Abstract

In the companion paper, [Appl. Opt. 46, 5853 (2007)] a highly accurate white light interference model was developed from just a few key parameters characterized in terms of various moments of the source and instrument transmission function. We develop and implement the end-to-endprocess of calibrating these moment parameters together with the differential dispersion of the instrument and applying them to the algorithms developed in the companion paper. The calibrationprocedure developed herein is based on first obtaining the standard monochromatic parameters at the pixel level: wavenumber, phase, intensity, and visibility parameters via a nonlinear least-squaresprocedure that exploits the structure of the model. The pixel level parameters are then combined to obtain the required “global” moment and dispersion parameters. Theprocess is applied to both simulated scenarios of astrometric observations and to data from the microarcsecond metrology testbed (MAM), an interferometer testbed that has played a prominent role in the development of this technology.

© 2007 Optical Society of America

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References

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  1. J. Marr, "Space Interferometry Mission(SIM) overview and current status," Proc. SPIE 4852, 1-15 (2003).
    [CrossRef]
  2. R. A. Laskin, "SIM technology development overview-light at the end of the tunnel," Proc. SPIE 4852, 16-32 (2003).
    [CrossRef]
  3. R. Goullioud and T. J. Shen, "MAM testbed detailed description and alignment," in Proceedings of IEEE Conference on Aerospace (IEEE, 2004), pp. 2191-2200.
  4. R. Goullioud and T. J. Shen, "SIM astrometric demonstration at 24 picometers on the MAM testbed," in Proceedings of IEEE Conference on Aerospace (IEEE, 2004), pp. 2179-2190.
  5. M. Milman, C. Zhai, and M. Regehr, "White-light interferometry using a channeled spectrum. 1. General models and fringe estimation algorithms," Appl. Opt. 46, 5853-5865 (2007).
    [CrossRef] [PubMed]
  6. X. Pan, F. Zhao, and M. Shao, "MAM testbed data analysis: cyclic averaging," Proc. SPIE 4852, 612-622 (2003).
    [CrossRef]
  7. R. C. M. Learner, A. P. Thorne, I. Wynee-Jones, J. W. Brault, and M. C. Abrams, "Phase correction of emission line Fourier transform spectra," J. Opt. Soc. Am. A 12, 2165-2171 (1995).
    [CrossRef]
  8. K. Rahmelow and W. Hubner, "Phase correction in Fourier transform spectroscopy," Appl. Opt. 36, 6678-6686 (1997).
    [CrossRef]
  9. C. D. Barnet, J. M. Blaisdell, and J. Susskind, "Practical method for rapid and accurate computation of interferometric spectra for remote sensing application," IEEE Trans. Geosci. Remote Sens 38, 169-183 (2000).
    [CrossRef]
  10. A. Ben-David and A. Ifarraguerri, "Computation of a spectrum from a single-beam Fourier-transform infrared interferogram," Appl. Opt. 41, 1181-1189 (2002).
    [CrossRef] [PubMed]
  11. M. Regehr and M. Milman, "Analysis and numerical modeling of error sources in SIM star light phase detection," in Proceedings of IEEE Conference on Aerospace (IEEE, 2005), pp. 1-8.
  12. M. Milman, "Accurately computing the optical pathlength difference for a Michelson interferometer with minimal knowledge of the source spectrum," J. Opt. Soc. Am. A 22, 2774-2785 (2005).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989).
  14. P. R. Lawson, "Phase and group delay estimation," in Principles of Long Baseline Stellar Interferometry, P. R. Lawson, ed., Course Notes from the 1999 Michelson Interferometry Summer School (Jet Propulsion Laboratory, Calif., 1999).
  15. D. G. Luenberger, Optimization By Vector Space Methods (Wiley, 1969).
  16. G. Golub and V. Pereyra, "The differentiation of pseudoinverse and nonlinear least squares problems whose variable separate," SIAM (Soc. Ind. Appl. Math) J. Numer. Anal. 10, 413-432 (1973).
  17. R. Fletcher, Practical Methods for Optimization (Wiley, 1987).
  18. M. Milman and S. G. Turyshev, "Observational model for miscroarcsecond astrometry with the Space Interferometer Mission," Opt. Eng. 42, 1873-1883 (2003).
    [CrossRef]
  19. A. J. Pickles, "A stellar spectral flux library: 1150-25000 Å," Publ. Astron. Soc. Pac. 110, 863-878 (1998).
    [CrossRef]

