Evaluating imaging quality of optical dome affected by aero-optical transmission effect and aero-thermal radiation effect

Wang Hui; Shouqian Chen; Wang Zhang; Fanyang Dang; Lin Ju; Xianmei Xu; Zhigang Fan

doi:10.1364/OE.373020

1. Introduction

When a missile with an infrared seeker flies at high speed, the flow field in front of the optical dome forms a strong shock wave, causing its temperature and pressure to change constantly. The flow field and optical dome is primarily convective heat transfer. The non-uniform temperature and pressure distribution of the flow field results in the non-uniform distribution of temperature, deformation and refractive index of a aerodynamically heated optical dome. The beam was broken up after transmitting through flow field and optical dome, resulting in the image blurring, jittering, offsetting and losing of resolution. These are referred to as aero-optical transmission effect. Intense infrared radiation emitted by the high temperature shock wave and aerodynamically heated optical dome produces background noise signal on the target image and reduces the signal-to-noise ratio of optical seeker. In severe cases, “thermal barrier” phenomenon is formed and the target signal is overwhelmed by noise signal. These are referred to as aero-thermal radiation effect. The aero-optical transmission effect and the aero-thermal radiation effect are collectively known as aero-optical effect [1]. In recent years, there has been increasing interest in aero-optical issues with the development of high-speed missiles and laser weapon systems. Wang [2] developed a numerical model for predicting the aero-optical performance of the supersonic flow field and computed phase variances brought from the density fluctuations. Xu [3] adopted ray tracing based on Runge-Kutta method to examine the influence of altitude on aero-optic imaging deviation, indicating that the imaging deviation decreases as altitude increasing. Guo [4] adopted the direct simulation Monte Carlo based on the Boltzmann Eq. and the ray-racing method to simulate beam transmission through the nonuniform flow field over the optical window. Xu et al. [5] proposed a backward ray-tracing method for aero-optics simulations of flow field, with the advantage of simplifing the ray-tracing computation markedly in aero-optics simulations. Fan et al.[6–8]employed an irregular grid model to describe the refractive index distribution of an aerodynamically heated optical dome, and investigated the joint influences exerted by the nonuniform aerodynamic flow field surrounding the different shapes of optical dome and the aerodynamic heating of the dome on imaging quality degradation. The optical dome profile includes: spherical dome, ellipsoidal dome, and Paraboloidal dome. Niu et al.[9] examined the infrared thermal radiation noise of a flying supersonic dome at near-ground altitudes, and the results showed that the radiance received by the dome surface was proportional to the dome radius. Wang et al. [10] investigated the influence of the radiation of the aerodynamically heated optical dome on the detector and indicated the effect exerted by the radiation of the aerodynamically heated dome on the detector should not be neglected. Zhang et al. [11] examined the relationship between the variation of the dome’s optical characteristics and the infrared image degradation, and the results showed that the thermo-optic effect, elastic-optic effect, thermal deformation, and variation of transmittance have little effect on the optical system, the thermal radiation degraded imaging quality severely. However, the imaging evaluation parameters were not presented in his paper. Although numerous papers on aero-optics have been published, most of them focus on the aero-optical effect of the flow field. Only a few have focused on comprehensive influence of aero-optical transmission effect and aero-thermal radiation effect on imaging quality degradation of optical dome.

The primary coverage of this study could summarize the following points: (1) examine the refractive index distribution of the optical dome based on the thermo-optical and develop a ray-tracing program to trace the ray from the target and the thermal radiation of the optical dome; (2) Three evaluation parameters (MTF, irradiance, PSNR of distorted images) were established to evaluate the aero-optical transmission effect and the aero-thermal radiation effect; (3) investigate the influence of flight speed and thickness of optical dome on aero-optical transmission effect and aero-thermal radiation effect.

