Enhanced Imaging Using Inverse Design of Nanophotonic Scintillators

Converting ionizing radiation into visible light is essential in a wide range of fundamental and industrial applications, such as electromagnetic calorimeters in high‐energy particle detectors, electron detectors, image intensifiers, and X‐ray imaging. These different areas of technology all rely on scintillators or phosphors, i.e., materials that emit light upon bombardment by high‐energy particles. In all cases, the emission is through spontaneous emission. The fundamental nature of spontaneous emission poses limitations on all these technologies, imposing an intrinsic trade‐off between efficiency and resolution in all imaging applications: thicker phosphors are more efficient due to their greater stopping power, which however comes at the expense of image blurring due to light spread inside the thicker phosphors. Here, the concept of inverse‐designed nanophotonic scintillators is proposed, which can overcome the trade‐off between resolution and efficiency by reshaping the intrinsic spontaneous emission. To exemplify the concept, multilayer phosphor nanostructures are designed and these nanostructures are compared to state‐of‐the‐art phosphor screens in image intensifiers, showing a threefold resolution enhancement simultaneous with a threefold efficiency enhancement. The enabling concept is applying the ubiquitous Purcell effect for the first time in a new context—for improving image resolution. Looking forward, this approach directly applies to a wide range of technologies, including X‐ray imaging applications.


Introduction
Converting ionizing radiation into visible light lies at the heart of a vast range of technologies: from medical imaging systems However, such approaches complicate the fabrication process of scintillators and still involve loss of photons and a limited resolution.
Here we present a new concept for overcoming the fundamental trade-off between conversion efficiency and resolution in scintillators and phosphor screens (Figure 1). Our concept improves both efficiency and resolution by exploiting the Purcell effect over the large volumes of the imaging screen. We leverage existing optimization tools to design the imaging characteristics of nanophotonic structures made from a combination of die-lectric materials and scintillators/phosphors. Specifically, our design optimizes the point-spread function (PSF), efficiency, and resolution. Our numerical results showcase that the optimized planar nanostructure achieves a 2.8 efficiency enhancement along with more than tripled resolution compared to naïve bulk scintillators. Our work applies nanophotonic principles such as the Purcell effect to enhance resolution, PSF features, and other image processing characteristics, paving the way for future imaging technologies based on spontaneous emission enhancement. Breaking the trade-off between efficiency and resolution in scintillators and phosphor screens. The screen converts input radiation (e.g., electrons) into visible photons, emitted isotropically and detected by a photodetector. The image formed by the object shown above the structure is depicted under the scintillator. a) For a thin screen (3 µm), we get high image resolution, but low conversion efficiency since part of the radiation is not stopped by the thin structure. b) For a moderate thickness screen (6 µm is considered the standard structure today [24] ), the efficiency is higher than the 3 µm bulk, but at the price of lower image resolution. The resolution deteriorates because the points of emission are now on average farther away from the detector, which causes light to spread further before reaching the detector. c) For a thick screen (9 µm in this example), we get even higher conversion efficiency, but at the price of even lower image resolution. d) An inverse-designed nanophotonic structure made from alternating layers of the scintillator (light blue) and another dielectric material (dark blue) with varying thicknesses. The structure contains 6 µm of scintillation material and an overall thickness of 12 µm. The thicknesses of the layers are optimized to achieve the highest resolution. e) The trade-off between efficiency and resolution in homogeneous phosphor screens (yellow line). The axes are normalized with respect to the efficiency and resolution of the standard structure. Our inverse-designed nanophotonic structure (orange dot) achieves simultaneous high efficiency and high resolution.

