text:
\section{Introduction}

\begin{figure}[t!]
	\centering
	\subfloat[]{\includegraphics[width=0.22\textwidth]{1a.png}}
    \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.22\textwidth]{1b.png}}
    \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.22\textwidth]{1c.png}}
	 \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.22\textwidth]{1d.png}}
	\caption{Example of depth guided image relighting. (a): Original input image. (b): Corresponding depth map. (c): Relighted image by our method. (d): Ground truth.}
\label{fig:fig1}
\end{figure}

Given an image, image relighting aims to relight this image into another image with different ambient conditions. In the NTIRE 2021 Depth Guide One-to-one Relighting Challenge, the depth maps are provided. We plot examples including the original image, the corresponding depth map, the relighted image by the proposed method, and the ground truth in \figref{fig:fig1}. As shown in \figref{fig:fig1}, there are two inherent challenges for image relighting. First, to generate the image with different ambient conditions, it is necessary to generate shadows into the relighted image. Second, similarly, the shadow from the original image needs to be removed. For example, in \figref{fig:fig1} (c), although the region shadows are removed, the texture of the grass cannot be recovered appropriately.

Previously, many methods \cite{debevec2000acquiring,matusik2004progressively,reddy2012frequency} based on developing visual priors or capture properties of relighted images have achieved impressive performance. Recently, some deep learning-based methods \cite{xu2018deep,puthussery2020wdrn,elhelou2020aim,yang2021S3Net} are proposed without explicit inverse rendering steps for estimating scene properties. However, these methods do not consider complex scenes and various ambient conditions. Moreover, in NTIRE 2021 Depth Guide One-to-one Relighting Challenge, there are extra difficulties needed to be addressed. First, the number of training data given in the VIDIT \cite{helou2020vidit} dataset is 300, which is not enough to train the network. To address these issues, our network is based on the backbone pre-trained from ImageNet. Furthermore, we leverage extra data from Track II: Any-to-Any relighting \cite{yang2021S3Net} and develop a strategy that can generate new image pairs to increase the number of the training data and the robustness of the network.

Second, depth and image features should be effectively extracted and fused. Depth maps and images contain the different attributes of features. Depth maps present spatial information while the images provide texture, light cues, and dark cues. Thus, we refer to the methods proposed in the RGB-D salient object detection \cite{pang2020hierarchical,chen2020progressively,zhang2020asymmetric} and propose a multi-modal bifurcated network to deal with this issue. This network contains encoder and decoder parts. Our encoder applies two branches without sharing weight to extract features, respectively. Additionally, we apply the dynamic dilated pyramid module to effectively integrate two features. In the decoder parts, we gradually magnify the feature maps and recover the image. Motivated by the U-Net \cite{ronneberger2015u,yang2019wavelet}, we apply skip connection to connect the feature maps with identical size from the encoder and decoder parts to obtain better relighted images.

We make the following contributions in this paper:
\begin{enumerate}
\item The multi-modal bifurcated network is proposed for depth guided image relighting. This structure can extract image and depth features by two branches. Then, these two features are fused by dynamic dilated pyramid modules effectively.
\item We propose a new strategy to leverage additional images from Track II and construct more input-output pairs as the training data.
\item Several experiments performed on the VIDIT \cite{helou2020vidit} dataset demonstrate that our solution achieves
the \textbf{1}$^{st}$ place in terms of MPS and SSIM in the NTIRE 2021 Depth Guided One-to-one Relighting Challenge.
\end{enumerate}

\section{Related Works}
Two tasks are very similar to depth guided image relighting: RGB-D saliency object detection and image relighting. In this section, we briefly describe several works related to these tasks.
\subsection{RGB-D Salient Object Detection}
Salient Object Detection (SOD) aims to imitate the human
visual system and detect certain regions or objects that attract human attention. RGB-D SOD is known as combining the extra depth maps to fulfill salient detection. Some models \cite{chen2020progressively,zhang2020asymmetric} extract features from images and depth maps independently, and conducts feature maps fusion of the two modalities in the decoder. In \cite{zhang2020asymmetric}, Asymmetric Two-Stream Architecture (ATST) is proposed which considers the inherent differences between the RGB and the depth data for the salient detection. In \cite{chen2020progressively}, Guided Residual (GR) blocks are proposed to feed the RGB image and the depth image alternately to reduce the mutual degradation. They also address progressive guidance in the stacked GR blocks within each side-output to remedy the false detection and the missing parts. On the other hand, in \cite{zhao2020single}, the single stream network is designed that concatenates the RGB image and depth image across the channel dimension and directly applies the depth map to guide both the early fusion and the middle fusion between the RGB information and the depth information, which saves the feature encoder of the depth stream. Additionally, in \cite{chen2021rgb}, authors propose the RD3D that 3D convolutional neural networks are introduced to address the RGB-D SOD. This network adopts the progressive fusion involving both the encoder and the decoder stages.
Since the multi-modal fusion methods from RGB-D SOD can be leveraged for the depth-guided image relighting, in this paper, the proposed MBNet is based on the HDFNet \cite{pang2020hierarchical}.
\begin{figure*}[t!]
  \centering
\includegraphics[width=0.8\textwidth]{architecture_bi_V2.png}

    \caption{{The architecture of the proposed multi-modal bifurcated network. The network consists of two streams as encoder parts: depth stream and RGB-image stream. We use the dense architecture ,DDPM and skip connection for better feature extraction as decoder parts.
}}
\label{fig:architecture}
\end{figure*}

\subsection{Image Relighting}
The algorithms for image relighting can be categorized as the physical-based method and the deep learning-based method. Traditional methods \cite{debevec2000acquiring,matusik2004progressively,reddy2012frequency} depend on the physical observation to further estimate the ambient conditions, reflectance, and lighting of the scene in the image and then re-render this scene by another illumination setting.
In \cite{matusik2004progressively}, an algorithm treating the complex scene as a linear system that transforms the original light into the reflected light is proposed. This algorithm progressively refines the approximation of the reflectance field until the required precision is reached. In \cite{reddy2012frequency}, authors prove that the light transport of diffuse scenes under the spatially varying illumination can be decomposed into the direct, near-range and far-range transports. They separate these three components in the frequency space.
On the other hand, many deep learning-based methods for image relighting \cite{xu2018deep,puthussery2020wdrn,yang2021S3Net} are proposed. Image relighting can be seen as an image-to-image translation problem. Several low-level image processing problems like haze/smoke removal \cite{yang2020net,chen2018color,chen2020pmhld}, underwater enhancement \cite{yang2021LAFFNet}, reflection removal \cite{tsai2018efficient},
image deraining \cite{yang2020wavelet} and image desnowing \cite{chen2020jstasr} are very similar to image relighting. Generally speaking, the encoder-decoder structure like U-Net \cite{ronneberger2015u,yang2019wavelet} can deal with these tasks. Some methods that deal with the ambient conditions especially for the image relighting are developed. Gray loss described in \cite{puthussery2020wdrn} can drive the network to learn the illumination gradient in target domain images. Xu \textit{et.al} propose a CNN-based method \cite{xu2018deep} to relight a scene under a new illumination based on five images captured under a pre-defined illumination setting. This method tries to estimate a non-linear function that generalizes the estimation from the optimal sparse samples.

\section{Proposed methods}
\subsection{Overall Neural Network}
As shown in \figref{fig:architecture}, to leverage the depth information for depth guided image relighting task effectively, we refer the dual-stream bifurcated architecture in \cite{pang2020hierarchical} as the backbone. Our network consists of two streams to extract the depth and the image features. We apply the two ResNet 50 \cite{he2016deep} pre-trained from the ImageNet as the backbones. In the network design, the depth and image features from three intermediate layers are fused to achieve the representative features. We fuse the features from the conv3, the conv4 and the conv5 layers to balance the effectiveness and the efficiency of the network. Specifically, for the features in the shallower layers, they are generally noisier and the high-resolution of these features may increase the computational burden. However, the features in the conv3 to the conv5 may still preserve the valid information \cite{pang2020hierarchical} and with lower resolution. To fuse these two features with multiple receptive fields, we leverage the densely connected architecture to generate the combined features with the fruitful texture and the structural information. Then, these features are fed to the Dynamic Dilated Pyramid Module (DDPM)\unskip~\cite{pang2020hierarchical} that can generate a more discriminative result. We will describe this module in the following section. The output of the DDPM is combined with the output of the decoder by convolving with the multi-scale convolution kernels\unskip~\cite{chen2019pms,chen2020jstasr,ren2016single}. In the decoder part, similar to the U-net \cite{yang2019wavelet,ronneberger2015u}, we gradually magnify the feature maps and implement the skip connection to concatenate the identical size feature maps. Furthermore, we make our network learn the residual \cite{he2016deep} instead of the whole images. That is, the final output is the difference between the original image and the relighted image.

\subsection{Dynamic Dilated Pyramid Module}

\begin{figure*}[t!]
  \centering
\includegraphics[width=0.8\textwidth]{DDPM.png}
    \caption{{The architecture of the dynamic dilated pyramid module (DDPM). There are two modules in the network, that is, the kernel generation units (KGUs) and the kernel transformation units (KTUs).}}
\label{fig:architecture_DDPM}
\end{figure*}

In this part, we illustrate the detail of the DDPM. As shown in \figref{fig:architecture_DDPM}, the input of DDPM is the fused feature $f_{E}$ and the feature $(f_{RGB})_{D}$ from encoders, respectively. First, the $(f_{RGB})_{D}$ passes through the convolution kernel to reduce the dimension of the feature which is termed as $^{*}(f_{RGB})_{D}$. Second, the kernel generation units (KGUs) \cite{pang2020hierarchical} are adopted on the fused feature $f_{E}$ to generate different weight tensors (i.e, $f_{KGU}^{1}$, $f_{KGU}^{2}$, $f_{KGU}^{3}$) which can cover three square neighborhoods (i.e., $3\times3$, $7\times7$, and $11\times11$) . It is noted that KGUs are with four densely connected layers \cite{huang2017densely} which can improve the feature propagation and feature reuse effectively. Then, we leverage the kernel transformation units (KTUs) to yield regular convolution kernels with various dilation rate (i.e., 1, 3, 5) by reorganizing kernel tensors and inserting various numbers of zeros. Then, the three parallel output features combine with the $^{*}(f_{RGB})_{D}$ by the convolution kernels, respectively. We term these combined features as $f_{K}^{1},f_{K}^{2},f_{K}^{3}$. Finally, we combine $f_{K}^{1},f_{K}^{2},f_{K}^{3}$ and $^{*}(f_{RGB})_{D}$ to generate the output of the DDPM $f_{C}$.

