# Data Privacy and Utility Trade-Off Based on Mutual Information Neural
Estimator

Qihong Wu, Jinchuan Tang, Shuping Dang, Gaojie Chen This work was supported by
Guizhou University through “The Secure Encryption Mechanisms of Spatially
Embedded Networks” under Grant No. 702957213301.Q. Wu and J. Tang is with the
School of Computer Science and Technology, Guizhou University, Guiyang, P.R.
China e-mail: {gs.qhwu<EMAIL_ADDRESS>J. Tang is the corresponding
author.S. Dang is with the Department of Electrical & Electronic Engineering,
University of Bristol, Bristol, UK e-mail<EMAIL_ADDRESS>Chen
is with the Department of Engineering, University of Leicester, Leicester, UK
e-mail<EMAIL_ADDRESS>
###### Abstract

In the era of big data and the Internet of Things (IoT), data owners need to
share a large amount of data with the intended receivers in an insecure
environment, posing a trade-off issue between user privacy and data utility.
The privacy utility trade-off was facilitated through a privacy funnel based
on mutual information. Nevertheless, it is challenging to characterize the
mutual information accurately with small sample size or unknown distribution
functions. In this article, we propose a privacy funnel based on mutual
information neural estimator (MINE) to optimize the privacy utility trade-off
by estimating mutual information. Instead of computing mutual information in
traditional way, we estimate it using an MINE, which obtains the estimated
mutual information in a trained way, ensuring that the estimation results are
as precise as possible. We employ estimated mutual information as a measure of
privacy and utility, and then form a problem to optimize data utility by
training a neural network while the estimator’s privacy discourse is less than
a threshold. The simulation results also demonstrated that the estimated
mutual information from MINE works very well to approximate the mutual
information even with a limited number of samples to quantify privacy leakage
and data utility retention, as well as optimize the privacy utility trade-off.

###### Index Terms:

Privacy utility trade-off, mutual information estimator, KL-divergence, neural
networks.

##  I Introduction

Data privacy involves publishing data efficiently to minimize risk and protect
sensitive data. The content of data and its associated metadata information
can be exploited to infer sensitive and personally identifiable information,
which can be harmful to individuals and organizations. The availability of
data publishing are credited with advancing solutions to complex problems in
data sharing, data acquisition, among others in the era of big data and IoT
[1, 2]. However, big data publishing comes with a massive privacy disclosure
where explosive sensitive data growth has been witnessed. The authors of [3]
have already demonstrated the impact of individual data loss by using examples
of privacy disclosure and listed the complexity of tackling technological and
legislative challenges for big data and individual privacy in the age of IoT.

Usually, the approaches to keep certain information private can be achieved by
distorting the information while disclosing relevant information [4, 5, 6, 7,
8]. The authors in [9, 10] established k-anonymity as the characteristic that
each record is indistinguishable from at least $k-1$ other records on the
quasi-identification to prevent the identity of the owner of public data from
being revealed. Generalization and suppression were utilized to obtain
k-anonymity. Although k-anonymity overcomes the problem of identity
disclosure, it does not preclude attribute disclosure or homogeneity attacks.
Individuals may be exposed to the relationship between identities and
sensitive attributes, which may jeopardize the distribution of individuals and
the entire dataset. To address this issue, the authors of [11] introduced
$l$-diversity, which requires sensitive characteristics to have at least $l$
well-represented values in each equivalence class to withstand the homogeneity
attack in $k$-anonymity. As noted in [11], $l$-diversity is faced with two
major attacks: the first is similarity attack, which ignores the danger posed
by semantic relationships between attributes; the second is skewness attack,
in which the adversary might deduce sensitive information based on the
distribution of sensitive qualities, which is a serious invasion of privacy.
As a result, preventing property disclosure issues is insufficient. To this
end, the $t$-closeness approach was introduced in [12] where the earth mover’s
distance was used to compute the $t$-closeness. But it has drawbacks such as
large loss of data utility and inability to discriminate semantic information.
The term utility refers to certain system properties and intelligibility which
represents the amount of useful information that can be extracted from the
protected data.

In an attempt to provide better data utility, the authors of [13] developed a
model which combines $k$-anonymity and differential privacy, where the latter
is capable of answering queries to statistical databases with provable privacy
guarantees by minimizing attackers’ opportunities to identify database records
[6, 7]. The requirement that information is disseminated only to a limited
extent while the data still meets the usability of certain desired recipients
has becomes critical nowadays as a large amount of personal information is
being widely disseminated, shared and openly accessible by anyone in big data
and IoT applications. Since distorting data too much will destroy the value of
data to the desired receipents while distorting too little will help the
adversaries to deduce the sensitive information of targeted individual,
finding an acceptable privacy utility trade-off between privacy protection and
data utility is the key in the development and deployment of privacy
protection methods. To make things work, the measurements and
characterizations on both the privacy and utility have to be achieved.

The privacy protection, data uitility as well as their trade-off can be
characterized mathematically with the help of information theory. In [4, 8],
the authors consider expected distortion as a measure of privacy and utility
where the collective privacy of all or subsets of database items are obtained
to offer a progressive conclusions of the fuzzy region of rate distortion as
the number of data samples rises arbitrarily. To simulate privacy leakage and
data distortion, the authors of [14] proposed a concept called the “privacy
funnel” to characterize the trade-off between data privacy disclosure and
utility. In simple terms, the privacy funnel used mutual information to
measure both data distortion and privacy disclosure. However, their proposed
greedy algorithm used merged elements in the context of unknown distribution
of data sets to approximate mutual information which runs fast at the expense
of accuracy. Precise calculations on mutual information only apply to discrete
variables because the sum can be precisely calculated, or to finite problems
where the probability distribution is known. For more general problems, this
is impossible. Furthermore, to process complex data when the data distribution
is unavailable, the authors of [15] added differential privacy guarantee noise
to the data compressed by the autoencoder to resist inference attacks in
motion-aware applications. Although differential privacy provided strong
protection for each identified entry, it failed to separate the information
associated with sensitive data from the information associated with non-
sensitive data. Thus, the increased noise level seriously compromised data
utility. Meanwhile, the authors of [16] proposed a model where the private
information is equal to zero when the first stage of the expected information
predictor is a linear operator. Although the model achieves a perfect utility
competitive trade-off when utility and private data are orthogonal, it
requires both an understanding of the model used to handle downstream utility
tasks and the first part of the model must be linear. However, in data
publishing, it is usually impossible to obtain the information about such
downstream task models. This encourages us to estimate mutual information
using a data-driven approach and optimize privacy funnel using mutual
information as a measure of privacy and utility.

Recently, a mutual information neural estimator (MINE) was proposed in [17] to
obtain the approximated mutual information from data samples. Unlike
traditional non-universal methods based on the partitioning of probability
space [18, 19], k-nearest neighbor statistics [20, 21], maximum likelihood
estimation [22] and variational lower bound [23], MINE does not require the
data distribution that is difficult to obtain in practice and it can also
match the sample size and dimension better. The simulation results show that
such a method can estimate mutual information through neural network training
in the case of unknown data distribution, and it is almost indistinguishable
from real mutual information. The authors of [24] applied MINE to train hidden
layer with information bottleneck loss, and froze it before moving onto the
next layer. The authors of [25] relaxed the Gaussian posterior assumption by
using the MINE [17] to train a variational information bottleneck to improves
its robustness to adversarial perturbations. The work in [26] investigated
unsupervised learning of representations by maximizing mutual information
based on MINE between an input and the output of a deep neural network
encoder, and MINE has been shown once again to estimate mutual information
more precisely than traditional methods. This inspires us to use mutual
information as a measure of both privacy and utility, which will be formulated
as optimization problems for privacy utility trade-off.

