Abstract

The theory of a large class of optical resonators has been developed in a compact form by the means of raising and lowering differential operators (commonly used in quantum mechanics). The theory is applicable to any cavity for which paraxial ray theory may be applied successfully and where losses, aperturing, and aberrations can be ignored. The resonator need not be planar (so that image rotation may occur), the optical elements may be astigmatic and the optic axis incompletely defined (such as when dispersive prisms are used). A discussion of existence and uniqueness of paraxial (pencillike) modes is provided, including the modification of the theory when degeneracies are present. It is proved that unstable cavities do not possess paraxial modes.

© 1972 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  2. W. K. Kahn, J. T. Nemit, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, 1967), pp. 501–526.
  3. P. Baues, Opto-Electron. 1, 103 (1969).
    [CrossRef]
  4. J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970). (The notation, although similar, is not identical to this paper.)
  5. S. A. Collins, Appl. Opt. 3, 1263 (1964).
    [CrossRef]
  6. A. Messiah, Quantum Mechanics (Wiley, New York, 1961), Vol. 1, Chap. 12.
  7. E. E. Bergmann, A. Holz, “Exact Solutions of an n-Dimensional Anisotropic Oscillator in a Uniform Magnetic Field,” Lehigh University preprint (1970), to appear in Il Nuovo Cimento, part B; E. E. Bergmann, A. Holz, Bull. Am. Phys. Soc., Ser. 2, 16, No. 000 (1971).
  8. Mention of such operators in connection with laser cavity modes is given by M. M. Popov, Opt. Spectrosc. 25, 170 (1968).
  9. Hörmander calls this class of functions φ, see L. Hörmander’Linear Partial Differential Operators, Die Grundlehren der mathematischen Wissenschaften (Springer-Verlag, New York, 1969), Band 116, p. 18.
  10. Defined in F. Riesz, B. Sz.-Nagy, Functional Analysis (Fredrick Ungar, New York, 1955), p. 280.
  11. H. Kogelnik in Lasers, A. K. Levine, Ed. (Marcel Dekker, New York, 1966), Vol. 1, Chap. 5.
  12. It should be emphasized that existence of paraxial solutions to the Fredholm problem with infinite limits of integration is not equivalent to the existence of arbitrary eigenfunction solutions derived for finite limits of integration by D. J. Newman, S. P. Morgan, Bell Syst. Tech. J. 43, 113 (1964) or by J. A. Cochran, Bell Syst. Tech. J. 44, 77 (1965).
  13. See, for example, L. D. Landau, E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory (Addison-Wesley, Reading, Mass., 1965), pp. 8ff.
  14. See Ref. 6, p. 173.
  15. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 494ff.
  16. See, for example, Ref. 11, p. 312.
  17. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  18. Ref. 11, pp. 310ff.
  19. R. V. Pole, R. A. Myers, H. Wieder, in Proceedings of the Symposium on Modern OpticsPolytechnic Press, (Brooklyn, 1967), p. 63.
  20. J. A. Arnaud, Appl. Opt. 8, 189 (1969).
    [CrossRef] [PubMed]
  21. Ref. 11, pp. 308ff.
  22. The most general treatment of optical resonators where apertures and losses can be neglected appears to be that of Ref. 4.
  23. Ref. 9.
  24. P. Connes, Rev. Opt. 35, 37 (1956).

1970 (1)

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970). (The notation, although similar, is not identical to this paper.)

1969 (2)

1968 (1)

Mention of such operators in connection with laser cavity modes is given by M. M. Popov, Opt. Spectrosc. 25, 170 (1968).

1965 (1)

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

1964 (2)

S. A. Collins, Appl. Opt. 3, 1263 (1964).
[CrossRef]

It should be emphasized that existence of paraxial solutions to the Fredholm problem with infinite limits of integration is not equivalent to the existence of arbitrary eigenfunction solutions derived for finite limits of integration by D. J. Newman, S. P. Morgan, Bell Syst. Tech. J. 43, 113 (1964) or by J. A. Cochran, Bell Syst. Tech. J. 44, 77 (1965).

1961 (1)

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

1956 (1)

P. Connes, Rev. Opt. 35, 37 (1956).

Arnaud, J. A.

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970). (The notation, although similar, is not identical to this paper.)

