# EpiRL: A Reinforcement Learning Agent
to Facilitate Epistasis Detection

Kexin Huang
Courant Institute
New York University
521 Mercer Street, New York, NY 10012
<EMAIL_ADDRESS>
Rodrigo Nogueira
Tandon School of Engineering
New York University
6 MetroTech Center, Brooklyn, NY 11201
<EMAIL_ADDRESS>
###### Abstract

Epistasis (gene-gene interaction) is crucial to predicting genetic disease.
Our work tackles the computational challenges faced by previous works in
epistasis detection by modeling it as a one-step Markov Decision Process where
the state is genome data, the actions are the interacted genes, and the reward
is an interaction measurement for the selected actions. A reinforcement
learning agent using policy gradient method then learns to discover a set of
highly interacted genes.

##  1 Introduction

The fundamental goal for studying genetics is to understand how certain genes
can incur disease and traits. Since the advent of Genome-Wide Association
Studies (GWAS) (Burton et al., 2007), thousands of SNP (Single Nucleotide
Polymorphism)s have been identified and associated with genetic diseases and
traits. These SNPs are discovered through one-SNP-at-a-time statistical
analysis. However, individual gene marker is insufficient to explain many
complex diseases and traits (Mackay & Moore, 2014). Now, most geneticists
believe that gene-gene interaction (epistasis) can explain the missing
heritability incurred by the traditional approach.

There has been a substantial amount of work on epistasis detection. Exhaustive
combinatorial search methods like Multifactor Dimensionality Reduction (MDR)
(Yang et al., 2018) have been shown successful, but only in small genome-scale
due to computational complexity. Later, attempts to reduce search spaces
exhibit efficiency, like ReliefF and Spatially Uniform ReliefF (Niel et al.,
2015). Besides, machine learning-based algorithms gain popularity. Most of the
machine learning based methods for epistasis detection model the epistatic
process as a non-linear neural network. It predicts if an input sequence is
disease or healthy. Then, they rely on examining the internal weights of the
models to find the interacting SNPs. A high weight means the corresponding
SNPs are contributing to predict the disease. If multiple high weight SNPs are
detected, then they are considered interacted. For example, Random Forest
models each node as an SNP and grows a classification tree and later examines
the decision trace for interpretation (Jiang et al., 2009); BEAM (Bayesian
Epistasis Association Mapping) uses MCMC algorithm to iteratively test each
marker’s probability of association with disease, dependent on other markers
(Zhang et al., 2011). However, most machine learning based algorithms suffer
from a limited number of input sequences compared to the size of the sequence
(#SNPs). Another interesting approach is ant colony optimization algorithm
(Wang et al., 2010), which finds a refined subset of SNPs by iteratively
updating a selection probability distribution.

Although there are efficient methods to measure if a given SNPs set interact,
previous works all suffer from the high computational cost of finding all
possible n-combinations of SNP. For example, for a standard GWAS dataset with
$10^{6}$ SNPs, a 2-locus exam requires $5*10^{11}$ searches, a 3-locus exam
asks for $1.6*10^{17}$, a 4-locus search needs $4*10^{22}$ iterations. Hence,
how to utilize these metrics to get an SNP set from a genome-scale data is the
challenging part. Another challenge is that all the algorithms above assume
and output fixed n-locus interactions (typically 2 or 3) where n is unknown
for real biological data. We tackle these two challenges by introducing a
novel model based on Reinforcement Learning to the task of epistasis
detection.

##  2 Method

###  2.1 Model

Figure 1: An illustration of our One-Step MDP Model. $S$ is genome data, and
the state has $l$ actions where a probability $p_{x}$ is associated with the
action $SNP_{x}$. All $SNP_{x}$ whose $p_{x}$ are larger than a threshold $p$
would be selected as an interaction set and reward $R$ is computed based on
the set. Then the One-Step MDP terminates.

A typical GWAS dataset contains examples of sequences with no disease
(control) and with disease (case), where both have $l$ SNPs. We denote $t_{1}$
and $t_{2}$ as the number of control and case sequences, respectively. Each
SNP has three genotypes $\\{aa,Aa,AA\\}$, which is encoded by $\\{0,1,2\\}$.
We want to find a set of highly interacted SNPs with the size from $2$ to $n$.

We model the epistasis process as a one-step Markov Decision Process (MDP)
(Figure 1). The state $S$ is a latent representation encoded from genome data;
The action space is all the SNPs, where highly interacted SNPs are selected by
a probability threshold so that it poses no constraint to fix the size of
interaction; the reward is efficient interaction measurements like MDR correct
classification rate (CCR) and Rule Utility (Yang et al., 2018). A
reinforcement learning agent will learn to select SNPs that have high rewards,
i.e., high interaction, by using the policy gradient method. Our approach
solves the challenges mentioned above because firstly since it optimizes over
iterations and chooses only a small set of actions, it is non-exhaustive,
which means computationally feasible. It also utilizes the efficacy of
interaction measurements like MDR CCR and Rule Utility. Second, it picks an
action as long as the action passes a probability threshold, which means it
can output a different size of interaction set every iteration.

###  2.2 Network

Figure 2: An illustration of our EpiRL agent. The agent first encodes D, a
mini-batch of genome data, and then it predicts action-values through $W$
where a set of actions are selected and reward calculated. Along with the
baseline reward, we compute the loss and iterate to learn the best actions
with Reinforcement Learning.

For the input $D$, we randomly sample $K$ sequences, half from the case and
the other half from the control data set (Figure 2). We then encode each
sequence using the output of Convolutional Neural Network (CNN) or last hidden
state of Recurrent Neural Network(RNN) to capture the spatial structure of the
genome. These $K$ latent representations will be the state for our EpiRL
agent.

We then feed the state into a two-layer neural network $W$, which serves as a
value function approximator. The neural network will output $l$ probabilities
$P(SNP_{m}|D)$ for every SNP. We determine the size of interactions $n$ as the
number of SNPs that have probabilities larger than $1/n$ to allow up to
$n$-locus interaction. We then sample $n$ SNPs based on the probability
distribution generated by the network to ensure exploration for our RL agent.
This filtering forms our interaction set
$I=\\{SNP_{a_{1}},SNP_{a_{2}},..,SNP_{a_{n}}\\}$.

###  2.3 Reward

Given this SNP set $I$, we calculate the reward, which measures the
interaction. Our method uses the sum of two metrics as a reward: MDR CCR and
Rule Utility (Yang et al., 2018). These two measures are based on MDR
(Motsinger & Ritchie, 2006), which is a procedure that collapses the selected
interacted data set into a four variable table. Then we perform two
statistical calculations on top of this table, described as follows.

We have a genome data with size $(t_{1}+t_{2})\times l$, where $t_{1}$ and
$t_{2}$ are the number of sequences in control and case, respectively, and $l$
is the total number of SNPs. Suppose our RL agent picks $n$ actions. We then
extract the genome data with these selected SNPs and form a sub data set $I$
with size $(t_{1}+t_{2})\times n$.

There are three genotypes for each SNP: $\\{0,1,2\\}$. Therefore, for $n$
SNPs, there are $3^{n}$ possible combinations of SNPs. We denote each
combination as $\alpha_{t}$ where $t\in\\{1,\cdots,3^{n}\\}$. For each
combination, we can count the number of control and case in $I$. We denote
them $\alpha_{t}^{1}$ and $\alpha_{t}^{2}$. We assign a binary category to
each combination: if $\frac{\alpha_{t}^{2}}{\alpha_{t}^{1}}<1$, then this
combination is in the low-risk group $LR$, and if $\geq 1$, it is called a
high-risk group $HR$. Basically, for this specific genotype, if the number of
the case exceeds the number of control, then it is high-risk, and vice versus.
Now, we can construct four variables:

| High Risk | Low Risk
---|---|---
Case | TP | FN
Control | FP | TN

where

$TP=\sum_{t\in HR}\alpha_{t}^{2}$ (1) $FP=\sum_{t\in HR}\alpha_{t}^{1}$ (2)
$FN=\sum_{t\in LR}\alpha_{t}^{2}$ (3) $FP=\sum_{t\in LR}\alpha_{t}^{1}$ (4)

The above equations mean that we first divide case and control sequences in
the low and high-risk group and then retrieve the number of cases and controls
in each group. Now, we can calculate the two measures. These two measures
together are shown to be effective in measuring epistasis (Yang et al., 2018).
MDR CCR is the correct classification rate and Rule Utility $U$ derives from
the chi-square statistics of rule relevance, which measures the interaction:

$CCR=0.5\cdot(\frac{TP}{TP+FN}+\frac{TN}{FP+TN})$ (5)
$U=\frac{(R-\delta)^{2}}{(1+\delta)(\gamma-\delta-1)}$ (6)

where

$R=\frac{FP+TN}{TP+FN}$ (7) $\delta=\frac{FP}{TP}$ (8)
$\gamma=\frac{TP+FP+TN+FN}{TP}$ (9)

We sum CCR and U as our reward. Note that the calculation is fast since $n$ is
usually a small number. In our preliminary study on a set with 100 SNPs, the
average running time for one iteration is $0.01s$, where an iteration consists
that the network predicts probabilities, calculates the reward and back-
propagates the gradient.

###  2.4 Training

We train the model using REINFORCE algorithm (Williams, 1992). Our objective
consists of three parts:

$J_{1}=(R-\hat{R})\sum_{t\in I}-\log P(t|D)$ (10) $J_{2}=||R-\hat{R}||^{2}$
(11) $J_{3}=\lambda\sum_{t\in L}p(t|D)\log P(t|D)$ (12)

$\hat{R}$ is a baseline reward computed by the value network $U$, a 2-layer
neural network that minimizes $J_{2}$. $J_{1}$ is the advantage policy
gradient. The advantage is the gap between reward and baseline, which ensures
the agent to prefer actions that output rewards higher than expected. $J_{3}$
is the entropy regularization across all SNPs $L$ to mitigate peaky
probability distribution, where $\lambda$ is the parameter to adjust the
intensity of the mitigation.

##  3 Experiment

We use simulated data from GAMETES software, which generates random, pure,
strict, n-locus epistasis model (Urbanowicz et al., 2012). To evaluate our
method, we record SNPs set with top $K$ rewards across $C$ generated datasets.
We compare this set with the ground truth labels and compute the recall
$R@K=\frac{L}{C}$ where the agent gets $L$ predictions right in $C$ data set.
We are also interested in the average time the agent takes to detect the right
interaction.

In our preliminary study, we experiment our agent in a simulated 2-locus
dataset with 600 sequences of the case and control set and with 100 SNPs. We
design our data with standard genome constraint: 0.2 heritability; 0.7
prevalence; 0.2 minor allele frequency for both of 2 interacting SNPs. We
minimize our objectives using the Adam optimizer (Kingma & Ba, 2014) with
learning rate $1e^{-3}$.

We experimented the RL agent 50 times on the same data set. In each round of
experiment, the RL agent is asked to find the interacted 2-locus SNPs under
5000 iterations. Out of the 50 trials, 34 times the agent finds the interacted
SNPs under 5000 iterations. In the 34 times that the agent successfully
predicts the interaction set, the average iteration is 2260.6 and the average
time to find the SNPs is 22.4 s. In comparison, the exhaustive search takes 51
s.

In the future, we will experiment on a larger dataset with various locus
interactions. We will compare the recalls and the average running time with
existing methods: MDR, BEAM, and Ant Optimization. At last, we will run the
agent on GWAS Coronary Artery Disease (CAD) dataset since CAD is shown under
epistasis effect and we will compare other study’s reported epistasis on CAD
with ours.

