# Noise in stimulated Raman scattering measurement: From basics to practice.

X. Audier  Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel,
Marseille, France  S. Heuke  Aix Marseille Univ, CNRS, Centrale Marseille,
Institut Fresnel, Marseille, France  P. Volz  APE Angewandte Physik &
Elektronik GmbH, Berlin, Germany  I. Rimke  APE Angewandte Physik & Elektronik
GmbH, Berlin, Germany  H. Rigneault<EMAIL_ADDRESS>Aix Marseille
Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France

###### Abstract

We revisit laser intensity noise in the context of stimulated Raman scattering
(SRS), which has recently proved to be a key technique to provide label free
images of chemical bonds in biological and medical samples. Contrary to most
microscopy techniques, which detect a weak photon flux resulting from light
matter interactions, SRS is a pump-probe scheme that works in the high flux
regime and happens as a weak modulation ($10^{-4}-10^{-6}$) in a strong laser
field. As a result, laser noise is a key issue in SRS detection. This
practical tutorial provides the experimentalists with the tools required to
assess the amount of noise and the ultimate SRS detection limit in a
conventional lock-in-based SRS system. We first define the quantities that are
relevant when discussing intensity noise, and illustrate them through a
conventional model of light detection by a photodiode. Stimulated Raman
Scattering is then introduced in its lock-in-based implementation, and the
model presented is adapted in this particular case. The power spectral density
(PSD), relative intensity noise (RIN), signal to noise ratio (SNR), and
sensitivity of the system are derived and discussed. Two complementary methods
are presented that allow measurement of the RIN and assessment of the
performance of a SRS system. Such measurements are illustrated on two
commercial laser systems. Finally, the consequences of noise in SRS are
discussed, and future developments are suggested. The presentation is made
simple enough for under-graduated, graduated students, and newcomers in the
field of stimulated Raman, and more generally in pump-probe based schemes.

Laser noise, Stimulated Raman scattering, Relative intensity noise.

††preprint: APL

##  I Introduction

Over the last two decades, coherent Raman imaging has evolved as a mature,
label-free, imaging technique with numerous applications in biology and
medicine Cheng and Xie (2015). The seminal work of Zumbusch in 1999 revived
coherent anti-Stokes Raman scattering (CARS) Maker and Terhune (1965) as a
vibrational microscopic imaging modality Zumbusch, Holtom, and Xie (1999).
Since then, the coherent Raman imaging field experienced a second revolution
in 2008 when stimulated Raman scattering (SRS) Bloembergen (1967) was also
demonstrated as a powerful vibrational imaging scheme Freudiger  _et al._
(2008); Nandakumar, Kovalev, and Volkmer (2009). Contrary to CARS, SRS is free
of nonresonant background and scales linearly with the molecular concentration
Rigneault and Berto (2018). These key features initiated the development of
SRS imaging technologies He  _et al._ (2017); Berto  _et al._ (2017); Alshaykh
_et al._ (2017); Liao  _et al._ (2016); Liao, Cheng, and Wang (2015); Liao
_et al._ (2015); Berto, Andresen, and Rigneault (2014); Freudiger  _et al._
(2014); Heuke  _et al._ (2018a, b) and facilitated their successful
applications in biology Lu  _et al._ (2015); Fu, Yang, and Xie (2017); Ozeki
_et al._ (2012); Wakisaka  _et al._ (2016), chemistry Wei  _et al._ (2017);
Winterhalder and Zumbusch (2015) and medicine Ji  _et al._ (2015); Orringer
_et al._ (2017); Sarri  _et al._ (2019); Cicerone and Camp (2018) as a
quantitative and label free chemical imaging modality.

Contrary to CARS, which detects faint generated photons at specific
wavelengths, SRS is a pump-probe scheme that works in the high photon flux
regime. It manifests itself as a weak modulation ($10^{-4}-10^{-6}$) that is
transferred from an amplitude modulated (AM) laser on an unmodulated (probe)
laser beam Freudiger  _et al._ (2008).

Because the modulation transferred to the probe beam is weak, the laser noise
and the detection of electronic noise are key components to achieve the
ultimate SRS detection level Ozeki  _et al._ (2009, 2010). For instance,
performing SRS with noisy fiber lasers requires the development of specific
balanced detection schemes Freudiger  _et al._ (2014); Crisafi  _et al._
(2017); Laptenok  _et al._ (2019), which are at best $3\text{\,}\mathrm{dB}$
above the shot noise limit. Although they are key to SRS, it appears that
considerations on laser and detection noise in SRS imaging systems are still
lagging behind when compared to the demonstrated technological and application
advances. For instance, it is often not clear in published papers how the
reported SRS detection ability compares to the shot noise limit.

This paper is intended to provide the SRS experimentalist with theoretical
basics and most importantly a reliable experimental method to characterize the
noise of an SRS system as compared to the ultimate shot noise. More generally,
our presentation applies also to any pump probe spectroscopy scheme. Although
no fundamental breakthrough is presented here, it is our understanding that
the required background to master noise physical description and measurement
is often not available to physical chemists that are building and using SRS
systems. The scope of this paper is to provide a tutorial on noise found in
SRS systems accessible for under-graduated, graduated students, and newcomers
in the field.

We start by defining noise in the context of light intensity detection, and
present a conventional model used to describe such noise. The important
quantities - power spectral density (PSD), relative intensity noise (RIN), and
signal to noise ratio (SNR) - are calculated for this model. We then present a
typical lock-in-based SRS measurement system and describe how the previous
model changes in this context. The SNR and sensitivity of such SRS system are
discussed and linked to the laser RIN. Two complementary measurements of the
RIN are presented, that can be used to characterize and optimize the
performance of a SRS system. These measurements are illustrated using two
commercial laser systems commonly used for SRS imaging. We finally discuss the
implications of such measurements and additional means to further increase the
signal to noise ratio in SRS in both current systems and in future
technological developments. The paper is complemented with a supplementary
information section that is intended to be used as a practical handbook to
perform RIN noise measurement of a SRS laser system. For a comprehensive
introduction to coherent Raman and SRS we invite the reader to refer to the
tutorials Rigneault and Berto (2018).

##  II Definitions and laser intensity noise model

Although various implementations of SRS have been developed, they mostly share
the same working principle, which is the detection of small intensity
variations in an intense laser beam. In this section, a standard model of
laser intensity measurement is presented, along with the definitions of the
power spectral density (PSD), relative intensity noise (RIN), and signal to
noise ratio (SNR). The RIN of a laser is calculated for this model, to
illustrate the contribution of classical intensity noise, quantum
fluctuations, and electrical noise.

###  II.1 Model

The optical intensity of a laser is modeled by a time-varying photon rate
$\mathcal{I}_{opt}(t)$. In this tutorial, the term _optical intensity_ is
preferred to optical power in order to avoid confusion when discussing
electrical power later. The laser is assumed monochromatic for simplification
and is detected with a photodiode.

The electrical current $I(t)$ at the output of this photodiode is modeled
using a semi-classical approach, consistent with the detection of coherent
states of light. Similar work has been done by Quinlan et al. Quinlan  _et
al._ (2013) and can be found in textbooks Saleh, Teich, and Saleh (1991).

The detection of a single photon at time $t=0$ produces an electrical current
$h(t)$ at the output of the photodiode. The temporal spread of $h(t)$ provides
the bandwidth of the photodiode, and the area under the curve $h(t)$ is equal
to a single electric charge $q$:

$\int_{-\infty}^{+\infty}h(t)\mathrm{d}t=q$ (1)

In this work, the contribution of the photodiode will often be simplified by
assuming that its bandwidth is greater than all electronic frequencies of
interest. In the time domain, $h(t)$ will be approximated by a Dirac
distribution. In the frequency domain, $\hat{h}(f)$ will therefore be assumed
to be flat $\hat{h}(f)\approx\hat{h}(0)=q$.

Because SRS is a nonlinear optical process that requires short optical pulses
Rigneault and Berto (2018), it is necessary to model pulsed lasers, consisting
of intense photon bursts occurring at time scales of femtoseconds to
picoseconds. Since the pulse duration is much shorter than the electronic
bandwidth of most measurement devices, all the photons in a single pulse are
considered to arrive simultaneously. For pulsed laser light (derived from
mode-locked oscillators, optical parametric oscillators and amplifiers) the
successive pulses are numbered with the integers $k\in\mathbb{Z}$. The time of
arrival of pulse $k$ is $t_{k}=k\times T_{r}$, where $T_{r}$ is the period of
the oscillator, the inverse of the repetition rate $f_{r}$. The number of
electrons generated by the optical pulse $k$ is a random variable $X_{k}$
which follows a Poisson distribution of mean $N(t_{k})$. The average number of
electron generated per pulse is linked to the optical intensity
$\mathcal{I}_{opt}(t)$ via the photon-to-electron conversion equation:

$N(t_{k})=\frac{\eta}{h\nu}\int_{pulse\,k}\mathcal{I}_{opt}(t)\mathrm{d}t$ (2)

Where $\eta$ is the detector quantum efficiency, $\nu$ the optical frequency,
and $h$ the Plank constant. In addition to the photo-detection events, the
detector current noise is modeled with a stochastic current $\epsilon(t)$.
This current noise encompasses the detector dark current, the Johnson-Nyquist
noise of the load resistor, and any other sources of electrical noise that is
generated at the output of the photodiode independently of the optical
intensity. With this model, the electrical current $I(t)$ can be expressed by:

$I(t)=\Big{[}\sum_{k}X_{k}\delta(t-t_{k})\Big{]}\otimes h(t)+\epsilon(t)$ (3)

where $\otimes$ is the convolution operator. The electrical noise
$\epsilon(t)$ is considered to be independent from the $\\{X_{k}\\}$, and for
simplification it will be assumed that the electrical noise has zero mean:
$\langle\epsilon(t)\rangle=0$. Here, $\langle\cdot\rangle$ stands for the
ensemble average, meaning the average over all possible realizations of the
measurement given the exact same system in the exact same state. This is
analogous to having a large number of identical systems all performing the
same measurement. For such ensemble of systems, each will record different
values for $X_{k}$, but these values will be distributed in a Poisson law of
average $N(t_{k})$. The laser intensity fluctuations over time have therefore
two origins. The first one is the fluctuation of $N(t_{k})$, which arises from
generation of optical pulses that are not perfectly identical, and is referred
to as classical noise here. The second is the randomness on the measurement
and is called shot noise. It causes a fluctuation of the value $X_{k}$ even
when pulses would be perfectly engineered ($N(t_{k})$ = constant). From the
model described by equation 3, one can compute the power contribution of the
DC component and frequency components of the electrical current $I(t)$ through
a load resistor R.

###  II.2 Definitions

Let us consider a quantity $A$ that is a function of time. In the following,
definitions are made using the general quantity $A$, although $A$ will be
replaced by photodiode current $I$ for calculations. Similar derivations could
be done using voltages but the discussion, in particular in terms of power,
signal, and noise, would be identical.

