# TorchXRayVision:
A library of chest X-ray datasets and models

Joseph Paul CohenS,M,Q,A Joseph D. VivianoQ Paul BertinM,Q
Paul MorrisonQ,F Parsa TorabianW Matteo GuarreraB,E,P Matthew P LungrenS,A
Akshay ChaudhariS,A Rupert BrooksN Mohammad HashirM,Q Hadrien BertrandQ
SStanford University MUniversité de Montréal QMila, Quebec AI Institute
ACenter for Artificial Intelligence in Medicine & Imaging (AIMI) NNuance
Communications FFontbonne University WUniversity of Waterloo BUniversity of
California, Berkeley PPolitecnico di Torino EEURECOM
Correspondance: https://github.com/mlmed/torchxrayvision/issues

###### Abstract

TorchXRayVision is an open source software library for working with chest
X-ray datasets and deep learning models. It provides a common interface and
common pre-processing chain for a wide set of publicly available chest X-ray
datasets. In addition, a number of classification and representation learning
models with different architectures, trained on different data combinations,
are available through the library to serve as baselines or feature extractors.

https://github.com/mlmed/torchxrayvision

NOT FOR MEDICAL USE

##  1 Introduction

Chest X-rays are the most common medical imaging test in the world and
represent the bulk of medical computer vision publications and open medical
imaging data in the deep learning community [UKNHS2019imagingstats]. Yet
despite the large number of datasets and publications, it can be difficult for
researchers to properly compare previous work and to investigate
generalization across different datasets. Even when data and code are
available, small but important differences in dataset organization,
processing, or training procedures can significantly impact the results. This
makes establishing meaningful baselines a strenuous task for researchers. In
addition, constantly re-implementing the same dataloaders is not the best use
of time. There is a need for common software infrastructure.

TorchXRayVision (XRV) was created to address this difficulty by establishing a
reusable framework for reproducible research and consistent baseline
experiments. A key design objective is to provide a clear interface and
separation between datasets and models. The library provides a common
interface to multiple available chest X-ray datasets, which can easily be
swapped out during model training and evaluation. Common pre- and post-
processing components are provided and the datasets are compatible with
torchvision [Paszke2019pytorch] components for augmentation. We include pre-
trained and easily downloadable models which can be used directly for baseline
comparisons or to generate feature vectors for downstream tasks. Three
specific use cases where the project has already proved useful include:

  * •

Evaluating models: It is important to rigorously evaluate the robustness of
models using multiple external datasets. However, associated clinical data
with each dataset can vary greatly which makes it difficult to apply methods
to multiple datasets. TorchXRayVision provides access to many datasets in a
uniform way so that they can be swapped out with a single line of code. These
datasets can also be merged and filtered to construct specific distributional
shifts for studying generalization.

  * •

Developing models: Making it easier for Deep Learning researchers to work on
medical tasks. Pre-trained models are useful for baseline comparisons as well
as feature extractors. These pre-trained models have already been used for
transfer learning to related chest X-ray tasks such as patient severity
scoring [Cohen2020severity; Gomes2020covidtransferseverity] and predicting
aspects about a patients clinical trajectory [cohen2020covidProspective;
Gomes2020ICU; Maurya2020Intubation]. As illustrated in Cherti2021fewshot
TorchXRayVision pre-trained models are used to study few-shot transfer
learning. In Delbrouck2021multimodal the pre-trained models are used as
feature extractors of images for multi-modal models, and in
Sundaram2021gantransfer the library is used for baseline models as well as to
explore methods of generative adversarial network-based (GAN) data
augmentation. The pre-trained models and training pipeline were used in
[Tetteh2021balancedbatch] to explore how balancing batches while training on
multiple datasets improves performance.

  * •

Studying model failures and limitations: The many pre-trained models provided
are not perfect and can be studied to determine how they fail, which can
inform the development of better models. Also, the many datasets available in
TorchXRayVision make it possible to study out-of-distribution generalization
when covariate and concept shifts are present. The library was initially
developed for this purpose in cohen2020limits. This library has already been
used in work by Robinson2021covidshortcut which focused on studying shortcut
learning caused by covariate shift in chest X-ray models. Work by Viviano2020
explored failures in saliency maps using this library and special utilities
are included to produce datasets with different types of covariate shift and
spurious correlations. Work by Cohen2021gif generated counterfactual
explanations for model predictions using the classifiers and autoencoder in
the library.

A key design consideration when developing this library was to follow an
object-oriented approach. This turns the various components of an experiment
into objects which can be easily swapped. The primary objects are datasets and
models (including pre-trained weights). A suite of utilities for working with
these objects are also included.

The library is available in Python via pip with the package name import
torchxrayvision as xrv. It is based on PyTorch [Paszke2019pytorch] and modeled
after the torchvision library (hence the name). TorchXRayVision’s
compatibility and conformity to established convention makes adoption
intuitive for practitioners.

##  2 Models

The library is composed of core and baseline classifiers. Core classifiers are
trained specifically for this library (initially trained for cohen2020limits)
and baseline classifiers come from other papers that have been adapted to
provide the same interface and work with the same input pixel scaling as our
core models. All models will automatically resize input images (higher or
lower using bilinear interpolation) to match the specified size they were
trained on. This allows them to be easily swapped out for experiments. See
§LABEL:sec:preprocessing for details on image pro-processing. Pre-trained
models are hosted on GitHub and automatically downloaded to the user’s local
`~/.torchxrayvision` directory.

###  2.1 Core Classifiers

Core pre-trained classifiers are provided as PyTorch Modules which are fully
differentiable in order to work seamlessly with other PyTorch code. Models are
specified using the “weights" parameter which have the general form of:

$\underbrace{\text{densenet121}}_{\text{Architecture}}\text{-}\overbrace{\text{res224}}^{Resolution}\text{-}\underbrace{\text{rsna}}_{\text{Training
dataset}}$

Each pre-trained model aims to have 18 independent output classes as defined
in NaN as the prediction of that pathology. The models indicating “all" as the
dataset have been trained on as many datasets were available at the time. More
details about each set of weights is available at the head of the
Cohen2019Chester, each image was randomly rotated up to 45 degrees, translated
up to 15% and scaled larger of smaller up to 10%.

⬇

# Generic function to load any core model

model = xrv.models.get_model(weights="densenet121-res224-all")

\par# DenseNet 224x224 model trained on multiple datasets

model = xrv.models.DenseNet(weights="densenet121-res224-all")

\par# DenseNet trained on just the RSNA Pneumonia dataset

model = xrv.models.DenseNet(weights="densenet121-res224-rsna")

\par# ResNet 512x512 model trained on multiple datasets

model = xrv.models.ResNet(weights="resnet50-res512-all")

\par\end{code}

\par\subsection{Baseline Classifiers}

\parCurrently there are two baseline classifiers from other research groups
which were added to make it easier to compare against TorchXRayVision core
models. The models adhere to the same interface as the core models and can be
easily swapped out.

\parThe first is a JFHealthcare model \citep{ye2020weaklychexpert} trained on
CheXpert data \citep{Irvin2019CheXpert} and the second is an official CheXpert
model \citep{Irvin2019CheXpert} trained by the CheXpert team.

\par\begin{code}

# DenseNet121 from JF Healthcare for the CheXpert competition

model = xrv.baseline_models.jfhealthcare.DenseNet()

\par# Official Stanford CheXpert model

model = xrv.baseline_models.chexpert.DenseNet()

\end{code}

\parThe weights for the CheXpert model are much larger than all the other
models (6GB) and have been hosted on Academic Torrents
\citep{Cohen2014academictorrents} as well as the Internet Archive.

\par\subsection{Classifier Interface}

\parEach classifier provides a field \codeinline{model.pathologies} which
aligns to the list of predictions that the model makes. Depending on the
weights loaded this list will change. The predictions can be aligned to
pathology names as follows:

\par\begin{code}

predictions = model(img)[0] # 0 is first element of batch

dict(zip(model.pathologies,predictions.detach().numpy()))

# output:

{’Atelectasis’: 0.3566849,

’Consolidation’: 0.72457345,

’Infiltration’: 0.8974177,

…}

\end{code}

\parGetting a specific output can be achieved as follows. The outputs remain
part of the computation graph and can therefore be embedded in a larger
network.

\par\begin{code}

prediction = model(img)[:,model.pathologies.index("Consolidation")]

\end{code}

\par\subsection{Feature Extraction}

\parThe pre-trained models can also be used as features extractors for semi-
supervised training or transfer learning tasks. A feature vector can be
obtained for each image using the \codeinline{model.features} function. The
resulting size will vary depending on the architecture and the input image
size. For some models there is a \codeinline{model.features2} method that will
extract features at a different point of the computation graph. Example UMAP
visualizations \citep{McInnes2018UMAP} of the features from different models
is shown in Figure \ref{fig:representations}.

\begin{code}

feats = model.features(img)

\end{code}

\par\par\par\par\begin{figure}[H]

\centering\includegraphics[width=0.49\textwidth]{reps/xrv-
densenet121-densenet121-res224-all__dim_1024_.png}

\includegraphics[width=0.49\textwidth]{reps/chexpert-
densenet121-ensemble__dim_3072_.png}

\\\

\includegraphics[width=0.49\textwidth]{reps/xrv-
resnetae-101-elastic__dim_18432_.png}

\includegraphics[width=0.49\textwidth]{reps/xrv-resnet-
resnet50-res512-all__dim_2048_.png}

\caption{UMAP visualizations of the representations from different models.
2048 images, each containing only one of the 4 pathologies listed, are
included in the UMAP.}

\label{fig:representations}

\end{figure}

\par\subsection{Model Calibration}

\parAs described in \cite{Cohen2019Chester} and \cite{cohen2020limits}, Eq.
\ref{eq:normalize} can be applied to calibrate the output of the model so that
they can be compared with a piecewise linear transformation. The goal is to
make a prediction of $0.5$ the estimated decision boundary based on held out
test data. For each disease, we computed the optimal operating point of the
ROC curve by maximizing the difference (\textit{True positive rate} -
\textit{False positive rate}). It corresponds to the threshold which maximizes
the informativeness of the classifier \citep{powersEvaluation2011}. This is
computed with the held out test subset of the dataset being used for training,
as the model is intended to be evaluated on one of the other datasets. Also
keep in mind that doesn’t change the AUC on the test set, it is just for a
nice calibrated probability output so you can use pred > 0.5 to get a
prediction.

\begin{equation}

\label{eq:normalize}

f_{opt}(x) = \begin{cases}

\frac{x}{2opt} & x\leqopt \\\

1-\frac{1-x}{2(1-opt)} & otherwise

\end{cases}

\end{equation}

\par\par\subsection{Autoencoders}

\parThe library also provides a pre-trained autoencoder that is trained on the
PadChest, NIH, CheXpert, and MIMIC datasets. This model was developed in
\cite{Cohen2021gif} and provides a fixed latent representation. The goal of
this model is to provide another representation extraction model which is not
trained using supervised labels and has a decoder to reconstruct images from
the latent representation.

\par\begin{code}

ae = xrv.autoencoders.ResNetAE(weights="101-elastic")

z = ae.encode(image)

image2 = ae.decode(z)

\end{code}

\par\section{Datasets}

\parThe datasets in this library aim to fit a simple interface where the
\codeinline{imgpath} and \codeinline{csvpath} are specified. Some datasets
require more than one metadata file and for some the metadata files are
packaged in the library so only the imgpath needs to be specified.

\parDocumentation for each dataset class provides citation information and
download links. When possible, metadata is included with the library so only
the \codeinline{imgpath} needs to be specified. When possible based on the
license, datasets have been uploaded to Academic Torrents
\citep{Cohen2014academictorrents}. Otherwise, the dataset can be downloaded
from its origin by following the links provided in the class documentation.

\par\par\par\par\par\par\par\par\par\par\par\par\par\begin{code}

dataset = xrv.datasets.VinBrain_Dataset(imgpath="./train",

csvpath="./train.csv")

\end{code}

\parEach dataset object also supports the standard \codeinline{transform}
argument which takes PyTorch transforms. While standard transforms will be
applied there may be issues in how they deal with the specific tensor shapes
returned by the dataloader (\codeinline{[1,RES,RES]} where RES is the image
resolution)

\begin{code}

transform = torchvision.transforms.Compose([xrv.datasets.XRayCenterCrop(),

xrv.datasets.XRayResizer(224)])

\end{code}

\parTables \ref{tab:datasets} and \ref{tab:labels_available} display the total
count of images and the counts of labels available per dataset.

