Terahertz electromagnetically induced optical limiter in asymmetric coupled quantum wells

Abstract

This paper explores the optical limiting (OL) characteristics of an input probe field within a closed-loop asymmetric coupled quantum well (ACQW) system. Our research reveals that the application of Terahertz (THz) coupling laser fields can induce the OL behavior within the THz domain in the ACQW. The study demonstrates that revers saturable absorption arises through cross-Kerr nonlinearity, contributing to the emergence of OL behavior. Additionally, it is shown that the self-defocusing is induced concurrently within the system, further enhancing nonlinear refractive OL effects. Furthermore, it is revealed that the properties of induced OL can be manipulated by adjusting the characteristics of the applied fields. The discovered outcomes hold potential utility in safeguarding sensors and detectors operating within the THz band.

Introduction

Optical limiters as a light power-limiting devices hold a significant position in the field of photonics, primarily due to their capacity to safeguard human eyes and optical instruments sensitive to high-intensity laser beams. The advancement of increasingly high-power lasers for various applications such military applications necessitates the consideration of optical limiters. A typical optical limiter comprises a nonlinear material that exhibits transparency at low light intensities and transitions to an opaque state when the light intensity surpasses a specific threshold. Despite the diverse range of limiting mechanisms explored to achieve OL in different materials, OL devices based on nonlinear mechanisms can be broadly categorized into two types: energy-spreading optical limiters and energy-absorbing optical limiters. Energy absorption typically occurs through multi-photon or RSA mechanisms. Meanwhile, the energy-spreading mechanism can manifest through processes such as self-focusing, self-defocusing, induced scattering, or induced refraction¹. Various passive OL behaviors have been introduced via multi-photon², RSA³ from the first category and nonlinear refraction^4,5, self-focusing and self-defocusing^6,7 from the second type. Notably, self-defocusing is favored over the self-focusing mechanism for OL, as self-focusing has the potential to inflict damage to the material due to the intense field at the focal spot¹. Passive systems employ a nonlinear optical material that serves as an integrated sensor, processor, and modulator. This configuration presents opportunities for achieving high-speed operation, simplicity, compactness, and cost-effectiveness. Nevertheless, passive systems impose stringent demands on the nonlinear medium utilized⁸. The passive limiting behavior is contingent upon the intrinsic physical properties of the material, making it impossible to manipulate optical limiter characteristics such as the linear transmission range, limiting threshold, limiting range, and limiting wavelength. In recent investigations, there has been considerable exploration of electromagnetically induced optical limiters utilizing nonlinear absorption, aiming to enable the coherent manipulation of optical limiter characteristics⁹. Furthermore, a recent breakthrough has demonstrated that OL at microwave wavelengths can be regulated in a nitrogen-vacancy center crystal through the laser fields¹⁰ or an acoustic field¹¹.

Over the recent years, there has been a significant focus on the nonlinear optical properties of low-dimensional semiconductor quantum systems, both in theoretical investigations and practical applications. Experimental findings from the most recent studies highlight a substantial enhancement in the nonlinear optical characteristics of low-dimensional quantum systems¹². This enhancement is attributed to the pronounced quantum confinement of electrons within these systems. In this regards, asymmetric multiple quantum wells present a conducive platform for the design of terahertz (THz) devices, owing to their tunable energy levels that enable intersubband transitions within the conduction band to fall within the THz range¹³. Furthermore, the values of these transitions can be customized by altering the geometry of the well and introducing a direct current (dc) electric field along the growth direction of the layers. The THz region delineates a segment of the electromagnetic spectrum extending between 100 GHz and 10 THz, situated intermediate to microwave and infrared frequencies. Terahertz radiations, imperceptible to the unaided eye, inherently embodies safety features, diverging from X-rays and exhibit inherent safety, non-destructiveness, and non-invasiveness, rendering them preferred for applications prioritizing human safety and material preservation. The THz band occupies a distinctive and pivotal role in the electromagnetic spectrum, presenting manifold benefits and prospects across scientific, technological, and industrial spheres¹⁴. In this study, we present an electromagnetically induced optical limiter implemented in an asymmetric double quantum well (ACQW) in THz region. Optical limiting in the THz band range refers to the control or suppression of high-intensity THz radiation to protect sensitive optical sensors, detectors, and components from damage caused by excessive power levels. This capability finds applications in various fields where THz radiation is utilized, such as spectroscopy, imaging, communication, and security screening. Our findings reveal that both nonlinear absorption, attributed to RSA, and refractive effects, facilitated by electromagnetically induced self-defocusing, contribute to inducing optical limiting behavior in a closed-loop transition within an ACQW. It is demonstrated that the linear transmission, OL threshold, OL and damage regions can be controlled by characteristics of the applied fields.

