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Study the Interference Pattern using MATLAB 

 September 9, 2022

By  Shubham

What is interference?

According to Wikipedia, interference is a phenomenon in which two waves combine by adding their displacement at every single point in space and time to form a resultant wave of greater, lower, or the same amplitude.

It is of two types constructive and destructive interference. These interferences result due to the interaction of waves that are correlated or coherent with each other. Interference effects can be observed with all waves, for example, light, radio, acoustic, surface water, gravity, or matter waves.

The instruments based upon the principle of interference are called interferometers. These are essential optical tools used to precisely measure wavelength, distance, refraction index, and optical beams' temporal coherence. The most famous interferometer used today is the Michelson Interferometer. It is an amplitude-splitting interferometer devised by Albert Michelson in 1890.

The Michelson Interferometer

 In the Michelson interferometer, light from a source is split into two beams at a beam splitter (partially reflecting mirror).
Michelson interferometer

Michelson Interferometer

Physical apparatus setup of Michelson Interferometer

Physical apparatus setup of Michelson Interferometer

The light beam travels to the beam splitter, where it is partially reflected and partially transmitted. M_1 and M_2 are plane mirrors where M_1 is a fixed mirror while M_2 is movable. In normal settings, the mirrors M_1 and M_2 are perpendicular to each other while the beam splitter is at an angle of  to the mirror. From the beam splitter, the light goes to a mirror M_1 and is reflected to the beam splitter while the other beam is reflected from the mirror M_2 back to the beam splitter. The two beams recombine and are then detected at the detector.

The two waves will interfere constructively or destructively as per the following conditions of path difference,

Destructive interference gives us a dark spot. It is given by,

2\Delta d=n\lambda

Constructive interference gives us a bright spot. It is given by,

2\Delta d=\left ( n+\frac{1}{2} \right ) \lambda

Interference pattern due to monochromatic plane wave

Assume the two mirrors M_1 and M_2 is illuminated by an incident plane wave. The mirror M_2 is titled at an angle \theta/2 for the X axis. Let the Origin of the coordinate system be the centre of mirror M_2. The waves are reflected normally by mirror M_1 (wave 1) and at an angle \theta to the Z axis by mirror M_2 (wave 2).

The electric fields for the two plane waves at the point \left (x_D,z_D \right ) on the screen of the detector are,

E_1=E_0e^{i(k(z_D+2\Delta d)+i\pi -\omega t)}

E_2=E_0e^{i(kz_Dcos\theta +kx_Dsin\theta -\omega t)}

The resultant electric field at the detector is given by,

E_t=E_1+E_2

E_t=E_0e^{-i\omega t}\left [ e^{ik(z_D+2\Delta d)+i\pi}+e^{ik(z_Dcos\theta +x_Dsin\theta)} \right ]

E_t=E_0e^{-i\omega t}\left [ cos(k(z_D+2\Delta d)+\pi)+cos(k(z_Dcos \theta +x_Dsin \theta)) \right ]

+iE_0e^{-i\omega t}\left [ sin(k(z_D+2\Delta d)+\pi)+sin(k(z_Dcos \theta +x_Dsin \theta)) \right ]

The intensity S of the combined beam is given at the detector screen as,

S=E_tE^{*}_t

which can be solved to get,

S_D=4E^{2}_0cos^{2}(k(\Delta d-(x_D/2)sin\theta +z_Dsin^{2}(\theta /2))+\pi/2)

S_D=S_0sin^{2}(k(\Delta d-(x_D/2)sin\theta +z_Dsin^{2}(\theta /2)))

hence,

S_0=4E^{2}_0

When Mirrors are perpendicular to the beam

If the two mirrors are precisely parallel (\theta =0), then the intensity on the screen is,

S_D=S_0sin^{2}(k\Delta d)

and the whole area of the screen will be uniformly illuminated. The image below shows how you can represent this condition.

Optical equivalent of MI

Optical equivalent of Michelson Interferometer

The screen will be dark for an integral multiple of a wavelength and bright when there is an odd multiple of a half-wavelength.

