1. Introduction

In recent years, the arena of wireless optical communication has emerged as a promising contender for high-speed data transmission, capturing the interest of both the scientific community and industry practitioners [1–5]. This communication modality, which relies on light waves to relay information, boasts high bandwidth capabilities [6,7] and enhanced security features [8,9] over conventional wireless radiofrequency (RF) communications, positioning it as a futuristic vector for data exchange. Wireless optical communication systems have demonstrated enormous potential across several domains, including data centers [10,11], urban networking [12,13], and intersatellite as well as satellite-to-ground links [14–16].

However, to achieve widespread adoption, a crucial challenge involves extending the transmission range of light waves through the atmosphere while maintaining robust communication quality [17,18]. The geometric spread of laser beams poses a limitation to the transmission distance, diminishing signal strength and fidelity. To overcome these constraints and to augment the feasible transmission range of wireless optical communications, researchers are exploring an array of solutions. These include enhancing the quality of laser beams [19–22], employing advanced modulation techniques [23,24], and achieving more proficient control over beam divergence [17,25,26] and propagation propagation [4,27]. The miniaturization of devices and energy efficiency optimization also play a pivotal role in steering the commercial viability of wireless optical communication technologies.

This paper delves into the modulation of the Rayleigh length of a Gaussian beam through the judicious application of lenses, and contemplates how this approach could potentially enhance the long-range transmission capacity of wireless optical communication systems. With meticulous consideration, our scholarly exposition seeks to contribute to the evolving dialogue on how optical beam control can serve as a linchpin in the advancement of wireless optical communication frontiers.

2. Concept

In this paper, we embark on a comprehensive analysis of the modulation of a Gaussian beam’s Rayleigh length by applying a lens with a specific focal length. The principal focus lies in the thorough computational investigation of how varying the lens characteristics can effectively modify the propagation properties of a Gaussian beam to achieve desirable communication distances. Through meticulously derived formulas and corresponding graphical illustrations in Fig. 1, we unpack the behavior of the Gaussian beam waist and its resulting Rayleigh length under the influence of a focusing optical lens.

 

Fig. 1. (a) The principle of extending Rayleigh length. (b) Relative Rayleigh lengths of a beam pre and post lens passage and their variation with lens focal length.

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Figure 1 illustrates the principle of extending the Rayleigh length and the variation of relative Rayleigh lengths before and after passing through a lens. In Fig. 1(a), the modulation of a Gaussian beam is achieved by employing a lens with a known focal length. It is common knowledge that a lens refracts light, causing the concentration of a Gaussian beam at a focal point and resulting in a reduced Gaussian beam spot. The lens depicted in Fig. 1(a) has a focal length, F and is positioned at the beam waist with a radius, ${\omega _0}$.

According to the definition of the Rayleigh length, which is the distance from the beam waist to the point where the cross sectional area of the beam is doubled due to diffraction, when the beam propagates a distance equal to its original Rayleigh length, ${z_0}$, the beam’s waist expands to $\sqrt 2 $ times its initial size. For a Gaussian beam emitted in a collimated fashion with a waist radius ${\omega _0}$, the Rayleigh length, ${z_0}$, can be quantified by the equation ${z_0} = {\omega _0}^2/\lambda $, where $\lambda $ represents the wavelength of the beam. This relationship underscores the inherent properties of a Gaussian beam, showing how the Rayleigh length is a function of both the waist radius and the beam’s wavelength. To ensure the generality of our simulation results, we have normalized the parameters in the subsequent simulations. This process allows us to focus specifically on the optimization of the Rayleigh length of the beam while keeping the simulation conditions consistent.

Furthermore, we have set the Gaussian beam quality factor, ${\textrm{M}^2}$, to 1, implying that for the purposes of our modeling, the beam is considered ideal. This idealization is crucial as it negates the influence of factors such as beam divergence and imperfections that are otherwise present in practical scenarios. The quality factor of ${\textrm{M}^2} = 1$ represents a perfect Gaussian beam profile and allows us to focus solely on the theoretical aspects of beam propagation in accordance with Gaussian beam optics.

When considering an optimally-focused system, after passing through the lens, the beam converges to a minimum waist radius, $\omega _0^\mathrm{^{\prime}}$, at a distance, l, given by:

Upon propagation over a certain distance, where the beam waist diverges to $\sqrt 2 {\omega _0}$, the transmission distance, $l^{\prime}$, can be calculated as:

At this point, the total distance from the beam’s source is:

The introduction of the lens extends the Rayleigh length to $z_0^{\prime}$, as per the definition of the Rayleigh length.

