# Type of Non-reciprocity in Fiber Sagnac Interferometer induced by Geometric
Phases

Dongzi Zhao  1 Jing-Zheng Huang  1 Tailong Xiao  1 Hongjing Li1  Xiaoyan Wu
and Guihua Zeng1  1State Key Laboratory of Advanced Optical Communication
Systems and Networks, Institute of Quantum Sensing and Information Processing,
Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

The non-reciprocity of Sagnac interferometer provides ultra-high sensitivity
for parameter estimation and offers a wide range of applications, especially
for optical fiber sensing. In this work, we study a new type of non-
reciprocity existed in optical fiber Sagnac interferometer where the
polarization dependent loss is taken into consideration. In particular, this
non-reciprocity is irrelevant to the physical effects that being considered in
previous studies, which originates from the geometric phases induced by
continuous-weak-measurement. In consequence, it has a unique phenomenon of
sudden phase transition, which may open a new way for the future design of
high precision optical fiber sensors.

††journal: osac††articletype: Research Article

##  1 Introduction

The optical fiber Sagnac interferometer plays important roles in various
applications, such as fiber optic gyroscope (FOG)[1, 2], current detection[3],
temperature sensing[4], strain detectors[5] and so on[6]. Standing in the
heart of all these applications, it is the non-reciprocity of Sagnac loop
induced by varying physical effects. However, the versatility of Sagnac
inteferomter sometimes becomes a curse for one to pursue high accuracy and
precision. For example, in the case of FOG, the irrelevant effects caused by
magnetic fields, thermal transits, acoustic fields and non-linearity of optic
fiber[7, 8] will all induce undesired non-reciprocal phases, which
dramatically limit the performance of the FOG and make constrains on the
design of optic fiber coil to minimize these irrelevant effects[6].

In this work, a new type of non-reciprocity exists in fiber Sagnac
interferometer is studied. Different from the previous studies, the non-
reciprocal phase is irrelevant to the physical effects we mentioned above, but
because the optical axes of the birefringence and polarization dependent loss
(PDL) in the optic fiber are in different directions. Moreover, when the PDL
strength reaches a certain level, there will be a sudden transition of this
non-reciprocal phase. In the beginning, the phenomenon is proposed and
analyzed based on the traditional transfer matrix method (TMM)[9, 10]. To
further explore its origin, we employ a recent proposed continuous-weak-
measurement model[11], and reveal its deep relation to the geometric phase.

This paper is organized as follows. The phenomenon of a new type non-
reciprocity in Sagnac interferometer is presented along with a theoretical
analysis based on TMM in Sec.II. Furthermore, the physical insight of this
non-reciprocity is explored by using the model of continuous weak measurement
and geometric phase in Sec.III, and the corresponding influences (both
negative effects and potential for sensing) in the context of optical sensing
are discussed Sec.IV. Finally, conclusions are made in Sec.V.

##  2 Principle

The basic idea of a fiber Sagnac interferometer is shown in Fig.1[6]. The
light ejected from a source passes through a beam splitter (Coupler 1) and a
polarizer, and then divided into two beams by another beam splitter (Coupler
2), and then traverse along the clockwise (CW) and counterclockwise (CCW)
directions of the fiber ring respectively. Consequently, any time varying
physical effects and inherently non-reciprocal phenomena can produce a non-
reciprocal phase-shift, which can be finally detected by the detector placed
on the other end of Coupler 1.

Figure 1: The Schematic diagram of Sagnac interferometer.

As a common feature in optical fiber, the PDL can be easily generated by a
number of physical effects. For examples, the pressure in the diametrical
direction of the fiber will make the transmission coefficients of the two
orthogonal polarization states being different, along with extra birefringence
causing by the elasto-optic effect[12]. Similarly, the bending of the optical
fiber will cause extra birefringence and extra PDL at the same time[13, 14,
15, 16, 17, 18]. Although the PDL and the bireferingence simultaneously appear
in the fiber, their principal axes may in general be not coincident [20, 22,
19, 21].

Figure 2: A simple situation where the principal axes of PDL and birefringence
in the fiber are differed by a fixed angle with respect to each other. Figure
3: Due to the geometric characteristics of the optical fiber loop, the
principal axes of PDL and birefringence in the fiber exhibits ‘symmetric’
characteristics at the two ports. Here, $\theta_{a}$ represents the angle
between the PDL axis and the birefringence axis, and $\theta_{l}$ represents
the angle between liner input polarization and the birefringence axis.

For simplicity, we assume the principal axes of the birefringence and PDL of
the fiber loop in Fig.1 are fixed and differed by an angle $\theta_{a}$ with
respect to each other, as is shown in Fig.2 and Fig.3. Here, we make the
principal axes of birefrigence as the laboratory coordinate. To analyze the
evolution of optical polarization when the light traverses through the fiber
loop, we divide the fiber loop into a series of small segments and employ the
transfer matrix method (TMM)[10]. First, the optical fiber loop is divided
into N segments, and the length ($l$) of each segment is made much smaller
than the beat length of the optical fiber. We can then consider each of the
segments as a combination of a birefringent wave plate and a non-ideal
polarizer, as is depicted in Fig.4. The effective birefringent wave plate and
non-ideal polarized of the $k$th segment can be described by Jones matrices
$\left[\begin{array}[]{cc}1&0\\\ 0&e^{-i\phi_{k}}\\\ \end{array}\right]$ and
$\left[\begin{array}[]{cc}1&0\\\ 0&\sqrt{\eta_{k}}\\\ \end{array}\right]$
respectively, with the overall phase shift and transmittance being omitted for
simplicity. Here, $\phi_{k}=2\pi(n_{o}-n_{e})l/\lambda$ is the phase
difference introduced by the effective birefringent wave plate, with $\lambda$
being the wavelength of the light, $n_{o}$ and $n_{e}$ being the refractive
indexes along the two orthogonal principal axes. $\eta=T_{min}/T_{max}\approx
1$ is the PDL strength of the effective non-ideal polarizer, of which the
polarization-dependent transmittance being denoted by $T_{max}$ and $T_{min}$
respectively. In consequence, the overall PDL strength of the optical fiber
loop is given by

$\Gamma_{dB}=-10\log\eta^{N}=-10N\log\eta.$ (1)

Figure 4: A divided segment of the optical fiber.

The Jones matrix of the $k$th segment can be expressed in the following form:

$\begin{array}[]{c}\left[\begin{array}[]{cc}cos(\theta_{k}/2)&\\!sin(\theta_{k}/2)\\\
sin(-\theta_{k}/2)&\\!cos(\theta_{k}/2)\\\
\end{array}\right]\left[\begin{array}[]{cc}1&\\!\\!0\\\
0&\\!\\!\sqrt{\eta_{k}}\\\
\end{array}\right]\left[\begin{array}[]{cc}cos(\theta_{k}/2)&\\!sin(-\theta_{k}/2)\\\
sin(\theta_{k}/2)&\\!cos(\theta_{k}/2)\\\
\end{array}\right]\left[\begin{array}[]{cc}1&\\!\\!0\\\
0&\\!\\!e^{-i\phi_{k}}\\\ \end{array}\right]\\\
=R^{-1}(\theta_{k})M_{\eta_{k}}R(\theta_{k})C(\phi_{k}),\end{array},$ (2)

where $\theta_{k}/2$ is the angle between the principal axes of the effective
non-ideal polarizer and effective phase plate.

We then consider an input light with liner polarization state of
$\ket{\psi_{in}}=\left[\begin{array}[]{c}cos(\theta/2)\\\ sin(\theta/2)\\\
\end{array}\right]$. For simplicity, we assume that the axes and strengths of
birefringence and PDL are time-invariant, which means $\theta_{k}=\theta$,
$\eta_{k}=\eta$, $\phi_{k}=\phi$. Here we assume that PDL axis and input
polarization are aligned, so the same $\theta/2$ is used to indicate the
direction of the PDL axis and the input state. For the light propagating in
the CW direction, the transfer matrix of the fiber loop evolves to be:

$\ket{\psi_{out+}}=\prod{R^{-1}(\theta)M_{\eta}R(\theta)C(\phi)\ket{\psi_{in}}}.$
(3)

On the other hand, the light propagating in the CCW direction evolves to be:

$\ket{\psi_{out-}}=\prod{C(\phi)R(\theta)M_{\eta}R^{-1}(\theta)\ket{\psi_{in}}}.$
(4)

By projecting both $\ket{\psi_{out+}}$ and $\ket{\psi_{out-}}$ back to the
initial polarization state $\ket{\psi_{in}}$, the phase difference between the
CW light and the CCW light can be obtained by:

$\braket{\psi_{out+}}{\psi_{in}}=\sqrt{I_{+}}e^{i\phi_{+}},\braket{\psi_{out-}}{\psi_{in}}=\sqrt{I_{-}}e^{i\phi_{-}}.$
(5)

The change of phase difference $\phi_{+}-\phi_{-}$ along with $\eta$ by
choosing specific values of $\theta$ is numerically simulated in Fig.5. Here
we choose $N=200$, $\psi=2\pi/N$, the variation range of $\eta$ is from 0.92
to 1, and the corresponding variation range of $\Gamma_{dB}$ is from 84.56dB
to 0dB. According to the numerical simulation results, we find that a phase
transition appears when $\eta$ reaches a certain value.

Figure 5: Sorted by the left end of the curve, from bottom to top in the
figure are the phase difference changes with $\eta$ when
$\theta=0,\pi/4,3\pi/8,7\pi/16,15\pi/32,31\pi/64,63\pi/128,$
$127\pi/256,255\pi/512$, $511\pi/1024,1023\pi/2048,2047\pi/4096,\pi/2$. It can
be seen that as $\theta$ approaches $\pi/2$, the slope of the curve neer
$\eta=\eta_{c}$ will gradually increase until a sudden phase transition
occurs, where $\eta_{c}\approx 0.96$. The variation range of $\eta$ in the
figure corresponds to the variation range of $\Gamma_{dB}$ from 0dB to
84.56dB. At the transition point, $\Gamma_{dB}\approx 42dB$.

##  3 Theory

###  3.1 Theoretical analysis based on measurement-induced geometric phase

With assistance of TTM, we have discovered a new type of non-reciprocal phase
induced by PDL, but its physical essence is not clear. In order to explain
this phenomenon, we introduce a measurement-induced geometric phase theory
that recently proposed by Gebhart, et al.[11]. In this theory, it was shown
that Geometric phases are not necessarily a consequence of adiabatic time
evolution, but can also be induced by a cyclic sequence of quantum
measurements. Moreover, the mapping between the measurement sequence and the
geometric phase undergoes a topological transition by varying the measurement
strength[11]. The $k$th variable strength measurement can be written by the
follow form:

$M_{k}^{r_{k}}(\boldsymbol{n}_{k},\eta_{k})=\boldsymbol{R}^{-1}(\boldsymbol{n}_{k})M_{k}^{r_{k}}(\boldsymbol{e}_{z},\eta_{k})\boldsymbol{R}(\boldsymbol{n}_{k}),$
(6)

where $r_{k}\in\\{+,-\\}$ is the measurement readout, $\eta_{k}$ is the
measurement strength, and

$M_{k}^{+}(\boldsymbol{e}_{z},\eta_{k})=\left[\begin{array}[]{cc}1&0\\\
0&\sqrt{\eta_{k}}\\\
\end{array}\right],M_{k}^{-}(\boldsymbol{e}_{z},\eta_{k})=\left[\begin{array}[]{cc}0&0\\\
0&\sqrt{1-\eta_{k}}\\\ \end{array}\right]$ (7)

are the operators of measurement on z direction, corresponding to the readouts
of $+$ and $-$ respectively. And the operator

$R(\boldsymbol{n}_{k})=\left[\begin{array}[]{cc}cos(\theta_{k}/2)&e^{-i\phi_{k}}sin(\theta_{k}/2)\\\
sin(-\theta_{k}/2)&-e^{-i\phi_{k}}cos(\theta_{k}/2)\\\ \end{array},\right]$
(8)

rotates the measurement orientation from $z$ to
$n_{k}=(sin(\theta_{k})cos(\phi_{k}),sin(\theta_{k})sin(\phi_{k}),cos(\theta_{k}))$.
Interestingly, the operator $M_{k}^{+}(\boldsymbol{n}_{k},\eta_{k})$ can be
effectively implemented by rotating an imperfect polarizers and adding phase
plate[11], which is perfectly matches the optical fiber segment model we have
proposed in the last section. In particular, the PDL matrix $M_{\eta}$ is
exactly the same with $M_{k}^{+}(\boldsymbol{e}_{z},\eta_{k})$ if we consider
the PDL strength as the measurement strength.

