11institutetext:  Korea University, Seoul, South Korea
11email: {sihwapark, iamsjune<EMAIL_ADDRESS>
22institutetext: Korea University Anam Hospital, Seoul, South Korea
22email<EMAIL_ADDRESS>

# 3D Teeth Reconstruction from Panoramic Radiographs using Neural Implicit
Functions

Sihwa Park 11 Seongjun Kim 11 In-Seok Song 22 Seung Jun Baek 1(✉)1(✉)✉✉

###### Abstract

Panoramic radiography is a widely used imaging modality in dental practice and
research. However, it only provides flattened 2D images, which limits the
detailed assessment of dental structures. In this paper, we propose Occudent,
a framework for 3D teeth reconstruction from panoramic radiographs using
neural implicit functions, which, to the best of our knowledge, is the first
work to do so. For a given point in 3D space, the implicit function estimates
whether the point is occupied by a tooth, and thus implicitly determines the
boundaries of 3D tooth shapes. Firstly, Occudent applies multi-label
segmentation to the input panoramic radiograph. Next, tooth shape embeddings
as well as tooth class embeddings are generated from the segmentation outputs,
which are fed to the reconstruction network. A novel module called Conditional
eXcitation (CX) is proposed in order to effectively incorporate the combined
shape and class embeddings into the implicit function. The performance of
Occudent is evaluated using both quantitative and qualitative measures.
Importantly, Occudent is trained and validated with actual panoramic
radiographs as input, distinct from recent works which used synthesized
images. Experiments demonstrate the superiority of Occudent over state-of-the-
art methods.

###### Keywords:

Panoramic radiographs 3D reconstruction Teeth segmentation Neural implicit
function

††footnotetext: Corresponding Author

##  1 Introduction

Panoramic radiography (panoramic X-ray, or PX) is a commonly used technique
for dental examination and diagnosis. While PX produces 2D images from
panoramic scanning, Cone-Beam Computed Tomography (CBCT) is an alternative
imaging modality which provides 3D information on dental, oral, and
maxillofacial structures. Despite providing more comprehensive information
than PX, CBCT is more expensive and exposes patients to a greater dose of
radiation [3]. Thus, 3D teeth reconstruction from PX is of significant value,
e.g., 3D visualization can aid clinicians with dental diagnosis and treatment
planning. Other applications include treatment simulation and interactive
virtual reality for dental education [12].

Previous 3D teeth reconstruction methods from 2D PX have relied on additional
information such as tooth landmarks or tooth crown photographs. For example,
[14] developed a model which uses landmarks on PX images to estimate 3D
parametric models for tooth shapes, while [1] reconstructed a single tooth
using a shape prior and reflectance model based on the corresponding crown
photograph. Recent advances in deep neural networks have significantly
impacted research on 3D teeth reconstruction. X2Teeth [13] performs 3D
reconstruction of the entire set of teeth from PX based on 2D segmentation
using convolutional neural networks. Oral-3D [22] generated 3D oral structures
without supervised segmentation from PX using a GAN model [8]. Yet, those
methods relied on synthesized images as input instead of real-world PX images,
where the synthesized images are obtained from 2D projections of CBCT [27].
The 2D segmentation of teeth from PX is useful for 3D reconstruction in order
to identify and isolate teeth individually. Prior studies on 2D teeth
segmentation [11, 28] focused on binary segmentation determining the presence
of teeth. However, this information alone is insufficient for the construction
of individual teeth. Instead, we leverage recent frameworks [17, 21] on multi-
label segmentation of PX into 32 classes including wisdom teeth.

In this paper, we propose _Occudent_ , an end-to-end model to reconstruct 3D
teeth from 2D PX images. Occudent consists of a multi-label 2D segmentation
followed by 3D teeth reconstruction using _neural implicit functions_ [15].
The function aims to learn the occupancy of dental structures, i.e., whether a
point in space lies within the boundaries of 3D tooth shapes. Learning
implicit functions is computationally advantageous over conventional encoder-
decoder models outputting explicit 3D representations such as voxels, e.g.,
implicit models do not require large memory footprints to store and process
voxels. Considering that 3D tooth shapes are characterized by tooth classes,
we generate embeddings for tooth classes as well as segmented 2D tooth shapes.
The combined class and shape embeddings are infused into the reconstruction
network by a novel module called Conditional eXcitation (CX). CX performs
learnable scaling of occupancy features conditioned on tooth class and shape
embeddings. The performance of Occudent is evaluated with actual PX as input
images, which differs from recent works using synthesized PX images [22, 13].
Experiments show Occudent outperforms state-of-the-art baselines both
quantitatively and qualitatively. The main contributions are summarized as
follows: (1) the first use of a neural implicit function for 3D teeth
reconstruction, (2) novel strategies to inject tooth class and shape
information into implicit functions, (3) the superiority over existing
baselines which is demonstrated with real-world PX images.

##  2 Methods

The proposed model, Occudent, consists of two main components: 2D teeth
segmentation and 3D teeth reconstruction. The former performs the segmentation
of 32 teeth from PX using UNet++ model [29]. The individually segmented tooth
and the tooth class are subsequently passed to the latter for the
reconstruction. The reconstruction process estimates the 3D representation of
the tooth based on a neural implicit function. The overall architecture of
Occudent is depicted in Fig. 1.

Figure 1: (a) The PX image is segmented into 32 teeth classes using UNet++
model. For each tooth, a segmented patch is generated by cropping input PX
with the predicted segmentation mask, which is subsequently encoded via an
image encoder. The tooth class is encoded by an embedding layer. The patch and
class embeddings are added together to produce the condition vector. (b) The
reconstruction process consists of $N$ ResBlocks to compute occupancy features
of points sampled from 3D space. The condition vector from PX is processed by
a Conditional eXcitation (CX) module incorporated in the ResBlocks. (c) The
Conv with CX sub-module is composed of batch normalization, CX, ReLU, and a
convolutional layer. The FC with CX is similar to the Conv with CX where
Conv1D layer is replaced by fully connected layer. (d) CX injects condition
information into the reconstruction network using excitation values. CX uses a
trainable weight matrix to encode the condition vector into an excitation
vector via a gating function. The input feature is scaled using the excitation
vector through component-wise multiplication.

###  2.1 2D Teeth Segmentation

The teeth in input PX are segmented into 32 teeth classes. The 32 classes
correspond to the traditional numbering of teeth, which includes incisors,
canines, premolars and molars in both upper and lower jaws. We pose 2D teeth
segmentation as a _multi-label segmentation_ problem [17], since nearby teeth
can overlap with each other in the PX image, i.e., a single pixel of the input
image can be classified into two or more classes.

The input of the model is $H\times W$ size PX image. The segmentation output
has dimension $C\times H\times W$, where channel dimension $C=33$ represents
the number of tooth classes: one class for the background and 32 classes for
teeth similar to [17]. Hence, the $H\times W$ output at each channel is a
segmentation output for each tooth class. The segmentation outputs are used to
generate tooth patches for reconstruction, which is explained later in detail.
For the segmentation, we adopt pre-trained UNet++ [29] as the base model.
UNet++ is advantageous for medical image segmentation due to its modified skip
pathways, which results in better performance compared to the vanilla UNet
[20].

