Wigner distribution function of Lorentz–Gauss beams: a note

Abstract

Stimulated by a recent investigation of the Wigner distribution function of a Lorentz–Gauss beam, we present closed-form expression for such a function at the initial plane, which is alternative to that deduced in the aforementioned investigation. Such an expression can be usefully exploited to fully account for the Wigner-plane dynamics of Lorentz–Gauss beams as far as the paraxial propagation is concerned.

1 Introduction

A definite interest for the Lorentz and Lorentz–Gauss beams clearly emerges from the literature. In fact, after the primary explicit introduction of the Lorentz beams [1] (see also [2, 3]) on the basis of the experimental observations reported in [4, 5], mainly regarding the radiation emitted by single-mode diode lasers [6], various properties of such beams as well as of their Gaussian-modulated version, have been widely investigated (see, for instance, [7–10]). Very recently, the Wigner distribution function (WDF) of Lorentz–Gauss beams has been considered [11]. In the quoted paper, in fact, resorting to the approximation of a Lorentzian function by a finite sum of (even-order) Hermite–Gaussian functions, as originally proposed in [12], the analytical expressions for the WDFs of Lorentz and Lorentz–Gauss beams have been worked out. Such expressions comprise, in accord with the WDF non-linearity, the auto-WDFs of each single-function and the cross-WDFs for each couple of functions in the relevant sums.

Let us recall that the WDF is a basic tool for the hybrid, i.e., space/spatial-frequency, representation of optical signals [14–20]. As is well known, the “quantum-like” pair of Fourier conjugate space/spatial-frequency variables span the wave-optical phase-plane. In it, following the replacement of light rays by optical wavefunctions as descriptors of optical disturbances, light signals with the relevant “dynamics” of the propagation are described by suitable phase–space distribution functions, among which the WDF is definitely the most important one. It represents the wave-optical tool closest to the geometric-optical concept of light ray, due to its localization properties and dynamical behavior, which under paraxial propagation through real optical systems is ruled by the same transfer law of ray optics. The WDF retains in fact the information about the optical signal as conveyed by the relative wave-optical description whilst obeying the simple rules of evolution according to the corresponding ray-optical approach, thus accommodating for both the formal and conceptual simplicity of geometrical optics and the completeness of wave optics.

It is our intent here to show that a closed-form (and hence more compact) expression for the WDF of Lorentz–Gauss beams can be deduced, which may accordingly parallel that presented in [11]. This is done in Sect. 2, where also, through appropriate limits of the obtained expression, the WDFs for a purely Lorentzian and a purely Gaussian beam are deduced. The general expression is so seen to retain the features of both the limit expressions. Then, in Sect. 3 we resort to the quoted transfer law obeyed by the WDF under paraxial propagation through real optical systems. As is well known, it strictly relates to the evolution of the “classical-like” pair of optical variables, namely the light-ray coordinates of geometrical optics, which evolution is in turn ruled by the system ray-matrix. Thus, the WDF retains its formal expression under propagation, the arguments being basically transformed as the ray-variables. For exemplificative purposes, we show the Wigner charts of the WDF for the Lorentz–Gauss beam at the initial plane and at a subsequent plane under free-propagation, thus explicitly picturing the expected shear along the space-coordinate axis. In general, the expression of the WDF at the initial plane retains its crucial role in determining the evolution of the WDF under free propagation through a generic, and hence not necessarily real, optical system, although through a more involved evolution law. This is also discussed in Sect. 3. Therefore, as far as the paraxial propagation is concerned, the expression of the WDF of the Lorentz–Gauss beam at the initial plane is all one needs to fully account for the Wigner-plane dynamics of such a beam.

Concluding notes are given in Sect. 4, where the WDF of a super Lorentz–Gauss beam (in particular, composed by two Lorentzians) is considered, the relevant explicit closed-form expression being accordingly deduced.

2 The Wigner distribution function of a Lorentz–Gauss beam: a closed form expression

The Lorentz–Gaussian beams can be represented by the wavefunction

$$\varphi _{_{\mathrm{LG}}}(\xi,0)=\frac{\varphi _0}{1+\frac{\xi ^2}{w_{_{ \mathrm{L}}}^2}}{\text{e}}^{-\frac{\xi ^2}{2w_{_{\mathrm{G}}}^2}},$$

(1)

at the initial plane conventionally taken at z = 0, ξ addressing a Cartesian coordinate in that plane as well as in any (transverse) plane along the z-direction of propagation. Here, $\varphi _0$ is an arbitrary (in general, complex) constant. It may be finalized, for instance, through a normalization condition; for the time being, we let it undetermined. Also, the parameters $w_{\mathrm{L}}$ and $w_{\mathrm{G}}$ relate to the widths of the Lorentzian and Gaussian parts of the wavefunction; in fact, the two limits of a purely Lorentzian or a purely Gaussian beam are recovered for $w_{\mathrm{G}} \rightarrow \infty$ and $w_{\mathrm{L}} \rightarrow \infty$, respectively.

Evidently, the above parameters might be combined into the ratio $\beta \equiv w_{\mathrm{L}}/w_{\mathrm{G}},$ thus suggesting the alternative expressions for the wavefunction: $\varphi _{_{\mathrm{LG}}}(\xi,0)\propto { 1\over {1+\xi ^2}}{\text{e}}^{-\beta ^2\xi ^2/2}$ or $\varphi _{_{\mathrm{LG}}}(\xi,0)\propto { 1\over {1+\xi ^2/\beta ^2}}{\text{e}}^{-\xi ^2/2},$ which clearly show that the two aforementioned limits may be equivalently gained with $\beta \rightarrow 0$ and $\beta \rightarrow \infty$, respectively. However, we prefer to deal with the expression (1) displaying both parameters, in view of the analysis of the super Lorentz–Gauss beams, that will be performed in Sect. 4; such beams are in fact characterized by various $w_{\mathrm{L}}$-like parameters.

Moreover, the variable ξ, entering (1), is to be intended as unitless, resulting from the scaling of the space-coordinate by some characteristic transverse width w ₀, i.e., ξ ≡ x/w ₀. This accordingly amounts to the scaling of the longitudinal (i.e., propagation) variable z by the relevant confocal parameter $z_{\mathrm{c}}=kw_0^2,$ namely $\zeta \equiv z/z_{\mathrm{c}}.$ In the light of such a scaling, $w_{\mathrm{L}}$ and $w_{\mathrm{G}}$ are to be intended as measured in units of w ₀ as well.

We finally mention that the expression (1) conforms to the assumption of a rectangular symmetry, which amounts to the possibility of expressing the 3D wavefield $\Upphi _{_{\mathrm{LG}}}(\xi,\eta,\zeta )$ as the product of 2D wavefields, separately depending on the transverse coordinates ξ and η, i.e., $\Upphi _{_{\mathrm{LG}}}(\xi,\eta,\zeta )=\varphi _{_{\mathrm{LG}}}(\xi,\zeta )\psi _{_{\mathrm{LG}}}(\eta,\zeta ),$ with $\psi _{_{\mathrm{LG}}}$ being of course like (1) and the variable η accordingly signifying η≡ y/w ₀; the parameters $w_{\mathrm{L}}$ and $w_{\mathrm{G}}$ inherent to the ξ- and η-dependence may in principle be different. Hence, the WDF of the 3D wavefunction turns out to be the product of the WDFs of the 2D wavefunctions.