Therefore, the process described in Sec.II can be described a quasicontinuous
measurement sequence with postselecting the results of $r_{k}=+$. According to
the analysis made in Ref.[11], after $N$ quasicontinuous measurements the
initial state $\ket{\psi_{i}n}$ becomes:

$\ket{\psi_{N}}=M^{+}_{N}(\boldsymbol{n}_{N},\eta_{N})M^{+}_{N-1}(\boldsymbol{n}_{N-1},\eta_{N-1})...M^{+}_{1}(\boldsymbol{n}_{1},\eta_{1})\ket{\psi_{in}}.$
(9)

Different measurement sequences induce state trajectories on the Bloch sphere.
For example, by taking $\theta_{k}=\pi/4$ and $\phi_{k}=2\pi k/N$, the state
trajectories under different measurement strength are depicted, as is shown in
Fig.6. If $\ket{\psi_{N}}$ is then projected back to $\ket{\psi_{in}}$, the
corresponding geometric phase induced by this process is
$\chi_{geom}=arg(\braket{\psi_{in}}{\psi_{N}})$. According to the geometric
phase theory, this measurement-induced geometric phase can be equivalently
expressed via the solid angle $\Omega$ of the trajectory on Bloch sphere by
$\chi_{geom}=\Omega/2$[23, 24].

Figure 6: When taking $\theta_{k}=\pi/4$ and $\phi_{k}=2\pi/N$, the state
trajectories under different measurement strength are depicted on the Bloch
sphere. The circle drawn by the black dashed line represents the measurement
orientation of the measurement sequences ({$n_{k}$}). The red curves in (a)
and (b) represent the state trajectories under different measurement strength.
And the green curves represent the final projection process. We can find that
the solid angle of trajectory may be vary different under different
measurement strength.

Back to our analysis of the fiber Sagnac loop. Let us split the effective
phase plate in the fiber segment by two parts (as is shown in Fig.6), and then
$C(\phi_{k})$ in Eq.(2) can be rewritten as

$C(\phi_{k})=\left[\begin{array}[]{cc}1&0\\\
0&e^{-i\phi_{k+1}}\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
0&e^{i\phi_{k}}\end{array}\right]=C(\phi_{k+1})C^{-1}(\phi_{k}),$ (10)

where $\phi_{k+1}-\phi_{k}=\phi$. Then Eq.(3) and Eq.(4) can be rewritten as:

$\begin{array}[]{lll}\ket{\psi_{out}}&=...C(\phi_{k+2})C^{-1}(\phi_{k+1})R^{-1}(\theta_{k+1})M_{\eta_{k+1}}R(\theta_{k+1})C(\phi_{k+1})\\\
&C^{-1}(\phi_{k})R^{-1}(\theta_{k})M_{\eta_{k}}R(\theta_{k})C(\phi_{k})C^{-1}(\phi_{k-1})...\ket{\psi_{in}}\\\
&=\sum
C^{-1}(\phi_{k})R^{-1}(\theta_{k})M_{\eta_{k}}R(\theta)C(\phi_{k})\ket{\psi_{in}}.\end{array}$
(11)

Consequently, the transfer matrix connecting $\ket{\phi_{k-1}}$ and
$\ket{\phi_{k}}$ is:

$\begin{array}[]{c}C^{-1}(\phi_{k})R^{-1}(\theta_{k})M_{\eta_{k}}R(\theta_{k})C(\phi_{k})\\\
=\\!\\!\left[\\!\begin{array}[]{cc}cos(\theta_{k}/2)&\\!\\!sin(\theta_{k}/2)\\\
e^{-i\phi_{k}}sin(\theta_{k}/2)&\\!\\!-e^{-i\phi_{k}}cos(\theta_{k}/2)\\\
\end{array}\\!\right]\\!\left[\\!\begin{array}[]{cc}1&\\!\\!\\!0\\\
0&\\!\\!\\!\sqrt{\eta_{k}}\\\
\end{array}\\!\right]\\!\left[\\!\begin{array}[]{cc}cos(\theta_{k}/2)&\\!\\!e^{-i\phi_{k}}sin(\theta_{k}/2)\\\
sin(-\theta_{k}/2)&\\!\\!-e^{-i\phi_{k}}cos(\theta_{k}/2)\\\
\end{array}\\!\right]\\!,\end{array}$ (12)

which is coincident with Eq.(6). Consequently, the non-reciprocal phase
difference between the light traverse along opposite directions in the fiber
Sagnac loop can be interpreted as a PDL-induced geometric phase.

Figure 7: Splitting the birefringent wave plate in each segment into two
parts.

###  3.2 Numerical simulation

In order to study the PDL-induced non-reciprocity in the sense of geometric
phase, we first recalculate the scenario that presented in Fig.5. By setting
$\phi_{k}=2k\pi/N(k=1,2,3...N)$, $\theta_{k}=\theta$ (with $0<\theta<\pi$),
$\eta_{k}=\eta$ (with $1-\eta<<1$), and making the initial polarization state
as $\ket{\psi_{in}}=\left[\begin{array}[]{c}cos(\theta/2)\\\
sin(\theta/2)\end{array}\right]$, we can plot the polarization state
trajectory on the Bloch sphere that similar to the one shown in Fig.6.

Figure 8: The phase difference between CW light and CCW light varies with
$\theta$ and $\eta$. Figure 9: The curve of the phase change with $\theta$
before(red) and after(blue) the phase transition point. Figure 10: The
trajectory under different $\theta$ when $\eta>\eta_{c}$(a) and
$\eta<\eta_{c}$(b).

Moreover, we find that the trajectory of polarization state on the Bloch
sphere changes along with the PDL strength. Because the final projection will
make the polarization state return to the initial state with the shortest
trajectory on the Bloch sphere, as is shown in Fig.6.

The PDL-induced non-reciprocal phases under different $\theta$ and $\eta$ are
calculated and depicted in Fig.8. In the case of weak PDL strength
($\eta>0.96$), there is only a small fluctuation of the non-reciprocal phase
with the change of $\theta$, as is depicted in Fig.9 in red. In this case, the
evolution trajectory of the polarization state (system state in the geometric
phase theory) on the Bloch sphere is shown in Fig.6(a). We can find that the
end point of the evolution trajectory cannot across the hemisphere. Therefore,
after the final projection, the flux of the curved surface surrounded by the
overall evolution trajectory yields a small value, corresponding to a small
geometric phase. As the PDL strength increases, the end point of the evolution
trajectory will finally crossed the hemisphere, and the solid angle of the
trajectory will suddenly change, as shown in Fig.10(b). This happens at the
point of $\eta=\eta_{c}\approx 0.96$ by setting $N=200$. in our numerical
simulation, and the sudden changed phase curve is depicted in Fig.9 in blue.
After all, this phase transition can be explained by the sudden change of the
solid angle of closed trajectory on the Bloch sphere.

##  4 Discussions

###  4.1 Negative effects

In practice, the PDL in an optical fiber is usually weak, thus the non-
reciprocal phase difference induced by PDL is also small. If the PDL-induced
non-reciprocal phase is small comparing to the signal, this effect can be
neglected. For instance, suppose the phase to be measured in the fiber Sagnac
interferometer is in the order of $2\pi\times 10^{-2}$. If we consider that
the PDL-induced non-reciprocal phase being an order of magnitude lower than
the signal is tolerable, then the overall PDL of the fiber loop $\Gamma_{dB}$
should be lower than 2.2dB. It is easily to be achieved in most of the optical
fiber coils for normal applications.

However for a much smaller signal, the requirement for PDL strength will
become more and more stringent. As is summarized in the following table, when
the phase precision of the fiber Sagnac interferometer reaches to $2\pi\times
10^{-6}$, the effect of PDL-induced non-reciprocity can not be neglected if
$\Gamma_{dB}$ is larger than 0.0201dB. In this case, some parameters such as
the radius of the fiber coil should be carefully considered when designing the
fiber Sagnac interferometer[20].

Phase precision | $2\pi\times 10^{-3}$ | $2\pi\times 10^{-4}$ | $2\pi\times 10^{-5}$ | $2\pi\times 10^{-6}$
---|---|---|---|---
Maximum tolerable $\Gamma_{dB}$ | 0.8002dB | 0.1603dB | 0.0501dB | 0.0201dB
Table 1: Maximum tolerable $\Gamma_{dB}$ under different sensitivity
requirements.

###  4.2 Potential for sensing

The sudden phase change happens at the phase transition point provides a good
opportunity for high sensitivity parameter estimation. Specifically, we can
make the new-type non-reciprocal phase very close to the phase transition
point by introducing a high strength PDL. At this point, a very slight change
on the PDL will cause an abrupt change on the phase, which indicates an
ultrahigh sensitivity.

The extra PDL can be effectively introduced in many ways. For example, one can
add an external pressure on a bent fiber from a different direction, as is
shown in Fig.11. Here $\theta_{s}$ represents the angle between the directions
of the pressure and the banding radius, but is not exactly equal to the one
between the axes of extra birefringence and PDL, which requires more rigorous
calculations[20, 19, 21]. According to the previous analysis and results shown
in Fig.5, we would need a PDL strength of about 42dB, which can in principle
be realized by a 210 turns of single mode fiber coil with a radius of 9mm,
according to the results presented in Ref.[20]. This simple example is just
for indicating the possibility. Many other possible ways, such as applying
polarizing optical fiber or fiber Bragg grating, are worth to be explored for
achieving the goal.

Figure 11: Tilting the angle between the birefringence axis and the PDL axis
by adding an external pressure on a bent fiber from a different direction.
Here, R is the radius of curvature and $\theta_{s}$ is the angle between
pressure direction and banding radius.

Moreover, in practice there are many other physical effects that can induce
non-reciprocal phase in a fiber Sagnac loop, such as the power reflectance,
Kerr effect, Faraday effect and so on, which will presented as noises in our
sensing application. For example, the reflected power from the interfaces of
devices may introduce so-called amplitude-type noise, and a broadband light
source is required for suppressing this noise[25]. A more rigorous analysis on
these kinds of noises and errors is an important issue in the future
investigation.

##  5 Conclusion

In summary, we have found a new type of non-reciprocity in the optical fiber
Sagnac interferometer, which is induced by the PDL (more precisely, the
difference between the princial axes of PDL and birefringence). To explore the
origin of this new non-reciprocity, we applied a recent proposed continuous-
weak-measurement theory model and interpret the non-reciprocal phase as a PDL-
induced geometric phase. Fortunately, this effect is so small that it can be
neglected for most of the common applications using fiber Sagnac
interferometers, unless the requirement for precision is extremely high.
Moreover, the numerical simulation indicates that a phase transition
phenomenon will happen when the PDL strength reaches a certain value, which
implies a new possibility for designing new type of sensors with ultra-high
sensitivity.

Funding This work was supported by the civil aerospace advance research
project (D020403) and the National Natural Science Foundation of China (Grants
No.62071298, No.61671287, No.61631014, and 193 No.61901258).

Disclosures The authors declare that there are no conflicts of interest
related to this article.

Data availabilityData underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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Let $V$ and $W$ be quiver representations over $\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of $\Hom_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.

§ INTRODUCTION

Some important classes of representations over bound quivers admit a very well-behaved basis. Namely one where the arrows send basis elements to other basis elements or zero and no arrow sends two different basis elements to the same basis element. Examples include tree and string modules, permutation representations, interval modules over incidence algebras, projective and injective modules over monomial algebras and laminated modules over preprojective algebras. The notion of quiver representations over $\bF_1$, as introduced by Szczesny [11], is designed to capture the combinatorial structure of these bases.

