Self-focusing propagation characteristics of a radially-polarized beam in nonlinear media

Lu Lu; Lu Lu; Zhiqiang Wang; Zhiqiang Wang; Zhiqiang Wang; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.456430

1. Introduction

In recent decades, filamentation propagation in transparent media [1], which can generate long plasma channels with very high intensities, has sparked extensive interest owing to its wide application in many fields [2–5], such as lighting control, remote diagnostics, and LIDAR. The self-focusing length is described as the self-focusing effect that overcomes diffraction to a collapse of the beam on itself [1], which may play a crucial role in the steerable manipulation of the filament length [6–10]. To date, the known methods for controlling the position of the filamentation domain are as follows: modulating the laser pulse power [6], adjusting the divergence angle of the initial laser [7], launching negatively chirped ultrashort pulses [8,9], and double-lens setup [10]. Recently, the U.S. Naval Research Laboratory evaluated the self-focusing length in atmospheric turbulence using theoretical and numerical methods [11,12].

Since 1961, the vector optical field [13] with a spatially inhomogeneous SoP at the field cross-section has attracted considerable attention, which can be widely used in nonlinear optics [14], optical tweezers [15], optical micromanipulation [16], super-resolution microscopy [17], optical information transmission [18], and atmospheric propagation [19–24]. In recent years, optical arrangements containing spatial light modulators have been widely presented [25–37]. As a typical type of cylindrical vector beam, the radially polarized beam (RPB) has been studied extensively both theoretically and experimentally owing to its interesting and unique focusing properties [38–41]. Although these studies have been extensive and seem comprehensive, nonlinear effects can significantly affect the essence of RPB propagation; particularly, the Kerr effect strongly exists when an intense laser beam is present in nonlinear media.

It is well known that the self-focusing length of a Gaussian beam can be described by the semi-empirical Marburger formula [42–43]. However, there is currently no analytical formula for the majority of laser beams (i.e., RPB or Gaussian) to quantitatively characterize the relationship between the self-focusing length and input power, which is worthy of further exploration. The self-focusing length of an RPB may spark extensive interest owing to its wide application in many fields, such as laser filamentation [10], high-power atmospheric propagation [10], optical communications [44], optical coherence encryption [45], and far-field imaging [46]. Thus, the evaluation of the self-focusing length of an RPB has potential application prospects. If a laser beam with a spatially inhomogeneous SoP can be used to control the self-focusing length, it may provide an alternative route for manipulating the filamentation domain.

In this study, the analytical formula of the self-focusing length of the RPB was derived, and the correctness was verified by numerical calculation. It was shown that the self-focusing length is inversely proportional to the input power, which is similar to the performance of the semi-empirical formula of the Gaussian beam. Furthermore, the relationship between the SoP at the field cross-section and the self-focusing length was studied. It was found that the RPB has a relatively long self-focusing length compared to a beam with a spatially homogenous SoP. Additionally, the influence of the topological charge on the self-focusing length was evaluated. Based on numerical calculations, it was found that a low topological charge can not only achieve a long self-focusing length but also obtain a high-power density at the target. With the aid of the RPB, a controllable self-focusing length and high optical power density on the target can be achieved, which may have practical application significance in the fine optical manipulation, optical information transmission, high-power long-range atmospheric propagation, and related areas.

2. Theory

We discuss the monochromatic (or continuous) regime, for which the intensity of the laser does not depend on time. The propagation nonlinear dynamics of a high-power laser beam in the Kerr medium were evaluated using nonlinear Schrödinger (NLS) equations. Under the slowly varying amplitude approximation, the NLS equation is [47,48]:

(1)$$2\textrm{i}k\frac{{\partial {\mathbf E}}}{{\partial z}} + {\nabla ^2}{\mathbf E} + \frac{{4{n_2}{k^2}}}{{3{n_0}}}[{2({{\mathbf E}\cdot {{\mathbf E}^ \ast }} ){\mathbf E} + ({{\mathbf E}\cdot {\mathbf E}} ){{\mathbf E}^ \ast }} ]= 0,$$

where ${\mathbf E} = {\mathbf E}({\mathbf r},z)$ is the amplitude of the electric field, the wavenumber related to the wavelength is $k = {{2\pi } / \lambda }$, ${\nabla ^\textrm{2}}$ is the transverse Laplacian, ${n_\textrm{0}}$ (${n_2}$) is the linear (nonlinear) refractive index, $\left\langle \bullet \right\rangle$ denotes the statistical ensemble average, and ${\ast} $ is the complex conjugate.

