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Two-dimensional photothermally induced grating

  • Tauqeer Khan , 1 ,
  • Muneeb ur Rahman 2 ,
  • Nusrat Riaz 3 ,
  • Ahsan Bin Asad 4 ,
  • Muhammad Imtiaz Khan 3 ,
  • Ziauddin , 4, 5,
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  • 1School of Physics and Mechanics, Wuhan University of Technology, Wuhan 430070, China
  • 2CECOS Unversity of IT and emerging Sciences, Peshawar Khyber Pakhtunkhwa, Pakistan
  • 3Abbottabad University of Science and Technology, Khyber Pakhtunkhwa, Pakistan
  • 4Quantum Optics Lab. Department of Physics, COMSATS University Islamabad, Pakistan
  • 5Research Center for Quantum Physics, Huzhou University, Huzhou 313000, China

Author to whom any correspondence should be addressed.

Received date: 2024-09-27

  Revised date: 2024-12-27

  Accepted date: 2024-12-30

  Online published: 2025-02-28

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© 2025 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing. All rights, including for text and data mining, AI training, and similar technologies, are reserved.
This article is available under the terms of the IOP-Standard License.

Cite this article

Tauqeer Khan , Muneeb ur Rahman , Nusrat Riaz , Ahsan Bin Asad , Muhammad Imtiaz Khan , Ziauddin . Two-dimensional photothermally induced grating[J]. Communications in Theoretical Physics, 2025 , 77(6) : 065101 . DOI: 10.1088/1572-9494/ada42a

1. Introduction

The phenomenon of electromagnetically induced grating (EIG) can be achieved when the traveling wave used in electromagnetically induced transparency (EIT) [1, 2] is replaced by a standing wave. EIG has been extensively studied both theoretically and experimentally [37]. The phenomenon of EIG has become a focal point in optical research due to its wide range of applications, including optical switching [6, 8, 9], bistability [10], light storage [11], imaging [12], and dipole solitons [13]. In EIG, the incident probe light undergoes diffraction into various orders due to a diffraction grating formed by the spatial modulation of the amplitude and phase of the transmission. This phenomenon was initially investigated theoretically [3, 14] and subsequently observed experimentally in a cold atomic medium [15, 16]. Since then, EIG has been extensively studied in various systems [17, 18], including quantum dots, quantum wells [19, 20], and artificial molecules [21]. Diffraction gratings have been widely utilized in advanced optical systems due to their ability to manipulate light propagation and energy distribution effectively. These systems benefit significantly from efficient energy management and optimal control, akin to hybrid microgrids for energy systems, as demonstrated in [22]. Such precise control mechanisms are further explored in load management strategies for photovoltaic-based electric vehicle charging stations [23] and the scheduling of solar-powered energy storage systems for domestic loads [24]. These principles of control and energy distribution parallel the dynamics of coupling strength adjustment in photothermally induced gratin (PTIG), where tunable diffraction enables enhanced energy transfer to higher diffraction orders, providing a robust framework for high-precision applications in sensing and spectroscopy.
The two-dimensional (2D) grating offers several advantages over one-dimensional (1D) gratings. For instance, a 2D grating can diffract light into multiple directions within the plane, resulting in enhanced diffraction orders, which is essential for various applications. Additionally, 2D gratings can achieve higher spatial resolution compared to 1D gratings, making them particularly valuable in imaging, sensing, and spectroscopy. The study of 2D gratings has been conducted in various atomic systems, including two-level atomic configurations [25], three-level systems [26], four-level configurations [27], and four-level N-type systems [28].
Furthermore, in optomechanical systems, a phenomenon analogous to EIT occurs, referred to as Optomechanically Induced Transparency (OMIT) [29, 30]. In OMIT, radiation pressure is generated inside the cavity due to the interaction between control and probe fields. This radiation pressure induces coherent oscillations of the cavity mirror. When the frequency difference between the probe and control fields matches the oscillation frequency of the mirror, transparency is generated through destructive interference. Recent developments have also explored a related phenomenon known as Photothermally Induced Transparency (PTIT). In PTIT, transparency is achieved via the photothermal effect in a cavity [31]. Here, radiation pressure is produced as a result of photon absorption by the cavity mirror, leading to thermal expansion of the mirror's coating. The photothermal effect can either increase or decrease the optical length of the cavity [32, 33]. Recently, the concepts of OMIT and PTIT have been extended to one-dimensional optomechanically induced grating (OMIG) [34, 35] and PTIG [36] and magnomechanical induced grating [37]. The development of advanced optical and sensing technologies has revolutionized scientific and industrial applications, with fiber-based and integrated photonic sensors providing high precision and sensitivity. For example, Chen et al [38] demonstrated simultaneous chemical and temperature detection using an all-fiber Michelson interferometer, while it has been achieved high-sensitivity hydrogen detection with a hybrid plasmonic waveguide [39]. Photonic devices have also excelled in structural monitoring, such as in [40] hybrid FBG-based vibration transducer and their soil settlement study [41]. Advances in optical materials, including PhOLEDs and waveguide devices, highlight innovation in photonics [42, 43]. Emerging technologies, such as nonlinear geometric phase-coded fluids [44] and high-Q silicon microring resonators [45], further expand possibilities, including subwavelength resonance phenomena [46].
In earlier investigation [35], 1D PTIG have been explored extensively, providing significant insights into light diffraction and beam manipulation. Building on this study, we are motivated to extend the phenomenon from 1D PTIG to 2D PTIG. This transition is not just a theoretical extension, but it introduces a host of practical advantages that are critical for advanced optical systems. The 2D grating offers enhanced control over light propagation, allowing diffraction in multiple directions within a plane. This multi-directional control enables more efficient manipulation of light, leading to higher diffraction orders, which is essential for a variety of cutting-edge applications. The ability to harness these enhanced diffraction orders has far-reaching implications in fields such as optical imaging, high-precision sensing, and advanced spectroscopy. Moreover, the spatial resolution provided by 2D gratings is superior to that of 1D gratings, making it an invaluable tool in systems requiring precise spatial light control. Whether in imaging systems, where fine detail is dominant, or in spectroscopy, where distinguishing between closely spaced spectral features is crucial, the advantages of 2D gratings are clear. A key aspect of the 2D PTIG system is the coupling strength, which governs the diffraction pattern. By adjusting the coupling strength in our proposed system, we can tune the diffraction orders and enhance the energy distribution across higher orders within the plane. This tuning ability is particularly beneficial in applications requiring controlled diffraction, as it provides flexibility in manipulating light energy for specific tasks. In conclusion, the shift from 1D to 2D PTIG represents not only a natural progression in research but also a leap forward in technological capabilities. Enhanced diffraction, spatial resolution, and tunable coupling strength make 2D gratings a powerful tool in optical sciences and engineering.