An $\bF_1$-vector space is a pointed set. The distinguished point is the zero element while the remaining elements form a basis of the vector space. The $\bF_1$-linear maps are pointed maps satisfying an injectivity condition. We let $Q$ be a quiver and $K$ be a field. Representations of $Q$ over $\bF_1$ are defined in analogy to representations over $K$. There is a scalar extension functor that turns a representation $V$ of $Q$ over $\bF_1$ into a representation $V^K$ of $Q$ over $K$. If $V$ was obtained from a basis as above then $V^K$ is isomorphic to the original representation. The category of quiver representations over $\bF_1$ behaves in many ways like a non-additive version of a module category: It has kernels, cokernels, direct sums and tensor products. In addition it satisfies versions of the Jordan-Hölder and Krull-Schmidt theorems. This was first observed by Szczesny [11].

The goal of this paper is to study aspects of the relation between the combinatorics of $\bF_1$-representations and the representation theory of their scalar extensions. Specifically we will describe a basis of the space of homomorphisms between two scalar extensions of $\bF_1$-representations and characterise when all scalar extensions of such a representation are indecomposable. In particular, this answers question 6.14 in [7].

Jun and Sistko [8] reinterpreted the data of an $\bF_1$-representation $V$ as a labeled graph $\Gamma_V$ called the coefficient quiver of $V$. This perspective provides a powerful dictionary from representation theory to graph theory. For example, the $\bF_1$-representation $V$ is indecomposable if and only if $\Gamma_V$ is connected. Further, $V$ is simple if and only if $\Gamma_V$ is strongly connected.

We let $V$ and $W$ be two representations of $Q$ over $\bF_1$. The first goal of the paper is to construct a basis of  $\Hom_{KQ}(V^K,W^K)$. This basis will be indexed by a subset of the direct summands of $V\otimes W$ that we call admissible components. If $\Gamma_V$ and $\Gamma_W$ are trees then this basis agrees with the one introduced by Crawley-Boevey [1] for tree modules. In particular, we obtain a combinatorial description of $\End_{KQ}(V^K)$.

An $\bF_1$-representation $V$ is absolutely indecomposable if $V^K$ is indecomposable for all fields $K$. Our second goal is to use the basis above to give a combinatorial characterisation of absolutely indecomposable representations. To that end, we define and study a special class of the admissible components in $\Gamma_{V\otimes V}$ that we call covering components. This yields the following theorem:

Let $V$ be an indecomposable representation of $Q$ over $\bF_1$. Then the following are equivalent:

* $V$ is absolutely indecomposable,

* $V^\bC$ is indecomposable,

* $\Gamma_{V\otimes V}$ contains exactly one covering component.

In this case, the unique covering component corresponds to the identity in our basis of $\End_{KQ}(V^K,V^K)$ for all fields $K$. This criterion can be tested efficiently in practice: All covering components can be found in $O(m^2)$ time where $m$ is the number of arrows in $\Gamma_V$.

Absolutely indecomposable representations over finite fields were first studied by Kac [9], see also [2]. Their main results describe how the number of absolutely indecomposable representations depend on the number of elements of the underlying field. Unfortunately, those formulas usually give the wrong result for $\bF_1$. A more detailed discussion of this can be found at the end of section 6 in [7].

Jun and Sistko [7] introduced the notion of finite nice length for representations over $\bF_1$. They showed that given a representation $V$ of finite nice length, computing the Euler characteristic of the quiver Grassmannians of $V^\bC$ becomes a combinatorial problem. To do so they adapted an approach developed by Cerulli-Irelli [3] and [6] to the language of $\bF_1$-representations. We apply our previous characterisation to obtain the following theorem.

Let $V$ be an indecomposable representation over $\bF_1$ with finite nice length. Then $V$ is absolutely indecomposable.

Sections 2 and 3 recall the basic theory of quiver representations over $\bF_1$ and their coefficient quivers. Section 4 constructs the basis of the homomorphism spaces between scalar extensions. Section 5 prepares the theory of covering components which we need to prove the characterisation of absolutely indecomposable representations in section 6. The final section contains the proof that indecomposable finite nice length representations are absolutely indecomposable.

§ QUIVER REPRESENTATIONS OVER F1

We start by recalling the category $\Rep(Q,\bF_1)$ and its basic properties following [11] and [7].

A vector space over $\bF_1$ is a finite pointed set $(V,0_V)$. Let $(V,0_V)$ and $(W,0_W)$ be $\bF_1$-vector spaces. An $\bF_1$-linear map is a pointed map $f:V\rightarrow W$ such that $f$ is injective on $V\backslash f^{-1}(0_W)$. This defines a category called $\Vect(\bF_1)$.

By abuse of notation, we will usually refer to an $\bF_1$-vector space $(V,0_V)$ just by $V$. The subset $V\backslash\{0_V\}$ should be interpreted as a basis of $V$. The category $\Vect(\bF_1)$ is not additive yet it shares many properties with the categories of vector spaces over actual fields. We adapt some of the terms from linear algebra.

Let $V$ and $W$ be $\bF_1$-vector spaces and let $f:V\rightarrow W$ be an $\bF_1$-linear map.

* The dimension of $V$ is given by $\dim_{\bF_1}(V) = |V\backslash\{0_V\}|$.

* $V$ is a subspace of $W$ if $V$ is a pointed subset of $W$.

* If $V$ is a subspace of $W$ then the quotient $W/V$ is the pointed set $(W\backslash V)\cup \{0_W\}$.

* The kernel of $f$ is the subspace $f^{-1}(0_W)$ of $V$. The cokernel of $f$ is the quotient $W/\Image(f)$.

* There is a unique $0$-map from $V$ to $W$ given by sending all elements in $V$ to $0_W$.

* The direct sum of $V$ and $W$ is the pointed set $V\oplus W =V\sqcup_{0_V\sim 0_W}W$.

* The tensor product  is the pointed set $V\otimes W =V\times W /(V\times \{0_W\}\cup\{0_V\}\times W )$

* The dual map $f^t:W\rightarrow V$ is given by

\[f^t(w) = \begin{cases}
v &\textnormal{if }w \neq 0_w\textnormal{ and }f^{-1}(w) = \{v\}\\
0_V &\textnormal{else.}
\end{cases}\]

This gives the category $\Vect(\bF_1)$ enough structure to define exact sequences. In fact it forms a proto-exact category in the sense of [4].

The direct sum $V\oplus W$ is not a categorical biproduct since $\Vect(\bF_1)$ admits neither categorical products nor coproducts.

Next we want to consider quiver representations over $\bF_1$.

A quiver is a quadruple $Q=(Q_0,Q_1,s,t)$ where $Q_0$ is the set of vertices and $Q_1$ is the set of arrows while $s$ and $t$ are maps from $Q_1$ to $Q_0$ sending an arrow to its source, respectively its target. We always assume that $Q_0$ and $Q_1$ are finite.

The $n$-loop quiver $\mathbb{L}_n$ has one vertex and $n$ loops at that vertex.

We fix a quiver $Q$ for the rest of the paper.

Let $K$ be a field or $\bF_1$. A quiver representation of $Q$ over $K$ is a tuple $V = ((V_i)_{i\in Q_0},(f_\alpha)_{\alpha\in Q_1})$. Here, $V_i$ is a finite dimensional $K$-vector space for each $i\in Q_0$ and $f_\alpha:V_{s(\alpha)}\rightarrow V_{t(\alpha)}$ is a $K$-linear map for each $\alpha\in Q_1$. Let $V$ and $W$ be representations of $Q$ over $K$. A homomorphism of quiver representations $f:V\rightarrow W$ is a family of maps $(f_i)_{i\in Q_0}$ where each $f_i$ goes from $V_i$ to $W_i$ and for each arrow $\alpha\in Q_1$ the following diagram commutes:

\[\begin{tikzcd}
{V_{s(\alpha)}} & {W_{s(\alpha)}} \\
{V_{t(\alpha)}} & {W_{t(\alpha)}.}
\arrow["{f_{s(\alpha)}}", from=1-1, to=1-2]
\arrow["{W_\alpha}", from=1-2, to=2-2]
\arrow["{V_\alpha}"', from=1-1, to=2-1]
\arrow["{f_{t(\alpha)}}"', from=2-1, to=2-2]
\end{tikzcd}\]

The quiver representations of $Q$ over $K$ form the category $\Rep(Q,K)$. If $K$  is not $\bF_1$ then we denote the space of homomorphisms by $\Hom_{KQ}(V,W)$.

As before many concepts from representation theory can be adapted to the $\bF_1$-setting. The category $\Rep(Q,\bF_1)$ satisfies versions of the Krull-Schmidt and Jordan-Hölder theorems, as proven in <cit.>. The goal of this paper is to study relations between $\bF_1$-representations and their scalar extensions which will be introduced next.

Let $K$ be a field. There is a faithful exact functor
\[-\otimes_{\bF_1}K:\Vect(\bF_1)\rightarrow\Vect(K)\]
called scalar extension. It sends an $\bF_1$-vector space $V$ to the free $K$-vector space on $V\backslash\{0_V\}$. We obtain a faithful exact functor
\[-\otimes_{\bF_1}K:\Rep(Q,\bF_1)\rightarrow\Rep(Q,K)\]
by applying the previous functor pointwise. To simplify notation we will write $V^K$ for $V\otimes_{\bF_1}K$. These functors are never full and rarely dense.

§ COEFFICIENT QUIVERS

Coefficient quivers for quiver representations over $\bF_1$ were introduced by Jun and Sistko in [8]. They reinterpret an $\bF_1$-representation $V$ of $Q$ as a quiver $\Gamma_V$ with a special map $c:\Gamma_V\rightarrow Q$ called a winding. This provides a powerful connection to graph theory.

Let $\Gamma$ be a quiver and $c:\Gamma\rightarrow Q$ a morphism of quivers. The map $c$ is called a winding if for all $\alpha\in Q_1$ and all $a,b\in c^{-1}(\alpha)$ with $a\neq b$ we have $s(a)\neq s(b)$ and $t(a)\neq t(b)$. This condition can be visualised as follows: Consider a subquiver of $\Gamma$ of one of the following forms.

\[\begin{tikzcd}
\bullet & \bullet & \bullet && \bullet & \bullet & \bullet
\arrow[from=1-1, to=1-2]
\arrow[from=1-3, to=1-2]
\arrow[from=1-6, to=1-5]
\arrow[from=1-6, to=1-7]
\end{tikzcd}\]

Then the two arrows in the subquiver must be mapped to different arrows in $Q_1$.

It is sometimes useful to interpret a winding $c:\Gamma \rightarrow Q$ as a colouring of $\Gamma$. An arrow $b$ with $c(b) = \alpha$ would be considered an $\alpha$-coloured arrow and a vertex $w$ with $c(w) = i$ would be considered an $i$-coloured vertex.

Let $V$ be an $\bF_1$-representation of $Q$. The coefficient quiver of $V$ is the pair $(\Gamma_V,c_V)$ where $\Gamma_V$ is the following quiver:

\begin{align*}
(\Gamma_V)_0 &=\coprod_{i\in Q_0} V_i\backslash\{0_{V_i}\}\\
(\Gamma_V)_1 &=\coprod_{\alpha\in Q_1} \left\{(v,w)\in ((\Gamma_V)_0)^2\mid V_\alpha(v) = w \right\}\\
s((v,w)) &= v\\
t((v,w)) &= w
\end{align*}

and $c_V$ is induced by the disjoint unions above.

The injectivity condition of $\bF_1$-linear maps ensures that $c_V$ is a winding.

We consider the two loop quiver

\[\bL_2 = \begin{tikzcd}
\arrow["a",color={rgb,255:red,92;green,92;blue,214},loop, in=160, out=200 ,distance=3em, from=1-1, to=1-1]
\arrow["b",color={rgb,255:red,214;green,92;blue,92},loop, in=-20, out=20 ,distance=3em, from=1-1, to=1-1]
\end{tikzcd}\]

and the $\bF_1$-representation $V=(V_*,V_a,V_b)$ given by $V_* = \{1,2\}\cup \{0\}$ and

\begin{align*}
V_a(1) &= 1& V_b(1) &= 2\\
V_a(2) &= 0& V_b(2) &= 1.
\end{align*}

Then the coefficient quiver of $V$ is given by

\[\Gamma_V =\begin{tikzcd}
1 & 2.
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-1, to=1-2]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-2, to=1-1]
\arrow[color={rgb,255:red,92;green,92;blue,214},loop, in=160, out=200 ,distance=3em, from=1-1, to=1-1]
\end{tikzcd}\]

Let $c:\Gamma \rightarrow Q$ be a winding. A full subquiver $H$ of $\Gamma$ is called

* successor closed if for every arrow $a\in \Gamma$ we have

\[s(a)\in H\Rightarrow t(a)\in H.\]

* predecessor closed if for every arrow $a\in \Gamma$ we have

\[t(a)\in H\Rightarrow s(a)\in H.\]

Let $V$ and $W$ be $\bF_1$-representations over $Q$.

