1. Introduction

Rare-earth ion-doped luminescent materials have gained widespread use in various rapidly evolving fields such as lasers, optical communications, and solar energy [1–3]. These materials have unique optical properties due to the presence of multiple emission bands, which can emit light located in the visible to infrared spectrum range when pumped at different frequencies or doped with different doping media [4]. The ability to modulate the multiband emission in the emission spectrum has become the focal point of our research efforts.

The spontaneous emission of rare-earth ions is not an inherent property of luminescent materials, but rather the result of the interaction between the rare-earth ions as the active medium and its electromagnetic environment [5]. The emission spectrum of rare-earth ions can be modulated by various internal and external conditions. Changing the dopant acceptor or dopant element is one of the internal modulation methods of emission spectrum. Silica-based glasses, with their superior optical, electromagnetic, and mechanical properties, have a wide range of applications in lasers and optical waveguides. They also hold an irreplaceable position as a common acceptor for rare-earth ion doping [6,7]. As an amorphous condensate, glass can provide a large number of sites for rare-earth ions, allowing them to emit light spontaneously as they enter their excited state [8]. The luminescence properties of rare earth doped glasses are controlled by a number of factors, the degree of polycondensation of the glass increases with doping, and the covalency of rare earth ions and the local crystal field in the glass change with differing amounts of ion injection and doping elements, which affect the luminescent properties of rare earth ions in the glass [9–11]. For instance, the rare-earth ion $Dy^{3+}$ has three emission bands in the visible spectrum, and it is possible to obtain emission bands of pure yellow light by doping it into $LiLuF_4$ and YAG [12]. In another example, by co-doping $Zn(PO_3)_2$ glass with $Dy^{3+}$ and $Eu^{3+}$, the ratio of blue and yellow light emission intensity can be modulated to obtain white light emission [13,14].

Surface-enhanced Raman scattering (SERS) has been a cornerstone in the study of light-matter interactions [15,16]. Inspired by SERS, the main means of controlling the emission enhancement of emitters include metal nanoparticles and photonic crystals (PhCs) cavities, which have already made great progress [5]. This study will focus on the modulation of emission by PhCs cavities, the cavity plays a pivotal role in the external modulation of rare-earth ions emission spectrum. The Purcell effect, which enhances the emissivity of atoms by resonance between the cavity mode and the dipole emission mode, provides a means of external modulation of rare-earth ion emission spectrum [17]. Photonic crystals, a fundamental component of integrated optics, are widely used to control the emission spectrum of rare-earth ions due to their ability to guide index and their photonic band gap structure that inhibits the propagation of electromagnetic modes that fall within the band gap [18]. PhCs cavities possess exceptional mode modulation capabilities and are distinguished by a high-quality factor and a low minimum mode volume threshold. Typically, the quality factor of a PhCs cavity can exceed $10^4$ [19]. Achieving precise modulation of the atomic emission spectrum through cavities was initially challenging. Early experiments such as metallic cavities and quantum dots embedded in speckle cavities validated the Purcell theory but only yielded small Purcell coefficients [20].

The modulation of the emission spectrum using PhCs was first proposed by Yablonovitch [21]. The atoms are located in a Fabry-Perot cavity composed of multiple layers of mirrors, which are now known as one-dimensional PhCs, enabling the inhibition of the spontaneous emission of atoms. Since then, more PhCs cavity structures have been developed for the emission enhancement modulation of LEDs, and the work on the calculation of the cavity spontaneous emission lifetimes by the finite difference time domain (FDTD) algorithm was also presented almost simultaneously [22–25]. K. Kounoike et al. achieved higher Purcell coefficients through the confinement of electromagnetic modes within PhCs cavities, as demonstrated in their research. The study employed both two-dimensional and three-dimensional PhCs cavities to modulate the spectrum of spontaneous emission. These cavities were produced by inducing defects in the PhCs. The quantum dots located within the defects are enveloped by a two-dimensional PhCs, resulting in a significant enhancement of the spontaneous emission emitted by the quantum dots [26]. According to Fujita et al.’s research, the implementation of two-dimensional PhCs can effectively inhibit and absorb spontaneous emission within the crystal, undergo modulation, and subsequently re-emit [27]. Joannopoulos et al. proposed a unique opto-fluidic platform that combines organic molecular solutions with PhCs to achieve a tremendous emission spectrum enhancement [28]. The above works illustrate in detail the significant position and distinctive function of PhCs in emission spectrum modulation.

Based on the characteristics of PhCs cavities, nonlinear effects are particularly significant in PhCs cavities, the design of PhCs and cavity topologies remains a challenging task, as the vast number of possibilities of geometric parameters cannot be fully enumerated in practical fabrication or theoretical calculations. As a result, a significant number of PhCs cavity structures with excellent properties remain undiscovered. To overcome this hurdle, we have turned to machine-learning techniques for feature extraction and regression analysis. By utilizing physics-based models of cavity properties and mapped geometric parameters of the cavity, this approach presents an optimal method for constructing PhCs cavities with desired properties. With the ability to rapidly analyze vast amounts of data, machine learning offers an unparalleled opportunity to unlock the full potential of PhCs cavities and drive progress in this exciting field [29,30].

