Spherical Gauss-Laguerre beam propagation in 4D space-time

Junhe Zhou; Junhe Zhou; Junhe Zhou; Qingsong Hu; Qingsong Hu

doi:10.1364/OE.504905

1. Introduction

Beams are usually considered as a spatial concept and exist in the spatial coordinates. Recently, the introduction of the time frame in the beam pattern has stimulated the discovery of various novel beams, such as the spatiotemporal optical vortex (STOV) beams, which possess the vortex on the hybrid spatiotemporal plane [1–11].

As is indicated by quite a few pioneering researchers, stable spatiotemporal beam propagation can occur in the media with anomalous dispersion, where the second order derivative with respect to t can be combined with the second order derivative with respect to the spatial coordinates and forms the spatiotemporal Laplacian operator [12–15]. Such treatment enables the discovery of several classes of spatiotemporal beams in the anomalous dispersive media, such as the spatiotemporal 2D Bessel STOV beams [12], the 3D spatiotemporal localized pulse [13,14], and the quasi-non-spreading spatiotemporal beams [15].

The 2D beam propagation in the anomalous dispersive media has been studied by the pioneering researchers [12], and the principle to describe the beam propagation in the x-z-t coordinates is very similar to the beam propagation in the pure 3D spatial coordinates. Therefore, the beam pattern can be directly transferred from the spatial coordinates to the combined spatiotemporal coordinates, which suggests the beam classes in the x-z-t coordinates resemble the conventional spatial beam classes in the x-y-z space, such as the Bessel beams, the Gaussian beams, the Hermite Gaussian beams, the Laguerre Gaussian beams, etc.

Unlike the 3D case, the spatiotemporal beam propagation in the 4D x-y-z-t space-time is based on the 4D wave equation, and the beam pattern is distinctive from those of the conventional spatial beams. For example, S. Longhi suggested the 3D diffraction free beam in the 4D space-time had the shape of the sinc function, which did not have any related counterparts in the 3D space [14]. Apart from the sinc function shaped beam, it was suggested by O. V. Borovkova et. al. that a more general class of quasi-non-spreading beams could be generated from a two-dimensional integral [15].

While many novel spatiotemporal 3D beams in the 4D space-time have been proposed and discussed [13–15], an important problem remains unsolved, i.e., the diffraction free spatiotemporal 3D beams, which have been derived with the close form expressions, have infinite powers [14]. For example, the integration of the power of the sinc function shaped beam results in an unlimited number, which makes the beam non-realizable in practice. The method to generate quasi-non-spreading beams might lead to the beams with limited powers, but the no close form expressions were derived [15]. Researchers might be particularly interested in obtaining a close form Gaussian shaped beam class in the 4D space-time, which is like the Hermite Gaussian beams and the Laguerre Gaussian beams in the 3D space, so that it will be more feasible for generation in practice.

In this work, we tend to derive a novel class of close form Gaussian shaped 3D beams in the 4D space-time, which are referred to as the spherical Gauss-Laguerre beams (indicating its differentiation from the cylindrical Laguerre Gaussian beams), whose expression includes the product of the spherical harmonic function and the spherical Gauss Laguerre function, which make the beams to be with limited powers. The concept of spherical Gauss-Laguerre function was previously proposed and discussed in the molecule science and was used as a basis function to find the solution to the free space Schrodinger equation [16]. Unlike the pure spatial spherical Gauss-Laguerre function in [16], the spherical Gauss-Laguerre beams proposed here evolve with respect to z. Particularly, it has a z dependent 3D beam waist. To the best of the author’s knowledge, such a concept has not been introduced in wave propagation and does not have any related counterparts in the conventional spatial beam studies. The derived expression is verified by the numerical simulations with excellent agreement.

2. Theory

Ignoring the polarization dependent effect, the optical beam propagation in the dispersive media under the paraxial approximation can be characterized by [12]:

(1)$$\frac{{\partial U}}{{\partial z}} = \frac{1}{{2jk}}\left( {\frac{{{\partial^2}U}}{{\partial {x^2}}} + \frac{{{\partial^2}U}}{{\partial {y^2}}}} \right) + j\frac{{{\beta _2}}}{2}\frac{{{\partial ^2}U}}{{\partial {t^2}}},$$

where j is the imaginary unit, U is the beam envelope, k is the wave number in the media, x, y and z are the Cartesian coordinates, and β₂ is the dispersion coefficient. t = t’-z/v_g is the converted time frame, where t’ is the original time frame, v_g is the group velocity.