2007 (1)

2005 (1)

2003 (4)

X. Pan, F. Zhao, and M. Shao, "MAM testbed data analysis: cyclic averaging," Proc. SPIE 4852, 612-622 (2003).
[CrossRef]

J. Marr, "Space Interferometry Mission(SIM) overview and current status," Proc. SPIE 4852, 1-15 (2003).
[CrossRef]

R. A. Laskin, "SIM technology development overview-light at the end of the tunnel," Proc. SPIE 4852, 16-32 (2003).
[CrossRef]

M. Milman and S. G. Turyshev, "Observational model for miscroarcsecond astrometry with the Space Interferometer Mission," Opt. Eng. 42, 1873-1883 (2003).
[CrossRef]

2002 (1)

2000 (1)

C. D. Barnet, J. M. Blaisdell, and J. Susskind, "Practical method for rapid and accurate computation of interferometric spectra for remote sensing application," IEEE Trans. Geosci. Remote Sens 38, 169-183 (2000).
[CrossRef]

1998 (1)

A. J. Pickles, "A stellar spectral flux library: 1150-25000 Å," Publ. Astron. Soc. Pac. 110, 863-878 (1998).
[CrossRef]

1997 (1)

1995 (1)

1973 (1)

G. Golub and V. Pereyra, "The differentiation of pseudoinverse and nonlinear least squares problems whose variable separate," SIAM (Soc. Ind. Appl. Math) J. Numer. Anal. 10, 413-432 (1973).

Appl. Opt. (3)

IEEE Trans. Geosci. Remote Sens (1)

C. D. Barnet, J. M. Blaisdell, and J. Susskind, "Practical method for rapid and accurate computation of interferometric spectra for remote sensing application," IEEE Trans. Geosci. Remote Sens 38, 169-183 (2000).
[CrossRef]

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

Opt. Eng. (1)

M. Milman and S. G. Turyshev, "Observational model for miscroarcsecond astrometry with the Space Interferometer Mission," Opt. Eng. 42, 1873-1883 (2003).
[CrossRef]

Proc. SPIE (3)

X. Pan, F. Zhao, and M. Shao, "MAM testbed data analysis: cyclic averaging," Proc. SPIE 4852, 612-622 (2003).
[CrossRef]

J. Marr, "Space Interferometry Mission(SIM) overview and current status," Proc. SPIE 4852, 1-15 (2003).
[CrossRef]

R. A. Laskin, "SIM technology development overview-light at the end of the tunnel," Proc. SPIE 4852, 16-32 (2003).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

A. J. Pickles, "A stellar spectral flux library: 1150-25000 Å," Publ. Astron. Soc. Pac. 110, 863-878 (1998).
[CrossRef]

SIAM (1)

G. Golub and V. Pereyra, "The differentiation of pseudoinverse and nonlinear least squares problems whose variable separate," SIAM (Soc. Ind. Appl. Math) J. Numer. Anal. 10, 413-432 (1973).

Other (7)

R. Fletcher, Practical Methods for Optimization (Wiley, 1987).

M. Regehr and M. Milman, "Analysis and numerical modeling of error sources in SIM star light phase detection," in Proceedings of IEEE Conference on Aerospace (IEEE, 2005), pp. 1-8.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989).

P. R. Lawson, "Phase and group delay estimation," in Principles of Long Baseline Stellar Interferometry, P. R. Lawson, ed., Course Notes from the 1999 Michelson Interferometry Summer School (Jet Propulsion Laboratory, Calif., 1999).

D. G. Luenberger, Optimization By Vector Space Methods (Wiley, 1969).

R. Goullioud and T. J. Shen, "MAM testbed detailed description and alignment," in Proceedings of IEEE Conference on Aerospace (IEEE, 2004), pp. 2191-2200.

R. Goullioud and T. J. Shen, "SIM astrometric demonstration at 24 picometers on the MAM testbed," in Proceedings of IEEE Conference on Aerospace (IEEE, 2004), pp. 2179-2190.

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Figures (12)

Fig. 1
Fig. 1

Flow chart of the end-to-end calibration and estimationprocedure.

Fig. 2
Fig. 2

(Color online) Schematic of a stellar interferometer.