* If $W$ is a subrepresentation of $V$ then $\Gamma_W$ is a successor closed subquiver of $\Gamma_V$.

* If $W$ is a quotient representation of $V$ then $\Gamma_W$ is a predecessor closed subquiver of $\Gamma_V$.

* $\Gamma_{V\oplus W} = \Gamma_V \sqcup \Gamma_W$.

* The indecomposable direct summands of $V$ correspond to the connected components of $\Gamma_V$.

One can define a category $\mathcal{C}_Q$ of windings over $Q$ that is equivalent to $\Rep(Q,\bF_1)$. The morphisms are based on the following observation: Let $f:V\rightarrow W$ be a morphism of quiver representations over $\bF_1$. Then $f$ factors as

\[V\rightarrow V/\ker(f)\xrightarrow{\overline{f}} \Image(f)\rightarrow W\]

where $\overline{f}$ is an isomorphism. The triple $(V/\ker(f),\Image(f),\overline{f})$ can be interpreted in the language of windings.

Let $c:\Gamma\rightarrow Q$ and $c':\Gamma'\rightarrow Q$ be two windings. A morphism of windings $\Phi:(\Gamma,c)\rightarrow(\Gamma',c')$ is a triple $\Phi=(F,U,\phi)$ where

* $F$ is a predecessor closed subquiver of $\Gamma$,

* $U$ is a successor closed subquiver of $\Gamma'$,

* $\phi:F\rightarrow U$ is an isomorphisms with $c'\circ\phi=c$.

Let $(F,U,\phi):\Gamma_1 \rightarrow \Gamma_2$ and $(F',U',\psi):\Gamma_2 \rightarrow \Gamma_3$ be two morphisms of windings. Their composition is given by $(F\cap \phi^{-1}(F'),U'\cap\psi(U),\psi\circ\phi)$ where $\psi\circ\phi$ is considered with the appropriate domain and codomain. We obtain a category $\mathcal{C}_Q$ of windings over $Q$.

The assignment $V\mapsto \Gamma_V$ defines an equivalence of categories

\[\Rep(Q,\bF_1)\rightarrow \mathcal{C}_Q.\]

This equivalence allows us to freely switch between an $\bF_1$-representation and its coefficient quiver depending on which one is more convenient. The tensor product of two representations will be central to this paper. We study it next.

Let $V$ and $W$ be in $\Rep(Q,\bF_1)$. The tensor product $V\otimes W$ is given by the vector spaces

\[(V\otimes W)_i = V_i \otimes W_i = V_i \times W_i/(V\times \{0_{W_i}\}\cup \{0_{V_i}\}\times W_i)\]

and the maps

\begin{align*}
(V\otimes W)_\alpha:(V\otimes W)_{s(\alpha)}&\rightarrow (V\otimes W)_{t(\alpha)}\\
v\otimes w &\mapsto V_\alpha(v)\otimes W_\alpha(w).
\end{align*}

Here we denote an element $(v,w)\in (V\otimes W)_i$ by $v\otimes w$.

Let $c:\Gamma \rightarrow Q$ and $c':\Gamma' \rightarrow Q$ be windings. We define the tensor product of windings $c\otimes c':\Gamma \otimes \Gamma' \rightarrow Q$ by

\begin{align*}
(\Gamma \otimes \Gamma')_0 &= \{(v,w)\in \Gamma_0 \times \Gamma'_0 \mid c(v) = c'(w)\}\\
(\Gamma \otimes \Gamma')_1 &= \{(a,b)\in \Gamma_1 \times \Gamma'_1 \mid c(a) = c'(b)\}\\
s((a,b)) &= (s(a),s(b))\\
t((a,b)) &= (t(a),t(b))\\
c\otimes c'((v,w)) &= c(v)\\
c\otimes c'((a,b)) &= c(a).
\end{align*}

We will give an example of the tensor product of two windings in Example <ref>. Comparing definitions yields:

Let $V,W\in \Rep(Q,\bF_1)$. Then

\[\Gamma_{V\otimes W} \cong \Gamma_V \otimes \Gamma_W.\]

§ MORPHISMS BETWEEN SCALAR EXTENSIONS

Let $V,W\in \Rep(Q,\bF_1)$ and let $K$ be a field. We want to use the combinatorial structure of the $\bF_1$-representations to obtain a basis of $\Hom_{KQ}(V^K,W^K)$. Our approach is inspired by [10]. By definition we have an inclusion

\[\Hom_{KQ}(V^K,W^K)\subset \prod_{i\in Q_0}\Hom_K(V^K_i,W^K_i).\]

The representations $V^K$ and $W^K$ carry distinguished bases given by the non-zero elements in $V$ respectively $W$. For $i\in Q_0$, $v\in V_i\backslash\{0_{V_i}\}$, $w\in W_i\backslash\{0_{W_i}\}$ we define the linear map $\mathbf{b}_{v,w}: V^K_i\rightarrow W^K_i$ by

\[\mathbf{b}_{v,w}(v')=\delta_{v,v'}w\]

for all $v'\in V_i\backslash\{0_{V_i}\}$. Then the set

\[\mathcal{B}_{V,W}=\left\{\mathbf{b}_{v,w} \in \Hom_K(V^K_i,W^K_i) \mid i\in Q_0, v\in V_i\backslash\{0_{V_i}\}, w\in W_i\backslash\{0_{W_i}\}\right\}\]

forms a basis of $\prod_{i\in Q_0}\Hom_K(V^K_i,W^K_i)$. For an element $f\in \Hom_{KQ}(V^K,W^K)$ we define its coefficients $f_{v,w}$ by

\[f = \sum_{\mathbf{b}_{v,w}\in \mathcal{B}_{V,W}} f_{v,w}\mathbf{b}_{v,w}.\]

To simplify notation we set $f_{0,w} = f_{v,0} = 0$.

We define an equivalence relation $\sim$ on $\mathcal{B}_{V,W}$ by $\mathbf{b}_{v,w} \sim\mathbf{b}_{v',w'}$ if and only if for all $f\in \Hom_{KQ}(V^K,W^K)$ we have $f_{v,w} = f_{v',w'}$.

There is at most one equivalence class consisting of all basis elements whose coefficient is always zero. We call it the zero class. We will show that the non-zero equivalence classes define a basis of $\Hom_{KQ}(V^K,W^K)$. To that end we will derive a combinatorial description of the equivalence relation using the tensor product in $\Rep(Q,\bF_1)$.  The connection is given by the bijection
\begin{align*}
(\Gamma_{V\otimes W})_0 &\rightarrow \mathcal{B}_{V,W}\\
(v,w)&\mapsto \mathbf{b}_{v,w}.
\end{align*}

We consider the two loop quiver

\[\bL_2 = \begin{tikzcd}
\arrow["a",color={rgb,255:red,92;green,92;blue,214},loop, in=160, out=200 ,distance=3em, from=1-1, to=1-1]
\arrow["b",color={rgb,255:red,214;green,92;blue,92},loop, in=-20, out=20 ,distance=3em, from=1-1, to=1-1]
\end{tikzcd}\]

and two $\bF_1$-representations $V$ and $W$ with coefficient quivers

\begin{align*}
\Gamma_V &=\begin{tikzcd}[ampersand replacement=\&]
1 \& 2 \& 3
\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-2]
\arrow[color={rgb,255:red,214;green,92;blue,92}, from=1-2, to=1-3]
\end{tikzcd},& \Gamma_W &=
\begin{tikzcd}[ampersand replacement=\&]
1' \& 2'.
\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-2]
\end{tikzcd}
\end{align*}

The tensor product $\Gamma_{V\otimes W}$ is given by

\[\begin{tikzcd}
{(1,1')} & {(2,1')} & {(3,1')} \\
{(1,2')} & {(2,2')} & {(3,2')}
\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=2-2]
\end{tikzcd}\]

The non-zero equivalence classes are $\{\mathbf{b}_{1,2'}\}$ and $\{\mathbf{b}_{1,1'},\mathbf{b}_{2,2'}\}$.

Let $(v,w)$ and $(v',w')$ be in the same connected component of $\Gamma_{V \otimes W}$. Then $\mathbf{b}_{v,w} \sim\mathbf{b}_{v',w'}$

Without loss of generality we can assume that there is an arrow $(a,b)\in (\Gamma_{V\otimes W})_1$ with

\begin{align*}
s((a,b)) &= (v,w)& t((a,b))&=(v',w').
\end{align*}

We set $i=c_V(v)=c_W(w)$, $j=c_V(v')=c_W(w')$ and $\alpha = c_V(a)=c_W(b)$. Let $f\in \Hom_{KQ}(V^K,W^K)$. We consider the commutative square

\[\begin{tikzcd}
{V^K_i} & {W^K_i} \\
{V^K_j} & {W^K_j}
\arrow["{f_i}", from=1-1, to=1-2]
\arrow["{W_\alpha}", from=1-2, to=2-2]
\arrow["{V_\alpha}"', from=1-1, to=2-1]
\arrow["{f_j}"', from=2-1, to=2-2]
\end{tikzcd}\]

and compute

\[W_\alpha(f_i(v)) = W_\alpha(\sum_{\overline{w}\in W_i}f_{v,\overline{w}}\cdot\overline{w})=\sum_{\overline{w}\in W_i}f_{v,\overline{w}}\cdot W_\alpha(\overline{w})\]

\[f_j(V_\alpha(v))=\sum_{\Tilde{w}\in W_j}f_{V_\alpha(v),\Tilde{w}} \cdot \Tilde{w}=\sum_{\Tilde{w}\in W_j}f_{v',\Tilde{w}} \cdot \Tilde{w}.\]

We want to examine the coefficient of $w'$. The injectivity condition for $\bF_1$-linear maps implies that $W_\alpha(\overline{w}) = w'$ if and only if $\overline{w} = w$. Hence the coefficient of $w'$ in  $W_\alpha(f_i(v))$ is $f_{v,w}$. The coefficient of $w'$ in $f_j(V_\alpha(v))$ is $f_{v',w'}$. The Lemma follows from commutativity of the square.

Next we will characterise the components of $\Gamma_{V\otimes W}$ that correspond to non-zero equivalence classes.

Let $C\subset \Gamma_{V\otimes W}$ be a connected component. We say that $C$

* reflects successors if for all vertices $(v,w)\in C_0$ and arrows $b\in (\Gamma_W)_1$ with $s(b)=w$, there is an arrow $a\in (\Gamma_V)_1$ with $s(a) = v$ and $c_V(a) = c_W(b)$. In this case the arrow $(a,b)\in C_1$ satisfies $s(a,b) = (v,w)$.

\[\begin{tikzcd}
v & w \\
{t(a)} & {t(b).}
\arrow[from=1-1, to=1-2]
\arrow["b", from=1-2, to=2-2]
\arrow[dashed, from=2-1, to=2-2]
\arrow["a"', dashed, from=1-1, to=2-1]
\end{tikzcd}\]

* induces predecessors if for all vertices $(v',w')\in C_0$ and arrows $a\in (\Gamma_V)_1$ with $t(a)=v'$ there is an arrow $b\in (\Gamma_W)_1$ with $t(b) = w'$ and $c_V(a) = c_W(b)$. In this case the arrow $(a,b)\in C_1$ satisfies $t(a,b) = (v',w')$.

\[\begin{tikzcd}
s(a) & s(b) \\
{v'} & {w'.}
\arrow[dashed, from=1-1, to=1-2]
\arrow["b", dashed, from=1-2, to=2-2]
\arrow[from=2-1, to=2-2]
\arrow["a"', from=1-1, to=2-1]
\end{tikzcd}\]

* is admissible if $C$ reflects successors and induces predecessors.

