Nonparaxial Focusing of Partially Coherent Gaussian Schell-Model and Bessel-Correlated Beams in Free Space

Abstract

The nonparaxial focusing of partially coherent beams in free space has been studied using the coherent-state and coherent-mode decomposition methods. Analytical expressions for the width and angular divergence of partially coherent Gaussian Schell-model (GSM) beams have been obtained using the coherent-state method. It has been shown that the focusing plane is shifted in the opposite axial direction compared to the geometric focusing plane. The influence of the nonparaxiality and spatial coherence of Bessel-correlated vortex beams on the intensity distribution and displacement of the focus plane has been analyzed. It has been shown that the shift of the focus plane increases with a decrease in the coherence radius of the source. A smaller diffraction spread has been shown for partially coherent Bessel-correlated beams compared to GSM beams.

1. Introduction

The theory of the propagation of partially coherent waves in free space, as well as in inhomogeneous media, has now been developed quite fully [1,2,3,4,5]. Partially coherent sources can be used to overcome speckles in imaging systems and increase the signal-to-noise ratio in free-space optical communication, remote sensing, etc. Gaussian Schell-model (GSM) beams are familiar partially coherent sources which represent natural sources [1]. The properties of these beams were studied extensively until recently [6,7,8,9,10,11,12,13]. Partially coherent sources with specific propagation properties that lead to the formation of highly directional light beams were considered in the last two decades [14,15,16,17,18,19,20,21,22,23].

Tightly focused beams find important applications in optical data storage, microscopy, lithography, particle trapping, etc. The focusing of partially coherent GSM and Laguerre–Gaussian-correlated Schell-model beams was studied theoretically and experimentally [24,25,26,27,28,29,30]. The focusing properties of partially coherent beams with specific correlation functions were also investigated. In [31], the focusing of partially coherent Bessel–Gaussian beams was considered.

The focusing of a Gaussian beam in free space has been long and well studied using various methods, including analytical, asymptotic, and numerical approaches to integration. Usually, the problem is solved by calculation of diffraction integrals. However, the numerical simulation of the diffraction integrals is often time-consuming, and asymptotic methods have been developed for the calculations. A hybrid integration method combining numerical integration with asymptotic methods was proposed in [32]. In [33], the accuracy and computational savings of this hybrid technology were examined. Because of the existence of high-frequency oscillatory terms in the Rayleigh–Sommerfeld integral, the calculations are usually performed numerically on a submicron scale. Therefore, other time-saving modeling approaches are needed to facilitate the solution of the problem. Coherent-mode decomposition is a powerful method to simulate the propagation of partially coherent beams [34]. Another effective analytical method for studying the propagation and focusing of light beams is the quantum theoretical method of coherent states [35,36,37]. Density matrix formalism is useful for describing partially coherent sources in inhomogeneous media [38]. Most publications are devoted to the analysis of beam characteristics in paraxial approximation. However, it is well known that nonparaxial effects significantly influence the characteristics of a tightly focused beam [23,39,40,41,42,43,44].

While the focusing of Gaussian Schell-model beams is well studied [27,28,29,30], the nonparaxial focusing of Bessel-correlated beams has not been considered yet.

In this paper, the nonparaxial focusing and spreading of partially coherent Gaussian Schell-model and Bessel-correlated vortex beams in free space are investigated using the coherent-state and -mode decomposition methods. The influence of the effects of nonparaxiality on the field intensity distributions in the axial and radial directions near the focusing plane is analyzed.

2. Focusing of Gaussian Schell-Model Beam

The cross-spectral density (CSD) function of the Gaussian Schell-model beam is given by [1]

$$\Gamma (x_1,x_2,0)=I_0\mathit{exp}\left\{-\frac{x_1^2+x_2^2}{a^2}-\frac{{\left(x_1-x_2\right)}^2}{r_0^2}-\frac{i{\alpha }_0}{2}\left(x_2^2-x_1^2\right)\right\}$$

(1)

where a is the width of the beam, $r_0$ is the coherence length, ${\alpha }_0=k/R_f$, $k=2\pi /\lambda$ is the wavenumber, and $R_f$ is the wavefront curvature radius.