2. Optical dome analysis

2.1 Refractive index of optical dome and ray transmission algorithms

Under the aero-thermal environment, the temperature of the optical dome was non-uniform gradient distribution. The thermal-structural coupling analysis of the optical dome was carried out by using the finite element software ANSYS, and temperature distribution $t(x, y, z)$ of the optical dome was obtained. Due to the non-uniform temperature field distribution of the optical dome, thermal stress and thermal strain are generated inside the dome [12,13]. According to the thermo-optical effect and elasto-optical effect, the refractive index of the optical dome was also non-uniform gradient distribution. Since the variation value of refractive index caused by thermo-optical effect was much more larger than that caused by elasto-optical effect, only the change value of refractive index caused by the thermo-optical effect was considered. The phenomenon that the refractive index of optical material changes with temperature is referred to as the thermo-optical effect, which can be expressed as [6]

(1)$$n[{\lambda, t(x, y, z)} ]= n(\lambda ,{t_0}) + \frac{{\textrm{d}(\lambda, t)}}{{\textrm{d}t}}\Delta t(x, y, z)$$

where $n(\lambda ,{t_0})$ is the refractive index of any grid node at reference temperature ${t_0}$, ${{dn(\lambda, t)} \mathord{\left/ {\vphantom {{dn(\lambda, t)} {dt}}} \right.} {dt}}$ is thermo-optical coefficient, $\Delta t(x, y, z)$ is temperature change value of $t(x, y, z)$ relative to ${t_0}$.

Since the refractive index of aerodynamically heated optical dome was non-uniform gradient distribution, rays transmitting through the dome were curved lines determined by the Fermat principle. It can be expressed as [3,14,15]

(2)$$\frac{\textrm{d}}{{\textrm{ds}}}\left[ {n(r )\frac{{\textrm{d}r}}{{\textrm{ds}}}} \right] = \nabla n(r )$$

where r is the position vector of ray propagation, $\textrm{d}s$ is a step size along propagation path, and $n(r )$ is the refractive index distribution.

Equation (2) described the relationship between the refractive index of medium and the change of optical propagation path. It's a second order differential equation. By introducing a new parameter t, the Eq. (2) was simplified to a first order differential Eq. Systems.

(3)$$t = \int {\frac{{\textrm{d}s}}{{n(r )}}} $$

Then Eq. (2) can be expressed as

(4)$$\frac{{{\textrm{d}^2}r}}{{d{t^2}}} = n(r )\nabla n(r )$$

Equation (4) can be expressed as first order differential Eq. systems.

(5)$$\left\{ {\begin{array}{l} {\frac{{\textrm{d}r}}{{\textrm{d}t}} = T}\\ {\frac{{\textrm{d}T}}{{\textrm{d}t}} = n(r)\nabla n(r)} \end{array}} \right.$$

The fourth order Runge-Kutta method was used to solve Eqs. (5), then the recursive equations were obtained as follows [16]:

(6)$$\left\{ {\begin{array}{l} {{r_{i + 1}} = {r_i} + \frac{h}{6}({K_1} + 2{K_2} + 2{K_3} + {K_4})}\\ {{T_{i + 1}} = {T_i} + \frac{h}{6}({L_1} + 2{L_2} + 2{L_3} + {L_4})} \end{array}} \right.$$

where ${K_j}$ and ${L_j}$ (j=1,2,3,4) can be expressed as Eqs. (7) and (8). h in Eq. (8) was the tracing step length.

(7)$$\left\{ {\begin{array}{l} {{K_1} = {T_i}}\\ {{K_2} = {T_i} + {{h{L_1}} \mathord{\left/ {\vphantom {{h{L_1}} 2}} \right.} 2}}\\ {{K_3} = {T_i} + {{h{L_2}} \mathord{\left/ {\vphantom {{h{L_2}} 2}} \right.} 2}}\\ {{K_4} = {T_i} + h{L_3}} \end{array}} \right.$$

(8)$$\left\{ {\begin{array}{l} {{L_1} = n(r)\nabla n(r) \quad (at {r_i})}\\ {{L_2} = n(r)\nabla n(r) \quad (at {r_i} + {{h{K_1}} \mathord{\left/ {\vphantom {{h{K_1}} 2}} \right.} 2},\;\textrm{in the direction of} \;{K_1})}\\ {{L_3} = n(r)\nabla n(r) \quad (at {r_i} + {{h{K_2}} \mathord{\left/ {\vphantom {{h{K_2}} 2}} \right.} 2},\;\textrm{in the direction of} \;{K_2})}\\ {{L_4} = n(r)\nabla n(r) \quad (at {r_i} + h{K_3},\;\textrm{in the direction of} \;{K_3})} \end{array}} \right.$$

By adopting the fourth order Runge-Kutta method to solve Fermat formula, the coordinates of ray propagate points r_i in the non-uniform refractive index medium can be obtained.