####  II.2.1 DC power

In order to study the contribution of different frequencies to the electrical
current, the finite-time Fourier transform Gardner (1986) of a quantity $A(t)$
is defined as:

$\mathcal{F}_{T}\\{A(t)\\}(f)\equiv\hat{A}_{T}(f)\equiv\int^{T/2}_{-T/2}A(t)e^{-2i\pi
ft}\mathrm{d}t$ (4)

The time average of $A(t)$ is defined as:

$A_{avg}\equiv\lim_{T\rightarrow\infty}\frac{1}{T}\langle\hat{A}_{T}(0)\rangle$
(5)

The total power of quantity $A$ is defined as the time average of $A^{2}$:

$P_{tot}[A]\equiv(A^{2})_{avg}$ (6)

Using the definition of the time variance
$Var[A]\equiv(A^{2})_{avg}-A_{avg}^{2}$, the total power can be split into two
components:

$P_{tot}[A]=A_{avg}^{2}+\mathrm{Var}[A]$ (7)

These two terms have two distinct origins, the first is the DC power, the
second is the power carried by the fluctuations of $A$. Typically when
measuring the value of $A$, the DC power corresponds to the signal power, and
the power of the fluctuations are referred to as noise power $P_{Noise}$.

Note that ”power” as defined here differs by a constant from the usual
electrical power expressed in Watts. The power defined by equation 6 has the
dimension of $[A]^{2}$, where $[A]$ is the unit of $A$. To obtain an
electrical power in $W$, one needs to multiply by the appropriate factor, such
as $R$ if $A$ is a current or $1/R$ if $A$ is a voltage. For instance, the DC
power and noise power of a voltage $V$ generating a current $I$ passing
through a load resistor R are:

$\displaystyle P_{DC}$ $\displaystyle=V_{avg}^{2}/R=R\,I_{avg}^{2}$ (8)
$\displaystyle P_{Noise}$ $\displaystyle=\mathrm{Var}[V]/R=R\,\mathrm{Var}[I]$
(9)

The noise power expressed through the variance does not provide information on
the frequency at which the fluctuations of $A$ are happening.

####  II.2.2 Power Spectral Density (PSD)

To study how the different frequencies contribute to $\mathrm{Var}[A]$, and
therefore to the noise power, the double-sided (positive and negative
frequencies) power spectral density of a quantity $A(t)$ is defined as:

$S_{A}(f)\equiv\lim_{T\rightarrow\infty}\frac{1}{T}\langle|\hat{A}_{T}(f)|^{2}\rangle$
(10)

Because $A(t)$ is usually a real quantity, $S_{A}$ is an even function of
frequency. For this reason, only the positive frequencies are usually
considered and the single-sided power spectral density of a quantity $A(t)$ is
defined as:

$S^{+}_{A}(f)\equiv 2S_{A}(f);f>0$ (11)

As mentioned above, the name ”power spectral density” is ambiguous, as it
usually does not have the dimension of a power density, but rather of
$[A]^{2}/Hz$, where $[A]$ is the unit of $A$. The PSD measures the amount of
electrical power (variance) in the signal per unit of bandwidth. For instance,
the amount of power coming from the frequency range with width $\Delta f$
centered around $f_{0}$ is given by:

$P_{\Delta f}(f_{0})=\int_{f_{0}\pm\Delta f/2}R\,S^{+}_{I}(f)\mathrm{d}f$ (12)

####  II.2.3 Relative Intensity Noise (RIN)

The power spectral density divided by $A_{avg}^{2}$ gives the relative
intensity noise (RIN) of quantity $A$:

$\mathcal{RIN}_{A}(f,A_{avg})\equiv\frac{S^{+}_{A}(f)}{A_{avg}^{2}}$ (13)

The RIN is expressed in ”per unit bandwidth” $1/Hz$ and quantifies the
relative contribution of each spectral component to the total signal power.

####  II.2.4 Signal to Noise Ratio (SNR)

When measuring DC value of the quantity $A(t)$, the signal power is given by
the DC electrical power $A_{avg}^{2}$. The power of the noise is given by
equation 12 where the integral covers the bandwidth $\Delta f$ of the
measurement system (typically $f\in[0,\Delta f]$, to measure the DC
component). The signal to noise ratio can therefore be expressed as:

$\mathcal{SNR}=\frac{A_{avg}^{2}}{\int_{0}^{\Delta
f}\,S^{+}_{A}(f)\mathrm{d}f}$ (14)

As a result, the SNR for a measurement of the average value of $A$ is
inversely related to the RIN:

$\mathcal{SNR}^{-1}=\int_{0}^{\Delta
f}\mathcal{RIN}_{A}(f,A_{avg})\mathrm{d}f$ (15)

###  II.3 Calculations

####  II.3.1 Average and relative intensity

Under the assumption that the electrical noise has a null time average, the
time average current $I_{avg}$ can be computed (equation 1, 3, 4, and 5) in
terms of the time average number of electrons generated $N_{avg}$:

$\displaystyle I_{avg}$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle\int^{T/2}_{-T/2}\Big{(}\Big{[}\sum_{k}X_{k}\delta(t-t_{k})\Big{]}\otimes
h(t)+\epsilon(t)\Big{)}e^{-2i\pi ft}\mathrm{d}t\rangle|_{f=0}$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\int^{T/2}_{-T/2}\sum_{k}\langle
X_{k}\rangle h(t-t_{k})\mathrm{d}t$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\int^{T/2}_{-T/2}\sum_{k}N(t_{k})h(t-t_{k})\mathrm{d}t$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\sum_{t_{k}\in[-T/2,T/2]}qN(t_{k})$
$\displaystyle=qN_{avg}$ (16)

The average photocurrent $I_{avg}$ delivered by the photodiode can either be
measured directly with an oscilloscope, or derived from the laser average
intensity $\mathcal{I}_{opt,avg}$ using the following relationship:

$I_{avg}=\frac{q\eta}{h\nu}\mathcal{I}_{opt,avg}$ (17)

The relative intensity $\alpha(t)$ is defined as the expected current divided
by the average current:

$\alpha(t)\equiv\frac{\langle I(t)\rangle}{I_{avg}}=\frac{N(t)}{N_{avg}}$ (18)

####  II.3.2 Relative Intensity Noise

The photocurrent power spectral density $S_{I}(f)$ can be computed by
considering the photodetection events statistically independent:

$\langle X_{k}X_{l}\rangle=N(t_{k})N(t_{l})+N(t_{k})\delta_{k,l}$ (19)

The PSD from the model defined here reads (Annex A):

$S_{I}(f)=S_{\epsilon}(f)+|\hat{h}(f)|^{2}\Big{[}N_{avg}+S_{N}(f)\Big{]}$ (20)

where $\hat{h}(f)$, the Fourier transform of $h(t)$, is the spectral response
of the detector. For frequencies within the detector bandwidth, one can assume
$\hat{h}(f)\approx\hat{h}(0)=q$. Using the relative intensity $\alpha(t)$ from
equation 18, one obtains a simple expression of the power spectral density for
a laser detected on a photodiode:

$\displaystyle S^{+}_{I}(f)$
$\displaystyle=2S_{\epsilon}(f)+2q^{2}N_{avg}+2q^{2}N_{avg}^{2}S_{N/N_{avg}}(f)$
$\displaystyle=S^{+}_{\epsilon}(f)+2qI_{avg}+I_{avg}^{2}S^{+}_{\alpha}(f)$
(21)

The relative intensity noise of the photocurrent $I(t)$ reads:

$\mathcal{RIN}_{I}(f,I_{avg})=\frac{S^{+}_{\epsilon}(f)}{I_{avg}^{2}}+\frac{2q}{I_{avg}}+S^{+}_{\alpha}(f)$
(22)

It is important to note that $\mathcal{RIN}_{I}$ covers all current
fluctuations while $S^{+}_{\alpha}(f)$ covers only the classical fluctuations
of the laser intensity:

$\mathcal{RIN}_{\langle I\rangle}=S^{+}_{\alpha}(f)$ (23)

Equation 21 illustrates the dependence of the electrical PSD at the output of
the detector with respect to both frequency and average current from the
detector. The three terms on the right of equations 21 and 22 are linked to
the electrical noise, the laser shot noise, and the laser excess (classical)
noise, respectively (Table 1). The impact of these three terms will be
developed further, and measured in the following sections.

Limitation |  Electronic |  Shot Noise |  Excess (Classical) Noise
---|---|---|---
$\mathcal{PSD}\approx$ | $S^{+}_{\epsilon}(f)$ | $2qI_{avg}$ | $S^{+}_{\alpha}(f)\,I_{avg}^{2}$
$\mathcal{RIN}\approx$ | $\frac{S^{+}_{\epsilon}(f)}{I_{avg}^{2}}$ | $\frac{2q}{I_{avg}}$ | $S^{+}_{\alpha}(f)$
Table 1: The three measurement regimes and associated dominant power spectral
densities and relative intensity noise.

As can be seen in equation 15, the SNR for laser intensity measurements is
inversely proportional to the RIN integrated over the photodiode bandwidth.
Equation 22 illustrates that the RIN depends on the frequency and average
laser intensity. At low frequencies ($f<1\,\mathrm{MHz}$) lasers usually
features fluctuations in intensity significantly above the shot noise
$S^{+}_{\alpha}\gg 2q/I_{avg}$. Frequently, this noise renders impossible the
direct measurements of small intensity fluctuations ($10^{-4}-10^{-6}$)
induced by SRS. For this reason, lock-in amplification is typically used to
increase the SNR by moving the measurement towards higher frequencies.

##  III Low noise SRS detection through lock-in amplification

###  III.1 System and model description

In a standard SRS system (Figure 1), a pump laser beam with electric field
intensity $\mathcal{I}_{p}$ and a Stokes beam with electric field intensity
$\mathcal{I}_{s}$ are sent through a sample. When collecting $\mathcal{I}_{p}$
or $\mathcal{I}_{s}$ in the far field after their interactions with the
sample, one can measure an intensity loss $\Delta\mathcal{I}_{p}$ on the pump
beam, and an intensity gain $\Delta\mathcal{I}_{s}$ on the Stokes beam
Rigneault and Berto (2018). Without loss of generality, the following will
assume that the SRS Stokes beam is collected by the photodiode (stimulated
Raman gain, SRG, modality), while the SRS Pump beam is discarded using an
optical filter. After interaction with the sample, the Stokes beam has gained
a relative intensity $\beta$, proportional to the number of molecular bonds
$N$ in the probed volume, their stimulated Raman cross section $\sigma$, and
the optical intensity of the SRS pump beam:

$\displaystyle\beta\equiv\frac{\Delta\mathcal{I}_{s}}{\mathcal{I}_{s}}$ (24)
$\displaystyle\beta\propto N\sigma\mathcal{I}_{p}$ (25)

Figure 1:  Typical Stimulated Raman Scattering Scanning Microscope system. The
SRS pump laser is modulated using an acousto-optic modulator (AOM). It
recombines with the Stokes beam using a dichroic mirror (DM) and both are sent
on the sample through a scanning microscope. The modulated beam is filtered
out using an optical filter and the intensity of the other beam is measured
with a photodiode. The signal from the photodiode $I$ is mixed with a
reference signal $r$, filtered and amplified using a lock-in amplifier, which
generated an output current $I_{m}$.

The amplitude of the relative SRS gain $\beta$ is typically $10^{-4}$ to
$10^{-6}$. The exact expression of the proportionality factor in equation 25
depends on many experimental parameters, and in-depth developments can be
found in the literature Rigneault and Berto (2018); Cheng  _et al._ (2012).

