# Sampling theorem for in-line digital holography

In digital holography, the image sensor must have a sufficient pixel density. The book New Techniques in Digital Holography by Pascal Picart and others, the following condition is given:

Considering the maximum angle $\theta_{max}$ between the two waves, the micro fringes locally produced by the two tilted wavefronts must be sampled so that the sampling pitch is at least equal to two times the fringe period. Thus, this leads to the maximum acceptable angle for the setup, according to the following equation: $\theta_{max} < 2sin^{-1}(\frac{\lambda}{4max(p_x,p_y)})$

Here, $p_x$ and $p_y$ denote the pixel pitch of the sensor, $\lambda$ the illumination wavelength and $\theta_{max}$ the maximum incident angle on the sensor like this:

Does this formula translate at all to an in-line setup like the following? How is the maximum angle between reference and object light determined? Considering the following drawing, it could approximate something like 85 degrees:

I'm sure the above beam interaction has a minimal influence on reconstruction, so how useful is the formula here?

In your first drawing, it's better to use two arrows: one from the object to a point on the sensor, and one from the reference source to the same point on the sensor. That's the way the object and reference light beams actually travel. The formula relates the angle between object and reference beams to the pixel density required to detect the resulting fringes at that point on the sensor.

In your second drawing, it's better to draw two arrows instead of a single line. The first arrow will go from the reference source to a point on the sensor just outside the shadow of the object. The other arrow (very short) will go from the object to the same point on the sensor. It will be obvious then that the two drawings are describing essentially the same situation: the formula relates the angle between object and reference beams to the pixel density required to detect the resulting fringes at that point on the sensor.

EDIT: Light does travel the way you've shown in your updated drawing. However, the larger the angle between object and reference, the closer together the sensor pixels need to be in order to detect the interference pattern, because the fringes in the interference pattern are closer together when the angle is larger. That just means that only fringes formed relatively close to the "shadow" of the object will be spaced far enough apart for most sensors to detect.

On the other hand, if the reference source is close to the object, the angle between object and reference rays at any point on the sensor can be much smaller, resulting in larger fringes. See this paper.

• Thank you. I updated the question and the graphics. Does anything prohibit light traveling like in the updated second drawing?
– smcs
Commented Apr 18, 2018 at 15:39
• Well this answers my original question, thanks. Unfortunately, now the theorem is basically useless to me, as it's impossible to reach this condition in an in-line setup. Are you maybe aware of other ways to determine a realistically sufficient resolution for this?
– smcs
Commented Apr 19, 2018 at 8:17
• I added a brief explanation to the answer, along with a link to a paper that tells how to do it. Commented Apr 19, 2018 at 15:21
• Thanks, I'm working with a large distance and paraxial approximation in reconstruction though. Still an interesting paper.
– smcs
Commented Apr 20, 2018 at 15:04