Singh et al. [31] utilized neural networks to establish a correlation between the topological characteristics of PhCs and their design parameters, thereby enabling the capture of the intricate and multi-dimensional design space. The utilization of machine learning techniques is significant in the process of inverse design engineering of nanophotonic devices. In the study conducted by Song et al. [32], a hybrid parametric model was developed to predict the various wavelengths and polarizations of arbitrary digital design region (DDR) building blocks. The model was composed of a DDR feature extractor, wavelength, and polarization controller. The inclusion of polarization prediction in the physics model represents a significant advancement in this field. Furthermore, the effectiveness of machine learning in analogous applications has been validated in the inverse design of topological photonic structures, Raman amplifiers, and optical fibers [33–35].

The primary objective of this study is to combine a sophisticated physics model with a machine learning algorithm to achieve the inverse design of a dual-cavity PhCs structure. The aim is to enhance specific emission bands and inhibit other bands in the multiband emission spectrum of rare-earth ions. While a single cavity can enhance long-wavelength emission, it is not effective in inhibiting short-wavelength emission. Therefore, the study utilizes a dual-cavity PhCs structure, which includes two cavities that work in together to modulate the emission spectrum of rare-earth ions. The effectiveness of this cavity in modulating the emission spectrum and the resulting outcomes are demonstrated through practical examples. The active medium is substituted with a multilevel model in the calculation model to approximate real modulation conditions. Compared with the work on this topic, this model has better universality and can be easily extended to more structures and can also be applied to different luminescent materials.

The subsequent sections of the paper are organized in the following manner: In Section 2, the authors present a detailed account of the cavity model and modulation theory, as well as the machine learning techniques and neural network models employed. Additionally, the authors provide a comprehensive explanation of the data collection procedures utilized in the dataset, with particular emphasis on their relevance to the underlying physics. The subsequent content pertains to the methodology and principles of the inverse design. Section 3 presents an analysis of the training and performance of the machine learning model. The generalization performance of the completed training model was verified. Section 4 presents a practical demonstration of the inverse design model and the modulation process of specific emission spectrum. This is achieved through the use of $Dy^{3+}$ ion-doped glass as an active medium, which serves as an illustrative example. The current study is summarized in Section 5.

2. Data collection and inverse design models

2.1 Cavity model and modulation theory

The present study analyzes fiber-like PhCs with a two-dimensional periodic structure. The PhCs has a central cavity surrounded by air holes arranged in a triangular lattice. This arrangement is intended to confine electromagnetic modes within the central cavity. The structure, shown in Fig. 1, was designed to enable unrestricted variation of the central cavity and surrounding air holes on a micron-scale level.

 

Fig. 1. The structural model of the fiber-shaped dual-cavity PhCs. The diameter of the fiber is $8{\mu }m$ and air holes are arranged in a triangular lattice in the fiber with a lattice constant $a=1{\mu }m$ which has an air hole radius of $r_1$. The interior of the fiber contains a two-cavity structure, indicated by the shaded parts, which are the central cavity and the second cavity formed by the PhCs defect. The radius of the central cavity is $r_2$, which is both the first cavity and the location where the active medium is located.

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The PhCs fiber exhibits a circular cross-sectional shape, with a diameter of $2R=8{\mu }m$, and is composed of silica as its default material. In the fiber model, silica is regarded as a pure dielectric material. The dielectric function has no imaginary part and no frequency dependence, and can be regarded as a lossless material. The fiber contains air holes arranged periodically in a triangular lattice at the center of a circular cross-section. The lattice constant is $a=1{\mu }m$ and the radius of the air hole is $r_1$. The central cavity present in the fiber cross-section is regarded as the primary cavity, characterized by a radius of $r_2$, which is indicated by dot shading in Fig. 1. The part shown by line shading is treated as the second cavity. Actually, it is also a defective structure formed due to the elimination of air holes within the PhCs. The two cavities operate in conjunction to modulate the electromagnetic mode within the cavity.

The PhCs cavity is designed to modify the electromagnetic modes within the visible frequency range, with a specified wavelength range of $400\sim 800 nm$. The process of modifying the geometric parameters of the cavity located within the fiber core involves modifying a defective structure within the PhCs, specifically the cavity situated at the center. The remaining air holes within the PhCs retain their periodic lattice configuration. In the event that the frequency of an electromagnetic mode falls within the photonic band gap of a given PhCs, the mode will undergo conversion into an evanescent wave. Consequently, the mode will be rendered incapable of propagating through the crystal, unless the periodicity of the PhCs structure is disrupted during the process. It is plausible to maintain a local mode in the PhCs despite the existence of an electromagnetic mode within the photonic band gap, through the alteration of the cavity’s width, owing to the presence of defects in the crystal. In some cases, specific rare-earth ions in their ground state display different photon emission modes upon returning to the ground state after being excited to higher energy levels by an external pump source. The specific dual-cavity PhCs are employed to modulate the emission spectrum of rare-earth ions in this study. An active medium of rare-earth ion doped glass is inserted into the central cavity, where emission modes at various wavelengths can be enhanced or inhibited. Recent advancements in fabrication technology have demonstrated the feasibility of constructing the proposed dual-cavity PhCs structure, indicating its practical manufacturability for experimental and applied use [36].