Model and equations

To commence the investigation, we consider an array comprising 50 modulation-doped ACQWs, grown on a semi-insulating GaAs substrate. Figure 1a illustrates a single period of the coupled well along with the corresponding excitation scheme. Each period of the ACQW is composed of two thick GaAs wells with thicknesses of 70 and 60 Å, respectively, separated by a 20-Å thin barrier of $A{l}_{0.33}G{a}_{0.77}$As. The separation between each coupled quantum well period is maintained by a barrier of $A{l}_{0.33}G{a}_{0.77}$As with a thickness of 950 Å, exhibiting structural similarities to the configurations which has been used to study Phase-sensitive modulation instability¹⁵, propagation of soliton¹⁶, Stabilization of modulation instability¹⁷, spatiotemporal-vortex four-wave mixing¹⁸ and realizing polarization quantum phase gate¹⁹. Another alternative is asymmetric-coupled Ge/SiGe quantum wells which has been used as electro-absorption modulator²⁰, intensity and phase modulator²¹ and second harmonic generator at THz region²². The two wells in the ACQWs are strongly coupled due to their close proximity, leading to a significant overlap of electronic wave functions within the coupled system. The characteristics of these wave functions and their corresponding energy levels are contingent upon the width, depth, and separation between the wells. The electronic energy levels and associated wave functions can be determined through the self-consistent solution of Schrödinger and Poisson’s equations within the envelope function formalism²³.

Figure 1.

(a) Schematic representation of a single-period asymmetric double quantum well structure, comprising two GaAs wells of differing thicknesses, specifically 70 Å and 60 Å respectively, with a 20 Å thin $A{l}_{0.33}G{a}_{0.77}As$ barrier separating them. (b) Diagram illustrating the energy level configuration for the quantum well system under investigation, consisting of three levels.

In this study, the system under investigation is a three-level atomic system that interacts with three distinct laser fields in THz region to generate a closed-loop configuration. The schematic representation of this system can be observed in Fig 1b. An input probe field with Rabi frequency ${\mathrm{\Omega }}_p=\frac{E_p{\mu }_{12}}{ħ}$ is applied to the $\mid 1>\to \mid 2>$ transition. Two stronger driving and coupling fields with Rabi frequencies ${\mathrm{\Omega }}_d=\frac{E_d{\mu }_{13}}{ħ}$ and ${\mathrm{\Omega }}_c=\frac{E_c{\mu }_{23}}{ħ}$ are applied to the transitions $\mid 1>\to \mid 3>$ and $\mid 2>\to \mid 3>$, respectively. Here, $E_k,{\omega }_k(k=p,c,d)$ refer to the amplitude and frequency of the electric fields. The parameter ${\mu }_{\mathit{ij}}$ stands for the atomic induced dipole moment of the corresponding transition. It is assumed that the input probe and coupling fields have a plane wavefront while a Gaussian profile is considered for the driving field.

Now, utilizing the dipole and the rotating wave approximations, the Hamiltonian of the system can be expressed as follows:

$$\begin{array}{c}\hfill \widehat{H}=-ħ{\mathrm{\Omega }}_p{e}^{-i{\mathrm{\Delta }}_{p}t}|2⟩⟨1|-ħ{\mathrm{\Omega }}_c{e}^{-i{\mathrm{\Delta }}_{c}t}|3⟩⟨2|-ħ{\mathrm{\Omega }}_d{e}^{-i{\mathrm{\Delta }}_{d}t}|3⟩⟨1|+H.C.\end{array}$$ 1

Here ${\mathrm{\Delta }}_p={\omega }_p-{\omega }_{21}$, ${\mathrm{\Delta }}_c={\omega }_c-{\omega }_{32}$, and ${\mathrm{\Delta }}_d={\omega }_d-{\omega }_{31}$ are detuning of the lase field frequency with the central frequancy of the corresponding transitions. The equations of motion for the density matrix of the given system can be written as follows:²⁴

