Abstract

We introduce a framework for modeling, analysis, and simulation of aero-optics wavefront aberrations that is based on spatial–temporal covariance matrices extracted from wavefront sensor measurements. Within this framework, we present a quasi-homogeneous structure function to analyze nonhomogeneous, mildly anisotropic spatial random processes, and we use this structure function to show that phase aberrations arising in aero-optics are, for an important range of operating parameters, locally Kolmogorov. This strongly suggests that the d5/3 power law for adaptive optics (AO) deformable mirror fitting error, where d denotes actuator separation, holds for certain important aero-optics scenarios. This framework also allows us to compute bounds on AO servo lag error and predictive control error. In addition, it provides us with the means to accurately simulate AO systems for the mitigation of aero-effects, and it may provide insight into underlying physical processes associated with turbulent flow. The techniques introduced here are demonstrated using data obtained from the Airborne Aero-Optics Laboratory.

© 2014 Optical Society of America

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References

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  1. D. L. Fried, “Statistics of a geometrical representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  2. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
  3. R. J. Hudgin, “Wavefront compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393–395 (1977).
    [CrossRef]
  4. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67, 390–393 (1977).
    [CrossRef]
  5. J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting, AIAA 2008-1076 (AIAA, 2008).
    [CrossRef]
  6. G. A. Tyler, “Analysis of boundary layer data,” (the Optical Sciences Company, 2009).
  7. L. Poyneer, M. van Dam, and J.-P. Veran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry,” J. Opt. Soc. Am. A 26, 833–846 (2009).
    [CrossRef]
  8. E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
    [CrossRef]
  9. D. J. Goorskey, R. Drye, and M. R. Whiteley, “Spatial and temporal characterization of AAOL flight test data,” Proc. SPIE 8395, 839509 (2012).
  10. S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
    [CrossRef]
  11. N. De Lucca, S. Gordeyev, and E. Jumper, “In-flight aero-optics of turrets,” Opt. Eng. 52, 071405 (2013).
    [CrossRef]
  12. T. J. Brennan and D. J. Wittich, “Statistical analysis of Airborne Aero-Optical Laboratory optical wavefront measurements,” Opt. Eng. 52, 071416 (2013).
    [CrossRef]
  13. O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems (SIAM, 2001).
  14. T. J. Brennan, P. H. Roberts, and D. C. Zimmerman, “WaveProp user’s guide, a wave optics simulation system,” (the Optical Sciences Company, 2007).
  15. T. J. Brennan and C. R. Vogel, “Spatial-temporal control applied to atmospheric adaptive optics,” (the Optical Sciences Company, 2013).
  16. C. M. Hardy, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
    [CrossRef]

2013 (3)

N. De Lucca, S. Gordeyev, and E. Jumper, “In-flight aero-optics of turrets,” Opt. Eng. 52, 071405 (2013).
[CrossRef]

T. J. Brennan and D. J. Wittich, “Statistical analysis of Airborne Aero-Optical Laboratory optical wavefront measurements,” Opt. Eng. 52, 071416 (2013).
[CrossRef]

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
[CrossRef]

2012 (1)

D. J. Goorskey, R. Drye, and M. R. Whiteley, “Spatial and temporal characterization of AAOL flight test data,” Proc. SPIE 8395, 839509 (2012).

2010 (1)

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[CrossRef]

2009 (1)

1999 (1)

1977 (2)

1965 (1)

Axelsson, O.

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems (SIAM, 2001).

Barker, V. A.

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems (SIAM, 2001).

Brennan, T. J.

T. J. Brennan and D. J. Wittich, “Statistical analysis of Airborne Aero-Optical Laboratory optical wavefront measurements,” Opt. Eng. 52, 071416 (2013).
[CrossRef]

T. J. Brennan and C. R. Vogel, “Spatial-temporal control applied to atmospheric adaptive optics,” (the Optical Sciences Company, 2013).

T. J. Brennan, P. H. Roberts, and D. C. Zimmerman, “WaveProp user’s guide, a wave optics simulation system,” (the Optical Sciences Company, 2007).

Cavalieri, D.

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
[CrossRef]

De Lucca, N.

