Superluminal light propagation in a normal dispersive medium

Zahra Amini Sabegh; Mohammad Mahmoudi

doi:10.1364/OE.424860

1. Introduction

A wide variety of theoretical and experimental studies have recently been conducted on the group velocity of structured light fields propagating in free space. Padgett et al. have experimentally explored how the transverse spatial structure of photons affects the dispersion of light beam in free space. It was observed that the group velocity of structured single photons is slower in both Bessel and focused Gaussian beams along the propagation axis than the speed of light in free space $c$ [1]. In another experimental work, it was demonstrated that the addition of orbital angular momentum (OAM) of the structured beam reduces the intrinsic delay of twisted photons with respect to the same beam without OAM [2]. It is noted that the group velocity reduction for the Bessel light beam near the critical frequency has been reported in Ref. [3]. On the other hand, the group velocity of the Laguerre-Gaussian (LG) beam with helical wavefront has been theoretically studied and shown that it is inversely proportional to the OAM value of light [4]. The authors declared that the slowing of light is due to the field confinement and the dispersion of light, which implies its intrinsic property in free space. Theoretical and experimental investigation of the evolution of the group velocity was performed for the focused Gaussian and LG beams along the propagation axis in Ref. [5]. It was presented that the group velocity of a Gaussian mode can be both subluminal and superluminal in different propagation distances on the optical axis, while a LG mode has the slow-light behavior for all propagation distances. However, a comment [6] found the results of Ref. [5] questionable in some aspects and suggested using the projection of group velocity vector onto the beam axis instead of its absolute value, particularly when one interprets the measured relatively large propagation delays. More recently, Saari resolved the contradictions between various definitions of group velocity, which led to the different versions of group velocity for Bessel-Gauss pulses depending on the type of pulse and method of recording it in the output plane [7].

Over the last three decades, the propagation of the plane-wave light beam through a dispersive medium has attracted much attention. It is well-known that the group velocity of such beams, interacting with matter, can be either subluminal [8,9] or superluminal [10,11]. Generally, the subluminal (superluminal) phenomenon occurs when the medium dispersion has normal (anomalous) behavior [12,13]. In recent years, both of these phenomena and also switching from one to another have been studied in different quantum systems such as atomic media [14–17], Fermi gases [18], quantum dot molecules [19], photonic crystals [20], superconducting quantum circuits [21], graphene [22], and nitrogen-vacancy centers [23]. The superluminal or subluminal group velocity of a light beam results from its interaction with matter or the structure of the wavefront. Recently, it has been shown that the transfer of OAM between light beams occurs through two-component slow light, under the EIT conditions [24]. In studying the OAM transfer, we have reported the preliminary results of the group velocity of the probe LG field in a dispersive medium [25]. More recently, Juzeliūnas et al. have reported the group velocity of an off-axis vortex beam in a double-Raman gain atomic medium for the superposition of two probe fields and studied the effect of the manipulation of the two-photon detuning on the behavior of the phase profile [26]. Here, we study the group velocity of an individual probe LG beam in a spatially dependent dispersive medium and show the strange relation between the group velocity and dispersion behavior of the system.

In this manuscript, we study the propagation of a LG beam through a dispersive medium and investigate how the group velocity of the LG beam depends on the dispersion of the medium. We calculate a general analytical expression for the group velocity of the LG beam in a dispersive medium which can reduce to its well-known formula in free space. We continue our study by considering two scenarios: uniformly and spatially dependent dispersive medium, in which the obtained analytical expressions help us to explore the behavior of the group velocity and its relation with the medium refractive index. The first scenario involves the study of the effect of the doughnut-like intensity profile of the probe LG field on the group velocity under the multi-photon resonance condition. Moreover, we investigate the group velocity and its $z$-component along the propagation direction (out of the optical axis) and demonstrate that there are some regions where the group velocity vector is along the optical axis. On the other hand, the effect of the OAM of strong coupling LG field on the group velocity of the probe LG field is studied through the second scenario. It is shown that the gain-assisted superluminal (absorption-assisted subluminal) group velocity with normal (anomalous) dispersion appears in some regions of the medium. Such a strange result is the effect of the classical interference of the coupling planar and LG fields on the helical phase front of the probe LG field, which has not yet been reported before in a dispersive medium.

