## Abstract

Here, an all-optical switch is modeled in the form of fiber Bragg grating (FBG) based on electromagnetically induced transparency (EIT) phenomena dealing with a three-level EIT silicon (Si) nanocrystalline medium. The EIT consists of two fields, namely a strong control and weak probe fields that interact with the medium. Fast-optical switching of the probe is achieved by applying a pulsed control field, which relies on both steady-state and transient density matrices. The effects of four significant parameters of interest, i.e., Rabi frequency of the control field, EIT nanocrystal density, FBG length, and spontaneous decay rate, have been extensively investigated to optimize the probe transmission. Subsequently, the time-dependent density matrix is solved to obtain the response time of the FBG-EIT switch. Eventually, FBG is modeled such that low index layers contain EIT material and the high index layers do not. The FBG-EIT switch satisfies the criteria of maximum transmission and minimum rise time to give out the optimal operational condition. Despite the optical switching is in OFF mode at Ω_{c}=0 THz demonstrating a high FBG reflectance, the reflectance suddenly vanishes at a certain $\mathrm{\Omega }_c^s$ to satisfy the equivalence of low and high refractive indices (switching ON). Therefore, a fast-optical switch is envisaged operating as large as 100 GHz in theory. The future applications of the FBG- EIT include the fast Q-switching of fiber lasers, ultra-fast modulation, optical quantum communication, and high speed optical processing and computation.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical achievements are often made by producing new materials with suitable properties. The coherent effects demonstrate a new way to the dramatic alteration in the optical properties of the medium. A modified optical response of an atomic medium, based on the laser induced coherence of the atomic states, results in the quantum interference between the excitation pathways to control the optical response. As a consequence, the absorbance suddenly drops and the optical material promptly switches to transparency at the resonant frequency of transition. The EIT attractive properties address the nonlinear quantum effect to demonstrate notable switching applications. The importance of the EIT arises from the fact that high non-linear susceptibility can be achieved within the spectral region of induced transparency associated with steep dispersion [1–3]. Regarding the EIT quantum interference phenomenon, the electromagnetic field can control the optical response of the material. In fact, the absorbance of the multiple-level atomic ensembles disappears by using an electromagnetic control (coupling) field [1,4,5]. This effect may enhance the nonlinear properties of the photonic crystal cavities, based on the optical characteristics of the nanocrystals [6,7]. Recently, there is a great deal of interest in the EIT context of three- and multiple-level systems [8]. A number of its nonlinear optical applications include slowing light [1,9], optical switches [2,4,10], data storage [11,12] and the wavelength converter [13]. Furthermore, one tunes the output power of the probe field at a certain wavelength in the master oscillator fiber power amplifier (MOFPA) using EIT based on FBG as an example of an all-optical adjustable fiber laser. The tunability of output power of a CW DC Yb:Silica laser fiber has been investigated, using a CW control field. The output coupler reflectance varies with control field from zero to high reflectance [14].

All-optical switching, as the main component of the network in data processing of optical signals, have been widely studied during the last decades. The capabilities of optical switches and storage by making use of the EIT effect of Rubidium (Rb) and Strontium (Sr) atomic gases have been previously reported [4,11]. Similarly, an all-optical EIT switch dealing with phase and control field regulations were reported in ^{85}Rb in a four-level structure [12,13]. Furthermore, EIT in some rare-earth ion doped crystals, metal composite nanocrystals and semiconductor multiple quantum wells have been also proposed for the implementation of all-optical switching [7,15–22].

On the other hand, the acousto-optic modulators (AOM) and various saturable absorbers are widely used as Q-switching elements in fiber lasers [23–29]. However, current Q-switched (Q-SW) lasers suffer from high cavity losses degrading the beam quality. Furthermore, the application of various saturable absorbers may enhance the complexity of the system. Therefore, it is a challenge to develop an all-fiber optical alternative for purpose of fast Q-switching performance. [30,31]

Moreover, the use bulk components for intra cavity switching elements in the fiber lasers cause a beam quality degradation alongside the addition of high cavity losses, resulting in a decrease of laser performance and efficiency reduction. Thus in order to scale up laser output, the use of larger pump powers are essential [32,33]. Moreover, the bulk components require fine alignment and good mechanical stability, which complicates the design of a practical device. Q-switching of a fiber laser using an all-fiber intensity modulator employs an all fiber acousto-optic-based attenuator as cavity loss modulator [34]. Furthermore, the application of piezoelectric transducer in AOM may increase the complexity against all optical FBG-EIT switch.