In the anomalous dispersion media with β₂ < 0, the second order derivative with respect to t can be absorbed into the 2D Laplacian operation and form a 3D Laplacian operator [12]:

(2)$$\frac{{\partial U}}{{\partial z}} = \frac{1}{{2jk}}\left( {\frac{{{\partial^2}U}}{{\partial {x^2}}} + \frac{{{\partial^2}U}}{{\partial {y^2}}} + \frac{{{\partial^2}U}}{{\partial {T^2}}}} \right),$$

where $T = t/\sqrt {|{{\beta_2}} |k}$ is the normalized time and has the unit of length.

One may convert the Cartesian coordinates (x, y, T) into the spherical coordinates (r, θ, φ) as is demonstrated in Fig. 1:

Fig. 1. The (r, θ, φ) coordinates

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and obtain the equation in the spherical coordinates:

(3)$$\frac{{\partial U}}{{\partial z}} = \frac{1}{{2jk}}\left( {\frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{\partial }{{\partial r}}} \right) + \frac{1}{{{r^2}\sin \theta }}\frac{\partial }{{\partial \theta }}\left( {\sin \theta \frac{\partial }{{\partial \theta }}} \right) + \frac{1}{{{r^2}{{\sin }^2}\theta }}\frac{{{\partial^2}}}{{\partial {\varphi^2}}}} \right)U.$$

As is demonstrated in the Appendix, the spherical Gauss-Laguerre beams are the solution to Eq. (3):

(4)$$\begin{aligned} &U \propto {\left( {\frac{{{w_0}}}{{w(\zeta )}}} \right)^{3/2}}{\left( {\frac{{\sqrt 2 r}}{{w(\zeta )}}} \right)^l}L_p^{l + 1/2}\left( {\frac{{2{r^2}}}{{{w^2}(\zeta )}}} \right){Y_{lm}}({\theta ,\varphi } )\exp \left( { - \frac{{{r^2}}}{{{w_0}^2 - 2jz/k}}} \right)\exp \left( {j\left( {2p + l + \frac{3}{2}} \right)\xi } \right),\\ &{Y_{lm}}({\theta ,\varphi } )= P_l^m({\cos \theta } ){e^{im\varphi }}, \end{aligned}$$

where w₀ is the initial beam waist, l is a non-negative integer number, and m is the vortex of the beam, which is an integer number to be selected between -l to l, indicating one l value may correspond to (2l + 1) possible m values ranging from -l to + l, $P_l^m$ is the associated Legendre function with the order of l and the degree of m, L_p^l^+ 1/2 is the generalized Laguerre polynomial with the radial order of p and the azimuth order of l + 1/2 (the reason for using l + 1/2 instead of l is detailed in the appendix), ζ is referred to as the reduced length, w(ζ) is the beam waist at the corresponding reduced length, and ξ is the Guoy phase which are defined as:

(5)$$\begin{aligned} &\zeta = \frac{z}{{{z_R}}},\\ &{z_R} = \frac{{k{w_0}^2}}{2},\\ &w(\zeta )= {w_0}\sqrt {1 + {\zeta ^2}} ,\\ &\xi = {\tan ^{ - 1}}(\zeta ). \end{aligned}$$

Very interesting properties can be inferred from Eq. (4). Firstly, the beam shape distribution in the radial direction is not associated with the number m, but the number l. This suggests that the beam structure does not expand as the topological charge increases, which is unlike the traditional spatial vortex beams. Secondly, the vortex exists on the φ plane, which is the x-y plane in Fig. 1. However, one may choose the x-T plane or y-T plane as the φ plane as well. This suggests that the beams are able to hold vortex either on the spatial plane or on the spatiotemporal plane, and the expression in Eq. (4) is a generalized expression for the spatial orbital angular momentum (OAM) beams or the STOV beams. Furthermore, the plane can be selected as any planes containing the original point and this creates a novel class of space-time interactive beams.