The local density of states (LDOS) of the cavity has been calculated to demonstrate the cavity’s modulation of electromagnetic modes within a specific range of wavelengths. The LDOS is utilized to determine the emitted power of a single dipole within a given system. Recent studies have optimized the magnetic Purcell factor in Europium ions through tailored silicon antennas and LDOS design, while others achieved spatial control of dipolar emissions in $Eu^{3+}$ ions using silicon nanoantennas and cylindrical vector beams, where the application of optimization algorithms and linking emission to LDOS is very enlightening [37,38]. In the context of PhCs cavities, the LDOS is employed to quantify the extent of overlap between the cavity’s inherent resonance modes and the source. In rare-earth ion-doped glass, the ions act as the emitters. When these ions are excited (typically by optical pumping), they absorb energy and then re-emit it as light. To use the way energy is emitted as a key analogy, just as a dipole source emits electromagnetic radiation when it oscillates, rare earth ions emit light when they transition between different energy levels. Therefore, the dipole source is used instead of the excited gain medium to obtain the LDOS of the cavity. We can easily find out whether the emission at that wavelength is enhanced or suppressed by observing the LDOS of the cavity correspond to different wavelengths.

Practically the same emitting source in different shapes or media in the cavity radiates different powers, and the value of LDOS is proportional to the emitting power of the source. In addition, the spontaneous emission rate of the atoms is also proportional to the LDOS. A reduced LDOS within a given band indicates inhibition of spontaneous emission from the atoms, whereas an elevated LDOS value implies an enhancement of spontaneous emission from the atoms [39]. Thus, an effective mapping of the LDOS of the cavity to the modulation of the spectrum of rare-earth ion emission can be established.

LDOS can be used as direct observation data of rare-earth ion emission spectrum in addition LDOS is also closely related to the Purcell effect. According to the cavity quantum electrodynamics, the active medium and the cavity are divided into the weakly coupled system and the strongly coupled system [40]. In the weakly coupled system, the cavity can significantly enhance the LDOS in the active medium, which in turn enhances the emission rate of the active medium in the cavity. The spontaneous emission rate at the $r$ point can be given by Fermi’s golden rule [24]:

Divide Eq. (1) by Eq. (2), and the level of spontaneous emission rate enhancement or Purcell effect enhancement factor can be defined by the following equation:

In Eq. (3), ${\omega _0}/{\Delta \omega _m}$ is reduced to $Q_m$, which is material quality factor. The remaining parts are reduced to the mode volume $V_{eff}$.In addition, the enhancement of the Purcell effect can also be seen in the LDOS of the cavity, as reflected by the appearance of a resonant cavity mode in the cavity, where a substantial enhancement of the LDOS on its resonance peak can be observed. In this case, the contribution of this resonant cavity mode to the LDOS at $\omega$ can be written as [41]:

Comparing Eq. (3) and Eq. (4), a linear relationship can be found between the Purcell effect enhancement factor and LDOS at a certain frequency. So far, the specific enhancement of the Purcell effect in the cavity can be observed through LDOS.

2.2 Deep learning model and inverse design methods

The PhCs contain two cavities, the first of which is situated in the central region encompassed by air holes and houses the active medium. The glass material of the PhCs, located in the defect structure of the PhCs, serves as the secondary cavity, which works in conjunction with the first cavity to modulate the active medium’s emission. The cavity’s local density of states (LDOS) is affected by differences in the cavities’ geometrical parameters. The model’s cavity exhibits a strong nonlinear effect under the combined influence of cavity electrodynamics and PhCs, so we performed finer geometry planning in the preparation of the data set. Two cavity geometrical parameters vary in the calculation, the first one is the geometrical parameter of the central cavity, which increases in size from 0 in steps of 0.001$\mu {m}$ until it is tangent to the air hole of the PhCs with a fixed geometrical size of the air hole. Secondly, the size of the air hole of the PhCs increases from 0.2 $\mu {m}$ to 0.4 $\mu {m}$ in steps of 0.001$\mu {m}$. PhCs with different air hole sizes exhibit different photonic band gaps in different frequency ranges and different dispersion relations at the same frequency. The size of the secondary cavity varies with the changing air hole size but does not require specific geometric parameters to be controlled.

The diameter of the central cavity when tangent to the neighboring air hole varies when the PhCs air hole is at different sizes, so the number of central cavity size variations at each air hole size varies. In order to build an inverse model with more predictive power, a total of 12,000 sets of PhCs cavity structure parameters with their corresponding cavity LDOS data were collected in the data collecting process and calculated by the FDTD method [42]. A larger calculation resolution was selected to obtain more accurate results in the FDTD calculating process, and the data collection process lasted about 50 hours, using HPC for the calculations.

