# End-to-End Jet Classification of Quarks and Gluons with the CMS Open Data

M. Andrews1, J. Alison1, S. An1,2, B. Burkle3, S. Gleyzer4, M. Narain3, M.
Paulini1, B. Poczos5, E. Usai3 1 Department of Physics, Carnegie Mellon
University, Pittsburgh, USA  2 CERN, Geneva, Switzerland  3 Department of
Physics, Brown University, Providence, USA  4 Department of Physics and
Astronomy, University of Alabama, Tuscaloosa, USA  5 Machine Learning
Department, Carnegie Mellon University, Pittsburgh, USA

###### Abstract

We describe the construction of novel end-to-end jet image classifiers to
discriminate quark- versus gluon-initiated jets using the simulated CMS Open
Data. These multi-detector images correspond to true maps of the low-level
energy deposits in the detector, giving the classifiers direct access to the
maximum recorded event information about the jet, differing fundamentally from
conventional jet images constructed from reconstructed particle-level
information. Using this approach, we achieve classification performance
competitive with current state-of-the-art jet classifiers that are dominated
by particle-based algorithms. We find the performance to be driven by the
availability of precise spatial information, highlighting the importance of
high-fidelity detector images. We then illustrate how end-to-end jet
classification techniques can be incorporated into event classification
workflows using Quantum Chromodynamics di-quark versus di-gluon events. We
conclude with the end-to-end event classification of full detector images,
which we find to be robust against the effects of underlying event and pileup
outside the jet regions-of-interest.

##  1 Introduction

The study of jet substructure at the CERN Large Hadron Collider (LHC) has
played an instrumental role in the understanding of the standard model (SM) of
particle physics through the analysis of jets produced from Quantum
Chromodynamics (QCD) [1, 2] and from the decay of boosted heavy resonances or
particles such as the top quark [3, 4, 5, 6] or the Higgs boson [7, 8].
Furthermore, jet substructure analysis remains a central tool in searches for
physics beyond the standard model (BSM) involving boosted heavy new resonances
(see [9, 10] for comprehensive reviews). A major component of understanding
jets arising from boosted heavy resonances, exotic or other BSM physics
processes, is a detailed understanding of the vastly more dominant QCD jet
background. For this reason, the characterization and discrimination of light
quark- versus gluon-initiated jets that comprise QCD jets has also been
extensively examined [11, 12]. Indeed, a number of BSM resonances involve
preferential decays to either quarks or gluons [13], making their study
valuable in their own right. The development of effective tools to identify
gluons from their light-quark counterparts—and to analyze jet substructure in
general—has become a popular topic of study as a result.

Differences in quark versus gluon jet radiation patterns arise from their
distinct QCD color charges: gluon initiated jets carry a larger QCD color
factor leading to a shower evolution with a higher branching probability [14,
15]. The gluon jet thus exhibits a slightly softer and broader radiation
pattern of higher particle multiplicity relative to a quark jet. To better
appreciate the recent experimental advances in jet substructure discrimination
that exploit this signature, it is instructive to sketch the jet finding and
reconstruction process at the typical LHC experiment. Fundamentally, data
recorded with the LHC experiments exists as detector-level information, either
as energy depositions in calorimeter cells or “hits” in tracking systems such
as pixel or strip detectors. Through a largely rule-based “particle-flow”
algorithm [16, 17], the detector data from the various subdetectors are
combined and processed to reconstruct the particle-level information
corresponding to the 4-momenta of all the resolved particles originating from
a proton-proton collision at the LHC. A jet clustering algorithm [18] is then
applied to group nearby particles into a jet object. A number of jet-level
features can be derived to characterize the jet as a whole, such as its shape
and particle multiplicity. At each stage of the jet reconstruction chain, a
level of abstraction occurs to idealize the representation of the data. While
this simplifies their interpretation, it potentially involves information loss
that may be of disadvantage for more exhaustive BSM searches, where exotic
decay signatures might be expected.

While earlier techniques in jet substructure analysis involving LHC-like data
began with jet-level variables [1, 2, 19], recent advances in machine learning
(ML)—particularly in deep neural networks (NN)—have permitted classification
algorithms to directly probe the particle-level data constituting the jet. For
a review of current state-of-the-art ML-based jet classification see [20, 21].
A number of non-trivial challenges have arisen from such approaches, for
example, the order in which the particle-level data should be presented to the
classifier, and how variable numbers of particles are handled. Especially for
heavy resonance decays, classifier performance may be particularly sensitive
to the choice of the ordering scheme. To address these issues, solutions have
been proposed involving artificial but physically-motivated ordering schemes
[6, 22, 23, 24, 25, 26, 27, 28], as well as those utilizing order-invariant
algorithms [29, 30]. Another highly popular approach has been to exploit the
spatial distribution of the particles in the detector as a natural solution to
the ordering problem. Inspired by the immense success of convolutional neural
networks (CNN) in computer vision [31], such “jet images” have been created by
pixelating the particle-level data into a grid that amounts to a coarse-
grained histogram-like image of the underlying detector geometry, which can
then be fed to a CNN [11, 32, 33, 34]. However, as the pixelation process
itself involves an additional, lossy operation, such attempts have been noted
to underperform relative to algorithms that directly use particle-level data,
even when artificial ordering is involved [12, 26].

However, the ultimate information bottleneck in all of the above particle-data
based approaches is the rule-based particle reconstruction algorithm itself.
Most of the breakthrough success in applying CNNs to computer vision have come
from bypassing rule-based “feature engineering” altogether and instead
allowing CNNs to learn relevant features directly from the raw camera data.
Therefore, this fact motivates the idea of applying “end-to-end physics
classification” whereby CNNs are used to directly train on maps of the true
detector-level data (or their simulated equivalents), in all their richness,
_before_ any particle processing is performed. In theory, this gives the
classifier full access to the maximum recorded event information at a level
not achievable with processed particle- or jet-level data, while avoiding the
particle ordering problem altogether. Thus, while the end-to-end approach
bears a resemblance to existing jet image techniques, the underlying
information content is fundamentally different. Among simpler neutrino
experiments, this end-to-end strategy has quickly found success [35, 36], but
among the more mature LHC experiments, adoption is still in its infancy. Its
success is by no means guaranteed, due to the much more complex detector
systems and workflows of hadron collider experiments.

In previous work [37, 38], we laid the basic foundation for performing end-to-
end physics classification on complex LHC detectors like the Compact Muon
Solenoid (CMS) using electromagnetic objects. In this paper, we apply the same
techniques to the problem of jet substructure classification, again using
simulated CMS Open Data taking advantage of this excellent public resource, as
others have [19, 39]. We study the classification of jets initiated by quarks
versus gluons from QCD dijet production and use image windows around the jet
region-of-interest (ROI), or jet-view images, as done by the cited work above.
While the image representation of the tracking information in Ref. [38] is
simplistic, for our purposes, these images are not expected to significantly
impact the classification of quark versus gluon jets that generally do not
contain displaced secondary track vertices. We thus benchmark our results
against one of the current state-of-the-art jet classifiers based on particle-
level data [26].

In addition, we consider the event classification of the full dijet process as
either di-quark or di-gluon initiated. A number of potential implementations
for this approach are explored: from more standard workflows, where the jet
classification is factorized from the event classification, to a unified event
classification workflow, where a single training is performed on the full
detector-view images, analogous to [38] (and to some degree, also [39, 40,
41]). While dijet decays are simple enough that jet ordering becomes trivial,
they are useful as a pedagogical example to highlight the ability of end-to-
end event classifiers on full detector-view images to completely capture
discriminating features at both the jet- and event-level scales in the limit
that the event topology can be fully and intuitively modeled by hand. A more
general application demonstrating the performance of this technique for the
case of complex multi-body decays in which both ordering and modeling are non-
trivial is left for future work.

This paper is arranged as follows. In Section 2, we introduce our data sample
and event selection, while in Section 3, we briefly describe the CMS geometry
and our jet image construction procedure. In Section 4, we outline our network
architecture and training strategy. Finally, in Sections 5 and 6, we present
the results of our jet and event classification studies, respectively, and
summarize our conclusions in Section 7.

##  2 Open Data Simulated Samples

We use simulated 2012 CMS Open Data [42] for all the studies presented in this
paper. The CMS Open Data are ideally suited to the end-to-end approach due to
their use of the Geant4 [43] package, which delivers the state-of-the-art in
first-principles detector simulation, together with the most detailed geometry
models of the CMS detector available [44].

Both quark and gluon samples are derived from the QCD dijet production dataset
with a generated invariant transverse momentum in the range
$\hat{p}_{T}=90-170\,\mathrm{GeV}$ [45]. The events are generated and
hadronized with the Pythia 6 [46] package with the Z2* tune, which accounts
for the differences in the quark versus gluon shower evolution. The samples
also account for the multi-parton interactions from the underlying event and
have run-dependent pileup (PU) ranging from a peak average PU of
$\langle\mathrm{PU}\rangle=18-21$.

We impose a basic event selection on the dijet events for them to be used and
categorized under the quark or gluon class label, for both jet and event
classification. First, we require that the two outgoing partons from the
Pythia hard-scatter event be either both light quarks $q_{l}$, where $l=u,d,s$
(or their antiparticles), or both gluons, otherwise the event is discarded. In
addition, each of the partons must be matched to a reconstructed jet to within
a cone of $\Delta R<0.4$, where $R$ is the angular separation in the
pseudorapidity-azimuthal ($\eta-\phi$) plane. Jets with wide-angle radiation
are thus excluded from this study, for simplicity. In turn, each of the
parton-matched reconstructed jets must have a minimum transverse momentum of
$p_{T}>70\,\mathrm{GeV}$ and pseudorapidity of $|\eta|<1.8$. For the jet
identification studies, only the leading-$p_{T}$ jet in each passing event is
used.

For computational convenience, we only use a subset of the total dijet dataset
of the CMS Open Data—we found no statistically significant advantage to
training on the full dataset. In addition, to minimize training bias due to
unbalanced class or PU representation, we truncate the dataset to contain a
balanced proportion of samples per class per PU run era (Run2012AB, Run2012C,
Run2012D), as summarized in Table 1. We therefore obtain a grand total of
793900 samples for training and validation, and 139306 samples for the final
test set. This thus gives our results in Sections 5 and 6 a statistical
uncertainty of about $0.4\%$, assuming Poisson statistics.

Category | Run2012AB | Run2012C | Run2012D
---|---|---|---
Train+Validation | 107778 | 148990 | 140182
Test | 18136 | 23770 | 27747
Table 1: Number of training events per class passing event selection, by PU
run era. Only the leading-$p_{T}$ jet in the event is used for the jet
classification studies.

##  3  CMS Detector and Images

The CMS detector is arranged as a series of concentric cylindrical sections
split into a barrel section and two circular endcap sections. The innermost
sections comprise the inner tracking system for identifying charged particle
tracks. This is then enclosed by the electromagnetic calorimeter (ECAL) which
measures energy deposits from electromagnetic particles, followed by the
hadronic calorimeter (HCAL) which measures energy deposits from hadrons.
Finally, the calorimeters are enclosed by the outer tracking system used to
identify muons.

The CMS Open Data contains the calibrated, reconstructed hits [47] of the ECAL
and HCAL at the crystal- and tower-level, respectively. Following the image
construction techniques in [38], we are able to form calorimeter images whose
pixels correspond exactly to physical crystals or towers. Due to the
complexity of the CMS tracking detectors and the absence of a reconstructed
hit collection for tracks in the CMS Open Data, the track information can only
be approximated with the results of the reconstructed track fits that are the
only track collection present in the CMS Open Data. An ECAL-like granularity
image grid is thus filled with the ($\eta$,$\phi$)-positions of each
reconstructed track with pixel intensity equal to the track’s $p_{T}$. A study
into more accurate representations of the tracking information is reserved for
future work. In practice, as long as the track image resolution is comparable
to the effective resolution of the reconstructed track position, any loss in
information from using a discrete image grid for the track positions is
expected to be negligible. The full detector-view images therefore consist of
three subdetector channels: one each for ECAL, HCAL, and the reconstructed
tracks.

