Comprehensive analysis of imaging quality degradation of an airborne optical system for aerodynamic flow field around the optical window Chenglong Hao, Shouqian Chen, Wang Zhang, Jinhan Ren, Chong Li, Hongjun Pang, Honghao Wang, Qian Liu, Chao Wang, Huiying Zou, and Zhigang Fan* Research Center for Space Optical Engineering, P.O. Box 307, Harbin Institute of Technology, Harbin 150001, China *Corresponding author: [email protected] Received 4 September 2013; accepted 30 September 2013; posted 8 October 2013 (Doc. ID 196869); published 13 November 2013
We investigated the influences exerted by the nonuniform aerodynamic flow field surrounding the optical window on the imaging quality degradation of an airborne optical system. The density distribution of flow fields around three typical optical windows, including a spherical window, an ellipsoidal window, and a paraboloidal window, were calculated by adopting the Reynolds-averaged Navier–Stokes equations with the Spalart–Allmaras model provided by FLUENT. The fourth-order Runge–Kutta algorithm based ray-tracing program was used to simulate the optical transmission through the aerodynamic flow field. Four kinds of imaging quality evaluation parameters were presented: wave aberration of the entrance pupil, point spread function, encircled energy, and modulation transfer function. The results show that the imaging quality of the airborne optical system was affected by the shape of the optical window and angle of attack of the aircraft. © 2013 Optical Society of America OCIS codes: (080.2710) Inhomogeneous optical media; (110.3000) Image quality assessment; (080.2720) Mathematical methods (general); (000.4430) Numerical approximation and analysis. http://dx.doi.org/10.1364/AO.52.007889
1. Introduction
In an aerodynamic environment, the aerodynamic flow field surrounding the dome is nonuniform. The optical path varies when light is transmitted through the nonuniform aerodynamic flow field surrounding the optical window. This varying optical path length (OPL) through the aerodynamic flow field leads to imaging blur, shift, jitter, loss of intensity of light, and loss of resolution of the system [1]. In recent years, there has been increasing interest in aero-optical issues on the aerodynamic flow field. Shi used numerical methods to simulate aero-optical effects for three typical optical windows, including 1559-128X/13/337889-10$15.00/0 © 2013 Optical Society of America
a cavity window, a raised window, and a sphere window [2]. Fei did some numerical calculation on aero-optical transmission effects of mean flow field for a cavity window, establishing the average flow field calculation method for optical transmission effects [3]. Gao et al. investigated the distorted wavefront caused by the density fluctuation of the supersonic mixing layer using the nanobased planar laser scattering technique, showing that the aero-optical aberrations can severely limit the performance of airborne laser systems [4]. Xu and Cai examined the influence of altitude on the imaging deviation of a two-dimensional aerodynamic flow field, showing that the aero-optical imaging deviation decreases as the altitude increases [5]. Sjoqvist and co-workers conducted laser beam propagation experiments to examine the influence of engine exhaust and plume 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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on the performance of airborne laser systems, showing that the performance of airborne laser systems is mostly affected by the beam wandering and beam broadening effects due to the engine exhaust and plume disturbance [6–8]. Although many reports have been published on the nonuniform aerodynamic flow field [1–10], only a few have focused on the imaging quality degradation of airborne optical systems caused by the flow field around the optical window with different shapes. The objectives of this study were to (1) scrutinize the mean density distributions of the aerodynamic flow field surrounding the three typical optical windows, (2) develop a ray-tracing program to simulate optical transmission through the aerodynamic flow field, (3) investigate the imaging quality evaluation parameters of the airborne optical system in the aerodynamic environment, and (4) compare the imaging quality degradation of airborne optical systems affected by the shape of the optical window. 2. Aerodynamic Flow Field Computation A. Density Computation of the Nonuniform Aerodynamic Flow Fields Surrounding the Optical Windows
The computational fluid dynamics (CFD) software FLUENT provides several kinds of turbulence models, such as the direct numerical simulation model, the large eddy simulation model, the Reynoldsaveraged Navier–Stokes equations model, and the Spalart–Allmaras model [11]. The Spalart–Allmaras model is especially designed for solving problems in aerospace aerodynamics and characterizing the mean flow quality [11], so it was adopted for this study. The three-dimensional computations were performed on the three typical round-head aircraft, as shown in Figs. 1–3. The nose radius of the aircraft with a spherical window was 100 mm; the bottom diameter of the ellipsoidal window was 200 mm and the length–diameter ratio of the ellipsoidal window was 1; and the bottom diameter of the
Fig. 1. Round-head high-speed aircraft with sphere optical window.
Fig. 3. Round-head high-speed aircraft with paraboloid optical window.
Fig. 4. Computational grid model for the CFD computation.
paraboloidal window was 200 mm, while the distance from apex to bottom surface was 200 mm. The geometry modeling and the grid generation were done by GAMBIT. The computational grid for all the CFD computations is shown in Fig. 4. The computational grid was an unstructured and nonuniform grid with multiblock topology. It was composed primarily of tetrahedral elements but might include hexahedral, pyramidal, and wedge elements [11]. The total number of grids for the aircraft with a spherical optical window, an ellipsoidal optical window, and a paraboloid optical window were 422,252, 653,580, and 511,399, respectively. The grids around the optical dome were sufficiently dense. The grid model was imported into FLUENT. The boundary conditions of the aerodynamic computation are listed in Table 1. The solver was set to be density based, implicit, and steady. Energy equations were added and solved simultaneously with the flow equations [5]. The standard wall functions were adopted to handle the interface of the flow field with the surface of the high-speed aircraft. The flux type was set to be the advection upstream splitting method. The under-relaxation factors for the turbulent viscosity, the modified turbulent viscosity, and the solid were set to be 0.6, 0.55, and 1.0, respectively. The difference scheme of the flow equations was set to be the second-order upwind scheme. The second-order upwind scheme was also employed to discretize the modified turbulent viscosity equations.