Convolutional neural network (CNN) is a feedforward neural network with convolutional computation and depth, which is one of the representative algorithms in machine learning [43]. As shown in the CNN schematic in Fig. 2(a), the LDOS data are used as input into the neural network training. the LDOS data obtained from the FDTD calculation correspond to the frequencies one by one, and data processing is performed on them before input into the CNN, and the LDOS corresponding to each set of geometric parameters is reshaped into a separate data matrix. Within the neural network, the features of the input data are extracted by the convolutional layer, and then the features obtained by the pooling layer are pooled. After several iterations of this process, the corresponding cavity geometry parameters are output by the fully connected layer neurons, and the output results in the geometry parameters of the PhCs.

 

Fig. 2. Schematic diagram of modulating the emission spectrum by the inverse design of PhCs cavities based on deep learning with FDTD methods. (a) The training process of the neural network model with the correspondence between input and output. (b) The core process in the inverse design of emission spectrum modulation.

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Upon identification of the requisite LDOS relations and their corresponding geometric parameters, we proceed to establish the necessary formulations for both forward and backward models in the relevant machine-learning context. In our model, the position of the air hole in the triangular lattice and the substrate material are unchanged; the variables are the radius of the air hole, $r_1$, versus the radius of the central cavity, $r_2$. If $F$ is specified as a one-to-one mapping function and the change of geometric parameters in the PhCs corresponds to its LDOS, then the forward model may be written as Eq. (5):

In the forward model, the process is constructed in order to find the numerical solution of its LDOS by physical relations, and the LDOS data is computed and precise enough under the physical formula using a rigorous demonstration. Here, $F$ is a formalization of the computational process.

Equation (6) is the inverse design approach we seek. $G$ is a mapping of the LDOS to the geometric parameters in the PhCs that cannot be explained by a self-consistent physical model, and the resulting function is not analytic. Despite the fact that the mapping relationship between the two is entirely nonlinear, the neural network is responsible for fitting many inputs of the LDOS to the geometric parameter relation, yielding a more accurate output. Notably, the mapping relationship between $F$ and $G$ is theoretically an inverse function, but it is impossible to discover an analytical solution for $F$’s inverse function. Focus is placed mostly on solving the inverse problem.

Figure 2 presents the flowchart of the inverse design. The LDOS data results obtained from the FDTD calculations for the PhCs cavity are collected and normalized to eliminate noisy data and ensure a common scale. This process excludes models lacking feature resonance peaks in some too-small-sized central cavities. The resulting dataset is then split into two subsets for training and validation. The CNN algorithm trains the dataset and predicts the geometric parameters of the cavity structure corresponding to the LDOS used as the input sample. The training process of the model also contains the optimization and improvement of the CNN model, which includes the tracking of underfitting and overfitting during the training process and the evaluation of the model’s generalization performance.

The core process of inverse design is shown in Fig. 2(b). It takes as input an artificially designed LDOS relation related to the modulation target, which is simplified and has the key features of the target emission band, and the details of the creation method are described in the next section. The neural network gives the corresponding geometric parameters by predicting this artificially designed LDOS, and then the FDTD algorithm calculates the target modulation result, i.e., the true LDOS distribution, which is marked by a red box in Fig. 2(b).

3. Numerical results and analysis

In Section 2.2, the data set was collected by varying the size of the central cavity and the size of the air hole of the PhCs inside the PhCs. It is worth noting that when calculating the LDOS of the cavity by FDTD, some of the results show that there are almost no resonant modes that can exist in the cavity when the central cavity size is too small. The internal space of the central cavity in case of very small size is not enough to resonate the electromagnetic modes and to create standing wave conditions in the cavity. The displayed LDOS values show a garbled waveform in the visible wavelength range and no resonant peaks can be observed. Modes in the cavity that cannot have resonant peaks cannot achieve enhancement or inhibition of a particular waveform, so the collected data set has a need for data cleaning. After eliminating the structures that cannot form resonance peaks, 8600 sets of PhCs cavity structure parameters corresponding to their LDOS are divided, 8590 sets are used as training sets to train the convolutional neural network, and 10 sets of data are divided into test sets to check the performance of CNN.

The root mean square error (RMSE) is used as an evaluation index for the degree of CNN training during the training process, and the half mean square error (HMSE) is used as a loss function to evaluate the training degree of the model, which is:

As a sequence-to-sequence regression network, $S$ in the loss function is the sequence length, $R$ is the number of the responses, and $t$ and $y$ are the target output and the network’s prediction for response $ij$, respectively.