In this paper, mutual information is adopted to measure both privacy discourse
and data utility at the same time. Different from previous works which used
merged elements to approach data distribution for privacy and utility
characterizations, this paper uses MINE [17] instead for accuracy. Then, by
interval training mutual information estimator and encoder to maximize mutual
information, we put mutual information into the privacy utility trade-off
framework to obtain the privacy utility model using only single mutual
information to measure data privacy and utility, making the model more
flexible. Simulations demonstrate that the model’s practicality and assess the
impact of various noise levels and privacy precisions on the maximum data
utility. The simple model allows us to take different parameters on the model
into consideration, which includes the influence of different noise levels on
data utility and the influence of privacy accuracy on the overall model.

This contribution can be summarized as follows:

  * •

The mutual information estimator is used to safeguard data privacy when it is
published. It overcomes the problem of unknown distributions being unable to
extract mutual information and maximizes data utility under the assumption of
fixed privacy budget.

  * •

Unlike other measuring methods, the privacy utility trade-off which we
proposed relies solely on mutual information. MINE is used throughout this
work to estimate mutual information for a better accuracy. And the proposed
trade-off framework is straightforward and simple to implement.

  * •

We perform the simulation to find the impacts of various parameters such as
noise levels and privacy budgets on the privacy utility trade-off framework
and offer some useful insights on choosing the parameters.

The notation is given in Table LABEL:Table.1. The remainder of the paper is
laid out as follows. In Section II, we describe the system model. Then we
formulate the privacy utility trade-off objective using neural estimator in
Section III, and Section IV is extensive simulation part, finally we summarize
our comments in Section V.

TABLE I: Notations Symbol | Meaning
---|---
$C(S,q)$ | Cost function
$d(X,Y)$ | Distortion measure
$D$ | Distortion level
$D_{KL}(\cdot||\cdot)$ | KL-divergence of two distributions
$E_{P_{X,Y}}[d(X,Y)]$ | Expected distortion
$\hat{G}$ | Estimated gradient
$H(\cdot)$ | Information entropy
$\hat{H}_{\theta}(\cdot)$ | Estimated cross-entropy with neural network $\theta$
$I(\cdot;\cdot)$ | Mutual information of two random variables
$\hat{I}_{\theta}(\cdot;\cdot)$ | Estimated mutual information with neural network $\theta$
$k$ | Minibatch size
$n$ | Number of samples from $Y$
$p(\cdot,\cdot)$ | Joint probability density of two random variables
$p(\cdot)$ | Marginal probability density
$\hat{P}^{\left(n\right)}$ |  | Empirical distribution associated to $n$ i.i.d.
---
samples for a given distribution $P$
$P(Y|X)$ | Conditional distribution, privacy mapping
$q$ | Belief distribution
$q_{0}^{*}$ | Optimal $q$ before observation of data
$q_{y}^{*}$ | After observing published data
$S$ | Sensitive dataset
$T_{\epsilon}(X;Y)$ | Characterization of privacy utility trade-off
$X$ | Dataset related to S
$Y$ | Published dataset
$\Delta C$ | Expected inference cost gain
$\epsilon$ | Privacy parameter, privacy budget
$\eta$ | Learning rate
$\theta$ | Deep neural network parameter
$\phi$ | Encoder weights

##  II System Model

As shown in Fig. 1, we consider a case where a data publisher has some
sensitive data set $S\in\mathcal{S}$ which is correlated with some non-
sensitive data set $X\in\mathcal{X}$, and the data publisher wishes to share
$X$ with data receiver. This correlation can be regarded as an auxiliary
information to infer the sensitive dataset $S$. To reduce the disclosure of
sensitive dataset $S$, the data publisher send a perturbed version of $X$
denoted by $Y\in\mathcal{Y}$, where $Y$ is the published dataset generated by
adding Gaussian or Laplacian noise to $X$. Dataset $Y$ that passes through the
privacy mechanism should be as irrelevant as possible to sensitive data $S$,
while retaining as much information as possible about $X$, since $Y$ will be
processed by the data receiver to provide utility. We assume that $S,X$ and
$Y$ follow a Markov chain $S\rightarrow X\rightarrow Y$.

Figure 1: Privacy mechanism model.

Under the assumptions above, we propose to design a privacy framework which
maps $X$ to a random variable $Y$ so that the information leakage from $S$ to
$Y$ is limited while the data utility of $X$ given $Y$ is maximum. Our
challenge, therefore, is to use a data-driven approach to find the privacy
utility trade-off for both discrete and continuous data. To fulfil such
vision, unlike [4], which used a greedy algorithm with merged elements, we
propose to use the mutual information nerual network estimator to obtain the
more accurate metrics of information leakage from $S$ to $Y$ as well as the
utility of $X$ given $Y$. The combined results on privacy and utility measures
will be used in the proposed privacy utility trade-off framework to optimize
utility while safeguarding privacy.

##  III Privacy utility trade-off

In this section, we describe the privacy and utility metrics in data
publishing separately first. Then, we introduce the estimator for mutual
information. Finally, we discuss the privacy utility trade-off problem which
firstly adopts mutual information neural estimator as a measure of both
privacy and utility.

###  III-A Privacy and Utility Metric

Here, we consider the inference threat model introduced in [27]. At this
moment, illegal data receiver as an attacker tries to infer sensitive
information $S$. More precisely, the data receiver chooses a belief
distribution $q$ from the set $P_{S}$ of all probability distributions over
$S$ to minimize the expected inference cost function $C(S,q)$. Under the
logarithmic loss $C(S,q)=-\log_{2}q(S)$, the optimal $q$ before observation of
data $Y$ is

$\displaystyle q_{0}^{*}$ $\displaystyle=\mathop{\arg\min}\limits_{q\in
P_{S}}E_{P_{S}}[C(S,q)]$ (1) $\displaystyle=\mathop{\arg\min}\limits_{q\in
P_{S}}E_{P_{S}}[-\log_{2}q(S)],$

and the Shannon entropy of $S$ is given by

$\displaystyle H(S)$ $\displaystyle=E_{P_{S}}[-\log_{2}q_{0}^{*}(S)].$ (2)

After observing published data $y\in Y$, we get

$\displaystyle q_{y}^{*}$ $\displaystyle=\mathop{\arg\min}\limits_{q\in
P_{S}}E_{P_{S|Y}}[C(S,q)|Y=y]$ (3)
$\displaystyle=\mathop{\arg\min}\limits_{q\in
P_{S}}E_{P_{S|Y}}[-\log_{2}q(S|Y)],$

and the conditional entropy of $S$ given $Y$ is written as

$\displaystyle H(S|Y)$ $\displaystyle=E_{P_{S|Y}}[-\log_{2}q_{y}^{*}(S|Y)].$
(4)

Because publishing data $Y$ is observable, so the data receiver obtains an
expected gain in inference cost of

$\displaystyle\Delta C=H(S)-H(S|Y)=I(S;Y),$ (5)

where the expected inference cost gain $\Delta C$ measures the improvement in
the inference quality from sensitive data $S$ after the observed $Y$. The
design goal of the privacy utility trade-off should aim at reducing $\Delta C$
to bring the inference with observing $Y$ closer to the initial inference
without observing $Y$, the formula can be written as

$\displaystyle\min\Delta C$ $\displaystyle=\min I(S;Y)$ (6)
$\displaystyle=\min\int_{S\times
Y}p(s,y)\log_{2}{\frac{p(s,y)}{{p(s)p(y)}}}{\rm d}s\,{\rm d}y.$

where $p(s,y)$ is the joint probability density of $S$ and $Y$, $p(s)$ and
$p(y)$ are the corresponding marginal probability density.

