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- Open Access

# Propagation algorithms for Wigner functions

- Minyi Zhong
^{1}Email author and - Herbert Gross
^{1, 2}

**12**:7

https://doi.org/10.1186/s41476-016-0008-6

© The Author(s) 2016

**Received:**3 April 2016**Accepted:**25 May 2016**Published:**28 June 2016

## Abstract

We propose an algorithm to remove a parabolic wavefront from a convergent or divergent beam in the Wigner function. Using this approach we numerically collimate the beam. This avoids a dense sampling in phase space to describe a convergent wavefront. Thereby we reduce the required computer memory, but maintain computational accuracy and physical effects. Furthermore, we compare two algorithms, shearing and Radon transform, to propagate the Wigner function in free space. We use the fast Fourier transform to accurately perform shearing. However, zero-padding is necessary to circumvent aliasing. We prove that the Radon transform is a more efficient approach for a long propagated distance.

## Keywords

- Phase space
- Parabolic wavefront
- Sampling
- Shearing
- Radon transform

## Background

The Wigner function is a helpful tool to analyze optical signals in phase space [1, 2]. It includes information about ray optics and wave optics [3]. In particular for partially coherent light, the Wigner function visualizes the coherence effects in a straightforward manner [4]. However, the computation of the Wigner function has certain difficulties. We require a two-dimensional Wigner function to describe light field with one transverse dimension. For a field with two transverse dimensions, the Wigner function spans four dimensions. Consequently, the question of how to efficiently implement the Wigner function becomes an important issue in practice. The main goal of this work is to propagate light by using Wigner functions, while saving computer memory and keeping computational accuracy.

In optical systems, we often encounter strongly convergent or divergent beams. To represent such a beam in phase space requires a dense sampling grid. At first we introduce a method based on numerical collimation, to represent a convergent or divergent beam in phase space with as few sampling points as possible. This preserves all the diffraction effects. The corresponding details are introduced in Removing a parabolic wavefront in phase space.

The Wigner function can be propagated by using the ABCD matrix formalism. The propagation in free space corresponds to a shearing of signals in phase space. However a straightforward implementation of this can severely burden the computer memory. In Shearing and Radon transform we describe another algorithm based on the Radon transform with a better computational efficiency.

In this paper we assume that all the light is within a small numerical aperture (NA). The propagation operators are paraxial. The incident light fields are partially coherent in space with one transverse dimension in this paper. All the algorithms can be extended to fields with two transverse dimensions.

## Method 1: Removing a parabolic wavefront in phase space

_{1},

*x*

_{2}) representing the correlation function between every two arbitrary transverse positions given by x

_{1}and

*x*

_{2}, x = (x

_{1}+

*x*

_{2})/2, Δx = x

_{1}-

*x*

_{2}. As the angle u and the spatial distance Δx are Fourier conjugated by their definition in Eq. 1, the samplings on the spatial and angular axes follow the following relation,

_{max}denotes the maximum value on the angular axis and λ is the wavelength of the light field, the symbols d(∆x) and d(x) defining the spatial distance between two adjacent sampling points on the ∆x and x axes respectively, N

_{x}representing the total sampling points on the x axis. Here we use the term (N

_{x}-2) because we define one less sampling point on the non-negative x values than on the negative x values. Based on Eq. 2, the sampling on the angular axis u is automatically defined once the sampling on the spatial axis x is chosen.

After the propagation we need to scale the collimated beam width by a factor of R/(R + L), so that it is comparable to the original beam width.

Figure 1b depicts a simple system as an example. In the original system, the kinoform lens with a focal length of 16 mm (i.e., R = −16 mm) focuses a quasi-collimated input beam into a convergent beam. The kinoform lens has a diameter of 2 mm and a uniform groove height Δz = 3.5λ, where λ denotes the wavelength. We define the refractive index of the lens as n = 2.0 with λ = 0.6328 μm. The phase difference generated by the groove height is 2π(n-1)Δz/λ = 7π, leading to destructive interference. Thus the outgoing convergent beam carries additional diffraction effects. The beam propagates in free space with a distance L = 10 mm after leaving the kinoform lens. In the transformed system, we place a paraxial negative lens right behind the kinoform lens to collimate the convergent beam. According to Eq. 3, the collimated beam propagates in free space with a transformed distance L_{trans} = 26.7 mm. After this propagation we desire the same diffraction effects as in the original system.

_{source}being the correlation function of the incident light onto the kinoform lens, t

_{j}representing the phase modulation function of a surface (i.e., either the kinoform lens or the paraxial negative lens), Δz

_{j}referring to the height of the corresponding surface. For the paraxial negative lens we define Δz

_{par}= exp[−iπx

^{2}/(λf

_{par})], where f

_{par}is its focal length. The profile of the kinoform lens is shown in Fig. 2a.

## Results and Discussion of Method 1

Figure 2 compares the phase space of the original and the transformed systems given by Fig. 1b. In the original system, the focusing effect produced by the kinoform lens is expressed by a general tilt of all the signals in Fig. 2c. The additional diffraction effects are indicated by the oscillatory ripples in Fig. 2c. In the transformed system the paraxial negative lens introduces an extra defocusing effect to the convergent beam. It brings the signals back to the horizontal orientation in Fig. 2e. Thus the convergent beam is collimated. We propagate this collimated beam with the distance L_{trans} = 26.7 mm and scale the transverse beam width after the propagation. Figure 2g shows the nearly identical transverse intensities, with a Pearson correlation coefficient [6] of 0.9996, between the original and the transformed systems. It indicates that the diffraction effects caused by the kinoform lens are preserved in the transformed system.