As is well known, the WDF of a 2D wavefunction is conveyed by the 1D Fourier-like integral

$${\mathcal{W}}(\xi,\kappa,\zeta )={ 1\over {2\pi }}\int\limits_{-\infty }^{+\infty }\psi ^{*}(\xi -\xi ^{\prime }/2,\zeta )\psi\; (\xi +\xi ^{\prime }/2,\zeta ){\text{e}}^{-i\kappa \xi ^{\prime }}d\xi ^{\prime },$$

(2)

where κ denotes the Fourier conjugate variable to ξ. Evidently, if ξ addresses a scaled space-variable, then κ addresses a similarly scaled spatial-frequency variable.

On evaluating the Wigner integral (2) for the Lorentz–Gaussian wavefunction (1), one might resort to the well-known property of the WDF, which, in the case of the product of two functions, turns out to be the convolution with respect to the spatial frequency of the WDFs of the two functions, in a certain analogy with the Fourier transform of the product of functions. Hence, regarding (1) as the product of a Gaussian and a Lorentzian function, one has

$${\mathcal{W}}_{_{\mathrm{LG}}}(\xi,\kappa,0)=\int {\mathcal{W}}_{_{\mathrm{G} }}(\xi,\kappa -\kappa ^{\prime },0){\mathcal{W}}_{_{\mathrm{L}}}(\xi,\kappa ^{\prime },0)d\kappa ^{\prime },$$

(3)

with $\mathcal{W}_{\mathrm{G}}$ and $\mathcal{W}_{\mathrm{L}}$ being the WDFs, respectively, of the Gaussian and Lorentzian part of (1). The former is well known, whilst the latter, to the author’s knowledge, has still not been considered in the literature (apart from, of course, in [11], the expression there presented having, as said, quite an involved structure). Therefore, we prefer to deal with the overall expression (1); $\mathcal{W}_{\mathrm{L}}$ will be deduced from $\mathcal{W}_{\mathrm{LG}}$ in the limit $w_{\mathrm{G}}\rightarrow \infty.$ We will reconsider the property (3) in the next section.

Then, on account of (1), the Wigner integral (2) specializes as

$${\mathcal{W}}_{\mathrm{LG}}(\xi,\kappa,0)=\frac{|\varphi _0|^2}\pi {\text{e}}^{- \frac{\xi ^2}{w_{\mathrm{G}}^2}}\int\limits_{-\infty }^{+\infty }{ 1\over {[1+(\xi -\xi ^{\prime })^2/w_{_{\mathrm{L}}}^2][1+(\xi +\xi ^{\prime })^2/w_{_{\mathrm{L}}}^2}}{\text{e}}^{-\frac{\xi ^{\prime 2}}{w_{_{\mathrm{G}}}^2} }{\text{e}}^{-2i\kappa \xi ^{\prime }}{\text{d}}\xi ^{\prime}.$$

(4)

By conveniently manipulating the Lorentz-like factors in the integral on the basis of the usually exploited identity (1 + a ²) = (1 + ia)(1 − ia), we end up with the expression

$${\mathcal{W}}_{\mathrm{LG}}(\xi,\kappa,0)=\frac{|\varphi _0|^2}{4i\pi \xi /w_{_{_{\mathrm{L}}}}^3}{\text{e}}^{-\frac{\xi ^2}{w_{_{\mathrm{G}}}^2} }\int\limits_{-\infty }^{+\infty }\left[{ 1\over {A_{_{_{\mathrm{L}}}}^2+y^2}}-{ 1\over {B_{_{_{\mathrm{L}}}}^2+y^2}}\right]{\text{e}}^{-\frac{y^2}{w_{_{\mathrm{G}}}^2} }{\text{e}}^{-2i\kappa y}{\text{d}}y,$$

(5)

where

$$A_{_{\mathrm{L}}}=w_{_{\mathrm{L}}}-i\xi, \; B_{_{\mathrm{L}}}=w_{_{ \mathrm{L}}}+i\xi =A_{_{\mathrm{L}}}^{*}\hbox {.}$$

We see that the WDF of a Lorentz–Gauss beam is expressible as the sum of two integrals of the same form; each of them, involving a Lorentz-like factor modulated by a complex quadratic exponential, looks quite similar to the Huygens–Fresnel integral of a Lorentzian or a Lorentz–Gaussian function, eventually yielding the explicit expressions for corresponding wavefunctions.

Accordingly, the evaluation of the above integrals can be approached by the Fourier transform method, which resorts to the well-known property that the Fourier transform of a convolution product $(f_1\otimes f_2)(y)=\int\nolimits_{-\infty }^{+\infty }f_1(y^{\prime })f_2(y-y^{\prime }){\text{d}}y^{\prime }$ results into the product of the Fourier transforms of the convoluted functions, i.e., ${(\widehat{\mathcal{F}}f_1\otimes f_2)(\kappa )=\sqrt{2\pi }(\widehat{\mathcal{F}}f_1)(\kappa )(\widehat{\mathcal{F}}f_2)(\kappa ). }$ Thus, one has

$$\int\limits_{-\infty }^{+\infty }{ 1\over {A_{_{_{\mathrm{L}}}}^2+y^2}}{\text{e}}^{- \frac{y^2}{w_{_{\mathrm{G}}}^2}}{\text{e}}^{-2i\kappa y}{\text{d}}y=\sqrt{2\pi }{\text{e}}^{-\kappa ^2w_{_{\mathrm{G}}}^2}\left[ \widehat{{\mathcal{F}}}^{-1}\left( \widehat{ {\mathcal{F}}}{ 1\over {A_{_{_{\mathrm{L}}}}^2+y^2}}\right) \left( \widehat{ {\mathcal{F}}}{\text{e}}^{-\frac{y^2}{w_{_{\mathrm{G}}}^2}}\right) \right] [-i\kappa w_{_{\mathrm{G}}}^2],$$

${ \widehat{\mathcal{F}}}$ denoting the Fourier transform operator, which is well known to operate as

$$(\widehat{{\mathcal{F}}}f)(\kappa )={ 1\over {\sqrt{2\pi }}}\int\limits_{-\infty }^{+\infty }{\text{\text{e}}}^{-i\kappa \xi }f(\xi ){\text{d}}\xi.$$

