In the actual trajectory, a MBW is represented as a spherical shock wave traveling at a specified velocity and passing through the indoor trajectory as a longitudinal wave [41]. The far-field propagation law of its peak pressure from the muzzle position in the ballistic trajectory can be expressed as [42, 43], and as shown in Figure 2A. P S γ , θ M = P m 0 θ M γ − α M θ M (1) where γ denotes the distance from the muzzle position in the ballistic trajectory; θ _M represents the angle between the direction of MBW propagation and the trajectory direction; a _M (θ _M ) indicates the attenuation rate of the MBW pressure; and P _m0 represents the initial pressure of the MBW.

At a certain distance, the attenuation of the MBW pressure gradually decreases [44]. Figure 2B displays the normalized MBW pressure curve at the O _M as a function of time [45].

The distributions of the MBW pressure in T _a and T _d [46–48] are, respectively expressed as P S t = P 0 + a t , t ∈ T a P S t = P 0 + P m 1 − t T + e − b t / T + , t ∈ T d (2) where the rate of pressure variation within T _a is denoted by a, P ₀ represents the environmental pressure, P _m represents the peak overpressure, and T ₊ represents the positive pressure duration of the MBW. The exponential decay rate is denoted by b.

The pressure variation of the MBW is nonlinear, displaying an “N”-shaped pattern. In an air medium, the density ρ is related to the temperature T and the pressure p via the ideal equation of state p = ρ R T . The ideal gas constant is denoted by R. The connection between refractive index and density can be derived using the Gladstone–Dale relationship [49–51], which is represented as: P S = ρ R T n − 1 = K ρ (3) where the Gladstone–Dale constant is denoted as K.

Thus, the relationship between refractive index and pressure can be expressed as [52, 53]. P S = n R T K − R T K (4)

Linear changes in both pressure and refractive index are observed.

During T _a , the corresponding spatial distance is termed the rise distance X _a . During T _d , the spatial distance is identified as the recovery distance X _d . The spatial length associated with T ₊ is referred to as the positive pressure duration distance X ₊.

Consequently, the gradient refractive index distribution within the spatial range corresponding to the MBW at a given moment is represented as n a x s = n 0 + α , x s ∈ X a n d x s = n 0 + n m 1 − t X + e − β x s / X + , x s ∈ X d (5)

The rate of refractive index variation within T _a is denoted by α, which is also defined as the rate of pressure variation within T _a . The exponential decay rate is denoted by β.

Assuming the coordinate system for the spatial region of the MBW gradient refractive index is denoted as o _s - x _s y _s z _s , the starting position of the gradient refractive index distribution is at o _s in the upper right corner of the MBW spatial region. The coordinate origin is located at o _s . The farthest right end of the MBW gradient refractive index spatial region corresponds to the y _s -axis. The x _s -axis is positioned above the trajectory line at a distance of l _D/2, parallel to the trajectory but oriented in the opposite direction. Rays originating from the uppermost region of the MBW spatial area enter the gradient refractive index space. Given the symmetrical distribution of MBWs on both sides of the trajectory, the distributions of the refractive indices n _a and n _d in the upper half of the gradient are analyzed, as illustrated in Figure 3.

A ray tracing technique is proposed to examine the signal sensing mechanism of muzzle blasts passing through a LSD. The scenario where the MBW fully enters the LSD is analyzed using the refracted optical path diagram from geometric optics. The ray traverses the MBW gradient refractive index medium from left to right, as shown in Figure 5. The main optical axis of the beam path within the MBW gradient refractive index medium aligns with the y _s -axis. Rays emitted from the o _s are refracted within each small segment of the MBW gradient refractive index medium. The initial conditions are: the angle of incidence is θ ₀, the initial position is x _s = 0, and the environmental refractive index is n ₀.

