Lab

What does the LiNbO3 lattice look like, and how do the hexagonal and rhombohedral cells, the octahedral columns (Nb, empty, Li along +c) and the cleavage plane sit in it?

cell setting: hexagonal, rhombohedral, both; layers: Nb octahedra, Li octahedra, empty sites, bonds, axes, cleavage plane (01-12), column highlight; supercell n × n × 1 (n = 1, 2); pick an atom or site

atom and octahedron geometry; picked species and Wyckoff site; fractional and Cartesian coordinates (Å); nearest-neighbour bond lengths (Å); lattice constants a, c (Å); rhombohedral a_R (Å) and α (°)

Static geometry from the room-temperature structure (a = 5.148 Å, c = 13.863 Å) expanded by its symmetry operations; the hexagonal-to-rhombohedral conversion gives a_R = 5.494 Å, α = 55.87°, and the (01-12) cleavage plane lies at 32.75° to c.(Hsu et al. 1997; Abrahams et al. 1966; Weis & Gaylord 1985; Volk & Wöhlecke 2009)

Which atoms move when the polarization reverses, by how much, and why is a single cell not how real crystals switch?

flip parameter s −1 to +1; reference frame: O fixed, Nb fixed, centre of mass; applied field E (kV/mm)

Nb, Li and O displacements along c (Å); polarization direction; double-well energy (mRy per 10-atom cell)

Li, Nb and a small O twist are interpolated linearly between the up structure (s = +1) and its inversion about the NbO6 centroid; s = 0 is the centrosymmetric R-3c reference. The inset well U = U0 (s² − 1)² − kEs is a display shape scaled to the reported 18.3 mRy (2858 K) depth.(Hsu et al. 1997; Weis & Gaylord 1985; Inbar & Cohen 1996; Volk & Wöhlecke 2009)

What changes, and by how much, as the Li2O content goes from congruent (about 48.4 mol%) to stoichiometric (50 mol%)?

Li2O content 48.0 to 50.0 mol% (step 0.01); defect model: Li-site vacancy, Nb-site vacancy

[Li]/[Nb] ratio; Nb_Li antisite and V_Li vacancy fractions (%); Curie temperature Tc band (°C); UV absorption edge λ_20, λ_15 (nm); coercive and internal field anchors (kV/mm); melting point (°C)

R1 = c/(100 − c) and antisite parameter x = (1 − 2R2)/(5 − 4R2) with R2 = c/100, placed per the chosen defect model; Tc from the Iyi and Bordui fits and the UV edge from its calibrated fit (48.0 to 50.0 mol% only); Ec and Eint shown only at sourced anchors.(Volk & Wöhlecke 2009; Iyi et al. 1992; Bordui et al. 1992; Kovács et al. 1997; Gopalan et al. 1998; Carruthers et al. 1971; Weis & Gaylord 1985)

Which intrinsic and hydrogen-related centres exist in LiNbO3, where are they in the lattice, and what light do they absorb?

centre: Nb_Li, V_Li, V_Nb, bound polaron, free polaron, bipolaron, O⁻ hole polaron, OH⁻; pump 532 nm: on, off; illuminate below 80 K: on, off

lattice site; absorption peak (eV, nm); OH⁻ stretch line (cm⁻¹); lifetime ranges where sourced; EPR g factors

Catalogue, no simulation: bound polaron Nb_Li⁴⁺ at 1.6 eV, free polaron Nb_Nb⁴⁺ 0.95 eV, bipolaron 2.5 eV, O⁻ hole polaron near 500 nm, OH⁻ at 3482/3467 cm⁻¹ (CLN) and 3466 cm⁻¹ (SLN). Peaks are markers; widths drawn only where sourced (OH⁻ FWHM 32 cm⁻¹).(Volk & Wöhlecke 2009; Herth et al. 2005; Furukawa et al. 2001)

What happens at the damage-resistance threshold of Mg (and Zn, In, Sc, Hf, Zr), and why is it lower in Li-rich crystal?

dopant: MgO, ZnO, In2O3, Sc2O3, HfO2, ZrO2; host: congruent, melt [Li]/[Nb] = 1.2, near-stoichiometric; concentration 0 to 9 mol% (step 0.1)

sourced threshold concentrations (mol%); OH⁻ absorption band below and above threshold (cm⁻¹); Mg site occupancy

Browser of sourced thresholds, no interpolation (nearest lower sourced state shown). In congruent host the MgO threshold is 4.5 to 5 mol% and the OH⁻ line moves from 3484 to 3534 cm⁻¹; Mg sites follow one proposed scheme; the two concentration bases differ by about 2×.(Volk & Wöhlecke 2009; Bryan et al. 1984; Furukawa et al. 2000; Zhong et al. 1980; Iyi et al. 1995)

How does point group 3m decide which components of each property tensor are zero, equal, or tied, and what appears when an applied field lowers the symmetry?

property: pyroelectric (rank 1), permittivity (2), piezoelectric d (3), electro-optic r (3), elastic c (4), photoelastic p (4); operations: C3, C3², mirror yz, mirror at +60°, mirror at −60°; applied field: none, x, y, z

Voigt matrix with each component zero, independent or related; number of independent components; residual point group (3m, m, 1)

Neumann's principle: a generic tensor is averaged over the selected operations and contracted with Voigt factors, and linear relations are found numerically. Full 3m leaves 4 independent d (d16 = −2 d22) and 6 c (c66 = (c11 − c12)/2); a field along y leaves m, along x leaves 1.(Weis & Gaylord 1985; Nye 1985; IEEE Std 176-1987)

How is the sign of the polar axis defined and found on a real crystal?

test: compress along c, cool, etch in acid

sign of charge on the +c face; cooling charge per kelvin (C/K per cm², range); compression charge (C)

Sign rules from the standards: compression along c makes the +c face negative, cooling makes it positive, and acid etches the −c face much faster. Optional charges Q = p3 A ΔT with |p3| 4 to 9.5 × 10⁻⁵ C/(m² K), and Q = d33 F.(Weis & Gaylord 1985; Volk & Wöhlecke 2009; IEEE Std 176-1987; IRE 1949)

How do coercive field, internal field and spontaneous polarization differ between congruent, near-stoichiometric and Mg-doped crystals?

crystal: CLN, SLN, CLN:5% MgO, SLN:1% MgO, near-SLN:2% MgO; wafer thickness 0.1 to 1 mm

polarization versus field loop (C/m², kV/mm); coercive field Ec (kV/mm); internal field Eint (kV/mm); spontaneous polarization Ps (C/m²); poling voltage (kV)

Loops built from sourced anchors only: Ps band 0.71 to 0.80 C/m², forward switching at Ec and reverse at Ec − 2Eint (loop shifted by Eint); the tanh loop shape is a display template. Poling voltage V = Ec t.(Volk & Wöhlecke 2009; Gopalan et al. 1998; Chen et al. 2001; Chen et al. 2003)

How does a field pulse through comb electrodes turn a single-domain crystal into a periodically poled one, and what does the switching current record?

field pulse (voltage, duration); electrode period and duty (template presets); hold at Ec/2 after the pulse: on, off

domain map (top view and cross-section); switched area fraction X(t); switching current I(t); integrated charge (µC/cm²); duty cycle; KAI exponent n

Stochastic cellular automaton on a triangular lattice: nucleation at the +z electrode edge rising with E − Ec, sideways growth along six Y-wall directions; all rates are templates. I = 2Ps dA/dt + C dV/dt with Ps = 0.71 C/m²; X(t) is fitted with the KAI model.(Volk & Wöhlecke 2009; Yamada et al. 1993; Myers et al. 1995; Gopalan et al. 1999; Lee et al. 2011; Shur 2006)

How wide is a domain wall, and on what length scales do its strain, birefringence, and (when charged) conduction live?

length scale 10 µm to 0.1 nm (log); wall tilt θ 0 to 5°

wall feature at the current scale and its width (µm, nm); bound charge σ_b (C/m²); sheet electron density (e/cm²)

Sourced widths per scale: X-ray shear strain up to 10 µm, birefringence within 2 µm, PFM 65 to 150 nm (probe-limited), ideal 1 to 2 lattice units (about 0.5 to 1.0 nm), TEM bound 0.28 nm (LiTaO3). Tilt gives σ_b = 2Ps sin θ; one measured conduction point at θ ≈ 1°.(Gopalan et al. 2007; Volk & Wöhlecke 2009; Werner et al. 2017; Schröder et al. 2012)

What does a z-cut LN plate look like between crossed polarizers, and what does a transverse field do to it?

plate thickness 0.2 to 2.0 mm; source 632.8 nm or white light; field E 0 to 5 kV/mm; field direction x or y; numerical aperture 0.55 (fixed)

conoscopic intensity pattern; induced Δn; optic-axis tilt in air (deg); state (uniaxial or biaxial); first dark-ring angle (deg)

