A2. The modes become the eigenfunctions of an Hermitian operator when N1 = ±N2, A1 = ∓A2; analytical solutions can then be written as generalized prolate spheroidal functions. The same solutions are derived for a resonator in which one mirror is replaced by an infinite phase conjugate mirror. Real nonsymmetric eigenvalue problems are associated with the condition N1A1 = −N2A2; such configurations can generate pairs of modes with same power losses but different oscillation frequencies. Extension to cavities with two internal apertures yields a system of four coupled integral equations with eight independent parameters; again the modes can be the solutions of Hermitian or real nonsymmetric eigenvalue problems under special conditions.

© 1983 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1965).
    [CrossRef]
  2. Y. Ananiev, Résonateurs Optiques et Probléme de Divergence du Rayonnement Laser (Mir, Moscow, 1982).
  3. G. D. Boyd, J. P. Gordon, Bell. Syst. Tech. J. 40, 489 (1961).
  4. G. D. Boyd, H. Kogelnik, Bell. Syst. Tech. J. 41, 1347 (1962).
  5. D. Slepian, H. O. Pollak, Bell. Syst. Tech. J. 40, 43 (1961).
  6. D. Slepian, Bell. Syst. Tech. J. 43, 3009 (1964).
  7. P. Horowitz, J. Opt. Soc. Am. 63, 1528 (1973).
    [CrossRef]
  8. J. Nagel, D. Rogovin, P. Avizonis, R. Butts, Opt. Lett. 4, 300 (1979).
    [CrossRef] [PubMed]
  9. A. G. Fox, T. Li, Bell. Syst. Tech. J. 40, 453 (1961).
  10. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 131 (1969).
    [CrossRef] [PubMed]
  11. A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
    [CrossRef]
  12. M. D. Feit, J. A. Fleck, Appl. Opt. 20, 2843 (1981).
    [CrossRef] [PubMed]
  13. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  14. W. D. Murphy, M. L. Bernabe, Appl. Opt. 17, 2358 (1978).
    [CrossRef] [PubMed]
  15. P. Lavigne, J. L. Lachambre, J. Gilbert, G. Otis, Can. J. Phys. 54, 816 (1976).
    [CrossRef]
  16. P. H. Sarkies, Opt. Commun. 31, 189 (1979).
    [CrossRef]
  17. F. Martin, B. Grek, H. Pépin, Rev. Sci. Instrum. 51, 1656 (1980).
    [CrossRef]
  18. T. Li, Bell Syst. Tech. J. 42, 2609 (1963).
  19. H. P. Kortz, R. Ifflander, H. Weber, Appl. Opt. 20, 4124 (1981).
    [CrossRef] [PubMed]
  20. S. A. Collins, J. Opt. Soc. Am. 60, 1168 (1970).
    [CrossRef]
  21. A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
    [CrossRef]
  22. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).
  23. P. M. Morse, H. Feshback, Methods of Mathematical Physics, Vol. 1 (McGraw-Hill, New York, 1950), Chap. 8.
  24. A. E. Siegman, Opt. Commun. 31, 369 (1979).
    [CrossRef]
  25. H. Kogelnik, Bell. Syst. Tech. J. 44, 455 (1965).
  26. J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).
  27. S. N. Vlasov, W. I. Talanov, Radio Eng. Electron. Phys. 10, 812 (1965).
  28. T. Li, Bell Syst. Tech. J. 44, 917 (1965).
  29. P. A. Bélanger, C. Paré, M. Piché, J. Opt. Soc. Am. 73, 567 (1983).
    [CrossRef]
  30. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).
  31. A. H. Paxton, T. C. Salvi, Opt. Comm. 26, 305 (1978).
    [CrossRef]
  32. D. S. Jones, Methods in Electromagnetic Wave Propagation (Clarendon, Oxford, 1979), Chap. 7.

