# The Far-/Near-Field Beam Squint and Solutions for THz Intelligent Reflecting
Surface Communications

Wanming Hao, _Member, IEEE_ , Xiaobei You, Fuhui Zhou, _Senior Member, IEEE_ ,
Zheng Chu, _Member, IEEE_ , Gangcan Sun, Pei Xiao, _Senior Member, IEEE_ W.
Hao, X. You and G. Sun are with the School of Information Engineering,
Zhengzhou University, Zhengzhou 450001, China. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>iegcsun@zzu.edu.cn)F. Zhou is with the College
of Electronic and Information Engineering, Nanjing University of Aeronautics
and Astronautics, Nanjing, 210000, China. (email: zhoufuhui@ieee.org)Z. Chu
and P. Xiao are 5GIC & 6GIC, Institute for Communication Systems (ICS),
University of Surrey, Guildford GU2 7XH, UK. (Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Terahertz (THz) and intelligent reflecting surface (IRS) have been regarded as
two promising technologies to improve the capacity and coverage for future 6G
networks. Generally, IRS is usually equipped with large-scale elements when
implemented at THz frequency. In this case, the near-field model and beam
squint should be considered. Therefore, in this paper, we investigate the far-
field and near-field beam squint problems in THz IRS communications for the
first time. The far-field and near-field channel models are constructed based
on the different electromagnetic radiation characteristics. Next, we first
analyze the far-field beam squint and its effect for the beam gain based on
the cascaded base station (BS)-IRS-user channel model, and then the near-field
case is studied. To overcome the far-field and near-field beam squint effects,
we propose to apply delay adjustable metasurface (DAM) to IRS, and develop a
scheme of optimizing the reflecting phase shifts and time delays of IRS
elements, which effectively eliminates the beam gain loss caused by beam
squint. Finally, simulations are conducted to demonstrate the effectiveness of
our proposed schemes in combating the near and far field beam squint.

###### Index Terms:

Terahertz, intelligent reflecting surface, beam squint, delay adjustable
metasurface.

##  I Introduction

Terahertz (0.1$\sim$10 THz) communication has been recognized as a promising
candidate technology for future B5G/6G networks owing to its ultra-wideband
bandwidth and ultra-high transmission rate [1], [2]. The 2019 World Radio
Communication Conference has proclaimed that the frequency band 275-450 GHz
with a total of 137 GHz bandwidth will be used for land mobile service
applications [3]. However, the attenuation of THz signals is large, and they
are also issues with diffraction and obstacles [4]. To tackle those problems,
intelligent reflecting surface (IRS) composed of a large number of energy-
efficient passive elements can be applied to improve the strength of THz
signals or generate new links by adjusting the reflecting elements [5, 6, 7,
8], and thus IRS technique constitutes a viable solution for future THz
communications.

Generally, the structure of IRS elements is relatively simple, and only the
amplitude and phase of reflection signals can be adjusted, which makes that
IRS elements are frequency-independent. As a result, there exists beam squint
in IRS communications. In fact, there have been several research focusing on
beam squint [9]. For example, Chen _et al_. [10] proposed a novel phase shift
design scheme for mitigating beam squint for the line-of-sight (LoS) and non-
Los (NLoS) scenarios, respectively. Specifically, for the former, they derived
the optimal phase shift for each subcarrier and common phase shift matrix for
all subcarriers. For the latter, they proposed a mean channel covariance
matrix (MCCM) scheme to maximize the achievable rate. Wan _et al._ [11]
designed a new set of basis vectors to approximate the optimal precoder, where
each new basis vector was devised to form a radiation pattern with a wide beam
for overcoming beam squint effect. Although the above proposed phase
optimization algorithms can reduce beam squint effects, the results are not
remarkably effective. Later, several new antenna structures are developed by
adding time delayers between radio frequency (RF) chains and phase shifters,
which effectively solving beam squint effects. For example, Tan and Dai [12]
designed a delay-phase precoding structure to solve beam squint effects by
converting frequency-independent beams into frequency-dependent beams,
obtaining the near-optimal beam gain at all subcarriers. Gao and Wang [13]
proposed a wideband hybrid beamforming technique based on the true-time-delay
lines and designed a new hardware implementation of analog combiner to achieve
the near-optimal performance. Zhai _et al._ [14] designed an antenna
architecture called THzPrism, where several true-time-delay lines were
inserted among phase shifters to achieve frequency-based beam spreading.

In addition, it is well known that the electromagnetic radiation field is
divided into far-field region and near-field region according to the
Fraunhofer distance [15], [16]. As IRS size increases, its near-field range
will be expanded, and thus the phase of array steering vector is nonlinear to
the antenna index [16]. There have been several works related to the near-
field model. Cui _et al._ [17] investigated the near-field channel estimation
problem in extremely large-scale multiple-input-multiple-output (XL-MIMO)
systems and proposed on-grid and off-grid near-field channel estimation
schemes to achieve a better normalized mean square error (NMSE) performance.
Later, they studied the near-field beam squint effect in THz communications,
and discovered a new phenomenon that the beam energy at different frequencies
focus on different locations [18]. On this basis, they proposed a phase-delay
focusing method to mitigate the near-field beam squint effect. Zhang _et al._
[19] investigated the performance of IRS-aided system in both the far-field
and near-field by deriving accurate closed-form approximations of the ergodic
capacity.

In summary, based on the above analysis, the beam squint and near-field model
have been studied. However, the works in [10]-[14] only consider the far-field
beam squint without IRS, and the works in [17]-[19] only consider either the
near-field beam squint without IRS or near-filed IRS communications without
beam squint. Generally, for THz IRS communications with large-scale array
antennas, the near-field, far-filed and beam squint all should be considered.
To our best knowledge, this has not yet been investigated to date.

To fill this gap, in this paper, we first analyze the far-field and near-field
channel models and beam squints in THz IRS communications, respectively.
Specifically, by theoretical derivation, the far-field beam squint makes that
the beams reflected by IRS at different subcarriers squint toward different
directions, while the near-field beam squint makes that the beams reflected by
IRS at different subcarriers aim at different locations and most of beams fail
to focus on the user. Next, to deal with the far-field and near-field beam
squint effects, we propose an effective scheme based on the delay adjustable
metasurface (DAM) [20] technique that can adjust the delays of signals
reflected by different IRS elements to solve severe beam gain loss caused by
beam squint. On this basis, we propose a scheme of jointly optimizing the
reflection phase shifts of IRS elements and the time delay of DAMs in far-
field and near-field, respectively, which makes the beams at all subcarriers
focusing on the desired direction or location.

The rest of this paper is organized as follows. The far-field and near-field
channel models of IRS-aided wideband THz communications are introduced,
respectively, in Section II. The far-field and near-field beam squint effects
are revealed, respectively, in Section III. In Section IV, we describe our
proposed schemes for overcoming the far-field and near-field beam squint
effects, respectively. Section V presents the simulation results to verify our
proposed schemes, and final conclusions are drawn in Section VI.