Figure 1(b) presents the results of analytical computations for the variation of ratio of altered Rayleigh length, denoted as $z_0^{\prime}$, to the original Rayleigh length, denoted as ${z_0}$, against different lens focal lengths, represented by F. The figure shows that the Rayleigh length progressively increases with increasing focal length, surpasses the initial length, reaches a peak, and subsequently decreases. Detailed calculations indicate that the Rayleigh length reaches its apex with $z_0^{\prime}/{z_0}$ = 1.4145 when $F/{z_0}$ ≈ 1.3761.

3. Method

3.1 Extension of the Rayleigh distance

Building upon the derived formulas, simulation-based transmission trials are conducted to validate the effectiveness of extending the Rayleigh length. These simulations employe both theoretical Gaussian spatial distribution models and diffractive integral methodologies for analysis. Within Fig. 2(a), we present the comparative decay profiles of the Gaussian beam’s power as a function of distance, computed under conditions of an absent lens and an optimally-focused lens using both aforementioned theoretical and diffractive integral approaches. The power loss mentioned is evaluated by measuring the decrease in power of the Gaussian beam within a specified area around its central transmission path. This area is a circle whose boundary extends to a distance equal to twice the radius of the beam’s waist. The attenuation of the beam’s power is calculated by comparing the power within this circular area to the power of the beam when it was initially emitted. Consistency is observed between the outcomes of the two computational strategies, with a discernible attenuation reduction following the application of the lens apparent in the results.

 

Fig. 2. (a) Simulation results of theoretical Gaussian spatial distribution models and diffractive integral methodologies (b) Cross-sectional distributions of the Gaussian beam’s propagation without lens. (c) Cross-sectional distributions of the Gaussian beam’s propagation with lens ($F/{z_0}$ = 1.3761).

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Figures 2(b) and (c) exhibit the power cross-sectional distributions of the Gaussian beam’s propagation with and without the lens, respectively, showcasing post-optimization improvements. A clear contraction in the beam divergence is observed when the beam passes through the lens, which is optimally focused with a focal length set at $F/{z_0}$ = 1.3761. This adjustment results in an increase in the Rayleigh length, which is a measure of beam quality, enhanced by 41% compared to its original value, demonstrating the realized enhancement in longitudinal propagation stability.

The experimental setup for measuring the extension of Rayleigh length is illustrated in Fig. 3(a). A laser beam with a wavelength of 532 nm outputs a linearly polarized Gaussian beam with an approximate diameter of 0.75 mm. The beam is expanded by a factor of six using a beam expander (BE) and its polarization direction is ensured by a polarizer. A half-wave plate (HWP) is employed to rotate the polarization direction to match the operational orientation of the spatial light modulator (SLM). The SLM is loaded with a holographic image possessing intensity modulation. After calculating the phase distribution for different lensed Gaussian beams, a blazed diffraction grating along the x-direction is added to create the final phase-modulated hologram. Via the grating, first-order diffraction energy is angularly separated from other orders, followed by blocking with a spatial filter composed of a 4f system (f1 = f2 = 400 mm). A CCD camera (Mako G-223B) mounted on a translation stage begins to move from near the imaging plane of the SLM to observe the intensity distribution during propagation.

 

Fig. 3. (a) Rayleigh distance extension experimental setup. BE. beam expander; M1∼M5, mirror; POL, polarizer; HWP, half-wave plate; BS, beam splitter; L1∼L2, lens; SLM, spatial light modulator; CCD, charge coupled device. (b) Experiment data and fitting curve of beam radius at ($1/{e^2}$) and transmission distance. (c) Emitter spots and the spot enlarged by $\sqrt 2 $ in Beam Radius without lens. (d) Emitter spots and the spot enlarged by $\sqrt 2 $ in Beam Radius with lens.

