Abstract

We analyze frequency conversion and its control among three light waves using a geometric approach that enables the dynamics of the waves to be visualized on a closed surface in three dimensions. It extends the analysis based on the undepleted-pump linearization and provides a simple way to understand the fully nonlinear dynamics. The Poincaré sphere has been used in the same way to visualize polarization dynamics. A geometric understanding of control strategies that enhance energy transfer among interacting waves is introduced, and the quasi-phase-matching strategy that uses microstructured quadratic materials is illustrated in this setting for both type I and II second-harmonic generation and for parametric three-wave interactions.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Yariv, Quantum Electronics (Wiley, New York, 1980).
  2. R. Boyd, Nonlinear Optics (Wiley, New York, 1988).
  3. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, 1992).
  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. K. C. Rustagi, S. C. Mehendale, and S. Menakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. QE-18, 1029 (1982).
    [CrossRef]
  6. C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Rev. A 127, 14 (1988).
  7. S. Trillo, S. Wabnitz, R. Chisari, and G. Cappellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
    [CrossRef] [PubMed]
  8. C. J. McKinstrie and X. D. Cao, “Nonlinear detuning of three-wave interactions,” J. Opt. Soc. Am. B 10, 898–912 (1993).
    [CrossRef]
  9. J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Vol. 17 of Texts in Applied Mathematics, 2nd ed. (Springer-Verlag, New York, 1999).
    [CrossRef]
  10. M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998).
    [CrossRef]
  11. M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, in Proceedings of the Fields Institute Conference in Honour of the 60th Birthday of Vladimir I. Arnol’d, Fields Institute Communications Series, E. Bierstone, B. Khesin, A. Khovanskii, and J. Marsden, eds. (Field Institute, Toronto, Ontario, Canada, 1999).
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  13. D. David, D. D. Holm, and M. V. Tratnik, “Integrable and chaotic polarization dynamics in nonlinear optical beams,” Phys. Lett. A 137, 355–369 (1989).
    [CrossRef]
  14. N. N. Akhmediev and A. Ankiewicz, Solitons (Chapman & Hall, London, 1997).
  15. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  16. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
    [CrossRef]
  17. A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities—an analytical approach,” Opt. Commun. 124, 184–194 (1996).
    [CrossRef]
  18. A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
    [CrossRef] [PubMed]
  19. C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990).
    [CrossRef]

1998

M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998).
[CrossRef]

1996

A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities—an analytical approach,” Opt. Commun. 124, 184–194 (1996).
[CrossRef]

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

1995

1993

1992

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

S. Trillo, S. Wabnitz, R. Chisari, and G. Cappellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
[CrossRef] [PubMed]

1990

1989

D. David, D. D. Holm, and M. V. Tratnik, “Integrable and chaotic polarization dynamics in nonlinear optical beams,” Phys. Lett. A 137, 355–369 (1989).
[CrossRef]

1988

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Rev. A 127, 14 (1988).

1982

K. C. Rustagi, S. C. Mehendale, and S. Menakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. QE-18, 1029 (1982).
[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]

Alber, M. S.

M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998).
[CrossRef]

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]

Batha, S. H.

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.

Byer, R. L.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Cao, X. D.

Cappellini, G.

Chisari, R.

David, D.

D. David, D. D. Holm, and M. V. Tratnik, “Integrable and chaotic polarization dynamics in nonlinear optical beams,” Phys. Lett. A 137, 355–369 (1989).
[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]

Eckardt, R. C.

Fejer, M. M.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Holm, D. D.

D. David, D. D. Holm, and M. V. Tratnik, “Integrable and chaotic polarization dynamics in nonlinear optical beams,” Phys. Lett. A 137, 355–369 (1989).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kobyakov, A.

A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities—an analytical approach,” Opt. Commun. 124, 184–194 (1996).
[CrossRef]

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Lederer, F.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities—an analytical approach,” Opt. Commun. 124, 184–194 (1996).
[CrossRef]

Luther, G. G.

M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998).
[CrossRef]

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Rev. A 127, 14 (1988).

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Marsden, J. E.

M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998).
[CrossRef]

McKinstrie, C. J.

Mehendale, S. C.

K. C. Rustagi, S. C. Mehendale, and S. Menakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. QE-18, 1029 (1982).
[CrossRef]

Menakshi, S.

K. C. Rustagi, S. C. Mehendale, and S. Menakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. QE-18, 1029 (1982).
[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]

Peschel, U.

A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities—an analytical approach,” Opt. Commun. 124, 184–194 (1996).
[CrossRef]

Pierce, J. W.

Robbins, J. M.

M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998).
[CrossRef]

Rustagi, K. C.

K. C. Rustagi, S. C. Mehendale, and S. Menakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. QE-18, 1029 (1982).
[CrossRef]

Tratnik, M. V.

D. David, D. D. Holm, and M. V. Tratnik, “Integrable and chaotic polarization dynamics in nonlinear optical beams,” Phys. Lett. A 137, 355–369 (1989).
[CrossRef]

Trillo, S.

Wabnitz, S.

IEEE J. Quantum Electron.

K. C. Rustagi, S. C. Mehendale, and S. Menakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. QE-18, 1029 (1982).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities—an analytical approach,” Opt. Commun. 124, 184–194 (1996).
[CrossRef]

Opt. Lett.

Phys. Lett. A

D. David, D. D. Holm, and M. V. Tratnik, “Integrable and chaotic polarization dynamics in nonlinear optical beams,” Phys. Lett. A 137, 355–369 (1989).
[CrossRef]

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. A

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Rev. A 127, 14 (1988).