_Notations:_ We use the following notations throughout this paper. The scalars
and vectors are denoted by italic letters $x$ and bold-face lower-case
$\bf{x}$, respectively; $(\cdot)^{T}$ denotes the transpose and
$\left|{\cdot}\right|$ stands for the absolute operator of a complex scalar.
min$\\{a,b\\}$ and max$\\{a,b\\}$ represent the minimum and maximum from the
real number $a$ to $b$, respectively.

##  II The Far-Field and Near-Field Channel Models

We consider a basic IRS-aided wideband THz communication as shown in Fig. 1.
In this paper, we mainly focus on investigating the beam squint caused by IRS,
and thus the BS and user are all assumed equipped with single antenna.
Meanwhile, we assume that there is no direct path between BS and user [6]. The
transmission bandwith, carrier frequency and subcarrier index are denoted as
$B$, $f_{c}$ and $M$, respectively. Thus, the $m$-th subcarrier frequency
$f_{m}$ can be written as

$\displaystyle f_{m}=f_{c}+\frac{B}{M}(m-1-\frac{M-1}{2}).$ (1)

Suppose there are $L_{1}$ propagation paths between BS and IRS, and $L_{2}$
propagation paths between IRS and user. Let $u(t)$ and $v(t)$ denote the
channel impulse response (CIR) spanning from BS to IRS and from IRS to user,
respectively [21]. Therefore, the CIR between BS and the $r$-th IRS element
can be expressed as [22]

$\displaystyle
u_{r}(t)=\sum\nolimits_{l_{1}=1}^{L_{1}}{\bar{\alpha}}_{l_{1}}e^{-j2\pi{f_{c}}\tau_{l1,r}^{\rm{BR}}}\delta(t-\tau_{l1,r}^{\rm{BR}}),$
(2)

where ${\bar{\alpha}_{l_{1}}}$ is the equivalent baseband complex gain of the
$l_{1}$-th path and $\tau_{l1,r}^{\rm{BR}}$ is the time delay of the
$l_{1}$-th path between BS and the $r$-th IRS element, where
$l_{1}\in\\{{1,2,\cdots,L_{1}}\\}$ and $r\in\\{{1,2,\cdots,R}\\}$. Similar to
$u_{r}(t)$, the CIR between the $r$-th IRS element and user can be expressed
as

$\displaystyle
v_{r}(t)=\sum\nolimits_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\tau_{l2,r}^{\rm{RU}}}\delta(t-\tau_{l2,r}^{\rm{RU}}),$
(3)

where ${\bar{\alpha}_{l_{2}}}$ is the equivalent baseband complex gain of the
$l_{2}$-th path and $\tau_{l2,r}^{\rm{RU}}$ is the time delay of the
$l_{2}$-th path between the $r$-th IRS element and user, where
$l_{2}\in\\{{1,2,\cdots,L_{2}}\\}$.

Denote $f_{r}(t)$ as the signal reflected by the $r$-th IRS element, we have

$\displaystyle
f_{r}(t)=\theta_{r}u_{r}(t)\ast{s(t)}=\theta_{r}\sum\nolimits_{l_{1}=1}^{L_{1}}{\bar{\alpha}}_{l_{1}}e^{-j2\pi{f_{c}}\tau_{l1,r}^{\rm{BR}}}s(t-\tau_{l1,r}^{\rm{BR}}),$
(4)

where $\boldsymbol{\theta}=[\theta_{1},\theta_{2},\cdots,\theta_{R}]^{T}$ and
denoting $\theta_{r}=\beta_{r}e^{j\phi_{r}}$ as the reflection coefficient of
the $r$-th IRS element, where $\beta_{r}\in[0,1]$ and $\phi_{r}\in[0,2\pi)$
are the amplitude reflection coefficient and phase shift of the $r$-th IRS
element, respectively. The default value of the amplitude reflection
coefficient is set as $\beta_{r}=1,\forall_{r}$ for maximizing the signal
power by IRS. Additionally, $s(t)$ is the signal transmitted by BS. Therefore,
$\boldsymbol{\theta}$ can be simplified as

$\displaystyle\boldsymbol{\theta}=[e^{j\phi_{1}},e^{j\phi_{2}},\cdots,e^{j\phi_{R}}]^{T}.$
(5)

Fig. 1: The system model of IRS-aided downlink wideband THz communications.

The $y_{r}(t)$ is the received signal of the user and denoted as

$\displaystyle y_{r}(t)=$ $\displaystyle v_{r}(t)\ast{f_{r}(t)}+n_{r}(t)$
$\displaystyle=$
$\displaystyle\sum\nolimits_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\tau_{l2,r}^{\rm{RU}}}f_{r}(t-\tau_{l2,r}^{\rm{RU}})+n_{r}(t),$
(6)

where $n_{r}(t)$ is the additive complex Gaussian noise. Substituting (4) into
(6), $y_{r}(t)$ can be expressed as

$\displaystyle y_{r}(t)=$
$\displaystyle\theta_{r}\sum\nolimits_{l_{1}=1}^{L_{1}}\sum\nolimits_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\tau_{l1,r}^{\rm{BR}}}e^{-j2\pi{f_{c}}\tau_{l2,r}^{\rm{RU}}}$
$\displaystyle\cdot{s(t-\tau_{l1,r}^{\rm{BR}}-\tau_{l2,r}^{\rm{RU}})}+n_{r}(t)$
$\displaystyle=$ $\displaystyle\theta_{r}h_{r}(t)\ast{s(t)}+n_{r}(t).$ (7)

According to (7), the CIR of the $r$-th cascaded BS-IRS-user channel can be
expressed as

$\displaystyle
h_{r}(t)=\sum_{l_{1}=1}^{L_{1}}\sum_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\tau_{l1,r}^{\rm{BR}}}e^{-j2\pi{f_{c}}\tau_{l2,r}^{\rm{RU}}}\delta(t-\tau_{l1,r}^{\rm{BR}}-\tau_{l2,r}^{\rm{RU}}).$
(8)

Based on the above, it is clear that the CIR of the $r$-th cascaded BS-IRS-
user channel is relevant to the time delay $\tau_{l1,r}^{\rm{BR}}$ and
$\tau_{l2,r}^{\rm{RU}}$. However, the time delay is different for the far-
field and near-field. Next, based on the constructed CIR, we will analyze the
far-field and near-field channel models, respectively.