Table 1.
Fig. 2. Round-head high-speed aircraft with ellipsoid optical window. 7890
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Boundary Conditions for Aerodynamic Computation
Altitude (km) Mach number (Mach) Inlet total pressure (Pa) Inlet total temperature (K) Angle of attack (°)
20 3.38 5466 216.65 0
B.
Gladstone–Dale Relation
The refractive index of the flow field can be computed from the density of the flow field, according to the Gladstone–Dale relation [12]: n 1 K GD ρ;
(1)
where ρ is the density of the flow field and K GD is the Gladstone–Dale coefficient. K GD is dependent on the wavelength λ, expressed as [13] 7.52 × 10−3 −4 K GD λ 2.23 × 10 1 : (2) λ2 In this study, the imaging sensor was considered to be InSb, which is sensitive to a wavelength ranging from 3 to 5 μm, so a 4 μm wavelength was computed. 3. Optical Transmission through the Aerodynamic Flow Field Surrounding the Optical Window
The optical transmission through the aerodynamic flow field was simulated using a ray-tracing program based on the fourth-order Runge–Kutta algorithm [1,5]. As shown in Fig. 5, the grid models for the optical transmission simulation of the flow field (in red) were different from the grid models for the aerodynamic computation (in black). The grid models for the optical transmission of the flow field were composed primarily of cuboid elements, while the grid models for the aerodynamic computation were composed primarily of tetrahedral elements. In addition, the grid models for the optical transmission simulation of the flow field were much finer compared with the grid models for the aerodynamic computation, so as to precisely simulate the optical transmission through the flow field. The unstructured and nonuniform nature of the grid models for the aerodynamic computation forced us to use an interpolation to determine the refractive index of the node in the grid models for the optical transmission [9]. The refractive index of the node in the grid model for the optical transmission was evaluated using the refractive indices of the eight nodes in the grid model for the aerodynamic computation. These eight nodes were nearest to the node in the grid model for the optical transmission. The refractive index of the node in the grid model for the optical transmission was obtained by P8 n d−2 i P8 iFlow nFlow i1 ; (3) −2 i1 di where niFlow is the refractive index of the ith node in the grid model for the aerodynamic computation, and di is the spatial distance from the node in the grid model for the optical transmission to the ith node in the grid model for the aerodynamic computation. For an arbitrary ray transmitting through the aerodynamic flow field, its entire OPL can be expressed as
Fig. 5. Cross-sectional view of the grid model for optical transmission simulation of the aerodynamic flow field (in red) and grid model for the aerodynamic computation (in black). (a) Spherical window, (b) ellipsoid window, and (c) paraboloid window.
OPL
X
OPLiFlow ;
(4)
i
where OPLiFlow is the OPL of the arbitrary ray transmitting through the flow field at the ith step of the ray-tracing procedure. OPLiFlow with respect to the average density of the local flow field can be expressed as [10] OPLiFlow 1 K GD ρi liFlow ; 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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where liFlow is the actual distance traversed by the ray at the ith step in the flow field and ρi is the average density of the local flow field. The wave aberration of an arbitrary ray transmitting through the flow field can be expressed as [1] W k x; y
2π OPLk − OPL0 ; λ
(6)
where OPLk is the entire OPL of the arbitrary ray transmitting through the flow field, and OPL0 is the ensemble averaged OPL, expressed as [1] OPL0
1X OPLk ; N k
4. Results A. Aerodynamic Computational Results of the Flow Fields Surrounding the Optical Windows
Figures 7–9 show the mean density contour of the aerodynamic flow fields surrounding the three typical optical windows (Mach number 3.38, 0° angle of attack, 20 km altitude). The mean density fluctuations next to the optical window became more and more intense along the flow direction (the x direction) [10]. In addition, the strong detached shock wave was located in front of the optical windows. Our finding is
(7)
where N is the number of rays transmitting through the flow field. The wave aberration of the entire entrance pupil can be expressed as [1] Wx; y
X k
W k x; y
X 2π k
λ
OPLk − OPL0 : (8) Fig. 7. Mean density contour of the aerodynamic flow field surrounding the spherical window at the angle of attack of 0°.
As for the methods in computing point spread function (PSF) and modulation transfer function (MTF), see Xiao and Fan [1] for detailed analyses. The diameters of the entrance pupil and exit pupil of the dome were 90 and 12 mm, respectively. Incident angles were defined with respect to the dome, as in Fig. 6 [1]. The location of the center of the dome specifies the origin of the coordinate system [1]. The azimuth incident angle is measured relative to the z axis [1]. In the plot, the azimuth incident angle increases in a clockwise direction with 0° at the z axis, while the elevation incident angle increases toward the center of the plot with the centermost circle representing the zenith angle [1]. The optical transmission axis is aligned with the x axis [1].
Fig. 8. Mean density contour of the aerodynamic flow field surrounding the ellipsoid window at the angle of attack of 0°.
Fig. 6. Definitions for azimuth and elevation incident angles. Curved arrows indicate positive angles [1].
