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This article presents the theory behind Young's Two-Pinhole experiment and simulates the setup in OpticStudio using geometrical ray tracing. The theoretical and simulated results are then compared.
Authored By Jeffrey P. Wilde
Introduction
Young's Two-Pinhole experiment is one of the most well-known experiments in physics. This experiment revealed the wave-like properties of light by presenting the transformation of a point source into an interference pattern. The results of Young's experiment can be interpreted qualitatively as a fringe pattern or quantitatively as the coherence factor given as a function of the width of the source. Each form will be analyzed.
In this article, the theory behind the double slit experiment will be discussed. Then, the experiment will be accurately modelled in OpticStudio's Non-Sequential Mode.
Young's Two-Pinhole experiment
Young’s Two-Pinhole experiment is a canonical setup for illustrating the role of spatial coherence in the formation of interference fringes. The general layout is shown below.
The fringe pattern formed at the observation plane depends on the spatial coherence of the light illuminating the pinhole plane, as well as the separation between the two pinholes and the propagation distance from the pinhole plane to the observation plane. Although the mathematical rigor of statistical optics as applied to this problem might seem daunting, a physical understanding of fringe formation from an extended source is actually fairly straightforward when it’s recognized that the observed fringe pattern is simply the sum of many elemental fringes arising from different points on the source [Ref. 1, Section 5.2.1]. Here we consider the case in which the source is incoherent, meaning that any two points on the source radiate randomly in an uncorrelated fashion. Such is the case for a thermal incandescent source.
This setup can be reasonably well simulated in OpticStudio Non-Sequential Mode using only geometric ray tracing combined with surface scattering and “Importance Sampling” of the scattered rays. We begin by again noting that an elemental fringe pattern at the observation plane is formed by each point on the extended source. In OpticStudio, such fringe patterns are found by using coherent detection of rays with a Detector Rectangle. A collection of elementary fringe patterns (from sample points taken across the whole source) is then summed on an intensity basis to yield a resultant fringe pattern. The visibility of the resultant fringe pattern is a measure of the partial coherence of the light at the pinholes. Before explaining the simulation details, we first briefly review the basic optical physics of Young’s experiment. The results of an analytical analysis will be used for comparison with the ray trace simulation.
For simplicity, we will assume the pinholes are restricted to lie along the x-axis, symmetrically disposed about the origin and separated by a distance Δp. The time-average intensity pattern formed in the observation plane is given by:
where I1 and I2 are the intensities at the pinholes, µ12 is the so-called “complex coherence factor” evaluated at the pinholes, and φ12 = arg{µ12} [Ref. 1, Eq. 5.2-36]. The fringe visibility:
is determined by the pinhole intensities and the modulus of the coherence factor. The value of |µ12| lies between zero and one and is simply a measure of the extent to which the optical fields at the two pinholes are correlated with one another on a time-average basis. For the case of I1 = I2 we see that V = |µ12|.
The coherence factor µ12 of the light illuminating the pinholes is a function of the source size, its spatial coherence function, and the propagation distance from the source to the pinhole plane. In addition to assuming the source is perfectly incoherent (with a delta function spatial coherence), we also restrict the light to be quasi monochromatic, i.e., it is sufficiently narrowband that any relative path length differences encountered across the beam’s wavefront are much shorter than the temporal coherence length c/Δν. In this way, spatial coherence effects only are considered. As the beam propagates away from the source, it acquires spatial coherence, meaning the coherence function width begins broadening [Ref. 2]. By invoking the van Cittert-Zernike theorem, we know the coherence function at the pinhole plane is given by a scaled Fourier transform of the source intensity profile, where the scaling factor depends on the wavelength and the propagation distance z1.
The source is often assumed to be circular with a uniform intensity, in which case the coherence function at the pinhole plane takes the form of a jinc function. For simplicity, we assume a uniform 1-D line source along the x-axis, centered at the origin, having a half-width of w. In this case the coherence function at the pinhole plane becomes
For a pinhole separation of Δx = Δp, the coherence factor is:
For small separations the fields emanating from the two pinholes are highly correlated (µ12 ~ 1), and we would expect to observe a fringe pattern having high visibility. In contrast, as the pinhole separation increases, the fields become less correlated and the fringe visibility diminishes. The coherence factor depends in a similar fashion on the source width 2w. As an example, the coherence factor for λ = 1.0 µm, z1 = 10 mm, and a pinhole separation Δp = 100 µm, is plotted as a function of source width below.
The OpticStudio simulation details are outlined in the following sections and the results compared with the curve in the figure above.
