# Channel Characterization of IRS-assisted Resonant Beam Communication Systems

Wen Fang, Wen Chen, , Qingqing Wu, , Xusheng Zhu,
Qiong Wu, , Nan Cheng,  W. Fang, W. Chen, Q. Wu, and X. Zhu are with the
Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai
200240, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>xushengzhu@sjtu.edu.cn). Q. Wu is with the School of
Internet of Things Engineering, Jiangnan University, Wuxi 214122, China
(e-mail: qiongwu@jiangnan.edu.cn). N. Cheng is with the State Key Lab. of ISN
and School of Telecommunications Engineering, Xidian University, Xi’an 710071,
China (e-mail: dr.nan.cheng@ieee.org).

###### Abstract

To meet the growing demand for data traffic, spectrum-rich optical wireless
communication (OWC) has emerged as a key technological driver for the
development of 6G. The resonant beam communication (RBC) system, which employs
spatially separated laser cavities as the transmitter and receiver, is a high-
speed OWC technology capable of self-alignment without tracking. However, its
transmission through the air is susceptible to losses caused by obstructions.
In this paper, we propose an intelligent reflecting surface (IRS) assisted RBC
system with the optical frequency doubling method, where the resonant beam in
frequency-fundamental and frequency-doubled is transmitted through both direct
line-of-sight (LoS) and IRS-assisted channels to maintain steady-state
oscillation and enable communication without echo-interference, respectively.
Then, we establish the channel model based on Fresnel diffraction theory under
the near-field optical propagation to analyze the transmission loss and
frequency-doubled power analytically. Furthermore, communication power can be
maximized by dynamically controlling the beam-splitting ratio between the two
channels according to the loss levels encountered over air. Numerical results
validate that the IRS-assisted channel can compensate for the losses in the
obstructed LoS channel and misaligned receivers, ensuring that communication
performance reaches an optimal value with dynamic ratio adjustments.

###### Index Terms:

Optical wireless communication, Resonant beam communication, Intelligent
reflecting surface, Near-field Fresnel diffraction

##  I Introduction

The wireless communication technology has undergone significant
transformations over the past few decades, evolving towards $6$G driven by the
ever-growing demand for higher data rates, enhanced reliability, and increased
capacity [1] [2]. The evolution from earlier radio communication to $5$G and
beyond has been marked by remarkable advancements in modulation schemes,
hardware capabilities, and spectrum utilization, which have collectively
enabled a myriad of applications, ranging from mobile internet access to the
Internet of Everything (IoE), connecting billions of devices worldwide [3].
However, as the number of connected devices continues to surge and data-
intensive applications like augmented reality (AR), virtual reality (VR), and
high-definition video streaming become more prevalent, the traditional radio
frequency (RF) communication systems face escalating challenges, including
spectrum scarcity, signal attenuation, and electromagnetic interference [4]
[5]. To address these limitations, alternative communication technologies are
being explored, with optical wireless communication (OWC) emerging as a
promising candidate benefiting from the spectrum-rich, spectrum license-free,
and electromagnetic interference-free [6].

The OWC leverages visible, infrared, and ultraviolet light for data
transmission, offering a compelling alternative to traditional RF
communication systems due to its high bandwidth, immunity to RF interference,
and enhanced security from the confined transmission channels [7]. The
existing OWC encompasses various technologies including visible light
communication (VLC) and free-space optical (FSO) communication, where the
former utilizes light-emitting diodes (LEDs) for both illumination and data
transmission, modulating light intensity at high speeds to transmit data
imperceptible to the human eye [8]. However, VLC data rates decrease rapidly
with extended coverage due to diminishing light intensity. FSO communication,
on the other hand, uses lasers to establish high-speed point-to-point links
over free space, providing gigabit data rates [9]. Nevertheless, FSO systems
require precise alignment of the transmitter and receiver, making installation
and maintenance more challenging and limiting the effective range [10].

To address the above limitations of VLC and FSO, a resonant beam-based OWC
system utilizing spatially separated laser cavities as the transmitter and
receiver, with retroreflectors installed at both ends, has been proposed in
[11] [12], allowing the beam to be collinearly reflected. Firstly, the
collinear properties of the retroreflector further facilitate self-alignment
without the need for tracking, thus enhancing automatic beamforming.
Additionally, due to resonant beams’ collimation properties, they exhibit
high-power and large-range characteristics like lasers. Besides, in a resonant
beam system (RBS), the gain medium in the transmitter (Tx) continuously
absorbs the energy from the pump source for amplification and excites the
resonant beam, which is then transmitted through space to the receiver (Rx)
[13]. This transmission over air is affected by various losses, including air
loss and diffraction loss. When the gain efficiency of excitation and
amplification compensates for these transmission losses, the resonant beam
transmission reaches a steady state, allowing stable information transmission
through the round-trip oscillation between the Tx and Rx [14]. However,
communication will automatically weaken or even be interrupted if the
transmission loss increases beyond a certain threshold, particularly when
external objects obstruct the RB channel. Consequently, the requirement for a
line-of-sight (LoS) link remains a significant limitation for the
applicability of RBS, similar to other OWC technologies.

Intelligent reflecting surfaces (IRS) represent a breakthbreakth technology
that enhances wireless communication by controlling the propagation
environment, providing a low-cost and easily deployable solution for achieving
coverage enhancement and eliminating blind spots [15]. An IRS is a planar
array of passive elements whose electromagnetic properties can be dynamically
adjusted to reflect incident signals in desired directions, which has recently
been applied to OWC systems to overcome some inherent limitations [16]. For
VLC, [17] provides a comprehensive tutorial on using IRSs in indoor VLC
systems to mitigate signal obstructions. It compares optical RISs with RF-RISs
and optical relays, and proposes solutions and research directions for
integrating optical RISs with emerging technologies to enhance VLC system
performance, including non-orthogonal multiple access (NOMA) and multiple-
input multiple-output (MIMO) systems. [18] and [19] introduce a novel approach
utilizing liquid crystal-based IRS-enabled VLC transmitters and receivers in
indoor VLC systems. The former demonstrates superior performance in data rate
and illumination compared to traditional VLC transmitters, while the latter
enhances signal strength and data rate by steering incident light and
amplifying intensity in the LC RIS-based receiver. [20] investigates the
temporal characteristics of IRS-based VLC channels, deriving impulse responses
and delay spreads for general and specific configurations, and provides exact,
bound, and asymptotic delay expressions. Besides, the communication
performance including spectral efficiency and sum rate is optimized in [21],
[22] and [23] by the joint resource management method and power-domain NOMA.
On the other hand, [24] investigates the use of IRSs to mitigate the LoS
requirement in FSO systems, designing a phase-shift distribution for IRSs to
control beam direction and developing a statistical model for geometric and
misalignment losses. [25] presents a practical approach to model and design an
IRS-assisted FSO communication system considering random misalignment, where a
geometric and misalignment loss (GML) model that accounts for the statistical
distribution of incidence and reflection angles is established for driving the
analytical closed-form expressions. A phase-shift design for IRSs to focus
beams in FSO systems is proposed in [26], where a new pointing error model and
an outage performance analysis with Monte Carlo simulations are conducted.
Then, [27], [28], and [29] study the co-channel interference, dual-hop effect,
and other communication performance aspects in IRS-assisted FSO-RF systems,
demonstrating significant performance enhancement compared to traditional
FSO/RF systems.

Therefore, to enhance the communication performance in the absence of LoS
channels, we propose using IRS to mitigate transmission losses over the
resonant cavity (i.e., the free space between the Tx and Rx) in a resonant
beam communication (RBC) system. Meanwhile, to address the issue of echo
interference caused by the round-trip propagation of the resonant beam, we
adopt an optical frequency doubling method, which maintains the excitation
process in the gain medium of the fundamental-frequency beam within the
resonant cavity, while using the frequency-doubled beam for information
loading and transmission. Moreover, in the RBC transmitter, the transmitted
power can be dynamically allocated between the direct LoS channel and the IRS-
assisted channel by controlling the beam-splitting ratio based on the loss
level over air, thereby ensuring that communication performance is maximized.
The main contributions of this work can be summarized as follows.

  * •

We propose the IRS-assisted RBC system design incorporating the optical
frequency doubling method, where the resonant beam can be transmitted in both
direct LoS and IRS-assisted channels. The frequency-doubled beam, generated by
second harmonic generation (SHG), is used for downlink communication, while
the fundamental-frequency beam is responsible for oscillations.

  * •

We establish an IRS-assisted RBC channel model using Fresnel diffraction under
near-field optical transmission conditions, where the channel gain is
influenced by the effective transfer aperture, transmission distance, and the
relative positions of the IRS and receivers.

  * •

Based on transmission loss caused by external object obstruction, receiver
misalignment, and changes in distance, the beam-splitting ratio in the Tx can
be dynamically adapted to maximize communication power. Numerical evaluation
reveals that channel capacity of RBC can be maximized using the ratio optimal
algorithm.

The remainder of this paper is organized as follows. The RBC system model and
signal models are presented in Section II. The transmission model in direct
LoS and IRS-assisted channels under near-filed propagation is established in
Section III. The power for the frequency-fundamental beam and frequency-
doubled beam, the channel capacity of the IRS-assisted RBC system are derived
in Section IV. Simulation results of communication performance with different
parameters are demonstrated in Section V. Finally, the conclusions are drawn
in Section VI.

Figure 1: The example system structure of IRS-assisted RBC system.

##  II System Model

In this section, we first introduce the system structure and work process of
the IRS-assisted RBC system. Then, we present the near-field characteristics
of resonant beam propagation. Next, we establish the signal model for resonant
beam propagation, considering free-space optical transmission.