J. A. Arnaud, Appl. Opt. 8, 189 (1969).
[CrossRef] [PubMed]

Baues, P.

P. Baues, Opto-Electron. 1, 103 (1969).
[CrossRef]

Bergmann, E. E.

E. E. Bergmann, A. Holz, “Exact Solutions of an n-Dimensional Anisotropic Oscillator in a Uniform Magnetic Field,” Lehigh University preprint (1970), to appear in Il Nuovo Cimento, part B; E. E. Bergmann, A. Holz, Bull. Am. Phys. Soc., Ser. 2, 16, No. 000 (1971).

Collins, S. A.

Connes, P.

P. Connes, Rev. Opt. 35, 37 (1956).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 494ff.

Fox, A. G.

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

Holz, A.

E. E. Bergmann, A. Holz, “Exact Solutions of an n-Dimensional Anisotropic Oscillator in a Uniform Magnetic Field,” Lehigh University preprint (1970), to appear in Il Nuovo Cimento, part B; E. E. Bergmann, A. Holz, Bull. Am. Phys. Soc., Ser. 2, 16, No. 000 (1971).

Hörmander’, L.

Hörmander calls this class of functions φ, see L. Hörmander’Linear Partial Differential Operators, Die Grundlehren der mathematischen Wissenschaften (Springer-Verlag, New York, 1969), Band 116, p. 18.

Kahn, W. K.

W. K. Kahn, J. T. Nemit, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, 1967), pp. 501–526.

Kogelnik, H.

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

H. Kogelnik in Lasers, A. K. Levine, Ed. (Marcel Dekker, New York, 1966), Vol. 1, Chap. 5.

Landau, L. D.

See, for example, L. D. Landau, E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory (Addison-Wesley, Reading, Mass., 1965), pp. 8ff.

Li, T.

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

Lifshitz, E. M.

See, for example, L. D. Landau, E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory (Addison-Wesley, Reading, Mass., 1965), pp. 8ff.

Messiah, A.

A. Messiah, Quantum Mechanics (Wiley, New York, 1961), Vol. 1, Chap. 12.

Morgan, S. P.

It should be emphasized that existence of paraxial solutions to the Fredholm problem with infinite limits of integration is not equivalent to the existence of arbitrary eigenfunction solutions derived for finite limits of integration by D. J. Newman, S. P. Morgan, Bell Syst. Tech. J. 43, 113 (1964) or by J. A. Cochran, Bell Syst. Tech. J. 44, 77 (1965).

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 494ff.

Myers, R. A.

R. V. Pole, R. A. Myers, H. Wieder, in Proceedings of the Symposium on Modern OpticsPolytechnic Press, (Brooklyn, 1967), p. 63.

Nemit, J. T.

W. K. Kahn, J. T. Nemit, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, 1967), pp. 501–526.

Newman, D. J.

It should be emphasized that existence of paraxial solutions to the Fredholm problem with infinite limits of integration is not equivalent to the existence of arbitrary eigenfunction solutions derived for finite limits of integration by D. J. Newman, S. P. Morgan, Bell Syst. Tech. J. 43, 113 (1964) or by J. A. Cochran, Bell Syst. Tech. J. 44, 77 (1965).

Pole, R. V.

R. V. Pole, R. A. Myers, H. Wieder, in Proceedings of the Symposium on Modern OpticsPolytechnic Press, (Brooklyn, 1967), p. 63.

Popov, M. M.

Mention of such operators in connection with laser cavity modes is given by M. M. Popov, Opt. Spectrosc. 25, 170 (1968).

Riesz, F.

Defined in F. Riesz, B. Sz.-Nagy, Functional Analysis (Fredrick Ungar, New York, 1955), p. 280.

Sz.-Nagy, B.

Defined in F. Riesz, B. Sz.-Nagy, Functional Analysis (Fredrick Ungar, New York, 1955), p. 280.

Wieder, H.

R. V. Pole, R. A. Myers, H. Wieder, in Proceedings of the Symposium on Modern OpticsPolytechnic Press, (Brooklyn, 1967), p. 63.

