nation in semiconductors exists, driven by device require- ..... The sideband cut ofi by the knife is redundant and even .... infinite transfer lens ...... twice higher if we automatically knew the fields at negative ... exit pupil phases estimated from the usual band pass as the ...... This is indicated by the heavy vertical line at.
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The knife edge test as a wavefront sensor
KenKnight, Charles Elman, Ph.D. The University of Arizona, 1987
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r,,'
THE KNIFE EDGE TEST AS A WAVEFRONT SENSOR
by Charles Elman KenKnight
A Dissertation Submitted to the Faculty of the DEPARTMENT OF GEOSCIENCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA
198 7
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Final Examination committee, we certify that we have read the dissertation prepared by entitled
Charles Elman KenKnight
The Kni fe Edge Test as a Wa ve:;..;f-,-r...;;;o.;.;n...;;;.t_S;;..;e~n_s...;;;o..;..r_ _ _ _ _ _ _ _ _ __
and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of ____...;;;D~o~c...;;;t...;;;o~r~o~f_P~h~'~·l~o~s~o~p~h~y________________________________
Date
/
~~Z
Date
6/S-/87 Date
Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation
r~;~y$~ ;6issertatio~Director
~s52~".'~
fDate
I
~~-.Q. ~J\'~(
STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission~ provided that accurate acknowledgement of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the materials is in the interests of scholarship. In all other instances~ hoYever~ permission must be obtained from the author.
SIGNED~_~_~~
IN MEMORY OF
Ida
Mu~arie
Krysto£ Serkowski
who were taken £rom those who could not help them because we did not understand what ravaged their molecules at the atomic level
iii
PREFACE
In
the
nineteen seventies the electron
microscope
began to be used xor a widening range ox problems in state chemistry and mineralogy.
solid-
Many important minerals are
not strictly periodic but contain local heterogeneities. the
usual xray crystallographic structure rexinements there
is no possibility ox taking into account submicron such as twins, boundaries. xraction
units.
dislocations, In
most
stacking xaults, or antiphase
structures
a
o£
structural
In these studies the capability o£ high
resolution
limits
composed o£ mixtures
the structure
resolution
periodicities
The microscope point resolu-
determination.
The
so-called
that reveals the presence o£
without being able to establish the
certain relative
location of those density £luctuations is to no avail. 100
tion
kV instruments ox 1980 or later attain a point o£
tiny
but increasing evidence exists xor
images is crucial to progress.
'lattice'
ieaturea
cases these xeatures occupy
ox the volume,
transition
tion
In
0.3 or 0.4 nm,
tantalizingly close to the
The
resolu0.2 nm
that would permit their application to most structural prob-
lems.
Considerable interest about dexect structure determi-
nation in
semiconductors exists,
ments £or industry.
driven by device require-
The semiconductor unit cell iv
dimensions
v
generally fall between those of
oxide and
and those of the close-packed metals.
sulfide minerals
They also approximate
the atomic separations in biological molecules, whose structures
are reported at the cost of many man-years per
cule.
The
comparison
known
biological structures are yet so
it were otherwise,
effects
and
few
in
to the number of medical importance that we must
admit an ignorance that is profound, If
mole-
if not nearly complete.
we might design drugs
without
having high specificity for a task.
side
We
might
understand and combat more kinds of cancer and design
tests
to identify its presence. In optics, above with up
a
what
might
be presumed to be another
area
of
in
the
large volume of literature has appeared
period which has been devoted to -beating the seeingimage data obtained by earth-based telescopes
looking
through a turbulent flow of air containing density fluc-
tuations usually called -heat waves·. point
From a
mathematical
of view the wavefront-perturbing screen just in front
of the telescope is equivalent to the specimen just in front of the objective lens of a microscope.
