Abstract

In stable resonators any given initially paraxial rays remain close to the axis of the structure and are, in fact, confined within well-defined contours—the envelope of the ray system. Previously, envelopes of rays in empty resonators had been found and their form identified with the variation of the spot size. This geometric optical approach is extended to general resonators, comprising arbitrary arrangements of lenses and convergent or divergent inhomogeneous focusing media. An invariant quadratic form involving parameters descriptive of any of the ray segments that result from a given initial ray segment leads to a differential equation satisfied by the ray segments and their envelope in portions of the resonator. A maximum–minimum problem for the envelope is formulated and solved. In convergent media the envelope function is found to be periodically modulated. The period of the modulation depends only on the properties of the convergent medium; the location of relative maxima and minima, as well as their ratio, depends on both the medium and associated optics. In special cases, results are compared with available solutions of the corresponding electromagnetic problem. A particularly simple resonator is analyzed, and envelope characteristics correlated with the stability limits.

© 1966 Optical Society of America

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References

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  1. J. R. Pierce, Theory and Design of Electron Beams (D. Van Nostrand Co., Inc., Princeton, N.J., 1954), 2nd ed., Chap. 11, pp. 194–213.
  2. G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
  3. W. K. Kahn, “Ray Theory of Optical Resonators Containing an Inhomogeneous Medium”, 1964 U.R.S.I. Spring Meeting, Abstract 6–1–6, 84 (April1964).
  4. M. Bertolotti, Nuovo Cimento 32, 1242 (1964).
    [CrossRef]
  5. B. Macke, J. Phys. Appl. 26, 104A (1965).
  6. G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, N.Y., 1964), pp. 379–395.
  7. A. E. Siegman, Proc. IEEE 53, 277 (1965).
    [CrossRef]
  8. W. K. Kahn, Appl. Opt. 5, 407 (1966).
    [CrossRef] [PubMed]
  9. V. P. Bykov, L. A. Vainstein, Soviet Phys.-JETP 20, 338 (1965).
  10. W. K. Kahn, in Ref. 6, pp. 399–402.
  11. W. K. Kahn, Appl. Opt. 4, 758 (1965).
    [CrossRef]
  12. W. K. Kahn, “A Ray Theory of Optical Resonators and Beam Waveguides,” P.I.B.-MRI-1285–65, Polytechnic Institute of Brooklyn (July1964). A summary of this report was presented at the 1965 G-MTT Symposium in Clearwater, Florida, 5–7 May 1965. Symp. Dig. Ref. 1–5, pp. 21–24.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 4.
  14. W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).
  15. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publishing Co. Inc., Reading, Mass., 1963).
  16. L. B. Felsen, W. K. Kahn, in Proceedings of the Symposium on Millimeter Waves (Polytechnic Press, Brooklyn, 1959) pp. 477–512.
  17. D. R. Herriott, H. Kogelnik, R. Kompfner, Appl. Opt. 3, 523 (1964).
    [CrossRef]
  18. G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).
  19. J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
    [CrossRef]
  20. S. A. Collins, Appl. Opt. 3, 1263 (1964).
    [CrossRef]
  21. H. Kogelnik, Bell System Tech. J. 44, 455 (1965).
  22. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, 129 (1965).
    [CrossRef]
  23. H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).
  24. E. A. Marcatili, Bell System Tech. J. 43, 2887 (1964).
  25. S. E. Miller, Bell System Tech. J. 44, 2017 (1965).
  26. D. Marcuse, Bell System Tech. J. 44, 2065 (1965).
  27. D. Marcuse, Bell System Tech. J. 44, 2083 (1965).
  28. D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).
  29. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  30. J. R. Pierce, Proc. Natl. Acad. Sci. U.S. 47, 1808 (1961).
    [CrossRef]
  31. E. L. Ince, Ordinary Differential Equations (Dover Publications, Inc., New York, 1956); first American ed.; Chap. 3.
  32. Reference 13, p. 121.
  33. S. Barone, W. K. Kahn, B. Lippmann, N. Marcuvitz, S. Schneider, Res. Rept. R–628–57, PIB 556 Polytechnic Institute of Brooklyn (October1957).

