Behavior of Finite-Energy Fresnel–Bessel Beams in Long Free-Space Optical Communication Links

Abstract

In this paper, we studied the propagation properties of a finite-energy Fresnel–Bessel beam (FEFBB) propagating in a turbulent atmosphere. We analyzed and investigated the system parameters’ impact on the kurtosis parameters and the beam size. The results show that the FEFBB profiles turn into Gaussian-like profiles as they propagate through longer links. In long links, higher-order FEFBBs are sharper than lower-order beams. However, FEFBBs with higher Fresnel numbers are flatter than those with lower Fresnel numbers. On the other hand, beams with higher orders and bigger radii spread less. Moreover, FEFBBs have the advantage of keeping the beam size smaller than that in other classical beams. We anticipate our results will be useful for free-space optics (FSO) designers.

1. Introduction

Bessel functions are an accurate solution to the Helmholtz equation. Therefore, the classic Bessel beams are diffraction-free and offer significant characteristics, namely anextraordinary depth of field and self-healing [1]. Classic Bessel beams have infinite energy. Additionally, Bessel beams inherently feature prominent side lobes that can carry a significant portion of the beam’s energy. However, generating such a beam can be challenging in practice [1]. To address this, combining Bessel and Gaussian features can result in beams with finite energy that are more practical to realize. Moreover, Bessel-like beams may find niche interest. Bessel-like beams are a broader category of beams engineered to approximate the properties of Bessel beams and overcome existing challenges. Beams with such characteristics are important for use in a wide range of applications employing FSO communication systems [2]. In these systems, the propagation of optical beams takes place in the atmosphere, which occur following a horizontal or slant path. Optical beams suffer from random fluctuations in the refractive index, which causes well-known turbulence phenomena. Turbulence affects the optical beam by causing temporal irradiance fluctuations and phase fluctuations. Thus, strong limitations need to be addressed in terms of propagating laser beams and the degradation of the bit error rate performance of FSO communication links [3]. Many approaches have been exploited to overcome the existing limitations, among which is the utilization of novel laser beams for FSO systems. Recently, different beams have been considered for FSO links, such as a cylindrical-sinc Gaussian beam, which is a half-order spherical Bessel beam [4], Airyprime [5], the Mathieu beam [6], and an Ince–Gaussian beam [7]. Besides this, Bessel beams of different types have been studied theoretically and realized experimentally [8,9,10,11]. Among the Bessel beam family, Bessel beams with angular orbital momentum have been considered. The beam wander of these beams propagating through atmospheric turbulence was studied theoretically and experimentally in [12]. This study revealed that the impact of a turbulent atmosphere on Bessel beams of a high order is less severe. On the other hand, the transformation of Bessel beams with angular orbital momentum as they propagate in a turbulent atmosphere has been analyzed as well [13]. Here, it was found that the shape of a Bessel vortex beam with a higher topological charge was more stable as the beam propagated in a turbulent atmosphere [13]. In turbulent media, Bessel-like beams suffer from degradations in capacity compared to Gaussian beams. Yet these beams have a superior capacity when compared to OAM-based single-photon links [14]. In another study, it was shown that the impact of atmospheric turbulence on Bessel–Gaussian beams is stronger for beams with an annular shape [15]. Besides this, in 2014, Lorenser et al. [16] demonstrated that in a low-Fresnel-number regime, energy-efficient Bessel beams with a small number of sidelobes could be generated. Here, the beam was shaped by utilizing a setup incorporating a liquid crystal spatial light modulator (SLM). Unlike the traditional axicon-generated Bessel–Gaussian beams, which have a limited propagation region and a weak central intensity, FEFBB beams overcome these drawbacks by utilizing a customized initial phase and amplitude distribution. This method improves the intensity and stability of the central lobe, making it significantly more suitable for free-space optical communication and other long-distance applications [16]. Afterward, Halba et al. examined the generation of generalized spiraling Bessel beams using Fresnel diffraction of Bessel–Gaussian beams [17]. We show that FEFBBs have a lower scintillation than beams with trigonometric multiples such as cosh, sinh, and sin Gaussian beams [18]. Thus, understanding the behavior of FEFBBs as they propagate in a turbulent atmosphere will shed light on the importance of optimizing the system parameters to improve the performance of FSO systems. Accordingly, this work examines the propagation properties, namely the beam size and kurtosis parameter, of FEFBBs.

Scientists have widely used random phase screens (RPSs) to analyze the propagation of beams propagating through turbulent environments. Accordingly, the propagation properties of a Gaussian beam were investigated in turbulence generated based on the Kolmogorov power spectrum [19]. Then, the scintillation index of the Gaussian beam was calculated numerically using a modified von-Karman phase screen [20]. Moreover, the RPS was modified to rainy conditions to study the propagation of a laser beam [21]. The RPS approach allows scientists to analyze non-traditional beams, too. Regarding this, the evolution of Ince–Gaussian beams into Gaussian beams using an RPS has been reported [22].

