4.1 Light that writes itself: photorefraction

L1 Phenomenon

optical damage, 1966

A beam of visible light passing through lithium niobate leaves behind a change in refractive index that outlasts the beam. Light frees electrons from impurity and defect sites; the crystal's polar structure pushes them along the c axis without an applied voltage (the bulk photovoltaic effect); they are trapped again in the dark edges of the beam. The separated charge sets up a space-charge field, and the Pockels effect turns that field into an index change.

It was first called optical damage (Ashkin et al. 1966 reported (title)): a focused laser beam is distorted and scattered by its own track. It is reversible; uniform light or heat erases it.

Write and erase: |E_sc| and Δn_e of a Gaussian beam trackopen circuit, local saturation

cross-section, x across beam (±2.4 w), z along beam (1 mm); |Δn_e| log levels 10⁻⁸..10⁻³ .:-=+*#%

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Intensity I 1.00e+00 W/cm²

Exposure t 1.00e+01 s

σ_d 1.00e−15 S/m

Time in dark 1.00e+05 s

Crystal

Light

I 1.00e+00 W/cm² |E_sat| 2.80e+06 V/m Δn_e now −4.73e−04 τ lit 2.48e+00 s τ dark 2.48e+05 s = 2.87 d

k_G(Fe) 2.8×10⁻⁹ cm/V P-comp σ_d range P-comp α, κ, w, 1 mm template MgO: κ ×100 reported ε₃₃ 28, n_e 2.2028, r₃₃^T 32.2 pm/V P-comp model parameter, no data: α, κ

Write a scar, turn the light off

532 nm beam into a 1 mm slab; bulk photovoltaic field, open circuit, local saturation

Intensity1.0e+00 W/cm²

Exposure 10.0 s

Crystal

Dark conductivity1.0e−15 S/m

Time in the dark 1.16 d

Index change now: −4.73e−04
Saturated field: 2.80e+06 V/m
Scar lifetime, lit: 2.48 s
Scar lifetime, dark: 2.87 d

k_G Fe P-comp σ_d range P-comp MgO: photoconductivity ×100 reported absorption, photoconductivity, beam size: model parameters, no data template ε₃₃ 28, n_e 2.2028, r₃₃^T 32.2 pm/V P-comp beam: 532 nm, computed spectral color

Quantity	Value	Conditions	Source
Lowest intensity at which photorefraction is observed	10 mW/cm²	blue-green light, LiNbO₃:Fe; no intensity threshold	Volk & Wöhlecke 2009, p. 52P-comp
Extraordinary index decrease	up to 10⁻³	focused Ar⁺ laser, 488 nm, 20 mW; ordinary change much smaller	Chen 1969reported
Bulk photovoltaic saturation voltage	> 1000 V (about 10⁵ V/cm)	doped LiNbO₃	Glass et al. 1974reported

L2 Charge transport

Volk & Wöhlecke Eqs. 3.2-3.19; Schwesyg Eqs. 2.23-2.25

Three currents (Volk & Wöhlecke Eq. 3.2, p. 52; Schwesyg Eq. 2.23 P-comp): the bulk photovoltaic (Glass) current, diffusion of conduction-band electrons, and drift in the total conductivity.

j = j_{pv} + j_{diff} + j_{drift}, j_{pv, i} = β_{i j k} e_{j} e_{k} I, j_{diff} = k_{B} T μ_{e} \nabla n_{e}, j_{drift} = (σ_{d} + κ I) E

(1)

For point group 3m the symmetric photovoltaic tensor has the pattern of the piezoelectric d tensor: nonzero β₃₁₁ = β₃₂₂, β₃₃₃, β₂₂₂ = −β₁₁₂ = −β₁₂₁ = −β₂₁₁, β₁₁₃ = β₂₂₃ (p. 53 P-comp). β₁₁₃ drives a current that oscillates along the beam; its coherence length (half period) is l_c = λ / (2|n_e − n_o|), about 5λ in LN, and the full period λ/|n_e − n_o| about 11λ; it matters only when the lit region is that small, as in waveguides.
Scalar form: j_pv = k_GαI with k_G the Glass constant (Eq. 3.5).
Open circuit, steady state: drift cancels the photovoltaic current. Once σ_ph = κI ≫ σ_d, the field no longer depends on intensity.