The SoP distribution of the radially polarized vector field at the initial cross-section is linearly polarized at any position, and the electric field can be expressed as [25,34,36]:

(2)$${\mathbf E}(r,\varphi ,z = 0) = A(r)({\cos \delta {{\vec{{\mathbf e}}}_h} + \sin \delta {{\vec{{\mathbf e}}}_v}} ),$$

where $A({\mathbf r}) = {A_0}({{\mathbf r}/{w_0}} )\exp [{ - {{\mathbf r}^2}/2w_0^2} ]$, ${A_0}$ is the normalized constant, ${w_0}$ is the beam waist associated with the Gaussian beam, $\delta = m\varphi + {\varphi _0}$, $m$ is the topological charge, $\varphi$ is the azimuthal angle, and ${\varphi _0}$ is the initial phase.

With the introduction of polarization, Eq. (1) can be divided into a pair of coupled NLS equations for the two orthogonal components.

(3)$$\begin{array}{l} \textrm{2i}k\frac{{\partial {E_h}}}{{\partial z}} + {\nabla ^2}{E_h} + \frac{{4{n_2}{k^2}}}{{3{n_0}}}[{3{{|{{E_h}} |}^2}{E_h} + 2{{|{{E_v}} |}^2}{E_h} + E_v^2E_h^ \ast } ]= 0\\ \textrm{2i}k\frac{{\partial {E_v}}}{{\partial z}} + {\nabla ^2}{E_v} + \frac{{4{n_2}{k^2}}}{{3{n_0}}}[{3{{|{{E_v}} |}^2}{E_v} + 2{{|{{E_h}} |}^2}{E_v} + E_h^2E_v^ \ast } ]= 0 \end{array}$$

where ${E_h}$ (${E_v}$) is the horizontal or vertical component of an optical field.

Owing to the hollow shape in the initial source plane, the RPB can retain its original shape for long-range propagation, and the electric field can be written as:

(4)$${\mathbf E}(r,z) = {A_z}\left( {\frac{r}{{{w_z}}}} \right)\exp \left( { - \frac{{{r^2}}}{{w_z^2}} + \textrm{i}{r^2}{\varphi_z}} \right)({\cos \delta {{\vec{{\mathbf e}}}_h} + \sin \delta {{\vec{{\mathbf e}}}_v}} )$$

By inserting the initial conditions (beam width ${w_0}$, phase ${\varphi _0} = 0$, and amplitude ${A_0} = 1$) and Eq. (4) into Eq. (3), a set of equations for these quantities is obtained as follows:

(5)$$\frac{{\textrm{d}{w_z}}}{{\textrm{d}z}} = \frac{2}{k}{\varphi _z}{A_z},$$

(6)$$\frac{{\textrm{d}{\varphi _z}}}{{\textrm{d}z}} = \frac{1}{{2kw_z^4}} - \frac{{2\varphi _z^2}}{k} + \frac{{2{n_2}kA_z^2}}{{{n_0}w_z^2}},$$

(7)$$\frac{{\textrm{d}{A_z}}}{{\textrm{d}z}} ={-} \frac{2}{k}{\varphi _z}{A_z}.$$

Combining Eqs. (5) and (6), the dynamics of the beam width of a RPB are

(8)$$\frac{{{\textrm{d}^2}{w_z}}}{{\textrm{d}{z^2}}} = \frac{1}{{{k^2}w_z^3}} + \frac{{4{n_2}A_z^2}}{{{n_0}{w_z}}},$$

then, using ${ {({{{\textrm{d}{w_z}} / {\textrm{d}z}}} )} |_{z = 0}} = 0$, Eq. (8) can also be expressed as,

(9)$${\left( {\frac{{\textrm{d}{w_z}}}{{\textrm{d}z}}} \right)^2} + \left( {\frac{4}{{w_z^2}} - \frac{4}{{w_0^2}}} \right) + \frac{{16{n_2}kA_z^2}}{{{n_0}}}\ln \left( {\frac{{{w_z}}}{{{w_0}}}} \right) = 0.$$

With boundary conditions ${w_{z = 0}} = {w_0}$ and ${ {({{{\textrm{d}{w_z}} / {\textrm{d}z}}} )} |_{z = 0}} = 0$, the analytical expression for the beam width is obtained,

(10)$$w_z^2 = w_0^2 + \frac{{{z^2}}}{{{k^2}w_0^2}} + \frac{{4{n_2}A_z^2{z^2}}}{{{n_0}}}\left( {1 - \ln \frac{{{A_z}}}{{{A_0}}}} \right).$$

In physical terms, the evolution of the beam width is determined by the competition between two factors: the diffraction and nonlinearity of the medium. Here, the focusing case with ${n_2} > 0$ is considered, where ${{{A_z}} / {{A_0}}}$ represents the normalized amplitude.