Fig. 3
Fig. 3

(Color online) Detection windows for pixels 39, 40, and 41 for the nominal 80 pixel SIM CCD over passband 400 1000 nm .

Fig. 4
Fig. 4

Cost functionals for stroke lengths of 2 and 10 μ m .

Fig. 5
Fig. 5

(Color online) Wavelength calibration error using different stroke lengths; the dashed curves show the standard deviation that is inversely proportional to the stroke length using the longest stroke length standard deviation as the scale.

Fig. 6
Fig. 6

(Color online) Calibration results with and without the envelope projection technique.

Fig. 7
Fig. 7

(Color online) OPD error from an end-to-end simulation for the guide interferometer observation scenario, calibration is performed with noise on.

Fig. 8
Fig. 8

(Color online) OPD error from an end-to-end simulation for the science observation scenario.

Fig. 9
Fig. 9

(Color online) Wavelength calibration error using different stroke lengths, the longest stroke contains N = 2201 measurements.

Fig. 10
Fig. 10

(Color online) MAM pixel calibration result using and without using the envelope projection technique.

Fig. 11
Fig. 11

(Color online) OPD errors using different algorithms for OPD range ± 80 nm .

Fig. 12
Fig. 12

(Color online) OPD chops for average delay over 30 s.

Tables (1)

Tables Icon

Table 1 Channel Effective Wavelengths and Second Moments for Different Star Spectra

Equations (119)