E_{pv} = \frac{k_{G} α I}{σ_{ph} + σ_{d}}, E (t) = - E_{sat} [1 - e^{- t / τ}], E (t) = E_{sat} e^{- t / τ}, τ = \frac{ε ε_{0}}{σ_{ph} + σ_{d}}

(2)

Writing (open circuit) and erasure (uniform light or dark) follow Eqs. 3.18-3.19 (p. 57). The dielectric relaxation time τ sets both. Index change through the linear electro-optic effect (Schwesyg Eq. 2.25 P-comp):

Δ n_{e} = - \frac{1}{2} n_{e}^{3} r_{33} E_{z}, Δ n_{o} = - \frac{1}{2} n_{o}^{3} r_{13} E_{z}

(3)

r₃₃ is about 3 r₁₃, so extraordinary light sees about three times the change (Schwesyg p. 19). The lit region shows a decrease of n_e (Chen 1969 reported); the figures plot |E| and Δn_e < 0, since Volk & Wöhlecke and Schwesyg do not give the sign of j_pv relative to +c.

A fringe pattern with modulation m and wave number K gives a diffusion field (monopolar electrons, Eq. 3.9) that is independent of intensity and shifted a quarter period from the fringes; the photovoltaic field is in phase with them (pp. 54-55):

E_{dif} (z) = \frac{k_{B} T}{e} \frac{K m \sin K z}{1 + m \cos K z}

(4)

Quantity	Value	Conditions	Source
Glass constant k_G, LiNbO₃:Fe	2.8×10⁻⁹ cm/V		Volk & Wöhlecke 2009, p. 59P-comp
Glass constant k_G, LiNbO₃:Cu	5.5×10⁻¹⁰ cm/V		Volk & Wöhlecke 2009, p. 59P-comp
Photoconductivity, as-grown doped crystals	10⁻⁹ to 10⁻¹⁰ S/m	moderate intensity (10⁻¹¹ to 10⁻¹² Ω⁻¹cm⁻¹)	Volk & Wöhlecke 2009, p. 59P-comp
Saturated photovoltaic field	10⁶ to 10⁷ V/m	LiNbO₃:Fe, LiNbO₃:Cu (10⁴ to 10⁵ V/cm)	Volk & Wöhlecke 2009, p. 59P-comp
Diffusion field amplitude	about 1.5×10⁵ V/m	grating period about 1 µm	Volk & Wöhlecke 2009, p. 58P-comp
Dark conductivity, undoped or lightly doped CLN	10⁻¹⁴ to 10⁻¹⁶ S/m	room temperature, extrapolated from high temperature	Volk & Wöhlecke 2009, p. 60P-comp
Dark storage time of gratings, low-doped LiNbO₃:Fe	up to about 1 year	hours or seconds at high Fe content	Volk & Wöhlecke 2009, p. 60P-comp
Electron Hall mobility	about 0.8 cm²/(V s)	strongly reduced LiNbO₃, photo-Hall, room temperature	Volk & Wöhlecke 2009, p. 59P-comp
Sensitivity S′₁, LiNbO₃:Fe	0.01-0.1 cm/J; 0.05-0.3 cm/J	633 nm; 488 nm	Volk & Wöhlecke 2009, p. 57P-comp

