Computer-Generated Holography (CGH)

Joel E. Tohline
tohline@physics.lsu.edu

Analytical Sinc Functions

Before we will be able to believe the technique that we develop to produce 3D images of, say, a cube viewed from different orientation angles, we need to check our numerically generated Fourier transform results against some analytically known results.

One-dimensional slit parallel to the image screen:

According to my derivation in class, the amplitude Atot that is produced at any point "y" along the image screen is

Atot = [ sinq ] /q ,
[Eq. 1]
where,
q = pyw/( lR) .
[Eq. 2]

Assignment #1: In your java program that utilizes the 1-D FFT from Numerical Recipes, create a 1-D array of Atot values as given by equation (1), above, then make a quantitative comparison between this array of numbers and the array of amplitudes that you have derived from the 1-D FFT of a 1-D slit having width "w". (In order to make this comparison, you will have to choose "reasonable" values of R and l.)

Answer

Two-dimensional, rectangular aperture parallel to the image screen:

According to derivations presented in a variety of different optics textbooks (for example, p. 393 of the 6th edition of Born and Wolf's "Principles of Optics", 1980), for a 2-D, rectangular aperture the amplitude Atot that is produced at any (x,y) location on the image screen is

Atot = [ sinq / q ] [ sina / a ] ,
[Eq. 3]
where,
q = pya/( lR) ,
a = pxb/( lR) ,
[Eq. 4]

and "a" and "b" are the width and height of the aperture.

Assignment #2: In your java program that utilizes the 2-D FFT from Numerical Recipes, create a 2-D array of Atot values as given by equation (3), then quantitatively compare this array of numbers with the array of amplitudes that you have derived from the 2-D FFT of a rectangular aperture having width "a" and height "b".



One-dimensional slit inclined to the image screen:

According to my derivation in class, the amplitude Atot that is produced at any point "h" along the image screen is

Atot = [ sinq ] /q ,
[Eq. 5]
where,
q = phw cosz/[ lR ( 1 + h sinz / R)] .
[Eq. 6]

Assignment #3: In your java program that utilizes the 1-D FFT from Numerical Recipes, create a 1-D array of Atot values as given by equation (5), then make a quantitative comparison between this array of numbers and the array of amplitudes that you derive from the 1-D FFT of a 1-D slit having width "w" but that is tipped at an angle z to the image screen. The slit that you use will have to be "foreshortened" by a well-defined amount, and you will have to somehow adjust the initial phases across your slit to account for the slit's tilt.

Two-dimensional, rectangular aperture inclined to the image screen:

Assignment #4: Combine the ideas covered in assignments #2 and 3 to figure out how to calculate/derive Atot values for a rectangular aperture that is tilted by an angle z to the image screen.

Interesting links regarding CGH: