Detecting the transverse spin density of light via electromagnetically induced transparency

Jinhong Liu; Jinze Wu; Jinze Wu

doi:10.1364/OE.463519

1. Introduction

The spin angular momentum (SAM) of light is associated with circular polarization [1]. For a beam of light that can be described within the paraxial approximation, the spin density, a quantity describing the local SAM, only has a longitudinal component aligned with the propagation direction. However, when light experiences strong spatial confinement and hence the paraxial approximation is invalid, the spin density can have transverse component [2–4]. Such transverse spin appears in tightly focused beams [5–7], evanescent waves [8], interference fields [9], scattering fields [10], surface plasmon polaritons [11,12], whispering gallery modes [13,14], and electromagnetic waves in bianisotropic media [15]. Recently, it was found that the transverse spin survives even in the focused or evanescent fields generated from unpolarized light [16,17]. In addition, the transverse spin is also discovered in sound waves [18,19] and gravitational waves [20].

One of the most important features of the transverse spin is the spin-momentum locking, which refers to the effect that the transverse spin changes its sign when the propagation direction of light is reversed [2,4,21]. The spin-momentum locking is a consequence of the time-reversal symmetry of Maxwell’s equations and is recognized as the quantum spin-Hall effect of light [22]. In the light-matter interaction that involves the transverse spin, the spin-momentum locking results in spin- and propagation-direction-dependent absorption, emission, and scattering of light. Such chiral interaction brings about the field of chiral quantum optics [23] and offers new functionalities including spin-dependent routing of photons [24], chiral nanophotonic interface [25], single-atom controlled optical isolator and circulator [26,27], to mention a few. Further applications of the transverse spin can be found in the fields of optical tweezer [28], microscopy [29,30], and sensing [31].

For the investigations and applications of the transverse spin, the detection of transverse spin density (TSD) is of crucial importance. Since the TSD is a near-field three-dimensional polarization quantity, its detection usually involves near-field and nanoprobing techniques [32]. The most widely used scheme to detect the TSD employs a subwavelength nanoparticle as a local field probe. Scanning the probe and analyzing the scattered far field, one can reconstruct the distribution of the TSD with a subwavelength resolution. By use of this scheme, the TSDs of focused beams and evanescent waves have been successfully measured [5,6,16].

In this article, we theoretically propose an alternative scheme to detect the TSD of light using a multilevel atomic medium. The underlying physics is the so-called electromagnetically induced transparency (EIT) [33–35], which is a quantum interference effect occurring in coherent systems [36,37] and has a wide range of applications in the fields of quantum information processing [38,39] and precision metrology [40,41]. Particularly, it enables the efficient modulation of the optical properties of atomic media by using external fields. In our scheme, the susceptibility of the atomic medium is modulated via EIT by a coupling field whose TSD is to be detected. The TSD encoded in the susceptibility is extracted by measuring the polarization distribution of a probe beam passing through the medium. To show the feasibility of the scheme, we present two case studies, in which the coupling field is provided by a tightly focused field and a two-beam interference field, respectively. We show that the key features of the TSD, such as the symmetry property and the spin-momentum locking, can be clearly observed in our scheme. Since the atomic medium responds to the coupling field in a fast way, our scheme offers the possibilities of detecting the TSD in real time. This is the major advantage of the present scheme in comparison with the existed nanoprobing scheme, which requires a sophisticated time-consuming procedure.

2. Detecting the TSD of a tightly focused field via EIT

We begin with the definition of the spin density of light [4],

(1)$$\mathbf{s}(\mathbf{r})=\frac{1}{4\omega}\mathrm{Im}[\epsilon_{0} \mathbf{E}^{{\ast}}(\mathbf{r})\times\mathbf{E}(\mathbf{r})+\mu_{0} \mathbf{H}^{{\ast}}(\mathbf{r})\times\mathbf{H}(\mathbf{r})],$$

where $\mathbf {r}$ is the position vector, $\omega$ is the angular frequency, $\epsilon _0$ and $\mu _{0}$ are the vacuum permittivity and permeability, respectively. Both the electric field $\mathbf {E}$ and the magnetic field $\mathbf {H}$ contribute to the spin density $\mathbf {s}$. Here we only consider the electric contribution, because the electric dipole interaction dominates in the light-atom interaction in the present scheme.

For an optical field that propagates mainly along the $z$ axis, $s_{z}\propto \mathrm {Im}(E_{x}^{\ast }E_{y})$ is referred to as the longitudinal spin density, which is aligned with the propagation direction, while both $s_{x}\propto \mathrm {Im}(E_{y}^{\ast }E_{z})$ and $s_{y}\propto \mathrm {Im}(E_{z}^{\ast }E_{x})$ are referred to as the TSD, which are transverse to the propagation direction. It can be seen that a necessary condition for the generation of the TSD $s_{x}$ or $s_{y}$ is the presence of the longitudinal field component $E_{z}$. In free space, a plane wave does not have the longitudinal field component due to the transverse condition $\mathbf {k}\cdot \mathbf {E}=0$ with $\mathbf {k}$ being the wave vector, and thus its TSD is zero. For a paraxial beam, the longitudinal field component and the TSD are negligible. Two typical examples of the fields exhibiting significant TSD are the tightly focused field and the interference field. The former is considered in the present section and the latter is discussed in the next section.

Figure 1(a) depicts the system for detecting the TSD of the tightly focused field $\mathbf {E}$, which can be generated via high-numerical-aperture (NA) focusing of an incident paraxial beam $\mathbf {E}^{(\mathrm {i})}$. The focused field can be expressed using the Richard-Wolf diffraction theory as [42]

(2)$$\begin{aligned} \mathbf{E}(\mathbf{r})=\mathbf{E}(r,\varphi,z)=&-\frac{\mathrm{i}fk}{2\pi}\int_{0}^{\theta_{\mathrm{m}}}\int_{0}^{2\pi}\hat{R}_{z}(\phi)\hat{R}_{y}(-\theta)\hat{R}_{z}(-\phi)\mathbf{E}^{(\mathrm{i})}(\theta,\phi)\\ &\times e^{-\mathrm{i}kr\sin\theta\cos(\phi-\varphi)+\mathrm{i}kz\cos\theta}\sqrt{\cos\theta}\sin\theta\mathrm{d}\phi\mathrm{d} \theta, \end{aligned}$$

where $(r,\varphi,z)$ is the cylindrical coordinate, $f$ is the focal length, $k$ is the wave number, $\theta _{\mathrm {m}}=\arcsin \mathrm {NA}$ with NA being the numerical aperture, $\mathbf {E}^{(\mathrm {i})}=\mathbf {e}_{x}\exp {(-f^{2}\tan ^{2}\theta /w^{2})}$ is the incident beam with the beam width $w$, $\hat {R}_{n}(\gamma )$ denotes the operator of the rotation about the $n$ axis by the angle $\gamma$, $\theta$ and $\phi$ are shown in Fig. 1(a).

Fig. 1. (a) Sketch of the system for detecting the TSD of the tightly focused coupling field via EIT. The left inset shows the intensity and polarization distributions of the coupling field in $xz$-plane ($y=0$). The red and blue ellipses and black lines denote the right- and left-elliptic and linear polarizations, respectively. Their scales represent the local amplitude of the field. The right inset depicts the schematic of the Stokes-polarimetry method to measure the polarization distribution of the probe beam. QWP, quarter-wave plate; P, polarizer. (b) The relevant energy levels in $^{133}$Cs D1 line. The atoms are prepared in the outermost Zeeman sublevels $|{g_{2},m_{F}=\pm {4}}\rangle$.

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For the linearly $x$-polarized incident beam considered in our discussion [see Fig. 1(a)], focusing results in the presence of the longitudinal field component $E_z$. The focused field can be written in the form

(3)$$\mathbf{E}(\mathbf{r})=E_{x}(\mathbf{r})\mathbf{e}_{x}+E_{z}(\mathbf{r})\mathbf{e}_{z}=E(\mathbf{r})\left[\alpha(\mathbf{r})\mathbf{e}_{x}+\beta(\mathbf{r})e^{\mathrm{i}\psi(\mathbf{r})}\mathbf{e}_{z}\right].$$

$E=\sqrt {|E_{x}|^{2}+|E_z|^{2}}$ is the amplitude, $\psi$ is the relative phase between $E_{x}$ and $E_{z}$, $\alpha =E_{x}/E$ and $\beta =E_{z}/E$ satisfy $\alpha ^{2}+\beta ^{2}=1$, $\mathbf {e}_{i}$ denotes the basis vector along the $i$ axis $(i=x,y,z)$. Here, the $y$ component of the field $E_{y}$ is much weaker than the $x$ and $z$ components, i.e., $|E_{y}|\ll |E_{x}|$ and $|E_{y}|\ll |E_{z}|$, and thus it can be neglected. The phase $\psi$ is in general nonzero, indicating that the field vector $\mathbf {E}$ spins in the $xz$ plane. Hence, the spin density is purely transverse along the $y$ axis,

(4)$$\mathbf{s}(\mathbf{r})=\frac{\epsilon_{0}}{2\omega}\mathrm{Im}[E_{z}^{{\ast}}(\mathbf{r})E_{x}(\mathbf{r})]\mathbf{e}_{y}={-}\frac{\epsilon_{0}}{2\omega}E^{2}(\mathbf{r})\alpha(\mathbf{r})\beta(\mathbf{r})\sin{\psi(\mathbf{r})}\mathbf{e}_{y},$$

i.e., the TSD. Figure 2 displays the distributions of the intensity $I\propto E^{2}$ and the TSD $s=|\mathbf {s}|$ of the focused field for different values of NA. The TSD exhibits an anti-symmetric two-lobe distribution along the $x$ axis: $s>0$, $s<0$, and $s=0$ in the regions of $x>0$, $x<0$, and $x=0$, corresponding to right- and left-elliptic and linear polarizations, respectively [also see the inset of Fig. 1(a)].

Fig. 2. The distributions of the intensity $I$ and the TSD $s$ of the tightly focused coupling field in $xz$-plane ($y=0$) and $xy$-plane ($z=0$) for $\mathrm {NA}=0.6$ ($f=1.8\,\mathrm {mm}$), $\mathrm {NA}=0.7$ ($f=1.4\,\mathrm {mm}$), $\mathrm {NA}=0.8$ ($f=1.0\,\mathrm {mm}$), and $\mathrm {NA}=0.9$ ($f=0.6\,\mathrm {mm}$). The other parameters are: $\lambda =895\, \mathrm {nm}$, $w=350\, \mathrm {\mu }\mathrm {m}$.

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The TSD of the focused field can be detected by using an atomic medium [see Fig. 1(a)], which consists of, e.g., $^{133}$Cs atoms with an excited state $|{e}\rangle =|{6\mathrm {P}_{1/2},F=3}\rangle$, and two ground states $|{g_{1}}\rangle =|{6\mathrm {S}_{1/2},F=3}\rangle$ and $|{g_{2}}\rangle =|{6\mathrm {S}_{1/2},F=4}\rangle$ in the D1 line [see Fig. 1(b)]. The atoms are assumed to be prepared in the outermost Zeeman sublevels $|{g_{2},m_{F}=\pm {4}}\rangle$ via $\pi$-polarized optical pumping by using, e.g., a resonant incoherent field linearly polarized along the $y$ axis, which is taken to be the quantization axis [43]. The focused field is resonant with the transition $|{e}\rangle \leftrightarrow |{g_{1}}\rangle$. It can be decomposed into its two spin components,

(5)$$\mathbf{E}(\mathbf{r})=\frac{E(\mathbf{r})}{\sqrt{2}}\left[\alpha(\mathbf{r})+\mathrm{i}\beta(\mathbf{r})e^{\mathrm{i}\psi(\mathbf{r})}\right]\mathbf{e}_{+}+\frac{E(\mathbf{r})}{\sqrt{2}}\left[\alpha(\mathbf{r})-\mathrm{i}\beta(\mathbf{r})e^{\mathrm{i}\psi(\mathbf{r})}\right]\mathbf{e}_{-},$$

with $\mathbf {e}_{\pm }=(\mathbf {e}_{x}\mp \mathrm {i}\mathbf {e}_{z})/\sqrt {2}$ being the spin basis vectors for right- and left-circular polarizations. They drive the $\sigma ^{\pm }$ transitions $|{e,m_{F}\pm {1}}\rangle \leftrightarrow |{g_{1},m_{F}}\rangle$.

