\section{Introduction}
	In face biometrics, the ability to extract well-defined facial features is of vital importance. For systems trained using only visual domain-based face images, the feature extraction process is highly specialized for such images. In addition, the performance of face recognition for images taken in low-light or no-light environments is heavily impacted. The use of sensors that operate in other spectra, such as infrared, rather than the visual band has been applied in the past to alleviate this problem.  The infrared spectrum is of particular interest due to its ability to capture ``heat signature'' and is not dependent on lighting.

	Face biometrics is the primary technology in contemporary automated physical access security systems. Current approaches to face verification and identification are reaching outstanding performance, it is heavily restricted to the use of the visual or RGB band of the electromagnetic radiation. Turning focus to other spectra gave rise to infrared face recognition, cross-spectral recognition, and cross-spectrum synthesis.

	Cross-spectral subject recognition leverages existing techniques optimized for visible images. It attempts to create or generate another latent representation in the targeted domain, in order to compare it to the ones obtained from the visible images. An approach that used the polarization-state information was proposed in \cite{gurton2014enhanced,short2015exploiting}. Another approach created multi-spectral images by encoding the magnitude and phase of the Gabor responses \cite{nicolo2012long}.  A method to transform thermal face images into the visible domain using generative adversarial networks (Thermal-to-Visible GAN) is proposed in \cite{zhang2018tv}.

	The purpose of cross-spectrum synthesis is to generate images in the desired domain using the information from another domain. The authors of \cite{juefei2015nir} describe a method to reconstruct visible images using near-infrared (NIR) images, and vice versa using joint dictionary learning. A synthesis technique proposed in \cite{riggan2018thermal} uses the global and local regions of the face to create highly discriminative visible-like faces that allow for great verification performance.

	An evaluation of using the existing Convolutional Neural Networks (CNNs) trained on visible images to extract features for the purpose of cross-spectral, or heterogeneous, face recognition was reported in \cite{saxena2016heterogeneous}. The experiments with varying metric learning strategies showed that the results from pre-trained CNNs are on par or better than the state-of-the-art techniques developed for NIR images \cite{saxena2016heterogeneous}.

	A deep network model proposed by \cite{guo2017face} used both NIR and visual images for face recognition. An adaptive score fusion strategy was used for the final classification, allowing for better performance in combating the problem of illumination in face recognition.

	\section{The Proposed Approach}\label{sec:framework}

	In this paper, the goal is to study the feasibility of face verification and identification in the visual spectrum providing only infrared face images as probes. We propose the following approach to reach this goal. We use a Generative Adversarial Network (GAN) to synthesize the visible images from the thermal images, and then apply face recognition techniques on the synthesized images. We start with a set of both the visual and thermal images, and normalize the pairs of such using the facial landmarks. After normalization, we use a GAN to learn and generate the mapping between the thermal and visible images. Finally, we evaluate the performance of face recognition using the synthesized visible images and three different CNNs, namely InceptionV3 \cite{szegedy2016rethinking}, Xception \cite{chollet2017xception}, and MobileNet \cite{howard2017mobilenets}.

	\subsection{Database}
	In this paper, the data was taken from the Carl database \cite{espinosa2013new,espinosa2010criterion} which consists of visible and thermal images collected using a thermographic camera TESTO 880-3. The database contains a total of 41 subjects. For each subject, four image acquisition sessions were performed, each with three different lighting settings (natural illumination, infrared illumination, artificial illumination), and five images for each setting. The resolution of the images is 640x480 and 160x120 for the visible and thermal images, respectively.

	\subsection{Normalization}
	Since the collection of visible and thermal images both require different sensors, the images taken are generally of different resolution and are misaligned. In this paper, we perform a normalization process that consists of both face alignment and image resizing across the thermal and visual domains.

	The normalization process starts with locating the face and then attempts to find the facial landmarks. This stage is only applied to the visual images, as the detector used for facial landmarking is not optimal for thermal images. Due to this challenge, we have developed a mapping function that translates the pixel coordinates from the visual to the thermal image, such that the facial landmarks found in one are imposed onto the other. This approach is explained below.

	We apply a pre-trained detector, \texttt{dlib} \cite{king2009dlib}, that extracts both the 68 landmarks and the location of the face from the visible and thermal images. The extracted faces are centered based on the eyes and nose landmarks, and are resized to the image resolution of 96x96 pixels. Figure \ref{fig:normalization} shows an example of the landmark extraction and centering process that results in the mapping, or alignment of the images in both spectra.
	\begin{figure}[!ht]
		\centering
		\includegraphics[width=0.45\textwidth,interpolate]{fig/normalization.png}
		\caption{Landmark extraction and centering of a face from the thermal and visible domain.}
		\label{fig:normalization}
	\end{figure}

	\subsection{Synthesis}
	To generate the synthetic images in the visible spectrum given the infrared images, we propose to use a CycleGAN structure \cite{zhu2017unpaired} to learn the mapping between the images in two different spectra. We modified the CycleGAN structures so it can learn and transform a thermal image into a visible image. One of the major benefits of using CycleGAN for image synthesis is that the pairing (based on the same subject whose images in both spectra are acquired) is not required. It means that the GAN can be trained on a set of thermal images and visual images that are not necessarily taken from the same subjects or in the same setup.  Figure \ref{fig:gan} illustrates the overall CycleGAN structure, consisting of two generators, $G_{V->T}$ and $G_{T->V}$, and two discriminators, $D_{V}$ and $G_{T}$.

	\begin{figure}[!ht]
		\centering
		\includegraphics[width=0.45\textwidth,interpolate]{fig/gan.png}
		\caption{The architecture of the CycleGAN. $v$ and $t$ are the original visual and thermal images, while $\hat{V}$ and $\hat{T}$ are the synthesized images, respectively.}
		\label{fig:gan}
	\end{figure}

	A generic generator, $G_{X->Y}$, enables the process of synthesizing an image in the domain $Y$ from an image in the domain $X$ based on the learned mapping $G: X->Y$. In this paper, the CycleGAN uses two generators, $G_{V->T}$ and $G_{T->V}$, that translates images from the visual domain, $V$, to the thermal domain, $T$, and vice versa. Figure \ref{fig:gen} illustrates the generator architecture of the CycleGAN. The architecture is based on a modified version of the U-Net CNN \cite{ronneberger2015u}. In this paper, we perform convolution with $stride=2$ in order to implement the downsampling processing, as opposed to U-Net which uses $2\times 2$ max pooling. Due to this change, the upsampling process is also slightly modified: the 2D upsampling is performed on the concatenated results between the convolution output and the skip connection.

	\begin{figure*}[!ht]
		\centering
		\includegraphics[width=0.85\textwidth,interpolate]{fig/generator.png}
		\caption{The generator of the CycleGAN.  The downsampling process is implemented by using  a stride of 2 during the convolution procedure.  Upsampling rescales the input by a factor of 2 using interpolation.  The blue dashed lines represent the skip connections.}
		\label{fig:gen}
	\end{figure*}

	A discriminator, $D$, attempts to determine whether the synthesized image $\hat{x}$ is real or fake. This is performed by learning to distinguish between a real image $x$ and a fake image $\hat{x}$. The discriminator chosen in this paper is shown in Figure \ref{fig:disc}. It is based on the patchGAN classifier \cite{li2016precomputed}. The benefit of using a patchGAN classifier is that the output is an $NxN$ matrix, as opposed to a single value (binary Yes or No), as each element in the $NxN$ matrix contains values indicating whether the patch $x_{ij}$ is real or fake.

	\begin{figure*}[!ht]
		\centering
		\includegraphics[width=0.85\textwidth,interpolate]{fig/discriminator.png}
		\caption{Discriminator of the CycleGAN.}
		\label{fig:disc}
	\end{figure*}

	Together, the generator and the discriminator form two players, $G$ and $D$, of a min-max game where $G$ attempts to generate an image that can fool $D$, while $D$ tries to correctly distinguish between the real and fake images. This process can be interpreted as an adversarial loss \cite{goodfellow2014generative}. In the case of CycleGAN, a new loss called a cycle consistency loss \cite{zhou2016learning}, is introduced that also needs to be minimized. The cyclic loss assumes that given two generators, $G_{V->T}$ and $G_{T->V}$, we are able to execute the approximations $G_{T->V}(G_{V->T}(v))\approx v$ and $G_{V->T}(G_{T->V}(t))\approx t$.