3. Results and discussions

Numerical simulations have been conducted to verify the correctness of the expression. The beam is discretized on the 3D 200 × 200 × 200 grid with the size of 100 µm × 100 µm × 100 µm. The numerical simulation of Eq. (2) is based on the Fourier transform. The beam is assumed to be with wavelength of 1550 nm and the initial beam width of 10 µm. The propagation distance is 500 µm, which is selected as 2.5 times the Rayleigh length. Without loss of generality, the vortex plane of the beam is chosen as the x-y plane, which can be easily converted to x-T plane to obtain the beam with the STOV.

Firstly, we beam parameters l = 1, m = 0, and p = 0. Since the beam is 3D, we plot the 2D amplitude and phase distributions of the beam when T = 0 µm in Fig. 2. Comparing the results by the simulations with the results by the analytical formula, i.e., Eq. (4), it can be concluded that the formula is exact and the newly discovered spherical Gauss-Laguerre beams indeed exist.

Fig. 2. Amplitude and phase distributions of the spherical Gauss-Laguerre beam (l = 1, m = 0, and p = 0) propagates for 500 µm in the z direction when T = 0 µm. (a) the amplitude of the initial beam. (b) the phase of the initial beam. (c) the amplitude by the analytical formula. (d) the phase by the analytical formula. (e) the amplitude by the numerical simulations. (f) the phase by the numerical simulations.

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The beam parameters are switched to l = 1, m = 1, and p = 0 in Fig. 3, which suggests that the spiral phase pattern occurs with the topological charge of 1. The agreement between the numerical and analytical results further verifies the exactness of the formula for the spherical Gauss-Laguerre beams. From Figs. 2,3, it can be further concluded that the beam radius does not expand much as the topological charge increases from 0 to 1, which is different from the traditional spatial vortex beams. The beam radius is impacted by the parameter l rather than m, so is the Guoy phase of the beam.

Fig. 3. Amplitude and phase distributions of the spherical Gauss-Laguerre beam (l = 1, m = 1, and p = 0) propagates for 500 µm in the z direction when T = 0 µm. (a) the amplitude of the initial beam. (b) the phase of the initial beam. (c) the amplitude by the analytical formula. (d) the phase by the analytical formula. (e) the amplitude by the numerical simulations. (f) the phase by the numerical simulations.

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Finally, the beam parameters are switched to l = 1, m = 1, and p = 1 in Fig. 4, which indicates that the radial order of the beam increases from 0 to 1 and results in multiple rings for the beam. The beam holds two rings with the topological charge of 1 as shown in figure. Again, the numerical results and the analytical results show exact agreement.

Fig. 4. Amplitude and phase distributions of the spherical Gauss-Laguerre beam (l = 1, m = 1, and p = 1) propagates for 500 µm in the z direction when T = 0 µm. (a) the amplitude of the initial beam. (b) the phase of the initial beam. (c) the amplitude by the analytical formula. (d) the phase by the analytical formula. (e) the amplitude by the numerical simulations. (f) the phase by the numerical simulations.

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The sliced output of the beam with respect to different values of T is shown in Fig. 5, which demonstrates the wave packet distribution of the beam shown in Fig. 4 in the 3D manner. It can be seen from Fig. 5 that the beam follows the Gaussian distribution in the T frame as T increases from 5 µm to 80 µm. It should be noted that the numerical and analytical results agree excellently in this case as well.

Fig. 5. Amplitude distributions of the spherical Gauss-Laguerre beam (l = 1, m = 1, and p = 1) propagates for 500 µm in the z direction when T have different values. (a-p) T = 5 µm to 80 µm.

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To summarize, the proposed spherical Gauss-Laguerre beams indeed exist. The propagation principle of the beam is distinctive from the conventional Laguerre Gaussian beam and their field distribution evolves in the 4D space-time according to the derived expression, i.e., Eq. (4). Topological charges can exist either in space or in space-time.

To realize such a beam, one may use an arbitrary spatiotemporal field generator as mentioned in [17], which consists of a spectral pulse shaper and a multi-plane light conversion (MPLC) device. During the experiment, it will be essential to keep the normalized time frame to be well scaled with the two spatial coordinates, i.e., x and y.

