Abstract

A nonlinear optical effect in which a linearly polarized wave propagating in a single quadratic medium is converted into a wave that is cross polarized to the input wave is investigated theoretically and observed experimentally in β-barium borate crystal. It is proved that this effect is a result of cascading of two different second-order processes. It starts with the generation of an extraordinary second-harmonic wave by type I interaction and is followed by type II difference-frequency mixing between the second-harmonic wave and the ordinary fundamental wave. The experiment was performed (a) for phase-matched type I interaction and non-phase-matched type II interaction and (b) for non-phase-matched type I interaction and phase-matched type II interaction. The observed generation of a cross-polarized wave is to our knowledge the only cubic effect whose first manifestation has been observed in quadratic crystal.

© 2002 Optical Society of America

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2001

2000

Y. Deyanova, S. Saltiel, and K. Koynov, “Optimization of the process of frequency tripling and quadrupling in double grating quasi-phase matched structures,” Opt. Commun. 178, 437–447 (2000).
[CrossRef]

S. M. Saltiel and Yu. S. Kivshar, “Phase-matching for nonlinear optical parametric processes with multistep-cascading,” Bulg. J. Phys. 27, 57–64 (2000).

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4358 (2000).
[CrossRef] [PubMed]

A. Chowdhury, S. C. Hagness, and L. McCaughan, “Simultaneous optical wavelength interchange with a two-dimensional second-order nonlinear photonic crystal,” Opt. Lett. 25, 832–834 (2000).
[CrossRef]

S. Saltiel, K. Koynov, Y. Deyanova, and Yu. S. Kivshar, “Nonlinear phase shift resulting from two-color multistep cascading,” J. Opt. Soc. Am. B 17, 959–965 (2000).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Quadratic solitons resulting from double-resonance wave mixing,” J. Opt. Soc. Am. B 17, 2018–2025 (2000).
[CrossRef]

V. V. Konotop and V. Kuzmiak, “Double-resonant processes in χ(2) nonlinear periodic media,” J. Opt. Soc. Am. B 17, 1874–1883 (2000).
[CrossRef]

X. Mu, X. Gu, M. V. Makarov, Y. J. Ding, J. Wang, J. Wei, and Y. Liu, “Third-harmonic generation by cascading second-order nonlinear processes in a cerium-doped KTiOPO4 crystal,” Opt. Lett. 25, 117–119 (2000).
[CrossRef]

J. P. Fève, B. Boulanger, and Y. Guillien, “Efficient energy conversion for cubic third-harmonic generation that is phase matched in KTiOPO4,” Opt. Lett. 25, 1373–1375 (2000).
[CrossRef]

Ch. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10, 688–10, 701 (2000).
[CrossRef]

V. V. Konotop and V. Kuzmiak, “Simultaneous second- and third-harmonic generation in one-dimensional photonic crystals,” J. Opt. Soc. Am. 16, 1370–1376 (2000).
[CrossRef]

Y. Takagi and S. Muraki, “Third-harmonic generation in a noncentrosymmetrical crystal: direct third-order or cascaded second-order process?” J. Lumin. 87–89, 865–867 (2000).
[CrossRef]

1999

P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation in beta barium borate through second-order and third-order susceptibilities,” Opt. Lett. 24, 4–6 (1999).
[CrossRef]

Y.-Y. Zhu and N.-B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
[CrossRef]

X. Liu, L. Qian, and F. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascade χ(2)(2) nonlinearity,” Opt. Lett. 24, 1777–1779 (1999).
[CrossRef]

B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. 24, 1410–1412 (1999).
[CrossRef]

Yu. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, “Two-color multistep cascading and parametric soliton-induced waveguides,” Phys. Rev. E 60, R5056–R5059 (1999).
[CrossRef]

Yu. S. Kivshar, T. J. Alexander, and S. Saltiel, “Spatial optical solitons resulting from multistep cascading,” Opt. Lett. 24, 759–761 (1999).
[CrossRef]

