# Uncovering Bias in Face Generation Models

Cristian Muñoz
Holistic AI
<EMAIL_ADDRESS>
& Sara Zannone
Holistic AI
<EMAIL_ADDRESS>
& Umar Mohammed
Holistic AI
<EMAIL_ADDRESS>
& Adriano Koshiyama
Holistic AI
<EMAIL_ADDRESS>

###### Abstract

Recent advancements in GANs and diffusion models have enabled the creation of
high-resolution, hyper-realistic images. However, these models may
misrepresent certain social groups and present bias. Understanding bias in
these models remains an important research question, especially for tasks that
support critical decision-making and could affect minorities. The contribution
of this work is a novel analysis covering architectures and embedding spaces
for fine-grained understanding of bias over three approaches: generators,
attribute channel modifier, and post-processing bias mitigators. This work
shows that generators suffer from bias across all social groups with attribute
preferences such as between 75%-85% for whiteness and 60%-80% for the female
gender (for all trained CelebA models) and low probabilities of generating
children and older men. Modifier and mitigators work as post-processor and
manipulate the attribute boundaries, these changes affect the generator
performance. For instance, attribute channel perturbation strategies modify
the embedding spaces. We quantify the influence of this change on group
fairness by measuring the impact on image quality and group features.
Specifically, we use the Fréchet Inception Distance (FID), the Face Matching
Error and the Self-Similarity score. For Interfacegan, we analyze one and two
attribute channel perturbations and examine the effect on the fairness
distribution and the quality of the image. Finally, we analyzed the post-
processing bias mitigators, which are the fastest and most computationally
efficient way to mitigate bias. We find that these mitigation techniques show
similar results on KL divergence and FID score, however, self-similarity
scores show a different feature concentration on the new groups of the data
distribution. The weaknesses and ongoing challenges described in this work
must be considered in the pursuit of creating fair and unbiased face
generation models.

_Keywords_ Bias Analysis $\cdot$ Face Generation $\cdot$ Bias Mitigatin
$\cdot$ GANs $\cdot$ Difussion Models

##  1 Introduction

Generative Adversarial Networks (GANs) and Diffusion Models (DM) are
relatively recent machine learning models that have been used, among other
things, as generative models Goodfellow et al. (2016). These models have been
used for a wide range of applications, such as image generation, style
transfer, and video synthesis Yang et al. (2022)Lievrouw and Pope (1994)Kawar
et al. (2022). One of the most recent and popular application of these models
is face generation, which involves generating new realistic facial images.
However, it is easy to see why implications around fairness and discrimination
may arise in these cases.

It is a well-known fact that the datasets used in machine learning
applications are not always representative of the whole population, and can in
fact under-represent minorities Buolamwini and Gebru (2018)Wolfe and Caliskan
(2022). When training datasets are biased, the generated datsets will also be
unbalanced. Since the generated datasets may be employed in a variety of
applications, including as inputs to other AI systems, it is important to
ensure that they display enough diversity, especially when it comes to
protected attributes like gender, ethnicity or age. It is essential to perform
a correct bias analyis of the generative models, and also select the best
mitigation strategy for each specific case.

Measuring bias in GANs and DMs can be a complex task, as there is not a clear
method or benchmark established yet, but is a crucial step to evaluate the
effectiveness of the bias correction methods and to guarantee fairness in the
generated images. Here, we propose a variety of ways to measure the diversity
of the output dataset, and investigate the mechanistic sources of bias in the
networks.

##  2 Related Works

The most powerful deep learning models are usually trained by ingesting an
incredible amount of information in an unsupervised fashion. Then, strategies
are created to extract and manipulate in an intentional manner the features
learned by the model. Understanding the model behavior helps us create more
efficient methods to control these large networks. This section covers the
current most popular AI systems for Face Generation, the strategies used to
control the face attributes, and mitigation techniques that improve the
fairness of the model’s outputs. Finally, we detail some previous works that
analyse bias in face generation.

###  2.1 Face Generators

In the last few years, face generation models have shown remarkable
performance on several tasks, particularly for high-resolution, hyperrealistic
images, face generation Karakas et al. (2022) and face manipulation
manipulation Kammoun et al. (2022). GAN-based and Diffusion-based models are
the most popular strategies that use adverbial approach and variational
inference during training. Diffusion models present a high cost for both
training and inference, however, results suggest that these models are better
at covering the data distribution Rombach et al. (2021).

####  2.1.1 GAN for Face Generation

GANs exploit adversarial training to build generators with excellent
performance. There are numerous strategies that can be used to build GAN-based
FG models, more details can be found Kammoun et al. (2022)Wang et al.
(2022)Ning et al. (2020). In Kammoun et al. (2022), models are organized in
three main types: Conditional, Controllable, and Progressive. Uncontrolled
approaches have shown the best performance due to the big amount of
unannotated data available. One of the most important families of GAN models
is the StyleGAN family Karras et al. (2019). StyleGAN2Karras et al. (2020), a
SOTA model in facial generation which applies effective strategies to improve
the model performance such as: generator normalization, revisiting progressive
growing and regularizing the generator. The model can be controlled modifying
the input vector (Z and W) or some style layer parameters. CIPS Anokhin et al.
(2020) is another GAN-Based model that uses random latent vectors and pixel
coordinates to independently compute the color value at each pixel,
eliminating the need for spatial convolutions or similar operations that
propagate information across pixels. This new architecture is trained in an
adversarial fashion and is said to produce similar image quality as SOTA
methods.

####  2.1.2 Diffusion Models for Face Generation (DM)

Diffusion models are a type of machine learning models that describe the
evolution of a quantity over time, inspired by diffusion processes in physics.
DMs define how information spreads and evolves over time, by modeling the
change over a small time step as a combination of a drift term and a random
fluctuation term. Denoising Diffusion Probabilistic Models (DDPM) Ho et al.
(2020) is a DM that uses a weighted variational bound to achieve high-quality
image synthesis through a connection with denoising score matching and
Langevin dynamics. These models also offer a generalization strategy for
autoregressive decoding. Diffusion Models have achieved state-of-the-art
performance for image generation task on benchmark datasets such as
CIFAR10Krizhevsky et al. (2009) and LSUNSong and Xiao (2015). Another
interesting strategy that merges ideas from DMs and adversarial training is
Latent Diffusion Rombach et al. (2021). Latent Diffusion Models (LDMs) apply a
latent space of pre-trained autoencoders to produce high-quality and flexible
results, while reducing computational requirements. They can achieve SOTA
performance on image in-painting, unconditional image generation, semantic
scene synthesis, and super-resolution, by introducing cross-attention layers
into the model architecture.