Appl. Opt. (2)

Bell Syst. Tech. J. (4)

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

It should be emphasized that existence of paraxial solutions to the Fredholm problem with infinite limits of integration is not equivalent to the existence of arbitrary eigenfunction solutions derived for finite limits of integration by D. J. Newman, S. P. Morgan, Bell Syst. Tech. J. 43, 113 (1964) or by J. A. Cochran, Bell Syst. Tech. J. 44, 77 (1965).

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

J. A. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970). (The notation, although similar, is not identical to this paper.)

Opt. Spectrosc. (1)

Mention of such operators in connection with laser cavity modes is given by M. M. Popov, Opt. Spectrosc. 25, 170 (1968).

Opto-Electron. (1)

P. Baues, Opto-Electron. 1, 103 (1969).
[CrossRef]

Rev. Opt. (1)

P. Connes, Rev. Opt. 35, 37 (1956).

Other (15)

W. K. Kahn, J. T. Nemit, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, 1967), pp. 501–526.

Hörmander calls this class of functions φ, see L. Hörmander’Linear Partial Differential Operators, Die Grundlehren der mathematischen Wissenschaften (Springer-Verlag, New York, 1969), Band 116, p. 18.

Defined in F. Riesz, B. Sz.-Nagy, Functional Analysis (Fredrick Ungar, New York, 1955), p. 280.

H. Kogelnik in Lasers, A. K. Levine, Ed. (Marcel Dekker, New York, 1966), Vol. 1, Chap. 5.

See, for example, L. D. Landau, E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory (Addison-Wesley, Reading, Mass., 1965), pp. 8ff.

See Ref. 6, p. 173.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 494ff.

See, for example, Ref. 11, p. 312.

Ref. 11, pp. 308ff.

The most general treatment of optical resonators where apertures and losses can be neglected appears to be that of Ref. 4.

Ref. 9.

Ref. 11, pp. 310ff.

R. V. Pole, R. A. Myers, H. Wieder, in Proceedings of the Symposium on Modern OpticsPolytechnic Press, (Brooklyn, 1967), p. 63.

A. Messiah, Quantum Mechanics (Wiley, New York, 1961), Vol. 1, Chap. 12.

E. E. Bergmann, A. Holz, “Exact Solutions of an n-Dimensional Anisotropic Oscillator in a Uniform Magnetic Field,” Lehigh University preprint (1970), to appear in Il Nuovo Cimento, part B; E. E. Bergmann, A. Holz, Bull. Am. Phys. Soc., Ser. 2, 16, No. 000 (1971).

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Equations (69)