Fig. 9. Mean density contour of the aerodynamic flow field surrounding the paraboloid window at the angle of attack of 0°.
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similar to that of previous computational and experimental studies [14,15], in which the strong detached shock wave is observed in front of the blunt nose in supersonic flight. B. Imaging Quality Evaluation of the Airborne Optical System for Three Typical Optical Windows
An ideal optical system with 150 mm focal length was fixed at the back of the flow field to single out imaging quality degradation caused by the aerooptical disturbance of the aerodynamic flow fields surrounding the optical windows. Four imaging quality evaluation parameters (i.e., the wave aberration of the entrance pupil, PSF, encircled energy, and MTF) were calculated at the incident angle of 0°∕90° (azimuth/elevation). The illustrative diagrams (Figs. 10–19) correspond to the results obtained using the ray-tracing program (see Xiao and Fan [1] and Xiao and Zuo [16] for validation of the ray-tracing program). Results on wave aberration for the 0°∕90° (azimuth/elevation) incident angle are shown in Figs. 10–12. The peak-to-valley (PV) values of the wave aberration for the aircraft with a spherical optical window, the aircraft with an ellipsoidal optical window, and the aircraft with a paraboloidal optical window were 0.8799λ, 0.3778λ and 0.3946λ, respectively. Moreover, there were discontinuities of wave aberrations around the edge of the entrance pupil,
Fig. 10. Wave aberration result of the airborne optical system for the spherical window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 12. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
which could be attributed to the relatively sparse refractive index grids of the nonuniform flow field. The refractive index grids might not be fine enough for tracing rays through the flow regions. The results of effect of the refractive index grid density on the ray-tracing results are listed in Table 2. Figures 13–15 show the PSF results for the 0°∕90° (azimuth/elevation) incident angle for three typical optical windows. A notable characteristic of the PSF results is the greater scattering of energy of the spherical window than the ellipsoidal window and the paraboloidal window. The MTF results for the 0°∕90° (azimuth/elevation) incident angle for three typical optical windows are shown in Figs. 16–18. The cutoff frequency of the airborne optical system is 17 lp∕mm. The MTF decreases were observed, as opposed to the diffraction-limited MTF [1]. In addition, a greater MTF decrease was observed in the optical system with a spherical window than that with an ellipsoidal window and a paraboloidal window. The reduction of MTF shows that the decrease in the contrast of imaging details is the major effect of the aero-optical disturbance of the nonuniform aerodynamic flow fields on the imaging quality degradation of the airborne optical system. Figure 19 shows the normalized encircled energy for the PSF as a function of encircling radius for the 0°∕90° (azimuth/elevation) incident angle. For the diffraction-limited PSF, nearly 50% of the energy was encircled with a bucket radius close to λf 0 ∕D (f 0 is the focal length and D is the diameter of the entrance pupil of the airborne optical system) [17]. For the PSF of the spherical window, nearly 50% of the energy was encircled with a bucket radius close to 2.5λf 0 ∕D. For the PSF of the ellipsoidal window, nearly 50% of the energy was encircled with a bucket Table 2.
Fig. 11. Wave aberration result of the airborne optical system for the ellipsoid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
PV Values of Wave Aberration for Three Typical Windows with Different Grid Density
Optical Window Type
Sparser Grid
Present Grid
Denser Grid
Densest Grid
Spherical window Ellipsoidal window Paraboloidal window
0.9234λ 0.3957λ 0.4225λ
0.8799λ 0.3778λ 0.3946λ
0.8767λ 0.3765λ 0.3912λ
0.8789λ 0.3770λ 0.3941λ
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Fig. 13. PSF result of the airborne optical system for the spherical window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 14. PSF result of the airborne optical system for the ellipsoid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 15. PSF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
radius close to 1.25λf 0 ∕D. Also for the PSF of the paraboloidal window, nearly 50% of the energy was encircled with a bucket radius close to 1.25λf 0 ∕D. Thus, the resolving power of the airborne optical system is affected by the aero-optical disturbance of the aerodynamic flow field. 5. Discussion
The imaging quality of the airborne optical system was also affected by fight parameters of the aircraft, for example, angle of attack [5]. There has been some 7894
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Fig. 16. MTF result of the airborne optical system for the spherical window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 17. MTF result of the airborne optical system for the ellipsoid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 18. MTF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
research focused on calculating aero-optical effects with a paraboloidal optical window, so the paraboloidal optical window was calculated as an example for the influences of angle of attack. Angle of attack of the aircraft has a significant influence on the imaging quality of the airborne optical system. Figures 9, 20, and 21 show the computed mean density contours for an altitude of 20 km and angle of attack of 0°, 5°, and 15°, respectively. It is seen in Fig. 9 that the flow field was a symmetrical distribution. And Figs. 20 and 21 show that as angle of attack increased, the flow fields were asymmetrical distributions.
Fig. 19. Encircled energy results of the airborne optical system.
Fig. 20. Mean density contour of the aerodynamic flow field surrounding the paraboloid window at the angle of attack of 5°.
Fig. 22. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 5°.
Fig. 23. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 15°.
Figures 15, 24, and 25 show the PSF results for angle of attack of 0°, 5°, and 15°, respectively. Figure 24 shows that the secondary peaks appeared around the central core of the PSF, and Fig. 25 shows that more secondary peaks appeared around the central core. The MTF results for angle of attack of 0°, 5°, and 15° are shown in Figs. 18, 26, and 27, respectively. A greater MTF decrease was observed in the optical system when the angle of attack was 5° compared to the angle of attack being 0°. In addition, the sagittal MTF decrease was more serious than the Fig. 21. Mean density contour of the aerodynamic flow field surrounding the paraboloid window at the angle of attack of 15°.