###  II-A The Far-Field Channel Model

In the far-field, it is usually assumed that the IRS array size is much
smaller than the distance between BS and IRS as well as the distance between
IRS and user. Based on this, the time delay $\tau_{l1,r}^{\rm{BR}}$ and
$\tau_{l2,r}^{\rm{RU}}$ can be written as

$\displaystyle\tau_{l1,r}^{\rm{BR}}=\tau_{l1}^{\rm{BR}}+(r-1)\frac{d\sin\chi_{l_{1}}}{c}=\tau_{l1}^{\rm{BR}}+(r-1)\frac{\varphi_{l1}^{\rm{BR}}}{f_{c}},$
(9)
$\displaystyle\tau_{l2,r}^{\rm{RU}}=\tau_{l2}^{\rm{RU}}-(r-1)\frac{d\sin\psi_{l_{2}}}{c}=\tau_{l2}^{\rm{RU}}-(r-1)\frac{\varphi_{l2}^{\rm{RU}}}{f_{c}},$
(10)

where $c$ and $d$ are the speed of light and the IRS element spacing,
respectively. We set $d$ as half wavelength, i.e., $d=\frac{\lambda_{c}}{2}$,
where $\lambda_{c}$ is the carrier wavelength. For notational simplicity, we
define $\tau_{l1}^{\rm{BR}}=\tau_{l1,1}^{\rm{BR}}$ and
$\tau_{l2}^{\rm{RU}}=\tau_{l2,1}^{\rm{RU}}$. Moreover, $\chi_{l_{1}}$ and
$\psi_{l_{2}}$ represent the angle of arrival (AOA) of the $l_{1}$-th path
between the BS and IRS as well as the angle of departure (AOD) of the
$l_{2}$-th path between the IRS and user, respectively. Denote
$\varphi_{l1}^{\rm{BR}}$ and $\varphi_{l2}^{\rm{RU}}$ as the normalized AOA
and AOD with the range $[-\frac{1}{2},\frac{1}{2}]$, respectively.

According to (9) and (10), the CIR of the $r$-th cascaded BS-IRS-user channel
in (8) can be obtained as

$\displaystyle h_{r}^{\rm{far}}(t)=$
$\displaystyle\sum\nolimits_{l_{1}=1}^{L_{1}}\sum\nolimits_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}e^{-j2\pi{f_{c}}\tau_{l1}^{\rm{BR}}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\tau_{l2}^{\rm{RU}}}$
$\displaystyle\cdot{e^{-j2\pi{f_{c}}(r-1)\frac{\varphi_{l1}^{\rm{BR}}}{f_{c}}}e^{j2\pi{f_{c}}(r-1)\frac{\varphi_{l2}^{\rm{RU}}}{f_{c}}}\delta(t-\tau_{l1,r}^{\rm{BR}}-\tau_{l2,r}^{\rm{RU}})}.$
(11)

Let
$\alpha_{l_{1}}={\bar{\alpha}}_{l_{1}}e^{-j2\pi{f_{c}}\tau_{l1}^{\rm{BR}}}$
and
$\alpha_{l_{2}}={\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\tau_{l2}^{\rm{RU}}}$,
we have

$\displaystyle h_{r}^{\rm{far}}(t)=$
$\displaystyle\sum\nolimits_{l_{1}=1}^{L_{1}}\sum\nolimits_{l_{2}=1}^{L_{2}}\alpha_{l_{1}}\alpha_{l_{2}}e^{-j2\pi(r-1)\varphi_{l1}^{\rm{BR}}}$
$\displaystyle\cdot{e^{j2\pi(r-1)\varphi_{l2}^{\rm{RU}}}\delta(t-\tau_{l1,r}^{\rm{BR}}-\tau_{l2,r}^{\rm{RU}})}.$
(12)

By applying Fourier transform to (12), the frequency response corresponding of
(12) can be expressed as

$\displaystyle h_{r}^{\rm{far}}(f)$
$\displaystyle=\int_{-\infty}^{+\infty}h_{r}^{\rm{far}}(t)e^{-j2\pi{f}t}\mathrm{d}t$
$\displaystyle=\sum_{l_{1}=1}^{L_{1}}\sum_{l_{2}=1}^{L_{2}}\alpha_{l_{1}}\alpha_{l_{2}}e^{-j2\pi(r-1)(\varphi_{l1}^{\rm{BR}}-\varphi_{l2}^{\rm{RU}})}e^{-j2\pi{f}(\tau_{l1,r}^{\rm{BR}}+\tau_{l2,r}^{\rm{RU}})}$
$\displaystyle=\sum_{l_{3}=1}^{L_{1}L_{2}}\alpha_{l_{3}}^{C}e^{-j2\pi(r-1)\varphi_{l3}^{\rm{C}}(1+\frac{f}{f_{c}})}e^{-j2\pi{f}\tau_{l3}^{\rm{C}}},$
(13)

where $\alpha_{l_{3}}^{C}$, $\varphi_{l3}^{\rm{C}}$ and $\tau_{l3}^{\rm{C}}$
are defined as the equivalent complex gain, angle and delay of the cascaded
BS-IRS-user channel, respectively, where
$\alpha_{l_{3}}^{C}=\alpha_{l_{1}}\alpha_{l_{2}}$,
$\varphi_{l3}^{\rm{C}}=\varphi_{l1}^{\rm{BR}}-\varphi_{l2}^{\rm{RU}}$ and
$\tau_{l3}^{\rm{C}}=\tau_{l1,r}^{\rm{BR}}+\tau_{l2,r}^{\rm{RU}}$.
Additionally, we have $\varphi_{l3}^{\rm{C}}\in(-1,1)$ and
$l_{3}\in{\\{1,2,\cdots,L_{1}L_{2}\\}}$.

###  II-B The Near-Field Channel Model

It is well known that as IRS array size increases, so does the Fraunhofer
distance. Based on this, the CIR of the $r$-th cascaded BS-IRS-user channel
can be written as

$\displaystyle h_{r}^{\rm{near}}(t)=$
$\displaystyle\sum\nolimits_{l_{1}=1}^{L_{1}}\sum\nolimits_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\tau_{l1,r}^{\rm{near,BR}}}$
$\displaystyle
e^{-j2\pi{f_{c}}\tau_{l2,r}^{\rm{near,RU}}}\delta(t-\tau_{l1,r}^{\rm{near,BR}}-\tau_{l2,r}^{\rm{near,RU}}),$
(14)

where $\tau_{l1,r}^{\rm{near,BR}}$ denotes the time delay of the $l_{1}$-th
path between BS and the $r$-th IRS element, and $\tau_{l2,r}^{\rm{near,RU}}$
denotes the time delay of the $l_{2}$-th path between the $r$-th IRS element
and the user, which can be defined as
$\tau_{l1,r}^{\rm{near,BR}}=\frac{d_{l1,r}^{\rm{BR}}}{c}$ and
$\tau_{l2,r}^{\rm{near,RU}}=\frac{d_{l2,r}^{\rm{RU}}}{c}$, where
$d_{l1,r}^{\rm{BR}}$ is the distance of the $l_{1}$-th path from the BS to the
$r$-th IRS element, and $d_{l2,r}^{\rm{RU}}$ is the distance of the $l_{2}$-th
path from the $r$-th IRS element to the user.

