The phase space of a monochromatic partially-coherent beam is given by the Wigner function,

$$ W\left(x,u\right)={\displaystyle \int}\varGamma \left(x+\frac{\Delta x}{2},x-\frac{\Delta x}{2}\right) \exp \left(-i\frac{2\pi }{\lambda }u\Delta x\right)d\Delta x $$

(1)

where x and u denote the spatial and angular variables in phase space, Γ(x_{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,

$$ {u}_{max}=\frac{1}{d\left(\Delta x\right)}\cdot \frac{\lambda }{2}=\frac{1}{2d(x)}\cdot \frac{\lambda }{2}=\frac{\left({N}_x-2\right)}{2\cdot 2{x}_{max}}\cdot \frac{\lambda }{2} $$

(2)

where u_{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.

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.

According to Siegman [5], a beam with a convergent wavefront with a radius of curvature R propagating for a distance of L generates the same diffraction effects as this beam without the convergent wavefront propagating for a transformed distance. The equation for the transformed distance is written as follows.

$$ {L}_{trans}=RL/\left(R+L\right) $$

(3)

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[−iπx^{2}/(λf_{par})], where f_{par} is its focal length. The profile of the kinoform lens is shown in Fig. 2a.