mAP

    Mean Average Precision

# Feature Mixing for Writer Retrieval and Identification on Papyri Fragments

Marco Peer<EMAIL_ADDRESS>0000-0001-6843-0830 Computer Vision Lab, TU
WienViennaAustria and  Robert Sablatnig  0000-0003-4195-1593
<EMAIL_ADDRESS>Computer Vision Lab, TU WienViennaAustria

(2023; 2023)

###### Abstract.

This paper proposes a deep-learning-based approach to writer retrieval and
identification for papyri, with a focus on identifying fragments associated
with a specific writer and those corresponding to the same image. We present a
novel neural network architecture that combines a residual backbone with a
feature mixing stage to improve retrieval performance, and the final
descriptor is derived from a projection layer. The methodology is evaluated on
two benchmarks: PapyRow, where we achieve a mAP of 26.6 % and 24.9 % on writer
and page retrieval, and HisFragIR20, showing state-of-the-art performance
(44.0 % and 29.3 % mAP). Furthermore, our network has an accuracy of 28.7 %
for writer identification. Additionally, we conduct experiments on the
influence of two binarization techniques on fragments and show that binarizing
does not enhance performance. Our code and models are available to the
community111https://github.com/marco-peer/hip23.

Writer Retrieval, Writer Identification, Document Analysis, Historical
Documents, Ancient Papyri

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: 7th International Workshop on Historical Document
Imaging and Processing; August 25-26, 2023; San José, California,
USA††journalyear: 2023††copyright: rightsretained††conference: 7th
International Workshop on Historical Document Imaging and Processing; August
25–26, 2023; San Jose, CA, USA††booktitle: 7th International Workshop on
Historical Document Imaging and Processing (HIP ’23), August 25–26, 2023, San
Jose, CA, USA††doi: 10.1145/3604951.3605515††isbn: 979-8-4007-0841-1/23/08
Figure 1. We propose a novel neural network architecture for the task of
writer retrieval and writer identification of papyri. It is based on mixing
feature maps followed by a learned aggregation layer.

##  1\. Introduction

The retrieval of fragmented documents is an ongoing challenge for scholars and
researchers in the field of cultural heritage (Wolf et al., 2011). However,
manual classification of these documents requires significant human effort and
is not scalable for larger databases. Fortunately, the advent of new
technologies and advancements in deep learning have made the task of fragment
retrieval more feasible and efficient. Although state-of-the-art methods
achieve 80 % Mean Average Precision (mAP) on writer retrieval on historical
databases (Fiel et al., 2017; Christlein et al., 2019), the task of fragment
retrieval presents unique challenges due to the small, damaged, or degraded
nature of many fragments, which can make it difficult to accurately identify
their contents and historical significance (Seuret et al., 2020; Cilia et al.,
2020).

The objective of this paper is to investigate a deep-learning-based approach
to fragment retrieval for papyri, with a particular emphasis on (a)
identifying all fragments associated with a specific writer using a query
fragment from that writer (_writer retrieval_), and (b) identifying all
fragments that correspond to the same image (_page retrieval_). For each
query, the similarities to each document of the database are calculated and a
ranked list is returned. We also present results for the task of _writer
identification_ for further research, where a network is trained on known
identities and fragments are classified according to their writer label. Our
methodology is evaluated on two benchmarks: PapyRow (Cilia et al., 2020), a
dataset of ancient Greek papyri handwriting with currently no existing
evaluation, and HisFragIR20 (Seuret et al., 2020), a large dataset for
fragments of historical documents.

We employ a novel neural network architecture that utilizes a residual
backbone combined with a feature mixing stage, resulting in improved retrieval
performance. The feature mixing is inspired by the attention-based transformer
architecture which proves to be beneficial for vision tasks (Yu et al.,
2022a). We obtain the final descriptor from a projection layer consisting of
fully-connected layers. An overview of the network is shown in Figure 1. Our
approach is supervised and therefore trained on the writer labels of the
fragments. It handles those fragments on an image level, eliminating the need
for interest point detection or local feature extraction. Furthermore, we
explore the impact of binarization by implementing two methods: Sauvola’s
algorithm (Sauvola et al., 1997) and the technique proposed by Christlein et
al. (Christlein et al., 2022) based on U-Net. Our findings indicate that
binarizing the images does not enhance performance, as background statistics
essential for retrieving fragments of the same pages are removed. On papyri
fragments, we achieve an identification rate of 28.7 % and a retrieval score
of 26.6 % mAP. In the end, we show that our network competes with state-of-
the-art methods on the HisFragIR20 dataset.

Our contributions are summarized as follows:

  * •

We propose a new network architecture that does not require a codebook or
attention, instead using a residual network to mix feature maps.

  * •

We provide baseline results for writer identification and writer/page
retrieval tasks on the PapyRow dataset, along with an analysis of the effect
of binarization.

  * •

We evaluate our algorithm on the HisFragIR20 dataset, demonstrating that our
method achieves state-of-the-art performance.

The paper is structured as follows: In Section 2, we cover relevant related
work in the field of writer identification and retrieval of fragments. We
describe our preprocessing steps and the network architecture in Section 3,
followed by details about the evaluation in Section 4 and our results in
Section 5. Finally, we conclude the paper in Section 6.

##  2\. Related Work

In the following, we highlight related work on fragment retrieval as well as
approaches proposed in particular for ancient papyri.

#### Fragment Retrieval

One of the first methods for the automated retrieval of fragments is proposed
by Wolf et al., who study the use of local descriptors of physical attributes
and handwriting of fragments included in the Cairo Genizah collection (Wolf et
al., 2011). Seuret et al. (Seuret et al., 2020) contribute the HisFragIR20
dataset introduced at the ICFHR 2020 Competition on Image Retrieval for
Historical Handwritten Fragments. The winner of this competition on writer
retrieval uses an unsupervised residual network inspired by the work of
Christlein et al. (Christlein et al., 2017) which clusters SIFT descriptors
and train a neural network on those cluster labels. For page retrieval, a
method based on training two different residual networks on writer labels won.
The networks are trained on the full image as well as a cropped view.
Currently, the approach by Ngo et al. (Ngo et al., 2021) using a modified
NetVLAD layer with attention is leading the performance for the HisFragIR20
dataset.

#### Writer Retrieval and Identification for Papyri

Pirrone et al. (Pirrone et al., 2021) investigate a self-supervised approach
for retrieving papyri fragments using a Siamese network with contrastive loss.
They evaluate their method on the Michigan Papyrus Collection and a subset of
HisFragIR20. Meanwhile, Christlein et al. (Christlein et al., 2022) apply a
previously published algorithm of training a network on clustered SIFT
descriptors (Christlein et al., 2017) on the GRK-Papyri (Mohammed et al.,
2019) dataset. They also study binarization on papyri and show that removing
degradation and background artifacts significantly improves their unsupervised
method. The authors of GRK-Papyri present baseline results using local NBNN
(Mohammed et al., 2017), a learning-free algorithm based on SIFT descriptors,
but encounter issues with document degradation. Nasir et al. (Nasir and
Siddiqi, 2020) focus on binarization and neural network training for writer
identification using $512\times 512$ patches. Finally, Cilia et al. published
PapyRow (Cilia et al., 2020) in 2020, which extends GRK-Papyri by adding
documents and providing fragments through line segmentation.

(a) Original

(b) Sauvola

(c) U-Net

Figure 2. Qualitative examples of binarization on PapyRow fragments. The last
two examples show hard samples.

##  3\. Methodology

In this section, we describe the relevant steps of preprocessing as well as
the neural network architecture used for retrieval.

###  3.1. Preprocessing

Due to the heavy degradation of the fragments in the PapyRow dataset, we
employ binarization as a preprocessing step. Our binarization approach
involves two different strategies. Firstly, we apply the traditional algorithm
by Sauvola et al. (Sauvola et al., 1997) to binarize the fragments. Secondly,
we follow the approach proposed by Christlein et al. (Christlein et al.,
2022), where a U-Net is trained with data augmentations based on TorMentor
(Nicolaou et al., 2022) using $512\times 128$ patches of the DIBCO2017 dataset
(Pratikakis et al., 2017). The augmentations are chosen to imitate the
degradation typically contained in the PapyRow fragments, we mainly apply the
so-called _plasma_ augmentations. We present qualitative examples of the two
binarization methods in Figure 2. The U-Net effectively removes major parts of
the degradation, resulting in visually cleaner images. However, Sauvola’s
binarization can still segment the handwriting, though artifacts of the
papyri’s structure are still visible. We analyze the impact of binarization in
Section 5.

###  3.2. Network Architecture

Our approach proposes a three-stage neural network for fragment retrieval,
with the first stage being feature extraction using a residual network, the
second stage being feature mixing with depthwise convolutions, and the third
stage being a projection stage consisting of two fully connected layers to
learn a discriminative fragment descriptor.

#### Feature extraction

To extract features from the input image, we use ResNet34 pretrained on
ImageNet as a residual backbone. We remove the average pooling and flattening
layer and instead use the feature maps of the last residual block.