N. De Lucca, S. Gordeyev, and E. Jumper, “In-flight aero-optics of turrets,” Opt. Eng. 52, 071405 (2013).
[CrossRef]

Drye, R.

D. J. Goorskey, R. Drye, and M. R. Whiteley, “Spatial and temporal characterization of AAOL flight test data,” Proc. SPIE 8395, 839509 (2012).

Fried, D. L.

Goorskey, D. J.

D. J. Goorskey, R. Drye, and M. R. Whiteley, “Spatial and temporal characterization of AAOL flight test data,” Proc. SPIE 8395, 839509 (2012).

Gordeyev, S.

N. De Lucca, S. Gordeyev, and E. Jumper, “In-flight aero-optics of turrets,” Opt. Eng. 52, 071405 (2013).
[CrossRef]

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
[CrossRef]

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[CrossRef]

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting, AIAA 2008-1076 (AIAA, 2008).
[CrossRef]

Greenwood, D. P.

Hardy, C. M.

Hudgin, R. J.

Johnston, R. A.

Jumper, E.

N. De Lucca, S. Gordeyev, and E. Jumper, “In-flight aero-optics of turrets,” Opt. Eng. 52, 071405 (2013).
[CrossRef]

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
[CrossRef]

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[CrossRef]

Jumper, E. J.

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting, AIAA 2008-1076 (AIAA, 2008).
[CrossRef]

Lane, R. G.

Poyneer, L.

Roberts, P. H.

T. J. Brennan, P. H. Roberts, and D. C. Zimmerman, “WaveProp user’s guide, a wave optics simulation system,” (the Optical Sciences Company, 2007).

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Siegenthaler, J. P.

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting, AIAA 2008-1076 (AIAA, 2008).
[CrossRef]

Tyler, G. A.

G. A. Tyler, “Analysis of boundary layer data,” (the Optical Sciences Company, 2009).

van Dam, M.

Veran, J.-P.

Vogel, C. R.

T. J. Brennan and C. R. Vogel, “Spatial-temporal control applied to atmospheric adaptive optics,” (the Optical Sciences Company, 2013).

Whiteley, M.

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
[CrossRef]

Whiteley, M. R.

D. J. Goorskey, R. Drye, and M. R. Whiteley, “Spatial and temporal characterization of AAOL flight test data,” Proc. SPIE 8395, 839509 (2012).

Wittich, D. J.

T. J. Brennan and D. J. Wittich, “Statistical analysis of Airborne Aero-Optical Laboratory optical wavefront measurements,” Opt. Eng. 52, 071416 (2013).
[CrossRef]

Zenk, M.

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
[CrossRef]

Zimmerman, D. C.

T. J. Brennan, P. H. Roberts, and D. C. Zimmerman, “WaveProp user’s guide, a wave optics simulation system,” (the Optical Sciences Company, 2007).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

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

Opt. Eng. (3)

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The Airborne Aero-Optics Laboratory, AAOL,” Opt. Eng. 52, 071408 (2013).
[CrossRef]

N. De Lucca, S. Gordeyev, and E. Jumper, “In-flight aero-optics of turrets,” Opt. Eng. 52, 071405 (2013).
[CrossRef]

T. J. Brennan and D. J. Wittich, “Statistical analysis of Airborne Aero-Optical Laboratory optical wavefront measurements,” Opt. Eng. 52, 071416 (2013).
[CrossRef]

Proc. SPIE (1)

D. J. Goorskey, R. Drye, and M. R. Whiteley, “Spatial and temporal characterization of AAOL flight test data,” Proc. SPIE 8395, 839509 (2012).

Prog. Aerosp. Sci. (1)

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[CrossRef]

Other (6)

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems (SIAM, 2001).

T. J. Brennan, P. H. Roberts, and D. C. Zimmerman, “WaveProp user’s guide, a wave optics simulation system,” (the Optical Sciences Company, 2007).

T. J. Brennan and C. R. Vogel, “Spatial-temporal control applied to atmospheric adaptive optics,” (the Optical Sciences Company, 2013).