Incorporation of fiber coupled FBGs with optical modulators enables us to achieve the facile, compact and robust Q-SW fiber laser cavities. Furthermore, the ability to tune the Bragg wavelength may enhance polarization selectivity and fast switching. There are few examples of all-fiber actively Q-SW lasers using FBG wavelength tuning according to the thermal or mechanical strain methods via electro-mechanical transducers. For instance, exposing the magnetic field to magnetostrictive material, the rod periodically stretches/relaxes to impose the grating wavelength shifting thereby altering the Q-factor of the cavity [35].

Furthermore, an all-fiber optical switch was demonstrated in an Yb based fiber laser. Strong stable pulses are developed to obtain Q-switching up to 200 kHz [36]. Moreover, the modeling of a typical Q-SW Yb:silica MOFPA has been carried out regarding the numerical solution of coupled time-dependent rate equations. Moreover, there are several articles available in the context of fiber laser modeling [37–39]. The master oscillator is modeled in various rise times to investigate its effect on the output peak power and pulse duration [27,40,41].

In this work, the effect of EIT in Si nanocrystals is investigated. There are several articles available to address the application of Si nanocrystals for EIT process [7,14,42–44]. An all-optical switch EIT implementation on FBG i.e., FBG-EIT structure is proposed via the significant change in transmission of probe field under the desired control field tuning. At first, the Hamiltonian/density matrix has been solved to demonstrate EIT susceptibility in both stationary-state and time-dependent modes. Consequently, the effect of four parameters of interest; i.e., control field Rabi frequency (Ω_{c}), EIT nanocrystal density (N), length of FBG medium (L) and spontaneous decay rate (${\mathrm{\gamma }_{31}}$) on the probe field transmittance are extensively inspected. According to the results, the optimal all-fiber switch condition is obtained using MATLAB software to find the maximum probe transmission with minimum rise time. In fact, the performance of optical switch at a high rate strongly depends on control field Rabi frequency to promptly change FBG reflectance to null at a certain Ω_{c}. This nature of FBG-EIT is extensively studied here. Consequently, this optical switch exhibits high potential to be employed in the fiber lasers. Despite there are a few articles available addressing the optical EIT switches [45,46], however, to the best of our knowledge, the context of EIT doped FBG has rarely been inspected so far.

## 2. Theory

#### 2.1 EIT mechanism

The theory of EIT can be simply described using quantum mechanics whose details are given in Supplement 1. Figure 1 illustrates three-energy-levels of $|1\rangle $, $|2\rangle $ and $|3\rangle $ respectively. In fact, $|1\rangle $ and $|2\rangle $ represent ground states, and $|3\rangle $ denotes the excited state. Transitions $|1\rangle \leftrightarrow |3\rangle $ and $|2\rangle \leftrightarrow |3\rangle $ are allowed, while $|1\rangle \leftrightarrow |2\rangle $ is a dipole forbidden. The three-level system is called Λ-type due to its shape. It is irradiated by a couple of laser pulses namely the probe and control fields, at angular frequencies ${\mathrm{\omega }_\textrm{p}}$ and $\; {\mathrm{\omega }_\textrm{c}}$, respectively. When the control field is turned on, a narrow transparency window promptly opens. The detuning of the probe and control frequencies, based on the corresponding transitions, are given by [47–50]:

where, ${\omega _{31}} = \frac{{{E_3} - {E_1}}}{\hbar }$ and ${\omega _{32}} = \frac{{{E_3} - {E_2}}}{\hbar }$, E_{j}and E

_{i}ascertain the energy of states $|\textrm{j}\rangle $ and $|\textrm{i}\rangle $, respectively. The ${\Delta _\textrm{p}}$ and ${\Delta _{\textrm{c}}}\; $ denote to be the angular frequency differences (detuning) between the corresponding states, accordingly.

