Electromagnetically Induced Grating of Surface Polaritons via Coherent Population Oscillation

Abstract

We propose a scheme to study the electromagnetically induced grating (EIG) of surface polaritons (SPs) in a negative index metamaterial/rare-earth-ion-doped crystal interface waveguide system, based on coherent population oscillation (CPO) modulating by a standing wave control field. Absorption grating can be formed via the large absorption modulation induced by the linear susceptibility of the system; the diffraction of SPs can be realized but with a very small first-order diffraction efficiency and the phase modulation in this case, is negligible. However, when the giant Kerr nonlinearity is taken into account, the phase modulation can be significantly enhanced and accompanied by high transmission at the same time, thus, a phase grating, which effectively diffracts SPs into a high-order direction, can be induced. For both the absorption and phase grating, the dependencies of the first-order diffraction efficiency on the Rabi frequency of the standing wave control field, optical detuning, and interaction length are discussed. The results obtained here have certain theoretical significance for spectral enhancements and precision measurements at the micro–nanoscales.

1. Introduction

Diffraction is the physical phenomenon of a wave traveling away from its original straight line when it encounters an obstacle and occurs if the wavefront of a wave is obstructed when encountering a (transparent or opaque) barrier, by which the wavefront is altered in amplitude, phase, or both [1]. Grating based on the diffraction effect is a kind of beam splitting element and the core element of spectral instruments, which has great significance in the fields of precision spectral analysis [2], optical holography [3], etc.

Recently, many fascinating phenomena based on quantum interference and coherence have attracted the attention of researchers (electromagnetically induced transparency (EIT) [4,5,6] is the most prominent one). By replacing the traveling wave of EIT with a standing wave, a grating called electromagnetically induced grating (EIG) is formed [7]. Due to the spatial modulation induced by the standing wave, the amplitude of a resonant probe field changes in a period, and then the probe field can be diffracted into high-order directions. In comparison with traditional optical grating, the EIG configuration is easier to reconstruct and can be flexibly tuned to control the dynamic behavior of light propagation due to the existence of various energy levels, selection rules, and giant Kerr nonlinearities, together with significant suppression of optical absorption due to the interplay of light-media resonance and EIT effect.

An atomic absorption grating was theoretically first proposed by Ling and coworkers [7], and then experimentally demonstrated in cold [8,9,10] and hot [11,12] atomic samples. Amplitude gratings, including absorption and gain gratings, in general, are well known for having low diffraction power, as opposed to phase grating, which disperses energy into higher diffraction orders much more efficiently [13]. Therefore, finding schemes to promote the diffraction efficiency of EIG is the main concern and lays a solid foundation for realizing the related applications. Then, EIG based on various physical mechanisms, such as EIT [13,14,15,16,17,18], spontaneously generated coherence [19,20], coherent population trapping [21], coherent population oscillation (CPO) [22], active Raman gain [23], $\mathcal{PT}$ symmetry [24,25], etc., in media, such as atomic gas [26,27,28,29,30], Rydberg atomic gas [31,32,33,34], solid material [35,36,37], semiconductor [38,39], etc., with and without nonlinearity [40,41,42], one- and two-dimensions [43,44,45], have been intensively studied. Recently, a series of experimental work on one- and two-dimensional EIGs has also been realized in atomic gas [46,47,48].

On the other hand, surface plasmon polariton (SPP) is a collective excitation mode generated by the interaction between photons and free electrons on the metal surface; the diffraction of SPPs and propagation modulation of SPPs passing through gratings have attracted attention [49], as they have significance for both deep understanding of SPPs theoretically and broad applications in micro–nano-optics. However, the propagation lengths of SPPs hugely limit the study of the diffraction of SPPs due to the Ohmic loss of the metals. In 2008, Sanders et al. proposed the concept of surface polaritons (SPs), which are excited at the surfaces of negative index metamaterials (NIMMs) and can propagate with low losses for a long distance; they investigated the coherent control of SPs via interaction between SPs and a coherence medium [50]. Such a scheme provides a good platform for studying the active control of SPs [51,52,53,54,55,56], SP solitons [57,58], and nonlinear propagation of SPs [59,60,61] as they can suppress Ohmic loss via quantum destructive interference between designable electric and magnetic absorption of NIMM. Thus, there is also the possibility of studying the diffraction of SPs.