Below we consider the coherence function (1) in the coherent-state (CS) representation [35,36,37,38,39,40]:

$$\Gamma \left(\alpha ,\beta ,0\right)=⟨\alpha |\widehat{\Gamma }|\beta ⟩=\iint ⟨\alpha |x⟩⟨x|\widehat{\Gamma }|x’⟩⟨x’|\beta ⟩dxdx’$$

(2)

The coherent state |α⟩ can be defined as an eigenfunction of the annihilation operator $\stackrel{̑}{a}$:

$$\stackrel{̑}{a}\left|\alpha \right.⟩=\alpha \left|\alpha \right.⟩,$$

where $\stackrel{̑}{a}=\frac{\widehat{x}}{w_0}+i\frac{k{w}_0}{2}{\widehat{p}}_x,{\stackrel{̑}{p}}_x=-\frac{i}{k}\frac{\partial }{\partial x}.$

The explicit form of CS is given by a Gaussian beam function

$$\left|\alpha \right.⟩={\left(\frac{2}{\pi w_0^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathit{exp}\left(-\frac{x^2}{w_0^2}+\frac{2x}{w_0}\alpha -\frac{{\alpha }^2}{2}-\frac{{\left|\alpha \right|}^2}{2}\right),$$

(3)

where the complex eigenvalues $\alpha =\frac{x_0}{w_0}+i\frac{k{w}_0}{2}{p}_{x0}=\left|\alpha \right|\mathit{exp}(i\vartheta )$ determine the initial coordinates x₀ of the center of the elementary beam and the angle of its inclination p_x0 = n₀sinθ₀ to the z-axis, n₀ is the refractive index of the medium, and w₀ is the elementary beam width.

The coherent states are the generating functions for the modes:

$${\Psi }_{\alpha }\left(x\right)=\left|\alpha \right.⟩=\mathit{exp}\left(-\frac{{\left|\alpha \right|}^2}{2}\right){\sum }_m\frac{{\alpha }^m}{{\left(m!\right)}^{1/2}}\left|m\right.⟩$$

(4)

where $|m⟩={\Psi }_m\left(x\right)={\left(\frac{k\omega }{\pi }\right)}^{1/4}\frac{1}{\sqrt{2^{n}m!}}exp\left(-\frac{k\omega }{2}{x}^2\right)H_m(\sqrt{k\omega }x)$ is the Gauss–Hermite function.

Note that the Gauss–Hermite functions are the modal solutions of the Helmholtz equation describing the propagation of light in a planar graded-index waveguide with the refractive index in the transverse direction $n^2\left(x\right)=n_0^2-{\omega }^2{x}^2$, where $n_0$ is the refractive index on the axis and $\omega$ is the gradient parameter.

The average of any beam value $\widehat{M}$ can be obtained using the following expression:

$$⟨M⟩=\frac{Tr\left(\Gamma M\right)}{Tr\left(\Gamma \right)}=\frac{\int ⟨\alpha |\widehat{\Gamma }|\beta ⟩⟨\beta |\widehat{M}|\alpha ⟩d^2\alpha d^2\beta }{\int ⟨\alpha |\widehat{\Gamma }|\beta ⟩d^2\alpha d^2\beta }$$

(5)

where $d^2\alpha =d\left(Re\alpha \right)d\left(Im\alpha \right)$, and Tr denotes the trace.

2.1. Beam Width

The average beam width is determined by

$$\Delta x^2=〈{\widehat{x}}^2〉-{〈\widehat{x}〉}^2=\iint d^2\alpha d^2\beta ⟨\alpha |\widehat{\Gamma }|\beta ⟩⟨\beta |{\widehat{x}}^2|\alpha ⟩/\iint d^2\alpha d^2\beta ⟨\alpha |\widehat{\Gamma }|\beta ⟩⟨\beta |\alpha ⟩$$

