Abstract

A comprehensive physical model of adiabatic three-wave mixing is developed for the fully nonlinear regime, i.e., without making the undepleted pump approximation. The conditions for adiabatic evolution are rigorously derived, together with an estimate of the bandwidth of the process. Furthermore, these processes are shown to be robust and efficient. Finally, numerical simulations demonstrate adiabatic frequency conversion in a wide variety of physically attainable configurations.

© 2013 Optical Society of America

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    [CrossRef]
  4. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
    [CrossRef]
  5. H. Suchowski, V. Prabhudesai, D. Oron, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
    [CrossRef]
  6. H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
    [CrossRef]
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  8. G. Porat, H. Suchowski, Y. Silberberg, and A. Arie, “Tunable upconverted optical parametric oscillator with intracavity adiabatic sum-frequency generation,” Opt. Lett. 35, 1590–1592 (2010).
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  9. G. Porat, Y. Silberberg, A. Arie, and H. Suchowski, “Two photon frequency conversion,” Opt. Express 20, 3613–3619 (2012).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. G. G. Luther, M. S. Alber, J. E. Marsden, and J. M. Robbins, “Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media,” J. Opt. Soc. Am. B 17, 932–941 (2000).
    [CrossRef]
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  20. S. Longhi, “Third-harmonic generation in quasi-phase-matched χ(2) media with missing second harmonic,” Opt. Lett. 32, 1791–1793 (2007).
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    [CrossRef]
  24. S. Meng, L. Fu, and J. Liu, “Adiabatic fidelity for atom-molecule conversion in a nonlinear three-level system,” Phys. Rev. A 78, 053410 (2008).
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  25. X. Zhou, Y. Zhang, Z. Zhou, and G. Guo, “Adiabatic evolution in nonlinear systems with degeneracy,” Phys. Rev. A 81, 043614 (2010).
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    [CrossRef]
  33. D. N. Nikogosyan, Nonlinear Optical Crystals (Springer, 2005).
  34. Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
    [CrossRef]
  35. S. Saltiel and Y. Deyanova, “Polarization switching as a result of cascading of two simultaneously phase-matched quadratic processes,” Opt. Lett. 24, 1296–1298 (1999).
    [CrossRef]
  36. A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
    [CrossRef]

2013 (1)

2012 (5)

2011 (1)

H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
[CrossRef]

2010 (4)

X. Zhou, Y. Zhang, Z. Zhou, and G. Guo, “Adiabatic evolution in nonlinear systems with degeneracy,” Phys. Rev. A 81, 043614 (2010).
[CrossRef]

Sh. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Phys. Rev. A 82, 013812 (2010).
[CrossRef]

G. Porat, H. Suchowski, Y. Silberberg, and A. Arie, “Tunable upconverted optical parametric oscillator with intracavity adiabatic sum-frequency generation,” Opt. Lett. 35, 1590–1592 (2010).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. 35, 3093–3095 (2010).
[CrossRef]

2009 (2)

H. Suchowski, V. Prabhudesai, D. Oron, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
[CrossRef]

2008 (3)

S. Meng, L. Fu, and J. Liu, “Adiabatic fidelity for atom-molecule conversion in a nonlinear three-level system,” Phys. Rev. A 78, 053410 (2008).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[CrossRef]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

2007 (3)

S. Longhi, “Third-harmonic generation in quasi-phase-matched χ(2) media with missing second harmonic,” Opt. Lett. 32, 1791–1793 (2007).
[CrossRef]

D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8, 180–198 (2007).
[CrossRef]

H. Pu, P. Maenner, W. Zhang, and H. Y. Ling, “Adiabatic condition for nonlinear systems,” Phys. Rev. Lett. 98, 050406 (2007).
[CrossRef]

2003 (1)

J. Liu, B. Wu, and Q. Niu, “Nonlinear evolution of quantum states in the adiabatic regime,” Phys. Rev. Lett. 90, 170404 (2003).
[CrossRef]

2000 (1)

1999 (1)

1997 (1)

1995 (1)

N. B. Baranova, M. A. Bolshtyanskiĭ, and B. Ya Zel’dovich, “Adiabatic energy transfer from a pump wave to its second harmonic,” Quantum Electron. 25, 638–640 (1995).
[CrossRef]

1993 (2)

1992 (1)

1991 (1)

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

1973 (1)

M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8, 2128–2135 (1973).
[CrossRef]

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]

1957 (1)

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[CrossRef]

Alber, M. S.