\par\subsection{Image pre-processing}

\label{sec:preprocessing}

\parBoth models and datasets expect the image pixel values to be between
\codeinline{[-1024,1024]}. The origin of this is arbitrary. The preprocessing
of images used scales from the possible range of the images and not the min
and max of the pixels. For example, if the image is 16-bit encoded then the
possible pixels values are between 0 and 65,536 so this range will be mapped;
not the min and max pixels of the specific image. The idea here is that we
don’t want to arbitrarily increase the contrast of an image because that could
be removing information.

\par\begin{table}[h!]

\caption{Details of datasets that are included in this library. Number of
images shows total images / usable frontal images. Useable frontal means
images that are readable, have all necessary metadata, and are in AP, PA, AP
Supine, or AP Erect view. }

\label{tab:datasets}

\resizebox{\columnwidth}{!}{\begin{tabular}{lrcrrlll}

\topruleName & \multicolumn{3}{c}{\\# Images (Total/Frontal)} & Citation &
Geographic Region \\\

\midruleNational Library of Medicine Tuberculosis & 800 &/& 800 &
\cite{Jaeger2014NLMTuberculosis} & USA+China\\\ OpenI (National Library of
Medicine) & 7,470 &/& 4,014 & \citet{Demner-Fushman2016} & USA \\\

ChestX-ray8 (NIH) & 112,120 &/& 112,120 & \citet{WangNIH2017} & Northeast USA
\\\

RSNA Pneumonia Challenge & 26,684 &/& 26,684 &\citet{Shih2019RSNAKaggle} &
Northeast USA \\\

CheXpert (Stanford University) & 223,414 &/& 191,010 &
\citet{Irvin2019CheXpert} & Western USA \\\ Google Labelling of NIH data &
4,376 &/& 4,376 & \citet{Majkowska2019} & Northeast USA \\\ MIMIC-CXR (MIT) &
377,095 &/& 243,324 & \citet{Johnson2019mimic-cxr} & Northeast USA \\\

PadChest (University of Alicante) & 158,626 &/& 108,722&
\citet{Bustos2019PadChest} & Spain \\\ SIIM-ACR Pneumothorax Challenge &
12,954 &/& 12,954& \citet{Filice2020SIIMPneumothorax} & Northeast USA\\\

COVID-19 Image Data Collection (CIDC) & 866 &/& 698&
\cite{cohen2020covidProspective} & Earth \\\

StonyBrook COVID-19 RALO Severity & 2,373 &/& 2,373& \cite{Cohen2021RALO} &
Northeast USA \\\

Object-CXR (JF Healthcare) & 9,000 &/& 9,000& - & China \\\ VinBrain VinDr-CXR
& 15,000 &/& 15,000& \cite{Nguyen2020vinbrain} & Vietnam \\\

\bottomrule\end{tabular}

}

\end{table}

\par\begin{table}[H]

\caption{\smallLabels available for each dataset, the total number of positive
examples for each indication across all datasets, and the total number of
example in each dataset, and the sum over each row in the right column. The
COVID-19 datasets are excluded from this table because they have many unique
pathologies.}

\label{tab:labels_available}

\centering\par\resizebox{\columnwidth}{!}{\input{datasets.tex}

}

\end{table}

\par\par\subsection{Dataset common fields}

\parEach dataset contains a number of common fields. These fields are
maintained when \codeinline{xrv.datasets.SubsetDataset} and
\codeinline{xrv.datasets.MergeDataset} are used.

\par\begin{itemize}[leftmargin=.3in]

\item\codeinline{dataset.pathologies} a list of strings identifying the
pathologies contained in this dataset. This list corresponds to the columns of
the \codeinline{.labels} matrix. Although it is called pathologies, the
contents do not have to be pathologies and may simply be attributes of the
patient.

\par\item\codeinline{dataset.labels} field is a NumPy matrix
\citep{harris2020array-numpy} which contains a \codeinline{1}, \codeinline{0},
or \codeinline{NaN} for each pathology. Each column is a pathology and each
row corresponds to an item in the dataset. A \codeinline{1} represents that
the pathology is present, \codeinline{0} represents the pathology is absent,
and \codeinline{NaN} represents no information.

\par\item\codeinline{dataset.csv} field which holds a Pandas DataFrame
\citep{mckinney-proc-scipy-2010-pandas} of the metadata \codeinline{.csv} file
that is included with the data. For some datasets multiple metadata files have
been merged together. It is largely a "catch-all" for associated data and the
referenced publication should explain each field. Each row aligns with the
elements of the dataset so indexing using \codeinline{.iloc} will work.
Alignment between the DataFrame and the dataset items will be maintained when
using tools from this library.

\end{itemize}

\parIf possible, each dataset’s \codeinline{.csv} will have some common fields
of the csv. These will be aligned when datasets are merged together.

\par\begin{itemize}[leftmargin=.3in]

\item\codeinline{dataset.csv.patientid} is a unique id that will uniquely
identify patients in the dataset. This is useful when trying to prevent
patient overlap between train and test sets or in conjunction with the next
field to observe patients over time.

\par\item\codeinline{dataset.csv.offset_day_int} is an integer time offset for
the image in the unit of days. This is expected to be for relative times and
has no guarantee to be an absolute time although for some datasets it is and
is formatted in unix epoch time.

\par\item\codeinline{dataset.csv.view} is a string indicating the
projection/view in which the chest X-ray was acquired. Most will be “PA”,
“AP”, or “AP Supine”. A good discussion of views is contained in
\citep{Bustos2019PadChest}.

\end{itemize}

\par\par\subsection{Dataset tools}

\par\subsubsection{Relabelling datasets}

\parWorking with dataset objects is a task that the library is designed to
help with. Tasks such as aligning, composing, or taking a subset of a dataset
are made easy using the functions discussed below. The function
\codeinline{xrv.datasets.relabel_dataset} will add, remove, and reorder the
\codeinline{.labels} field to have the same order as the pathologies argument
passed to it. If a pathology is specified but doesn’t exist in the dataset
then a \codeinline{NaN} will be put in place of the label.

\begin{code}

# Note: dataset is directly changed, no return value

xrv.datasets.relabel_dataset(xrv.datasets.default_pathologies, dataset)

\end{code}

\par\subsubsection{Filtering based on views}

\parSpecific views can be specified in the constructor to select only those
views. This is only supported on datasets which have view information. Common
views have been standardized to “PA”, “AP”, or “AP Supine”, but other non-
standardized ones may exist. It is best to first load the dataset without
filtering based on view and call \codeinline{dataset.csv.view.unique()} to see
what is available.

\begin{code}

dataset = xrv.datasets.PC_Dataset(…, views=["PA","AP","AP Supine"])

\end{code}

\newpage\subsubsection{Ensuring one image per patient}

\parThe \codeinline{unique_patients} argument will tell the dataset to only
allow 1 image per patient. This only works on datasets which provide a
patientid.

\begin{code}

dataset = xrv.datasets.PC_Dataset(…, unique_patients=True)

\end{code}

\par\subsubsection{Obtaining summary statistics on a dataset}

\parSimply printing the object will return counts for the available labels and
their classes. This is also returned as a dictionary with the function
\codeinline{dataset.totals()}.

\begin{code}

print(d_chex)

# Output:

CheX_Dataset num_samples=191010 views=[’PA’, ’AP’]

{’Atelectasis’: {0.0: 17621, 1.0: 29718},

’Cardiomegaly’: {0.0: 22645, 1.0: 23384},

’Consolidation’: {0.0: 30463, 1.0: 12982},

…}

\end{code}

\par\subsubsection{Merging datasets together}

\par\parThe class \codeinline{xrv.datasets.MergeDataset} can be used to merge
multiple datasets together into a single dataset. This class takes in a list
of dataset objects and assembles the datasets in order. This class will
correctly maintain the \codeinline{.labels}, \codeinline{.csv}, and
\codeinline{.pathologies} fields and offer pretty printing.

\par\begin{code}

dmerge = xrv.datasets.MergeDataset([dataset1, dataset2, …])

# Output:

MergeDataset num_samples=261583

├0 PC_Dataset num_samples=94825 views=[’PA’, ’AP’]

├1 RSNA_Pneumonia_Dataset num_samples=26684 views=[’PA’, ’AP’]

├2 NIH_Dataset num_samples=112120 views=[’PA’, ’AP’]

├3 SIIM_Pneumothorax_Dataset num_samples=12954

└4 VinBrain_Dataset num_samples=15000 views=[’PA’, ’AP’]

\end{code}

\par\subsubsection{Taking a subset of a dataset}

\parWhen you only want a subset of a dataset the \codeinline{SubsetDataset}
class can be used. A list of indexes can be passed in and only those indexes
will be present in the new dataset. This class will correctly maintain the
\codeinline{.labels}, \codeinline{.csv}, and \codeinline{.pathologies} fields
and offer pretty printing.

\par\begin{code}

dsubset = xrv.datasets.SubsetDataset(dataset, [0,5,60])

# Output:

SubsetDataset num_samples=3

└ of PC_Dataset num_samples=94825 views=[’PA’, ’AP’]

\end{code}

\parFor example this class can be used to create a dataset of only female
patients by selecting that column of the csv file and using
\codeinline{np.where} to convert this boolean vector into a list of indexes.

\par\begin{code}

idxs = np.where(dataset.csv.PatientSex_DICOM=="F")[0]

dsubset = xrv.datasets.SubsetDataset(dataset, idxs)

# Output:

SubsetDataset num_samples=48308

└ of PC_Dataset num_samples=94825 views=[’PA’, ’AP’] data_aug=None

\end{code}

\par\par\subsection{Pathology and semantic masks}

\parMasks for pathologies or semantic regions are also included for some
datasets. These are useful for segmentation or for validating that a model is
attributing importance to the correct area (as explored in \cite{Viviano2020}
and \cite{Cohen2021gif}).

\parMasks are not returned by the datasets by default; the constructor for the
dataset must specify \codeinline{pathology_masks=True} and/or
\codeinline{semantic_masks=True} for them to be returned. These are treated
differently because pathology masks are associated with the
\codeinline{dataset.pathologies} while semantic masks are unrelated (such as
segmentations of the lungs). If no pathology masks exist the data will not
have those arguments available and won’t be constructed. If images do not have
masks available then the key \codeinline{"pathology_masks"} on the sample will
be empty.

\par\par\begin{table}[h]

\caption{\smallCounts of pathology and semantic masks available for each
dataset.}

\label{tab:mask_counts}

\centering\rowcolors{2}{gray!15}{white}

\begin{tabular}{lrrrrr}

\topruleTask & NIH & RSNA & SIIM-ACR & VinBrain & CIDC \\\

\midrulePneumothorax & (BBox) 98 & & (Seg) 3,576 & (BBox) 96 & \\\

Lung Opacity & & (Seg) 6,012 & & (BBox) 1,322 & \\\

Atelectasis & (BBox) 180 & & & (BBox) 186 & \\\

Effusion & (BBox) 153 & & & (BBox) 1,032 & \\\

Cardiomegaly & (BBox) 146 & & & (BBox) 2,300 & \\\

Infiltration & (BBox) 123 & & & (BBox) 613 & \\\

Pneumonia & (BBox) 120 & & & & \\\

Mass & (BBox) 85 & & & & \\\

Nodule & (BBox) 79 & & & & \\\

Nodule/Mass & & & & (BBox) 826 & \\\

Aortic enlargement & & & & (BBox) 3,067 & \\\

Pleural\_Thickening & & & & (BBox) 1,981 & \\\

Calcification & & & & (BBox) 452 & \\\

Interstitial lung disease & & & & (BBox) 386 & \\\

Consolidation & & & & (BBox) 353 & \\\

Lung & & & & & (Seg) 425 \\\

\bottomrule\end{tabular}

\par\end{table}

\par\begin{figure}[t]

\centering\vspace{-20pt}

\includegraphics[width=1\textwidth]{docs/masks.png}\caption{Example images and
corresponding mask information available from multiple datasets. Some are
bounding boxes and some are more exact segmentations.}

\label{fig:my_label}

\end{figure}

\parExample usage:

\par\begin{code}

dataset =
xrv.datasets.RSNA_Pneumonia_Dataset(imgpath="stage_2_train_images_jpg",

views=["PA","AP"],

pathology_masks=True)

\end{code}

\parEach sample will have a \codeinline{pathology_masks} dictionary where the
index of each pathology (in \codeinline{dataset.pathologies}) will correspond
to a mask of that pathology (if it exists). There may be more than one mask
per sample, but only one per pathology. For simplicity, if there are multiple
masks for a single pathology they will be merged together using a logical or.
The resulting mask will be values between \codeinline{0} and \codeinline{1} of
type \codeinline{float}. Data augmentations will be performed to these masks
as well using the same seed to ensure an identical transformation is applied.

\begin{code}

sample["pathology_masks"][dataset.pathologies.index("Lung Opacity")]

\end{code}

The \codeinline{has_masks} column in \codeinline{.csv} will let you know if
any masks exist for that sample:

\begin{code}

dataset.csv.has_masks.value_counts()

# Output:

False 20672

True 6012

\end{code}

\par\par\par\par\subsection{Distribution shift tools}

\par\parA \textit{covariate shift} between two data distributions arises when
some extraneous variable confounds with the variables of interest in the first
dataset differently than in the second \citep{Moreno-Torres2012datasetshift}.
Covariate shifts between the training and test distribution in a machine
learning setting can lead to models which generalize poorly, and this
phenomenon is commonly observed in CXR models trained on a small dataset and
deployed on another one \citep{Zhao2019,DeGrave2020shortcut}. We provide tools
to simulate covariate shifts in these datasets so researchers can evaluate the
susceptibility of their models to these shifts, or explore mitigation
strategies.

\par\begin{code}

d = xrv.datasets.CovariateDataset(d1 = # dataset1 with a specific condition.

d1_target = # target label to predict.

d2 = # dataset2 with a specific condition.

d2_target = #target label to predict.

mode="train", # train, valid, or test.

ratio=0.75)

\par\end{code}

\par\begin{figure}[h]

\begin{center}

\includegraphics[width=0.85\textwidth]{docs/CovariateDataset-Diagram.png}

\end{center}

\caption{A CovariateDataset example of how the data would be split with a
ratio of 0.75 specified. The target label will be balanced 50/50 in each split
but the ratio of the origin dataset will be varied.}

\label{fig:covariatediagram}

\end{figure}

\parThe class \codeinline{xrv.datasets.CovariateDataset} takes two datasets
and two arrays representing the labels. It returns samples for the output
classes with a specified ratio of examples from each dataset, thereby
introducing a correlation between any dataset-specific nuisance features and
the output label. This simulates a covariate shift. The test split can be set
up with a different ratio than the training split; this setup has been shown
to both decrease generalization performance and exacerbate incorrect feature
attribution \citep{Viviano2020}. See Figure \ref{fig:covariate} for a
visualization of the effect the ratio parameter has on the mean class
difference when correlating the view (each dataset) with the target label. The
effect seen with low ratios is due to the majority of the positive labels
being drawn from the first dataset, where in the high ratios, the majority of
the positive labels are drawn from the second dataset. With any ratio, the
number of samples returned will be the same in order to provide controlled
experiments. The dataset has 3 modes, train sampled using the provided ratio
and the valid and test dataset are sampled using $1-$ratio.

\par\begin{figure}[t]

\begin{center}

\includegraphics[width=\textwidth]{docs/covariate.png}

\end{center}

\caption{An example of the mean class difference drawn from the COVID-19
dataset at different covariate ratios. Here, the first COVID-19 dataset
consisted of only AP images, whereas the second dataset consisted of only PA
images. The third row shows, for each ratio, the difference in the class
means, demonstrating the effect of sampling images from the two views on the
perceived class difference. The fourth row shows the difference between each
ratio’s difference image, and the difference image with a ratio of 0.5
(balanced sampling from all views).}

\label{fig:covariate}

\end{figure}

\par\newpage\section{Transfer Learning Example}

\parThis is an example of fine-tuning a pre-trained model on a dataset which
is in the TorchXRayVision format. This code is available
online\footnote{{\url{https://github.com/mlmed/torchxrayvision/blob/master/scripts/transfer_learning.ipynb}}}.

\par\begin{code}

# Use XRV transforms to crop and resize the images

transforms = torchvision.transforms.Compose([xrv.datasets.XRayCenterCrop(),

xrv.datasets.XRayResizer(224)])

\par# Load Google dataset and PyTorch dataloader

dataset = xrv.datasets.NIH_Google_Dataset(imgpath=dataset_dir,

transform=transforms)

dataloader = torch.utils.data.DataLoader(dataset, batch_size=8)

\par# Load pre-trained model and erase classifier

model = xrv.models.DenseNet(weights="densenet121-res224-all")

model.op_threshs = None # prevent pre-trained model calibration

model.classifier = torch.nn.Linear(1024,1) # reinitialize classifier

\paroptimizer = torch.optim.Adam(model.classifier.parameters()) # only train
classifier

criterion = torch.nn.BCEWithLogitsLoss()

\par# training loop

for batch in dataloader:

outputs = model(batch["img"])

targets = batch["lab"][:, dataset.pathologies.index("Lung Opacity"), None]

loss = criterion(outputs, targets)

loss.backward()

optimizer.step()

\par\end{code}

\par\par\section{Acknowledgements}

\label{sec:acknowledgements}

\parWe thank the following organizations for supporting this project in
various ways: CIFAR (Canadian Institute for Advanced Research), Mila (Quebec
AI Institute), University of Montreal, Stanford University’s Center for
Artificial Intelligence in Medicine \& Imaging, and Carestream Health. We
thank AcademicTorrents.com for making data available for our research.

\par\par\bibliographystyle{iclr2022-nopagenum}

\bibliography{refs}

\par\end{document}


# Essential norm of generalized Integral type Operators from Mixed-Norm to
Zygmund-Type Spaces

Y. Estaremi, M. Hassanlou and M. S. Al Ghafri Y. Estaremi<EMAIL_ADDRESS>Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan,
Iran.  M. Hassanlou<EMAIL_ADDRESS>Engineering Faculty of Khoy, Urmia
University of Technology, Urmia, Iran  M. S. Al Ghafri<EMAIL_ADDRESS>Department of Mathematics, Sultan Qaboos University, Muscat, Oman

###### Abstract.

In this paper, we consider the generalized integration operator from mixed-
norm space into Zygmund-type and Bloch-type spaces and find an estimation for
the essential norm of this operator.

###### Key words and phrases:

essential norm, generalized integration operator, mixed-norm space Zyymund-
type space, weighted Bloch space.

###### 2020 Mathematics Subject Classification:

47B33; 47B38; 30H05.

##  1\.  Introduction and Preliminaries

We denote by $\mathbb{D}$ the open unit disc in the complex plan $\mathbb{C}$
and $H(\mathbb{D})$ the space of analytic functions on $\mathbb{D}$. Every
analytic function (self-map) $\varphi:\mathbb{D}\rightarrow\mathbb{D}$ induces
a linear operator via composition called composition operator $C_{\varphi}$,
$C_{\varphi}f=f\circ\varphi$. A complete history of these operators on the
spaces of analytic functions can be found in [3]. Throughout the recent
decades there has been a significant interest on investigating the operator
properties of composition operator and generalization of it while they act on
some spaces of analytic or measurable functions. The above operator can be
generalized by

$\left(C_{\varphi}^{g}f\right)(z)=\int_{0}^{z}f^{\prime}(\varphi(\xi))g(\xi)\
d\xi,\quad f\in H(\mathbb{D}),z\in\mathbb{D},$

where $g\in H(\mathbb{D})$, see [5]. Also, if we replace n-th derivative with
1-th derivative, then we have

$\left(C_{\varphi,g}^{n}f\right)(z)=\int_{0}^{z}f^{(n)}(\varphi(\xi))g(\xi)\
d\xi,\quad f\in H(\mathbb{D}),z\in\mathbb{D},$

where $n\in\mathbb{N}_{0}$. If $n=0$, then $C_{\varphi,g}^{0}$ is the Volterra
composition operator. Also for $\varphi(z)=z$ and $g^{\prime}$ we have
Volterra type operator $T_{g}$,

$T_{g}f(z)=\int_{0}^{z}f(w)g^{\prime}(w)dw.$

$C_{\varphi,g}^{n}$ is called generalized integration operator, see for
example [4, 12] and references therein. A natural question dealing with this
kind of operators or other classes of operators is that, under which
conditions they are bounded, compact, have closed range and $\cdots$. Another
aspect is computing or estimating norm, essential norm, numerical range,
$\cdots$ of the operator. All characterizations are in terms of symbols induce
the operators.

The aim of this paper is to estimate the essential norm of generalized
integration operators from mixed-norm spaces, which include some well-known
spaces of analytic functions, into weighted Zygmund space. As a result,
compactness criteria for the operator, has been proved in [4], is obtained.

A positive continuous function $\phi$ on $[0,1)$ is called normal if there are
two constants $b>a>0$ such that

  * (i)

$\frac{\phi(r)}{(1-r)^{a}}$ is non-increasing in $[0,1)$ and
$\frac{\phi(r)}{(1-r)^{a}}\downarrow 0$

  * (ii)

$\frac{\phi(r)}{(1-r)^{b}}$ is non-decreasing in $[0,1)$ and
$\frac{\phi(r)}{(1-r)^{b}}\uparrow\infty$

as $r\rightarrow 1^{-}$. For $0<p,q<\infty$ and $\phi$ normal, the mixed-norm
space $H(p,q,\phi)$ consists of all analytic functions in $\mathbb{D}$ for
which

$\|f\|_{p,q,\phi}^{p}=\int_{0}^{1}M_{q}^{p}(f,r)\frac{\phi^{p}(r)}{1-r}dr<\infty$

where

$M_{q}(f,r)=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{q}d\theta\right)^{1/q}.$

If $1\leq p<\infty$, then $H(p,q,\phi)$ becomes a Banach space equipped with
the norm $\|.\|_{p,q,\phi}$. In the case $0<p<1$, $\|.\|_{p,q,\phi}$ is a
quasi-norm on $H(p,q,\phi)$ and this space is a Fr$\acute{e}$chet space but is
not a Banach space. For $\alpha>-1$, $\phi(r)=(1-r)^{(\alpha+1)/p}$ is a
normal function. If $p=q$, then $H(p,p,(1-r)^{(\alpha+1)/p})$ is weighted
Bergman space $A_{\alpha}^{p}$ which is defined by

$A_{\alpha}^{p}=\\{f\in
H(\mathbb{D}):\|f\|_{A_{\alpha}^{p}}^{p}=\int_{\mathbb{D}}|f(z)|^{p}(1-|z|^{2})^{\alpha}dA(z)<\infty\\},$

where $dA$ is the normalized area measure on $\mathbb{D}$.

A positive continuous function $\mu$ on $\mathbb{D}$ is called weight. For a
weight $\mu$, the weighted Zygmund space $\mathcal{Z}_{\mu}$, Zygmund-type
space, is the space of $f\in H(\mathbb{D})$ such that

$\sup_{z\in\mathbb{D}}\mu(|z|)|f^{\prime\prime}(z)|<\infty.$

The weighted Zygmond space $\mathcal{Z}_{\mu}$ is a Banach space with respect
to the norm:

$\|f\|_{\mathcal{Z}_{\mu}}=|f(0)|+|f^{\prime}(0)|+\sup_{z\in\mathbb{D}}\mu(|z|)|f^{\prime\prime}(z)|.$

The weighted Bloch space $\mathcal{B}_{\mu}$ defined by

$\mathcal{B}_{\mu}=\\{f\in
H(\mathbb{D}):\|f\|_{\mathcal{Z}_{\mu}}=|f(0)|+\sup_{z\in\mathbb{D}}\mu(|z|)|f^{\prime}(z)|<\infty\\}.$

Note that if $\mu(z)=(1-z^{2})$, then we obtain classical Zygmund space
$\mathcal{Z}$ and Bloch space $\mathcal{B}$. For a complete information about
classical Bloch space, see [13].

Hu in [6] has characterized those holomorphic symbols on the unit ball for
which the induced extended ces$\grave{a}$ro operator is bounded (compact) on
mixed norm space. The boundedness of ces$\grave{a}$ro operators on mixed norm
space is studied in [7]. Boundedness and compactness of the generalized
integration operator from a mixed norm space into a weighted Zygmund space
investigated by Gu and Liu in [4]. More results on integral type operators
acting on mixed norm, Zygmond and Bloch spaces can be found in [1, 5, 8, 9,
10, 11, 12].

Gu and Liu in [4] have characterized compact generalized integration operators
$C_{\varphi,g}^{n}$ between mixed norm and weighted Zygmund spaces. Indeed,
they have proved that
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is compact if and
only if

$A=\lim_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n+1}}=0$
$B=\lim_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}=0$

In this paper, motivated by the above results, we find an approximation of the
essential norm of the generalized integration operators
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ and prove that

$\|C_{\varphi,g}^{n}\|_{e}\approx\max\\{A,B\\},$

it means that there exists constants $C,D$ such that $CB\leq A\leq DB$ or
equivalently $CA\leq B\leq DA$.

