# Unveiling the Power of Wavelets: A Wavelet-based Kolmogorov-Arnold Network
for Hyperspectral Image Classification

Seyd Teymoor Seydi<EMAIL_ADDRESS>
###### Abstract

Hyperspectral image classification is a crucial but challenging task due to
the high dimensionality and complex spatial-spectral correlations inherent in
hyperspectral data. This paper employs Wavelet-based Kolmogorov-Arnold Network
(wav-kan) [1] architecture tailored for efficient modeling of these intricate
dependencies. Inspired by the Kolmogorov-Arnold representation theorem, Wav-
KAN incorporates wavelet functions as learnable activation functions, enabling
non-linear mapping of the input spectral signatures. The wavelet-based
activation allows Wav-KAN to effectively capture multi-scale spatial and
spectral patterns through dilations and translations. Experimental evaluation
on three benchmark hyperspectral datasets (Salinas, Pavia, Indian Pines)
demonstrates the superior performance of Wav-KAN compared to traditional
multilayer perceptrons (MLPs) and the recently proposed Spline-based KAN
(Spline-KAN) model [2]. Across the datasets, Wav-KAN achieved an average
overall accuracy of 92.62% and kappa coefficient of 0.9157, outperforming
Spline-KAN (avg. accuracy 89.85%, kappa 0.8793) and MLP (avg. accuracy 77.69%,
kappa 0.7119). On the challenging Indian Pines dataset, Wav-KAN’s significant
performance gains (85.54% accuracy, 0.8348 kappa) over Spline-KAN (77.31%,
0.7395) and MLP (35.13%, 0.2984) underscore its ability to model complex
spectral-spatial dependencies effectively. The competitive results of Spline-
KAN, especially on Salinas, highlight the potential of theory-driven kernels
like splines and wavelets in hyperspectral classification tasks. Wav-KAN’s
wavelet-based architecture enables interpretable, non-linear mapping while
capturing multi-scale patterns, demonstrating the advantages of incorporating
representation theorems into deep learning models for enhanced performance on
high-dimensional, correlated data like hyperspectral imagery. In this work, we
are: (1) conducting more experiments on additional hyperspectral datasets
(Pavia University, WHU-Hi, and Urban Hyperspectral Image) to further validate
the generalizability of Wav-KAN; (2) developing a multiresolution Wav-KAN
architecture to capture scale-invariant features; (3) analyzing the effect of
dimensional reduction techniques on classification performance; (4) exploring
optimization methods for tuning the hyperparameters of KAN models; and (5)
comparing Wav-KAN with other state-of-the-art models in hyperspectral image
classification.

###### Index Terms:

Kolmogorov-Arnold Networks (KAN), Hyperspectral Image Classification, Wavelet,
Wav-KAN, Neural Networks.

##  I Introduction

Hyperspectral imaging is a remote sensing technique that captures the
reflectance or emission of light from an object across a wide range of
contiguous spectral bands, typically ranging from visible to infrared regions
of the electromagnetic spectrum [3]. Unlike traditional RGB or multispectral
images, which have a limited number of broad bands, hyperspectral images
consist of hundreds of narrow, closely spaced spectral bands, providing
detailed spectral information for each pixel in the image. This rich spectral
information enables the identification and discrimination of various
materials, surface features, and objects based on their unique spectral
signatures [4].

The high spectral resolution of hyperspectral data allows researchers to
precisely characterize the electromagnetic spectrum of an object.
Mathematically, a hyperspectral image can be represented as a 3D cube, with
two spatial dimensions and one spectral dimension. However, most hyperspectral
sensors have a low spatial resolution, so each pixel corresponds to a mixture
of materials observed over a surface area. This spectral mixing poses
challenges for accurate classification and requires advanced techniques to
handle the mixed pixel problem, such as spectral unmixing [5].

Deep learning techniques have shown remarkable success in hyperspectral image
classification. These models can be categorized into three main groups based
on their learning approach: Supervised Learning, Semi-Supervised Learning, and
Unsupervised Learning. In supervised learning, deep neural networks are
trained using labeled data, where both input (hyperspectral image pixels) and
corresponding output labels (classes) are provided during the training
process. Examples of supervised models include Convolutional Neural Networks
(CNNs), Recurrent Neural Networks (RNNs), and fully connected deep neural
networks. These models learn to extract discriminative features from the input
data and map them to the desired output labels through a hierarchical network
architecture [6]. CNNs are particularly well-suited for hyperspectral data due
to their ability to capture local spatial and spectral patterns through
convolutional and pooling operations. The convolutional layers apply learnable
filters to the input data, generating feature maps that capture different
patterns. These feature maps are then subsampled by pooling layers, reducing
their spatial dimensions while retaining the most important features. This
process is repeated in multiple layers, with each layer learning increasingly
abstract and complex representations [7]. RNNs, on the other hand, are
designed to handle sequential data and can model long-range dependencies in
hyperspectral pixels by treating them as a sequence of spectral values. RNNs
have been used for hyperspectral classification by incorporating recurrent
layers that can capture spectral-spatial contexts [8].

Hyperspectral datasets pose several challenges for deep learning models due to
their inherent complexity. Hyperspectral images have hundreds of spectral
bands, resulting in a high-dimensional feature space. This high dimensionality
can lead to the ”curse of dimensionality,” where the number of training
samples required for accurate modeling increases exponentially with the number
of dimensions [9]. Deep learning models may struggle to effectively learn and
generalize from high-dimensional data, especially when the available training
samples are limited. Hyperspectral images exhibit complex spatial-spectral
correlations, where neighboring pixels and spectral bands are often highly
correlated. Capturing and modeling these correlations is crucial for accurate
classification but can be challenging for deep learning models [10].
Traditional models may treat spatial and spectral information separately,
failing to capture the intricate relationships between them. Due to the
relatively coarse spatial resolution of hyperspectral sensors, a single pixel
may contain a mixture of materials or spectra, known as mixed pixels [11].
This spectral mixing poses challenges for accurate classification, as the
observed spectrum is a combination of multiple endmembers (pure material
spectra). Additionally, spectral variability within the same material class
can arise due to factors like illumination conditions, atmospheric effects,
and sensor noise, further complicating the classification task [12]. Despite
these challenges, deep learning techniques have shown promising results in
hyperspectral image classification, often outperforming traditional machine
learning methods. Ongoing research focuses on developing more robust and
efficient deep learning models tailored specifically for hyperspectral data,
along with strategies for dealing with limited labeled data, high
dimensionality, and spatial-spectral correlations.

Kolmogorov-Arnold Networks (KANs), grounded in the Kolmogorov-Arnold
representation theorem [13], provide notable benefits including enhanced
interpretability and accuracy [2]. These networks feature univariate learnable
activation functions on their edges, with nodes summing these activation
functions. Most previous studies on KANs have concentrated on depth-2
representations, with the exception of Liu et al. [2], who expanded KANs to
support arbitrary widths and depths, a variant known as B-Spline KAN.

Within the realm of Universal Approximation Theory (UAT), wavelets have been
investigated in neural networks. This study applies wavelets to the
Kolmogorov-Arnold representation theorem for neural networks with arbitrary
widths and depths [1], targeting hyperspectral image classification through
both continuous wavelet transform (CWT) and discrete wavelet transform (DWT).

Section II discusses the methodologies that we applied to our datasets.
Section III introduces benchmark hyperspectral datasets inclduing Indian
Pines, Pavia, and Salinas 111We will add more simulations in next versions..
Section IV presents Experiment and Results. The comparison of Wav-KAN, Spline-
KAN and MLPs is given. Finally, Section V concludes the paper.

##  II Methods

Wavelet Kolmogorov-Arnold Networks (Wav-KAN) offer several advantages over
Spline Kolmogorov-Arnold Networks (Spline-KAN) and traditional multilayer
perceptrons (MLPs). First and foremost, Wav-KAN enhances interpretability and
performance by leveraging wavelet functions within the Kolmogorov-Arnold
network structure. This allows Wav-KAN to efficiently capture both high-
frequency and low-frequency components of the input data. Wavelets, with their
ability to perform multiresolution analysis, ensure that the network can
isolate significant patterns while discarding irrelevant noise, providing a
more accurate and nuanced representation of the original data without the
pitfalls of noise overfitting. This feature makes Wav-KAN superior in terms of
interpretability and robustness compared to Spline-KAN and MLPs.

In terms of computational efficiency and training speed, Wav-KAN also holds an
edge. Unlike Spline-KAN, which requires smooth functions and grid spaces for
better performance—often resulting in cumbersome and computationally expensive
operations—Wav-KAN utilizes the inherent scaling properties of wavelets. This
reduces the need for additional terms in activation functions and speeds up
training. Additionally, the wavelet-based approach in Wav-KAN obviates the
need for recalculating previous steps when capturing more details, as discrete
wavelet transform (DWT) can efficiently combine local detailed information
with broader trends. This results in faster and more efficient training
compared to both Spline-KAN and MLPs.