A well-collimated $x$-polarized probe beam, which is resonant with the transition $|{e}\rangle \leftrightarrow |{g_{2}}\rangle$, passes through the medium along the $y$ axis. It can be approximately treated as a plane wave,

(6)$$\mathbf{E}_{\mathrm{p}}=E_{\mathrm{p}}\mathbf{e}_{x}=\frac{1}{\sqrt{2}}E_{\mathrm{p}}\mathbf{e}_{+}+\frac{1}{\sqrt{2}}E_{\mathrm{p}}\mathbf{e}_{-},$$

where $E_{\mathrm {p}}$ is the amplitude. Its two spin components drive the $\sigma ^{\pm }$ transitions $|{e,m_{F}\pm {1}}\rangle \leftrightarrow |{g_{2},m_{F}}\rangle$. Since the atoms are prepared in the sublevels $|{g_{2},m_{F}=\pm {4}}\rangle$, the coupling and probe fields are coupled to two $\Lambda$-type three-level atomic systems $|{g_{1},m_{F}=\pm {2}}\rangle \leftrightarrow |{e,m_{F}=\pm {3}}\rangle \leftrightarrow |{g_{2},m_{F}=\pm {4}}\rangle$, as shown by the red and blue solid lines in Fig. 1(b). Without the coupling field, the probe beam undergoes a two-level susceptibility, which results in a strong resonant absorption. The coupling field dramatically modifies the susceptibility and significantly suppresses the absorption due to the EIT effect. In the approximation that the probe field is much weaker than the coupling field, i.e., $|E_{\mathrm {p}}|\ll |E|$, the susceptibilities for the two spin components of the probe beam are given by [35,36]

(7)$$\chi_{{\pm}}(\mathbf{r})=\mathrm{i}\frac{7Nd^{2}}{12\epsilon_{0}\hbar}\frac{\gamma_{g_{1}g_{2}}\rho_{g_{2,\mp{4}}}}{\gamma_{g_{1}g_{2}}\gamma_{eg_{2}}+|\Omega_{{\mp}}(\mathbf{r})|^{2}}.$$

Here $N$ is the atomic number density, $d$ is the reduced dipole matrix element, $\rho _{g_{2,\mp {4}}}=1/2$ is the population of the sublevel $|{g_{2},m_{F}=\mp {4}}\rangle$. $\gamma _{eg_{2}}=\Gamma /2+\gamma$ is the decoherence rate between $|{e,m_{F}=\pm {3}}\rangle$ and $|{g_{2},m_{F}=\pm {4}}\rangle$, and $\gamma _{g_{1}g_{2}}=\gamma$ is the decoherence rate between $|{g_{1},m_{F}=\pm {2}}\rangle$ and $|{g_{2},m_{F}=\pm {4}}\rangle$, where $\Gamma$ and $\gamma$ denote the spontaneous decay rate and the dephasing rate, respectively. $\Omega _{\pm }(\mathbf {r})$ are the Rabi frequencies corresponding to the two spin components of the coupling field, and can be written as

(8)$$\Omega_{{\pm}}(\mathbf{r})=\frac{\Omega}{4\sqrt{2}}\left[\alpha(\mathbf{r})\pm\mathrm{i}\beta(\mathbf{r})e^{\mathrm{i}\psi(\mathbf{r})}\right]u(\mathbf{r}),$$

where $\Omega =dE_{0}/\hbar$ with $E_{0}=E(\mathbf {r}=\mathbf {0})$, and $u(\mathbf {r})=E(\mathbf {r})/E_{0}$ is the normalized amplitude of the coupling field. The factors $7/12$ in Eq. (7) and $1/4$ in Eq. (8) are calculated from Clebsch-Gordan coefficients. From Eq. (8) and Eq. (4), we can obtain

(9)$$|\Omega_{{\pm}}(\mathbf{r})|^{2}=\frac{\Omega^{2}}{32}u^{2}(\mathbf{r})\pm\frac{\omega d^{2}}{8\epsilon_{0}\hbar^{2}}s(\mathbf{r}).$$

As we can seen, the TSD of the coupling field $s$ is encoded into the susceptibility $\chi _{\pm }$ via the Rabi frequency $\Omega _{\mp }$. The susceptibilities $\chi _{+}$ and $\chi _{-}$ are different, i.e., $\chi _{+}\neq \chi _{-}$, except for $x=0$. In Fig. 3 we plot the distribution of $\mathrm {Im}(\chi _{+}-\chi _{-})$. It exhibits an anti-symmetric two-lobe pattern along the $x$ axis, which is similar to that of the TSD.