In this work, combining the active controllability of EIG and the urgent need to study the diffraction of SPs, we propose a scheme to study the EIG of low loss SPs in a NIMM/rare-earth-ion-doped crystal interface waveguide system, based on CPO modulating by a standing wave control field. Based on the multi-scale method, we calculated the linear and nonlinear susceptibilities, respectively, and investigated the giant Kerr nonlinearity due to the confinement enhancement effect of the system. Then, we studied EIG of SPs based on systems under linear and nonlinear excitations, respectively. For both cases, the dependencies of the first-order diffraction efficiency on the Rabi frequency of the control field, optical detuning, and interaction length are discussed.

The rest of the article is arranged as follows: In Section 2, we introduce our theoretical model. In Section 3, the derivation of the main equation is given. In Section 4, we investigate the absorption grating and phase grating, respectively. In the final section, we summarize the main results obtained in this work.

2. Theoretical Model

The planar waveguide system under study consisted of a layer of NIMM occupying the lower half plane $z<0$, and a Kerr medium covering part of the NIMM in the upper half plane $z>0$, as shown in Figure 1a. The low-loss transverse magnetic (TM) guide mode SPs can be excited on the surface of the NIMM and propagated along the positive x direction. The Kerr medium here can be selected as rare-earth-ion doped crystals or other materials with third-order nonlinearity. In this case, we selected Pr${}^{3+}$:Y${}_2$SiO${}_5$ (Pr:YSO) as the Kerr medium. Two strong electric fields propagating with an angled overlap in the crystal, and the two counterpropagating components of the strong fields in the y direction can form a standing wave along the y direction, which can lead to intensity-dependent light-matter interaction. The SPs pass through the standing wave region and acquire periodical modulation (including absorption, gain, or phase modulation); therefore, SP-grating was formed.

The Pr:YSO was excited as a typical two-level CPO scheme and the energy level configuration is shown in Figure 1b, which contains a ground state $|1\rangle$ and an excited state $|2\rangle$, the transition $|1\rangle \to |2\rangle$ was chosen from ${}^3$H${}_4$→${}^1$D${}_2$ of Pr:YSO. The ground state $\left|1\right.〉$ couples to the excited state $\left|2\right.〉$ with the weak probe field ${\mathbf{E}}_p$ with center angle frequency ${\omega }_p$ (which propagates along x direction and is chosen from guide modes of SPs at the interface of NIMM) and the strong control field ${\mathbf{E}}_c$ with center angle frequency ${\omega }_c$ (which is the standing wave field formed in the system), respectively. ${\Delta }_p$ (${\Delta }_c$) is the optical detuning of the probe (control) field, $\Delta ={\Delta }_p-{\Delta }_c$. The atoms occupy the state $\left|2\right.〉$ can spontaneously radiate to the ground state with the rate ${\Gamma }_2$.

The total electric field in the interaction region can be expressed as $\mathbf{E}={\mathbf{E}}_p+{\mathbf{E}}_c$. The probe field ${\mathbf{E}}_p$ was chosen as the lowest order TM mode SP, which reads

$${\mathbf{E}}_p={\mathcal{E}}_p{\mathbf{u}}_p\left(z\right)e^{i(k_{p}x-{\omega }_{p}t)}+c.c.,$$

(1)

with ${\mathcal{E}}_p$ being the amplitude of the probe field, ${\mathbf{u}}_p\left(z\right)$ being the mode function in the z direction, $k_p=k\left({\omega }_p\right)$ being the propagate constant of the probe field; specific expressions are the results from the dielectric-NIMM interface, and are shown in Appendix A. The standing wave control field reads

$${\mathbf{E}}_c={\mathcal{E}}_c\sin (\pi y/\Lambda )e^{i(k_{cx}x-{\omega }_{c}t)}+c.c.,$$

(2)

where ${\mathcal{E}}_c$ is assumed as a real constant for simplicity, and $\Lambda =\pi /k_{cy}$ represents the spatial period of the standing wave in y direction, which can be controlled by the angle at which the two components intersect and satisfy $k_c^2=k_{cx}^2+k_{cy}^2$.

