Abstract

The higher-order singular value decomposition (HOSVD) was used to reconstruct the three-dimensional (3D) refractive index field of an aerodynamically heated window. The numerical 3D optical distortion was evaluated for both the reconstructed and the exact refractive index fields of the window, excluding the influence of the elasto-optical effect. The method based on the HOSVD truncation was shown to reduce the refractive index information required to capture the major optical distortion of the window. The refractive index information was reduced by reconstructing the refractive index field of the window using the truncated n-mode singular matrices. The method can also be used to evaluate the optical distortion of the window.

© 2012 Optical Society of America

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References

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    [CrossRef]
  2. J. M. Cicchiello and E. J. Jumper, “Low-order representation of fluid-optic interactions associated with a shear layer,” presented at the 39th AIAA Aerospace Sciences Meeting & Exhibit, Reno, Nevada, 1–11January2001.
  3. J. M. Cicchiello and E. J. Jumper, “Addressing the oblique-viewing aero-optic problem with reduced order methods,” presented at the 32nd AIAA Plasmadynamics and Lasers Conference, Anaheim, California, 1–9June2001.
  4. J. M. Cicchiello, “Low-order representation of dynamic aero-optic distortions,” Ph.D. dissertation (University of Notre Dame, 2001).
  5. L. Wu, J. C. Fang, and Z. H. Yang, “Proper orthogonal decomposition applied in the analysis of simulating aero-optical distortions,” J. Infrared Millim. W. 26, 312–316 (2007).
    [CrossRef]
  6. D. Xu, H. W. Liu, L. Wu, and Y. Q. An, “High-order singular value decomposition applied in aero-optical effects analysis,” Acta Opt. Sin. 30, 3367–3372 (2010).
    [CrossRef]
  7. L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000).
    [CrossRef]
  8. L. De Lathauwer, B. De Moor, and J. Vandewalle, “Dimensionality reduction in higher-order-only ICA,” in Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics (IEEE, 1997), pp. 316–320.
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    [CrossRef]
  14. L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank- (r1, r2,…, rN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (1)

2010 (3)

2007 (3)

L. Wu, J. C. Fang, and Z. H. Yang, “Proper orthogonal decomposition applied in the analysis of simulating aero-optical distortions,” J. Infrared Millim. W. 26, 312–316 (2007).
[CrossRef]

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).
[CrossRef]

Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik 118, 557–560 (2007).
[CrossRef]

2000 (2)

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank- (r1, r2,…, rN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000).
[CrossRef]

An, Y. Q.

D. Xu, H. W. Liu, L. Wu, and Y. Q. An, “High-order singular value decomposition applied in aero-optical effects analysis,” Acta Opt. Sin. 30, 3367–3372 (2010).
[CrossRef]

Bellman, R. E.

R. E. Bellman, Matrix Analysis (McGraw-Hill, 1978).

Cai, W. W.

Cicchiello, J. M.

J. M. Cicchiello and E. J. Jumper, “Low-order representation of fluid-optic interactions associated with a shear layer,” presented at the 39th AIAA Aerospace Sciences Meeting & Exhibit, Reno, Nevada, 1–11January2001.

J. M. Cicchiello and E. J. Jumper, “Addressing the oblique-viewing aero-optic problem with reduced order methods,” presented at the 32nd AIAA Plasmadynamics and Lasers Conference, Anaheim, California, 1–9June2001.

J. M. Cicchiello, “Low-order representation of dynamic aero-optic distortions,” Ph.D. dissertation (University of Notre Dame, 2001).

De Lathauwer, L.

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank- (r1, r2,…, rN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “Dimensionality reduction in higher-order-only ICA,” in Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics (IEEE, 1997), pp. 316–320.

De Moor, B.

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank- (r1, r2,…, rN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “Dimensionality reduction in higher-order-only ICA,” in Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics (IEEE, 1997), pp. 316–320.

Fan, Z. G.

Fang, J. C.

L. Wu, J. C. Fang, and Z. H. Yang, “Proper orthogonal decomposition applied in the analysis of simulating aero-optical distortions,” J. Infrared Millim. W. 26, 312–316 (2007).
[CrossRef]

Harris, D. C.

D. C. Harris, Materials for Infrared Windows and Domes (SPIE, 1999).

Jumper, E. J.