2. Group velocity of the LG field in a dispersive medium

The group velocity of the LG field can be obtained employing the well-known formula for the magnitude of the group velocity vector, $v_g=1/|\nabla \partial _\omega \Phi |$, where $\partial _\omega \Phi$ is the phase profile derivation of beam, $\Phi$, with respect to $\omega$ [27]. The phase profile of the LG field has the following form

(1)$$\Phi(r,\varphi,z)=\frac{n\omega}{c}z(1+\frac{r^2}{2(z^2+z_R^2)}) -(2p+|l|+1)tan^{{-}1}(z/z_R)+l\varphi,$$

where the refractive index of the medium, laser frequency, waist of the laser beam, the Rayleigh range, azimuthal and radial indices are indicated by $n$, $\omega$, $w_0$, $z_R=n\omega w_0^2/2c$, $l$, and $p$, respectively. In a dispersive medium, the refractive index relates to the susceptibility of the medium as $n\approx 1+Re(\chi )/2$. Using Eq. (1), one can calculate the group velocity of the probe LG field in the dispersive medium, through the propagation axis, $r=0$, and at $z=0$ as below

(2)$$v_g=\frac{c}{(n+\omega\partial n/\partial\omega)[1+\frac{2c^2}{n^2\omega^2w_0^2}(2p+|l|+1)]}.$$

Under the mentioned conditions, the vector of group velocity related to the LG field has just the $z$-component, which is given by $v_z=\partial _z\partial _\omega \Phi /|\nabla \partial _\omega \Phi |^2$ [7]. Note that if $w_0\rightarrow \infty$, the above equation turns into the well-known formula of the group velocity $v_g=c/(n+\omega \partial n/\partial \omega )$ related to a group of plane waves in a dispersive medium [12]. It can also be deduced from Eq. (2) that the group velocity of the Gaussian field ($l=0$ and $p=0$) is smaller than $c$ in a medium with normal dispersion. However, its value undergoes a great reduction for the higher modes of the LG field. The Gaussian field propagates with a superluminal group velocity when anomalous dispersion dominates the medium. It is also expected from this equation that the higher-order LG modes experience a smaller increase in the group velocity. It is worth noting that our analytical expression for the group velocity of the LG beam in a dispersive medium is in good agreement with the group velocity expression of the LG beam in free space, $n=1$ and $\partial n/\partial \omega =0$, [4]

(3)$$v_g=\frac{c}{1+\frac{2c^2}{\omega^2w_0^2}(2p+|l|+1)},$$

which is smaller than $c$ for a Gaussian beam and all LG modes. Notice that the analytical expression for the group velocity at $z=0$ and out of the propagation axis, $r\neq 0$, reads

(4)$$v_g(r,\varphi)=\dfrac{c}{(n+\omega\partial n/\partial\omega)[1+\frac{2c^2}{n^2\omega^2w_0^2}(-\frac{r^2}{w_0^2}+(2p+|l|+1))]},$$

which can be reduced to Eq. (2) on the optical axis, $r=0$.

In order to study the behavior of the group velocity out of the optical axis in a dispersive medium, we recalculate Eq. (2) for $r\neq 0$ and $z\neq 0$. Therefore, a general expression for the group velocity magnitude takes the following form:

(5)$$v_g(r,z)=\frac{cB_1^3}{(n+\omega\partial n/\partial\omega)\sqrt{B_2^2+(B_3+B_4-B_5)^2}},$$

in which

\begin{aligned} B_1&=4c^2z^2+n^2\omega^2w_0^4,\\ B_2&=4c^2rz({-}16c^4z^4+n^4\omega^4w_0^8),\\ B_3&=48c^4n^2\omega^2w_0^4z^2(r^2+z^2)+n^6\omega^6w_0^{12},\\ B_4&=2c^2n^4\omega^4w_0^8[{-}r^2+6z^2+w_0^2(2p+|l|+1)],\\ B_5&=32c^6z^4[r^2-2z^2+w_0^2(2p+|l|+1)]. \end{aligned}

It can be observed that Eq. (5) reduces to Eq. (2) at $r=z=0$. Here the group velocity vector has a radial component as

(6)$$v_r(r,z)=\frac{\partial_r\partial_\omega\Phi}{|\nabla\partial_\omega\Phi|^2}= \frac{4c^2rz(4c^2z^2-n^2\omega^2w_0^4)B_1}{\sqrt{B_2^2+(B_3+B_4-B_5)^2}}v_g(r,z),$$

in addition to the $z$-component, which is given by

(7)$$v_z(r,z)=\frac{\partial_z\partial_\omega\Phi}{|\nabla\partial_\omega\Phi|^2}= \frac{B_3+B_4-B_5}{\sqrt{B_2^2+(B_3+B_4-B_5)^2}}v_g(r,z).$$

Thus the group velocity vector is not exactly along the propagation direction. The coefficient $(B_3+B_4-B_5)/\sqrt {B_2^2+(B_3+B_4-B_5)^2}$ in Eq. (7) is exactly equal to one at $r=0$, $z=0$, and $z=z_R$. It means that the group velocity vector has just the $z$-component on the optical axis ($r=0$), at the waist of the beam ($z=0$) and the Rayleigh range ($z=z_R$). However, the mentioned coefficient takes the value of about one at other points leading to a difference between $v_z(r,z)$ and $v_g(r,z)$ in this case. In the following, we are going to find out how the dispersive medium can affect the propagation of the LG beam.