In Example <ref> the components $\{(1,2')\}$ and $\{(1,1'),(2,2')\}$ are admissible, the component $\{(3,2')\}$ only reflects successors and the other two components satisfy neither condition.

Let $f:V\rightarrow W$ be a morphism of quiver representations over $\bF_1$ and $(F,U,\phi):\Gamma_V\rightarrow \Gamma_W$ the corresponding morphism of windings. We assume that $\Image(f)$ is indecomposable. Then the set

\[\{(v,\phi(v))\mid v\in F_0\}\]

is the vertex set of an admissible component which we will call $C_f$. In this case the projections yield isomorphisms $F\cong C_f\cong U$. An example of an admissible component that is not of this form will be given in Example <ref>.

Let $C\subset \Gamma_{V\otimes W}$ be an admissible component. We consider the linear map from $V^K$ to $W^K$ given by

\[\mathbf{b}^C=\sum_{(v,w)\in C_0} \mathbf{b}_{v,w}.\]

Then $\mathbf{b}^C\in \Hom_{KQ}(V^K,W^K)$.

We verify this on basis vectors. Let $v\in V_i\backslash \{0_{V_i}\}$ and let $\alpha\in Q_1$ with $s(\alpha)=i$. We once more consider the diagram

\[\begin{tikzcd}
{V^K_i} & {W^K_i} \\
{V^K_j} & {W^K_j}
\arrow["{\mathbf{b}^C_i}", from=1-1, to=1-2]
\arrow["{W_\alpha}", from=1-2, to=2-2]
\arrow["{V_\alpha}"', from=1-1, to=2-1]
\arrow["{\mathbf{b}^C_j}"', from=2-1, to=2-2]
\end{tikzcd}\]

and compute

\[W_\alpha(\mathbf{b}^C_i(v))=W_\alpha(\sum_{(v,\overline{w})\in C_0}\overline{w}) = \sum_{(v,\overline{w})\in C_0} W_\alpha(\overline{w})\]

\[\mathbf{b}^C_j(V_\alpha(v)) = \sum_{(V_\alpha(v),\Tilde{w})\in C_0} \Tilde{w}.\]

The former sum has no repeated non-zero summands due to $\bF_1$-linearity. Thus we only have to show that the following two sets are equal:

\begin{align*}
&A :=\{W_\alpha(\overline{w})\mid (v,\overline{w})\in C_0\}\backslash\{0_{W_j}\}& &\textnormal{and}& &B:=\{\Tilde{w}\mid (V_\alpha(v),\Tilde{w})\in C_0\}.
\end{align*}

Let $W_\alpha(\overline{w})$ be in $A$. Then there exists an arrow $b\in (\Gamma_W)_1$ with $s(b) = \overline{w}$ and $c_W(b) = \alpha$. As $C$ reflects successors there exists an arrow $a$ in $(\Gamma_V)_1$ with $s(a) = v$ and $c_V(a) = c_W(b) = \alpha$. Together they form an arrow $(a,b)\in C_1$ satisfying $t((a,b)) = (V_\alpha(v),W_\alpha(\overline{w}))\in C_0$. This proves $W_\alpha(\overline{w})\in B$ and hence $A\subset B$. Similarly $B\subset A$ is a consequence of $C$ inducing predecessors.

Let $f:V\rightarrow W$ be a morphism of quiver representations over $\bF_1$ and let $C_f$ be the component constructed from $f$ in Example <ref>. Then $\mathbf{b}^{C_f} = f^K$.

Let $C\subset \Gamma_{V\otimes W}$ be a component that is not admissible. Then $C$ is part of the zero class.

By Lemma <ref> it suffices to prove this for one element of $C_0$. We will only consider the case where $C_0$ does not reflect successors. A similar argument works when $C_0$ does not induce predecessors. Since $C$ does not reflect successors there is $(v,w)\in C_0$ and $b\in (\Gamma_W)_1$ such that $s(b)=w$ but there is no arrow $a\in (\Gamma_V)_1$ with $s(a) = v$ and $c_V(a) = c_W(b)$. Let $f\in \Hom_{KQ}(V^K,W^K)$. We set $\alpha = c_W(b)$, $i=s(\alpha)$ and $j=t(\alpha)$. From the proof of Lemma <ref> we know

\[\sum_{\overline{w}\in W_i}f_{v,\overline{w}}\cdot W_\alpha(\overline{w}) = \sum_{\Tilde{w}\in W_j}f_{V_\alpha(v),\Tilde{w}} \cdot \Tilde{w}.\]

The right hand side vanishes because $V_\alpha(v) = 0$. Since $W_\alpha(w) = t(b) \neq 0$, we must have $f_{v,w} = 0$.

The map

\begin{align*}
\phi:\{\textnormal{admissible components of }\Gamma_{V\otimes W}\}&\rightarrow \{\textnormal{non-zero equivalence classes in } \mathcal{B}_{V,W}\}\\
C&\mapsto \{\mathbf{b}_{v,w} \mid (v,w)\in C_0\}
\end{align*}

is a bijection.

We check that it is well-defined first. Let $C$ be an admissible component. Then $\phi(C)$ is a subset of an equivalence class by Lemma <ref>. The map $\mathbf{b}^C$ witnesses that this equivalence class is not the zero class and that it is contained in $\phi(C)$. This shows that $\phi$ is well-defined. It is injective by construction. Let $\mathbf{b}_{v,w}$ be part of a non-zero class. Then the component of $(v,w)$ must be admissible by Lemma <ref>. This component is a preimage of the class of $\mathbf{b}_{v,w}$ under $\phi$.

Let $V$ and $W$ be $\bF_1$-representations of $Q$ and let $K$ be a field. Then the set

\[\mathscr{B}_{V,W} = \{\mathbf{b}^C \mid C \textnormal{ admissible component of } \Gamma_{V\otimes W}\}\]

is a basis of $\Hom_{KQ}(V^K,W^K)$.

The $\mathbf{b}^C$ are linearly independent because they are sums over disjoint sets of basis vectors in $\mathcal{B}_{V,W}$. Let $f\in \Hom_{KQ}(V^K,W^K)$. By definition we have

\[f = \sum_{\mathbf{b}_{v,w}\in \mathcal{B}_{V,W}} f_{v,w}\mathbf{b}_{v,w}.\]

We drop all summands from the zero class. Then we group the remaining summands according to their equivalence classes and apply Corollary <ref>. This yields

\[f = \sum_C\sum_{(v,w)\in C_0} f_{v,w}\mathbf{b}_{v,w} = \sum_C f_C \mathbf{b}^C\]

where the sums are over all admissible components and $f_C = f_{v,w}$ for any $(v,w)\in C_0$. Thus $\mathscr{B}_{V,W}$ is a generating set and hence a basis.

The special case where $\Gamma_V$ and $\Gamma_W$ are trees or aperiodic cycles was studied by Crawley-Boevey [1] and Krause [10]. The latter used so-called admissible triples to construct a basis of $\Hom_{KQ}(V^K,W^K)$. There is a bijection

\begin{align*}
\{\textnormal{admissible components of }\Gamma_{V\otimes W}\}&\rightarrow\{\textnormal{admissible triples connecting }\Gamma_V \textnormal{ and }\Gamma_W\} \\
C&\mapsto (C,\pi_1,\pi_2)
\end{align*}

where the maps $\pi_1$ and $\pi_2$ are given by

\begin{align*}
\pi_1:C &\rightarrow \Gamma_V& \pi_2:C &\rightarrow \Gamma_W\\
(v,w)&\mapsto v& (v,w)&\mapsto w.
\end{align*}

In this way we recover the standard basis for morphisms between tree modules.

§ COVERING COMPONENTS

As an application we want to study when a scalar extension $V^K$ of an indecomposable $\mathbb{F}_1$-representation $V$ remains indecomposable. We will relate this to the presence or absence of special admissible components of $\Gamma_{V\otimes V}$ which we will call covering components. They encode an unusual kind of symmetry of $V$.

Let $V,W\in \Rep(Q,\bF_1)$. A component $C$ of $\Gamma_{V\otimes W}$ is called a covering component if the two projections

\begin{align*}
\pi_1:C &\rightarrow \Gamma_V& \pi_2:C &\rightarrow \Gamma_W\\
(v,w)&\mapsto v& (v,w)&\mapsto w
\end{align*}

are covering maps in the topological sense.

Let $V$ be indecomposable. Then the component $C_\sigma$ induced by an automorphism $\sigma:V\rightarrow V$ is a covering component. In this case both projections are isomorphisms.

We consider the following representation over $\mathbb{L}_2$:

\[\begin{tikzcd}
1 & 2 & 3
\arrow[color={rgb,255:red,92;green,92;blue,214},loop, in=160, out=200 ,distance=3em, from=1-1, to=1-1]
\arrow[color={rgb,255:red,214;green,92;blue,92},loop, in=-20, out=20 ,distance=3em, from=1-3, to=1-3]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-1, to=1-2]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=1-2, to=1-3]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=1-3, to=1-2]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-2, to=1-1]
\end{tikzcd}\]

Here the colours blue and red correspond to the two arrows of $\mathbb{L}_2$. Its tensor square is given by

\[\begin{tikzcd}
&&&&& {(2,1)} & {(3,1)} \\
{(1,1)} & {(2,2)} & {(3,3)} && {(1,2)} && {(3,2).} \\
&&&& {(1,3)} & {(2,3)}
\arrow[color={rgb,255:red,92;green,92;blue,214},loop, in=160, out=200 ,distance=3em, from=2-1, to=2-1]
\arrow[color={rgb,255:red,214;green,92;blue,92},loop, in=-20, out=20 ,distance=3em, from=2-3, to=2-3]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-6, to=2-5]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=2-5, to=1-6]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=3-6, to=2-7]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=2-7, to=3-6]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=3-6, to=3-5]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=3-5, to=3-6]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=3-5, to=2-5]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=2-5, to=3-5]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=1-6, to=1-7]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=1-7, to=1-6]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-7, to=2-7]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=2-7, to=1-7]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=2-1, to=2-2]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=2-2, to=2-3]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=2-2, to=2-1]
\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, from=2-3, to=2-2]
\end{tikzcd}\]

Both components are covering components. The left one is induced by the identity. This is the smallest representation with two covering components but no non-trivial automorphism.

The next proposition will give alternative characterisations of covering components. Let
\begin{align*}
\tau:V\otimes W &\xrightarrow{\cong} W\otimes V\\
v\otimes w &\mapsto w\otimes v
\end{align*}

be the standard braiding.

Let $C$ be a component of $\Gamma_{V\otimes W}$. Then the following are equivalent:

* $C$ is a covering component.

* $C$ and $\tau(C)$ are admissible components.

* Let $(v,w)\in C_0$ and $\alpha\in Q_1$ be an arrow with $s(\alpha) = c_V(v)$. Then the following are equivalent:

* There is an arrow $a\in (\Gamma_V)_1$ with $s(a) = v$ and $c_V(a) = \alpha$.

* There is an arrow $(a,b)\in C_1$ with $s((a,b))=(v,w)$ and $c_{V\otimes W}((a,b)) = \alpha$.

* There is an arrow $b\in (\Gamma_W)_1$ with $s(b) = w$ and $c_W(b) = \alpha$.

Dually let $\beta \in Q_1$ with $t(\beta) = c_V(v)$. Then the following are equivalent:

* There is an arrow $a\in (\Gamma_V)_1$ with $t(a) = v$ and $c_V(a) = \beta$.

* There is an arrow $(a,b)\in C_1$ with $t((a,b))=(v,w)$ and $c_{V\otimes W}((a,b)) = \beta$.

* There is an arrow $b\in (\Gamma_W)_1$ with $t(b) = w$ and $c_W(b) = \beta$.

$(i)\Leftrightarrow (iii)$: We note that $\pi_1$ and $\pi_2$ are windings. Now $\pi_1$ is a covering map if and only if the equivalences $(1)\Leftrightarrow(2)$ and $(1')\Leftrightarrow(2')$ hold. Similarly $\pi_2$ is a covering map if and only if the equivalences $(2)\Leftrightarrow(3)$ and $(2')\Leftrightarrow(3')$ hold.