I. Towers, R. Sammut, A. V. Buryak, and B. A. Malomed, “Soliton multistability as a result of double-resonance wave mixing in media,” Opt. Lett. 24, 1738–1740 (1999).
[CrossRef]

K. Gallo and G. Assanto, “Analysis of lithium niobate all-optical wavelength shifters for the third spectral window,” J. Opt. Soc. Am. 16, 741–753 (1999).
[CrossRef]

S. Saltiel and Y. Deyanova, “Polarization switching as a result of cascading of two simultaneously phase-matched processes,” Opt. Lett. 24, 1296–1298 (1999).
[CrossRef]

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).
[CrossRef]

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999).
[CrossRef]

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1656 (1999).
[CrossRef]

G. I. Petrov, S. M. Saltiel, and A. B. Ivanova, “Measurement of χ(2) components by comparing polarization resolved second-order cascade processes,” in ICONO’98: Nonlinear Optical Phenomena, S. Chesnokov, V. Kandidov, and N. Koroteev, eds. Proc. Proc. SPIE 3733, 112–120 (1999).
[CrossRef]

I. Shoji, H. Nakamura, K. Ohdaira, T. Kondo, R. Ito, T. Oka-moto, K. Tatsuki, and S. Kubota, “Absolute measurement of second-order nonlinear-optical coefficients of β-BaB2O4 for visible to ultraviolet second-harmonic wavelengths,” J. Opt. Soc. Am. B 16, 620–624 (1999).
[CrossRef]

1998

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[CrossRef]

M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-mm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).
[CrossRef]

A. D. Boardman, P. Bontemps, and K. Xie, “Vector solitary optical beam control with mixed type I–type II second-harmonic generation,” Opt. Quantum Electron. 30, 891–905 (1998).
[CrossRef]

G. P. Banfi, P. K. Datta, V. Degiorgio, G. Donelli, and D. Fortusini, and J. N. Sherwood, “Frequency shifting through cascaded second-order processes in an N-(4-nitrophenyl)-L-prolinol crystal,” Opt. Lett. 23, 439–441 (1998).
[CrossRef]

K. Koynov and S. Saltiel, “Nonlinear phase shift via χ(2) multistep cascading,” Opt. Commun. 152, 96–100 (1998).
[CrossRef]

1997

S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, “Quasi-phase matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

A. D. Boardman and K. Xie, “Vector spatial solitons influenced by magneto-optic effects in cascadable nonlinear media,” Phys. Rev. E 55, 1899–1909 (1997).
[CrossRef]

O. Pfister, J. S. Wells, L. Hollberg, L. Zink, D. A. Van Baak, M. D. Levenson, and W. R. Bosenberg, “Continuous-wave frequency tripling and quadrupling by simultaneous three-wave mixing in periodically poled crystals: application to a two-step 1.19–10.71-μm frequency bridge,” Opt. Lett. 22, 1211–1213 (1997).
[CrossRef] [PubMed]

I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of nonlinear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
[CrossRef]

M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes in periodically poled LiNbO3,” Opt. Lett. 22, 274–276 (1997).
[CrossRef] [PubMed]

M. A. Krumburel, J. N. Sweetser, D. N. Fittinghoff, K. W. DeLong, and R. Trebino, “Ultrafast optical switching by use of fully phase matched cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Lett. 22, 245–247 (1997).
[CrossRef]

J. N. Sweetser, M. A. Krumburel, and R. Trebino, “Amplified ultrafast optical switching by cascading cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Commun. 142, 269–272 (1997).
[CrossRef]

M. Sheik-Bahae and M. Ebrahimzadeh, “Measurement of nonlinear refraction in the second-order χ(2) materials KTiOPO4, KNbO4, β-BaB2O4, and LiB3O5” Opt. Commun. 142, 294–298 (1997).
[CrossRef]

