On the Stability of Spatially Distributed Cavity Laser and Boundary of Resonant Beam SLIPT

1 On the Stability of Spatially Distrib uted Ca vity Laser and Boundary of Resonant Beam SLIPT Mingliang Xiong, Member , IEEE , Zeqian Guo, Qingqing Zhang, Member , IEEE , Qingwen Liu, Senior Member , IEEE , Gang W ang, Senior Member , IEEE , Gang Li, Member , IEEE , and Bin He, Senior Member , IEEE Abstract —Spatially distributed ca vity (SDC) lasers are a promising technology for simultaneous light inf ormation and power transfer (SLIPT), offering benefits such as increased mobility and intrinsic safety , which are adv antageous for various Internet of Things (IoT) devices. However , achievi ng beam transmission over meter -lev el long working distances presents significant challenges from cavity stability constraints, manu- facturing/assembly tolerances, and diffraction losses. This pa- per conducts a theoretical in vestigation of the fundamental restrictions limiting long-range resonant beam generation. W e in vestigate cavity stability and beam characteristics, and pr opose a binary-search-based Monte Carlo simulation algorithm as well as a linear approximation algorithm to quantify the maximum acceptable tolerances for stable operation. Numerical results indicate that the stable region contracts sharply as distance increases. For fixed-component systems, an acceptable tolerance of 0 . 01 mm r estricts the achie vable transmission distance to less than 2 m. T o address this limitation, we also pr ove the feasi- bility of long-range beam f ormation using precision adjustable elements, paving the way f or advanced engineering applications. Experimental results v erified this assumption, demonstrating that by tuning the stable region during assembly , the transmission distance could be extended to 2 . 8 m. This work provides essential theoretical insights and practical design guidelines for realizing stable, long-range SDC systems. Index T erms —Spatially distributed cavity (SDC), resonant beam system (RBS), laser stability , tolerance analysis, SLIPT , self-alignment. I . I N T RO D U C T I O N W ITH the pervasi ve growth of Internet of Things (IoT) devices such as mobile phones, smart speakers, elec- tronic watches, and unmanned aerial vehicles (UA Vs), the efficient deli very of both power and data has become increas- ingly critical [1–8]. Simultaneous light information and power transfer (SLIPT) is a promising technology to address these demands, le veraging light’ s inherent adv antages including high This work was supported in part by the National Natural Science Foundation of China under Grant 62305019 and U23B2059. M. Xiong, Z. Guo and Q. Liu are with the School of Computer Sci- ence and T echnology , T ongji University , Shanghai 201804, China (e-mail: mlx@tongji.edu.cn; guozeqian@tongji.edu.cn; qliu@tongji.edu.cn) Q. Zhang is with the College of Information Engineering, Zhe- jiang Univ ersity of T echnology , Hangzhou 310014, China (e-mail: qingqingzhang@zjut.edu.cn) G. W ang is with the State K ey Lab of Intelligent Autonomous Systems, School of Automation, Beijing Institute of T echnology , Beijing 100081, China (e-mail: gangwang@bit.edu.cn) G. Li and B. He are with Shanghai Research Institute for Autonomous Intelligent Systems, T ongji Uni versity , Shanghai 201804, China. (e-mail: lig@tongji.edu.cn; hebin@tongji.edu.cn) Fig. 1. SDC application scenario diagram (a) indoor smart living (b) industrial intelligence. power density , large bandwidth, and immunity to electromag- netic interference (EMI) [9–11]. Spatially distributed cavity (SDC)-based laser systems have emer ged as a feasible solution for improving laser alignment mobility and safety in a variety of applications, including wireless po wer transfer (WPT), optical wireless communications (O WC), and SLIPT . Since the carrier is an intracavity laser that resonates within the dis- tributed ca vity structure, these systems, which include resonant beam charging (RBC), resonant beam communications (RB- Com), and resonant beam SLIPT (RB-SLIPT), are also known as resonant beam systems (RBS). As illustrated in Fig. 1, the application scenario diagram of the SDC demonstrates its capability to provide power supply and data communication for indoor intelligent mobile terminals. Furthermore, RBS has also been explored for high-accuracy position and 3D pose estimation applications [12]. As depicted in Fig. 2, a fundamental SDC-based SLIPT system configuration typically employs two retroreflectors positioned at the transmitter (Tx) and the receiv er (Rx), respectiv ely , forming a laser resonator . An gain medium (laser crystal), pumped by a suitable source, is placed within the transmitter to provide optical po wer gain for the oscillating light inside the cavity . Partial of the resonant beam is released by the output mirror; and then, the information and the power is decoupled to the photon detector (PD) and the photov oltaic cells (PV), respectively . A key advantage of SDCs is their self-alignment characteristic, facilitated by the retroreflective property at both ends. By obviating the need for sophisticated acquisition, point- ing, and tracking (APT) modules that are indispensable to traditional laser systems, this self-alignment feature signifi- cantly enhances system mobility , thereby rendering it highly suitable for mobile IoT devices such as U A Vs and autonomous mobile robots (AMRs) that demand dynamic locomotion dur - ing charging or communication processes. Moreover , given that the resonant beam is confined within the cavity , any obstruction or foreign object intruding into the beam path will immediately disrupt the oscillation, thereby forming an 2 M1 L1 L2 M2 L3 L4 M3 L5 L6 M4 D PD PV EOM2 Receiver Resonant beam Second harmonic beam M5 Gain Resonant Laser Beam (Information and Power Carrier) Move Spatially Distributed Cavity Retroreflector (Tx) Retroreflector (Rx) Decouple PV PD Modulate T=10% Lens Mirror Pump Fig. 2. Spatially distributed cavity laser for mobility enhanced SLIPT . (Tx: transmitter; Rx: receiver; PV : photov oltaic cells; PD: photodiode) intrinsic safety mechanism. This mechanism is not only par- ticularly crucial for high-power tra nsmission scenarios but also essential for IoT deployments in smart cities or industrial en vironments where high-power operation is mandated to be hazard-free. The receiv er , which integrates a compact cat’ s-eye retroreflector and photovoltaic devices, can be embedded into wireless sensors, drones, and other mobile IoT devices that are constrained by limited space for bulk y batteries or dedicated charging interfaces. This feature enables such devices to operate continuously without the need for frequent battery replacement or wired char ging, which is particularly valuable for high-power and electromagnetically sensiti ve applications. The concept of SDC dates back to 1974, when G. J. Linford proposed a system with two corner cube retrore- flectors to support very long laser cavities for en vironmental sensing [13]. In 1980, Linford further designed a secure laser communication system utilizing SDC to prev ent un- intended detection [14]. More recently , in 2016, W i-Charge Company disclosed a patent describing a distributed coupled resonator laser based on two telecentric cat’ s eye retroreflectors (CER) [15]. The same year , Liu et al. presented comprehensi ve in vestigations into SDC-based WPT systems, wherein the technology’ s salient advantages were elucidated, including mobility , intrinsic safety , multiple-receiv er charging capability , high power deliv ery , compact size, and EMI-free operation. Collectiv ely , these features were shown to effecti vely meet the core technical requirements of IoT application scenarios. [16]. Subsequently , in 2018, Q. Zhang et al . de veloped an ana- lytical model for a distributed laser char ging (DLC) system employing two corner cube retroreflectors [17]. M. Xiong et al . proposed an RBCom system using SDC to mitigate challenges related to beam attenuation and tracking [18, 19]. Addressing the need for multiple receiv ers, J. Lim et al . proposed a wireless optical po wer transfer system that utilized spatial wa velength di vision and distributed laser ca vity resonance, achieving 1 . 