###  2.2 Face Attribute Controllers

Attribute Controllers, also called mixed styles generation Kammoun et al.
(2022), refer to the process of modifying multiple facial attributes
simultaneously to generate faces in mixed styles using GANs, turning
unconditional GANs into controllable GANs. This can be achieved through
various methods, some of which will be cover below. These methods can be used
to control a wide range of facial attributes, such as facial expressions,
poses, hair styles, ages, and demographic characteristics such as race, gender
and even emotions. A controller can modify the input vector space or space
parameter to variate the face attributes. One of the best-known methods is
Interfacega Shen et al. (2020). This framework is used for semantic face
editing by interpreting the latent semantics learned by GANs. Interfacegan
finds that well-trained GANs have a disentangled representation after linear
transformations. It allows for precise control of facial attributes, and it
can be used for real image manipulation. Another space modifier is
GANSpaceHärkönen et al. (2020), this technique analyzes GANs and create
interpretable controls for image synthesis by using PCA on ‘latent space‘, and
applying layer-wise perturbations along the principal directions. The method
also allows for control of multiples GANs in a similar way as StyleGAN with
good results. StyleSpaceWu et al. (2021) analyzes the ‘latent style space‘ of
StyleGAN2, which consist of channel-wise style parameters and is found to be
more disentangled than previous strategies. This methodology enables the
discovery of style channels that control specific visual attributes by using a
pretrained classifier or a small number of example images.

One common problem with this approach is to disentangle attribute
perturbation. For example, if we perturb the latent space that controls the
gender attribute, we could also accidentally modify other undesired
attributes, such as age or ethnic origin. These modifications are usually
proportional to the scale of perturbation. InterfaceGAN and StyleGAN have
guidelines for modifying the latent space or style latent to disconnect
dependencies between facial attributes. The fact that there are alternatives
that allow for the disconnection of these dependencies indicates the existence
of multiple local optima that tend to demand of defining the boundary at which
an attribute change. Therefore, if there are multiple solutions, what
criterion allows me to define the optimal boundary for fairness in face
generation?

###  2.3 Bias in Machine Learning

Generating synthetic images can help improve the accuracy of various
classification tasks Xu et al. (2018)Sattigeri et al. (2019), but it’s also
known that machine learning models can amplify biases present in the training
data. This can lead to the creation of images with unintended biases. To
address this issue, there has been a growing focus on analyzingWolfe and
Caliskan (2022), detecting and mitigating bias in deep learning modelsYu et
al. (2020)Choi et al. (2020)McDuff et al. (2019). Bias mitigators con grouped
in three categories: pre-processor, in-processor and post-processor. In this
work we will focus on post-processor strategies.

###  2.4 Post-processing Bias Mitigation for Face Generation

Bias mitigation for face generation can lead to inaccuracies and unfair
representations of certain groups of people. There are post-processing
techniques that can help reduce these biases. One such technique is
StyleFlowAbdal et al. (2021), which focuses on this problem by formulating
conditional exploration as a conditional continuous normalizing flow in the
GAN latent space, conditioned by the attribute features. This method enables
fine-grained disentangled edits along various attributes and has been
evaluated using the face and car latent spaces of StyleGAN. Another well-known
strategy is FairGenTan et al. (2020), which utilizes GAN interpretation and a
Gaussian Mixture Model (GMM) to produce images with a more balanced attribute
distribution. By controlling the sampling of latent codes, FairGen has been
shown to effectively reduce biases in generated images and can even be used to
uncover biases in commercial face classifiers and super-resolution models. A
different path to removing biases from generated images is to modify the model
parameters themselves. This is where FairStyle Karakas et al. (2022) comes in
- it directly modifies a pre-trained StyleGAN2 model to generate images with a
more representative distribution of face attributes. The style space of the
StyleGAN2 model is leveraged to control these attributes and reduce biases. It
is important to note that bias mitigation for face generation is a complex
issue and there’s no one size fits all solution. It requires a combination of
technical, ethical, and social considerations and ongoing research.
Additionally, there’s not a standard methodology or evaluation protocol for
fairness in FG, and there are trade-offs between diversity and the realism of
the generated images.

##  3 Methodology

In this section, we describe the methodology employed for the analysis of
fairness in attribute controllers and bias mitigators. The analysis is
conducted at three levels. Firstly, by examining the presence of bias in
generator models. Secondly, by analyzing the presence of bias when perturbing
the attribute boundaries defined by the controllers. And thirdly, by
evaluating the effects on data generation caused by the mitigation strategy.
This analysis allows us to gain a deeper understanding of the subject and
identify potential areas for further research in this field.

###  3.1 Attribute Classifier

In order to evaluate the various characteristics of images generated by GANs,
we employed the FaceNet model for face matchingSerengil and Ozpinar (2020) and
face attribute analysis Serengil and Ozpinar (2021) from the Deepface library.
This library is equipped with both facial recognition capabilities and the
ability to analyze facial attributes such as age, gender, emotion, and
ethnicity. The facial recognition component of the model performed
exceptionally well in experiments using the Label Faces from the Wild (LFW)
data, achieving an accuracy rate of 99.2%. For more information about the
model, please refer to the following link
111https://github.com/serengil/deepface. The model is also capable of
classifying facial variations, including gender (female, male), ethnicity
(white, Latino/Hispanic, black, Asian, Indian, Middle Eastern), and emotion
(happy, angry, neutral, surprised, fearful, sad, disgusted). To evalute the
interfacegan with 2 channel perturbation and bias mitigators we use
classifier222https://catlab-team.github.io/fairstyle/ which was used for the
FairStyle Karakas et al. (2022) strategy.

##  4 Results

In this work, we conduct an in-depth examination of bias in face generation
models. First, we examine whether the generated dataset is balanced for people
of different gender, race and age. We also include an analysis of the emotions
displayed by different subjects. Then, we alter the latent space of the
network models to get further insight into the mechanistic causes of bias. To
accomplish this, we analyse various attribute controllers and bias mitigation
techniques. Through this investigation, we are able to gain a deeper
understanding of the crucial considerations that must be taken into account
for improving the effectiveness of the bias correction.

###  4.1 Bias in Face Generation Models

If the datasets used to train face generation models are not a representative
sample of the population, the output images will also contain a similar bias.
As a first step, we analyse how the generated faces are distributed when it
comes to gender, ethnicity, age and emotion. For our analysis, we aggregated
the results according to the training dataset. The StyleGAN2-FFHQ and CIPS
models were trained on the FFHQ dataset, while StyleGAN2-CelebA, LDM and DDPM
models were trained on the CelebA dataset. Figure 1 (a) and (b) displays the
results of our analysis for the models trained on the CelebA and FFHQ dataset,
respectively. The generated datasets seem to display bias for virtually every
considered attribute.

