# Phase Retrieval for Imaging and WaveFront Sensing by dfgh4bnmu

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```									          Phase Retrieval
for Imaging and WaveFront Sensing

Jim Fienup
Robert E. Hopkins Professor of Optics

University of Rochester
Institute of Optics
Presented at EECS Dept., University of Michigan
General Dynamics Distinguished Lecture Series
February 9, 2006
JRF 2/06-1
Outline

•   Image reconstruction for stellar speckle interferometry
! Phase retrieval, hybrid input-output algorithm

! Finding bounds on object support

•   Image reconstruction from far-field diffraction patterns
! Optical

! X-ray

•   Wavefront Sensing for the Hubble Space Telescope
! Phase objects (wave fronts) can be reconstructed

! Gradient search algorithms

•   Wavefront Sensing for James Webb Space Telescope

JRF 2/06-2
Passive Imaging of Space Objects

•   Problem: atmospheric turbulence causes phase errors, limits resolution
!   ! 1 arc-sec ! 5*10–6rad. ! !/ ro for ! = 0.5 microns and ro = 10 cm
– Best resolution as though through only 10 cm aperture (vs. Keck 10 m)

•   Solutions:
! Hubble Space Telecope (2.4 m diam.), \$2 B

! Adaptive optics + laser guide star, \$10"s M

! Optical interferometry, \$10"s M

! Stellar speckle interferometry, < \$1 M

aberrated optical system

}
blurred
object                                                          image

JRF 2/06-3
Labeyrie"s Stellar Speckle Interferometry

1. Record blurred images: gk ( x , y ) = f (x, y ) ! sk (x , y ),    k = 1 . . ., K
,
— where sk(x, y) is kth point-spread function due to atmospheric turbulence
2. Fourier transform: Gk (u, v ) = F (u ,v )Sk (u,v ),       k = 1, . . ., K
— where Sk(u, v) is kth optical transfer function
K                           K
3. Magnitude square and average: 1                         2 1
! Gk (u, v ) = F (u ,v ) K ! Sk (u ,v )
2                           2
K k =1                        k =1
1 K
! Sk (u ,v )
2
4. Measure or determine transfer function
K k =1
— atmospheric model or measure reference star
1 K
!
2
5. Divide by        Sk (u, v )   to get F (u ,v ) 2
K k =1

Amplitude interferometry (Michelson) can also yield Fourier magnitude data
Reference: A. Labeyrie, "Attainment of Diffraction Limited Resolution in Large
Telescopes by Fourier Analysing Speckle Patterns in Star Images," Astron. and
Astrophys. 6, 85-87 (l970).

JRF 2/06-4
Phase Retrieval Basics
#
Fourier transform: F (u ,v ) =               \$\$   !#
f (x , y )e !i 2" (ux +vy )dx dy

= F (u ,v ) e         v
= F [f (x , y )]
i% (u , )

#
Inverse transform: f (x , y ) =            \$\$   "#
F (u ,v )e i 2! (ux +vy )dudv    = F "1[F (u ,v )]

Phase retrieval problem:
Given F (u,v ) and some constraints on f (x , y ),
Reconstruct f (x , y ), or equivalently retrieve ! (u,v )

[                               ]       [
F (u ,v ) = F [f (x , y )] = F e ic f (x ! x o , y ! y o ) = F e ic f * (!x ! x o , ! y ! y o )                  ]
(Inherent ambiguities: phase constant, images shifts, twin image all result in same data)

Autocorrelation:
rf (x , y ) =   \$\$
#
"#
f (x                                                    [
!, y ! )f * (x ! " x , y ! " y )dx !dy ! = F "1 F (u ,v ) 2   ]
•    Patterson function in crystallography is an aliased version of the autocorrelation
•    Simply need Nyquist sampling of the Fourier intensity to avoid aliasing
JRF 2/06-5
Phase of an Optical Field

•     Monochromatic EM radiation propagates as a sinusoidal wave
•     Phase is the relative position of the peaks and troughs of the wave
•     Phase also plays an important role in imaging via F.T.
2"
!=    OPD
"2                                                         #

Field = A2 = A2 e i!2
|A2|
2
Intensity = A2

Im

|A|
"
|A1|                                                                                        Re

JRF 2/06-6
Constraints in Phase Retrieval

•   Nonnegativity constraint: f(x, y) # 0
! True for ordinary incoherent imaging, x-ray diffraction, MRI, etc.