$$\begin{array}{cc}\hfill {\dot{\rho }}_{11}=& i{\mathrm{\Omega }}_p\left({\rho }_{21},-,{\rho }_{12}\right)+i{\mathrm{\Omega }}_d\left({\rho }_{31},-,{\rho }_{13}\right)+{\gamma }_2{\rho }_{22},\hfill \\ \hfill {\dot{\rho }}_{22}=& i{\mathrm{\Omega }}_p\left({\rho }_{12},-,{\rho }_{21}\right)+i{\mathrm{\Omega }}_c\left({\rho }_{32},e^{-i\varphi },-,{\rho }_{23},e^{i\varphi }\right)-{\gamma }_2{\rho }_{22}+{\gamma }_3{\rho }_{33},\hfill \\ \hfill {\dot{\rho }}_{21}=& -i{\mathrm{\Omega }}_p\left({\rho }_{22},-,{\rho }_{11}\right)-i{\mathrm{\Omega }}_d{\rho }_{23}+i{\mathrm{\Omega }}_c{\rho }_{31}{e}^{-i\varphi }+i{\mathrm{\Delta }}_p{\rho }_{21}-\frac{{\gamma }_{21}}{2}{\rho }_{12},\hfill \\ \hfill {\dot{\rho }}_{31}=& -i{\mathrm{\Omega }}_d\left({\rho }_{33},-,{\rho }_{11}\right)-i{\mathrm{\Omega }}_p{\rho }_{32}+i{\mathrm{\Omega }}_c{\rho }_{21}{e}^{i\varphi }+i{\mathrm{\Delta }}_d{\rho }_{31}-\frac{{\gamma }_{31}}{2}{\rho }_{13},\hfill \\ \hfill {\dot{\rho }}_{32}=& -i{\mathrm{\Omega }}_c\left({\rho }_{33},-,{\rho }_{22}\right)e^{i\varphi }-i{\mathrm{\Omega }}_p{\rho }_{31}+i{\mathrm{\Omega }}_d{\rho }_{12}+i{\mathrm{\Delta }}_c{\rho }_{32}-\frac{({\gamma }_{32}+{\gamma }_{21})}{2}{\rho }_{32},\hfill \\ \hfill {\dot{\rho }}_{33}=& -\left({\dot{\rho }}_{11},+,{\dot{\rho }}_{22}\right),\hfill \end{array}$$ 2

where $\varphi ={\varphi }_p+{\varphi }_c-{\varphi }_d$ shows the relative phase of applied fields. It is noteworthy that the optical characteristics of the closed-loop quantum system depend solely on the relative phase of the applied field under conditions of multi-photon resonance²⁵. The parameter ${\gamma }_{\mathit{ij}}\left(=,{\gamma }_i,+,{\gamma }_{\mathit{ij}}^{\mathit{dph}}\right)$ denotes overall population decay rate of the transition $\mid i>\to \mid j>$, respectively. This decay rate encompasses two components: the decay rate resulting from low-temperature longitudinal optical phonon emission and a second term arising from electron-electron scattering and electron-phonon scattering at the interfaces. It is assumed that the multi-photon resonance conditions, i.e. ${\mathrm{\Delta }}_p+{\mathrm{\Delta }}_c={\mathrm{\Delta }}_d$ and ${\overrightarrow{k}}_p+{\overrightarrow{k}}_c={\overrightarrow{k}}_d$ are fulfilled. Consequently, the explicit time-dependent exponential terms have been dropped in Eq. (2)²⁶.

The relationship governing the Gaussian driving laser beam is expressed as follows:

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_d={\mathrm{\Omega }}_{d0}exp\left(\frac{-r^2}{w_0^2}\right)\end{array}$$ 3

where the beam waist is represented by $w_0$.

In the forthcoming discussion, we will delve into the mathematical equations and relationships that serve to articulate the principles of OL theory. Firstly, let’s consider the vector polarization relation in the atomic medium, provided as follows:

$$\begin{array}{c}\hfill \overrightarrow{P}(z,t)={\chi }_p\overrightarrow{E_p}{e}^{-i\left({\omega }_p,t,-,k_p,z\right)}+c.c\end{array}$$ 4

Here, ${\chi }_p$ is referred to as the electric susceptibility, signifying the medium’s response to the probe field. The corresponding relationship is expressed by

$$\begin{array}{c}\hfill {\chi }_p=\frac{N{\mu }_{21}^2{\rho }_{21}}{ħ{\mathrm{\Omega }}_p}.\end{array}$$ 5

where, N and ${\mu }_{21}$ stand for density of atoms and electric dipole moment of $\mid 1>-\mid 2>$ transition.

To attain the amplitude of the output probe’s field, we initially define the wave equation in the following form

$$\begin{array}{c}\hfill {\mathrm{\nabla }}^2{\overrightarrow{E}}_p-{\mu }_0{\epsilon }_0\frac{{\partial }^2{\overrightarrow{E}}_p}{\partial t^2}-{\mu }_0\frac{{\partial }^2\overrightarrow{P}}{\partial t^2}=0,\end{array}$$ 6

In this equation, we incorporate the linearly probe field, ${\overrightarrow{E}}_p(z,t)=\overrightarrow{{\epsilon }_p}(z)e^{-i\left({\omega }_p,t,-,k_p,z\right)}$, and Eq. (4). Subsequently, by employing the method of gradual change approximation, we solve the wave equation and simplify it to a more manageable form²⁷.

$$\begin{array}{c}\hfill \frac{\partial {\epsilon }_p}{\partial z}=2\pi i{k}_p{\epsilon }_p{\chi }_p\end{array}$$ 7

