# Signum phase mask differential microscopy

Jeeban Kumar Nayak<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur, India,
741246  Niladri Modak  Tampere University, Photonics Laboratory, Physics Unit,
Tampere, FI-33720, Finland  Sayan Ghosh  Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur, India,
741246  Nirmalya Ghosh<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur, India,
741246

###### Abstract

We propose and experimentally demonstrate a differential microscopy method to
obtain simultaneous amplitude, phase, and quantitative polarization gradient
imaging in a single experimental embodiment. A full-field optical spatial
differentiator is achieved in a relatively simple setup by placing a glass
cover slip as a Signum phase mask in the Fourier plane of a standard $4-f$
imaging system and accordingly named Signum phase mask differential
microscopy. The longstanding requisite of polarized light to obtain the
spatial differentiation of the field at the object plane is eliminated in our
scheme and, hence, leads to the emergence of quantitative differential
polarization contrast imaging by integrating polarization degree of freedom as
an additional contrast agent in the framework of differential microscopy.
Implementation of the proposed differential imaging scheme in high-resolution
microscopy is experimentally demonstrated alongside its functionality for a
broad wavelength range. Simultaneous acquisition of differential phase,
amplitude, and polarization (anisotropy) gradient imaging in a rather
elementary optical setup enables a low-cost multi-functional differential
microscopy system that is anticipated to emerge as a revolutionary tool in
label-free imaging and optical image processing.

††preprint: APS/123-QED

Interaction of light with inhomogeneous natural objects provides the intrinsic
mechanism of image contrast through the spatial modulation of the amplitude,
phase, and polarization of the light field, which is traditionally utilized
for optical imaging and microscopy for a wide range of applications in various
disciplines [1, 2, 3, 4, 5]. One of the major challenges in the early days of
microscopy was to visualize the so-called phase objects (nearly transparent)
under a conventional bright-field microscope, as these objects influence the
amplitude of the incident light marginally. In such cases, phase and
polarization imaging can capture essential morphological details about the
internal structure of the object, hence, have paramount importance in label-
free microscopy [6, 7, 8, 9, 10]. Many of the natural objects of biological or
non-biological origin possess intrinsic polarization anisotropy effects, i.e.,
birefringence (phase anisotropy) and dichroism (amplitude anisotropy) due to
their anisotropic molecular or structural organizations [3, 11, 12, 13].
Therefore, quantitative probing of the vectorial transformation of light with
desirable spatial resolution provides a wealth of morphological, structural,
and functional information on the objects and has been a key tool in
biomedical and clinical research [3]. On the other hand, in the realm of phase
imaging, phase contrast [14, 15, 16, 17, 18] and differential interference
contrast (DIC) [19, 20] are the first-generation techniques, and are still
widely adopted to produce contrast images of phase objects such as biological
specimen and etc. Owing to their characteristic working principles, some
longstanding limitations are associated with these traditional phase and
differential microscopy methods, e.g., incapability for quantitative phase
imaging, complexity and bulkiness of the setup, polarization dependency, etc.
[7, 19].

In recent years, there has been a renewed engagement towards the development
of next-generation differential microscopy and image processing methods with
an aim to achieve versatile and compact optical microscopy systems [21, 22,
23]. A wide range of optical systems, starting from reflection in glass
surfaces [24, 25, 26, 27, 28, 29] to multi-functional meta-surfaces [21, 30,
31, 32, 33, 34, 35, 36], have been utilized to realize various optical spatial
differentiator. On the basis of the corresponding working principles, these
recently proposed differential imaging methods can be primarily divided into
two types. One, where the spatial differentiation is obtained incorporating
the polarization-dependent phase or beam shift and can be considered as a DIC-
inspired microscopy technique [21, 30, 25, 26]. In the other approach, the
spatial dispersion of the meta-surfaces is appropriately tailored to make them
perform as a spatial differentiator [37, 38, 23, 32, 22, 36]. Although these
recently developed spatial differentiaton methods have shown great promises in
optical computing and the possible intersection with optical microscopy
systems, most of these techniques utilize polarized light to achieve the
spatial differentiation[21, 30, 36, 25]. This requisite of differential
imaging system makes it impossible to capture simultaneous phase and
polarization images of a specimen in a single experimental embodiment. In this
context, it is pertinent to note that the traditional polarization microscopic
approaches are often severely compromised while probing low level spatial
fluctuations of polarization anisotropy effects, which is desirable for most
practical applications [3]. Thus, encompassing polarization as an contrast
mechanism within the framework of conventional differential microscopy (which
considers only amplitude and phase) can be a revolutionary tool in the realm
of label-free microscopy.

In this work, we propose and experimentally demonstrate a Signum phase mask
differential (SPMD) microscopy scheme, which enables simultaneous amplitude,
phase, and quantitative differential polarization imaging/polarization
gradient imaging in a simple experimental framework. The proposed microscopy
scheme exploits the principle of one-dimensional (1D) Hilbert transformation
[39, 40], where a full-field spatial differentiator is obtained by placing a
Signum phase mask in the Fourier plane of a conventional $4-f$ imaging setup
[41]. This simple technique enables us to realize a multi-functional
differential imaging system built by using just a pair of lenses and a
commercially available glass cover slip. Importantly, unlike most of the
existing differential microscopy methods [19, 30, 21], the proposed spatial
differentiator does not require polarized light to obtain the differentiation
of the electric field distribution at the object plane. This not only reduces
the complexity and bulkiness of the microscopy system but also enables
utilizing the polarization of light as an additional intrinsic contrast
mechanism in the framework of differential microscopy. As a consequence, two
new kinds of imaging systems are emerged in the domain of differential
microscopy: polarization-resolved differential imaging and quantitative
polarization (anisotropy) gradient imaging. It is worth mentioning that,
although the Hilbert transformation has been previously used for image
processing, its utilization in the realm of differential microscopy and more
specifically its potential for merging the polarization and differential
imaging was not touched upon [39, 42]. Usually, the polarization properties of
an object are described using the polarization anisotropy parameters
(birefringence and dichroism) [3, 12, 11, 13]. Being quantitative in nature,
our proposed SPMD imaging system can quantify the spatial gradient of these
anisotropy parameters throughout a specimen and possess the ability to probe
and subsequently produce high contrast images of pure polarization objects
even with weak spatial variation of polarization anisotropy effects. The
polarization-resolved differential microscopy also holds certain advantages to
significantly enhance the image contrast of phase-polarization objects, which
will be discussed subsequently. The adaptability of the SPMD imaging scheme in
high numerical aperture (NA $\sim 0.8$) microscopy is validated alongside its
applicability for broad wavelength regions by imaging various biological
cells. Demonstration of such a low-cost multi-functional setup that can be
easily integrated with the conventional commercial bright-field microscope is
expected to have a significant impact on the next generation of label-free
microscopy and optical image processing techniques.

The proposed differential imaging scheme and experimental setup are
illustrated in Fig. 1(a). A commercially available glass cover slip placed in
the Fourier plane of a $4-f$ imaging setup imparts different phases $(\phi)$;
to the negative $(\phi=\pi:k_{y}<0)$ and positive $(\phi=0:k_{y}>0)$ spatial
frequencies on the Fourier plane. Also, due to the high optical scattering at
the edge of the cover slip, the transmittance of the zero-order spatial
frequencies is reduced to a great extent. In this way, a simple cover slip
placed at an appropriate orientation effectively acts as a Signum mask in the
Fourier plane [41]. The 1D Signum function is chosen for the sake of
mathematical simplicity and corresponding experimental feasibility (see Fig.
1(a)). With a 1D Signum function (in the $y$-direction) modulating the spatial
frequencies $(k_{x},k_{y})$ in the Fourier plane, a full-field spatial
differentiator is achieved where the output intensity distribution at the
image plane can be described as (see Supporting information Sec. S.1 for
details)

$\begin{split}&\mathbf{E}_{image}(x^{\prime},y^{\prime})\propto\int\frac{\partial\mathbf{E}_{obj}(x,y)}{\partial
y}\ln|y^{\prime}-y|dy\\\
&\propto\ln|y^{\prime}-a_{1}|+\ln|y^{\prime}-a_{2}|+\ln|y^{\prime}-a_{3}|+......+\ln|y^{\prime}-a_{n}|\end{split}$
(1)

Here $\mathbf{E}_{obj}(x,y)$, describes the spatial variation of the Electric
field at the object plane, which corresponds to the complex vectorial
transmittance function of the input object, and
$y=a_{1},a_{2},a_{3},\ldots,a_{n}$ are the point of spatial
discontinuities/jumps in terms of either the spatial variation of the
amplitude, phase, or polarization in the $\mathbf{E}_{obj}$. As evident from
Eq. (1), the spatial differentiation of the object field distribution
$(\mathbf{E}_{obj}(x,y))$ is directly encoded in the captured intensity
distribution $(I_{out})$ at the image plane. It is also implied that there
will be intensity localization/enhancement at the point of discontinuities in
the object field, which leads to the production of high-contrast images and
serves as the foundation for the SPMD imaging and microscopy scheme. More
importantly, the achieved spatial differentiator considers the vectorial
nature of light, which is described by the Jones vector field $[E_{1}\
E_{2}]^{T}$ [3, 11]. As a consequence, alongside amplitude and phase objects,
differential images of pure polarization objects can be produced in the SPMD
imaging scheme by recording the polarization-resolved intensity distribution
(Stokes parameters $(S)$ [3, 43]) at the image plane. The detailed theoretical
treatment, particularly, the differential polarization imaging, is provided in
the Supporting information Sec. S.1.