A couple of laser fields are applied considering the transitions $|1\rangle \leftrightarrow |3\rangle $ and $|2\rangle \leftrightarrow |3\rangle $. The interaction of atom-laser is expressed by Rabi frequency. Here, Ω_{c} and Ω_{p} denote to be Rabi frequency of control and probe lasers respectively, written as below [47,48,50–52]:

Here, ${\mathrm{\mu }_{\textrm{ij}}}$, E_{p} and E_{c} ascertain the electric dipole of ij transition in EIT ions, the amplitude of probe and control electric fields, respectively.

${\mathrm{\gamma }_{\textrm{ij}}} = {\mathrm{\gamma }_{\textrm{ji}}}$ are the decay rates between the states i and j, composed of the radiative decay rates due to the spontaneous emission and the dephasing rates. The spontaneous decay rate ${\mathrm{\gamma }_{\textrm{ij}}}$ is obtained from the quantized theory [53,54] where the electromagnetic field is also treated quantum mechanically. For N similar species, the time-dependent population of excited state 3 is given by:

and, the corresponding spontaneous decay rate is written as below [53]:Note that ${\mathrm{\gamma }_{31}}$ is a cubic function of probe frequency (ω_{p}).

The complex susceptibility through the matrix elements is written as follows [47,51,55]:

Therefore, the time-dependent susceptibility according to is obtained, namely [50]:

Thus, the susceptibility holds almost all information regarding the propagation of light in the medium. The absorption coefficient, transmission and refractive index can be obtained according to the following Eqs. [51,54,56]:

_{0}are the absorbing medium length and initial refractive index when no control field is applied. Moreover, it is worth noting that α and T, in Eqs. above are determined via linear susceptibility χ to envisage steady state and transient EIT behaviors.

The model is based on three-level lambda system of atomic vapor (where n∼1), however it is developed to Si nanocrystals in SiO_{2} host having large refractive indices [7,14,42–44]. The proposed structure of EIT based modulator is analyzed here. The element is designed according to all-optical switching of FBG basis relying on the EIT phenomenon. The sequential modeling steps of EIT doped FBG are given in the Supplement 1.

FBG as a periodic microstructure with typically few millimeters-length can be photo inscribed along the core of a single mode fiber. This is done by transverse illumination of the fiber using a UV laser beam via a phase mask to generate an interference pattern along the core. This consists of spatial periodic modulation in the form of low (n_{L}) and high (n_{H}) refractive index-layers, respectively. The effective refractive index leads the reflection of light along the core of an optical fiber within a narrow spectral range for which a Bragg condition is satisfied. The central Bragg wavelength is given by [57]:

_{eff}denote to be vacuum wavelength, grating period and effective refractive index of the fiber, respectively. In fact, the effective refraction index is dependent n

_{L}/n

_{H}and the central wavelength slightly varies with n

_{eff}and grating period. Figure 2 illustrates (a) a three-energy-level diagram of Λ-type EIT nanocrystal and (b) EIT material incorporated in FBG as a number of periodic (n

_{L}, n

_{H}) layers.

Figure 2(a) depicts the energy diagram of a three level Λ-type EIT nanocrystal. The silicon nanocrystals were previously used for EIT process [7,14,42–44]. Note that n_{L}/n_{H} layers are considered along FBG length where n_{L} layers are made up of (Si nanocrystal: SiO_{2}) with refractive indices ranging 1.4415-1.4589. This varies against control field intensity (control Rabi frequency), whereas n_{H} layers are not made of EIT material to be invariant under the control field having a certain refractive index (typically 1.4589). Despite, the n_{L} is sensitive to the control field, however it would nonlinearly increase to become equal n_{H} at a certain Rabi frequency. Similarly, the absorption (imaginary part of the refractive index) approaches to negligible value at large control fields. Furthermore, the periodic n_{L}/n_{H} layers look like a grating alongside FBG length according to Eq. (12), allowing high transmission of the narrow band probe field at a desire wavelength. Note that, the beam profile of control field covers whole FBG length through a cylindrical lens to attain the adequate intensity requires for EIT process.

