Computer-Generated Holography (CGH)

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

This discussion combines details from Chapter I.B in which we discussed a two-dimensional aperture oriented parallel to the image screen, and Chapter II.A in which a one-dimensional aperture was tilted with respect to the image screen.

General Concept

Consider the amplitude (and phase) of light that is incident at a location (x₁,y₁) on an image screen whose center is located a distance Z from a rectangular aperture of width w and height h but which is tilted about its x-axis by an angle y relative to the image screen. By analogy with equations (I.B-1) and (II.A-2), the complex number A representing the light amplitude and phase at (x₁,y₁) will be,

where, now, the distance D_j,k is given by the expression,

Comparison with Leseberg & Frère

Here we depart somewhat from the sequence of steps that have been taken in earlier chapters in order to compare our result for the tilted 2-D, rectangular aperture with the published result of Leseberg & Frère (1988). Considering the situation when | X_j / L | << 1 and | Y_k / L | << 1, we will use the binomial expansion to approximate the square root in equation (2), but will not yet drop the quadratic terms in favor of the linear ones. That is, we will for the moment employ the approximation,

With this approximation, equation (1) becomes,

where A₀ = exp[i(2p L / l ) ]. Now, as in Chapter I.B, we would like to consider the case where the rectangular aperture is divided into an infinite number of divisions in both the X and Y dimensions and convert the summations in equation (5) into integrals whose limits in both directions are given by the edges of the aperture. Without assuming that the aperture is uniformly bright [i.e., setting a_j,k = a₀(X,Y) dX dY], and by not yet prescribing the phase f across the aperture, equation (5) becomes,

In order to see the correspondence between this expression and the Leseberg & Frère (1988) discussion, it is important to realize that Leseberg & Frère used a slightly different coordinate notation than has been used here in the derivation of equation (6). As illustrated earlier in our Figure 2, we have chosen to measure the separation (Z) between the aperture and the image screen along a coordinate axis that runs through the center of the aperture and lies perpendicular to the aperture, whereas Leseberg & Frère have measured this separation (they called it z₀) along a coordinate axis that runs through the center of the aperture but lies perpendicular to the image screen. Hence, the following transformations must be used:

It is also necessary to utilize the following notational relationships:

Making these substitutions, equation (6) becomes,

Finally, we note that Leseberg and Frère considered the specific case where "the object is illuminated by an on-axis plane wave,"¹ and in so doing, adopted the following phase function f(x',y') for the light leaving the aperture:

In this particular case, then, equation (7) becomes,

which is identical to equation (2) of Leseberg & Frère (1988).

Now the principal point of this derivation has been to demonstrate that the form of our integral expression (6) [or of Leseberg & Frère's equivalent expression shown here as equation (9)], is the same as the form of equation (I.B-6) for the untilted rectangular aperture. That is, in both cases, the last exponential term under the integrals can be written as,

where n and m are coefficients that do not depend on X or Y (i.e., on the intrinsic shape or extent of the aperture). Hence, as Leseberg & Frère put it, "... the Fourier transform and the coordinate transformation can be implemented independent of each other." A similar point has been made by Rabal, Bolognini, & Sicre (1985), although their comments were targeted towards experimental applications. As our discussion at the end of Chapter II.A points out, this idea can even begin to be appreciated in the context of a one-dimensional slit.

Discussion

It does not appear as though Leseberg & Frère actually viewed their result from this perspective, however. They were not looking for ways to speed up the calculation of a "fly-by" sequence, but rather for ways to speed up the calculation of the holographic pattern in the first place! What they discussed, therefore, were ways in which the extended surface of a nonsimple structure might be broken down into a series of (presumably connected) small, rectangular apertures that are each oriented at a different angle to the image screen.

Here we will complete the double integral in equation (6) analytically and offer that result as a mechanism for speeding up the calculation of the holographic pattern, then return to the Leseberg & Frère result later as a mechanism to implement rapid "fly-by" capabilities once multiple apertures have been pieced together to construct the surface of a three-dimensional object.