Recall that for any operator $T$ between two Banach spaces $X$ and $Y$, the
essential norm of $T$ is denoted by $\|T\|_{e,X\to Y}$ and is defined as
follows

$\|T\|_{e,X\rightarrow Y}=\inf\\{\|T-S\|:\ S\ \text{is a compact operator
from}\ X\ \text{to}\ Y\\}.$

The operator $T$ is compact if and only if $\|T\|_{e,X\to Y}=0$.

##  2\.  Essential norm of
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ or
$\mathcal{B}_{\mu}$

In this section we give an estimation for the essential norm of the operator
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ or
$\mathcal{B}_{\mu}$. First we recall a fundamental lemma for later use in the
paper.

######  Lemma 2.1.

[10] Assume that $p,q\in(0,\infty)$, $\phi$ is normal and $f\in H(p,q,\phi)$.
Then for each $n\in\mathbb{N}_{0}$, there is a positive constant $C$ in
dependent of $f$ such that

$|f^{(n)}(z)|\leq C\frac{\|f\|_{p,q,\phi}}{\phi(|z|)(1-|z|^{2})^{1/q+n}},\ \ \
z\in\mathbb{D}.$

The following lemma can be obtained by a standard argument, see [2].

######  Lemma 2.2.

Let $\varphi$ be a analytic self-map of $\mathbb{D}$. Then
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is compact if and
only if $C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is bounded
and for any bounded sequence $\\{f_{k}\\}$ in $H(p,q,\phi)$ which converges
uniformly to zero on compact subsets of $\mathbb{D}$ as $k\rightarrow\infty$,
we have $\|C_{\varphi,g}^{n}f_{k}\|_{\mathcal{Z}_{\mu}}\rightarrow 0$ as
$k\rightarrow\infty$.

By the above observations, in the next Proposition, we get that every bounded
generalized integration operator $C_{\varphi,g}^{n}$ with
$\|\varphi\|_{\infty}<1$, is compact.

######  Proposition 2.3.

Let $p,q\in(0,\infty)$, $\phi$ is normal, $g\in H(\mathbb{D})$ and $\varphi$
be a analytic self-map of $\mathbb{D}$. If $\|\varphi\|_{\infty}<1$ and the
operator $C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is
bounded then it is compact

###### Proof.

Since $C_{\varphi,g}^{n}$ is bounded, then there exists a positive constant
$C$ such that for

(2.1) $\|C_{\varphi,g}^{n}f\|_{\mathcal{Z}_{\mu}}\leq C\|f\|_{p,q,\phi},$

for all $f\in H(p,q,\phi)$. Let $f_{1}(z)=\frac{z^{n}}{n!}$. Then we have

(2.2)
$\displaystyle\sup_{z\in\mathbb{D}}\mu(|z|)|g^{\prime}(z)|=\|C_{\varphi,g}^{n}f_{1}\|_{\mathcal{Z}_{\mu}}\leq
C\|f_{1}\|_{p,q,\phi}<\infty.$

Similarly for $f_{2}(z)=\frac{z^{n+1}}{(n+1)!}$, by using triangle inequality
and the fact that $|\varphi(z)|<1$, we get that

(2.3)
$\displaystyle\sup_{z\in\mathbb{D}}\mu(|z|)|\varphi^{\prime}(z)g(z)|<\infty.$

Now assume that ${\\{f_{k}\\}}_{k\in{\mathbb{N}}}$ is a bounded sequence in
$H(p,q,\phi)$ such that converges to $0$, uniformly on compact subsets of
$\mathbb{D}$, as $k\to\infty$. This implies that
$\\{f_{k}^{\prime}\\}_{k=1}^{\infty}$ and
$\\{f_{k}^{\prime\prime}\\}_{k=1}^{\infty}$ converge uniformly to $0$, on
compact subsets of $\mathbb{D}$, as $k\to\infty$. By the Lemma (2.2) and
Proposition (2.3) we obtain

$\displaystyle\|C_{\varphi,g}^{n}f_{k}\|_{\mathcal{Z}_{\mu}}=$
$\displaystyle|(C_{\varphi,g}^{n}f_{k})^{\prime}(0)|+\sup_{z\in\mathbb{D}}\mu(z)|(C_{\varphi,g}^{n}f_{k})^{\prime\prime}(z)|$
$\displaystyle\leq$
$\displaystyle|f_{k}^{(n)}(\varphi(0))g(0)|+\sup_{z\in\mathbb{D}}\mu(|z|)|(C_{\varphi,g}^{n}f_{k})^{\prime\prime}(z)|$
$\displaystyle\leq$
$\displaystyle|f_{k}^{(n)}(\varphi(0))g(0)|+\sup_{z\in\mathbb{D}}\mu(|z|)|f_{k}^{(n+1)}(\varphi(z))\varphi^{\prime}(z)g(z)|$
$\displaystyle+\sup_{z\in\mathbb{D}}\mu(|z|)|f_{k}^{(n)}(\varphi(z))g^{\prime}(z)|$
$\displaystyle=$
$\displaystyle|f_{k}^{(n)}(\varphi(0))g(0)|+\sup_{|\varphi(z)|\leq\rho}\mu(|z|)|f_{k}^{(n+1)}(\varphi(z))\varphi^{\prime}(z)g(z)|$
$\displaystyle+\sup_{|\varphi(z)|\leq\rho}\mu(|z|)|f_{k}^{(n)}(\varphi(z))g^{\prime}(z)|,$

which tends to $0$, as $k\rightarrow\infty$. Here $\rho=\|\varphi\|_{\infty}$,
$\rho\in(0,1)$. Hence by Lemma 2.2, $C_{\varphi,g}^{n}$ is compact. ∎

Now, in the next Theorem we estimate the essential norm of
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$. We use
$\|C_{\varphi,g}^{n}\|_{e}$ for the essential norm of
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ instead of
$\|C_{\varphi,g}^{n}\|_{e,H(p,q,\phi)\to\mathcal{Z}_{\mu}}$.

######  Theorem 2.4.

Let $p,q\in(0,\infty)$, $\phi$ is normal and $\varphi$ be a analytic self-map
of $\mathbb{D}$. If the operator
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is bounded then

$\displaystyle\|C_{\varphi,g}^{n}\|_{e}\approx\max\\{$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n+1}},$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}\\}.$

###### Proof.

If $\|\varphi\|_{\infty}<1$, then by the Proposition 2.3, the operator
$C_{\varphi,g}^{n}$ is compact and therefore $\|C_{\varphi,g}^{n}\|_{e}=0$.
Also by Theorem 2.7 of [4], we have

$\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n+1}}=0$

and

$\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}=0.$

Now assume that $\|\varphi\|_{\infty}=1$. Let $\\{z_{j}\\}_{j\in\mathbb{N}}$
be a sequence in $\mathbb{D}$ such that $|\varphi(z_{j})|\rightarrow 1$, as
$j\rightarrow\infty$. Since $\phi$ is a normal function, then we have a
positive constant $b$ from the definition. Put $\alpha=1/q+b+1$ and define the
sequence $\\{f_{k}\\}$ as follows

$\displaystyle
f_{k}(z)=\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\left(\frac{1}{(1-\overline{\varphi(z_{k})}z)^{\alpha}}-\frac{\alpha(1-|\varphi(z_{k})|^{2})}{(\alpha+n)(1-\overline{\varphi(z_{k})}z)^{\alpha+1}}\right).$

Direct calculations shows that

$\displaystyle f_{k}^{(n)}(z)=$
$\displaystyle\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n-1)\overline{\varphi(z_{k})}^{n}}{(1-\overline{\varphi(z_{k})}z)^{\alpha+n}}$
$\displaystyle-\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n-1)(\alpha+n)(1-|\varphi(z_{n})|^{2})\overline{\varphi(z_{n})}^{n}}{(\alpha+n)(1-\overline{\varphi(z_{k})}z)^{\alpha+n+1}}$
$\displaystyle f_{k}^{(n+1)}(z)=$
$\displaystyle\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n)\overline{\varphi(z_{k})}^{n+1}}{(1-\overline{\varphi(z_{k})}z)^{\alpha+n}}$
$\displaystyle-\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n-1)(\alpha+n)(1-|\varphi(z_{k})|^{2})\overline{\varphi(z_{k})}^{n}}{(\alpha+n)(1-\overline{\varphi(z_{k})}z)^{\alpha+n+1}}.$

It is not difficult to show that $\\{f_{k}\\}$ is a bounded sequence in
$H(p,q,\phi)$, i.e., there exists a positive constant $C$ such that

(2.4) $\sup_{k}{\|f_{k}\|_{p,q,\phi}}\leq C.$

Also, it is convergent to zero uniformly on compact subsets of $\mathbb{D}$.
Moreover, we have

$f_{k}^{(n)}(\varphi(z_{k}))=0,\ \
f_{k}^{(n+1)}(\varphi(z_{k}))=-\frac{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)\overline{\varphi(z_{k})}^{n+1}}{\phi(|\varphi(z_{k})|)(1-|\varphi(z_{k})|^{2})^{1/q+n+1}}.$

Therefore for any compact operator $T:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$
we have

$\lim_{n\rightarrow\infty}\|Tf_{k}\|_{\mathcal{Z}_{\mu}}=0.$

By these observations we get that

$\displaystyle\|C_{\varphi,g}^{n}-T\|$
$\displaystyle\succeq\limsup_{k\rightarrow\infty}\|C_{\varphi,g}^{n}f_{k}-Tf_{k}\|_{\mathcal{Z}_{\mu}}$
$\displaystyle\succeq\limsup_{k\rightarrow\infty}\|C_{\varphi,g}^{n}f_{k}\|_{\mathcal{Z}_{\mu}}-\limsup_{k\rightarrow\infty}\|Tf_{k}\|_{\mathcal{Z}_{\mu}}$
$\displaystyle\succeq\limsup_{k\rightarrow\infty}\frac{\mu(|z_{k}|)|\varphi^{\prime}(z_{k})g(z_{k})||\varphi(z_{k})|^{n+1}}{\phi(|\varphi(z_{k})|)(1-|\varphi(z_{k})|^{2})^{1/q+n+1}}.$

And so

(2.5)
$\displaystyle\|C_{\varphi,g}^{n}\|_{e}=\inf_{T}\|C_{\varphi,g}^{n}-T\|\succeq\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n+1}}.$

Now we define the sequence $\\{h_{k}\\}$ as follows

$\displaystyle
h_{k}(z)=\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\left(\frac{\alpha+n+1}{(1-\overline{\varphi(z_{k})}z)^{\alpha}}-\frac{\alpha(1-|\varphi(z_{k})|^{2})}{(1-\overline{\varphi(z_{k})}z)^{\alpha+1}}\right).$

It is easy to see that

$\displaystyle h_{k}^{(n)}(z)=$
$\displaystyle\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n-1)(\alpha+n+1)\overline{\varphi(z_{k})}^{n}}{(1-\overline{\varphi(z_{k})}z)^{\alpha+n}}$
$\displaystyle-\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n)(1-|\varphi(z_{n})|^{2})\overline{\varphi(z_{n})}^{n}}{(\alpha+n)(1-\overline{\varphi(z_{k})}z)^{\alpha+n+1}}$
$\displaystyle h_{k}^{(n+1)}(z)=$
$\displaystyle\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n+1)\overline{\varphi(z_{k})}^{n+1}}{(1-\overline{\varphi(z_{k})}z)^{\alpha+n+1}}$
$\displaystyle-\frac{(1-|\varphi(z_{k})|^{2})^{b+1}}{\phi(|\varphi(z_{k})|)}\frac{\alpha(\alpha+1)\cdots(\alpha+n+1))(1-|\varphi(z_{k})|^{2})\overline{\varphi(z_{k})}^{n+1}}{(1-\overline{\varphi(z_{k})}z)^{\alpha+n+2}}.$

Then $\\{h_{k}\\}$ is a bounded sequence in $H(p,q,\phi)$ converging to zero
uniformly on compact subsets of $\mathbb{D}$ and

$h_{k}^{(n)}(\varphi(z_{k}))=\frac{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)\overline{\varphi(z_{k})}^{n}}{\phi(|\varphi(z_{k})|)(1-|\varphi(z_{k})|^{2})^{1/q+n}},\
\ h_{k}^{(n+1)}(\varphi(z_{k}))=0.$

Hence

$\lim_{n\rightarrow\infty}\|Th_{k}\|_{\mathcal{Z}_{\mu}}=0,$

for any compact operator $T:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$. By
definition of the operator norm we have

$\displaystyle\|C_{\varphi,g}^{n}-T\|$
$\displaystyle\succeq\limsup_{k\rightarrow\infty}\|C_{\varphi,g}^{n}h_{k}-Th_{k}\|_{\mathcal{Z}_{\mu}}$
$\displaystyle\succeq\limsup_{k\rightarrow\infty}\|C_{\varphi,g}^{n}h_{k}\|_{\mathcal{Z}_{\mu}}-\limsup_{k\rightarrow\infty}\|Th_{k}\|_{\mathcal{Z}_{\mu}}$
$\displaystyle\succeq\limsup_{k\rightarrow\infty}\frac{\mu(|z_{k}|)|g(z_{k})||\varphi(z_{k})|^{n}}{\phi(|\varphi(z_{k})|)(1-|\varphi(z_{k})|^{2})^{1/q+n+}}.$

Therefore, using the definition of the essential norm we obtain

(2.6)
$\displaystyle\|C_{\varphi,g}^{n}\|_{e}=\inf_{T}\|C_{\varphi,g}^{n}-T\|\succeq\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}.$

From (2.5) and (2.6), the lower estimate of the essential norm is achieved.

