PHY294H - Lecture 44

Interference due to a thin oil film

When interference occurs due to a thin film with refractive index n, we have to take into account the fact that the wavelength of light is shorter inside the thin film. [ Recall that the speed of light inside a material with refractive index n is, v_n = c/n. The frequency however is the same, so we have, $c = f \lambda$ and $v_n = f \lambda_n$. Taking the ratio of these equations yields $\lambda_n = \lambda/n$. ] The number of wavelengths of light inside a thickness 2 l is then, $2l/\lambda_n$. Notice that the only modification to our previous analysis is that $\lambda \rightarrow \lambda_n$.

Interferometers

We have seen that the wave nature of light leads to easily observed effects in a variety of situations, but typically when there is a length scale which is of order the wavelength of light or smaller. The typical wavelength of light is $0.5\mu m$, so light can be sensitive to rather small lengths. This sensitivity is used as a tool in interferometers.

Michelson interferometer

Michelson invented the interferomenter near the end of the 19th century. It was used in the famous Michelson-Morley experiment which indicated the absence of an ``ether''.

The Michelson interferometer produces interference fringes by splitting a beam of monochromatic light so that one beam strikes a fixed mirror (this beam has pathlength l₁) and the other a movable mirror (this beam has pathlength l₂. When the reflected beams are brought back together, an interference pattern results. As usual the condition for constructive interference is $l_2-l_1 = n\lambda$. Note that if the media through which the beams travel is different for the two paths, then the path lengths l₂ and l₁ must be modified. For example if path l₁ is through air and path l₂ is through water, then the length of path 2 must be multiplied by the refractive index of water.

Fabry-Perot interferometer

The Fabry-Perot interferometer uses multiple reflections of the two beams to enhance the interference pattern. This is achieved by using two parallel mirrors one of which is a partially transmitting mirror.

Diffraction

Consider breaking up the single slit diffraction process up into 2N+1 sources equally spaced in a slit of width a. The spacing between adjacent sources is a/2N, as we place one source at each edge of the slit. The electric field at a position on the screen at angle to the center is then,

$$E = E_0 \sum_{i=-N}^N sin(\omega t + \alpha {i\over N})$$ (1)

where

$$\alpha = {\pi a sin(\theta) \over \lambda}$$ (2)

Now we define x=i/N and change the sum to an integral, then

$$E = E_0N \int_{-1}^1 sin(\omega t + \alpha x) = -{E_0N\over \alpha}[cos(\omega t + \alpha) - cos(\omega t - \alpha)]$$ (3)

Using the trigonometric identity

$$cos(x\pm y) = cos(x)cos(y) \mp sin(x) sin(y)$$ (4)

we find,

$$E = 2 N E_0 sin(\omega t) {sin(\alpha)\over \alpha}$$ (5)

The intensity is then,

$$I = c\epsilon_0<E^2> = I_0 {sin^2(\alpha)\over \alpha^2}$$ (6)

Diffraction minima then occur at $\alpha = n \pi$ where $n = \pm 1, \pm 2, ...$ or,

$$a sin(\theta) = n \lambda\ \ \ \ minima\ \ n= \pm 1, \pm 2, .....$$ (7)

In the limit $\theta \rightarrow 0$, there is a diffraction maximum so that the central maximum is twice the size of all of the other maxima.

Multiple slit interference pattern

The intensity of a multiple slit interference pattern, where there are N slits separated by distance d is given by,

$$I = I_0 {sin^2(N \beta) \over sin^2(\beta)}$$ (8)

where

$$\beta = {\pi d sin(\theta) \over \lambda}$$ (9)

This expression can be found by summing up the contributions of the site to the total electric field, as we did in the diffraction case. In fact if $d\rightarrow a/N$ with $N\rightarrow \infty$, we recover the diffraction case. Principle maxima occur when $\beta = n\lambda$ or,

$$d sin(\theta) = n \lambda \ \ \ principal\ maxima \ \ n = 0, \pm 1, \pm 2 ...$$ (10)

which is the same as the condition for maxima in the double slit case. However the size of the principal maxima increases as N², which implies that the angular width of these peaks is proportional to 1/N. This is very important as it allows high precision measurements of spacings by using systems with a large number of slits, i.e. $N\rightarrow \infty$. Systems with many slits are often called diffraction gratings. If we find the location of a principal maxima, and we know d to high precision, then we can find the wavelength of the light. This was a key method in finding precise values of the wavelengths of light emitted in early spectroscopic studies of atoms and molecules.