Results on wave aberration for angle of attack of 0°, 5°, and 15° are shown in Figs. 12, 22, and 23, respectively. When the angle of attack was 5°, it is seen in Fig. 22 that the PV value of the wave aberration for the upwind was 0.4056λ; the PV value of the wave aberration for the downwind was 0.2132λ. When the angle of attack was 15°, it is seen in Fig. 23 that the PV value of the wave aberration for the upwind was 0.9214λ; the PV value of the wave aberration for the downwind was 0.4558λ. The PV value of the wave aberration increased as the angle of attack of the aircraft increased.
Fig. 24. PSF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 5°. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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angles are shown in Figs. 12, 28, and 29, respectively. The PV values of the wave aberration for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident angles were 0.3946λ, 0.5021λ, and 2.0321λ, respectively. Figures 15, 30, and 31 show the PSF results for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident angles, respectively. The secondary peaks appeared around the central core of the PSF as the elevation incident angle decreased and azimuth angle increased, indicating that the beam was broken
Fig. 25. PSF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 15°.
Fig. 28. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕80° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 26. MTF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 5°.
Fig. 29. Wave aberration result of the airborne optical system for the paraboloid window at 15°∕75° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 27. MTF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 15°.
meridional MTF decrease. When the angle of attack was 15°, the MTF decreased to 0.4 at the cutoff frequency of the optical system, as is shown in Fig. 27. The incident angle also affects the imaging quality of the airborne optical system [16]. The paraboloidal window was calculated as an example for the influences of incident angle. Results (Mach number 3.38, 0° angle of attack, 20 km altitude) on wave aberration for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident 7896
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Fig. 30. PSF result of the airborne optical system for the paraboloid window at 0°∕80° (azimuth/elevation) incident angle and the angle of attack of 0°.
MTF decreased to 0.4 at the cutoff frequency of the optical system with 15°∕75° (azimuth/elevation) incident angle. 6. Conclusion
Fig. 31. PSF result of the airborne optical system for the paraboloid window at 15°∕75° (azimuth/elevation) incident angle and the angle of attack of 0°.
up after transmitting through the nonuniform aerodynamic flow field. Our finding is similar to that of previous experimental studies conducted by Sjoqvist and co-workers [6–8], in which the laser beam is severely broken up after transmitting through the nonuniform engine exhaust and plume. The MTF results for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident angles are shown in Figs. 18, 32, and 33, respectively. The MTF decreased to 0.6 at the cutoff frequency of the optical system with 0°∕80° (azimuth/elevation) incident angle; the sagittal MTF decreased to 0.05 and the meridional
We employed the wavefront data obtained from the ray-tracing program based on the fourth-order Runge–Kutta algorithm to assess the imaging quality degradation of the airborne optical system due to the aero-optical disturbance of the aerodynamic flow field surrounding the optical window. The results for the three typical different shapes of the optical windows were compared. The wavefront data were computed at 20 km altitude on an airborne optical system at Mach 3.38 with a 90 mm entrance aperture. A significant contribution to wide-angle scattering in the PSF was observed. The MTF results showed that the loss of contrast for higher spatial frequencies is the major effect exerted by the aerooptical disturbance of the aerodynamic flow field on the imaging quality degradation of the airborne optical system. The experimental validation of the mathematical model of the optical transmission through the aerodynamic flow field will be pursued in future research. This research was jointly supported by the National Natural Science Foundation of China (Grant No. 61275020) and the Aeronautical Science Fund of China (Grant No. 20080177003). The authors are grateful to the reviewers and editors for their helpful and invaluable comments. References
Fig. 32. MTF result of the airborne optical system for the paraboloid window at 0°∕80° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 33. MTF result of the airborne optical system for the paraboloid window at 15°∕75° (azimuth/elevation) incident angle and the angle of attack of 0°.
