Modeling and Design of IRS-Assisted FSO System Under Random Misalignment

The strong dependency of the free-space optical (FSO) communication systems on the line-of-sight link put forward an inevitable challenge. Motivated by this fact, in this work, we aim to present and analyze a practical approach to model and design the intelligent reflecting surface (IRS)-assisted FSO communication system incorporating the impact of random misalignment. Contrary to the existing literature, an insightful IRS-assisted FSO model is proposed to envision the impact of the crucial system parameters. To this end, we propose a geometric and misalignment loss (GML) model that incorporates the random statistical modeling of the angle of incidence and the angle of reflection corresponding to the transmitted laser beam. Therefore, the proposed GML model allows us to inspect a realistic transmission and reception scenario of an IRS-assisted FSO system. Moreover, unlike the existing works, we consider independent fading statistics between the laser source (transmitter)-IRS link and IRS-photodetector (receiver) link. All these considerations allow us to envision the performance of a practical IRS-assisted FSO system. Several analytical closed-form expressions corresponding to the statistical distribution functions concerning the proposed GML model are also derived. Our derived closed-form expressions can be used as mathematical tools to examine the performance of an IRS-assisted FSO system under random misalignment. Further, our key findings indicate that under the impact of the random misalignment, the non-orthogonality of the lens with respect to the beam-line significantly impacts the system performance. Furthermore, simulation results validate the presented analysis and reveal unprecedented insights for the proposed IRS-assisted FSO system design.

(CSI), diverse weather conditions, etc., [2], [3]. Atmospheric turbulence (AT) refers to the difference in the temperature and the pressure of the atmosphere caused by solar heating and wind, which leads to variation in the air refractive index along the transmission path. To address the aforestated challenges, certain approaches have been put forth in the literature. For instance, in [2], two novel diversity-improving transmit laser selection schemes under imperfect CSI have been proposed to enhance the overall performance of FSO systems. Statistics of an unmanned aerial vehicle (UAV)-based FSO system have been investigated in [3]. To this end, the authors in [3] have proposed different fluctuation models for the position and orientation of the UAV under diverse weather conditions. In recent studies [4], [5], [6], [7], [8], the application of intelligent reflecting surfaces (IRSs) in FSO has been put forward.

A. Related Works
Motivated by these recent studies on IRS, we put light on the manifold benefits of the IRS. IRS is characterized as an array-like structure made up of planar reflecting elements which can significantly alter the properties of the incident beam [9]. The variation so caused in the incident beam may correspond to the polarisation, phase, amplitude, or frequency [10]. According to [4], IRS can be classified into two categories, i.e., mirrorbased designs and meta-surface-based designs. Compared to the existing methods such as [3], the IRS proves to be a robust and cost-effective alternative for maintaining line-of-sight (LOS) [5]. In [6], the IRS unit cell is partitioned into several subsets and the impact of each unit on the wireless channel has been optimized. Impact based upon physical parameters of the IRS, such as its size, position, and orientation, on the end-to-end FSO channel has been studied in [7]. Further, considering the Huygens-Fresnel principle, a point-to-point IRS-assisted FSO link is assumed in [8], and the IRS is configured into tiles such that both linear phase (LP) and quadrature-phase (QP) profiles are taken into account. The current research on IRS-assisted FSO systems accounts for the turbulence-induced optical signal fading in a manner comparable to that of the non-IRS-assisted FSO system.
Several other works have also drawn our attention to the study of IRS-assisted FSO systems. In [11], a UAV-equipped IRS is proposed for a laser path controllable FSO system. To this end, the system performance is analyzed using ergodic capacity, which determines the impact of atmospheric conditions, configurations of laser devices, and steady ability of UAV on the considered system. Contrary to this, a UAV-based IRS-assisted hybrid FSO/radio frequency (RF) wireless communication system has been studied in [12], which provides system link redundancy. Additionally, the work in [12] also demonstrates the impact of reflecting elements on the phase shift of the optical beam along with the pointing error due to the position and orientation fluctuation of the IRS mounted on the UAV. In [13], the impact of cloud coverage on the high-altitude platform-assisted FSO links has been optimized by the implementation of multiple UAVs equipped with an IRS array. Further, the phase-shift design of an IRS has been proposed in [14] for focusing the optical beam at the receiver (Rx). Also, the authors in [14] introduced a new pointing error model applicable for a scenario with a beam width lesser than the Rx aperture size. In [15], the effect of jamming due to a malicious UAV on the IRS-assisted UAV-based dual-hop FSO is studied. To this end, the authors in [15] demonstrated that the impact of the malicious UAV jammer at the legitimate UAV (utilized as a relay) is more severe than the case where the jammer attacks the destination (Rx). The authors in [16] proposed a power scaling law, according to which the power collected by the Rx lens may be scaled quadratically or linearly depending upon the IRS size or may even saturate at a constant value. In [17], an optical IRS-aided mixed dual-hop FSO and RF system is proposed, wherein polar codes have been introduced to overcome turbulence and building sway induced fading in the FSO links.