Recalling that

$$\left(\widehat{{\mathcal{F}}}{ 1\over {A_{_{_{\mathrm{L}}}}^2+y^2}}\right)(\kappa )=\sqrt{ { \pi\over 2}}{ 1\over {A_{_{_{\mathrm{L}}}}}}{\text{e}}^{-A_{_{_{\mathrm{L}}}}|\kappa |},$$

for $\Re (A_{_{\mathrm{L}}})>0,$ as it is the case here, the expression (5) explicitly evaluates to

$${\mathcal{W}}_{_{\mathrm{LG}}}(\xi,\kappa,0)=\frac{|\varphi _0|^2}{4\xi /w_{_{_{\mathrm{L}}}}^2}{\text{e}}^{-\frac{\xi ^2}{w_{_{\mathrm{G}}}^2}-\kappa ^2w_{_{ \mathrm{G}}}^2}\Im \left\{ { 1\over {1-i\xi /w_{_{_{\mathrm{L}}}}}}\left[ {\mathcal{E}}(\xi,\kappa )+{\mathcal{E}}(\xi,-\kappa )\right] \right\},$$

(6)

where $\Im$ signifies imaginary part and

$${\mathcal{E}}(\xi,\kappa )={\text{e}}^{{\mathcal{X}}(\xi,\kappa )^2}\mathrm{erfc}( {\mathcal{X}}(\xi,\kappa )),$$

(7)

being in turn

$${\mathcal{X}}(\xi,\kappa )=\frac{w_{_{\mathrm{L}}}-i\xi +\kappa w_{_{_{ \mathrm{G}}}}^2}{w_{_{_{\mathrm{G}}}}}=\beta (1-i\xi /w_{_{_{\mathrm{L} }}})+\kappa w_{_{\mathrm{G}}},$$

(8)

with β introduced above as: $\beta \equiv w_{_{\mathrm{L}}}/w_{_{\mathrm{G}}}.$ Finally, $\mathrm{erfc}$ denotes the complementary error function. Note also that [13]

$$\begin{aligned} {\mathcal{E}}(\xi,\kappa ) &= _1F_1({ 1\over 2};{ 1\over 2};{\mathcal{X}}(\xi,\kappa )^2)-{ 2\over {\sqrt{\pi }}}{\mathcal{X}}(\xi,\kappa ) _1F_1(1;{ 3\over 2}; {\mathcal{X}}(\xi,\kappa )^2)\\ &={\text{e}}^{{\mathcal{X}}(\xi,\kappa )^2}-{ 2\over {\sqrt{\pi }}}{\mathcal{X}}(\xi,\kappa ) _1F_1(1;{ 3\over 2};{\mathcal{X}}(\xi,\kappa )^2), \end{aligned}$$

₁ F ₁ representing the Kummer function. Evidently, $\mathcal{E}^{*}(\xi,\kappa )=\mathcal{E}(-\xi,\kappa ).$

Expression (6) represents the main result of this note. As said, it parallels the expression presented in [11], offering indeed a closed-form expression for the WDF of a Lorentz–Gauss wavefunction.

It is worth noting that, notwithstanding the factor ${ 1\over \xi},$ $\mathcal{W}_{_{\mathrm{LG}}}(\xi,\kappa,0)$ has no singularity at ξ = 0. This can be verified through the Taylor series expansion of $\mathcal{W}_{_{\mathrm{LG}}}(\xi,\kappa,0),$ yielding

$${\mathcal{W}}_{_{\mathrm{LG}}}(\xi,\kappa,0)\sim \frac{|\varphi _0|^2}{ 4/w_{_{\mathrm{L}}}}{\text{e}}^{-\kappa ^2w_{\mathrm{G}}^2}\left\{ \frac{4\beta }{ \sqrt{\pi }}+{\mathcal{E}}(0,\kappa )[1-2\beta {\mathcal{X}}(0,\kappa )]+ {\mathcal{E}}(0,-\kappa )[1-2\beta {\mathcal{X}}(0,-\kappa )]\right\} \quad+{\mathcal{O }}(\xi ),$$

which evidently shows that $\mathcal{W}_{_{\mathrm{LG}}}(\xi,\kappa,0)$ takes on finite values at ξ = 0.

Also, a closer inspection of the expression (6) allows to recognize the WDF of the Gaussian part in $\varphi _{_{\mathrm{LG}}},$ being indeed $\mathcal{W}_{_{\mathrm{G}}}(\xi,\kappa,0)\propto {\text{e}}^{-\frac{\xi ^2}{w_{_{\mathrm{G}}}^2}-\kappa ^2w_{_{\mathrm{G}}}^2}.$ So, one can presume that the remaining part of $\mathcal{W}_{_{\mathrm{LG}}}$ arises from the Lorentzian part.

As said in fact, one can easily deduce the expression for the WDF of a purely Lorentzian beam from (6) in the limit $w_{_{\mathrm{G}}}$ $\rightarrow \infty,$ amounting to $\beta \rightarrow 0.$ In that case, we see that

$${\mathcal{E}}(\xi,\kappa )\sim {\text{e}}^{2(1-i\xi /w_{\mathrm{L}})\kappa w_{_{ \mathrm{L}}}}{\text{e}}^{\kappa ^2w_{\mathrm{G}}^2}\mathrm{erfc}(\kappa w_{_{\mathrm{G }}})\simeq {\text{e}}^{2(1-i\xi /w_{_{\mathrm{L}}})\kappa w_{_{\mathrm{L}}}}{\text{e}}^{\kappa ^2w_{_{\mathrm{G}}}^2}(1-{ \kappa\over {|\kappa |}}),$$

the latter expression following from the asymptotic formula for the Kummer function, i.e., [13]

$$_1F_1(a;c;z)\sim \frac{\Upgamma (c)}{\Upgamma (a)}z^{a-c}{\text{e}}^z,$$

for fixed a, c, and $\Re (z)\rightarrow \infty.$

As a result, one can write

$$\begin{aligned} {\mathcal{E}}(\xi,\kappa ) + {\mathcal{E}}(\xi, - \kappa ) &\mathop \rightarrow \limits_{w_{G} \to \infty} {\text{e}}^{\kappa ^{2} w_{\hbox{G}}^{2}} \left\{{\text{e}}^{2(1 - i\xi /w_{\hbox{L}})\kappa w_{\hbox{L}}} (1 - \frac{\kappa}{|\kappa|}) + {\text{e}}^{- 2(1 - i\xi /w_{{\text{L}}} )\kappa w_{\hbox{L}}} (1 + \frac{\kappa }{|\kappa|})\right\}\\ &= 2{\text{e}}^{\kappa^{2} w_{\hbox{G}}^{2}} {\text{e}}^{- 2(1 - i\xi /w_{\hbox{L}}) |\kappa |w_{\hbox{L}}}, \end{aligned}$$