In addition, the angular axis in Fig. 2e has half the range of the angular axis in Fig. 2c. An angular range of −0.04 ≤ u ≤ 0.04 (radian) is sufficient for the collimated beam, whereas a range of −0.08 ≤ u ≤ 0.08 (radian) is required for the convergent beam. Thus the grid in the Wigner function is decreased from 1024 × 1024 pixels to 512 × 512 pixels by using the transformed system. This method saves a factor of 4 of the computer memory.

_{kino}and t

_{par}includes a sum of the surface heights given by the kinoform lens and the paraxial negative lens, seen in Eq. 5.

The remaining height in Eq. 6 (red curve in Fig. 3a) is step jumps with a uniform height given by the kinoform lens. The paraxial negative lens eliminates the fast-oscillatory phase (Fig. 3c) generated by the focusing effect of the kinoform lens. In the end only the phase jumps produced by individual groove height of the kinoform lens are preserved in the correlation function, shown as quadrilaterals of constant phase in Fig. 3d. The edge of each quadrilateral indicates the groove position in the kinoform lens at corresponding location given by the x and Δx axes. If the step height equals to a multiple of the wavelength, the quadrilaterals will vanish. The absolute values of the correlation function do not change between the surfaces, because the surfaces only modify the phase of the incident beam.

Furthermore, our method offers an alternative to magnify the convergent beam near the focus region. As the collimated beam in the transformed system has a larger transverse width, we obtain a beam profile of more pixels filled with non-zero values. When we are interested in the exact focus of the convergent beam, we need to propagate the collimated beam to infinity (i.e., far field). A far-field propagation in phase space is performed by a 90° rotation, i.e., a Fourier transform.

It is worth noting that the Wigner function can describe non-paraxial light [7]. However, the numerical collimation method presented in this section has a limitation. Since we use the thin element approximation to process surfaces and apply ABCD matrices to propagate light in free space, our propagation operators are paraxial. Therefore, the algorithms are only valid for systems within a small NA. If the condition of a small NA is fulfilled, the convergent beam is allowed to contain a wavefront deviating from a parabola, e.g., a non-perfect focusing beam.

## Method 2: Shearing and Radon transform

_{o}= u

_{o}z, where x

_{o}marks the shifted distance and u

_{o}is the angular coordinate of a horizontal row. A shift in space corresponds to multiplying a phase factor in the Fourier domain. Thus the spatial shift of each row can be represented by two Fourier transforms,

The advantage of employing a Fourier transform in shearing is that it returns an accurate value at each shifted pixel. However, shearing with a Fourier transform also has a disadvantage. A large propagated distance z in free space leads to a large shifted distance of x_{o} in phase space. Some signals in phase space are sheared outside the original region. The discrete Fourier transform mirrors these signals back into the original region and causes aliasing. To avoid this error, one must make sure that the computational region in phase space is wide enough to support shearing. Otherwise zero-padding should be used to increase the computational region.

Figure 4d and e depict the schematic diagrams of the geometries for shearing and for the Radon transform. Assume there are two points A = (0, u_{o}) and B = (0, −u_{o}) in phase space at the propagated distance 0. For the propagated distance z, the shearing angle is defined as θ = arctan(z). Points A and B in phase space are sheared to coordinates of (u_{o}z, u_{o}) and (−u_{o}z, −u_{o}) respectively. The distance between these two points in the transverse intensity at the propagated distance z is denoted by Δx_{AB} = 2u_{o}z. The total width of the transverse intensity is Δx_{max} = 2x_{m} + u_{m}z, where x_{m} and u_{m} define the maximum values on the spatial and angular axes in phase space at the propagated distance 0.

_{m}/x

_{m}, so that the two axes in phase space have the same unit and remain uniform during rotation. The projected intensity after a rotation of angle α yields a maximum width of Δx

_{max}’ = 2x

_{m}cosα + 2(u

_{m}/ϒ)sinα. Inside the projected intensity, the distance between A and B is Δx

_{AB}’ = 2(u

_{o}/ϒ)sinα. In order to implement the Radon transform, we have to find out the relation between the rotation angle α and the propagated distance z. This is achieved by keeping the ratio between Δx

_{max}and Δx

_{AB}, and the ratio between Δx

_{max}’ and Δx

_{AB}’ equal.

From Eq. 9 we derive the rotation angle α = arctan(ϒz). The ratio factor for the transverse intensity width between shearing and the Radon transform is ϒz/sinα.

## Results and Discussion of Method 2

## Conclusions

Our methods contribute to a fast implementation to paraxially propagate beams in phase space, particularly for partially coherent light.

We show how to apply a method known for working with coherent light to problems of partial coherence. A parabolic wavefront is subtracted from a convergent beam. The beam is converted into quasi-collimated form. The required sampling density to represent this beam in phase space is thus reduced. The diffraction effects in propagation are preserved after the parabolic wavefront is removed. Furthermore, this approach offers an alternative to magnify the convergent beam near the focus region by observing the collimated beam at the physically equivalent distance.

Besides, we compare the efficiency of two algorithms, shearing and Radon transform, for propagating the Wigner function in free space. For a large propagated distance, the shearing method requires zero-padding to avoid aliasing. Its demand on the computer memory grows linearly with the propagated distance. In contrast to this, the Radon transform keeps the computer memory within a finite limit.

The advantages of our proposed methods are even greater with the operation of a four dimensional phase space, even though this is not specifically shown in this paper.

## Declarations

### Acknowledgements

This work is partially supported by the German Federal Ministry of Education and Research within the project fo + (03WKCK1D). We thank Martin Kielhorn for comments and discussions which improved this manuscript. We are grateful to the reviewer for the constructive input.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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