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Optical performance evaluation of an infrared system of a hypersonic vehicle in an aero-thermal environment

Wenzhi Zhang, Lin Ju, Zhigang Fan, Wenwen Fan, and Shouqian Chen

Open Access Open Access

Abstract

At hypersonic velocities, the turbulent flow field generated by an aircraft, along with its temperature distribution, leads to significant aerodynamic optical effects that severely impede the performance of internal optical systems. This study proposes a method for analyzing the temporal characteristics of imaging degradation in a detector window infrared imaging system under different field angles of hypersonic velocity. Based on heat transfer theory, a method for solving the transient temperature field in the optical window of a high-speed aircraft is derived and established, considering unsteady thermal conduction-radiation coupling. Additionally, an optical window radiation tracing method is introduced, which directly determines the initial direction vector of light reaching the detector. This method reduces the workload of radiation transmission, significantly enhancing the efficiency of radiation calculations. The time characteristics of image degradation caused by aero-optical effects in high-speed aircraft are analyzed using metrics such as peak signal-to-noise ratio, wave aberration, and point diffusion function. The results demonstrate that as working time increases and the viewing angle widens, the impact of aero-optics on the aircraft imaging system becomes more severe. Moreover, compared to the aerodynamic light transmission effect, the aerodynamic thermal radiation effect has a more detrimental influence on imaging quality.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

When the aircraft carrying the infrared detection system operates at high speed, the flow field around the high-speed aircraft will form shock waves, which will induce the separation of the boundary layer and make the inviscid fluid interfere with the boundary layer. As a consequence, the density gradient of the fluid changes irregularly, and the refractivity of the flow field around the detector window presents a three-dimensional heterogeneous gradient distribution. At the same time, a strong heterogeneous convective heat transfer occurs between the detector window and the fluid, resulting in a rapid increase in the detector window temperature. The refractivity of the detector window presents a three-dimensional heterogeneous gradient distribution as well due to the thermal-optical effect of the optical materials. When the rays of target pass through the flow field with a three-dimensional non-uniform gradient distribution of refractivity and the detector window with a three-dimensional non-uniform gradient distribution of refractivity by aerodynamic heating, the direction of the rays will change and the wave front will be distorted, resulting in degraded image quality of the target [1,2], which is called aerodynamic optical transmission effect [3–5].

Meanwhile, the detector window continuously heated by aerodynamics produces intense radiation across the full spectrum. The radiation emitted by the detector window passes through the internal optical system of the high-speed aircraft and reaches the detector, producing background noise and interfering with target imaging. When the temperature of the optical window is too high, the detector target signal will be drowned by noise, this phenomenon is called aerothermal radiation effect [6].

As shown in Fig. 1, the aerodynamic optical transmission effect and the aerodynamic thermal radiation effect exist at the same time when the high-speed aircraft is working. The target image acquired by high-speed aircraft will have image migration, image jitter, image blur, energy attenuation and background noise interference. This will further impair the imaging quality of high-speed aircraft carrying infrared optical system, and seriously weaken the ability of infrared precision-guided weapons to detect, identify, track and strike targets. Therefore, thermal control of the dome, windows, and infrared optical system is of utmost importance for high-speed aircraft. Numerous researchers have devoted their efforts to mitigating the influence of aerodynamic thermal radiation effects on imaging by incorporating innovative aerospace nanomaterials into the design and implementation of infrared optical systems. In line with the proposed thermal control approach, Zhang successfully developed an onboard infrared optical system featuring a wide field of view and a single lens, ensuring consistent image quality across a wide temperature range [7]. Additionally, Kumar highlights the exceptional thermal properties of novel nanomaterials, emphasizing their significant advantages for aviation applications [8].

 figure: Fig. 1.

Fig. 1. Schematic diagram of high-speed aircraft's aerodynamic optical effects.

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In the last two decades, a growing number of researchers became interested in the effects of aero-optics on the performance of satellite-sensitive navigation, target tracking and airborne laser systems [9]. Zhao et al. put forward the model for analyzing the effects of aero-optics to calculate the phase changes caused by supersonic flow fields through the derived ray tracing recursive algorithm [10]. Chang et al. proposed the method for predicting the tracking point flow field data by applying the GRNN method to the interpolation of the flow field as well as its refractive index data. The aerodynamic light transmission effect of hemispherical dome under different working conditions is studied by using three dimensional ray tracing method [11]. Guo et al. established an aerodynamic optical effect analysis model in two-dimensional space. The direct simulation Monte Carlo method was proposed to calculate the flow field of two-dimensional hypersonic vehicle, and the imaging deviation and phase deviation were calculated by ray tracing [12]. Xie et al. proposed using the back propagation neural network to interpolate the refractivity and gradient of the flow field respectively. The refractive index distribution modified by this method is of great significance to the improvement of the accuracy of the results of aero-optics effects [13]. Xu et al. applied a reverse ray tracing model to the study of aero-optics effects, which laid the foundation for the development of active laser lighting [14]. Xiao et al. established a comprehensive influence analysis model for imaging degradation of high-speed aircraft optical system due to aerodynamic optical transmission effect under the influence of aerodynamic flow field and aerodynamic heating [15–17]. Yang et al. proposed a simulation method of starlight transmission under hypersonic conditions, based on which the effects of aero-optics on platform satellite navigation were analyzed [18]. Fan et al. established an integrated aero-optical effect simulation analysis method to further analyze the imaging degradation characteristics of optical systems in time-varying flow fields [19–20].