Only deflection in the x _s direction is considered. According to Snell’s law, at the interface where refraction takes place within a small path segment n x s sin x s = n 0 ⁡ sin ⁡ θ 0 (9)

The length of an arc on a ray trajectory can be expressed as d s 2 = d x s 2 + d y s 2 d x s d s 2 = sin ⁡ θ → d y s d x s 2 = n x s 2 n 0 2 ⁡ sin 2 ⁡ θ 0 − 1 (10)

By designing N ₀ = sinθ ₀, the above equation can be simplified. d y s d x s 2 = n x s 2 − N 0 2 N 0 2 → d y s d x s = n x s 2 N 0 2 − 1 (11)

The above equation represents the trajectory of light propagation.

The rays propagate through the gradient refractive index medium of the MBW, and the angle between the refracted rays and the optical axis is termed the deflection angle, dθ. d θ x s , y s = d y s d x s (12)

The width, Δd, of the radiation flux reduced on the photosensitive surface at the receiver corresponds to the distance of the ray deflected in the x _s direction. This can be expressed as Δd = x _s .

The LSD is affected by MBW, leading to the refraction of parallel rays. Consequently, the area of radiation flux on the photosensitive surface at the receiver is modified. The change in the radiation flux, Δφ′, observed on the photosensitive surface of the photoelectric detection sensor can be expressed as. Δ φ ′ = E L Δ S ′ = E L w D Δ d (13) where ΔS' is the altered area of the radiation flux on the photosensitive surface and where Δd represents the width of the radiation flux reduced on the photosensitive surface.

The distributions of the gradient refractive indices within X _a and X _d of the MBW are expressed as n _a and n _d , as shown in Equation 5.

In linear gradient refractive index media, n _a , deriving an analytical solution for ray trajectories is intricate and does not reveal detailed characteristics of light paths. Consequently, the fourth-order Runge–Kutta method, which offers superior accuracy compared to the Eulerian method, is utilized for trajectory analysis to obtain numerical solutions.

The conversion of dy _s to y _s is achieved through numerical integration using the fourth-order Runge–Kutta method. This method involves a four-step piecewise approximation of differentials. By decreasing the step size and increasing the order, the error range can be better managed. Complex curves can be approximated within a specified step length. Consequently, under conditions of convergent computation, results that are relatively satisfactory can frequently be obtained.

In Figure 5, the initial state of the parallel light is d y s d x s = n 0 + a x s 2 n 0 2 ⁡ sin 2 ⁡ θ 0 − 1 y s 0 = 0 , x s ∈ X a d y s d x s = n 0 + n m 1 − t X + e − β x s / X + 2 n 0 2 ⁡ sin 2 ⁡ θ 0 − 1 y s 0 = 0 , x s ∈ X b (14)

By choosing a specific step length h, the function values over the interval are subjected to iterative solving in a single step. The formula used for these calculations is as follows: k 1 = f x s n , y s n k 2 = f x s n + h 2 , y s n + h 2 k 1 k 3 = f x s n + h 2 , y s n + h 2 k 2 k 4 = f x s n + h , y s n + h k 3 y s n + 1 = y s n + h 6 k 1 + 2 k 2 + 2 k 3 + k 4 y s 0 = 0 (15)

Within X _a , three ray propagation trajectories are selected to illustrate light propagation conditions, designated as Ray ₁ (x _s = 0 nm), Ray ₂ (x _s = 4 nm), and Ray ₃ (x _s = 9 nm). The refractive trajectories of these rays in X _a are illustrated in Figure 7A, obtained through a simulation using the fourth-order Runge-Kutta method.

As illustrated in Figure 7A, the light propagation trajectory is predominantly characterized by straight-line movement. However, the deflection angles of different rays reveal more detailed features of light propagation, with the deflection of rays progressively decreasing as they travel.

Within X _d , three ray propagation paths are selected to demonstrate light transmission conditions, identified as Ray ₄ (x _s = 0 nm), Ray ₅ (x _s = 80,000 nm), and Ray6 (x _s = 400,000 nm). The refractive trajectories of these rays in X _d are presented in Figure 7B, obtained through a simulation using the fourth-order Runge-Kutta method.