B. Motivation and Contributions
In light of the above literature review, we observe that none of these works have considered the impact of the random misalignment (by modeling all the statistical angles to be random) on the considered IRS-assisted FSO system. Note that, the strong dependency of the FSO systems on the LOS path is significantly impacted by the random misalignment. Although the authors in [7] have considered the impact of misalignment, however, it is only in terms of modeling the fluctuation coefficients. Moreover, all the statistical angles (on which the FSO communication is highly dependent), such as angle of incidence, angle of reflection, beam non-orthogonality, etc., are assumed to be deterministic (state-of-the-art) [7], which is impractical. However, our system design approach is unique and more flexible, which provides interesting insights for future investigations of IRSassisted FSO systems. To this end, in this work, an IRS-assisted FSO system considering the impact of random misalignment is proposed. Moreover, a geometric and misalignment loss (GML) model incorporating the impact of random misalignment is also proposed. Motivated by the aforesaid discussion, we highlight the novel contributions of this work as follows: r A practical IRS-assisted FSO system has been modeled and designed considering the random misalignment at the laser source (LS), IRS, and the photodetector (PD). Specifically, we consider the angle of incidence and the angle of reflection of the transmitted laser beam with respect to (w.r.t.) the IRS to be uniformly distributed within a certain range. r Moreover, differing from the existing literature [7], [8], we consider independent fading statistics between LS-IRS and IRS-PD. Therefore, a more practical IRS-FSO channel model is investigated in this work.
r Motivated by the practical characteristics mentioned in the first bullet point, we also consider h IRS to be random which accounts for the fraction of the power received at the PD after getting reflected from the IRS w.r.t. the transmitted power of the LS. Therefore, a more practical investigation corresponding to h IRS is examined in this work.
r Further, a GML model is also proposed which takes into account the random statistical modeling of the angle of incidence, the angle of reflection, beam non-orthogonality,etc. Therefore, the proposed GML model allows us to inspect a realistic transmission and reception scenario of an IRSassisted FSO system. Thus contrary to [7], a practical GML model is proposed in this work.
r For the first time, several analytical closed-form expressions corresponding to the statistical distribution functions concerning the proposed GML model are also derived. These derived closed-form expressions can be used as mathematical tools to examine the performance of an IRSassisted FSO system under random misalignment.
r Furthermore, for the considered set-up, some interesting findings have been marked in this work that have not been previously reported in the literature.

C. Technical Challenge of the Analysis and New Insights
Contrary to [8] wherein deterministic GML is considered, in this work, we model the GML to be random, i.e., a random GML model is proposed. Moreover, differing from the existing works such as [3], [4], [5], [6], [7], [8], we consider independent fading statistics between LS-IRS and IRS-PD for the first time. These realistic considerations put forward an inevitable challenge to develop several closed-form expressions for the distribution functions as investigated in this work. Consequently, an inevitable mathematical challenge to develop closed-form expressions for the performance metrics comes into account.
According to our findings, the proposed IRS-assisted FSO system is significantly impacted by the random beam nonorthogonality (a scenario of random misalignment). To be more precise, h IRS , which accounts the percentage of the power received at the PD after being reflected from the IRS w.r.t. the transmitted power of the LS, is significantly impacted by the random misalignment.
The rest of the paper is organized as follows. Section II discusses the system and channel models of the proposed IRSassisted FSO system. IRS channel characterization accounting a random GML model has been detailed in Section III. In Section IV, we emphasize the error performance of the proposed IRS-assisted FSO system. Section V aims to investigate the numerical results based on the derived theoretical results and draws interesting insights related to the proposed system. Section VI draws concluding remarks. Finally, the paper contains three appendices.