###  II-A IRS-assisted RBC system

The system structure of the IRS-assisted RBC system, depicted in Fig. 1,
encompasses a spatially separated transmitter and receiver, constituting a
resonant cavity for free space transmission. The input reflector including
plane mirror M$1$ and confocal lens L$1$, second harmonic generation (SHG),
electro-optic modulator (EOM), beam splitter, and controller make up the
transmitter, while the beam combiner, output reflector including confocal lens
L$2$ and plane mirror M$2$, and detector are comprised of the receiver. The
photons, amplified by the pump source in the gain medium, propagate forth and
back through free space, reflect off the output reflector, and then undergo
another free-space transmission back to the transmitter. Moreover, the
amplification in the gain medium compensates for the transfer loss in the
resonant cavity, enabling the system to reach a stable state and forming the
standing wave.

Additionally, the SHG crystal serves as an electromagnetic wave frequency-
doubling device that takes in a fundamental frequency beam and generates a
second frequency-doubled beam as output. For instance, the SHG crystal is
precisely embedded in M$1$, facilitating the doubling of the $1064$ nm
resonance beam to a $532$ nm beam. After the completion of information loading
by the EOM, the frequency-doubled beam is transmitted to the receiver via the
optical channel of the fundamental beam for communication. Furthermore,
reflective coatings with varying reflectivity for different frequency bands
are applied to M$1$, EOM, and the gain medium to facilitate the reflection and
transmission of beams at both fundamental and doubled frequencies. In the
receiver, the output reflector selectively outputs the frequency-doubled beam,
which is photoelectrically converted in the detector to transfer the signal.
Due to the co-linear reflection characteristics of retro-reflectors, the
resonant beam can establish self-alignment and achieve a stable state.

According to the above introduction, if an external object blocks the LoS
optical channel, the stable state for both fundamental and doubled frequency
beams can no longer be maintained due to the sharply increased loss, resulting
in a degradation in communication performance. To ensure optimal performance,
we propose adopting an IRS to adjust the optical phase for NLoS transmission
in free space. As illustrated in Fig. 1, if a possible obstruction occurs in
the LoS transmission, the beam controller transmits a portion of the resonant
beam in the LoS channel and the remainder in the IRS-assisted channel via the
beam splitter.

In summary, take Fig. 1 as an example, the work process of IRS-assisted RBC
system can be expressed by the following principles: i) the resonant beam at
fundamental frequency, i.e. wavelength at $1064$ nm, propagates back and forth
in the resonant cavity; ii) $95\%$ of the fundamental-frequency beam passes
through M$1$ and then through SHG to form doubled-frequency beam, i.e.
wavelength at $532$ nm; iii) EOM loads information onto the doubled-frequency
beam; iv) the beams at fundamental and doubled frequencies transfer through
the direct LoS and IRS-assisted channels to receiver under the control of beam
splitter in a certain ratio; v) the output reflector in the receiver outputs
the doubled-frequency beam and reflects the fundamental-frequency beam to the
transmitter; and vi) the detector performs photoelectric conversion and
completes the downlink information transmission. Moreover, the steady-state
and retro-reflective properties of the resonant cavity allow the resonant beam
to maintain radiation safety by blocking interruptions and achieve mobility
through self-tracking.

###  II-B Near Field Characteristic

As a type of electromagnetic field, the resonant beam exhibits both near-field
and far-field propagation behaviors, which are manifested in phase difference.
In far-field propagation, the resonant beam is modeled as a planar wave,
whereas in near-field propagation, it is modeled as a spherical wave [30]. The
Rayleigh distance is used to demarcate the near-field from the far-field, and
it can be expressed as

$D_{\mathrm{r}}=\frac{2D^{2}}{\lambda}.$ (1)

where $D$ is the diameter of transfer aperture, and $\lambda$ is the
wavelength of transfer beam, which can be calculated by $\lambda=u/v$ with $u$
and $v$ representing the light speed and operating frequency. As shown in Fig.
2, when the transfer distance between the Tx/Rx and the IRS far exceeds
$D_{\mathrm{r}}$, the transfer beam exhibits the propagation specificities of
planar wave [31]. Conversely, the near-field property is considered if the
transfer distance is less than $D_{\mathrm{r}}$ [32].

It is evident that the Rayleigh distance increases as the beam wavelength
decreases. Consequently, if the beam wavelength $\lambda$ is sufficiently
small with a specific operating aperture, near-field propagation occurs more
frequently than far-field propagation, as observed in OWC. For instance, in
the case of a RBC system, if the transfer aperture of retro-reflectors in
Tx/Rx is $2.5$ mm, the Rayleigh distance for the fundamental-frequency beam
(i.e., wavelength at $1064$ nm) is approximately $47$ m, while for the
doubled-frequency beam (i.e., wavelength at $532$ nm), it is approximately
$93$ m. Given that the application scenarios for the IRS-assisted RBC system
typically involve indoor environments, priority is given to the performance of
near-field communication.

Figure 2: The schematic of the near-field and far-field according to Rayleigh
distance.

###  II-C Signal Model

In this subsection, we develop the signal model of the IRS-assisted RBC system
in both direct LoS and IRS-assisted channels. We denote
$\mathbf{H}_{\rm{los}}$ and $\mathbf{H}_{\rm{irs}}$ to specify the direct and
IRS-reflected LoS channels between Tx and Rx, respectively. Additionally, due
to the high transmission speed of the resonant beam, the hardware devices in
the RBC system can hardly discern the delays of the multibeam paths, rendering
RBC channels frequency-flat.

In general, OWC channels are influenced by transfer losses, misalignment
losses, and atmospheric turbulence [33]. However, in common application
scenarios for RBC systems, which are predominantly indoor environments with
meter-level distances, atmospheric turbulence can be disregarded. Then, the
channel function $\mathbf{H}$ can be modeled as
$\mathbf{H}=\mathbf{H}_{\rm{p}}\mathbf{H}_{\rm{g}}$, where
$\mathbf{H}_{\rm{g}}$ represents the misalignment losses. Benefiting from the
use of retro-reflectors in transceivers, the RBC system can achieve perfect
beam tracking. Thus, the positions of the transmitter, IRS, and receiver
remain stable at specific locations and the misalignment loss in transmission
is mainly caused by the movement of Rx. Besides, $\mathbf{H}_{\rm{p}}$, which
indicates the propagation function over air, is characterized by two stages in
an IRS-assisted channel, namely Tx-IRS and IRS-Rx, whereas in a direct LoS
channel, it is simply represented by Tx-Rx. Thus, in an IRS-assisted channel,
$\mathbf{H}_{\rm{p}}$ can be expressed as
$\mathbf{H}_{\rm{ir}}\mathbf{H}_{\rm{r}}\mathbf{H}_{\rm{ti}}$, where
$\mathbf{H}_{\rm{ti}}$ and $\mathbf{H}_{\rm{ir}}$ denote the channel functions
for Tx-IRS and IRS-Rx, respectively, and $\mathbf{H}_{\rm{r}}$ represents the
transfer function within the IRS. The transfer function $\mathbf{H}_{\rm{r}}$
can be substituted by $\bm{\Phi}\in\mathbb{C}^{N\times M}$, where
$[\Phi]{n,m}=me^{j\phi{n,m}}$, with $m$ representing the amplitude reflection,
which accounts for absorption and scattering, and $\phi_{n,m}$ being the phase
shift argument of the element $(n,m)$.

##  III IRS-Assisted Transmission Model

In this section, we first determine the power density distribution of the
resonant beam based on near-field lightwave propagation. Following this, we
investigate the transfer model of both direct LoS and IRS-assisted channels to
understand how the IRS influences the propagation of the resonant beam.

###  III-A Near-field Propagation Model

To determine the impact of free-space propagation and the IRS on the resonant
beam, we employ scalar field theory to derive the changes in density
distribution when the diffracting aperture and transfer distance are larger
than the wavelength. As the resonant beam propagates from point $\dot{S}$ in
the wave-source plane to point $\dot{P}$ in the observation plane, the wave
distribution at point $\dot{P}$ results from a sub-wave that forms after the
wave at point $\dot{S}$ reaches point $\dot{Q}$ and is then emitted towards
point $\dot{P}$. This propagation process can be accurately described by
Kirchhoff’s diffraction formula

$\centering\tilde{U}(P)=\frac{A}{i\lambda}\iint_{\Sigma}\frac{e^{ikl}}{l}\frac{e^{ikr}}{r}\left[\frac{\cos(\vec{n}\cdot\vec{r})-\cos(\vec{n}\cdot\vec{l})}{2}\right]d\sigma,\@add@centering$
(2)

where $l$ and $r$ represent the transfer distances between points $\dot{S}$,
$\dot{Q}$, and $\dot{P}$, respectively. The wave-number $k$ is defined as
$k=2\pi/\lambda$, where $\lambda$ is the wavelength. The symbol $\Sigma$
denotes the set of all sub-waves. For a resonant beam with paraxial
propagation, the wave satisfies

$\centering\begin{cases}r=z_{1}\\\ \frac{Ae^{ikl}}{l}\rightarrow
U\left(x_{1},y_{1}\right)\\\ \cos(\vec{n}\cdot\vec{r})=1\\\
\cos(\vec{n}\cdot\vec{l})=-1\end{cases},\@add@centering$ (3)

Thus, the diffraction transfer formula can be transformed as

$\centering\tilde{U}(P)=\frac{1}{i\lambda
z_{1}}\iint_{\Sigma}U\left(x_{1},y_{1}\right)e^{ikr}d\sigma,\@add@centering$
(4)

where $U\left(x_{1},y_{1}\right)$ is the density distribution on the source
aperture. $kr$ represents the phases of spherical sources with $r$ denoting
the transfer distance from source point to observe point. For near-field
lightwave propagation, the Fresnel approximation can be used to deduce $r$,

$\centering
r=\sqrt{z_{1}^{2}+\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}},\@add@centering$
(5)

In addition, the distance of propagation beam axis $z_{1}$ is larger than
transfer aperture, i.e.
$z_{1}\gg\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}$, the binomial
expansion of (5) is

$\displaystyle r$
$\displaystyle=z_{1}+\frac{1}{2z_{1}}\left[\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}\right]-\frac{1}{8z_{1}^{3}}$
(6)
$\displaystyle\left[\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}\right]^{2}+\ldots.$

