Abstract

A paraxial resonance equation is derived. This gives the mirror separation as a function of the radii of curvature of the mirrors and an integer N which is the number of return transits necessary to form a closed path of rays. Differentiating the paraxial resonance equation gives a formula for the relative mode density as a function of mirror separation. It is shown that the output power from a laser incorporating solid mirrors is inversely proportional to the mode density. In the case of hole coupling, the output power follows the same general profile but dips in power occur at the mirror separations corresponding to the resonance configurations of modes characterized by low values of N. Further confirmation of the paraxial resonance equation is obtained from passive resonators in which conic interference fringes and sudden increases in transmitted intensity are found to occur at the predicted mirror separations for low values of N corresponding to mode-degenerate configurations. The positions of the vertices of the ray traces are found to correspond to the patterns of discrete spots which are obtained in the output of a CO2 laser incorporating Brewster angle windows and a solid germanium mirror. The laser configurations which give maximum output power are plotted as a cliff of constant height above the g1g2 plane of the stability diagram, where g1 and g2 are the configuration coordinates. The relative merits of all possible cavity configurations having one mirror in common are shown as a set of equipower contours, and the hyperbolic curves of constant N are also superimposed on the stability diagram. The advantages of simplicity and directness in using the ray model are made clear.

© 1970 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. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  3. G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).
  4. S. A. Collins, Appl. Opt, 3, 1263 (1964).
    [Crossref]
  5. M. Bertolotti, Nuovo Cimento 32, 1262 (1964).
  6. B. Macke, J. Phys. (Paris) 26, 104A (1965).
  7. M. Pauthier, Quantum Electron. 2, 1253 (1963).
  8. J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964).
    [Crossref]
  9. D. Herriott, H. Kogelnik, R. Kompfner, Appl. Opt. 3, 523 (1964).
    [Crossref]
  10. V. P. Vykov, L. A. Vainstein, Sov. Phys.-JETP 20, 338 (1965).
  11. W. K. Kahn, Appl. Opt. 4, 758 (1965).
    [Crossref]
  12. W. K. Kahn, Appl. Opt. 5, 407 (1966).
    [Crossref] [PubMed]
  13. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  14. N. Kurauchi, W. K. Kahn, Appl. Opt. 5, 1023 (1966).
    [Crossref] [PubMed]
  15. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [Crossref] [PubMed]
  16. D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).
  17. J. H. McElroy, H. E. Walker, Appl. Opt. 7, 1235 (1968).
    [Crossref] [PubMed]
  18. J. R. Johnson, Appl. Opt. 6, 1930 (1967).
    [Crossref] [PubMed]
  19. D. C. Sinclair, Spectra-Physics Laser Technical Bulletin Number 6 (April1968).
  20. A. G. Fox, T. Li, Quantum Electron. 2, 1263 (1963).
  21. D. C. Sinclair, Appl. Opt. 3, 1067 (1964).
    [Crossref]
  22. R. J. Freiberg, A. S. Halsted, Laser Focus 21, 111 (1968).
  23. R. J. Freiberg, A. S. Halsted, Appl. Opt. 8, 355 (1969).
    [Crossref] [PubMed]

1969 (1)

1968 (3)

R. J. Freiberg, A. S. Halsted, Laser Focus 21, 111 (1968).

D. C. Sinclair, Spectra-Physics Laser Technical Bulletin Number 6 (April1968).

J. H. McElroy, H. E. Walker, Appl. Opt. 7, 1235 (1968).
[Crossref] [PubMed]

1967 (1)

1966 (3)

1965 (5)

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

V. P. Vykov, L. A. Vainstein, Sov. Phys.-JETP 20, 338 (1965).

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

D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).

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

1964 (5)

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964).
[Crossref]

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

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

M. Bertolotti, Nuovo Cimento 32, 1262 (1964).

D. C. Sinclair, Appl. Opt. 3, 1067 (1964).
[Crossref]

1963 (2)

A. G. Fox, T. Li, Quantum Electron. 2, 1263 (1963).

M. Pauthier, Quantum Electron. 2, 1253 (1963).

1962 (1)

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

1961 (2)

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).

Bertolotti, M.

M. Bertolotti, Nuovo Cimento 32, 1262 (1964).

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).

Collins, S. A.

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

Fox, A. G.

A. G. Fox, T. Li, Quantum Electron. 2, 1263 (1963).

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

Freiberg, R. J.

R. J. Freiberg, A. S. Halsted, Appl. Opt. 8, 355 (1969).
[Crossref] [PubMed]

R. J. Freiberg, A. S. Halsted, Laser Focus 21, 111 (1968).

Gerardo, J. B.

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964).
[Crossref]

Gordon, J. P.

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

Halsted, A. S.

