# Lasers and Diffraction

This practical work aims to study the diffraction phenomenon through the observation and analysis of the laser beam deflected by different obstacles.

## Elements of diffraction

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of light around the corners of an obstacle or aperture into the region of the geometrical shadow of the obstacle. These characteristic behaviors are exhibited when a wave encounters an obstacle or an aperture that is comparable in size to its wavelength.

The scalar theory of diffraction was established by Augustin Fresnel. The Fraunhofer diffraction is an approximation which applies to the far field; that is when the diffraction pattern is viewed at a long distance from the diffracting aperture.

The theoretical results will be provided without demonstration.

## First example: diffraction by a single-slit

Consider a single slit which is illuminated by a light plane wave (a laser beam is a good approximation). The intensity profile of the diffracted light can be calculated using the Fraunhofer diffraction equation as \[ I(\theta)=I_{0}\,\text{sinc}\left(\dfrac{\pi\,\ell\,\sin \theta}{\lambda}\right)^{2} \quad\text{with}\quad \text{sinc}(x)=\frac{\sin x}{x} \] where

- $I(\theta)$ is the intensity in a direction given by the angle $\theta$;
- $\ell$ is the width of the slit;
- $\lambda$ is the wavelength of the plane wave.

Let us compute the minima of the intensity pattern (dark fringes corresponding to destructive interference), defined by $I(\theta)=0$. This gives \[ \dfrac{\pi\,\ell\,\sin \theta}{\lambda}=k\,\pi \quad\Longrightarrow\quad \sin\theta_k=k\frac{\lambda}{\ell} \quad\text{with}\quad k\in \mathbb{Z^*} \]

If the interference pattern is recorded by a CCD sensor at a distance $D$, these minima correspond to the positions $x_k$ given by \[ \tan\theta_k=\frac{x_k-x_0}{D} \] where $x_0$ is the position of the central maximum. If we assume small angles, we have $\tan \theta_k\simeq \sin\theta_k$ and we find \begin{equation} \boxed{x_{k}=k\frac{\lambda D}{\ell}+x_0} \label{eq:TP2Minima} \end{equation}

We note $i_{\text{d}}$ the distance between two darks-fringes : \[ i_{\text{d}}=\lambda\,\frac{D}{\ell} \] The distance between $k=-1$ and $k=1$ dark fringes is $2i_\text{d}$. The central fringe is two times wider than the others fringes

## Second example: diffraction by a circular aperture

When a plane wave falls on a plate with a circular hole, this aperture diffracts light and produces a divergent beam. The narrower hole is, the more divergent the beam is. The intensity profile shows circular dark fringes. The central spot is named the **Airy disc** and its diameter is given by
\begin{equation}
\boxed{\phi=a\frac{\lambda D}{d}}
\label{eq:TP2_formule_airy}
\end{equation}
where $d$ is the diameter of the hole and $a$ is a numerical constant that you will have to measure.

## Equipment

### CCD sensor

The intensity profile is recorded by horizontal CCD sensor directly interfaced with a PC computer. The sensor is composed of 2048 pixels, 14 μm in width. The optical device fixed on an optical bench can be horizontally and vertically adjusted.

The sensor is very sensitive, and it is often necessary to attenuate the intensity of the beam in order to avoid saturation. This is why a filter is screwed on the sensor. To reduce the intensity further, two crossed polarizers (Malus's law) can be used.

Once adjusted, luminous pictures can be sampled and operated with the help of software, named CALIENS™.

### CALIENS software

**Turn on the sensor before turning on the CALIENS™ software**

The software displays in real time the distribution of the light intensity along the CCD sensor.
For measurements, inactivate `Temps réel`

and record the signal with the button `Acquisition`

. Then, you can zoom in and display cursors and simulations.
Note that you can add a theoretical model. For this, you have to activate the `Interference`

tab then click on `Param.`

to choose the model.

The figure below shows a screenshot of a CALIENS window.

### Laser

For the experiments, we use a 1 mW HeNe laser, which emits in the red part of the visible spectrum.

