Abstract

The asymptotic analysis technique for calculating the modes of unstable strip resonators is extended to include the effects of a saturable but otherwise uniform gain. Utilizing simultaneous forward and backward (in time) propagation and the Rigrod gain formula, an iterative algorithm is employed to find the intensities inside a resonator. In contrast with Fox-Li type iterations, this scheme converges rapidly and gives all the higher modes. Mode properties at critical Fresnel numbers are examined as a function of gain saturation, which is seen to reduce mode degeneracy.

© 1981 Optical Society of America

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References

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  1. A. E. Siegman, Proc. IEEE 53, 277 (1965).
    [CrossRef]
  2. A. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1966).
  3. W. Streiffer, IEEE J. Quantum Electron. QE-4, 229 (1968).
    [CrossRef]
  4. L. Bergstein, Appl. Opt. 7, 495 (1968).
    [CrossRef] [PubMed]
  5. L. Chen, L. Felsen, IEEE J. Quantum Electron. QE-9, 1102 (1973).
    [CrossRef]
  6. P. Horwitz, J. Opt. Soc. Am. 63, 1528 (1973).
    [CrossRef]
  7. G. Moore, R. McCarthy, J. Opt. Soc. Am. 67, 228 (1977).
    [CrossRef]
  8. G. Moore, R. McCarthy, J. Opt. Soc. Am. 67, 221 (1977).
    [CrossRef]
  9. M. M. Weiner, IEEE J. Quantum Electron. QE-13, 803 (1977).
    [CrossRef]
  10. M. M. Weiner, Appl. Opt. 18, 1828 (1979). The basic resonator equation is described in numerous papers, but this work is the only one to express it explicitly in terms of the effective Fresnel number.
    [CrossRef] [PubMed]
  11. From the waveguide theory of resonator analysis, e.g., G. Vinukurov, V. Lyubimov, I. Orlova, Opt. Spectrosk. 34, 427 (1973), the next to lowest-loss eigenvalue has a geometric optics asymptote of M−0.50 but is closer to M−0.25 for Feff’s less than 50 or so.
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 752–754.
  13. Y. Ananev, Sov. J. Quantum Electron. 5, 615 (1975).
    [CrossRef]
  14. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
    [CrossRef]
  15. This formula assumes no interference between forward and backward traveling waves. If such interference exists, the effect is to shift the overall value of intensity by a small amount. See G. P. Agrawal, M. Lax, J. Opt. Soc. Am. 69, 1717 (1979).
    [CrossRef]
  16. W. H. Louisell, M. Lax, G. P. Agrawal, H. W. Gatzke, Appl. Opt. 18, 2730 (1979).
    [CrossRef] [PubMed]
  17. E. A. Sziklas, A. E. Siegman, Appl. Opt. 14, 1874 (1975). The computer algorithm described in this paper is the archetype of a number of simulation codes in use today.
    [CrossRef] [PubMed]
  18. D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973).
    [CrossRef] [PubMed]
  19. C. Santana, L. B. Felsen, Appl. Opt. 17, 2239 (1978).
    [CrossRef] [PubMed]
  20. G. McAllister, W. Steier, W. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
    [CrossRef]
  21. R. Butts, P. Avizonis, J. Opt. Soc. Am. 68, 1072 (1978).
    [CrossRef]

1979 (3)

1978 (2)

1977 (3)

1975 (2)

1974 (1)

G. McAllister, W. Steier, W. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

1973 (4)

From the waveguide theory of resonator analysis, e.g., G. Vinukurov, V. Lyubimov, I. Orlova, Opt. Spectrosk. 34, 427 (1973), the next to lowest-loss eigenvalue has a geometric optics asymptote of M−0.50 but is closer to M−0.25 for Feff’s less than 50 or so.

D. B. Rensch, A. N. Chester, Appl. Opt. 12, 997 (1973).
[CrossRef] [PubMed]

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

P. Horwitz, J. Opt. Soc. Am. 63, 1528 (1973).
[CrossRef]

1968 (2)

W. Streiffer, IEEE J. Quantum Electron. QE-4, 229 (1968).
[CrossRef]

L. Bergstein, Appl. Opt. 7, 495 (1968).
[CrossRef] [PubMed]

1966 (1)

A. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1966).

1965 (2)

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

W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

Agrawal, G. P.

Ananev, Y.

Y. Ananev, Sov. J. Quantum Electron. 5, 615 (1975).
[CrossRef]

Avizonis, P.

Bergstein, L.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 752–754.

Butts, R.

Chen, L.

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

Chester, A. N.

Felsen, L.

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

Felsen, L. B.

Fox, A.

A. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1966).

Gatzke, H. W.

Horwitz, P.

Lacina, W.

G. McAllister, W. Steier, W. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

Lax, M.

Li, T.

A. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1966).

Louisell, W. H.

Lyubimov, V.

