Defocus-Aware Modeling and Control Analysis of a QD-Based Optical Tracking System: Experimental and Simulated Evaluation Using the DOLCE Terminal Hideki Takamoto Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan takamoto@space.t.u-tokyo.ac.jp Kuna Shitara Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan shitara@space.t.u-tokyo.ac.jp Vinicius Ferreira Nery Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan viniciusfnery@space.t.u-tokyo.ac.jp Kazuki Takashima Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan takashima@space.t.u-tokyo.ac.jp Yuki Kusano Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan kusano@space.t.u-tokyo.ac.jp Norihide Miyamura School of Science and Engineering Meisei University Tokyo, Japan norihide.miyamura@meisei-u.ac.jp Kota Kakihara Arkedge Space Inc. Tokyo, Japan kakihara-kota@arkedgespace.com Toshihiro Suzuki Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan suzuki t@space.t.u-tokyo.ac.jp Takayuki Hosonuma JAXA Kanagawa, Japan hosonuma.takayuki@gmail.com Satoshi Ikari Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan ikari@space.t.u-tokyo.ac.jp Ryu Funase Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan funase@space.t.u-tokyo.ac.jp Shinichi Nakasuka Department of Aeronautics and Astronautics The University of Tokyo Tokyo, Japan nakasuka@space.t.u-tokyo.ac.jp Abstract—Free-space optical communication (FSOC) is at- tracting significant attention as a core technology for future space communication infrastructure, owing to its capability of provid- ing wide bandwidth and high antenna gain with narrow beam divergence. However, the small divergence angle of optical beams makes the link highly sensitive to pointing errors, requiring highly accurate tracking sensors. The quadrant detector (QD), with its simple structure and fast response, is a strong candidate for fine tracking, but its linear response range is severely limited when placed at the focal plane. To address this limitation, this study investigates a defocus-based approach in which the QD is intentionally placed out of focus to expand its field of view. Using wave-optics simulations and laboratory experiments, the response characteristics of QDs under defocus conditions are systematically evaluated. Furthermore, acquisition and tracking experiments were conducted with a numerical terminal simulator to verify the validity of QD defocus. The results clarify the trade-off between expanded linear range and tracking accuracy, providing valuable insights for the design of compact and cost- effective optical communication terminals. Index Terms—Free-space optical communication (FSOC), Pointing, Acquisition, and Tracking (PAT), Quadrant Detector (QD), Defocus, Terminal Simulator I. BACKGROUND In recent years, driven by the growing demand for high- capacity data transmission between satellites and between satellites and the ground, research and development of free- space optical communication (FSOC) has been actively pur- sued [1]. Compared with microwave communication, optical communication offers significant advantages such as a wider bandwidth and higher antenna gain resulting from its narrow beam divergence [1], making it a promising core technology for next-generation space communication infrastructure. How- ever, due to the small divergence angle of optical beams, the system is highly sensitive to even slight attitude errors of the transmitter and receiver as well as to their relative motion. Therefore, the implementation of highly accurate pointing, acquisition, and tracking (PAT) functions is indispensable [1]. In many previous studies, fine tracking within PAT systems has employed sensors such as camera-based imagers and position-sensitive detectors (PSDs). Among these, the quadrant detector (QD) has attracted particular attention as a candidate sensor for spaceborne optical communication terminals, owing to its simple structure and fast response. A QD consists of a photodiode divided into four quadrants, where the incident spot position can be estimated by computing differential ratios of photocurrent outputs from the quadrants. This enables fast and low-latency acquisition of angular error signals, which is a major advantage for compensating attitude fluctuations and platform vibrations. One of the major limitations of the QD, however, is its extremely narrow linear response range. When the QD is placed at the focal plane, the region over which the output signal maintains an approximately proportional relationship is constrained by the diffraction limit of the lens, and typically extends to only several tens of microradians. Consequently, its effective operating range is limited in situations requiring wide fields of view (FOV), such as initial acquisition or cases with large platform pointing errors. Conventional optical terminal designs have addressed this issue by combining coarse tracking sensors with wide FOV and fine tracking sensors with high precision but narrow FOV. Nevertheless, for future optical communication systems aiming at miniaturization and cost reduction, simplification of sensor configuration and relaxation of design constraints are strongly desired. To address this challenge, the present study focuses on the “defocus” method, in which the QD is intentionally placed out of the focal plane. In such a configuration, the spot size ex- pands and the linear response range has the potential to extend. At the same time, however, the intensity distribution deviates from a simple flat-spot approximation or Airy-disk profile and becomes nonlinear. Most previous studies have relied on idealized distribution models, and detailed evaluations of QD response