Propagation of Gaussian vortex beams in electromagnetically induced transparency media

Yong Wang; Yong Wang; Miaojun Guo; Miaojun Guo; Jinze Wu; Jinze Wu; Jinhong Liu; Xudong Yang; Jinhong Li

doi:10.1364/OE.472845

1. Introduction

As an important branch of quantum optics, the study of the coherent interaction between light and atoms has made great progress in the past decades owing to its widespread applications in the fields of quantum information processing [1,2] and precision metrology [3,4]. One of the most remarkable phenomena of the coherent light-atom interaction is the electromagnetically induced transparency (EIT), which renders an atomic medium transparent for a resonant probe field when an additional coupling field is applied to the medium [5–7]. A probe pulse propagating in an EIT medium experiences an extremely steep anomalous dispersion, which results in an ultraslow group velocity down to a few meters per second [8]. This fascinating effect has been used to implement light storage and quantum memory [9–12]. Experimental and theoretical studies show that EIT can substantially enhance the nonlinear parametric processes in atoms [13–15], enabling the efficient generation of non-classical light, including narrowband biphotons [16,17] and twin beams [18,19], which are crucial resources in quantum communications [20–23]. Besides, owing to the flexibly controllable properties, EIT media provide a powerful platform to study various physical phenomena, such as optical non-reciprocity [24–26], parity-time symmetry [27–29], superradiance lattice [30–33], Goos-Hänchen and Imbert-Fedorov shifts [34,35], to mention a few. In addition to atomic systems, EIT has also been widely studied in superconducting circuits [36], quantum dots [37], optomechanical systems [38], nanoplasmonics [39], and so on, making it possible to realize integrated EIT-based components for quantum engineering.

Early studies and applications of EIT are concerned with the frequency spectral properties of the media and treat the involved optical fields as plane waves with uniform spatial distributions of amplitude, phase, and polarization [5]. However, practical light beams usually have nonuniform spatial structures. Over the past few years, the study of light beams with diverse spatial or spatiotemporal structures has rapidly advanced [40,41], bringing breakthroughs to the fields of optical sensing [42,43], optical communications [44], super-resolution imaging [45,46], optical tweezers [47], etc. Such structured light can be generated by utilizing spatial light modulators, digital micromirror devices, geometric phase plates, metasurfaces, etc., on various platforms including free space, fiber, and integrated devices [48]. An example of structured light with fundamental and practical interest is vortex beams, which are characterized by a helical phase factor $e^{\mathrm {i}l\theta }$ ($l$ is an integer number) with a phase singularity that appears as an isolated dark spot and that possesses a certain topological charge [49]. More significantly, it has been proven that one photon of a vortex beam carries quantized orbit angular momentum equivalent to $l\hbar$ [50–52], offering the possibility to encode and process information in high-dimensional Hilbert spaces and enabling higher information capacity [53,54]. Due to these intriguing features, the study of vortex beams has attracted enormous interests, bringing about the field of singular optics [55,56].

In recent years, considerable attention has been devoted to the study of EIT with vortex beams, and a variety of novel functionalities and applications have been proposed or implemented. It has been demonstrated that the vortex of a beam can be converted into atomic coherence and thus stored in an atomic medium by using the dynamic EIT protocol [57–59]. Due to the topological stability of the vortex, the retrieved beam still maintains the phase singularity [60]. Another example is the so-called spatially dependent EIT, which manifests as the spatially structured absorption profile by probing an EIT medium with a vortex beam [61–63]. Very recently, EIT with vortex beams is employed to realize a spatial atomic compass, which enables the precise detection of 3D magnetic field alignment [64]. In all these investigations and applications, the light propagation within the atomic media is an important research topic [65–68]. It is expected that the study of the propagation of vortex beams in EIT media holds the promise of opening up possibilities for developing new EIT-based techniques and applications.

In this article, we focus on the study of the propagation and evolution of Gaussian vortex beams in an EIT medium with $\Lambda$-type three-level atoms. Utilizing the generalized Huygens-Fresnel principle, we derive the analytical expressions of fully and partially coherent Gaussian vortex beams propagating in the EIT medium, which is characterized by an $ABCD$ transfer matrix. Based on the analytical expressions, we show the periodic focusing and diverging of the intensity distributions of the beams. For the fully coherent beam, the location and the topological charge of its phase singularity keep unchanged during propagation. However, for the partially coherent beam, the phase singularity undergoes splitting and recombination, and meanwhile new phase singularities with opposite topological charge are generated. These novel features offer the capability to manipulate the phase singularities of vortex beams via EIT, and may find applications in singular optics and topological photonics.

2. Propagation of fully coherent Gaussian vortex beams in EIT media

We consider the system schematically shown in Fig. 1. The atomic medium is comprised by, e.g., cesium atoms with a $\Lambda$-type three-level system including an excited state $|{e}\rangle =|{6\mathrm {P}_{1/2},F'=3}\rangle$, and two ground states $|{m}\rangle =|{6\mathrm {S}_{1/2},F=4}\rangle$ and $|{g}\rangle =|{6\mathrm {S}_{1/2},F=3}\rangle$ in the D1 line. A fundamental Gaussian beam $E_{\rm {c}}$ and a Gaussian vortex beam $E_{\rm {p}}$ propagate in the medium. They are referred to as the coupling and probe beams, and drive the transitions $|{e}\rangle \leftrightarrow |{m}\rangle$ and $|{e}\rangle \leftrightarrow |{g}\rangle$, respectively. The coupling beam dramatically modifies the optical properties of the medium. Here, we study the propagation of the probe beam, a Gaussian vortex beam, in such a modulated medium.