##  4 Conclusion

Our work proposes a novel approach to model epistasis detection as a one-step
MDP and introduces reinforcement learning to address this problem. We believe
this will lead a new path to tackle the computational challenge in gene-gene
interaction detection.

## References

  * Burton et al. (2007) Paul R Burton, David G Clayton, Lon R Cardon, et al.  Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls.  2007\.
  * Jiang et al. (2009) Rui Jiang, Wanwan Tang, Xuebing Wu, and Wenhui Fu.  A random forest approach to the detection of epistatic interactions in case-control studies.  _BMC bioinformatics_ , 10(1):S65, 2009.
  * Kingma & Ba (2014) Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  _arXiv preprint arXiv:1412.6980_ , 2014.
  * Mackay & Moore (2014) Trudy FC Mackay and Jason H Moore.  Why epistasis is important for tackling complex human disease genetics.  _Genome medicine_ , 6(6):42, 2014.
  * Motsinger & Ritchie (2006) Alison A Motsinger and Marylyn D Ritchie.  Multifactor dimensionality reduction: an analysis strategy for modelling and detecting gene-gene interactions in human genetics and pharmacogenomics studies.  _Human genomics_ , 2(5):318, 2006.
  * Niel et al. (2015) Clément Niel, Christine Sinoquet, Christian Dina, and Ghislain Rocheleau.  A survey about methods dedicated to epistasis detection.  _Frontiers in genetics_ , 6:285, 2015.
  * Urbanowicz et al. (2012) Ryan J Urbanowicz, Jeff Kiralis, Nicholas A Sinnott-Armstrong, Tamra Heberling, Jonathan M Fisher, and Jason H Moore.  Gametes: a fast, direct algorithm for generating pure, strict, epistatic models with random architectures.  _BioData mining_ , 5(1):16, 2012.
  * Wang et al. (2010) Yupeng Wang, Xinyu Liu, Kelly Robbins, and Romdhane Rekaya.  Antepiseeker: detecting epistatic interactions for case-control studies using a two-stage ant colony optimization algorithm.  _BMC Research Notes_ , 3(1):117, Apr 2010.  ISSN 1756-0500.  doi: 10.1186/1756-0500-3-117.  URL https://doi.org/10.1186/1756-0500-3-117.
  * Williams (1992) Ronald J. Williams.  Simple statistical gradient-following algorithms for connectionist reinforcement learning.  _Machine Learning_ , 8(3):229–256, May 1992.  ISSN 1573-0565.  doi: 10.1007/BF00992696.  URL https://doi.org/10.1007/BF00992696.
  * Yang et al. (2018) Cheng-Hong Yang, Yu-Da Lin, and Li-Yeh Chuang.  Multiple-criteria decision analysis-based multifactor dimensionality reduction for detecting gene-gene interactions.  _IEEE Journal of Biomedical and Health Informatics_ , 2018.
  * Zhang et al. (2011) Yu Zhang, Bo Jiang, Jun Zhu, and Jun S Liu.  Bayesian models for detecting epistatic interactions from genetic data.  _Annals of human genetics_ , 75(1):183–193, 2011\.


††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Ultrahigh Dynamic Range and Low Noise Figure Programmable Integrated
Microwave Photonic Filter

Okky Daulay  Nonlinear Nanophotonics Group, MESA+ Institute of Nanotechnology
University of Twente, Enschede, Netherlands  Gaojian Liu  Nonlinear
Nanophotonics Group, MESA+ Institute of Nanotechnology University of Twente,
Enschede, Netherlands  China Academy of Space Technology, Xi’an, China
Kaixuan Ye  Nonlinear Nanophotonics Group, MESA+ Institute of Nanotechnology
University of Twente, Enschede, Netherlands  Roel Botter  Nonlinear
Nanophotonics Group, MESA+ Institute of Nanotechnology University of Twente,
Enschede, Netherlands  Yvan Klaver  Nonlinear Nanophotonics Group, MESA+
Institute of Nanotechnology University of Twente, Enschede, Netherlands
Qinggui Tan  China Academy of Space Technology, Xi’an, China  Hongxi Yu  China
Academy of Space Technology, Xi’an, China  Marcel Hoekman  LioniX
International BV, Enschede, Netherlands  Edwin Klein  LioniX International BV,
Enschede, Netherlands  Chris Roeloffzen  LioniX International BV, Enschede,
Netherlands  Yang Liu  Institute of Physics, Swiss Federal Institute of
Technology Lausanne (EPFL), Lausanne, Switzerland  David Marpaung
<EMAIL_ADDRESS>Nonlinear Nanophotonics Group, MESA+ Institute of
Nanotechnology University of Twente, Enschede, Netherlands

###### Abstract

Microwave photonics (MWP) has adopted a number of important concepts and
technologies over the recent pasts, including photonic integration, versatile
programmability, and techniques for enhancing key radio frequency performance
metrics such as the noise figure and the dynamic range. However, to date,
these aspects have not been achieved simultaneously in a single circuit. Here,
we demonstrate, for the first time, a multi-functional integrated microwave
photonic circuit that enables on-chip programmable filtering functions while
achieving record-high key radio frequency metrics of $>$120 dB.Hz dynamic
range and 15 dB of noise figure that are previously unreachable. We unlock
this unique feature by versatile complex spectrum tailoring using an all
integrated modulation transformer and a double injection ring resonator as a
multi-function optical filtering component. This work breaks the conventional
and fragmented approach of integration, functionality and performance that
currently prevents the adoption of integrated MWP systems in real
applications.

## Introduction

Integrated microwave photonic (MWP) systems can offer significant advantages
in future radio frequency (RF) and microwave systems to realize advanced
concepts such as multi-band, all-spectrum communications Akyildiz2020  and
broadband programmable front-ends Chappell2014 , which are important for next
generation communications (for example 6G). To play a key role in real RF
applications, integrated MWP circuits need to simultaneously show advanced
programmability and exceptional performance in terms of losses, noise figure,
and dynamic range in a reduced footprint capmany2007microwave ;
yao2009microwave ; capmany2012microwave ; marpaung2013integrated ;
marpaung2019integrated . In recent pasts, a number of integrated MWP functions
have widely been demonstrated, such as for filtering marpaung2013si ;
shahnia2015independent ; fandino2017monolithic ; liu2020integrated ;
tao2021hybrid , phase shifting li2011broadband ; burla2014onchip ;
porzi2018photonic ; chew2019integrated ; mckay2019brillouin , delay line
zhuang2011low ; liu2017gigahertz ; ji2019onchip ; tsokos2021silicon , waveform
generator khan2010ultrabroad ; weiner2011ultrafast ; marpaung2011impulse ;
wang2015reconfigurable ; zhu2020si ; falconi2022  and beamforming
yi2011photonic ; yi2011photonic ; zhuang2014chip ; miller2020large ;
zhu2020silicon . Typically, these functions were achieved in dedicated, or
application specific circuits, and the measured RF metrics were only sparsely
reported. The values of the RF gain, noise figure (NF), and spurious-free
dynamic range (SFDR) in these circuits are usually far-off from the
requirements of practical RF systems.

Recently, new ways of building a programmable integrated MWP circuit have been
extensively explored, for example through mesh-waveguide circuits that can
synthesize a large-number of functionalities through programming
zhuang2015programmable ; perez2017multipurpose ; perez2020multipurpose ;
bogaerts2020 . While versatile, the typical functional performance of these
circuits, such as filter extinction, or passband quality, and the RF metrics
were significantly lower than that of application specific circuits liu2017all
; liu2018link ; zhu2019positive ; daulay2021microwave . Thus, at present, the
unique and well-sought combination of high integration-density, versatile
programmability, and high RF performance remains elusive. In particular, there
is a bottleneck in co-integrating components that support NF reduction and
linearization with components that provide functionalities.

Achieving simultaneous on-chip programmability and high RF performance
requires integration of new types of photonic devices. A versatile modulation
transformer (MT) that can tailor the phases and amplitudes of optical carrier
and RF sidebands has recently identified as a key element for enhancement of
dynamic range guo2021versatile ; daulay2021tutorial ; liu2021integrated . On
the other hand, a programmable resonator such as a double-injected ring
resonator (DI-RR) cohen2018response  can provide a large number of filtering
functions in a device.

Fig. 1: Artistic impression of an ultra-high dynamic range programmable
integrated MWP circuit. The circuit contains of a versatile modulation
transformer (MT) to independently tailor the phase and amplitude of optical
modulation spectrum and an equally versatile double-injection ring resonator
(DI-RR) to synthesize a variety of responses, including programmable RF
filters. The combination of MT and DI-RR also allows for linearization through
cancellation of intermodulation distortion (IMD) and noise figure (NF)
reduction through low biasing and carrier suppression technique, leading to
ultra-high dynamic range. SFDR: spurious-free dynamic range.

In this work, we experimentally demonstrate a programmable integrated MWP
circuit with a unique combination of a versatile MT device and a DI-RR,
realized in a low-loss silicon nitride platform. With this circuit, we show
for the first time, an array of RF filtering functions simultaneously with
record-low NF of 15 dB, achieved using low-biasing technique, and ultra-high
dynamic range of $>$ 120 dB.Hz, achieved using on-chip linearization. Our
results, point a significant step forward towards the realization of practical
programmable integrated MWP circuit in a real life applications.

Fig. 2: Programmable basic building blocks. (a) Picture of the photonic chip,
containing of a modulation transformer (MT) and a double injection ring
resonator (DI-RR). The MT consists of a spectral de-interleaver used to
isolate one RF sideband from the entire modulated spectrum. The phase shifter
and tunable coupler are used to control the phase and amplitude of the
isolated sideband. Modulation transformation is achieved after recombination
at the output. See Supp. Info. A for details of the MT. (b) Experimental
result of intensity-to-phase modulation (IM-PM) conversion with 25 dB
extinction, achieved using the MT with IM input. (c) Measured phase-to-
intensity modulation (PM-IM) conversion with 64 dB extinction, achieved using
the MT with PM input. The DI-RR circuit is used to synthesize multiple filter
functionalities. (d-g) Simulated (blue line) and measured (black line)
selected responses of the silicon nitride DI-RR when tuned to exhibit notch
filter (d), bandpass filter (e), Fano-like response (f), and all-pass response
(g). See Supp. Info. B for details of the DI-RR.

## Results

### Programmable Microwave Photonics

The concept of ultrahigh dynamic range and low NF programmable integrated MWP
filter explored here is illustrated in Fig. 1. The MT shapes the input
modulated signal from an optical modulator to a desired modulation format that
suited the targeted filtering function. Further, the spectral shaping can be
used to create bespoke optical carrier and sidebands phase and amplitude
distribution leading to linearization and NF reduction. The DI-RR is used as a
filtering element that can provide notch or bandpass filtering from the same
output port. In this way, high dynamic range and programmable filtering can be
achieved simultaneously.

The MT is implemented as an asymmetric Mach-Zehnder interferometer (aMZI)
loaded with 3 ring resonators as a spectral de-interleaver luo2010high ;
guo2021versatile , combined with a tunable attenuator and a phase shifter (For
the details of the MT, see Supplementary information A). The DI-RR is an add-
drop ring resonator that is being double-injected from single input, hence
capable of synthesizing a large number of responses (see Supplementary
information B for the details of the DI-RR). The schematic of the entire
circuit overlaid on a photograph of the fabricated silicon nitride chip is
shown in Fig. 2a. The waveguide propagation loss in the circuit is 0.1 dB/cm,
and the fiber-to-chip coupling loss is 1.1 dB/facet.

