# Lecture 37 - Diffraction gratings and polarization

In this lecture will we discuss diffraction gratings and polarizers.

We will then go outside to use these to investigate the properties of sunlight and skylight.

## A two slit grating

$E_{1}=E_{0}e^{i\omega t}$

$E_{2}=E_{0}e^{i(\omega t + \delta)}$

$E_{\theta}=E_{1}+E_{2}=E_{0}e^{i\omega t}(1+e^{i\delta})$

and the intensity is

$I_{\theta}=E_{\theta}^{*}E_{\theta}=E_{0}^2(1+e^{-i\delta})(1+e^{i\delta})=E_{0}^2(2+2\cos\delta)=4E_{0}^2\cos^{2}{\frac{\delta}{2}}$

$\frac{I_{\theta}}{I_{0}}=\frac{(2+2\cos\delta)}{4}$

## A three slit grating

$E_{1}=E_{0}e^{i\omega t}$

$E_{2}=E_{0}e^{i(\omega t + \delta)}$

$E_{3}=E_{0}e^{i(\omega t + 2\delta)}$

$E_{\theta}=E_{1}+E_{2}+E_{3}=E_{0}e^{i\omega t}(1+e^{i\delta}+e^{2i\delta})$

$I_{\theta}=E_{0}^{2}(1+e^{i\delta}+e^{2i\delta})(1+e^{-i\delta}+e^{-2i\delta})$

$=E_{0}^{2}(3+2(e^{i\delta}+e^{-i\delta})+1(e^{2i\delta}+e^{-2i\delta})$

$=E_{0}^{2}(3+4\cos\delta+2\cos 2\delta)$

$\frac{I_{\theta}}{I_{0}}=\frac{3+4\cos\delta+2\cos 2\delta}{9}$

## A four slit grating

$E_{1}=E_{0}e^{i\omega t}$

$E_{2}=E_{0}e^{i(\omega t + \delta)}$

$E_{3}=E_{0}e^{i(\omega t + 2\delta)}$

$E_{4}=E_{0}e^{i(\omega t + 3\delta)}$

$E_{\theta}=E_{1}+E_{2}+E_{3}+E_{4}=E_{0}e^{i\omega t}(1+e^{i\delta}+e^{2i\delta}+e^{3i\delta})$

$I_{\theta}=E_{0}^{2}(1+e^{i\delta}+e^{2i\delta}+e^{3i\delta})(1+e^{-i\delta}+e^{-2i\delta}+e^{-3i\delta})$

$=E_{0}^{2}(4+3(e^{i\delta}+e^{-i\delta})+2(e^{2i\delta}+e^{-2i\delta})+(e^{3i\delta}+e^{-3i\delta})$

$=E_{0}^{2}(4+6\cos\delta+4\cos 2\delta+2\cos 3\delta)$

$\frac{I_{\theta}}{I_{0}}=\frac{4+6\cos\delta+4\cos 2\delta+2\cos 3\delta}{16}$

## n slits

n=2 $\frac{I_{\theta}}{I_{0}}=\frac{(2+2\cos\delta)}{4}$

n=3 $\frac{I_{\theta}}{I_{0}}=\frac{3+4\cos\delta+2\cos 2\delta}{9}$

n=4 $\frac{I_{\theta}}{I_{0}}=\frac{4+6\cos\delta+4\cos 2\delta+2\cos 3\delta}{16}$

For any $n$

$\frac{I_{\theta}}{I_{0}}=\frac{n+\sum\limits_{k=1}^{n-1}2(n-k)\cos(k\delta)}{n^{2}}$

## Maxima

We can thus see that irrespective of the number of slits $n$ the condition for maxima is the same as for the double slit

$d\sin\theta=m\lambda$       (m=0,1,2,..)

but that the larger the number of slits from which diffraction occurs the sharper the maxima will be.

## Spectroscopy

Here is the basic idea:

A diffraction grating can be used to separate light into it's different colors, as you can see in the video for this lab where a diffraction grating is used to study the spectral lines of hydrogen.

Later we will go outside and use the diffraction grating to separate the spectra of the sun.

## X-ray diffraction

In order for diffraction effects to be observed the size of the features from which diffraction occurs need to be comparable to the wavelength of light.