$(ii)\Leftrightarrow (iii)$: The implications $(1)\Leftarrow (2) \Rightarrow (3)$ and $(1')\Leftarrow (2') \Rightarrow (3')$ hold for every tensor product of windings. The implication $(2)\Leftarrow (3)$ is equivalent to $C$ reflecting successors and the implication $(1')\Rightarrow (2')$ is equivalent to $C$ inducing predecessors. Similarly the implications $(2')\Leftarrow (3')$ and $(1)\Rightarrow (2)$ are equivalent to $\tau(C)$ being an admissible component.

We now restrict our focus to endomorphisms of an indecomposable representation. In this case there is a much simpler criterion.

Let $V$ be indecomposable and $C$ a component of $\Gamma_{V\otimes V}$. Then the following are equivalent:

* $C$ is a covering component,

* $C$ is admissible and $\pi_1:C\rightarrow \Gamma_V$ is surjective on vertices.

$(i)\Rightarrow(ii)$: The coefficient quiver $\Gamma_V$ is connected since $V$ is indecomposable. Any non-empty covering of a connected quiver  is surjective.

$(ii)\Rightarrow(i)$: By Proposition <ref> we need to show that $\tau(C)$ is an admissible component. This is equivalent to the implications $(2')\Leftarrow (3')$ and $(1)\Rightarrow (2)$ from the same proposition. We will only prove $(1)\Rightarrow (2)$. The other implication follows from a similar argument.

Let $(v,w)\in C_0$ and $a\in (\Gamma_V)_1$ with $s(a) = v$. Assume that there is a sequence $(v_0,\dots,v_n)$ of vertices of $\Gamma_V$ with $v_0=v_n=v$, $v_1=w$ and $(v_{i-1},v_i)\in C_0$ for $1\leq i\leq n$. Then there is an arrow $b\in(\Gamma_V)_1$ with $s(b) = w$ and $c_V(a)=c_v(b)$.

We consider the vertex $(v_{n-1},v_n)$. The arrow $a$ starts at $v_n=v$ and $C$ reflects successors, hence there must be an arrow $a_{n-1}\in(\Gamma_V)_1$ with $s(a_{n-1}) = v_{n-1}$ and $c_V(a_{n-1}) = c_V(a)$. By descending induction there is an arrow $a_1\in(\Gamma_V)_1$ with $s(a_1)=v$ and $c_V(a_1) = c_V(a)$. The desired arrow is given by $b:=a_1$ since $v_1= w$.

\[\begin{tikzcd}
v & w & {v_2} & {...} & {v_{n-1}} & v \\
{t(a)} & {t(b)} & {t(a_2)} & {...} & {t(a_{n-1})} & {t(a)}
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=1-5, to=1-6]
\arrow["a"', from=1-1, to=2-1]
\arrow["b"', dashed, from=1-2, to=2-2]
\arrow["a"', from=1-6, to=2-6]
\arrow[dashed, from=2-5, to=2-6]
\arrow["{a_{n-1}}"', dashed, from=1-5, to=2-5]
\arrow[dashed, from=2-4, to=2-5]
\arrow[dashed, from=2-3, to=2-4]
\arrow["{a_2}"', dashed, from=1-3, to=2-3]
\arrow[dashed, from=2-1, to=2-2]
\arrow[dashed, from=2-2, to=2-3]
\end{tikzcd}\]

If we can find such a sequence for each vertex of $C$ then this proves $(1)\Rightarrow (2)$. We start by finding just one vertex that admits such a sequence. Let $v_1\in(\Gamma_V)_0$ be any vertex. Because $\pi_1$ is surjective on vertices there must be another vertex $v_2\in(\Gamma_V)_0$ with $(v_1,v_2)\in C_0$. Using induction we obtain an infinite sequence $(v_i)_{i\in \bN}$ of vertices of $\Gamma_V$ with $(v_i,v_{i+1})\in C_0$ for all $i\in \bN$. By the pigeonhole principle there must be some $i<j\in \bN$ with $v_i = v_j$. The subsequence $(v_i,v_{i+1},\dots,v_j)$ is of the desired form for the vertex $(v_i,v_{i+1})$.

Now let $(v,w)$ and $(v',w')$ be adjacent vertices in $C$ such that $(v,w)$ admits such a sequence. If we can show that $(v',w')$ also admits such a sequence then every vertex admits one and we are done. Let $(a,b)\in(\Gamma_V)_1$ be the arrow connecting $(v,w)$ and $(v',w')$. We assume $s((a,b)) = (v,w)$ and $t((a,b))=(v',w')$. A similar argument works should $(a,b)$ have the opposite orientation. Now let $(v_0,\dots,v_n)$ be a sequence for $(v,w)$. Let $\alpha = c_V(a)$. Because $C$ reflects successors and $v=v_n$ there is an arrow $a_{n-1}\in (\Gamma_V)_1$ with $s(a_{n-1})=v_{n-1}$ and $c_V(a_{n-1}) = \alpha$. By induction there is an arrow $a_i$ with $s(a_i)=v_i$ for all $0\leq i\leq n$. The winding condition gives $a_0=a_n=a$ and $a_1=b$. We obtain a sequence $((a_0,a_1),(a_1,a_2),\dots,(a_{n-1},a_n))$ of arrows in $C_1$. Now the sequence $(t(a_0),t(a_1)\dots,t(a_n))$ satisfies $t(a_0) =t(a)=v'$, $t(a_1)=t(b)=w'$, $t(a_n) = t(a)=v'$ and $(t(a_{i-1}),t(a_i)) = t((a_{i-1},a_i))\in C_0$ for all $1\leq i \leq n$. Hence it is the desired sequence for $(v',w')$ completing the proof.

§ ABSOLUTELY INDECOMPOSABLE REPRESENTATIONS

An indecomposable $\bF_1$-representation $V$ is called absolutely indecomposable if $V^K$ is indecomposable for all fields $K$.

In this section we will apply our previous considerations to obtain the following combinatorial characterisation of absolutely indecomposable representations.

Let $V$ be an indecomposable quiver representation over $\bF_1$. Then the following are equivalent:

* $V$ is absolutely indecomposable,

* $V^\mathbb{C}$ is indecomposable,

* $\Gamma_{V\otimes V}$ contains exactly one covering component.

$(i)\Rightarrow(ii)$: This is clear.

$(ii)\Rightarrow(iii)$: This will be Proposition <ref>.

$(iii)\Rightarrow(i)$: This will be Proposition <ref>.

The unique covering component in $(iii)$ is $C_\identity$. We start by showing that any representation with more than one covering component becomes decomposable over $\bC$. To this end we use the following criterion for decomposable modules.

Let $K$ be a field, $A$ a $K$-algebra and $V$ a finite dimensional $A$-module. If there is an endomorphism $f$ of $V$ that has at least two different eigenvalues then $V$ is decomposable.

Let $f$ be such an endomorphism and $\lambda \neq \mu$ two of its eigenvalues. By the fitting lemma every endomorphism of an indecomposable module is either nilpotent or an isomorphism. The endomorphism $f-\lambda \identity_V$ is neither.

We also need the following version of the Perron-Frobenius theorem.
Let $M$ be the adjacency matrix of a connected, regular directed graph $G$. Then the characteristic polynomial of $M$ has a simple root.

See [5] for a graph theoretic treatment of the Perron-Frobenius theorem. The theorem is stated there using the weaker assumption that the graph is strongly connected.

Let $V\in \Rep(Q,\bF_1)$ be indecomposable and let $C \neq C_\identity$ be a covering component in $\Gamma_{V\otimes V}$. Then $V^\bC$ is decomposable.

We will show that the endomorphism $\mathbf{b}^C$ has multiple eigenvalues by interpreting it as the adjacency matrix of a graph and then applying the Perron-Frobenius theorem. Let $G = (G_0,G_1,s,t)$ be the directed graph given by

\begin{align*}
G_0 &= (\Gamma_V)_0\\
G_1 &= C_0\\
s((v,w)) &= v\\
t((v,w)) &= w.
\end{align*}

The adjacency matrix of $G$ is equal to the matrix representation of $\mathbf{b}^C$ with respect to the standard basis of $V^\bC$. Let $v\in (\Gamma_V)_0$ be a vertex. A vertex $(v',w')$ of $C_0$ becomes an edge starting at $v$ if and only if $\pi_1((v',w')) = v$. Thus the outdegree of $v$ is the degree of the covering $\pi_1$. This proves that $G$ is $k$-regular for $k = \degree(\pi_1)$. The graph $G$ might not be connected. If it is not then it is a disjoint union of $k$-regular connected graphs and its adjacency matrix decomposes as a block diagonal matrix where each block is the adjacency matrix of a connected component. Let $D$ be such a connected component and let $(v,w)$ be an edge in $D$. Such an edge must exist because $D$ is $k$-regular. If $v=w$ then $C$ intersects $C_{\identity}$. So we must have $v\neq w$ and consequently $D$ consists of multiple vertices. Applying the Perron-Frobenius theorem to the adjacency matrix of $D$ shows that it has multiple different eigenvalues. Then $\mathbf{b}^C$ must have multiple different eigenvalues, too. In particular $V^\bC$ is decomposable.

This completes one direction of our characterisation of absolutely indecomposable representations. For the other direction we need to prove that scalar extensions of representations with just one covering component are indecomposable. We will use the following criterion for indecomposable representations.

Let $K$ be a field, $A$ a $K$-algebra and $V$ a finite dimensional $A$-module. Let $I\subset \End_A(V)$ be a subspace consisting entirely of nilpotent endomorphisms such that $K\cdot\identity_V \oplus I = \End_A(V)$. Then $V$ is indecomposable.

Let $f = \lambda\identity + f_{nil}\in \End_A(V)$ with $f_{nil}\in I$. Then the only eigenvalue of $f$ is $\lambda$. Let $V=V_1 \oplus V_2$ be a non-trivial direct sum decomposition. The projection onto $V_1$ has eigenvalues 0 and 1, a contradiction.

Let $V\in \Rep(Q,\bF_1)$ be an indecomposable representation with just one covering component. Let $K$ be a field. Then $V^K$ is indecomposable.

We will apply the previous proposition to the subspace

\[I =\Kspan(\mathscr{B}_{V,V}\backslash \{\mathbf{b}^{C_\identity}\}).\]

The claim $K\cdot\identity_{V^K} \oplus I = \End_{KQ}(V^K)$ follows from $\mathbf{b}^{C_\identity} = \identity_{V^K} $ and Theorem <ref>. So we only need to check that $I$ consists of nilpotent morphisms. Let $f\in I$ be non-nilpotent and let $n>|(\Gamma_V)_0|$. The endomorphism $f^n$ is a linear combination of elements of the form

\[\mathbf{b}^{C^n}\cdot\dotso\cdot\mathbf{b}^{C^1}.\]

for some sequence $(C^i)$ of admissible components in $\Gamma_{V\otimes V}$ none of which are covering components. We choose a non-zero element of this form. Let $v\in V$ be non-zero. Its image is given by

\begin{equation}\label{abs:expand}
\mathbf{b}^{C^n}\cdot\dotso\cdot\mathbf{b}^{C^1}(v) = \sum_{(v_{n-1},v_n)\in C^n_0}\dots\sum_{(v_1,v_2)\in C^2_0}\sum_{(v,v_1)\in C^1_0} v_n.
\end{equation}

Because $\mathbf{b}^{C^n}\cdot\dotso\cdot\mathbf{b}^{C^1}$ is not zero there must be a sequence $(v_0,\dots,v_n)$ with $v_i\in V$ not zero and $(v_{i-1},v_i)\in C^i_0$ for $1\leq i \leq n$. By the pigeonhole principle there are $0\leq i<j\leq n$ with $v_i = v_j$. We now restrict our attention to the subsequence $\mathbf{b}^{C^j}\cdot\dotso\cdot\mathbf{b}^{C^{i+1}}$ interpreted as an endomorphism of $V^\bC$ instead of $V^K$. We consider its basis expansion

\[\mathbf{b}^{C^j}\cdot\dotso\cdot\mathbf{b}^{C^{i+1}} = \sum_{\mathbf{b}^C\in \mathscr{B}_{V,V}} \lambda_C \mathbf{b}^C.\]

Using (<ref>) the sequence $(v_i,\dots,v_j)$ witnesses that the coefficient of $v_i$ in $\mathbf{b}^{C^j}\cdot\dotso\cdot\mathbf{b}^{C^{i+1}}(v_i)$ is not zero and hence $\lambda_{C_\identity}\neq0$. In particular none of the standard basis vectors of $V^\bC$ lie in the kernel of $\mathbf{b}^{C^j}\cdot\dotso\cdot\mathbf{b}^{C^{i+1}}(v_i)$. Then none of them lie in the kernel of $\mathbf{b}^{C^{i+1}}$ either. Finally this implies that $\pi_1:C^{i+1}\rightarrow V$ is surjective and hence that $C^{i+1}$ is a covering component by Proposition <ref>, a contradiction.