More specifically, for the barrel sections, the images are resolved in the
finer ECAL barrel (EB), with the HCAL barrel (HB) hits up-sampled to match.
The reconstructed track hits span the width of a single EB crystal. The
difference in segmentation between the ECAL endcaps (EE) ($iX,iY$) and the
HCAL endcaps (HE) ($i\eta,i\phi$) imposes a constraint on the construction of
high-fidelity, multi-channel detector images. As explained in [38], we devise
two image geometry strategies: one where the EE segmentation is preserved and
the HE hits are projected onto an ($iX,iY$) grid (ECAL-centric), and another
where the HE segmentation is preserved and the ECAL hits are projected onto an
($i\eta,i\phi$) grid but at a finer EB-like granularity (HCAL-centric). In
either case, the track hits appear as isolated pixels in the corresponding
segmentation. As illustrated in Figure 1(a), this gives us a contiguous
detector image of resolution $\Delta i\eta\times\Delta i\phi=280\times 360$ of
ECAL barrel-like granularity, spanning the pseudorapidity range $|\eta|<3$.
For the entirety of this paper, we use only the HCAL-centric geometry strategy
to simplify the construction of jet image windows. We found no significant
benefit to appending additional wrap-around pixels along the $\phi$ edges of
these full detector-view images. For the jet-view images, this is described
below.

(a) Composite full detector-view image. Image resolution: 280 $\times$ 360.

(b) Composite individual jet-view images for leading jet (left) and sub-
leading jet (right) in a representative dijet event. Image resolution: 125
$\times$ 125.

Figure 1: Representative dijet event in HCAL-centric geometry at EB-like
granularity: full detector-view image (1(a)) spanning the range $|\eta|<3$,
and individual jet-view images (1(b)) spanning $\Delta R\sim 1$. Images are
multi-channel composites of information from tracks (orange), ECAL (blue), and
HCAL (gray), all in log-scale.

The process of creating a jet-view image window given a reconstructed jet
passing the event selection is as follows: from the full multi-subdetector
image described above, we localize the jet by first taking the centroid of the
reconstructed jet, then scan for the HCAL tower with the highest energy
deposit within a window of $9\times 9$ HCAL towers ($45\times 45$ image pixels
or $\Delta R\lesssim 0.4$). The most energetic HCAL tower defines the center
of the jet-view image around which we crop a window of $125\times 125$ image
pixels ($\Delta R\lesssim 1$), as illustrated in Figure 1(b). For jets which
fall near the $\phi$ edge of the detector image, we do a wrap-around padding
so that the jet shower appear seamless and the jet-view image dimensions are
preserved. We do not pad in the $\eta$ direction although this is, of course,
a possibility. This imposes an effective pseudorapidity cut on the
reconstructed jet of about $|\eta|<1.57$. While the jet-view window sizes here
were chosen to be fairly generous, future applications may choose to optimize
the window sizes further to extend the jet pseudorapidity range.

To gain a better intuition for the jet images, we present a number of
visualizations: Figure 2 shows the various subdetector image overlays averaged
over the full test set of about 70k jets for each class (described in Table
1), while Figure 3 shows subdetector images for a single jet. Visually, one
notes two main differences from previous, particle-based jet images (e.g. [34,
11]). First, the end-to-end images appear noticeably more “raw” in that they
contain more noise and stray hits. This is, of course, expected given that we
are looking at the (simulated) physical detector deposits in all their
richness. Importantly, this provides a prime setting for the classifier to
learn an internal noise-mitigation strategy, potentially allowing it to
maintain performance even under higher PU conditions. This PU robustness would
be greatly diminished if one were to instead train on highly pruned jet
images, which could themselves be stripped of meaningful hits. Second, the
end-to-end jet images are rendered in the finer ECAL-like granularity as
opposed to the coarser HCAL-like granularity more commonly used in previous
work. As the results in Section 5 show, quark versus gluon discrimination is
essentially dominated by precise spatial resolution. Moreover, while each
subdetector channel has the same $125\times 125$ resolution, the effective
size of the particle features differs dramatically across the subdetector
images due to differences in the underlying detector resolution. That is, in
the tracks image, particles appear as individual, isolated pixels, while in
the ECAL image, as roughly $3\times 3$ pixel showers, and in the HCAL image,
as clusters of $5\times 5$ pixel blocks. Such a classification task,
therefore, poses a unique, multi-scale feature extraction task for the CNN. As
we will find, CNNs do not disappoint.

(a) Tracks overlay. Left: gluon jet, Right: quark jet.

(b) ECAL overlay. Left: gluon jet, Right: quark jet.

(c) HCAL overlay. Left: gluon jet, Right: quark jet.

Figure 2: Jet-view image overlays split by subdetector: tracks (2(a)), ECAL
(2(b)), and HCAL (2(c)) over 70k jets each, all in log-scale. Gluon jets
appear on the left, quark jets on the right. Gluon jet showers are visibly
more dispersed in all channels. Note the presence of horizontal bands in the
ECAL (2(b)) overlays, highlighting the image fidelity to capture the energy
leakage across the ECAL barrel-endcap boundary. Image resolution: $125\times
125$.

(a) Tracks channel. Left: gluon jet, Right: quark jet.

(b) ECAL channel. Left: gluon jet, Right: quark jet.

(c) HCAL channel. Left: gluon jet, Right: quark jet.

(d) Composite jet image. Left: gluon jet, Right: quark jet.

Figure 3: Representative jet-view image for a single jet split by sub-
detector: tracks (3(a)), ECAL (3(b)), and HCAL (3(c)) and combined into a
composite image (3(d)), all in log-scale. Gluon jets appear on the left, quark
jets on the right. Gluon jet showers are visibly more dispersed in all
channels. Image resolution: 125 $\times$ 125.

As a reference to establish the maximum performance that can be obtained with
the ECAL-like granularity detector images in the limit that particle position
resolution saturates image resolution, we construct detector images filled
with only isolated pixels (i.e. with no lateral shower width) corresponding to
the generator-level particle positions. Specifically, we take all the stable
particles from the Pythia generator’s particle table and construct a two-
channel image with pixels corresponding to the ($\eta$,$\phi$)-positions of
the stable particles weighted by their $p_{T}$. These include particles from
both the hard-scatter and the underlying event but not from PU. We place all
electrons and photons in one image channel and all the hadrons in the other.
We then train on jet-view and full detector-view images as before.

##  4 Network and Training

We use the same training strategy both for the classification of quark versus
gluon jets, and for the classification of di-quark versus di-gluon QCD events.
We employ the ResNet-15 [48] CNN architecture found in [38], where the same
hyper-parameters were found to be optimal. The ADAM adaptive learning rate
optimizer [49] is used to minimize the binary cross-entropy loss in batches of
32 samples. We use an initial learning rate of $5\times 10^{-4}$, and
explicitly reduce it by half every 10 epochs for a total of 30 training
epochs. We reserve about 26k out of the 768k training samples for our
validation set (see Table 2). All training was done using PyTorch [50] running
on a single NVIDIA Titan X GPU with PyArrow for data I/O [51].

Training Samples | Validation Samples | Test Samples
---|---|---
per class | per class | per class
384000 | 12950 | 69653
Table 2: Number of events in training, validation, and final test set _for each class_. The total training+validation and test sets contain a balanced proportion of class samples. Algorithm | Inputs | Architecture
---|---|---
A | 2 $\times$ $q/g$ score | FCN128$\times$2
B | 2 $\times$ $q/g$ score, jet 4-momenta | FCN128$\times$2
C | Full detector image | ResNet-15
Table 3: List of end-to-end event classification algorithms.

For the jet classification studies, we use the multi-subdetecor channel jet-
view images as constructed in Section 3 and feed these directly to the CNN. To
aid in the interpretation of the results, we also experiment with different
combinations of subdetector channels in the input image, as discussed in
greater detail in Section 5.

For the event classification studies, we explore a number of approaches for
integrating jet classification into a dijet event classification workflow, as
summarized in Table 3. Naively, the simplest way to construct a dijet
classifier would be to simply concatenate the outputs of the individual jet
classifiers, ordered by $p_{T}$, and feed these into a fully-connected neural
network (FCN) which acts as the actual event classifier. The jet classifier
output used as input to the event classifier could either be the final feature
vector of the CNN, or just the final quark versus gluon prediction ($q/g$
score). While the latter tends to be more standard among LHC analyses, the
former could potentially be beneficial if the event discrimination were driven
by some other internal property of the jet unrelated to its $q/g$ score. In
this case, it is possible to either train the jet classifier in-place
concurrently with the event classifier, or use a pre-trained, pre-calibrated
jet classifier. Since such a scenario is not expected for dijet
classification, we simply use the $q/g$ score from the best end-to-end jet
classification algorithm as input to a FCN-based dijet event classifier
(algorithm A). This, however, does not account for the event-level kinematics
of the jet. Therefore, we can additionally augment the dijet classifier inputs
with the 4-momenta of the reconstructed jets (or the coordinates of the jet
image centers), also ordered by $p_{T}$ (algorithm B).

The above event classifiers collectively follow a factorized workflow, where
the jet identification is done first, followed by a separate event classifier
that is engineered to exploit the topology of the underlying physics process.
However, one could also construct a unified end-to-end event classifier that
takes in the full detector-view image (see Figure 1(a)) all at once (algorithm
C) to perform dijet classification directly from detector date. This avoids
the particle ordering and numbering issues introduced in Section 1 since the
natural spatial distribution of the jets in the full detector is exploited. We
found no significant benefit to augmenting the full detector-view dataset with
images randomly “rotated” in $\phi$ to enforce azimuthal symmetry, suggesting
this had been learned by the classifier implicitly.

Algorithm C may be particularly suited to complex, multi-body decays where
engineering an effective event classifier is non-trivial. Indeed, because
algorithm C does not depend on one’s ability to model the event topology, it
can potentially serve as a guide for engineering a specialized one that does.
One challenge in implementing algorithm C in an analysis is calibrating and
deriving uncertainties for it, since jet- and event-level sources of
uncertainty become coupled into a single classifier response. A possible
solution is to first derive the jet-level sources by selectively masking out
the detector image outside the object-of-interest. These studies can then be
used to correct the response of the classifier before deriving the event-level
sources on the fully unmasked detector image. More detailed, analysis-specific
work is needed to demonstrate this approach.

To evaluate classifier performance, we use the Receiver Operating
Characteristic (ROC) curve, which can be interpreted in terms of the signal
efficiency (true positive rate) versus background rejection (true negative
rate), as is commonly used in high-energy physics. The area under the ROC
curve (AUC) is used to select the best algorithm based on the validation set.
In addition, we also present the inverse of the false positive rate (FPR) at a
fixed true positive rate (TPR) of 70%. For an unbiased estimate of
performance, all final performance metrics presented in our Results (Sections
5,6) are calculated from the test set which is statistically independent from
the validation set.

##  5 Jet ID Results

The end-to-end jet classification results lend themselves well to a detector
performance interpretation. To this end, we first perform the quark versus
gluon tagging using single subdetector-channel images, to understand the
relative importance of each subdetector. The results are presented in the last
three rows of Table 4. The best single subdetector performance comes from the
reconstructed tracks image followed by ECAL, then HCAL, correlating strongly
with the resolution of the underlying physical detector. Given that quark
versus gluon jets differ primarily in how broadly their jet constituents are
spaced, increased detector resolution would mean we are better able to resolve
the positions of the constituents. In fact, a close visual inspection of the
core of the single-jet subdetector images in Figure 3 suggests this pattern of
broader constituent distribution is indeed more apparent with the finer tracks
image, followed by the ECAL image where the constituents are just barely
resolved for the gluon jet but not for the quark jet. By the HCAL image, it is
no longer so obvious. This reinforces the importance of building high-
fidelity, full-granularity detector images that are able to capture all the
nuances in the energy deposition patterns. Lastly, we note the remarkable
ability of the CNNs to extract meaningful information even from the highly-
sparse tracks images which is purely composed of isolated pixels if not empty
space.

|  | 1/FPR
---|---|---
Jet image | ROC AUC | @TPR=0.7
Generated EM+Had | 0.854 | 6.46
Tracks+ECAL+HCAL | 0.807 | 4.45
Tracks+ECAL | 0.804 | 4.35
ECAL+HCAL | 0.781 | 3.76
Tracks | 0.782 | 3.73
ECAL | 0.760 | 3.28
HCAL | 0.682 | 2.27
Table 4: End-to-end jet classification results. Statistical uncertainties are
in the third significant figure.