\subsection{Extra Data Usage}
In this section, we propose two strategies to increase the training data so that our model can learn the mapping function of the depth guided image relighting robustly. In this challenge, the input image (6500-N) is with 6500K color temperature and the north illumination angle while the output image (4500-E) is with 4500K color temperature and with the east illumination angle. In order to improve the robustness of our network, we leverage the images in Depth Guided Image Relighting: Track II Any-to-Any relighting \cite{yang2021S3Net}. This track provides the images with various illumination temperature and different illumination angles. Specifically, we apply the images with the different illumination angle (6500-NE) but the identical color temperature as the input and the ground truths are the same ones (4500-E). As shown in \figref{fig:aug}, the additional image (6500-NE) is very similar to the original one (6500-N). With this additional data, the model can understand the direction information comprehensively.

Moreover, We adopt additional images which contain the same scene but with the different illuminating angle (4500-W). We can flip horizontally the image with the west illumination angle to achieve the new image with the east illumination angle. Thus, we develop a new strategy to further increase training data and illustrate it in \figref{fig:aug2}. As shown in \figref{fig:aug2} (d) and \figref{fig:aug2} (e), we flip the original RGB (6500-N) images and the corresponding depth map horizontally. The horizontally flipped image of (4500-W) is the output of the flipped input (6500-N). With this operation, the training data can be increased.

\subsection{Loss Functions}
In this paper, we leverage three loss functions to measure the differences between the relighted images and the ground truth. The three loss functions are the Charbonnier loss \cite{barron2019general}, the SSIM loss \cite{zhao2016loss}, and the perceptual loss \cite{johnson2016perceptual}. The Charbonnier loss can be expressed as:
\begin{equation}
L_{Cha}(x, \hat x) = \frac{1}{T}\sum_{i}^{T}\sqrt[]{(x_i-\hat{x_i})^2+\epsilon ^2}
\label{eq:res}
\end{equation}
where $x$ and $\hat x$ are the ground truth and relighted images, respectively. $e$ is a tiny constant for the stable and the robust convergence.

\begin{figure*}[t!]
	\centering
	\subfloat[]{\includegraphics[width=0.24\textwidth]{4-1.png}}
    \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.24\textwidth]{4-2.png}}
    \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.24\textwidth]{4-3.png}}
	 \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.24\textwidth]{4-4.png}}
	\caption{Additional images are used in the training phase. (a) Original image (6500-N). (b) Different illuminating angle image (6500-NE). (c) Guided depth map. (d) Output image.}
	\label{fig:aug}
\end{figure*}

\begin{figure*}[t!]
	\centering
	\subfloat[]{\includegraphics[width=0.16\textwidth]{5-1.png}}
    \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.16\textwidth]{5-2.png}}
    \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.16\textwidth]{5-3.png}}
	 \hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.16\textwidth]{5-4.png}}
	\hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.16\textwidth]{5-5.png}}
	\hspace{0.01em}
	\subfloat[]{\includegraphics[width=0.16\textwidth]{5-6.png}}
	\caption{Using (4500-W) images to increase the diversity of the training data. (a): Original input image. (b): Original depth map. (c): Output image. (d): Flipped image of (a). (e): Flipped depth map of (b). (f): Flipped image of (4500-W).}
\label{fig:aug2}
\end{figure*}

The second loss function is the SSIM loss function. SSIM loss is expressed as:
\begin{equation}
  {L_{SSIM}}(x,\hat x) = -SSIM(x,\hat x)
  \label{eq:SSMI}
\end{equation}
SSIM loss is beneficial to reconstruct local structures and details.

Finally, the perceptual loss is written as:

\begin{equation}
L_{Per}(x,\hat{x})= |(VGG(x)-VGG(\hat{x})|
\label{eq:res}
\end{equation}
where $VGG$ means the VGG19 network \cite{simonyan2014very}.  In our work, we use the features from conv3-3 layer.
The overall loss function containing three terms is expressed as:
\begin{equation}
L_{Total}= \lambda_{1} L_{cha}+ \lambda_{2} L_{SSIM}+ \lambda_{3} L_{Per}
\label{loss}
\end{equation}
where $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ are weights to control the final objective functions. These three weights are empirically adjusted as hyper-parameters.

\section{Experiments}

\subsection{Experimental Setting}
In the NTIRE 2021 Depth Guide One-to-one Relighting Challenge, the novel Virtual Image Dataset for Illumination Transfer (VIDIT) \cite{helou2020vidit} is provided as the training and the validation data. This dataset consists of 390 various scenes that are captured at 40 different illumination conditions including 8 different azimuthal angles and five color temperatures such as 2500K, 4500K, etc. Furthermore, the corresponding depth maps are provided. In track I - depth guided one-to-one relighting, the input images are depth map and a pre-defined illumination condition $\theta$ = North, temperature = 6500K (e.g., (6500-N)) and the output image is set at a different illumination setting $\theta$ = East, temperature = 4500K. Though in track I, only two conditions of images are used as the input and the output pairs, it is allowed to utilize the extra data to improve the accuracy of the model. During the training and the evaluation phases, the image size is 1024 $\times$ 1024, and we do not use any data augmentation like random flip and random crop. The Adam optimizer \cite{kingma2014adam} is utilized with a batch size of 3 to train the network. We train the network for 200 epochs with the momentum $\beta_{1}$ = 0.5, $\beta_{2}$ = 0.999. The learning rate is initialed as $10^{-4}$ and divided by ten after 50 epochs. The $\lambda_1$, $\lambda_2$ and $\lambda_3$ in \eqref{loss} are set as 1, 1.1 and 0.1, respectively. We perform our experiments on a single Nvidia V100 graphic card and the PyTorch platform. We spend about 11 hours finishing the model training. In the testing phase, we take 2.8867 seconds to predict a single image. The source code will be available in our project page.

\begin{table}[ht]
\small
    \centering
    \caption{{The ablation experiment of applying the different data and the residual learning.}}
\begin{tabular}{l|cc}
\hline
Description   & PSNR & SSIM \\ \hline\hline
Baseline &     18.0215  &  0.6834 \\
+ Extra data  &  18.9677    &  0.7103    \\
+ Extra data + Residual learning  &   \textcolor[rgb]{ 1,  0,  0}{19.3558}   &  \textcolor[rgb]{ 1,  0,  0}{0.7175}
\end{tabular}
\label{tab:ablation}
\end{table}

\begin{table}[ht]
\small
  \centering
  \caption{Relighted results by some state-of-the-art RGB-D salient object methods.}
    \begin{tabular}{l|cc}
    \hline
    Methods & PSNR & SSIM \\ \hline \hline
    ATST \cite{zhang2020asymmetric}  & 18.2678 & 0.665 \\
    DANET \cite{zhao2020single} & 18.3341 & 0.6805 \\
    RD3D \cite{chen2021rgb} & 17.7763 & 0.6668 \\
    PGAR \cite{chen2020progressively}  & 17.6436 & 0.6748 \\
    Ours  & \textcolor[rgb]{ 1,  0,  0}{19.3558} & \textcolor[rgb]{ 1,  0,  0}{0.7175} \\
    \end{tabular}%
  \label{tab:addlabel}%
\end{table}%

\begin{figure*}[t]
  \centering
\includegraphics[width=\linewidth]{vis_cameraready.png}
    \caption{{Visual comparison for the relighted results recovered by the MBNet and other solutions.
}}

\label{fig:vis}
\end{figure*}

\begin{table*}[tbp]
  \centering
  \caption{The average SSIM, PSNR, MPS and LPIPS  of some submissions over NTIRE 2021 depth guided image relighting validation and testing dataset.}
    \begin{tabular}{r|cc|cccc}
    \multicolumn{1}{l|}{User name} & \multicolumn{2}{c|}{Validation} & \multicolumn{4}{c}{Testing} \\
    \hline
          & SSIM  & PSNR  & MPS   & SSIM  & LPIPS & PSNR \\ \hline
    auy200& 0.6937&18.4492&0.7620 & 0.6874 & 0.1634 & 18.8358 \\
    aics &0.7069&19.1026& 0.7601 & 0.6799 & 0.1597 & 18.8639  \\
    lifu &0.7104&19.0048& 0.7600 & 0.6903 & 0.1702 & 19.8645 \\
    jimmy3505090 &0.7069&18.2937& 0.7551 & 0.6772 & 0.1670 & 18.2766\\
    DeepBlueAI & 0.6757&18.6800&0.7494 & 0.6879 & 0.1891 & 19.8784\\
    Ours &0.7175 & 19.3558& 0.7663 & 0.6931 & 0.1605 & 19.1469 \\
    \end{tabular}%
  \label{tab:addlabel}%
\end{table*}%

\subsection{Ablation Experiments}
We conduct the ablation experiment to verify that each module applied in this paper can benefit the proposed relighting network. In all experiments, the image size is set as 1024$\times$ 1024. We test each module and report the results under the validation set. We select the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM) as objective metrics for the quantitative evaluation. Overall, the ablation studies consist of three different experimental scenarios: 1) We apply the original training data to train the MBNet as the baseline. 2) We apply the method described in Section 3 to increase the training data to train our MBNet. 3) We increase the training data and also apply the residual learning \cite{he2016deep} strategy. We summary the ablation experiments in Table \ref{tab:ablation}. One can see that both PSNR and SSIM scores of setting 2 are increased compared with setting 1. It can show that increasing training data is beneficial for better performance and robustness. Additionally, compared with setting 2, the performance of setting 3 is improved effectively. It demonstrates that the residual learning can further improve the accuracy of the relighting.