Suppose that $\\{r_{j}\\}$ be a sequence in $(0,1)$ and $r_{j}\rightarrow 1$
as $j\rightarrow\infty$. Since
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow{{\mathcal{Z}}}_{\mu}$ is bounded,
then it is clear that
$C_{r_{j}\varphi,g}^{n}:H(p,q,\phi)\rightarrow{{\mathcal{Z}}}_{\mu}$ is also
bounded. Also, since $\|r_{j}\varphi\|_{\infty}<1$, then by Proposition 2.3,
we get that $C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow{{\mathcal{Z}}}_{\mu}$ is
a compact operator for any $j$. Thus

$\displaystyle\|C_{\varphi,g}^{n}\|_{e}\leq\|C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n}\|=\sup_{\|f\|_{p,q,\phi}\leq
1}\|(C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n})f\|_{\mathcal{Z}_{\mu}},\ \ \
j\in\mathbb{N}.$

Now let $f\in H(p,q,\phi)$ and $\|f\|_{Q_{K}(p,q)}\leq 1$. Then

$\displaystyle\|(C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n})f\|_{\mathcal{Z}_{\mu}}=$
$\displaystyle|((C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n})f)(0)|+|((C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n})f)^{\prime}(0)|$
$\displaystyle+\sup_{z\in\mathbb{D}}\mu(|z|)|{((C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n})f)^{\prime\prime}}(z)|$
$\displaystyle\leq$
$\displaystyle|(f^{(n)}(\varphi(0))-f^{(n)}(r_{j}\varphi(0)))g(0)|$
$\displaystyle+\sup_{z\in\mathbb{D}}\mu(|z|)|\varphi^{\prime}(z)g(z)(f^{(n+1)}(\varphi(z))-r_{j}f^{(n+1)}(r_{j}\varphi(z)))|$
$\displaystyle+\sup_{z\in\mathbb{D}}\mu(|z|)|g^{\prime}(z)(f^{(n)}(\varphi(z))-f^{(n)}(r_{j}\varphi(z)))|$
$\displaystyle=$
$\displaystyle|(f^{(n)}(\varphi(0))-f^{(n)}(r_{j}\varphi(0)))g(0)|$
$\displaystyle+\sup_{|\varphi(z)|\leq\delta}\mu(|z|)|\varphi^{\prime}(z)g(z)(f^{(n+1)}(\varphi(z))-r_{j}f^{(n+1)}(r_{j}\varphi(z)))|$
$\displaystyle+\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|\varphi^{\prime}(z)g(z)(f^{(n+1)}(\varphi(z))-r_{j}f^{(n+1)}(r_{j}\varphi(z)))|$
$\displaystyle+\sup_{|\varphi(z)|\leq\delta}\mu(|z|)|g^{\prime}(z)(f^{(n)}(\varphi(z))-f^{(n)}(r_{j}\varphi(z)))|$
(2.7)
$\displaystyle+\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|g^{\prime}(z)(f^{(n)}(\varphi(z))-f^{(n)}(r_{j}\varphi(z)))|,$

where $\delta\in(0,1)$, is fixed. Since $f_{r_{j}}\rightarrow f$ uniformly on
compact subsets of $\mathbb{D}$, where $f_{r}(z)=f(rz)$, then

$|(f^{(n)}(\varphi(0))-f^{(n)}(r_{j}\varphi(0)))g(0)|\rightarrow 0$

as $j\rightarrow\infty$. Also

$\limsup_{j\rightarrow\infty}\sup_{|\varphi(z)|\leq\delta}\mu(|z|)|\varphi^{\prime}(z)g(z)(f^{(n+1)}(\varphi(z))-r_{j}f^{(n+1)}(r_{j}\varphi(z)))|=0$
$\limsup_{j\rightarrow\infty}\sup_{|\varphi(z)|\leq\delta}\mu(|z|)|g^{\prime}(z)(f^{(n)}(\varphi(z))-f^{(n)}(r_{j}\varphi(z)))|=0.$

Here we use (2.2) and (2.3) and the fact that $f_{r_{j}}^{(n+1)}\rightarrow
f^{(n+1)}$ and $f_{r_{j}}^{(n)}\rightarrow f^{(n)}$ uniformly on compact
subsets of $\mathbb{D}$. Also by 2 and Lemma 2.1 we have

$\displaystyle\|(C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n})f\|_{\mathcal{Z}_{\mu}}\leq$
$\displaystyle\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|\varphi^{\prime}(z)g(z)(f^{(n+1)}(\varphi(z))-r_{j}f^{(n+1)}(r_{j}\varphi(z)))|$
$\displaystyle+\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|g^{\prime}(z)(f^{(n)}(\varphi(z))-f^{(n)}(r_{j}\varphi(z)))|$
$\displaystyle\leq$
$\displaystyle\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|\varphi^{\prime}(z)g(z)f^{(n+1)}(\varphi(z))|$
$\displaystyle+\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|\varphi^{\prime}(z)g(z)r_{j}f^{(n+1)}(r_{j}\varphi(z))|$
$\displaystyle+\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|g^{\prime}(z)f^{(n)}(\varphi(z)))|$
$\displaystyle+\sup_{\delta<|\varphi(z)|<1}\mu(|z|)|g^{\prime}(z)f^{(n)}(r_{j}\varphi(z))|$
$\displaystyle\leq$ $\displaystyle
2C\sup_{\delta<|\varphi(z)|<1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n+1}}$
(2.8)
$\displaystyle+2C\sup_{\delta<|\varphi(z)|<1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}.$

So

$\displaystyle\|C_{\varphi,g}^{n}\|_{e}\leq$
$\displaystyle\sup_{\|f\|_{p,q,\phi}\leq
1}\|(C_{\varphi,g}^{n}-C_{r_{j}\varphi,g}^{n})f\|_{\mathcal{Z}_{\mu}}$
$\displaystyle\leq$ $\displaystyle
2C\sup_{\delta<|\varphi(z)|<1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n+1}}$
$\displaystyle+2C\sup_{\delta<|\varphi(z)|<1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}.$

Therefore as $\delta\rightarrow 1$ we obtain

$\displaystyle\|C_{\varphi,g}^{n}\|_{e}\preceq\max\\{$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n+1}},$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}\\}.$

This completes the proof. ∎

If we set $n=1$ in the Theorem 2.6, then we have the following corollary.

######  Corollary 2.5.

Let $p,q\in(0,\infty)$, $\phi$ be normal and $\varphi$ be a analytic self-map
of $\mathbb{D}$. If the operator
$C_{\varphi}^{g}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is bounded, then

$\displaystyle\|C_{\varphi}^{g}\|_{e}\approx\max\\{$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+2}},$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+1}}\\},$

and as a result, $C_{\varphi}^{g}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is
compact if and only if

$\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+2}}=0,\
\ \limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+1}}=0.$

When $n=1$ and $g(z)=\varphi^{\prime}(z)$, then
$C_{\varphi,\varphi^{\prime}}^{1}=C_{\varphi}$, the well-known composition
operator, and we have the next corollary.

######  Corollary 2.6.

Let $p,q\in(0,\infty)$, $\phi$ is normal and $\varphi$ be a analytic self-map
of $\mathbb{D}$. If the operator
$C_{\varphi}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is bounded then

$\displaystyle\|C_{\varphi}\|_{e}\approx\max\\{$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime
2}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+2}},$
$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+1}}\\},$

and as a result, $C_{\varphi}:H(p,q,\phi)\rightarrow\mathcal{Z}_{\mu}$ is
compact if and only if

$\limsup_{|\varphi(z)|\rightarrow 1}\frac{\mu(|z|)|\varphi^{\prime
2}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+2}}=0,\ \
\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|\varphi^{\prime}(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+1}}=0.$

With different choices on $\phi$ and $g$ and the other parameters of the
operator and space, we can obtain some other results.

######  Theorem 2.7.

Let $p,q\in(0,\infty)$, $\phi$ is normal and $\varphi$ be a analytic self-map
of $\mathbb{D}$. If the operator
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{B}_{\mu}$ is bounded then

$\displaystyle\|C_{\varphi,g}^{n}\|_{e}\approx\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}.$

The proof is similar to the proof of Theorem 2.6.

######  Corollary 2.8.

Let $p,q\in(0,\infty)$, $\phi$ is normal and $\varphi$ be a analytic self-map
of $\mathbb{D}$. Then the operator
$C_{\varphi,g}^{n}:H(p,q,\phi)\rightarrow\mathcal{B}_{\mu}$ is compact if and
only if

$\displaystyle\limsup_{|\varphi(z)|\rightarrow
1}\frac{\mu(|z|)|g(z)|}{\phi(|\varphi(z)|)(1-|\varphi(z)|^{2})^{1/q+n}}=0.$

Declarations
Conflict of interest. On behalf of all authors, the corresponding author
states that there is no conflict of interest.
Acknowledgement. Our manuscript has no associate data.

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# Chiral topological light for detecting robust enantio-sensitive observables

N. Mayer1, D. Ayuso2, M. Ivanov1,2,3, E. Pisanty4, O. Smirnova 1,5 1Max-Born-
Institute, Max-Born str. 2A, 12489 Berlin, Germany
2Department of Physics, Humboldt University, Newtonstr. 15, D-12489 Berlin,
Germany
3Blackett Laboratory, Imperial College London, SW7 2AZ London, United Kingdom.
4 Department of Physics, Technical University Berlin, Berlin, Germany

###### Abstract

Topological response of matter to electromagnetic fields is the property in
demand in materials design Yan:2012aa  and metrology gupta_saxena_2014  due to
its robustness against the noise or decoherence, stimulating recent advances
in ultrafast photonics. Galan:2020aa ; Silva:2019aa . Similar topological
robustness is desirable in enantio-sensitive detection but is currently
missing. We introduce the concept of a chiral topological light - a hybrid
between the vortex beam and synthetic chiral light, which interacts with
chiral molecules extremely efficiently due to the electric dipole nature of
this interaction. An integer topological charge is encoded in the azymuthal
evolution of light’s handedness. The topological protection arises due to
mapping of topological charge into the intensity of the far-field profile of
non-linear optical response, where enantio-sensitivity arises as its spatial
rotation. The intensity maxima are robust against noise and enable detection
of percent level enantiomeric excesses in mixtures as we demonstrate
numerically for randomly oriented fenchone molecules. Next we show that using
the topological charge as a parameter one can reconstruct the chiral and
achiral components of non-linear response. Our work opens a way to the new
extremely efficient and robust chiroptical spectroscopies with attosecond time
resolution.