A. System Model
A single-input-single-output (SISO)-IRS-assisted FSO system is considered. The transmitter (Tx) is equipped with a LS which transmits a Gaussian laser beam to the Rx via an optical IRS. For the detection of the transmitted laser beam, the Rx is equipped with a PD and a lens.
The IRS is embedded on the facade of a building which is referred to as the IRS plane, and it is assumed to be centered at the origin in the xy−plane. The IRS plane is configured into T number of tiles, i.e., T = T x T y , where T x and T y are the number of tiles along x− and y−direction, respectively. Each tile has length D x with tile spacing d x in x−direction and width D y with tile spacing d y in y−direction. Therefore, the complete size of the IRS is given by D x,total × D y,total , where Dî , total = TîDî + (Tî − 1)dî,î ∈ {x, y}. In this work, we assume that each tile acts as one of the reflection elements of the IRS. Considering the Tx plane in Fig. 1, the LS positioned at the origin of the x i l y i l z i l −coordinate system emits a laser beam which is incident on one of the tiles at the IRS plane. However, certain degradation factors such as atmospheric loss, AT, and GML caused by building sway, result in deviation of the incident beam w.r.t. its original propagation path. This is also explicitly demonstrated in Fig. 1 by dash-dotted lines. Now, the incident beam forms a beam footprint on the tile with its center at the point p i0 l = (x i0 l , y i0 l , 0), where l ∈ {0, 1, 2} depending upon the beam deviation due to misalignment. To be more specific, when l = 0, the beam follows the original path, i.e., no misalignment. However, when l = 1 or 2, the beam gets deviated to the region R 1 or R 2 , respectively, as shown in Fig. 1. The distance between the LS and the center of the beam footprint is denoted by d LS l . The angle of incidence so formed by the laser beam w.r.t. the IRS plane is denoted by θ i l . The angle formed between the projection of the beam axis on the IRS plane and the x−axis is denoted by φ i l , and it is assumed to be zero, considering that LS and IRS are positioned at the same height. Note that, φ i 0 , φ i 1 and φ i 2 are the projection angles corresponding to l ∈ {0, 1, 2}.
On the IRS plane, all the tiles are centered at point p t = (x t , y t , 0) and are assumed to have a continuous surface [6]. The phase shift profile of the tile is denoted by φ t (p l , p q t ), where p l = (x l , y l , 0) is an arbitrary point on the IRS plane, and p q t = (x q t , y q t , 0) is the point on which the continuous phase shift profile of the tile is centered. Further, a lens with radius r is located at a distance d r l from the IRS plane receiving the reflected beam. The received beam is then observed at an arbitrary point O l located at point p 0 l = (x 0 l , y 0 l , z 0 l ), on the PD deployed at the Rx plane. The center of the lens lies at the origin of the x r l y r l z r l −coordinate system, i.e., at point p r 0 = (x r 0 , y r 0 , z r 0 ). Now, as mentioned previously, the reflected beam may also deviate from its original path, like the incident beam, and therefore, the observation point O l may lie either in R 1 or R 2 . The normal vector along the lens plane makes an angle θ r l with the IRS plane and intersects it at a point p r0 l = (x r0 l , y r0 l , 0) on the IRS plane. The angle between the projection of normal vector on the IRS plane, and the x−axis is denoted by φ r l . Moreover, due to the impact of the random GML on the considered system, we assume the respective angles, viz. θ i l , θ r l , and φ r l to be uniformly distributed, i.e., (θ i l , θ r l , φ r l ) ∼ U [a, b]. Note that, the Fig. 1 is used to demonstrate the position and orientation of the laser beam at LS, IRS, and PD. The other relevant system parameters are discussed in the subsequent sections. Contrary to the non-IRS-assisted FSO system, the performance of an IRS-assisted FSO system is highly dependent on the h IRS . To this end, the incident beam and the reflected beam gets deviated from the center of its beam footprint when we model θ i l and θ r l as random. Thus the terminology random misalignment is emphasized in this work. The implementation of the proposed system model will be beneficial for building-tobuilding communication. Therefore, the proposed system model will be highly applicable for transmitting data at high speed for inter-building communications.

