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Abstract
Vector beam propagation through a four-level tripod atomic system has been investigated. The three transitions of the tripod atomic system are coupled by a strong control field and the two constituent orthogonally polarized components of a weak probe vector beam. An external magnetic field induces anisotropy, creating a difference in the refractive indices of the two polarization components of the beam. This difference in refractive indices varies with the magnetic field strength and directly relates to the polarization orientation at any transverse plane. Thus, the transverse polarization structure can be rotated as desired with appropriate magnetic field strength. We further study the effect of nonlinearity and inhomogeneous broadening on the vector beam’s polarization rotation. Therefore, the mechanism of efficient polarization control and manipulation of a vector beam can open up a new avenue for high-resolution microscopy and high-density optical communications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vector beams (VBs) [1–3] are generated by the vector superposition of two orthogonally polarized, orbital angular momentum (OAM) carrying Laguerre-Gaussian (LG) modes [4]. This results in a heterogeneous distribution of polarization on the transverse plane. The polarization distribution is radially symmetric namely radial, spiral, and azimuthal when the two constituent LG modes have equal and opposite OAM. These types of VBs have a net-zero OAM and are called cylindrical vector (CV) beams [1]. The polarization distribution has both azimuthal and radial variation, e.g., lemon, star, and web polarization distribution when the two LG modes have a zero and a non-zero OAM, respectively. Such beams are called full Poincaré (FP) beams [2].
VBs have drawn great attention for their varied applicability, e.g., CV beams can produce a strong longitudinal field component [5] at the focus of a high numerical aperture (NA) lens, producing a spot size as small as $0.16 \lambda ^2$[6]. This focusing property of CV beams has applications in single-molecule spectroscopy, STED [7], confocal microscopy [8–10], efficient laser cutting [11],optical trapping of particles [12,13], and atomic magnetometry [14,15]. A high dimensional Hilbert space based on the polarization and OAM of a VB [16] can be leveraged to encode single-photon qubits for applications in quantum information [17–19]. In free space optical communication, the transverse polarization distribution of vector modes can be used to increase the transmission data rate [20]. The spatial degree of freedom of scaler OAM beams has been used to increase information content [21] in optical communication. However, scaler OAM beams bifurcate into multiple spatial solitons during nonlinear propagation [22,23]. This fragmentation [24–26] can be prevented by using VBs, e.g., CV beams can propagate in a saturable Kerr nonlinear medium for longer distances than scalar beams, without breakup [27]. In many of the aforementioned applications, it is important to maintain a stable intensity and transverse polarization distribution of the VB [28]. Therefore, it is preferable to have a controllable VB polarization rotation.
In this work, we illustrate a scheme to control the rotation of a VB’s polarization structure, using the magneto-optic effect [29] in a four-level tripod atomic system [30]. The three transitions of the four-level tripod atomic system are driven by a strong control field and the two left and right circular polarization components of a weak VB. In presence of an external magnetic field, the magneto-optic effect creates a difference in the refractive indices of the two VB polarization components. We have derived an expression relating this difference in refractive indices with the polarization orientation at any point on the VB’s transverse plane. The difference between refractive indices of the two VB components can be varied by changing the magnetic field strength. Thus, providing a means to control the rotation of the polarization structure of a VB. This method also makes it possible to switch between radial, azimuthal, and spiral polarization states of a CV beam, by varying the magnetic field strength. Some related works on vector beam polarization rotation can be found in [28,31–33]. The proposed technique is a simple way to fully control the polarization rotation of a weak VB at any propagation length, by merely varying the magnetic field. Due to the Electromagnetically induced transparency (EIT), [34] of the tripod system, possible absorption during the rotation process is also avoided. At high intensities of VB, polarization rotation per unit change of magnetic field decreases, and nonlinear effects such as “self-focusing” [35] become prominent. In the end, we discuss the effect of inhomogeneous broadening on polarization rotation.
This paper is organized into the following sections. Section 2 gives the theoretical formalism for the four-level tripod system and the expression for the polarization rotation angle of the VB. Section 3 collates all the results with their explanations. Finally, Section 4 presents our conclusions.