###  2.2 3D Teeth Reconstruction

####  2.2.1 Neural Implicit Representation.

Typical representations of 3D shapes are point-based [19, 7], voxel-based [4,
26], or mesh-based methods [25]. These methods represent 3D shapes explicitly
through a set of discrete points, vertices, and faces. Recently, implicit
representation methods based on a continuous function which defines the
boundary of 3D shapes have become increasingly popular [16, 15, 18]. Occupancy
Networks [15] is a pioneering work which utilizes neural networks to
approximate the implicit function of an object’s occupancy. The term occupancy
refers to whether a point in space lies in the interior or exterior of object
boundaries. The occupancy function maps a 3D point to either 0 or 1,
indicating the occupancy of the point. Let $o_{\textrm{A}}$ denote the
occupancy function for an object A as follows:

$o_{\textrm{A}}:\rm I\\!R^{3}\rightarrow\\{0,1\\}$ (1)

In practice, $o_{\textrm{A}}$ can be estimated only by a set of observations
of object A, denoted by $\mathcal{X}_{\textrm{A}}$. Examples of observations
are projected images or point cloud data obtained from the object. Our
objective is to estimate the occupancy function conditioned on
$\mathcal{X}_{\textrm{A}}$. Specifically, we would like to find function
$f_{\theta}$ which estimates the occupancy probability of a point in 3D space
based on $\mathcal{X}_{\textrm{A}}$ [15]:

$f_{\theta}:\rm I\\!R^{3}\times\mathcal{X}_{\textrm{A}}\rightarrow[0,1]$ (2)

Inspired by the aforementioned framework, we leverage segmented tooth patch
and tooth class as observations denoted by condition vector $\bm{c}$.
Specifically, the input to the function is a set of $T$ randomly sampled
locations within a unit cube, and the function outputs the occupancy
probability of the input. Thus, the function is given by
$f_{\theta}:(x,y,z,\bm{c})\rightarrow[0,1]$.

The model for $f_{\theta}$ is depicted in Fig. 1 (b). The sampled locations
are projected to 128 dimensional feature vectors using 1D convolution. Next,
the features are processed by a sequence of ResNet blocks followed by FC
(fully connected) layers. Conditional vector $\bm{c}$ is used for each block
through Conditional eXcitation (CX) which we will explain later.

####  2.2.2 Class-specific Conditional Features.

A distinctive feature of the tooth reconstruction task is that teeth with the
same number share properties such as surface and root shapes. Hence, we
propose to use tooth class information in combination with a segmented tooth
patch from PX. The tooth class is processed by a learnable embedding layer
which outputs a class embedding vector.

Next, we create a square patch of the tooth using the segmentation output as
follows. A binary mask of the segmented tooth is generated by applying
thresholding to the segmentation output. A tooth patch is created by cropping
out the tooth region from the input PX, i.e., the binary mask is applied
(bitwise AND) to the input PX to obtain the patch. The segmented tooth patch
is subsequently encoded using a pre-trained ResNet18 model [9], which outputs
a patch embedding vector. The patch and class embeddings are added to yield
the condition vector for the reconstruction model. This process is depicted in
Fig. 1 (a).

Our approach differs from previous approaches, such as Occupancy Networks [15]
which uses only single-view images for 3D reconstruction. X2Teeth [13] also
addresses the task of 3D teeth reconstruction from 2D PX. However, X2Teeth
only uses segmented image features for the reconstruction. By contrast,
Occudent leverages a class-specific encoding method to boost the
reconstruction performance, as demonstrated in ablation analysis in
Supplementary Materials.

####  2.2.3 Conditional eXcitation.

To effectively inject 2D observations into the reconstruction network, we
propose Conditional eXcitation (CX) inspired by Squeeze-and-Excitation Network
(SENet) [10]. In SENet, excitation refers to scaling input features according
to their importance. In Occudent, the concept of excitation is extended to
incorporating conditional features into the network. Firstly, the condition
vector is encoded into excitation vector $\bm{e}$. Next, the excitation is
applied to input feature by scaling the feature components by $\bm{e}$. The CX
procedure can be expressed as:

$\bm{e}=\alpha\cdot\sigma(W\bm{c}),$ (3)
$\bm{y}=F_{\textrm{ext}}(\bm{e},\bm{x})$ (4)

where $\bm{c}$ is the condition vector, $\sigma$ is a gating function, $W$ is
a learnable weight matrix, $\alpha$ is a hyperparameter for the excitation
result, and $F_{\textrm{ext}}$ is the excitation function. We use sigmoid
function for $\sigma$, and component-wise multiplication for the excitation,
$F_{\textrm{ext}}(\bm{e},\bm{x})=\bm{e}$
$\mathbin{\mathchoice{\vbox{\hbox{\scalebox{0.8}{$\displaystyle\otimes$}}}}{\vbox{\hbox{\scalebox{0.8}{$\textstyle\otimes$}}}}{\vbox{\hbox{\scalebox{0.8}{$\scriptstyle\otimes$}}}}{\vbox{\hbox{\scalebox{0.8}{$\scriptscriptstyle\otimes$}}}}}$
$\bm{x}$. The CX module is depicted in Fig. 1 (d). CX differs from SENet in
that CX derives the excitation from the condition vector, whereas SENet
derives it from input features. Our approach also differs from Occupancy
Networks which used Conditional Batch Normalization (CBN) [6, 5] which
combines conditioning with batch normalization. However, the conditioning
process should be independent of input batches because those components serve
different purposes in deep learning models. Thus, we propose to separate
conditioning from batch normalization, as is done by CX.

##  3 Experiments

####  3.0.1 Dataset.

The pre-training of the segmentation model was done with a dataset of 4000 PX
images, sourced from ‘The Open AI Dataset Project (AI-Hub, S. Korea)’. All
data information can be accessed through ‘AI-Hub (www.aihub.or.kr)’. The
dataset consisted of two image sizes, 1976 $\times$ 976 and 2988 $\times$
1468, which were resized to 256 $\times$ 768 to train the UNet++ model.

For the main experiments for reconstruction, we used a set of 39 PX images and
matched CBCT images, obtained from Korea University Anam Hospital. This study
was approved by the Institutional Review Board at Korea University (IRB
number: 2020AN0410). The panoramic radiographs were of dimensions 1536
$\times$ 2860 and were resized to 600 $\times$ 1200 and randomly cropped of
592 $\times$ 1184 size for the segmentation training. The CBCT images were of
size 768 $\times$ 768 $\times$ 576, capturing cranial bones. The teeth labels
for 2D PX and CBCT were manually annotated by two experienced annotators and
subsequently verified by a board-certified dentist. To train and evaluate the
model, the dataset was partitioned into training (30 cases), validation (2
cases), and testing (7 cases) subsets.

####  3.0.2 Implementation Details.

For the pre-training of the segmentation model, we utilized a combination of
cross-entropy and dice loss. For the main segmentation training, we used only
dice loss. The segmentation and reconstruction models were trained separately.
Following the completion of the segmentation model training, we fixed this
model to predict its output for the reconstruction model.