Substituting $\tau_{l1,r}^{\rm{near,BR}}$ and $\tau_{l2,r}^{\rm{near,RU}}$
into (14), we have

$\displaystyle h_{r}^{\rm{near}}(t)=$
$\displaystyle\sum\nolimits_{l_{1}=1}^{L_{1}}\sum\nolimits_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}e^{-j2\pi{f_{c}}\frac{d_{l1,r}^{\rm{BR}}}{c}}$
$\displaystyle{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\frac{d_{l2,r}^{\rm{RU}}}{c}}\delta(t-\frac{d_{l1,r}^{\rm{BR}}}{c}-\frac{d_{l2,r}^{\rm{RU}}}{c}).$
(15)

Similar to (13), we take Fourier transform to (15), the frequency response of
(15) can be expressed as

$\displaystyle h_{r}^{\rm{near}}(f)$
$\displaystyle=\int_{-\infty}^{+\infty}h_{r}^{\rm{near}}(t)e^{-j2\pi{f}t}\mathrm{d}t$
$\displaystyle=\sum_{l_{1}=1}^{L_{1}}\sum_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}\frac{d_{l1,r}^{\rm{BR}}}{c}}e^{-j2\pi{f_{c}}\frac{d_{l2,r}^{\rm{RU}}}{c}}e^{-j2\pi{f}(\frac{d_{l1,r}^{\rm{BR}}}{c}+\frac{d_{l2,r}^{\rm{RU}}}{c})}$
$\displaystyle=\sum_{l_{1}=1}^{L_{1}}\sum_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}{\bar{\alpha}}_{l_{2}}e^{-j2\pi{f_{c}}(\frac{d_{l1,r}^{\rm{BR}}}{c}+\frac{d_{l2,r}^{\rm{RU}}}{c})}e^{-j2\pi{f}(\frac{d_{l1,r}^{\rm{BR}}}{c}+\frac{d_{l2,r}^{\rm{RU}}}{c})}$
$\displaystyle=\sum_{l_{1}=1}^{L_{1}}\sum_{l_{2}=1}^{L_{2}}{\bar{\alpha}}_{l_{1}}{\bar{\alpha}}_{l_{2}}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f}{f_{c}})(d_{l1,r}^{\rm{BR}}+d_{l2,r}^{\rm{RU}})}.$
(16)

It is clear that the phase
$-\frac{2\pi}{\lambda_{c}}(1+\frac{f}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})$
is non-linear to the IRS element index $r$ in the near-field, which is in
contrast to the phase in the far-field, i.e.,
$-2\pi(r-1)\frac{d}{\lambda_{c}}(1+\frac{f}{f_{c}})\upsilon$ is linear to the
IRS element index $r$.

##  III The Far-field and Near-Field Beam Squint Effects

In this section, the far-field and near-field beam squint effects are
analyzed, respectively. Specially, the far-field beam squint makes that the
beams at different subcarriers squint toward different directions, which is
distinct from that in the near-field, where the beam squint generates a new
effect as the beams at different subcarriers spread to different locations.

###  III-A The Far-Field Beam squint Effect

For analyze the far-field beam squint effect, we consider a single path and
set as $L_{1}=1$, $L_{2}=1$ for sake of simplicity. Thus, the far-field
channel can be rewritten as

$\displaystyle
h_{r}^{\rm{far}}(f)=\alpha{e^{-j2\pi(r-1)\varphi(1+\frac{f}{f_{c}})}}e^{-j2\pi{f}\tau_{0}},$
(17)

where $\varphi=\frac{d}{\lambda_{c}}(\sin\chi-\sin\psi)$ denotes the
equivalent angle of the cascaded BS-IRS-user channel and
$\tau_{0}=\tau_{0}^{\rm{BR}}+\tau_{0}^{\rm{RU}}$ is the equivalent time delay.
Let $\upsilon=\sin\chi-\sin\psi$, we have

$\displaystyle
h_{r}^{\rm{far}}(f)=\alpha{e^{-j2\pi(r-1)\frac{d}{\lambda_{c}}(1+\frac{f}{f_{c}})\upsilon}}e^{-j2\pi{f}\tau_{0}}.$
(18)

Based on (18), the cascaded channel vector
${\bf{h}}(f)=[h_{1}(f),h_{2}(f),\cdots,h_{R}(f)]^{T}$ can be expressed as

$\displaystyle{\bf{h}}^{\rm{far}}(f)=\alpha{{\bf{a}}((1+\frac{f}{f_{c}})\varphi)}e^{-j2\pi{f}\tau_{0}},$
(19)

where

$\displaystyle{\bf{a}}((1+\frac{f}{f_{c}})\varphi)=[1,e^{-j2\pi\frac{d}{\lambda_{c}}(1+\frac{f}{f_{c}})\upsilon},\cdots,e^{-j2\pi(R-1)\frac{d}{\lambda_{c}}(1+\frac{f}{f_{c}})\upsilon}]^{T},$

is the spatial steering vector. We define
$g(f_{m},\nu,\phi_{r})=\left|{\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi)\boldsymbol{\theta}^{\rm{far}}\right|$
as the gain at frequency $f_{m}$ and it can be expressed as

$\displaystyle g(f_{m},\nu,\phi_{r})=$
$\displaystyle\left|{\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi)\boldsymbol{\theta}^{\rm{far}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j2\pi(r-1)\frac{d}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})\nu}e^{j\phi_{r}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j[\phi_{r}-\pi(r-1)(1+\frac{f_{m}}{f_{c}})\nu]}\right|.$
(20)

By setting $f_{m}=f_{c}$ and $\varphi=\varphi_{0}$ with
$\varphi_{0}=\frac{d}{\lambda_{c}}(\sin\chi_{0}-\sin\psi_{0})$, one can
observe when $\phi_{r}-\pi(r-1)(1+\frac{f_{c}}{f_{c}})\nu_{0}=0$, (20) can
reach its maximum value, and thus we have

$\displaystyle\phi_{r,c}=2\pi(r-1)\nu_{0}.$ (21)

According to (21), the reflection phase shift vector
$\boldsymbol{\theta}_{c}^{\rm{far}}$ can be written as

$\displaystyle\boldsymbol{\theta}_{c}^{\rm{far}}=[1,e^{j2\pi\nu_{0}},\cdots,e^{j2\pi(R-1)\nu_{0}}]^{T}.$
(22)

Therefore, the beam gain achieved by the user under an arbitrary direction
$\varphi$ at frequency $f_{m}$ can be denoted as

$\displaystyle g(f_{m},\nu,\phi_{r,c})=$
$\displaystyle\left|{\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi)\boldsymbol{\theta}_{c}^{\rm{far}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j2\pi(r-1)\frac{d}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})\nu}{e^{j2\pi(r-1)\nu_{0}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\pi(r-1)[2\nu_{0}-(1+\frac{f_{m}}{f_{c}})\nu]}\right|.$
(23)

To maximize the beam gain, it is observed that
$2\nu_{0}-(1+\frac{f_{m}}{f_{c}})\nu=0$ should be satisfied. In this case, the
beam direction can be expressed as

$\displaystyle\nu=\frac{2}{1+{f_{m}}/{f_{c}}}\nu_{0}.$ (24)