When $z_{1}$ is significantly larger, the higher-order terms can be ignored.
Thus, the transfer distance $r$ can be expressed as:

$\centering r\approx
z_{1}+\frac{1}{2z_{1}}\left[\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}\right].\@add@centering$
(7)

Thus, the density at any point $P$, whose coordinates are $(x,y)$ (where the
surface perpendicular to the optical axis and parallel to the transfer
aperture is the $xoy$ surface), on the observation aperture can be determined
using the Fresnel-Kirchhoff formula for near-field propagation:

$\displaystyle\tilde{U}(P)$ $\displaystyle=\tilde{U}(x,y)$ (8)
$\displaystyle=\frac{e^{ikz_{1}}}{i\lambda
z_{1}}\iint_{\Sigma}\tilde{U}\left(x_{1},y_{1}\right)e^{\frac{ik}{2z_{1}}\left[\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}\right]}dx_{1}dy_{1}$

where $\tilde{U}\left(x_{1},y_{1}\right)$ represents the density distribution
on the source aperture. Furthermore, (8) can be written as the convolution of
two-dimensional functions

$\centering U(x,y)=\iint
U\left(x_{1},y_{1}\right)h\left(x-x_{1},y-y_{1}\right)\mathrm{d}x_{1}\mathrm{d}y_{1}\@add@centering$
(9)

and

$\centering h(x,y)=\frac{e^{ikz}}{i\lambda
z}e^{\frac{ik}{2z}\left(x^{2}+y^{2}\right)}.\@add@centering$ (10)

Therefore, the diffraction propagation formula for resonant beam in
convolution reads

$\centering U(x,y)=U(x_{1},y_{1})\ast h(x,y),\@add@centering$ (11)

where $(x_{1},y_{1})$ and $(x,y)$ denote the coordinates of the propagation
point on the source and observation apertures.

Furthermore, the density distribution on the observation aperture can be
obtained through Fourier and inverse Fourier transformations

$\displaystyle U(x,y)$ $\displaystyle=\mathcal{H}[U(x,y)]$ (12)
$\displaystyle=\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x_{1},y_{1}\right)\\}\cdot\mathscr{F}\\{h(x,y)\\}\\}.$

The Fourier transformed transfer function is given by

$\displaystyle H(\nu_{x},\nu_{y},z)=\mathscr{F}\\{h(x,y)\\}=\exp\left(i2\pi
z\sqrt{\frac{1}{\lambda^{2}}-\nu_{x}^{2}-\nu_{y}^{2}}\right),$ (13)

where $\nu_{x}$ and $\nu_{y}$ represent the spatial frequency coordinates, and
$z$ is the transfer distance between the wave-source plane and the observation
plane.

###  III-B Direct LoS Channel Model

In this subsection, we establish the channel model for RB direct LoS
propagation, and the misalignment loss due to receiver mobility is derived
from the changes in density distribution.

####  III-B1 Channel Model

In both Rx and Tx, the resonant beam must propagate through retro-reflectors,
with TX also passing through a gain medium. For the input/output retro-
reflectors, the interval between the plane mirrors (M$1$ or M$2$) and the
convex lenses (L$1$ or L$2$) is denoted as $l_{\rm{c}}$. To ensure maximum
pumping efficiency, the gain medium is located at the focal plane of the input
reflector, thus they are separated by $f_{\rm{c}}$. The propagation in space
of M$1$/M$2$-L$1$/L$2$, L$1$-gain medium can be derived from (12) and (13)

$\begin{aligned}
&\mathcal{H}_{l_{\rm{c}}}[U(x,y),l_{\rm{c}}]=\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x,y\right)\\}\cdot
H(v_{x},v_{y},l_{\rm{c}})\\}\\\
&\mathcal{H}_{f_{\rm{c}}}[U(x,y),f_{\rm{c}}]=\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x,y\right)\\}\cdot
H(v_{x},v_{y},f_{\rm{c}})\\}\end{aligned},$ (14)

where $U\left(x,y\right)$ represents the density distribution of the incident
resonant beam.

Moreover, the field changes when the incident beam propagates through the
apertures, including M$1$, M$2$ L$1$, L$2$, and the gain medium, can be
expressed as:

$\displaystyle\mathcal{P}_{M}[U(x,y),M]=U(x,y)\cdot\begin{cases}1,x^{2}+y^{2}\leq
r_{\rm{M}}^{2}\\\ 0,else\end{cases}$ (15)
$\displaystyle\mathcal{P}_{L}[U(x,y),L]=U(x,y)\cdot\begin{cases}e^{\frac{-i\pi}{\lambda
f_{\rm{c}}}(x^{2}+y^{2})},x^{2}+y^{2}\leq r_{\rm{L}}^{2}\\\ 0,else\end{cases}$
$\displaystyle\mathcal{P}_{G}[U(x,y),G]=U(x,y)\cdot\begin{cases}1,x^{2}+y^{2}\leq
r_{\rm{G}}^{2}\\\ 0,else\end{cases}$

$r_{\rm{M}}$, $r_{\rm{L}}$, and $r_{\rm{G}}$ represent the radii of M$1$/M$2$,
L$1$/L$2$, and the gain medium, respectively. The significance of the above
equation lies in the fact that the region within the aperture is the effective
transmission region, and the action of the focusing mirror $M$ on the optical
field is related to its focal length. Hence, the incident resonant beam
propagation in transmitter is
$\mathcal{T}_{T}\triangleq\mathcal{P}_{G}\mathcal{H}_{f_{\rm{c}}}\mathcal{P}_{L}\mathcal{H}_{l_{\rm{c}}}\mathcal{P}_{M}$,
while the propagation in the output reflectors can be defined as
$\mathcal{T}_{R}\triangleq\mathcal{P}_{L}^{-1}\mathcal{H}_{l_{\rm{c}}}^{-1}\mathcal{P}_{M}\mathcal{H}_{l_{\rm{c}}}\mathcal{P}_{L}$
with reverse transmission function $(\cdot)^{-1}$. Furthermore, the
propagation in free space between Tx and Rx is

$\displaystyle\mathcal{H}_{D}[U(x,y),D]=\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x,y\right)\\}\cdot
H(v_{x},v_{y},D)\\}$ (16)

where $D$ is the transmission distance. Thus, the direct LoS channel function
for resonant beam propagation over a round trip is

$\displaystyle\mathbf{H}_{\rm{los}}=\mathcal{T}_{T}^{-1}\mathcal{H}_{D}^{-1}\mathcal{T}_{R}\mathcal{H}_{D}\mathcal{T}_{T}.$
(17)

If an external object obstructs the direct LoS transmission, it is equivalent
to introducing a transfer aperture of decreasing size at the object’s location
in free space. The impact of external object can be expressed as

$\displaystyle\mathcal{P}_{B}[U(x,y),B]=$ (18) $\displaystyle
U(x,y)\cdot\begin{cases}1,x^{2}+y^{2}\leq r_{\rm{B}}^{2}\ \&\&\ x\ |\ y\leq
r_{\rm{B}}-d\\\ 0,x^{2}+y^{2}>r_{\rm{B}}^{2}\ ||\ x\ |\
y>r_{\rm{B}}-d\end{cases},$

where $r_{\rm{B}}$ represents the radius of the object aperture, which is
always set equal to $r_{\rm{M}}$ or $r_{\rm{L}}$, and $d$ denotes the
obstruction depth. The piecewise function in (18) indicates that the
unobstructed area is the effective transmission aperture, expressed as $1$.
Based on this premise, the free space propagation can be calculated as
$\mathcal{H}_{D}=\mathcal{H}_{D2}\mathcal{P}_{B}\mathcal{H}_{D1}$, where $D1$
and $D2$ represent the distances in Tx-object and object-Rx, respectively.
Consequently, the direct LoS channel model with an external object can also be
derived in (17).

####  III-B2 Misalignment Transmission

Denoting the plane input reflector located on the $xoy$-plane, with the
director along the optical axis perpendicular to the plane being the $z$-axis.
When the transceivers are strictly aligned, i.e., the output reflector plane
is perpendicular to the optical axis, the changes in the density distribution
of direct LoS transmission can be calculated using (17). However, the movement
of Rx (translations along the $xoy$ plane and rotations about the axes) will
result in misalignment loss.

Firstly, the transformation relationship before and after translation along
the $x$, $y$, or $z$ axes is:

$(x^{\prime},y^{\prime},z^{\prime},1)^{T}=M_{\mathrm{t}}(x,y,z,1)^{T},$ (19)

where $(x,y,z)$ and $(x^{\prime},y^{\prime},z^{\prime})$ represent the
coordinates before and after translation. $M_{\mathrm{t}}$ is the translation
matrix,

$M_{\mathrm{t}}=\left[\begin{array}[]{cccc}{1}&{0}&{0}&{\Delta x}\\\
{0}&{1}&{0}&{\Delta y}\\\ {0}&{0}&{1}&{\Delta z}\\\ {0}&{0}&{0}&{1}\\\
\end{array}\right],$

$\Delta x$, $\Delta y$ and $\Delta z$ represent the translation distances
along the $x$, $y$, and $z$ axes, respectively. On the one hand, when the
receiver moves along the $z$-axis, the propagation channel can be calculated
using (16), with $D$ replaced by $D+\Delta z$. On the other hand, if the
receiver translates along the $xoy$ plane as depicted in Fig. 3(a), the
misalignment transfer function must incorporate the additional parameters
$\exp[i2\pi(v_{x}\Delta x+v_{y}\Delta y)]$. Hence, the function of a round-
trip transmission with such translation can be expressed as