Each 3D tooth label was fit in 144 $\times$ 80 $\times$ 80 size tensor which
was then regarded as $[-0.5,0.5]^{3}$ normalized cube in 3D space. For the
training of the neural implicit function, a set of $T=100,000$ points was
sampled from the unit cube. The preprocessing was consistent with that used in
[23]. We trained all the other baseline models with these normalized cubes.
For example, for 3D-R2N2 [4], we voxelized the cube to $128^{3}$ size. For a
fair comparison, the final meshes produced by each model were extracted and
compared using four different metrics. The detailed configuration of our model
is provided in Supplementary Materials.

####  3.0.3 Baselines.

We considered several state-of-the-art models as baselines, including 3D-R2N2
[4], DeepRetrieval [24, 13], Pix2Vox [26], PSGN [7], Occupancy Networks
(OccNet) [15], and X2Teeth [13]. To adapt the 3D-R2N2 model to single-view
reconstruction, we removed its LSTM component, following the approach in [15].
As for the DeepRetrieval method, we employed the same encoder architecture as
3D-R2N2, and utilized the encoded feature vector of the test image query.
Subsequently, we compared each encoded vector from the test image to the
encoded vectors from the training set, and retrieved the tooth with the
minimum Euclidean distance of encoded vectors from the training set.

Table 1: Comparison with baseline methods. The format of results is _mean_ $\pm$_std_ obtained from 10 repetitions of experiments. Method | IoU | Chamfer-$L_{1}$ | NC | Precision
---|---|---|---|---
3D-R2N2 | 0.585$\pm$0.005 | 0.382$\pm$0.008 | 0.617$\pm$0.009 | 0.634$\pm$0.010
PSGN | 0.606$\pm$0.015 | 0.342$\pm$0.016 | 0.829$\pm$0.012 | 0.737$\pm$0.018
Pix2Vox | 0.562$\pm$0.005 | 0.388$\pm$0.008 | 0.599$\pm$0.007 | 0.664$\pm$0.009
DeepRetrieval | 0.564$\pm$0.005 | 0.394$\pm$0.006 | 0.824$\pm$0.003 | 0.696$\pm$0.005
X2Teeth | 0.592$\pm$0.006 | 0.361$\pm$0.017 | 0.618$\pm$0.002 | 0.670$\pm$0.009
OccNet | 0.611$\pm$0.006 | 0.353$\pm$0.008 | 0.872$\pm$0.003 | 0.691$\pm$0.011
Occudent (Ours) | 0.651$\pm$0.004 | 0.298$\pm$0.006 | 0.890$\pm$0.001 | 0.739$\pm$0.008

####  3.0.4 Evaluation Metrics.

The evaluation of the proposed method was conducted using the following
metrics: volumetric Intersection over Union (IoU), Chamfer-$L_{1}$ distance,
and Normal Consistency (NC), as outlined in prior work [15]. In addition, we
used volumetric precision [13] as a metric given by $|D\cap G|/|D|$ where $G$
denotes the ground-truth set of points occupied by the object, and $D$ denotes
the set of points predicted as the object.

####  3.0.5 Quantitative Comparison.

Table 1 presents a quantitative comparison of the proposed model with several
baseline models. The results demonstrate that Occudent surpasses the other
methods across all the metrics, and the methods based on neural implicit
functions (Occudent and OccNet) perform better compared to conventional
encoder-decoder approaches, such as Pix2Vox and 3D-R2N2. The performance gap
between Occudent and X2Teeth is presumably because real PX images are used as
input data. X2Teeth used _synthesized_ images generated from the 2D
projections of CBCT in [27]. Thus, both the input 2D shape and the target 3D
shape come from the same modality (CBCT). However, the distribution of real PX
images may differ significantly from that of 2D-projected CBCT. Explicit
methods can be more sensitive to such differences than implicit methods,
because typically in explicit methods, input features are directly encoded and
subsequently decoded to predict the target shapes [4, 26, 13]. Overall, the
differences in the IoU performances among the baselines are somewhat small.
This is because all the baselines are moderately successful in generating
coarse tooth shapes. However, Occudent is significantly better at generating
details such as root shapes, which will be shown in the subsequent section.

Figure 2: Visual representation of sample outputs. Boxes are used to highlight
the incisors and molars in the upper jaw.

####  3.0.6 Qualitative Comparison.

Fig. 2A illustrates the qualitative results of the proposed method in
generating 3D teeth mesh outputs. From our model, each tooth is generated, and
generated teeth are combined along with an arch curve based on a beta function
[2]. Fig. 2A demonstrates that our proposed method generates the most similar-
looking outputs compared to the ground truth. For instance, our model can
reconstruct a plausible shape for all tooth types including detailed shapes of
molar roots. 3D-R2N2 produces larger and less detailed tooth shapes. PSGN and
OccNet are better at generating rough shapes than 3D-R2N2, however, lack in
detailed root shapes.

As illustrated in Fig. 2B, Occudent produces a more refined mesh of tooth
shape representation than voxel-based methods like 3D-R2N2 or X2Teeth. One of
the limitations of voxel-based methods is that they heavily depend on the
resolution of the output. For example, increasing the output size leads to an
exponential increase in model size. By contrast, Occudent employs continuous
neural implicit functions to represent shape boundaries, which enables us to
generate smoother output and to be robust to the target size.

##  4 Conclusion

In this paper, we present a framework for 3D teeth reconstruction from a
single PX. To the best of our knowledge, our method is the first to utilize a
neural implicit function for 3D teeth reconstruction. The performance of our
proposed framework is evaluated quantitatively and qualitatively,
demonstrating its superiority over state-of-the-art techniques. Importantly,
our framework is capable of accommodating two distinct modalities, PX, and
CBCT. Our framework has the potential to be valuable in clinical practice and
also can support virtual simulation or educational tools. In the future,
further improvements can be made, such as incorporating additional imaging
modalities or exploring neural architectures for more robust reconstruction.

Acknowledgements. This work was supported by the Korea Medical Device
Development Fund grant funded by the Korea Government (the Ministry of Science
and ICT, the Ministry of Trade, Industry and Energy, the Ministry of Health &
Welfare, the Ministry of Food and Drug Safety) (Project Number: 1711195279,
RS-2021-KD000009); the National Research Foundation of Korea (NRF) Grant
through the Ministry of Science and ICT (MSIT), Korea Government, under Grant
2022R1A5A1027646; the National Research Foundation of Korea (NRF) grant funded
by the Korea government (MSIT) (No. 2021R1A2C1007215); the MSIT, Korea, under
the ICT Creative Consilience program (IITP-2023-2020-0-01819) supervised by
the IITP (Institute for Information & communications Technology Planning &
Evaluation)

