Abstract

We present a simple analytical solution for a singly resonant optical parametric oscillator (OPO). The analysis permits calculating the depletion efficiency of the OPO even when the resonated signal (or idler) suffers from strong intracavity losses. The main results are (1) that the maximum efficiency as a function of the pump power is highly dependent on the mirror reflectivity and on the intracavity losses; (2) that the threshold intensity and the pump intensity at which maximum efficiency is reached depend on the reflectivity, and the ratio between the two intensities maintains the relation 1I3max/Ithπ2/4; and (3) that the maximum efficiency for a Gaussian beam is 70%.

© 2000 Optical Society of America

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  1. S. A. Akhmanov and R. V. Khokhlov, “Concerning one possibility of amplification of light waves,” Zh. Eksp. Teor. Fiz. 43, 351 (1962) [Sov. Phys. JETP 16, 252–257 (1963)].
  2. R. H. Kingston, “Parametric amplification and oscillation at optical frequencies,” Proc. IRE 50, 472 (1962).
  3. N. M. Kroll, “Parametric amplification in spatially extended media and the application to the design of tunable oscillators at optical frequencies,” Phys. Rev. 127, 1207–1211 (1962).
    [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. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
    [CrossRef]
  6. A. E. Siegman, “Nonlinear optical effects: an optical power limiter,” Appl. Opt. 1, 739–744 (1962).
    [CrossRef]
  7. L. B. Kreuzer, “Single and multi-mode oscillation of the singly resonant optical parametric oscillator, in Proceedings of the Joint Conference on Lasers and Opto-Electronics,” (Institution of Electronic and Radio Engineers, London, 1969) pp. 52–63; S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
    [CrossRef]
  8. J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. QE-7, 109–117 (1971).
    [CrossRef]
  9. D. D. Lowenthal, “Cw periodically poled LiNbO3 optical parametric oscillator model with strong idler absorption,” IEEE J. Quantum Electron. 34, 1356–1366 (1998).
    [CrossRef]
  10. T. G. Giallorenzi and C. L. Tang, “Cw parametric scattering in ADP with strong absorption in the idler band,” Appl. Phys. Lett. 12, 376–378 (1968).
    [CrossRef]
  11. T. G. Giallorenzi and C. L. Tang, “Effect of idler attenuation on the spontaneous parametric scattering of intense light in LiNbO3 and NH4H2PO4,” Phys. Rev. 184, 353–355 (1969).
    [CrossRef]
  12. A. Yariv and W. H. Loisell, “Theory of the optical parametric oscillator,” IEEE J. Quantum Electron. QE-2, 418–424 (1966).
    [CrossRef]
  13. E. S. Cassedy and M. Jain, “A theoretical study of injection tuning of optical parametric oscillators,” IEEE J. Quantum Electron. QE-15, 1290–1301 (1979).
    [CrossRef]
  14. K. D. Shaw, Opt. Commun. 144, 134 (1997).
    [CrossRef]
  15. P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator–upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
    [CrossRef]
  16. R. Baumgartner and R. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
    [CrossRef]
  17. A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).
  18. L. R. Marshall, A. Kaz, and O. Aytur, “Multimode pumping of optical parametric oscillators,” IEEE J. Quantum Electron. 32, 177–182 (1996).
    [CrossRef]

1998 (1)

D. D. Lowenthal, “Cw periodically poled LiNbO3 optical parametric oscillator model with strong idler absorption,” IEEE J. Quantum Electron. 34, 1356–1366 (1998).
[CrossRef]

1997 (1)

K. D. Shaw, Opt. Commun. 144, 134 (1997).
[CrossRef]

1996 (1)

L. R. Marshall, A. Kaz, and O. Aytur, “Multimode pumping of optical parametric oscillators,” IEEE J. Quantum Electron. 32, 177–182 (1996).
[CrossRef]

1979 (2)

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

E. S. Cassedy and M. Jain, “A theoretical study of injection tuning of optical parametric oscillators,” IEEE J. Quantum Electron. QE-15, 1290–1301 (1979).
[CrossRef]

1972 (1)

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator–upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

1971 (1)

