TECHNICAL RESEARCH REPORT
Analysis of a high-resolution optical wave-front control system
by E.W. Justh, P.S. Krishnaprasad
CDCSS TR 2002-2
(ISR TR 2002-5)
CENTER FOR DYNAMICS
AND CONTROL OF
SMART STRUCTURES
C
S
D
+
-
The Center for Dynamics and Control of Smart Structures (CDCSS) is a joint Harvard University, Boston University, University of Maryland center,
supported by the Army Research Office under the ODDR&E MURI97 Program Grant No. DAAG55-97-1-0114 (through Harvard University). This
document is a technical report in the CDCSS series originating at the University of Maryland.
Web site  http://www.isr.umd.edu/CDCSS/cdcss.html
2001 Conference on Information Sciences and Systems, The Johns Hopkins University,March 21#7B23, 2001
Analysis of a high-resolution optical wave-front control system
E. W. Justh and P. S. Krishnaprasad
1
Institute for Systems Research and
Department of Electrical and Computer Engineering
University of Maryland
College Park, MD 20742, USA
e-mail: justh@isr.umd.edu, krishna@isr.umd.edu
Abstract | We consider the formulation and anal-
ysis of a problem of automatic control: correcting for
the distortion induced in an optical wavefrontdueto
propagation through a turbulent atmosphere. It has
recently been demonstrated that high-resolution opti-
cal wave-front distortion suppression can be achieved
using feedback systems based on high-resolution spa-
tial light modulators and phase-contrast techniques.
We examine the modeling and analysis of such sys-
tems, for the purpose of re#0Cning their design. The
approachtaken here might also be applicable to other
problems involving feedbackcontrol of physical #0Celds,
particularly if the #0Celd sensing is performed opti-
cally.
I. Introduction
Correcting for the distortion induced in an optical wave front
due to propagation through a turbulent atmosphere can be
formulated as problem of automatic control. Thermal gradi-
ents in the air produce index-of-refraction variations experi-
enced by light which passes through it, leading to wave-front
distortion. Wave-front correction is achieved byapplying #28e.g.,
using an array of micromirrors#29 conjugate distortions to pro-
duce net null distortion. Adaptive optics is the discipline con-
cerned with feedback compensation of wave-front distortion in
real-time.
Akey criterion for an adaptive optic system is the resolu-
tion required for wave-front control #28or conjugation#29, whichin
turn depends on the optical wavelength and on the strength of
the turbulence. Stronger turbulence also requires an adaptive
optic system to have a faster response #28i.e., a higher frame
rate#29. Recentadvances in high-resolution liquid-crystal #28LC#29
and microelectromechanical #28MEMS#29 devices have led to spa-
tial light modulators #28SLMs#29 that meet both the speed and
resolution requirements for high-resolution #28#3E 10
4
actuators#29
adaptive optics #5B1, 2#5D. However, in addition to the devices,
there is also a need for control laws which scale appropriately
in this high-resolution, high-speed regime. Suitable control
laws based on high-resolution SLMs and phase-contrast tech-
niques have recently been demonstrated #5B3, 4#5D. Our focus here
is on the modeling and analysis of these systems, extending
earlier work #5B5#5D.
In the next section we brie#0Dy review the high-resolution
wave-front control system architecture and its origins. In Sec-
1
This researchwas supported in part bygrants from the Army
Research O#0Ece under the ODDR&E MURI97 Program Grant
No. DAAG55-97-1-0114 to the Center for Dynamics and Control
of Smart Structures #28through Harvard University#29, and by the Na-
tional Science Foundation under the Learning and IntelligentSys-
tems Initiative Grant CMS 9720334.
Fig. 1: High-resolution wave-frontcontrol system block diagram.
tion III, weintroduce the mathematical models and analyze
them. In Section IV, we digress from the analysis to clarify
certain practical issues and provide context for the mathe-
matical work. In Section V, weconsiderthewave-front con-
trol system as a wave-front estimator. Concluding remarks
appear in Section VI.
II. High-resolution wave-front control system
The general system architecture we consider is shown in
#0Cgure 1. The distorted beam enters the wave-front corrector,
which modulates the wavefront with the objective of cancel-
ing the distortion. The corrected beam is then #0Cltered using
acontrolled optical two-dimensional spatial Fourier #0Clter, and
the resulting #0Cltered beam is used to update the wave-front
corrector. The control system design problem involves choos-
ing the wave-front corrector update law and the Fourier #0Clter
so as to ensure stability, and convergence to a wave-front-
distortion-free corrected beam in as few iterations as possible.
