Abstract

A theoretical threshold model based on the spherical wave assumption for a pulsed double-pass pumped singly resonant confocal positive-branch unstable optical parametric oscillator (OPO) has been proposed. It is demonstrated that this model is also applicable to the plane-parallel resonator in the special case. The OPO threshold as a function of important parameters such as the cavity magnification factor, cavity physical length, crystal length, pump pulsewidth, output coupler reflectance and crystal position inside the resonator has been presented. Experimental data show the good agreement with the results obtained from the theoretical model.

©2004 Optical Society of America

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References

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  1. “Feature issue on optical parametric oscillation and amplification,” J. Opt. Soc. Am. B10, 1656–1791 (1993).
  2. “Feature issue on optical parametric oscillators and amplifiers,” J. Opt. Soc. Am. B10, 2148–2243 (1993).
  3. “Feature issue on optical parametric devices,” J. Opt. Soc. Am. B12, 2084–2320 (1995).
  4. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp.858–913.
  5. W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.
  6. M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE2986, 113–122 (1997).
  7. J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.
  8. B. C. Johnson, V. J. Newell, J. B. Clark, and E. S. McPhee, “Narrow-bandwidth low-divergence optical parametric oscillator for nonlinear frequency-conversion applications,” J. Opt. Soc. Am. B 12, 2122–2127 (1995).
    [Crossref]
  9. G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” Applied Optics 40, 5446–5451 (2001).
    [Crossref]
  10. S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” Applied Optics 42, 1048–1051 (2003).
    [Crossref] [PubMed]
  11. S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM non-confocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.
  12. S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE Journal of Quantum Electronics QE-15, 415–431 (1979).
    [Crossref]
  13. E. Granot, S. Pearl, and M. M. Tilleman, “Analytical solution for a lossy singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 17, 381–386 (2000).
    [Crossref]
  14. R. L. Byer, “Optical parametric oscillator,” in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), pp. 587–702.
  15. W. J. Alford and A. V. Smith, “Wavelength variation of the second-order nonlinear coefficients of KNbO3, KTiOPO4, KTiOAsO4, LiNbO3, LiIO3, β-BaB2O4, KH2PO4, and LiB3O5 crystals: a test of Miller wavelength scaling,” J. Opt. Soc. Am. B 18, 524–533 (2001).
    [Crossref]

2003 (1)

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” Applied Optics 42, 1048–1051 (2003).
[Crossref] [PubMed]

2001 (2)

2000 (1)

1995 (1)

1979 (1)

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE Journal of Quantum Electronics QE-15, 415–431 (1979).
[Crossref]

Alford, W. J.

Allik, T. H.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM non-confocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Bowers, M. S.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM non-confocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE2986, 113–122 (1997).

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

Brosnan, S. J.

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE Journal of Quantum Electronics QE-15, 415–431 (1979).
[Crossref]

Brown, M. K.

M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE2986, 113–122 (1997).

Byer, R. L.

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE Journal of Quantum Electronics QE-15, 415–431 (1979).
[Crossref]

R. L. Byer, “Optical parametric oscillator,” in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), pp. 587–702.

Chandr, S.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM non-confocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Clark, J. B.

Ehrlich, Y.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” Applied Optics 42, 1048–1051 (2003).
[Crossref] [PubMed]

Farmer, J. N.

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

Fastig, S.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” Applied Optics 42, 1048–1051 (2003).
[Crossref] [PubMed]

Granot, E.

Hansson, G.

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” Applied Optics 40, 5446–5451 (2001).
[Crossref]

Hutchinson, J. A.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM non-confocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Johnson, B. C.

Karlsson, H.

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” Applied Optics 40, 5446–5451 (2001).
[Crossref]

Laurell, F.

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” Applied Optics 40, 5446–5451 (2001).
[Crossref]

McPhee, E. S.

Neuman, W. A.

W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Newell, V. J.

Pearl, S.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” Applied Optics 42, 1048–1051 (2003).
[Crossref] [PubMed]

E. Granot, S. Pearl, and M. M. Tilleman, “Analytical solution for a lossy singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 17, 381–386 (2000).
[Crossref]

Rosenwaks, S.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” Applied Optics 42, 1048–1051 (2003).
[Crossref] [PubMed]

Schaprf, W. S.

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp.858–913.

Smith, A. V.

Tilleman, M. M.