## References

  * [1] Abdelrehim, A.S., Farag, A.A., Shalaby, A.M., El-Melegy, M.T.: 2d-pca shape models: Application to 3d reconstruction of the human teeth from a single image. In: Medical Computer Vision. Large Data in Medical Imaging: Third International MICCAI Workshop, MCV 2013, Nagoya, Japan, September 26, 2013, Revised Selected Papers 3. pp. 44–52. Springer (2014)
  * [2] Braun, S., Hnat, W.P., Fender, D.E., Legan, H.L.: The form of the human dental arch. The Angle Orthodontist 68(1), 29–36 (1998)
  * [3] Brooks, S.L.: Cbct dosimetry: orthodontic considerations. In: Seminars in Orthodontics. vol. 15, pp. 14–18. Elsevier (2009)
  * [4] Choy, C.B., Xu, D., Gwak, J., Chen, K., Savarese, S.: 3d-r2n2: A unified approach for single and multi-view 3d object reconstruction. In: Proceedings of the European Conference on Computer Vision (ECCV) (2016)
  * [5] De Vries, H., Strub, F., Mary, J., Larochelle, H., Pietquin, O., Courville, A.C.: Modulating early visual processing by language. Advances in Neural Information Processing Systems 30 (2017)
  * [6] Dumoulin, V., Belghazi, I., Poole, B., Mastropietro, O., Lamb, A., Arjovsky, M., Courville, A.: Adversarially learned inference. arXiv preprint arXiv:1606.00704 (2016)
  * [7] Fan, H., Su, H., Guibas, L.J.: A point set generation network for 3d object reconstruction from a single image. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 605–613 (2017)
  * [8] Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial networks. Communications of the ACM 63(11), 139–144 (2020)
  * [9] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 770–778 (2016)
  * [10] Hu, J., Shen, L., Sun, G.: Squeeze-and-excitation networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 7132–7141 (2018)
  * [11] Koch, T.L., Perslev, M., Igel, C., Brandt, S.S.: Accurate segmentation of dental panoramic radiographs with u-nets. In: 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019). pp. 15–19. IEEE (2019)
  * [12] Li, Y., Ye, H., Ye, F., Liu, Y., Lv, L., Zhang, P., Zhang, X., Zhou, Y.: The current situation and future prospects of simulators in dental education. Journal of Medical Internet Research 23(4), e23635 (2021)
  * [13] Liang, Y., Song, W., Yang, J., Qiu, L., Wang, K., He, L.: X2teeth: 3d teeth reconstruction from a single panoramic radiograph. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd International Conference, Lima, Peru, October 4–8, 2020, Proceedings, Part II 23. pp. 400–409. Springer (2020)
  * [14] Mazzotta, L., Cozzani, M., Razionale, A., Mutinelli, S., Castaldo, A., Silvestrini-Biavati, A.: From 2d to 3d: Construction of a 3d parametric model for detection of dental roots shape and position from a panoramic radiograph—a preliminary report. International journal of dentistry 2013 (2013)
  * [15] Mescheder, L., Oechsle, M., Niemeyer, M., Nowozin, S., Geiger, A.: Occupancy networks: Learning 3d reconstruction in function space. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 4460–4470 (2019)
  * [16] Mildenhall, B., Srinivasan, P.P., Tancik, M., Barron, J.T., Ramamoorthi, R., Ng, R.: Nerf: Representing scenes as neural radiance fields for view synthesis. In: ECCV (2020)
  * [17] Nader, R., Smorodin, A., De La Fourniere, N., Amouriq, Y., Autrusseau, F.: Automatic teeth segmentation on panoramic x-rays using deep neural networks. In: 2022 26th International Conference on Pattern Recognition (ICPR). pp. 4299–4305. IEEE (2022)
  * [18] Park, J.J., Florence, P., Straub, J., Newcombe, R., Lovegrove, S.: Deepsdf: Learning continuous signed distance functions for shape representation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (June 2019)
  * [19] Qi, C.R., Su, H., Mo, K., Guibas, L.J.: Pointnet: Deep learning on point sets for 3d classification and segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 652–660 (2017)
  * [20] Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation. In: Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18. pp. 234–241. Springer (2015)
  * [21] Silva, B., Pinheiro, L., Oliveira, L., Pithon, M.: A study on tooth segmentation and numbering using end-to-end deep neural networks. In: 2020 33rd SIBGRAPI Conference on Graphics, Patterns and Images (SIBGRAPI). pp. 164–171. IEEE (2020)
  * [22] Song, W., Liang, Y., Yang, J., Wang, K., He, L.: Oral-3d: Reconstructing the 3d structure of oral cavity from panoramic x-ray. In: Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2-9, 2021. pp. 566–573. AAAI Press (2021), https://ojs.aaai.org/index.php/AAAI/article/view/16135
  * [23] Stutz, D., Geiger, A.: Learning 3d shape completion under weak supervision. CoRR abs/1805.07290 (2018), http://arxiv.org/abs/1805.07290
  * [24] Tatarchenko, M., Richter, S.R., Ranftl, R., Li, Z., Koltun, V., Brox, T.: What do single-view 3d reconstruction networks learn? In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 3405–3414 (2019)
  * [25] Wang, N., Zhang, Y., Li, Z., Fu, Y., Liu, W., Jiang, Y.G.: Pixel2mesh: Generating 3d mesh models from single rgb images. In: Proceedings of the European conference on computer vision (ECCV). pp. 52–67 (2018)
  * [26] Xie, H., Yao, H., Sun, X., Zhou, S., Zhang, S.: Pix2vox: Context-aware 3d reconstruction from single and multi-view images. In: Proceedings of the IEEE/CVF international conference on computer vision. pp. 2690–2698 (2019)
  * [27] Yun, Z., Yang, S., Huang, E., Zhao, L., Yang, W., Feng, Q.: Automatic reconstruction method for high-contrast panoramic image from dental cone-beam ct data. Computer methods and programs in biomedicine 175, 205–214 (2019)
  * [28] Zhao, Y., Li, P., Gao, C., Liu, Y., Chen, Q., Yang, F., Meng, D.: Tsasnet: Tooth segmentation on dental panoramic x-ray images by two-stage attention segmentation network. Knowledge-Based Systems 206, 106338 (2020)
  * [29] Zhou, Z., Siddiquee, M.M.R., Tajbakhsh, N., Liang, J.: Unet++: A nested u-net architecture for medical image segmentation. In: Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support, pp. 3–11. Springer (2018)

## Model Configuration

Table S1: Detailed model configuration of Occudent. # of ResBlocks with CX | 5
---|---
Hidden feature size | 128
Condition vector dimension | 128
Optimizer | Adam optimizaer
Learning rate | 1e-4
Batch size | 10
Training epochs | 200 to 300
Usage of early stopping method | Yes
GPU for pre-training | NVIDIA Tesla V100
GPU for main experiments | NVIDIA A100 and NVIDIA RTX A6000

## Ablation Study

Table S2: Ablation study on Conditional eXcitation (CX) and tooth class embeddings. We demonstrate the effectiveness of two components of our model: CX and the inclusion of tooth class embeddings. We compare CX with the case where Conditional Batch Normalization (CBN) [6, 5] is used instead, as in the Occupancy Network [15]. A comparison of the 1st and 2nd rows and the 3rd and 4th rows shows that CX outperforms CBN irrespective of using tooth class embeddings. By comparing the 1st and 3rd rows and the 2nd and 4th rows, we observe that incorporating tooth class embeddings into reconstruction improves the model’s performance. | IoU | Chamfer-$L_{1}$ | NC | Precision
---|---|---|---|---
CBN Only | 0.611$\pm$0.006 | 0.353$\pm$0.008 | 0.872$\pm$0.003 | 0.691$\pm$0.011
CX Only | 0.621$\pm$0.003 | 0.336$\pm$0.005 | 0.875$\pm$0.001 | 0.701$\pm$0.006
CBN+Tooth Class | 0.641$\pm$0.010 | 0.311$\pm$0.013 | 0.886$\pm$0.005 | 0.728$\pm$0.011
CX+Tooth Class | 0.651$\pm$0.004 | 0.298$\pm$0.006 | 0.890$\pm$0.001 | 0.739$\pm$0.008