Note that, the Gaussian and Sech^{2} profiles can be used as well as step functions. The Sech^{2} function has exponential wings and the wings of the Gaussian one plunge steeply downwards. Naturally, in any experimental situation, the wings ultimately end in the noise background and Sech^{2} function fits better that Gaussian one. Here, the nature of EIT time response is estimated in order of 10 ps, hence the pulses with several femtosecond durations, act as a step function against the EIT time response.

## 3. Results and Discussion

#### 3.1 EIT properties

There are several methods available to implement the EIT medium to realize a typical FBG based optical switch coupled with the fiber laser. One may rely on the doping of semiconductor quantum dots/nanocrystals as multi-level atoms into a selected region of FBG. The latter is made up of typical fiber with a certain grating period. Here, Si nanocrystals are doped within periodic low index layers in silica (SiO_{2}) core as the FBG host. The transmission of probe pulse is modulated by the control (coupling) field according to Eqs. (9,10). Moreover, Table 1 tabulates the typical value of the parameters involved in the model [14]. Subsequently, the equations are numerically solved in MATLAB software according to the given parameters. Finally, FBG architecture is predesigned to simulate the transient (switching) effect.

The optical properties of species are fundamentally tied to their intrinsic energy-level structures. The linear response of an atom to the resonant light is described by the first order susceptibility including the dominant EIT features. Figure 3 depicts the real part (phase shift) and the imaginary part (absorption) of the linear susceptibility versus probe pulse detuning ${\mathrm{\Delta }_\textrm{p}}$. In the absence of the control field (Ω_{c}=0), the susceptibility χ obeys a Lorentzian function. In contrast, at the attendance of the control field for instance typically (Ω_{c}=1 THz), Autler-Townes splitting takes place and the profile splits into two parts [49]. It is supposed small Rabi frequency of the control field leads to a sharp spectral transmission window whose linewidth approaches to that of EIT at Ω_{c}=0 [4]. One recognizes that Im($\mathrm{\chi }$) undergoes a destructive interference over the region such that the coherently driven medium becomes transparent to the probe field according to Eq. (9). The results resemble to be in good agreement with the findings given in Refs. [17,19,22,45,47,48,50,54,58].

#### 3.2 Steady-state EIT transmission

At first, the steady-state property of the optical switch is studied. Four involving parameters are examined at various probe pulse detuning, demonstrating an EIT-FBG optical switch in the presence of the control field. The parameters include the Rabi frequency of control pulse (Ω_{c}), Si nanocrystals density (N), length of FBG medium (L) and the spontaneous decay rate (${\mathrm{\gamma }_{31}}$) to achieve the maximum transmission of the probe. Let assume a grating period of 375 nm; hence, in the case, the number of FBG pair layers changes from 100 to 700, then the FBG length would vary from 37.5 to 262.5 µm.

Figure 4(a) displays the transmission of the probe field in terms of detuning. Note that the variation of Rabi frequency of control pulse is carried out ranging 0–3 THz, regarding data given in Table 1. When Rabi frequency elevates, then this parameter enhances and the corresponding bandwidth enlarges too. Regarding EIT, the transmission of the probe field at line center increases, in favor of larger control field intensity. Figure 4(b) plots this parameter versus the Si nanocrystals concentration. In fact, EIT nanocrystal density notably affects the probe transmission. It is elucidated that this parameter undergoes large values at low EIT concentrations. Figure 4(c) depicts the variation of transmittance versus different FBG lengths. Not only the transmission peak reduces with EIT concentration, but also it decreases against FBG length. However, the width of spectral window in which an EIT medium appears transparent remains invariant in terms of N and L parameters. These parameters play an important role in the applications of EIT comprising between fiber length and EIT concentration. In fact, a long fiber can be used in low concentration or a short fiber requires an adequate dense concentration. Finally, Fig. 4(d) demonstrates the transmission of the probe in terms of the spontaneous decay rate. This slightly levels down at higher decay rates, as if it is less sensitive to ${\mathrm{\gamma }_{31}}$. This arises from the fact that transmittance usually takes small values regarding $|1 \mathop \leftrightarrow \limits_{} |3 $ transition. Furthermore, the transmission coefficient obtains maximum value at center line due to the constructive interference.