Both sets of models show some gender bias. However, models trained on the FFHQ
dataset demonstrate a preference for male face generation, while models
trained on the CelebA dataset prefer female faces. In the case of CelebA, the
preference is more pronounced, reaching over 70% for the DDPM model. The
disparity seems to be even more striking when we examine ethnicity: all models
show a strong preference for the whiteness category, reaching, in the case of
CelebA, values that exceed 80%. This means that less than $20\%$ of the
generated dataset will belong to an ethnicity other than white. The output
dataset is unbalanced when it comes to age too. All models generate mostly
faces categorized in the age range of 20 to 40 years. The CIPS model shows
stronger preference for the 20-29 age group. When it comes to emotions, it
appears that most of the images generated display people looking happy, or
neutral, with little variety when it comes to other, comparatively negative,
types of emotion.

((a))

((b))

Figure 1: Face Attribute Distribution of models for dataset (a) CelebA dataset
and (b) FFHQ dataset.

Overall, the output datasets seem to uncover the presence of some serious
bias, especially when it comes to ethnicity.

####  4.1.1 Representative Similarity in Face Generation Models

The Self-Similarity score quantifies the degree to which face features are
shared within a group. A high value indicates that the images share many
characteristics. This value could be used as an indicator to compare the
diversity between groups. It should be considered that this is only a relative
value which should be examined together with other metrics, since distortion
or poor image quality can lead to false positives under this approach.

In Figure 7, we observe no obvious relationship between diversity and bias.
The CIPS, DDPM and StyleGAN-CelebA models all display a generally high score,
which would indicate many similar features in the generated images. On the
other hand, the StyleGAN2 and LDM model tend to score in the lower values for
gender, race, and emotion. This suggests that the output images are relatively
diverse for all of these attributes. However, every single one of the models
examined seems to display a high self-simillarity score for the age attribute.
What could cause this lack of diversity when it comes to age?

A key factor is the diversity of examples. In the Figure 8 we observe the
results of our intersectional analysis for gender and age. We can see, for
example, that the LDM model was not able to generate images of males in the
10-19 age range (you can review Appendix 1.3). All images that were generated
in this age region were categorized as women. Indeed, equality at the
intersection of multiple face attributes is one of the problems that most
hinders mitigation strategies.

Next, we performed a global analysis of the generated data, Table 1 presents
the following image quality metrics: FID Score, Face Matching Error, and
Global Self-Similarity. We considered both the CelebA and FFHQ training
datasets. Interestingly, the StyleGAN models present the lowest self-
similarity score and the highest image quality (lowest FID score), when
trained on the FFHQ dataset. However, for the CelebA dataset, it is DDPM that
displays the best image quality (lowest FID Score). Conducting a review of the
generated datasets, it was observed that even with a low FID value, LDM
creates many images with faces easily identifiable as fake. We show some
example images from the CelebA dataset, and from the datasets generated by the
Latent Diffusion Model, StyleGAN2 and Denoising Diffusion Probabilitic Model
(Figure 2 (a)(b)(c)(d) respectively).

Table 1: Model Statistics Model | Dataset | Image Size | FID | %Error FM | Self-Similarity
---|---|---|---|---|---
DDPM | CelebA | 256x256 | 22.66 | 4.48% | 0.378
LDM | CelebA | 256x256 | 27.14 | 5.38% | 0.057
StyleGAN2 | CelebA | 256x256 | 30.28 | 1.16% | 0.354
StyleGAN2 | FFHQ | 1024x1024 | 22.54 | 11.22% | 0.061
CIPS | FFHQ | 1024x1024 | 68.39 | 0.22% | 0.267

((a))

((b))

((c))

((d))

Figure 2: Image examples from: (a) CelebA dataset. (b) Latent Diffusion Model.
(c) StyleGAN2. (d) Denoising Diffusion Probabilistic model.

###  4.2 Face Attribute Controller

For this analysis, we used controllers in various vector spaces to alter the
preference of the attributes. For interfacegan, this is observed in the Z and
W spaces. For GANSPace, we chose component 2 and 13, which refer to Gender and
Age respectively. This selection was carried out manually, identifying the
best components to control these attributes. For interfacegan and GANSpace
controllers, the perturbation domain was between -2 and 2, while it was
between -30 and 30 for StyleSpace. All controllers modify the StyleGAN2 model
trained on the FFHQ dataset. For each perturbation, we generated 10000 output
images, which were then analyzed and used to produced the statistics presented
here. Figure 9 and Figure 10 show the effect of perturbing the control channel
of an attribute using different strategies. We can notice that changing one
attribute often affects all the other attributes. This implies that the
attributes are correlated in the model. For example, positively scaling the
gender attribute reduces the preference for the happy characteristic for both
interfacegan (Z and W) and stylespace. When the same perturbation is applied
to GANSpace, the happiness feature increases slightly. Similarly, negatively
perturbing the gender attribute increasaes the preference for the white
category for StyleGAN (Z), GANSpace and StyleSpace, but for GANSpace this
preference decreases.

Interestingly, we can see in Figure 9 that perturbing the Age attribute for
stylespace, causes more complex changes in the race, gender and emotion
attributes. This gives us an idea of why it is difficult to mitigate bias in
this attribute.

####  4.2.1 Representational Similarity, Face Match Error and FID Score

For each of the perturbations introduced in the previous section, we also
investigated the impact on the new data distribution. Figure 3 (a) and (b)
show that perturbing the gender attribute quickly increases the self-
similarity score for the interfacegan and GANSpace controllers. This becomes
more evident when comparing these results to the stylespace gender attribute
curve in Figure 3 (c), which, when negatively perturbed, manages to reduce the
self-similarity score even further. If we compare it to the Match Error
results in Figure 3 (d),(e) and (f), we can see that the error curve for the
stylespace age attribute controller is much higher compared to the other
controllers. To better understand the reason behind this increase in error,
Figure 4 shows a list of 5 images extracted for each perturbation that was
performed using stylespace. It can be seen that for values outside the [-6 ,
6] domain, the controller performs changes in the eyes that distort the
generated image. This distortion cannot be spotted using the FID metric
(Figure 3 (i)). However, we can notice an anomaly in the self-similarity score
( Figure 3 (c)) and Match Error (Figure 3 (f)). This is because the eye
distortion is a characteristic that all generated images will tend to share,
and this would increase the similarity of the generated images.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

Figure 3: Performance metrics: The rows show the behaviour Self-Similaty, Face
Matching error, FID Score. Each columns represent the Attribute Controller:
Interfacegan, GANSpace and Stylespace. Figure 4: Image samples perturbing
Young channel attribute from StyleSpace. The scale used was from -30 to 30.
Can be oserved face distortion outside the interval of -6 to 6.