\subsection{Comparison with State-of-the-art Methods}
First, we compare the MBNet with four state-of-the-art RGB-D SOD methods including ATST \cite{zhang2020asymmetric}, DANET \cite{zhao2020single}, RD3D \cite{chen2021rgb} and PGAR \cite{chen2020progressively} as described in the Section 2. Note that, we use the same training set to train these methods. We replace the final convolutional layers of RGB-D SOD networks, so their output is three-channel tensors as relighted images. As shown in Table \ref{tab:addlabel}, the MBNet outperforms other methods with a large margin. Our method achieves the best performance on both PSNR and SSIM, which surpasses the second place 1.02 dB and 0.037 in SSIM.

Furthermore, we report the performances of some submissions in the NTIRE 2021 Depth Guide One-to-one Relighting Challenge \cite{elhelou2021ntire}. The performance is evaluated on the validation and the test dataset and the results are shown in Table \ref{tab:addlabel}. Additionally, The Mean Perceptual Score (MPS) is used for final evaulation. The MPS is defined as the average of the normalized SSIM and LPIPS \cite{zhang2018perceptual} scores. The MBNet produces moderate quality outputs and the \textbf{$1^{st}$} place performance in both MPS and SSIM metrics. We also plot some relighted images generated by our method and other participants in \figref{fig:vis}. Compared to other methods, our images can remove more shadows and present clear outlines of objects, though some results are not satisfactory.

\section{Conclusion}
In this paper, to address depth guided image relighting, we develop the multi-modal bifurcated network. This network extracts both depth and image features by a dual bifurcated backbone. To fuse multi-modal features, the dynamic dilated pyramid module is introduced. This module contains densely connected layers and multi-scale kernels to fuse and refine features from a dual bifurcated backbone. Furthermore, to improve the robustness and performance, we propose a new strategy to increase the training image pairs by leveraging extra images. Several experiments implemented on the novel VIDIT \cite{helou2020vidit} dataset proves that our solution achieves the \textbf{$1^{st}$} place in terms of MPS and SSIM in the NTIRE 2021 Depth Guided One-to-one Relighting Challenge.

\section{Acknowledgement}
We thank to National Center for High-performance Computing (NCHC) for providing computational and storage resources.

{\small
\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction and Summary}

In recent years, F-theory compactifications have been studied for the construction of the realistic models \cite{arXiv:0803.1194, arXiv:1001.0577, arXiv:1009.3497}. A four dimensional low energy effective field theory realized as a non-Abelian gauge theory on 7-branes wrapping on a surface can be a candidate for a supersymmetric Grand Unified Theory (GUT). Since some of important properties of low-energy effective theory in the
visible sector are determined only by geometry of local neighbourhood along
the GUT surface in Type IIB / F-theory compactifications \cite{Local},
correspondence between geometry around the GUT complex surface and
low-energy physics can be established to some extent without referring
to global geometry.
Some class of complex surfaces with certain topology may even be ruled
out \cite{Beasley:2008kw}, when various phenomenological requirements
are imposed.
Reference \cite{Beasley:2008kw} argued, for example, that the condition
of no exotic matter fields in a hypercharge flux scenario of
$SU(5)_{\GUT}$ breaking can exclude Hirzebruch surfaces for the GUT
7-brane. References \cite{Caltech-0904, Hayashi:2009bt, Cordova:2009fg} also
made it clear that the low-energy Yukawa matrices receive independent
contributions from enhanced singularity points whose numbers are
determined by intersection numbers on the GUT complex surface.
In this article, we further pursue this program, and report some
constraints on the topology of GUT complex surfaces following from
phenomenological conditions.

In section \ref{sec:exotic}, we take a moment to have a brief look at
the exotic matter free condition in the hypercharge flux scenario of
$SU(5)_{\GUT}$ breaking.
Requiring that chiral multiplets in the off-diagonal components of the
$5 \times 5$ matrix of $SU(5)_{\GUT}$ be absent below the Kaluza--Klein
scale, Ref. \cite{Beasley:2008kw} already concluded that all the
Hirzebruch surfaces $\IF_n$ (except $\IF_0$) are ruled out in this scenario.
In fact, there is an error in the calculation. We see that the proof
can still be fixed by {\it fully} exploiting the exotic matter free
condition.

In section \ref{sec:flavor}, we move on to study consequences of a new
topological condition motivated by phenomenology. Because the number
of $E_6$ enhancement points is always even in $SU(5)$ models in F-theory \cite{Hayashi:2009bt, Cordova:2009fg},
the low-energy up-type Yukawa matrix consists of contributions from
two or more $E_6$ type points, and the approximately rank-1 structure
of individual contributions \cite{Hayashi:2009ge, arXiv:0811.2417} is lost
generically. One of the solutions to this problem is to impose a
discrete symmetry $\Gamma$ \cite{Hayashi:2009bt}, so that the contributions
from multiple enhanced singularity points are aligned and the
approximate rank-1 structure is maintained even after all these
contributions are summed up.
This discrete symmetry solution sets constraints on the number
of enhanced singularity points.
We find that both del Pezzo surfaces and a rational elliptic surface are
excluded by the constraints when $\Gamma=\IZ_2$ (which can be used as
$R$-parity simultaneously). We also argue that a less general condition for the number of codimension-three singularity points can eliminate del Pezzo surfaces and a rational elliptic surface for all the discrete symmetry $\Gamma$.

To summarize,
the $SU(5)_{\GUT}$ breaking by the hypercharge flux excludes Hirzebruch
surfaces (except $\IF_0$), and the discrete symmetry solution for the
alignment problem of the Yukawa matrices excludes del Pezzo surfaces and
a rational elliptic surface. In other words, if one chooses Hirzebruch
surfaces as the $SU(5)_{\GUT}$ GUT surface, one cannot use the
hypercharge flux scenario for $SU(5)_{\GUT}$ breaking, and if one
chooses del Pezzo surfaces or a rational elliptic surface as the
$SU(5)_{\GUT}$ GUT surface, one cannot use a discrete symmetry solution
for the alignment problem of the low-energy Yukawa matrices. Note that
those results hold also when one embeds the GUT surface in a global
Calabi--Yau fourfold. The global embedding of the $SU(5)_{\GUT}$ GUT
surface has been recently discussed in \cite{Caltech-0904, arXiv:0906.4672, arXiv:0908.1784}, and the geometry search has been performed
\cite{arXiv:1101.4908, arXiv:1103.3358}. Our results can constrain the
possible choices for the GUT surface in the geometry search if one
relies on either one of [or on both of] the two scenarios.

\section{No go theorems for SU(5) GUT models}

F-theory compactified on an elliptically fibered Calabi--Yau fourfold realizes a four-dimensional $\cN=1$ supersymmetric theory as a low energy effective field theory. When the elliptic fiber degenerates and generates a singularity of a type $G$ at a codimension one locus $S$ in the base manifold $B_3$, a number of 7-branes wrap on the divisor $S$ and realizes a non-Abelian gauge theory with the gauge group $G$ \cite{hep-th/9602022, hep-th/9602114}. Since we are interested in constructing phenomenological models from F-theory compactifications, we focus on $SU(5)_{\GUT}$ for the gauge group $G$. We will argue that phenomenological requirements indeed restrict the possible candidates for the GUT divisor $S$.

\subsection{Exotic matter free conditions in $SU(5)_{\GUT}$ breaking}
\label{sec:exotic}

The $SU(5)_{\GUT}$ gauge group should be broken to obtain the Standard Model gauge group $SU(3) \times SU(2) \times U(1)_Y$. As for the breaking scenario\footnote{Two other $SU(5)$ breaking scenarios were also referred to in \cite{Beasley:2008kw}. One is to use a Wilson line on a GUT divisor $S$ that is not simply connected, and the other is four-dimensional GUT breaking scenario involving $SU(5)_{\rm GUT}$-${\bf adj}$ chiral multiplets in the effective theory.}, we turn on a hypercharge flux $\cL_Y \equiv (L_Y)^{-5/6} \in H^{2}(S, \IZ)$ \cite{GUT-breaking-via-non-surj-flux-F, arXiv:0806.0634} where the charge against $L_Y$ gives the hypercharge for matter fields.
The flux $\cL_Y$ should satisfy the BPS condition \cite{arXiv:0802.2969, Beasley:2008dc, Hayashi:2008ba}
\beq
\ox \wedge c_1(\cL_Y) = 0,
\label{BPS}
\eeq
where $\ox$ is a K\"ahler form on the GUT divisor $S$. The hypercharge flux $\cL_Y$ yields chiral multiplets in the off-diagonal blocks of the 5 $\times$ 5 matrix of $SU(5)_{\GUT}$. They are in the representation of ${\bf (3, 2)_{-5/6}}$ and ${\bf (\bar{3}, 2)_{+5/6}}$ under $SU(3) \times SU(2) \times U(1)_Y$.
Since the presence of those matter fields under the GUT scale breaks the $SU(5)$ gauge coupling unification, we require that there are no zero modes for the matter fields in the representation of ${\bf (3, 2)_{-5/6}}$ and ${\bf (\bar{3}, 2)_{+5/6}}$.
Then, the exotic matter free conditions are \cite{arXiv:0802.2969, Beasley:2008dc, Hayashi:2008ba}
\bea
h^{0}(S, \cL_Y^{\pm 1}) &=& 0, \qquad h^{1}(S, \cL_Y^{\pm 1}) = 0, \label{exotic1}\\
\kernel d_2 &=& 0, \qquad  \coker d_2 = 0,\label{exotic2}
\eea
where $\kernel d_2$ and $\coker d_2$ are understood as their dimension. Here, $d_2$ is a map
\beq
d_2\; : \; H^{0}(S, \cL_Y \otimes \cO(K_S)) \rightarrow H^{2}(S, \cL_Y).
\label{map}
\eeq
The case where the map $d_2$ is non-trivial has been discussed in \cite{hep-th/0208104} in the context of Type IIB string theory.