# Alignment-Free Coupling to Arrays of Diamond Microdisk Cavities for Scalable
Spin-Photon Interfaces

Helaman R. Flores  1 Samuel R. Layton  2  Dirk Englund  3 and Ryan M. Camacho1
1Department of Electrical and Computer Engineering, Brigham Young University,
450 EB, Provo, UT 84604, USA
2Department of Mathematics, Brigham Young University, 275 TMCB, Provo, UT
84604, USA
3 Department of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We propose a scalable design for a spin-photon interface to a color center in
a diamond microdisk. The design consists of a silicon oxynitride hexagonal
lattice overlaid on a diamond microdisk to enable vertical emission from the
microdisk into low-numerical aperture modes, with quantum efficiencies as high
as 45% for a tin vacancy (SnV) center. Our design is robust to manufacturing
errors, potentially enabling large scale fabrication of quantum emitters
coupled to optical collection modes. We also introduce a novel approach for
optimizing the free space performance of a complex structure using a dipole
model, achieving comparable results to full-wave finite difference time domain
simulations with a 650,000 times reduction in computational time.

##  1 Introduction

Efficiently interfacing optical fields with numerous quantum memories is a
fundamental requirement for quantum networks. Solid-state systems, especially
color centers in diamond, have emerged as promising candidates for such
interfaces. They are well suited for building nodes in quantum repeater
networks due to their ability to transmit quantum information via spin-
entangled photons [1, 2, 3]. However, a significant limitation of these color
centers, including Nitrogen-Vacancy (NV), Silicon-Vacancy (SiV), and Tin
Vacancy (SnV) centers, is their suboptimal emission efficiency into preferred
spatial and spectral modes, particularly into the zero-phonon line (ZPL) [4,
5]. Enhancing the photonic emission of color centers in the ZPL and
effectively coupling these photons into network channels (such as fibers or
free space) is crucial for improving entanglement generation rates, a current
bottleneck in quantum network performance [5].

For spin-photon interfaces in such systems, the figure of merit, denoted as
$\eta$, represents the probability that an excited state of the emitter leads
to a detected photon in the ZPL. This efficiency, $\eta$, can be
conceptualized as a product of two efficiencies: $\eta_{ZPL}$, the proportion
of emissions into the ZPL, and $\eta_{col}$, the collection efficiency of
these emissions into the desired mode [4]. Optimizing both these components is
vital for maximizing $\eta$, thereby enhancing the interface efficiency
essential for scalable and efficient quantum networks.

We define the figure of merit $\eta$ as the probability that an excited state
of the emitter results in a photon detected in the zero-phonon-line (ZPL),

$\displaystyle\eta=\eta_{ZPL}\eta_{col},$ (1)

where $\eta_{ZPL}$ is the fraction of emission of the color center into the
ZPL [6], and $\eta_{col}$ represents the fraction of emission coupled into the
desired output mode [7]. The factor $\eta_{ZPL}$ can be determined using the
Purcell enhancement $F$ of the device at the targeted wavelength, and depends
on the emission ratios $\Gamma_{\textrm{total},0}$ and
$\Gamma_{\textrm{ZPL},0}$, which represent the total unmodified emission rate
and ZPL emission rate of the chosen color center, respectively [4]:

$\displaystyle\eta_{ZPL}=\frac{F}{F+(\frac{\Gamma_{\textrm{total},0}}{\Gamma_{\textrm{ZPL},0}}-1)}.$
(2)

The parameter $\eta_{col}$ is defined as the radial power in the far field
collected into numerical apertures (NA) less than 0.7. Prior research has
demonstrated that both $\eta_{ZPL}$ and $\eta_{col}$ can be simultaneously
enhanced by positioning a color center in specially designed optical cavities
[7, 4]. For silicon vacancy (SiV) centers, simulations predict free-space
collection quantum efficiencies up to 41% [4]. However, fabricating such
devices necessitates intricate diamond etching, a process challenged by
diamond’s material properties. Empirical evaluations of individually designed
devices have shown average quantum efficiencies up to 5.5% for nitrogen
vacancy centers [7]. Recent studies suggest that SnV centers may be more
advantageous for quantum networking applications [8, 9, 10, 11, 12], with
$\Gamma_{\textrm{total},0}/\Gamma_{\textrm{ZPL},0}-1\approx 0.25$, and are
thus employed for computations in this work.

Here, we introduce a design for the vertical emission of spin-polarization
entangled photons from diamond color centers, achieving quantum efficiencies
up to 45% for a SnV center. This approach mitigates the precision requirements
for diamond etching, leading to a design that is more scalable and tolerant of
manufacturing variances.

Additionally, we present a numerically efficient optimization strategy for
this multilayered diamond emission device. Our approach yields results
comparable to those from a comprehensive full-wave finite difference time
domain (FDTD) parameter sweep, which can be computationally intensive,
particularly for resonant structures where electric fields remain trapped in
the cavity for extended durations [13]. We propose a novel method for
designing emitter devices that uses a simplified dipole representation of
scattering from a near-field perturbation [4, 14], coupled with a single
control simulation performed using FDTD. This method facilitates rapid
optimization of key parameters in complex designs, and may be applicable to
photonic device design more generally.

##  2 Design

Microdisk resonators have high quality factors, and thus provide the Purcell
enhancement necessary for $\eta_{ZPL}$ to approach unity. However, radiation
loss is dominated by radial emission in the plane of the disk. In order to
achieve out-of-plane emission in the vertical direction, and thus high
$\eta_{col}$, our design builds on previous work showing that carefully placed
perturbations in the near field mode of the microdisk can scatter photons
vertically instead of radially. The difficulty of accurately etching the
necessary patterns into diamond (n=2.4) prompts the exploration of alternative
device designs and materials [15]. To avoid etching the diamond more than
necessary to create resonance, we propose a design in which a grating layer of
silicon oxynitride (Si3OxNy) is etched with a triangular lattice of holes and
then placed on top of an array of fabricated diamond microdisks, as shown in
1. A bulk silicon dioxide glass substrate (n=1.4) supports the grating layer,
also shown in Figure 1. The lattice holes scatter light from the microdisk
mode so as to constructively interfere within the target numerical aperture.

Figure 1: Design concept. (a) Perspective from above disk, with a 60°cutout to
show the cross-section and bulk oxide removed. The purple mark represents the
point dipole source used to excite the disk mode. (b) Coordinate system for
the far field emission analysis (Eqs 9-10). A dipole location is highlighted
in yellow, and a point in the far field in black. (c) $x-z$ cross-section of a
microdisk array couple to numerical apertures. Disks would most likely be
fabricated with an undercut [15], part of which is included in the FDTD
simulations. (d) Microdisk with design parameters labeled.

One pivotal aspect of our design is the capacity to overlay the grating layer
onto the microdisk while accommodating reasonable alignment tolerances between
the grating holes and the microdisk. This alignment is defined by a vector
$(u,v)$ in the $x-y$ plane, signifying the relative position of a lattice
hole’s center to the disk’s center. Due to the hexagonal symmetry of the
lattice, we can constrain the domains of $u$ and $v$ to $0\leq u\leq a/4$ and
$0\leq v\leq u\tan(\pi/6)$, as illustrated in Fig. 2.