characteristics under actual defocused conditions have been limited. Therefore, accurately characterizing the QD response under defocus is essential to provide important insights for designing tracking systems that aim to achieve both wide FOV and high accuracy. The objective of this study is to systematically evaluate the sensitivity characteristics and linear response range of QDs under defocus conditions through theoretical modeling, numerical simulation, and experimental validation. In the sim- ulation, a wave-optics-based propagation model is employed to generate spot intensity distributions for various defocus distances, which are then applied to calculate the QD voltage outputs, thereby constructing a nonlinear QD response model. In the experimental study, a QD is mounted on an optical bench, and the relative distance from the focal plane is varied while injecting the optical spot, so that the response variation with defocus can be directly measured. These experiments validate the simulation model and quantitatively assess the extent of linear FOV expansion under defocus. Furthermore, PAT experiments were conducted using a numerical termi- nal simulator to verify the effectiveness of the QD defocus method. Originally, the research plan included conducting tracking control experiments using a QD within the DOLCE terminal. However, due to constraints in research resources and avail- able experiment time, the present work focuses on defocus experiments with a standalone QD, emphasizing its output characteristics and their correlation with the simulation model. Although control experiments are limited to simulator analysis, a deeper understanding of the QD sensor behavior is expected to provide design guidelines that can be extended to full terminal systems in the future. In summary, this study comprehensively analyzes QD defo- cus behavior from both optical system and sensor response per- spectives, thereby providing fundamental knowledge toward the simultaneous realization of wider FOV and high-precision tracking performance in optical communication terminal de- sign. II. PRINCIPLE OF QD Fig. 1: Schematic of a QD-based fine tracking sensor A QD consists of four photodiodes arranged in a quadrant configuration. The photocurrents generated in each segment are converted into voltages by operational amplifiers, allowing the incident beam position to be detected. Compared to other optical sensors, the QD is characterized by its high angular resolution and fast response speed. Figure 1 illustrates a typical configuration when a QD is employed as a fine-tracking sensor. Here, the focal length of the front lens is denoted as f , and the defocus, defined as the distance from the focal plane to the detector surface, is denoted as d. As shown in the figure, the incident beam is focused by the front lens and directed onto the QD, which is placed near the focal plane. When the incident beam deviates in angle, the spot position on the QD shifts accordingly. This displacement changes the ratio of the optical power received by the four quadrants, from which the beam spot position can be estimated. When the spot is located near the center of the QD, the output signals of the quadrants are approximately proportional to the displacement of the spot from the center. If we denote the output voltages corresponding to the four quadrants A–D as va–vd, the beam spot position (px, py ) can be expressed as follows [2]: px = P0 (va + vd) − (vb + vc) va + vb + vc + vd , (1) py = P0 (va + vb) − (vc + vd) va + vb + vc + vd (2) Here, P0 represents the region in which the QD provides a linear response as a tracking sensor, referred to as the linear field of view (FOV). When the spot on the QD is assumed to be a circular flat-top spot with diameter ds, simple geometric analysis shows that the proportionality constant is given by Kf lat = π/8 ≃ 0.393, i.e., P0 = Kf lat · ds = π 8 ds (3) On the other hand, when the QD is placed exactly at the focal plane of the lens, the tracking accuracy is ultimately limited by the diffraction condition. In this case, the spot forms an Airy distribution, and using the Airy disk diameter (first dark ring), defined as dA = 1.22λf /R (where R is the aperture radius and λ is the wavelength), P0 can be expressed as [3]: P0 = 3π 32 · λf R = 1 4.14 dA (4) In practice, however, when the QD is placed away from the focal plane, the spot does not become flat but exhibits a non-uniform intensity distribution. Thus, the response does not fully agree with either of the above expressions. To address this, we define a conversion coefficient K under defocused conditions, and P0 is expressed in terms of the geometric spot diameter ds as: P0 = K · ds (5) Finally, the beam directions (qx, qy ) are obtained by dividing the estimated spot positions by the focal length of the lens. qx = px f , qy = py f (6) III. QD NUMERICAL SIMULATOR A. Analysis Method In this section, we describe a numerical simulator developed to model the output characteristics of a QD under various spot conditions. The simulator calculates the relative output voltages of each quadrant cell by summing the optical intensity incident on each cell when the spot image is placed at a given position on the QD surface. By applying the position estimation algorithm described in the previous section to these simulated outputs, the behavior of the linear FOV as well as the estimated spot position can be evaluated. All parameter values are set identical to those used in the experimental configuration presented in Table 2. Fig. 2: Simulated spot intensity distributions under different defocus conditions. In the simulation experiment, the first step is to