! Not true for wavefront sensing or coherent imaging

•   The support of an object is the set of points over which it is nonzero
! Meaningful for imaging objects on dark backgrounds

! Wavefront sensing through a known aperture

•   A good support constraint is essential for complex-valued objects
! Coherent imaging or wave front sensing

•   Atomiticity when have angstrom-level resolution
! For crystals -- not applicable for coarser-resolution, single-particle

•   Object intensity constraint (wish to reconstruct object phase)
! E.g., measure wavefront intensity in two planes (Gerchberg-Saxton)

! If available, supercedes support constraint

JRF 2/06-7
Phase Caries More Information
than Amplitude

G   |F[G]|      phase{FT[G]}   F –1[| F[G]| exp[i phase{F[M]}]]

M   | F[M]|     phase{F[M]}
F –1[| F[M]| exp[i phase{F[G]}]]

JRF 2/06-8
Why
Phase is More Important than Amplitude

•   Wave front = surface of constant phase: ! ( x , y , z ) = const
•   Light travels in direction perpendicular to wave front

Light spreads out – dimmer

Light is concentrated – brighter

! ( x , y , z ) = const

•   Where the light is concentrated, after propagation,
depends on the phase

JRF 2/06-9
Is Phase Retrieval Possible?

JRF 2/06-10
First Phase Retrieval Result

(a) Original object, (b) Fourier modulus data, (c) Initial estimate
(d) – (f) Reconstructed images — number of iterations: (d) 20, (e) 230, (f) 600

Reference: J.R. Fienup, Optics Letters, Vol 3., pp. 27-29 (1978).
JRF 2/06-11
Iterative Transform Algorithm

F

Measured intensity

F –1

Hybrid Input-Output version
\$g ! ( x , y )                 , gk ( x , y ) satisfies constraints
!
gk +1 ( x , y ) = % k
& gk ( x , y ) " # gk ( x , y ) , gk ( x , y ) violates constraints
!               !

JRF 2/06-12
Error-Reduction and HIO
\$g k ( x ) , x "S & g k (x ) # 0
!                   !
•   Error reduction algorithm        ER: g k +1(x ) = %
&          0 , otherwise
!   Satisfy constraints in object domain
!   Equivalent to projection onto (nonconvex) sets algorithm
!   Proof of convergence (weak sense)
!   In practice: slow, prone to stagnation, gets trapped in local minima

& g ! ( x ) , x "S & gk (x ) # 0
!
•   Hybrid-input-output algorithm HIO: g k +1(x ) = ' k
(g k ( x ) \$ %g k ( x ) , otherwise
!

!   Uses negative feedback idea from control theory
– # is feedback constant
!   No convergence proof (can increase errors temporarily)
!   In practice: much faster than ER, can climb out of local minima

JRF 2/06-13
Image Reconstruction from
Simulated Speckle Interferometry Data

Labeyrie"s
stellar speckle
interferometry
gives this

J.R. Fienup, "Phase Retrieval Algorithms: A Comparison," Appl. Opt. 2l, 2758-2769 (1982).
JRF 2/06-14
Error Metric versus Iteration Number

JRF 2/06-15
Autocorrelation Support
Autocorrelation:
rf (x , y ) =   \$\$
#
"#
f (x                                                 [
!, y ! )f * (x ! " x , y ! " y )dx !dy ! = F "1 F (u ,v ) 2   ]

Object
Support                      Forming Autocorrelation Support

0

Autocorrelation Support
JRF 2/06-16
Bounds on Object Support

Triple-Intersection Rule: [Crimmins, Fienup, & Thelen, JOSA A 7, 3 (1990)]
JRF 2/06-17
Triple Intersection for Triangle Object

Support
Object (a)           Autocorrelation Support
(b)

Alternative
(d)
Object Support
Support Constraint
Triple Intersection –(c)

\$• Family of solutions for object support from autocorrelation support
\$•\$Use upper bound for support constraint in phase retrieval
JRF 2/06-18
PROCLAIM 3-D Imaging Concept
Phase Retrieval with Opacity Constraint LAser IMaging

tunable laser

direct-detection
array

!1, !2, . . ., ! n
initial
estimate      phase     3-D
from     retrieval
locator    algorithm   FFT
set

collected                                     reconstructed
data set                                          object
JRF 2/06-19
Object for Laboratory Experiments

ST Object. The three concentric discs forming a pyramid can be seen as
dark circles at their edges. The small piece on one of the two lower legs
was removed before this photograph was taken.

