Polarization Linear circular and elliptical polarization Mathematics of

Polarization Linear, circular, and elliptical polarization Mathematics of polarization Uniaxial crystals Birefringence Polarizers Prof. Rick Trebino Georgia Tech www. physics. gatech. edu/frog/lectures

45° Polarization Here, the complex amplitude, ~E 0, is the same for each component. So the components are always in phase.

Arbitrary-Angle Linear Polarization y Here, the y-component is in phase with the x-component, but has different magnitude. E-field variation over time (and space) a x

The Mathematics of Polarization Define the polarization state of a field as a 2 D vector— Jones vector—containing the two complex amplitudes: For many purposes, we only care about the relative values: (alternatively normalize this vector to unity magnitude) Specifically: 0° linear (x) polarization: E /E = 0 ~y ~x 90° linear (y) polarization: E /E = ¥ ~y ~x 45° linear polarization: E ~ y /E ~x= 1 Arbitrary linear polarization:

Circular (or Helical) Polarization Or, more generally, Here, the complex amplitude of the y-component is -i times the complex amplitude of the xcomponent. So the components are always 90° out of phase. The resulting E-field rotates counterclockwise around the k-vector (looking along k).

Right vs. Left Circular (or Helical) Polarization Or, more generally, E-field variation over time (and space) y x Here, the complex amplitude of the y-component is +i times the complex amplitude of the x-component. So the components are always 90° out of phase, but in the other direction. kz-wt = 90° kz-wt = 0° The resulting E-field rotates clockwise around the k-vector (looking along k).

Unequal arbitrary-relative-phase components yield elliptical polarization E-field variation over time (and space) y where Or, more generally, x The resulting E-field can rotate clockwise or counterclockwise around the kvector (looking along k).

The mathematics of circular and elliptical polarization Circular polarization has an imaginary Jones vector y-component: Right circular polarization: Left circular polarization: Elliptical polarization has both real and imaginary components: We can calculate the eccentricity and tilt of the ellipse if we feel like it.

When the phases of the x- and y-polarizations fluctuate, we say the light is unpolarized. where qx(t) and qy(t) are functions that vary on a time scale slower than 1/w, but faster than you can measure. The polarization state (Jones vector) will be: In practice, the amplitudes vary, too! As long as the time-varying relative phase, qx(t) - qy(t), fluctuates, the light will not remain in a single polarization state and hence is unpolarized.

Light with very complex polarization vs. position is also unpolarized. Light that has passed through “cruddy stuff” is often unpolarized for this reason. We’ll see how this happens later. The polarization vs. position must be unresolvable, or else, we should refer to this light as locally polarized.

An intentionally spatially varying polarization can be interesting. Imagine a donut-shaped beam with radial polarization. Focus Longitudinal electric field at the focus! Lens This type of beam can also focus to a smaller spot than a beam with a spatially uniform polarization!

A complex polarization spatial dependence… An optical vortex y x

Birefringence The molecular "spring constant" can be different for different directions.

Birefringence The x- and ypolarizations can see different refractive index curves.

Uniaxial crystals have an optic axis Uniaxial crystals have one refractive index for light polarized along the optic axis (ne) and another for light polarized in either of the two directions perpendicular to it (no). Light polarized along the optic axis is called the extraordinary ray, and light polarized perpendicular to it is called the ordinary ray. These polarization directions are the crystal principal axes. Light with any other polarization must be broken down into its ordinary and extraordinary components, considered individually, and recombined afterward.

Birefringence can separate the two polarizations into separate beams o-ray no e-ray ne Due to Snell's Law, light of different polarizations will bend by different amounts at an interface.

Calcite is one of the most birefringent materials known.

Birefringent Materials

How to make a polarizer We’ll use birefringence, and we’ll take advantage of the Fresnel equations we derived last time. Perpendicular polarization 1. 0 Parallel polarization 1. 0 R R . 5 T 0 0° 30° T 60° Incidence angle, qi 90° 0 0° 30° 60° Incidence angle, qi When ni > nt, there’s a steep variation in reflectivity with incidence angle. This corresponds to a steep variation with refractive index. 90°

Polarizers take advantage of birefringence, Brewster's angle, and total internal reflection. The Glan-Thompson polarizer uses two prisms of calcite (with parallel optical axes), glued together with Canada balsam cement (n = 1. 55). Optic axis Cement n = 1. 55 no = 1. 66 ne = 1. 49 Optic axis Alternatively, the optical axis points out of the screen, and the polarizations are also rotated. Snell’s Law separates the beams at the entrance. The perpendicular polarization then goes from high index (1. 66) to low (1. 55) and undergoes total internal reflection, while the parallel polarization is transmitted near Brewster's angle. If the gap is air, it’s called the Glan-Taylor polarizer.

Even better, rotate the second prism: The Nicol polarizer. Optic axis no = 1. 66 TIR ne = 1. 49 Brewster’s Angle Optic axis (out of page) Combine two prisms of calcite, rotated so that the ordinary polarization in the first prism is extraordinary in the second (and vice versa). The perpendicular polarization goes from high index (no) to low (ne) and undergoes total internal reflection, while the parallel polarization is transmitted near Brewster's angle.

Real Polarizers Air-spaced polarizers

Wollaston Polarizing Beam Splitter The Wollaston polarizing beam splitter uses two rotated birefringent prisms, but relies only on refraction. The ordinary and extraordinary rays have different refractive indices and so diverge.

The Pile-of-Plates Polarizer After numerous Brewster-angle transmissions, mainly the parallel polarization remains. Unpolarized input light Unfortunately, only about 10% of the perpendicular polarization is reflected on each surface, so you need a big pile of plates!

Dielectric polarizers A multi-layer coating (which uses interference; we’ll get to this later) can also act as a polarizer. It still uses the same basic ideas of TIR and Brewster’s angle. Glass

Wire Grid Polarizer Input light contains both polarizations The light can excite electrons to move along the wires, which then emit light that cancels the input light. This cannot happen perpendicular to the wires. Such polarizers work best in the IR. Polaroid sheet polarizers use the same idea, but with long polymers.

Wire grid polarizer in the visible Using semiconductor fabrication techniques, a wire-grid polarizer was recently developed for the visible. The spacing is less than 1 micron.

The wires need not be very long. Hoya has designed a wire-grid polarizer for telecom applications that uses small elongated copper particles. Extinction coefficient > 10, 000 Transmission > 99%

The Measure of a Polarizer The ideal polarizer will pass 100% of the desired polarization and 0% of the undesired polarization. It doesn’t exist. 0° Polarizer The ratio of the transmitted irradiance through polarizers oriented parallel and then crossed is the Extinction ratio or Extinction coefficient. We’d like the extinction ratio to be infinity. Type of polarizer Ext. Ratio 0° Polarizer 90° Polarizer Transmission Cost Calcite: 106 > 95% $1000 - 2000 Dielectric: 103 > 99% $100 - 200 Polaroid sheet: 103 ~ 50% $1 - 2

Visible wire-grid polarizer performance Transmission A polarizer’s performance can vary with wavelength and incidence angle. Extinction ratio The overall transmission is also important, as is the amount of the wrong polarization in each beam.