Title: ETH_Sem3 / Physik_III / Woche2

Q: What is a lens useful for?
A: It makes many paths extremal and have the same optical length, thus allowing more light to travel from A to B.

Q: What is the paraxial approximation?
A: Consider only rays that make a small angle with respect to some optical axis.

Q: In the case of parallel incoming rays, what is the focal length? (Equation)
A: \[f=\frac{Rn_2}{n_2-n_1}\]The lens will focus the rays at this distance on the side the light goes to.

Q: What is a virtual image?
A: A point from where the rays appear to arrive to an observer (generally) behind a diverging lens.

Q: What are spherical aberrations?
A: Due to the approximations made, the spherical surface of a lens is not the perfect surface. This means that rays far from the optical axis of the lens are imperfectly redirected and are not focused at the focal length.

Q: Thin lens formula / lensmaker's equation
\[[...] = [...]\]
A: \frac{1}{f}; \frac{1}{d_1} + \frac{1}{d_2}

Sign convention
Focal length \(f\):
- \(f>0\): converging lens, light is bent towards the optical axis.
- \(f<0\): diverging lens, light is bent away from the optical axis
Object location \(d_1\):
- \(d_1>0\): a real object that is on the side the light comes from.
- \(d_1<0\): a virtual object that is on the side the light goes to.
Image location \(d_2\):
- \(d_2>0\): a real image that is on the side the light goes to.
- \(d_2<0\): a virtual image that is on the side the light comes from.

Q: The three rules for geometrically finding the image of an object:
A: 1. An incoming ray that goes through the center of the lens passes through it undeflected.
2. An incoming ray that travels parallel to the optical axis will be deflected by the lens so that it "passes" through the focus.
3. An incoming ray that "passes" through the focus will be deflected so that it travels parallel to the optical axis after the lens.

Q: Def: magnification factor \(M\)
A: \(M = \frac{h'}{h} = -\frac{d_2}{d_1}\)
Where \(h\) is the size of the object and \(h'\) the size of the image.

Q: Does the focal length of a lens depend on the medium it is in?
A: Yes! A lens in water will have a bigger focal length than in air.

Q: How does a simple telescope look like?
Answer for backside:
\[\frac{f_1}{f_2}\]
A: What is this?
By what factor is the angular separation magnified?

Q: Why do we need telescopes?
A: To look at objects far away that have a small angular separation.
Our eyes only have an angular resolution of about \(0.008\) degrees.

Q: Why do we need microscopes ?
A: To make a small object take up a bigger part of our field of vision without bringing it closer.
Our eyes can't focus on objects closer than about 15 cm.

Q: How does a simple microscope look like ?
A: What is this?

Q: What is the superposition principle when talking about a wave equation?
A: Any linear combination \(E = aE_1 + bE_2\) of solutions \(E_1\) and \(E_2\) is itself also a solution.

Q: What is the wave equation for \(\vec{E}\)?
A: \[\nabla^2\vec{E} = \mu_0\varepsilon_0 \frac{\partial^2\vec{E}}{\partial t^2} = \frac{1}{c^2} \frac{\partial^2\vec{E}}{\partial t^2}\]

Q: What is Huygen's principle?
A: Every point on the wavefront of a wave emits a hemispherical wave that propagates in the forward direction.
Forward direction: the direction normal to the wavefront at that point, away from the source

Q: What is a wavefront?
A: A surface in space where the wave has the same phase.

Q: Equation for a plane wave
A: \[E(\vec{r},t) = E_0e^{i(\vec{k}\cdot\vec{r} - \omega t)}\]

Q: Equation for a spherical wave
A: \[E(\vec{r},t) = \frac{A}{r} e^{i(kr-\omega t)} = E_0(r)\, e^{i(kr-\omega t)}\]

Q: How is \(\text{sinc}(x)\) defined?
A: \[\text{sinc}(x) = \begin{cases}\frac{\sin(x)}{x} & x\neq 0 \\ 1 & x=0 \end{cases}\]

Q: What is the intensity and what is it proportional to?
A: The intensity is the optical power per unit area and is proportional to the magnitude squared of the electric field.\[I\propto \vert E(\vec{r},t)\vert^2\]

Q: What is a transmission function?
A: A function \(t:\mathbb{R}^2 \to [0,1]\) that is multiplied to the electric field to describe how a certain planar object lets light travel through it.
For example, for a square aperture of side length \(a\) centered at \((0,0)\), the transmission function is\[t(x,y) = \begin{cases} 1 & x,y \in (-a/2, a/2) \\ 0 & \text{otherwise} \end{cases}\]