Physics and Computation of Aero-Optics - University of Notre Dame

48 Pages·2011·1.78 MB·English

Aero-Optics 1
Physics and Computation of Aero-Optics
Meng Wang
Department of Aerospace and Mechanical Engineering, University of Notre
Dame, Notre Dame, IN 46556; email: [email protected]
Ali Mani
Department of Chemical Engineering, Massachusetts Institute of Technology,
Cambridge, MA 02139; email: [email protected]
Stanislav Gordeyev
Department of Aerospace and Mechanical Engineering, University of Notre
Dame, Notre Dame, IN 46556; email: [email protected]
Key Words wavefront distortion, turbulent flow, high-fidelity simulation, opti-
cal mitigation
Abstract
This article provides a critical review of aero-optics with an emphasis on recent developments
in computational predictions and the physical mechanisms of flow-induced optical distortions.
Followingabriefintroductionofthefundamentaltheoryandkeyconcepts,computationaltech-
niquesforaberratingflowfieldsandopticalpropagation arediscussedalongwithabriefsurvey
Annu.Rev. Fluid Mech. 2012. 44
ofwavefrontsensorsusedinexperimentalmeasurements. Newphysicalunderstandinggenerated
through numerical and experimental investigations are highlighted for a numbers of important
aero-optical flows including turbulent boundary layers, separated shear layers and flow over
optical turrets. Approachesfor mitigating aero-optical effects are briefly discussed.
1 INTRODUCTION
Distortions of optical signals by turbulentflow are widely observed in nature and
in technological applications. The twinkling of stars, or stellar scintillation, is
the result of refraction of light (electro-magnetic waves) by turbulent fluctua-
tions in the Earth’s atmosphere. Shadowgraphs and Schlieren are popular flow
visualization techniques which exploit the optical distortions to visualize the flow
field that produced them. Over the past four decades, much attention has been
paid to the aero-optical effects on the performance of airborne laser systems for
communication, target tracking and directed-energy weapons. In these systems
optical distortions produced by turbulent flows surrounding the projection aper-
ture pose a serious problem, causing beam distortion, jitter and much reduced
effective range. The performance of airborne and ground-based imaging systems
is likewise impaired by turbulent flows in the vicinity of the viewing aperture.
Thedirect cause of optical distortions is the density variations in theflow field.
For air and many other fluids, the index of refraction is linearly related to the
density of the fluid by the Gladstone-Dale relation (Wolfe & Zizzis 1978). When
a beam of an initially planar wavefront is transmitted through a variable density
field,differentpartsofthebeampropagateatdifferentlocalspeedoflight, result-
ing in distortions of the wavefront. An optical beam emitted from a projection
aperture, or received by a viewing aperture, typically transmits/receives through
two distinct flow regions: the active turbulence region induced by solid objects
2
Aero-Optics 3
near the optical window, and atmospheric turbulence. The propagation through
atmospheric turbulence has been studied extensively (Chernov 1960,Tatarski
1961) andisrelatively wellunderstood. Becauseofthelargetemporalandspatial
scales associated with atmospheric turbulence, its aberrating effects are of low
frequency (< 100 Hz) and can be largely corrected using Adaptive-Optic (AO)
systems (e.g. Lloyd-Hart 2003, Hardy 1998). In contrast, the turbulent flow in-
duced by solid surfaces near the aperture, which may be comprised of turbulent
boundary layers, free shear-layers, wakes, and shock waves for supersonic and
transonic flows, is characterized by much smaller turbulence scales. The size of
the optically-active flow region is typically thinner than or comparable to the
aperturesize. Aero-optics, to follow the conventional definition (Gilbert & Otten
1982,Sutton 1985,Jumper & Fitzgerald 2001), is concerned with the aberrating
effects of compressible turbulence in this region. Compared to the atmospheric
boundary layer, the aero-optical flow generates stronger optical aberrations at
smaller scales and higher frequencies, which are beyond the capability of today’s
AOtechnology. Mitigationoftheseaberrationsviaactiveandpassiveflow-control
has been actively pursued in recent years (e.g. Gordeyev et al. 2010a, 2010b).
