Question
Show (for normally incident plane waves) that if an aperture has a center of symmetry (i.e., if the aperture function is even), then the diffracted field in the Fraunhofer case also possesses a center of symmetry.
Step 1
Step 1: First, we define the aperture function as a complex quantity, which can be expressed as follows: $$u(y,z) = A(y,z)e^{i\phi(y,z)} \tag{1}$$ where $A(y,z)$ is the amplitude and $\phi(y,z)$ is the phase. Show more…
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Show that Fraunhofer diffraction patterns have a center of symmetry [i.e., $I(Y, Z)=I(-Y,-Z)]$, regardless of the configuration of the aperture, as long as there are no phase variations in the field over the region of the hole. Begin with Eq. (10.41). We'll see later (Chapter 11) that this restriction is equivalent to saying that the aperture function is real.
Suppose we have a single slit along the $y$ -direction of width $b$ where the aperture function is constant across it at a value of $\mathscr{A}_{0}.$ What is the diffracted field if we now apodize the slit with a cosine function amplitude mask? In other words, we cause the aperture function to go from $\mathscr{A}_{0}$ at the center to 0 at $\pm b / 2$ via a cosinusoidal dropoff.
Suppose we have a single slit along the $y$ -direction of width $b$ where the aperture function is constant across it at a value of $\mathscr{A}_{0} .$ What is the diffracted field if we now apodize the slit with a cosine function amplitude mask? In other words, we cause the aperture function to go from $\mathscr{A}_{0}$ at the center to 0 at $\pm b / 2$ via a cosinusoidal dropoff
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