We next consider the effect of combining two subdetector images in a single
multi-channel image, as presented in rows 3-4 of Table 4. We can combine the
tracks and ECAL images to incorporate information about the photons that are
absent from the tracks image (Tracks+ECAL). Alternatively, we could swap out
the tracks image for the HCAL (ECAL+HCAL), which amounts to taking the charged
hadron information from the coarser HCAL image, to form a purely calorimetric
image. The former approach achieves the best discrimination so far, while the
latter is less performant due to having sacrificed the precise spatial
information of the tracks. Indeed, despite the ECAL+HCAL image having the
advantage in terms of neutral hadron information, we see that the tracks-only
image performs as well as the calorimeter-only image, highlighting again the
importance of precise spatial resolution.

Of course, the best overall performance is obtained when all three subdetector
images are combined together into a single input image (Tracks+ECAL+HCAL), as
shown in the second row of Table 4. Although the three subdetector images have
identical image resolution, the effective feature scales among them differ
dramatically (see Figure 3). Therefore, that the CNN can extract meaningful
features at these different feature scales and deliver robust performance
notwithstanding serves as a testament to their power and versatility.

We can then compare these results to the generator-level images to understand
how much better a jet classifier could perform in the limit of maximal
detector resolution and no PU. From the top row of Table 4 (Generated EM+Had),
we find that detector resolution effects and PU together account for about 5%
of the performance loss at the jet level.

To put the end-to-end jet classification results into context, we can
benchmark against one of the current state-of-the-art jet classifiers, the
QCD-aware recursive neural network (RecNN) algorithm [26, 27]. We use the
default architecture, hyper-parameters, and training strategy implemented in
[52] but with the training split and evaluation frequency modified for
consistency with the strategy used here (see Section 4). Using the exact same
jet samples as used for the end-to-end studies, we find the following results
between the different pre-processing schemes in Table 5. For the top-scoring
pre-processing scheme, we calculate the mean and standard deviation over 5
random seeds of the training set shuffling (as in [26]), and compare this with
the Tracks+ECAL+HCAL jet image results. These appear in the top two rows of
Table 5 with the corresponding ROC curves plotted in Figure 4.

|  | 1/FPR
---|---|---
Jet ID Algorithm | ROC AUC | @TPR=0.7
Jet image, Tracks+ECAL+HCAL | 0.8076 $\pm$ 0.0002 | 4.473 $\pm$ 0.013
RecNN, ascending-$p_{T}$ | 0.8017 $\pm$ 0.0002 | 4.281 $\pm$ 0.013
RecNN, descending-$p_{T}$ | 0.802 | 4.30
RecNN, anti-$k_{T}$ | 0.800 | 4.25
RecNN, Cambridge/Aachen | 0.801 | 4.26
RecNN, no rotation/re-clustering | 0.800 | 4.23
RecNN, $k_{T}$ | 0.800 | 4.24
RecNN, $k_{T}$-colinear10-max | 0.799 | 4.23
RecNN, random | 0.797 | 4.15
Table 5: End-to-end versus RecNN jet classification results. Top scores for
each algorithm represent mean and standard deviation over 5 trials.
Statistical uncertainties are in the third significant figure. Figure 4: Jet
classification ROC curves.

We find the end-to-end jet image algorithm to be highly competitive with the
top performing RecNN, even after taking into account statistical uncertainties
and variations due to the random number seed. While the ascending-$p_{T}$ pre-
processing gives the best RecNN results, most of the other schemes fare
comparably. As previous studies [12, 26] have shown image-based approaches to
underperform relative to direct particle-data based algorithms, our results
suggest this discrepancy can be attributed to limitations in the jet image
construction rather than to the use of CNNs themselves. Since previous jet
image work involved pixelating the particle-level jet constituents into HCAL-
like granularity images, this is not unexpected. High-fidelity, high-
granularity detector images that are as minimally processed and as information
rich as possible, therefore, are essential to effective jet image tagging. For
heavy jet flavor tagging, in particular, further progress is likely to be
found in more sophisticated treatments of the tracking detectors.

##  6 Event ID Results

We can extend quark versus gluon tagging to QCD di-quark versus di-gluon event
classification to compare the different ways one might build a dijet event
classifier using end-to-end jet classification, as described in Section 4. The
results are summarized in Table 6 with the corresponding ROC curves in Figure
5. The result of using generator-level particle images (algorithm C-Gen) is
also included for reference.

|  | 1/FPR
---|---|---
Event ID Algorithm | ROC AUC | @TPR=0.7
A: 2 $\times$ $q/g$ score | 0.882 | 8.44
B: 2 $\times$ $q/g$ score $+$ jet 4-momenta | 0.888 | 9.04
C: Full detector image | 0.889 | 9.05
C-Gen: Full detector image, |  |
generated particle inputs | 0.911 | 12.92
Table 6: End-to-end event classification results. Statistical uncertainties
are in the third significant figure. Figure 5: Event classification ROC
curves.

These indicate that event classification performance is dominated by jet-level
differences (algorithm A), with negligible gain from including the jet
4-momenta (algorithm B). To first approximation, this is expected given that
both di-quark and di-gluon production channels have similar non-resonant
kinematics. Although di-quark events are slightly differentiated in their
angular distribution due to spin correlation and polarization effects, these
effects are not expected to be of significance. The results for algorithm B
were not sensitive to the choice of using the reconstructed jet centroids or
the image coordinates of the jet image centers. While the full detector-view
approach (algorithm C) performs just as well as the topology-specific approach
of B, this confirms C’s ability to learn the same information as B at both the
jet- and event-level, without prior knowledge of the event topology. The
approach of C, therefore, may prove illuminating when tackling more complex
decays which are difficult to model. Lastly, looking at the results of the
generator-level images with no PU, we find the difference relative to the
physical detector images to be approximately 3% at the event level versus 5%
at the jet level, likely owing to the complementary information that two jets
provide.

To gain a better understanding of the impact of the underlying event and PU in
the full detector-view approach (algorithm C), we train a separate event
classifier on full detector images but with the pixel intensities outside of
the jet windows masked or zeroed out (algorithm C-Zero). We can then perform a
transfer learning test by evaluating the classifier trained on the original
scenario (algorithm C) on the zeroed-out detector images and vice-versa. The
results are presented in Table 7.

|  | 1/FPR
---|---|---
Event ID Algorithm | ROC AUC | @TPR=0.7
C: Full detector image | 0.889 | 9.05
C-Zero: Full detector image, |  |
jet windows only | 0.887 | 8.95
C, evaluated on C-Zero | 0.883 | 8.56
C-Zero, evaluated on C | 0.884 | 8.58
Table 7: Event classification supplementary results. Statistical uncertainties
are in the third significant figure.

As these indicate, the loss in performance from either training starting point
is minimal, showing that the end-to-end algorithm is largely insensitive to
the underlying event and pileup outside of the jet region-of-interest,
suggesting the classifier is focusing solely on the pertinent features of the
event. This suggests the end-to-end technique is effective at PU mitigation
and detector images should be as minimally processed as possible when
presented to the CNN in order to maximize the information that can be
extracted.

##  7 Conclusions

In this paper, we demonstrated the application of end-to-end classification
techniques to the tagging of light quark jets versus gluon jets using
simulated CMS Open Data. We constructed jet-view images, which were high-
fidelity maps of the physical detector deposits, to give the jet classifiers
direct access to the maximum recorded event information. The resulting multi-
subdetector images showed rich features spanning various length scales. Using
a ResNet-15 CNN, we achieved effective feature extraction to obtain
performance competitive with current state-of-the-art quark versus gluon
taggers based on traditional particle-level inputs. We found that precise
spatial resolution was of paramount importance, highlighting the importance of
high-fidelity detector images and especially the critical role played by the
track information. We also explored classifying di-quark versus di-gluon QCD
events to illustrate ways in which end-to-end jet classifiers can be used to
build event classifiers. We found the discriminating performance to be largely
dominated by jet-level differences and thus noted similar performance across
the different dijet classifier architectures.

Finally, we showed that full detector-view event classifiers were robust and
versatile against underlying event and pileup outside the jet region-of-
interest, making them a compelling tool for complex, multi-body event
topologies, where event classifier engineering becomes non-trivial. In future
work, we plan to pursue more sophisticated representations of the tracking
information in the context of heavy flavor jet tagging and boosted decays.

## Acknowledgments

We thank the entire CMS Collaboration for successfully recording LHC proton-
proton collision data as well as producing and releasing high quality
simulated data used in this paper. We also congratulate all members in the
CERN accelerator departments for the excellent performance of the LHC and
thank the technical and administrative staffs at CERN and at other CMS
institutes for their contributions to the success of the CMS effort. In
addition, we gratefully acknowledge the computing centres and personnel of the
Worldwide LHC Computing Grid for delivering so effectively the computing
infrastructure essential to CMS analyses. Finally, we acknowledge the enduring
support for the construction and operation of the LHC and the CMS detector.

We would like to thank the CMS Collaboration and the CERN Open Data group for
releasing their simulated data under an open access policy. We strongly
support initiatives to provide such high-quality simulated datasets that can
encourage the development of novel but also realistic algorithms, especially
in the area of machine learning. We believe their continued availability will
be of great benefit to the high energy physics community in the long run.

Finally, M.A. and M.P. are supported by the Office of High Energy Physics of
the U.S. Department of Energy (DOE) under award DE-SC0010118. S.A. is
supported by the Marie Skłodowska-Curie Innovative Training Network Fellowship
of the European Commission’s Horizon 2020 Programme under contract number
765710 INSIGHTS.

## References

  * [1] G. Aad, B. Abbott, J. Abdallah, A. A. Abdelalim, A. Abdesselam, O. Abdinov, B. Abi, M. Abolins, H. Abramowicz, H. Abreu, et al., Study of jet shapes in inclusive jet production inppcollisions at $\sqrt{s}=7\;tev$ using the atlas detector, Physical Review D 83 (5) (Mar 2011).  doi:10.1103/physrevd.83.052003.
URL http://dx.doi.org/10.1103/PhysRevD.83.052003

  * [2] S. Chatrchyan, V. Khachatryan, A. M. Sirunyan, A. Tumasyan, W. Adam, T. Bergauer, M. Dragicevic, J. Ero, C. Fabjan, et al., Shape, transverse size, and charged-hadron multiplicity of jets in pp collisions at $\sqrt{s}=7\;tev$, Journal of High Energy Physics 2012 (6) (Jun 2012).  doi:10.1007/jhep06(2012)160.
URL http://dx.doi.org/10.1007/JHEP06(2012)160

  * [3] M. Aaboud, et al., Measurement of jet-substructure observables in top quark, $W$ boson and light jet production in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, JHEP 08 (2019) 033.  arXiv:1903.02942, doi:10.1007/JHEP08(2019)033.
  * [4] A. Sirunyan, A. Tumasyan, W. Adam, F. Ambrogi, E. Asilar, T. Bergauer, J. Brandstetter, E. Brondolin, M. Dragicevic, J. Ero, et al., Measurement of jet substructure observables in $t\bar{t}$ events from proton-proton collisions at $\sqrt{s}=13\;tev$, Physical Review D 98 (9) (Nov 2018).  doi:10.1103/physrevd.98.092014.
URL http://dx.doi.org/10.1103/PhysRevD.98.092014

  * [5] G. Aad, et al., Performance of top-quark and $W$-boson tagging with ATLAS in Run 2 of the LHC, Eur. Phys. J. C 79 (arXiv:1808.07858. 5) (2018) 375. 79 p, 79 pages in total, author list starting page 63, 39 figures, 6 tables, submitted to The European Physical Journal C. All figures including auxiliary figures are available at http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/JETM-2018-03.  doi:10.1140/epjc/s10052-019-6847-8.
URL http://cds.cern.ch/record/2635896

  * [6] A. Sirunyan, et al., Identification of heavy-flavour jets with the CMS detector in pp collisions at 13 TeV. Identification of heavy-flavour jets with the CMS detector in pp collisions at 13 TeV, JINST 13 (CMS-BTV-16-002. CMS-BTV-16-002-004. 05) (2017) P05011. 114 p.  doi:10.1088/1748-0221/13/05/P05011.
URL https://cds.cern.ch/record/2298594

