Propagation algorithms for Wigner functions

If a beam is strongly convergent or divergent, a large angular range on the angular axis is required to fully describe the beam of a large angle cone. Thus, according to Eq. 2, we are forced to have a dense sampling on the spatial axis x in order to fulfill the range requirement on the angular axis. Eventually this leads to aliasing (Fig. 1a). However, if the optical component does not need so many sampling points to describe its spatial structure, a dense sampling on the spatial axis x is a waste of computer memory. Therefore we convert the convergent or divergent beam into a quasi-collimated beam by removing a parabolic wavefront. For a quasi-collimated beam, the angular axis does not need a big range any more. Then the sampling density on the spatial axis x can be reduced.

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.

To apply this transformation in the Wigner function, we insert a paraxial negative lens directly into Eq. 1. This lens only removes a convergent curvature from the wavefront, without changing any other phase effects from the kinoform lens. It is expressed in an equation as follows.

$$ W\left(x,u\right)={\displaystyle \int }{\varGamma}_{source}\left(x+\frac{\Delta x}{2},x-\frac{\Delta x}{2}\right)\cdot {t}_{kino}\left(x+\frac{\Delta x}{2}\right)\cdot {t}_{kino}^{*}\left(x-\frac{\Delta x}{2}\right)\cdot {t}_{par}\left(x+\frac{\Delta x}{2}\right)\cdot {t}_{par}^{*}\left(x-\frac{\Delta x}{2}\right) \exp \left(-i\frac{2\pi }{\lambda }u\Delta x\right)d\Delta x $$

(4)

$$ {t}_j= \exp \left[-\frac{i2\pi }{\lambda}\left(n-1\right)\Delta {z}_j\right]\;\mathrm{where}\kern0.55em j = kino,\kern0.5em par $$

(5)

where W denotes the Wigner function of light leaving the paraxial negative lens, Γ _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[−iπx²/(λf_par)], where f_par is its focal length. The profile of the kinoform lens is shown in Fig. 2a.

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.

Figure 3 depicts more intermediate results to explain the phase effects introduced by the paraxial negative lens. Based on Eq. 4 and 5, the product between t_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.

$$ {t}_{kino}(x){t}_{par}(x)= \exp \left[-i\frac{2\pi }{\lambda}\left(n-1\right)\Delta {z}_{kino}(x)\right] \exp \left[-i\frac{2\pi }{\lambda}\left(n-1\right)\Delta {z}_{par}(x)\right]= \exp \left\{-i\frac{2\pi }{\lambda}\left(n-1\right)\left[\Delta {z}_{kino}(x)+\Delta {z}_{par}(x)\right]\right\} $$

(6)

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.

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.

As we are interested in light intensity along propagation, another method called Radon transform [1] reaches the same goal as shearing. A Radon transform projects an image along a radial line oriented at a specific angle to derive the intensity distribution. Figure 4a-c give a qualitative illustration. Figure 4a is the phase space at the propagated distance z generated by shearing the phase space at the propagated distance 0. The vertical lines in Fig. 4a indicate the integration direction of signals for the transverse intensity at the propagated distance z. Figure 4b results from an inverse shearing of Fig. 4a in the directions of the red arrows. This corresponds to a back propagation from the distance z to 0. Thus Fig. 4b returns the phase space at the propagated distance 0. The integration lines in Fig. 4a are sheared into a tilted orientation in Fig. 4b. They still imply the integration direction for the transverse intensity at the propagated distance z. Alternatively we may perform a rotation of the phase space of Fig. 4b into Fig. 4c, and integrate the signals along the integration lines vertically in Fig. 4c to derive the transverse intensity at the propagated distance z. This is what we call as a Radon transform.

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_oz, u_o) and (−u_oz, −u_o) respectively. The distance between these two points in the transverse intensity at the propagated distance z is denoted by Δx_AB = 2u_oz. The total width of the transverse intensity is Δx_max = 2x_m + u_mz, where x_m and u_m define the maximum values on the spatial and angular axes in phase space at the propagated distance 0.

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.

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.