Performance of finite energy Airy Hermite Gaussian beam in strong atmospheric turbulence

This paper presents intensity evolution of finite energy Airy Hermite Gaussian (FEAHG) beam and its scintillation performance under strong atmospheric conditions. Atmosphere is modeled utilizing split step propagation. Our results reveal that beam evolves into elliptic shape during propagation. Point like scintillation reduction is obtained for all settings of FEAHG beam. For selected apertures, less scintillation index is provided by FEAHG beam. We think that our results are used in free space optical communication systems.


Introduction
Some of the main benefits of 5G and beyond mobile technologies are that they provide low latency and high bit rates. Because of these low latency and higher bit rate need, importance of optical wireless communication (OWC) systems increases. Comparing with thermal and shot noise, scintillation is approximately 10 7 times stronger [1]. It is shown in [2] that higher received power can be obtained since scintillation is less for far infrared wavelengths. Doubinary modulation shows better performance than on-off keying under scintillating conditions [3]. Regarding with this, we investigate that scintillation index of cylindricalsinc Gaussian beam is inversely proportional to sinc argument. Since scintillation index is low, this type of beam has low bit error rate [4]. Recently, we have showed that hypergeometric sinusoidal beam with larger source size has less scintillation index than Gauss beam [5]. Furthermore, scintillation mitigates when Bessel argument of Mathieu-Gauss beam increases [6]. Crosstalk probability of scalar mode of Laguerre Gaussian beam is than vector mode [7]. In addition to this, scintillation index of Laguerre Gauss vortex beam is less than Gauss beam. Besides variations in intensity, effect of scintillation on phase of the propagating field is studied in [8]. On the other hand, radial polarized phase-locked multi-Gaussian Schell-Model vortex beam array evolves into flat topped Gaussian shape after propagation in turbulence [9]. Similarly, optical bottle beam turns into Gaussian beam and its Kurtosis parameter is less than Gauss beam at close distance [10]. Opposed to this, cylindrical-sinc Gaussian beam is more resistive to turbulence since it protects its shape for long distances [11].
Selected beam in this article is composed of Airy and Hermite-Gaussian beams. Airy beam is firstly defined in [12] and Hermite-Gaussian modes are shown in [13] by solving paraxial wave equation in Cartesian coordinates. Airy beam behaves like in free space if propagation distance is short in an atmospheric turbulence [14]. Both area and point like scintillation of Airy beam are less than Gauss beam especially Gaussian source size is nearly 1.6 cm [15]. On the other hand, higher propagation factor is obtained in stronger turbulence and Hermite-Gaussian beam with low Gaussian source size is used [16]. Airy-Hermite Gaussian beam is introduced to the literature in [17] by examining propagation properties in free space. Intensity distribution of FEAHG beam is given after propagation in ABCD optical system [18]. In addition, propagation behavior of three dimensional Airy-Hermite Gaussian beam is presented [19]. FEAHG beam loses its initial profile in a short propagation distance in uniaxial crystal and beam evolves into an arc-like shape [20]. Moreover, for FEAHG beam propagating in photorefractive media, x-direction soliton components protect their * Mert Bayraktar mert.bayraktar@tai.com.tr 1 Turkish Aerospace Industries, Havacılık Bulvarı No: 17, Ankara, Turkey position while y-directions ones slowly enlarges [21]. It is found in [22] that intensity of spatiotemporal self-accelerating Airy-Hermite Gaussian beam has a periodic variation along propagation axis in strongly nonlocal nonlinear media. Lately, Airy beam and Hermite Gaussian beam is combined by establishing an experimental setup involving spatial light modulator [23].
In this paper, we study intensity distribution, point like and aperture averaged scintillation of finite energy Airy Hermite Gaussian beam propagating in atmospheric turbulence. We benefit from split step propagation model to simulate atmospheric turbulence. This original paper is the first research in the literature which analyzes the performance of finite energy Airy Hermite Gaussian beam in atmospheric turbulence. Intensity evolution of finite energy Airy Hermite Gaussian beam is plotted at different distances. Point like scintillation is compared with Gaussian beam which is evaluated using the same method and analytical spherical wave expression. Effect of aperture averaging is analyzed and scintillation performance is plotted against aperture radius. It is expected that results of this study will be beneficial for OWC system designers.