	\subsection{Identification}\label{sec:identification}

	For identification, we use a four-fold cross-validation process where each fold represents a different acquisition session. Identification is based on a $1:N$ comparison where we attempt to find the identity of the subject. For evaluation, we apply transfer learning on three different CNNs: InceptionV3, Xception, and MobileNet.

	We train each network with the set of real (baseline) visual images acquired in three separate sessions, and validate using the images from the remaining session. After selecting the optimal parameters for the particular validation set, we test the final performance of the system with the synthesized visual images. We compare the performance between the baseline and the synthesized images.

	The process of transfer learning for each model involves loading the pre-existing weights optimized for the ImageNet challenge. The top fully-connected and classification layers from each model are replaced with an average pooling layer, two fully-connected layers with 512 units, and a classification layer. Two sequential training process is applied to each model. Initially, the model is trained with the bottom layers' weights frozen, allowing only the newly attached classification layers to update at a higher learning rate. After updating the classification layers, the entire model is finely-tuned with a small learning rate.

	\subsection{Verification}\label{sec:verification}

	For verification, we also apply a four-fold cross-validation process where each fold represents a different acquisition session. Verification is defined as a $1:1$ comparison where we attempt to confirm the claimed identity is correct.  In addition, the evaluation is based on the output of all three networks: InceptionV3, Xception, and MobileNet. In contrast to identification, the verification system uses the networks to generate a feature template for matching.

	In this paper, we examine the features obtained from the fully-connected and classification layer. From the fully-connect layer, we can extract 512 features, since there are 512 units in the fully-connected layer, while the classification layer produces 41 score-level features. Using these features as a basis of a template, we compare the similarity between the enrolled and the probe features. We evaluate the performance of common metrics that can be used to compute the similarity between the features.

	Table \ref{tab:metrics} denotes the selected metrics used in this paper for calculating the similarity between the features. The evaluation metrics are based on the distance measures that have been used for comparing different probability distributions. A survey performed in \cite{cha2007comprehensive} shows the common metrics used for calculating similarity/dissimilarity between such distributions. A more exhaustive list of distances can be found in \cite{deza2006dictionary}. In this paper, we have chosen the best performing metric from each family to illustrate their performance.

	\begin{table}[!htb]
		\begin{footnotesize}
			\centering
			\caption{Evaluation Metrics}\label{tab:metrics}
			\begin{tabular}{@{}|ll|@{}}
				\hline
				City Block	&	\parbox{6cm}{\begin{equation}	\sum\limits_{i = 1} ^d\left|p_i-q_i\right|	\end{equation}}	\\
				\hline
				Kulczynski d	&	\parbox{6cm}{\begin{equation}	\frac{\sum\limits_{i = 1} ^d\left|p_i-q_i\right|}{\sum\limits_{i = 1} ^d\min(p_i,q_i)}	\end{equation}}	\\
				\hline
				Czekanowski	&	\parbox{6cm}{\begin{equation}	1-\frac{2\sum\limits_{i = 1} ^d\min(p_i,q_i)}{\sum\limits_{i = 1} ^d (p_i+q_i)}	\end{equation}}	\\
				\hline
				Dice	&	\parbox{6cm}{\begin{equation}\label{eq:dice}	1-\frac{2\sum\limits_{i = 1} ^d p_i q_i}{\sum\limits_{i = 1} ^d {p_i}^2 + \sum\limits_{i = 1} ^d {q_i}^2}	\end{equation}}	\\
				\hline
				Squared	&	\parbox{6cm}{\begin{equation}	\sum\limits_{i = 1} ^d \frac{(p_i-q_i)^2}{p_i+q_i}	\end{equation}}	\\
				\hline
				Squared-Chord	&	\parbox{6cm}{\begin{equation}	\sum\limits_{i = 1} ^d (\sqrt{p_i}-\sqrt{q_i})^2	\end{equation}}	\\
				\hline
				Jensen-Shannon	&	\parbox{6cm}{\begin{equation}	\frac{1}{2}\left[\sum\limits_{i = 1} ^d p_i \ln\frac{2p_i}{p_i+q_i} + \sum\limits_{i = 1}^d q_i \ln\frac{2q_i}{p_i+q_i}\right]	\end{equation}}	\\
				\hline
			\end{tabular}
		\end{footnotesize}
	\end{table}

	%

	%

	\section{Experiments and Results}\label{sec:experiments}

	To evaluate the performance of CycleGAN used for visual image synthesis for both the identification and verification. A four-fold cross-validation technique is used, and the real visual images are used for training and validation, and the synthesized images are used for testing.

	\subsection{Image Synthesis}

	The original images are taken from the Carl database, cropped and normalized based on the facial landmarks. Each of the thermal and visual images is used to train the CycleGAN such that visual images can be synthesized from the thermal images while the thermal images can be generated from the visual images. Figure \ref{fig:visual} illustrates the original and synthesized faces in both thermal and visual domains.
	\begin{figure}[!ht]
		\centering
		\includegraphics[width=0.48\textwidth,interpolate]{fig/created.png}
		\caption{Samples of real visual and thermal images of three different subjects from the Carl database (left panel), and the generated images using a CycleGAN (right panel).}
		\label{fig:visual}
	\end{figure}

	\subsection{Face Identification}

	The performance of face identification ($1:N$) is evaluated in this work using the rank performance and the true acceptance rate at the targeted 1\% and 0.1\% False Acceptance Rate (FAR).  For evaluation, each of the targeted FARs corresponds to a specific acceptance threshold. The testing set was formed based on visual images synthesized from the real thermal images using the GAN trained on both visual and thermal images. The validation set was formed from the original images in the same domain (visual) as the training images.

	Table \ref{tab:id} reports the performance for the validation and test sets, for the three network models. Validation set (Valid) shows high and consistent performance because the validation images come from the same domain as the training images. The test set (Test) produces low performance due to the fact that the test sets are the results of synthesis from thermal images. For testing, the Xception model shows the best performance in terms of acceptance rate at the targeted FAR. However, when evaluating the top rank or rank-1 performance, the MobileNet model demonstrates the highest performance. The difference in the performance indicates that Xception provides a wider spread of confident scores which are better for selecting a specific operational point, whereas MobileNet is better for a ranked-based approach.

	\begin{table}[!htb]
		\centering
		\begin{footnotesize}
			\caption{Average identification performance (\%) and standard deviation across four-fold cross-validation}\label{tab:id}
			\begin{tabular}{@{}ll||c|c|c@{}}
				\multicolumn{2}{c||}{\multirow{2}{*}{Models}}	&	\multicolumn{2}{c|}{True Acceptance Rate}	&	\multicolumn{1}{c}{\multirow{2}{*}{Rank-1}}	\\
				&& \multicolumn{1}{c|}{@ 1\% FAR} & \multicolumn{1}{c|}{@ 0.1\% FAR} & \\
				\hline
				\hline
				\multirow{3}{*}{\rotatebox[origin=c]{90}{Test}}	&	InceptionV3	&	37.95	$\pm$	1.92	&	14.80	$\pm$	1.54	&	45.03	$\pm$	0.76	\\
				&	Xception	&	\textbf{43.83	$\pm$	1.38}	&	\textbf{20.80	$\pm$	1.34}	&	35.12	$\pm$	2.09	\\
				&	MobileNet	&	28.45	$\pm$	1.37	&	9.90	$\pm$	0.94	&	\textbf{48.91	$\pm$	1.99}	\\
				\hline
				\multirow{3}{*}{\rotatebox[origin=c]{90}{Valid}}	&	InceptionV3	&	\textbf{94.20	$\pm$	2.43}	&	\textbf{85.31	$\pm$	3.18}	&	94.41	$\pm$	1.74	\\
				&	Xception	&	\textbf{94.20	$\pm$	3.04}	&	84.74	$\pm$	6.95	&	91.79	$\pm$	2.87	\\
				&	MobileNet	&	92.53	$\pm$	3.63	&	80.74	$\pm$	6.09	&	\textbf{95.35	$\pm$	1.95}	\\
			\end{tabular}
		\end{footnotesize}
	\end{table}

	Figure \ref{fig:cmc} illustrates the performance of different network models when accepting identities at increasing ranks. The performance is evaluated in terms of the Cumulative Matching Characteristic (CMC) curve which characterizes the identification rate at different rank levels. The CMC curve shows that each of the models operates at a similar performance, with the Xception model performing the worst for the test set.
	\begin{figure}[!ht]
		\centering
		\includegraphics[width=0.48\textwidth,interpolate]{fig/cmc41-eps-converted-to.pdf}
		\caption{CMC curve for InceptionV3, Xception, and MobileNet using the Carl database. Test case indicates the use of the synthesized visual images, and validation case indicates the use of real visual images.}
		\label{fig:cmc}
	\end{figure}

	\subsection{Face Verification}
	For verification, we evaluate the performance using the Receiver Operating Characteristic (ROC) curves, the True Acceptance Rate at targeted FAR, equal error rate (EER), and Area Under the Curve (AUC). First, we examine the performance between the scores obtained from the classification layer and the features extracted from the Fully-Connected (FC) layers. Next, we examine the influence of different similarity metrics for comparing the enrolled and probe feature templates.

