Abstract

The numerical technique of Fox and Li for computing laser resonator modes is applied to the case of three-dimensional laser resonators without circular or rectangular symmetries. The computation techniques are explained, and results are presented for several specific resonators, both stable and unstable. The effect of laser medium shock waves on the refractive index of the optical cavity is approximated by a thin sheet near one resonator mirror. The near-field burn pattern of the laser output beam exactly follows the phase pattern of the shock fronts, in good qualitative agreement with experimental results reported for gas dynamic lasers. The far-field output beam demonstrates pronounced astigmatism, being considerably broadened at right angles to the flow direction, and it suggests a breakup of the far-field pattern into several separate intensity spots. The optical phase of the resonator mode is quite smooth, even in the worst cases studied, suggesting the possibility of phase compensation by suitable optics.

© 1973 Optical Society of America

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References

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  1. For example, see Refs. 2 through 5 and articles cited therein.
  2. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  3. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  4. P. F. Checcacci, A. Consortini, A. Scheggi, IEEE Trans. Microwave Theory Tech. MTT-16, 103 (1968).
    [CrossRef]
  5. L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, 1969).
  6. G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).
  7. Leonard Bergstein, Appl Opt. 7, 495 (1968).
    [CrossRef] [PubMed]
  8. Yu. A. Kalinin, A. A. Mak, A. I. Stepanov, A. V. Folomeev, V. A. Fromzel’, Sov. Phys.-JETP 29, 624 (1969).
  9. H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).
  10. Yu. A. Anan’ev, V. E. Sherstobitov, Sov. J. Quantum Electron. 1, 263 (1971).
    [CrossRef]
  11. A. N. Chester, Appl. Opt. 11, 2584 (1972).
    [CrossRef] [PubMed]
  12. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  13. T. Li, Bell Syst. Tech. J. 44, 917 (1965).
  14. Walter K. Kahn, Appl. Opt. 5, 407 (1966).
    [CrossRef] [PubMed]
  15. S. R. Barone, Appl. Opt. 6, 861 (1967).
    [CrossRef] [PubMed]
  16. Anthony E. Siegman, Raymond Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
    [CrossRef]
  17. William Streifer, IEEE J. Quantum Electron. QE-4, 229 (1968).
    [CrossRef]
  18. Robert L. Sanderson, William Streifer, Appl. Opt. 8, 131 (1969).
    [CrossRef] [PubMed]
  19. R. L. Sanderson, William Streifer, Appl. Opt. 8, 2129 (1969).
    [CrossRef] [PubMed]
  20. Robert L. Sanderson, William Streifer, Appl. Opt. 8, 2241 (1969).
    [CrossRef] [PubMed]
  21. A. N. Chester, IEEE J. Quantum Electron. QE-9, 209 (1973).
    [CrossRef]
  22. In particular, unstable resonators can be so sensitive to mirror misalignment or aberration that a perturbation treatment may not converge rapidly enough to model many experimentally interesting cases.21
  23. Since the mirrors are curved, they cannot exactly lie in a plane of constant z. However, as long as the phase shifts introduced into the reflected beam are correctly included, this is not a source of significant error.
  24. See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).
  25. D. B. Rensch, A. N. Chester, to be published.
  26. M. A. Lintner, “PROJ—Algorithm and Computer Programs for the Hidden Line Problem for Single Valued Surfaces,” Idaho Nuclear Corporation, Idaho Falls, Idaho.
  27. P. O. Clark, “Design considerations for high power laser cavities,” AIAA Paper 72-708, AIAA 5th Fluid and Plasma Dynamics Conference, Boston, Massachusetts, 26–28 June 1972.
  28. E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-7, 581 (1971); erratum: IEEE J. Quantum Electron. QE-8, 389 (1972).
    [CrossRef]
  29. E. T. Gerry, IEEE Spectrum 7, 51 (1970).
    [CrossRef]

1973

A. N. Chester, IEEE J. Quantum Electron. QE-9, 209 (1973).
[CrossRef]

1972

1971

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-7, 581 (1971); erratum: IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

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

1970

H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).

E. T. Gerry, IEEE Spectrum 7, 51 (1970).
[CrossRef]

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

1969

1968

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

P. F. Checcacci, A. Consortini, A. Scheggi, IEEE Trans. Microwave Theory Tech. MTT-16, 103 (1968).
[CrossRef]

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

1967

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

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

1966

1965

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

1962

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

1961

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

Anan’ev, Yu. A.

