Photorefractive Effect
Light-induced refractive index changes via space-charge fields in electro-optic materials. Key for holographic storage, phase conjugation, and beam coupling.
LiNbO₃ with e-polarized beams and the grating along c: r₃₃ = 30.8, r₁₃ = 8.6 pm/V, ε₃₃ = 28 (Weis & Gaylord 1985), n_o, n_e (Zelmon et al. 1997). N_eff depends on doping and is an input, not a material constant.
Grating period Λ = λ/(2 sin θ)
1.532 µm
E_D = E_q at Λ = 0.394 µm
Diffusion field E_D
104.3 kV/m
k_B T K / q
Trap-limited field E_q
1576 kV/m
q N_eff / (ε₀ ε_K K), ε_K = 28
Effective coefficient r_eff
30.18 pm/V
n = 2.2336, θ_in = 4.46°, fringe contrast cos 2θ_in included
Space-charge field E_sc (per unit m)
97.8 kV/m
E_D / (1 + E_D/E_q), a quarter period from the fringes
Index grating ½n³r_eff E_sc (per unit m)
1.645e-5
multiply by m = 2√(I₁I₂)/(I₁ + I₂); the polarization overlap is in r_eff
Two-beam coupling gain Γ
3.897 cm⁻¹
2π n³ r_eff E_sc / (λ cos θ_in); the c-axis sense sets which beam gains
Signal gain after L
6.975×
(1 + β_I)/(1 + β_I e^(−ΓL)), no absorption; e^(ΓL) = 7.017
Gain against crossing angle
Model
E_sc = E_D / (1 + E_D/E_q), Γ = 2π n³ r_eff E_sc / (λ cos θ)
Kukhtarev's band-transport model in steady state, with one carrier type, diffusion only (no applied field), small fringe modulation and no photovoltaic current. The grating is shifted a quarter period from the fringes, so the beams exchange energy: dI₁/dz = Γ I₁I₂/(I₁ + I₂).
Small angles give long periods and a weak diffusion field; large angles give short periods, where the trap density caps the field. Γ peaks near Λ = 2π√(ε₀ε_Kk_BT/(q²N_eff)).
n⁴r_eff = cos 2θ · cos β · [n_o⁴r₁₃(sin²β − sin²θ) + 2n_o²n_e²r₄₂ sin²β + n_e⁴r₃₃(cos²β − sin²θ)]
For e-polarized beams at ±θ inside, the grating couples them through ê₁·ε(R·K̂)ε·ê₂ times the fringe contrast ê₁·ê₂ = cos 2θ, and E_q uses the permittivity along K, ε_K = ε₁₁ sin²β + ε₃₃ cos²β (Feinberg et al. 1980). With K along c (β = 0, LiNbO₃) only r₃₃ and r₁₃ act. In BaTiO₃ a tilted grating brings in the large r₄₂, and r_eff peaks near β ≈ 53°, close to where r₄₂ sin²β cos β is largest. But ε₁₁ ≫ ε₃₃, so tilting K also lowers E_q: when the trap density limits the field (short periods, low N_eff), Γ peaks at a smaller β, down to β = 0. n is the e-index along the bisector; walk-off (≈ 1° in BaTiO₃) and anisotropic carrier mobility are neglected.
A grating's strain is neither zero nor free: its effective r and ε add piezoelectric and elasto-optic terms that depend on K (Zgonik et al. 1994). The clamped and unclamped BaTiO₃ sets are the two limiting cases, not the grating's own values. The r are 633 nm values used at every wavelength.
Fe:LiNbO₃ is dominated by its photovoltaic current, which this model leaves out: its diffusion gain is a lower bound. SBN and the sillenites (BSO, BGO) need the effective coefficient of their cut and polarization, so enter them as a custom crystal.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Steady-state Kukhtarev diffusion grating, small m: E_sc = E_D/(1 + E_D/E_q), Γ = 2πn³r_eff E_sc/(λ cos θ). e-beams: Feinberg's tensor r_eff with cos 2θ contrast and ε along K. BaTiO₃ clamped and free sets as limits; no grating piezo/elasto-optic terms, walk-off, photovoltaic current or applied field.
- Kukhtarev N. V., Markov V. B., Odulov S. G., Soskin M. S., Vinetskii V. L. (1978). Holographic storage in electrooptic crystals. I. Steady state. Ferroelectrics 22, 949–960. doi:10.1080/00150197908239450
- Yeh P. (1989). Two-wave mixing in nonlinear media. IEEE J. Quantum Electron. 25, 484–519. doi:10.1109/3.18564
- Feinberg J., Heiman D., Tanguay A. R. Jr., Hellwarth R. W. (1980). Photorefractive effects and light-induced charge migration in barium titanate. J. Appl. Phys. 51, 1297–1305. Eq. 7 (ε along K), eqs. 13 and 19 (tensor coupling of the two beams); the e-wave r_eff(β, θ) on this page is contracted from them. doi:10.1063/1.327824
- Zgonik M., Bernasconi P., Duelli M., Schlesser R., Günter P., Garrett M. H., Rytz D., Zhu Y., Wu X. (1994). Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO₃ crystals. Phys. Rev. B 50, 5941–5949. Table II (clamped r₁₃, r₃₃, r₄₂, ε₁₁, ε₃₃) and Table III (stress-free values), 633 nm, 23 °C. doi:10.1103/PhysRevB.50.5941
- Wemple S. H., DiDomenico M. Jr., Camlibel I. (1968). Dielectric and optical properties of melt-grown BaTiO₃. J. Phys. Chem. Solids 29, 1797–1803 (n_o, n_e single-oscillator fits, 0.4–0.7 µm, via the refractiveindex.info database). doi:10.1016/0022-3697(68)90164-9
- Weis R. S., Gaylord T. K. (1985). Lithium niobate: summary of physical properties and crystal structure. Appl. Phys. A 37, 191–203. doi:10.1007/BF00614817
- Zelmon D. E., Small D. L., Jundt D. (1997). Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol.% magnesium oxide–doped lithium niobate. J. Opt. Soc. Am. B 14, 3319. doi:10.1364/JOSAB.14.003319