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Physica D

M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998).
[CrossRef]

Other

M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, in Proceedings of the Fields Institute Conference in Honour of the 60th Birthday of Vladimir I. Arnol’d, Fields Institute Communications Series, E. Bierstone, B. Khesin, A. Khovanskii, and J. Marsden, eds. (Field Institute, Toronto, Ontario, Canada, 1999).

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

N. N. Akhmediev and A. Ankiewicz, Solitons (Chapman & Hall, London, 1997).

A. Yariv, Quantum Electronics (Wiley, New York, 1980).

R. Boyd, Nonlinear Optics (Wiley, New York, 1988).

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, 1992).

J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Vol. 17 of Texts in Applied Mathematics, 2nd ed. (Springer-Verlag, New York, 1999).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Orbit of the reduced three-wave equations is constructed by intersection of a three-wave surface with a plane that is one level set of the reduced Hamiltonian. Here p=(-1,-1, 1), s=1, ΔΓ=40, and the Manley–Rowe relations are defined by (q1(0), q2(0), q3(0))=(0.1, 0.6, 1.0).

Fig. 2
Fig. 2

Reduced three-wave phase space for type II second-harmonic generation or parametric frequency conversion on a three-wave surface. Here, p=(-1,-1, 1), s=1, ΔΓ=10, and the Manley–Rowe relations are defined by (q1(0), q2(0), q3(0))=(1.0, 0.5, 2.0).

Fig. 3
Fig. 3

Reduced three-wave phase space for type I second-harmonic generation on a three-wave surface, where p=(-1,-1, 1), s=1, and ΔΓ=2. The Manley–Rowe relations are defined by (q1(0), q2(0), q3(0))=(1.0, 1.0, 2.0), so K2=K3.

Fig. 4
Fig. 4

Composite trajectory with 30 segments of length lc=π/Δk for type II quasi-phase matching or parametric frequency conversion is plotted on a three-wave surface. Here p=(-1, 1, 1), ΔΓ=40, and initially s=1 and (q1(0), q2(0), q3(0))=(0.0, 1.0, 1.0).

Fig. 5
Fig. 5

Composite trajectory with 30 segments of length lc=π/Δk for type I quasi-phase-matching second-harmonic generation. Here p=(1, 1,-1), ΔΓ=40, and initially s=1, and (q1(0), q2(0), q3(0))=(1.0, 1.0, 0.0).

Fig. 6
Fig. 6

Composite trajectory for the zig-zag strategy for type II second-harmonic generation. The phase mismatch is modulated after lc=π/(2Δk) and (q1(0), q2(0), q3(0))=(0.1, 0.6,exp(i * π/5)), so the initial relative phase is near π/5. Here s=1, ΔΓ=10, and the Z coordinate is defined by p=(-1,-1, 1).

Equations (48)

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

dA1dz=-iγ1A2*A3 exp(-iΔkz),
dA2dz=-iγ2A1*A3 exp(-iΔkz),
dA3dz=-iγ3A1A2 exp(iΔkz),
dq1dξ=iΔΓq1-isq2*q3,
dq2dξ=iΔΓq2-isq1*q3,
dq3dξ=iΔΓq3-isq1q2,
dqjdz={qj, H}=-2i Hqj*,
H=s2(q1*q2*q3+q1q2q3*)-ΔΓ2j=13|qj|2,
{qi, qj}=0,
{qi, qj*}=-2iδij.
K1=12(|q1|2-|q2|2),
K2=12(|q2|2+|q3|2),
K3=12(|q1|2+|q3|2),
(q1, q2, q3)(q1 exp(-iθ1), q2 exp(iθ1), q3),
(q1, q2, q3)(q1, q2 exp(-iθ2), q3 exp(-iθ2)),
(q1, q2, q3)(q1 exp(-iθ3), q2, q3 exp(-iθ3)),
X+iY=q1q2q3*,
Z=j=13pj|qj|2,
ϕ=-η(X2+Y2)+κ(Z-Z1)(Z+Z2)(Z-Z3),
κ=1/η2,Z1=2(p3K2+p1K1),
Z2=2(p1K3+p2K2),
Z3=-2(p2K1-p3K3),
dWjdξ={Wj, Hr},j=1, 2, 3,
Hr=sX-ΔΓ2η(Z+2δ),
{X, Y}=-ϕ/Z,
{Z, X}=-2ηY,
{Y, Z}=-2ηX,
dXdξ=-ΔΓY,
dYdξ=ΔΓX+s ϕZ,
dZdξ=-2sηY.
dWdξ=rHr×rϕ,
12dZdξ2+U=E,
U=2η[ϕ(X=0, Y=0, Z)+rZ]+12ΔΓ2Z2,
ξ-ξ0= dZ2[E-U(Z)],
ξ-ξ0=12k dθ1-M sin2(θ),
Z(ξ)=ζ2-(ζ2-ζ3)cn2{[(ζ1-ζ3)/2]1/2(ξ-ξ0)|M},
L= dZ2[E-U(Z)]=K(M)k,
Z=mX-(smHr+2δ),
dXdT=-ΓY,
dYdT=ΓX+s ϕZ,
dZdT=-2sηY.
dX0dT=-ΓY0,
dY0dT=ΓX0,
dZ0dT=0.
dδXdT=-ΓδY,
dδYdT=+ΓδX+s ϕZZ=Z0,
dδZdT=-2sηδY.
L=πk1+121/2M+

Metrics