(9)

thus amounting to

$$\begin{aligned} {\mathcal{W}}_{_{\mathrm{LG}}}(\xi,\kappa,0)\mathop \rightarrow \limits_{w_{G} \to \infty} {\mathcal{W}}_{_{\mathrm{L}}}(\xi,\kappa,0) &=\frac{|\varphi _0|^2}{2\xi /w_{_{\mathrm{L}}}^2}{\text{e}}^{-2|\kappa |w_{ \mathrm{L}}}\Im \left\{ { 1\over {1-i\xi /w_{_{\mathrm{L}}}}}{\text{e}}^{2i\xi |\kappa |}\right\}\\ &=\frac{|\varphi _0|^2}{2/w_{_{\mathrm{L}}}}\tfrac 1{(1+\xi ^2/w_{_{\mathrm{ L}}}^2)}{\text{e}}^{-2|\kappa |w_{\mathrm{L}}}[\cos (2\xi |\kappa |)+2|\kappa |w_{_{ \mathrm{L}}}\hbox {sinc}(2\xi |\kappa |)]. \end{aligned}$$

(10)

In the opposite limit: i.e., $\beta \rightarrow \infty,$ we see that

$${\mathcal{E}}(\xi,\kappa )+{\mathcal{E}}(\xi,-\kappa )\mathop \rightarrow \limits_{w_{_{\mathrm{L}}} \to \infty}{ 2\over {\sqrt{\pi }\beta }}{\text{e}}^{\beta ^2(1-i\xi /w_{\mathrm{L}})^2}{ 1\over {(1-i\xi /w_{_{\mathrm{L}}})}}{\text{e}}^{-\beta ^2(1-i\xi /w_{\mathrm{L}})^2}={ 2\over {\sqrt{\pi }\beta }}{ 1\over {(1-i\xi /w_{_{\mathrm{L}}})}},$$

which eventually yields the very expression of the WDF of a Gaussian beam,

$${\mathcal{W}}_{_{\mathrm{LG}}}(\xi,\kappa,0)\mathop \rightarrow \limits_{w_{_{\mathrm{L}}} \to \infty} {\mathcal{W}}_{_{\mathrm{G}}}(\xi,\kappa,0)=|\varphi _0|^2\frac{w_{_{\mathrm{G}}}}{\sqrt{\pi }}{\text{e}}^{-\frac{\xi ^2}{w_{ \mathrm{G}}^2}-\kappa ^2w_{\mathrm{G}}^2}.$$

As said, the WDF of the 3D wavefunction $\Upphi _{_{\mathrm{LG}}}(\xi,\eta,\zeta )$ turns out to be the product of the WDFs $\mathcal{W}_{_{\mathrm{LG}}}(\xi,\kappa,0)$ pertaining to the two 2D wavefunctions like (1), i.e., in symbols

$${\mathcal{W}}_{_{\mathrm{LG}}}(\xi,\kappa,\eta,\nu,0)={\mathcal{W}}_{_{ \mathrm{LG}}}(\xi,\kappa,0){\mathcal{W}}_{_{\mathrm{LG}}}(\eta,\nu,0).$$

3 Wigner charts for Lorentz–Gauss beams

Expression (6) allows us to determine the WDF of a Lorentz–Gaussian beam paraxially propagating through any optical system, thus fully accounting for the Wigner-plane dynamics of such a beam under paraxial propagation.

As is well known, in fact, the WDF obeys the transfer law [15, 16, 33]

$${\mathcal{W}}({\mathbf{v}},\zeta _o)={\mathcal{W}}({\mathbf{M}}^{-1}{\mathbf{v}},\zeta _i),$$

(11)

under the paraxial propagation of the corresponding wavefield through a real optical system. Here, the array $\mathbf{v}$ conveniently gathers the WDF variables, namely the dimensionless Fourier conjugate pair: $\mathbf{v}^{\mathbf{\top }}=(\xi,\kappa );$ ζ _i and ζ _o identify the ζ-locations, respectively, of the input and output planes, between which the optical system is understood to operate, transporting the wavefunction on the plane at ζ _i to the wavefunction on the plane at ζ _o . Finally, $\mathbf{M}= \left( \begin{array}{ll} A & B\\ C & D \end{array} \right)$ represents the 2 × 2 ray transfer matrix, which according to the geometrical optics formalism synthesizes the paraxial propagation through an optical system in terms of a linear transformation of the ray variables, namely the position q and the momentum p (i.e., the angle of the propagating ray relative to the z-axis). Since we are dealing here with dimensionless space variables, the ray vector $\left( \begin{array}{ll} q\\ p \end{array} \right)$ turns into the unitless form $\left( \begin{array}{ll} q/w_0\\ kw_0p \end{array} \right)$ just conveying the Fourier pair $\left( \begin{array}{ll} \xi\\ \kappa \end{array} \right)$. The ray optical momentum p comes in fact to be normalized to the far-field divergence angle ϑ = 1/kw ₀, associated with a beam having w ₀ as a characteristic width. Consequently, the entries of $\mathbf{M}$ are dimensionless as well, the off-diagonal entries B and C (which usually have the dimensions of length and length⁻¹) being normalized, respectively, to $z_{\mathrm{c}}$ and $z_{\mathrm{c}}^{-1}.$

In particular, the free propagation by ζ from ζ _i  = 0 signifies $\mathbf{M}=\left( \begin{array}{ll} 1 & \zeta\\ 0 & 1 \end{array} \right)$ and hence the WDF for a Lorentz–Gauss beam propagating in free-space is simply given by

$${\mathcal{W}}_{_{\mathrm{LG}}}(\xi,\kappa,\zeta )={\mathcal{W}}_{_{\mathrm{LG} }}(\xi -\zeta \kappa,\kappa,0),$$

amounting, as is well known, to a ξ-shear of the Wigner chart in the Wigner phase-plane (ξ, κ).

For exemplificative purposes, Fig. 1 shows the Wigner charts of the WDF $\mathcal{W}_{_{\mathrm{LG}}}$ of the Lorentz–Gauss beam (1) at $\zeta =0$ and $\zeta =1$ for some values of β; precisely, for β = 0, amounting to a purely Lorentzian beam, β = 0.2, β = 1 and β = 2, which is near to the purely Gaussian limit. We see in fact that with increasing β the contour lines in the Wigner-plane (ξ, κ) tend to become ellipses, which are well known to characterize the Wigner chart of the WDF of a Gaussian beam. Note that, using β as a parameter for the graphs amounts to scaling the space-coordinate by $w_{_{\mathrm{L}}}$ (for β = 0) or $w_{_{\mathrm{G}}}$ (for β ≠ 0).