(3 of 10)
© 2023 The Authors. Advanced Optical Materials published by Wiley-VCH GmbH

The Underlying Mechanism: Effective Spontaneous Emission Rate
The Purcell effect describes the spontaneous emission rate enhancement of point-like localized (dipole) emitters (atoms, molecules, and quantum dots) due to their electromagnetic environment. [25,26] Typically, the emitters are located inside a cavity with a maximized quality factor Q and a reduced light mode volume V, since the Purcell factor (the emission rate enhancement) F P scales as Q/V. The Purcell effect was introduced into nanophotonics by Yablonovitch. [26] In the case studied in this work, the emission points are distributed over a large volume compared to the radiation's wavelength, leading to a large mode volume. Additionally, an efficient light outcoupling (from the device to air) requires a low Q. These requirements are exactly the opposite of the standard situations in which the Purcell effect is applied, which motivates a different design strategy and figure of merit. [18] In this section, we employ the concept of effective spontaneous emission enhancement, which captures global features of the structure rather than local ones.
Recently, a few studies have introduced the concept of nanophotonic scintillators, [15][16][17][18][19][27][28] showing how the Purcell effect could be used to enhance and shape spontaneous emission upon excitations by energetic particles. [18][19]28] In particular, nanophotonic scintillators can control the photonic localdensity-of-states (LDOS) to enhance the spontaneous emission rate in scintillation and in cathodoluminescence. [17][18]28] Our work develops nanophotonic scintillators for the context of scintillation imaging-showing how nanophotonics can improve the important characteristics of the imaging system.
We start the derivation using the formula of the spontaneous emission rate of a single point dipole under the dipole approximation [29] , 2 c Im , ; where ε 0 is the vacuum permittivity, ℏ is the reduced Planck's constant, c is the speed of light in vacuum, ℏω is the energy difference between the ground and excited states of the dipole, and p is the dipole moment. In addition, G ij (r,r′; ω) is the dyadic Green's function, which incorporates the whole electromagnetic environment; it describes the ith component of the electric field at location r as a result of a dipole excitation in r′ oriented along j. The spontaneous emission rate enhancement, that is, the Purcell factor of a local dipole, is defined by is the emission rate of the same dipole in free space. We aim to calculate the spontaneous emission rate from a multilayered planar structure, uniform in the x-y directions and alternating on the z axis. As a first step, we follow [18] to consider the Green's function from three-layered structures, where the dipole emitters are located in the central layer. The spontaneous emission rate enhancement for three-layered structures relies on Fresnel's reflection and transmission coefficients between the layers. For a general multilayer structure, this result can be generalized by considering the effective Fresnel's coefficients, which accumulate the contribution of the Fresnel's coefficients from multiple layers to describe the effective reflection and transmission, as discussed in [29] .
Describing the three-layer structure, we denote the central layer as region 1, which has a thickness d and permittivity ε 1 , while the lower and upper layers are denoted by regions 2 and 3, respectively, with permittivities ε 2 and ε 3 . The emission rate of light outcoupling to region 3 from a single dipole (averaged over all orientations), with frequency ω, and depth z, is [30] where k i is the total wave vector amplitude in region i; l i and u are the z-components and in-plane wave vector components, s\p represent the TE\TM polarizations, respectively. Moreover, , ; where r ij , t ij are the Fresnel's coefficients between layers i, j ∈ {1, 2, 3}. This result is extended to a general multilayered structure using the effective Fresnel coefficients from a multilayered structure. [30] We note that using Equation 2, we can identify the spontaneous emission rate for light outcoupling at solid angle θ, frequency ω and location z inside the multilayer structure: where d is the list of structure layers' thicknesses, and T ∥s (d),T ∥p (d),T ⊥p (d) denote the structure's effective Fresnel coefficients. To demonstrate the dependence of the emission on the emitter's location, we present in Figure 2 the values obtained by γ for three emitters located at different locations along the z axis of a multilayer structure.
To characterize the entire nanostructure, we convert the local (z dependent) emission properties to a global property. The effective angle-dependent emission rate Γ eff is defined as the number of photons per second that couple out of the nanostructure for each angle θ and frequency ω: [18] , where z is the depth and G(z) is the emitter absorption profile in the structure for the incoming radiation (which depends on the absorption coefficients of the materials, the radiation energy, and particle type). Y(ω) is the emitter's spectral distribution (normalized by ( )d 1 ). We note that the integrand is non-vanishing only in layers with phosphor material. The non-emitting dielectric layers have Y(ω) = 0 but still contribute to the electromagnetic environment of the phosphor layers and absorb a fraction of the incoming radiation.