1996

A. Dubietis, G. Valiulis, R. Danielius, and A. Piskarkas, “Fundamental-frequency pulse compression through cascaded second-order processes in a type II phase-matched second-harmonic generator,” Opt. Lett. 21, 1262–1264 (1996).
[CrossRef] [PubMed]

G. Stegeman, D. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking, pulse compression, and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

1995

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

L. Lefort and A. Barthelemy, “Intensity-dependent polarization rotation associated with type II phase-matched second-harmonic generation: application to self-induced transparency,” Opt. Lett. 20, 1749–1751 (1995).
[CrossRef] [PubMed]

L. Lefort and A. Barthelemy, “All-optical transistor action by polarization rotation during type-II phase-matched second harmonic generation,” Electron. Lett. 31, 910–911 (1995).
[CrossRef]

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 on bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
[CrossRef]

1994

1992

1988

P. Qiu and A. Penzkofer, “Picosecond third harmonic light generation in β-BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[CrossRef]

1985

1980

A. I. Kovrigin, D. V. Yakovlev, B. V. Zhdanov, and N. I. Zheludev, “Self-induced optical activity in crystals,” Opt. Commun. 35, 92–95 (1980).
[CrossRef]

Albert, O.

Alexander, T. J.

Angelis, C. De

Arbore, M. A.

Arie, A.

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1656 (1999).
[CrossRef]

Asobe, M.

Assanto, G.

Banfi, G. P.

Bang, O.

Banks, P. S.

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M. A. Krumburel, J. N. Sweetser, D. N. Fittinghoff, K. W. DeLong, and R. Trebino, “Ultrafast optical switching by use of fully phase matched cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Lett. 22, 245–247 (1997).
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J. N. Sweetser, M. A. Krumburel, and R. Trebino, “Amplified ultrafast optical switching by cascading cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Commun. 142, 269–272 (1997).
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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).
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Ch. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10, 688–10, 701 (2000).
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Y.-Y. Zhu and N.-B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
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G. I. Petrov, S. M. Saltiel, and A. B. Ivanova, “Measurement of χ(2) components by comparing polarization resolved second-order cascade processes,” in ICONO’98: Nonlinear Optical Phenomena, S. Chesnokov, V. Kandidov, and N. Koroteev, eds. Proc. Proc. SPIE 3733, 112–120 (1999).
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N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4358 (2000).
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N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4358 (2000).
[CrossRef] [PubMed]

B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. 24, 1410–1412 (1999).
[CrossRef]

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S. Saltiel, K. Koynov, Y. Deyanova, and Yu. S. Kivshar, “Nonlinear phase shift resulting from two-color multistep cascading,” J. Opt. Soc. Am. B 17, 959–965 (2000).
[CrossRef]

Y. Deyanova, S. Saltiel, and K. Koynov, “Optimization of the process of frequency tripling and quadrupling in double grating quasi-phase matched structures,” Opt. Commun. 178, 437–447 (2000).
[CrossRef]

S. Saltiel and Y. Deyanova, “Polarization switching as a result of cascading of two simultaneously phase-matched processes,” Opt. Lett. 24, 1296–1298 (1999).
[CrossRef]

Yu. S. Kivshar, T. J. Alexander, and S. Saltiel, “Spatial optical solitons resulting from multistep cascading,” Opt. Lett. 24, 759–761 (1999).
[CrossRef]

K. Koynov and S. Saltiel, “Nonlinear phase shift via χ(2) multistep cascading,” Opt. Commun. 152, 96–100 (1998).
[CrossRef]

I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of nonlinear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
[CrossRef]

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G. I. Petrov, O. Albert, J. Etchepare, and S. M. Saltiel, “Cross-polarized wave generation by effective cubic nonlinear optical interaction,” Opt. Lett. 26, 355–357 (2001).
[CrossRef]

M. de Sterke, S. M. Saltiel, and Yu. S. Kivshar, “Efficient collinear fourth-harmonic generation by two-channel multistep cascading in a single two-dimensional nonlinear photonic crystal,” Opt. Lett. 26, 539–541 (2001).
[CrossRef]

S. M. Saltiel and Yu. S. Kivshar, “Phase-matching for nonlinear optical parametric processes with multistep-cascading,” Bulg. J. Phys. 27, 57–64 (2000).