Now, by solving the differential equation at the point $z=l$, we attain the domain of the output probe’s field

$$\begin{array}{c}\hfill {\epsilon }_p(z=l)={\epsilon }_p(z=0)e^{i2\pi l{k}_p{\chi }_p}\end{array}$$ 8

with the consideration of the resonant absorption as $\left(\alpha ,l,=,\frac{4\pi n{\mu }_{21}^2{k}_{p}l}{ħ\gamma }\right)$ and the normalized susceptibility of the medium $\left(S_p,=,\frac{{\rho }_{21}\gamma }{{\mathrm{\Omega }}_p}\right)$, Eq. (8) can be written as

$$\begin{array}{c}\hfill {\epsilon }_p(z=l)={\epsilon }_p(z=0)e^{i\frac{\alpha l}{2}{S}_p}\end{array}$$ 9

where l is the length of the sample. Finally the transmission of the input probe field is given by

$$\begin{array}{c}\hfill T=\frac{|{\epsilon }_p{(z=l)|}^2}{|{\epsilon }_p{(0)|}^2}=e^{-\alpha lIm[S_p]}.\end{array}$$ 10

Hence, Utilizing the subsequent analytical relation, the imaginary part of the probe field coherence can be articulated in the following manner:

$$\begin{array}{c}\hfill Im[{\rho }_{21}]=\frac{{\mathrm{\Omega }}_p\left[\left({\mathrm{\Omega }}_p^4,+,2,{\mathrm{\Omega }}_c^4,+,3,{\mathrm{\Omega }}_c^2,{\mathrm{\Omega }}_p^2\right),-,\left({\mathrm{\Omega }}_d^4,+,5,{\mathrm{\Omega }}_c^2,{\mathrm{\Omega }}_d^2\right)\right]}{2\left(2,{\mathrm{\Omega }}_c^6,+,3,{\mathrm{\Omega }}_d^6,+,2,{\mathrm{\Omega }}_p^6,+,4,{\mathrm{\Omega }}_p^4,{\mathrm{\Omega }}_d^2,+,5,{\mathrm{\Omega }}_p^2,{\mathrm{\Omega }}_d^4,+,{\mathrm{\Omega }}_c^4,\left(7,{\mathrm{\Omega }}_p^2,+,3,{\mathrm{\Omega }}_d^2\right),+,{\mathrm{\Omega }}_c^2,\left(7,{\mathrm{\Omega }}_p^4,-,40,{\mathrm{\Omega }}_p^2,{\mathrm{\Omega }}_d^2,+,7,{\mathrm{\Omega }}_d^4\right)\right)}.\end{array}$$ 11

Under the assumption that ${\gamma }_{21}$ = ${\gamma }_{32}$ = $\gamma$, all parameters are normalized to the value of $\gamma$. By analyzing Eq. (11), it becomes clear that the nonlinear cross-Kerr effect plays a critical role in shaping the optical properties of the system^25,28. Each parameter, as defined in the analytical relationship, has a specific role in the realm of multi-photon transision. This underscores the importance of comprehending the unique contributions of these parameters to achieve a comprehensive understanding of the overall optical behavior of the system. The terms ${\mathrm{\Omega }}_p^5$, ${\mathrm{\Omega }}_p{\mathrm{\Omega }}_d^4$, ${\mathrm{\Omega }}_p{\mathrm{\Omega }}_c^4$, ${\mathrm{\Omega }}_p{\mathrm{\Omega }}_c^2{\mathrm{\Omega }}_d^2$ and ${\mathrm{\Omega }}_p^3{\mathrm{\Omega }}_c^2$ represent five-photon transitions through $|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩\stackrel{\mathrm{\Omega }_p^{\ast }}{\to }|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩\stackrel{\mathrm{\Omega }_p^{\ast }}{\to }|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩$, $|1⟩\stackrel{{\mathrm{\Omega }}_d}{\to }|3⟩\stackrel{\mathrm{\Omega }_d^{\ast }}{\to }|1⟩\stackrel{{\mathrm{\Omega }}_d}{\to }|3⟩\stackrel{\mathrm{\Omega }_d^{\ast }}{\to }|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩$, $|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩\stackrel{{\mathrm{\Omega }}_c}{\to }|3⟩\stackrel{\mathrm{\Omega }_c^{\ast }}{\to }|2⟩\stackrel{{\mathrm{\Omega }}_c}{\to }|3⟩\stackrel{\mathrm{\Omega }_c^{\ast }}{\to }|2⟩$, $|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩\stackrel{{\mathrm{\Omega }}_c}{\to }|3⟩\stackrel{\mathrm{\Omega }_c^{\ast }}{\to }|2⟩\stackrel{\mathrm{\Omega }_d^{\ast }}{\to }|1⟩\stackrel{{\mathrm{\Omega }}_d}{\to }|2⟩$ and $|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩\stackrel{\mathrm{\Omega }_p^{\ast }}{\to }|1⟩\stackrel{{\mathrm{\Omega }}_p}{\to }|2⟩\stackrel{{\mathrm{\Omega }}_c}{\to }|3⟩\stackrel{\mathrm{\Omega }_c^{\ast }}{\to }|2⟩$, respectively. Other multi-photon transitions with minor rule have been ignored in calculation of Eq. (11). The first term is generated via Kerr nonlinearity while other terms are the contributions of the cross Kerr nonlinearity in absorption behavior of the system. In the regime of weak probe fields, an electromagnetically induced transparency (EIT) window emerges when the following condition is satisfied:

$$\begin{array}{c}\hfill 2{\mathrm{\Omega }}_c^4+{\mathrm{\Omega }}_p^4-{\mathrm{\Omega }}_d^4={\mathrm{\Omega }}_c^2\left(5,{\mathrm{\Omega }}_d^2,-,3,{\mathrm{\Omega }}_p^2\right).\end{array}$$ 12

The terms featuring positive (negative) signs in the numerator of Eq. (11) are accountable for inducing absorption (gain) of the input probe field through the depletion of population in state $\mid 2⟩$ ($\mid 1⟩$). When the depletion of state $\mid 1⟩$ predominates, the system exhibits gain.

It is worth noting that, as per Eq. (11), for low values of the input Rabi frequency, absorption, transparency, and potentially gain can occur depending on the Rabi frequency of the driving and coupling fields. Within the regime of weak input probe fields, the presence of two coupling and driving fields with identical detunings establishes a $\mathrm{\Lambda }$-type coherent population trapping (CPT), resulting in the population being trapped within levels $\mid 1⟩$ and $\mid 2⟩$²⁹. When the Rabi frequency of the driving field is small, the majority of the population is trapped in level $\mid 1⟩$ due to CPT, leading to absorption of the input probe field. As the driving Rabi frequency increases, the populations of the two states can equalize, creating a transparent state for the input probe field and inducing Electromagnetically Induced Transparency (EIT). With higher values of the driving Rabi frequency, population transfer to state $\mid 2⟩$ via the excited state $\mid 3⟩$ occurs, resulting in amplification of the input probe field.

Results and discussions

In this section, we present the findings and research related to OL clearly, utilizing mathematical expressions and numerical calculations, with a special emphasis on density matrix calculations. The results are transparently displayed and analyzed separately, focusing on both nonlinear absorption and nonlinear refraction. This approach aims to provide a straightforward understanding of the research outcomes in the realm of OL, emphasizing the distinct contributions of nonlinear absorption and nonlinear refraction through rigorous mathematical analyses.

Nonlinear absorption

Optical limiting is a specific application of nonlinear absorption where the goal is to limit the amount of transmitted light when the intensity becomes too high, protecting sensitive optical components. The main type of nonlinear absorption processes involved in OL is the RSA. Our analytical result shows that the multi-photon transitions play a major rule in establishing the RSA in this system. The initial positive terms in the numerator of Eq. (11) correspond to an absorption phenomenon, whereas the subsequent negative terms contribute to system amplification. Notably, the impact of these terms is contingent upon the strength of the applied fields. Consequently, variations in the absorption characteristics of the system give rise to the potential for either SA or RSA under different parameter settings. To investigate this, we employ a Gaussian distribution to model the intensity of the driving field, analyzing its effect on the absorption behavior of the ACQW across various points within the beam’s cross-section. Figure 2 illustrates the absorption behavior of the input probe field as a function of the probe field intensity. All parameters are normalized by ${\gamma }_{31}={\gamma }_{32}=\gamma =3.03\times {10}^{12}$ Hz^30,31. The parameters utilized are as follows: ${\gamma }_{21}=0.05\gamma$, $\varphi =0$, ${\mathrm{\Delta }}_p={\mathrm{\Delta }}_c={\mathrm{\Delta }}_d=0$, ${\mathrm{\Omega }}_c=10\gamma$, $\alpha l=500\gamma$, $w_0=0.5\times {10}^{-3}m$, ${\mathrm{\Omega }}_{d0}=3\gamma$ (a), ${\mathrm{\Omega }}_{d0}=5\gamma$ (b), ${\mathrm{\Omega }}_{d0}=7\gamma$ (c), and ${\mathrm{\Omega }}_{d0}=8\gamma$ (d). Analysis of Fig. 2 reveals distinct absorption behaviors across various points within the beam’s cross-section, indicating spatially-dependent absorption within the system. Furthermore, at the center of the beam, an increase in the intensity of the driving field causes a transition from SA to RSA. Conversely, at points farther from the beam axis with lower intensities, RSA cannot be induced within the ACQW. This behavior stems from the fact that heightened intensity of the driving field amplifies the contribution of the negative term in numerator of Eq. (11), particularly for small input probe field values, thereby facilitating RSA generation, even in the presence of gain. As the intensity of the driving field escalates, the central region of the input probe field undergoes amplification, surpassing the EIT condition delineated in Eq. (12). This phenomenon can be harnessed for amplifying weak input probe fields, thereby extending the range of radars or seekers.