Inverse Design of 1D Multilayer Nanostructures
A variety of optimization techniques were proved useful in the design of passive photonic devices such as photonic crystals, [32,33] waveguide elements, [34,35] plasmonics, [36,37] and more. [38][39][40] In active photonic devices, optimization techniques were used to design cavities that enhance emission rate from point-like localized emitters. [41] Approaches based on electromagnetic reciprocity, [17] frequency-averaged local density of states, [42] and trace formulations [43] were recently proposed to optimize emission from an ensemble of dipoles. In this section, we formulate the inverse-design problem and exemplify the approach for improving radiation conversion efficiency. We observe that the optimized structures strongly depend on the figures of merit, emphasizing that different applications will benefit from different nanophotonic scintillators.
In the case studied, the structure's thicknesses constitute the degrees of freedom in the optimization procedure. Formally, a planar nanostructure can be represented by a vector of layers' thicknesses, , where N is the maximal number of layers to be considered. The design is done by numerically solving a nonlinear constrained optimization problem over the structure's geometrical features, with equality and inequality constraints. The constraints may arise from manufacturing limitations, such as a lower bound on the layers' thicknesses. We denote by d * a local solution to the above problem. For large structures, the vector of thicknesses lies in a high dimensional space, which further challenges the optimization procedure. To solve the optimization problem we leverage existing optimization algorithms such as the interior-point method. [44,45] As an example, phosphors are used in many applications as converters of light wavelength. In such applications, the phosphor absorbs light of a particular wavelength and then emits light at a longer wavelength. An important figure of merit for many of these applications is conversion efficiency. In our context, we will define the efficiency of our multilayer nanostructure as the amount of outcoupled photons emitted from the multilayered structure compared to the number of outcoupled photons emitted by a bulk structure where d bulk is the thickness of the corresponding bulk structure. Figure 3 shows the optimization map of multilayered periodic structures and illustrates how the optimization procedure converges to thicknesses that maximize the conversion efficiency (defined in Equation 6). This optimization map allows us to visualize the dependence of the effective spontaneous emission rate on the structures' geometrical features and guide the design process of periodic structures. For this specific example, we chose the structure to be a 20-layer periodic structure with only two degrees of freedom, thus allowing us to visualize the optimization map. In Figure 3b, we observe that the structures obtained along the optimization path have different emission characteristics, such as the frequency integrated effective emission rate. Generally, the structures have more than two degrees of freedom, and therefore cannot be visualized on a map. This simple example already shows key features from the more general optimization problem, for instance, that multiple local maxima exist. We validate our theoretical results with an independent numerical Finite-difference time-domain (FDTD) simulation, shown in Figure 3b

Improving Imaging Performance Using Inverse-Designed Nanostructures
We use the framework of effective spontaneous emission to overcome the fundamental trade-off between resolution and efficiency in scintillators and phosphor screens. We propose Adv. Optical Mater. 2023, 11, 2202318 Figure 2. The underlying mechanism: outcoupled spontaneous emission of emitters in a multilayer structure. a) An example of a multilayer structure with three emitters in different locations. The structure is composed of alternating layers of P43 phosphor (light blue, with a refractive index of 2.3 [31] ) and silica (dark blue, with a refractive index of 1.454). The outcoupled radiation is the light that couples to free space below the structure. b) The spontaneous emission rate at angle zero (downward) of each emitter in (a) as a function of free-space wavelength. The values are normalized by the maximum emission of an identical emitter in a bulk material. It is shown that the spontaneous emission of the red emitter is enhanced, and for the other emitters, it is suppressed. c) The spontaneous emission rate at the central wavelength of the spectrum 450 λ =  nm, of each emitter in (a) as a function of the outcoupled angle. Both enhancement and suppression can be seen for different emitters that only differ by their location within the multilayer structure.