As a consequence, the control field intensity causes to elevate the transmission, whereas conversely the other parameters such as concentration, length and ${\mathrm{\gamma }_{31}}$ level down the probe transmittance. Therefore, the higher transmission is envisaged using large Ω_{c} at low EIT concentration in short FBG length according to the present model. Furthermore, the results of Figs. 4(a)-(d) attest to be consistent with Refs. [19,45,48,50,59,60].

#### 3.3 Time-dependent EIT transmission

When the control field is switched on (Ω_{c}≠0), the instantaneous transmission response of EIT material against Ω_{c}, N, L and ${\mathrm{\gamma }_{31}}$ would sensibly change to demonstrate a fast optical switch property through parametric characterization. The rise time (t_{rise}) as an indication of EIT temporal response demonstrates the time required to reach the maximum transmission peak value (T_{max}) according to Eqs. (8) and (10).

Figure 5 shows the transient variation of probe transmission in terms of (a) control Rabi frequencies, (b) EIT concentrations, (c) FBG lengths and (d) spontaneous decay rates, accordingly. After a few picoseconds, transmission reaches the maximum value. According to Figs. 6(a)-(d) insets, the rise time linearly decreases with Ω_{c} ranging 4 −11 ps, following a nonlinear growth versus N and L as well as a plateau in terms of ${\mathrm{\gamma }_{31}}$ that slightly drops to another plateau at larger ${\mathrm{\gamma }_{31}}$. In addition, one reminds that ${\mathrm{\gamma }_{31}}$ is a cubic function of ω_{p} according to Eq. (6), assuring reliable tunability of probe frequency takes place.

We elucidate despite t_{rise} is shortened with Ω_{c} and ${\mathrm{\gamma }_{31}}$, however, the rise time of probe increases at larger values of N and L. The peak transmission usually approaches to steady-state conditions after ∼ 10 ps. T_{max} and minimum t_{rise} are considered as the criteria for optimal parameters in transient conditions of an FBG-EIT based optical switch. According to Fig. 2, four values of interest are examined within a selected range i.e., Ω_{c} (0–3 THz), N (1×10^{21}−6×10^{21} m^{−3}), L (37.5−262.5 µm) and ${\mathrm{\gamma }_{31}}$ (0.001–0.3 THz). In fact, it is worth noting a variety of thousands of cases have been examined to find the optimal parameters among them. According to the criteria, two typical extreme cases are chosen to represent graphically. In order to inspect the effects on probe transmission as well as probe rise time, it is essential to vary a single parameter and keeping the others fixed. As a consequence, the optimal values are assessed for the reliable operation. Figure 6 indicates the behavior of T_{max} and t_{rise} in terms of those parameters considering steady-state and time-dependent modes.

Figure 6(a) demonstrates the transmission and rise time in terms of control Rabi-frequency such that T_{max} increases and t_{rise} decreases at higher Rabi frequency. This assures a short rise time takes place at maximum transmission and large Rabi frequency. Figures 6(b) and (c) illustrate the same transmission behavior as to T_{max} takes places at low concentration and short FBG length leading to desirable rise time. Finally, Fig. 6(d) displays the transmission and rise time versus decay rate which both linearly reduce at larger decay rate. Note that maximum transmission and minimum rise time meet the optimum criteria as expected.

In summary; the coupling field must be strong enough to separate the two peaks of the Autler-Townes splitting [19]. So, the elevation of Ω_{c} provides the possibility to increase the peak and width of spectral of transmission, thus the time required to reach the maximum transmission peak value (rise time) reduces. On the other hand, the absorption notably changes with concentration which can significantly affect on the EIT window. Finally, the dephasing rate of the spin arises from the photon interactions. A higher dephasing rate leads to the smaller ground state coherency and a weaker EIT effect [59,60].