Substituting the critical power of the Gaussian beam ${P_{cr}} = {{{\varepsilon _0}n_0^2\pi c} / {{n_2}{k^2}}}$ and the input power of the RPB ${P_{in}} = {\varepsilon _0}{n_0}\pi c{I_0}w_0^2$ into Eq. (10), the self-focusing length can be expressed as follows:

(11)$${z_f} = \frac{{{z_R}}}{{\sqrt {[{ - 1 + \ln ({{{{A_z}} / {{A_0}}}} )} ]{{{P_{\textrm{in}}}} / {{P_{\textrm{cr}}}}} - 0.25} }}.$$

where the critical power of the RPB should be satisfied [31]: $P_{cr}^R = 2.4({m^2} + 1){P_{cr}}$. The self-focusing length corresponds to the input power and topological charge.

3. Numerical calculation and analysis

3.1 Comparison of analytical and numerical results

To verify the accuracy of the analytical formula for the self-focusing length of the RPB, a comparison of the analytical formula in Eq. (11), and the numerical result should be obtained firstly. In the numerical calculation, the parameters were chosen as follows: wavelength $\lambda = 0.532$ µm, step number $M = \textrm{2}000$, transverse size $8{w_0}$, grid number $N = 512$, Rayleigh length is ${z_R} = \pi w_0^2/\lambda$, linear refractive index of Kerr medium ${n_0} = 1$, nonlinear refractive index ${n_2} = \textrm{5} \times {10^{ - 23}}$ m²/W, initial phase ${\varphi _0}\textrm{ = 0}$, and the range of input power for PRB is $[27{P_{cr}},50{P_{cr}}]$ (or $[68{P_{cr}},91{P_{cr}}]$) with the topological charge $m = 1$ (or $m = 2$). The numerical results were obtained by the fast Fourier transform Runge-Kutta split-step method [49]. In Fig. 1, a comparison of the numerical calculations shows that the analytical and numerical results are in good agreement. This will facilitate the quantitative and qualitative analysis, which may have a significant theoretical basis in optical field collapse, filament propagation, and related areas. It is well known that the self-focusing length for a Gaussian beam with moderate input powers can be described by the semi-empirical Marburger formula [42–43], $z = {{0.367{z_R}} / {\sqrt {{{[\sqrt {P/{P_{cr}}} - 0.852]}^2} - 0.0219} }}$, and it is obvious that the self-focusing length for the RPB exhibits a similar phenomenon, that is, the self-focusing length is inversely proportional to the input power. The analytical relation between the self-focusing length and input power is beneficial for the selection of the initial parameters for practical applications, and it is convenient to further analyze the physical characteristics in detail.

Fig. 1. ${{{z_f}} / {{z_R}}}$ as a function of the relative input laser power ${P / {{P_{cr}}}}$ for RPB.

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3.2 Self-focusing length with SOP

The self-focusing length can be adjusted by the input power, which is a common method for manipulating the position of the filamentation domain [6]. In practical experimental scenarios, as an alternative or combination regimen to input power manipulation, achieving a controllable self-focusing length remains a great challenge, which may significantly broaden the manipulation dimensionality of the self-focusing length. Owing to the intrinsic nature of light, polarization is important to serve as a degree of freedom for manipulating light. The self-focusing property of the RPB under fixed input power was evaluated with the aid of SoP. The input power $P = 30{P_{cr}}$ is considered as an example, and the extension to other input powers is straightforward because of the deduction of the analytical expression of the self-focusing length for RPB (Eq. (11)). When the initial optical field cross-section has a spatially homogeneous SoP, that is, a hollow Gaussian beam (HGB), there is no analytical expression for the self-focusing length because of the variable propagation shape. Therefore, the numerical calculation mentioned in Section 3.1 is used. In Fig. 2, it is shown that the self-focusing length can be extended to 5.44 times the Rayleigh length, and that of the HGB is only 1.58 times the Rayleigh length, which may have a practical application significance for increasing the self-focusing length when the laser beam has a spatially inhomogeneous SoP at the field cross-section, and it may be a potential feasible solution to fill the shortage up with the method of the input power as well.