1983 (1)

1981 (2)

1980 (1)

F. Martin, B. Grek, H. Pépin, Rev. Sci. Instrum. 51, 1656 (1980).
[CrossRef]

1979 (3)

A. E. Siegman, Opt. Commun. 31, 369 (1979).
[CrossRef]

P. H. Sarkies, Opt. Commun. 31, 189 (1979).
[CrossRef]

J. Nagel, D. Rogovin, P. Avizonis, R. Butts, Opt. Lett. 4, 300 (1979).
[CrossRef] [PubMed]

1978 (2)

1976 (2)

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

P. Lavigne, J. L. Lachambre, J. Gilbert, G. Otis, Can. J. Phys. 54, 816 (1976).
[CrossRef]

1973 (1)

1970 (2)

1969 (1)

1968 (1)

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
[CrossRef]

1965 (4)

S. N. Vlasov, W. I. Talanov, Radio Eng. Electron. Phys. 10, 812 (1965).

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

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

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1965).
[CrossRef]

1964 (2)

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

D. Slepian, Bell. Syst. Tech. J. 43, 3009 (1964).

1963 (1)

T. Li, Bell Syst. Tech. J. 42, 2609 (1963).

1962 (1)

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

1961 (3)

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

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

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

Ananiev, Y.

Y. Ananiev, Résonateurs Optiques et Probléme de Divergence du Rayonnement Laser (Mir, Moscow, 1982).

Avizonis, P.

Bélanger, P. A.

Bernabe, M. L.

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).

Butts, R.

Collins, S. A.

Feit, M. D.

Feshback, H.

P. M. Morse, H. Feshback, Methods of Mathematical Physics, Vol. 1 (McGraw-Hill, New York, 1950), Chap. 8.

Fleck, J. A.

Fox, A. G.

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
[CrossRef]

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

Gilbert, J.

P. Lavigne, J. L. Lachambre, J. Gilbert, G. Otis, Can. J. Phys. 54, 816 (1976).
[CrossRef]

Gordon, J. P.

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

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

Grek, B.

F. Martin, B. Grek, H. Pépin, Rev. Sci. Instrum. 51, 1656 (1980).
[CrossRef]

Horowitz, P.

Ifflander, R.

Jones, D. S.

D. S. Jones, Methods in Electromagnetic Wave Propagation (Clarendon, Oxford, 1979), Chap. 7.

Kogelnik, H.

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

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1965).
[CrossRef]

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

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

Kortz, H. P.

Lachambre, J. L.

P. Lavigne, J. L. Lachambre, J. Gilbert, G. Otis, Can. J. Phys. 54, 816 (1976).
[CrossRef]

Lavigne, P.

P. Lavigne, J. L. Lachambre, J. Gilbert, G. Otis, Can. J. Phys. 54, 816 (1976).
[CrossRef]

Li, T.

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
[CrossRef]

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1965).
[CrossRef]

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

T. Li, Bell Syst. Tech. J. 42, 2609 (1963).

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

Martin, F.

F. Martin, B. Grek, H. Pépin, Rev. Sci. Instrum. 51, 1656 (1980).
[CrossRef]

Miller, H. Y.

Morse, P. M.

P. M. Morse, H. Feshback, Methods of Mathematical Physics, Vol. 1 (McGraw-Hill, New York, 1950), Chap. 8.

Murphy, W. D.

Nagel, J.

Otis, G.

P. Lavigne, J. L. Lachambre, J. Gilbert, G. Otis, Can. J. Phys. 54, 816 (1976).
[CrossRef]

Paré, C.

Paxton, A. H.

A. H. Paxton, T. C. Salvi, Opt. Comm. 26, 305 (1978).
[CrossRef]

Pépin, H.

F. Martin, B. Grek, H. Pépin, Rev. Sci. Instrum. 51, 1656 (1980).
[CrossRef]

Piché, M.

Pollak, H. O.

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

Rogovin, D.

Salvi, T. C.

A. H. Paxton, T. C. Salvi, Opt. Comm. 26, 305 (1978).
[CrossRef]

Sanderson, R. L.

Sarkies, P. H.

P. H. Sarkies, Opt. Commun. 31, 189 (1979).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Opt. Commun. 31, 369 (1979).
[CrossRef]

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

Slepian, D.

D. Slepian, Bell. Syst. Tech. J. 43, 3009 (1964).

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

Streifer, W.

Talanov, W. I.

S. N. Vlasov, W. I. Talanov, Radio Eng. Electron. Phys. 10, 812 (1965).

Vlasov, S. N.

S. N. Vlasov, W. I. Talanov, Radio Eng. Electron. Phys. 10, 812 (1965).

Weber, H.

Wilkinson, J. H.