This criterion yields a practically fast algorithm.

Let $V\in \Rep(Q,\bF_1)$ be indecomposable. Whether $V$ is absolutely indecomposable can be tested in $O(m^2)$ time where $m$ is the number of arrows in $\Gamma_V$.

Let $n$ be the number of vertices in $\Gamma_V$. The quiver $\Gamma_V$ is connected because $V$ is indecomposable. In particular $n\leq m+1$. The tensor product $\Gamma_{V\otimes V}$ can be computed in $O(m^2)$ time. We can identify all covering components by finding all admissible components $C$ such that $\tau(C)$ is admissible. Across all components there are $nm$ test cases each for reflecting successors, inducing predecessors and their dual properties. Each of those cases can be checked in constant time.

§ REPRESENTATIONS WITH FINITE NICE LENGTH

Cerulli-Irelli [3] and Haupt [6]  developed a combinatorial technique to compute the Euler characteristic of certain quiver Grassmannians. Jun and Sistko [7] adapted these results to the context of $\bF_1$-representations. They introduced the notion of finite nice length as a criterion for when this approach can be applied. We will show that all indecomposable representations with finite nice length are absolutely indecomposable.

Finite nice length representations are defined via a sequence of increasingly fine gradings satisfying certain compatibility conditions.

Let $V\in \Rep(Q,\bF_1)$ be a representation. A grading of $V$ is a map $\partial:(\Gamma_V)_0\rightarrow \bZ$. Let $\underline{\partial}=(\partial_0,\dots,\partial_n)$ be a sequence of gradings for $V$. A grading $\partial_{n+1}$ is $\underline{\partial}$-nice if for each pair of arrows $a,b\in (\Gamma_V)_1$ satisfying

* $c_V(a) = c_V(b)$,

* $\partial_i(s(a)) = \partial_i(s(b))$ for all $0\leq i \leq n$,

* $\partial_i(t(a)) = \partial_i(t(b))$ for all $0\leq i \leq n$,

we have

\[\partial_{n+1}(s(a))-\partial_{n+1}(t(a))=\partial_{n+1}(s(b))-\partial_{n+1}(t(b)).\]

The sequence $\underline{\partial}$ is nice if $\partial_i$ is $(\partial_0,\dots,\partial_{i-1})$-nice for all $0\leq i \leq n$. The representation $V$ has finite nice length if there exists a nice sequence $\underline{\partial}=(\partial_0,\dots,\partial_n)$ for $V$ that distinguishes vertices. That is for each pair of vertices $v\neq w\in (\Gamma_V)_0$ there exists $0\leq i \leq n$ with $\partial_i(v)\neq \partial_i(w)$. In this case the nice length of $V$ is the smallest $n$ such that there exists a nice sequence $(\partial_0,\dots,\partial_n)$ for $V$ that distinguishes vertices.

We will prove that indecomposable finite nice length representations are absolutely indecomposable by studying certain strings in the coefficient quiver.

Let $\Gamma$ be a quiver. We consider an alphabet consisting of the two symbols $a$ and $a^-$ for each arrow $a\in \Gamma_1$. We define $s(a^-):=t(a)$ and $t(a^-):= s(a)$. A string in $\Gamma$ is a non-empty word $a_n\dots a_1$ over this alphabet such that $s(a_{i+1}) = t(a_i)$ for all $1\leq i\leq n-1$. We set $s(a_n\dots a_1) := s(a_1)$ and $t(a_n\dots a_1) := t(a_n)$. If $C = a_n\dots a_1$ and $D = b_m\dots b_1$ are two strings with $t(D)=s(C)$ then we define their composition by $CD=a_n\dots a_1b_m\dots b_1$. A string $C$ is called a cycle if $s(C) = t(C)$. In this case we denote the $k$-fold composition of $C$ with itself by $C^k$. Let $c:\Gamma\rightarrow Q$ be a winding. We extend $c$ to a map on strings by setting $c(a^-) = c(a)^-$ and $c(a_n\dots a_1) = c(a_n)\dots c(a_1)$.

We will describe a certain type of string within a coefficient quiver that prevents finite nice length. Then we will show that such a string always exists for not absolutely indecomposable representations.

Let $V\in \Rep(Q,\bF_1)$ be a representation. If there is a string $E$ in $\Gamma_V$ that is not a cycle but can be completed to a cycle by a string $D$ in such a way that $c(D) = c(E)^s$ for some $s\geq 1$, then $V$ does not have finite nice length.

We expand $E = a_m\dots a_1$, $D = a_{m(s+1)}\dots a_{m+1}$. We interpret all indices on these arrows modulo $m(s+1)$. Let $\underline{\partial}=(\partial_0,\dots,\partial_n)$ be a nice sequence for $V$. We will prove by induction that $\partial_i(s(a_k)) = \partial_i(s(a_{k+m}))$ for $0\leq i \leq n$ and $1\leq k \leq m(s+1)$. Then $\underline{\partial}$ does not distinguish vertices since $s(a_1) = s(E)\neq t(E) = s(a_{m+1})$. We assume this holds for all $i<r$. For $r = 0$ this is an empty condition. Our assumptions are chosen such that for any $1\leq k \leq m(s+1)$ the pair of arrows $a_k$ and $a_{k+m}$ satisfies conditions $(i)$-$(iii)$ in the the niceness condition for $\partial_r$. For $1\leq k \leq m(s+1)$ this implies

\[\partial_r(s(a_k))-\partial_r(s(a_{k+1}))=\partial_r(s(a_{k+m}))-\partial_r(s(a_{k+m+1})).\]

A telescope sum along the cycle $DE$ yields

\[\partial_r(s(a_k))-\partial_r(s(a_{k+m}))=\partial_r(s(a_{k+m}))-\partial_r(s(a_{k+2m})).\]

We call this difference $\Delta_k$. A second telescope sum shows

\[0 =\partial_r(s(a_k))-\partial_r(s(a_{k+m(s+1)}))=(s+1)\Delta_k.\]

In particular, $\Delta_k = 0$ for all $k$ as required. This completes the induction.

Let $V\in \Rep(Q,\bF_1)$ be indecomposable and of finite nice length. Then $V$ is absolutely indecomposable.

We prove the contraposition. Let $V$ be an indecomposable but not absolutely indecomposable representation. By <ref> there is a covering component $C\neq C_\identity$ in $\Gamma_{V\otimes V}$. We have to find a string as in the previous proposition. There are two cases to consider depending on the degree of the covering $\pi_1:C \rightarrow \Gamma_V$.

If $\degree(\pi_1) = 1$ then $V$ does not have finite nice length.

In this case the projections $\pi_1$ and $\pi_2$ are isomorphisms. Thus $C$ must be of the form $C_\sigma$ for the non-identity automorphism $\sigma =\pi_2\circ(\pi_1)^{-1}:V\rightarrow V$. Let $v\in(\Gamma_V)_0$ be a vertex that is not fixed by $\sigma$. Because $\Gamma_V$ is connected there is an undirected path from $v$ to $\sigma(v)$ and hence a string $E$ with $s(E) = v$ and $t(E) = \sigma(v)$. Let $n>1$ be minimal such that $\sigma^n(v) = v$. We set $D = \sigma^{n-1}(E)\dots\sigma(E)$. Then $E$ and $D$ satisfy the conditions of the previous proposition. Hence, $V$ does not have finite nice length.

If $\degree(\pi_1) \geq 2$ then $V$ does not have finite nice length.

We fix an element $v\in (\Gamma_V)_0$ and two different elements $(v,w),(v,w')\in \pi_1^{-1}(v)$. Since $C$ is connected there is an undirected path and hence a string $\Tilde{E}$ from $(v,w)$ to $(v,w')$. We set $E = \pi_2(\Tilde{E})$. In particular $s(E) = w$ and $t(E) = w'$. The path $\pi_1(\Tilde{E})$ is a cycle at $v$. It acts on $\pi_1^{-1}(v)$ because $\pi_1$ is a covering map. Let $n$ be minimal such that $(\pi_1(\Tilde{E}))^n$ lifts to a path from $(v,w')$ to $(v,w)$. We call this path $D$. Then the strings $E$ and $D$ satisfy the conditions of the previous proposition. Hence $V$ does not have finite nice length.

The converse does not hold as the following example shows.

An example of an absolutely indecomposable representation with infinite nice length was given by Jun and Sistko in [7]. We give a different example here. We consider the following representation of $\bL_2$:

\[\begin{tikzcd}[ampersand replacement=\&]
1 \& 2.
\arrow["b"{description}, color={rgb,255:red,214;green,92;blue,92}, curve={height=-9pt}, from=1-1, to=1-2]
\arrow["a"{description}, color={rgb,255:red,214;green,92;blue,92}, curve={height=-9pt}, from=1-2, to=1-1]
\arrow["c"{description}, color={rgb,255:red,92;green,92;blue,214}, from=1-2, to=1-1]
\end{tikzcd}\]

Here the two colours correspond to the two arrows of $\bL_2$. The string $ab$ satisfies the conditions of Proposition <ref> hence $V$ does not have finite nice length. On the other hand its tensor square is given by

\[\begin{tikzcd}[ampersand replacement=\&]
{(1,1)} \& {(2,2)} \& {(1,2)} \& {(2,1).}
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-1, to=1-2]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-2, to=1-1]
\arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-2, to=1-1]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-3, to=1-4]
\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-6pt}, from=1-4, to=1-3]
\end{tikzcd}\]

In particular there is just one covering component and $V$ is absolutely indecomposable.

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# Depth-multiplexing spectral domain OCT for full eye length imaging with a
single modulation unit

Guanghan Meng*  Andrew Zhang  Fabio Feroldi  Austin Roorda  and Laura Waller
<EMAIL_ADDRESS><EMAIL_ADDRESS>University of California
Berkeley, Berkeley, California, USA, 94720
*The authors contributed equally to this research.

††journal: opticajournal††articletype: Research Article

Measuring the axial length of the eye is emerging as a crucial approach to
measure progression and monitor management of myopia. The high cost of current
swept-source OCT devices, the preferred method for such measurements, limits
their broad use, especially in lower-income communities.While spectral domain
(SD) OCT is a more affordable option, its limited imaging range falls short
for full eye length measurement. Existing depth-multiplexing (DM) techniques
for SD-OCT provide a workaround by capturing images at multiple depths within
the eye. However, these methods typically require multiple light modulation
units or detectors for simultaneous imaging across depths, adding complexity
and cost. In response, we propose a novel DM-SD-OCT approach that utilizes a
single light modulation unit for depth encoding. This innovative method
facilitates the capture of images at multiple depths within the eye using a
single line scan camera, with subsequent computational demixing. Our
implementation of this system successfully enabled simultaneous acquisition
and demixing of signals from three distinct depths within the eye. The
system’s effectiveness was demonstrated using a model eye, confirming its
potential as a cost-effective solution for comprehensive eye length
measurement in clinical myopia research.

Myopia stands as a pervasive global issue that is progressively worsening [1].
Despite its widespread prevalence, a definitive cure remains elusive. Axial
eye length measurement offers heightened sensitivity to track the eye’s
elongation, a characteristic of myopia progression, and holds the potential to
detect changes within the eye even before pediatric patients begin to exhibit
visual consequences. Swept-source OCT devices are the gold standard for axial
eye length measurement in clinics, but the costs limit its availability to
resource-rich clinics only. Introducing an affordable OCT device for axial eye
length measurement, accessible across a spectrum of eye clinics including
community healthcare centers, has the potential to reshape routine eye
examinations. This becomes all the more pivotal considering the global
prevalence of myopia.