Under the electric-dipole and rotating-wave approximations, the Hamiltonian of the system reads

$$\widehat{\mathcal{H}}=\sum _{j=1}^2{E}_j|j\rangle \langle j|-\hslash \left[{\Omega }_c^{*}\left(y\right)e^{-i\left(k_{cx}x-{\omega }_{c}t\right)}\right.\left.+{\zeta }^{*}\left(z\right){\Omega }_p^{*}{e}^{-i\left(k_{p}x-{\omega }_{p}t\right)}\right]|1\rangle \langle 2|+\mathrm{H}.\mathrm{c}.,$$

(3)

where $E_j$ is the eigenenergy of the jth levels, $\zeta \left(z\right)={\mathbf{e}}_{12}·{\mathbf{u}}_p\left(z\right)$, ${\Omega }_p=\left|{\mathbf{p}}_{12}\right|{\mathcal{E}}_p/\hslash$ is the half Rabi frequency of the probe field, ${\Omega }_c\left(y\right)=\left|{\mathbf{p}}_{12}\right|{\mathcal{E}}_c\sin (\pi y/\Lambda )/\hslash ={\Omega }_{c0}\sin (\pi y/\Lambda )$ is the spatial dependent Rabi frequency of the control field, and $e_{12}$ is the unit vector of the electric-dipole matrix element ${\mathbf{p}}_{12}$ associated with the transition from $|1\rangle$ to $|2\rangle$, i.e., ${\mathbf{p}}_{12}$ = ${\mathbf{e}}_{12}{p}_{12}$.

The interaction information of the system is given by the evolution of the density matrix $\sigma$, which is governed by the optical Bloch equation

$$\begin{array}{c}i\frac{\partial }{\partial t}{\sigma }_{11}-i{\Gamma }_2{\sigma }_{22}+\left({\Omega }_c^{*}\left(y\right)+{\zeta }^{*}\left(z\right){\Omega }_p^{*}{e}^{-i\Theta }\right){\sigma }_{21}-\left({\Omega }_c\left(y\right)+\zeta \left(z\right){\Omega }_p{e}^{i\Theta }\right){\sigma }_{21}^{*}=0,\hfill \\ i\frac{\partial }{\partial t}{\sigma }_{22}+i{\Gamma }_2{\sigma }_{22}+\left({\Omega }_c\left(y\right)+\zeta \left(z\right){\Omega }_p{e}^{i\Theta }\right){\sigma }_{21}^{*}-\left({\Omega }_c^{*}\left(y\right)+{\zeta }^{*}\left(z\right){\Omega }_p^{*}{e}^{-i\Theta }\right){\sigma }_{21}=0,\hfill \\ \left(i\frac{\partial }{\partial t}+d_{21}\right){\sigma }_{21}-\left({\Omega }_c\left(y\right)+\zeta \left(z\right){\Omega }_p{e}^{i\Theta }\right)\left({\sigma }_{22}-{\sigma }_{11}\right)=0,\hfill \end{array}$$

(4)

and satisfies ${\sum }_{j=1}^2{\sigma }_{jj}=1$ ($j=1,2$). Here, $\Theta =\left(k_p-k_{cx}\right)x-\Delta t$ and $d_{21}=-\Delta {\omega }_{21}+\Delta +i{\gamma }_{21}$, in which, $\Delta {\omega }_{21}$ is the energy-level shift due to the inhomogeneous broadening resulting from the solid environment, and the decoherence rate of the system is defined as ${\gamma }_{21}={\Gamma }_2/2+{\gamma }_{{}_{21}}^{dph}$, ${\gamma }_{{}_{21}}^{dph}$ is the decoherence rate due to processes that are not associated with the population transfer.

Under the framework of the semi-classical theory, the dynamic evolution of the probe field passing through the crystal is described by the Maxwell wave equation. By applying the slowly varying envelope approximation and paraxial approximation, and taking the steady state response of the crystal into account, the Maxwell equation can be reduced to

$$\frac{\partial }{\partial x}{\mathbf{E}}_p=i\frac{\pi }{{\epsilon }_0{\lambda }_p}\mathbf{P},$$

(5)

with the electric polarization intensity of the system being $\mathbf{P}={\mathbf{P}}_{host}+{\mathcal{N}}_{ion}\mathrm{Tr}\left({\mathbf{p}}_{jk}{\rho }_{kj}\right)$, where ${\rho }_{jk}={\sigma }_{jk}{e}^{i(k_{p}x-{\omega }_{p}t)}$, ${\mathcal{N}}_{ion}$ is the ion density, and ${\mathbf{P}}_{host}={\epsilon }_0{\chi }_{host}{\mathbf{E}}_{\mathbf{p}}$ is the electric polarization intensity in the absence of the ions (assumed as a constant background in our study for simplicity).

