Abstract

The distorted wavefront due to the density fluctuation of the supersonic mixing layer is measured by the nano-based planar laser scattering technique, with the spatiotemporal resolutions as 101mm and 1μs (time interval between two exposures), respectively. The optical path difference (OPD) is analyzed from the viewpoints of correlation and structure functions. Depending on the structure of the flow field, the type of the correlation function is Gaussian or exponential. The power index of the structure function is about 1.1, and the consistency over many cases is quite good. The far-field propagation of the Gaussian beam is simulated by Fourier transform, and the Strehl ratio is calculated with the rms of the OPD. The beam center and beam spread about the center in the far field are calculated, and are dominated by the statistics of the tilt of the distorted wavefront. The results indicate that aero-optical aberrations can severely limit the performance of airborne laser systems.

© 2010 Optical Society of America

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References

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  1. K. G. Gilbert and L. J. Otten, eds., Aero-Optical Phenomena (American Institute of Aeronautics and Astronautics, 1980).
  2. E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
    [CrossRef]
  3. J. D. Trolinger, “Aero-optical characterization of aircraft optical turrets by holography, interferometry and shadowgraph,” in Aero-Optical Phenomena, K.G.Gilbert and L.J.Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), pp. 200–217.
  4. M. Malley, G. W. Sutton, and N. Kincheloe, “Beam-jitter measurements of turbulent aero-optical path differences,” Appl. Opt. 31, 4440–4443 (1992).
    [CrossRef] [PubMed]
  5. R. J. Hugo and E. J. Jumper, “Experimental measurement of a time-varying optical path difference using the small-aperture beam technique,” Appl. Opt. 35, 4436–4447 (1996).
    [CrossRef] [PubMed]
  6. S. Abado, S. Gordeyev, and E. J. Jumper, “Designing and testing a high bandwidth 2-D wave front sensor for aero-optics,” Proc. SPIE 7466, 746602 (2009).
    [CrossRef]
  7. P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).
  8. F. R. Zubair and H. J. Catrakis, “Aero-optical interactions along laser beam propagation paths in compressible turbulence,” AIAA J. 45, 1663–1674 (2007).
    [CrossRef]
  9. S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).
  10. Y. X. Zhao, “Experimental study of the spatiotemporal structure of supersonic mixing layer,” Ph.D. dissertation (College of Aerospace and Material Engineering, National University of Defense Technology, 2008).
  11. L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
    [CrossRef]
  12. Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
    [CrossRef]
  13. E. J. Fitzgerald and E. J. Jumper, “The optical distortion mechanism in a nearly incompressible free shear layer,” J. Fluid Mech. 512, 153–189 (2001).
  14. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  15. V. N. Mahajan, “Strehl ratio for aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 860–861 (1983).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  17. A. Mani, M. Wang, and P. Moin, “Statistical description of the free-space propagation of highly aberrated optical beams,” J. Opt. Soc. Am. A 23, 3027–3035 (2006).
    [CrossRef]
  18. A. Mani, P. Moin, and M. Wang, “Computational study of optical distortions by separated shear layers and turbulent wakes,” J. Fluid Mech. 625, 273–298 (2009).
    [CrossRef]

2010 (1)

Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
[CrossRef]

2009 (3)

A. Mani, P. Moin, and M. Wang, “Computational study of optical distortions by separated shear layers and turbulent wakes,” J. Fluid Mech. 625, 273–298 (2009).
[CrossRef]

S. Abado, S. Gordeyev, and E. J. Jumper, “Designing and testing a high bandwidth 2-D wave front sensor for aero-optics,” Proc. SPIE 7466, 746602 (2009).
[CrossRef]

L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
[CrossRef]

2007 (2)

F. R. Zubair and H. J. Catrakis, “Aero-optical interactions along laser beam propagation paths in compressible turbulence,” AIAA J. 45, 1663–1674 (2007).
[CrossRef]

S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).

2006 (1)

2001 (3)

E. J. Fitzgerald and E. J. Jumper, “The optical distortion mechanism in a nearly incompressible free shear layer,” J. Fluid Mech. 512, 153–189 (2001).

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[CrossRef]

1996 (1)

1992 (1)

1983 (1)

Abado, S.

S. Abado, S. Gordeyev, and E. J. Jumper, “Designing and testing a high bandwidth 2-D wave front sensor for aero-optics,” Proc. SPIE 7466, 746602 (2009).
[CrossRef]

Catrakis, H. J.

F. R. Zubair and H. J. Catrakis, “Aero-optical interactions along laser beam propagation paths in compressible turbulence,” AIAA J. 45, 1663–1674 (2007).
[CrossRef]

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).

Cheng, Y. Z.

Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
[CrossRef]

L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
[CrossRef]

S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).

Dimotakis, P. E.

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).

Fitzgerald, E. J.

E. J. Fitzgerald and E. J. Jumper, “The optical distortion mechanism in a nearly incompressible free shear layer,” J. Fluid Mech. 512, 153–189 (2001).