Analytical Prescription

If we drop from equation (6) [or eq. (7) or eq. (9)] the terms that are quadratic in X (= x') and Y (= y') in favor of the terms that are linear in these variables, respectively, and assume that the aperture is uniformly illuminated (i.e., set u₀ = a₀ cosy = constant),² then the integrals can be completed analytically. (As Leseberg & Frère [1988] point out, even when the Fourier transform is completed numerically, "it is usual to neglect the additional quadratic phase factor [on the image screen]." If the intensity across the image screen is then turned into a transparent hologram, when light is shown back through the hologram, "the complex conjugate of this phase appears in the reconstructed image." But, as Leseberg & Frère put it, "this is of no concern for an intensity recording as performed by most detecting devices.")

Employing these approximations in equation (9), for example, the expression to be integrated is,

Using the shorthand notation, k º 2p/l, p º x/r₀, and q º y/r₀, equation (11) may also be written in the form,

which is precisely the same as Patorski's (1983) equation (2).³ Finally, writing this equation in the form of expression (10), we obtain,

and the subscript "P" is being used to indicate that these are the wave numbers derived by Patorski (1983). Now, as in Chapter I.B, it seems clear that the limits of integration should be from X₂ to X₁ in the x' (= X) coordinate, and in the y' (= Y) coordinate the limits should be from Y₂ to Y₁. Employing these limits, then, both of these integrals can be completed in the same fashion as described in Chapter I.A for the 1D slit, giving,

where,

Assuming the aperture is centered on the image screen -- i.e., Y₂ = - Y₁ and X₂ = - X₁ -- the two phase angles Q₁ = J₁ = 0, and equation (15) becomes,

which is identical to Patorski's (1983) equation (6) in all but one respect. In the y' (= Y) coordinate direction, Patorski set his integration limit to Y₁ = (w/2)cosy whereas we have set ours to Y₁ = (w/2). Hence the argument of the second sinc function in Patorski's expression differs from ours by a factor of cosy. We believe that our limits are the correct ones because, independent of the direction from which you look, the integration is done in the coordinate system of the aperture and therefore should always have the same limits. This aspect of the problem is, in part, what permits us to implement the Fourier transform and the coordinate transformation independent of one another, as discussed above. As explained in footnote 2, below, we agree with Patorski that a factor of cosy should enter the calculation when considering the brightness per unit area of the light that is incident on the aperture, but this factor should not be introduced into the limits of integration because tilting the aperture does not alter the aperture's inherent size or shape.

Having shown the correspondence between our derivation and similar ones presented previously by Leseberg & Frère (1988) and Patorski (1983), we now return to our original notation in equation (6), and apply the same conditions regarding the property of light leaving the surface of the aperture as we have employed in earlier chapters. That is to say, we will assume that the aperture is uniformly illuminated and that the phase f = 0 across the entire aperture. As we also drop the terms that are quadratic in X and quadratic in Y, equation (6) becomes,

where,

In the spirit of the discussion at the end of Chapter II.A, we also note that this result may be presented in the following general form:

where,

and,

Then we simply note that for any point (x₁,y₁) on a flat image plane that is tilted with respect to the rectangular aperture as illustrated in Figure II.2, x_a = x₁, y_a = ( y₁ cosy ), and z_a = ( Z - y₁ siny ) .

¹Initially, it seemed a bit odd to me that Leseberg & Frère would prescribe f(x',y') in this particular way. This phase function makes sense if the light that is illuminating the rectangular surface (the aperture) originates as a coherent source from the image screen; but then one would not expect the light to bounce off of the surface and head directly back to the image screen. Eventually I realized that Leseberg & Frère picked up this particular prescription for f from the article by Patorski (1983). As an experimentalist, Patorski was pointing his coherent source of light directly at the image screen, then inserting a tilted, transparent aperture between the light source and the screen. Hence, the object [aperture] was being "illuminated by an on-axis plane wave."

² If a₀ is the brightness per unit area of the coherent plane wave whose propagation vector is perpendicular to the image screen, as assumed by Patorski (1983), then when the tilted transparent aperture is inserted between the light source and the screen, the brightness per unit area on the aperture will be u₀(x,y) = a₀ cosy. In conjunction with equation (8), therefore, the complex amplitude of light that is incident (and, in this case, also leaving) the aperture will be,

³In order to see the correspondence between our equation (12) and Patorski's equation (2), note that Patorski uses the angle b instead of y, and that the ratio (z₀/r₀) = [ 1 - (p² + q²) ]^1/2. Also, there are two typesetting mistakes in Patorski's equation (2): in one place the coefficient "g" appears instead of a "q", and a factor of y' has been inadvertently omitted from the last term inside the exponential.