Title: ETH_Sem3 / Physik_III / Woche3

Q: What does the Fraunhofer approximation corresponds to?
A: Consider that the wavefronts are flat when arriving at the aperture.

How can you explain that the Fraunhofer diffraction applies in general?

Q: What is the Rayleigh criterion?
A: Two objects are just resolved if the intensity maximum of one lines up with the first intensity minimum of the other.

The angle between two objects that satisfies the Rayleigh criterion is given by \(\varphi\):\[ \sin(\varphi) = \frac{\lambda}{a} \]where \(\lambda\) is the wavelength and \(a\) the size of the aperture / lens.
Back Extra: Note that since \(\varphi \ll1\), \(\sin(\varphi) \approx \varphi\) can be used.

Q: When are two light sources incoherent / coherent ?
A: Incoherent: they do not have a definite phase relationship and the intensity patterns can be considered separately.
Coherent: they have a definite phase relationship and interference can occur.

Q: What is the Abbé limit?
A: Light with a wavelength \(\lambda\), travelling in a medium with refractive index \(n\) and converging to a spot with half-angle \(\theta\) will have a minimal resolvable distance of\[d = \frac{\lambda}{2n\sin(\theta)} = \frac{\lambda}{2NA}\]where \(NA = n\sin(\theta)\) is the numerical aperture.

Q: What is the polarization of an EM-wave?
A: The direction of the electromagnetic fields.
Usually, only the direction of the electric field is specified.

Q: What are the different types of polarization?
A: - Linear polarization
  - The field always points in the same direction
  - \(\vec{E}(\vec{r},t) = (E_x\hat{x} + E_y\hat{y}) e^{i(kz-\omega t)}\)
- Elliptical / circular polarisation
  - There is a phase difference between the \(x\) and \(y\) components
  - \(\vec{E}(\vec{r},t) = (E_x\hat{x} + E_ye^{i\phi}\hat{y}) e^{i(kz-\omega t)}\)
  - Special cases:
    - \(\phi = 0\): linear
    - \(\phi = \pm \frac{\pi}{2}\): circular

Q: What is a birefringent material and what can it do?
A: A material that has different indices of refraction for the \(x\) and \(y\) components of the electric fields.
This can change the polarization of EM-waves by changing the phase difference between the field components.

Q: When is a length \(L\) of a birefringent material called
1. a half-wave plate?
2. a quarter-wave plate?
A: 1. \((n_x-n_y)k_0L = \pi\)
2. \((n_x-n_y)k_0L = \pi/2\)

Q: Refraction and reflection at a surface
What is p-polarization and s-polarization?
A: p-polarization: (in-plane), the electric field is in the plane of refraction/reflection
s-polarization: (senkrecht), the electric field is perpendicular to the plane of refraction/reflection

The boundary conditions at an interface are:
- \( B_{\bot1} = B_{\bot2} \)
- \( D_{\bot1} = D_{\bot2} \)
- \( H_{\parallel1} = H_{\parallel2} \)
- \( E_{\parallel1} = E_{\parallel2} \)

Fresnel equations
(for p-polarization)
P-polarization:\[\begin{align} t_p &= \frac{E_t}{E_i} = \frac{2n_1\cos(\theta_i)}{n_2\cos(\theta_i) + n_1\cos(\theta_t)}  \\ r_p &= \frac{E_r}{E_i} = \frac{n_2\cos(\theta_i) - n_1\cos(\theta_t)}{n_2\cos(\theta_i) + n_1\cos(\theta_t)}  \end{align}\]S-polarization:\[\begin{align} t_s &= \frac{E_t}{E_i} = \frac{2n_1\cos(\theta_i)}{n_1\cos(\theta_i) + n_2\cos(\theta_t)}  \\ r_s &= \frac{E_r}{E_i} = \frac{n_1\cos(\theta_i) - n_2\cos(\theta_t)}{n_1\cos(\theta_i) + n_2\cos(\theta_t)}  \end{align}\]

Q: What is Brewster's angle?
A: For p-polarized light, \(\theta_B\) is the angle where no light is reflected (\(r_p\to0\) in the Fresnel equations)
\[\theta_B = \arctan\bigg(\frac{n_2}{n_1}\bigg)\]