3. Results and discussion

3.1 Group velocity behavior in a uniformly dispersive medium

Let us consider a dispersive medium, whose four selected transitions are driven by four external fields. As shown in Fig. 1, we establish a four-level double $V$-type system in the suggested model. In our notation, the Rabi frequency $\Omega _{ij}=\vec {\mu }_{ij}\cdot \vec {E}_{ij}/\hbar$ is defined as the scale of the applied field strength $E_{ij}$ in which $\mu _{ij}$ and $\hbar$ are the induced dipole moment of the transition $|i\rangle \leftrightarrow |j\rangle$ and Planck’s constant, respectively. It is assumed that the $|1\rangle \leftrightarrow |4\rangle$ transition is excited by a weak probe LG field with the frequency $\omega _{41}$, and the Rabi frequency in cylindrical coordinates

(8)$$\Omega_{41}(r,\varphi,z)=\Omega_{41_0}(\frac{\sqrt{2}r}{w_0})^{|l_{41}|}e^{-\frac{r^2}{w_0^2}}L_{p_{41}}^{|l_{41}|}(2r^2/w_0^2)e^{i\Phi(r,\varphi,z)}.$$

Here $L_{p_{41}}^{|l_{41}|}$, $l_{41}$, $p_{41}$, $\Omega _{41_0}$, and $\Phi (r,\varphi ,z)$ stand for the associated Laguerre polynomial, OAM value, radial index, constant Rabi frequency, and phase profile of the probe LG field, respectively. The suggested dispersive medium is considered so that the central frequency of the probe transition equals its spontaneous emission rate, $\bar {\omega }_{41}=\gamma _{41}$. Three strong coupling fields are also applied to three transitions of the system, $|1\rangle \leftrightarrow |3\rangle$, $|2\rangle \leftrightarrow |3\rangle$, and $|2\rangle \leftrightarrow |4\rangle$, with the Rabi frequencies ($\Omega _{31}$, $\Omega _{32}$, and $\Omega _{42}$) and frequencies ($\omega _{31}$, $\omega _{32}$, and $\omega _{42}$) included. All of the three coupling fields are considered to be planar at the first scenario, while at the second scenario, one of them, $\Omega _{42}$, has a LG form. Furthermore, spontaneous emission rates from the upper energy states, $|3\rangle$ and $|4\rangle$, to the lower states, $|1\rangle$ and $|2\rangle$, are denoted by $\gamma _{31}$, $\gamma _{32}$, $\gamma _{41}$, and $\gamma _{42}$.

Fig. 1. Schematic diagram of the four-level double $V$-type system driven by a weak probe field ($\Omega _{41}$) and three strong coupling fields ($\Omega _{31}$, $\Omega _{32}$, and $\Omega _{42}$).

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In order to describe the interaction of the applied fields with the considered system, we can use the von Neumann equation for the density matrix of the system, $i\hbar \partial \rho /\partial t=[H,\rho ]$. Considering the electric-dipole moment and rotating-wave approximations in the interaction picture, the Hamiltonian can be written as

(9)$$\begin{aligned} H={-}\hbar[\Omega_{31}^\ast e^{i\Delta_{31}t}|1\rangle\langle3|+\Omega_{41}^\ast e^{i\Delta_{41}t}|1\rangle\langle4| +\Omega_{32}^\ast e^{i\Delta_{32}t}|2\rangle\langle3| +\Omega_{42}^\ast e^{i(\Delta_{42}t-\phi_0)}|2\rangle\langle4|+C.C.],\\ \end{aligned}$$

where $\phi _0$ and $\Delta _{ij}=\omega _{ij}-\bar {\omega }_{ij}$ are the relative phase of the applied fields and frequency detuning between laser frequency ($\omega _{ij}$) and central frequency of the corresponding transition ($\bar {\omega }_{ij}$), respectively. The Bloch equations, which are similar to those of the double-$\Lambda$ type atomic system [28], are then obtained for the density matrix elements by substituting Eq. (9) into the von Neumann equation