  * [7] G. Aad, B. Abbott, D. C. Abbott, O. Abdinov, A. A. Abud, K. Abeling, D. K. Abhayasinghe, S. H. Abidi, O. S. AbouZeid, et al., Identification of boosted higgs bosons decaying into b-quark pairs with the atlas detector at 13 tev, The European Physical Journal C 79 (10) (Oct 2019).  doi:10.1140/epjc/s10052-019-7335-x.
URL http://dx.doi.org/10.1140/epjc/s10052-019-7335-x

  * [8] CMS Collaboration, Inclusive search for a highly boosted Higgs boson decaying to a bottom quark-antiquark pair at $\sqrt{s}=13~{}\mathrm{TeV}$ with $137~{}\mathrm{fb}^{-1}$ (4 2020).
  * [9] R. Kogler, B. Nachman, A. Schmidt, L. Asquith, E. Winkels, M. Campanelli, C. Delitzsch, P. Harris, A. Hinzmann, D. Kar, et al., Jet substructure at the large hadron collider, Reviews of Modern Physics 91 (4) (Dec 2019).  doi:10.1103/revmodphys.91.045003.
URL http://dx.doi.org/10.1103/RevModPhys.91.045003

  * [10] A. Abdesselam, et al., Boosted Objects: A Probe of Beyond the Standard Model Physics, Eur. Phys. J. C 71 (2011) 1661.  arXiv:1012.5412, doi:10.1140/epjc/s10052-011-1661-y.
  * [11] Quark versus Gluon Jet Tagging Using Jet Images with the ATLAS Detector, Tech. Rep. ATL-PHYS-PUB-2017-017, CERN, Geneva (Jul 2017).
URL https://cds.cern.ch/record/2275641

  * [12] New Developments for Jet Substructure Reconstruction in CMS (Jul 2017).
URL https://cds.cern.ch/record/2275226

  * [13] J. Gallicchio, M. D. Schwartz, Quark and gluon tagging at the lhc, Physical Review Letters 107 (17) (Oct 2011).  doi:10.1103/physrevlett.107.172001.
URL http://dx.doi.org/10.1103/PhysRevLett.107.172001

  * [14] M. Tanabashi, et al., Review of Particle Physics, Phys. Rev. D 98 (3) (2018) 030001\.  doi:10.1103/PhysRevD.98.030001.
  * [15] A. J. Larkoski, E. M. Metodiev, A Theory of Quark vs. Gluon Discrimination, JHEP 10 (2019) 014.  arXiv:1906.01639, doi:10.1007/JHEP10(2019)014.
  * [16] M. Aaboud, G. Aad, B. Abbott, J. Abdallah, O. Abdinov, B. Abeloos, S. H. Abidi, O. S. AbouZeid, N. L. Abraham, et al., Jet reconstruction and performance using particle flow with the atlas detector, The European Physical Journal C 77 (7) (Jul 2017).  doi:10.1140/epjc/s10052-017-5031-2.
URL http://dx.doi.org/10.1140/epjc/s10052-017-5031-2

  * [17] A. M. Sirunyan, et al., Particle-flow reconstruction and global event description with the CMS detector, JINST 12 (10) (2017) P10003.  arXiv:1706.04965, doi:10.1088/1748-0221/12/10/P10003.
  * [18] M. Cacciari, G. P. Salam, G. Soyez, The anti-ktjet clustering algorithm, Journal of High Energy Physics 2008 (04) (2008) 063–063.  doi:10.1088/1126-6708/2008/04/063.
URL http://dx.doi.org/10.1088/1126-6708/2008/04/063

  * [19] A. Tripathee, W. Xue, A. Larkoski, S. Marzani, J. Thaler, Jet substructure studies with cms open data, Physical Review D 96 (7) (Oct 2017).  doi:10.1103/physrevd.96.074003.
URL http://dx.doi.org/10.1103/PhysRevD.96.074003

  * [20] A. J. Larkoski, I. Moult, B. Nachman, Jet substructure at the large hadron collider: A review of recent advances in theory and machine learning, Physics Reports 841 (2020) 1–63.  doi:10.1016/j.physrep.2019.11.001.
URL http://dx.doi.org/10.1016/j.physrep.2019.11.001

  * [21] A. Butter, et al., The Machine Learning Landscape of Top Taggers, SciPost Phys. 7 (2019) 014.  arXiv:1902.09914, doi:10.21468/SciPostPhys.7.1.014.
  * [22] Identification of Jets Containing $b$-Hadrons with Recurrent Neural Networks at the ATLAS Experiment, Tech. Rep. ATL-PHYS-PUB-2017-003, CERN, Geneva (Mar 2017).
URL https://cds.cern.ch/record/2255226

  * [23] D. Guest, J. Collado, P. Baldi, S.-C. Hsu, G. Urban, D. Whiteson, Jet flavor classification in high-energy physics with deep neural networks, Physical Review D 94 (11) (Dec 2016).  doi:10.1103/physrevd.94.112002.
URL http://dx.doi.org/10.1103/PhysRevD.94.112002

  * [24] J. Pearkes, W. Fedorko, A. Lister, C. Gay, Jet constituents for deep neural network based top quark tagging (2017).  arXiv:1704.02124.
  * [25] S. Egan, W. Fedorko, A. Lister, J. Pearkes, C. Gay, Long short-term memory (lstm) networks with jet constituents for boosted top tagging at the lhc (2017).  arXiv:1711.09059.
  * [26] G. Louppe, K. Cho, C. Becot, K. Cranmer, QCD-Aware Recursive Neural Networks for Jet Physics, JHEP 01 (2019) 057.  arXiv:1702.00748, doi:10.1007/JHEP01(2019)057.
  * [27] T. Cheng, Recursive Neural Networks in Quark/Gluon Tagging (2018).  arXiv:hep-ph/1711.02633v2.
  * [28] G. Kasieczka, N. Kiefer, T. Plehn, J. Thompson, Quark-gluon tagging: Machine learning vs detector, SciPost Physics 6 (6) (Jun 2019).  doi:10.21468/scipostphys.6.6.069.
URL http://dx.doi.org/10.21468/SciPostPhys.6.6.069

  * [29] P. T. Komiske, E. M. Metodiev, J. Thaler, Energy flow networks: deep sets for particle jets, Journal of High Energy Physics 2019 (1) (Jan 2019).  doi:10.1007/jhep01(2019)121.
URL http://dx.doi.org/10.1007/JHEP01(2019)121

  * [30] H. Qu, L. Gouskos, Particlenet: Jet tagging via particle clouds (2019).  arXiv:1902.08570.
  * [31] A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, Commun. ACM 60 (6) (2017) 84–90.  doi:10.1145/3065386.
URL http://doi.acm.org/10.1145/3065386

  * [32] L. de Oliveira, M. Kagan, L. Mackey, B. Nachman, A. Schwartzman, Jet-images — deep learning edition, JHEP 07 (2016) 069.  arXiv:1511.05190, doi:10.1007/JHEP07(2016)069.
  * [33] G. Kasieczka, T. Plehn, M. Russell, T. Schell, Deep-learning Top Taggers or The End of QCD?, JHEP 05 (2017) 006.  arXiv:1701.08784, doi:10.1007/JHEP05(2017)006.
  * [34] P. T. Komiske, E. M. Metodiev, M. D. Schwartz, Deep learning in color: towards automated quark/gluon jet discrimination, JHEP 01 (2017) 110.  arXiv:1612.01551, doi:10.1007/JHEP01(2017)110.
  * [35] A. Aurisano, et al., A Convolutional Neural Network Neutrino Event Classifier, JINST 11 (09) (2016) P09001.  arXiv:1604.01444, doi:10.1088/1748-0221/11/09/P09001.
  * [36] C. Adams, et al., Deep neural network for pixel-level electromagnetic particle identification in the microboone liquid argon time projection chamber, Phys. Rev. D 99 (2019) 092001\.  doi:10.1103/PhysRevD.99.092001.
URL https://link.aps.org/doi/10.1103/PhysRevD.99.092001

  * [37] M. Andrews, M. Paulini, S. Gleyzer, B. Poczos, End-to-End Event Classification of High-Energy Physics Data, J. Phys. Conf. Ser. 1085 (4) (2018) 042022.  doi:10.1088/1742-6596/1085/4/042022.
  * [38] M. Andrews, M. Paulini, S. Gleyzer, B. Poczos, End-to-End Physics Event Classification with CMS Open Data: Applying Image-Based Deep Learning to Detector Data for the Direct Classification of Collision Events at the LHC, Comput. Softw. Big Sci. 4 (1) (2020) 6.  arXiv:1807.11916, doi:10.1007/s41781-020-00038-8.
  * [39] C. F. Madrazo, I. H. Cacha, L. L. Iglesias, J. M. de Lucas, Application of a Convolutional Neural Network for image classification to the analysis of collisions in High Energy Physics (2017).  arXiv:1708.07034.
  * [40] W. Bhimji, et al., Deep Neural Networks for Physics Analysis on low-level whole-detector data at the LHC, J. Phys. Conf. Ser. 1085 (4) (2018) 042034.  arXiv:1711.03573, doi:10.1088/1742-6596/1085/4/042034.
  * [41] Convolutional Neural Networks with Event Images for Pileup Mitigation with the ATLAS Detector, Tech. Rep. ATL-PHYS-PUB-2019-028, CERN, Geneva (Jul 2019).
URL https://cds.cern.ch/record/2684070

  * [42] Getting Started with CMS 2011 Open Data, cERN Open Data Portal (2017).
URL http://opendata.cern.ch/docs/cms-getting-started-2011

  * [43] S. Agostinelli, et al., GEANT4: A Simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250–303.  doi:10.1016/S0168-9002(03)01368-8.
  * [44] S. Banerjee, CMS Simulation Software, J. Phys.: Conf. Ser. 396 (2012) 022003.  doi:10.1088/1742-6596/396/2/022003.
  * [45] Simulated dataset GJet_Pt40_doubleEMEnriched_TuneZ2star_8TeV_ext-pythia6 in AODSIM format for 2012 collision data, cERN Open Data Portal (2017).  doi:10.7483/OPENDATA.CMS.2W51.W8AT.
  * [46] T. Sjostrand, S. Mrenna, P. Skands, PYTHIA 6.4 Physics and Manual, JHEP 0605 (2006) 026.  doi:10.1088/1126-6708/2006/05/026.
  * [47] S. Chatrchyan, et al., The CMS Experiment at the CERN LHC, JINST 3 (2008) S08004.  doi:10.1088/1748-0221/3/08/S08004.
  * [48] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in: 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016, Las Vegas, NV, USA, June 27-30, 2016, 2016, pp. 770–778.  doi:10.1109/CVPR.2016.90.
URL https://doi.org/10.1109/CVPR.2016.90

  * [49] D. P. Kingma, J. Ba, Adam: A Method for Stochastic Optimization (2014).  arXiv:1412.6980.
  * [50] A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, A. Lerer, Automatic differentiation in pytorch, in: NIPS-W, 2017.
  * [51] Apache Software Foundation, Apache Arrow, https://arrow.apache.org/docs/python/index.html, accessed 16-Apr-2020.
  * [52] G. Louppe, Repository for the code of “QCD-Aware Recursive Neural Networks for Jet Physics”, https://github.com/glouppe/recnn (October 2018).


# Steering, Splitting and Cloning of Optical Beam in a Coherently Driven Raman
Gain System

Onkar N. Verma<EMAIL_ADDRESS>Tarak N. Dey<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Guwahati, Guwahati- 781
039, Assam, India

###### Abstract

We propose an all-optical anti-waveguide mechanism for steering, splitting,
and cloning of an optical beam beyond the diffraction-limit. We use a
spatially inhomogeneous pump beam to create an anti-waveguide structure in a
Doppler broadened $\mathcal{N}$-type four-level Raman gain medium for a co-
propagating weak probe beam. We show that a transverse modulated index of
refraction and gain due to the spatially dependent pump beam hold the keys to
steering, splitting and cloning of an optical beam. We have also shown that an
additional control field permits the propagation of an optical beam through an
otherwise gain medium without diffraction and instability. We further discuss
how finesse of the cloned images can be increased by changing the detuning of
the control field.

