Gauss-Bessel Beam Shaper [Power Control]¶

Description¶

This twin implements a quantized radial phase mask that converts a collimated Gaussian beam into a Bessel-like beam with extended depth of focus. In a \(2f\) configuration, the mask placed at the front focal plane produces the exact Fourier transform of the shaped field. The ideal Bessel beam requires a linear radial phase \(\phi_{\text{ideal}}(\rho) = k_\rho \rho\); here the phase is quantized into \(Q\) levels. Lower \(Q\) (e.g., binary) yields multiple diffraction orders, while higher \(Q\) approaches a clean Bessel ring with a \(J_0^2\) profile.

A key feature is the phase scaling factor \(\alpha\) (modulation depth):

• \(\alpha = 0\): pure Gaussian (all energy in central spot).
• \(0 < \alpha < 1\): energy split between spot and rings.
• \(\alpha = 1\): pure Bessel generator (energy ideally in rings).

Thus \(\alpha\) provides simple control over the energy distribution. The output polarization is unchanged.

Model Parameters¶

Design parameters:

• Ring separation ratio (\(M\)): Controls the spacing between rings relative to the diffraction-limited focal spot size (default: 2).

  • \(M = 1\): First ring just touches central spot (minimum resolution)
  • \(M = 2\): Clear separation between rings (recommended default)
  • \(M = 3\): Multiple rings well resolved
  • Warning issued if \(M < 1\) (rings not resolved)

• Phase scaling factor (\(\alpha\)): Modulation depth (default: 1.0). Scales the applied phase:

  • \(\alpha = 0\): No modulation – pure Gaussian output.
  • \(0 < \alpha < 1\): Partial modulation – energy splits between central spot and rings.
  • \(\alpha = 1\): Full modulation – all energy ideally goes into rings.

• Quantization levels (\(Q\)): Number of discrete phase levels (2, 4, 8, 16, 0 for continuous phase). Default is 0 (continuous phase). The beam structure depends strongly on \(Q\):

  • \(Q = 0\) (continuous phase): Ideal Bessel phase – a single ring with \(J_0^2\) profile (for \(\alpha = 1\)).
  • \(Q = 8\): Close approximation to the ideal; most energy in the first ring, faint higher orders.
  • \(Q = 4\): Moderate approximation; the first ring is dominant but may have slightly broadened shape.
  • \(Q = 2\): Binary phase mask – produces multiple rings (odd orders) and each ring may exhibit a double-peak structure due to the abrupt phase jumps and the nature of the Hankel transform. No central spot appears for \(\alpha = 1\) because the average transmission is zero.

• Sampling accuracy (\(S\)): Multiplicative factor that increases the number of samples per \(2\pi\) phase period beyond the base value (default: 1.0). The base number of samples per \(2\pi\) is automatically set to \(\max(16, Q)\) to ensure at least one sample per quantization level. The actual number of samples per period is then \(S \times \max(16, Q)\). Increase \(S\) if you observe numerical artifacts or if the ring structure appears distorted.

• Export Designed Phase: When enabled, system simulation pauses at the shaper plane and an export dialogue opens and allows for the export of the designed phase.

Simulation Model¶

The shaper applies a quantized radial phase mask, scaled by \(\alpha\). Within the paraxial Fourier optics framework, the field in the focal plane is the Fourier transform of the shaped field.

Ideal Radial Phase¶

The ideal continuous phase for a Bessel beam is linear in the radial coordinate:

where \(\rho = \sqrt{x^2+y^2}\) is the radial coordinate in the shaper plane. In practice, the phase is taken modulo \(2\pi\) because only the value modulo \(2\pi\) affects the field.

From Fourier optics, the transverse wavenumber \(k_{\rho}\) determines the first ring radius in the focal plane:

The focal spot size for a collimated Gaussian beam with radius \(w_0\) is:

The ring separation ratio \(M = r_0 / \Delta r_{\text{focus}}\) yields the simple design equation:

Quantized Radial Phase with Scaling¶

For a given quantization level \(Q > 0\), the ideal phase modulo \(2\pi\) is first quantized to the nearest of \(Q\) equally spaced levels, then scaled by \(\alpha\). The scaled quantized phase is:

where \(\operatorname{round}\) rounds to the nearest integer. For the continuous case (\(Q = 0\)), we simply take \(\Phi_{\text{Bessel}}(\rho) = \alpha \cdot (k_\rho \rho \bmod 2\pi)\).

The complex transmission of the mask is \(t(\rho) = \exp(i \Phi_{\text{Bessel}}(\rho))\).

