Abstract

Modes and losses of the planar Fabry-Perot and confocal resonators have been computed using various kernels obtained by different approximations of diffraction theory. For Fabry-Perot resonators no practical influence was found of the shape of the kernel on the results, at least for Fresnel numbers ranging up to 42.25. For the confocal resonator large differences were found even at small Fresnel numbers N ≳ 1. In particular, a qualitative difference was found between the results obtained by the so-called parabolic approximation and the kernel of Rayleigh-Luneberg diffraction theory. These differences confirm the influence of the amplitude in the approximations.

© 1978 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. G. Toraldo di Francia, La diffrazione della luce (Ed. Sci. Einaudi, Torino, 1958).
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  4. A. Consortini, F. Pasqualetti, Appl. Opt. 11, 2381 (1972).
    [CrossRef] [PubMed]
  5. G. Toraldo di Francia, “Theory of Optical Resonators,” in Proceedings of the International School of Physics Enrico Fermi, Course 31, Quantum Electronics and Coherent Light, Varenna, 1963 (Academic, New York, 1964).
  6. F. Schwering, IRE Trans. Antennas Progag. AP-10, 99 (1962).
    [CrossRef]
  7. A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
    [CrossRef]
  8. L. A. Vainshtein, Qh. Tekh. Fiz. 34, 193 (1964) [Sov. Phys. Tekh. Phys. 9, 157 (1964)].
  9. D. Slepian, E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965).
  10. L. W. Chen, L. B. Felsen, IEEE J. Quantum Electron. QE-9, 1102 (1973).
    [CrossRef]
  11. C. Santana, L. B. Felsen, Appl. Opt. 15, 1470 (1976).
    [CrossRef] [PubMed]
  12. See, for example, R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2129 (1969); A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970); D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973); P. Horwitz, J. Opt. Soc. Am. 63, 1528 (1973); A. E. Siegman, Appl. Opt. 13, 353 (1974).
    [CrossRef] [PubMed]

1976 (1)

1973 (2)

A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
[CrossRef]

L. W. Chen, L. B. Felsen, IEEE J. Quantum Electron. QE-9, 1102 (1973).
[CrossRef]

1972 (1)

1969 (1)

1965 (1)

D. Slepian, E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965).

1964 (1)

L. A. Vainshtein, Qh. Tekh. Fiz. 34, 193 (1964) [Sov. Phys. Tekh. Phys. 9, 157 (1964)].

1962 (1)

F. Schwering, IRE Trans. Antennas Progag. AP-10, 99 (1962).
[CrossRef]

1961 (1)

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Chen, L. W.

L. W. Chen, L. B. Felsen, IEEE J. Quantum Electron. QE-9, 1102 (1973).
[CrossRef]

Consortini, A.

Felsen, L. B.

C. Santana, L. B. Felsen, Appl. Opt. 15, 1470 (1976).
[CrossRef] [PubMed]

L. W. Chen, L. B. Felsen, IEEE J. Quantum Electron. QE-9, 1102 (1973).
[CrossRef]

Fox, A. G.

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

Li, T.

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

Pasqualetti, F.

Sanderson, R. L.

Santana, C.

Schwering, F.

F. Schwering, IRE Trans. Antennas Progag. AP-10, 99 (1962).
[CrossRef]

Slepian, D.

D. Slepian, E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965).

Sonnenblick, E.

D. Slepian, E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965).

Streifer, W.

Toraldo di Francia, G.

G. Toraldo di Francia, “Theory of Optical Resonators,” in Proceedings of the International School of Physics Enrico Fermi, Course 31, Quantum Electronics and Coherent Light, Varenna, 1963 (Academic, New York, 1964).

G. Toraldo di Francia, La diffrazione della luce (Ed. Sci. Einaudi, Torino, 1958).

Vainshtein, L. A.

