In wave optics, light propagation through free space is modeled as a linear, shift-invariant (LTI) system, where lenses and phase masks act as spatial modulators.
1. Free-Space Propagation (The LTI System View)
Propagation between two parallel planes can be calculated either by spatial convolution (impulse response) or frequency domain multiplication (transfer function).
Rayleigh-Sommerfeld Diffraction (Spatial Domain) Free space acts as a spatial filter. The field at distance $z$ is the 2D convolution of the initial field with the free-space impulse response $h(x,y,z)$.
$$U(x,y,z) = U(x,y,0) * h(x,y,z)$$$$h(x,y,z) = \frac{z}{j\lambda} \frac{\exp(j k r)}{r^2}$$- $r = \sqrt{x^2 + y^2 + z^2}$ is the distance between source and observation points.
- $k = \frac{2\pi}{\lambda}$ is the wavenumber.
- When to use: Theoretical formulations for exact scalar diffraction without paraxial restrictions. Rarely used directly in numerical computation due to the highly oscillatory integral.
Angular Spectrum of Plane Waves (Frequency Domain)
This is the spatial frequency equivalent of Rayleigh-Sommerfeld. The Fourier transform decomposes the wavefront into a set of plane waves traveling at different angles. The transfer function $H(f_X, f_Y)$ applies a phase delay to each spatial frequency component based on the propagation distance.
$$U(x,y,z) = \mathcal{F}^{-1}\left\{ \mathcal{F}\{U(x,y,0)\} \cdot H(f_X, f_Y) \right\}$$$$H(f_X, f_Y) = \exp\left(j z \sqrt{k^2 - (2\pi f_X)^2 - (2\pi f_Y)^2}\right)$$- $f_X, f_Y$ are spatial frequencies (cycles per unit length).
- When to use: Numerical simulations of propagation (like in
torchoptics). It is exact for scalar fields and highly efficient due to the Fast Fourier Transform (FFT).
2. Paraxial Approximations
When the propagation distance $z$ is large compared to the transverse dimensions (the paraxial approximation), the exact distance $r$ is approximated using a binomial expansion.
Fresnel Diffraction (Near-Field)
The impulse response simplifies to a quadratic phase factor (a spatial chirp). The field is the convolution of the source with this chirp.
$$U(x,y,z) = \frac{\exp(jkz)}{j\lambda z} \iint U(\xi,\eta,0) \exp\left( j\frac{k}{2z} [(x-\xi)^2 + (y-\eta)^2] \right) d\xi d\eta$$- When to use: Analytical calculations in the near-field, or evaluating light immediately after it passes through an aperture or lens.
Fraunhofer Diffraction (Far-Field) As $z$ becomes extremely large, the quadratic phase term across the aperture approaches unity. The diffraction pattern becomes the exact Fourier transform of the aperture, scaled by the propagation distance.
$$U(x,y,z) = \frac{\exp(jkz) \exp\left[j\frac{k}{2z}(x^2+y^2)\right]}{j\lambda z} \mathcal{F}\{U(\xi,\eta,0)\}\Bigg|_{f_X=\frac{x}{\lambda z}, f_Y=\frac{y}{\lambda z}}$$- When to use: Analyzing far-field patterns (e.g., the Airy disk of a circular aperture) or propagation over macroscopic distances without lenses.
3. Optical Elements and Operations
Thin Element Approximation A thin optical element modulates the complex amplitude of the incoming wave without spatially shifting the rays.
$$U_{out}(x,y) = U_{in}(x,y) \cdot A(x,y) \exp(j\phi(x,y))$$- $A(x,y)$ represents amplitude attenuation (e.g., an amplitude grating or a binary mask).
- $\phi(x,y)$ represents phase modulation.
- When to use: Modeling Spatial Light Modulators (SLMs), coded apertures, or custom refractive optics where physical thickness $h(x,y)$ dictates the phase via $\phi(x,y) = \frac{2\pi}{\lambda} (n_{element} - n_{medium}) h(x,y)$.
Thin Lens Amplitude Transmittance A spherical lens imparts a quadratic phase shift, effectively applying a spatial chirp to the wavefront that either converges or diverges the field.
$$t_{lens}(x,y) = \exp\left(-j \frac{k}{2f} (x^2 + y^2)\right)$$- $f$ is the focal length. Positive $f$ indicates a converging (convex) lens.
- When to use: Representing ideal lenses in an optical path.
The Fourier Transforming Property of a Lens
If an object is placed at the front focal plane of a convex lens, the field at the back focal plane is the exact Fourier Transform of the object, without the residual quadratic phase curvature seen in Fraunhofer diffraction.
$$U_f(x_f,y_f) = \frac{1}{j\lambda f} \mathcal{F}\{U_0(x_0,y_0)\}\Bigg|_{f_X=\frac{x_f}{\lambda f}, f_Y=\frac{y_f}{\lambda f}}$$- When to use: Designing 4f correlators, optical filters, or computational optical systems that manipulate light in the frequency domain.
4. Observables
Intensity Optical sensors operate at timescales far slower than optical frequencies (approx. $10^{14}$ Hz). Therefore, phase is lost, and sensors integrate the absolute square of the complex amplitude over time.
$$I(x,y) = |U(x,y)|^2 = U(x,y) U^*(x,y)$$- $U^*$ is the complex conjugate.
- When to use: The final step of any forward model before applying sensor noise, quantization, or thresholding constraints.