Moreover, Wav-KAN benefits from having fewer parameters while maintaining high
performance. By employing wavelets, which can capture both low-frequency and
high-frequency functions, Wav-KAN networks require fewer parameters than their
Spline-KAN and MLP counterparts. This not only enhances the model’s
generalization ability but also leads to significant improvements in accuracy
and robustness. The adaptive nature of wavelets allows Wav-KAN to conform to
the data structure, akin to how water conforms to the shape of its container,
thus achieving higher accuracy and robustness compared to Spline-KAN and MLPs.
Overall, Wav-KAN provides a powerful and efficient neural network model with
applications across various fields.

##  III Dataset

In this study we used three benchmark hyperspectral datasets.

|
---|---
(a) Color composite Indian Pine dataset | (b) Ground Truth
|
(c) Color composite Pavia dataset | (d) Ground Truth
|
(e) Color composite Salinas dataset | (f) Ground Truth
Figure 1: Incorporate hyperspectral dataset in this study

###  III-A Indian Pines

The Indian Pines dataset, acquired by the Airborne Visible/Infrared Imaging
Spectrometer (AVIRIS) sensor on June 12, 1992, covers an agricultural area in
Northwestern Indiana, USA. The dataset consists of 224 spectral bands ranging
from 400-2500 nm, with a spatial resolution of 20 m per pixel. The scene,
comprising 145 x 145 pixels Figure 1a, contains 16 distinct land cover classes
Figure 1b, primarily focusing on different crop types and vegetation species
[14].

###  III-B Pavia

The Pavia Centre dataset is a hyperspectral image acquired by the ROSIS sensor
during a flight campaign over Pavia, northern Italy. It consists of 102
spectral bands with a spatial dimension of 1096 x 715 pixels Figure 1c. The
geometric resolution of the dataset is 1.3 meters. The ground truth data for
the Pavia Centre scene differentiates 9 classes Figure 1d [15].

###  III-C Salinas

The Salinas dataset is a hyperspectral image acquired over the Salinas Valley,
California. The study area covers 512 x 217 samples with a spatial resolution
of 3.7-meter pixels Figure 1e. The dataset originally consisted of 224
spectral bands acquired by the AVIRIS sensor, but 20 water absorption bands
were removed, leaving 204 bands. The ground truth data contains 16 classes
representing different land cover types in the region Figure 1f [16].

##  IV Experiment and Results

###  IV-A Visual Analysis

The provided Figure 2 showcases the results of hyperspectral classification
for the Salinas. A qualitative evaluation indicates that all three
methodologies exhibit robust performance in discriminating the major land
cover classes, as demonstrated by the strong spatial and spectral
correspondence between the classified images and the ground truth.
Nevertheless, close inspection reveals minor discrepancies, suggesting
potential misclassifications or class confusion. The MLP classifier appears to
have a slightly higher tendency for misclassification (Figure 2a), especially
in border regions between classes, when compared to the Spline-KAN (Figure 2b)
and Wav-KAN approaches (Figure 2c).

|
---|---
(a) | (b)
|
(c) | (d)
Figure 2: Result of hyperspectral classification for Salinas dataset. (a) MLP,
(b) Spline-KAN, (c) Wav-KAN, (d) Ground Truth

Based on the visual assessment of the classification results presented in the
Figure 3, the MLP Figure 3a approach appears to achieve the best overall
performance among the three methods compared for the Pavia dataset. The MLP
classifier demonstrates a higher degree of spatial and spectral consistency
with the ground truth data, suggesting more accurate discrimination of urban
land cover classes. The Wav-KAN (Figure 3c) exhibits better performance than
the Spline-KAN (Figure 3b), as evidenced by fewer misclassifications and a
more coherent representation of complex spatial patterns.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 3: Result of hyperspectral classification for Pavia dataset. (a) MLP,
(b) Spline-KAN, (c) Wav-KAN, (d) Ground Truth

The MLP model Figure 4(a) displays significant noise, fragmentation, and
misclassifications, with small, scattered regions and poorly defined edges,
indicating its inability to effectively capture the intricate spectral-spatial
patterns. The Spline-KAN model Figure 4(b) demonstrates an improvement, with
larger, more homogeneous regions and reduced noise, but still suffers from
some misclassifications and jagged edges. In contrast, the Wav-KAN model
Figure 4(c) and the unspecified model Figure 4(d) produce visually superior
results, characterized by well-defined, smooth edges, minimal noise or
fragmentation, and consistent, large homogeneous regions, suggesting effective
capture of the complex spectral-spatial information. These models exhibit
fewer misclassifications and better delineation of boundaries between
different classes, contributing to their enhanced visual quality and
classification accuracy on this challenging hyperspectral dataset.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 4: (a) Indian Pines MLP, (b) Indian Pines Spline-KAN, (c) Indian Pines
Wav-KAN, (d) Indian Pines Ground Truth

###  IV-B Numerical Results

Table I presents the results of hyperspectral classification using three
different models: Spline-KAN, Wav-KAN, and MLP, on three datasets: Salinas,
Pavia, and Indian Pine. The performance metrics used for evaluation are
Overall Accuracy (OA) and Kappa coefficient ($\kappa$).

Salinas Dataset: On the Salinas dataset, the Wav-KAN model demonstrates
superior performance, achieving the highest OA of 0.9341 and $\kappa$ of
0.9264. This indicates that the Wav-KAN model effectively captures the
spectral and spatial features of the Salinas dataset, resulting in accurate
classification. The Spline-KAN model closely follows with an OA of 0.9261 and
$\kappa$ of 0.9178, suggesting that the spline-based kernel also provides a
good representation of the data. The MLP model, however, lags behind with an
OA of 0.8655 and $\kappa$ of 0.8499, indicating that the non-linear mapping of
the MLP may not be as effective in capturing the complex relationships within
the Salinas dataset.

Pavia Dataset: For the Pavia dataset, all three models exhibit excellent
performance, with OA and $\kappa$ values surpassing 0.98. The MLP model
slightly outperforms the others, achieving an OA of 0.9910 and $\kappa$ of
0.9873. This suggests that the non-linear mapping of the MLP is particularly
suitable for the Pavia dataset, effectively capturing the underlying patterns.
The Wav-KAN and Spline-KAN models also perform remarkably well, with OA values
of 0.9901 and 0.9863, and $\kappa$ values of 0.9860 and 0.9806, respectively.
The high performance across all models indicates that the Pavia dataset may
have more distinguishable features that are easier to classify.

Indian Pine Dataset: The Indian Pine dataset presents a more challenging
classification task, as evidenced by the lower OA and $\kappa$ values compared
to the other datasets. The Wav-KAN model significantly outperforms the other
models, achieving an OA of 0.8554 and $\kappa$ of 0.8348. This suggests that
the wavelet-based kernel in the Wav-KAN model is more effective in capturing
the complex spectral and spatial dependencies in the Indian Pine dataset. The
Spline-KAN model follows with an OA of 0.7731 and $\kappa$ of 0.7395,
indicating that the spline-based kernel still provides a reasonable
representation of the data. However, the MLP model struggles on this dataset,
with a substantially lower OA of 0.3513 and $\kappa$ of 0.2984. This poor
performance suggests that the non-linear mapping of the MLP may not be
suitable for the intricate characteristics of the Indian Pine dataset.

TABLE I: Results on Different Datasets Dataset | Model | Overall Accuracy | Kappa
---|---|---|---
Salinas | Spline-KAN | 0.9261 | 0.9178
Wav-KAN | 0.9341 | 0.9264
MLP | 0.8655 | 0.8499
Pavia | Spline-KAN | 0.9863 | 0.9806
Wav-KAN | 0.9901 | 0.9860
MLP | 0.9910 | 0.9873
Indian Pine | Spline-KAN | 0.7731 | 0.7395
Wav-KAN | 0.8554 | 0.8348
MLP | 0.3513 | 0.2984

##  V Conclusion

This study evaluated the Spline-KAN, Wav-KAN, and MLP models for hyperspectral
image classification on the Salinas, Pavia, and Indian Pines datasets. The
Wav-KAN model performed best on Salinas and Indian Pines, indicating its
effectiveness in capturing complex spectral-spatial features. The MLP excelled
on Pavia, suggesting its suitability for urban areas. On Salinas, Spline-KAN
closely followed Wav-KAN, showing the spline kernel’s potential. However, MLP
struggled with Indian Pines, highlighting its limitations for intricate
datasets. Overall, Wav-KAN emerged as the most robust and versatile approach,
but MLP’s strengths on Pavia suggest scenario-specific advantages. These
findings underscore the importance of model selection based on dataset
characteristics and the need for advanced techniques to handle diverse
hyperspectral data effectively. Exploring ensemble or hybrid models combining
multiple approaches could further improve classification accuracy and
robustness.

