Abstract

An experiment is described that investigates certain interference effects for second-harmonic generation within a resonant cavity. By employing a noncollinear geometry, the phases of two fundamental beams from a frequency-stabilized dye laser can be controlled unrestricted by the boundary conditions imposed in an optical cavity containing a KDP crystal and resonant at the second harmonic. The fundamental beams are either traveling or standing waves and generate either one or two coherent sources of ultraviolet radiation within the cavity. The experiment demonstrates explicitly the dependence of second-harmonic phase on the fundamental phases and the dependence of coupling efficiency on the overlap of the harmonic polarization wave with the cavity-mode function. The measurements agree well with a simple theory.

© 1985 Optical Society of America

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  1. P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation I. Semiclassical theory,” Opt. Acta 27, 321 (1980);C. M. Savage, D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557 (1983).
    [CrossRef]
  2. P. Mandel, T. Erneux, “Amplitude self-modulation of in-tracavity second harmonic generation,” Opt. Acta 29, 7 (1982).
    [CrossRef]
  3. P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation II. Quantum theory,” Opt. Acta 28, 211 (1981);G. Milburn, D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401 (1981).
    [CrossRef]
  4. B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408 (1984).
    [CrossRef]
  5. L. A. Lugiato, G. Strini, F. de Martini, “Squeezed states in second harmonic generation,” Opt. Lett. 8, 256 (1982).
    [CrossRef]
  6. R. H. Kingston, A. L. McWhorter, “Electromagnetic mode mixing in nonlinear media,” Proc. IEEE 53, 4 (1965).
    [CrossRef]
  7. A. Ashkin, G. D. Boyd, J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109 (1966);G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]
  8. M. K. Oshman, S. E. Harris, “Theory of optical parametric oscillation internal to the laser cavity,” IEEE J. Quantum Electron. QE-4, 491 (1968).
    [CrossRef]
  9. R. G. Smith, J. V. Parker, “Experimental observation of and comments on optical parametric oscillation internal to the laser cavity,” J. Appl. Phys. 41, 3401 (1970).
    [CrossRef]
  10. R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. QE-6, 215 (1970).
    [CrossRef]
  11. D. G. Gonzalez, S. T. K. Nieh, W. H. Steier, “Two-pass-internal second-harmonic generation using a prism coupler,” IEEE J. Quantum Electron. QE-9, 23 (1973).
    [CrossRef]
  12. V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
    [CrossRef]
  13. R. G. Smith, “Optical parametric oscillators,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), p. 837.
  14. R. L. Byer, “Optical parametric oscillators,” in Quantum Electronics: A Treatise, H. Rabin, C. L. Tang, eds.(Academic, New York, 1975), Vol. 1, Part B, p. 587.
  15. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  16. Y. R. Shen, The Principles of Nonlinear Optics (WileyNew York, 1984).
  17. G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]
  18. F. Zernike, J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).
  19. For boundless plane waves, the integration length would be the crystal length. In actual practice for finite beam widths, L would be the length of the interaction region, and correction for the finite width in the x direction would have to be made.
  20. H. Fery, F. Hermann, “Noncollinear second harmonic generation in KDP,” Opt. Commun. 8, 291 (1973).
    [CrossRef]
  21. F. Zernike, “Refractive Indices of ADP and KDP between 2000 Å and 1.5 μm,” J. Opt. Soc. Am. 54, 1215 (1964).
  22. J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70 (1971).
    [CrossRef]

1984 (1)

B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408 (1984).
[CrossRef]

1982 (2)

L. A. Lugiato, G. Strini, F. de Martini, “Squeezed states in second harmonic generation,” Opt. Lett. 8, 256 (1982).
[CrossRef]

P. Mandel, T. Erneux, “Amplitude self-modulation of in-tracavity second harmonic generation,” Opt. Acta 29, 7 (1982).
[CrossRef]

1981 (1)

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation II. Quantum theory,” Opt. Acta 28, 211 (1981);G. Milburn, D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401 (1981).
[CrossRef]

1980 (1)

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation I. Semiclassical theory,” Opt. Acta 27, 321 (1980);C. M. Savage, D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557 (1983).
[CrossRef]