R. J. Freiberg, A. S. Halsted, Appl. Opt. 8, 355 (1969).
[Crossref] [PubMed]

R. J. Freiberg, A. S. Halsted, Laser Focus 21, 111 (1968).

Herriott, D.

Johnson, J. R.

Kahn, W. K.

Kogelnik, H.

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

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

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

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

Kompfner, R.

Kurauchi, N.

Li, T.

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

A. G. Fox, T. Li, Quantum Electron. 2, 1263 (1963).

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

Macke, B.

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

McCumber, D. E.

D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).

McElroy, J. H.

Pauthier, M.

M. Pauthier, Quantum Electron. 2, 1253 (1963).

Sinclair, D. C.

D. C. Sinclair, Spectra-Physics Laser Technical Bulletin Number 6 (April1968).

D. C. Sinclair, Appl. Opt. 3, 1067 (1964).
[Crossref]

Vainstein, L. A.

V. P. Vykov, L. A. Vainstein, Sov. Phys.-JETP 20, 338 (1965).

Verdeyen, J. T.

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964).
[Crossref]

Vykov, V. P.

V. P. Vykov, L. A. Vainstein, Sov. Phys.-JETP 20, 338 (1965).

Walker, H. E.

Appl. Opt (1)

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

Appl. Opt. (9)

Bell Syst. Tech. J. (5)

D. E. McCumber, Bell Syst. Tech. J. 44, 333 (1965).

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

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).

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

J. Phys. (Paris) (1)

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

Laser Focus (1)

R. J. Freiberg, A. S. Halsted, Laser Focus 21, 111 (1968).

Nuovo Cimento (1)

M. Bertolotti, Nuovo Cimento 32, 1262 (1964).

Proc. IEEE (1)

J. B. Gerardo, J. T. Verdeyen, Proc. IEEE 52, 690 (1964).
[Crossref]

Quantum Electron. (2)

A. G. Fox, T. Li, Quantum Electron. 2, 1263 (1963).

M. Pauthier, Quantum Electron. 2, 1253 (1963).

Sov. Phys.-JETP (1)

V. P. Vykov, L. A. Vainstein, Sov. Phys.-JETP 20, 338 (1965).

Spectra-Physics Laser Technical Bulletin Number 6 (1)

D. C. Sinclair, Spectra-Physics Laser Technical Bulletin Number 6 (April1968).

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Figures (10)

Fig. 1
Fig. 1

The coordinates used in the ray analysis.

Fig. 2
Fig. 2

Ecliptic and nonecliptic ray traces for N = 3 and 4. The positions of vertices of one ray trace on one mirror are shown with full dots. When an equivalent ray trace exists, the corresponding vertices on the same mirror are shown with open dots. The equivalent ray traces to planar ecliptics are sketched with dashed lines. In the case of planar nonecliptics, the ray trace is its own equivalent. For the sake of clarity, equivalent ray traces have been omitted from the sketches of nonplanar ray traces.

Fig. 3
Fig. 3

Comparison of output power (experimental) with reciprocal mode density (theoretical) as a function of mirror separation in the case of solid mirrors. The experimental data are due to Pauthier.7

Fig. 4
Fig. 4

Measured output power from a hole-coupled CO2 laser as a function of mirror separation. The numbers under each dip correspond to the value of N at that mirror separation.

Fig. 5
Fig. 5

A sequence of interference fringes found at the resonance configurations of modes characterized by low values of N.

Fig. 6
Fig. 6

Power histograms showing the resonance configurations for low values of N in three different passive resonators: (a) b1 = b2 = 32 cm; (b) b1 = 26 cm, b2 = 32 cm; (c) b1 = 32 cm, b2 = ∞.

Fig. 7
Fig. 7

Predicted and observed mode patterns from a CO2 laser. The vertices of the equivalent ray traces for even N are not shown (see text).

Fig. 8
Fig. 8

Contour graphs showing the relative output power as a function of configuration coordinates with one mirror fixed. The line corresponds to Pauthier’s experiment.7

Fig. 9
Fig. 9

Superposition of equipower and equimode contours on the stability diagram.

Fig. 10
Fig. 10

The ray traces for N = 2. O1 and O2 are the centers of curvature of mirrors 1 and 2, respectively.