###### pacs:

42.50.Gy, 32.80.Qk, 42.65.-k

##  I Introduction

Optical beam guiding, deflection and cloning has attracted a great deal of
attention due to its tremendous applications in optical imaging, optical
switching, optical lithography, laser machining, and free-space communication
technologies. The guiding and steering of an optical beam is made possible by
virtue of a refractive index of the medium. Several techniques such as
mechanical I_Cindrich ; D_H_McMahon , thermal W_B_Jackson , electrical
J_D_Zook , acousto-optical R_W_Dixon  and all optical g_p_agrawal ; a_t_ryan ;
y_li ; a_j_stentz  have been proposed to control the refractive index for beam
deflection. However, all-optical methods have been paid much effort owing to
many striking features such as high speed, efficiency, and quick nonlinear
response time.

The nonlinear optical interactions between light and matter creates a new
avenue to control over beam propagation dynamics through a medium. This is
feasible as the absorptive and dispersive properties of the medium can be
modified by the strength of the interactions. Such manipulation of dispersion
and absorption leads to many novel phenomena including electromagnetically
induced transparency (EIT) harris ; michael , coherent population
trapping(CPT) e_arimondo , saturated absorption techniques t_w_hansch ;
g_s_agarwal_09  or lasing without inversion(LWI) olga_kocha . The sharp
refractive index changes near the centre of the transparency window for the
EIT medium is the key concept for beam deflection sautenkov ; zhou ; sun . The
ability to control of light deflection is also possible by use of transverse
magnetic field through an atomic medium schlesser ; holzner ; Karpa_Nat_06 .
Further a suitable spatially dependent control field can be used to modulate
the refractive index along the transverse direction. This spatially modulated
refractive index generates several effects such as induced focusing Moseley1 ;
Moseley2 ; focusing3 ; focusing4 ; mitsunaga , waveguiding truscott ; kapoor ;
andersen ; mukund  and anti-waveguiding defocusing .

Most of the EIT-based schemes for producing beam deflection and guiding have
low transmission due to presence of medium absorption sautenkov ; zhou ; sun .
Therefore, finding an alternative medium which displays gain with the desired
variation of refractive index is a challenging task. In this context, active
Raman gain(ARG) media have attracted a lot of attention agarwal . Recently Zhu
et al. C_Zhu_13  have theoretically studied the beam deflection in an ARG
medium. They have used spatially inhomogeneous pump beam to deflect a weak
probe beam. They have found that the deflection angle is increased by an order
of magnitude as compared to EIT medium. Nonetheless the probe field
experiences a large amount of gain during the propagation through a
$\Lambda$-type ARG medium C_Zhu_13 ; agarwal . This large gain makes the probe
beam propagation unstable and thus limits the practical application agrawal1 ;
agrawal2 ; Gisin . Moreover, the input spot sizes for individual Gaussian
profiles of pump and probe beams are equal to $1.4$ cm and $1$ mm,
respectively. Hence, the diffraction spreading of such beams are not relevant
since Rayleigh length is much larger than the length of the medium. Focusing
laser beams into smaller spots dey  and increasing the spatial resolution of
arbitrary images om2  is a fundamental problem in all-optical image processing
Lantz_Nat_08 ; Cohen_Nat_10 . Distortion and absorption holds the fundamental
limitation for the creation, detection, or propagation of small images. This
limitation affects the applications such as efficient transfer and conversion
of small images Li ; om1 ; ding ; cao , steering wang ; Hang ; lida1  or
optical manipulation of light beams lida2 . Here we address these issues by
considering the propagation of diffraction-limited beams and arbitrary images
through a controllable ARG medium.

In this paper we exploit an anti-waveguide mechanism defocusing  to show beam
steering, splitting, and cloning of an arbitrary images in an inhomogeneously
broadened medium. To facilitate these processes, we use spatial inhomogeneous
pump beam to write an anti-waveguide inside the medium for co-propagating
probe beam. At two-photon Raman detuning condition, the refractive index and
gain of the probe susceptibility are high at the peak of the Gaussian pump
beam whereas at wings both are very small. The high refractive index together
with gain allow to deflect the probe beam when it is launched at the wings of
the pump beam. The control field parameters such as detuning and intensity can
be used to control the transmission intensity and width of the deflected probe
beam. Next, we reveal splitting of a single super-Gaussian probe beam into two
Gaussian beams by use of two-peak pump beam structure. The bright(cladding)
and dark(core) regions of the pump field profile induces a high(cladding) and
low(core) refractive index of the probe field which lead to formation of an
anti-waveguide structures inside the medium. More specifically, the super-
Gaussian probe beam guided out from the core where it was injected. The
diffraction-limited probe beam gets focus in the cladding due to the
converging refractive index. We also observe that the transmitted probe beam
gets the shape of the pump beam with finesse two times larger than the initial
finesse of the pump beam. Further, we demonstrate the cloning of a doughnut-
shaped pump beam structure onto the probe beam. Our numerical simulation shows
that the cloned probe has a controllable gain with high finesse. Furthermore,
our scheme can be employed for cloning the arbitrary pump images to the probe
beam even though the pump images are severely distorted due to diffraction. It
follows that our findings can greatly improve the device performance on beam
steering, splitting and image cloning beyond the diffraction limit.

The article is organized as follows. In the next section, we introduce the
physical model and basic equations of motion for a four-level system. In Sec.
III, an approximate expression for a linear susceptibility of a weak probe
field is derived using perturbative approach. We include the thermal motion of
the atoms by averaging the susceptibility over Maxwell-Boltzmann velocity
distribution. In Sec. IV, we describe the beam propagation equations for the
evolution of both pump and probe fields under paraxial approximations. In Sec.
V, we discuss our results based on numerical simulation. We first explain the
spatially dependent susceptibility for different shapes of the pump beam and
advantage of a uniform control beam. We then perform numerical integration of
the beam propagation equations in order to demonstrate steering, splitting and
cloning of an optical beam. Obtained results are summarised in the final
section.

Figure 1:  (Color online) (a)Schematic illustration to produce steering,
splitting and cloning of the optical beam. The beam shaped pump, probe and a
plane wave control fields are co-propagating with the thermal 87Rb atoms.
(b)Energy-level diagram of a four-level 87Rb atomic system in $\mathcal{N}$
configuration. The atomic transition $|3\rangle\leftrightarrow|1\rangle$ is
coupled by a pump field of Rabi frequency $G$. The weak probe field of Rabi
frequency $g$ interacts with the atomic transition
$|3\rangle\leftrightarrow|2\rangle$. A control field of Rabi frequency
$\Omega$ connects the transition $|4\rangle\leftrightarrow|2\rangle$ to
produce controllable gain of the system.

##  II Physical Model and Basic Equations

The schematic of the system under consideration for the generation of
steering, splitting and cloning of an optical beam is illustrated in Fig.1(a)
where three co-propagating fields interact within the inhomogeneously
broadened medium. The electrical dipole allowed transitions
$|1\rangle\leftrightarrow|3\rangle$, $|3\rangle\leftrightarrow|2\rangle$, and
$|2\rangle\leftrightarrow|4\rangle$ form a four-level $\mathcal{N}$-type
atomic system as shown in Fig.1(b). The transitions
$|1\rangle\leftrightarrow|2\rangle$, $|3\rangle\leftrightarrow|4\rangle$ and
$|1\rangle\leftrightarrow|4\rangle$ are generally forbidden electric dipole
transitions. The atomic transitions $|3\rangle\leftrightarrow|1\rangle$,
$|3\rangle\leftrightarrow|2\rangle$, and $|4\rangle\leftrightarrow|2\rangle$
are driven by a pump field with frequency $\omega_{{}_{1}}$, a weak probe
field with frequency $\omega_{{}_{2}}$ and a control field with frequency
$\omega_{{}_{3}}$, respectively. This generic level configuration can be found
for example in energy levels of 87Rb which contain ground levels
$|1\rangle=|5S_{1/2},F=2\rangle$, $|2\rangle=|5S_{1/2},F=3\rangle$ and excited
levels $|3\rangle=|5P_{1/2},F^{\prime}=2\rangle$ and
$|4\rangle=|5P_{3/2},F^{\prime}=4\rangle$, respectively min_yan ; Li_prl .

We define three co-propagating electric fields as follows:

$\displaystyle{\vec{E}_{j}}(\vec{r},t)=\hat{e}_{j}\mathcal{E}_{j}(\vec{r})~{}e^{-i\left(\omega_{j}t-k_{j}z\right)}+{c.c.},$
(1)

where, $\mathcal{E}_{j}(\vec{r})$ are slowly varying envelopes, $\hat{e}_{j}$
is the unit polarization vector, and $k_{j}$ is the wave number of electric
fields. The index $j\in\\{1,2,3\\}$ denotes the pump, probe, and control
fields, respectively. Under the action of three coherent fields, the
interaction Hamiltonian of the system in the dipole and rotating wave
approximation is given by

$\displaystyle{\mathcal{H}_{I}}/{\hbar}=(\Delta_{{}_{2}}-\Delta_{{}_{1}}-\Delta_{{}_{3}})|4\rangle\langle
4|-(\Delta_{{}_{1}}-\Delta_{{}_{2}})|2\rangle\langle 2|$
$\displaystyle-\Delta_{{}_{1}}|3\rangle\langle 3|-(g|3\rangle\langle
2|+G\,|3\rangle\langle 1|\,+\Omega\,|4\rangle\langle 2|+\text{H.c.})\,,$ (2)

where $\Delta_{{}_{1}}=\omega_{{}_{1}}-\omega_{{}_{31}}$,
$\Delta_{{}_{2}}=\omega_{{}_{2}}-\omega_{{}_{32}}$,
$\Delta_{{}_{3}}=\omega_{{}_{3}}-\omega_{{}_{42}}$ are the single-photon
detunings of the pump, probe, and control fields, respectively. The atomic
transition frequencies are denoted by $\omega_{ij}$. The Rabi frequencies of
pump, probe and control fields are defined as

$\displaystyle
G=\frac{\vec{d}_{13}\cdot\vec{\mathcal{E}}_{\rm{1}}}{\hbar},~{}~{}g=\frac{\vec{d}_{23}\cdot\vec{\mathcal{E}}_{\rm{2}}}{\hbar},~{}{\textrm{and}}~{}~{}\Omega=\frac{\vec{d}_{24}\cdot\vec{\mathcal{E}}_{\rm{3}}}{\hbar}\,,$
(3)

where the $d_{ij}$ are the corresponding dipole moment matrix elements of
transitions $|i\rangle\leftrightarrow|j\rangle$.