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In the experiment, Gaussian beams with a waist radius of 2.9 mm are modulated, both in the simple form and with an overlaid lens phase characterized by $F/{z_0}$ = 1.3761. The beam profiles are recorded as the propagation distance changes. Subsequent analysis of these profiles included fitting them to the ideal Gaussian beam intensity distribution to determine the beam radius. A further investigation involved analyzing the data correlating the beam’s transport distance with its radius. By fitting the curve of the beam radius variation during transmission, the experimental results are presented in Fig. 3(b). This figure marks, with a red dashed line, the distances at which the spot size of the Gaussian beam, both with and without the added lens phase, expanded to $\sqrt 2 $ times its initial radius; the propagation distance without the lens phase was 0.501 m, and with the lens phase, it increased to 0.714 m. This indicates a relative enhancement of the Rayleigh length by 40.02%, which is in close alignment with the simulated results. Figure 3(c) and (d) depict the Gaussian beam profiles at the transmission distance ‘z’ of 0 (where the camera is positioned at the imaging plane of the SLM), and after propagation, where the beam radius has expanded to $\sqrt 2 $ times the original radius under both conditions, with and without the lens phase.

Based on the calculations presented above, it is necessary to use a lens with a similar order of magnitude as the original Rayleigh length to extend it. However, this is a challenging task to achieve using a conventional single lens, especially considering the transmission distance in practical long-distance wireless optical communication, usually reaching kilometer scales. To address this challenge, Fig. 4(a) illustrates a conceptual double-lens combination method. This approach employs two lenses with identical focal lengths, and by adjusting the distance between the two lenses, the effect of a lens with a long focal length can be obtained. The parameter d is defined as the distance between the right focal point of the first lens and the left focal point of the second lens. In this case, the focal lengths of both lenses are f, the distance between the two lenses is $L = 2f - d$, and the equivalent focal length of the combination is $F = {f^2}/d$.

 

Fig. 4. (a) Concept of adjusting Rayleigh length with a dual lens system. (b) Relationship between equivalent focal length and lens spacing in a dual lens system.

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Figure 4(b) shows the relationship between the equivalent focal length of the double-lens system and the distance between the two lenses. As the value of $d/f$ increases, the ratio $F/f$ of the equivalent focal length to the single lens focal length also increases, with both exhibiting an inverse relationship. Utilizing this relationship, it is comparably easier to obtain an equivalent long-focal-length lens by adjusting the distance between two short-focal-length lenses. This approach, therefore, satisfies the lens focal length requirement needed for Rayleigh length adjustment.

3.2 Optimization of Gaussian light with limited emission aperture

In contemporary wireless optical communication systems, the transmission Rayleigh length of a beam is significantly influenced by the finite dimensions of the emission aperture. This study examines the Rayleigh length under circumstances where the beam is truncated due to the constraints of aperture size. Illustrated in Fig. 5(a) are several Gaussian beams with varying waist radii that experience peripheral power attenuation as a result of the aperture’s occlusion. The beam’s energy within the shaded area is effectively blocked by the obstruction presented by the aperture. It is evident that as the ratio of the beam’s waist radius (${\omega _0}$) to the aperture radius (${\textrm{r}_a}$) shifts, the degree of energy loss associated with this obstruction alters. For instance, when the ratio ${\omega _0}/{\textrm{r}_a}$ is at 0.667, it is estimated that approximately 99% of the beam’s energy is transmitted through the aperture, whereas this figure reduces to about 86% when the ratio ${\omega _0}/{\textrm{r}_a}$ equals 1.

 

Fig. 5. (a) Illustration of the impact of emission aperture on Gaussian beam. (b) Correlation between light power transmittance (solid line) and relative Rayleigh length (dashed line) with relative beam size. (c) Correlation between transmission loss and relative beam size at various transmission distances. (d) Comparison of transmission loss over relative beam size at various transmission distances with and without optimization.

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It is generally postulated that when the aperture radius is at minimum 1.5 times greater than the waist radius, equating to a ${\omega _0}/{\textrm{r}_a}$ value of 0.667, the beam is deemed to have traversed the aperture with negligible losses. Consequently, the Rayleigh length corresponding to this specific waist radius configuration, expressed as ${\textrm{z}_a} = \pi {({0.667{\textrm{r}_a}} )^2}/\lambda $, is recognized as the theoretical Rayleigh length for the given emission aperture diameter. Nonetheless, the attainment of maximum transmission distance by the beam, even when it is completely transmitted through the aperture, is not assured. Figure 5(b) delineates the interplay between the transmission rate and the relative Rayleigh length (${\textrm{z}_0}/{\textrm{z}_a}$) as a function of variations in the ratio of waist radius to aperture radius (${\omega _0}/{\textrm{r}_a}$), where ${\textrm{z}_a}$ denotes the aforementioned theoretical Rayleigh length for the emission aperture diameter in question.