#### 3.3 FBG-EIT architecture

FBG-EIT can be coupled with the fiber laser to operate both as a fast modulator and an output coupler at the same time. According to Fig. 2(b), the high refracting index layers are chosen to be made up of a non-EIT material, invariant under various control fields, whereas the low index layers contain EIT material as we discussed in previous sections. Notice that maximum transmission and minimum rise time are criteria for better performance of an all optical rapid switch [61,62]. One may carry out a desired FBG-EIT architecture according to the optimal conditions to meet the criteria above. For instance, let FBG runs at λ_{p} = 1080 nm (ω_{p}=1.746×10^{15} Hz), so maximum reflectance would naturally take place at the Bragg wavelength. Subsequently, the optimal values of ${\mathrm{\gamma }_{31}}$ would be determined at λ_{p} whose optimal value gives out to be ${\mathrm{\gamma }_{31}}$=0.002378 THz (point P). Figure 7(a) plots the probe wavelength in terms of ${\mathrm{\gamma }_{31}}$ according to Eq. (6): ${\mathrm{\omega }_\textrm{p}}\sim \mathrm{\gamma }_{31}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}} \right.}\!\lower0.7ex\hbox{$3$}}}$. Then, the FBG length is obtained based on the following relation, namely L=Λ×pair layers. Here, 200 pair layers are selected and for a grating period of Λ=375 nm, then FBG length is determined to be L=75 µm. Subsequently, the optimal dopant concentration is chosen N=1×10^{21} m^{−3}.

On the other hand, by making use of a pulsed control field, the probe transmission can be modulated regarding transient behavior. Hence, the control pulse duration adjusts the switching time. As a consequence, in the absence of a control field Ω_{c}=0 (switching OFF mode), then the refractive low/high index layers are typically set to 1.4415/1.4589, respectively. In the presence of a control field (Ω_{c}≠0), the low index of FBG is elevated in terms of the external coupling field (Ω_{c}) according to Eqs. (7) and (11). When n_{L}=n_{H}, then FBG reflectance becomes nearly null and the whole probe field is allowed to traverse through FBG-EIT switch. According to the Eq. (11) and using the optimal parameters, N=1×10^{21} m^{−3}, L=75 µm and ${\mathrm{\gamma }_{31}}$=0.002378 THz, the optimal control field at switch mode is determined to be $\mathrm{\Omega }_\textrm{c}^\textrm{s}$ = 1.1 THz such that the condition ${\textrm{n}_\textrm{L}} = {\textrm{n}_\textrm{H}}$ holds.

Moreover, according to Eqs. (3) and (13);

and Rabi frequency $\mathrm{\Omega }_{\textrm{c}}^{\textrm{s}}$ required to trigger the switch is estimated I_{c}∼ 10

^{5}W/cm

^{2}[14,51]:

On the other hand, the FBG reflectance can be given by the following relations [63,64]:

where L, $\kappa $, $\mathrm{\Delta }\beta $, $\beta $, and n_{eff}ascertain to be the total length of FBG, coupling coefficient, wave vector detuning, fiber core propagation constant, and effective refractive index, respectively. When λ= λ

_{p}, the probe wavelength is set to be equal to center Bragg wavelength, thus no wave vector detuning takes place, i.e., $\mathrm{\Delta }\beta = 0$.

Figure 7(b) depicts the variation of n_{L} and FBG reflectance as a function of Ω_{c}, indicating the optimal switching condition occurs at point Q. This attests the reflectance levels down to vanish at ${\textrm{n}_\textrm{L}} = {\textrm{n}_\textrm{H}}$. Fig. 7(b) inset shows the refractive index in terms under whole range of Ω_{c}. A portion of n_{L} is highlighted in the interval of 0.7–1.2 THz. Figure 7(c) demonstrates corresponding EIT absorbance versus Ω_{c}. Note that the absorbance becomes negligible at $\mathrm{\Omega }_c^s$. Moreover, Fig. 7(d) represents the probe wavelength λ_{p} in terms of Ω_{c} regarding Eq. (12). In fact, probe wavelength shifts of ∼ 10 nm takes place at $\mathrm{\Omega }_c^s$.

