Abstract

A theoretical investigation has been undertaken to study the transverse modes of two-dimensional positive branch, confocal unstable resonators. Mode amplitude and phase information is obtained from a numerical-iterative type calculation that uses the Fresnel integral for propagating the cavity radiation back and forth between resonator mirrors. Near- and far-field distributions for empty cavity resonators are presented for various resonator Fresnel numbers and magnifications, along with results of resonator mode stability and diffraction losses when cavity perturbations such as mirror misalignment and/or a uniformly saturable gain medium are included. In addition, the diffractive calculations are compared with results obtained from geometric models.

© 1973 Optical Society of America

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References

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  1. A. E. Siegman, Proc. IEEE 53, 277 (1965).
    [CrossRef]
  2. William F. Krupke, Walter R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
    [CrossRef]
  3. S. R. Barone, Appl. Opt. 6, 861 (1967).
    [CrossRef] [PubMed]
  4. Walter K. Kahn, Appl. Opt. 5, 407 (1966).
    [CrossRef] [PubMed]
  5. A. E. Siegman, Laser Focus (May1971), p. 42.
  6. Anthony E. Siegman, Raymond Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
    [CrossRef]
  7. William Streifer, IEEE J. Quantum Electron. QE-4, 229 (1968).
    [CrossRef]
  8. R. L. Sanderson, William Streifer, Appl. Opt. 8, 2129 (1969).
    [CrossRef] [PubMed]
  9. A. N. Chester, Appl. Opt. 11, 2584 (1972).
    [CrossRef] [PubMed]
  10. A. N. Chester, IEEE J. Quantum Electron. QE-9, Feb. (1973).
  11. Yu. A. Ananev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
    [CrossRef]
  12. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  13. Leonard Bergstein, Appl. Opt. 7, 495 (1968).
    [CrossRef] [PubMed]
  14. Robert L. Sanderson, William Streifer, Appl. Opt. 8, 131 (1969).
    [CrossRef] [PubMed]
  15. Robert L. Sanderson, William Streifer, Appl. Opt. 8, 2241 (1969).
    [CrossRef] [PubMed]
  16. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  17. D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 62, 718A (1972).
  18. When the gain of the laser medium is not appreciably saturated by the radiation intensity, the constant gain of the homogeneous medium may be factored out of the wave equation (see Ref. 19). In this case, the thin-sheet approximation is accurate even though the transverse field distribution changes significantly from one end of the resonator to the other.
  19. A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
    [CrossRef]
  20. D. B. Rensch, A. N. Chester, Chemical Laser Mode Control Program, Final Report, Contract DAAH01-72-C-0067, Hughes Research Laboratories, Malibu, California, 31July1972.
  21. R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “CO2 Unstable Confocal Ring Resonators,” 7th International Quantum Electronics Conference, 8–11 May 1972, Montreal; abstract in IEEE J. Quantum Electron. (Digest of Technical Papers) QE-8, 555 (1972).
  22. Similar departures from the geometric results for small m’s were previously found for an exactly soluble unstable resonator (see Ref. 10).
  23. An alternate formulation of this limitation is given in Ref. 10.
  24. It could be that the presence of saturable gain perturbed the eigenvalues sufficiently that our calculations were no longer, exactly at the mode crossings. However, we did not see increased difficulty in obtaining convergence near the mode-crossing positions. Moreover, the cavity fields appear to be uniphase on both sides of such mode crossings, so that if two modes have indeed crossed, the physics somehow conspires to make the lowest loss field distribution always have nearly uniform optical phase.The entire treatment of modes in unstable resonators is somewhat clouded by the fact that the propagation operator is non-Hermitean. [The implications of this fact have also been considered by E. R. Peressini (see Ref. 25).] Thus the eigenmodes of the empty unstable resonator are not orthogonal in the usual sense (see, for example, Ref. 9) and conceivably not even complete. Thus field distributions cannot necessarily be resolved into an orthogonal set of cavity modes in the unstable case. This state of affairs is further complicated by the fact that modes are basically solutions of a linear problem. For the empty cavity case, any linear combination of modes is still a solution of the wave equation with the resonator’s boundary conditions. As soon as saturable gain is introduced (particularly the large gain required to sustain oscillation with greater than 50% output coupling), superposition of fields is no longer valid, and there are no eigenmodes in the usual sense.The approach to the geometric limit, which presumably occurs as the Fresnel number increases, is also poorly understood in mathematical treatments of unstable resonators. Barone’s classical derivation (Ref. 3) of the geometrical limit appears to involve divergent integrals and may require further justification. Wallace’s (Ref. 26) recent expansion of the resonator integral equation in the vicinity of the geometrical limit (which raises similar questions about convergence) does not appear to be consistent with Siegman’s 3–D results showing continued mode crossings up to the highest Fresnel number that he studied.To our knowledge, the mathematical points mentioned above have never been considered in detail. (However, see Ref. 27) Until they are, any attempts to relate zero gain cavity calculations to the behavior of real devices rests on rather shaky ground.
  25. E. R. Peressini, Hughes Aircraft Co., Culver City, Calif., private communication.
  26. J. Wallace, Avco Everett Research Laboratories, Everett, Mass., private communication.
  27. J. D. Reichert, “Modes of Unstable Optical Resonators,” National Meeting, Optical Society of America, 13–16 Mar. 1973, Denver, Colo.