2. Theoretical formulation
2.1 Level system
We consider a four level tripod system as shown in Fig. 1. A weak probe VB, traveling along the positive $z$ axis, couples both, $|1\rangle \leftrightarrow |4\rangle$ and $|3\rangle \leftrightarrow |4\rangle$ transitions, with its right circularly polarized component, $\vec {E}_R$ and left circularly polarized component, $\vec {E}_L$, respectively. Consequently, by selection rules, the strong control field, $\vec {E}_c$ coupling $|2\rangle \leftrightarrow |4\rangle$ transition is a $\pi$ polarized wave traveling along $x$ or $y$ axis. The configuration in Fig. 1 can be realized with Rubidium (Rb) atomic vapour in presence of an external magnetic field, by choosing the Zeeman sublevels of $^{87}$Rb $D_2$ $(5^2S_{1/2} \rightarrow 5^2P_{3/2})$ transition hyperfine structure as: $|1\rangle = |5^2S_{1/2}, F = 1, m_F = -1\rangle$, $|2\rangle = |5^2S_{1/2}, F = 1, m_F = 0\rangle$, $|3\rangle = |5^2S_{1/2}, F = 1, m_F = +1\rangle$, and the excited state $|4\rangle = |5^2P_{3/2}, F = 0, m_F = 0\rangle$. The probe and control fields are defined as:
Fig. 1. Schematic diagram of a four level tripod system. A longitudinal magnetic field, $\vec {B}= B_z\hat {z}$ generates the Zeeman sublevels, $|1\rangle$, $|2\rangle$, and $|3\rangle$ with an energy separation of $\hbar \beta _L$ between them. The energy of $|2\rangle$ is indiscriminately set to zero and $|4\rangle$ is taken as the excited state. The right circularly polarized component, $\vec {E}_R$ and the left circularly polarized component, $\vec {E}_L$, of a weak probe VB couples the transitions, $|1\rangle \leftrightarrow |4\rangle$ and $|3\rangle \leftrightarrow |4\rangle$, respectively. The transition, $|2\rangle \leftrightarrow |4\rangle$ is coupled by a $\pi$ polarized, strong control field, $\vec {E}_c$. The spontaneous emission decay rate from $|4\rangle$ to $|j\rangle$ $(j= 1,2,3)$ is denoted by $\gamma _{4j}$. The detunings of the probe and control fields are denoted by $\delta _p$ and $\delta _c$, respectively.
The time-dependent Hamiltonian of the four-level tripod system in Fig. 1, under the electric dipole approximation, is given as:
The effective Hamiltonian obeying the Schrödinger equation, in the transformed basis is given by $\boldsymbol {\mathcal {H}} = \boldsymbol {\hat {U}}^\dagger \boldsymbol {H}\boldsymbol {\hat {U}} - i\hbar \boldsymbol {\hat {U}}^\dagger \partial _t\boldsymbol {\hat {U}}$, which under the rotating wave approximation gives:
In Eq. (5), the one photon detunings, $\delta _{p(c)}$ and Rabi frequencies, $\Omega _i$ of the fields are given by:
The dynamics of atomic state populations and coherences is governed by the following Liouville equation:
2.2 Propagation equations
The beam propagation equations for $\vec {E}_{R(L)}$ are derived from the Maxwell’s equation, under the slowly varying envelope and paraxial wave approximations. The propagation equations in terms of the respective field envelopes are given as:
2.3 Refractive index of medium
Under the weak probe approximation $\left (|\vec {E}_c|>>|\vec {E}_{R(L)}|\right )$, at steady state, $\rho _{11} = \rho _{33} = 1/2$ and $\rho _{22} = \rho _{44} = \rho _{42} = \rho _{24} = 0$. Under this steady state condition, the coherences, $\rho _{41}$ and $\rho _{43}$, are given as:
In Eq. (9), $\Delta _{1(3)4} = \delta _p \mp \beta _L + i\Gamma _{1(3)4}$ and $\Delta _{24} = \delta _c + i\Gamma _{24}$, $\Delta _{12} = \beta _L -(\delta _p - \delta _c) + i\Gamma _{12}$, $\Delta _{13} = 2\beta _L + i\Gamma _{13}$, and $\Delta _{23} = \beta _L +(\delta _p - \delta _c) + i\Gamma _{23}$. Here $\Gamma _{ij}$ and $\Gamma _{i4}$ are the decoherence rates of $\rho _{ij}$ and $\rho _{i4}$, respectively ($i < j; \;i,j= 1,2,3$). The refractive indices corresponding to $\vec {E}_{R(L)}$ can then be written in terms of medium susceptibility as:
2.4 Polarization rotation angle
A VB can be represented as a vector superposition of two orthogonally polarized, Laguerre-Gaussian modes. In the circular polarization basis, a VB is written as:
where,From Eq. (14) and Eq. (15), orientation, $\xi$ is given by:
Substituting Eqns. [(12), (13)] in Eq. (16) gives:
3. Results and analysis
3.1 Free space propagation
In free space, where $n_L= n_R = 1$ or inside any medium where $n_L = n_R$, Eq. (18) gives:
The above equation suggests that VBs with $|l_R| = |l_L|$, e.g., CV beams, are polarization invariant in a media where $n_R = n_L$. On the other hand, VBs with $|l_R| \neq |l_L|$, e.g., FP beams, undergo a Gouy phase dependent rotation of their polarization structure. In free space, at $z = z_r$, $\eta (z_r) = \pi /4$, thus for a VB with “lemon” polarization distribution $l_{R}=1, l_{L}=0$, $\theta = 0$, $\alpha = \pi /4$), the polarization rotation angle would be: The negative sign in Eq. (20) represents clockwise rotation. Figure 2 shows the polarization rotation of a VB with “lemon” polarization distribution at different propagation lengths. In Fig. 2(b), after a free space propagation of one Rayleigh length, the entire polarization structure has rotated by an angle $\pi /4$ instead of $\pi /8$ as predicted in Eq. (20). This is because the rotation of the polarization structure as a whole depends on the rotational symmetry of the polarization pattern. Nonetheless, the polarization at each point on the transverse plane does rotate by $\pi /8$. In the far field regime ($z \rightarrow \infty$), for the “lemon” polarization distribution, Eq. (19) gives, $\Delta \xi \rightarrow -\pi /4$. Figure 2(c) shows the transverse polarization distribution at $z = 20z_R$, representing the far field regime. In Fig. 2(c), the polarization structure has rotated clockwise by $\pi /2$, and not $\pi /4$, for the same reason as Fig. 2(b). The results shown in Fig. 2 are consistent with some of the related works [28,31]. In the following sections, we will discuss how the four level tripod system (Fig. 1) can be used to achieve full control over the polarization rotation.Fig. 2. (a), (b), (c): Transverse intensity and polarization distribution for a VB with “lemon” polarization structure ($l_{R}=1, l_{L}=0$, $\theta = 0$, $\alpha = \pi /4$) at $z = 0$, $z = z_R$, and $z = 20 z_R$, respectively. Throughout the paper, the beam waist is taken as, $w_0 = 60$ $\mu$m and the white, red, blue colors of polarization correspond to right circular, linear, and left circular polarizations, respectively. Also, beam diffraction doesn’t affect polarization rotation, hence it has been ignored in all figures similar to Fig. 2 for visual clarity. The field intensity has been normalized to unity.