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting, AIAA 2008-1076 (AIAA, 2008).
[CrossRef]

G. A. Tyler, “Analysis of boundary layer data,” (the Optical Sciences Company, 2009).

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Figures (11)

Fig. 1.
Fig. 1.

Laser beam pointing angles for data sets from the AAOL November 2011 campaign in which the Mach number was 0.65. Data sets 20, 37, 41, 43, and 52 are employed in analyses in this paper and are highlighted in red.

Fig. 2.
Fig. 2.

Schematic diagram showing line-of-sight (LOS) angle and its relationship to the turret and flow geometry.

Fig. 3.
Fig. 3.

Sequence of static-removed phases obtained from AAOL data set 52. Increasing time frame number corresponds to left-to-right followed by top-to-bottom ordering of the phases. Units of the color bar on the right are radians.

Fig. 4.
Fig. 4.

Static-removed phase variance σ2(x) for AAOL data sets 20 (on left) and 43 (on right).

Fig. 5.
Fig. 5.

Structure functions for AAOL data set 43, plotted against r/d, where r=|Δx| and d is subaperture spacing. The left subplot shows the normalized structure function without quasi-homogenization (red curve), together with 1 standard deviation error bars (plotted in green). Normalization means dividing by the aperture-averaged rms phase. The right subplot shows the quasi-homogeneous structure function.

Fig. 6.
Fig. 6.

Quasi-homogeneous structure functions for the selected AAOL data sets, plotted on a log scale. In the plot on the right, structure functions D(r/d) are scaled by dividing through by D(1), and the horizontal axis is restricted to the range 1r/d4. The dashed yellow line indicates a 5/3 power law.

Fig. 7.
Fig. 7.

Time-lagged prediction errors. The plot on the left shows scaled lagged-prediction error Elag2(s)/σ¯2 plotted against discrete time shift s. The dashed yellow line near the top indicates where E2(s)=2σ¯2. The right plot shows normalized lagged-prediction error Elag2(s)/Elag2(1). The dashed yellow line indicates a 5/3 power law.

Fig. 8.
Fig. 8.

Prediction error comparison. The Marechal approximation [Eq. (17)] to the Strehl ratio is plotted against the LOS angle. Red curves marked “MV” in the plot legend indicate minimum variance prediction, while blue curves marked “lag” stand for time-lagged prediction. s denotes shift in number of time frames.

Fig. 9.
Fig. 9.

Vector fields associated with flow matrices F. Plots in the left column show vector flow fields; the right column shows the magnitudes of the flow fields (speed). Displayed from top to bottom are AAOL flights 20, 37, 43, 41, and 52, which have LOS angles of 90°, 100°, 110°, 120°, and 130°. The flow field magnitudes are normalized relative to the free-stream flow speed.

Fig. 10.
Fig. 10.

Eigenvalues for flow matrices F obtained from AAOL data sets. The dashed red circle in each of the plots indicates the boundary of the unit disk in the complex plane. Represented from left to right and then top to bottom are data sets 20, 37, 43, 41, and 52. These have corresponding LOS angles of 90°, 100°, 110°, 120°, and 130°.

Fig. 11.
Fig. 11.

Performance of AO control schemes to mitigate simulated aero-optic disturbances during beam slewing. The green curve, labeled “open-loop,” corresponds to no AO correction; the blue curve represents conventional integral control; the red curve represents tOSC’s adaptive predictive control.

Tables (1)

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Table 1. Power Law Exponents of Lagged Prediction Error for Small Time Lags

Equations (73)