Differential image formation of a rectangular slit utilizing the SPMD imaging
scheme is illustrated in Fig. 1(a). Owing to the amplitude gradient at the
boundaries, intensity localization at the edges of the slit is observed at the
output image plane. To further validate the proposed spatial differentiation
scheme, we numerically simulate the experimental setup and obtain the
differential image of an input Gaussian beam (Fig.1(b)I., II.), where the
spatial intensity distribution at the image plane conforms to the spatial
gradient of a Gaussian distribution. Next, both numerically and experimentally
recorded differential images of a circular phase object are shown Fig. 1(c).
The phase difference between the background and the object leads to an
intensity localization at the circumference of the circle (Fig. 1(c)II., IV.).
The extracted intensity profiles in each pixel of the magenta dashed line
across both the bright field (Fig. 1(c)i.,iii.), and differential image (Fig.
1(c)ii.,iv.) provide compelling proof of the ability of the SPMD imaging
scheme to produce high contrast images of pure phase objects. Note that, the
spatial variation of the intensity distribution at the image plane (Fig.
1(c)ii.) emulates the Logarithmic behaviour as obtained in Eq. (1). Similar
differential images of conventional amplitude objects, e.g., slit, and
circular aperture, have also been recorded, and corresponding theoretical and
experimental results are given in Supporting information Sec. S.2. The
direction along which the spatial differentiation is carried out can be
conveniently tuned by steering the orientation of the Signum phase mask (glass
cover-slip) in the Fourier plane. A rectangular phase object (bright field
image: Fig. 1(d)I.) is utilized to illustrate this, where spatial
differentiation along $\hat{y}$ (Fig. 1(d)II.), $\hat{x}$ (Fig. 1(d)II.),
$\hat{x}+\hat{y}$ (Fig. 1(d)III.) are performed, and consequent intensity
localization along different edges are observed in the corresponding
differential images. Bright-field and differential images of several phase
objects are additionally provided to substantiate the production of high-
contrast images of pure phase objects with the SPMD imaging setup (Fig. 1(e)).
A spatial light modulator (SLM) is employed to prepare the phase objects, that
are used as imaging targets, and the subsequent differential images are
recorded using the SPMD imaging setup (Details regarding optical properties of
the used SLM are provided in the method section).

Figure 1: Demonstrating the working principle of the proposed SPMD imaging.
(a) $4-f$ experimental arrangement with a $633$ nm laser illumination, where a
commercially available glass coverslip (acting as a Signum phase mask) placed
in the Fourier plane leads to the formation of the differential image of an
input object (I.-IV.). The spatial differentiation scheme is illustrated by
considering a rectangular slit as an input object (I.-III.). (b) With an input
Gaussian beam (I.), a two-lobed intensity pattern is observed in the output
image plane (II.), and the corresponding spatial variation of the intensity
profiles (II., IV.) validate formation of differential images with random
input objects. (c) Theoretical (I., II., i., ii. ) and experimental (III.,
IV., iii., iv.) differential images of a circular phase object with a phase
discontinuity of $\pi$ at its boundary. The respective intensity profiles
(ii., iv.) along the Magenta dashed lines confirm the intensity localizations
in the output images at the points of phase discontinuities. (d) Spatial
differentiation along a preferred direction is achieved by mechanically
rotating the coverslip in the Fourier plane. The differential images of a
rectangular phase object (bright field image I.) with spatial differentiation
along horizontal $(\hat{x})$ (II.), vertical $(\hat{y})$ (d III.), and
$45^{o}$ oriented line (IV.) are presented. (e) Differential images of some
additional phase objects prepared by SLM are also shown. All the white scale
bars correspond to $1600\mu m$ length ((b)-(e)).

Next, we demonstrate the adaptability of the SPMD scheme in the domain of
microscopy. To do so, commercial microscope objectives are integrated with the
existing spatial differentiator setup and various biological cells are used as
imaging specimens (Fig. 2). While the bright-field images of the unstained
onion and human cheek cells show low contrast due to their pure phase nature
(Fig. 2(a)I.,(b)I.), corresponding differential images exhibit high contrast
with clearly visible cell structures (Fig. 2(a)II.,(b)II)(intensity profile
corresponding to the white dashed line is given in (Fig. 2(a)III.,IV).
Moreover, we have recorded the differential images of an Aglaonema commutatum
leaf cell using three different microscope objectives with 10X (numerical
aperture, NA $\approx 0.3$), 20X (NA $\approx 0.4$), and 50X (NA $\approx
0.8$) magnification (Fig.2c). The observed differential images authenticate
the ability of the SPMD microscope to produce high-contrast images even with
high spatial resolution. Thus, it is evident that the proposed SPMD microscopy
scheme holds great potential for real-time label-free imaging of transparent
biological objects. In addition to this, we also note that the spatial
differentiation process in the SPMD scheme is intrinsically independent of the
wavelength of the incident light, and thus, the same experimental arrangement
can be utilized for a broad range of wavelengths. Indeed, high-contrast
differential images of pure phase objects are obtained with laser illumination
at different wavelengths, keeping the optical spatial differentiator unchanged
(see Sec S.5 of SI).

Figure 2: Imaging unlabeled biological cells using the SPMD microscopy. Bright
field (top row), and the corresponding differential images (bottom row) are
recorded using a $633$ nm laser illumination source. (a) Onion cell (bright
field image I.) imaged by the SPMD microscope (II.) with 10X objective shows
contrast improvement that is apparent also from the obtained intensity
profiles (III.,IV.) white dotted lines of I.,II. (b) Similarly, the invisible
cell structures, i.e., cell boundary and nucleus, of a human cheek cell
(bright field image I.) are revealed in the obtained differential images (II.)
(c) Validating the performance of high numerical aperture (NA) imaging using
the SPMD microscope. The cross-section of a plant (Aglaonema commutatum) leaf
is imaged in the SPMD microscope using the objective lenses of different
magnifications/NA (II. 10X $\approx$ 0.25, IV. 20X $\approx$ 0.4, VI. 50X
$\approx$ 0.8). Corresponding bright field images are presented in I., III.,
and V. The clearly visible stomata and guard cells (spatial dimension $\sim
50\mu m$) in the captured differential images demonstrate the formation of
high-contrast images with high spatial resolution in the proposed SPMD
microscopy. White scale bars correspond to $1600\mu m$ length

As discussed earlier, one of the biggest advantages of the SPMD imaging scheme
is that it can utilize the polarization degree of freedom as an additional
contrast mechanism in the domain of differential microscopy. We first
demonstrate, how the integration of polarization-resolved measurements with
differential imaging can significantly enhance the image contrast. For this
purpose, a phase-polarization object is prepared using the SLM, and the
polarization-resolved intensity patterns at the output image plane are
recorded by inserting conventional polarizing optical elements, such as linear
polarizers and quarter wave plates (Fig. 3(a)). Even in the case of bright-
field imaging, the polarization-resolved measurement (Fig. 3(a)II.) reveals
the structure of the object (by reducing the overwhelmed background signal),
which is not visible in the unpolarized intensity (Fig. 3(a)I.). Consequently,
the differential image corresponding to the polarization-resolved intensity
pattern (Fig. 3(a)ii.) exhibits much higher contrast as compared to that with
the unpolarized light (Fig. 3(a)i.). This observation highlights the
utilization of polarization as an additional handle to improve the image
contrast in SPMD microscopy.

Next, we describe the quantitative polarization gradient imaging capability of
the proposed SPMD scheme. As Eq. (1) suggests, the proposed spatial
differentiator can transfer the information of polarization anisotropy [44]
gradient at the object plane to the polarization-resolved intensity
distribution at the image plane. This is illustrated considering a pure
polarization object, where although there is no change in the absolute
amplitude or phase of the electric field transmitted through the object
$\mathbf{E_{obj}}$, the polarization of the transmitted light gets modified
due to the anisotropic nature of the object, and differ with respect to the
background signal. The effect of such discontinuity in the polarization
anisotropy parameter, e.g., circular retardance (optical rotation) around the
circumference of a circle-shaped object, is studied (Fig. 3(b)I.). As a
consequence of the polarization discontinuity, differential polarization-
contrast images are obtained by recording the Stokes vector elements ($[I\ Q\
U\ V]^{T}$) at the image plane (Fig. 3(b)I.). The spatial gradient of the
polarization anisotropy effect (rotation) is also quantified using standard
polarization algebra $(\Psi=atan(U/Q))$ and presented in Fig. 3(b)i. (see
Supporting information Sec. S.1 for details). In the case of natural
polarization objects, that exhibit multiple anisotropy effects simultaneously,
such as biological cells and tissues [3], recording the $4\times 4$ Mueller
matrix (in the same experimental set up- see Supporting information Sec. S.4)
and subsequent Mueller matrix decomposition analysis can provide a complete
quantitative polarimetric information throughout the specimen. More
importantly, this kind of polarization differential microscopy can be
advantageous while investigating the morphology of an object with weak spatial
variation of polarization anisotropy effects, thus showing promise in
biomedical and clinical research.

Figure 3: Demonstration of the (a) polarization-resolved differential imaging,
and (b) polarization gradient imaging in the SPMD imaging set up. (a) A phase-
polarization object, shaped as a snowman, prepared using the SLM, is used as
an imaging target. Both the bright field (I.,II.), and differential (i.,ii)
images of the input object are recorded with unpolarized (I.,i.) light, and
also with the polarization projective measurements (II.,ii.). The
polarization-resolved bright field (II.) and differential (ii.) images exhibit
significantly greater contrast with respect to their unpolarized counterparts.
(b) Differential image of a circle-shaped pure polarization object is
presented, where the polarization inside a circle is slightly rotated (due to
the circular retardance effect) with respect to the background’s polarization.
The discontinuity in the polarization anisotropy parameter leads to an
intensity localization around the circumference of the circle (Stokes
parameter I). The corresponding Stokes parameters and the calculated
polarization rotation (i.) demonstrate the quantitative polarization gradient
imaging. The white scale bars correspond to $800\mu m$ spatial distance.