From the waveguide theory of resonator analysis, e.g., G. Vinukurov, V. Lyubimov, I. Orlova, Opt. Spectrosk. 34, 427 (1973), the next to lowest-loss eigenvalue has a geometric optics asymptote of M−0.50 but is closer to M−0.25 for Feff’s less than 50 or so.

McAllister, G.

G. McAllister, W. Steier, W. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

McCarthy, R.

Moore, G.

Orlova, I.

From the waveguide theory of resonator analysis, e.g., G. Vinukurov, V. Lyubimov, I. Orlova, Opt. Spectrosk. 34, 427 (1973), the next to lowest-loss eigenvalue has a geometric optics asymptote of M−0.50 but is closer to M−0.25 for Feff’s less than 50 or so.

Rensch, D. B.

Rigrod, W.

W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

Santana, C.

Siegman, A. E.

Steier, W.

G. McAllister, W. Steier, W. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

Streiffer, W.

W. Streiffer, IEEE J. Quantum Electron. QE-4, 229 (1968).
[CrossRef]

Sziklas, E. A.

Vinukurov, G.

From the waveguide theory of resonator analysis, e.g., G. Vinukurov, V. Lyubimov, I. Orlova, Opt. Spectrosk. 34, 427 (1973), the next to lowest-loss eigenvalue has a geometric optics asymptote of M−0.50 but is closer to M−0.25 for Feff’s less than 50 or so.

Weiner, M. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 752–754.

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

A. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1966).

IEEE J. Quantum Electron. (4)

W. Streiffer, IEEE J. Quantum Electron. QE-4, 229 (1968).
[CrossRef]

M. M. Weiner, IEEE J. Quantum Electron. QE-13, 803 (1977).
[CrossRef]

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

G. McAllister, W. Steier, W. Lacina, IEEE J. Quantum Electron. QE-10, 346 (1974).
[CrossRef]

J. Appl. Phys. (1)

W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Spectrosk. (1)

From the waveguide theory of resonator analysis, e.g., G. Vinukurov, V. Lyubimov, I. Orlova, Opt. Spectrosk. 34, 427 (1973), the next to lowest-loss eigenvalue has a geometric optics asymptote of M−0.50 but is closer to M−0.25 for Feff’s less than 50 or so.

Proc. IEEE (1)

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

Sov. J. Quantum Electron. (1)

Y. Ananev, Sov. J. Quantum Electron. 5, 615 (1975).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 752–754.

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Figures (16)

Fig. 1
Fig. 1

Nomenclature for confocal unstable resonator. When the confocal condition is met, the output is a collimated beam.

Fig. 2
Fig. 2

Absolute value of resonator equation eigenvalue. These curves are for an empty, untilted (ɛ = 0) resonator with magnification M = 2.9.

Fig. 3
Fig. 3

First virtual images of feedback mirror edges. These virtual images are in turn reimaged by the resonator mirrors, forming a set of successively further removed virtual images.

Fig. 4
Fig. 4

Multiply reflected rays emanating from the confocal virtual Source. If the source angle θn is small enough, it will share the same divergence as the output beam.

Fig. 5
Fig. 5

Nomenclature for geometric resonator analysis (after Ref. 8).

Fig. 6
Fig. 6

(a) Forward and (b) backward propagation. They proceed in opposite time sense but the same spatial sense.

Fig. 7
Fig. 7

Empty resonator results from (a) Rench and Chester and (b) our calculations. Here M = 2.9, Feff = 0.64.

Fig. 8
Fig. 8

Loaded resonator calculation for the conditions of Fig. 7. Here G0l = 5.0.

Fig. 9
Fig. 9

Resonator intensity profile calculations for Feff = 8.892, M = 2.9. The three cases are (a) empty resonator, (b) G0l = 1.065, and (c) G0l = 5.0.

Fig. 10
Fig. 10

Resonator phase profile calculations for the same conditions as Fig. 9. Although quite similar, the profiles show slight differences when superimposed.

Fig. 11
Fig. 11

Resonator saturated gain profile for the conditions of Fig. 9(b). The medium extends from ¼ to ¾ the distance from the primary to the feedback mirrors.

Fig. 12
Fig. 12

Resonator intensity profile calculations for Feff = 9.390, M = 2.9. The three cases are (a) empty resonator, (b) G0l = 1.065, and (c) G0l = 5.0.

Fig. 13
Fig. 13

Resonator intensity profile calculations for Feff = 9.863, M = 2.9. The three cases are (a) empty resonator, (b) G0l = 1.065, and (c) G0l = 5.0.

Fig. 14
Fig. 14

Portion of Fig. 2 (solid curve) showing results from saturated gain calculations. Here M = 2.9. It will be seen that the presence of gain saturation splits the empty resonator mode degeneracy.

Fig. 15
Fig. 15

Critical Feff for onset of mode degeneracy as a function of magnification. For Feff’s greater than the data points, there are no mode degeneracies.

Fig. 16
Fig. 16

(a) Degenerate and (b) nondegenerate mode patterns for Feff = 11.883 and G0l ~ 2. The effect of the degeneracy on intensity profiles is subtle.