Amiranashvili, Sh.

Sh. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Phys. Rev. A 82, 013812 (2010).
[CrossRef]

Arie, A.

G. Porat, Y. Silberberg, A. Arie, and H. Suchowski, “Two photon frequency conversion,” Opt. Express 20, 3613–3619 (2012).
[CrossRef]

G. Porat and A. Arie, “Efficient two-process frequency conversion through a dark intermediate state,” J. Opt. Soc. Am. B 29, 2901–2909 (2012).
[CrossRef]

H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
[CrossRef]

G. Porat, H. Suchowski, Y. Silberberg, and A. Arie, “Tunable upconverted optical parametric oscillator with intracavity adiabatic sum-frequency generation,” Opt. Lett. 35, 1590–1592 (2010).
[CrossRef]

A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[CrossRef]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

G. Porat and A. Arie, “Efficient broadband frequency conversion via simultaneous three wave mixing processes,” Appl. Phys. Lett.102 (to be published).

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]

Arnol’d, V. I.

V. I. Arnol’d, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1978).

Bahabad, A.

A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
[CrossRef]

Baranova, N. B.

N. B. Baranova, M. A. Bolshtyanskiĭ, and B. Ya Zel’dovich, “Adiabatic energy transfer from a pump wave to its second harmonic,” Quantum Electron. 25, 638–640 (1995).
[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]

Bolshtyanskii, M. A.

N. B. Baranova, M. A. Bolshtyanskiĭ, and B. Ya Zel’dovich, “Adiabatic energy transfer from a pump wave to its second harmonic,” Quantum Electron. 25, 638–640 (1995).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Bruner, B. D.

H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
[CrossRef]

Cao, X. D.

Cappellini, G.

Caspani, L.

Chisari, R.

Clerici, M.

Crisp, M. D.

M. D. Crisp, “Adiabatic-following approximation,” Phys. Rev. A 8, 2128–2135 (1973).
[CrossRef]

Demircan, A.

Sh. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Phys. Rev. A 82, 013812 (2010).
[CrossRef]

Deyanova, Y.

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.

Feynman, R. P.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[CrossRef]

Fu, L.

S. Meng, L. Fu, and J. Liu, “Adiabatic fidelity for atom-molecule conversion in a nonlinear three-level system,” Phys. Rev. A 78, 053410 (2008).
[CrossRef]

Furukawa, Y.

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

Galun, E.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Ganany-Padowicz, A.

H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
[CrossRef]

A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
[CrossRef]

Gayer, O.

A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
[CrossRef]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Guo, G.

X. Zhou, Y. Zhang, Z. Zhou, and G. Guo, “Adiabatic evolution in nonlinear systems with degeneracy,” Phys. Rev. A 81, 043614 (2010).
[CrossRef]

Hellwarth, R. W.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[CrossRef]

Hum, D. S.

D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Phys. 8, 180–198 (2007).
[CrossRef]

Ito, R.

Juwiler, I.

H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
[CrossRef]

A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
[CrossRef]

Kaplan, A. E.

Kärtner, F. X.

Kitamoto, A.

Kondo, T.

Ling, H. Y.

H. Pu, P. Maenner, W. Zhang, and H. Y. Ling, “Adiabatic condition for nonlinear systems,” Phys. Rev. Lett. 98, 050406 (2007).
[CrossRef]

Liu, J.

S. Meng, L. Fu, and J. Liu, “Adiabatic fidelity for atom-molecule conversion in a nonlinear three-level system,” Phys. Rev. A 78, 053410 (2008).
[CrossRef]

J. Liu, B. Wu, and Q. Niu, “Nonlinear evolution of quantum states in the adiabatic regime,” Phys. Rev. Lett. 90, 170404 (2003).
[CrossRef]

Longhi, S.

Luther, G. G.

Maenner, P.