The turbulent screen
moves by rapidly so it must be sensed with pr.cious and photons borrowed from the faint scene;
few
similarly the biolo-
gical
specimen alters rapidly due to chemical readjustments
after
an ionization event caused by a fast electron
so
must be senaed with a few quanta per projected atom area.
it
vi Having
gained some proficiency in computer
tions of the optics of turbulence, electron microscope specimens.
simula-
I turned to the optics of
What follows are some sugges-
tions aimed mostly for microscopists and aimed for quantitative high resolution work. zero depth, surface.
For simplicity the specimen
which makes the sample into a surface,
has
a rough
I appeal to the reader if a given section seems not
to be in his specialty area.
The central theme of wavefront
sensing has many applications. The
central
interference. tering, matter
method
in wave£ront sensing
is
wave
A crucial instrument function is spatial fil-
i.e., tailoring of those interferences. The subject necessarily
involves
mathematics at the
level
of
Fourier transforms, vhich the reader may usually assume is a kind
of
applied trigonometry that vill be cared for
computer routine. go along.
The
The required language is developed as usual
£ormalism is algebra just
enough to serve our bookkeeping needs for the of interest.
by
a we
elaborate
interferences
The aim here is instrument design and instru-
ment use. Those hoping to find a simple relationship between a high-resolution image of a general specimen and its internal structure will be disappointed - again.
TABLE OF CONTENTS Page TITLE
i
STATEMENT BY AUTHOR
ii
DEDICATION
iii iv
PREFACE TABLE OF CONTENTS
vii
LIST OF ILLUSTRATIONS
ix
LIST OF TABLES
xiii
ABSTRACT 1.
INTRODUCTION
2.
SSB HOLOGRAPHY THEORY
2. 1 2.1.1 2.1.2 2.2
2.3 2.4 3.
xiv 1
15
Algorithms for a Phase Object Application to Microscopy Application to Other Problems Second Order Theory for an Amplitude Object Understanding the Second Order Terms Super-Resolution and the Knife Edge
SIMULATION OF A ROUGH SURFACE
3.1
3.2 3.3 3.4 3.5 3.5.1 3.5.2 3.5.3
3.5.4 3.5.5
Experiments on and Theories of Light Scattered from Rough Surfaces Pover Lava and Fourier Transforms The Transition from Inverse Cube to Inverse Square Lavs A Rough Surface in the Computer Simulation Results Surveying the Main Effects Optoelectronic Playback Optimum Subtraction of Correlated Waveforms The Approach to High Roughness Attempts to Use an Off-Axis Focal Plane Aperture vii
15 15 29 32 34 36 39 40
51 56
62 68 74 111
114 118 139
viii
Resolution Limits in Relat~on to Detector Noise Extending the Method to Atomic Phase Screens Controlling Edge Diffraction Effects
3.5.6 3.5.7
3.5.8
144 149 154
4.
FINDING THE DEFOCUS
169
5.
TILTED ILLUMINATION IN ELECTRON MICROSCOPY
179
5.1 5.2 5.3
Aberration Effects The sse Advantage The Challenge of Charging Aberrations
6.
CONCLUSION
7.
APPENDICES
7.A 7.B 7.C
B.
179 IB4 IBB 191
Exponential ACF in a Circular Pupil Normalizing Power Spectra from the FFT Matching Two Correlated Waveforms REFERENCES
194 195 197 199
LIST OF ILLUSTRATIONS Page Figure 1.
Schematic
geo~etry
o£ kni£e edge test.
2
2. Data o:f Kormendy [35] on slcattered light at the Palomar 48 inch Schmidt telescope.
42
3. Simulation o:f light scattering :from the edge o:f a lens having sur:face roughness.
47
4. Result o:f one dimensional simulation o:f e:f:fects o:f a kni£e edge a) centered on the central Airy disk or b) tangent to the :first dark ring.
75
5. Simulation o:f light scattering £rom a sur:face with initial rms roughness 0 = 0.2 radian and corrected in response to errors inferred from a kni£e edge test and the algorithm c:f Misell.
77
6. Scattered light :for initial roughness 0 = 0.2 radian a£ter no correction :for a second order imaging term CT(u).
82
7. Simulation with the same conditions as in Figure 4 but no correction was made for the second order imaging term CT(u).