B. Channel Model
We consider an intensity modulation and direct detection (IM/DD) for the SISO-IRS-assisted FSO system, the signal intensity y received by the PD can be modeled as follows where x s is the on-off keying (OOK) modulated symbol transmitted by the LS with average transmit power P i l , ω ∼ (0, σ 2 ω ) is the additive white Gaussian noise (AWGN) with zero mean and variance σ 2 ω impairing the PD and h o is the overall channel gain between LS-PD which is given by where h IRS represents the fraction of power of the LS which is reflected by the IRS and collected by the lens at PD, h LS-IRS = h p 1 h a 1 and h IRS-PD = h p 2 h a 2 are the independent channel gains between LS-IRS and IRS-PD, respectively. We discuss about these as below 1) Atmospheric Loss: It corresponds to the attenuation of the transmitted optical beam over a propagation path which mainly results from the phenomenons such as absorption and scattering by the particles present in the atmosphere. The atmospheric loss is given by [3] and where h p 1 and h p 2 are the components of h o which represent independent atmospheric loss corresponding to LS-IRS and IRS-PD, respectively, and κ is the attenuation coefficient.
2) AT: The AT-induced fading is modeled using Gamma-Gamma (GG) turbulence model [18]. In this work, we consider independent GG turbulence models between LS-IRS and IRS-PD. To this end, h a 1 and h a 2 represent the independent fading channel coefficients corresponding to LS-IRS and IRS-PD, respectively. The probability density function (PDF) of h a 1 and h a 2 based on the GG model is given by [18]  and where K α (·) is the modified Bessel function of the second kind of order α and Γ(·) is the Gamma function. Further, A and B represent the turbulence parameters for a particular turbulence regime.

3) GML Model:
Recall that we position the IRS in the xy−plane. Moreover, considering Fig. 1, z− axis is orthogonal to the xy−plane. Therefore, without loss of generality, we assume that positions of LS, IRS, and lens at PD fluctuate due to the building sway in the x and z directions. The characterization of fluctuation coefficients which represent the impact of GML is illustrated using the coordinate system in Fig. 2. Further, this representation considerably helps us to obtain a simplified analysis of the proposed GML model. The fluctuation coefficients characterization is fundamentally similar to [7]. However, since the transmitted laser beam forms the angle of incidence (θ i l ) and the angle of reflection (θ r l ) w.r.t. x−axis on the IRS plane, as shown in Fig. 1. Therefore, we neglect the fluctuations at the positions of LS, IRS, and PD in y−direction. To this end, we discuss the fluctuation coefficients at LS, IRS, and PD below r At LS: The position of LS fluctuates in the two possible directions, i.e., one parallel to the beam-line (along z i l ) and the other in the direction orthogonal to it (along x i l ), as shown in Fig. 2. So, let us denote the fluctuation coefficients corresponding to LS by z LS and x LS for the two cases, respectively. Now, as the fluctuations of LS are considered to be smaller than the distance, d LS l , so the impact of z LS on h IRS is neglected. PD by z PD and x PD . Hence, z PD is considered in the direction of reflected beam, i.e., along z r l and x PD is taken along x r l , i.e., the direction orthogonal to the reflected beam. Since, d r l is larger than the fluctuation in the direction of the reflected beam, we consider that the impact of z PD on h IRS is negligible. In the following lemma, the misalignment between the center of beam footprint and center of lens at PD is modeled by g incorporating the fluctuation coefficients ( x LS , z IRS , x PD ). Moreover, we specify g in the direction of x r l −axis, as shown in the Fig. 2.
Lemma 1: The misalignment g can be modeled as a function of fluctuation coefficients [7] and other related angles as where cos(θ rp l ) ∈ [0, 1] depicts the non-orthogonality of the lens w.r.t. the beam-line. 1 Moreover, we assume θ rp l to be uniformly distributed, i.e., θ rp l ∼ U[a , b ], as considered for θ i l and θ r l . Proof: The proof of (7) is analogous to the proof of [7,Eq. 19].
Remark 1: Since θ i l and θ r l are modeled as random in (7), therefore, the misalignment g is highly dependent on these two angle parameters. Consequently, h IRS is significantly impacted.
As discussed in Appendix A, the PDF of g is derived as: , u m is the node of the Hermite polynomial [19, p. 924].
Proof: Refer to Appendix A. The channel gain due to GML, h IRS is given by [3] h IRS = 1 where P i l = π 4η |E 0 | 2 w 2 0 is the total power emitted by the LS, η is the free space impedance, E 0 is the electric field at origin of the x i l y i l z i l coordinate system, w 0 is the Gaussian beam waist, A r is the area of the lens at Rx, I IRS (p r l ) is the power intensity of the laser beam emitted by LS and reflected by IRS plane at lens on the PD [20] and point, p r l = (x r l , y r l , z r l ) denotes a point on the lens plane, E IRS (p r l ) is the electric field emitted by the 1 Beam-line refers to the line that connects LS to the center of the beam footprint on IRS plane which is then reflected towards the PD and received by the lens on the PD. Now, for the collection of the maximum fraction of the total power emitted by the LS which is reflected by the IRS on the lens, it is assumed that the tile on the IRS plane is considerably large. Therefore, the lens collects the maximum fraction of the total power when the reflected beam is orthogonal to the lens. However, considering the impact of misalignment at LS, IRS, and PD, respectively, and the configuration of the IRS plane into T number of tiles, the condition of orthogonality of the reflected beam may not always be satisfied. Therefore, to show the impact of misalignment on the reflected beam at the lens on PD, we assume that the reflected beam is non-orthogonal to the lens and orthogonality of the same is considered as a special case. LS and reflected by the IRS and observed by the lens. Hence, we have where, E t (p r l ) is the component of the electric field of the LS which is reflected by the t th tile.