(6)

where $\widehat{x}=\frac{1}{\sqrt{2k\omega }}\left(\widehat{a}+{\widehat{a}}^{+}\right)$; ${\widehat{x}}^2=\frac{1}{2k\omega }\left(1+2{\widehat{a}}^{+}\widehat{a}+{\widehat{a}}^2+{\widehat{a}}^{+2}\right)$, $\widehat{a}{\widehat{a}}^{+}={\widehat{a}}^{+}\widehat{a}+1$;

$$⟨\beta |{\widehat{x}}^2\left(z\right)|\alpha ⟩=\frac{1}{2k\omega }\left\{1+e^{-2i\Omega z}exp\left[\left(e^{-2\chi }-1\right){\beta }^{*}\alpha \right]e^{-\chi }{\alpha }^2+e^{2i\Omega z}exp\left[\left(e^{2\chi }-1\right){\beta }^{*}\alpha \right]e^{\chi }{{\beta }^{*}}^2+{\beta }^{*}\alpha \right\}⟨\beta |\alpha ⟩,$$

(7)

$$\chi =\frac{i{\omega }^2}{k{n}_0^3}z,⟨\beta |\alpha ⟩=exp\left\{-\frac{{\left|\alpha \right|}^2}{2}-\frac{{\left|\beta \right|}^2}{2}+\alpha {\beta }^{*}\right\},$$

$$⟨\alpha |\widehat{\Gamma }|\beta ⟩=I_0{\left(\frac{k\omega }{\pi }\right)}^{1/2}exp\left\{-\frac{{\alpha }^{*2}+{\left|\alpha \right|}^2+{\beta }^2+{\left|\beta \right|}^2}{2}\right\}\sqrt{\frac{\pi }{\tilde{p}}}exp\left[\frac{{\check{q}}^2}{4\tilde{p}}\right]\sqrt{\frac{\pi }{p}}exp\left[\frac{k\omega {\alpha }^{*2}}{2p}\right],$$

(8)

where $p=\frac{k\omega }{2}+\frac{1}{a^2}+\frac{1}{r_0^2}-\frac{i{\alpha }_0}{2},$ $\tilde{p}=\frac{k\omega }{2}+\frac{1}{a^2}+\frac{1}{r_0^2}+\frac{i{\alpha }_0}{2}-\frac{1}{p{r}_0^4},$

$$\tilde{q}=-\sqrt{2k\omega }\beta -\frac{\sqrt{2k\omega }{\alpha }^{*}}{p{r}_0^2},w_0^2=\frac{2}{k\omega }.$$

Calculating the integral (6), we obtain

$${\Delta x}^2=\frac{1}{2k\omega }\left\{1+\frac{2\left(\gamma -{\gamma }^2+4\xi \eta \right)}{{\left(1-\gamma \right)}^2-4\xi \eta }-\frac{2\xi {\left[{\left(1-\gamma \right)}^2-4\xi \eta \right]}^{1/2}\exp \left[-i\left(\frac{2\omega }{n_0}+\frac{3{\omega }^2}{k{n}_0^3}\right)z\right]}{{\left[1-4\xi \eta \exp (2\chi )+{\gamma }^2\exp (2\chi )-2\gamma \exp (\chi )\right]}^{3/2}}-\frac{2\eta {\left[{\left(1-\gamma \right)}^2-4\xi \eta \right]}^{1/2}\exp \left[i\left(\frac{2\omega }{n_0}+\frac{3{\omega }^2}{k{n}_0^3}\right)z\right]}{{\left[1-4\xi \eta \exp (-2\chi )+{\gamma }^2\exp (-2\chi )-2\gamma \exp (-\chi )\right]}^{3/2}}\right\},$$

(9)

where $\eta =\left[\frac{1}{2}-\frac{k\omega }{2}\frac{\kappa }{{\left|\kappa \right|}^2-1/r_0^4}\right]$, $\xi ={\eta }^{\mathrm{*}}$, $\kappa =\frac{k\omega }{2}+\frac{1}{a^2}+\frac{1}{r_0^2}-\frac{i{\alpha }_0}{2}$, $\gamma =\frac{k\omega }{r_0^2\left({\left|\kappa \right|}^2-1/r_0^4\right)}$,

$$\chi =-2i\frac{{\omega }^2}{k{n}_0^3}z,w_0^2=\frac{2}{k\omega }$$

Note that the third and fourth terms on the right side of Equation (9) are complex values. However, these two values are complex conjugate quantities, so the sum of these two values is a real quantity. If we put $n_0$ = 1 and $\omega \to 0$, then the expressions obtained will describe the focusing of the light beam in free space.