7 mW over 1 m transmission distance with optical gratings and corner cubes [20]. In the same year , W . W ang et al . demonstrated an RBC system based on a quasi-SDC struc- ture (dual-mirror ca vity with spatially separated deployment), experimentally achieving 2 -W charging power over 2 . 6 m. Howe ver , this specific implementation, based on a dual-mirror cavity , lacked the inherent mobility of typical SDC designs due to its alignment requirements [21]. Significant adv ancements continued in subsequent years. In 2021, W . W ang et al . e xperimentally v alidated a Nd:YV O 4 thin disk SDC laser employing two CERs, demonstrating an improv ed alignment-free range of − 13 ◦ to 13 ◦ [22]. Further enhancing performance, M. Xiong et al . proposed a low- diffraction-loss SDC design based on focusing CERs, which provide both retroreflection and focusing capabilities, and established an analytical model for output power calcula- tion [23]. For simulating the intracavity light field, M. Liu et al . introduced a Fox-Li-based algorithm to compute light field distribution and dif fraction loss in SDCs [24]. Since 2022, experimental research on SDC-based systems has become more prev alent and practical. Q. Liu et al . de vel- oped a complete RBC system utilizing focusing CERs and a thin disk Nd:YVO 4 crystal, demonstrating mobile charging of a smartphone over 2 m with receiv ed optical po wer up to 5 W , resulting in 0 . 6 W of electrical charging power [25]. N. Javed et al . presented a WPT system incorporating an erbium-doped fiber amplifier (EDF A) and ball lens retroreflectors [26]. Q. Sheng et al . explored the use of Nd:GdV O 4 crystal in an SDC and successfully enhanced the recei ver’ s field of vie w (FoV) by compensating the field-curvature [27]. Z. Zhang et al . demonstrated 86 . 3 -mW optical power delivery ov er 200 -cm using vertical external-cavity surface-emitting lasers (VEC- SELs) as the gain medium, paving the way for semiconductor- based SDCs with potential for improved electro-optical ef- ficiency [28]. In 2023, Q. Sheng et al . reported an SDC system achie ving 5 -W output optical power over 1 to 5 m with a receiving CER FoV up to ± 30 ◦ [29]. They also proposed a telescope-based SDC design in the same year , demonstrating 1 . 3 -W electrical power output at the receiver ov er a 5 -m transmission distance [30]. Inv estigating receiver’ s FO V , Y . Zuo et al . designed a ball-lens CER, showing rotation tolerances up to 38 ◦ in an SDC configuration. Furthermore, two-coupled SDC designs have been inv estigated in [30, 31] to potentially mitigate safety risks and reduce transmission loss ov er e xtended distances. Theoretical in vestigations hav e also developed signifi- cantly . Studies have analyzed safety aspects related to foreign object intrusion into the optical cavity [32] and proposed sim- ulation methods for analyzing stable light intensity distribution within SDCs [24]. Optimization algorithms hav e been de vel- oped for asymmetric SDC designs [33], and analytical models were established for optimizing the FoV of SDC systems [34]. While these studies cov er various aspects, existing research, particularly experimental demonstrations, primarily focuses on meter-le vel transmission (working) distances. There remains a notable lack of theoretical guidance specifically addressing the fundamental limitations of SDC technology for stable resonant beam formation over longer transmission distance. Unlike traditional external-cavity lasers, the spatially separated nature of SDCs means that the stability of the resonator imposes strict limits on the maximum achiev able cavity length. Although sta- ble resonators can be theoretically designed under ideal condi- tions, practical manufacturing tolerances and alignment errors inevitably introduce deviations from the optimal operational point, consequently limiting the achiev able ca vity length and thus the maximum transmission distance. Moreov er, the trade- off between the cross-sectional area of the gain medium and the magnification of any integrated telescope optics represents 3 M1 L1 L2 M2 L3 L4 EO M1 M3 L5 L6 SHG Gain M4 Pum p PD P D PV EO M2 Transmitter R e ce i v er Resonant beam Second ha rmon ic b eam M5 (a) (b) L2 Gain M1 M2 L1 d 1 Working distance z f 2 d w 0 d 2 f 1 L2 Gain M1 M2 L1 L3 L4 d 1 f 3 d t Working distance z 0 Telescope f 4 f 1 f 2 d 2 d g Retroreflector Retroreflector Retroreflector Retroreflector (d) (c) d 1 f 1 Pupil f 1 d 1 =f 1 Pupil d 1 f 1 d 1 >f 1 d w Fig. 3. System design of the duplex simultaneous light information and power transfer system based on a coupled spatially distrib uted ca vity: (a) fundamental deployment, (b) cavity with internal telescope, (c) common telecentric cat’ s eye retroreflector, (d) retroreflector with focusing ability . M1, M2: mirrors; L1–L4: lenses; d 1 , d 2 , f 1 – f 4 , d w : intervals between elements; f 1 – f 4 are also focal lenses of L1–L4, respectively another critical, yet underexplored, determinant factor in long- range resonant beam formation. Motiv ated by these challenges and the prospect of e x- tending SDC applications to longer ranges, this work conducts a rigorous theoretical and analytical in vestigation. The main contributions of this paper are summarized as follows: • W e establish a comprehensiv e theoretical framework for analyzing the stability of telescope-enhanced spatially distributed cavity (SDC) laser systems. This framew ork identifies critical parameters influencing resonator stabil- ity and defines the boundaries for stable resonant beam formation across varying operational distances. • W e propose analytical methods, including a binary- search-based Monte Carlo simulation and a linear approx- imation technique, to quantify the maximum allowable manufacturing and assembly tolerances. These methods provide practical design guidelines crucial for achie ving stable SDC operation at extended ranges. • W e identify se veral significant factors requiring careful consideration in the design of long-range SDC systems, including a detailed in vestigation into the relationship between the gain medium cross-sectional area and the magnification of the telescope. • W e established an experimental platform to validate the performance of the telescope-enhanced SDC system. W e provide practical design and assembly guidelines for de- veloping long-range SDC systems, specifically addressing parameter selection and adjustment procedures. The remainder of this paper is structured as follo ws: Section II describes the system structure of a SDC, detailing both a basic SDC configuration and a telescope-based design. Section III presents the analytical system model and introduces the proposed algorithms for tolerance analysis. Numerical results and detailed discussions are provided in Section IV. Finally , Section V concludes the paper . I I . S Y S T E M S T RU C T U R E This section details the architecture of the SDC, outlining its constituent components and the underlying mechanism enabling self-alignment and self-tracking. These features are critical for facilitating mobile WPT , OWC, and passi ve posi- tioning. W e first describe the basic SDC configuration, which utilizes two retroreflectors, and then introduce an enhanced version incorporating a telescope to potentially extend the transmission range or increase the field of vie w . As illustrated in Fig. 3 (a), the fundamental SDC archi- tecture comprises two retroreflectors, each consisting of a lens- mirror pair (M1-L1 and M2-L2), and a gain medium. T ypi- cally , in a SDC-based WPT or O