Fig. 3. The distribution of $\mathrm {Im}(\chi _{+}-\chi _{-})$ in $xz$-plane ($y=0$) and $xy$-plane ($z=0$). The parameters are: $\lambda =895\, \mathrm {nm}$, $w=350\, \mathrm {\mu }\mathrm {m}$, $\mathrm {NA}=0.8$, $f=1.0 \,\mathrm {mm}$, $N=5.0\times 10^{18}\,\mathrm {m}^{-3}$, $\Omega =2\pi \times 5.0\,\mathrm {MHz}$, $\Gamma =2\pi \times 2.67\,\mathrm {MHz}$, $\gamma =2\pi \times 100\,\mathrm {kHz}$, and $d=2.7\times 10^{-29}\,\mathrm {C\cdot m}$.

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After transmitting through the atomic medium, the probe beam can be expressed as

(10)$$\mathbf{E}_{\mathrm{p}}^{(\mathrm{t})}=\mathbf{E}_{\mathrm{p,+}}^{(\mathrm{t})}+\mathbf{E}_{\mathrm{p,-}}^{(\mathrm{t})}=\frac{1}{\sqrt{2}}E_{\mathrm{p}}e^{\mathrm{i}\int_{{-}L/2}^{L/2}n_{+}(\mathbf{r})\mathrm{d}y}\mathbf{e}_{+}+\frac{1}{\sqrt{2}}E_{\mathrm{p}}e^{\mathrm{i}\int_{{-}L/2}^{L/2}n_{-}(\mathbf{r})\mathrm{d}y}\mathbf{e}_{-},$$

with $L$ being the length of the medium along the $y$ axis. $n_{\pm }=\sqrt {1+\chi _{\pm }}\approx 1+\chi _{\pm }/2$ are the complex refractive indexes experienced by the two spin components of the probe beam. The real part $\mathrm {Re}(n_{\pm })=1+\mathrm {Re}(\chi _{\pm })/2$ and the imaginary part $\mathrm {Im}(n_{\pm })=\mathrm {Im}(\chi _{\pm })/2$ represent the refractive indexes and the absorption coefficients. Since the susceptibilities are pure imaginary, i.e., $\mathrm {Re}\chi _{\pm }=0$, the two spin components experience the same phase shift while different absorptions.

This spin-dependent absorption leads to the spin splitting of the transmitted probe beam, as shown in Fig. 4, in which we plot the distributions of the transmitted intensity $I\mathrm {_{p}^{(t)}}\propto |\mathbf {E}\mathrm {_{p}^{(t)}}|^{2}$ and the third Stokes parameter $S_{3}^{(\mathrm {t})}\propto |\mathbf {E}_{\mathrm {p,+}}^{(\mathrm {t})}|^{2}-|\mathbf {E}_{\mathrm {p,-}}^{(\mathrm {t})}|^{2}$, which describes the handedness or helicity of light and can be readily measured using the standard Stokes-polarimetry method schematically shown in the inset of Fig. 1(a) [44]. In the region of $x>0$, one has $\mathrm {Im}\chi _{+}>\mathrm {Im}\chi _{-}$ (see Fig. 3). The right-circular-polarization (RCP) component undergoes a stronger absorption than that of the left-circular-polarization (LCP) component. Consequently, the transmitted probe beam is left-elliptically polarized, i.e., $S_{3}^{(\mathrm {t})}<0$. In the region of $x<0$, $\mathrm {Im}\chi _{+}<\mathrm {Im}\chi _{-}$ results in a weaker absorption of the RCP component than that of the LCP component, and thus the transmitted probe beam is right-elliptically polarized, i.e., $S_{3}^{(\mathrm {t})}>0$. For $x=0$, the equal absorption ($\mathrm {Im}\chi _{+}=\mathrm {Im}\chi _{-}$) for both the RCP and LCP components leads to the linear polarization, i.e., $S_{3}^{(\mathrm {t})}=0$, the same as the incident polarization. Here, it should be emphasized that the underlying physics of the spin splitting discussed above is rather different from that of the spin-Hall effect of light [45,46]. The spin splitting in the spin-Hall effect of light arises from the interaction between the spin and the orbit angular momenta, i.e., the spin-orbit interaction [3].

Fig. 4. The distributions of (a) the intensity $I\mathrm {_{p}^{(t)}}$ (normalized to the incident intensity) and (b) the third Stokes parameter $S\mathrm {_{3}^{(t)}}$ of the transmitted probe beam for the case of the tightly focused coupling field with $L=2.2\,\mathrm {\mu }\mathrm {m}$. The other parameters are as in Fig. 3.