#### Feature Mixing

Our feature mixing stage is inspired by the work of Yu et al. (Yu et al.,
2022a), who show that the strength of transformer-based architectures does not
lie solely in their attention mechanism but rather in their network structure
(_MetaFormer_). As an alternative to self-attention, they propose various
token mixing layers, such as ConvFormer (Yu et al., 2022b), which incorporate
depthwise convolutions. We suggest using ConvFormer on the feature maps as
tokens to enhance the learned representations. Moreover, token mixing on
feature maps is computationally efficient since we have already extracted the
primary visual elements of the input.

Figure 3. Feature mixing block using separable convolutions.

The feature mixing design, shown in Figure 3, follows the implementation
recommended by Yu et al. (Yu et al., 2022b), and comprises two skip
connections and normalization layers. The first stage is token mixing, which
we refer to as _separable convolution_ : Initially, we expand the channels
using a fully-connected layer followed by StarReLU as an activation function
defined by

(1) $\mathrm{StarReLU}(x)=a\mathrm{ReLU}(x)^{2}+b,$

where $a$ and $b$ are learnable parameters. During depthwise convolution, each
input channel is convolved with its own set of filters instead of all input
filters being aggregated into the output. Subsequently, the channels are
reduced to their original size using a second fully-connected layer. In the
final stage of the block, a scaling layer is included, which learns a
weighting parameter for each output dimension. The feature mixer is composed
of $L$ feature mixing blocks, with typically $L=4$.

#### Projection Layer

To aggregate the feature maps in a meaningful manner, the final stage of the
network consists of two fully-connected layers: Assume the representation
$\boldsymbol{X}\in\mathbb{R}^{C\times(HW)}$, where $C$ corresponds to the
number of channels and $HW$ to the flattened feature map dimension. To reduce
the number of channels, the first linear layer is determined by

(2)
$\boldsymbol{X}_{c}=\boldsymbol{\mathrm{W}}_{c}\boldsymbol{X}+\boldsymbol{\mathrm{b}}_{c},$

with learnable parameters $\boldsymbol{\mathrm{W}}\in\mathbb{R}^{k\times
C},\boldsymbol{\mathrm{b}}\in\mathbb{R}^{k}$. The second layer operates on the
feature map dimension ($HW$) and is described by

(3)
$\boldsymbol{X}_{\mathrm{out}}=\boldsymbol{\mathrm{W}}_{r}\boldsymbol{X}_{c}^{\mathrm{T}}+\boldsymbol{\mathrm{b}}_{r},$

with $\boldsymbol{\mathrm{W}}\in\mathbb{R}^{n\times
HW},\boldsymbol{\mathrm{b}}\in\mathbb{R}^{n}$. Therefore, we obtain a
flattened final representation
$\boldsymbol{\mathrm{X}}_{\mathrm{out}}\in\mathbb{R}^{kn}$ which is
subsequently $l_{2}$-normalized.

###  3.3. Writer Identification and Retrieval

To identify a writer, we forward $\boldsymbol{\mathrm{X}}_{\mathrm{out}}$ to a
classifier consisting of a dropout and a single linear layer. The network is
trained with cross-entropy loss. For writer retrieval, we remove the
classifier during inference and use the descriptor
$\boldsymbol{\mathrm{X}}_{\mathrm{out}}$ to rank the fragments based on the
cosine similarity.

##  4\. Evaluation

In the following, details about the datasets used, our implementation and the
hyperparameters are given. In the end, the evaluation for writer
identification and retrieval is described.

###  4.1. Datasets

We evaluate our algorithm proposed on two datasets: PapyRow(Cilia et al.,
2020) and HisFragIR20 (Seuret et al., 2020).

#### PapyRow

Introduced by Cilia et al. (Cilia et al., 2020), the PapyRow dataset consists
of 6498 fragments of ancient papyri written by 23 scribes. The number of
samples per writer greatly varies, e.g., Kyros2 only contributed 11 samples
while more than 700 fragments are assigned to Dios. PapyRow extends the GRK-
Papyri dataset with an additional corpus provided by different institutions.
The authors preprocess the papyri by applying a line segmentation to ensure
each sample contains a minimum amount of text. Afterwards, image enhancing
methods such as background smoothing are used to reduce the heavy degradation
of the original images. The width of the fragments is standardized to 1200.
There is no official train/test split available.

#### HisFragIR20

Seuret et al. (Seuret et al., 2020) propose the HisFragIR20 dataset,
introduced at the ICFHR 2020 Competition on Image Retrieval for Historical
Handwritten Fragments. While the training set consists of 100k fragments based
on the HisIR19 test dataset, the test set is new and includes 20k fragments
from the 9th to the 15th century. Each sample is generated by applying a
fragment algorithm. The authors provide writer and page labels. Examples are
shown in Figure 4.

Figure 4. Samples of the HisFragIR20 dataset.

###  4.2. Setup

#### Training

We train for a maximum of 50 epochs with a learning rate of $l_{r}=10^{-4}$,
Adam optimizer and a batch size of 128. A linear warmup scheduler is used
during the first two epochs. Afterward, we apply cosine annealing to
$l_{r}/10$. We resize the PapyRow fragments to a size of $512\times 128$ and
the HisFragIR20 images to $512\times 512$ preserving the aspect ratio. For
experiments on the HisFragIR20 dataset, we train our network with the triplet
loss using a margin $m=0.15$ and hard triplets. This is mainly due to the
large size of the training set, where the triplet loss outperforms
classification-based training. All of our networks are trained on the
corresponding writer labels.

#### Network architecture

As a residual backbone, we use ResNet34 where the average pooling and
flattening layer are removed. The depth of our feature mixing network is four.
The channel projection layer reduces the number of channels to $512$, and we
project each feature map to four-dimensional space, yielding a flattened
fragment descriptor of dimension 2048. As a classifier head, we use a linear
layer with softmax and dropout ($p=0.5$).

###  4.3. Evalution Protocol

Our approach is evaluated on two different tasks - writer identification and
writer retrieval.

#### Writer Identification

We cut the PapyRow dataset regarding their page label to avoid bias due to
similarities of the background structures, in particular when using color
images. For each scribe, approximately 30%, 20%, and 50% of the pages are used
for training/validation and testing. The network is trained with the
classification head and we report the accuracy. Each result reported is an
average of five runs with a different split.

#### Writer Retrieval

Writer retrieval is evaluated in a _leave-one-image-out_ validation: Each
fragment of the test set is once used as a query. A ranked list of the
remaining fragments of the test set is returned. For calculating the
similarity score of fragments, we drop the classifier head and use the
$l2$-normalized output of the projection layer as a fragment descriptor. In
our experiments, we whiten the fragment descriptors and reduce their dimension
to 256 via PCA. The ranking is based on the cosine similarity. We provide
results on two tasks: writer and page retrieval. The only difference is the
corresponding label of the fragment (writer ID versus page ID). The
performance is reported in terms of mAP and Top-1 accuracy.

Since no official training and test set is available for the PapyRow dataset,
we perform $k$-fold cross-validation. We split the dataset into four different
parts based on the writers as shown in Table 1. The split is determined by
considering Kyros{1,2} and Victor{1,2,3} as one scribe each to ensure a fair
evaluation, but not for training or inference. Each ID is then used as a
training set and the remaining three parts are used for evaluation. The final
performance is the average of all models trained.

Table 1. PapyRow, writer and page retrieval: $k$-fold validation split. ID |  Writers |  Fragments
---|---|---
1 |  Aparhasios, Ieremias, Konstantinos, Kyros, Philotheos |  1694
2 |  Amais, Dios, Hermauos, Kollouthos, Menas |  1619
3 |  Daueit, Dioscorus, Theodosius, Pilatos, Victor |  1599
4 |  Abraamios, Andreas, Anouphis, Isak, Psates |  1586

##  5\. Experiments

In this section, we provide our experiments. Firstly, we start with the
preprocessing, afterwards we describe our results on writer identification
followed by writer and page retrieval for PapyRow and HisFragIR20.

###  5.1. Preprocessing

Since no ground truth for binarized PapyRow fragments is available, we
validate our U-Net-based approach on a similarly augmented version of the
DIBCO2018 dataset. We use the mean squared error as a loss function. We obtain
a training loss of $\mathcal{L}_{\mathrm{train}}=0.027$ and a validation loss
of $\mathcal{L}_{\mathrm{validation}}=0.028$. The effectiveness is shown by
qualitative samples on PapyRow in Figure 2.

###  5.2. Writer Identification

We evaluate the task of writer identification on the PapyRow dataset.
Additionally, we study the influence on binarization. The accuracies for each
version of the dataset are shown in Table 2.

Table 2. Writer Identification on the PapyRow dataset. | Accuracy
---|---
Color | 28.7
Binarized (Sauvola) | 28.2
Binarized (U-Net) | 27.8

Color images of the PapyRow fragments achieve the best performance with
$28.7~{}\%$, but Sauvola’s algorithm as well as U-Net-binarized images are
close and within $1~{}\%$. Therefore, the network seems to be able to extract
relevant characteristics of handwriting to distinguish the writers.

Figure 5. Qualitative results of writer retrieval. The model used is trained
on split ID 4. The queries are shown on the left. The five most similar
fragments included in the test set are illustrated, from left to right. Red is
a different writer, while green indicates that the fragment is written by the
same scribe. Writer, page, and fragment ID as named in the PapyRow database
are given on top of the fragment.

###  5.3. Writer Retrieval

Next, we investigate the performance of writer retrieval. The writers of the
training and test set are disjunct, so no writer is included in both sets.