## References

  * [1] Z. Bozorgasl and H. Chen, “Wav-kan: Wavelet kolmogorov-arnold networks,” _arXiv preprint arXiv:2405.12832_ , 2024.
  * [2] Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou, and M. Tegmark, “Kan: Kolmogorov-arnold networks,” _arXiv preprint arXiv:2404.19756_ , 2024.
  * [3] M. T. Eismann, _Hyperspectral remote sensing_. SPIE Press, 2012.
  * [4] J. Bioucas-Dias, A. Plaza, G. Camps-Valls, P. Scheunders, N. Nasrabadi, and J. Chanussot, “Hyperspectral remote sensing data analysis and future challenges,” _IEEE Geoscience and Remote Sensing Magazine_ , vol. 1, no. 2, pp. 6–36, 2013.
  * [5] J. M. Bioucas-Dias, A. Plaza, N. Dobigeon, M. Parente, Q. Du, P. Gader, and J. Chanussot, “Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches,” _IEEE journal of selected topics in applied earth observations and remote sensing_ , vol. 5, no. 2, pp. 354–379, 2012.
  * [6] Y. Chen, H. Jiang, C. Li, X. Jia, and P. Ghamisi, “Deep feature extraction and classification of hyperspectral images based on convolutional neural networks,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 54, no. 10, pp. 6232–6251, 2016.
  * [7] W. Hu, Y. Huang, L. Wei, F. Zhang, and H. Li, “Deep convolutional neural networks for hyperspectral image classification,” _Journal of Sensors_ , vol. 2015, 2015.
  * [8] L. Mou, P. Ghamisi, and X. X. Zhu, “Deep recurrent neural networks for hyperspectral image classification,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 55, no. 7, pp. 3639–3655, 2017.
  * [9] G. F. Hughes, “On the mean accuracy of statistical pattern recognizers,” _IEEE transactions on information theory_ , vol. 14, no. 1, pp. 55–63, 1968.
  * [10] L. Fang, S. Li, J. A. Benediktsson, and X. Kang, “Spectral–spatial classification of hyperspectral imagery with object-based machine learning algorithms,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 54, no. 8, pp. 4941–4959, 2015.
  * [11] B. Somers, G. P. Asner, L. Tits, and P. Coppin, “Endmember variability in spectral mixture analysis: A review,” _Remote Sensing of Environment_ , vol. 115, no. 7, pp. 1603–1616, 2011.
  * [12] X. Chen, Y. Yamaguchi, H. Tsu, and Z. Pan, “Spectral variability study for endmember selection,” _IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing_ , vol. 4, no. 1, pp. 205–214, 2011.
  * [13] A. N. Kolmogorov, _On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables_. American Mathematical Society, 1961.
  * [14] M. F. Baumgardner, L. L. Biehl, and D. A. Landgrebe, “220 band aviris hyperspectral image data set: June 12, 1992 indian pine test site 3,” Sep 2015. [Online]. Available: https://purr.purdue.edu/publications/1947/1
  * [15] G. Licciardi, P. Marpu, J. Chanussot, and J. Benediktsson, “Linear versus nonlinear pca for the classification of hyperspectral data based on the noise-adjusted principal components,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 9, no. 3, pp. 447–451, 2012.
  * [16] F. Melgani and L. Bruzzone, “Classification of hyperspectral remote sensing images with support vector machines,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 42, no. 8, pp. 1778–1790, 2004.


# The Anatomy of Geometric Phase for an Optical Vortex Transiting a Lens

Mark T. Lusk<EMAIL_ADDRESS>Department of Physics, Colorado School of Mines,
Golden, CO 80401, USA  Andrew A. Voitiv  Department of Physics and Astronomy,
University of Denver, 2112 E. Wesley Avenue, Denver, CO 80208, USA  Chuanzhou
Zhu  Department of Physics, Colorado School of Mines, Golden, CO 80401, USA
Mark E. Siemens<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Denver, 2112 E. Wesley Avenue, Denver, CO 80208, USA

###### Abstract

We present an analytical means of quantifying the fractional accumulation of
geometric phase for an optical vortex transiting a cylindrical lens. The
standard fiber bundle of a Sphere of Modes is endowed with a Supplementary
Product Space at each point so that the beam waists and their positions can be
explicitly tracked as functions of lens transit fraction. The method is
applied to quantify the accumulation of geometric phase across a single lens
as a function of initial state and lens position within the beam. It can be
readily applied to a series of lenses as well.

##  I Introduction

The interplay of geometry and information is at the heart of our quest to
understand Nature, a perspective that has materialized from the merging of two
fronts: our evolving understanding that general relativity [1] and quantum
field theory [2] are based on the geometric nature of gauge fields; and, that
the interpretation of any measurement is an application of information theory
[3]. Essentially all of the scientific and technological progress of the last
century might be summed up as milestones in our understanding of the
relationship between these two concepts.

A particularly rich manifestation of this interplay is the geometric holonomys
exhibited by fields as they evolve [4, 5]. In simple settings, there is a
geometric phase that is just the angle between initial and final orientations
of a vector field when parallel transported in a closed loop—e.g. the holonomy
of Foucault’s pendulum [6]. At the other extreme, the holonomy is
characterized by non-Abelian gauge fields that form the basis of the standard
model [2] and also underlie current research in topological quantum computing
[7].

Classical and quantum optics comprise an accessible common ground for the
study and harnessing of geometric information. Few-photon states embrace
higher-dimensional parameter spaces and admit the study of entanglement in
geometric phase, while classical optics can still exhibit non-Abelian behavior
through synthetic gauge fields [8, 9, 10]. With an eye towards these richer
settings, the transit of classical electromagnetic beams through linear
optical elements offers a particularly illuminating arena for elucidating how
and why geometric phase accumulates. There 3D electromagnetic modes can be
viewed as 2D fields with the propagation axis interpreted as time.

Laser beams composed of linear combinations of Laguerre-Gaussian (LG) modes
may accumulate geometric phase as they transit linear optical elements [11,
12]. The associated parameter space can be projected onto a Sphere of Modes
(SOM) (sometimes referred to as the modal or optical Poincaré Sphere) so that
the geometry of SU(2) underlies such geometric information. Experimental
progress in this area has focused on the use of optical elements that
collectively trace out closed circuits composed of geodesic arcs [13].

An explicit determination of non-geodesic trajectories has not been undertaken
in the SOM setting, and the calculation of geometric phase for non-cyclic
processes [14] has yet to be considered as well. Previous calculations of
geometric phase lean heavily on the results from the more studied setting in
which it is polarization that evolves [15]. The analogy is limited, though,
since projection onto the SOM necessarily disregards information about basis
vectors which are themselves evolving [11]. As result, the way in which
geometric phase is actually accumulated is obscured by drawing on the
polarization analogy.

These issues are addressed in the present work, and one immediate dividend of
the approach is that it allows geometric phase to be evaluated as a function
of lens transit fraction—explicitly showing that geometric phase accumulation
arises from light-matter interactions within the lens rather than free-space
propagation after the lens. The SOM is extended to a fiber bundle, shown in
Fig. 1, so that the total phase of the beam can be tracked. Optical elements
are decomposed into differential constituents allowing phase to be calculated
as a function of transit fraction. Each point on the fibers is additionally
endowed with a Supplementary Product Space (SPS) so that the beam waists and
their relative positions can also be explicitly traced as beams move through
optical elements.

Figure 1:  _Fiber Space_. The Sphere of Modes (brown) is the projection of a
3D product space (pink), $\\{w_{1},w_{2},z_{2}\\}$, onto its associated fiber
(green) followed by a projection of the fiber position, $\chi$, down to its
base. Subscripts ”1” and ”2” refer to principle axes of an elliptical beam
cross-section. Here $z=0$ is defined as the position of waist $w_{1}$.

The method is applied to the transit of a beam composed of $LG_{0,+1}$ and
$LG_{0,-1}$ modes—i.e. with zero radial mode number and opposite charges—with
an optical vortex through a single cylindrical lens placed at a specified
position relative to the beam waist, as shown in Fig. 2. The full trajectory
across the SOM and through the SPS is calculated as a function of transit
fraction. The geometric phase is then calculated for each lens using two
distinct approaches using a geometric notion of distant parallelism that does
not require that the SOM trajectory be closed.