85
8. Multiple sur:face.
86
loops o£ correction o£ a
rougher
9. Rms residual roughness (RRR) a:fter myltiple loops o:f correction under varied high noise conditions.
89
10. Rms residual roughness (RRR) a:fter multiple loops o:f correction under varied high noiae conditions, not using the second order .ubtraction o:f CT ( u ) •
92
11. Power screens.
95
spectra
for
two sine
ix
wave
ph •••
x Figure
Page
12. Testing the estimated ACF of the focal plane fields against the ACF calculated directly from the -true- fields.
97
13. One loop of optimum correction to surfaces with varied initial roughness and with a fixed and high (lQ~) noise due to the detector.
100
14. Progress in one loop of correction measured by rms residual roughness divided by initial roughness as a function of initial roughness and for 3 levels of detector noise.
102
15. Correlation of estimates of focal plane phases with true phases calculated as an unweighted average over all frequencies in the focal plane aperture and expressed as a -factor of improvement-. 16. Decreases in scattered light at the lowest (and strongest) spatial frequency after one loop of roughness correction using the knife edge algorithm with various initial roughnesses. 17. Caption as in Figure 16 except that tilts 56 to 64 are described as an average.
104
107
the 108
18. Comparison of two digital data reduction methods with an optoelectronic playback of knife edge test data.
113
19. The optimum multiplier Q of the estimated detector plane phases when measuring ~ phase object having various initial roughnesses.
116
20. Trends in the approach to high roughness. The incident light distributions in the focal plane are for various initial roughnesses.
120
21. Saturation effects in the approach to very rough surfaces.
124
22. Saturation of 4 moments of the irradiance distribution at the pupil plane detector during knife edge tests with very rough surfaces.
126
23. Shearing interferometer for thick supported near a phase grating.
131
samples
xi Figure
Page
24. A multi-wavelength interferometer suited to atudy of thick samples at selected separations of rays passing through the sample volume, schematically presented.
134
25. A scanning microscope for phase difference measurements.
137
26. Study of a system that makes successive corrections to a rough surface based upon data obtained with a pinhole on axis and a focal plane aperture well off axis.
141
27. Same roughness.
143
as
Figure
26
for
low
initial
28. The scattering lay for fast electrons diffracting from atomic phaae screens: the elastic differential cross-sections for light atoms.
148
29. Focal plane fields given as the moduli squared when edge diffraction ia strong and surface errors have been corrected to a low level.
155
30. Irradiance in pupil number 2 when an aperture edge occulted some of the low-angle diffracted light in the preceding focal plane.
157
31. Light distribution in a second focal plane when amplitude jumps existed at the edges of pupils 1 and 2, but the edge of pupil 2 was moved inwards to occult the bright edge there.
158
32. Light distribution in first focal plane when the preceding pupil is tapered linearly to zero at its edge.
161
33. Irradiance distributions in a second pupil after the first pupil vas tapered linearly to zero at its edge and a focal plane aperture removed low- and high-angle scattered light.
162
34. Light distribution in a second focal plane after tapers in two preceding pupils decreased the edge diffraction effects relative to surface scattering.
164
xii Figure
Page
35. Light distribution in a £irst £oca1 p1ane after a 1arger taper vas used at the edges o£ pupil 1.
165
36. Light distribution in a second £oca1 plane when preceding pupi1s were ~apered linearly to zero and the inner edge of the preceding £ocal plane aperture was placed at a zero o£ the beat pattern arising from the 1inear taper.
167
37. Effects of de£ocus on the spectrum of the pupil phases a£ter one loop of correction £or roughness.
170
38. Results o£ e££orts to estimate .S cycle of defocus using a cross-correlation of estimated fields in the detector plane and a priori known fields at the entrance plane, at various 1evels of detector noise.
176
39. The £ocal p1ane aberration phase £unction for electron microscopy in genera1ized units and defocus 0 = 0.7 Sch.
181
LIST OF TABLES Page Table 1.
O£ten-occurring Fourier trans£orms
2.