1966 (1)

1965 (12)

V. P. Bykov, L. A. Vainstein, Soviet Phys.-JETP 20, 338 (1965).

W. K. Kahn, Appl. Opt. 4, 758 (1965).
[CrossRef]

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

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

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

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, 129 (1965).
[CrossRef]

H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).

S. E. Miller, Bell System Tech. J. 44, 2017 (1965).

D. Marcuse, Bell System Tech. J. 44, 2065 (1965).

D. Marcuse, Bell System Tech. J. 44, 2083 (1965).

D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

1964 (5)

E. A. Marcatili, Bell System Tech. J. 43, 2887 (1964).

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

D. R. Herriott, H. Kogelnik, R. Kompfner, Appl. Opt. 3, 523 (1964).
[CrossRef]

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

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

1962 (1)

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

1961 (2)

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

J. R. Pierce, Proc. Natl. Acad. Sci. U.S. 47, 1808 (1961).
[CrossRef]

Barone, S.

S. Barone, W. K. Kahn, B. Lippmann, N. Marcuvitz, S. Schneider, Res. Rept. R–628–57, PIB 556 Polytechnic Institute of Brooklyn (October1957).

Berreman, D. W.

D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).

Bertolotti, M.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 4.

Boyd, G. D.

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

Brouwer, W.

W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).

Bykov, V. P.

V. P. Bykov, L. A. Vainstein, Soviet Phys.-JETP 20, 338 (1965).

Collins, S. A.

Deschamps, G. A.

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, N.Y., 1964), pp. 379–395.

Felsen, L. B.

L. B. Felsen, W. K. Kahn, in Proceedings of the Symposium on Millimeter Waves (Polytechnic Press, Brooklyn, 1959) pp. 477–512.

Fukatsu, Y.

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, 129 (1965).
[CrossRef]

Goubau, G.

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

Herriott, D. R.

Hirano, J.

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

Ince, E. L.

E. L. Ince, Ordinary Differential Equations (Dover Publications, Inc., New York, 1956); first American ed.; Chap. 3.

Kahn, W. K.

W. K. Kahn, Appl. Opt. 5, 407 (1966).
[CrossRef] [PubMed]

W. K. Kahn, Appl. Opt. 4, 758 (1965).
[CrossRef]

S. Barone, W. K. Kahn, B. Lippmann, N. Marcuvitz, S. Schneider, Res. Rept. R–628–57, PIB 556 Polytechnic Institute of Brooklyn (October1957).

W. K. Kahn, “A Ray Theory of Optical Resonators and Beam Waveguides,” P.I.B.-MRI-1285–65, Polytechnic Institute of Brooklyn (July1964). A summary of this report was presented at the 1965 G-MTT Symposium in Clearwater, Florida, 5–7 May 1965. Symp. Dig. Ref. 1–5, pp. 21–24.

W. K. Kahn, in Ref. 6, pp. 399–402.

L. B. Felsen, W. K. Kahn, in Proceedings of the Symposium on Millimeter Waves (Polytechnic Press, Brooklyn, 1959) pp. 477–512.

W. K. Kahn, “Ray Theory of Optical Resonators Containing an Inhomogeneous Medium”, 1964 U.R.S.I. Spring Meeting, Abstract 6–1–6, 84 (April1964).

Kogelnik, H.

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

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

D. R. Herriott, H. Kogelnik, R. Kompfner, Appl. Opt. 3, 523 (1964).
[CrossRef]

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

Kompfner, R.

Lippmann, B.

S. Barone, W. K. Kahn, B. Lippmann, N. Marcuvitz, S. Schneider, Res. Rept. R–628–57, PIB 556 Polytechnic Institute of Brooklyn (October1957).

Macke, B.

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

Marcatili, E. A.

E. A. Marcatili, Bell System Tech. J. 43, 2887 (1964).

Marcuse, D.

D. Marcuse, Bell System Tech. J. 44, 2065 (1965).

D. Marcuse, Bell System Tech. J. 44, 2083 (1965).

Marcuvitz, N.

S. Barone, W. K. Kahn, B. Lippmann, N. Marcuvitz, S. Schneider, Res. Rept. R–628–57, PIB 556 Polytechnic Institute of Brooklyn (October1957).

Mast, P. E.

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, N.Y., 1964), pp. 379–395.

Miller, S. E.