An increase in the relative Rayleigh length (${\textrm{z}_0}/{\textrm{z}_a}$) is observed alongside an escalation in the waist-to-aperture ratio (${\omega _0}/{\textrm{r}_a}$), accompanied by a concomitant decrement in the beam transmission rate. This trend exposes a fundamental trade-off within the confines of a fixed emission aperture diameter, whereby enhancing the beam’s transmission efficiency relative to the aperture serves to decrease the beam’s waist radius, which in turn compromises the beam’s theoretical Rayleigh length.

Figure 5 presents a nuanced analysis of the emission aperture’s influence on the spatial characteristics of a Gaussian beam and its subsequent impact on the transmission efficacy. Panel (a) offers a visual representation of how an emission aperture modulates the Gaussian beam’s profile. Panel (b) charts the relationship between the emitted light power (illustrated by the solid line) and the relative Rayleigh length (indicated by the dashed line) as a function of the beam’s dimensions relative to the aperture. Panel (c) explores the interdependence of transmission loss on the beam size, evaluated across different propagation distances. Panel (d) illustrates the reduction in power loss and concomitant increase in received power when employing the optimal ${\omega _0}/{\textrm{r}_a}$ ratios at transmission distances of ${\textrm{z}_a}$, $2{\textrm{z}_a}$, and $3{\textrm{z}_a}$.

The conventional theoretical formulations for Rayleigh length prediction become inapplicable in instances where the Gaussian beam’s spatial distribution is perturbed due to aperture-induced truncation. Addressing this computational void, we engaged in a detailed simulation using a diffractive integral approach to model the far-field transmission characteristics of Gaussian beams with diverse waist radii as they negotiate an aperture. The simulation quantifies the received power post-aperture at incremental transmission distances. Specifically, Fig. 5(c) depicts the long-length dissemination of Gaussian beams with varying waist radii through a consistent aperture radius (${\textrm{r}_a}$), and records the power emanating into a circular aperture of radius ${\textrm{r}_a}$ situated at the transmission axis center, at distances of ${\textrm{z}_a}$, $2{\textrm{z}_a}$, and $3{\textrm{z}_a}$. The graphical output underscores three distinct power loss trajectories correlating with these distances.

The simulation’s revelations pinpoint that power loss was optimally mitigated at transmission lengths of ${\textrm{z}_a}$, $2{\textrm{z}_a}$, and $3{\textrm{z}_a}$ when the ${\omega _0}/{\textrm{r}_a}$ ratios stood at 0.8245, 0.8779, and 0.8872, respectively. With these optimized ratios, Fig. 5(d) indicates that there is an increase in received power by 6.54%, 12.47%, and 13.74% when compared to the non-optimized standard. In the non-optimized state, corresponding to a ${\omega _0}/{\textrm{r}_a}$ ratio of 0.667, there is a baseline level of power loss that is experienced during transmission. These ratios ingeniously encapsulate the impact of aperture-induced optical power attenuation. Drawing from these insights, it becomes evident that maximizing transmission efficiency in real-world applications is not a straightforward consequence of merely ensuring the entirety of the optical power traverses the aperture unimpeded over extended distances. Contrarily, a slight upscaling in the Gaussian beam dimension—which consequentially elevates the theoretical Rayleigh length—can bring about an enhancement in transmission performance albeit at the expense of a finite portion of the optical power. This strategic trade-off provides a compelling avenue for optimizing the interplay between beam size, aperture constraints, and transmission range to achieve superior communication system performance.

3.3 Simulation of optical transmission under actual link conditions

In the final simulation series, we meticulously adjust both the lens configuration antecedent to the light beam and the emergent beam’s waist radius to calculate the maximum achievable Rayleigh length for a given aperture diameter. At this stage, due to the alterations in the spatial distribution of the beam, it was not feasible to directly determine the correlation between aperture size and Rayleigh length based on the beam waist radius variation alone. For a reference point, we designate the Rayleigh length associated with the waist radius where (${\omega _0}/{\textrm{r}_a} = \; 0.667$) as the benchmark Rayleigh length for the aperture radius (${\textrm{r}_a}$). This is followed by computing the power attenuation within a circular area of radius (${\textrm{r}_a}$) centered around the transmission axis as the beam with said waist radius traversed the stipulated emission aperture. If variations of Gaussian beams with different waist radii cross this aperture and propagate a certain distance to reach the specific attenuation threshold, that distance is identified as their respective Rayleigh length.

