Stray light refers to unintended light outside the designed optical path that reaches the detector in an optical system, negatively impacting image quality through contrast reduction, ghosting, and other artifacts. Effectively analyzing and controlling stray light during the optical-design stage is essential to maximize system performance [1–3]. Currently, ray tracing is the most widely used technique in stray-light analysis. However, achieving high accuracy requires a large number of rays, significantly increasing computational time and system load [4].

To address these challenges, various technical approaches have been developed to enhance the efficiency of stray-light analysis. These include designs that inherently reduce stray light [5–10], as well as algorithmic methods that minimize stray-light paths, thus decreasing computational workload [11]. Additionally, alternative methods have been explored to directly improve analysis efficiency, such as stray-light analysis using models based on the bidirectional scattering distribution function (BSDF) [12], integrated ray tracing (IRT) [13], and forward-backward-forward ray tracing [4]. Other strategies to reduce ray-tracing time include grid optimization [14, 15], normalized detector irradiance (NDI) evaluation [16, 17], importance sampling [18–20], radiative models that account for orbital characteristics [20, 21], and characteristic rays [22]. Backward ray-tracing optimization is another approach that has been used to enhance the efficiency of ray tracing itself [23].

Backward ray tracing is particularly effective in stray-light analysis, as it focuses on identifying critical stray-light paths and key surfaces that impact image quality. Unlike forward ray tracing, which traces light from the source, backward ray tracing traces from the detector toward the light source, enabling selective analysis of specific stray-light paths that affect the detector [1]. This feature makes backward ray tracing especially useful for stray-light analysis.

However, traditional backward ray tracing, including the optimization method proposed by Yang et al. [23], requires rays to originate from the entire detector area, necessitating a large number of rays for accurate analysis, and resulting in significant computational demands. In contrast, this study significantly improves the efficiency of backward ray tracing by restricting ray origins to specific points and lines on the detector. We propose the corner-departure and line-departure backward ray-tracing methods based on this approach, which reduces the number of rays needed while maintaining accuracy.

In Section 2, we discuss widely used ray-tracing methods and introduce the line- and corner-departure backward ray-tracing methods, emphasizing their differences from conventional techniques. Section 3 describes the optical payload used in the analysis, and the analysis sequence. In Section 4, the simulation results obtained by applying the three techniques are compared and analyzed, to describe the effectiveness of the corner-departure backward ray-tracing method.

2.1. Conventional Methods

Among the various methods for analyzing stray light, the most widely used technique in optical-analysis programs is ray tracing. Ray tracing is generally divided into the sequential and nonsequential approaches. The Sequential ray tracing method traces rays in a specified order across surfaces, focusing primarily on intended rays, such as those reaching the focal point. To model unintended rays, like stray light, nonsequential ray tracing method is applied.

Nonsequential ray tracing includes both forward and backward techniques, as illustrated in Fig. 1. In forward ray tracing, rays are traced from the light source to the detector along their actual path, providing accurate quantitative predictions of radiometric values at specific field angles. Conversely, backward ray tracing traces rays from the detector to the light source, allowing efficient identification of critical ray paths or surfaces that could potentially produce stray light, although it does not provide precise radiometric data [3, 18]. Often a combination of forward and backward ray tracing is employed: Backward ray tracing identifies critical paths, which are then quantitatively evaluated using forward ray tracing.

Figure 1. Illustration of nonsequential ray tracing: (a) Forward ray tracing, (b) backward ray tracing.

The random nature of nonsequential ray tracing requires a significant number of rays, leading to substantial time and memory demands. Yang et al. [23] introduced an improvement to backward ray tracing that enhances accuracy while reducing computational load by half. Cha et al. [4] proposed the forward-backward-forward ray-tracing method, which significantly shortens the overall time for stray-light analysis. Nonetheless, backward ray tracing still requires considerable time and system resources.

2.2. Proposed Method: Corner-departure

In conventional backward ray tracing, light is traced from the detector plane as an area source, typically treated as a Lambertian emitter that radiates rays across a full hemispherical angular space. The proposed corner-departure method operates on the same principle, but treats only the corners of the detector as point sources, hence the term corner-departure backward ray tracing. Additionally, we introduce an intermediate method using two diagonal lines across the detector as line sources, referred to as the line-departure method. Figure 2 illustrates the light source configurations for each approach.

