Abstract

A method of building a dynamical model for temporal evolution of an aero-optical aberration is presented. Based on the proper orthogonal decomposition (POD) of the optical aberration, the minimum description length (MDL) principle is used to select a dynamical model for the POD coefficients. The MDL principle can balance the model error against model size so as to prevent the model from overfitting. This strategy is validated with the numerical simulation of the aero-optical aberration caused by the supersonic mixing layer, and the model obtained can accurately predict the time evolution of the POD coefficients in short time. The potential application of our method in extending the bandwidth of the adaptive optics system is also discussed.

© 2014 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, 1982).
  2. E. J. Jumper and E. J. Fitzgerald, Prog. Aerosp. Sci. 37, 299 (2001).
    [CrossRef]
  3. W. Wang, A. Mani, and S. Gordeyev, Annu. Rev. Fluid Mech. 44, 299 (2012).
    [CrossRef]
  4. S. Gordeyev and E. J. Jumper, Prog. Aerosp. Sci. 46, 388 (2010).
    [CrossRef]
  5. Q. Gao, S. H. Yi, Z. F. Jiang, H. Lin, and Y. X. Zhao, Appl. Opt. 49, 3786 (2010).
    [CrossRef]
  6. Q. Gao, S. H. Yi, Z. F. Jiang, H. Lin, and Y. X. Zhao, Opt. Express 20, 16494 (2012).
    [CrossRef]
  7. Q. Gao, Z. Jiang, S. Yi, and X. Wang, Opt. Lett. 38, 751 (2013).
    [CrossRef]
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  9. P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University, 1996).
  10. Q. Gao, Z. Jiang, S. Yi, W. K. Xie, and T. H. Liao, Appl. Opt. 51, 3922 (2012).
    [CrossRef]
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  13. K. Judd and A. Mees, Physica D 82, 426 (1995).
    [CrossRef]
  14. T. Nakamura, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1263 (2003).
    [CrossRef]
  15. T. Nakamura, D. Kilminster, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1129 (2004).
    [CrossRef]
  16. J. Rissanen, Complexity in Statistical Inquiry (World Scientific, 1989).
  17. M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).
  18. R. M. Rennie, D. A. Duffin, and E. J. Jumper, AIAA J. 46, 2787 (2008).
    [CrossRef]
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    [CrossRef]
  20. T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, IEEE J. Sel. Top. Quantum Electron. 5, 83 (1999).
    [CrossRef]
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  22. S. Abado, S. Gordeyev, and E. Jumper, Opt. Eng. 49, 064403 (2010).
    [CrossRef]

2013 (1)

2012 (5)

2010 (3)

S. Abado, S. Gordeyev, and E. Jumper, Opt. Eng. 49, 064403 (2010).
[CrossRef]

S. Gordeyev and E. J. Jumper, Prog. Aerosp. Sci. 46, 388 (2010).
[CrossRef]

Q. Gao, S. H. Yi, Z. F. Jiang, H. Lin, and Y. X. Zhao, Appl. Opt. 49, 3786 (2010).
[CrossRef]

2009 (1)

L. A. Aguirre and C. Letellier, Math. Probl. Eng. 2009, 238960 (2009).

2008 (1)

R. M. Rennie, D. A. Duffin, and E. J. Jumper, AIAA J. 46, 2787 (2008).
[CrossRef]

2004 (2)

T. Nakamura, D. Kilminster, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1129 (2004).
[CrossRef]

C. W. Rowley, T. Colonius, and R. W. Murray, Physica D 189, 115 (2004).
[CrossRef]

2003 (1)

T. Nakamura, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1263 (2003).
[CrossRef]

2001 (1)

E. J. Jumper and E. J. Fitzgerald, Prog. Aerosp. Sci. 37, 299 (2001).
[CrossRef]

1999 (1)

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, IEEE J. Sel. Top. Quantum Electron. 5, 83 (1999).
[CrossRef]

1995 (1)

K. Judd and A. Mees, Physica D 82, 426 (1995).
[CrossRef]

Abado, S.

S. Abado, S. Gordeyev, and E. Jumper, Opt. Eng. 49, 064403 (2010).
[CrossRef]

Aguirre, L. A.