In recent years, there has been an increasing amount of research on predicting image quality in high-speed aircraft. However, most of these studies have primarily focused on the aerodynamic optical transmission effects in the flow field, neglecting the aerodynamic optical transmission and aerodynamic heat radiation effects of optical dome or windows. Additionally, there is a lack of research on the combined effects of airborne optical transmission and airborne thermal radiation on the imaging degradation of high-speed aircraft, as well as limited studies on imaging quality degradation under different field of view angles.

This paper aims to analyze the temporal characteristics of imaging degradation in a detector window infrared imaging system under various field angles at hypersonic velocities. By utilizing an unsteady thermal conduction-radiation coupling finite element model, the time characteristics of the temperature field in the detection window of high-speed aircraft are determined. Furthermore, the proposed optical window radiation transmission method is used to analyze the impact of optical window radiation noise on imaging degradation. The temperature of the detector window is converted into refractive index using the thermo-optical effect. The wave aberration caused by the flow field and the optical window at the entrance pupils within the 0° to 20° field of view is analyzed using the Gladstone-Dale formula and the fourth-order Runge-Kutta method.

2. Aero-thermal radiation of optical windows

2.1 Temperature distribution of an optical window

In an aero-thermal environment, the solution of optical window temperature field of high speed aircraft is a problem of unsteady thermal conductively-radiation coupling heat transfer. The law of conservation of energy is applied to the heat transfer problem of three-dimensional variable refractive index solid media under Cartesian coordinates, and the differential equation of the transient temperature field variable T(x, y, z, τ) of the three-dimensional variable refractive index solid medium is obtained. It can be expressed as

$$\rho {c_p}\frac{{\partial T}}{{\partial \tau }} = k\left( {\frac{{{\partial^2}T}}{{\partial {x^2}}} + \frac{{{\partial^2}T}}{{\partial {y^2}}} + \frac{{{\partial^2}T}}{{\partial {z^2}}}} \right) - {q^{\rm R}}$$
where ρ is the density of the detector window, cp is the specific heat capacity of the detector window, k is the thermal conductivity of the detector window, and qR is the radiant heat flow. In this study, the boundary conditions corresponding to Eq. (1) can be expressed as
$$\left\{ \begin{array}{l} T(x,y,z) = {T_0}\\ q = k\frac{{\partial T}}{{\partial x}} + k\frac{{\partial T}}{{\partial y}} + k\frac{{\partial T}}{{\partial z}} \end{array} \right.$$

Figure 2(a) shows the schematic diagram of optical window discretization and Fig. 2(b) shows unit control body direction. The energy control differential Eq. (1) can be expressed as

$${a_p}T_p^{m + 1} = {a_N}T_N^{m + 1} + {a_S}T_S^{m + 1} + {a_W}T_W^{m + 1} + {a_E}T_E^{m + 1} + {a_T}T_T^{m + 1} + {a_B}T_B^{m + 1} + b$$
where
$$\left\{ \begin{array}{l} {a_N} = k\Delta {x^2}\Delta {z^2}\Delta \tau \quad \quad {a_W} = k\Delta {y^2}\Delta {z^2}\Delta \tau \quad \quad {a_T} = k\Delta {x^2}\Delta {y^2}\Delta \tau \\ {a_S} = k\Delta {x^2}\Delta {z^2}\Delta \tau \quad \quad {a_E} = k\Delta {y^2}\Delta {z^2}\Delta \tau \quad \quad \textrm{ }{a_B} = k\Delta {x^2}\Delta {y^2}\Delta \tau \\ {a_P} = {a_N} + {a_S} + {a_W} + {a_E} + {a_T} + {a_B} + \rho {c_p}\Delta {x^2}\Delta {y^2}\Delta {z^2}\\ b = \rho {c_p}\Delta {x^2}\Delta {y^2}\Delta {z^2}T_P^m - {q^R}\Delta {x^2}\Delta {y^2}\Delta {z^2}\Delta \tau \end{array} \right.$$

 figure: Fig. 2.

Fig. 2. Finite element model of thermal conduction-radiation coupling heat transfer: (a) refractive index grid model of the window, (b) hexahedral orientation diagram, (c) unit control solid Angle ΔΩm and radiation intensity Im passing through its center.