Generally, the susceptibility in the interaction region is a complex function depending on the mode of distribution, inhomogeneous broadening, etc. In order to obtain an average effective susceptibility with periodic modulation in the y direction, we applied the average field approximation and took the inhomogeneous broadening of the crystal into account; then, the electric polarization intensity of the system became

$$\mathbf{P}={\mathbf{P}}_{\mathrm{host}}+{\mathcal{N}}_{ion}{\int }_{-\infty }^{\infty }d\left(\Delta {\omega }_{21}\right)f\left(\Delta {\omega }_{21}\right)〈{\mathbf{p}}_{12}{\sigma }_{21}{e}^{i\left(k_{p}x-{\omega }_{p}t\right)}+c.c.〉,$$

(6)

where $f(\Delta {\omega }_{21})$ is the inhomogeneous broadening distribution function, which, for simplicity, is assumed to have a Lorentzian form, i.e., $f(\Delta {\omega }_{21})=W_{21}/\left[\pi (\Delta {\omega }_{{}_{21}}^2+W_{21}^2)\right]$, with $2{\mathrm{W}}_{21}$ being the full width at half maximum (FWHM), and the expectation operator $〈\dots 〉$ is defined as $〈\psi \left(z\right)〉\equiv {\int }_{-\infty }^{\infty }dz{\zeta }^{*}\left(z\right)\psi \left(z\right)/{\int }_{-\infty }^{\infty }dz{\left|\zeta \left(z\right)\right|}^2$, i.e., the average over mode distribution function in the z-direction.

Equations (4) and (5) can totally describe the interaction and propagation properties of our system.

3. Derivation of the Main Equation

3.1. The Solutions of the Bloch Equation

Note that, the Rabi frequencies of the two fields in our system satisfy the condition $|{\Omega }_p|\ll |{\Omega }_{c0}|$, and in order to study the contributions of the linear and nonlinear responses of the system, respectively, we solved Equation (4) by the multi-scale method [51,62] in the following.

Before solving, we first needed to know the base state of the system, which corresponded to the state in the absence of the probe field, and also as the zeroth order solution of the system, i.e., ${\Omega }_p=0$ and $\partial /\partial t=0$. Under this condition, by solving Equation (4), we can obtain that

$$\begin{array}{cc}\hfill {\sigma }_{11}^{\left(0\right)}& =\frac{{\Gamma }_2{\left|d_{21}\right|}^2+2{\gamma }_{21}{\left|{\Omega }_c\left(y\right)\right|}^2}{{\Gamma }_2{\left|d_{21}\right|}^2+4{\gamma }_{21}{\left|{\Omega }_c\left(y\right)\right|}^2},\hfill \\ \hfill {\sigma }_{22}^{\left(0\right)}& =\frac{2{\gamma }_{21}{\left|{\Omega }_c\left(y\right)\right|}^2}{{\Gamma }_2{\left|d_{21}\right|}^2+4{\gamma }_{21}{\left|{\Omega }_c\left(y\right)\right|}^2},\hfill \\ \hfill {\sigma }_{21}^{\left(0\right)}& =-\frac{{\Omega }_c\left(y\right)}{d_{21}}\frac{{\Gamma }_2{\left|d_{21}\right|}^2}{{\Gamma }_2{\left|d_{21}\right|}^2+4{\gamma }_{21}{\left|{\Omega }_c\left(y\right)\right|}^2}.\hfill \end{array}$$

(7)

It is helpful to make a qualitative analysis of the property on the base state and its implication for SP propagation. ${\sigma }_{jj}^{\left(0\right)}$ represents the population occupying the state $|j\rangle$. Except for vanishing points in ${\Omega }_c\left(y\right)$, if the condition $|{\Omega }_{c0}{|}^2\gg {\gamma }_{21}^2$ is satisfied, we can obtain that ${\sigma }_{11}^{\left(0\right)}\approx {\sigma }_{22}^{\left(0\right)}=1/2$; it means that for nearly half of the ions occupying the excited state, the system shows saturated absorption, which can largely suppress the absorption of the system for the probe field when the probe field turns on, or even generates the optical gain for the probe field [63]. In any case, it can help the low-loss SPs propagate for a long distance stably.

