PHYS 415: OPTICS Polarization (from Trebino’s lectures)

1. PHYS 415: OPTICSPolarization (from Trebino’s lectures) F. ÖMER ILDAY Department of Physics, Bilkent University, Ankara, Turkey

2. Here, the complex amplitude, E0 is the same for each component. So the components are always in phase. ~ 45° Polarization

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

4. The Mathematics of Polarization Define the polarization state of a field as a 2D 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: Ey /Ex = 0 90° linear (y) polarization: Ey /Ex= ¥ 45° linear polarization: Ey /Ex= 1 Arbitrary linear polarization:

5. Circular (or Helical) Polarization Or, more generally, 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. The resulting E-field rotates counterclockwise around the k-vector (looking along k).

6. E-field variation over time (and space) y x kz-wt = 0° kz-wt = 90° Right vs. Left Circular (or Helical) Polarization Or, more generally, 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. The resulting E-field rotates clockwise around the k-vector (looking along k).

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

8. 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.

9. 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: 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. In practice, the amplitudes vary, too!

10. Light with very complex polarizationvs. 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.

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

12. Birefringence The x- and y-polarizations can see different refractive index curves.

13. 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.

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

15. Calcite Calcite is one of the most birefringent materials known.

16. Polarizers take advantage of birefringence, Brewster's angle, and total internal reflection. Here’s one approach: 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.

17. Polarizers Air-spaced polarizers

18. 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.

19. Dielectric polarizers A multi-layer coating (which uses interference; we’ll get to this later) can also act as a polarizer. Melles-Griot catalog Glass Ealing Optics catalog

20. 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 perpen- dicular to the wires. Such polarizers work best in the IR. Polaroid sheet polarizers use the same idea, but with long polymers.

21. 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.

22. 0°Polarizer 90° Polarizer 0°Polarizer 0° Polarizer 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. The ratio of the transmitted irradiance through polarizers oriented parallel and then crossed is the Extinction ratioor Extinction coefficient. We’d like the extinction ratio to be infinity. Type of polarizerExt. RatioCost Calcite: 106 $1000 - 2000 Dielectric: 103 $100 - 200 Polaroid sheet: 103 $1 - 2

23. x Wave plate z Optic axis y +45° Polarization -45° Polarization Wave plates When a beam propagates through a birefringent medium, one polarization sees more phase delay than the other. This changes the relative phase of the x and y fields, and hence changing the polarization. Polarization state: } Input: } Output:

24. Wave plates (continued) Wave plate output polarization state: (45-degree input polarization) Quarter-wave plate Half-wave plate A quarter-wave plate creates circular polarization, and a half-wave plate rotates linear polarization by 90. We can add an additional 2mp without changing the polarization, so the polarization cycles through this evolution as d increases further.

25. Half-wave plate When a beam propagates through a half-wave plate, one polarizationexperiences half of a wavelength more phase delay than the other. If the incident polarization is +45° to the principal axes, then the output polarization is rotated by 90° to -45°.

26. If a HWP, this yields 45° polarization. If a QWP, this yields circular polarization. ± Wave plates and input polarization Remember that our wave plate analysis assumes 45° input polarization relative to its principal axes. This means that either the input polarization is oriented at 45°, or the wave plate is. 0° or 90° Polarizer ±45° Polarizer Wave plate w/ axes at 0° or 90° Wave plate w/ axes at ±45° If a HWP, this yields 90° or 0° polarization. If a QWP, this yields circular polarization.

27. How not to use a wave plate If the input polarization is parallel to the wave plate principal axes, no polarization rotation occurs! 0° or 90° Polarizer ±45° Polarizer Wave plate w/ axes at 0° or 90° Wave plate w/ axes at ±45° This arrangement can, however, be useful. In high-power lasers, we desire to keep the laser from lasing and then abruptly allow it to do so. In this case, we switch between this and the previous case.

28. d Thickness of wave plates When a wave plate has less than 2p relative phase delay, we say it’s a zero-order wave plate. Unfortunately, they tend to be very thin. Solve for d to find the thickness of a zero-order quarter-wave plate: Using green light at 500 nm and quartz, whose refractive indices are ne – no = 1.5534 – 1.5443 = 0.0091, we find: d = 13.7 mm This is so thin that it is very fragile and very difficult to manufacture.

29. d Multi-order wave plates A multi-order wave plate has more than 2p relative phase delay. We can design a twentieth-order quarter-wave plate with 20¼ waves of relative phase delay, instead of just ¼: d = 561 mm This is thicker, but it’s now 41 times more wavelength dependent! It’s also temperature dependent due to n’s dependence on temperature.

30. Optic axes Output beam x z y Input beam d1 d2 First plate Second plate A thick zero-order wave plate The first plate is cut with fast and slow axes opposite to those of the second one. The Jones vector becomes: Now, as long as d1 – d2 is equal to the thin zero-order wave plate, this optic behaves like the prohibitively thin one! This is ideal.

31. Polarization state at receiver = Distance Polarization Mode Dispersion plagues broadband optical-fiber communications. Imagine just a tiny bit of birefringence, Dn, but over a distance of 1000 km… If l = 1.5 mm, then Dn ~ 10-12 can rotate the polarization by 90º! Newer fiber-optic systems detect only one polarization and so don’t see light whose polarization has been rotated to the other. Worse, as the temperature changes, the birefringence changes, too.

32. ±45° Polarizer Quarter wave plate (QWP) Additional QWP and linear polarizer comprise a circular "analyzer." 45° Polarizer QWP -45° polarized light Circular polarizers A circular polarizer makes circularly polarized light by first linearly polarizing it and then rotating it to circular. This involves a linear polarizer followed by a quarter wave plate Unpolarized input light 45° Polarizer QWP 45° polarized light Circularly polarized light

33. 90 polarizer 0 polarizer Yellow filter (rejects red) Polarization Spectroscopy The 45°-polarized Pump pulse re-orients molecules, which induces some birefringence into the medium, which then acts like a wave plate for the Probe pulse until the molecules re-orient back to their initial random distribution.