S. E. Miller, Bell System Tech. J. 44, 2017 (1965).

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publishing Co. Inc., Reading, Mass., 1963).

Pierce, J. R.

J. R. Pierce, Proc. Natl. Acad. Sci. U.S. 47, 1808 (1961).
[CrossRef]

J. R. Pierce, Theory and Design of Electron Beams (D. Van Nostrand Co., Inc., Princeton, N.J., 1954), 2nd ed., Chap. 11, pp. 194–213.

Schneider, S.

S. Barone, W. K. Kahn, B. Lippmann, N. Marcuvitz, S. Schneider, Res. Rept. R–628–57, PIB 556 Polytechnic Institute of Brooklyn (October1957).

Schwering, F.

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

Siegman, A. E.

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, 129 (1965).
[CrossRef]

Unger, H. G.

H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).

Vainstein, L. A.

V. P. Bykov, L. A. Vainstein, Soviet Phys.-JETP 20, 338 (1965).

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, 129 (1965).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 4.

Appl. Opt. (5)

Arch. Elek. Übertrag. (1)

H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).

Bell System Tech. J. (7)

E. A. Marcatili, Bell System Tech. J. 43, 2887 (1964).

S. E. Miller, Bell System Tech. J. 44, 2017 (1965).

D. Marcuse, Bell System Tech. J. 44, 2065 (1965).

D. Marcuse, Bell System Tech. J. 44, 2083 (1965).

D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).

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

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

J. Phys. Appl. (1)

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

Nuovo Cimento (1)

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

Proc. IEEE (3)

A. E. Siegman, Proc. IEEE 53, 277 (1965).
[CrossRef]

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE, 53, 129 (1965).
[CrossRef]

Proc. Natl. Acad. Sci. U.S. (1)

J. R. Pierce, Proc. Natl. Acad. Sci. U.S. 47, 1808 (1961).
[CrossRef]

Soviet Phys.-JETP (1)

V. P. Bykov, L. A. Vainstein, Soviet Phys.-JETP 20, 338 (1965).

Trans. Inst. Radio Engrs. (1)

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

Other (12)

W. K. Kahn, “A Ray Theory of Optical Resonators and Beam Waveguides,” P.I.B.-MRI-1285–65, Polytechnic Institute of Brooklyn (July1964). A summary of this report was presented at the 1965 G-MTT Symposium in Clearwater, Florida, 5–7 May 1965. Symp. Dig. Ref. 1–5, pp. 21–24.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), Chap. 4.

W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publishing Co. Inc., Reading, Mass., 1963).

L. B. Felsen, W. K. Kahn, in Proceedings of the Symposium on Millimeter Waves (Polytechnic Press, Brooklyn, 1959) pp. 477–512.

W. K. Kahn, in Ref. 6, pp. 399–402.

G. A. Deschamps, P. E. Mast, in Proceedings of the Symposium on Quasi-Optics (Polytechnic Press, Brooklyn, N.Y., 1964), pp. 379–395.

J. R. Pierce, Theory and Design of Electron Beams (D. Van Nostrand Co., Inc., Princeton, N.J., 1954), 2nd ed., Chap. 11, pp. 194–213.

W. K. Kahn, “Ray Theory of Optical Resonators Containing an Inhomogeneous Medium”, 1964 U.R.S.I. Spring Meeting, Abstract 6–1–6, 84 (April1964).

E. L. Ince, Ordinary Differential Equations (Dover Publications, Inc., New York, 1956); first American ed.; Chap. 3.

Reference 13, p. 121.

S. Barone, W. K. Kahn, B. Lippmann, N. Marcuvitz, S. Schneider, Res. Rept. R–628–57, PIB 556 Polytechnic Institute of Brooklyn (October1957).

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Figures (6)

Fig. 1
Fig. 1

Definition of the ray parameters at the Nth reference plane.

Fig. 2
Fig. 2

General two-mirror resonator and equivalent beam waveguide.

Fig. 3
Fig. 3

Convergent medium separated by gaps and equivalent resonator.

Fig. 4
Fig. 4

A simple resonator filled with an inhomogeneous focusing medium and equivalent beam waveguide.

Fig. 5
Fig. 5

Stable regions for the resonator shown in Fig. 4.