When the anti-canonical divisor $-K_{S}$ is an effective divisor, $h^0(S, \cO(D))=0$ for a divisor $D$ always implies $h^0(S, \cO(D+K_S)) = h^2(S, \cO(-D)) = 0$ \cite{Beasley:2008dc}. Hence, the exotic free conditions \eqref{exotic1}--\eqref{exotic2} are equivalent to
\beq
h^{i}(S, \cL_Y^{\pm 1}) = 0\quad{\rm for}\;i=0,\cdots, 2,
\label{exotic_free}
\eeq
if $-K_S \geq 0$. This includes the cases of Hirzebruch surfaces, del
Pezzo surfaces and rational elliptic surfaces. In such
situations, it follows from \eqref{exotic_free} that
\beq
\chi( S, \cL_Y ) = 0, \qquad \chi( S, \cL^{-1}_Y ) = 0.
\label{eq:chi=0}
\eeq

In fact, Ref.~\cite{Beasley:2008kw} has used the conditions \eqref{eq:chi=0} to set constraints on
Hirzebruch surfaces $\IF_n$.
In the following combinations, Ref~\cite{Beasley:2008kw}
has obtained
\bea
0 = \chi(\IF_n, \cL_Y) - \chi(\IF_n, \cL_Y^{-1}) &=& c_1(\IF_n) \cdot c_1(\cL_Y) = b(n+2)+2a-2bn^2?,\label{hir1}\\
0 = \frac12 (\chi(\IF_n, \cL_Y) + \chi(\IF_n, \cL_Y^{-1}) ) &=&
 1 + \frac{1}{2} c_1(\cL_Y)^2 = 1+\frac12(2ab-b^2 n^2)?\label{hir2}
\eea
Here, $\cL_Y$ is chosen as
\beq
\cL_Y=\cO(af+b\sigma),
\label{hypercharge1}
\eeq
where $a$ and $b$ are integers, and $f$ and $\sigma$ are generators of the effective curves in $\IF_n$ which satisfy the intersection
\beq
f \cdot f = 0, \qquad f \cdot \sigma = 1, \qquad \sigma \cdot \sigma =-n. \label{int_hir}
\eeq
Under the results of the calculation \eqref{hir1} and
\eqref{hir2}, only Hirzebruch surfaces $\IF_n$ with $n = 0, 1$ are
allowed. All the other Hirzebruch surfaces, $n \geq 2$, were excluded for
this reason.

However, the correct results for the computation are
\bea
0 &=&\chi(\IF_n, \cL_Y) - \chi(\IF_n, \cL_Y^{-1}) = c_1(\IF_n) \cdot c_1(\cL_Y) = 2a+2b-bn  ,\label{hir3}\\
0 &=& \frac12 (\chi(\IF_n, \cL_Y) + \chi(\IF_n, \cL_Y^{-1}) ) = 1 + \frac{1}{2} c_1(\cL_Y)^2 = 1+\frac12(2ab-b^2 n) , \label{hir4}
\eea
from the intersection numbers \eqref{int_hir}. Then, the only
constraints that we can derive from \eqref{hir3} and \eqref{hir4} are
$n={\rm even}$, $b=\pm 1$ and $a=b (n-2)/2$. Therefore, the conditions \eqref{hir3} and \eqref{hir4} are not enough to set a bound on the ``$n$'' of Hirzebruch surfaces $\IF_n$.

However, we can indeed exclude all the Hirzebruch surfaces except
for $\IF_0$ by exploiting all of \eqref{exotic_free}, not
just (\ref{eq:chi=0}). It will be straightforward to see
that\footnote{
One will also arrive at the same conclusion by using the BPS
condition (\ref{BPS}) with a K\"ahler form strictly in the interior
of the K\"ahler cone, instead of using the condition
$h^0(S, \cL^{\pm 1}_Y ) = 0$.
If the K\"ahler form is on the boundary of the cone,
at least one has to be careful in discussing the validity of the
Katz--Vafa style effective field theory description on 7+1 dimensions \cite{hep-th/9606086, arXiv:0802.2969, Beasley:2008dc}. }
\begin{equation}
 h^0( S , \cL_Y) > 0
\end{equation}
for $b = 1$ and $a = (n-2)/2$ for $S = \IF_n$ with an even $n \geq 2$.
After using all of the exotic matter free condition (\ref{exotic_free}),
\beq
\IF_0\quad {\rm with} \quad \cL_Y = f - \sigma,\quad {\rm or} \quad -f + \sigma
\eeq
is the only solution in the $S= \IF_n$ series in the hypercharge flux
scenario of $SU(5)_{\GUT}$ breaking.

\subsection{Discrete symmetry for the alignment of Yukawa matrices}
\label{sec:flavor}

As we have already explained in Introduction, the desirable flavor structures can also set constraints on the GUT divisor in a certain scenario.
If one could achieve a configuration such that there are only one $E_6$ and one $D_6$ singularity points,
the up-type and the down-type Yukawa couplings have an approximate rank one structure \cite{Hayashi:2009ge, arXiv:0811.2417}. In generic F-theory compactifications, however, one can show that the number of $E_6$ singularity points are always even and the approximate rank one structure is generically broken \cite{Hayashi:2009bt, Cordova:2009fg}. Hence, one has to set some constraints which realize a situation where only one Yukawa point dominantly contributes to a Yukawa coupling in a low energy effective field theory.

One way to achieve the desired configuration
is to impose a discrete symmetry globally \cite{Hayashi:2009bt}\footnote{There are two other solutions to recover the approximate rank one Yukawa matrices. One is to factorize matter curves and make completely independent wavefunctions localize along the separated matter curves \cite{Caltech-0904, Hayashi:2009bt, Cordova:2009fg}. Then, one could realize a situation such that only single $E_6$ codimension-three singularity point contributes to the up-type Yukawa coupling. In this case,
the matter wavefunctions need to be separated at the level of factorization of spectral surfaces, not at the level of matter curves \cite{hep-th/0602238, Tatar:2009jk}. In models without an $SO(10)$ or $SU(6)$ GUT divisors, this factorization condition needs to be lifted to a constraint on the global 4-fold geometry \cite{Tatar:2009jk, Hayashi:2010zp, arXiv:1006.0226, arXiv:1006.0483}. The other is to tune the complex structure moduli of the matter curve for {\bf 10} representation and make the wavefunctions of {\bf 10} matter fields localize along the matter curve \cite{Hayashi:2009bt}. Then, one codimension-three singularity point makes a dominant contribution to the Yukawa matrix in the four-dimensional effective theory, and recovers the approximate rank one structure of the Yukawa matrices in the Standard Model.}.
If the discrete symmetry relates all the codimension-three singularity
points of the same type, all the Yukawa matrices generated from the codimension-three singularities are aligned and have the same structures. Hence, one can recover the approximate rank one structure in that situation.

Introducing a discrete symmetry has another phenomenological
motivation. Imposing $\IZ_2$ symmetry has been considered in
\cite{Tatar:2009jk, Hayashi:2009bt} for a solution for the prohibition
of the renormalizable R-parity violating operators. In order to forbid
the dimension-5 proton decay operators also, the $\IZ_2$ symmetry is not
enough and one has to extend it to a larger discrete symmetry (e.g., \cite{320886}). Therefore, imposing a discrete symmetry may lead to realize the desirable flavor structure and also forbid the dangerous proton decay operators simultaneously.

Then, let us see how the discrete symmetry for the desirable flavor
structure sets topological constraints on the GUT divisor $S$ of $SU(5)$
GUT models. Let the symmetry group be $\Gamma$. Imagine a case where all the elements of the symmetry group $\Gamma$ acts on the global geometry non-trivially so that they act on
the $E_6$ singularity points [and $D_6$ type points] faithfully and
transitively. It then follows that
\beq
\# E_6 = |\Gamma|, \qquad \# D_6 = |\Gamma|.
\label{flavor1}
\eeq
More generally, there could be a normal subgroup $H_{E_6}$ or $H_{D_6}$
of $\Gamma$ which acts non-trivially on the {\it neighborhood} of the
$E_6$ or $D_6$ singularity points but acts trivially on the {\it points} respectively. In that case, the requirement \eqref{flavor1} is relaxed to
\beq
\# E_6 = \frac{|\Gamma|}{|H_{E_6}|}, \qquad \# D_6 = \frac{|\Gamma|}{|H_{D_6}|}.
\label{flavor2}
\eeq

\subsubsection{Del Pezzo surfaces}

First, we apply the constraint \eqref{flavor1} to del Pezzo surfaces, $dP_{n}$ and $\IP^1 \times \IP^1 (= \IF_0)$, assuming that $H_{E_6} = H_{D_6} = \phi$.
The numbers of the $E_6$ and $D_6$ singularity enhancement points on the GUT divisor $S$ are \cite{hep-th/9908193, Hayashi:2008ba}
\beq
\# E_6 = (5K_S + \eta) \cdot (4K_S + \eta), \qquad \# D_6 = (5K_S + \eta) \cdot (3K_S + \eta),
\eeq
where $\eta$ is a divisor in $S$ which satisfies $c_1(N_{S|B_3}) = 6K_S + \eta$. Then, we have
\beq
0 = \# D_6 - \# E_6 = (5K_{S} + \eta) \cdot (-K_{S}), \label{flavor_dp1}
\eeq
because of \eqref{flavor1}. Since $|5K_{S} + \eta|$ is a class of the matter curve for the {\bf 10} representation, $5K_S + \eta$ is an effective divisor. In the case of del Pezzo surfaces, then, one can see that Eq.~\eqref{flavor_dp1} contradicts the ampleness of $-K_{dP}$. This follows from Nakai's ampleness criterion, which states that the necessary and sufficient conditions for an ample divisor $D$ on a non-singular projective surface $S$ are
\beq
D \cdot D > 0, \qquad C \cdot D > 0
\eeq
for any curve $C$ in $S$. Therefore, any del Pezzo surfaces do not
satisfy the constraints (\ref{flavor1}), (\ref{flavor_dp1}).