As illustrated in Figure 7B, the trajectory of light propagation is predominantly characterized by curve propagation. The deflection angle of rays reveals finer details of light propagation, with ray deflection initially increasing, then decreasing, and eventually stabilizing as they continue to propagate. Specifically, in Figure 7B, a characteristic line y _s = 500 mm is parallel to the x _s -axis. Ray ₄ begins deflection from the initial position x _s = 0 mm, intersects the characteristic line at (0.88 mm, 500 mm), and its deflection reaches (1 mm, 561 mm). Ray ₅ begins deflection from x _s = 0.08 mm, intersects the characteristic line at (1 mm, 500 mm), and its deflection reaches (1 mm, 500 mm). Ray6 starts deflection from x _s = 0.40 mm and concludes its deflection at (1 mm, 341 mm).

In the gradient refractive index distribution within X _d of the MBW, the peak refractive index (n _m), positive pressure duration distance X ₊, and exponential decay rate β directly influence the distribution of n _d . A detailed analysis is required to understand how variations in these three parameters affect n _d and their impact on the deflection of parallel rays. In the analysis, if one characteristic variable is altered, the other two are held constant.

The peak refractive coefficients are set at n _m1 = 1.0005, n _m2 = 1.0010, and n _m3 = 1.0020. A simulation analysis is performed for Ray ₄ and Ray ₅, and the refractive trajectories of the rays are presented in Figure 8A.

As shown in Figure 8A, with only variations in the peak refractive index (n _m), the refractive trajectories of Ray ₄₁, Ray ₄₂, and Ray ₅₁ are nearly identical, as are those of Ray ₅₁, Ray _52, and Ray ₅₃. The refraction of rays in the gradient refractive index medium of the MBW is minimally affected. Therefore, variations in the peak refractive index do not significantly impact ray refraction. The intersection of Ray ₅ with the characteristic line occurs at (1 mm, 500 mm). For rays with x _s > 0.08 mm, the intersection points with the characteristic line are all greater than 1 mm. A significant factor preventing rays from reaching the photosensitive surface of the photodetector at the receiver of the LSD is that x _s > 0.08 mm for the parallel light from the light source.

The positive pressure duration distance is set at X ₊₁ = X _d /15, X ₊₂ = X _d /5, and X ₊₃ = X _d /25. A simulation analysis is conducted for Ray ₄ and Ray ₅. The refractive trajectories of the rays for different X ₊ values are illustrated in Figure 8B.

As shown in Figure 8B, variations in the positive pressure duration distance X ₊ significantly impact the refraction of Ray ₄₁, Ray ₄₂, Ray ₄₃, Ray ₅₁, Ray ₅₂, and Ray ₅₃ in the gradient refractive index medium of the MBW. Ray ₄ and Ray ₅ exhibit different behaviours. For the same X ₊, the deflections of Ray ₄₂ and Ray ₅₂, as well as those of Ray ₄₃ and Ray ₅₃, are nearly parallel. As X ₊ increases, the intersection points of Ray ₄₂ and Ray ₅₂ with the characteristic line shift to the right from their previous positions, with minimal effect on ray deflection. Conversely, as X ₊ decreases, the intersection points of Ray ₄₃ and Ray ₅₃ with the characteristic line shift to the left from their previous positions, with a more significant impact on ray deflection. Notably, the refractive trajectories of Ray ₄₁ and Ray ₅₁ align with those of Ray ₄ and Ray ₅ in Figure 7B.

Regardless of variations in X ₊, the intersection point of Ray ₄ with the characteristic line remains below 1 mm. However, Ray ₅ traverses the MBW gradient refractive index medium at X ₊ = X _d /15, with the intersection point of Ray ₅ with the characteristic line equaling 1 mm. As X ₊ increases, Ray ₅ experiences varying degrees of refraction, designated as Ray _α–1, Ray _α–2, and Ray _α–3. All these intersection points with the characteristic line exceed 1 mm. Conversely, as X ₊ decreases, the refractions of Ray ₅, labeled as Ray _α+1, Ray _α+2, and Ray _α+3, increase, with all intersection points with the characteristic line being less than 1 mm. The refraction trajectories are illustrated in Figure 8C.

