2.1 Birefringence and the index ellipsoid

LiNbO₃ is optically uniaxial and negative: light polarized along the c axis sees the extraordinary index n_e, light polarized perpendicular to it sees the ordinary index n_o, and n_e < n_o. At 1550 nm, n_o = 2.2111 and n_e = 2.1376, a birefringence of −0.07355 P-db derived.

Between crossed polarizers in a cone of convergent light (conoscopy), a z-cut plate shows a dark cross (isogyres) and concentric dark rings (isochromes). For a 1.00 mm plate at 632.8 nm the first three dark rings sit at 11.1, 15.7 and 19.4° from the axis, in air derived. A field across the plate, along x or y, makes the crystal biaxial through r₂₂: the single optic axis splits in two and the cross opens into two hyperbolic brushes around the melatopes.

Index ellipsoid

With no field, in the crystal frame (IEEE Std 176-1987, z = c) P, the index ellipsoid is

A wave whose wave vector makes the angle θ with c has one ordinary index n_o and one extraordinary index

The extraordinary Poynting vector walks off the wave vector by ρ, tan ρ = (n_e(θ)²/2)(1/n_e² − 1/n_o²) sin 2θ. Largest walk-off: 2.150° at θ = 43.92° (632.8 nm), 2.000° at 44.00° (1064 nm), 1.938° at 44.03° (1550 nm) derived.

                                c                               
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                                |                 k             
                                |              ***              
                                |             **                
                                |           **                  
                    --·····;;;;;;;;:::::,,**          ++ S      
                  =|·=====·····;;;;:::::***,,.    ++++          
                ===|========;;····;:::***,,,,.++++              
              =====||======;;;;;;···***:,,++++...y              
           --=======\\====;;;;;;;:***·++++,,.......             
             ========\\==;;;;;;;:*++++··,,.........             
            ;=========\\\;;;;;::++::,,,···..........            
             ;;;;;;;;;;;\\·::::::,,,,,...··........             
             :;;;;;-;;;:::\\·:,,,,,.......··...------           
              ,----::::::,,,\\·,...........··.....    x         
                ,,,,,,,,,,,...\\\·..........·...                
                  ...............\\\··.....··.                  
                      ...............-------                    
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Dispersion

Zelmon et al. fitted congruent LN (CLN) at 21 °C over 0.4 to 5.0 µm with a three-term Sellmeier equation P-db:

Coefficients transcribed from the paper by refractiveindex.info (CC0) P-db; the paper itself is not held.

Computed indices, 21 °C derived

The birefringence shrinks by a factor 2.2 from 0.4 to 5 µm. The group index has its minimum where material group-velocity dispersion (GVD) changes sign: 1.913 µm (e) and 1.899 µm (o) derived.

Check against measurement: Nelson and Mikulyak measured n_o = 2.2866, n_e = 2.2028 at 632.8 nm and 24.5 °C, and 2.2323, 2.1561 at 1060 nm P-comp. Zelmon gives 2.2865, 2.2022 at 632.8 nm: n_o differs by −0.0001, n_e by −0.0006.

Models and errata

At 1550 nm: Jundt n_e = 2.13786 at 24.5 °C vs Zelmon 2.13756 at 21 °C derived. Gayer 5 mol% MgO: n_e = 2.13057, n_o = 2.20881 at 24.5 °C derived. Composition moves n_e far more than n_o: near-stoichiometric LN (SLN) has n_o = 2.288, n_e = 2.190 at 633 nm P-comp, so n_e drops by 0.012 relative to CLN while n_o changes by +1.5×10⁻³; the generalized Sellmeier gives 2.2859 and 2.1867 at 50.0 mol% Li₂O S.

Thermo-optic coefficients

n_e is about ten times more temperature-sensitive than n_o, so heating a plate changes its birefringence almost entirely through n_e. Moretti's measured range is 320 to 515 K in the abstract and 300 to 515 K in the title reported.

Conoscope

Each pixel is one propagation direction (u, v) = sin θ_air (cos φ, sin φ). Each eigenwave is refracted with its own index (the extraordinary angle by fixed-point iteration of eq. 2.2), so the retardation Γ = (2πd/λ)(n_slow cos θ_slow − n_fast cos θ_fast) is exact for plane waves. Between crossed polarizers I = sin²(2ψ) sin²(Γ/2), ψ the azimuth of the eigen-polarization relative to the polarizer: dark rings at Γ = 2πm, isogyres where ψ = 0 or 90°. Fresnel transmission differences between the eigenwaves are neglected. White light: I(λ) summed over eleven Gaussian-weighted samples from 420 to 680 nm [Born & Wolf]. The ring angles in L1 come from the exact phase, not the small-angle formula.

Field-induced biaxiality

For E along x or y on a z-cut plate, the in-plane impermeability splits by r₂₂E, so the birefringence for propagation along z is n_o³r₂₂E. At 632.8 nm, E = 1 kV/mm and r₂₂ = 3.4 pm/V (clamped; Turner 1966 and Kaminow et al. 1980 agree P-comp), this is 4.064×10⁻⁵, a retardation of 0.404 rad for d = 1 mm derived. To first order the optic axes tilt from c by θ with (n_o − n_e) sin²θ = n_o³r₂₂E: 1.26° inside, 2.88° in air. The exact eigen-decomposition with the full tensor gives 1.27° / 2.91° for E along x and 1.22° / 2.80° for E along y (Kaminow set, r₅₁ = 18.2 pm/V); the Turner set (r₅₁ = 28 pm/V) gives 1.30° / 2.98° for E along x derived. The difference between x and y comes from r₅₁, which the first-order formula omits; it reaches about 3%. A static field on an unclamped plate sees the free coefficients (Zook et al.: r₂₂ = 6.81 pm/V P-comp), which roughly double the effect.

Evaluating Sellmeier models

Inputs are vacuum wavelength in µm and temperature in °C; every model refuses input outside its stated window. Tests: n_e < n_o over the whole window (catches the 5 mol% MgO swap); Zelmon CLN at 1.55 µm: n_o = 2.21111, n_e = 2.13756. Group index and GVD from analytic derivatives of n²(λ).

Error budget at 1550 nm derived

Zelmon (21 °C) and Jundt (24.5 °C) differ by 3.0×10⁻⁴ in n_e; 3.5 K × dn_e/dT gives 1.3×10⁻⁴ of that, the remaining 1.7×10⁻⁴ is model-to-model spread. This spread moves the first-order SHG QPM period at 1550 nm by 13 nm (18.989 vs 18.976 µm). Composition and doping dominate every other uncertainty for n_e (0.012 between CLN and SLN at 633 nm); indices here are for CLN unless another variant (SLN, 5 mol% MgO:CLN, 1% MgO:SLN) is named.

Primary-data caveats

Weis & Gaylord Table 2 misprints n_o at 472.7 nm as 3.3597; the neighbouring rows give 2.3597 P. Machine ingestion of that table needs the fix. The Zelmon and Jundt papers are not held: their coefficients are database (P-db) and secondary (S) transcriptions.