Abstract

By using multiple crystals of periodically poled lithium niobate arranged in a fan-fold configuration with mirrors between the crystals to periodically refocus the light and retroreflect it after the final crystal, one can achieve very long interaction lengths and efficient frequency conversion at low input power. We present the theory and numerical simulations of devices using this concept, including devices for singly resonant optical parametric oscillation and second-harmonic generation. We find quantitative agreement between plane-wave analysis of these devices and numerical simulations in three spatial dimensions.

© 1999 Optical Society of America

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References

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  1. W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion in a continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 21, 1336–1338 (1996).
    [CrossRef] [PubMed]
  2. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
    [CrossRef]
  3. 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]
  4. G. T. Moore, K. Koch, and E. C. Cheung, “Theory of multi-stage intracavity frequency conversion in optical parametric oscillators,” in Solid State Lasers and Nonlinear Crystals, G. J. Quarles, L. Esterowitz, and L. K. Cheng, eds., Proc. SPIE 2379, 84–94 (1995).
    [CrossRef]
  5. G. T. Moore and K. Koch, “The tandem optical parametric oscillator,” IEEE J. Quantum Electron. 32, 2085–2094 (1996).
    [CrossRef]
  6. G. T. Moore, K. Koch, and M. E. Dearborn, “Gain enhancement of multi-stage parametric intracavity frequency conversion,” IEEE J. Quantum Electron. 33, 1734–1742 (1997).
    [CrossRef]
  7. S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  8. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  9. D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
    [CrossRef]
  10. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [CrossRef]
  11. 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]
  12. R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
    [CrossRef]
  13. Notation and properties of elliptic integrals and Jacobi elliptic functions taken from M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New, York, 1965), Chap. 16 and 17, pp. 576–626.
  14. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22, 1553–1555 (1997).
    [CrossRef]
  15. R. G. Smith, “Effects of momentum mismatch on parametric gain,” J. Appl. Phys. 41, 4121–4124 (1970).
    [CrossRef]
  16. Elliptic integrals and Jacobi elliptic functions were computed using programs in W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

1998 (1)

1997 (3)

1996 (2)

1995 (1)

G. T. Moore, K. Koch, and E. C. Cheung, “Theory of multi-stage intracavity frequency conversion in optical parametric oscillators,” in Solid State Lasers and Nonlinear Crystals, G. J. Quarles, L. Esterowitz, and L. K. Cheng, eds., Proc. SPIE 2379, 84–94 (1995).
[CrossRef]

1992 (1)

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

1979 (1)

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

1970 (2)

R. G. Smith, “Effects of momentum mismatch on parametric gain,” J. Appl. Phys. 41, 4121–4124 (1970).
[CrossRef]

S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1966 (1)

1962 (1)

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]

Alexander, J. I.

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]

Batchko, R. G.

Baumgartner, R. A.

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

Bloembergen, N.

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]

Bosenberg, W. R.

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Byer, R. L.

Cheung, E. C.

G. T. Moore, K. Koch, and E. C. Cheung, “Theory of multi-stage intracavity frequency conversion in optical parametric oscillators,” in Solid State Lasers and Nonlinear Crystals, G. J. Quarles, L. Esterowitz, and L. K. Cheng, eds., Proc. SPIE 2379, 84–94 (1995).
[CrossRef]

Collins Jr., S. A.

Dearborn, M. E.

G. T. Moore, K. Koch, and M. E. Dearborn, “Gain enhancement of multi-stage parametric intracavity frequency conversion,” IEEE J. Quantum Electron. 33, 1734–1742 (1997).
[CrossRef]

Drobshoff, A.

Ducuing, J.

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]

Fejer, M. M.

Imeshev, G.

Jundt, D. H.

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Koch, K.

G. T. Moore, K. Koch, and M. E. Dearborn, “Gain enhancement of multi-stage parametric intracavity frequency conversion,” IEEE J. Quantum Electron. 33, 1734–1742 (1997).
[CrossRef]

G. T. Moore and K. Koch, “The tandem optical parametric oscillator,” IEEE J. Quantum Electron. 32, 2085–2094 (1996).
[CrossRef]

G. T. Moore, K. Koch, and E. C. Cheung, “Theory of multi-stage intracavity frequency conversion in optical parametric oscillators,” in Solid State Lasers and Nonlinear Crystals, G. J. Quarles, L. Esterowitz, and L. K. Cheng, eds., Proc. SPIE 2379, 84–94 (1995).
[CrossRef]

Kogelnik, H.

Li, T.

Miller, G. D.

Moore, G. T.