Fig. 2. Intensity distribution located at various propagation planes for (a) RPB and (b) HGB, where the corresponding propagation dynamics are (c) and (d), respectively.

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3.3 Self-focusing length with topological charge

In this section, the influence of the topological charge on the self-focusing length is evaluated using Stokes parameters. Herein, the topological charges were selected as $m = 1$ and $m = 2$. At the source plane $z = 0$ with the input power $P = 30{P_{cr}}$, the Stokes parameters for the case of $m = 1$ are shown in Fig. 3(a); as the propagation distance increases, the Stokes parameters are redistributed and the hollow beam converges into four filaments at the location of $z = 5.44{z_R}$ in Fig. 3(b). Similarly, for the $m = 2$ case, the Stokes parameters at the source plane are shown in Fig. 3(c), which requires a much higher input power to collapse, that is, $P = 90{P_{cr}}$, and converges to eight filaments at the plane of $z = 1.30{z_R}$ in Fig. 3(d). In physics, the axial symmetry of the light-matter nonlinear interaction is broken by the designable symmetrical polarization structure at the initial optical field cross-section and the self-focusing property of the propagation medium; there is no circular polarization (${S_3} = 0$) at the source plane in Fig. 3(a) and Fig. 3(c), while the circular polarization states occur on the receiver plane in Fig. 3(b) and Fig. 3(d) because the Stokes parameter ${S_3}$ is not equal to zero. It is clear that the hybrid polarization spatial structure is formed during the propagation process, so the symmetry is broken by refractive index change and self-focusing spots appear subsequently. As shown in Figs. 3(b) and (d), it is observed that a lower topological charge induces a longer self-focusing length; thus, the topological charge can also be regarded as a controllable parameter to manipulate the length of self-focusing. Additionally, the effective radius of the filaments of $m = 1$ is smaller than that of $m = 2$, which indicates that the power density of the lower topological charge is higher. It is possible that an RPB with a low topological charge as a laser source may be of great advantage for achieving long-range and high-power laser atmospheric propagation.

Fig. 3. Stokes parameters with different topological charges $m = 1$ (i.e., (a) $P = 30{P_{cr}}$ at the source plane $z = 0$, (b)$P = 30{P_{cr}}$ at the location of the self-focusing length $z = 5.44{z_R}$) and $m = 2$ (i.e., (c) $P = 90{P_{cr}}$ at the source plane $z = 0$, (d) $P = 90{P_{cr}}$ at the location of the self-focusing length $z = 1.30{z_R}$.

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4. Conclusion and discussion

In summary, an analytical expression of the self-focusing length of the RPB with respect to the input power was obtained. It was demonstrated that the self-focusing length with the input power has a similar performance to the semi-empirical Marburger formula for a Gaussian beam. Furthermore, the influence of the SoPs and topological charge on the self-focusing length were evaluated. It was found that the self-focusing length could be controlled by adjusting the SoP at the initial cross-sectional optical field and topological charge. Additionally, two characteristics of the RPB should be mentioned: (1) the self-focusing length of the RPB appears at a relatively long-range propagation distance compared with the HGB; (2) for a lower topological charge, the target optical power density of the RPB is larger than that of a higher topological charge. Therefore, the RPB as a laser source may not only extend the self-focusing length but also improve the power density of the target. These analytical and numerical findings may open a new window for realizing applications such as the multi-manipulation of optical fields, optical information transmission, and high-power long-range atmospheric propagation.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11804234, 11903062, 11974218, 12192254, 91750201); Innovation Group of Jinan (2018GXRC010); Local Science and Technology Development Project of the Central Government (YDZX20203700001766).

Disclosures

The authors declare that they have no conflicts of interest.

Data availability

The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Self-focusing propagation characteristics of a radially-polarized beam in nonlinear media

Abstract

1. Introduction

2. Theory

3. Numerical calculation and analysis

3.1 Comparison of analytical and numerical results

3.2 Self-focusing length with SOP

3.3 Self-focusing length with topological charge

4. Conclusion and discussion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (3)

Equations (11)

Optics Express