As a direct result of equations 24 and 25, if an intensity modulation at
frequency $f_{0}$ is applied on the SRS pump beam (by the AOM), the change in
intensity $\Delta\mathcal{I}$ of the Stokes beam will also be modulated at the
same frequency. Here, the first laser (SRS Pump) is referred to as the
amplitude modulated (AM) laser, and the second (SRS Stokes) as the demodulated
(probe) laser. Note that the roles (AM/probe) of the two lasers would be
switched in the stimulated Raman loss (SRL) modality. The modulation of the AM
beam is assumed to be total:

$\mathcal{I}_{AM}=\mathcal{I}_{p}\frac{1+\cos(2\pi f_{0}t)}{2}$ (26)

The probe laser intensity after interaction on the sample reads:

$\mathcal{I}_{probe}=\mathcal{I}_{s}+\Delta\mathcal{I}_{s}=\mathcal{I}_{s}\Big{(}1+\frac{\beta}{2}+\frac{\beta}{2}\cos(2\pi
f_{0}t)\Big{)}$ (27)

As described previously, the contribution of SRS to the average optical
intensity on the probe beam is not easily detectable. In order to simplify the
mathematical derivations, the DC contribution of the SRS process is discarded,
and only on the modulation at frequency $f_{0}$ is considered. With this
simplification, equation 27 becomes:

$\mathcal{I}_{probe}=\mathcal{I}_{s}\Big{(}1+\frac{\beta}{2}\cos(2\pi
f_{0}t)\Big{)}$ (28)

The probe laser intensity is detected by the photodiode which generates a
current, modeled with equation 3. The difference with the previous derivation
(section II) is that the expected number of electrons generated by the pulse
$k$ is now modulated:

$N(t_{k})=N_{avg}\alpha(t_{k})m(t_{k})$ (29)

Where $\alpha(t)$ is the relative intensity fluctuation in the absence of the
modulated beam (equation 18) and $m(t)$ the modulation transferred from the
modulated beam:

$m(t)=1+\frac{\beta}{2}\cos(2\pi f_{0}t)$ (30)

With this definition $I_{avg}m(t)$ fluctuates between $I_{avg}-\frac{\Delta
I}{2}$ and $I_{avg}+\frac{\Delta I}{2}$. Where $\Delta I$ is linked to $\beta$
through the following relation:

$\beta=\frac{\Delta I}{I_{avg}}$ (31)

The current $I(t)$ flowing from the photodiode is then sent to a lock-in
amplifier which mixes it with the reference signal $r(t)=g\,\cos(2\pi
f_{0}t)$. The amplitude $g$ of this reference signal comes as an overall gain
and does not affect the discussion on SNR. The phase of the reference signal
is also chosen to be in phase with the modulation signal for optimal
demodulation. The mixed current $I_{m}$ can be expressed as:

$I_{m}(t)=\Big{[}[\sum_{k}X_{k}\delta(t-t_{k})]\otimes
h(t)+\epsilon(t)\Big{]}\cdot r(t)$ (32)

In terms of Fourier components:

$\displaystyle\hat{I}_{mT}(f)$
$\displaystyle\mathrel{\mathop{=}\limits_{T\to+\infty}}\hat{I}_{T}(f)\otimes\hat{r}_{T}(f)$
(33) $\displaystyle\hat{m}_{T}(f)$
$\displaystyle=1+\frac{\beta}{4}(\delta(f-f_{0})+\delta(f+f_{0}))$ (34)
$\displaystyle\hat{r}_{T}(f)$
$\displaystyle=\frac{g}{2}(\delta(f-f_{0})+\delta(f+f_{0}))$ (35)
$\displaystyle\hat{m}_{T}(f)\otimes\hat{r}_{T}(f)$
$\displaystyle\mathrel{\mathop{=}\limits_{T\to+\infty}}g[\frac{\beta}{4}\delta(f)+\frac{1}{2}(\delta(f-f_{0})+\delta(f+f_{0}))+\frac{\beta}{8}(\delta(f-2f_{0})+\delta(f+2f_{0}))]$
(36)

###  III.2 Average current and DC power

The average current at the output of the lock-in amplifier $I_{m,avg}$ can
readily be computed using equation 5, 18 33, and 36:

$\displaystyle I_{m,avg}$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle\hat{I}_{mT}(0)\rangle$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle\hat{I}_{T}(f)\rangle\otimes\hat{r}_{T}(f)|_{f=0}$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}I_{avg}\hat{\alpha}_{T}(f)\otimes\hat{m}_{T}(f)\otimes\hat{r}_{T}(f)|_{f=0}$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}gI_{avg}[(\frac{\beta}{4})\hat{\alpha}_{T}(0)+\frac{1}{2}(\hat{\alpha}_{T}(-f_{0})+\hat{\alpha}_{T}(f_{0}))+\frac{\beta}{8}(\hat{\alpha}_{T}(-2f_{0})+\hat{\alpha}_{T}(2f_{0}))]$
(37)

The terms $\frac{1}{T}\hat{\alpha}_{T}(\pm f_{0})$ and
$\frac{1}{T}\hat{\alpha}_{T}(\pm 2f_{0})$ will vanish when averaged over all
pulses ($T\rightarrow\infty$). In the specific case where the modulation
frequency $f_{0}$ is half of the laser repetition rate, the last term will not
vanish anymore, and the SNR will effectively be doubled. Annex B discusses
this scenario, which has been highlighted before by Ozeki and collaborators
Ozeki  _et al._ (2010). In the conventional SRS case, however, the only
relevant term is $\lim_{T\rightarrow\infty}\frac{1}{T}\hat{\alpha}_{T}(0)=1$,
and the average mixed current at the output of the lock-in amplifier reads:

$I_{m,avg}=gI_{avg}\frac{\beta}{4}$ (38)

From this current, the DC power at the output of the lock-in with a load
resistor $R$ can be derived as:

$P_{DC}=RI_{avg}^{2}g^{2}\frac{\beta^{2}}{16}$ (39)

###  III.3 Power spectral density

From equation 33 one can calculate the photocurrent power spectral density
$S_{I_{m}}(f)$ (equation 10), for the mixed current at the output of the lock-
in amplifier:

$\displaystyle S_{I_{m}}(f)$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle|\hat{I}_{mT}(f)|^{2}\rangle$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle|\hat{I}_{T}(f)\otimes\hat{r}_{T}(f)|^{2}\rangle$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle|q\sum_{k}X_{k}e^{-2i\pi
kf/f_{r}}\otimes\hat{r}_{T}(f)|^{2}\rangle+\lim_{T\rightarrow\infty}\frac{1}{T}\langle|\hat{\epsilon}_{T}(f)\otimes\hat{r}_{T}(f)|^{2}\rangle$
(40)

Detailed calculations can be found in Annex C. Assuming no correlation between
noises at different frequencies, small values of the relative SRS gain
($\beta\ll 1$), and frequencies small compared to the modulation frequency
($f\ll f_{0}$), the PSD of the lock-in output current can be simplified as:

$S^{+}_{I_{m}}(f)=\frac{g^{2}}{2}\Big{(}S^{+}_{\epsilon}(f_{0})+2qI_{avg}+I_{avg}^{2}S^{+}_{\alpha}(f_{0})\Big{)}=\frac{g^{2}}{2}S^{+}_{I}(f_{0})$
(41)

Equation 41 illustrates the advantage of using lock-in detection. Due to the
modulation/demodulation scheme, the low frequency noise features at the output
of the lock-in device correspond to the high frequency noise features of the
laser. By choosing the $f_{0}$ that minimizes these features, one can recover
the minimum noise allowed by the laser system. In particular, by having the
modulation frequency $f_{0}$ sufficiently high, one can avoid the noise at low
frequency that is inherent to laser systems. The factor $\frac{g^{2}}{2}$ is
the gain $G$ of the lock-in amplifier system:

$G\equiv\frac{g^{2}}{2}$ (42)

###  III.4 Signal to noise ratio

Integrating the PSD over the lock-in bandwidth $\Delta f$ (equation 12), the
electrical power of the noise (equation 9) at the output of the lock-in
amplifier can be expressed as:

$P_{Noise}=G\,R\,\Delta
f\,S^{+}_{I}(f_{0})=G\,RI_{avg}^{2}\,\mathcal{RIN}_{I}(f_{0},I_{avg})\,\Delta
f$ (43)

Equation 43 assumes for simplification that the PSD is constant around
$f_{0}$, which is typically a good approximation. Using equations 39 and 43,
the signal to noise ratio as the output of the lock-in amplifier can
ultimately be expressed as:

$SNR=\frac{P_{DC}}{P_{Noise}}=\frac{\beta^{2}}{8\Delta
f\,\mathcal{RIN}_{I}(f_{0},I_{avg})}$ (44)

The result of equation 44 is that, assuming all other sources of noise are
negligible, the SNR in a lock-in-based SRS setup is given by the RIN of the
laser around the modulation frequency. Another way to interpret equation 44 is
to express the sensitivity of the system, which is the smallest SRS gain
$\beta_{min}$ the system can detect with an SNR of 1.

$\beta_{min}=\sqrt{8\Delta f\,\mathcal{RIN}_{I}(f_{0},I_{avg})}$ (45)

As a direct result of equations 44 and 45, any deviation from the minimal RIN
results in a sub-optimal measurement that impairs the SNR, the sensitivity, or
slows down signal acquisition. The time between two successive measurement is
inversely proportional to the lock-in bandwidth. As a result, for a RIN that
is doubled, the acquisition time also has to be doubled to maintain the same
acquisition SNR.

###  III.5 SNR optimization

Combining equations 44 and 22, the SNR of the system can be written as:

$\large SNR(f_{0},\Delta f,I_{avg})=\frac{\frac{\beta^{2}}{8\Delta
f}}{\frac{S^{+}_{\epsilon}(f_{0})}{I_{avg}^{2}}+\frac{2q}{I_{avg}}+S^{+}_{\alpha}(f_{0})}$
(46)

The different terms of the denominator in equation 46 are precisely the RIN
associated with electronic noise, shot noise, and laser excess noise
respectively (Table 1). Using this equation, one can identify all the
parameters that can maximize the SNR in a lock-in-based SRS system:

(i) Maximizing $\beta$, the relative modulation of the probe beam. This goes
beyond the scope of this tutorial, but $\beta$ increases linearly with the
number of molecular bonds probed, as well as with the laser intensity in the
AM beam (equation 25). It is important to note that $\beta$ is constant with
respect to the laser intensity in the probed beam, and therefore does not
depend on $I_{avg}$. It does, however, depend linearly on the AM laser average
intensity, further details are discussed in section V.2.

(ii) Minimizing $\Delta f$, the lock-in bandwidth. In principle an arbitrarily
high SNR can be achieved by reducing $\Delta f$. In practice the bandwidth has
to be large enough to allow changes in the SRS signal. In a SRS point scanning
microscope the signal is expected to vary from one pixel to the next. In this
case $\Delta f$ should match the pixel acquisition rate to allow fluctuations
from one pixel to the next, while filtering out fluctuations within a pixel.

