Abstract

A simple plane-wave model of pulsed, singly resonant, optical-parametric-oscillator and optical-parametric-oscillator–amplifier operation leads to a description of such systems in terms of a discrete dynamical system. The theoretical limits on conversion efficiencies derivable from this model were explored. Analysis of the model for an optical parametric oscillator–amplifier (OPOA) indicates that the effect that backconversion has in limiting efficiency can be avoided if one precisely shapes the time profile of the pump pulse and combines it with an OPOA that is Q switched. For a case of type I phase matching with β-barium borate with a specific pump profile and a 65-mJ input pulse, under the assumption of small absorption, the following are demonstrated: (1) the theoretical possibility of amplification to a few joules at quantum efficiencies higher than 90% and (2) the possibility of amplification to approximately 1 J at an energy efficiency near 45% in a configuration satisfying realistic stress constraints. Pulse widths are in the nanosecond range, and spot sizes are in the millimeter range. Issues of implementation are discussed.

© 1996 Optical Society of America

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  1. J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).
  2. A. Fix, T. Schroder, R. Wallenstein, J. G. Haub, M. J. Johnson, B. J. Orr, “Turnable β-barium borate optical parametric oscillator: operating characteristics with and without injection seeding,” J. Opt. Soc. Am. B 10, 1744–1750 (1993).
  3. M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE J. Quantum Electron. 28, 739–749 (1992).
  4. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
  5. P. P. Bey, C. L. Tang, “Plane-wave theory of a parametric oscillator and a coupled oscillator-upconverter,” IEEE J. Quantum Electron. 8, 361–369 (1972).
  6. R. L. Byer, “Parametric oscillators-linear materials,” in Non-linear Optics, P. G. Harper, B. S. Wherrett, eds. (Academic, New York, 1977), pp. 91–160.
  7. R. A. Baumgartner, R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).
  8. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, New York, 1990).
  9. S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990).
  10. R. A. Holmgren, A First Course in Dynamical Systems (Springer-Verlag, New York, 1994).
  11. D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
  12. V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).
  13. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).
  14. Y. L. Luke, Mathematical Functions and their Approximations (Academic, New York, 1975), pp. 376–379.
  15. H. Nakatani, W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Laser-induced damage in beta-barium metaborate,” Appl. Phys. Lett. 53, 2587–2589 (1988).
  16. P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).
  17. B. R. Suydam, “Self-focusing of very powerful laser beams,” in Laser-Induced Damage in Materials, A. J. Glass, A. H. Guenther, eds., Natl. Bur. Stand. (U.S.) Special Pub.387, 42–48 (1973).
  18. S. J. Brosnan, R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. 15, 415–431 (1979).
  19. S. T. Yang, R. C. Eckardt, R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).

1994 (1)

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).

1993 (2)

1992 (2)

M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE J. Quantum Electron. 28, 739–749 (1992).

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).

1988 (1)

H. Nakatani, W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Laser-induced damage in beta-barium metaborate,” Appl. Phys. Lett. 53, 2587–2589 (1988).

1987 (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).

1979 (2)

S. J. Brosnan, R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. 15, 415–431 (1979).

R. A. Baumgartner, R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).

1972 (1)

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

1962 (1)

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

Armstrong, J. A.

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

Baumgartner, R. A.

R. A. Baumgartner, R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).

Bey, P. P.

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

Bloembergen, N.

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

Bosenberg, W. R.

H. Nakatani, W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Laser-induced damage in beta-barium metaborate,” Appl. Phys. Lett. 53, 2587–2589 (1988).

Brosnan, S. J.

S. J. Brosnan, R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. 15, 415–431 (1979).

Byer, R. L.

S. T. Yang, R. C. Eckardt, R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).

S. J. Brosnan, R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. 15, 415–431 (1979).

R. A. Baumgartner, R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).

R. L. Byer, “Parametric oscillators-linear materials,” in Non-linear Optics, P. G. Harper, B. S. Wherrett, eds. (Academic, New York, 1977), pp. 91–160.

Cheng, L. K.

H. Nakatani, W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Laser-induced damage in beta-barium metaborate,” Appl. Phys. Lett. 53, 2587–2589 (1988).

Cui, Y.

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

Ducuing, J.

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

Dunn, M. H.

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).

Eckardt, R. C.

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).

Fix, A.

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

Haub, J. G.

Holmgren, R. A.

R. A. Holmgren, A First Course in Dynamical Systems (Springer-Verlag, New York, 1994).

Johnson, M. J.

Luke, Y. L.

Y. L. Luke, Mathematical Functions and their Approximations (Academic, New York, 1975), pp. 376–379.

Malyak, P. H.

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).

Milton, M. J. T.

M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE J. Quantum Electron. 28, 739–749 (1992).

Nakatani, H.