The dynamical evolution of the atomic system can be described by the density
matrix equations om2 ,

$\displaystyle\dot{\rho}=-\frac{i}{\hbar}\left[\mathcal{H}_{I},\rho\right]+\mathcal{L}\rho\,.$
(4)

where the Liouvillian matrix $\mathcal{L}\rho$, defined in Eq. (9), describes
the relaxation by radiative and non-radiative decay

$\displaystyle\mathcal{L}\rho=\left[\begin{array}[]{cccc}\gamma_{{}_{13}}\rho_{{}_{33}}+\gamma_{{}_{14}}\rho_{{}_{44}}&-\gamma_{{}_{c}}\rho_{{}_{12}}&-\Gamma_{{}_{13}}\rho_{{}_{13}}&-\Gamma_{{}_{14}}\rho_{{}_{14}}\\\
-\gamma_{{}_{c}}\rho_{{}_{21}}&\gamma_{{}_{23}}\rho_{{}_{33}}+\gamma_{{}_{24}}\rho_{{}_{44}}&-\Gamma_{{}_{23}}\rho_{{}_{23}}&-\Gamma_{{}_{24}}\rho_{{}_{24}}\\\
-\Gamma_{{}_{31}}\rho_{{}_{31}}&-\Gamma_{{}_{32}}\rho_{{}_{32}}&-(\gamma_{{}_{13}}+\gamma_{{}_{23}})\rho_{{}_{33}}&-\Gamma_{{}_{34}}\rho_{{}_{34}}\\\
-\Gamma_{{}_{41}}\rho_{{}_{41}}&-\Gamma_{{}_{42}}\rho_{{}_{42}}&-\Gamma_{{}_{43}}\rho_{{}_{43}}&-(\gamma_{{}_{14}}+\gamma_{{}_{24}})\rho_{{}_{44}}\\\
\end{array}\right].$ (9)

The radiative decay rates from the excited states $|3\rangle$ and $|4\rangle$
to ground states $|1\rangle$ and $|2\rangle$ are labeled by $\gamma_{{}_{i3}}$
and $\gamma_{{}_{i4}}$, $i\in\\{1,2\\}$ and the collisions dephasing rate
$\gamma_{{}_{c}}$ describes redistribution of populations between ground
levels. The decay rate of the atomic coherence is defined as

$\displaystyle\Gamma_{{}_{\alpha\beta}}=\frac{1}{2}\left[\displaystyle\sum_{i}\gamma_{{}_{i\alpha}}+\displaystyle\sum_{i}\gamma_{{}_{i\beta}}\right]+\gamma_{{}_{c}},~{}i\notin\\{\alpha,\beta\\}\,.$
(10)

Substituting the interaction Hamiltonian of Eq.(II) and the Liouvillian matrix
of Eq.(9) in the density matrix Eq.(4), the equations of motion for the four-
level atomic system can be described as

$\displaystyle\dot{\rho}_{{}_{11}}$
$\displaystyle=\gamma_{{}_{13}}\rho_{{}_{33}}+\gamma_{{}_{14}}\rho_{{}_{44}}+{i}G^{*}\rho_{{}_{31}}-{i}G\rho_{{}_{13}}\,,$
$\displaystyle\dot{\rho}_{{}_{22}}$
$\displaystyle=\gamma_{{}_{23}}\rho_{{}_{33}}+\gamma_{{}_{24}}\rho_{{}_{44}}+{i}g^{*}\rho_{{}_{32}}-{i}g\rho_{{}_{23}}+{i}\Omega^{*}\rho_{{}_{42}}-{i}\Omega\rho_{{}_{24}}\,,$
$\displaystyle\dot{\rho}_{{}_{21}}$
$\displaystyle=-\left[\gamma_{c}-i\Delta_{{}_{R}}\right]\rho_{{}_{21}}-{i}G\rho_{{}_{23}}+{i}g^{*}\rho_{{}_{31}}+{i}\Omega^{*}\rho_{{}_{41}}\,,$
$\displaystyle\dot{\rho}_{{}_{33}}$
$\displaystyle=-(\gamma_{{}_{13}}+\gamma_{{}_{23}})\rho_{{}_{33}}+{i}G\rho_{{}_{13}}-{i}G^{*}\rho_{{}_{31}}+{i}g\rho_{{}_{23}}-{i}g^{*}\rho_{{}_{32}}\,,$
$\displaystyle\dot{\rho}_{{}_{31}}$
$\displaystyle=-\left[\Gamma_{{}_{31}}-i\Delta_{{}_{1}}\right]\rho_{{}_{31}}+{i}g\rho_{{}_{21}}+{i}G(\rho_{{}_{11}}-\rho_{{}_{33}})\,,$
$\displaystyle\dot{\rho}_{{}_{32}}$
$\displaystyle=-\left[\Gamma_{{}_{32}}-i\Delta_{{}_{2}}\right]\rho_{{}_{32}}+{i}G\rho_{{}_{12}}-{i}\Omega\rho_{{}_{34}}+{i}g(\rho_{{}_{22}}-\rho_{{}_{33}})\,,$
$\displaystyle\dot{\rho}_{{}_{34}}$
$\displaystyle=-\left[\Gamma_{{}_{34}}-i(\Delta_{{}_{2}}-\Delta_{{}_{3}})\right]\rho_{{}_{34}}+{i}G\rho_{{}_{14}}+{i}g\rho_{{}_{24}}-{i}\Omega^{*}\rho_{{}_{32}}\,,$
$\displaystyle\dot{\rho}_{{}_{41}}$
$\displaystyle=-\left[\Gamma_{{}_{41}}-i(\Delta_{{}_{R}}+\Delta_{{}_{3}})\right]\rho_{{}_{41}}+i\Omega\rho_{{}_{21}}-{i}G\rho_{{}_{43}}\,,$
$\displaystyle\dot{\rho}_{{}_{42}}$
$\displaystyle=-\left[\Gamma_{{}_{42}}-i\Delta_{{}_{3}}\right]\rho_{{}_{42}}+{i}\Omega(\rho_{{}_{22}}-\rho_{{}_{44}})-{i}g\rho_{{}_{43}}\,,$
(11)

together with population conservation condition
$\rho_{{}_{11}}+\rho_{{}_{22}}+\rho_{{}_{33}}+\rho_{{}_{44}}=1$ and two-photon
Raman detuning $\Delta_{{}_{R}}=\Delta_{{}_{1}}-\Delta_{{}_{2}}$. In the next
section, we obtain the analytical expression for the linear susceptibility of
the probe field in a compact form with the assumption of equal decay rates
from excited states, i.e.,
$\gamma_{{}_{13}}=\gamma_{{}_{23}}=\gamma_{{}_{14}}=\gamma_{{}_{24}}=\gamma/2$
.

##  III  Probe Susceptibility for Hot atomic medium

In this section, we derive an approximate solution of linear susceptibility of
the probe field in a hot atomic medium. The analytical solution of the atomic
coherence ${\rho}_{{}_{32}}$ for the probe field can be obtained by solving
the density matrix Eqs.(II) in the steady state condition.

We assume that all atoms are prepared initially in the ground state
$|1\rangle$. Due to the presence of large detuning of the strong pump and weak
probe fields, most of the atoms populate at their ground state $|1\rangle$
while other states $|j\rangle(j\neq 1)$, remain empty at later time. Hence the
system turns to an ARG configuration for the probe field. Since Raman gain
process is basically a second-order process, we therefore expand the density
matrix elements to first order in the probe field $g$ and to second order in
the pump field $G$ but all orders in the control field $\Omega$ in the weak
probe field limit. The perturbation expansion of the density matrix can be
expressed as

$\displaystyle\rho_{{}_{ij}}$
$\displaystyle=\rho_{{}_{ij}}^{(0)}+G\rho_{{}_{ij}}^{(1)}+G^{*}\rho_{{}_{ij}}^{(2)}+g\rho_{{}_{ij}}^{(3)}+g^{*}\rho_{{}_{ij}}^{(4)}+G^{2}\rho_{{}_{ij}}^{(5)}$
$\displaystyle+|G|^{2}\rho_{{}_{ij}}^{(6)}+{G^{*}}^{2}\rho_{{}_{ij}}^{(7)}+gG\rho_{{}_{ij}}^{(8)}+gG^{*}\rho_{{}_{ij}}^{(9)}$
$\displaystyle+g^{*}G\rho_{{}_{ij}}^{(10)}+g^{*}G^{*}\rho_{{}_{ij}}^{(11)}+g|G|^{2}\rho_{{}_{ij}}^{(12)},$
(12)

where, $\rho_{{}_{ij}}^{(0)}$ describes the solution in the absence of all
three optical fields and $\rho_{{}_{ij}}^{(k)}$ denotes the $k$-th order
solution. Now we substitute the above expression in the Eqs.(II) and equate
the coefficients of $g$, $g^{*}$, $G^{n}$ ($n\in 1,2$), and constant terms. As
a result, we obtain a set of 12 coupled simultaneous linear algebraic
equations to determine the expression of ${\rho}_{{}_{32}}^{(12)}$. We use
back substitution method to solve these algebraic equations in order to derive
the probe coherence ${\rho}_{{}_{32}}$. The different terms in the expression
of the probe coherence are given in Appendix A. The atomic coherence
${\rho}_{{}_{32}}$ will yield the probe susceptibility $\chi$ at frequency
$\omega_{{}_{2}}$

${\chi}(\Delta_{{}_{2}})=\frac{N|d_{{}_{32}}|^{2}}{{\hbar}}{\rho}_{{}_{32}}\,,$
(13)

where $N$ is the atomic density of the medium. The above analysis is valid for
stationary atoms. While for a hot atomic system, the thermal motion of the
atoms causes inhomogeneous broadening of the atomic spectra. The thermal
velocity $v$ of the atom can be included in the susceptibility expression(13)
by introducing velocity-dependent field detunings
$\Delta_{j}(v)=\Delta_{j}-k_{j}v,~{}j\in\\{1,2,3\\}$. The term $k_{j}v$ is the
Doppler shift experienced by an atom with a velocity component $v$ in the
direction of the beam propagation of the fields. We have assumed the wave
vectors of the three fields are nearly equal ($k_{j}\approx k$). The negative
sign in the velocity-dependent field detuning $\Delta_{j}(v)$ indicate that
atom and field are co-propagating. The susceptibility of a hot atomic vapour
system needs to be averaged over the entire velocity distribution of atoms and
it is given by

$\langle\chi\rangle=\int_{-\infty}^{\infty}\chi(kv)P(kv)d(kv)\,.$ (14)

The velocity distribution of the atom is assumed to obey the Maxwell-Boltzmann
distribution

$P(kv)d(kv)=\frac{1}{\sqrt{2\pi\mathcal{D}^{2}}}e^{-\frac{(kv)^{2}}{2{\mathcal{D}}^{2}}}d(kv)\,.$
(15)

The Doppler width $\mathcal{D}$ at temperature $T$ defined by

$\mathcal{D}=\sqrt{\frac{k_{B}T\omega^{2}}{Mc^{2}}}\,,$ (16)

where $M$ is the atomic mass and $k_{B}$ is the Boltzmann constant. Doppler
broadening plays a crucial role to control the width of the absorption or gain
window of the thermal media Agarwal ; Kash ; Javan ; Peng . The spectral
features of window become narrower in a Doppler broadened medium as compared
with the homogeneous medium. The steepness of the refractive index due to the
narrowing of resonance window can be useful in many applications such as slow
light, storage of light and high resolution spectroscopy. Thus we include
atomic velocity effect on the beam propagation dynamics through ARG medium by
considering Doppler averaging in the susceptibility expression.

##  IV Beam propagation equations and beam profiles

The propagation of co-propagating pump and probe fields with amplitudes
$\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ along the $z$-direction are governed
by Maxwell’s wave equations. Under slowly varying envelope and paraxial wave
approximations, the beam propagation equations for pump and probe field can be
expressed in the following form

$\displaystyle\frac{\partial G}{\partial z}$
$\displaystyle=\frac{i}{2{k_{1}}}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)G\,,$ (17a)
$\displaystyle\frac{\partial g}{\partial z}$
$\displaystyle=\frac{i}{2{k_{2}}}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)g+2i{\pi}k_{1}\langle{\chi}\rangle{g}\,.$ (17b)

The velocity-averaged susceptibility $\langle\chi\rangle$ is included only in
the probe beam equation, whereas this effect is very negligible on the pump
beam propagation under the weak probe field om1 . The second partial
derivatives in the transverse directions $(x,y)$ represent a paraxial
diffraction. The diffraction of beam or image is inevitable since its
constituent plane wave components acquire different phases during its
propagation. The spatially dependent refractive index of the fields can be
used to suppress or even reverse due to diffraction. We use a suitable
spatially dependent pump field to produce spatially dependent refractive index
for the probe field. For this purpose, we choose the transverse spatial
profile of the pump beam as a Laguerre-Gaussian with charge $m$, denoted by
LGPm. The profile of pump beam can be written as

$\displaystyle G(x,y,z)=G_{\rm
0}\frac{{w_{0}}}{{w_{z}}}\Bigg{(}\frac{r\sqrt{2}}{{w_{z}}}\Bigg{)}^{m}\exp\left[{\frac{ikr^{2}}{2R_{z}}-\frac{r^{2}}{w_{z}^{2}}}\right]$
$\displaystyle\times\exp\left[-i(m+1)\tan^{-1}\Bigg{(}{\frac{z}{z_{R}}}\Bigg{)}+im\theta\right]\,,$
(18)

where $G_{0}$ is an initial peak amplitude, $m$ is the azimuthal index. The
beam width is defined as
$w_{z}=w_{0}\sqrt{1+\left({{z}/{{z_{R}}}}\right)^{2}}$, where $w_{0}$ is the
beam waist at $z=0$, and $z_{R}={{\pi w_{0}^{2}}}/{\lambda}$ is the Rayleigh
length. The radial distance from the axis of the beam is given by
$r=\sqrt{x^{2}+y^{2}}$. Note that for the azimuthal index $m=0$, the Laguerre-
Gaussian pump (LGP${}_{m})$ beam reduces to a Gaussian pump beam (GP${}_{0})$.
Figure 2 shows the intensity distribution of the pump field against radial
position $x$ at different lengths of the medium. The LGPm beam exhibits a dark
spot in the centre and a bright profile in the annular region. This makes
intensity profile in contrast to GP0 beam. It is clearly shown in Fig.2 that
diffraction induced distortion of the pump beam profile is not severe even
after 5 cm of propagation. Therefore, phase modulation imposed on the probe
beam due to the spatially varying pump beam is effective throughout the length
of the medium. The probe beam possess a Gaussian profile

$\displaystyle g(x,y)=$
$\displaystyle{g}_{0}\>e^{-[\frac{(x-a)^{2}+y^{2}}{w_{p}^{2}}]^{f}}\,,$ (19)

at an entry face of the medium. The initial peak amplitude and the width of
the probe field are denoted by $g_{0}$ and $w_{p}$ and $a$ is the initial
location of the centre of the probe beam along the $x$ direction. We have
chosen the initial intensity of the probe beam such that it gets absorbed
inside the medium without pump and control fields. The integer values of $f$
decides the input profile of the probe beam - either a Gaussian $(f=1)$ or a
super-Gaussian $(f>1)$.