Fig. 1. (a) Sketch of the system. (b) Relevant energy levels in the cesium D1 line.

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The fundamental Gaussian coupling beam can be expressed as

(1)$${E_{\rm{c}}} = {E_{{\rm{c}},0}}\frac{{{z_{\rm{R}}}}}{{{z_{\rm{R}}} - {\rm{i}}z}}\exp\left({-\frac{k_0}{2}\frac{{{r^2}}}{{{z_{\rm{R}}} - {\rm{i}}z}}}\right) ,$$

where $E_{{\rm {c}},0}$ is a constant, $z_{\rm {R}}=k_0w_{\rm {c},0}^2/2$ is the Rayleigh range, $w_{{\rm {c}},0}$ is the radius of the beam waist, $k_0=2\pi /\lambda$ is the wave number with $\lambda$ being the wavelength, and $r=\sqrt {x^2+y^2}$. We assume that the Rayleigh range $z_{\rm {R}}$ is much larger than the length of the medium, i.e., ${z_R}\gg z$ in the medium. Under this assumption, the diffraction of the coupling beam can be neglected, and its field distribution can be approximately written as ${E_{\mathrm {c}}}\approx E_{{\mathrm {c,0}}}\exp ({-{r^2}/w_{{\mathrm {c}},0}^2})$. Furthermore, this is a necessary condition to obtain the analytical results in the following calculation.

In the approximation that the probe beam is much weaker than the coupling beam, i.e., $|E_{\rm {p}}|\ll |E_{\rm {c}}|$, and thus almost all the atoms are populated in the state $|{g}\rangle$ via optical pumping, the refractive index experienced by the weak probe beam is given by [69,70]

(2)$$n(r) = \sqrt{1 + \chi(r)} \approx 1 + \frac{1}{2}\chi(r),$$

with the susceptibility

(3)$$\chi(r) = \frac{{N{d^2}}}{{{\epsilon_0}\hbar}}\frac{{{\Delta_{\rm{p}}} - {\Delta_{\rm{c}}} - {\rm{i}}\gamma}}{{({{\Delta_{\rm{p}}} - {\rm{i}}\Gamma {\rm{/2}}})({{\Delta_{\rm{p}}} - {\Delta_{\rm{c}}} - {\rm{i}}\gamma}) - \Omega _{\rm{c}}^2(r)}},$$

where $N$ is the atomic number density, $d$ is the dipole matrix element, $\epsilon _0$ is the vacuum permittivity, $\hbar$ is the reduced Planck’s constant, $\Gamma$ is the spontaneous decay rate of the excited state $|{e}\rangle$, $\gamma$ is the dephasing rate between the two ground states $|{m}\rangle$ and $|{g}\rangle$, ${\Omega _{\rm {c}}}(r)=dE_{\rm {c}}(r)/\hbar ={\Omega _{{\rm {c}},0}}\exp (-r^2/w_{\rm {c},0}^2)$ is the coupling Rabi frequency with ${\Omega _{{\rm {c}},0}}=d{E_{{\rm {c,0}}}}/\hbar$, ${\Delta _{\rm {c}}}$ and ${\Delta _{\rm {p}}}$ are the coupling and probe frequency detuning, respectively. If the coupling beam is absent, i.e., ${\Omega _{\rm {c}}}=0$, the probe beam undergoes a homogeneous refractive index $n=1+\frac {{N{d^2}}}{{2{\epsilon _0}\hbar }}\frac {1}{{{\Delta _{\rm {p}}}-{\rm {i}}\Gamma {\rm {/2}}}}$. In the presence of the coupling beam, which spatially modulates the refractive index $n(r)$ via the spatially dependent Rabi frequency ${\Omega _{\rm {c}}}(r)$, the atomic medium becomes a gradient-index medium for the probe beam.

The spatial modulation of the refractive index is a joint effect of EIT and Autler-Townes splitting (ATS), two similar phenomena but with different underlying physics [71]. They coexist in the atom-light interaction of the three-level system with relative weights determined by the Rabi frequency $\Omega _{\rm {c}}$ [72–74]. As $\Omega _{\rm {c}}$ increases, the EIT effect becomes weak while the ATS effect becomes strong. When $\Omega _{\rm {c}}<\Gamma /2+\gamma$, EIT plays a dominant role while ATS can be ignored. When $\Omega _{\rm {c}}\gg \Gamma /2+\gamma$, the former is negligible while the later dominates. In the intermediate region, both of them make contributions. In the present article, we use the term of EIT medium for convenience though ATS also occurs for the parameters used in the calculation.

In the following, we consider the case that the width of the probe beam is much smaller than that of the coupling beam [see Fig. 1(a)], and therefore, keeping ${r^2}$ up to the first order in Eq. (2), the refractive index can be approximately expressed as

(4)$$n(r) = n_0 - \frac{1}{2}{n_0}{\beta^2}{r^2},$$

with $n_0 = 1 + \chi (0)/2$ and

(5)$$\beta = \sqrt{-\frac{1}{{{n_0}}}{{\left.{\frac{{\partial\chi}}{{\partial{r^2}}}}\right|}_{r = 0}}} = 2\sqrt{\frac{{\alpha\tilde\gamma}}{{2{{({\tilde\Gamma\tilde\gamma - \Omega_{{\rm{c}},0}^2})}^2} + \alpha\tilde\gamma({\tilde\Gamma\tilde\gamma - \Omega_{{\rm{c}},0}^2})}}\frac{{\Omega_{{\rm{c}},0}^2}}{{w_{{\rm{c}},0}^2}}} ,$$

where $\alpha = N{d^2}/({{\epsilon _0}\hbar })$, $\tilde \Gamma = {\Delta _{\rm {p}}} - {\rm {i}}\Gamma /2$, and $\tilde \gamma = {\Delta _{\rm {p}}} - {\Delta _{\rm {c}}} - {\rm {i}}\gamma$. For an optical system with a parabolic refractive-index profile as expressed by Eq. (4), the $ABCD$ transfer matrix for paraxial rays reads [75]