Values o£ optimum multiplier £or updating phases
xiii
16 117
ABSTRACT
An algorithm to reduce data £rom the kni£e edge test is given. The method is an extension o£ the theory o£ single sideband holography to second order e££ects.
Application to
phase microscopy is especially use£ul because a second
order
attenuate The
vanishes when the kni£e edge
the unscattered radiation probing
does
the
that
wavelength) quadratic The
nal.
trans£orm
sensed and corrected small (less than wave£ront
Convergence to a
optics quarter
null
until limited by detector-injected noise in best
£orm o£ the algorithm used only
o£ the smoothed detector record,
trans£orm,
solving
errors.
an
inverse trans£orm,
£or the phase o£ the input
Iterations were
help£ul only £or a
not
specimen.
algorithm was tested by simulation o£ an active
system
the
term
troublesome
a
was sig-
Fourier
a £iltering
and
wave£ront
an
o£
arctangent de£ormation.
Wiener £iltering of the
data record that smoothed high £requency noise be£ore analysis. The simplicity and sensitivity o£ this wave£ront sensor makes it a candidate £or active optic control o£ small-angle light scattering in space. a
two
In real time optical processing
dimensional signal can be applied as a voltage to
a
de£ormable mirror and be received as an intensity modulation xiv
xv
at an output plane. Combination of these features may permit a real time null test. should
allow
the
Application to electron
finding
of
defocus,
microscopes
astigmatism,
and
spherical aberrations for single micrographs at 0.2 nm resolution.
For
some
thin
specimens (up to nearly
100
layers thick) the left-right symmetry of diffraction measurements front
in a bandpass and reconstruction of the
deformations caused by the specimen with double
bandpass.
atom allows wavethat
1.0
INTRODUCTION The
kni£e edge test introduced in 1858 by Foucault,
like all the schlieren techniques, optical testing; the
test
a review is available (1).
geometry is in Figure 1.
corrugations, and
has been very use£ul
then
plane or spherical,
The essence o£
A wave£ront
£ree
£rom
encounters a test object
a lens that causes a real £ocus.
itself is the test object.
in
O£ten the
lens
At the focus a kni£e edge blocks
hal£ o£ the Fourier transform spectrum o£ the object and, desired,
half o£ the central Airy disk.
Behind the next lens
at or near the image o£ the object comes the detector, may
be
the
testing.
retina
eye
£or
qualitative
a television system,
and digitized.
which
optical
Here we will assume that the detector is an
o£ sensors, brated
o£ the
i£
array
or.£ilm that will be cali-
Phase perturbations o£ the wave£ront
due to the test object are changed into amplitude variations at the
the detector plane when hal£ o£ the light is blocked £ocal
record
plane.
would
make
Quantitative
decoding of
the knife edge test
into
the a
in
detector wave£ront
sensor. For Figure
1,
test
objects
placed at the pupil plane
Lohmann (2) introduced a method which
single-sideband (558) holography. 1
he
1
o£
named
In the £irst demonstration
2
)
PI LI
FP
P2 L2
Figure 1. Schematic geometry of knife edge test. PI entrance pupil, Ll lens under test or an objective lens about 1 focal length after a wavefront-perturbing specimen; L2 field lens; FP focal plane with knife and ray-limiting stop; P2 exit pupil and detector. For some of this study we assume lens Ll is deformable.
1.0
o£
3
method by Bryngdahl & Lohmann [3] the weakly
the
tering
object was
detector
illuminated by a plane wave and the £ilm
was not quite at the pupil plane 2 where the
would be in £ocus.
objec~
scat-
test
Playback (decoding) involved
re-
placing the £ilm plate by a positive image and reversing all light
rays through the system.
holography,
unlike
method,
but
versed
light
When
the more £amiliar
lensless
holography
the imper£ections o£ the lenses would limit the
O£ten
[4),
Lenses are essential to SSB
playing back through the same lenses with paths allows a high
quality
this is not possible or desirable,
re-
reconstruction.
the
imper£ections
would have to be measured in the course o£ taking
data.