2. Model and formalism

We propose a photothermal cavity of length L, which can be simultaneously driven by both a strong pump field and a weak probe field, as illustrated in figure 1(a). The frequency of the weak probe field is denoted by ωp, while the strong pump field has a frequency ωl. The weak probe field propagates in the x-direction and lacks spatial dependence. In contrast, the strong field is applied symmetrically through the cavity, intersecting to generate a standing wave (SW), as depicted in figure 1(b). The dispersion and absorption of the probe field are now dependent on the strength of the strong field. Our aim is to investigate the periodic changes in these coefficients as the SW shifts from antinodes to nodes along the z-direction. When the probe field propagates perpendicularly through the proposed cavity, it will diffract into various orders. It is important to note that the left mirror's coating is made of fused silica. In this setup, the SW field can be absorbed by the cavity mirror, causing the thermal expansion of the mirror surfaces. To analyze the dynamics of the proposed system, we calculate the equations of motion, as discussed in [31].
$\begin{eqnarray}\dot{{x}_{\rm{th}\,}}=-\frac{{\gamma }_{\,\rm{th}}}{m}({x}_{\rm{th}\,}+\hslash \beta {g}_{\,\rm{cp}}{a}^{\dagger }a),\end{eqnarray}$
$\begin{eqnarray}\dot{a}=-[\kappa -\,\rm{i}\,({{\rm{\Delta }}}_{c}+{g}_{\rm{cp}}{x}_{\rm{th}})]a+{{\rm{E}}}_{{l}}+{E}_{p}{{\rm{e}}}^{-\rm{i}\delta t}.\end{eqnarray}$
Here, γth gives us the photothermal relaxation rate, $\beta =\frac{\rm{d}\,{z}_{\rm{th}}}{\,\rm{d}{P}_{c}}$ is known as the photothermal coefficient, where κ is the cavity decay. The power can be represented by Pc = ℏωca2/τc, where τc = 2L/c is known as the cavity round trip time. Also, Δc = ωc − ωl is the detuning of the cavity and δ = ωp − ωl. The annihilation (creation) operator is a (and a), respectively. In the above equations, gcp = ωc/L represents coupling strength whereas xth is the change occur under the photothermal effect. In the above equation, the last terms represents the weak probe field and strong field. Here, the probe field amplitude is $| {\,{\rm{E}}}_{p}| =\sqrt{\tfrac{2\kappa {P}_{p}}{\hslash {\omega }_{p}}}$ whereas the SW field is by El = Emsin[πxx] with an amplitude ∣Em∣ =$\sqrt{\frac{2\kappa {P}_{l}}{\hslash {\omega }_{l}}}$. To explore the effect of photothermal cavity on the output probe field, we must find the steady state solution of the above equations. We use the approximation as ⟨xtha⟩ ≈ ⟨xth⟩⟨a⟩ we can get
$\begin{eqnarray}\begin{array}{rcl}\langle \dot{{x}_{\rm{th}}}\rangle & = & -\displaystyle \frac{{\gamma }_{\rm{th}}}{m}(\langle {x}_{\rm{th}}\rangle +\hslash \beta {g}_{\rm{cp}}\langle {a}^{\dagger }\rangle \langle a\rangle ),\\ \langle \dot{a}\rangle & = & -[\kappa -\,\rm{i}\,({{\rm{\Delta }}}_{c}+{g}_{\rm{cp}}\langle {x}_{\rm{th}}\rangle )]\langle a\rangle +{{\rm{E}}}_{l}+{{\rm{E}}}_{p}{{\rm{e}}}^{-\rm{i}\delta t}.\end{array}\end{eqnarray}$
