Hankel Transform

The Hankel transform (also called Fourier-Bessel transform) is the natural transform for circularly symmetric functions in two dimensions.

Definition

The zero-order Hankel transform of a radially symmetric function $u(r)$ is:

$$ \tilde{U}(\rho) = 2\pi \int_0^{\infty} u(r) J_0(2\pi\rho r) \, r \, dr $$

where $J_0$ is the zero-order Bessel function.

Relationship to 2D Fourier Transform

For a circularly symmetric function $u(r)$ where $r = \sqrt{x^2 + y^2}$:

• The 2D Fourier transform is also circularly symmetric
• The Hankel transform gives the radial dependence directly

Key Transform Pair

$$ \text{circ}(r) \xleftrightarrow{\mathcal{H}} \frac{J_1(2\pi\rho)}{\rho} $$

This is the Airy disk pattern.

Properties

• Self-inverse (like the Fourier transform for even functions)
• Reduces 2D problems with circular symmetry to 1D
• Used extensively in optics for circular apertures and lenses

Related Topics

• Bessel function
• Airy disk
• Circular symmetry