Three back-propagation optimizers were used to update the parameters of a neural network. Figure 3 shows the convergence of the RMSE and loss of the model with the different optimizers, as well as the prediction of the neural network on the test set data during the training process. When the Adam optimizer was used, the training state curve in Fig. 3(a) indicates a substantial decrease in both RMSE and loss at the beginning of the training, followed by stabilization after 40 epochs. The three figures after the training state curve show the relationship between the true and predicted values at different training epochs (5, 10, and 100 epochs). The bar graph at the bottom shows the RMSE values between the predicted and true values of a single sample, which was used to evaluate the prediction accuracy. After 50 epochs of training, the RMSE between the true and predicted values tends to stabilize, and although there are still a few samples with poor predictions, the overall deviation of the predicted values is low.

 

Fig. 3. CNN training states under different optimization solvers: (a) Adam, (b) RMSProp, (c) SGDM. From left to right, the loss and RMSE of training 50 epochs under the optimization solver are shown. The values obtained from the neural network predictions in the test set for training 5 epochs, 10 epochs, and 50 epochs are compared with the true values. The blue histogram shows the RMSE values of the predicted and true values for each sample.

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The RMSProp solver, as depicted in Fig. 3(b), exhibits a rapid convergence of RMSE and loss at the initial progress of training, leading to a high degree of conformity between predicted and actual values by the 10th epoch. By the 50th epoch, the RMSE between predicted and actual values for the 10 test samples is better than that of the Adam and SGDM solvers. Unlike the Adam optimizer, the SGDM solver has a slow convergence rate at the outset, as demonstrated in Fig. 3(c), and does not attain convergence within the first 10 epochs. However, it ultimately converges to a lower level at the 50th epoch. Notably, all three optimization solvers are enhancements of the stochastic gradient descent (SGD) algorithm, with SGDM adding first-order momentum and RMSProp incorporating second-order momentum. In contrast, the Adam optimization solver combines the SGD algorithm with both first- and second-order momentum. Taking into account the training and post-processing time expenses, the neural network model trained to the 50th epoch utilizing the RMSProp algorithm surpasses the performance of the models produced by the other two algorithms in terms of both training and prediction accuracy. In the process of calculating the cavity LDOS by the FDTD method, the LDOS did not change excessively when the geometric parameters of the cavity were changed in a small range due to the setting of the computational resolution. However, to further explore the effect of training epochs on the prediction accuracy, the training epoch was increased to 100 epochs, and ten sets of samples from the same test set as the previous model were used to test the accuracy of the training model. As shown in Fig. 4, the RMSE between the predicted and true values of the Adam solution optimizer and the RMSProp solution optimizer did not change much in more epochs of training, and even some samples showed an increase in error. The best-performing RMSProp solver in 50 epoch training has a partial increase in RMSE between the predicted and true values after more epoch training, indicating a possible overfitting problem. After training for 100 epochs, the SGDM optimization solver is the best, and the RMSE between the predicted and true values for all ten sets of samples is reduced and at a low level, and the error values for some of the samples are extremely low and have basically exceeded the limit of what can be discriminated as an FDTD calculation. Therefore, the model after 100 training epochs using the SGDM optimization solver was used as the final version in the subsequent application. A special artificial LDOS relation is designed to examine the generalization performance of the trained CNN model as well as the emission spectrum modulation performance in realistic situations. In the PhCs cavity, the resonance mode shows a curve similar to a Gaussian distribution. In the structure designed in this paper, the central cavity is surrounded by a second cavity formed by PhCs defects, and the modes in the cavity can be modulated by the joint action of the inner and outer cavities. Since the wavelength distribution of the target electromagnetic mode is in the visible range with a small wavelength, the cavity size is relatively close to the wavelength of the electromagnetic mode, and multiple resonance peaks appear for the electromagnetic mode in this range. For the modulation of the rare earth ion emission spectrum, the main purpose is for the enhancement of a certain emission peak and the inhibition of other emission peaks. In the ideal condition, there is emission only in the emission band, and the emission in the rest of the band range is completely inhibited. In reality, the resonance peaks of the cavity and the emission spectrum of rare-earth ions are distributed similarly to a Gaussian curve, so a situation that does not exist at all in the data set was artificially designed, as shown by the orange curve in Fig. 5. The use of the Gaussian curve as input data for the CNN model was intended as a simplified representation of the expected emission profile for initial testing and training of the network. The Gaussian curve provides a smooth, continuous function that is mathematically tractable and is often used as an idealized model for various physical processes, including emission profiles. Additionally, the Gaussian function can better characterize the peak details and avoid using a single maximum value as a prediction sample, which will affect the generalization performance of the neural network model. The LDOS data with a resonant peak at $500 nm$ were manually designed with a maximum value of 9 and an FWHM of $25 nm$, described by Eq. (8):

 

Fig. 4. The RMSE between the predicted and true values of the test set samples trained at 50 epoch and 100 epoch with different optimization solvers.

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Fig. 5. The yellow line is a simplified version of LDOS used to test the generalization performance of the CNN model in the form of a Gaussian curve representing the resonance peaks in the cavity. The blue line is the real LDOS calculated by substituting the geometric parameters obtained from the CNN prediction into the FDTD algorithm using the LDOS represented by the yellow line as input.