H. Nakatani, W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Laser-induced damage in beta-barium metaborate,” Appl. Phys. Lett. 53, 2587–2589 (1988).

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

Orr, B. J.

Pershan, P. S.

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

Rasband, S. N.

S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990).

Schroder, T.

Sibbett, W.

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).

Suydam, B. R.

B. R. Suydam, “Self-focusing of very powerful laser beams,” in Laser-Induced Damage in Materials, A. J. Glass, A. H. Guenther, eds., Natl. Bur. Stand. (U.S.) Special Pub.387, 42–48 (1973).

Tang, C. L.

H. Nakatani, W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Laser-induced damage in beta-barium metaborate,” Appl. Phys. Lett. 53, 2587–2589 (1988).

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

Terry, J. A. C.

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).

Velsko, S.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).

Verhulst, F.

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, New York, 1990).

Wallenstein, R.

Yang, S. T.

Yang, Y.

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).

Zalkin, A.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).

Appl. Opt. (1)

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).

Appl. Phys. Lett. (1)

H. Nakatani, W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Laser-induced damage in beta-barium metaborate,” Appl. Phys. Lett. 53, 2587–2589 (1988).

IEEE J. Quantum Electron. (1)

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

IEEE J. Quantum Electron. (3)

M. J. T. Milton, “General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves,” IEEE J. Quantum Electron. 28, 739–749 (1992).

R. A. Baumgartner, R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).

S. J. Brosnan, R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. 15, 415–431 (1979).

J. Appl. Phys. (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).

J. Opt. Soc. Am. B (1)

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, M. H. Dunn, “Low-threshold opration of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).

J. Opt. Soc. Am. B (2)

Phys. Rev. (1)

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

Other (8)

R. L. Byer, “Parametric oscillators-linear materials,” in Non-linear Optics, P. G. Harper, B. S. Wherrett, eds. (Academic, New York, 1977), pp. 91–160.

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, New York, 1990).

S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990).

R. A. Holmgren, A First Course in Dynamical Systems (Springer-Verlag, New York, 1994).

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).

Y. L. Luke, Mathematical Functions and their Approximations (Academic, New York, 1975), pp. 376–379.

B. R. Suydam, “Self-focusing of very powerful laser beams,” in Laser-Induced Damage in Materials, A. J. Glass, A. H. Guenther, eds., Natl. Bur. Stand. (U.S.) Special Pub.387, 42–48 (1973).

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Figures (6)

Fig. 1
Fig. 1

Graphs of the mapping G E p for the OPO parameters given in Section 4. The graphs are for energy-equivalent pump intensities E p = 50, 100, 140, 150, 200, 250 mJ. Each application of G E p represents the evolution of the intracavity signal intensity over a signal round trip. G E p takes ρ values to ρ values, where ρ is the signal-to-pump ratio of photon fluxes at the beginning of a round trip. For E p < 140 mJ, G E p has one fixed point f s1, which is asymptotically stable, and orbits of points in the interval 0 < ρ ≤ 2 converge to f s1. At a value E pB of E p between 140 and 150 mJ, the system bifurcates: for E p slightly above E pB , G E p has a pair of fixed points f u and f s2 that are considerably greater than f s1. As the parameter E p increases still further, f u and f s2 decrease. The graphs for all E p are contained in a region bounded by the two lines having slope 0.6. The point P where the higher of these lines intersects the diagonal gives the bound on the steady-state values of ρ that corresponds to 100% conversion efficiency in the Bey–Tang model.

Fig. 2
Fig. 2

Mapping G = G 250 mJ in the parameterized family G E p of mappings referred to in the caption of Fig. 1. The sequence G n (x 0), which is called the orbit of x 0, is shown for x 0 = 0.60, 0.61. These orbits have limit points f s1 and f s2, respectively. The orbit is unstable with respect to the initial point x 0 because the unstable fixed point f u is between 0.60 and 0.61, and f u separates the domains of attraction of the stable fixed points f s1 and f s2.

Fig. 3
Fig. 3

Schematic diagram of the OPOA cavity for the specific oscillator–amplifier design described in Section 5. A signal round trip involves the signal's passing through two pairs of walk-off-compensated BBO crystals (represented by the rectangles on the left side of the illustration), and a change in direction between them is mediated by several reflections from the high-reflectivity (HR) mirrors (represented by the line segments at the turns of the pump-path beam and the signal-path beam). With this arrangement, the incidence of the signal at each HR mirror is at an angle that at least doubles the area of incidence of the signal beam on the HR mirror over the area for normal incidence and so reduces fluence there by at least one-half. This geometry helps to reduce that fluence to a level below or closer to the damage threshold of each HR mirror in the paths of the signals. Because the input coupler (IC) doubles as the output coupler (OC), it has a higher damage threshold that do the HR mirrors. The pump is single passed along a path that goes through the two pairs of crystals but that makes a wider loop than does the signal beam. The pump beam goes through beam-expansion optics and is reflected from its own HR mirrors, so that the fluence that is incident on the pump beam's HR mirrors is reduced by both the beam-expansion and the incidence angles. The performance analysis for this design permits flexibility in the size of the pump-beam loop. Inside the cavity, the resonated signal does not go through optics, except possibly for some that make minor adjustments to compensate for beam divergence.