1979 (1)

V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
[CrossRef]

1973 (2)

D. G. Gonzalez, S. T. K. Nieh, W. H. Steier, “Two-pass-internal second-harmonic generation using a prism coupler,” IEEE J. Quantum Electron. QE-9, 23 (1973).
[CrossRef]

H. Fery, F. Hermann, “Noncollinear second harmonic generation in KDP,” Opt. Commun. 8, 291 (1973).
[CrossRef]

1971 (1)

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70 (1971).
[CrossRef]

1970 (2)

R. G. Smith, J. V. Parker, “Experimental observation of and comments on optical parametric oscillation internal to the laser cavity,” J. Appl. Phys. 41, 3401 (1970).
[CrossRef]

R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. QE-6, 215 (1970).
[CrossRef]

1968 (2)

M. K. Oshman, S. E. Harris, “Theory of optical parametric oscillation internal to the laser cavity,” IEEE J. Quantum Electron. QE-4, 491 (1968).
[CrossRef]

G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1966 (1)

A. Ashkin, G. D. Boyd, J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109 (1966);G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1965 (1)

R. H. Kingston, A. L. McWhorter, “Electromagnetic mode mixing in nonlinear media,” Proc. IEEE 53, 4 (1965).
[CrossRef]

1964 (1)

Ashkin, A.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109 (1966);G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Boyd, G. D.

G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

A. Ashkin, G. D. Boyd, J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109 (1966);G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Byer, R. L.

R. L. Byer, “Optical parametric oscillators,” in Quantum Electronics: A Treatise, H. Rabin, C. L. Tang, eds.(Academic, New York, 1975), Vol. 1, Part B, p. 587.

de Martini, F.

Drummond, P. D.

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation II. Quantum theory,” Opt. Acta 28, 211 (1981);G. Milburn, D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401 (1981).
[CrossRef]

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation I. Semiclassical theory,” Opt. Acta 27, 321 (1980);C. M. Savage, D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557 (1983).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109 (1966);G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Erneux, T.

P. Mandel, T. Erneux, “Amplitude self-modulation of in-tracavity second harmonic generation,” Opt. Acta 29, 7 (1982).
[CrossRef]

Falk, J.

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70 (1971).
[CrossRef]

Fery, H.

H. Fery, F. Hermann, “Noncollinear second harmonic generation in KDP,” Opt. Commun. 8, 291 (1973).
[CrossRef]

Gonzalez, D. G.

D. G. Gonzalez, S. T. K. Nieh, W. H. Steier, “Two-pass-internal second-harmonic generation using a prism coupler,” IEEE J. Quantum Electron. QE-9, 23 (1973).
[CrossRef]

Harris, S. E.

M. K. Oshman, S. E. Harris, “Theory of optical parametric oscillation internal to the laser cavity,” IEEE J. Quantum Electron. QE-4, 491 (1968).
[CrossRef]

Hermann, F.

H. Fery, F. Hermann, “Noncollinear second harmonic generation in KDP,” Opt. Commun. 8, 291 (1973).
[CrossRef]

Hitz, C. B.

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70 (1971).
[CrossRef]

Kingston, R. H.

R. H. Kingston, A. L. McWhorter, “Electromagnetic mode mixing in nonlinear media,” Proc. IEEE 53, 4 (1965).
[CrossRef]

Kleinman, D. A.

G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Kornienko, N. E.

V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
[CrossRef]

Krylov, V. N.

V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
[CrossRef]

Lugiato, L. A.

Mandel, P.

P. Mandel, T. Erneux, “Amplitude self-modulation of in-tracavity second harmonic generation,” Opt. Acta 29, 7 (1982).
[CrossRef]

McNeil, K. J.

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation II. Quantum theory,” Opt. Acta 28, 211 (1981);G. Milburn, D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401 (1981).
[CrossRef]

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation I. Semiclassical theory,” Opt. Acta 27, 321 (1980);C. M. Savage, D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557 (1983).
[CrossRef]

McWhorter, A. L.