The types of aero-optical distortions and their far-field impact depend on a
number of physical and geometric parameters, including the optical wavelength
λ, aperture or beam size a, turbulence length scale ℓ of the aberrating field, and
distanceofpropagationL. Incrudeterms,small-scaleturbulence,witheddysizes
less than the optical aperture (ℓ < a), causes optical scattering, beam spread,
and consequent attenuation of intensity, whereas turbulent eddies larger than
or comparable in size to the aperture (ℓ a) are mostly responsible for the un-
≥
steady tilt of thebeam(beam jitter) (Cassady etal. 1989). Inaddition, themean
4 WANG, MANI & GORDEYEV
density gradient in the beam path causes steady wavefront distortions known as
lensing effect. The magnitude of wavefront distortions OPD (its precise def-
rms
inition will be given later) is generally small in an absolute sense but can be a
significant fraction of, or even exceed the optical wavelength. In other words,
the optical phase distortion 2πOPD /λ, which determines the far-field beam
rms
quality, can be easily of O(0.1) to O(1). For the same distortion magnitude, the
phase distortion of an aberrated beam is inversely proportional to the optical
wavelength, making the aero-optics problem particularly acute for short wave-
length beams (Jumper & Fitzgerald 2001). This poses a significant impediment
to using shorter wavelength laser systems, which are preferable in the absence of
aero-optical aberrations; for an unaberrated beam the diffraction-limited optical
intensity scales with a2/(L2λ2) (Born & Wolf 2002).
A number of review articles have been written on aero-optics in the past. Re-
search in the pre-1980 era is documented in Gilbert & Otten (1982). Progress
since then has been surveyed by Sutton (1985), Jumper & Fitzgerald (2001),
and more recently Gordeyev & Jumper (2010) with a focus on the aero-optics of
turrets. Among recent advances in this field are high-speed and high-resolution
wavefront sensors which allow measurements of instantaneous wavefront errors
in unprecedented detail, and the increasing role played by numerical simula-
tions. Application of computational fluid dynamics (CFD) to aero-optics prob-
lems has historically lagged most other areas because of the challenging nature
of the computations, which must be time-accurate, compressible, and capture
optically important flow scales. However, this has begun to change with recent
advances in high-fidelity simulation tools and the concomitant increase in com-
puting power. A new generation of computational techniques including direct
Aero-Optics 5
numerical simulation (DNS), large-eddy simulation (LES), and hybrid methods
combining LES with Reynolds-averaged Navier-Stokes (RANS) approaches, have
been employed hand-in-hand with experimental and theoretical approaches to
elucidate the physics of aero-optics, predict aberration effects, and develop tech-
niques for their mitigation. This article is an attempt to provide a critical review
of some recent progress in the understanding and prediction of aero-optical ef-
fects. The discussion is primarily from a computational perspective, although
important experimental findings are also included.
2 THEORETICAL BACKGROUND
2.1 Basic Equations
The theoretical foundation for electromagnetic wave propagation in a turbulent
mediumisdiscussedextensivelyinMonin&Yaglom(1975), andisbrieflyoutlined
here in the context of aero-optics. In the most general sense, the propagation of
electromagneticwavesisgovernedbytheMaxwellequationsalongwithcompress-
ible Navier-Stokes equations. Various simplifications can be made depending on
relevant physical parameters and length and time scales. For aero-optical prob-
lems, the time scale for optical propagation is negligibly short relative to flow
time scales, and hence optical propagation can be solved in a frozen flow field at
each time instant. If the optical wavelength is much shorter than the smallest
flow scale (Kolmogorov scale), which is generally the case, the effect of depolar-
ization is negligible, and the Maxwell equations reduce to a vector wave equation
inwhichallthreecomponentsoftheelectromagnetic fieldaredecoupled. Ascalar
6 WANG, MANI & GORDEYEV
component of the electric field at frequency ω is governed by
ω2n2
2U + U = 0, (1)
∇ c2
0
where c is the speed of light in vacuum and n is the index of refraction. The
0
latter is related to the density of air via the Gladstone-Dale relation: n(x,y,z) =
1+K (λ)ρ(x,y,z), whereK istheGladstone-Dale constantandisingeneral
GD GD
weakly dependent on the optical wavelength (Wolfe & Zizzis 1978). In aero-
optics, fluctuations in index of refraction are small ( 10−4) and have scales
∼
much larger than the optical wavelength, and hence an optical beam propagates
predominantly in the axial (z) direction with slowly varying amplitude. It is
customary to invoke the paraxial approximation, which assumes U(x,y,z) =
A(x,y,z)exp( ikz) and ∂2A/∂z2 k∂A/∂z , where k = ω/c∞ is the optical
− | | ≪ | |
wavenumber in the free stream. This leads to the parabolized wave equation for
the complex amplitude
∂A n2
2ik + 2A+k2 1 A= 0, (2)
⊥
− ∂z ∇ n2∞ −
(cid:18) (cid:19)
where 2 is the Laplacian operator in the transverse directions. Effects not in-
⊥
∇
cluded in the paraxial wave equation include large-angle scattering and energy
dissipation/absorption, which are insignificant for aero-optical problems. A de-
tailed discussion of the approximations and limitation of applicability is given by
Monin & Yaglom (1975).