Figure 2. Light-source configurations for backward ray tracing. The squares outlined with long dashed lines represent detectors, and the black surfaces, lines, and ‘X’ positions indicate light-source locations: (a) Conventional backward ray tracing, using the entire area as the source, (b) line-departure method, applying line sources along the detector diagonals, (c) corner-departure method, using only the center and four corner points as sources.

In the line- and corner-departure methods, light sources are positioned at or along the detector’s center and corners. These points serve as critical reference points for calculating the relative illumination of the optical system. Additionally, they act as key locations for normal-light entry and stray-light blockage when designing baffles for stray-light suppression [8, 9]. Thus the center and corners of the detector are critical for both stray-light suppression and the overall optical-system design.

3.1. LWIR Optical Payload

In this study, the methods were validated using a long-wavelength-infrared (LWIR) optical payload. The payload is a Cassegrain-type catadioptric system with dimensions of approximately 185 mm × 185 mm × 153 mm. The primary parameters of the optical payload are summarized in Table 1.

TABLE 1. Primary parameters of the optical payload.

Parameter	Value
Wavelength (μm)	7.4–9.4
F-number	1.4
Full Field of View (degrees)	2.4 × 2.4
Detector Array (pixels)	512 × 512
Detector Pitch (μm)	15

The optical payload includes a primary mirror (PM), a secondary mirror (SM), and four field-correction lenses, along with a window and filter in front of the detector. Figure 3(a) shows an isometric view of the optical system with labeled x, y, and z coordinates to indicate spatial orientation, while Fig. 3(b) provides a cross-sectional view in the yz plane.

Figure 3. Optical payload view: (a) Three-dimensional (3D) model, with labeled x, y, z coordinate axes indicating spatial arrangement, (b) cross-sectional view in the yz plane.

This study does not focus on the payload design itself but highlights the compact nature of the system, with the external baffle length nearly matching the distance between the primary and secondary mirrors, which slightly limits stray-light blockage. Consequently, it is essential to assess the impact of stray light and mitigate any problematic light paths accordingly.

3.2. Analysis Model

Stray-light analysis is conducted using the commercial software LightTools, with three models based on the geometry shown in Fig. 3 representing the conventional, line-departure, and corner-departure methods. These models are identical except for their light-source configurations, as outlined in Section 2.2. The conventional method uses the entire detector plane as a full-area source, while the corner-departure method reduces computational load by treating only the detector’s corners as point sources. The line-departure method, an intermediate approach, positions line sources along the detector’s diagonals. These targeted configurations enable efficient stray-light analysis by focusing on critical illumination points, as shown in Fig. 2. Table 2 provides a summary of the three analysis models.

TABLE 2. Lights sources and detector for the three analysis models.

Parameters		Conventional Backward	Line-departure	Corner-departure
Source	Shape	Square Surface	2 lines	5 points
	Position	Detector Surface	Detector Diagonal Lines	4 Corners and 1 Center
	Angular Distribution	Lambertian
	Divergence Angle	Hemispherical
	Spectrum (μm)	7.4–9.4
Detector	Shape	Circle (Φ 91 mm)
	Position	Entrance Pupil
	Type	Radiant Intensity

In the models, surface properties such as partial reflection/transmission and scattering affect stray-light behavior, and are measured and modeled as outlined in Table 3. In particular, a black coating is applied to all mechanical housings to minimize scattering, with its BSDF modeled using the ABg scattering model [24, 25] with parameters A = 0.015, B = 0.100, and g = 0.200 as follows:

TABLE 3. Average surface properties of the payload across the longwavelength-infrared (LWIR) spectrum.

Parts	Material/Coating	Transmittance (%)	Reflectance (%)	Absorption (%)
L1	IRG 26/ARa)	98	1.46	0.54
L2	ZnS/AR	98	1.36	0.64
L3	IRG 26/AR	98	1.58	0.42
L4	IRG 26/AR	98	1.40	0.60
PM, SM	SiC/HRb)	0	99	1
Window	Ge/AR	97	2.75	0.25
Filter	Ge/AR	93	6.87	0.13
Housing	Black Paint	0	2	98

^a)AR, anti-reflection; ^b)HR, high-reflection..

where β = sin(θ_scatter), β₀ = sin(θ_specular).

3.3. Analysis Sequence

Overall, we follow an enhanced stray-light analysis procedure called the forward-backward-forward stray-light analysis sequence [4], as illustrated in Fig. 4. This method does not perform forward ray-tracing for all directions, but instead complements it with omnidirectional analysis through backward ray tracing, achieving high detection of stray-light paths, i.e., critical paths. Despite this, it requires less time, making it an efficient method for stray-light analysis. In this paper, we aim to improve further efficiency by modifying the analysis method used in sequence 2, which is the backward ray-tracing step in this process.