Yu. A. Anan’ev, 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

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

Born, M.

See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Boyd, G. D.

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

Checcacci, P. F.

P. F. Checcacci, A. Consortini, A. Scheggi, IEEE Trans. Microwave Theory Tech. MTT-16, 103 (1968).
[CrossRef]

Chester, A. N.

A. N. Chester, IEEE J. Quantum Electron. QE-9, 209 (1973).
[CrossRef]

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

D. B. Rensch, A. N. Chester, to be published.

Clark, P. O.

P. O. Clark, “Design considerations for high power laser cavities,” AIAA Paper 72-708, AIAA 5th Fluid and Plasma Dynamics Conference, Boston, Massachusetts, 26–28 June 1972.

Consortini, A.

P. F. Checcacci, A. Consortini, A. Scheggi, IEEE Trans. Microwave Theory Tech. MTT-16, 103 (1968).
[CrossRef]

Folomeev, A. V.

Yu. A. Kalinin, A. A. Mak, A. I. Stepanov, A. V. Folomeev, V. A. Fromzel’, Sov. Phys.-JETP 29, 624 (1969).

Fox, A. G.

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

Fromzel’, V. A.

Yu. A. Kalinin, A. A. Mak, A. I. Stepanov, A. V. Folomeev, V. A. Fromzel’, Sov. Phys.-JETP 29, 624 (1969).

Gerry, E. T.

E. T. Gerry, IEEE Spectrum 7, 51 (1970).
[CrossRef]

Hella, R.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-7, 581 (1971); erratum: IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

Kahn, Walter K.

Kalinin, Yu. A.

Yu. A. Kalinin, A. A. Mak, A. I. Stepanov, A. V. Folomeev, V. A. Fromzel’, Sov. Phys.-JETP 29, 624 (1969).

Kogelnik, H.

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

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

Li, T.

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

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

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

Lintner, M. A.

M. A. Lintner, “PROJ—Algorithm and Computer Programs for the Hidden Line Problem for Single Valued Surfaces,” Idaho Nuclear Corporation, Idaho Falls, Idaho.

Locke, E. V.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-7, 581 (1971); erratum: IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

Mak, A. A.

Yu. A. Kalinin, A. A. Mak, A. I. Stepanov, A. V. Folomeev, V. A. Fromzel’, Sov. Phys.-JETP 29, 624 (1969).

Miller, H. Y.

Rensch, D. B.

D. B. Rensch, A. N. Chester, to be published.

Sanderson, R. L.

Sanderson, Robert L.

Scheggi, A.

P. F. Checcacci, A. Consortini, A. Scheggi, IEEE Trans. Microwave Theory Tech. MTT-16, 103 (1968).
[CrossRef]

Sherstobitov, V. E.

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

Siegman, A. E.

Siegman, Anthony E.

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

Stepanov, A. I.

Yu. A. Kalinin, A. A. Mak, A. I. Stepanov, A. V. Folomeev, V. A. Fromzel’, Sov. Phys.-JETP 29, 624 (1969).

Streifer, William

Weinstein, L. A.

L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, 1969).

Westra, L.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-7, 581 (1971); erratum: IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

Wolf, E.

See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Zeiders, G.

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-7, 581 (1971); erratum: IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

Zucker, H.

H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).

Appl Opt.

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

Appl. Opt.

Bell Syst. Tech. J.

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

H. Zucker, Bell Syst. Tech. J. 49, 2349 (1970).

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

T. Li, Bell Syst. Tech. J. 44, 917 (1965).

IEEE J. Quantum Electron.

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, 209 (1973).
[CrossRef]

E. V. Locke, R. Hella, L. Westra, G. Zeiders, IEEE J. Quantum Electron. QE-7, 581 (1971); erratum: IEEE J. Quantum Electron. QE-8, 389 (1972).
[CrossRef]

IEEE Spectrum

E. T. Gerry, IEEE Spectrum 7, 51 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

P. F. Checcacci, A. Consortini, A. Scheggi, IEEE Trans. Microwave Theory Tech. MTT-16, 103 (1968).
[CrossRef]

Sov. J. Quantum Electron.

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

Sov. Phys.-JETP

Yu. A. Kalinin, A. A. Mak, A. I. Stepanov, A. V. Folomeev, V. A. Fromzel’, Sov. Phys.-JETP 29, 624 (1969).

Other

L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, 1969).

For example, see Refs. 2 through 5 and articles cited therein.