optimized designs of multilayer nanophotonic structures to improve the eventual image. Specifically, we show that the inverse-designed structure improves the light's PSF, resulting in simultaneous higher efficiency and higher resolution. We compare our phosphor designs to the phosphors used in current state-of-the-art image intensifiers. Since each layer's thickness is optimized independently, the optimal designs are generally aperiodic (Figure 1d), in contrast to one-dimensional photonic crystals scintillators. [18] We further developed our effective spontaneous emission theoretical model to produce the system's PSF, which is used to quantify and analyze imaging devices. The PSF describes the structure's impulse response to input radiation as a function of the radial distance on the detector, r: PSF = PSF(r; d). When normalized, it represents the probability distribution of the input radiation to be projected at a certain point onto a plane adjacent to the device interface. The details for the analytical calculation of a structure's PSF are presented in Appendix A (Supporting Information).
The detected images are obtained by applying the structure's PSF as a convolution filter on the incoming image. For a numerical comparison between the physical images and the detected/processed images, we use the mean squared error (MSE) image quality where Y i and i Y  are the vectors of grayscale values of the original image and the detected image, and n is the total number of pixels in the image.
Moreover, we compute the structure's detection efficiency and resolution from the PSF. We define the detection efficiency of a structure as the total emission that couples out, normalized by that of a corresponding homogeneous media where d bulk is the bulk's thickness. The resolution of an imaging system can be calculated from the Fourier transform of its PSF, [46] that is, the modulation-transfer-function (MTF) which depends on spatial frequencies MTF = MTF(k; d). In our case, we assume the PSF to be circularly symmetric, such that the MTF is given by the Hankel transform of the PSF's radial component. The MTF plays an important role, as the detected image depends on its behavior in low and high spatial frequencies. [4] Specifically, the resolution of the structure is defined by the spatial frequency, where the MTF reaches 3% of its zero-frequency amplitude. [46,47] An additional important characteristic of the imaging system is the detective quantum efficiency (DQE). The DQE captures how noise propagates through an imaging system relative to the signal and is defined as the ratio of the square of the output image SNR to that of the input. [48] Following the analysis in [22] , we described the entire imaging system as a series of cascaded stages, where each stage corresponds to an increase or a loss of optical quanta. The stages substantially limiting the DQE are called quantum sinks. At zero frequency, the principal quantum sink is the quantum efficiency (resulting from limited absorption). Due to light's generation efficiency and optical spreading, an optical quantum sink becomes dominant for nonzero frequencies. As a result, enhancing the conversion via spontaneous emission for nonzero frequencies and steering scintillation light can prevent the sharp decrease of the DQE at nonzero spatial frequencies.
Assuming a purely Poissonian noise, the DQE of an optically coupled multilayer nanophotonic scintillator can be expressed using the MTF [22] DQE ; 1 1 MTF ;  scintillator). Such derivations are commonly applied for both the MTF and DQE, to describe various imaging detectors. However, to the best of our knowledge, here is the first time that the Purcell effect and its considerations are used to enhance these imaging quantities.