Further approximations can be made to obtain closed-form solutions to Equa-
tion 2. As depicted schematically in Figure 1, the propagation domain consists
of two distinct regions: the optically active turbulent near-field (from z = 0 to
z ) and the free space extending to the distant far field (from z = z to L, with
1 1
L z ). Fortransmissionacrosstheaero-optical region,itcanbeshownthatthe
1
≫
Aero-Optics 7
contribution from theLaplacian term inEquation 2, which representsthediffrac-
tion effect, is negligible relative to the last term for z1 ℓ2⊥/λ, where ℓ⊥ is the
≪
transverse length scale of the optical wave. If the refractive index is expressed as
n = n∞+n′,andthelinearapproximationn2/n2∞ 1+2n′/n∞ ismadeaboutthe
≈
index-of-refraction deviation from the free-stream value, Equation 2 can be inte-
grated from 0 to z to obtain A(x,y,z )= A(x,y,0)exp ik z1n′(x,y,z)dz ,
1 1 −n∞ 0
(cid:16) (cid:17)
R
or
z1
U(x,y,z )= U(x,y,0)exp ik n(x,y,z)dz , (3)
1 0
−
(cid:18) Z0 (cid:19)
where k0 = k/n∞ = ω/c0 = 2π/λ0 is the optical wavenumber in vacuum.
Oncethesolutionpasttheaero-opticalregion,U(x,y,z ),hasbeendetermined,
1
itcanbeusedastheinitialconditiontopropagatetheopticalbeamtothefarfield
using the wave equation (Equation 1) or paraxial wave equation (Equation 2).
′
By setting n = n∞ and n = 0, either equation can be solved analytically using
Fourier transform techniques, which are the basis for Fourier optics (Goodman
2004).
2.2 Near-Field Distortion Measures
Equation 3 suggests that the dominant aero-optical effect after transmission
through the turbulence region is a phase distortion of the optical wavefront;
the amplitude is approximately unchanged. The integral in Equation 3 is known
as the optical path length (OPL). It is most commonly derived from geometric
opticsbyassumingstraightraypathsandisgenerally dependentontheflow-time
scale (but not optical time scale):
z1
OPL(x,y,t) = n(x,y,z,t)dz. (4)
Z0
8 WANG, MANI & GORDEYEV
Ray bending can be accounted for by solving the eikonal equation (Born & Wolf
2002)butisgenerallynegligibleforaero-optics. Inpractice,therelativedifference
in the OPL over the aperture is a more relevant representation of wavefront
distortions. It is called the optical path difference (OPD) and is defined as
OPD(x,y,t) = OPL(x,y,t) OPL(x,y,t) , (5)
−h i
where the angle brackets denote spatial averaging over the aperture. The optical
phasedistortionis then2πOPD/λ. Itshouldbenotedthattheoptical wavefront,
defined as the locus of constant phase, is (after removing the spatial mean) the
conjugate (negative) of the OPD: W(x,y,t) = OPD(x,y,t).