Jones calculus per pixel with exact Snell angles per eigenwave: Γ = (2πd/λ)(n_o cos θ_o − n_e(θ_e) cos θ_e), I = sin²(2ψ) sin²(Γ/2), ψ from the field-perturbed impermeability; white light integrated against CIE 1931 2° color-matching functions.(Zelmon et al. 1997; Turner 1966; Born & Wolf 2019)

What are n_o, n_e and n_g of LN at any wavelength and temperature, how much do the published models disagree, and what do the errata change?

wavelength 0.4 to 5.0 µm (log axis); temperature 20 to 250 °C (models outside validity greyed out); material: congruent LN, 5 mol% MgO, 1% MgO stoichiometric LN; quantity: n, n_g, n_e − n_o, dn/dT; 5 mol% MgO table: as printed in 1997 or corrected

n_o, n_e per model; group index n_g; birefringence n_e − n_o; dn/dT (1/K); maximum model spread at the cursor

Sellmeier models (Zelmon, Jundt, Gayer) each within its validity window; n_g and group-velocity dispersion by finite differences, dn/dT by central difference over ±1 K; the 1997 5 mol% MgO table has its o/e headings swapped, which the erratum corrects.(Zelmon et al. 1997; Jundt 1997; Gayer et al. 2008; Nelson & Mikulyak 1974; Weis & Gaylord 1985)

Which index does a wave see for a given direction and polarization, and how does a field reshape the ellipsoid?

propagation polar angle θ 0 to 90°; azimuth φ; wavelength; field state: field direction x, y or z, optical polarization x, y or z, r-coefficient set, E = 1 V/µm

n_o; extraordinary index n(θ); walk-off angle ρ (deg); index along polarization; first-order δn; principal indices; largest principal-axis tilt (deg)

Impermeability B = B0 + ΔB with ΔB_I = r_Ij E_j, exact 3 × 3 symmetric eigen-decomposition; the ellipsoid section normal to k gives the two eigen-indices, 1/n_e(θ)² = cos²θ/n_o² + sin²θ/n_e², and the Poynting vector walks off from k by ρ.(Weis & Gaylord 1985; Turner 1966; Kaminow et al. 1980; Zook et al. 1967; Lenzo et al. 1966; Bernal et al. 1966; Zelmon et al. 1997)

How much does a volt change the index, and what does that do to a real modulator?

drive voltage −6 to +6 V; bulk calculator: wavelength, field E_z

transmission T (linear, dB); phase (units of π); |Δn_eff| per arm, measured-derived and computed

Δn_e = −½ n_e³ r33 E_z and Δφ = (2π/λ) Δn_e L; T = cos²(πV/(2V_π)) with the measured V_π of 2.90 V at 1550 nm, compared with a dashed curve from the FEM-computed V_π L over 0.9 cm.(Li et al. 2026; Zelmon et al. 1997; Jiang 2026)

Where in the cross-section does the phase shift come from, and why does the textbook formula overestimate V_π L by 20%?

integrand: total or r33 only; hover position

|E_x|² map; electrode field e_x map; overlap integrand map; depth profile; position x, y (µm); region; integrand value; dn_eff/dV (1/V); region split (%); Γ; V_π L (V cm)

Power-normalized perturbation dn_eff/dV = −(n_e⁴ r33/(2 n_eff)) ∫_LN e_x |E_x|² dA / ∫|E_x|² dA, integrated on exported FEM fields; the textbook V_π L = λg/(2 n_e³ r33 Γ) carries n_e³ instead of n_e⁴/n_eff, a factor n_e/n_eff = 1.208 here.(Li et al. 2026; Jiang 2026)

Which nonlinear coefficient does a given cut and polarization use, and why does birefringent phase matching give up d33?

cut x, y or z; in-plane propagation direction; input polarization TE/TM (o/e for bulk); coefficient set: symbolic or numeric

nonlinear polarization vector P_NL/(2ε0) in units of d E²; active coefficient; coefficient value (pm/V) for a numeric set

P_i(2ω) = 2ε0 Σ d_il [E_x², E_y², E_z², 2E_yE_z, 2E_xE_z, 2E_xE_y]_l with the 3m Kleinman-symmetric d (d15 = d24 = d31); input fields projected onto crystal axes, output projected on the chosen mode. o-polarized inputs never reach d33.(Volk & Wöhlecke 2009; Zhu et al. 2021; Shoji et al. 1997; Kitamoto et al. 1995; Kleinman 1962)

What period or angle phase-matches a process, how tolerant is it, and where does an OPO tune?

process: SHG, SFG, DFG, OPO; wavelengths; temperature; dispersion model; QPM order m and duty D; length L; mode: bulk or waveguide

QPM period Λ (µm); coherence length L_c (µm); type-I BPM angle θ_pm (deg); walk-off ρ (deg); FWHM acceptance in λ (nm) and T (K); tuning dλ/dT (nm/K); GVM (fs/mm); OPO signal and idler vs period and temperature

Δk = k3 − k1 − k2 − 2πm/Λ with bulk Sellmeier indices; type-I BPM angle from the extraordinary index n_e(θ); acceptance FWHM = 2.783/(L |dΔk/dq|); OPO signal/idler by bisection on Δk = 0 at fixed period. Thermal expansion of the period is not included.(Zelmon et al. 1997; Jundt 1997; Gayer et al. 2008; Fejer et al. 1992)

Why does second-harmonic power oscillate without poling, grow as L² with it, and saturate at high pump?

poling: none, QPM, QPM with duty error; pump power 1 mW to 10 W (log); length 0.1 to 50 mm; normalized efficiency η_norm (%/W/cm²); loss

SH power P_2ω(z) (W); conversion (%); η_norm (%/W/cm²); efficiency for the chosen length (%/W); absolute conversion (%)

Coupled-mode equations dA_ω/dz = −iκ A_2ω A_ω* e^(−iΔkz), dA_2ω/dz = −iκ A_ω² e^(+iΔkz), κ² = η_norm, integrated by RK4 with poling d(z) = ±d_eff; undepleted L² and depleted tanh² limits overlaid.(Armstrong et al. 1962; Boyd 2008; Wang et al. 2018)

What do %/W/cm², %/W and absolute % each measure, and how do reported devices compare when the normalization is respected?

series filter: waveguide, ring, DFG; log axis toggle; converter: η_norm (%/W/cm²), length L, ring Q

reported efficiency by year (%/W/cm², %/W or absolute %); SH wavelength; platform; date; converted efficiency (%/W)

Records plotted by year with each normalization on its own panel; converter %/W = η_norm L²; ring efficiency 16 g²/(ħω κ_a² κ_b), which scales as Q³ at fixed g.(Yamada et al. 1993; Wang et al. 2018; Zhao et al. 2020; Rao et al. 2019; Li et al. 2024; Park et al. 2022; Mishra et al. 2021; Lu et al. 2019; Lu et al. 2020)

How much gain does a pump give, and how much pump does a resonator need to oscillate?

OPA: η_norm (%/W/cm²), pump peak power, length, Δk; OPO: intrinsic Q at pump and signal, coupling ratios, g/2π, pump power

gain per length (dB/cm); total gain G (dB); threshold P_th (W); signal plus idler output P_s+i (W); conversion efficiency (%)

Gain Γ = √(η_norm P_p), G_dB/L = 8.686 Γ, G = cosh²(ΓL) at Δk = 0; threshold P_th = ħω_p κ_p² κ_s κ_i/(16 g² κ_p,ex), above it the pump clamps and P_s+i ∝ √(P/P_th) − 1, efficiency peaking at 4 P_th.(Ledezma et al. 2022; Jankowski et al. 2022; Lu et al. 2021; McKenna et al. 2022; Gao et al. 2021; Ledezma et al. 2023; Roy et al. 2023; Hwang et al. 2023; Hwang et al. 2026; Stokowski et al. 2024)

When does a phase-mismatched χ⁽²⁾ process out-pull the intrinsic Kerr effect, and with which sign?

phase mismatch Δk (signed, via period detuning or directly); power P; length L; η_norm; effective area A_eff; n2 source: DeSalvo 1996 or Zhu 2021

Kerr phase φ_Kerr (rad); cascaded phase φ_casc (rad); ratio; z-averaged SH power fraction; sign (focusing or defocusing)

Cascaded phase Δφ_NL = −η_norm P L/Δk against γ_Kerr = 2π n2/(λ A_eff); Δk > 0 gives a negative (defocusing) phase; full RK4 coupled-mode integration where |Δk| L is small and the formula fails.(DeSalvo et al. 1996; Zhu et al. 2021; DeSalvo et al. 1992)

How do soliton steps form in a detuning scan, and what sets the comb span?

normalized pump f; detuning scan range and rate; normalized dispersion d2; photorefractive term on or off

intracavity power vs detuning; spectrum (dB, log); intracavity waveform; soliton number; detuning ζ; span at −40 dB (lines)

