# Experimental Implementation of the Fractional Vortex Hilbert’s Hotel

Xi Chen  Hubei Key Laboratory of Optical Information and Pattern Recognition,
Wuhan Institute of Technology, Wuhan 430205, China  Shun Wang  Hubei Key
Laboratory of Optical Information and Pattern Recognition, Wuhan Institute of
Technology, Wuhan 430205, China  Chenglong You<EMAIL_ADDRESS>Quantum
Photonics Laboratory, Department of Physics & Astronomy, Louisiana State
University, Baton Rouge, LA 70803, USA  Omar S. Magaña-Loaiza
<EMAIL_ADDRESS>Quantum Photonics Laboratory, Department of Physics &
Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA  Rui-Bo Jin
<EMAIL_ADDRESS>Hubei Key Laboratory of Optical Information and Pattern
Recognition, Wuhan Institute of Technology, Wuhan 430205, China  Guangdong
Provincial Key Laboratory of Quantum Science and Engineering, Southern
University of Science and Technology, Shenzhen 518055, China

###### Abstract

The Hilbert hotel is an old mathematical paradox about sets of infinite
numbers. This paradox deals with the accommodation of a new guest in a hotel
with an infinite number of occupied rooms. Over the past decade, there have
been many attempts to implement these ideas in photonic systems. In addition
to the fundamental interest that this paradox has attracted, this research is
motivated by the implications that the Hilbert hotel has for quantum
communication and sensing. In this work, we experimentally demonstrate the
fractional vortex Hilbert’s hotel scheme proposed by G. Gbur [Optica 3,
222-225 (2016)]. More specifically, we performed an interference experiment
using the fractional orbital angular momentum of light to verify the Hilbert’s
infinite hotel paradox. In our implementation, the reallocation of a guest in
new rooms is mapped to interference fringes that are controlled through the
topological charge of an optical beam.

††preprint: APS/123-QED

##  I Introduction

The origin of the Hilbert hotel can be traced back to the winter of 1924, when
David Hilbert gave a series of lectures on the infinity of mathematics and
physics at the University of Göttingen Kragh (2014). In these lectures, he
used a hotel as an example to illustrate the difference between finite and
infinite countable sets. He considered a situation in which a full hotel with
a limited number of rooms cannot host new residents. Following this reasoning,
he pointed out that a full hotel with an unlimited number of rooms can still
host new visitors if every current guest in the hotel is asked to move up one
room. This process would lead to a new vacant room, which can be
mathematically described with the formula $\infty+1\leftrightarrow\infty$. As
such, this scheme can also be generalized to accommodate any number of guests.
Nowadays, this thought experiment is known as the Hilbert’s infinite hotel
paradox Kragh (2014). Remarkably, this paradoxical tale has triggered debate
and research in other fields of science including cosmology, philosophy, and
theology Hedrick (2014); Loke (2014).

The Hilbert’s hotel paradox has been used to explore the weirdness of
infinity, a subject of utmost importance in mathematics. Interestingly,
quantum mechanical systems have served as a relevant platform to explore and
model the counter-intuitive aspects of this paradox. For example, cavity
quantum electrodynamics (QED) platforms have been used to conduct experimental
investigations of this paradox Oi  _et al._ (2013). Specifically, Oi and co-
workers experimentally proved that the Hilbert’s hotel paradox can be emulated
using a cavity QED system with an infinite number of modes. In this case, all
the quantum amplitudes were moved up by one level, leaving an unoccupied
vacuum state that corresponds to the model of $\infty+1\leftrightarrow\infty$
Oi  _et al._ (2013). In addition, the orbital angular momentum (OAM) of light
has been used to perform experimental implementations of the Hilbert hotel
paradox Potoček  _et al._ (2015). In this regard, Berry predicted the
possibility of generating an infinite number of vortex pairs by passing a beam
through a half-integer spiral phase plate Berry (2004). This prediction was
experimentally verified by Leach and colleagues who generated an alternating
vortex streamline that confirmed the presence of an infinite number of OAM
modes Leach  _et al._ (2004). However, no transitions in-between the half-
integer topological charge was reported. Therefore, one cannot relate this
experiment with the verification of the Hilbert’s Hotel paradox. Motivated by
these ideas, Potoček and colleagues experimentally investigated the Hilbert’s
hotel paradox by mapping an OAM state to three times the original quantum
number Potoček  _et al._ (2015). More recently, Gbur identified the
possibility of exploiting the propagation of optical beams through fractional
vortex plates to perform a direct implementation of the Hilbert’s paradox Gbur
(2016). The difference between Potoček’s scheme and Gbur’s proposal resides on
the mechanisms used to accommodate a new guest. It is worth mentioning that
the work by Potoček utilizes a quantum state mapping function to represent the
room and the guest. In contrast, in Gbur’s scheme, the room and the guest are
represented by optical vortices with opposite topological numbers. However,
Gbur’s scheme has not yet been experimentally verified due to the challenges
of performing direct measurements of the phase distribution at different
spatial locations Magaña Loaiza  _et al._ (2014); Mirhosseini  _et al._
(2016); Wang and Gbur (2017); Noce (2020); Magaña-Loaiza and Boyd (2019); Lv
_et al._ (2022).