$\displaystyle\mathcal{H}_{T}[U(x,y),M_{t}]=$
$\displaystyle\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x,y\right)\\}\cdot
H(v_{x},v_{y},D+\Delta z)$ (20) $\displaystyle\times$
$\displaystyle\exp[i2\pi(v_{x}\Delta x+v_{y}\Delta y)]\\}$
$\displaystyle\mathcal{H}_{T}^{-1}[U(x,y),M_{t}]=$
$\displaystyle\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x,y\right)\\}\cdot
H(v_{x},v_{y},D+\Delta z)$ $\displaystyle\times$
$\displaystyle\exp[i2\pi(-v_{x}\Delta x-v_{y}\Delta y)]\\}$

Based on (17), the channel model of RBC system with translating receivers
reads

$\displaystyle\mathbf{H}_{\rm{los,t}}=\mathcal{T}_{T}^{-1}\mathcal{H}_{T}^{-1}\mathcal{T}_{R}\mathcal{H}_{T}\mathcal{T}_{T}.$
(21)

Moreover, the movement of the receiver also includes the rotation of the
output reflector around the $x$, $y$ and $z$ axes as shown in Fig. 3(b). Based
on (13), the wave vector of density distribution as

$\varpi=2\pi[v_{x},v_{y},\omega(v_{x},v_{y})]$ (22)

with
$\omega(v_{x},v_{y})=\left(\lambda^{-2}-v_{x}^{2}-v_{y}^{2}\right)^{1/2}$.
Defining the rotation matrix as
$\mathcal{M}=\mathbf{R_{m_{1}}}(\theta_{1})\mathbf{R_{m_{2}}}(\theta_{2})\cdots\mathbf{R_{m_{n}}}(\theta_{n})$,
where $\mathbf{R_{m_{i}}}(\theta_{i})$ represents the rotation matrix around
$x$, $y$, or $z$-axis, $[m_{1},m_{2},\cdots,m_{n}]\in\\{x,y,z\\}$, and
$\theta_{i}$ is the rotation angle. The relationship between the wave vectors
before and after rotation can be given by
$\varpi=\mathcal{M}^{-1}\hat{\varpi}$. If the rotation matrix is
$\begin{aligned}
\mathcal{M}^{-1}=\left[\begin{array}[]{lll}a_{1}&a_{2}&a_{3}\\\
a_{4}&a_{5}&a_{6}\\\ a_{7}&a_{8}&a_{9}\end{array}\right],\end{aligned}$ the
frequency-domain coordinates $(v_{x},v_{y})$ before rotation can be calculated
as

$\begin{aligned}
\begin{cases}v_{x}=\mathcal{M}_{x}(\hat{v_{x}},\hat{v_{y}})=a_{1}\hat{v_{x}}+a_{2}\hat{v_{y}}+a_{3}\hat{w}(\hat{v_{x}},\hat{v_{y}})\\\
v_{y}=\mathcal{M}_{y}(\hat{v_{x}},\hat{v_{y}})=a_{2}\hat{v_{x}}+a_{5}\hat{v_{y}}+a_{6}\hat{w}(\hat{v_{x}},\hat{v_{y}})\end{cases}\end{aligned}.$
(23)

Then, the density distribution on the rotated receiver is:

$\displaystyle\mathcal{H}_{O}[U(x,y),\mathcal{M}]=$
$\displaystyle\mathscr{F}^{-1}\\{\hat{G}(\hat{v_{x}},\hat{v_{y}})|J(\hat{v_{x}},\hat{v_{y}})|\\}$
(24) $\displaystyle=$
$\displaystyle\mathscr{F}^{-1}\\{G(\mathcal{M}_{x}(\hat{v_{x}},\hat{v_{y}}),\mathcal{M}_{y}(\hat{v_{x}},\hat{v_{y}}))$
$\displaystyle\;|J(\hat{v_{x}},\hat{v_{y}})|\\}$

where $G(x,y)$ is the angular spectrum of density distribution $U(x,y)$, and
$G(x,y)=\mathcal{F}\\{U(x,y)\\}\cdot H(v_{x},v_{y},D)$.
$J(\hat{v_{x}},\hat{v_{y}})$ represents a Jacobian function for compensating
the non-linearity introduced by rotational transformation,

$\displaystyle J(\hat{v_{x}},\hat{v_{y}})=$
$\displaystyle\left(a_{2}a_{6}-a_{3}a_{5}\right)\frac{\hat{v_{x}}}{\hat{w}(\hat{v_{x}},\hat{v_{y}})}+\left(a_{3}a_{4}-a_{1}a_{6}\right)$
(25)
$\displaystyle\frac{\hat{v_{y}}}{\hat{w}(\hat{v_{x}},\hat{v_{y}})}+\left(a_{1}a_{5}-a_{2}a_{4}\right).$

For the resonant beam propagating in the reverse direction, i.e., reflected
from the receiver back to the transmitter, a similar counter-rotation process
as described above should be performed, where the rotation angle takes on the
negative value of $\theta_{i}$ in $\mathcal{M}$.

Hence, the channel model for resonant beam propagation a round-trip with
rotating receivers can be expressed as

$\displaystyle\mathbf{H}_{\rm{los,r}}=\mathcal{T}_{T}^{-1}\mathcal{H}_{O}^{-1}\mathcal{T}_{R}\mathcal{H}_{O}\mathcal{T}_{T}.$
(26)

Here, the direct LoS channel model without and with misalignment has been
depicted in (17), (21) and (26).

(a) Translation along $xoy$ plane.

(b) Rotation around $x/y/z$-axis.

Figure 3: The schematic illustration of LoS channel with misalignment
receivers.

###  III-C IRS-assisted Channel Model

Unlike direct LoS propagation, the resonant beam passes through the IRS in
free space propagation. Therefore, based on (16), the transfer function for
Tx-IRS and IRS-Rx can be depicted as

$\begin{aligned}
\mathcal{H}_{D_{ti}}[U(x,y),D_{ti}]=\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x,y\right)\\}\cdot
H(v_{x},v_{y},D_{ti})\\}\\\
\mathcal{H}_{D_{ir}}[U(x,y),D_{ir}]=\mathscr{F}^{-1}\\{\mathscr{F}\\{U\left(x,y\right)\\}\cdot
H(v_{x},v_{y},D_{ir})\\}\end{aligned},$ (27)

where $D_{\rm{ti}}$ and $D_{\rm{ir}}$ are the distances of Tx-IRS and IRS-Rx,
$U(x,y)$ represents the incident field distribution. Importantly, the impact
of the IRS on resonant beam propagation is primarily expressed through changes
in the propagation phase. Fig. 4 introduces a sample coordinate system for the
IRS-assisted RBC system, wherein the resonant beam is incident along the
$z$-axis of the Tx’s coordinate system onto the IRS plane, and then reflected
towards the Rx. The IRS lies in the $xoy$ plane of its coordinate system, with
its center at point $(x_{i},y_{i},0)$, and the $z$-axis is perpendicular to
the $xoy$ plane. Let $\Phi_{i}=(\theta_{i},\phi_{i})$ and
$\Phi_{r}=(\theta_{r},\phi_{r})$ denote the angles of incidence and reflection
at the IRS, respectively, where $\theta_{k}$ ($k\in{i,r}$) is the elevation
angle (i.e., the angle between the beam path and the $z$-axis), and $\phi_{k}$
($k\in{i,r}$) is the azimuth angle (i.e., the angle between the projection of
the beam path on the $xoy$ plane and the $x$-axis).

To ensure the consistency of coordinate systems, we rotate and transform the
coordinate system of Tx. Assume that the coordinates of the emission point in
the input reflector are $(x,y,z)$, and the conversion matrix is
${}^{t}_{i}\mathbf{R}$. The rotated coordinates in the coordinate system of
the IRS plane $(x^{\prime},y^{\prime},z^{\prime})$ are

$\displaystyle(x^{\prime},y^{\prime},z^{\prime})^{T}$
$\displaystyle=^{t}_{i}\mathbf{R}(x,y,z)^{T}$ (28)
$\displaystyle=\left[\begin{array}[]{ccc}{a_{x}}&{a_{y}}&{a_{z}}\\\
{b_{x}}&{b_{y}}&{b_{z}}\\\
{c_{x}}&{c_{y}}&{c_{z}}\end{array}\right]\left[\begin{array}[]{c}{x}\\\ {y}\\\
{z}\end{array}\right],$

where the conversion matrix can be calculated by
$\mathbf{R}=\mathbf{R_{x}}\mathbf{R_{y}}\mathbf{R_{z}}$, $\mathbf{R_{x}}$ with
$\mathbf{R_{y}}$ and $\mathbf{R_{z}}$ representing the counter-clockwise
rotation matrix around $x$, $y$ and $z$ axes.

Figure 4: The example coordinate system of IRS-assisted RBC system.

For the incident beam with angle $\Phi_{i}=(\theta_{i},\phi_{i})$, we
transform the Tx coordinate to the Cartesian coordinate as

$(x_{t}^{\prime},y_{t}^{\prime},z_{t}^{\prime})^{T}=\mathbf{R_{y}}(-\theta_{i})\mathbf{R_{z}}(-\phi_{i})(x_{t},y_{t},z_{t})^{T}.$
(29)

Then, if the distances of Tx-IRS and IRS-Rx along IRS plane are known, the
transmission distancess for the two stages can also be calculated as

$\displaystyle
d_{\mathrm{ti}}=\frac{D_{x}^{i}}{cos(\pi-\phi_{i})}\frac{1}{cos(\frac{\pi}{2}-\theta_{i})}=\frac{D_{x}^{i}}{-cos(\phi_{i})sin(\theta_{i})},$
(30a) $\displaystyle
d_{\mathrm{ir}}=\frac{D_{x}^{r}}{cos(\phi_{r})}\frac{1}{cos(\frac{\pi}{2}-\theta_{r})}=\frac{D_{x}^{r}}{cos(\phi_{r})sin(\theta_{r})},$
(30b)

where $D_{x}^{i}$ and $D_{x}^{r}$ represent the distances of Tx-IRS and IRS-Rx
along the $x_{i}$-axis in the IRS plane.