#### PapyRow

We train four models using each set of Table 1 once. The average performance
for writer and page retrieval is given in Table 3.

Table 3. Writer and page retrieval on the PapyRow dataset. | Writer | Page
---|---|---
| mAP | Top-1 | mAP | Top-1
Color | 26.6 | 74.0 | 24.9 | 61.8
Binarized (Sauvola) | 20.3 | 47.8 | 12.3 | 31.2
Binarized (U-Net) | 14.0 | 28.2 | 6.0 | 14.2

While the color images perform best for both tasks, we notice a significant
drop in performance (26.6 % to 20.3 % in terms of mAP) when using binarized
versions of the fragments, and Sauvola’s algorithm even outperforms our U-Net.
We think this is mainly due to degradation removed during binarizing which
influences, and improves, the retrieval of color images. The mAP and Top-1
accuracy for page retrieval drop by more than 50 % on the binarized versions,
indicating that fragment descriptors of the same page are more similar than
fragments of one scribe but different pages. Therefore, we argue that our
network relies on background structures for retrieval. In comparison to writer
identification, we train the network on fewer writers but a similar amount of
fragments, which may also affect the generalization ability to handwriting.
The dataset also includes scribes who have contributed only a few papyri (such
as five writers with only one page each), which could potentially skew the
learned representations towards features that are present in the degradation.

#### Qualitative results for Writer Retrieval of PapyRow

Figure 5 presents selected results from our best model, which is trained on
color images, where we illustrate the top five nearest fragments retrieved.
The retrieval performance is good for some fragments, such as Konstantinos or
Dios, with all of the five retrieved fragments belonging to the same or
different pages of the scribe. However, degradation in some fragments, like
for the fragment written by Victor, presents difficulties for the model.

#### Ablation Study

Additionally, we provide an ablation study to give insights into the influence
of each stage of our network. We define the stages as _ResNet34_ as a feature
extractor, _Feature Mixer_ as an additional step to learn meaningful features,
and the _Projection_ to aggregate the feature maps. We use the HisFragIR20
(Seuret et al., 2020) dataset trained on hard triplets. We apply an average
pooling layer for networks without the projection layer to obtain the final
descriptor. In Table 4, our results for writer retrieval are presented.

Table 4. Ablation study: Influence of each stage of our proposed network on the writer retrieval performance (HisFragIR20). Network architecture | mAP | Top-1
---|---|---
ResNet34+AvgPool | 41.0 | 79.6
ResNet34+Feature Mixer+AvgPool | 41.9 | 81.0
ResNet34+Projection | 40.8 | 80.4
ResNet34+Feature Mixer+Projection | 44.0 | 81.9

The ResNet34 with average pooling demonstrated strong retrieval performance,
achieving 41.0 % mAP. Adding our projection layer yields a slightly lower
performance, while plugging in the feature mixing stage results in a
significant gain. Our proposed network performs best, with 44.0 % mAP and 81.9
% Top-1 accuracy.

#### Comparison to state of the art

Finally, we evaluate the effectiveness of our proposed network by comparing it
to state-of-the-art methods on the HisFragIR20 dataset. Our approach
outperforms the competition winner in writer retrieval, but is surpassed by
A-VLAD (Ngo et al., 2021), which uses NetVLAD with an additional attention
layer. Note that the competition winner on writer retrieval uses the same
approach as Christlein et al. (Christlein et al., 2022) for the GRK-Papyri
dataset. However, we set a new state of the art for page retrieval with a mAP
of $29.3~{}\%$ and a Top1-accuracy of $45.0~{}\%$. It is interesting to note
that even a simple supervised ResNet34 (as shown in Table 4) trained on full
fragments surpasses the competition winner by a considerable margin
($41.0~{}\%$ versus $33.7~{}\%$). Table 5 summarizes our comparison to state-
of-the-art methods.

Table 5. Comparison to state of the art on the HisFragIR20 dataset. | Writer | Page
---|---|---
| mAP | Top-1 | mAP | Top-1
Winner ICFHR2020 (Seuret et al., 2020) | 33.7 | 68.9 | 22.6 | 36.4
A-VLAD (Ngo et al., 2021) | 46.6 | 85.2 | n.a | n.a
Ours | 44.0 | 81.9 | 29.3 | 45.0

##  6\. Conclusion

In this paper, we presented an approach for writer retrieval and
identification particularly designed for fragments. We evaluated our method on
two datasets, PapyRow, for which we provided baseline results, and
HisFragIR20. For the latter, we showed that we compete with the state of the
art. The influence of binarization on both tasks was studied, and we conducted
an ablation study on the stages of our network, demonstrating its
effectiveness. Our main findings include that for the task of writer and page
retrieval, binarization harms the performance since background characteristics
are lost. However, further studies are necessary to investigate the features
learned by deep-learning-based methods when training on color images.

To guide future research, we suggest exploring the potential of un- and self-
supervised methods in the context of writer retrieval, as current approaches
are only focused on small patches. Additionally, the limited size of the
PapyRow dataset may contribute to the relatively modest performance scores,
and expanding to more extensive papyri databases could be a solution. Lastly,
we encourage further investigation into different network architectures for
the domain of writer retrieval.

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  *[mAP]: Mean Average Precision


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

# Resonant two-photon ionization of helium atoms studied by attosecond
interferometry

L. Neoričić<EMAIL_ADDRESS>Department of Physics, Lund
University, Box 118, 22100 Lund, Sweden  D. Busto<EMAIL_ADDRESS>Department of Physics, Lund University, Box 118, 22100 Lund, Sweden  Institute
of Physics, Albert Ludwig University, Stefan-Meier-Strasse 19, 79104 Freiburg,
Germany  H. Laurell  Department of Physics, Lund University, Box 118, 22100
Lund, Sweden  R. Weissenbilder  Department of Physics, Lund University, Box
118, 22100 Lund, Sweden  M. Ammitzböll  Department of Physics, Lund
University, Box 118, 22100 Lund, Sweden  S. Luo  Department of Physics, Lund
University, Box 118, 22100 Lund, Sweden  J. Peschel  Department of Physics,
Lund University, Box 118, 22100 Lund, Sweden  H. Wikmark  Department of
Physics, Lund University, Box 118, 22100 Lund, Sweden  J. Lahl  Department of
Physics, Lund University, Box 118, 22100 Lund, Sweden  S. Maclot  Department
of Physics, Lund University, Box 118, 22100 Lund, Sweden  R. J. Squibb
Department of Physics, University of Gothenburg, Origovägen 6B, 41296
Gothenburg, Sweden  S. Zhong  Department of Physics, Lund University, Box 118,
22100 Lund, Sweden  P. Eng-Johnsson  Department of Physics, Lund University,
Box 118, 22100 Lund, Sweden  C. L. Arnold  Department of Physics, Lund
University, Box 118, 22100 Lund, Sweden  R. Feifel  Department of Physics,
University of Gothenburg, Origovägen 6B, 41296 Gothenburg, Sweden  M.
Gisselbrecht  Department of Physics, Lund University, Box 118, 22100 Lund,
Sweden  E. Lindroth  Department of Physics, Stockholm University, AlbaNova
University Center, SE-106 91 Stockholm, Sweden  A. L’Huillier  Department of
Physics, Lund University, Box 118, 22100 Lund, Sweden
<EMAIL_ADDRESS>
###### Abstract

We study resonant two-photon ionization of helium atoms via the $1s3p$, $1s4p$
and $1s5p^{1}$P1 states using the 15th harmonic of a titanium-sapphire laser
for the excitation and a weak fraction of the laser field for the ionization.
The phase of the photoelectron wavepackets is measured by an attosecond
interferometric technique, using the 17th harmonic. We perform experiments
with angular resolution using a velocity map imaging spectrometer and with
high energy resolution using a magnetic bottle electron spectrometer. Our
results are compared to calculations using the two-photon random phase
approximation with exchange to account for electron correlation effects. We
give an interpretation for the multiple $\pi$-rad phase jumps observed, both
at and away from resonance, as well as their dependence on the emission angle.

##  I Introduction

Attosecond techniques using single pulses Goulielmakis  _et al._ (2004);
Schultze  _et al._ (2010) or pulse trains Paul  _et al._ (2001); Klünder  _et
al._ (2011) combined with a synchronized laser field have become essential
tools for the study of photoionization processes in atoms and molecules. The
phase information provided by techniques such as streaking Kienberger  _et
al._ (2004) or RABBIT (Reconstruction of Attosecond Beating By Interference of
Two-photon transitions) Paul  _et al._ (2001) complements cross-section
measurements, giving insight into the photoionization temporal dynamics
Schultze  _et al._ (2010); Klünder  _et al._ (2011). Experiments have been
performed in different atomic Ossiander  _et al._ (2017); Zhong  _et al._
(2020); Alexandridi  _et al._ (2021) and molecular Haessler  _et al._ (2009);
Huppert  _et al._ (2016); Nandi  _et al._ (2020) systems, and in broad photon
energy regions, typically between $20$ and $100$ eV Isinger  _et al._ (2017);
Ossiander  _et al._ (2017). When atoms or molecules are ionized into “flat”,
structureless continua, the phase does not vary much with energy, apart from
the threshold region, which may be affected by the variation of the so-called
Coulomb phase and a laser-induced “continuum-continuum” phase Pazourek  _et
al._ (2015); Dahlström  _et al._ (2013). In contrast, pronounced phase
variations are observed in the vicinity of Cooper minima Klünder  _et al._
(2011); Alexandridi  _et al._ (2021) and broad shape resonances in atomic or
molecular systems Zhong  _et al._ (2020); Nandi  _et al._ (2020).