1973 (1)

A. N. Chester, IEEE J. Quantum Electron. QE-9, Feb. (1973).

1972 (2)

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 62, 718A (1972).

A. N. Chester, Appl. Opt. 11, 2584 (1972).
[CrossRef] [PubMed]

1971 (2)

A. E. Siegman, Laser Focus (May1971), p. 42.

Yu. A. Ananev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

1970 (1)

1969 (4)

1968 (2)

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

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

1967 (2)

S. R. Barone, Appl. Opt. 6, 861 (1967).
[CrossRef] [PubMed]

Anthony E. Siegman, Raymond Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

1966 (2)

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

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

1965 (1)

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

1961 (1)

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

Ananev, Yu. A.

Yu. A. Ananev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

Arrathoon, Raymond

Anthony E. Siegman, Raymond Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

Barone, S. R.

Bergstein, Leonard

Buczek, C. J.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “CO2 Unstable Confocal Ring Resonators,” 7th International Quantum Electronics Conference, 8–11 May 1972, Montreal; abstract in IEEE J. Quantum Electron. (Digest of Technical Papers) QE-8, 555 (1972).

Chenausky, P. P.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “CO2 Unstable Confocal Ring Resonators,” 7th International Quantum Electronics Conference, 8–11 May 1972, Montreal; abstract in IEEE J. Quantum Electron. (Digest of Technical Papers) QE-8, 555 (1972).

Chester, A. N.

A. N. Chester, IEEE J. Quantum Electron. QE-9, Feb. (1973).

A. N. Chester, Appl. Opt. 11, 2584 (1972).
[CrossRef] [PubMed]

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 62, 718A (1972).

D. B. Rensch, A. N. Chester, Chemical Laser Mode Control Program, Final Report, Contract DAAH01-72-C-0067, Hughes Research Laboratories, Malibu, California, 31July1972.

Fox, A. G.

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

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

Freiberg, R. J.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “CO2 Unstable Confocal Ring Resonators,” 7th International Quantum Electronics Conference, 8–11 May 1972, Montreal; abstract in IEEE J. Quantum Electron. (Digest of Technical Papers) QE-8, 555 (1972).

Kahn, Walter K.

Krupke, William F.

William F. Krupke, Walter R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

Li, T.

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

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

Miller, H. Y.

Peressini, E. R.

E. R. Peressini, Hughes Aircraft Co., Culver City, Calif., private communication.

Reichert, J. D.

J. D. Reichert, “Modes of Unstable Optical Resonators,” National Meeting, Optical Society of America, 13–16 Mar. 1973, Denver, Colo.

Rensch, D. B.

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 62, 718A (1972).

D. B. Rensch, A. N. Chester, Chemical Laser Mode Control Program, Final Report, Contract DAAH01-72-C-0067, Hughes Research Laboratories, Malibu, California, 31July1972.

Sanderson, R. L.

Sanderson, Robert L.