We assume that the probe field is weaker than strong field and then we consider xth = xs + δxth, a = as + δa that leads to linearized equations (1) and (3), gives the steady-state solution as
$\begin{eqnarray}{x}_{s}=-\beta {g}_{\mathrm{cp}}| {a}_{s}{| }^{2},\end{eqnarray}$
$\begin{eqnarray}{a}_{s}=\displaystyle \frac{{{\rm{E}}}_{l}}{\kappa -\,\rm{i}({{\rm{\Delta }}}_{c}+{g}_{\rm{cp}}{x}_{s})},\end{eqnarray}$
with the linearized equations
$\begin{eqnarray}\delta \dot{{x}_{\rm{th}\,}}=-\frac{{\gamma }_{\,\rm{th}}}{m}\left[\delta {x}_{\rm{th}\,}+\hslash \beta {g}_{\,\rm{cp}}({a}_{s}\delta {a}^{* }+{a}_{s}^{* }\delta a)\right],\end{eqnarray}$
$\begin{eqnarray}\delta \dot{a}=-\kappa \delta a+\,\rm{i}\,({{\rm{\Delta }}}_{c}+{g}_{\rm{cp}}{x}_{s})\delta a+i{g}_{\rm{cp}}{a}_{s}{x}_{\rm{th}}+{{\rm{E}}}_{p}{{\rm{e}}}^{-\rm{i}\delta t}.\end{eqnarray}$
Figure 1. (a) Schematic configuration of the suggested cavity where probe and pump fields are applied. (b) Schematic of the two dimensional grating when a probe field diffracts through a standing wave.
In equation (7), Δ0 = Δc + gcpxs and is known as the effective detuning. To find the dynamical behavior of the proposed system next we consider the ansatz as
$\begin{eqnarray}\delta a=\delta {a}_{-}{{\rm{e}}}^{-\,\rm{i}\delta t}+\delta {a}_{+}{{\rm{e}}}^{\rm{i}\,\delta t},\end{eqnarray}$
$\begin{eqnarray}\delta {x}_{\,\rm{th}}={x}_{-}{{\rm{e}}}^{-\rm{i}\delta t}+{x}_{+}{{\rm{e}}}^{\rm{i}\,\delta t}.\end{eqnarray}$
The solution can be obtained in the first order when we substitute equations (8) and (9) into equations (6) and (7) such as
$\begin{eqnarray}\delta {a}_{-}=\frac{m{\alpha }_{2}{\alpha }_{3}-\rm{i}\,| {a}_{s}{| }^{2}{g}_{\rm{cp}}^{2}{\gamma }_{\rm{th}}\beta }{m{\alpha }_{1}{\alpha }_{2}{\alpha }_{3}-\rm{i}| {a}_{s}{| }^{2}{g}_{\rm{cp}}^{2}{\alpha }_{1}\beta {\gamma }_{\rm{th}}+\rm{i}| {a}_{s}{| }^{2}{g}_{\rm{cp}}^{2}{\alpha }_{3}\beta {\gamma }_{\,\rm{th}}},\end{eqnarray}$
where
$\begin{eqnarray*}\begin{array}{rcl}{\alpha }_{1} & = & \kappa -\rm{i}\,\delta -\,\rm{i}{{\rm{\Delta }}}_{0},\\ {\alpha }_{2} & = & {\gamma }_{\rm{th}}/m-\,\rm{i}\delta ,\\ {\alpha }_{3} & = & \kappa -\rm{i}\,\delta +\,\rm{i}{{\rm{\Delta }}}_{0}.\end{array}\end{eqnarray*}$
For any cavity system one can write the input-output relation as [47]
$\begin{eqnarray}{\,{\rm{E}}}_{\rm{out}\,}(t)+{\,{\rm{E}}}_{{\rm{p}}}{{\rm{e}}}^{-\rm{i}\delta t}+{{\rm{E}}}_{{\rm{l}}\,}=\sqrt{2\kappa }\delta a,\end{eqnarray}$
where
$\begin{eqnarray}{\,{\rm{E}}}_{\rm{out}\,}(t)={\,{\rm{E}}}_{\rm{out}}^{0}+{{\rm{E}}}_{\rm{out}}^{+}{{\rm{E}}}_{{\rm{p}}}{{\rm{e}}}^{-\rm{i}\delta t}+{{\rm{E}}}_{\rm{out}}^{-}{{\rm{E}}}_{{\rm{p}}}{{\rm{e}}}^{\rm{i}\,\delta t}.\end{eqnarray}$
When we solve the equations (11) and (12) then it leads to
$\begin{eqnarray}{\,{\rm{E}}}_{\rm{out}}^{+}=\displaystyle \frac{\sqrt{2\kappa }\delta {a}_{-}}{{{\rm{E}}}_{{\rm{p}}\,}}-1.\end{eqnarray}$
For suitability, we must define
