Phys 322 Chapter 8 Lecture 22 Polarization

Reminder: Exam 2, October 22nd (see webpage) Dichroism = selective absorption of light of certain polarization

Linear dichroism - selective absorption of one of the two P-state (linear) orthogonal polarizations

Circular dichroism - selective absorption of L-state or R-state circular polarizations

Using dichroic materials one can build a polarizer Dichroic crystals

Anisotropic crystal structure: one polarization is absorbed more than the other

Example: tourmaline

Elastic constants for electrons may be different along two axes Polaroid

1928: dichroic sheet polarizer, J-sheet long tiny crystals of herapathite aligned in the plastic sheet

Edwin Land 1938: H-sheet 1909-1991

Attach Iodine molecules to polymer molecules - molecular size iodine wires Presently produced: HN-38, HN-32, HN-22 Birefringence

Elastic constants for electrons may be different along axes

Resonance frequencies will be different for light polarized Refraction index depends on along different axes polarization: birefringence

Dichroic crystal - absorbs one of the orthogonal P-states, transmits the other Optic axis of a crystal: the direction of linear polarization along which the resonance is different from the other two axes (assuming them equal) Calcite (CaCO3)

Ca C O Image doubles Ordinary rays (o-rays) - unbent Extraordinary rays (e-rays) - bend Calcite (CaCO3)

emerging rays are orthogonaly polarized

Principal plane - any plane that contains optical axis Principal section - principal plane that is normal to one of the cleavage surfaces Birefringence and Huygens’ principle Crystal structure and birefringence

NaCl - cubic crystal

Four 3-fold symmetry axes - optically isotropic Anisotropic, uniaxial birefringent crystals: hexagonal, tetragonal, trigonal

Uniaxial crystal: atoms are arranged symmetrically around optic axis

O-wave - E everywhere is perpendicular to the optic axis, no c/v

When E is parallel to optic axis: ne c/v||

Birefringence  ne - no principal indices of refraction

Calcite: (ne - no) = -0.172 (negative uniaxial crystal) Crystal structure and birefringence

Negative uniaxial crystal Positive uniaxial crystal Biaxial crystals

Two optic axes and three principal indices of refraction Orthorhombic, monoclinic, triclinic

Example: mica, KH2Al3(SiO4)3 Birefringent polarizers

Nicol prism Glan-Thompson Wollaston prism polarizer

1828, William Nicol Polarization by scattering

Scattering can polarize light

Scattering can depolarize light Polarization by reflection

Reminder: Fresnel equations show r||=0 at B - Brewster angle

1  B  tan n2 n1  At Brewster angle reflected and transmitted rays form right angle

B

Reflected light will be fully

polarized at B Fresnel equations  tan2    2 R  i t R  r || 2  Fresnel equations tan i t  2  sin  sin  i  t  i t R   r    2  sin i t sin i t

tani t r||    tan … i t

Reflectance of unpolaized light: R  R R  ||  2 Fresnel equations

Multiple plate polarizer

Degree of polarization:

V  I p Itot

analyzing polarizer I  I V  max min Imax  Imin detector Glare is horizontally polarized

Puddle reflection viewed Puddle reflection viewed through polarizer that through polarizer that transmits only horizontally transmits only vertically polarized light polarized light

Light reflected into our eyes from the puddle reflects at about Brewster's Angle. So parallel (i.e., vertical) polarization sees zero reflection.

Polarizer sunglasses transmit only vertically polarized light. Polarizers are very useful in photography.

Without a polarizer With a polarizer

The effect of a polarizer is probably the one “filter” effect that you can’t reproduce later using Photoshop! Retarders

The two orthogonal P-states develop mutual phase shift as they pass through a retarder

After propagating through:    E  E||  E polarized light    phase shift: E  E||  E birefringence extraordinary  2 in-phase ordinary     dne  no 0 in vacuum direction of polarization: E||  E0|| coskx t fast axis - the one that propagates faster  slow axis - the one that propagates slower E  E0 cos kx  t  Full-wave plate  2     dn  n  e o  = m.2, m=…,-1,0,+1,… 0 E||  E0|| coskx t  Alternatively: dne  no  m0 E  E0 cos kx  t 

full wave - no observable effect

Note: n=n(), and it is generally true only for one wavelength Half-wave plate   =  +m.2, m=…,-1,0,+1,… 2     dne  no 0 dne  no  0 / 2  m0 E||  E0|| coskx t  E  E0 cos kx  t 

Polarization is rotated around the optic axis

Special case: =45o Polarization is rotated by 90o Quarter-wave plate  . 2  = /2 +m 2, m=…,-1,0,+1,…     dne  no 0 dne  no  0 / 4  m0 E||  E0|| coskx t  E  E coskx  t  Linear polarization is converted into  0 circular/elliptical and vice versa Reflective retarders

Total internal reflection: phase shift between the two components. Glass - n=1.51, and 45o shift occurs at incidence angle 54.6o.