H. Pu, P. Maenner, W. Zhang, and H. Y. Ling, “Adiabatic condition for nonlinear systems,” Phys. Rev. Lett. 98, 050406 (2007).
[CrossRef]

Marsden, J. E.

McKinstrie, C. J.

Meng, S.

S. Meng, L. Fu, and J. Liu, “Adiabatic fidelity for atom-molecule conversion in a nonlinear three-level system,” Phys. Rev. A 78, 053410 (2008).
[CrossRef]

Messiah, A.

A. Messiah, Quantum Mechanics (North Holland, 2005).

Morandotti, R.

Moses, J.

Nikogosyan, D. N.

D. N. Nikogosyan, Nonlinear Optical Crystals (Springer, 2005).

Nitanda, F.

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

Niu, Q.

J. Liu, B. Wu, and Q. Niu, “Nonlinear evolution of quantum states in the adiabatic regime,” Phys. Rev. Lett. 90, 170404 (2003).
[CrossRef]

Oron, D.

H. Suchowski, V. Prabhudesai, D. Oron, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[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]

Phillips, C. R.

Porat, G.

Prabhudesai, V.

Pu, H.

H. Pu, P. Maenner, W. Zhang, and H. Y. Ling, “Adiabatic condition for nonlinear systems,” Phys. Rev. Lett. 98, 050406 (2007).
[CrossRef]

Rangelov, A. A.

A. A. Rangelov and N. V. Vitanov, “Broadband sum-frequency generation using cascaded processes via chirped quasi-phase-matching,” Phys. Rev. A 85, 045804 (2012).
[CrossRef]

Robbins, J. M.

Sacks, Z.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Saltiel, S.

Sasaki, T.

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

Sato, M.

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

Shirane, M.

Shoji, I.

Silberberg, Y.

G. Porat, Y. Silberberg, A. Arie, and H. Suchowski, “Two photon frequency conversion,” Opt. Express 20, 3613–3619 (2012).
[CrossRef]

H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
[CrossRef]

G. Porat, H. Suchowski, Y. Silberberg, and A. Arie, “Tunable upconverted optical parametric oscillator with intracavity adiabatic sum-frequency generation,” Opt. Lett. 35, 1590–1592 (2010).
[CrossRef]

H. Suchowski, V. Prabhudesai, D. Oron, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[CrossRef]

Suchowski, H.

Trillo, S.

Vernon, F. L.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[CrossRef]

Vidal, F.

Vitanov, N. V.

A. A. Rangelov and N. V. Vitanov, “Broadband sum-frequency generation using cascaded processes via chirped quasi-phase-matching,” Phys. Rev. A 85, 045804 (2012).
[CrossRef]

Wabnitz, S.

Wu, B.

J. Liu, B. Wu, and Q. Niu, “Nonlinear evolution of quantum states in the adiabatic regime,” Phys. Rev. Lett. 90, 170404 (2003).
[CrossRef]

Ya Zel’dovich, B.

N. B. Baranova, M. A. Bolshtyanskiĭ, and B. Ya Zel’dovich, “Adiabatic energy transfer from a pump wave to its second harmonic,” Quantum Electron. 25, 638–640 (1995).
[CrossRef]

Yaakobi, O.

Yokotani, A.

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

Yoshida, H.

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

Yoshida, K.

Y. Furukawa, A. Yokotani, T. Sasaki, H. Yoshida, K. Yoshida, F. Nitanda, and M. Sato, “Investigation of bulk laser damage threshold of lithium niobate single crystals by Q-switched pulse laser,” J. Appl. Phys. 69, 3372–3374 (1991).
[CrossRef]

Zhang, W.

H. Pu, P. Maenner, W. Zhang, and H. Y. Ling, “Adiabatic condition for nonlinear systems,” Phys. Rev. Lett. 98, 050406 (2007).
[CrossRef]

Zhang, Y.

X. Zhou, Y. Zhang, Z. Zhou, and G. Guo, “Adiabatic evolution in nonlinear systems with degeneracy,” Phys. Rev. A 81, 043614 (2010).
[CrossRef]

Zhou, X.

