16. More About Polarization

Transcription

1 16. More About Polarization Polarization control Wave plates Circular polarizers Reflection & polarization Scattering & polarization

2 Birefringent materials have more than one refractive index A special case of birefringence is a uniaxial crystal, where two of the three indices are the same. A light wave with polarization along the optic axis experiences one value for n: the extraordinary index n e while orthogonal polarizations experience the other value: the ordinary index n o.

3 Most materials are not birefringent. But sometimes birefringence can be induced. Birefringence requires some form of anisotropy at the atomic or molecular level. This can be due to anisotropy in the electron binding, or to more macroscopic features. Materials with random atomic or molecular structure (e.g., glasses, liquids, gases) do not exhibit birefringence. However, external factors, such as applied mechanical stress, can lead to a net orientation, and therefore can induce birefringence in otherwise non-birefringent materials. This can be observed when the object is placed between crossed polarizers. This is known as stress birefringence.

4 Birefringence for polarization control If both polarizations are present, this has the effect of changing the relative phase of the x and y fields, and hence altering the polarization. y E in Suppose we illuminate a slab of birefringent material with a wave that has equal parts of ordinary and extraordinary polarization: x r E = E xˆ + E yˆ in optic axis { } Input: Ex ( z, t) = Re E0 exp [ j( kz ωt) ] % E ( z, t) = Re E exp [ j( kz ωt) ] y { } 0 % } Input polarization state: 1 1

5 Wave plates The output wave acquires a phase that is different for the two polarization components: y x E in Output: [ + knod ω ] [ + kn d ] E { } x ( z, t) = Re E0 exp j( kz t) } % E (, ) = Re{ } y z t E0 exp j( kz e ω t) % Here, λ is the wavelength in empty space. thickness d E0 exp( jknod ) % E0 exp( jkned ) % 1 2π exp j [ no ne] d λ A device that changes the polarization of a light wave in this manner is called a wave plate.

6 Wave plates (continued) Wave plate output polarization state: (assuming 45-degree input polarization) exp 1 2π j [ n ] o ne d λ 2 π [ ] exp 2 π n [ ] o ne d j no ne d λ λ output polarization state Quarter-wave plate Half-wave plate linear π/2 j right circular π 1 45 linear 3π/2 j left circular 2π 1 45 linear A quarter-wave plate creates circular polarization from linear polarization, and a half-wave plate rotates 45 linear polarization to its orthogonal state. We can add an additional 2mπ without changing the polarization, so the polarization cycles through this evolution as d increases further.

7 Half-Wave Plate When a beam propagates through a half-wave plate, one polarization experiences half of a wavelength more phase delay than the other. -45 polarization at output +45 polarization at input Vertical (green): 4 cycles Horizontal (blue): 3.5 cycles If the incident polarization is 45 to the optic axis, then the output polarization is rotated by 90. If the incident polarization is parallel or perpendicular to the optic axis of the plate, then no polarization rotation occurs.

8 Half-wave plate for arbitrary angle linear input polarization Polarization state: E } x( z, t) = Re{ E0 cos( α) exp [ j( kz ωt) ]} 1 % tan E (, ) = Re{ ( ) [ ]} α y z t E0 sin α exp j( kz ωt) % For a half-wave plate, so the output state is: [ ] k n n d = π = = j tan tan tan e π α α ( α) o e y input α x α output If the incident polarization is at an angle α to the optic axis, then the output polarization remains linear, and is rotated to α.

9 Circular polarizers Unpolarized input light A circular polarizer makes circularly polarized light by first linearly polarizing it and then rotating it to circular. This uses a linear polarizer followed by a quarter wave plate linear polarizer quarter-wave plate Circularly polarized light

10 Light beams can have complicated polarization dependence Here are a few examples. Azimuthal polarization Radial polarization An optical vortex y Polarization can also be different for different frequencies in a beam. x

11 How does the LCD projector work? white light is split into red, green, blue by dichroic mirrors liquid crystal displays (LCDs) impose images on each of the three color beams three colors recombined in a dichroic beam combiner lens projects image onto external screen

12 Dichroic beam combiner Blue light and red light are reflected. Green light is transmitted. This requires the green light to have a different polarization from the red and blue components.