Lugiato-Lefever equation dψ/dt = −(1 + iζ)ψ + i|ψ|²ψ + i(d2/2)∂²ψ/∂θ² + f solved by split-step Fourier (512 modes default, 2048 expert); all parameters are display values, not material constants, with the conversion from Q, FSR, D2, γ and P_in printed.(Lugiato & Lefever 1987; Herr et al. 2014; Chembo & Menyuk 2013)

How do drive voltage and finesse set the width of an electro-optic comb?

modulation depth β = V_p/V_π,rt; intrinsic Q and coupling (or round-trip amplitude r); FSR; D2 (expert); line-counting threshold (dB)

finesse F; β (rad); envelope slope (dB/line); lines above −60 dB; span (nm); line power spectrum (dB)

Steady state E(t) = t_c E_in/(1 − r e^(iβ cos Ωt)) sampled on 4096 points and Fourier transformed; asymptotic slope 8.686 asinh(|ln r|/β) dB/line, 27.29/(βF) at high finesse; dispersion and detuning via coupled-mode equations over the sidebands.(Zhang et al. 2019; Hu et al. 2022; Zhang et al. 2023; Yu et al. 2022; Zhang et al. 2025)

How does a microwave move light between two frequency bins of a photonic molecule?

ring coupling μ (supermode splitting 2μ); microwave frequency; drive voltage (coupling G); intrinsic and external Q

frequency shift (GHz); conversion efficiency (%); G/κ; residual carrier (dB); efficiency vs G

Two-mode coupled-mode theory in the supermode basis: |s_b,out/s_in|² = κ_a,ex κ_b,ex G²/(κ_a κ_b/4 + G²)², reaching 1 at G = κ/2 with no intrinsic loss; time-domain view shows Rabi-like transfer for pulsed input.(Hu et al. 2021)

How do film thickness, etch depth, width and cut set the mode, where do TE and TM hybridize, and how does an x-cut ring see a rotating crystal?

film thickness 300 to 900 nm; etch fraction 0.2 to 1.0; width 0.6 to 3.0 µm; sidewall 60, 70 or 80°; wavelength 0.775, 1.064 or 1.55 µm; cut x or z; polarization TE or TM; ring tab: propagation angle φ

effective index n_eff; group index n_g; effective area A_eff (µm²); LN power fraction; TE and TM hybrid fractions; interpolation error; ring index n(φ)

Interpolated lattice of precomputed full-vector FEM mode solutions, with held-out solves giving the interpolation error; ring tab uses the bulk approximation 1/n(φ)² = cos²φ/n_o² + sin²φ/n_e², spanning n_e to n_o, mode confinement ignored.(Jiang 2026; Li et al. 2026; Zelmon et al. 1997; Kaushalram et al. 2020; Tan et al. 2024)

How do Q and dB/m convert, and how close are TFLN devices to the material floor?

intrinsic Q_i; group index n_g; wavelength; log axis toggle

loss α (dB/m, 1/m); intrinsic Q_i; reported loss by year (dB/m); material and absorption floors (dB/m)

α = 2π n_g/(Q_i λ), α_dB = 4.343 α; records plotted by year against the estimated material limit of annealed ion-sliced film and the lower bound of the bulk absorption range.(Zhang et al. 2017; Luke et al. 2020; Zhu et al. 2024; Gao et al. 2022; Gao et al. 2021; Desiatov et al. 2019; Shams-Ansari et al. 2022; Leidinger et al. 2015; Schwesyg 2011)

Which way do you press LN to get the most charge, and why does the answer have threefold symmetry?

press direction θ 0 to 180°, φ 0 to 360°; data set: Kushibiki 1999, Smith & Welsh 1971

longitudinal coefficient d_L (pC/N, signed); face charge density at 1 MPa (µC/m²); displacement per volt along the press direction (pm/V)

d = e s^E and d_L(n) = d_ijk n_i n_j n_k = d33 cos³θ + (d31 + d15) cos θ sin²θ − d22 sin³θ sin 3φ, so the sign flips every 60° around the equator; d_L is 20.30 pC/N along Y, 8.69 along Z, and |d_L| peaks at 38.80 pC/N at θ = 118.9°.(Kushibiki et al. 1999; Smith & Welsh 1971; Weis & Gaylord 1985)

How do e, d, g, h and the stiff/compliant, clamped/free constants convert into each other, and where do factors of 2 come from?

given form: e, d, g, h; Voigt view: IEEE engineering strain, full tensor index; matrix cell selection

piezoelectric matrices e, d (pC/N), g (V m/N), h (V/m); stiffness c^E, c^D (GPa); compliance s^E, s^D; permittivity ε^S, ε^T; full-tensor components and factor (1, 2 or 4) for a selected cell

IEEE 176 relations between the four piezoelectric forms with the Voigt maps, so c66 = (c11 − c12)/2 but s66 = 2(s11 − s12), and d16 = −2 d22. Computed ε^T 83.34 / 28.48 against measured 83.3 / 28.5.(IEEE Std 176-1987; Kushibiki et al. 1999; Graham 1976)

How fast does sound travel in each direction of LN, and how much does piezoelectricity stiffen it?

piezoelectric stiffening: on, off; view: XY, YZ, XZ plane or 3D; mode: slow shear, fast shear, quasi-longitudinal; data set

phase velocity (m/s); slowness (s/m); group velocity (m/s); power-flow angle (°); polarization vector; stiffening increment (m/s)

Christoffel equation with piezoelectric stiffening, group velocity from its solution; velocities span 3474.0 to 7356.9 m/s. Stiffening lifts the Z-longitudinal velocity from 7102 to 7329 m/s (measured 7328.20) and leaves the Y-shear polarized along X unchanged.(IEEE Std 176-1987; Jaeken & Cottenier 2016; Kushibiki et al. 1999)

Why are 128° Y-X, 41° Y-X and 64° Y-X the standard LN SAW cuts?

rotated Y-cut angle θ 0 to 180° (step 0.5°), X propagation; data set: Kushibiki 1999, Ogi 2002, Warner 1967; branch: Rayleigh, leaky SAW

free and metallized surface velocity (m/s); coupling k² (%); leaky-SAW loss, open and shorted (dB per wavelength); slow and fast shear thresholds (m/s)

Partial-wave SAW solver for Rayleigh (real v) and leaky (complex v) waves on open and shorted surfaces, k² = 2(v_free − v_met)/v_free. Rayleigh k² peaks at 5.42% at 129°, where the leaky wave has faded; near 60 to 70° the leaky wave reaches 9 to 11%.(Campbell & Jones 1968; Ingebrigtsen 1969; Kushibiki et al. 1999; Ogi et al. 2002; Warner et al. 1967)

What does a surface wave look like inside the crystal, and how is a leaky wave different?

cut: Y-Z, 128° Y-X, 64° Y-X (leaky), X-Z, X-Y; surface: open, shorted; wave phase 0 to 360°

normalized u1, u2, u3 and potential φ versus depth (wavelengths); surface particle orbit and its sense; energy within 1 wavelength (%); power-flow angle (°); wavelength at 1 GHz (µm)

Eigenvector of the partial-wave SAW solution: four partial waves give displacement and potential versus depth from 0 to −3λ, matched at the surface to a vacuum potential decaying as exp(−k x3).(Campbell & Jones 1968; Kushibiki et al. 1999)

How does a comb of electrodes pick one frequency out of a radio signal?

cut and surface (velocity from the cut-angle sweep); electrode period p 0.5 to 20 µm; finger pairs N 1 to 200; metallization ratio (display only)

centre frequency f0 (MHz, GHz); finger width (µm); null-to-null fractional bandwidth (%); array-factor response |H(f)|² (dB)

f0 = v/p; array factor H = sin X/(N sin(X/N)) with X = Nπ(f − f0)/f0, conductance shape (sin X/X)², null-to-null bandwidth 2f0/N. Absolute conductance is not modelled; a measured TFLN point (1.9 mS, 3.36 MHz at 1.85 GHz) is a marker.(Kushibiki et al. 1999; Jiang et al. 2020)

How does a sound wave diffract light, and what sets the angle and the frequency shift?

optical wavelength λ0 400 to 1700 nm; RF frequency 10 MHz to 5 GHz (log); acoustic mode: X-longitudinal 6544.6 m/s, Z-shear 3590.4 m/s, 128° Y-X SAW 3978.4 m/s; optical polarization: o, e

acoustic wavelength Λ (µm); Bragg angle, internal and external (°); optical frequency shift (GHz); wavevector diagram; figures of merit M1, M2, M4

Λ = v/f and sin θ_B = λ0/(2nΛ) inside the crystal, external angle by Snell's law, n from the Zelmon Sellmeier; the diffracted order is shifted by +f. At 1550 nm and 1 GHz on X-longitudinal, Λ = 6.545 µm and θ_B,ext = 6.80°. Efficiency is not modelled.(Weis & Gaylord 1985; Zelmon et al. 1997; Kushibiki et al. 1999)