Velsko, S. P.

W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Applied Optics (2)

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” Applied Optics 40, 5446–5451 (2001).
[Crossref]

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” Applied Optics 42, 1048–1051 (2003).
[Crossref] [PubMed]

IEEE Journal of Quantum Electronics (1)

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE Journal of Quantum Electronics QE-15, 415–431 (1979).
[Crossref]

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

Other (9)

R. L. Byer, “Optical parametric oscillator,” in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), pp. 587–702.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM non-confocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

“Feature issue on optical parametric oscillation and amplification,” J. Opt. Soc. Am. B10, 1656–1791 (1993).

“Feature issue on optical parametric oscillators and amplifiers,” J. Opt. Soc. Am. B10, 2148–2243 (1993).

“Feature issue on optical parametric devices,” J. Opt. Soc. Am. B12, 2084–2320 (1995).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp.858–913.

W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE2986, 113–122 (1997).

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

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Figures (8)

Fig. 1.
Fig. 1. Confocal unstable singly resonant OPO. The Input Mirror M1 is a concave mirror, which is highly reflecting at the signal wavelength and highly transmitting at the pump and idler wavelengths. The output coupler M2 is a convex mirror, which is highly reflective at the pump wavelength, highly transmitting at the idler wavelength, and has signal reflectance R.

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Fig. 2.
Fig. 2. Threshold energy versus cavity physical length. Solid line shows results of our model for various cavity magnification factors. Dashed line shows results of BB model for plane-parallel resonator. lc =20 mm, 2rp =4 mm, R=0.82, T=13.5 ns.

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Fig. 3.
Fig. 3. Threshold energy versus crystal length. Solid line shows results of our model for various cavity magnification factors. Dashed line shows results of BB model. L=60 mm, 2rp=4 mm, R=0.82, T=13.5 ns.

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Fig. 4.
Fig. 4. Threshold energy versus reflectance to signal of output mirror. Solid line shows results of our model for various cavity magnification factors. Dashed line shows results of BB model. L=60 mm, lc =20 mm, 2rp =4 mm, T=13.5 ns.

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Fig. 5.
Fig. 5. Threshold energy versus pump pulsewidth. Solid line shows results of our model for various cavity magnification factors. Dashed line shows results of BB model. L=60 mm, lc =20 mm, 2rp =4 mm, R=0.82.

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Fig. 6.
Fig. 6. Threshold energy versus crystal position relative to output mirror. Solid line shows results of our model for various cavity magnification factors. Dashed line shows results of BB model. L=60 mm, lc =20 mm, 2rp =4 mm, R=0.82, T=13.5 ns.

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Fig. 7.
Fig. 7. Threshold energy versus cavity physical length for a plane-parallel resonator. Solid line shows results of our model for M=1. Dashed line shows results of BB model. lc =20 mm, 2rp =4 mm, R=0.82, T=13.5 ns.

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Fig. 8.
Fig. 8. Threshold energy versus crystal position. Solid line shows results of our model for M=1.33, L=36 mm, lc =20 mm, 2rp =4 mm, R=0.82, T=13.5 ns.

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Tables (1)

Tables Icon

Table 1. The threshold energy values of the unstable resonators with three different cavity magnifications from experiments and theoretical model.

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Equations (30)