The FID score could not detect this image distortion, but due to the detection
error and the self-similarity score, we could note an anomaly in the generated
data. This indicates that calculating one metric may not always be sufficient.
Instead, our analysis suggests that a variety of quality and diversity metrics
should be calculated in order to correctly evaluate the models.

####  4.2.2 Disambiguation of face attributes

Interfacegan proposes an alternative to disambiguating disturbance in
attribute channels using conditional boundaries. This alternative allows us to
explore the effect of perturbing more than one attribute space. For this, we
used the boundary for gender alone, and the boundary for gender conditioned on
the age attribute. The conditional boundary keeps the age vector from
modifying the gender attribute, or at least it reduces its effect. The
perturbation on the gender channel acts from [-3,3]. For this experiment, the
CelebFaces333https://catlab-team.github.io/fairstyle/ classifier was used,
following Karakas et al. (2022), which has the binary category Young/Adult and
Male/Female Interfacegan. The optimal distribution for group equity: [Young-
Man, Adult-Man,Young-Woman, Adult-Woman] is [1/4, 1/4, 1/4, 1/4]. For each
perturbation on the attribute channel, the distribution of categories was
analyzed using 3 performance metrics: FID score, Self-Similarity and the KL
divergence, which calculate how close distribution of categories is to a
uniform distribution. Our results show that for a small neighborhood the FID
score and Self-Similarity increase rapidly, while the increment is smaller in
the direction where KL decreases (Figure 5). The image was interpolated using
Bi-linear sampling: a detailed plot using a 5x5 matrix is showed in Figure 11.
This related idea could help finding more effective and efficient strategies
to mitigate bias.

Figure 5: KL, Self-Similarity and FID score metrics in 2 attribute
perturbation: gender and age. The age attribute used is the age boundary
conditioned with gender attribute.

From the top exposed image the best setting is a scale of Age=3 and a scale of
Gender=0. In this case the attribute distribution was: [0.045, 0.318, 0.389,
0.247].

###  4.3 Bias Mitigation

Mitigation strategies, in most cases, manage to adjust the data distribution
to a proportion of unbiased attributes. However, for the case of Gender and
Young, the mitigation still does not achieve an optimal distribution of
attributes. In Figure 6, the results are presented by replicating the
FairStyle experiments and using the FairGen trained model. The KL results are
very similar to those described in the literature Karakas et al. (2022). An
important analysis is to understand how the self-similarity value changes
after the mitigation. In Figure 2 it can be observed that, even if the overall
value changes a little, the values by fairness group save important
information about the new distributions. In Figure 6 (b) it can be seen that
the two algorithms decrease the score for the “Young Woman” group, but the
FairStyle algorithm also increases the “Young Man” and decreases the other
categories. This is probably due to the reduction of very similar examples
leaving the images with different invoices. Unlike the FairGen model,
apparently the groups generated by the algorithm have a lot of concentration
and therefore more similar images per group. Error detection is also an
anomaly detection strategy, but it has a high cost since you have to run a
detector model to perform the evaluation.

((a))

((b))

Figure 6: Bias Performance Metrics for Mitigators 2 fairness group Young-Gender: (a) Data Distribution. (b) Self-Similarity. Table 2: Bias Performance Metrics for mitigators. Metrics | StyleGAN2 | FairGEN | FairStyle
---|---|---|---
Self-Similarity | 0.377 | 0.364 | 0.402
FID | - | 26.267 | 24.737
KL | - | 3.00E-02 | 2.08E-02

##  5 Conclusion

Face generation models have managed to recreate super-realistic images, but
the bias problem involves more criteria than just the proportion of generated
images. Models show a much greater preference for groups such as White (Race),
and age attributes between 20 to 40 years. Models trained with the FFHQ
dataset have a preference for male characteristics and models trained with
CelebA have a preference for females. Most models fail to generate female
gender attributes outside the 20-40 age ranges. Image semantic quality and
feature diversity persistence in interest groups are necessary to improve our
models. We also observed that very large perturbations in the style space add
unwanted features that distort the image. These distortions are not
necessarily detected by the FID score, so it is important to accompany the
evaluations with additional metrics such as self-similarity, or the use of
face detectors to detect anomalies in the newly fitted model.

##  6 Appendices

##  Appendix A Face Generation Model

###  A.1 Distribution of face attributes of generation models

Table 3: Data distribution for Face Attribute on each Generator Model. Attribute | Category | Models
---|---|---
|  | DDPM-CelebA | LDM-CelebA | StyleGAN2-CelebA | StyleGAN2-FFHQ | CIPS-FFHQ
Gender | Woman | 0.4397 | 0.1035 | 0.4582 | 0.1015 | 0.3081
| Man | 0.4054 | 0.0434 | 0.4708 | 0.0575 | 0.3183
Race | asian | 0.4582 | 0.1261 | 0.4194 | 0.2255 | 0.3869
| indian | 0.3846 | 0.0747 | 0.4150 | 0.1395 | 0.2004
| black | 0.5184 | 0.1914 | 0.5447 | 0.2166 | 0.3664
| white | 0.4021 | 0.0711 | 0.3777 | 0.0691 | 0.2742
| middle eastern | 0.4126 | 0.0853 | 0.4770 | 0.0987 | 0.2247
| latino hispanic | 0.3865 | 0.0843 | 0.3789 | 0.1129 | 0.3402
Emotion | angry | 0.3891 | 0.0653 | 0.3789 | 0.0541 | 0.2239
| disgust | 0.3662 | 0.0987 | 0.4712 | 0.1067 | —
| fear | 0.4433 | 0.0710 | 0.4311 | 0.0705 | 0.2132
| happy | 0.3835 | 0.0697 | 0.3906 | 0.0678 | 0.2814
| sad | 0.4165 | 0.0628 | 0.3688 | 0.0622 | 0.2565
| surprise | 0.4222 | 0.0614 | 0.3432 | 0.1192 | 0.2078
| neutral | 0.3988 | 0.0613 | 0.3727 | 0.0685 | 0.2822
Age | 10-19 | 0.5386 | 0.4829 | 0.7442 | 0.4383 | 0.5461
| 20-29 | 0.3911 | 0.0627 | 0.4025 | 0.0814 | 0.3001
| 30-39 | 0.3993 | 0.0727 | 0.4030 | 0.0627 | 0.2172
| 40-49 | 0.4588 | 0.0987 | 0.4809 | 0.0870 | 0.2943
| 50-inf | 0.5685 | 0.1813 | 0.6338 | 0.1197 | 0.3502