Figure 2:  (Color online)  Pump intensities profile for two different shapes
namely Gaussian (GP0) and Laguerre-Gaussian(LGP3) is plotted against $x$ at
$y=0$ plane. The initial amplitude and width of profiles are $G_{0}=2\gamma$
and ${w_{0}}=100~{}{\mu}$m, respectively.

##  V Results and Discussions

###  V.1 Spatial modulation of the probe field susceptibility

Figure 3: (Color online)  Real and imaginary part of the averaged
susceptibility is plotted against the transverse co-ordinate $x$ at $y=0$
plane. The spatial probe gain profile (solid black line) is reduced by a
factor of ten to visualise it with $\Omega(x,y)=\gamma$. The common parameters
are fixed as follows: single photon detuning of pump and probe fields
$\Delta_{{}_{1}}=\Delta_{{}_{2}}=180{\gamma}$, Doppler width
$\mathcal{D}=140\gamma$, density $N=2.5\times 10^{10}$ atoms/cm3 and atomic
coherence decay rate $\gamma_{{}_{c}}=0.01\gamma$. The other parameters are
same as in Fig.2

In order to elucidate the effect of position dependent characteristic of the
pump field on the probe beam dynamics, we first numerically explore the
behaviour of velocity-averaged probe susceptibility under different detuning
and intensity of the control field. Fig. 3 shows the spatial variation of
probe dispersion and gain plotted against transverse axis $x$ at $y=0$ plane.
Here two different transverse profiles of pump beam namely Gaussian
(GP${}_{0})$ and Laugerre-Gaussian (LGP${}_{3})$ have been used. We begin with
Gaussian pump beam and study the usefulness of uniform control field $\Omega$
on the spatially modulated probe susceptibility. In absence of a control
field, the position dependent refractive index of the probe is zero under two-
photon Raman condition whereas the spatial gain profile of the probe field
takes the shape of the pump beam profile. The spatially dependent pump
structure generates a probe gain profile which is one of the key components in
realising the deflection of the probe beam if it is off-centered with respect
to the pump beam. In absence of the control field $\Omega=0$, the gain profile
of the probe field is fifteen times larger than in case of control field
$\Omega=\gamma$. This large gain can create modulation instability of system
agrawal1 ; agrawal2 ; Gisin . Therefore a controllable gain of the medium is
required to avoid the modulation instability. It is clear from Fig. 3 that the
position dependent probe gain can be substantially suppressed by a uniform
control field with $\Omega(x,y)=\gamma$. This restricted probe gain is
accompanied by a Gaussian shaped spatial refractive index. The gradient of the
refractive index is dependent on the sign of the control field detuning. At
red control field detuning, the slope of the spatial refractive index attains
maximum at the line centre and decreases gradually toward the wings. Hence, a
convex lens like refractive index can be mimiced in the ARG medium for
$\Delta_{{}_{3}}\geq\Delta_{{}_{2}}$. On the contrary the blueshifted the
control field detuning $\Delta_{{}_{3}}<\Delta_{{}_{2}}$ can generate a
concave refractive index profile onto the medium. Therefore the refractive
index gradient allows us to focus or defocus the probe beam towards the centre
of the pump beam. As a result the probe field propagates through the gain
window with narrowing or broadening, respectively. Hence a control field can
prepare a gain medium with suitable spatial refractive index for encompass the
probe beam deflection to a great extent.

Next we consider higher order LGP3 mode to investigate the spatial
inhomogeneous character of $\langle\chi\rangle$ in the presence of uniform
control beam. The grey double dashed dotted line in Fig. 3 and 3 shows the
transverse variation of probe refractive index as well as gain, respectively.
The position dependent refractive index and gain both increase in the bright
region whereas it decreases at the dark region of the doughnut shaped pump
beam. In other words, LGP3 induces a diverging gradient index in the region
$|r|\leq$ 0.005 cm whereas a converging gradient index exists in regions 0.005
cm$\leq|r|\leq$ 0.02 cm of the medium. Thus bright and dark regions of LGP3
resembles a waveguide and anti waveguide structure inside the atomic medium.
Fig.3 is also show that the waveguide and anti-waveguide features are
accompanied with gain and absorption, respectively. As a result, the probe
beam is guided out from dark region and confined at bright region in the
course of propagation inside the medium. Hence the shape of the pump beam
profile can be efficiently transfered to the transmitted probe beam.

Figure 4: (Color online) (a) The transverse probe beam intensity is plotted at
different propagation distances within the medium. The initial amplitude,
width and peak position of Gaussian probe beam are $g_{0}=10^{-3}\gamma$,
${w_{p}}=70~{}{\mu}$m and $a=1.7$ mm, respectively. Single photon detuning of
pump, probe and control fields are
$\Delta_{{}_{1}}=\Delta_{{}_{2}}=\Delta_{{}_{3}}=180\gamma$. (b) depict the
transmitted probe beam width can be controllable by changing the detuning of
control field at $z=4$ cm under two-photon Raman resonance condition
$\Delta_{{}_{1}}=\Delta_{{}_{2}}=180\gamma$. The other parameters are as in
Fig. 3.

###  V.2  Numerical simulation of paraxial beams equations

We have simulated numerically the propagation equations for pump (17a) and
probe (17b) beams by split step operator method Shen  to demonstrate the
spatial susceptibility as well as diffraction effects on the beam’s
propagation dynamics.

####  V.2.1 Optical beam steering

First, we study how the deflection of a probe beam can be controlled by a
spatial dependence of the pump Rabi frequency. The shape and position of a
probe beam is given by Eq (19) at the entry face of the medium. Fig. 4 shows
the spatial evolutions of the probe beam with $a=1.7$ mm and $w_{p}=70~{}\mu$m
when the peak of pump beam is centered at the origin $(0,0)$ with
$w_{c}=100\mu$m. Initially the overlap area between the probe and pump beam is
very negligible. The overlap area is gradually increased due to the broadening
of both the beams during propagation. It is evident from this figure that
after a propagation of one Rayleigh length, the probe beam progressively
enters the pump region. The bright region of the pump beam tends to refract
the probe beam into it and subsequently enhances the probe beam amplitude. As
a result, the probe beam is focused towards the high intensity region of the
pump and remains confined there. Ii is noteworthy that the probe beam gains
the initial shape of the pump beam and retains this shape as it propagates
along the $z$ axis. Similarly, if the peak position of the probe beam is
shifted along the positive $x$-direction then it can be dragged by the pump
beam towards the pump line centre.

Fig. 4 exhibits the effect of control field detuning onto the propagation
dynamics of a probe pulse at $z=4$ cm. It is seen that the deflected probe
beam becomes narrower at redshifted detuning as compared to a blue-shifted
detuning. Therefore the sign of the detuning of the control field gives an
additional flexibility to control the width of the deflected probe beam. Thus
the ARG medium not only acts as an effective beam deflector but also can act
like a lens with a wide focal length tunability.

Figure 5:  (Color online) Propagation dynamics of single super-Gaussian probe
beam in presence of double Gaussian pump beam. The parameters are as in Fig.4
except that the Gaussian probe beam is injected at centre $(0,0)$ with width
$w_{p}=80\mu$m and the double Gaussian pump beam has width $100\mu$m.

####  V.2.2 Optical beam splitting

Next, we demonstrate the spatial evolution of single super-Gaussian probe as
well as double Gaussian pump beam with different propagation distance $z$. At
the entrance face of the medium, the probe beam is launched in the dark region
of the double Gaussian pump beam as shown in Fig. 5. The position dependent
pump beam creates two gain peaks together with converging refractive index in
the probe susceptibility which is similar to the grey double dashed dotted
line in Fig.3. The gain and spatial inhomogeneity of the refractive index is
accountable for this splitting of a single super-Gaussian probe beam into two
Gaussian beams. The converging lens effect in the intense regions of the pump
leads to focusing of the cloned probe beam towards it. The finesse of the
transmitted probe beam can be defined as the ratio of the spacing between
peaks to the width of peaks. The transmitted probe beam width is reduced by a
factor of 1.5 and the peaks separation are increased by 0.7 mm as compared to
the initial shape of the pump beam. Hence the finesse of the cloned image has
doubly enhanced as compared with initial pump image. Noticeably from Fig. 5
the transmitted probe beam structure is preserved even though the pump beam
suffers distortion due to diffraction.

####  V.2.3 Optical beam cloning

In this section, we investigate the efficient transfer of images between two
co-propagating orthogonal polarized optical beams. We adopt all-optical anti-
waveguide mechanism to clone the images from pump to probe beam. An all-
optical anti-waveguide structure can be formed inside the medium with use of
LGP3 beam which has zero intensity at the beam centre. The dark and bright
regions of LGP3 beam give rise to minimum and maximum refractive index
gradient on the probe susceptibility. As a results a diverging and converging
refractive index is formed in the core and cladding region of the anti-wave
guide structure. Thus an all-optical anti-waveguide for a probe beam is
generated by the co-propagating doughnut-shaped strong pump beam. In order to
demonstrate the cloning mechanism in ARG medium, the centre of the dark region
of the doughnut pump beam is the initial location of probe beam. The diverging
refractive index gradient and diffraction leads to the probe beam leaving the
core region and slowly enter the high intensity regions of the pump beam.
Therefore each wing of the probe beam profile experiences gain and converging
gradient of refractive index in the cladding region. Thus the probe energy is
guided into the annular ring of the doughnut-shaped beam and leaving a zero
intensity in the dark region. Hence the transmitted probe beam profile
acquires a doughnut-shaped profile as shown in Fig. 6. We have found that the
transmitted structure of probe beam is twice as sharp compared to the LGP3
beam structure. The spatial evolution of probe beam at different propagation
distances are similar to that in Fig. 5.

Figure 6: (Color online)Image transfer from doughnut-shaped pump structure to
the probe beam via anti-waveguiding mechanism. In (a), the 3D intensity
profile of pump beam at the output of 5-cm-long medium. In (b), the cloned 3D
probe intensity profile at the exit face of the rubidium vapour cell. The
other parameters are same as in Fig. 4 except atomic density $N=2.5\times
10^{11}$ atoms/cm3 and ground state atomic coherence decay rate
$\gamma_{{}_{c}}=0.001\gamma$, $G_{0}=2.5\gamma$ and $\Omega=5\gamma$. Figure
7: (Color online) (a) Three letters “ARG” are imprinted on pump beam. (b) The
efficiently transferred image onto the probe beam after 2 cm length of
propagation inside the atomic medium. (c) The transmitted pump beam image
which is completely blurred at the exit face of medium. The parameters are
same as in Fig.6.