(iii) Maximizing $I_{avg}$ to minimize electrical noise and shot noise.
Depending on the photodiode noise features, one can increase the probed laser
beam intensity (and therefore $I_{avg}$) to reach the regime where the
electronic noise $S^{+}_{\epsilon}(f_{0})/I_{avg}^{2}$ becomes negligible with
respect to classical and quantum laser fluctuations. In the systems described
below, this regime was reached for a few milliwatts of average intensity in
the probe beam. When the SNR is limited by the laser, one of two scenarios
have to be considered.

a) The RIN is limited by the shot noise, in which case the SNR will increase
linearly with $I_{avg}$. The SNR is then limited by how much $I_{avg}$ can be
increased. This limit is set by the laser maximum output power, the
photodamage threshold, or the laser excess noise.

b) The RIN is limited by the laser excess noise, in which case increasing
$I_{avg}$ will not change the SNR. The SNR limit allowed by the laser is
reached and cannot be improved by increasing the probe laser beam intensity.

In this last scenario, the laser excess noise ultimately sets the limit on the
SNR in a lock-in-based SRS system. By using a modulation frequency that
minimizes this excess RIN one can achieve the optimal SNR. A proper
measurement of the laser RIN is therefore necessary to quantify and optimize
the performance of such SRS system. Two experimental procedures used for laser
RIN measurement are detailed in the following section, (i) a complete RIN
characterization for different frequencies and optical intensities, and (ii) a
faster RIN assessment on an existing lock-in-based SRS microscope. Both
methods are illustrated on two commercial laser sources.

##  IV Laser intensity noise measurements

As described previously the RIN of a photodiode output current is composed of
three different terms (equation 22) that come from electronic noise, quantum
fluctuations, and classical fluctuations (Table 1). The electronic
contribution is due only to the photodiode and can be made negligible by
choosing the adequate device and laser intensity. The quantum and classical
fluctuations are inherent to the laser source. Its understanding and
characterization is crucial to assess the performance of a SRS system. The RIN
is a function of both frequency and average current provided by the
photodiode. Here, we describe a way to reconstruct the full
$\mathcal{RIN}_{I}(f,I_{avg})$ surface over a large range of frequencies
($500\,\mathrm{kHz}$ to $30\,\mathrm{MHz}$) and low laser intensities (0.1mW
to 50mW) suitable for SRS imaging applications.

Figure 2: Schematics of the RIN measurement methods. In both cases,
measurement of the average photocurrent $I_{avg}$ can be performed either
directly using an oscilloscope after the photodiode, or estimated by measuring
the optical average power using a power meter and using the photoconvertion
equation (eq. 17). Proper characterization of the gain and bandwidth of the
measurement electronics is key to properly estimating the laser RIN.

###  IV.1 Measured laser sources

Device 1 is a Ti:Saph femtosecond laser (Chameleon, Coherent Inc) operating at
a repetition rate of $80\text{\,}\mathrm{MHz}$ with a pulse width duration of
$150\text{\,}\mathrm{fs}$ and a wavelength of $800\text{\,}\mathrm{nm}$. This
beam acts as the (unmodulated) SRS pump beam of a lock-in-based SRS system
(SRL modality) whereas the (AM) Stokes beam is provided by a tunable fs
optical parametric oscillator OPO (Compact OPO, APE & Coherent Inc) pumped by
the Ti:Saph laser. Here, only the $800\text{\,}\mathrm{nm}$ beam is
considered, as it is the one monitored by the SRS photodiode.

Device 2 is a picosecond OPO (Emerald, APE) operating at a repetition rate of
$80\text{\,}\mathrm{MHz}$ with a pulse duration of $2\text{\,}\mathrm{ps}$ and
a wavelength of $800\text{\,}\mathrm{nm}$. This OPO is pumped by a 20W
frequency doubled ytterbium (Yb) fiber laser (Emerald engine, APE)
($80\text{\,}\mathrm{MHz}$ rep rate, $2\text{\,}\mathrm{ps}$ pulse duration)
that provides also the Stokes beams at $1030\text{\,}\mathrm{nm}$. As
explained previously, only the $800\text{\,}\mathrm{nm}$ beam is considered in
the RIN measurement. Note that device 2 is similar to the commercially
available one box SRS laser system PicoEmerald (APE), although the PicoEmerald
uses an 10W frequency doubled Yb fiber laser.

###  IV.2 Method 1: frequency and intensity dependence

To properly quantify the noise level in devices 1 and 2, fast biased
photodiode (Thorlabs DET10A/M, $1\,\mathrm{ns}$ rise time) is placed at the
position of the forward SRS detector (Figure 1). The average photocurrent
generated by the photodiode was measured with a fast oscilloscope (HP 54111D
$500\,\mathrm{MHz}$ bandwidth, $50\,\mathrm{\Omega}$ input). The typical
photocurrent was several milliamperes. The associated shot noise PSD can be
calculated using Table (1) and is on the order of $-190\,\mathrm{dBW/Hz}$
which is far below the input noise ($-180\,\mathrm{dBW/Hz}$) of the spectrum
analyzer used here (Zurich instrument HF2LI). For this reason, a filtering and
amplification step was added. The photodiode was loaded with a
$50\,\mathrm{\Omega}$ resistor and the resulting voltage was filtered with a
$30\,\mathrm{MHz}$ low pass filter (Mini-Circuits BLP$-$30$+$) to damp the 80
MHz repetition rate and harmonics of the lasers. The signal was then amplified
with a $25\text{\,}\mathrm{dB}$ low noise preamplifier (Ref 153579, APE)
working in the range of $100\,\mathrm{kHz}$ to $100\,\mathrm{MHz}$. The
overall effect of this filtering and amplification stage was to amplify by
more than $20\,\mathrm{dB}$ the electrical components between
$500\,\mathrm{kHz}$ and $30\,\mathrm{MHz}$ (Supplementary Figure 6). Then, the
electrical power spectral density was acquired with a spectrum analyzer
(Zurich instrument HF2LI). The PSD was measured for 1000 frequencies from
$500\,\mathrm{kHz}$ to $30\,\mathrm{MHz}$ with a bandwidth of
$1.1\text{\,}\mathrm{kHz}$ to $70\text{\,}\mathrm{kHz}$. The effect of the
filter and amplifier was numerically subtracted from the obtained data.

The power spectral density associated with electronic noise was measured by
blocking the laser beam. The noise level was $-173\,\mathrm{dBW/Hz}$ measured
at the input of the spectrum analyzer. This noise is a combination of
amplified dark current, input and output noise of the amplifier, and input
noise of the spectrum analyzer. All electronic noises are merged in an
equivalent power spectral density at the detector $S^{+}_{\epsilon}(f)$. The
obtained value after compensating for the preamplifier stage gives:
$S^{+}_{\epsilon}(f)\approx-173-23=-196\mathrm{dBW/Hz}$, with very little
dependence in frequency (Supplementary Figure 7). After numerically accounting
for filtering and amplification, the contribution of the electronic noises to
the power spectral density was removed numerically to keep only the laser RIN:

$\mathcal{RIN}_{laser}(f,I_{avg})=\mathcal{RIN}_{I}(f,I_{avg})-\frac{S^{+}_{\epsilon}(f)}{I_{avg}^{2}}=\frac{2q}{I_{avg}}+S^{+}_{\alpha}(f)$
(47)

###  IV.3 Method 1: Results

For an average photocurrent of $I_{avg}=5\,\mathrm{mA}$ the recovered laser
RIN are plotted in Figure 3a for devices 1 and 2. For device 1 (Ti:Saph
laser), the measured data matches the expected shot noise limit for
frequencies above $2\,\mathrm{MHz}$. Consequently, any SRS related modulation
of the laser above this frequency can be detected down to the shot noise
limit. For device 2, the measured data is within $5\,\mathrm{dB}$ of the shot
noise limit for frequencies above $20\,\mathrm{MHz}$. This corresponds to a
SNR for the system - at such modulation frequencies and photocurrent - that is
only a factor of 3 below the shot noise limit, meaning that a measurement
requires a bandwidth 3 times smaller than that of a shot noise limited system
to receive the same SNR (equation 15). Note that for device 2, pumping the OPO
with a 10W Yb fiber laser (PicoEmerald case) would result in a
$3\,\mathrm{dB}$ lower RIN, bringing the OPO 2dB above the shot noise for
frequencies above $20\,\mathrm{MHz}$.

Figure 3:  a) Relative intensity noise measured as a function of frequency for
device 1 (Ti:Saph) and 2 (Fiber laser pumped OPO) with average photodiode
current $I_{avg}=5\,\mathrm{mA}$. The solid line are the filtered data
(Savitzky–Golay filter, 51 points, 1st order). b) Relative intensity noise
(RIN) as a function of the average photodiode current for device 1 (Ti:Saph,
left triangles) and 2 (Fiber laser pumped OPO, right triangles). The RIN was
averaged between 10 and $30\,\mathrm{MHz}$. The excess noise on device 2
(dashed line) is fitted to match the measurements. The electronic noise was
removed numerically.

Complementary to the frequency analysis, one needs to measure the RIN for
different photodiode currents to characterize the optimal setup working
parameters. This analysis was performed by recording the RIN as a function of
frequency for photocurrents ranging from $0.5\,\mathrm{mA}$ to
$7\,\mathrm{mA}$, after which the detector saturates. This saturation comes in
the form of a distorted signal from the photodiode output current visualized
on the oscilloscope. The surfaces $\mathcal{RIN}(f,I_{avg})$ are plotted in
Figures 4 for both devices. To better illustrate
$\mathcal{RIN}(f=20\mathrm{MHz},I_{avg}$), its value was averaged from 10 to
$30\,\mathrm{MHz}$ in the frequency domain (Figure 3b).

Figure 4:  RIN of device 1 (left) and 2 (right) as a function of frequency
ranging from $500\,\mathrm{kHz}$ to $30\,\mathrm{MHz}$ and photocurrents
ranging from $0.5\,\mathrm{mA}$ to $7\,\mathrm{mA}$. The noise floor set by
the shot noise is represented by the green bottom surface. At low
photocurrents the electronic contribution to the RIN diverges, at low
frequencies the classical noise dominates. The peak shown on device 1 at 20MHz
corresponds to a SRS induced modulation at this frequency.

Device 1 is shot noise limited around $20\,\mathrm{MHz}$ for all photodiode
current, and therefore all laser intensities, allowed by the photodiode.
Device 2 shows a small excess noise of $5\,\mathrm{dB}$ above the shot noise
for $5\,\mathrm{mA}$ photocurrent. As expected from the model, device 2 is
shot noise limited for small enough photocurrents
($<$$500\text{\,}\mathrm{\SIUnitSymbolMicro A}$) and reaches a plateau at
$S_{\alpha}(f_{0}=20\,\mathrm{MHz})=-159\,\mathrm{dBc/Hz}$. Supplementary
information provides a practical handbook and software matlab script to
implement Method 1.

