Talbot Near-Field Revivals

Theory of Periodic Self-Imaging

Consider an amplitude grating with period d and wavelength λ. We can express the transmitted field using a Fourier series. For our finite grating model, we consider five equal modes (m ∈ {-2, -1, 0, 1, 2}), each with a coefficient of 1/5. The initial field at z = 0 is:

E(x, 0) = ∑_m (1/5) · exp(i · 2π · m · x / d)

As the light propagates in the paraxial approximation along the z direction, each mode picks up a phase shift proportional to the square of its transverse spatial frequency m/d. The field at distance z is given by:

E(x, z) = ∑_m (1/5) · exp(i · 2π · m · x / d) · exp(-i · π · λ · z · m² / d²)

Notice the quadratic phase term exp(-i · π · λ · z · m² / d²). Because m is an integer, when z reaches a specific distance z_R where all these phase factors are integer multiples of 2π, the entire field is reconstructed identically to the z = 0 plane.

In our convention, we define the full unshifted revival length z_R as:

z_R = 2 d² / λ

At z = z_R (or integer multiples of z_R), the grating image recurs exactly. At half the revival length, z = z_R / 2, the image also recurs but is shifted laterally by half a period, d/2. Some literature refers to d² / λ as the Talbot distance, denoting this half-period shifted image.

Worked Example

Let's calculate the expected field and intensity for specific inputs.

1. Inputs: Period d = 20 μm, Wavelength λ = 500 nm (0.5 μm), Propagation z/z_R = 0.25, Probe x/d = 0.
2. Unshifted Revival Length:
   z_R = 2 · d² / λ = 2 · (20 μm)² / 0.5 μm = 1600 μm = 1.6 mm.
3. Propagation Distance:
   z = 0.25 · z_R = 0.4 mm = 400 μm.
4. Field Calculation:
   At x = 0, the spatial phase exp(i · 2π · m · x / d) is 1 for all m.
   The propagation phase is -i · π · λ · z · m² / d².
   Since z = (0.25) · (2 d² / λ) = 0.5 d² / λ, the phase becomes:
   -i · π · λ · (0.5 d² / λ) · m² / d² = -i · π · 0.5 · m² = -i · π · m² / 2.
   Summing over m ∈ {-2, -1, 0, 1, 2}:
   • m = 0: exp(0) = 1
   • m = ±1: exp(-i · π/2) = -i
   • m = ±2: exp(-i · 2π) = 1
   E = (1/5) · [ 1 + 2(-i) + 2(1) ] = (3 - 2i) / 5 = 0.6 - 0.4i.
5. Probe Intensity:
   Intensity = |E|² = (0.6)² + (-0.4)² = 0.36 + 0.16 = 0.52.
6. Period-Averaged Intensity:
   Sum of |c_m|² = 5 · (1/5)² = 5/25 = 0.2. This is conserved for all z.

Unshifted revival length mm: 1.6 Propagation distance mm: 0.4 Field real: 0.6 Field imaginary: -0.4 Probe intensity: 0.52 Period averaged intensity: 0.2

Experiment

Use the interactive module to visualize the intensity carpet and the reconstructed field profiles. The interactive diagram dynamically calculates the Talbot effect based on your selected parameters.

1. Change the Period micrometers to observe its quadratic effect on the revival length z_R.
2. Vary the Propagation z over zR to see how the intensity profile evolves. Note the exact reconstruction at z/z_R = 1 and the half-period shifted reconstruction at z/z_R = 0.5.
3. Adjust the Probe x over period to inspect the complex field and intensity at a specific lateral position.

Limitations

This module uses a highly simplified, explicitly finite harmonic grating model containing only five Fourier modes (m ∈ {-2, -1, 0, 1, 2}). It does not simulate an infinite binary-slit grating, which would require an infinite sum of modes. Furthermore, the model uses the paraxial approximation, assuming the wavelength λ is small compared to the period d. Therefore, large diffraction angles and non-paraxial effects are not captured in this visualization.

References

For further reading and a deeper dive into the mathematical formulations of integer, fractional, and fractal Talbot effects, please consult the primary reference below.

• Berry, M. V., & Klein, S. (1996). Integer, fractional and fractal Talbot effects. Journal of Modern Optics. Available at: https://cris.technion.ac.il/en/publications/integer-fractional-and-fractal-talbot-effects/