J. M. Cicchiello and E. J. Jumper, “Addressing the oblique-viewing aero-optic problem with reduced order methods,” presented at the 32nd AIAA Plasmadynamics and Lasers Conference, Anaheim, California, 1–9June2001.

J. M. Cicchiello and E. J. Jumper, “Low-order representation of fluid-optic interactions associated with a shear layer,” presented at the 39th AIAA Aerospace Sciences Meeting & Exhibit, Reno, Nevada, 1–11January2001.

Liu, H. W.

D. Xu, H. W. Liu, L. Wu, and Y. Q. An, “High-order singular value decomposition applied in aero-optical effects analysis,” Acta Opt. Sin. 30, 3367–3372 (2010).
[CrossRef]

Liu, W. Y.

W. H. Yu and W. Y. Liu, Crystal Physics (University of Science and Technology of China, 1998).

Ma, L.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford University, 1985).

Vandewalle, J.

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank- (r1, r2,…, rN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “Dimensionality reduction in higher-order-only ICA,” in Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics (IEEE, 1997), pp. 316–320.

Wang, T.

Wang, Z. L.

Wu, L.

D. Xu, H. W. Liu, L. Wu, and Y. Q. An, “High-order singular value decomposition applied in aero-optical effects analysis,” Acta Opt. Sin. 30, 3367–3372 (2010).
[CrossRef]

L. Wu, J. C. Fang, and Z. H. Yang, “Proper orthogonal decomposition applied in the analysis of simulating aero-optical distortions,” J. Infrared Millim. W. 26, 312–316 (2007).
[CrossRef]

Xiao, H. S.

Xu, D.

D. Xu, H. W. Liu, L. Wu, and Y. Q. An, “High-order singular value decomposition applied in aero-optical effects analysis,” Acta Opt. Sin. 30, 3367–3372 (2010).
[CrossRef]

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).
[CrossRef]

Yang, Q. Y.

Yang, Z. H.

L. Wu, J. C. Fang, and Z. H. Yang, “Proper orthogonal decomposition applied in the analysis of simulating aero-optical distortions,” J. Infrared Millim. W. 26, 312–316 (2007).
[CrossRef]

Yu, W. H.

W. H. Yu and W. Y. Liu, Crystal Physics (University of Science and Technology of China, 1998).

Zhang, Y. P.

Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik 118, 557–560 (2007).
[CrossRef]

Zhao, Y.

Acta Opt. Sin. (1)

D. Xu, H. W. Liu, L. Wu, and Y. Q. An, “High-order singular value decomposition applied in aero-optical effects analysis,” Acta Opt. Sin. 30, 3367–3372 (2010).
[CrossRef]

Appl. Opt. (4)

J. Infrared Millim. W. (1)

L. Wu, J. C. Fang, and Z. H. Yang, “Proper orthogonal decomposition applied in the analysis of simulating aero-optical distortions,” J. Infrared Millim. W. 26, 312–316 (2007).
[CrossRef]

Optik (1)

Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik 118, 557–560 (2007).
[CrossRef]

SIAM J. Matrix Anal. Appl. (2)

L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank- (r1, r2,…, rN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000).
[CrossRef]

L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000).
[CrossRef]

Other (8)

L. De Lathauwer, B. De Moor, and J. Vandewalle, “Dimensionality reduction in higher-order-only ICA,” in Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics (IEEE, 1997), pp. 316–320.

R. E. Bellman, Matrix Analysis (McGraw-Hill, 1978).

D. C. Harris, Materials for Infrared Windows and Domes (SPIE, 1999).

J. F. Nye, Physical Properties of Crystals (Oxford University, 1985).

W. H. Yu and W. Y. Liu, Crystal Physics (University of Science and Technology of China, 1998).

J. M. Cicchiello and E. J. Jumper, “Low-order representation of fluid-optic interactions associated with a shear layer,” presented at the 39th AIAA Aerospace Sciences Meeting & Exhibit, Reno, Nevada, 1–11January2001.

J. M. Cicchiello and E. J. Jumper, “Addressing the oblique-viewing aero-optic problem with reduced order methods,” presented at the 32nd AIAA Plasmadynamics and Lasers Conference, Anaheim, California, 1–9June2001.