Topological properties of electronic response to electromagnetic fields in
solid state systems as well as in photonic structures are being actively
harvested to obtain robust observables, such as e.g. edge currents protected
with respect to material imperfections in topological insulators
RevModPhys.82.3045  or topologically protected light propagation pathways in
their photonic analogs Khanikaev:2013aa . Similar robustness of enantio-
sensitive optical signals in chiral molecular gases or liquids is much desired
for analytical purposes, but is currently missing. While the first ideas
connecting topological and chiral properties of electronic
responseOrdonez:2022aa ; Ordonez:2021arxiv  or microwave signals in molecular
gasesKai:topological ; Kai:microwave  are starting to emerge, similar ideas
can not be mapped to optical response. Topologically non-trivial optical
signals can be achieved due to vortex beams – beams carrying orbital angular
momentum (OAM) – and characterised by an integer topological charge
representing the number of spiral revolutions of light’s wavefront in space
within one wavelength Shen:2019aa . Vortex light has been successfully used
for chiral detection in the hard X-ray region Rouxel:2022aa ; Rouxel:2022ab ,
however its natural enantio-sensitivity in optical domain is weak: vortex
light is not enantio-sensitive in the electric dipole regime. Here we
introduce the chiral topological light which takes advantage of the global
topological structure of vortex light and high enantio-sensitivity of
synthetic chiral light, for obtaining topologically robust and highly enantio-
sensitive ultrafast optical response.

Synthetic chiral light Ayuso:2019aa  encodes chirality in the temporal
evolution of its three dimensional polarization vector: the Lissajous figure
drawn by the tip of the electric field during one cycle of evolution is
chiral. Its high enantio-sensitivity is due to local, i.e. electric dipole,
nature of induced chiral response. The handedness of light can be controlled
by the phase delay between its several (at least two) frequency components
both locally in every point in space and globally in the interaction region,
offering a unique opportunity to ignite structured in space chiral emitters
with controlled handedness in initially amorphous chiral medium, such as
molecular gas Ayuso:2021aa ; Khokhlova:2021aa . Here we show that this
opportunity can be used to create topologically robust enantio-sensitive
observables.

Our key idea is to imprint the topological parameter of the vortex beam, its
topological charge C, on the azimuthal phase of the chiral correlation
function $h$ characterising the handedness of the synthetic chiral light:
$\arg{h(\theta)}=C\theta+\phi_{L}$, where $\theta$ is azimuthal angle,
$\phi_{L}$ is the enantio-sensitive phase of synthetic chiral light. We show
that the intensity of emission triggered by such light depends on both chiral
and topological phases of $h$ as well as the enantio-sensitive phase
$\phi_{M}$ introduced by molecular medium:
$I(\theta)\propto\cos(\phi_{M}-\phi_{L}+C\theta)$. Using numerical simulations
we demonstrate that azimuthal intensity profile is modulated in a
topologically robust and molecular specific way leading to large enantio-
sensitive offset between intensity maxima (or minima) in opposite enantiomers
$\Delta\theta=\pi/C$. What’s more, the topologically controlled angular off-
set persists for small amounts of enantiomeric excess and can be detected even
in dilute mixtures.

To demonstrate these ideas we now focus on specific realization of chiral
topological light which involves Laguerre-Gaussian beams with bicircular
counter-rotating polarization propagating along the z-axis with frequencies
$\omega$ and $2\omega$ and orbital angular momentum $\ell_{\omega}$ and
$\ell_{2\omega}$. At the focus of the beams $z=0$, the Cartesian components of
the fields in the orthogonal plane $(x,y)$ are:

$\displaystyle\mathbf{E}^{\perp}_{\pm,r\omega}=$
$\displaystyle\mathcal{E}_{r\omega}e^{-\frac{\rho^{2}}{W^{2}_{0}}}\left(\frac{\sqrt{2}\rho}{W_{0}}\right)^{|\ell_{r\omega}|}e^{\text{i}\ell_{r\omega}\theta}e^{\text{i}\phi_{r\omega}}$
(1)
$\displaystyle\frac{(\mathbf{e}_{x}-\text{i}m_{r\omega}\mathbf{e}_{y})}{\sqrt{2}}$

where $\mathcal{E}_{r\omega}=\sqrt{I_{r\omega}}$ is the field strength,
$W_{0}$ is the beam waist, $\phi_{r\omega}$ is the CEP phase,
$\rho=\sqrt{x^{2}+y^{2}}$, $\theta=\arctan(y/x)$ are the polar and azimuthal
coordinates and $m_{r\omega}$ indicates right ($m_{r\omega}=1$) or left
($m_{r\omega}=-1$) circular polarization. Near the focus this field develops a
longitudinal component along the z-axis
$E_{z}=-(\text{i}/k)\nabla_{\perp}\cdot\mathbf{E}_{\perp}$ (see Fig. 1a):

$\displaystyle\mathbf{E}^{z}_{\pm,r\omega}=$
$\displaystyle-\frac{\text{i}\mathcal{E}_{r\omega}}{\sqrt{2}k_{r\omega}}e^{-\frac{\rho^{2}}{W^{2}_{0}}}\left(\frac{\sqrt{2}}{W_{0}}\right)^{|\ell_{r\omega}|}\rho^{|\ell_{r\omega}|-1}$
$\displaystyle
e^{\text{i}(l_{r\omega}+m_{r\omega})\theta}\left(|\ell_{r\omega}|-m_{r\omega}\ell_{r\omega}-\frac{2\rho^{2}}{W^{2}_{0}}\right)\mathbf{e}_{z}$

taking the light polarization vector out of polarization plane (x,y) - a
prerequisite for creating synthetic chiral light. Indeed, the total
bichromatic electric field
$\mathbf{E}(x,y)=\mathbf{E}_{\pm,\omega}+\mathbf{E}_{\pm,2\omega}$, combining
the longitudinal and transverse field components for each color
$\mathbf{E}_{\pm,r\omega}=\mathbf{E}^{\perp}_{\pm,r\omega}+\mathbf{E}^{z}_{\pm,r\omega}$
($r=1,2$, is an example of a synthetic chiral light Ayuso:2019aa .

Figure 1: The concept of chiral vortex light for bicircular counter-rotating
$m_{\omega}=-m_{2\omega}=1$ beams carrying OAMs
$\ell_{\omega}=-\ell_{2\omega}=1$. a) Tight-focusing of bicircular counter-
rotating Gaussian beams induces a longitudinal field, resulting in a synthetic
chiral field whose polarization vector draws a chiral Lissajous over one laser
cycle. b) Evolution of the chiral Lissajous curves with respect to the
azimuthal angle $\phi=\arctan(y/x)$ at a given radial position
$\rho=\sqrt{x^{2}+y^{2}}$ at $z=0$ for a chiral vortex with
$\ell_{\omega}=-\ell_{2\omega}=1$. c) Slices through the electric field
distribution at $z=0$. The figures show the total intensity of the electric
field $|\mathbf{E}|^{2}$, the absolute value of the chiral correlation
function $|h^{(5)}|$ and its phase distribution $\arg\left[h^{(5)}\right]$.
The phase distribution of $h^{(5)}$ describes the spatial distribution of the
handedness of light and is characterized by a topological charge $C=-6$. The
$x$ and $y$ coordinates are scaled to the width $W_{0}$ of the beams at the
focus.

The chiral Lissajous figure drawn by the polarization vector of such light
over a laser cycle is chiral (see inset in Fig. 1a) and its handedness is
controlled by the relative phase between the two colors
$\phi_{2\omega,\omega}=2\phi_{\omega}-\phi_{2\omega}$ Ayuso:2019aa , which in
turn depends on radial and azimuthal coordinates forming a chiral vortex with
the topological charge (see Methods):

$C=\ell_{2\omega}-2\ell_{\omega}+m_{2\omega}-2m_{\omega}.$ (3)

Indeed, the Lissajous curve drawn by the polarization vector of the electric
field over one laser cycle changes with the azimuthal angle, switching
handedness $2C$ times as the azimuthal angle changes by one period (Fig. 1b).
Hence, the superposition of two tightly-focused OAM beams at commensurate
frequencies gives rise to a chiral vortex, i.e. a vortex beam displaying
chirality locally at each given point with an azimuthally varying handedness
characterized by an integer topological charge $C$.

Figure 1c visualises the chiral vortex by displaying the beam total intensity
$|\mathbf{E}(x,y)|^{2}$, the absolute value $|h^{(5)}(x,y)|$ and the phase
$\arg[h^{(5)}(x,y)|$ of the chiral correlation function for OAM
$(\ell_{\omega},\ell_{2\omega})=\left(1,-1\right)$ and SAM
$m_{\omega}=-m_{2\omega}=1$. Both the chirality and the total intensity
maximize along rings (see Figure 1c), typical for vortex beams, while the
topological charge $C=-6$ characterizes the azimuthal phase distribution of
the light’s handedness quantified by the chiral correlation function.

The chiral topological charge $C$ is highly tunable thanks to its dependence
on the OAM of the two beams, which can take any integer value from $-\infty$
to $\infty$, enabling chiral vortices with arbitrarily high chiral topological
charge. OAM- controlled synthetic chiral light offers additional benefit: by
balancing orbital angular momenta of the two beams such that $C=0$, one can
create globally chiral light with a well-defined handedness bypassing the
somewhat limiting requirement of non-collinear beam geometry Ayuso:2019aa .

Figure 2: Enantio-sensitive High-Harmonic spectroscopy using chiral vortices.
a) Schematic depiction of the experimental setup. Two counter-rotating
circularly polarized beams with frequencies $\omega$ and $2\omega$ carrying
OAM $\ell_{\omega}$ and $\ell_{2\omega}$ are tightly-focused on a thin gas jet
of chiral molecules. The resulting synthetic chiral field posses an
azimuthally evolving handedness (inset); the resulting XUV radiation emitted
through the HHG process is detected in the far-field. Chiral discrimination
can be accomplished by measuring the offset angles of the azimuthal
interference patterns in the spatial profile of the high harmonics. b) Near-
field (top row) and far-field (bottom row) spatial profiles of H18 in
(+)Fenchone (left column) or (-)Fenchone (right column), displaying the
enantio-sensitive rotation of the harmonic profiles.

Fig. 2b shows the near-field intensity of harmonic 18 for randomly oriented
(+)Fenchone and (-)Fenchone driven by the chiral topological light with
fundamental frequency $\omega=0.044$ a.u. (1033 nm), peak intensity
$I_{0}=E_{0}^{2}=5\cdot 10^{14}$ W/cm2 and a beam waist of $W_{0}=2.5$
$\mu\text{m}$ at the jet position $z=0$, simulated using our DFT-based
S-matrix approach ( see Methods and Refs.Neufeld:2019aa ; Ayuso:2021aa ). The
peaks in azimuthal direction are due to the constructive interference between
chiral and achiral emission channels which occurs at the angles
$\theta=\left[2\pi n+(\phi_{L}-\phi_{M})\right]/C$; the angular position of
the peaks is therefore enantio-sensitive, as swapping molecular enantiomer
corresponds to $\pi$ shift of the molecular phase
$\phi_{M}\rightarrow\phi_{M}+\pi$. The number of peaks is controlled by the
topological charge $C=-6$ and is a topologically protected quantity.

The far field signal offers additional unique benefits due to spatial
filtering of chiral and achiral signals. The opportunity to efficiently
separate chiral and achiral signal in space arises due to different selection
rules for chiral and achiral quantum pathways, which in chiral topological
light translates into different orbital angular momentum content of respective
signals (Methods). The chiral signal has low orbital angular momentum content,
while the achiral signal has high orbital momentum content: the difference
between the two is controlled by the topological charge and can be large as
e.g. in our case. As a result, chiral signal with low orbital angular momentum
content ($OAM$=1,-1) is only weakly divergent and appears in the center of the
far field image (Fig. 2b), while highly divergent due to high orbital momentum
content ($OAM$=5,-5) achiral signal resides in the outer ring (Fig. 2b). Thus,
in a racemic or achiral medium the inner signal will not be present (Fig. 3b)
offering topologically robust approach to detecting chirality in mixtures with
low enantiomeric excess, which we demonstrate below. Robust detection of small
enantiomeric excess in chiral mixtures relies on the fact that the far-field
images for racemic and non-racemic mixtures have topologically distinct
shapes. The racemic signal is structureless in the inner ring, but even small
enantiomeric excess triggers the $|C|$-fold symmetry of the intensity
distribution (see, 3a,c, where $|C|=6$, for enantiomeric excess $\pm 4\%$ ).

Figure 3: Chiral discrimination using chiral vortices. a,b,c) Far-field
profiles of H18 for a mixture of right-handed and left-handed molecules with
enantiomeric excess $ee=C_{R}-C_{L}$ of -4%, 0 and 4% (a, b and c
respectively), where $C_{R}$ and $C_{L}$ are the concentration of the right-
and left-handed molecules. d) Radially integrated around the inner ring,
angle-resolved far-field H18 signal as a function of the enantiomeric excess
for a chiral vortex with $C=-6$. For each enantiomeric excess we normalize the
profile to its maximum for better visibility. e) Radially integrated around
the outer ring, angle-resolved far-field H18 signal as a function of the
enantiomeric excess for a chiral vortex with $C=-6$. For each enantiomeric
excess we normalize the profile to its maximum for better visibility. f) Phase
of the Fourier component $f_{6}$ of the inner and outer ring as a function of
enantiomeric excess for simulations excluding (solid red line) and including
(solid blue line) noise.

