# 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 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

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