Angular Spectrum
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The angular spectrum is a method for representing optical fields as a superposition of plane waves traveling in different directions. It provides an exact solution to the wave equation and is particularly useful for analyzing diffraction and propagation.
Definition
For a monochromatic field $U(x, y, z)$ at a plane $z = 0$, the angular spectrum is defined as the 2D Fourier transform:
$$ A(f_x, f_y; 0) = \mathcal{F}\{U(x, y, 0)\} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} U(x, y, 0) \, e^{-j2\pi(f_x x + f_y y)} \, dx \, dy $$where $f_x$ and $f_y$ are spatial frequencies (cycles per unit length).
Physical Interpretation
Each point $(f_x, f_y)$ in the angular spectrum represents a plane wave traveling in a specific direction. The spatial frequencies relate to propagation angles by:
$$ \sin\theta_x = \lambda f_x, \quad \sin\theta_y = \lambda f_y $$where $\lambda$ is the wavelength and $\theta_x$, $\theta_y$ are the angles from the optical axis.
| Spatial Frequency | Physical Meaning |
|---|---|
| $(0, 0)$ | On-axis plane wave (straight through) |
| Large $f_x$ or $f_y$ | Steeply angled plane waves |
| $\sqrt{f_x^2 + f_y^2} > 1/\lambda$ | Evanescent waves (don’t propagate) |
Propagation
The angular spectrum at a distance $z$ is:
$$ A(f_x, f_y; z) = A(f_x, f_y; 0) \cdot e^{j k_z z} $$where the $z$-component of the wave vector is:
$$ k_z = \begin{cases} 2\pi\sqrt{\frac{1}{\lambda^2} - f_x^2 - f_y^2} & \text{if } f_x^2 + f_y^2 < \frac{1}{\lambda^2} \text{ (propagating)} \\ j2\pi\sqrt{f_x^2 + f_y^2 - \frac{1}{\lambda^2}} & \text{if } f_x^2 + f_y^2 > \frac{1}{\lambda^2} \text{ (evanescent)} \end{cases} $$The field at any plane $z$ is obtained by inverse Fourier transform:
$$ U(x, y, z) = \mathcal{F}^{-1}\{A(f_x, f_y; z)\} $$Advantages
- Exact: No paraxial or Fresnel approximations required
- Intuitive: Decomposes fields into plane wave components
- Computational: Efficiently implemented via FFT algorithms
- Unified: Handles both Fresnel and Fraunhofer regimes
Connection to Diffraction
For an aperture with transmission function $t(x, y)$ illuminated by a unit-amplitude plane wave:
$$ A(f_x, f_y) = \mathcal{F}\{t(x, y)\} $$This directly gives the diffracted field’s angular distribution.