Figure 1 shows the change in beam width during focusing for different coherence lengths of the incident beam at a wavefront curvature radius $R_f=-100$ μm. Here and below on the graphs, the wavelength λ = 0.63 μm.

It follows that the beam width in the plane of focus increases with decreasing coherence length. The beam width is 0.72 μm for $r_0=100\mathsf{\mu }\mathrm{m}$, 1.547 μm for $r_0=10\mathsf{\mu }\mathrm{m}$, and 2.704 μm for $r_0=5\mathsf{\mu }\mathrm{m}$. In addition, the focus plane shifts $\delta z=0.5\mathsf{\mu }\mathrm{m}$ at $r_0=100\mathsf{\mu }\mathrm{m}$, $\delta z=4\mathsf{\mu }\mathrm{m}$ at $r_0=10\mathsf{\mu }\mathrm{m}$, and $\delta z=14\mathsf{\mu }\mathrm{m}$ at $r_0=5\mathsf{\mu }\mathrm{m}$, which occurs due to the effects of nonparaxiality and coherence.

In Figure 2, the spreading of a partially coherent focused beam with $R_f=\infty$ (plane wavefront) is shown for different initial coherence lengths. In contrast to Figure 1, where beam spreading occurs due to the wavefront curvature and diffraction, here only diffraction affects the beam width.

The beam width increases faster with distance with a shorter coherence length of the incident beam. In addition, the refractive index of the medium affects the spreading of the beam (Figure 2b). The spreading of the beam weakens with an increase in the refractive index of the medium.

2.2. Angular Divergence

The angular divergence of the partially coherent Gaussian Schell-model beam (1) in the initial plane z = 0 can be represented in the form

$${\left(\Delta \theta \right)}^2={\Delta p}^2=\frac{1}{k^2}\left(\frac{1}{a_0^2}+\frac{2}{r_0^2}+\frac{k^2{a}_0^2}{4R_f^2}\right),$$

(10)

where $a_0$ is the beam radius, $r_0$ is the coherence radius, and $R_f$ is the wavefront curvature radius.

In Figure 3, the angular divergence dependence on the beam waist width is presented for different coherence lengths.

It can be seen that the angular divergence of the GSM beam is higher for a source with less coherence.

The change in the angular divergence with the distance is determined by the formula

$$\Delta p^2=〈{\widehat{p}}^2〉-{〈\widehat{p}〉}^2=\frac{Tr\left({\Gamma }_0{p}^2\left(z\right)\right)}{Tr{\Gamma }_0}-\frac{{Tr}^2\left({\Gamma }_{0}p\left(z\right)\right)}{Tr{\Gamma }_0},$$

(11)

where $\widehat{p}=-i\sqrt{\frac{\omega }{2k}}\left(\widehat{a}-{\widehat{a}}^{+}\right)$; ${\widehat{p}}^2=\frac{\omega }{2k}\left(1+2{\widehat{a}}^{+}\widehat{a}-{\widehat{a}}^2-{\widehat{a}}^{+2}\right)$.

Calculating the traces in (11), we obtain

$${\Delta p}^2=\frac{\omega }{2k}\left\{1+\frac{2\left(\gamma -{\gamma }^2+4\xi \eta \right)}{{\left(1-\gamma \right)}^2-4\xi \eta }+\frac{2\xi {\left[{\left(1-\gamma \right)}^2-4\xi \eta \right]}^{1/2}\exp \left[-i\left(\frac{2\omega }{n_0}+\frac{3{\omega }^2}{k{n}_0^3}\right)z\right]}{{\left[1-4\xi \eta \exp (2\chi )+{\gamma }^2\exp (2\chi )-2\gamma \exp (\chi )\right]}^{3/2}}+\frac{2\eta {\left[{\left(1-\gamma \right)}^2-4\xi \eta \right]}^{1/2}\exp \left[i\left(\frac{2\omega }{n_0}+\frac{3{\omega }^2}{k{n}_0^3}\right)z\right]}{{\left[1-4\xi \eta \exp (-2\chi )+{\gamma }^2\exp (-2\chi )-2\gamma \exp (-\chi )\right]}^{3/2}}\right\}$$

(12)

2.3. Uncertainty Relationship

The threshold for the maximum focusing of the light is determined by the following uncertainty relationship:

$${\Delta x}^2\ge \frac{1}{4k^2{\Delta p}^2},$$

(13)

where the angular divergence $\Delta p$ is given by (12).