$\begin{eqnarray}{\,{\rm{E}}}_{\rm{out}}^{+}+1=\displaystyle \frac{\sqrt{2\kappa }\delta {a}_{-}}{{{\rm{E}}}_{p}}={{\rm{E}}}_{T}.\end{eqnarray}$
As we know the quadratures of the field can be defined as ET = Re[ET] + iIm[ET] such as Re[ET] (Im[ET]) is the out of phase (in phase) quadrature of the output field.
Now the phase dispersion of the probe field may be written as
$\begin{eqnarray}{\phi }_{1}({\omega }_{p})=\,\rm{arg}\,[{t}_{p}({\omega }_{p})].S.\end{eqnarray}$
The response of the proposed medium to a probe field can be described by Maxwell's equations. In the analysis that follows, we ignore the transverse component of the Maxwell equations. By applying the slowly varying envelope approximation and considering the steady-state regime, the expression can be written as follows:
$\begin{eqnarray}\displaystyle \frac{\partial {\,{\rm{E}}}_{p}}{\partial x}=[-\alpha +\,\rm{i}\,\eta ]{\,{\rm{E}}}_{p},\end{eqnarray}$
where $\alpha =(\tfrac{2\pi }{\lambda })\,\rm{Re}\,[{\,{\rm{E}}}_{{\rm{T}}\,}]$ and $\eta =\left(\tfrac{2\pi }{\lambda }\right)\,\rm{Im}\,[{\,{\rm{E}}}_{{\rm{T}}\,}]$ gives us the absorption and dispersion properties of the probe field, respectively. In this case, ${{ \mathcal E }}_{l}(z)$ is a function of z and we must say that α and η will be a function of z. By solving equation (16) one can find the transmission function of the proposed cavity will be
$\begin{eqnarray}T(y,z)={{\rm{e}}}^{-\alpha (y,z)L+\,\rm{i}\,\eta (y,z)L},\end{eqnarray}$
where, ∣T(yz)∣ = eα(y,z)L and φ2(yz) = η(yz)L represent the amplitude and phase modulation of the cavity, respectively. We now aim to extend the dynamics from a one-dimensional to a two-dimensional problem. In this case, the pump El(zy) must consist of two orthogonal standing wave fields with similar frequencies along the z- and y-axes. Therefore, the pump field must take the form El(zy) = E0[sin(πzz) + sin(πyy)]. In this situation, the absorption and dispersion are modulated periodically along both the z- and y-axes. By taking the Fourier transform of T(y, z) and using the Fraunhofer diffraction the intensity distribution may be written as [48, 49]
$\begin{eqnarray}\begin{array}{rcl}{I}_{p}({\theta }_{z},{\theta }_{y}) & = & {|{{\rm{E}}}_{p}({\theta }_{z},{\theta }_{y})|}^{2}\displaystyle \frac{{\rm{sin}}^{2}(N\pi {{\rm{\Lambda }}}_{z}\,\rm{sin}\,({\theta }_{z})/\lambda )}{{N}^{2}{\rm{sin}}^{2}(\pi {{\rm{\Lambda }}}_{z}\,\rm{sin}\,({\theta }_{z})/\lambda )}\\ & & \times \,\displaystyle \frac{{\rm{sin}}^{2}(N\pi {{\rm{\Lambda }}}_{y}\,\rm{sin}\,({\theta }_{y})/\lambda )}{{N}^{2}{\rm{sin}}^{2}(\pi {{\rm{\Lambda }}}_{y}\,\rm{sin}\,({\theta }_{y})/\lambda )},\end{array}\end{eqnarray}$
where
$\begin{eqnarray*}{{\rm{E}}}_{p}({\theta }_{z},{\theta }_{y})={\int }_{0}^{1}{\int }_{0}^{1}T(z,y)A{{\rm{e}}}^{-2\pi \iota {{\rm{\Lambda }}}_{y}y\rm{sin}({\theta }_{y})/\lambda }{\rm{d}}z{\rm{d}}y,\end{eqnarray*}$
whereas $A={{\rm{e}}}^{-2\pi \,\rm{i}{\Lambda }_{z}z\rm{sin}\,({\theta }_{z})/\lambda }$.