The propagation properties of such Fresnel–Bessel beams have not yet been studied, and they propagate in any random media. Since Bessel-type beams are important for FSO communication systems, this study considers the propagation of FEFBBs in weak, turbulent atmospheres considering the high attitude of aircraft. Optical systems on aircraft such as laser range finders or laser designators can be modeled. As is well known, the amount of power the receiver captures is a critical indicator in evaluating FSO communication link performance. Therefore, the received intensity, the beam size, and the kurtosis parameter of finite-energy Fresnel–Bessel beams as they propagate through long-link FSO links are examined. Since the selected beam has degrees of freedom in phase and amplitude, amplitude and I/Q modulation is applicable.

2. Theoretical Formulations

The electric field of an FEFBB at the source plane ($L=0$) can be described as [16]

$$U\left(\tilde{\rho },\tilde{z}\right)={\displaystyle \frac{i2\sqrt{2\pi }}{\omega }}N\tilde{z}\mathrm{e}\mathrm{x}\mathrm{p}\left[i\pi N\left({\displaystyle \frac{{\tilde{\rho }}^2}{\tilde{z}}}-\tilde{z}\right)\right]\underset{0}{\overset{\infty }{\int }}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\tilde{\rho }\tilde{z}\right)}^2\right]\mathrm{e}\mathrm{x}\mathrm{p}\left[i\pi N{\left({\tilde{\rho }}^2-1\right)}^2\right]J_n\left(2\pi N\tilde{\rho }{\tilde{\rho }}_0\right){\tilde{\rho }}_{0}d{\tilde{\rho }}_0$$

(1)

where $\tilde{\rho }$ = ${\displaystyle \frac{\rho }{\omega }}$ and $\tilde{z}$ = ${\displaystyle \frac{z}{L_{\mathrm{⍵}}}}$ are normalized coordinates, and $\left(\rho ,z\right)$ are the cylindrical coordinates. Here, ⍵ and L_⍵ are the beam radius and the approximate axial dimension, respectively, and N is the Fresnel number. Further, ${\tilde{\rho }}_0$ is the normalized input plane integration variable. $J_n$ is the Bessel function of order n. Accordingly, the average intensity of the FEFBB can be obtained as [16]

$$I(\tilde{\rho },\tilde{z})=U(\tilde{\rho },\tilde{z}) U^{*}(\tilde{\rho },\tilde{z})={\displaystyle \frac{8\pi }{{\omega }^2}}N\tilde{z} \exp (-2{\tilde{z}}^2) J_0^2 \left(2\pi N\tilde{\rho }\right)$$

(2)

where * refers to a complex conjugate. The average intensity of the FEFBB is illustrated in Figure 1. We see in Figure 1 that the beam enlarges according to the Bessel function order. In addition, more rings can be observed in the transverse plane as the Fresnel number increases. Fresnel diffraction theory can simplify the propagation of laser beams. Thus, the distribution of the beam in the frequency domain can be obtained as [23]

$$U_{l-1}\left(f_x,f_y\right)=\mathfrak{I}\left[U\left(x_{l-1},y_{l-1}\right)\right]$$

(3)

where $\mathfrak{I}$ is the Fourier transform. The beam after it passes through a diffractive plane at layer $l-1$ is as follows [24]:

$$U_{r_{l-1}}\left(f_x,f_y\right)=\mathfrak{I}\left[U_{l-1}\left(x_{l-1},y_{l-1}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[j{\phi }_i(x_{l-1},y_{l-1})\right]\right]$$

(4)

where $\left(x_{l-1},y_{l-1}\right)$ is the Cartesian coordinates in layer $l-1$, and ${\phi }_i\left(r\right)$ is the random phase fluctuations due to the turbulent atmosphere.