Next, the density distribution of the resonant beam will be altered by IRS to
achieve NLOS propagation,

$\mathcal{P}_{R}[U(x,y),R]=U(x,y)\cdot e^{j\Gamma(x,y)},$ (31)

where $(x,y)$ is the coordinate of an IRS element, $\Gamma(x,y)$ denotes the
phase shift introduced by the IRS, and
$\Gamma(x,y)=\frac{2\pi}{\lambda}(\Gamma_{x}x+\Gamma_{y}y)$ with a linear
phase-shift profile. Given the incident angle $\Phi_{i}=(\theta_{i},\Phi_{i})$
and reflection angle $\Phi_{r}==(\theta_{r},\Phi_{r})$, the phase-shift
profile follows the well-known constant phase-shift gradient design [33],

$\displaystyle\Gamma_{x}=cos(\frac{\pi}{2}-\theta_{i})cos(\Phi_{i})+cos(\frac{\pi}{2}-\theta_{r})cos(\Phi_{r}),$
(32a)
$\displaystyle\Gamma_{y}=cos(\frac{\pi}{2}-\theta_{i})sin(\Phi_{i})+cos(\frac{\pi}{2}-\theta_{r})sin(\Phi_{r}).$
(32b)

Thus, the IRS-assisted channel function for resonant beam propagation over a
round trip can be expressed as

$\displaystyle\mathbf{H}_{\rm{irs}}=\mathcal{T}_{T}^{-1}\mathcal{H}_{D_{ti}}^{-1}\mathcal{P}_{R}^{-1}\mathcal{H}_{D_{ir}}^{-1}\mathcal{T}_{R}\mathcal{H}_{D_{ir}}\mathcal{P}_{R}\mathcal{H}_{D_{ti}}\mathcal{T}_{T},$
(33)

where $\mathcal{P}_{R}^{-1}$ represents the reverse propagation in the IRS
with the opposite angle. Furthermore, the IRS-assisted channel model with
receiver misalignment can be calculated based on (20), (21), (24), and (26),

$\displaystyle\mathbf{H}_{\rm{irs},t}=\mathcal{T}_{T}^{-1}\mathcal{H}_{D_{ti}}^{-1}\mathcal{P}_{R}^{-1}\mathcal{H}_{T}^{-1}\mathcal{T}_{R}\mathcal{H}_{T}\mathcal{P}_{R}\mathcal{H}_{D_{ti}}\mathcal{T}_{T},$
(34a)
$\displaystyle\mathbf{H}_{\rm{irs},o}=\mathcal{T}_{T}^{-1}\mathcal{H}_{D_{ti}}^{-1}\mathcal{P}_{R}^{-1}\mathcal{H}_{O}^{-1}\mathcal{T}_{R}\mathcal{H}_{O}\mathcal{P}_{R}\mathcal{H}_{D_{ti}}\mathcal{T}_{T},$
(34b)

where $\mathbf{H}_{\rm{irs},t}$ and $\mathbf{H}_{\rm{irs},r}$ denote the
misalignment of translation and rotation, respectively. Besides,
$\mathcal{H}_{T}$ and $\mathcal{H}_{O}$ are used to replace
$\mathcal{H}_{D_{ir}}$ to characterize the propagation from the IRS to Rx,
with the transmission distance $D$ substituted by $D_{ir}$ in
$\mathcal{H}_{T}$ and $\mathcal{H}_{O}$.

##  IV Channel Capacity Characterization

Based on the developed resonant beam channel model, this section derives the
output power and channel capacity of the IRS-assisted RBC system.

###  IV-A Transfer Factor

The density distribution $U(x,y)$ on the transfer apertures during a round-
trip propagation in the resonant cavity can be obtained from the channel model
under different propagation restrictions in (17), (21), (26), (33), and (34).
However, the distribution varies during successive round-trip propagations due
to the presence of diffraction loss. A steady-state distribution will emerge
once the propagation time reaches a sufficient number of round-trips, where
the amplitude and phase of the optical field on the same aperture remain
unchanged during two consecutive transmissions, also known as the self-
reproducing mode. The transfer factor can then be derived from the difference
between the steady-state distributions at the $n-$th and $(n+1)-$th round-trip
propagations,

$\begin{aligned} \rho=\frac{\|U_{n+1}(x,y)\|_{1}}{\|U_{n}(x,y)\|_{1}}\\\
\end{aligned},$ (35)

where $\|U\|_{1}$ is the $L1$ norm of $U$, representing the sum of the
amplitudes of each element in the aperture. Furthermore, the transmission loss
can be calculated using the transfer factor as $\delta=1-|\rho|^{2}$, where
$|\cdot|^{2}$ denotes the beam intensity. The transfer efficiency can be
further calculated as $\eta=1-\delta$.

###  IV-B System Output Power

Besides, the steady-state condition of the resonator is related to the
transfer factor and gain, which can be written as

$\displaystyle\mathcal{G}^{2}\mathcal{R}_{i}\mathcal{R}_{o}\mathcal{\eta}_{s}^{2}\eta_{ig}\eta_{go}\eta_{og}\eta_{gi}=1,$
(36)

where $\eta_{ig}$, $\eta_{go}$, $\eta_{og}$, and $\eta_{gi}$ denote the
backward and forward transfer efficiencies of the input reflector-gain medium
and gain medium-output reflector, respectively. $\mathcal{G}$ is the gain
factor representing the light intensity amplified during one transit. For
homogeneously broadened lasers,

$\displaystyle\mathcal{G}=\exp\left[\frac{\textsl{g}_{0}\ell}{1+I/I_{g}}\right]$
(37)

with $\textsl{g}_{0}\ell$ being the small-signal gain, a parameter dependent
on the pump power and the material of the gain medium, and $I_{s}$ being the
saturation intensity. $\mathcal{R}_{\rm{i}}$ and $\mathcal{R}_{\rm{o}}$ denote
the reflectivity of input and output reflectors. Due to the frequency
multiplier SHG and information loading module embedded in M$1$, the equivalent
reflectivity of the input reflector can be written as
$\mathcal{R}_{\rm{i}}=R_{\rm{i}}\mathcal{T}_{\rm{S}}^{2}(1-\eta_{\rm{S}})^{2}R_{\rm{E}}$,
where $\mathcal{T}_{\rm{S}}$ and $\eta_{\rm{S}}$ are the transmittance and
transmission efficiency of SHG. $R_{\rm{E}}$ is the reflectivity of EOM.
Meanwhile, $\mathcal{R}_{\rm{o}}$ can be determined by the physical
reflectivity of the output reflector.

Furthermore, the output optical power can be calculated based on (36) and (37)
as

$\displaystyle P_{\rm{o}}$
$\displaystyle=A_{\rm{b}}\eta_{\rm{go}}I(1-\mathcal{R}_{\rm{o}})$ (38)
$\displaystyle=\frac{A_{\rm{b}}I_{\rm{s}}\left(1-\mathcal{R}_{\rm{o}}\right)\eta_{\rm{go}}}{2\mid\ln\sqrt{\mathcal{R}_{\rm{i}}\mathcal{R}_{\rm{o}}\eta_{\rm{g}}^{2}\eta_{\rm{o}}}\mid}\left(\textsl{g}_{0}\ell-\mid\ln\sqrt{\mathcal{R}_{\rm{i}}\mathcal{R}_{\rm{o}}\eta_{\rm{g}}^{2}\eta_{\rm{o}}}\mid\right)$

where $A_{b}$ and $I$ are the cross-sectional area and intensity of the
resonant beam on gain medium, and their product corresponds to the beam power
on the gain medium. The small-signal gain is determined by the pump power
$P_{\rm{i}}$ and the excitation efficiency $\eta_{\rm{e}}$, with the
relationship $\textsl{g}0\ell=\eta_{\rm{e}}P_{\rm{i}}/(A_{\rm{g}}I_{\rm{g}})$,
where $A_{\rm{g}}$ and $I_{\rm{s}}$ denote the cross-sectional area and
saturated intensity of the gain medium.
$\eta_{\rm{o}}=\eta_{ig}\eta_{go}\eta_{og}\eta_{gi}$ represents the transfer
efficiency of a round-trip transmission. For IRS-assisted channel, the
transfer efficiency in free space from Tx to Rx is divided into Tx-IRS and
IRS-Rx, $\eta_{\rm{go}}=\eta_{\rm{gr}}\eta_{irs}\eta_{\rm{ro}}$.
$\eta_{\rm{gr}}$ and $\eta_{\rm{ro}}$ are the forward transfer efficiency of
the two stages. $\eta_{irs}$ is the transfer efficiency of IRS.

Next, the output power of frequency-doubled beam $P_{o}^{2v}$ for
communication can be derived as

$\displaystyle
P_{t}^{v}=\frac{P_{\rm{o}}}{(1-\mathcal{R}_{\rm{o}})\eta_{ig}\eta_{go}},$
(39a) $\displaystyle
P_{t}^{2v}={P_{t}^{v}}^{2}(1-R_{i}^{v})\eta_{\rm{S}}R_{E},$ (39b)
$\displaystyle
P_{o}^{2v}=P_{t}^{2v}R_{i}^{2v}\eta_{lg}\eta_{go}(1-R_{o}^{2v}),$ (39c)

where $R_{i}^{v}$, $R_{i}^{2v}$ and $R_{o}^{2v}$ are the physical reflectivity
for the frequency-fundamental beam of M$1$, frequency-doubled beam of M$1$ and
M$2$. $P_{t}^{v}$, $P_{t}^{2v}$, and $P_{o}^{2v}$ denote the beam power of
frequency-fundamental in Tx, frequency-doubled in Tx and Rx, respectively.
Additionally, the transmission efficiency through the SHG $\eta_{\rm{S}}$ is
determined by the physical characteristics,

$\displaystyle\eta_{\rm{S}}=I_{b}\frac{8\pi^{2}C^{2}_{n}l^{2}_{\rm{s}}}{\varepsilon
c\lambda^{2}n^{3}},$ (40)

where $I_{b}$ is the beam intensity at the SHG crystal, expressed as
$\frac{2P_{i}}{\pi\omega^{2}_{b}}$ with $\omega_{b}$ being the beam radius.
$\varepsilon$ is the vacuum permeability, $c$ is the lightspeed, $C_{n}$,
$l_{\rm{s}}$, and $n$ represent the effective nonlinear coefficient,
thickness, and refractive index of the SHG crystal, respectively.

