Obscured Circular Aperture Diffraction
| $\lambda$ | Wavelength | $z$ | Aperture To Image/Observation Distance |
| $U_1$ | Aperture Field | $U_2$ | Image/Observation Field |
| $x_1,y_1$ | Aperture Plane Coordinates | $x_2,y_2$ | Image/Observation Plane Coordinates |
| $\Im$ | Fourier Transform Operator | $\Im ^{-1}$ | Inverse Fourier Transform Operator |
Fraunhofer Diffraction
Exact Analytic Solution
The Fraunhofer diffraction pattern of an obscured circular aperture using the exact analytic solution is given by $$I(\rho) \propto \frac{1}{\left ( 1-\varepsilon ^2 \right )^2}\left(\left ( \frac{2J_1(\pi D \rho / \lambda z)}{\pi D \rho / \lambda z} \right ) - \varepsilon^2 \left (\frac{2J_1(\varepsilon \pi D \rho / \lambda z)}{\varepsilon \pi D \rho / \lambda z} \right )\right )^2$$ where $\rho$ is the radial distance from the optical axis.
Fourier Transform
The Fraunhofer diffraction pattern is calculated using the Fourier transform method $$U_2(x_2,y_2) = \left | \Im\left ( U_1\left ( x_1,y_1\right )\right ) h\left ( x_2,y_2\right ) \right |^2$$ where $h$ is the impulse response $$h(x,y) = \frac{e^{ikz}}{i \lambda z}e^{\frac{ik}{2z}(x^2+y^2)}$$
Fresnel Diffraction
Transfer Function Method
The Fresnel diffraction pattern is calculated using the transfer function method $$U_2(x,y) = \left | \Im ^{-1}\left ( \Im \left ( U_1(x,y) \right )H(x,y) \right ) \right |^2$$ where $H$ is the transfer function $$H(x,y) = e^{ikz}e^{-i \pi \lambda z (x^2 + y^2)}$$
Impulse Response Method
The Fresnel diffraction pattern of is calculated using the impulse response method $$U_2(x,y) = \left | \Im ^{-1}\left ( \Im \left ( U_1\left (x,y\right ) \right )\Im \left ( h\left (x,y\right ) \right ) \right ) \right |^2$$ where $h$ is the impulse response $$h(x,y) = \frac{e^{ikz}}{i \lambda z}e^{\frac{ik}{2z}(x^2+y^2)}$$
Diffraction Limited Imaging
Coherent Imaging
A diffraction-limited coherent camera system with the specified aperture produces an image based on the equation $$U_2(x_2,y_2) = \Im ^{-1}\left (H(x_2,y_2) \Im \left (I_g(x_2,y_2) \right ) \right)$$ where $I_g(x_2,y_2)$ is the ideal image field and $H(x_2,y_2)$ is the transfer function $$H(x_2,y_2)=P(-\lambda z x_2,-\lambda z y_2)$$ where $P$ is the pupil function
Incoherent Imaging
A diffraction-limited incoherent camera system with the specified aperture produces an image based on the equation $$U_2(x_2,y_2) = \Im ^{-1}\left (\Im(|H(x_2,y_2)|^2) \Im \left (I_g(x_2,y_2) \right ) \right)$$ where $I_g(x_2,y_2)$ is the ideal image field and $H(x_2,y_2)$ is the transfer function $$H(x_2,y_2)=P(-\lambda z x_2,-\lambda z y_2)$$ normalized by the area of the transfer function, where $P$ is the pupil function
2. Numerical Simulation of Optical Wave Propagation with Examples in MATLAB by Jason D. Schmidt, SPIE Press, 2010