R. H. Kingston, A. L. McWhorter, “Electromagnetic mode mixing in nonlinear media,” Proc. IEEE 53, 4 (1965).
[CrossRef]

Midwinter, J. E.

F. Zernike, J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

Nieh, S. T. K.

D. G. Gonzalez, S. T. K. Nieh, W. H. Steier, “Two-pass-internal second-harmonic generation using a prism coupler,” IEEE J. Quantum Electron. QE-9, 23 (1973).
[CrossRef]

Oshman, M. K.

M. K. Oshman, S. E. Harris, “Theory of optical parametric oscillation internal to the laser cavity,” IEEE J. Quantum Electron. QE-4, 491 (1968).
[CrossRef]

Parker, J. V.

R. G. Smith, J. V. Parker, “Experimental observation of and comments on optical parametric oscillation internal to the laser cavity,” J. Appl. Phys. 41, 3401 (1970).
[CrossRef]

Ryzhkov, A. I.

V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (WileyNew York, 1984).

Smith, R. G.

R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. QE-6, 215 (1970).
[CrossRef]

R. G. Smith, J. V. Parker, “Experimental observation of and comments on optical parametric oscillation internal to the laser cavity,” J. Appl. Phys. 41, 3401 (1970).
[CrossRef]

R. G. Smith, “Optical parametric oscillators,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), p. 837.

Steier, W. H.

D. G. Gonzalez, S. T. K. Nieh, W. H. Steier, “Two-pass-internal second-harmonic generation using a prism coupler,” IEEE J. Quantum Electron. QE-9, 23 (1973).
[CrossRef]

Strini, G.

Strizhevskii, V. L.

V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
[CrossRef]

Volosov, V. D.

V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
[CrossRef]

Walls, D. F.

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation II. Quantum theory,” Opt. Acta 28, 211 (1981);G. Milburn, D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401 (1981).
[CrossRef]

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation I. Semiclassical theory,” Opt. Acta 27, 321 (1980);C. M. Savage, D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557 (1983).
[CrossRef]

Yarborough, J. M.

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70 (1971).
[CrossRef]

Yurke, B.

B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408 (1984).
[CrossRef]

Zernike, F.

F. Zernike, “Refractive Indices of ADP and KDP between 2000 Å and 1.5 μm,” J. Opt. Soc. Am. 54, 1215 (1964).

F. Zernike, J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

Appl. Phys. Lett. (1)

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70 (1971).
[CrossRef]

IEEE J. Quantum Electron. (4)

A. Ashkin, G. D. Boyd, J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109 (1966);G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

M. K. Oshman, S. E. Harris, “Theory of optical parametric oscillation internal to the laser cavity,” IEEE J. Quantum Electron. QE-4, 491 (1968).
[CrossRef]

R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. QE-6, 215 (1970).
[CrossRef]

D. G. Gonzalez, S. T. K. Nieh, W. H. Steier, “Two-pass-internal second-harmonic generation using a prism coupler,” IEEE J. Quantum Electron. QE-9, 23 (1973).
[CrossRef]

J. Appl. Phys. (2)

G. D. Boyd, D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

R. G. Smith, J. V. Parker, “Experimental observation of and comments on optical parametric oscillation internal to the laser cavity,” J. Appl. Phys. 41, 3401 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (3)

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation I. Semiclassical theory,” Opt. Acta 27, 321 (1980);C. M. Savage, D. F. Walls, “Optical chaos in sub/second harmonic generation,” Opt. Acta 30, 557 (1983).
[CrossRef]

P. Mandel, T. Erneux, “Amplitude self-modulation of in-tracavity second harmonic generation,” Opt. Acta 29, 7 (1982).
[CrossRef]

P. D. Drummond, K. J. McNeil, D. F. Walls, “Nonequilibrium transitions in sub/second harmonic generation II. Quantum theory,” Opt. Acta 28, 211 (1981);G. Milburn, D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401 (1981).
[CrossRef]

Opt. Commun. (1)

H. Fery, F. Hermann, “Noncollinear second harmonic generation in KDP,” Opt. Commun. 8, 291 (1973).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