Figure 2: Hexagonal lattice unit cell. The blue symmetry lines identify a
reduced symmetry area for $u$ and $v$, highlighted in pink. Points A and B are
key alignment points ($u,v$) exhibiting high radial symmetry.

Additionally, fabrication variances will inevitably lead to disparities in the
actualized devices. Consequently, we aim for a design that exhibits robustness
to these variations, focusing particularly on parameters such as $a$, $r_{h}$,
$t$, and $(u,v)$. In the design proposed, the utilization of a large microdisk
array against a singular triangular lattice facilitates vertical free-space
emission, even without precise alignment of the layers.

To narrow the parameter space for the lattice constant $a$, we compare the
overlap between a given lattice and the mode of the microdisk. In a triangular
lattice, the perimeter of the $n$th regular hexagon centered on any lattice
point will intersect with $6n$ other lattice points, as shown in Fig. 3. If
the center of the microdisk is aligned with a lattice point and radius of the
disk $r_{d}$ and the lattice constant $a$ are chosen properly, the disk mode
will strongly overlap with one of these hexagons.

Based on previous work demonstrating that perturbations in a microdisk
structure can be modeled as an array of dipole scatterers [4, 14],
constructive interference into lower numerical apertures will occur when the
number of equally spaced azimuthal scatterers is close to the azimuthal mode
number $m$ of the disk [4, 16]. We choose a disk mode of $m=18$ to couple to
hexagonal trace 3 in a triangular lattice, shown in Fig. 3. This choice has
the distinct advantage of a having a high ratio of equidistant holes from the
center (2/3) in the target hexagonal trace, creating a consistent overlap with
the concentrated electric field at the edge of the disk when $u,v<<a$. For the
$m=18$ mode, $r_{d}$ is also large enough to maintain a high Purcell
enhancement.

Figure 3:  (a) Expanding hexagonal traces from a center lattice point in a
triangular lattice. For $n=3$, 18 lattice holes form the perimeter of the
hexagon. Two-thirds (12) of these holes are equidistant from the center, and
the circumference of the microdisk is circled in red. (b) Overlap between the
lattice holes and the optical mode of the bare microdisk, with the
circumference circled in red. The mode is calculated in FDTD without the
lattice etched in the grating layer. Holes are highlighted in proportion to
the electric field magnitude incident at their center.

##  3 Simulation and Modeling

To decrease simulation time, we apply a dipole model as an efficient
alternative to computationally expensive finite-difference time-domain (FDTD)
parameter sweeps. This approach is grounded in existing research indicating
that refractive index variations in the vicinity of a microdisk’s near field
can effectively be represented as dipole scatterers [4, 14].

The emission characteristics of the structure are derived through far-field
projections from electromagnetic currents captured on a two-dimensional
surface $s$. These projections are facilitated by calculating the magnetic
($A$) and electric ($F$) vector potentials, derived from the surface electric
($J_{s}$) and magnetic ($M_{s}$) currents [17]:

$A(R)=\mu\oint_{s}J_{s}(r_{c})\frac{e^{-jk|r-r_{c}|}}{4\pi|r-r_{c}|}ds^{\prime}$
(3)

$F(R)=\epsilon\oint_{s}M_{s}(r_{c})\frac{e^{-jk|r-r_{c}|}}{4\pi|r-r_{c}|}ds^{\prime}.$
(4)

where $r$ is the location of the point in the far field and $r_{c}$ is the
location of the currents.

The electric far field $E_{ff}$, magnetic far field field $H_{ff}$ and far
field poynting vector $S_{ff}$ may then be calculated as [17]

$E_{ff}=-j\omega[A+\frac{1}{k^{2}}\nabla(\nabla\cdot
A)]-\frac{1}{\epsilon}\nabla\times F,$ (5)

$H_{ff}=-j\omega[F+\frac{1}{k^{2}}\nabla(\nabla\cdot
F)]-\frac{1}{\mu}\nabla\times A,$ (6)

$S_{ff}=E_{FF}\times H_{ff},$ (7)

where $\mu$ and $\epsilon$ represent the permeability and the permittivity of
the medium, respectively. The angular frequency $\omega$ and wave vector $k$
are calculated based on the ZPL emission wavelength.

We now examine the dipole approximation to this equation. The fields collected
by our FDTD simulations will appear to have been sourced by dipoles centered
on refractive-index perturbations. This is because holes in the grating layer
cause large scattering in comparison to the natural losses of a microdisk with
high a quality factor. Approximating each hole as a Hertzian dipole source of
length $l$ leads to a magnetic vector potential of an array of Hertzian
dipoles [18]:

$A(R)=\frac{\mu}{4\pi}l\sum\frac{e^{-jk|r-r^{\prime}_{n}|}}{|r-r^{\prime}_{n}|}[I_{nx}\hat{x}+I_{ny}\hat{y}+I_{nz}\hat{z}],$
(8)

where $r_{n}$ is the position of a point in the far field and $r^{\prime}_{n}$
is the position of the dipole. In this equation, each dipole is represented as
a sum of three excitation currents along Cartesian directions. The dipole
excitation current $I_{nx,ny,nz}$ is a coefficient proportional to the complex
electric field incident at the location of the hole in the lattice, which can
be obtained from an FDTD simulation. However, obtaining a specific near-field
for every possible lattice constant and alignment is computationally
expensive. We therefore propose a simplified model in which the electric field
for all possible lattice parameters is approximated by a single near-field
calculation of the microdisk in which no lattice holes are present. To
calculate the far-field profile of a particular lattice geometry, we find the
phase and amplitude for $I_{nx,ny,nz}$ in the simplified near field at the
center of each lattice hole. The corresponding far field for the layered
microdisk structure can then be modeled by

$\displaystyle E_{ff}=\frac{\eta lk}{4\pi
j}\sum_{n=0}^{N}\frac{e^{-jk|r-r^{\prime}_{n}|}}{|r-r^{\prime}_{n}|}[\hat{\theta}(I_{nx}cos(\theta)cos(\phi)$
(9) $\displaystyle+I_{ny}cos(\theta)sin(\phi)-I_{nz}sin(\theta))$
$\displaystyle+\hat{\phi}(-I_{nx}sin(\phi)+I_{ny}cos(\phi))]$

$S_{ff}=\frac{|E_{ff}|^{2}}{\eta}\hat{r},$ (10)

where $\theta$ and $\phi$ are defined as the angles between the far field
point and the origin, and $\eta$ is the impedance of light in glass. We assume
the dipole length $l$ is the same for every lattice hole, and the wave vector
$k$ is the spatial frequency of light in the glass substrate.

In FDTD and with the simplified dipole model, we sum up the poynting vector
over a sphere centered on the origin with a large enough radius such that
$r<<r^{\prime}$. For the dipole model, we sum the power collected in the glass
by a numerical aperture of 0.7 and normalize it by the total power passing
through the sphere in order to obtain $\eta_{col}$. In FDTD, the surface
currents used in the far-field projection must be evaluated in a homogeneous
medium. To apply this appropriately to our layered microdisk structure, we
project the surface currents collected by monitors placed only in the glass
substrate. This will capture light escaping from the edge of the microdisk to
around $\theta=70\degree$. We then normalize the resulting far-field
projection by the ratio of the power through the far field monitors to the
total power that exits the simulation. The FDTD near and far fields are
calculated using commerical software[19], and Eqs. 9-10 of the dipole model
are implemented in Python.

