Abstract

Closed or stable optical cavities, used frequently to determine the efficiency of high performance chemical laser nozzles, are designed primarily for maximum multimode power extraction from the medium. The very large (>500) Fresnel numbers associated with such cavities have in the past necessitated their analytical modeling by representing them as plane-parallel Fabry-Perot or rooftop cavities. In this paper, a rigorous 2-D scalar diffraction formalism of the closed cavity is presented in which quasi-monochromatic partially coherent fields in the space-frequency domain are used to obtain quasi-steady state but stable solutions using a simplified gain model. Small power fluctuations in the numerical iterative solution history that displays no monotonic increasing or decreasing trends are interpreted as the redistribution of energy from one degenerate set of high-order transverse modes into another. The degree of coherence in the second-order spatial correlation function (or the mutual coherence function) required of the input fields which permit such solutions is presented. Further, it is shown that the upstream/downstream coupling in this closed cavity occurs as a natural consequence of the physical model itself rather than through some artificial geometrical means, such as that introduced in the rooftop model. The axial variation in the resulting mode width is in excellent agreement with the Hermite-Gaussian distribution predicted for the particular geometry of interest. The computed closed-cavity power variation with mode width using a simplified gain model shows qualitative agreement with experimentally observed trends; quantitative agreement is poor and is ascribed to the rudimentary nature of the gain model. In the limiting case of small Fresnel numbers (NF ∼ 1) this procedure yields, in the bare cavity, the well-known fundamental mode of the cavity when appropriate symmetry constraints are applied.

© 1983 Optical Society of America

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References

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  1. D. J. Spencer, D. A. Durran, H. A. Bixler, Appl. Phys. Lett. 20, 164 (1972).
    [CrossRef]
  2. H. Mirels, D. J. Spencer, IEEE J. Quantum Electron. QE-7, 501 (1971).
    [CrossRef]
  3. D. J. Spencer, H. Mirels, T. A. Jacobs, Appl. Phys. Lett. 16, 384 (1970).
    [CrossRef]
  4. L. Forman, AFWL-TR-77-131, Rocketdyne Report RI/RD77-171 (Dec.1977).
  5. D. L. Hook et al., AFWL-TR-76-295, TRW Report 27351-6002-RU-00 (Apr.1977).
  6. G. W. Tregay et al., Bell Aerospace Report9276-928001 (Jan.1978).
  7. J. E. Broadwell, Appl. Opt. 13, 962 (1974).
    [CrossRef] [PubMed]
  8. S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
    [CrossRef]
  9. T. T. Yang, R. E. Swanson, AIAA Paper 79-1490, Williamsburg, Va. (1979).
  10. J. Theones, A. W. Ratcliff, AIAA Paper 73-644, Palm Springs, Calif. (1973).
  11. R. W. F. Gross, J. F. Bott, Eds., Handbook of Chemical Lasers (Wiley, New York, 1976), p. 110.
  12. A. W. Ratcliff, J. Theones, AIAA Paper 74-225, Washington, D.C. (1974).
  13. R. J. Hall, IEEE J. Quantum Electon. QE-12, 453 (1976).
    [CrossRef]
  14. W. L. Rushmore, S. W. Zelazny, AIAA Paper IV-5, Cambridge, Mass. (1978).
  15. T. T. Yang, J. Phys. C9, 51 (1980).
  16. On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
    [CrossRef]
  17. R. R. Mikatarian, AIAA Paper 74-547, Palo Alto, Calif. (1974).
  18. R. Tripodi, L. J. Coulter, B. R. Bronfin, L. S. Cohen, AIAA J. 13, 776 (1975).
    [CrossRef]
  19. A. Bhowmik, T. T. Yang, J. J. Vieceli, W. D. Chadwick, Appl. Opt. 22, 3347 (1983), same issue.
    [CrossRef] [PubMed]
  20. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 509.
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sees. 10.4.2 and 10.4.3.
  22. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  23. Ref. 21, p. 119.
  24. E. O. Brigham, Fast Fourier Transform (Prentice-Hill, Engle-wood Cliffs, N.J., 1964).
  25. The propagation Fresnel number NF is defined as NF = G2a1a2/λz, where Ga1 and Ga2 are the radial extent of the field at the input and output stations, respectively, λ is the wavelength, and z is the spacing between the two stations.
  26. E. A. Sziklas, A. E. Siegman, Appl. Opt. 14, 1874 (1975).
    [CrossRef] [PubMed]
  27. The periodicity can easily be found by computing the reentrant condition for the cavity; see, for example, D. R. Herriott, H. Kogelnik, R. Kompfner, Appl. Opt. 3, 523 (1964).
    [CrossRef]
  28. Yu. A. Ananev, Sov. J. Quantum Electron. 5, 615 (1975).
    [CrossRef]
  29. Ref. 20, p. 509.
  30. Note that, when K ≠ 0, the input function ψ(x) in Eq. (8) is neither symmetric nor antisymmetric about the optic axis as a result of the random spatial distribution of the phase function R(xn).
  31. L. Mandel, E. Wolf, Opt. Commun. 36, 247 (1981).
    [CrossRef]
  32. J. K. Cawthra, “Chemical Laser Nozzle Technology,” Final Report, (Apr.1983),in preparation.