Topological transition in the enantiosensitive signal as a function of
enantiomeric excess and azimuthal angle in the far-field is illustrated in
Figs. 3d,e. Fig. 3d shows the radially integrated signal of the inner ring,
while Fig. 3e shows the radially integrated signal of the outer ring. Both
signals show periodic oscillations as a function of azimuthal angle, with
minima (maxima) occuring $|C|=6$ times. The maxima (minima) of signal
intensity versus azimuthal angle are shifted by $2\pi/|C|=30^{o}$ for positive
and negative enantiomeric excess. Thus, both in inner and outer rings at
$ee=0$ we observe a jump in signal intensity signifying a topological
transition. These topological transitions can be easily detected e.g. by
performing a Fourier analysis of the signal with respect to the azimuthal
angle as a function of enantiomeric excess. The solid red lines in Fig. 3f
show the phase of Fourier component $f_{6}$ oscillating at the $C=6$ frequency
of the outer (top plot) and inner (bottom plot) signals as a function of the
enantiomeric excess. A clear $\pi$ phase jump is observed at $ee=0\%$,
indicating the switch in the handedness of the mixture.

To verify the topological robustness of our signal we have included noise in
our simulations (see the Methods section) via intensity fluctuations of the
driving fields. The blue solid lines Fig. 3f show the phase of the Fourier
component $f_{6}$ when noise is included. It is clear that the $\pi$ jump of
the phase is robust against noise, as it is topologically protected by the
properties of our chiral topological light.

Figure 4: Complete reconstruction of achiral and chiral components of the
right-circularly polarized component (SAM $m=1$) of H18 in Fenchone using a
chiral vortex with $C=-6$ topological charge. a) Far-field profile obtained
from the reconstruction algorithm. b) Far-field profile obtained from the SFA
simulations. c) Normalized difference between the reconstructed and SFA
profiles. d) Phase of the far-field dipole moment obtained from the
reconstruction algorithm. e) Phase of the chiral component obtained from the
reconstruction algorithm. f) Relative phase between the achiral components
obtained from the reconstruction algorithm.

Chiral topological light also offers an exciting opportunity to reconstruct
the phases and amplitudes of both chiral and achiral pathways contributing to
a given harmonic order in a topologically robust way. Such reconstruction
enables complete characterization of underlying chiral electron or vibrational
dynamics on attosecond to femtosecond timescales – the holy grail of
attosecond molecular spectroscopy. Indeed, the far-field signal for a mixture
of right-handed and left-handed molecules is given by

$\displaystyle S_{ee}(\Omega,\mathbf{k})$ $\displaystyle=$
$\displaystyle|A_{ach}(\Omega,\mathbf{k})+eeA_{ch}(\Omega,\mathbf{k})|^{2}=$
(4) $\displaystyle|A_{ach}|^{2}+ee^{2}|A_{ch}|^{2}+$ $\displaystyle
ee\left(A^{*}_{ach}A_{ch}+A_{ach}A_{ch}^{*}\right)$

First we reconstruct the amplitudes of chiral and achiral components and then
we reconstruct the relative phases between them. The amplitude $|A_{ch}|$ of
the chiral component can be found via
$|A_{ch}|^{2}=(S_{ee=1}+S_{ee=-1})/2-S_{ee=0}$.
The amplitude $|A_{ach}|$ of the achiral component can be extracted from the
racemic mixture signal $|A_{ach}|^{2}=S_{ee=0}$.

The analysis of multiphoton pathways (see Methods) shows that while the chiral
component of emission is dominated by a single pathway, two pathways, one of
which is stronger than the other, contribute to the achiral component. For
example for a final SAM co-rotating with the $2\omega$ field, one achiral
pathway (the dominant one) includes the absorption of a circularly polarized
photon $2\omega$ together with a longitudinally polarized $2\omega$ photon and
emission of a longitudinally polarized $\omega$ photon. The second achiral
pathway involves the absorption of 4 $\omega$ longitudinally polarized photons
and the emission of a circularly polarized $\omega$ photon. For the OAM and
SAM of the driving fields considered here ($\ell_{\omega}=m_{\omega}=1$,
$\ell_{2\omega}=m_{2\omega}=-1$) one achiral pathway has final OAM
$\ell_{ach,1}=\ell_{ch}+C$ and the other one has final OAM
$\ell_{ach,2}=\ell_{ch}-C$; thus $\ell_{ach,1}-\ell_{ach,2}=2C$. Thus, the
relative phase between these two achiral pathways oscillates as
$2(C\psi+\phi_{L})$, where $C$ is the topological charge, $\psi$ is the far-
field angle and $\phi_{L}=2\phi_{\omega}-\phi_{2\omega}$ is the laser field
phase, yielding the overall achiral signal:

$\displaystyle|A_{ach}|^{2}$ $\displaystyle=$
$\displaystyle|A_{ach,1}|^{2}+|A_{ach,2}|^{2}+$ (5) $\displaystyle
2\sqrt{|A_{ach,1}||A_{ach,2}|}\cos\left(2(C\psi+\phi_{L})+\phi_{ach}\right)$

Here $\phi_{ach}=\phi_{ach,1}-\phi_{ach,2}$ is the (molecular-specific)
relative phase between the two achiral pathways. In order to find this phase,
we set the phase of the laser field to $\phi_{L}=0$ and fit the signal of the
outer ring (see Fig. 3b) with respect of the azimuthal angle to the fit
function (see SI) $f_{fit}(\psi)\propto\cos\left(2C\psi+\phi_{ach}\right)$.
Armed with the knowledge of $|A_{ach}|^{2}$, $|A_{ch}|^{2}$ and $\phi_{ach}$,
we can now reconstruct the phase of the chiral component $\phi_{ch}$ from the
total signal:

$\displaystyle S_{ee}(\Omega,\mathbf{k})$ $\displaystyle=$
$\displaystyle|A_{ach}(\Omega,\mathbf{k})+eeA_{ch}(\Omega,\mathbf{k})|^{2}=$
$\displaystyle|A_{ach}|^{2}+ee^{2}|A_{ch}|^{2}+ee\left(A^{*}_{ach}A_{ch}+A_{ach}A_{ch}^{*}\right)$
$\displaystyle=$ $\displaystyle|A_{ach,1}|^{2}+|A_{ach,2}|^{2}+|A_{ch}|^{2}+$
$\displaystyle 2\sqrt{|A_{ach,1}||A_{ach,2}|}\cos(2C\psi+\phi_{ach})+$
$\displaystyle 2ee\sqrt{|A_{ach,1}||A_{ch}|}\cos(C\psi+\phi_{ch})+$
$\displaystyle 2ee\sqrt{|A_{ach,2}||A_{ch}}\cos(-C\psi+\phi_{ch}+\phi_{ach})$

To reconstruct $\phi_{ch}$, we first rewrite the achiral magnitudes as
$|A_{ach,2}|^{2}=\lambda|A_{ach,1}|^{2}$, where $\lambda\geq 0$ corresponds to
the ratio between the strength of the two pathways. Next, we minimize the
difference between the numerically-found signal and the fit function (Eq.
Chiral topological light for detecting robust enantio-sensitive observables)
taking the strength of the achiral pathway $|A_{ach,1}|^{2}$, the ratio
$\lambda$ and the phase $\phi_{ch}$ as free parameters. Successful fitting
requires good initial guess for the free parameters. The initial guess for the
the strength of the achiral component is $S_{ee=0}/2$, the ratio of two
achiral channels can be easily estimated from the ratio of longitudinal and
transversal light components, yielding $\lambda=0.01$. As we have no prior
knowledge on $\phi_{ch}$, we repeated the procedure for different initial
guesses looking to obtain the best result.

Fig. 4 shows the results of the reconstruction for H18 and the SAM component
$m=1$ (the results for SAM $m=-1$ are shown in the Supplementary Material).
Figs. 4a,b show respectively the reconstructed and numerically obtained far-
field profile for H18, while the top right figure shows the difference between
the two maps normalized by
$\int\,|f_{rec}(\mathbf{k})+f_{target}(\mathbf{k})|d\mathbf{k}$. The
reconstructed images are in excellent agreement, with errors is on the order
of $10^{-5}$, i.e. 0.01 %. The reconstructed phase of the far-field dipole
moment ($d=d_{ach}+d_{ch}$) is shown in Fig. 4d, while Fig. 4e shows the phase
of the chiral component and Fig. 4f the relative phase between the achiral
components. As shown in the Supplementary Material, the main differences
between the reconstructed and numerically obtained images are noticeable
either on the outer edge of the spatial profile (where the signal intensity is
very weak) or near the center of the vortex at $\mathbf{r}=0$, where its phase
is not well defined. Note that we allow ourselves to change a global overall
phase $\phi_{0}$ in order to match the reconstructed and numerical dipole
moment such that $d_{rec}=d_{num}e^{\text{i}\phi_{0}}$.

We note that the uniqueness of the solution given by the reconstruction
algorithm relies on the analysis of the involved pathways. The hierarchy of
pathways emerges due to weak intensity of longitudinal component: the more
longitudinally polarized photons are involved, the weaker is the pathway with
respect to the dominant ones that include at most two longitudinally polarized
photons.

Chiral topological light opens the way to obtaining robust, topologically
protected enantio-sensitive observables in randomly oriented chiral molecules.
While here we focused on non-linear optical response other exciting
applications include photoionization, transient absorption and enantio-
manipulation using optical forces as well as the ambitious goal of obtaining
quantized enantio-sensitive observables.

## Acknowledgments

Funded by the European Union (ERC, ULISSES, 101054696). Views and opinions
expressed are however those of the author(s) only and do not necessarily
reflect those of the European Union or the European Research Council. Neither
the European Union nor the granting authority can be held responsible for
them. D.A. acknowledges funding from Royal Society (URF/R1/201333) supporting
his research on chirality. E.P. acknowledges financial support by Royal
Society University Research Fellowship URF\R1\211390. M.I. acknowledge support
of the H2020 European Research Council Optologic grant (899794).

## Methods

###  .1 Chiral correlation function

We report here the analytical expression for the chiral correlation function
$h^{(5)}(-2\omega,-\omega,\omega,\omega,\omega)=\mathbf{E}^{*}(2\omega)\cdot\left[\mathbf{E}^{*}(\omega)\times\mathbf{E}(\omega)\right]\left(\mathbf{E}(\omega)\cdot\mathbf{E}(\omega)\right)$
for the general case of two OAM-carrying beams with frequencies $\omega$ and
$2\omega$, SAM $m_{\omega}$ and $m_{2\omega}$ and OAM $\ell_{\omega}$ and
$\ell_{2\omega}$.

$\displaystyle
h^{(5)}(\rho,\theta)=-\frac{\mathcal{E}_{2\omega}\mathcal{E}^{4}_{\omega}}{\sqrt{2}2k^{2}_{\omega}}e^{-5\frac{\rho^{2}}{W^{2}_{0}}}\left(\frac{\sqrt{2}}{W_{0}}\right)^{4|\ell_{\omega}|+|\ell_{2\omega}|}$
$\displaystyle\rho^{4|\ell_{\omega}|+|\ell_{2\omega}|-3}\left(|\ell_{\omega}|-m_{\omega}\ell_{\omega}-\frac{2\rho^{2}}{W^{2}_{0}}\right)^{2}$
$\displaystyle\left\\{\frac{|\ell_{\omega}|-m_{\omega}|\ell_{\omega}|-2\frac{\rho^{2}}{W^{2}_{0}}}{2k_{\omega}}\left[e^{im_{\omega}\theta}(m_{\omega}-m_{2\omega})-\right.\right.$
$\displaystyle\left.e^{-im_{\omega}\theta}(m_{\omega}+m_{2\omega})\right]$
$\displaystyle\left.+\frac{m_{2\omega}}{k_{2\omega}}(|\ell_{2\omega}|-m_{2\omega}\ell_{2\omega}-\frac{2\rho^{2}}{W^{2}_{0}})e^{-im_{2\omega}\theta}\right\\}$
$\displaystyle
e^{i(2\phi_{\omega}-\phi_{2\omega})}e^{i(2\ell_{\omega}+2m_{\omega}-\ell_{2\omega})\theta}$
(7)

It is easy to verify that both in the counter-rotating
$m_{\omega}=-m_{2\omega}$ and co-rotating case $m_{\omega}=m_{2\omega}$ the
azimuthal dependence of the chiral correlation function is given by $C\theta$,
where $C=2(\ell_{\omega}+m_{\omega})-(\ell_{2\omega}+m_{2\omega})$.

