Optics 079 · Polarization, anisotropy, and modulation

LCD Pixel Microscope

How can three voltage-controlled liquid-crystal shutters make one colored pixel? Each subpixel is a liquid-crystal cell between crossed polarizers, with a red, green or blue filter. With no voltage, the twisted molecules carry the polarization around with them, so light passes the second polarizer. Voltage tilts the molecules toward the beam; birefringence drops, the polarization is no longer turned, and the subpixel darkens. Three brightness levels mix the color.

A 3D cutaway of a twisted-nematic (TN) pixel. The director field relaxes to equilibrium and light is propagated layer by layer with Jones matrices; compare the power through each RGB aperture and the 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 $\text{valid model regime}$

Model checked against analytic results

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 shutters, each controlled by a voltage. Follow light through the layer stack, drive red, green and blue separately, and watch the liquid-crystal director and the polarization change together. Record a voltage step to see why a thicker cell responds more slowly.

The model is a one-dimensional Frank model at normal incidence, with an imposed uniform RMS field, assumed material parameters and separate LED color channels. Off-axis color, the transistor circuit and backflow are not modeled, and no calibrated panel specification is reproduced.

Physics tutorial

Audit the physics of LCD Pixel Microscope

BackgroundA liquid crystal emits no light. A backlight, two polarizers and three separately filtered shutters turn an electric bias into a colored pixel. Each thin slice in depth 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 found first, by minimizing a one-dimensional, one-constant Frank energy with fixed perpendicular anchoring at the two surfaces and an imposed uniform RMS field. The optical propagation then uses that same director; no separate voltage-dependent rotation angle is assumed.

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 turns a field that is usually elliptical into an intensity: $T_A=|\mathbf a^\mathsf T\mathbf E|^2+\epsilon_A|\mathbf a_\perp^\mathsf T\mathbf E|^2$ . At low voltage the twisted cell is not an exact ninety-degree rotator at every wavelength, because finite birefringence produces the Gooch–Tarry oscillations. Driving the molecules toward the beam direction lowers 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}

The unit-length director is fixed pointing up at the rear surface and along positive Z at the front. A sixty-four-segment energy is minimized on the sphere, and the step records integrate its projected gradient. The uniform RMS field is imposed; the model does not solve for the electric displacement self-consistently.

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 whose extraordinary axis is the director’s projection onto the transverse plane. The refractive indices are teaching assumptions without dispersion, not values for a calibrated mixture.

An independent optical limit and where the power goes

\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 gives an independent check of the product of slice matrices at zero field. Of the backlight power, the ideal input polarizer passes one half, the matching spectral line carries one third, the subpixel covers one third of the area, and sixty-four percent of that area is open.

Common difficulties

More voltage means more light?

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

Better mental modelThis normally-white TN pixel goes dark when driven. The backlight supplies the energy, and the analyzer absorbs the power it rejects. Orbiting the camera only changes your view of the schematic. Off-axis optics would need anisotropic boundary matching, not a cosine brightness factor, and they 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 rods at each end with the gold field loops.
What to observe: Light travels along positive X. The transverse input axis points up, along positive Y, and the second transverse axis is positive Z. A positive analyzer angle turns from Y toward Z. Seen from downstream, looking at the output face, the RGB stripes run left to right from positive Z to negative Z.
02
Scene 2: Independent RGB shutters
Choose independent RGB shutters. Select a stripe, then drag its voltage curve. Only that subpixel’s drive changes; the other channels stay fixed.
What to observe: The cell thickness is drawn enlarged, at six scene units per micrometre. The lateral scale, the rods and the exploded gaps are schematic. Changing the separation cannot change the brightness or the response.
03
Scene 3: Driven dark state
Move the depth cursor from entrance to exit. Compare the local director tilt, its azimuth and the electric-field ellipse, drawn at equal scale.
What to observe: Three spectral lines of equal power and isolated ideal filters stand in for RGB. The image shows fixed channel-normalized powers. The absolute total-power readout separately includes the input polarization loss, the spectral rejection and the aperture area.

LCD Pixel Microscope

Audit the physics of LCD Pixel Microscope

Core mathematical model

The same energy drives equilibrium and response

Optical eigenvalue at normal incidence

An independent optical limit and where the power goes

Common difficulties

More voltage means more light?

Run the experiment

Scene 1: Normally white pixel

Scene 2: Independent RGB shutters

Scene 3: Driven dark state

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