The programmability of our circuit is demonstrated by measurement results
depicted in Fig. 2b-2g. An optical modulated spectrum at the input of the MT
circuit can be transformed into an output spectrum consisting of spectral
components with designer phase and amplitude profiles. As examples, when
operated with intensity modulator input, our silicon nitride MT circuit is
capable of intensity-to-phase modulation (IM-PM) conversion with 25 dB
extinction (Fig. 2b), and with a phase modulator input, the circuit can
achieve PM-IM conversion with $>$ 60 dB of extinction (Fig. 2c). The DI-RR, on
the other hand, can synthesize a variety of responses from a single output
port, including notch filter, bandpass filter, Fano-like response, and an all-
pass response okky2021mwp  as shown in Fig. 2d-2g. Because of the low
propagation loss of the silicon nitride waveguide, narrow linewidth of the
filtering function of around 400 MHz can be achieved in a device with 20 GHz
free spectral range (FSR).

Fig. 3: Noise figure reduction of the programmable MWP filters. a Experiment
setup of programmable microwave photonics with enhanced performance using low-
biasing Mach-Zehnder Modulator (MZM). b High-rejection RF notch filter with 58
dB rejection. c, The measured RF gain and noise figure (NF) of the RF notch
filter. Maximum RF gain of 10 dB and a minimum NF of 15 dB are achieved
through the low-biasing of the MZM. d, The measured spurious-free dynamic
range (SFDR) of the RF notch filter at 1 GHz, reaching 116
$\rm{dB}\cdot\rm{Hz}^{2/3}$. e, Single-sideband (SSB) RF bandpass filter with
15 dB rejection, obtain using optical carrier re-insertion with the MT and a
bandpass response from the DI-RR. f, The measured RF gain and NF of the RF
bandpass filter. Maximum RF gain of -3 dB and minimum noise figure of 22 dB
are achieved using the low-biasing MZM. g, The measured SFDR of the RF
bandpass filter at 7 GHz, reaching a high value of 112
$\rm{dB}\cdot\rm{Hz}^{2/3}$. EDFA: erbium-doped fiber amplifier, MZM: Mach-
Zehnder modulator, PC: polarization controller, RF: radio frequency, PD:
photodetector Fig. 4: Linearization of RF notch filter. a Experiment setup of
simultaneous RF notch filter with linearization in a phase modulator
(PM)-based system. b, Measured RF notch filter with 55 dB rejection. c, The
measured two-tone RF spectra at the output of the photodetector for the
linearized RF notch filter (blue) and single-sideband (SSB) RF notch filter
without linearization (yellow) with 29 dB reduction of IMD3 power was
achieved. d, The measured spurious-free dynamic range (SFDR) of the proposed
linearized RF notch filter and SSB RF notch filter without linearization at RF
frequency of 9 GHz. The proposed linearized RF notch filter has a record-high
SFDR of 123 $\rm{dB}\cdot\rm{Hz}^{4/5}$. EDFA: erbium-doped fiber amplifier,
PM: Phase modulator, PC: polarization controller, RF: radio frequency, PD:
photodetector

### High Gain and Low Noise Figure MWP Filter

We carried out a series of MWP filtering experiments using our circuit, with a
setup diagrammed in Fig. 3a. A high power optical source implemented with a
external cavity laser and an erbium-doped fiber amplifier (EDFA) is sent to a
low-biased IM, creating an intensity modulated spectrum. We then configure our
MT to shape the IM spectrum into a dedicated output spectrum suitable for the
filtering tasks provided by the DI-RR (see Supplementary information C for
additional details of the extended experimental scenarios). In particular, we
synthesized two RF filters with simultaneously high RF performance metrics.
The first filter is an RF notch filter with intensity-to-asymmetric dual
sidebands (IM-aDSB) modulation transformation using the MT and a notch
response from the DI-RR. The second filter is an RF bandpass filter using
combination of optical carrier re-insertion technique using the MT and a
bandpass response from the DI-RR.

Figure 3b shows the measured RF notch filter with 58 dB rejection, 3-dB
bandwidth of 400 MHz at the central frequency of 7 GHz. The high rejection in
the filter was achieved through phase cancellation marpaung2015low  enabled by
adjusting the phase and amplitude of the isolated sideband in IM-aDSB
modulation conversion. Through low biasing the IM, we then achieved an optimum
measured RF gain of 10 dB and a lowest measured NF of 15 dB (Fig. 3c). These
high gain and low noise figure translated into a high spurious-free dynamic
range (SFDR) of 116 $\rm{dB}\cdot\rm{Hz}^{2/3}$ when we performed a two-tone
test at the frequency of 1 GHz (Fig. 3d). These results represent the best RF
gain, noise figure, and SFDR combination for an MWP notch filter. The RF notch
filter central frequency can be tuned over 20 GHz, which is the full FSR of
the DI-RR.

We then program the MT and the DI-RR to exhibit RF bandpass filter with 15 dB
rejection, as shown in Fig. 3e. We measured RF gain of -3 dB, NF of 22 dB
(Fig. 3f), and SFDR of 112 $\rm{dB}\cdot\rm{Hz}^{2/3}$ at 7 GHz (Fig. 3g). The
versatility of the MT allows for the combination of low biasing and single-
sideband (SSB)-based RF bandpass filtering, which would have not been possible
only with an IM or a dual-parallel MZM. As a result, we can demonstrate the
best combination of measured NF and SFDR for an RF bandpass filter.
Importantly, these results were achieved without any electrical amplification
in the system.

Table 1: Performance Comparison of Programmable Microwave Photonic Circuits

Year | Technology | Type of | Number of | Type of | Tuning | Performance | Gain | Noise | SFDR
---|---|---|---|---|---|---|---|---|---
| platform | devices | functions | functions | range | enhancement |  | figure |
|  |  |  |  | (GHz) |  | (dB) | (dB) | ($\rm{dB}\cdot\rm{Hz}^{2/3}$)
2017 fandino2017monolithic  | InP | Las., Mod., | 1 | LPF | 0-6 | No | -20 | N/A | 81.4
|  | RR, PD |  |  |  |  |  |  |
2018 perez2017multipurpose  | SOI | MZI mesh | 20 | BPF, notch | N/A | No | N/A | N/A | N/A
2017 liu2017all  | Si3N4 | RR | 1 | Notch | 0-12 | NF:LB | 8 | 15.6 | 116
2018 Zhang18OL  | SOI | Mod., RR, PD | 1 | BPF | 3-10 | No | -39 | N/A | 92.4
2019 liu2019integrationbrillouin  | As2S3 | RR, SBS | 1 | Notch | 0-15 | NF:LB | -10 | 27.1 | 96.5
2019 Zou19LPR  | InP | Las., Mod., | 7 | BPF, notch | 8-15 | No | N/A | N/A | N/A
|  | MMI |  | IFM, RFG |  |  |  |  |
2020 Gertler20APL  | SOI | SBS | 1 | BPF | 4-10 | No | -17 | 56.7 | 90.3
2020 daulay2020chip  | Si3N4 | RR | 1 | BPF | 2-7 | NF:CS | -10 | 27 | N/A
2021 daulay2021microwave  | Si3N4 | RR | 1 | Notch | 3-10 | NF:CS | 3 | 31 | 100
2021 guo2021versatile  | SOI | Mod., MT, | 2 | Notch, BPF | 5-25 | No | N/A | N/A | N/A
|  | RR, PD |  |  |  |  |  |  |
2021 tao2021hybrid  | InP+SOI | Las., Mod., | 2 | Notch, BPF | 3-25 | No | -28 | 51 | 99.7
|  | RR, PD |  |  |  |  |  |  |
2021 garret2021ultradeep  | Si3N4+As2S3 | RR, SBS | 1 | Notch | 2-12 | No | N/A | N/A | 92.2
2021 Li2021OpeX  | Si3N4+LiNbO3 | Mod., RR | 1 | Downconversion | 4-20 | No | -10 | 45 | 105
(this work) | Si3N4 | MT, DI-RR | 6 | Notch, BPF | 4-20 | NF:LB | 10 | 15 | 116
(this work) | Si3N4 | MT, DI-RR | 6 | Notch | 6-18 | SFDR:Lin. | -26 | 35 | 123

RR: ring resonator, PD: photodetector, Mod: modulator, MMI: multi-mode
interference, SBS: stimulated Brillouin scattering, MT: modulation
transformer, DI-RR: double-injection ring resonator, LPF: low pass filter,
BPF: bandpass filter, IFM: instantaneous frequency measurement, PS: phase
shifter, NF: noise figure, LB: low biasing, CS: carrier suppression, SFDR:
spurious-free dynamic range, Lin: linearization.

### On-Chip Linearization for Ultrahigh Dynamic Range

Some demanding applications of MWP require ultra-high SFDR beyond 120 dB.Hz
marpaung2019integrated . Such a metric can be achieved through linearization
of the modulator transfer function ZhangOoptica2016 ; ZhangOL20 ; MortonOE21 ;
feng2022ultrahighlinearity . But achieving linearization simultaneously with
functionalities has been proven challenging. Exploiting the versatility of our
MT and DI-RR circuit, we demonstrate, for the first time, simultaneous RF
notch filter and linearization in the same photonic chip.

Here, we implement third-order intermodulation distortion (IMD3) cancellation
technique using complex multi-order sidebands spectral shaping as we
previously reported liu2020linearized ; liu2021integrated ;
liu2021simultaneous  (see Supplmentary Notes D and E for the details of the
linearization technique). In this experiment, light is sent to a phase
modulator (PM) (Fig. 4a) that generates the optical carrier and first and
higher order sidebands. The MT is used to convert the PM to aDSB modulation
format, and to shape the amplitudes and phases of the first and second order
sidebands yielding to IMD3 suppression. The DI-RR is then used to filter the
first order sideband to create a phase cancellation RF notch filter. In this
way, simultaneous RF notch filter and linearization can be achieved, leading
to significant enhancement of the SFDR (For additional details of
linearization theoretical analysis and working principle, see Supplementary
information D and E respectively).

A conventional SSB RF notch filter without any linearization was used as a
benchmark for the proposed linearized RF notch filter. Figure 4b shows the
proposed linearized RF notch filter with $>$ 45 dB rejection improvement
compared to the conventional SSB RF notch filter. With the linearization, we
achieved 29 dB of IMD3 suppresion (Fig. 4c) and $>$ 20 dB SFDR improvement
(Fig. 4d). With a noise floor of -164 dBm/Hz, the conventional SSB RF notch
filter shows an SFDR of 103 $\rm{dB}\cdot\rm{Hz}^{2/3}$, while the linearized
RF notch filter exhibits a record-high SFDR of 123
$\rm{dB}\cdot\rm{Hz}^{4/5}$. It is clear from the slope of the measured IMD3
powers that the fifth order distortion becomes the dominant factor due to
cancellation of the third-order distortion. This result marked the highest
SFDR simultaneously achieved with on-chip signal functionality.

## Discussion

Table I summarizes the performance comparison of existing programmable MWP
circuits reported in the past 5 years. The route to largest number of
functions is still provided through MZI-mesh circuits perez2017multipurpose .
Virtually all demonstrations, except the application specific RF notch filter
reported in liu2017all , show relatively low RF performance, notably
characterised by very high NF ($>$25 dB) and low SFDR ($<$ 100 dB.Hz). These
results highlight the challenge in creating multifunctional and high-
performance MWP circuit. Our results overcome this challenge by delivering
high-level of reconfigurability with variety of functions in combination with
high gain, low NF, and ultrahigh SFDR. Importantly, in the context of
functional circuits, our demonstrations represent record-low NF and record-
high SFDR to date.