	Verification is a process of matching a probe template to a previously enrolled template ($1:1$ matching). We created two cohorts of ROC curves (Figure \ref{fig:roc41} and \ref{fig:roc512}) for each of the three CNN models: InceptionV3, Xception, and MobileNet, using the Dice metric (Table \ref{tab:metrics}, Equation \ref{eq:dice}). The first group is based on using the score-level features extracted from the classification layer. The second group is generated based on the features from the FC layer. Table \ref{tab:veriCmp} reports the performance between the score-level features and FC features.

	As shown in Table \ref{tab:veriCmp}, the FC features outperform the score-level features regardless of the network model, test vs. validation set, and performance metrics. This result indicates that in the process of combining the 512 FC features, the classification layer loses some important information when translating the FC features into the 41 scores. For validation using the score-level features, the Xception model offers the best performance while InceptionV3 slightly outperforms Xception when using FC features. For the testing, both Xception and InceptionV3 offers similar performance for both the score-level and FC features.

	\begin{figure}[!ht]
		\centering
		\includegraphics[width=0.48\textwidth,interpolate]{fig/roc41dice-eps-converted-to.pdf}
		\caption{ROC curve using 41 score-level features evaluated using Dice similarity metric.}
		\label{fig:roc41}
	\end{figure}

	\begin{figure}[!ht]
		\centering
		\includegraphics[width=0.48\textwidth,interpolate]{fig/roc512dice-eps-converted-to.pdf}
		\caption{ROC curve created using Dice similarity metric and 512 fully-connected features.}
		\label{fig:roc512}
	\end{figure}

	\begin{table*}[!htb]
		\centering
		\begin{footnotesize}
			\caption{Averaged verification performance (\%) and standard deviation using Dice metric across four-fold cross-validation} \label{tab:veriCmp}
			\begin{tabular}{@{}ll||r|r|r|r||r|r|r|r@{}}
				\multicolumn{2}{c||}{\multirow{3}{*}{Models}}		&	\multicolumn{4}{c||}{41 Score-Level Features}		&	\multicolumn{4}{c}{512 Features From the Fully-Connected Layer}											\\
				&& \multicolumn{2}{c|}{True Acceptance Rate} & \multicolumn{1}{c|}{\multirow{2}{*}{EER}} & \multicolumn{1}{c||}{\multirow{2}{*}{AUC}} & \multicolumn{2}{c|}{True Acceptance Rate} & \multicolumn{1}{c|}{\multirow{2}{*}{EER}} & \multicolumn{1}{c}{\multirow{2}{*}{AUC}} \\
				&&\multicolumn{1}{c|}{@ 1\% FAR}&\multicolumn{1}{c|}{@ 0.1\% FAR}&&&\multicolumn{1}{c|}{@ 1\% FAR}&\multicolumn{1}{c|}{@ 0.1\% FAR}&&\\
				\hline																														\hline
				\multirow{3}{*}{\rotatebox[origin=c]{90}{Test}}	&	InceptionV3	&	23.79	$\pm$	2.24	&	7.46	$\pm$	1.15	&	23.90	$\pm$	1.39	&	83.58	$\pm$	1.68	&	30.25	$\pm$	2.14	&	10.21	$\pm$	1.34	&	18.80	$\pm$	0.52	&	89.37	$\pm$	0.49	\\
				&	Xception	&	23.98	$\pm$	1.18	&	7.36	$\pm$	0.72	&	23.26	$\pm$	1.49	&	84.59	$\pm$	1.69	&	31.16	$\pm$	1.08	&	10.98	$\pm$	1.17	&	18.80	$\pm$	0.65	&	89.25	$\pm$	0.63	\\
				&	MobileNet	&	15.70	$\pm$	0.93	&	4.04	$\pm$	0.28	&	29.32	$\pm$	1.67	&	77.22	$\pm$	2.21	&	20.60	$\pm$	1.30	&	5.44	$\pm$	0.55	&	23.40	$\pm$	1.17	&	84.68	$\pm$	1.18	\\
				\multirow{3}{*}{\rotatebox[origin=c]{90}{Valid}}	&	InceptionV3	&	70.97	$\pm$	5.12	&	44.62	$\pm$	7.64	&	8.73	$\pm$	1.60	&	96.84	$\pm$	1.38	&	85.43	$\pm$	2.55	&	62.90	$\pm$	6.87	&	4.95	$\pm$	0.92	&	98.93	$\pm$	0.50	\\
				&	Xception	&	73.44	$\pm$	5.72	&	47.24	$\pm$	6.82	&	7.90	$\pm$	1.97	&	97.43	$\pm$	1.21	&	82.71	$\pm$	4.74	&	58.85	$\pm$	6.48	&	5.31	$\pm$	1.38	&	98.73	$\pm$	0.63	\\
				&	MobileNet	&	58.45	$\pm$	3.38	&	33.37	$\pm$	2.35	&	13.66	$\pm$	1.99	&	93.10	$\pm$	1.79	&	76.20	$\pm$	4.99	&	49.68	$\pm$	4.00	&	6.74	$\pm$	1.43	&	98.20	$\pm$	0.58	\\
			\end{tabular}
		\end{footnotesize}
	\end{table*}

	Table \ref{tab:idmetrics} reports the best performing metrics from each family using the FC features. As shown in Table \ref{tab:idmetrics}, Dice and Squared-Chord metrics offer on average the best performance for all three network models. Both InceptionV3 and Xception perform similarly, with InceptionV3 performing marginally better on the validation set and Xception performing better on the test set.

	\begin{table*}[!htb]
		\centering
		\begin{footnotesize}
			\caption{Averaged verification performance (\%) and standard deviation for various metrics using 512 features from the fully-connected layer across four-fold cross-validation} \label{tab:idmetrics}
			\begin{tabular}{@{}l||r|r|r|r||r|r|r|r@{}}
				\multirow{3}{*}{Metrics}	&	\multicolumn{4}{c||}{Test Set}		&	\multicolumn{4}{c}{Validation Set}											\\
				& \multicolumn{2}{c|}{True Acceptance Rate} & \multicolumn{1}{c|}{\multirow{2}{*}{EER}} & \multicolumn{1}{c||}{\multirow{2}{*}{AUC}} & \multicolumn{2}{c|}{True Acceptance Rate} & \multicolumn{1}{c|}{\multirow{2}{*}{EER}} & \multicolumn{1}{c}{\multirow{2}{*}{AUC}} \\
				&\multicolumn{1}{c|}{@ 1\% FAR}&\multicolumn{1}{c|}{@ 0.1\% FAR}&&&\multicolumn{1}{c|}{@ 1\% FAR}&\multicolumn{1}{c|}{@ 0.1\% FAR}&&\\
				\hline																														\hline
				\multicolumn{9}{c}{InceptionV3}\\
				City Block	&	27.61	$\pm$	2.79	&	8.80	$\pm$	1.58	&	19.52	$\pm$	0.68	&	88.72	$\pm$	0.64	&	84.42	$\pm$	2.38	&	60.44	$\pm$	6.45	&	4.91	$\pm$	0.86	&	98.91	$\pm$	0.55	\\

				Kulczynski d	&	\textbf{30.25	$\pm$	2.16}	&	10.06	$\pm$	1.16	&	18.72	$\pm$	0.51	&	89.46	$\pm$	0.49	&	85.89	$\pm$	2.53	&	63.22	$\pm$	6.44	&	4.88	$\pm$	0.89	&	98.97	$\pm$	0.46	\\

