Efficient diffraction control using a tunable active-Raman gain medium

In this work, we consider a tunable ARG system in a $\mathcal{N}$-type configuration as shown in Fig.1. The proposed configuration can be experimentally realized by considering the energy levels of ${}^{87}Rb$, with the two metastable ground states designated as $|1\rangle=|5S_{\frac{1}{2}},F=2\rangle$, and $|2\rangle=|5S_{\frac{1}{2}},F=3\rangle$, and the remaining two excited states as $|3\rangle=|5P_{\frac{1}{2}},F^{{}^{\prime}}=2\rangle$ and $|4\rangle=|5P_{\frac{3}{2}},F^{{}^{\prime}}=4\rangle$. As depicted in Fig.1, we consider three laser fields acting on the electric-dipole allowed transitions $|3\rangle\leftrightarrow|1\rangle$, $|3\rangle\leftrightarrow|2\rangle$ and $|4\rangle\leftrightarrow|2\rangle$, thereby forming a $\mathcal{N}$-type configuration. The transition $|3\rangle\leftrightarrow|1\rangle$ is driven by a strong Raman field$~{}(\mathcal{E}_{r})$ with frequency $\omega_{r}$, while a weak probe field$~{}(\mathcal{E}_{p})$ with frequency $\omega_{p}$ couples the transition $|3\rangle\leftrightarrow|2\rangle$. A strong control field$~{}(\mathcal{E}_{c})$ with frequency $\omega_{c}$, drives the transitions $|4\rangle\leftrightarrow|2\rangle$. We define the electric field of all the three laser fields as

where, $\hat{e}_{j}$ is the unit polarization vector, $\mathcal{E}_{j}(\vec{r})$ is the slowly varying envelope, $\omega_{j}$ is the frequency of the laser field and $k_{j}$ is the wave number, respectively. The index $j\in\{r,p,c\}$ indicate the Raman, probe and control field, respectively.

In presence of these three laser fields, the time-dependent Hamiltonian of the system under the electric dipole approximation can be expressed as

Here $\bm{d}_{ij}=\langle i|\bm{d}|j\rangle$ represents the dipole moment of the transition $|i\rangle\leftrightarrow|j\rangle$. To remove the time dependencies from the Hamiltonian in Eq. (2), we make use of the following unitary transformation

and rewrite the Hamiltonian into a time-independent form under the rotating wave approximation as

where $\Delta_{r}=\omega_{r}-\omega_{31}$, $\Delta_{p}=\omega_{p}-\omega_{32}$, and $\Delta_{c}=\omega_{c}-\omega_{42}$ are the detuning of the Raman, probe and control fields, respectively. The Rabi frequencies of the Raman($\Omega_{r}$), probe($\Omega_{p}$) and the control fields($\Omega_{c}$) are defined as

To study the effect of the three laser fields on the atomic populations and coherences of the system, we use the following Liouville equation

Here, the last term $\mathcal{L}\rho$ is Liouvillian operator which describes the effect of relaxation by radiative decay and is expressed as

where $\gamma_{ij}$ corresponds to radiative decay rates from excited states $|i\rangle$ to ground states $|j\rangle$. Next, by incorporating Eq. (4) and Eq. (7) in Eq. (6), we can write the equation for density matrix elements corresponding to the atomic populations and coherences of the system as

where, the time derivatives are denoted by overdots and complex conjugate by $``^{*}"$. The remaining equations for the density matrix elements can be found from the property of conjugate $\dot{\rho}_{ji}^{*}=\dot{\rho}_{ij}$ and the population conservation condition $\sum\limits_{i=1}^{4}\rho_{ii}=1$.

In this section, we provide an analytical expression for the linear susceptibility of the system derived in the steady state limit using a perturbative method. In this method, we assume all the atoms to be in state $|1\rangle$, i.e. $\rho_{11}\approx 1$. Now, as a strong Raman field is applied to $|1\rangle\leftrightarrow|3\rangle$ transition, the condition for atoms to remain in state $|1\rangle$ is made possible by considering the Raman field to be highly detuned from excited state $|3\rangle$. In addition to this, we also consider the weak probe field, which connects the adjacent transition $|3\rangle\leftrightarrow|2\rangle$ to be highly detuned. This arrangement of both the fields turns the system into an active Raman gain system where the probe sees Raman gain around the two-photon resonance condition [21]. Apart from this, we also apply a strong control field to modulate this induced Raman gain in probe field. Further, it is necessary to mention here that the detuning of this control field has a very negligible effect on the population in state $|1\rangle$. So, for this configuration, we expand the density matrix to first order in probe as it is considered weak, and keep only upto the second order terms corresponding to the Raman field as the Raman process is a second-order process [23]. With these assumption, the density matrix is then expanded in the following manner:

here $\rho_{{}_{ij}}^{(0)}$ is the zeroth-order solution determined in the absence of all fields and $\rho_{{}_{ij}}^{(k)}$ with $k\in\{1-12\}$ is the $k^{th}-$order solution of the density matrix equation. We now substitute the above equation in Eq. (8) and equate the coefficients of zeroth-order and each of the $k^{th}-$order terms. As a result, we obtain a total of 13 different sets of coupled density matrix equations corresponding to each of the $\rho_{{}_{ij}}^{(k)}$ and $\rho_{{}_{ij}}^{(0)}$ terms. Next, we solve these 13 different sets of equations for $\rho_{{}_{ij}}^{(k)}$ and $\rho_{{}_{ij}}^{(0)}$ in the steady state limit and substitute its expression in Eq. (II.2) to obtain the full expression of the atomic populations and coherences $\rho_{ij}$. In this work, we are interested in studying the effect of medium on the probe field, so we will focus on the atomic coherence $\rho_{32}$ which yields the linear response of the medium. We then write this linear response (susceptibility) $\chi_{p}$ of the medium at probe frequency $\omega_{p}$ in terms of the atomic coherence $\rho_{32}$ as

where $\Gamma_{a}=\frac{(\gamma_{31}+\gamma_{32}+\gamma_{41}+\gamma_{42})}{2}-i(\Delta_{p}-\Delta_{c})$, $\Gamma_{b}=\frac{(\gamma_{41}+\gamma_{42})}{2}-i(\Delta_{r}-\Delta_{p}+\Delta_{c})$, and $\Gamma_{c}=\gamma_{c}+i(\Delta_{r}-\Delta_{p})$. The susceptibility expression in Eq. (10), clearly depicts its dependence on both Raman and control field intensities. Hence, by suitably manipulating both these fields, we can control the susceptibility of the medium and ultimately control the propagation dynamics of the probe field. In the following section, we present the equation which shows how this susceptibility regulates the propagation of probe beam traveling through the atomic medium.

In order to study the propagation of the probe field through the atomic medium, we use the Maxwell’s wave equation under slowly varying envelope and paraxial wave approximation and write it in terms of the Rabi frequency of the probe field into the following form

The first terms on the right hand side of Eq. (11) leads to diffraction of the probe beam propagating either in free space or through any medium. The second term on the right hand side of Eq. (11) represents the effect of the medium on the propagation of probe beam. It is interesting to notice from Eq. (11) that, if we suitably modulate the medium susceptibility in the transverse direction then we can eventually overcome the effect of diffraction. In the next section, we discuss on the manipulation of the medium susceptibility in the transverse direction and its effect on the probe beam propagation.

In this subsection, we first analyze the effect of continuous wave(cw) Raman and control field on the medium susceptibility. In this regard, we numerically evaluate the medium susceptibility as in Eq. (10), first in presence of only cw Raman field and then in presence of both cw Raman and cw control field. The result of our simulation is presented in Fig. (2), where the plot shows the variation of imaginary and real parts of susceptibility with probe detuning. In Fig. 2, it can be seen that in presence of Raman field the probe sees a Raman gain around the two photon resonance condition $\Delta_{p}-\Delta_{r}=0$. This is due to the fact that a strong Raman field acts between the levels $|3\rangle\leftrightarrow|1\rangle$ that has population in lower state and thus induces gain for the weak probe field which acts on the unpopulated levels $|3\rangle\leftrightarrow|2\rangle$, under high detuned condition [21]. Further, when a strong control field is applied to $|4\rangle\leftrightarrow|2\rangle$ transition, then the probe sees significant reduction in gain at the two photon resonance condition because of the splitting of the single Raman gain peak into two, as shown in Fig. 2. This splitting can be attributed to the fact that a strong control field dresses the levels $|4\rangle$ and $|2\rangle$ into the superposition states $|\pm\rangle=(|4\rangle\pm|2\rangle)/\sqrt{2}$. As a result, the probe sees two Raman gain peaks corresponding to the transition $|3\rangle\leftrightarrow|+\rangle$ and $|3\rangle\leftrightarrow|-\rangle$, respectively. From above discussion, it is evident that a strong control field plays an important role in controlling the Raman gain seen by the probe. So, in the next section, we discuss on how this Raman gain modulation by the strong control field can be a major factor in creation of tunable waveguide like structure inside the atomic medium.