We note that the simplified dipole model matches more accurately with emission
profile of FDTD simulations when we assume that $I_{nz}=0$. This may be
attributed to the fact that, owing to boundary conditions, light traveling in
the plane of the lattice and polarized in the out-of-plane $\hat{z}$ direction
will not experience a discontinuity anywhere along the boundary of a lattice
hole, leading to reduced scattering relative to the in-plane components of the
near field.

A single emission pattern can be calculated using the simplified dipole model
in 3.6 s on one core of a Intel® Core™ i7-10750H processor. In contrast,
evaluating a single emission pattern in FDTD requires 5.1 hours on a cluster
of 128 cores of AMD EPYC 7763 processors. The simplified model may therefore
be used to search a much larger parameter space to identify optimal emission
patterns. FDTD simulations may then be run to verify the pattern and optimize
if necessary. The simplified dipole model predicts an optical lattice constant
$a=0.524/\lambda$, which is consistent with FDTD simulations, as shown in Fig.
5.

Table 1 shows the parameters used to obtain the optimal far-field emission.
With a Purcell enhancement of 52.6 and $\eta_{col}=0.47$, we find an upper
$\eta$ of 46% for a SnV center. We also list in Table 2 the fraction of light
naturally emitted into the ZPL for various color centers and the associated
spectral efficiency $\eta_{ZPL}$.

Parameters | $a$ | $d$ | $r_{h}$ | $r_{d}$ | $t$ | $r_{u}$ | $u$ | $v$
---|---|---|---|---|---|---|---|---
Units of $\lambda_{0}$ | $0.5168$ | $0.2931$ | $0.2$ | $1.5427$ | $0.9411$ | $1.45$ | $0$ | $0$
Table 1: Parameters of the optimized layered microdisk obtained from FDTD Color Center | NV | SiV | SnV
---|---|---|---
Estimated ZPL Emission | 0.03 | 0.7 | 0.8
Spectral Efficiency $\eta_{ZPL}$ | 0.62 | 0.99 | 0.995
Table 2: Spectral Efficiency for various color centers in diamond.

The best correspondence between the dipole scattering model and FDTD
simulation occurrs when the center of the disk is aligned over symmetry points
A and B on the lattice, as shown in Fig. 4. The correspondence between the
calculated far-field emission patterns of the two models is reduced when the
alignment is not near one of these symmetry points. This may be because each
simulation in the dipole model uses the near field mode of a bare disk with no
grating to calculate $I_{nx,ny,nz}$, which is then used in Eq. 9 to calculate
the far field. When the lattice and the microdisk are not aligned at a high-
symmetry point, the non-uniformites in the near-field caused by the lattice
holes make this approximation less valid. Nonetheless, we find by numerical
experimentation that optimization near the high symmetry points using the
simplified dipole model is sufficient to optimize the far-field verified by
FDTD even away from these high-symmetry points.

Another physical effect missing from the dipole model is the consequence of
the refractive index differences between the layers. The difference between
the diamond microdisk and the silicon dioxide lattice is much less than the
difference between the disk and the air below it, leading to preferential
emission into the bulk silicon dioxide. To include this effect in our model,
we phenomenologically multiply all calculated far-field emission patterns for
a given geometry by a constant parameter $\alpha$. Because the monitors in
FDTD simulation are placed to capture up to a 70°cone of light exiting the
simulation region from the microdisk, we perform a least-squares fit between
the dipole model and the FDTD model up to 70°to find $\alpha$, which has a
root mean squared error of $0.05$ and $0.03$ for symmetry points A and B
respectively.

Figure 4:  Comparison of emission of the optimized layered microdisk using
FDTD and the simplified dipole model. Near fields are displayed inside the
grating layer, and the far field in the upper hemisphere is depicted for each
model. a) High-symmetry points A and B in a hexagonal lattice. b) FDTD
simulation omitting the etch in the grating layer. The dashed red line depicts
the boundary of the microdisk. The near field inside of the grating layer is
used to find $I_{nx,ny,nz}$ in the dipole model. Without scatterers, the
farfield is directed radially away from the microdisk. c) FDTD simulation with
the disk center aligned with point A on the lattice. FDTD near field and
resulting far field. d) Geometry of the dipole model with the center of the
disk aligned on point A. Dipole near field and resulting far field. e) FDTD
simulation with the disk center aligned over point B on the lattice. FDTD near
field and resulting far field. f) Geometry of the dipole model with the center
of the disk aligned on point B. Dipole near field and resulting far field. g)
Comparison of collection efficiency ($\eta_{col}$) as the target numerical
aperture increases for alignment A according to FDTD simulation and the dipole
model h) Comparison of collection for alignment B.

It is notable that the dipole model demonstrates reasonable accuracy even
without accounting for variations in parameters such as disk thickness $t$,
hole radius $r_{h}$, lattice depth $d$, and disk radius $r_{d}$. As noted
above, the model’s primary utility is in rapidly searching for optimal values
of the lattice constant $a$. Initialization of the dipole model requires just
one FDTD simulation to calculate the microdisk’s mode, influenced by $d$,
$r_{d}$, and $t$. This helps isolate the lattice constant as the key variable
for optimization. It accurately identifies the optimal lattice constant with a
precision of approximately 1.5%, demonstrating the model’s potential in
streamlining the design process for complex photonic structures like the
layered diamond microdisk resonator.

To demonstrate the robustness of the optimized design to fabrication
variations, we swept parameters such as lattice constant, hole radius, and
disk thickness and monitored the decrease in efficiency. Statistical sampling
of 205 potential random variations indicates that the design consistently
yields high performance, as shown in Figure 5. A significant percentage of
samples ( 5%) maintain far field efficiencies above 40%, and the average
Purcell enhancement is 49.5. Experimentally, many color centers can be
simultaenously implanted in a diamond sample and only those with desirable
spectral properties may be addressed, which may increase the yield to near
unity for this design.

Figure 5 illustrates the robustness to manufacturing variations, showing the
impact of different parameters. Panel (a) presents results from FDTD
simulations for disk thickness $t$, while panel (b) shows the effects of
variations in hole radius $r_{h}$. Panel (c) compares the optimizations of the
lattice constant $a$ from both the dipole model and FDTD simulations. Panel
(d) shows the cumulative fraction of samples exceeding a certain $\eta_{col}$
threshold for 205 random $(u,v)$ alignments, as sampled from FDTD simulations.

Figure 5:  Manufacturing robustness according to the following parameters: (a)
Disk thickness $t$ from FDTD simulation. (b) Hole radius $r_{h}$ from FDTD.
(c) Lattice constant $a$, with results from the dipole model plotted alongside
FDTD. Peak performance is denoted by a dot. (d) Cumulative fraction of samples
above $\eta_{col}$ for random $(u,v)$ alignments sampled from FDTD.

##  4 Discussion

The layered microdisk resonator presented here achieves a high coupling
efficiency ($\eta_{col}$) of 0.46, attributed to the oxynitride grating layer
that enhances vertical emission. This layer directs most far-field emission
into the upper hemisphere. However, the collection efficiency ($\eta_{col}$)
is capped due to constructive interference into higher numerical apertures, a
consequence of the symmetric lattice structure. Optimizing hole locations in
the nitride layer could enhance lower numerical aperture interference, but
would require precise fabrication alignment.