According to the model, we have proposed an FBG-EIT micro-structure including L=75 µm, N=1×10^{21} m^{−3}, ${\mathrm{\gamma }_{31}}$=0.002378 THZ and $\mathrm{\Omega }_c^s$=1.1 THz to trigger the optical switch at point Q efficiently.

Figure 8(a) reveals time-dependent transmission at the optimal values of four parameters of interest to activate switch ON mode ON at $\mathrm{\Omega }_\textrm{c}^\textrm{s}$ alongside a minimum rise time of 10 ps. The duration of the control pulse (${\mathrm{\tau }_{\textrm{cp}}}$) may vary from ns to fs according to the type of laser. Figure 8(b) displays the temporal response of EIT doped FBG switch assuming identical rise/fall times, total switch duration at ON mode is given by; $\mathrm{\tau } \cong 2{\textrm{t}_{\textrm{rise}}} + {\mathrm{\tau }_{\textrm{cp}}}$. However, in practice depending on the pulse repetition rate of the control field ($\textrm{P}.\textrm{R}.\textrm{R} = \frac{1}{\mathrm{\tau }}$), it is feasible to achieve 100 MHz at 1080 nm employing a nominal mode locked laser as the control pulse. However, the switch is capable of operating up to 100 GHz in theory.

## 4. Conclusion

All-optical switches have attracted much attention as key devices for high-speed optical data processing and optical communication systems. In fact, ultrafast all-optical switching also yields notable progress in other research areas such as nanophotonics, integrated optics, nonlinear optics, material science, and optical communications. Here, a model is developed to realize the optical switching of an FBG based on the EIT effect. An optical fast switch is designed based on the assumption where the low index layers of FBG are filled with EIT nanocrystal dopant and the high index layers do not contain EIT material. The cascade-type three-level scheme of Si is modeled by solving the Liouville equation including Hamiltonian/density matrices. At first, the maximum transmission of probe pulse is obtained against different parameters of interest such as Ω_{c}, N, L and ${\mathrm{\gamma }_{31}}$ in a steady-state condition. The transmission peak of the probe decreases at larger concentration, longer FBG length and high values of the spontaneous decay rate. Conversely, the transmittance elevates with Rabi frequency of control. Secondly, the time-dependent transmission is determined, under these parameters of interest. Despite the rise time levels down in terms of Ω_{c} and ${\mathrm{\gamma }_{31}}$, however it increases with N and L.

A periodic n_{L}/n_{H} FBG micro-structure is envisaged such that n_{H} index layers contain ordinary material, whereas n_{L} index layers are made up of EIT nanocrystals. Typical FBG-structure, characterizing n_{L}/n_{H} repetitive stacks, operates as both an output coupler and an FBG-EIT switch. Since n_{L} material obeys EIT properties, at the onset of applying a certain control Rabi frequency ($\mathrm{\Omega }_\textrm{c}^\textrm{s}$), n_{L} value notably increases to reach n_{H}, let initially assume a high reflectance FBG-EIT. Subsequently, the total reflectance of FBG-EIT undergoes a prompt changeover to maximum transmission of the probe signal at the optimal control field $\mathrm{\Omega }_\textrm{c}^\textrm{s}$. Simultaneously, the FBG reflectance becomes null just to realize a fast optical switch with a prompt time response. Despite the imaginary part of the EIT refractive index (absorbance) suddenly disappears, however the real part (dispersion) elevates to reach n_{H} activating to switch ON.

As an example, for the operational wavelength 1080 nm as the probe signal, then the optimal parameters are obtained to be $\mathrm{\Omega }_\textrm{c}^\textrm{s}$=1.1 THz (equivalent to I_{c} ∼ 10^{5} W/cm^{2}), N=1×10^{21} m^{−3}, L=75 µm and ${\mathrm{\gamma }_{31}}$=0.002378 THz. This benefits the maximum peak of probe transmittance 99.9999 and minimum rise time of 10 ps according to the model.

As a fast switch, the temporal response of FBG-EIT transmittance is modulated by the transient properties of the control field to runs up to 100 GHz theoretically.

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement 1 for supporting content.

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