Fig. 1

Wigner charts of the WDF $\mathcal{W}_{\rm LG}$ of the Lorentz–Gauss beam (1) at $\zeta =0$ (on the left) and $\zeta=1$ (on the right) for a, b β = 0, c, d β = 0.2, e, f β = 1 and g, h β = 2

More in general, the evolution of the WDF under paraxial propagation through an optical system is conveyed by the relation [15, 16, 33]

$${\mathcal{W}}(\xi,\kappa,\zeta _o)=\iint {\mathsf{G}}(\xi,\kappa,\xi ^{\prime },\kappa ^{\prime }){\mathcal{W}}(\xi ^{\prime },\kappa ^{\prime },\zeta _i)d\xi ^{\prime }{\text{d}}\kappa ^{\prime },$$

(12)

whose kernel $\mathsf{G}(\xi,\kappa,\xi ^{\prime },\kappa ^{\prime})$ is given by

$${\mathsf{G}}(\xi,\kappa,\xi ^{\prime },\kappa ^{\prime })=\tfrac 1{2\pi }\iint g^{*}(\xi -\tfrac{\xi ^{\prime \prime }}2,\xi ^{\prime }- \tfrac{\xi ^{\prime \prime \prime }}2)g(\xi +\tfrac{\xi ^{\prime \prime }}2,\xi ^{\prime }+\tfrac{\xi ^{\prime \prime \prime }}2){\text{e}}^{-i(\kappa \xi ^{\prime \prime }-\kappa ^{\prime }\xi ^{\prime \prime \prime })}{\text{d}}\xi ^{\prime \prime }{\text{d}}\xi ^{\prime \prime \prime }.$$

(13)

It is completely determined by the line-spread function $g(\xi,\xi ^{\prime })$ of the system of concern, which in turn rules the evolution of the wavefunction propagating through the system, being the kernel of the Huygens–Fresnel diffraction integral [21, 22].

Indeed, the phase-plane transfer relation (12) for the WDF is paralleled by the space-domain transfer relation for the wavefunction,

$$\varphi (\xi,\zeta _o)=\int g(\xi,\xi ^{\prime })\varphi (\xi ^{\prime },\zeta _i){\text{d}}\xi ^{\prime },$$

where $g(\xi,\xi ^{\prime })$ is determined by the ray-matrix $\mathbf{M}$ according to

$$g(\xi,\xi ^{\prime })=\sqrt{\tfrac 1{2\pi iB}}{\text{e}}^{i{ 1\over {2B}}\left[ A\xi ^{\prime 2 }+D\xi ^2-2\xi \xi ^{\prime }\right] },$$

the principal root being conventionally taken for $\sqrt{iB},$ i.e., $\sqrt{iB}=\sqrt{|B|}{\text{e}}^{i{ \pi\over 4}\hbox {sgn}(B)}.$ Also, $g(\xi,\xi ^{\prime })$ reduces to

$$g(\xi,\xi ^{\prime })\mathop \rightarrow \limits_{B=0}\sqrt{\tfrac 1A} {\text{e}}^{i{ A\over {2C}}\xi ^2}\delta (\xi ^{\prime }-\xi),$$

for “imaging” systems.

The ray-spread function $\mathsf{G}(\xi,\kappa,\xi ^{\prime },\kappa ^{\prime })$ is seen to be the symplectic Fourier transform of the product $g^{*}(\xi -\tfrac{\xi ^{\prime \prime }}2,\xi ^{\prime }-\tfrac{\xi ^{\prime \prime \prime }}2)g(\xi +\tfrac{\xi ^{\prime \prime }}2,\xi ^{\prime }+\tfrac{\xi ^{\prime \prime \prime }}2)$ regarded as a function of $\xi ^{\prime \prime }$ and $\xi ^{\prime \prime \prime }.$ Notably, it has the structure of a “double” WDF, and hence it has all the properties of a WDF.

When real entries (A, B, C, D) are involved, (12)–(13) yield (11), being $\mathsf{G}(\xi,\kappa,\xi ^{\prime },\kappa ^{\prime })=\delta (A\xi ^{\prime }+B\kappa ^{\prime }-\xi )\delta (C\xi ^{\prime }+D\kappa ^{\prime }-\kappa ).$

Evidently, this is not the case when complex entries are involved. However, complex matrices that are meaningful in optics are those yielding Gaussian aperturing and Gaussian convolution. The former is conveyed by the lens-like ray-matrix with purely imaginary “focal length” f = iw, namely

$${\mathbf{G}}(w)= \left( \begin{array}{ll} 1 & 0\\ i/w & 1 \end{array} \right), \quad w>0,$$

(14)

the parameter w signifying the characteristic width of the aperture. Accordingly, it amounts to the Gaussian apodization of the wavefunction by ${\text{e}}^{-{1\over{2w}}\xi ^2}.$ In that case, the ray-spread function turns out to be

$${\mathsf{G}}_{\hbox {\rm{ga}}}(\xi,\kappa,\xi ^{\prime },\kappa ^{\prime })={ w\over {\sqrt{\pi }}}{\text{e}}^{-{ 1\over w}\xi ^2}{\text{e}}^{-w(\kappa -\kappa ^{\prime })^2}\delta (\xi ^{\prime }-\xi ),$$

(15)

thus signifying, in accord with the general property (3), that the WDF of a Gaussian apertured wavefunction results from the convolution in the spatial frequency domain of the WDFs of the aperture transmission ${\text{e}}^{-{ 1\over {2w}}\xi ^2}$ and the wavefunction. This applies to the Lorentz–Gauss wavefunction (1) as well; it can be seen as a Gaussian-apertured Lorentz wavefunction, so that the corresponding WDF is also deducible from (3) with $\mathcal{W}_{_{\mathrm{L}}}$ given by (10), or equivalently from (12) through (15) with $w=w_{_{\mathrm{G}}}^2,$ ζ _i  = 0 and $\mathcal{W}(\xi,\kappa,0)=\mathcal{W}_{_{\mathrm{L}}}(\xi,\kappa,0).$ In the light of this, the plots in the frames on the left of Fig. 1 from a to g can be seen as displaying the effect of an even more severe Gaussian aperturing of a Lorentz beam.

For completeness’ sake, let us finally recall that the Gaussian convolution is conveyed by the free-space section-like matrix with purely imaginary “length” d =  −iτ :

$${\mathbf{P}}(\tau )= \left( \begin{array}{ll} 1 & -i\tau\\ 0 & 1 \end{array} \right),\quad \tau >0,$$

(16)

which yields the Weierstrass–Gauss or Poisson transform. Such transform is central to the theory of the 1D HE, whose solutions are in fact obtained from the Poisson transform of definite initial conditions [23]. Gaussian convolution is optically implementable by the propagation through a Gaussian aperture (whose characteristic width is conveyed by the parameter τ ⁻¹), enclosed between a direct and inverse Fourier transform.

Paralleling (3), a further property of the WDF allows one to state that the WDF of a convolution is in turn the convolution of the WDFs of the convoluting functions with respect to the space variable, as also conveyed by (12) with the pertinent ray-spread function being indeed

$${\mathsf{G}}_{\hbox {\rm{gc}}}(\xi,\kappa,\xi ^{\prime },\kappa ^{\prime })=\sqrt{\pi \tau }{\text{e}}^{-{ 1\over \tau} (\xi -\xi ^{\prime })^2}{\text{e}}^{-\tau \kappa ^2}\delta (\kappa ^{\prime }-\kappa ).$$

(17)

In both cases, one deal with a WDF smoothed by the Gaussian kernels (15) or (17), the smoothing involving the spatial frequency or the space variable in accord with the Fourier complementarity of the inherent ray-matrices $\mathbf{G}(w)$ and $\mathbf{P}(\tau ),$ which in fact are the purely imaginary counterparts of the thin-lens and free-space section ray-matrices. We know that the WDF smoothing is usually performed to reduce the influence of the cross-terms when dealing with multi-component signals or to limit the negativity of the WDF, reducing both the zones over the Wigner-plane where the WDF is negative and the entity of the negative values. Figure 2 makes evident such an effect, comparing the negative-valued portions of the WDF over the (ξ, κ)-plane for a Lorentz beam (with, in particular, $w_{_{\mathrm{L}}}=1$) and a Lorentz–Gauss beam (with $w_{_{\mathrm{L}}}=1,$ β = 0.5).