				Czekanowski	&	30.22	$\pm$	2.19	&	9.95	$\pm$	1.30	&	18.72	$\pm$	0.51	&	89.47	$\pm$	0.49	&	85.74	$\pm$	2.42	&	63.22	$\pm$	6.44	&	4.89	$\pm$	0.90	&	98.97	$\pm$	0.46	\\
				Dice	&	\textbf{30.25	$\pm$	2.14}	&	\textbf{10.21	$\pm$	1.34}	&	18.80	$\pm$	0.52	&	89.37	$\pm$	0.49	&	85.43	$\pm$	2.55	&	62.90	$\pm$	6.87	&	4.95	$\pm$	0.92	&	98.93	$\pm$	0.50	\\

				Squared-Chord	&	29.54	$\pm$	2.66	&	9.35	$\pm$	1.41	&	\textbf{18.30	$\pm$	0.48}	&	\textbf{89.92	$\pm$	0.49}	&	\textbf{86.74	$\pm$	2.64}	&	63.31	$\pm$	6.40	&	4.41	$\pm$	0.90	&	\textbf{99.13	$\pm$	0.42}	\\

				Squared	&	29.67	$\pm$	2.72	&	9.67	$\pm$	1.40	&	18.43	$\pm$	0.56	&	89.78	$\pm$	0.52	&	86.48	$\pm$	2.10	&	\textbf{63.90	$\pm$	6.38}	&	4.46	$\pm$	0.90	&	99.10	$\pm$	0.46	\\

				Jensen-Shannon	&	29.66	$\pm$	2.70	&	9.48	$\pm$	1.53	&	18.34	$\pm$	0.51	&	89.87	$\pm$	0.50	&	86.66	$\pm$	2.32	&	63.73	$\pm$	6.37	&	\textbf{4.40	$\pm$	0.85}	&	99.12	$\pm$	0.44	\\
				\hline
				\multicolumn{9}{c}{Xception}\\
				City Block	&	24.14	$\pm$	1.05	&	7.17	$\pm$	0.91	&	20.99	$\pm$	0.79	&	87.32	$\pm$	0.84	&	80.55	$\pm$	7.31	&	50.79	$\pm$	9.77	&	5.31	$\pm$	1.56	&	98.77	$\pm$	0.63	\\

				Kulczynski d	&	30.86	$\pm$	1.24	&	10.86	$\pm$	1.09	&	18.84	$\pm$	0.74	&	89.24	$\pm$	0.69	&	82.49	$\pm$	4.79	&	58.19	$\pm$	6.40	&	5.37	$\pm$	1.38	&	98.72	$\pm$	0.63	\\

				Czekanowski	&	30.85	$\pm$	1.24	&	10.94	$\pm$	1.00	&	18.84	$\pm$	0.74	&	89.24	$\pm$	0.69	&	82.51	$\pm$	4.79	&	58.21	$\pm$	6.40	&	5.35	$\pm$	1.40	&	98.73	$\pm$	0.63	\\

				Dice	&	\textbf{31.16	$\pm$	1.08}	&	\textbf{10.98	$\pm$	1.17}	&	18.80	$\pm$	0.65	&	89.25	$\pm$	0.63	&	82.71	$\pm$	4.74	&	\textbf{58.85	$\pm$	6.48}	&	5.31	$\pm$	1.38	&	98.73	$\pm$	0.63	\\

				Squared-Chord	&	29.37	$\pm$	1.00	&	9.23	$\pm$	0.83	&	\textbf{18.54	$\pm$	0.65}	&	\textbf{89.65	$\pm$	0.59}	&	84.05	$\pm$	5.28	&	56.45	$\pm$	8.58	&	\textbf{4.60	$\pm$	1.28}	&	\textbf{99.04	$\pm$	0.46}	\\

				Squared	&	28.81	$\pm$	1.03	&	9.14	$\pm$	0.84	&	18.97	$\pm$	0.66	&	89.24	$\pm$	0.61	&	84.33	$\pm$	5.83	&	56.98	$\pm$	8.90	&	4.63	$\pm$	1.36	&	99.03	$\pm$	0.50	\\

				Jensen-Shannon	&	29.16	$\pm$	0.99	&	9.23	$\pm$	0.83	&	18.73	$\pm$	0.65	&	89.47	$\pm$	0.59	&	\textbf{84.50	$\pm$	5.91}	&	56.80	$\pm$	8.72	&	4.61	$\pm$	1.29	&	\textbf{99.04	$\pm$	0.48}	\\
				\hline
				\multicolumn{9}{c}{MobileNet}\\
				City Block	&	15.27	$\pm$	0.54	&	3.68	$\pm$	0.20	&	25.96	$\pm$	0.73	&	81.93	$\pm$	0.85	&	69.48	$\pm$	3.99	&	39.49	$\pm$	3.72	&	8.05	$\pm$	1.25	&	97.55	$\pm$	0.65	\\

				Kulczynski d	&	20.50	$\pm$	1.42	&	\textbf{5.50	$\pm$	0.49}	&	23.35	$\pm$	1.24	&	84.73	$\pm$	1.28	&	76.64	$\pm$	5.10	&	49.89	$\pm$	4.48	&	6.66	$\pm$	1.44	&	98.25	$\pm$	0.58	\\

				Czekanowski	&	20.46	$\pm$	1.42	&	5.37	$\pm$	0.60	&	23.35	$\pm$	1.24	&	84.74	$\pm$	1.28	&	76.64	$\pm$	5.08	&	49.89	$\pm$	4.48	&	6.66	$\pm$	1.45	&	98.25	$\pm$	0.58	\\

				Dice	&	\textbf{20.60	$\pm$	1.30}	&	5.44	$\pm$	0.55	&	23.40	$\pm$	1.17	&	84.68	$\pm$	1.18	&	76.20	$\pm$	4.99	&	49.68	$\pm$	4.00	&	6.74	$\pm$	1.43	&	98.20	$\pm$	0.58	\\

				Squared-Chord	&	19.51	$\pm$	0.96	&	4.87	$\pm$	0.31	&	22.99	$\pm$	1.05	&	\textbf{85.15	$\pm$	1.11}	&	\textbf{78.78	$\pm$	4.27}	&	\textbf{50.25	$\pm$	4.97}	&	\textbf{5.96	$\pm$	1.12}	&	\textbf{98.60	$\pm$	0.43}	\\

				Squared	&	19.12	$\pm$	0.83	&	4.84	$\pm$	0.31	&	23.48	$\pm$	0.94	&	84.66	$\pm$	1.03	&	77.92	$\pm$	4.23	&	49.62	$\pm$	4.65	&	6.21	$\pm$	1.19	&	98.46	$\pm$	0.48	\\

				Jensen-Shannon	&	19.35	$\pm$	0.93	&	4.86	$\pm$	0.39	&	\textbf{23.19	$\pm$	0.98}	&	84.95	$\pm$	1.07	&	78.44	$\pm$	4.31	&	50.08	$\pm$	4.81	&	6.04	$\pm$	1.15	&	98.54	$\pm$	0.45	\\

			\end{tabular}
		\end{footnotesize}
	\end{table*}

	\section{Conclusions}\label{sec:conclusions}

	This paper proposes to use GANs to generate visual images from thermal images that can be used by a typical CNN for face verification or identification. We used a specific type of GAN, CycleGAN, to generate synthetic visual images from thermal images. We showed we're capable of fine-tuning pre-trained CNNs using the original visual images to recognize synthesized visual images. We examined three common CNNs, Inception V3, Xception, and MobileNet, and achieved a rank-1 face identification accuracy of 95.35\% and 48.91\% on the real and synthesized visual images, respectively. For face verification, an EER of 4.41\% and 18.30\% was achieved, using the Squared-Chord metric on the real and synthesized visual images, respectively.

	Further study is required using diverse data sets and better fine-tuning approaches in order to improve the performance of the classifiers.

	\section*{Acknowledgment}
	This research was partially supported by the Natural Sciences and Engineering Research Council Canada (NSERC SPG grant ``Biometric-enabled Identity Management and Risk Assessment for Smart Cities'').