At present, our approach enables NF reduction through low biasing of MZM
intensity modulator. This translates into moderate SFDR enhancement as seen in
Table I. Our linearization method, on the other hand, is working for PM-based
MWP filter where maximum IMD3 suppression is achieved when the optical carrier
is partially suppressed (See Supplementary Info D). Due to the limitation in
maximum optical amplification in our experimental setup, the measured ultra-
high SFDR is still accompanied by relatively low link gain and high NF (Table
1). Achieving the gain, NF, and SFDR advantages simultaneously is feasible in
the PM link when higher optical amplification and higher power handling
components are used. Alternatively. this can also be achieved by developing
similar linearization strategy for low-biased MZM link.

In summary, we have designed, fabricated, and experimentally demonstrated, for
the first time, programmable integrated MWP circuit based on a unique
combination of versatile MT and equally versatile DI-RR. We reconfigure the
circuit to synthesize an array of RF filtering functions with high RF gain,
low NF, and ultrahigh SFDR concurrently. Hence, this work open a new paradigm
and an important step forward towards the realization of programmable
integrated MWP circuit with versatile functions, low NF, and ultra-high
dynamic range for real life applications.

## Methods

### Silicon Nitride Circuit Fabrication

The waveguides in our circuit are fabricated using LioniX standard TriPleX
Asymmetric Double-Stripe (ADS) geometry worhoff2015triplex ; roeloffzen2018low
. First, a $\rm SiO_{2}$ layer is grown from wet thermal oxidation of single-
crystal silicon substrate with temperatures equal to or above 1000 ℃. Then,
low-pressure chemical vapor deposition (LPCVD) is used for the Si3N4 layers
and together with the gas tetraethylorthosilicate (TEOS) for intermediate SiO2
layer. Next, the waveguides are patterned using contact lithography, and
processed with dry etching. Last, the waveguides are covered with an
additional SiO2 layer through LPCVD TEOS. Because typical top cladding
thickness cannot achieved only by LPCVD TEOS, plasma-enhanced chemical vapor
deposition (PECVD) is used to increase the SiO2 top cladding thickness to a
total of 8 µm. A thicker layers can be achieved because of the stress in PECVD
SiO2 layers is much less than LPCVD layers.

### Details of Programmable Microwave Photonics Experiments

An optical carrier from low relative-intensity noise (RIN) laser (Pure
Photonics PPCL550) with RIN of -155 dB/Hz and wavelength set at 1550 nm is
amplified with a low-noise erbium-doped fiber amplifier (EDFA, Amonics). Then,
the output of amplified optical carrier is optically modulated using a MZM
(Thorlabs, LN05S-FC 40 GHz) with bias point set at quadrature
($\theta_{\rm{B}}$ = $\pi$/2). The MZM is driven by RF signal from a vector
network analyzer (VNA, Keysight P5007A). The output of the MZM then sent to
another low-noise EDFA (Amonics) before being injected into a programmable
silicon nitride chip (LioniX International BV) fabricated using low-loss
TriPleX (Si3N4/SiO2) technologyworhoff2015triplex ; roeloffzen2018low  with
propagation loss of the optical waveguide at 0.1 dB/cm. The chip can be tuned
through thermo-optic tuning using custom made heater controller software,
while being stabilized by a thermoelectric cooler (TEC) controller. The
processed optical signal is sent to a photodetector (PD, APIC 40 GHz) and the
converted RF signal is measured with a VNA, while the RF spectrum analyzer
(RFSA, Keysight N9000B) is used to measure the RF filter’s noise and
linearity.

### Details of Linearization Experiments

A low RIN laser (Pure Photonics PPCL550) is modulated using a PM (EOSpace 20
GHz) using a sweeping RF signal with -3 dBm power from VNA (Keysight P5007A)
for RF notch filter experiment. For the dynamic range experiment, the two-tone
RF signal with power of 10 dBm, centered at 9 GHz with a space of 10 MHz from
signal generators (Wiltron 69147A and Rohde-Schwarz SMP02) is used to drive
the PM. The modulated signal is then amplified by a low-noise EDFA (Amonics),
before coupling to a programmable silicon nitride chip (LioniX International
BV). The processed optical spectrum is sent to a PD (APIC 40 GHz) to retrieve
the RF signal and measured using the VNA for the filter’s response and the
RFSA (Keysight N9000B) for linearity.

## Author Contributions

O.D. and G.L. contributed equally in this work. O.D., G.L.,and D.M. developed
the concept and proposed the physical system. O.D. designed the photonic
circuits, O.D. and G.L. developed and performed numerical simulations. O.D.
and G.L. performed the experiments with input from K.Y., R.B., and Y.K. E.K.,
M.H., and C.R. layout, developed, and fabricated the silicon nitride circuits.
D.M., O.D. and G.L. wrote the manuscript with input from all authors. D.M. led
and supervised the entire project.

##  Funding Information

This work was supported by Netherlands Organisation for Scientific Research
NWO Vidi (15702) and NWO Start Up (740.018.021).