(6)$$\left[{\begin{array}{cc}A & B\\C & D\end{array}}\right] = \left[{\begin{array}{cc}{\cos({\beta z})} & {\sin({\beta z})/\beta}\\ {-\beta\sin({\beta z})} & {\cos({\beta z})}\end{array}}\right].$$

The $ABCD$ matrix describes the transformation of the paraxial rays through the optical system, i.e., the EIT medium herein. It is a powerful tool to study the propagation of light beams in various optical systems within the paraxial region. The parameter $\beta$, which uniquely determines the $ABCD$ transfer matrix, plays a key role in the study of the propagation of the Gaussian vortex probe beam in the EIT medium. In Fig. 2 we plot the parameter $\beta$ as a function of the probe detuning ${\Delta _{\rm {p}}}$ and the coupling Rabi frequency ${\Omega _{{\rm {c}},0}}$. It is seen that $\beta$ is generally complex, and its real and imaginary parts get larger values in the region near ${\Delta _{\rm {p}}} \pm {\Omega _{{\rm {c}},0}} = 0$ [see Fig. 2(a)]. Furthermore, $\beta$ is approximately real, i.e., $\rm {Im}(\beta ) \approx 0$, when ${\Delta _{\rm {p}}} + {\Omega _{{\rm {c}},0}} = 0$ [see the dashed lines in Figs. 2(a) and 2(b)]. This can be more clearly seen in Fig. 2(c), where $\beta$ as a function of ${\Omega _{{\rm {c}},0}}$ with ${\Delta _{\rm {p}}} = -{\Omega _{{\rm {c}},0}}$ is plotted. Under such condition, the probe beam experiences focusing and diverging periodically during propagation, as will be discussed in the following. However, when this condition is not satisfied, i.e., ${\Delta _{\rm {p}}} \ne -{\Omega _{{\rm {c}},0}}$, the calculation diverges and no meaningful result is obtained.

Fig. 2. (a) $\beta$ versus $\Delta _\mathrm {p}$ and $\Omega _{\mathrm {c},0}$. The white dashed lines denote $\Delta _\mathrm {p}+\Omega _{\mathrm {c},0}=0$. (b) $\beta$ versus $\Delta _\mathrm {p}$ with different $\Omega _{\mathrm {c},0}$. (c) $\beta$ versus $\Omega _{\mathrm {c},0}$ with $\Delta _\mathrm {p}=-\Omega _{\mathrm {c},0}$. The parameters are: $\lambda =894.6\,\mathrm {nm}$, $w_{\mathrm {c},0}=100\,\mathrm {\mu }\mathrm {m}$, $\Delta _{\mathrm {c}}=0$, $N=5\times 10^{17}\,\mathrm {m}^{-3}$, $d=2.70\times 10^{-29}\,\mathrm {C}\cdot \mathrm {m}$, $\Gamma =2\pi \times 4.58\,\mathrm {MHz}$, $\gamma =2\pi \times 0.10\,\mathrm {MHz}$, $\hbar =1.05\times 10^{-34}\,\mathrm {J}\cdot \mathrm {s}$, and $\epsilon _0=8.85\times 10^{-12}\,\mathrm {F}\cdot \mathrm {m}^{-1}$.

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The Gaussian vortex probe beam at the entrance of the atomic medium ($z=0$) can be written as

(7)$${E_{\rm{p}}}({r,\theta,0}) = {\left({\frac{{\sqrt2r}}{{{w_{{\rm{p}},0}}}}}\right)^m}\exp\left({-\frac{{{r^2}}}{{w_{{\rm{p}},0}^2}}}\right)\exp({{\rm{i}}m\theta}),$$

where ${w_{\mathrm {p},0}}$ is the radius of the beam waist, $(r,\theta,z)$ is the cylindrical coordinate, $m$ is the topological charge, and $\exp (\mathrm {i}m\theta )$ is the helical phase. Here the beam waist is assumed to be at $z=0$. By utilizing the formula [76,77]

(8)$$\exp({{\rm{i}}m\theta}){r^m} = \frac{1}{{{2^m}}}\sum_{s = 0}^m{{{\rm{i}}^s}}\left( {\begin{array}{c}m\\s\end{array}}\right){H_{m - s}}(x){H_s}(y),$$

Eq. (7) can be rewritten in the Cartesian coordinate system as

(9)$$E_{\rm{p}}(x,y,0) = \frac{1}{{{2^m}}}\sum_{s = 0}^m {{{\rm{i}}^s}}\left( {\begin{array}{c}m\\s\end{array}}\right){H_{m - s}}\left( {\frac{{\sqrt2x}}{{{w_{{\rm{p}},0}}}}}\right){H_s}\left({\frac{{\sqrt2y}}{{{w_{{\rm{p}},0}}}}}\right)\exp\left({- \frac{{{x^2} + {y^2}}}{{w_{{\rm{p}},0}^2}}}\right),$$

where ${H_\cdot }(\cdot )$ and $\left ({\begin {array}{c}\cdot \\\cdot \end {array}}\right )$ represent the Hermitian polynomial and the binomial coefficient, respectively.

