Abstract

Changing the relative phase Δϕ between the fundamental and harmonic waves produces periodic power modulation of a laser intracavity second-harmonic generated wave. The periodic power modulations of the two counterpropagating second-harmonic waves are at exactly opposite phases. This behavior is different from that in the case of intracavity second-harmonic generation in a passive cavity. A phenomenological numerical model developed for a laser consists of double-pass second-harmonic generation. The steady-state plane-wave model incorporates second-order nonlinear interaction, laser gain, and linear dispersion that contribute to the phase difference Δϕ. The model predictions are in good agreement with the experimental results. The model is useful for optimization of laser intracavity second-harmonic generation, and it may be applied to different types of intracavity nonlinear interaction.

© 1999 Optical Society of America

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References

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  1. J. K. Wright, “Enhancement of second harmonic power generated by a dielectric crystal inside a laser cavity,” Proc. IEEE 51, 1663 (1963).
    [CrossRef]
  2. R. G. Smith, K. Nassau, and M. F. Galvin, “Efficient continuous optical second harmonic generation,” Appl. Phys. Lett. 7, 256–258 (1965).
    [CrossRef]
  3. R. Polloni and O. Svelto, “Optimum coupling for intracavity second harmonic generation,” IEEE J. Quantum Electron. QE-4, 528–530 (1968).
    [CrossRef]
  4. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  5. J. Falk and J. M. Yarborough, Department of Electrical Engineering, University of Pittsburgh, 348 Benedum Hall, Pittsburgh, Penn. 15261 (personal communication, 1997).
  6. J. M. Yarborough, J. Falk, and C. B. Hitz, “Enhancement of second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70–73 (1971).
    [CrossRef]
  7. K. A. Stankov and J. Jethwa, “A new mode-lock technique using a nonlinear mirror,” Opt. Commun. 66, 41–46 (1988).
    [CrossRef]
  8. Z. A. Tagiev, R. Zh. Kasumova, and Sh. Sh. Amirov, “Theory of intracavity second-harmonic-generation in the prescribed-intensity approximation,” Opt. Spectrosc. 75, 535–537 (1993).
  9. S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).
  10. A. Yariv, Quantum electronics, 2nd ed. (Wiley, New York, 1975), Chap. 16.
  11. W. Kochner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, Berlin, 1992), Chap. 10.
  12. MathCad program, Mathsoft, Inc., 101 Main Street, Cambridge, Mass. 02142.
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    [CrossRef]
  14. R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurement in nonlinear optics,” Appl. Phys. B: Lasers Opt. 63, 491–498 (1996).
    [CrossRef]
  15. K. A. Stankov, “A mirror with an intensity dependent reflection coefficient,” Appl. Phys. B: Lasers Opt. 45, 191–195 (1988).
    [CrossRef]
  16. G. Imeshev, M. Proctor, and M. M. Fejer, “Phase correction in double-pass quasi-phase-matched second-harmonic generation with a wedged crystal,” Opt. Lett. 23, 165–167 (1998).
    [CrossRef]
  17. Raicoll Crystals, Ltd., P.O. Box 3, Ariel, Israel.

1998

1997

S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).

1996

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurement in nonlinear optics,” Appl. Phys. B: Lasers Opt. 63, 491–498 (1996).
[CrossRef]

1993

Z. A. Tagiev, R. Zh. Kasumova, and Sh. Sh. Amirov, “Theory of intracavity second-harmonic-generation in the prescribed-intensity approximation,” Opt. Spectrosc. 75, 535–537 (1993).

1988

K. A. Stankov, “A mirror with an intensity dependent reflection coefficient,” Appl. Phys. B: Lasers Opt. 45, 191–195 (1988).
[CrossRef]

K. A. Stankov and J. Jethwa, “A new mode-lock technique using a nonlinear mirror,” Opt. Commun. 66, 41–46 (1988).
[CrossRef]

1971

J. M. Yarborough, J. Falk, and C. B. Hitz, “Enhancement of second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70–73 (1971).
[CrossRef]

1968

R. Polloni and O. Svelto, “Optimum coupling for intracavity second harmonic generation,” IEEE J. Quantum Electron. QE-4, 528–530 (1968).
[CrossRef]

1965

R. G. Smith, K. Nassau, and M. F. Galvin, “Efficient continuous optical second harmonic generation,” Appl. Phys. Lett. 7, 256–258 (1965).
[CrossRef]

R. K. Chang, J. Ducuing, and N. Bloembergen, “Relative phase measurement between fundamental and second-harmonic light,” Phys. Rev. Lett. 15, 6–8 (1965).
[CrossRef]

1963

J. K. Wright, “Enhancement of second harmonic power generated by a dielectric crystal inside a laser cavity,” Proc. IEEE 51, 1663 (1963).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Amirov, Sh. Sh.

Z. A. Tagiev, R. Zh. Kasumova, and Sh. Sh. Amirov, “Theory of intracavity second-harmonic-generation in the prescribed-intensity approximation,” Opt. Spectrosc. 75, 535–537 (1993).

Angert, N.

S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Berkovic, G.