G. T. Moore, K. Koch, and M. E. Dearborn, “Gain enhancement of multi-stage parametric intracavity frequency conversion,” IEEE J. Quantum Electron. 33, 1734–1742 (1997).
[CrossRef]

G. T. Moore and K. Koch, “The tandem optical parametric oscillator,” IEEE J. Quantum Electron. 32, 2085–2094 (1996).
[CrossRef]

G. T. Moore, K. Koch, and E. C. Cheung, “Theory of multi-stage intracavity frequency conversion in optical parametric oscillators,” in Solid State Lasers and Nonlinear Crystals, G. J. Quarles, L. Esterowitz, and L. K. Cheng, eds., Proc. SPIE 2379, 84–94 (1995).
[CrossRef]

Myers, L. E.

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]

Proctor, M.

Roberts, D. A.

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

Smith, R. G.

R. G. Smith, “Effects of momentum mismatch on parametric gain,” J. Appl. Phys. 41, 4121–4124 (1970).
[CrossRef]

Tulloch, W. M.

Weise, D. R.

Appl. Opt. (1)

IEEE J. Quantum Electron. (4)

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

G. T. Moore and K. Koch, “The tandem optical parametric oscillator,” IEEE J. Quantum Electron. 32, 2085–2094 (1996).
[CrossRef]

G. T. Moore, K. Koch, and M. E. Dearborn, “Gain enhancement of multi-stage parametric intracavity frequency conversion,” IEEE J. Quantum Electron. 33, 1734–1742 (1997).
[CrossRef]

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

J. Appl. Phys. (2)

R. G. Smith, “Effects of momentum mismatch on parametric gain,” J. Appl. Phys. 41, 4121–4124 (1970).
[CrossRef]

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (4)

Phys. Rev. (1)

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]

Proc. SPIE (1)

G. T. Moore, K. Koch, and E. C. Cheung, “Theory of multi-stage intracavity frequency conversion in optical parametric oscillators,” in Solid State Lasers and Nonlinear Crystals, G. J. Quarles, L. Esterowitz, and L. K. Cheng, eds., Proc. SPIE 2379, 84–94 (1995).
[CrossRef]

Other (2)

Elliptic integrals and Jacobi elliptic functions were computed using programs in W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Notation and properties of elliptic integrals and Jacobi elliptic functions taken from M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New, York, 1965), Chap. 16 and 17, pp. 576–626.

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Figures (13)

Fig. 1
Fig. 1

Scale diagram of a four-period POPO.

Fig. 2
Fig. 2

Compensation of a shift in relative optical phase by a shift in the PPLN grating phase.

Fig. 3
Fig. 3

Contour plot of the overlap integral I as a function of the air gap D and phase mismatch Q for the example discussed in the text. The black dot shows the values of D and Q used in the simulations.

Fig. 4
Fig. 4

Effect of equal phase detunings ΔΦ after each period on POPO small-signal gain. Curves are derived from plane-wave theory. Black dots are obtained from three-dimensional simulations with a resonant idler.

Fig. 5
Fig. 5

Calculated threshold for a 400-mW POPO in (Tn, Tp) space for Tr=0.99. A POPO with N periods is above threshold above the corresponding curve.

Fig. 6
Fig. 6

Solid curves show plane-wave calculations of efficiencies ηn(zˆ) and ηr(zˆ) and residual pump fraction ηp for a pump power of 400 mW and distributed losses corresponding to Tj=0.99 for all fields. Symbols show results from three-dimensional modeling with a resonant idler. The dotted curve for ηr is obtained by plane-wave modeling with discrete losses.

Fig. 7
Fig. 7

Curves show plane-wave calculations of efficiencies and residual pump for Tr=Tn=0.99 and Tp=0.74. Symbols show results obtained from three-dimensional modeling with a resonant idler.

Fig. 8
Fig. 8

Curves show plane-wave calculations of efficiencies and residual pump for Tr=Tp=0.99 and Tn=0.74. Symbols show results obtained from three-dimensional modeling with a resonant idler.

Fig. 9
Fig. 9

Steady-state NGF efficiency and residual pump as a function of pump power (or g2) is shown when N=4, Tj=0.99 for all fields and when there is no useful RGF outcoupling.

Fig. 10
Fig. 10

Steady-state NGF efficiency was computed by iterative plane-wave calculations, assuming no useful RGF outcoupling [ηr(N)=0]. The symbols show the maximum ηn(N) obtained as g2 was varied for various choices of N and Tj. The connecting lines are guides to the eye.

Fig. 11
Fig. 11

Threshold power (or g2) and the power for which maximum ηn(N) occurs in steady state form the lower and the upper boundaries of the three polygonal regions shown. These powers are determined from plane-wave iterative calculations for several even values of N under conditions of no useful RGF outcoupling. The region bounded by heavy lines is for the low-loss case where Tj=0.99 for all fields. The region shaded by horizontal lines is for Tn=0.74. The region shaded by vertical lines is for Tp=0.74. We use log-log scaling to make the boundaries straighter.