\section{Introduction}
Atomic coherence induced by coherent light matter interaction has significant role for precise
control of the optical property of a medium. The observation of atomic coherence in an atomic
system leads us to uncover many spectacular optical effects. Among the most well known effect
is Electromagnetically Induced Transparency (EIT) \cite{RMP2005,GSA2012} in which an opaque
medium becomes eminently transparent for the probe field with the support of a control field.
Diverse applications such as slowing and stopping of light \cite{Juzeli2002,Liu2017,Arsenovi2018},
coherent storage and retrieval of light \cite{Patnaik2004,Xu2016,Rui2016,Katz2018},
Rydberg blockade induced interactions \cite{JDP2010},
diffraction control and guiding of light \cite{Evers2011, Cui2017}, structured beam generation \cite{Radwell,Sharma}, etc.
have been documented using EIT.
In multi-level atomic system, EIT enhances the nonlinear susceptibility
which conducts us to use nonlinear optical regime in the investigation many nonlinear optical phenomena such as,
Kerr nonlinearity \cite{Hamedi2015,Rajitha2015,Wu2015}, self-phase modulation (SPM) \cite{Wong2001,Kasparian2018},
cross-phase modulation (XPM)\cite{Sun2007,Dmochowski2016}, four-wave mixing (FWM) \cite{Lezama2000,Kang2004,Jeong2016}, etc.
In FWM process, three electro-magnetic fields interact in a nonlinear optical system and generate
electro-magnetic field with a new frequency.
Numerous experiments have been carried out to demostrate the enhanced FWM process
in multilevel atomic systems \cite{Epple2012,Lee2016,ZiYu2017,Juo2018,Zubairy2018}.
FWM process using EIT has been observed in both cold \cite{Lezama2000,Danielle2004,Kang2004,Lee2016,ZiYu2017,Juo2018,Zubairy2018} and
room temperature \cite{Li1996,Epple2012} atomic system.
Besides FWM process, enhanced higher order multi-wave mixing processes have been studied \cite{Zhang2007,Nie2008}.
Hoonsoo $et$ $al.$ experimentally demonstrated EIT based six-wave mixing (SWM) signal in a $N$-type cold atomic system
\cite{Hoonsoo2004}. Recently an experimental observation of FWM signal in a $N$-type cold atomic system has been studied
by Chang-Kai $et$ $al.$ at low light levels \cite{Chang-Kai2014}.

The four-level system not only generates new signal but also it permits the generated signal to
propagate through the nonlinear medium along with the probe field under the condition of EIT \cite{Kang2004,Chang-Kai2014}.
The effect of enhanced nonlinearity on these propagating signals through the atomic medium remains unexplored.
Also the shape of the generated FWM signal and the progression during its propagation through the inhomogeneously broadened medium
is not completely elucidated.
In this paper, we theoretically investigate all those questions
in a simple four-level $N$-type atomic configuration as shown in Fig \ref{Fig1}.
A weak probe field and a strong control field resonantly drive the system.
In order to efficiently generate a FWM signal,
it is essential that the phase-matching condition, $2\vec{K}_p=\vec{K}_c+\vec{K}_g$ is strictly fulfilled \cite{Epple2012}.
Here we conceive this by using a collinear geometry, which generates the FWM signal
in the same direction as the probe field propagates. The steady state of the optical Bloch equations
are numerically solved to study the atomic coherence created by the nonlinear atom-field interaction.
The numerical result clearly shows gain in the FWM spectrum which indicates the possibility of
generation of a $\sigma^+$-polarised ($\Delta m=+1$) signal. The frequency of the signal,
$\omega_g$ related to the frequency of the probe field, $\omega_p$ and the frequency of the control field,
$\omega_c$  by $\omega_g=2\omega_p-\omega_c$.

Apart from the generation and control of the FWM signal, the storage and retrieval of this signal along with the
probe has captivated enormous attention due to its potential application as an optical memory
\cite{Camacho2009,Zaremba2002,Irina2011,Jinghui2013}.
Recently many experiments demonstrate  simultaneous storage and retrieval of both the signals in multi-level
atomic system \cite{Irina2011,Jinghui2013}. But the correlation between the input and the output pulse shape has not yet been explored in details.
It is noticeable from the previous demonstrations that the retrieved FWM signal does not preserve its predifined
shape and also its intensity is reduced considerably. We overcome these limitations by considering a much
simpler degenerate atomic system in which only one probe and one control field is interacting nonlinearly.
In this paper we demonstrate the shape preserving storage and retrieval of the FWM signal without
compromising its intensity. We also find that the storage and retrieval process is robust with
both the adiabatiac and non-adiabatic switching of the control field.

The arrangement of the article is as follows. In section \ref{Model}, we configure the physical model and describe
the system using a semiclassical theory.
In section \ref{Dynamical}, we discuss the dynamical equations of motion for the $N$-type system using Liouville's equation.
In section \ref{Pulse}, we derive the pulse propagation equations for the optical fields.
In the next section \ref{Generation}, we investigate the generation and control of the FWM signal.
In section \ref{Storage}, storage and retrieval of the FWM signal is demonstrated.
In section \ref{analysis}, we derive the analytical expression of the nonlinear coherence under weak probe approximation
in order to explain the FWM scheme.
Finally in section \ref{Conclusion}, we briefly conclude our work.
\section{THEORETICAL MODEL}
\subsection{Model Configuration}
\label{Model}
In this work, the four wave mixing mechanism has been exploited for the generation and control of an optical signal.
The model system consists of an inhomogeneously broadened four level $^{85}$Rb atomic system interacting with two co-propagating optical fields as shown in Fig \ref{Fig1} .
The atomic transitions $\ket1\leftrightarrow\ket3$ and $\ket2\leftrightarrow\ket4$ are coupled by a $\pi$-polarised ($\Delta m=0$) probe field of frequency $\omega_p$, whereas $\sigma^-$-polarised ($\Delta m=-1$) control field with frequency $\omega_c$ couple $\ket1\leftrightarrow\ket4$ transition.
These optical fields are defined as
\begin{equation}\label{eq:field}
\vec{E}_{j}(z,t)=\hat e_{j}\mathcal E_{0j}(z,t)e^{i(k_j z-\omega_j t)}+c.c.,
\end{equation}
where $\mathcal E_{0j}(z,t)$ is the space-time dependent amplitude, $k_{j}=\omega_{j}/c $ is the propagation constant along $z$-direction and $\hat e_{j}$ is the polarisation unit vector of the optical field.
The subscript, $j\in \{p,c\}$ indicates the probe field and control field respectively.
The Hamiltonian describing the interaction between the four atomic level system with the two optical fields under electric-dipole approximation is as given below
\begin{equation}\label{eq:1}
\begin{aligned}
H^{'}=&\hbar \omega_{13}\ket1\bra1+\hbar(\omega_{13}-\omega_{14})\ket4\bra4\\
+&\hbar(\omega_{13}-\omega_{14}+\omega_{24})\ket2\bra2-\hbar\Omega_{p}e^{-i\omega_{p}t}\ket1\bra3\\
-&\hbar\Omega_{c}e^{-i\omega_{c}t}\ket1\bra4-\hbar\Omega_{p}e^{-i\omega_{p}t}\ket2\bra4 + h.c.,\\
\end{aligned}
\end{equation}
where the Rabi frequencies of the probe and control are defined by
\begin{equation}\label{eq:rabi}
\Omega_p=\frac{\vec{d}_{13}.\hat e_p}{\hbar} \mathcal E_{0p} \quad \textrm{and} \quad \Omega_c=\frac{\vec{d}_{14}.\hat e_c}{\hbar} \mathcal E_{0c}.
\end{equation}
The dipole moments for the atomic transitions between states $\ket1\leftrightarrow\ket3$, $\ket2\leftrightarrow\ket4$ and $\ket1\leftrightarrow\ket4$ are denoted by $\hat {d}_{13}$, $\hat {d}_{24}$, and $\hat {d}_{14}$, respectively.
Note that $\hat {d}_{13}$ = $\hat {d}_{24}$ according to the Clebsch-Gordan coefficients of the considered level scheme.
We perform the following unitary transformation in order to remove the explicit time dependence from the Hamiltonian
\begin{equation}
H=U^\dagger H^{'} U -i\hbar U^\dagger \frac{\partial U}{\partial t},
\end{equation}
where $U$ is defined as
\begin{equation}
U=e^{-i\left(\omega_{p}\ket1\bra1 + (\omega_{p}-\omega_{c})\ket4\bra4 + (2\omega_{p}-\omega_{c})\ket2\bra2\right)t}.
\end{equation}
Now the transformed Hamiltonian $H$  takes the following form
\begin{equation}\label{eq:2}
\begin{aligned}
H=-&\hbar\Delta_p\ket1\bra1-\hbar(\Delta_p-\Delta_c)\ket4\bra4\\
-&\hbar(\Delta_p-\Delta_c+\Delta_q)\ket2\bra2-\hbar\Omega_p\ket1\bra3\\
-&\hbar\Omega_c\ket1\bra4-\hbar\Omega_p\ket2\bra4 + h.c.,\\
\end{aligned}
\end{equation}
where $\Delta_p=\omega_p-\omega_{13}$, and $\Delta_q=\omega_p-\omega_{24}$  are the detunings of the probe field with $\ket1\leftrightarrow\ket3$ and $\ket2\leftrightarrow\ket4$ transitions, respectively.
The detuning due to the control field corresponding to  $\ket1\leftrightarrow\ket4$ transition is given by $\Delta_c=\omega_c-\omega_{14}$.
\begin{figure}[t]
\includegraphics[scale=0.35]{Figure1.eps}
\caption{(a) A simple block diagram of the the model system. (b) Schematic representation of the four-level $N$-type
atomic system. We use degenerate hyperfine sublevels of
$5^2 S_{\frac{1}{2}}$ and $5^2 P_{\frac{1}{2}}$. Two metastable ground states are defined as
$\ket 3=\ket {F=3,m_F=1}$ and $\ket 4=\ket {F=3,m_F=2}$.
Two excited states are defined as $\ket 1=\ket {F^{'}=2,m_{F^{'}}=1}$
and $\ket 2=\ket {F^{'}=2,m_{F^{'}}=2}$.}
\label{Fig1}
\end{figure}
\subsection{Dynamical Equations}
\label{Dynamical}
We use Liouville's equation to account for the various radiative and non-radiative decay processes of the atomic system.
The decay of the atomic system is caused by various mechanisms such as flight-through broadening, population exchange, and atom-atom and atom-wall collisions.
To govern the response of the atomic populations and coherences of the four-level atomic system, the following density matrix equations are employed

