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I know that when light from a laser is reflected from a rough surface onto a screen, it produces a speckle pattern.

I believe, a Fourier transform of the intensity distribution of the speckle pattern can be used to study the different interference patterns that make up the speckle pattern.

I was told that any two points on the rough surface separated by a distance $d$ contributes to the interference pattern that has a fringe spacing $$y=\frac{\lambda L}{d},$$ where $\lambda$ is the wavelength of the laser and $L$ is the distance from the rough surface to the screen.

However, because two points on any rough surface separated by a distance $d$ are also at different heights, say a difference in height of $x$, wouldn't the fringe spacing for two points points be $$y=\frac{(\lambda-x)L}{d}$$

In the image below, the red line illustrates the distance from the slit to the second point on the rough surface.

enter image description here

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To first order, light scattered from a slightly raised point on a surface is phase shifted in proportion to the ratio of the height difference to the light wavelength. When fringes are formed by the interference between light scattered from two separated points, the primary effect of raising one of the points is not to change the spacing of the fringes, but to shift their position. On the other hand, moving one of the points laterally along the scattering surface will change the spacing (and/or orientation) of the fringes.

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    $\begingroup$ Thanks so much for the answer! Do you have a source for this at all? $\endgroup$
    – zld123
    Commented Dec 31, 2018 at 17:11
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    $\begingroup$ I think you already have a source: you can do a Fourier transform of two point scatterers, then see how the Fourier transform changes when you shift the phase or position of one of the scatterers. $\endgroup$
    – S. McGrew
    Commented Dec 31, 2018 at 17:37

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