Both the wave-front corrector and the controlled Fourier
#0Clter use a high-resolution SLM and a high-resolution imager,
as shown in #0Cgures 2 and 3. The key feature of the system
of #0Cgure 1 is that the subblocks shown in #0Cgures 2 and 3 can
use parallel, distributed processing between the imagers and
SLMs. That is, each SLM pixel is driven by the correspond-
ing camera pixel #28with any additional controls being common
to all pixels#29. This feature, refered to in adaptive optics as
#5Cdirect control," enables the resolution to scale without im-
pacting system speed.
The system of #0Cgure 1 has its roots in the phase-contrast
technique developed by Frits Zernike during the 1930s #28for
which he was later awarded a Nobel Prize in physics#29 #5B6#5D.
Zernike observed that the system of #0Cgure 3 #28without the im-
ager and with the SLM replaced by a #0Cxed phase plate hav-
ing a centered phase-shifting dot to shift only the zero-order
Fourier component#29 is capable of producing an intensity image
related to the input beam wavefront#5B7#5D.Zernike's technique
has found application in phase-contrast microscopes for many
Fig. 2: Opto-electronically controlled wave-front corrector.
Fig. 3: Opto-electronically controlled spatial Fourier #0Clter.
years #5B8#5D. Recent advances in high-resolution SLMs have led
to renewed interest in phase-contrast techinques for various
applications, including adaptive optics #5B3, 4, 9, 10#5D.
The system of #0Cgure 1 is a nonlinear feedback system, be-
cause the wave-front phase of the corrected beam enters non-
linearly into the #0Cltered beam intensity image. The Fourier
#0Clter operator, i.e., the mapping from the Fourier-domain im-
ager to the Fourier #0Clter, introduces further nonlinearity,and
this additional source of nonlinearity turns out to be essen-
tial to producing a practical wave-frontcontrol system. Ad-
dressing these nonlinear e#0Bects is the main contribution of the
analysis presented here and in #5B5#5D.
III. Mathematical models
The key to successful analysis of this type of system is to
capture the underlying physics with su#0Ecient #0Cdelity, while
keeping the nonlinear modeling simple enough to extract
qualitative insights beyond what a linearized approximation
to the dynamics can provide. To describe the optical #0Celd
#28for a monochromatic beam#29, we introduce a complex en-
velope A#28x;y;z#29, describing a single componentofthe elec-
tric or magnetic #0Celd. We distinguish the z direction as
the #5Coptical axis," and denote the transverse coordinates as
r =#28x;y#29. The underlying electromagnetic #0Celd component
that A#28r;z#29 represents is then obtained by taking the real part
of A#28r;z#29e
i#28!t,kz#29
, where k =2#19=#15 and ! = kc #28with #15 the
optical wavelength, and c the speed of light#29.
The complex envelope A#28r;z#29 can be expressed in polar
form as
A#28r;z#29=a#28r;z#29e
i#1E#28r;z#29
; #281#29
where #5Ba#28r;z#29#5D
2
is the intensityand#1E#28r;z#29 is the phase #28the
quantityweareinterested in measuring and controlling#29. The
intensityiswhatacamerawould measure at the point z along
the optical axis.
In the wave-frontcontrol setting, weareinterested in how
the phase at a particular point z
0
along the optical axis evolves
in time. Therefore, we drop the argument z from equation #281#29,
and weallow #1E to depend on a time variable t. #28This time vari-
able corresponds to quasi-static changes in the complex enve-
lope, not the time scale of electromagnetic #0Celd oscillations.#29
Because we assume that a#28r#29 = 0 outside of a bounded region,
we can use a Fourier series representation:
A#28r;t#29 =
X
p
a
p
#28t#29e
i
2#19
#0D
p#01r
;
a
p
#28t#29 =
1
#0D
2
Z
A#28r;t#29e
,i
2#19
#0D
p#01r
dr; #282#29
where p is an ordered pair of integers #28i.e., p takes values in
the integer lattice in the plane#29, and #0D is a beam-size param-
eter determining the spectral resolution #28assumed su#0Eciently
#0Cne to avoid aliasing#29. Without loss of generality,we assume
a#28r#29 = 0 outside of a square region #0A with sides of length #0D.