These studies require broadband radiation, single attosecond pulses or
attosecond pulse trains, typically spanning a few tens of eV in energy, in
order to resolve ultrafast dynamics with time scales in the attosecond regime.
Attosecond techniques may also be applied to the study of two-photon resonant
processes, where the first, extreme ultraviolet (XUV) photon comes into
resonance with a bound Swoboda  _et al._ (2010); Villeneuve  _et al._ (2017);
Drescher  _et al._ (2022); Autuori  _et al._ (2022) or quasi-bound
autoionizing state Haessler  _et al._ (2009); Barreau  _et al._ (2019). The
study of resonant two-photon ionization usually demands a higher spectral
resolution than that of non-resonant processes, and the time scales are in the
femtosecond range. In the early measurements Swoboda  _et al._ (2010); Kotur
_et al._ (2016), the XUV frequency was tuned the across the resonance.
Recently, powerful energy-resolved methods like “rainbow” RABBIT have been
developed Gruson  _et al._ (2016). In particular, the nontrivial phase
variation around the $2s2p$ doubly excited state in helium Gruson  _et al._
(2016); Busto  _et al._ (2018), and the $3s^{-1}4p$ window resonance in argon
Turconi  _et al._ (2020) have been extensively studied. In simple cases like
helium, the temporal dynamics of the wavepacket created by resonant
photoionization can be recovered Gruson  _et al._ (2016); Busto  _et al._
(2018).

Angle detection brings a new dimension, momentum, to this type of measurement,
allowing the reconstruction of coherent electron wavepackets in time and
space. This has been beautifully shown for two-photon resonant ionization of
Ne via the $3p^{5}3d^{1}\text{P}_{1}$ state Villeneuve  _et al._ (2017), and
recently of He via the $1s3p$ and $1s4p^{1}\text{P}_{1}$ states Autuori  _et
al._ (2022). Photoionization studies with angular resolution Heuser  _et al._
(2016); Cirelli  _et al._ (2018); Busto  _et al._ (2019); Autuori  _et al._
(2022) have pointed out that additional phase jumps as a function of emission
angle appear due to the interference of angular channels, beside the phase
jumps as a function of energy.

Here, we study two-photon resonant ionization of helium through intermediate
states in the 1snp ${}^{1}\text{P}_{1}$ Rydberg series with $n=3-5$. We
generate odd high-order harmonics of an infrared (IR) laser field, and tune
the IR laser frequency in order to reach the $1s3p$, $1s4p$ or a coherent
superposition of $1s4p$ and $1s5p$ states with the 15th harmonic. We further
ionize by absorption of an additional IR photon. We analyze the phase of the
created wavepacket by interferometry using another, non-resonant two-photon
process leading to the same final state, namely, absorption of the 17th
harmonic and emission of an IR photon. We perform two series of experiments,
one with angular resolution and moderate energy resolution using a velocity
map imaging spectrometer (VMIS) and one with high energy resolution and no
angular resolution using a magnetic bottle electron spectrometer (MBES). In
the latter case, we also rotate the polarization of the probe field in order
to eliminate one of the angular channels. We discuss the theory of “below-
threshold” interferometry in the angle-integrated and angle-resolved cases and
we compare our experimental results with simulations using the two-photon
random phase approximation with exchange (RPAE) Vinbladh  _et al._ (2019) to
account for electron correlation effects. This allows us to interpret the
multiple phase jumps observed both as a function of energy and emission angle.

##  II Principle of the experiment

Figure 1: Energy levels and excitation processes involved in the present
study. The violet (red) arrows denote excitation by the $15^{\text{th}}$ and
$17^{\text{th}}$ harmonics (fundamental). The spherical harmonics
corresponding to the different angular momentum channels are indicated in
cyan.

Our experiment is based on the RABBIT technique, with a focus on a narrow
spectral region just below the ionization threshold ($I_{p}$) of helium, as
illustrated in Fig. 1. The $15^{\text{th}}$ harmonic (H15) of the fundamental
laser field is tuned into resonance with one or several Rydberg states.
Harmonics of order 17 and above have enough energy to ionize the system and
generate a comb of photoelectron peaks spaced by $2\hbar\omega$. In the
presence of a dressing field with photon energy $\hbar\omega$, two quantum
paths lead to the same final state in the continuum, resulting in the
formation of sidebands (SBs). Varying the relative time-delay between the XUV
and IR pulses causes periodic modulations in the SB16 signal, which can
generally be written as

$I_{\mathrm{SB}}=|A^{+}|^{2}+|A^{-}|^{2}+2|A^{+}||A^{-}|\cos({2\omega\tau-\Delta\varphi)},$
(1)

where $A^{\pm}$ denotes the amplitude of the absorption ($+$) and emission
($-$) paths. The oscillation phase can be decomposed as
$\Delta\varphi=\Delta\varphi_{\text{XUV}}+\Delta\varphi_{\text{A}}$, where
$\Delta\varphi_{\text{XUV}}$ is the group delay of the attosecond pulses and
$\Delta\varphi_{\text{A}}$ originates from the two-photon ionization process.
SB16 differs from the other sidebands as the absorption path is below the
ionization threshold. In a first set of experiments, we record the SB16 signal
as a function of energy and angle. In a second one, we perform an angle-
integrated measurement with high spectral resolution. In both cases, we
extract both the amplitude of the signal oscillating at frequency $2\omega$
and the phase of the oscillation by applying a spectrally resolved RABBIT
analysis Paul  _et al._ (2001); Müller and Laubscher (2001); Gruson  _et al._
(2016). In this implementation of the RABBIT technique, sometimes called
“below-threshold RABBIT” Swoboda  _et al._ (2010); Kheifets and Bray (2021);
Drescher  _et al._ (2022); Autuori  _et al._ (2022), the interesting
information is the phase change due to resonant two-photon ionization, which
is “read” by interferometry with the emission path (absorption of the 17th
harmonic and emission of an IR photon).

##  III Experimental method

A Ti:Sa-based laser operating at 1 kHz provided 3 mJ pulses with a central
wavelength tunable from 780 to 820 nm and a FWHM bandwidth of approximately 40
nm. The pulses were split using a 70-30 beamsplitter, so that the intense
fraction could be used as the pump beam and the weaker one as the probe, as
illustrated in Figure 2(a). The pump was focused into an 8 mm long pulsed gas
cell containing a rare gas, thereby generating odd harmonics of the
fundamental IR field. The resulting spectrum was filtered with a 200 nm thick
aluminium filter so as to suppress the transmission of the pump beam into the
detection region. The probe pulses were directed onto a piezomotor-driven
delay stage and recombined with the XUV pulses in a helium jet. The intensity
of the probe in the interaction region was reduced in order to avoid two-IR-
photon absorption.

Two complementary experimental measurements were performed, as indicated in
Figure 2(b,c). In the angle-resolved experiment (b), a velocity map imaging
spectrometer Eppink and Parker (1997); Rading  _et al._ (2018) was installed
for the electron detection and the helium atoms were provided by an Even-Lavie
valve pulsed at 500 Hz Even (2014). To optimize the 15th and 17th harmonic
signal, the gas target was chosen to be xenon or krypton. The photoelectrons
were detected with a microchannel plate leading to a phosphor screen, recorded
by a camera. The resulting images captured the projection of the momentum of
electrons in the plane perpendicular to their travel. The momentum maps were
recorded as a function of the delay between the pump and probe pulses, in
steps of 250 as. The photoelectron momentum distribution was obtained at each
delay by performing an inverse Abel transform on the VMIS images Smith  _et
al._ (1988). In order to minimize the appearance of mathematical artefacts in
the centre of the inverted images, a well-known problem of inversion methods
based on Legendre polynomial decomposition, an iterative inversion algorithm
based on Vrakking (2001) was applied. The oscillation phase for each energy
and angle bin was extracted by fitting a cosine function to the recorded time-
delay signal.

Figure 2: Experimental setup: (a) Optical interferometer allowing us to create
an attosecond pulse train and a delayed dressing field; (b) Velocity map
imaging spectrometer; (c) Magnetic bottle electron spectrometer. We use the
following abbreviations: BS, beam splitter; A, aperture; FM, focusing mirror;
GC, gas cell; RTS, rough translation stage; PTS, piezo-driven translation
stage; HM, holey mirror; F, filter; TM, toroidal mirror; ELV, Even-Lavie
valve; MCP, microchannel plates.

The high spectral resolution measurements were carried out using an MBES for
the electron detection, with a 2 m long flight tube and a 4$\pi$ sr collection
angle [see Figure 2(c)]. The energy resolution was better than 80 meV at low
kinetic energies. He atoms were provided by an effusive gas jet. The delay
between the two beams was stabilized with an RMS error of less than 20 as.
High-order harmonics were generated in argon. In this experiment, the laser
central wavelength was chosen to be 782 nm and the bandwidth 30 nm, so that
the spectral region covered by the 15th harmonic included the $1s4p^{1}$P1
resonance. The relative contributions of s and d angular channels to the two-
photon transition amplitude were varied by rotating the polarization of the
probe in a direction either parallel or perpendicular to the polarization of
the pump pulse.