The constraint \eqref{flavor1} is in fact stronger than the general constraint \eqref{flavor2}. However, in the case of $\Gamma=\IZ_2$, one can show that any del Pezzo surfaces cannot satisfy the general constraint \eqref{flavor2}. Note that the number of the $D_6$ singularity points is always larger than the number of the $E_6$ singularity points, since
\beq
\# D_6 - \# E_6 = (5K_{dP} + \eta) \cdot (-K_{dP}) > 0.
\eeq
Here, we use the fact that $5K_{dP} + \eta$ is an effective divisor and $-K_{dP}$ is an ample divisor. Therefore, the constraint \eqref{flavor2} is satisfied if and only if $\# D_6 = 2$ and $\# E_6 = 1$. However, the number of the $E_6$ singularity points is always even \cite{Hayashi:2009bt, Cordova:2009fg}. Then, the only possibility of $\# D_6 = 2$ and $\# E_6 =1$ cannot be realized. Namely, it is impossible to satisfy the constraint \eqref{flavor2} when $\Gamma = \IZ_2$.

\subsubsection{Rational elliptic surface}

We next apply the constraint \eqref{flavor1} to a rational elliptic surface. Since the anti-canonical divisor of a rational elliptic surface is not ample, one cannot use the discussion of del Pezzo surfaces for the constraints from the discrete symmetry solution in the case of a rational elliptic surface.
The condition \eqref{flavor_dp1} in the case of a rational elliptic surface implies that the matter curve of the class $5K_S + \eta$ is parallel to the elliptic fiber {\bf (i)}. In the case of special complex structure moduli of a rational elliptic surface, the elliptic fiber can degenerate into smaller irreducible pieces at some special points on the base $\IP^1$. In such a case, the matter curve can be one of the components ({\bf ii}). In these cases, the number of $E_6$ singularity points are
\bea
\# E_6 &=& (5K_S + \eta) \cdot (- 2K_S) + (5K_S + \eta) \cdot (6K_S + \eta), \nonumber \\
&=& {\rm deg}K_{\bar{c}_{{\bf 10}}} = 2g-2, \nn \\
&=& 0 \quad {\rm for}\; {\bf (i)},\\
&{\rm or}& -2 \quad  {\rm for}\; {\bf (ii)},
\eea
where $\bar{c}_{{\bf 10}}$ is a curve in the class
$|5K_S + \eta|$.
Hence, neither case realizes the configuration
$\# E_6 > 0$.
The negative result for case {\bf (ii)} means that the condition (\ref{flavor1}) will never be satisfied.  In case {\bf (i)}, the condition (\ref{flavor1}) means that there is no $E_6$ type singularity point in the GUT complex surface $S$, and the up-type Yukawa couplings need to be generated through D-brane instantons \cite{arXiv:0707.1871, arXiv:0811.2936}.

The general constraint \eqref{flavor2} cannot be
satisfied in the case of $\Gamma = \IZ_2$ for a rational elliptic
surface, either. Given the fact that $\# E_6$ is even,
while $\#E_6 = \#D_6$ cannot be realized, as we have seen above,
the only possibility
under $\Gamma = \IZ_2$ is $\# E_6 = 2$ and $\# D_6 = 1$.
This implies that
\beq
\# D_6 - \# E_6 = (5K_S + \eta) \cdot (-K_{S}) = -1.
\label{flavor3}
\eeq
Since $5K_{S}+ \eta$ is an effective divisor, the curve class can be written as $a f + \sum_i b_i C_i$ where $a$ and $b_i$ are positive integer. $f$ is the elliptic fiber class and $C_i$'s are curves which satisfy $C_i^2=-1, C_i \cdot f = 1$. Those are the generators of the cone for the effective curves. Due to the intersection number, $(af + \sum_i b_i C_i) \cdot (-K_{S}) = \sum_i b_i > 0$, it is impossible to satisfy \eqref{flavor3}.

\vspace*{1cm}
\noindent
{\bf Acknowledgments}: We would like to thank Kenji Hashimoto for discussions. T.K. and T.W. thanks the organizers of a program ``Branes,
Strings and Black holes'' in YITP, Kyoto University, where a part of the
work was done. This work was supported in part by a Grant-in-Aid
\# 23540286 (T.K.), by a Grant-in-Aid for Scientific Research on Innovative
Areas 2303 and by WPI Initiative from MEXT, Japan (T.W.).

\vspace*{1.5cm}


text:
\section{\label{sec:level1}Introduction}

	Investigation of graphene, a two-dimensional sheet of carbon atoms packed in a hexagonal lattice,
has clearly demonstrated unique opportunities of this material for both fundamental and applied physics.
The Dirac model is of primary importance for understanding the physical properties of graphene.
According to this model, at energies below one or two~eV~the quasiparticles in graphene are either
massless or very light and obey the linear dispersion relation although move with the
Fermi velocity~$\upsilon_F$~which is less than the speed of light $c$ by approximately a factor
of 300 (see \cite{1,2,3}).

	Theoretical description of the electrical conductivity and reflectances of graphene has been
developed in the framework of the Kubo response theory, the Boltzmann transport equation, the
formalism of the correlation functions in the random phase approximation and some other,
more phenomenological approaches, such as the two-dimensional Drude model. This provided insight
into the nature of optical properties of graphene and offered a clearer view on its universal
conductivity expressed via the electric charge $e$ and the Planck constant $\hbar$
 \cite{4,5,6,7,8,9,10,11,12,13,14}. Some of the theoretical approaches mentioned above have been
 applied also to calculate the Casimir (Casimir-Polder) force between two graphene sheets and
 between an atom and graphene which is caused by the zero-point and themal fluctuations of the electromagnetic field ~\cite{15,16,17,18,19,20,21,22,22a,23,24,25,26,27}. It should be mentioned,
 however, that the cited theoretical approaches are not based on the first principles of quantum electrodynamics at nonzero temperature which should be directly applicable to a simple system
 described by the Dirac model. In this connection, due to some ambiguities in the statistical
 description and limiting transitions, several conflicting results related to the conductivity,
 optical properties and Casimir forces in graphene systems have been obtained
 (see a discussion \cite{28,29,30}).

	The most fundamental description of the optical properties, Casimir effect and electrical
conductivity of graphene is given by the polarization tensor in (2+1)-dimensional space-time.
Although in some specific cases this quantity was considered by many authors (see, e.g., \cite{31,32,33}), the exact expressions valid at any temperature for graphene
with nonzero energy gap have been derived only recently \cite{34,35}.
The application region of the obtained polarization tensor was, however, restricted to pure
imaginary Matsubara frequencies. Because of this, it was possible to apply it for detailed
investigation of the Casimir effect in graphene systems ~\cite{30,36,37,38,39,40,41,42,43,44,45},
but not for calculations of the reflectances and conductivities of graphene which are defined
at real frequencies.

	The polarization tensor valid for a gapped graphene over the entire plane of complex
frequencies, including the real frequency axis, was derived in \cite{46,46a}.
In \cite{47,47a} the results of \cite{46,46a} were generalized for the case of graphene
with nonzero chemical potential. The novel form of the polarization tensor
\cite{46,46a,47,47a} has been used for a more detailed study of the Casimir effect in
graphene systems ~\cite{48,49,50,51,52,53,54} and for investigation of the electrical
conductivity of graphene on the basis of first principles of quantum electrodynamics
at nonzero temperature ~\cite{55,56,57,58}. Special attention was devoted to the
field-theoretical description of the reflectivity properties of graphene.
In \cite{46,46a} and ~\cite{59} they were investigated in detail for both pure (gapless)
and gapped graphene, respectively, and in \cite{60} the joint action of nonzero
energy gap and chemical potential was analyzed.

	Of particular interest is the configuration of a dielectric plate coated with a
graphene sheet. In addition to the fundamental interest in this subject, graphene coatings
find prospective technological applications to optical detectors, various optoelectronic
devices, solar cells, biosensors  etc.~\cite{61,62,63,64}. Using the polarization tensor
of \cite{46,46a}, the reflectivity properties of material plates coated with pure~\cite{65,65a}
and gapped~\cite{66} graphene have been investigated. The case of thin films coated with
gapped graphene has been considered as well~\cite{67}. However, the joint action of nonzero
energy gap and chemical potential on the reflectivity properties of graphene-coated plates,
which is of major interest for condensed matter physics and its applications,
remains to date unexplored.