J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. QE-7, 109–117 (1971).
[CrossRef]

1969 (1)

T. G. Giallorenzi and C. L. Tang, “Effect of idler attenuation on the spontaneous parametric scattering of intense light in LiNbO3 and NH4H2PO4,” Phys. Rev. 184, 353–355 (1969).
[CrossRef]

1968 (1)

T. G. Giallorenzi and C. L. Tang, “Cw parametric scattering in ADP with strong absorption in the idler band,” Appl. Phys. Lett. 12, 376–378 (1968).
[CrossRef]

1966 (1)

A. Yariv and W. H. Loisell, “Theory of the optical parametric oscillator,” IEEE J. Quantum Electron. QE-2, 418–424 (1966).
[CrossRef]

1965 (1)

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
[CrossRef]

1963 (1)

S. A. Akhmanov and R. V. Khokhlov, “Concerning one possibility of amplification of light waves,” Zh. Eksp. Teor. Fiz. 43, 351 (1962) [Sov. Phys. JETP 16, 252–257 (1963)].

1962 (4)

R. H. Kingston, “Parametric amplification and oscillation at optical frequencies,” Proc. IRE 50, 472 (1962).

N. M. Kroll, “Parametric amplification in spatially extended media and the application to the design of tunable oscillators at optical frequencies,” Phys. Rev. 127, 1207–1211 (1962).
[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]

A. E. Siegman, “Nonlinear optical effects: an optical power limiter,” Appl. Opt. 1, 739–744 (1962).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov and R. V. Khokhlov, “Concerning one possibility of amplification of light waves,” Zh. Eksp. Teor. Fiz. 43, 351 (1962) [Sov. Phys. JETP 16, 252–257 (1963)].

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]

Aytur, O.

L. R. Marshall, A. Kaz, and O. Aytur, “Multimode pumping of optical parametric oscillators,” IEEE J. Quantum Electron. 32, 177–182 (1996).
[CrossRef]

Baumgartner, R.

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

Bey, P. P.

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator–upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. QE-7, 109–117 (1971).
[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]

Byer, R.

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

Cassedy, E. S.

E. S. Cassedy and M. Jain, “A theoretical study of injection tuning of optical parametric oscillators,” IEEE J. Quantum Electron. QE-15, 1290–1301 (1979).
[CrossRef]

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]

Giallorenzi, T. G.

T. G. Giallorenzi and C. L. Tang, “Effect of idler attenuation on the spontaneous parametric scattering of intense light in LiNbO3 and NH4H2PO4,” Phys. Rev. 184, 353–355 (1969).
[CrossRef]

T. G. Giallorenzi and C. L. Tang, “Cw parametric scattering in ADP with strong absorption in the idler band,” Appl. Phys. Lett. 12, 376–378 (1968).
[CrossRef]

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
[CrossRef]

Jain, M.

E. S. Cassedy and M. Jain, “A theoretical study of injection tuning of optical parametric oscillators,” IEEE J. Quantum Electron. QE-15, 1290–1301 (1979).
[CrossRef]

Kaz, A.

L. R. Marshall, A. Kaz, and O. Aytur, “Multimode pumping of optical parametric oscillators,” IEEE J. Quantum Electron. 32, 177–182 (1996).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov and R. V. Khokhlov, “Concerning one possibility of amplification of light waves,” Zh. Eksp. Teor. Fiz. 43, 351 (1962) [Sov. Phys. JETP 16, 252–257 (1963)].

Kingston, R. H.

R. H. Kingston, “Parametric amplification and oscillation at optical frequencies,” Proc. IRE 50, 472 (1962).

Kroll, N. M.

N. M. Kroll, “Parametric amplification in spatially extended media and the application to the design of tunable oscillators at optical frequencies,” Phys. Rev. 127, 1207–1211 (1962).
[CrossRef]

Loisell, W. H.

A. Yariv and W. H. Loisell, “Theory of the optical parametric oscillator,” IEEE J. Quantum Electron. QE-2, 418–424 (1966).
[CrossRef]

Lowenthal, D. D.

