Obscured Circular Aperture Diffraction

The equations listed below use the following symbols:

 $\lambda$ 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

References:
1. Computational Fourier Optics by David Voelz, SPIE Press, 2011
2. Numerical Simulation of Optical Wave Propagation with Examples in MATLAB by Jason D. Schmidt, SPIE Press, 2010

Fresnel Diffraction

Fresnel Transfer Function Fresnel Impulse Response Fraunhofer Diffraction

Fraunhoffer Exact Analytic Fraunhoffer FFT Diffraction Limited Imaging

Original Image Coherent Imaging Incoherent Imaging 