###  A.2 Self-Similarity of generation models by face attribute.

Table 4: Self-Similarity of Generator Model for each Face Attribute group Attribute | Category | Models
---|---|---
|  | DDPM-CelebA | LDM-CelebA | StyleGAN2-CelebA | StyleGAN2-FFHQ | CIPS-FFHQ
Gender | Woman | 0.43974 | 0.10348 | 0.45822 | 0.1015 | 0.30807
| Man | 0.40538 | 0.04343 | 0.47082 | 0.05753 | 0.31827
Race | asian | 0.45825 | 0.1261 | 0.41942 | 0.22546 | 0.38692
| indian | 0.38458 | 0.07469 | 0.41502 | 0.13946 | 0.20037
| black | 0.51843 | 0.19144 | 0.54473 | 0.21656 | 0.3664
| white | 0.40211 | 0.07112 | 0.37773 | 0.0691 | 0.27415
| middle eastern | 0.41255 | 0.08532 | 0.47704 | 0.09866 | 0.22468
| latino hispanic | 0.38651 | 0.0843 | 0.37886 | 0.11288 | 0.34016
Emotion | angry | 0.38908 | 0.06532 | 0.37888 | 0.05412 | 0.22391
| disgust | 0.3662 | 0.09867 | 0.47117 | 0.10672 | —
| fear | 0.44335 | 0.07102 | 0.43112 | 0.07046 | 0.21316
| happy | 0.38351 | 0.06974 | 0.39061 | 0.06778 | 0.28135
| sad | 0.4165 | 0.06284 | 0.36884 | 0.06224 | 0.25652
| surprise | 0.42216 | 0.06138 | 0.3432 | 0.11917 | 0.20775
| neutral | 0.39875 | 0.06131 | 0.37274 | 0.06851 | 0.28215
Age | 10-19 | 0.53858 | 0.48292 | 0.74418 | 0.4383 | 0.54614
| 20-29 | 0.39113 | 0.06271 | 0.40254 | 0.08135 | 0.30008
| 30-39 | 0.39931 | 0.07266 | 0.40302 | 0.06266 | 0.21719
| 40-49 | 0.45882 | 0.09872 | 0.48089 | 0.08697 | 0.29432
| 50-inf | 0.56848 | 0.18127 | 0.63376 | 0.11967 | 0.35018

###  A.3 Data Distribution grouped by gender and age attribute.

Table 5: Data Distribution grouped by Gender and Age Range Modelo | Dataset | <,19] | 20-29 | 30-39 | 40-49 | [50,>
---|---|---|---|---|---|---
|  | Man | Woman | Man | Woman | Man | Woman | Man | Woman | Man | Woman
DDPM | CelebA | 0.001 | 0.000 | 0.152 | 0.261 | 0.094 | 0.433 | 0.013 | 0.012 | 0.003 | 0.002
LDM | CelebA | 0.001 | 0.000 | 0.162 | 0.223 | 0.124 | 0.416 | 0.027 | 0.021 | 0.007 | 0.004
StyleGAN2 | CelebA | 0.000 | 0.000 | 0.290 | 0.152 | 0.084 | 0.456 | 0.012 | 0.003 | 0.002 | 0.000
CIPS | FFHQ | 0.004 | 0.003 | 0.452 | 0.288 | 0.118 | 0.126 | 0.007 | 0.001 | 0.001 | 0.000
StyleGAN2 | FFHQ | 0.017 | 0.015 | 0.253 | 0.158 | 0.215 | 0.222 | 0.077 | 0.023 | 0.017 | 0.003

###  A.4 Model Self-Similarity

Figure 7: Model Self-Similarity by face attribute. Figure 8: Model Self-
Similarity grouped by gender and age.

##  Appendix B Attribute Controllers

###  B.1 One channel attribute perturbation for different embedding spaces.

Figure 9: Different embeddings spaces for each attribute controller: Z-space
(row 1) and W-space (row 2) for interfacegan, PCA space (row 3 ) for GANSpace
and style space (row 4) for stylespace algorithm.

Figure 10: Image examples from: (a) CelebA. (b) Latent Diffusion Model. (c)
StyleGAN2. (d) DDPM model

###  B.2 Two Channel Perturbation for Intergacegan

Figure 11: KL divergence, Sel-Similarity and FID score for two channel face
attribute space.

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††thanks: Corr.author

# Stationary striations in plasma, created by a short microwave pulse in a
waveguide filled with a neutral gas

Y. Bliokh  Physics Department, Technion, Israel Institute of Technology, Haifa
320003, Israel  Y. Cao<EMAIL_ADDRESS>Physics Department, Technion,
Israel Institute of Technology, Haifa 320003, Israel  V. Maksimov  Physics
Department, Technion, Israel Institute of Technology, Haifa 320003, Israel  A.
Haim  Physics Department, Technion, Israel Institute of Technology, Haifa
320003, Israel  J.G. Leopold  Physics Department, Technion, Israel Institute
of Technology, Haifa 320003, Israel  A. Kostinsky  Physics Department,
Technion, Israel Institute of Technology, Haifa 320003, Israel  Ya.E. Krasik
Physics Department, Technion, Israel Institute of Technology, Haifa 320003,
Israel

###### Abstract

It was observed experimentally that after crossing a waveguide filled with a
neutral gas, a short powerful microwave pulse leaves a periodic glow of plasma
along the waveguide, persisting several tens of nanoseconds. A theoretical
model is presented which in combination with numerical simulations proposes a
possible explanation of this phenomenon.

##  I Introduction

Interaction of strong electromagnetic waves with plasma has always attracted
considerable attention. Charged particles acceleration in the wake wave exited
by a powerful laser pulse propagating in plasma [1] is at present of
particular interest because of significant experimental and theoretical
achievements and practical importance (see, e.g., [2, 3, 4]).

The non-linear interaction of electromagnetic pulses with plasma when the
microwave and plasma frequencies are of the same order of magnitude, remains
permanently in researchers’ field of view. Recent progress in generating
extremely intense, hundreds of MW, sub-nanosecond microwave pulses [5, 6]
allows one to observe experimentally and study such phenomena as ionization-
induced self-channeling of a microwave beam,[7], frequency shift in the wake
excitated in a plasma-filled waveguide[8], complete absorption of a microwave
pulse in plasma,[10] frequency up-shift and pulse compression in a propagating
self-generated ionization front [9].

In this paper a new phenomenon is described which accompanies a neutral gas
ionization by intense sub-nanosecond microwave pulse propagating in a
waveguide. It is suggested that a periodic stratification of the plasma forms
near the waveguide wall and its glow is observed (see Fig. 1), for helium
within a pressure range of 7-20 torr and for air within 1-3 torr. The plasma
glow appears approximately 5 ns after the microwave pulse enters the gas tube
and persists for 70 ns.