In this subsection, we investigate the effect of spatially dependent Raman and control field on the susceptibility of the atomic medium. For this, we consider Raman field as a Gaussian beam and the control field as a Laguerre-Gaussian(LG) beam. The spatial profile of both the beams are expressed as

where $r=\sqrt{x^{2}+y^{2}}$ and $\phi=\tan^{-1}\left(\frac{y}{x}\right)$. Also, $w_{j}(z)=w_{0j}\sqrt{1+((z-q)/z_{0j})^{2}}$ and $R_{j}(z)=z+(z^{2}_{0j}/z)$ are the spot size and radius of curvature of the beams, respectively, with $w_{0j}$ being the minimum beam waist at focusing point $q$ and $z_{0j}=\pi w^{2}_{0j}/\lambda$ the Rayleigh length of the beams. The index $j\in\{r,c\}$ which represents the Raman and control beam, respectively.

Initially, we consider the case when only Raman field is present and thus examine the behavior of the susceptibility in the transverse direction using Eq. (10). We present the result of our simulation in Fig. (3). Figure 3 shows that in presence of a Gaussian Raman beam [See Fig. 3 for spatial profile], the probe beam sees a spatial gain profile in the transverse direction x at $y=0$ plane. This can be explained using the fact that Raman beam induces gain in the probe beam as explained in precedent subsection, also shown in Fig. 2. Hence, the probe beam will see gain only at the spatial position where Raman beam is present, leading to the formation of a gain window in the transverse direction similar to the profile of Raman beam. Interestingly, when the probe is slightly blue-detuned from the two photon resonance, it sees a waveguide like feature inside the medium as shown in Fig. 3. The variation of the refractive index in the transverse direction as shown in Fig. 3 depicts a variation similar to waveguide where the central region has higher index (core) than the regions away from center (cladding). Further, it can be seen in Fig. 3 that the refractive index profile also has a similar spatial shape in the transverse direction like the Raman Gaussian beam. As a result of this spatial similarity, the width of the core region can be considered to be equal to the width of the Raman beam i.e., $w_{r}(z)$. Accordingly, the length of the waveguide can then be attributed to be equivalent to the Rayleigh length $z_{0r}=\pi w^{2}_{0r}/\lambda$ of the Raman beam. Within this length, the width of the Raman beam doesn’t change much and hence the properties of the induced waveguide remains intact. So, from above discussion, it is apparent that a spatially dependent Raman beam plays a major role in creating a waveguide like feature inside the medium, which in turn can be able to support diffraction-less propagation of optical beams [18, 19].

Nevertheless, in the next section we show that this Raman induced optical waveguide(RIW) is unable to control efficiently the diffraction of narrow optical beams. To control diffraction of narrow optical beams, a high contrast waveguide with narrow core region is required. For the case of RIW, a high contrast waveguide with narrow core region can be achieved by increasing the atomic density of the system as well as by narrowing the width $w_{r}(z)$ of the Raman beam. This configuration, however, increases gain significantly and also reduces the length of the waveguide ($z_{0r}$) making it less efficient for diffraction control.

To overcome this problem, we make use of an additional spatial dependent control beam which can control the features of the waveguide. In this regard, we consider the control beam to be a LG beam as shown in Fig. 3. This particular spatial profile of the control beam creates a very narrow gain profile as well as a waveguide with a very narrow core region, which is shown in Fig. 3 and Fig. 3, respectively. Such spatial modulation in the gain as well as refractive index of the medium can be attributed to the fact that the presence of control beam decreases gain in the system, as explained in precedent subsection. Now, since the intensity distribution for the LG beam is a donought shape with zero intensity at the center and maximum intensity in the annular region, the spatial gain will be reduced in this annular region, resulting in the narrowing of the spatial gain profile. Similar reduction in the refractive index in the annular region is noticed due to the presence of LG control beam, which leads to the creation of a waveguide with narrow core.