3.2 Polarization rotation control using medium susceptibility
In this section, we present a scheme to control the polarization rotation of a VB using the tripod atomic system, shown in Fig. 1. In Fig. 1, the probe carrier frequency, $\omega _p$, is made equal to the transition frequency, $\omega _{42}$, of $|2\rangle \leftrightarrow |4\rangle$ transition (i.e., $\delta _p = 0$). In this way, $\vec {E}_R$ and $\vec {E}_L$ will be red and blue detuned, respectively by an amount $\beta _L$. The parameter $\beta _L$ is the frequency shift between the Zeeman triplets, $|1\rangle$, $|2\rangle$, and $|3\rangle$. The equal and opposite detunings experienced by $\vec {E}_{R(L)}$ causes them to have unequal refractive indices ($n_R \neq n_L$). This asymmetry in refractive index is the crux of achieving the desired polarization rotation of a VB. Note that the control field is considered resonant ($\delta _c = 0$) throughout the paper. With the configuration in Fig. 1, increasing $\beta _L$ by varying the magnetic field strength ($|\vec {B}|$), increases the detunings of $\vec {E}_R$ and $\vec {E}_L$ by equal amount, but opposite sign. This inturn increases the difference between the refractive indices, $n_R$ and $n_L$, causing the polarization at each point on the VB’s transverse plane to rotate according to Eq. (18). Figures, 3(a) – 3(e) show the evolution of transverse intensity and polarization distribution of a “lemon” VB with $\beta _L$, at a propagation distance of one Rayleigh length inside the medium. In Figs. 3(a) – 3(e), the magnetic field strength, $|\vec {B}|$ is increased to raise $\beta _L$ from $0$ to $0.23\gamma$. Thus, increasing the red (blue) detuning of $\vec {E}_{R(L)}$ by an amount of $\beta _L$, as mentioned earlier. This correspondingly, increases the difference between $n_R$ and $n_L$, causing the polarization at each point on the VB’s transverse plane to rotate anticlockwise, in accordance with Eq. (18). Decreasing $|\vec {B}|$ so as to reduce $\beta _L$ from $0.23\gamma$ to $0$ would reverse the process, undoing the rotation. The sense of rotation (clockwise or anticlockwise) in moving from Fig. 3(a) to 3(e), can be altered by increasing $\beta _L$ in negative direction, i.e., from $0$ to $-0.23\gamma$. This can be achieved by inverting the magnetic field direction. The process of polarization rotation as shown in Fig. 3, can be understood by studying the medium susceptiblity, $\chi _{41(3)}$ associated with $\vec {E}_{R(L)}$ as a function of $\beta _L$. Figures 4(a) and 4(b) show the variation of real and imaginary parts of $\chi _{4i}$ with $\beta _L$. In Fig. 4, as well as throughout the paper, $\delta _c = 0$ is maintained. The imaginary and real part of $\chi _{4i}$ represents absorption and dispersion, respectively. Figures 4(a) and 4(b) are laterally inverted mirror images of each other due to the equal and opposite detuning experienced by $\vec {E}_{R(L)}$ as mentioned earlier. In Figs. 4(a) and 4(b), at any given value of $\beta _L$ within the EIT window, “$\Delta$”, Re$[\chi _{41}] =-\text {Re}[\chi _{43}]$, making $n_R\neq n_L$ according to Eq. (10). Due to $n_R\neq n_L$, there is an additional polarization rotation given by the curly bracketed terms of Eq. (18), together with the polarization rotation due to free space propagation, $-\frac {1}{2}\Delta |l_{L,R}|\eta (z)$ [see Eq. (19)]. In Figs. 4(a) and 4(b), at $\beta _L = 0$ (i.e., $\vec {B} = 0$) both the real and imaginary part of $\chi _{4i}$ is zero, making $n_{R(L)} = 1$ [see Eq. (10)] and absorption zero, respectively, i.e., the medium behaves like free space. Hence, the polarization distribution in Fig. 3(a) at one Rayleigh length, is same as Fig. 2(b) that shows the polarization distribution after a free space propagation of one Rayleigh length. In Figs. 4(a) and 4(b), as $\beta _L$ is raised from $0$ to $0.23\gamma$, real parts of both $\chi _{41}$, $\chi _{43}$ increases by equal amount, with opposite signs; consequently increasing $\Delta (n_{R,L})$. This causes a polarization rotation at each point on the VB’s transverse plane, in accordance with Eq. (18), leading to the rotation shown in Figs. 3(a) – 3(e). To check the fidelity of Eq. (18), we measured the rotation angles at random points on the numerically obtained plots such as Fig. 3 and then using Eq. (18). The numerical and analytical results are in good agreement.