1983 (1)

1981 (1)

L. Mandel, E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

1980 (1)

T. T. Yang, J. Phys. C9, 51 (1980).

1979 (1)

On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
[CrossRef]

1978 (1)

S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
[CrossRef]

1976 (1)

R. J. Hall, IEEE J. Quantum Electon. QE-12, 453 (1976).
[CrossRef]

1975 (3)

R. Tripodi, L. J. Coulter, B. R. Bronfin, L. S. Cohen, AIAA J. 13, 776 (1975).
[CrossRef]

E. A. Sziklas, A. E. Siegman, Appl. Opt. 14, 1874 (1975).
[CrossRef] [PubMed]

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

1974 (1)

1972 (1)

D. J. Spencer, D. A. Durran, H. A. Bixler, Appl. Phys. Lett. 20, 164 (1972).
[CrossRef]

1971 (1)

H. Mirels, D. J. Spencer, IEEE J. Quantum Electron. QE-7, 501 (1971).
[CrossRef]

1970 (1)

D. J. Spencer, H. Mirels, T. A. Jacobs, Appl. Phys. Lett. 16, 384 (1970).
[CrossRef]

1964 (1)

1961 (1)

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

Ananev, Yu. A.

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

Behrens, W.

On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
[CrossRef]

Bhowmik, A.

Bixler, H. A.

D. J. Spencer, D. A. Durran, H. A. Bixler, Appl. Phys. Lett. 20, 164 (1972).
[CrossRef]

Blauer, J. A.

S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 509.

Brigham, E. O.

E. O. Brigham, Fast Fourier Transform (Prentice-Hill, Engle-wood Cliffs, N.J., 1964).

Broadwell, J. E.

Bronfin, B. R.

R. Tripodi, L. J. Coulter, B. R. Bronfin, L. S. Cohen, AIAA J. 13, 776 (1975).
[CrossRef]

Bullock, D.

On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
[CrossRef]

Cawthra, J. K.

J. K. Cawthra, “Chemical Laser Nozzle Technology,” Final Report, (Apr.1983),in preparation.

Chadwick, W. D.

Cohen, L. S.

R. Tripodi, L. J. Coulter, B. R. Bronfin, L. S. Cohen, AIAA J. 13, 776 (1975).
[CrossRef]

Coulter, L. J.

R. Tripodi, L. J. Coulter, B. R. Bronfin, L. S. Cohen, AIAA J. 13, 776 (1975).
[CrossRef]

Dee, D.

On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
[CrossRef]

Driscoll, R. J.

S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
[CrossRef]

Durran, D. A.

D. J. Spencer, D. A. Durran, H. A. Bixler, Appl. Phys. Lett. 20, 164 (1972).
[CrossRef]

Forman, L.

L. Forman, AFWL-TR-77-131, Rocketdyne Report RI/RD77-171 (Dec.1977).

Fox, A. G.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sees. 10.4.2 and 10.4.3.

Hall, R. J.

R. J. Hall, IEEE J. Quantum Electon. QE-12, 453 (1976).
[CrossRef]

Herriott, D. R.

Hook, D. L.

D. L. Hook et al., AFWL-TR-76-295, TRW Report 27351-6002-RU-00 (Apr.1977).

Jacobs, T. A.

D. J. Spencer, H. Mirels, T. A. Jacobs, Appl. Phys. Lett. 16, 384 (1970).
[CrossRef]

Kogelnik, H.

Kompfner, R.

Li, T.

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

Mandel, L.

L. Mandel, E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

Mikatarian, R. R.

R. R. Mikatarian, AIAA Paper 74-547, Palo Alto, Calif. (1974).

Mirels, H.

H. Mirels, D. J. Spencer, IEEE J. Quantum Electron. QE-7, 501 (1971).
[CrossRef]

D. J. Spencer, H. Mirels, T. A. Jacobs, Appl. Phys. Lett. 16, 384 (1970).
[CrossRef]

O'Keefe, D.

On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
[CrossRef]

Ratcliff, A. W.

J. Theones, A. W. Ratcliff, AIAA Paper 73-644, Palm Springs, Calif. (1973).

A. W. Ratcliff, J. Theones, AIAA Paper 74-225, Washington, D.C. (1974).

Raymonda, J. W.

S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
[CrossRef]

Rushmore, W. L.