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurement in nonlinear optics,” Appl. Phys. B: Lasers Opt. 63, 491–498 (1996).
[CrossRef]

Bloembergen, N.

R. K. Chang, J. Ducuing, and N. Bloembergen, “Relative phase measurement between fundamental and second-harmonic light,” Phys. Rev. Lett. 15, 6–8 (1965).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Chang, R. K.

R. K. Chang, J. Ducuing, and N. Bloembergen, “Relative phase measurement between fundamental and second-harmonic light,” Phys. Rev. Lett. 15, 6–8 (1965).
[CrossRef]

Ducuing, J.

R. K. Chang, J. Ducuing, and N. Bloembergen, “Relative phase measurement between fundamental and second-harmonic light,” Phys. Rev. Lett. 15, 6–8 (1965).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Falk, J.

J. M. Yarborough, J. Falk, and C. B. Hitz, “Enhancement of second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70–73 (1971).
[CrossRef]

Fejer, M. M.

Galvin, M. F.

R. G. Smith, K. Nassau, and M. F. Galvin, “Efficient continuous optical second harmonic generation,” Appl. Phys. Lett. 7, 256–258 (1965).
[CrossRef]

Hitz, C. B.

J. M. Yarborough, J. Falk, and C. B. Hitz, “Enhancement of second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70–73 (1971).
[CrossRef]

Imeshev, G.

Jethwa, J.

K. A. Stankov and J. Jethwa, “A new mode-lock technique using a nonlinear mirror,” Opt. Commun. 66, 41–46 (1988).
[CrossRef]

Kasumova, R. Zh.

Z. A. Tagiev, R. Zh. Kasumova, and Sh. Sh. Amirov, “Theory of intracavity second-harmonic-generation in the prescribed-intensity approximation,” Opt. Spectrosc. 75, 535–537 (1993).

Lotem, H.

S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).

Marowsky, G.

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurement in nonlinear optics,” Appl. Phys. B: Lasers Opt. 63, 491–498 (1996).
[CrossRef]

Nassau, K.

R. G. Smith, K. Nassau, and M. F. Galvin, “Efficient continuous optical second harmonic generation,” Appl. Phys. Lett. 7, 256–258 (1965).
[CrossRef]

Pearl, S.

S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Polloni, R.

R. Polloni and O. Svelto, “Optimum coupling for intracavity second harmonic generation,” IEEE J. Quantum Electron. QE-4, 528–530 (1968).
[CrossRef]

Proctor, M.

Roth, M.

S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).

Schwarzberg, E.

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurement in nonlinear optics,” Appl. Phys. B: Lasers Opt. 63, 491–498 (1996).
[CrossRef]

Shimony, Y.

S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).

Smith, R. G.

R. G. Smith, K. Nassau, and M. F. Galvin, “Efficient continuous optical second harmonic generation,” Appl. Phys. Lett. 7, 256–258 (1965).
[CrossRef]

Stankov, K. A.

K. A. Stankov, “A mirror with an intensity dependent reflection coefficient,” Appl. Phys. B: Lasers Opt. 45, 191–195 (1988).
[CrossRef]

K. A. Stankov and J. Jethwa, “A new mode-lock technique using a nonlinear mirror,” Opt. Commun. 66, 41–46 (1988).
[CrossRef]

Stolle, R.

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurement in nonlinear optics,” Appl. Phys. B: Lasers Opt. 63, 491–498 (1996).
[CrossRef]

Svelto, O.

R. Polloni and O. Svelto, “Optimum coupling for intracavity second harmonic generation,” IEEE J. Quantum Electron. QE-4, 528–530 (1968).
[CrossRef]

Tagiev, Z. A.

Z. A. Tagiev, R. Zh. Kasumova, and Sh. Sh. Amirov, “Theory of intracavity second-harmonic-generation in the prescribed-intensity approximation,” Opt. Spectrosc. 75, 535–537 (1993).

Wright, J. K.

J. K. Wright, “Enhancement of second harmonic power generated by a dielectric crystal inside a laser cavity,” Proc. IEEE 51, 1663 (1963).
[CrossRef]

Yarborough, J. M.

J. M. Yarborough, J. Falk, and C. B. Hitz, “Enhancement of second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70–73 (1971).
[CrossRef]

Appl. Phys. B: Lasers Opt.

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurement in nonlinear optics,” Appl. Phys. B: Lasers Opt. 63, 491–498 (1996).
[CrossRef]

K. A. Stankov, “A mirror with an intensity dependent reflection coefficient,” Appl. Phys. B: Lasers Opt. 45, 191–195 (1988).
[CrossRef]

Appl. Phys. Lett.

R. G. Smith, K. Nassau, and M. F. Galvin, “Efficient continuous optical second harmonic generation,” Appl. Phys. Lett. 7, 256–258 (1965).
[CrossRef]

J. M. Yarborough, J. Falk, and C. B. Hitz, “Enhancement of second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18, 70–73 (1971).
[CrossRef]

IEEE J. Quantum Electron.

R. Polloni and O. Svelto, “Optimum coupling for intracavity second harmonic generation,” IEEE J. Quantum Electron. QE-4, 528–530 (1968).
[CrossRef]

Opt. Commun.