We now shift our discussion to the possible quantitative phase imaging using
the SPMD scheme. Similar to the conventional differential imaging methods, in
the SPMD imaging scheme, the spatial field gradient at the object plane
dictates the intensity distribution at the image plane. Although the
theoretical expression obtained for the output intensity pattern $(I_{out})$
implies a blow-up of intensity values at the point of field discontinuity (Eq.
1), it is never the case for practical situations due to the non-ideal
behaviour of real step functions, and its corresponding derivatives. Thus, the
magnitude of the intensity distribution at the image plane can be incorporated
to quantitatively extract the spatial phase gradient distribution, which is
illustrated next. The behaviour of the localized intensity values with
different phase gradients is both numerically (Fig. 4(b)) calculated and
experimentally (Fig. 4(c)) studied in a circular phase object, where the phase
difference $\delta$ between the object and the background is increased
gradually. In both cases, the maximum magnitude of the localized intensity at
the boundary (edge of the circle) exhibits a monotonically increasing behavior
for $\delta=0$ to $\pi$ (Fig. 4(b), (c)). This indicates the possibility of
establishing a direct relation between the phase gradient and the magnitude of
the localized intensity, which can be utilized to quantify the local spatial
phase gradient itself (see Supporting information Sec. S.3). To experimentally
demonstrate the quantitative phase imaging capability of the SPMD scheme,
circular phase objects are prepared using the SLM with varying phases
(achieved by varying the grey levels). The phase step is gradually increased,
which is manifested in the maxima of the localized intensities in the
corresponding differential images (Fig. 4(d)I.). Now, the priorly obtained
relation (calibration) between the phase gradient and the magnitude of the
localized intensity (Fig. 4(c) is exploited to quantify the phase gradient at
the boundaries of the objects, and consequently, the phase distributions of
the circular objects are derived (see Supporting information Sec. S.3). Also,
the utilized SLM in the experiment modulates the polarization properties
(linear retardance) of light (hence the projected object), and quantitative
polarimetric imaging of the object is also obtained by recording the
polarization-resolved Stokes vectors (Fig. 4(d)II.). Through this
illustration, an imaging system with the ability of simultaneous quantitative
phase and polarization imaging is realized in a single experimental
embodiment.

Figure 4: Demonstration of simultaneous quantitative phase and polarization
imaging using the SPMD imaging setup. (a) A circular phase object with a phase
change of $\delta$ with respect to the background is considered. (b) The
theoretical results describing the variation of localized intensity with
$\delta$ from $0$ to $\pi$ is plotted (corresponding differential images are
presented in Sec. S.3 of SI) (c) The corresponding experimental results are
obtained using the SLM, where the full dynamic range of the SLM is used. The
reported variation of phase with changing grey levels[45] is exploited to
establish a direct relation between the phase difference and intensity
localization at the circumference. (d) The corresponding quantified phase
images of the circle are presented. (e) The quantitative polarization-resolved
measurement is carried out to find the linear retardance parameter $(\alpha)$
corresponding to different grey values. The white scale bar corresponds to
$1600\mu m$ spatial distance.

In summary, we have demonstrated a Signum phase mask differential (SPMD)
imaging and microscopy protocol with the capability of simultaneous
differential phase, differential amplitude, and quantitative polarization
gradient imaging in a simple experimental embodiment. The longstanding
requisite of using polarization degree of freedom of light to obtain a spatial
differentiation of image is eliminated in our scheme, enabling us to integrate
polarization as a contrast mechanism in the framework of differential imaging.
Importantly, the SPMD imaging scheme leads to a novel quantitative
polarization gradient imaging system with the potential to probe low levels
and small spatial variations of polarization anisotropy in a sample. The
utilization of the proposed SPMD scheme in the domain of microscopy is
validated by producing high contrast images of biological samples with high
spatial resolution by integrating high numerical aperture objectives.
Simultaneous demonstration of amplitude, phase, and polarization contrast
differential microscopy in a simple experimental embodiment, imaging
capability with high spatial resolution, functionality over a broad wavelength
region, quantitative and semi-quantitative extraction of polarization and
phase gradient information, all together pave the way for new generation cost-
effective multi-functional microscopy in label-free imaging and image
processing.

## Methods: Preparation of phase and polarization objects using a spatial
light modulator.

The spatial light modulator (SLM) is generally used to modulate the amplitude,
phase, and polarization of light in space and time [46]. Our work uses a
twisted nematic liquid crystal-based SLM [45] to generate several phase and
polarization objects that are used as imaging targets in the SPMD imaging
setup. A specific grey level value in the SLM represents a defined average
voltage across the liquid crystal cell. This voltage leads to a variable tilt
of the anisotropy liquid crystal molecules. Due to the inherent anisotropy
properties of the liquid crystal molecules, the SLM induces a particular phase
according to the tilt angle of the liquid crystal layer. In the case of our
twisted SLM, the net anisotropy effects are manifested as linear/circular
retardance and the accumulation of dynamic phases. The complete polarization
and phase modulation properties of the utilized SLM were obtained through
rigorous calibration [45]. Consequently, the accumulated phase and
polarization effects as the function of grey levels are obtained, which are
used to prepare regulated phase and polarization objects.

## References

  * Mertz [2019] J. Mertz,  _Introduction to optical microscopy_ (Cambridge University Press, 2019).
  * Evanko  _et al._ [2009] D. Evanko, A. Heinrichs, and C. Rosenthal, Milestones in light microscopy, Nature Cell Biol 11, S5 (2009).
  * He  _et al._ [2021] C. He, H. He, J. Chang, B. Chen, H. Ma, and M. J. Booth, Polarisation optics for biomedical and clinical applications: a review, Light: Science & Applications 10, 194 (2021).
  * Guo  _et al._ [2020] S.-M. Guo, L.-H. Yeh, J. Folkesson, I. E. Ivanov, A. P. Krishnan, M. G. Keefe, E. Hashemi, D. Shin, B. B. Chhun, N. H. Cho, _et al._ , Revealing architectural order with quantitative label-free imaging and deep learning, elife 9, e55502 (2020).
  * Mennel  _et al._ [2018] L. Mennel, M. M. Furchi, S. Wachter, M. Paur, D. K. Polyushkin, and T. Mueller, Optical imaging of strain in two-dimensional crystals, Nature communications 9, 516 (2018).
  * Park  _et al._ [2018] Y. Park, C. Depeursinge, and G. Popescu, Quantitative phase imaging in biomedicine, Nature photonics 12, 578 (2018).
  * Yin  _et al._ [2012] Z. Yin, T. Kanade, and M. Chen, Understanding the phase contrast optics to restore artifact-free microscopy images for segmentation, Medical image analysis 16, 1047 (2012).
  * Choi  _et al._ [2007] W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Tomographic phase microscopy, Nature methods 4, 717 (2007).
  * Kim  _et al._ [2014] T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, White-light diffraction tomography of unlabelled live cells, Nature Photonics 8, 256 (2014).
  * Cotte  _et al._ [2013] Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Marker-free phase nanoscopy, Nature Photonics 7, 113 (2013).
  * Chipman  _et al._ [2018] R. A. Chipman, W. S. T. Lam, and G. Young,  _Polarized light and optical systems_ (CRC press, 2018).
  * Goldstein [2017] D. H. Goldstein,  _Polarized light_ (CRC press, 2017).
  * Gil and Ossikovski [2022] J. J. Gil and R. Ossikovski,  _Polarized light and the Mueller matrix approach_ (CRC press, 2022).
  * Zernike [1934] F. Zernike, Diffraction theory of the knife-edge test and its improved form, the phase-contrast method, Monthly Notices of the Royal Astronomical Society, Vol. 94, p. 377-384 94, 377 (1934).
  * Zernike [1942a] F. Zernike, Phase contrast, a new method for the microscopic observation of transparent objects, Physica 9, 686 (1942a).
  * Zernike [1942b] F. Zernike, Phase contrast, a new method for the microscopic observation of transparent objects part ii, Physica 9, 974 (1942b).
  * Burch and Stock [1942] C. Burch and J. Stock, Phase-contrast microscopy, Journal of Scientific Instruments 19, 71 (1942).
  * Martin [1947] L. Martin, Phase-contrast methods in microscopy, Nature 159, 827 (1947).
  * Nomarski [1954] G. Nomarski, Dispositif interférentiel à polarisation pour l’étude des objets transparents ou opaques appartenant à la classe des objets de phase, Patent FR 1, 124 (1954).
  * Françon [1957] M. Françon, Polarization apparatus for interference microscopy and macroscopy of isotropic transparent objects, JOSA 47, 528 (1957).
  * Kwon  _et al._ [2020] H. Kwon, E. Arbabi, S. M. Kamali, M. Faraji-Dana, and A. Faraon, Single-shot quantitative phase gradient microscopy using a system of multifunctional metasurfaces, Nature Photonics 14, 109 (2020).
  * Zhou  _et al._ [2020] Y. Zhou, H. Zheng, I. I. Kravchenko, and J. Valentine, Flat optics for image differentiation, Nature Photonics 14, 316 (2020).
  * Kwon  _et al._ [2018] H. Kwon, D. Sounas, A. Cordaro, A. Polman, and A. Alù, Nonlocal metasurfaces for optical signal processing, Physical review letters 121, 173004 (2018).
  * Wang  _et al._ [2022a] R. Wang, S. He, and H. Luo, Photonic spin-hall differential microscopy, Physical Review Applied 18, 044016 (2022a).
  * Shou  _et al._ [2023] Y. Shou, J. Liu, and H. Luo, When optical microscopy meets all-optical analog computing: A brief review, Frontiers of Physics 18, 42601 (2023).
  * Zhu  _et al._ [2019] T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, _et al._ , Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection, Physical Review Applied 11, 034043 (2019).
  * He  _et al._ [2020a] S. He, J. Zhou, S. Chen, W. Shu, H. Luo, and S. Wen, Wavelength-independent optical fully differential operation based on the spin–orbit interaction of light, APL Photonics 5 (2020a).
  * He  _et al._ [2020b] S. He, J. Zhou, S. Chen, W. Shu, H. Luo, and S. Wen, Spatial differential operation and edge detection based on the geometric spin hall effect of light, Optics Letters 45, 877 (2020b).
  * Wang  _et al._ [2022b] R. Wang, S. He, S. Chen, and H. Luo, Brewster differential microscopy, Applied Physics Letters 121 (2022b).
  * Zhou  _et al._ [2022] J. Zhou, Q. Wu, J. Zhao, C. Posner, M. Lei, G. Chen, J. Zhang, and Z. Liu, Fourier optical spin splitting microscopy, Physical review letters 129, 020801 (2022).
  * Zhou  _et al._ [2021] J. Zhou, H. Qian, J. Zhao, M. Tang, Q. Wu, M. Lei, H. Luo, S. Wen, S. Chen, and Z. Liu, Two-dimensional optical spatial differentiation and high-contrast imaging, National science review 8, nwaa176 (2021).
  * Cordaro  _et al._ [2019] A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, High-index dielectric metasurfaces performing mathematical operations, Nano letters 19, 8418 (2019).
  * Zhu  _et al._ [2017] T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, Plasmonic computing of spatial differentiation, Nature communications 8, 15391 (2017).
  * Zhou  _et al._ [2019] J. Zhou, H. Qian, C.-F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, Optical edge detection based on high-efficiency dielectric metasurface, Proceedings of the National Academy of Sciences 116, 11137 (2019).
  * Engay  _et al._ [2021] E. Engay, D. Huo, R. Malureanu, A.-I. Bunea, and A. Lavrinenko, Polarization-dependent all-dielectric metasurface for single-shot quantitative phase imaging, Nano Letters 21, 3820 (2021).
  * Cotrufo  _et al._ [2022] M. Cotrufo, A. Arora, S. Singh, and A. Alù, Dispersion engineered metasurfaces for broadband, high-na, high-efficiency, dual-polarization analog image processing, arXiv preprint arXiv:2212.03468 (2022).
  * Silva  _et al._ [2014] A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, Performing mathematical operations with metamaterials, Science 343, 160 (2014).
  * Hwang and Davis [2016] Y. Hwang and T. J. Davis, Optical metasurfaces for subwavelength difference operations, Applied Physics Letters 109 (2016).
  * Davis  _et al._ [2000] J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, Image processing with the radial hilbert transform: theory and experiments, Optics Letters 25, 99 (2000).
  * Davis and Nowak [2002] J. A. Davis and M. D. Nowak, Selective edge enhancement of images with an acousto-optic light modulator, Applied optics 41, 4835 (2002).
  * Goodman and Sutton [1996] J. W. Goodman and P. Sutton, Introduction to fourier optics, Quantum and Semiclassical Optics-Journal of the European Optical Society Part B 8, 1095 (1996).
  * Davis  _et al._ [2001] J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, and J. Campos, Fractional derivatives—analysis and experimental implementation, Applied Optics 40, 5943 (2001).
  * Gupta  _et al._ [2015] S. D. Gupta, N. Ghosh, and A. Banerjee,  _Wave optics: Basic concepts and contemporary trends_ (CRC Press, 2015).
  * Modak  _et al._ [2021] N. Modak, B. Athira, A. K. Singh, and N. Ghosh, Generalized framework of weak-value amplification in path interference of polarized light for the enhancement of all possible polarization anisotropy effects, Physical Review A 103, 053518 (2021).
  * Pal  _et al._ [2016] M. Pal, C. Banerjee, S. Chandel, A. Bag, S. K. Majumder, and N. Ghosh, Tunable spin dependent beam shift by simultaneously tailoring geometric and dynamical phases of light in inhomogeneous anisotropic medium, Scientific reports 6, 39582 (2016).
  * Efron [1994] U. Efron,  _Spatial light modulator technology: materials, devices, and applications_ , Vol. 47 (CRC press, 1994).


