# Multibeam interference simulation using Zemax OpticStudio

The article presents a simulation of multibeam interference in the Zemax OpticStudio environment in Non-Sequential Mode. Potential applications, theory, and implementation of the optical system in OpticStudio and simulation results are discussed. The presented numerical models are highly flexible, allowing the user to easily introduce changes in terms of their application and perform qualitative and quantitative analyses.

Authored By Maciej Traczyk, Zemax Consultant Partner
Company: Systemy Optoelektroniczne Maciej Traczyk
website: www.optoelektronika.com.pl
email: kontakt@optoelektronika.com.pl

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## Introduction

Multi-beam interference can be used to modify coatings of various materials, such as ceramics, metals, polymers, and others. The interference patterns created on the surface allow the surface to have properties that it did not have before. Examples of applications could be:

• Production of hydrophobic coatings on metal sheets, glasses, and bathroom fittings
• Anti-counterfeit security area
• Improvement of the biocompatibility of surgical implants
• Increased abrasion resistance
• Scaffolds for cell and tissue culture in bioengineering

## Theory

A single laser beam can be described by the equation:

$$\overrightarrow{E}=\overrightarrow{E_0} \exp[-i(\overrightarrow{k}\overrightarrow{r}-wt)] \tag{1}$$

Where:

• $$\overrightarrow{E}$$ is the amplitude of the electric field intensity of the beam
• $$\overrightarrow{k} = \frac{2π}{λ}$$ is the wave vector
• ω is the circular wave frequency
• λ is the wavelength

The interference image of multiple laser beams is obtained by applying the superposition principle of each individual field $$\overrightarrow{E_j}$$ derived from each beam:

$$\overrightarrow{E}= \sum_{j=1}^{n} \overrightarrow{E_j} = \sum_{j=1}^{n} \overrightarrow{E_{0_j}} \exp[-i(\overrightarrow{k}\overrightarrow{r}-wt)] \tag{2}$$

The power density for n beams is proportional to:

$$I_n \approx \langle |\overrightarrow{E_n}|^2 \rangle \tag{3}$$

In the simplest case - interference of two laser beams - it is assumed that:

• the beams have the intensity $$I_1$$ and $$I_2$$
• they propagate in one plane
• they fall at the same angle θ to the normal to the surface
• the polarization effect is not considered

Then the intensity of the resultant field can be described by:

$$I(x) = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos(2 \cdot k \cdot x \sin \theta ) \tag{4}$$

And the spatial period of the fringe field is:

$$d= \frac{\lambda}{2 \cdot \sin \theta} \tag{5}$$

## Interference of 2 laser beams - comparison with theory

The following picture is a simulation of the interference of 2 laser beams incident at an angle of θ = 2.5 degrees to the surface normal for λ = 1064 nm.      Figure 1 - Laser Beam Interference

The cross-section shows that the spatial period of the resulting structure is about 12 µm.
The spatial period defined by equation (5) is:

$$d= \frac{\lambda}{2 \cdot \sin \theta} = \frac{1064 nm}{2 \cdot \sin (2.5)} =12.19 µm$$

## Simulations

In all simulations for the presented methods, the following assumptions were made:

• Non-Sequential Mode
• wavelength λ = 1064 nm
• source power P = 1 W
• the laser beam has a Gaussian profile
• diameter of the laser beam d = 10 mm
• number of rays for analysis is 1e9
• the result of the interference is observed on the rectangle type detector in the (coherent intensity) mode
• z = 1 - height of the roof prism / pyramid
• r = 12 mm - radius of the circle describing the base of the prism / pyramid
• glass from which the N-BK7 prism is made

## Optical System

The optical system consists of:

1.  light source with Gaussian profile
2.  beam splitter (roof prism or pyramid) - defined with Polygon Object in .POB file
3.  detector no.1 (large) and detector no.2 (small) Figure 2 - Optical Path

## Macro - construction of a roof prism and pyramids

A macro has been written to automate the construction of beam splitting elements. It enables the creation of roof prisms and pyramids. The user enters the number of faces (n = 2 for a prism, n> 2 for a pyramid), the height of the prism/pyramid and the radius of the circle on which the base of the prism/pyramid is described. Figure 3 - Fragment of the ZPL macro

The result of the macro operation is a file with the .POB extension, containing the positions of vertices and the definition of joining them into a solid.

An example file containing the definition of points and the created pyramid is shown in Figure 4.