Fig. 6
Fig. 6

Typical envelope contours as functions of the parameter Z ¯ 0 2.

Equations (75)

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R N = ( a N b N ) .
( 1 0 - c 1 ) ,
( 1 l 0 1 ) .
R N + 1 = T R N ,
T = ( t 11 t 12 t 21 t 22 )
( cos θ Z sin θ - Y sin θ cos θ ) .
- 1 1 2 τ + 1
T ˜ ζ T = ζ ,
ζ = ( 1 0 0 Z 2 ) .
R ˜ N ζ R N = a N 2 + Z 2 b N 2 = r 2 ,
( cos θ sin θ - sin θ cos ) ,
a ¯ N 2 + b ¯ N 2 = r 2
R N ( z ) = ( a N ( z ) b N ( z ) )
R N ( z ) = U R N ,
R N Lim z 0 R ( z ) .
R ˜ N ( z ) U ˜ - 1 ζ U - 1 R N ( z ) = r 2 .
R N ( z ) = ( f ( z ) d d z f ( z ) )
( u 22 2 + Z 2 u 21 2 ) f 2 - 2 ( u 12 u 22 + Z 2 u 11 u 21 ) f ( d f / d z ) + ( u 12 2 + Z 2 u 11 2 ) ( d f / d z ) 2 = r 2 ( det U ) 2 .
a N = r cos ϕ
b N = Y r sin ϕ ,
a N ( z ) = r ( u 11 cos ϕ + Y u 12 sin ϕ ) .
f ( z ) [ envelope = a N ( z ) ] max min ,
( / ϕ ) a N ( z ) = r ( - u 11 sin ϕ + Y u 12 cos ϕ ) = 0 ,
f ( z ) = ± r ( u 11 2 + Y 2 u 12 2 ) 1 2 ,
U = ( 1 z 0 1 ) ,
f ( z ) = ± r [ 1 + ( z / Z ) 2 ] 1 2 ,
[ f - z ( d f / d z ) ] 2 + Z 2 ( d f / d z ) 2 = r 2 .
( a ( z ) b ( z ) ) = T ( z ) ( a ( 0 ) b ( 0 ) )
T ( z ) = ( cos γ z Z 0 sin γ z - Y 0 sin γ z cos γ z ) ,
γ = Z 0 - 1 = Y 0 = ( c / l ) 1 2 .
n ( y ) = n 0 + n 2 y 2 , n 2 y 2 n 0 .
[ ( d 2 / d z 2 ) + γ 2 ] a ( z ) = 0 ,
γ 2 = - 2 n 2 / n 0 .
P = σ P - 1 σ ,
σ = ( 1 0 0 - 1 ) .
P P = ( cos θ p Z p sin θ p - Y p sin θ p cos θ p ) ,
Q Q = ( cos θ q Z q sin θ q - Y q sin θ q cos θ q ) ,
S = S = ( cos γ l Z 0 sin γ l - Y 0 sin γ l cos γ l ) ,
P ^ 2 = P P
Q ^ 2 = Q Q .
P ^ = ( cos 1 2 θ p Z p sin 1 2 θ p - Y p sin 1 2 θ p cos 1 2 θ p )
Q ^ = ( cos 1 2 θ q Z q sin 1 2 θ q - Y q sin 1 2 θ q cos 1 2 θ q )
T = P S Q Q S P .
P T P - 1 = P P S Q Q S = P ^ 2 S Q ^ 2 S .
T ^ = P ^ - 1 P T P - 1 P ^ = P ^ S Q 2 S P ^ ,
T ^ = ( cos θ ^ Z ^ sin θ ^ - Y ^ sin θ ^ cos θ ^ ) .
cos θ ^ = t ^ 11 = t ^ 22 = 1 2 τ
Z = ( - t ^ 12 / t ^ 21 ) 1 2 = t ^ 12 / [ 1 - ( τ / 2 ) 2 ] 1 2 .
R N ζ ^ R N = r ^ 2 .