Based on the base state, we make the asymptotic expansion as ${\sigma }_{jj}-{\sigma }_{jj}^{\left(0\right)}=ϵ{\sigma }_{jj}^{\left(1\right)}+{ϵ}^2{\sigma }_{jj}^{\left(2\right)}+{ϵ}^3{\sigma }_{jj}^{\left(3\right)}+···,(j=1,2),{\sigma }_{21}-{\sigma }_{21}^{\left(0\right)}=ϵ{\sigma }_{21}^{\left(1\right)}+{ϵ}^2{\sigma }_{21}^{\left(2\right)}+{ϵ}^3{\sigma }_{21}^{\left(3\right)}+···,$ where $ϵ$ is a dimensionless small parameter characterizing the typical amplitude ratio of the probe and the control field; therefore, ${\Omega }_p$ is also treated as a first-order small quantity in the following study. During the propagation of the probe field, we can assume that the probe can keep being the TM mode [50]. Substituting the above expansion and assumption into the Bloch equation, we can obtain a series of linear but inhomogeneous equations, order by order (see Appendix B for more details).

In the first-order solution, we can obtain the linear excitations of the system, which read

$${\sigma }_{21}^{\left(1\right)}={\mathcal{B}}_1\zeta \left(z\right){\Omega }_p{e}^{i\Theta }+{\mathcal{B}}_2{\zeta }^{*}\left(z\right){\Omega }_p^{*}{e}^{-i\Theta },$$

(8)

the other first-order density matrix elements ${\sigma }_{jj}^1$ and the explicit expressions of the related parameters $\mathcal{A}$, ${\mathcal{B}}_1$, and ${\mathcal{B}}_2$ are given in Appendix B. It needs to be pointed out that ${\sigma }_{21}^{\left(1\right)}$ contains two terms that are proportional to $\exp (i\Theta )$ and $\exp (-i\Theta )$. As a result, a new field may be generated via a four-wave mixing process, as shown in Reference [64]. However, due to the unsaturation feature, the new field has a very significant absorption and, hence, can be neglected safely in our system. Thus, the linear susceptibility of the system reads

$${\chi }_p^{\left(1\right)}=\frac{{\mathcal{N}}_{ion}{\left|{\mathbf{p}}_{12}\right|}^2}{{\epsilon }_0\hslash }{\int }_{-\infty }^{\infty }d\left(\Delta {\omega }_{21}\right)f\left(\Delta {\omega }_{21}\right)〈\zeta \left(z\right){\mathcal{B}}_1〉.$$

(9)

Based on the first and second solutions, we can obtain the nonlinear excitations in the third order, which reads

$${\sigma }_{21}^{\left(3\right)}={\zeta \left(z\right)\left|\zeta \left(z\right)\right|}^2{a}_{21}^{\left(3\right)}{\left|{\Omega }_p\right|}^2{\Omega }_p{e}^{i\Theta },$$

(10)

with $a_{21}^{\left(3\right)}$ given in Appendix B. Note that, we only keep the terms that are proportional to the third-order susceptibility during the calculation in the third-order for simplicity. Thus, the third-order susceptibility can be obtained as

$${\chi }_p^{\left(3\right)}=\frac{{\mathcal{N}}_{ion}{\left|{\mathbf{p}}_{12}\right|}^4}{{\epsilon }_0{\hslash }^3}{\int }_{-\infty }^{\infty }d\left(\Delta {\omega }_{21}\right)f\left(\Delta {\omega }_{21}\right)〈{\zeta \left(z\right)\left|\zeta \left(z\right)\right|}^2{a}_{21}^{\left(3\right)}〉.$$

(11)

By comparing Equations (9) and (11), we can find that there is an additional factor ${\left|\zeta \left(z\right)\right|}^2$ in Equation (11), ${\left|\zeta \left(z\right)\right|}^2$ is the enhancement factor of the SPs due to the confinement of the waveguide system [65], it can contribute a huge amplification in the third-order process, i.e., the nonlinear interaction between the SPs and the crystal can be enhanced hugely. Therefore, the giant Kerr nonlinearity can be realized in our system, a non-negligible (or even comparable to the linear susceptibility) ${\chi }_p^{\left(3\right)}$ can be generated. In the next section, we will discuss the contribution of ${\chi }_p^{\left(3\right)}$ for the EIG of SPs in detail.

3.2. Fraunhofer Diffraction of the Low Loss SPs

Based on the above results, the average effective susceptibility of the system reads $\chi ={\chi }_{host}+{\chi }_p^{\left(1\right)}+{\chi }_p^{\left(3\right)}{\left|{\mathcal{E}}_p\right|}^2$, and the electric polarization intensity can be written as the classic form, $\mathbf{P}={\epsilon }_0\chi {\mathbf{E}}_p$.