V. D. Volosov, N. E. Kornienko, V. N. Krylov, A. I. Ryzhkov, V. L. Strizhevskii, “Phase effects in intracavity optical second harmonic generation 1. The case of free harmonic output from the cavity,” Opt. Spectrosc. (USSR) 46, 64 (1979);V. D. Volosov, S. G. Karpenko, N. E. Kornienko, V. N. Krylov, A. A. Man’ko, V. L. Strizhevskii, “Second-harmonic generation in a resonator,” Sov. J. Quantum Electron. 5, 500 (1975).
[CrossRef]

Phys. Rev. A (1)

B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408 (1984).
[CrossRef]

Proc. IEEE (1)

R. H. Kingston, A. L. McWhorter, “Electromagnetic mode mixing in nonlinear media,” Proc. IEEE 53, 4 (1965).
[CrossRef]

Other (6)

R. G. Smith, “Optical parametric oscillators,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), p. 837.

R. L. Byer, “Optical parametric oscillators,” in Quantum Electronics: A Treatise, H. Rabin, C. L. Tang, eds.(Academic, New York, 1975), Vol. 1, Part B, p. 587.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Y. R. Shen, The Principles of Nonlinear Optics (WileyNew York, 1984).

F. Zernike, J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

For boundless plane waves, the integration length would be the crystal length. In actual practice for finite beam widths, L would be the length of the interaction region, and correction for the finite width in the x direction would have to be made.

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Figures (10)

Fig. 1
Fig. 1

Noncollinear second-harmonic generation of 3 from two fundamental traveling waves (1, 2).

Fig. 2
Fig. 2

Generation and superposition of two second-harmonic waves [r(2ω), l(2ω)] from two fundamental standing waves.

Fig. 3
Fig. 3

Unidirectional generation of a single second-harmonic beam within a resonant cavity.

Fig. 4
Fig. 4

Generation of two second-harmonic beams from two fundamental standing waves within a cavity resonant at the second harmonic.

Fig. 5
Fig. 5

Collinear frequency doubling in a Fabry–Perot cavity in which both the fundamental (solid line) and the second harmonic (dashed line) are resonant. (a) Reflection at cavity mirror showing relative phase of fundamental and harmonic standing waves. (b) Phase of generated second-harmonic field relative to the fundamental standing wave within the nonlinear crystal. Dispersive elements at the mirror itself or within the cavity must be used to compensate (a) relative to (b), otherwise little second harmonic will be coupled into the cavity mode.

Fig. 6
Fig. 6

Schematic of optical arrangement. Mirrors M1 and M2 are used to create two fundamental standing waves within the KDP crystal. Mirrors MA and MB form an optical resonator for the generated second harmonic at 295 nm.

Fig. 7
Fig. 7

Variation of intensity of second harmonic for the configuration illustrated in Fig. 2 and Eq. (12). In (a)–(c) the horizontal axis corresponds to voltage applied to a piezoelectric transducer (the diagonal line in the figure is the voltage versus time). The ordinate in each photograph is the UV intensity, with different sensitivities in (a)–(c). (a) Mirror MA is scanned, thus varying ψ in Eq. (12) through the dependence on s2 (ordinary two-beam interference), (b) Mirror M1 is scanned varying ψ in Eq. (12) through the dependence on h1 (phase of one fundamental beam), (c) Mirrors M1 and M2 are scanned simultaneously varying ψ in Eq. (12) through the dependence on (h1 + h2) (phases of both fundamental beams).

Fig. 8
Fig. 8

Fabry–Perot resonances produced by a variation of s2 and hence of the cavity length for the situation illustrated in Fig. 3 and expression (14).

Fig. 9
Fig. 9

Fabry–Perot resonances for the arrangement of Fig. 4 and expression (15). Second-harmonic power transmitted through mirror MB is displayed as a function of voltage applied to transducer of MA. (a) (h1, h2) tuned for maximum UV signal, (b) (h1, h2) tuned for minimum UV signal, (c) Resonance signal from a single UV wave, as in Fig. 8. The only distinction between (a) and (b) is the position of the mirrors M1, M2.