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

Appl. Opt. (6)

Bell Syst. Tech. J. (3)

T. Li, Bell Syst. Tech. J. 42, 2609 (1963).

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

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

Bell. Syst. Tech. J. (6)

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).

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

D. Slepian, Bell. Syst. Tech. J. 43, 3009 (1964).

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

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

Can. J. Phys. (1)

P. Lavigne, J. L. Lachambre, J. Gilbert, G. Otis, Can. J. Phys. 54, 816 (1976).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
[CrossRef]

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Comm. (1)

A. H. Paxton, T. C. Salvi, Opt. Comm. 26, 305 (1978).
[CrossRef]

Opt. Commun. (2)

P. H. Sarkies, Opt. Commun. 31, 189 (1979).
[CrossRef]

A. E. Siegman, Opt. Commun. 31, 369 (1979).
[CrossRef]

Opt. Lett. (1)

Radio Eng. Electron. Phys. (1)

S. N. Vlasov, W. I. Talanov, Radio Eng. Electron. Phys. 10, 812 (1965).

Rev. Sci. Instrum. (1)

F. Martin, B. Grek, H. Pépin, Rev. Sci. Instrum. 51, 1656 (1980).
[CrossRef]

Other (5)

Y. Ananiev, Résonateurs Optiques et Probléme de Divergence du Rayonnement Laser (Mir, Moscow, 1982).

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

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

P. M. Morse, H. Feshback, Methods of Mathematical Physics, Vol. 1 (McGraw-Hill, New York, 1950), Chap. 8.

D. S. Jones, Methods in Electromagnetic Wave Propagation (Clarendon, Oxford, 1979), Chap. 7.

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Figures (8)

Fig. 1
Fig. 1

(a) Conventional two-mirror laser resonator limited by two apertures; (b) and (c) laser resonators with an internal aperture.

Fig. 2
Fig. 2

ABCD matrix representation of laser resonators: (a) configuration with single internal aperture; (b) configuration with two internal apertures.

Fig. 3
Fig. 3

Curves of the eigenvalues γ as a function of Fresnel number N for a resonator with Hermitian kernel. The azimuthal index l is equal to 0, 1, or 2.

Fig. 4
Fig. 4

(a) Concentric resonator with quasi-Hermitian kernel (type 1); (b) confocal resonator with Hermitian kernel (type 2); (c) negative branch confocal resonator with real nonsymmetric kernel.

Fig. 5
Fig. 5

Curves of mode losses as a function of A2 for N1 = −1.0, N2 = 1.0, A1 = 0. The labels (l,p) refer to the mode azimuthal index l and radial index p.

Fig. 6
Fig. 6

Curves of mode losses as a function of N2 for N1 = −1.0, A1 = A2 = −1.0.

Fig. 7
Fig. 7

Curves of mode losses as function of N2 for N1 = 1.0, A1 = A2 = 0.

Fig. 8
Fig. 8

Curves of the l = 0 mode losses and of the imaginary part of the eigenvalues as a function of N2 for N1 = 1.5, A1 = A2 = 0.

Equations (41)

