Riad Haïdar, Nicolas Forget, Philippe Kupecek, and Emmanuel Rosencher, "Fresnel phase matching for three-wave mixing in isotropic semiconductors," J. Opt. Soc. Am. B 21, 1522-1534 (2004)

We deal with phase matching of three-wave mixing by total internal reflection in isotropic semiconductors. This technique makes use of the large relative phase lag between the three interacting waves at total internal reflection, as described by Augustin Fresnel. This is why we denote this technique as Fresnel phase matching. The theory of Fresnel phase matching is developed with a propagation matrix method: It allows us to describe the conditions (sample thickness, polarization, tuning angles, etc.) for phase matching, the influence of surface roughness, and the walk-off effects due to Goos–Hänchen shifts. Moreover, we show that nonresonant phase matching strongly alleviates the phase-matching tolerance while keeping good conversion yields. The potential of this technique is demonstrated by largely tunable mid-infrared generation (between 7 and 13 µm with a single sample) by use of difference-frequency mixing of two near-infrared sources. Excellent agreement between the presented theory and experiments is demonstrated both in GaAs and ZnSe samples.

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$\mathrm{\Delta}{\mathrm{\Phi}}_{F}$
is the Fresnel phase shift at total internal reflection, δϕ is the crystallographic phase shift, φ and θ are the internal incidence angles of the waves, ${\theta}_{C}$
is the critical angle calculated from the Snell–Descartes law, t
is the semiconductor-plate thickness, and ${d}_{\mathrm{eff}}$
is the effective nonlinear susceptibility calculated from ${d}_{\mathrm{GaAs}}=150\mathrm{pm}{\mathrm{V}}^{-1}$
(Ref. 18) and ${d}_{\mathrm{ZnSe}}=78\mathrm{pm}{\mathrm{V}}^{-1}$
(Ref. 43) for the given values of φ and θ.

Table 2

Some Typical Cases of Nonresonant Quasi-Phase-Matching Situations in a GaAs Plate and Comparison with a Resonant Case at $\theta =45\xb0$a

δφ is the crystallographic phase shift at reflection, ${d}_{\mathrm{eff}}^{2}/d_{\mathit{pss},45\xb0}{}^{2}$
(respectively, ${N}^{2}/N_{45\xb0}{}^{2})$
is the effective nonlinear susceptibility (respectively, number of bounces) ratio between the nonresonant and the resonant quasi-phase-matching situations. The nonresonant cases are calculated for three polarization combinations and three $\mathrm{\Delta}\mathit{kL}$
propagation phase shifts. Comparisons between theoretical and experimental conversion efficiencies is convincing in the $\mathit{psp}$
and $\mathit{spp}$
cases. This table shows that high efficiencies can be obtained in nonresonant quasi-phase matching, i.e., even for nonoptimal conversion yield over one single coherence length.
For polarization configurations psp, pss, spp, and pss.

Table 3

Roughness Standard Deviation Measurements of the ZnSe and GaAs Plates Used for Parametric Conversion in the Resonant Quasi-Phase-Matching Situationsa

Measurement

ZnSe

GaAs

σ (nm)

11

4

θ (°)

27

45

25

45

$R(\%)$

98

98.6

99.4

99.6

The Strehl ratios are estimated for $\mathrm{\lambda}=2\mathit{\mu}\mathrm{m}.$

Table 4

Conversion Efficiency Measurements of ZnSe and GaAs Plates in the Resonant Quasi-Phase-Matching Configurationsa

Measurement

ZnSe

GaAs

θ (°)

26.5

45

27

N

23

19

56

R

98%

98.6%

99.4%

${N}_{\mathrm{eff}}$

16

15

43

${N}_{\mathrm{experimental}}$

12

15

41

is the geometrical number of bounces calculated from the sample length, R
is the reflectivity at total internal reflection (or the Strehl ratio deduced from the roughness measurements), and ${N}_{\mathrm{eff}}$
is the number of efficient bounces calculated in our model with the determined Strehl ratio. The impact of the surface residual roughness is estimated in terms of efficient number of bounces ${N}_{\mathrm{eff}}$
and then compared with the experimental results derived from the conversion efficiency.

Tables (4)

Table 1

Theoretically Determined Resonant Fresnel Phase-Matching Conditions in (110) GaAs and ZnSe Platesa

$\mathrm{\Delta}{\mathrm{\Phi}}_{F}$
is the Fresnel phase shift at total internal reflection, δϕ is the crystallographic phase shift, φ and θ are the internal incidence angles of the waves, ${\theta}_{C}$
is the critical angle calculated from the Snell–Descartes law, t
is the semiconductor-plate thickness, and ${d}_{\mathrm{eff}}$
is the effective nonlinear susceptibility calculated from ${d}_{\mathrm{GaAs}}=150\mathrm{pm}{\mathrm{V}}^{-1}$
(Ref. 18) and ${d}_{\mathrm{ZnSe}}=78\mathrm{pm}{\mathrm{V}}^{-1}$
(Ref. 43) for the given values of φ and θ.

Table 2

Some Typical Cases of Nonresonant Quasi-Phase-Matching Situations in a GaAs Plate and Comparison with a Resonant Case at $\theta =45\xb0$a

δφ is the crystallographic phase shift at reflection, ${d}_{\mathrm{eff}}^{2}/d_{\mathit{pss},45\xb0}{}^{2}$
(respectively, ${N}^{2}/N_{45\xb0}{}^{2})$
is the effective nonlinear susceptibility (respectively, number of bounces) ratio between the nonresonant and the resonant quasi-phase-matching situations. The nonresonant cases are calculated for three polarization combinations and three $\mathrm{\Delta}\mathit{kL}$
propagation phase shifts. Comparisons between theoretical and experimental conversion efficiencies is convincing in the $\mathit{psp}$
and $\mathit{spp}$
cases. This table shows that high efficiencies can be obtained in nonresonant quasi-phase matching, i.e., even for nonoptimal conversion yield over one single coherence length.
For polarization configurations psp, pss, spp, and pss.

Table 3

Roughness Standard Deviation Measurements of the ZnSe and GaAs Plates Used for Parametric Conversion in the Resonant Quasi-Phase-Matching Situationsa

Measurement

ZnSe

GaAs

σ (nm)

11

4

θ (°)

27

45

25

45

$R(\%)$

98

98.6

99.4

99.6

The Strehl ratios are estimated for $\mathrm{\lambda}=2\mathit{\mu}\mathrm{m}.$

Table 4

Conversion Efficiency Measurements of ZnSe and GaAs Plates in the Resonant Quasi-Phase-Matching Configurationsa

Measurement

ZnSe

GaAs

θ (°)

26.5

45

27

N

23

19

56

R

98%

98.6%

99.4%

${N}_{\mathrm{eff}}$

16

15

43

${N}_{\mathrm{experimental}}$

12

15

41

is the geometrical number of bounces calculated from the sample length, R
is the reflectivity at total internal reflection (or the Strehl ratio deduced from the roughness measurements), and ${N}_{\mathrm{eff}}$
is the number of efficient bounces calculated in our model with the determined Strehl ratio. The impact of the surface residual roughness is estimated in terms of efficient number of bounces ${N}_{\mathrm{eff}}$
and then compared with the experimental results derived from the conversion efficiency.