Question

3) Diffraction Theorems. The Fresnel diffraction pattern $u(x, y)$ of an aperture function $A(x, y)$ in the source plane is given by the convolution $u(x, y) = A(x, y) * h(x, y)$, where $h(x, y)$ is the impulse response func- tion. The corresponding transfer function relationship is $\tilde{u}(\nu_x, \nu_y) = \tilde{A}(\nu_x, \nu_y) \tilde{h}(\nu_x, \nu_y)$. (1) Now, given that an 'impulse' in optics is simply a point source, we can surmise right away that the impulse response function for the Fresnel diffraction problem is a spherical wave whose intensity falls off as $1/z^2$, $h(x, y) = \beta \frac{e^{-ikz}}{z} \exp\left[-ik\frac{x^2 + y^2}{2z}\right]$. (2) In this problem, we wish to determine the normalization constant $\beta$ from first principles. 1. In the near field, the diffraction pattern $u(x, y)$ must reduce to the aperture function $A(x, y)$ as $z \to 0$. Therefore, the impulse response and transfer functions must reduce to the limits $h(x, y) \to \delta(x, y)$ and $\tilde{h}(\nu_x, \nu_y) \to 1$ as $z \to 0$. Calculate the transfer function $\tilde{h}(\nu_x, \nu_y)$ directly from eq. (2), and show that the required limit yields the value $\beta = i/\lambda$. 2. In the far field, the intensity diffraction pattern (using the correct value of $\beta$ derived above) is given in the Fraunhofer approximation by $I(x, y) = |u(x, y)|^2 = \frac{1}{\lambda^2 z^2} \left| \tilde{A}\left(\frac{x}{\lambda z}, \frac{y}{\lambda z}\right) \right|^2$. (3) Using the two-dimensional Parseval theorem, $\iint |A(x, y)|^2 dx \, dy = \iint |\tilde{A}(\nu_x, \nu_y)|^2 d\nu_x \, d\nu_y$, (4) show that the total power in the diffracted field is conserved from aperture to screen, i.e. $\iint |u(x, y)|^2 dx \, dy = \iint |A(x, y)|^2 dx \, dy$. (5)

          3) Diffraction Theorems. The Fresnel diffraction pattern $u(x, y)$ of an aperture function $A(x, y)$ in the
source plane is given by the convolution $u(x, y) = A(x, y) * h(x, y)$, where $h(x, y)$ is the impulse response func-
tion. The corresponding transfer function relationship is
$\tilde{u}(\nu_x, \nu_y) = \tilde{A}(\nu_x, \nu_y) \tilde{h}(\nu_x, \nu_y)$. (1)
Now, given that an 'impulse' in optics is simply a point source, we can surmise right away that the impulse
response function for the Fresnel diffraction problem is a spherical wave whose intensity falls off as $1/z^2$,
$h(x, y) = \beta \frac{e^{-ikz}}{z} \exp\left[-ik\frac{x^2 + y^2}{2z}\right]$. (2)
In this problem, we wish to determine the normalization constant $\beta$ from first principles.
1. In the near field, the diffraction pattern $u(x, y)$ must reduce to the aperture function $A(x, y)$ as $z \to 0$.
Therefore, the impulse response and transfer functions must reduce to the limits $h(x, y) \to \delta(x, y)$ and
$\tilde{h}(\nu_x, \nu_y) \to 1$ as $z \to 0$. Calculate the transfer function $\tilde{h}(\nu_x, \nu_y)$ directly from eq. (2), and show that
the required limit yields the value $\beta = i/\lambda$.
2. In the far field, the intensity diffraction pattern (using the correct value of $\beta$ derived above) is given in
the Fraunhofer approximation by
$I(x, y) = |u(x, y)|^2 = \frac{1}{\lambda^2 z^2} \left| \tilde{A}\left(\frac{x}{\lambda z}, \frac{y}{\lambda z}\right) \right|^2$. (3)
Using the two-dimensional Parseval theorem,
$\iint |A(x, y)|^2 dx \, dy = \iint |\tilde{A}(\nu_x, \nu_y)|^2 d\nu_x \, d\nu_y$, (4)
show that the total power in the diffracted field is conserved from aperture to screen, i.e.
$\iint |u(x, y)|^2 dx \, dy = \iint |A(x, y)|^2 dx \, dy$. (5)

$3) Diffraction Theorems. The Fresnel diffraction pattern u(x, y) of an aperture function A(x, y) in the source plane is given by the convolution u(x, y) = A(x, y) * h(x, y), where h(x, y) is the impulse response func- tion. The corresponding transfer function relationship is ũ(, ) = Ã(, ) h̃(, ). (1) Now, given that an 'impulse' in optics is simply a point source, we can surmise right away that the impulse response function for the Fresnel diffraction problem is a spherical wave whose intensity falls off as 1/z^2, h(x, y) = β(e^-ikz)/(z)[-ik(x^2 + y^2)/(2z)]. (2) In this problem, we wish to determine the normalization constant β from first principles. 1. In the near field, the diffraction pattern u(x, y) must reduce to the aperture function A(x, y) as z → 0. Therefore, the impulse response and transfer functions must reduce to the limits h(x, y) →δ(x, y) and h̃(, ) → 1 as z → 0. Calculate the transfer function h̃(, ) directly from eq. (2), and show that the required limit yields the value β = i/λ. 2. In the far field, the intensity diffraction pattern (using the correct value of β derived above) is given in the Fraunhofer approximation by I(x, y) = |u(x, y)|^2 = (1)/(λ^2 z^2)| Ã((x)/(λ z), (y)/(λ z)) |^2. (3) Using the two-dimensional Parseval theorem, ∬ |A(x, y)|^2 dx dy = ∬ |Ã(, )|^2 d d, (4) show that the total power in the diffracted field is conserved from aperture to screen, i.e. ∬ |u(x, y)|^2 dx dy = ∬ |A(x, y)|^2 dx dy. (5)$

Added by Megan A.

Question

Please give Ace some feedback