# Closed geodesics on hyperbolic surfaces with few intersections

Wujie Shen  Yau Mathematical Science Center, Tsinghua University
<EMAIL_ADDRESS>
###### Abstract.

We prove that, if a closed geodesic $\Gamma$ on a complete finite-type
hyperbolic surface has at least 2 self-intersections, then the length of
$\Gamma$ has an lower bound $2\log(5+2\sqrt{6})$, and the lower bound is
sharp, attained on a corkscrew geodesic on a thrice punctured sphere.

##  1\. Introduction

The study of nonsimple closed geodesics on hyperbolic surfaces is an
interesting question in 2-dimensional hyperbolic geometry. An oriented finite
type hyperbolic surface is an oriented surface $\Sigma=\Sigma_{g,n}$ without
boundary, with genus $g$ and $n$ points removed(puncture), and a complete
metric of constant curvature -1 with finite volume. The hyperbolic structure
near each removed point is a cusp tends to infinity. For positive integer $k$,
let $M_{k}$ be the infimum of lengths of closed geodesics of self-intersection
number at least $k$ among all oriented finite-type hyperbolic surfaces. When
$k\geqslant 1$, we call the closed geodesic is _nonsimple_ , otherwise we call
it _simple_.

There has been lots of works on the problem till now. When $k=1$, Hempel
showed in [7] that a nonsimple closed geodesic has a universal lower bound
$2\log(1+\sqrt{2})$ and Yamada showed in [9] that
$2\cosh^{-1}(3)=4\log(1+\sqrt{2})$ is the best possible lower bound and is
attained on a pair of pants with ideal punctures. Basmajian showed in [3] that
a nonsimple closed geodesic has a similar _stable neighborhood_ , and the
length of a closed geodesic gets arbitrarily large as its self-intersection
number gets large ([3, Corollary 1.2]). In [2], Baribaud computed the minimal
length of geodesics with given self-intersection number or given homotopy
types on pairs of pants.

It is interesting to study the minimal length of non-simple closed geodesics
with respect to its self-intersection number $k$ . Let $\omega$ be a closed
geodesic or a geodesic segment on a hyperbolic surface $\Sigma$ which can be
expressed as a local isometry $\phi$ from $S^{1}$ or segment $I$ to $\Sigma$.
We denote the length as $\ell(\omega)$.

######  Definition 1.1.

Its self-intersection number is denoted by $\left|\omega\cap\omega\right|$.
$\left|\omega\cap\omega\right|$ counts the intersection points of $\omega$
with multiplicity that an intersection point with $n$ preimages of $\phi$
contribute $\binom{n}{2}$ to $\left|\omega\cap\omega\right|$.

First, when $k\rightarrow\infty$, Basmajian showed ([4, Corollary 1.4]) that

(1) $\tfrac{1}{2}\log\frac{k}{2}\leqslant M_{k}\leqslant
2\cosh^{-1}(2k+1)\asymp 2\log k$

The notation $f(k)\asymp g(k)$ means that ${f(k)}/{g(k)}$ is bounded from
above and below by positive constants.

######  Conjecture 1.2.

When $k\geqslant 1$,

(2) $M_{k}=2\cosh^{-1}(1+2k)=2\log(1+2k+2\sqrt{k^{2}+k})$

and the equality holds when $\Gamma$ is a corkscrew geodesic(See definition
below) on a thrice-punctured sphere. In other words, any nonsimple closed
geodesic of self-intersection number at least $k$ on any finite-type
hyperbolic surface has length no less than $2\cosh^{-1}(1+2k)$ and the bound
is sharp.

In [10, Theorem 1.1] Shen-Wang improved the lower bound of $M_{k}$, that
$M_{k}$ has explicit growth rate $2\log{k}$, and for a closed geodesic of
length $L$, the self intersection number is no more than
$9L^{2}e^{\frac{L}{2}}$. The exact value for $M_{k}$ for sufficiently large
$k$ is computed in [11, Theorem 1.1], proved Conjecture 1.2 holds when
$k>10^{13350}$. In [13] the lower bound $k>10^{13350}$ is refined to $k>1750$
using another method.

However, to the best of the author’s knowledge, for small $k$, even $k=2$ we
cannot compute the exact value of $M_{k}$. In the present paper we give an
answer:

######  Theorem 1.3.

When $k=2$, $M_{2}=2\log(5+2\sqrt{6})$, and the lower bound is sharp and
attained on a corkscrew geodesic on a thrice punctured sphere.

Note that Theorem 1.3 can be generalized to general orientable finite-type
hyperbolic surfaces, possibly with holes and geodesic boundaries, since they
can be doubled to get a surface as in Theorem 1.3.

### Plan of the paper

The idea of the proof is based on the conclusion of [8], that is
$4\log(1+\sqrt{2})$ is the best possible lower bound of nonsimple closed
geodesic on hyperbolic surfaces, i.e. self-intersection number $k=1$.

Section 2 to section 4 is the preparation of the proof consisting of 3 parts.
First is the collar lemma, a basic lemma in 2-dimensional hyperbolic geometry,
and we will give a generalization, the collar neighborhood is not equal on the
different side, which will be used in section 5. Second is about geodesics on
pair of pants, we will state the result in [2] and as a simple corollary, we
prove Theorem 1.3 for the case that geodesic $\Gamma$ is contained in a pair
of pants(or ideal pair of pants) of given homotopy type. Third is about
winding number of a geodesic segment in a collar and we can compute the length
based on the winding number and the collar.

Section 5 is the main part of the proof. If $\ell(\Gamma)<2\log(5+2\sqrt{6})$,
then choose the shortest closed loop $\gamma$ in $\Gamma$, we prove the
theorem in 2 cases, by whether
$\ell(\Gamma\setminus\gamma)<4\log(1+\sqrt{2})$. If it holds then both
$\gamma$ and $\Gamma\setminus\gamma$ are freely homotopic to a multiple of a
simple closed geodesic(or cusp), hence $\Gamma$ contained in a pair of
pants(or ideal pair of pants). Otherwise $\ell(\gamma)$ is small, hence
contained in a collar in section 2 where the generalized collar lemma is used.
Then we complete the proof by contradiction since either the length of
$\Gamma$ is big or $\Gamma$ contained in a pair of pants.

### Acknowledgement

We would be thankful to many people, especially Professor Yi Huang and Zhongzi
Wang for reading the first draft of the paper and giving some helpful
discussions.