Finally, it is also worth noting that the Gaussian aperturing of the initial field manifests into a definite symmetry transformation between the wavefunctions corresponding to the unapertured and Gaussian-apertured fields, the latter being the Gaussianly modulated version of the former [1, 24, 25]. Therefore, the above considerations concerning the Gaussian-smoothing of the WDF apply not only to the Lorentz–Gauss beams of interest here, but more in general to any wavefunction, which, like the Lorentz–Gauss beams, can be understood as arising from a Gaussian-apertured source function. In fact, by virtue of the evolution law (12) through (15), we see that the WDF of the Gaussianly modulated version of a solution of the paraxial wave equation turns out to be the freely propagated Gaussian-smoothed version of the WDF of the original solution.

4 Concluding notes

We have deduced a closed-form expression for the WDF of a Lorentz–Gauss beam at the initial plane. As recalled, the WDF of Lorentz–Gauss beams has been the object of a detailed analysis in [11], where it has been evaluated resorting to an approximation of the Lorentzian function by a finite sum of (even-order) Hermite–Gaussian functions, as suggested in [12]. The expression (6), deduced here, parallels that worked out in the quoted paper.

According to the evolution law obeyed in general by the WDF, expression (6) provides a complete description of the Wigner-plane dynamics of Lorentz–Gauss beams paraxially propagating through optical systems. In particular, as far as real optical systems are concerned, one deals with the simple transfer law (11), which directly relates to that of the light ray-variables in geometrical optics.

As is well known, within the context of signal representations in the space-frequency domain, the Wigner distribution function is paralleled by the ambiguity function (AF). These functions are in fact regarded, within the general class of the bilinear space-frequency signal representations [26, 27], as the “founders” of two dual subclasses [28]. The WDF manifests an energetic nature as it seeks to combine the spatial and spectral energy densities of the signal, one can recover through the relevant marginal distributions, whilst the AF manifests a correlative nature, combining the spatial and spectral autocorrelations of the signal, that in turn can be recovered from the AF profiles along the inherent phase-plane axes.

The ambiguity function was first applied to optical problems by Papoulis [29]. Later, a simple relationship between the AF and the defocused optical transfer function (OTF) when plotted in polar coordinates was shown in [30]; for further developments connected with the concept of generalized OTF, we address the reader to [31].

Let us recall that the AF $\mathcal{A}(\widetilde{\xi},\widetilde{\kappa },\zeta )$ is defined by

$${\mathcal{A}}(\widetilde{\xi },\widetilde{\kappa },\zeta )={ 1\over {2\pi }}\int\limits_{-\infty }^{+\infty }\psi ^{*}(\xi ^{\prime }-\widetilde{\xi } /2,\zeta )\psi \;(\xi ^{\prime }+\widetilde{\xi }/2,\zeta ){\text{e}}^{-i\widetilde{ \kappa }\xi ^{\prime }}{\text{d}}\xi ^{\prime },$$

(18)

its inherent phase-plane being composed by the difference coordinates $(\widetilde{\xi },\widetilde{\kappa })$ as opposite to the center coordinates (ξ, κ), associated indeed with the WDF and accordingly interpretable as the space and spatial frequency Fourier conjugate variables. The AF relates to the WDF through the symplectic 2D Fourier transform,

$${\mathcal{A}}(\widetilde{\xi },\widetilde{\kappa },\zeta )={ 1\over {2\pi }}\int\limits_{-\infty }^{+\infty }\int\limits_{-\infty }^{+\infty }{\mathcal{W }}(\xi,\kappa,\zeta ){\text{e}}^{i(\kappa \widetilde{\xi }-\widetilde{\kappa }\xi )}{\text{d}}\xi {\text{d}}\kappa.$$

(19)

It is therefore an easy task to deduce the expression for the AF for the Lorentz–Gaussian wavefunction $\varphi _{_{\mathrm{LG}}}(\xi,0)$ at the plane $\zeta =0$ through (18) or (19). Even more simply, one can exploit the property that, for even functions, the WDF and the AF relates as

$${\mathcal{A}}(\widetilde{\xi },\widetilde{\kappa })={ 1\over 2}{\mathcal{W}}\left( \widetilde{{ \xi\over 2}},\frac{\widetilde{\kappa }}2\right),$$

which evidently applies to (1). It shows that in the case of the $\varphi _{_{\mathrm{LG}}},$ as in general of even functions, the AF turns out to be real; as is well known, the WDF is always a real function.

We also recall that the AF obeys the same transfer law (11) as the WDF under paraxial propagation of the corresponding wavefield through real optical systems.

We close the paper, by considering the super-Lorentz–Gauss wavefunction [32–34], which, being represented by

$$\varphi _{_{\mathrm{SLG}}}(\xi,0)=\varphi _0\prod_{j=1}^M{ 1\over {1+\frac{ \xi ^2}{w_j^2}}}{\text{e}}^{-\frac{\xi ^2}{2w_{\mathrm{G}}^2}},$$

(20)

at the input plane, displays the product of M Lorentz-like functions, each being characterized by the relevant parameter w _j .

The evaluation of the WDF of $\varphi _{_{\mathrm{SLG}}}(\xi,0)$ demands the rather cumbersome repeated application of the procedure outlined in Sect. 2 when evaluating the WDF of $\varphi _{_{\mathrm{LG}}}(\xi,0).$ As an exemplificative case, let us consider in detail the case M = 2, thus yielding, in some analogy with (5), the Wigner integral

$${\mathcal{W}}_{_{\mathrm{SLG}}}(\xi,\kappa,0)=\frac{|\varphi _0|^2w_1^3w_2^3 }{16\pi \xi ^2}{\text{e}}^{-\frac{\xi ^2}{w_{\mathrm{G}}^2}}\int\limits_{-\infty }^{+\infty }\left[{ 1\over {A_1^2+y^2}}-{ 1\over {B_1^2+y^2}}\right]\left[{ 1\over {B_2^2+y^2}}-{ 1\over {A_2^2+y^2}}\right]{\text{e}}^{-\frac{y^2}{w_{\mathrm{G}}^2}}{\text{e}}^{-2i\kappa y}{\text{d}}y,$$