K. A. Stankov and J. Jethwa, “A new mode-lock technique using a nonlinear mirror,” Opt. Commun. 66, 41–46 (1988).
[CrossRef]

Opt. Lett.

Opt. Spectrosc.

Z. A. Tagiev, R. Zh. Kasumova, and Sh. Sh. Amirov, “Theory of intracavity second-harmonic-generation in the prescribed-intensity approximation,” Opt. Spectrosc. 75, 535–537 (1993).

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. Lett.

R. K. Chang, J. Ducuing, and N. Bloembergen, “Relative phase measurement between fundamental and second-harmonic light,” Phys. Rev. Lett. 15, 6–8 (1965).
[CrossRef]

Proc. IEEE

J. K. Wright, “Enhancement of second harmonic power generated by a dielectric crystal inside a laser cavity,” Proc. IEEE 51, 1663 (1963).
[CrossRef]

Proc. SPIE

S. Pearl, Y. Shimony, H. Lotem, M. Roth, and N. Angert, “Intracavity second harmonic generation in passively Q-switched Nd: YAG laser,” in 10th Meeting on Optical Engineering in Israel, S. R. Rotman and I. Shladov, eds., Proc. SPIE 3110, 232–237 (1997).

Other

A. Yariv, Quantum electronics, 2nd ed. (Wiley, New York, 1975), Chap. 16.

W. Kochner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, Berlin, 1992), Chap. 10.

MathCad program, Mathsoft, Inc., 101 Main Street, Cambridge, Mass. 02142.

J. Falk and J. M. Yarborough, Department of Electrical Engineering, University of Pittsburgh, 348 Benedum Hall, Pittsburgh, Penn. 15261 (personal communication, 1997).

Raicoll Crystals, Ltd., P.O. Box 3, Ariel, Israel.

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Figures (6)

Fig. 1
Fig. 1

Schematic description of double-pass intracavity laser second-harmonic generation. With the reflectance R2(2ω)=0 of mirror M2, harmonic waves are limited to the region bonded by mirrors M1 and M2.

Fig. 2
Fig. 2

Theoretical front and back harmonic power and fundamental power as a function of the relative phase term Λ with nonlinear coupling factor κl=0.2 and mirror M1 reflection coefficient ρ2ω=0.45. For clarity the 2ω curves have been multiplied by 2.

Fig. 3
Fig. 3

Theoretical harmonic output power as a function of relative phase term Λ and reflection coefficient ρ2ω of mirror M1: (a) Front power with weak nonlinear coupling factor κl=0.05, (b) back power with weak nonlinear coupling factor κl=0.05, (c) front power with strong nonlinear coupling factor κl=0.6.

Fig. 4
Fig. 4

Theoretical back harmonic power as a function of relative phase Λ for ρ2ω1 and various values of phase-matching term Δkl.

Fig. 5
Fig. 5

Experimental results of front and back harmonic output power as a function of the air gap between M1 and the KTP crystal, with a reflection coefficient ρ2ω0.45 of M1.

Fig. 6
Fig. 6

Experimental results of back fundamental and harmonic power as a function of glass plate angle θ for a reflection coefficient ρ2ω0.45 of M1. The fundamental power is a small leakage measured behind folding mirror M2.

Equations (22)

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

Eω(z, t)=ω/nωaω(z)×exp[i(ωt-ωnωz/c+ϕω)],
E2ω(z, t)=2ω/n2ωa2ω(z)×exp[i(2ωt-2ωn2ωz/c+ϕ2ω)],
daω,1/dz=κaω,2a2ω sin(Δϕ),
daω,2/dz=κaω,1a2ω sin(Δϕ),
da2ω/dz=-κaω,1aω,2 sin(Δϕ),
dΔϕ/dz=-κ(aω,1aω,2/a2ω)-(aω,1a2ω/aω,2)-(aω,2a2ω/aω,1)cos(Δϕ)-Δk,
a2ω(z)=aω(0)tanhκaω(0)z/2/2,
aω(z)=aω(0)/coshκaω(0)z/2,
Δϕ(z)=π/2.
a2ω(z2)=aω(z1)tanhaω(z1)κl/2/2,
aω(z2)=aω(z1)/coshaω(z1)κl/2,
ΔϕR(z2)=π/2,
Λ=Δϕdispersion+Δϕreflection,
aωL(z2)=ρωaω(z2),
a2ωL(z2)=ρ2ωa2ω(z2),
ΔϕL(z2)=ΔϕR(z2)+Λ=π/2+Λ,
dI/dz=g0I/[1+(I/Is)],
ln G=g0L-(Iout-Iin)/Is.
ln(aout2/ain2)=2 ln(aout/ain)=g02L-(aout2-ain2)/as2,
ln[aω(z1)/aωL(z1)]=g0L-aω(z1)2-aωL(z1)2/2as2.
Λ(θ)=4ωcng(ω)cos{sin-1[sin θ/ng(ω)]}-ng(2ω)cos{sin-1[sin θ/ng(2ω)]}t+Δϕreflection,
Δ(lair)=4ωcΔnairlair+Δϕreflection,

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