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In the far-field region ($y\gg \lambda$), the diffracted probe field can be calculated by the Huygens-Fresnel diffraction integral [47],

(11)$$\mathbf{E}\mathrm{_{p}^{(f)}}(x,y,z)={-}\frac{\mathrm{i}k}{2\pi y}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathbf{E}\mathrm{_{p}^{(t)}}(x',z')e^{\mathrm{i}k\frac{(x-x')^{2}+(z-z')^{2}}{2y}}\mathrm{d}x'\mathrm{d}z'.$$

Fig. 5 presents the far-field distributions of the intensity $I\mathrm {_{p}^{(f)}}\propto |\mathbf {E}\mathrm {_{p}^{(f)}}|^{2}$ and the third Stokes parameter $S\mathrm {_{3}^{(f)}}\propto |\mathbf {E}_{\mathrm {p,+}}^{(\mathrm {f})}|^{2}-|\mathbf {E}_{\mathrm {p,-}}^{(\mathrm {f})}|^{2}$. In spite of the strong zero-order spot, the first- and higher-order diffraction spots are too weak to be clearly seen [Fig. 5(a)]. The diffracted probe field in the far-field region still exhibits spin splitting as in the near-field region [Fig. 5(b)]. For $x>0$ and $x<0$, it has the elliptic polarization with opposite handedness. In addition, the polarization handedness of the diffraction spots flips its sign alternatively along both the $x$ and $z$ directions.

Fig. 5. The far-field distributions of (a) the intensity $I\mathrm {_{p}^{(f)}}$ and (b) the third Stokes parameter $S\mathrm {_{3}^{(f)}}$ of the transmitted probe beam at $y=1\,\mathrm {mm}$ for the case of the tightly focused coupling field. The other parameters are as in Fig. 3.

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From the above discussions, we can conclude that there is a corresponding relationship between the TSD distribution of the tightly focused coupling field (Fig. 2) and the distribution of the third Stokes parameter of the transmitted probe beam [Figs. 4(b) and 5(b)]. This relationship provides a feasible approach to detect the TSD, whose key features can be inferred from the transmitted probe beam. In particular, the anti-symmetry of the TSD can be clearly seen from the spin splitting of the probe beam. The spin-momentum locking, which manifests as the change of the TSD’s sign for the coupling field that propagates along the opposite direction, can also be readily confirmed by observing the change of the orientation of the spin splitting of the probe beam. Furthermore, since the atomic medium can respond to the variation of the coupling field within a few tens of nanoseconds, which is determined by the lifetime of the atoms, the TSD can be detected in real-time using this scheme.

3. Detecting the TSD of a two-beam interference field via EIT

In addition to high-NA focusing, interference is another way to produce fields that exhibit transverse spin. In this section, we consider the system for detecting the TSD of the interference field $\mathbf {E}$ generated from two Gaussian beam $\mathbf {E}_{1}$ and $\mathbf {E}_{2}$ (see Fig. 6),

(12)$$\mathbf{E}(\mathbf{r})=\mathbf{E}_{1}(\mathbf{r})+\mathbf{E}_{2}(\mathbf{r})=\boldsymbol{\mathrm{\epsilon}}_{1}\mathcal{E}(\mathbf{r}_{1})e^{\mathrm{i}kz_{1}}+\boldsymbol{\mathrm{\epsilon}}_{2}\mathcal{E}(\mathbf{r}_{2})e^{\mathrm{i}kz_{2}}.$$

$\mathcal {E}(\mathbf {r}_{1,2})=\mathcal {E}_{0}z_{\mathrm {R}}/(z_{\mathrm {R}}+\mathrm {i}z_{1,2})\exp {[-k(x_{1,2}^{2}+y_{1,2}^{2})/(2z_{\mathrm {R}}+2\mathrm {i}z_{1,2})]}$ are the complex amplitude distributions written in the $x_{1,2}y_{1,2}z_{1,2}$ coordinate frames with the constant amplitude $\mathcal {E}_{0}$, the Rayleigh range $z_{\mathrm {R}}=kw_0^{2}/2$, and the beam waist $w_{0}$. $\boldsymbol{\mathrm{\epsilon}}_{1}=-\mathbf {e}_{z}\sin {\theta }+\mathbf {e}_{x}\cos {\theta }$ and $\boldsymbol{\mathrm{\epsilon}}_{2}=\mathbf {e}_{z}\sin {\theta }+\mathbf {e}_{x}\cos {\theta }$ are the polarization vectors. The $x_{1}y_{1}z_{1}$ and $x_{2}y_{2}z_{2}$ frames are obtained by rotating the $xyz$ frame with the angles of $\theta$ and $-\theta$ around the $y$ axis, and hence one has $x_{1}=-z\sin {\theta }+x\cos {\theta }$, $x_{2}=z\sin {\theta }+x\cos {\theta }$, $y_{1}=y_{2}=y$, $z_{1}=z\cos {\theta }+x\sin {\theta }$, and $z_{2}=z\cos {\theta }-x\sin {\theta }$.