(10)\begin{align} \dot{\rho}_{11}&=i[\Omega_{31}^{*}\rho_{31}-\Omega_{31}\rho_{13}+\Omega_{41}^{*}\rho_{41}-\Omega_{41}\rho_{14}] +\gamma_{31}\rho_{33}+\gamma_{41}\rho_{44},\notag\\ \dot{\rho}_{22}&=i[\Omega_{32}^{*}\rho_{32}-\Omega_{32}\rho_{23}+\Omega_{42}^{*}e^{{-}i\phi_0}\rho_{42}-\Omega_{42}e^{i\phi_0}\rho_{24}]{+} \gamma_{32}\rho_{33}+\gamma_{42}\rho_{44},\notag\\ \dot{\rho}_{33}&=i[\Omega_{31}\rho_{13}-\Omega_{31}^{*}\rho_{31}+\Omega_{32}\rho_{23}-\Omega_{32}^{*}\rho_{32}] -(\gamma_{31}+\gamma_{32})\rho_{33},\notag\\ \dot{\rho}_{12}&=i[\Omega_{31}^{*}\rho_{32}-\Omega_{32}\rho_{13}+\Omega_{41}^{*}\rho_{42}-\Omega_{42}e^{i\phi_0}\rho_{14} -(\Delta_{31}-\Delta_{32})\rho_{12}],\notag\\ \dot{\rho}_{13}&=i[\Omega_{31}^{*}(\rho_{33}-\rho_{11})+\Omega_{41}^{*}\rho_{43}-\Omega_{32}^{*}\rho_{12}-\Delta_{31}\rho_{13}] -\frac{(\gamma_{31}+\gamma_{32})}{2}\rho_{13},\notag\\ \dot{\rho}_{14}&=i[\Omega_{41}^{*}(\rho_{44}-\rho_{11})+\Omega_{31}^{*}\rho_{34}-\Omega_{42}^{*}e^{{-}i\phi_0}\rho_{12}-\Delta_{41}\rho_{14}] -\frac{(\gamma_{41}+\gamma_{42})}{2}\rho_{14},\notag\\ \dot{\rho}_{23}&=i[\Omega_{32}^{*}(\rho_{33}-\rho_{22})+\Omega_{42}^{*}e^{{-}i\phi_0}\rho_{43}-\Omega_{31}^{*}\rho_{21}-\Delta_{32}\rho_{23}] -\frac{(\gamma_{31}+\gamma_{32})}{2}\rho_{23},\notag\\ \dot{\rho}_{24}&=i[\Omega_{42}^{*}e^{{-}i\phi_0}(\rho_{44}-\rho_{22})+\Omega_{32}^{*}\rho_{34}-\Omega_{41}^{*}\rho_{21}-\Delta_{42}\rho_{24}] -\frac{(\gamma_{41}+\gamma_{42})}{2}\rho_{24},\notag\\ \dot{\rho}_{34}&=i[\Omega_{31}\rho_{14}-\Omega_{41}^{*}\rho_{31}+\Omega_{32}\rho_{24}-\Omega_{42}^{*}e^{{-}i\phi_0}\rho_{32} -(\Delta_{41}-\Delta_{31})\rho_{34}]\notag\\ &-\frac{(\gamma_{31}+\gamma_{32}+\gamma_{41}+\gamma_{42})}{2}\rho_{34},\notag\\ \dot{\rho}_{44}&=-(\dot{\rho}_{11}+\dot{\rho}_{22}+\dot{\rho}_{33}). \end{align}

Note that the terms of the spontaneous emissions have been phenomenologically added to the Bloch equations.

Then, we analytically solve Eq. (10) considering the steady-state condition and assumption of $\Delta _{31}=\Delta _{32}=\Delta _{42}=0$ and $\gamma _{31}=\gamma _{32}=\gamma _{41}=\gamma _{42}=\gamma$, for better understanding how the dispersive medium reacts to the probe field. The coherence term of the $|1\rangle \leftrightarrow |4\rangle$ transition $\rho _{41}$ corresponding to the third-order susceptibility of the system determines the response of the system to the probe field. Since the analytical expression of the susceptibility is too long, we present the susceptibility and its derivation with respect to the probe detuning, taking $\Delta _{41}=0$ for simplicity as below

(11)$$\chi=\frac{-i\gamma\Omega_{31}\Omega_{32}^\ast\Omega_{42}e^{i\phi_0}}{A_1},$$

and

(12)$$\frac{\partial\chi}{\partial\omega}=\frac{\Omega_{31}\Omega_{32}^\ast\Omega_{42}e^{i\phi_0}A_2}{A_1 A_3},$$

in which

\begin{aligned} A_1&=\gamma^2(|\Omega_{31}|^2+|\Omega_{32}|^2+|\Omega_{42}|^2)+4|\Omega_{31}|^2|\Omega_{42}|^2,\\ A_2&=2\gamma^4+3\gamma^2(|\Omega_{32}|^2+|\Omega_{42}|^2)+(|\Omega_{32}|^2+|\Omega_{42}|^2)^2 +|\Omega_{31}|^2(\gamma^2+|\Omega_{32}|^2-|\Omega_{42}|^2),\\ A_3&=2\gamma^2(|\Omega_{31}|^2+|\Omega_{32}|^2+|\Omega_{42}|^2)+(|\Omega_{32}|^2+|\Omega_{42}|^2)^2 +|\Omega_{31}|^2(|\Omega_{31}|^2+2|\Omega_{32}|^2-2|\Omega_{42}|^2). \end{aligned}