Figure 2.

Absorption behavior of the probe field versus the corresponding Rabi frequency at different spatial distances from the center of beam. Each graph corresponds to a distinct Gaussian field, denoted as follows: (a) ${\mathrm{\Omega }}_{d0}=3\gamma$, (b) ${\mathrm{\Omega }}_{d0}=5\gamma$, (c) ${\mathrm{\Omega }}_{d0}=7\gamma$, and (d) ${\mathrm{\Omega }}_{d0}=8\gamma$. The utilized parameters include ${\gamma }_{21}=0.05\gamma$, ${\mathrm{\Delta }}_p$ = ${\mathrm{\Delta }}_c$ = ${\mathrm{\Delta }}_d$ = 0, ${\mathrm{\Omega }}_c=10\gamma$, $\alpha l=500\gamma$, $w_0=0.5\times {10}^{-3}m$ and $\varphi =0$.

It is expected that the various values for the intensity of output probe field can be obtained in different points of the beam cross section passing through the ACQW. Figure 3 depicts the output intensity via input probe field intensity for ${\mathrm{\Omega }}_{d0}=3\gamma$ (a), ${\mathrm{\Omega }}_{d0}=5\gamma$ (b), ${\mathrm{\Omega }}_{d0}=7\gamma$ (c), and ${\mathrm{\Omega }}_{d0}=8\gamma$ (d). Other used parameters are same as in Fig. 2. Three linear, OL and damage regoins are indicated in plots. In OL region, the output decreases by increasing the intensity of the input probe field which can protect the sensors against the intense THz sources. The OL region does not appear for small values of the driving field intensities, however it can be appeared by increasing the intensity of driving field. The transparent line is shown by thick dashed line.It is evident that under the influence of a strong driving field, the output surpasses the Rabi frequency of the input probe field due to induced gain within the system. This gain proves beneficial for amplifying weak input probe fields, consequently expanding the operational range of radar or seekers.

Figure 3.

The output intensity via input probe field intensity for ${\mathrm{\Omega }}_{d0}=3\gamma$ (a), ${\mathrm{\Omega }}_{d0}=5\gamma$ (b), ${\mathrm{\Omega }}_{d0}=7\gamma$ (c), and ${\mathrm{\Omega }}_{d0}=8\gamma$ (d). Other used parameters are same as in Fig. 2. Three linear, optical limiting and damage regions are separated by vertical dot lines. The transparent line (TL) is shown by thick dot-dashed line.

As depicted in Figures 2 and 3, when the probe field passes through the ACQW system which interacts with a Gaussian driving and plane coupling fields, a collection of SA and RSA phenomena are established inside the medium. However, it seems that the total average transmission over the beam cross section can play a major rule in determining the OL behaviour of the system. Thus, to construct an OL system utilizing our findings, it is imperative to incorporate an aperture of appropriate radius in the external plane of the ACQW system or at the entrance window of the detectors. In this context, it is necessary to define the averaged transmission as:

$$\begin{array}{c}\hfill T(z)=\frac{\underset{0}{\overset{a}{\int }}2\pi Trdr}{\pi a^2},\end{array}$$ 13

where the upper limit of the integral, denoted as a, represents the radius of the aperture.

Our numerical findings underscore the significant role played by the magnitude of a in shaping the OL behavior. In Fig. 4, we present the averaged transmission as a function of the input probe field Rabi frequency for both closed (a) and open (b) aperture conditions. The parameters employed are as follows, ${\mathrm{\Omega }}_{d0}=7\gamma$ (solid), $6\gamma$ (dashed), $5\gamma$ (dash-dotted) and $3\gamma$ (dotted), with $a=0.2w_0$ (a) and $5w_0$ (b). Other parameters remain consistent with those in Fig. 2. In Fig. 4a, the aperture radius is assumed to be comparable to the waist of the driving Gaussian beam. Notably, we observe that OL behavior can be established by augmenting the intensity of the driving field in the closed aperture configuration. Conversely, when considering larger values for the aperture radius, referred to as an open aperture, the optical limiting behavior diminishes, as illustrated in Fig. 4b, even under intense driving fields. In Fig. 4c, the average transmission is depicted as a function of the input probe field Rabi frequency for ${\mathrm{\Omega }}_{d0}=7\gamma$, considering various aperture radii: $a=0.2w_0$ (solid), $0.5w_0$ (dashed), $w_0$ (dash-dotted), and $5w_0$ (dotted). It is evident that reducing the aperture size contributes constructively to the generation of optical limiting (OL) behavior. However, passing through the aperture leads to a decrease in the total energy of the output. Consequently, the minimum permissible aperture radius is dictated by the dynamic range of the detectors, defined as the disparity between their noise floor and saturation intensity.