D. D. Lowenthal, “Cw periodically poled LiNbO3 optical parametric oscillator model with strong idler absorption,” IEEE J. Quantum Electron. 34, 1356–1366 (1998).
[CrossRef]

Marshall, L. R.

L. R. Marshall, A. Kaz, and O. Aytur, “Multimode pumping of optical parametric oscillators,” IEEE J. Quantum Electron. 32, 177–182 (1996).
[CrossRef]

Miller, R. C.

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
[CrossRef]

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]

Shaw, K. D.

K. D. Shaw, Opt. Commun. 144, 134 (1997).
[CrossRef]

Siegman, A. E.

Tang, C. L.

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator–upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

T. G. Giallorenzi and C. L. Tang, “Effect of idler attenuation on the spontaneous parametric scattering of intense light in LiNbO3 and NH4H2PO4,” Phys. Rev. 184, 353–355 (1969).
[CrossRef]

T. G. Giallorenzi and C. L. Tang, “Cw parametric scattering in ADP with strong absorption in the idler band,” Appl. Phys. Lett. 12, 376–378 (1968).
[CrossRef]

Yariv, A.

A. Yariv and W. H. Loisell, “Theory of the optical parametric oscillator,” IEEE J. Quantum Electron. QE-2, 418–424 (1966).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. G. Giallorenzi and C. L. Tang, “Cw parametric scattering in ADP with strong absorption in the idler band,” Appl. Phys. Lett. 12, 376–378 (1968).
[CrossRef]

IEEE J. Quantum Electron. (7)

A. Yariv and W. H. Loisell, “Theory of the optical parametric oscillator,” IEEE J. Quantum Electron. QE-2, 418–424 (1966).
[CrossRef]

E. S. Cassedy and M. Jain, “A theoretical study of injection tuning of optical parametric oscillators,” IEEE J. Quantum Electron. QE-15, 1290–1301 (1979).
[CrossRef]

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator–upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

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

L. R. Marshall, A. Kaz, and O. Aytur, “Multimode pumping of optical parametric oscillators,” IEEE J. Quantum Electron. 32, 177–182 (1996).
[CrossRef]

J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. QE-7, 109–117 (1971).
[CrossRef]

D. D. Lowenthal, “Cw periodically poled LiNbO3 optical parametric oscillator model with strong idler absorption,” IEEE J. Quantum Electron. 34, 1356–1366 (1998).
[CrossRef]

Opt. Commun. (1)

K. D. Shaw, Opt. Commun. 144, 134 (1997).
[CrossRef]

Phys. Rev. (3)

T. G. Giallorenzi and C. L. Tang, “Effect of idler attenuation on the spontaneous parametric scattering of intense light in LiNbO3 and NH4H2PO4,” Phys. Rev. 184, 353–355 (1969).
[CrossRef]

N. M. Kroll, “Parametric amplification in spatially extended media and the application to the design of tunable oscillators at optical frequencies,” Phys. Rev. 127, 1207–1211 (1962).
[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]

Phys. Rev. Lett. (1)

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
[CrossRef]

Proc. IRE (1)

R. H. Kingston, “Parametric amplification and oscillation at optical frequencies,” Proc. IRE 50, 472 (1962).

Sov. Phys. JETP (1)

S. A. Akhmanov and R. V. Khokhlov, “Concerning one possibility of amplification of light waves,” Zh. Eksp. Teor. Fiz. 43, 351 (1962) [Sov. Phys. JETP 16, 252–257 (1963)].

Other (2)

L. B. Kreuzer, “Single and multi-mode oscillation of the singly resonant optical parametric oscillator, in Proceedings of the Joint Conference on Lasers and Opto-Electronics,” (Institution of Electronic and Radio Engineers, London, 1969) pp. 52–63; S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).

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Figures (2)

Fig. 1
Fig. 1

Comparison between the nondepleted-signal approximation (solid curve) and the present model [Eq. (15)] for two cases: 1, R=0.95 and l=0.01 (dashed curve), where the maximum efficiency is ηmax0.84; and 2, R=0.99 and l=0.05 (dotted curve), where the maximum efficiency is ηmax0.168.