WC system, one retroreflector is situated at the transmitter and the other at the recei ver . The gain medium is integrated within the transmitter , precisely positioned in the pupil of the retroreflector . As depicted in Fig. 3 (c) and (d), the pupil is defined as an aperture located at the focal point of the lens, which is essential for the retroreflection property . A ray passing through the pupil will be reflected directly back towards its source. The distance between the lens and the mirror , denoted as d 1 , governs the focusing behavior . When d 1 equals the lens focal length f 1 , the retroreflector exhibits a non-focusing characteristic. Con versely , if d 1 > f 1 , the retroreflector can activ ely focus the incident beam. These behaviors will be theoretically v alidated in the subsequent section. Due to the retroreflection occurring at both the transmitter and receiv er ends, photons generated within the pumped gain medium are effecti vely confined and oscillate within this dual-retroreflector cavity . The gain media utilized in SDC systems are typically crystals; ho wever , alternati ve materials such as glasses, or ganic compounds, and semiconductors are also viable. In this study , we utilize an Nd:YVO 4 crystal as a representative exam- ple [22, 25, 35]. This crystal absorbs pump light at 880 nm and amplifies light at 1064 nm, which is called in-band pumping. Consequently , the optical power circulating within the SDC is progressiv ely increased as photons trav erse the gain medium 4 during each round trip. The remarkable self-aligning capability of the system stems directly from the inherent retroreflectiv e characteristic. An intuitive, though not entirely precise, explanation is that photons originating from the gain medium are consistently reflected back to the gain medium by both retroreflectors, forming the oscillation. Howe ver , a more rigorous under- standing requires considering that retroreflectors inherently in vert the input image: photons generated at the top of the gain medium, for instance, will arriv e at the bottom after reflection. Therefore, a comprehensi ve analysis must consider the collecti ve beha vior of all photons within the cavity . W a ve optics theory provides an ef ficient tool for studying beam propagation in such systems. Pre vious research has conclu- siv ely demonstrated the existence of an oscillating optical path between the two retroreflectors that exhibits exceptionally low diffraction loss [24, 25]. A laser beam spontaneously forms along this resonant path e ven when the retroreflectors are not perfectly aligned along the same optical axis, thereby enabling the mobility of both the transmitter and recei ver ends. As illustrated in Fig. 3 (b), the enhanced SDC incor- porates a telescope within the transmitter . This telescope, composed of two lenses (L3 and L4), serv es to either compress the beam spot size at the gain medium or enlarge the effecti ve field of view of the transmitter . Under the assumption that a plane mirror (represented by the green dashed line in Fig. 3) is hypothetically placed at the beam waist between lenses L3 and L4, the combination of the telescope and the retroreflectors can be conceptually viewed as two coupled SDCs. The left portion of the telescope (L3) paired with the first retroreflector (L1- M1) forms one effecti ve SDC, while the right portion (L4) paired with the second retroreflector (L2-M2) forms another . This analogy clarifies why the inclusion of the telescope does not compromise the self-aligning property . This assumption holds because the wa vefront curv ature radius at the beam waist is infinite, effecti vely resembling a plane mirror . The telescope configured in this manner ef fectively possesses two pupils at both sides. It functions by compressing the light spot at its right-side pupil and projecting a corresponding image onto its left-side pupil. W e will learn in the next section that a small light spot in the gain medium is required to improve the output power . In the subsequent sections, we will quantitatively in vesti- gate the influence of v arious system parameters, including the distances between components and the focal lengths of the lenses. It is important to note that while distance parameters such as d 1 , d 2 , d g , d t , and d w are adjustable, their acceptable ranges for stable operation vary significantly [36]. Specifically , the gain medium is nominally positioned at the focal image plane of both L1 and L3, implying that the ideal interval between L1 and L3, d g , should equal f 1 + f 3 . Howe ver , accounting for potential fabrication tolerances, we introduce a variable δ d g to represent the deviation from this ideal distance in our subsequent analysis. The interval between lenses L3 and L4, d t , should be close to f 3 + f 4 . This distance is adjustable and critical for finding a stable operating point for the SDC with the telescope. W e will further explore the characteristics of d t ’ s adjustment range. The working distance, d w , is defined as the separation between the transmitter’ s pupil and the receiver’ s pupil. The total cavity length, represented by the interval between L4 and L2 (for the version with a telescope) or L1 and L2 (for the basic version), is thus gi ven by f 4 + d w + f 2 or f 1 + d w + f 2 , respectively . In the next section, we will establish a comprehensiv e system model to analyze the stability re gion of this resonator configuration under variations in these distance parameters and determine their boundaries for stable operation. Furthermore, we will in vestigate the principles guiding parameter selection and examine the tolerance le vels acceptable during fabrication. I I I . T H E O RY A N D A L G O R I T H M Optical resonators can be effecti vely analyzed using the matrix method, also known as the ABCD matrix or ray transfer matrix method [37]. In this approach, different optical components such as plane mirrors, concave mirrors, conv ex lenses, and free-space propagation are represented by specific 2 × 2 matrices. The transformation of a ray’ s position and angle as it passes through the system is calculated by multiplying these optical matrices. Crucially , the ov erall system matrix is obtained by multiplying the individual component matrices in the order opposite to the beam propagation direction. A. Stable Re gion W idth For a plane mirror , the ray transfer matrix M M is the identity matrix, representing no change in ray position or angle upon reflection (considering propagation along the optical axis). A common retroreflector that simply rev erses the beam path without focusing is represented by the matrix − M M . This matrix effecti vely rev erses both the ray’ s angle and its displacement from the optical axis. Using these fundamental matrices, the matrix for a specific focusing retroreflector system, consisting of a mirror between two free-space sections and lenses, is deduced as [38]: M RR = M T ( f ) M L ( f ) M T ( d ) M M M T ( d ) M L ( f ) M T ( f ) =  1 0 − 1 /f eqL 1   − 1 0 0 − 1   1 0 − 1 /f eqL 1  , (1) where M L ( f ) is the matrix of lens with focal length f , M T ( d ) represents a translation or drift of length d , f eqL is the equiv alent focal length of the focusing retroreflector , defined by: f eqL = f 2 d − f . (2) From the structure of M RR , it can be seen that this retrore- flector configuration is equi valent to a common retroreflector (represented by the matrix  − 1 0 0 − 1  ) combined with a lens of focal length f eqL . If d > f , f eqL is positi ve, indicating the matrix exhibits the characteristic of a focusing retroreflector . Con versely , if d < f , f eqL is negati ve, and the matrix performs as a defocusing retroreflector . By appropriately adjusting the equiv alent focal length f eqL (which depends on f and d ), the SDC can be configured to operate in a stable state, minimizing diffraction losses. 