3. Results and discussion

To begin, we examine the characteristics of the output probe field, with the real part of this field, Re[ET], indicating the probe field's absorption properties. Figure 2(a) presents the spectrum of Re[ET] as a function of the normalized probe field detuning, δ/κ. In the absence of coupling strength gcp, the spectrum displays a Lorentzian shape, signaling significant absorption of the probe field within the cavity. However, as shown in figure 2(b), when gcp is nonzero, a narrow transparency window appears in the Re[ET] spectrum, characteristic of PTIT [31]. This phenomenon arises because, with the coupling strength gcp activated (i.e. when the control field is switched on), the cavity mirror absorbs intracavity photons, raising its temperature. The resultant temperature change induces thermal expansion of the mirror, altering the cavity length. This effect is sensitive to the incident field's power on the mirror surface. The expansion in cavity length generates Stokes and anti-Stokes scattering within the cavity, leading to PTIT. Notably, PTIT emerges when the probe field frequency closely aligns with that of the pump field, distinguishing it from OMIT, where the beat frequency of the pump and probe matches the mechanical frequency.
Figure 2. The real part of Eout against δ/κ for (a) gcp = 0 and (b) gcp/2π = 1.8 MHz. The other parameters are Em/2π = 0.5 MHz, κ/2π = 530 kHz, Δc/2π = 10 MHz, Δ0 = 0.28κ, β = −1.8 pm W−1 and γth/2π = 15.9 Hz.
In figure 3, we present the spectrum of amplitude modulation of the transmission function for various values of the coupling strength gcp. It is evident from figure 3 that the amplitude modulation depends on the coupling strength gcp. Specifically, by setting gcp/2π = 0. MHz, we illustrate the spectrum of ∣T(zy)∣ as a function of Λz and Λy, as shown in figure 3(a). The plot reveals a minor amplitude modulation at this coupling strength. Furthermore, we increase the coupling strength gcp from 0.6 to 1 MHz and re-plot the amplitude modulation, as depicted in figure 3(b). The results indicate that increasing the coupling strength enhances the amplitude modulation, making it more pronounced. Similarly, we examine the phase modulation for different values of pgcp, as illustrated in figure 4. It is observed that for a high value of gcp/2π = 1 MHz, the phase modulation becomes more pronounced compared to a smaller value of gcp/2π = 0.6 MHz. Comparing figures 3 and 4, it is clear that the amplitude modulation is more significantly affected by higher values of the coupling strength gcp. This suggests that the diffraction system is influenced by both the amplitude and phase modulation of the probe field.
Figure 3. (a) The amplitude ∣T(zy)∣ of the transmission function when gcp/2π = 0.6 MHz (b) The amplitude ∣T(zy)∣ of the transmission function when gcp/2π = 1 MHz. The other parameters are Em/2π = 0.5 MHz, κ/2π = 530 kHz, Δc/2π = 10 MHz, Δ0 = 0.28κ, β = −1.8 pm W−1 and γth/2π = 15.9 Hz.
Figure 4. (a) The phase φ(zy) of the transmission function gcp/2π = 0.6 MHz (b) The phase φ(xy) of the transmission function gcp/2π = 1 MHz. The other parameters are Em/2π = 0.5 MHz, κ/2π = 530 kHz, Δc/2π = 10 MHz, Δ0 = 0.28κ, β = −1.8 pm W−1 and γth/2π = 15.9 Hz.
To gain a comprehensive understanding of the phenomenon of PTIG, it is essential to first discuss PTIT. When a strong pump field interacts with the proposed cavity, the mirrors of the cavity absorb photons, leading to thermal expansion. This thermal expansion alters the refractive index of the mirrors, thereby modifying the optical path length of the cavity through the photothermal effect. As a result, the cavity length can be altered due to the coupling between intracavity power and the cavity length. In the presence of the photothermal effect, even a slight difference between the probe and pump fields can generate interference, which subsequently modifies the optical path length. This modification leads to the creation of anti-Stokes and Stokes optical sidebands. The process can give rise to transparency, known as PTIT. Next, we consider a system that interacts with a standing wave field in both the z- and y-directions, as depicted in figure 1(b). When the standing wave field is reproduced within the proposed cavity, the absorption of the cavity changes periodically, resulting in the formation of nodes and anti-nodes, as illustrated in figure 1(b). Consequently, when the probe field propagates through the cavity, it experiences periodic absorption and transmission. This periodic modulation of the cavity's absorption enables the occurrence of PTIG within the suggested cavity structure.
We analyze equation (18) and present the diffraction intensity distribution Ip(θzθy) as a function of both the θz- and θy-directions for two different values of the coupling strength gcp, as illustrated in figure 5. For a coupling strength of gcp/2π = 0.6 MHz, the intensity distribution is shown in figure 5(a). Here, we observe that the maximum energy is transferred to the first diffraction order, with minimal energy transferred to higher orders. Increasing the coupling strength from gcp/2π = 0.6 MHz to gcp/2π = 1 MHz reveals that energy is also transferred to the second and third diffraction orders, indicating enhanced energy distribution across multiple orders as shown in figure 5(b).