Algorithm 1 Optimizing Communication Power

1:$z$, $l_{\rm{s}}$, $P_{i}$

2:initialize $R_{\rm{i}}$, $R_{\rm{o}}$, $D_{x}^{i}$, $D_{x}^{r}$,
$r_{\rm{M}}$, $r_{\rm{L}}$, $r_{\rm{B}}$, $M_{\rm{t}}$, $\mathcal{M}$,
$d\leftarrow 0$

3:$U(x,y)\leftarrow\mathbf{H}_{\rm{los}}/\mathbf{H}_{\rm{irs}}U^{\prime}(x,y)$

4:do (35) to get $\eta$

5:do (39a) to get $P_{\rm{t}}^{v}$

6:while $0<d<r_{\rm{B}}$ and $\exists M_{\rm{t}}$ and $\exists\mathcal{M}$ do

7: $\eta\leftarrow$(17), (21), (26), (33), (34)

8: $P_{\rm{oc}}^{d}\leftarrow\eqref{eq:sumoutpowera}\ |_{\gamma\leftarrow 1}$

9: $P_{\rm{oc}}^{i}\leftarrow\eqref{eq:sumoutpowerb}\ |_{\gamma\leftarrow 0}$

10: $P_{\rm{oc}}^{m}\leftarrow max(P_{\rm{oc}}^{d},P_{\rm{oc}}^{i})$

11: $P_{\rm{oc}}\leftarrow P_{\rm{oc}}^{d}+P_{\rm{oc}}^{i}\
|_{\gamma\in[0,1]}$

12: $\gamma\leftarrow 1$

13: if $P_{\rm{oc}}\leq P_{\rm{oc}}^{m}$ then

14: $\gamma\leftarrow\gamma+\Delta\gamma$

15: $P_{\rm{oc}}^{d}$, $P_{\rm{oc}}^{i}$ $\leftarrow$ (41)

16: $P_{\rm{oc}}\leftarrow P_{\rm{oc}}^{d}+P_{\rm{oc}}^{i}$

17: end if

18:end while

19:return $P_{\rm{oc}}$, $\gamma$

Furthermore, the communication power is the sum of frequency-doubled beam
power through direct LoS channel and IRS-assisted channel, i.e.
$P_{\rm{oc}}=P_{\rm{oc}}^{\rm{d}}+P_{\rm{oc}}^{\rm{i}}$, reads

$\displaystyle P_{\rm{oc}}^{\rm{d}}=\gamma
P_{t}^{2v}(1-R_{i}^{2v})\eta_{\rm{ig}}\eta_{\rm{go}}(1-R_{o}^{2v}),$ (41a)
$\displaystyle
P_{\rm{oc}}^{\rm{i}}=(1-\gamma)P_{t}^{2v}(1-R_{i}^{2v})\eta_{\rm{ig}}\eta_{\rm{gr}}\eta_{\rm{r}}\eta_{\rm{ro}}(1-R_{o}^{2v}),$
(41b)

where $\gamma$ represents the power-splitting (PS) ratio of the two channels
relating to the transfer efficiency of the direct LoS channel, and
$0\leq\gamma\leq 1$.

###  IV-C Power Maximization and Channel Capacity

From (39) and (41), the total power of the frequency-doubled beam $P_{t}^{2v}$
at the transmitter is determined by fixed parameters, capping the output power
available for communication from both channels. To ensure that the output
power remains at its peak value when transfer efficiency decreases due to
object obstructions, the energy between the two channels must be optimally
distributed, i.e.

$\displaystyle\left.\max_{d,{\forall{M_{t}}},{\forall{\mathcal{M}}}}P_{\rm{oc}}(\gamma)\right|_{\begin{subarray}{c}z=z_{\text{ini}}\\\
l_{\rm{s}}=l_{{\rm{s}},\text{ini}}\\\
P_{\rm{i}}=P_{{\rm{i}},\text{ini}}\end{subarray}},$ (42a) $\displaystyle s.t.\
\ $ $\displaystyle P_{\rm{oc}}\leq P_{\rm{oc}}^{d}(\gamma=1,\Delta d=0),$
(42b) $\displaystyle d\in[0,r_{\rm{B}}],$ (42c) $\displaystyle\gamma\in[0,1],$
(42d)

where $\forall{M_{t}}$, $\forall{\mathcal{M}}$ in (42a) denote that the
misalignment parameters, including translation and rotation, are randomized.
Besides, the system parameters $z$, $l_{\rm{s}}$ and $P_{\rm{i}}$ are preset.
(42a) represents that the maximum power is less than the output power in the
direct LoS channel when the total power is transmitted through the channel
without external obstructions. (42b) restricts the obstruction depth $d$ in
the LoS channel to the aperture radius, which is equal to the radii of
input/output reflectors. (42c) states that $\gamma$ must be between 0 and 1.

Under the preset system parameters and PS ratios, the total output power
$P_{\rm{oc}}$ needs to be monitored in real-time in response to external
object obstruction and receiver movement. The monitored power should then be
compared with the output power fully allocated to the direct LoS channel
($\gamma=1$) and the IRS-assisted channel ($\gamma=0$) under these conditions.
If the total output power is less than the output power of the two independent
channels, the PS ratio should be dynamically adjusted to maximize the output
power. We adopt the gradient ascent method to solve the maximization power,
the algorithm is depicted as Algorithm 1.

(a) Transmission distance

(b) Invasion depth

(c) Translation distance (z=5m)

(d) Rotation angle (z=5m)

Figure 5: The changes of E2E efficiency $\eta$ in direct LoS and IRS-assisted
channels under different parameters ($P_{\rm{in}}=200$ W).

Based on the optimized output power under different conditions, the signal-to-
noise ratio (SNR) and channel capacity can be derived from the received
communication power of two channels,

$\begin{aligned} \mathrm{SNR}=\frac{(\eta_{c}P_{\rm{oc}})^{2}}{2\pi
e\left[2q\left(\eta_{c}P_{\mathrm{oc}}+I_{\mathrm{k}}\right)B+\frac{4KTB}{L_{\mathrm{r}}}\right]}\end{aligned},$
(43)

where $\eta_{c}$ represents the photo-detector responsivity in $A/W$, and $e$
is the natural constant. $[\cdot]$ denotes the total noise, expressed as
$\sigma^{2}$, which includes background current $I_{\rm{k}}$, noise bandwidth
$B$, load resistance $L_{r}$, electric charge $q$, Boltzmann’s constant $K$,
and temperature in Kelvin $T$.

Then, the channel capacity, represented by the spectral efficiency
$S_{\rm{E}}$, can be expressed as

$\begin{aligned}
S_{\rm{E}}=B_{\rm{c}}\log_{2}\left(1+\mathrm{SNR}\right)\end{aligned},$ (44)

where $B_{\rm{c}}$ denotes the channel bandwidth.

##  V Numerical Evaluation

In this section, we evaluate the performance of RIS-assisted RBC system,
including end-to-end (E2E) efficiency, output power, the frequency-doubled
beam power, and channel capacity for communication under different parameters.

The structural parameters of the system are as follows: the radii of the
input/output reflectors and the gain medium are $2.5$ mm, the distance between
M$1$/M$2$ and L$1$/L$2$ is $5$ cm, and the saturation intensity for the gain
medium with Nd:YVO4 crystal is $1260$ $\rm{W/cm^{2}}$ [34]. We adopt the
LiNbO3 crystal as the SHG medium for frequency doubling, which has a high
nonlinear coefficient of $4.7$ pm/V and a walk-off angle of $0^{\circ}$ at
$1064$ nm; the refractive index of LiNbO3 SHG is $2.23$ [35]. For channel
capacity calculation, the values of the parameters in (43) are:
$\eta{\rm{c}}=0.6$ A/W, $I{\rm{c}}=5100$ $\mu$A, $B=811.7$ MHz,
$L{\rm{r}}=5100$ K$\Omega$, and $K=300$ K. Finally, to maintain the self-
reproducing model between transceivers, we assume the reflectivity of M$1$ for
the fundamental frequency beam is $5\%$, allowing the remaining $95\%$ of the
transmitted beam to be frequency-doubled. The wavelength of the fundamental
frequency is $1064$ nm, resulting in a frequency-doubled wavelength of $532$
nm.

###  V-A End-to-End Efficiency

In the IRS-assisted RBC system, the primary factor affecting communication
performance is channel transmission efficiency. Thus, we first analyze the
variations in E2E (Tx-Rx) efficiency $\eta$ ($=\sqrt{\eta_{\rm{o}}}$).

Figure 5(a) illustrates the impact of spatial distance $z$ on $\eta$ and
output power $P_{o}$ in both the direct LoS channel and the IRS-assisted
channel. It is observed that both efficiency and output power decrease with
increasing transmission distance, with $P_{o}$ decreasing from its peak value
to nearly $0$ W. Moreover, compared to the direct LoS channel, $\eta$ and
$P_{\rm{o}}$ in the IRS-assisted channel are notably lower due to the longer
path of the resonant beam. For instance, at a distance of $z=5$ m, $\eta$ is
$90.14\%$ with $P_{\rm{o}}=22.54$ W in the direct LoS channel, whereas in the
IRS-assisted LoS channel, $\eta$ is $85.18\%$ with $P_{\rm{o}}=13.01$ W.
Furthermore, in the direct LoS channel, obstruction by external objects
decreases transfer efficiency, as illustrated in Fig. 5(b), due to the
narrowing effective transfer aperture with increasing obstruction depth $d$.
The E2E efficiency in the direct LoS channel noticeably decreases as $d$
increases from $0$ to $2.5$ mm, regardless of transmission distance.
Conversely, active modulation of IRS enables the IRS-assisted channel to
circumvent object occlusion effects, maintaining constant $\eta$ regardless of
$d$.