	In this paper, we present the formalism using the polarization tensor of \cite{46,46a,47,47a} and
calculate the reflectance of the dielectric plate coated with real graphene sheet which possesses
nonzero energy gap
in the spectrum of quasiparticles and chemical potential.
Thus, here we exploit the same formalism, as was used in Refs.~\cite{48,49,50,51,52,53,54}
in the Casimir configurations of two parallel graphene sheets (graphene-coated plates),
but in quite different physical situation.
All computations are performed at
the room temperature $T=300~$K, at the liquid-nitrogen temperature $T=77~$K, and at the
liquid-helium temperature $T=4.4~$K for the plate made of silica glass SiO$_2$. It is shown
that there is an interesting interplay between the chemical potential, energy gap, frequency
and temperature in their impact on the
reflectance of a graphene-coated plate.

	Another result obtained is that at the fixed frequency $\omega$ the increasing chemical
potential and energy
gap affect the reflectance in opposite directions. Specifically, an increase of the chemical potential
either increases the reflectance or leaves it unchanged. On the other hand, an increase of
the energy gap either decreases the reflectance or leaves it constant. It is shown that for
any values of the chemical potential and energy gap the reflectance of a graphene-coated plate
approaches unity at sufficiently
low frequencies. With increasing frequency, the reflectance drops to that of an uncoated plate.
According to our results, for larger chemical potential the frequency region, where the
reflectance of
a graphene-coated plate depends on the presence of coating, becomes wider. However,
at frequencies $\hbar\omega>20~$meV the graphene coating makes no impact on the reflectance
of a SiO$_2$ plate any
more. An impact of temperature on the reflectance is more pronounced for graphene
coating with
smaller chemical potentials. For sufficiently high chemical potential, temperature
no longer affects
the reflectance. By and large, the presented formalism allows computation of the
impact of graphene
coating on the reflectance of a plate made of any material.

	The paper is organized as follows. In Section 2 we present the formalism of the
Dirac model expressing
the reflectance of a graphene-coated plate via the polarization tensor.
Section 3 contains the computational results for the reflectance as a function of
frequency and chemical potential under the
varying energy gap and temperature. In Section 4 the reflectance is considered
as a function of energy
gap under different values of the chemical potential and temperature.
In Section 5 the reader will find our conclusions and a discussion.

	\section{\label{sec:level1}General formalism for the reflectance}

We consider sufficiently thick dielectric plate, which can be treated as a semispace,
coated with gapped graphene possessing some chemical potential $\mu$. It is well known
that for a pure freestanding graphene the energy gap in the spectrum of quasiparticles
$\Delta=0$, but under an influence of the defects of structure, electron-electron
interaction and in the presence of a material substrate the nonzero gap
$\Delta\leq 0.1$
or $0.2~$eV arises~\cite{1,3,33,67aa,67ab}.
As to the chemical potential, it describes the doping
concentration, which is nonzero for any real graphene sample and takes typical values
$\mu\sim 0.1~$eV~\cite{3}. Below we consider only the case of  normal incidence
so that the projection of the wave vector on a graphene plane $k=0$
(a dependence on the incident angle is discussed in \cite{60,65,66}).
Because of this, it is sufficient to consider only the transverse magnetic (TM), i.e.,
$p$-polarized, electromagnetic waves of frequency $\omega$ incident on a graphene-coated plate.

	The TM reflection coefficient on the dielectric plate coated with a graphene sheet is given by~\cite{45}
	\begin{equation}
	r_{\rm TM}^{(g,p)}(\omega,0)=
	\frac{r_{\rm TM}^{(g)}(\omega, 0)+r_{\rm TM}^{(p)}(\omega, 0)
[1-2r_{\rm TM}^{(g)}(\omega, 0)]}{1-r_{\rm TM}^{(g)}(\omega, 0)r_{\rm TM}^{(p)}(\omega, 0)}, \label{eq1}
	\end{equation}
	where $r_{\rm TM}^{(g)}$ and $r_{\rm TM}^{(p)}$ are the TM reflection coefficients on graphene and on a plate taken singly, respectively (see Appendix for more details).
The magnitude of the wave vector projection on the plane of graphene is put equal to zero because we consider the normal incidence.

	The TM reflection coefficient of the electromagnetic waves on a freestanding graphene sheet is expressed as ~\cite{46,67}
	\begin{equation}
	r_{\rm TM}^{(g)}(\omega,0)=\frac{\tilde{\Pi}_{00}(\omega,0)}
	{2+\tilde{\Pi}_{00}(\omega,0)}, \label{eq2}
	\end{equation}
	where the 00 component of the normalized polarization tensor
of graphene  $\tilde{\Pi}_{00}(\omega,0)$ is connected with the conventional definition
 of the 00 component of this tensor by~\cite{67}
	\begin{equation}
	\tilde{\Pi}_{00}(\omega,0)=-\frac{i\omega}{\hbar c} \lim_{k \to 0} \frac{\Pi_{00} (\omega,k)}{k^2}. \label{eq3}
	\end{equation}
	The explicit expression for the quantity $\Pi_{00}$ for graphene with nonzero $\Delta$ and $\mu$ at any temperature $T$ is presented in \cite{57} (see Appendix concerning different but equivalent expressions for the reflection coefficients on a freestanding graphene sheet in terms of the polarization tensor, electric susceptibility and density-density correlation functions).

	For us it is now important that the 00 component of the polarization tensor is directly connected with the in-plane conductivity of graphene ~\cite{44,55,56,57}
	\begin{equation}
	\Pi_{00} (\omega,k)=\frac{4 \pi i \hbar k^2}{\omega} \sigma (\omega,k). \label{eq4}
	\end{equation}
	Combining this equation with (\ref{eq3}), one obtains
	\begin{equation}
	\tilde{\Pi}_{00}(\omega,0)=\frac{4 \pi }{c} \sigma (\omega,0). \label{eq5}
	\end{equation}
	As a result, the TM reflection coefficient on graphene (\ref{eq2}) is expressed in terms
of the in-plane conductivity
	\begin{equation}
	r_{\rm TM}^{(g)}(\omega,0)=\frac{2 \pi \sigma (\omega, 0)}{c+2 \pi \sigma (\omega, 0)}. \label{eq6}
	\end{equation}

	Now we return to the reflection coefficient $r_{\rm TM}^{(p)}$ on an uncoated dielectric plate. At the normal incidence it is given by the commonly known expression
	\begin{equation}
	r_{\rm TM}^{(p)}(\omega,0)=\frac{\sqrt{\varepsilon (\omega)}-1}{\sqrt{\varepsilon (\omega)}+1}=\frac{n(\omega)-1+ik(\omega)}{n(\omega)+1+ik(\omega)}, \label{eq7}
	\end{equation}
	where $\varepsilon (\omega)$ is the frequency-dependent dielectric permittivity of the plate material
and $n(\omega)$ and $k(\omega)$ are the real and imaginary parts of its complex index of
refraction, respectively. Representing the complex conductivity of graphene as
	\begin{equation}
	\sigma (\omega, 0)={\rm Re}\sigma (\omega)+i {\rm Im}\sigma (\omega, 0), \label{eq8}
	\end{equation}
	and substituting (\ref{eq6})--(\ref{eq8}) in (\ref{eq1}),
one finds the reflection coefficient on a graphene-coated plate
	\begin{equation}
	r_{\rm TM}^{(g,p)}(\omega,0)=
\frac{n(\omega)-1+\frac{4\pi}{c}{\rm Re}\sigma (\omega,0)+
i[k(\omega)+\frac{4\pi}{c} {\rm Im} \sigma (\omega,0)]} {n(\omega)+1+
\frac{4\pi}{c}{\rm Re}\sigma (\omega,0)+i[k(\omega)+
\frac{4\pi}{c} {\rm Im}\sigma (\omega,0)]}. \label{eq9}
	\end{equation}
{}	From (\ref{eq9}) one easily obtains the reflectance of the plate coated with a graphene sheet
	\begin{eqnarray}
&&
	{\cal R}(\omega)=[r_{\rm TM}^{(g,p)}(\omega,0)]^2
\label{eq10}	\\
&&
=\frac{[n(\omega)-1+\frac{4\pi}{c}{\rm Re}\sigma (\omega,0)]^2+
[k(\omega)+\frac{4\pi}{c} {\rm Im}\sigma (\omega,0)]^2} {[n(\omega)+1+
\frac{4\pi}{c}{\rm Re}\sigma (\omega,0)]^2+[k(\omega)+\frac{4\pi}{c} {\rm Im}\sigma (\omega,0)]^2}.
\nonumber	\end{eqnarray}

The exact expressions for real and imaginary parts of the in-plane conductivity of
graphene $\sigma(\omega,0)$ have been found in \cite{57} by using the polarization tensor.
For the real part it was obtained
\begin{equation}
\hspace*{-1cm}
{\rm Re}\sigma (\omega,0)=\sigma_0 \theta(\hbar \omega - \Delta)
	\frac{(\hbar \omega)^2+\Delta^2}{2(\hbar \omega)^2}
\left(\tanh\frac{\hbar \omega+2 \mu}{4 k_B T}+\tanh\frac{\hbar \omega-2 \mu}{4 k_B T}\right),
\label{eq11}
	\end{equation}
	where the universal conductivity of graphene is $\sigma_0={e^2}/(4 \hbar)$, the step function takes the values $\theta(x)=1$, $x\geq 0$ and $\theta(x)=0$, $x<0$, and $k_B$ is the Boltzmann constant.