Sherstobitov, V. E.

Yu. A. Ananev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Laser Focus (May1971), p. 42.

A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

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

Siegman, Anthony E.

Anthony E. Siegman, Raymond Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

Sooy, Walter R.

William F. Krupke, Walter R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

Streifer, William

Wallace, J.

J. Wallace, Avco Everett Research Laboratories, Everett, Mass., private communication.

Appl. Opt. (8)

Bell Syst. Tech. J. (1)

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

IEEE J. Quantum Electron. (5)

Anthony E. Siegman, Raymond Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

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

A. N. Chester, IEEE J. Quantum Electron. QE-9, Feb. (1973).

William F. Krupke, Walter R. Sooy, IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

D. B. Rensch, A. N. Chester, J. Opt. Soc. Am. 62, 718A (1972).

Laser Focus (1)

A. E. Siegman, Laser Focus (May1971), p. 42.

Proc. IEEE (1)

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

Sov. J. Quantum Electron. (1)

Yu. A. Ananev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
[CrossRef]

Other (9)

D. B. Rensch, A. N. Chester, Chemical Laser Mode Control Program, Final Report, Contract DAAH01-72-C-0067, Hughes Research Laboratories, Malibu, California, 31July1972.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “CO2 Unstable Confocal Ring Resonators,” 7th International Quantum Electronics Conference, 8–11 May 1972, Montreal; abstract in IEEE J. Quantum Electron. (Digest of Technical Papers) QE-8, 555 (1972).

Similar departures from the geometric results for small m’s were previously found for an exactly soluble unstable resonator (see Ref. 10).

An alternate formulation of this limitation is given in Ref. 10.

It could be that the presence of saturable gain perturbed the eigenvalues sufficiently that our calculations were no longer, exactly at the mode crossings. However, we did not see increased difficulty in obtaining convergence near the mode-crossing positions. Moreover, the cavity fields appear to be uniphase on both sides of such mode crossings, so that if two modes have indeed crossed, the physics somehow conspires to make the lowest loss field distribution always have nearly uniform optical phase.The entire treatment of modes in unstable resonators is somewhat clouded by the fact that the propagation operator is non-Hermitean. [The implications of this fact have also been considered by E. R. Peressini (see Ref. 25).] Thus the eigenmodes of the empty unstable resonator are not orthogonal in the usual sense (see, for example, Ref. 9) and conceivably not even complete. Thus field distributions cannot necessarily be resolved into an orthogonal set of cavity modes in the unstable case. This state of affairs is further complicated by the fact that modes are basically solutions of a linear problem. For the empty cavity case, any linear combination of modes is still a solution of the wave equation with the resonator’s boundary conditions. As soon as saturable gain is introduced (particularly the large gain required to sustain oscillation with greater than 50% output coupling), superposition of fields is no longer valid, and there are no eigenmodes in the usual sense.The approach to the geometric limit, which presumably occurs as the Fresnel number increases, is also poorly understood in mathematical treatments of unstable resonators. Barone’s classical derivation (Ref. 3) of the geometrical limit appears to involve divergent integrals and may require further justification. Wallace’s (Ref. 26) recent expansion of the resonator integral equation in the vicinity of the geometrical limit (which raises similar questions about convergence) does not appear to be consistent with Siegman’s 3–D results showing continued mode crossings up to the highest Fresnel number that he studied.To our knowledge, the mathematical points mentioned above have never been considered in detail. (However, see Ref. 27) Until they are, any attempts to relate zero gain cavity calculations to the behavior of real devices rests on rather shaky ground.

E. R. Peressini, Hughes Aircraft Co., Culver City, Calif., private communication.

J. Wallace, Avco Everett Research Laboratories, Everett, Mass., private communication.

J. D. Reichert, “Modes of Unstable Optical Resonators,” National Meeting, Optical Society of America, 13–16 Mar. 1973, Denver, Colo.

When the gain of the laser medium is not appreciably saturated by the radiation intensity, the constant gain of the homogeneous medium may be factored out of the wave equation (see Ref. 19). In this case, the thin-sheet approximation is accurate even though the transverse field distribution changes significantly from one end of the resonator to the other.