Figure 4. Overall analysis sequence: Forward-backward-forward.

In sequence 1, a quick stray-light survey is conducted to obtain the relative illuminance of stray light. Although analyzing stray light with a narrower range of incidence angles provides a more accurate assessment of its impact, it also requires more time. Therefore, in sequence 1 forward ray tracing is performed at wider intervals to roughly understand the distribution of stray light, and the data later is updated with data from sequence 3 to obtain the final stray-light distribution.

In sequence 2, backward ray tracing is performed. Figure 5 shows the completion of the corner-departure backward ray tracing, where rays originate from the detector position of the payload and reach the entrance pupil.

Figure 5. 3D view of corner-departure backward ray tracing in progress.

The purpose of backward ray tracing is to identify critical paths, which are the primary stray-light paths. By examining the radiant-intensity distribution map at the detector located at the entrance pupil, critical paths can be identified. To achieve this, the radiant-intensity graph is analyzed to determine the angles of incidence where high peaks are observed. Additionally, stray-light paths that have a significant impact at these angles are identified. In sequence 3, forward ray tracing is performed at the newly identified angles of incidence from sequence 2.

Then, in sequence 4 the relative-illuminance data from the newly identified angles in sequence 3 are updated to sequence 1, to finally confirm the stray-light impact.

3.4. Ray-tracing Conditions

We employed Monte Carlo ray tracing with probabilistic ray splitting. Due to the inherent randomness of the Monte Carlo method, each detector cell is subject to statistical error. In this approach the maximum error of the analysis is estimated as the first standard deviation of the receiver cell with the highest illuminance or intensity, calculated as:

where f represents the illuminance (or intensity) of each ray, and N is the total number of rays traced from a single source. As N increases, the error converges according to:

To achieve an error of approximately 1% (or 99% accuracy), more than 10,000 rays must reach each bin. When fewer rays are used, fluctuations in results increase; Therefore, simulation conditions are carefully set to maintain low error rates and reliability across the three methods. Table 4 provides a summary of the parameters and performance comparisons for each ray-tracing method, with the total number of rays determined to achieve a peak error rate of approximately 1%.

TABLE 4. Summary of ray-tracing conditions.

Variables	Conventional Backward	Line-departure Backward	Corner-departure Backward
Total Rays	20,000,000	2,500,000	400,000
Relative Ray Power Threshold	10−6
Total Peak Error Rate (%)	1.84	1.50	1.16
Stray Light Peak Error Rate (%)	1.01	1.02	0.95

Additionally, a relative-ray-power threshold of 10⁻⁶ is applied to limit the number of rays traced, thereby preventing an excessive number of ray paths [22]. With this threshold, approximately 11 critical paths are equally identified in each case.

Table 5 summarizes the number of ray paths detected when the normalized-power threshold is set to 10⁻⁶. While the conventional method detects additional low-power paths compared to the proposed corner-departure method, these paths are found to have negligible impact on the analysis results. This indicates that the proposed method maintains its efficiency by focusing on key stray-light contributors, without unnecessary computational overhead.

TABLE 5. Number of ray paths according to normalized-power range.

Normalized-power Range	Conventional Backward	Line-departure Backward	Corner-departure Backward
10−6	106	102	102

4.1. Critical-path Finding

This section describes the results of the forward-backward-forward analysis sequence explained in section 3.3, with a particular focus on the results based on the analysis method used in sequence 2.

Figure 6 shows a graph of the relative illuminance obtained in sequence 1. The angle of incidence (AOI) and relative illuminance values on the graph are represented on a logarithmic scale. The analysis is conducted in intervals of 0.2° for the 0°–3° range, 1° for the 3°–10° range, and 10° for the 10°–50° range, resulting in a total of 27 forward ray-tracing runs.

Figure 6. Quick survey: Relative-illuminance graph of stray light, analyzed through simplified forward ray-tracing.

Figure 7 shows the radiant-intensity distribution map obtained in sequence 2. The distribution map is represented in a vertical and horizontal polar-coordinate format, allowing the observation of stray-light effects based on the angle of incidence. For angles of incidence above 52°, almost all light is blocked by the outer baffle, resulting in no intensity being displayed. Therefore, the horizontal axis of the chart is shown up to 52°.

Figure 7. Normalized-radiant-intensity distribution maps for all paths obtained through three backward ray-tracing methods: (a) Conventional backward, (b) line-departure method, and (c) corner-departure method.