The exponential decay rates are established as β ₁ = 0.6, β ₂ = 0.8, and β ₃ = 0.98. A simulation analysis is performed for Ray ₄ and Ray ₅; the refractive trajectories of these rays for various β values are depicted in Figure 8D.

As illustrated in Figure 8D, varying only the exponential decay rate β resulted in more pronounced effects on ray refraction, with Ray ₄ and Ray ₅ demonstrating the following observations: For the same β, the deflections of Ray ₄₁ and Ray ₅₁, as well as those of Ray ₄₃ and Ray ₅₃, are nearly parallel. When β increases, the intersection points of Ray ₄₃ and Ray ₅₃ with the characteristic line move leftward from their initial positions, though the ray deflections remain largely unchanged.

Conversely, a decrease in β causes the intersection points of Ray ₄₁ and Ray ₅₁ with the characteristic line to shift rightward from their previous positions, with a more significant impact on the ray deflections. Additionally, the refractive paths of Ray ₄₂ and Ray ₅₂ align with those of Ray ₄ and Ray ₅ depicted in Figure 7B.

When β ₀ = 0.506, the intersection point of Ray _4β0 with the characteristic line is 1 mm. When β ₁ = 0.800 > 0.506, the intersection point of Ray _4β1 with the characteristic line is less than 1 mm. When β ₂ = 0.400 < 0.506, the intersection point of Ray _4β2 with the characteristic line is greater than 1 mm. A critical factor preventing rays from parallel light from entering the photosensitive surface of the photodetector at the LSD receiver is β < 0.506. The refraction trajectories described are illustrated in Figure 8E.

Based on the propagation characteristics of the MBW, the distribution n _d corresponding to three features is assessed at varying propagation distances of the MBW. The refraction of rays resulting from parallel light passing through the MBW is analyzed.

The gradient refractive index n _d , peak refractive index n _m, positive pressure duration distance X ₊, and exponential decay rate β are assumed for the MBW at different propagation distances, as listed in Table 1.

Ray ₄ and Ray ₅ are selected from within X _d . The analysis of ray refraction is conducted across the gradient refractive index media n _d1 ∼ n _d4 at different MBW, propagation distances. Ray ₄₁ and Ray ₅₁ represent the refractions in n _d1. Ray ₄₂ and Ray ₅₂ illustrate the refractions in n _d2, Ray ₄₃ and Ray ₅₃ show the refractions in n _d3, and Ray ₄₄ and Ray ₅₄ indicate the refractions in n _d4. This is depicted in Figure 8F.

The intersections of the ray refraction trajectories depicted in Figure 8F with the characteristic line are presented in Table 2.

First, the right edge AB of the gradient refractive index medium of the MBW contacting the left edge F 1 F 1 ′ of the LSD is referred to as the beginning. When the MBW enters the LSD, the distance is about L _sl = 0.08 mm, which is referred to as cessation.

This procedure is designated as process 1, during which an MBW enters the LSD. The termination of process 1 is referred to as another initiation. The distance about denoted as L _sl = 0.89 mm marks this cessation. This subsequent phase is known as process 2, wherein the MBW re-enters the LSD. The end of process 2 is identified as a new beginning. This cycle continues until the MBW fully overlaps with the LSD, known as process 3, where the MBW again enters the LSD. The three phases of the MBW entering the LSD are illustrated in Figure 9, with a total duration of T _s .

Process 1 of the MBW entering the LSD is depicted in Figure 9A. The distance covered by the MBW within the LSD is L _sl = 0.08 mm. This instance is denoted as T _e1. The parallel light from the LSD is refracted by the MBW, leading to a reduced width on the photosensitive surface at the receiver. This refraction results in an ineffective reception area, as illustrated. The remaining light reaches the photosensitive surface at the receiver, creating the effective reception area.