Relationship (13) is valid for a beam with a flat wavefront. In the case of beams with a spherical wavefront, the following uncertainty relationship can be obtained [39,40]:

$$R_f^2\ge \frac{{\Delta x}^4}{{\Delta x}^2{\Delta p}^2-1/4k^2}$$

(14)

and because $\Delta p\le 1$, $R_f\ge \Delta x$, i.e., the minimum value of the wavefront curvature radius is diffraction-limited.

This indicates that the numerical aperture (convergence angle) is also diffraction-limited.

Relationship (14) can be obtained from the following Schrodinger–Robertson uncertainty relation [42]:

$${\Delta x}^2{\Delta p}^2-{\sigma }_{xp}^2\equiv {\sigma }_x^2{\sigma }_p^2\left(1-r^2\right)\ge 1/4k^2,$$

(15)

where $r=\frac{{\sigma }_{xp}}{{\sigma }_x{\sigma }_p}$, $R_f=-\frac{{\sigma }_x^2}{{\sigma }_{xp}}$, ${\sigma }_{xp}=\frac{1}{2}\left(〈xp+px〉\right)-〈x〉〈p〉$,

$${\sigma }_{xp}=\frac{1}{2}〈xp+px〉=-\frac{i}{2k}〈{\widehat{a}}^2-{\widehat{a}}^{+2}〉,〈x〉〈p〉=0,$$

$${\sigma }_{xp}=\frac{i}{2k}\left\{\frac{2\xi {\left[{\left(1-\gamma \right)}^2-4\xi \eta \right]}^{1/2}\mathit{exp}\left[-i\left(\frac{2\omega }{n_0}+\frac{3{\omega }^2}{k{n}_0^3}\right)z\right]}{{\left[1-4\xi \eta \mathit{exp}(2\chi )+{\gamma }^2\mathit{exp}(2\chi )-2\gamma \mathit{exp}(\chi )\right]}^{3/2}}-\frac{2\eta {\left[{\left(1-\gamma \right)}^2-4\xi \eta \right]}^{1/2}\mathit{exp}\left[i\left(\frac{2\omega }{n_0}+\frac{3{\omega }^2}{k{n}_0^3}\right)z\right]}{{\left[1-4\xi \eta \mathit{exp}(-2\chi )+{\gamma }^2\mathit{exp}(-2\chi )-2\gamma \mathit{exp}(-\chi )\right]}^{3/2}}\right\}$$

(16)

In Figure 4, the dependence of the uncertainty values on the propagation distance is shown.

The uncertainty changes with distance, taking a minimum value in the plane of focus.

3. Focusing of Bessel-Correlated Beams

Consider now a partially coherent Bessel-correlated vortex beam with the wavefront curvature radius R_f in the plane z = 0 [23]:

$$\Gamma \left(r_1,r_2\right)=\frac{A{\xi }^{-l/2}}{(1-\xi )}exp\left[-\frac{1+\xi }{1-\xi }\frac{r_1^2+r_2^2}{2w^2}-\frac{i{k}_{}}{2R_f}(r_2^2-r_1^2)\right]I_l\left(\frac{4\sqrt{\xi }}{1-\xi }\frac{r_1{r}_2}{w^2}\right)exp\left[-il({\phi }_1-{\phi }_2)\right],$$

(17)

where $\xi =\frac{r_0^4}{w^4}{\left[{\left(1+\frac{w^4}{r_0^4}\right)}^{1/2}-1\right]}^2$, $w$ is the spot size at the waist of the beam, $r_0$ is the coherence length, $R_f$ is the wavefront curvature, and l is the topological charge.

Note that when $r_0\to \mathrm{\infty }$ (fully coherent case) $\xi \to 0$, and when $r_0\to 0$ (completely incoherent case) $\xi \to 1$.