Which in-plane direction on an X-cut or Y-cut film gives the largest strain-optic response, and what happens if the p14/p41 signs are paired with the wrong frame?

cut: X, Y, Z; in-plane angle φ 0 to 180°; data set: Avakyants 1976 (Set B), Andrushchak 2009, Krupych 2014, Coquin 1971; flip p14 and p41 signs: on, off

longitudinal photoelastic coefficient p′_bbbb(φ) (dimensionless); its maximum and angle (°); frame of the data set

p′_bbbb = p_ijkl b_i b_j b_k b_l from the Voigt p (no factors). Flipping p14, p41 moves the X-cut optimum from about 45° to about 135° at unchanged size (Andrushchak: +0.304) and leaves Y-cut unchanged; on Z-cut p′_bbbb = p11 for all φ.(Avakyants et al. 1976; Andrushchak et al. 2009; Krupych et al. 2014; Coquin 1971; Weis & Gaylord 1985)

Where is the energy lost when a microwave signal is turned into light through a phonon?

intracavity photon number n_c 10² to 10⁵ (log); detuning: red, blue; C0 and η_m within one decade of the reported values

total efficiency η_total; cooperativity C; internal gain; efficiency per stage (microwave, IDT and waveguide, optomechanical, chip, fiber)

η_total = η_oc η_o η_m 4C/(1 ∓ C)² with C = C0 n_c (upper sign red, lower blue detuning), using reported η_oc = 0.652, η_o = 0.66, η_m = 9.9 × 10⁻⁴, C0 = 1.2 × 10⁻⁵. The blue side diverges at C = 1; the red side peaks at η_e = 4.24 × 10⁻⁴.(Jiang et al. 2020)

How can detecting one optical photon add exactly one photon to a microwave resonator?

protocol step (5 steps); data panel (published source-data files); heralded or unconditional, where the data provide both

published measured curves (units as in the source data); CW efficiency (%); heralding rate (Hz); thermal microwave photon number; thermal occupation at 2 GHz versus temperature

Step-through of the protocol: a blue-detuned pulse makes a photon-phonon pair, an optical click heralds it, and the phonon is swapped into the microwave resonator by piezoelectric coupling. Plots are rebuilt from the open source data (5% CW efficiency, 15 Hz heralding, < 2 thermal photons).(Jiang et al. 2023)

How can a piezoelectric crystal whose bulk coefficient is about 0.01 nm/V move a photonic cavity by tens of nm per volt?

aspect ratio R = L/w 1 to 100 (log); voltage U 0 to 4 V; actuator: plate, transverse extension, bender

displacement per volt for each actuator (nm/V, order of magnitude); reported optical tuning slope (nm/V) and span (nm)

Scaling law with prefactor 0.01 nm/V: plate Δ ~ U, transverse extension Δ ~ R U, bender Δ ≈ ½ d (dE/dz) L² ~ R² U, with R = L/w. Reported tuning: about 5 nm/V small-signal slope below 0.5 V and 1520 to 1560 nm within 4 V.(Jiang et al. 2020)

How large an index scar does a beam write, how fast, and how long does it last after the light goes off?

intensity 1e-2 to 1e6 W/cm²; exposure time 0.1 to 1e4 s; crystal: LiNbO3:Fe, congruent undoped, 5 mol% MgO, near-stoichiometric; dark conductivity 1e-16 to 1e-14 S/m; lights off

space-charge field |E| (V/m) versus time 1e-3 to 1e7 s; saturation field (V/m); Δn_e; relaxation time under light and in the dark (s); index profile across the beam

Reduced band transport: σ_ph = κI, E_sat = k_G α I/(σ_ph + σ_d); the field builds as E_sat(1 − exp(−t/τ)), τ = εε₀/(σ_ph + σ_d), and decays in the dark with τ = εε₀/σ_d; Δn_e = −½ n_e³ r33 E. α and κ are model parameters, no data.(Volk & Wöhlecke 2009; Weis & Gaylord 1985; Bryan et al. 1984; Chen 1969)

Why is a photovoltaic grating in phase with the light while a diffusion grating is shifted by a quarter period?

grating period 0.2 to 20 µm; modulation 0 to 0.95; photovoltaic weight 0 to 1× the Fe:LN Glass constant; temperature 77 to 400 K

intensity I(z); charge density ρ(z); space-charge field E_sc(z) (V/m); Δn(z); phase shift between I and Δn (deg); diffusion field amplitude (V/m)

1D transport over two fringe periods with photovoltaic current k_G α I, diffusion and drift σE, plus Gauss law and continuity, stepped implicitly to steady state; closed form E_dif = (k_B T/e) K m sin Kz/(1 + m cos Kz) overlaid.(Volk & Wöhlecke 2009; Weis & Gaylord 1985)

How much MgO removes photorefractive damage, and how does stoichiometry change it?

y axis log or linear; series: CLN, NSLN

photorefractive damage threshold intensity (kW/cm²) versus MgO concentration (mol%); GRIIRA and MgO threshold markers (mol%, at.%)

Data, no model: seven points (cw green, beam distortion after 10 min), lower bounds > 8000 kW/cm² drawn as arrows, dashed guides not fits; markers at GRIIRA thresholds about 5 % (CLN) and 1.5 % (NSLN), ≥ 4.5 at.% MgO and > 0.78 mol% (near-stoichiometric).(Volk & Wöhlecke 2009; Furukawa et al. 2001; Bryan et al. 1984; Furukawa et al. 2000)

Why does a TFLN ring's resonance deform into a triangle in one sweep direction, and how does photorefraction fight it?

pump power 0.01 to 100 mW; sweep direction: increasing or decreasing wavelength; sweep speed 1e-3 to 1e3 nm/s; photorefraction on/off; photorefractive strength; cladding: SiO2-clad, air-clad; dn_e/dT 3.34e-5 or 3.75e-5 /K

transmission versus laser wavelength for both sweep directions; detuning; cavity temperature rise (K) and space-charge field (V/m) versus time; thermal and photorefractive shifts (pm)

Coupled-mode amplitude da/dt = (iΔ − κ_tot/2)a + √κ_ex s_in; Δ shifts with cavity temperature and with an opposite-sign photorefractive term, each relaxing with its own time constant, τ_pr = εε₀/(σ_d + κI). Q, n_g, Γ and thermal and PR parameters are model parameters, no data.(Carmon et al. 2004; Lin et al. 2021; Xu et al. 2021; Moretti et al. 2005; Jundt 1997)

Where does green-induced infrared absorption come from, and how does it cut second-harmonic output?

green intensity 0 to 10 kW/cm²; MgO 0 to 6 %; crystal: CLN, NSLN; IR power

polaron absorption spectrum versus photon energy 0.3 to 3.5 eV; band center (eV, nm); HWHM (eV); polaron population, IR absorption, temperature rise and relative SHG (model values)

Bands drawn as Gaussians in photon energy at tabulated centers and HWHM, O⁻ as a marker. Toy chain (model parameters, no data): dN/dt = g I_g(1 − N/N_max) − N/τ_p, α_IR = σN, ΔT ∝ α_IR P_IR L, SHG_rel = sinc²(Δk(ΔT)L/2); N_max → 0 at the MgO threshold (5 % CLN, 1.5 % NSLN).(Schwesyg 2011; Volk & Wöhlecke 2009; Furukawa et al. 2001)

At what intensity does each kind of damage start?

regime filter: cw, pulsed; crystal highlight: CLN, MgO:CLN, NSLN, Fe:LN

threshold intensities on a log axis 1e-2 to 1e10 W/cm²; damage fluence on a separate axis 1e-1 to 1e2 J/cm²; vendor thresholds (MW/cm²) in a separate row

Data, no model: each sourced threshold placed by regime and definition, from photorefraction in Fe:LN at 10 mW/cm² to nanosecond breakdown in the GW/cm² range; fluence is never converted to intensity. Vendor 100-300 MW/cm² values are not a material constant.(Volk & Wöhlecke 2009; Furukawa et al. 1991)

How much charge and voltage does a temperature change put on a lithium niobate wafer or film, and does it matter next to the thermo-optic shift?

temperature step −50 to +200 K; cut: z-cut, x-cut; thickness: 0.5 mm wafer, 1 mm wafer, 600 nm film; |p3|: 4.0e-5, 6.4e-5, 8.3e-5, 9.5e-5 C/(m² K) or spread band; screening conductivity 1e-16 to 1e-6 S/m

bound surface charge (µC/cm²); unscreened field (V/m); voltage (kV); Δn_e; coercive field reference (kV/mm); screened field versus time