Figure 1: The simulated phase distribution of a Gaussian beam after adding a
phase. The simulation is performed according to Eq. (2). The corresponding
topological charge $L$ in (a-e) are 1, 1.44, 1.5, 1.53, and 1.7, respectively.
The “+” sign (blue to red, counterclockwise, positive vortex) indicates a
room, whereas the “-” sign (blue to red, clockwise, reverse vortex) represents
a guest (blue=0, red=2$\pi$). In the panels (f), (g), (h), we illustrate the
vortex dynamics simulated in (c), (d), (e).

Here, we designed an interference experiment using fractional vortex light to
verify the Hilbert hotel paradox. Our apparatus enables us to demonstrate the
first experimental implementation of the fractional vortex Hilbert’s hotel
proposed by Gbur Gbur (2016). By changing the topological charge $L$ from 1.5
to 1.7 in our experiment, we produce different interference structures that
enable us to move a guest one room up in the hotel.

##  II Theory

Following Gbur’s proposal Gbur (2016), we estimate the propagated optical
field reflected by a spatial light modulator. For this purpose, we consider a
fractional spiral phase $t(\phi)$ encoded on the spatial light modulator
(SLM),

$t(\phi)=e^{i\mu\phi},$ (1)

here $\mu$ represents the topological charge, which can be any rational number
(fractional or integer), and $\phi$ is the azimuth angle. Following the
calculation reported in the Supplementary Material Gbur (2017); Gradshteyn and
Ryzhik (2007); Zhao  _et al._ (2012); Li  _et al._ (2015); Wang  _et al._
(2019), it can be shown that the light field at the prorogation distance $z$
and radial coordinate $\rho$ is given by

$U_{\mu}(\rho,\phi,z)=\frac{{\exp\left[{i\pi\mu}\right]\sin(\pi\mu)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{{U_{n}\left({\rho,\phi,z}\right)}}{{(\mu-n)}}},$
(2)

where,

$\displaystyle
U_{n}\left({\rho,\phi,z}\right)=\sqrt{\frac{\pi}{8}}\exp\left(ikz\right)\exp\left(in\phi\right)\exp\left({\frac{ik\rho^{2}}{4z}}\right)$
(3)
$\displaystyle\times\left({-i}\right)^{\frac{n}{2}}\sqrt{\frac{k\rho^{2}}{z}}\times\left[J_{\frac{n-1}{2}}\left(\frac{k\rho^{2}}{4z}\right)-iJ_{\frac{n+1}{2}}\left(\frac{k\rho^{2}}{4z}\right)\right].$

In this case, $U_{n}\left({\rho,\phi,z}\right)$ is the propagation field of an
integer-order vortex beam, $n$ is an arbitrary integer associated to
topological charge, $J_{n}(x)$ is the Bessel function of the first kind,
$\rho$ is the polar position, and $k$ is the wave number. These parameters
enable one to define the distribution field of $U_{\mu}(\rho,\phi,z)$. This
can be characterized by interfering the field with a plane wave
$E\left({\rho,\phi,z}\right)$ Bazhenov  _et al._ (1992); Basistiy  _et al._
(1993); Liu  _et al._ (2019). The intensity distribution associated with the
interference structure can be expressed as

$I=\left|{U_{\mu}(\rho,\phi,z)+E\left({\rho,\phi,z}\right)}\right|^{2},$ (4)

where

$E(\rho,\phi,z)=A_{0}\exp[-ik\rho\cos\phi]\exp[-\frac{{\rho^{2}-\beta\rho\cos\phi}}{{w^{2}(z)}}].$
(5)

The parameter $A_{0}$ describes the amplitude of the plane wave. Furthermore,
$\exp[-\frac{{\rho^{2}-\beta\rho\cos\phi}}{{w^{2}(z)}}]$ defines the range of
the plane wave, $w(z)$ corresponds to the radius of the beam at position $z$,
and $\beta$ represents a constant related to a horizontal shift.

##  III Simulation

Figure 2: The experimental setup. A laser beam at a wavelength of 632.8 nm is
expanded by two lenses (the focal lengths of Lens 1 and Lens 2 are 30 mm and
200 mm, respectively) to achieve a diameter of 6 mm. Then, the beam passes
through a polarization beam splitter (PBS) to purify the polarization, and
injects into a Michelson interferometer, which is composed of a beam splitter
(BS), a plane mirror, and a spatial light modulator (SLM). The fork-shaped
fringe is loaded onto the SLM to prepare the desired vortex beam. The
resulting interference fringes are recorded by a charge-coupled device (CCD)
camera.