Unlike the CSD proposed in [15], here we have introduced a new parameter—the wavefront curvature radius R_f. In contrast to the conventional LG modes, here we consider the generalized Laguerre–Gauss modes with spherical wavefronts as the eigenfunctions

$${\Psi }_{nl}\left(r,\phi \right)=\frac{1}{w}{\left(\frac{2n!}{\pi (n+l)!}\right)}^{1/2}{\left(\frac{\sqrt{2}r}{w}\right)}^l{e}^{-r^2/w^2-i{k}_{}{r}^2/2R_f}{L}_n^l\left(\frac{2r^2}{w^2}\right)e^{il\phi }.$$

(18)

The CSD (17) can be represented in the form of coherent mode decomposition as [23]

$$\Gamma (r_1,r_2,0)={\sum }_n{\lambda }_n{\Psi }_n^{*}\left(r_1\right){\Psi }_n\left(r_2\right)={\sum }_n{\lambda }_n{\sum }_{pp’}{a}_p^{*}{a}_{p’}{\psi }_p^{*}{\psi }_{p’}$$

(19)

where ${\lambda }_n=\frac{n!}{(n+l)!}{\xi }^n$; $\xi =\frac{r_0^4}{w^4}{\left[{\left(1+\frac{w^4}{r_0^4}\right)}^{1/2}-1\right]}^2$; ${\psi }_{pl}\left(r,\phi \right)=\frac{J_l\left({\mu }_{pl}\frac{r}{R_0}\right)\mathit{exp}(il\phi )}{\left(\sqrt{\pi }{R}_0{J}_{l+1}\left({\mu }_{pl}\right)\right)}$;

$$a_p=B_0{(-1)}^n{\left(\frac{{\mu }_{pl}}{2R_0}\right)}^l\frac{q^{*n}}{q^{n+l+1}}exp\left(-\frac{{\mu }_{pl}^2}{4R_0^{2}q}\right)L_n^l\left(\frac{{\mu }_{pl}^2}{2{w^{2}R}_0^2{\left|q\right|}^2}\right);B_0={\left(\frac{2}{w^2}\right)}^{(l+1)/2}\sqrt{\frac{n!}{(n+1)!}}\frac{1}{R_0{J}_{l+1}\left({\mu }_{pl}\right)};$$

$q=\frac{1}{w^2}+\frac{i{k}_{}}{2R_f}$; $q^{*}=\frac{1}{w^2}-\frac{i{k}_{}}{2R_f}$; and ${\mu }_{pl}$ are the positive zeros of the Bessel function $J_l\left(z\right)$.

The evolution of the modified CSD (17) can be presented as

$$\Gamma \left(r_1,r_2,z\right)={\sum }_{nl}{\lambda }_{nl}{\sum }_{pp’}{c}_{pl}{c}_{p^{’}l}{J}_l({\mu }_p{r}_1/R_0)J_l({\mu }_{p^{’}}{r}_2/R_0)e^{i({\beta }_{pl}-{\beta }_{p^{’}l})z},$$

(20)

where $c_{pl}=\frac{a_p}{\sqrt{\pi }{R}_0{J}_{l+1}\left({\mu }_{pl}\right)}$, ${\beta }_{pl}=k_{}{\left[1-{\left(\frac{{\mu }_{pl}}{k_{}{R}_0}\right)}^2\right]}^{1/2}$.

3.1. Average Beam Radius

The average value of the beam radius can be expressed as the mean square deviation $\Delta r$:

$$\Delta r^2=〈{\widehat{r}}^2〉-{〈\widehat{r}〉}^2=\frac{Tr\left({\Gamma }_0{r}^2\left(z\right)\right)}{Tr{\Gamma }_0}-\frac{{Tr}^2\left({\Gamma }_{0}r\left(z\right)\right)}{Tr{\Gamma }_0}$$

(21)

Due to the axial symmetry of the incident beam, the average value ${〈\widehat{r}〉}^2=0$.