Figure 1: Plasma glow at 18 torr helium pressure: (a) broad field-of-view and
(b) detailed close-up view. Frame duration of 1.5 ns. The time delay of 12 ns
with respect to the microwave pulse entering the gas chamber.

There is nothing unexpected in that perturbations remain in the plasma after
the electromagnetic pulse’s passage. The wake excited by a short laser pulse
is an example of such perturbation which has received most of the attention.
This perturbation propagates together with the pulse and behind it, that is,
in the pulse’s frame of reference, it does not move. The periodic
stratification demonstrated in Fig. 1, is static in the laboratory frame of
reference and appears in the same positions along the waveguide from shot to
shot. The purpose of the present article is to propose a physical model which
explains this phenomenon.

##  II Experimental setup

The experimental setup (see Fig. 2) used in this research is similar to that
described in [9].

Figure 2: Experimental setup

The high-power microwave (HPM) pulse ($\sim 250-300$ MW, $\sim 0.35$ ns, $\sim
9.6$ GHz, $TM_{01}$ mode) is generated by a super-radiant backward wave
oscillator (SR-BWO) (see [5, 6] for details). The output of the SR-BWO is
connected to a 2.8 cm diameter circular waveguide with a calibrated coupler
(#1) at its center. This waveguide is followed by the gas tube, filled with
helium or air at the desired pressure. Inside the gas tube, the HPM pulse
propagates along a circular waveguide consisting of 12 parallel stainless
steel cylindrical wires (1 mm diameter) distributed uniformly (5-mm apart)
along its 2.8 cm perimeter. Electromagnetic simulations confirm that the wave
propagating along this waveguide is practically indistinguishable from the
wave propagating along a solid-wall cylindrical waveguide. A second calibrated
coupler (#2) is connected at the exit from the gas tube followed by an
impedance-matched horn antenna. Couplers #1 and #2 measure the incident,
transmitted, and reflected wave forms acquired by an Agilent DSO81204B
oscilloscope (12 GHz, 40 Gs/s). At the entrance and exit of the gas tube, 0.3
mm thick Mylar interface windows are installed to separate the
vacuum/gas/vacuum media. A fast framing intensified 4QuikE ICCD camera
(Stanford Computer Optics) operating with a frame duration of 1.5 ns was used
to capture the light emission from the plasma formed by the HPM pulse.

The plasma light emission patterns are shown in Fig. 1. One can see periodic
plasma glow (striations) near wires which appears approximately 5 ns after the
HPM pulse has left the wire waveguide and this glow is present for 70 ns. Such
striations were obtained in helium within the pressure range of 7-20 torr, and
in air at pressures of 1 - 3 Torr, which corresponds to the transparency
window of the HPM pulse in air. By capturing frames at different time delays
between the HPM pulse registered by coupler #1 and the 4QuilE camera frame, it
can be stated that the striations are stationary and do not move along the
path of the HPM pulse propagation. A typical longitudinal period of these
striations is 1.9 cm.

##  III Ionization

The electric field of a symmetric TM microwave mode propagating in a
cylindrical waveguide and the energy of electrons, oscillating in the wave
fields, are maximal near the axis. When the microwave power is moderate (a few
MW) the plasma is created mostly near and its density is maximal on the axis.
When the microwave power is sufficiently high, the region of maximal
ionization rate shifts from the axis toward the waveguide wall. The reason is
a non-monotonic dependence of the electron impact ionization cross section
$\sigma(w_{e})$ on the electron energy $w_{e}$. The cross section is maximal
when the electron energy is of the order of 100 eV and decreases with
increasing electron energy. For microwave power of the order of tens of MW or
higher, the electron oscillating energy near the axis is so large that the
ionization cross sections is smaller than its maximal value. In contrast, near
the waveguide wall, where the wave electric field is smaller than near the
axis, the ionization cross section is close to its maximal value.

The time evolution of the electron density, $n_{e}(r,t)$, is described by the
equation

$\frac{d}{dt}\ln n_{e}(t)=n_{g}\sigma[w(r,t)]|v(r,t)|,$ (1)

where $n_{g}$ is the neutral gas density, $v$ is the electron velocity, and
$\sigma[w(r,t)]$ is the ionization cross section, which depends on the
electron kinetic energy $w=mv^{2}/2$. If the duration of the electromagnetic
pulse, $t_{\rm pulse}$, is large compared to the field oscillation period
$T=2\pi/\omega$, $\omega t_{\rm pulse}\gg 1$ (where $\omega$ is the wave
frequency), it is convenient to use the ionization rate
$I(r,t)=\sigma[w(r,t)]|v(r,t)|$ averaged over a period of the oscillations:

$\bar{I}(r;P)=\frac{\omega}{2\pi}\int_{\pi/\omega}^{\pi/\omega}dt\,\sigma[w(r,t)]|v(r,t)|$
(2)

The dependence of the ionization rate $\bar{I}(r;P)$ on the radius for various
values of the microwave power $P$ are shown in Fig. 3.

Figure 3: Ionization rate $\bar{I}(r;P)$ as function of radius for various
values of microwave power $P$ (in GW).

The radial profile of the plasma density, which remains in the waveguide
behind the pulse, can be calculated as

$n_{e}(r,t)=n_{0}\exp\left[n_{g}\int_{-\infty}^{t}\bar{I}(r;P(t^{\prime}))dt^{\prime}\right],$
(3)

where $n_{0}$ is the initial electron density, and $P(t)$ is the pulse power’s
temporal profile. An example of the electron density evolution is presented in
Fig. 4. One can see that the electron density in a narrow layer near the
waveguide wall is almost an order of magnitude larger than near the axis. This
difference increases rapidly with the neutral gas density growth.

Figure 4: The temporal and radial evolution of the plasma density growth
factor (DGF) $n_{e}(r,t)/n_{0}$. For helium, at 2.5 torr pressure, pulse power
300 MW, pulse duration 0.35 ns. The temporal profile of the normalized pulse
power is shown as a shaded area in (b).

Note that the plasma density grows rapidly at the rear tail (late time) of the
pulse, and the characteristic rise time of the plasma density is much smaller
than the pulse duration. The importance of this will be discussed below.