Figure 5. The diffraction intensity distribution Ip(θzθy) versus θz and θy (a) gcp/2π = 0.6 MHz (b) gcp/2π = 1 MHz where the parameters are N = 5, Λz = 4λ and L = 50 μm The other parameters are Em/2π = 0.5 MHz, κ/2π = 530 kHz, Δc/2π = 10 MHz, Δ0 = 0.28κ, β = −1.8 pm W−1 and γth/2π = 15.9 Hz.
Further, we set gcp/2π = 1.1 MHz and provide a contour plot of Ip(θzθy), shown in figure 6(a). The plot reveals a pronounced intensity at the zeroth order, with some energy transferred to the first order, while the second and third orders exhibit lower intensity. As depicted in figure 6(a), energy transfer occurs to the (±m, 0) and (0, ±n) orders, where m and n are integers, with the dominant energy concentrated in the (0,0) order. Additionally, some energy is transferred to the (±1, ±1) order. The coupling strength gcp plays a fundamental role in the behavior of our proposed system. By increasing gcp from 1 to 1.5 MHz, we observe a redistribution of the probe field's energy across various diffraction orders, as depicted in figure 6(b). Specifically, the energy is transferred to the zeroth order (0, 0), the first order (±1, ±1), the second order (±2, ±2), and the third order (±3, ±3). In general, the energy can be transferred to any higher-order mode denoted as (±m, ±n). A particularly noteworthy observation is the dominance of the probe field energy in higher orders compared to the zeroth order (0, 0). As the coupling strength gcp is further increased, the energy transfer becomes increasingly pronounced in higher orders, as shown in figure 6(c). This indicates that the probe field energy in the higher orders becomes more prominent than in the lower orders. Remarkably, as gcp increases further, the probe field energy can be almost entirely transferred to higher orders. This behavior is illustrated in figure 6(d), where for gcp/2π = 5 MHz, the majority of the probe field energy is transferred to the third order (±3, ±3), with only a small amount of energy still present in the second order. However, the energy in the second order is significantly lower compared to that in the third order.
Figure 6. (a) The diffraction intensity distribution Ip(θzθy) versus θz and θy when (a) gcp/2π = 1.1 MHz (b) gcp/2π = 1.5 MHz (c) gcp/2π = 3 MHz (d) gcp/2π = 5 MHz. The parameters are N = 5, Λz = 4 λand L = 50 μm The other parameters remains the same as shown in figure 5.
The effective detuning, Δ0, can also influence the group delay of the weak probe field, as discussed in [31], where it was demonstrated that a significant group delay can be achieved for small values of Δ0. In our system, we are also interested in understanding the effect of Δ0 on the diffraction grating. To this end, we fixed θx = 0.52 radian and present a contour plot of the intensity distribution as a function of θy and Δ0/κ for two different values of the coupling strength, gcp, as illustrated in figure 7. In figure 7(a), the spectrum is shown for a coupling strength of gcp/2π = 1.5 MHz. It is observed that for small values of Δ0/κ, the intensity of the probe field is high for the zeroth, first, and second diffraction orders, and gradually decreases as the value of the effective detuning increases. Subsequently, we increased the value of gcp from 1.5 to 5 MHz and plotted the intensity distribution versus θy and Δ0/κ again, as depicted in figure 7(b). Notably, the probe field intensity becomes larger in the third-order, indicating that maximum energy is transferred to higher orders, while the intensity decreases with increasing values of Δ0/κ.
Figure 7. (a) The diffraction intensity distribution Ip(θzθy) versus θy and Δ0/κ when sinθz = 0.52 radian (a) gcp/2π = 1.5 MHz (b) gcp/2π = 5 MHz. The parameters are N = 5, Λy = Λz = 4λ and L = 50 μm The other parameters are Em/2π = 0.5 MHz, κ/2π = 530 kHz, Δc/2π = 10 MHz, β = −1.8 pm W−1 and γth/2π = 15.9 Hz.
For a comprehensive theoretical analysis, it is essential to provide experimental validation. Several parameters can be employed in our theoretical investigation, including the cavity length L, Λz = π/kz, Λy = π/ky, the mirror substrate thickness e, and the photon lifetime in the cavity. In our setup, we assume a mirror thickness of 6 mm, while the thickness of fused silica is approximately 130 nm. The pump power and beam waist are 90 mW and 9 mm, respectively. The cavity quality factor is 541 × 106 and the finesse is 5760, as reported earlier [31]. It is important to highlight that the phenomenon of PTIT has been experimentally observed by recording the weak output field [31]. In our theoretical model, anti-nodes and nodes are generated within the cavity when the pump field forms a standing wave. The nodes block the probe field, while the anti-nodes scatter it. The intensity of the scattered field can be measured, allowing one to deduce the presence of a diffraction grating. Therefore, the PTIG can be experimentally observed in our suggested system.