The average value $〈{\widehat{r}}^2〉$ is determined by the expression

$$〈r^2\left(z\right)〉=\frac{Tr\Gamma r^2}{Tr\Gamma }={\sum }_{pp’}⟨{\psi }_p|\widehat{\Gamma }|{\psi }_{p’}⟩⟨{\psi }_{p’}|r^2|{\psi }_p⟩,$$

(22)

where $\widehat{\Gamma }={\sum }_n{\lambda }_n\left({\sum }_p{a}_p|J_l({\mu }_{p}r/R_0)⟩e^{i{\beta }_{p}z}\right)\left({\sum }_{p\mathrm{’}}{a}_{p\mathrm{’}}^{\mathrm{*}}⟨J_l({\mu }_{p^{\mathrm{’}}}r/R_0)|e^{-i{\beta }_{p\mathrm{’}}z}\right)$,

$$\begin{gathered} ⟨{\psi }_{p’}|r^2|{\psi }_p⟩={\int }_0^{R_0}{r}^2{J}_0({\mu }_{p}r/R_0\left)J_0\right({\mu }_{p’}r/R_0)rdr=\frac{4{\mu }_p{\mu }_{p’}}{{\Delta }^2}\left[J_1\left({\mu }_p\right)J_1\left({\mu }_{p’}\right)\right],\mathrm{if}l=0, \\ ⟨{\psi }_{p’}|r^2|{\psi }_p⟩={\int }_0^{R_0}{r}^2{J}_1({\mu }_{p}r/R_0\left)J_1\right({\mu }_{p’}r/R_0)rdr=\frac{2R_0^2}{{\Delta }^2}\left[\sigma J_1\left({\mu }_p\right)J_1\left({\mu }_{p’}\right)\right],\mathrm{if}l=1, \end{gathered}$$

$\mathsf{\Delta }={\left(\frac{{\mu }_p}{R_0}\right)}^2-{\left(\frac{{\mu }_{p’}}{R_0}\right)}^2$, $\sigma ={\left(\frac{{\mu }_p}{R_0}\right)}^2+{\left(\frac{{\mu }_{p’}}{R_0}\right)}^2$. Analytical expressions can also be obtained for $l>1$. Note that expression (22) describes the nonparaxial radius of the beam when the propagation constants are given by ${\beta }_{pl}=k{\left[1-{\left(\frac{{\mu }_{pl}}{k{R}_0}\right)}^2\right]}^{1/2}$. Expression (22) describes the paraxial radius of the beam when the propagation constants are given by ${\beta }_{pl}=k\left[1-\frac{1}{2}{\left(\frac{{\mu }_{pl}}{k{R}_0}\right)}^2\right]$.

In Figure 5, the dependences of the nonparaxial (1) and paraxial (2) beam radii $w=\Delta r$ on the propagation distance are presented.

It can be seen that the focusing plane corresponding to the minimum value of the beam radius is shifted in the opposite axial direction compared to the geometric focusing plane $z_{fg}=R_f$, i.e., the distance to the focusing plane decreases with decreasing coherence radius. The focus plane shift and the average beam radius increase with a decrease in the radius of coherence. Note that the focusing plane in the paraxial approximation coincides with the geometric focusing plane.

3.2. Diffraction Spreading

In Figure 6, the diffraction spreading values of the beam radii of the focused plane wavefront GSM and the Bessel-correlated beams with distance are shown.

It can be seen that a partially coherent Bessel-correlated beam has less beam spread than the Gaussian Schell-model beam. This advantage of Bessel-correlated beams can be useful for free-space optical communication.

3.3. Beam Intensity Profile

The intensity distribution is determined with the following expression (20):

$$I\left(r,z\right)=\Gamma \left(r,r,z\right)={\sum }_{nl}{\lambda }_{nl}{\Psi }_{nl}^{*}{(r,z)\Psi }_{nl}(r,z)={\sum }_n{\lambda }_n{\sum }_{pp’}{a}_p^{*}{a}_{p’}{\psi }_p^{*}{\psi }_{p’}{e}^{i({\beta }_{pl}-{\beta }_{p^{’}l})z}$$

(23)

This expression describes the effects of nonparaxiality when the propagation constants are given by ${\beta }_{pl}=k{\left[1-{\left(\frac{{\mu }_{pl}}{k{R}_0}\right)}^2\right]}^{1/2}$ and it does not take into account nonparaxial effects when the propagation constants are given by ${\beta }_{pl}=k\left[1-\frac{1}{2}{\left(\frac{{\mu }_{pl}}{k{R}_0}\right)}^2\right]$.