From (24), one can observe $\nu\thickapprox\nu_{0}$ for any subcarrier in low
frequency narrowband system due to $\frac{f_{m}}{f_{c}}\thickapprox 1$. This
means that the beams at all subcarriers will focus on the same direction and
the maximum beam gain of each subcarrier can be obtained at this direction.
However, for the ultra-wide band THz system, ${f_{m}}/{f_{c}}\thickapprox 1$
no longer holds, specifically for edge subcarriers, it indicates that angle
$\nu$ changes with the frequency $f_{m}$. This means that the beams at
different subcarriers will squint toward different directions. Moreover, if we
set
${\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi)={\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi_{0})$,
for subcarrier frequency $f_{m}$, (20) should be

$\displaystyle g(f_{m},\nu_{0},\phi_{r,c})=$
$\displaystyle\left|{\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi_{0})\boldsymbol{\theta}_{c}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\pi(r-1)[(1-\frac{f_{m}}{f_{c}})\nu_{0}]}\right|.$
(25)

Similarly, it is clear that the beam gain $g(f_{m},\nu_{0},\phi_{r,c})$
reaches its maximum value since $f_{m}\thickapprox{f_{c}}$ in narrowband
systems, while (25) can only obtain its maximum value around the central
subcarriers in wideband THz IRS systems. This means that the beams at
different subcarriers will be steered towards different directions. We will
show the above phenomenon in simulation results section.

###  III-B The Near-Field Beam squint Effect

In the previous section, we have analyzed the characteristic of the beam
squint in THz IRS communications based on far-field, and next we study the
near-far beam squint.

We assume that the location axis of BS and user are, respectively, $(0,0)$ and
$(x,y)$, and $(x_{R},y_{R})$ denotes the axis of the first IRS element. Thus,
the axis of the $r$-th IRS element is $(x_{R}+(r-1)d,y_{R})$, and we assume
that IRS is an ($1\times{R}$) ULA for the convenience. Here, we still consider
a single path and set $L_{1}=1$, $L_{2}=1$ to ease explanation of the beam
squint. Thus, the near-field channel can be expressed as

$\displaystyle
h_{r}^{\rm{near}}(f)={\alpha}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}.$
(26)

According to (26), the cascaded channel vector
${\bf{h}}^{\rm{near}}(f)=[h_{1}^{\rm{near}}(f),h_{2}^{\rm{near}}(f),\cdots,h_{R}^{\rm{near}}(f)]^{T}$
can be written as

$\displaystyle{\bf{h}}^{\rm{near}}(f)=\alpha{{\bf{b}}((1+\frac{f}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}))}e^{-j2\pi{f}\tau_{0}},$
(27)

where

$\displaystyle{\bf{b}}((1+\frac{f}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}))=$
$\displaystyle[1,e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f}{f_{c}})(d_{1}^{\rm{BR}}+d_{1}^{\rm{RU}})},\cdots,$
$\displaystyle
e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f}{f_{c}})(d_{R}^{\rm{BR}}+d_{R}^{\rm{RU}})}]^{T},$

is the spatial steering vector. Based on the definition of the beam gain in
the previous section, for subcarrier frequency $f_{m}$ at location
$(x^{\prime},y^{\prime})$, we have

$\displaystyle g(f_{m},d_{r}^{\rm{RU^{\prime}}},\phi_{r}^{\rm{near}})=$
$\displaystyle\left|{\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}}))\boldsymbol{\theta}^{\rm{near}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})}e^{j\phi_{r}^{\rm{near}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j[\phi_{r}^{\rm{near}}-\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})]}\right|,$
(28)

where $d_{r}^{\rm{RU^{\prime}}}$ is the distance between the $r$-th IRS
element and the location $(x^{\prime},y^{\prime})$, while the distance from BS
to the $r$-th IRS element $d_{r}^{\rm{BR}}$ is fixed. Moreover, we set
$f_{m}=f_{c}$ and $d_{r}^{\rm{RU^{\prime}}}=d_{r}^{\rm{RU}}$, (28) can be re-
expressed as

$\displaystyle g(f_{c},d_{r}^{\rm{RU}},\phi_{r}^{\rm{near}})=$
$\displaystyle\left|{\bf{b}}^{T}((1+\frac{f_{c}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}))\boldsymbol{\theta}^{\rm{near}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{c}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}e^{j\phi_{r}^{\rm{near}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j[\phi_{r}^{\rm{near}}-\frac{4\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})]}\right|.$
(29)

When $g(f_{c},d_{r}^{\rm{RU}},\phi_{r}^{\rm{near}})$ reaches its maximum
value, we have

$\displaystyle\phi_{r}^{\rm{near}}-\frac{4\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})=0,$
(30)

and

$\displaystyle\phi_{r}^{\rm{near}}=\frac{4\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}).$
(31)

Therefore, the optimal reflecting phase shift vector
$\boldsymbol{\theta}_{c}^{\rm{near}}$ can be expressed as

$\displaystyle\boldsymbol{\theta}_{c}^{near}=[e^{j\frac{4\pi}{\lambda_{c}}(d_{1}^{\rm{BR}}+d_{1}^{\rm{RU}})},e^{j\frac{4\pi}{\lambda_{c}}(d_{2}^{\rm{BR}}+d_{2}^{\rm{RU}})},\cdots,e^{j\frac{4\pi}{\lambda_{c}}(d_{R}^{\rm{BR}}+d_{R}^{\rm{RU}})}]^{T}.$
(32)

Similar to (20), the beam gain obtained with the reflecting phase shift vector
$\boldsymbol{\theta}_{c}^{near}$ at location $(x^{\prime},y^{\prime})$ and
subcarrier frequency $f_{m}$ can be denoted as

$\displaystyle g(f_{m},d_{r}^{\rm{RU^{\prime}}},\phi_{r}^{\rm{near}})=$
$\displaystyle\left|{\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})){\boldsymbol{\theta}_{c}^{near}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})}e^{j\frac{4\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\frac{2\pi}{\lambda_{c}}[2(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})-(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})]}\right|.$
(33)

It is obvious that the maximum value of (33) can be obtained when
$2(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})-(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})=0$,
and then the distance $d_{r}^{\rm{RU^{\prime}}}$ between the $r$-th IRS
element and location $(x^{\prime},y^{\prime})$ can be written as

$\displaystyle
d_{r}^{\rm{RU^{\prime}}}=\frac{2(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}{1+{f_{m}}/{f_{c}}}-d_{r}^{\rm{BR}}.$
(34)

From (34), one can observe that the distance $d_{r}^{\rm{RU^{\prime}}}$ can be
approximated as $d_{r}^{\rm{RU^{\prime}}}\thickapprox{d_{r}^{\rm{RU}}}$ over a
given frequency band in narrowband systems due to ${f_{m}}/{f_{c}}\thickapprox
1$. This means that the beams at all subcarriers will focus on the same
location $(x,y)$ so that the high beam gain of all subcarriers can be obtained
at location $(x,y)$. On the contrary, one can observe that the beams at
different subcarriers will be spread over different locations in wideband THz
IRS systems, which is a new phenomenon referred to as the beam squint in the
near-field. We will show the above phenomenon in simulation results section.