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For the unit shown in Fig. 2(c), the radiation transfer equation takes radiation intensity I as its variable, and the radiation transfer equation along any direction can be expressed as

$$\frac{{\textrm{d}{I^m}({\mathbf r},{\mathbf \Omega ^m})}}{{\textrm{d}s}} ={-} ({\kappa _a} + {\sigma _s})I({\mathbf r},{{\Omega }^m}) + {\kappa _a}\frac{{\sigma {T^4}(\vec{r})}}{\pi } + \frac{{{\sigma _s}}}{{4\pi }}\int\limits_{\Omega ^{\prime} = 4\pi } {I({\mathbf r},{{\mathbf \Omega }^{{m_i}}})\varPhi ({{\mathbf \Omega }^{{m_i}}},{{\mathbf \Omega }^m})d{{\mathbf \Omega }^{{m_i}}}}$$
where ${I^m}({\mathbf r},{{\mathbf \Omega }^m})$ is the radiation intensity in the direction of ds microsegment, ${\kappa _a}$, ${\sigma _s}$ is the absorption coefficient and scattering coefficient, and $\varPhi ({{\mathbf \Omega }^{{m_i}}},{{\mathbf \Omega }^m})$ and is the scattering phase function. The equation can be expressed as
$$\frac{{\textrm{d}I}}{{\textrm{d}s}} ={-} \beta I + S$$
where $\beta = {\kappa _a} + {\sigma _s}$ is extinction coefficient, S is the source term of the equation, which can be expressed as
$$S = {\kappa _a}\frac{{\sigma {T^4}({\mathbf r})}}{\pi } + \frac{{{\sigma _s}}}{{4\pi }}\int\limits_{{\Omega ^{{m_i}}} = 4\pi } {I({\mathbf r},{{\mathbf \Omega }^{{m_i}}})\varPhi ({{\mathbf \Omega }^{{m_i}}},{{\mathbf \Omega }^m})d{{\mathbf \Omega }^{{m_i}}}}$$

The finite volume method is used to expand Eq. (6) in the Cartesian coordinate system, and integrate in discrete element Vp and discrete control solid angle $\Delta {\Omega ^m}$. The equation in the following form can be obtained.

$$\frac{{\partial {I^m}}}{{\partial x}}D_x^m + \frac{{\partial {I^m}}}{{\partial y}}D_y^m + \frac{{\partial {I^m}}}{{\partial z}}D_z^m ={-} \beta {I^m}\Delta {\Omega ^m} + {S^m}\Delta {\Omega ^m}$$
where ${\mathbf n}$ is the normal direction outside the control plane, and Dm can be expressed as
$${D^m} = \int\limits_{\Delta {\Omega ^m}} {({\mathbf n} \cdot {{\mathbf s}^m})\textrm{d}\Omega }$$

The radiation equation is discretized, the radiation intensity in any direction can be expressed as

$$I_p^m = \frac{{{A_W}D_{xW}^m + {A_S}D_{yS}^m + {A_B}D_{zB}^m + VS_p^m\Delta {\Omega ^m}}}{{{A_E}D_{xE}^m + {A_N}D_{yN}^m + {A_F}D_{zF}^m + \beta V\Delta {\Omega ^m}}}$$
where A is the area of the control surface, V is the volume of the control body, and the subscript p represents the P unit control body. In equation, the source term S can be expressed as
$$S = {\kappa _a}\frac{{\sigma {T^4}}}{\pi } + \frac{{{\sigma _s}}}{{4\pi }}\sum\limits_{k = 1}^{{M_\varphi }} {\sum\limits_{l = 1}^{{M_\theta }} {{I^m}(\theta _l^m,\varphi _k^m)2\sin \theta _l^m\sin (\frac{{\Delta \theta _l^m}}{2})\Delta \varphi _k^m} }$$
where Mθ and Mφ correspond to the number of units with discrete polar Angle θ(0 ≤ θ ≤ π) and horizontal angle φ(0 ≤ φ ≤ π) in 4π space respectively. After obtaining the radiation intensity distribution, the radiation heat source term in the governing Eq. (4) of unsteady thermal conductivity can be expressed as
$${q^R} = (1 - {\sigma _s})(4\pi \frac{{\sigma {T^4}}}{\pi } - \sum\limits_{k = 1}^{{M_\varphi }} {\sum\limits_{l = 1}^{{M_\theta }} {{I^m}(\theta _l^m,\varphi _k^m)2\sin \theta _l^m\sin (\frac{{\Delta \theta _l^m}}{2})\Delta \varphi _k^m} } )$$

In this study, given the temperature field T 0(x, y, z) at the initial moment of the optical window, the radiation transfer model is used to solve the radiation heat source term, which is then substituted into the energy equation to obtain a new temperature field. The process is repeated until the temperature field converges.

2.2 Model of aero-thermal radiation effect

Each element of the detector window is regarded as a Lambert radiator with the same radiance at all directions, which radiates infrared light to 4π space. In the window with refractive index distribution irregularly, the fourth-order Runge-Kutta method is used to trace the rays, and the rays outside the optical window propagate in a straight line. The radiation flux received by each pixel of the detector is simulated to calculate the irradiance distribution of the detector surface.