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The artificially designed Gaussian curve data is fed into the trained CNN as a prediction sample, and the neural network gives a radius of $0.4215{\mu }m$ for the central cavity and $0.3319{\mu }m$ for the air hole in the PhCs. After obtaining the geometric parameters of the PhCs cavity, the LDOS of the cavity is calculated by FDTD and is shown in the blue curve in Fig. 5. In the $200 nm$ range centered at $500 nm$, the LDOS curve profile in the PhCs cavity is very close to the artificially designed LDOS and fits well with the predicted sample. However, there is an obvious depression in LDOS at 490nm. This is due to the formation of resonance mode or interference effect caused by the configuration of the dual-cavity structure. The depression here indicates the area where destructive interference or mode suppression occurs, resulting in a reduction in LDOS. The number of modes present in the $550{\sim }700 nm$ range is low and consistent with the predicted sample characteristics. The number of modes at $700 nm$ increases gradually toward larger wavelengths due to the proximity of the cavity size to the mode wavelengths, but does not affect the enhancement of the main peak and the inhibition of modes at nearby wavelengths.

So far, a CNN model has been created that exhibits excellent generalization performance and prediction accuracy. In the next section, we will analyze how the inverse design modulation of the CNN model’s emission spectrum performs with actual rare earth ions.

4. Rare earth ion emission spectrum modulation with $Dy^{3+}$ as an example

Applying the completed CNN model trained in Section 3, the dual-cavity structure of the PhCs can theoretically enhance or inhibit the modes in any visible range. Before applying the PhCs cavity to modulate the emission spectrum of rare-earth ions, the emission spectrum of the target rare-earth ion to be modulated in a specific doped glass is first selected, and it is worth noting that the intensity of the emission spectrum of rare-earth ions doped in different glass types, the peak position will change, and even two emission summits at adjacent wavelengths will merge into one peak [44]. After the modulation target is selected, its emission spectrum is analyzed to determine the parameters of the emission spectrum such as band, peak, and FWHM. At the stage of neural network intervention to predict the geometric parameters, the emission peaks of the target to be enhanced are fitted by the method in the previous section and substituted into the neural network in the form of a Gaussian curve to predict the geometric parameters of the cavity, which is for the case of a single emission peak. If there is more than one band of emission peaks in the emission spectrum of rare-earth ions, one can choose to enhance one emission peak alone and inhibit the other, or enhance them together.

$Dy^{3+}$ rare-earth ion was chosen as a modulation case for the PhCs dual-cavity model to practically apply the structure to the modulation of the rare-earth ion emission spectrum. The fabrication of white light diodes and white light lasers has attracted a lot of attention, as an important rare earth ion, $Dy^{3+}$ plays an important role in the fabrication of light emitting materials, and its emission spectrum in 1.0mol${\% }$ $Dy^{3+}$ doped $TeO_2-ZnO-PbO-PbF_2-Na_2O$ (TZPPN) glass has three emission peaks, as shown in Fig. 6. A general model with a four-energy system and three emission bands were developed to study the emission spectrum of $Dy^{3+}$-doped glass [45]. The model utilizes a generalized form of the Maxwell-Bloch equation by taking a system with an initial population density of $N_0$ and pumping some of the ground-state atoms to the excited state under the action of a pumping light source, and the excited-state atoms undergo luminescence transitions to each energy level below the excited state when emission occurs, and the emission process of $Dy^{3+}$ ions satisfies this form. We apply the emission parameters of $Dy^{3+}$ in our model, and the ground-state particles produce three visible spectrum by spontaneous emission under the action of $454 nm$ pump light. Two strong emission bands correspond to the $^4F_{9/2}{\sim }^6H_{15/2}$ (blue) transition and the $^4F_{9/2}{\sim }^6H_{13/2}$ (yellow) transition, and one red emission band, corresponding to the $^4F_{9/2}{\sim }^6H_{11/2}$ transition.

 

Fig. 6. Partial energy level of TZPPN glass doped 1 $mol$ $Dy_2O_3$, three visible light emissions at a $454 nm$ pump light source [45].

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The experiment value of the emission spectrum of 1 mol${\% }$ $Dy^{3+}$ doped TZPPN glass as shown in Fig. 7, with a yellow line, two strong emission peaks are located at $482 nm$ and $574 nm$, one red light emission peak is located at $660 nm$, and the FWHM of the emission peak is about $10 nm$. In order to simulate the same emission spectrum of $Dy^{3+}$ as the real case, the multilevel model is chosen to act as an active medium in the cavity. By defining the pump rate from the ground state to the excited state as $1.53*10^6/s$, the initial population density of the ground state as $4.363*10^{26}/m^3$, and the radiative rate $1/\tau _R$ or non-radiative rate $1/\tau _{NR}$ between energy levels, where $\tau _R=374{\mu }s$ and $\tau _{NR}=100 {\mu }s$ [45,46].The obtained emission spectrum of $Dy^{3+}$ is shown in the blue curve of Fig. 7. The emission intensity is related to the position and size of the detector, and the peak value of each emission peak is close to the experimental value, and this multilevel model is applied to the PhCs cavity as a modulation object.