Protecting the privacy of data publishing is considered essential, at the same
time, the framework of privacy utility trade-off should maintain the utility
of the perturbed data $Y$. In [28], the key issue is that utility is
determined not only by the data modifications made, but also by the
anticipated data uses. Since potential data uses are truly diverse and it may
even be hard to identify them all now of the data release, privacy protection
can seldom be performed in a data use-specific manner. As a result, it is more
common to speak about information loss rather than utility. Measures of
information loss provide a basic approach for the data protector to determine
how much harm a particular masking technique is causing to the data.

As the discrimination between two distribution of mean information, KL-
divergence can be used to measure information loss, which formula is as (7).
Meanwhile, we know that KL-divergence is equal to mutual information by one
form as (8). Consequently, we can measure data utility by mutual information.

###  III-B Mutual Information Neural Estimator (MINE)

The theoretical mutual information defined above depends on probability
density functions which are difficult to obtain in practice. Here, we focus on
an estimator named MINE for its good performance in approximating the mutual
information from data samples with high accuracy [17]. It utilizes the
Donsker-Varadhan representation (DV-representation) of the Kullback-Leibler
divergence (KL-divergence), which is related to the mutual information. In
MINE, mutual information is estimated by parameterizing the lower bound of KL-
divergence and improving the lower bound by continuous training.

The KL-divergence is a measurement of the dissimilarity between distributions
$P$ and $Q$, which can be written as

$D_{KL}(P||Q)=\int{P(x)\log_{2}{\frac{P(x)}{Q(x)}}}{\rm d}x,$ (7)

where $D_{KL}(P||Q)\lg 0$, with $D_{KL}(P||Q)=0$ if and only if $P(x)=Q(x)$.
Consequently, we get

$\displaystyle I(S;Y)$ $\displaystyle:=\int_{S\times
Y}p(s,y)\log_{2}{\frac{p(s,y)}{{p(s)p(y)}}}{\rm d}s\,{\rm d}y$ (8)
$\displaystyle=D_{KL}(p(s,y)||p(s)p(y))$
$\displaystyle=E_{p(S,Y)}\left[\log_{2}\left(\frac{p(s,y)}{p(s)p(y)}\right)\right],$

where, given samples of $S$ and $Y$, we can estimate $I(S;Y)$ at the cost of
accuracy when the number of samples disobey the Law of Large Numbers. In this
work, we pay attention to DV-representation [29], which could result in a more
accurate estimator.

######  Theorem 1.

(DV-representation). The KL-divergence admits representation given by

$D_{KL}(P||Q)=\sup_{T:\Omega\rightarrow R}E_{P}[T]-\log_{2}(E_{Q}[e^{T}]),$
(9)

where the supremum is taken over all functions $T$ such that the two
expectations are finite.

###### Proof.

See Appendix. ∎

Let $\mathcal{F}$ be any class of functions $T:\Omega\rightarrow R$ satisfying
the integrability constraints of the theorem, we then have a lower-bound given
by

$D_{KL}(P||Q)\geq\sup_{T\in\mathcal{F}}E_{P}[T]-\log_{2}(E_{Q}[e^{T}]).$ (10)

For an MINE, the approach here is to choose $\mathcal{F}$ to be the family of
functions $T_{\theta}:\mathcal{S}\times\mathcal{Y}\rightarrow\mathbb{R}$
parametrized by a deep neural network with parameter $\theta\in\Theta$ so that

$I(S;Y)\geq
I_{\Theta}(S;Y)=\sup_{\theta\in\Theta}E_{P_{SY}}[T_{\theta}]-\log_{2}\left(E_{P_{S}P_{Y}}[e^{T_{\theta}}]\right),$
(11)

the expectations above are estimated using empirical samples from $P_{SY}$ and
$P_{S}\times P_{Y}$ or by shuffling the samples from the joint distribution
along the batch axis, which is defined by Definition 1. As a result, the MINE
algorithm to train the estimator is given by Algorithm 1, and the structure of
a four-layer structured of neural network, as shown in Fig. 2, is adopted in
this work.

######  Definition 1.

(Mutual Information Neural Estimator (MINE)). Let
$\mathcal{F}=\left\\{T_{\theta}\right\\}_{\theta\in\Theta}$ the set of
functions parametrized by a neural network. MINE is defined as [17]

${\hat{I}(S;Y)_{n}}=\sup_{\theta\in\Theta}{E_{P_{SY}^{\left(n\right)}}}[T_{\theta}]-\log_{2}\left({E_{P_{S}^{\left(n\right)}\hat{P}_{Y}^{\left(n\right)}}}[e^{T_{\theta}}]\right),$
(12)

where $n$ is the number of samples, $\hat{P}^{\left(n\right)}$ is the
empirical distribution associated to $n$ independent and identically
distributed (i.i.d.) samples by given distributions $P_{SY},P_{S},P_{Y}$.

Figure 2: A four-layer neural network approximated $T_{\theta}$. Algorithm 1
Mutual information neural estimator [17]

0:
$\left(s^{\left(1\right)},y^{\left(1\right)}\right),...,\left(s^{\left(k\right)},y^{\left(k\right)}\right)$,
$k$, $\eta$, where $(s,y)$ is a given dataset, $k$ is minibatch size, and
$\eta$ is learning rate.

0: $\hat{I}_{\theta}(S;Y)$

1: repeat

2: Draw $k$ minibatch samples from the joint distribution:
$\left(s^{\left(1\right)},y^{\left(1\right)}\right),...,\left(s^{\left(k\right)},y^{\left(k\right)}\right)\sim
P_{SY}$

3: Draw $n$ samples from the $Y$ marginal distribution:
$\overline{y}^{\left(1\right)},...,\overline{y}^{\left(n\right)}\sim P_{Y}$

4: Evaluate the mutual information lower-bound:
$\displaystyle\hat{I}_{\theta}(S;Y)\leftarrow$
$\displaystyle{\frac{1}{k}{\sum_{i=1}^{k}T_{\theta}{\left(s^{\left(i\right)},y^{\left(i\right)}\right)}}}$
$\displaystyle-\log_{2}{\left(\frac{1}{k}\sum_{i=1}^{k}e^{T_{\theta}{\left(s^{\left(i\right)},y^{\left(i\right)}\right)}}\right)}$

5: Evaluate bias corrected gradients with moving average:
$\hat{G}(\theta)\leftarrow\widetilde{\nabla}_{\theta}\,\hat{I}_{\theta}(S;Y)$

6: Update the model parameters: $\theta\leftarrow\theta+\eta\hat{G}(\theta)$

7: until convergence

###  III-C Privacy Utility Trade-off

To model the privacy disclosure and data utility, the authors of [14] proposed
a concept called the “privacy funnel” to represent the trade-off between data
utility and user privacy. Briefly, the privacy funnel uses mutual information
to evaluate both data utility and privacy disclosure. And the optimization
problem of privacy utility trade-off for a given distortion level $D$ is
characterized as follow:

$\min_{P_{Y|X}:E_{P_{X,Y}}[d(X,Y)]\leq D}\Delta C,$ (13)

where $P_{Y|X}$ is privacy mapping, $d(X,Y)$ is the distortion measure,
$E_{P_{X,Y}}[d(X,Y)]$ represents the expected distortion, $D$ represents the
level of distortion, $\Delta C$ describes expected inference cost gain. Since
the data distribution is often inaccessible and only a proportion of the
sampled data is available, we propose to use MINE to obtain estimated mutual
information as a measure of both privacy and utility.