2.2 Aerodynamic optical transmission of the optical dome

By tracing the target ray transmitting through the optical dome and accumulating the optical path length within each tracing step length, optical path length (OPL) of an arbitrary ray at the exit pupil can be obtained. It can be expressed as [6,7]

(9)$$\textrm{OPL}_{k} = \sum\limits_i {{n_i}(\rm {r} )} {l_i}$$

where ${n_i}(r)$ is the refractive index of the $i$th step of ray tracing at r, ${l_i}$ is the optical transmission distance of the $i$th step of ray tracing.

Then the wave aberration at the entire exit pupil can be expressed as [8]

(10)$$\textrm{W(x, y)} = \sum\limits_k {\frac{{2\pi }}{\lambda }} (\textrm{OPL}_{k} - \overline {\textrm{OPL}})$$

where $\overline {OPL}$ denotes the average OPL.

The pupil function of the optical dome can be expressed as [6]

(11)$$A(x^{\prime}, y^{\prime}) = \left\{ {\begin{array}{ll} {a(x, y) \exp [{jW(x, y)}] {x^2} + {y^2} \le {{({D \mathord{\left/ {\vphantom {D 2}} \right.} 2})}^2}}\\ { 0 \qquad \qquad \qquad {x^2} + {y^2} \ge {{({D \mathord{\left/ {\vphantom {D 2}} \right.} 2})}^2}} \end{array}} \right.$$

Where $a(x, y)$ is the amplitude distribution of the pupil function and D is the pupil diameter.

The target imaging process of a supersonic aircraft usually satisfies the far-field approximation condition. According to Huygens principle, the amplitude distribution of the light field at the image plane can be obtained by Fourier transform as follows [17]

(12)$$U(x^{\prime}, y^{\prime}) = \int\!\!\!\int {A(x, y)\exp \left[ {\textrm{ - }j\frac{{2\pi }}{{\lambda f}}(xx^{\prime} + yy^{\prime})} \right]} \textrm{d}x\textrm{d}y$$

where $f$ is the focal length of the optical system.

The optical system is generally considered as a linear system. When the rays emitted from the object transmits through the linear system to the image space, the frequency remains unchanged, but the contrast decreases and phase shifts occur simultaneously.

The relationship between contrast, phase shift and frequency is refer as optical transfer function (OTF), which can be expressed as

(13)$$\textrm{OTF}({f_{x^{\prime}}},{f_{y^{\prime}}}) = \int\!\!\!\int U (x^{\prime}, y^{\prime}){U^\ast }(x^{\prime}, y^{\prime})\exp [{ - j2\pi ({f_{x^{\prime}}}x^{\prime} + {f_{y^{\prime}}}y^{\prime})} ]\textrm{d}x^{\prime}\textrm{d}y^{\prime}$$

The modulation transfer function (MTF) of the optical dome can be obtained from the module of OTF.

2.3 Aerodynamic thermal radiation of the optical dome

The material of optical dome has a certain transmittance for infrared radiation, and its internal thermal radiation rays can pass through the dome and cause radiation interference to the detector. Therefore, the aerodynamic thermal radiation of the optical dome can be seen as the radiation of the semitransparent medium. According to Beer Lambor's law and Kirchhoff's law, the radiant energy emitted by the semitransparent medium is the result of the comprehensive effect of the emission and absorption of the semitransparent medium [18]. In this study, the surface and internal medium of optical dome was regarded as the radiative sources.

In the aerodynamic thermal environment, the temperature and refractive index of the dome were non-uniform. In thermal-structural coupling analysis, the dome was divided by grid element. As long as the surface element and its thickness was small enough, the temperature distribution in each small element can be considered as uniform. The emissivity of surface elements is not only related to the surface condition of the material, but also to the thickness of the material. It can be expressed as [19]:

(14)$$\varepsilon ({\lambda ,T} )= \frac{{(1 - \rho )(1 - \exp ( - \alpha (\lambda ,T)b)}}{{1 - \rho \cdot \exp ( - \alpha (\lambda ,T)b)}}$$

Where $\rho $ is the material reflectivity, $b$ is thickness of calculated position, $\alpha ({\lambda ,T} )$ is the material absorption coefficient.