JRF 2/06-20
3-D Laser Fourier Intensity
Laboratory Data

Data cube:
1024x1024 CCD pixels
\$\$\$x 64 wavelengths
Shown at right:
128x128x64 sub-cube
(128x128 CCD pixels at
each of 64 wavelengths)

JRF 2/06-21
Imaging Correlography

•    Get incoherent-image information from coherent speckle pattern
•    Estimate 3-D Incoherent-object Fourier squared magnitude
!   Like Hanbury-Brown Twiss intensity interferometry
! [Dk (u ,v , w ) " I o ] # [Dk (u, v , w ) " Io ]
2
FI (u, v , w )                                                            k
(autocovariance of speckle pattern)

•    Easier phase retrieval: have nonnegativity constraint on incoherent
image
•    Coarser resolution since correlography SNR lower
References:

P.S. Idell, J.R. Fienup and R.S. Goodman, "Image Synthesis from Nonimaged Laser Speckle
Patterns," Opt. Lett. 12, 858-860 (1987).

J.R. Fienup and P.S. Idell, "Imaging Correlography with Sparse Arrays of Detectors," Opt.
Engr. 27, 778-784 (1988).

J.R. Fienup, R.G. Paxman, M.F. Reiley, and B.J. Thelen, “3-D Imaging Correlography and
Coherent Image Reconstruction,” in Proc. SPIE 3815-07, Digital Image Recovery and
Synthesis IV, July 1999, Denver, CO., pp. 60-69.
JRF 2/06-22
Image Autocorrelation from Correlography

JRF 2/06-23
Thresholded Autocorrelation

JRF 2/06-24
Triple Intersection
of Autocorrelation Support

JRF 2/06-25
Locator Set, Slices 50-90

JRF 2/06-26
Dilated Locator Set
used as Support Constraint

JRF 2/06-27
Fourier Modulus Estimate
from Correlography

JRF 2/06-28
Fourier Magnitude, DC Slice

Before Filtering          After Filtering

JRF 2/06-29
Incoherent Image
Reconstructed by ITA

JRF 2/06-30
Support Constraint from Thresholded
Incoherent Image

JRF 2/06-31
Dilated Support Constraint
from Thresholded Incoherent Image

JRF 2/06-32
Coherent Image Reconstructed by
ITA from One 128x128x64 Sub-Cube

JRF 2/06-33
PROCLAIM Example System Parameters

Parameter          Symbol   Microscopic     Example      Megascopic
Center wavelength         !         0.5 µm       0.773 µm       0.5 µm
Range              R          10 cm         89 cm       1,000 km
Detector width         N"u         2 cm         1.23 cm         5m
Number of linear detectors   N          1,000          1,024          50
Detector pitch         "u        20 µm          12 µm         10 cm
Lateral resolution      #xy        2.5 µm         56 µm         10 cm
Lateral ambiguity interval N#xy        2.5 mm         57 mm          5m
Center frequency          \$     600x1012 Hz   388x1012Hz    600x1012 Hz
Bandwidth           M"\$      30x1012 Hz   3.12x1012Hz    1.5x109 Hz
Fractional bandwidth     M"\$/\$         0.05        0.00828      2.5x10–6
Number of frequencies       M           100             64          100
Frequency interval        "\$     0.3x1012 Hz   50.2x109Hz     15x106 Hz
Range resolution         #z         5 µm         46.7 µm        10 cm
Range ambiguity interval    M#z        0.5 mm         3 mm          10 m

JRF 2/06-34
New Interest in Non-Crystallographic
X-Ray Diffraction

• Non-crystallographic x-ray diffraction made possible by high-intensity,
highly coherent x-ray sources -- synchrotron radiation

JRF 2/06-35
Image Reconstruction from X-Ray
Diffraction Intensity

Target

Coherent
X-ray beam
Detector
array
(CCD)