Figure 6(a) portrays the optimized Rayleigh lengths obtained for various emitting apertures using dual optimization tactics vis-à-vis the baseline lengths prior to optimization, indicating a substantial improvement. Through specific numerical analysis, the Rayleigh length has increased by 40.63% compared to the original length. Additionally, we simulate the optimization process for prospective transmission links in pragmatic scenarios. The results are depicted in Fig. 6(b), which compares the transmission power decay both before and after optimization for links with distances of 3 km, 5 km, and 10 km, and an aperture diameter of 10 cm for both emission and reception. Through these optimization simulations, a measurable enhancement in transmission performance is invariably observed. Specifically, at the respective link distances of 3 km, 5 km, and 10 km, the power loss decreased significantly after optimization when compared to the original settings. The optimizations led to improvements in received power of 14.23%, 14.68%, and 14.75%, respectively. By tailoring the transmission system parameters to better suit the propagation distance, the adjustments have yielded a considerable increase in transmission efficiency at each respective distance. The benefits from the optimizations affirm the potential for significantly improved optical link performance through strategic engineering of the system’s operational parameters.

 

Fig. 6. (a) Correlation of optimized and non-optimized Rayleigh length with aperture radius. (b) Comparison of transmission loss over various distances for 10 cm aperture with and without optimization.

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4. Conclusion

This research paper provides an exhaustive analysis of Gaussian beam propagation with an emphasis on its implications for wireless optical communication (WOC) systems. Gaussian beams are the linchpin of high-speed optical data links, where beam divergence and energy retention across transmission distances necessitate refined control over their spatial characteristics. We delve into the theoretical construct of Gaussian beams and scrutinize the effects of lens-induced focusing on their waist radius and Rayleigh length—a critical determinant of beam quality over propagation. This study employs advanced optical modeling, leveraging diffraction integral simulations to predict the behavior of Gaussian beams as they transit through optical lens systems and the attendant apertures. By systematically varying lens focal lengths and appraising the impact on the beam’s far-field spread, we identify optimal conditions that diminish power losses and augment the effective transmission range. The analysis elucidates the complex interaction between beam waist, aperture dimensions, and Rayleigh length, thereby enabling the engineering of WOC systems for maximal transmission efficiency and reliability.

The findings from this research could have considerable relevance for the refinement of wireless optical communication (WOC) systems, particularly those with constraints on their emission and reception aperture dimensions. Our findings provide pragmatic insights on how to utilize the available transmission aperture to its fullest potential to minimize link losses. The innovative aspect of our study, which sets it apart from other systems that control beam divergence [17,25,26], is our exploration into optimizing transmission efficiency through the intentional convergence of the beam rather than the traditional approach of collimation.

The study commences from a foundational model of beam transmission, delving into the ability of lenses to refine beam convergence for improved transmission outcomes. A particular innovation of this work is the proposed dual-lens system that allows for efficient focal length adjustments while maintaining equal focal distances for both lenses. Such an arrangement is particularly advantageous in practical optical link scenarios where beam expansion may be required for extended range transmission; varying the focal length ratio of the dual lenses could theoretically accomplish both focusing and beam widening concurrently. In pursuit of minimal loss in the optical link, our research suggests that modifying the ratio between the Gaussian beam’s waist radius and the emission aperture dimension, coupled with an appropriate truncation of the beam, can significantly minimize transmission losses.

Whilst not directly addressing atmospheric turbulence and scattering particles—factors that significantly impact WOC system performance—our study’s insights could be integrated with adaptive optics strategies [28,29] to enhance system robustness and efficiency. Therefore, our research contributes to the development of more reliable and efficient WOC systems by addressing the intricate relationships between beam shape, aperture sizing, and transmission efficiency.

To conclude, the comprehensive investigation put forth in this paper details the imperative optimization of a Gaussian beam’s propagation within wireless optical communication frameworks, contextualized within the theoretical confines of Rayleigh criterion. The results underscore the significance of meticulously engineered lens arrangements—both single and dual configurations—to finetune the beam’s waist and extend the Rayleigh length. The study further illustrates the pertinence of the beam waist-aperture diameter ratio and its direct influence on the minimization of diffraction-induced power loss. The simulation outcomes encapsulate the pivotal enhancements achievable through prescribed alterations, thereby substantiating their implementation for boosted beam longevity and power at operative distances. Ultimately, this research acts as a springboard for future advancements in optical communication, enabling practitioners to further refine systems for even greater efficacy in overcoming the challenges of high-speed data transmission across vast expanses.