Polarization and devices Feb. 1, 2018

• Polarization: vec e = hat x E_x cos ( omega t -k z ) + hat y E_y cos (omega t -k z + phi ), phi = phi_y - phi_x is the relative phase between the 2 components.
Unpolarized -phi is a random variable.
Linearly polarized - phi = 0 or pi
Circularly polarized - E_x = E_y , phi = pm pi/2
Elliptically polarized - E_x ne Ey , phi ne 0 or pi
Polarization devices - polarizer, birefringent polarizers, rotator.
Ref. 2 Sec. 6.1

Rotator: based on optical activity or Faraday effect
Put in terms of right circular unit vector hat e_R = {hat x - j hat y}/sqrt(2) and left circular unit vector hat e_R = {hat x + j hat y}/sqrt(2), vec E = E_R e^{-jk_R z) hat e_R + E_L e^{-jk_L z) hat e_L;
angle of rotation = rho z = (k_L - k_R ) z /2; rho is rotatory power. For example, Faraday effect rho = V H where H is magnetic field intensity and V is the Verdet constant, i.e. magneto-optics.
Ref. 2 Sec. 6.6

• Anisotropic materials: causes birefringence since refractive indices are different along different axes, i.e. directional dependent. P_i = sum_j epsilon_o chi_{ij} E_j
In general, the refractive indices are different in all 3 axes (x, y, z) in biaxial crystals. More common are uniaxial crystals which have one axis with refractive index different from those of the other two axes. This unique axis is called optic axis and refractive index along this axis is called n_e or n_{||} The refractive index along the other 2 axes is n_o or n_{_|_}
They are used to make polarization devices, e.g.
quarter wave plate causes a pi/2 phase difference between x and y components, i.e. Delta phi = pi/2 (2 m + 1 )
It can be a polarization converter.
Half wave plate causes a pi phase difference between x and y components, i.e. Delta phi = pi (2 m + 1 )
• Inhomogeneous media: refractive index is a function of position, P = epsilon_o chi (r) E
Applications - grade index fiber, mirage.
• Nonlinear media: P = epsilon_o ( chi ^{(1)} E + chi ^{(2)} E^2 + chi ^{(3)} E^3 )
Nonlinear effects result in harmonic generation.
Four-wave mixing (FWM), Stimulated Raman Scattering (SRS), and Stimulated Brillouin Scattering (SBS) rob energy from signal into different freq. channels.
• Dispersive medium: Different frequency components travel with different speeds, e.g. Prisms.