**Beam profile**

The profile intensity of a laser is Gaussian: \[ I(r)=I_\text{max}\,\mathrm{e}^{-2r^2/w^2} \] where $r$ is the radial distance from the centre of the axis beam. The $w$ parameter measures the beam radius.**Beam divergence**

The beam diameter increase along the optical path because of the beam divergence. If we knows the beam radius size at two separate points ($w_1$ and $w_2$) and the distance between these points, we can calculate the beam divergence by the angle \begin{equation} \boxed{\theta\simeq \frac{w_2-w_1}{d}} \label{eq:TP2divergence} \end{equation}

## Manipulations

**SAFETY INSTRUCTIONS**

- Before turning on the laser, be sure that it is pointed away from yourself and others.
- Never look directly into the laser.
- Never direct the laser beam at another person.

### Divergence of the laser beam

- Place the laser and the CCD sensor on the optical bench. Fix the CCD sensor (with the filter) about 50 cm from the laser. Turn on the laser.
- Turn on the CCD sensor then the CALIENS software.
- Carefully align(horizontally and vertically) the laser beam with the centre of the sensor to obtain an optimum signal.
- Cross the polarisers to reduce the intensity if necessary. Once adjusted, acquire the signal.
- Click on
`Interference`

then`Simulation`

and choose*Gaussian beam*. Fit the model curve with the experimental curve, using the spot size parameter. Measure $w$. - Fix the sensor CCD at about 2 m and measure $w$ again.

Calculate the beam divergence of the laser and express the result in arc minutes.

### Diffraction by a single slit

We analyse the diffraction of the laser beam by a single slit of $\ell=280\,\mathrm{\mu m}$ in width.

**To prepare**: what should the value of the $w$ parameter be if we impose that the light intensity is uniform on the diffracting object with an accuracy better than 1%?

- Set the slit on the optical bench so that the laser beam is centred over the diffracting object.
- Place the CCD sensor at a distance $D=1,0\;\mathrm{m}$ from the slit. NB: CCD sensor is 15 mm offset from the support axis.
- Acquire the intensity profile of diffracted light and accurately measure the $x_{k}$ position of the minima (use the zoom function).
- Fill in a table in REGRESSI™, with $k$ and $x_k$ data.
- Plot $x_k$ versus $k$ and choose the model to adjust with your data.
- Is the \eqref{eq:TP2Minima} law verified ? Determine the wavelength with uncertainty at the 95% confidence level

### Diffraction by a circular aperture

**To prepare**: Find the theoretical value of the constant $a$ relative to the circular aperture diffraction.

In this experiment we explore the light diffracted by a circular hole. For this, take the support perforated with 3 holes with diameters $d$ of 120, 160 and 400 μm. The procedure is the same as for single slit diffraction. To centre the beam on the hole, you can illuminate the back of the hole with a lamp then adjust the laser spot over the hole.

If the intensity is too low, you can unscrew the filter from the sensor. Then, since the sensor is extremely sensitive, it is necessary to reduce the intensity by means of the polarisers. Finally, the sensor is protected from stray light by the PVC tube supplied.

- Observe the diffraction pattern on a screen behind the hole and then observe the signal in CALIENS™. Identify the Airy disc and the rings.
- In order to measure the Airy disc diameter, adjust the vertical position of the CCD sensor.
- Acquire the intensity profile of diffracted light and accurately measure $\phi$, the Airy disc diameter.
- Repeat these operations for the 3 holes.
- Fill in a table in REGRESSI, with $\phi$ and $d$ data.
- Plot $x_k$ versus $x=\lambda D/d$ and choose the model to adjust with your data.
- Is the \eqref{eq:TP2_formule_airy} law verified? Determine the $a$ constant with uncertainty at the 95% confidence level. Comment on your result.

### Equipment

- a laser on an optical bench;
- a CCD sensor;
- a PC computer with CALIENS™ installed;
- two polarisers to modulate intensity;
- diffracting apertures : holes (120-160-400 μµm) and a single slit (340 μm).