Computer-Generated Holography (CGH)

PHYS4412 (Computational Science II)
Department of Physics & Astronomy
Louisiana State University

General Concept

Consider the amplitude (and phase) of light that is incident at a location y₁ on an image screen that is located a distance Z from a slit of width w. First, as illustrated in Figure I.1, consider the contribution due only to two rays of light: one coming from location Y₁ at the top edge of the slit ( a distance D₁ from point y₁ on the screen) and another coming from location Y₂ at the bottom edge of the slit (a distance D₂ from the same point on the screen).

Figure I.1

The complex number A representing the light amplitude and phase at y₁will be,

where l is the wavelength of the light, a( Y_j ) is the brightness of the light at point Y_j on the aperture, and f_j is the phase of the light as it leaves point Y_j.

Now, if we consider for the moment that at all locations on the aperture Y_j , the light has a brightness a( Y_j ) = 1.0 and a phase f_j = 0.0, then,

where A₀ = exp[i(2p D₁ / l ) ]. So the question of whether the amplitude at y₁ on the image screen will be bright due to constructive interference or faint as a result of destructive interference comes down to a question of what the phase difference is between the two distances D₁ and D₂ where,

Utility of FFT Techniques

Now, let's rewrite equation (1) to take into account many more than just two points along the aperture. That is,

or, in the more restrictive case when we assume that everywhere along the aperture the phase f_j = 0,

where, in both equations, the summation is taken over all "j" elements of light along the aperture, and to simplify typing we have replaced a(Y_j) with a_j .   If | Y_j / L | << 1, we can drop the quadratic term in favor of the linear one in expression (3) and deduce that,

Hence, equation (6) becomes,

where, now, A₀ = exp[i(2p L / l ) ]. When written in this form, it should immediately be apparent why discrete Fourier transform techniques (specifically FFT techniques) are useful tools for evaluation of the complex amplitude A.

(If it is not immediately apparent to you how expression [8] can be adapted to FFT techniques, click here for an explicit demonstration.)

Analytical Result

If we consider the limit where the aperture shown in Figure I.1 is divided into an infinite number of divisions, then we can convert the summation in equation (6) or (8) into an integral running between the limits, Y₂ and Y₁. Specifically, equation (8) becomes,

where, again, we have returned to the case where the aperture is assumed to be uniformly bright and set a(Y) = a₀ dY. (It should be understood that a₀ is the brightness of the aperture per unit length.) If we now make the substitution,

equation (9) takes the form,

where the lower limit of integration is q₂ = 2 p y₁ Y₂ / (lL ) = J₁ - b₁, the upper limit of integration is q₁ = 2 p y₁ Y₁ / (lL ) = J₁ + b₁, and

The integral in equation (11) can be readily evaluated, giving,

The last two lines of equation (13) have been derived by realizing that, for any angle c,

Notice that if the center of the aperture is used to define the origin of the Y (and y) coordinate axis so that Y₂ = - Y₁, as illustrated in Figure I.1, then J₁ = 0 and the amplitude A has no complex component. If the aperture is not centered as depicted in Figure I.1, then the angle J₁ simply serves to introduce a phase shift in the evaluation of A.

Although this problem was solved for a specific location y₁ on the image screen, we see that the sinc function solution can readily be evaluated for any other location on the screen without having to redo the integral.