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f ( q ) = λ K ( q , q ) f ( q ) d 2 q .
x l y m ( / x ) n ( / y ) ν f c l m n p
K ( q , q ) = N exp i [ qUq + qVq + q Wq + d · q + d · q ] ,
K ( q , q ) = N exp i [ qUq + qVq + q Wq ] .
f k l ( q ) = λ k l N exp i ( qUq + qVq + q Wq ) f k l ( q ) d 2 q .
and             b j ( q ) = v j · q + w j · q b j ( q ) = v j * · q - w j * · q ,
b j ( q ) f k l ( q ) = λ j λ k l K ( q , q ) b j ( q ) f k l ( q ) d 2 q .
f k l ( q ) b j ( q ) K ( q , q ) d 2 q = f k l λ j ( v j · q - w j · q ) K ( q , q ) d 2 q .
[ ( v j · q + w j · ) - λ j ( v j · q - w j · ) ] K ( q , q ) = 0.
v j + 2 i Ww j = - λ j ( i V T w j ) ,
+ i Vw j = λ j ( v j - 2 i Uw ) .
Xu j ( I 2 W 0 V ) ( v j i w j ) = λ j ( 0 - V T I - 2 U ) u j λ j Yu j .
[ λ j V T + 2 ( U + W ) + λ j - 1 V ] w j = 0.
0 = η j 2 det V + η j [ det ( V + E ) - det ( V - E ) ] / 2 + det E + det V A .
η j λ j + λ j - 1 ,
E 2 ( U + W ) ,
V A ( V - V T ) / 2.
[ b j , b k ] = v k * · w j + w k * · v j .
[ b j , b k ] = i w k ( λ k * V - λ j V T ) w j .
[ b j , b k ] = i w k ( λ j - 1 V - λ k * - 1 V T ) w j .
( λ k * - λ j - 1 ) [ b j , b k ] = 0.
( λ j - λ k - 1 ) [ b j , b k ] = 0 ,
( λ j - λ k - 1 ) [ b j , b k ] = 0.
[ b j , b j ] c j ;             [ b j , b j ] - c j .
a i a i f = n i c i f ,             i = 1 , 2 ,
[ a i ( q ) a i ( q ) - n i c i ] K ( q , q ) f k l ( q ) d 2 q = 0 ,             i = 1 , 2 ,
b i ( q ) K ( q , q ) f ( q ) d 2 q = K ( q , q ) λ i b i ( q ) f ( q ) d 2 q
K ( q , q ) [ λ i * λ i a i ( q ) a i ( q ) - n i c i ] f k l ( q ) d 2 q .
c j = + 1.
P ( v 1 , v 2 ) ;             Q ( w 1 , w 2 ) .
f 00 ( q ) = N 00 exp [ - q ( PQ - 1 ) q / 2 ] ,
f 00 ( q ) = N 00 exp ( - P q 2 ) .
f k l ( q ) = N k l [ a 1 ( q ) ] k [ a 2 ( q ) ] l f 00 ( q )
λ k l = λ 1 - k λ 2 - l λ 00 .
ν k l m = ν 00 m + ( 2 π i ) - 1 ( ν 00 , m + 1 - ν 00 m ) ( k ln λ 1 + l ln λ 2 ) .
λ 1 m λ 2 n = 1.
( f k + p m , l + p n ; all integer p with k + p m 0 and l + p n 0 ) .
λ 1 = λ 2 .
[ det ( V + E ) - det ( V - E ) ] 2 = 16 det V ( det E - det V A ) < 16 ( det V ) 2 .
( α a 1 + β a 2 ,             α * a 1 + β * a 2 ) = α 2 c 1 + β 2 c 2 0.
λ 1 m = 1 = λ 2 n .
F j 1 p l = 1 p exp ( - 2 π i j l / p ) T l ( F ) λ 00 - l .
λ = ± 1.
b 1 = b 1 .
K ( q , q ) = K 2 ( q , q ) K 1 ( q , q ) d 2 q .
K i ( q , q ) = N i exp i ( q U i q + q V i q + q W i q + d i · q + d i · q ) ,
N i = ( 2 π ) - 1 ( det V i ) 1 2 .
lim 0 exp ( - q 2 ) .
U = U 1 - 1 4 V 1 ( W 1 + U 2 ) - 1 V 1 T ,
V = - 1 2 V 1 ( W 1 + U 2 ) - 1 V 2 ,
W = W 2 - 1 4 V 2 T ( W 1 + U 2 ) - 1 V 2 ,
d = d 1 - 1 2 V 1 ( W 1 + U 2 ) - 1 ( d 1 - d 2 ) ,
d = d 2 - 1 2 V 2 T ( W 1 + U 2 ) - 1 ( d 1 + d 2 ) ,
N = N 1 N 2 ( i π ) [ det ( W 1 + U 2 ) ] - 1 2 .
K ( q ) ( i π ) - 1 exp i ( q 2 - 1 ) .
f ( q ) = lim 0 K ( q - q ) f ( q ) d 2 q .
Lim 0 K ( q ) = δ ( q ) .
T 1 T 2 T 3 = ( T 1 T 2 ) T 3 = T 1 ( T 2 T 3 ) .
K j ( q , q ) N j exp ( i qSq ) ,             j = 1 , 2 , 3.
K 12 ( q , q ) = N 12 exp i ( - qS ϕ - 1 Sq / 8 - qS ϕ - 1 Sq / 4 - q S ϕ - 1 Sq / 8 ) .
K 123 ( q , q ) = N 123 exp ( - i qSq ) .
U = - W ; V = - V T , W = - U , N = N * ; d = - d .
g ( r ) = λ N exp i ( rUr + rVr + r Wr + d · r + d · r ) g ( r ) d 2 r .
g ( r ) = exp ( i k · q ) f ( q + q 0 ) .
k + 2 Uq 0 + Vq 0 + d = 0 = - k + 2 W q 0 + V T q 0 + d .
q 0 = [ V T + 2 ( U + W ) + V ] - 1 ( d + d ) .
λ = 1
b 1 f = 0 = b 2 f .
( b 1 , b 2 ) = c 0.

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