4. Conclusion

In summary, we have proposed a novel class of 3D spatiotemporal beams which propagate in the 4D x-y-z-t space-time with the Gaussian field distribution. The beams can have either the spatial vortex or the STOV depending on the choice of the vortex plane, which do not resemble any of the existing beams in the spatial coordinates. Furthermore, unlike the spatial vortex beams, which have the beam diameters and Guoy phases growing linearly with respect to the increase of the topological charge, the proposed spherical Gauss-Laguerre beams have their beam diameters and Guoy phases independent of it. The close form expression for the newly discovered beams is verified by the numerical simulations with excellent agreement.

Since the beam operates in the 4D x-y-z-t space time, which differs from the traditional space-time beams, it suggests that the beams can offer a new degree of freedom for light control. For example, the beam can be simultaneously multiplexed on the spatial vortex and the STOV and thus can carry more information in the transmission of a potential free space optical communication system.

Appendix

In the appendix, the derivation of the spherical Gauss-Laguerre beam is provided in detail. We may assume:

(6)$$U = R({z,r} )\Psi ({\theta ,\varphi } )$$

Substituting Eq. (6) into Eq. (3) and using the variable separation method, we may derive the equations for R and Ψ as:

(7)$$\left\{ \begin{array}{l} \frac{1}{{\sin \theta }}\frac{\partial }{{\partial \theta }}\left( {\sin \theta \frac{\partial }{{\partial \theta }}} \right)\Psi + \frac{1}{{{{\sin }^2}\theta }}\frac{{{\partial^2}}}{{\partial {\varphi^2}}}\Psi + \lambda \Psi = 0,\\ \frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{\partial }{{\partial r}}} \right)R + \left( { - 2jk\frac{{\partial R}}{{\partial z}} - \frac{\lambda }{{{r^2}}}R} \right) = 0. \end{array} \right.$$

where λ is a constant to be determined. The first equation for Ψ(θ, φ) is well known and its solution is the spherical harmonic function. It can be solved by expanding the function as:

(8)$$\Psi = \Theta (\theta )\Phi (\varphi ).$$

One may obtain:

(9)$$\left\{ {\begin{array}{c} {\frac{1}{{\sin \theta }}\frac{d}{{d\theta }}\left( {\sin \theta \frac{d}{{\partial \theta }}} \right)\Theta - \frac{{{m^2}}}{{{{\sin }^2}\theta }}\Theta + \lambda \Psi = 0,}\\ {\frac{{{d^2}\Phi }}{{d{\varphi^2}}} + {m^2}\Phi = 0,} \end{array}} \right.$$

where m² is a constant. Since Φ(φ) is a periodical function of φ, m must be an integer. Furthermore, we have λ=l(l + 1) to ensure the convergence of Θ, where l is a non-negative integer. The integer m can be selected as an integer number between -l to l. Equation (9) yields the well-known spherical harmonic function as the solution for Ψ:

(10)$$\Psi \propto {Y_{lm}}({\theta ,\varphi } )= P_l^m({\cos \theta } ){e^{im\varphi }},$$

where P_l^m stands for the Legendre function with the order of l and the degree of m, while m is also the vortex or the topological charge of the beam.

With the computed constant λ=l(l + 1), we have the radial equation for the function R as

(11)$$\frac{{{\partial ^2}R}}{{\partial {r^2}}} + \frac{2}{r}\frac{{\partial R}}{{\partial r}} + \left( { - 2jk\frac{{\partial R}}{{\partial z}} - \frac{{l({l + 1} )}}{{{r^2}}}R} \right) = 0.$$

Introducing the transform $R = u{r^{ - 1/2}}$, we have

(12)$$\frac{{{\partial ^2}}}{{\partial {r^2}}}u + \frac{1}{r}\frac{\partial }{{\partial r}}u + \left( { - 2jk\frac{\partial }{{\partial z}} - \frac{{{{({l + 1/2} )}^2}}}{{{r^2}}}} \right)u = 0.$$