We
return to this point below.
The
character o£ the object in£luences the way that
the data should be obtained. o£
a
£ew
When the test object
opaque or absorbing £eatures,
spectrum about the optical
a~is
consists
symmetry
o£
the
allows a good reconstruction
because no essential in£ormation is lost on the kni£e
edge.
A more elaborate setup wes deemed necessary for test objects that altered only the phase of the wave£ront, since symmetry about the optical axis is then lost in general and the knife edge obliterates some in£ormation needed for faith£ul reconstruction. ter
Bryngdahl & Lohmann therefore used a beam spl1t-
at the focal plane and arranged to detect the two
s1g-
1.0
4
nals occurring near tvo output pupils.
Again the light rays
were all reversed in the analog reconstruction. We
hoped that a more complete study o£ SSB hologra-
phy would £urniah the method needed to decode the kni£e edge teat.
We wanted to achieve a digital reconstruction o£
detector
record,
iterations,
in
pre£erably
by
a method with £ew
the interest o£ rapid data
the
or
no
processing.
We
hoped that attention to second-order e££ects might suggest a geometry
to pre£er during a measurement so that the
o£ use£ulness might be expanded, more rapid, o£
and so on. i. e. ,
de£ocus,
domain
or the reductions be
made
We also wanted to study the e££ects
placing the detector in the plane where
the object is £ocused by the trans£er lens(es). Figure small Ll.
1
aa drawn represents
a
telescope
having
£igure imper£ections on the large primary lens, Figure
1 £or that case is not to scale;
lens
the ratio
o£
£ocal length Fl to £ocal length F2 is apt to be 100 or more; then
the e££ect o£ imper£ections o£ lens L2 can
dered
negligible.
the primary lens.
edge,
Small-scale in£ormation about the lens Ll measured £rom the
The limit is set by the size o£ lens L2 or a
purposely introduced in FP because the detector has
limited
at
beyond which light approaching the £oca1 plane
FP cannot pass. stop
consi-
We would like high linear resolution
is lost because there is a largest angle, kni£e
be
number o£ samples across its width.
Loss o£
a
light
1.0
5
beyond
a
~imiting
angle de%ines a
be resolved on the lens L1. well
We
smal~est
especia~ly
d~tail
that
can
wanted to see how
we could estimate the location and height 0% the
bumps on axis
~ens
many
L1 using light on only one side 0% the optical
despite
the xact that some inxormation is
light is blocked by the knife edge. (a telescope in space,
lost
when
In several applications
electron microscopy) a second
expo-
sure vith the knife xlipped about the beam is not practical. A
central
aspect of this study is the limitation posed
the information loss.
We vill find that wavefront
by
de%orma-
tions should not be higher than about 1/4 0% a wavelength 0% the radiation probing the object because of the loss. Focusing of light by lens L1 causes a Fourier transform spectrum of the test object to occur in the focal plane FP.
We will be interested only in the interferences between
this
"scattered light" of the object and light in the
tral
Airy
disk.
interferences trum.
O%f-a~is
interferences 0%
Unavoidably,
there are also going to
between dif%erent parts of the holography
cen-
avoids difficulty
object
spec-
with
mutual
between dif%erent components of the
spectrum
the object by shifting the desired interferences into
high
frequency
passband that avoids problems with the
wanted interferences;
a un-
SSB holography must plan to deal with
the mutual interferences. scopy
be
In the context 0% electron micro-
Misell [5] asserted that the
unwanted
inter%erences
1.0
6
could
be removed easily.
there
is an observing geometry for which the usual
We
argue in section 2
sha~l
square-
detector can be nearly blind to the most harmful
~aw
interferences between Fourier components of a phase The the
condition centra~
is that the knife edge is
p~aced
Airy disk but not attenuating it.
We
mutua~
object.
tangent
in section 3 that the second order corrections
gested
by Misell give poorer
tions.