###  IV.4 Method 2: using an existing lock-in SRS system

Although the measurement procedure described above provides a complete
measurement of the laser RIN, it requires low noise filtering and
amplification steps, as well as a sensitive spectrum analyzer. When
characterizing the performance of a lock-in-based SRS system, such measurement
can be simplified by making use of the existing lock-in amplifier. As can be
seen from equation 9 and 43, the electrical power
$P_{Noise}=\mathrm{Var}[V_{m}]/R$ of the voltage fluctuations at the output of
a lock-in amplifier is a direct measurement of the photodiode current PSD (and
therefore RIN) around the reference frequency of the lock-in:

$\displaystyle S^{+}_{I}(f_{0})$ $\displaystyle=\frac{P_{Noise}}{G\,R\,\Delta
f}=\frac{\mathrm{Var}[V_{m}]}{G\,R^{2}\,\Delta f}$ (48)
$\displaystyle\mathcal{RIN}_{I}(f_{0},I_{avg})$
$\displaystyle=\frac{\mathrm{Var}[V_{m}]}{G\,\Delta
f\,R^{2}\,I_{avg}^{2}}=\frac{S^{+}_{\epsilon}(f_{0})}{I_{avg}^{2}}+\frac{2q}{I_{avg}}+S^{+}_{\alpha}(f_{0})$
(49)

The voltage variance can be measured directly using an oscilloscope with a
bandwidth larger than that of the lock-in. Proper characterization of the
lock-in gain $G$ and bandwidth $\Delta f$ is important to measure the
parameters in equation 49 and ensure a quantitative comparison of the model
with experimental data. In this study, a commercially available detector and
lock-in amplifier (APE LIA, Berlin) was used. The integration $\tau$ time was
set to $20\text{\,}\mathrm{\SIUnitSymbolMicro s}$ which is linked to its
bandwidth $\Delta f$ by:

$\Delta f=\frac{1}{2\pi\tau}=8\,\mathrm{kHz}$ (50)

The gain $G$ of the lock-in amplifier was measured experimentally using a
reference signal at $f_{0}=20\,\mathrm{MHz}$. The input signal entering the
lock-in was a sinusoidal voltage of amplitude $300\,\mathrm{uV}$ and frequency
$f_{0}$. The output signal from the lock-in resulted in a DC voltage of
amplitude $1.8\,\mathrm{V}$. The associated gain in power was therefore
measure to be $G=20\log(V_{out}/V_{in})=75.56\,\mathrm{dB}$. The reference,
lock-in input and output signals were generated and recorded using a
multipurpose system (Zurich Instrument HF2LI).

Given $G$ and $\Delta f$, and following equation 49, the RIN of the photodiode
current $I$ can be measured experimentally using the lock-in, by recording the
variance of the lock-in output voltage. This RIN measurement is made at the
lock-in frequency $f_{0}$ and varying laser average intensity (and therefore
$I_{avg}$).

More specifically, by plotting the RIN as a function of $I_{avg}$ on a log-log
plot, one can identify the different regimes (Electronic noise, Shot noise,
Classical noise) of limitation of the measurement.

The electronic noise $S^{+}_{\epsilon}(f_{0})$ can be measured directly by
blocking the laser beam before the photodiode ($I_{avg}=0$). The measured
lock-in output voltage standard deviation in the absence of the laser was
$\sqrt{\mathrm{Var}[V_{m}]}=1.1\,\mathrm{mV}$. This corresponds to an
electrical noise power in a $R=50\,\mathrm{Ohm}$ resistance of:

$P_{Noise}=\mathrm{Var}[V_{m}]/R=2.42\times
10^{-8}\,\mathrm{W}=-76\,\mathrm{dBW}$ (51)

Following equation 48, the electrical noise prior to amplification is
therefore:

$S^{+}_{\epsilon}(f_{0})=-190\,\mathrm{dBW/Hz}$ (52)

The PSD and RIN associated to laser shot noise are given by Table 1. The
average photocurrent was measured through the average optical intensity using
equation 17, knowing the quantum efficiency $\eta=0.8$ of the photodiode at
$800\text{\,}\mathrm{nm}$. As a result for a laser at $800\,\mathrm{nm}$
wavelength $I_{avg}=0.5\\_\mathcal{I}_{opt,avg}$ where $\mathcal{I}_{opt,avg}$
is expressed in Watts and $I_{avg}$ in Amperes. The RIN was measured on the
two commercial laser systems described in Section IV.1, using the lock-in
voltage output standard deviation, for laser average optical intensities
ranging from $0.6$ to $50\,\mathrm{mW}$ on the photodiode (Figure 5). The
contribution of the electronic noise was removed numerically, as described in
section IV.2.

###  IV.5 Method 2: Results

The measured RIN from device 1 matches the model (dotted orange line), with no
measurable classical (excess) laser noise. For photocurrents $I_{avg}$ higher
than $5\,\mathrm{mA}$ (i.e. laser intensities on the photodiode above
$10\,\mathrm{mW}$) the electronic noise is negligible and the SRS measurement
is shot noise limited.

Excess laser noise ($S_{\alpha}$) has to be introduced to fit the experimental
data from device 2 with the model (equation 49). This noise is estimated using
the RIN measurements for high $I_{avg}$ values:

$S_{\alpha,dev2}(f=20\,\mathrm{MHz})=-161\,\mathrm{dBc/Hz}$ (53)

Using this value of excess laser RIN, the measured RIN from device 2 can be
fitted with the model (Figure 5).

Figure 5:  Photodiode current RIN (expressed in dBW), for fixed frequency
$f=20\,\mathrm{mW}$ and increasing average laser photodiode current $I_{avg}$
(and therefore average laser intensity). The figure shows measurements for
device 1 (Ti:Saph - orange left triangles) and device 2 (Fiber laser pumped
OPO - blue right triangles). The shot noise (solid line) and excess laser
noise (dash line) are added to provide the modeled noise for device 2 (dotted
line). The electronic noise was removed numerically from all measurements but
is shown (dash-dotted line) for comparison.

The RIN measurement obtained using Method 2 (existing lock-in amplifier) is
slightly discordant with the more complete study using Method 1. The excess
laser RIN obtained is $2\,\mathrm{dB}$ below the previously measured value.
This difference can be attributed to changes in the OPO alignment that
introduced additional excess laser noise in the first case. It is noted that
Method 2 measurement of the RIN, for device 2, presents relatively strong
discrepancies with the model, in the form of fluctuations above and below the
expected value. These fluctuations are also attributed to changes in the OPO
alignment during the time of the measurement, which introduced up to
$2\,\mathrm{dB}$ of fluctuations in the RIN measurement.

##  V Discussion

###  V.1 Consequences of RIN on SRS measurements

Excess laser RIN has a direct and major impact on the sensitivity of lock-in-
based SRS measurements. As illustrated by equation 46, the SNR in such
measurement is limited by either electronic noise, shot noise, or laser excess
intensity noise. While electronic noise can usually be avoided, and shot noise
sets the physical limit, excess laser RIN is more difficult to control and
minimize. Such excess noise in lasers can usually be explained by excess noise
in the pump laser or amplified stimulated emission, which are harder to
address.

Since the RIN is a function of frequency, the choice of the laser modulation
frequency used in the lock-in detection is an important parameter to minimize
the laser RIN and maximize the SNR. As illustrated in Figure 3a, the RIN can
be lowered by an order of magnitude by simply changing the modulation
frequency from 5MHz to 20MHz, therefore increasing the SNR by a factor of 10.

It is crucial to note that the SNR is limited by the laser that is used as a
probe and detected by the photodiode. In the case of SRS this is particularly
important, since only one of the laser used (pump or stokes) is the probe and
has to exhibit a low noise. The other laser is usually amplitude modulated and
its contribution to noise is negligible. Complementary to this, once the SNR
is limited by excess laser RIN, there is no advantage of increasing the laser
power at the sample plane (and therefore the current of the photodiode). Since
the SRS signal scales as the laser noise in this case, the SNR will not be
improved by using higher laser power, as detailed in section III.5. Although
it is important to understand how the SNR can be improved by increasing the
intensity of the SRS pump or Stokes lasers, this analysis should be done
considering the photodamage on the sample, that usually sets the experimental
limit to the laser intensities.

###  V.2 Optimal pump probe ratio

It has been demonstrated that if the SNR of a lock-in-based system is shot
noise limited, and if the photodamage is linear with the intensity of both
laser beams, the optimal SNR is achieved for constant photodamage when the AM
beam has twice the intensity of the probe beamMoester, Ariese, and De Boer
(2015).

This result can be extracted from equation 46. The total optical intensity on
the sample is:

$\mathcal{I}_{tot}=\mathcal{I}_{probe}+\mathcal{I}_{AM}$ (54)

and the SNR of the measurement is, for a certain proportionality constant
$\kappa$:

$\mathcal{SNR}(\mathcal{I}_{AM},\mathcal{I}_{probe})=\kappa\mathcal{I}_{AM}^{2}(\mathcal{RIN}_{I}(\mathcal{I}_{probe}))^{-1}$
(55)

The optimal ratio between the probe and AM laser intensities is obtained by
deriving equation 55 (or it’s logarithm) with respect to $\mathcal{I}_{AM}$,
while keeping $\mathcal{I}_{tot}$ constant. Omitting electronic noises, the
resulting optimal relationship between the intensities becomes:

$\mathcal{I}_{AM}=2\,\mathcal{I}_{probe}+S^{+}_{\alpha}(f_{0})\frac{\eta}{h\nu}\mathcal{I}_{probe}^{2}$
(56)

Note that this relationship is expressed in terms of average intensity, and
that the peak intensity of the AM beam is twice as high due to modulation. The
result is consistent with the 2:1 (AM/probe) ratio in the case of shot noise
limited measurements ($S^{+}_{\alpha}(f_{0})\ll 2q/I_{avg}$), and an even more
biased ratio towards the AM beam in the case of a noisy probe beam
($S^{+}_{\alpha}(f_{0})\geq 2q/I_{avg}$), as discussed above. Equation 56
generalizes the result obtained by Moester and collaboratorsMoester, Ariese,
and De Boer (2015), in the case of excess laser intensity noise.

###  V.3 Sensitivity and performance report

In the work presented here, using device 2 as the probe beam, there is no need
to push the intensity above few milliwatts (on the photodiode). In this case
it is interesting to increase the optical intensity on the second (modulated)
SRS laser to increase the SNR.

With a lock-in integration time of $\tau=20\,\mathrm{\mu s}$ and average
photocurrent of $I_{avg}=5\,\mathrm{mA}$ (i.e. $10\,\mathrm{mW}$ of laser
power on the photodiode) the measured RIN (Figure 3) is -162 and -157
$\mathrm{dBc/Hz}$ for devices 1 and 2, respectively. The SRS detection limit
for device 1 is therefore predicted to be (equation 45):

$\frac{\Delta
I_{min}}{I_{avg}}=\beta_{min}=\sqrt{\frac{8}{2\pi\tau}10^{-16.2}}=2.0\times
10^{-6}$ (57)

And for device 2:

$\frac{\Delta
I_{min}}{I_{avg}}=\beta_{min}=\sqrt{\frac{8}{2\pi\tau}10^{-15.7}}=3.6\times
10^{-6}$ (58)

To reach such sensitivity with a fiber laser usually requires noise
cancellation using balanced detection schemes Crisafi  _et al._ (2017); Nose
_et al._ (2012); Freudiger  _et al._ (2014). Because the sensitivity of the
system also depends on its bandwidth, the proper way to evaluate and assess
the performance of a SRS system with respect to laser noise is to specify:

(i) The average photocurrent from the detector, which sets the shot noise
limited RIN and SNR (Table 1).

