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 a circular aperture using the exact analytic solution is given by $$I(\rho ) \propto \left (\frac{2J_1(\pi \rho D/\lambda z)}{\pi \rho D/\lambda z} \right )^{2}$$ where $\rho$ is the radial distance from the optical axis, $D$ is the aperture diameter, and $J_1$ is the First Bessel Function.
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 $$U_2(x_2,y_2) = |\Im ^{-1}\left (H(x_2,y_2) \Im \left (I_g(x_2,y_2) \right ) \right)|^2$$ 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
3. Thanks to Michael Ashley for pointing out incorrect spellings of Fraunhofer.