Fig. 6. Sketch of the system for detecting the TSD of the two-beam interference coupling field via EIT. The inset shows the intensity and polarization distributions of the coupling field in $xz$-plane ($y=0$).

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As with the tightly focused field, the interference field can also be written in the form of $\mathbf {E}=E_{x}\mathbf {e}_{x}+E_{z}\mathbf {e}_{z}$ [see Eq. (3)] with the transverse and longitudinal field components calculated from Eq. (12),

(13a)$$E_{x}(\mathbf{r})=\cos{\theta}\left[\mathcal{E}(\mathbf{r}_{1})e^{\mathrm{i}kx\sin{\theta}}+\mathcal{E}(\mathbf{r}_{2})e^{-\mathrm{i}kx\sin{\theta}}\right]e^{\mathrm{i}kz\cos{\theta}},$$

(13b)$$E_{z}(\mathbf{r})={-}\sin{\theta}\left[\mathcal{E}(\mathbf{r}_{1})e^{\mathrm{i}kx\sin{\theta}}-\mathcal{E}(\mathbf{r}_{2})e^{-\mathrm{i}kx\sin{\theta}}\right]e^{\mathrm{i}kz\cos{\theta}}.$$

The interference field can be regarded as a field that propagates along the $z$ axis with the wave number of $k \cos {\theta }$. The relative phase $\psi =\arg E_{z}-\arg E_{x}$ is generally nonzero, leading to the field vector $\mathbf {E}$ spinning in the $xz$ plane and hence the TSD along the $y$ axis. Figures 7(a) and 7(b) display the distributions of the intensity $I\propto |E_{x}|^{2}+|E_{z}|^{2}$ and the TSD $s=|\mathbf {s}|$ [Eq. (4)] of the interference field. The transverse spin occurs in the interference region and vanishes in the region where only one of the two beams is present.

Fig. 7. The distributions of (a) the intensity $I$ and (b) the TSD $s$ of the two-beam interference field, and (c) $\mathrm {Im}(\chi _{+}-\chi _{-})$ in $xz$-plane ($y=0$) and $xy$-plane ($z=0$). The parameters are: $\lambda =895\, \mathrm {nm}$, $w_{0}=10\, \mathrm {\mu }\mathrm {m}$, $\theta =15^{\circ }$, $N=1\times 10^{18}\,\mathrm {m}^{-3}$, $\Omega =2\pi \times 5\,\mathrm {MHz}$, $\Gamma =2\pi \times 2.67\,\mathrm {MHz}$, $\gamma =2\pi \times 100\,\mathrm {kHz}$, and $d=2.70\times 10^{-29}\,\mathrm {C\cdot m}$.

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In order to see the features of the transverse spin in greater detail, we examine the TSD in the region near the origin, i.e., $|x_{1,2}|\ll w_{0}$, $|y_{1,2}|\ll w_{0}$, and $|z_{1,2}|\ll z_{R}$. Under this approximation, one has $\mathcal {E}(\mathbf {r}_{1})\approx \mathcal {E}(\mathbf {r}_{2})\approx \mathcal {E}_0$ and hence

(14a)$$E_{x}(\mathbf{r})=2\mathcal{E}_0\cos{\theta}\cos{(kx\sin{\theta})}e^{\mathrm{i}kz\cos{\theta}},$$

(14b)$$E_{z}(\mathbf{r})={-}2\mathrm{i}\mathcal{E}_0\sin{\theta}\sin{(kx\sin{\theta})}e^{\mathrm{i}kz\cos{\theta}},$$

and

(15)$$\mathbf{s}(\mathbf{r})=\frac{\epsilon_{0}}{2\omega}\mathcal{E}^{2}_0\sin{2\theta}\sin(2kx\sin\theta)\mathbf{e}_{y}.$$