III. CHARACTERIZATION OF IRS CHANNEL
This section determines the electric field emitted by LS and the electric field incident on the tile at the IRS plane. These are then used to determine the total electric field reflected from the tile in Section III-A and the channel gain due to GML between the LS and the PD, h IRS in Section III-B.
The electric field of the Gaussian laser beam emitted by the LS is given by [21] where is the wave number, λ is the wavelength of the laser beam, w(z i l ) = w 0 1 + ( is the radius of curvature of the beam wavefront, and z 0 = πw 2 0 λ is the Rayleigh range. Lemma 2: Considering thatd LS l D x , D y , the electric field emitted by LS which is incident on the IRS is given by , and j denotes the imaginary unit. Proof: The proof of (12) is similar to proof of [8, eq. 7]. However, the rotation matrix used to obtain (12) is different.
Remark 2: The standard form of the rotation matrix used for obtaining (12) is given by, R y i l (θ i l ) = ⎡ ⎣ cos(θ i l ) 0 − sin(θ i l ) 0 1 0 sin(θ i l ) 0 cos(θ i l ) ⎤ ⎦ , where R y i l (θ i l ) denotes the counter-clockwise rotation of θ i l ∼ U[a, b] around y i l −axis. Therefore, using the above matrix for θ i l ∈ [a, b], we obtain the rotation matrix, which is used to derive (12), i.e., R T To this end, as proposed in [8], we have also implemented Huygens-Fresnel Principle to study the impact of IRS on incident beam. According to this principle, every point on the tile at the IRS plane is treated as secondary point source which emits a spherical wave, so the electric field at the Rx plane is given by the sum of the waves originating from all the secondary sources [21]. Therefore, the electric field reflected by the tile on IRS plane to the observation point O l positioned at an arbitrary point p 0 l = (x 0 l , y 0 l , z 0 l ) is given by [21] E t (p 0 l ) = 1 jλ represents a spherical wave, is the tile response, ζ t denotes the tile efficiency factor given by using [8, eqs. 45 and 46] as, ζ t = √ δ 1 δ 2 and Σ t corresponds to the tile area. For simplifying (13), the phases of all the spherical sources, k|p 0 l − p l |, is approximated to |p 0 l − p l | as described in [8], i.e., For practical link distances, the approximation of (14) in (13) by the first two terms of (14), i.e., T 1 and T 2 are considered to be appropriate [8].