Equation (12) resembles to the radial equation for the 2D beams in the polar coordinates:

(13)$$\frac{{{\partial ^2}}}{{\partial {\rho ^2}}}v + \frac{1}{\rho }\frac{\partial }{{\partial \rho }}v + \left( { - 2jk\frac{\partial }{{\partial z}} - \frac{{{m^2}}}{{{\rho^2}}}} \right)v = 0,$$

where v is the radial function of the 2D beams in the spatial cylindrical coordinates, and ρ is the radius of the polar coordinates. It is already well-known that the Laguerre Gaussian function is the solution to Eq. (13), which yields the 2D spatial Laguerre Gaussian beam’s radial distribution function:

(14)$$\begin{aligned} &{v_{pn}}({\rho ,z} )\propto {\left( {\frac{1}{{w(\zeta )}}} \right)^2}{\left( {\frac{{\sqrt 2 \rho }}{{w(\zeta )}}} \right)^n}L_p^n\left( {\frac{{2{\rho^2}}}{{{w^2}(\zeta )}}} \right)\exp \left( { - \frac{{{\rho^2}}}{{{w_0}^2({1 - j\zeta } )}} + j{\Psi _{pn}}} \right),\\ &{\rho ^2} = {x^2} + {y^2},\\ &\varphi = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right),\\ &{z_R} = \frac{{k{w_0}^2}}{2},\\ &\zeta = \frac{z}{{{z_R}}},\\ &w(\zeta )= {w_0}\sqrt {1 + {\zeta ^2}} ,\\ &{\Psi _{pn}} = ({1 + n + 2p} )\xi ,\\ &\xi = {\tan ^{ - 1}}(\zeta ), \end{aligned}$$

where w₀ is the initial beam waist, L_pⁿ is the generalized Laguerre polynomial with the radial order of p and the azimuth order of n, ζ is referred to as the reduced length, w(ζ) is the beam waist at the corresponding reduced length, and ξ is the Gouy phase. Comparing Eq. (12) with Eq. (13), the only difference is that m has been replaced with l + 1/2. Therefore, one may replace m with l + 1/2 in Eq. (14), and we have:

(15)$$u \propto {\left( {\frac{1}{{w(\zeta )}}} \right)^2}{\left( {\frac{{\sqrt 2 r}}{{w(\zeta )}}} \right)^{l + 1/2}}L_p^{l + 1/2}\left( {\frac{{2{r^2}}}{{{w^2}(\zeta )}}} \right)\exp \left( { - \frac{{{r^2}}}{{{w_0}^2 - 2jz/k}}} \right)\exp \left( {j\left( {2p + l + \frac{3}{2}} \right)\xi } \right).$$

Combining the solutions for R and Ψ, we have

(16)$$\begin{aligned} &U = R({z,r} )\Psi ({\theta ,\varphi } )= {r^{ - 1/2}}u\Psi ({\theta ,\varphi } )\\ &\propto {\left( {\frac{1}{{w(\zeta )}}} \right)^{3/2}}{\left( {\frac{{\sqrt 2 r}}{{w(\zeta )}}} \right)^l}L_p^{l + 1/2}\left( {\frac{{2{r^2}}}{{{w^2}(\zeta )}}} \right){Y_{lm}}({\theta ,\varphi } )\exp \left( { - \frac{{{r^2}}}{{{w_0}^2 - 2jz/k}}} \right)\exp \left( {j\left( {2p + l + \frac{3}{2}} \right)\xi } \right). \end{aligned}$$

By multiplying a factor of w₀^3/2, we have the lowest order beam to be with the unit peak value, and we get Eq. (4), which describes the novel 3D spherical Gauss-Laguerre beam propagation in the 4D space-time.

Funding

National Key Research and Development Program of China (2022ZD0119302); National Natural Science Foundation of China (62375206); Science and Technology Commission of Shanghai Municipality (2021SHZDZX0100); Fundamental Research Funds for the Central Universities.

Disclosures

The authors declare no conflicts of interest.

Data availability

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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Spherical Gauss-Laguerre beam propagation in 4D space-time

Abstract

1. Introduction

2. Theory

3. Results and discussions

4. Conclusion

Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (16)

Optics Express