Using Eq. (2), we can simulate the propagation of a plane wave reflected by an
SLM. Figure 1 reports the predictions from our simulation for $\lambda$ =
623.8 nm and $z$ = 0.1 m. Fig. 1(a) corresponds to a situation in which $L=1$.
In this case, we produce a field distribution containing an integer vortex
with a phase from 0 to 2$\pi$ rotating clockwise. Fig. 1(b) shows our results
for $L=1.44$. This situation produces a fractional vortex field distribution
with a gap located to the right. Interestingly, there is a new pair of
vortices formed on the far left of the gap. One rotates clockwise, marked as
“+”, whereas the other shows a counterclockwise rotation, marked as “-”.

As the value of $L$ increases, the vortex pairs in the opening increase. When
$L=1.5$, as shown in Fig. 1(c), the opening area extends to infinity, and an
infinite number of vortex pairs are generated. This particular situation
enables the direct implementation of the Hilbert’s infinite hotel model.
Namely, a hotel with an unlimited number of rooms, where the rooms are fully
occupied by guests. As illustrated in Fig. 1(f), each “+” corresponds to a
room, whereas each “-” represents a guest. Thus, room #1 and guest #1 are
generated together. In general, room #N and guest #N are generated together.
Consequently, there are infinite pairs of guests and rooms. This is equivalent
to a one-to-one correspondence between two infinite number sets, namely
$\infty\leftrightarrow\infty$.

As the value of $L$ continues to increase, for example, from $L=1.5$ to
$1.55$, it is possible to observe that the vortex begins to annihilate at the
infinity position. Specifically, the “-” vortex in the previous pair will
continuously move to the position of the “+” vortex in the back pair. As a
result, the $N^{th}$ negative vortex and the $(N+1)^{th}$ positive vortex are
gradually connected, overlapped, and finally annihilated. Fig. 1(d) shows the
situation when $L=1.53$, in this case, the negative vortex is moving close to
the positive vortex. For the first pair of vortices, there is still some
distance between the negative and positive vortices, i.e., there is no
annihilation. However, other infinite vortex pairs have been overlapped and
annihilated. This process corresponds to the model in the Hilbert hotel
paradox shown in Fig. 1(g). Interestingly, each guest (positive vortex) moves
one step to the next room (negative vortex). Here, guest #1 moves into room
#2, guest #2 moves into room #3, and , guest #N moves into room #(N+1), while
room #1 on the far left is vacant. In this case, the vortex dynamics
represents a one-to-one correspondence between an infinite number set plus one
number and an infinite number set, namely $\infty+1\leftrightarrow\infty$.

As $L$ increases, the “-” vortex in the first pair continues to move to the
position of the “+” vortex in the second pair. Figure 1(e) shows the case in
which $L=1.7$. Under this condition, the positive and negative vortices on the
right side have completely overlapped and annihilated. Only an empty room (the
positive vortex) remains. The corresponding relationship of
$\infty+1\leftrightarrow\infty$ is also illustrated in Fig. 1(h).

##  IV Experiment and Results

Figure 3: Experimental interference fringes obtained by using different values
of $L$. The upward-opening-fork shape pattern (red) represents the room, which
is obtained by the interference with the positive vortex. The downward-
opening-fork shape structure (green) represents the guest, which is obtained
by interference with the negative vortex. The tilted blue lines represent
tilted interference fringes formed after the overlap and annihilation of
positive and negative vortices.

We have simulated the phase distribution in Fig. 1 according to Gbur’s theory.
However, it is challenging to perform a direct observation of the phase
patterns in the experiment Zhang  _et al._ (2021); Kotlyar  _et al._ (2020).
For this reason, we characterize the phase distribution by performing an
interference experiment as indicated in Fig. 2 Jesus-Silva  _et al._ (2012);
Wen  _et al._ (2019, 2020); Matta  _et al._ (2020); Kotlyar  _et al._ (2020).
We expanded a laser beam at a wavelength of 632.8 nm by using two lenses
(focal length L1 = 30 mm; L2 = 200 mm) to achieve a diameter of 6 mm. Then,
the beam passes through a polarization beam splitter (PBS) that we use to
control the initial polarization state. This beam is then injected into a
Michelson interferometer, which is composed of a beam splitter (BS), a plane
mirror, and a spatial light modulator (SLM, Holoeye, Pluto-2-NIR-011). The
fork-shaped fringe is loaded on the SLM to prepare the desired vortex beam.
Finally, the beam reflected by the SLM interferes with a beam reflected by the
plane mirror. In our experiment, we deliberately displaced our interfering
beams to observe denser interference patterns. The resulting interference
fringes are recorded by a CCD camera (Newport, laser beam profiler, LBP2-HR-
VIS2). In Fig. 3, we report the experimental interference fringes.