G. I. Petrov, S. M. Saltiel, and A. B. Ivanova, “Measurement of χ(2) components by comparing polarization resolved second-order cascade processes,” in ICONO’98: Nonlinear Optical Phenomena, S. Chesnokov, V. Kandidov, and N. Koroteev, eds. Proc. Proc. SPIE 3733, 112–120 (1999).
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M. Sheik-Bahae and M. Ebrahimzadeh, “Measurement of nonlinear refraction in the second-order χ(2) materials KTiOPO4, KNbO4, β-BaB2O4, and LiB3O5” Opt. Commun. 142, 294–298 (1997).
[CrossRef]

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G. Stegeman, D. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking, pulse compression, and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
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R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–31 (1992).
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W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
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Yu. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, “Two-color multistep cascading and parametric soliton-induced waveguides,” Phys. Rev. E 60, R5056–R5059 (1999).
[CrossRef]

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M. A. Krumburel, J. N. Sweetser, D. N. Fittinghoff, K. W. DeLong, and R. Trebino, “Ultrafast optical switching by use of fully phase matched cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Lett. 22, 245–247 (1997).
[CrossRef]

J. N. Sweetser, M. A. Krumburel, and R. Trebino, “Amplified ultrafast optical switching by cascading cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Commun. 142, 269–272 (1997).
[CrossRef]

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Y. Takagi and S. Muraki, “Third-harmonic generation in a noncentrosymmetrical crystal: direct third-order or cascaded second-order process?” J. Lumin. 87–89, 865–867 (2000).
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J. N. Sweetser, M. A. Krumburel, and R. Trebino, “Amplified ultrafast optical switching by cascading cascaded second-order nonlinearities in a polarization-gate geometry,” Opt. Commun. 142, 269–272 (1997).
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S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, “Quasi-phase matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
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Y.-Y. Zhu and N.-B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
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S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, “Quasi-phase matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
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Figures (9)

Fig. 1
Fig. 1

Sketch for generation of a XPW based on cascading of two different second-order processes: NLC, nonlinear crystal.

Fig. 2
Fig. 2

Phase-matched first step. Conversion efficiency for the SH wave (dashed curve), XPW efficiency magnified 5(Δk2L)2 times (solid curve), and depletion of the fundamental intensity (dotted curve) as functions of dimensionless input amplitude parameter σ1A0L. Ratio |σ2/σ1|=0.53.

Fig. 3
Fig. 3

Phase-matched first step. Conversion efficiency for the SH wave (dotted curves) and the XPW (solid curves) as functions of the detuning of the first step from exact phase matching for two values of the normalized input amplitude: (a) σ1A0L=1, (b) σ1A0L=2. The XPW efficiency is magnified 3(Δk2L)2 times. Ratio |σ2/σ1|=0.53.

Fig. 4
Fig. 4

Phase-matched second step. Theoretical predictions for XPW conversion efficiency, ηB=IB/IA (darker curves) and SH signal, ISHG/IA (lighter curves) for deviation Δθ about the PM angle for type II SHG for three input intensities. Ratio |σ2/σ1|=0.53; length of the BBO crystal, L=1.5 mm. The curves for the conversion into SH wave S are divided by 8.

Fig. 5
Fig. 5

Experimental arrangement: NDF, neutral-density filters; L’s, lenses; P, polarizer; RT, three-axis rotational table; A, analyzer; C, color filter; PD’s, photodiodes.