The dual form of (13) is given by

$T_{\epsilon}(X;Y):=\max_{\begin{subarray}{c}P_{Y|X}:\hat{I}(S;Y)\leq\epsilon\\\
S\rightarrow X\rightarrow Y\end{subarray}}\hat{I}(X;Y),$ (14)

where $T_{\epsilon}(X;Y)$ is characterization of privacy utility trade-off;
$P_{Y|X}$ represents privacy mapping; $\hat{I}(X;Y)$ and $\hat{I}(S;Y)$ is the
estimated mutual information of $I(X;Y)$ and $I(S;Y)$, respectively;
$\epsilon$ stands for privacy budget.

The privacy utility trade-off with a privacy budget can be used as a solution
to the optimization problem of (14). The remained task is to design a trade-
off algorithm. From a communication theoretic perspective, privacy mechanism
is similar to a noisy channel, and the optimal transmission rate can be
regarded as a function of the mutual information $I(X;Y)$ between input $X$
and output $Y$ of a channel $P_{Y|X}$. However, the mutual information also
depends on the channel probability distribution, i.e. added noise. Rather than
approximating the channel probability distribution itself, we will approximate
the mutual information $I(X;Y)$ between the samples of the channel input and
output. Then we optimize the mutual information due to added noise so that the
adversaries will be diffcult to infer the sensitive data, which can be
regarded as an encoding process, and also optimize the infering capability of
legal receiver by minimizing cross-entropy mutual information between $Y$ and
$X$ so that the data utility is guaranteed, which can be regarded as a
decoding process. Thus, we give our optimization framework as Algorithm 2.

Algorithm 2 Privacy Utility Trade-off

0: $S,x,y,n_{0},\epsilon,\eta,\theta_{1},\theta_{2}$ and $\theta_{3}$, where
$S$ is sensitive datasets, $x,y$ are data samples, $n_{0}$ denotes additive
nosie, $\epsilon$ is privacy budget; $\eta$ is learning rate, $\theta_{1}$ is
the network for estimation training process, $\theta_{2}$ is the network for
an encoder tuned by $\theta_{1}$, and $\theta_{3}$ is the decoding network.

0:
$\hat{I}_{\theta_{1}}(X;Y),\hat{I}_{\theta_{2}}(X;Y),\hat{H}_{\theta_{3}}(x),\hat{x}$

1: $y\leftarrow x+n_{0}$

2: Calculate the approximated $I(S;Y)$ with Algorithm 1

3: while $\hat{I}(S;Y)\leq\epsilon$ do

4: Encoding training process:

5: repeat

6: Evaluate the estimated mutual information lower-bound:
$\displaystyle\hat{I}_{\theta_{1}}(X;Y)\leftarrow$
$\displaystyle{\frac{1}{k}{\sum_{i=1}^{k}T_{\theta_{1}}{\left(x^{\left(i\right)},y^{\left(i\right)}\right)}}}$
$\displaystyle-\log_{2}{\left(\frac{1}{k}\sum_{i=1}^{k}e^{T_{\theta_{1}}{\left(x^{\left(i\right)},y^{\left(i\right)}\right)}}\right)}$

7: To maximize $\hat{I}_{\theta_{1}}(X_{\theta_{2}};Y)$ in the encoding
process,
gradients:$\hat{G}_{xy}(\theta_{1})\leftarrow\widetilde{\nabla}_{\theta_{1}}\,\hat{I}_{\theta_{1}}(X_{\theta_{2}};Y)$$\hat{G}_{xy}(\theta_{2})\leftarrow\widetilde{\nabla}_{\theta_{2}}\,\hat{I}_{\theta_{1}}(X_{\theta_{2}};Y)$,where
$X_{\theta_{2}}$ is the output of neural network $\theta_{2}$.

8: Update the model
parameters:$\theta_{1}\leftarrow\theta_{1}+\eta\hat{G}(\theta_{1})$$\theta_{2}\leftarrow\theta_{2}+\eta\hat{G}(\theta_{2})$

9: until convergence

10: Decoder training process:

11: repeat

12: Evaluate the cross-entropy of estimated $\hat{x}$:
$\hat{H}_{\theta_{3}}(X,\hat{X})\leftarrow-\frac{1}{k}{\sum_{i=1}^{k}T_{\theta_{3}}{\left(x^{\left(i\right)}\right)}}\log_{2}T_{\theta_{3}}{\left(x^{\left(i\right)}\right)}$

13: Evaluate bias corrected gradients with moving average:
$\hat{G}_{x}(\theta_{3})\leftarrow\widetilde{\nabla}_{\theta_{3}}\hat{H}_{\theta_{3}}(X)$

14: Update the model parameters:
$\theta_{3}\leftarrow\theta_{3}+\eta\hat{G}(\theta_{3})$

15: until convergence

16: end while

##  IV Simulation Results

In this section, we conduct simulations to verify and evaluate the performance
of our proposed model. We first compare true mutual information with estimated
mutual information based on MINE for a known data distribution. Then, the
proposed privacy utility trade-off framework is assessed from a variety of
angles, including the influence of the privacy utility trade-off under various
noise levels and privacy budget $\epsilon$. Finally, the effect of noise
parameters is taken into account.

###  IV-A The Accuracy of the Estimated Mutual Information

We compare the mutual information with and without MINE. The estimated results
are obtained by Algorithm 1, and the true MI is calculated using a mutual
information formula. In the simulations, $S$ is a dataset of random numbers
that follow a standard normal distribution. $S$ and $Y$ follow a joint normal
distribution with zero mean and a variance $0.2$. The true mutual information
with (8) is $0.6586$, and the maximum epochs is $500$, then we import the data
into MINE for iterative estimation. The parameters of the neural network are
described in Table LABEL:Table.2. As shown in Fig. 3, the estimated mutual
information converges to the actual mutual information as the number of
training epochs increase. At epoch $250$, the mutual information obtained by
MINE is indistinguishable from the true mutual information.

TABLE II: Simulation Parameters Parameter | Symbol | Value
---|---|---
Learning rate | $\eta$ | 0.0005
Neural network layers | $L$ | 3
Number of learning sessions | epoch | 500
Privacy budget | $\epsilon$ | [0.5, 0.75, 1.0, 1.25]
Small batch data | minibatch | 20000
Figure 3: Mutual information estimation and true mutual information.

###  IV-B Trade-Off Performance for Different Privacy Budgets

The privacy utility trade-off under different privacy budgets is explored in
this subsection. Batches of datasets $S$ and $X$ are generated from the same
distribution. In addition to this, $Y$ is obtained by adding Gaussian noise to
$X$. For mutual information estimation, we use the Adam optimizer and a
learning rate of $\eta=0.0005$. Note that we do not have access to the true
joint distribution $p(s,y)$, $p(x,y)$ and the marginal distributions $p(s)$,
$p(x)$, $p(y)$, We therefore utilize samples of these distributions and
approximated the expectations by the sample average.