σ_b = p3 ΔT, E = −p3 ΔT/(ε33 ε₀) for an isolated unscreened plate, V = E d, Δn_e = −½ n_e³ r33 E, compared with (dn_e/dT) ΔT; screening E0 exp(−t/τ), τ = εε₀/σ. Linear in ΔT with the 25 °C coefficient; cooling makes the +c face positive.(Savage 1966; Popescu et al. 2013; Bartholomäus et al. 1994; Crystal Technology Inc. 1999; Weis & Gaylord 1985; Volk & Wöhlecke 2009)

Why does the bias point of a modulator walk, which way, and on what time scale?

preset: bulk-like, TFLN clad, TFLN air-clad; per-layer ε_r and conductivity 1e-18 to 1e-4 S/m; interface conductance on/off; temperature 77 to 400 K; waveform: step or square wave, period 1e-3 to 1e4 s

LN layer voltage V_LN(t) (V) over 1e-6 to 1e6 s; relaxation time (s); bias point on the MZM transfer curve; |V_LN/V| versus frequency 1 mHz to 1 MHz; drift sign

Cladding, LN and buried oxide in series, each C = ε_r ε₀/t and G = σ/t per unit area; V_LN moves from capacitive to resistive division, τ = (C1+C2)/(G1+G2) for two layers; bias φ = φ0 + πV_LN/V_π, T = cos²(φ/2); σ scaled by an Arrhenius factor (E_a a model parameter).(Li et al. 2026; Holzgrafe et al. 2024; Volk & Wöhlecke 2009)

How much do the indices move with temperature, how strong is the thermal lens, and how far does a ring resonance shift per kelvin?

wavelength 0.4 to 5 µm; temperature 20 to 200 °C; polarization o/e; power 0.1 to 100 W; beam radius 10 to 1000 µm; absorption 1e-4 to 1e-1 cm⁻¹; length 1 to 50 mm; thermal conductivity 3.92 (CLN) or 5.97 (near-SLN) W/(m K); Γ 0.5 to 1; n_g 2.0 to 2.4

dn_o/dT and dn_e/dT (1/K) versus wavelength; thermal lens focal length (mm) with f ≫ L validity flag; ring resonance shift (pm/K)

dn_e/dT from the Jundt temperature-dependent Sellmeier; dn_o/dT from the Crystal Technology Sellmeier and the Moretti 632 nm formula; thermal lens f = πw²K_th/(αPL dn/dT); ring shift λΓ(dn/dT)/n_g. Gayer 2008 MgO:LN coefficients are excluded from dn/dT.(Jundt 1997; Crystal Technology Inc. 1999; Moretti et al. 2005; Schwesyg 2011; Volk & Wöhlecke 2009)

What does losing lithium at the surface do to the index and the Curie temperature, and why does the depleted skin guide light?

surface Li2O deficit 0 to 1.5 mol%; depth scale √(Dt) 0.1 to 20 µm; wavelength: 632.8 or 1064 nm

Li2O deficit (mol%) versus depth; Δn_e and Δn_o from bulk versus depth; Curie temperature (°C) versus depth; guiding flags for e and o light

Deficit Δc(z) = Δc_s erfc(z/(2√(Dt))) with D and Δc_s as model parameters; indices from the Schlarb generalized Sellmeier (valid 400 to 1200 nm); Curie temperature T_c = 39.064c − 746.73 (Bordui fit, Iyi alternate); a skin guides when n(surface) > n(bulk).(Volk & Wöhlecke 2009)

What goes wrong in lithium niobate fabrication, why, and what fixes were used?

card: hotplate bake, 4 K cooldown, e-beam discharge layer, a-Si hard mask under ion milling, HF etch of −c and +c faces, z-cut interferometer drift; pyroelectric coefficient shared with the heated-wafer figure

per card: cause, mechanism, fix and status; unscreened bake voltage (kV) as a range over the pyroelectric coefficient spread

Practitioner notes paired with mechanism sources; the hotplate card computes V = |p3| ΔT d/(ε33 ε₀) for a 25 to 120 °C bake on a 0.5 mm z-cut wafer (7.5 to 17.8 kV unscreened); the cooldown card shows no number because p3 below room temperature is not in held sources.(Jiang 2025; Jiang 2024; Jiang 2018; Bulmer et al. 1986; Volk & Wöhlecke 2009; Savage 1966; Popescu et al. 2013)

What has been tried against bias drift since 1981, and what stability did each report?

platform: bulk Ti:LN, TFLN, TFLT; mitigation class: buffer/cladding, interface and anneal, surface treatment, material swap, temperature, heterogeneous, reset protocol, thermal bias

timeline 1980 to 2026 with one marker per reported result, grouped by mitigation class and shaped by platform; reported stability metric as text

Data, no model: rows from the chapter's drift table with year, platform, mitigation class and the metric as reported (e.g. < 0.3 % in 8 h, drift-free 12 h, < 1 dB over 46 h); metrics are not comparable, so there is no y axis.(Yamada & Minakata 1981; Gee et al. 1985; Bulmer et al. 1986; Nagata et al. 1996; Holzgrafe et al. 2024; Li et al. 2026; Yeh et al. 2026; Shi et al. 2025; Powell et al. 2024; Al Sayem et al. 2026; Lomonte et al. 2021; Warner et al. 2025)

What happens, in order and to scale, between a powder charge and a fibered thin-film LN modulator, and which numbers are known?

device route: ridge with DUV and SiO2 mask, or strip with DUV and DLC mask; process step 1 to 30; playback speed 0.5x to 4x, reverse, scrub; scale: illustrative or true scale

cross-section, die, wafer and boule views per step; process time (s, min, h); temperature (°C); film thickness (nm); implant energy (keV); fluence (cm⁻²); etch rate (nm/min); per-step parameters with provenance grade

Thirty-step charge-to-chip sequence with per-step models: pull length L = v·t, Gaussian He implant profile centred at 0.68 µm, Preston CMP dz/dt = Kp·P·v, etch depth = R·t with sidewall tan α_LN = S·tan α_mask; true scale is 600 nm LN on 4.7 µm SiO2 on 500 µm Si.(Luke et al. 2020; Li et al. 2023; Kharel et al. 2021; Guarino et al. 2007)

How fast does a 150 mm boule grow, and why is the melt not LiNbO3 of formula composition?

growth time 0 to 110 h; pull rate 1 to 2 mm/h; rotation 5 to 10 rpm; crystal composition 48 to 50 mol% Li2O

boule body length (mm); growth time (h); pull time (h); body mass (kg); Curie temperature Tc (°C) vs composition (mol% Li2O)

Body length L = v·t for a 153 mm × 110 mm boule (55 to 110 h pull, 9.4 kg body). Tc(c) from two linear fits, 39.064c − 746.73 (Bordui) and 39.26c − 760.67 (Iyi), against the congruent 48.45 mol% and the 1255 °C melting point.(Bordui et al. 1992; Iyi et al. 1992)

Which crystal axis is normal to an x-, y-, z-, or 128° Y-X cut wafer, and which electro-optic coefficient does a TE mode use on each?

cut: x, y, z or 128° Y-X; cut angle θ swept continuously from Y to 128° Y-X

wafer-normal unit vector; in-plane axes; Euler angles (deg); electro-optic coefficient engaged for TE and TM with in-plane or vertical field

Crystal axes are rotated into wafer axes in the IEEE 176-1987 frame, with one stated Euler convention; the engaged r coefficient follows from the rotated electro-optic tensor. Flat positions are vendor-specific and drawn as illustrative.(IEEE Std 176-1987)

How deep do Ti and proton-exchange waveguides form for a given temperature and time?

temperature 200 to 250 °C; exchange time (h, log toggle); rate convention: d²/t or D; polarization: TE or TM

proton-exchange depth d (µm) vs time; extraordinary index step; Ti recipe points (Ti thickness nm, temperature °C, time h)

Proton exchange d² = r(T)·t with r = 1.37 µm²/h at 249 °C and 0.37 µm²/h at 217 °C (x-cut); Arrhenius interpolation gives Eₐ = 0.90 eV, valid only from 217 to 249 °C. Only nₑ rises (+0.12); Ti is shown as recipe points, with no diffusion curve.(Jackel et al. 1982; Luo et al. 2019; Chung et al. 2017)

How does implant energy set film thickness, and what are the thermal steps from implant to split?

implant energy (keV)

projected range Rp (µm) vs implant energy; process temperature (°C) per step from implant to split and anneal

Range is shown only at reported points (He⁺ 195 keV gives a 0.68 µm peak; 680 nm film); no Rp(E) curve without a stored SRIM calculation. Thermal steps: implant at 100 °C, room-temperature bond, < 200 °C strengthening, about 230 °C split, 520 °C O2 anneal.(Guarino et al. 2007; Rabiei & Günter 2004; Wang et al. 2022; You et al. 2019; Shams-Ansari et al. 2022)

How much is polished off after the split, and how uniform is a 600 nm film across a 100 mm wafer?

