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\begin{document}

%\preprint{APS/123-QED}

\title{Sgnal-to-noise ratio gain by shot noise separation\\using interferometric spatial mode filtering}
%\title{Dark-Field Separation of Optical Modes in a Thermal Point Source}
%\title{Towards Luminosity Transit Coronagraphy by Image Inversion}

%\author{Zhengxin Liang$^{1}$}
%\author{Gillian Si Ying Foo$^{1}$}
\author{Peng Kian Tan$^{1}$}
\email{cqttpk@nus.edu.sg}
\author{Christian Kurtsiefer$^{1,2}$}
\affiliation{$^{1}$Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\affiliation{$^{2}$Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore, 117551}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
  Dark-field techniques are successfully used in microscopy for increasing
  the contrast of almost transparent objects, and for edge detection by
  removing image components with low spatial frequencies. A removal of image
  components with little interesting information but high intensity yields a
  higher signal-to-noise ratio for the image components of interest.
  Here, we present a technique to enhance the signal-to-noise ratio of an
  image signature contained in a spatial asymmetry. While being an
  interferometric technique based on image inversion, it will work with
  incoherent light sources, and thus be applicable in many practical imaging
  scenarios. We experimentally demonstrate an increase of the signal-to-noise
  ratio in edge asymmetry detection by an order of magnitude in a
  proof-of-principle experiment.
\end{abstract}

\keywords{Dark-Field, Edge Detection, Image Inversion, Incoherent Light, Spatial Asymmetry}%Use showkeys class option if keyword display desired

\maketitle


\section{Spatial Mode Filter}
A century ago, dark-field imaging in microscopy enabled the convenient observation of objects that scatter very little light by removing illumination components from the observation path that are not scattered by a specimen in a smart optical configuration \cite{lister:30}. As a result, the image formed by the light scattered from an object has a much higher contrast, or signal-to-noise ratio, than the corresponding image in a bright-field illumination.

In a more modern description, one could view this as a spatial filtering technique, that removes low frequency components of an image that contain little image information, but contribute to the background and thus noise in an observation. Dedicated spatial optical high pass filters can be used to enhance the edges of an object like the border of an almost transparent cell. Especially for objects with very little scattering, the removal of unscattered light might be the only way to have the signal level exceed the the noise level, ultimately given by the photon shot noise, of an unscattered background \cite{guyon:99, itoh:17, currie:20, miller:21}.

In the same spirit, we present an image inversion technique to enhance the signal-to-noise ratio in images where the information of interest is contained in a deviation from a spatial symmetry. The technique relies on an interferometric removal of symmetric image components by splitting the image information into two paths,
and inverting the image in one of the paths. The interferometer can be arranged such that symmetric image components leave the interferometer through one port, while asymmetric image components are passed to the other port. 

Similar techniques based on image inversion \cite{mankei:16} have been predicted to outperform direct imaging in resolving sub-Rayleigh sources \cite{mankei:19, tang:16}, including for thermal light with a faint secondary source \cite{zixin:21, allouche:09}. Implementations have also been suggested in super-resolution microscopy \cite{pushkina:21}, and towards astronomical coronagraphs \cite{ruane:24, tavrov:07, tavrov:11, tavrov:18}.



\section{Interferometer Setup}
In our proof-of-principle demonstration illustrated in Fig.~\ref{fig:setup}, we show that this interferometric removal of symmetric image content works for non-laser illumination sources, and thus might be of interest in applications seeking to identifiy an exoplanet passing by a much brighter star in astronomy, or in precision manufacturing process control.

\begin{figure}[h!]
\centering\includegraphics[width=8.6cm]{setup.eps}
\caption{\label{fig:setup} Experimental setup for a Mach-Zehnder interferometer with radial image inversion. SMF: single-mode fiber, Pol: linear polariser, SPAD: single-photon avalanche detector.  The setup is aligned to an interferometric visibility of $\mathrm{V=96\,\%}$.}
\end{figure} %Fig.~\ref{fig:setup}. 
%, with an input light of $N_{S}-N_{A}$ intensity, and the bright-field and dark-field outputs at $N_{S}-1.5N_{A}$ and $0.5N_{A}$ intensities respectively.

The non-laser incoherent illumination is provided by a thermal light source generated from a low-pressure Mercury vapor gas discharge lamp. The light is spectrally filtered by a Fabry-P\'{e}rot etalon at the 546.1\,nm emission line to increase its temporal coherence to around 0.5\,ns for a corresponding coherent length of about 15\,cm. This relaxes the requirement for zero optical path difference in the interferometer. 

The light beam is then transmitted through a single-mode optical fiber to project into the fundamental Gaussian mode for spatial coherence, which is then collimated to a beam diameter around 2\,mm.
The beam is passed through a polariser to increase its interferometric visibility as orthogonal polarisation states do not interfere.

A non-polarising beamsplitter separates the incident beam into two, with one half passing through a pair of identical achromatic plano-convex lenses which perform the radial image inversion.
The other half is passed through a 6.35\,mm thick glass block to compensate for the optical path increase due to the two lens substrates.

The two beams are recombined at a second beamsplitter, with a 0.3\,mm glass cover slip rotated to optimise for constructive and destructive interference fringes at the bright-field and dark-field output ports respectively, resulting in an interferometric visibility $\mathrm{V=96\,\%}$.