Here, the values of privacy budget $\epsilon$ are taken as $0.5$, $0.75$, $1$
and $1.25$, respectively. The maximum mutual information $I(X;Y)$ of different
privacy budgets is displayed in Fig. 4. As shown in Fig. 4, the estimated
mutual information value of each epoch gradually grows and eventually
converges to near a maximum value, indicating that the threshold of privacy
disclosure rises within a specific range. The more sensitive information which
is published, the more utility recipients receive, and the more stable the
data becomes after training. The greatest value of mutual information
collected in the last epochs for varied privacy budgets. Fig. 5 shows the
comparison results between the various precision of privacy disclosure. It
demonstrates that a larger privacy budget would lead to both more privacy
disclosure and higher data utility.

Figure 4: Trade-off performance for different privacy budgets with respect to
number of epochs. Figure 5: Trade-off performance for privacy budgets $0.5$,
$0.75$, $1$ and $1.25$.

###  IV-C Trade-Off Performance for Different Gaussian and Laplacian Noise
Levels

In this subsection, we consider the influence of adding noise to dataset for
different noise levels. We studied privacy utility trade-off using Gaussian
noise and Laplacian noise, respectively. In order to assess the influence of
different noise levels, we set $\epsilon$ to $0.75$ and the other
hyperparameters as in Subsection IV-B. After that, we choose to add either
Gaussian noise or Laplacian noise and perform training for $100$ epochs. We
evaluate the privacy utility trade-off at the end of each epoch, and collect
the budget of the best utility for different noise levels. As demonstrated in
Fig. 6, the impact of different noise levels on the privacy utility trade-off
varies. It showcases the scenario when the estimated mutual information
changes with different noise levels. It shows that the mutual information
obtained by adding Gaussian noise is more stable and larger than that obtained
by adding Laplacian noise while it tends to converge.

Figure 6: Trade-off performance for different Gaussian (GA) and Laplacian (LP)
noise levels. Figure 7:  Maximum utility of adding different noise levels with
privacy budgets $0.5$, $0.75$, $1$ and $1.25$. Figure 8: Trade-off performance
for different Gaussian parameters with privacy budgets $0.5$, $0.75$, $1$ and
$1.25$.

Fig. 7 illustrates the maximum value of mutual information, namely the maximum
utility when the privacy budgets are $0.5$, $0.75$, $1.0$ and $1.25$,
respectively, by adding Gaussian noise and Laplacian noise to the dataset. It
can be clearly observed that when the same privacy budget is taken, the
maximum utility obtained by adding Gaussian noise is greater than that
obtained by adding Laplacian noise. The maximum data utility rises in tandem
with the level of privacy budget.

###  IV-D Trade-Off Performance for Different Gaussian Parameters

In this subsection, we consider the influence of setting different Gaussian
parameters for maximum mutual information. The privacy utility trade-off of
Gaussian noise with the standard deviation $0.1117$ versus noise generated by
standard Gaussian distribution are investigated respectively. The privacy
utility trade-off is affected heavily by different noise parameters. The
maximum mutual information $I(X;Y)$ as shown in Fig. 8. To evaluate the
effects of different Gaussian noise parameters, we set $\epsilon$ to $0.75$
and set the other hyperparameters to the parameters in Subsection IV-B. Then,
we modified the standard deviation in the Gaussian noise from the value
$0.1117$ to the deviation of the standard Gaussian distribution, i.e. $1$, and
ran the framework for $100$ epochs. We evaluated the privacy utility trade-off
at the end of each period, and collect the maximum utility for different
parameter settings. It can be seen that different mean values have little
influence on maximum utility. However, a smaller standard deviation will lead
to greater maximum mutual information, in other words, we can set a small
variance to obtain a greater utility.

##  V Conclusion

In this paper, we consider the privacy utility trade-off in data publishing,
which is very important to IoT and big data applications. Specifically, mutual
information was used as a measure of both privacy disclosure and data utility,
and MINE is adopted to estimate mutual information for a better accuracy. A
privacy utility trade-off framework is proposed by incoperating MINE. Our
simple but effective model does not require other training strategies, thus
making the model more stable and less sensitive to hyperparameters. Simulation
results demonstrated the effectiveness of the framework with regard to
different parameter settings.

[Proof of (10)] For completeness, similar to [17], we prove it as follows.

###### Proof.

For a given function T, consider the Gibbs distribution
$G(x)=\frac{1}{Z}e^{T(x)}Q(x)$, where $Z=E_{Q}[e^{T(x)}]$, then

$\displaystyle E_{P}[T(x)]-\log_{2}(Z)$
$\displaystyle=E_{P}[T(x)-\log_{2}(Z)]$ (15)
$\displaystyle=E_{P}\left[\log_{2}{e^{T(x)}}-\log_{2}(Z)\right]$
$\displaystyle=E_{P}\left[\log_{2}\frac{e^{T(x)}}{Z}\right]$
$\displaystyle=E_{P}\left[\log_{2}\frac{e^{T(x)}Q(x)}{ZQ(x)}\right]$
$\displaystyle=E_{P}\left[\log_{2}\frac{G(x)}{Q(x)}\right],$

let $\Delta$ be subtract between KL-divergence and above, then

$\displaystyle\Delta$
$\displaystyle:=D_{KL}(P||Q)-\left(E_{P}\left[T(x)\right]-\log_{2}\left(E_{Q}\left[e^{T(x)}\right]\right)\right)$
(16)
$\displaystyle=E_{P}\left[\log_{2}\frac{P(x)}{Q(x)}-\log_{2}\frac{G(x)}{Q(x)}\right]$
$\displaystyle=E_{P}\left[\log_{2}\frac{P(x)}{G(x)}\right]$
$\displaystyle=\int{P(x)\log_{2}{\frac{P(x)}{Q(x)}}}{\rm d}x$
$\displaystyle=D_{KL}(P||G),$

because of the non-negative of KL-divergence, $\Delta\geq 0$, so that

$D_{KL}(P||Q)\geq E_{P}[T]-\log_{2}\left(E_{Q}[e^{T}]\right).$ (17)

Thus, we conclude our proof. ∎

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††thanks: These two authors contributed equally to this work.††thanks: These
two authors contributed equally to this work.

# Nonlinear optical radiation of a lithium niobate microcavity

Yuan-Hao Yang  CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, Anhui 230026, P. R. China.  CAS Center
For Excellence in Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, P. R. China.  Xin-Biao
Xu  CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei, Anhui 230026, P. R. China.  CAS Center For
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, P. R. China.  Jia-Qi Wang  CAS
Key Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.  Mai Zhang  CAS Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei,
Anhui 230026, P. R. China.  CAS Center For Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei,
Anhui 230026, P. R. China.  Ming Li<EMAIL_ADDRESS>CAS Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei,
Anhui 230026, P. R. China.  CAS Center For Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei,
Anhui 230026, P. R. China.  Zheng-Xu Zhu  CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei, Anhui
230026, P. R. China.  CAS Center For Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, P. R. China.  Zhu-Bo Wang  CAS Key Laboratory of Quantum Information,
University of Science and Technology of China, Hefei, Anhui 230026, P. R.
China.  CAS Center For Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, P. R.
China.  Chun-Hua Dong  CAS Key Laboratory of Quantum Information, University
of Science and Technology of China, Hefei, Anhui 230026, P. R. China.  CAS
Center For Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, P. R. China.  Wei
Fang  Interdisciplinary Center for Quantum Information, State Key Laboratory
of Modern Optical Instrumentation, College of Optical Science and Engineering,
Zhejiang University, Hangzhou 310027, China.  Intelligent Optics $\&$
Photonics Research Center, Jiaxing Research Institute Zhejiang University,
Jiaxing 314000, China.  Huakang Yu<EMAIL_ADDRESS>School of Physics and
Optoelectronics, South China University of Technology, Guangzhou 510460, P. R.
China.  Zhiyuan Li  School of Physics and Optoelectronics, South China
University of Technology, Guangzhou 510460, P. R. China.  Guang-Can Guo  CAS
Key Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.  CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, P. R. China.  Chang-Ling Zou<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei, Anhui 230026, P. R. China.  CAS Center For
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, P. R. China.