 n=2, h=1 mm, r=12 mm V 1.0000 -8.4853 -8.4853 0.0000V 2.0000 8.4853 -8.4853 0.0000V 3.0000 8.4853 8.4853 0.0000V 4.0000 -8.4853 8.4853 0.0000V 5.0000 -8.4853 0.0000 1.0000V 6.0000 8.4853 0.0000 1.0000R 1.0000 2.0000 3.0000 4.0000 0.0000 1.0000R 4.0000 3.0000 6.0000 5.0000 0.0000 0.0000R 1.0000 2.0000 6.0000 5.0000 0.0000 0.0000T 1.0000 4.0000 5.0000 0.0000 0.0000T 3.0000 2.0000 6.0000 0.0000 0.0000 n=3, h=1 mm, r=12 mm V 1.0000 12.0000 0.0000 0.0000V 2.0000 -6.0000 10.3923 0.0000V 3.0000 -6.0000 -10.3923 0.0000V 4.0000 0.0000 0.0000 1.0000V 5.0000 0.0000 0.0000 0.0000T 1.0000 2.0000 4.00000.00000.0000T 2.0000 3.0000 4.00000.00000.0000T 3.0000 1.0000 4.00000.00000.0000T 1.0000 2.0000 5.0000T 2.0000 3.0000 5.0000T 3.0000 1.0000 5.0000 n=4, h=1 mm, r=12 mm V 1.0000 12.0000 0.0000 0.0000V 2.0000 0.0000 12.0000 0.0000V 3.0000 -12.0000 0.0000 0.0000V 4.0000 -0.0000 -12.0000 0.0000V 5.0000 0.0000 0.0000 1.0000V 6.0000 0.0000 0.0000 0.0000T 1.0000 2.0000 5.00000.00000.0000T 2.0000 3.0000 5.00000.00000.0000T 3.0000 4.0000 5.00000.00000.0000T 4.0000 1.0000 5.00000.00000.0000T 1.0000 2.0000 6.0000T 2.0000 3.0000 6.0000T 3.0000 4.0000 6.0000T 4.0000 1.0000 6.0000 n=6, h=1 mm, r=12 mm V 1.0000 12.0000 0.0000 0.0000V 2.0000 6.0000 10.3923 0.0000V 3.0000 -6.0000 10.3923 0.0000V 4.0000 -12.0000 0.0000 0.0000V 5.0000 -6.0000 -10.3923 0.0000V 6.0000 6.0000 -10.3923 0.0000V 7.0000 0.0000 0.0000 1.0000V 8.0000 0.0000 0.0000 0.0000T 1.0000 2.0000 7.00000.00000.0000T 2.0000 3.0000 7.00000.00000.0000T 3.0000 4.0000 7.00000.00000.0000T 4.0000 5.0000 7.00000.00000.0000T 5.0000 6.0000 7.00000.00000.0000T 6.0000 1.0000 7.00000.00000.0000T 1.0000 2.0000 8.0000T 2.0000 3.0000 8.0000T 3.0000 4.0000 8.0000T 4.0000 5.0000 8.0000T 5.0000 6.0000 8.0000T 6.0000 1.0000 8.0000 Figure 4 - Different geometries of the beam splitter element and content of .POB files obtained with macro

## Method 1 - detector rotation

A roof prism was used as the beam splitting element (n = 2).

The beam, after passing through the optical system, creates an interference pattern in the detector plane. Then the detector is rotated by a given angle and the radiation propagates again (without clearing the detector). The sample can be rotated by any angle and any number of propagations can be performed. Figure 5 - Detector Viewer settings for the Detector Rectangle Figure 6 - Editor Rotation of the Detector Alpha angle = 90 degrees
 Table 1 - Sample simulation results Propagation=1, Alpha=0 degree  Propagation=2, Alpha=60 degrees    Table 2 - Sample simulation results Propagation=1, Alpha=0 degree  Propagation=2, Alfa=90 degrees  ## Method 2 - pyramid

In this method, the beam splitting element is a pyramid. The number of walls and the geometric dimensions of the pyramid can be defined by the user using a macro.
Examples of results are presented in Table 3:

 Table 3 - Sample simulation results n=3 n=4 n=5 n=6 It is also possible to simulate the imperfections of the beam splitter by editing the .POB file.

By changing the position of the vertices, you can deliberately introduce a defect – see Table 4.

 Table 4 - Sample simulation results n=4 - ideal V 1.0000 12.0000 0.0000 0.0000V 2.0000 0.0000 12.0000 0.0000V 3.0000 -12.0000 0.0000 0.0000V 4.0000 -0.0000 -12.0000 0.0000V 5.0000 0.0000 0.0000 1.0000V 6.0000 0.0000 0.0000 0.0000T 1.0000 2.0000 5.00000.00000.0000T 2.0000 3.0000 5.00000.00000.0000T 3.0000 4.0000 5.00000.00000.0000T 4.0000 1.0000 5.00000.00000.0000T 1.0000 2.0000 6.0000T 2.0000 3.0000 6.0000T 3.0000 4.0000 6.0000T 4.0000 1.0000 6.0000 n=4 - disturbed V 1.0000 12.0000 0.0000 0.0000V 2.0000 0.0000 12.0000 0.0000V 3.0000 -12.0000 0.0000 0.0000V 4.0000 -0.0000 -12.0000 0.0000V 5.0000 0.5000 0.5000 1.0000V 6.0000 0.0000 0.0000 0.0000T 1.0000 2.0000 5.00000.00000.0000T 2.0000 3.0000 5.00000.00000.0000T 3.0000 4.0000 5.00000.00000.0000T 4.0000 1.0000 5.00000.00000.0000T 1.0000 2.0000 6.0000T 2.0000 3.0000 6.0000T 3.0000 4.0000 6.0000T 4.0000 1.0000 6.0000 ## Conclusion

This article presents simulations of multibeam interference in OpticStudio in non-sequential mode. The presented solution is very flexible and can be easily adapted to the user's needs. Thanks to ZPL Macro, it is possible to create various geometries of pyramids and roof prisms. By editing files in which splitting elements have been defined, it is also possible to simulate imperfections created during production.

## References

An article about a potential biomedical application:
https://www.sciencedirect.com/science/article/abs/pii/S0925963516300139