In this paper, the optical system and detector are imaging into the optical window space. By calculating the field angle of the object, the rays radiated by each element that can enter the pupil are screened out, greatly reducing the amount of calculation of ray tracing. As shown in Fig. 3, the field angle of the optical system can be expressed as

$$\tan 2\omega = \frac{{\sqrt {l_y^2 + l_z^2} }}{{{n_e}f^{\prime}}}$$
where ly and lz are the length and width of the detector, ne is the refractivity of the window glass, and f’ is the focal length of the ideal optical system. At this time, the sampling range on the entry pupil image plane can be expressed as
$$p = 1.2{n_e}d\tan 2\omega$$
where ned is the distance between the plane of the detector window and the image plane of the entry pupil. For uniform sampling in this region, the coordinates of m × n uniform sampling points corresponding to the micro-element P(x1, y1, z1) can be expressed as
$$({x_0} + Nd,\textrm{ }(\frac{{m - 1}}{2} - i)\frac{p}{{m - 1}} + {y_1},\textrm{ }(\frac{{n - 1}}{2} - j)\frac{p}{{n - 1}} + {z_1})\textrm{ }\left\{ \begin{array}{l} i = 0,1,2,\ldots ,m - 1\\ j = 0,1,2,\ldots ,n - 1 \end{array} \right.$$
where x0 is the coordinate of the surface behind the window. The direction vector of the ray radiated by the microelement P(x1, y1, z1) that needs to be tracked by the ray can be expressed as
$${\mathbf T} = ({x_0} + Nd - {x_1},\textrm{ }(\frac{{m - 1}}{2} - i)\frac{p}{{m - 1}},\textrm{ (}\frac{{n - 1}}{2} - j\textrm{)}\frac{p}{{n - 1}})\textrm{ }\left\{ \begin{array}{l} i = 0,1,2,\ldots ,m - 1\\ j = 0,1,2,\ldots ,n - 1 \end{array} \right.$$

 figure: Fig. 3.

Fig. 3. Schematic diagram of unit element radiation transmission of optical window.

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The radiation flux carried by each ray within the sampling range can be expressed as

$$\varPhi = \frac{{{\varPhi _{{x_1},{y_1},{z_1}}}{p^2}}}{{4\mathrm{\pi }mn{{({{x_0} - {x_1} + {n_e}d} )}^2}}}$$

The window radiation model of detector introduced above is used to trace the ray carrying radiation flux of detector element radiation to the position on the detector plane. The radiation flux received by each pixel is integrated in the detector plane to obtain the radiation noise distribution generated by the detector window on the detector. Thus, the response voltage of the detector can be obtained according to formula (18).

$${V_i}_j = G \cdot {R_i}_j \cdot {\varPhi _i}_j + {V_{{N_{ij}}}}$$
where Vij is the voltage generated by the (i, j) pixel due to the radiation noise of the detector window, G is the gain of the preamplifier, Rij is the response coefficient of the pixel (i, j), and Φij is the radiation flux on the (i, j) pixel. VNij is the root-mean-square noise of (i, j) pixel.

3. Calculation model of the effect of aerodynamic optical transmission

3.1 Ray tracing algorithm

At high speeds, the detector window interacts continuously with the airflow around it. The external flow field creates a large density gradient around the detector window. The temperature distribution of detector window which is heating continuously by the flow field is uneven. Before arriving at the infrared detector, the beam from the target passes through the flow field and detector window with heterogeneous refractive index distribution successively, and most focuses on the detector form a distorted target image. In aero-thermal environment, the fluid density near the detector window and the refractive index of the detector window are non-rotationally symmetric with respect to the optical axis, so that the path and optical path of the rays with the same angle (field angle) from the optical axis are different. Therefore, in aero-thermal environment, the evaluation of imaging quality of optical system under different field angles cannot simply be defined by the angle between rays and optical axis.

In this paper, we propose the definition of ray incidence angle as shown in the Fig. 4. The center of the detector window is the origin of coordinates, and the x-axis is the optical axis. The incidence direction of rays is defined by zenith angle θ and azimuth φ. Zenith angle θ is defined as the angle between the ray direction and the positive direction of the x-axis. The zenith angle increases in the clockwise direction and the positive direction of the x-axis is 0 degrees. The ray azimuth φ is defined as the angle between the ray direction and the positive direction of the z-axis, which increases counterclockwise along the z-axis, and the positive direction of the z-axis is 0 degrees.

 figure: Fig. 4.

Fig. 4. Schematic diagram of defining the Angle of ray incidence.

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The refractive index of a flow field is affected by its density in aero-thermal conditions. According to the Gladstone-Dale formula, the refractive index distribution of the outflow field can be calculated from its density. The refractivity nF of the outflow field is expressed as:

$${n_F} = 1 + {K_{GD}}{\rho _F}$$
$${K_{\textrm{GD}}}(\lambda )= 2.23 \times {10^{ - 4}}(1 + \frac{{7.52 \times {{10}^{ - 3}}}}{{{\lambda ^2}}})$$
where ρF is the external fluid density, λ is the wavelength.

The change of temperature and stress has a serious effect on the refractive index of infrared materials. The phenomenon that the refractive index of an optical material varies with its temperature is called the thermo-optical effect. The result of research shows that the influence of thermal-optical effect on refractive index of detector window is dominant. The refractive index at (x, y, z) in the detector window can be expressed as

$${n_{x,y,z}}[{\lambda ,T({x,y,z} )} ]= {n_0}({\lambda ,{T_0}} )+ \frac{{\partial n({\lambda ,T} )}}{{\partial T}}\Delta T({x,y,z} )$$
where, nx,y,z[λ, T(x, y, z)] is the refractive index of the optical window (x, y, z) at the reference temperature, $\partial n({\lambda ,T} )/\partial T$ is the thermo-optical coefficient, and $\Delta T({x,y,z} )$ is the temperature change relative to the reference temperature $T({x,y,z} )$.