Fig. 4
Fig. 4

Time profile of energy equivalents E p i of the pump intensities of the specially shaped pump pulse in the example that is given in Section 5 and defined in Table 1. These are intensities at the input face of the crystal. The intensity is a step function, and it is constant within each of the 15 individual segments J i of the pulse and has the same values on J 14 and J 15; between these segments, the OPOA cavity is Q switched. The intensities were chosen to be constant within each individual signal round trip so that the theory of discrete dynamical systems could be applied. However, if the generated signal intensity is not too sensitive to changes in the pump intensity, then the piecewise constant profile can be replaced by a sawtooth profile on segments J 1J 13.

Fig. 5
Fig. 5

Mappings for G i = G E pJi that represent the evolution if the intracavity signal intensities over the segments J i , where i = 1, … , 10, of the duration of the specially shaped pump pulse shown in Fig. 4. The mappings are shown together with the graphical-analysis staircase that shows the development of the signal intensity for the particular case of a 65.5-mJ input signal. The graph of each G E p is labeled by E p , which is expressed as a multiple of 100 mJ. The corners of the staircase alternate between the diagonal and the graphs of the G i , and each round trip is represented by a point on the graph of the mapping G i that represents the development of the signal during that round trip. The round-trip signal power-loss coefficient R = 0.80.

Fig. 6
Fig. 6

Mappings G i = G E p J i that represent the evolution of the intracavity signal intensities over segments J i : (a) Mappings of the evolution of the intracavity signal intensities over segments J i for i = 11, …, 14 of the duration of the specially shaped pump pulse shown in Fig. 4 are shown together with the graphical-analysis staircase, which shows the development of the signal intensity for the particular case of a 65.5-mJ input signal. The graph of each G E p is labeled by E p , which is expressed as a multiple of 100 mJ. The same information for segments J 1J 10 is shown in Fig. 5. The corners of the staircase alternate between the diagonal and the graphs of the G i , and each signal round trip is represented by a point on the graph of the mapping G i that represents the development of the signal during that round trip. The orbit represented by the staircase comes near a period-2 limit cycle of the mapping G 14 , which consists of points p 1 and p 2. Points on the graph of G 14 that represent this limit cycle are shown. The round-trip signal power-loss coefficient R during this part of the pulse is 0.80. (b) Mappings of G 14 and G 15 that represent the evolution of the signal intensity during segments J 14 and J 15. The points presenting the period-2 limit cycle are shown again. Going from J 14 to J 15, the pump intensity stays the same, but the round-trip signal power-loss coefficient R is switched from 0.80 to 0.65, with the result that the energy equivalent of the signal intensity, ρE p15, is near the stable fixed point f s of G 15 at the beginning of J 15.

Tables (1)

Tables Icon

Table 1 Time Profile of the Pump Pulse and the Loss Coefficient R from Section 5 a

Equations (11)

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

u p 2 ( ξ ) = u p 2 ( 0 ) sn 2 ( U , γ 2 ) , u s 2 ( ξ ) = u s 2 ( 0 ) + u p 2 ( 0 ) [ 1 sn 2 ( U , γ 2 ) ] ,
γ 2 = u p 2 ( 0 ) u p 2 ( 0 ) + u s 2 ( 0 ) , ξ 0 = K ( γ 2 ) / [ u p 2 ( 0 ) + u s 2 ( 0 ) ] 1 / 2 ,
l = C × crystal length ,
I s [ n + 1 ] = G ( I s [ n ] ) , G  independent of  n .
G ( ρ ) = R { ρ + 1 sn 2 [ u p ( 0 ) ( l ξ 0 ) / γ , γ 2 ] } ,
E p J = I p J ( 7  ns ) ( pulse cross-section area ) ,
G i ( ρ E p J i ) = G i ( ρ ) E p J i , i = 1 , 2 ,
E s J = ( ρ E p J ) ( λ p / λ s ) ( width of  J / 7 ns ) .
η = 1 R 2 R 1 R 2 ρ ,
E s 0 = ρ 0 E p J 1 ( 25 / 7 ) ( λ p / λ s ) ,
| d G 14 / d x ( p 2 ) | × | d G 14 / d x ( p 1 ) | = | d ( G 14 G 14 ) / d x ( p 1 ) | .

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