The integral in equation #282#29 can then be considered an integral
over #0A, and A#28r;t#29canbeintepreted as a spatially periodic
function satisfying
A#28x+ m
x
#0D;y+ m
y
#0D;t#29=A#28x;y;t#29; #283#29
where m
x
and m
y
are arbitrary integers. This interpreta-
tion of the Fourier series representation gives rise to periodic
boundary conditions for the PDE systems introduced below.
A. Single-pixel Fourier phase #0Clter
Suppose only the zero-order Fourier componentisphase-
shifted by the SLM in #0Cgure 3. Let the distorted beam com-
plex envelopein#0Cgure1berepresented as a#28r#29e
i#1E#28r#29
, and let
the wave-front-correcting SLM impose a phase distribution
u#28r;t#29 on the distorted beam. The corrected beam is then
represented by a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
. We denote the #0Cltered im-
age by#5Bf#28u + #1E#29#5D#28r;t#29 to emphasize that it is a functional of
the phase of the corrected wave front. The evolution equation
we assume for u is
@u
@t
= #11
#02
l
2
#01u ,f#28u + #1E#29
#03
; #284#29
where the gain function #11 sati#0Ces #11#28r;t#29 #3E 0, 8r;t. The dif-
fusion term can be considered strictly as an aid to analysis,
with the di#0Busion length l#3E0 being arbitrary small.
The dynamics are thus determined by f, which captures
the e#0Bects of the Fourier phase #0Cltering of the corrected beam.
Besides the phase-shift of the zero-order Fourier component,
there is also an intensity measurement included in f. Letting #12
represent the phase-shift of the zero-order Fourier component,
wethus obtain the following conventional Zernikewave-front
sensor model:
#5Bf
conv
#28u + #1E#29#5D#28r;t#29
=
#0C
#0C
#0C
#0C
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
+#28e
i#12
,1#29
1
#0D
2
Z
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
dr
#0C
#0C
#0C
#0C
2
: #285#29
#28Wehave ignored #0Cnite aperature e#0Bects by failing to truncate
the Fourier series at some #0Cnite frequency.#29
Some of the undesirable nonlinearities presentinf
conv
can
be cancelled by taking the di#0Berence of two such images, cor-
responding to oppositely-directed Fourier phase-shifts #5B3, 4, 5#5D
#28a related idea can be found in #5B11#5D#29. The resulting #5Cdi#0Beren-
tial" Zernikewave-front sensor model is
#5Bf
di#0B
#28u + #1E#29#5D#28r;t#29
=
#0C
#0C
#0C
#0C
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
+#28e
i#12
,1#29
1
#0D
2
Z
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
dr
#0C
#0C
#0C
#0C
2
,
#0C
#0C
#0C
#0C
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
+#28e
,i#12
,1#29
1
#0D
2
Z
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
dr
#0C
#0C
#0C
#0C
2
=,4sin#12 Im
#1A
a#28r#29e
,i#5Bu#28r;t#29+#1E#28r#29#5D
1
#0D
2
Z
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
dr
#1B
: #286#29
The image subtraction required for the di#0Berential wave-front
sensor can potentially be incorporated into the circuitry of
the imager in #0Cgure 2 #5B12#5D. The di#0Berential Zernike wave-
front sensor image given by equation #286#29 is straightforward
to interpret. At each point r #28and for #0Cxed t#29, the value of
the operator f
di#0B
is a periodic function of the corrected beam
phase. However, there is also global coupling through the
zero-order Fourier component
1
#0D
2
R
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
dr.
The dynamics given by equation #284#29, with l = 0 and f
given by equation #286#29, are #28formally#29 gradient dynamics with
respect to the energy functional
#5BV #28u#29#5D#28t#29=,2#0D
2
sin#12
#0C
#0C
#0C
#0C
1
#0D
2
Z
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
dr
#0C
#0C
#0C
#0C
2
; #287#29
which is proportional to #28the negativeof#29theintensityinthe
zero-order Fourier component of the corrected beam #5B4, 5#5D.
Remark on notation: We will generally drop the r and t argu-
ments of u and f#28u + #1E#29, as well as the r argumentofa and
#1E, in the remainder of the development. The equations are
then more compact and easy to read, but a;#1E, and u must be
interpreted as functions, and f as an operator.