The obtained susceptibility, Eq. (11), is one of 44 different terms of the third order susceptibility, which is corresponding to $\Omega _{31}\Omega _{32}^\ast \Omega _{42}$ called the cross-Kerr effect as a result of the scattering of the coupling fields into the probe field frequency independent on the probe field. In this case, the frequency ($\omega _{31}+\omega _{42}-\omega _{32}$) equals to the probe frequency, $\omega _{41}$, as the multi-photon resonance condition, $(\Delta _{31}+\Delta _{42})-(\Delta _{32}+\Delta _{41})=0$, is fulfilled. The imaginary (real) part of the susceptibility describes the absorption (dispersion) behavior of the system toward the probe field. Contractually, the positive (negative) value for the imaginary part of the susceptibility is assigned to the absorption (gain) of the medium. However, the positive (negative) value of the dispersion slope defines the normal (anomalous) dispersion for the probe field. The susceptibility of the dispersive medium helps to obtain the group velocity through the calculation of the refractive index.

At this stage, one can study the behavior of the group velocity, medium absorption and dispersion by the obtained analytical results [Eqs. (2), (11), and (12)]. The imaginary (a) and real (b) parts of the susceptibility, $\chi$, are numerically plotted as a function of the dimensionless detuning of the probe field, $\Delta _{41}/\gamma$, in Fig. 2. The values of parameters are considered to be $\Omega _{31}=\Omega _{32}=\Omega _{42}=\gamma$, $\gamma _{31}=\gamma _{32}=\gamma _{41}=\gamma _{42}=\gamma$, $\Delta _{31}=\Delta _{32}=\Delta _{42}=0$, $\Omega _{41}=0.01\gamma$, and $\phi _0=0$. As exhibited in Fig. 2, a gain dip is accompanied by a normal dispersion around zero probe detuning $\Delta _{41}=0$. Moreover, the results are consistent with the Kramers-Kronig dispersion relations which are described in both linear [29,30] and nonlinear optics [31]. More precisely, the normal (anomalous) dispersion is accompanied by the gain dip (absorption peak) or absorption (gain) doublet in the absorption spectrum. [32].

Fig. 2. The imaginary (a) and real (b) parts of the susceptibility, $\chi$, versus the dimensionless detuning of the probe field, $\Delta _{41}/\gamma$. The applied parameters are considered to be $\Omega _{31}=\Omega _{32}=\Omega _{42}=\gamma$, $\gamma _{31}=\gamma _{32}=\gamma _{41}=\gamma _{42}=\gamma$, $\Delta _{31}=\Delta _{32}=\Delta _{42}=0$, $\Omega _{41}=0.01\gamma$, and $\phi _0=0$.

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On the other hand, Fig. 3 presents the behavior of group velocity, $(v_g/c)-1$, versus $\Delta _{41}/\gamma$ for the different values of OAM, $l_{41}=0$ (solid), $1$ (dashed), $2$ (dash-dotted), $5$ (dotted), at $r_{max}=w_0\sqrt {|l_{41}|/2}$. The frequency of probe transition, $\omega =\bar {\omega }_{41}$, is considered to be about $10^{14}~Hz$. The values of the radial index and beam waist of probe LG field are taken as $p_{41}=0$ and $w_0=20\mu m$, respectively, while other parameters are the same as those in Fig. 2. It is obvious that the group velocity is smaller than $c$ with the multi-photon resonance condition, which is called subluminal light propagation. As deduced from Eq. (2), the probe Gaussian field has a subluminal group velocity when the dispersion slope is positive. Moreover, the probe LG field including higher values of the OAM propagates along $z$-direction slower than the Gaussian mode. Since the presented results are valid just around $\Delta _{41}=0$, a magnified view around zero probe detuning is provided in the inset of Fig. 3. Our next step is to study the difference of the group velocity magnitude and its projection onto the propagation axis via Eqs. (7), (11), and (12).