## References

  * (1) I. F. Akyildiz, A. Kak, and S. Nie, “6g and beyond: The future of wireless communications systems,” _IEEE Access_ , vol. 8, pp. 133 995–134 030, 2020\.
  * (2) W. J. Chappell, E. J. Naglich, C. Maxey, and A. C. Guyette, “Putting the radio in “software-defined radio”: Hardware developments for adaptable rf systems,” _Proceedings of the IEEE_ , vol. 102, no. 3, pp. 307–320, 2014\.
  * (3) J. Capmany and D. Novak, “Microwave photonics combines two worlds,” _Nature photonics_ , vol. 1, no. 6, p. 319, 2007.
  * (4) J. Yao, “Microwave photonics,” _Journal of lightwave technology_ , vol. 27, no. 3, pp. 314–335, 2009.
  * (5) J. Capmany, J. Mora, I. Gasulla _et al._ , “Microwave photonic signal processing,” _Journal of Lightwave Technology_ , vol. 31, no. 4, pp. 571–586, 2012.
  * (6) D. Marpaung, C. Roeloffzen, R. Heideman _et al._ , “Integrated microwave photonics,” _Laser & Photonics Reviews_, vol. 7, no. 4, pp. 506–538, 2013\.
  * (7) D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” _Nature photonics_ , vol. 13, no. 2, pp. 80–90, 2019.
  * (8) D. Marpaung, B. Morrison, R. Pant _et al._ , “Si 3 n 4 ring resonator-based microwave photonic notch filter with an ultrahigh peak rejection,” _Optics express_ , vol. 21, no. 20, pp. 23 286–23 294, 2013\.
  * (9) S. Shahnia, M. Pagani, B. Morrison, B. J. Eggleton, and D. Marpaung, “Independent manipulation of the phase and amplitude of optical sidebands in a highly-stable rf photonic filter,” _Optics express_ , vol. 23, no. 18, pp. 23 278–23 286, 2015.
  * (10) J. S. Fandiño, P. Muñoz, D. Doménech, and J. Capmany, “A monolithic integrated photonic microwave filter,” _Nature Photonics_ , vol. 11, no. 2, pp. 124–129, 2017.
  * (11) Y. Liu, A. Choudhary, D. Marpaung, and B. J. Eggleton, “Integrated microwave photonic filters,” _Advances in Optics and Photonics_ , vol. 12, no. 2, pp. 485–555, 2020.
  * (12) Y. Tao, H. Shu, X. Wang _et al._ , “Hybrid-integrated high-performance microwave photonic filter with switchable response,” _Photonics Research_ , vol. 9, no. 8, pp. 1569–1580, 2021.
  * (13) W. Li, N. H. Zhu, L. X. Wang, and H. Wang, “Broadband phase-to-intensity modulation conversion for microwave photonics processing using brillouin-assisted carrier phase shift,” _Journal of lightwave technology_ , vol. 29, no. 24, pp. 3616–3621, 2011.
  * (14) M. Burla, L. R. Cortes, M. Li _et al._ , “On-chip programmable ultra-wideband microwave photonic phase shifter and true time delay unit,” _Optics letters_ , vol. 39, no. 21, pp. 6181–6184, 2014.
  * (15) C. Porzi, G. Serafino, M. Sans _et al._ , “Photonic integrated microwave phase shifter up to the mm-wave band with fast response time in silicon-on-insulator technology,” _Journal of Lightwave Technology_ , vol. 36, no. 19, pp. 4494–4500, 2018.
  * (16) S. X. Chew, D. Huang, L. Li _et al._ , “Integrated microwave photonic phase shifter with full tunable phase shifting range ($>$ 360°) and rf power equalization,” _Optics express_ , vol. 27, no. 10, pp. 14 798–14 808, 2019\.
  * (17) L. McKay, M. Merklein, A. C. Bedoya _et al._ , “Brillouin-based phase shifter in a silicon waveguide,” _Optica_ , vol. 6, no. 7, pp. 907–913, 2019\.
  * (18) L. Zhuang, D. Marpaung, M. Burla _et al._ , “Low-loss, high-index-contrast si 3 n 4/sio 2 optical waveguides for optical delay lines in microwave photonics signal processing,” _Optics express_ , vol. 19, no. 23, pp. 23 162–23 170, 2011.
  * (19) Y. Liu, A. Choudhary, D. Marpaung, and B. J. Eggleton, “Gigahertz optical tuning of an on-chip radio frequency photonic delay line,” _Optica_ , vol. 4, no. 4, pp. 418–423, 2017.
  * (20) X. Ji, X. Yao, Y. Gan _et al._ , “On-chip tunable photonic delay line,” _APL Photonics_ , vol. 4, no. 9, p. 090803, 2019.
  * (21) C. Tsokos, E. Andrianopoulos, A. Raptakis _et al._ , “True time delay optical beamforming network based on hybrid inp-silicon nitride integration,” _Journal of Lightwave Technology_ , vol. 39, no. 18, pp. 5845–5854, 2021.
  * (22) M. H. Khan, H. Shen, Y. Xuan _et al._ , “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” _Nature Photonics_ , vol. 4, no. 2, pp. 117–122, 2010\.
  * (23) A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” _Optics Communications_ , vol. 284, no. 15, pp. 3669–3692, 2011.
  * (24) D. Marpaung, L. Chevalier, M. Burla, and C. Roeloffzen, “Impulse radio ultrawideband pulse shaper based on a programmable photonic chip frequency discriminator,” _Optics express_ , vol. 19, no. 25, pp. 24 838–24 848, 2011.
  * (25) J. Wang, H. Shen, L. Fan _et al._ , “Reconfigurable radio-frequency arbitrary waveforms synthesized in a silicon photonic chip,” _Nature communications_ , vol. 6, no. 1, pp. 1–8, 2015.
  * (26) Z. Zhu, Y. Liu, M. Merklein _et al._ , “Si 3 n 4-chip-based versatile photonic rf waveform generator with a wide tuning range of repetition rate,” _Optics letters_ , vol. 45, no. 6, pp. 1370–1373, 2020.
  * (27) F. Falconi, C. Porzi, A. Malacarne _et al._ , “Uwb fastlytunable 0.550 ghz rf transmitter based on integrated photonics,” _Journal of Lightwave Technology_ , pp. 1–1, 2021.
  * (28) X. Yi, T. X. Huang, and R. A. Minasian, “Photonic beamforming based on programmable phase shifters with amplitude and phase control,” _IEEE Photonics Technology Letters_ , vol. 23, no. 18, pp. 1286–1288, 2011.
  * (29) L. Zhuang, M. Hoekman, C. Taddei _et al._ , “On-chip microwave photonic beamformer circuits operating with phase modulation and direct detection,” _Optics express_ , vol. 22, no. 14, pp. 17 079–17 091, 2014.
  * (30) S. A. Miller, Y.-C. Chang, C. T. Phare _et al._ , “Large-scale optical phased array using a low-power multi-pass silicon photonic platform,” _Optica_ , vol. 7, no. 1, pp. 3–6, 2020.
  * (31) C. Zhu, L. Lu, W. Shan _et al._ , “Silicon integrated microwave photonic beamformer,” _Optica_ , vol. 7, no. 9, pp. 1162–1170, 2020.
  * (32) L. Zhuang, C. G. Roeloffzen, M. Hoekman, K.-J. Boller, and A. J. Lowery, “Programmable photonic signal processor chip for radiofrequency applications,” _Optica_ , vol. 2, no. 10, pp. 854–859, 2015.
  * (33) D. Pérez, I. Gasulla, L. Crudgington _et al._ , “Multipurpose silicon photonics signal processor core,” _Nature communications_ , vol. 8, no. 1, pp. 1–9, 2017.
  * (34) D. Pérez, A. Lopez, P. DasMahapatra, and J. Capmany, “Multipurpose self-configuration of programmable photonic circuits,” _Nature communications_ , vol. 11, no. 6359, pp. 1–11, 2020.
  * (35) W. Bogaerts, D. Pérez, J. Capmany _et al._ , “Programmable photonic circuits,” _Nature_ , vol. 586, no. 7828, pp. 207–216, Oct 2020.
  * (36) Y. Liu, J. Hotten, A. Choudhary, B. J. Eggleton, and D. Marpaung, “All-optimized integrated rf photonic notch filter,” _Optics letters_ , vol. 42, no. 22, pp. 4631–4634, 2017.
  * (37) Y. Liu, D. Marpaung, A. Choudhary, J. Hotten, and B. J. Eggleton, “Link performance optimization of chip-based si 3 n 4 microwave photonic filters,” _Journal of lightwave technology_ , vol. 36, no. 19, pp. 4361–4370, 2018\.
  * (38) Z. Zhu, Y. Liu, M. Merklein _et al._ , “Positive link gain microwave photonic bandpass filter using si 3 n 4-ring-enabled sideband filtering and carrier suppression,” _Optics express_ , vol. 27, no. 22, pp. 31 727–31 740, 2019.
  * (39) O. Daulay, G. Liu, and D. Marpaung, “Microwave photonic notch filter with integrated phase-to-intensity modulation transformation and optical carrier suppression,” _Optics letters_ , vol. 46, no. 3, pp. 488–491, 2021.
  * (40) X. Guo, Y. Liu, T. Yin _et al._ , “Versatile silicon microwave photonic spectral shaper,” _APL Photonics_ , vol. 6, no. 3, p. 036106, 2021.
  * (41) O. Daulay, G. Liu, X. Guo, M. Eijkel, and D. Marpaung, “A tutorial on integrated microwave photonic spectral shaping,” _Journal of Lightwave Technology_ , vol. 39, no. 3, pp. 700–711, 2021.
  * (42) G. Liu, O. Daulay, Y. Klaver _et al._ , “Integrated microwave photonic spectral shaping for linearization and spurious-free dynamic range enhancement,” _Journal of Lightwave Technology_ , 2021.
  * (43) R. Cohen, O. Amrani, and S. Ruschin, “Response shaping with a silicon ring resonator via double injection,” _Nature Photonics_ , vol. 12, no. 11, pp. 706–712, 2018.
  * (44) L.-W. Luo, S. Ibrahim, A. Nitkowski _et al._ , “High bandwidth on-chip silicon photonic interleaver,” _Optics express_ , vol. 18, no. 22, pp. 23 079–23 087, 2010.
  * (45) O. Daulay, G. Liu, R. Botter _et al._ , “Reconfigurable double-injection ring resonator for integrated microwave photonic signal processing,” in _2021 International Topical Meeting on Microwave Photonics (MWP)_ , Nov. 2021, pp. 1–4.
  * (46) D. Marpaung, B. Morrison, M. Pagani _et al._ , “Low-power, chip-based stimulated brillouin scattering microwave photonic filter with ultrahigh selectivity,” _Optica_ , vol. 2, no. 2, pp. 76–83, 2015.
  * (47) W. Zhang and J. Yao, “On-chip silicon photonic integrated frequency-tunable bandpass microwave photonic filter,” _Opt. Lett._ , vol. 43, no. 15, pp. 3622–3625, Aug 2018.
  * (48) Y. Liu, A. Choudhary, G. Ren _et al._ , “Integration of brillouin and passive circuits for enhanced radio-frequency photonic filtering,” _APL Photonics_ , vol. 4, no. 106103, pp. 106 103–1–106 103–12, 2019.
  * (49) X. Zou, F. Zou, Z. Cao _et al._ , “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” _Laser & Photonics Reviews_, vol. 13, no. 6, p. 1800240, 2019\.
  * (50) S. Gertler, E. A. Kittlaus, N. T. Otterstrom, and P. T. Rakich, “Tunable microwave-photonic filtering with high out-of-band rejection in silicon,” _APL Photonics_ , vol. 5, no. 9, p. 096103, 2020.
  * (51) O. Daulay, R. Botter, and D. Marpaung, “On-chip programmable microwave photonic filter with an integrated optical carrier processor,” _OSA Continuum_ , vol. 3, no. 8, pp. 2166–2174, 2020.
  * (52) M. Garret, Y. Liu, D.-Y. Choi _et al._ , “Ultra-deep multi-notch microwave photonic filter utilising on-chip brillouin processing and microring resonators,” in _2021 Conference on Laser and Electro-Optics (CLEO) Europe_ , Jun. 2021, p. 1.
  * (53) J. Li, S. Yang, H. Chen, and M. Chen, “Hybrid microwave photonic receiver based on integrated tunable bandpass filters,” _Opt. Express_ , vol. 29, no. 7, pp. 11 084–11 093, Mar 2021.
  * (54) C. Zhang, P. A. Morton, J. B. Khurgin, J. D. Peters, and J. E. Bowers, “Ultralinear heterogeneously integrated ring-assisted mach–zehnder interferometer modulator on silicon,” _Optica_ , vol. 3, no. 12, pp. 1483–1488, Dec 2016.
  * (55) Q. Zhang, H. Yu, P. Xia _et al._ , “High linearity silicon modulator capable of actively compensating input distortion,” _Opt. Lett._ , vol. 45, no. 13, pp. 3785–3788, Jul 2020.
  * (56) P. A. Morton, J. B. Khurgin, and M. J. Morton, “All-optical linearized mach-zehnder modulator,” _Opt. Express_ , vol. 29, no. 23, pp. 37 302–37 313, Nov 2021.
  * (57) H. Feng, K. Zhang, W. Sun _et al._ , “Ultra-high-linearity integrated lithium niobate electro-optic modulators,” _Arxiv_ , pp. 1–16, 2022.
  * (58) G. Liu, O. Daulay, Q. Tan, H. Yu, and D. Marpaung, “Linearized phase modulated microwave photonic link based on integrated ring resonators,” _Optics express_ , vol. 28, no. 26, pp. 38 603–38 615, 2020.
  * (59) G. Liu, O. Daulay, Q. Tan _et al._ , “Simultaneous notch filtering and linearization in an integrated microwave photonic circuit,” in _2021 International Topical Meeting on Microwave Photonics (MWP)_ , Nov. 2021, pp. 1–4.
  * (60) K. Worhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “Triplex: a versatile dielectric photonic platform,” _Advanced Optical Technologies_ , vol. 4, no. 2, pp. 189–207, 2015.
  * (61) C. G. Roeloffzen, M. Hoekman, E. J. Klein _et al._ , “Low-loss si3n4 triplex optical waveguides: Technology and applications overview,” _IEEE journal of selected topics in quantum electronics_ , vol. 24, no. 4, pp. 1–21, 2018.
  * (62) W. Bogaerts, P. De Heyn, T. Van Vaerenbergh _et al._ , “Silicon microring resonators,” _Laser & Photonics Reviews_, vol. 6, no. 1, pp. 47–73, 2012\.

## Supplementary Information A: Modulation Transformer

Fig. S1: Operational principle of Modulation Transformer. Schematic and
operational principle of the modulation transformer (MT). The spectral de-
interleaver is used to isolate one sideband from the optical carrier and the
other sideband. A cascaded of tunable attenuator and a phase shifter is used
to tailor the phase and amplitude of the isolated sideband. A combination with
the optical carrier and unprocessed sideband leads to versatile spectral
shaping, synthesizing variety of modulation format, such as single-sideband
(SSB) modulation, phase modulation, and arbitrary modulation. INT: Spectral
de-interleaver, TA: tunable attenuator, PS: phase shifter. Fig. S2: Spectral
de-interleaver responses. a, Simulation of amplitude and phase of the bar port
from a spectral de-interleaver constructed using an asymmetric Mach-Zehnder
Interferometer (aMZI) and three ring resonators. b, Simulation of amplitude
and phase from the cross port of spectral de-interleaver. c, Fitting of
simulation (dashed lines) and experimental results of spectral de-interleaver.
(d) The rejection of spectral de-interleaver is 20 dB with stopband 47% and
transition band 1.15% of spectral de-interleaver’s FSR. e, The transition
between bar and cross port of spectral de-interleaver is 3.7 GHz.

A modulation transformer (MT) is an optical device designed to independently
shape and synthesize arbitrary optical modulation spectrum. Such device
constructed from multiple optical elements, namely spectral de-interleaver
built from an asymmetric Mach-Zehnder Interferometer (aMZI) with three ring
resonators topology luo2010high , a tunable attenuator, a phase shifter and a
combiner (Fig. S1). This device takes any conventional phase or intensity
modulation spectrum as an input and synthesize an output optical modulation
spectrum with components (optical carrier and sidebands) entirely independent
in phases and amplitudes guo2021versatile . This technique is different to
prior techniques which mainly focus on PM-IM or IM-PM transformation by
tailoring the phase of optical carrier.

The two output ports of spectral de-interleaver consist of one isolated
sideband in one channel and an optical carrier with the remaining sideband in
the other channel. Then, the isolated sideband is routed to cascaded phase
shifter and tunable attenuator to independently tailor its phase and
amplitude. Last, we re-combine this tailored isolated sideband with the rest
of the optical spectrum component (optical carrier and unprocessed sideband)
to synthesize different modulation spectrum with designer phase and amplitude
relations between its spectrum elements.