J. M. Cicchiello, “Low-order representation of dynamic aero-optic distortions,” Ph.D. dissertation (University of Notre Dame, 2001).

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Figures (9)

Fig. 1.
Fig. 1.

Unfolding of the (I1×I2×I3)-tensor A to the (I1×I2I3)-matrix A(1), (I2×I3I1)-matrix A(2), and (I3×I1I2)-matrix A(3) (I1=I2=I3=4) [7].

Fig. 2.
Fig. 2.

Visualization of the HOSVD for the third-order tensor A [7].

Fig. 3.
Fig. 3.

Simplified heat flux distribution on the outside surface of the window obtained from the hypersonic arc wind tunnel experiment [1].

Fig. 4.
Fig. 4.

Temperature profiles mapped onto the path between the inside and the outside surface centers of the window at different working times: (a) 5, (b) 10, and (c) 15 s.

Fig. 5.
Fig. 5.

Refractive index grid model of the window [1].

Fig. 6.
Fig. 6.

Color scale maps of (a) the reconstructed refractive index field of the window obtained from the HOSVD truncation and (b) the exact refractive index field of the window at 15 s.

Fig. 7.
Fig. 7.

Definitions of the azimuth and elevation incident angles. The curved arrows indicate positive angles [1].

Fig. 8.
Fig. 8.

Ray tracing on the deformed surface of the window. A, B, C, and D are nodes of the deformed surface grid encircling intersection point E of the incident ray and the deformed surface. A, B, C, and D are also counterparts of the nondeformed surface grid encircling intersection point E [1].

Fig. 9.
Fig. 9.

Wave aberration results of the window for (a) the reconstructed refractive index field obtained from the HOSVD truncation and (b) the exact refractive index field at 15 s.

Tables (3)

Tables Icon

Table 1. Main Physical Properties of the Standard Zinc Sulfide Crystal Near 300 K

Tables Icon

Table 2. Refractive Index Variations of the Point with Maximum Temperature Variation and the Point with Maximum Equivalent Von Mises Strain Variation at 15 sa [1]

Tables Icon

Table 3. First Three-Order n-Mode (1n3) Singular Values Obtained from the Matrix SVDs of the Mode-n Matrix Unfoldings B(1), B(2), and B(3) of the Third-order Tensor Ba

Equations (26)

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

A=UVZ;
(A×Un)i1i2jniN=inai1i2iniNujnin,
(i11)I2I3In1In+1IN+(i21)I3In1In+1IN+(in11)In+1In+2IN+(in+11)In+2In+3IN++iN.
B(n)=UA(n).
A,B=i1i2iN(A)i1i2iN(B)i1i2iN.
A=A,A,
A=S×U(1)1×U(2)2×U(n)n×U(N)N,
Sin=α,Sin=β=0whenαβ.
Sin=1Sin=2Sin=In0,
A(n)=U(n)S(n)(U(n+1)U(n+2)U(N)U(1)U(2)U(n1))T,
FG=(fi1i2G)1i1I1;1i2I2.
A(n)=U(n)Σ(n)V(n)T.
S(n)=(n)V(n)T(U(n+1)U(n+2)U(N)U(1)U(2)U(n1)).
S=A×U(1)T1×U(2)T2×U(n)Tn×U(N)TN.
S(n)=U(n)TA(n)(U(n+1)U(n+2)U(N)U(1)U(2)U(n1)).
F=i=1rλiUiViT,
F^=i=1kλiUiViT.
AA^2i1=I1+1r1σi1(1)2+i2=I2+1r2σi2(2)2++in=In+1rnσin(2)2++iN=IN+1rNσiN(N)2,
S^=A×W(1)T1×W(2)T2×W(3)T3×W(n)Tn×W(N)TN.
A^=S^×W(1)1×W(2)2×W(3)3×W(n)n×W(N)N.
e=i1=181i2=181i3=19|(B^)i1i2i3(B)i1i2i3|i1=181i2=181i3=19|(B)i1i2i3|.
OPLi=t0t0+Δt(nilit+linit)dt+ε0ε0+Δε(niliε+liniε)dε,
OPL=iOPLi.
gk(x,y)=2πλ(OPLkOPL0),
OPL0=1NrkOPLk,
G(x,y)=kgk(x,y)=k2πλ(OPLkOPL0).

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