Waves reflected by M2 at some angle

This condition is the commonly studied interference pattern formed by an interferometer. The wavefronts reflected from mirror M_1 are parallel to the screen, while the wavefronts reflected from mirror M_2 are tilted at an angle to the screen. The path difference along the line of intersection is zero and, therefore, is the same for all the wavelengths. When a source of white light is used, we get a central achromatic bright fringe. On either side of the central fringe, few coloured straight fringes are observed.

The fringes are given by following equation,

S_D=S_0sin^{2}(k(\Delta d-(x_D/2)sin\theta +z_Dsin^{2}(\theta/2)))

Using above equation, the spacing \Delta x_D between two adjacent bright and dark fringes at positions x_D_1 and x_D_2 is given by,

\Delta x_D=\left|x_D_1-x_D_2 \right|

k(\Delta d-(x_D_1/2)sin\theta +z_Dsin^{2}(\theta /2))=\pi +k(\Delta d-(x_D_2/2)sin\theta +z_Dsin^{2}(\theta /2))

\Delta x_D=\frac{\lambda }{sin\theta}

When,

\Delta x_D=\left|x_D_1-x_D_2 \right|=\theta \to 0

sin\theta \to 0

\Delta x_D \to \infty

The fringes disappear, and the screen becomes uniformly illuminated, as described above.

The fringe separation becomes smaller as the wavelength decreases and the tilt angle becomes larger. When mirror M_2 is moved, the fringes move horizontally across the detector screen.

Modelling the interferometer in MATLAB

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Study the interference pattern formed by two light waves and model a MATLAB version of the Michelson Interferometer; Developed in MATLAB R2018a

To model the Michelson interferometer in MATLAB and generate the interference pattern, we first need to write a For example, if the wavelength is 400 nanometres. We will write the code as follows:

function that will convert the input wavelength to its corresponding colour. It is easy and can be achieved using the if – and condition several times.

if (lambda>=380)&&(lambda<440)

   thiscolor = [(440-lambda)/(440-380),0,1];

Be sure to change the value of lambda to the nanometre scale. Next, we need to initialise the several variables that will contain information to be used in modelling. It will include data like wavelength, mirror separation distance, angle theta, detector screen dimensions, propagation constants etc.

Now we will use the electric field equations that we described earlier (E_1 and E_2). The interference can be performed by simply adding them together as we did earlier with the equation to get E_t. We will also use the detector screen intensity equation obtained by E_t. The final part is to plot the detector screen intensity. It can be achieved using pcolor() and colorMap() function of MATLAB. You might also want to plot a line graph to study the variation in the intensity of the detector screen. We will also print out useful information in the command window like wavelength, mirror separation distance, angle theta and fringe spacing.

Now that we have created the model let's run it and check our result.

result

Result

Inputs:

wavelength = 650.00 nm 

mirror separation distance = 1202.50 nm 

X tilt angle = 30.00 deg 

Outputs:

fringe spacing = 1.30 um 

As you can see, I am using a wavelength of 650 nanometres which corresponds to the wavelength of red light. The mirror separation distance I used is around 1202.5 nanometres, while the angle theta is 30 degrees. The fringe spacing, we got in our plot is approximately 1.3 micrometres. The plot is quite self-explanatory. The intensity is maximum whenever there is constructive interference, and we get bright maxima. In contrast, we get minima whenever there is destructive interference.

Application of wave interference

Interferometry has played an essential role in the advancement of physics and has a wide range of applications in physical and engineering measurement. Young's double slit interferometer experiment significantly influenced the general acceptance of the wave theory of light. In quantum mechanics, this experiment demonstrates the inseparability of light's wave and particle natures and other quantum particles (wave-particle duality).

It is also used in acoustic interferometers. An acoustic interferometer is an instrument for measuring the physical characteristics of sound waves in a gas or liquid, such as velocity, wavelength, absorption, or impedance.

The Michelson interferometer can be used to measure small displacement accurately. For example, it is possible to measure a plant's growth rate.

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About the author 

Shubham

I have done my bachelor degree in electronics from University of Delhi. I like to tinker with the hardware to understand how it actually works.

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