(ii) The achieved experimental RIN by either its absolute value or its
distance in dB from the shot noise limit.

###  V.4 Future developments

When imaging biological samples and other diluted species, the number of Raman
active molecules in the focal spot is such that to detect a signal, one needs
to stay on a pixel from several to tens of microseconds. The associated
bandwidth is therefore limited to tens of kilohertz, and the fastest one can
expect to perform imaging is tens of thousands of pixels per second. This
acquisition rate corresponds to a few frames per second with 100 by 100 pixel
images. This limit could be overcome in two ways.

First, by reducing the laser repetition frequency and increasing the pulse
peak power. This would further increase the efficiency of the SRS process and
render modulation at half of the repetition frequency more practical. In this
case low frequency noise would be extremely important to characterize in order
to preserve the SNR. The limitation set by nonlinear photodamage will however
put a limit of the amount of peak power that can be delivered on the sample.

Second and complementary to this first point, as the time spent per pixel
cannot be diminished further, major developments are to be expected in
spatially multiplexed SRS. Either by having multiple foci at once in the
sampleHeuke  _et al._ (2018a), or with more robust spatial multiplexing, one
can increase the amount of information collected from the sample and further
increase the imaging speed. In this case, a lot of power will be necessary to
enable the SRS process to take place at multiple locations. This high power
can be achieved with fiber lasers, or optical parametric amplifiers, but the
question of noise in such system still needs to be addressed to ensure optimal
- shot noise limited - SNR.

##  VI Conclusion

This work presented a description and ways to characterize laser noise in the
context of stimulated Raman scattering. The laser excess RIN ($S_{\alpha}(f)$)
was introduced as well as the shot noise RIN. The relation between RIN and
signal to noise ratio in SRS was derived (equation 44), as well as the optimal
ratio of pump and Stokes intensities to maximize the SNR for non shot noise
limited systems. Two methods for measuring the RIN were presented, either with
a full characterization using commercially available electronic parts, or a
more accessible implementation using the SRS lockin amplifier. Two laser
systems were studied. One was a solid state Ti-Sapphire laser which was shot
noise limited around $20\,\mathrm{MHz}$ for laser power up to
$70\,\mathrm{mW}$. The second system was an OPO pumped by a 20W fiber laser
which showed excess RIN of $-160\pm 1\,\mathrm{dBc/Hz}$ at $20\,\mathrm{MHz}$.
Future developments in SRS will likely involve an improvement in the noise
performance of lasers, and a systematic characterization of the system’s
noise.

###### Acknowledgements.

We would like to acknowledge Professor Randy A. Bartels for his useful input
on this work. Dr. Ingo Rimke has financial interest in APE.

##  VII Annex A

A more detailed derivation of the photocurrent power spectral density for the
light detection model presented in this work has been done by Quinlan and
collaborators Quinlan  _et al._ (2013). The concise calculation using the
notations used here gives:

$\displaystyle S_{I}(f)$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle|\hat{I}_{T}(f)|^{2}\rangle$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\langle|\hat{h}_{T}(f)[\sum_{k}X_{k}e^{-2i\pi
kf/f_{r}}]+\hat{\epsilon}_{T}(f)|^{2}\rangle$
$\displaystyle=S_{\epsilon}(f)+\lim_{T\rightarrow\infty}\frac{|\hat{h}_{T}(f)|^{2}}{T}\sum_{k}\sum_{l}\langle
X_{k}X_{k}\rangle e^{-2i\pi(k-l)f/f_{r}}$
$\displaystyle=S_{\epsilon}(f)+\lim_{T\rightarrow\infty}\frac{|\hat{h}_{T}(f)|^{2}}{T}[\sum_{k}\sum_{l}N(t_{k})N(t_{l})e^{-2i\pi(k-l)f/f_{r}}+\sum_{k}N(t_{k})]$
$\displaystyle=S_{\epsilon}(f)+|\hat{h}(f)|^{2}[S_{N}(f)+N_{avg}]$

##  VIII Annex B

As explained in more details in the work of Ozeki and collaborators Ozeki  _et
al._ (2010), using a modulation frequency that is half of the repetition rate
of the laser can increase by two fold the SNR. In this particular case, we
obtain:

$\displaystyle\frac{1}{T}\hat{\alpha}_{T}(\pm 2f_{0})$
$\displaystyle=\frac{1}{T}\int^{T/2}_{-T/2}\frac{\langle
I(t)\rangle}{I_{avg}}e^{\pm 4i\pi f_{0}t}\mathrm{d}t$
$\displaystyle=\frac{1}{T}\int^{T/2}_{-T/2}\sum_{k}\frac{N(t_{k})}{qN_{avg}}h(t-t_{k})e^{\pm
4i\pi f_{0}t}\mathrm{d}t$

With the approximating that $h(t)$ is a Dirac distribution of area q:

$\displaystyle\mathrel{\mathop{=}\limits_{T\to+\infty}}\frac{1}{T}\sum_{t_{k}\in[-T/2,T/2]}\frac{N(t_{k})}{N_{avg}}e^{\pm
2ik\pi(\frac{2f_{0}}{f_{r}})}$

The sum does not vanish when $2f_{0}=f_{r}$, instead the terms add up
constructively and the limit give:

$\lim_{T\rightarrow\infty}\frac{1}{T}\hat{\alpha}_{T}(\pm
2f_{0})=\lim_{T\rightarrow\infty}\frac{1}{T}\hat{\alpha}_{T}(0)=1$ (59)

Adding the contribution of this terms to equation 37, the mixed average
current $I_{m,avg}$ at the output of the lockin amplifier is doubled, and the
DC power is quadrupled. The terms $\hat{\alpha}_{T}(\pm 2f_{0})$ are also
involved in the calculation of the noise Power (Annex C, equation 64). Taking
those into account, the noise power is doubled as well, resulting in a net
improvement of the SNR of a factor of 2.

##  IX Annex C

Equation 40 is valid when the electronic noise has a zero expected value and
is independent from the laser intensity noise. The second term of equation 40
correspond to electronic noise and can be expressed as:

$\lim_{T\rightarrow\infty}\frac{1}{T}\langle|\hat{\epsilon}_{T}(f)\otimes\hat{r}_{T}(f)|^{2}\rangle=\frac{g^{2}}{4}[S_{\epsilon}(f-f_{0})+S_{\epsilon}(f+f_{0})]$
(60)

The first term of equation 40 relates to laser noise and requires more
computation:

$\displaystyle\langle|q\sum_{k}X_{k}e^{-2i\pi
kf/f_{r}}\otimes\hat{r}_{T}(f)|^{2}\rangle$ $\displaystyle=\,$
$\displaystyle\langle\mathcal{F}_{T}\\{q\sum_{k}X_{k}\delta(t-t_{k})r(t)\\}(f)\mathcal{F}_{T}^{\star}\\{q\sum_{l}X_{l}\delta(t-t_{l})r(t)\\}(f)\rangle$
$\displaystyle=\,$ $\displaystyle q^{2}\sum_{k}\sum_{l}\langle
X_{k}X_{l}\rangle\mathcal{F}_{T}\\{\delta(t-t_{k})r(t)\\}(f)\mathcal{F}_{T}^{\star}\\{\delta(t-t_{l})r(t)\\}(f)$

Using equations 19 and 29:

$\displaystyle=\,$ $\displaystyle
q^{2}\sum_{k}\sum_{l}\Big{(}N_{avg}^{2}\alpha(t_{k})m(t_{k})\alpha(t_{l})m(t_{l})+N_{avg}\alpha(t_{k})m(t_{k})\delta_{k,l}\Big{)}r(t_{k})e^{-2i\pi
kf/f_{r}}r(t_{l})e^{2i\pi lf/f_{r}}$ $\displaystyle=\,$ $\displaystyle
I_{avg}^{2}|\sum_{k}\alpha(t_{k})m(t_{k})r(t_{k})e^{-2i\pi
kf/f_{r}}|^{2}+qI_{avg}\sum_{k}\alpha(t_{k})m(t_{k})|r(t_{k})|^{2}$

Using the property that the Fourier Transform of a product is the convolution
of the Fourier Transforms:

$\displaystyle=\,$ $\displaystyle
I_{avg}^{2}|\hat{\alpha}_{T}(f)\otimes\hat{m}_{T}(f)\otimes\hat{r}_{T}(f)|^{2}+qI_{avg}\hat{\alpha}_{T}(f)\otimes\hat{m}_{T}(f)\otimes\hat{r}_{T}(f)\otimes\hat{r}_{T}(f)|_{f=0}$

Using the definition of $\hat{m}_{T}(f)$ and $\hat{r}_{T}(f)$ (equations 34
and 35):

$\displaystyle=\,$ $\displaystyle
g^{2}I_{avg}^{2}|\hat{\alpha}_{T}(f)\otimes\Big{(}\frac{\beta}{4}\delta(f)+\frac{1}{2}(\delta(f-f_{0})+\delta(f+f_{0}))+\frac{\beta}{8}(\delta(f-2f_{0})+\delta(f+2f_{0})\Big{)}|^{2}$
$\displaystyle\hskip
28.45274pt+qg^{2}I_{avg}\hat{\alpha}_{T}(f)\otimes\Big{(}\frac{\beta}{8}(\delta(f-f_{0})+\delta(f+f_{0}))+\frac{1}{2}\delta(f)+\frac{1}{4}(\delta(f-2f_{0})+\delta(f+2f_{0}))$
$\displaystyle\hskip
28.45274pt+\frac{\beta}{16}(\delta(f-f_{0})+\delta(f+f_{0}))+\frac{\beta}{16}(\delta(f-3f_{0})+\delta(f+3f_{0}))\Big{)}|_{f=0}$
$\displaystyle=\,$
$\displaystyle\frac{g^{2}I_{avg}^{2}}{4}|\frac{\beta}{2}\hat{\alpha}_{T}(f)+\hat{\alpha}_{T}(f+f_{0})+\hat{\alpha}_{T}(f-f_{0})+\frac{\beta}{4}(\hat{\alpha}_{T}(f+2f_{0})+\hat{\alpha}_{T}(f-2f_{0}))|^{2}$
$\displaystyle\hskip
28.45274pt+\frac{qg^{2}I_{avg}}{4}\Big{(}2\hat{\alpha}_{T}(0)+\frac{3\beta}{4}(\hat{\alpha}_{T}(f_{0})+\hat{\alpha}_{T}(-f_{0}))+\hat{\alpha}_{T}(2f_{0})+\hat{\alpha}_{T}(-2f_{0})+\frac{\beta}{4}(\hat{\alpha}_{T}(3f_{0})+\hat{\alpha}_{T}(-3f_{0}))\Big{)}$
(64)

Equation 41 then needs to be re-injected in equation 40 to take the time
average. It is expected that the terms $\frac{1}{T}\hat{\alpha}_{T}(f)$ tend
to zero when T becomes large ($T\gg 1/f$). The exception is when $f$ is a
multiple of the laser repetition rate $f_{r}$. In this case
$\lim_{T\rightarrow\infty}\frac{1}{T}\hat{\alpha}_{T}(f)=1$ (Annex B). For
small values of the relative SRS modulation $\beta$, the terms containing
$\beta$ in equation 41 can be neglected, and the equation becomes:

$S_{I_{m}}(f)=\frac{g^{2}I_{avg}^{2}}{4}[S_{\alpha}(f-f_{0})+S_{\alpha}(f+f_{0})]+g^{2}\frac{qI_{avg}}{2}+\frac{g^{2}}{4}[S_{\epsilon}(f-f_{0})+S_{\epsilon}(f+f_{0})]$
(65)

Equation 65 is valid assuming no correlations between different frequencies in
the laser intensity noise $\alpha$, or the electric noise $\epsilon$.
Additionally, the lock-in also applies a low pass filter, meaning the only
relevant frequencies will be for $f<\Delta f\ll f_{0}$, where $\Delta f$ is
the filter bandwidth defined in equation 50. One can expect $S_{I}$ to be
slowly varying around $f_{0}$, and as a result this function will be
approximated it by its value in $f_{0}$. The single sided power spectral
density for the mixed current then reads:

$S^{+}_{I_{m}}(f)=\frac{g^{2}}{2}\Big{(}I_{avg}^{2}S^{+}_{\alpha}(f_{0})+2qI_{avg}+S^{+}_{\epsilon}(f_{0})\Big{)}=\frac{g^{2}I_{avg}^{2}}{2}\mathcal{RIN}_{I}(f_{0},I_{avg})$
(66)

##  X Supplementary figures

Figure 6:  Supplementary figure: Measured transfer function of the filter and
preamplification step. Figure 7:  Supplementary figure: Equivalent electronic
noise power spectral density $S^{+}_{\epsilon}(f)$.