The mathematical description of each optical building block in the MT can be
described as follows,

First, a phase shifter is written as

$H_{ps}\left(z\right)=e^{-j\phi}\vspace{-0.0cm}$ (1)

then, a tunable coupler used for a tunable attenuator can be described as

$H_{tc}\left(z\right)=\begin{bmatrix}\sqrt{0.5}&-j\sqrt{0.5}\\\
-j\sqrt{0.5}&\sqrt{0.5}\end{bmatrix}\begin{bmatrix}1\\\
e^{-j\phi}\end{bmatrix}\begin{bmatrix}\sqrt{0.5}&-j\sqrt{0.5}\\\
-j\sqrt{0.5}&\sqrt{0.5}\end{bmatrix}\vspace{-0.0cm}$ (2)

last, a spectral de-interleaver built from three ring resonators assisted aMZI
structure can be expressed as

$H_{bar}=A_{11}A_{12}H_{11}\left(z\right)-A_{13}A_{14}H_{22}\left(z\right)$
(3)

$H_{cross}=A_{21}A_{22}H_{11}\left(z\right)-A_{23}A_{24}H_{22}\left(z\right)$
(4)

where,

$\displaystyle H_{11}\left(z\right)$
$\displaystyle=T_{1}\left(\omega\right)T_{3}\left(\omega\right)$
$\displaystyle H_{22}\left(z\right)$
$\displaystyle=T_{2}\left(\omega\right)H_{\Delta
L}\left(z\right)H_{ps}\left(z\right)$ $\displaystyle H_{\Delta
L}\left(z\right)$ $\displaystyle=\gamma z^{-1}$ $\displaystyle A_{11}$
$\displaystyle=c_{3}c_{4}e^{-j\phi_{2}}-s_{3}s_{4}$ $\displaystyle A_{12}$
$\displaystyle=c_{1}c_{2}e^{-j\phi_{1}}-s_{1}s_{2}$ $\displaystyle A_{13}$
$\displaystyle=c_{4}s_{3}e^{-j\phi_{2}}-c_{3}s_{4}$ $\displaystyle A_{14}$
$\displaystyle=c_{1}s_{2}e^{-j\phi_{1}}-c_{2}s_{1}$ $\displaystyle A_{21}$
$\displaystyle=c_{3}c_{4}e^{-j\phi_{2}}-s_{3}s_{4}$ $\displaystyle A_{22}$
$\displaystyle=c_{2}s_{1}e^{-j\phi_{1}}-s_{1}s_{2}$ $\displaystyle A_{23}$
$\displaystyle=c_{3}c_{4}-s_{3}s_{4}e^{-j\phi_{2}}$ $\displaystyle A_{24}$
$\displaystyle=c_{1}s_{2}e^{-j\phi_{1}}-c_{2}s_{1}$

Here, c and s are the cross-coupling coefficient and the self-coupling
coefficient of tunable element in the spectral de-interleaver respectively.
Then, the amplitude response ($T\left(\omega\right)$) of the ring resonator
used in the spectral de-interleaver can be expressed as bogaerts2012silicon

$T\left(\omega\right)e^{-j\theta}={\frac{a-c_{r}e^{-j\psi(\omega)}}{1-ac_{r}e^{-j\psi(\omega)}}}e^{-j(\pi+\psi(\omega))}$
(5)

Fig. S3: Experimental results of intensity-to-phase modulation (IM-PM) and
phase-to-intensity modulation (PM-IM) conversions. a, RF magnitude response of
the intensity modulation (IM) link without the photonic integrated circuit
(PIC) as a benchmark. b, RF magnitude response with the PIC containing MT and
DI-RR (solid blue line). Here, the PIC response is set to be similar with the
benchmark. c, Conversion from IM (blue) to phase modulation (PM) (red) with
optimized extinction of 25 dB. d, RF magnitude response of the PM-based
photonic link without the PIC as benchmark.  (e) RF magnitude response of the
PM link with the PIC (solid red line). f, Conversion from PM (red) to IM
(blue) with optimized extinction of 64 dB.

.

where $c_{r}=\sqrt{1-k}$, $a=10^{{-}\alpha L/20}$, k, $\alpha$, L, and $\psi$
are the self-coupling coefficient, the single-pass amplitude of the optical
ring resonator, the coupling coefficient, the propagation loss of optical
waveguide (dB/cm), the round trip length of ring, and the round-trip phase
respectively. Similarly, we can define $s_{r}=\sqrt{k}$ as the cross-coupling
coefficient.

Figure S2(a) and (b) show the simulated phase and amplitude responses from bar
and cross port of spectral de-interleaver respectively. It is critical to
synthesize a ”box-shaped” response, flat-top to precisely isolate optical
modulation spectrum elements (optical carrier and sidebands). The spectral de-
interleaver has a linear phase response in the passband, while in the stopband
the phase shows a nonlinear response, which may cause signal distortion in the
MT. Figure S2(c) shows 160 GHz of spectral de-interleaver’s free spectral
range (FSR) with 80 GHz of bandwidth and rejection of up to 20 dB. The
transition band in this response is $1.15\%$ of spectral de-interleaver’s FSR
or 1.85 GHz as shown in Fig. S2(d) with 3.7 GHz of transition bandwidth
between bar and cross port as shown in Fig. S2(e). These results show the
narrowest transition band reported in a spectral de-interlever device to date
zhuang2011low . The bar port then connected to a cascaded tunable attenuator
and phase shifter. Last, a combiner at the output of the MT is used to re-
combine the output of cascaded tunable attenuator and phase shifter with the
output of cross port of spectral de-interleaver to synthesize different
modulation spectrum.

In this work, two examples are given to show modulation transformation
process, such as intensity-to-phase modulation (IM-PM) transformation and
phase-to-intensity modulation (PM-IM) transformation. Such modulation
transformation can be mathematically described as

$\displaystyle E_{up}\left(t\right)$ $\displaystyle=H_{tc}\begin{bmatrix}0\\\
H_{ps}H_{11}E_{in}\left(t\right)\end{bmatrix}$
$\displaystyle=0.5\begin{bmatrix}e^{-j\pi}-1&-j\left(e^{-j\pi}+1\right)\\\
-j\left(e^{-j\pi}+1\right)&1-e^{-j\pi}\end{bmatrix}\begin{bmatrix}0\\\
H_{ps}H_{11}E_{in}\left(t\right)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}-1&0\\\ 0&1\end{bmatrix}\begin{bmatrix}0\\\
H_{ps}H_{11}E_{in}\left(t\right)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0\\\
H_{ps}H_{11}E_{in}\left(t\right)\end{bmatrix}$
$\displaystyle=H_{ps}H_{11}E_{in}\left(t\right)$
$\displaystyle=e^{-j\pi}H_{11}E_{in}\left(t\right)$
$\displaystyle=-H_{11}E_{in}\left(t\right)$ $\displaystyle
E_{low}\left(t\right)$ $\displaystyle=H_{12}H_{\Delta L1}E_{in}\left(t\right)$
$\displaystyle E_{comb}\left(t\right)$
$\displaystyle=E_{up}\left(t\right)+e^{-j\pi/2}E_{low}\left(t\right)$
$\displaystyle=-\left(H_{11}+jH_{\Delta L1}H_{12}\right)E_{in}\left(t\right)$

where $E_{up}\left(t\right)$ and $E_{low}\left(t\right)$ are the output field
of upper and lower sidebands respectively.

We controlled the MT in our circuit to synthesis two different modulation
schemes from a conventional PM and IM input. In the first experiment, we sent
an IM spectrum directly to a photodetector in a simple photonic link setup and
measured the RF transmission for a reference. Then, we process the IM spectrum
using the MT in our circuit and convert into PM spectrum. Figure S3(a) shows
the IM spectrum in the RF domain as the reference signal in a simple photonic
link setup. Then, the IM spectrum is sent to our circuit containing of a MT as
shown in Fig. S3(b). Here, we try to match the RF spectrum (with photonic
chip) with the reference spectrum (without photonic chip). While both of the
sidebands in IM spectrum are in-phase, the measured RF transmission is high
(solid blue line in Fig. S3(b) and (c)). Later, we tailored the phase of
isolated sideband by tuning the phase shifter in the MT. Last, the processed
isolated sideband is re-combined with the optical carrier and remaining
sideband to synthesized the PM spectrum (solid red line in Fig. S3(c)) with 25
dB extinction.

Next, in second experiment, we used the MT for PM-IM transformation with the
same principle as prior experiment, but with the PM spectrum as an input.
Figure S3(d) shows the transmission of PM spectrum as a benchmark in a simple
photonic link setup with direct detection. Then, we sent the PM spectra into
the un-activated MT in our circuit as shown in Fig. S3(e). When both sidebands
are out of phase, the measured RF transmission is low (solid red line in Fig.
S3(e) and (f)). Then we tuned the phase shifter in the MT to rotate the phase
of isolated sideband. Eventually, an IM spectrum is synthesized after the
signal recombination, and it is corresponding to a maximum RF transmission
(solid blue line in Fig. S3(f)) with 64 dB extinction. In these two
experiments, the operational bandwidth and extinction are mainly limited by
the roll-off and the dispersion of the spectral de-interleaver, notably at the
transition band (operational frequency of $<$ 5 GHz).

## Supplementary Information B: Double Injection Ring Resonator

The use of an optical circuit named double-injection ring resonator (DI-RR) is
aimed to provide various unique responses in a ring resonator-based optical
circuit from single output. The underlying idea of this circuit is to inject
two, mutually coherent, optical signals of the same wavelength into a single
add-drop ring resonator from single input cohen2018response  as illustrated in
Fig. S4(a). To optimize such circuit for our device, we further investigate
the variety of unique phase and amplitude response synthesized by the DI-RR,
which is a key parameter for programmable integrated MWP circuit.

Mathematically, the model describing the transmitted electric field dependence
on the wavelength of the DI-RR is given by

$E_{t1}\left(\lambda\right)=\frac{\left(\tau_{1}+\tau_{2}^{*}\alpha
e^{-j\theta}\right)}{1-\tau_{1}\tau_{2}^{*}\alpha
e^{-j\theta}}\big{|}E_{j1}\left(\lambda\right)\big{|}e^{-j\phi_{j1}}-\frac{K_{1}K_{2}^{*}\sqrt{\alpha}e^{-j\theta}}{1-\tau_{1}\tau_{2}^{*}\alpha
e^{-j\theta}}\big{|}E_{j2}\left(\lambda\right)\big{|}e^{-j\phi_{j2}}$ (6)

where $\tau=\big{|}\tau\big{|}e^{-j\psi_{\tau}}$ is the transmission of the
directional coupler, $K=\big{|}K\big{|}e^{-j\psi_{K}}$ is the coupling
coefficients of the directional couplers, $\alpha$ is the loss coefficient of
the ring, $E_{j}$ is the injected fields, and $\phi_{j}$ is the injected
fields phases. $\theta$ is the phase accumulated by the light traversing the
ring at steady state that described as

$\theta\left(\lambda\right)=\frac{2\pi}{\lambda}n_{eff}\left(\lambda\right)L_{ring}$
(7)

with $\lambda$ being the wavelength, $L_{ring}$ the length perimeter of the
ring, and $n_{eff}$ the effective index of the propagating mode. Fig. S4(b–g)
depict the measured RF phase and amplitude responses of the six different
functions synthesized from the DI-RR, such as notch filter (Fig. S4(b)),
bandpass filter (Fig. S4(c)), triangular response (Fig. S4(d)), sawtooth
response (Fig. S4(e)), all-pass response (Fig. S4(f)) and Fano-like response
(Fig. S4(g)). The inset of each figure depict the response obtained from high
resolution optical spectrum analysis. The characterization is important to
have a full understanding of the DI-RR’s potential in the system.

Fig. S4: RF phase and magnitude responses of the DI-RR. a, Schematic of double
injection ring resonator. b, Notch filter. (c) Bandpass filter. d, Triangular
response e, Sawtooth response. f, All-pass response. g, Fano-like response.
The inset of each figure shows the corresponding optical response.

## Supplementary Information C: Extended Experiments

Fig. S5: Experiment scenarios of programmable integrated MWP circuit. (a)
Experiment setup used for demonstrating the four different filtering
scenarios. The modulation transformer (MT) and the double-injection ring
resonator (DI-RR) are programmed to exhibit different functions using a common
intensity modulation (IM) input. (b) Scenario 1: Single sideband notch filter.
The MT is used to synthesize single sideband modulation (SSB) and the DI-RR to
synthesize 5 dB-deep notch response. The measured filter response is show on
the right panel. (c) Scenario 2: Cancellation notch filter. MT is used to
create asymmetric dual-sideband modulation conversion while DI-RR shows 5 dB-
deep notch response. Destructive interference at the otch frequency amplifies
the RF notch filter response to 58 dB. (d) Scenario 3: PM-IM bandpass filter.
The MT is used to create phase modulation. The over-coupled (OC) notch
response from DI-RR breaks the PM condition and creates a bandpass filter
response. (e) Scenario 4: Single-sideband bandpass filter. The MT is used to
re-insert an optical carrier to mix with a single sideband response that has
been filtered using the DI-RR. The DI-RR was tuned to exhibit a sharp bandpass
filter response.