Equations (46)

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λ f ( x ) = ( 2 i M 2 M 2 1 F eff ) 1 / 2 × 1 + + 1 + exp [ i π 2 M 2 M 2 1 F eff ( x x M ) 2 ] × f ( x ) d x ,
λ f ( x ) = f ( x / M ) ,
L c = 1 ( 1 / M ) ,
L c ( i ) = 1 ( | λ i 2 | / M ) .
f i ( x ) = n = 1 N { f n , i ( x ) exp [ i k r n ( x ) ] r n ( x ) ± f n , i ( x ) exp [ i k r n ( x ) ] r n ( x ) } + f ˆ i ( x ) .
f i ( x ) n = 1 N ( 1 w n ) 1 / 2 × { exp [ 1 2 i k ( x x n ) 2 w n ] f n , i ( x ) ± exp [ 1 2 i k ( x + x n ) 2 w n ] f n ( x ) } + f ˆ i ( x ) .
w n = d ( M 2 n 1 ) ,
x n = a M n .
1 = 1 2 π ( i 2 F eff ) 1 / 2 × { n = 1 N + 1 [ exp ( 2 π i F eff β n ) β n 1 / 2 ± exp ( 2 π i F eff β n 1 ) β n + 1 / 2 ] μ n × ( 1 1 μ for symmetric modes when n = N + 1 ) } .
β n = 1 M n 1 + M n ,
λ = 1 / μ * .
f n , i ( x ) = ( μ M ) n ( 1 M 2 n ) ( 1 M 2 ) ( a + M n x ) ,
f ˆ i ( x ) = μ N + 1 1 μ [ a ( 1 M 2 ) ] 1 × 2 exp ( 2 π i F eff ) / d
θ 1 = D / f 2 ,
θ n = D / M n 1 f 2 ,
θ c = 2 λ / D .
D M N 1 f 2 = 2 λ D ,
N = ln ( 4 M F eff ) ln M + 1.
N = ln ( 250 F eff ) ln M .
1 2 a a + a I ( x ) d x = I g .
E = exp [ i k ( z + L + d ) ] f ( z , r ) 1 ρ exp ( i k ρ ) g ( r , θ ) .
f ( 0 , r ) = 1 ( L + d ) 1 / 2 g ( L + d , r L + d ) ,
f ( d , r ) = 1 d 1 / 2 g ( d , r d ) exp ( 2 i k L ) .
c 2 2 E = 2 E t 2 2 c G E t
f z = G f ,
g ρ = G g .
G = G 0 1 + I / I s ,
I = | f ( z , r ) | 2 + 1 ρ | g ( r , θ ) | 2 .
f ( z , r ) f g exp [ H ( z ) ] .
H ( 0 ) = 0 ,
H ( L ) = 1 4 ln M .
g ( L + d z ) = ( L + d ) 1 / 2 exp [ H ( z ) ] f 0 .
G = d H d z = G 0 1 + [ exp ( 2 H ) + ( L + d d + L z ) exp ( 2 H ) ] f g 2 .
f fwd ( x , w ) = a + a K ( x , x 1 ) f 0 ( x 1 ) exp [ i k 2 d x 1 2 ] d x 1 .
K ( x , y ) = [ i λ L ] 1 / 2 exp [ i k 2 L ( x y ) 2 ] .
f bkw ( x , w ) = + K * ( x , x 1 ) f 0 ( x 1 ) d x 1
G ( x , w ) = G 0 1 + | f fwd ( x , w ) | 2 + | f bkw ( x , w ) | 2 .
G e x ( x , w , x 1 ) = exp [ G ( l ) d l ] ,
G bkw ( x , w , x 1 ) = exp [ g ( l ) d l ]
1 = 1 2 π ( i 2 F eff ) 1 / 2 × { n = 1 N + 1 Γ n ( a ) exp [ ( 2 π i F eff β n ) β n 1 / 2 ± Γ n ( a ) exp ( 2 π i F eff β n 1 ) β n + 1 / 2 ] μ n × ( 1 1 μ for symmetric modes when n = M + 1 ) } .
Γ n ( x ) = exp ( m = 1 n { 0 L G [ X 1 ( w ) , w ] d w + 0 L G [ X 2 ( w ) , w ] d w } 2 n 0 L G ( 0 , w ) d w ) ,
X 1 ( w ) = M n 1 M 2 n { [ 1 + w D ( M 1 ) ] ( M m 1 ) ( x M n a ) ( M m 1 ) 1 ( x M n a ) } ,
X 2 ( w ) = M n 1 M 2 n × { M m ( x M n a ) M m [ 1 + w D ( M 1 ) ] × ( x M n a ) } ,
Δ λ 0 < δ .
1 M exp { 2 G 0 l / [ 1 + ( I / I s ) ] } = 1 ,
G 0 l = 1 2 [ 1 + ( I / I s ) ] ln M .

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