Then, Equation (5) becomes

$$\frac{\partial {\mathbf{E}}_p}{\partial x}=\left(-\alpha +i\beta \right){\mathbf{E}}_p,$$

(12)

where $\alpha =\left(\pi /{\lambda }_p\right)\mathrm{Im}\left(\chi \right)$ and $\beta =\left(\pi /{\lambda }_p\right)\mathrm{Re}\left(\chi \right)$ are the absorption and dispersion coefficients of the probe field, respectively.

We assume that the interaction length of the crystal experienced by the probe field in the x direction is L (thickness of the crystal in x direction), and then, we obtain the transmission function of the probe field at $x=L$,

$$T\left(y\right)=e^{-\alpha \left(y\right)L}{e}^{i\beta \left(y\right)L},$$

(13)

where the first and second terms in the exponential correspond to the absorption and phase modulation, respectively.

For the low-loss SPs, normalizing the probe field to its initial amplitude, and by the Fourier transformation of T, we can obtain the Fraunhofer diffraction equation of the SPs, which is given by

$$I_p\left(\theta \right)=|F\left(\theta \right){|}^2\frac{{sin}^2\left(N\pi \Lambda sin\theta /{\lambda }_p\right)}{N^2{sin}^2\left(\pi \Lambda sin\theta /{\lambda }_p\right)},$$

(14)

where

$$F\left(\theta \right)={\int }_0^{\Lambda }T\left(y\right)exp\left(-i2\pi ysin\theta /{\lambda }_p\right)dy,$$

(15)

represents the Fraunhofer diffraction of a single space period, N is the number of spatial periods of the grating, and $\theta$ is the diffraction angle with respect to the x direction. The nth-order diffraction angle is determined by the grating equation, $\Lambda \sin \theta =j{\lambda }_p$. Therefore, we can obtain the jth-order diffraction intensity as

$$I_p\left({\theta }_j\right)=|F\left({\theta }_j\right){|}^2={\left|{\int }_0^{\Lambda }T\left(y\right)exp\left(-i2n\pi y/\Lambda \right)dy\right|}^2.$$

(16)

For a normalized incident probe field, $I_p\left({\theta }_j\right)$ also equals to the nth-order diffraction efficiency.

4. Results and Discussion

Next, we chose realistic system parameters to carry out our study. The NIMM was selected as a sliver-based material; the relative permittivity and permeability of the NIMM can be described by the Drude model in the optical region, i.e., $\epsilon \left(\omega \right)={\epsilon }_{\infty }-{\omega }_e^2/\left[\omega \left(\omega +i{\gamma }_e\right)\right]$, $\mu \left(\omega \right)={\mu }_{\infty }-{\omega }_m^2/\left[\omega \left(\omega +i{\gamma }_m\right)\right]$, where ${\omega }_e$ (${\omega }_m$) is the electric (magnetic) plasma frequency, ${\gamma }_e$ (${\gamma }_m$) is the electric (magnetic) decay rate, and ${\epsilon }_{\infty }$ and ${\mu }_{\infty }$ are the background constants. The related parameters are ${\omega }_e=1.37\times {10}^{16}{\mathrm{s}}^{-1}$, ${\gamma }_e=2.37\times {10}^{13}{\mathrm{s}}^{-1}$, ${\omega }_m={10}^{15}{\mathrm{s}}^{-1}$, ${\gamma }_m={10}^{12}{\mathrm{s}}^{-1}$ (as for Ag) [50,63]. The two-level system was chosen from ${}^3$H${}_4$→${}^1$D${}_2$ of Pr:YSO [66], thus, ${\omega }_p=4.95\times {10}^{14}{\mathrm{s}}^{-1}$, which is slightly off the lossless point in eigen dispersion spectrum of the TM SPs (to obtain an acceptable suppression of the Ohmic loss and a required SP confinement [50]), ${\lambda }_c\simeq {\lambda }_p=606\mathrm{nm}$, $\left|{\mathbf{p}}_{12}\right|=2.6\times {10}^{-30}\mathrm{C}·\mathrm{cm}$, ${\Gamma }_2=10\mathrm{kHz}$, ${\gamma }_{21}^{dph}=0.1$ kHz, the inhomogeneous broadening width is $W_{21}=2\mathrm{GHz}$, and the number density of the doped ions is ${\mathcal{N}}_{ion}=5\times {10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$, the refractive index of Pr:YSO in our system is $n\approx 1.33$, the corresponding ${\chi }_{host}=0.77$ [67].