5 For a specific telescope configuration utilized within this SDC system, characterized by lens focal lengths f 3 and f 4 , and an interval d t between the lenses, the ray transfer matrix describing propagation from one pupil plane to another is deriv ed as: M TS = M T ( f 4 ) M L ( f 4 ) M T ( d t ) M L ( f 3 ) M T ( f 3 ) =   − M tel 0 − f 3 + f 4 − d t f 3 f 4 − 1 M tel   , (3) where M tel = f 4 /f 3 represents the angular magnification of the telescope. For a telescope-based SDC, the round-trip transfer matrix, incorporating the telescope module, is expressed as: M SDC , t = M T ( d 1 ) M L ( f 1 ) M T ( d g ) M L ( f 3 ) M T ( d t ) M L ( f 4 ) M T ( f 4 ) M T ( d w ) M T ( f 2 ) M L ( f 2 ) M T ( d 2 ) M M M T ( d 2 ) M L ( f 2 ) M T ( f 2 ) M T ( d w ) M T ( f 4 ) M L ( f 4 ) M T ( d t ) M L ( f 3 ) M T ( d g ) M L ( f 1 ) M T ( d 1 ) M M . (4) Here, d 1 , d 2 , f 1 , f 2 , d w are as defined for the general SDC, while f 3 and f 4 are the focal lengths of the lenses in the telescope module, d t is the distance between f 3 and f 4 , and d g is the distance between the first lens L1 and the telescope module’ s first lens L3. The ideal design distance d g is set as d g = f 1 + f 3 . If there is a manufacturing or assembly error δ g , the actual distance becomes d g = f 1 + f 3 + δ d g . W e will analyze the impact of this error on system stability . Next, we deduce the fundamental mode (TEM 00 ) radius, which characterizes the spatial extent of the lo west-order Gaussian beam within the resonator . For a Gaussian beam, the complex number q encapsulates both the wav efront radius of curvature ρ and the TEM 00 mode radius w 00 ; that is [37]: 1 q = 1 ρ − j λ π w 2 00 , (5) where λ is the wav elength of the stimulated laser and j is the imaginary unit ( j 2 = − 1 ). From (5), we can see that w 00 of a Gaussian beam at a giv en location can be obtained from the q -parameter by extracting the imaginary part of its reciprocal [23]: w 00 = s − λ π ℑ [1 /q ] , (6) where ℑ [ · ] denotes the operator for extracting the imaginary part of a complex number . F or a physically meaningful beam radius, ℑ [1 /q ] must be negati ve. If a Gaussian beam with an initial comple x beam param- eter q in enters an optical system with a ray transfer matrix  A B C D  , the transformed complex beam parameter q out at the output plane is obtained as [37]: q out = Aq in + B C q in + D . (7) By applying the ABCD law , we can compute the transformed q -parameter of the beam after it completes a full round trip within the resonator . Let  A 0 B 0 C 0 D 0  be the ray-transfer matrix representing one complete round trip within the SDC, starting and ending at mirror M1. For a stable optical resonator , according to the self-consistent mode theory , the q -parameter of the circulating mode at any given plane must remain unchanged after one round trip. This means the light field profile is reproduced after each round trip at the same location. Thus, for the q - parameter q 0 at M1, we must have q 0 = q out after one round trip starting with q in = q 0 . This self-consistency condition yields the following equations [37]:    q 0 = A 0 q 0 + B 0 C 0 q 0 + D 0 , A 0 D 0 − B 0 C 0 = 1 . (8) The optical transfer matrix satisfies A 0 D 0 − B 0 C 0 = 1 because its unit determinant ensures the system’ s reversibility , allowing light rays in vacuum to trace back from output to input. Solving the quadratic equation for q 0 from the first line of (8), we obtain two possible solutions. Howe ver , to ensure the mode radius w 00 is a real value, the imaginary part of 1 /q 0 in (6) must be a negati ve number . This condition selects the physically acceptable solution for 1 /q 0 [37]: 1 q 0 = − A 0 − D 0 2 B 0 − j 2 | B | p 4 − ( A 0 + D 0 ) 2 . (9) Then, the fundamental mode radius at z = 0 (the location of M1) can be deduced from (6) by substituting the deriv ed q 0 : w 00 ( z = 0) = s 2 λ | B 0 | π p 4 − ( A 0 + D 0 ) 2 . (10) From equation (10), we can see that a real solution for the mode radius exists only when the term under the square root in the denominator is positiv e, i.e., 4 − ( A 0 + D 0 ) 2 > 0 . This inequality forms the basis of the resonator stability criterion. Generally , a stability parameter g is defined for simplification, relating to the trace of the round-trip matrix: g = A 0 + D 0 2 . (11) Using this parameter, the formula for the fundamental mode radius at M1 can be rewritten in a simplified form: w 00 ( z = 0) = s λ | B 0 | π p 1 − g 2 . (12) For the mode radius to be a real value, the term 1 − g 2 must be positiv e, which leads to the follo wing condition [39]: − 1 < g < 1 . (13) The formula (13) is the most important equation for this study . It’ s the rule that determines whether the laser system can produce a stable, usable beam. Based on the formula deriv ations abov e, it is clear that both the distance parameters d = ( d 1 , d 2 , d g , d t , d w ) and the focal length parameters f = ( f 1 , f 2 , f 3 , f 4 ) influence the stability of the SDC. T ypically , the focal lengths f are fixed by design and component selection, while the distance parameters d can be adjusted during assembly . A critical question in the 6 design and implementation of such a system is to determine the acceptable range for an adjustable parameter while all other parameters are fixed, such that the SDC remains in a stable state. For instance, if all parameters except d t are fixed, we can calculate the stable region width for d t . This width, denoted as D d t , is the difference between the values of d t at the boundaries of the stable region where g = 1 and g = − 1 : D d t = d t | g =1 − d t | g = − 1 . (14) This equation tells us the range of d t that will still allow for a stable cavity . It helps designers determine how much tolerance they hav e for manufacturing and assembly errors. Similarly , we can obtain the stable region width D d 1 for the parameter d 1 . Calculating these boundary values inv olves finding the roots of the polynomial equations g ( d i ) = ± 1 . This calculation can be performed using numerical equation-solving functions av ailable in software like Matlab . B. Beam Radius Evolution Follo wing the determination of the resonator’ s stability and the fundamental mode properties at M1, the next step is to calculate the fundamental mode radius w 00 ( z ) at an arbitrary location z along the beam path within the cavity . As established earlier, the mode radius at any point can be calculated from the complex beam parameter q ( z ) using formula (6). Therefore, the primary objective is to obtain q ( z ) . Having already obtained the self-consistent q 0 at M1 (which we define as z = 0 ), we can calculate q ( z ) by applying the ABCD law (7) using q 0 as the input parameter q in and the ray- transfer matrix of the optical system situated between z = 0 and the location z . Let M t ( z ) =  A t B t C t D t  be the ray transfer matrix describing the propagation from the plane z = 0 (M1) to the arbitrary plane z . This system matrix can be expressed as the following piece wise function, accounting for the sequence of optical elements: M t ( z ) =            M T ( z − z k ) , for k = 1 M T ( z − z k ) k − 1 Y i =1 [ M L ( f i ) M T ( z i − z i − 1 )] , for k ≥ 2 and z k < z ≤ z k +1 (15) where z k represents the cumulative distance from the starting plane (M1) to the end of the k -th complete segment of the optical system, and f i ( i = 1 , 2 , . . . , k ) is the focal length of the lens within the i -th segment. In this formulation, the SDC is conceptually divided into several segments, each typically containing a free-space propagation section follo wed by a lens or other optical element. The multiplicative operation denotes the ordered product of the matrices for all completed segments the beam passes through before reaching z . The final segment, within which the point z is located, is not complete, and its free-space propagation matrix M T ( z − z k ) is written on the left side of the product, consistent with the rev erse- order multiplication rule of matrix optics. The cumulati ve distance z k represents the total length covered by all segments completed before the beam reaches the k -th optical element. It is calculated recursiv ely by: z k =  0 , for k = 1 z k − 1 + z k − 1 , for k ≥ 2 (16) where z k represents the length of the k -th segment, defined as the interv al between the optical elements (mirror-lens or lens-lens) bounding that segment in the direction of beam propagation starting from M1. For the telescope-based SDC configuration described by (4), the component intervals can be defined as: ( z k ) 5 k =1 = ( d 1 , d g , d t , f 4 + d w + f 2 , d 2 ) . (17) Once the matrix M t ( z ) =  A t B t C t D t  from z = 0 to z is determined, we can obtain the complex beam parameter q ( z ) at location z using the ABCD law (7) with q 0 . Subsequently , it is straightforwar d to calculate the TEM 00 mode radius at z , w 00 ( z ) , by substituting q ( z ) into formula (6). It is important to note that w 00 ( z ) represents the radius of the fundamental Gaussian mode. The actual beam in the resonator may be a superposition of multiple transverse modes. The ratio of the ov erall beam radius to the fundamental mode radius at any plane z is characterized by a constant v alue, which is defined as the beam propagation factor M 2 . For a pure TEM 00 mode, M 2 = 1 . For multimode beams, M 2 > 1 . C. T olerance Analysis The performance and stability of a real SDC system depend on the precise values of the distance parameters d = ( d 1 , d 2 , d g , d t , d w ) and focal length parameters f = ( f 1 , f 2 , f 3 , f 4 ) . In practical manufacturing and assembly , there are inherent tolerances. Let τ d and τ f represent the tolerance associated with distance parameters and lens focal lengths, respectiv ely . Due to the existence of these tolerances, the actual physical parameters ( d ′ i and f ′ k ) in a fabricated system may de viate from their nominal design v alues ( d i and f k ). These deviations can potentially cause the real system to operate in an unstable re gion. T o guarantee that the SDC operates within the stable region despite these manufacturing and assembly variations, the stability criterion | g | < 1 , as shown in (13), must be satisfied for all possible combinations of parameters within their respective tolerance ranges:   g ( d ′ 1 , d ′ 2 , d ′ g , d ′ t , d ′ w , f ′ 1 , f ′ 2 , f ′ 3 , f ′ 4 )   < 1 , (18) where the actual parameters d ′ i and f ′ k are allowed to vary within the intervals defined by: d ′ i ∈ [ d i − τ d , d i + τ d ] , for i ∈ { 1 , 2 , g , t , w } , f ′ k ∈ [ f k − τ f , f k + τ f ] , for k ∈ { 1 , 2 , 3 , 4 } . (19) Generally , the tolerance of the lenses τ f is specified by the manufacturer or determined by design requirements, denoted here as a fixed value τ ∗ f . The objective then becomes deter- mining the maximum acceptable tolerance for the distance parameters, τ max d , under the giv en design parameters and the fixed lens tolerance τ ∗ f , such that the stability criterion is always met. W e propose a binary-search-based Monte Carlo 7 (BMC) simulation method that uniquely integrates the effi- ciency of binary search with the statistical rob ustness of Monte Carlo simulation, for long-range SDC systems to ov ercome the limitations of con ventional tolerance analysis methods in complex resonant beam systems. The pseudocode for the BMC algorithm is depicted in Algorithm 1. The process begins by calculating the nominal stability parameter g 0 for the system with the ideal design parameters d and f . Subsequently , an iterati ve loop is e xecuted to perform a binary search for the maximum acceptable τ d . W ithin each iteration of the binary search loop, an internal loop conducts a Monte Carlo simulation (MCS). The MCS inv olves generating N sets of random de viations ( δ d i and δ f k ) for the distance and focal length parameters, respectiv ely . These deviations are drawn from uniform distributions U [ − τ d , τ d ] for distances and U [ − τ ∗ f , τ ∗ f ] for focal lengths, where τ d = ( l + u ) / 2 is the current test value in the binary search (with l and u being the lower and upper bounds of the search interval for τ d ). For each generated set of de viations, the actual parameters d ′ i = d i + δ d i and f ′ k = f k + δ f k are computed, the corresponding round-trip matrix M SDC , t is determined, and the resulting stability parameter g is calculated. Here d ′ , δ d , f ′ , and δ f represent the combination of d ′ i δ d i , f ′ k and δ f k , respectively . The maximum absolute value of g encountered across all N samples in the current MCS run, denoted g max , is tracked. After completing the MCS, the algorithm checks if g max is less than or equal to 1, indicating that all tested configurations within the current τ d range are stable. If g max ≤ 1 , the current τ d is a possible maximum, so the lower bound l of the binary search is updated to τ d . Con versely , if g max > 1 , the current τ d is too large, and the upper boundary u is reduced to τ d . After a specified number of iterations, I , the v alue of τ d con verges to wards the maximum acceptable tolerance τ max d . Algorithm 1 Binary-Search-Based Monte Carlo Simulation for T olerance Analysis 1: Input: Parameters d and f , τ ∗ f , samples N , iterations I , τ d bounds [ l, u ] 2: Output: Max τ max d 3: for iter = 1 to I do 4: τ d ← ( l + u ) / 2 5: g max ← 0 6: f or n = 1 to N do 7: δ d ∼ U [ − τ d , τ d ] , δ f ∼ U [ − τ ∗ f , τ ∗ f ] 8: d ′ ← d + δ d , f ′ ← f + δ f 9: M SDC , t ← sdc matrix ( d ′ , f ′ ) 10: g ← g pa rameter ( M SDC , t ) 11: g max ← max ( g max , | g | ) 12: end f or 13: if g max ≤ 1 then 14: l ← τ d 15: else 16: u ← τ d 17: end if 18: end f or 19: return τ max d ← τ d Howe ver , the BMC simulation can be computationally expensi ve and slow , especially when a lar ge number of sam- ples, N , is required to achiev e suf ficient statistical confidence. Hence, we propose an alternati ve linear appr oximation method to estimate τ max d more quickly . The actual stability parameter g ′ for a system with tolerances ( τ d , τ f ) can be related to the nominal stability parameter g ( d , f ) and a de viation term ∆ g ( τ d , τ f ) induced by these tolerances. The absolute value of the perturbed stability parameter | g ′ | must satisfy the inequality: | g ′ ( d , f , τ d , τ f ) | ≤ | g ( d , f ) | + | ∆ g ( τ d , τ f ) | . (20) The goal is to find the maximum possible τ max d such that for any combination of deviations within the ranges [ − τ max d , τ max d ] for distances and [ − τ ∗ f , τ ∗ f ] for focal lengths, the resulting resonator is stable. Under the assumption of small deviations, the worst-case variation of g , | ∆ g ( τ d , τ f ) | , can be approximated using a linear T aylor expansion around the nominal design point. The total worst-case deviation is approximately the sum of the absolute contributions from each parameter’ s maximum deviation: | ∆ g ( τ d , τ f ) | ≈ X i     ∂ g ∂ d i     τ d + X k     ∂ g ∂ f k     τ f . (21) Here, the partial deriv ativ es ∂ g ∂ d i and ∂ g ∂ f k represent the sensitiv- ity of the stability parameter g to changes in each distance and focal length parameter , respectiv ely , ev aluated at the nominal design values. Considering the stable condition restriction | g ′ | < 1 , we can approximate the boundary of the stable region in the presence of tolerances by: | g ( d , f ) | + X i     ∂ g ∂ d i     τ d + X k     ∂ g ∂ f k     τ f < 1 . (22) Since the focal length tolerance τ ∗ f is given and fixed, we can solve this inequality for τ d to find the maximum acceptable tolerance τ max d under this linear approximation: τ max d = 1 − | g ( d , f ) | − S f τ ∗ f S d , (23) where S d = X i     ∂ g ∂ d i     , S f = X k     ∂ g ∂ f k     . (24) In practice, the partial deri vati ves ∂ g ∂ d i