A. Electric Field Received From Tile at Lens
In this subsection, we consider the tile phase shift profile as QP profile which is analogous to [8]. Thus the laser beam incident on the tile is redirected to the desired direction and the phase of the desired beam is compensated by the applying additional phase shift. Therefore, the tile phase shift profile is given by Since (θ i l , θ r l , φ r l ) ∼ U[a, b], therefore, the phase components of the incident beam and the desired beam are exploited to obtain the QP profile in (15) explicitly from the considered range, where the constants are given by ) . Therefore, the electric field emitted from LS, reflected by the tile at IRS plane and received at the lens is obtained as the rotation matrix R y r l (θ r l ) is similar to R y i l (θ i l ) and it denotes the counter-clockwise rotation of θ r l ∼ U[a, b] around y r l −axis. Therefore, to express p 0 l in terms of p r l for deriving (16), we use the above rotation matrices to obtain, Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

B. The Channel Gain Due to GML
In the preceding subsection, the reflected electric field from a tile on the IRS plane centered at point p t is determined. So, the channel gain due to GML on the lens w.r.t. the laser beam incident on the IRS and reflected by it on the lens, is obtained by employing (16) and (10) in (9) as Now, as for the size of lens d r l , i.e., 1 holds, so we can approximate x r l = y r l = r 2 in the erf(.) terms in (16) and substitute C dt and C dt . Thus, we get , and (.) * denotes the complex conjugate of a complex number.
Lemma 3: To examine the impact of random misalignment on h IRS , we can express h IRS as a function of g as where ). Now, the closed-form expression of the channel gain due to GML in (19) is given by where , and Proof: Refer to Appendix B. Remark 4: It can be clearly observed from (20) that the parameters b g and c g are dependent on the misalignment g which is further dependent on random θ i l and θ r l . Thus, h IRS (g) is strongly impacted by the parameters b g and c g .
Moreover, the closed-form expression corresponding to the PDF of h IRS (g) is derived as where , ω GLag m and u GLag m are the weights and nodes of the Gauss-Laguerre quadrature, respectively and 1 ≤ m ≤ n. The first hundred values of ω GLag m and u GLag m are well tabulated in [22].
Proof: Refer to Appendix C. Now, the average of misalignment (g avg ) is obtained by integrating the (7) w.r.t. θ i l , so as to see the impact of θ r l , cos(θ rp l ) and Gaussian fluctuations at LS, IRS, and PD, respectively as − (a cos(θ r l ) + sin(θ r l ) log(sin(a)))) + x where g avg is also zero mean Gaussian distributed as it depends upon Gaussian fluctuation coefficients, so g avg ∼ N (0, σ 2 g avg ), with zero mean and variance, − a 2 cos 2 (θ r l ) + sin 2 (θ r l ) log(sin 2 (a)) Therefore, the channel gain due to GML can also be represented as a function of g avg as where b g avg = ρ 2 x l y l g avg +ρ x l y l y l 2ρ y l − ρ x l g avg + x l , and c g avg = −ρ 2 x l y l g 2 avg + 2 y −2ρ x l y l y l g avg 4ρ y l + ρ x l g 2 avg − x l g avg . The proof of (24) is similar to that of (20). Similarly, the PDF of h IRS (g avg ) is given by where ω avg m and u avg m are the weights and nodes of the Gauss-Laguerre quadrature, respectively and 1 ≤ m ≤ n. Note that, The expression of f h IRS (g avg ) (h IRS (g avg )) given in (25) can be obtained by exploiting the relation between g avg and h IRS (g avg ) in (24) and considering that g avg follows a zero-mean Gaussian distribution. Note that, the final PDF of h IRS (g avg ) is given by Gauss-Laguerre quadrature and the proof is analogous to the proof of (21).

IV. ERROR PERFORMANCE ANALYSIS
The error performance of an OOK-modulated IRS-assisted FSO system is obtained taking into account the fading due to AT and GML. The instantaneous bit error rate (BER) is given by where γ = P i l σ 2 n |h p 1 h p 2 | 2 , is the signal-to-noise-ratio (SNR). Now, the average BER of the system is given by We determine the ensemble average of (26) over independent GG distributed random values using MATLAB. Remark 5: A novel BER expression considering independent fading between LS-IRS and IRS-PD with random statistics of h IRS (g) is mathematically formulated in (27). All these practical considerations bring forward a tedious task to derive the closedform expression of (27). However, we will develop a closed-form expression of (27) in the future extension of this work.