Fig. 10
Fig. 10

Integrated second-harmonic power versus voltage on transducers of M1, M2 as discussed in the text, (a) (h1, h2) are scanned simultaneously, and (b) h1 is scanned alone to produce a modulation of coupling efficiency. The horizontal scale is such that 1 major division ≈ 58 nm. When the nonlinear responses of the transducers are included, the peak-to-peak spacing is λ3/2 in (a) and λ3 in (b).

Equations (19)

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1 ( r , t ) = E 1 ( r ) ê 1 exp [ i ( ω t + K 1 x sin β K 1 z cos β ϕ 1 ] + c . c . , 2 ( r , t ) = E 2 ( r ) ê 2 exp [ i ( ω t + K 2 x sin β K 2 z cos β ϕ 2 ] + c . c . ,
| E 1 | = | E 2 | E , | K 1 | = | K 2 | K .
P i ( 2 ω ) ( z , t ) = 2 j k χ i j k e 1 j e 2 k E 1 E 2 × exp [ i ( 2 ω t 2 K z cos β ϕ 1 ϕ 2 ) ] + c . c . ,
d d z E i ( 2 ω ) ( z ) = 8 π i ω n 2 ω c j k exp [ i ( K 3 2 K cos β ) z ] × χ i j k e 1 j e 2 k E 2 exp [ i ( ϕ 1 + ϕ 2 ) ] .
E i ( 2 ω ) ( L / 2 ) = i D i E 2 exp [ i ( ϕ 1 + ϕ 2 ) ] sin ( Δ K L / 2 ) Δ K L / 2 ,
E i ( 2 ω ) ( L / 2 ) = i D i E 2 exp [ i ( ϕ 1 + ϕ 2 ) ] ,
D i 8 π ω L n 2 ω c j k χ i j k e 1 j e 2 k .
| E ( 2 ω ) ( L / 2 ) | 2 = D 2 E 4 .
E r ( 2 ω ) ( L / 2 ) = i D E 2 exp [ i ( ϕ 1 + ϕ 2 ) ] ,
E l ( 2 ω ) ( L / 2 ) = i D E 2 exp [ i ( θ 1 + θ 2 ) ] .
θ 1 = ϕ 1 + 2 k h 1 + ϕ r , θ 2 = ϕ 2 + 2 k h 2 + ϕ r .
E l ( 2 ω ) ( L / 2 ) = i D E 2 exp [ i ( ϕ 1 + ϕ 2 + 2 k H + 2 ϕ r ) ] .
E l ( 2 ω ) ( L / 2 ) = i D E 2 exp [ i ( ϕ 1 + ϕ 2 + 2 k H + 2 k 3 s 2 + 2 ϕ r + γ ) ] ,
E r + l ( 2 ω ) ( L / 2 ) = i D E 2 exp [ i ( ϕ 1 + ϕ 2 ) ] × { 1 + exp [ 2 i ( k H + k 3 s 2 + Φ r / 2 ] } ,
| E r + l ( 2 ω ) ( L / 2 ) | 2 = 2 ( 1 + cos ψ ) D 2 E 4 ,
E R ( 2 ω ) ( L / 2 ) = E r ( 2 ω ) ( L / 2 ) ( 1 + r 2 exp [ α L + 2 i k 3 ( s 1 + s 2 ) ] + { r 2 exp [ α L + 2 i k 3 ( s 1 + s 2 ) ] } 2 + ) = E r ( 2 ω ) ( L / 2 ) ( 1 R exp [ α L + i δ ) ] ,
I R ( 2 ω ) T | E R ( 2 ω ) | 2 T D 2 E 4 / [ ( ( 1 Re α L ) 2 + 4 Re α L sin 2 ( k 3 s + γ ) ] ,
I R + L ( 2 ω ) 2 ( 1 + cos ψ ) T D 2 E 4 ( 1 Re α L ) 2 + 4 Re α L sin 2 δ / 2 .
I R + L ( 2 ω ) 2 { 1 + cos [ 2 k ( h 1 + h 2 ) + 2 k 3 s 2 + Φ r ] } T D 2 E 4 ( 1 Re α L ) 2 + 4 Re α L sin 2 [ k 3 ( s 1 + s 2 ) + γ ] .

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