Phase Mask [Stitching-Error]¶

Description¶

The fabrication of a linear grating with period \(d\) can be affected by positional errors when writing successive grating rows. This results in stripes of height \(h\) with slight horizontal displacements, described by a stitching error parameter \(s\), where \(-s_{\rm max} \le s \le +s_{\rm max}\). The Stitching-Error Phase Mask is positioned directly behind a nominal grating component (without stitching error) to model this effect, as illustrated in Figure 1.

Simulation Model¶

The simulation model is based on the detour phase principle [1,2]. A horizontal displacement \(s\) of a grating stripe results in a constant phase shift in its response. This detour phase \(\phi_{\rm s}\) is given by

The incident field from the grating is decomposed according to the stripe structure of height \(h\). Each subfield is then assigned a random stitching error \(s\) (drawn uniformly from the range \(-s_{\rm max} \le s \le +s_{\rm max}\)), and the corresponding detour phase \(\phi_{\rm s}\) is applied, thereby simulating the stitching error.

Figure 1: Illustration of parameters for simulating stitching errors in grating fabrication.

In VirtualLab Fusion, the field decomposition uses smooth edges of width \(\Delta w_{\rm e}\) between subfields. This ensures proper numerical sampling and allows control over the physical sharpness of the edges to accurately model diffraction effects.

Model Parameters¶

• Period: \(d\) of the linear grating.
• Height: \(h\) of the grating rows (without stitching error).
• Maximum Stitching Error: \(s_{\rm max}\), expressed in meters ([\(s_{\rm max}\)] = m).
• Beam Diameter: \(D\) of the incident beam; this information is used for proper field decomposition.
• Seed: \(a \in \mathbb{Z}\) determines the random number generation for the stitching error distribution.
  • \(a > 0\): A single, initially generated random distribution is reused for all simulations, ensuring reproducibility.
  • \(a < 0\): A new random distribution is generated for each simulation run.
• Edge Width: \(\Delta w_{\rm e}\), specified as a percentage of \(h\) (default: \(\Delta w_{\rm e} = 5\%\)).

Channel Information¶

The digital twin is function-based and defined on a single plane. Light can strike the plane from either side and be either transmitted or reflected. Consequently, four channels must be specified:

• Channel \(+/+\) – In this channel, the intended functionality is provided.
• Channel \(+/-\) – Not activated.
• Channel \(-/-\) – This channel can be selected, but it does not alter the incident light.
• Channel \(-/+\) – Not activated.

Recommended Usage¶

The stripe pattern with height \(h\) is aligned along the component's x-axis. By rotating the component relative to the grating, the stripes can be aligned orthogonally to the grating grooves.

The beam diameter can be initially determined by the Beam Size Detector. Select a larger beam size if you are unsure about the actual beam size.

The edge width \(\Delta w_{\rm e}\) is given as a percentage of the height \(h\). Reducing \(\Delta w_{\rm e}\) allows for more accurate modeling of diffraction effects in the y-direction but requires a higher spatial sampling rate, which increases computational cost. For faster simulations, the edge width can be increased. Adjust this parameter carefully to balance accuracy and performance.

References¶

[1] A. Lohmann and B. P. Paris, "Binary Fraunhofer holograms generated by computer," Appl. Opt., vol. 6, no. 10, pp. 1739–1748, 1967.

[2] J. Bucklew and N. C. Gallagher Jr., "Comprehensive error models and a comparative study of some detour-phase holograms," Appl. Opt., vol. 18, no. 16, pp. 2861–2869, 1979.