generate the intensity distributions of the spot under different defocus conditions. At zero defocus, the spot follows an Airy distribu- tion, while under defocused conditions, the spot does not form a flat-top distribution but instead exhibits a specific intensity profile. Next, for each spot distribution, the simulator computes the cell output voltages of quadrants A–D as the spot is shifted along the QD sensor’s X-axis, as illustrated in Fig. 3. The position estimation algorithm is then applied to the calculated outputs. This procedure enables the evaluation of both the estimation accuracy and the extent of the linear FOV. Fig. 3: Illustration of spot displacement across the QD sensor in the simulation. B. Simulation Results This section presents representative outputs from the QD simulator and discusses the estimation of the optimal conver- sion coefficient K under defocused conditions. The aim is to achieve position estimation with higher accuracy than that provided by the conventional flat-spot assumption. Fig. 4 presents examples of the QD output voltage ratios as a function of the spot center position for several defocus condi- tions. The simulated responses clearly reproduce the behavior near the linear FOV as well as the influence of noise at low received optical power. The modeled noise sources include dark noise, shot noise, and amplifier noise. As the defocus increases, the spot size becomes larger, resulting in a shallower slope in the response curves and an expanded linear FOV. At the same time, changes in the spot intensity distribution induced by defocus manifest as additional inflection points in the response curves. (a) received optical power levels = −18.6dBm (b) received optical power levels = −55.0dBm Fig. 4: QD output voltage ratios. We then examine the conversion from voltage ratio to estimated position. Figure 5 shows the estimated spot positions px and estimation errors (px − xtrue) when applying the conversion defined by Eq. 4 to the voltage ratio at d = 0. For d̸ = 0, the estimated positions and errors obtained with conversion coefficients K = 0.3, 0.35, π/8(≃ 0.392) are shown in Fig. 6. For the Airy distribution case (d = 0), Eq. 4 provided an excellent match to the simulated response. Under defocused conditions, however, the optimal coefficient depends on the degree of defocus: for d = 2 mm, where the spot shape remains close to an Airy distribution, K = 0.35 provided the best agreement, while for d = 10 mm, where the spot shape resembles a flat-top distribution, K = π/8 matched more closely. These results demonstrate that the optimal conversion coefficient varies with the amount of defocus. Table I summarizes the conversion coefficients that min- imize the root-mean-square (RMS) error of the estimated (a) Estimated position (b) Estimation error Fig. 5: Simulation results at d = 0 mm (The red dashed lines indicate the linear FOV range). (a) d = 2 mm (b) d = 6 mm (c) d = 10 mm Fig. 6: Estimated positions and errors under defocused condi- tions with various conversion coefficients K. position within the linear FOV for several defocus values. The results confirm that larger defocus leads to larger optimal K. Although the linear FOV increases with defocus, the RMS error within the linear region also increases, revealing a trade- off between linear FOV and estimation accuracy. While the table lists the coefficients that minimize RMS error, in practice, if a tolerance for estimation error is specified, one can instead select K to maximize the linear FOV while still satisfying the error requirement. TABLE I: Optimal conversion coefficient K, linear FOV, and RMS error under different defocus values. Defocus [mm] 2 4 6 8 10 Optimal K 0.343 0.390 0.395 0.398 0.406 Linear FOV [mm] 0.027 0.062 0.095 0.13 0.16 RMS [mm] 0.0027 0.0045 0.0046 0.0046 0.0048 IV. QD EXPERIMENT A. Experimental Configuration To validate the results obtained from the simulations in the previous section, an experiment using an actual QD was conducted. The experimental configuration is shown in Fig. 7. In the setup, a 1550 nm laser beam with a 2.27 mm beam diameter emitted from a fiber collimator was focused by a front lens and then incident onto the QD, which was mounted on a three-axis stage positioned near the focal plane. Fig. 7: Experimental configuration of the QD setup. First, with the defocus position fixed, the QD was translated horizontally in steps of 0.01 mm, similar to the procedure em- ployed in the simulation, and its output voltage corresponding to the spot position was read out and recorded by Arduino. By repeating this measurement while varying the defocus distance, the variation of the QD response under different defocus conditions was characterized. The specifications of the QD (KPDE150Q-H15) and the inverting amplifier used in the experiment are summarized in Table II. TABLE II: Specifications of the QD and inverting amplifier. Quadrant Detector (QD) Active area diameter 1.5 mm Element gap 0.10 or 0.04 mm Responsivity (@1550 nm) 1.0 A/W I/V conversion resistance 200 kΩ Dark current 0.3 nA Inverting Amplifier R1 22 kΩ R2 39 kΩ C 0.0001 μF The focal length of the front lens for the QD was set to 75 mm, consistent with the simulation described later. The laser power was adjusted to 18.6 dBm using an optical attenuator, such that the QD would just reach saturation when all the laser power was incident on a single quadrant. The entire experimental apparatus was placed on a vibration-isolated optical table. During the measurements, the surrounding environment was darkened by switching off nearby electrical lighting and covering windows with blackout curtains to minimize background light.