1. H. S. Xiao and Z. G. Fan, “Imaging quality evaluation of aerodynamically heated optical dome using ray tracing,” Appl. Opt. 49, 5049–5058 (2010). 2. K. Shi, “Numerical simulation of aero-optical effects for the flow field around the optical window,” J. Infrared Laser Eng. 39, 7–11 (2010), in Chinese. 3. J. Fei, “Study on aero-optical effect technology for high speed missile infrared image guide,” J. Infrared Laser Eng. 27, 42–44 (1998), in Chinese. 4. Q. Gao, Z. F. Jiang, S. H. Yi, and Y. X. Zhao, “Optical path difference of the supersonic mixing layer,” Appl. Opt. 49, 3786–3792 (2010). 5. L. Xu and Y. L. Cai, “Influence of altitude on aero-optic imaging deviation,” Appl. Opt. 50, 2949–2957 (2011). 6. L. Sjoqvist, O. Gustafsson, and M. Henriksson, “Laser beam propagation in close vicinity to a downscaled jet engine exhaust,” Proc. SPIE 5615, 137–148 (2004). 7. M. Henriksson, L. Sjoqvist, D. Seiffer, N. Wendelstein, and E. Sucher, “Laser beam propagation experiments along and across a jet engine plume,” Proc. SPIE 7115, 71150E (2008). 8. M. Henriksson, O. Gustafsson, L. Sjoqvist, D. Seiffer, and N. Wendelstein, “Laser beam propagation through a full scale aircraft turboprop engine exhaust,” Proc. SPIE 7836, 78360L (2010). 9. E. Frumker and O. Pade, “Generic method for aero-optic evaluations,” Appl. Opt. 43, 3224–3228 (2004). 10. T. Wang, Y. Zhao, D. Xu, and Q. Y. Yang, “Numerical study of evaluating the optical quality of supersonic flow fields,” Appl. Opt. 46, 5545–5551 (2007). 11. FLUENT 6.3 User’s Guide (2004), http://www.ansys.com. 12. W. Merzkirch, Flow Visualization, 2nd ed. (Academic, 1987). 13. X. L. Yin, Principle of Aero-Optics (China Astronautics, 2003). 14. D. C. Harris, Materials for Infrared Windows and Domes (SPIE, 1999). 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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15. T. Zarutski, E. Arad, and R. Arieli, “Experimental and computational study on the effects of bumps on the aerodynamics of missile’s noses,” presented at the 42nd Israeli Conference on Aerospace Sciences, Haifa, Israel, January 1–18, 2002. 16. H. S. Xiao and B. J. Zuo, “Joint influences of aerodynamic flow field and aerodynamic heating of the dome on imaging
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quality degradation of airborne optical systems,” Appl. Opt. 51, 8625–8636 (2012). 17. M. R. Whiteley and D. J. Goorskey, “Influence of aero-optical disturbances on acquisition, tracking, and pointing performance characteristics in laser systems,” Proc. SPIE 8052, 805206 (2011).
We investigated the influences exerted by the nonuniform aerodynamic flow field surrounding the optical window on the imaging quality degradation of an airborne optical system. The density distribution of flow fields around three typical optical windows, including a spherical window, an ellipsoidal window, and a paraboloidal window, were calculated by adopting the Reynolds-averaged Navier–Stokes equations with the Spalart–Allmaras model provided by FLUENT. The fourth-order Runge–Kutta algorithm based ray-tracing program was used to simulate the optical transmission through the aerodynamic flow field. Four kinds of imaging quality evaluation parameters were presented: wave aberration of the entrance pupil, point spread function, encircled energy, and modulation transfer function. The results show that the imaging quality of the airborne optical system was affected by the shape of the optical window and angle of attack of the aircraft. © 2013 Optical Society of America OCIS codes: (080.2710) Inhomogeneous optical media; (110.3000) Image quality assessment; (080.2720) Mathematical methods (general); (000.4430) Numerical approximation and analysis. http://dx.doi.org/10.1364/AO.52.007889
1. Introduction
In an aerodynamic environment, the aerodynamic flow field surrounding the dome is nonuniform. The optical path varies when light is transmitted through the nonuniform aerodynamic flow field surrounding the optical window. This varying optical path length (OPL) through the aerodynamic flow field leads to imaging blur, shift, jitter, loss of intensity of light, and loss of resolution of the system [1]. In recent years, there has been increasing interest in aero-optical issues on the aerodynamic flow field. Shi used numerical methods to simulate aero-optical effects for three typical optical windows, including 1559-128X/13/337889-10$15.00/0 © 2013 Optical Society of America
a cavity window, a raised window, and a sphere window [2]. Fei did some numerical calculation on aero-optical transmission effects of mean flow field for a cavity window, establishing the average flow field calculation method for optical transmission effects [3]. Gao et al. investigated the distorted wavefront caused by the density fluctuation of the supersonic mixing layer using the nanobased planar laser scattering technique, showing that the aero-optical aberrations can severely limit the performance of airborne laser systems [4]. Xu and Cai examined the influence of altitude on the imaging deviation of a two-dimensional aerodynamic flow field, showing that the aero-optical imaging deviation decreases as the altitude increases [5]. Sjoqvist and co-workers conducted laser beam propagation experiments to examine the influence of engine exhaust and plume 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
7889
on the performance of airborne laser systems, showing that the performance of airborne laser systems is mostly affected by the beam wandering and beam broadening effects due to the engine exhaust and plume disturbance [6–8]. Although many reports have been published on the nonuniform aerodynamic flow field [1–10], only a few have focused on the imaging quality degradation of airborne optical systems caused by the flow field around the optical window with different shapes. The objectives of this study were to (1) scrutinize the mean density distributions of the aerodynamic flow field surrounding the three typical optical windows, (2) develop a ray-tracing program to simulate optical transmission through the aerodynamic flow field, (3) investigate the imaging quality evaluation parameters of the airborne optical system in the aerodynamic environment, and (4) compare the imaging quality degradation of airborne optical systems affected by the shape of the optical window. 2. Aerodynamic Flow Field Computation A. Density Computation of the Nonuniform Aerodynamic Flow Fields Surrounding the Optical Windows
The computational fluid dynamics (CFD) software FLUENT provides several kinds of turbulence models, such as the direct numerical simulation model, the large eddy simulation model, the Reynoldsaveraged Navier–Stokes equations model, and the Spalart–Allmaras model [11]. The Spalart–Allmaras model is especially designed for solving problems in aerospace aerodynamics and characterizing the mean flow quality [11], so it was adopted for this study. The three-dimensional computations were performed on the three typical round-head aircraft, as shown in Figs. 1–3. The nose radius of the aircraft with a spherical window was 100 mm; the bottom diameter of the ellipsoidal window was 200 mm and the length–diameter ratio of the ellipsoidal window was 1; and the bottom diameter of the
Fig. 1. Round-head high-speed aircraft with sphere optical window.