Fig. 3. (a)-(e): Transverse intensity and polarization distribution for a VB with “lemon” polarization structure ($l_{R}=1, l_{L}=0$, $\theta = 0$, $\alpha = \pi /4$) at $z = z_R$, for different values of $\beta _L$. All figures from (a) to (e) are obtained, keeping $\delta _c = 0$. The rabi frquencies of the fields at $z = 0$ are taken as $\Omega _c = 4\gamma$, $\Omega _{R(L)} = 0.05\gamma$. The decoherence rates are taken as $\Gamma _{ij}\approx 10^{-3}\gamma$ and $\Gamma _{i4} = 3/2 \gamma$ $(i<j,\; i,j = 1,2,3)$. The density of atoms, $\mathcal {N} = 2\times 10^{11}$ cm$^{-3}$. The field intensity has been normalized with respect to the field intensity at $z= 0$. The negative sign on the polarization rotation angle shown on top of figure (a) signifies clockwise rotation.
Fig. 4. (a): Real and imaginary part of $\chi _{41}$ vs $\beta _L$. (b): Real and imaginary part of $\chi _{43}$ vs $\beta _L$. The width of EIT window, $\Delta \approx \gamma$. All other relevant parameters used are same as mentioned in Fig. 3.
Fig. 5. (a)-(c): Transverse intensity and polarization distribution for a CVB with azimuthal polarization distribution ($l_L = -1$, $l_R = 1$, $\theta = \pi$, $\alpha = 3\pi /8$) at $z = z_R$ for different values of $\beta _L$. The plot shows the transformation of CVB’s polarization state from azimuthal at $\beta _L = 0$ [Fig. 5(a)], to spiral at $\beta _L = 0.05\gamma$ [Fig. 5(b)], and radial at $\beta _L = 0.1\gamma$ [Fig. 5(c)]. All other relevant parameters used are same as mentioned in Fig. 3.
Additionally, it is also possible to transform the polarization state of CV beams to another desired state. Figure 5 shows how by varying $\beta _L$ from $0$ to $0.1\gamma$, the polarization structure of the CV beam changes from azimuthal [Fig. 5(a)] to spiral [Fig. 5(b)] and then to radial [Fig. 5(c)]. This sort of transformation is however limited to CV beams and full Poincaré beams or noncylindrical VBs such as lemon, web, and star, don’t undergo such transformation.
3.3 Effect of control field amplitude on polarization rotation
In this section, we discuss the role of control field amplitude in rotating the VB’s polarization structure. The choice of $\Omega _c$ is determined by the fact that the EIT window width, “$\Delta$” (see Fig. 4), should be wide enough to accommodate for a $360^0$ polarization rotation with minimal intensity reduction, due to absorption. Figure 6(a) shows a plot of polarization rotation angle, $\Delta \xi$ vs $\beta _L/\gamma$ for a “lemon” VB, at one Rayleigh length and Fig. 6(b) shows the corresponding absorption (Im[$\chi _{41}$]) profile of $\vec {E}_R$ near resonance, for different control field Rabi frequencies. In Fig. 6(a), at $\beta _L/\gamma = 0$, all the straight lines intersect at $\Delta \xi = -\pi /8$, which is the polarization rotation angle for free space propagation of one Rayleigh length [see Eq. (20)]. This is because at $\beta _L/\gamma = 0$, both Re[$\chi _{41}]=$ Re[$\chi _{43}]=0$ [see Fig. 4], making the refractive indices, $n_R = n_L = 1$, according to Eq. (10). Thus, causing a polarization rotation of $\Delta \xi = -\pi /8$ after propagation of one Rayleigh length as shown earlier in Eq. (20). The rotation angle from $\Delta \xi = -\pi /8$ to $7\pi /8$ on the left vertical axis of Fig. 6(a) represents a $2\pi$ anticlockwise rotation of the polarization structure of a “lemon” VB at one Rayleigh length. In Fig. 6, as $\Omega _c/\gamma$ is increased from $2$ to $4$, the slope of the straight lines ($\Delta \xi$ per unit $\beta _L/\gamma$) decreases in Fig. 6(a) and the probe absorption within the range, $\beta _L/\gamma \in [0,0.4]$ becomes flatter in Fig. 6(b). We found $\Omega _c = 4\gamma$ to be the optimum value for providing a wide enough EIT window to enable $2\pi$ polarization rotation without significant intensity drop due to absorption. For values smaller than $\Omega _c= 4\gamma$, although the rotation rate ($\Delta \xi$ per unit $\beta _L/\gamma$) is higher, the absorption rate (Im$[\chi _{41}]$ per unit $\beta _L/\gamma$) is also substantial as seen in Fig. 6(b). Hence, as $\beta _L/\gamma$ is increased to achieve $2\pi$ polarization rotation, field intensity would noticeably deplete towards higher values of $\beta _L/\gamma$. Increasing $\Omega _c$ beyond $4\gamma$ to get a wider EIT window, would be redundant for achieving the $2\pi$ polarization rotation. Therefore, we chose the input control Rabi frequency to be around $\Omega _c = 4\gamma$. In the next section, we discuss the effect of nonlinearity on polarization rotation and VB propagation.