The path of rays passing through the flow field and detector window with irregular refractive index gradient can be determined by the ray equation. The ray equation derived from Fermat's principle can be expressed as

$$\frac{\textrm{d}}{{\textrm{d}s}}\left[ {n({\mathbf r})\frac{{\textrm{d}{\mathbf r}}}{{\textrm{d}s}}} \right] = \nabla n({\mathbf r})$$
where, r is the position vector at the trace point of the ray, $\nabla n({\mathbf r})$ is the refractive index gradient at position r, and ds step size of the ray tracing. The ray equation is transformed into a system of first-order differential equations, which can be expressed as
$$\left\{ \begin{array}{l} \frac{{\textrm{d}{\mathbf r}}}{{\textrm{d}s}} = {\mathbf T}\\ \frac{{\textrm{d}{\mathbf T}}}{{\textrm{d}s}} = n\nabla n \end{array} \right.$$
where, T is the direction vector of the ray at position r. The fourth-order Runge-Kutta method with higher accuracy is used to numerically solve the above first-order differential equations, and the above equation can be expressed as
$$\left\{ \begin{array}{l} {{\mathbf r}_{i + 1}} = {{\mathbf r}_i} + \frac{h}{6}({{\mathbf K}_1} + 2{{\mathbf K}_2} + 2{{\mathbf K}_3} + {{\mathbf K}_4})\\ {{\mathbf T}_{i + 1}} = {{\mathbf T}_i} + \frac{h}{6}({{\mathbf L}_1} + 2{{\mathbf L}_2} + 2{{\mathbf L}_3} + {{\mathbf L}_4}) \end{array} \right.$$
where
$$\begin{array}{l} \left\{ \begin{array}{l} {{\mathbf K}_1} = {{\mathbf T}_i}\\ {{\mathbf L}_1} = n\nabla n\textrm{ (at }{{\mathbf r}_i}\textrm{)} \end{array} \right.\textrm{ }\\ \left\{ \begin{array}{l} {{\mathbf K}_2} = {{\mathbf T}_i} + \frac{h}{2}{{\mathbf L}_1}\\ {{\mathbf L}_2} = n\nabla n\textrm{ (at }{{\mathbf r}_i}\textrm{ + h}{{\mathbf K}_\textrm{1}}\textrm{)} \end{array} \right.\\ \left\{ \begin{array}{l} {{\mathbf K}_3} = {{\mathbf T}_i} + \frac{h}{2}{{\mathbf L}_2}\\ {{\mathbf L}_3} = n\nabla n\textrm{ (at }{{\mathbf r}_i}\textrm{ + h}{{\mathbf K}_\textrm{2}}\textrm{)} \end{array} \right.\\ \left\{ \begin{array}{l} {{\mathbf K}_4} = {{\mathbf T}_i} + \frac{h}{2}{{\mathbf L}_3}\\ {{\mathbf L}_4} = n\nabla n\textrm{ (at }{{\mathbf r}_i}\textrm{ + h}{{\mathbf K}_\textrm{3}}\textrm{)} \end{array} \right. \end{array}$$

3.2 Image quality evaluation of an optical window

For a beam of light from infinity passes through the inhomogeneous medium and reaches the pupil plane, the optical path length (OPL) can be expressed as

$$\textrm{OPL} = \sum\limits_i {{n_i}{l_i}}$$
where, ni is the index of refraction at the point where the ray falls in the tracing process, and li is the distance between ray landing points in step i of ray tracing.

The wave aberration of any ray passing through the window can be calculated by the following equation

$$W(x,y) = \frac{{2\pi }}{\lambda }(OP{L_i} - OP{L_0})$$
where OPL0 is the average value of OPL.

The generalized pupil function can be expressed by the following formula

$$A(x,y) = a(x,y)\exp [jW(x,y)]\textrm{ }$$
where a (x, y) is the function of pupil. According to Huygens’ principle, the amplitude distribution on the image plane can be represented by the Fourier transform of pupil function, which can be expressed as
$$U(x^{\prime},y^{\prime}) = \int\!\!\!\int {A(x,y)\exp [ - j\frac{{2\pi }}{{\lambda f^{\prime}}}(xx^{\prime} + yy^{\prime})]} \textrm{d}x\textrm{d}y$$
where f ‘ is the focal length of the optical system. The intensity of the light is proportional to the square of the amplitude, so the point spread function of the optical system can be expressed as
$$\textrm{PSF}(x^{\prime},y^{\prime}) = |U { {(x^{\prime},y^{\prime})} |^2}$$

The image distortion degradation caused by aerodynamic optical transmission effect can be obtained by using the convolution point diffusion fuzzy function of input image. In this paper, the peak signal-to-noise ratio, which is widely used in imaging evaluation, is used to evaluate image distortion. It can be expressed as [,21,22]

$$\textrm{PSNR} = 20\lg 10\left[ {\frac{{G - 1}}{{\textrm{MSE}}}} \right]$$
where, G is the max gray level of the image and MSE is the mean square error of the image.