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Y ( x ) = k k + I ( k ) { 1 + V ( k ) cos ( k x + ψ ( k ) ) } d k ,
I ¯ = k k + I ( k ) d k ,     V ¯ = 1 I ¯ k k + I ( k ) V ( k ) d k ,
p ( k ) = I ( k ) V ( k ) I ¯ V ¯ ,
μ j = k k + ( k k ¯ ) j p ( k ) d k , j = 0 , 1 , 2 ,   …   ,
k ¯ = k k + k p ( k ) d k .
ψ ( k ) = a 0 + a 1 k + r ( k ) .
Y ( x ) = I ¯ { 1 + V ¯ [ 1 μ 2 2 u 2 ] cos ( k ¯ u + a 0 ) } ,
Y ( x ) = I ¯ { 1 + V ¯ [ 1 μ 2 2 u 2 + μ 4 24 u 4 ] × cos ( k ¯ u + a 0 1 2 δ u 2 μ 3 6 u 3 ) } ,
δ = k k + ( k k ¯ ) 2 r ( k ) p ( k ) d k .
Y j ( x ) = 0 I j ( k ) { 1 + V j ( k ) cos ( k x + ψ ( k ) ) } d k ,
y = A ( k ¯ ) X + η .
X = [ I ¯ , I ¯ V ¯ cos ( k ¯ d + ψ ¯ ) , I ¯ V ¯ sin ( k ¯ d + ψ ¯ ) ] T ,
A ( k ) = [ 1 cos ( k u 1 ) sin ( k u 1 ) 1 cos ( k u N ) sin ( k u N ) ] .
min k , X y A ( k ) X 2 .
X 0 = A ( k 0 ) y ,
min k | y A ( k ) A ( k ) y | 2 ,
max k | Q T ( k ) y | 2 .
ϕ ¯   j = k ¯ j u j + a 0 ,
a 1 = 1 μ 2 ( k k ¯ j ) ψ ( k ) p j ( k ) d k ,
a 0 = ψ ( k ) p j ( k ) d k a 1 k ¯ j ,
ϕ ¯   j = k ¯ j d + ψ ¯ j ;     ψ ¯ j = ψ ( k ) p j ( k ) d k .
ψ ( k ) p ( k ) d k = ψ ( k ¯ ) + 1 2 ψ ( k ) ( k k ¯ ) 2 p ( k ) d k ,
| ψ ( k ) p ( k ) d k ψ ( k ¯ ) | 1 2 | ψ ( k ) ( k k ¯ ) 2 p ( k ) | d k 1 2 ψ μ 2 .
k ¯ = j I j ( k ) V j ( k ) k d k j I j ( k ) V j ( k ) d k = j k ¯ j I j ( k ) V j ( k ) d k j I j ( k ) V j ( k ) d k = j k ¯ j I ¯ j V ¯ j I ¯ j V ¯ j ,
μ 2 = j ( k k ¯ ) 2 I j ( k ) V j ( k ) d k j I j ( k ) V j ( k ) d k .
         ( k k ¯ ) 2 I j ( k ) V j ( k ) d k = k 2 I j ( k ) V j ( k ) d k 2 k ¯ k ¯ j I j ( k ) V j ( k ) d k + k ¯ j 2 I j ( k ) V j ( k ) d k ,
k 2 I j ( k ) V j ( k ) d k = ( k k ¯ j ) 2 I j ( k ) V j ( k ) d k + 2 k k ¯ j I j ( k ) V j ( k ) d k k ¯ j 2 I j ( k ) V j ( k ) d k = μ 2 j I ¯ j V ¯ j + k ¯ j 2 I ¯ j V ¯ j ,
μ 2 = j ( k ¯ k ¯ j ) 2 I ¯ j V ¯ j j I ¯ j V ¯ j + j μ 2 j I ¯ j V ¯ j j I ¯ j V ¯ j .
μ 3 = j ( k ¯ k ¯ j ) 3 I ¯ j V ¯ j j I ¯ j V ¯ j + O ( | μ 3 j | + max j μ 2 j | k ¯ k ¯ j | ) ,
μ 4 = j ( k ¯ k ¯ j ) 4 I ¯ j V ¯ j j I ¯ j V ¯ j + O ( | μ 4 j | + max j | μ 3 j | | k ¯ k ¯ j | + max j μ 2 j | k ¯ k ¯ j | 2 ) ,
a 0 = ψ ( k ) p ( k ) d k k ¯ a 1 = j ψ ( k ) I j ( k ) V j ( k ) d k j I j ( k ) V j ( k ) d k k ¯ a 1
= j I ¯ j V ¯ j ψ ¯ j j I ¯ j V ¯ j k ¯ a 1 ,