For
an absorbing object,
than no
however,
such
to
demon-
wil~
strate
resu~ts
that
sug-
correc-
corrections for
mutual interferences are appropriate. In contains
a
preceding finite
our
studies
the knife edge.
a~gorithm
of
approach possible, the
We
shou~d
optica~
expect to make
on~y
data can be used to correct Ll
iteratively.
is suitable even in the case that no iteration such as electron microscopy, ~oop
sets are possible, in
a one
But we might hope and do show that successive
input to a second
error
path
makes errors of second and higher order in the
because
of data-taking closely residua~
need to assess the accuracy of the
the
system
however perfect those data. This is because
mates the calculation of the
data
the
improvement in our knowledge of the object from
object phases. sets
that
deformable lens or mirror in the
detector record, our
we have assumed
signa~
is
ca~ulating
approxi-
phase function that we
a~gorithm.
When multiple
we only need to be concerned
a correction
This
sent to
the
that
deformable
1.0
7
element
becomes
remain.
In
small when
that
small
waveiront
context the algorithm need not have
accuracy ior large bumps. Accordingly, oi
deiormations
we emphasize a method
analyzing the detector record £or the needed
correction
signal that does not have lengthy iteration loops. we
high
However,
discuss whether iterations might be use£ul ior the
one-
oi-a-kind data represented by a lucky electron micrograph. We demonstrate in section 3 that a weak phase object is
recoverable
unlikely
case
£rom
kniie-edge image data,
o£ a rough sur£ace,
even
for which
in
the
the
Fourier
amplitudes of the scattered light have no symmetry about the optical
axis.
The ef£ect o£ £inite detector noise
measurement sequence is explicitly considered. have
of
the
We would not
to restrict ourselves to an object causing very
deformations, ment
on
small
according to information theory. The measure-
the image moduli at the detector but oi no
yields hali o£ the available in£ormation.
phases
That will allow us
in the SSB geometry to estimate all the moduli and phases in hal£
o£ the £ocal plane (to a tilt angle fixed by the
sity
of samples at the detector).
about
no
With a priori
absorption in the object,
estimste all the phases in the object, half the required information. irom
knowledge
we should be
able
however large,
to
again
It might be help£ul to begin
a ray optics solution [6] with
this was not studied.
den-
large
phases,
though
The present algorithm makes estimates
1.0
8
of
the fields until the very last step,
of
phase continuity and adequate sampling
This
system
atmosphere
finally
appear.
is not intended for imaging through the since
such a scene (the atmosphere)
weakly scattering object. dity
at which questions
is
whole not
a
We study the upper limit of vali-
of our algorithm and make some suggestions that should
help a study of very rough surfaces. If we have a priori knowledge about the scene, the
e.g.,
existence of one or more strongly scattering points
some
other recognizable pattern,
or
we will find that we
also estimate a few parameters of the transfer optics
can with-
out degrading any of the estimates of the object phases.
In
particular, defocus in light microscopy is easily found when a known pattern is included in the scene [7J. a
deformable
will
In the case of
lens we could put in special deformations
by defining a calibration mode to be used as
to find out about the transfer optics. reductions detector fields,
plane phases,
required
In general the
can be separated into two parts, or equivalently,
the
at
data
A) finding the focal
plane
B) correcting the focal plane fields for effects of
the transfer optics.
Problem A is the concern of sections 2
and 3. How to find data for problem B is attacked in section 4.
One of the aberrations of special interest is defocus.
will taking
present is
results that show that defocus
harmless.
during
Successive planes of a dilute
We
datathree-
9
1.0
dimensional object can be reconstructed as though a throughiocus series oi data had been recorded, not
to
be known to
object
microscopists.
a remark that seems
For
a
volume-iilling
only the iields at the output iace can be estimated,
however. In
SSB holography of an amplitude (light absorbing)
object the image irradiance distribution is, oi
aberrations
amplitude The
and to iirst order,
cut
Double-sideband
interference
between
[3). even
marred to
by
opposite
which causes overtones oi
each
component and destructive interierence of some comWhen the object alters only the phase oi parts
the plane wave,
oi
the resulting SSB image [3,8) hardly resem-
the object because the position of each Fourier compo-
nent in the image is shifted by a quarter cycle.