Using variational calculus, for equation #284#29 with l =0,we
obtain
dV
dt
=
#0EV
#0Eu
#01
@u
@t
=,4sin#12 Re
#1AZ
a#28r#29e
,i#28u+#1E#29
dr
1
#0D
2
Z
iae
i#28u+#1E#29
@u
@t
dr
#1B
=,
Z #12
4sin#12 Im
#1A
ae
,i#28u+#1E#29
1
#0D
2
Z
ae
i#28u+#1E#29
d^r
#1B#13
@u
@t
dr
=,
Z
1
#11
#10
@u
@t
#11
2
dr; #288#29
where #28#0EV=#0Eu#29 #01 v = lim
#0F!0
#5BV #28u + #0Fv#29 , V #28u#29#5D=#0F. Observe
that dV=dt #14 0, and dV=dt = 0 only at equilibria of the
dynamics. The feedback system thus evolves to maximize the
power in the zero-order Fourier componentofthe corrected
beam. It is clear that u#28r;t#29=,#1E#28r#29 minimizes V #28u#29, so that
phase correction #28or phase conjugation#29 corresponds to energy
functional minimization. A standard performance metric for
adaptive optic systems is the Strehl ratio, St, which is the
normalized zero-order Fourier componentintensity,
#5BSt#28u#29#5D#28t#29=
#0C
#0C
#0C
1
#0D
2
R
a#28r#29e
i#5Bu#28r;t#29+#1E#28r#29#5D
dr
#0C
#0C
#0C
2
#10
1
#0D
2
R
a#28r#29dr
#11
2
: #289#29
The Lyapunov functional for the single-pixel Fourier #0Clter is
thus proportional to the Strehl ratio #5B4, 5#5D.
B. General Fourier phase #0Clter with common phase shift
If instead of a single Fourier component, multiple Fourier
components experience a common phase shift, the #28di#0Beren-
tial#29 wave-front sensor image #28in the absence of any correction#29
becomes
#5Bf
common
#28#1E#29#5D#28r#29
=
#0C
#0C
#0C
#0C
#0C
ae
i#1E
+
,
e
i#12
,1
#01
X
p2I
#12
1
#0D
2
Z
ae
i#1E
e
,i
2#19
#0D
p#01r
dr
#13
e
i
2#19
#0D
p#01r
#0C
#0C
#0C
#0C
#0C
2
,
#0C
#0C
#0C
#0C
#0C
ae
i#1E
+
,
e
,i#12
,1
#01
X
p2I
#12
1
#0D
2
Z
ae
i#1E
e
,i
2#19
#0D
p#01r
dr
#13
e
i
2#19
#0D
p#01r
#0C
#0C
#0C
#0C
#0C
2
= ,4sin#12
X
p2I
Im
#1A
ae
,i#28#1E,
2#19
#0D
p#01r#29
1
#0D
2
Z
ae
i#28#1E,
2#19
#0D
p#01r#29
dr
#1B
; #2810#29
where I is a #0Cnite index set that mayormay not contain 0.
To state rigorous results for the system of #0Cgure 1, wemake
the following hypotheses:
#0F periodic boundary conditions on #0A, a square region with
sides of length #0D;
#0F initial conditions u#28r;0#29;Du#28r;0#29;#1E#28r#29;D#1E#28r#29 2 L
2
#28#0A#29;
#0F #11 = #11#28r#29 #3E 0 and
1
#11
2 L
2
#28#0A#29;
#0F 0 #3C#12#3C#19and l#3E0;
#0F I is a #0Cnite set;
#0F
R
#5Ba#28r#29#5D
2
dr is bounded; and
#0F all integrals are understood to be integrals over #0A.
Proposition 1. Weak solutions for equation #284#29, with f given
by equation #2810#29, exist and are unique.
Proof: Straightforward application of methods in #5B14#5D.
Proposition 2 #28from #5B5#5D#29: System #284#29, with f#28#1E#29 be given by
equation #2810#29, is a gradient system with respect to the energy
functional
V =
Z
l
2
2
jruj
2
dr
,2#0D
2
sin#12
X
p2I
#0C
#0C
#0C
#0C
1
#0D
2
Z
a#28r#29e
i#28u#28r;t#29+#1E#28r#29,
2#19
#0D
p#01r#29
dr
#0C
#0C
#0C
#0C
2
:#2811#29
Speci#0Ccally, @u=@t = ,r
u
V #28with respect to the inner prod-
uct #3Cg;h#3E=
R
1
#11#28r#29
g#28r#29h#28r#29dr on L
2
#28#0A#29#29, and
dV
dt
= ,
Z
1
#11
#10
@u
@t
#11
2
dr: #2812#29
Thus, V also serves as a Lyapunov functional for the dynam-
ics; i.e., dV=dt #14 0, with dV=dt = 0 only at equilibria.