####  V.2.4 Arbitrary image cloning

Fig. 7 shows the cloning of an arbitrary images and its diffraction effects
through an ARG medium. In order to elucidate the arbitrary image cloning, we
consider the probe beam as a plane wave whereas the pump beam carries complex
image such as three letters “ARG” structured at the entrance face of the
medium. The two-dimensional transverse profile of the pump beam creates gain
for the probe beam wherever two-photon Raman condition is satisfied. Hence the
transverse pattern of the pump beam can be efficiently transferred to the
probe beam. The cloned probe beam also experiences focusing effects at the
high intensity regions of the pump beam. Thus the transmitted probe beam has
better resolution than the original pump beam images as can be seen in Fig.
7b. Fig. 7c illustrates that the diffraction induced distortion severely
affects the pump beam images and is completely distorted after a propagation
of 2 cm.

##  VI Conclusion

In conclusion, we have studied diffractionless steering, splitting and cloning
of an optical beam in a Doppler broadened four level $\mathcal{N}$-type Raman
gain medium using a spatially inhomogeneous pump beam. The spatial pump beam
profile gives rise to transverse modulation in the refractive index and gain
for the probe beam. The modulated refractive index along with gain can
optically form anti-waveguide structure inside the medium. The properties of
anti-waveguide structure such as refractive index and gain can be controlled
by the application of control field which lead to steering of the probe beam
very efficiently. We further demonstrated that a single probe beam can be
split into two Gaussian modes when it is injected at the centre between two
Gaussian modes of pump beam. We found that the probe beam profile has acquired
the shape of the pump beam and propagates without usual diffraction. We next
show that the transfer of doughnut-shaped pump image onto a low power
Gaussian-shaped probe beam can be possible with high finesse. Finally, by
numerical simulations we have established that an arbitrary image with three
letters “ARG” imprinted on pump beam can be cloned on to the transmission
profile of the probe. The finesse of cloned image has increased twice as
compared to the initial resolution of pump images. Thus this scheme might be
useful in optical switching, optical lithography, and optical imaging
processing.

###### Acknowledgements.

TND acknowledges Science and Engineering Board of India for financial support
(SR/S2/LOP-0033/2010).

##  Appendix A Expressions of probe susceptibility

The related $(12)$th-order contributions for $\rho_{{}_{32}}$are obtained as

$\displaystyle{\rho}_{{}_{32}}^{{}^{(12)}}$
$\displaystyle=\frac{i({\rho}_{{}_{22}}^{{}^{(6)}}-{\rho}_{{}_{33}}^{{}^{(6)}})+i{\rho}_{{}_{12}}^{{}^{(9)}}-i{\Omega}{\rho}_{{}_{34}}^{{}^{(12)}}}{\left[\gamma_{{}_{32}}-i\Delta_{{}_{2}}\right]}$
(20) $\displaystyle{\rho}_{{}_{34}}^{{}^{(12)}}$
$\displaystyle=\frac{i{\rho}_{{}_{14}}^{{}^{(9)}}+i{\rho}_{{}_{24}}^{{}^{(6)}}-i{\Omega^{*}}{\rho}_{{}_{32}}^{{}^{(12)}}}{\left[\Gamma_{{}_{34}}-i(\Delta_{{}_{2}}-\Delta_{{}_{3}})\right]}$
(21) $\displaystyle{\rho}_{{}_{12}}^{{}^{(9)}}$
$\displaystyle=\frac{i{\rho}_{{}_{32}}^{{}^{(3)}}-i{\rho}_{{}_{13}}^{{}^{(2)}}-i{\Omega}{\rho}_{{}_{14}}^{{}^{(9)}}}{\left[\gamma_{{}_{c}}+i\Delta_{{}_{R}}\right]}$
(22) $\displaystyle{\rho}_{{}_{14}}^{{}^{(9)}}$
$\displaystyle=\frac{i{\rho}_{{}_{34}}^{{}^{(1)}}-i{\Omega^{*}}{\rho}_{{}_{12}}^{{}^{(9)}}}{\left[\Gamma_{41}+i(\Delta_{{}_{R}}+\Delta_{{}_{3}})\right]}$
(23) $\displaystyle{\rho}_{{}_{33}}^{{}^{(6)}}$
$\displaystyle=\frac{i({\rho}_{{}_{13}}^{{}^{(2)}}-{\rho}_{{}_{31}}^{{}^{(1)}})}{\left[\gamma_{{}_{13}}+\gamma_{{}_{23}}\right]}$
(24) $\displaystyle{\rho}_{{}_{24}}^{{}^{(6)}}$
$\displaystyle={\rho}_{{}_{44}}^{{}^{(6)}}={\rho}_{{}_{22}}^{{}^{(6)}}=0$ (25)
$\displaystyle{\rho}_{{}_{13}}^{{}^{(2)}}$
$\displaystyle={\rho}_{{}_{31}}^{{}^{(1)}}=\frac{-i}{\left[\Gamma_{{}_{13}}+i\Delta_{{}_{1}}\right]}$
(26) $\displaystyle{\rho}_{{}_{32}}^{{}^{(3)}}$
$\displaystyle={\rho}_{{}_{34}}^{{}^{(1)}}=0$ (27)
$\displaystyle{\rho}_{{}_{32}}$
$\displaystyle={A}\left[\frac{{2\Gamma_{{}_{31}}[\Gamma_{{}_{34}}-i(\Delta_{{}_{2}}-\Delta_{{}_{3}})]}}{(\gamma_{{}_{13}}+\gamma_{{}_{23}})(\Gamma_{{}_{31}}^{2}+\Delta_{{}_{1}}^{2})}+\frac{{[\Gamma_{{}_{34}}-i(\Delta_{{}_{2}}-\Delta_{{}_{3}})][\Gamma_{{}_{41}}+i(\Delta_{{}_{R}}+\Delta_{{}_{3}})]}-|\Omega|^{2}}{(\Gamma_{{}_{31}}+i\Delta_{{}_{1}})\left[(\gamma_{{}_{c}}+i\Delta_{{}_{R}})(\Gamma_{{}_{41}}+i(\Delta_{{}_{R}}+\Delta_{{}_{3}}))+|\Omega|^{2}\right]}\right]$
(28)

with

$\displaystyle{A}$
$\displaystyle=\frac{-ig|G|^{2}}{({\gamma_{32}-i\Delta_{2}})\\{\gamma_{34}-i(\Delta_{2}-\Delta_{3})\\}+{|\Omega|^{2}}}$
(29)

## References

  * (1) I. Cindrich, Appl. Opt. 6, 1531 (1967).
  * (2) D. H. McMahon, A. R. Franklin, and J. B. Thaxter, Appl. Opt. 8, 399 (1969).
  * (3) W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, Appl. Opt. 20, 1333 (1981).
  * (4) J. D. Zook, Appl. Opt. 13, 875 (1974).
  * (5) R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
  * (6) G. P. Agrawal, Phys. Rev. Lett. 64, 2487 (1990).
  * (7) A. T. Ryan and G. P. Agrawal, Opt. Lett. 18, 1795 (1993).
  * (8) Y. Li, D. Y. Chen, L. Yang, and R. R. Alfano, Opt. Lett. 16, 438 (1991).
  * (9) A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, Opt. Lett. 17, 19 (1992).
  * (10) S. E. Harris, Phys. Today 50(7), 36 (1997).
  * (11) M. Fleischhauer, and A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 663 (2005).
  * (12) E. Arimondo, Progress in Optics, 35, 257(1996).
  * (13) T. W. Hansch, M. D. Levenson, and A. L. Schawlow, Phys. Rev. Lett. 26, 946 (1971).
  * (14) G. S. Agarwal and T. N. Dey, Laser & Photonics Reviews 3, 287 (2009).
  * (15) O. Kocharovskaya, Phys. Rep. 219, 175 (1992).
  * (16) V. A. Sautenkov, H. Li, Y. V. Rostovtsev, and M. O. Scully, Phys. Rev. A81, 063824 (2010).
  * (17) D. L. Zhou, L. Zhou, R. Q. Wang, S. Yi, and C. P. Sun, Phys. Rev. A76, 055801 (2007).
  * (18) Q. Sun, Y. V. Rostovtsev, and M. S. Zubairy, Phys. Rev. A74, 033819 (2006).
  * (19) R. Schlesser and A. Weis, Opt. Lett. 17, 1015 (1992).
  * (20) R. Holzner, P. Eschle, S. Dangel,R. Richard, H. Schmid, U. Rusch, B. Rohricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, Phys. Rev. Lett. 78, 3451(1997).
  * (21) L. Karpa and M. Weitz, Nat. Phys. 2, 332 (2006).
  * (22) R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, Phys. Rev. Lett. 74, 670 (1995).
  * (23) R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, Phys. Rev. A53, 408 (1996).
  * (24) D. R. Walker, D. D. Yavuz, M. Y. Shverdin, G. Y. Yin, A. V. Sokolov, and S. E. Harris, Opt. Lett. 27, 2094 (2002).
  * (25) N. A. Proite, B. E. Unks, J. T. Green, and D. D. Yavuz, Phys. Rev. A 77, 023819 (2008).
  * (26) M. Mitsunaga, M. Yamashita, and H. Inoue Phys. Rev. A62, 013817 (2000).
  * (27) A. G. Truscott, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 82, 1438 (1999).
  * (28) R. Kapoor and G. S. Agarwal, Phys. Rev. A61, 053818 (2000).
  * (29) J. A. Andersen, M. E. J. Friese, A. G. Truscott, Z. Ficek, P. D. Drummond, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A63, 023820 (2001).
  * (30) M. Vengalattore and M. Prentiss, Phys. Rev. Lett. 95, 243601 (2005).
  * (31) D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, Phys. Rev. A63, 031801(R) (2001).
  * (32) G. S. Agarwal and S. Dasgupta, Phys. Rev. A70, 023802 (2004).
  * (33) C. Zhu, L. Deng, and E. W. Hagley, Phys. Rev. A88, 013841 (2013).
  * (34) G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
  * (35) G. P. Agrawal, J. Opt. Soc. Am. B7, 1072 (1990).
  * (36) B. V. Gisin, A. A. Hardy, and B. A. Malomed, Phys. Rev. E50, 3274 (1994).
  * (37) T. N. Dey and J. Evers, Phys. Rev. A84, 043842 (2011).
  * (38) O. N. Verma and T. N. Dey, Phys. Rev. A89, 033830 (2014).
  * (39) B. E. Cohen, Nature (London)467, 407 (2010).
  * (40) E. Lantz, Nat. Photonics 2, 71 (2008).
  * (41) H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A78, 013803 (2008).
  * (42) O. N. Verma, L. Zhang, J. Evers, and T. N. Dey, Phys. Rev. A88, 013810 (2013).
  * (43) D. Ding, Z. Zhou and B. Shi, Opt. Lett. 39, 240 (2014).
  * (44) M. Cao,L. Zhang,Y. Yu,F. Ye, D. Wei, W. Guo,S. Zhang, H. Gao, and F. Li, Opt. Lett. 39, 2723 (2014).
  * (45) H. Wang and X. Peng, J. Opt. Soc. Am. B29, 429 (2005).
  * (46) C. Hang and V. V. Konotop, Phys. Rev. A81, 053849 (2010).
  * (47) L. Zhang, T. N. Dey, and J. Evers, Phys. Rev. A87, 043842 (2013).
  * (48) L. Zhang, and J. Evers, Phys. Rev. A89, 013817 (2014).
  * (49) M. Yan, E. G. Rickey, and Y. Zhu, Phys. Rev. A64, 041801(R) (2001).
  * (50) R. B. Li, L. Deng, and E. W. Hagley, Phys. Rev. Lett. 110, 113902 (2013).
  * (51) G. Vemuri, G. S. Agarwal, and B. D. Nageswara Rao, Phys. Rev. A53, 2842 (1996).
  * (52) M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, Phys. Rev. Lett. 82, 5229 (1999).
  * (53) A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, Phys. Rev. A66, 013805 (2002).
  * (54) Y. Peng, Y. Niu, L. Zhang, A. Yang, L. Jiang, and S. Gong, Opt. Lett. 37, 3333 (2012).
  * (55) A. D. Bandrauk, H. Shen, Journal of Physics A: Mathematical and General 27, 7747 (1994).