Experimental and theoretical investigation of generation of a cross-polarized wave by cascading of two different second-order processes

G. I. Petrov; O. Albert; N. Minkovski; J. Etchepare; S. M. Saltiel

doi:10.1364/JOSAB.19.000268

Journal of the Optical Society of America B
Vol. 19,
Issue 2,
pp. 268-279
(2002)
•https://doi.org/10.1364/JOSAB.19.000268

Experimental and theoretical investigation of generation of a cross-polarized wave by cascading of two different second-order processes

G. I. Petrov, O. Albert, N. Minkovski, J. Etchepare, and S. M. Saltiel

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G. I. Petrov, O. Albert, N. Minkovski, J. Etchepare, and S. M. Saltiel, "Experimental and theoretical investigation of generation of a cross-polarized wave by cascading of two different second-order processes," J. Opt. Soc. Am. B 19, 268-279 (2002)

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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Nonlinear optics, devices (190.4360)
- Nonlinear optics, four-wave mixing (190.4380)
- Polarization-selective devices (230.5440)
- Optical processing (200.4740)

About this Article
History
- Original Manuscript: April 12, 2001
- Revised Manuscript: September 13, 2001
- Published: February 1, 2002

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.

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Case Designation	Input Wave	Step I SHG	Step II DFM	Equivalent Cubic Interaction
O1a	o	$o_{1} o_{1} \to e_{2}$	$e_{2} o_{1} \to e_{1}$	$o_{1} o_{1} o_{1} \to e_{1}$
O2	o	$o_{1} o_{1} \to o_{2}$	$o_{2} o_{1} \to e_{1}$	$o_{1} o_{1} o_{1} \to e_{1}$
E1	e	$e_{1} e_{1} \to o_{2}$	$o_{2} e_{1} \to o_{1}$	$e_{1} e_{1} e_{1} \to o_{1}$
E2	e	$e_{1} e_{1} \to e_{2}$	$e_{2} e_{1} \to o_{1}$	$e_{1} e_{1} e_{1} \to o_{1}$

Point Group	Optimal Azimuthal Angle for the Processes $o_{1} o_{1} - e_{2}, e_{2} o_{1} - e_{1};$ $e_{1} e_{1} - o_{2}, o_{2} e_{1} - o_{1};$ $e_{1} e_{1} - e_{2}, e_{2} e_{1} - o_{1}$ a	Optimal Azimuthal Angle for the Process $o_{1} o_{1} - o_{2}, o_{2} o_{1} - e_{1}$
$\bar{4} m 2$	22.5°, 67.5°	Not possible
$3 m$	Depends on ratio $d_{22} / d_{31}$	Depends on ratio $d_{22} / d_{31}$
$\bar{4}$	Depends on ratio $d_{36} / d_{31}$	Not possible
3	Depends on ratios $d_{11} / d_{31}$ and $d_{22} / d_{31}$	Depends on ratio $d_{11} / d_{31}$ and $d_{22} / d_{31}$
32	15°, 45°, 75°	15°, 45°, 75°
$\bar{6}$	Depends on ratio $d_{22} / d_{11}$	Depends on ratio $d_{22} / d_{11}$
$\bar{6} m 2$	15°, 45°, 75°	15°, 45°, 75°