In particular, unstable resonators can be so sensitive to mirror misalignment or aberration that a perturbation treatment may not converge rapidly enough to model many experimentally interesting cases.21

Since the mirrors are curved, they cannot exactly lie in a plane of constant z. However, as long as the phase shifts introduced into the reflected beam are correctly included, this is not a source of significant error.

See, for example, Eqs. (8.2.1) and (8.3.20) in M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

D. B. Rensch, A. N. Chester, to be published.

M. A. Lintner, “PROJ—Algorithm and Computer Programs for the Hidden Line Problem for Single Valued Surfaces,” Idaho Nuclear Corporation, Idaho Falls, Idaho.

P. O. Clark, “Design considerations for high power laser cavities,” AIAA Paper 72-708, AIAA 5th Fluid and Plasma Dynamics Conference, Boston, Massachusetts, 26–28 June 1972.

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Figures (15)

Fig. 1
Fig. 1

Gaussian input beam. (a) Contour plot of intensity (10% contours). (b) Hidden-line plot of intensity. (c) Contour plot of phase. (d) Hidden-line plot of phase.

Fig. 2
Fig. 2

Gaussian beam of Fig. 1 after one round trip through the stable resonator.

Fig. 3
Fig. 3

Pseudorandom input beam. (a) Intensity (20% contours). (b) Intensity. (c) Phase (1-rad contours). (d) Phase.

Fig. 4
Fig. 4

The beam of Fig. 3 after one round trip through the stable resonator.

Fig. 5
Fig. 5

The beam of Fig. 3 after eight round trips through the stable resonator. Contour separations as in Fig. 3.

Fig. 6
Fig. 6

Far-field intensity and phase pattern of the beam shown in Fig. 5.

Fig. 7
Fig. 7

Near-field intensity and phase distribution incident at the output mirror of a positive branch confocal unstable resonator with circular mirrors. (a) Intensity (20% contours). (b) Intensity. (c) Phase (1-rad contours). (d) Phase.

Fig. 8
Fig. 8

Far-field intensity and phase distribution of the beam of Fig. 7 after transmission around the edge of the output mirror.

Fig. 9
Fig. 9

Typical shock-wave patterns in a high power gas dynamic laser (redrawn from Ref. 27); η0 denotes the refractive index, is a small increase in refractive index, and ρ is the gas density.

Fig. 10
Fig. 10

Phase of an initial beam after passing through the edge-shock pattern.

Fig. 11
Fig. 11

Near-field mode distribution in an unstable resonator containing the edge-shock pattern of Fig. 10 (compare with Fig. 7).

Fig. 12
Fig. 12

Far-field distribution of the output beam for the mode pattern of Fig. 11 (compare with Fig. 8).

Fig. 13
Fig. 13

Phase of an initial beam after passing through the centered shock pattern.

Fig. 14
Fig. 14

Near-field mode distribution in an unstable resonator containing the centered shock pattern of Fig. 13 (compare with Figs. 7 and 11).

Fig. 15
Fig. 15

Far-field distribution of the output beam for the mode pattern of Fig. 14 (compare with Figs. 8 and 12).

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y , z , t ) = i Re [ E ^ ( x , y , z ) exp ( i k z - i ω t ) ] ,
E ^ ( x , y , 0 ) = [ R 2 ( x , y ) ] 1 / 2 exp [ - i k ( x 2 + y 2 ) / r 2 ] E ^ ( x , y , 0 ) .
E ^ ( x , y , L ) = ( - i / 2 k ) d x d y ( 1 + L / ρ ) × exp ( i k ρ - i k L ) E ^ ( x , y , 0 ) ,
ρ = [ ( x - x ) 2 + ( y - y ) 2 + L 2 ] 1 / 2
x = m Δ
y = n Δ ,
E ^ ( j , l , L ) = m n P ( j , l , m , n ) E ^ ( m , n , 0 ) ,
P ( j , l , m , n ) = ( - i / 2 k ) ( 1 + L / ρ ) exp ( i k ρ - i k L )
ρ = { Δ 2 [ ( j - m ) 2 + ( l - n ) 2 ] + L 2 } 1 / 2 .
u = min ( j - m , l - n )
v = max ( j - m , l - n ) .
y = ± ( x - x 0 ) tan θ .
Δ ± ( x , y ) = sin θ x - x 0 ) y cot θ .
ϕ ( x , y ) = ϕ 0 [ exp ( - Δ + 2 / Δ 0 2 ) + exp ( - Δ - 2 / Δ 0 2 ) ]

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