###### Abstract

The nonlinear optical radiation of an integrated lithium niobate microcavity
is demonstrated, which has been neglected in previous studies of nonlinear
photonic devices. We find that the nonlinear coupling between confined optical
modes on the chip and continuum modes in free space can be greatly enhanced on
the platform of integrated microcavity, with feasible relaxation of the phase-
matching condition. With an infrared pump laser, we observe the vertical
radiation of second-harmonic wave at the visible band, which indicates a
robust phase-matching-free chip-to-free-space frequency converter and also
unveils an extra energy dissipation channel for integrated devices. Such an
unexpected coherent nonlinear interaction between the free-space beam and the
confined mode is also validated by the different frequency generation.
Furthermore, based on the phase-matching-free nature of the nonlinear
radiation, we build an integrated atomic gas sensor to characterize Rb
isotopes with a single telecom laser. The unveiled mechanism of nonlinear
optical radiation is universal for all dielectric photonic integrated devices,
and provides a simple and robust chip-to-free-space as well as visible-to-
telecom interface.

_Introduction.-_ Cavity-enhanced nonlinear photonic devices offer a compact
solution for efficient frequency conversion under low pump power owing to the
small mode volume and high quality-factor Shen  _et al._ (2020); Strekalov
_et al._ (2016); Kippenberg  _et al._ (2018); Bruch  _et al._ (2021); Xiao
_et al._ (2020). In particular, the chip-integrated optical microcavity, which
benefits from the high-flexibility in dispersion engineering, enables
significant nonlinear optics effects among multiple resonant modes with
distinct wavelengths by $\chi^{(2)}$ and $\chi^{(3)}$ processes Bruch  _et
al._ (2019); Del’Haye  _et al._ (2007), as well as Raman and Brillouin
scattering Takahashi  _et al._ (2013); Eggleton  _et al._ (2019); Safavi-
Naeini  _et al._ (2019). During last decades, the preparation and micro/nano-
fabrication techniques of thin-film optical materials with high nonlinear
susceptibility, such as aluminum nitride Xiong  _et al._ (2012); Jung  _et
al._ (2013); Soltani  _et al._ (2016); Guo  _et al._ (2016); Wang  _et al._
(2021), gallium nitride Bruch  _et al._ (2015); Mohamed  _et al._ (2017);
Stassen  _et al._ (2019); Zheng  _et al._ (2022), gallium arsenide Kuo  _et
al._ (2014); Midolo  _et al._ (2015); Dietrich  _et al._ (2016), and lithium
niobate (LN) Lin  _et al._ (2019a); Pan  _et al._ (2019); Lin  _et al._
(2019b); Liu  _et al._ (2021); Qi and Li (2020); Zhou  _et al._ (2021); Xue
_et al._ (2021); Lin  _et al._ (2020); Yuan  _et al._ (2021); Liu  _et al._
(2020) have been well developed, making integrated microcavities with quality
(Q) factor higher than $10^{7}$ obtainable. As a result, an unprecedented
conversion efficiency $5\times 10^{4}\ \mathrm{W^{-1}}$ of second-harmonic
generation (SHG) and ultra-low-threshold optical parametric oscillators are
demonstrated in thin-film LN microcavities Lu  _et al._ (2020, 2021).

Despite the excellent performance of on-chip nonlinear optics devices, the
experimental studies are restricted to the guided modes within the integrated
photonic circuits. From the fundamental light-matter interaction point of
view, the enhanced nonlinear optical processes should not be limited to the
in-plane guided modes. It is anticipated that the potential coupling from the
on-chip modes to free-space continuum modes would lead to additional optical
radiation channels Li and et al. (2022), which might impose ultimate
performance limitations of integrated devices. For practical applications, the
in-plane nonlinear frequency conversion has several drawbacks: Firstly,
special geometry designs are required for matching the frequencies and
momentums of target optical modes with very different frequencies Guo  _et
al._ (2016); Wang  _et al._ (2021); Lu  _et al._ (2019). Secondly, the
collection and routing of the converted light for specific optical
applications, such as the spectroscopy of gas sample in free space Hummon  _et
al._ (2018); Niffenegger  _et al._ (2020), demands stable and specially
designed chip-to-free-space linear mode converters, such as grating,
holographic, and meta-surface structures Lee  _et al._ (2007); Jiang  _et al._
(2016); Kim  _et al._ (2018); Calafiore  _et al._ (2014); Meng  _et al._
(2020). Moreover, these devices usually have a limited working bandwidth.

In this Letter, the nonlinear optical radiation of an integrated LN microring
due to the second-harmonic and sum-frequency emissions are experimentally
observed. By simultaneously exciting counter-propagating resonant modes with
17.6 mW telecom pump laser, the radiation to vertical direction of 6.57 pW is
collected. Compared to previous doubly-resonant SHG process where strict
structure design is needed for the phase-matching conditions, the second-
harmonic radiation is mode- and pump wavelength-independent. Furthermore, the
broadband second-harmonic radiation is applied to the spectroscopy studies of
visible transition lines of rubidium atoms by an telecom probe laser, which
manifests an unique chip-to-free-space and also visible-to-telecom interface.
Our work reveals the unexplored regime of cavity nonlinear photonics, merges
the gap between integrated photonics devices and free-space applications.

Figure 1: Schematic illustration of the nonlinear optical radiation (NOR)
process. The blue arrows indicate counter-propagating telecom pump lasers
(frequency $\omega$) while the red arrows indicate vertically radiated second-
harmonice light (frequency $2\omega$). The inset shows the detail of the
nonlinear interaction between guided modes and free-space continuum modes at
the cross-section of the microring.

Figure 1 schematically illustrates the nonlinear optical radiation (NOR) from
an on-chip microring cavity through second-harmonic process. Two telecom pump
lasers (the blue arrows) with the same frequency ($\omega$) and polarization
are coupled into the microring cavity from opposite directions, which excite
the degenerate clockwise (CW) and counter-clockwise (CCW) modes, respectively.
As the inset shows, along the microring the standing-wave pump field creates
the second-harmonic (SH) nonlinear polarization through the $\chi^{(2)}$
susceptibility of the dielectric material, which is equivalent to
nanoparticles oscillating at a frequency of $2\omega$ and radiating light into
the free-space Yu  _et al._ (2014). Therefore, by selecting an appropriate
crystal axis and tuning the polarization of the pump laser to match the
nonlinear susceptibility tensor, the SH radiation wave can be collected from
the direction perpendicular to the chip. Reversely, an input light from free-
space can also couple with the guided modes in the microring by exciting a
localized spot of the nonlinear polarizability. It is also anticipated that
radiation due to the sum-frequency generation also exists Li and et al. (2022)
when selectively driving the counter-propagating pump modes with different
frequencies. Comparing to the conventional SH generation between on-chip
guided modes, where special geometry design Surya  _et al._ (2019); Luo  _et
al._ (2019); Briggs  _et al._ (2021) or periodically poling Fejer  _et al._
(1992); Wang  _et al._ (2018); Jankowski  _et al._ (2020); Lu  _et al._
(2020); Hao  _et al._ (2020); Chen  _et al._ (2021) is required for strict
phase-matching or quasi-phase-matching conditions, the NOR is expected to be
ultra-broadband and free from the dispersion engineering of the microcavities.