\begin{equation}
\dot \rho=-\frac{i}{\hbar}[H,\rho]+ \mathcal{L}_\rho
\end{equation}

where the second term represents the decay processes that can be determined by
\begin{align}
\mathcal{L}\rho = \mathcal{L}_{r}\rho+\mathcal{L}_{c}\rho
 \label{decay1}
\end{align}
with
\begin{align}
\mathcal{L}_{r}\rho = &-\sum\limits_{i=1}^2\sum\limits_{j=3}^4 \frac{\gamma_{ji}}{2}\left(\ket i \bra i \rho-2\ket j \bra j \rho_{ii}+\rho\ket i \bra i \right)\,.
\label{decay2}
\end{align}
The spontaneous decay rates from the excited state $\ket i$, $(i \in 1,2)$ to the ground state $\ket j$, $(j \in 3,4)$ are denoted by $\gamma_{ji}$  in Eq.(\ref{decay2}).
The dephasing in the ground state [$\mathcal{L}_{c}\rho$ in Eq.(\ref{decay1})] is due to collision at a rate $\gamma_c$.
Now the dynamics of the model system can be obtained in the following form

\begin{equation}\label{eq:r}
\begin{aligned}
\dot\rho_{11}&=-(\gamma_{31}+\gamma_{41})\rho_{11}+i\Omega_p\rho_{31}+i\Omega_c\rho_{41}-i\Omega_p^*\rho_{13}\\
&-i\Omega_c^*\rho_{14},\\
\dot\rho_{12}&=-[i(\Delta_q-\Delta_c)+\frac{1}{2}(\gamma_{31}+\gamma_{41}+\gamma_{32}+\gamma_{42})]\rho_{12}\\
&+i\Omega_p\rho_{32}+i\Omega_c\rho_{42}-i\Omega_p^*\rho_{14},\\
\dot\rho_{13}&=[i\Delta_p-\frac{1}{2}(\gamma_{31}+\gamma_{41})]\rho_{13}+i\Omega_p(\rho_{33}-\rho_{11})\\
&+i\Omega_c\rho_{43},\\
\dot\rho_{14}&=[i\Delta_c-\frac{1}{2}(\gamma_{31}+\gamma_{41})]\rho_{14}+i\Omega_p(\rho_{34}-\rho_{12})\\
&+i\Omega_c(\rho_{44}-\rho_{11}),\\
\dot\rho_{22}&=-(\gamma_{32}+\gamma_{42})\rho_{22}+i\Omega_p\rho_{42}-i\Omega_p^*\rho_{24},\\
\dot\rho_{23}&=[i(\Delta_p-\Delta_c+\Delta_q)-\frac{1}{2}(\gamma_{32}+\gamma_{42})]\rho_{23}\\
&+i\Omega_p(\rho_{43}-\rho_{21}),\\
\dot\rho_{24}&=[i\Delta_q-\frac{1}{2}(\gamma_{32}+\gamma_{42})]\rho_{24}+i\Omega_p(\rho_{44}-\rho_{22})\\
&-i\Omega_c\rho_{21},\\
\dot\rho_{33}&=\gamma_{31}\rho_{11}+\gamma_{32}\rho_{22}+i\Omega_p^*\rho_{13}-i\Omega_p\rho_{31}\\
\dot\rho_{34}&=-[i(\Delta_p-\Delta_c)+\gamma_{c}]\rho_{34}+i\Omega_p^*\rho_{14}-i\Omega_p\rho_{32}\\
&-i\Omega_c\rho_{31},\\
\dot\rho_{44}&=-(\dot\rho_{11}+\dot\rho_{22}+\dot\rho_{33})\\
\dot\rho_{ij}&=\dot\rho_{ji}^*
\end{aligned}
\end{equation}
where the overdot stands for the time derivative and star $(*)$ denotes the complex conjugate.
For an inhomogeneously broadened medium, frequency shift due to finite atomic motion is inhibited.
In presence of Doppler shift, the probe and control field detuning are modified to
$\Delta^{'}_{p}=\Delta_{p}-k_{p}v$, $\Delta^{'}_{q}=\Delta_{p}-k_{p}v$, $\Delta^{'}_{c}=\Delta_{c}-k_{c}v$, respectively.
The sign of frequency shift $\pm k_{i}v, (i\in{p,q,c})$ indicates the motion of atoms either counter propagating or co-propagating.
Therefore, these effects can be incorporated into the equations of motion (\ref{eq:r}) by taken into account the avearging over a Maxwell velocity distribution.
Hence the velocity averaging of the atomic coherences $\langle\rho_{ij}(z,t)\rangle$ can be expressed as
\begin{equation}
\langle\rho_{ij}(z,t)\rangle= \int \rho_{ij}(z,v,t)\mathcal{P}(kv)d(kv),
\end{equation}
where $\mathcal{P}(kv)d(kv)$ is probability that an atom has a velocity between $v$ and $v + dv$ and obeyed by the Maxwell-Boltzmann velocity distribution
\begin{equation}\label{eq:distribution}
\mathcal{P}(kv)d(kv)=\frac{1}{\sqrt{2\pi D^{2}}}  e^{-\frac{(kv)^2}{2D^2}} d(kv).
\end{equation}
The Doppler line width D  is given by $D=\sqrt{k_{B}T\nu^2_{c}/Mc^2}$, where $M$ is the atomic mass, $k_{B}$ is the Boltzmann constant and $T$ is the thermal equilibrium temperature.
At room temperature (T=300K), the Doppler width, $D$ is 37$\gamma$ for $^{85}$Rb atoms.
\subsection{Pulse Propagation Equations}
\label{Pulse}
In this section, we use Maxwell's equations to govern the spatio-temporal evaluation of an optical field through a nonlinear medium.
The wave equation for the probe, control and generated fields can be expressed as
\begin{equation}\label{eq:wave_eq}
\left(\nabla^2+\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\vec{E}=\frac{4\pi}{c^2}\frac{\partial^2 \vec{\mathcal P}}{\partial t^2},
\end{equation}
where $\vec{E}=\vec{E}_p+\vec{E}_c+\vec{E}_g$ is the total field and its induced polarization  is $\vec{\mathcal P}$.
The source term  $\vec{\mathcal P}$ that appears in the right hand side of Eq.(\ref{eq:wave_eq}) is the origin of  linear and non-linear response of the medium.
The macroscopic polarization $\vec{\mathcal P}$ can be defined in terms of the atomic coherences as
\begin{equation}\label{eq:polarization}
\begin{aligned}
\vec{\mathcal P}&=\mathcal N((\vec{d}_{13}\rho_{13}+\vec{d}_{24}\rho_{24})e^{-i\omega_p t}\nonumber\\
&+\vec{d}_{14}\rho_{14}e^{-i\omega_c t}+\vec{d}_{23}\rho_{23}e^{-i\omega_g t}+c.c.,)
\end{aligned}
\end{equation}
where $\mathcal{N}$ is the atomic density.
Note that the atomic density under consideration is weak, otherwise local field correction need to be taken into account.
Under slowly-varying envelope approximation (SVEA), Eqs.(\ref{eq:wave_eq}) can be reformulated as
\begin{equation}\label{eq:14}
\begin{aligned}
&\left(\frac{\partial }{\partial z}+\frac{1}{c}\frac{\partial }{\partial t}\right)\Omega_p=i\eta_p\left\langle \rho_{13}(z,t)+\rho_{24}(z,t) \right\rangle,\\
&\left(\frac{\partial }{\partial z}+\frac{1}{c}\frac{\partial }{\partial t}\right)\Omega_c=i\eta_c\langle\rho_{14}(z,t)\rangle,\\
&\left(\frac{\partial }{\partial z}+\frac{1}{c}\frac{\partial }{\partial t}\right)\Omega_g=i\eta_g\langle\rho_{23}(z,t)\rangle.\\
\end{aligned}
\end{equation}
The coupling constant for the probe, control and generated fields are defined as
$\eta_p=\eta_c=\eta_g=\eta=3 \mathcal N \gamma \lambda^2/8 \pi$.
For simplicity, we consider $\gamma=\gamma_{j1}=\gamma_{j2}$ $(j \in 3,4)$, and $\lambda_{13}=\lambda_{14}=\lambda_{23}=\lambda$.
The angular bracket denotes a statistical average over the velocity distribution of the atom.
We introduce the following co-moving coordinate system in order to perform numerical computation:
\begin{equation}
\tau=t-\frac{z}{c},\hspace{0.25cm}\xi=z.
\end{equation}
 Therefore, the expressions within the round bracket of Eq.(\ref{eq:14}) can be easily substituted by $\partial/\partial \xi$ in  the frame of moving coordinate system,
Subsequently, simultaneous solutions of Bloch Eqs.(\ref{eq:r}) and Maxwell's Eqs.(\ref{eq:14})  in space-time coordinate explore the dynamical
progression of the optical fields inside the medium.