Split step propagation
Source field expression of FEAHG beam is taken from [18] as where s x , s y denote transverse source plane coordinates, sx = sy = 0.7 cm are Gaussian source sizes in x and y directions, respectively. Ai and H m are Airy function and m th order Hermite polynomial in order. In split step propagation approach, received field is calculated numerically as Here, L refers to propagation distance, r x , r y being receiver transverse plane coordinates, indicate Fourier transform, − refers to its inverse, N s refers to number of screens, and k is the wave number and it is evaluated as 2 ∕ where operating wavelength is selected as = 1550 nm . Random phase fluctuations due to atmospheric turbulence are hidden in exp j i . Turbulence is generated based on modified von-Karman power spectral density [24] as where f in being the inner scale, f 0 refers to outer scale of turbulence, and f is the spatial frequency. Fried parameter, r 0 , consists of refractive index structure constant C 2 n and it is taken as 10 −12 m −2∕ 3 in order to satisfy strong turbulent conditions. After obtaining received field, point like scintillation can be computed as where intensity on the receiver plane is calculated as u r r x , r y , L × u r r x , r y , L * . Here * refers to complex conjugate and ⟨⟩ indicates ensemble average. While calculating point like scintillation, we take into the consideration that √ r 2 x + r 2 y < √ 0.5 L . Similarly, aperture averaged scintillation is written as Our simulations are carried out in MATLAB by setting variables as follows: • N = 500, • Source plane dimensions:10 × 10 cm • Transverse plane number of grids: 512 × 512 • Output power: 1W

Results and discussions
This part of the manuscript involves comments about the numerical results that are obtained from MATLAB based upon above formulations. Figure 1 presents intensity evolution of FEAHG beam with a = 1, m = 2 . Beam has elliptical  Gaussian intensities placed on positive x-axis. As the propagation distance increases, beam evolves into Gaussian like distribution by decreasing its peak intensity by approximately 1000 times. As compared to Fig. 1, a is selected as 0.1 in Fig. 2. Peak intensity is observed on the origin and peak of outer intensities decreases going through the edge of the transverse plane. Additionally, peak intensity on the transverse source plane decreases 400 times by changing a . During propagation, outer intensities vanish and beam enlarges more in y-axis than x-axis. Comparing with Fig. 1,   Fig. 3 Intensity evolution of FEAHG beam with a = 1, m = 5 when C 2 n = 10 −12 m −2∕ 3 and = 1550 nm  Fig. 3. We see that beam concentrates on positive x-axis and number of inner intensities increases. When propagation distance increases, beam evolves into single elliptical distribution and loses its peak intensity by 3000 times at long distance. Figure 4 shows intensity evolution of FEAHG beam with a = 0.1, m = 5 . We investigate that large number of outer peaks is observed on the source plane. At long propagation distances, beam turns into elliptical Gaussian like shape. Furthermore, beam size in x-direction is larger than the one in y-direction. Scintillation performance of FEAHG beam is plotted in Figs. 5 and 6. We compare our results with Gaussian beam because lasers radiates in Gaussian distribution. We see that all kinds of FEAHG beam provide scintillation mitigation as compared to theoretical spherical wave and numerical Gaussian beam couple. Among FEAHG beams, beam with a = 0.1, m = 5 has the least scintillation index. Moreover, beams with a = 0.1, m = 2 and a = 1, m = 5 show similar performance to each other. Lastly, FEAHG beam with a = 1, m = 2 provides the highest scintillation index but it is still better than Gaussian beam. As a general investigation, rising trend is observed in weak turbulent conditions, decaying profile is seen in moderate turbulence, and stable behavior under strong turbulent conditions. This way, performance of FEAHG beam can be analyzed under these atmospheric conditions. From here, we can investigate performance of FEAHG beam in weak and moderate turbulence. In weak turbulence conditions, rising slope will be observed at long distances as 5 km [4]. In addition, peak of scintillation curve will shift to longer distances under moderate turbulence conditions as in [6]. Regarding with this, except FEAHG beam with a = 1, m = 2 all beams show better performance in the beginning of weak turbulence region when we focus on the part until nearly 0.75 km. On the other hand, beam with a = 0.1, m = 5 has the best performance like in weak and strong turbulence region. We see that scintillation increases by increasing a . Lastly, beam with a = 0.1, m = 2 with slight raise. Extended form of scintillation curves until 1.5 km is expected at longer distances  in moderate turbulence by considering above references. The same observations in weak turbulence are valid also for moderate turbulence region. In case of aperture averaged scintillation, all settings of FEAHG beam have less scintillation index than Gaussian beam. However, beam having a = 1, m = 2 has the highest scintillation index at small aperture openings. Additionally, scintillation index of beam with a = 0.1, m = 5 is the least for small apertures. When the radius increases, performance of FEAHG beam seem close to each other.

Conclusion
We study propagation and scintillation behavior of FEAHG beam in atmospheric turbulence. Using split step propagation, we calculate the received field and scintillation index. We investigate that less beam size can be obtained for low a and m values. Along propagation axis, beam evolves into elliptic Gaussian distribution. Scintillation index of FEAHG beam is inversely proportional to m . Furthermore, scintillation index increases when a increases. Satisfying aperture averaging conditions, we show that aperture averaged scintillation index of FEAHG beam is less than Gaussian beam. We hope that these results will be used in next generation free space optics systems.

Data availability
The data that support the findings of this study are available from Mert BAYRAKTAR, upon reasonable request.