In Figure 7, the intensity distributions along the propagation axis $I\left(0,z\right)=\Gamma (\mathrm{0,0},z)$ are presented for nonparaxial and paraxial beams with different coherent radii and $R_f=-100\mathsf{\mu }\mathrm{m}$, $w=30\mathsf{\mu }\mathrm{m}$. The simulations show that intensity oscillations in front of the focus plane occur for coherent radiation (Figure 7a). However, these oscillations disappear if the coherence radius becomes smaller than the incident beam radius (Figure 7b). Note that there are no oscillations in the paraxial approximation. This indicates that nonparaxiality causes oscillations in the intensity distribution. However, the oscillations disappear for sources with low coherence (Figure 7b, curve 1). The focus plane shift increases, and the intensity decreases with a decrease in the radius of coherence.

In Figure 8, the intensity distributions in the initial plane and the focus plane are presented for a Bessel-correlated beam with different coherence radii and topological charge l = 0.

In the initial plane $z=0$, the intensity distribution becomes Gaussian when the radius of coherence $r_0\to \infty$, i.e., the source is completely coherent. The intensity distribution at $r_0\to 0$ (a completely incoherent source) becomes asymmetric with a long tail whose length increases with decreasing coherence radius.

In Figure 9, the intensity distributions in the focal plane are presented for a Bessel-correlated vortex beam with l = 1.

An increase in the coherence radius leads to an annular distribution with sharp boundaries (Figure 9a,b). The thickness of the ring is equal to FWHM = 0.86 μm in the focus plane (Figure 9b). Long tails in the intensity distribution appear with a decrease in the coherence radius (Figure 9c,d). In this case, the value FWHM = 0.78 μm in the focus plane. Although the bright central part of the ring is smaller in the case of a low coherence beam, a significant amount of the power of the focused beam is outside this bright ring.

4. Conclusions

Thus, the nonparaxial focusing of partially coherent Gaussian Schell-model and Bessel-correlated vortex beams in free space has been investigated using the coherent-state and the mode decomposition methods.

It is found that the average radii in the focusing plane increase with decreasing coherence radius. However, calculations of the intensity profiles show that bright spots and rings of smaller sizes can be obtained using the sources with low-coherence Bessel-correlated sources. The beam width estimated on the basis of the intensity distribution, i.e., the FWHM (full width at half maximum) value, shows a behavior different from that based on the calculation of the standard deviation. Indeed, the FWHM value of the focusing spot of a low-coherent light beam determined by the intensity distribution can be smaller than that of a coherent light beam. It is shown that the effect of shifting the focusing plane in the axial direction opposite to the geometric focusing plane is enhanced with a decrease in coherence.

It follows from the simulation results that the partially coherent Bessel-correlated beams have less spread than the Gaussian Schell-model beams. This preferred property can be useful in specific fields, especially in optical communication in free space.

Future research can be related to the consideration of focusing partially coherent and partially polarized vector beams [45]. Of practical interest is the consideration of structured vortex beams with spin angular momentum (SAM) and orbital angular momentum (OAM) [46,47,48], partially coherent accelerating beams [49], beams possessing self-healing abilities in a turbulent media [50], and spatiotemporal optical vortex beams [51].

In summary, the nonparaxial focusing of the GSM beam has been studied using the CS decomposition method. The nonparaxial focusing and spreading of a Bessel-correlated beam in free space were analyzed using the coherent-mode decomposition method. It is shown that partially coherent Bessel-correlated beams have a smaller spatial spread than GSM beams. The average spot size in the focusing plane increases with a decrease in the degree of coherence. However, the FWHM value of the focusing spot of a low-coherence light beam, determined by the intensity distribution, can be less than that of a coherent light beam. It is shown that the shift of the focusing plane increases with decreasing coherence.

The results obtained can be useful for microparticle trapping, transmitting information, optical visualization, and optical communication in free space.