\section{Generation and control of FWM signal}
\label{Generation}
In this section, we proceed with numerical simulation of Maxwell-Bloch equations to reveal
the generation of optical field  and its control by exploiting FWM mechanism.
We have adopted Cash-Karp Runge-Kutta method in order to solve coupled partial differential equations.
We begin with lay out of the initial and boundary conditions of the atoms and fields.
The time dependent envelope of the probe field at the entry face of the medium can be written as
\begin{equation}\label{eq:probeshape}
\Omega_p(\xi=0,\tau)=\Omega_p^0 e^{-(\frac{\tau-\tau_0}{\sigma_p})^2},
\end{equation}
where $\Omega_p^0$, $\sigma_p$ and $\tau_0$ are the amplitude, temporal width and peak location of the Gaussian shaped probe field, respectively.
In our entire computation, we have assumed that the amplitude of control field ($\Omega^{0}_{c}=3.0\gamma$) is larger than the amplitude of the probe field ($\Omega^{0}_{p}=0.1\gamma$) and
subsequently the dynamical evolution of control field can be neglected.
Initially all atoms are occupied in the $\ket{3}$ state whereas all other states are unoccupied for $\xi\in(0,L)$.
\begin{figure}
\includegraphics[scale=0.725]{Figure2.eps}
\caption{Propagation dynamics of the generated FWM signal as function of position and time.
The parameters are $\Omega_p^0=0.1\gamma$, $\Omega_c^0=3.0\gamma$, $\mathcal N=1\times 10^{12}$atoms/cm$^3$, $\lambda=7.95\times 10^{-5}$cm,
$\Delta_p=\Delta_c=\Delta_{q}=\Delta_g=0$,
$\sigma_p=75/\gamma$, $\gamma\tau_0$=300, $\gamma_{c}=0$.}
\label{Fig2}
\end{figure}

Our first finding on the generation of optical pulse in the presence of a continuous wave (cw) control field is shown in Fig.\ref{Fig2} .
Figure (\ref{Fig2}) shows the temporal variation of generated field at different propagation distances.
It is clear from  Fig. (\ref{Fig2}) that the amplitude of the generated field increases gradually and takes the probe field shape while propagating along the medium.
Note that temporal shape of the generated field remains unchanged after it attains saturation intensity.
For the sake of generality, we compare the temporal shape and width of the generated field with the probe field as in Fig. \ref{Fig3}.
It can be seen that the time dependent envelope of the generated field is same as the input probe field, except its width is narrowed by a factor of $\sqrt{2}$.
We also notice that the temporal shapes of both generated and probe fields propagate through the medium without absorption and distortion.
Therefore, only the cw control field efficiently clones the temporal shape of probe field to generate signals and protect them during propagation.
\begin{figure}
\includegraphics[scale=0.35]{Figure3.eps
\caption{Normalised intensity profile, $I/I_{peak}$ of the Gaussian shaped probe pulse and FWM signal at
input ($\eta\xi/\gamma=0$) and output ($\eta\xi/\gamma=20$) boundary of the medium. Temporal peak position of the
output pulses are shifted at $\gamma\tau$=300. Inset figure shows the actual peak location.
All other parameters are same as in Fig. (\ref{Fig2}).}
\label{Fig3}
\end{figure}

In order to prove the efficiency and the robustness of the model system, we next consider input probe field to be an amplitude modulated Gaussian pulse.
For this purpose, the input envelope for the probe field is expressed as follows:
\begin{equation}\label{eq:dprobeshape}
\Omega_p(\xi=0,\tau)=\Omega_p^0\left(1+m_a\cos \omega_m t \right) e^{-(\frac{\tau-\tau_0}{\sigma_p})^2}
\end{equation}
where $m_a$ and $\omega_m$ are termed as the depth of modulation and frequency of the modulating signal, respectively.
In practice,  $\omega_m$ is small compared to the carrier frequency $\omega_p$ of the probe pulse.
Figure \ref{Fig4} depicts the intensity profile of the probe and generated fields as a function of $\gamma\tau$ at propagation distance $\eta\xi/\gamma = 20$.
This figure confirms precisely that the generated FWM signal gets its shape from the envelope of the probe field and propagates as a shape preserving pulse.
However the time resolution of the generated envelope is higher as compared to the probe pulse.
\begin{figure}
\includegraphics[scale=0.35]{Figure4.eps}
\caption{Intensity profile of the amplitude modulated Gaussian shaped probe pulse and generated FWM signal
at output ($\eta\xi/\gamma=20$) boundary of the medium.
The parameters for the probe pulse are $\Omega_p^0=0.06\gamma$, $m_a=0.65$, $w_m=\gamma/5$, $\sigma_p=75/\gamma$ and $\gamma\tau_0$=300.
Other parameters are same as in Fig. (\ref{Fig2}).}
\label{Fig4}
\end{figure}

\section{Storage and Retrieval of EM radiations}
\label{Storage}
In the previous section, we have demonstrated how the time independent control field efficiently generates and controls the propagaton of FWM signal.
Here we explore the dynamics of  the generated field in presence of time dependent profile of control field.
We address this issue by considering the temporal profile of control field to be of inverted super-Gaussian shape which is defined as:
\begin{equation}
\Omega_c(\xi=0,\tau)=\Omega_c^0 \Big[1-e^{-\big(\frac{\tau-\tau_c}{\sigma_c}\big)^{\alpha}}\Big],
\end{equation}
where the parameter $\alpha$ regulates the rapidity of the switching action of the control field.
Switching off and on of control field intensity in time domain holds the key of storage and retrieval process.
Figure \ref{Fig5} displays the spatio-temporal characteristics of probe and FWM fields.
The initial profile of the probe field  is chosen as a Gaussian.
As seen from Fig. \ref{Fig5} the probe field intensity gradually diminishes due to dynamical reduction of control intensity.
Simultaneously FWM signal is formed and its spatio-temopral evolution follows the same dynamical behaviour as the probe field.

