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The complex amplitude of a wave is $E(x,y) = A(x,y) e^{i\phi(x,y)}$ , where $A$ is the amplitude and $\phi$ the phase.

Sets: $x \in \R$ , $z \in \C$ , $n \in \N$ , $k \in \Z$ .

The Fourier transform:

\mathcal{F}\{f(x)\} = \int_{-\infty}^{\infty} f(x) e^{-2\pi i \xi x} \, dx

Fresnel diffraction:

U(x', y') = \frac{e^{ikz}}{i\lambda z} \iint U(x, y) \exp\left[\frac{ik}{2z}\left((x'-x)^2 + (y'-y)^2\right)\right] dx \, dy

The Helmholtz equation:

\nabla^2 U + k^2 U = 0

where $k = \frac{2\pi}{\lambda}$ is the wavenumber.

The ray-transfer matrix of a thin lens:

\begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix}

\begin{aligned} E_{out} &= t(x,y) \cdot E_{in} \\ &= |t| e^{i\phi_t} \cdot A_{in} e^{i\phi_{in}} \\ &= |t| A_{in} e^{i(\phi_t + \phi_{in})} \end{aligned}

The angular spectrum transfer function:

H(f_x, f_y) = \exp\left(ikz\sqrt{1 - \lambda^2 f_x^2 - \lambda^2 f_y^2}\right)

The point spread function:

I(x,y) = \left|\mathcal{F}^{-1}\{H \cdot \mathcal{F}\{U_0\}\}\right|^2