13 Depolarization by reflection or transmission Suppose that 45 polarization is incident on an interface, which has different parallel (x) and perpendicular (y) reflection coefficients. Incident light fields: [ ω ] [ ω ] { } Ex ( z, t) = Re E0 exp % j( kz t) Ey ( z, t) = Re E0 exp % j( kz t) Reflected light fields: { } [ ω ] [ ω ] { r } Ex ( z, t) = Re xe0 exp j( kz t) % E ( z, t) = Re E exp j( ) y { r kz t } y 0 % y Incident polarization Reflected polarization (if r x >r y ) x Unless light is purely parallel or perpendicularly polarized (or incident at 0 ), some polarization rotation will occur (also true for transmitted light).

14 Most real stuff depolarizes Real stuff, like a piece of clear plastic, is very non-uniform: a series of interfaces at random angles. plastic baggie crossed polarizers

15 Fresnel Reflection and Depolarization Fresnel reflections are a common cause of polarization rotation. This effect is particularly strong near Bewster's angle. For a wide range of angles, R > R

16 Glare is polarized Window reflection viewed through polarizer that transmits only s polarized light Window reflection viewed through polarizer that transmits only p polarized light Polarizing sunglasses transmit only vertically polarized light, because for objects like puddles on the ground or car windows, the glare is largely horizontally polarized (s polarized).

17 Depolarization by unintended birefringence (polarization mode dispersion) Imagine an optical fiber with just a tiny bit of birefringence, n, but over a distance of 1000 km Pipes containing fiber optic cables Polarization state at receiver = 1 2π exp j n d λ Distance Because d is large, even n as small as can rotate the polarization by 90º! (recall: in fibers, λ = 1.5 µm) Many fiber-optic systems detect only one polarization and so don t see light whose polarization has been rotated by π/2. Worse, as the temperature changes, the birefringence changes, too.

18 Scattering by molecules is not spherically symmetric. It has a "dipole pattern." The field emitted by an oscillating dipole excited by a vertically polarized light wave: Direction of light excitation E-field and electron oscillation Emitted intensity pattern Directions of scattered light E-field No light is emitted along direction of oscillation! Directions of scattered light E-field

19 Scattering of polarized light Vertically polarized input light No light is scattered along the input field direction, i.e. with k out parallel to E input. Horizontally polarized input light

20 Scattering of unpolarized light Again, no light is scattered along the input field direction, i.e. with k out parallel to E input. Horizontally polarized scattered light Atmosphere Unpolarized input light Vertically polarized Sun's rays Vertically polarized scattered light Partially polarized Unpolarized We should therefore expect the blue sky to be polarized in certain directions (at right angles to the sun).

21 Skylight is polarized in certain directions Right-angle scattering is polarized In clouds, light is scattered multiple times. So the light emerging from a cloud has its polarization randomized. vertical polarizer horizontal polarizer This polarizer transmits horizontal polarization (of which there is little).

22 Don t use a polarizer on a wide-angle lens. A polarizer on a wide-angle lens necessarily sees both polarized and unpolarized regions of the sky. It s difficult to imagine this effect being useful!

23 Brewster's Angle Revisited A trigonometric calculation reveals that the reflection coefficient for parallel-polarized light goes to zero for Brewster's angle incidence, tan(θ i ) = n t / n i When the reflected beam makes a right angle with the transmitted beam, and the polarization is parallel, then no scattering can occur, due to the scattered dipole emission pattern. n sin( θ ) = n sin( θ ) i i t t n i n t direction of motion of oscillating molecules at the surface (along the direction of the E-field in the transmitted beam) θ i θ i θ t θ i +θ t = 90 But our right-angle assumption implies that θ i + θ t = 90. So: n o sin( θ ) = n sin(90 θ ) i i t i = n t cos( θ ) nt tan( θ i ) = n i i