Abstract

Based on the ripple transfers of electric-field amplitude and phase in frequency tripling, simple formulas are derived for the harmonic laser’s beam-quality factor M3ω2, with an arbitrary fundamental incidence to ideal nonlinear crystals. Whereas the harmonic beam’s quality is generally degraded, the beam’s divergence is similar to that of the fundamental after nonlinear frequency conversion. For practical crystals with periodic surface ripples that are caused by their machining, a multiorder diffractive model is presented with which the focusing properties of harmonic beams can be studied. Predictions of the theories are shown to be in excellent agreement with full numerical simulations.

© 2002 Optical Society of America

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References

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  1. A. E. Siegman, “New developments in laser resonators,” in, Optical Resonators, D. A. Holmes ed., Proc. SPIE 1224, 2–14 (1990).
    [CrossRef]
  2. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893–5901 (1993).
    [CrossRef] [PubMed]
  3. M. W. Sasnett, “Propagation of multimode laser beams: the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, London, 1989), pp. 132–142.
  4. R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1652 (1992).
    [CrossRef] [PubMed]
  5. R. Martinez-Herrero and P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
    [CrossRef]
  6. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  7. P. A. Belanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1992).
    [CrossRef]
  8. A. E. Siegman, “Binary optics cannot improve beam quality,” Opt. Lett. 18, 675–677 (1993).
    [CrossRef] [PubMed]
  9. A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
    [CrossRef]
  10. J. T. Hunt, P. A. Renard, and R. G. Nelson, “Focusing properties of an aberrated laser beams,” Appl. Opt. 15, 1458–1464 (1976).
    [CrossRef] [PubMed]
  11. D. A. Holmes and P. V. Avizonis, “Approximate optical system model,” Appl. Opt. 15, 1075–1082 (1975).
    [CrossRef]
  12. P. Pliszka and P. P. Banjerjee, “Nonlinear transverse effects in second harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
    [CrossRef]
  13. M. A. Drager and J. K. McIver, “Second harmonic generation in a nonlinear anisotropic medium with diffraction and depletion,” J. Opt. Soc. Am. B 7, 776–784 (1990).
    [CrossRef]
  14. S. C. Sheng and A. E. Siegman, “Non-linear optical calculations using fast-transform methods: second harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1990).
    [CrossRef]
  15. D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
    [CrossRef]
  16. M. H. Dunn and M. Ebrahimzaden, “Parametric generation of tunable light from continuous-wave to femtosecond pulses,” Science 286, 1513–1517 (1999).
    [CrossRef] [PubMed]
  17. W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
    [CrossRef]
  18. C. E. Barker, D. Milam, and R. Boyd, ICF Quarterly Rep. 3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1993), p. 55.
  19. C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
    [CrossRef]
  20. J. A. Paisner, E. M. Campbell, and W. J. Hogan, “The National Ignition Facility Project,” Rep. UCRL-JC-117397 Rev. 1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1994).
  21. R. S. Craxton, “High efficiency frequency tripling schemes for high power Nd:glass lasers,” IEEE J. Quantum Electron. 17, 1771–1782 (1981).
    [CrossRef]
  22. J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
    [CrossRef] [PubMed]
  23. L. J. Qian and X. M. Deng, “Approximate model on transform of beam characteristics in frequency tripling,” Acta Opt. Sin. 16, 1223–1226 (1996).
  24. G. D. Boyd and D. A. Kleinman, “Second harmonic generations of Gaussian laser beams,” J. Appl. Phys. 39, 3597–3618 (1968).
    [CrossRef]
  25. T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
    [CrossRef]

2001 (1)

T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
[CrossRef]

1999 (1)

M. H. Dunn and M. Ebrahimzaden, “Parametric generation of tunable light from continuous-wave to femtosecond pulses,” Science 286, 1513–1517 (1999).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

L. J. Qian and X. M. Deng, “Approximate model on transform of beam characteristics in frequency tripling,” Acta Opt. Sin. 16, 1223–1226 (1996).

1995 (3)

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
[CrossRef]

1993 (4)

1992 (2)

1991 (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

1990 (3)

A. E. Siegman, “New developments in laser resonators,” in, Optical Resonators, D. A. Holmes ed., Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

M. A. Drager and J. K. McIver, “Second harmonic generation in a nonlinear anisotropic medium with diffraction and depletion,” J. Opt. Soc. Am. B 7, 776–784 (1990).
[CrossRef]

S. C. Sheng and A. E. Siegman, “Non-linear optical calculations using fast-transform methods: second harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1990).
[CrossRef]

1981 (1)

R. S. Craxton, “High efficiency frequency tripling schemes for high power Nd:glass lasers,” IEEE J. Quantum Electron. 17, 1771–1782 (1981).
[CrossRef]

1980 (1)

W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
[CrossRef]

1976 (1)

1975 (1)

1968 (1)

G. D. Boyd and D. A. Kleinman, “Second harmonic generations of Gaussian laser beams,” J. Appl. Phys. 39, 3597–3618 (1968).
[CrossRef]

Auerbach, J. M.