The experimental fork-shaped interference fringes in Fig. 3 correspond to the
simulated phase vortex shown in Fig. 1. In particular, Fig. 3(a) reports
experimental interference fringes measured for $L=1$. The original positive
vortex in Fig. 1(a) changed to an upward-opening-fork shape pattern.
Remarkably, more pairs of downward and upward forks are formed as one
increases $L$. Fig. 3(b) shows the interference fringes for $L=1.44$.
Similarly, the case for $L=1.5$ is shown in Fig. 3(c) and illustrated in Fig.
3(f). These panels show countless upward (red) and downward (green) forked
patterns, which are alternately distributed. In this case, we can use the
bright upward-opening-fork shape pattern to represent the room, and the bright
downward-fork pattern to represent the guest. This is equivalent to the
situation of the Hilbert’s infinite hotel model. Indeed, there is a one-to-one
correspondence between an infinite number of rooms and an infinite number of
guests, namely, $\infty\leftrightarrow\infty$.

Figure 3(d) and (g) show the case for $L=1.53$. This condition forces the
downward forks (guest) to move close to the upward forks (room). The downward
and the upward-fork patterns slowly overlap to form a long tilted stripe (in
blue). Fig. 3(e) and (h) show the case in which $L=1.7$. Here, guest
(downward-fork pattern) #1 moves into the room (upward-fork pattern) #2, guest
#2 moves into the room #3, and, consequently, guest #N moves into the room
#(N+1), leaving room #1 vacant. This relationship can be described as
$\infty+1\leftrightarrow\infty$. For sake of completeness, we also performed a
theoretical simulation of the interference fringes, see Supplementary
Materials S1(c). It can be seen that the experimental interferogram is in
agreement with the simulated interferogram.

##  V Discussions

Our work reveals that the counter-intuitive concepts associated with infinity
can be explored in optical platforms. Furthermore, in the theoretical scheme
proposed by Gbur in ref. Gbur (2016), the Hilbert’s infinite hotel paradox was
studied using $L=4.5$. However, our experiment was carried out using $L=1.5$.
As such, it should be emphasized that both values enable demonstrating the
correspondence $\infty+1\leftrightarrow\infty$. The difference resides in the
fact that the interference patterns produced for $L=1.5$ are simpler for
experimental observations.

In our experiment, we explored the case in which
$\infty+1\leftrightarrow\infty$. However, this platform can be used to
implement the Hilbert’s infinite hotel paradox for different correspondences
such as $\infty+N\leftrightarrow\infty$, or $2\infty\leftrightarrow\infty$.
Recently, new theoretical schemes for the Hilbert’s hotel paradox were
proposed Wang and Gbur (2017); Noce (2020), notably, these schemes may also be
experimentally demonstrated using our approach.

##  VI Conclusion

In conclusion, we demonstrated the first experimental implementation of the
fractional vortex Hilbert’s hotel. In contrast to previous platforms, our
scheme enables a direct implementation of the Hilbert’s infinite hotel
paradox. We believe that our experiment can be used to investigate other
counter-intuitive effects associated to infinity.

## Acknowledgments

We thank Dr. Le Wang (Nanjing University of Posts and Telecommunications) and
Dr. Shi-Long Liu (The University of Ottawa) for the helpful discussions. This
work was supported by the National Natural Science Foundations of China (Grant
Numbers 91836102, 12074299, and 11704290) and by the Guangdong Provincial Key
Laboratory (Grant No. GKLQSE202102). C.Y. and O.S.M.L. acknowledge support
from Cisco.

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##  VII Supplementary Information

## S1: Calculation of Eq.(3)

In this section, we show the derivation of Eq. (3) in the main text.

### S1-1. Calculation of $A_{n}(\kappa_{x},\kappa_{y})$ at the position of
$z=0$

In the space domain, at the position of $z=0$, assume there is a light wave
with the distribution of $U(x,y)$ on the $xoy$ plane.

$U(x,y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A(\kappa_{x},\kappa_{y})e^{i(\kappa_{x}x+\kappa_{y}y)}d\kappa_{x}d\kappa_{y},$
(S1)

where $A(\kappa_{x},\kappa_{y})$ is spatial frequency distribution and
$\kappa$ is the spatial frequency. The inverse Fourier transformation of
$U(x,y)$ is

$A(\kappa_{x},\kappa_{y})=\frac{1}{{(2\pi)^{2}}}\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{}}U(x,y)e^{-i(\kappa_{x}x+\kappa_{y}y)}dxdy.$
(S2)

We can change the above formula from a Cartesian coordinate system to a polar
coordinate system using $x=\rho\cos\varphi$, $y=\rho\sin\varphi$,
$\kappa_{x}=\kappa\cos\phi_{\kappa}$ , $\kappa_{y}=\kappa\sin\phi_{\kappa}$
and obtain