Fig. 3. Round-head high-speed aircraft with paraboloid optical window.
Fig. 4. Computational grid model for the CFD computation.
paraboloidal window was 200 mm, while the distance from apex to bottom surface was 200 mm. The geometry modeling and the grid generation were done by GAMBIT. The computational grid for all the CFD computations is shown in Fig. 4. The computational grid was an unstructured and nonuniform grid with multiblock topology. It was composed primarily of tetrahedral elements but might include hexahedral, pyramidal, and wedge elements [11]. The total number of grids for the aircraft with a spherical optical window, an ellipsoidal optical window, and a paraboloid optical window were 422,252, 653,580, and 511,399, respectively. The grids around the optical dome were sufficiently dense. The grid model was imported into FLUENT. The boundary conditions of the aerodynamic computation are listed in Table 1. The solver was set to be density based, implicit, and steady. Energy equations were added and solved simultaneously with the flow equations [5]. The standard wall functions were adopted to handle the interface of the flow field with the surface of the high-speed aircraft. The flux type was set to be the advection upstream splitting method. The under-relaxation factors for the turbulent viscosity, the modified turbulent viscosity, and the solid were set to be 0.6, 0.55, and 1.0, respectively. The difference scheme of the flow equations was set to be the second-order upwind scheme. The second-order upwind scheme was also employed to discretize the modified turbulent viscosity equations.
Table 1.
Fig. 2. Round-head high-speed aircraft with ellipsoid optical window. 7890
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Boundary Conditions for Aerodynamic Computation
Altitude (km) Mach number (Mach) Inlet total pressure (Pa) Inlet total temperature (K) Angle of attack (°)
20 3.38 5466 216.65 0
B.
Gladstone–Dale Relation
The refractive index of the flow field can be computed from the density of the flow field, according to the Gladstone–Dale relation [12]: n 1 K GD ρ;
(1)
where ρ is the density of the flow field and K GD is the Gladstone–Dale coefficient. K GD is dependent on the wavelength λ, expressed as [13] 7.52 × 10−3 −4 K GD λ 2.23 × 10 1 : (2) λ2 In this study, the imaging sensor was considered to be InSb, which is sensitive to a wavelength ranging from 3 to 5 μm, so a 4 μm wavelength was computed. 3. Optical Transmission through the Aerodynamic Flow Field Surrounding the Optical Window
The optical transmission through the aerodynamic flow field was simulated using a ray-tracing program based on the fourth-order Runge–Kutta algorithm [1,5]. As shown in Fig. 5, the grid models for the optical transmission simulation of the flow field (in red) were different from the grid models for the aerodynamic computation (in black). The grid models for the optical transmission of the flow field were composed primarily of cuboid elements, while the grid models for the aerodynamic computation were composed primarily of tetrahedral elements. In addition, the grid models for the optical transmission simulation of the flow field were much finer compared with the grid models for the aerodynamic computation, so as to precisely simulate the optical transmission through the flow field. The unstructured and nonuniform nature of the grid models for the aerodynamic computation forced us to use an interpolation to determine the refractive index of the node in the grid models for the optical transmission [9]. The refractive index of the node in the grid model for the optical transmission was evaluated using the refractive indices of the eight nodes in the grid model for the aerodynamic computation. These eight nodes were nearest to the node in the grid model for the optical transmission. The refractive index of the node in the grid model for the optical transmission was obtained by P8 n d−2 i P8 iFlow nFlow i1 ; (3) −2 i1 di where niFlow is the refractive index of the ith node in the grid model for the aerodynamic computation, and di is the spatial distance from the node in the grid model for the optical transmission to the ith node in the grid model for the aerodynamic computation. For an arbitrary ray transmitting through the aerodynamic flow field, its entire OPL can be expressed as
Fig. 5. Cross-sectional view of the grid model for optical transmission simulation of the aerodynamic flow field (in red) and grid model for the aerodynamic computation (in black). (a) Spherical window, (b) ellipsoid window, and (c) paraboloid window.
OPL
X
OPLiFlow ;
(4)
i
where OPLiFlow is the OPL of the arbitrary ray transmitting through the flow field at the ith step of the ray-tracing procedure. OPLiFlow with respect to the average density of the local flow field can be expressed as [10] OPLiFlow 1 K GD ρi liFlow ; 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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where liFlow is the actual distance traversed by the ray at the ith step in the flow field and ρi is the average density of the local flow field. The wave aberration of an arbitrary ray transmitting through the flow field can be expressed as [1] W k x; y
2π OPLk − OPL0 ; λ
(6)
where OPLk is the entire OPL of the arbitrary ray transmitting through the flow field, and OPL0 is the ensemble averaged OPL, expressed as [1] OPL0
1X OPLk ; N k
4. Results A. Aerodynamic Computational Results of the Flow Fields Surrounding the Optical Windows
Figures 7–9 show the mean density contour of the aerodynamic flow fields surrounding the three typical optical windows (Mach number 3.38, 0° angle of attack, 20 km altitude). The mean density fluctuations next to the optical window became more and more intense along the flow direction (the x direction) [10]. In addition, the strong detached shock wave was located in front of the optical windows. Our finding is
(7)
where N is the number of rays transmitting through the flow field. The wave aberration of the entire entrance pupil can be expressed as [1] Wx; y
X k
W k x; y
X 2π k
λ
OPLk − OPL0 : (8) Fig. 7. Mean density contour of the aerodynamic flow field surrounding the spherical window at the angle of attack of 0°.