X. Zhou, Y. Zhang, Z. Zhou, and G. Guo, “Adiabatic evolution in nonlinear systems with degeneracy,” Phys. Rev. A 81, 043614 (2010).
[CrossRef]

Zhou, Z.

X. Zhou, Y. Zhang, Z. Zhou, and G. Guo, “Adiabatic evolution in nonlinear systems with degeneracy,” Phys. Rev. A 81, 043614 (2010).
[CrossRef]

Appl. Phys. B (2)

H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B 105, 697–702 (2011).
[CrossRef]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

A. Ganany-Padowicz, I. Juwiler, O. Gayer, A. Bahabad, and A. Arie, “All-optical polarization switch in a quadratic nonlinear photonic quasicrystal,” Appl. Phys. Lett. 94, 091108 (2009).
[CrossRef]

C. R. Phys. (1)

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Figures (12)

Fig. 1.
Fig. 1.

Normalized photon flux of each wave of the two stationary states with |q1|2=|q2|2. (a) Minus state and (b) plus state.

Fig. 2.
Fig. 2.

Normalized photon flux of each wave of the two stationary states with K2/(K1+K3)=0.3. (a) Minus state and (b) plus state.

Fig. 3.
Fig. 3.

Phase space portrait with normalized phase-mismatch ΔΓ=0.6P3 and (a) P2=0, (b) P2=0.3P3. Arrows indicate motion of fixed points with increasing ΔΓ.

Fig. 4.
Fig. 4.

Normalized reduced phase-space canonical momentum for the two stationary states with (a) P2=0, (b) P2=0.3P3.

Fig. 5.
Fig. 5.

Numerical solutions of Eq. (1) with |q1|2=|q2|2. ΔΓ is linearly chirped from 10P3 to 10P3. The system always starts in the minus stationary state. The normalized interaction length is (a)–(c) ΔξP3=1, (d)–(f) ΔξP3=10, (g)–(i) ΔξP3=100. The dashed curves correspond to the minus stationary state calculated using Eq. (22). The nonlinear adiabatic condition rnl in (c), (f), and (i) was calculated using Eq. (26). Only the bottom row, in which rnl1, satisfies the adiabatic condition.

Fig. 6.
Fig. 6.

Numerical solution of Eq. (1) with the same parameters as in Fig. 5(g), except that P2=0.3P3. The dashed curves correspond to the minus stationary state.

Fig. 7.
Fig. 7.

Numerical solution of Eq. (1) with the same parameters as in Fig. 5(g), but assuming a strong pump at ω2, a weak signal at ω1 and no input energy at ω3, i.e., the approximate linear dynamics regime. The dashed curves correspond to the minus stationary state. The inset of (a) shows |q2|2/P3 and has the same horizontal axis.

Fig. 8.
Fig. 8.

Numerically calculated conversion efficiency with (a) P2=0 and (b) P2=0.3P3, and all other parameters the same as in Fig. 5(g), for various values of the normalized phase-mismatch at the center of the interaction medium, ΔΓ(0)/P3. The chirp rate and interaction length were kept constant. The dashed lines indicate the values of ΔΓ(0)/P3 where the estimation yields η=1/2.

Fig. 9.
Fig. 9.

SFG simulation results: (a) Intensities of the three waves along the crystal for input wavelength λ1=1550nm and input intensity 400MW/cm2. (b) Conversion efficiency versus λ1 for different input intensities, which are indicated in units of MW/cm2.

Fig. 10.
Fig. 10.

SHG simulation results: (a) Intensities of the two waves along the crystal for input wavelength λ1=1550nm and input intensity 200MW/cm2. (b) Conversion efficiency versus λ1 for different input intensities, which are indicated in units of MW/cm2.

Fig. 11.
Fig. 11.

DFG simulation results: (a) Intensities of the three waves along the crystal for input wavelength λ2=1550nm and input intensity 200MW/cm2. (b) Conversion efficiency versus λ2 for different input intensities, which are indicated in units of MW/cm2.

Fig. 12.
Fig. 12.

OPA simulation results: (a) Intensities of the three waves along the crystal for input wavelength λ2=1550nm and input intensity 400MW/cm2. (b) Conversion efficiency versus λ2 for different input intensities, which are indicated in units of MW/cm2.