##  2\. A generalization of the collar lemma

The collar lemma ([6, Lemma 13.6]) is as follows:

######  Lemma 2.1.

for any simple closed geodesic $c$ in $\Sigma$,

$N(c)=\left\\{p\in\Sigma\,:\,d(p,c)<w(\ell(c))\right\\}$

is an embedded annulus, where $w(x)$ is defined by

$w(x):=\sinh^{-1}\left(\frac{1}{\sinh({x}/{2})}\right)$

Moreover if $c_{1},c_{2}$ are two disjoint simple closed geodesics in
$\Sigma$, $N(c_{1})\cap N(c_{2})=\emptyset$.

Let $\Sigma^{\prime}$ be the surface with boundary components
$c^{\prime},c^{\prime\prime}$ constructed by cutting along curve
$c\subseteq\Sigma$, together with gluing map
$\phi^{\prime}:\Sigma^{\prime}\rightarrow\Sigma$ that gluing
$c^{\prime},c^{\prime\prime}$ to the curve $c\subseteq\Sigma$. There is a
_maximal collar_ $N^{\prime}(c^{\prime})$ in $\Sigma^{\prime}$ that
$\delta(c^{\prime})>0$ is the maximum real number satisfies the collar

$\widetilde{N}^{\prime}(c^{\prime})=\left\\{p\in\Sigma^{\prime}\,:\,d(p,c^{\prime})<\delta(c^{\prime})\right\\}$

is an embedded annulus in $\Sigma^{\prime}$.
$N^{\prime}(c)=\phi^{\prime}(N^{\prime}(c^{\prime}))$ is an embedded annulus(a
half collar) with $c$ as one of its boundary components. We define

$w_{1}(x)=\log\left(\frac{e^{x/4}+1}{e^{x/4}-1}\right)$

for $x>0$. Note that for $x\leqslant\log(5+2\sqrt{6})<2.3$ we have
$w_{1}(x)<2w(x)$. For simple closed geodesic $c\in\Sigma$ we have the
generalized collar lemma:

######  Lemma 2.2.

If $\ell(c^{\prime})<2.3$, then we have $\delta(c^{\prime})\geqslant
w_{1}(\ell(c^{\prime}))$. In other words, the collar

$N^{\prime}(c^{\prime})=\left\\{x\in\Sigma\,:\,d(x,c)<w_{1}(\ell(c))\right\\}$

is an embedded annulus.

###### Proof.

We know $\widetilde{N}^{\prime}(c^{\prime})$ has two boundary components. If
$\widetilde{N}^{\prime}(c^{\prime})\cap c^{\prime\prime}=\emptyset$, then the
one is $c$ and the other is a curve tangent to itself by the maximality of
$\delta(c^{\prime})$, assume $q\in\partial N^{\prime}(c)$ is one of the
tangent points. (Note that if $\widetilde{N}^{\prime}(c^{\prime})\cap
c^{\prime\prime}\neq\emptyset$, then since
$w_{1}(\ell(c^{\prime}))<2w(\ell(c^{\prime}))$, we have
$N^{\prime}(c^{\prime})\cap c^{\prime\prime}=\emptyset$, the lemma holds.)
Hence there exists a shortest simple geodesic $\gamma$ which contained in
$\overline{N^{\prime}(c)}\subseteq\Sigma$, contains two different shortest
geodesic segments of length $\delta(c^{\prime})$ connecting $q$ and $c$ in
$N^{\prime}(c)$. $\gamma$ has endpoints $r_{1},r_{2}\in c$, and perpendicular
to $c$ at $r_{1},r_{2}$.

Suppose $c\setminus\\{r_{1},r_{2}\\}$ contains two arcs $c_{1}^{\prime}$ and
$c_{2}^{\prime}$, and $c_{1}^{\prime}\cup\gamma$, $c_{2}^{\prime}\cup\gamma$
are simple closed curves, hence freely homotopic to simple closed geodesics
$c_{1}^{\prime\prime}$ and $c_{2}^{\prime\prime}$, then
$c,c_{1}^{\prime\prime},c_{2}^{\prime\prime}$ are boundary components of a
pair of pants $\Sigma_{0}\subseteq\Sigma$, since
$c_{1}^{\prime\prime}\cup\gamma$ or $c_{2}^{\prime\prime}\cup\gamma$ could not
create bigons,
$c_{1}^{\prime\prime}\cap\gamma=c_{2}^{\prime\prime}\cap\gamma=\emptyset$, we
have $\gamma\subseteq\Sigma_{0}$.

Next we prove $\ell(\gamma)\geqslant w_{1}(x)$. $\Sigma_{0}$ can be
constructed by gluing 2 hexagons along 3 nonadjacent boundary segments. In the
Poincaré disk model of $\mathbb{H}^{2}$, let $\mathbb{H}^{1}$ be the
horizontal line. Let $A_{1},A_{2}\subseteq\mathbb{H}^{1}$,
$A_{1}(-r,0),A_{2}(r,0)$ satisfying $\log\frac{1+r}{1-r}=\frac{\ell(c)}{4}$.
$\omega_{1},\omega_{2}$ are geodesics in $\mathbb{H}^{2}$ passes $A_{1},A_{2}$
and perpendicular to $\mathbb{H}^{1}$.
$P_{1}(-\frac{2r}{1+r^{2}},\frac{1-r^{2}}{1+r^{2}})$,
$P_{2}(\frac{2r}{1+r^{2}},\frac{1-r^{2}}{1+r^{2}})$ are one of endpoints of
$\omega_{1},\omega_{2}$. $\omega_{3}^{\prime}$ is the geodesic connecting
$P_{1},P_{2}$. Let $\omega_{3}$ be the geodesic $Q_{1}Q_{2}$ where
$Q_{1},Q_{2}$ are both on the infinity boundary,
$\omega_{3}\cap\omega_{1}=\emptyset$ and $\omega_{3}\cap\omega_{2}=\emptyset$.
For $j=1,2$ assume $\omega_{j}^{\prime}$ be the unique geodesic segment
perpendicular to $\omega_{j}$ and $\omega_{3}$. Then
$A_{1}A_{2},\omega_{1},\omega_{1}^{\prime},\omega_{3},\omega_{2}^{\prime},\omega_{2}$
compose the boundary of a hexagon, and suppose $\Sigma_{0}$ is constructed by
gluing 2 copies of the hexagon along $\omega_{1},\omega_{2},\omega_{3}$.

Since $\gamma\subseteq\Sigma_{0}$, then

$\displaystyle\ell(\gamma)$ $\displaystyle\geqslant
2d(\mathbb{H}^{1},\omega_{3})\geqslant
2d(\mathbb{H}^{1},\omega_{3}^{\prime})=2\log\left(1+\frac{1-r}{1+r}\right)-2\log\left(1-\frac{1-r}{1+r}\right)$
$\displaystyle=2\log\frac{e^{{\ell(c)}/4}+1}{e^{{\ell(c)}/4}-1}=w_{1}(\ell(c))$

∎

$\omega_{3}^{\prime}$$\omega_{3}$$Q_{1}$$Q_{2}$$A_{1}$$A_{2}$$P_{1}$$P_{2}$$O$$\omega_{1}$$\omega_{2}$
Figure 1.  A hexagon of $\Sigma_{0}$

Since $w_{1}(x)>w(x)$ for $x>0$, we have

$N^{\prime\prime}(c^{\prime\prime})=\left\\{p\in\Sigma^{\prime}\,:\,d(p,c^{\prime\prime})<2w(\ell(c))-w_{1}(\ell(c))\right\\}\subseteq\Sigma^{\prime}$

is either empty or an embedded annulus, and define
$N^{\prime\prime}(c)=\phi^{\prime}(N^{\prime\prime}(c^{\prime\prime}))\subseteq\Sigma$,
either empty or an embedded annulus also. Define

$N_{1}(c)=N^{\prime}(c)\cup N^{\prime\prime}(c)$

######  Lemma 2.3.

$N_{1}(c)$ is an embedded annulus in $\Sigma$.

###### Proof.

If not, then there exists $y\in N^{\prime}(c)\cap N^{\prime\prime}(c)$,
connecting $c$ and $y$ by shortest geodesic $\zeta$ in $N^{\prime}(c)$ and
$\zeta^{\prime}$ in $N^{\prime\prime}(c)$, and
$\ell(\zeta)+\ell(\zeta^{\prime})<2w(\ell(c))$, hence for all
$z\in\zeta\cup\zeta^{\prime}$,
$d(z,c)\leqslant\frac{1}{2}(\ell(\zeta)+\ell(\zeta^{\prime}))<w(\ell(c))$,
hence $\zeta\cup\zeta^{\prime}\subseteq N(c)$. $\zeta\cup\zeta^{\prime}$ only
intersects $c$ at endpoints so it lies on one side of $N(c)$, this is
impossible since $N^{\prime}(c)$ and $N^{\prime\prime}(c)$ are on the
different side of $c$. ∎

Similarly, for the ideal punctures of $\Sigma$ we have the similar result.
When $\Sigma$ has punctures, we consider the universal covering
$p:\mathbb{H}^{2}\to\Sigma$, where $\mathbb{H}^{2}$ is the hyperbolic plane.
Each puncture has a neighborhood whose boundary lifts to a union of horocycles
that can intersect in at most points of tangency. Such a neighborhood is
called a _cusp_ of the surface.

In the upper half-plane model for $\mathbb{H}^{2}$. Let $\Gamma$ be a cyclic
group generated by a parabolic isometry of $\mathbb{H}^{2}$ fixing the point
$\infty$. Let $H_{c}=\left\\{(x,y)\in\mathbb{H}^{2}\,\big{|}\,y\geqslant
c\right\\}$ be a horoball. Each cusp can be modelled as $H_{c}/\Gamma$ for
some $c$ up to isometry, and is diffeomorphic to $S^{1}\times[c,\infty)$ so
that each circle $S^{1}\times\left\\{t\right\\}$ with $t\geqslant c$ is the
image of a horocycle under $p$. Each circle is also called a _horocycle_ by
abuse of notation. The circle $S^{1}\times\left\\{t\right\\}$ with $t\geqslant
c$ is called an _Euclidean circle_. A cusp is _maximal_ if it lifts to a union
of horocycles with disjoint interiors such that there exists at least one
point of tangency between different horocycles.