##  IV Eigenmodes of a waveguide filled partially with plasma

The neutral gas ionization process induced by a powerful microwave pulse
described in Sect. III assumes that the structure of the microwave fields
remains unchanged by the presence of plasma, which is correct for plasma
density smaller than critical $n_{\rm crit}=\omega^{2}m/4\pi e^{2}$. When the
plasma density is close to critical, the structure of the wave’s fields vary
considerably. In order to trace this variation, let us consider the simple
model of assuming a step-wise three-layer density distribution:

$n(r)=\left\\{\begin{array}[]{clc}n_{1},&r<r_{1}&\mbox{(region I)},\\\
n_{2},&r_{1}<r<r_{2}&\mbox{(region II)},\\\ n_{3},&r_{2}<r<R&\mbox{(region
III)}\end{array}\right.$ (4)

where $R$ is the waveguide radius. This plasma density profile is a rough
description of the expected density distribution described in Sect. III.

Matching tangential components of the electromagnetic fields at the interfaces
between the different regions, one can find the spectrum $\omega(k)$ of the
layered waveguide eigenmodes as the solution of a boundary value problem.

Even such a simple form of the plasma density distribution contains many
parameters. Below, only such values of the parameters will be considered which
correspond to the experimental conditions. In particular, because the plasma
density in region II can exceed the ones in regions I and III, the model can
be further simplified, by assuming $n_{1}=n_{3}=0$.

An example of the eigenmodes spectrum is shown in Fig. 5.

Figure 5: Spectrum of the eigenmodes for $R=1.4$ cm, $r_{1}=1.0$ cm,
$r_{2}=1.2$ cm, plasma density $n_{e}=8\cdot 10^{11}\,{\rm cm}^{-3}$. Red
thick lines – volumetric electromagnetic wave (EMW), backward surface wave
(BSW), and forward surface wave (FSW). The dashed horizontal line marks the
experimental frequency. The dashed arrow points out the frequency $f$ and the
wave vector $k$ of the incoming pulse, and line “a” is tangential to the
dispersion curve at this point. Line “b” is parallel to line “a”. The
intersection between lime “b” and the dispersion curve BSW defines the
frequency and wave vector of the excited backward surface wave.

The presence of the plasma with a ring-shaped transversal profile enriches the
spectrum by two surface waves: a backward- and a forward-propagating mode. The
spectrum of the electromagnetic wave changes only little if the plasma layer’s
thickness is small compared to the waveguide radius. On the contrary, the
change in the structure of the wave’s fields are very strong in Fig. 6.

Figure 6: Amplitudes of the electromagnetic fields and electric field squared
in the empty waveguide (a) and in the presence of the plasma (b).

Compared to an empty waveguide, in the plasma-filled waveguide the amplitudes
of the longitudinal component of the electric field, $E_{z}(r)$, and the
azimuthal component of the magnetic field, $H_{\phi}(r)$, decrease over all
cross sections, but not that much. In contrast, the variation in the radial
electric field, $E_{r}(r)$, profile is considerably stronger. This component
of the field increases strongly in region II, occupied by the plasma which
means that the oscillatory energy of the plasma electrons $w\propto
E_{z}^{2}+E_{r}^{2}$ exceeds the value calculated ignoring the change in the
eigenmode fields. Because the ionization cross section is a decreasing
function of the electron energy (in the energy range of interest), changes in
the eigenmode fields structure suppresses ionization and the region with the
most effective ionization is shifted closer to the waveguide axis, widening
the plasma layer.

In the numerical example relevant to Fig. 5, the plasma density was chosen so
that the Langmuir frequency $\omega_{p}$ is equal to the empty waveguide cut-
off frequency $\omega_{c}$. This is an arbitrary choice, but the analysis
allows one to assume that the plasma density in the expanded layer is close to
this value. Indeed, if the plasma density is small, so that
$\omega_{p}\ll\omega_{c}$, the deformation of the eigenmode fields is small,
and also the ionization in this region continues. When the density is large
enough, so that $\omega_{p}\simeq\omega_{c}$, the ionization slows down. The
density cannot be so large that $\omega_{p}\gg\omega_{c}$, because the
waveguide becomes opaque for the incoming microwaves if the frequency
$\omega_{\rm EMW}$ is close to the cut-off frequency, $\omega_{c}$, as is the
case in the experiments.

Thus, a high-power microwave pulse, propagating in the waveguide filled with a
neutral gas, can create a hollow, tubular plasma. Note that a similar hollow
plasma column can be created in an unbounded system by a microwave pulse with
a bell-shaped radial profile when the required power density can be achieved
by focusing the microwave beam [7, 11], so that the hollow plasma is created
in the vicinity of the focal plane only. A hollow plasma can also be created
by a tubular electron beam, propagating in a neutral gas along a strong
guiding magnetic field [12]. Ionization of a neutral gas by a powerful tubular
laser beam can also create a plasma with specified density distribution [13].

##  V Backward surface wave excitation and plasma glow striations

Propagation of a powerful electromagnetic pulse in a preliminarily prepared
plasma is accompanied by the excitation of a wake, which is a slow eigenmode
of either an unbounded plasma or a plasma-filled waveguide. Phase velocity of
this eigenmode coincides with the group velocity of the pulse. It seems
reasonable to assume that a similar wake is excited and remains in the plasma
behind the pulse even though the plasma was created by the pulse itself.

Indeed, for the case considered, the perturbation, produced by an
electromagnetic pulse, remains in a tubular plasma in the form of a backward
surface wave. The wave number and the frequency of the exited surface wave are
defined by Čherenkov’s resonance condition $\omega_{\rm BSW}(k)=k_{\rm
BSW}v_{g}$, as shown in Fig. 5. Here $\omega_{\rm BSW}$ and $k_{\rm BSW}$ are
assigned to the backward surface wave and $v_{g}$ is the group velocity of the
electromagnetic pulse. This wave can persist in the plasma for a long time
after the pulse passage. Radial distribution of the electric and magnetic
fields of this backward propagating surface wave is shown in Fig 7.

Figure 7: Radial distribution of the electric ($E_{r}$, $E_{z}$), magnetic,
($H_{\varphi}$) fields, and longitudinal power flux density $P\sim
E_{r}H_{\varphi}$ of the excited backward propagating surface wave.

Axially-symmetric surface waves with negative group velocity in waveguides
containing a plasma column in a dielectric tube, separated from the conducting
wall, were considered in [14]. Backward surface waves in a waveguide with a
step-wise annular plasma column, similar to that considered in this article,
were studied in [15]. It was shown here that the backward and forward
propagating modes are associated with the inner and outer plasma surfaces,
respectively. Surface waves in waveguides filled with smoothly distributed
inhomogeneous plasma were studied in [16, 17]. It was shown that the absence
of sharp boundaries between different layers leads to collision-less
dissipation of the surface waves. The reason is the presence of resonant
electrons in the smooth transition plasma layer.