Moreover, it is interesting to note that, unlike RIW case where the waveguide narrowing is only possible by narrowing the width of the Raman beam, which leads to a decrease in waveguide length, here the narrowing of waveguide is mainly due to the spatial shape of the control beam, which leaves the waveguide length unchanged. Hence, the spatial profile of the control beam plays a decisive role in controlling the features of the waveguide and in creating high contrast tunable waveguide in the medium. In the following subsection, we show how this control beam tunable Raman induced waveguide(CRIW) holds a greater advantage over RIW in controlling diffraction of narrow optical beams.

In this subsection, we explore on the possibility of diffractionless propagation of arbitrary modes of narrow optical beams in RIW and CRIW created inside the atomic medium. For this, we consider the initial profile of the probe beam as different Hermite-Gaussian modes, which are expressed as

where $H_{n}$ and $H_{m}$ are the Hermite polynomials of order $n$ and $m$, respectively. With above mentioned different initial profile, we study the propagation of probe beam by numerically solving Eq. (11) using Fourier split step method(FSSM) [25]. We first consider the propagation of $HG_{10}$ probe beam with a width ($w_{p}$) of 10 $\mu$m in RIW and present the result of our simulation in Fig. 4. Figure 4 shows the intensity distribution of the output probe beam in the transverse plane after propagating a distance of $z=2$mm in RIW. It is clear from Fig. 4 that probe beam gets diffracted after propagating through this induced wavegide. This is because, RIW has a wide core and is unable to control the diffraction of the probe beam. In addition to this, we also study the output probe power $P(z)=\iint\left|\Omega_{p}(x,y,z)\right|^{2}dx~{}dy$, and found that the probe beam is highly amplified by a factor of 100 after propagating a distance of $z=2$mm, as shown in Fig. 4. So, due to its inability to control diffraction along with creation of such large gain in the probe beam, which can lead to instability in the medium [20], the practical applicability of the RIW is severely limited.

However, this problem is eventually overcome by utilizing an additional LG control beam, which can efficiently control both diffraction as well as gain in the probe beam, as shown in Figure 4 and Figure 4, respectively. Such control over diffraction by the control beam can be attributed to the creation of waveguide with narrow core, which is explained in detail in precedent subsection. In addition to diffraction control, the probe beam is also slightly focused as shown in Fig. 4. Such focusing behaviour have been observed earlier in atomic based waveguides [19] and can be controlled by suitable modulating the features of the waveguide. Further, it is interesting to notice that the probe beam propagates without diffraction in this narrow waveguide [See Fig. 4] upto a distance of $2$mm, which is 5 times the Rayleigh length of the probe beam $z_{0p}$. Also, unlike earlier case, here the probe beam attains a reasonable amount of gain at a distance of $2$mm, as shown in Fig. 4. Our result on diffraction control of narrow optical beams to several Rayleigh lengths, may have potential application in high density lossless information transfer to macroscopic lengths and high contrast imaging.

Further, in this work, we are dealing with propagation of optical beams in a waveguide induced inside a gain system. So, it is necessary to verify the validity of both weak field approximation and paraxial approximation used in the calculation. In this regard, for our numerical calculations on beam propagation, we start with low probe power and throughout the propagation we found that the maximum intensity of probe is always very much smaller than the maximum intensity of Raman and Control beam. This condition of probe intensity validates our weak field approximation. Also, the width of the probe beam $w_{p}$ doesn’t change much throughout it’s propagation and the condition for paraxiality $\lambda/2\pi w_{p}<0.1$ is always satisfied [26], thus validating our paraxial approximation.

Next, to establish that CRIW can support arbitrary modes of the probe beam, we also study the propagation of $HG_{20}$ probe beam in it and present our result in Fig. 5. From Fig. 5, it is noticeable that CRIW is able to control the diffraction of the probe beam to length of $2.5z_{0p}$ with reasonable amount of gain, while RIW fails to do so. Apart from this, we also performed numerical simulation on propagation of other higher order modes of the probe beam and found diffractionless propagation of such modes in CRIW. Hence, from above discussions, it becomes evident that control beam plays an important role in creation of CRIW and subsequently in controlling the diffraction of arbitrary modes of narrow optical beams to several Rayleigh lengths.