$A(\kappa,\phi_{\kappa})=\frac{1}{{(2\pi)^{2}}}\int_{0}^{\infty}{\int_{0}^{2\pi}\rho
U(\rho,\varphi)}e^{-i\kappa\rho\cos(\phi-\phi_{\kappa})}d\rho d\phi.$ (S3)

Note in the space domain, $\rho=\sqrt{x^{2}+y^{2}}$ and $\phi$ is the polar
angle. In the frequency domain, $\kappa=\sqrt{\kappa_{x}^{2}+\kappa_{y}^{2}}$
and $\phi_{\kappa}$ is the polar angle. For simplicity in the calculation, we
assume $U(\rho,\varphi)=1$, i.e., this is a plane wave with each frequency
having an equal amplitude of 1.

$A(\kappa,\phi_{\kappa})=\frac{1}{{(2\pi)^{2}}}\int_{0}^{\infty}{\int_{0}^{2\pi}\rho}e^{-i\kappa\rho\cos(\phi-\phi_{\kappa})}d\rho
d\phi.$ (S4)

In the space domain, let assume there is a spatial light modulator (SLM) (or
other phase component, such as a spiral phase plate) with a transfer function
of

$t(\phi)=e^{i\alpha\phi},$ (S5)

where $\phi$ is the azimuth angle; $\alpha$ is the phase delay produced by the
SLM, and $\mu$ can be an integer or a fraction. When $n$ is an integer, we
get:

$t(\phi)=e^{in\phi},$ (S6)

If the phase $e^{in\phi}$ is added to the plane wave, we obtain

$A_{n}(\kappa,\phi_{\kappa})=\frac{1}{{(2\pi)^{2}}}\int_{0}^{\infty}{\int_{0}^{2\pi}\rho}e^{-i\kappa\rho\cos(\phi-\phi_{\kappa})}e^{in\phi}d\rho
d\phi.$ (S7)

Let $\phi-\phi_{\kappa}=\phi^{\prime}$, and

$A_{n}(\kappa,\phi_{\kappa})=\frac{1}{{2\pi}}e^{in\phi_{\kappa}}\int_{0}^{\infty}{\rho\frac{1}{{2\pi}}\int_{0}^{2\pi}{e^{-i\kappa\rho\cos\phi^{\prime}}}}e^{in\phi^{\prime}}d\rho
d\phi^{\prime}.$ (S8)

Considering the standard form of m-order Bessel function:

$J_{m}\left(x\right)=\frac{1}{{2\pi
i^{m}}}\int_{0}^{2\pi}{e^{-im\phi}}e^{ix\cos\left(\phi\right)}d\phi,$ (S9)

Eq. (S8) can be rewritten as

$A_{n}(\kappa,\phi_{\kappa})=\frac{{i^{\left|n\right|}}}{{2\pi}}e^{in\phi_{\kappa}}\int_{0}^{\infty}{J_{\left|n\right|}\left({\kappa\rho}\right)}\rho
d\rho.$ (S10)

Using the following equation from the standard integral tables (Eq. (14) of
Section 6.561 in Ref. Gradshteyn and Ryzhik (2007), also see Ref. Gbur (2017)

$\int_{0}^{\infty}{{\rm{x}}^{\mu}J_{\nu}}\left({\alpha
x}\right)dx=2^{\mu}\alpha^{-\mu-1}\frac{{\Gamma\left({\frac{1}{2}+\frac{1}{2}\nu+\frac{1}{2}\mu}\right)}}{{\Gamma\left({\frac{1}{2}+\frac{1}{2}\nu-\frac{1}{2}\mu}\right)}},$
(S11)

we obtain

$A_{\rm{n}}\left({\kappa,\phi_{\kappa}}\right)=\frac{{i^{\left|n\right|}}}{{2\pi}}e^{in\phi_{\kappa}}2\cdot\kappa^{-2}\frac{{\Gamma\left({\frac{1}{2}+\frac{1}{2}n+\frac{1}{2}}\right)}}{{\Gamma\left({\frac{1}{2}+\frac{1}{2}n-\frac{1}{2}}\right)}}.$
(S12)

This equation can be further simplified as

$A_{n}\left({\kappa,\phi_{\kappa}}\right)=\frac{{i^{\left|n\right|}}}{{2\pi}}e^{in\phi_{\kappa}}2\cdot\kappa^{-2}\frac{{\Gamma\left({1+\frac{1}{2}n}\right)}}{{\Gamma\left({\frac{1}{2}n}\right)}}=\frac{{i^{\left|n\right|}}}{{2\pi}}e^{in\phi_{\kappa}}2\cdot\kappa^{-2}\frac{{\rm{n}}}{2}.$
(S13)

The integration result is:

$A_{n}(\kappa_{x},\kappa_{y})=\frac{{\left|n\right|i^{\left|n\right|}}}{{2\pi\kappa^{2}}}e^{in\phi_{\kappa}}.$
(S14)

Note, in the definition of Gamma function $\Gamma(x)$, $x>0$, therefore, we
can change $n$ to be $\left|n\right|$ in the above equation.