Fig. 6. (a): Polarization rotation angle, $\Delta \xi$ vs $\beta _L/\gamma$, of a “lemon” VB for different input control Rabi frequencies, at a propagation distance of one Rayleigh length, with a fixed input probe field amplitude. (b): Zoomed plot of Im[$\chi _{41}$] vs $\beta _L/\gamma$ at the corresponding input control Rabi frequencies. The plot shows the variation of $\Delta \xi$ and Im[$\chi _{41}$], with $\beta _L/\gamma$ for input control field Rabi frequencies, $\Omega _c/\gamma = 2, 3, 4$. All other relevant parameters and conditions remain same as Fig. 3. The negative sign on the vertical axis of figure (a) represents the clockwise rotation of polarization.
3.4 Effect of high probe intensity on polarization rotation
At VB intensities, comparable to the control field intensity, the coherences [see Eqs. (9a), (9b)] calculated using a weak probe approximation $\left (|\vec {E}_{R(L)}|<<|\vec {E}_c|\right )$, become invalid. To incorporate the effect of high VB intensity on polarization rotation and beam propagation, the relevant coherences are evaluated numerically by using Gaussian elimination. A comparison between the propagation results for low and high probe intensities is curated in Fig. 7. Figures 7(a) and 7(b) shows a comparison between the transverse intensity profile and polarization rotation of a “lemon” VB for input Rabi frequencies, $\Omega _{R(L)} = 0.1\gamma$ and $\Omega _{R(L)} = 2\gamma$, respectively, with $\Omega _c = 4\gamma$ and $\beta _L = 0.23\gamma$. In Fig. 7(b), the polarization rotation for $\Omega _{R(L)} = 2\gamma$ is less compared to $\Omega _{R(L)} = 0.1\gamma$ in Fig. 7(a). Also, the beam spot of the high intensity VB [Fig. 7(b)] is slightly squeezed as compared to the low intensity VB [Fig. 7(a)], indicating an emergence of the $3$rd order nonlinear effect of “self-focusing”. The “self-focusing” becomes conspicuous upon lowering the control field intensity, since that leads to a steeper dispersion profile near resonance, and hence higher nonlinear probe ($\vec {E}_{R(L)}$) susceptibility. Figure 7(a$^\prime$) – 7(d$^\prime$) shows the transverse intensity and polarization distribution along with the longitudinal intensity profile of a “lemon” VB at one Rayleigh length for an input control field Rabi frequency of $\Omega _c= 2\gamma$, compared to $\Omega _c = 4\gamma$ in Fig. 7(a) and 7(b). In Fig. 7(b$^\prime$) for a higher intensity VB, its beam spot is distinctively brighter and smaller in diameter than a low intensity VB in Fig. 7(b$^\prime$). Correspondingly, the longitudinal profiles in Figs. 7(c$^\prime$) and 7(d$^{\prime }$), highlights the “self focusing" in case of the high VB intensity. Therefore, the higher the intensity of VB; the lesser is the polarization rotation at a given, $\beta _L$. Moreover, nonlinear effects such as “self-focusing” become evident at high intensities of VB.