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[CrossRef]

Fourguette, D. C.

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).

Gilbert, K. G.

K. G. Gilbert and L. J. Otten, eds., Aero-Optical Phenomena (American Institute of Aeronautics and Astronautics, 1980).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gordeyev, S.

S. Abado, S. Gordeyev, and E. J. Jumper, “Designing and testing a high bandwidth 2-D wave front sensor for aero-optics,” Proc. SPIE 7466, 746602 (2009).
[CrossRef]

He, L.

Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
[CrossRef]

L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
[CrossRef]

S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).

Hugo, R. J.

Jumper, E. J.

S. Abado, S. Gordeyev, and E. J. Jumper, “Designing and testing a high bandwidth 2-D wave front sensor for aero-optics,” Proc. SPIE 7466, 746602 (2009).
[CrossRef]

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[CrossRef]

E. J. Fitzgerald and E. J. Jumper, “The optical distortion mechanism in a nearly incompressible free shear layer,” J. Fluid Mech. 512, 153–189 (2001).

R. J. Hugo and E. J. Jumper, “Experimental measurement of a time-varying optical path difference using the small-aperture beam technique,” Appl. Opt. 35, 4436–4447 (1996).
[CrossRef] [PubMed]

Kincheloe, N.

Mahajan, V. N.

Malley, M.

Mani, A.

A. Mani, P. Moin, and M. Wang, “Computational study of optical distortions by separated shear layers and turbulent wakes,” J. Fluid Mech. 625, 273–298 (2009).
[CrossRef]

A. Mani, M. Wang, and P. Moin, “Statistical description of the free-space propagation of highly aberrated optical beams,” J. Opt. Soc. Am. A 23, 3027–3035 (2006).
[CrossRef]

Moin, P.

A. Mani, P. Moin, and M. Wang, “Computational study of optical distortions by separated shear layers and turbulent wakes,” J. Fluid Mech. 625, 273–298 (2009).
[CrossRef]

A. Mani, M. Wang, and P. Moin, “Statistical description of the free-space propagation of highly aberrated optical beams,” J. Opt. Soc. Am. A 23, 3027–3035 (2006).
[CrossRef]

Otten, L. J.

K. G. Gilbert and L. J. Otten, eds., Aero-Optical Phenomena (American Institute of Aeronautics and Astronautics, 1980).

Sutton, G. W.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tian, L. F.

Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
[CrossRef]

L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
[CrossRef]

S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).

Trolinger, J. D.

J. D. Trolinger, “Aero-optical characterization of aircraft optical turrets by holography, interferometry and shadowgraph,” in Aero-Optical Phenomena, K.G.Gilbert and L.J.Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), pp. 200–217.

Wang, M.

A. Mani, P. Moin, and M. Wang, “Computational study of optical distortions by separated shear layers and turbulent wakes,” J. Fluid Mech. 625, 273–298 (2009).
[CrossRef]

A. Mani, M. Wang, and P. Moin, “Statistical description of the free-space propagation of highly aberrated optical beams,” J. Opt. Soc. Am. A 23, 3027–3035 (2006).
[CrossRef]

Xi, S. H.

Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
[CrossRef]

L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
[CrossRef]

Yi, S. H.

S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).

Zhao, Y. X.

Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
[CrossRef]

L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
[CrossRef]

S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).

Y. X. Zhao, “Experimental study of the spatiotemporal structure of supersonic mixing layer,” Ph.D. dissertation (College of Aerospace and Material Engineering, National University of Defense Technology, 2008).

Zubair, F. R.

F. R. Zubair and H. J. Catrakis, “Aero-optical interactions along laser beam propagation paths in compressible turbulence,” AIAA J. 45, 1663–1674 (2007).
[CrossRef]

AIAA J. (1)

F. R. Zubair and H. J. Catrakis, “Aero-optical interactions along laser beam propagation paths in compressible turbulence,” AIAA J. 45, 1663–1674 (2007).
[CrossRef]

Appl. Opt. (2)

Chin. Conf. Theoret. Appl. Mech. Beijing (1)

S. H. Yi, Y. X. Zhao, L. He, Y. Z. Cheng, and L. F. Tian, “NPLS technique and its application in visualization of the supersonic turbulence,” Chin. Conf. Theoret. Appl. Mech. Beijing (2007).

J. Fluid Mech. (3)

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).

E. J. Fitzgerald and E. J. Jumper, “The optical distortion mechanism in a nearly incompressible free shear layer,” J. Fluid Mech. 512, 153–189 (2001).