Derived numbers

Quantity	Value	Formula and inputs	Status
Δn_e at the saturated photovoltaic field	1.7×10⁻⁴ to 1.7×10⁻³	½ n_e³ r₃₃ E; n_e = 2.2028 at 632.8 nm, r₃₃^T = 32.2 pm/V (Weis & Gaylord Tables 2, 4); E = 10⁶ to 10⁷ V/m	derived
Dielectric relaxation time, dark	6.9 h to 29 d	εε₀/σ_d; ε₃₃ = 28; σ_d = 10⁻¹⁴ to 10⁻¹⁶ S/m	derived
Dielectric relaxation time, light	0.25 s to 2.5 s	ε₃₃ = 28; σ_ph = 10⁻⁹ to 10⁻¹⁰ S/m	derived
Diffusion field per unit modulation, 1 µm fringes	1.6×10⁵ V/m	(k_BT/e)(2π/Λ), T = 295 K; agrees with p. 58	derived

Conflict. The computed dark relaxation times (6.9 h to 29 d) disagree with the 5 to 500 days printed on p. 159 of the same compilation for the same conductivity range; the upper Δn_e bound matches the 10⁻³ reported by Chen 1969.

L3 Mechanism and measurement

dark conduction, transport schemes, holography

Dark conductivity in undoped LN is protonic: one activation energy of about 1.1 eV covers 200 to 1000 °C and the conductivity scales with hydrogen concentration; Li⁺ transport has an activation energy near 0.4 eV, which does not match the measured conductivity (pp. 59-60 P-comp). At 0.2 to 0.25% Fe the activation energy drops to 0.3 to 0.4 eV and electron tunneling between Fe sites takes over (pp. 60-61).
Mixed proton-electron dark conduction gives stretched-exponential decay, δΔn ∝ exp[−(t/τ)^β]; purely ionic conduction gives a single exponential (p. 61). The write-and-erase figure uses a single exponential.
Transport schemes: the one-center model (Fe²⁺ donor, Fe³⁺ trap) fails to fit several data sets; the two-center model adds the Nb antisite (Nb_Li) as a shallow electron trap (Sec. 3.2.2-3.2.3, pp. 62-73). n-type photoconductivity is confirmed only for congruent LN doped with Fe and Cu (p. 59).
Full band-transport model: Kukhtarev et al. reported (title) (Crossref issue date 1978, often cited as 1979). The figures use the reduced single-time-constant form and a 1D grating solver.
Measurement: two-beam holographic recording with diffraction efficiency η = exp(−αL/cos θ) sin²(πΔnL/(λ cos θ)) (Eq. 3.11); light-induced birefringence between crossed polarizers; beam distortion and far-field scattering (fanning). In waveguides and resonators: resonance shift versus power and time (4.3).
Error sources: dark and photoconductivity vary by orders of magnitude between samples, so absolute write and erase times are crystal-specific.

Fringes to field: 1D grating along c, two periods, steady staterows normalized to their amplitude

phase, Δn vs I: +3.3 deg
E_dif per unit m: 1.60e+05 V/m
|E₁|, fundamental: 1.50e+06 V/m
Λ: 1.00 µm
m: 0.50

Photovoltaic field: local, in phase with the fringes. Diffusion field: from bright to dark, a quarter period off. Set k_G to 0 to see 90 deg; diffusion off gives 0 deg.

Period Λ 1.00 µm

Modulation m 0.50

Photovoltaic k_G1.00 × Fe

Temperature 295 K

Diffusion

dashed: closed form Eq. 3.9, diffusion only P-compk_G(Fe) P-compI₀, κ, σ_d, α: model parameters, no data templatephase: fundamental Fourier components of Δn and I

Fringes to field

1D grating along c, two periods, steady state

Photovoltaic current1.00 × Fe

Fringe period 1.00 µm

Fringe contrast0.50

Temperature295 K

Diffusion current

Index peak behind fringe peak: +3.3°
Diffusion field per unit contrast: 1.60e+05 V/m
Field amplitude, fundamental: 1.50e+06 V/m

Photovoltaic field: local, in phase with the fringes. Diffusion field: from bright to dark, a quarter period off. Set k_G to 0 to see 90 deg; diffusion off gives 0 deg.