 

Fig. 7. Experimentally determined emission intensity of 1 mol${\% }$ $Dy^{3+}$ doped TZPPN glass, indicated by yellow line [45]. The blue line indicates the multi-energy emission model built by FDTD, whose emission intensity in free space is close to the real situation.

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The three emission bands of $Dy^{3+}$ ions located in the visible range are most prominent at $574 nm$ for a yellow light, about half the intensity of yellow light at $482 nm$ for blue light, and lower at $660 nm$ for a red light. The yellow light emission was chosen to demonstrate the modulation performance of the dual-cavity PhCs, and the LDOS of the target cavity should theoretically have a higher number of modes present near $574 nm$ and a lower number of modes present in the remaining two emission bands. Based on the morphology of the target LDOS, a prediction sample that can be processed by the neural network is artificially designed, as shown in the yellow line in Fig. 8, and a Gaussian curve with the center at $574 nm$ and an FWHM of $25 nm$ is used as the prediction sample to obtain the cavity geometry data by the neural network prediction. The radius of the central cavity is $0.4841{\mu }m$ and the radius of the PhCs air hole is $0.2296{\mu }m$ in the predicted results. The real LDOS data obtained by substituting this parameter into the FDTD calculation is shown in the blue line part in Fig. 8, and it can be seen that the number of modes in the range of $100nm$ with $570nm$ as the center is large and the general shape is similar to the predicted sample. In the blue and red light emission, the number of modes at $482nm$ and $660nm$ in the band is relatively small. In addition, it has been verified that including the radius of the central area and the peripheral air holes, a fabrication tolerance of 20nm will not lead to qualitative changes in the modulation results. However, the structural parameters mentioned above should be considered one of the optimal solutions within this range.

 

Fig. 8. The yellow line indicates the LDOS data fitted by Gaussian curves are used as input to the CNN, with peaks located near the yellow light emission spectrum, and blue and red light are present in the cavity in a smaller number of modes. The blue line indicates the LDOS of the cavity calculated from the geometric parameters predicted by the CNN.

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The cavity geometry parameters predicted by the neural network model are then calculated by the physics model based on FDTD to obtain an LDOS distribution that meets the expectation of modulation of the $Dy^{3+}$ ion emission spectrum. Applying the geometric parameter predicted by CNN, the PhCs cavity structure is designed as shown in the Fig. 9(c), where the central cavity is filled with fused silica glass doped by $Dy^{3+}$ ions with a radius of $0.4841{\mu }m$. The outside of the central cavity consists of a triangular lattice of air-hole PhCs, and the third cycle of the PhCs forms a defect to become the second cavity. Outside the cavity, the PML layer absorbs electromagnetic waves of all frequencies and incident angles without any reflection, thus avoiding interference. Flux monitoring region surrounds entire cavity, the blue line indicates the flux monitor region, used here to accumulate the appropriate Fourier-transformed fields to compute a flux spectrum, the value of which is positively correlated with the accumulated time positively correlated. The blue line in Fig. 9(a) represents the emission spectrum collected in the active medium in free space (without PhCs structure), running from the beginning to $t=400$. The ratio of each peak intensity of the emission spectrum collected in free space is consistent with that of $Dy^{3+}$ defined by the multi-level system in Fig. 7, and subsequent regulation results will be based on this benchmark. After applying the PhCs cavity structure, also keeping the running time from the beginning to $t=400$, the intensity of the collected emission spectrum is demonstrated in the yellow line in Fig. 9(a), where the modulation effect of the cavity on the emission spectrum is very obvious, and the blue light emission is further inhibited based on the original one, and the intensity of the yellow emission is greatly enhanced compared with the weak emission in the original active medium.

 

Fig. 9. Results of modulation of $Dy^{3+}$ ion emission spectrum by PhCs cavities. (a) The emission intensity of $Dy^{3+}$ ion-doped fused quartz in free space and under cavity modulation. (b) Comparison of the total emission intensity before and after the modulation of the three emission peaks. (c)Field distribution in the cavity, the blue line is the flux monitoring region.

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The integrals of each peak region of the emission spectrum are calculated to quantify the modulating effect of the cavity on the emission spectrum of rare earth ions, as demonstrated in Fig. 9(b), it can be observed that the emission of blue light is significantly inhibited, with the cumulative emission intensity of blue light corresponding to about 29.9${\% }$ of the active medium in free space and the intensity of the red light emission is 31.1${\% }$ that in free space. the emission of yellow light is significantly enhanced, with the cumulative emission intensity in free space of 5654.63, and the cumulative emission intensity reaches 15790 under the modulation of the cavity, and the emission is enhanced by nearly 2.79 times as much. The modulation target was in accordance with the expectation and the emission of the active medium approximates pure yellow light emission. Figure 9(c) shows the field distribution of the cavity structure that modulates yellow light emission. The mode in the cavity is dominated by 574nm, which has an absolute advantage. The other emission bands are greatly suppressed, but there are still residues, which have been greatly weakened compared to free space.