	The imaginary part for the in-plane conductivity of graphene is somewhat more complicated~\cite{57}
	\begin{equation}
	{\rm Im}\sigma (\omega,0)=\frac{\sigma_0}{\pi}
\left[\frac{2 \Delta}{\hbar \omega}-\frac{(\hbar \omega)^2+\Delta^2}{(\hbar \omega)^2}
\ln\left|\frac{\hbar \omega +\Delta}{\hbar \omega -\Delta}\right|
+Y(\omega, \Delta, \mu)\right], \label{eq12}
	\end{equation}
	where the quantity $Y$ is defined as
	\begin{eqnarray}
&&
	Y(\omega, \Delta, \mu)=2\int_{\Delta/(\hbar \omega)}^\infty \!dt \sum_{\kappa=\pm 1}
\left(\exp\frac{\hbar \omega t+2\kappa \mu}{2k_BT}+1\right)^{-1}
\nonumber \\
&&~~~~~~~~~~~~~
\times\left[1+\frac{(\hbar \omega)^2+
\Delta^2}{(\hbar \omega)^2}\frac{1}{t^2-1}\right]. \label{eq13}
	\end{eqnarray}

Note that equations (\ref{eq11})--(\ref{eq13}) are exact in the framework of the Dirac model.
They are derived on the basis of first principles of quantum electrodynamics at nonzero
temperature. Taking into account that the Dirac model is applicable at sufficiently low
frequencies ($\hbar\omega<1-2~$eV), one arrives at the conclusion that these equations
give the proper description of all physical properties dealing with electrical conductivity
and reflection of light from graphene in this frequency region.
We remind that the imaginary part of the dielectric permittivity of graphene,  which is
proportional to the real part (\ref{eq11}) of its conductivity, is nonzero
over the wide frequency region.
This means that the concept of free Dirac quasiparticles allows the proper account of
energy losses by conduction electrons in graphene due to scattering processes
without resort to any additional parameters like in the 2-dimensional Drude model \cite{14}
or Kubo approach \cite{67j}.

Although physics behind
(\ref{eq11})--(\ref{eq13}) was already discussed in the literature \cite{57},
we briefly recall the meaning of different contributions to the conductivity.
Thus, the real part of conductivity (\ref{eq11}) is interpreted as originating from the
interband transitions \cite{57,67a}. The imaginary part of conductivity (\ref{eq12})
contains contributions from both the interband and intraband transitions.
To separate them analytically, one needs, however, to consider several distinct
asymptotic regions of parameters in the quantity $Y$ defind in (\ref{eq13}),
as is done in detail in \cite{57}. For one example, under the conditions
$k_BT\ll\mu$, $\hbar\omega\ll2\mu$, and $\Delta<2\mu$ one arrives at \cite{57}
\begin{equation}
{\rm Im}\sigma(\omega,0)=\frac{\sigma_0}{\pi}\left[
\vphantom{\frac{(\hbar \omega)^2+\Delta^2}{(\hbar \omega)^2}
\ln\left|\frac{\hbar \omega +\Delta}{\hbar \omega -\Delta}\right|}
\frac{4\mu}{\hbar\omega}-4\ln 2\frac{k_BT}{\hbar\omega}
-\frac{(\hbar \omega)^2+\Delta^2}{(\hbar \omega)^2}
\ln\frac{2\mu+\hbar \omega}{2\mu-\hbar \omega}\right].
\label{eq13a}
\end{equation}
\noindent
Here, the first two terms on the right-hand side correspond to the intraband transitions,
whereas the third term originates from the interband transitions \cite{7,57}
(see Ref.~\cite{57} for other relationships between $\hbar\omega$, $T$, $\mu$, and
$\Delta$ in (\ref{eq12}) and (\ref{eq13})).

	Using (\ref{eq10})--(\ref{eq13}) and the optical data for the complex index of
refraction of the plate material (see, for instance, \cite{68}), one can calculate
the reflectance of any material plate coated with real graphene sheet possessing nonzero
energy gap and chemical potential. Such computations are presented in the next sections.

	\section{\label{sec:level1}Reflectance as the function of frequency and chemical
potential at a varying energy gap}

	We apply the formalism of Sec.~II to describe the reflectance of a fused
	silica (SiO$_2$) plate coated with a graphene sheet. It has been known that
	fused silica plates are of frequent use as graphene
substrates \cite{69,70,71,72,73}.
	This material is characterized by the static dielectric permittivity
	$\varepsilon(0)\approx3.82$ which remains unchanged up to rather high frequency
	$\hbar \omega=10~$meV~\cite{68}. At higher frequencies the detailed optical
data for $n(\omega)$ and $k(\omega)$ for SiO$_2$ are contained in ~\cite{68}.

	Numerical computations of the reflectance of graphene-coated SiO$_2$
	plate are performed by using (\ref{eq10})--(\ref{eq13}). We start with the
dependence of reflectance on frequency for three different values of the chemical
potential and varying energy gap. The computational results are shown
	in figures~\ref{fig1}(a),~\ref{fig1}(b) and ~\ref{fig1}(c) at three temperatures $T=300~$K, $77$~K, $4.4$~K,
respectively.
	In each of the the gray regions from left
	to right are plotted for the values of the chemical potential
	$\mu=0$, $0.1$, and $0.8~$eV, respectively. The width of the gray regions
	is caused by the values of the energy gap which varies from
	$\Delta=0.1$~eV (the left boundary line of each region)
to $\Delta=0$ (the right boundary line of each region).
The bottom
	lines in ~\ref{fig1}(a),~\ref{fig1}(b) and ~\ref{fig1}(c), which are horizontal at
$\hbar\omega\leq 10~$meV, demonstrate the
	reflectance of an uncoated SiO$_2$ plate. It is obtained from
	~(10) by putting $\sigma(\omega,0)=0$
	\begin{equation}
	{\cal R}_0(\omega)=\frac{[n(\omega)-1]^2+k^2(\omega)}{[n(\omega)+1]^2+k^2(\omega)}. \label{eq14}
	\end{equation}
	In the frequency region  $\hbar\omega\leq 10~$meV
we have $k(\omega)=0$ \cite{68} and (\ref{eq14}) reduces to
	\begin{equation}
	{\cal R}_0(\omega)=
\left[\frac{\sqrt{\varepsilon(0)}-1}{\sqrt{\varepsilon(0)}+1}\right]^2
\approx0.1044. \label{eq15}
	\end{equation}

	As is seen in figure~\ref{fig1}, at the very low frequencies the reflectance of a
graphene-coated SiO$_2$ plate is close to unity and drops to the reflectance of an
uncoated plate with increasing frequency.
Physically this means that at low frequencies the graphene coating behaves like a metallic one,
whereas at high frequencies becomes transparent.
The frequency region, where this drop takes place,
shifts to higher frequencies with increasing chemical potential of graphene.
The width of the gray regions (i.e., the range of frequencies where the energy gap of
graphene sheet varies between zero and $0.1~$eV) quickly decreases with increasing $\mu$.
This means that for larger $\mu$ an impact of the energy gap on the reflectance quickly
decreases and for $\mu=0.8~$eV there is no impact of $\Delta$ on the reflectance at
any temperature. With decreasing temperature, some of the gray regions become wider.
This feature is especially pronounced for the undoped graphene ($\mu=0$).
Thus, at $T=77~$K and $4.4~$K the left boundaries of the left gray regions are missing
because they appear at much lower frequencies than those ones shown in figures~\ref{fig1}(b) and ~\ref{fig1}(c).
Note that for any reflectance smaller than unity the widths of all gray regions
increase with decreasing frequency, but decrease when the reflectance approaches
unity.
The right boundary lines of the left gray regions in figures~\ref{fig1}(a)
and \ref{fig1}(b), plotted for
the case $\mu=\Delta=0$ at $T=300~$K and $77~$K, coincide with the lines 3 and 2,
respectively, obtained earlier for this particular case in \cite{65}.

It is interesting that at a fixed frequency the increase of $\mu$ typically leads
to a larger (or the same) reflectance, whereas the increase of $\Delta$ results in a
smaller (or the same) reflectance, i.e., both quantities act in opposite directions.
This effect has simple physical explanation. Thus, increasing $\mu$ leads to an increased
density of charge carriers and increased conductivity resulting in the higher reflectance.
Just to the opposite, an increased energy gap leads to the decreased mobility of charge
carriers, decreased conductivity and, finally, to the lower reflectance.
At $\hbar \omega\geq 20~$meV the graphene coating does not make any impact on the
reflectance. Note that here and below we do not show the unobservable
infinitesimally narrow peaks ${\cal R}=1$ which appear under the condition
$\hbar \omega=\Delta$ (see \cite{66} for a discussion of this artefact).

	Now we consider a dependance of the reflectance on the chemical potential.
This is done separately at three different temperatures. In figure~\ref{fig2}, the
reflectance of a graphene-coated plate is shown as a function of $\mu$ at $T=300~$K.
The four gray regions from top to bottom are plotted for the frequency
$\hbar\omega=0.1$, $1$, $5$, and $10~$meV, respectively. In each of these regions,
the upper boundary line is plotted for the energy gap $\Delta=0$ and the lower
boundary line for $\Delta=0.1~$eV. The dashed lines are computed for $\Delta=0.05~$eV.
They are depictured only for sufficiently wide gray regions related to
$\hbar\omega=0.1$ and 1~meV. As is seen in figure~\ref{fig2}, with increasing
$\mu$ an impact of the energy gap on the reflectance quickly decreases in line with
figure~\ref{fig1}. The reflectance is the monotonously increasing function of $\mu$.
The maximum values of reflectance are reached faster for the light of lower frequency.

