Liquid Crystal Polarizer

Model transmission through twisted nematic (TN), super-twisted nematic (STN), vertically aligned (VA), and electrically controlled birefringence (ECB) LC cells.

Model: Textbook approximation

Γ = 2πΔn·d/λ, X = √(Φ² + (Γ/2)²), J = R(−Φ)·[cos X·I + (sin X/X)·[[−iΓ/2, Φ], [−Φ, iΓ/2]]]

90° TN, polarizer along the entrance director, parallel analyzer: T∥ = sin²X/(1 + u²), u = 2Δn·d/λ (Gooch & Tarry)

VA: Δn_eff = n(θ_tilt) − n_o, 1/n(θ)² = cos²θ/n_o² + sin²θ/n_e²

CellWavelength (nm)Δn = nₑ − nₒCell thickness (μm)Twist angle Φ (°)Polarizer (° from entrance director)Analyzer (° from entrance director)

Retardance Γ

11.424 rad

1.818 waves

Δn·d/λ

1.818

first minimum at √3/2 = 0.866: d = 2.382 μm

Transmission

0.87 %

of the light after the first polarizer

Configuration

twist 90°

P 0°, A 0°

Transmission against wavelength (Δn held constant)

Transmission against Δn·d/λ

Angles are counter-clockwise looking into the beam, from the entrance director (VA: from the plane the director leans in). Not modelled: pretilt, the field-dependent director profile, dispersion of Δn, oblique viewing, reflections.

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

Uniformly twisted layer in the cell plane (Jones matrix, Gooch-Tarry), or a uniformly tilted VA layer; ideal polarizers, normal incidence. Not modelled: pretilt, field-driven director profile, Δn dispersion.

Gooch C. H., Tarry H. A. (1975). The optical properties of twisted nematic liquid crystal structures with twist angles ≤90 degrees. J. Phys. D: Appl. Phys. 8, 1575–1584. doi:10.1088/0022-3727/8/13/020
Yeh P., Gu C. (1999). Optics of Liquid Crystal Displays: Jones matrix of a twisted nematic layer. Wiley.
Born M., Wolf E. (1999). Principles of Optics, 7th ed., ch. 15 Optics of crystals, pp. 790–852. Cambridge University Press. doi:10.1017/CBO9781139644181.024

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