Figure 2: (a) Experimental setup. The pump laser is split by a 50:50 BS and
launched to the chip from two directions. The radiated light is collected by
an objective lens with N.A. of 0.47. IR laser, infrared laser; EDFA, erbium
doped fiber amplifier; BS, beam splitter; PC, polarization controller; PD,
photo-detector; SPCM, single photon counting module; 780 SMF, 780 nm single
mode fiber. The NOR signal is coupled into a standard 780 nm single mode fiber
which can naturally filter the scattering of telecom pump. The axes on the
chip denote crystal orientations of the thin-film lithium niobate, which is
corresponding to the axes in (b). (b) The CCD (Basler acA1920-155um) image of
the microring when the bi-directional pumps are on-resonance with the cavity,
showing the SH radiation. (c) The SEM picture of the device. (d) and (e) show
the detected NOR signal and transmission spectrum of pump respectively. The
orange line in (e) is fitting of cavity resonance with
$Q_{\text{load}}=5.15\times 10^{4}$.

_Experiment setup.-_ The NOR mechanism is investigated by an integrated LN
microcavity with the experimental setup shown in Fig. 2(a). The LN microring
is fabricated with 500 nm thick X-cut thin-film LN on a sapphire substrate
(NanoLN), and Fig. 2(c) shows the SEM image of the waveguide-coupled microring
device. The device patterns are defined by hydrogen silses-quioxane (HSQ)
resist via electron-beam lithography, and then transferred to LN through the
optimized $\mathrm{Ar^{+}}$ etching process by inductively coupled plasma
reactive ion etching (ICP RIE) tool. A thickness of 220 nm of LN is etched.
The radius of the microring is 70$\>\mu$m, and the widths of the coupling
waveguide and microring cavity are $1.72\,\mathrm{\mu m}$ and
$1.80\,\mathrm{\mu m}$, respectively. The coupling gap between the waveguide
and microring is about $600\,\mathrm{nm}$. Figure 2(e) presents the
transmission spectrum of the device, with the loaded and intrinsic quality
factors of a typical resonance around 1550 nm [the orange dips in Fig. 2(e)]
are $0.52\times 10^{5}$ and $1.0\times 10^{5}$, respectively.

Figure 3: Relationship between powers of pump and collected signals. (a), (b)
The input-output power dependency of NOR and DF generation, respectively. The
dots show the experimental results while the solid lines represent quadratic
and linear fitting results. The insets of (a) and (b) are illustrations of the
input pump and output signals for NOR and DF generation, respectively. (c) The
power of NOR signal when fixing $P_{\text{pump}}=15.8~{}\mathrm{mW}$ and
sweeping pump frequency across cavity resonance. (d) Detected signals DF
signals for visible input ($P_{\text{input}}=60.0~{}\mathrm{mW}$) focused on
microring cavity (blue line) and on waveguide (red line), respectively. The
purple area denotes the background noise level after filtering while visible
input is not injected.

A continuous-wave telecom laser is amplified by an erbium-doped fiber
amplifier (EDFA) and divided into two paths by a 50:50 fiber beam-splitter.
Consequently, the chip is driven by the lasers in two opposite directions
using the fiber lens, with an estimated single-side coupling efficiency of
$27.4\%$. Two fiber circulators are introduced in each path for separating and
detecting transmitted and reflected signals. Fiber polarization controllers
are adjusted for maximum coupling efficiencies to the TE modes of the
integrated device. The free-space radiation of the microcavity at visible
wavelength ($780\,\mathrm{nm}$) is collected by an objective lens with a
$\text{N.A.}=0.47$ and a working distance around 2$\,\mathrm{cm}$. We use a
90:10 beam-splitter to separate the free-space signal into two parts, with 10%
of the signal sending to the CCD for real-time imaging, while the rest (90%)
of the signal is directed to a single mode fiber through a collimator and then
fed into a single photon counting module (SPCM).

NOR.- When scanning the pump lasers, we observed the periodic shining of the
microring from the CCD which only responds to the visible wavelengths. Shown
in Fig. 2(b) is the image of the device obtained from the CCD, which is lit
obviously only when the pump laser is on-resonance with the microring, while
the waveguide is invisible. Besides, the radiated visible photons from the
microring, which are collected from a small section of waveguide in the
microring as the blue circle shown in Fig. 2(b), are counted by SPCM when the
wavelength of the telecom laser scans from 1540 nm to 1570 nm with a scanning
speed of $0.5\,\mathrm{nm/s}$. The resulting spectra of the counts and the
transmitted telecom pump laser are recorded [Figs. 2(d) and (e)], which show
aligned peaks of the visible photons and the dips of the pump laser. The
spectra confirm that the radiation occurs for all microring modes when the
pump laser is on-resonance with them, and indicate the resonance-enhanced
nonlinear frequency up-conversion from telecom to visible wavelengths.

The visible radiation from the chip might be induced by the scattering of the
unexpected SH generation between guided modes in the microring and waveguide,
however, the scattering process can not explain the following phenomena: (i)
The radiation is broadband over a bandwidth exceeding $30\,nm$. (ii) The
radiation by the microring is not uniform along the azimuthal angle $\phi$.
The large bandwidth indicates the NOR is free of strict phase-matching
condition, in contrast to the narrow bandwidth of SH generation between on-
chip guided modes. The non-uniform radiation from the microring is attributed
to the dependence of the radiation intensity on the $\chi^{(2)}$ nonlinear
susceptibility tensor. To be specific, for the X-cut LN and TE microring modes
used in this work, the radiation intensity is symmetrical about the Y-crystal
axis [labelled in Fig. 2(b)] as $\propto\cos^{4}\phi$ and is the brightest in
the upper and lower parts where the largest second-order nonlinear
coefficients ($d_{33}=-27\,\mathrm{pm}/\mathrm{V}$) could be accessed. These
results agree with our theoretical predictions of NOR Li and et al. (2022).

To further verify the physical mechanism of the NOR, we study the power-
dependent relationship between the input pump and visible radiation. As shown
in Fig. 3(a), the power of visible radiation signal increases near
quadratically with the pump power. Each data point in Fig. 3(a) represents the
fitted peak value of the collected radiation spectra for a given pump power.
One of the typical spectra of the NOR is shown in Fig. 3(c), with the fixing
pump power of 15.8 mW. By fitting the experimental data with
$P_{\mathrm{NOR}}=A+B\times P_{\mathrm{pump}}^{2}$, we get $A=0.0879\pm
0.0274\,\mathrm{pW}$ and $B=0.0209\pm 2.21\times
10^{-4}\,\mathrm{pW}\cdot\mathrm{mW}^{-2}$ with high coefficient of
determination $R^{2}=0.998$, which agrees with the expected SH generation
process.

Reversal of NOR.- According to the intrinsic coherent nonlinear optical
effect, the theory of NOR also predicts the different-frequency (DF)
conversion from the free-space photons to on-chip guided photons, as a
reversal process of the sum-frequency generation in $\chi^{(2)}$ dielectric
material. Therefore, we design and carry out the experiment by injecting
visible laser to the microring, as schematically illustrated by the inset of
Fig. 3(b). The DF conversion is realized by reversing the setup of the
collection of visible radiation in Fig. 2(b), and a vertically input visible
photon could be converted to two counter-propagating telecom photons in the
microring, with the conversion stimulated by a CCW pump laser. To avoid the
noisy background of the generated CW signal due to the backscattering of CCW
pump in the micoring, we investigate the non-degenerate case of the NOR by
employing microring modes with different mode orders.