4. Summary

In summary, we have conducted a theoretical investigation of 2D PTIG using a weak probe field to explore the diffraction of probe energy through a photothermal cavity driven by a standing wave pump field. Photothermal effects cause the beating of the probe and standing wave to generate a 2D diffraction grating initiated by PTIT. By varying the cavity's coupling strength, different diffraction orders of the probe energy can be achieved in the plane, with a significant transfer of energy to higher orders, especially the third order. The ability to manipulate the coupling strength effectively allows it to act as a switching mechanism, which is crucial for a wide range of applications.
1
Fleischhauer M, Imamoglu A, Marangos J P 2005 Electromagnetically induced transparency: optics in coherent media Rev. Mod. Phys. 77 633

DOI

2
Badshah F, Abbas M, Zhou Y, Huang H, Rahmatullah 2025 Coherent control of Surface Plasmon Polaritons Excitation via tunneling-induced transparency in quantum dots Opt. Laser Technol. 182 112078

DOI

3
Ling H Y, Li Y-Q, Xiao M 1998 Electromagnetically induced grating: homogeneously broadened medium Phys. Rev. A 57 1338

DOI

4
Kuang S-Q, Wan R-G, Du P, Jiang Y, Gao J-Y 2008 Transmission and reflection of electromagnetically induced absorption grating in homogeneous atomic media Opt. Express 16 15455

DOI

5
Wen J, Zhai Y-H, Du S, Xiao M 2010 Engineering biphoton wave packets with an electromagnetically induced grating Phys. Rev. A 82 043814

DOI

6
Brown A W, Xiao M 2005 All-optical switching and routing based on an electromagnetically induced absorption grating Opt. Lett. 30 699

DOI

7
Bae I-H, Moon H S, Kim M-K, Lee L, Kim J B 2010 Transformation of electromagnetically induced transparency into enhanced absorption with a standing-wave coupling field in an Rb vapor cell Opt. Express 18 1389

DOI

8
Chen Y Y, Liu Z Z, Wan R G 2017 Beam splitter and router via an incoherent pump-assisted electromagnetically induced blazed grating Appl. Opt. 56 5736 5744

DOI

9
Badshah F, Zhou Y, Shi Z, Huang H, Ullah Z, Idrees M 2024 Modulating V-type ultracold atomic tunneling via coherent injection in a mazer cavity Chin. J. Phys. 92 1674

DOI

10
Zhai P W, Su X M, Gao J Y 2001 Optical bistability in electromagnetically induced grating Phys. Lett. A 289 27 33

DOI

11
Tabosa J, Lezama A 2007 Light grating storage in cold atoms J. Phys. B: At. Mol. Opt. Phys. 40 2809 2815

DOI

12
Wen J, Du S, Chen H, Xiao M 2011 Electromagnetically induced Talbot effect Appl. Phys. Lett. 98 081108

DOI

13
Zhang Y et al 2011 Four-wave mixing dipole soliton in laser-induced atomic gratings Phys. Rev. Lett. 106 093904

DOI

14
Badshah F, Ammara, Asghar S, Ziauddin, Dong S-H 2024 Opt. Laser Technol. 177 111104

DOI

15
Mitsunaga M, Imoto N 1999 Observation of an electromagnetically induced grating in cold sodium atoms Phys. Rev. A 59 4773 4776

DOI

16
Cardoso G, Tabosa J 2002 Electromagnetically induced gratings in a degenerate open two-level system Phys. Rev. A 65 033803

DOI

17
de Araujo L E 2010 Electromagnetically induced phase grating Opt. Lett. 35 977 979

DOI

18
Carvalho S A, de Araujo L E 2011 Electromagnetically induced blazed grating at low light levels Phys. Rev. A 83 053825

DOI

19
Zhou F et al 2013 Electromagnetically induced grating in asymmetric quantum wells via Fano interference Opt. Express 21 12249 12259

DOI

20
Cheng G L, Zhong W X, Chen A X 2015 Phonon induced phase grating in quantum dot system Opt. Express 23 9870 9880

DOI

21
Xiao Z H, Zheng L, Lin H Z 2012 Photoinduced diffraction grating in hybrid artificial molecule Opt. Express 20 1219 1229

DOI

22
Ullah Z, Wang S, Lai J, Azam M, Badshah F, Wu G, Elkadeem M R 2023 Implementation of various control methods for the efficient energy management in hybrid microgrid system Ain Shams Eng. J. 14 101961

DOI

23
Zhou Y, Wang J, Gao L, Wang G, Shi Z, Lu D, Huang H, Hu C 2024 Realization of chiral two-mode Lipkin–Meshkov–Glick models via acoustics Rep. Prog. Phys. 87 100502

