Abstract

An optical cavity is degenerate when an arbitrary ray retraces its own path after a single round trip. The condition for degeneracy is given for ring type cavities incorporating internal lenses, using geometrical optics methods. The simplest linear configurations require a spherical mirror or a corner cube, a thin lens, and a plane mirror. Planar rings with four plane mirrors require at least three thin focusing elements. A nonplanar ring is discussed which requires only two thin lenses. The alignment of degenerate cavities is, in general, as critical as the alignment of plane Fabry-Perot.

© 1969 Optical Society of America

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References

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  1. W. A. Hardy, IBM J. Res. Dev. 9, 31 (1965); R. V. Pole, H. Wieder, E. S. Barrekette, Appl. Opt. 6, 1571 (1967); H. Wieder, R. V. Pole, Appl. Opt. 6, 1761 (1967).
    [CrossRef] [PubMed]
  2. R. A. Myers, R. V. Pole, IBM J. Res. Dev. 11, 502 (1967); M. L. Dakss, C. G. Powell, Quantum Electronics Conference, Miami, Fla., May 1968.
    [CrossRef]
  3. R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965). The cavity discussed by this author has a spherical symmetry.
    [CrossRef]
  4. J. S. Wilczynski, R. E. Tibbetts, J. Opt. Soc. Amer. 55, 1574 (1965).
  5. G. Toraldo di Francia, Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963), p. 157.
  6. P. Connes, Rev. Opt. 35, 1 (1956).
  7. H. Kogelnik, Bell System Tech. J. 44, 455 (1965).
  8. M. Bertolotti, Nuovo Cimento 32, 1242 (1964).
    [CrossRef]
  9. B. Macke, J. Phys. Appl. Paris, 26, 104A (1965).
  10. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  11. For the properties of isometric transformations, see, for instance, G. Girard, A. Lentin, in Géométrie. Mécanique, Hachette, Ed. (Paris, 1964).
  12. D. A. Berkowitz, J. Opt. Soc. Amer. 55, 1464 (1965).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, (1965), 148.
  14. A. Fletcher, T. Murphy, A. Young, Proc. Roy. Soc. A223, 216 (1959).
  15. J. A. Arnaud, “Enhancement of Optical Receiver Sensitivities by Amplification of the Carrier,” to be published in the J. Quantum Electron, Nov.1968.

1967 (1)

R. A. Myers, R. V. Pole, IBM J. Res. Dev. 11, 502 (1967); M. L. Dakss, C. G. Powell, Quantum Electronics Conference, Miami, Fla., May 1968.
[CrossRef]

1966 (1)

1965 (6)

D. A. Berkowitz, J. Opt. Soc. Amer. 55, 1464 (1965).
[CrossRef]

R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965). The cavity discussed by this author has a spherical symmetry.
[CrossRef]

J. S. Wilczynski, R. E. Tibbetts, J. Opt. Soc. Amer. 55, 1574 (1965).

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

W. A. Hardy, IBM J. Res. Dev. 9, 31 (1965); R. V. Pole, H. Wieder, E. S. Barrekette, Appl. Opt. 6, 1571 (1967); H. Wieder, R. V. Pole, Appl. Opt. 6, 1761 (1967).
[CrossRef] [PubMed]

B. Macke, J. Phys. Appl. Paris, 26, 104A (1965).

1964 (1)

M. Bertolotti, Nuovo Cimento 32, 1242 (1964).
[CrossRef]

1959 (1)

A. Fletcher, T. Murphy, A. Young, Proc. Roy. Soc. A223, 216 (1959).

1956 (1)

P. Connes, Rev. Opt. 35, 1 (1956).

Arnaud, J. A.

J. A. Arnaud, “Enhancement of Optical Receiver Sensitivities by Amplification of the Carrier,” to be published in the J. Quantum Electron, Nov.1968.

Berkowitz, D. A.

D. A. Berkowitz, J. Opt. Soc. Amer. 55, 1464 (1965).
[CrossRef]

Bertolotti, M.

M. Bertolotti, Nuovo Cimento 32, 1242 (1964).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, (1965), 148.

Connes, P.

P. Connes, Rev. Opt. 35, 1 (1956).

Fletcher, A.

A. Fletcher, T. Murphy, A. Young, Proc. Roy. Soc. A223, 216 (1959).

Girard, G.

For the properties of isometric transformations, see, for instance, G. Girard, A. Lentin, in Géométrie. Mécanique, Hachette, Ed. (Paris, 1964).

Hardy, W. A.

W. A. Hardy, IBM J. Res. Dev. 9, 31 (1965); R. V. Pole, H. Wieder, E. S. Barrekette, Appl. Opt. 6, 1571 (1967); H. Wieder, R. V. Pole, Appl. Opt. 6, 1761 (1967).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

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

Lentin, A.

For the properties of isometric transformations, see, for instance, G. Girard, A. Lentin, in Géométrie. Mécanique, Hachette, Ed. (Paris, 1964).

Li, T.

Macke, B.

B. Macke, J. Phys. Appl. Paris, 26, 104A (1965).

Murphy, T.

A. Fletcher, T. Murphy, A. Young, Proc. Roy. Soc. A223, 216 (1959).

Myers, R. A.

R. A. Myers, R. V. Pole, IBM J. Res. Dev. 11, 502 (1967); M. L. Dakss, C. G. Powell, Quantum Electronics Conference, Miami, Fla., May 1968.
[CrossRef]

Pole, R. V.