(a) Invasion depth

(b) Translation distance

(c) Ratation angle ($d=1$ mm)

Figure 6: The changes of output power $P_{\rm{o}}$ in direct LoS and IRS-
assisted channels with different parameter ($z=5$ m, $P_{\rm{in}}=200$ W).

(a) Input power ($l_{s}=2$ mm)

(b) SHG thickness

(c) Translation distance

(d) Rotation angle

Figure 7: The changes of the output frequency-doubled beam power
$P_{\rm{o}}^{2v}$ in direct LoS and IRS-assisted channels with different
parameters.

Moreover, the effects of receiver misalignment on efficiency are depicted in
Figs. 5(c) and 5(d). Under different obstruction depths $d$, E2E efficiency in
both channels gradually decreases as the translation distance $\Delta x$
increases. For the direct LoS channel, deeper obstructions lead to lower
transmission efficiency; for example, at $d=1$ mm, $\eta$ is significantly
lower than at $d=2$ mm. Specifically, at $\Delta x=2$ mm, $\eta$ is $52.92\%$
at $d=1$ mm and $6.08\%$ at $d=2$ mm. Importantly, when the direct LoS channel
is free of obstacles or has minimal interference ($d\leq 0.5$ mm), the IRS-
assisted channel performs worse than the direct LoS channel at short
translation distances ($\Delta\leq 2$ mm). Conversely, if the translation
distance exceeds $2.5$ mm, the efficiency of the IRS-assisted channel
surpasses that of the direct LoS channel because IRS phase adjustments can
compensate for losses at greater distances. Moreover, the superiority of the
IRS-assisted channel becomes more pronounced with increasing translation
distance; for example, the efficiency difference between IRS-assisted and
direct LoS channels without obstruction is $5.29\%$ at $\Delta x=3$ mm and
increases to $19.06\%$ at $\Delta x=4$ mm. Finally, regarding misalignment in
rotation, $\eta$ decreases gradually with increasing rotation angle due to the
diminishing effective aperture. At the same obstruction depth, $\eta$ is
significantly lower at $d=2$ mm than at $d=1$ mm for the same rotation angle,
indicating that greater obstruction depth results in lower efficiency.
Meanwhile, the invasion position along the optical axis $z_{\rm{o}}$ (defined
as the distance between Tx and external object divided by $z$) has a
relatively minor impact on efficiency; curves of $\eta$ at $z_{\rm{o}}=0.3$,
$0.5$, and $0.7$ with $d=1$ mm and $2$ mm are nearly identical. Moreover,
under the same rotation angle, $\eta$ in the direct LoS channel without
obstruction ($d=0$ mm) is higher than in the IRS-assisted channel due to the
shorter transmission distance. For example, at $\theta_{y}=10^{\circ}$ and
$20^{\circ}$, $\eta$ is $91.88\%$ and $86.21\%$ in the direct LoS channel with
$d=0$ mm, compared to $89.13\%$ and $83.63\%$ in the IRS-assisted channel.

###  V-B Power for Communication

To analyze the communication power with the frequency-doubled beam, we first
evaluate the power in the fundamental frequency under different parameters, as
depicted in Fig.6. In Fig.6(a), the output power $P_{\rm{o}}$ in the direct
LoS channel decreases with increasing obstruction depth $d$, whereas in the
IRS-assisted channel, it remains constant regardless of transmission distance.
Similar to the trend in $\eta$ shown in Fig. 5(b), longer distances result in
lower output power. Specifically, $P_{\rm{o}}$ for both channels at $z=3$ m
surpasses that at $z=5$ m, which in turn exceeds $P_{\rm{o}}$ at $z=7$ m.
Furthermore, when $d$ reaches a certain value in the direct LoS channel, the
output power drops to zero, as the system gain at Tx cannot compensate for the
transmission losses. The invasion depth at which $P_{\rm{o}}$ reaches zero is
inversely related to transmission distance, i.e., $d$ decreases at
$P_{\rm{o}}=0$ as $z$ increases. For example, $d\approx 1.3$ mm at $z=7$ m,
$d\approx 1.6$ mm at $z=5$ m, and $d\approx 1.8$ mm at $z=3$ m. Figure 6(b)
shows the changes in output power with translation misalignment. As $\Delta x$
increases, the output power decreases. In the direct LoS channel, $P_{\rm{o}}$
drops with increasing depth at the same translation distance. Then,
$P_{\rm{o}}$ of the direct LoS channel is superior to that of the IRS-assisted
channel when $\Delta x$ and $d$ are small. It can be seen that $P_{\rm{o}}$ at
$\Delta x\leq 2.25$ mm and $1$ mm in direct LoS channel with $d=0$ mm and
$0.5$ mm is greater than that in IRS-assisted channel. The variation of output
power with rotation angle is illustrated in Fig. 6(c), where $P_{\rm{o}}$
diminishes as $\theta_{y}$ rises from $0$ to $50^{\circ}$. The output power in
the direct LoS channel without invasion is significantly higher than in the
IRS-assisted channel, regardless of the rotation angle. For instance,
$P_{\rm{o}}$ is $19.65$ W and $10.68$ W at $\theta_{y}=10^{\circ}$ and
$20^{\circ}$ in the direct LoS channel, compared to $12.75$ W and $7.65$ W in
the IRS-assisted channel.

(a) SHG thickness

(b) Invasion depth

(c) PS ratio with translation

(d) PS ratio with rotation

Figure 8: The changes of the communication power in two channels
$P_{\rm{oc}}^{d}$ and $P_{\rm{oc}}^{i}$ with different PS ratios ($P_{i}=200$
W, $z=5$ m in b, c, d).

(a) Invasion depth

(b) Translation distance

(c) Rotation angle

Figure 9: The impact of dynamic power-splitting ratio on the communications
under different parameters ($z=5$ m, $P_{\rm{i}}=200$ W).

Then, the output power for frequency-doubled beam $P_{\rm{o}}^{2v}$ can be
derived from (39) based on transfer efficiency and output power for frequency-
fundamental beam. Figure 7 illustrates the variations in frequency-doubled
power under different input power $P_{\rm{i}}$, SHG thickness $l_{\rm{s}}$,
and misalignment parameters $\Delta x$ and $\theta_{y}$, assuming all power is
allocated to one channel.

Firstly, the output power for frequency-fundamental $P_{\rm{o}}$ and
frequency-doubled $P_{\rm{o}}^{2v}$ gradually enhance as the input power
$P_{\rm{i}}$ rises from $0$ to $300$ W as illustrated in Fig. 7(a), and the
output power remains essentially $0$ W when $P_{i}$ is less than $50$ W since
the pumping power threshold is not yet satisfied. $P_{\rm{o}}^{2v}$ and
$P_{\rm{o}}$ in the direct LoS channel are greater than those in the IRS-
assisted channel under the same input power and distance, due to lower
transmission losses. For example, $P_{\rm{o}}^{2v}$ is $1.16$ W, $17.24$ W,
$69.74$ W in direct LoS channel with $P_{\rm{i}}=100$ W, $200$ W, $300$ W and
$z=3$ m, compared to $0.26$ W, $5.41$ W, and $23.54$ W in the IRS-assisted
channel. As shown in Fig. 7(b), the thickness of the SHG crystal is another
significant factor affecting the frequency-doubled power. The power increases
with the SHG thickness regardless of transmission distance and pump power. For
the same $l_{\rm{s}}$, $P_{\rm{o}}^{2v}$ in the direct LoS channel is also
greater than in the IRS channel. At $l_{\rm{s}}=2$ mm, $3$ mm, and $4$ mm,
with $z=3$ m and $P_{\rm{i}}=200$ W, $P_{\rm{o}}^{2v}$ is $17.25$ W, $38.80$
W, and $68.98$ W in the direct LoS channel, compared to $5.41$ W, $12.18$ W,
and $21.66$ W in the IRS channel. Afterwards, the frequency-doubled power with
misalignment receiver at $l_{\rm{s}}=5$ mm, $P_{\rm{i}}=200$ W and $z=5$ m is
depicted in Figs. 7(c) and 7(d), where the power drops from the maximum with
the increase in $\Delta x$ and $\theta_{y}$. Additionally, in the direct LoS
channel, the power decreases with increasing obstruction depth $d$, dropping
to $0$ more rapidly with greater $d$. For example, $P_{\rm{o}}^{2v}$ is
$10.72$ W, $4.66$ W, $0.2$ W at $d=0$ mm, $0.5$ mm, and $1.0$ mm under $\Delta
x=0.5$ mm. When $d=0.5$ mm and $1.5$ mm, the translation distances at which
$P_{\rm{o}}^{2v}$ drops to $0$ are approximately $2$ mm and $1.5$ mm,
respectively. Similar to the fundamental frequency power in Fig.6(b), the
power in the IRS-assisted channel is lower than in the direct LoS channel if
the obstruction depth and translation distance are small. Finally,
$P_{\rm{o}}^{2v}$ in the direct LoS channel is greater than in the IRS-
assisted channel if the rotation angle is less than a certain value, i.e.,
$\theta_{y}<30^{\circ}$. For instance, $P_{\rm{o}}^{2v}$ is $9.50$ W, $4.47$
W, and $1.84$ W at $\theta_{y}=5^{\circ}$, $10^{\circ}$, and $15^{\circ}$ in
the direct LoS channel, compared to $2.98$ W, $1.64$ W, and $0.78$ W in the
IRS channel. The invasion location $z_{\rm{o}}$ has a relatively minor impact
on $P_{\rm{o}}^{2v}$, as shown in Fig.7(d), where $P_{\rm{o}}^{2v}$ exhibits a
similar trend with rotation angle at $z_{\rm{o}}=0.3$, $0.5$, and $0.7$.