Point group	$3 m$ negative uniaxial crystal
Transparency range	0.189–3.5 µm
Linear absorption coefficient at 532 nm	0.01 cm^-1
$d^{(2)}$ coefficients at 850 nm	$d_{22} = 2.3 pm / V,$ $d_{31} = 0.04 pm / V$
$χ^{(3)}$ coefficient at 850 nm	$χ^{(3)} = 3.3 \times 10^{- 22} m^{2} / V^{2}$
Type I SHG interaction
Length of crystal	1 mm
Effective nonlinearity in phase-matching direction	$d_{eff, ooe} = d_{31} sin θ - d_{22} sin (3 φ) cos θ$
Phase matching and azimuthal angles	$θ_{PM} = 38.83 °,$ $φ = 15 °$
Calculated refractive indices	$n_{o, 620} = n_{e, 310} = 1.6679273$
Calculated spectral width of the phase-matched curve (FWHM)	$Δ λ_{PM} = 1.5 nm$
Calculated internal angular width of the phase-matched curve (FWHM)	$Δ θ_{PM} = 0.12 °$
Walk-off angle	$ρ = 4.62 °$
Aperture length	$L_{a} = d / ρ = 2.46 mm$
Group-velocity delay	$ν_{1 o - 2 e} = 378 fs / mm$
Nonstationary length	$L_{ν, 1 o - 2 e} = τ / ν_{1 o - 2 e} = 0.264 mm$
Type II SHG interaction
Length of crystal	1.5 mm
Effective nonlinearity in phase matching direction	$d_{eff, eoe} = d_{eff, oee} = d_{22} cos (3 φ) {cos}^{2} θ$
Phase matching and azimuthal angles	$θ_{PM} = 58.33 °,$ $φ = 15 °$
Calculated refractive indices	$n_{e, 610} = 1.58006,$ $n_{e, 310} = 1.62399$
Calculated spectral width of the phase-matched curve (FWHM)	$Δ λ_{PM} = 1.1 nm$
Calculated internal angular width of the phase-matched curve (FWHM)	$Δ θ_{PM} = 0.16 °$
Walk-off angles	$ρ_{1 o - 1 e} = 3.62 °,$ $ρ_{1 o - 2 e} = 4.01 °$
Aperture lengths	$L_{a, 1 o - 1 e} = 3.2 mm,$ $L_{a, 1 o - 2 e} = 2.9 mm$
Group-velocity delays	$ν_{1 o - 2 e} = 172 fs / mm,$ $ν_{1 e - 2 e} = 488 fs / mm,$ $ν_{1 o - 1 e} = 316 fs / mm$
Nonstationary lengths	$L_{ν, 1 o - 2 e} = 0.58 mm,$ $L_{ν, 1 e - 2 e} = 0.21 mm,$ $L_{ν, 1 o - 1 e} = 0.32 mm$

Case Designation	Input Wave	Step I SHG	Step II DFM	Equivalent Cubic Interaction
O1a	o	$o_{1} o_{1} \to e_{2}$	$e_{2} o_{1} \to e_{1}$	$o_{1} o_{1} o_{1} \to e_{1}$
O2	o	$o_{1} o_{1} \to o_{2}$	$o_{2} o_{1} \to e_{1}$	$o_{1} o_{1} o_{1} \to e_{1}$
E1	e	$e_{1} e_{1} \to o_{2}$	$o_{2} e_{1} \to o_{1}$	$e_{1} e_{1} e_{1} \to o_{1}$
E2	e	$e_{1} e_{1} \to e_{2}$	$e_{2} e_{1} \to o_{1}$	$e_{1} e_{1} e_{1} \to o_{1}$

Point Group	Optimal Azimuthal Angle for the Processes $o_{1} o_{1} - e_{2}, e_{2} o_{1} - e_{1};$ $e_{1} e_{1} - o_{2}, o_{2} e_{1} - o_{1};$ $e_{1} e_{1} - e_{2}, e_{2} e_{1} - o_{1}$ a	Optimal Azimuthal Angle for the Process $o_{1} o_{1} - o_{2}, o_{2} o_{1} - e_{1}$
$\bar{4} m 2$	22.5°, 67.5°	Not possible
$3 m$	Depends on ratio $d_{22} / d_{31}$	Depends on ratio $d_{22} / d_{31}$
$\bar{4}$	Depends on ratio $d_{36} / d_{31}$	Not possible
3	Depends on ratios $d_{11} / d_{31}$ and $d_{22} / d_{31}$	Depends on ratio $d_{11} / d_{31}$ and $d_{22} / d_{31}$
32	15°, 45°, 75°	15°, 45°, 75°
$\bar{6}$	Depends on ratio $d_{22} / d_{11}$	Depends on ratio $d_{22} / d_{11}$
$\bar{6} m 2$	15°, 45°, 75°	15°, 45°, 75°

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