(electron micrograph)
Collection of gold balls         Far-field diffraction pattern
(has complex index of refraction)       (Fourier intensity)
JRF 2/06-36
Example on Real X-Ray Data
(Data from M. Howells/LBNL and H. Chapman/LLNL)

(a) X-ray data          (b) Autocorrelation from (a)     (b) Triple Intersection

(c) Initial Support constraint   (d) Electron micrograph
computed from (b)                  of object             JRF 2/06-37
Active Conformal Thin Imaging System
(ACTIS) without Imaging Optics

Conformal Array
Coherent                             of Detectors
Illuminator

Modification of figure from Brad Tousley (DARPA/TTO)

•   Coherent illumination, sensed by a conformal array of detectors in far field
! Angle-angle (not range) image

•   Phase retrieval needed to form an image
! Heterodyne detection over large arrays is beyond the state of the art

! Support constraint formed by laser illumination system

•   With no imaging optics, a wider aperture fits on a platform
! Giving finer resolution, wider total field-of-view, thin system

JRF 2/06-38
Determine HST Aberrations from PSF

Measurements & Constraints:
Pupil plane: known aperture shape
phase error fairly smooth function
Focal plane: measured PSF intensity

JRF 2/06-39
Nonlinear Optimization Algorithms
Employing Gradients
i! ( x )
pupil model: g ( x ) = g ( x ) e          , G(u ) = F [ g ( x )]

!W (u ) [ G(u ) – F (u ) ]
2
Minimize Error Metric, e.g.: E =
u
Contour Plot of Error Metric                         Repeat three steps:

1. Compute gradient:
!E !E
!!!!    ,    ,…
!p1 !p2
Parameter 2

a
c
2. Compute direction of
search
b                        3. Perform line search

Parameter 1
Gradient methods:
(Steepest Descent)
Conjugate Gradient
BFGS/Quasi-Newton
…                                                                             JRF 2/06-40
Analytic Gradients
with Phase Values as Parameters

!W (u ) [ G(u ) – F (u ) ]             G (u ) = P [ g ( x )]
2
E=
u                                                                                         J
i! ( x )
Optimizing over g (x ) = gR ( x ) + i gI ( x ) = mo ( x ) e                       , ! (x ) =   " a j Z j (x )
j =1
For point-by-point pixel (complex) value, g(x),
!E
!g (x )
= 2 Im g    {   W*
}
(x )

For point-by-point phase map, \$(x),
!E
!" (x )
{
= 2 Im g ( x ) gW * ( x )           }
!E       #                             &
For Zernike polynomial coefficients,                 = 2 Im \$" g ( x ) gW * ( x )Z j ( x ) '
!a j     %x                            (
where                      "                        %
G(u )
GW      (u ) = W (u ) \$ F (u )        ! G (u ) ' , and gW (x ) = P † "GW (u ) %
#           &
#        G (u )          &

P [•]\$\$can\$be\$a\$single\$FFT\$or                                Analytic\$gradients\$very\$fast
multiple-plane\$Fresnel\$transforms                                       compared\$with
with\$phase\$factors\$and\$obscurations                             calculation\$by\$finite\$differences
J.R. Fienup, “Phase-Retrieval Algorithms for a Complicated Optical System,” Appl. Opt. 32, 1737-1746 (1993).
J.R. Fienup, J.C. Marron, T.J. Schulz and J.H. Seldin, “Hubble Space Telescope Characterized by Using
Phase Retrieval Algorithms,” Appl. Opt. 32 1747-1768 (1993).
JRF 2/06-41
Sources of Obscurations in HST

JRF 2/06-42
Hubble Telescope Retrieval Approach

• Pupil (support constraint) was known imperfectly
• Phase was relatively smooth and dominated by low-order Zernike"s
— Use boot-strapping approach

1. With initial guess for pupil, fit Zernike polynomial coefficients
(parametric phase retrieval by gradient search)
2. With initial guess for Zernike polynomials, estimate pupil by ITA
(retrieve magnitude, given an estimate of phase)
3. Redo steps 1 and 2 until convergence (2 iterations)
4. Estimate phase map by ITA, starting with Zernike polynomial phase
(nonparametric phase retrieval by G-S or gradient search)
5. Refit Zernike coefficients to phase map
6. Redo steps 2 - 5