##  IV Theoretical method

###  IV.1 Two-photon matrix element

The two-photon transition matrix element connecting the initial state
$\ket{a}$ with the continuum state $\ket{q}$ via all dipole-allowed
intermediate states can be written in the lowest order of perturbation theory
as

$\displaystyle M^{\pm}_{qa}(\Omega,\omega)=-i\lim_{\xi\rightarrow
0^{+}}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{p}\frac{\langle
q\mid ez\mid p\rangle\langle p\mid ez\mid
a\rangle}{\epsilon_{a}+\hbar\Omega-\epsilon_{p}+i\xi}E_{q}E^{\pm}_{1}.$ (10)

Here $\epsilon_{a}$ and $\epsilon_{p}$ represents the energies of the initial
and intermediate states, $E_{q}$, $E_{1}^{+}=E_{1}$ the $q^{\mathrm{th}}$
harmonic and laser fields, with frequencies $\Omega$ and $\omega$,
respectively. In this expression, electron correlation is neglected and
lowest-order perturbation theory for the interaction with the radiation fields
is assumed. In addition, both fields are described as monochromatic and
linearly polarized along the same direction ($z$). Energy conservation implies
that $\epsilon_{q}=\epsilon_{a}+\hbar\Omega\pm\hbar\omega$, where the sign
$\pm$ refers to absorption or emission of the IR photon. Note that in the
latter case, $E^{-}_{1}$ is the complex conjugate of $E_{1}^{+}$. Since the IR
laser field is delayed by $\tau$ relative to the XUV field,
$E_{1}^{\pm}\propto e^{\mp i\omega\tau}$. When the second photon absorption or
emission is above the threshold ($\hbar\Omega>I_{p}$), the two-photon matrix
element is intrinsically complex. When $\hbar\Omega<I_{p}$, which is the case
when the 15th harmonic is absorbed, the lowest-order contribution can be
chosen to be real, but if electron correlation is accounted for, as discussed
in section IV.3, the matrix element is again intrinsically complex.

###  IV.2 Angular momentum channels

The next step consists in using spherical coordinates and introducing the
different angular momentum channels (see Figure 1). Two-photon ionization of
He leads to $s$\- or $d$\- final states, and the angle-resolved two-photon
amplitude can be written as

$\displaystyle
A_{\parallel}^{\pm}(\theta)\propto\sum_{L=0,2}Y_{L,0}\left(\theta,0\right)\left(\begin{array}[]{ccc}L&1&1\\\
0&0&0\end{array}\right)e^{i\left(\eta_{L}-\tfrac{L\pi}{2}\right)}\mathbb{M}_{L}^{\pm}$
(13)

Here $L={0,2}$ is the angular momentum of the final state, $\eta_{L}$ the
energy-dependent scattering phase, sum of the Coulomb phase and a contribution
from the short-range potential. The phase $L\pi/2$ describes the effect of the
centrifugal potential. $Y_{L,0}$ denotes a spherical harmonic, and
$\mathbb{M}_{L}^{\pm}$ is the reduced two-photon matrix element Saha  _et al._
(2021), which depends on the photon energies. The amplitude varies with the
polar angle $\theta$, but due to the cylindrical symmetry, not with the
azimuth. Since the ionization is from an $s$-orbital, the photoelectron can
only occupy an $m=0$ state with respect to the XUV-field polarization.

If the polarization of the laser field is turned 90∘ with respect to the XUV
field, the interaction with the IR field will change the magnetic quantum
number, as defined with respect to the XUV field, by $\Delta m=\pm 1$. Since
the ionization is from an $s$-orbital, only one angular momentum channel, the
$d$-channel, survives.

$\displaystyle A_{\perp}^{\pm}(\theta,\phi)\propto-
Y_{2,1}\left(\theta,\phi\right)\left(\begin{array}[]{ccc}2&1&1\\\
-1&1&0\end{array}\right)e^{i\eta_{2}}\mathbb{M}_{2}^{\pm}$ (16)

In this case, the angular dependence will be that of a $d$-wave.

###  IV.3 Electron correlation

The two-color two-photon RPAE approach Vinbladh  _et al._ (2019) is used to
calculate correlated two-photon matrix elements that include channel coupling,
after both one- and two-photon interaction, as well as ground state
correlation. The method, which is described in detail in Ref. Vinbladh  _et
al._ (2019), is gauge independent and was originally developed for above
threshold ionization, but is here used in a scenario where, in the absorption
path, no individual photon is able to ionize the atom by itself. To facilitate
the comparison with experimental results, experimental energies are used for
the bound excited states that are encountered in the sum over intermediate
states $p$: $1s3p^{1}\text{P}_{1}$, $1s4p^{1}\text{P}_{1}$, and
$1s5p^{1}\text{P}_{1}$. Both time orders of the photon interaction are
considered, but the result is completely dominated by the time order where the
XUV photon is absorbed in the first step, as described by Eq.(10). The
emission path of the RABBIT scan is over a smooth spectral region in the
continuum and the phase change over the small energy interval discussed here
is small. In the absorption path, the situation is very different. The bound
state resonances are narrow, with only light-induced broadening possible (see
below). If monochromatic XUV light is tuned over such an excited state
resonance, the denominator in Eq. (10) abruptly changes sign and a sharp
$\pi$-rad jump of the phase is expected at the resonance energy. Phase jumps
are also to be expected when (and if) the matrix elements in the numerator
change sign.

Figure 3: Absorption matrix elements to the $s$ (red) and $d$ (blue)
continuum. The real and imaginary parts are shown respectively in the solid
and dashed lines. The matrix elements in (a) do not include electron
correlation while the matrix elements in (b) are obtained using RPAE.

Figure 3 presents the real and imaginary parts of the two-photon transition
amplitudes in the absorption path for the $s$ and the $d$ channels without (a)
and with (b) electron correlation. The transition matrix elements from the
ground state to the bound excited states and from these states to the
continuum can be chosen to be real. Therefore, the amplitudes are real-valued
in (a). At resonance, where the amplitude diverges, the phase variation is
simply due to a sign change, while in between resonances it happens because
the amplitude goes to zero. The phase jump in between the resonances happens
at more or less the same energy for both the $s$ and the $d$ channel.

Through the introduction of electron correlation into the ionization process,
the matrix elements acquire an imaginary part leading to a phase shift of the
outgoing photoelectron. The phase variation at resonance is slightly smoother,
while the energy where the phase jumps takes place between two resonances is
now different for the $s$ and the $d$ channels. When incoherently adding the
contributions from the two channels, this leads to a smoother phase variation
between resonances. On the other hand, when these contributions are added
coherently, the resulting phase jump depends on the angle of emission, as
discussed more in detail below.

###  IV.4 Ionization-induced broadening and finite pulse effects

It is possible to go beyond lowest-order perturbation theory for the dressing
field close to resonance in a simple way, by adding an intensity-dependent AC-
Stark shift ($\delta_{p}$) and ionization-induced width ($\gamma_{p}$) to the
resonance energy
($\epsilon_{p}\rightarrow\epsilon_{p}+\delta_{p}+i\gamma_{p}$). The added
complex term in the denominator will induce a change in the phase dependence
across the resonance. Changing $\epsilon_{p}$ into
$\epsilon_{p}+\delta_{p}+i\gamma_{p}$ into the denominator of Eq. (10) leads
to an additional phase term,

$\chi_{p}=\mathrm{arctg}\left[\frac{\gamma_{p}}{\epsilon_{a}+\hbar\Omega-\epsilon_{p}-\delta_{p}}\right].$
(17)

Close to the resonance, the phase varies as
$\mathrm{arctg}(\gamma_{p}/\delta\epsilon)$, where $\delta\epsilon$ goes
through zero. The derivative of this function close to zero (and hence the
slope of the phase variation with energy) is equal to $-1/\gamma_{p}$. Since
$\gamma_{p}$ is proportional to the probe intensity, the absolute value of the
slope of the phase variation across the resonance is expected to decrease as
the intensity increases. In some of the simulations presented below, we
include an ionization-induced width equal to 10 meV, corresponding to a probe
intensity of $6\times 10^{11}$ W/cm2. We do not include any Stark shift,
however, since the origin of the energy axis in the experiment is not
precisely known. Our experimental data are adjusted to the unshifted position
of the resonant states.

To compare with experimental results, we also include bandwidth effects
through convolution with appropriate line profiles. Using Gaussian profiles
defined by the central frequencies $\Omega_{0}$, $\omega_{0}$ and bandwidths
$\sigma_{q}$, $\sigma_{1}$ for the XUV and IT fields respectively, the matrix
elements leading to the same final energy $\epsilon_{q}$ are summed up as

$\displaystyle\\!M^{\pm}_{qa}\\!\\!=\\!\\!\tfrac{1}{2\pi\sigma_{1}\sigma_{q}}\\!\int\\!M^{\pm}_{qa}(\omega_{qa}\\!\\!\mp\\!\omega,\omega)e^{-\frac{(\omega-\omega_{0})^{2}}{2\sigma^{2}_{1}}-\frac{(\omega_{qa}\mp\omega-\Omega_{0})^{2}}{2\sigma^{2}_{q}}}\\!d\omega,$
(18)

where $\hbar\omega_{qa}=\epsilon_{q}-\epsilon_{a}$.