polishing pressure P; pad velocity v

film thickness h (nm); thickness standard deviation σ (nm); removal target (nm); film thickness map; film interference colour

Preston removal dz/dt = Kp·P·v, with Kp a display value, not a material constant; target removal 0.68 − 0.60 µm = 80 nm. Map is illustrative with the reported σ of 5.9 nm (8 mm edge exclusion, 4-inch wafer); colour from thin-film interference with LN dispersion.(Preston 1927; Luke et al. 2020)

Why did wafer-scale TFLN need optical steppers?

pattern-area fraction; EBL preset: 7600 µC/cm² at 100 nA, 1900 µC/cm² × 4 passes at 8 nA, 300 µC/cm² at 2 nA; wafer diameter 100 or 150 mm

EBL write time (h/cm², h per wafer); DUV exposure time per wafer (min); exposure field count

EBL time t = D·A/I, giving 21.1 h/cm² at 7600 µC/cm² and 100 nA over the full area, against under 1 min per wafer for DUV. 22 × 22 mm fields tiled inside an 8 mm edge exclusion give 6 fields on 100 mm and 21 on 150 mm.(Kaufmann et al. 2023; Jiang & Lin 2016; Luke et al. 2020)

What sets the sidewall angle of an Ar-etched LN waveguide, and when does redeposition stop?

etch selectivity S = R_LN/R_mask, 1 to 3; mask sidewall angle (deg, default 85°); SC-1 clean on or off

model LN sidewall angle (deg, upper bound); nearest measured sidewall angle (deg); redeposition regime map vs DC bias (100 to 800 V) and pressure (0.3 to 15 mTorr)

Mask-edge recession gives tan α_LN = S·tan α_mask (S = 3 for DLC, 1 for SiO2), compared with measured about 60° (SiO2), about 75° (Cr) and 80° (DLC). The regime map plots reported redeposition-free points (5 to 7 mTorr at 600 V, about 11 mTorr at 400 V) with no fitted surface.(Li et al. 2023; Kaufmann et al. 2023; Krasnokutska et al. 2018)

Which fabrication routes produced the lowest loss, and when?

log axis toggle; route marker legend

propagation loss (dB/m) vs year; intrinsic Q; fabrication route; source and status per point

Best reported loss per route (EBL with Ar etch, DUV with Ar etch, DUV with DLC mask, PLACE, CMP from bulk) vs year; Q converted to loss at 1550 nm with the stated group index. Material limit about 0.2 dB/m, reached by annealing from 1.5 dB/m.(Zhang et al. 2017; Zhu et al. 2024; Luke et al. 2020; Li et al. 2023; Kaufmann et al. 2023; Gao et al. 2022; Shams-Ansari et al. 2022)

How does a comb electrode on an x-cut film flip domains, and what limits the period?

poling period presets 4.38, 5.51, 0.747, 0.6 µm; threshold field (default 21 kV/mm, marks at 28.2 kV/mm and 3 Ec); voltage pulse

electric field vs depth (kV/mm); peak field (kV/mm); duty cycle D; poled depth fraction; effective nonlinear coefficient d_eff

Electrostatics of coplanar comb electrodes on LN/SiO2/Si; domains grow where |E| exceeds the threshold. d_eff = (2/π)·d33·sin(πD) × depth overlap; nucleation and wall-motion kinetics are illustrative (no thin-film parameters sourced).(Chen et al. 2024; Zhao et al. 2020; Nagy & Reano 2020)

Where do the atoms in a modulator come from, and how much niobium does a wafer hold?

wafer: 76.2 mm × 0.5 mm, 100 mm × 0.35 mm, 150 mm × 0.5 mm, 200 mm × 0.5 mm

LN mass per wafer (g); Nb mass per wafer (g); world Nb mine production (t/yr); world Li production (t/yr)

Wafer mass = volume × 4647 kg/m³ (flats ignored); Nb is 62.84 wt% of LiNbO3, so a 200 mm wafer holds 45.9 g Nb against 112,000 t of world output. Chain: pyrochlore and brine or spodumene, Nb2O5, Li2CO3, boule, wafer, LNOI, foundry, chip, module.(USGS 2026)

Which LN effect does each device family use, and how mature is it?

effect column (hover highlights the families that use it); device family (click opens its section)

11 × 6 grid of device families against effects: Pockels, χ⁽²⁾, χ⁽³⁾, piezoelectric, acousto-optic, heterogeneous gain/absorption; maturity tag: commercial, sampling or in development, research

No physics: a lookup of 11 device families against six physical effects. A commercial or sampling tag appears only where a dated company statement exists; otherwise the family is tagged research.(Exail; AFR 2025; Summit Partners 2026; Murata 2026; Optica 2025; The Quantum Insider 2025; Aghaee Rad et al. 2025)

How does a voltage on a Mach-Zehnder become bits, and what do Vπ and bandwidth do to the eye?

Vπ: 2.0 V template or an atlas device; bias phase φB −π to π; drive Vpp 0.1 to 2 Vπ; symbol rate 10 to 250 GBd; format NRZ or PAM4

transfer T(V) with operating point and swing; optical eye; Vπ (V); bias (deg); Vpp/Vπ; static extinction ratio (dB); eye height (normalized)

T = ½(1 + cos(πV/Vπ + φB)). A pseudo-random bit sequence drive, band-limited at the 3 dB bandwidth, passes through T(V) and is folded over two symbol periods to form the eye.(eo-modulator-atlas 2026)

What sets VπL, and what mode-field overlap do measured devices imply?

electrode gap 1 to 15 µm; overlap Γ 0.1 to 1.0; wavelength 0.4 to 4 µm; r33: 30.8 pm/V (Turner 1966) or 28.8 pm/V (Kaminow 1980)

VπL (V·cm); n_e; Γ; implied overlap Γ_implied vs electrode gap (µm) for 71 x-cut TFLN push-pull MZM rows

Push-pull lateral-gap model VπL = λg / (2 n_e³ r33 Γ), with n_e from the congruent LN Sellmeier at 21 °C and the clamped 633 nm r33. Inverting it with each device's reported VπL gives Γ_implied; Γ > 1 marks devices outside the model.(Weis & Gaylord 1985 (Turner 1966; Kaminow 1980); Zelmon et al. 1997; eo-modulator-atlas 2026)

How do velocity mismatch, RF loss, and impedance mismatch set the bandwidth of a traveling-wave modulator?

electrode length 0.5 to 40 mm; RF index 1.8 to 4.5; optical group index 2.0 to 4.5; RF loss 0 to 15 dB/cm at 67 GHz; loss exponent 0.5 (skin effect) or 1.0 (dielectric); Z0 20 to 80 Ω; load 20 to 80 Ω or = Z0; source 50 Ω; atlas device presets

EO S21 (dB) vs frequency 0 to 300 GHz; f3dB (GHz); |nRF − ng|; total RF loss at f3dB (dB); velocity-limited and loss-limited frequencies (GHz)

Overlap of a lossy RF wave, α ∝ f^k, with the co-propagating optical wave, including source and load reflections. Single-effect limits: f_vel = 0.443c / (L|nRF − ng|) and the frequency where total RF loss reaches 6.4 dB.(Chung et al. 1991; eo-modulator-atlas 2026)

Which devices are both efficient and fast, on which platform, and when?

year 2017 to 2026; platform toggles; all bands; all VπL conventions (hollow markers); design rows (crosses)

VπL (V·cm, log 0.01 to 200) vs 3 dB bandwidth (GHz, log 0.5 to 400); bandwidth lower bounds as right-pointing arrows

Data plot of published telecom push-pull MZM and IQ device rows. VπL is the reported DC value, else Vπ × length, else the RF value, never converted between conventions; no frontier fit is drawn.(eo-modulator-atlas 2026)

How long must a modulator be for a given drive voltage, and how much easier is modulation at shorter wavelengths?

view: footprint or wavelength; electrode gap and overlap Γ for the model curve (template, shared with the VπL calculator)

footprint: Vπ (V, log 0.1 to 30) vs electrode length (mm, log 0.01 to 50) with iso-VπL lines 0.1 to 10 V·cm; wavelength: VπL (V·cm, log 0.03 to 30) vs wavelength (nm, log 300 to 5000)

Atlas push-pull device rows; iso lines Vπ (V) = 10 VπL (V·cm) / L (mm). The wavelength view (TFLN, TFLT) overlays VπL(λ) = λg / (2 n_e(λ)³ r33 Γ).(eo-modulator-atlas 2026; Zelmon et al. 1997; Weis & Gaylord 1985)

Why do resonant modulators trade speed for efficiency?

loaded Q 2×10³ to 2×10⁵ (log); tuning 1 to 30 pm/V (template); voltage −5 to 5 V; laser detuning (linewidths)

transmission vs wavelength offset (pm); loaded Q; linewidth (GHz); photon lifetime (ps); resonance shift (pm); transmission at the laser; atlas 3 dB bandwidth (GHz) vs loaded Q

