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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