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S-Dual of Maxwell Chern-Simons Theory

Adi Armoni

Department of Physics, Faculty of Science and Engineering
Swansea University, SA2 8PP, UK

<EMAIL_ADDRESS>

Abstract

We discuss the dynamics of three dimensional Maxwell theory coupled to a level
$k$ Chern-Simons term. Motivated by S-duality in string theory we argue that
the theory admits an S-dual description. The S-dual theory contains a non-
gauge 1-form field, previously proposed by Deser and Jackiw [1] and a level
$k$ $U(1)$ Chern-Simons term, ${\cal Z}_{\rm MCS}={\cal Z}_{\rm DJ}{\cal
Z}_{\rm CS}$. The couplings to external electric and magnetic currents and
their string theory realisations are also discussed.

##  1 Introduction

Chern-Simons theory is vastly used in mathematical physics, in condensed
matter physics and in string theory [2]. It was studied intensively in the
past three decades, yet the dynamics of Yang-Mills Chern-Simons theory is not
fully understood at the strong coupling regime.

Four dimensional S-duality is an exact duality between two ${\cal N}=4$ super
Yang-Mills theories, enabling us to calculate quantities in the strong
coupling regime using a dual weakly coupled theory. In the Abelian case it
reduces to the old electric-magnetic duality which swaps electric and magnetic
fields

$F\longleftrightarrow*F\,.$ (1.1)

In 3d Abelian S-duality relates the electric field to a dual scalar

$f\longleftrightarrow*d\phi\,.$ (1.2)

The purpose of this note is to extend S-duality to 3d Maxwell Chern-Simons
(MCS) theory, with either a compact or non-compact $U(1)$ gauge group. It
should hold on any spin manifold. The Lagrangian of the theory is given by

$L=-\frac{1}{2g^{2}}da_{e}\wedge*da_{e}+\frac{k}{4\pi}a_{e}\wedge da_{e}\,.$
(1.3)

MCS theory contains a vector boson of mass $M=\frac{g^{2}k}{2\pi}$. At low
energies the kinetic term is irrelevant and the theory flows to a pure level
$k$ Chern-Simons theory. As explained in section (3) the theory admits a
global $\mathbb{Z}_{k}$ 1-form symmetry generated by

$G\equiv\exp\left(i\oint(a_{e}-\frac{1}{M}*da_{e})\right)\,.$ (1.4)

When the theory is compactified on a torus, the global ${\mathbb{Z}_{k}}$
1-form symmetry is spontaneously broken, resulting in $k$ degenerate vacua.

Several attempts were made to find the S-dual of (1.3). In [1] Deser and
Jackiw proposed a ’self-dual model’ (SDM) which describes a massive vector.
While SDM describes a massive vector it does not admit a ${\mathbb{Z}_{k}}$
1-form symmetry neither does it flow to a pure Chern-Simons theory at low
energies, hence it cannot be an exact dual of MCS theory.

A closely related problem concerns the open string dynamics on a certain
Hanany-Witten brane configuration. It is well known [3, 4] that MCS theory
lives on the left brane configuration of fig.(1). Type IIB S-duality maps the
left configuration into the right configuration. Thus, knowing the field
theory that lives on the right configuration will solve the problem of finding
the S-dual. In early attempts [3, 4, 5] the authors found gauge theories with
a fractional level Chern-Simons term. While the theories they found are
classically equivalent to MCS, it cannot be the full answer, as it does not
admit the symmetries nor the same dynamics as the electric theory.

Fig. 1: The electric theory on the left brane configuration is Maxwell Chern-
Simons. The magnetic theory, obtained by type IIB S-duality, lives on the
right brane configuration.

##  2 Derivation of the duality

We may use 4d S-duality between Maxwell theories to derive the 3d duality. In
4d a pure Maxwell theory with a coupling $g$ is dual to a pure Maxwell theory
with a coupling $1/g$.

Consider the following partition function

${\cal Z}=\int DF_{m}DA_{e}\exp
i\int(-\frac{g^{2}}{2}F_{m}\wedge*F_{m}+F_{m}\wedge dA_{e})$ (2.1)

$A_{e}$ is the ’electric’ gauge field, $F_{m}$ is a ’magnetic’ gauge invariant
2-form. $g$ is the ’electric’ gauge coupling.

Upon integrating over $F_{m}$ we obtain the electric theory

${\cal Z}=\int DA_{e}\exp
i\int\left(-\frac{1}{2g^{2}}dA_{e}\wedge*dA_{e}\right)\,.$ (2.2)

If instead we integrate over $A_{e}$ we obtain

${\cal Z}=\int DF_{m}\delta(dF_{m})\exp
i\int\left(-\frac{g^{2}}{2}F_{m}\wedge*F_{m}\right)\,,$ (2.3)

hence it can be written in terms of $A_{m}$ such that $F_{m}=dA_{m}$.

${\cal Z}=\int DA_{m}\exp
i\left(-\frac{g^{2}}{2}dA_{m}\wedge*dA_{m}\right)\,.$ (2.4)

This is the magnetic theory dual to the electric theory.

Let us use dimensional reduction of (2.1) in order to derive the 3d duality.
Upon reducing to 3d the 4d 2-form $F_{m}$ becomes a 3d 2-form $f_{m}$ and
1-form $a_{m}$. The 4d gauge field $A_{e}$ becomes a 3d gauge field $a_{e}$
and a scalar $\phi_{e}$. The the 2-form $f_{m}$ and the scalar $\phi_{e}$
decouple from the rest of the action and admit

${\cal Z}=\int Df_{m}D\phi_{e}\exp
i\int(-\frac{g^{2}}{2}f_{m}\wedge*f_{m}+f_{m}\wedge d\phi_{e})\,,$ (2.5)

which leads to the well known S-duality

$d\hat{a}_{m}\longleftrightarrow*d\phi_{e}\,,$ (2.6)

where $f_{m}=d\hat{a}_{m}$.

Let us focus on the duality between $a_{e}$ and $a_{m}$, which is the prime
purpose of this note. We add to the action a Chern-Simons term111The Chern-
Simons term can be obtained by a dimensional reduction of a space dependent
theta term $\int d^{4}x\,\theta(x)F_{e}\wedge F_{e}$, such that
$\theta(x)=\frac{k}{4\pi}H(x^{3})$, where $H(x^{3})$ is the Heaviside step
function.. Our proposal is the following ’master’ partition function

${\cal Z}=\int Da_{m}Da_{e}\exp
i\int(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}+a_{m}\wedge
da_{e}+\frac{k}{4\pi}a_{e}\wedge da_{e})$ (2.7)

Note that $a_{m}$ is a gauge invariant 1-form. Upon integration over $a_{m}$
we obtain the electric theory

${\cal Z}=\int Da_{e}\exp
i\int(-\frac{1}{2g^{2}}da_{e}\wedge*da_{e}+\frac{k}{4\pi}a_{e}\wedge
da_{e})\,,$ (2.8)

namely Maxwell Chern-Simons theory.

In order to derive the magnetic theory we should use (2.7) and integrate over
$a_{e}$. This is a subtle point. Instead, let us use a change of variables
$a_{e}=b-\frac{2\pi}{k}a_{m}$, to obtain the following partition function

${\cal Z}=\int Da_{m}Db\exp
i\int(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}-\frac{\pi}{k}a_{m}\wedge
da_{m}+\frac{k}{4\pi}b\wedge db).$ (2.9)

Equation (2.9) is our proposal for the S-dual of Maxwell Chern-Simons theory.
The partition function of the magnetic theory is a product of the Deser-Jackiw
theory and a level $k$ Chern-Simons term

${\cal Z}_{\rm MCS}={\cal Z}_{\rm DJ}{\cal Z}_{\rm CS}\,.$ (2.10)

Note that $a_{m}$ is not a gauge field and therefore the term
$\frac{\pi}{k}a_{m}\wedge da_{m}$ is not ill-defined.

Both the electric and the magnetic theories describe a massive vector of mass
$M=\frac{g^{2}k}{2\pi}$ and a decoupled level $k$ Chern-Simons theory. Both
theories exhibit a 1-form ${\mathbb{Z}_{k}}$ symmetry.

Let us now provide another argument in favour of our proposal (2.9). We begin
with the magnetic brane configuration of fig.(1). It was argued by Gaiotto and
Witten [6] that the theory which lives on the intersection of the 3-brane and
the tilted 5-brane (without a D5 brane) is

${\cal Z}=\int DaDc\exp i\int(\frac{1}{2\pi}a\wedge dc+\frac{k}{4\pi}c\wedge
dc)\,.$ (2.11)

In order to understand what happens when we add a D5 brane, let us assume that
the terms that we need to add to the action are $k$ independent. Indeed, the
information about $k$ is encoded in the tilted fivebrane, not in the
threebrane. Let us use $k=0$, since in this case the duality is well
understood: the electric theory is pure Maxwell and the magnetic (mirror)
theory is a massless scalar. The brane realisation of the duality was provided
in the seminal work of Hanany and Witten [7].

We may write the theory of a free massless scalar as follows

${\cal Z}=\int DaDc\exp i\int(a\wedge*a+\frac{1}{2\pi}a\wedge dc)\,,$ (2.12)

with $a$ being a gauge invariant 1-form. The equation of motion for $c$ is
$*da=0$, namely that $a=d\chi$. Thus, for $k=0$ we obtain a theory of a free
scalar $(d\chi)^{2}$, as expected.