\section{Gain in Signal-to-Noise Ratio}An edge asymmetry is introduced into the pupil plane of the optical path by a knife edge obstruction. The knife is linearly translated into the beam path in $10\,\mu$m steps from 0\,mm to 2\,mm, increasing the asymmetric image components, or optical modes, by $-N_{A}$ intensity. (Negative sign due to obstruction of light intensity)

The light intensities at the bright-field and dark-field output ports are measured by a pair of Silicon actively quenched single-photon avalanche detectors with a dark count rate of 1000\,photoevents per second, and recorded by a field programmable gate array timestamp module.

\begin{figure}[h!]
\centering\includegraphics[width=8.6cm]{obstruction.eps}
\caption{\label{fig:obstruction}Top: the output photoevents rate at the dark-field port $0.5N_{A}$ (red), bright-field port $N_{S}-1.5N_{A}$ (blue), and combined intensity of $N_{S}-N_{A}$ (black). The combined intensity begins with only symmetric $N_{S}$ modes due to the Gaussian beam from the single-mode optical fiber being radially symmetric. Bottom: the corresponding signal-to-noise ratios (SNR) for the edge asymmetry measurements with (red) versus without (black) image inversion.}
\end{figure} %Fig.~\ref{fig:obstruction}.

When the obstruction position of the knife edge increases as shown in Fig.~\ref{fig:obstruction}, the asymmetric spatial optical modes increasingly populate the dark-field output as the incident light decreases in radial symmetry about the optical axis of the image inversion lens pair.

The dark-field output increases at a rate of $0.5\,N_{A}$ due to the beamsplitting ratio of $50:50$, because asymmetric spatial optical modes do not overlap after image inversion.
The bright-field output decreases at a rate of $1.5\,N_{A}$ due to the other half of the beamsplitting ratio presenting $0.5\,N_{A}$ as before, in addition to the knife edge obstruction itself contributing another $1.0\,N_{A}$ factor.

At the half-way obstruction position around 1\,mm, the two output intensities balance and then start to decrease at the same rate, as there is no longer any interferometric overlap by image inversion, and the light is therefore just transmitting through the second beamsplitter directly.

\begin{figure}[h!]
\centering\includegraphics[width=8.6cm]{shotnoise.eps}
\caption{\label{fig:obstruction}Shot noise separation. The signal of the asymmetric modes is separated from the shot noise contribution by the symmetric modes, thereby increasing its signal-to-noise ratio.}
\end{figure} %Fig.~\ref{fig:shotnoise}.

The expected gain $G$ in the signal-to-noise ratio (SNR) for measuring the asymmetric optical modes $N_{A}$ contribution may be described by the ratio between the signal-to-noise ratio with against without image inversion, or SNR$_{A}$ against SNR$_{S+A}$ respectively, as described by:
\begin{equation} \label{eqn:snr}
\begin{split}
G & = \frac{SNR_{A}}{SNR_{S+A}}\\
& = \frac{(0.5 \times N_{A})/\sqrt{0.5 \times N_{A}}}{N_{A}/\sqrt{N_{S} \pm N_{A}}}\\
%& = \frac{\sqrt{0.5 \times (N_{S} \pm N_{A})}}{\sqrt{N_{A}}}\\
& = \sqrt{0.5 \times \left( \frac{N_{S}}{N_{A}} \pm 1 \right)}
\end{split}
\end{equation}

The asymmetric spatial optical modes do not overlap with themselves upon image inversion, and so transmit through the $50:50$ beamsplitter without interference, resulting in the factor 0.5. The sign of $\pm$ depends on whether the edge asymmetry in image components is due to a light source (+), or an obstruction (--) as demonstrated in our proof-of-principle experiment.

\begin{figure}[h!]
\centering\includegraphics[width=8.6cm]{gain.eps}
\caption{\label{fig:gain}Gain G in the signal-to-noise ratio of edge asymmetry detection in the image components (red points) in the experiment, compared to the model described by Eqn.~\ref{eqn:snr} (black line).}
\end{figure} %Fig.~\ref{fig:gain}.

As shown in Eqn.~\ref{eqn:snr}, the gain $G$ in SNR increases inversely with the population of asymmetric modes $N_{A}$, and is thus more significant for small asymmetric contributions $N_{A}$ against a bright point source $N_{S}$. The gain in signal-to-noise ratio reduces to unity when the asymmetric-to-symmetric modes ratio is as follows:
\begin{equation} \label{eqn:ratio}
N_{A}= -\frac{1}{3}N_{S}, \frac{1}{2}N_{S}
\end{equation}

The measured gain in the signal-to-noise ratio for detecting asymmetric optical modes is shown in Fig.~\ref{fig:gain}. It reduces to 1 as predicted by Eqn.~\ref{eqn:ratio} when the asymmetric modes $N_{A}$ by obstruction is about $1/3$ of the source original intensity $N_{S}$ as a radially symmetric point of light emitting from a single-mode optical fiber.



\section{Outlook}
This work may be improved towards zero-path optical difference and lens-less configuration to be achromatic, which would allow to input blackbody radiation and other broadband light sources.  This could be useful for increasing the signal-to-noise ratio of follow-up measurements, e.g. spectroscopy, of very faint light sources embedded in a much brighter and noisy environment.


\section{Acknowledgments}
This research is supported by the Quantum Engineering Programme through NRF2021-QEP2-03-P02, the Ministry of Education and the National Research Foundation, Prime Minister's Office, Singapore.


\bibliography{HBT}



%\section{Appendix}
%$0.5 \times N_{S}/N_{A} \pm 1 = 1$\\
%$N_{S}/N_{A} \pm 1 = 2$\\
%$N_{S}/N_{A}= 3$ or $2$\\
%$N_{S}= 3N_{A}$ or $2N_{A}$


\end{document}
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