The Fourier Transform at Work: Young's Experiment

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From the previous page, we know that a plane wave of light incident upon an aperture [A(x)] will produce the Fourier Transform of A(x) on the image plane. As an example, let A(x) be given by:

square aperture
[Equation 1]

We already know what the Fourier Transform of the square box function is - it's the sinc function. Using equation [1] from the box function page, we can the Fourier Transform of A(x) {this page's equation [1]}:

fourier transform of single square aperture
[Equation 2]

Equation [2] gives us the E-fields that are on the image plane. In real life, our eyes see the magnitude squared of the E-field, which is the intensity of light (this is because the eye doesn't know what a negative or positive E-field is, it just sees light). Hence, the intensity of light (I) on the image plane for the single aperture of Equation [1] is given by:

fourier transform squared gives intensity
[Equation 3]

Young's Experiment

Now we are ready to talk about the famous experiment of Thomas Young in 1803. This experiment is the original "proof" that light behaved as a wave, shattering the Newtonian view of light as a particle. Young let sunshine in through a small pinhole at one end of the room. He then allowed it to pass through two apertures, separated by a distance D, and each of width W. The aperture in question is illustrated in Figure 1.

Young's famous two slit experiment

Figure 1. The Setup for Young's Light Diffraction Experiment.

The Aperture Function A(x) corresponding to Figure 1 is given in Equation [4], and plotted in Figure 2.

A(x) for Young's Experiment
[Equation 4]

Young's famous two slit experiment, the plot of the aperture

Figure 2. Plot of the Aperture Function A(x).

Given A(x), we can now take the Fourier Transform to get the image. First, let's get the Fourier Transform of one of the rectangles functions of Equation [4]. This can be done simply, using the Fourier Transform Shift Property, along with the fact that we already know the Fourier Transform of the rect function is the sinc:

Fourier Transform of shifted rectangle function
[Equation 5]

Similarly, we can find the Fourier Transform of the other rectangle function that makes up A(x):

Fourier Transform of the other shifted rectangle function
[Equation 6]

Recall the cosine relationship:

cosine identity rectangle function
[Equation 7]

Finally, using the linearity of the Fourier Transform, and Equations [4-7], we can get the Fourier Transform of A(x):

young's fourier transform
[Equation 8]

Here is an example of the diffraction pattern (the image intensity) for W=3*lambda, and D = 10*lambda:

Young's famous two slit experiment, the plot of the aperture

Figure 3. Example Image Plane Intensity for the Double Slit Aperture.

The actual image Young saw would have looked somewhat similar to Figure 4:

Young's famous two slit experiment, the plot of the aperture and the fourier transform

Figure 4. Image Plane for Double Slit Light Intensity Experiment.

There you have it. The above is the process Young went through in order to prove light behaved as a wave, at least in some cases. The image of Figure 4 resembles what Young originally saw, moving the world of Physics in a new direction. The Fourier Transform makes the math here very simple, and allows for an underlying understanding (intuition) on how the light is behaving. If you understand the Fourier Transform, you understand the diffraction through the aperture.

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