4. Results and discussion

4.1 Result for effects of aero-optical

The initial temperature of the detector window was 298 K, and the high speed aircraft operated for 15 seconds (Mach number 5, angle of attack 0°, altitude 10 km). The temperature distribution of detector window is calculated by thermal conduction-radiation coupling. The Fig. 5 shows the optical window temperature distribution for 1-15s. Under the influence of aerodynamic heating, the temperature of the detector window distribution is irregular. The peak temperature is on the outer surface of the detector window, near the optical axis. After the flight time of the aircraft t > 5s, the temperature gradient distribution of the detector window gradually becomes more irregular, and the surface temperature inside the detector window increases gradually. At the fifteenth second of flight time, the internal surface temperature of the window is 840 K.

 figure: Fig. 5.

Fig. 5. Temperature distribution at different time in optical window.

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In this paper, the effect of aerodynamic optical transmission and aerodynamic thermal radiation on the imaging quality of high-speed aircraft is studied. The detailed parameters are shown in Table 1.

Tables Icon

Table 1. The parameters of infrared detection system

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The thermal noise generated by the optical window is radiated by ray tracing method. The position and energy of each discrete ray beam reaching the detector receiving surface are calculated. In addition, the total irradiance distribution of aerodynamic heat radiation of the optical window on the detector receiving surface is obtained by statistics.

In this paper, according to the thermal radiation calculation model in Section 2, the illuminance distribution of optical window thermal noise on the detector under aerothermal environment is obtained as shown in the Fig. 6. Because the temperature distribution of the detector window is heterogeneous, and the maximum temperature of the detector window deviates from the optical axis, the distribution of irradiance on the detector is heterogeneous, and the maximum irradiance deviates from the center of the detector. Besides, the radiation distribution detected by the detector is basically stable over time, but the irradiance value becomes larger and larger. At the fifteenth second, the maximum irradiance on the detector is 32W/m2.

 figure: Fig. 6.

Fig. 6. Detector irradiance distribution in 15 seconds.

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When the radiation noise reaches the receiving surface of the detector, it will cause great interference to the target image. In order to give an intuitive comparison, the thermal noise distribution generated by the optical window is converted into the gray distribution corresponding to the target image. Finally, the distortion target of the target image under the effect of the aerodynamic thermal radiation of the detector window is obtained. In aero-thermal environment, imaging degradation caused by radiation noise caused of detector window is simulated as shown in the Fig. 7. In comparison to the original image Fig. 7(a), as time goes by, the background noise of the image is gradually enhanced, which causes the outline and features of the image to fade away. The characters of the image do not become fuzzy and dithering, the overall gray of the image increases, and the image is in the state of overexposure, which results in the degradation of the image quality. After the high-speed aircraft operated for 10s, the thermal radiation noise continued to increase, leading to blindness of the detector. Meanwhile the details of the image information of image was drown by the radiation noise. If the image recovery algorithm is not adopted, the infrared optical system of high-speed aircraft will lose its guidance ability.

 figure: Fig. 7.

Fig. 7. Degradation image under aerodynamic thermal radiation effect with the working time: (a) original image; (b)∼(f) degradation image.

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The optical transmission effect of variable refractive index medium also has an important influence on the infrared detection accuracy of high-speed aircraft in complex thermal environment. In this study, the fourth-order Runge-Kutta method is applied to study the flow field with heterogeneous gradient of refractive index and the optical transmission effect of the detector window. When the ray incidence angles are 0°/0° (azimuth/zenith), 0°/15° and 0°/21°, the internal wave aberrations of the high-speed aircraft detector window system are calculated respectively. Figure 8 shows the infrared detection system wave aberration at exit pupil when the incidence angle is 0°/0° (azimuth/zenith angle) during the operation of the high-speed aircraft for 15s. The infrared detection system wave aberration at exit pupil is similar to the distribution of the temperature field of the optical window in a complex thermal environment, showing a trend of large center and small surrounding. This is because the temperature field distribution of detector window in complex thermal environment shows that the temperature around the optical axis is higher and the edge is lower. Because of thermo-optical effect, the refractive index around the optical axis of the detector window is larger. With the increase of zenith incidence angle, the peak-valley difference of wave aberration (PV) increases, but the increase is not significant.

 figure: Fig. 8.

Fig. 8. The incidence angle is 0°/0° detector window infrared detection system wave aberration at exit pupil.

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The peak-valley (PV) difference of wave aberration increases with zenith incidence angle rising, but the increase is not significant as shown in the Fig. 9. Therefore, with the increase of the incidence angle of zenith, the image quality of optical window will decrease, but the decrease is not significant. In addition, in the same field angle within 15s of high-speed aircraft operation, PV value of infrared detection system wave aberration at exit pupil gradually increases with working time, but the distribution of wave aberration is almost unchanged.

 figure: Fig. 9.

Fig. 9. PV value of infrared detection system wave aberration at exit pupil at different incidence angles.

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Figure 10 shows the point spread function (PSF) of different field angles of infrared detection system under aero-optics effect. At each angle of view, the point spread function is diffused and secondary peaks appear at each incidence angle. And the diffusion of point diffusion function will lead to the deterioration of image quality of detector window optical system. With the working time of infrared detection system, the dispersion of point diffusion will be gradually serious. It can be predicted that the problem of image blurring and offsetting caused by aerodynamic optical transmission effect will become more and more serious with the advance of working time.

 figure: Fig. 10.