Figure 4.

The averaged transmission of the probe field versus the input probe field across different amplitude values of the driving Gaussian field, specifically in the (a) closed, (b) open and (c) for ${\mathrm{\Omega }}_{d0}=7\gamma$ considering different values of the the aperture radius, i.e. $a=0.2w_0$ (solid), $0.5w_0$ (dashed), $w_0$ (dash-dotted) and $5w_0$ (dotted). The specific parameters required for these analyses are elaborated in Fig. 2.

A comprehensive examination of Fig. 4 elucidates the pivotal role of the driving field intensity in electromagnetically induced closed aperture OL within the ACQW. To precisely delineate the OL region, we present a density plot illustrating the slope of averaged transmission versus the input probe field Rabi frequency in Fig. 5, across different intensity ranges of the driving field, namely ${\mathrm{\Omega }}_{d0}=0-3\gamma$ (a), $3\gamma -6\gamma$ (b), and $6\gamma -10\gamma$ (c). The parameters utilized are consistent with those in Fig. 4. An analysis of Fig. 5 reveals that for lower values of driving field intensities, OL behavior does not manifest, but as the intensity increases, the system exhibits OL behavior.

Figure 5.

The density plot for the slope of averaged transmission versus the input probe field Rabi frequency for different intensity ranges of the driving field, i.e. ${\mathrm{\Omega }}_{d0}=0-3\gamma$ (a), $3\gamma -6\gamma$ (b) and $6\gamma -10\gamma$ (c) for ${\mathrm{\Omega }}_c=10\gamma$. The specific parameters required for these analyses are elaborated in Fig. 4.

The underlying physics of this phenomenon lies in the spatially-dependent absorption induced by the applied fields, which influences the intensity of the output probe field. We consider a scenario wherein absorption in the center of the input probe field increases with the intensity of the input probe field. Consequently, transmission decreases as the input field intensity rises in the central region of the beam. To illustrate this, we plot the output intensity profile for ${\mathrm{\Omega }}_{d0}=3\gamma$ (a) and $7\gamma$ (b) for various values of the input field intensity in Fig. 6. The parameters used are consistent with those in Fig. 4. Beneath each intensity profile, the central cross-section is presented to facilitate comparison across different intensities of the input probe field. For low values of driving field intensity, the absorption in the center of the beam decreases with increasing input probe field intensity, resulting in an increase in the output intensity within the central region of the beam. Consequently, the averaged transmission does not exhibit OL behavior,leads to establish of the linear region. However, as the intensity of the driving field increases, the behavior transitions towards a RSA-like behavior in the central region of the beam, notably occurring within the optical limiting region. Finally, the damage region is started from ${\mathrm{\Omega }}_p=9\gamma$ which is in good agreement with the results presented in Figs. 3, 4 and 5. Subsequently, OL is induced within the system in THz band by selecting appropriate values for the radius of the aperture with respect to the waist of the driving field.

Figure 6.

The intensity profile of the output probe field for (a) ${\mathrm{\Omega }}_{d0}=3\gamma$, and (b) ${\mathrm{\Omega }}_{d0}=7\gamma$ for differnt values of the Rabi frequency of the input probe field. Below each intensity profile the central cross section has been plotted for comparing the results for different intensities of input probe field. The relevant parameters are specified in Fig. 2.

Nonlinear refraction

In the previous section, we effectively demonstrated the formation of the OL phenomenon, primarily by considering the spatially-dependent absorption of the input probe field while incorporating an aperture in the output. It is noteworthy that the Kerr nonlinearity can induce nonlinear refraction, resulting in the focusing or defocusing of the input probe field as it passes through the ACQW. In materials exhibiting self-defocusing behavior, the nonlinear refractive index, often induced by mechanisms like the Kerr effect, leads to a decrease in the refractive index with increasing light intensity. As a result, the medium tends to spread out the light beam, causing it to diverge or defocus. When a high-intensity laser beam encounters such a medium, the defocusing action effectively redistributes the light energy over a larger area, preventing the light from reaching excessively high intensities. This prevents damage to sensitive optical components or detectors downstream of the medium, thus serving as a protective mechanism. Consequently, we aim to investigate the induced nonlinear refractive index to establish OL behavior in the THz band, specifically through defocusing induced by applied fields. Unlike in linear optics, where the refractive index remains constant ($n_0$), nonlinear refraction causes a shift in the refractive index ($n_1$) that is proportional to the square or higher powers of the light’s intensity. The nonlinear refractive index can be expressed as:

$$\begin{array}{c}\hfill n=n_0+n_{1}I\left(I_c,,,I_p,,,I_d\right).\end{array}$$ 14