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γ V ( s ) = 0 1 K ( s , s ) V ( s ) d s ,
- 1 A 1 A 2 + M 2 ( A 2 2 - 1 ) + 1 2 M ( A 1 2 - 1 ) 1 ,
M = B 1 / B 2 .
γ 1 E 1 ( r 1 ) = j l + 1 k B 1 0 a 0 r 2 d r 2 J l ( k r 1 r 2 B 1 ) × exp [ - j k A 1 2 B 1 ( r 1 2 + r 2 2 ) ] E 2 ( r 2 ) ,
γ 2 E 2 ( r 2 ) = j l + 1 k B 2 0 a 0 r 1 d r 1 J 1 ( k r 1 r 2 B 2 ) × exp [ - j k A 2 2 B 2 ( r 1 2 + r 2 2 ) ] E 1 ( r 1 ) ,
γ 1 E 1 ( s ) = j l + 1 π N 1 0 1 d s exp [ - j π N 1 A 1 ( s + s ) ] × J l ( 2 π N 1 s s ) E 2 ( s ) ,
γ 2 E 2 ( s ) = j l + 1 π N 2 0 1 d s exp [ - j π N 2 A 2 ( s + s ) ] × J l ( 2 π N 2 s s ) E 1 ( s ) .
N 1 = ( a 0 2 ) / ( λ B 1 ) , N 2 = ( a 0 2 ) / ( λ B 2 ) .
γ E 1 ( s ) = 0 1 d s K 1 ( s , s ) E 1 ( s ) ,
γ E 2 ( s ) = 0 1 d s K 2 ( s , s ) E 2 ( s ) ,
K 1 ( s , s ) = ( - 1 ) l + 1 π 2 N 1 N 2 exp [ - j π ( N 1 A 1 s + N 2 A 2 s ) ] × 0 1 d s exp [ - j π ( N 1 A 1 + N 2 A 2 ) s ] × J l ( 2 π N 1 s s ) J l ( 2 π N 2 s s ) ,
K 2 ( s , s ) = K 1 ( s , s ) .
γ E 1 ( s ) = 0 1 H ( s , s ) E 1 ( s ) d s ,
H ( s , s ) = π 2 N 2 exp [ - j π N A ( s - s ) ] 0 1 d s J l ( 2 π N s s ) × J l ( 2 π N s s ) ,
Γ ϕ l ( r ) = 0 1 d r c r r J l ( c r r ) ϕ l ( r ) .
E 1 ( s ) = ϕ l ( s ) s exp ( - j π N s ) ,
γ = ϕ l ( 1 ) = Γ 2 ,
c = 2 π N .
B 1 = 2 R ( R - ) 2 ,
B 2 = - 2 R ( R + ) - 2 ,
A 1 = - 1 - 2 R - 1 ,
A 2 = - 1 + 2 R - 1.
A 1 = 1 - 2 d 1 R = d 2 - d 1 d 2 + d 1 ,
A 2 = 1 - 2 d 2 R = d 1 - d 2 d 1 + d 2 ,
B 1 = 2 d 1 ( 1 - d 1 R ) = 2 d 1 d 2 d 1 + d 2 ,
B 2 = 2 d 1 ( 1 - d 2 R ) = 2 d 1 d 2 d 1 + d 2 .
γ 1 E 1 ( s ) = j l + 1 π N 1 0 1 d s exp [ - j π N 1 A 1 ( s + s ) ] × J l ( 2 π N 1 s s ) E 2 ( s ) ,
E 2 ( s ) = E 1 * ( s ) .
γ 1 2 E 1 ( s ) = 0 1 d s H 1 ( s , s ) E 1 ( s ) ,
γ E ˜ 1 ( s ) = 0 1 d s R 1 ( s , s ) E ˜ 1 ( s ) ,
E ˜ 1 ( s ) = E 1 ( s ) exp [ - j N 1 A 1 s ] ,
R 1 ( s , s ) = ( - 1 ) l + 1 π 2 N 1 N 2 0 1 d s J l ( 2 π N 1 s s ) × J l ( 2 π N 2 s s ) ,
γ 1 E 1 l ( s 1 ) = j l + 1 π N 1 0 1 d s 4 J l ( 2 π N 1 s 4 s 1 ) × exp [ - j π N 1 A 1 ( s 4 + s 1 ) ] E 4 l ( s 4 ) ,
γ 2 E 2 l ( s 2 ) = j l + 1 π N 2 0 1 d s 1 J l ( 2 π N 0 N 2 s 1 s 2 ) × exp [ - j π N 2 ( A 2 s 1 + A 0 s 2 ) ] E 1 l ( s 1 ) ,
γ 3 E 3 l ( s 3 ) = j l + 1 π N 3 0 1 d s 2 J l ( 2 π N 3 s 3 s 2 ) × exp [ - j π N 3 A 3 ( s 2 + s 2 ) ] E 2 l ( s 2 ) ,
γ 4 E 4 ( s 4 ) = j l + 1 π N 0 0 1 d s 3 J l ( 2 π N 0 N 2 s 4 s 3 ) × exp [ - j π N 2 ( A 0 s 3 + A 2 s 4 ) E 3 l ( s 3 ) ,
A 0 = ( a 2 a 1 ) 2 D 2 , N 0 = a 2 2 λ B 2 , N 1 = a 1 2 λ B 1 , N 2 = a 1 2 λ B 2 , N 3 = a 2 2 λ B 3 .
N 3 A 3 = - N 2 A 0 ,
N 1 A 1 = - N 2 A 2 ,
N 3 = ± N 0 N 2 ,
N 1 = ± N 0 N 2 .

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