The length of the photosensitive surface at the receiver is W_D = 330 mm. The area Δ S e 1 ′ where the irradiance is reduced on the photosensitive surface, where Δ s e 1 ′ = W D Δ d D = 330 mm × 0.08 mm (16)

Process 2 of the entering procedure is illustrated in Figure 9B. The distance covered by the MBW within the LSD is L _sl = 0.89 mm. This instance is denoted as T _e2. The reduced width on the photosensitive surface at the receiver is Δd = 0.89 mm. The area Δ S e 2 ′ where the irradiance is reduced on the photosensitive surface, where Δ s e 2 ′ = W D Δ d D = 330 mm × 0.89 mm (17)

Process 3 of the entering procedure is depicted in Figure 9C. The distance traveled by the MBW within the LSD is L _sl = 1 mm. This instance is denoted as T _e3. At this point, the MBW fully overlaps the LSD. The reduced width on the photosensitive surface at the receiver is Δd = 0.89 mm. The area Δ S e 3 ′ where the irradiance is reduced on the photosensitive surface, where Δ s e 3 ′ = W D Δ d D = 330 mm × 0.89 mm (18)

Initially, the contact of the right edge AB of the MBW gradient refractive index medium with the right edge of the LSD is referred to as the start. When the MBW exits the LSD, the distance is about L _sl = 1.22 mm, which denotes the cessation. This phase is identified as process 1, in which the MBW leaves the LSD. Additionally, the end of process 1 is termed another initiation. The distance about termed L _sl = 1.22 mm marks this cessation. This is represented by process 2, where the MBW continues to leave the LSD. The three phases of the MBW leaving the LSD are illustrated in Figure 10, with a total duration of T _s .

Process 1 of the MBW leaving the LSD is illustrated in Figure 10A. The distance travelled by the MBW within the LSD is L _sl = 1.08 mm. This instance is denoted as T _o1. The part of the parallel light from the LSD is refracted by the MBW, resulting in a reduced width Δd = 0.92 mm on the photosensitive surface at the receiver. The area ΔS’ _o1 where the irradiance is reduced on the photosensitive surface, where Δ s o 1 ′ = W D Δ d D = 330 mm × 0.92 mm (19)

Process 2 of leaving is depicted in Figure 10B. The distance travelled by the MBW within the LSD is L _sl = 2 mm. This instance is denoted as T _o2. The reduced width on the photosensitive surface at the receiver is Δd = 0 mm. The area ΔS’ _o2 where the irradiance is reduced on the photosensitive surface, where Δ s o 2 ′ = W D Δ d D = 330 mm × 0 mm (20)

The process of MBW entering and exiting the LSD was analyzed using a discrete method. Sampling points selected of the method to fully represent the passage of MBW through the LSD. In future work, we will increase the number of LSDs to facilitate the collection of more MBW signals for analysis, thereby enhancing the precision of our proposed discretization analysis method.

The variation in the radiation flux Δφ′ is linearly related to the width Δd of the reduced radiation flux on the photosensitive surface. Assuming E _L = 1, the scatter data of Δφ′ for the five positional states and other instances of L _sl are fitted using a third-degree polynomial method, as shown in Figure 11.

As shown in Figure 11, during the MBW passing through the LSD, the variation of the radiation flux Δφ′ on the photosensitive surface at the receiver first increases and then decreases.

Based on the intersections listed in Table 2, the curve of the sensing variation in radiative flux is shown in Figure 12.

A quantitative calculation is conducted for the variation in radiative flux at the light screen detector receiver. During the MBW passing through the LSD, the peak value of the variation in radiative flux Δφ′ on the photosensitive surface at the receiver decreases as the distance of MBW propagation increases. However, the overall trend of the radiative flux variation remains constant.