(21)

being indeed

$$A_1=w_1-i\xi =B_1^{*}, \quad A_2=w_2-i\xi =B_2^{*}\hbox {.}$$

Accordingly, if $w_1\neq w_2, \mathcal{W}_{_{\mathrm{SLG}}}(\xi,\kappa,0)$ comes to result from the superposition of four integrals of the type

$$\int\limits_{-\infty }^{+\infty }{ 1\over {[a^2+y^2][b^2+y^2]}}{\text{e}}^{-\frac{y^2}{ w_{\mathrm{G}}^2}}{\text{e}}^{-2i\kappa y}{\text{d}}y={ 1\over {(b^2-a^2)}}\int\limits_{-\infty }^{+\infty }[{ 1\over {a^2+y^2}}-{ 1\over {b^2+y^2}}]{\text{e}}^{-\frac{y^2}{w_{\mathrm{G} }^2}}{\text{e}}^{-2i\kappa y}{\text{d}}y, \quad a\neq b,$$

(22)

which, as indicated, can again be recast as the sum of two integrals having the form of the Fourier transform of a Lorentz–Gaussian-like function. Therefore, after some algebra, we end up with the closed-form, albeit rather involved, expression

$${\mathcal{W}}_{_{\mathrm{SLG}}}(\xi,\kappa,0)=\frac{|\varphi _0|^2w_1^3w_2^3 }{16(w_1^2-w_2^2)\xi }{\text{e}}^{-\frac{\xi ^2}{w_{\mathrm{G}}^2}-\kappa w_{\mathrm{G }}^2}\Im \left\{ \sum_{j=1,2}B_j(\xi )\left[ {\mathcal{E}}_j(\xi,\kappa )+ {\mathcal{E}}_j(\xi,-\kappa )\right] \right\}$$

(23)

where $\mathcal{E}_j(\xi,\kappa )$ is defined as (7) with $w_{_{\mathrm{L}}}$ replaced by w _j , j = 1, 2, and hence β by $\beta _j\equiv w_j/w_{_{\mathrm{G}}},$ j = 1, 2. Also,

$$\begin{aligned} &B_1(\xi )=\frac{w_2}{(w_1-i\xi )(w_s-i\xi )(w_d-i\xi )}, \\ &{B_2(\xi )=\frac{w_1}{(w_2-i\xi )(w_s-i\xi )(w_d+i\xi )},} \end{aligned}$$

where

$$w_s=\frac{w_1+w_2}2, \quad w_d=\frac{w_1-w_2}2$$

Again, as seen for the $\mathcal{W}_{_{\mathrm{LG}}},$ notwithstanding the presence of the factor 1/ξ, $\mathcal{W}_{_{\mathrm{SLG}}}$ takes on finite values at ξ = 0, being in fact

$$\begin{aligned} {\mathcal{W}}_{_{\mathrm{SLG}}}(0,\kappa ,0) =&\frac{|\varphi _0|^2w_1w_2}{ 4(w_1^2-w_2^2)^3}{\text{e}}^{-\kappa w_{_{\mathrm{G}}}^2}\left\{ \frac{ 4w_1w_2(w_1^4-w_2^4)}{\sqrt{\pi }w_{_{\mathrm{G}}}}\right.\\ &\left.\vphantom{\left\{ \frac{ 4w_1w_2(w_1^4-w_2^4)}{\sqrt{\pi }w_{_{\mathrm{G}}}}\right.}-w_2^3[{\mathcal{A}}_1(\kappa )+{\mathcal{A}}_1(-\kappa )]-w_1^3[ {\mathcal{A}}_2(\kappa )+{\mathcal{A}}_2(-\kappa )]\right\}, \end{aligned}$$

(24)

with

$$\begin{aligned} {\mathcal {A}}_{1} \left( \kappa \right) &= {\mathcal{E}}_{1} \left( {0,\kappa} \right)\left[ {2\left( {w _{1}^{2} - w _{2}^{2} } \right)\beta _{1} \left( {\beta _{1} +{\kappa}w _{G} } \right) + w _{2}^{2} - 5w _{1}^{2} } \right] \\ {\mathcal {A}}_{2} \left( \kappa \right) &= {\mathcal{E}}_{2} \left( {0,\kappa} \right)\left[ {2\left( {w _{1}^{2} - w _{2}^{2} } \right)\beta _{2} \left( {\beta _{2} + \kappa w _{G} } \right) - w _{1}^{2} + 5w _{2}^{2} } \right]. \\ \end{aligned}$$

Needless to say, the expression (23) is valid for w ₁ ≠ w ₂. In the case where $w_1=w_2\equiv w_{_{\mathrm{L}}},$ the decomposition exemplified in (22) is no longer the only trick to apply in order to evaluate the relevant Wigner integral, since w ₁ = w ₂ amounts to $A_1= A_2=A_{_{\mathrm{L}}}.$ Then, when dealing with the Wigner integral

$${\mathcal{W}}_{_{\mathrm{SLG}}}(\xi ,\kappa ,0)=-\frac{|\varphi _0|^2w_{_{ \mathrm{L}}}^6}{16\pi \xi ^2}{\text{e}}^{-\frac{\xi ^2}{w_{_{\mathrm{G}}}^2} }\int\limits_{-\infty }^{+\infty }\left[{ 1\over {A_{_{\mathrm{L}}}^2+y^2}}-{ 1\over {B_{_{ \mathrm{L}}}^2+y^2}}\right]^2{\text{e}}^{-\frac{y^2}{w_{_{\mathrm{G}}}^2}}{\text{e}}^{-2i\kappa y}{\text{d}}y,$$

(25)

with, as before,

$$A_{_{\mathrm{L}}}=w_{_{\mathrm{L}}}-i\xi =B_{\rm{L}}^* ,$$

one can note that

$$\left[ { 1\over {a^2+y^2}}-{ 1\over {b^2+y^2}}\right]^2=-{ 1\over {2a}}{ d\over {da}}{ 1\over {a^2+y^2}}-{ 1\over {2b}}{ d\over {db}}{ 1\over {b^2+y^2}}-2{ 1\over {a^2+y^2}}{ 1\over {b^2+y^2}}.$$

Accordingly, the evaluation of (25) can once more proceed through the evaluation of integrals resembling the Fourier transform of a Lorentz–Gaussian-like function.

