🗒️ Note: We will assume $\mu=\mu_0$ ie. the material doesn't affect the magnetic field of the light wave

States of polarization

Linear polarization properties:

• For a wave travelling in the $z$ direction

  $$ \hat {\mathrm{x}}E_{0x}\cos(kz-\omega t+\phi)+\hat {\mathrm y}E_{0y}\cos(kz-\omega t+\phi) $$

• Electric field remains in the same plane as the waves propagates

• The plane containing electric field vectors is at an angle $\arctan(E_{0y}/E_{0x})$ to the $x$-axis

• The maximum $E$-vector amplitude is $\sqrt{E^2_{0x}+E^2_{0y}}$

• There is no phase shift between $\hat{\mathrm x}$ and $\hat {\mathrm y}$ components of the wave

Circular polarization

• The electric field vector remains the same size but rotates around the direction of propagation as the wave advances

• There is a $\pm90\degree$ between $x$ and $y$

• For a wave travelling in the $z$ direction

  $$ \hat {\mathrm x}E_{0x} \cos(kz-\omega t+\phi)+\hat {\mathrm y}E_{0y}\cos(kz-\omega+[\phi\pm \pi/2]) $$

  🗒️ Note: one can also use $\sin$ or $-\sin$ functions for $x$ and $y$

• Right hand circular (RHC) when looking at the photon coming towards you

  🗒️ Note: this convention is not used by everyone

• If we add LHC and RHC waves of the same amplitude, we get linear polarization with twice the original amplitude

Elliptical Polarization

• Travelling in the $z$ direction

  $$ \hat {\mathrm x}E_{0x} \cos(kz-\omega t+\phi_1)+\hat {\mathrm y}E_{0y}\cos(kz-\omega+\phi_2) $$

• Linear polarization if $\phi_1=\phi_2$

• Circular polarization if $E_{0x}=E_{0y}$ and $\phi_1-\phi_1=\pm \pi/2$

• $E$-vector describes an ellipse with time at any given point

Unpolarized light

• Wave train composed of very small fragments of polarized light where $E_{0y}$, $E_{0x}$ and $\phi_2-\phi_1$ vary rapidly and randomly

Polarized light by reflection

• 🍺 Brewster’s angle
• 🏂 Fresnel coefficients
• 🧁 Polarized light by scattering
• 🐳 Polarized light by dichroism

Brewster’s angle

($\theta_i=\theta_b)$ is known as the Brewster angle →

1. Incoming light is unpolarized
2. Light excites dipoles in denser medium
3. refracted ray and weak reflected ray formed

💼 Consider: If the angle between the reflected and refracted rays is $90\degree$ then there is no reflected ray in one of the polarizations because dipoles do not radiate along their length

• Since $\theta_i+\theta_r=\pi/2$ then $\sin(\theta_r)=\cos(\theta_i)$ combining with Snell we get

  $$ \tan(\theta_i)=n_r/n_i $$

Fresnel coefficients

🗒️ Notes:

• Incoming wave ($1$), refracting into a denser medium. It is drawn twice for the two different polarizations (each orthogonal to the direction of propagation)
• Amplitude of the reflected and refracted are $b$ and $a$ for one polarization and $B$ and $A$ for the other