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Figures (17)

Fig. 1
Fig. 1

Definition of unstable resonator geometry.

Fig. 2
Fig. 2

Geometric wavefront pattern for the type I, positive branch unstable resonators.

Fig. 3
Fig. 3

Normalized near- and far-field intensity distributions for N0 = 5, m = 10.

Fig. 4
Fig. 4

Normalized near- and far-field intensity distributions for N0 = 5, m = 5.

Fig. 5
Fig. 5

Normalized near- and far-field intensity distributions for N0 = 5, m = 2.5.

Fig. 6
Fig. 6

Normalized near- and far-field intensity distributions for N0 = 5, m = 1.42.

Fig. 7
Fig. 7

Normalized near- and far-field intensity distributions for N0 = 30, m = 1.42.

Fig. 8
Fig. 8

Normalized near- and far-field intensity distributions for N0 = 60, m = 10.

Fig. 9
Fig. 9

Normalized near- and far-field intensity distributions for N0 = 60, m = 5.

Fig. 10
Fig. 10

Normalized near- and far-field intensity distributions for N0 = 60, m = 2.5.

Fig. 11
Fig. 11

Normalized near- and far-field intensity distributions for N0 = 60, m = 1.42.

Fig. 12
Fig. 12

Ratio of computed 2-D loss to the geometric loss for various values of geometric loss and outer Fresnel number.

Fig. 13
Fig. 13

Near- and far-field intensity distributions for N0 = 5, m = 2.5, and gain at ten times threshold.

Fig. 14
Fig. 14

Near- and far-field intensity distributions for N0 = 30, m = 1.42, and gain at ten times threshold.

Fig. 15
Fig. 15

Unstable resonator showing mirror misalignment.

Fig. 16
Fig. 16

Ratio of near-field area with mirror tilt/near-field area without mirror tilt. (Asterisk indicates the angle β where time-varying modes appear.)

Fig. 17
Fig. 17

Beam-steering angle ϕ vs mirror misalignment angle β.

Tables (4)

Tables Icon

Table I Summary of Mode Calculations for Empty Confocal Resonators

Tables Icon

Table II Far-Field Half-Angles Containing 50% and 90% of the Power, Comparing Calculated Unstable Resonator Output with a Uniformly Illuminated Aperture

Tables Icon

Table III Confocal Unstable Resonator Parameters Used to Analyze Saturable Gain

Tables Icon

Table IV Unstable Resonator Configurations Chosen for Mirror Misalignment Study

Equations (19)

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g i = 1 - L / R i , i = 1 , 2 ,
g 1 g 2 < 0             or             g 1 g 2 > 1.
g 1 + g 2 = 2 g 1 g 2 .
m = g 2 / g 1 ,
C = 1 - ( 1 / m 2 ) , 3 - D ( spherical mirrors ) ,
C = 1 - ( 1 / m ) , 2 - D ( cylindrical mirrors ) .
a 1 = m a 2 .
g 1 = ( m + 1 ) / 2 m ;             g 2 = ( m + 1 ) / 2.
N eq = N 0 ( m - 1 ) / 2 m 2 ,
N 0 = ( m a 2 ) 2 / L λ .
U n + 1 ( X 2 ) = 1 - i ( 2 λ L ) 1 / 2 - a 1 a 1 U n ( X 1 ) exp [ + i K ( X 1 - X 2 ) 2 / 2 L ] d X 1 ,
E ( x , t ) = Re U ( X ) exp ( - i w t ) .
g = g 0 / [ 1 + ( 1 + r 1 ) I / I 0 ] ,
ϕ = 2 K ( X - X 0 ) sin ( β ) .
( 1 - geometric loss ) exp ( 2 q s L ) = ( 1 / m ) exp ( 2 q s L ) = 1.
ϕ = ( c d ¯ - R 2 β ) / e d ¯ = β ( R 1 - R 2 ) / ( R 1 + R 2 - L ) .
ϕ = M β .
M = 2 ( m + 1 ) / ( m - 1 ) .
a b ¯ = M R 2 β = 4 ( m + 1 ) L β / ( m - 1 ) 2 .

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