**Table of Contents:**

**YOUNG’S EXPERIMENT**

As you may have noted already, in the days before lasers or even in the present day if one doesn’t have access to a laser, it would be difficult to perfectly interfere two waves of light from different sources (especially as most of the readily available light sources were white light). Dr. Thomas Young’s brilliant double slit experiment provided a solution to this problem, and today is one of the most important and well-known experiments in optics. Dr. Thomas Young, not having access to handy-dandy lasers when he first conducted this experiment two hundred years ago, used sunlight and first had the light pass through a single slit in order to create harmonic plane waves, and then used two separate slits to create an interference pattern. This experiment is one of the first in our discussion of “wave-front splitting” interferometry, as opposed to amplitude splitting which we will discuss further later in this section.

The first slit to create plane waves acts as a spatial filter, and only passes through light of wavelength(s) that would match to the slit width which inherently creates a spatially coherent wave, even if it is not temporally coherent. In the configuration of the slits, if the single slit is positioned directly between the double slits, the phase of each wave passing from the single slit to the double slits will be equal. Of course, today we can just use a laser to directly illuminate the double slits to bypass the first slit, since the laser already emits coherent waves. In any case, the configuration of the experiment is then given in the Figure below.

Looking at this image, recall from the last section that the condition for constructive or destructive interference depends on the path length difference between the rays. Clearly, we can find the conditions for interference by finding

\begin{equation} \delta = \overline{S_2P} – \overline{S_1P} = hsin(\theta)\end{equation}

where the last equivalence was taken using similar angles to find the path length difference. In the limit that $D>>h$, which is normally a good approximation for this experiment, we can approximate $\theta_1 = \theta_2 = \theta$. In this case, we would have that “$\theta$” is given as:

\begin{equation} \theta = tan(\frac{x}{D}) \end{equation}

such that then the interference condition would be:

\begin{equation} \delta = \frac{2\pi}{\lambda}\frac{hx}{D} \end{equation}

If we don’t want to make the approximation that $D>>h$, we can also find this condition by placing a lens directly after the slits. In this case, we would then have that the angles between each of the rays would be equal, and that the screen is effectively “at infinity” compared to the placement of the double slits.

**EFFECTS OF FINITE COHERENCE LENGTH**

It is now interesting to consider the interference that occurs when the source has a finite coherence length, which corresponds to when the source is not monochromatic. In this case, a few different situations can occur depending on the degree of incoherence in comparison to the path length difference. To begin the analysis, we will define the spectral bandwidth as $\Delta\lambda_0$. If this width is small compared to the wavelength i.e. $\Delta\lambda_0 << \lambda_0$, then the source is quasi-monochromatic. We can then perhaps intuitively see that the m=0 order would be unaffected by the presence of multiple wavelengths and each of the wavelengths would constructively interfere here, but the higher orders would begin to “wash out” as the contrast between the constructive and destructive interference would fade with the addition of a random assortment of colors.

The maxima are spread by an amount given by:

\begin{equation} |\Delta{x}| = |m|\frac{D}{h}\Delta\lambda_0 \end{equation}

where we can then also compare this to the mean spacing of the fringes

\begin{equation} |\delta{x}| = \frac{D}{h}\lambda_0 \end{equation}

The spacing between the maxima fringes is small, or approximately negligible, when the following conditions are met:

\begin{equation} \frac{|\Delta{x}|}{|\delta{x}|} << 1 \end{equation}

or

\begin{equation} |m| << \frac{\lambda_0}{\Delta\lambda} \end{equation}

What this physically means is that the spacing between maxima is negligible i.e. there are many interference fringes when the bandwidth of the source is much less than the wavelength of the source. In other words, highly monochromatic sources will exhibit many fringes while broadband sources may only exhibit a few.

For example, with white light, the fringe corresponding to the m=0 order would just be white light, as we expected from above. However, as the orders increase, different colors will interfere constructively/destructively in accordance to their wavelength, which then leads to different orders and positions of fringes on the screen. These will begin to “wash out” more and more as the wavelengths are more and more out of phase with respect to each other.

Another way to look at this, is to compare the path length difference and the coherence length. For the following analysis, refer to the figure below (taken from Hecht).

As can be seen in the figure, the corresponding wave groups are labeIed. Let’s first start with the case where the coherence length is comparable to or longer than the coherence length. This situation is depicted in the top image above. In this case, the corresponding wave groups more or less align with one another, meaning that a few distinguishable fringes could be seen. As the image above has three total peaks and valleys, there would be approximately 3 fringes seen. The second possible case occurs when the path length difference is shorter than the coherence length. In this case, the interfering wavegroups do not match up, meaning that the contrast of the fringes would degrade and the interference only lasts a short time as then further wave groups propagate through.

**OTHER WAVEFRONT-SPLITTING INTERFEROMETERS**

Briefly, we will describe here some other types of wavefront-splitting interferometers. The main types are Fresnel’s double mirror, the Fresnel double prism, and Lloyd’s mirror.