Fresnel rhomb

Unlike devices that use birefringent materials this is achromatic Circular polarizer

Can you make a polarizer that transmits right circularly polarized light but not the left circularly polarized light? Phys 322 Chapter 8 Lecture 23 Polarization

Reminder: Exam 2, October 22nd (see webpage) Optical activity

Discovered in 1811, Dominique F. J. Arago

Optically active material: any substance that cause a plane of polarization to appear to rotate

Dextrorotatory (d-rotatory) - polarization rotates right (CW) Levorotatory (l-rotatory) - polarization rotates left (CCW) Latin: dextro - right levo -left Optical activity

Depending on the structure, quartz can be d-or l-rotatory Optical activity

1895, Fresnel: linearly polarized light can be represented at the sum of R- and L-states that propagate at different speeds (circular birefringence):  E E  0 ˆi cosk z  t  ˆjsin k z t  R 2 R  R  E E  0 ˆi cosk z  t  ˆjsin k z t  L 2 L L

  kR  kL ˆ  kR  kL  ˆ  kR  kL  E  E0 cos z ti cos z  jsin z  2   2   2  Optical activity

  kR  kL ˆ  kR  kL  ˆ  kR  kL  E  E0 cos z ti cos z  jsin z  2   2   2 

 ˆ z = 0: E  E0i cost Optical activity

  kR  kL ˆ  kR  kL  ˆ  kR  kL  E  E0 cos z ti cos z  jsin z  2    2   2    ˆ ˆ z =d (kR > kL): E  E0i cos jsin cost



d   nL  nR 0

nL > nR : d-rotatory nL < nR : l-rotatory Nature of optical activity Optical activity and biology

-helix DNA (protein) Induced optical effects:

Photoelasticity = stress birefringence

Faraday effect = magnetic field induced birefringence Faraday isolator

/4 /4 Induced optical effects:

Kerr effect = birefringence induced by electric field (n~ E2)

Pockels effect = birefringence induced by electric field (n~ E) Pockels cell

Laser amplifier Liquid crystals

- possesses properties of liquid and solid Nematic liquid crystal: molecules spontaneously arrange in parallel manner

Optical modulator: Prearrange crystals using windows with microgroves

http://www.cs.sandia.gov/~sjplimp/lammps/movies.html

Modeling a nematic liquid crystal, S. S. Patnaik, S. J. Plimpton, R. Pachter, W. W. Adams, Liquid Crystals, 19, 213-220 (1995). Liquid crystal display

Screen Liquid crystal display Liquid crystal modulators Mathematical description of polarization

1852: Stokes parameters Assume we have a set of 4 filters, each changes the intensity of unpolarized incident light by 1/2: George G. Stokes 1819-1903 1. Isotropic

2. Linear polarizer at 0o

3. Linear polarizer at +45o

4. Circular polarizer (transmits R) Stokes parameters

Position each of the filters one by one in front of I0 any beam and record transmitted irradiance

I1 Stokes parameters: I

S0  2I0 incident radiation I2

S1  2I1  2I0 I S2  2I2  2I0 state of polarization 3

S3  2I3  2I0

S0  Typically is normalized to set S0=1   S1 2 2 2 2 Stokes vector:   S0  S1 S2 S3 S2    S3  Stokes parameters

I0 unpolarized polarized polarized polarized light: at 0o at +45o at -45o I1 1 1 1 1 I

0 1 0 0 I2 0 0 1 -1 0 0 0 0 I3 polarized S  2I at 90o R -state L -state 0 0 S  2I  2I 1 1 1 1 1 0 S 2I 2I -1 0 0 2  2  0

0 0 0 S3  2I3  2I0 0 1 -1 Stokes parameters

I0 Mixing two noncoherent beams: ' '' S0  S0 S0

... I1 I

I2

I unpolarized polarized 3 light: at 90o S0  2I0 2 1 3 S  2I  2I 0 -1 -1 1 1 0 += S 2I 2I 0 0 0 2  2  0

0 0 0 S3  2I3  2I0 Jones vectors applicable only to polarized waves  i x  Ex t  E e  E  Jones vector: ~ 0x   E   i  Ey t y   E0 ye  Normalize: E ei x  1 Horizontal polarization: ~ 0x ~ Eh    Eh     0  0 ~  0  ~ 0 Vertical polarization: E  v  i y  Ev    E0 ye  1 ~ E ei x  1 ~ 1 1 Sum of two coherent 0x i x E   E0xe E45   i x      beams: E0xe  1 2 1 R-state:  E ei x   ~ 0x i  1   1  ~ 1  1   x i x ER     E0xe  E e ER  i y  / 2 ei / 2  0x    i E0xe     i 2   Jones vectors: superposition ~ 1 Eh    0 ~ 0 Ev    1 twice the amplitude Mixing waves: ~ 1 1 E45    2 1 ~ ~ 1  1  1 1 2 1 ER  EL          2  i 2 i 2 0 ~ 1  1  ER    2  i horizontal ~ 1 1 EL    2 i Jones vectors: optical elements ~ 1 Eh    A 0 ~ ~ ~ a11 a12  Ei E A E A  ~ 0 t i   E  a21 a22  v   1 1 0 1 Example: horizontal polarizer A    ~ 1   0 0 E45      2 1 ~ ~ 1 01 1 ~ 1.  E ~ 1  1  Et A Eh        h 0 0 0 0 ER         2  i ~ ~ 1 00 0 2. ~ 1 1 Et A Ev        0 0 1 0 EL         2 i ~ ~ 1 0 1  1  1 1 3. E A E   1 ~ t R        Eh 0 0 2  i 2 0 2 Jones vectors: multiple optical elements

~ E ~ ~ ~ i Et A 3A 2A 1Ei A Ei

A1 A2 A2

See table 8.6 (page 378) for matrices