τ = [ 2 cos θ p cos θ q - 1 2 ( Y 0 Z p + Z 0 Y q ) ( Y 0 Z q + Z 0 Y q ) sin θ p sin θ q ] cos 2 γ l - [ ( Y 0 Z p + Z 0 Y p ) sin θ p cos θ q + ( Y 0 Z q + Z 0 Y q ) cos θ p sin θ q ] sin 2 γ l + ( Y 0 Z p - Z 0 Y q ) ( Y 0 Z q - Z 0 Y q ) sin θ p sin θ q
R N ( z ) = U R N ,
U = T ( z ) P ^ = ( cos γ z cos 1 2 θ p - Z 0 Y p sin γ z sin 1 2 θ p Z p cos γ z sin 1 2 θ p + Z 0 sin γ z cos 1 2 θ p - Y 0 sin γ z cos 1 2 θ p - Y p cos γ z sin 1 2 θ p - Y 0 Z p sin γ z sin 1 2 θ p + cos γ z cos 1 2 θ p ) .
f ( z ) - ± r ^ [ ( cos γ z cos 1 2 θ p - Z 0 Y p sin γ z sin 1 2 θ p ) 2 + Y ^ 2 ( Z p cos γ z sin 1 2 θ p + Z 0 sin γ z cos 1 2 θ p ) 2 ] 1 2 .
f ( z ) = ± r ^ / 2 Z ¯ p { [ - 2 Z ¯ 0 Z ¯ p ( 1 - Z ¯ p 2 ) sin θ p ] sin 2 γ z [ ( Z ¯ p 2 - Z ¯ 0 2 ) ( 1 + Z ¯ p 2 ) + ( Z ¯ p 2 + Z ¯ 0 2 ) ( 1 - Z ¯ p 2 ) cos θ p ] cos 2 γ z ( Z ¯ p 2 + Z ¯ 0 2 ) ( 1 + Z ¯ p 2 ) + ( Z ¯ p 2 - Z ¯ 0 2 ) ( 1 - Z ¯ p 2 ) cos θ p } 1 2 ,
Z ¯ 0 = Y ^ Z 0 , Z ¯ p = Y ^ Z p .
f ( z ) = ± r ^ [ 1 2 ( 1 + Z ¯ 0 2 ) + 1 2 ( 1 - Z ¯ 0 2 ) cos 2 γ z ] 1 2 .
T = T ^ = ( cos 2 γ l - c Z 0 sin 2 γ l Z 0 sin 2 γ l - c Z 0 2 ( 1 - cos 2 γ l ) - Y 0 sin 2 γ l - c ( 1 + cos 2 γ l ) cos 2 γ l - c Z 0 sin 2 γ l ) .
- 1 cos 2 γ l - c Z 0 sin 2 γ l + 1 ,
Z ^ 2 = Z 0 sin 2 γ l - c Z 0 2 ( 1 - cos 2 γ l ) / Y 0 sin 2 γ l + c ( 1 + cos 2 γ l )
Z ^ 2 = - Z 0 2 tan 2 γ l ( c Z 0 - cot γ l ) / ( c Z 0 + tan γ l ) ,
γ j β , Y 0 = Z 0 - 1 j β ,
1 - ( τ / 2 ) 2 0 ,
1 - ( τ / 2 ) 2 = - 4 sin 2 γ l cos 2 γ l ( c / λ + tan γ l ) ( c / γ - cot γ l ) .
- tan γ l c / γ cot γ l , n π γ l ( 2 n + 1 ) π / 2 ,
cot γ l c / γ - tan γ l , ( 2 n - 1 ) π / 2 γ l n π ,
tanh β l c / β coth β l . β 0 ,
θ p 0 ,             Z p Z 0 ;
θ q 0 ,             θ q Y q 2 c .
f ( z ) = ± r ^ [ 1 2 ( 1 + Z ¯ 0 2 ) + 1 2 ( 1 - Z ¯ 0 2 ) cos 2 γ z ] 1 2 .
f ( 0 ) is a relative minimum when Z ¯ 0 2 > 1 ;
f ( 0 ) is a relative maximum when 1 > Z ¯ 0 2 > 0 ,
f ( 0 ) is a minimum and 0 > Z ¯ 0 2 .
f ( z ) = ± r cos γ z when Z ¯ 0 2 = 0 ;
f ( z ) , cos γ z 1 f ( z ) = ± r ^ , cos 2 γ z = 1 } when Z ¯ 0 2 .
c = [ ( d / d z ) l n f ( z ) ] z = l ,

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