##  V Theory of below-threshold RABBIT in helium

We now generalize the theory of RABBIT to the particular case where absorption
of the lowest harmonic takes place in the discrete spectrum Kheifets and Bray
(2021). We consider two-photon ionization of He, with only a single angular
channel for the first XUV photon absorption. In the angle-resolved case, the
sideband signal is

$I_{\text{SB}}(\theta,\tau)=|A_{\parallel}^{+}(\theta,\tau)+A_{\parallel}^{-}(\theta,\tau)|^{2},$
(19)

where $\tau$ denotes the delay between the XUV and the IR fields, and $\theta$
is the emission angle. The amplitudes for the absorption and emission paths
are defined as

$\\!A_{\parallel}^{\pm}(\theta,\tau)\\!\propto\\!-Y_{00}\sqrt{\tfrac{1}{3}}e^{i\eta_{0}}\mathbb{M}_{0}^{\pm}\\!+\\!Y_{20}(\theta,0)\sqrt{\tfrac{2}{15}}e^{i(\eta_{2}-\pi)}\mathbb{M}_{2}^{\pm}.$
(20)

The coherent addition of the four terms included in Eqs. (19,20) depends on
the phases $\text{arg}[e^{i(\eta_{L}-L\pi/2)}\mathbb{M}_{L}^{\pm}]$. For the
emission path, as in the ordinary RABBIT description with the intermediate
state of the two-photon transition in the continuum, in the asymptotic limit,
we have Dahlström  _et al._ (2013),

$\text{arg}\left(e^{i\left[\eta_{L}-\tfrac{L\pi}{2}\right]}\mathbb{M}_{L}^{-}\right)=\eta_{1}^{-}+\pi+\phi_{cc}^{L-}+\phi_{\text{XUV}}^{-}-\omega\tau.$
(21)

Here $\eta_{1}^{-}$ denotes the scattering phase in the $p$-continuum, at the
energy corresponding to the absorption of the 17${}^{\text{th}}$ harmonic,
$\phi_{cc}^{L-}$ is the additional phase due to the continuum-continuum
emission process, which slightly depends on the angular momentum Fuchs  _et
al._ (2020); Peschel  _et al._ (2021), $\phi_{\text{XUV}}^{-}$ is the phase of
the 17${}^{\text{th}}$ harmonic and $\tau$ is the delay between the XUV and IR
fields. For the absorption path,

$\text{arg}\left(e^{i\left[\eta_{L}-\tfrac{L\pi}{2}\right]}\mathbb{M}_{L}^{+}\right)=\eta_{L}-\tfrac{L\pi}{2}+\phi_{L}+\phi_{\text{XUV}}^{+}+\omega\tau.$
(22)

$\eta_{L}$ is here the scattering phase of the final state ($s$ or $d$),
$\phi_{L}$ reflects the sign changes of the two-photon matrix element at or in
between resonances, as well as electron correlation (see Figure 3) and
$\phi_{\text{XUV}}^{+}$ is the phase of the 15${}^{\text{th}}$ harmonic. In
contrast to the case where both paths are above the threshold, the asymmetry
between the two paths does not allow us to eliminate the scattering phase of
the final state and extract the Wigner time delay Dahlström  _et al._ (2013).
A RABBIT measurement allows, however, the study of the phase variation
($\phi_{L}$) of the two-photon matrix element across the resonance.

###  V.1 Angle-integrated below-threshold RABBIT

Figure 4: (a) Phase $\eta_{L}-L\pi/2$ (blue) and $\phi_{\text{cc}}^{L-}$ (red)
as function of energy for $L=0$ (solid) and $L=2$ (dashed); (b) Reference
phase
$\varphi_{\text{ref}}^{L}=\eta_{L}-\eta_{1}^{-}-\phi_{\text{cc}}^{L-}-L\pi/2-\pi$
for $L=0$ (solid) and $L=2$ (dashed).

In an angle-integrated measurement, the $\theta$-integration of
$I_{\text{SB}}(\theta,\tau)$ eliminates the cross-products between different
angular momentum channels (due to the orthogonality of the spherical
harmonics) which now add incoherently,

$\displaystyle I_{\mathrm{SB}}(\tau)$
$\displaystyle\propto\frac{1}{3}\left[|\mathbb{M}_{0}^{+}|^{2}+|\mathbb{M}_{0}^{-}|^{2}+2|\mathbb{M}_{0}^{+}||\mathbb{M}_{0}^{-}|\cos(\Delta\varphi_{0})\right]$
$\displaystyle+\frac{2}{15}\left[|\mathbb{M}_{2}^{+}|^{2}+|\mathbb{M}_{2}^{-}|^{2}+2|\mathbb{M}_{2}^{+}||\mathbb{M}_{2}^{-}|\cos(\Delta\varphi_{2})\right],$
(23)

where

$\displaystyle\Delta\varphi_{L}=\text{arg}\left(e^{i\left[\eta_{L}-\tfrac{L\pi}{2}\right]}\mathbb{M}_{L}^{+}\right)-\text{arg}\left(e^{i\left[\eta_{L}-\tfrac{L\pi}{2}\right]}\mathbb{M}_{L}^{-}\right).$
(24)

In ordinary (above-threshold) RABBIT,

$\Delta\varphi_{L}=2\omega\tau-\Delta\phi_{\text{XUV}}-\Delta\eta_{1}-\Delta\phi_{\text{cc}}^{L},$
(25)

where $\Delta\eta_{1}=\eta_{1}^{-}-\eta_{1}^{+}$,
$\Delta\phi_{\text{XUV}}=\phi_{\text{XUV}}^{-}-\phi_{\text{XUV}}^{+}$ and
$\Delta\phi_{\text{cc}}^{L}=\phi_{\text{cc}}^{L-}-\phi_{\text{cc}}^{L+}$.
Neglecting the small $L$-dependence of $\Delta\phi_{\text{cc}}$ Peschel  _et
al._ (2021), the two oscillatory terms in Eq. (23) are in phase, meaning that
RABBIT measurement allows to unambiguously determine the phase
$\Delta\eta_{1}$ Klünder  _et al._ (2011); Dahlström  _et al._ (2013) and the
Wigner delay approximated by $\Delta\eta_{1}/2\omega$. When one path is below
threshold, however,

$\Delta\varphi_{L}=2\omega\tau-\Delta\phi_{\text{XUV}}+\varphi_{\text{ref}}^{L}+\phi_{L},$
(26)

with
$\varphi_{\text{ref}}^{L}=\eta_{L}-\eta_{1}^{-}-\phi_{\text{cc}}^{L-}-L\pi/2-\pi$.
Eq. 23 describes the incoherent addition of two terms oscillating with the
same frequency $2\omega$, but with different phase variation as a function of
energy ($\phi_{L}$) and different reference phase
($\varphi_{\text{ref}}^{L}$), complicating the interpretation of the extracted
phase. Although both $\eta_{L}$ and $\phi_{\text{cc}}^{L-}$ vary significantly
close to the threshold due to the influence of the Coulomb potential, as shown
in Figure 4 (a), the reference phase $\varphi_{\text{ref}}^{L}$ does not
depend much on the energy or on $L$ in the range investigated in the present
work [Fig. 4 (b)]. This implies that when $\phi_{0}\simeq\phi_{2}$, for
example at resonance, both terms in Eq. (23) vary with a similar phase offset.
However, when $\phi_{L}$ changes sign because the $L$-dependent matrix element
goes to zero, which happens at different energies for the $s$ and $d$
channels, the interpretation of the phase extracted from the experimental
measurement is not clear. This is where angle-resolved measurements or the use
of different polarizations for the excitation fields become useful.

###  V.2 Angle-resolved below-threshold RABBIT

We first rewrite Eq. (20) as

$A_{\parallel}^{\pm}(\theta)\propto
e^{i\eta_{0}}\mathbb{M}_{0}^{\pm}+\sqrt{2}P_{2}(\cos\theta)e^{i\eta_{2}}\mathbb{M}_{2}^{\pm},$
(27)

where we have introduced $P_{2}$, the Legendre polynomial of order 2, equal to
$(3x^{2}-1)/2$. Using Eqs. (21,22), we have

$\displaystyle A_{\parallel}^{-}(\theta)+A_{\parallel}^{+}(\theta)\propto$
(28) $\displaystyle\ \
e^{i(-\omega\tau+\phi_{\text{XUV}}^{-}+\eta_{1}^{-})}\left[e^{i\phi_{\text{cc}}^{0-}}|\mathbb{M}_{0}^{-}|-\sqrt{2}P_{2}(\cos\theta)e^{i\phi_{\text{cc}}^{2-}}|\mathbb{M}_{2}^{-}|\right]$
$\displaystyle-e^{i(\omega\tau+\phi_{\text{XUV}}^{+})}\left[e^{i(\eta_{0}+\phi_{0})}|\mathbb{M}_{0}^{+}|+\sqrt{2}P_{2}(\cos\theta)e^{i(\eta_{2}+\phi_{2})}|\mathbb{M}_{2}^{+}|\right].$

We define $\chi^{\pm}(\theta)$ as the argument of the quantities inside the
brackets. The angle-resolved RABBIT signal can be written as

$\displaystyle I_{\text{SB}}$
$\displaystyle(\theta)=|A_{\parallel}^{+}(\theta)+A_{\parallel}^{-}(\theta)|^{2}\propto|A_{\parallel}^{+}(\theta)|^{2}+|A_{\parallel}^{-}(\theta)|^{2}$
(29)
$\displaystyle+2|A_{\parallel}^{+}(\theta)||A_{\parallel}^{-}(\theta)|\cos[2\omega\tau-\Delta\phi_{\text{XUV}}-\eta_{1}^{-}-\Delta\chi(\theta)],$

where $\Delta\chi(\theta)=\chi^{-}(\theta)-\chi^{+}(\theta)$. In contrast to
the angle-integrated result, we now have a single oscillatory signal. The
price to pay is that the phase depends not only on the energy but also on the
emission angle $\theta$. Eq. (29) can be expanded into a Legendre polynomial
decomposition as in Fuchs  _et al._ (2020); Peschel  _et al._ (2021). We will
not do that in the present work, but concentrate on the understanding of the
phase variation with emission angle.

