Optics 079 · Polarization, anisotropy, and modulation

LCD Pixel Microscope

A 3D TN pixel cutaway with relaxed director fields, layer-by-layer Jones propagation, RGB aperture power and frozen switching records.

Selected subpixel transmission

T_j

$0.50$

Total pixel / backlight power

P_{\rm pixel}/P_{\rm back}

$50\%$

Continuum instability threshold

V_{\rm th}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

LCD Pixel Microscope

Red subpixel RMS drive

V_R

0\,\mathrm{V_{rms}}

Green subpixel RMS drive

V_G

0\,\mathrm{V_{rms}}

Blue subpixel RMS drive

V_B

0\,\mathrm{V_{rms}}

Physical cell gap

d

4\,\mu\mathrm m

Analyzer axis from upward toward positive Z

\theta_A

90^\circ

Analyzer blocked/pass intensity ratio

\epsilon_A

0.001

Exploded assembly spacing; visual only

g

0.70

Selected subpixel

j

1

Probe fractional depth

s

0.50

Split one pixel into three voltage-controlled shutters. Follow light through the real layer stack, drive RGB independently, and watch the director and polarization change together. Record a voltage step to see why a thicker cell responds more slowly.

Normal-incidence one-dimensional Frank model with an imposed uniform RMS field, assumed material and isolated LED channels. No off-axis color, transistor circuit, backflow or calibrated panel specification.

Physics tutorial

Audit the physics of LCD Pixel Microscope

BackgroundA liquid crystal does not emit light. A backlight, two polarizers and three separate filtered shutters turn electric bias into a colored pixel. Every depth slice changes the optical field through $\mathbf E_{\rm out}=\mathbf P_A\left(\prod_j\mathbf J_j\right)\mathbf E_{\rm in}$ . The director is first obtained by minimizing a one-dimensional, one-constant Frank energy with fixed perpendicular surface anchors and an imposed uniform RMS field. Optical propagation then uses that same director, not an invented voltage-dependent rotation angle.

Why it mattersHow can three voltage-controlled liquid-crystal shutters make one colored pixel?

Start with the essentials

Focus question: How can three voltage-controlled liquid-crystal shutters make one colored pixel?
One-sentence intuition: The analyzer converts a generally elliptical field into intensity: $T_A=|\mathbf a^\mathsf T\mathbf E|^2+\epsilon_A|\mathbf a_\perp^\mathsf T\mathbf E|^2$ . A low-voltage twist is not an exact ninety-degree optical rotator at every wavelength. Finite birefringence produces the Gooch–Tarry oscillations. Driving the molecules toward the beam reduces their effective birefringence and darkens a crossed stack.

Core mathematical model

The same energy drives equilibrium and response

\begin{aligned}E&=V/d\\F&=\int_0^d\left[\frac K2|\partial_x\mathbf n|^2-\frac{\epsilon_0\Delta\epsilon}2E^2n_x^2\right]dx\\\gamma_1\partial_t\mathbf n&=(\mathbf I-\mathbf n\mathbf n^\mathsf T)\mathbf h\\\mathbf h&=K\partial_x^2\mathbf n+\epsilon_0\Delta\epsilon E^2n_x\mathbf e_x\end{aligned}

Unit-length directors are fixed upward at the rear surface and along positive Z at the front. A sixty-four-segment energy is minimized on the sphere; step records integrate its projected gradient. The uniform RMS field is imposed, not a self-consistent electric displacement calculation.

Optical eigenvalue at normal incidence

\begin{aligned}n_{\rm eff}^{-2}&=\frac{\sin^2\beta}{n_o^2}+\frac{\cos^2\beta}{n_e^2}\\\delta_j&=\frac{2\pi}{\lambda}(n_{{\rm eff},j}-n_o)\Delta x\\n_o&=1.50,\quad n_e=1.65\end{aligned}

Tilt is measured from the substrate plane toward the beam. Each slice is a lossless retarder with the projected director as its extraordinary axis. The refractive indices are nondispersive teaching assumptions, not a calibrated mixture.

An independent optical limit and power budget

\begin{aligned}T_\perp(0)&=1-\frac{\sin^2(\frac\pi2\sqrt{1+u^2})}{1+u^2}\\u&=\frac{2d(n_e-n_o)}{\lambda}\\\frac{P_{\rm pixel}}{P_{\rm back}}&=\frac{0.64}{18}(T_R+T_G+T_B)\end{aligned}

The closed-form Gooch–Tarry solution independently checks the zero-field layer product. The budget has one half from the ideal input polarizer, one third from the matching spectral line, one third subpixel area and a sixty-four percent open fraction.

Common difficulties

More voltage means more light?

Typical misconceptionAn LCD behaves like an LED, and moving the camera measures its viewing angle.

Better mental modelThis normally-white TN pixel becomes dark under drive. Its backlight supplies the energy; the analyzer absorbs rejected power. Camera orbit only changes the view of the schematic. Off-axis optics would need anisotropic boundary matching, not a cosine brightness multiplier, and are not modeled here.

Run the experiment

01
Scene 1: Normally white pixel
Start with the normally-white pixel. Open the liquid-crystal view and compare the anchored end rods with the gold field loops.
What to observe: Light travels along positive X. The transverse input axis is upward, positive Y; the second axis is positive Z. Positive analyzer angle turns from Y toward Z. RGB runs from positive Z to negative Z, left to right when viewing the output face from downstream.
02
Scene 2: Independent RGB shutters
Choose independent RGB shutters. Select a stripe, then drag its voltage curve. Only that subpixel drive changes; the other channels remain fixed.
What to observe: The physical cell thickness is enlarged by six scene units per micrometre. Lateral scale, rods and exploded gaps are schematic. Changing separation cannot alter brightness or response.
03
Scene 3: Driven dark state
Move the depth cursor from entrance to exit. Compare the local director tilt, azimuth and equal-scale electric-field ellipse.
What to observe: Three equal-power spectral lines and isolated ideal filters stand in for RGB. The image displays fixed channel-normalized powers; the absolute total-power readout separately includes input polarization loss, spectral rejection and the aperture area.