and ∂ g ∂ f k can be es- timated numerically by introducing a small perturbation to each parameter individually while holding others constant and observing the resulting change in g . All the contributions to the absolute v ariation of g induced by these de viations are accumulated to model the worst-case condition. D. Output P ower The resonant beam circulates within the optical cavity , experiencing amplification from the gain medium and suf fer- ing various power losses. The output power extracted from a specific mirror (in this case, M2) is achiev ed at a steady state 8 where the total optical gain within the ca vity exactly balances the total cavity losses. Based on the classic Rigrod analysis for solid-state lasers, the output power can be approximated by the following equation [23]: P out = T 2o π a 2 g I s  1 + q R 2 R 1   1 − √ R 1 R 2   l g η c P in I s V g − ln 1 √ R 1 R 2  . (25) In this equation, P out is the extracted output power and should always be a non-ne gativ e number . T 2o represents the ef fectiv e transmissivity from the gain medium interface towards the output side through mirror M2. I s is the saturation intensity of the gain medium, a material property . a g is the effecti ve radius of the acti ve region (pumped volume) within the gain medium, assuming a cylindrical geometry . R 1 and R 2 are the equiv alent round-trip power reflectivities experienced by the resonant beam at the left side and right side of the gain medium, respectiv ely , accounting for all losses and reflections encountered during a round trip to these interfaces. l g is the physical thickness of the gain medium along the propagation axis. η c is a combined ef ficiency of the pump po wer into the gain medium. P in is the incident pump po wer . V g is the v olume of the active gain medium, calculated as V g = π a 2 g l g . The activ e area is the cross-sectional region of the gain medium that is effecti vely pumped and provides optical amplification to the circulating resonant beam. The equiv alent parameters T 2o , R 1 , and R 2 are computed by considering the series of optical elements and interfaces the beam encounters in a round trip: the y are gi ven by: T 2o = T ARs , 2o T air T M2 , R 1 = T diff , 1 T ARs , 1 R M1 , R 2 = T diff , 2 T ARs , 2 T air R M2 . (26) Here, T ARs , 2o = T 7 ar quantifies the total power transmission through all anti-reflection (AR) coatings traversed by the beam in the output path from the gain medium to the exterior of the cavity . T ar = 0 . 995 is the specified power transmissi vity of a single AR coating. Similarly , T ARs , 1 = T 6 ar and T ARs , 2 = T 14 ar represent the cumulativ e transmission through AR coatings on the path from the gain medium interface to M1 and M2, respectiv ely , and back to the interf ace. R M1 and R M2 are the power reflectivities of mirrors M1 and M2, respectively . T ypically , R M1 = 0 . 999 represents a high reflectivity (HR) coating, while R M2 < 1 acts as the output coupler, with its specific value depending on the desired output po wer coupling. T M2 = 1 − R M2 is the power transmissivity of the output coupling mirror M2. T air = exp( − α air d w ) is the attenuation of air transmission, where α air = 10 − 4 m − 1 is the absorption coefficient of clear air . T diff represents the power transmission factor associated with diffraction losses incurred by the resonant beam as it propagates through finite apertures of optical de vices within the ca vity . W e use the following empirical formula to estimate the dif fraction loss factor occurring near the gain medium at both interfaces (accounted for in R 1 and R 2 via T diff , 1 and T diff , 2 ) [23]: T diff = 1 − exp " − 2  a g w 00 ( z g )  2 # , (27) where w 00 ( z g ) is the fundamental mode radius at the longitu- dinal position, z g , of the gain medium. The ratio a g /w 00 ( z g ) is a measure of ho w well the beam is clipped by the gain medium’ s physical extent. Note that this formula provides an estimation and may not be strictly accurate for all cases, particularly for complex mode profiles or strong clipping. Howe ver , it is computationally efficient and provides a useful approximation for ev aluating the impact of the gain medium aperture on beam transmission. This is particularly relev ant because optical resonators are highly sensitiv e to intracavity losses, the diffraction loss calculated by this formula typically remains near zero, as expected for a well-aligned stable cavity . I V . R E S U LT S A N D D I S C U S S I O N In this section, we in vestig ate key properties of the SDC system with emphasis on the characteristics of the stable oper- ating re gion and the constraints on the working distance. Based on this analysis, we provide guidelines for system design and assembly . Since we have chosen Nd:YVO 4 as the gain medium, a material commonly used in practical resonant beam systems, the saturation intensity I s = 1 . 1976 × 10 7 W / m 2 . The radius of the acti ve gain medium is set to a g = 2 . 5 mm, and its thickness is l g = 1 mm. The combined pump coupling efficienc y is η c = 0 . 439 [40]. The pump power is set to P in = 65 W at a wa velength of 880 nm, which stimulates a resonant beam at a w avelength of 1064 nm. The reflectivity of the output coupling mirror M2 is set to R M2 = 0 . 95 . Unless otherwise specified, the analysis uses the fol- lowing default distance parameters: ( d 1 , d 2 , d g , d t , d w ) = (30 , 30 , 55 , 85 . 4 , 6000) mm. The default focal length param- eters are ( f 1 , f 2 , f 3 , f 4 ) = (30 , 30 , 25 , 60) mm. In the sub- sequent analysis, parameters are either held at their default values, scaled by a stated factor relative to the basic configu- ration, or varied as independent v ariables as indicated. A. Stable Re gion W idth Figure 4 illustrates how the stability parameter g varies with different values of d t and d 1 . The color coding represents different levels of the g parameter . From Fig. 4 (a)–(d), we can observe that the width of the stable region in terms of d t adjustment (i.e., D d t ) remains relati vely constant for v arying d 1 , provided d w is fixed. Howe ver , as d w increases, D d t be- comes narrower . This indicates that ev en if the parameter d 1 is not precisely manufactured or assembled in a real system, it is still possible to adjust d t to achieve operation within the stable region. Ne vertheless, achie ving stable operation becomes more challenging and requires more rigorous adjustment precision for longer d w . Figure 4 (e) – (h) show that small deviation in d g ( δ d g < 10 mm), have a negligible impact on stability . Howe ver , a very large deviation in d g will distorts the pattern of the stable region. In practice, δ d g is expected to be much smaller than 10 mm, so this impact can typically be ignored. A notable characteristic observed in Fig. 4 is that the boundary g = 1 remains constant despite significant changes in d w . As depicted in Fig. 5, the width of the stable region de- creases as the working distance d w increases. W e consider four 9 29 30 31 84 85 86 -1 0 1 29 30 31 84 85 86 -1 0 1 29 30 31 84 85 86 -1 0 1 29 30 31 84 85 86 -1 0 1 29 30 31 84 85 86 -1 0 1 29 30 31 84 85 86 -1 0 1 29 30 31 84 85 86 -1 0 1 29 30 31 84 85 86 -1 0 1 Fig. 4. Stability parameter g under different ( d 1 , d t ): (a)–(d) shows different working distance d w , and (e)–(f) shows different deviation δd g . 10 0 10 1 10 2 10 3 10 -3 10 -2 10 -1 10 0 10 1 10 2 Fig. 5. Stable region width under different working distance d w (solid line: D d 1 ; dash line: D d t ; d g = f 1 + f 3 ). different configurations in this analysis. The basic configura- tion uses focal lengths ( f 1 , f 2 , f 3 , f 4 ) = (30 , 30 , 25 , 60) mm, and the other three conditions scale these focal lengths by factors of 1, 2, and 3. W e observ e that at kilometer-le vel working distances, the stable region width is on the order of 0 . 