Proof: See #5B5#5D. 2
The interpretation of the energy functional #2811#29 is analo-
gous to that of equation #287#29 for the single-pixel Fourier #0Cl-
ter. The second term of equation #2811#29 represents the sum of
the intensities within the Fourier components of the corrected
beam that are phase-shifted by the Fourier #0Clter. The system
therefore evolves to maximize the total intensity within the
collection of phase-shifted pixels.
C. General Fourier phase #0Clter with arbitrary phase shifts
The #28di#0Berential#29 wave-front sensor image for arbitrary
Fourier component phase shifts is given by
f
arb
#28#1E#29
=
#0C
#0C
#0C
#0C
#0C
ae
i#1E
+
X
p2I
,
e
i#12
p
,1
#01
#12
1
#0D
2
Z
ae
i#1E
e
,i
2#19
#0D
p#01r
dr
#13
e
i
2#19
#0D
p#01r
#0C
#0C
#0C
#0C
#0C
2
,
#0C
#0C
#0C
#0C
#0C
ae
i#1E
+
X
p2I
,
e
,i#12
p
,1
#01
#12
1
#0D
2
Z
ae
i#1E
e
,i
2#19
#0D
p#01r
dr
#13
e
i
2#19
#0D
p#01r
#0C
#0C
#0C
#0C
#0C
2
= ,4
X
p2I
sin#12
p
Im
#1A
ae
,i#28#1E,
2#19
#0D
p#01r#29
1
#0D
2
Z
ae
i#28#1E,
2#19
#0D
p#01r#29
dr
#1B
+g#28#1E#29; #2813#29
where the operator g#28#1E#29isgiven by
g#28#1E#29=
#0C
#0C
#0C
#0C
#0C
X
p2I
,
e
i#12
p
,1
#01
#12
1
#0D
2
Z
ae
i#1E
e
,i
2#19
#0D
p#01r
dr
#13
e
i
2#19
#0D
p#01r
#0C
#0C
#0C
#0C
#0C
2
,
#0C
#0C
#0C
#0C
#0C
X
p2I
,
e
,i#12
p
,1
#01
#12
1
#0D
2
Z
ae
i#1E
e
,i
2#19
#0D
p#01r
dr
#13
e
i
2#19
#0D
p#01r
#0C
#0C
#0C
#0C
#0C
2
:#2814#29
By taking g#28#1E#29 #11 0 in equation #2813#29, we obtain a nonlinear
approximation to the wave-front sensor image. Simulation and
experimental work suggest that this nonlinear approximation
adequately captures the system's qualitativebehavior #5B4#5D.
Proposition 3: Under the same hypotheses as Proposition
2, system #284#29, with f = f
arb
, g, is a gradient system with
respect to the energy functional
V =
Z
l
2
2
jruj
2
dr
,2#0D
2
X
p2I
sin#12
p
#0C
#0C
#0C
#0C
1
#0D
2
Z
a#28r#29e
i#28u#28r;t#29+#1E#28r#29,
2#19
#0D
p#01r#29
dr
#0C
#0C
#0C
#0C
2
:#2815#29
Also, V serves as a Lyapunov functional for the dynamics.
Proof: Analogous to the proof of Proposition 2. 2
IV. Practical implementation issues
For a single-pixel Fourier #0Clter, image contrast su#0Bers when
the beam is highly aberrated, a problem that can be overcome
by using a multi-pixel Fourier #0Clter. The theoretical work
presented here and in #5B5#5D indicates that even for a multi-pixel
Fourier #0Clter, a gradient dynamics property can hold, so that
phase-shifting multiple Fourier components need not upset the
convergence behavior of the system of #0Cgure 1.