Furthermore, based on the power-splitting ratio $\gamma$, the frequency-
doubled beam in the transmitter can be divided into two streams, transmitting
in the direct LoS channel and the IRS-assisted channel. The relationships
between the output frequency-doubled power for the two channels and the PS
ratio are depicted in Fig. 8.

Firstly, as the SHG thickness $l_{\rm{s}}$ increases, the communication power
in both channels, $P_{\rm{oc}}^{\rm{d}}$ and $P_{\rm{oc}}^{\rm{i}}$, rises
from zero as shown in Fig. 7(a). Similarly, the increase in distance results
in a decrease in $P_{\rm{oc}}^{\rm{d}}$ and $P_{\rm{oc}}^{\rm{i}}$. More
importantly, $P_{\rm{oc}}^{\rm{d}}$ increases while $P_{\rm{oc}}^{\rm{i}}$
decreases as $\gamma$ gets greater under the same $l_{\rm{s}}$ and $z$. In
other words, $P_{\rm{oc}}^{\rm{d}}$ and $P_{\rm{oc}}^{\rm{i}}$ are inversely
proportional to $\gamma$. Then, the changes of $P_{\rm{oc}}^{\rm{d}}$ and
$P_{\rm{oc}}^{\rm{i}}$ with the invasion depth and PS ratio are depicted in
Fig. 8(b), where $P_{\rm{oc}}^{\rm{d}}$ and $P_{\rm{oc}}^{\rm{i}}$ drop from
the peak power as the obstruction deepens. It is evident that as $\gamma$
changes from $0.3$, $0.5$ to $0.7$, $P_{\rm{oc}}^{\rm{d}}$ gradually increases
while $P_{\rm{oc}}^{\rm{i}}$ gradually decreases under the same $d$. The sum
of the power in the two channels, $P_{\rm{oc}}$, is shown by the dashed lines.
When $d$ is small, the total power $P_{\rm{oc}}$ with higher $\gamma$ is
greater, while $P_{\rm{oc}}$ with smaller $\gamma$ is greater if the invasion
depth deepens. That is, when $d\leq 0.75$ mm, $P_{\rm{oc}}$ at $\gamma=0.7$ is
higher than that at $\gamma=0.5$ and $0.3$. Conversely, when $d>0.75$ mm,
$P_{\rm{oc}}$ at $\gamma=0.3$ is greater than that at $\gamma=0.5$ and $0.7$.
This is because when $d$ is small, $P_{\rm{oc}}^{\rm{d}}$ with higher $\gamma$
is greater than $P_{\rm{oc}}^{\rm{i}}$, hence the total power with higher
$\gamma$ is more prominent. Conversely, as $d$ increases,
$P_{\rm{oc}}^{\rm{i}}$, which constitutes a larger proportion when $\gamma$ is
small, becomes greater, hence $P_{\rm{oc}}$ with smaller $\gamma$ generates a
greater value. Besides, if all transmitted power is allocated to one of the
channels, i.e., $P_{\rm{oc}}^{\rm{d}}$ with $\gamma=1$ or
$P_{\rm{oc}}^{\rm{i}}$ with $1-\gamma=1$, the power in the direct LoS channel
or IRS channel is greater than the power partially allocated, such as
$\gamma=0.3$, $0.5$, or $0.7$. Figures 8(c) and 8(d) show the changes in
communication power with $\gamma$ under misalignment. $P_{\rm{oc}}^{\rm{d}}$
increases while $P_{\rm{oc}}^{\rm{i}}$ decreases as $\gamma$ rises from $0$ to
$1$. The increased invasion depth results in a significant decrease in
$P_{\rm{oc}}^{\rm{d}}$. Simultaneously, as the translation distance $\Delta x$
and rotation angle $\theta_{y}$ increase, the power gradually diminishes.
Moreover, there are intersection points between $P_{\rm{oc}}^{\rm{d}}$ and
$P_{\rm{oc}}^{\rm{i}}$ at different values of $\gamma$, which demonstrates
that $\gamma$ needs to be adjusted to maximize the communication power under
different misalignment and invasion parameters.

(a) SNR vs. Invasion depth.

(b) Spectral Efficiency vs. Invasion depth.

Figure 10: The changes of SNR, channel capacity for communication with the
optimal PS ratios.

###  V-C Power Maximization

To ensure optimal communication performance, the channel PS ratio between
direct and IRS-assisted channels needs to be dynamically adjusted under
various parameters. Next, we analyze how the maximum performance can be
achieved by regulating the PS ratio.

Figure 9 depicts the impact of dynamic $\gamma$ on communication power under
$z=5$ m, $l_{\rm{s}}=5$ mm, and $P_{\rm{i}}=200$ W, where the green solid line
represents the maximum power after dynamic adjustment. Based on Fig. 8(b), the
power allocated entirely to the direct or IRS channel exceeds the partially
allocated power under a certain invasion depth. Therefore, based on the
changes in $d$ in Fig. 9(a), the power is entirely allocated to the direct LoS
channel if $d$ is small, while the power is allocated to the IRS LoS channel
as $d$ increases. For example, $P_{\rm{oc}}=P_{\rm{oc}}^{\rm{d}}$ with
$\gamma=1$ when $d\leq 0.75$ mm, and $P_{\rm{oc}}=P_{\rm{oc}}^{\rm{i}}$ with
$\gamma=0$ when $0.75<d<2.5$ mm. Similarly, as shown in Fig. 9(b), if the
translation distance is small, i.e., $\Delta x\leq 0.5$ mm,
$P_{\rm{oc}}=P_{\rm{oc}}^{\rm{d}}$ with $\gamma=1$, while
$P_{\rm{oc}}=P_{\rm{oc}}^{\rm{i}}$ with $\gamma=0$ at larger translation
distances, i.e., $\Delta x>0.5$ mm. Finally, when the receiver rotates with an
increasing angle, the power in the IRS-assisted channel is always greater than
the power in the direct LoS channel and the combined power of both channels
($0<\gamma<1$) as shown in Fig. 9(c). Therefore, all transmitted power is
allocated to the IRS-assisted channel under dynamic regulation, i.e.,
$P_{\rm{oc}}=P_{\rm{oc}}^{\rm{i}}$ with $\gamma=0$ at
$\theta_{\rm{y}}\in[0^{\circ},50^{\circ}]$. Similarly, $\gamma$ can be
dynamically adjusted to maximize power depending on different system
parameters, channel parameters, and receiver conditions.

###  V-D Channel capacity

According to (43), (44) and the optimal communication power shown in Fig.9,
the $\mathrm{SNR}$ and spectral efficiency of the IRS-assisted RBC system
under specific parameters can be calculated numerically. The variations of
$\mathrm{SNR}$ and spectral efficiency $S_{\rm{E}}$ with different invasion
depths, translation distances, and rotation angles are illustrated in Fig.10.
As depicted in Fig.10(a), when the transceiver is aligned, i.e., $\Delta x=0$
mm and $\theta_{\rm{y}}=0^{\circ}$, $\mathrm{SNR}$ initially experiences a
slight decrease but then stabilizes, maintaining around $92.95$ dB.As $\Delta
x$ and $\theta_{\rm{y}}$ increase, the $\mathrm{SNR}$ shows a significant
decrease under the same invasion depth due to the reduction in output power,
as shown in Figs. 8(c) and 8(d), with the invasion location at $z_{o}=0.5$.
For example, $\mathrm{SNR}$ is approximately $87.60$ dB and $73.96$ dB at
$\Delta x=1$ mm and $2$ mm, respectively, and approximately $87.48$ dB and
$79.31$ dB at $\theta_{\rm{y}}=10^{\circ}$ and $\theta_{\rm{y}}=20^{\circ}$,
respectively, with $d>1$ mm. Additionally, when the invasion depth is small
($d\leq 1$ mm), the $\mathrm{SNR}$ experiences a slight decrease but then
remains constant despite the receiver misalignment.

Similarly, the spectral efficiency $S_{\rm{E}}$ follows the same trend as
$\mathrm{SNR}$, as shown in Fig. 10(b). It exhibits a slight downward trend
before stabilizing in both the aligned and misaligned cases. $S_{\rm{E}}$
remains at $10.09$ bps/Hz and $8.51$ bps/Hz at $\Delta x=1$ mm and $2$ mm,
respectively, and approximately $10.07$ bps/Hz and $9.13$ bps/Hz at
$\theta_{\rm{y}}=10^{\circ}$ and $\theta_{\rm{y}}=20^{\circ}$, respectively,
with $d>1$ mm. The maximum spectral efficiency is approximately $11$ bps/Hz
when the transceiver is aligned.

This numerical analysis of $\mathrm{SNR}$ and spectral efficiency reveals that
the assistance of the IRS enables the RBC system to maintain a high level of
operating efficiency despite external object invasion and transceiver
misalignment.

##  VI Conclusion

In this paper, we propose an IRS-assisted resonant beam communications (RBC)
system utilizing a frequency-doubling method to achieve highly efficient
downlink communication in the presence of external object invasion and
receiver misalignment. By leveraging near-field optical propagation, we
establish a transmission model for both direct line-of-sight (LoS) and IRS-
reflected channels, and then derive the output frequency-doubled beam power
based on the transfer factor. To optimize the communication performance, we
maximized the frequency-doubled power by dynamically adjusting the power-
splitting ratio between the two channels, taking into account different
obstruction depths and misalignment parameters. Finally, numerical evaluations
demonstrated that with IRS assistance and power ratio regulation, the signal-
to-noise ratio (SNR) and spectral efficiency can be maintained at optimal
values, with the maximum spectral efficiency reaching approximately $11$
bps/Hz.

Additionally, several issues need to be addressed in future research,
including: 1) analyzing uplink communication using resonant beams with IRS
assistance; 2) evaluating communication performance metrics such as bit error
rate (BER), Q factor, and modulation methods for RBC; and 3) optimizing system
design to further enhance communication performance.

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