a 1 = 1 μ 2 j I ¯ j V ¯ j ( k ¯ k ¯ j ) ψ ¯ j j I ¯ j V ¯ j ,
δ = ( k k ¯ ) 2 r ( k ) p ( k ) d k j I ¯ j V ¯ j ( ψ ¯ j a 0 a 1 k ¯ j ) ( k ¯ j k ¯ ) 2 i I ¯ j V ¯ j .
δ y i = δ I + δ ( I V ) cos ( k ¯ u i + ϕ ¯ ) I ¯ V ¯ δ k sin ( k ¯ u i + ϕ ¯ ) u i I ¯ V ¯ sin ( k ¯ u i + ϕ ¯ ) δ ϕ .
D [ 1 cos ( k ¯ u 1 + ϕ ¯ ) sin ( k ¯ u 1 + ϕ ¯ ) sin ( k ¯ u 1 + ϕ ¯ ) u 1 1 cos ( k ¯ u 2 + ϕ ¯ ) sin ( k ¯ u 2 + ϕ ¯ ) sin ( k ¯ u 2 + ϕ ¯ ) u 2 1 cos ( k ¯ u N + ϕ ¯ ) sin ( k ¯ u N + ϕ ¯ ) sin ( k ¯ u N + ϕ ¯ ) u N ] ,
δ X [ δ I , δ ( I V ) , I ¯ V ¯ δ ϕ , I ¯ V ¯ δ k ] T ,
δ y = D δ X .
D = V Λ ,
Λ = diag ( N , N / 2 , N / 2 , N 3 Δ u 2 / 24 ) ,
D = ( D T D ) 1 D T Λ 1 V T .
δ X = D δ y err .
E ( δ X δ X T ) = D Cov ( δ y err ) ( D ) T Λ 1 V T Cov ( δ y err ) V Λ 1 ,
E ( δ X i 2 ) = Λ i i 2 v i T Cov ( δ y err ) v i ,
Cov ( δ y err ) = σ read 2 I N × N + I ¯ diag ( 1 + V ¯ cos ( k ¯ u 1 + ϕ ¯ ) ,   …   ,   1 + V cos ( k ¯ u N + ϕ ¯ ) ) ,
v i T Cov ( δ y err ) v i = σ read 2 + I ¯ + I ¯ V ¯ v i T diag ( cos ( k ¯ u 1 + ϕ ¯ ) ,   …   , cos ( k ¯ u N + ϕ ¯ ) ) v i .
v i T diag ( cos ( k ¯ u 1 + ϕ ¯ ) ,   …   ,   cos ( k ¯ u N + ϕ ¯ ) ) v i = O ( 1 / N ) .
E ( δ k 2 ) = 1 [ I ¯ V ¯ ] 2 [ 24 N L 2 ] { σ read 2 + I ¯ } = 24 σ read 2 I tot 2 V ¯ 2 L Δ u + 24 I tot V ¯ 2 L 2 ,
E ( δ ϕ 2 ) = 2 N σ read 2 ( I tot V ¯ ) 2 + 2 I tot V ¯ 2 ,
E ( δ ( I V ) 2 ) = 2 σ read 2 N + 2 I tot N 2 ,
E ( δ I 2 ) = σ read 2 N + I tot N 2 .
Y ( x ) I ¯ { 1 + V ¯ [ 1 1 2 μ 2 ( x + a 1 ) 2 ] cos ( k ¯ x + ψ ¯ ) }
min k , X | y [ A ( k ) X + δ y env ] | 2 ,
δ y i env = 1 2 I ¯ V ¯ μ 2 ( u i + d + a 1 ) 2 cos ( k ¯ u i + k ¯ d + ψ ¯ ) ,
i = 1 , 2 ,   …   ,   N ,
min k , X | P y P A ( k ) X | 2 ,
e 1 u i cos ( k ¯ u i + ϕ ¯ ) , e 2 u i 2 cos ( k ¯ u i + ϕ ¯ ) ,
i = 1 , 2 ,   …   ,   N .
e 3 u i 2 sin ( k ¯ u i + ϕ ¯ ) ,
Q = [ e 1 e 2 e 3 ] ,
P = I Q Q T .
P δ y = P D δ X ,
δ X = ( P D ) P δ y .
C l ( k , Δ u ) i = 1 N u i l cos ( k u i ) ,
S l ( k , Δ u ) i = 1 N u i l sin ( k u i ) .
C l = S l + 1 = 0 for odd l .
C 0 ( k , Δ u ) = i = 1 N cos ( k u i ) = i = 1 N sin [ ( i N / 2 ) k Δ u ] sin [ ( i N / 2 1 ) k Δ u ] 2 sin ( k Δ u / 2 ) = sin ( N k Δ u / 2 ) sin ( k Δ u / 2 ) .
S 2 l + 1 ( k , Δ u ) = C 2 l ( k , Δ u ) k ,
C 2 l + 2 ( k , Δ u ) = S 2 l + 1 ( k , Δ u ) k ,
