Theory
A. Poisson Distribution
The detection of photons follows Poisson distribution, which is an effective tool describing the occurrence of a random event. For example, if the occurrence of a random event X follows Poisson distribution, the probability of it happening k times can be expressed as:
where λ is the expectation value of X.
Although we could not guarantee all the events of photon detection to be single-photon events, we could control the ratio of unwanted multi-photon events to be small enough to neglect. Quantitatively, the probability inequality Pr(X=1) >> Pr(X>1) by using Eq. (1) should be satisfied.
B. Single Slit Diffraction
When a beam of light with intensity I0 passes through a slit, its intensity on a screen with a separation of s in one dimension will follow the expression:
where d is the width of slit, λ is the light wavelength and x is the distance from the center of the intensity pattern.
C. Coherence
Two lights need to be both temporally and spatially coherent to form a detectable interference pattern. Temporal coherence is a measure of the correlation of light wave’s phase at different points along the direction of propagation [2]. A light is described as perfect temporally coherent if it is exactly monochromatic. The more different frequency lights it contains, the less temporally coherent it will be.
Similarly, spatial coherence is a measure of the correlation of a light wave’s phase at different points transverse to the direction of propagation, which shows how uniform the phase of a wave front is [3]. If two lights are from an exact point source before they reach the double slits,
they are perfectly spatially coherent.
Figure 1. Diagram of young’s Experiment with an extended slit source [2]: this diagram gives a real situation for our experiment when the light source is not an ideal point source
Quantitatively, the width b of point source S’S’’, the wavelength of light λ, the distance a between the double slits S1 and S2, and the distance l between the point source and double slits should satisfy: then final intensity function can be described as
Otherwise, if the light is not perfectly spatially coherent, the intensity function will change to:
Due to the equipment restriction, the point source condition was not ideally satisfied in our experiment. Therefore, we would expect our interference pattern to follow Eq. (4).
D. Visibility
The visibility of an interference pattern is a quantitative description of the quality of the fringes, which is represented by [2]:
where Imax and Imin are the irradiances corresponding to the maximum and adjacent minimum in the fringe system [2], which are expressed as:
The value of visibility varies between 0 to 1. The more closed visibility is to 1, the more obvious the double slit interference will be.