When $(x^{\prime},y^{\prime})=(x,y)$, we have
${\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}}))={\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}))$.
In this case, (35) can be expressed as

$\displaystyle g(f_{m},d_{r}^{\rm{RU}},\phi_{r}^{\rm{near}})=$
$\displaystyle\left|{\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})){\boldsymbol{\theta}_{c}^{near}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\frac{2\pi}{\lambda_{c}}[(1-\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})]}\right|.$
(35)

Based on the above analysis, when $(x^{\prime},y^{\prime})=(x,y)$, the phase
$e^{j\frac{2\pi}{\lambda_{c}}[(1-\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})]}$
will change with the IRS element $r$, which suffers the beam gain loss for
several subcarriers. We will show the above phenomenon in simulation results
section.

##  IV Proposed Schemes for Overcoming Far-field and Near-Field Beam Squint
Effects

In Section III, we have analyzed the far-field and near-field beam squint
effects, respectively. The results have shown exist serious beam gain loss due
to the fact that the reflecting beams by IRS can not focus on the desired
direction or location at all subcarriers. To address the above issue, inspired
by the frequency-dependent characteristic of time delay line, we propose an
effective scheme based on the DAM technique. Specifically, we apply the DAM to
IRS so that the delay of the reflected signals by IRS can be adjusted, which
can effectively mitigate the far-field and near-field beam squints effects by
optimizing the reflection coefficients and delays of the IRS elements.

###  IV-A The Proposed Scheme for Overcoming Far-Field Beam squint Effect

Based on the DAM-based IRS structure, the frequency response of the $r$-th
cascaded BS-IRS-user channel with DAM can be expressed as

$\displaystyle
h_{r}^{\rm{D,far}}(f)=\alpha{e^{-j2\pi(r-1)\frac{d}{\lambda_{c}}(1+\frac{f}{f_{c}})\nu}}e^{-j2\pi{f}\tau_{r}^{\rm{D,far}}}e^{-j2\pi{f}\tau_{0}},$
(36)

where
${\boldsymbol{\tau}^{\rm{D,far}}}=[\tau_{1}^{\rm{D,far}},\tau_{2}^{\rm{D,far}},\cdots,\tau_{R}^{\rm{D,far}}]^{T}$
denotes time delays at IRS. We denote
$g(f_{m},\nu,\phi_{r},\tau_{r}^{\rm{D,far}})=\left|{\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi)\boldsymbol{\theta}_{c}{\boldsymbol{\tau}^{\rm{D,far}}}\right|$
as the beam gain at subcarrier frequency $f_{m}$, which can be represented as

$\displaystyle g(f_{m},\nu_{0},\phi_{r},\tau_{r}^{\rm{D,far}})$
$\displaystyle=$
$\displaystyle\left|{\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi_{0})\boldsymbol{\theta}_{c}^{\rm{far}}{\boldsymbol{\tau}^{\rm{D,far}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j2\pi(r-1)\frac{d}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})\nu_{0}}{e^{j2\pi(r-1)\nu_{0}}}e^{-j2\pi{f_{m}}\tau_{r}^{\rm{D,far}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j[2\pi(r-1)\nu_{0}-\pi(r-1)(1+\frac{f_{m}}{f_{c}})\nu_{0}-2\pi{f_{m}}\tau_{r}^{\rm{D,far}}]}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j[\pi(r-1)\nu_{0}-\pi(r-1)\frac{f_{m}}{f_{c}}\nu_{0}-2\pi{f_{m}}\tau_{r}^{\rm{D}}]}\right|.$
(37)

where $\tau_{r}^{\rm{D,far}}$ characterizes the time delay imposed by the
$r$-th DAM. To maximize the beam gain, we have

$\displaystyle-2\pi{f_{m}}\tau_{r}^{\rm{D,far}}=$
$\displaystyle\pi(r-1)\frac{f_{m}}{f_{c}}\nu_{0}-\pi(r-1)\nu_{0}$
$\displaystyle=$ $\displaystyle-\pi(r-1)\nu_{0}(1-\frac{f_{m}}{f_{c}}).$ (38)

From (38), one can observe that $\tau_{r}^{\rm{D,far}}$ is decided by angle
$\nu_{0}$ and subcarrier frequencies $f_{m}$. Next, we divide
$\pi(r-1)\frac{f_{m}}{f_{c}}\nu_{0}-\pi(r-1)\nu_{0}$ into
$\pi(r-1)\frac{f_{m}}{f_{c}}\nu_{0}$ and $-\pi(r-1)\nu_{0}$. Considering
$\pi(r-1)\frac{f_{m}}{f_{c}}\nu_{0}$ is frequency-dependent, we set
$\tau_{r}^{\rm{D}}$ as

$\displaystyle-2\pi{f_{m}}\tau_{r}^{\rm{D,far}}=\pi(r-1)\frac{f_{m}}{f_{c}}\nu_{0},$
(39)

then, $\tau_{r}^{\rm{D,far}}$ can be written as

$\displaystyle\tau_{r}^{\rm{D,far}}=-(r-1){\nu_{0}}/{2f_{c}}.$ (40)

From (40), one can see that $\tau_{r}^{\rm{D,far}}$ is a function of $r$,
$f_{c}$ and $\nu_{0}$, which can be applied to all subcarriers due to its
frequency independent nature.

Next, the second part $-\pi(r-1)\nu_{0}$ can be realized by adding an extra
phase shift at IRS. Therefore, the optimal reflecting coefficient
$\boldsymbol{\theta}^{\rm{far}}_{c}$ can be written as

$\displaystyle\boldsymbol{\theta}^{\rm{far}}_{c}=[1,e^{j2\pi\nu_{0}}e^{-j\pi\nu_{0}},\cdots,e^{j2\pi(R-1)\nu_{0}}e^{-j\pi(R-1)\nu_{0}}]^{T}.$
(41)

Consequently, the phase shift of the $r$-th IRS element is

$\displaystyle\phi^{\rm{far}}_{r,c}=\pi(r-1)\nu_{0}.$ (42)

Considering $\tau_{r}^{\rm{D}}$ is larger than $0$, the time delay of the
$r$-th IRS DAM can be rewritten as

$\tau_{r}^{\rm{D,far}}=\left\\{\begin{aligned}
-(r-1)\frac{\nu_{0}}{2f_{c}}&&{\nu_{0}\leqslant 0},\\\
(R-1)\frac{\nu_{0}}{2f_{c}}-(r-1)\frac{\nu_{0}}{2f_{c}}&&{\nu_{0}>0}.\end{aligned}\right.$
(43)

Based on the obtained IRS reflecting coefficients and time delays, the beam
gain of the $m$-th subcarrier can be expressed as