Fig. 12
Fig. 12

Power conversion efficiency as a function of period number is shown for SHG of 1.064 µm radiation at two power levels of the incident first harmonic. Circles show results of three-dimensional simulations. Curves are determined from plane-wave theory.

Fig. 13
Fig. 13

This curve shows how dephasing by an angle ΔΦ after each period affects the conversion efficiency of a four-period PSHG device with 400-mW incident first-harmonic power. The curve is calculated with plane-wave theory. The filled circles are obtained from three-dimensional simulations.

Equations (36)

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uj(z, r)=2π1wj sj(z)exp-r2wj2sj(z),
u˜j(z, K)=wj2πexp-K2wj2sj(z)4.
AjBjCjDj.
σsz+ws24iZs2σs=iκσpσi* exp(iqz),
σiz+wi24iZi2σi=iκσpσs* exp(iqz),
σpz+wp24iZp2σp=iκσsσi exp(-iqz),
κ=dcωsωpωinsnpni.
σj(z, r)=σˆj(zˆ)uj(z, r),
σiz+wi24iZi2σi=iκ exp(iqz)σˆpσˆs*2πwpwssp(z)ss*(z)×exp-r2wp2sp(z)-r2ws2ss*(z).
dσ˜idz-K2wi24iZiσ˜i
=iκ exp(iqz)σˆpσˆs*1π1wswpsp(z)+wpwsss*(z)×exp-K24sp(z)wp2+ss*(z)ws2.
Δσ˜i(L/2, K)
=iκσˆpσˆs*1π-L/2L/2dz exp(iqz)×expK2wi2(L/2-z)4iZi 1wswpsp(z)+wpwsss*(z)×exp-K24sp(z)wp2+ss*(z)ws2.
dσˆidzˆ=d2Ku˜i*(L/2, K)Δσ˜i(L/2, K)=iκIσˆpσˆs*,
I=8π -L/2L/2dzexp(iqz)S(z),
S(z)=wpwswisp(z)ss*(z)+wpwiwssp(z)si*(z)+wiwswpsi*(z)ss*(z).
S(z)=4wps*(z)/1-ωs-ωiωp2
dσˆrdzˆ+αrσˆr=iκIσˆpσˆn*,
dσˆndzˆ+αnσˆn=iκIσˆpσˆr*,
dσˆpdzˆ+αpσˆp=iκIσˆrσˆn.
σˆr(zˆ)=σˆr(0)exp(-αr zˆ)cosh{g[1-exp(-αpzˆ)]/αp}.
σˆr(zˆ)=σˆr(0)exp(-α+zˆ)[cosh(γzˆ)-(α-/γ)sinh(γzˆ)],
ρr(zˆ)=exp(-α zˆ)ρr(0)nd(Λ|m),
ρn(zˆ)=exp(-α zˆ)1-mρp(0)sd(Λ|m),
ρp(zˆ)=exp(-α zˆ)ρp(0)cd(Λ|m),
B(zˆ)=Tr00Tn×cosh gzˆexp(izˆΔΦ)sinh gzˆexp(-izˆΔΦ)sinh gzˆcosh gzˆ,
cosh gexp(izˆΔΦ)sinh gexp(-izˆΔΦ)sinh gcosh g=exp(izˆΔΦ/2)00exp(-izˆΔΦ/2)cosh gsinh gsinh gcosh g×exp(-izˆΔΦ/2)00exp(izˆΔΦ/2)
G0(N)=12H12[exp(iΔΦ/2)Tn-exp(-iΔΦ/2)Tr]cosh g+H×12[exp(iΔΦ/2)Tn+exp(-iΔΦ/2)Tr]cosh g-HN-12[exp(iΔΦ/2)Tn-exp(-iΔΦ/2)Tr]cosh g-H×12[exp(iΔΦ/2)Tn+exp(-iΔΦ/2)Tr]cosh g+HN2,
H2=12[exp(iΔΦ/2)Tn+exp(-iΔΦ/2)Tr]cosh g2-TrTn.
G0=1+gˆ2g2-Qˆ2sinh2(gˆ2-Qˆ2),
g=αp1-exp(-αp N)log[exp(αr N)+exp(2αrN)-1].
dρ1dzˆ=-κIρ2ρ1,
dρ2dzˆ=12κIρ12.
ρ1(zˆ)=ρ1(0)sech(gzˆ/2),
ρ2(zˆ)=12ρ1(0)tanh(gzˆ/2),
2η2(zˆ)=exp(-2αzˆ)tanh2{g[1-exp(-αzˆ)]/(α2)}.

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