Beam Characteristics for Different \(\alpha\) and \(Q\)¶

The Fourier transform (Hankel transform) of the scaled phase mask yields an output that depends on both \(\alpha\) and \(Q\):

• For \(\alpha = 0\), the mask has no phase modulation, so the output is simply the focused Gaussian beam (central spot only).
• For \(0 < \alpha < 1\), the output contains both a central Gaussian component and ring structures. The central spot intensity decreases as \(\alpha\) increases, while the ring energy grows.
• For \(\alpha = 1\) and \(Q = 0\) (continuous), the output is a single Bessel ring with intensity proportional to \(J_0^2(k_\rho r)\).
• For \(\alpha = 1\) and \(Q = 2\) (binary), the mask acts as a radial square wave, producing multiple odd orders; each order may exhibit double-peak structures.
• For \(\alpha = 1\) and \(Q \ge 4\), the output progressively approaches the ideal single ring, with higher \(Q\) giving cleaner results.

Key Physical Principles¶

• Fourier optics foundation: With collimated input and shaper at distance \(f\) before the lens, the focal plane field is the exact Fourier transform with no quadratic phase distortion---essential for extended depth of focus.
• Parameter \(\alpha\): Controls the energy split between central spot and rings; a simple and intuitive power control.
• Parameter \(Q\): Determines how faithfully the output approximates an ideal Bessel beam. Higher \(Q\) reduces artifacts (multiple orders, double peaks) and concentrates energy into the first ring.

Typical Application Scenarios¶

1. Tunable laser processing: Adjust \(\alpha\) to control how much energy remains in the central spot vs. rings for applications requiring variable spot-to-ring energy ratios in drilling or cutting.

2. Optical trapping with variable confinement: Use \(\alpha\) to tune between tight central trapping (\(\alpha\) near 0) and extended ring trapping (\(\alpha\) near 1) for multi-particle manipulation.

3. Microscopy with adjustable background: Control ring brightness to optimize the balance between central resolution and background illumination in imaging systems.

4. Material processing parameter optimization: Systematically explore \(\alpha\) values to find the optimal energy distribution for specific material interactions.

5. Beam shaping research: Study how energy distribution between Gaussian and Bessel components affects propagation and self-reconstruction properties.

6. Educational demonstrations: Visually demonstrate the transition from pure Gaussian to pure Bessel-Gauss beams through continuous \(\alpha\) variation.

Software Usage¶

This twin is available in the Digital Twin Hub. To achieve the optimal configuration for extended depth of focus, follow these steps:

System Setup¶

1. Generate a Gaussian beam: Place a Gaussian Beam Mode twin (SF-GAUS01) in your system.

2. Collimate the beam (if needed): If your source is not already collimated, add a collimation element:

   • Use an Ideal Lens [Collimation] twin (CF-ILCO01) for perfect aberration-free collimation, OR
   • Use a Spherical Lens twin (CS-SLEN01) with appropriate curvature to achieve collimation.

   The beam at the shaper plane must have infinite wavefront curvature (\(R_{\text{in}} \to \infty\)).

3. Add the beam shaper: Place the Gauss-Bessel Beam Shaper [Power Control] twin (CF-BESP01) at a distance exactly equal to the focal length \(f\) before the Fourier lens.

4. Configure the shaper: Set the design parameters (\(M\), \(\alpha\), \(Q\), Sampling accuracy \(S\)).

5. Add the Fourier lens: Place a lens with focal length \(f\) in a 2-f configuration:

   • Use an Ideal Lens [2f-Setup] twin (CF-ILSU02) for perfect aberration-free Fourier transformation, OR
   • Use a Thin Lens twin (CF-THLE01) with focal length \(f\) placed at a distance \(f\) from the element plane.

6. Observe the result: Place field monitors (DF-FMON01) in the focal plane and beyond to observe the extended depth of focus of the ring-shaped Bessel-like beam.

Exporting the designed phase for fabrication:¶

• Check the Export Designed Phase option in the shaper's dialogue.
• When enabled, system simulation pauses at the shaper plane and an export dialogue opens.
• The dialogue displays the current pixel size \(\Delta x = \pi w_0/(M N)\) where \(N = S \cdot \max(16, Q)\). The user can specify the desired number of sampling points (i.e., the grid size) for the exported mask. The minimum allowed number of points corresponds to the current \(N\) (i.e., the sampling that would be used internally if the simulation continued). Larger grid sizes increase resolution but also file size.
• Users can adjust the number of points based on fabrication constraints; the software automatically resamples the phase data accordingly. However, reducing the number of points below the minimum would degrade the mask and is not permitted.
• After closing the export dialogue, the simulation continues normally.