J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
[CrossRef] [PubMed]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Avizonis, P. V.

Banjerjee, P. P.

Barker, C. E.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Belanger, P. A.

Boni, R.

W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
[CrossRef]

Bowers, M. S.

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Second harmonic generations of Gaussian laser beams,” J. Appl. Phys. 39, 3597–3618 (1968).
[CrossRef]

Caird, J. A.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Campbell, J. H.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Craxton, R. S.

R. S. Craxton, “High efficiency frequency tripling schemes for high power Nd:glass lasers,” IEEE J. Quantum Electron. 17, 1771–1782 (1981).
[CrossRef]

Craxton, S. D.

W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
[CrossRef]

Deng, X. M.

L. J. Qian and X. M. Deng, “Approximate model on transform of beam characteristics in frequency tripling,” Acta Opt. Sin. 16, 1223–1226 (1996).

Drager, M. A.

Dunn, M. H.

M. H. Dunn and M. Ebrahimzaden, “Parametric generation of tunable light from continuous-wave to femtosecond pulses,” Science 286, 1513–1517 (1999).
[CrossRef] [PubMed]

Ebrahimzaden, M.

M. H. Dunn and M. Ebrahimzaden, “Parametric generation of tunable light from continuous-wave to femtosecond pulses,” Science 286, 1513–1517 (1999).
[CrossRef] [PubMed]

Eimerl, D.

J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
[CrossRef] [PubMed]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Foley, R. J.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Holmes, D. A.

Hou, Z.

T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
[CrossRef]

Hunt, J. T.

Jacobs, S. D.

W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Second harmonic generations of Gaussian laser beams,” J. Appl. Phys. 39, 3597–3618 (1968).
[CrossRef]

Li, F. M.

T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
[CrossRef]

Martinez-Herrero, R.

R. Martinez-Herrero and P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1652 (1992).
[CrossRef] [PubMed]

McIver, J. K.

Mejias, P. M.

R. Martinez-Herrero and P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1652 (1992).
[CrossRef] [PubMed]

Milam, D.

Milonni, P. W.

J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
[CrossRef] [PubMed]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Murray, J. R.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Nelson, R. G.

Piquero, G.

Pliszka, P.

Qian, L. J.

T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
[CrossRef]

L. J. Qian and X. M. Deng, “Approximate model on transform of beam characteristics in frequency tripling,” Acta Opt. Sin. 16, 1223–1226 (1996).

Renard, P. A.

Rizzo, J. E.

W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
[CrossRef]

Seka, W.

W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
[CrossRef]

Sheng, S. C.

S. C. Sheng and A. E. Siegman, “Non-linear optical calculations using fast-transform methods: second harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1990).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Binary optics cannot improve beam quality,” Opt. Lett. 18, 675–677 (1993).
[CrossRef] [PubMed]

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893–5901 (1993).
[CrossRef] [PubMed]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

S. C. Sheng and A. E. Siegman, “Non-linear optical calculations using fast-transform methods: second harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1990).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” in, Optical Resonators, D. A. Holmes ed., Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Smith, A. V.

Speck, D. R.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Wang, T.

T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
[CrossRef]

Wonterghem, B. M.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Woods, B. W.

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Zhu, H.

T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
[CrossRef]

Acta Opt. Sin. (1)

L. J. Qian and X. M. Deng, “Approximate model on transform of beam characteristics in frequency tripling,” Acta Opt. Sin. 16, 1223–1226 (1996).

Appl. Opt. (4)

IEEE J. Quantum Electron. (2)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

R. S. Craxton, “High efficiency frequency tripling schemes for high power Nd:glass lasers,” IEEE J. Quantum Electron. 17, 1771–1782 (1981).
[CrossRef]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, “Second harmonic generations of Gaussian laser beams,” J. Appl. Phys. 39, 3597–3618 (1968).
[CrossRef]

J. Mod. Opt. (1)

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

R. Martinez-Herrero and P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
[CrossRef]

W. Seka, S. D. Jacobs, J. E. Rizzo, R. Boni, and S. D. Craxton, Opt. Commun. 34, 469 (1980).
[CrossRef]

T. Wang, L. J. Qian, H. Zhu, Z. Hou, and F. M. Li, “Beam-quality improved efficient second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Commun. 188, 213–217 (2001).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

S. C. Sheng and A. E. Siegman, “Non-linear optical calculations using fast-transform methods: second harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1990).
[CrossRef]

Proc. SPIE (2)

A. E. Siegman, “New developments in laser resonators,” in, Optical Resonators, D. A. Holmes ed., Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

C. E. Barker, B. M. Wonterghem, J. M. Auerbach, R. J. Foley, J. R. Murray, J. H. Campbell, J. A. Caird, D. R. Speck, and B. W. Woods, in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 398 (1995).
[CrossRef]

Science (1)

M. H. Dunn and M. Ebrahimzaden, “Parametric generation of tunable light from continuous-wave to femtosecond pulses,” Science 286, 1513–1517 (1999).
[CrossRef] [PubMed]

Other (3)

C. E. Barker, D. Milam, and R. Boyd, ICF Quarterly Rep. 3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1993), p. 55.