The field distribution of the probe beam propagating in the EIT medium can be calculated by using the generalized Huygens-Fresnel integral formula [75]

(10)$$\begin{aligned} {E_{\rm{p}}}({x,y,z}) = &\frac{{{\rm{i}}k}}{{2{\rm{\pi}}B}}\exp(-{\rm{i}}kz)\int{\int{{\rm{d}}x'{\rm{d}}y'}}{E_{\rm{p}}}({x',y',0})\\ &\times\exp\left\{{-\frac{{{\rm{i}}k}}{{2B}}\left[{A({{{x'}^2}+{{y'}^2}})-2({x'x + y'y}) +D({{x^2}+{y^2}})}\right]}\right\}, \end{aligned}$$

with $k=n_0k_0$ being the wave number in the medium. Substituting Eq. (9) into Eq. (10), we obtain

(11)$$\begin{aligned} {E_{\rm{p}}}({x,y,z}) = &\frac{{{\rm{i}}k}}{{2{\rm{\pi}}B}}\exp\left({ - {\rm{i}}kz}\right)\frac{1}{{{2^m}}}\sum_{s = 0}^m {{{\rm{i}}^s}}\left( {\begin{array}{*{20}{c}}m\\s\end{array}}\right)\exp\left[{ - \frac{{{{\rm{i}}kMD + {k^2}w_{{\rm{p}},0}^2}}}{{2MB}}({{x^2} + {y^2}})}\right]\\ &\times \frac{{4{\rm{\pi}}B}}{M}{\left({1 - \frac{{4B}}{M}}\right)^{\frac{m}{2}}}{H_{m - s}}\left({\frac{{\sqrt2{\rm{i}}k{w_{{\rm{p}},0}}x}}{{\sqrt{{M^2} - 4MB}}}}\right){H_s}\left({\frac{{\sqrt2{\rm{i}}k{w_{{\rm{p}},0}}y}}{{\sqrt{{M^2} - 4MB}}}}\right), \end{aligned}$$

with $M = 2B + {\rm {i}}kAw_{{\rm {p}},0}^2$. Equation (11) is the main result of this section. The distributions of the intensity and the phase during propagation are given by $I({x,y,z}) = |{{E_{\rm {p}}}({x,y,z})}|^2$ and $\phi ({x,y,z}) = \arg {E_{\rm {p}}}({x,y,z})$, respectively.

Fig. 3. The distribution of the intensity $I$ in the $xz$-plane ($y$ = 0) of the coherent Gaussian vortex beam propagating in the EIT medium with ${w_{{\rm {p}},0}} = 10\,\mathrm {\mu } \rm {m}$ ($z_{\rm {R}}=0.35\,\rm {mm}$) and ${\Delta _{\rm {p}}} = -{\Omega _{{\rm {c}},0}}$. The other parameters are as in Fig. 2.

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Fig. 4. The distributions of the intensity $I$ and the phase $\phi$ in the $xy$-plane of the coherent Gaussian vortex beam propagating in the EIT medium with ${\Omega _{\mathrm {c},0}} = 2\pi \times 10\,\mathrm {MHz}$. The other parameters are as in Fig. 3.

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Figure 3 shows the intensity distribution in the $xz$-plane ($y=0$) of the probe beam propagating in the EIT medium. The beam experiences focusing and diverging periodically, and meanwhile the intensity becomes weak due to the atomic absorption. The period (denoted by $\Lambda$) is determined by the parameter $\beta$, i.e., $\Lambda = 2\pi /\rm {Re}(\beta )$ and thus depends on the coupling Rabi frequency ${\Omega _{{\rm {c}},0}}$ [see Fig. 2(c)]. The period $\Lambda$ decreases as the Rabi frequency ${\Omega _{{\rm {c}},0}}$ increases. It is to be noted that the intensity in the beam center is always zero during propagation. This fact can be also seen in Fig. 4, which plots the distributions of the intensity and the phase of the beam propagating at different $z$. The zero intensity of the beam center is intimately associated with the phase singularity. The intensity at the position of the phase singularity is exactly zero. This is the fundamental property of the coherent vortex beam, and is still true when the beam propagates in the EIT medium. Additionally, Fig. 4 shows the periodic evolution of the phase distribution. For the odd topological charge, the phase at $z = 0,\Lambda /2,\Lambda,\ldots$ has the same distribution; while for the even topological charge, the phase at $z = 0,\Lambda,2\Lambda,\ldots$ has the same distribution. The phase singularity keeps unchanged during propagation. As will be discussed in the following section, this is not the case for the partially coherent vortex beam.