To demonstrate the fundamental trade-off and show how a multilayer structure overcomes it, we focused on the example of phosphor screens used today in image intensifier tubes. [24] Similar phosphor screens are at the heart of many applications, such as night vision devices and medical imaging devices. [20] The same concept also applies to scintillators used in X-ray imaging. We focus on bulk structures used in image intensifiers tubes, composed of P43 phosphor screens (Gd2O2S:Tb) with a refractive index of 2.3, [31] and an absorption coefficient of 25 µm −1 (extracted using numerical simulations [49] ). We considered three thicknesses of phosphor screens: 3 µm (thin structure, yields high resolution), 6 µm (standard structure in use today), and 9 µm (thick structure, yields high efficiency). [24] To overcome the trade-off of these conventional bulk phosphors, we designed a 12 µm multilayer structure, which combines layers of silica (with a refractive index of 1.454) between the phosphor layers. This structure contains 6 µm of silica and 6 µm of P43 phosphor, distributed across 90 alternating layers with varying thicknesses. The structure maximizes the effective emission rate by solving the following optimization problem, where we maximize the emission in central angles ( where d l denotes the lth layer in the structure for l ∈ [1, N]. We require that all layers have thicknesses in the range [0, d max ] for a given d max > 0, and that the combined thickness of phosphor layers equals the amount of phosphor in the standard structure, d bulk = 6 µm. The inverse-designed nanostructure presented in Figure 1d, is obtained by solving the problem in Equation 10. We compare the MTF and DQE of the structures as a function of the spatial frequency (line-pair (LP)/mm) in Figure 4a,b, and the  (3,6, and 9 µm) are compared with those of the inverse-designed structure. The resolution of the structure is defined as the spatial frequency for which the MTF intersects with the 3% line. [46] As expected for homogeneous structures, the thinner the materials, the higher the resolution, but the lower the DQE(k = 0). The optimized structure achieves the highest resolution-1.6 times higher compared to today's state-of-the-art high-resolution structure. c) The original image before being processed by the structures. d-g) The processed images for the respective PSFs ((d)-(f) correspond to the bulk structures and (g) corresponds to the optimized design). We used the MSE image quality index to compare the processed images to the original image. As we can see, the optimized structure achieves the lowest MSE due to its high resolution, even compared to the 3 µm bulk structure (with the highest resolution today). Figure 5a. Indeed, the optimal optically coupled multilayer has a highly concentrated PSF that enables extracting high spatial frequencies and reconstructing sharper images (see the comparison in Figure 4c-f), by overcoming the quantum sink arising from optical spreading. As a result, although the optimized and standard structures contain the same amount of phosphor, the efficiency of the optimized structure is higher by a factor of 2.8. Furthermore, its resolution is enhanced by 1.6 compared to today's state-of-the-art phosphor screen. [24] Thus, our multilayer structure can create high-resolution images with less radiation exposure time, or alternatively, increase lifetimes for portable devices, while achieving higher image quality. Finally, the optimized structure's DQE increased substantially, benefiting both from the enhancements in efficiency and MTF.

structures' PSF in
We evaluate the quality of detected images using the MSE for various added noise levels. The detected images are quantized with a fixed number of quanta set by the sensor bit depth and a fixed dynamic range. As a result, detection with lower efficiencies darken the images, which increases the MSE with respect to the original image. Importantly, we find that our optimized design outperforms the bulk phosphor screens on all tested images by achieving the lowest MSE.
Since the multilayer structure affects the emission rate, it also improves the image production timing. Figure 5b shows results for our optimization approach, taking the emission time as the figure of merit, essential for applications such as time-of-flight PET scans. [50] We present the number of detected photons per eV of the incoming electron beam. In our results, we consider an electron beam with 120 eV. We find that our approach can improve the timing in which an image is produced and increase the total number of detected photons.
We now consider the effect of an additive independent uniform noise on the images. The values of the original image are in the range [0, 1]. In Figure 6, we added noises with different standard deviations and show how the noise affect the detection process. As the additive noise increases, the detection of the image deteriorates. The inverse-designed structure leads to more accurate detections, even in the presence of noise. In Supplementary Figure 1, we repeat this result for a sensor with a dynamic range adjusted to the signal amplitude. In that case, the images are not darkened at lower efficiencies, yet their MSE deteriorates relative to the original image.