Fig. 10. The PSF of detector window affected by aero-optical transmission effect.

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Figure 11 shows the target distortion image of infrared optical system of high-speed aircraft under the effect of aero-optics transmission. The simulation results agree with the above analysis results. As time goes on, the impact of aerodynamic optical transmission on imaging is increasing, the distorted image becomes fuzzy, and the image quality gradually declines.

 figure: Fig. 11.

Fig. 11. Degradation image of optical system of high speed aircraft with incidence angle of 0°/0° affected by aerodynamic optical transmission effect.

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Figure 12 shows the PSNR change of degraded image with working time. The PSNR of the image decreases due to the influence of aero-optics transmission. Compared with the simulation degradation imaging results under the aerodynamic heat radiation effect above, it can be seen that the aerodynamic heat radiation effect has a more severe effect on the infrared detection system performance.

 figure: Fig. 12.

Fig. 12. PSNR of degraded images caused by aerodynamic optical transmission effect.

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4.2 Comprehensive aerodynamic influence analysis of optical window

Figure 13 is a simulation image of image degradation under the combined action of aerodynamic optical transmission effect and thermal radiation of a high-speed aircraft carrying an infrared detection system within 15s of operation. The results of simulation show that when the working time of high-speed aircraft with infrared guidance system is t > 5s, the aerodynamic thermal environment adversely affects the imaging quality of high-speed aircraft infrared detection system. After t > 10s, the infrared detection system does not have the ability to identify the target accurately without any algorithm processing.

 figure: Fig. 13.

Fig. 13. Comprehensive aero-optics effect affects simulation imaging.

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The change curve of PSNR of target image over time is shown in Fig. 14. The peak signal-to-noise ratio of simulation imaging decreases rapidly along with time. In aero-thermal environment, aero-thermal radiation effect dominates the degraded of detection performance of infrared detection system carried by high-speed aircraft. As the working day progresses, the infrared thermal radiation of the detector window seriously restricts the detection and recognition of the target by the imaging system of high-speed aircraft.

 figure: Fig. 14.

Fig. 14. PSNR of images distorted by comprehensive aero-optics effects.

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5. Summary

In this study, the imaging degradation characteristics of a detector window infrared detection system operating at 5 Ma and 10 km are analyzed using the proposed time-characteristic analysis method. The unsteady heat conduction-radiation coupled heat transfer model established in this paper is employed to solve the temperature field distribution in the optical window. The optical window radiation tracking method proposed in this study reduces the number of radiated rays to only 2.8% of the total. Within 10 seconds of operation under these specific conditions, the thermal radiation noise rapidly increases to 26 W/m2, resulting in the submergence of image details and a rapid decrease in the image's peak signal-to-noise ratio from 25 dB to 12 dB. Simultaneously, the image experiences blurring due to the aerodynamic optical transmission effect, but the image's peak signal-to-noise ratio remains above 16 dB throughout the working time. The results indicate that, in comparison to the aerodynamic light transmission effect, the aerodynamic thermal radiation effect plays a predominant role in reducing image quality. Moreover, the temperature distribution of the detector window determines both the aerodynamic optical transmission effect and the aerodynamic thermal radiation effect, highlighting the urgency to reduce the optical window temperature in high-speed aircraft.

Disclosures

The authors declare no conflicts of interest.

Data availability

The underlying data for the results presented in this paper are not yet publicly available, but can be obtained by email from the corresponding author.

References

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Data availability

The underlying data for the results presented in this paper are not yet publicly available, but can be obtained by email from the corresponding author.

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Figures (14)
Fig. 1.
Fig. 1. Schematic diagram of high-speed aircraft's aerodynamic optical effects.

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Fig. 2.
Fig. 2. Finite element model of thermal conduction-radiation coupling heat transfer: (a) refractive index grid model of the window, (b) hexahedral orientation diagram, (c) unit control solid Angle ΔΩm and radiation intensity Im passing through its center.

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Fig. 3.
Fig. 3. Schematic diagram of unit element radiation transmission of optical window.

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Fig. 4.
Fig. 4. Schematic diagram of defining the Angle of ray incidence.

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Fig. 5.
Fig. 5. Temperature distribution at different time in optical window.

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Fig. 6.
Fig. 6. Detector irradiance distribution in 15 seconds.

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Fig. 7.
Fig. 7. Degradation image under aerodynamic thermal radiation effect with the working time: (a) original image; (b)∼(f) degradation image.

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Fig. 8.
Fig. 8. The incidence angle is 0°/0° detector window infrared detection system wave aberration at exit pupil.

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Fig. 9.
Fig. 9. PV value of infrared detection system wave aberration at exit pupil at different incidence angles.

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Fig. 10.
Fig. 10. The PSF of detector window affected by aero-optical transmission effect.

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Fig. 11.
Fig. 11. Degradation image of optical system of high speed aircraft with incidence angle of 0°/0° affected by aerodynamic optical transmission effect.

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Fig. 12.
Fig. 12. PSNR of degraded images caused by aerodynamic optical transmission effect.

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Fig. 13.
Fig. 13. Comprehensive aero-optics effect affects simulation imaging.