W. L. Rushmore, S. W. Zelazny, AIAA Paper IV-5, Cambridge, Mass. (1978).

Siegman, A. E.

Solomon, W. C.

S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
[CrossRef]

Spencer, D. J.

D. J. Spencer, D. A. Durran, H. A. Bixler, Appl. Phys. Lett. 20, 164 (1972).
[CrossRef]

H. Mirels, D. J. Spencer, IEEE J. Quantum Electron. QE-7, 501 (1971).
[CrossRef]

D. J. Spencer, H. Mirels, T. A. Jacobs, Appl. Phys. Lett. 16, 384 (1970).
[CrossRef]

Sugimura, T.

On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
[CrossRef]

Swanson, R. E.

T. T. Yang, R. E. Swanson, AIAA Paper 79-1490, Williamsburg, Va. (1979).

Sziklas, E. A.

Theones, J.

J. Theones, A. W. Ratcliff, AIAA Paper 73-644, Palm Springs, Calif. (1973).

A. W. Ratcliff, J. Theones, AIAA Paper 74-225, Washington, D.C. (1974).

Tregay, G. W.

G. W. Tregay et al., Bell Aerospace Report9276-928001 (Jan.1978).

Tripodi, R.

R. Tripodi, L. J. Coulter, B. R. Bronfin, L. S. Cohen, AIAA J. 13, 776 (1975).
[CrossRef]

Vieceli, J. J.

Wolf, E.

L. Mandel, E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 509.

Yang, T. T.

A. Bhowmik, T. T. Yang, J. J. Vieceli, W. D. Chadwick, Appl. Opt. 22, 3347 (1983), same issue.
[CrossRef] [PubMed]

T. T. Yang, J. Phys. C9, 51 (1980).

T. T. Yang, R. E. Swanson, AIAA Paper 79-1490, Williamsburg, Va. (1979).

Zelazny, S. W.

S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
[CrossRef]

W. L. Rushmore, S. W. Zelazny, AIAA Paper IV-5, Cambridge, Mass. (1978).

AIAA J. (2)

S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, W. C. Solomon, AIAA J. 16, 297 (1978).
[CrossRef]

R. Tripodi, L. J. Coulter, B. R. Bronfin, L. S. Cohen, AIAA J. 13, 776 (1975).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

D. J. Spencer, D. A. Durran, H. A. Bixler, Appl. Phys. Lett. 20, 164 (1972).
[CrossRef]

D. J. Spencer, H. Mirels, T. A. Jacobs, Appl. Phys. Lett. 16, 384 (1970).
[CrossRef]

Bell Syst. Tech. J. (1)

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

IEEE J. Quantum Electon. (1)

R. J. Hall, IEEE J. Quantum Electon. QE-12, 453 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Mirels, D. J. Spencer, IEEE J. Quantum Electron. QE-7, 501 (1971).
[CrossRef]

J. Phys. (1)

T. T. Yang, J. Phys. C9, 51 (1980).

Opt. Commun. (1)

L. Mandel, E. Wolf, Opt. Commun. 36, 247 (1981).
[CrossRef]

Opt. Eng. (1)

On one occasion a negative branch unstable resonator was used. See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bullock, D. Dee, Opt. Eng. 18, 363 (1979).
[CrossRef]

Sov. J. Quantum Electron. (1)

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

Other (17)

Ref. 20, p. 509.

Note that, when K ≠ 0, the input function ψ(x) in Eq. (8) is neither symmetric nor antisymmetric about the optic axis as a result of the random spatial distribution of the phase function R(xn).

Ref. 21, p. 119.

E. O. Brigham, Fast Fourier Transform (Prentice-Hill, Engle-wood Cliffs, N.J., 1964).

The propagation Fresnel number NF is defined as NF = G2a1a2/λz, where Ga1 and Ga2 are the radial extent of the field at the input and output stations, respectively, λ is the wavelength, and z is the spacing between the two stations.

R. R. Mikatarian, AIAA Paper 74-547, Palo Alto, Calif. (1974).

J. K. Cawthra, “Chemical Laser Nozzle Technology,” Final Report, (Apr.1983),in preparation.

W. L. Rushmore, S. W. Zelazny, AIAA Paper IV-5, Cambridge, Mass. (1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 509.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sees. 10.4.2 and 10.4.3.