### S1-2. Calculation of $U_{n}\left({\rho,\phi,z}\right)$ at the position of
$z$

Next, we consider the distribution of the light field at the position of $z$.

$U_{\rm{z}}({\rm{x}},y)\equiv
U({\rm{x}},y,z)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}}A_{\rm{z}}(\kappa_{x},\kappa_{y})e^{i(\kappa_{x}x+\kappa_{y}y)}d\kappa_{x}d\kappa_{y},$
(S15)

where $A_{\rm{z}}$ is the distribution of $A$ at the position of $z$.
According to the solution of the Helmholtz equation,

$A_{\rm{z}}(\kappa_{x},\kappa_{y}){\rm{=}}A_{\rm{0}}(\kappa_{x},\kappa_{y})\exp(i\kappa_{z}z),$
(S16)

where $A_{\rm{0}}(\kappa_{x},\kappa_{y})\equiv A(\kappa_{x},\kappa_{y})$ is
the distribution of $A$ at the position of $0$. Therefore, the general
distribution of $U(x,y,z)$ is (Note, $U(x,y,0)=U(x,y)$)

$U(x,y,z)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}}A(\kappa_{x},\kappa_{y})e^{i(\kappa_{x}x+\kappa_{y}y+\kappa_{z}z)}d\kappa_{x}d\kappa_{y},$
(S17)

The above equation can also be intuitively understood as the diffraction from
U(x,y,0) to U(x,y,z). In the polar coordinate, the equation for the case of
$A_{n}$ can be written as:

$U_{n}\left({\rho,\phi,z}\right)=\int_{0}^{\infty}\int_{0}^{2\pi}{A_{n}}\left({\kappa_{x},\kappa_{y}}\right)e^{i\kappa\rho\cos\left({\phi_{\kappa}-\phi}\right)}e^{i\kappa_{z}z}\kappa
d\kappa d\phi_{\kappa}.$ (S18)

In order to simplify the calculation, we ignore the evanescent wave and only
consider the field propagating near the z-axis. So,

$\begin{array}[]{l}k^{2}=\kappa_{x}^{2}+\kappa_{y}^{2}+\kappa_{z}^{2}\\\
\kappa^{2}=\kappa_{x}^{2}+\kappa_{y}^{2}.\\\ \end{array}$ (S19)

Therefore,

$\kappa_{z}=\sqrt{{k}^{2}-\kappa^{2}}\approx k-\frac{{\kappa^{2}}}{{2k}}.$
(S20)

After substituting Eq.(S20) into Eq.(S18), we obtain:

$U_{n}\left({\rho,\phi,z}\right)\approx
e^{ikz}\int{A_{n}}\left({\kappa_{x},\kappa_{y}}\right)e^{i\kappa\rho\cos\left({\phi_{\kappa}-\phi}\right)}e^{-i\kappa^{2}z/2k}\kappa
d\kappa d\phi_{\kappa}.$ (S21)

After substituting Eq.(S14) and Eq.(S9) into Eq.(S21), we obtain

$U_{n}\left({\rho,\phi,z}\right)=e^{ikz}e^{in\phi}\left|n\right|\int_{0}^{\infty}{\frac{{J_{\left|n\right|}\left({\kappa\rho}\right)}}{\kappa}}e^{-i\kappa^{2}z/2k}d\kappa.$
(S22)

In order to simplify the calculation, we set $b=z/2k$ and substitute it into
the above formula to get:

$U_{n}\left({\rho,\phi,z}\right)=\frac{{\rho
e^{ikz}e^{in\phi}\left|n\right|}}{2}\int_{0}^{\infty}{\frac{{2J_{\left|n\right|}\left({\kappa\rho}\right)}}{{\kappa\rho}}}e^{-ib\kappa^{2}}d\kappa.$
(S23)

Utilize the standard Bessel equation

$\frac{{2nJ_{n}\left(x\right)}}{x}=J_{n-1}\left(x\right)+J_{n+1}\left(x\right),$
(S24)

we obtain

$U_{n}\left({\rho,\phi,z}\right)=\frac{{\rho
e^{ikz}e^{in\phi}}}{2}\int_{0}^{\infty}{\left[{J_{\left|n\right|-1}\left({\rho\kappa}\right)+J_{\left|n\right|+1}\left({\rho\kappa}\right)}\right]}e^{-ib\kappa^{2}}d\kappa.$
(S25)