DOI

24
Ullah Z, Qazi H S, Rehman A U, Hasanien H M, Wang S, Elkadeem M R, Badshah F 2024 Efficient energy management of domestic loads with electric vehicles by optimal scheduling of solar-powered battery energy storage system Electr. Power Syst. Res. 234 110570

DOI

25
Cheng G-L, Cong L, Chen A-X, Phys. J 2016 Two-dimensional electromagnetically induced grating via gain and phase modulation in a two-level system J. Phys. B 49 085501

DOI

26
Wu J, Hu T-T 2018 Two-dimensional electromagnetically induced gain-phase grating with an incoherent pump field Laser Phys. Lett. 15 065202

DOI

27
Wang L, Zhou F, Hu P, Niu Y, Gong S 2014 Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system J. Phys. B 47 225501

DOI

28
Qiu T, Yang G 2015 Two-dimensional electromagnetically induced cross-grating in a four-level N-type atomic system J. Phys. B 48 115504

DOI

29
Weis S, Riviére R, Deléglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Optomechanically induced transparency Science 330 1520 1523

DOI

30
Safavi-Naeini A H, Alegre T P M, Chan J, Eichenfield M, Winger M, Lin Q, Hill J T, Chang D E, Painter O 2011 Electromagnetically induced transparency and slowlight with optomechanics Nature 472 69 73

DOI

31
Ma J et al 2020 Photothermally induced transparency Sci. Adv. 6 eaax8256

DOI

32
Konthasinghe K, Velez J G, Hopkins A J, Peiris M, Profeta L T M, Nieves Y, Muller A 2017 Self-sustained photothermal oscillations in high-finesse Fabry–Perot microcavities Phys. Rev. A 95 013826

DOI

33
An K, Sones B A, Fang-Yen C, Dasari R R, Feld M S 1997 Optical bistability induced by mirror absorption: measurement of absorption coefficients at the sub-ppm level Opt. Lett. 22 1433 1435

DOI

34
Abbas M, Asadpour S H, Hamedi H R, Ziauddin 2021 Optomechanically induced grating Opt. Express 29 42306

DOI

35
Rasul K, Hussain A, Asghar S, Ziauddin, Dong S-H 2024 The effect of Casimir force on a diffraction grating in an optomechanical system Phys. Scr. 99 095118

DOI

36
Rasul K, Hussain A, Badshah F, Abbas M, Ziauddin, Dong S-H 2023 Investigation of diffraction grating in photothermal cavity Opt. Commun. 549 129934

DOI

37
Liu W, Abbas M, Asadpour S H, Hamedi H R, Zhang P, Sanders B C 2024 Generating grating in cavity magnomechanics New J. Phys. 26 093042

DOI

38
Chen Y, Li C, Yang X 2023 Simultaneous measurement of trace dimethyl methyl phosphate and temperature using all fiber Michaelson interferometer cascaded FBG Opt. Express 31 6203 6216

DOI

39
Wang G, Feng W 2023 On-chip Mach–Zehnder interferometer sensor with a double-slot hybrid plasmonic waveguide for high-sensitivity hydrogen detection Opt. Express 31 39500 39513

DOI

40
Xu D, Su Z, Lalit B, Qin Y 2022 A hybrid FBG-based load and vibration transducer with a 3D fused deposition modelling approach Meas. Sci. Technol. 33 065106

DOI

41
Xu D, Jiang L, Qin Y, Shen H, Ji B 2024 High-precision FBG-based sensor for soil settlement monitoring: A comparative study with magnetic settlement gauges and PIV technique Sensors and Actuators A 366 114935

DOI

42
Zhang J et al 2019 Deep red PhOLED from dimeric salophen Platinum(II) complexes Dyes Pigments 162 590 598

DOI

43
Chen S et al 2024 Mid-infrared hyperuniform disordered solids waveguide devices with morphology engineering and wall-network regulation Laser Photon. Rev. 2400469

DOI

44
Pan J et al 2024 Nonlinear geometric phase coded ferroelectric nematic fluids for nonlinear soft-matter photonics Nat. Commun. 15 8732

DOI

45
Guo R et al 2024 High-Q silicon microring resonator with ultrathin sub-wavelength thicknesses for sensitive gas sensing Appl. Phys. Rev. 11 021417

DOI

46
Wan L, Raveh D, Yu T, Zhao D, Korotkova O 2023 Optical resonance with subwavelength spectral coherence switch in open-end cavity Sci. China Phys., Mech. Astron. 66 274213

DOI

47
Walls D F, Milburn G J 2007 Quantum Optics Vol. 127 Springer

48
Arkhipkin V G, Myslivets S A 2018 One- and two-dimensional Raman-induced diffraction gratings in atomic media Phys. Rev. A 98 013838

DOI

49
Shui T, Li L, Wang X, Yang W-X 2020 One- and two-dimensional electromagnetically induced gratings in an Er3+ - doped yttrium aluminum garnet crystal Sci. Rep. 10 4019

DOI

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