V. NUMERICAL RESULTS
In this section, we emphasize a detailed discussion related to the findings of the proposed IRS-assisted FSO system under random misalignment. The analytical plots are obtained by using the derived theoretical results examined in the previous sections; the simulation results are obtained using Monte Carlo simulations averaged over 10 7 channel realizations. The key system and channel parameters are also provided in Table I. Moreover, the statistics of the angles are considered to be uniformly distributed, i.e., (θ i l , θ r l , φ r l ) ∼ U [a, b], where a = π 36 and b = π 3 . Further, . The tile configuration is considered to be 4 × 4, if otherwise stated. Also, an avalanche photodiode is used at the Rx with a gain of ϑ = 250. In Fig. 3, we validate the channel gain due to GML, i.e., h IRS (g) using its exact form (19) and approximated form (20). Recall that h IRS accounts for the fraction of the power received at the PD after getting reflected from the IRS w.r.t. the transmitted power of the LS. The trend of the curves presented in Fig.  3 represents a Gaussian-distributed curve with a very large variance. The large variance results from the random misalignment considered in this work. Moreover, this Gaussian trend is because of the Gaussian fluctuation coefficients at LS, IRS, and PD, as discussed in the GML model of Section II. Therefore, it verifies the correctness of the approximated form. We partition the overall considered range of normalized misalignment into two regimes, i.e., negative values and positive values, as shown in Fig. 3. The negative values correspond to the lesser impact of misalignment, and consequently, h IRS (g) shows an increasing trend. On the other side, when there is no misalignment (g nm = g r = 0), h IRS (g) attains a maximum value. Further, the positive values correspond to more impact of the misalignment, and therefore, h IRS (g) decreases. Moreover, we can observe that as the lens radius increases, the performance curves of h IRS (g) shift upward, which eventually corresponds to the more fraction of the power received at PD. Fig. 4 illustrates the h IRS (g) vs. the θ r l (angle of reflection) performance curves considering different values of θ rp l . We use (20) to plot the performance curves of h IRS (g). The impact of the angle of reflection and number of tiles (T ) can be clearly seen from the figure. Further, it is noted that for different tile configurations, corresponding to the sub-plots illustrated in Fig. 4, i.e., for T = 4 × 4 and T = 6 × 6, the h IRS (g) varies for the proposed system model. To this end, as θ r l increases, h IRS (g) decreases, as expected intuitively. It indicates that at higher values of θ r l , the probability of the received laser beam getting out of the field-of-view of the PD becomes very high. The negligible difference in the values of h IRS (g) when θ r l ranges from 5 • to 10 • is due to the exp(.) and erf(.) terms involved in the expression of h IRS (g). These functions along with other terms in the h IRS (g) given in (20) leads to the small difference in the values of h IRS (g) when the angle ranges from 5 • to 10 • . Further, we consider two different communication scenarios, i.e., beam orthogonality (θ rp l = 0 • as a special case) and beam non-orthogonality (θ rp l = 0 • ). It can be observed from the figure that when θ rp l = 0 • , h IRS (g) takes the maximum value. It is worth mentioning that the scenario of beam orthogonality (θ rp l = 0 • ) corresponds to the maximum power collected by the lens since it corresponds to no misalignment. However, with an increase in the non-orthogonality (impact of misalignment) of the beam, h IRS (g) decreases, which shows the reduction in the received power collected by the lens. Moreover, the impact of the non-orthogonality of the laser beam is dominant when θ rp l = 60 • . In this scenario, h IRS (g) is noted to be minimum.
In Fig. 5, we demonstrate the BER vs. SNR performance curves under different AT regimes for the proposed IRS-assisted FSO system. Moreover, we have also compared the BER performance with the fading scenario considered in [8] for strong turbulence regime. To this end, for the first time, we demonstrate the impact of the independent fading between LS-IRS and IRS-PD on the BER performance as given in (26), denoted by h a 1 and h a 2 , respectively. Therefore, we consider the combined impact of the AT and the random misalignment (incorporated in the GML model, i.e., h IRS (g)) on the BER performance of the proposed IRS-assisted FSO system. Thus the BER performance curves shown in Fig. 5 correspond to a practical IRS-assisted FSO scenario. Further, it is clear from the plot that when independent fading is considered the system performance deteriorates. It can be seen from the figure that the BER performance is the worst for the strong turbulence regime (A = 4.2, B = 1.4), whereas the BER performance is the best for the weak AT regime (A = 11.6, B = 10.1).    Fig. 6 are obtained by replacing h IRS (g) by h IRS (g avg ) in (26). We consider a moderate AT-based scenario. At θ rp l = 0 • (a special case of beam orthogonality), we observe that the considered IRS-assisted FSO system attains a minimum BER ≈ 10 −7 . This is because θ rp l = 0 • corresponds to a no misalignment scenario, and hence the lens collects the maximum received power. However, for any other value of θ rp l (θ rp l = 0 • ), i.e., considering the beam non-orthogonality (impact of misalignment), the BER performance deteriorates in such a way that the BER increases as θ rp l increases. Another interesting observation is that at higher values of θ r l , the BER performance saturates as the received laser beam goes out of the field-of-view of the PD and leading to almost no contribution to the received power.