As a result, one obtains

$${\mathcal{W}}_{_{\mathrm{SLG}}}(\xi ,\kappa ,0)=\frac{|\varphi _0|^2w_{_{ \mathrm{L}}}^4}{32w_{_{\mathrm{G}}}\xi ^3}{\text{e}}^{-\frac{\xi ^2}{w_{_{\mathrm{G} }}^2}-\kappa w_{_{\mathrm{G}}}^2}\left\{ \Im \left\{{ 1\over {(1-i\xi /w_{_{\mathrm{ L}}})^3}}[{\mathcal{V}}(\xi ,\kappa )+{\mathcal{V}}(\xi ,-\kappa )]\right\}-4\frac{w_{_{ \mathrm{L}}}^2\xi }{\sqrt{\pi }}\frac{w_{_{\mathrm{L}}}^2-\xi ^2}{(w_{_{ \mathrm{L}}}^2+\xi ^2)^2}\right\}$$

(26)

where

$${\mathcal{V}}(\xi ,\kappa )={\mathcal{E}}(\xi ,\kappa )\left[ w_{_{\mathrm{G} }}(1-i\xi /w_{_{\mathrm{L}}})^2+2i\xi {\mathcal{X}}(\xi ,\kappa )(1-i\xi /w_{_{ \mathrm{L}}})-i\xi /\beta \right].$$

Again, (25) takes on finite values along the line ξ = 0, being specifically

$${\mathcal{W}}_{_{\mathrm{SLG}}}(0,\kappa ,0)=\frac{|\varphi _0|^2w_{_{\mathrm{L }}}}{96w_{_{\mathrm{G}}}^5}{\text{e}}^{-\kappa w_{\mathrm{G}}^2}\left\{ \frac{4w_{_{ \mathrm{L}}}}{\sqrt{\pi }}[4w_{_{\mathrm{L}}}^4-4w_{_{\mathrm{L}}}^2w_{_{ \mathrm{G}}}^2(2-\kappa ^2w_{_{\mathrm{G}}}^2)+15w_{_{\mathrm{G}}}^4]+ {\mathcal{G}}(\kappa )+{\mathcal{G}}(-\kappa )\right\} ,$$

with

$${\mathcal{G}}(\kappa )={\mathcal{E}}(0,\kappa )\left\{ w_{_{\mathrm{G}}}^2\left( w_{_{\mathrm{G}}}-2w_{_{\mathrm{L}}}{\mathcal{X}}(0,\kappa )\right) ^2-12w_{_{ \mathrm{L}}}w_{_{\mathrm{G}}}^4\left[ \kappa (w_{_{\mathrm{G}}}-w_{_{\mathrm{ L}}}{\mathcal{X}}(0,\kappa ))+{\mathcal{X}}(0,\kappa )\right] +14w_{_{\mathrm{G} }}^5\right\}.$$

It is an easy task to verify that (23) turns into the expression of the WDF for a purely Gaussian (w ₁ and $w_2\rightarrow \infty$), a purely super-Lorentzian ($w_{_{\mathrm{G}}}\rightarrow \infty )$ or a purely Lorentzian ($w_1\rightarrow \infty ,$ $w_{_{\mathrm{G}}}\rightarrow \infty$ and w ₂ finite, for instance) beam.

Specifically, being new, we write down the expression of the WDF for a super-Lorentz beam (with M = 2) that can be evaluated through the specific definition or, as said, from the limit of (23) for $w_{\mathrm{G}}\rightarrow \infty ,$ thus getting for w ₁ ≠ w ₂,

$${\mathcal{W}}_{_{\mathrm{SLG}}}(\xi ,\kappa ,0) \mathop \rightarrow \limits_{w_{G} \to \infty} {\mathcal{W}}_{_{\mathrm{SL}}}(\xi ,\kappa ,0)=\frac{|\varphi _0|^2w_1^3w_2^3}{8(w_1^2-w_2^2)(w_s^2+\xi ^2)(w_d^2+\xi ^2)}[{\mathcal{L}}(\xi ,\kappa ,w_1,w_2)-{\mathcal{L}}(\xi ,\kappa ,w_2,w_1)],$$

where $\mathcal{L}(\xi ,\kappa ,w_1,w_2)$ strictly resembles the form of the WDF (10) of the pure Lorentz beam, being

$${\mathcal{L}}(\xi ,\kappa ,w_1,w_2)=\frac{w_2{\text{e}}^{-2|\kappa |w_1}}{(w_1^2+\xi ^2) }[a(\xi ,w_1,w_2)\cos (2\xi |\kappa |)+2|\kappa |w_1b(\xi ,w_1,w_2)\hbox {sinc }(2\xi |\kappa |)]$$

and

$$a(\xi ,w_1,w_2)=\frac 14(5w_1^2-w_2^2)-\xi ^2,\,\,\,\,\,\,b(\xi ,w_1,w_2)=\frac 14(w_1^2-w_2^2)-2\xi ^2.$$

Use of the result (9), in relation to both $\mathcal{E}_1(\xi ,\kappa )$ and $\mathcal{E}_2(\xi ,\kappa )$, has been done in the above limit.

As said, the obtained expression reproduces, even if through a more involved structure, the basic form of the Lorentz-beam WDF (10), which is in fact recovered in the limit for $w_1\rightarrow \infty$ or $w_2\rightarrow \infty$.

For completeness’ sake, we finally report the expression of the WDF of a super-Lorentz beam (with M = 2) in the case that $w_1=w_2\equiv w_{_{\mathrm{L}}}$,

$$\begin{aligned} {\mathcal{W}}_{\rm SL}(\xi ,\kappa ,0)&=\frac{|\varphi_0|^2w_{\rm L}^3{\text{e}}^{-2|\kappa |w_{\rm L}}}{16(1+\xi ^2/w_{\rm L}^2)^3\xi^2} \left\{ \left[\frac{5\xi ^2}{w_{\rm L}^2}+\frac{\xi^4}{w_{\rm L}^4}+\frac{2\xi^4 |\kappa|}{w_{\rm L}^3} -2|\kappa |w_{\rm L}\right]\cos (2\xi |\kappa |)\right.\\ &\quad\left.+2|\kappa |w_{\rm L}\left[1+\frac{5\xi^2}{w_{\rm L}^2}+\frac{4\xi^2|\kappa|}{w_{\rm L}}+\frac{4\xi^4|\kappa|} {w_{\rm L}^3}\right]\hbox{sinc}(2\xi |\kappa |)\right\}. \end{aligned}$$

Figures 2 and 3 closes the paper. It shows the Wigner charts of the WDF for a super-Lorentz–Gauss beam for some values of the relevant parameters, seen at the planes $\zeta =0$ and $\zeta =1$ after free propagation, displaying in the latter case the expected ξ-shear. According to the values of the inherent parameters, as indicated in the figure caption, we start from a situation where the Lorentzian parts dominate over the Gaussian one to reach, through an intermediate situation where the various components almost equally contribute, the opposite situation where the Gaussian component tends to dominate over the Lorentzian ones.

Fig. 2

a, b Wigner charts of the WDFs, $\mathcal{W}_{\mathrm{L}}$ and $\mathcal{W}_{\mathrm{LG}}$, of a Lorentz and a Lorentz-Gauss wavefunction, respectively, with w_L = 1, and w_L = 1, β = 0.5, c, d respective negative-valued portions over the (ξ,κ)-plane, with minimum values of −2 × 10⁻³ and −10⁻³

Fig. 3

Wigner charts of the WDF $\mathcal{W}_{\mathrm{SLG}}$ of the super-Lorentz-Gauss beam (20) with M = 2 at $\zeta$ = 0 (on the left) and $\zeta$ = 1 (on the light) for a, b β₁ = 0.33, β₂ = 1.7, c, d β₁ = 1.25, β₂ = 0.5, e, f β₁ = 2, β₂ = 4