As for the methods in computing point spread function (PSF) and modulation transfer function (MTF), see Xiao and Fan [1] for detailed analyses. The diameters of the entrance pupil and exit pupil of the dome were 90 and 12 mm, respectively. Incident angles were defined with respect to the dome, as in Fig. 6 [1]. The location of the center of the dome specifies the origin of the coordinate system [1]. The azimuth incident angle is measured relative to the z axis [1]. In the plot, the azimuth incident angle increases in a clockwise direction with 0° at the z axis, while the elevation incident angle increases toward the center of the plot with the centermost circle representing the zenith angle [1]. The optical transmission axis is aligned with the x axis [1].
Fig. 8. Mean density contour of the aerodynamic flow field surrounding the ellipsoid window at the angle of attack of 0°.
Fig. 6. Definitions for azimuth and elevation incident angles. Curved arrows indicate positive angles [1].
Fig. 9. Mean density contour of the aerodynamic flow field surrounding the paraboloid window at the angle of attack of 0°.
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similar to that of previous computational and experimental studies [14,15], in which the strong detached shock wave is observed in front of the blunt nose in supersonic flight. B. Imaging Quality Evaluation of the Airborne Optical System for Three Typical Optical Windows
An ideal optical system with 150 mm focal length was fixed at the back of the flow field to single out imaging quality degradation caused by the aerooptical disturbance of the aerodynamic flow fields surrounding the optical windows. Four imaging quality evaluation parameters (i.e., the wave aberration of the entrance pupil, PSF, encircled energy, and MTF) were calculated at the incident angle of 0°∕90° (azimuth/elevation). The illustrative diagrams (Figs. 10–19) correspond to the results obtained using the ray-tracing program (see Xiao and Fan [1] and Xiao and Zuo [16] for validation of the ray-tracing program). Results on wave aberration for the 0°∕90° (azimuth/elevation) incident angle are shown in Figs. 10–12. The peak-to-valley (PV) values of the wave aberration for the aircraft with a spherical optical window, the aircraft with an ellipsoidal optical window, and the aircraft with a paraboloidal optical window were 0.8799λ, 0.3778λ and 0.3946λ, respectively. Moreover, there were discontinuities of wave aberrations around the edge of the entrance pupil,
Fig. 10. Wave aberration result of the airborne optical system for the spherical window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 12. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
which could be attributed to the relatively sparse refractive index grids of the nonuniform flow field. The refractive index grids might not be fine enough for tracing rays through the flow regions. The results of effect of the refractive index grid density on the ray-tracing results are listed in Table 2. Figures 13–15 show the PSF results for the 0°∕90° (azimuth/elevation) incident angle for three typical optical windows. A notable characteristic of the PSF results is the greater scattering of energy of the spherical window than the ellipsoidal window and the paraboloidal window. The MTF results for the 0°∕90° (azimuth/elevation) incident angle for three typical optical windows are shown in Figs. 16–18. The cutoff frequency of the airborne optical system is 17 lp∕mm. The MTF decreases were observed, as opposed to the diffraction-limited MTF [1]. In addition, a greater MTF decrease was observed in the optical system with a spherical window than that with an ellipsoidal window and a paraboloidal window. The reduction of MTF shows that the decrease in the contrast of imaging details is the major effect of the aero-optical disturbance of the nonuniform aerodynamic flow fields on the imaging quality degradation of the airborne optical system. Figure 19 shows the normalized encircled energy for the PSF as a function of encircling radius for the 0°∕90° (azimuth/elevation) incident angle. For the diffraction-limited PSF, nearly 50% of the energy was encircled with a bucket radius close to λf 0 ∕D (f 0 is the focal length and D is the diameter of the entrance pupil of the airborne optical system) [17]. For the PSF of the spherical window, nearly 50% of the energy was encircled with a bucket radius close to 2.5λf 0 ∕D. For the PSF of the ellipsoidal window, nearly 50% of the energy was encircled with a bucket Table 2.
Fig. 11. Wave aberration result of the airborne optical system for the ellipsoid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
PV Values of Wave Aberration for Three Typical Windows with Different Grid Density
Optical Window Type
Sparser Grid
Present Grid
Denser Grid
Densest Grid
Spherical window Ellipsoidal window Paraboloidal window
0.9234λ 0.3957λ 0.4225λ
0.8799λ 0.3778λ 0.3946λ
0.8767λ 0.3765λ 0.3912λ
0.8789λ 0.3770λ 0.3941λ
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Fig. 13. PSF result of the airborne optical system for the spherical window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 14. PSF result of the airborne optical system for the ellipsoid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 15. PSF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
radius close to 1.25λf 0 ∕D. Also for the PSF of the paraboloidal window, nearly 50% of the energy was encircled with a bucket radius close to 1.25λf 0 ∕D. Thus, the resolving power of the airborne optical system is affected by the aero-optical disturbance of the aerodynamic flow field. 5. Discussion
The imaging quality of the airborne optical system was also affected by fight parameters of the aircraft, for example, angle of attack [5]. There has been some 7894
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Fig. 16. MTF result of the airborne optical system for the spherical window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 17. MTF result of the airborne optical system for the ellipsoid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 18. MTF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 0°.
research focused on calculating aero-optical effects with a paraboloidal optical window, so the paraboloidal optical window was calculated as an example for the influences of angle of attack. Angle of attack of the aircraft has a significant influence on the imaging quality of the airborne optical system. Figures 9, 20, and 21 show the computed mean density contours for an altitude of 20 km and angle of attack of 0°, 5°, and 15°, respectively. It is seen in Fig. 9 that the flow field was a symmetrical distribution. And Figs. 20 and 21 show that as angle of attack increased, the flow fields were asymmetrical distributions.