Equations (76)

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T = ( 1 2 d b 1 2 d 2 d 2 b 1 2 b 1 2 b 2 + 4 d b 1 b 2 1 2 d b 1 4 d b 2 + 4 d 2 b 1 b 2 ) ,
( x l α l ) = T l ( x 0 α 0 ) ,
T l ( A B C D ) l = 1 sin θ × ( A sin l θ sin ( l 1 ) θ B sin l θ C sin l θ D sin l θ sin ( l 1 ) θ )
cos θ 1 2 ( A + D ) = 1 2 d b 1 2 d b 2 + 2 d 2 b 1 b 2 .
T N = I the identity matrix .
θ = 2 K π / N ,
d ± = b 1 + b 2 2 ± 1 2 ( b 1 2 + b 2 2 + 2 b 1 b 2 cos θ ) 1 2 θ = 2 K π / N ,
d = b [ 1 ± cos ( K π / N ) ] .
d = ( b 1 / 2 ) [ 1 cos ( 2 K π / N ) ] .
α 0 = w 2 / b 2 ,
α 0 = υ 2 d { 1 ± [ b 1 ( b 2 d ) b 2 ( b 1 d ) ] 1 2 } ,
x l ( 2 ) = w 2 cos l θ
x l ( 1 ) = ± w 2 [ b 1 ( b 2 d ) b 2 ( b 1 d ) ] 1 2 cos ( l + 1 2 ) θ ,
x l ( 2 ) = υ 2 cos ( θ / 4 ) cos ( l 1 4 ) θ
x 1 ( 1 ) = ± υ 2 cos ( θ / 4 ) [ b 1 ( b 2 d ) b 2 ( b 1 d ) ] 1 2 cos ( l + 1 4 ) θ .
( y 0 β 0 ) = T M ( x 0 α 0 ) ,
x l ( 2 ) 2 + y l ( 2 ) 2 2 x l ( 2 ) y l ( 2 ) cos M θ = w 2 2 sin 2 M θ .
x l 2 w 2 2 ( 1 + cos M θ ) + y l 2 w 2 2 ( 1 cos M θ ) = 1 ,
x l ( 1 2 ) 1 2 ( x l ( 2 ) y l ( 2 ) )
y l ( 1 2 ) 1 2 ( x l ( 2 ) + y l ( 2 ) ) .
x l ( 2 ) = ± w 1 [ b 2 ( b 1 d ) b 1 ( b 2 d ) ] 1 2 cos ( l + 1 2 ) θ .
y l ( 2 ) = ± w 1 [ b 2 ( b 1 d ) b 1 ( b 2 d ) ] 1 2 cos ( l + M + 1 2 ) θ .
x l ( 2 ) 2 + y l ( 2 ) 2 2 x l ( 2 ) y l ( 2 ) cos M θ = w 1 2 b 2 ( b 1 d ) b 1 ( b 2 d ) sin 2 M θ .
( w 1 / w 2 ) 2 = b 1 ( b 2 d ) / b 2 ( b 1 d ) .
E · ( v × n ˆ 1 ) = 0 , E · ( v × n ˆ 2 ) = 0 ,
1 ρ = [ d ( b 1 + b 1 d ) ( d b 1 ) ( d b 2 ) ] 1 2 | 2 d b 1 b 2 | .
d = [ ( b 1 + b 2 ) / 2 ] ± 1 2 ( b 2 2 b 1 2 ) , 1 2 | b 2 | | b 1 | .
1 ρ = 2 b 1 [ d ( b 1 + b 2 d ) ( d b 1 ) ( d b 2 ) ] 1 2 | 2 d b 1 b 2 | .
K / N = 1 2 π cos 1 ( b 1 / b ) , | b 2 | | b 1 | .
ν 0 = c 2 d { q + 1 π ( 1 + m + n ) cos 1 [ ( 1 d b 1 ) ( 1 d b 2 ) ] 1 2 } .
cos 2 θ 2 1 + cos θ 2 = ( 1 d b 1 ) ( 1 d b 2 ) .
ν 0 = ( c / 2 d ) [ q + ( 1 / π ) ( 1 + m + n ) ( θ / 2 ) ] .
ν 0 = ( c / 2 N d ) [ N q + K ( 1 + m + n ) ] .
I I 0 = { 1 N [ 1 + ( A / T ) ] 2 1 1 + [ 4 π d / c ( 1 R ) ] 2 ( ν ν 0 ) 2 } ,
P = ( I / I 0 ) P 0 ,
P 0 I 0 π w 0 2 = I 0 λ [ d ( d b 1 ) ( d b 2 ) ( b 1 + b 2 d ) ] 1 2 | 2 d b 1 b 2 | .
P = I 0 λ [ d ( d b 1 ) ( d b 2 ) ( b 1 + b 2 d ) ] 1 2 | 2 d b 1 b 2 | × { 1 N 1 [ 1 + ( A / T ) ] 2 1 1 + [ 4 π d / c ( 1 R ) ] 2 ( ν ν 0 ) 2 } .
P = I 0 λ [ d ( 2 b d ) ] 1 2 2 [ 1 N 1 [ 1 + ( A / T ) ] 2 × 1 1 + [ 4 π d / c ( 1 R ) ] 2 ( ν ν 0 ) 2 ] ( b 1 = b 2 = b )