In this case, the wavenumber of the probe field without modulation of the crystal $k\left({\omega }_p\right)=(7.57\times {10}^4+10.85i)$ cm${}^{-1}$. Thus, the typical absorption length $L_A=1/\mathrm{Im}\left[k\left({\omega }_p\right)\right]$ = 0.92 mm, which is much larger than L (usually tens of times the wavelength ${\lambda }_p$). Therefore, during the diffraction procession, SPs pass through the crystal, and Ohmic loss from the NIMM can be neglected safely.

For insight into the contributions of the giant nonlinearity in the system, we discuss the characteristics of SP EIG generated in our system, with and without nonlinearity, separately, in the following.

4.1. Absorption Grating of SPs Based on Linear Excitations of the System

When the third-order nonlinearity is absent, the system is in a linear excitation state, i.e., $\chi ={\chi }_{host}+{\chi }_p^{\left(1\right)}$. The probe field undergoes absorption and phase modulation in the y direction due to the standing wave control field.

Figure 2a shows the amplitude $\left|T\right(y\left)\right|$ (blue dashed line, characterizing the absorption modulation) and phase $\varphi \left(y\right)/\pi$ (red solid line, characterizing the phase modulation) of the transmission function $\mathrm{T}\left(y\right)$, as functions of y within two spatial periods (y is in units of $\Lambda$ for convenience). System parameters are given by ${\Omega }_{c0}=7$ kHz, $\Delta =-4.5$ kHz, $\Lambda /{\lambda }_p=4$, $N=5$, and $L=x_0$, with defining $x_0={\lambda }_p/\pi$ as the typical length in the propagation direction. The standing wave control field can lead to transparency for the probe field at the antinodes of the standing wave due to the CPO effect, and huge absorption at the nodes, corresponding to peaks ($\left|T\right(y\left)\right|\approx 1$) and dips ($\left|T\right(y\left)\right|\approx 0.1$) on the blue dashed line, respectively. We can see that the absorption modulation is close to $0.9$. On the contrary, the phase modulation is only about $0.12\pi$, which is neglected, as the red solid line in Figure 2a. Thus, an absorption grating was formed in the system. The corresponding diffraction intensity $I_p\left(\theta \right)$ as a function of the diffraction angle sin$\left(\theta \right)$ is shown in Figure 2b. We can see that the diffraction of the probe field was realized, but the first-order diffraction efficiency was quite small, which was only about $5.6\%$ (intensity at sin$\left(\theta \right)=0.25$); the second- and higher-order diffraction efficiencies were almost negligible.

For a detailed understanding of the absorption grating, we studied the influence of Rabi frequency ${\Omega }_{c0}$, detuning $\Delta$, and the interaction length L to the first-order diffraction efficiency. Figure 3a shows the first-order diffraction intensity $I_p\left({\theta }_1\right)$ as a function of ${\Omega }_{c0}$ and $\Delta$. System parameters are the same as that given in Figure 2. We can see a band region where the first-order diffraction intensity is higher than $5\%$, near this band; $I_p\left({\theta }_1\right)$ decayed quickly, especially in the decrease of ${\Omega }_{c0}$ or increase of $\Delta$. The physical reason is that when ${\Omega }_{c0}$ decreased, the system was under an unsaturation state, and the influence of the CPO effect became weak; an increase of $\Delta$ could lead the system to an off-resonant state, thus, the first-order diffraction efficiency decayed quickly. In addition, it was easy to find that the first-order diffraction efficiency of the absorption grating was relatively small, no matter the parameter region. Figure 3b shows the relation between the first-order diffraction intensity $I_p\left({\theta }_1\right)$ and the interaction length $L/x_0$. The system parameters are the same as those given in Figure 2a. $I_p\left({\theta }_1\right)$ increased first and then decreased with the increase of the interaction length $L/x_0$.

The reason for the rapid growth and then slow decline of the first-order diffraction intensity is that, with the increase of $L/x_0$, the absorption modulation first increased (see from Equation (13)), and then the absorption of the system gradually came into play when keeping the increased interaction length, which caused the slow decline of the first-order diffraction intensity.