A. Thresholding Fourier #0Clter operator
An example of a Fourier #0Clter operator that would give
rise to Fourier #0Clters of the type described by equation #2810#29 is
to compare the Fourier-domain intensity, on a pixel-by-pixel
basis, to a threshold, and phase-shift the pixel by #12 if that
threshold is exceeded. This type of wave-front sensor has been
studied numerically and found to yield a higher-contrast image
than a single-pixel Fourier #0Clter for highly aberrated beams
#5B3#5D. However, when incorporated into the feedback system
of #0Cgure 1, this thresholding wave-front sensor fails to evolve
into a single-pixel sensor as wave-front correction proceeds.
Instead, the system generally approaches an equilibrium with
the corrected beam intensity distributed among a number of
Fourier components.
B. Proportional Fourier #0Clter operator
A simple Fourier #0Clter operator which is observed to pro-
duce wave-front correction is the proportional Fourier #0Clter
operator, in which Fourier components are phase-shifted in
proportion to their power #5B3#5D. Since di#0Berent Fourier com-
ponents experience di#0Berent phase shifts, for a given Fourier
intensity distribution, the wave-front sensor image can be de-
scribed by equation #2813#29.
Comparing equations #2811#29 and #2815#29 suggests why the sys-
tem with a proportional Fourier #0Clter operator outperforms
the system with the thresholding operator. For the #28approxi-
mate#29 proportional Fourier #0Clter operator, the Lyapunov func-
tional is minimized by concentrating all of the intensityinthe
Fourier component #28or components#29 with the greatest phase
shift #28provided #12
p
#14 #19=2; 8p#29. However, for the thresholding
Fourier #0Clter operator, there is no preference #28in terms of Lya-
punov functional minimization#29 for any particular distribution
of intensity among the Fourier components phase-shifted by
#12.
Note that Propositions 2 and 3 apply to systems with
#0Cxed Fourier #0Clters, even though in the system of #0Cgure 1,
the Fourier #0Clter evolves in time as phase correction proceeds.
We are inferring how the system with an evolving Fourier #0Clter
will behave from the analysis for #0Cxed Fourier #0Clters.
V. Wave-front estimator analysis
The analysis of Section III demonstrates that high-
resolution wave-frontcontrol is feasible, in the sense that there
exists a parallel, distributed feedback control approach which
#28with some simpli#0Ccations and approximations#29 tends to con-
verge to an equilibrium representing wave-front correction.
Furthmore, the analysis is nonlinear and global, but is de-
terministic. To properly design and assess the performance
of an adaptive optic system for atmospheric turbulence com-
pensation, it is imperative to consider the statistical nature of
atmospheric turbulence and photodetector noise.
General problem formulation: Subject to constraints of
realizability, how can atmospheric turbulence compensation
be performed optimally,given stochastic models for the wave-
front distortion and photodetector noise?
Instead of this general problem, we consider the following.
Weaker problem formulation: Subject to constraints of
realizability, how can atmospheric turbulence compensation
be performed nearly optimally when the residual distortion
is small, and adequately when the residual distortion is large,
given simpli#0Ced stochastic models for the wave-front distortion
and photodetector noise?
Toinvestigate this weaker formulation, we discretize both
the time and spatial variables, and consider the system of
#0Cgure 1 to be a nonlinear estimator: based on noisy, nonlinear
measurements of the corrected beam wave front, the system
attempts to estimate the #28conjugate of the#29 input beam wave
front.
A. Modeling wave-front distortion and photon noise
Very detailed statistical models of atmospheric turbulence
and photon noise can be found in the literature, and have been
used to study and design adaptive optic systems #5B13#5D. Statis-
tical models for atmospheric turbulence generally assume par-
ticular forms for the spatial power spectra, with origins in the
study of thermal energy transfer and #0Duid motion across vari-
ous length scales. Motion of the turbulent structures produces
the temporal dependence of the wave-front distortion.
A standard approach for modeling photon noise is the semi-
classical model #5B13#5D. Electromagnetic #0Celd theory is used to
describe the system up to the plane where the intensitymea-
surementismade. The intensity measurement is then taken
tobeaPoisson process with its rate function determined by
the classical electromagnetic #0Celd irradiance. #28Integrating the
rate function over the area of a single photodetector gives the
average number of photons incident on the detector per unit
time.#29 The electromagnetic #0Celd irradiance depends on both
the deterministic optical system #28i.e., the operator f of Section
III#29, and the random atmospheric turbulence.