Fig. 7. Transverse intensity and polarization distribution of a “lemon” VB ($l_R = 1$, $l_L = 0$, $\theta = 0$, $\alpha = \pi /4$) at $z = z_r$, for (a) $\Omega _{R(L)} =0.1\gamma$ and (b) $\Omega _{R(L)} =2\gamma$ with $\beta _L = 0.23\gamma$, $\delta _c = 0$, and $\Omega _c = 4\gamma$. Figures (a$^\prime$) and (b$^\prime$) are for $\Omega _{R(L)} =0.1\gamma$ and $\Omega _{R(L)} =1\gamma$, respectively with $\beta _L = 0.11\gamma$, $\delta _c = 0$, and $\Omega _c = 2\gamma$. Figures (c$^\prime$) and (d$^\prime$) are longitudinal intensity profile corresponding to figures (a$^\prime$) and (b$^\prime$), respectively. All other parameters, assumptions remain same as Fig. 3
3.5 Effect of inhomogeneous broadening
In this section, we discuss the effect of “inhomogeneous broadening” [36] on the VB polarization rotation. For an atom moving with velocity $\vec {v}$, the laser frequencies are Doppler shifted in the atom’s rest frame by $\vec {k}_i.\vec {v}$, introducing a laser detuning of, $\delta _i + \vec {k}_i.\vec {v}$ $(i=p, c)$. Since different atoms have different $\vec {v}$, they interact inhomogeneously with the laser. This causes an increase in the linewidth of an atomic transition; a phenomenon known as “inhomogeneous broadening”. With the modified detuning, the steady-state coherences, $\rho _{ij}$ are given by their average over the one-dimensional Maxwell-Boltzmann velocity distribution [36]:
where $T$ is the temperature of the gas, $M$ is the mass of each atom, and $k_B$ is the Boltzmann constant. The velocity averaged coherences are then given by $<\rho _{ij}> = \int ^{\infty }_{-\infty }\rho _{ij}(v)f(v)dv$.Figure 8 shows the real and imaginary part of $<\rho _{41}>$ vs $\beta _L/\gamma$ at a room temperature of $T=300$ K. All parameters and assumptions used in Fig. 8 are same as mentioned in Fig. 3. As seen in Fig. 8, the width of EIT window for inhomogeneous broadening, $\Delta _{\text {in}} \approx 0.2\gamma$ (see Fig. 8), is narrower compared to $\Delta \approx \gamma$ (see Fig. 4), in case of homogeneous broadening. Also the absorption (Im[$<\rho _{41}>$]) profile within the EIT window is flatter compared to homogeneous broadening (see Fig. 4). In Fig. 8, “$\Delta _{\text {in}}$” can be made wide enough to achieve a $360^0$ polarization rotation by simply increasing the input control Rabi frequency, $\Omega _c$. The presence of inhomogeneous broadening doesn’t seem to have any detrimental effect on the flatness of absorption profile within the EIT window as shown in Fig. 8. Therefore, it should be possible to perform experiments without the use of cold/ultracold atoms.
Fig. 8. Real and imaginary part of Doppler averaged coherence, $<\rho _{41}>$ vs $\beta _L/\gamma$. The temperature is taken as $300$ K. All other parameters, assumptions remain same as Fig. 3.
4. Conclusion
A four-level tripod atomic system exhibits anisotropy in presence of an external magnetic field. This anisotropy creates a difference in the refractive indices of the vector beam’s two constituent polarization components. This difference in refractive indices can be varied by changing the magnetic field strength. We show that the polarization orientation at any point on the vector beam’s transverse plane has a direct dependence on the difference between the refractive indices of its two polarization components, at that point. Thus, the transverse polarization distribution of a vector beam can be rotated as desired by controlling the external magnetic field strength. This method can also be used to transform the polarization state of cylindrical vector beams between azimuthal, spiral, and radial polarization distributions at any given propagation length inside the medium. During nonlinear propagation, higher intensity of the vector beam leads to a lesser rotation of its polarization structure at a given magnetic field strength and Kerr nonlinearity causes beam focusing. Finally, we show that the polarization rotation procedure is not adversely affected by inhomogeneous broadening and is experimentally viable at room temperature.
Funding
Science and Engineering Research Board (CRG/2018/000054).
Acknowledgments
N.D. and T.N.D. gratefully acknowledges funding by the Science and Engineering Board (Grant No. CRG/2018/000054).
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 and may be obtained from the authors upon reasonable request.
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