A. Mani, P. Moin, and M. Wang, “Computational study of optical distortions by separated shear layers and turbulent wakes,” J. Fluid Mech. 625, 273–298 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Proc. SPIE (1)

S. Abado, S. Gordeyev, and E. J. Jumper, “Designing and testing a high bandwidth 2-D wave front sensor for aero-optics,” Proc. SPIE 7466, 746602 (2009).
[CrossRef]

Prog. Aerosp. Sci. (1)

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[CrossRef]

Sci. China Ser. G (1)

L. F. Tian, S. H. Xi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G 52, 1357–1363(2009).
[CrossRef]

Sci. China Tech. Sci. (1)

Y. X. Zhao, S. H. Xi, L. F. Tian, L. He, and Y. Z. Cheng, “Multiresolution analysis of density fluctuation in supersonic mixing layer,” Sci. China Tech. Sci. 53, 584–591 (2010).
[CrossRef]

Other (5)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

K. G. Gilbert and L. J. Otten, eds., Aero-Optical Phenomena (American Institute of Aeronautics and Astronautics, 1980).

Y. X. Zhao, “Experimental study of the spatiotemporal structure of supersonic mixing layer,” Ph.D. dissertation (College of Aerospace and Material Engineering, National University of Defense Technology, 2008).

J. D. Trolinger, “Aero-optical characterization of aircraft optical turrets by holography, interferometry and shadowgraph,” in Aero-Optical Phenomena, K.G.Gilbert and L.J.Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), pp. 200–217.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (5)

Fig. 1
Fig. 1

Sketch of the experimental setup used in the present study. The main components are the supersonic mixing layer wind tunnel and the NPLS system. Find more detail in the text.

Fig. 2
Fig. 2

Two time-correlated NPLS images and their OPLs of the supersonic mixing layer with M c = 0.12 . The exposure time is 5 ns , and the time interval between the two exposures is 5 μs . Find more details in the text.

Fig. 3
Fig. 3

OPD of the mixing layer in Fig. 3 (solid curve, Fig. 3a; dashed curve, Fig. 3b). The average is over the whole streamwise part.

Fig. 4
Fig. 4

Correlation and structure functions for the OPDs in regions (a), (c) B1 and (b), (d) B2.

Fig. 5
Fig. 5

Normalized far-field intensity distribution. (a) σ = 32 δ , planar wavefront; (b) σ = 32 δ , distorted wavefront due to region B3; (c) σ = 32 δ , distorted wave-front due to region B4; (d) σ = 64 δ , planar wavefront; (e) σ = 64 δ , distorted wavefront due to region B3; and (f) σ = 64 δ , distorted wavefront due to region B4.

Tables (3)

Tables Icon

Table 1 Calibration Parameters of the Supersonic Mixing Layer Wind Tunnel with M c = 0.12

Tables Icon

Table 2 Parameters of the Correlation and Structure Functions of the OPD in Fig. 4

Tables Icon

Table 3 Beam Center and Beam Spread about the Center

Equations (20)

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M c = ( U 2 U 1 ) / ( a 1 + a 2 ) = 0.12 , 0.24 , 0.50 ,
n ( x , z ) n ( x , z ) 1 = K GD ρ ( x , z ) ,
OPL ( x ) = z 0 z 1 n ( x , z ) d z ,
OPD ( x ) = OPL ( x ) OPL ( x ) ,
ϕ ( x ) = OPD ( x 0 ) OPD ( x + x 0 ) x 0 .
ϕ ( x ) = C 2 exp ( ( x x 0 ) 2 / l 2 ) ,
ϕ ( x ) = C 2 exp ( x / l ) ,
D ( x ) = ( OPD ( x + x 0 ) OPD ( x 0 ) ) 2 x 0 ,
D ( x ) = C 2 ( | x | / l ) μ ,
A 1 : x / δ = 300 811 of Fig . 3 a ; A 2 : x / δ = 900 1411 of Fig . 3 a ; B 1 : x / δ = 300 811 of Fig . 3 b ; B 2 : x / δ = 900 1411 of Fig . 3 b .
A ( x ) = 1 2 π σ exp ( x 2 2 σ 2 ) .
U ( x , z 1 ) = A ( x ) exp ( j k L ( x ) ) ,
L ( x ) = z 0 z 1 n ( x , z ) d z .
SR = exp ( ( k L rms ) 2 ) ,
A 3 : x / δ = 600 727 of Fig . 3 a ; A 4 : x / δ = 1200 1327 of Fig . 3 a ; B 3 : x / δ = 600 727 of Fig . 3 b ; B 4 : x / δ = 1200 1327 of Fig . 3 b .
x ¯ = L ¯ x | z 1 ( z z 1 ) + x ¯ | z 1 α ( z z 1 ) + x ¯ | z 1 ,
x 2 ¯ = L x 2 ¯ | z 1 ( z z 1 ) 2 + 2 x L x ¯ | z 1 ( z z 1 ) + x 2 ¯ | z 1 β ( z z 1 ) 2 + 2 γ ( z z 1 ) + x 2 ¯ | z 1 ,
α = L x ¯ | z 1 , β = L x 2 ¯ | z 1 , γ = x L x ¯ | z 1 .
f ( x ) ¯ = f ( x ) I ( x ) d x I ( x ) d x ,
x = x x ¯ , L x = L x L x ¯ .

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