L. Forman, AFWL-TR-77-131, Rocketdyne Report RI/RD77-171 (Dec.1977).

D. L. Hook et al., AFWL-TR-76-295, TRW Report 27351-6002-RU-00 (Apr.1977).

G. W. Tregay et al., Bell Aerospace Report9276-928001 (Jan.1978).

T. T. Yang, R. E. Swanson, AIAA Paper 79-1490, Williamsburg, Va. (1979).

J. Theones, A. W. Ratcliff, AIAA Paper 73-644, Palm Springs, Calif. (1973).

R. W. F. Gross, J. F. Bott, Eds., Handbook of Chemical Lasers (Wiley, New York, 1976), p. 110.

A. W. Ratcliff, J. Theones, AIAA Paper 74-225, Washington, D.C. (1974).

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Figures (13)

Fig. 1
Fig. 1

(a) Experimental closed cavity of interest in which very large propagation Fresnel numbers NF ≃ 550 are encountered. The mutual coherence of an input field is evaluated at the curved mirror plane following its reflection and propagation to the focal plane, where the field now acts as an extended quasi-monochromatic partially coherent source. (b) Lens equivalent shows the predominantly geometric nature of the upstream–downstream coupling in this cavity, wherein Fresnel diffraction effects are now incorporated by the present formalism.

Fig. 2
Fig. 2

Result of propagating a sum of spatially periodic functions, coherently summed with random relative phases, shows (a) that it acts as a high spatial frequency phase grating, and (b) its high degree of correlation.

Fig. 3
Fig. 3

(a) Diffraction-limited image is obtained when a plane wave [K = 0 in Eq. (8)] is propagated to the focal plane of the curved mirror, and (b) its high degree of correlation, which leads to a beam-train solution with periodic imaging, when a resonator computation is attempted with it.

Fig. 4
Fig. 4

(a) K = 0.5 in Eq. (8) shows the appearance of small amounts of off-axis energy following a single propagation step; (b) its degree of correlation is very high, |μ1,2| ≥ 0.75.

Fig. 5
Fig. 5

(a) With K = 1.0 in Eq. (8) the propagated field shows increased amounts of off-axis energy accompanied by a reduction in peak intensity. Imaging characteristic is still strongly evident. (b) Correlation of off-axis points with a point on-axis is reduced but still high, |μ1,2| ≳ 0.30.

Fig. 6
Fig. 6

(a) When K = 2.0 in Eq. (8) the imaging characteristic completely disappears; (b) its mutual coherence is now |μ1,2| ≤ 0.2 for points lying outside a highly coherent region of width Rλ/2d.

Fig. 7
Fig. 7

Parabolic approximation of the small signal gain profile used in the resonator computations presented in this paper.

Fig. 8
Fig. 8

Typical input field random phase distribution.

Fig. 9
Fig. 9

With mirror reflection coefficients near unity (ρ1 = 0.992, ρ2 = 0.998) the iteration history shows slow convergence to quasi-steady state but stable solution.

Fig. 10
Fig. 10

Irradiance distribution at curved mirror (Xc = 1.5 cm) in quasi-steady-state conditions.

Fig. 11
Fig. 11

Quasi-steady-state irradiance distribution at plane mirror (Xc = 1.5 cm) shows waist formation (compare with Fig. 10).

Fig. 12
Fig. 12

Quasi-steady-state two-way average intensity distribution I used in Eq. (10) for computing the loaded gain (Xc = 1.5 cm).

Fig. 13
Fig. 13

(a) Predicted closed-cavity power variation with Xc for two sets of mirror reflectivities; (b) experimental data.

Equations (13)

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N 1 > 4 G a 2 / λ R ,
N 1 > 1310 .
N = 4 ( N F + N 1 ) 7440 .
μ 1 , 2 = μ ( P 1 , P 2 ) = 1 [ I ( P 1 ) I ( P 2 ) ] 1 / 2 x min x max × dx I ( x ) exp [ 2 π i ( ρ 1 ρ 2 ) / λ ] ρ 1 ρ 2 ,
ψ ( x n ) = m = 1 N F A m sin [ m x n + ϕ R ( m ) ]
ψ ( x ) = K 1 ψ 0 ( x ) exp [ i ϕ ( x ) K 2 ] ,
ψ ( x ) = ψ ( O ) exp [ i ϕ ( x ) K ] .
ψ ( x n ) = ψ ( O ) exp { i π [ R ( x n ) 1 / 2 ] K } , ( O R ( x n ) 1 ) ,
g 0 ( x n ) = { 4 g max ( x n + X c x 0 / 2 ) 2 / x 0 2 + g max , x > X c , 0 , x X c ,
g ( x n ) = g 0 ( x n ) / [ 1 + I ( x n ) / I s ] ,
I ¯ ( x n ) = [ I inc ( + ) ( x n ) + I inc ( ) ( x n ] { exp [ g ( x n ) L g ] 1 } / g ( x n ) .
ψ tran ( ± ) ( x n ) = ψ inc ( ± ) ( x n ) exp [ g ( x n ) L g / 2 ] ,
P cc = ( 1 ρ 1 ) n | ψ tran ( ) ( x n ) | 2 + ( 1 ρ 2 ) n | ψ tran ( + ) ( x n ) | 2

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