001 mm, which is e xtremely narrow . If the focal lengths of the lenses are increased, for instance, scaled by a factor of 3 times the basic configuration, the stable region width increases to the order of 0 . 01 mm. At this operational scale, the SDC system proposed in this work demonstrates theoretical feasibility for supporting kilometer-scale SLIPT applications. In all cases presented, the stable region width for d t ( D d t ) is greater than that for d 1 ( D d 1 ) at any giv en d w . This supports the conclusion that d t related to the telescope structure is a more effecti ve parameter for adjusting the operating point into the stable region. B. Relation Between w 00 and g P arameters Figure 6 illustrates ho w the TEM 00 mode radii at v arious optical elements (L1, the gain medium, L3, L4, and L2) change -1 -0.5 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 L1 Gain L3 L4 L2 Choice 1 Choice 2 L1, Gain, L3 Fig. 6. TEM 00 mode radius at dif ferent optical devices varies with stable parameter g . as the stability parameter g varies. W e observe that the w 00 values at L1, the gain medium, and L3 are nearly identical and ov erlap as g changes, indicating that the beam propagating between L1 and L3 is approximately collimated. The beam radius reaches its minimum v alue when g = 0 . This suggests that g = 0 is an optimal design criterion because, as implied by equation (27), a smaller mode radius at the gain medium location results in lo wer diffraction loss. From this figure, we also see that w 00 at L4 follows a similar trend to that at L3 as g increases. In contrast, w 00 at L2 increases monotonically with g , a behavior different from the other locations. This characteristic should be considered in the design process, as a small beam spot at L2 is generally desired to minimize the size requirements for the recei ver optics. Consequently , another possible design choice could be to set g at or near the intersection point of the curves for w 00 at L2 and the gain medium, although this strategy might impose stricter requirements on component tolerances and assembly . Based on the analysis presented abov e, we understand 10 10 0 10 1 10 2 10 3 83.5 84 84.5 85 85.5 86 86.5 87 87.5 Fig. 7. d ∗ t (obtained by setting g = 0 ) varies with working distance d w . Fig. 8. 3D perspective of the g = 0 and g = ± 1 surfaces in the coordinate space defined by ( d 1 , d 2 , d t ) (pink surface: g = 0 ; blue surface: g = ± 1 ). that d t can serv e as an ef fective adjustment parameter to enable the system to operate at g = 0 , which is an optimal operating point for minimizing the mode size at the gain medium. Hence, we next in vestigate the behavior of d t under the condition g = 0 (denoted as d ∗ t ), as depicted in Fig. 7. This figure shows that d ∗ t decreases as the working distance d w increases, and it appears to approach a fixed asymptotic v alue. When different values for d 1 are considered, we observe that this asymptotic line shifts upwards with decreasing d 1 . This trend can be attributed to the gradual shrinking of the stable zone and the fixed g = 1 boundary , a phenomenon also intuitiv ely demonstrated in Fig. 4. Here, we plot the surfaces corresponding to g = 0 and g = ± 1 within the three-dimensional coordinate system defined by ( d 1 , d 2 , d t ) , as shown in Fig. 8. The pink surface represents the set of parameter combinations yielding g = 0 , while the blue surface delineates the boundaries of the stable region where g = ± 1 . It can be clearly observed that the stable region boundary ( g = ± 1 ) encloses the g = 0 surface. This visualization helps to understand that if manufacturing or assembly tolerances lead to deviations in ( d 1 , d 2 , d t ) , the M1 L1 Gain L3 L4 L2 M2 0 0.5 1 1.5 2 2.5 Fig. 9. TEM 00 mode radius at dif ferent optical devices changes with the adjustment of d t ( d w = 6 m). actual operating point of the system might fall within the region bounded by g = ± 1 (i.e., within the stable zone), or it might fall outside, leading to an unstable resonator . Figure 9 pro vides an intuitiv e visualization of the mode radius distribution at key optical components as d t is adjusted. W e see that the beam spot size is very small at mirrors M1 and M2. The beam is collimated between lenses L1 and L3 and is magnified by the telescope (formed by L3 and L4) as it propagates to wards L2. Con versely , considering the rev erse direction, the beam coming from free space is compressed upon passing through the telescope. Furthermore, as d t is increased, the mode radius changes according to the behavior described in relation to Fig. 6. When d t = 85 . 05 mm, the stability parameter g approaches zero, resulting in the smallest mode radius at the gain medium. In practice, one might choose to set d t slightly off the exact g = 0 point, for example, at d t = 85 . 4 mm, to achiev e a small w 00 at L2 while maintaining the w 00 at the gain medium almost unchanged. C. The Maximum Acceptable T olerance Figure 10 presents the maximum acceptable distance tolerance τ max d , calculated using both the linear approximation method and the BMC algorithm, as a function of the working distance d w . Here the fixed tolerance τ ∗ f is set to be 1% of the maximum focal length in each configuration. It is evident that τ max d deteriorates rapidly as d w increases. Even when the ov erall focal lengths are scaled up by a factor of 3 compared to the basic configuration, the working distance limit where τ max d drops belo w 0 . 01 mm improves by only approximately 1 m, remaining belo w the 2-m mark. This result provides a crucial guideline: it is not advisable to rely solely on non-adjustable distance parameters set to their predetermined operating points for long d w , as typical manufacturing and assembly tolerances do not support such a design approach. Fortunately , Fig. 5 suggests a feasible strategy for achie ving long d w . Specifically , for a kilometer-le vel SDC system, one should incorporate ad- justability into components like the retroreflector or telescope, ensuring that the adjustment precision is within the acceptable range indicated by Fig. 5. For instance, for a 10-m SDC, the 11 0 0.5 1 1.5 2 10 -2 10 -1 10 0 10 1 10 2 Fig. 10. The maximum acceptable tolerance τ max d varies with work- ing distance d w obtained through linear appropriation and binary-search Monte Carlo simulation (BMC) algorithm { Config 1: ( f 1 , f 2 , f 3 , f 4 ) = (30 , 30 , 25 , 60) mm); d 1 = f 1 , d 2 = f 2 ; d g = f 1 + f 3 ; d t is set upon g = 0 ; Config 2 and Config 3 are twice and three times of Config 1, respectiv ely } . 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 11. Dif fraction loss factor T diff varies with working distance d w ( g = 0 ). required adjustment precision is on the order of 0 . 