Figure 2:  _Lens Transit_. A thin, converging, cylindrical lens changes the
waist and waist position along its active axis. Subscripts ” _p_ ” and ” _a_ ”
refer to the passive and active axes of the cylindrical lens.

##  II State Space for Tilted Vortices

Under a paraxial approximation for the monochromatic electromagnetic vector
potential [16], free-space electrodynamic wave propagation is governed by a 2D
Schrödinger equation [16] with axial position playing the role of time:

$i\partial_{z}u=-\frac{1}{2k}\nabla^{2}_{\perp}u.$ (1)

This governs the motion of a tilted Gaussian vortex [17] whose transit through
cylindrical lenses is most easily carried out by expressing it in terms of
linear combinations of products of normalized 1D Hermite-Gaussian (HG) modes:

$\displaystyle u_{m}(x,z)$ $\displaystyle=$ $\displaystyle
A_{p,m}(z)e^{-\frac{x^{2}}{B_{p}^{2}(z)}}$ (2)
$\displaystyle\mathbb{H}_{m}\biggl{(}\frac{x\sqrt{2}}{B_{p}(z)}\biggr{)}e^{i\frac{kx^{2}}{2R_{p}(z)}}e^{-i\psi_{p}(z,m)}.$

Here

$A_{p,m}(z):=\frac{1}{(2\pi)^{1/4}\sqrt{2^{m-1}m!B_{p}(z)}}$ (3)

is a normalization factor, $\mathbb{H}_{m}$ is a Hermite polynomial of order
m, $B_{p}(z)$ is the beam radius, $R_{p}(z)$ is the radius of curvature of the
beam, and $\psi_{p}(z,m)$ is the Gouy phase:

$\displaystyle B_{p}(z)$ $\displaystyle=$ $\displaystyle
w_{p}\sqrt{1+\frac{(z-z_{p})^{2}}{z_{\scriptscriptstyle Rp}^{2}}}$
$\displaystyle R_{p}(z)$ $\displaystyle=$
$\displaystyle(z-z_{p})\biggl{(}1+\frac{z_{\scriptscriptstyle
Rp}^{2}}{(z-z_{p})^{2}}\biggr{)}$ (4) $\displaystyle\psi_{p}(z,m)$
$\displaystyle=$ $\displaystyle\biggl{(}m+\frac{1}{2}\biggr{)}{\rm
tan}^{-1}\biggl{(}\frac{z-z_{p}}{z_{\scriptscriptstyle Rp}}\biggr{)}.$

The beam waist, $w_{p}$, is located at $z_{p}=0$, and $z_{\scriptscriptstyle
Rp}:=kw_{p}^{2}/2$ is the Rayleigh length. Subscripted ” _p_ ” labels indicate
the assignment of the x-axis as the _passive_ axis of the cylindrical lenses
that will be considered shortly. For prescribed wavenumber, $k$, the mode of
Eq. 2 can be expressed as a ket,

$u_{m}(x,z)\equiv\braket{x,z}{m^{w_{p},z_{p}}},$ (5)

where the superscripts completely characterize the 1D beam by identifying its
waist and corresponding axial position. Although bra-ket notation is more
frequently associated with quantum mechanics, the features that make it
convenient there hold for any Hilbert space with complex scalars, including
the present setting.

Two-dimensional modes can then be constructed in which HG modes are defined on
both x and y axes:

$u_{m}(x,z)u_{n}(y,z)\equiv\braket{x,y,z}{m^{w_{p},z_{p}}n^{w_{a},z_{a}}}.$
(6)

Note that the y-dependent HG mode is generalized to allow for an _active_ ”
_a_ ” waist and waist position that differs from the mode that depends on x.
With these 2D modes as basis vectors, general first-order modes can be
expressed in terms of the components of a spinor, $\\{v_{1},v_{2}\\}\in
SU(2)$:

$\ket{u_{\rm
vortex}}=v_{1}\ket{1^{w_{p},z_{p}}0^{w_{a},z_{a}}}+v_{2}\ket{0^{w_{p},z_{p}}1^{w_{a},z_{a}}}$
(7)

with

$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle
e^{i\chi}\cos(\theta/2)e^{-i\phi/2}$ $\displaystyle v_{2}$ $\displaystyle=$
$\displaystyle e^{i\chi}\sin(\theta/2)e^{i\phi/2}.$ (8)

Here $\phi\in[0,2\pi]$ and $\theta\in[0,\pi]$. It is important to keep in
mind, though, that the spinor itself is an incomplete description of the state
because all information on the beam waists and their posisitions has been
flensed.

###  II.1 Sphere of Modes Fiber Bundle and Its Supplementary Product Space

The spinor can be projected onto the SOM using a Hopf map to obtain the
following SOM coordinates for the requisite Cartesian coordinates:

$\displaystyle x_{\scriptscriptstyle\\!S\\!O\\!M}$ $\displaystyle=$
$\displaystyle|v_{1}|^{2}-|v_{2}|^{2}=\cos\theta$ $\displaystyle
y_{\scriptscriptstyle\\!S\\!O\\!M}$ $\displaystyle=$ $\displaystyle 2{\rm
Re}(v_{1}v_{2}^{*})=\cos\phi\sin\theta$ (9) $\displaystyle
z_{\scriptscriptstyle\\!S\\!O\\!M}$ $\displaystyle=$ $\displaystyle-2{\rm
Im}(v_{1}v_{2}^{*})=\sin\phi\sin\theta.$

A star, *, denotes complex conjugation. We have chosen the polar angle,
$\theta$, so that it is measured relative to the x-axis in anticipation of the
fact that transits through a cylindrical lens will then lie on arcs of
constant $\theta$. Intersections of the SOM with the z-axis ($\theta=\pi/2$,
$\phi=\pm\pi/2$) are associated with Laguerre-Gaussian modes, the standard
convention [12].

Position on the SOM is an even less complete description of the state than the
original spinors because the spinor phase, $\chi$, has been lost. This can be
remedied, though, and the process will point the way towards a means of re-
introducing the beam waist information as well. The spinor phase, $\chi$, can
be accounted for by generalizing the SOM to a fiber bundle in which a fiber
extends out from each point on its surface with $\chi$ measured along such
rays. As illustrated in Fig. 1, we can now imagine physical processes that
result in two types of motion: transit across the surface of the SOM but also
_horizontal lifts_ along the fibers [18] which track the change in total phase
of the beam.

The generalization from SOM to fiber bundle motivates us to endow each point
on the fibers with a 3D SPS in which the beam is assumed to have waists,
$w_{p}$ and $w_{a}$, aligned with the x and y axes of the original beam, with
waist positions of $z_{p}$ and $z_{a}$, respectively. Using $z_{p}=0$ as a
datum, the spinors are endowed with a three-element pedigree:

$v_{j}=v_{j}(\phi,\theta,\chi;w_{p},w_{a},z_{a}),\quad j=1,2.$ (10)

The generalized state space shown in Fig. 1 allows state evolution to be
represented as a trajectory in which beam waists, waist orientations, and
waist positions can all evolve. For a single cylindrical lens oriented with
its active axis in the y-direction, $w_{1}=w_{x}=w_{p}$ and
$w_{2}=w_{y}=w_{a}$.

##  III State Evolution as a Fraction of Lens Transit

###  III.1 Beam Propagation Through a Cylindrical Lens

Consider the propagation of a converging 1D $H\\!G_{m}$ mode, Eq. 2, through a
thin cylindrical lens of focal length, $f$, positioned at $z_{\ell}$ (Fig. 2).
The active axis of the lens is aligned with the y-axis, and the mode that
emerges will have the same form as Eq. 2 but with a new waist, $w_{a}$, and
waist position, $z_{a}$. The mode radii along the passive and active axes of
the lens have a simple relationship that is sometimes referred to as the
Lensmaker’s Equation [19]:

$\frac{1}{f}=\frac{1}{R_{p}}-\frac{1}{R_{a}}.$ (11)

Using Eq. II2 and its counterpart for the active axis of the lens, this can be
expressed in terms of Rayleigh lengths:

$\frac{1}{f}=\frac{1}{(z_{\ell}-z_{p})\bigl{(}1+\frac{z_{\scriptscriptstyle
Rp}^{2}}{(z_{\ell}-z_{p})^{2}}\bigr{)}}-\frac{1}{(z_{\ell}-z_{a})\bigl{(}1+\frac{z_{\scriptscriptstyle
Ra}^{2}}{(z_{\ell}-z_{a})^{2}}\bigr{)}}.$ (12)

Continuity of the beam radius at the lens provides a second equation that also
relates the two Rayleigh lengths:

$\frac{z_{\scriptscriptstyle
Rp}^{2}+(z_{\ell}-z_{p})^{2}}{z_{\scriptscriptstyle
Rp}}=\frac{z_{\scriptscriptstyle
Ra}^{2}+(z_{\ell}-z_{a})^{2}}{z_{\scriptscriptstyle Ra}}.$ (13)

Keeping in mind that we have set $z_{p}=0$, Eqs. 12 and 13 can be solved
together to give the active-axis Rayleigh length as well as positions in the
Supplementary Parameter Space as functions of the lens position, $z_{\ell}$:

$\displaystyle z_{a}$ $\displaystyle=$ $\displaystyle
f+z_{\ell}-\frac{f^{2}(f-z_{\ell})}{z_{\scriptscriptstyle
Rp}^{2}+(f-z_{\ell})^{2}}$ $\displaystyle z_{\scriptscriptstyle Ra}$
$\displaystyle=$ $\displaystyle\frac{f^{2}z_{\scriptscriptstyle
Rp}}{z_{\scriptscriptstyle Rp}^{2}+(f-z_{\ell})^{2}}$ (14) $\displaystyle
w_{a}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2z_{\scriptscriptstyle
Ra}}{k}}.$

It is also convenient to define a _natural lens offset_ ,
$-d_{\scriptstyle\rm{nat}}$, and associated _natural focal length_ ,
$f_{\scriptstyle\rm{nat}}$, for which $z_{a}=0$—i.e. a lens position such that
the active and passive waists are both at $z=0$:

$\displaystyle f_{\scriptstyle\rm{nat}}$ $\displaystyle=$ $\displaystyle
z_{\scriptscriptstyle
Rp}\csc(\alpha/2)\biggl{(}-1+\frac{2}{1+\tan(\alpha/4)}\biggr{)}$
$\displaystyle d_{\scriptstyle\rm{nat}}$ $\displaystyle=$ $\displaystyle
z_{\scriptscriptstyle Rp}(-1+\frac{2}{1+\tan(\alpha/4)}).$ (15)

Here we have borrowed notation from the theory of mode converters [20] by
replacing $d_{\scriptstyle\rm{nat}}/f_{\scriptstyle\rm{nat}}$ with
$\sin(\alpha/2)$. In mode converters, the spacing between lenses is
$d_{\scriptstyle\rm{nat}}$, and $\alpha$ is the difference in Gouy phase
between active and passive axes after propagation through the lens, the
difference that changes the mode of the beam. This connection is still useful
even though our attention is focused on a single lens.

Without making any such restriction on the position of the active waist, the
1D beam that emerges from the lens is

$\ket{u^{\rm(out)}}=\ket{m^{w_{a},z_{a}}}e^{i\psi_{a}(z_{\ell},m)}e^{-i\psi_{p}(z_{\ell},m)}.$
(16)

This implies that a cylindrical lens generates the following change in the
total phase of the 1D $HG_{m}$ mode, a difference of Gouy phases:

$\Delta_{\scriptscriptstyle\rm
Gouy}(m,f,z_{\ell})=\psi_{a}(z_{\ell},m)-\psi_{p}(z_{\ell},m).$ (17)

This phase change is an explicit function of lens position, $z_{\ell}$, and by
virtue of Eqs. II2 and III.1, a function of lens focal length, $f$, as well.

Now extend the approach to quantify the result of sending in a linear
combination of 2D HG modes,

$\ket{u_{\rm
in}}=v_{1}^{\rm(in)}\ket{1^{w_{p},z_{p}}0^{w_{p},z_{p}}}+v_{2}^{\rm(in)}\ket{0^{w_{p},z_{p}}1^{w_{p},z_{p}}},$
(18)

through a cylindrical lens with its active axis aligned in the y-direction.
The output beam is then

$\ket{u_{\rm
out}}=v_{1}^{\rm(out)}\ket{1^{w_{p},z_{p}}0^{w_{a},z_{a}}}+v_{2}^{\rm(out)}\ket{0^{w_{p},z_{p}}1^{w_{a},z_{a}}},$
(19)

where the output spinor is

$\underline{v}^{\rm(out)}=[G]\underline{v}^{\rm(in)}.$ (20)

The lens action on the input spinor can be expressed algebraically [21]:

$[G]:=\begin{pmatrix}e^{-i\Delta_{\scriptscriptstyle\rm
Gouy}(0,f,z_{\ell})}&0\\\ 0&e^{-i\Delta_{\scriptscriptstyle\rm
Gouy}(1,f,z_{\ell})}\end{pmatrix}.$ (21)

The coordinates, $\underline{v}^{\rm(out)}$, can then be converted to SOM
coordinates using Eq. II.1.

It is important to note, though, that this matrix representation does not
capture evolution in the SPS.

###  III.2 Fractional Lens Transit

We are now in a position to study beam transit through a cylindrical lens by
tracking its trajectory through Supplementary Product Space, along fibers, and
across the surface of the Sphere of Modes. This intra-lens description draws
on the form of the Thin Lens Approximation [22], in which the effect of the
lens on the beam is taken to be a multiplicative factor of $e^{-i\mu y^{2}}$
with $\mu=z_{R}/(2f)$. This implies that passage through N successive lenses
would generate a total phase factor of

$\prod_{n=1}^{N}e^{-i\mu_{n}y^{2}}\equiv e^{-i(\sum_{n=1}^{N}\mu_{n})y^{2}}.$
(22)

In this same vein, we may view a single lens as a stack of differentially thin
lenses and imagine the beam output after passing through a fraction, $\zeta$,
of these as

$u_{\rm out}(\zeta)=e^{-\frac{y^{2}}{2f/\zeta}}u_{\rm in},\quad\zeta\in[0,1].$
(23)

Jumps in active-axis waist, $w_{a}$, waist position, $z_{a}$, fiber phase,
$\chi$, and beam structure, $\ket{u_{\rm out}}$, can therefore be transformed
to smooth functions of lens transit fraction, $\zeta$, by replacing $f$ with
$f/\zeta$ in Eqs. III.1, 17, and 19. Parametric plots as a function of $\zeta$
then produce trajectories through the SPS, along fibers and across the surface
of the SOM.

##  IV Detailed Analysis of Two Different Lens Transits

Consider the transit of a first-order Gaussian mode through a cylindrical lens
with a mode converter phase parameter of $\alpha$, natural focal length,
$f_{\scriptstyle\rm{nat}}$, and general lens position, $z_{\ell}$. Assume that
the initial fiber phase, $\chi$, is zero. We now have the tools to map out
paths across the SOM, along fibers, and through the SPS as the lens transit
fraction, $\zeta$, increases from 0 to 1.

A circular arc will be generated on the SOM for which the polar angle,
$\theta$, is constant—a counterclockwise rotation about the x-axis. The change
in azimuthal angle, $\Delta\phi$, is obtained from the normalized inner
product of the Stokes coordinates just before and just after the lens. Eqs.
II, II.1, and 20 then imply that

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\cos^{-1}\biggl{[}\frac{\underline{v}^{\rm(out)}.\underline{v}^{\rm(in)}}{|\underline{v}^{\rm(out)}||\underline{v}^{\rm(in)}|}\biggr{]}$
(24) $\displaystyle=$
$\displaystyle\tan^{-1}(z_{\ell})+\tan^{-1}\biggl{(}\frac{1+z_{\ell}(z_{\ell}-f)}{f}\biggr{)}.$

The arc length is a function of the lens position, $z_{\ell}$. Placements
further to the left of the passive waist position ($z=0$) will decrease the
arc length; likewise the arc length increases as the lens is moved further to
the right. Eq. 24 implies that the maximum arc length for any lens is $\pi$.

For the sake of making a concrete comparison, two sets of single-lens transits
are considered in Fig. 3. Each set is for a particular lens so they are
labeled as Lens 1 $\Leftrightarrow\alpha=\pi/2$ and Lens 2
$\Leftrightarrow\alpha=\pi/4$. For each lens, the SOM trajectory is calculated
for a range of lens positions. For both sets, the pre-lens mode of the beam
($\zeta=0$) is associated with position $\\{\phi=\pi/2,\theta=\pi/4\\}$,
indicated by the small black sphere in Fig. 3. All trajectories, for both
lenses, are shown to lie on the small-circle, for which $\theta=\pi/4$. As a
given lens is moved further from the source, the arc length increases, and the
trajectory angle subtended approaches the maximum value of $\pi$. Note, in
particular, that each lens can be positioned so as to generate an arc angle of
$\pi/2$, but that the lenses must be placed in distinctly different positions
for this to occur: $z_{\ell}=0.59z_{\scriptscriptstyle
Rp}\,(z_{\scriptscriptstyle\rm rel}=1.41)$ and
$z_{\ell}=1.77z_{\scriptscriptstyle Rp}\,(z_{\scriptscriptstyle\rm
rel}=2.61)$, respectively.