The control field not only generates the FWM signal but also enables the storage and retrieval of the signal along with the
probe pulse by temporal variation of its own intensity.
The intensity lowering (rising) to zero (maximum) with time produces switching off (on) of the control field as shown in inset of Fig.\ref{Fig5}.
The spatiotemporal evolution of probe and generated signals in the presence of super Gaussian shaped control field is shown in Fig \ref{Fig5}.
The signal and probe fields are depicted by solid red and dashed blue lines, respectively.
Figure \ref{Fig5} shows that the generation of signal field is accomplished within a  short length of medium in the presence of both control and probe fields.
It is also evident from Fig.\ref{Fig5} that storage of signal and probe fields start as soon as both peaks experience the falling intensity of the control field.
Hence the intensities of both signal and probe fields are gradually stored by adiabatic switching off the control field during the propagation through the atomic medium.
The storing of the fields can be completed by making intensity of control field zero.
The stored pulses can be retrieved on demand by switching on the control field.
The retrieval process of stored fields can be initiated by switching on of the control field at a later time.
The maximum intensity of the control field leads to retrieval of both probe and signal from the medium without loss of generality.
\begin{figure}
\includegraphics[scale=0.725]{Figure5.eps}
\caption{Storage and retrieval of Gaussian probe pulse (dashed blue line, $i=p$) and generated FWM signal (solid red line, $i=g$)
are demonstrated. We multiply $10^{2}$ and $10^{5}$ with the probe field intensity ($|\Omega_p/\gamma|^2$) and FWM signal intensity
($|\Omega_g/\gamma|^2$) respectively. Inset figure shows intensity profile the control field (solid green).
The parameters are $\sigma_c=180/\gamma$, $\gamma\tau_c=550$, $\alpha=4$ and all other parameters are same as in Fig. (\ref{Fig2}).}
\label{Fig5}
\end{figure}

Next we investigate the ground state atomic coherence which is solely responsible for the storage and retrieval of electromagnetic radiation in an atomic medium \cite{LukinPRL2000}.
Subsequently we have plotted the atomic coherence $\rho_{43}$ as a function of time at different propagation distances
as shown in Figure \ref{Fig6}.
As depicted from Figure \ref{Fig6}, at entry face of the medium $\eta\xi/\gamma=0$,
$\rho_{43}$ takes the temporal shape of probe field in presence of constant control.
The atomic coherence attains its maximum value when control field is switched off.
The probe and signal pulses are stored inside the medium in the form of ground atomic coherence
by switching off the control field.
This coherence ($\rho_{43}$) is well preserved inside the medium and  efficiently can be retrieved before it decays at a rate of $\Gamma_{43}$.
The atomic coherence starts generating the replica of the stored pulses after the control field is switched on.
The Raman scattering between stored atomic coherence and control field intensity produces back the stored signal.
Therefore, the atomic coherence plays the main role
behind the storage and retrieval of the probe as well as the FWM signal.

\begin{figure}
\includegraphics[scale=0.35]{Figure6.eps}
\caption{The temporal profile of ground state atomic coherence, $\rho_{43}$ is plotted as
a function of time for different propagation distances. All parameters are same as in figure \ref{Fig5}.}
\label{Fig6}
\end{figure}

\section{ANALYSIS AND DISCUSSIONS}
\label{analysis}
\subsection{Perturbative Analysis}
In this section, we derive an analytical expression for the atomic coherence which can successfully explain the generation of signal due to four wave mixing in four level atomic system.
The perturbative expression for the coherence and population is determined under a  weak probe approximation ($\Omega_p\ll\Omega_c$) that are correct in all orders for the control field of Rabi frequencies $\Omega_c$ and $2^{nd}$ order in probe Rabi frequencies $\Omega_p$.
The solutions of the density matrix equations can be approximated as
\begin{equation}
\begin{aligned}
\rho_{ij}&=\rho_{ij}^{(0)}+\Omega_p\rho_{ij}^{(1)}+\Omega_p^* \rho_{ij}^{(2)}+\Omega_p^2 \rho_{ij}^{(3)}+|\Omega_p|^2 \rho_{ij}^{(4)}\nonumber\\
&+\Omega_p^{*2} \rho_{ij}^{(5)}\\
\end{aligned}
\end{equation}
where $\rho_{ij}^{(0)}$ describes the solution in the absence of $\Omega_p$ and $\rho_{ij}^{(k)}$, $k\in \{1,2,3,4,5\}$ is
the higher order solution in the presence of weak $\Omega_p$.
The steady-state value of the atomic coherence, $\rho_{23}$ can be expressed
by the following expression
\begin{equation}\label{eq:10}
\rho_{23}=\frac{\frac{i\Omega_p^2\Omega_{c}^*}{[i\Delta_{p^{'}}-\frac{\gamma_{31}+\gamma_{41}}{2}][i(\Delta_{p^{'}}-\Delta_{c^{'}}+\Delta_{q^{'}})-\frac{\gamma_{32}+\gamma_{42}}{2}]}}{\bigg[i(\Delta_{p^{'}}-\Delta_{c^{'}})-\Gamma_{34}+\frac{\Omega_{c}^2}{i\Delta_{p^{'}}-\frac{\gamma_{31}+\gamma_{41}}{2}}\bigg]}.
\end{equation}
The above expression corresponds to the four wave mixing in a system of four level atoms and produces a frequency  $\omega_g=2\omega_p-\omega_c$.
The induced nonlinear atomic polarization $\vec{P}^{NL}$ is expressed as
$\vec{P}^{NL} =\chi^{(3)} (\omega_g) \vec{E}_p^2 \vec{E}^*_c$ where $\chi^{(3)}(\omega_g)$ is a $3^{\textrm{rd}}$ order nonlinearity.
Under the Doppler-broaden, the nonlinear susceptibility  $\chi^{(3)}(\omega_g)$  can be written as
\begin{equation}
\langle \chi^{(3)} (\omega_g) \rangle=\frac{\mathcal N |\vec{d}_{23}| |\vec{d}_{13}|^2 |\vec{d}_{14}|}{\hbar^3 |\Omega_p|^2 \Omega^*_c} \langle \rho_{23} \rangle,
\end{equation}
where $\mathcal N$ is the atomic density.

\subsection{Nonlinear Susceptibility}
We next study the medium with Kerr nonlinearity by using two different approaches:
firstly we use weak-approximated analytical solution (\ref{eq:10}) to obtain response of the generated signal coherence.
Secondly, we solve full  density matrix equations (\ref{eq:r}) numerically at steady state limit
to find the nonlinear susceptibility of signal.
The results from both approaches are in good agreement as shown in Fig.\ref{Figure7}.
It ensures that a new optical field generation is possible even in the weak probe regime.
The gain of the generated signal can be enhanced by reducing the control field intensity.
This is possible due to the dark state ($\Omega_p \ll \Omega_c$) becoming feeble as the control field intensity is reduced significantly, as
causing the population leaves from dark state and populate in the excited states.
Hence the efficiency of nonlinear signal generation can be enhanced with a suitable value of control field intensity.

\begin{figure}
\includegraphics[scale=0.35]{Figure7.eps}
\caption{Imaginary part of nonlinear Doppler averaged coherence, $\langle\rho_{23}\rangle$ as a function of $\Delta_p$ in units of $\gamma$
for two different approach and for various values of $\Omega_c$. The parameters are $\Omega_p=0.1\gamma$,
$\gamma=2\pi\times5.75\times10^6$ Hz, $D=37\gamma$, T=300K.}
\label{Figure7}
\end{figure}

\section{Conclusion}
\label{Conclusion}
In conclusion, we have demonstrated an efficient generation and control of a degenerate FWM signal by optical cloning
in a $N$-type inhomogeneously broadened atomic system.
A strong control field and a weak probe field tailored the medium susceptibility with a gain profile which enable us to
generate the FWM signal. The control field can be utilized as a parameter to manage the efficiency of
the FWM signal. The propagation dynamics of the generated FWM signal and the effect of enhanced
non-linearity are studied in detail. The FWM signal clones the temporal shape of the probe field and travels through
the atomic medium along with the probe field without changing its shape and intensity.
This scheme allows us to generate an efficient FWM signal of any arbitrary shape. We have also demonstrated the coherent
storage and retrieval of the signal with the assistance of a time dependent profile of the control field.
The generation, control, storage and retrieval of such higher order nonlinear optical signals in multilevel atomic systems
can have potential applications in nonlinear optical spectroscopy, quantum information processing and optical communication systems.
\section*{Acknowledgments}
N. S. Mallick would like to thank MHRD, Government of India for financial support.