Modeling of Young’s experiment with a point source
The first step is to create an elemental fringe pattern for a single point source. The key is to make very efficient use of the rays so that a high-fidelity fringe pattern is formed without tracing rays that don’t contribute to the interference process. This is done with a straightforward Non-Sequential layout as shown in the figure below. The two pinholes are circular disks of radius 5 µm separated by 100 µm (the disk centers are displaced along the x-axis by +/- 50 µm). The distance from the source plane to the pinhole plane is 10 mm as is the distance from the pinhole plane to the observation plane. The wavelength is taken to be 1 µm.
The corresponding Non-Sequential Component Editor entries are shown below.
The source is just a single on-axis ray polarized along the y-axis; however, this ray is traced repeatedly by setting the # of Analysis Rays to be one million. After propagating a short distance, taken here to be 10 um, the source ray encounters a Standard Surface. The purpose of this surface is to convert the incoming on-axis ray into two scattered rays directed at the two pinholes. This is done by first setting the scattering parameters as shown in the figure below. The choice of Scatter Model isn’t particularly important as long as the scattered ray distribution is sufficiently large to fill the solid angle subtended by a pinhole.
Next, the two scattered rays are directed at the two pinholes by using Importance Sampling as shown below. The Size parameter, which determines the radius of the target sphere used by Importance Sampling, is set to be slightly smaller than the radius of the pinholes. Doing so ensures that essentially all rays will pass through the pinholes.
When the rays arrive at the pinholes, Scattering and Importance Sampling are used once again (see below). This time each ray striking a pinhole is converted into five scattered rays, and Importance Sampling directs these rays to a circular region with a 400 µm radius on the observation plane.
While Importance Sampling is effective at tracing only rays of interest, these rays are weighted as though they are a subset of a broader solid angle of rays. For Gaussian scattering, the overall solid angle is determined by the Sigma parameter, while for Lambertian scattering the solid angle is 2π. So, the individual ray intensity can become quite low. To ensure that the rays are allowed to trace, the relative ray intensity threshold should be reduced, and can in fact just be set to its minimum value in the Non-Sequential section of the System Explorer:
We can now trace the system using the following settings which turn on the ray splitting and scattering features:
The resulting fringe pattern is found by viewing the coherent irradiance distribution as shown below.
Note that the number of Total Hits is very close to 10 million, which is consistent with 1 million source rays being doubled by the first scattering surface, then increased by a factor of five when scattering from the pinholes. This number of rays is seen to produce a good high-contrast elemental fringe pattern.
Extending the simulation for an incoherent line source
To model an incoherent line source, a point source is spatially scanned in discrete steps along the x-axis, and the corresponding elemental fringe patterns are sequentially calculated. The elemental fringes are all basically the same apart from a phase shift that depends on the point source location. These elemental fringes are then added in intensity to arrive at a resultant fringe pattern for the entire source. Here we use a source sampling period of 5 µm. A Matlab script was written that links to OpticStudio using DDE. The elemental fringes are generated in a loop. For each loop iteration, the point source position is set, a raytrace is conducted, and the coherent data from the Rectangle Detector are written to file using the SaveDetector command. The file is then read, the complex amplitude image is extracted, and the image intensity is computed from the amplitude squared. A cumulative sum of the elemental fringes is maintained, so that at the end of the loop, the final resultant fringe intensity pattern is obtained.
Simulated fringe patterns for different source widths
Below, the simulated resultant fringe patterns are shown for source widths ranging from 10 µm to 350 µm. The visibility is calculated from a central region of the detector, and the value for each source width is reported in the title of the corresponding image. Note the phase reversal of the fringes as the source is increased in size from 50 µm to 150 µm, and then again in going from 150 µm to 250 µm.
Comparison of simulation with theory
We can now overlay the simulated fringe visibility points (taken with an appropriate choice of sign to represent the coherence factor) on the theoretical curve from the beginning of the article. Doing so yields very good correspondence between theory and simulation.
We conclude by noting that ray tracing has been used to model interference of light emanating from two small pinholes. The angular distribution of the rays propagating from each pinhole is set by the scattering model. In reality, the light will exhibit diffraction, so at the observation plane it is the overlap of two diffracted beams that is ultimately detected [Ref. 1, Section 5.2.5]. However, this level of detail is ignored here.
References
- J. W. Goodman, Statistical Optics (Wiley, 1985).
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge, 2007), Section 3.2, “Generation of spatial coherence from an incoherent source. The van Cittert-Zernike theorem”
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