Figure 8 is the radiant-intensity graph along the vertical axis. It is normalized based on the highest intensity on the detector, excluding the focal point. By checking the angles of incidence at the peaks of the plot, major stray-light paths are identified. The numbers at the arrow tails in Fig. 8 represent the major-path numbers, which correspond to the order of the radiant-intensity distribution maps for the major paths shown in Fig. 9.

Figure 8. Peak finding: Normalized-radiant-intensity graph of stray light. Arrow markers indicate peaks: (a) Conventional backward method, (b) line-departure method, and (c) corner-departure method.

Figure 9. Normalized-radiant-intensity distribution map of newly identified stray-light paths (from left to right: path 1, path 2, path 3, path 4, path 5, paths 6–10, path 11): (a) Conventional backward method, (b) line-departure method, and (c) corner-departure method.

The comparison of methods in Figs. 7 and 9 reveals key differences. While the line-departure method exhibits a denser intensity distribution in Fig. 7, the proposed method highlights higher intensity values along critical paths in Fig. 9. This indicates that the proposed method efficiently identifies significant stray-light contributors by focusing computational resources on impactful paths, enhancing efficiency for a chosen level of accuracy.

Figure 10 presents updated sequence 1 data, incorporating forward ray-tracing results from sequence 3 at newly identified angles. This iterative process allows the proposed method to refine its findings and align closely with conventional backward ray tracing.

Figure 10. Relative-illuminance graph with sequence-3 results updating sequence 1 data. Peak plots are observed at the blue arrow markers. Line types, colors, and symbols are as follows: Solid line with red triangles, corner-departure method; Dotted line with blue diamonds, line-departure method; Dotted line with green squares, conventional backward method; black dotted line with circles, quick survey.

Overall, the proposed method successfully replicates the results of traditional backward ray tracing, while identifying the same 11 critical paths and revealing additional stray-light impacts at 6.4° and 12.4°. These results demonstrate its reliability and its ability to deliver efficient, consistent, and comprehensive analysis.

4.2. Comparison of Simulation Time and Memory Usage

First, we compare simulation time and system load (memory usage) across the three backward ray-tracing methods, all set to achieve similar error rates. The line-departure and corner-departure backward methods complete the simulations faster and with lower system load than conventional backward ray tracing, as detailed in Table 6. For reference, the desktop used for analysis is equipped with four 32-gigabyte memory modules. In the case of conventional backward ray tracing, the high system load requires partial use of virtual memory on the storage device.

TABLE 6. Comparison of simulation time and memory usage.

Variables	Conventional Backward	Line-departure Backward	Corner-departure Backward
Simulation Time (minutes)	66.0	8.2	2.5
Memory Usage (GB)	135	20	9

Among the various methods used for stray-light analysis, we propose the corner-departure backward ray-tracing method, designed to reduce both analysis time and system load—common limitations of conventional backward ray-tracing techniques. To validate this approach, we apply the forward-backward-forward analysis method to an LWIR optical system under development. Three different backward ray-tracing methods are evaluated, each with an equivalent error rate, and simulations are conducted to compare accuracy, required time, and system load.

Our analysis demonstrates that the proposed corner-departure backward ray-tracing method detects stray light as effectively as the conventional approach, while reducing system load by an order of magnitude and performing the analysis over 26 times as quickly. Due to its significantly shorter analysis time, this method is expected to be highly beneficial for efficiently examining scattered-light distributions in optical systems. The efficacy of this method is further demonstrated in its application to baffle optimization in LWIR catadioptric payloads detailed in another paper [26].

Another key observation is the reduced number of ray paths identified by the proposed corner-departure method, compared to the conventional approach (Table 5). While the conventional method detects additional low-power paths, these do not impact critical performance metrics, as Figs. 7 and 8 demonstrate. This highlights the proposed method’s ability to focus computational resources on significant stray-light contributors, ensuring both efficiency and accuracy.

An additional advantage of the proposed method is its ability to focus on critical paths with higher intensity, as observed in Figs. 7 and 9. By emphasizing key stray-light contributors, the proposed method ensures effective analysis within critical paths while maintaining computational efficiency.

Future work will investigate strategies for selectively including boundary regions or other detector areas, to further improve the coverage of critical paths without significantly increasing analysis time or system load.

This research was supported by the Agency for Defense Development and funded by the government (Defense Acquisition Program Administration) in 2024, as part of the Defense Research and Development Program (Grant no. 912984301).

The authors declare no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.