###  V.3 Angular phase jumps

In contrast to what we have discussed so far, which is the phase difference
between the absorption and the emission paths [Eqs. (29,26)], we now consider
the phase difference between the angular channels for the absorption and the
emission paths individually,

$\displaystyle\Delta\varphi^{+}=\eta_{0}-\eta_{2}+\phi_{0}-\phi_{2}$ (30)
$\displaystyle\Delta\varphi^{-}=\phi_{cc}^{0-}-\phi_{cc}^{2-}-\pi.$ (31)

Factorizing the phase terms, the quantities inside the bracket in Eq. (28) are
proportional to
$\exp(i\Delta\varphi^{\pm})|\mathbb{M}_{0}^{\pm}|+\sqrt{2}P_{2}|\mathbb{M}_{2}^{\pm}|$.
There will be a phase jump in the path considered (emission or absorption) as
a function of the emission angle whenever the real part of this expression
changes sign. Introducing

$\Upsilon^{\pm}=\frac{|\mathbb{M}_{0}^{\pm}|}{\sqrt{2}|\mathbb{M}_{2}^{\pm}|}\cos\Delta\varphi^{\pm},$
(32)

a phase jump will happen at an angle $\theta$ such that

$P_{2}(\cos\theta)=-\Upsilon^{\pm}.$ (33)

This equation has a solution only if

$-1\leq\Upsilon^{\pm}\leq\frac{1}{2}.$ (34)

These two inequalities set conditions for a sign change as a function of
angle, and hence a phase jump, in the absorption or emission paths. We also
note that when $|\mathbb{M}_{0}^{\pm}|\ll\sqrt{2}|\mathbb{M}_{2}^{\pm}|$, or
when $\Delta\varphi^{\pm}\approx\pm\pi/2$, a phase jump will happen at an
angle such that $P_{2}(\cos\theta)$ is close to zero, i.e. at a node of
$Y_{20}$, $54.7^{\circ}$ or $125.3^{\circ}$ (often referred to as “magic”
angles). In contrast, if
$|\mathbb{M}_{0}^{\pm}|\gg\sqrt{2}|\mathbb{M}_{2}^{\pm}|$ and
$\Delta\varphi^{\pm}\neq\pm\pi/2$, the inequality (34) is not fulfilled,
leading to an absence of a phase jump as a function of angle.
$\Delta\varphi^{-}$ is positive and, in general, small, increasing close to
the threshold Fuchs  _et al._ (2020). We find that $\Upsilon^{-}\geq 1/2$ so
that no phase jump can be found in the emission path. The absorption path will
in contrast exhibit different energy regions with or without a phase jump,
depending on whether Eq. (34) is fulfilled or not. Because of the non-zero
imaginary part of
$\exp(i\Delta\varphi^{\pm})|\mathbb{M}_{0}^{\pm}|+\sqrt{2}P_{2}|\mathbb{M}_{2}^{\pm}|$,
the phase jump as a function of angle is smaller than $\pi$ rad. The direction
of the phase jump depends on the sign of $\Delta\varphi^{+}$. This is
illustrated in Figure 5, showing both $\Upsilon^{\pm}$ as a function of
energy, and the phase variation of the absorption path as a function of angle
(vertical axis) and energy (horizontal axis). The black dashed lines indicate
the different energy regions, where a phase jump as a function of angle is to
be or not be expected.

Figure 5: Phase variation of the absorption path as a function of angle and
energy. The top figure shows the variation of $\Upsilon^{\pm}$ with energy
together with the limits of the inequality in Eq. 34 (dashed horizontal black
lines). The absorption path is shown in blue and the emission path, which does
not lead to any phase jump, in magenta. The black vertical dotted lines
indicate the energies at which $\Upsilon^{+}$ intersects the black dashed
lines. This indicates the limits of the spectral region displaying phase jumps
in angle. The vertical red dotted lines indicate the energies at where
$\Upsilon^{+}$ changes sign, leading to a change in the direction of the phase
jump in angle. At these energies, the phase jump occurs at the magic angle.
The cyclic color bar is shown in the top left corner of the bottom figure.

###  V.4 Delay-integrated asymmetry parameters

The sideband signal can be expressed using the angle-integrated signal,
$I_{\text{SB}}(\tau)$, and two delay-dependent asymmetry parameters
($\beta_{i}(\tau)$, $i=2,4$) as Cooper and Zare (1968); Busto  _et al._
(2019); Joseph  _et al._ (2020)

$I_{\text{SB}}\left(\theta,\tau\right)=\frac{I_{\text{SB}}(\tau)}{4\pi}\left[1+\sum_{i=2,4}\beta_{i}(\tau)P_{i}(\cos\theta)\right]$
(35)

Another way to parameterize the sideband angular dependence Saha  _et al._
(2021) is to extract the angle-integrated signal and asymmetry parameters,
$I_{\text{SB}}^{\pm}$ and $\beta_{i}^{\pm}$ for the absorption and emission
processes separately, and then similar quantities for the oscillating cross
term $A_{\parallel}^{+*}A_{\parallel}^{-}$ (or its complex conjugate). The
delay-integrated angular $\beta_{i}$-parameters Joseph  _et al._ (2020) can be
simply obtained as Saha  _et al._ (2021)

$\displaystyle\beta_{i}=\frac{{\beta_{i}^{+}I_{\text{SB}}^{+}+\beta_{i}^{-}I_{\text{SB}}^{-}}}{I_{\text{SB}}^{+}+I_{\text{SB}}^{-}}.$
(36)

They are the sum of the ordinary $\beta$-parameters for the absorption and
emission paths, weighted with the relative strength of each path.

##  VI Experimental results

###  VI.1 Angle-integrated measurements

Figure 6: Two-photon ionization via $1snp$ states with $n=3,4,5$. (a) Angle-
integrated intensity. Red line: measurement, black line: RPAE-based
simulations. Relative heights of individual peaks are not to scale. (b) Angle-
integrated phase. Red solid line: measured data, shaded areas: error of the
weighted average. RPAE-based simulations excluding (grey dashed line) and
including (black solid line) the effect of the pulse bandwidth and power-
induced broadening ($\gamma_{p}=10$ meV). Vertical grey lines indicate the
energies corresponding to the $1snp^{1}\text{P}_{1}$ resonances with
$n=3,4,5$.

Figure 6 presents results obtained with the velocity map imaging spectrometer.
We show the angle-integrated intensity (a) and phase (b) for sideband 16, as a
function of the total energy absorbed in the two-photon ionization process.
Several measurements have been performed, such that the 15th harmonic spans a
large energy region across the $1snp^{1}\text{P}_{1}$ (n=3-5) series. Figure
6(a) presents three broad maxima, approximately centered at the position of
the resonances. The relative intensities of the maxima is arbitrary, since the
three measurements have been performed with different central laser
frequencies. Therefore, we normalize the maxima in Figure 6(a). The broadening
is partly due to the XUV and IR bandwidths leading to a convolution of the
amplitudes (see Eq. 18), to the ionization-induced additional width, as well
as to the spectrometer resolution. The first peak corresponding to the
$1s3p^{1}\text{P}_{1}$ resonance exhibits a double structure, that could be
due to a continuum structure induced by the interference between direct non-
resonant two-photon ionization and ionization from the $1s3p^{1}\text{P}_{1}$
bound state Knight  _et al._ (1990). Such a feature has been previously
observed in photoelectron spectroscopy Autuori  _et al._ (2022), transient
absorption measurements Chini  _et al._ (2013), as well in numerical
simulations Swoboda  _et al._ (2010).

Figure 6(b) shows the phase variation of the angle-integrated sideband 16
intensity. The presented curves are obtained by performing a weighted
averaging over a few measurements, as detailed in the SM. Phase jumps of
approximately $-\pi$ rad can be observed when the 15th harmonic energy becomes
equal to that of the $np^{1}\text{P}_{1}$ states. Likewise, phase jumps of
opposite sign and similar magnitude can be seen in between each two successive
resonant jumps Autuori  _et al._ (2022); Drescher  _et al._ (2022). The
splitting of the $1s3p^{1}\text{P}_{1}$ state is also accompanied by a phase
variation.