Using the equation from the standard integral tables

$\int_{0}^{\infty}{{\rm{e}}^{-\alpha x^{2}}J_{\nu}}\left({\beta
x}\right)dx=\frac{1}{2}\sqrt{\frac{\pi}{\alpha}}e^{-\beta^{2}/8\alpha}I_{\nu/2}\left({\beta^{2}/8\alpha}\right),$
(S26)

where

$I_{m}\left({-ix}\right)=\left({-i}\right)^{m}J_{m}\left(x\right),$ (S27)

we achieve

$\begin{array}[]{lll}U_{n}\left({\rho,\phi,z}\right)&=&\frac{{1}}{2}\rho
e^{ikz}e^{in\phi}\left\\{{\int_{0}^{\infty}{\left[{J_{\left|n\right|-1}\left({\rho\kappa}\right)}\right]}e^{-ib\kappa^{2}}d\kappa{\rm{+}}\int_{0}^{\infty}{\left[{J_{\left|n\right|+1}\left({\rho\kappa}\right)}\right]}e^{-ib\kappa^{2}}d\kappa}\right\\}\\\
&=&\frac{{1}}{2}\rho e^{ikz}e^{in\phi}\sqrt{\frac{{\pi k}}{{i2z}}}\times
e^{ik\rho^{2}/4z}\left[{\left({-i}\right)^{\frac{{\left|n\right|-1}}{2}}J_{\frac{\left|n\right|-1}{2}}\left({\frac{k\rho^{2}}{4z}}\right)+\left({-i}\right)^{\frac{{\left|n\right|+1}}{2}}J_{\frac{\left|n\right|+1}{2}}\left({\frac{k\rho^{2}}{4z}}\right)}\right].\\\
\end{array}$ (S28)

After simplification, finally we obtain

$U_{n}\left({\rho,\phi,z}\right)=\sqrt{\frac{\pi}{8}}\exp\left(ikz\right)\exp\left(in\phi\right)\exp\left({\frac{ik\rho^{2}}{4z}}\right)\times\left({-i}\right)^{\frac{\left|n\right|}{2}}\sqrt{\frac{k\rho^{2}}{z}}\times\left[J_{\frac{\left|n\right|-1}{2}}\left(\frac{k\rho^{2}}{4z}\right)-iJ_{\frac{\left|n\right|+1}{2}}\left(\frac{k\rho^{2}}{4z}\right)\right].$
(S29)

If we only consider the case of positive integers, i.e. $\left|n\right|$ can
be changed to $n$, then, the above equation is just the Eq. (3) in the main
text.

## S2: Calculation of Eq. (2)

In this section, we show how to derive the Eq. (2) in the main text. Consider
the plane light passing through a fractional-order spiral phase plate. The
transfer function of the fractional-order spiral phase plate can be expanded
by an integer order, i.e.,

$\exp({i\alpha\phi})=\frac{{\exp({i\pi\alpha})\sin\left({\pi\alpha}\right)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{{\exp({in\phi})}}{{\phi-n}}}.$
(S30)

Use Fourier expansion to expand the above formula

$\exp({i\alpha\phi})=\sum\limits_{n=-\infty}^{\infty}{c_{k}}e^{in\phi},$ (S31)

where,

$\begin{array}[]{lll}c_{k}&=&\frac{1}{{2\pi}}\int\limits_{0}^{2\pi}{e^{i\alpha\phi}e^{-in\phi}}d\phi\\\
&=&\frac{1}{{2\pi}}\frac{1}{{i(\alpha-n)}}e^{i\phi(\alpha-n)}|_{0}^{2\pi}\\\
&=&\frac{1}{{2\pi}}\frac{1}{{i(\alpha-n)}}[e^{i2\pi(\alpha-n)}-1]\\\
&=&\frac{1}{{2\pi}}\frac{1}{{i(\alpha-n)}}{\rm{e}}^{i\pi(\alpha-n)}[e^{i\pi(\alpha-n)}-e^{-i\pi(\alpha-n)}]\\\
&=&\frac{1}{{2\pi}}\frac{1}{{i(\alpha-n)}}e^{i\pi\alpha}e^{-i\pi n}e^{i\pi
n}2i\sin(\pi\alpha)\\\
&=&\frac{{\sin(\pi\alpha)}}{\pi}\frac{{e^{i\pi\alpha}}}{{(\alpha-n)}}.\\\
\end{array}$ (S32)

Note, in the above calculation, $e^{-i\pi n}=\cos(-\pi n)+i\sin(-\pi
n)=\cos(\pi n)$, and $e^{i\pi n}=\cos(\pi n)+i\sin(\pi n)=\cos(\pi n)$, so
$e^{-i\pi n}=e^{i\pi n}$

Therefore,

$e^{i\alpha\phi}=\frac{{e^{i\pi\alpha}\sin(\pi\alpha)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{{e^{in\phi}}}{{(\alpha-n)}}}.$
(S33)