The blue light emission at $482 nm$ and the red light emission at $660 nm$ of $Dy^{3+}$ ions were modulated using the same method as that for modulating yellow light, respectively. The artificially designed LDOS relations are fed into the CNN to obtain the geometric parameters of the cavity. The geometrical parameters of the cavity used to modulate the blue light emission are $0.2693{\mu }m$ for the radius of the central cavity and $0.6582{\mu }m$ for the radius of the air hole, the geometrical parameters of the cavity used to modulate the red light emission are $0.3302{\mu }m$ for the radius of the central cavity and $0.5581{\mu }m$ for the radius of the air hole. The modulation results are shown in Fig. 10, as in Fig. 10(a), (b), under the modulation of the cavity, the emission intensity of blue light is increased by 2.41 times, and the emission intensity of yellow light and red light suppressed significantly. In Fig. 10(d),(e), the red light modulation results show that the emission intensity of red light is increased by 2.81 times compared with that of free space, while the emission intensity of blue and yellow light emission intensities are inhibited to 23.7${\% }$ and 22.1${\% }$ in free space. The emission of blue light and yellow light is greatly suppressed. Figure 10(c) and (f) show the field distribution of the cavity modulating blue light and red light respectively. Similar to the structure of modulating yellow light, the distribution of the field is dominated by the pattern of the modulated band, while there are still other suppressed patterns.

 

Fig. 10. Results of modulation of the blue and red light emission spectrum of $Dy^{3+}$ ions by a PhCs cavity. (a) The emission intensity of blue light modulation results and emission intensity in free space.(b) Comparison of emission intensity quantification before and after modulation of blue light. (c) Field distribution in the cavity for modulating 482 nm emission. (d) Comparison of emission intensity of red light modulation results and emission intensity in free space. (e) Comparison of emission intensity quantification before and after modulation of red light. (f) Field distribution in the cavity for modulating 660 nm emission.

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In addition, if the flux monitoring area is only set on the left side of the cavity and the other structural parameters remain unchanged, the result will be anisotropy in the control of the cavity. As shown in Fig. 11, compared with the results of flux collection of the overall structure, the flux collected on one side shows a more obvious emission enhancement effect. Among them, as shown in Fig. 11(a),(b),(e),(f) when regulating 574nm and 660nm emission, the enhancement of yellow light reaches 16.28 times and the enhancement of red light reaches 30.79 times, which is similar to pure color light emission. For the control of 482nm emission, blue light is enhanced by 17.78 times, but yellow light is also enhanced to a certain extent, as shown in Fig. 11(c),(d), failing to achieve the effect of pure light control.

 

Fig. 11. Results of modulation of the yellow, blue and red light emission spectrum of $Dy^{3+}$ ions by a PhCs cavity. The flux monitoring region is set on the left side of the cavity with a linear structure. The bar graph below represents the total emission within the emission zone compared to the total emission in free space. (a),(b) Result for modulating 574nm emission, the red line in the inset is the flux collection range. (c),(d) Result for modulating 482nm emission. (a),(b) Result for modulating 660nm emission.

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A complete set of the inverse design flows of rare-earth ion emission spectrum modulation is shown in this section. For rare-earth ions with different emission bands, the modulating values of the target emission bands are predicted using a neural network to obtain the geometric parameters of the cavity. The physics-driven FDTD algorithm yields modulating results for photonic devices with predicted geometrical parameters, which are exemplified in this section for $Dy^{3+}$ ions and can be extended to other rare-earth ions as well. Given the current mature manufacturing process of PhCs fibers, it is believed that it can provide a reference for the parameter planning of physical fabrication.

5. Conclusion

In this paper, we present a pioneering approach to modulating the emission spectrum of rare-earth ions by utilizing PhCs cavities with inverse design. Our dual-cavity structure, combined with a convolutional neural network-based approach, allows us to precisely predict geometric parameters that enhance or inhibit specific spectrum features. We validate our approach numerically using $Dy^{3+}$ ions and demonstrate that our designed cavity effectively enhances the intensity of the target emission spectrum while inhibiting unwanted spectrum components. By modulating the emission spectrum of $Dy^{3+}$ ions in the visible range, we achieve a significant increase in the intensity of yellow light emission by 2.79 times that in free space, while the emission of blue and red light is inhibited, resulting in nearly pure yellow light emission. We are also able to modulate red light emission, inhibiting blue and yellow light emissions, while enhancing the red light emission intensity by 2.81 times compared to free space, and blue light and yellow light are greatly suppressed. If a linear area is used to collect flux on the left side of the cavity, yellow light can be enhanced by 16.28 times and red light can be enhanced by 30.79 times.

Our ability to modulate the spectrum properties of rare-earth ions offers the potential for developing advanced optical materials with tailored emission characteristics for specific applications. In the future, we plan to investigate the scalability of our method by exploring other rare-earth element ions in PhCs cavities. Additionally, we believe that our approach can be extended to other types of cavities, opening up new possibilities for controlling the emission spectrum of rare-earth elements.