Rather than use the detailed, realistic models for turbu-
lence and photon noise, we instead use simple models with
parameters derived from the more detailed models. These are
the simplifying assumptions wemake:
#0F The phase-correcting SLM spatial discretization is
matched to the smallest feature size present in the dis-
tortion, so that the wave front can be approximated
as piecewise constant #28i.e., constantover the area of a
phase-correcting SLM pixel#29. The Fried parameter, de-
pendentonwavelength and the strength of turbulence,
is proportional to this smallest feature size #5B13#5D.
#0F For each phase-correcting SLM pixel, the change in in-
put beam wave front from one time step to the next is
independent and normally distributed.
#0F The input beam intensity distribution is quasi-static,
so that for purposes of the analysis it is taken to be
constant, although it mayvary spatially.
#0F The Poisson distribution for photon noise is approxi-
mated as a Gaussian distribution with the same mean
and variance #28a reasonable approximation as long as
the intensity is not too low#29, so that the photon noise
process is modeled as independent and normally dis-
tributed.
B. Nonlinear estimator
Weletk denote the discrete time variable, and welets,a
two-dimensional vector of integers, denote the discrete trans-
verse coordinates. For the wave-front estimation problem we
use a discrete Fourier transform:
A
s
=
X
p
a
p
e
i
2#19
n
p#01s
;
a
p
=
1
n
2
X
s
A
s
e
,i
2#19
n
p#01s
; #2816#29
where the a
p
now represent discrete Fourier transform coe#0E-
cients. The total number of grid points #28in both the spatial
and Fourier domains#29 is n
2
.
The state of the distortion #28speci#0Ccally, the e#0Bect of the at-
mospheric turbulence on input beam phase#29 is denoted #1E
s
#28k#29.
We assume that the update equation for the atmosphere is
#1E
s
#28k +1#29=#1E
s
#28k#29+w
s
#28k#29; #2817#29
where the w
s
#28k#29 are independent and normally distributed,
with #28scalar#29 covariances q
s
#28k#29 #28which are reasonable to as-
sume are equal for all s#29.
The estimate of the state of the distortion at time k #28given
information up to time k , 1#29 is denoted
^
#1E
s
#28kjk , 1#29. The
error signal,
"
s
#28k#29=#1E
s
#28k#29,
^
#1E
s
#28kjk ,1#29; #2818#29
is produced by the phase-correcting SLM, and is what we
would like the wave-frontcontrol system to minimize #28in an
appropriately weighted mean-square sense#29. Welet"#28k#29de-
note the matrix f"
s
#28k#29g. In the absense of measurement
noise, the image measured by, e.g., the single-pixel di#0Berential
Zernikewave-front sensor can then be expressed as
#5Bf#28"#29#5D
s
#28k#29=,4sin#12 Im
#28
a
s
e
,i"
s
#28k#29
1
n
2
X
#16s
a
#16s
e
i"
#16s
#28k#29
#29
: #2819#29
The measured image including photon noise is then
#5Bf#28"#29#5D
s
#28k#29+v
s
#28k#29, where we assume the v
s
#28k#29 are independent
and normally distributed, with #28scalar#29 covariances r
s
#28k#29.
A natural update equation for the estimator is
^
#1E
s
#28k +1jk#29=
^
#1E
s
#28kjk ,1#29 + c
s
#28k#29#28#5Bf#28"#29#5D
s
#28k#29+v
s
#28k#29#29; #2820#29
where fc
s
g is a gain matrix #28replacing the gain function #11#28r;t#29
of Section III#29. Equation #2820#29 is a discrete-time #28and spatially
discretized#29 version of the deterministic gradient#0Dow #284#29, with
l = 0, and with additional noise terms. The evolution equation
for the error is
"
s
#28k +1#29="
s
#28k#29,c
s
#28k#29#28#5Bf#28"#29#5D
s
#28k#29+v
s
#28k#29#29 +w
s
#28k#29: #2821#29
We thus have a nonlinear update equation for the estima-
tion error, involving both process and measurement noise. We
would like to analyze the stability and convergence properties
of equation #2821#29.
C. Relationship between Strehl ratio and error covariance
A natural performance metric for wave-front estimation is
the weighted sample error covariance
'#28"#29=
1
n
2
P
s
a
s
#28"
s
, #16"#29
2
1
n
2
P
s
a
s
; #2822#29
where
#16" =
1
n
2
P
s
a
s
"
s
1
n
2
P
s
a
s
: #2823#29
The relationship between the Strehl ratio and the wave-front
error covariance is well-known in the optics literature #5B13#5D. For
concreteness, we outline a derivation.