Recently, thermally tuned vertical cavity surface emitting lasers (VCSEL) have
been employed for low-cost full-eye swept-source OCT [2, 3], but the low-cost
VCSEL originally designed for lidar is limited in optical output power and
tuning bandwidth. Spectral domain (SD) OCT is more cost-effective than SS-OCT,
but it faces the limitation of confined imaging range beyond the reference
mirror depth. The integration of multiple reference arms targeting different
depths within the eye has been explored to overcome this constraint of SD-OCT,
and related approaches fall into two categories: sequential acquisition and
simultaneous acquisition. Sequential acquisition alternates between different
arms for data collection [4, 5, 6, 7, 8], thus sacrificing temporal
resolution. In contrast, simultaneous acquisition methods generally integrate
multiple interferometers or optical modulation devices, one for each reference
arm [9, 10, 11, 12, 13, 14]. As a result, these methods amplify costs and
bulkiness, thus compromising accessibility and portability. Although a single-
interferometer-based depth-multiplexing strategy has been developed for SS-OCT
[15, 16, 17], it relies on coherence revival of the SS-laser and is not
applicable to SD-OCT. Furthermore, the method lies with the polarization
division of light in the sample arm, introducing additional loss of the signal
reflected from the eye and demanding more power to be delivered to the
subject’s eye.

Here, we propose a novel depth-multiplexing technique that aims to address all
the above issues and holds broad applicability across all spectral domain OCT
systems seeking enhanced imaging depth in ophthalmology applications. The
single-interferometer-based technique employs a solitary scanning mirror to
encode distinct phase ramps for multiple reference arms, facilitating
simultaneous data acquisition of the entire eye length with a single scan.
Furthermore, as the light division happens in the reference arm and is
polarization independent, reflected signal from the sample arm does not
experience light loss on its way back, in contrast to the existing
polarization-dependent depth-multiplexing strategy. Additionally, although
previous depth-multiplexing strategies generally involve two reference arms
only, our power-efficient and single-modulation unit-based method enables
simultaneous capturing and computational demixing of signal from three
different depths.

Previously, a phase-shift method achieved by deviating the light beam from the
pivot axis of the galvanometer scanner [18, 19, 20] was developed to remove
the complex conjugate artifact in OCT image reconstruction. The off-center
distance of the incident beam from the pivot axis induces an optical path
length (OPL) change as the galvanometer scans around its rotation axis,
resulting in a linear phase ramp along the slow axis of the B-scan image. By
applying a Hilbert transform (HT) to the Fourier-domain signal along the
lateral-scanning direction (the slower axis in a B-scan), both amplitude and
phase components can be recovered, which are sufficient to suppress the
complex conjugate artifact in the reconstruction.

Figure 1: Principle of frequency-encoded depth multiplexing demonstrated by
simulation. (A) Model eye image input into the simulation pipeline.
Arrowheads: the depths of three reference arms. (B) Simulated frequency-
encoded B scan. Left: simulated cross-correlation term in the B scan. Right:
Fourier transform of the B scan along the transverse axis. Boxes: the spectrum
peaks corresponding to the three depths in (A). (C) Reconstructed images from
the three cropped spectra in (B)

Our depth-multiplexing (DM) SD-OCT method is an extension of this phase-
modulation approach. Three reference arms targeting the front, middle, and
back of the eye, are simultaneously modulated by a separate galvanometer
introduced to the reference arm set (called reference scanner from now) and
imaged with the same interferometer (Fig. 1). The reference scanner is
synchronized with the fast scanner in the sample arm, and reference beams with
varying OPLs are directed onto the reference scanner at different positions,
resulting in distinct phase ramps along the transverse axis of the B scan. The
phase ramps allow signal from different depths to be demixed via a Fourier
transform of the B scan along the transverse axis [9] (Fig. 1B,C).

The modulated OPL of the reference arm can be given by $\Delta
z_{r}=2s\tan(\theta)$, where $\theta$ is the angle between the incident beam
and the galvo normal direction, $s$ is the off-center distance of the incident
beam. When the reference scanner is centered around $\theta=45^{\circ}$, we
get $\Delta z_{r}\approx 2s(1+2(\theta-\pi/4))\approx
4s\Delta\theta=4s\omega/r$, where $\omega$ is the angular speed of the galvo,
and $r$ the line scan rate of the system. Therefore, the slope of the phase
ramp between each pair of adjacent line scans is

$\phi=k\Delta z_{r}=4ks\frac{\omega}{r},$ (1)

where $k$ is the central wavenumber. The specific phase shifts for the
multiple reference arms should be selected based on the spectrum width and
shape of the spectrum after the HT. If the spectra are expected to be clean
and uniform, and the $n$ reference arms each occupy $1/n$ of the full $\pi$
phase range in the quadrant, then the phase shift for the $i$th reference arm
($1$st arm is closest to the rotation axis) is:

$\phi_{i}=\frac{(2i-1)\pi}{2n}.$ (2)

Based on equations 1 and 2, the off-center distance for the $i$th reference
arm is:

$s_{i}=\frac{\phi_{i}}{4k}\frac{r}{\omega}=\frac{(2i-1)\pi}{8kn}\frac{r}{\omega}=\frac{(2i-1)\lambda}{16n}\frac{r}{\omega}.$
(3)

Therefore, when there are 3 reference arms, we will have:

$s_{i}=\frac{(2i-1)\lambda}{48}\frac{r}{\omega}.$ (4)

We used a low-cost line camera (Spyder3 1k, Teledene Dalsa) along with a
holographic grating (HD 1800, Wasatch Photonics) to build a spectrometer, and
off-the shelf optics from Thorlabs Inc. for all other parts in our system. The
key components in the reference arm set includes two cube beam splitters
(BS005 and BS053), a 5x beam reducer (GBE05-B), a scanner (GVS011), and a
lenslet array (MLA12) (Fig. 2). The lenslet array is critical to reflect light
with different OPLs back to the correct path, and has a pitch of
$1.0\text{mm}$ by $1.4\text{mm}$. We used the $1.0\text{mm}$ separation
distance between beams on the reference scanner to match the $1.0\text{mm}$
pitch, i.e., the off-center distances for the three beams are
$s_{1}=0.5\text{mm}$ (retina arm), $s_{2}=1.5\text{mm}$ (lens arm), and
$s_{3}=2.5\text{mm}$ (cornea arm). Given the $830\text{nm}$ central wavelength
for our light source (SLD830S-A20W, 55 nm bandwidth, Thorlabs Inc.), the
optimal step size of our galvo is determined as $\omega/r=34.58\
\mu\text{rad}/\text{line}$. We thus chose to use a mechanical scanning angle
of $\pm 0.8^{\circ}=\pm 0.01396\ \text{rad}$, which gives an actual step size
of $\omega/r=35.80\ \mu\text{rad}$. This results in an expected phase ramp of
$0.542\ \text{rad}/\text{line}$, $1.63\ \text{rad}/\text{line}$, and $2.71\
\text{rad}/\text{line}$ for arms $1$, $2$, and $3$, respectively. These phase
ramps shift the spectra centers of the three reference arms to $\pi/6$,
$\pi/2$, and $5\pi/6$, respectively.

Figure 2: Design of the DM-SD-OCT setup.

Signal fall-off and reference arm depths characterization were performed using
a protected silver mirror as the sample. The home-built spectrometer using
off-the-shelf optics and a low-cost line scan camera gives a 0.11 nm spectral
resolution, resulting in a 1.4 mm imaging range within 6dB signal fall-off
(Fig. 3). The depths of the three reference arms are centered around 0 $\mu
m$, 4531 $\mu m$, and 36512 $\mu m$, respectively (Fig. 3). With a 3 $\mu s$
camera exposure time (corresponding to a 300 kHz A-scan rate), we achieved a
peak signal-to-noise (SNR) above 90 $dB$ at all three depths (Fig. 3). The
axial resolution of our system is 14.4 $\mu m$.

Figure 3: Characterization of the DM-SD-OCT system.

As the proof-of-concept demonstration, we imaged a model eye (Rowe
Technologies Inc.) with our DM-SD-OCT system. As in Fig. 4, using two
reference arms, we successfully captured and demixed the images of the cornea
and the retina (Fig. 4 A) using the frequency demixing method (Fig. 1). More
specifically, by equally dividing the $0-\pi$ phase quadrant of the B-scan’s
Fourier transform along the transverse axis into two segments, the retina
image was reconstructed from the $0-\pi/2$ range while the cornea from the
$\pi/2-\pi$ range. This setting is sufficient for axial eye length
measurement, but does not provide information about the anterior chamber depth
(ACD). By adding an additional reference arm whose OPL matches the pupil, such
information becomes available (Fig. 4B). However, due to noise introduced by
galvanometer vibration, in practice, the frequency spectra from the three
depths are not as isolated as in the simulation (Fig. 1), leading to cross-
talk among images after frequency demixing ($0-\pi/3$ range for the retina,
$\pi/3-2\pi/3$ for the pupil, and $2\pi/3-\pi$ for the cornea, respectively).
To suppress the cross-talk, we minimized the correlations, i.e., dot products,
among the three images obtained from the initial frequency demixing. This was
achieved by reassigning the energy of each pixel across all images to the
image exhibiting the highest brightness at that particular location, as
determined after the initial demixing process. (Fig. 4C). Dimensions of the
eye structures measured with our DM-SD-OCT match those from Zeiss IOL Master
700 (1). For example, the OPL of the axial length (AL) is measured as 35.47
mm, corresponding to a geometrical length of 25.99 mm [21], in agreement with
the 26.04 mm value measured with IOL Master 700. Central corneal thickness
(CCT), aqueous depth (AQD) and anterior chamber depth (ACD) also match the IOL
Master readings [22].

Figure 4: Experimental validation using a model eye. (A) DM SD-OCT result
after frequency demixing, with 2 reference arm depths. (A) DM SD-OCT result
using 3 reference arm depths, after frequency demixing (left), and after an
additional decorrelation step (right). (C) Flow chart for the decorrelation
step. $I_{i}$: input images obtained with initial frequency demixing as in
(B). $M$: the binary mask where the correlation (dot product) of any pair of
images is above a user-defined value $v$. $I_{s}um$: the sum of all input
images. $m_{i}$: the mask for the $i^{th}$ decorrelated image.
$I_{i_{decorr}}$: the $i^{th}$ decorrelated output image. Table 1:
Meausurement comparison between DM SD-OCT and Zeiss IOL Master 700.

In summary, we proposed a novel depth-multiplexing method for SD-OCT using a
single phase modulation unit, without the need for additional detector. The
method offers a low-cost alternative for full-eye length measurement, and is
widely applicable to any SD-OCT devices where an extended imaging depth is
desired. The current drawbacks include reduced dynamic range of the
spectrometer when multiple arms are simultaneously captured by the same
sensor, although this was not a concern for our experiments. As a proof of
concept demonstration, we adopted a free-space optics setup for the reference
arm set, which offers high degrees of freedom in alignment and adjustable arm
lengths. To make the system more compact, a fiber-based module for beam
splitting and delaying can be adopted, with a miniaturized lenslet array
collimator. For in vivo measurements, where the subjects’ eye lengths vary
from patient to patient, mirrors controlling the optical delay of the retina
reference arm can be translated during the initial alignment process to locate
the retina, which will be demonstrated in future work. Additionally,
considering the well-defined shapes and sparsity of eye structures, deep
learning algorithms can potentially be developed to directly demix signals
from different depths, completely ruling out the need for frequency modulation
and demixing.

Funding. This work was funded by the Center for Innovation in Vision and
Optics, University of California, Berkeley.

Acknowledgement. G. Meng was funded by the Center for Innovation in Vision and
Optics.

Disclosures. G. Meng and L. Waller filed a provisional patent: "Single
modulation unit-based depth-multiplexing strategy for spectral domain optical
coherence tomography (OCT)", B24-019.

Data availability. Data and code will be available upon reasonable request to
correspondence authors.

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