3. Propagation of partially coherent Gaussian vortex beams in EIT media

In this section, we consider the partially coherent Gaussian vortex probe beam propagating in the EIT medium. The statistical properties of the partially coherent beam are well described by the cross-spectral density function (CSDF) ${W_{\rm {p}}}({{x_1},{y_1},{x_2},{y_2},z}) = \langle {E_{\rm {p}}^*({{x_1},{y_1},z}){E_{\rm {p}}}({{x_2},{y_2},z})}\rangle$, where $\langle \cdot \rangle$ represents the ensemble average. Substituting Eq. (9) into the definition of the CSDF and assuming that the beam has Gaussian degree of coherence, we obtain

(12)$$\begin{aligned} W_{\mathrm{p}}(x_1,y_1,x_2,y_2,0) = &\frac{1}{2^{2m}}\sum^m_{s_1=0}\sum^m_{s_2=0}(-\mathrm{i})^{s_1}\mathrm{i}^{s_2}\left(\begin{array}{c}m\\s_1\end{array}\right)\left(\begin{array}{c}m\\s_2\end{array}\right)H_{m-s_1}\left(\frac{\sqrt{2}x_1}{w_{\mathrm{p},0}}\right)H_{s_1}\left(\frac{\sqrt{2}y_1}{w_{\mathrm{p},0}}\right)\\ &\times H_{m-s_2}\left(\frac{\sqrt{2}x_2}{w_{\mathrm{p},0}}\right)H_{s_2}\left(\frac{\sqrt{2}y_2}{w_{\mathrm{p},0}}\right)\exp\left(-\frac{x_1^2+y_1^2+x_2^2+y_2^2}{w_{\mathrm{p},0}^2}\right)\\ &\times \exp\left[-\frac{(x_1-x_2)^2+(y_1-y_2)^2}{2\sigma^2}\right], \end{aligned}$$

where $\sigma$ denotes the transverse coherence width. When $\sigma \rightarrow \infty$, Eq. (12) represents a fully coherent Gaussian vortex beam.

The generalized Huygens-Fresnel integral formula to calculate the CSDF of the probe beam propagating in the EIT medium is [75]

(13)$$\begin{aligned} W_{\mathrm{p}}(x_1,y_1,x_2,y_2,z) = &\frac{|k|^2}{(2{\pi}B)^2}\exp[2\mathrm{Im}(k)z]\int{\int{\int{\int{\mathrm{d}{x'_1}\mathrm{d}{y'_1}\mathrm{d}{x'_2}\mathrm{d}{y'_2}}}}}W_{\mathrm{p}}(x'_1,y'_1,x'_2,y'_2,0)\\ &\times \exp\left\{{\frac{{{\mathrm{i}}{k^*}}}{{2B}}\left[{A({{x'_1}^2 + {y'_1}^2}) - 2(x'_1 x_1 + y'_1 y_1) + D(x_1^2 + y_1^2)}\right]}\right.\\ &\left.{ - \frac{\mathrm{i}k}{2B}\left[A({x'_2}^2 + {y'_2}^2) - 2(x'_2 x_2 + y'_2 y_2) + D(x_2^2 + y_2^2)\right]}\right\}. \end{aligned}$$

Substituting Eq. (12) into Eq. (13), we obtain

(14)$$\begin{aligned} W_{\mathrm{p}}(x_1,y_1,x_2,y_2,z)=&\frac{|k|^2}{(2{\pi}B)^2} \exp{[2\mathrm{Im}(k)z]}P\\ &\times\sum_{s_1=0}^m\sum_{s_2=0}^m \left(\begin{array}{c}m\\s_1\end{array}\right) \left(\begin{array}{c}m\\s_2\end{array}\right) \sum_{c_1=0}^{\left[\frac{m-s_1}{2}\right]} \sum_{d_1=0}^{m-s_2} \sum_{e_1=0}^{\left[\frac{d_1}{2}\right]} \sum_{c_2=0}^{\left[\frac{s_1}{2}\right]} \sum_{d_2=0}^{s_2} \sum_{e_2=0}^{\left[\frac{d_2}{2}\right]}Q_1Q_2, \end{aligned}$$

with

\begin{aligned} P = &\frac{1}{2^{2m}}\frac{\pi^2}{M_1M_2}\left(\frac{1-G^2}{2}\right)^{\frac{m}{2}} \exp\left[-\frac{\mathrm{i}k^*D}{2B}(x_1^2+y_1^2)+\frac{\mathrm{i}kD}{2B}(x_2^2+y_2^2)\right]\\ &\times\exp\left[-\left(\frac{k^*\sqrt{x_1^2+y_1^2}}{2B\sqrt{M_2}} - \frac{k\sqrt{x_2^2+y_2^2}}{4M_1\sqrt{M_2}B\sigma^2}\right)^2 - \frac{k^2(x_2^2+y_2^2)}{4M_1B^2}\right], \end{aligned}

\begin{aligned} Q_1 = &\mathrm{i}^{s_2-s_1}({-}1)^{c_1+c_2+e_1+e_2} \left(\begin{array}{c}m-s_2\\d_1\end{array}\right) \left(\begin{array}{c}s_2\\d_2\end{array}\right) \frac{d_1!d_2!}{e_1!e_2!(d_1-2e_1)!(d_2-2e_2)!}\\ &\times\frac{(m-s_1)!}{c_1!(m-s_1-2c_1)!} \frac{s_1!}{c_2!(s_1-2c_2)!} \left(\frac{2\sqrt{2}}{w_{\mathrm{p},0}}\right)^{m-2c_1-2c_2}\\ &\times\left[\frac{\sqrt{2}G}{\sigma^2\sqrt{M_1(1-G^2)}}\right]^{d_1+d_2-2e_1-2e_2} \left(\frac{1}{2\mathrm{i}\sqrt{M_2}}\right)^{m-2c_1-2c_2+d_1+d_2-2e_1-2e_2}, \end{aligned}

\begin{aligned} Q_2 = & H_{m-s_2-d_1}\left[-\frac{\mathrm{i}k\sqrt{2}Gx_2}{2B\sqrt{M_1(1-G^2)}}\right] H_{m-s_1-2c_1+d_1-2e_1}\left(-\frac{k^*x_1}{2\sqrt{M_2}B}+\frac{kx_2}{4M_1\sqrt{M_2}B\sigma^2}\right)\\ &\times H_{s_2-d_2}\left[-\frac{\mathrm{i}k\sqrt{2}Gy_2}{2B\sqrt{M_1(1-G^2)}}\right] H_{s_1-2c_2+d_2-2e_2}\left(-\frac{k^*y_1}{2\sqrt{M_2}B}+\frac{ky_2}{4M_1\sqrt{M_2}B\sigma^2}\right). \end{aligned}