According to diffraction theory, the electric field of the beam after it propagates to layer $l$ can be represented as

$$U_{r_l}\left(x_l,y_l\right)={\mathfrak{I}}^{-1}\left[U_{r_{l-1}}\left(f_x,f_y\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[jk(L_l-L_{l-1})\right]\exp \left[-j\pi \lambda (L_l-L_{l-1})\left({f_x}^2,{f_y}^2\right)\right]\right]$$

(5)

The average intensity of the beam at the receiving side can be expressed as in Equation (6) by multiplying the electric field of the beam at the receiving side by its complex conjugate:

$$〈I_{r }(\mathrm{r}, L)〉 ={\displaystyle \frac{1}{N_r}}{\sum }_{i=1}^{N_r}{U}_{r_l}(\mathrm{r}, L)U_{r_l}^{*} (\mathrm{r}, L)$$

(6)

Here, < > means the average, and r = ($r_x,r_y$) is the Cartesian coordinates in the receiver plane, $L$ is the propagation distance, and $N_r$ represents the number of realizations. To achieve a theoretical average, 500 realizations are conducted

The kurtosis parameter is one important propagation property that is used to evaluate beams propagating in atmospheric turbulence. It refers to the sharpness of the beam and indicates whether the beam becomes flatter or sharper as it propagates through the atmosphere. Therefore, this parameter is calculated using the fourth, second, and zero moments in the x-direction of the receiver plane as follows [22,25];

$$K_x={\displaystyle \frac{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} r_x^4⟨I_r\left(r_x,r_y\right)⟩\mathrm{d}{r}_x\mathrm{d}{r}_y{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} ⟨I_r\left(r_x,r_y\right)⟩\mathrm{d}{r}_x\mathrm{d}{r}_y}{{\left[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} r_x^2⟨I_r\left(r_x,r_y\right)⟩\mathrm{d}{r}_x\mathrm{d}{r}_y\right]}^2}}$$

(7)

Due to the symmetrical nature of the beam, the behavior of the kurtosis parameter in the y-direction is the same as that in the x-direction, so in the remainder of this work, the kurtosis parameter including the x-y directions is considered.

Beam size is another parameter that helps to understand the propagation of laser beams as they propagate through the atmosphere. Beam size is related to the kurtosis parameter and can be expressed as follows [22,25]:

$$\mathrm{w}(z)={\left[{\displaystyle \frac{2\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} r_x^2⟨I_r\left(r_x,r_y\right)⟩\mathrm{d}{r}_x\mathrm{d}{r}_y+{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} r_y^2⟨I_r\left(r_x,r_y\right)⟩\mathrm{d}{r}_x\mathrm{d}{r}_y\right)}{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} ⟨I_r\left(r_x,r_y\right)⟩\mathrm{d}{r}_x\mathrm{d}{r}_y}}\right]}^{0.5}$$

(8)

3. Numerical Results

The behavior of an FEFBB that is propagating in a 6 km FSO communication link operating at 1550 nm is investigated. A split-step approach is utilized to model the turbulent atmosphere channel. To obtain reliable results, we evaluate the model against the constraints established in the literature [20,22]. In this context, the communication link is divided into 25 screens each with a grid size of $1024\times 1024$ grids. The atmosphere structure constant $C_n^2$ is set to ${10}^{-17}{\mathrm{m}}^{-2/3}, {2\times 10}^{-15}{\mathrm{m}}^{-2/3},$ and ${10}^{-14}{\mathrm{m}}^{-2/3}$. In this scenario, the structure constant is calculated using the Hufnagel–Valley model by providing the attitude of the laser designator installed on an aircraft or a helicopter [23,26]. Consequently, the intensity of the FEFBB after it propagates for 6 km in a turbulent atmosphere is presented in Figure 2 and Figure 3. Figure 2 illustrates the effect of the Bessel beam order on the received intensity. On the other hand, Figure 3 presents the effect of the Fresnel number on the received intensity, where it is clear that the Fresnel number does not affect the FEFBB profiles as it propagates in a turbulent atmosphere. Thus, it is obvious that as the beam propagates for longer, it evolves into a single petal, losing the central dark hollow region and decreasing the bright rings significantly. Here, as the FEFBBs propagate for $3 \mathrm{k}\mathrm{m},$ the beam profile tends to have a diamond shape at the center. Due to this beam evolution, the outer rings come together and generate a diamond shape first and then evolve into a Gaussian shape. So, the major factor in this evolution is the random changes in the refractive index in the atmosphere. Then, as the propagation distance increases to 6 km, it turns into a circular shape. Moreover, it is clear that FEFBBs with higher beam orders keep their intensity levels higher as they propagate. However, as the Fresnel number decreases, the beams preserve higher intensity as they propagate.

Figure 4 shows the impact of the fundamental beam waist on the variation in the beam size against the propagation distance for FEFBBs that propagate under different turbulence levels. It is observed that as the turbulence level increases, the FEFBBs spread more quickly. As the beam order increases, under different turbulence levels, the FEFBBs with bigger beam waists have the advantage of spreading less than those with small beam waists, as observed in Figure 4. Moreover, FEFBBs with higher beam orders have more greatly reduced beam sizes than those with lower beam orders. Here, Figure 4 illustrates that beams with a beam order of 10 and a beam waist of 5 cm have a lower beam size under both strong and weak turbulence.