Following the similar procedure as before, we obtain

$\begin{array}[]{lll}A_{\alpha}(\kappa,\phi_{\kappa})&=&\frac{1}{{(2\pi)^{2}}}\int_{0}^{\infty}{\int_{0}^{2\pi}\rho}e^{-i\kappa\rho\cos(\phi-\phi_{\kappa})}e^{i\alpha\phi}d\rho
d\phi\\\
&=&\frac{1}{{(2\pi)^{2}}}\int_{0}^{\infty}{\int_{0}^{2\pi}\rho}e^{-i\kappa\rho\cos(\phi-\phi_{\kappa})}\frac{{\exp\left[{i\pi\alpha}\right]\sin\left({\pi\alpha}\right)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{{\exp\left[{in\phi}\right]}}{{\alpha-n}}}d\rho
d\phi\\\
&=&\frac{{\exp\left[{i\pi\alpha}\right]\sin\left({\pi\alpha}\right)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{1}{{\alpha-n}}}\frac{1}{{(2\pi)^{2}}}\int_{0}^{\infty}{\int_{0}^{2\pi}\rho}e^{-i\kappa\rho\cos(\phi-\phi_{\kappa})}\exp\left[{in\phi}\right]d\rho
d\phi\\\
&=&\frac{{\exp\left[{i\pi\alpha}\right]\sin\left({\pi\alpha}\right)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{1}{{\alpha-n}}}\times
A_{n}(\kappa_{x},\kappa_{y}),\\\ \end{array}$ (S34)

where,

$A_{n}(\kappa_{x},\kappa_{y})=\frac{{\left|n\right|i^{\left|n\right|}}}{{2\pi\kappa^{2}}}e^{in\phi_{\kappa}}.$
(S35)

Change from frequency domain to space domain,

$\begin{array}[]{lll}U_{\alpha}\left({\rho,\phi,z}\right)&=&\frac{{\exp\left[{i\pi\alpha}\right]\sin\left({\pi\alpha}\right)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{1}{{\alpha-n}}}\times
e^{ikz}\int{A_{n}}\left({\kappa_{x},\kappa_{y}}\right)e^{i\kappa\rho\cos\left({\phi_{\kappa}-\phi}\right)}e^{-i\kappa^{2}z/2k}\kappa
d\kappa d\phi_{\kappa}\\\
&=&\frac{{\exp\left[{i\pi\alpha}\right]\sin\left({\pi\alpha}\right)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{1}{{\alpha-n}}}\times
U_{n}\left({\rho,\phi,z}\right),\\\ \end{array}$ (S36)

Finally, we obtain

$U_{\alpha}(\rho,\phi,z)=\frac{{\exp\left[{i\pi\alpha}\right]\sin(\pi\alpha)}}{\pi}\sum\limits_{n=-\infty}^{\infty}{\frac{{U_{n}\left({\rho,\phi,z}\right)}}{{(\alpha-n)}}}.$
(S37)

This is just the Eq. (2) in the main text.

## S3: Calculation of Eq. (S20)

Here we derive Eq. (S20):

$\sqrt{k^{2}-\kappa^{2}}\approx k-\frac{{\kappa^{2}}}{{2k}}.$ (S38)

According to the Newton’s generalized binomial theorem:

$\begin{array}[]{lll}\left({x+y}\right)^{r}&=&\sum\limits_{k=0}^{\infty}{\left({\begin{array}[]{*{20}c}r\\\
k\\\ \end{array}}\right)}x^{r-k}y^{k}\\\
&=&x^{r}+rx^{r-1}y+\frac{{r(r-1)}}{{2!}}x^{r-2}y^{2}+\frac{{r(r-1)(r-2)}}{{3!}}x^{r-3}y^{3}+...,\\\
\end{array}$ (S39)

we obtain,

$\sqrt{k^{2}-\kappa^{2}}=(k^{2}-\kappa^{2})^{1/2}=\left({k^{2}}\right)^{1/2}+\frac{1}{2}\left({-k^{2}}\right)^{-1/2}\cdot\kappa^{2}+O\approx
k-\frac{{\kappa^{2}}}{{2k}}.$ (S40)

## S4: Comparison of the simulation and experimental results

In Fig. S1, we compare the phase simulation graphs by using Eq. (2), the
interference graph simulated by using Eq. (4), and the interference experiment
results. It can be seen that the theoretical simulation of interference in
Fig. S1(b) corresponds well to the experimental interference patterns in Fig.
S1(b) .

Figure S1:  Comparison of the numerical simulations and experimental results
at different L values of 1, 1.44, 1.5, 1.53, and 1.7: (a1-e1) The simulations
of phase distribution by using Eq. (2); (a2-e2) The simulations of
interference patterns by using Eq. (4); (a3-e3) The experimental interference
patterns.