S 1 ( k , Δ u ) = C 0 k = Δ u [ sin ( N k Δ u / 2 ) cos ( k Δ u / 2 ) 2 sin 2 ( k Δ u / 2 ) N cos ( N k Δ u / 2 ) 2 sin ( k Δ u / 2 ) ] ,
C 2 ( k , Δ u ) = S 1 ( k , Δ u ) k = Δ u 2 [ ( N 2 + 1 ) sin ( N k Δ u / 2 ) 4 sin ( k Δ u / 2 ) + N cos ( N k Δ u / 2 ) cos ( k Δ u / 2 ) 2 sin 2 ( k Δ u / 2 ) sin ( N k Δ u / 2 ) 2 sin 3 ( k Δ u / 2 ) ] .
C l ( k , Δ u ) N l Δ u l for even l ,
S l ( k , Δ u ) N l Δ u l for odd l .
C 2 l ( 0 , Δ u ) = ( 1 ) l 2 l k 2 l C 2 l ( k , Δ u ) | k = 0 .
1 ( 2 l + 1 ) ! ( 1 ) l N 2 l + 1 ( k Δ u 2 ) 2 l .
C 2 l ( 0 , Δ u ) ( 2 l ) ! 1 ( 2 l + 1 ) ! ( 1 ) l N 2 l + 1 ( Δ u 2 ) 2 l = N 2 l + 1 Δ u 2 l ( 2 l + 1 ) 2 2 l .
C 2 l ( 0 , Δ u ) = i = 1 N u i 2 l ,
i = 1 N u i 2 l N 2 l + 1 Δ u 2 l ( 2 l + 1 ) 2 2 l .
γ ( k ) = | P ( k ) y | 2 = y T P ( k ) 2 y ,
γ ( k ) = y T [ P ( k ) P ( k ) + P ( k ) P ( k ) ] y = 2 y T P ( k ) P ( k ) y ,
γ ( k ) = 2 y T [ P ( k ) P ( k ) + 2 P ( k ) P ( k ) ] y .
γ ( k 0 ) = 2 y T P ( k 0 ) P ( k 0 ) y .
P ( k 0 ) y = [ A ( k 0 ) A ( k 0 ) A ( k 0 ) + A ( k 0 ) d d k ( A ( k ) ) | k = k 0 A ( k 0 ) ] X 0 = ( 1 A ( k 0 ) A ( k 0 ) ) A ( k 0 ) X 0 = ( 1 P ( k 0 ) ) A ( k 0 ) = P ( k 0 ) A ( k 0 ) X 0 ,
A ( k 0 ) A ( k 0 ) = I 3 × 3 ,
d d k ( A ( k ) ) | k = k 0 A ( k 0 ) + A ( k 0 ) A ( k 0 ) = 0.
γ ( k 0 ) = | P ( k 0 ) z | 2 ,
z = A ( k 0 ) X 0 = I ¯ V ¯ ( u 1 sin ( k 0 u 1 + ϕ ¯ ) ,   …   , u N sin ( k 0 u N + ϕ ¯ ) ) .
| z | 2 = O ( I ¯ 2 V ¯ 2 N 3 Δ u 2 ) = O ( I tot 2 V ¯ 2 N Δ u 2 ) ,
v 1 i = 1 N , v 2 i = 2 N cos ( k ¯ u i + ϕ ¯ ) ,
v 3 i = 2 N sin ( k ¯ u i + ϕ ¯ ) ,
v 4 i = 24 N 3 Δ u 2 u i sin ( k ¯ u i + ϕ ¯ )
D = V Λ ,
V [ v 1 , v 2 , v 3 , v 4 ] ,
Λ = [ N 0 0 0 0 N 2 0 0 0 0 N 2 0 0 0 0 N 3 Δ u 2 24 ] .
i = 1 N ( v 1 i ) 2 = 1 ,
i = 1 N ( v 2 i ) 2 = 1 N i = 1 N [ 1 + cos ( 2 k ¯ u i + 2 ϕ ¯ ) ] = 1 + 1 N i = 1 N cos ( 2 k ¯ u i ) cos ( 2 ϕ ¯ ) = 1 + O ( 1 N ) ,
i = 1 N ( v 3 i ) 2 = 1 N i = 1 N [ 1 cos ( 2 k ¯ u i + 2 ϕ ¯ ) ] = 1 1 N i = 1 N cos ( 2 k ¯ u i ) cos ( 2 ϕ ¯ ) = 1 + O ( 1 N ) ,
i = 1 N ( v 4 i ) 2 = 12 N 3 Δ u 2 i = 1 N u i 2 [ 1 cos ( 2 k ¯ u i + 2 ϕ ¯ ) ] = 12 N 3 Δ u 2 i = 1 N u i 2 [ 1 cos ( 2 k ¯ u i ) cos ( 2 ϕ ¯ ) ] = 1 + O ( 1 N ) ,
i = 1 N v 1 i v 2 i = 2 N i = 1 N cos ( k ¯ u i + ϕ ¯ ) = 2 N i = 1 N cos ( k ¯ u i ) cos ( ϕ ¯ ) = O ( 1 N ) ,