All-pass ring T = (a² − 2ar cos φ + r²) / (1 − 2ar cos φ + (ar)²), resonance shifted by tuning × V. Linewidth ν0/Q_L and photon lifetime Q_L/(2πν0) set the speed limit.(Wang et al. 2018; eo-modulator-atlas 2026)

How much of a pump does a TFLN PPLN waveguide convert, and when does it saturate?

normalized efficiency 100 to 10,000 %/W/cm² (log); length 1 to 30 mm; pump power 0.1 to 1000 mW (log); pump wavelength 1.40 to 1.70 µm

second-harmonic power vs pump power (mW, log-log), undepleted and depleted; ηL² (%/W); P2 (mW); conversion (%); bulk QPM period (µm); phase-mismatch sinc² vs pump wavelength

Undepleted P2 = ηL²P1²; depleted, lossless and phase matched P2 = P1 tanh²(√(ηP1) L). QPM period Λ = λ1 / (2(n2 − n1)) from bulk Sellmeier indices or a template effective-index offset.(Wang et al. 2018; Chen et al. 2024; Li et al. 2024)

Why does a parametric oscillator switch on abruptly, and how low can its threshold go on a chip?

intrinsic Q 10⁵ to 10⁸ (log); coupling regime; nonlinear coupling g (template); pump power 1 µW to 10 mW (log)

signal power vs pump power (log-log); threshold power (µW); reported device thresholds (µW)

Coupled-mode model of a doubly or triply resonant OPO: output is exactly zero below threshold (classical, no spontaneous emission) and follows the steady-state solution above. Threshold scales as 1/Q² (doubly resonant, equal Q, critical coupling).(Lu et al. 2021; Gao et al. 2021; McKenna et al. 2022)

How does a microwave drive on a low-loss ring make hundreds of evenly spaced lines?

modulation index β = πV/Vπ, 0 to π; intrinsic Q (log); drive frequency = free spectral range, 10 GHz (template); dispersion on or off

comb line power (dB) vs frequency offset; lines above −60 dB; span (nm)

Steady state of a ring modulated at its free spectral range, from a coupled-mode recursion of sideband amplitudes with round-trip loss, detuning and dispersion; on a dB axis the comb envelope is triangular.(Zhang et al. 2019)

How far have LN microwave-to-optical transducers come, and at what temperature?

total efficiency only (toggle); log scale (default on)

conversion efficiency (log, 6.6×10⁻⁶ to 8.7%) vs year; mechanism; temperature class (cryogenic or room); efficiency definition (total, internal, on-chip)

Data plot of 8 reported transducers (cavity EO, piezo-optomechanical, bulk LN EO). Total, internal and on-chip efficiencies are tagged separately and never merged.(Holzgrafe et al. 2020; McKenna et al. 2020; Jiang et al. 2020; Xu et al. 2021; Sahu et al. 2022; Jiang et al. 2023; Weaver et al. 2024; Warner et al. 2025)

How does a voltage on comb electrodes make a sound wave that modulates light?

frequency 0.5 to 6 GHz; crystal cut (values from the crystal chapter)

acoustic wavelength (µm); IDT finger period (µm); top view of fingers and wave crests crossing a waveguide

λ_ac = v/f; a single-electrode IDT period equals λ_ac (two fingers per period). v is the SAW velocity of the chosen cut, or a template display value, not a material constant. The photoelastic index-change sign is qualitative.

When did each dated event in the history of lithium niobate happen, and how well is each one sourced?

chapter period: all or one of ten; status: V, P, S, derived; region: USSR, Japan, Europe, China, Ti:LN; sort by date ascending or descending

event date and date precision; title; people; institution; DOI or URL; status grade; event counts by status; per-year event strip 1801 to 2026

Chronology of 162 dated events, each with date precision, people, institution, citation and a status grade (V confirmed by the record itself, P via a secondary link, S secondary only, derived from cited data); counts recompute from the filtered set.

How are the names tantalum, niobium and lithium related, and where was each one coined?

select a node: Tantalus, Niobe, Pelops, Ta, Nb, Li, LiNbO3, LiTaO3

coining sentence (original language and English gloss); author; year; page; source status

Tree of mythic and chemical relations: Tantalus to his children Niobe and Pelops, the elements Ta and Nb named after them, Li from Greek lithos, and the compounds LiNbO3 and LiTaO3; each name links to the page where it was coined.(Rose 1844; Rose 1846; Arfwedson 1818; Hatchett 1802; Ekeberg 1802)

How is a crystal boule pulled from a melt on a rotating seed?

pull rate (display range); crystal rotation on or off

animated boule growing from a seed out of the melt (illustration, no material data)

Schematic Czochralski growth: a seed drawn up from the melt grows into a boule. Growth rates and diameters are not sourced in this chapter, so the pull rate is a display value, not a material constant.

What signal and idler wavelengths did the 1965 and 1995 LN optical parametric oscillators reach, and how does a computed tuning curve compare?

mode: 1965 (0.529 µm pump, birefringent phase matching) or 1995 (1.064 µm pump, quasi-phase matching); crystal temperature; QPM period 26 to 32 µm (1995, model extension)

signal wavelength (µm); idler wavelength (µm); computed tuning curve; reported tuning bands (µm)

Signal and idler from energy conservation and phase matching with LN Sellmeier dispersion, compared with reported bands: 0.97 to 1.15 µm (1965, not an energy-conjugate pair), 1.66 to 2.95 µm at 31 µm period and 1.4 to 4 µm overall (1995).(Bell Laboratories Record 1965; Myers et al. 1995)

How did reading, thermal fixing and MgO doping change a hologram written in LN?

532 nm brush drawn on the plate; read (1968); fix (1971); MgO (1980); reset; diffraction preview on or off

refractive-index map (phase image); diffraction preview

One-trap rate equation for the photorefractive index change, with the decay time a display value. Reading erases, fixing stops decay, MgO raises photoconductivity a hundredfold so the trace is weaker and fades faster; Δn about 10⁻³ is quoted, not simulated.(Chen et al. 1968; Chen 1969; Amodei & Staebler 1971; Bryan et al. 1984)

How does the IDT finger period set the centre frequency of a surface acoustic wave on quartz versus LN?

IDT finger period p; substrate: quartz or LN

centre frequency f0 (Hz); SAW velocity v_SAW (m/s); electromechanical coupling K²

Centre frequency f0 = v_SAW / p, with SAW velocity and coupling taken from the material-property data with their source grades; the 1965 White and Voltmer device is labelled as quartz.(White & Voltmer 1965)

What index profile and guided mode do Li2O out-diffusion, Ti in-diffusion and proton exchange produce for a given time and temperature?

process: Li2O out-diffusion (1973), Ti in-diffusion (1974), proton exchange (1982); diffusion time; temperature; Δn axis linear or log

index change Δn vs depth (0 to 20 µm); fundamental mode intensity

Model diffusion profiles between sourced anchors: a 12 µm single-mode out-diffused guide, about 1 µm Ti confinement, and a Δnₑ = 0.12 proton-exchange step with diffusion constants 1.37 and 0.37 µm²/h.(Carruthers et al. 1974; Schmidt & Kaminow 1974; Jackel et al. 1982)

How do a 1973 bulk LN waveguide and a 2018 thin-film rib compare in size, drive voltage and bandwidth?

field of view 50 to 1 µm

guide size (µm); half-wave voltage Vπ (V); 3 dB bandwidth (GHz)

Same-scale cross-sections of a graded diffused skin in bulk LN and a sub-micrometre rib on SiO2 on Si: guide 12 µm vs 0.68 µm, Vπ about 24 V (derived from 0.13 rad/V) vs 1.4 V, bandwidth 3.2 GHz vs N/A.(Kaminow et al. 1973; Carruthers et al. 1974; Rabiei & Günter 2004; Wang et al. 2018)

How did periodic domain reversal evolve from proposal to nanophotonic PPLN, and why does it make second-harmonic power grow?

year 1962 to 2018 (snaps at 1962, 1980, 1990, 1993, 2018); fundamental wavelength; log axis toggle

domain pattern; phasor sum of SH contributions; SH power vs number of domains; first-order QPM period (µm)

Each coherence length adds one SH phasor; with domain reversal the sum grows and power scales as N², without it the sum stays bounded and oscillates. First-order period from Sellmeier dispersion, about 6.8 µm at 1064 nm and 18.99 µm at 1550 nm.(Armstrong et al. 1962; Feng et al. 1980; Magel et al. 1990; Yamada et al. 1993; Wang et al. 2018)

What line rates did dated sources state for LN modulators from 1997 to 2022?