Here $M_1=\frac {1}{w^2_{\mathrm {p},0}}+\frac {1}{2\sigma ^2}-\frac {\mathrm {i}kA}{2B}$, $M_2=\frac {1}{w^2_{\mathrm {p},0}}+\frac {1}{2\sigma ^2}+\frac {\mathrm {i}k^*A}{2B}-\frac {1}{4M_1\sigma ^4}$, and $G=\frac {\sqrt {2}}{{w^2_{\mathrm {p},0}}\sqrt {M_1}}$. Equation (14) is the main result of this section, from which we can obtain the distributions of the intensity and the phase. When $x_1=x_2=x$ and $y_1=y_2=y$, the CSDF represents the intensity distribution, i.e., $I(x,y,z) = W_{\mathrm {p}}(x,y,x,y,z)$. The spectral degree of coherence is defined as $\mu (x_1,y_1,x_2,y_2,z)=W_{\mathrm {p}}(x_1,y_1,x_2,y_2,z)/\sqrt {W_{\mathrm {p}}(x_1,y_1,x_1,y_1,z)W_{\mathrm {p}}(x_2,y_2,x_2,y_2,z)}$. The phase distribution is $\phi (x,y,z)=\arg {\mu (x,y,x_{\mathrm {r}},y_{\mathrm {r}})}$ with $(x_{\mathrm {r}},y_{\mathrm {r}})$ being the reference point.

In Fig. 5, we plot the intensity distribution in the $xz$-plane ($y=0$) of the partially coherent probe beam propagating in the EIT medium with different topological charge $m$ and coherence width $\sigma$. The partially coherent beam also experiences periodic focusing and diverging during propagation. It is to be noted that the intensity of the beam center is not zero in the focal plane for the beam with small topological charge $m$ and coherence width $\sigma$. This is very different from the case of the fully coherent beam, in which the center intensity is always zero. As the topological charge $m$ or the coherence width $\sigma$ increases, the center intensity gradually decreases. For the beam with a large topological charge $m$ and a finite coherence width $\sigma$, our calculation shows that the center intensity can be very small but not zero. When the coherence width $\sigma$ goes to infinity, i.e., the beam becomes fully coherent, the center intensity tends to exactly zero, as discussed in Sec. 2.

Fig. 5. The distribution of the intensity $I$ in the $xz$-plane ($y = 0$) of the partially coherent Gaussian vortex beam propagating in the EIT medium. The parameters are as in Fig. 3.

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Figure 6 displays the distributions of the intensity and the phase of the partially coherent probe beam propagating in the EIT medium at different $z$ with the coherence width $\sigma = 2{w_{\rm {p},0}}$ and different topological charge $m$. It is seen that the evolution of the phase distribution during propagation exhibits interesting behaviors. Firstly, the phase singularity of the beam with positive (negative) topological charge of $m$ splits into $|m|$ phase singularities with topological charge of +1 (-1). (The case of negative topological charge is not shown in Fig. 6.) Secondly, new $|m|$ phase singularities with opposite topological charge are generated. It is the influence of the opposite topological charge that results in the nonzero center intensity. Thirdly, the multiple phase singularities recombine into one and the new generated phase singularities vanish at $z = \Lambda /2,\Lambda,\ldots$.

Fig. 6. The distributions of the intensity $I$ and the phase $\phi$ in the $xy$-plane of the partially coherent Gaussian vortex beam propagating in the EIT medium with $\sigma = 2w_{\rm {p},0}$ and different $m$. The other parameters are as in Fig. 3.

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The splitting and generation of the phase singularities are strongly dependent on the coherence width $\sigma$. Figure 7 shows the evolution of the intensity and the phase of the partially coherent probe beam with the topological charge $m=2$ and different coherence width $\sigma$. As the coherence width $\sigma$ increases, the splitting effect of the phase singularity becomes weak, and the distance between the original and the generated phase singularities gets large. Meanwhile, the intensity in the beam center decreases. For the fully coherent beam, i.e., $\sigma \rightarrow \infty$, the splitting and generation of the phase singularity vanishes, and the center intensity is exactly zero. This result is further confirmed by the calculation of the phase distribution for different $\sigma$ up to $100w_{\rm {p},0}$ (not shown Fig. 7) with a resolution of $256\times 256$.

Fig. 7. The distributions of the intensity $I$ and the phase $\phi$ in the $xy$-plane of the partially coherent Gaussian vortex beam propagating in the EIT medium with $m=2$ and different $\sigma$. The other parameters are as in Fig. 3.