Fig. 3. The behavior of group velocity, $(v_g/c)-1$, as a function of $\Delta _{41}/\gamma$ for the different modes of probe LG field with $\omega \sim 10^{14}~Hz$, $p_{41}=0$, and $w_0=20\mu m$, at $r_{max}=w_0\sqrt {|l_{41}|/2}$

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Now, for the best understanding of the difference between the magnitude of the group velocity and its $z$-component, the coefficient of $v_g$ in Eq. (7) is plotted as a function of dimensionless propagation distance $z/z_R$ (horizontal axis) and the radial coordinate $r/w_0$ (vertical axis) for $l_{41}=1$ in Fig. 4. Other used parameters are the same as those in Fig. 3. Since the coefficient has a negligible dependence on the OAM of the probe field ($l_{41}$), it is presented just for $l_{41}=1$. According to Fig. 4, the value of $v_g$ is different from that of $v_z$ everywhere except at the waist of the beam, the Rayleigh range, and on the optical axis; however, the most differences have appeared at the points away from the optical axis. So, the group velocity is exactly along the propagation axis just at $z=0$, $r=0$, and $z=z_R$.

Fig. 4. The coefficient of $v_g$ in Eq. (7) as a function of $z/z_R$ and $r/w_0$ for $l_{41}=1$. Other used parameters are the same as those in Fig. 3.

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3.2 Group velocity behavior in a spatially dependent dispersive medium

Finally, we prepare a dispersive medium with a spatially dependent refractive index by considering one of the strong coupling fields as a LG field. As a result, the magnitude of the group velocity vector, which has three components, cannot be expressed by a simple mathematical expression. Notice that due to the complexity and length of the resulting expression, it is not presented here. In this section, we would like to investigate the group velocity of the probe LG field at $z=0$. It is intriguing to note that the group velocity has only $z$-component, and Eq. (4) describes the magnitude of the group velocity vector.

We now proceed by regarding the strong coupling field of the $|2\rangle \leftrightarrow |4\rangle$ transition, $\Omega _{42}$, as a LG field with zero radial index, $p_{42}=0$, which takes the following form

(13)$$\Omega_{42}(r,\varphi)=\Omega_{42_0} (\frac{\sqrt{2}r}{w_{LG}})^{|l_{42}|}e^{-\frac{r^2}{w_{LG}^2}}e^{il_{42}\varphi}.$$

Here, the constant Rabi frequency, waist and OAM value of the strong coupling LG field are described by $\Omega _{42_0}$, $w_{LG}$, and $l_{42}$, respectively. Consequently, the spatially dependent behavior of the medium absorption for zero probe detuning is given by

(14)$$Im(\chi)=\frac{-\gamma\Omega_{31}\Omega_{32}^\ast\Omega_{42}(r)\cos(l_{42}\varphi+\phi_0)}{A_1}.$$

In addition, one can obtain the refractive index of the medium and its dispersion slope in the case of zero probe detuning which are respectively given by

(15)$$n=1+\frac{\gamma\Omega_{31}\Omega_{32}^\ast\Omega_{42}(r)\sin(l_{42}\varphi+\phi_0)}{2A_1},$$

and

(16)$$\frac{\partial n}{\partial\omega}\approx\dfrac{1}{2}\frac{\partial Re(\chi)}{\partial\omega}=\frac{\Omega_{31}\Omega_{32}^\ast\Omega_{42}(r) A_2 \cos(l_{42}\varphi+\phi_0)}{2A_1 A_3}.$$

We now explore how the helical wavefront of the strong coupling LG field, which is indicated by its OAM, affects the response of the medium to the probe LG field [Eqs. (14) and (16)], and its group velocity [Eq. (4)] at $z=0$. At first, we study the behavior of the probe LG field group velocity as a function of dispersion slope at $r_{max}=w_0\sqrt {|l_{41}|/2}$, in Fig. 5. Used parameters are considered to be $w_0=w_{LG}=100\mu m$, $\omega \sim 10^{14}~Hz$, $p_{41}=0$, $l_{41}=l_{42}=1$, and $\phi _0=0$ . Other used parameters are the same as those in Fig. 2. Four regions are introduced for the probe LG beam propagation. Region (I) stands for the superluminal light propagation in normal dispersion which is an unusual result of our suggested model. Region (III) defines the subluminal light propagation in anomalous dispersion. We introduce these two concepts for the first time which have not yet been reported in the previous documents. The non-structured beam does not show such interesting behaviors due to passing through the dispersive medium and can propagate only in the regions (II) and (IV).

Fig. 5. Probe LG field group velocity as a function of dispersion slope at $r_{max}=w_0\sqrt {|l_{41}|/2}$. Used parameters are considered to be $w_0=w_{LG}=100\mu m$, $\omega \sim 10^{14}~Hz$, $p_{41}=0$, $l_{41}=l_{42}=1$, and $\phi _0=0$. Other used parameters are the same as those in Fig. 2.