The spacing of atoms in a crystal is normally on the order of a few angstroms, so diffraction with visible light will not occur.

However, xrays have wavelengths of this size and so the structure of crystals can be studied by x-ray diffraction.

In x-ray diffraction what is measured is the spacing between planes of atoms $d$.

The condition for constructive interference is

$2d\sin\theta=m\lambda$     (m=0,1,2,..)

This formula is known as Bragg's Law.

## Modern Diffraction Applications

Very intense x-ray beams, with a broad ranges of wavelengths can be obtained at a synchrotron.

The closest to here is the NSLS at Brookhaven National Laboratory.

This year this will be replaced by the NSLS-II.

Neutron diffraction is another technique: it has advantages to study biological systems over x-rays, see for example here.

## Polarization

When we first considered electromagnetic waves we considered a wave which only had an oscillating electric field in single direction, and the radio wave we generated from the dipole antenna was such a wave.

Such a wave is said to be plane polarized.

More generally, light can be made up of randomly oriented oscillating E-fields, in which case we say it is unpolarized.

The first thing that determines the polarization of light is the way in which it is generated, but subsequently the polarization can be changed by the light's passage through certain kinds of materials, reflection and scattering.

## Polarizers

Polarizers are materials which change the polarization of light, an ideal polarizer allows electric field to pass only in a given direction.

The polarization of the light after it has passed through the polarizer is the direction defined by the polarizer.

The way to approach the calculation of the magnitude of the intensity that passes through the polarizer is by finding the component of the electric field which is in that direction

$E=E_{0}\cos\theta$

the intensity is then

$I=E^{2}=E_{0}^{2}\cos^{2}\theta=I_{0}\cos^{2}\theta$

This formula is known as Malus' Law.

In the diagrams below the lines indicate the axis of the polarizer, which should not be confused with the direction of the wires in a wire-grid polarizer, which would be at 90o to the depicted lines.

## Crossed Polarizers

If we start with unpolarized light, and pass it through a polarizer, the polarization of the transmitted light will be in the direction of the polarizer and the intensity will be reduced by half.

(We can consider the average angle of the polarization of the light with the polarizer to be $\theta=45^{o}$ and hence $I=I_{0}\cos^{2}(45^{o})=\frac{I_{0}}{2}$.)

A subsequent polarizer with polarization direction at right angles to the first will then allow no light to pass as the polarization direction of the incoming light is at $90^{o}$ to the polarizer direction.

## Three polarizers

Addition of a third polarizer between the crossed polarizers actually allows some light to pass!

If the angle of the added polarizer is at $45^{o}$ to the other two then we can reason that the final light will have

$I=I_{0}\cos^{2}(45^{o})\cos^{2}(45^{o})\cos^{2}(45^{o})=\frac{I_{0}}{8}$

## Polarization from reflection and scattering

When light is reflected from a surface the component of polarization perpendicular to surfaces is preferentially transmitted or absorbed.

This means that when we look at the reflected light the polarization is preferentially polarized parallel to the surface.

Light can also be polarized by scattering from molecules.

Scattering occurs more strongly for shorter wavelengths of light, hence blue is scattered more than red, giving the sky it's blue color.

The scattering also results in a horizontal polarization for skylight, which we will now go an observe!

## Outdoor activities!

Before saying anything else: Do not look at the sun

Instead, do these things (and others you may come up with)

• Before going outside, use the diffraction grating to look at the fluorescent lighting in the building.
• Also before going outside use the polarizer to look at the projector light and any LCD screens you have handy.
• Use the polarizers to filter direct sunlight on to a surface and check whether direct sunlight has a polarization.
• Use the diffraction grating to separate the direct sunlight in to a spectra (again, projecting on a surface). Also check with the polarizer to see if any wavelength of the direct sunlight is polarized.
• If you are very careful and hold the diffraction grating so that the field of view is close to, but does not include the sun, you can see a very nice spectra.
• Look at the sky through a polarizer (away from the sun) and rotate it to check whether the skylight is polarized.
• Use the polarizer to look at reflected light from a flat surface (for example, a road) and see if this has a polarization.
• Make sure I get all my polarizers and gratings back before you disperse!