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4. Conclusion

In conclusion, employing the generalized Huygens-Fresnel principle, we have derived the analytical expressions of the fully and partially coherent Gaussian vortex beams propagating in the EIT medium. Based on the analytical expressions, we have studied the evolution of the intensity and phase distributions of the beams. Both the fully and partially coherent beams experience focusing and diverging periodically with a period which can be tuned by the coupling Rabi frequency. For fully coherent beam, the location and the topological charge of the phase singularity keeps unchanged during propagation. However, the phase singularity of the partially coherent beam with the positive (negative) topological charge of $m$ splits into $|m|$ phase singularities with topological charge of +1 (−1) during propagation. Moreover, new phase singularities with opposite topological charge are generated. As the coherence width goes to infinity, the splitting and generation of the phase singularities vanish. Our results provide the possibility of using EIT media to study the propagation of various structured light in gradient-index media. Compared with the commonly used gradient-index fiber with a fixed refractive-index profile, the EIT medium has a configurable refractive-index profile that can be flexibly modulated by the coupling field. Furthermore, the EIT medium can be employed to manipulate the phase singularity of vortex beams, and holds promise in singular optics and topological photonics.

Funding

National Natural Science Foundation of China (12004334, 12104332); Taiyuan University of Science and Technology Scientific Research Initial Funding (20212076, 20222061); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2020L0368); China Postdoctoral Science Foundation (2020M671686).

Disclosures

The authors declare no conflicts of interest.

Data Availability

No data were generated or analyzed in the presented research.

References

1. C. Monroe, “Quantum information processing with atoms and photons,” Nature 416(6877), 238–246 (2002). [CrossRef]

2. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008). [CrossRef]

3. D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007). [CrossRef]

4. N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10⁻¹⁸ instability,” Science 341(6151), 1215–1218 (2013). [CrossRef]

5. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]

6. K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66(20), 2593–2596 (1991). [CrossRef]

7. J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A 51(1), 576–584 (1995). [CrossRef]

8. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

9. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432(7016), 482–486 (2004). [CrossRef]

10. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3(12), 706–714 (2009). [CrossRef]

11. M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010). [CrossRef]

12. G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111(3), 033601 (2013). [CrossRef]

13. M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77(21), 4326–4329 (1996). [CrossRef]

14. D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004). [CrossRef]

15. Z.-P. Wang and S.-X. Zhang, “High-efficiency four-wave mixing in a five-level atomic system based on two-electromagnetically induced transparency in the ultraslow propagation regime,” Phys. Scr. 81(3), 035401 (2010). [CrossRef]

16. S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100(18), 183603 (2008). [CrossRef]

17. C. Shu, P. Chen, T. K. A. Chow, L. Zhu, Y. Xiao, M. M. T. Loy, and S. Du, “Subnatural-linewidth biphotons from a Doppler-broadened hot atomic vapour cell,” Nat. Commun. 7(1), 12783 (2016). [CrossRef]

18. C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32(2), 178–180 (2007). [CrossRef]

19. M. Guo, H. Zhou, D. Wang, J. Gao, J. Zhang, and S. Zhu, “Experimental investigation of high-frequency-difference twin beams in hot cesium atoms,” Phys. Rev. A 89(3), 033813 (2014). [CrossRef]

20. L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001). [CrossRef]

21. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423(6941), 731–734 (2003). [CrossRef]

22. T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96(9), 093604 (2006). [CrossRef]

23. N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1(3), 165–171 (2007). [CrossRef]

24. D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110(9), 093901 (2013). [CrossRef]

25. S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018). [CrossRef]

26. G. Lin, S. Zhang, Y. Hu, Y. Niu, J. Gong, and S. Gong, “Nonreciprocal amplification with four-level hot atoms,” Phys. Rev. Lett. 123(3), 033902 (2019). [CrossRef]

27. P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016). [CrossRef]

28. Y. Jiang, Y. Mei, Y. Zuo, Y. Zhai, J. Li, J. Wen, and S. Du, “Anti-parity-time symmetric optical four-wave mixing in cold atoms,” Phys. Rev. Lett. 123(19), 193604 (2019). [CrossRef]

29. Y. He, J. Wu, Y. Hu, J.-X. Zhang, and S.-Y. Zhu, “Unidirectional reflectionless anti-parity-time-symmetric photonic lattices of thermal atoms,” Phys. Rev. A 105(4), 043712 (2022). [CrossRef]

30. D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, “Superradiance lattice,” Phys. Rev. Lett. 114(4), 043602 (2015). [CrossRef]

31. D.-W. Wang, H. Cai, L. Yuan, S.-Y. Zhu, and R.-B. Liu, “Topological phase transitions in superradiance lattices,” Optica 2(8), 712–715 (2015). [CrossRef]

32. H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122(2), 023601 (2019). [CrossRef]

33. Y. He, R. Mao, H. Cai, J.-X. Zhang, Y. Li, L. Yuan, S.-Y. Zhu, and D.-W. Wang, “Flat-band localization in Creutz superradiance lattices,” Phys. Rev. Lett. 126(10), 103601 (2021). [CrossRef]

34. Ziauddin, S. Qamar, and M. S. Zubairy, “Coherent control of the Goos-Hänchen shift,” Phys. Rev. A 81(2), 023821 (2010). [CrossRef]

35. X.-J. Zhang, H.-H. Wang, Z.-P. Liang, Y. Xu, C.-B. Fan, C.-Z. Liu, and J.-Y. Gao, “Goos-Hänchen shift in a standing-wave-coupled electromagnetically-induced-transparency medium,” Phys. Rev. A 91(3), 033831 (2015). [CrossRef]

36. Z. Dutton, K. V. R. M. Murali, W. D. Oliver, and T. P. Orlando, “Electromagnetically induced transparency in superconducting quantum circuits: Effects of decoherence, tunneling, and multilevel crosstalk,” Phys. Rev. B 73(10), 104516 (2006). [CrossRef]

37. S. Marcinkevičius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett. 92(4), 041113 (2008). [CrossRef]

38. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010). [CrossRef]

39. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef]