VI. CONCLUSION
In this work, a practical IRS-assisted FSO system has been modeled and designed considering the impact of random misalignment. To this end, we have done the random statistical modeling of the transmitted laser beam, the reflected laser beam from the IRS, and the received laser beam at an arbitrary observation point on the Rx plane. Moreover, contrary to the existing works, we consider independent fading between the LS-IRS and IRS-PD, which makes the proposed system a realistic IRS-assisted FSO system. Also, a GML model is proposed (given by h IRS (g)), which incorporates the impact of the random misalignment at LS, IRS, and PD. All the aforestated considerations allow us to examine the impact of the beam nonorthogonality on the proposed IRS-assisted FSO system. We also derive several analytical closed-form expressions corresponding to the statistical distribution functions concerning the proposed GML model. These derived closed-form expressions can be used as mathematical tools to examine the performance of an IRS-assisted FSO system under random misalignment. As far as our findings are concerned, we observe that the random beam non-orthogonality (a scenario of random misalignment) has a significant impact on the proposed IRS-assisted FSO system. To be more specific, under the impact of the random misalignment, h IRS which accounts for the fraction of the power received at the PD after getting reflected from the IRS w.r.t. the transmitted power of the LS, is significantly impacted. Consequently, the BER performance degrades. Another insightful observation is that at higher values of θ r l , the BER performance saturates as the received laser beam goes out of the field-of-view of the PD and results in almost no contribution to the received power. Moreover, we have also studied the impact of reflection elements or tiles on the system performance. Simulation results support the analysis that has been presented and provide crucial information for the proposed system design.

APPENDIX A
The closed-form of the PDF of g is obtained as described below From (7), the misalignment g can be written as where I 1 = sin(θ r l ) sin(θ i l ) , I 2 = x LS cos(θ rp l ) , I 3 = sin(θ r l +θ r l ) sin(θ i l ) , I 4 = z IRS cos(θ rp l ) , and I 5 = x PD cos(θ rp l ) . Now, let I 2 = I 4 = I 5 = cos(θ rp l ) , such that x LS = z IRS = x PD = , ∀ ∼ N (0, σ 2 ), is Gaussian-distributed with zero mean and variance σ 2 .
Since (θ i l , θ r l ) ∼ U [a, b], and θ rp l ∼ U[a , b ], therefore, by exploiting the relationship between the variables in I 1 by using [23, p. 271], the PDF of I 1 is given as sin(a) sin (θ i l ) f sin(θ i l ) (sin(θ i l )) f sin(θ r l ) (I 1 sin (θ i l )) d (sin (θ i l )) , where f sin(θ i l ) (sin(θ i l )) = × r −r e ⎛ ⎝ (ρx l y l (x r l −g) + y l ) 2 4ρ y l −ρ x l (x r l −g) 2 − x l (x r l −g) where C e = erf √ ρ y l πr