It is interesting to consider the implications of our approach in the special cases that contain a reflective layer (often Al), as in modern image intensifiers tubes (starting from Generation 3). [20,21] One of the purposes of this layer is to reflect all the backward emission and thus increase the screen's efficiency-as more photons reach the detector. However, due to the spread of light, this increase comes at the price of lowering the screen's resolution. The objective function in Equation 10 can be modified to design an optimized structure with negligible backward emission. This way, the Al layer can be removed, which improves certain aspects of the screen's performance, including its efficiency, resolution, and lifetime. In particular, the applied voltage used in image intensifiers can be reduced. [47]

Discussion
Our work presented specially designed nanophotonic scintillators that control the effective spontaneous emission rate and overcome the fundamental trade-off in today's phosphors/ scintillators. We design these nanostructures and compare our designs to modern phosphor screens in image intensifiers, showing threefold resolution and efficiency enhancements, along with a faster emission. The Purcell effect, which is the key to our study, is already present in a vast range of applications in science and engineering. Yet, our work is the first to demonstrate its application in improving imaging systems. The framework developed in this manuscript applies to scintillation, but more generally to structural optimization in the context of many-body emission problems. Therefore, recently developed topology optimization techniques in similar contexts (e.g., for surface-enhanced Raman scattering [51] ) could also be applied to extend our work in the future.
The choice to present the concept using 1D multilayer structures has several practical advantages. From the fabrication perspective, such structures can be fabricated with existing methods (for example, spin coating, [52] sputtering, [53] and epitaxial growth [54] ). The requirement for thin layers (in the order  of hundreds of nanometers), poses practical challenges on these fabrication methods which introduce noises to the resulting structures. From the design perspective, the emission rate from each point in this structure can be described analytically, using the solution for a dipole embedded in a 3-layer structure, [30] with the effective reflection and transmission coefficients calculated recursively. [18,55] Generalizing this approach to complex geometries of nanophotonic scintillators could be done using electromagnetic reciprocity. [17] A substantial part of the improvement in performance that we showed arises from the fact that we consider aperiodic multilayer structures rather than periodic ones. It has been shown in various disciplines that aperiodic topologies can overcome the limitations posed by periodic arrangements. [56,57] A key advantage of aperiodic structures is their extremely large parameter space to optimize over, which became feasible in recent years thanks to advances in optimization techniques and the ability to describe such structure with a partially analytical formalism. [45] Our methodology can be used for arbitrary emitters, optimized separately for any emission spectrum, material combination, size, and type of incoming energetic particle. Moreover, other optimization algorithms can be used depending on the application's figure of merit and the constraints. Each case is expected to lead to a different optimal structure. What makes our approach versatile is that the final step of the energy con-version process-spontaneous emission-is independent of the type of input radiation (electron beam, X-ray, gamma-ray, electrical current, UV, or visible light). This is why the nanophotonic design only depends on this final step. Nevertheless, future work should investigate additional effects of the multilayer structure on other physical processes involved in the energy conversion (e.g., the creation and diffusion of e-h pairs, scattering and energy loss of incoming particles, etc.). We note that, in general, better performances of the inverse-designed nanophotonic scintillator are reached with a narrower emission spectrum Y(ω) since the local density of photonic states can be concentrated in a narrower spectral range. [58] From a practical point-of-view, manufacturing challenges impede the fabrication of thick nanophotonic scintillators. As a result, applications which require less radiation absorption (and a smaller number of layers) are more plausible to benefit more from inverse design in the short term. Such applications include light-intensifiers, electron cameras, and projectional radiography machines. The approach we developed in this work is applicable beyond radiation detection and is especially promising for enhancing phosphor screens that convert other types of input radiation into UV/visible light. Specifically, phosphor screens are commonly used in light sources such as LEDs, [59] electroluminescent displays, [60] and image intensifiers [4,61] -all now becoming promising candidates for novel enhancements using nanophotonic scintillators/phosphors. Figure 6. The effect of noise on the detected images. Comparison between the bulks and inverse-designed structures in the presence of additive independent uniform noise. The MSE between the detected and the original images deteriorates with increased noise. We note that the images detected with the thin bulk of 3 µm contain more of the features of the original image, compared to the thicker bulk, although being less efficient. The optimal structure achieves the lowest MSE for all levels of noise.