To demonstrate the feasibility of the MT together with the versatility of DI-
RR in programmable integrated MWP circuit, we conducted experiments with an
intensity modulation (IM) spectrum as the input optical spectrum. Four
different scenarios were applied to demonstrated the versatility of the
circuit. Figure S5(a) shows the experiment setup and Fig. S5(b - e) show the
operation principle and application of four different scenarios of the
proposed device. In the system, the input RF signal is upconverted to optical
domain with intensity modulator generating single frequency optical carrier
and two identical amplitude sidebands that are in-phase.

In the first scenario, we convert an intensity-to-single sideband (SSB)
modulation to, later on, create a SSB RF notch filter as shown in Fig. S5(b).
An IM signal is sent to pass through a spectral de-interleaver in the MT,
which spatially isolated one sideband in the spectrum from another sideband
and optical carrier. Then, the isolated sideband was sent to tunable
attenuator to be fully attenuated, where the other sideband and optical
carrier were sent to a DI-RR which synthesized a notch response. Last, the
output of DI-RR was combined with the fully attenuated sideband and exhibited
a SSB RF notch filter.

Then, in the second scenario, an IM signal is sent to the MT, perform
intensity-to-asymmetric dual sideband (aDSB) modulation conversion and
synthesize high rejection RF notch filter. Here, the attenuation of isolated
sideband is similar with the rejection of notch response synthesized by the
DI-RR. Then, the phase shifter was used to create phase difference (0 - $\pi$)
between two sidebands relative to optical carrier. In parallel, the other
sideband and optical carrier were processed by the same notch response as in
the first scenario. It is important that the rejection of notch response from
the ring to be equal with the amplitude’s difference between two sidebands.
The idea is to have a desctructive interference between aDSB together with
notch response from DI-RR to amplify the rejection of synthesized RF notch
filter as shown in Fig. S5(c). By precisely tailor the phase of the DI-RR, we
can tune the central frequency of the RF notch filter as shown in Fig. S6(a).

Next, in the third scenario, we conducted phase-to-intensity modulation (PM-
IM) conversion technique to create RF bandpass filter. Here, we converted the
modulation spectrum from intensity-to-phase modulation (IM-PM). During the
process, we only changed the phase of the isolated sideband by controlling the
interconnected phase shifter from 0 - $\pi$. While, the other sideband and
optical carrier were processed using notch response of DI-RR set at over-
coupling (OC) regime. After IM-PM conversion using MT, due to OC state in the
DI-RR, a $\pi$-phase shift is introduced at the desired notch frequency,
creating constructive interference for phase-to-intensity (PM-IM) modulation
conversion and synthesized RF bandpass filter as shown in Fig. S5(d).
Previously, this approach can only be done in a phase modulator-based MWP
system.

Last, another RF bandpass filter is exhibited using optical carrier re-
insertion technique with the MT. In this approach, a SSB modulation is sent to
the spectral de-interleaver in the MT and split into two outputs containing
only optical carrier in one output and one sideband in the other. Then, the
sideband was sent to a DI-RR which tuned and synthesized a bandpass response.
The RF bandpass filter is constructed while the optical carrier is re-injected
to the processed sideband using a combiner at the output of MT as shown in
Fig. S5(e). Figure S6(b) shows the tuning range of the RF bandpass filter
achieved by tailoring the phase of DI-RR.

Fig. S6: RF Filters Tunability. (a) Tuning range of RF notch filter with low-
biasing MZM. (b) Tuning range of RF bandpass filter with low-biasing MZM.

## Supplementary Information D: Theoretical Analysis of Linearization Method

A theoretical analysis is performed to find the requirements for linearity
improvement in the proposed RF notch filter. We first analyze the relation
between optical spectrum and the beating products after photodetection to find
the main contributors of the IMD3. When the optical carrier is modulated by a
two-tone RF signal with angular frequency of $\omega_{1}$, $\omega_{2}$, and
voltage of $V_{RF}$ via a phase modulator (PM), the optical spectrum at the
output of PM can be expressed as

$E_{out}\left(t\right)=\sqrt{P_{i}}e^{j\omega_{c}t}\sum_{\scriptscriptstyle
n=-\infty}^{\scriptscriptstyle+\infty}\sum_{\scriptscriptstyle
k=-\infty}^{\scriptscriptstyle+\infty}J_{n}\left(m\right)J_{k}\left(m\right)e^{j\left(n\omega_{1}+k\omega_{2}\right)t}\vspace{-0.0cm}$
(8)

where $\omega_{c}$, $P_{i}$, $J_{n}$, $m=\pi\cdot V_{RF}/V_{\pi,RF}$, and
$V_{\pi,RF}$ is the angular frequency of the optical carrier, input optical
power, the n-th order Bessel function of the first kind, the modulation index
of PM, and the RF half-wave voltage of the PM respectively. When the two-tone
signal can be regarded as small signal, we can only take zero to second order
sidebands into account to simplify the analysis.

The optical spectrum after spectral shaping can be written as

$\displaystyle E_{p}\left(t\right)$
$\displaystyle=\sqrt{P_{i}}e^{j\omega_{c}t}$ (9)
$\displaystyle\cdot\left\\{\begin{array}[]{ll}\sqrt{A}\cdot
J_{0}J_{0}\vspace{1ex}\\\ +\sqrt{A}\cdot
J_{-1}J_{1}\left[e^{-j\left(\omega_{1}+\omega_{2}\right)t}+e^{j\left(\omega_{1}+\omega_{2}\right)t}\right]\vspace{1ex}\\\
+J_{0}J_{1}\left(e^{j\omega_{1}t}+e^{j\omega_{2}t}\right)\vspace{1ex}\\\
+J_{-1}J_{2}\left[e^{j\left(2\omega_{1}-\omega_{2}\right)t}+e^{j\left(2\omega_{2}-\omega_{1}\right)t}\right]\vspace{1ex}\\\
+\sqrt{A}\cdot
J_{-1}J_{0}\left(e^{-j\omega_{1}t}+e^{-j\omega_{2}t}\right)\vspace{1ex}\\\
+\sqrt{A}\cdot
J_{-2}J_{1}\left[e^{-j\left(2\omega_{1}-\omega_{2}\right)t}+e^{-j\left(2\omega_{2}-\omega_{1}\right)t}\right]\vspace{1ex}\\\
+J_{0}J_{2}\left(e^{j2\omega_{1}t}+e^{j2\omega_{2}t}\right)\vspace{1ex}\\\
+J_{1}J_{1}e^{j\left(\omega_{1}+\omega_{2}\right)t}\vspace{1ex}\\\
+\sqrt{A}\cdot
J_{-2}J_{0}\left(e^{-j2\omega_{1}t}+e^{-j2\omega_{2}t}\right)\vspace{1ex}\\\
+\sqrt{A}\cdot J_{-1}J_{-1}e^{-j\left(\omega_{1}+\omega_{2}\right)t}\\\
\end{array}\right\\}$

where $J_{n}=J_{n}(m)(n=0,\pm 1,\pm 2)$, $J_{-n}=(-1)^{n}J_{n}$. A is the
power suppression to the optical carrier and the lower sideband.

The photocurrent of the RF signal detected from the processed optical spectrum
can be expressed as

$\displaystyle I_{PD}(t)$ $\displaystyle=R_{PD}|E_{p}(t)|^{2}$ (10)
$\displaystyle=I_{1}\cos{\omega_{1,2}t}+I_{3}\cos{\left(2\omega_{1,2}-\omega_{2,1}\right)t}$

where $R_{PD}$ is responsivity of photodetector, $I_{1}$ and $I_{3}$ are the
amplitude coefficients for fundamental signal and IMD3 components, which can
be written as

$\displaystyle
I_{1}\propto(\sqrt{A}-A)J_{0}^{3}J_{1}+(1-\sqrt{A})J_{0}J_{1}^{3}+(1-A)J_{0}^{2}J_{1}J_{2}$
(11)

$\displaystyle
I_{3}\propto(A-\sqrt{A})J_{0}J_{1}^{3}+(1-\sqrt{A})J_{0}^{2}J_{1}J_{2}$ (12)

when $m\ll 1$, $J_{n}(m)\approx m^{n}/(2^{n}n!)$, $I_{1}$ and $I_{3}$ can be
expressed as

$\displaystyle I_{1}\propto(\sqrt{A}-A)m$ (13)

$\displaystyle I_{3}\propto(2A-3\sqrt{A}+1)m^{3}$ (14)

## Supplementary Information E: Operational Principle of Simultaneous Filter
and Linearization

Fig. S7: Linearization of RF notch filter.(a) Experiment setup used for
demonstrating simultaneous RF notch filter with linearization. (b) Signal flow
for RF notch filter formation. The filter is formed through RF cancellation at
the notch frequency. (c) Signal flow for the simultaneous RF notch filter and
linearization. The amplitude and phase of the optical carrier and first and
second order sidebands are adjusted to achieve cancellation of IMD3 products
at the output.

In this experiment, we experimentally demonstrated, for the first time,
simultaneous RF notch filter and linearization using a combination of the MT
and the DI-RR in a single photonic chip. The DI-RR is set to create a notch
response, while The MT works for two purposes, synthesize desired modulations
spectrum needed for RF notch filter and linearization. We conducted the
experiment with setup as depicted in Fig. S7(a) with the operating principle
of simultaneous RF notch filter and linearization as illustrated in Fig. S7(b)
and (c).

The RF notch filter here is synthesized with the basis of phase cancellation
marpaung2015low . First, the RF signal (I) is phase modulated into the optical
carrier generating single frequency optical carrier with multiple sidebands
that are out-of-phase (II). The phase modulation spectrum then coupled into
the programmable integrated MWP circuit containing of the MT and DI-RR. The
spectral de-interleaver in the MT spatially separates the phase modulation
spectrum into two parts, where the optical carrier and lower sideband are
attenuated by the tunable attenuator (III). The upper sideband is processed by
notch response from the DI-RR set at under-coupling (UC) regime to equalize
the amplitudes of two sidebands at designate frequency (IV). Next, the optical
spectrum are recombined at the output of the chip, creating an asymmetric dual
sideband (aDSB) modulation with anti-phase relation (V). After photodetection,
a high rejection RF notch filter is achieved, as the mixing products of the
sidebands and the optical carrier are destructively interfered at the notch
frequency (VI).

Next, the operating principle of simultaneous RF notch filter and
linearization can be explained as, two-tone RF signal located at the passband
of the RF notch filter (I) is phase modulated into the optical carrier. Here,
the multi-order sidebands in phase-modulated spectrum are taken into account
for IMD3 reduction (II). Then, the optical spectrum is coupled to the photonic
chip, where the spectral de-interleaver then split the optical modulation
spectrum into two channels. one channel contains the optical carrier, the -1
order and -2 order sidebands, which are manipulated by a cascaded tunable
attenuator and phase shifter to create aDSB modulation for RF notch filter and
meet the linearization condition (III). The other channel consists of the +1
order and +2 order sidebands. The +1 order sideband is processed by a notch
response from the DI-RR (IV). As the two-tone signal lies in the passband and
the DI-RR is used to tailor the frequency components at stopband, the DI-RR’s
response will not influence much to the linearity. Then, the manipulated
spectrum are recombined at the output of the chip (V) and sent to the
photodetector, resulting in an RF spectrum with suppressed IMD3 terms due to
the destructive interference between the mixing products of the multi-order
sidebands and the optical carrier (VI).