Figure 3:  _Sphere of Modes Trajectories for Lenses 1 and 2_. A Gaussian beam,
Eq. 7, with spinor state Eq. II and SOM angles of $\phi=\pi/2,\,\theta=\pi/4$
(black sphere, $\zeta=0$), transits two individual cylindrical lenses in
separate experiments. Their focal lengths are described by Eq. III.11 with
$\alpha_{1}=\pi/2$ and $\alpha_{2}=\pi/4$. As a reminder, $\alpha$ is the
difference in Gouy phase between active and passive axes after propagation
through the lens, the difference that changes the mode of the beam. The spinor
evolves according to Eq. 20, with $f$ replaced by $f/\zeta$, generating the
blue curve shown. The length of this trajectory depends on lens position,
$z_{\ell}=z_{\scriptscriptstyle\rm rel}d_{\scriptstyle\rm{nat}}$, with
$d_{\scriptstyle\rm{nat}}$ given by Eq. III.12 for each lens. In all cases,
the endpoint of an arc is associated with $\zeta=1$.

The blue arc in Fig. 3 shows the trajectory traced out on the SOM, in
association with each lens, as a function of lens transit fraction. The arc
endpoints ($\zeta=1$) shown with red (darker, smaller) spheres are for Lens 1
($\alpha=\pi/2$) while the green (larger, lighter) spheres are for lens 2
($\alpha=\pi/4$). These relative position labels, $z_{\scriptscriptstyle\rm
rel}$ are given in terms of the respective natural lens displacements,
$d_{\scriptstyle\rm{nat}}$. Fig. 4 shows how these arc lengths change as a
function of lens position, now in units of the passive Rayleigh length,
$z_{\scriptscriptstyle Rp}$.

Figure 4:  _Arc Length of Sphere of Modes Trajectories_. As shown in Fig. 3,
the complete trajectory arc length is a function of lens position, $z_{\ell}$.
The total angle subtended is given by Eq. 24 and is plotted here for Lens 1
and Lens 2. The dashed line is at an arc angle of $90^{\circ}$, and the
associated lens positions are $z_{\ell 1}=0.59z_{\scriptscriptstyle Rp}$ and
$z_{\ell 2}=1.77z_{\scriptscriptstyle Rp}$. These correspond to
$z_{\scriptscriptstyle\rm rel,1}=1.41$ and $z_{\scriptscriptstyle\rm
rel,2}=2.61$.

A trajectory along fibers and through the SPS accompanies the SOM trajectory
of Fig. 3. The evolving fiber phase, $\chi$, is obtained using Eqs. II, II.1,
and 20:

$\chi(\zeta)=\tan^{-1}z_{\ell}+\tan^{-1}\biggl{[}\frac{-fz_{\ell}+\zeta+z_{\ell}^{2}\zeta}{f}\biggr{]}.$
(25)

The resulting fiber phase is plotted in Fig. 5(a) for the same parameters as
Figs. 3 and 4.

Figure 5:  _Trajectory Through the Supplementary Product Space_. Figs. 3 and 4
show the Sphere of Modes trajectories associated with beam transit through two
lenses, and here the corresponding trajectories along fibers and through the
Supplementary Product Space are plotted using Eqs. III.12,3 and 25. The lens
positions, $z_{\scriptscriptstyle\rm rel,1}=1.41$ and
$z_{\scriptscriptstyle\rm rel,2}=2.61$, result in the same SOM trajectories
with an arc angle of $90^{\circ}$, a convenient reference point for comparing
the geometric phase accumulated in each case. Here
$z_{\ell}=z_{\scriptscriptstyle\rm rel}d_{\scriptstyle\rm{nat}}$ with
$d_{\scriptstyle\rm{nat}}$ given by Eq. III.12.

For each lens, a trajectory through the SPS can be generated using Eqs.
III.12,3. As noted previously, the focal length, $f$, must be replaced by
$f/\zeta$ in these equations. The resulting progress through the SPS is
plotted in panels (b) and (c) of Fig. 5.

The combination of spinor coefficients, $v_{1}$ and $v_{2}$, along with fiber
phase, $\chi$, active beam waist, $w_{a}$, and active waist position, $z_{a}$,
is sufficient to generate beam cross-sections before and after each lens. The
results shown in Figs. 6 and 7 are based on the same information as was used
to produce Figs. 3, 4, and 5, but it is useful to see that identical start and
end points of the SOM trajectory can be associated with such distinctly
different beam cross-sections for lenses 1 and 2.

As an important verification step, the beam that emerges from each lens can
also be described using the Thin Lens Equation [22]:

$u_{\rm out}=u_{\rm in}e^{-iy^{2}/(2f)}.$ (26)

Cross-section plots using this equation are plotted in panels (c) and (d) of
Fig. 7, and it is clear that they are identical to the results shown in panels
(e) and (f), respectively. Of course, the form of Eq. 26 does not allow any
conclusions to be obtained concerning the evolution of basis modes as a
function of lens transit, nor does it provide any information about evolution
along fibers and through the SPS.

Figure 6:  _Beam Cross-Sections Before Each Lens_. The information used to
plot the trajectories on the Sphere of Modes (Figs. 3 and 4), motion along the
fibers (Fig. 5(a)), and trajectories through the Supplementary Product Space
(Fig. 5(b,c)), can all be combined in Eq. 7 to produce beam cross-sections
that depend on the transit fraction, $\zeta$. The beam magnitude has a cross-
section that does not depend on transit fraction, so a single magnitude cross-
section is plotted for each lens in panels (a) and (b). Panels (c) and (d)
give the cross-sectional phase of the beam entering the lens. These look
similar, but they are clearly not the same. This is because the lens positions
are different, $z_{\scriptscriptstyle\rm rel,1}=1.41$ and
$z_{\scriptscriptstyle\rm rel,2}=2.61$, the values used to produce Fig. 5.
Figure 7:  _Beam Cross-Sections After Each Lens_. The information used to plot
the trajectories on the Sphere of Modes (Figs. 3 and 4), motion along the
fibers (Fig. 5(a)), and trajectories through the Supplementary Product Space
(Fig. 5(b,c)), can all be combined in Eq. 7 to produce beam cross-sections
that depend on the transit fraction, $\zeta$. Panels (a) and (b) are the
corresponding cross-sectional phases for the beams exiting the lenses, and
these are clearly different. Panels (c) and (d) are verification plots in
which the Thin Lens Equation, Eq. 26, is used to plot the phase cross-sections
directly. These plots are identical to mode-based plots of panels (a) and (b).

##  V Geometric Phase

It is now a well-told story that an experimentally measurable geometric phase
[23, 24] can be identified with non-adiabatic dynamics [25] and does not
require that trajectories on the SOM be closed [26, 14, 27]. A gauge-invariant
geometric phase associated with the _non-geodesic_ arcs, such as the small-
circle trajectories of Fig. 3, can be obtained by closing the arc with a
geodesic, as shown in Fig. 8 and evaluating the difference between the total
and dynamic phases associated with the complete circuit:

$\Phi_{\rm geom}=\Phi_{\rm tot}-\Phi_{\rm dyn}.$ (27)

The geodesic arc is the Stokes manifestation of a projection operation carried
out on the beam that emerges from our lens:

$\ket{u_{\rm proj}}=\hat{P}\ket{u_{\rm out}}.$ (28)

where $\ket{u_{\rm out}}$ is given in Eq. 19. The projection operator,
$\hat{P}=\ket{u_{\rm proj}}\bra{u_{\rm proj}}$, is defined in terms of a
vector with the same SOM position as the input to the lens, but it has the SPS
coordinates of the beam coming out of the lens:

$\ket{u_{\rm
proj}}=v_{1}^{\rm(in)}\ket{1^{w_{p},z_{p}}0^{w_{a},z_{a}}}+v_{2}^{\rm(in)}\ket{0^{w_{p},z_{p}}1^{w_{a},z_{a}}}.$
(29)

The projection operation therefore produces a final state vector that results
in a closed SOM circuit while not returning the beam to its original state:

$\ket{u_{\rm
final}}=\underline{v}^{\rm(in)*}\cdot\underline{v}^{\rm(out)}\ket{u_{\rm
proj}}.$ (30)

Figure 8:  _Intersection of Small-Circle Arc and Geodesic_. The geometric
phase accumulated by beam transit through a lens is quantified using Eq. 35.
This requires that the solid angle subtended by the moon-shaped sliver, shown
here, between geodesic and small-circle arcs be evaluated. Eq. 36 gives its
solid angle.