R. A. Myers, R. V. Pole, IBM J. Res. Dev. 11, 502 (1967); M. L. Dakss, C. G. Powell, Quantum Electronics Conference, Miami, Fla., May 1968.
[CrossRef]

R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965). The cavity discussed by this author has a spherical symmetry.
[CrossRef]

Tibbetts, R. E.

J. S. Wilczynski, R. E. Tibbetts, J. Opt. Soc. Amer. 55, 1574 (1965).

Toraldo di Francia, G.

G. Toraldo di Francia, Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963), p. 157.

Wilczynski, J. S.

J. S. Wilczynski, R. E. Tibbetts, J. Opt. Soc. Amer. 55, 1574 (1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, (1965), 148.

Young, A.

A. Fletcher, T. Murphy, A. Young, Proc. Roy. Soc. A223, 216 (1959).

Appl. Opt. (1)

Bell System Tech. J. (1)

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

IBM J. Res. Dev. (2)

W. A. Hardy, IBM J. Res. Dev. 9, 31 (1965); R. V. Pole, H. Wieder, E. S. Barrekette, Appl. Opt. 6, 1571 (1967); H. Wieder, R. V. Pole, Appl. Opt. 6, 1761 (1967).
[CrossRef] [PubMed]

R. A. Myers, R. V. Pole, IBM J. Res. Dev. 11, 502 (1967); M. L. Dakss, C. G. Powell, Quantum Electronics Conference, Miami, Fla., May 1968.
[CrossRef]

J. Opt. Soc. Amer. (3)

R. V. Pole, J. Opt. Soc. Amer. 55, 254 (1965). The cavity discussed by this author has a spherical symmetry.
[CrossRef]

J. S. Wilczynski, R. E. Tibbetts, J. Opt. Soc. Amer. 55, 1574 (1965).

D. A. Berkowitz, J. Opt. Soc. Amer. 55, 1464 (1965).
[CrossRef]

J. Phys. Appl. Paris (1)

B. Macke, J. Phys. Appl. Paris, 26, 104A (1965).

Nuovo Cimento (1)

M. Bertolotti, Nuovo Cimento 32, 1242 (1964).
[CrossRef]

Proc. Roy. Soc. (1)

A. Fletcher, T. Murphy, A. Young, Proc. Roy. Soc. A223, 216 (1959).

Rev. Opt. (1)

P. Connes, Rev. Opt. 35, 1 (1956).

Other (4)

G. Toraldo di Francia, Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963), p. 157.

J. A. Arnaud, “Enhancement of Optical Receiver Sensitivities by Amplification of the Carrier,” to be published in the J. Quantum Electron, Nov.1968.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, (1965), 148.

For the properties of isometric transformations, see, for instance, G. Girard, A. Lentin, in Géométrie. Mécanique, Hachette, Ed. (Paris, 1964).

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Figures (4)

Fig. 1
Fig. 1

Three-lens cavity: (a) shows the notations used for the calculation of the lens ray matrix; (b) is the degenerate linear configuration which may be obtained from (a) when f1f2f, and c31 = c23 = f3R/2.

Fig. 2
Fig. 2

Four-lens cavity: (a) shows the notations used for the calculation of the lens ray matrix; (b) shows a ring configuration. By adjusting the axial position of the lenses, an exact fulfillment of the degeneracy condition can be obtained, in general; (c) and (d) show the two degenerate linear configurations that a general four-lens cavity may take.

Fig. 3
Fig. 3

These figures represent three nonplanar degenerate cavities. In (a) and (b), linear type cavities. In (c), ring type cavity.

Fig. 4
Fig. 4

(a) Maxwell fish-eye cavity. From a geometrical optics point of view, this cavity is rigorously degenerate; (b) rigorously degenerate cavity using the properties of a Luneburg lens.

Equations (18)

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[ x 1 x ˙ 1 ] = [ A B C D ] [ x 0 x ˙ 0 ] ;
B = C = 0 , A = D = 1.
[ A B C D ] = [ a b c d ] [ d b c a ] = [ a d + b c 2 a b 2 c d a d + b c ] .
k L - ( 2 p + l + 1 ) cos - 1 A + D 2 = 2 K π ,
x = A x + B x ˙ + δ , x ˙ = C x + D x ˙ + γ ,
q = ( A q + B ) / ( C q + D ) ,
k L - π μ + i log ( ψ j ) = 0 ( mod 2 π ) ,             j = 1 , 2 ,
[ 0 f - ( 1 / f ) 0 ] [ 1 c 0 1 ] = [ 0 f - ( 1 / f ) ( c / f ) ] .
c 12 = f 1 f 2 f 3             c 23 = f 2 f 3 f 1             c 31 = f 3 f 1 f 2 .
L = 2 ( f 1 + f 2 + f 3 ) + c 12 + c 23 + c 31 = ( f 1 f 2 + f 2 f 3 + f 3 f 1 ) 2 f 1 f 2 f 3 .
f 4 f 2 c 12 c 23 = - f 1 f 3 c 23 c 34 = f 2 f 4 c 34 c 41 = - f 3 f 1 c 41 c 12 = f 2 f 4 - f 1 f 3 .
R 2 cos φ + [ ( R / 2 ) cos φ ] 2 l = R 2 cos φ - ( R / 2 cos φ ) 2 l .
c / f = - 2 cos ( μ π / M ) ,
L ( n 2 / n 0 ) 1 2 = μ π ,
cos ( Ω / 2 ) = cos θ 1 cos θ 3 - cos ν sin θ 1 sin θ 3 .
cos ν = cot θ 1 cot θ 3 ,
L 2 = - 4 cos ( θ 1 + θ 3 ) cos ( θ 1 - θ 3 ) .
Ω = β 12 - β 23 + β 34 - β 2 N 1 ,

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