Thus Good-
man [9) dismissed the kniie edge test irom serious ration
as
simple
Zernike phase contrast geometry in which
flected in
the
and
imaging is
difiracted waves tilted
oi the optical axis,
ponents.
bles
(DSB)
to
object
ofi by the knife is redundant
undesirable.
Fourier
proportional
distribution immediately behind the
sideband
sides
in the absence
a
wavefront sensor,
preferring the
much the
less unde-
light (dc Fourier component) is delayed or advanced
phase by a quarter wavelength through use of a
some material. highly
conside-
A Zernike phase contrast image,
step "in
however,
desirable for a detector with a fog background
is
like
1.0
10
film,
because such an image is approximately proportional to
the phase variations in a phase object (ignoring the average irradiance) narrow
and
therefore the image has good contrast
features.
Film is the preferred detector
electron microscopy.
for
most
In section 5 we consider the feasibili-
ty
of
We
will discuss how to approximate the Zernike
~xtending
for
the present results to electron microscopy. phase
con-
trast condition in section 5. Arguments pro and con must be weighed concerning the use of a knife edge in electron microscopy.
In conventional
transmission electron microscopy we allegedly deal with weak phase what fully
objects [10, is
but see 11 or 12 and section 3.5.7]
here called pupil plane 1.
The 5SB
appropriate for a weak phase object.
algorithm
in is
Inelastic scat-
tering events are probable in a larger portion of the atomic volume than elastic scattering, deflections
near the nucleus.
energy-changing appreciably. and
general,
here,
angle scattering modifies the
atoms, image
scattering
is less simple
which is for elastic
than
scattering
the
study
only.
In
image data must be compared to synthetic images for
structural However,
Especially for light
Image interpretation with a mixture of elastic
inelastic
reported
small
which is dominated by large
interpretation
as
outlined in
section
3.5.7.
the SSB geometry is decisively attractive in compa-
rison to the usual D5B geometry for its reduction of defocus
1.0
and
11 chromatic
order
aberrations by a factor of 4
spherical
limiting
and" of
aberration by a factor of 16 at
resolution.
third
the
same
The defocus and spherical aberrations
must be estimated in the presence of noise; it helps greatly that they are smaller.
Also,
tude objects or phase
objects,
attractive
DSB
SSB
imaging is
decisively
because the focus may be changed at will
digital playback of the data. for
whether one deals with ampli-
This feature is not
during possible
imaging of a phase object because certain
components
are
image
of
a
A practical matter is that fast electrons are
phase object. especially
permanently lost from the DSB
Fourier
effective at polymerizing impurities within
the
vacuum chamber,
fastening them to electrodes that the elec-
trons
These
strike.
voltage Since
semiconducting
residues
develop
drop in order to conduct away
arriving
electrons.
the microscope is an electron interferometer,
exceedingly potential. edge But
sensitive
to these minor changes
in
it
is
electric
The only practical solution is to warm the knife
so that the conductivity of the residues is kept since
specimen
a
the knife edge is only millimeters away that
may be cryogenically cooled,
this
low.
from
a
solution
poses some engineering problems. In scenes ration.
space a telescope will attempt
observations
of
having large intensity ratios at small angular sepaAn
example might be a search for planets around
a
1.0
12
nearby, than
but unresolvable, star. Random zigure errors larger X/IOOO would cause an extended point spread
around the image oz the star that would prevent of
zunction
observation
planets even as £avorable as our Jupiter and Venus [13).
