Computer-Generated Holography (CGH)
PHYS4412 (Computational Science II)
Department of Physics & Astronomy
Louisiana State University
PHYS4412 (Computational Science II)
Department of Physics & Astronomy
Louisiana State University
| I.B | Two-Dimensional, Rectangular Aperture |
|---|---|
| Oriented Parallel to the Image Screen |
This discussion parallels the somewhat more detailed one presented in Chapter I.A on the one-dimensional aperture oriented parallel to the image screen.
Utility of FFT Techniques
Consider the amplitude (and phase) of light that is incident at a location (x1,y1) on an image screen that is located a distance Z from a rectangular aperture of width w and height h. By analogy with equation (I.A-5), the complex number A representing the light amplitude and phase at (x1,y1) will be,
| A (x1,y1) | = | S | S | aj,k exp[ i ( 2p Dj,k/ l + fj,k ) ] , |
| j | k | |||
where, here, the summations are taken over all "j,k" elements of light across the entire 2-D aperture, and now the distance Dj,k is given by the expression,
| Dj,k | º | [ ( Xj - x1 )2 + ( Yk - y1 )2 + Z2 ]1/2 |
| = | [ Z2 + x12 + y12 - 2 ( x1 Xj + y1 Yk ) + Xj2 + Yk2 ]1/2 | |
| = | L [ 1 - 2 ( x1 Xj + y1 Yk ) / L2 + ( Xj2 + Yk2) / L2 ]1/2 , |
| L | º | [ Z2 + x12 + y12 ]1/2 . |
If | Xj / L | << 1 and | Yk / L | << 1, we can drop the quadratic terms in favor of the linear ones in expression (2) and deduce that,
| Dj,k | » | L [ 1 - 2 ( x1 Xj + y1 Yk ) / L2 ]1/2 |
| » | L [ 1 - ( x1 Xj + y1 Yk ) / L2 ] . |
Hence, equation (1) becomes,
| A (x1,y1) | » | A0 | S | S | aj,k exp{i fj,k } exp{ - i [ 2 p ( x1 Xj + y1 Yk ) / ( lL ) ] } , |
| j | k | ||||
where A0 = exp[i(2p L / l ) ]. When written in this form, it should be apparent why discrete Fourier transform techniques (specifically 2D-FFT techniques) are useful tools for evaluation of the complex amplitude A.
Analytical Result
Now, as in Chapter I.A, 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. If we specifically consider the case where the aperture is assumed to be uniformly bright (i.e., aj,k = a0 dX dY, and a0 is the brightness per unit area), and the phase f = 0 at all location on the aperture, equation (5) becomes,
| A (x1,y1) | » | A0 a0 | ò ò | exp{ - i [ 2p ( x1 X + y1 Y ) / ( lL ) ] } dX dY , | ||
| = | A0 a0 | ò | exp{ - i [2p x1X / (lL)]} dX | ò | exp{ - i [2p y1Y / (lL)]} dY . | |
| A (x1,y1) | » | A0 a0 (w h) e( - i Q1 ) e( - i J1 ) sinc ( a1 ) sinc ( b1 ) . | |
where,
| a1 | º | p x1 ( l L )-1 [ X1 - X2 ] = p x1 h ( l L )-1 , |
| Q1 | º | p x1 ( l L )-1 [ X1 + X2 ] , |
| b1 | º | p y1 ( l L )-1 [ Y1 - Y2 ] = p y1 w ( l L )-1 , |
| J1 | º | p y1 ( l L )-1 [ Y1 + Y2 ] . |
It is worth noting that this derivation closely parallels the one presented in § 8.5.1 (p. 393) of Born & Wolf. Specifically, our equation (6) is identical to the "Fraunhofer diffraction integral" written down by Born & Wolf at the beginning of their § 8.5.1; and if we follow Born & Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then Q1 = J1 = 0, and the intensity I(P) at point P given by their eq. (1) precisely matches the expression for A2 that is obtained using our equation (7).