######  Lemma 2.4 (Adams, [1]).

For an orientable, metrically complete hyperbolic surface, the area with a
maximal cusp is at least $4$. The lower bound $4$ is realized only in an ideal
pair of pants.

If $c$ is a cusp, we define $N^{\prime}(c)$ is a union of horocycles of $c$
that length less than 4. It is an embedded cylinder.

##  3\. geodesics on pair of pants

######  Definition 3.1.

By _corkscrew geodesic_ we mean a geodesic in the homotopy class as described
in Figure 2. that is a curve consisting of the concatenation of a simple arc
and another that winds $k$ times around a boundary. The name ”corkscrew” comes
from [11].

Figure 2.  A corkscrew geodesic of $k=2$

There is a unique hyperbolic structure in the thrice punctured sphere
$\Sigma_{0,3}$, the corkscrew geodesic consisting of the concatenation of a
simple arc and another that winds 2 times around a cusp has 2 self-
intersections, see Figure 2, which has length
$2arccosh^{-1}(5)=2\log(5+2\sqrt{6})$.

If the geodesic $\Gamma$ is contained in a pair of pants, then we use the
following results in [2]. Assume a pair of pants $P$ with geodesic boundaries
$\gamma_{1},\gamma_{2},\gamma_{3}$. For $i=1,2,3$, define

$c_{i}=\cosh\frac{\ell(\gamma_{i})}{2}\ \ \
c_{i,n}=\cosh\frac{n\ell(\gamma_{i})}{2}\qquad
s_{i}=\sinh\frac{\ell(\gamma_{i})}{2}\ \ \
s_{i,n}=\sinh\frac{n\ell(\gamma_{i})}{2}$

Let $\Gamma_{m,n}$ is the unique closed geodesic in $P$ that has homotopy type
of a curve winding around $\gamma_{1}$ $m$ times and then winding around
$\gamma_{2}$ $n$ times. The length of $\Gamma_{m,n}$ can be computed as
follows:

######  Theorem 3.2.

(3)
$\displaystyle\cosh\frac{\ell(\Gamma_{m,n})}{2}=\frac{s_{1,m}}{s_{1}}\frac{s_{2,n}}{s_{2}}(c_{3}+c_{1}c_{2})+c_{1,m}c_{2,n}$

Note that when $P$ is an ideal pair of pants, i.e. some of
$\gamma_{1},\gamma_{2},\gamma_{3}$ become cusps as their lengths tend to 0,
then $c_{i},c_{i,n},s_{i},s_{i,n}$ can be defined, and if $\gamma_{1}$ becomes
cusp, $\frac{s_{1,m}}{s_{1}}$ can be altered by

$\lim_{\ell(\gamma_{1})\rightarrow
0+}\frac{\cosh\frac{m\ell(\gamma_{1})}{2}}{\cosh\frac{\ell(\gamma_{1})}{2}}=m$

in (3). And if $\gamma_{2}$ becomes cusp, similarly $\frac{s_{2,n}}{s_{2}}$
can be altered by $n$ in (3).

######  Corollary 3.3.

Suppose $m,n$ are positive integers and $m+n\geqslant 3$. $P$ is a pair of
pants or an ideal pair of pants. $\Gamma_{m,n}$ is a closed geodesic in $P$ as
above. Then

$\ell(\Gamma_{m,n})\geqslant 2\log(5+2\sqrt{6})$

The equality holds if and only if $P$ is a thrice punctured sphere and $(m,n)$
is equal to $(1,2)$ or $(2,1)$.

###### Proof.

Since $c_{i}=\cosh\frac{\ell(\gamma_{i})}{2}\geqslant 1$ and similarly
$c_{i,m}\geqslant 1$, and when $\ell(\gamma_{1})>0$

$\displaystyle\frac{s_{1,m}}{s_{1}}$
$\displaystyle=\frac{\sinh\frac{m\ell(\gamma_{1})}{2}}{\sinh\frac{\ell(\gamma_{1})}{2}}=\frac{\alpha^{m}-\alpha^{-m}}{\alpha-\alpha^{-1}}=\sum_{k=0}^{m-1}\alpha^{m-1-2k}$
$\displaystyle=\frac{1}{2}\sum_{k=0}^{m-1}(\alpha^{m-1-2k}+\alpha^{2k+1-m})>m$

Here $\alpha=e^{\ell(\gamma_{1})/2}$. Similarly when $\ell(\gamma_{2})>0$ we
have $\frac{s_{2,n}}{s_{2}}>n$. Hence we have

$\cosh\frac{\ell(\Gamma_{m,n})}{2}=\frac{s_{1,m}}{s_{1}}\frac{s_{2,n}}{s_{2}}(c_{3}+c_{1}c_{2})+c_{1,m}c_{2,n}\geqslant
2mn+1\geqslant 5$

Hence $\ell(\Gamma_{m,n})\geqslant 2\log(5+2\sqrt{6})$. The equality holds if
and only if $\frac{s_{1,m}}{s_{1}}=m$, $\frac{s_{2,n}}{s_{2}}=n$, and
$c_{3}=c_{1}=c_{2}=c_{1,m}=c_{2,n}=1$, that is,
$\ell(\gamma_{1})=\ell(\gamma_{2})=\ell(\gamma_{3})=0$, and $mn=2$. ∎

##  4\. Winding number of geodesic segments in a collar

Let $c\in\Sigma$ be a simple closed geodesic, and $w>0$. This section we
always assume the collar

$N(c)=\left\\{x\in\Sigma:d(x,c)<w\right\\}$

is an embedded annulus. And for cusps, assume $c$ is a puncture and $N(c)$ is
defined to be the cusp neighborhood with boundary horocycle of length 4.
$\delta$ is a geodesic segment in $N(c)$ with endpoints $x_{1},x_{2}$ on the
same component of $\partial N(c)$.

Next we define the _winding number_ $W(\delta)$ of the arc $\delta$. The
definitions are similar as [12].

  1. (1)

When $c$ is a closed geodesic, every point of $\delta$ projects orthogonally
to a well-defined point of $c$. The winding number of $\delta$ is given by the
quotient of the length of the projection of $\delta$ divided by $\ell(c)$.

  2. (2)

Similarly when $c$ is a cusp, every point of $\delta$ projects orthogonally to
a well-defined point of the length $h$ horocycle. The winding number of
$\delta$ is given by the quotient of the length of the projection of $\delta$
divided by $h$.

######  Theorem 4.1.

If $c$ is a closed geodesic, then

$\ell(\delta)=2\sinh^{-1}\left(\sinh\frac{W(\delta)\ell(c)}{2}\cdot\cosh{w}\right)$

###### Proof.

The universal covering $p:\mathbb{H}^{2}\to\Sigma$ from the Poincaré disk
$\mathbb{H}^{2}$ to $\Sigma$ is locally isometric. Let $\widetilde{\delta}$,
$\widetilde{c}$ be a lift of $\delta$ and $c$. The connected component
$\widetilde{N(c)}$ of $p^{-1}(N(c))$ containing $\widetilde{\delta}$ is a
universal cover of the annulus $N(c)$. Let $\widetilde{x}_{1}$ and
$\widetilde{x}_{2}$ be lifts of $x_{1}$,$x_{2}$ in $\widetilde{N(c)}$. Let
$\widetilde{\eta}_{1}$ and $\widetilde{\eta}_{2}$ be the shortest geodesics
from $\widetilde{x}_{1}$ and $\widetilde{x}_{2}$ to $\widetilde{c}$
respectively. Then $\eta_{1}:=p\circ\widetilde{\eta}_{1}$,
$\eta_{2}=p\circ\widetilde{\eta}_{2}$ is a geodesic connecting $x$ and $c$ and
$\widetilde{\eta}_{1}$ and $\widetilde{\eta}_{2}$ are both perpendicular to
$c$. Let $\widetilde{y}_{1}$ and $\widetilde{y}_{2}$ be the two feet. Without
loss of generality, we may assume $\widetilde{c}$ is the horizontal diameter
of $\mathbb{H}^{2}$ and the origin $O$ is the middle point of
$\widetilde{y}_{1}$ and $\widetilde{y}_{2}$, as illustrated in Figure 3.

The geodesic $\widetilde{\delta}$ between $\widetilde{x}_{1}$ and
$\widetilde{x}_{2}$, $\widetilde{\eta}_{1}$, $\widetilde{\eta}_{2}$ and the
geodesic $\delta_{1}$ between $\widetilde{y}_{1}$ and $\widetilde{y}_{2}$ form
a Saccheri quadrilateral, and half of it is a Lambert quadrilateral. Note that
$d(\widetilde{y}_{1},\widetilde{y}_{2})=W(\delta)\ell(c)$. The property of the
Lambert quadrilateral gives

$\sinh\left(\frac{\ell(\delta)}{2}\right)=\sinh\left(\frac{W(\delta)\ell(c)}{2}\right)\cosh\left(\ell\left(\widetilde{\eta}_{1}\right)\right)$

∎

$\widetilde{c}$$\widetilde{\delta}$$\widetilde{\eta}_{1}$$\widetilde{\eta}_{2}$$\widetilde{y}_{1}$$\widetilde{y}_{2}$$\widetilde{x}_{1}$$\widetilde{x}_{2}$$O$
Figure 3.  A covering of the annulus.