	Next, we present the computational results for the reflectance as a function of
$\mu$ at $T=77~$K. In figure~\ref{fig3} it is plotted (a) at $\hbar\omega=0.1$~meV and
(b) at $\hbar\omega=1.0$~meV. The upper and lower boundary lines of the gray regions
are plotted for the energy gap $\Delta=0$ and $0.1$~eV, respectively, whereas the
dashed middle lines for $\Delta=0.05~$eV. In an inset to figure~\ref{fig3}(b) the
region of small values of $\mu\leq 0.12~$eV is shown on an enlarged scale.
As can be seen in figure~\ref{fig3}, the reflectance is again a monotonously increasing
function of $\mu$. At $\omega=1.0~$meV [see figure~\ref{fig3}(b)] the lower boundary line
of the dashed region ($\Delta=0.1~$eV) is very similar to the respective line in
figure~\ref{fig2} plotted at $T=300~$K and differs from it by only a more sharp angle.
{}From figure~\ref{fig3}(a) it is seen that for $\mu\geq 0.12~$eV the reflectance
${\cal R}=1$ and does not depend on $\mu$. At $\hbar\omega=1.0~$meV the reflectance
approaches to unity at much larger $\mu\approx0.8~$eV [see figure~\ref{fig3}(b)].
We do not present
the computational results for a reflectance at $\hbar\omega=5~$and $10~$meV because
at $T=77~$K they are nearly the same as at $T=300~$K (see figure~\ref{fig2}).

	Finally, we repeat computations of the reflectance of a graphene-coated SiO$_2$
plate as a function of $\mu$ at $T=4.4~$K. The computational results are shown in
figures~\ref{fig4}(a) and \ref{fig4}(b) for the values of frequency $\hbar\omega=0.1~$ and $1.0$~meV,
respectively. All notations are the same as already explained in figures~\ref{fig3}(a) and \ref{fig3}(b).
It is seen that figure~\ref{fig4}(a) differs from figure~\ref{fig3}(a) plotted at $\omega=0.1~$meV,
$T=77~$K by a sharper angle of the lower line restricting the gray region.
It is seen also that at $\hbar\omega=0.1~$meV, $T=4.4~$K the graphene coating with
$\mu=0$~does not influence on the reflectance at any value of the energy gap from 0
to 0.1~eV (this is not the case at $\hbar\omega=0.1~$meV, $T=77~$K).
Figure~\ref{fig4}(b) plotted at $\hbar\omega=1.0~$meV, $T=4.4~$K differs from
figure~\ref{fig3}(b) plotted at the same frequency but $T=77~$K by only a sharper
angle of the lower line restricting the gray region (see the inset).
In this case at $\mu=0$ the graphene coating does not make an impact on the reflectance
at both $77~$K and $4.4~$K. At $T=4.4~$K the computational results for the reflectance
at the frequencies $\hbar\omega=5~$ and $10~$meV, are again nearly the same as
at $T=300~$K. They are shown by the bottom line and by the next to it gray region
in figure~\ref{fig2}.

	\section{\label{sec:level1}Reflectance as the function of energy gap under
 different chemical potentials and frequencies}

	In this section we present the computational results for the reflectance of
a graphene-coated SiO$_2$
plate as a function of the energy gap. In so doing, the chemical potential
$\mu$ may at times be fixed,
whereas the frequency takes several values, and at other times the fixed quantity
is the frequency and
the chemical potential takes different values. All computations are
again performed by using (\ref{eq10})--(\ref{eq13}).
In figure~\ref{fig5} the reflectance is shown as a function of
$\Delta$ for graphene coating with $\mu=0.01~$eV  at (a) $T=300~$K and
(b) $T=77~$K.
In figure~\ref{fig5}(a) the seven lines from top to bottom are plotted
for the values of
$\hbar\omega=~0.01$, $0.05$, $0.1$, $0.25$, $0.5$, $1$, and $\geq5$~meV,
 respectively.
In figure~\ref{fig5}(b) there are five lines for the values of
$\hbar\omega=0.01$, $0.05$, $0.1$, $0.5$
and $\geq 1$~eV. In both figures 5a and 5b the computational
results do not change when
$\hbar\omega$ becomes larger than $5$ and $1$~meV, respectively.
When decreasing temperature to below
$77$~K, the computational results do not change, so that figure~\ref{fig5}(b)
remains applicable.
As is seen in figure~\ref{fig5}, the reflectance decreases monotonously
with increasing $\Delta$.
At lower temperature this decrease goes on faster.

	In figure~\ref{fig6} similar computational results for the reflectance as
a function of $\Delta$ at $T=300~$K for graphene coating with $\mu=0.1~$eV.
It is interesting that for the relatively large $\mu$ the reflectance is
already temperature-independent, when decreasing $T$ to below $300$~K.
Because of this, the computational results at $T=77~$K and $4.4$~K are not shown.
The seven lines from top to bottom are plotted for $\hbar\omega=0.01$,
 $0.1$,  $0.5$, $1$, 2.5, 5, and 10~meV, respectively.
As is seen in figure~\ref{fig6}, in this case the reflectance varies with $\mu$ very slowly.

	Now we consider the dependence of the reflectance on energy gap at the fixed
frequency and different values of $\mu$. In figure~\ref{fig7} the reflectance of a
graphene-coated plate is shown as a function of the energy gap at
$\hbar\omega=0.1~$meV and (a) $T=300~$K, (b) $T=77~$K, and (c) $T=4.4~$K.
The seven lines from bottom to top are plotted for the values of chemical
potential $\mu=0$, $0.01$, $0.03$, $0.05$, $0.07$, $0.1$, and $\geq 0.5~$eV,
respectively. As is seen in figure~\ref{fig7}, the reflectance is again a decreasing
function of $\Delta$. At fixed $\Delta$ it takes either larger or the same values
for larger chemical potential. With decreasing temperature, a decrease of the
reflectance with increasing $\Delta$ becomes faster. At $T=4.4~$K [figure~\ref{fig7}(c)]
the graphene coating with $\mu=0$ does not influence on the reflectance shown by the bottom line.
At fixed $\Delta$ the impact of temperature on the reflectance is more pronounced for graphene
coating with smaller chemical potential. This is because at zero temperature the properties of
graphene with small $\mu$ are almost the same as for an undoped graphene.

	Finally, in figure~\ref{fig8} the computational results for the reflectance are shown
as the function of $\Delta$ at $\hbar\omega=1~$meV and (a) $T=300~$K, (b) $T=77~$K
(the values of the reflectance computed at $T=77~$K do not change with further decreasing
temperature). The seven lines from bottom to top are plotted for $\mu\leq 0.01~$eV,
$\mu=0.03$, $0.07$, $0.1$, $0.2$, $0.5$, and $0.8~$eV, respectively. As can be seen in
figure~\ref{fig8}, the top two lines corresponding to the largest values of the chemical
potential are nearly flat, i.e., the reflectance does not depend on $\Delta$ at any
temperature $T\leq 300$~K. It is seen also that the respective lines in figures~\ref{fig8}(a)
and ~\ref{fig8}(b) are very similar, i.e., the impact of temperature is very minor for
all considered values of the chemical potential.

	\section{\label{sec:level1}Conclusions and discussion}

	In this paper, we have presented the formalism, based of the first principles of
quantum electrodynamics at nonzero temperature, which allows to compute the reflectance of
a dielectric plate coated with real graphene sheet possessing nonzero energy gap and chemical
potential. In doing so, the graphene sheet is described by the polarization tensor in
(2+1)-dimensional space-time, and the dielectric plate by the measured optical data for
the complex index of refraction of the plate material. This formalism was applied to the
case of a graphene-coated plate made of silica glass at room, liquid-nitrogen, and
liquid-helium temperatures.

The results of numerical computations show that there is a nontrivial interplay in the
joint action of the energy gap, chemical potential, frequency of the incident light,
and temperature on the reflectance. According to our results, an increase of the chemical
potential and the energy gap of graphene coating influences the reflectance in opposite
directions by making it larger and smaller, respectively.
The physical explanation for this result is provided.
It is shown that at sufficiently
low frequencies the reflectance of a graphene-coated plate approaches unity, but goes
down to the reflectance of an uncoated plate at sufficiently
high frequencies.
In this respect graphene coating is qualitatively similar to a metallic film.
At $\hbar\omega\geq  20~$meV the graphene coating does not affect
the reflectance of a plate made of silica glass independently of the values of the energy
gap and chemical potential. Temperature makes major impact on the reflectance for graphene
coatings with relatively small chemical potential. For sufficiently high chemical potential,
a decrease from the room temp45erature down to the liquid-helium value does not influence
on the reflectance.

We underline that the graphene coating significantly
affects the reflectance only at frequencies
below approximately $\hbar\omega=20~$meV, i.e., fully in the application region of the
Dirac model which extends up to 1--2~eV. This means that the developed formalism can be
reliably used for controlling the reflectance of graphene-coated plates, e.g.,
for making it larger or smaller, by modifying the frequency, temperature, energy gap or
the doping concentration.

It would be of much interest to compare the predicted effects with experimental data.
Currently, however, available experiments are performed at higher frequencies.
Thus, the reflectance of graphene-coated SiO${}_2$ substrates was measured at
$\hbar\omega$ above 60~meV \cite{84}, 130~meV \cite{85}, and 500~meV \cite{67a}.
In these frequency regions, an impact of graphene on the reflectance of a substrate
is rather small and
was found in agreement with calculations using the polarization tensor with
$\Delta=\mu=0$ \cite{65,65a} (note  that with increasing frequency the influence of
nonzero $\Delta$ and $\mu$ on the graphene reflectance decreases).
The relative infrared reflectance of graphene deposited
on a SiC substrate, investigated for different values of $\mu$ at
$\hbar\omega\geq 12~$meV \cite{86}, is in qualitative agreement with  the computational
results presented in Fig.~\ref{fig1}.
We conclude that large impact of graphene coating on a substrate reflectance in
the far infrared region and
the possibilities to control it predicted above might be beneficial
for numerous applications of graphene-coated plates mentioned in section 1.

	\ack{V.~M.~M. was partially funded by the Russian Foundation for Basic
Research Grant No. 19-02-00453 A.
The work of V.~M.~M. was also partially supported by the Russian Government Program of
Competitive Growth of Kazan Federal University.}

{