$\displaystyle g(f_{m},\nu,\phi_{r}^{\rm{far}},\tau_{r}^{\rm{D,far}})$
$\displaystyle=$
$\displaystyle\left|{\bf{a}}^{T}((1+\frac{f_{m}}{f_{c}})\varphi)\boldsymbol{\theta}^{\rm{far}}_{c}{\boldsymbol{\tau}^{\rm{D,far}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j2\pi(r-1)\frac{d}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})\nu}e^{j\pi(r-1)\nu_{0}}e^{-j2\pi{f_{m}}\tau_{r}^{\rm{D,far}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j\pi(r-1)(1+\frac{f_{m}}{f_{c}})\nu}e^{j\pi(r-1)(1+\frac{f_{m}}{f_{c}})\nu_{0}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\pi(r-1)[(1+\frac{f_{m}}{f_{c}})(\nu_{0}-\nu)]}\right|.$
(44)

It is obvious that (44) can always reach its maximum value for any subcarrier
as long as $\nu_{0}=\nu$. This means that the beam squint effects can be
eliminated by optimizing the reflecting coefficient and time delays to satisfy
$\nu_{0}=\nu$.

###  IV-B The Proposed Scheme for Overcoming Near-Field Beam squint Effect

Similarly, the near-field frequency response of the $r$-th cascaded BS-IRS-
user channel at location $(x,y)$ can be expressed as

$\displaystyle
h_{r}^{\rm{D,near}}(f)={\alpha}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}e^{-j2\pi{f}\tau_{r}^{\rm{D,near}}},$
(45)

where
${\boldsymbol{\tau}^{\rm{D,near}}}=[\tau_{1}^{\rm{D,near}},\tau_{2}^{\rm{D,near}},\cdots,\tau_{R}^{\rm{D,near}}]^{T}$
denotes the time delays at IRS, and $\tau_{r}^{\rm{D,near}}$ characterizes the
time delay imposed by the $r$-th IRS element. Additionally, the beam gain
$g(f_{m},d_{r}^{\rm{RU}},\phi_{r}^{\rm{near}})=\left|{\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}))\boldsymbol{\theta}_{c}^{\rm{near}}e^{-j2\pi{f_{m}}\tau_{r}^{\rm{D,near}}}\right|$
at location $(x,y)$ can be rewritten as

$\displaystyle g(f_{m},d_{r}^{\rm{RU}},\phi_{r}^{\rm{near}})$ $\displaystyle=$
$\displaystyle\left|{\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}))\boldsymbol{\theta}_{c}^{\rm{near}}e^{-j2\pi{f_{m}}\tau_{r}^{\rm{D,near}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}e^{j\phi_{r,c}^{\rm{near}}}e^{-j2\pi{f_{m}}\tau_{r}^{\rm{D,near}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\frac{2\pi}{\lambda_{c}}[(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})(1-\frac{f_{m}}{f_{c}})]-j2\pi{f_{m}}\tau_{r}^{\rm{D,near}}}\right|.$
(46)

Similarly, to obtain the maximum beam gain, we have

$\displaystyle\frac{2\pi}{\lambda_{c}}[(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})(1-\frac{f_{m}}{f_{c}})]-2\pi{f_{m}}\tau_{r}^{\rm{D,near}}=0.$
(47)

According to (47), the phase adjusted by DAM can be written as

$\displaystyle-2\pi{f_{m}}\tau_{r}^{\rm{D,near}}=\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})\frac{f_{m}}{f_{c}}-\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}).$
(48)

Similar to the previous analysis, we divide
$-2\pi{f_{m}}\tau_{r}^{\rm{D,near}}$ into
$\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})\frac{f_{m}}{f_{c}}$
and $-\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})$. It is
obvious that the former is frequency-dependent and can be achieved by setting
$\tau_{r}^{\rm{D,near}}$ as

$\displaystyle-2\pi{f_{m}}\tau_{r}^{\rm{D,near}}=\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})\frac{f_{m}}{f_{c}}.$
(49)

Fig. 2: (a) The normalized beam gain versus the angle in narrowband IRS
systems. (b) The beam model in narrowband IRS systems. (c) The normalized beam
gain versus the angle in wideband systems. (d) The far-field beam squint model
in THz IRS systems.

Thus, $\tau_{r}^{\rm{D,near}}$ can be expressed as

$\displaystyle\tau_{r}^{\rm{D,near}}=-\frac{d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}}{c}.$
(50)

From (50), one can observe that the $r$-th IRS time delay in the near-field is
only relevant to the distance from BS to the $r$-th IRS and from the $r$-th
IRS to the location $(x,y)$ and the speed of light $c$, which is
straightforward to implement for all subcarriers.

Next, the second part
$-\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})$ can be obtained
by DAM, namely setting an extra phase shift to the IRS element. Thus, the
phase shifts provided by IRS can be written as

$\displaystyle\boldsymbol{\theta}_{c}^{near}=$
$\displaystyle[e^{j\frac{4\pi}{\lambda_{c}}(d_{1}^{\rm{BR}}+d_{1}^{\rm{RU}})}e^{-j\frac{2\pi}{\lambda_{c}}(d_{1}^{\rm{BR}}+d_{1}^{\rm{RU}})},e^{j\frac{4\pi}{\lambda_{c}}(d_{2}^{\rm{BR}}+d_{2}^{\rm{RU}})}e^{-j\frac{2\pi}{\lambda_{c}}(d_{2}^{\rm{BR}}+d_{2}^{\rm{RU}})},$
$\displaystyle\cdots,e^{j\frac{4\pi}{\lambda_{c}}(d_{R}^{\rm{BR}}+d_{R}^{\rm{RU}})}e^{-j\frac{2\pi}{\lambda_{c}}(d_{R}^{\rm{BR}}+d_{R}^{\rm{RU}})}]^{T}.$
(51)

The phase shift of the $r$-th IRS element is

$\displaystyle\phi_{r}^{\rm{near}}=\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}).$
(52)

However, one can observe that the time delay
$-\frac{d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}}{c}$ is less than 0, we can add a
common delay $T$ to satisfy $min\\{\tau_{r}^{\rm{D,near}}\\}\geqslant 0$.
Finally, the optimal time delay of the $r$-th IRS can be represented as

$\displaystyle\tau_{r}^{\rm{D,near*}}=T-\frac{d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}}{c}.$
(53)

Based on the above analysis, we investigate the beam gain at location
$(x^{\prime},y^{\prime})$ based on the near-field. By substituting (52) and
(53) into
$g(f_{m},d_{r}^{\rm{RU^{\prime}}},\phi_{r}^{\rm{near}},\tau_{r}^{\rm{D,near*}})=\left|{\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}}))\boldsymbol{\theta}_{c}^{\rm{near}}e^{-j2\pi{f_{m}}\tau_{r}^{\rm{D,near*}}}\right|$
over a given frequency band, the beam gain can be expressed as