Fig. 6
Fig. 6

Dependence of XPW signal (Ie,out-Ibg) on intensity of the input pump. Solid curve, quadratic fit to the experimental points that are recorded for Io,in<350 GW/cm2.

Fig. 7
Fig. 7

Experimentally measured XPW signal and non-phase-matched SH signal as a function of the deviation Δθ from the phase-matched angle for type II SHG. Input power for these two curves, Io,in300 GW/cm2. The lowest curve, taken with Io,in30 GW/cm2, represents a phase-matched type II SHG signal measured in a separate experiment when the input polarizer was misaligned (both o and e waves enter the BBO crystal).

Fig. 8
Fig. 8

Normalized XPW signal for three different sets of input power and crystal length.

Fig. 9
Fig. 9

Experimentally measured XPW signal and attenuated phase-matched SH signal as functions of deviation Δθ from the phase-matched angle for type I SHG. Input power, Io,in350 GW/cm2.

Tables (3)

Tables Icon

Table 1 Schemes for XPW Generation in a Quadratic Medium

Tables Icon

Table 2 Point Groups of Uniaxial Quadratic Crystals in Which Single-Crystal XPW Generation Is Possible

Tables Icon

Table 3 Parameters of BBO Crystal Used for XPW Generationa

Equations (30)

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dAdz=0,
dSdz=-iσ1A2 exp(iΔk1z),
dBdz=-iσ2SA* exp(-iΔk2z)-iγAAA* exp(-iΔk3z),
deff,I step=2πm e2χ(2):e1Ae1A2,
deff,II step=2πm e2χ(2):e1Ae1B2,
χeff(3)=2πme1Bχ(3):e1Ae1Ae1A.
B=|A|2Aσ1σ2Δk1 [exp(-iΔk2L)-1]Δk2+γtot [exp(-iΔk3L)-1]Δk3,
B(Δk10)=-i σ1σ2Δk2|A|2AL sin(Δk1L/2)(Δk1L/2)×exp(iΔk1L/2-iΔk2L)
B(Δk20)=-i σ1σ2Δk1|A|2AL sin(Δk2L/2)(Δk2L/2)×exp(-iΔk2L/2).
B(Δk30)=-iγtot|A|2AL sin(Δk3L/2)(Δk3L/2)×exp(-iΔk3L/2).
B(Δk1=0, Δk2=0)=-|A|2ALσ1σ22L+iγ.
Az=-iσ1SA* exp(-iΔk1z)-iσ2SB* exp(-iΔk2z),
Sz=-iσ1A2 exp(iΔk1z)-i2σ2AB exp(iΔk2z)
Bz=-iσ2SA* exp(-iΔk2z).
|B/A0|2=σ2A0 tanh(σ1A0L)sech(σ1A0L)Δk22.
2z2+iΔk2 z+2σ22A2B(z)
=-σ1σ2A3 exp(-iΔk3z),
2z2-iΔk2 z+2σ22|A|2S(z)
=-σ1Δk3A2 exp(iΔk1z).
ηB=-σ1σ2A02Q2δkq sinqL2-sinδkL22+cosqL2-cosδkL22,
ηS=σ1A0Δk3Q2TqΔk3 sinqL2+sinδkL22+cosqL2-cosδkL22,
|B2|12 σ12A4Δk12 sin2(2σ2|A0|L).
|S2|4 σ12A04Δk12 sin212Δk1L.
|B2/B1|2=12 Δk2Δk1 σ1σ2 cosh(σ1A0L)tanh(σ1A0L)2.
|B2/B1|2=Δk2Δk1 σ1σ22 exp(2|σ1A0|L).
deff,ooe-d22 sin(3φ)cos θ,
deff,oee=d22 cos(3φ)cos2 θ.
2z2-iΔk2 zS+2iσ2A Bz exp(iΔk2z)
=-σ1Δk3A2 exp(iΔk1z),
2z2+iΔk2 zB=-iσ2A Sz exp(-iΔk2z).

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