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Fig. 14.
Fig. 14. PSNR of images distorted by comprehensive aero-optics effects.

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Tables (1)

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Table 1. The parameters of infrared detection system

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Equations (31)

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ρ c p ∂ T ∂ τ = k ( ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 + ∂ 2 T ∂ z 2 ) − q R
{ T ( x , y , z ) = T 0 q = k ∂ T ∂ x + k ∂ T ∂ y + k ∂ T ∂ z
a p T p m + 1 = a N T N m + 1 + a S T S m + 1 + a W T W m + 1 + a E T E m + 1 + a T T T m + 1 + a B T B m + 1 + b
{ a N = k Δ x 2 Δ z 2 Δ τ a W = k Δ y 2 Δ z 2 Δ τ a T = k Δ x 2 Δ y 2 Δ τ a S = k Δ x 2 Δ z 2 Δ τ a E = k Δ y 2 Δ z 2 Δ τ   a B = k Δ x 2 Δ y 2 Δ τ a P = a N + a S + a W + a E + a T + a B + ρ c p Δ x 2 Δ y 2 Δ z 2 b = ρ c p Δ x 2 Δ y 2 Δ z 2 T P m − q R Δ x 2 Δ y 2 Δ z 2 Δ τ
d I m ( r , Ω m ) d s = − ( κ a + σ s ) I ( r , Ω m ) + κ a σ T 4 ( r → ) π + σ s 4 π ∫ Ω ′ = 4 π I ( r , Ω m i ) Φ ( Ω m i , Ω m ) d Ω m i
d I d s = − β I + S
S = κ a σ T 4 ( r ) π + σ s 4 π ∫ Ω m i = 4 π I ( r , Ω m i ) Φ ( Ω m i , Ω m ) d Ω m i
∂ I m ∂ x D x m + ∂ I m ∂ y D y m + ∂ I m ∂ z D z m = − β I m Δ Ω m + S m Δ Ω m
D m = ∫ Δ Ω m ( n ⋅ s m ) d Ω
I p m = A W D x W m + A S D y S m + A B D z B m + V S p m Δ Ω m A E D x E m + A N D y N m + A F D z F m + β V Δ Ω m
S = κ a σ T 4 π + σ s 4 π ∑ k = 1 M φ ∑ l = 1 M θ I m ( θ l m , φ k m ) 2 sin ⁡ θ l m sin ⁡ ( Δ θ l m 2 ) Δ φ k m
q R = ( 1 − σ s ) ( 4 π σ T 4 π − ∑ k = 1 M φ ∑ l = 1 M θ I m ( θ l m , φ k m ) 2 sin ⁡ θ l m sin ⁡ ( Δ θ l m 2 ) Δ φ k m )
tan ⁡ 2 ω = l y 2 + l z 2 n e f ′
p = 1.2 n e d tan ⁡ 2 ω
( x 0 + N d ,   ( m − 1 2 − i ) p m − 1 + y 1 ,   ( n − 1 2 − j ) p n − 1 + z 1 )   { i = 0 , 1 , 2 , … , m − 1 j = 0 , 1 , 2 , … , n − 1
T = ( x 0 + N d − x 1 ,   ( m − 1 2 − i ) p m − 1 ,  ( n − 1 2 − j ) p n − 1 )   { i = 0 , 1 , 2 , … , m − 1 j = 0 , 1 , 2 , … , n − 1
Φ = Φ x 1 , y 1 , z 1 p 2 4 π m n ( x 0 − x 1 + n e d ) 2
V i j = G ⋅ R i j ⋅ Φ i j + V N i j
n F = 1 + K G D ρ F
K GD ( λ ) = 2.23 × 10 − 4 ( 1 + 7.52 × 10 − 3 λ 2 )
n x , y , z [ λ , T ( x , y , z ) ] = n 0 ( λ , T 0 ) + ∂ n ( λ , T ) ∂ T Δ T ( x , y , z )
d d s [ n ( r ) d r d s ] = ∇ n ( r )
{ d r d s = T d T d s = n ∇ n
{ r i + 1 = r i + h 6 ( K 1 + 2 K 2 + 2 K 3 + K 4 ) T i + 1 = T i + h 6 ( L 1 + 2 L 2 + 2 L 3 + L 4 )
{ K 1 = T i L 1 = n ∇ n  (at  r i )   { K 2 = T i + h 2 L 1 L 2 = n ∇ n  (at  r i  + h K 1 ) { K 3 = T i + h 2 L 2 L 3 = n ∇ n  (at  r i  + h K 2 ) { K 4 = T i + h 2 L 3 L 4 = n ∇ n  (at  r i  + h K 3 )
OPL = ∑ i n i l i
W ( x , y ) = 2 π λ ( O P L i − O P L 0 )
A ( x , y ) = a ( x , y ) exp ⁡ [ j W ( x , y ) ]  
U ( x ′ , y ′ ) = ∫ ∫ A ( x , y ) exp ⁡ [ − j 2 π λ f ′ ( x x ′ + y y ′ ) ] d x d y
PSF ( x ′ , y ′ ) = | U ( x ′ , y ′ ) | 2
PSNR = 20 lg ⁡ 10 [ G − 1 MSE ]
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