# Anisotropic flow measurements in Pb–Pb collisions at 5.02 TeV with ALICE

You Zhou (for the ALICE Collaboration)  Niels Bohr Institute, University of
Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark<EMAIL_ADDRESS>
###### Abstract

Anisotropic flow is a sensitive probe of the initial conditions and the
transport properties of the Quark Gluon Plasma (QGP) produced in heavy-ion
collisions. In these proceedings, we present the first results of elliptic
($v_{2}$), triangular ($v_{3}$) and quadrangular flow ($v_{4}$) of charged
particles in Pb–Pb collisions at $\sqrt{s_{{}_{\rm NN}}}=$ 5.02 TeV with the
ALICE detector. In addition, the comparison of experimental measurements to
various theoretical calculations will be discussed. This provides a unique
opportunity to test the validity of the hydrodynamic picture and discriminates
between various possibilities for the temperature dependence of shear
viscosity to entropy density ratio of the produced QGP.

##  1 Introduction

Collisions of high-energy heavy-ions, at the Brookhaven Relativistic Heavy Ion
Collider (RHIC) and the CERN Large Hadron Collider (LHC), allow us to create
and study the properties of Quark Gluon Plasma (QGP) in the laboratory. The
azimuthal anisotropy in particle production is, at these energies, an
observable which provides experimental information on the Equation of State
and the transport properties of the QGP. This anisotropy is usually
characterized by the Fourier flow-coefficients [1],

$\displaystyle v_{n}$ $\displaystyle=$
$\displaystyle\langle\cos[n(\varphi-\Psi_{n})\rangle,\;\;\;{\rm or\
equivalently}$ $\displaystyle v_{n}$ $\displaystyle=$ $\displaystyle\langle
e^{in\varphi}e^{-in\Psi_{n}}\rangle,$ (1)

where $\varphi$ is the azimuthal angle of the particles, $\Psi_{n}$ is the
$n^{\rm{th}}$-order flow plane (or named final state symmetry plane) angle and
$\langle~{}\rangle$ denotes an average over the selected particles and events.
Experimental measurements together with comparisons to theoretical
calculations show that anisotropic flow sheds new light on the initial
conditions and is sensitive to the shear viscosity to entropy density ratio
$\eta/s$ of the QGP. Recent anisotropic flow measurements [2, 3, 4] in Pb–Pb
collisions at $\sqrt{s_{\rm NN}}=$ 2.76 TeV provide additional information to
improve the theoretical descriptions of data. In these proceedings, we present
the latest anisotropic flow measurements in Pb–Pb collisions at $\sqrt{s_{\rm
NN}}=$ 5.02 TeV in ALICE [5]. The comparisons to hydrodynamic and AMPT model
calculations will be discussed below.

##  2 Analysis Details

The data used in this analysis were recorded with the ALICE detector in
November 2015 in run 2 at the LHC with Pb-Pb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV. About 140 k Pb–Pb events were recorded with a minimum-bias
trigger, based on signals from two VZERO detectors
(-3.7$\textless\eta\textless$-1.7 for VZERO-C and
2.8$\textless\eta\textless$5.1 for VZERO-A) and on the Silicon Pixel Detector.
The VZERO detectors were also used for the determination of the collision
centrality in Pb–Pb collisions. Charged particles are reconstructed using the
Inner Tracking System and the Time Projection Chamber with full azimuthal
coverage for pseudo-rapidity range $|\eta|\textless$0.8.

##  3 Results and Discussions

Figure 1: (color online) (a) Anisotropic flow $v_{n}$ integrated over the
$p_{\rm T}$ range 0.2 $<p_{\rm T}<$ 5.0 GeV/$c$, as a function of event
centrality, for the two–particle (with $|\Delta\eta|>1$) and multi-particle
cumulant methods. The ratios of $v_{n}$ from Pb–Pb collisions at 5.02 TeV and
2.76 TeV, are presented in Fig. 1 (b) and (c). Various hydrodynamic
calculations are also presented [7, 9].

The centrality dependence of $v_{2}$, $v_{3}$ and $v_{4}$ are presented in
Fig. 1. Both two-particle correlations with pseudorapidity and multi-particle
cumulants methods are used in these measurements. It is seen that
$v_{2}\\{2,|\Delta\eta|>1\\}$ increases from central to peripheral collisions,
saturates in the 40–50% centrality class with a maximum value of 0.104 $\pm$
0.001 (stat.) $\pm$ 0.002 (syst.). Similar centrality dependence is observed
in the measurements of multi-particle cumulants $v_{2}\\{4\\}$, $v_{2}\\{6\\}$
and $v_{2}\\{8\\}$ and results are agree within 1%. Assuming that non-flow
effects are suppressed by the pseudorapidity gap, the remaining differences
between two- and multi-particle cumulants of $v_{2}$ can be attributed to the
different sensitivities to elliptic flow fluctuations. Compared to $v_{2}$,
higher harmonics i.e. $v_{3}$ and $v_{4}$ are smaller and the centrality
dependence is much weaker. In addition, the hydrodynamic calculations [7]
which combine the changes in initial spatial anisotropy and the hydrodynamic
response are presented in Fig. 1. (a) for comparison. The calculations are
found to be compatible with the measured anisotropic flow $v_{n}$
coefficients. Furthermore, using a new set of Lund string fragmentation
parameters and partonic cross section 3 mb, the AMPT model with string melting
scenario can quantitatively reproduce the anisotropic flow measurements [8].

To better illustrate the energy evolution of anisotropic flow, Fig 1. (b) and
(c) shows the ratios of $v_{n}$ measured at 5.02 TeV to 2.76 TeV. It is
observed that the increase of $v_{2}$ and $v_{3}$ from the two energies is
rather moderate, while for $v_{4}$ it is more pronounced. Considering none of
the ratios of flow harmonics from two energies exhibit a significant
centrality dependence in the centrality range 0–50%, a fit with a constant
value are performed for these ratios. An increase of (3.0$\pm$0.6)%,
(4.3$\pm$1.4)% and (10.2$\pm$3.8)% over the centrality range 0–50% in Pb–Pb
collisions is obtained for $v_{2}$, $v_{3}$ and $v_{4}$, respectively. This
increase of anisotropic flow is in agreement with theoretical predictions in
Ref [9]. The data seems to support a low value of $\eta/s$ for the QGP created
in Pb–Pb collisions at $\sqrt{s_{{}_{\rm NN}}}=$ 5.02 TeV and indicates that
$\eta/s$ does not change significantly with respect to Pb–Pb collisions at
$\sqrt{s_{{}_{\rm NN}}}=2.76$ TeV.

Figure 2: (color online) $v_{n}(p_{\rm T})$ for the centrality 10–20%, 20–30%
and 30–40%. Measurements for Pb–Pb collisions at $\sqrt{s_{{}_{\rm NN}}}=$
2.76 TeV are presented as shading. (d) The ratio of $v_{2}\\{4\\}$ in 20–30%
from two collision energies is also shown here.

.

The transverse momentum ($p_{\rm T}$) dependent $v_{2}\\{2,|\Delta\eta|>1\\}$,
$v_{3}\\{2,|\Delta\eta|>1\\}$ and $v_{4}\\{2,|\Delta\eta|>1\\}$ are presented
for the 0–5% and 30–40% centrality classes. In 0–5% centrality class,
$v_{3}\\{2\\}$ is larger than $v_{2}\\{2\\}$ for $p_{\rm T}>$ 2 GeV/$c$, and
$v_{4}\\{2\\}$ is roughly compatible with $v_{2}\\{2\\}$. For the 30–40%
centrality class, $v_{2}\\{2\\}$ is higher than $v_{3}\\{2\\}$ and
$v_{4}\\{2\\}$ for the entire $p_{\rm T}$ range measured, with no crossing of
the different order flow coefficients. The $p_{\rm T}$-differential
$v_{2}\\{4\\}$ decreases from mid-central to central collisions over the
$p_{\rm T}$ range measured. The results are consistent with the previous
measurements from 2.76 TeV, as also illustrated by the ratio of $v_{2}\\{4\\}$
for the two energies in Fig. 2 (d). This suggests that the increase observed
in the $p_{\rm T}$ integrated flow results seen in Fig. 1 could be due to
stronger radial flow produced at 5.02 TeV. It leads to an increase of mean
transverse momentum $\langle p_{\rm T}\rangle$ from 2.76 to 5.02 TeV. The
measurements of $p_{\rm T}$-differential flow are expected to be more
sensitive to initial conditions and $\eta/s$ of the QGP. The results are found
to be roughly compatible with AMPT calculations, meanwhile no hydrodynamic
calculations are available yet. The future comparisons of $p_{\rm
T}$-differential flow between experimental measurements and hydrodynamic
calculations will provide important information to constrain further details
of the theoretical calculations, e.g. determination of radial flow and freeze-
out conditions.

Figure 3: (color online) $p_{\rm T}$ integrated $v_{2}$ in centrality 20-30%
as a function of collisions energy.

Figure 3 presents the collisions-energy dependence of the fully $p_{\rm
T}$-integrated $v_{2}$ measured in 20–30% centrality with different collision
systems and energies. A continuous increase of anisotropic flow for this
centrality has been observed from SPS/RHIC to LHC energies. For these fully
$p_{\rm T}$ integrated coefficients, an increase of $4.9\pm 1.9\%$ is observed
going from $\sqrt{s_{{}_{\rm NN}}}=2.76$ to 5.02 TeV, which is close to values
of the previously-mentioned hydrodynamic calculations [7, 9].

##  4 Summary

In summary, the first anisotropic flow measurements of charged particles in
Pb–Pb collisions at $\sqrt{s_{{}_{\rm NN}}}=$ 5.02 TeV at the LHC are
presented in this proceedings. An average increase of (3.0$\pm$0.6)%,
(4.3$\pm$1.4)% and (10.2$\pm$3.8)%, is observed for the transverse momentum
integrated elliptic, triangular and quadrangular flow, respectively, over the
centrality range 0–50% going from 2.76 TeV to 5.02 TeV. The $p_{\rm
T}$-differential flow has also been investigated, it does not change
appreciably between the two LHC energies. Therefore, the increase in
integrated flow coefficients could be mainly due to an increase in average
$p_{\rm T}$. This agrees with predictions from hydrodynamic models [7, 9].
Further comparisons of $p_{\rm T}$-differential flow measurements and
theoretical calculations will provide extra constraints on the initial
conditions and the transport properties of the QGP.

## References

## References

  * [1] S. Voloshin and Y. Zhang, Z. Phys. C 70, 665 (1996)
  * [2] J. Adam et al. [ALICE Collaboration], arXiv:1604.07663 [nucl-ex].
  * [3] J. Adam et al. [ALICE Collaboration], arXiv:1605.02035 [nucl-ex].
  * [4] J. Adam et al. [ALICE Collaboration], arXiv:1606.06057 [nucl-ex].
  * [5] J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett. 116, no. 13, 132302 (2016)
  * [6] K. Aamodt et al. [ALICE Collaboration], JINST 3, S08002 (2008).
  * [7] J. Noronha-Hostler, M. Luzum and J. Y. Ollitrault, Phys. Rev. C 93, no. 3, 034912 (2016)
  * [8] Z. Feng, G. M. Huang and F. Liu, arXiv:1606.02416 [nucl-ex].
  * [9] H. Niemi, K. J. Eskola, R. Paatelainen and K. Tuominen, Phys. Rev. C 93, no. 1, 014912 (2016)