JRF 2/06-43
Pupil Function Reconstruction

Pupil Reconstructed from ITA           Inferred Model of Pupil

JRF 2/06-44
James Webb Space Telescope
(Next Generation Space Telescope)

http://ngst.gsfc.nasa.gov/

See farther back towards the beginnings of the universe
Light is red-shifted into infrared
JRF 2/06-45
James Webb Space Telescope (JWST)

•   See red-shifted light from early universe
!   0.6 µm to 28 µm
!   L2 orbit for passive cooling,
avoiding light from sun and earth
!   6 m diameter primary mirror
– Deployable, segmented optics
– Phase retrieval to align segments

JRF 2/06-46
Fienup Group Visits “JWST”

JRF 2/06-47
Phase Retrieval for JWST

J. Green (JPL), B. Dean (GSFC) et al.,
Proc. SPIE (Glasgow 2004)

R. Lyon et al., (GSFC)

NASA has chosen phase retrieval
as the fine phasing approach for JWST.

D.S Acton et al.( Ball Aerospace),
Proc. SPIE (Glasgow 2004)
JRF 2/06-48
JWST at UofR
WaveFront Sensing Improvements

•   Develop improved WFS (phase retrieval) algorithms
!   Faster, converge more reliably, less sensitive to noise, 2% jumps
!   Work with larger aberrations, broadband illumination, jitter
– refining iterative transform, gradient search algorithms
Ideal PSF
– Increase robustness and accuracy
!   Extended objects
– Phase diversity
Phase retrieval performance

•   Experiments with UofR JWST laboratory simulator
! Adaptive optics MEMS deformable mirror

– 18 hex. segments
!   Interferometer measure wavefront independently
!   Put in misalignment, reconstruct wavefronts,
compare with interferometer “truth”

JRF 2/06-49
Optical Testing Using Phase Retrieval

Optical wave fronts (phase) can be measured by many forms of interferometry
Novel wave front sensor: a bare CCD detector array, detects reflected intensity
Wave front reconstructed in the computer by phase retrieval algorithm

Approach:
Simulation Results
PSF at z=333µm                      µm
PSF at z=f=500

Illumination
Part                    Wavefront
under
test
CCD Array              PSF1                      PSF2
measures              Actual wrapped phase   Reconstructed wrapped phase

Display of                  intensity of
measured                    reflected field
wavefront
Computer                   True                  Retrieved
Phase                   Phase

JRF 2/06-50
Phase Diversity
•    Phase retrieval works when have one measurement and one unknown
! Image reconstruction from Fourier magnitude

! Wavefront sensing through a known aperture, from a point source

•    Some problems have two unknowns: finite source and aberrations
! Wavefront sensing from extended object

! Need at least two measurements to determine two unknowns

aberrated optical system               conventional

}
unknown                                                                        image
extended object

known
diversity image
}   defocus
References:
length
R.G. Paxman and J.R. Fienup, "Optical Misalignment Sensing and Image Reconstruction Using Phase
Diversity," J. Opt. Soc. Am. A 5, 914-923 (1988).
R.G. Paxman, T.J. Schulz and J.R. Fienup, "Joint Estimation of Object and Aberrations Using Phase
Diversity," J. Opt. Soc. Am. A 9, 1072-85 (1992).
M.R. Bolcar and J.R. Fienup,“Method of Phase Diversity in Multi-Aperture Systems Utilizing Individual Sub-
Aperture Control,” in Unconventional Imaging, Proc. SPIE 5896-14 (July 2005).
JRF 2/06-51
Summary of Phase Retrieval

•   Phase retrieval can give diffraction-limited images
! \$Despite atmospheric turbulence or aberrated optics
! \$Despite lack of Fourier phase measurements

•   Can work for many imaging modalities
!   \$Passive, incoherent or laser imaging correlography
— Nonnegativity constraint
— Support constraint — may be loose and derived from given data
!   Active, coherent imaging
— Complex-valued image, so NOT nonnegative
— Support constraint: must be tight and of special type
— or helped by 3-D, or by opaque object, or by correlography
•   Can be used for wavefront sensing, using simple hardware
!   For atmospheric turbulence, telescope aberrations, eye aberrations
•   Current hot areas: JWST WFSC, Non-crystallographic x-ray imaging,
Unconventional laser imaging, Characterization of ultrafast pulses, etc.

JRF 2/06-52
JRF 2/06-53

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