Assuming the temperature of a surface element was ${T_i}$, according to Planck's radiation law, the spectral radiance of the surface element can be expressed as [10]

(15)$${L_\lambda } = \frac{{2\varepsilon h{c^2}{\lambda ^{ - 5}}}}{{\exp ({{{hc} \mathord{\left/ {\vphantom {{hc} {k{T_i}\lambda }}} \right.} {k{T_i}\lambda }}} )- 1}}$$

where $\varepsilon $ is the emissivity of the surface element, h is the Planck constant, c is the light speed, k is the Boltzmann constant, and $\lambda $ is the wavelength.

Any radiation surface element can be considered as Lambert radiator, which had the same radiant luminance in all directions. It emitted infrared radiation rays in all directions of space. The rays transmits with radiative energy. The sampled rays with limited radiative energy were adopted to stand for the continuous radiation energy of the radiative source. Assuming that the direction of infrared radiated rays were at an angle $\theta$ with the normal direction of the surface elememt, and the area of surface element was ds, the radiation power $\textrm{d}W$ of infrared radiated rays emitted by the surface element within the spatial angle ${\mathop{\rm d}\nolimits} \varOmega $ and within the waveband ${\mathop{\rm d}\nolimits} \lambda $ can be expressed as [10]

(16)$$\textrm{d}W = {L_\lambda }\textrm{cos}\theta \textrm{d}\;\lambda\; \textrm{d}\;s\;\textrm{d}\varOmega $$

Tracing infrared radiation rays adopted fourth order Runge-Kutta method. The radiation noise distribution on the detector can be obtained by integrating the radiation power.

According to the radiation power of the optical dome, the output voltage of each pixel on the detector can be obtained by Eq. (17). The output voltage of each pixel was converted into the gray value of each pixel by a pre-processing circuit, thereby the distorted image was obtained under the thermal radiation interference of the optical dome.

(17)$${V_{ij}} = G \cdot {R_{ij}} \cdot \textrm{d}{W_{ij}} + {V_{Nij}}$$

where ${V_{ij}}$ is the output voltage of the radiation power received by the pixel (i, j), G is the gain of the detector preamplifier, ${R_{ij}}$ is the response rate of the pixel (i, j), $\textrm{d}{W_{ij}}$ is the radiative power of the optical dome received by the pixel (i, j), ${V_{Nij}}$ is the root mean square noise at the pixel (i, j).

Peak Signal-to-Noise Ratio (PSNR) was used to evaluate distorted images quality. It is a kind of the most common and widespread objective imaging evaluation parameters. It can be expressed as [20]

(18)$$\textrm{PSNR}(f,G) = 10\log 10\left[ {\frac{{{{(L - 1)}^2}}}{{\textrm{MSE}(f,G)}}} \right]$$

where L is the pixel number of the image, MSE is mean squared error of the image.

3. Results and dicussion

3.1 The results of aero-optical transmission effect and aero-thermal radiation effect

The hemispherical shell-shaped optical dome in this study took the form of the nose of a maneuvering missile, its structure parameters were shown in Fig. 1. The bottom diameter of the optical dome was 200 mm. The fineness ratio F was defined as the ratio of L to D, which primarily determined the aerodynamic drag [21]. The fineness ratio F was 0.5. The optical dome was made up of sapphire crystals.

Fig. 1. Schematic diagram of optical dome’s structural parameters:D was 200mm; L was 100mm.

Optical System Parameters	value
Focal length(mm)	150
Wavelength(µm)	3∼5
Diameters of the entrance pupil(mm)	90
Diameters of the exit pupil(mm)	6.6
Detector pixels length(µm) × width(µm)	50×50
Number of detector pixels	128×128
Responsivity of the infrared photodetector(V/W)	6×10⁶
Output dynamic range (V)	0.4∼2.1
Pre-amplification	78
Maximum RMS (root mean square) noise (V)	0.1
Working time of optical dome(s)	10

Abstract

1. Introduction

2. Optical dome analysis

2.1 Refractive index of optical dome and ray transmission algorithms

2.2 Aerodynamic optical transmission of the optical dome

2.3 Aerodynamic thermal radiation of the optical dome

3. Results and dicussion

3.1 The results of aero-optical transmission effect and aero-thermal radiation effect

3.2 Influence of flight speed on aero-optical imaging quality

3.3 Influence of thickness of optical dome on aero-optical imaging quality

4. Summary

Funding

Acknowledgments

References

Cited By

Figures (20)

Tables (1)

Equations (18)

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