Fig. 6 further shows the results of a two-photon RPAE calculation, not
including (dotted black curve, only shown in (b)) and including (black curve)
bandwidth effects, with $\sigma_{1}$= 25 meV, $\sigma_{q}$= 150 meV, as well
as an additional width due to ionization from the Rydberg state
($\gamma_{p}=10$ meV). These simulations reproduce the behavior of the
intensity and phase variation across the resonances.

###  VI.2 Parallel and perpendicular polarizations

High-spectral-resolution angle-integrated RABBIT measurements were performed
in the energy region around the $1s4p^{1}\text{P}_{1}$ resonance. Our aim was
to compare measurements with the dressing field polarization parallel
($0^{\circ}$) and perpendicular ($90^{\circ}$) to that of the XUV, while
keeping the probe intensity at the same value. Both s and d channels coexist
in the parallel configuration, while only the d channel is allowed when the
probe polarization is perpendicular to that of the pump, since only $\Delta
m=\pm 1$ transitions are possible. In these measurements, we normalize the
phase variation to that of the non-resonant sideband 18, which exhibits a flat
phase. A comparison between parallel and perpendicular IR polarizations is
shown in Figure 7. The black and red symbols and solid curves correspond to
the parallel and perpendicular case, respectively. We observe a smooth phase
variation across the resonance rather than a sudden $\pi$ jump, in agreement
with the results shown in Figure 23. In order to reproduce this behavior in
theoretical calculations, it is necessary to include both ionization
broadening and folding due to finite pulse effects. In the region
investigated, close to the $1s4p$ resonance, the phase variation of the $s$
and $d$ channels are very similar, so that the parallel or perpendicular
configurations do not yield very different phase measurements.

Figure 7: Angle-integrated SB16 phase measured with the magnetic bottle
electron spectrometer. Data points: parallel (red) and perpendicular (black)
XUV and IR polarizations; solid curves: calculated total phase for parallel
XUV and IR polarizations (red) and d-wave only phase (black).

###  VI.3 Delay-integrated photolectron angular distributions

Figure 8(a,b) presents the variation of the $\beta_{2}$ and $\beta_{4}$
parameters for the delay-integrated signal. These coefficients vary
significantly across the observed energy range, with maxima close to the
$1s3p^{1}\text{P}_{1}$, $1s4p^{1}\text{P}_{1}$ and $1s5p^{1}\text{P}_{1}$
resonances. The experimental behavior is reproduced by the calculations, here
shown without including any folding, i.e. assuming a monochromatic dressing
field, at a wavelength of 782 nm. The strong variation of the $\beta_{2}$ and
$\beta_{4}$ parameters between two resonances indicates a varying angular
distribution, due to a change in the relative amplitude and phase of the $s$
and $d$ angular channels.

Figure 8: Energy variation of the delay-integrated asymmetry parameters
$\beta_{2}$ (blue) and $\beta_{4}$ (red); (a) Experimental results; (b)
Theoretical calculations obtained at a given dressing field wavelength, equal
to 782 nm. The grey lines indicate the energies corresponding to the $1snp$
1P1 resonances with n=4,5.

###  VI.4 Angle-resolved phase measurements

We here concentrate on the spectral region around the $1s4p^{1}\text{P}_{1}$
resonance. Figure 9 displays the phase variation as a function of angle and
energy. The phase is indicated in color, following a cyclic representation,
thus avoiding unwrapping issues. The phase decreases (increases) when moving
clockwise (anti-clockwise) in the cyclic color bar. Fig. 9(a,right) shows the
measured values in the angle and energy region, delimited by a dashed contour,
where the signal is sufficiently high for a reliable phase and amplitude
retrieval. Fig. 9(a,left) shows the results of the two-photon-RPAE
calculation. The two results agree quite well, and in the following, we focus
on the interpretation of the theoretical results, which give a more complete
picture of the phase variation as a function of both energy and angle. In
order to gain a deeper insight into the interplay of the different angular
channels, we show the sideband variation where we selectively suppress the
resonant path to the $s$ continuum (b,left) or to the $d$ continuum (b,right).

Figure 9:  Angle-and-energy resolved relative phase of SB16 around the 4p
resonance. (a) Comparison between experiment (right) and theory (left). Only
the experimental results with significant signal-over-noise ratio are
presented. This region is indicated by the black dashed line. The radial
coordinate is the total energy, while the polar angle is the electron emission
angle relative to the polarization axis. The cyclic color bar is shown in the
top right of the figure. In (b), we present the contribution of the $s$
(right) and $d$ (left) waves in the absorption path to the total phase
variation.

In general, the phase variation observed in Fig. 9(a,left) is very similar to
that in Fig. 9(b,left), indicating that the overall phase variation in angle
and energy is dominated by that of the resonant path to the $d$ wave. In the
following we focus on interpreting the phase variation in the vicinity of
three spectral regions indicated with coloured half circles in Fig. 9(a).

First at the position of the 4$p$ resonance, indicated by a red half circle
close to 25.31 eV, a sharp $\pi$ rad phase jump along the energy axis is
observed at all emission angles. In this region both the $s$ and $d$ channels
experience a $\pi$ jump due to the resonance such that angle resolved
measurements do not provide additional information compared to angle
integrated measurements.

Another phase jump along the energy axis can observed in the vicinity of the
green half circle. In this region, the matrix element $\mathbb{M}_{2}^{+}$
crosses 0 and changes sign such that the relative weight of the $s$ and $d$
resonant paths varies very fast as a function of energy. As a result, a strong
angular dependence of the position of the phase jump can be observed in Fig
9(a,left): at angle above 54.7 degrees the phase jump appears above the green
line, while at angles below the magic angle, the phase jump appears at
energies below the green line. This is consistent with the fact that
$\mathbb{M}_{2}^{+}$ and $Y_{20}$, should have opposite signs in order for the
real part of $A^{+}_{\parallel}(0)$ to cancel.

At 54.7 degrees, where $Y_{20}$ has a node, the phase variation as a function
of energy is fully dominated by the $s$ channel as can be observed by
comparing Fig. 9(a,left) with Fig. 9(b,right). In addition, in the close
vicinity of the spectral region where $\mathbb{M}_{2}^{+}$ cancels no phase
variation as a function of emission angle is observed in (a,left), since the
$s$ channel dominates. This corresponds to the spectral region shown in Fig.
5, where the inequalities in Eq. 34 are not fulfilled.

In contrast to the region close to the 4$p$ resonance, here angle resolved
measurements provide a lot of information on the competition between the $s$
and $d$ resonant channels that cannot be obtained from angle-integrated
measurements.

Finally, over a large energy range, from 25 eV to 25.3 eV, a sharp phase jump
can be observed as a function of angle close to 54.7 degrees. In this spectral
region the $d$ resonant channel strongly dominates and the phase jump as a
function of angle reflects the change of sign of $Y_{20}$. The exact angle at
which the phase jumps and the magnitude of the phase jump depends on the
relative weight between $s$ and $d$. When $\mathbb{M}_{0}^{+}=0$ (blue
circle), the phase exhibits a sharp pi jump exactly at the magic angle, while
the phase jumps occurs at lower(higher) angles below(above) the blue line.
Interestingly, although the d channel dominates, the change of sign of
$\mathbb{M}_{0}^{+}$ across the blue line manifests as a change of direction
of the angular phase jump.

This analysis shows that by examining the positions of the phase jumps as a
function of angle and energy, it is possible to infer not only the position of
the resonance, but also the positions of the sign changes of
$\mathbb{M}_{L}^{+}$, even in the case where two-photon ionization is very
much dominated by one contribution, the $d$ channel. This also shows that
phase measurements as a function of angle and energy give a lot of information
about the relative contributions of the two angular channels.

##  VII Conclusion

In conclusion, we have studied the phase and intensity of two-photon
ionization through the $1s3p^{1}\text{P}_{1}$, $1s4p^{1}\text{P}_{1}$ and
$1s5p^{1}\text{P}_{1}$ resonance series, using an interferometric method.
While the intensity variation presents a series of peaks at the positions of
the resonance, the phase undergoes $\pi$-rad phase changes both at and in
between the resonances. Going through the resonance, the phase decreases
linearly with energy with a total phase variation of $\pi$ rad at all emission
angles. The smaller the ionization induced by the dressing field, the more
rapid this linear decrease in phase is. The phase variation remains similar
when varying the relative polarization of the XUV and dressing fields. A
second sharp phase variation of $\pi$ rad is observed between two resonances.
This phase variation is strongly dependent on the emission angle and reflects
the angle- and polarization- dependent interplay between the two angular
channels contributing to the two-photon ionization process.

We have shown both experimentally and theoretically that, even for a simple
atomic system like helium, angle-resolved measurements, as well as
measurements with different polarizations, provide invaluable insight to the
physics of resonant two-photon ionization. They should become an essential
tool to the investigation of more complex atoms or molecules.

## Acknowledgements

The authors acknowledge support from the Swedish Research Council (2013-8185,
2016-04907, 2017-04106, 2018-03731, 2020-0520, 2020-03315, 2020-06384), the
Swedish Foundation for Strategic Research (FFL12-0101), the European Research
Council (advanced grant QPAP, 884900) and the Knut and Alice Wallenberg
Foundation. AL and MA are partly supported by the Wallenberg Center for
Quantum Technology (WACQT) funded by the Knut and Alice Wallenberg foundation.
JL acknowledges financial support by the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie Grant
Agreement 641789 (MEDEA).

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