$\displaystyle
g(f_{m},d_{r}^{\rm{RU^{\prime}}},\phi_{r}^{\rm{near}},\tau_{r}^{\rm{D,near*}})$
$\displaystyle=$
$\displaystyle\left|{\bf{b}}^{T}((1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}}))\boldsymbol{\theta}_{c}^{\rm{near}}e^{-j2\pi{f_{m}}\tau_{r}^{\rm{D,near*}}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})}e^{j\frac{2\pi}{\lambda_{c}}(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}e^{-j2\pi{f_{m}}(T-\frac{d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}}}{c})}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{-j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})}e^{j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})[(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU}})-(d_{r}^{\rm{BR}}+d_{r}^{\rm{RU^{\prime}}})]}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{r=1}^{R}e^{j\frac{2\pi}{\lambda_{c}}(1+\frac{f_{m}}{f_{c}})(d_{r}^{\rm{RU}}-d_{r}^{\rm{RU^{\prime}}})}\right|.$
(54)

From (54), it is apparent that the maximum beam gain can be obtained for any
subcarrier at location $(x^{\prime},y^{\prime})$, namely
$d_{r}^{\rm{RU}}=d_{r}^{\rm{RU^{\prime}}}$. This means that the beam squint
effect can also be elimated by optimizing the reflecting coefficients and time
delays of the IRS. We will present the performance analysis in Section V.

##  V Numerical Results

In this section, numerical results are presented to illustrate the beam squint
effects and the effectiveness of our proposed DAM approaches in the IRS-aided
THz systems, respectively. Unless otherwise stated, we set $L_{1}=L_{2}=1$,
$M=128$, $f_{c}=200$ GHz and $B=6$ GHz.

###  V-A Far-Field Beam squint Effects and the Proposed Solution

Fig. 3: The normalized beam gain versus subcarrier under different bandwidths.

To better account for the far-field beam squint effect, Fig. 2 shows the
normalized beam gain at different subcarriers against an arbitrary angle
$\upsilon$ with a fixed distance from BS to IRS in narrowband systems and
wideband systems, respectively. We denote $f_{1}$, $f_{c}$ and $f_{M}$ as the
lowest, the central and the highest subcarrier frequency, respectively.
Obviously, Fig. 2(a) illustrates that the beam from IRS to the user at
subcarrier frequency $f_{1}$ and $f_{M}$ can achieve the highest gain in
narrow systems. In other words, beams at all subcarriers can point at the same
direction as shown in Fig. 2(b).

On the contrary, from Fig. 2(c), we can observe that the far-field beam squint
effect makes that the beams at different subcarriers steer toward different
directions. We also plot the beam squint model as shown in Fig. 2(d), where
the user is located at the far-field region and the beams reflected by IRS
fail to focus on the user’s direction, leading to the serious gain loss over
the given bandwidth as discussed in Section III.

Fig. 4: The normalized beam gain versus the subcarriers under different IRS
elements. Fig. 5: The normalized beam gain and beam pattern against the angle
under the proposed scheme. Fig. 6: (a) The location of maximum beam gain at
subcarriers $f_{1}$,$f_{c}$ and $f_{M}$, respectively. (b) The near-field beam
squint model in THz IRS communications. Fig. 7: The near-field beam squint
effect under different IRS elements.

According to (23), we further evaluate the beam squint effect under different
frequencies, bandwidth and IRS elements in Figs. 3 and 4. It can be observed
from Fig. 3 that the maximum gain can be achieved almost across all
subcarriers when the bandwidth and frequency are relatively small. Whereas,
with the bandwidth and frequency increases, the beam gain loss rapidly
increases at different subcarriers. For example, 70$\%$ subcarriers obtain
only its 20$\%$ beam gain when $B=30$ GHz, $f_{c}=200$ GHz. This means that as
the bandwidth $B$ increases, the beam squint effect becomes more pronounced.
In addition, Fig. 4 shows the normalized beam gain versus subcarries with
different IRS elements. Similarly, one can observe that with the IRS elements
increase, the beam squint effect becomes more obvious. For example, the united
beam gain at all subcarries almost the same when $R=10$. However, when $R=64$,
we find that only 23$\%$ subcarriers can obtain a relatively high beam gain.
Fig. 5 illustrates the normalized beam gain and beam pattern against the angle
under the proposed scheme. Obviously, we can find that the beams at other
subcarriers are focused on the spatial direction $\nu_{0}$ and the generated
beams can achieve nearly 98$\%$ gain at the desired direction, which show the
effectiveness of our proposed scheme.

###  V-B Near-Field Beam Squint Effect and The Proposed Solution

In the near-field case, we set the locations of BS, user and the first IRS
element as $(0,0)$, $(3,0)$ and $(1,1)$, respectively. To better explain near-
field beam squint effect, we estimate and draw the locations of beams at
subcarrier frequency $f_{1}$, $f_{c}$ and $f_{M}$.

Fig. 6 plots the near-field beam squint phenomenon in THz IRS communications.
Similarly, we first set IRS phase shifters to place the maximum beam gain at
subcarrier $f_{c}$ on the user. Then, we plot the locations of maximum beam
gains at subcarriers $f_{1}$ and $f_{M}$, as shown in Fig. 6(a). It is clear
that the near-field beam squint causes the maximum beam gains for different
subcarriers to occur different locations, which is different from the far-
field beam squint. To show the near-field beam squint more clearly, we also
plot the beam transmission pattern in Fig. 6(b). In addition, we plot the
near-field beam squint effect under different IRS elements in Fig. 7 by
showing the normalized beam gains at different subcarriers. On can observe the
beam gain loss for most of subcarries. Furthermore, the beam gain loss
increases as the IRS elements increase.

Fig. 8 shows the beam pattern after applying proposed scheme. It can be
clearly seen that the beams at all subcarriers nealy focus on the same
location as shown in Fig. 8(a), which show the effectiveness of the proposed
scheme. Meanwhile, to clearly demonstrate that the beam squint effect has been
eliminated, we also plot the beam model in Fig. 8(b). Thus, we can conclude
from Fig. 8 that the near-field beam squint effect also can be overcome by our
proposed scheme.

Fig. 8: (a) The location of maximum beam gain at subcarriers $f_{1}$,$f_{c}$
and $f_{M}$, respectively. (b)The near-field beam squint model after applying
proposed scheme.

##  VI Conclusion

In this paper, we first pointed out the far-field and near-field beam squint
effects in an IRS-aided wideband THz communications. Specially, the far-field
beam squint makes that beams at different subcarriers squint toward different
directions, while the near-field beam squint spreads beams at different
subcarriers point over different locations. Next, we proposed a scheme that
applying DAM to IRS for overcoming the far-field and near-field beam squint
effects, through joint optimization of the time delays and reflection phase
shifts of IRS elements. Simulations results showed that our proposed schemes
are effective to deal with beam squint effects in THz IRS communications. The
far-field and near-field beam squint issues in multi-user MIMO systems will be
considered in our future work.

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