−
To facilitate analysis and mitigation of distortions, the time-dependent OPD
is often decomposed into a time-averaged spatial component, called the steady-
lensing term, OPD (x,y), and an unsteady component. The unsteady part
steady
canbefurthersplitintoaspatiallylinearcomponent,calledunsteadytiltorbeam
jitter, and the rest, usually called high-order distortions (Gordeyev & Jumper
2010). In other words,
OPD(x,y,t) = OPDsteady(x,y)+[A(t)x+B(t)y]+OPDhigh−order(x,y,t). (6)
Physically,thesethreecomponentsaffectanoutgoingbeamindifferentways. The
steady-lensing term, OPD (x,y), is a function of the time-averaged density
steady
field only and imposes a steady distortion like defocus, coma and so on. The tilt
or jitter, represented by the second term on the right-hand side, does not change
thespatialdistributionof theoutgoingbeambutsimplyre-directs itindirections
defined by functions A(t) and B(t). The specific forms of A(t) and B(t) depend
on the definition of tilt. For the so-called G-tilt (Tyson 2000), A(t) and B(t)
are the spatially-average gradient components of the OPD in x- and y-directions,
Aero-Optics 9
respectively, whereas for the Z-tilt (Sasiela 2007), they are defined such that the
magnitude of OPDhigh−order in Equation 6 is minimized in the least square sense
at each time instant. Finally, the high-order term causes the beam to change its
shape and intensity distribution.
The decomposition in Equation 6 is particularly useful when an adaptive optic
system is used to correct for aberrating wavefronts. The purpose of an adaptive
optic system is to apply a conjugate wavefront to the outgoing beam so that
it will negate optical aberrations from the flow and the beam will become re-
collimated after passing through the turbulent media (Tyson 1997). Among the
distortion components given in Equation 6, steady lensing is easily corrected by
a deformable mirror with a large range of motion, the tilt component is removed
using a fast-steering mirror, and the high-order term can be compensated for by
using a high-bandwidth deformable mirror (Tyson 1997).
2.3 Statistical Theory
An important equation relating the statistical properties of a turbulent medium
and those of aero-optical aberrations, the so-called linking equation, was derived
by Sutton (1969) (see also Sutton 1985, Steinmetz 1982, and Havener 1992). In
the most general form, the linking equation can be written as,
z1 z1
hOPD2i = KG2D Covρ′(z,z′)dz′dz, (7)
Z0 Z0
where the overbar denotes time averaging, Covρ′ is the covariance function of
density fluctuations, and z is the integration distance along the traversing beam
1
through the turbulence region. In the case of homogeneous turbulent flows,
the density covariance is most commonly modeled by an exponential function
(Steinmetz 1982), ρ2 exp( z z′ /Λ), or a Gaussian function (Wolters 1973),
rms −| − |
10 WANG, MANI & GORDEYEV
ρ2 exp( z z′ 2/Λ2), where Λ is the characteristic length scale for density
rms −| − |
fluctuations. Substituting these models into Equation 7 leads to
z1
OPD2 = αK2 ρ2 (z)Λ(z)dz, (8)
h i GD rms
Z0
where α = 2 for the exponential covariance function and √π for the Gaussian
covariance function. Both the density fluctuation magnitudeand length scale are
allowed to vary slowly along the beam path.
Both forms of the linking equation allow one to calculate aero-optical distor-
tions indirectly from statistical properties of the turbulent flow. Since the full
covariance matrix in Equation 7, is difficult to measure experimentally, the sim-
plified equation, Equation 8 is commonly used instead. However, since the sim-
plified linking equation is derived for homogenous and isotropic turbulent flows,
its applicability to inhomogeneous flows, such as shear layers (Hugo & Jumper
2000) and boundary layers (Gilbert & Otten 1982,Tromeur et al. 2006b), has
been questioned. It has been shown that with appropriate choice of the length
scaleΛ , thesimplifiedlinkingequation canbeusedtoobtainaccurate resultsfor
ρ
flow fields with anisotropic and inhomogeneous turbulence. The key is to use the
correct density correlation length defined based on Equation 7 (Wang & Wang
2011).
3 PREDICTION OF FAR-FIELD DISTORTIONS
GiventheOPDprofilesafterthebeampassestheturbulenceregion,itsfree-space
propagation from z = z to L can be solved using Fourier optics to obtain the
1
exact far-field projection. Figure 2 shows an example of instantaneous far-field
irradiance of a Gaussian beam subject to strong aero-optical distortions by the
turbulent wake behind a circular cylinder of diameter D (Mani et al. 2009). It