4.2. Phase Grating of SPs Based on the Nonlinear Response of the System

We now investigate the nonlinear excitations of the system. With the presence of the third-order nonlinearity, the susceptibility became $\chi ={\chi }_{host}+{\chi }_p^{\left(1\right)}+{\chi }_p^{\left(3\right)}{\left|{\mathcal{E}}_p\right|}^2$.

Figure 4a shows the amplitude $\left|T\right(y\left)\right|$ (blue dashed line) and phase (red solid line) of the transmission function $\mathrm{T}\left(y\right)$ as functions of $y/\Lambda$ within two spatial periods when the third-order susceptibility was taken into account. System parameters were given by ${\Omega }_{c0}=6.5$ kHz, $\Delta =-14$ kHz, and $L=x_0$. Other system parameters were the same as that given in Figure 2. In this case, we can see that the transmission was around $0.94$, and changed slightly with $y/\Lambda$; however, the phase modulation could be $1.14\pi$, thus, a high-quality phase-grating was generated in the system. This proves that the ’showing up’ of a giant Kerr nonlinearity changes the properties of the system hugely. The corresponding diffraction intensity $I_p\left(\theta \right)$ as a function of the diffraction angle sin$\left(\theta \right)$ is shown in Figure 4b. We can see that the modulated phase deflected a significant portion of the probe field out of the central diffraction pattern into two additional side patterns located at $sin\theta =\pm 0.25$, and a relatively small amount of energy was also shifted into the higher-order diffraction. The first-order diffraction efficiency could be up to nearly $25\%$ (much larger than that in the linear case), and even the second- and third-order diffraction efficiencies were enhanced due to the third-order nonlinearity.

As shown in Figure 4c,d, when slightly increasing the interaction length to $L=1.2x_0$, the transmission amplitude $\left|T\right(y\left)\right|$ does not change, but the phase modulation increases to $1.36\pi$, we can see that the second- and third-order diffraction intensities increased, but the first-order diffraction intensity decreased due to the conservation of energy, such a feature can be used to study high-order diffraction of SPs.

Figure 5a shows the first-order diffraction intensity $I_p\left({\theta }_1\right)$ as a function of ${\Omega }_{c0}$ and $\Delta$. System parameters were the same as those given in Figure 4a. Similar to Figure 3a, we also found a band region where the first-order diffraction intensity was higher than $20\%$, the parameter range was about $\Delta$ = −14.5∼−13.5kHz and ${\Omega }_{c0}$ = 6∼7 kHz. Near this band, $I_p\left({\theta }_1\right)$ also decayed quickly. Compared to the linear case, it was easy to find that the first-order diffraction efficiency of the absorption grating was relatively large. Figure 5b shows the relation between the first-order diffraction intensity $I_p\left({\theta }_1\right)$ and the interaction length $L/x_0$. The system parameters are the same as those given in Figure 2a. $I_p\left({\theta }_1\right)$ increased first and then decreased with the increase of the interaction length $L/x_0$, due to the increase of the phase modulation, the first-order diffraction intensity had a certain degree of oscillation.

Finally, as shown in Figure 3a and Figure 5a, the peak areas of the linear and nonlinear cases mainly worked in different parameter regions only with small partial area overlaps; thus, experimentally, one can adjust the detuning and intensity of the control field to obtain the absorption grating (dominantly induced by linear susceptibility) and phase grating (dominantly induced by nonlinear susceptibility), respectively.

5. Conclusions

In summary, we proposed a scheme to study the EIG of low loss SPs in a NIMM/rare-earth-ion-doped crystal interface waveguide system, based on CPO-modulating via a standing wave control field. Based on the multi-scale method, we obtained linear and nonlinear susceptibilities, respectively, and a giant Kerr nonlinearity was realized due to the confinement enhancement effect of the system. In the linear region, an absorption grating was generated via the large absorption modulation induced by the linear susceptibility of the system, but first-order diffraction efficiency was very small, and the phase modulation was negligible in this case. In the nonlinear region, the phase modulation was significantly enhanced via the third-order nonlinearity, and accompanied by a high transmission at the same time; thus, phase grating was induced, which effectively diffracted SPs into a high-order direction. For the absorption and phase grating, the dependencies of the first-order diffraction efficiency on the Rabi frequency of the control field, optical detuning, and interaction length were discussed. These results not only provide a theoretical basis for the study of the optical properties of SPs, but also have broad application prospects in the field of SP-guiding, spectral analysis, and measurements at the micro–nanoscales.