L. A. Aguirre and C. Letellier, Math. Probl. Eng. 2009, 238960 (2009).

Berkooz, G.

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University, 1996).

Bifano, T. G.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, IEEE J. Sel. Top. Quantum Electron. 5, 83 (1999).
[CrossRef]

Colonius, T.

C. W. Rowley, T. Colonius, and R. W. Murray, Physica D 189, 115 (2004).
[CrossRef]

Duffin, D. A.

R. M. Rennie, D. A. Duffin, and E. J. Jumper, AIAA J. 46, 2787 (2008).
[CrossRef]

Fitzgerald, E. J.

E. J. Jumper and E. J. Fitzgerald, Prog. Aerosp. Sci. 37, 299 (2001).
[CrossRef]

Gao, Q.

Gibson, S.

Gordeyev, S.

W. Wang, A. Mani, and S. Gordeyev, Annu. Rev. Fluid Mech. 44, 299 (2012).
[CrossRef]

S. Gordeyev and E. J. Jumper, Prog. Aerosp. Sci. 46, 388 (2010).
[CrossRef]

S. Abado, S. Gordeyev, and E. Jumper, Opt. Eng. 49, 064403 (2010).
[CrossRef]

Holmes, P.

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University, 1996).

Horenstein, M. N.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, IEEE J. Sel. Top. Quantum Electron. 5, 83 (1999).
[CrossRef]

Jiang, Z.

Jiang, Z. F.

Judd, K.

T. Nakamura, D. Kilminster, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1129 (2004).
[CrossRef]

T. Nakamura, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1263 (2003).
[CrossRef]

K. Judd and A. Mees, Physica D 82, 426 (1995).
[CrossRef]

Jumper, E.

S. Abado, S. Gordeyev, and E. Jumper, Opt. Eng. 49, 064403 (2010).
[CrossRef]

Jumper, E. J.

S. Gordeyev and E. J. Jumper, Prog. Aerosp. Sci. 46, 388 (2010).
[CrossRef]

R. M. Rennie, D. A. Duffin, and E. J. Jumper, AIAA J. 46, 2787 (2008).
[CrossRef]

E. J. Jumper and E. J. Fitzgerald, Prog. Aerosp. Sci. 37, 299 (2001).
[CrossRef]

Kilminster, D.

T. Nakamura, D. Kilminster, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1129 (2004).
[CrossRef]

Lempert, W.

B. Thurow, M. Samimy, and W. Lempert, “Simultaneous MHz rate flow visualization and wavefront sensing for aero-optics,” (American Institute of Aeronautics and Astronautics, 2003).

Letellier, C.

L. A. Aguirre and C. Letellier, Math. Probl. Eng. 2009, 238960 (2009).

Liao, T. H.

Lin, H.

Lumley, J. L.

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University, 1996).

Mali, R. K.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, IEEE J. Sel. Top. Quantum Electron. 5, 83 (1999).
[CrossRef]

Mani, A.

W. Wang, A. Mani, and S. Gordeyev, Annu. Rev. Fluid Mech. 44, 299 (2012).
[CrossRef]

Mees, A.

T. Nakamura, D. Kilminster, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1129 (2004).
[CrossRef]

T. Nakamura, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1263 (2003).
[CrossRef]

K. Judd and A. Mees, Physica D 82, 426 (1995).
[CrossRef]

Murray, R. W.

C. W. Rowley, T. Colonius, and R. W. Murray, Physica D 189, 115 (2004).
[CrossRef]

Nakamura, T.

T. Nakamura, D. Kilminster, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1129 (2004).
[CrossRef]

T. Nakamura, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1263 (2003).
[CrossRef]

Perreault, J.

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, IEEE J. Sel. Top. Quantum Electron. 5, 83 (1999).
[CrossRef]

Rennie, R. M.

R. M. Rennie, D. A. Duffin, and E. J. Jumper, AIAA J. 46, 2787 (2008).
[CrossRef]

Rissanen, J.

J. Rissanen, Complexity in Statistical Inquiry (World Scientific, 1989).

Rowley, C. W.

C. W. Rowley, T. Colonius, and R. W. Murray, Physica D 189, 115 (2004).
[CrossRef]

Samimy, M.

