In a crystal without inversion symmetry there exist two-step indirect contributions to third-order nonlinear optical processes (cascading). Contributions to optical four-wave mixing occur through optical rectification and linear electro-optic effects. In contrast to cascading by second-harmonic generation, which has to satisfy strict phase-matching conditions, optical rectification is always allowed. In polar KNbO3 crystals we measured four-wave mixing in several geometries to evaluate the direct contribution of the third-order polarizabilities and the cascaded contribution. We present a theoretical model and show experimentally that the cascading effect is large and that contributing polarization gratings must be transversely polarized.
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Some Nonlinear Optical Phenomena Relevant for Describing Cascaded Nonlinearities in Four-Wave Mixinga
Process and Generated Polarization
Optical rectification: single beam produces homogeneous static polarization
Optical rectification: two beams produce static polarization grating
Pockels effect (linear EO effect): homogeneous static field
Pockels effect (linear EO effect): diffraction on a static grating
Second-harmonic generation: collinear type I
Second-harmonic generation: noncollinear type I and type II
Four-wave mixing: general case, four distinguishable waves
Intensity-dependent refractive index: single eigenpolarization
Special care was taken to distinguish between the cases in which the static electric fields are spatially homogeneous or modulated and to determine the numerical factors correctly. The susceptibilities χ(n) are invariant to any permutation of waves in their arguments.
Table 2
Effective Third-Order Susceptibilities in KNbO3 Crystals in the Orthorhombic Phase for Some Characteristic Geometriesa
Line
Effective
Direct Contribution
Contribution of Cascading
Polarization, Direction
1
T, z T, x
2
(L, y) T, x
3
(L, z) T, x
4
T, y T, x
5
(L, z) T, y
6
T, x T, y
7
T, x T, y
8
T, y T, x
9
T, x T, y
10
0
0 0
The direct contribution is related to the values of
in the cubic phase, where two independent tensor elements exist. Optical rectification can produce two static polarization waves with longitudinal (L) transverse (T) polarization and with the direction of the grating vector parallel to one of the crystallographic axes, as noted in the table. Because of a strong depolarizing field in the case of longitudinal polarization, only the transversely polarized static gratings contribute to cascading, and the contribution of longitudinal fields (in parentheses) are neglected. The cascaded contribution is calculated from Eqs. (14) and (8).
Table 3
Effective Third-Order Susceptibilities in KNbO3 at a Wavelength of 532 nm at Room Temperaturea
Line
Effective
Measured
Calculated
Direct Contribution
Contribution of Cascading
1
580
690
40
650
2
290
360
40
320
3
290
360
40
320
4
330
390
40
350
5
170
150
110
40
6
260
210
110
100
7
330
350
210
140
8
360
310
290
20
9
300
280
210
70
10
90
40
40
0
Different components of
that correspond to those listed in Table 2 are measured by changing the sample orientation and polarizations of the beams. KTaO3 is taken as a reference sample with a value of
. The relative accuracy of the values is estimated to be within ±30%. The results may still contain a systematic error that is due to uncertainties of the nonlinear susceptibility of the reference material. The direct contribution is estimated from the values of
in KTaO3, which are taken to be the same as in KNbO3. The values of the cascaded contributions are calculated from the clamped values of the electro-optic tensor and clamped dielectric constants presented in Table 4.
Table 4
Room-Temperature Refractive Indices ni, Clamped Dielectric Tensor
, and Clamped Electro-Optic Tensor
KNbO3 Crystal at a Wavelength of 532 nma
Some Nonlinear Optical Phenomena Relevant for Describing Cascaded Nonlinearities in Four-Wave Mixinga
Process and Generated Polarization
Optical rectification: single beam produces homogeneous static polarization
Optical rectification: two beams produce static polarization grating
Pockels effect (linear EO effect): homogeneous static field
Pockels effect (linear EO effect): diffraction on a static grating
Second-harmonic generation: collinear type I
Second-harmonic generation: noncollinear type I and type II
Four-wave mixing: general case, four distinguishable waves
Intensity-dependent refractive index: single eigenpolarization
Special care was taken to distinguish between the cases in which the static electric fields are spatially homogeneous or modulated and to determine the numerical factors correctly. The susceptibilities χ(n) are invariant to any permutation of waves in their arguments.
Table 2
Effective Third-Order Susceptibilities in KNbO3 Crystals in the Orthorhombic Phase for Some Characteristic Geometriesa
Line
Effective
Direct Contribution
Contribution of Cascading
Polarization, Direction
1
T, z T, x
2
(L, y) T, x
3
(L, z) T, x
4
T, y T, x
5
(L, z) T, y
6
T, x T, y
7
T, x T, y
8
T, y T, x
9
T, x T, y
10
0
0 0
The direct contribution is related to the values of
in the cubic phase, where two independent tensor elements exist. Optical rectification can produce two static polarization waves with longitudinal (L) transverse (T) polarization and with the direction of the grating vector parallel to one of the crystallographic axes, as noted in the table. Because of a strong depolarizing field in the case of longitudinal polarization, only the transversely polarized static gratings contribute to cascading, and the contribution of longitudinal fields (in parentheses) are neglected. The cascaded contribution is calculated from Eqs. (14) and (8).
Table 3
Effective Third-Order Susceptibilities in KNbO3 at a Wavelength of 532 nm at Room Temperaturea
Line
Effective
Measured
Calculated
Direct Contribution
Contribution of Cascading
1
580
690
40
650
2
290
360
40
320
3
290
360
40
320
4
330
390
40
350
5
170
150
110
40
6
260
210
110
100
7
330
350
210
140
8
360
310
290
20
9
300
280
210
70
10
90
40
40
0
Different components of
that correspond to those listed in Table 2 are measured by changing the sample orientation and polarizations of the beams. KTaO3 is taken as a reference sample with a value of
. The relative accuracy of the values is estimated to be within ±30%. The results may still contain a systematic error that is due to uncertainties of the nonlinear susceptibility of the reference material. The direct contribution is estimated from the values of
in KTaO3, which are taken to be the same as in KNbO3. The values of the cascaded contributions are calculated from the clamped values of the electro-optic tensor and clamped dielectric constants presented in Table 4.
Table 4
Room-Temperature Refractive Indices ni, Clamped Dielectric Tensor
, and Clamped Electro-Optic Tensor
KNbO3 Crystal at a Wavelength of 532 nma