M. W. Sasnett, “Propagation of multimode laser beams: the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, London, 1989), pp. 132–142.

J. A. Paisner, E. M. Campbell, and W. J. Hogan, “The National Ignition Facility Project,” Rep. UCRL-JC-117397 Rev. 1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1994).

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Figures (3)

Fig. 1
Fig. 1

Amplitude distributions of a third-harmonic beam in (a) the angular domain and (b) the spatial domain calculated by numerical simulations based on the nonlinear coupled-wave equations (solid curves) with strong pump depletion for ideal tripling with perfect phase matching, ΔKd=ΔKt=0, and polarization matching with θp=35.3°. KDP crystals were used, and fundamental beam size w=115 mm with an intensity of 2 GW/cm2 and crystal lengths Ld=Lt=11.5 mm were used in the simulations (the corresponding conversion efficiency is ∼80%, and the conversion process is then in the region of strong pump depletion). A profile of the incident fundamental is also shown for comparison (dotted curves). To show the beam background clearly, we have plotted all curves for amplitude instead of intensity.

Fig. 2
Fig. 2

Calculated amplitude distributions of a third-harmonic beam in the angular domain for various conditions: (a) strong pump (2 GW/cm2) and nonideal tripling with phase mismatch ΔKdLd=ΔKtLt=0.5, (b) strong pump (2 GW/cm2) and nonideal tripling with polarization mismatch Δθp=2°, (c) strong pump (2 GW/cm2) with nonideal tripling ΔKdLd=ΔKtLt=0.5 but without walk-off, (d) weak pump depletion (0.02 GW/cm2) and ideal tripling (the corresponding conversion efficiency is much less than 1%, and conversion then occurs in the region where there is no pump depletion).

Fig. 3
Fig. 3

Amplitude distributions of a third-harmonic beam in the angular domain with surface ripples for Γ=180 nm and Λ=5 mm calculated by numerical simulations based on nonlinear coupled equations (solid curves) with (a) strong pump depletion and (b) weak pump depletion. Perfect phase-matching and polarization-matching conditions have been assumed in the simulations. The focusing profile of the incident fundamental is also shown (dotted curves).

Equations (22)

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

E1ω(r)=E0[1+ρ1ω(r)].
I1ω=I0[1+2ρ1ω(r)].
η(I1ω)=η0(I0)+2I0ρ1ω η0I0,
ρ3ω=γρ1ω,
γ=1η0 (η0I0)I0.
φ3ω=3φ1ω.
ρ3ωφ3ω=γ003ρ1ωφ1ω.
E1ω=E0[1+p1ω+(fx)exp(i2πfxx)+p1ω-(fx)exp(-i2πfxx)],
E3ω=E0[1+p3ω+(fx)exp(i2πfxx)+p3ω-(fx)exp(-i2πfxx)].
p3ω+(fx)p3ω-(fx)=ABBAp1ω+(fx)p1ω-(fx),
σ3ω2= fx2{[p3ω+(fx)]2+[p3ω-(fx)]2}= fx2{(A2+B2)[p1ω+(fx)]2+[p1ω-(fx)]2+4ABp1ω+(fx)p1ω-(fx)}.
(A-B)2σ3ω2/σ1ω2(A+B)2.
(A-B)M3ω2/M1ω2(A+B).
3M3ω2/M1ω2γ.
φ1ω(x)=2πΓλ1ω(nω-1)sin2πxΛ,
φ3ω(x)=2πΓ(Λ1ω/3)(n1ω-1)sin2(πx/Λ).
E3ω(x, y)=E03ω(x, y)exp[iφ3ω(x)],
E3ω(x, y)=E03ω exp-i πΓλ3ω/(n1ω-1) cos(2πx/Λ)=E03ω exp-i πΓ2λ3ω/(n1ω-1)×[exp(i2πx/Λ)+exp(-i2πx/Λ)]E03ω1-iπΓ2λ3ω/(n1ω-1)×[exp(i2πx/Λ)+exp(-i2πx/Λ)]-12 πΓ2λ3ω/(n1ω-1)2[exp(i4πx/Λ)+exp(-i4πx/Λ)]+.
I(±N)=1N!2πΓ(n1ω-1)2λ3ω2N,
θx(±N)=±Nλ3ωΛ,
E1ω=E0 exp[-(x/w)8+iφ1ω(x)],
φ1ω(ω)=α exp[-(x/w)2]+β exp[-(x/w)4],

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