5 mm, which is a feasible v alue. Even extending the working distance to 100 m remains achiev able with larger focal length parameters, although the precision in need is v ery strict. D. Output P ower and W orking Distance Boundary The finite radius of the acti ve gain medium is also a critical f actor limiting the working distance d w . Since the aperture of the gain medium is typically smaller than other optical components in the cavity , it often contributes the dominant source of diffraction loss. As d w increases, the w 00 within the gain medium also increases and eventually approaches the physical edge of the gain medium aperture. Because the resonator is highly sensitiv e to intracavity losses, this increasing diffraction loss causes the output po wer to deteriorate rapidly . As sho wn in Fig. 11, the dif fraction loss factor T diff remains close to 1 (implying minimal diffraction loss) over a 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 3.5 Fig. 12. Output power P out varies with working distance d w ( f 3 = 25 mm, g = 0 ). significant range of working distances. Ho wever , it decreases sharply to wards zero as d w approaches a certain threshold. This threshold often corresponds to the stability parameter approaching the boundary g = − 1 , as can be inferred from Fig. 4. In Fig. 11, f 4 is varied between 60 mm, 80 mm, and 100 mm, while f 3 is fixed at 25 mm. This configuration, where the telescope compresses the beam, provides efficient beam spot size reduction at the gain medium location. The parameter d ∗ t here refers to the value of d t optimized to achiev e g = 0 at a specific working distance d ∗ w . T wo groups of configurations are compared: the dashed lines represent results where d ∗ t is obtained for a preset working distance d ∗ w = 25 m, while the solid lines correspond to d ∗ t obtained for d ∗ w = 50 mm. W e can infer that if d ∗ t is designed for a gi ven d ∗ w , the diffraction loss remains low until d w approximately reaches 2 d ∗ w , the transmission distance can reach 100 m. From this figure, we also learn that incorporating a telescope generally impro ves the diffraction loss performance. Furthermore, using a telescope with larger magnification ( M tel = f 4 /f 3 ) allows the system to perform well ov er a longer range of working distances before diffraction loss becomes significant. Figure 12 supports the observ ations regarding dif fraction loss and its impact on output power . T wo main conclusions can be drawn : Firstly , if a desired working distance d w is targeted, d t is recommenced to be adjusted to satisfy the g = 0 condition at that distance; Secondly , to mitigate diffraction loss and improv e performance at longer working distances, the magnification of the telescope should be increased. Si- multaneously , an appropriate radius for the gain medium, a g , must be selected. When f 4 is 100 mm and d ∗ t is 125.2 mm, the SDC system achiev es a maximum transmission distance of 76.8 m. Figure 13 intuitiv ely demonstrates the combined impact of a g and M tel on the output power . For a designed working distance d w , a larger M tel is generally preferred to maximize output power . The gain medium radius a g requires careful selection, as both very large and very small v alues can lead to reduced output power . E. Experimental Setup and Results 12 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 Fig. 13. Output power P out varies with gain medium radius a g and telescope magnification M tel ( d w = 100 m, g = 0 ). Fig. 14. Schematic Diagram of the RBS Experiment T o verify the feasibility of adjusting the telescope- enhanced SDC system for greater working distance, we es- tablished the experimental platform as illustrated in Fig. 14. Both the transmitter and recei ver adopted a CER structure with a focal length of f = 30 mm (see Config 1 in Fig. 10). An Nd:YV O 4 crystal was employed as the gain medium, which absorbed a 20 W pump laser at 880 nm for optical amplification. Based on the operating mechanism of the laser resonator , a resonant beam at 1064 nm was generated, with a telescope structure embedded in the resonator cavity . T o pre- vent performance instability of the crystal caused by excessiv e temperature, a cooling system was attached to the crystal to maintain it at a relati vely lo w temperature. The optical po wer receiv ed was detected by an optical po wer meter at the recei ver end. T o ensure experimental safety , the researchers wore OD4- lev el protectiv e goggles throughout the experiment. The initial configuration restricted the beam initialization to within 0.5 m; howe ver , follo wing the adjustment of the stable region, the system demonstrated a capability of op- erating at distances greater than 2 m. Figure 15 illustrates the v ariation characteristic of the output optical po wer P out Fig. 15. Output power P out varies with transmission distance d w in experiment. of the SDC system with the transmission distance d w . It can be observed that when d w is in the range of 0 – 1.2 m, the output optical power P out remains around 3 W , showing good stability; as d w continues to increase, P out exhibits a significant downw ard trend. When d w = 2 m, the output optical power drops to about 2.18 W , it can still achiev e a transmission efficienc y of 10.9%. When d w = 2 . 4 m, the output optical power drops to about 1.47 W , at this point, the power has dropped to half its maximum value. Achieving resonance becomes dif ficult beyond 2.8 m, primarily because the required tuning precision e xceeds the limits, thereby prev enting the system from reaching a stable resonant state. F . Discussion From the analysis of Fig. 9, it is e vident that the maxi- mum acceptable tolerance τ max d can only ensure the feasibility of the system within a 0 . 5 -m transmission range, which is insufficient to support long-distance SLIPT applications, but the stable region of the proposed SDC scheme exhibits superior adaptability , enabling the system to reliably exceed existing distance limitations. As further illustrated in T able I, the increase in transmission distance not only af fects the output power of the laser but also leads to a notable variation in diffraction loss factor T diff , these two parameters determine the ov erall ef ficiency and stability of the SLIPT system. T ABLE I P A R A M ET E R T R A DE - O FF S O F L A SE R C A V I T Y A T D I FF E RE N T W O RK I N G D I STA N CE S distance d t Stable Region Width T diff Output Po wer (m) (mm) (%) (W) 20 4.147 1 3.314 30 2.764 1 3.246 40 2.074 0.9991 3.112 50 1.659 0.9973 2.833 60 1.382 0.9919 2.264 70 1.185 0.9793 1.173 ... ... ... ... 13 The insights deriv ed from this analysis inform a design framew ork aimed at optimizing SDC system performance and extending its effecti ve working distance. It is clear that the maximum acceptable tolerance τ max d in component manufac- turing and assembly is a critical limiting factor for achieving long-distance resonant beam transmission. Due to inherent limitations in manufacturing processes, including the fabrica- tion of optical tubes and lenses, it is challenging to produce fixed components with sufficient accuracy to maintain stability for working distances exceeding approximately 2 meters using fixed component interv als. Fortunately , our analysis reveals that the stable region width in terms of adjustable parameters like d t is often significantly wider than the overall required tolerance for fixed systems, as shown in Fig. 5. This implies that incorporating adjustability into either the d 1 or d t parameter is a viable approach. In this scenario, the primary limitation shifts from manufacturing tolerance to the precision of the adjustment mechanism. For instance, as illustrated in Fig. 5, designing a 10-meter SDC system would require an adjustment precision on the order of 0 . 5 mm, which is technically feasible. Even extending the working distance to 100 meters remains achiev- able by utilizing lar ger parameter configurations and ensuring higher adjustment precision. Finally , it is crucial to remember the importance of selecting an adequate combination of gain medium radius, a g , and telescope magnification, M tel , as this significantly impacts the ef ficiency and feasibility of long- range resonant beam formation. V . C O N C L U S I O N S This study presents a theoretical in vestigation and ex- perimental verification into the properties and limitations of SDC system.W e analyzed cavity stability , beam parameters, and output power , with particular emphasis on the influence of manufacturing and assembly tolerances on system stabil- ity . W e dev eloped and applied two complementary methods: a binary-search-based Monte Carlo simulation and a linear approximation method to determine the maximum accept- able distance tolerance. Numerical results demonstrate that achieving stable operation ov er long working distances is sev erely constrained by typical tolerances, as the stable region width decreases with the working distances. 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