Figure 5 shows the variation in the kurtosis parameter against the propagation distance for FEFBBs propagating under different turbulence levels. Under weak and moderate turbulence, as the propagation distance increases, the kurtosis parameter increases. On the other hand, under strong turbulence, the kurtosis parameter starts to increase as the propagation distance increases up to about 2.5 km and then starts to decrease. Moreover, as the beam order increases, FEFBBs with a higher beam waist have a higher kurtosis parameter at longer propagation distances. However, the beam waist’s impact on increasing the kurtosis parameter of the FEFBBs is more noticeable for beams that propagate under strong turbulence than those that propagate under moderate and weak turbulence.

On the other hand, the behavior of the FEFBBs as they propagate in a turbulent atmosphere is compared with the behavior of different laser beams. In this regard, both the beam size and the kurtosis parameter of FEFBBs are compared with the beam sizes and the kurtosis parameters of a Gaussian beam (GB), a Gaussian vortex beam (GVB), cosine and cosine hyperbolic Gaussian beams (CosGB and CoshGB), sine and sine hyperbolic Gaussian beams (SinGB and SinhGB), and a Bessel vortex beam (BVB). In this context, Figure 6 shows the variation in the beam size of these beams as they propagate under strong turbulence. It is observed that FEFBBs are effectively reduced in beam size, especially for long links. Furthermore, FEFBBs with a smaller beam waist have the advantage of reduced beam sizes compared with those of the other beams.

Besides this, the variation in the beam size as the beams propagate under moderate turbulence is examined in Figure 7. According to the general trend, the beam sizes increase as the propagation distance increases for all beam types. This ratio of the increase in beam size for beams with bigger beam waists is higher than the increase in the beam size of beams with small beam waists. Moreover, as observed from Figure 7, FEFBBs have reduced beam sizes as they propagate as compared with those of the other beams. This advantage is more dominant for beams with small beam waists and comparable at long propagation distances. On the other hand, BVBs with bigger beam waists have the lowest beam sizes for propagation distances up to 3 km, and then the FEFBBs start to lower in their beam sizes.

Furthermore, the behavior of the kurtosis parameter of the FEFBBs is compared with the behavior of the kurtosis parameter of a GB, GVB, CosGB, CoshGB, SinGB, SinhGB, and BVB under both strong and moderate turbulence, as shown in Figure 8 and Figure 9. Figure 8 shows the variation in the kurtosis parameter with the propagation distance as the beams propagate under strong turbulence. The general behavior of the kurtosis parameter of the GB, GVB, CosGB, CoshGB, SinGB, SinhGB, and BVB is almost saturated as the propagation distance increases. However, the kurtosis parameter of the FEFBBs increases and then decreases as the propagation distance increases. On the other hand, the results reveal that FEFBBs with small beam waists have the highest kurtosis parameter and exhibit a sharper profile than those of the other beams, as seen in Figure 8a. Besides this, for links longer than 3.5 km, FEFBBs with higher beam waists still have the highest kurtosis parameter values, as observed from Figure 8b. Furthermore, the FEFBBs with an order of 10 have the highest kurtosis parameter values, as illustrated in Figure 8b.

Finally, Figure 9 compares the variation in the kurtosis parameter of different beam types propagating in moderate turbulence. It is observed that the FEFBBs with a small beam waist have higher kurtosis parameters than those of the other beams for links longer than 1 km from Figure 9a. However, this behavior of the FEFBBs with bigger beam waists starts to be dominant for distances longer than 2.5 km since FEFBBs have kurtosis parameters lower than those of a GB, CosGB, CoshGB, and BVB, as shown in Figure 9b.

4. Conclusions

The propagation properties of FEFBBs propagating in turbulent FSO communication links are assessed in detail here. The impact of the beam order, beam radius, and Fresnel number on both the kurtosis parameter and the beam size of FEFBBs is investigated. In this regard, our numeric results illustrate that the profiles of FEFBBs that propagate for 6 km in turbulence lose their central spots and bright rings as they propagate. On the other hand, higher-order FEFBBs diverge less, so they spread less as they propagate. Besides this, FEFBBs with a smaller beam waist have reduced beam sizes compared to those of a GB, GVB, CosGB, CoshGB, SinGB, SinhGB, and BVB. Moreover, FEFBBs have the highest kurtosis parameters for 6 km links under different turbulence levels. This presentation of both the beam sizes and the kurtosis parameters explains the benefit of utilizing FEFBBs for FSO communication links where high intensity, precision, or security is the prior requirement. In this regard, the results obtained will pave the way for the use of FEFBBs to improve the performance of directed infrared counter-measure systems, 5G backhaul or data centers, and aircraft-to-aircraft or ground-to-UAV links.