In the current setting, where dynamic phase [25] accumulates as a function of
azimuthal angle subtended, $\phi$, the phase can be defined with respect to
the Jones vector as

$\Phi_{\rm
dyn}(\phi,\theta)=-i\int_{\phi_{\scriptstyle\rm{i}}}^{\phi_{\scriptstyle\rm{i}}+\Delta\phi}d\phi^{\prime}\underline{v}^{*}(\phi^{\prime},\theta)\cdot\partial_{\phi^{\prime}}\underline{v}(\phi^{\prime},\theta).$
(31)

Here the fiber phase has been removed. Using Eq. II, this implies that

$\Phi_{\rm dyn}(\Delta\phi,\theta)=-\frac{\Delta\phi}{2}\cos\theta,$ (32)

a useful result already in the literature for polarization [28]. Within the
thin-lens approximation, this dynamic phase accumulates because of a change of
basis and not in association with any physical propagation. For transit
through a single lens, Eq. 24 can be used to express this as a function of
lens position, $z_{\ell}$, and the result is plotted in Figure 9.

Figure 9:  _Dynamic Phase Accumulation as a Function of Lens Position_. Eqs.
32 and 24 are combined to generate the dynamic phase accumulated by beam
transits through Lens 1 and Lens 2, followed by a projection operation, as
plotted on the SOM in Fig. 3. This phase depends on lens position, $z_{\ell}$.
For lens positions of $z_{\ell 1}=0.59z_{\scriptscriptstyle Rp}$ and $z_{\ell
2}=1.77z_{\scriptscriptstyle Rp}$, the SOM trajectories are the same for each
lens ($90^{\circ}$ arc angles) and the dynamic phase is evaluated to be the
same, $-31.8^{\circ}$, for the two lenses.

The same expression can also be used to show that the dynamic phase
accumulated over any portion of a geodesic is zero since the associated polar
angle, $\theta$, is equal to $\pi/2$ in Eq. 32. This has the geometric
interpretation that the requisite closure of a geodesic arc simply re-traces
the arc. Eq. 32 therefore gives the total dynamic phase accumulated over the
closed circuit shown in Fig. 8.

Total phase, $\Phi_{\rm tot}$, is typically expressed [25] so that the state
at the end of a closed circuit is the same as that at the beginning except for
a phase factor of $e^{i\Phi_{\rm tot}}$. With a SPS, though, the two state
need only have the same SOM projection, so that the total phase can be
generalized by expressing it in terms of Jones vectors of the state going into
the lens, $\ket{u_{\rm in}}$, and the state produced by the projection
operation, $\ket{u_{\rm proj}}$:

$\Phi_{\rm tot}=\arg(\underline{v}^{*}_{\rm in}\cdot\underline{v}_{\rm out}).$
(33)

As with the dynamic phase, the evolving fiber phase has been removed. When
applied to beam transit through a single lens, this is

$\Phi_{\rm tot}^{\rm
lens}=-\tan^{-1}\bigl{(}\cos(\Delta\phi/2),\cos(\theta)\sin(\Delta\phi/2)\bigr{)}.$
(34)

Since the geodesic arc represents a projection operation, though, it does not
contribute any total phase accumulation. The total phase of Eq. 34 therefore
applies to the entire closed circuit shown in Fig. 8. As with the dynamic
phase, Eq. 24 can be used to plot this as a function of lens position, carried
out in Fig. 10 for Lens 1 and Lens 2.

Figure 10:  _Total Phase Accumulation as a Function of Lens Position_. Eqs. 34
and 24 are combined to generate the total phase accumulated by beam transits
through Lens 1 and Lens 2, followed by a projection operation, as plotted on
the SOM in Fig. 3. This phase depends on lens position, $z_{\ell}$. For lens
positions of $z_{\ell 1}=0.59z_{\scriptscriptstyle Rp}$ and $z_{\ell
2}=1.77z_{\scriptscriptstyle Rp}$, the SOM trajectories are the same for each
lens ($90^{\circ}$ arc angles) and the total phase is evaluated to be the
same, $-35.3^{\circ}$, for the two lenses.

Geometric phase is defined as the difference between the total and dynamic
phase [25], and it can be immediately calculated using Eqs. 32 and 34.

As a check on this approach for dissecting geometric phase, the reliance on
differential parallelism [24] can be replaced with Pancharatnam’s method [26]
of quantifying the degree of parallelism between two arbitrarily separated
states. The geometric phase derived in this way has a nice geometric
interpretation; it is negative one-half the solid angle,
$\Omega_{\rm\scriptstyle moon}$, subtended by the moon-shaped sliver of Fig.
8:

$\Phi_{\rm geom,lens}^{\rm Panch}=-\frac{\Omega_{\rm\scriptstyle moon}}{2}.$
(35)

Here

$\displaystyle\Omega_{\rm\scriptstyle moon}$ $\displaystyle=$
$\displaystyle\Delta\phi(1-\cos\theta)$ (36) $\displaystyle-2$
$\displaystyle\,\cot^{-1}\biggl{[}\bigl{(}\cot\Delta\phi+\frac{(\cot\theta+\csc\theta)^{2}}{\sin\Delta\phi}\bigr{)}\biggr{]}.$

Although equivalent to using Pancharatnam’s _In-Phase Rule_ , this expression
was obtained by simply calculating the intersection of a curvilinear
tetrahedron [29] and a polar cap of constant latitude. As usual, $\Delta\phi$
and $\theta$ refer to the azimuthal span and polar angle with respect to the
x-axis, respectively.

Now we can compare our derivation of geometric phase, using Eqs. 32 and 34, to
the Pancharatnam result of Eq. 35. This is carried out for the two different
lens transits shown in Fig. 3 to produce the comparisons of geometric phase
accumulation plotted in Fig. 11. The results match perfectly. Note that the
two lenses produce the same geometric phase, $3.44^{\circ}$, if Lens 1 is
positioned at $z_{\ell 1}$ and Lens 2 is positioned at $z_{\ell 2}$. This is
expected since the geometric phase does not depend on the fiber phase, $\chi$,
or position within the SPS.

Figure 11:  _Geometric Phase Accumulation as a Function of Lens Position_.
Eqs. Eqs. 32 and 34 or, equivalently, Eq. 35, can be used to generate the
geometric phase accumulated by beam transits through lenses 1 and 2, as
plotted on the SOM in Fig. 3. This phase depends on lens position, $z_{\ell}$.
For lens positions of $z_{\ell 1}=0.59z_{\scriptscriptstyle Rp}$ and $z_{\ell
2}=1.77z_{\scriptscriptstyle Rp}$, the SOM trajectories are the same for each
lens ($90^{\circ}$ arc angles) and the geometric phase is evaluated to be the
same for the two lenses.

The nonlinear relationship between SOM arc subtended and geometric phase is
shown in Fig. 12, where it is shown that the maximum magnitude of geometric
phase that can be accumulated for Lens 1 or Lens 2 is $-26.36^{\circ}$.
However, Eq. 35 can be used to calculate the geometric phase of an arbitrary
small-circle arc beyond the maximum arc angle of $\pi$ that can be generated
with a single lens. This is relevant to a series of mode converters for
instance. As shown in the figure, Eq. 35 has the expected asymptote of
negative one-half the solid angle of a spherical cap centered on the x-axis:

$\Omega_{\rm cap}=4\pi\sin^{2}(\theta/2),$ (37)

where $\theta$ is the polar angle as measured from the x-axis.

Figure 12:  _Geometric Phase as a Function of Arc Subtended_. The geometric
phase accumulated by beam transit through a lens, Eq. 35, can be used to plot
the geometric phase accumulated as a function of the SOM arc subtended. The
maximum magnitude for a single lens, $-26.4^{\circ}$, occurs for an arc angle
of $180^{\circ}$, while a complete small circle would generate twice as much
geometric phase, $-52.7^{\circ}$. This is equal to negative one-half the solid
angle enclosed by a polar cap of polar angle $\pi/4$, as given by Eq. 37.

Eqs. 35 and 36 can be readily applied to quantify the geometric phase
accumulated across any cylindrical lens. To calculate the geometric phase of
one or more mode converters, for instance, we only need to know the input and
output spinor for each lens. The geometric phase of the converter is then the
sum of that of its components. In the same way, a closed circuit composed of a
series of mode converters can be calculated in this way as well. Of course, it
is usually easier to identify the geometric phase as being negative one-half
the solid angle subtended by the closed circuit, but this comes at the cost of
losing all information about modes.