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line rate (Gb/s, log) vs year 1985 to 2025; source status per point

Step chart of rates stated in dated sources only, with no interpolation or trend: 10 Gb/s (1997), 20 Gb/s (1998), 40 Gb/s (2000), 100 Gb/s (2009, vendor), 1960 Gb/s single-wavelength net (2022, research).(Lin & Sanders 1997; Lightwave 1998; Wooten et al. 2000; Fujitsu 2009; Xu et al. 2022)

How does crystal ion slicing lift a thin LN film off a bulk crystal?

implant energy (illustrative scale)

implant depth (µm); final film thickness (µm, nm)

Three steps on one cross-section: implant, bond, split and polish. The depth scale is illustrative except the sourced pair 3.8 MeV to about 10 µm; sourced films are 9 µm (1998) and 680 nm (2004).(Levy et al. 1998; Levy, Osgood & Radojevic, US 6540827; Rabiei & Günter 2004)

How did thin-film LN modulators change the trade-off between VπL and bandwidth relative to bulk Ti:LN?

publication-date cutoff (year slider, rule at 2018-10); log axes toggle

VπL (V·cm); 3 dB bandwidth (GHz); device and source per point

Scatter of modulator devices from the electro-optic modulator compilation, bulk Ti:LN and thin-film LN by marker shape, filtered by publication date.(Wang et al. 2018)

What has happened in LN photonics since 2023, by date, kind and source grade?

kind: paper, foundry, funding, product, standard; region

event date; title; status grade; kind; source link; vendor-claim tag

Newest-first list of dated events from the chronology, with vendor claims tagged separately from independently confirmed results.

Who did it first in each of twelve contested priority claims in LN history?

expand a dispute

question; competing dated claims with status grades; verdict

Twelve disputes from the history text, each laid out as dated competing claims with their evidence grades; the side with stronger evidence is labelled, not visually weighted.

Who are the people named in the history of lithium niobate?

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one cell per person: portrait where licensed, otherwise initials

Grid from the history's people table; portraits only where the image licence allows (Rose, Berzelius, Hatchett, Czochralski, Ashkin), and no generated portraits.

Where does an LN chip come from, step by step, and who does each step?

node selection (10 nodes, ore to system)

directed flow: ore (Nb, Li) → Nb2O5, Li2CO3 → crystal → bulk wafer → thin-film wafer → foundry → chip → module → system; node cards with 2 to 4 graded facts; Nb per 6-inch wafer (g)

No physics: a graded fact map; each fact carries an evidence grade (F fact, V vendor claim, T trade press, E estimate) and a source. Nb in a 150 mm × 0.5 mm wafer is an estimate (25.8 g).(USGS 2026; Sumitomo Metal Mining; Shin-Etsu Chemical; TDG Holding 2026; NanoLN; Soitec; X-FAB 2026; AFR 2025; Murata 2026; Business Wire China 2026; Kushibiki et al. 1999)

Who grows crystals, makes thin-film wafers, runs foundries, and builds devices, by region?

stage (hover highlights companies in several stages); company tag (opens its fact card)

matrix of stage (crystal, thin-film wafer, foundry, device, PPLN, quantum/computing) by region (US, EU/CH, CN, JP, TW, UK, CA); company tags with evidence grade; dated facts

No physics: companies with dated facts graded F (fact), V (vendor claim) or T (trade press), from pages opened and read; snippet-only (C) entries are excluded.(Sumitomo Metal Mining; Shin-Etsu Chemical; TDG Holding 2026; NanoLN; Soitec; X-FAB 2026; Summit Partners 2026; AFR 2025; Murata 2026; Exail; optics.org 2025)

How large have LN wafers become, and who said so?

product filter: bulk or thin film

to-scale wafer outlines 76.2, 100, 150, 200 mm (300 mm dashed, low grade); 12 dated, graded milestones; gross die count of a 5 × 1 mm die on 150 and 200 mm wafers

Milestones placed on their wafer size by date. Area ratio (200/150)² = 1.778; die count is a derived gross-die estimate with a 3 mm edge exclusion (template).(Luke et al. 2020; Liu et al. 2025; Zhang et al. 2026; optics.org 2025; C114 2025; Yicai 2024; X-FAB 2026; UMC 2026; TDG Holding 2026; NanoLN)

How fast are LN transmitters in research, and what is shipping?

hover on a product milestone (highlights the research points behind it)

yearly maximum symbol rate (GBd), TFLN and TFLT, 2017 to 2026; graded product milestones: 200 G/lane available, 400 G/lane sampling, 130 GBd coherent driver modulator volume shipment, 1.6T reference transceiver

Research panel: highest atlas symbol rate per year. Product panel: dated milestones graded F, V or T. PAM4 at about 200 GBd carries 400 Gb/s gross.(eo-modulator-atlas 2026; AFR 2025; Business Wire China 2026; Summit Partners 2026)

How does LN compare with other electro-optic platforms in the atlas sample?

tab: VπL vs bandwidth, symbol and line rate, papers per year, wafer suppliers, integration mode, footprint; year 2017 to 2026; metric: symbol, line or net rate; IM/DD or coherent; exclude OFC 2026 papers

VπL (V·cm) vs 3 dB bandwidth (GHz); yearly maximum symbol rate (GBd), line and net rate (Gb/s); papers per year by platform; wafer-supplier shares (papers); papers per year by integration mode; electrode length (mm)

Counts, maxima and medians over the atlas sample at a pinned commit (233 papers, 486 device rows); bounds drawn as arrows, VπL in push-pull convention only. A curated sample, not a census.(eo-modulator-atlas 2026; photonics-atlas 2026)

What did LN and LT look like at OFC 2026?

theme filter: 400G/lane IM/DD; manufacturing (200 mm, wafer scale); co-design with electronics; beyond C band; TFLT

15 papers: code, short title, affiliations, abstract claim, DOI; postdeadline marker (Th4A.2, Th4A.5, Th4A.6, Th4B.2)

No physics: a table of 15 OFC 2026 papers with the authors' abstract claims and Crossref-verified DOIs, tagged by five themes.(OFC 2026 proceedings (15 papers))

What do published market forecasts say, and how much do they disagree?

base year only (toggle)

one bar per publisher from base value to forecast value (US$, log 10 M to 10 B) with base and forecast years; min-max bracket per product definition; China-only value in CNY

No model: teaser-page figures plotted as published, grouped by definition (TFLN modulator, TFLN devices, bulk LN crystal, China only), all low grade; no sum, average, consensus or currency conversion; the GII outlier sits apart.(Valuates Reports 2025; 24marketreports 2025; Intel Market Research 2026; QYResearch; Astute Analytica 2025; Dataintelo; Zhiyan Consulting 2025; GII)

Where does a non-specialist meet LN?

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10 cards: object, hook, caveat, evidence grade, source

No physics: ten everyday uses of LN, each with an always-visible caveat and its grade. Most SAW filters use LT, and BAW filters use AlN, not LN.(Microwave Journal 2018; Soitec; Sumitomo Metal Mining; Business Wire China 2026; Exail; Laser Focus World 2010; The Quantum Insider 2025; Evertiq 2025; Korth Kristalle; USGS 2026)

What happened in the LN industry, when, and how sure are we?

evidence grade (C rows hidden by default); region; stage

dated event marks with grade glyphs on a 2020 to 2026 time axis; event table: date, event, entities, region, source, access date, grade

No physics: 25 dated events graded F (fact), V (vendor claim) or T (trade press), each with source URL and access date; snippet-only (C) events appear only by toggle.(eeNews Europe 2020; Murata 2022; Wang et al. 2024; Lightwave 2024; AFR 2025; optics.org 2025; PIC Magazine 2025; X-FAB 2026; Murata 2026; UMC 2026; Business Wire China 2026; Summit Partners 2026; CSEM 2026; TDG Holding 2026)

What are LN's competing modulator platforms, and what does each do better?

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per platform (TFLN, TFLT, silicon photonics, InP, barium titanate, polymer): median VπL (V·cm); median 3 dB bandwidth (GHz) with lower-bound share; median electrode length (mm); row counts

Medians over telecom MZM and IQ device rows in the atlas sample; VπL in push-pull convention only (silicon photonics has none); medians that include lower bounds print as ≥.(eo-modulator-atlas 2026; Wang et al. 2024; The Quantum Insider 2025)

How much niobium does an LN wafer contain, and how much of world supply would an industry need?

wafer diameter 76.2 to 300 mm (3, 4, 6, 8, 12 inch presets); thickness 0.25 to 1.0 mm (template); wafers per year 10³ to 10⁷ (log); loss factor 1 to 5 (template)

LN mass per wafer (g); Nb mass per wafer (g); Nb per year (t); share of world mine output (%)

m_LN = ρπ(D/2)²t with ρ = 4,642.8 kg/m³; m_Nb = m_LN × M_Nb/M_LiNbO3 (62.8%); share = N × m_Nb × loss factor / 112,000 t world mine production (2025e). Outputs are estimates.(Kushibiki et al. 1999; USGS 2026; TDG Holding 2026)