Talbot Effect

The Talbot effect is a near-field diffraction phenomenon in which a periodic object illuminated by a coherent plane wave produces perfect self-images at regular distances downstream, without any lens.

Self-Imaging Condition

A periodic object with period $L$ produces exact self-images at the Talbot distances:

$$ z_m = \frac{2mL^2}{\lambda}, \quad m = 1, 2, 3, \ldots $$

The fundamental Talbot distance is:

$$ z_T = \frac{2L^2}{\lambda} $$

Mechanism

Each Fourier harmonic $e^{j2\pi nx/L}$ of the periodic object acquires a propagation phase $e^{-j\pi\lambda z n^2/L^2}$ under Fresnel propagation. At the Talbot distance, $\lambda z_T/L^2 = 2$, so the phase of the $n$th harmonic is $e^{-j2\pi n^2} = 1$ for all integers $n$. All harmonics return to their original phase relationship, producing an exact replica of the input field.

Fractional Talbot Images

At fractions of the Talbot distance, interesting intermediate patterns appear:

Distance	Pattern
$z = z_T$	Exact self-image
$z = z_T/2$	Self-image shifted by half a period ($L/2$)
$z = z_T/4$	Doubled frequency pattern
$z = z_T/p$ ($p$ integer)	Superposition of $p$ shifted copies

Applications

• Lithography: Self-imaging allows mask patterns to be replicated without a lens
• Interferometry: Talbot interferometers use the self-imaging property for wavefront sensing
• X-ray imaging: Talbot-Lau interferometry enables phase-contrast X-ray imaging