The TSD of the interference field exhibits a sinusoidal distribution along the $x$ axis with a period of $\pi /(k\sin {\theta })$. One has $s>0$, $s<0$, and $s=0$ in the regions of $m\pi /(k\sin \theta )<x<(m+1/2)\pi /(k\sin \theta )$, $(m+1/2)\pi /(k\sin \theta )<x<(m+1)\pi /(k\sin \theta )$, and $x=m\pi /(2k\sin \theta )$, respectively, with $m=0,\pm {1},\pm {2},\ldots$, as shown in the insets of Fig. 6 and Fig. 7(b).

The TSD of the interference field can also be detected via EIT with the atomic medium. With the interference field as the coupling field of EIT, we calculate and plot the distribution of $\mathrm {Im}(\chi _{+}-\chi _{-})$, as shown in Fig. 7(c). It can be seen that the primary characteristics such as the anti-symmetry and the period of the TSD are encoded in the susceptibility of the medium. As a result, the probe beam experiences a spin-dependent absorption after passing through the medium. Figure 8 presents the distributions of the intensity $I\mathrm {_{p}^{(t)}}$ and the third Stokes parameter $S\mathrm {_{3}^{(t)}}$ of the transmitted probe beam. They exhibit similar overall structures to the distributions of the intensity and the TSD of the coupling field [see Figs. 8(a) and 8(b)], respectively. Particularly, the anti-symmetry and the period of the TSD can be clearly observed in the $S\mathrm {_{3}^{(t)}}$ distribution of the transmitted probe beam, as shown in the inset of Fig. 8(b). Figure 9 presents the far-field distributions of the intensity $I\mathrm {_{p}^{(f)}}$ and the third Stokes parameter $S\mathrm {_{3}^{(f)}}$ of the transmitted probe beam, which are calculated from the Huygens-Fresnel diffraction integral [Eq. (11)]. The atomic medium periodically modulated by the interference field acts as a spin-dependent transmission grating for the probe beam. The features of the TSD such as the period and symmetry can be inferred from the diffracted far-field $S\mathrm {_{3}^{(f)}}$ distribution. Therefore, these results further validate the feasibility of our scheme for detecting the TSD.

Fig. 8. The distributions of (a) the intensity $I\mathrm {_{p}^{(t)}}$ (normalized to the incident intensity) and (b) the third Stokes parameter $S\mathrm {_{3}^{(t)}}$ of the transmitted probe beam for the case of the two-beam interference coupling field with $L=22\,\mathrm {\mu }\mathrm {m}$.The other parameters are as in Fig. 7.

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Fig. 9. The far-field distributions of (a) the intensity $I\mathrm {_{p}^{(f)}}$ and (b) the third Stokes parameter $S\mathrm {_{3}^{(f)}}$ of the transmitted probe beam at $y=1\,\mathrm {mm}$ for the case of the two-beam interference coupling field. The other parameters are as in Fig. 7.

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4. Conclusion

In conclusion, we have presented a scheme to detect the TSD of light via EIT with a multilevel atomic medium. We have shown through a theoretical study that the TSD of a coupling field coupled to the medium can be mapped to the polarization distribution of a probe beam passing through the medium. To verify the feasibility of the scheme, we have theoretically studied two cases, in which the coupling field is provided by a tightly focused field and a two-beam interference field, respectively. We have shown that the key features of the TSD, such as the spatial distribution, the symmetry property, and the spin-momentum locking, can be inferred from the transmitted probe beam both in near- and far-field regions. Our study provides the possibility of using EIT to detect the near-field three-dimensional quantities like the TSD of light. This approach may find applications in the fields of nanophotonics and plasmonics. Furthermore, the cooperation of the transverse spin and the EIT in atoms offers a powerful way to realize chiral manipulation of light, and holds promise in chiral quantum optics.

Funding

National Natural Science Foundation of China (12104332, 12004334); Taiyuan Institute of Technology Scientific Research Initial Funding (20020116); 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.

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Detecting the transverse spin density of light via electromagnetically induced transparency

Abstract

1. Introduction

2. Detecting the TSD of a tightly focused field via EIT

3. Detecting the TSD of a two-beam interference field via EIT

4. Conclusion

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (9)

Equations (17)

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