######  Theorem 4.2.

If $c$ is a cusp with boundary length 4(it is embedded by Lemma 2.4), then

$\ell(\delta)=2\log\left(2W(\delta)+\sqrt{4W^{2}(\delta)+1}\right)$

###### Proof.

When $c$ is a cusp, consider the projection map $p$ from the upper half plane
model of $\mathbb{H}^{2}$ to $\Sigma$, where
$p^{-1}(N(c))=\\{(x,y)\in\mathbb{H}^{2}:y\geqslant 1\\}$, and $A(-2,0)$ and
$B(2,0)$ be two adjacent points of a same point $x\in\partial N(c)$, $\ell$ be
the line $\\{(x,y)\in\mathbb{H}^{2}:y=1\\}$. Without loss of generality assume
$\widetilde{\delta}$ is a lift of the arc $\delta$ and is an arc of a circle
centered at the origin $O$ with endpoints $P_{1},P_{2}\in\ell$. The hyperbolic
length of the arc $P_{1}P_{2}$ is $d(P_{1},P_{2})=\ell(\delta)$, and
$d_{E}(P_{1},P_{2})=4W(\delta)$ where $d_{E}$ means the Euclidean distance in
$\mathbb{H}^{2}$, as in Figure 4 .Hence

$\ell(\delta)=d(P_{1},P_{2})=2\log\left(2W(\delta)+\sqrt{4W^{2}(\delta)+1}\right)$

∎

$O$$A$$B$$P_{1}$$P_{2}$$\widetilde{\delta}$$\infty$ Figure 4.  A covering of
$N_{0}(c_{i})$ when $c_{i}\in\mathcal{Y}$

##  5\. Proof of theorem 1.3

The notations are as above, $\Gamma$ is a closed geodesic in $\Sigma$ with at
least 2 self intersections, with length $\ell(\Gamma)$. Let
$|\Gamma\cap\Gamma|$ be the self intersection number of $\Gamma$. Let $\gamma$
be the shortest arc contained in $\Gamma$ that is a closed curve in $\Sigma$,
and $P\in\Sigma$ be its endpoint. Then $\gamma$ is simple since the length of
$\delta_{0}$ is minimal. We finish the proof based on whether the length
$\ell(\gamma)$ of $\gamma$ satisfies

$\ell(\gamma)\leqslant 2\log(5+2\sqrt{6})-4\log(1+\sqrt{2})<1.06$

###  5.1. Case 1: $\ell(\gamma)\geqslant 1.06$

######  Theorem 5.1.

If $\ell(\gamma)\geqslant 1.06$ and $\ell(\Gamma)<2\log(5+2\sqrt{6})$, then
$|\Gamma\cap\Gamma|\leqslant 1$.

###### Proof.

Assume $\gamma^{\prime}=\Gamma\setminus\gamma$, then $\gamma^{\prime}$ is a
geodesic arc with same endpoints and
$\ell(\gamma^{\prime})<4\log(1+\sqrt{2})$. Using Yamada’s result([9]) we have
$\gamma$ and $\gamma^{\prime}$ are freely homotopic to a multiple of simple
closed geodesics(or cusp) $\beta$ and $\beta^{\prime}$, and

$\ell(\beta)\leqslant\ell(\gamma)\qquad\ell(\beta^{\prime})\leqslant\ell(\gamma^{\prime})<4\log(1+\sqrt{2})<3.5255$

Using the collar lemma(2.1) we have

$N(\beta)=\left\\{p\in\Sigma\,:\,d(p,\beta)<w(\ell(\beta))\right\\}$

is an embedded annulus.

  1. (1)

If $\beta\cap\beta^{\prime}=\emptyset$, then since $\gamma^{\prime}$ freely
homotopic to a multiple of simple closed geodesic $\beta^{\prime}$ with
multiplicity $k\in\mathbb{Z}_{+}$, say $k\beta^{\prime}$. We have
$\gamma^{\prime}$ homotopic to a curve with basepoint $P$ with self-
intersection number $k-1$, assume $\gamma^{\prime\prime}$ is the curve with
infimum length satisfying this property. Then $\gamma^{\prime\prime}$ is a
geodesic arc with same endpoint $P$, since it is freely homotopic to a
multiple of $\beta^{\prime}$ and $\gamma^{\prime\prime}\cup\beta^{\prime}$
could not create bigons, we have
$\gamma^{\prime\prime}\cap\beta^{\prime}=\emptyset$. Hence $\Gamma$ is freely
homotopic to a curve starting at $P$, winding around $\beta$ one time, then
goes to $\beta^{\prime}$ then winding around it $k$ times, finally goes back
to $P$. The curve has self intersection number $k$.

We can construct a pair of pants $P$ with $\beta$ and $\beta^{\prime}$ as two
of three boundary components as follows.Connect $P$ and $\beta$ by shortest
geodesic $\delta_{1}$ in the annulus bounded by $\beta$ and $\gamma$, and
connect $P$ and $\beta^{\prime}$ by shortest geodesic $\delta_{2}$ in the
annulus bounded by $\beta^{\prime}$ and a part of $\gamma^{\prime}$. Consider
a closed curve
$\beta_{1}:=\delta_{1}\beta\delta_{1}^{-1}\delta_{2}\beta^{\prime}\delta_{2}^{-1}$
(by connecting the endpoints) we can choose an orientation of $\beta$ and
$\beta^{\prime}$ such that $\beta_{1}$ is freely homotopic to a simple closed
geodesic(or cusp) $\beta^{\prime\prime}$, which is the third boundary
component of $P$. Since the homotopy type of $\Gamma$ can be represented by a
closed geodesic $\Gamma^{\prime}$ contained in $P$, hence
$\Gamma=\Gamma^{\prime}\subseteq P$ by the uniqueness of closed geodesic with
given homotopy type. Using Corollary 3.3 we have $\ell(\Gamma)\geqslant
2\log(5+2\sqrt{6})$, a contradiction.

  2. (2)

If $\beta\cap\beta^{\prime}\neq\emptyset$, then $\beta$ is a simple closed
geodesic, not a cusp. Assume $t=\frac{1}{2}\ell(\beta)>0$. Since $\gamma$ and
$\beta$ freely homotopic, they bound an annulus $A$, and assume
$N^{\prime}(\beta)$ and $A$ are on the same side of $N(\beta)$. First note
that if $\beta\cap\beta^{\prime}\neq\emptyset$, since every arc
$\beta^{\prime\prime}$ of $\beta^{\prime}\cap N_{1}(\beta)$ satisfying
$\beta^{\prime\prime}\cap\beta=\emptyset$ is a geodesic segment get in and out
$N_{1}(\beta)$ on different sides(in fact the function $d(x,\beta^{\prime})$
first decreases to 0 and then increases when $x$ goes along the arc), we have

$\ell(\gamma^{\prime})\geqslant\ell(\beta^{\prime})\geqslant\ell(\beta^{\prime\prime})\geqslant
2w(\ell(\beta))=2\log\frac{e^{t}+1}{e^{t}-1}$

Let $\widetilde{\gamma}\subseteq\Gamma$ is the unique arc in $\Gamma\cap
N^{\prime}(\beta)$ containing or contained in $\gamma^{\prime}$. The winding
number $W(\widetilde{\gamma})$ can be defined above.

If $W(\widetilde{\gamma})\geqslant 1$, then using Theorem 4.1 we have

$\displaystyle\ell(\Gamma\setminus{\beta^{\prime\prime}})$
$\displaystyle\geqslant\ell(\widetilde{\gamma})=2\sinh^{-1}\left(\sinh(tW(\widetilde{\gamma}))\cosh
w_{1}(\ell(\beta))\right)$ $\displaystyle\geqslant 2\sinh^{-1}\left(\sinh
t\cosh\log\frac{e^{t/2}+1}{e^{t/2}-1}\right)$

If $w(\widetilde{\gamma})<1$, then in annulus $A$,
$d(P,\beta)>w_{1}(\ell(\beta))$, hence

$\displaystyle\ell(\Gamma\setminus{\beta^{\prime\prime}})$
$\displaystyle\geqslant\ell(\gamma)=2\sinh^{-1}\left(\sinh t\cosh
d(P,\beta)\right)$ $\displaystyle\geqslant 2\sinh^{-1}\left(\sinh
t\cosh\log\frac{e^{t/2}+1}{e^{t/2}-1}\right)$
$\displaystyle=2\log\left(\frac{(e^{t}+1)^{2}}{2e^{t}}+\sqrt{\left(\frac{(e^{t}+1)^{2}}{2e^{t}}\right)^{2}+1}\right)$

In both two cases, let $T=\frac{(e^{t}+1)^{2}}{2e^{t}}>2$ we have

$\ell(\Gamma)\geqslant\ell(\beta^{\prime\prime})+\ell(\Gamma\setminus\beta^{\prime\prime})\geqslant
2w_{1}(\ell(\beta))+\ell(\Gamma\setminus\beta^{\prime\prime})\geqslant H$

where

$\displaystyle H=$ $\displaystyle
2\log\frac{e^{t}+1}{e^{t}-1}+2\log\left(\frac{(e^{t}+1)^{2}}{2e^{t}}+\sqrt{\left(\frac{(e^{t}+1)^{2}}{2e^{t}}\right)^{2}+1}\right)$
$\displaystyle=$
$\displaystyle\log\frac{T}{T-2}+2\log\left(T+\sqrt{T^{2}+1}\right)$
$\displaystyle\frac{dH}{dT}=\frac{2}{\sqrt{T^{2}+1}}-\frac{2}{T(T-2)}$

There exists $T_{0}>2$ such that when $2\leqslant T\leqslant T_{0}$,
$\frac{dH}{dT}\leqslant 0$, and when $T\geqslant T_{0}$,
$\frac{dH}{dT}\geqslant 0$. When $T=3$ $\frac{dH}{dT}<0$, and when
$T=\frac{25}{8}$ $\frac{dH}{dT}>0$, hence $3<T_{0}<\frac{25}{8}$. Hence for
$T>2$ we have a contradiction by

$H(T)\geqslant
H(T_{0})>\log\frac{25}{9}+2\log\left(3+\sqrt{10}\right)>2\log(5+2\sqrt{6})$

∎

###  5.2. Case 2: $\ell(\gamma)<1.06$

The idea of the proof is to find the homotopy type of $\Gamma$ and prove that
$\Gamma$ is in a pair of pants, then we use the conclusions in Section 3 to
finish the proof.

######  Theorem 5.2.

If $\ell(\gamma)<1.06$ and $\ell(\Gamma)<2\log(5+2\sqrt{6})$, then we can find
a pair of pants $P\subseteq\Sigma$, each boundary component is a simple closed
geodesic or a cusp, and $\beta$ is one of the boundary components.