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Dspace(r)[ϕatmos(x+Δx)ϕatmos(x)]2
=6.88(r/r0)5/3.
σfit21|A|A[ϕ(x)PDMϕ(x)]2dx=const×(d/r0)5/3.
ϕ(x,t+s)=ϕ(xsv,t),
Dtime(s)=1|A|A[ϕatmos(x,t+s)ϕatmos(x,t)]2dx=6.88(s|v|/r0)5/3.
ϕQHomog(x)=ϕ(x)/σ(x).
ϕk+1=Fϕk+Swk,k=0,1,.
F=ϕk+1ϕkTϕkϕkT1,
Eaero=A0eiϕaero,
ϕaero=2πOPD/λ,
OPL(x,t)=n(x(s),y(s),z(s),t)ds,
ϕaero(x,t)=ϕstatic(x)+ϕ(x,t).
ϕstatic(x)=ϕaero(x,t).
Covϕ(x,x)ϕ(x)ϕ(x),
σ2(x)Covϕ(x,x)=ϕ2(x).
σ¯2=Aσ2(x)dx/|A|.
SMarechal(ϕ)exp(σ¯2).
Dϕ(x,x)[ϕ(x)ϕ(x)]2=σ2(x)+σ2(x)2Cov(x,x).
Dϕ/σ(x,x)[ϕ(x)σ(x)ϕ(x)σ(x)]2=22Cov(x,x)σ(x)σ(x).
Dϕ(r)mean{Dϕ(x,x):|xx|=randx,xA}.
XCovϕ(x,x,s)=ϕ(x,t+s)ϕ(x,t).
Elag2(s)1|A|A[ϕ(x,t+s)ϕ(x,t)]2dx=2[σ¯21|A|AXCovϕ(x,x,s)dx].
ϕ^(·,t+s)=Fϕ(·,t),
F=argminF˜[F˜ϕ(x,t)ϕ(x,t+s)]2dx=XCovϕCovϕ1.
EminVar2(s)1|A|A[Fϕ(·,t)ϕ(·,t+s)]2.
ϕstatic=1nframesk=1nframesψkψk.
ϕk=ψkϕstatic,k=1,2,,nframes.
ϕkϕkT=1nframesk=1nframesϕkϕkT.
ϕk+1ϕkT=1nframes1k=1nframes1ϕk+1ϕkT.
FargminF˜F˜ϕkϕk+12
=ϕk+1ϕkTϕkϕkT1.
ϕkϕkT1ϕkϕkT(ϕkϕkT2+βI)1.
V(β)=1nframes/2k=nframes/2nframes1Fβϕkϕk+12.
R=rk+1rk+1T,
R=ϕkϕkTϕk+1ϕkTFT=ϕkϕkTFϕkϕkTFT,
Dϕ=σ1T+1σT2C.
Dϕ/σ=2[11Tdiag(σ1)Cdiag(σ1)].
D(r˜i,j)=mean{Di,jsuch that|xixj|=r˜i,j}.
s=ΓAψ.
ψext=argminψΓAψs2+ΓCψ2=(ΓATΓA+ΓCTΓC)1ΓATs.
s¯x(xi)=i=1Nsx(xi)/N,s¯y(xi)=i=1Nsy(xi)/N.
Elag2(s)=ϕk+sϕk2,s=1,2,,
EminVar2(s)=ϕk+sFϕk2,s=1,2,,
Elag2(s)=const×sp
ϕk+1=F(x(tk))ϕk+S(x(tk))wk,
ψk+1=ϕstatic(x(tk))+ϕk+1.
F(x)=λ1F(x1)+λ2F(x2)+λ3F(x3),
Δϕk=ψkϕDM(tk).
ϕDM(tk)=Hak,
sk=ΓΔϕk+ηk,
ak+1=ak+γRsk,
x(t)=x0+tt0tendt0(xendx0),t0ttend.
ϕk+1=Fϕk+Swk,k=0,1,,
ϕ^n+1=Fϕ^n+Swn,n=0,1,,N
C^nϕ^nϕ^nT,
X^nϕ^n+1ϕ^nT,
C^limNC^n,
X^limNX^n.
ϕ^n=0,n=0,1,,
ϕkϕkT=C^,
ϕk+1ϕkT=X^.
ϕ^n=Fnϕ^0+j=0n1Fnj1Swj.
C^n=Fnϕ0ϕ0T(Fn)Tj=0n1Fnj1R(Fnj1)T.
C^=k=0FkR(Fk)T.
X^n=FC^n.
X^=FC^.
R=CFCFT,
C=ϕkϕkT.
R=(Swn)(Swn)T=C^n+1FX^nTX^nFT+FC^nFT.
R=C^FC^FT.
E=FEFT,
V1EVT=Λ(V1EVT)Λ.
e˜i,j=λiλje˜i,j,

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