4.3. Consider a Gaussian beam with its waist at plane 1 that gets displaced laterally as shown in Fig. P4.3.
$u_0(x_0, y_0) = A \exp \left[ - \frac{(x_0 - x_d)^2 + y_0^2}{w^2} \right]$
Displaced
axis
$f$
$f$
$x, y$
$x', y'$
$1_xd$
$-2f$
$f$
Plane (1)
(2)
(3)
Figure P4.3
$u_0(x_0, y_0)$
(a) What is the expression for the beam in plane 2 as a function of x and y? Can you put an iris in plane 2 that will act on the beam amplitude independent of its displacement $x_d$?
(b) Is there a ray-optical explanation of what happens?
(c) What is the emerging beam in plane 3? Is there a scale change?
(d) Suppose that we had a general illumination function $u_0(x_0, y_0)$ at plane 1. Without necessarily repeating the algebra, can you state $u(x', y')$ in plane 3?