###  .2 DFT-based SFA simulations in Fenchone

The dipole moment for the given enantiomer and harmonic order is computed as

$\mathbf{D}(N\omega)=\int d\Omega\int
d\beta\,\mathbf{D}_{\Omega\beta}(N\omega)$

where

$\displaystyle\mathbf{D}_{\Omega\beta}(N\omega)=$ $\displaystyle e^{iN\Omega
t^{\prime}_{r}}a_{rec}\,\mathbf{d}(U_{\Omega\beta}Re[\mathbf{k}(t^{\prime}_{r})])a_{prop}\,$
(9) $\displaystyle
e^{-\text{i}S(\mathbf{p}_{s},t_{i},t_{r})}\,a_{ion}\,\tilde{\Psi}_{\Omega\beta}(U_{\Omega\beta}\text{Re}[\mathbf{k}(t^{\prime}_{i})])$

where $\mathbf{d}(k)$ is the recombination matrix element in the laboratory
frame. Here $U_{\Omega\beta}$ is the rotation matrix that transform the
laboratory to the molecular frame. The integral over the solid angle
$d\Omega=d\alpha d\beta\sin(\beta)$ is performed by Lebedev quadrature, while
the integral over the $\beta$ angle is done by trapezoid method. In order to
find the rotation matrix, we first assume that the x-axis of the molecular
frame $x_{M}$ points toward the given Lebedev point and then rotate by an
angle $\beta$ around the x-axis. For all simulations we used a 17th order
Lebedev quadrature (for a total of 110 points) and 40 $\beta$ angles evenly
distributed on the $[0,2\pi]$ interval.

In the dipole expression, $\mathbf{p}$,
$t_{i}=t^{\prime}_{i}+\text{i}t^{\prime\prime}_{i}$,
$t_{r}=t^{\prime}_{r}+\text{i}t^{\prime\prime}_{r}$ are the complex saddle
momenta and complex times of ionization and recombination.
$S(\mathbf{p},t_{i},t_{r})=2^{-1}\int_{t_{i}}^{t_{r}}dt^{\prime}\,\left[\mathbf{p}+\mathbf{A}(t^{\prime})\right]^{2}+I_{p}(t_{r}-t_{i})$
is the action from the (complex) times of ionization and recombination. The
terms associated with the saddle point method on $(t_{i},t_{r},\mathbf{p})$
are given by

$a(\mathbf{p},t_{i},t_{r})=a_{ion}a_{prop}a_{rec}$
$a_{ion}=\sqrt{\frac{2\pi}{\partial^{2}_{t_{i}}S}}$
$a_{rec}=\sqrt{\frac{2\pi}{\partial^{2}_{t_{r}}S}}$
$a_{prop}=\left(\frac{2\pi}{\text{i}(t_{r}-t_{i})}\right)^{3/2}$

where the second derivatives of the action are given explicitly by

$\partial^{2}_{t_{i}}S=-\mathbf{E}(t_{i})\cdot\mathbf{k}(t_{i})$
$\partial^{2}_{t_{i}}S=\mathbf{E}(t_{r})\cdot\mathbf{k}(t_{r})$

where $\mathbf{E}(t)$ is the electric field. and all expressions for the
prefactor are calculated at the complex times.

The transition matrix elements of the right and left-handed molecules are
related by

$\mathbf{D}_{R}(k)=-\mathbf{D}_{L}(-k)$ (10)

while for the Dyson orbital we use the relation

$\Psi^{R}_{D}(\mathbf{k})=\Psi^{L}_{D}(-\mathbf{k})$ (11)

The matrix elements for Fenchone are calculated using DFT methods described in
Ayuso:2021aa ; TOFFOLI200225 .

###  .3 Noise simulations

In order to include the effect of noise on HHG driven by chiral vortex light,
we take the following approach. For a given electric field strength $E_{0}$
(which we assume to be the same for both fields) the Laguerre-Gaussian beam is
given in the near-field by
$\mathbf{E}(\mathbf{r})=E_{0}\mathbf{LG}_{l,p}(\mathbf{r})$, where
$\mathbf{LG}_{l,p}=LG_{l,p}(\mathbf{r})\mathbf{e}_{L}(\mathbf{r})$. Here
$LG_{l,p}$ is a Laguerre-Gaussian mode and $\mathbf{e}_{L}$ is the
polarization vector of the field. The corresponding laser intensity is
$I_{0}=|E_{0}|^{2}$. We then pick a value for the laser intensity from a
normal distribution of noise centered at $I_{0}$ with width $\gamma$. We call
this electric field intensity $I_{1}$. Then, for each point $\mathbf{r}$ in
the focus, we introduce intensity fluctuations such that at a given position
the electric field strength is given by

$\mathcal{I}(\mathbf{r})=I_{1}\,LG_{l,p}(\mathbf{r})(1+\delta_{I}(\mathbf{r}))$
(12)

where $\delta_{I}(\mathbf{r})=C\lambda(\mathbf{r})$. $\lambda(\mathbf{r})$ is
chosen from a Gaussian distribution centered at zero with width 1 and $C=0.1$
is a constant. There is therefore 68.2% probability that the fluctuation will
be below 0.1% of the signal at the given point. We produce 50 electric fields
using this approach, choosing a central intensity of $I_{0}=5\cdot 10^{14}$
W/cm2 with width $\gamma=3.51\cdot 10^{13}$ W/cm2.

For the results show in Fig. 3f, we scan the enantiomeric excesses $ee$
between $-100\%$ and $100\%$ with steps of $\Delta ee=0.05\%$, and pick
randomly one of the 50 shots for each single $ee$ value. After performing the
Fourier analysis for each $ee$ value, we obtain the phase of the Fourier
component $f_{6}$ by performing a moving average with window size of $\Delta
ee=5\%$.
For the phase reconstruction algorithm, we apply it to a far-field image
obtained by averaging the 50 noisy shots $S_{avg}=\sum_{i=1}^{50}S_{i}$. The
results of the reconstruction algorithm is shown in Figs. 4,5,6 of the
Supplementary Information.

##  I Multiphoton picture

The alternative multiphoton picture of enantiosensitive HHG driven by chiral
vortices can be understood by analyzing the contributing chiral and achiral
multiphoton pathways. To do so, we classify the multiphoton pathways by
indicating with a subscript the SAM of the photon, so that e.g.
$(N)\omega_{+}$ indicate the absorption of $N$ $\omega$ photons with SAM $m=1$
and $(-1)\omega_{z}$ indicates the emission of one $\omega$ photon with SAM
$m=0$.

In the specific case of bicircular counter-rotating fields, if the field has
no longitudinal component along its direction of propagation (i.e. if we
consider an achiral field in the dipole approximation), conservation of SAM
results in an harmonic spectrum with doublets at $3N+1$ and $3N+2$ harmonic
frequencies, where the $3N+1$ harmonics ($3N+2$) co-rotate with the $\omega$
($2\omega$) field Hickstein:2015aa . $3N$ harmonic orders are forbidden in
achiral media, since their generation requires absorption of an equal number
of photons from both drivers. In chiral media, the $3N$ harmonic orders can
instead be generated due to the broken parity of the medium, but are polarized
along the direction of propagation of the fields (the z-axis in our case), and
thus are not detectable in the far-field. We label this pathway as

$C_{z}=\left[(N)\omega_{+},(N)2\omega_{-}\right].$ (13)

Focusing on the specific case of 3N harmonic orders, if the field is chiral
(i.e. if it posses a longitudinal component along the propagation direction)
in the case of achiral media the following multiphoton pathways can now lead
to symmetry-allowed HHG:

$\displaystyle AC_{+}$ $\displaystyle=$
$\displaystyle\left[(N-2)\cdot\omega_{+},(2)\omega_{z},(N-1)\cdot
2\omega_{-}\right]$ (14) $\displaystyle AC_{-}$ $\displaystyle=$
$\displaystyle\left[(N-1)\cdot\omega_{+},(-1)\omega_{z},(N)\cdot
2\omega_{-},(1)2\omega_{z}\right]$ (15)

corresponding respectively to the emission of a photon with SAM $m=1$ and
$m=-1$. We label these pathways as achiral pathways (i.e. $AC_{m}$, with $m$
the SAM of the harmonic photon), since they occur already in achiral media
driven by a chiral field as they require the absorption and emission of an odd
number of photons. If the medium is chiral, two new pathways including
absorption of an equal number of $\omega$ and $2\omega$ photons open, i.e.

$\displaystyle C_{+}$ $\displaystyle=$
$\displaystyle\left[(N)\cdot\omega_{+},(N-1)\cdot
2\omega_{-},(1)2\omega_{z}\right]$ (16) $\displaystyle C_{-}$ $\displaystyle=$
$\displaystyle\left[(N-1)\cdot\omega_{+},(1)\omega_{z},(N)\cdot
2\omega_{-}\right]$ (17)

corresponding again respectively to the emission of a photon with SAM $m=1$
and $m=-1$. We label these pathways as chiral pathways ($C_{m}$) since they
can occur only in chiral media. Finding the corresponding OAM of all pathways
indicated above is straightforward, once we remember that the longitudinal
components of the fields carry OAMs of
$\ell_{\omega_{z}}=\ell_{\omega_{+}}+m_{\omega}$ and
$\ell_{2\omega_{z}}=\ell_{2\omega_{-}}+m_{2\omega}$. Obviously, other chiral
and achiral pathways including the absorption of a larger number of
z-polarized photons from either drivers are also in principle accessible: yet,
since the longitudinal component is relatively weak, we restrict ourselves
here to the photon pathways that include the absorption or emission of the
fewest number of z-polarized photons. Fig. 1a of the Supplementary
Information) shows schematically the multiphoton pathways $C_{z}$, $AC_{m}$
and $C_{m}$ for the case of a 3N harmonic order.

The results from the SFA simulations confirm the considerations above; in Fig.
1b of the Supplementary Information) we show the near-field OAM distributions
for H18 in (+)Fenchone driven by a field with
$\ell_{\omega}=-\ell_{2\omega}=1$ and $m_{\omega}=-m_{2\omega}=1$. For
comparison, we also report the OAM content for an artificial atom with
ionization potential equal to Fenchone driven by the same chiral field and the
OAM content in Fenchone for an achiral field with same OAM of the driving
beams, obtained by manually setting the longitudinal component of the field to
zero.
When the field is achiral, H18 in an atom is absent, while in the case of
Fenchone we observe a $\ell=0$ component polarized along the z-axis: this
corresponds to the pathway $C_{z}$ denoted above. When the field is chiral,
circularly polarized components with $\ell=\pm 5$ are observed for both the
atom and the molecule: these are the achiral pathways $AC_{+}$ and $AC_{-}$
denoted above. Finally, the chiral pathways $C_{+}$ and $C_{-}$ correspond to
the OAMs $\ell=\pm 1$ and are only seen in a chiral molecule, since they
require the absorption of an even number of photons. Note that in the far-
field only the SAM $m=\pm 1$ components are going to be observed, since $m=0$
polarization (corresponding to the $C_{z}$ pathway in black in Fig. 1 of the
Supplementary Information) will propagate in a direction orthogonal with
respect to the propagation axis of the beams.

The different OAM content of an atom and chiral molecule driven by a chiral
bicircular field is directly reflect in the far-field profile of H18, shown in
Fig. 1c of the Supplementary Information). In an atom (left figure of Fig. 1
of the Supplementary Information), where for a given SAM there is only one
contributing OAM, the far-field profile of H18 is a ring where the intensity
is mostly constant, while in Fenchone we observe an azimuthal interference
pattern with periodicity determined by the topological charge $C$,
corresponding in modulus to the net difference between the OAMs of chiral and
achiral pathways. The enantio-sensitive rotation of the spatial profile can be
understood from the perspective of the multiphoton pathways by accounting a
shift by $\pi$ of the phase of the chiral pathways $C_{\pm}$ when changing the
molecular enantiomer. The enantio-sensitive rotation of the spatial profile of
the high-harmonics in the far-field allows one also to use HHG driven by
chiral vortices as a highly-sensitive method to infer the enantiomeric excess
in a mixture of right and left molecular enantiomers.

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