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Figure 6 illustrates the imaginary part of the susceptibility (left column), dispersion slope (middle column), and dimensionless group velocity (right column) profiles as a function of $x$ (horizontal axis) and $y$ (vertical axis) for three different values of the OAM, i.e. $l_{42}=0,1,2$. In addition to the constant Rabi frequency of $\Omega _{42_0}=\gamma$, we have fixed the size of the medium cross-section at $0.5 mm\times 0.5 mm$ and the waist of the strong coupling LG field at $w_{LG}=100\mu m$. The waist and OAM value corresponding to the probe LG field are taken to be $w_0=100\mu m$ and $l_{41}=1$, respectively, whereas the other used parameters are the same as those in Fig. 3. It is clear that the increase in the OAM value of probe LG field cannot have a considerable impact on the pattern of group velocity, while the OAM value of the strong coupling LG field can significantly change the behavior of group velocity. On the other hand, the first row of Fig. 6 shows that the medium has a Gaussian-like gain response to the probe LG field with a normal dispersion and subluminal group velocity when $\Omega _{42}$ has a Gaussian function, $l_{42}=0$. In the second row of Fig. 6, a petal-like pattern has appeared whose right (left) petal displays a gain (absorption) region with normal (anomalous) dispersion, however, the group velocity profile behaves in an unusual manner. Although the subluminal (superluminal) group velocity exists in a medium characterized by normal (anomalous) dispersion, there is a region of normal (anomalous) dispersion in which the probe field experiences the gain-assisted superluminal (absorption-assisted subluminal) group velocity. It should be mentioned that the main reason for this strange behavior lies in the counterclockwise rotation of the petal-like pattern related to the group velocity. As exhibited in the third row of Fig. 6, the number of petals and regions with unusual group velocity behavior has been doubled by $l_{42}=2$. Generally, one can conclude that the number of petals equals $2l_{42}$ in all panels.

Fig. 6. The imaginary part of the susceptibility (left column), dispersion slope (middle column), and dimensionless group velocity (right column) profiles as a function of $x$ (horizontal axis) and $y$ (vertical axis) for three different values of the OAM, i.e. $l_{42}=0,1,2$. In addition to the constant Rabi frequency of $\Omega _{42_0}=\gamma$, the size of the medium cross-section is fixed at $0.5 mm\times 0.5 mm$ and the waist of the strong coupling LG field is considered to be $w_{LG}=100\mu m$. The waist and OAM value corresponding to the probe LG field are taken to be $w_0=100\mu m$ and $l_{41}=1$, respectively, whereas the other used parameters are the same as those in Fig. 3.

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Let us proceed with the interference perspective to find a physical interpretation for the obtained results in Fig. 6. Here two coupling planar fields ($E_{31}$ and $E_{32}$) and a coupling LG field ($E_{42}$) experience an interference inside the dispersive medium. The spatially dependent interference pattern resulting from the helical phase front of the coupling LG field can change the properties of a probe LG field propagating through this pattern. Figure 7 is plotted in order to study the mentioned changes. In this figure, the interference pattern of the three coupling fields (left column), intensity (middle column) and phase (right column) profiles of the probe LG field passing through the interference pattern are plotted as a function of $x$ and $y$ for three different values of the OAM, i.e. $l_{42}=0,1,2$. The constant amplitudes of the coupling and probe fields are considered to be $E_{31}=E_{32}=E_{42}$ and $E_{41}=0.01E_{31}$, respectively. Other parameters of the coupling and probe LG fields are the same as those in Fig. 6. The comparison between the left columns of Figs. 6 and 7 illustrates that the gain (absorption) regions originate from the constructive (destructive) interference of the three coupling fields in the dispersive medium. In the middle column of Fig. 7, it can easily be seen that the probe LG field propagating through the spatially dependent interference pattern is amplified (attenuated) in the constructive (destructive) interference regions. The first row of the right column shows that the phase front of the probe LG field has an undisturbed helical shape considering the Gaussian mode for $E_{42}$. Note that the probe LG field undergoes a distortion in its phase front as $E_{42}$ has the helical phase front with nonzero OAM value ($l_{42}=1$ or $2$). The evolution of the probe LG field can also be investigated using the susceptibility of the medium, calculated through the density matrix formalism, and wave propagation equation ($\partial E_{41}(z)/\partial z=ik_p\chi E_{41}(z)/2$) at $L=0.5\mu m$. Employing this manner yields the same results as those of the middle and right columns of Fig. 7 for the intensity and phase profiles of the probe LG field ($E_{41}(z=L)$). Actually, the output probe LG field is given by

