Theory
Theory
Single Slit Diffraction
Suppose a beam of light of wavelength λ travels through a single slit of width a. The intensity pattern a distance L from the single slit is given by
where I0 is a proportionality constant,
,
and x is the distance from the center of the intensity pattern [2].
This only works if the Fraunhofer condition
is satisfied. The graph below shows an ideal single slit diffraction pattern with λ = 546 nm, L = 23.5 cm, and a = 34 μm. These values satisfy the Fraunhofer condition.
Double Slit Diffraction
Instead suppose the light hits N slits each separated by distance d that satisfy the Fraunhofer condition. The intensity pattern a distance L from the single slit is given by
.
All other variables are the same as the single slit diffraction case [2]. For a double slit, N = 2 and the equation simplifies to
where Iss comes from the single slit diffraction equation and
.
The graph below shows an ideal double slit diffraction pattern with λ = 546 nm, L = 23.5 cm, a = 34 μm, and d = 500 μm.
Coherence
In order for single or double slit diffraction to occur, the incident light beam must be monochromatic and coherent when it hits the slits.
Light is considered monochromatic if every wave has a single wavelength.
Light is considered coherent if all of its individual waves are in phase with each other.
There are two forms of coherence to consider for the double slit experiment, temporal coherence, and spatial coherence. A temporally coherent light beam is in phase in time, while a spatially coherent light source is in phase in space. Temporal coherence is measured using the coherence time tc which is
where Δν is the frequency band of the light source [3]. Because frequency and wavelength are related for light by
where c is the speed of light, the range of wavelengths is proportional to the frequency bandwidth.
If a beam of light is perfectly monochromatic, there is no variation in wavelength or frequency, so tc is infinity.
In real life, no light source is perfectly monochromatic, and all light sources become temporally incoherent with time. The length light can travel without becoming temporally incoherent called the coherence length. This is given by
If a light source is not spatially coherent, passing it through a sufficiently narrow single slit will improve spatial coherence by disrupting the path of incoherent light. Rueckner and Peidle
state that a light source that passes through a single slit of width w will act as a point source as long as
where r is the distance from the single slit to the double slit. This condition must be satisfied to ensure that the beam is from a point source, and thus spatially coherent, which is why this equation is called the “coherence condition” [1]. As the single slit narrows, the condition is better satisfied, and the spatial coherence of the light is improved.
Visibility
The best way to quantify spatial coherence is through a quantity called visibility. Visibility is defined by Hecht as,
where Imax and Imin are the maximum and minimum intensities of the an interference pattern respectively [3].
When looking at double slit interference fringes, a visibility of 0 indicates that no destructive interference has occurred while a visibility of 1 indicates that perfect destructive interference has occurred. Consider the visibility of this double slit diffraction pattern.
The minima of the interference pattern do not reach zero, meaning that the visibility is greater than zero but less than one. If the intensity of the minima rose, the visibility of the pattern would fall to zero and the pattern would look like single slit diffraction. If the intensity of the minima fell, the visibility of the pattern would rise to 1 and the pattern would look like double slit diffraction pattern. The image above has a visibility of 0.8 [4].
Malus' Law
A polarizing filter controls the intensity of the light that passes through it. Malus’ Law states that the intensity is
where I0 is the intial intensity and ψ is the angle between the transmission axes of the light and polarizer [2].
When the polarization components of two waves of light are perfectly perpendicular, the waves cannot interfere with each other. Therefore when the light passes through the first polarizing filter and subsequently through quantum markers, the double-slit interference pattern will be destroyed because the waves from the slits will not be able to add either constructively or destructively. Instead, a single-slit pattern will be produced. Adding the quantum eraser immediately after the quantum markers will allow the light to interfere again, restoring the double-slit interference pattern because it is angled 45° from the two markers [1].
According to Rueckner and Piedle, polarizing the initial light beam parallel to the polarization of the quantum eraser is necessary to fully reproduce the interference pattern with the eraser. The light needs to have an initial polarization angle that is equidistant from the polarization angles of each of the quantum markers, otherwise the intensity of the light coming out of each slit will be different as seen by Malus' law and the interference pattern will not be fully restored by the quantum eraser [1].