Negative group velocity of the surface wave means that the wave, excited along
the plasma-filled section of the waveguide, propagates in the opposite
direction (upstream) towards the input window. Here the wave is reflected and
propagates towards the downstream window. Because the phase velocity of the
reflected wave is negative, these two incident and reflected waves form a
standing wave near the input window. Electrons, oscillating in the electric
field of the standing wave antinodes, can excite and/or ionize molecules of
the neutral gas, producing a periodically distributed plasma glow.

It is worth noting that, under certain conditions, stationary striations in a
glow discharge are the result of interference between two ionization waves
[18, 19, 20]. Unlike backward-propagating surface waves, ionization waves are
excited only in the discharge current and disappear immediately after current
interruption.

As shown in the Sect. IV, the process of the plasma creation by the microwave
pulse can be roughly divided into two phases. Initially, plasma is created in
a narrow layer near the waveguide boundary. When the electron Langmuir
frequency $\omega_{p}(r)$ in this layer approaches the cut-off frequency
$\omega_{c}$, the plasma density growth slows down, and the second phase of
the process begins. Now, ionization leads to widening of this layer with
approximately constant plasma density. In the step-wise model, Eq. (4), it
means that $n_{2}(t)\simeq{\rm const}$ and $r_{1}(t)$ decrease with time.
Simultaneously, the cut-off frequency of the plasma-filled waveguide grows
and, if the pulse duration is long enough, it reaches the incoming wave
frequency $\omega_{\rm EMW}$ (which, in the experiment, is close to the cut-
off frequency of the empty waveguide, $\omega_{\rm EMW}\simeq 1.17\omega_{c}$)
while the retained part of the pulse is reflected from the waveguide input
window.

When the layer thickness $d=r_{2}-r_{1}$ varies with time, the pulse group
velocity $v_{g}$ and the backward surface wave dispersion characteristics
$\omega_{\rm BSW}(k_{\rm BSW})$ vary too. In this case, effective excitation
of the surface wave is possible if the resonant frequency $\omega_{\rm BSW}$
and wave number $k_{\rm BSW}$ remain essentially unchanged during the pulse
duration. In Fig. 8 the dependence of the resonant surface wave frequency and
wavelength on the plasma layer thickness demonstrate relatively small
dispersion. The characteristic wavelength is about 2 cm, so the distance
between nodes of the standing wave is about 1 cm. Taking into account the
roughness of the model, one can consider that this value agrees well with the
experimentally observed period of the glow ( 1.8 cm).

Figure 8: Resonant wavelength (closed circles) and resonant frequency (closed
circles) of the backward surface wave as functions of the plasma layer
thickness.

##  VI Discussion

The resonant excitation of a wave of any nature by a moving perturbation is
effective when the characteristic spatial scale of the perturbation is small
compared with its wavelength, or otherwise, the amplitude of the wake behind
the perturbation is small. As seen in Fig. 5, the expected wavelength of the
backward surface wave is smaller than the electromagnetic pulse wavelength and
much smaller than the longitudinal dimension of the pulse. Thus, the
electromagnetic pulse with parameters corresponding to the experimental
conditions, moving through a tubular plasma, can excite a surface wave of a
small amplitude. Results of numerical simulation, presented in Fig. 9, confirm
this conclusion.

Figure 9: a – The time dependence of the radial electric field in the plasma
layer. Long-living wake remains in plasma after the pulse passage. b – Spectra
of the electromagnetic pulse (red line) and the wake (filled grey). Note that
the wake spectrum is narrower and shifted to lower frequencies. (See also Fig.
5.

The main peak of the wake spectrum, depicted in Fig. 9, is located in the
region of 8-9 GHz, that is in good agreement with Fig. 8. Note that the
spectral width of the wake is smaller than that of the pulse because of the
following reason. A short pulse (such as that considered) is characterized by
a wide spread of group velocity values rather than a single velocity.
Consequently, the resonant condition $\omega_{\rm BSW}(k_{\rm BSW})=k_{\rm
BSW}v_{g}$ is satisfied for a set of surface waves, but, due to the weak
dependence $\omega_{\rm BSW}(k_{\rm BSW})$ (see Fig. 5), the spectrum of the
excited surface waves remains narrow. This property of the wake spectrum and
its position strongly supports the backward flowing surface wave description.

Despite that the excitation of the backward flowing surface wave by the
electromagnetic pulse is confirmed in numerical simulations, the assumption
that this wave is responsible for the observed spatially-periodic plasma glow
casts doubt, because of the small amplitude of the surface wave. This
discrepancy can be resolved in the following way.

In the performed numerical simulation it was assumed that the plasma density
is stationary and homogeneously distributed along the waveguide. However, in
the experiment the plasma is created by the propagating electromagnetic pulse
itself. Then, the temporal (and longitudinal) profile of plasma density, which
is shown in Fig. 4, looks like shock wave and propagates along the waveguide
together with the electromagnetic pulse. In other words, the surface wave
propagates in the media with time-dependent, traveling parameters. The
characteristic spatial scale of the plasma density variation is much smaller
than the pulse length and can be of the order of the surface wave wavelength.
The amplitude of the resonant wave, excited by such abrupt traveling variation
of the medium parameters, is much larger compared to the case, when the medium
parameters are varied smoothly [21, 22]. Due to this reason, one can assume
that the stationary periodic plasma glow is indeed associated with the
excitation of the backward surface wave.

##  VII Conclusions

To summarize, it was shown that a sub-nanosecond microwave pulse of hundreds
MW power propagating in a neutral gas filled cylindrical waveguide produces a
tubular plasma, with the density mainly concentrated in a thin layer near the
waveguide wall. A surface wave with negative group velocity can be excited in
such plasma-filled waveguide. The wake in the form of this backward
propagating surface wave remains in the waveguide long after the pulse’s
passage. A backward propagating surface wave is reflected from the input
window and in combination with the incident wave forms a standing wave. Plasma
electrons, oscillating in the electric field of this standing wave’s
antinodes, excite and/or ionize molecules of the neutral gas, producing the
experimentally observed periodically distributed plasma glow along the
waveguide.

This scenario is based on a theoretical model supported by numerical
simulations and analysis. We cannot present indisputable experimental facts
that the observed spatially-periodic plasma glow appeared exactly in that way
as it is described in the paper. Should this be the case then this phenomenon
would be the first non-direct observation of the backward surface wave,
propagating along a tubular plasma in the absence of magnetic field. In future
experiments we will attempt to confirm the mechanism behind this interesting
phenomenon.

## Acknowledgements

The authors are grateful to E. Flyat for his technical support. This study was
supported by Pazy Foundation Grant No. 2032056.

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