L. A. Vainshtein, Qh. Tekh. Fiz. 34, 193 (1964) [Sov. Phys. Tekh. Phys. 9, 157 (1964)].

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Appl. Opt. (3)

Bell Syst. Tech. J. (2)

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

D. Slepian, E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965).

IEEE J. Quantum Electron. (1)

L. W. Chen, L. B. Felsen, IEEE J. Quantum Electron. QE-9, 1102 (1973).
[CrossRef]

IRE Trans. Antennas Progag. (1)

F. Schwering, IRE Trans. Antennas Progag. AP-10, 99 (1962).
[CrossRef]

Opt. Acta (1)

A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
[CrossRef]

Qh. Tekh. Fiz. (1)

L. A. Vainshtein, Qh. Tekh. Fiz. 34, 193 (1964) [Sov. Phys. Tekh. Phys. 9, 157 (1964)].

Other (3)

G. Toraldo di Francia, La diffrazione della luce (Ed. Sci. Einaudi, Torino, 1958).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

G. Toraldo di Francia, “Theory of Optical Resonators,” in Proceedings of the International School of Physics Enrico Fermi, Course 31, Quantum Electronics and Coherent Light, Varenna, 1963 (Academic, New York, 1964).

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Figures (6)

Fig. 1
Fig. 1

Infinite strip resonators: (a) normal cross section of an arbitrary geometry resonator, (b) normal cross section of the flat-roof resonator, (c) normal cross section of the Fabry-Perot resonator, and (d) normal cross section of the confocal resonator.

Fig. 2
Fig. 2

Fabry-Perot resonator; relative amplitude patterns of some modes in the case d = 100 λ. (a) mode m = 10 for N = 3 and (b) m = 20 for N = 12.25. Dashed lines refer to parabolic approximation; and solid lines refer to kernels KA, KK, and KR.

Fig. 3
Fig. 3

Confocal resonator. Power losses per transit of the fundamental mode plotted vs N for two values of the mirror distance d. The solid line refers to the Rayleigh-Luneberg diffraction formula; the dashed line refers to the Helmholtz-Kirchhoff formula; the dot–dash line refers to KA; and the dotted line refers to the parabolic approximation.

Fig. 4
Fig. 4

Confocal resonator. Power losses per transit of the four lowest order even modes plotted vs N in the case d = 100 λ as given by the different kernels.

Fig. 5
Fig. 5

Confocal resonator. Relative amplitude patterns of the mode having the lowest losses among those labeled m = 4 and m = 6 in Fig. 4. The solid line refers to the Rayleigh-Luneberg formula, and the dashed line refers to the Helmholtz-Kirchhoff formula. (a) refers to N = 2.4 and (b) refers to N = 2.5, before and after the crossing, respectively.

Fig. 6
Fig. 6

Power losses per transit of the fundamental mode for two values of d as obtained by using KRP (circles) and KPR (stars), respectively. The lines for comparison are taken from Fig. 3.

Equations (8)

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σ m u m ( P 1 ) = M 2 K ( P 1 , P 2 ) u m ( P 2 ) d l 2 ,
lim P 1 P ¯ M 2 υ ( P 2 ) K ( P 1 , P 2 ) d l 2 = υ ( P ¯ ) ,
K A = K ( P 1 , P 2 ) = exp [ i ( π / 4 k ρ ) ] ( λ ρ ) 1 / 2 ,
K K = 1 + cos θ 2 K A ,
K R = K A cos θ ,
K P P = exp { i π / 4 i k d [ 1 + 1 2 ( x 1 x 2 d ) 2 ] } ( λ d ) 1 / 2 ,
K P = exp ( i π / 4 + i k x 1 x 2 / d ) ( λ d ) 1 / 2
K R P = cos θ ( λ ρ ) 1 / 2 exp ( i π / 4 + i k x 1 x 2 / d ) K P R = 1 ( λ d ) 1 / 2 exp ( i π / 4 i k ρ ) ,

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