# Serial of year 23

You can find the serial also in the yearbook.

*We are sorry, this serial has not been translated.*

## Tasks

### 1. Series 23. Year - S. Petrin mirror maze

- What will you see when standing between two vertical mirrors connected at right angle?

Lets have plane mirror inclined at angle 45°, moving to the left at the speed $v$. From the right there is a light ray of speed $c$ (e.g. angle of incidence is 45°) and is reflected upwards. Using Huygens principle calculate angle between incoming and reflected ray, e.g. correct the law of reflection for mowing mirrors.

Z dílny Dalimilovy.

### 2. Series 23. Year - S. mystery of overhead projector and fish-eye

You have maybe noticed, that in overhead projectors is often used very special lens, which looks more like a grooved plate. It is created in such way, that normal planoconvex lens is cut in concentric circles, the „end“ is kept and the result is again assembled. Finally we have axially symmetrical hilly glass (see figure).

Such lens has identical curvature to the original lens, and, according to Snell law we would expect, that will focus the light in the same way as original lens. However, looking at the situation using Fermat principle, the different beam paths do not experience the same time, as we have removed in different places different glass thickness. For example the shortest time is represented by the light beam travelling along the optical axis. It seems, that Fermat principle is failing, according it the lens should focus only the light following the optical axis and will not function as it should. Decide, who is correct: Snell or Fermat? And why?

Find the path of beams in two-dimensional situation, when the dependence of refractive index is described by a function ($r$ is distance from the origin):

$n(r)=n_{0}⁄(1+(r⁄a))$.

- (
**Bonus:**If the point source of light is placed into the environment with varying refractive index, then some light can be focused into a single point, similar as in the situation of converging lens. This point is then called image of original point source. Describe the geometrical transformation from source to the image, which is induced by the environment from the previous question.

Z Kroniky Dalimilovy.

### 3. Series 23. Year - S. game with shadows

- In serial we discussed discrete distribution of point sources on a line and its projection onto another line. Now assume, that the points are distributed on a plane and the screen is the plane parallel to it. Describe the distribution of intensity on the screen in case, that light sources:

- lie at one line with equal spacing $d$.

- are as two parallel lines, where the distance between two of them is $d$.

- lie in the corners of rectangle network, where rectangles have sides $a$, $b$.

- Assume following situation: before the screen, presented by a plane $xy$ is a disc of radius $R$, parallel to the plane. The screen is illuminated from the side of the disc by a beam of parallel beams perpendicular to the screen $xy$. Explain, why this situation can be described using point light sources located continuously in the same plane as disc is located, excluding the disc itself. Find the intensity distribution in plane $xy$ as function of $x$ and $y$ (you can supply just integral, not it solution) and show, that point opposite the center of disc shows strange behaviour, which we would not expect from ray optics

### 4. Series 23. Year - S. Maxwell

### 5. Series 23. Year - S. a light in the matter

- The index of refraction in a nonlinear medium varies with the intensity of light $I$ as $n=n_{1}+n_{2}I$, where $n_{1}$ and $n_{2}$ are positive constants. Describe the behaviour of a ray of light of given width passing through this medium, assuming the light intensity decreases as we go from the centre to the edges of the ray. (Qualitative description is acceptable, an analytic model and solution will obtain extra credit.)

- A slab of width $a$ consists of 2$N$ parallel neigbouring slabs (with no gaps) with alternating indices of refraction $n_{1}$ and $n_{1}$. A light wave is incident perpendicularly on the front slab. What is the effective index of refraction of this composite slab for $N→∞?$ Can you think of a physical reason why?

**Hint:** for any real matrix $A$
<p style=„text-align:center;“>
lim_{$N→∞}(I+A/N)^{N}=\exp(A)$,

where $I$ is the identity matrix and exp($A)=I+A+A^{2}/2!+A^{3}/3!+\ldots]$.