Equations (35)

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dA1dz=iγ1A2*A3exp(i0zΔk(z)dz)dA2dz=iγ2A1*A3exp(i0zΔk(z)dz)dA3dz=iγ3A1A2exp(i0zΔk(z)dz),
dq1dξ=iΔΓq1iq2*q3dq2dξ=iΔΓq2iq1*q3dq3dξ=iΔΓq3iq1q2,
dqjdξ=2iHqj*,
H=12(q1*q2*q3+q1q2q3*)ΔΓ2j=13|qj|2
{qi,qj}=0,{qi*,qj*}=0,{qi,qj*}=2iδij.
(q1,q2,q3)(q1exp(iθ1),q2,q3exp(iθ1)),
(q1,q2,q3)(q1exp(iθ2),q2exp(iθ2),q3),
(q1,q2,q3)(q1,q2exp(iθ3),q3exp(iθ3)),
K1=|q1|2+|q3|2K2=|q1|2|q2|2K3=|q2|2+|q3|2,
(q1,q2,q3)(q1exp(iθ1ξ),q2exp(iθ2ξ),q3exp[i(θ1+θ2)ξ]),
dqjdξ=iθjqj,j=1,2dq3dξ=i(θ1+θ2)q3,
q1+=q2+={(ΔΓθ+)(ΔΓ2θ+)exp(iθ+ξ),ΔΓ>2P30,ΔΓ<2P3q3+={(ΔΓθ+)exp(2iθ+ξ),ΔΓ>2P3P32·exp(iΔΓξ),ΔΓ<2P3
q1=q2={(ΔΓθ)(ΔΓ2θ)exp(iθξ),ΔΓ<2P30,ΔΓ>2P3q3={(ΔΓθ)exp(2iθξ),ΔΓ<2P3P32·exp(iΔΓξ),ΔΓ>2P3,
θ±=5ΔΓ±ΔΓ2+6P36P3K1+K3.
Q1=18arg(q1)18arg(q2)+18arg(q3)Q2=14arg(q1)+14arg(q2)Q3=18arg(q1)18arg(q2)18arg(q3),
P1=|q1|2+|q2|22|q3|2P2=K2=|q1|2|q2|2P3=K1+K3=|q1|2+|q2|2+2|q3|2.
dQjdξ=HPj,dPjdξ=HQj
{Qi,Qj}=0,{Pi,Pj}=0,{Qi,Pj}=δij
H=18(P1+2P2+P3)(P12P2+P3)(P1+P3)cos(8Q1)ΔΓ8(P1+3P3).
dQ1dξ|(Q1±,P1±)=HP1|(Q1±,P1±)=0dP1dξ|(Q1±,P1±)=HQ1|(Q1±,P1±)=0.
Q1=0,Q1+=π8.
P1±=2|(ΔΓθ±)(ΔΓ2θ±)|2(ΔΓθ±)2,
P2±=0P3±=2|(ΔΓθ±)(ΔΓ2θ±)|+2(ΔΓθ±)2.
δP11νdP1±dξsin(νξ),
rnl|[12(|q1|2+|q2|2)|q3|2][12(|q1±|2+|q2±|2)|q3±|2]12(|q1|2+|q2|2)+|q3|2|=|δP1P3|1.
|d(P1±/P3)dξ|=|d(P1±/P3)dΔΓdΔΓdξ|ν,
2271P3|dΔΓdξ|1.
ηP32(|P2|P3)(P1P31).
ΔΓBW=ΔΓ(Δξ/2)ΔΓ(Δξ/2).
H(Qj,Pj)H(Qj±,Pj±)+122HQ12|(Qj±,Pj±)δQ12+122HP12|(Qj±,Pj±)δP12,
ddξ[δP1δQ1]=[02HQ12|Q1±,P1±2HP12|Q1±,P1±0][δP1δQ1][dP1±dξdQ1±dξ].
δP1=0ξcos[ν(ξξ)]dP1±dξdξ,
ν=2HQ12|Q1±,P1±2HP12|Q1±,P1±
δP11νdP1±dξsin(νξ),
ΔΓBW=ΔΓ(Δξ/2)ΔΓ(Δξ/2).

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