Fig. 19. Encircled energy results of the airborne optical system.
Fig. 20. Mean density contour of the aerodynamic flow field surrounding the paraboloid window at the angle of attack of 5°.
Fig. 22. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 5°.
Fig. 23. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 15°.
Figures 15, 24, and 25 show the PSF results for angle of attack of 0°, 5°, and 15°, respectively. Figure 24 shows that the secondary peaks appeared around the central core of the PSF, and Fig. 25 shows that more secondary peaks appeared around the central core. The MTF results for angle of attack of 0°, 5°, and 15° are shown in Figs. 18, 26, and 27, respectively. A greater MTF decrease was observed in the optical system when the angle of attack was 5° compared to the angle of attack being 0°. In addition, the sagittal MTF decrease was more serious than the Fig. 21. Mean density contour of the aerodynamic flow field surrounding the paraboloid window at the angle of attack of 15°.
Results on wave aberration for angle of attack of 0°, 5°, and 15° are shown in Figs. 12, 22, and 23, respectively. When the angle of attack was 5°, it is seen in Fig. 22 that the PV value of the wave aberration for the upwind was 0.4056λ; the PV value of the wave aberration for the downwind was 0.2132λ. When the angle of attack was 15°, it is seen in Fig. 23 that the PV value of the wave aberration for the upwind was 0.9214λ; the PV value of the wave aberration for the downwind was 0.4558λ. The PV value of the wave aberration increased as the angle of attack of the aircraft increased.
Fig. 24. PSF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 5°. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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angles are shown in Figs. 12, 28, and 29, respectively. The PV values of the wave aberration for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident angles were 0.3946λ, 0.5021λ, and 2.0321λ, respectively. Figures 15, 30, and 31 show the PSF results for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident angles, respectively. The secondary peaks appeared around the central core of the PSF as the elevation incident angle decreased and azimuth angle increased, indicating that the beam was broken
Fig. 25. PSF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 15°.
Fig. 28. Wave aberration result of the airborne optical system for the paraboloid window at 0°∕80° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 26. MTF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 5°.
Fig. 29. Wave aberration result of the airborne optical system for the paraboloid window at 15°∕75° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 27. MTF result of the airborne optical system for the paraboloid window at 0°∕90° (azimuth/elevation) incident angle and the angle of attack of 15°.
meridional MTF decrease. When the angle of attack was 15°, the MTF decreased to 0.4 at the cutoff frequency of the optical system, as is shown in Fig. 27. The incident angle also affects the imaging quality of the airborne optical system [16]. The paraboloidal window was calculated as an example for the influences of incident angle. Results (Mach number 3.38, 0° angle of attack, 20 km altitude) on wave aberration for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident 7896
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Fig. 30. PSF result of the airborne optical system for the paraboloid window at 0°∕80° (azimuth/elevation) incident angle and the angle of attack of 0°.
MTF decreased to 0.4 at the cutoff frequency of the optical system with 15°∕75° (azimuth/elevation) incident angle. 6. Conclusion
Fig. 31. PSF result of the airborne optical system for the paraboloid window at 15°∕75° (azimuth/elevation) incident angle and the angle of attack of 0°.
up after transmitting through the nonuniform aerodynamic flow field. Our finding is similar to that of previous experimental studies conducted by Sjoqvist and co-workers [6–8], in which the laser beam is severely broken up after transmitting through the nonuniform engine exhaust and plume. The MTF results for the 0°∕90° (azimuth/elevation), 0°∕80°, and 15°∕75° incident angles are shown in Figs. 18, 32, and 33, respectively. The MTF decreased to 0.6 at the cutoff frequency of the optical system with 0°∕80° (azimuth/elevation) incident angle; the sagittal MTF decreased to 0.05 and the meridional
We employed the wavefront data obtained from the ray-tracing program based on the fourth-order Runge–Kutta algorithm to assess the imaging quality degradation of the airborne optical system due to the aero-optical disturbance of the aerodynamic flow field surrounding the optical window. The results for the three typical different shapes of the optical windows were compared. The wavefront data were computed at 20 km altitude on an airborne optical system at Mach 3.38 with a 90 mm entrance aperture. A significant contribution to wide-angle scattering in the PSF was observed. The MTF results showed that the loss of contrast for higher spatial frequencies is the major effect exerted by the aerooptical disturbance of the aerodynamic flow field on the imaging quality degradation of the airborne optical system. The experimental validation of the mathematical model of the optical transmission through the aerodynamic flow field will be pursued in future research. This research was jointly supported by the National Natural Science Foundation of China (Grant No. 61275020) and the Aeronautical Science Fund of China (Grant No. 20080177003). The authors are grateful to the reviewers and editors for their helpful and invaluable comments. References
Fig. 32. MTF result of the airborne optical system for the paraboloid window at 0°∕80° (azimuth/elevation) incident angle and the angle of attack of 0°.
Fig. 33. MTF result of the airborne optical system for the paraboloid window at 15°∕75° (azimuth/elevation) incident angle and the angle of attack of 0°.
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