(17)$$E_{41}(z=L)=E_{41}(z=0)e^{k_p[{-}Im(\chi)+iRe(\chi)]L/2},$$

in which $E_{41}(z=0)$ indicates the amplitude of the incident probe LG field. Substituting imaginary and real parts of the susceptibility in the above equation leads to a new phase term for the probe LG field as $\exp \{i[l_{41}\varphi +k_pL(\gamma \Omega _{31}\Omega _{32}^\ast \Omega _{42}(r)\sin (l_{42}\varphi +\phi _0))/4A_1]\}$. Therefore, the phase profile of the output probe LG field has another spatial dependence in addition to its common phase term, $e^{il_{41}\varphi }$. It is also expected that the distortion of the phase front occurs just for the nonzero OAM values of the coupling LG field, for $\phi _0=0$; so that increasing the order of the OAM value provides more distortion parts in the phase profile of the output probe LG field. However, the probe LG field keeps its own helical phase front for the case of Gaussian mode of $E_{42}$. It means that the strange behavior related to the group velocity of the probe LG field inside the dispersive medium lies in the distortion of the helical phase front of the probe LG field due to the classical interference pattern of the coupling planar and LG fields.

Fig. 7. The interference pattern of the coupling fields (left column), intensity (middle column) and phase (right column) profiles of the probe LG field passing through the interference pattern as a function of $x$ and $y$ for three different values of the OAM, i.e. $l_{42}=0,1,2$. The constant amplitudes of the coupling and probe fields are considered to be $E_{31}=E_{32}=E_{42}$ and $E_{41}=0.01E_{31}$, respectively. Other parameters of the coupling and probe LG fields are the same as those in Fig. 6.

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In Fig. 8, we replot thegroup velocity profiles of Fig. 6 for the strong coupling LG field with reverse helical wavefront, i.e. the negative value of the OAM. According to Eqs. (14) and (16), it is expected that the imaginary part of the susceptibility and dispersion slope of the dispersive medium do not change for the negative values of the OAM. However, the group velocity has been affected by negative OAMs, and its petals rotate clockwise, which results in the change of the position of regions with unusual behaviors.

Fig. 8. The dimensionless group velocity profiles versus $x$ and $y$ for two negative values of the OAM, i.e. $l_{42}=-1,-2$. The values of parameters are the same as those in Fig. 6.

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Since the optical properties of a closed-loop system are highly dependent on the relative phase of the applied fields [14], we are going to investigate the effect of the changing the relative phase on the previous results of Fig. 6. Figure 9 is just the same as Fig. 6, but for the shift of the relative phase from $0$ to $\pi$. It is apparent from Eqs. (14) and (16) that the absorption and anomalous dispersion regions turn into the gain and normal dispersion ones, respectively, and vise versa. It is also shown that a simple change in the relative phase of applied fields leads to the interchange of the regions with unusual behavior of group velocity. However, the rotation direction of the petal-like patterns of the group velocity remains counterclockwise, which means that it is directly dependent on the sign of the OAM of the strong coupling LG field.

Fig. 9. The imaginary part of the susceptibility (left column), dispersion slope (middle column), and dimensionless group velocity (right column) profiles as a function of $x$ and $y$ for three different values of the OAM, i.e. $l_{42}=0,1,2$, and $\phi _0=\pi$ with the same parameters of Fig. 6.

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4. Conclusion

In summary, we have investigated the group velocity of the probe LG beam in a dispersive medium, by deriving two general analytical expressions for the group velocity on and out of the optical axis. The presented results, which are obtained with the aid of the well-known formula of the group velocity ($v_g=|\nabla \partial _\omega \Phi |^{-1}$), are consistent with the group velocity of a LG beam in free space. In the first scenario, the effect of the doughnut-like intensity profile of the probe LG beam is studied on the group velocity magnitude and its $z$-component along the propagation direction. It has been shown that they both are of equal value just at the waist of the beam and the Rayleigh range. Through the second scenario, we have demonstrated how the helical phase front of another LG field can affect the group velocity of the probe LG field. In this case, the gain-assisted superluminal group velocity may exist in the normal dispersion regions of a spatially dependent dispersive medium, while the absorption-assisted subluminal group velocity can be found in the anomalous dispersion regions simultaneously. We have proved that this strange behavior is physically rooted in the classical interference of two coupling planar and a LG fields inside the dispersive medium, which leads to the distortion in the phase front of the probe LG field. It would be valuable to find some applications of our reported results in the improvement of the information transmission in optical communications.

Funding

Z. Amini Sabegh acknowledges financial support from Iran’s National Elites Foundation (Shahid Chamran’s Scientific Prize, (Grant No. 15/10597).

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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Superluminal light propagation in a normal dispersive medium

Abstract

1. Introduction

2. Group velocity of the LG field in a dispersive medium

3. Results and discussion

3.1 Group velocity behavior in a uniformly dispersive medium

3.2 Group velocity behavior in a spatially dependent dispersive medium

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (19)

Optics Express