We found that adding a term $a\wedge*a$ to the action yields a theory that
describes the correct dual of Maxwell theory. We propose that $a\wedge*a$ is
the missing term in (2.11), namely that by adding it to (2.11) we obtain the
dual of MCS for any $k$. Note that

${\cal Z}=\int DaDc\exp i\int(a\wedge*a+\frac{1}{2\pi}a\wedge
dc+\frac{k}{4\pi}c\wedge dc)\,$ (2.13)

is almost identical to (2.7). An important difference is that Gaitto and
Witten introduced a gauge field $a$, whereas in (2.13) we added a term that
breaks gauge invariance. We may re-introduce gauge invariance in (2.13) by
transforming the fixed gauge vector $a$ into a gauge invariant term by adding
a scalar $\eta$ as follows

${\cal Z}=\int DaDcD\eta\exp
i\int((a-d\eta)\wedge*(a-d\eta)+\frac{1}{2\pi}a\wedge dc+\frac{k}{4\pi}c\wedge
dc)\,\,,$ (2.14)

such that under a gauge transformation $a\rightarrow
a+d\lambda,\eta\rightarrow\eta+\lambda$, with $a$ a $U(1)$ gauge field.
Eq.(2.13) may be viewed as the fixed gauge version of eq.(2.14) with
$d\eta=0$.

Our proposal (2.9) passes all the requirements from a dual theory: it admits a
${\mathbb{Z}_{k}}$ global symmetry, it flows to pure Chern-Simons theory in
the IR, it contains a massive vector of mass $M$ and, finally, when $k=0$ it
agrees with the results of Hanany and Witten [7]. As we shall see the brane
realisations of both electric and magnetic theories predict the existence of
$k$ degenerate vacua.

We summarize this section by writing the precise map between the electric and
magnetic variables using (2.7)

$\displaystyle-g^{2}a_{m}=*da_{e}$ (2.15) $\displaystyle
b=a_{e}-\frac{1}{M}*da_{e}$ (2.16)

or

$a_{e}=b-\frac{2\pi}{k}a_{m}$ (2.17)

##  3 Comments on ${\mathbb{Z}_{k}}$

Let us introduce a Wilson loop in MCS theory. We wish to measure the
${\mathbb{Z}_{k}}$ charge of the loop, namely the number of fundamental
strings, $n$, that pass through a a certain contour $C$. We will define an
operator $G$ such that

$GW_{n}=\exp(i\frac{2\pi n}{k})W_{n}\,,$ (3.1)

with $W_{n}$ a Wilson loop of charge $n$, $W_{n}=\exp(in\oint a_{e})$.

In order to the define $G$, let us consider the equation of motion in MCS

$d(\frac{1}{g^{2}}*da_{e}-\frac{k}{2\pi}a_{e})=j_{e}\equiv dJ_{e}\,,$ (3.2)

where $J_{e}$ is the integral of the electric current $j_{e}$ over a disc $D$
such that $C=\partial D$. The setup is depicted in fig.(2). By integrating
(3.2) we learn that

$\frac{1}{g^{2}}*da_{e}-\frac{k}{2\pi}a_{e}=J_{e}\,.$ (3.3)

Fig. 2: A Wilson loop passing through a domain $D$ (shaded region) whose
boundary is $C=\partial D$.

We can therefore define a generator of a ${\mathbb{Z}_{k}}$ symmetry as
follows

$G=\exp\left(i\frac{2\pi}{k}\oint_{C}(\frac{k}{2\pi}a_{e}-\frac{1}{g^{2}}*da_{e})\right)=\exp\left(i\oint_{C}b\right).$
(3.4)

Note that the implication of ${\mathbb{Z}_{k}}$ symmetry is a symmetry
$n\rightarrow n+k$, namely that a collection of $k$ strings is topologically
isomorphic to a singlet, namely to no strings at all. This is supported by
string theory: suppose that we attempt to place the endpoints of $k$
coincident strings on the worldvolume of the D3 brane. The collection of $k$
fundamental strings can transform itself into an anti D-string and a $(k,1)$
string. Instead of ending on the worldvolume of the D3 brane, the D-string can
end on an NS5 brane and a $(k,1)$ string can end on a $(1,k)$ fivebrane. Thus
string theory predicts that a collection of $k$ strings can be removed from
the worldvolume of the 3d gauge theory. A similar phenomenon happens in the
magnetic dual if we attempt to introduce $k$ coincident D-strings in the
worldvolume of the magnetic theory.

When the theory is defined on the torus the ${\mathbb{Z}_{k}}$ symmetry is
broken, resulting in $k$ vacua [8][9]. An intuitive explanation is as follows:
The level $k$ $U(1)$ Chern-Simons theory is equivalent (using level-rank
duality) to a level $1$ $SU(k)$ theory, that admits a ${\mathbb{Z}_{k}}$
centre symmetry. When it is defined on the torus the $SU(k)$ theory
deconfines, resulting in $k$ degenerate vacua, parameterized by the
eigenvalues of the ’t Hooft loop.

The $k$ vacua manifests themselves in both the electric and magnetic brane
configurations as follows: the D3 brane may end on any of the $k$ constituents
of the fivebranes. Each one of the $k$ choices corresponds to a vacuum.

##  4 Coupling to external sources, Wilson and magnetic loops

Consider the coupling of the electric gauge field to a source $j_{e}$, namely
$a_{e}j_{e}$. It translates to the coupling $(b-\frac{\pi}{k}a_{m})j_{e}$ in
the magnetic side. We therefore suggest that the Wilson loop

$W_{e}=\exp i\oint a_{e}$ (4.1)

in the electric side is mapped into a magnetic loop of the form

$M_{m}=\exp i\oint(b-\frac{2\pi}{k}a_{m})$ (4.2)

in the magnetic side.

We may use the above map between the Wilson loop in the electric side and its
magnetic counterpart to study the dynamics of 3d QED-CS. Using the worldline
formalism [10] we can write the partition function of MCS theory coupled to
$N_{f}$ massless fields as follows

${\cal Z}_{\rm QED-CS}=\int Da_{e}\exp(iS_{\rm
MCS})\sum_{n}\frac{(N_{f}\Gamma_{e})^{n}}{n!}$ (4.3)

with

$\Gamma_{e}=\int\frac{dt}{t^{\frac{5}{2}}}\int
Dx\exp(-\int_{0}^{t}d\tau(\dot{x})^{2})\exp i\oint a_{e}$ (4.4)

The duality transformation yields the following partition function

${\cal Z}_{\rm magnetic}=\int Da_{m}Db\exp(iS_{\rm DJ-
CS})\sum_{n}\frac{(N_{f}\Gamma_{m})^{n}}{n!}$ (4.5)

with

$\Gamma_{m}=\int\frac{dt}{t^{\frac{5}{2}}}\int Dx\exp(-\int
d\tau(\dot{x})^{2})\exp i\oint(b-\frac{2\pi}{k}a_{m})\,.$ (4.6)

It suggests that the dynamics of QED with $N_{f}$ massless flavours is
captured by a dual DJ-CS theory coupled to $N_{f}$ massless ’monopoles’. The
precise coupling of $a_{m}$ and $b$ to the monopoles is given by (4.6). We may
write the dual magnetic theory in a more ’standard’ form

${\cal Z}=\int Da_{m}DbD\bar{\psi}_{m}D\psi_{m}\exp iS_{\rm magnetic}\,,$
(4.7)

with $S_{magnetic}$ given by

$S_{magnetic}=\int\left(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}-\frac{\pi}{k}a_{m}\wedge
da_{m}+\frac{k}{4\pi}b\wedge
db+\bar{\psi}_{m}\gamma\wedge\star(i\partial+b-\frac{2\pi}{k}a_{m})\psi_{m}\right)\,.$
(4.8)

It is interesting to note that the QED-CS theory is mapped to a theory of
interacting massless magnetic ’monopoles’, with a coupling $1/gk$. Thus, when
the electrons couple strongly to $a_{e}$, the ’monopoles’ couple weakly to
$a_{m}$ and we may use perturbation in theory in the magnetic side to study
the strongly coupled electric theory.

Following Itzhaki[11] let us define a magnetic (“disorder”) loop in the
electric theory

$M_{e}=\exp\left(i\oint_{C}(ka_{e}-\frac{2\pi}{g^{2}}*da_{e})\right)\,,$ (4.9)

which is mapped into the electric loop in the magnetic side

$W_{m}=\exp\left(ik\oint_{C}b\right).$ (4.10)

The magnetic loop in the electric side and the electric (Wilson) loop in the
magnetic side are trivial [11].

Fig. 3: Rectangular Wilson loops can be realised in string theory by ending a
pair of F-string and anti F-string on the the threebrane. Similarly
Rectangular ’t Hooft loops can be realised by ending a pair of D-string and
anti D-string on the three brane. The end of the F-string represents a heavy
quark, whereas the end of the D-string represents a heavy monopole.

We suggest that a rectangular Wilson loop (or magnetic loop) should be
identified with the end points of an F-string and anti F-string (or a D-string
and anti D-string) that end on the threebranes of fig.(3).

A D-string can end on an NS5 brane instead of a threebrane, hence the magnetic
loop in the electric theory should be trivial. Similarly, a F-string can end
on a D5 brane instead of a threebrane, hence a Wilson loop in the magnetic
theory should be trivial. This is consistent with our definitions of the
magnetic loop (4.9) and the Wilson loop (4.10).

##  5 Summary

The purpose of this note is to find the S-dual of MCS theory. We found that
the dual theory (2.9) contains a non-gauge vector of mass $M$ and a decoupled
pure TQFT. The magnetic theory nicely captures the dynamics of the electric
theory: a theory with a mass gap that flows in the IR to a TQFT. The duality
we uncovered in this note is a precise manifestation of the duality between a
topological insulator and a topological superconductor outlined in ref.[12].

It will be interesting to find the S-dual of the non-Abelian $U(N)$ theory
that lives on a collection of $N$ coincident $D3$ branes, suspended between
tilted fivebranes. The master field of that theory may be obtained by
replacing the Abelian 1-forms of (2.7) by non-Abelian 1-forms as follows222I
thank Shigeki Sugimoto for suggesting that.

${\cal Z}=\int Da_{m}Da_{e}\exp i\,{\rm
tr}\int(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}+a_{m}\wedge(da_{e}+a_{e}\wedge
a_{e})+\frac{k}{4\pi}(a_{e}\wedge da_{e}+\frac{2}{3}a_{e}\wedge a_{e}\wedge
a_{e}))$ (5.1)

together with $a_{e}=b-\frac{2\pi}{k}a_{m}$. Other dualities that involve
$SO/Sp$ (and an orientifold in string theory) could also be derived. The
generalisation to supersymmetric QED/QCD theories with a CS term [5] is also
interesting and can be written down using the worldline formalism, as in
section (4). The duality found in this letter is useful to study the strong
coupling regime of those theories.

Finally, it is well known that MCS theory admits Seiberg duality. The
manifestation of the duality using an exchange of fivebranes in the magnetic
theory, might teach us about fivebranes dynamics.

### Acknowledgements

A.A. would like to thank the Yukawa institute for theoretical physics, where
part of this work was done, for warm hospitality. I would also like to thank
Mohammad Akhond and Shigeki Sugimoto for numerous discussions and
collaboration. I am grateful to Anton Kapustin and Zohar Komargodski for
reading and commenting on a draft version of this paper.

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