P = I 0 λ [ d ( b 1 d ) ] 1 2 [ 1 N 1 [ 1 + ( A / T ) ] 2 × 1 1 + [ 4 π d / c ( 1 R ) ] 2 ( ν ν 0 ) 2 ] ( b 2 = ) .
1 ρ = | 2 ( 1 g < ) g 1 + g 2 2 g 1 g 2 | [ g 1 g 2 ( 1 g 1 g 2 ) ] 1 2 ,
g 1 1 ( d / b 1 ) , g 2 1 ( d / b 2 ) ,
g < = g 1 if | 1 g 1 | > | 1 g 2 | ,
g < = g 2 if | 1 g 1 | < | 1 g 2 | .
g 1 g 2 = cos 2 ( θ / 2 ) ,
( 2 d / λ ) q 1 + m + n = ( 2 d / λ ) q 1 + 2 p + l = K N = 1 π cos 1 × [ ( 1 d b 1 ) ( 1 d b 2 ) ] 1 2 .
A has coordinates [ b 2 ( 1 cos i ) + b 1 d , b 2 sin i ] , B has coordinates [ b 1 cos ϕ , b 1 sin ϕ ] ,
C has coordinates [ b 2 ( 1 cos i ) + b 1 d , b 2 sin i ] .
b 1 sin ϕ b 2 sin i = [ d b 2 ( 1 cos i ) b 1 ( 1 cos ϕ ) ] tan ( ϕ i )
b 1 sin ϕ + b 2 sin i = [ d b 2 ( 1 cos i ) b 1 ( 1 cos ϕ ) ] tan ( ϕ + i ) .
sin ϕ [ b 2 ( b 1 + b 2 d ) cos i ] = 0.
sin i [ b 1 ( b 1 + b 2 d ) cos ϕ ] = 0.
i = 0 in which case d = b 1 ,
ϕ = 0 in which case d = b 2 ,
cos ϕ = b 1 b 1 + b 2 d and cos i = b 2 b 1 + b 2 d .
d min b ( a 2 / 2 b ) .
A is the point [ O , b 2 sin α , b 1 d + b 2 ( 1 cos α ) ] , B is the point [ b 1 sin β , O , b 1 cos β ] , C is the point [ O , b 2 sin α , b 1 d + b 2 ( 1 cos α ) ] , D is the point [ b 1 sin β , O , b 1 cos β ] ,
O 2 is the point [ O , O , b 1 + b 2 d ] .
d = b 2 ( 1 cos α ) + b 1
d = b 1 ( 1 cos β ) + b 2 .
d max b + ( a 2 / 2 b ) .
b 1 = b 10 [ 1 + ( e 1 2 / 2 ) cos 2 ( ψ θ 1 ) ]
b 2 = b 20 [ 1 + ( e 2 2 / 2 ) cos 2 ( ψ θ 2 ) ] ,
( b 10 + b 20 ) / 2 ± 1 2 [ ( b 10 2 + b 20 2 + 2 b 10 b 20 cos θ ) ] 1 2 .
d N = d N 0 [ 1 + ( E 1 2 / 2 ) cos 2 ( ψ θ 1 ) + ( E 2 2 / 2 ) cos 2 ( ψ θ 2 ) ] ,
E 1 2 ( d N 0 b 20 ) e 1 2 2 d N 0 b 10 b 20
E 2 2 ( d N 0 b 10 ) e 2 2 2 d N 0 b 10 b 20 .
tan 2 ψ p r = E 1 2 sin 2 θ 1 + E 2 2 sin 2 θ 2 E 1 2 cos 2 θ 1 + E 2 2 cos 2 θ 2 .
d N min max = d N 0 [ 1 + ( E 1 2 / 2 ) sin 2 ( ψ 0 θ 1 ) + ( E 2 2 / 2 ) sin 2 ( ψ 0 θ 2 ) ]
d N max min = d N 0 [ 1 + ( E 1 2 / 2 ) cos 3 ( ψ 0 θ 1 ) + ( E 2 2 / 2 ) cos 2 ( ψ 0 θ 2 ) ]
d N = d N min [ 1 + ( d N max d N min ) d N min cos 2 ( ψ ψ 0 ) ] .
L = 2 N d N [ 1 + w 1 2 ( b 1 + b 2 2 d N ) 2 d N b 1 ( b 2 d N ) + fourth and higher order terms ]
= 2 N d N [ 1 + w 2 2 ( b 1 + b 2 2 d N ) 2 d N b 2 ( b 1 d N ) + fourth and higher order terms ] .
w N [ ( L / 2 N ) d N ] 1 2 .
w N | d d N | 1 2 ,
d d N min ( 1 + ) ,
w N | d N max d N min d N min cos 2 ( ψ ψ 0 ) | 1 2 .

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