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E j ( r , z , t ) = 1 2 [ E j ( r , z ) · e i ( k z z + k r r − ω t ) + c . c ]
E j ( r , z ) = A j ( r 0 , z ) M j ′ ( z ) j = p , s , i
∇ 2 E j ( r , z , t ) − μ 0 σ ∂ E j ( r , z , t ) ∂ t − μ 0 ε 0 ∂ 2 E j ( r , z , t ) ∂ t 2 = μ 0 ∂ 2 P j ( r , z , t ) ∂ t 2
{ ∂ E s ( r , z ) ∂ z + α s E s ( r , z ) = i N s E p ( r , z ) E i * ( r , z ) e i Δ k r r e i Δ k z z ∂ E i ( r , z ) ∂ z + α i E i ( r , z ) = i N i E p ( r , z ) E s * ( r , z ) e i Δ k r r e i Δ k z z ∂ E p ( r , z ) ∂ z + α p E p ( r , z ) = i N p E s ( r , z ) E i ( r , z ) e − i Δ k r r e − i Δ k z z
α j = μ 0 σ j ω j 2 k j 1 + θ j 2 + i θ j 2 k jz 2
N j = ω j d eff n j c 1 + θ j 2
{ ∂ A s ( r , z ) ∂ z + α s A s ( r , z ) = i N s A p ( r ) A i * ( r , z ) e i Δ k r r 0 e i Δ k ′ z ∂ A i ( r , z ) ∂ z + α i A i ( r , z ) = i N i A p ( r ) A s * ( r , z ) e i Δ k r r 0 e i Δ k ′ z
A s ( r 0 , z ) = e − α z e i Δ k ′ z ⁄ 2 A s ( r 0 , 0 ) [ cosh ( g ( r 0 ) z ) − i Δ k ′ 2 g ( r 0 ) sinh ( g ( r 0 ) z ) ]
g ( r 0 ) = N s N i ∣ A p ( r 0 ) ∣ 2 − ( Δ k ′ ⁄ 2 ) 2
E s ( M s ′ ( l c ) r , l c ) = e − α l c E s ( r , 0 ) M s ′ ( l c ) cosh ( β 0 l c e − ( r 2 r p ) 2 )
β 0 = 2 N s N i n p c ε 0 I p 0 e − ( t τ p ) 2
E s f ( r , l c ) = E s f ( r , 0 ) · e − α f l c · cosh ( β f l c e − ( r ⁄ 2 r p ) 2 )
β f = 2 N s f N i f n p c ε 0 I p 0 e − ( t ⁄ τ p ) 2
N j f = ω j d eff c n j , j = s , i
E sr b ( r ) = R · E s f ( r , l c )
E pr b ( r ) = γ 0 · E p f ( r )
E j b ( M 1 r , 0 ) = E jr b ( r ) M 1
M 2 j ( z ) = 1 + 2 n j ( ∣ R 2 ∣ + 2 L 2 ) z
E s b ( M 2 s ( l c ) r , l c ) = e − α b l c e i Δ k ' l c 2 E s b ( r , 0 ) M 2 s ( l c ) cosh ( β b l c e − ( r 2 M 1 r p ) 2 )
β b = 2 N s b N i b n p c ε o · γ 0 2 I p 0 M 1 2 e − ( t / τ p ) 2
N j b = ω j d eff n j c 1 + θ j 2 , j = s , i
E sround ( M 3 s r ) = E s b ( r , l c ) M 3 s
E sround ( r ) = R e − ( α f + α b ) l c E start 0 e − ( r ⁄ 2 Mr s ) 2 M cosh ( β f l c e − ( r 2 Mr p ) 2 ) cosh ( β b l c e − ( r 2 Mr p ) 2 )
β b = γ β f · [ 1 − 2 ( M R 2 n s ) 2 r 2 ]
P m − 1 = ∫ 0 ∞ ( 1 2 n s c ε 0 ) · ∣ E start ( r ) ∣ 2 · 2 π r dr = ( 1 2 n s c ε 0 ) · ∣ E start 0 ∣ 2 · π r s 2
P m = ∫ 0 ∞ ( 1 2 n s c ε 0 ) · ∣ E sround ( r ) ∣ 2 2 π rdr
= R M 2 e − 2 ( α f + α b ) l c ( 2 π 2 n s c ε 0 ) · ∣ E start 0 ∣ 2 · ∫ 0 ∞ e − ( r Mr s ) 2 cosh 2 ( β f l c e − ( r 2 Mr p ) 2 ) cosh 2 ( β b l c e − ( r 2 Mr p ) 2 ) rdr
P m = P m − 1 { Re − 2 ( α f + α b ) l c [ 1 16 ( r s 1 r s ) 2 e 2 β f l c ( 1 + γ ) + 1 8 ( r s 2 r s ) 2 e 2 β f l c γ + 1 8 ( r s 3 r s ) 2 e 2 β f l c + 1 4 ] }
{ 1 r s 1 2 = 1 r s 2 + β f l c ( 1 + γ ) r p 2 + 4 β f l c γ ( R 2 n s ) 2 1 r s 2 2 = 1 r s 2 + β f l c γ r p 2 + 4 β f l c γ ( R 2 n s ) 2 1 r s 3 2 = 1 r s 2 + β f l c r p 2
Q = ∫ 0 ∞ ∫ 0 ∞ I p ( r , t ) dt 2 π rdr = I p 0 π 1.5 r p 2 τ p

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