Glass constant at 1 × the Fe value P-compdiffusion field, Eq. 3.9 P-compintensity, conductivities: model parameters, no data templatephase: fundamental Fourier components of Δn and I

Dated steps

Year	Step	Source
1966	optical damage reported in LiNbO₃ and LiTaO₃	Ashkin et al. 1966reported (title)
1967	annealing in an electric field made LiTaO₃ resistant up to 500 W/cm²; helped LN less	Levinstein et al. 1967reported
1968	holograms written into LN and erased thermally	Chen et al. 1968reported
1969	charge-drift model of the index change	Chen 1969reported
1974	bulk photovoltaic current as the driver	Glass et al. 1974reported
1980, 1984	MgO doping: conference abstract, then journal report (4.2)	Zhong et al. 1980S Bryan et al. 1984reported

Sources

Ashkin et al. 1966A. Ashkin, G. D. Boyd, J. M. Dziedzic et al., Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3. Applied Physics Letters 9 72-74 (1966), doi:10.1063/1.1754607.Read: title
Chen et al. 1968F. S. Chen, J. T. LaMacchia, D. B. Fraser, Holographic storage in lithium niobate. Applied Physics Letters 13 223-225 (1968), doi:10.1063/1.1652580.Read: abstract
Chen 1969F. S. Chen, Optically Induced Change of Refractive Indices in LiNbO3 and LiTaO3. Journal of Applied Physics 40 3389-3396 (1969), doi:10.1063/1.1658195.Read: abstract
Levinstein et al. 1967H. J. Levinstein, A. A. Ballman, R. T. Denton, A. Ashkin, J. M. Dziedzic, Reduction of the Susceptibility to Optically Induced Index Inhomogeneities in LiTaO3 and LiNbO3. Journal of Applied Physics 38 3101-3102 (1967), doi:10.1063/1.1710070.Read: abstract
Glass et al. 1974A. M. Glass, D. von der Linde, T. J. Negran, High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3. Applied Physics Letters 25 233-235 (1974), doi:10.1063/1.1655453.Read: abstract
Kukhtarev et al. 1978N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Holographic storage in electrooptic crystals. I. Steady state. Ferroelectrics 22 949-960 (1978), doi:10.1080/00150197908239450.Read: title
Zhong et al. 1980G. G. Zhong, J. Jin, Z. K. Wu, Measurements of optically induced refractive-index damage of lithium niobate doped with different concentrations of MgO. 11th International Quantum Electronics Conference (IQEC), 1980; J. Opt. Soc. Am. 70, 631 (1980) abstract (1980).Read: not read
Bryan et al. 1984D. A. Bryan, R. Gerson, H. E. Tomaschke, Increased optical damage resistance in lithium niobate. Applied Physics Letters 44 847-849 (1984), doi:10.1063/1.94946.Read: abstract
Volk & Wöhlecke 2009Tatyana Volk, Manfred Wöhlecke, Lithium Niobate: Defects, Photorefraction and Ferroelectric Switching. Springer Series in Materials Science 115 (2009), doi:10.1007/978-3-540-70766-0.Read: selected chapters: Ch. 2.2 (pp. 14-19), Ch. 3 (pp. 51-61), Ch. 4 (pp. 75-77, 82-85, 94-97), Sec. 7.1.3 (p. 168)
Weis & Gaylord 1985R. S. Weis, T. K. Gaylord, Lithium niobate: Summary of physical properties and crystal structure. Applied Physics A Solids and Surfaces 37 191-203 (1985), doi:10.1007/BF00614817.Read: full
Schwesyg 2011Judith R. M.-L. Schwesyg, Interaction of light with impurities in lithium niobate crystals. Dr. rer. nat. dissertation, Universität Bonn (2011).Read: selected: Sec. 2.2.2-2.3.2 (pp. 10-21), Ch. 5 Sec. 5.1 (pp. 69-72), Sec. 5.4-5.5 (pp. 99-101)