B. Thurow, M. Samimy, and W. Lempert, “Simultaneous MHz rate flow visualization and wavefront sensing for aero-optics,” (American Institute of Aeronautics and Astronautics, 2003).

Tesch, J.

Thurow, B.

B. Thurow, M. Samimy, and W. Lempert, “Simultaneous MHz rate flow visualization and wavefront sensing for aero-optics,” (American Institute of Aeronautics and Astronautics, 2003).

Verdult, V.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

Verhaegen, M.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

Wang, W.

W. Wang, A. Mani, and S. Gordeyev, Annu. Rev. Fluid Mech. 44, 299 (2012).
[CrossRef]

Wang, X.

Xie, W. K.

Yi, S.

Yi, S. H.

Zhao, Y. X.

AIAA J. (1)

R. M. Rennie, D. A. Duffin, and E. J. Jumper, AIAA J. 46, 2787 (2008).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

W. Wang, A. Mani, and S. Gordeyev, Annu. Rev. Fluid Mech. 44, 299 (2012).
[CrossRef]

Appl. Opt. (2)

IEEE J. Sel. Top. Quantum Electron. (1)

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, IEEE J. Sel. Top. Quantum Electron. 5, 83 (1999).
[CrossRef]

Int. J. Bifurcation Chaos Appl. Sci. Eng. (2)

T. Nakamura, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1263 (2003).
[CrossRef]

T. Nakamura, D. Kilminster, K. Judd, and A. Mees, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1129 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Math. Probl. Eng. (1)

L. A. Aguirre and C. Letellier, Math. Probl. Eng. 2009, 238960 (2009).

Opt. Eng. (1)

S. Abado, S. Gordeyev, and E. Jumper, Opt. Eng. 49, 064403 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Physica D (2)

K. Judd and A. Mees, Physica D 82, 426 (1995).
[CrossRef]

C. W. Rowley, T. Colonius, and R. W. Murray, Physica D 189, 115 (2004).
[CrossRef]

Prog. Aerosp. Sci. (2)

S. Gordeyev and E. J. Jumper, Prog. Aerosp. Sci. 46, 388 (2010).
[CrossRef]

E. J. Jumper and E. J. Fitzgerald, Prog. Aerosp. Sci. 37, 299 (2001).
[CrossRef]

Other (5)

K. G. Gilbert and L. J. Otten, eds. Aero-Optical Phenomena (American Institute of Aeronautics and Astronautics, 1982).

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University, 1996).

J. Rissanen, Complexity in Statistical Inquiry (World Scientific, 1989).

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

B. Thurow, M. Samimy, and W. Lempert, “Simultaneous MHz rate flow visualization and wavefront sensing for aero-optics,” (American Institute of Aeronautics and Astronautics, 2003).

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Figures (4)

Fig. 1.
Fig. 1.

Example of optical path difference (OPD) obtained in the simulation.

Fig. 2.
Fig. 2.

(a) POD eigenvalues and (b) cumulative energy computed with ensemble A.

Fig. 3.
Fig. 3.

Prediction of POD coefficients in ensemble B (black solid lines) and accurate data (red-dashed line) [(a)  a 1 ( t ) and (b)  a 5 ( t ) ].

Fig. 4.
Fig. 4.

(a) VAF for each a j in the four methods with time interval Δ T = 20 and (b) dependence of VAF for a 1 , a 5 , and a 10 on Δ T with method MDL2.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

R = DD T ,
R ψ = λ ψ .
ε k = i = 1 k λ i / i λ i
OPD ( x , t ) = j a j ( t ) ψ j ( x ) ,
a ( t ) = ( a 1 ( t ) , a 2 ( t ) , , a P ( t ) ) T , a ˙ ( t ) = ( a ˙ 1 ( t ) , a ˙ 2 ( t ) , , a ˙ P ( t ) ) T ,
a ˙ ( t ) = f ( a ( t ) ) + e ,
S = ( N 2 1 ) ln e T e N + ( M + 1 ) ( 1 2 + ln γ ) i = 1 M ln δ i ,
VAF j = max { 0 , ( 1 a j a ^ j 2 2 / a j 2 2 ) } × 100 % ,
a ˙ = La + n + e ,

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