i = 1 N v 1 i v 3 i = 2 N i = 1 N sin ( k ¯ u i + ϕ ¯ ) = 2 N i = 1 N cos ( k ¯ u i ) sin ( ϕ ¯ ) = O ( 1 N ) ,
i = 1 N v 2 i v 3 i = 1 N i = 1 N sin ( 2 k ¯ u i + 2 ϕ ¯ ) = O ( 1 N ) ,
i = 1 N v 1 i v 4 i = 24 N 2 Δ u i = 1 N u i sin ( k ¯ u i + ϕ ¯ ) = O ( 1 N ) ,
i = 1 N v 2 i v 4 i = 12 N 2 Δ u i = 1 N u i sin ( 2 k ¯ u i + 2 ϕ ¯ ) = O ( 1 N ) ,
i = 1 N v 3 i v 4 i = 12 N 2 Δ u i = 1 N u i [ 1 cos ( 2 k ¯ u i + 2 ϕ ¯ ) ] = 12 N 2 Δ u i = 1 N u i sin ( 2 k ¯ u i ) sin ( 2 ϕ ¯ ) = O ( 1 N ) .
D T D Λ 2 = [ N 0 0 0 0 N 2 0 0 0 0 N 2 0 0 0 0 N 3 Δ u 2 24 ] .
D Λ 1 V T .
δ X i env = ( D δ y env ) i Λ i i 1 v i T δ y env ,   i = 1 , 2 , 3 , 4.
δ X env μ 2 I ¯ V ¯ [ 1 2 N i u i 2 cos ( k ¯ u i + ϕ ¯ ) 1 N i u i 2 cos 2 ( k ¯ u i + ϕ ¯ ) 1 2 N i u i 2 cos ( k ¯ u i + ϕ ¯ ) sin ( k ¯ u i + ϕ ¯ ) 12 N 3 Δ u 2 i u i 3 cos ( k ¯ u i + ϕ ¯ ) sin ( k ¯ u i + ϕ ¯ ) ] μ 2 I ¯ V ¯ [ O ( N ) Δ u 2 O ( N 2 ) Δ u 2 O ( N ) Δ u 2 O ( 1 ) Δ u ] ,
δ I env = O ( N μ 2 Δ u 2 I ¯ V ¯ ) , δ ( I V ) env = O ( N 2 μ 2 Δ u 2 I ¯ V ¯ ) ,
δ ϕ env = O ( N μ 2 Δ u 2 ) , δ k env = O ( μ 2 Δ u ) .
e ˜ 1 = 24 N 3 Δ u 2 e 1 , e ˜ 2 = 160 N 5 Δ u 4 e 2 , e ˜ 3 = 160 N 5 Δ u 4 e 3 .
D T P D = Λ V T P V Λ Λ 2 i = 1 3 Λ V T e ˜ i e ˜ i T V Λ .
V T e ˜ 1 0 , V T e ˜ 2 ( 0 , 5 / 3 , 0 , 0 ) T , V T e ˜ 3 ( 0 , 0 , 5 / 3 , 0 ) T .
D T P D Λ [ 1 0 0 0 0 4 / 9 0 0 0 0 4 / 9 0 0 0 0 1 ] Λ = [ N 0 0 0 0 2 N 9 0 0 0 0 2 N 9 0 0 0 0 N 3 Δ u 2 24 ] .
E i v i T diag ( cos ( k ¯ u 1 + ϕ ¯ ) ,   …   ,   cos ( k ¯ u N + ϕ ¯ ) ) v i = O ( 1 / N ) ,
E 1 = 1 N i = 1 N cos ( k ¯ u i + ϕ ¯ ) = 1 N i = 1 N cos ( k ¯ u i ) cos ϕ ¯ ) = O ( 1 / N ) ,
E 2 = 2 N i = 1 N cos ( k ¯ u i + ϕ ¯ ) cos 2 ( k ¯ u i + ϕ ¯ ) = 1 N i = 1 N [ 1 + cos ( 2 k ¯ u i + 2 ϕ ¯ ) ] cos ( k ¯ u i + ϕ ¯ ) = 1 2 N i = 1 N [ 3 cos ( k ¯ u i + ϕ ¯ ) + cos ( 3 k ¯ u i + 3 ϕ ¯ ) ] = O ( 1 / N ) ,
E 3 = 2 N i = 1 N cos ( k ¯ u i + ϕ ¯ ) sin 2 ( k ¯ u i + ϕ ¯ ) = E 1 E 2 = O ( 1 / N ) ,
E 4 = 24 N 3 Δ u 2 i = 1 N u i 2 cos ( k ¯ u i + ϕ ¯ ) sin 2 ( k ¯ u i + ϕ ¯ ) = 6 N 3 Δ u 2 i = 1 N u i 2 i = 1 N u i 2 [ cos ( k ¯ u i + ϕ ¯ ) cos ( 3 k ¯ u i + 3 ϕ ¯ ) ] = 6 N 3 Δ u 2 i = 1 N u i 2 [ cos ( k ¯ u i ) cos ϕ ¯ cos ( 3 k ¯ u i ) cos ( 3 ϕ ¯ ) ] = O ( 1 / N )

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