###### Proof.

Notations as before, let $\beta$ be the simple closed geodesic or cusp free
homotopy to $\gamma$, choose the generalized collar $N_{1}(\beta)$ such that
$\gamma$ and the bigger halfcollar $N^{\prime}(\beta)$ is on the same side of
$\beta$, since $\ell(\gamma)<1.06$, $\gamma\subseteq N^{\prime}(\beta)$. Let
$\widetilde{\gamma}$ be the arc in $\Gamma\cap N_{1}(\beta)$ containing
$\gamma$. Assume $|\widetilde{\gamma}\cap\widetilde{\gamma}|=k$, and the
endpoints of $\widetilde{\gamma}$ are $y_{1},y_{2}\in\partial N_{1}(\beta)$.
Let $\alpha\in(k,k+1]$ be the winding number of $\widetilde{\gamma}$, and
assume
$\widetilde{\gamma}\cap\widetilde{\gamma}=\\{P=P_{1},P_{2},...,P_{k}\\}$, and
in $N_{1}(\beta)$, $d(P_{i},\beta)$(or $d(P_{i},h(\beta))$ when $\beta$ is a
cusp and $h(\beta)$ is a sufficiently small horocycle) is increasing on
$i\in\\{1,...,k\\}$. Let $\zeta\subseteq N_{1}(\beta)$ be the unique simple
geodesic with endpoints $y_{1},y_{2}$ and homotopy to the curve
$\widetilde{\zeta}$ starting from $y_{1}$ and goes along $\widetilde{\gamma}$
to $P_{k}$, then goes along $\widetilde{\gamma}$ from $P_{k}$ to $y_{2}$.
$\zeta$ has winding number $\alpha-k\in(0,1]$.

  1. (1)

If $\beta$ is a closed geodesic, assume $\ell(\beta)=2t>0$, let

$H_{1}(s,t)=\sinh^{-1}\left(\sinh{(st)}\cosh(w_{1}(2t))\right)$

Let $u=2(\cosh\frac{t}{2})^{2}$ we have

$\displaystyle\ell(\widetilde{\gamma})-\ell(\zeta)=2H_{1}(k+\alpha,t)-2H_{1}(\alpha,t)$
$\displaystyle\geqslant$ $\displaystyle
2H_{1}(1+\alpha,t)-2H_{1}(\alpha,t)\geqslant 2H_{1}(2,t)-2H_{1}(1,t)$
$\displaystyle=$ $\displaystyle
2\sinh^{-1}\left(\sinh{2t}\cosh(w_{1}(2t))\right)-2\sinh^{-1}\left(\sinh{t}\cosh(w_{1}(2t))\right)$
$\displaystyle=$ $\displaystyle 2\sinh^{-1}(u(2u-2))-2\sinh^{-1}(u)\geqslant
2\sinh^{-1}(2u)-2\sinh^{-1}u$ $\displaystyle\geqslant$ $\displaystyle
2\sinh^{-1}4-2\sinh^{-1}2>1.06$

The second inequality using the fact that $H_{1}(\alpha,t)$ is a concave
function on $\alpha$ by taking second derivate of $\alpha$. The third
inequality holds since $2u-2>2$ when $u>2$. The fourth inequality using
$2\sinh^{-1}(2u)-2\sinh^{-1}u$ is an increasing function on $u>2$. Hence we
have

$\ell(\Gamma\setminus\widetilde{\gamma})+\ell(\zeta)<4\log(1+\sqrt{2})$

The closed curve $(\Gamma\setminus\widetilde{\gamma})\cup\zeta$ is freely
homotopic to a multiple of a simple closed geodesic, say $\beta^{\prime}$, and
so does $(\Gamma\setminus\widetilde{\gamma})\cup\widetilde{\zeta}$. Hence if
$\beta\cap\beta^{\prime}=\emptyset$, $\Gamma$ is freely homotopic to a curve
starting at $P_{k}$, winding around $\beta$ finitely many times, then goes to
$\beta^{\prime}$ then winding around finitely many times, finally goes back to
$P_{k}$. Similar as case (1) in the proof of Theorem 5.1, the theorem holds.
If $\beta\cap\beta^{\prime}\neq\emptyset$ then same as case (2) in Theorem 5.1
to finish the proof.

  2. (2)

If $\beta$ is a cusp, similar as the previous case, let
$H_{1}(\alpha,t)=\log(2\alpha+\sqrt{4\alpha^{2}+1})$ and
$\ell(\widetilde{\gamma})-\ell(\zeta)=2H_{1}(k+\alpha,t)-2H_{1}(\alpha,t)$
also holds, similarly we can prove the case.

∎

###### Proof of Theorem 1.3.

If $\ell(\gamma)\geqslant 1.06$, then Theroem 5.1 implies that
$\ell(\Gamma)\geqslant 2\log(5+2\sqrt{6})$. If $\ell(\gamma)<1.06$, then
Theroem 5.2 implies that $\Gamma$ lies in a pair of pants and using Corollary
3 we get the conclusion. ∎

## References

  * [1] Colin C Adams.  Maximal cusps, collars, and systoles in hyperbolic surfaces.  _Indiana University Mathematics Journal_ , Volume 47, Issue 2, 419–437, 1998.
  * [2] Claire M Baribaud.  Closed geodesics on pairs of pants.  _Israel Journal of Mathematics_ , Volume 109, 339–347, 1999.
  * [3] Ara Basmajian.  The stable neighborhood theorem and lengths of closed geodesics.  _Proceedings of the American Mathematical Society_ , Volume 119, Number 1, 217–224, 1993.
  * [4] Ara Basmajian.  Universal length bounds for non-simple closed geodesics on hyperbolic surfaces.  _Journal of Topology_ , Volume 6, Issue 2, 513–524, 2013.
  * [5] Viveka Erlandsson and Hugo Parlier.  Short closed geodesics with self-intersections.  _Mathematical Proceedings of the Cambridge Philosophical Society_ , Volume 169, Issue 3, 623–638, 2020.
  * [6] Benson Farb and Dan Margalit.  A Primer on Mapping Class Groups.  _Princeton Unviersity Press_ , 2012.
  * [7] John Hempel.  Traces, lengths, and simplicity for loops on surfaces.  _Topology and its Applications_ , Volume 18, Issue 2-3, 153–161, 1984.
  * [8] Akira Yamada.  On Marden’s universal constant of Fuchsian groups.  _Kodai Mathematical Journal_ , Volume 4, Issue 2, 266–277, 1981.
  * [9] Akira Yamada.  On Marden’s universal constant of Fuchsian groups, II.  _Journal d’Analyse Mathèmatique_ , Volume 41, 234–248, 1982.
  * [10] Wujie Shen and Jiajun Wang.  Minimal length of nonsimple closed geodesics in hyperbolic surfaces,  arXiv:2207.08360v1.
  * [11] Ara Basmajian, Hugo Parlier and Hanh Vo.  The shortest non-simple closed geodesics on hyperbolic surfaces,  arXiv:2210.12966v2, preprint.
  * [12] Hanh Vo.  Short closed geodesics on cusped hyperbolic surfaces,  _Pacific Journal of Mathematics_ , Volume 318, 127-151, 2022.
  * [13] Wujie Shen.  Nonsimple closed geodesics with given intersection number on hyperbolic surfaces,  arXiv2305.00638, preprint.


The reproducing kernel Hilbert space $H$ is a subspace of
$L^{2}(\mathcal{X},\mathbb{P})$.

  2. 2.

The $\mathcal{H}$ norm $\|\cdot\|_{\mathcal{H}}$ is stronger than infinity
norm.

  3. 3.

Both $\Sigma,\Sigma_{n}$ are Hilbert-Schmidt from $\mathcal{H}$ to
$\mathcal{H}$.

  4. 4.

All eigenvalues of $\Sigma$ (counting multiplicity) can be arranged in a
decreasing (possibly infinite) sequence of non-negative real numbers
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{K}>\lambda_{K+1}\geq\ldots\geq
0$ with a positive gap between $\lambda_{K}$ and $\lambda_{K+1}$.

  5. 5.

The top $K$ eigenfunctions $\\{f_{i}\\}_{i=1}^{K}\subset\mathcal{H}$ and can
be picked to form an orthonormal set of functions in
$L^{2}(\mathcal{X},\mathbb{P})$.

  6. 6.

$\Sigma_{n}$ has a sequence of non-increasing, real, non-negative eigenvalues.

###### Proof.

Recall that the general conditions entail that
$M:=\sqrt{\sup_{x\in\mathcal{X}}k(x,x)}<\infty$. The kernel $k(\cdot,\cdot)$
is therefore a Mercer’s kernel, which satisfies the condition
$\int_{\mathcal{X}\times\mathcal{X}}k^{2}(x,y)d\mathbb{P}(x)d\mathbb{P}(y)<\infty$.
Then item 1 is an implication of Mercer’s theorem.

For item 2, for any $f\in\mathcal{H}$, by the reproducing property and Cauchy-
Schwarz inequality, we have $|f(x)|^{2}=\langle
f,k(\cdot,x)\rangle_{\mathcal{H}}^{2}\leq|k(x,x)|^{2}\|f\|_{\mathcal{H}}^{2}\leq
M^{2}\|f\|_{\mathcal{H}}^{2}$. Therefore $\|f\|_{\infty}\leq
M\|f\|_{\mathcal{H}}$.

Item 3 has been checked in the intermediate step of proof of Lemma B.3.

Both item 4 and item 5 are ensured by Mercer’s theorem.

Item 6 is true because of the relationship between the spectrum of
$\Sigma_{n}$ and that of the symmetric positive semi-definite kernel matrix
$K_{n}$, which has been checked in Lemma B.1. ∎

###  B.5 Part four: putting all ingredients together

To complete the proof of Theorem B.2, we have to deal with the error induced
by operator $\widehat{V}_{1}$ only having orthonormal columns in
$L^{2}(\mathcal{X},\mathbb{P}_{n})$ but not in
$L^{2}(\mathcal{X},\mathbb{P})$. This can be accomplished using the same trick
as part five of the proof of Theorem 3.2. So we omit the technical redundance
here.