40. D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic, 2008).

41. A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nat. Photonics 15(4), 253–262 (2021). [CrossRef]

42. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef]

43. V. D’Ambrosio, N. Spagnolo, L. D. Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013). [CrossRef]

44. A. E. Willner, K. Pang, H. Song, K. Zou, and H. Zhou, “Orbital angular momentum of light for communications,” Appl. Phys. Rev. 8(4), 041312 (2021). [CrossRef]

45. Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5(2), 86–92 (2018). [CrossRef]

46. M. Yoshida, Y. Kozawa, and S. Sato, “Subtraction imaging by the combination of higher-order vector beams for enhanced spatial resolution,” Opt. Lett. 44(4), 883–886 (2019). [CrossRef]

47. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]

48. J. Wang and Y. Liang, “Generation and detection of structured light: A review,” Front. Phys. 9, 688284 (2021). [CrossRef]

49. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]

50. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]

51. D. L. Andrews and M. Babiker, The Angular Momentum of Light (Cambridge University, 2013).

52. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015). [CrossRef]

53. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]

54. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef]

55. G. J. Gbur, Singular optics (CRC, 2017).

56. M. Soskin, S. V. Boriskina, Y. Chong, M. R. Dennis, and A. Desyatnikov, “Singular optics and topological photonics,” J. Opt. 19(1), 010401 (2017). [CrossRef]

57. D. Moretti, D. Felinto, and J. W. R. Tabosa, “Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79(2), 023825 (2009). [CrossRef]

58. L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, “Reversible optical memory for twisted photons,” Opt. Lett. 38(5), 712–714 (2013). [CrossRef]

59. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014). [CrossRef]

60. R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98(20), 203601 (2007). [CrossRef]

61. N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114(12), 123603 (2015). [CrossRef]

62. H. R. Hamedi, V. Kudriašov, J. Ruseckas, J. Ruseckas, and G. Juzeliūnas, “Azimuthal modulation of electromagnetically induced transparency using structured light,” Opt. Express 26(22), 28249–28262 (2018). [CrossRef]

63. X. Yang, Y. Chen, J. Wang, Z. Dou, M. Cao, D. Wei, H. Batelaan, H. Gao, and F. Li, “Observing quantum coherence induced transparency of hybrid vector beams in atomic vapor,” Opt. Lett. 44(11), 2911–2914 (2019). [CrossRef]

64. F. Castellucci, T. W. Clark, A. Selyem, J. Wang, and S. Franke-Arnold, “Atomic compass: Detecting 3D magnetic field alignment with vector vortex light,” Phys. Rev. Lett. 127(23), 233202 (2021). [CrossRef]

65. M. Mahmoudi and J. Evers, “Light propagation through closed-loop atomic media beyond the multiphoton resonance condition,” Phys. Rev. A 74(6), 063827 (2006). [CrossRef]

66. S. Kajari-Schröder, G. Morigi, S. Franke-Arnold, and G.-L. Oppo, “Phase-dependent light propagation in atomic vapors,” Phys. Rev. A 75(1), 013816 (2007). [CrossRef]

67. C. J. Gibson, P. Bevington, G.-L. Oppo, and A. M. Yao, “Control of polarization rotation in nonlinear propagation of fully structured light,” Phys. Rev. A 97(3), 033832 (2018). [CrossRef]

68. Z. Li, S. Franke-Arnold, T. W. Clark, J. Wang, D. Zhang, and C. Wang, “Transfer and evolution of structured polarization in a double-V atomic system,” Opt. Express 30(11), 19812–19823 (2022). [CrossRef]

69. J. Wu, J. Liu, Y. He, Y. Zhang, J. Zhang, and S. Zhu, “Quantum interference manipulation and enhancement with fluctuation-correlation-induced dephasing in an atomic system,” Phys. Rev. A 98(4), 043829 (2018). [CrossRef]

70. J. Liu, J. Wu, Y. Hu, Y. Zhang, and J. Zhang, “Experimental demonstration of quantum interference modulation via precise dephasing control in atoms,” Opt. Commun. 466, 125655 (2020). [CrossRef]

71. T. Y. Abi-Salloum, “Electromagnetically induced transparency and Autler-Townes splitting: Two similar but distinct phenomena in two categories of three-level atomic systems,” Phys. Rev. A 81(5), 053836 (2010). [CrossRef]

72. P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning Autler-Townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107(16), 163604 (2011). [CrossRef]

73. L. Giner, L. Veissier, B. Sparkes, A. S. Sheremet, A. Nicolas, O. S. Mishina, M. Scherman, S. Burks, I. Shomroni, D. V. Kupriyanov, P. K. Lam, E. Giacobino, and J. Laurat, “Experimental investigation of the transition between Autler-Townes splitting and electromagnetically-induced-transparency models,” Phys. Rev. A 87(1), 013823 (2013). [CrossRef]

74. J. Liu, J. Wu, Y. Zhang, Y. He, and J. Zhang, “Influence of dephasing on the Akaike-information-criterion distinguishing of quantum interference and Autler-Townes splitting in coherent systems,” J. Opt. Soc. Am. B 37(1), 49–55 (2020). [CrossRef]

75. A. E. Siegman, Lasers (University Science Books, 1986).

76. I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

77. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef]

Propagation of Gaussian vortex beams in electromagnetically induced transparency media

Abstract

1. Introduction

2. Propagation of fully coherent Gaussian vortex beams in EIT media

3. Propagation of partially coherent Gaussian vortex beams in EIT media

4. Conclusion

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (7)

Equations (17)

Optics Express