An Introduction to The Wigner Distribution in Geometric Optics

From Light Fields

An Introduction to the Wigner Distribution Using Geometric Optics Principles

Ramesh Raskar, MIT
Tom Cuypers, Hasselt University
Se Baek Oh, MIT
Roarke Horstmeyer, MIT

In computer science light propagation is often simulated using geometric optics for simplicity and efficiency reasons. Using this representation of light traveling in a straight line, we are able to model various phenomena including light reflection, transmission and refraction. For example, raytracers are commonly used to simulate light in computational photography and computer graphics. However, due to its wave property, light is also able to produce wave phenomena, such as diffraction and interference. An intuitive way of reproducing these effects is by performing phase tracking.

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Alternatively, The Wigner Distribution Function(WDF) is a popular representation of light used in Fourier Optics. It is a 4D function in position (2D) and spatial frequency(2D), and can be used to model diffraction and interference phenomena. As the spatial frequency axis of the WDF can be directly related to the angular direction of a plane wave, this function can be seen as a 4D light field, also called Augmented Light Fields. It can be used to model and simulate light propagation through diffractive elements. Constructive and destructive interference, an immediate consequence of wave propagation, is acquired from the WDF through integration of both positive and negative values. Note, however, that although the WDF contains negative values, the resulting intensity distributions, which is a projection of the WDF along the spatial frequency axis, is always non-negative.

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Non-negativeness

Although the WDF contains negative values, the projection of the WDF along any arbitrary direction in the x-u plane is always non-negative. This can be understood as follows: Propagation through a Graded Index (GRIN) medium with an elliptical index profile corresponds to the Fractional Fourier Transform, which is equivalent to rotation of the WDF with respect to the origin. Since the intensity inside the medium is always non-negative, so is the projection of the WDF.

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Paraxial vs. Non-paraxial region

The WDF is guaranteed to be valid only in the paraxial zone. This is the region where light is propagating close to the normal of the diffracting surface. Outside of the paraxial region, as the region of interest moves away from the paraxial region, the error of the WDF increases. However, this deviation is slow since there is no definite boundary between the paraxial and non-paraxial region. If the small error of the WDF is not tolerable, then more rigorous functions such as angle-impact WDF can be used.

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Near field vs. Far field

The WDF is conserved along rays in the paraxial region. Hence, it is valid in both the near-field and far-field, provided that we limit our analysis to the paraxial region. (Here, near-field is not the near-zone in optics where evanescent field is still strong) In the far-field, the observed wave at a single point is only dependent on angle, and is essentially independent of the distance from the grating. The near-field is the region close to the grating where the wave's distance to the grating also influences the observed pattern.

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Coherent vs. incoherent light

Additionally, the WDF is able to model both coherent and incoherent light. Coherence refers to the cross-correlation of a wavefront of light, and essentially describes how well light is able to interfere (produce constructive bright spots and destructive dark spots). The video above shows that even when different random sources create arbitrary wavefronts, the total wave at a distance approaches a nicely propagating coherent wave. The waves closer to the random sources can be thought of as being partially coherent.

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The WDF allows to simulate and understand applications such as a Gaussian Beam and the Rotating_PSF in terms of geometric optics.

Alternative Rendering techniques for wave phenomena in Computer Graphics

There are several alternative techniques used in computer graphics and audio rendering for simulating wave phenomena. Each with their own advantages and disadvantages. We can categorize them in three groups:

Phase tracking: a naive and inefficient technique where we track the phase of for each ray and take this into account when integrating over several rays.
Diffraction Shaders: by reducing the problem to the far-field, the render calculations can be reduced to parallel rays and there is no need for phase tracking.
Edge Diffraction: used in interactive audio rendering to reduce diffraction calculations, diffraction is only performed at edges.

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