The zero-order Fourier componentintensityis
i
0
#28"#29=#28a
0
#29
2
=
#0C
#0C
#0C
#0C
#0C
1
n
2
X
s
a
s
e
i"
s
#0C
#0C
#0C
#0C
#0C
2
: #2824#29
The Strehl ratio, St#28"#29=i
0
#28"#29=i
0
#28#16"#29, can be shown to satisfy
St#28"#29=1,'#28"#29 + h.o.t. #2825#29
To see this relationship between Strehl ratio and error covari-
ance, simply take the Taylor series expansion of i
0
#28"#29 around
#16",
i
0
#28"#29=i
0
#28#16"#29+
@i
0
@"
#0C
#0C
#0C
#0C
#16
"
#01#28",#16"#29+
1
2
@
@"
#10
@i
0
@"
#11
#0C
#0C
#0C
#0C
#16
"
#01#5B#28",#16"#29;#28",#16"#29#5D+#01#01#01;
#2826#29
where #16" denotes the n #02 n matrix whose elements are all #16".
Computing the partial derivatives in equation #2826#29 gives
i
0
#28"#29=i
0
#28#16"#29#281,'#28"#29#29 + h.o.t. #2827#29
There is thus a straightforward relationship between Strehl
ratio maximization for the deterministic wave-front control
problem and error covariance minimization for the wave-front
estimation problem. The Strehl ratio can be measured us-
ing the Fourier-domain imager in #0Cgure 3, which raises the
possibility of adapting the gain matrix on-line.
D. Implications for system design
Our general strategy for system design is then to start with
the basic architecture of #0Cgure 1, and make good choices for
the Fourier #0Clter operator and the gain matrix. Wehaveseen
from the deterministic analysis that the proportional Fourier
#0Clter operator #28with the peak phase shift constrained to be at
most #19=2#29 has favorable properties. For the stochastic system,
it is critical that the Fourier #0Clter provide a #0Cltered image
with good signal-to-noise ratio #28i.e., good constrast#29, so that
the system can start compensating phase distortion even when
the initial distortion is severe.
The deterministic analysis of Section III places no con-
straints on the gain function #28other than a sign condition and
the L
2
#28#0A#29 property of 1=#11#29. However, in the discrete-time
setting, the dynamical equation #284#29 is replaced by a forward-
Euler method, and it is evidentthatthereisanupperlimiton
each element of the gain matrix. For linear estimation prob-
lems, the gains start out large, and decrease as the estimate is
re#0Cned. A similar behavior is expected for the nonlinear esti-
mator described above. Our strategy is then to make sure the
gain matrix elements stay bounded appropriately #28to ensure
stabilityofthe nonlinear system#29 when the estimation error
is large, and to reduce the gain matrix #28e.g., using a common
scale factor#29 as the estimation error decreases. In the small-
estimation-error regime, a linear approximation to the nonlin-
ear estimator may be useful. By scaling the gain matrix based
on the measured Strehl ratio, the system can transition nat-
urally on its own from the nonlinear, large-distortion regime
to the linear, small-distortion regime.
VI. Summary and conclusions
Large arrays of sensors and actuators are becoming feasi-
ble to build. However, using sucharrays for feedbackcontrol
of physical #0Celds depends critically on our ability to devise
control schemes for such systems. Optics is a natural context
in which to investigate such control schemes, because there
has been considerable experimental work in adaptive optics
to draw upon, and because optics can be useful for measure-
ment in other engineering contexts, as well.
However, in considering optical wave-front measurement
and control, nonlinearityenters in an intrinsic way. Weneed
to be able to deal with the nonlinearity, as well as with the
constraints arising from having to use a parallel, distributed
control scheme rather than a centralized control law. The
strategy of identifying a nonlinear approximation to the dy-
namics which captures its essential features even far from equi-
librium can be simpler to carry out, and more revealing, than
a linearized analysis about the equilibrium. Furthermore, the
results can be used to guide design.
Acknowledgments
The authors would like to thank Mikhail Vorontsov of the
Army Research Laboratory, Adelphi, MD; Ralph Etienne-
Cummings of Johns Hopkins University, Baltimore, MD; and
Steve Serati and Teresa Ewing of Boulder Nonlinear Systems,
Inc., Lafayette, CO, for valuable discussions.
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