##  VI Lens Trajectories are Circular Arcs

In the examples we considered, lens trajectories on the SOM were found to lie
on circles, and this will be true for any lens transit. Consider an abstracted
setting in which the input beam to a lens is described by the following
2-parameter spinor:

$\underline{v}^{in}=\begin{pmatrix}\cos(\mu/2)\\\
\sin(\mu/2)e^{-i\nu}\end{pmatrix}.$ (38)

The effect of the lens on this state is to change the phase along one axis.
Rather than relating it to lens parameters, the change can be parametrized as
a linear function of lens transit fraction, $\zeta$:

$[G]:=\begin{pmatrix}1&0\\\ 0&e^{-i\zeta}\end{pmatrix}.$ (39)

Within this abstract setting, we can also account for a misalignment of the
lens axes with respect to the principal axes of the beam by simply considering
a different basis to describe a beam; specifically, we can always choose a
basis that is aligned with the lens axes. The spinor, as a function of lens
transit fraction, $\zeta$, is therefore just
$\underline{v}(\zeta)=[G]\underline{v}^{\rm(in)}$. Eq. II.1 then gives the
following SOM coordinates for the trajectory:

$\displaystyle x$ $\displaystyle=$ $\displaystyle\cos(\mu)$ $\displaystyle y$
$\displaystyle=$ $\displaystyle\cos(\zeta+\nu)\sin(\mu)$ (40) $\displaystyle
z$ $\displaystyle=$ $\displaystyle-\sin(\mu)\sin(\zeta+\nu).$

Angles $\mu$ and $\nu$ are clearly polar and azimuthal spherical coordinates,
and the lens fraction, $\zeta$, maps out an azimuthal trajectory—i.e. an arc
of constant latitude on a circle.

Now apply this reasoning to a cylindrical lens misoriented by an angle of
$\eta$ with respect to the y-axis of the pre-lens beam. The incoming beam can
be decomposed into active and passive components which are then subjected to
the lens operator of Eq. 21. The resulting spinor can subsequently be
converted back to the original basis. With lens orientation described by

$[R_{\eta}]:=\begin{pmatrix}\cos\eta&&-\sin\eta\\\
\sin\eta&&\cos\eta\end{pmatrix}.$ (41)

The output spinor is then [12]

$\underline{v}^{\rm(out)}=[R_{\eta}]^{T}[G][R_{\eta}]\underline{v}^{\rm(in)}.$
(42)

Several lens-transit trajectories are plotted in Fig. 13 to show how different
combinations of input beam and lens orientation map out circular arcs on the
SOM. Each arc is obtained by plotting the SOM coordinates as functions of lens
transit fraction, $\zeta$.

Figure 13:  _SOM Trajectories for Cylindrical Lens Transits_. Trajectories
generated by a lens with $\alpha=\pi/2$ are shown for several sets of initial
positions (colors) and lens orientations, $\eta$. The start of each trajectory
is indicated with a small sphere of the same color. Dashing is a guide to the
eye to more easily see that each trajectory lies on a circle. Red:
$\underline{r}_{\rm init}=\\{0,0,1\\}$, $\eta=\pi/4$; Green:
$\underline{r}_{\rm init}=\\{0,1,0\\}$, $\eta=\pi/2$; Magenta:
$\underline{r}_{\rm init}=\\{0.861,0.357,0.362\\}$, $\eta=0$; and Blue:
$\underline{r}_{\rm init}=\\{0.612,-0.353,0.707\\}$, with $\eta=j\pi/8$,
$j\in[0,7]$.

##  VII Discussion

First-order, elliptical, optical vortices accumulate geometric phase as they
propagate through astigmatic (cylindrical) lenses. While this has been
considered for almost thirty years [11], a number of outstanding issues have
been addressed in this paper by explicitly considering fiber bundles for
overall phase and by introducing a Supplementary Product Space that allows
beam waists and their positions to be explicitly tracked. Each lens can be
decomposed into a stack of differentially thin components, allowing the beam
to be described in terms of functions of lens transit fraction. While this
decomposition is not unique, it is arguably the simplest and most physically
grounded deconstruction of a lens. It also provides a clear explanation for
how the mode transits the Sphere of Modes while propagating through the lens.

A prescribed first-order Gaussian beam was applied to two distinctly different
cylindrical lenses to show how SOM arcs, fiber position, and location in the
SPS can be traced out as functions of transit fraction through each lens. It
was shown that identical transits on the SOM can be associated with
substantially different input and output beams.

A projection operator was then constructed that allowed us to determine gauge
invariant measures of dynamic phase, total phase, and geometric phase for beam
transits through a single lens. The resulting expression for geometric phase
was shown to be equal to that obtained using the standard Pancharatnam
connection.

The insights and methodology offered in this work are expected to prove useful
in a variety of applications more general than the single-lens focus of the
present work. For instance, it is much more common to quantify geometric phase
using mode converters that each utilize a pair of lenses. While this makes the
measurement of geometric phase more straightforward, it obscures the fact that
geometric phase is accumulated as beams propagate through each lens. The
approach developed here can be applied equally well to the second lens of a
mode converter to determine how its position, relative to the first,
influences the associated arc length traced out on the SOM.

Experimental realization of the predictions of sections I-IV are
straightforward since the measurement of lens-modified modes is possible with
modern interferometry [30]. Direct measurements of geometric phase provide a
greater experimental challenge, though, because of the need to remove dynamic
phase contributions [28, 13, 31], and this is even more challenging for non-
geodesic trajectories [28]. A key element of future work will be to
experimentally realize the projection operation for tracing out geodesics on
the Sphere of Modes while preserving position within the Supplementary Product
Space.

The single-lens setting of this work also seems ideal for re-visiting the
relationship between geometric phase and the change in orbital angular
momentum imposed on the beam by the lens [11]. As a final note, perhaps
important in future applications, the framework established in this analysis
applies equally well to single-photon settings.

##  VIII Acknowledgments

The authors acknowledge useful discussions with P. Ford and support from both
the W.M. Keck Foundation and the NSF (DMR 1553905).

## References

  * [1] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison Wesley, San Francisco, 2004).
  * [2] C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions (Princeton University Press, Princeton, New Jersey, 2013).
  * [3] M. Nielsen, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge New York, 2010).
  * [4] F. Wilczek and A. Shapere, Geometric Phases in Physics (World Scientific, New Jersey, USA, 1989).
  * [5] M. Berry, Physics Today 43, 34 (1990).
  * [6] J. Oprea, The American Mathematical Monthly 102, 515 (1995).
  * [7] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * [8] T. Iadecola, T. Schuster, and C. Chamon, Phys. Rev. Lett. 117, 073901 (2016).
  * [9] Y. Shen, Z. Wang, X. Fu, D. Naidoo, and A. Forbes, Phys. Rev. A 102, 031501 (2020).
  * [10] Y. Chen, Y.-L. Zhang, Z. Shen, C.-L. Zou, G.-C. Guo, and C.-H. Dong, Phys. Rev. Lett. 126, 123603 (2021).
  * [11] S. van Enk, Optics Communications 102, 59 (1993).
  * [12] M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999).
  * [13] E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
  * [14] J. Samuel and R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988).
  * [15] J. Courtial, Optics Communications 171, 179 (1999).
  * [16] M. Lax, W. H. Louisell, and W. B. McKnight, Physical Review A 11, 1365 (1975).
  * [17] J. M. Andersen, A. A. Voitiv, M. E. Siemens, and M. T. Lusk, Phys. Rev. A 104, 033520 (2021).
  * [18] S. Kobayashi, Foundations of Differential Geometry (Wiley, New York, 1996).
  * [19] J. Verdeyen, Laser Electronics (Prentice Hall, Englewood Cliffs, N.J, 1995).
  * [20] M. Beijersbergen, L. Allen, H. van der Veen, and J. Woerdman, Optics Communications 96, 123 (1993).
  * [21] D. Stoler, J. Opt. Soc. Am. 71, 334 (1981).
  * [22] B. Saleh, Fundamentals of Photonics (Wiley-Interscience, Hoboken, N.J, 2007).
  * [23] M. V. Berry, Journal of Physics A: Mathematical and General 18, 15 (1985).
  * [24] B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
  * [25] Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987).
  * [26] S. Pancharatnam, Proceedings of the Indian Academy of Sciences - Section A 44, 247 (1956).
  * [27] N. Mukunda and R. Simon, Annals of Physics 228, 205 (1993).
  * [28] R. Bhandari, Physics Letters A 138, 469 (1989).
  * [29] F. Eriksson, Mathematics Magazine 63, 184 (1990).
  * [30] J. M. Andersen, S. N. Alperin, A. A. Voitiv, W. G. Holtzmann, J. T. Gopinath, and M. E. Siemens, Appl. Opt. 58, 404 (2019).
  * [31] S. Maji, P. Jacob, and M. M. Brundavanam, Phys. Rev. Applied 12, 054053 (2019).