Our focus on light collection into a specific numerical aperture, like a
cryogenic objective, might differ from scalable designs involving microdisk
arrays coupling into optical fibers. In such designs, a spatial light
modulator could adjust free-space collection to fiber mode coupling, with
grating hole arrangements tailored for Gaussian beam profiles.

Our control FDTD simulation-based dipole model accurately predicts the optimal
lattice constant to within 5 nm. This model’s potential extends to designing
devices needing free-space coupling from near-field modes, though it currently
does not fully account for asymmetric perturbations, preferential upper
hemisphere emission, and lattice reflections. Refinements, such as modeling
scattering at hole edges, could enhance accuracy, making this approach a
valuable tool for solving various resonator scattering problems and reducing
dependence on extensive FDTD simulations.

##  5 Conclusion

We presented a scalable alignment-free design for free-space coupling from
diamond microdisks, offering a solution that alleviates the challenges of
precision nanofabrication in diamond. This design demonstrates robustness
across several parameters: the thickness of the diamond microdisk, the
alignment of the grating and microdisk layers, the lattice constant of the
nitride layer, and the dimensions of etches in the nitride. We also developed
a rapid optimization method using a single FDTD simulation and a dipole model
instead of comprehensive full-wave parameter sweeps. This method significantly
accelerates the design process for complex structures, especially for resonant
devices with high quality factors. The resulting optimized layered microdisk
design allows for creating arrays of quantum emitters with high alignment
yields and a maximum quantum efficiency $\eta$ of 0.45 for silicon vacancy
(SiV) centers in diamond. This advance in quantum emitter design holds promise
for enhancing the efficiency and scalability of quantum information systems.

##  6 Acknowledgements

The authors thank Yuqin Duan and Matthew Trusheim for valuable discussions
during the design process. This work was partially supported by the National
Science Foundation (NSF) award number 1941583.

## References

  * [1] M. Ruf, N. H. Wan, H. Choi, D. Englund, and R. Hanson, “Quantum networks based on color centers in diamond,” Journal of Applied Physics 130, 070901 (2021).
  * [2] M. Pompili, S. L. N. Hermans, S. Baier, H. K. C. Beukers, P. C. Humphreys, R. N. Schouten, R. F. L. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse, S. Wehner, and R. Hanson, “Realization of a multinode quantum network of remote solid-state qubits,” Science 372, 259–264 (2021).
  * [3] L. Orphal-Kobin, K. Unterguggenberger, T. Pregnolato, N. Kemf, M. Matalla, R.-S. Unger, I. Ostermay, G. Pieplow, and T. Schröder, “Optically coherent nitrogen-vacancy defect centers in diamond nanostructures,” Phys. Rev. X 13, 011042 (2023).
  * [4] Y. Duan, K. C. Chen, D. R. Englund, and M. E. Trusheim, “A vertically-loaded diamond microdisk resonator spin-photon interface,” Opt. Express 29, 43082–43090 (2021).
  * [5] P. C. Humphreys, N. Kalb, J. P. J. Morits, R. N. Schouten, R. F. L. Vermeulen, D. J. Twitchen, M. Markham, and R. Hanson, “Deterministic delivery of remote entanglement on a quantum network,” Nature 558, 268–273 (2018).
  * [6] L. Li, T. Schröder, E. H. Chen, M. Walsh, I. Bayn, J. Goldstein, O. Gaathon, M. E. Trusheim, M. Lu, J. Mower, M. Cotlet, M. L. Markham, D. J. Twitchen, and D. Englund, “Coherent spin control of a nanocavity-enhanced qubit in diamond,” Nature Communications 6, 6173 (2015).
  * [7] M. Gould, S. Chakravarthi, I. R. Christen, N. Thomas, S. Dadgostar, Y. Song, M. L. Lee, F. Hatami, and K.-M. C. Fu, “Large-scale gap-on-diamond integrated photonics platform for nv center-based quantum information,” J. Opt. Soc. Am. B 33, B35–B42 (2016).
  * [8] C. Bradac, W. Gao, J. Forneris, M. E. Trusheim, and I. Aharonovich, “Quantum nanophotonics with group IV defects in diamond,” Nature Communications 10, 5625 (2019).
  * [9] M. E. Trusheim, B. Pingault, N. H. Wan, M. Gündoğan, L. De Santis, R. Debroux, D. Gangloff, C. Purser, K. C. Chen, M. Walsh, J. J. Rose, J. N. Becker, B. Lienhard, E. Bersin, I. Paradeisanos, G. Wang, D. Lyzwa, A. R.-P. Montblanch, G. Malladi, H. Bakhru, A. C. Ferrari, I. A. Walmsley, M. Atatüre, and D. Englund, “Transform-limited photons from a coherent tin-vacancy spin in diamond,” Phys. Rev. Lett. 124, 023602 (2020).
  * [10] E. Takou and S. E. Economou, “Optical control protocols for high-fidelity spin rotations of single $\mathrm{Si}{\mathrm{v}}^{-}$ and $\mathrm{Sn}{\mathrm{v}}^{-}$ centers in diamond,” Phys. Rev. B 104, 115302 (2021).
  * [11] J. Arjona Martínez, R. A. Parker, K. C. Chen, C. M. Purser, L. Li, C. P. Michaels, A. M. Stramma, R. Debroux, I. B. Harris, M. Hayhurst Appel, E. C. Nichols, M. E. Trusheim, D. A. Gangloff, D. Englund, and M. Atatüre, “Photonic indistinguishability of the tin-vacancy center in nanostructured diamond,” Phys. Rev. Lett. 129, 173603 (2022).
  * [12] E. I. Rosenthal, C. P. Anderson, H. C. Kleidermacher, A. J. Stein, H. Lee, J. Grzesik, G. Scuri, A. E. Rugar, D. Riedel, S. Aghaeimeibodi, G. H. Ahn, K. Van Gasse, and J. Vučković, “Microwave spin control of a tin-vacancy qubit in diamond,” Phys. Rev. X 13, 031022 (2023).
  * [13] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Computer Physics Communications 181, 687–702 (2010).
  * [14] J. Zhu, X. Cai, Y. Chen, and S. Yu, “Theoretical model for angular grating-based integrated optical vortex beam emitters,” Opt. Lett. 38, 1343–1345 (2013).
  * [15] M. Mitchell, D. P. Lake, and P. E. Barclay, “Realizing $Q>$ 300 000 in diamond microdisks for optomechanics via etch optimization,” APL Photonics 4, 016101 (2019).
  * [16] J. Xie, J. Qian, T. Wang, L. Zhou, X. Zang, L. Chen, Y. Zhu, and S. Zhuang, “Integrated terahertz vortex beam emitter for rotating target detection,” Advanced Photonics 5, 066002 (2023).
  * [17] J. B. Schneider, _Understanding the Finite-Difference Time-Domain Method_ (www.eecs.wsu.edu/ schneidj/ufdtd, 2010).
  * [18] F. T. Ulaby and U. Ravaioli, _Fundamentals of Applied Electromagnetics_ (Pearson, 2023).
  * [19] ANSYS, “Lumerical,” https://www.lumerical.com/ p. 2023 R2 (2023).