Here, $I(I_c,I_p,I_d)$ denotes a function of the intensity of all applied fields. This indicates that the nonlinear change in the refractive index occurs through both Kerr and cross-Kerr nonlinearities. Negative values of $n_1$ signify that the input probe field may undergo defocusing as it traverses the medium, leading to alterations in energy distribution. These changes can be leveraged to induce OL behavior via the nonlinear refractive index. Based on the analytical results obtained from density matrix equations, the expression for the nonlinear refractive index can be further elaborated as follows:

$$\begin{array}{c}\hfill n=1+\frac{1}{2}Re[{\chi }_p]=1-\frac{N{\mu }_{21}^2}{2ħ{\mathrm{\Omega }}_p}\frac{{\mathrm{\Omega }}_c{\mathrm{\Omega }}_d\left({\mathrm{\Omega }}_c^2,-,{\mathrm{\Omega }}_p^2,)(,2,{\mathrm{\Omega }}_c^2,+,{\mathrm{\Omega }}_d^2,-,3,{\mathrm{\Omega }}_p^2\right)}{D},\end{array}$$ 15

where $D=\left(2,{\mathrm{\Omega }}_c^6,+,3,{\mathrm{\Omega }}_d^6,+,2,{\mathrm{\Omega }}_p^6,+,4,{\mathrm{\Omega }}_p^4,{\mathrm{\Omega }}_d^2,+,5,{\mathrm{\Omega }}_p^2,{\mathrm{\Omega }}_d^4,+,{\mathrm{\Omega }}_c^4,\left(7,{\mathrm{\Omega }}_p^2,+,3,{\mathrm{\Omega }}_d^2\right),+,{\mathrm{\Omega }}_c^2,\left(7,{\mathrm{\Omega }}_d^4,+,7,{\mathrm{\Omega }}_p^4,-,40,{\mathrm{\Omega }}_p^2,{\mathrm{\Omega }}_d^2\right)\right)$.

An examination of Eq. (15) reveals that the nonlinear refractive response is contingent upon the intensity of the applied fields. Consequently, distinct focusing or defocusing phenomena may manifest under specific parameter configurations. Taking into account the assumption that the driving field conforms to a Gaussian beam profile, one anticipates a spatially-varying nonlinear refractive index. We proceed to investigate the spatially dependent nonlinear refractive index generated by cross Kerr nonlinearity within the ACQW. Figure 7 depicts the two-dimensional profile of the nonlinear refractive index for ${\mathrm{\Omega }}_{d0}=3\gamma$ (a) and $7\gamma$ (b), with parameters identical to those in Fig. 6. For ease of comparison across different intensities of the input probe field, the central cross-section is provided beneath each profile. Examination of Fig. 7a reveals that, for weak intensities of the driving field, the refractive index at the center of the input probe field is smaller than that at the wings of the beam. Consequently, the medium defocuses the input probe field as it propagates through, even for weak probe field intensities, rendering it unsuitable for OL. Ideally, a weak input probe field should focus to enhance aperture transmission. However, this scenario changes significantly as the intensity of the driving field increases. In Fig. 7b, we observe that focusing happens for low input probe field intensities. However, as the intensity of the input probe field rises, there’s a notable shift in the refractive index profile towards defocusing, particularly noticeable with intense input probe fields. Moreover, the degree of defocusing amplifies with increasing input probe field intensity. This suggests that the defocusing effect induced by the nonlinear refractive index plays a significant role in establishing optical lattice (OL) behavior within this system.

Figure 7.

The refractive index response of the system to the probe field has been plotted for (a) ${\mathrm{\Omega }}_{d0}=3\gamma$ and (b) ${\mathrm{\Omega }}_{d0}=7\gamma$ . The graphs in the first and second rows correspond, respectively, to the two-dimensional and one-dimensional modes.

conclusion

The nonlinear absorption and refraction properties of a closed-loop asymmetric quantum well system in the THz band have been investigated. It was shown that through the application of THz laser fields to the system, OL behavior was induced via Kerr and cross-Kerr nonlinearity. It was observed that both reverse saturable absorption, arising from spatially-dependent nonlinear absorption, and electromagnetically induced defocusing via nonlinear refraction, contributed concurrently to the generation of OL behavior in the THz band. Furthermore, it was demonstrated that the characteristics of the induced optical limiter could be controlled by adjusting either the intensity of applied fields.

Author contributions

M. Mahmoudi conceived the idea of the research and directed the project. Both authors developed the research conceptions, analysed, and discussed the obtained results. N. Parkan performed the calculations and wrote the paper with major input from M. Mahmoudi.

Data availibility

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.