Due to the energy conservation condition, the frequency of the generated DF
signal is
$\omega_{\mathrm{DF}}=\omega_{\mathrm{input}}-\omega_{\mathrm{pump}}$, so the
process is only efficient when $\omega_{\mathrm{DF}}$ meets a resonant mode.
Denote the free spectrum range (FSR) of the microring cavity as $\Omega$, thus
we would expect an efficient DF process for
$\Delta=\omega_{\mathrm{input}}-2\omega_{\mathrm{pump}}$ being integer
multiples of $\Omega$ up to a small cavity dispersion, with the corresponding
integer being the mode order with respect to the telecom pump. For our device,
$\Delta\approx 600\,\mathrm{GHz}$ is large enough for high performance
filtering, corresponding to two FSRs of the cavity. We cascade four band-pass
filters to achieve high signal-to-noise ratio (SNR), with the signal and
background noise showing in Fig. 3(d). For the visible input laser, it was
generated by a PPLN frequency doubler pumped by a continuous-wave telecom
laser. A half wave plate is applied to control the polarization of visible
pump, which can tune the nonlinear interaction to match the strongest
nonlinear susceptibility $d_{33}$ by optimizing DF signal.

Figure 3(b) shows the linear relationship between the powers of non-degenerate
DF signal and the visible input while fixing both the power and frequency of
telecom pump on resonance, manifesting the characterization of DF conversion.
Figure 3(d) shows the DF signals obtained by fixing the on-chip pump power at
10.5 mW, and sweep the detuning of visible input (60.0 mW and 0 mW) with the
speed of 1 nm/s. Each data point in Figure 3(b) represents the highest value
of the peak minus the average of background noise when there is no visible
input. The coherent frequency conversion between the free-space light and the
on-chip guided modes not only validates the mechanism of NOR, but also
promises potential applications for detecting free-space radiations.

Application and discussion.- The demonstrated NOR directly opens up the
broadband nonlinear chip-to-free-space conversion channel without specially-
designed photonic structures, which offers new opportunities for practical
applications of photonic chip. For instance, by leveraging the well-developed
on-chip optical soliton microcomb Kippenberg  _et al._ (2018); Bruch  _et al._
(2021), the NOR could be applied to radiate visible soliton pulses into free
space and probe the gas or liquid sample around the chip. Here, we conduct a
proof-of-principle application of the NOR for sensing atomic gas with a
commercial telecom laser. As shown in Fig. 4(a), we insert a cylinder glass
cell filling with sample atoms above the chip, and measure the transmitted NOR
signal. There are two isotopes of Rubidium atom, i.e.
${}^{\text{85}}\text{Rb}$ and ${}^{\text{87}}\text{Rb}$, in the cell which
have been extensively investigated in quantum information processing, atomic
clock, and precision measurement Hummon  _et al._ (2018); Stern  _et al._
(2020); Gundavarapu  _et al._ (2019); Xie  _et al._ (2019). When NOR signal
passes through the cell, it would be absorbed by rubidium atoms if its
frequency matches the D2 atomic transition lines shown in Fig. 4(b).

Figure 4: (a) Illustration of multi-component gas sensor based on NOR. (b) D2
transitions of ${}^{\text{85}}\text{Rb}$ and ${}^{\text{87}}\text{Rb}$ atoms.
(c) Saturation absorption spectrum (SAS) of rubidium gas. (d) Tunability of
NOR under thermal control of the microcavity. The bandwidth of the device
covers all transition lines, shown by the vertical dashed lines.

The broadband nature of NOR makes it possible by simply tuning the frequency
of the resonant mode, which otherwise would be difficult for SHG between
guided modes and also requires additional chip-fiber or chip-to-free-space
coupling apparatus. Here, we use an off-chip heater to finely tune the
resonance frequency of the microring cavity to match the transition line at
around 780.25 nm. The normalized transmission spectra are shown in Fig. 4(d),
with four dips corresponding to each two transitions of
${}^{\text{85}}\text{Rb}$ and ${}^{\text{87}}\text{Rb}$, which agree with the
reference saturated absorption spectrum of Rubidium shown in Fig. 4(c).
According to the measured spectra, it is possible to distinguish the type of
atoms and their concentrations. The tunability and controllability of the
system are characterized with the following method: while gradually changing
the temperature of the heater, we find that the relative positions of the NOR
signal peak shift step by step, which scans across all the absorption dips of
Rb atoms. By utilizing the four-wave mixing in the microring cavity for
optical frequency comb generation, the NOR can be incorporated to realize
high-precision spectroscopy and sensing of various types of gas covering a
wide range of wavelengths Tan  _et al._ (2021); Bao  _et al._ (2021), as well
as optical clock by referencing to visible-wavelength atomic transitions
Hummon  _et al._ (2018); Stern  _et al._ (2020) based photonic chip.

It is worth noting that the collected NOR from a focus area with a radius of
only about $2\,\mathrm{\mu m}$ is 6.57 pW for only a 17.6 mW on-chip pump
power. Considering the radiation of the whole microring and taking account of
the loss of our apparatus, the NOR conversion efficiency of our device is
estimated to be above $1.50\times 10^{-7}$. It is anticipated that the
conversion efficiency could be further increased with a higher Q-factor and a
higher $P_{\mathrm{pump}}$. For example, with the state-of-art Q-factor of
$10^{7}$ Gao  _et al._ (2022) and an on-chip pump power of 100 mW, we predict
a conversion efficiency of $4.75\%$. Therefore, the NOR effect could
significantly modify the performance of high-quality factor microcavities, and
the NOR effect could be more profound for interfacing the free space and
photonic chip.

_Conclusion._ \- The nonlinear optical radiation of integrated photonic device
is experimentally revealed by investigating the free-space second-harmonic
radiation of a LN microring resonator. The NOR process as well as its reversal
free-space to chip different-frequency conversion process are experimentally
verified. The NOR is significant for integrated photonic devices due to the
enhanced light-matter interactions, and the principle is extensible to other
nonlinear optics processes. Thus, we expect similar phenomena could be
observed in photonic chips made by other materials. In additional to potential
applications for interfacing free-space and photonic chips, the revealed NOR
also appeals for more careful investigation and design of photonic devices,
especially for quantum devices and precision measurement instruments that NOR
might introduce non-negligible and un-avoidable noise photons.

###### Acknowledgements.

This work was funded by the National Key Research and Development Program
(Grant Nos. 2017YFA0304504, 2018YFA0306200), the National Natural Science
Foundation of China (Grant Nos. 11874342, 11904316, 11922411, 12104441,
11934012, 91850107, 12174116 and U21A6006), Anhui Provincial Natural Science
Foundation (Grant Nos. 2008085QA34 and 2108085MA22), Guangdong Innovative and
Entrepreneurial Research Team Program (Grant No. 2016ZT06C594), Science and
Technology Project of Guangdong (2020B010190001). It was also supported by the
Fundamental Research Funds for the Central Universities, and the State Key
Laboratory of Advanced Optical Communication Systems and Networks, China. This
work was partially carried out at the USTC Center for Micro and Nanoscale
Research and Fabrication.

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