Fig. 2
Fig. 2

OPO efficiency (η) as a function of the ratio N=I3/I3th for four cases: solid curve, plane wave, cw beam; dotted curve, Gaussian cross section, cw beam; dashed curve, plane wave with a temporally Gaussian shape (pulse); dashed–dotted curve, Gaussian cross section and temporally Gaussian shape (pulse). All were calculated for ηmax=1 (no loss).

Equations (49)

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η=2N(N-1).
η=sin2 ηN.
Ej(z)=uj(z)(4πW/0λjnj)1/2 exp[iϕj(z)]
(forj=1, 2, 3),
u1=(0λ1n1/4πW)1/2|E1|=[I1/(ω1W)]1/2,
u2=(0λ2n2/4πW)1/2|E2|=[I2/(ω2W)]1/2,
u3=(0λ3n3/4πW)1/2|E3|=[I3/(ω3W)]1/2.
u12ω1+u22ω2+u32ω3=1.
ξ4deff π(πW)1/2(0λ1λ2λ3n1n2n3)1/2z
du1dξ=-u2u3 sin θ,
du2dξ=-u1u3 sin θ,
du3dξ=u2u1 sin θ,
dθdξ=ΔS+u1u2u3-u3u2u1-u1u3u2cos θ,
m1=u12(0)+u32(0),
m2=u32(0),
m3=u12(0).
u12(ξL)=m1-{u3a2+(u3b2-u3a2)×sn2[(u3c2-u3a2)1/2(ξL+ξ0), γ]},
u32(m2-u32)(m1-u32)-[u32(0)-u32]2ΔS2/4.
u3a2, u3c2=½ {(m1+ΔS2/4)±[(m1+Δs2/4)2-ΔS2u32(0)]1/2},
u3b2u32(0),
γ2(u3b2-u3a2)/(u3c2-u3a2),
ξ0F(sin-1[{[u32(0)-u3a2]/(u3b2-u3a2)}1/2],γ)(u3c2-u3a2)1/2,
u12(0)=lu12(L),
ξu3(0)=u3(0)(u3c2-u3a2)1/2×Fcos-1(l-1-1)u12(0)u3b2-u3a21/2, γ+K(γ),
u3a2=0,u3b2=u32(0),u3c2=u32(0)+u12(0).
xu12(0)/u32(0).
I3I01/2ξu3(0)=1(1+x)1/2{F(cos-1[(l-1-1)x]1/2, γ)+K(γ)},
I0λ1λ20cn1n2n38(deff πL)2.
I3thI01/2=limx0 1x-[x(1/l-1)]1/20dφ1+1xsin2 φ-1/2.
I3thI01/2=sinh-11-ll1/2,
I3th=I0sinh-11-ll1/22.
η=u12(ξL)u32(0)(1-R)=x 1-Rl.
ηmax=xmax (1-R)l=1-R1-R(1-l).
ηmax=1-l/(1-R).
I3max=I0(l-l)K2(1-l)1/2.
1I3maxI3th=Nmax=(1-l)K2(1-l)1/2{sinh-1[(1-l)/l]1/2}2π242.46.
I3I01-RηR(1-l)sin-1η 1-R(1-l)1-R1/22.
I3th=I0 1-R(1-l)R(1-l),I3max/I3th=(π/2)2
I3=I30 exp[-(r/ρ)2],
ηGaus=2π0rd rI30 exp[-(r/ρ)2]η{I30 exp[-(r/ρ)2]}2π0rd rI30 exp[-(r/ρ)2].
ηGaus=1I30I3tI30d I3η(I3).
η(N)2.67ηmax (ln N)0.83N,
ηGaus=1.459ηmax (ln N)1.83N.
ηGausmax0.7ηmax
Nmax6.2.
I3(r, t)=I30(r)f(t)=I30(r)exp[-(t/τ)2],
η=-dtf(t)ηGaus[I3(t)]-dtf(t)=1.459ηmax -dtf(t)[ln(I3th/I3)]1.83/(I3th/I3)-dtf(t).
η=1.459ηmax (ln N)2.33Nτ-11dμ(1-μ2)1.83-dtf(t)0.9ηmax (ln N)2.33N.
η=2.11ηmax (ln N)1.33N,

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