The Hubble space telescope is hoped to have zigure errors in the
~/20
range.
cal
zigure
Thus a planet search requires active opti-
control and,
in turn,
a wavezront
sensor
of
adequate sensitivity, rapidity, accuracy, stability, robustness and, above all, One find
simplicity.
tedious
aspect oz this method is the
need
and remove carezully any trends in sensitivity
the
detector
and also those
array
trends
in
to
across
irradiance
across the image caused by any amplitude jumps in the scene. For example, tion,
a lens 01 radius R without a central
obstruc-
no figure errors, and a knife edge along the y direcin an inzinite focal plane causes an SSB image
tion
tude of the form,
adapted from Ojeda-Castaneda [1],
A (x, y) = const x log I (x'" (RII _ya ) 1 This
expression
diverges
infinite transfer lens pupil was 6ssumed. zrom an adequate
edge;
zor
The
at the
I
II ) / (x- (RII _ya ) I
circ~e
edge
I
II
I)
because
an
between the focal plane and the exit
actual
polynomials of approximation.
these trends have been removed. should
ampli-
image
even
trends are slow away order to
In section 2 we
order 4 seem assume
that
Errors in the trend removal
mimic the detector noise that is studied in
section
1.0
3,
13
but
more study is needed.
Thus the work reported
here
will emphasize the geometry o£ microscopy,
in which the ex-
tent o£ the
smaller than the
specimen o£
width o£ the
interest is much
illuminating
used to capture the
wave or
o£ the
objective
(slightly>
scattered radiation and the
attenuated illuminating wave.
lens
This paper is not complete as
i t stands £or the purpose o£ lens testing, despite our opening remarks. A paper just be£ore that o£ to Wilson [14]
attempted to find a
Ojeda-Casteneda [1) due quantitative
method o£
analyzing the photometry £rom the kni£e-edge test of a lens. He considered only the case o£ a
kni£e-edge exactly
ting the point-like image o£ the light source, by
Bryngdahl & Lohmann [3).
is
attractive
£or a
under test. tion
as suggested
That choice £or kni£e position
simpli£ication o£ the
maximizes the impact of
split-
£ormulas,
but
di£fraction by the edge o£ the lens
As we shall show in section 3.5.8, edge di£frac-
effects are
smallest i£ the edge o£ the stop inserted
in the
focal plane
£ields.
Thus Wilson needed to confine himsel£ to data at the
lens center. reached below. plane
He did,
coincides with a zero of the
however,
A simple
existed at the input pupil, very small.
He
conclusion like one
linear filtering of the
photometry can yield the
tion vas
make a
wavefront
detector-
de£ormation that
but clnly in case that mentioned an
radiation
upper
de£orma-
limit on the
1.0
14 0.1~,
order o£ conclusion
which is to be compared with 0.25
below.
di££raction-limited
Because the image of the
(ultimately due to Lin£oot [15)
kni£e
touched the
light source,
of the A
£ull
scattered
two-dimensional
our
central
seems to be use£ul mainly
radiation acts as a radiat~on
~n
his method
on a chord passing through or near the lens center. central spot o£
~
But the
re£erence beam £or all
passed through the £ocal plane.
projection o£ a
three-dimensional
specimen volume can be studied by the knife-edge test.
2.0
SSB HOLOGRAPHY THEORY A
£irst-order
theory o£ SSB holography
was
given
by Bryngdahl & Lohmann [31, but here we extend the theory to second
order.
The casual reader may want to note the rela-
tionships summarized in Table 1 and then proceerl immediately to tests o£ this theory in section 3. exists
between
radiation object),
objects
A crucial
that absorb none
o£
di££erence the
(a phase object> and those that do (an
amplitude
Our "object"
so they are given separate sections.
may include the unwanted bumps on a lens,
probing
but we sball begin
with microscopy in mind.
2.1
Algorithms £or e Phase Object
2.1.1
Application to Microscopy A
monochromatic plane wave
ting along z is described by exp i(kz - Qt), is
a wavenumber
and
propaga-
without tilt
where k The
Q is an angular £requency.
o£ no interest here,
= 2n/~
so we drop the
time £actor
dependence
is
exp(-iQt>.
A£ter interacting with the object, the wave£ront
is altered so that a wave sur£ace o£ £ixed phase ia given by z
+
W(x,y> = constant,
parallel
to
2
where
W(x,y> describes
by which the wave£ront
is
a
distance
distorted.
The
product kW(x,y> is a phase £unction that we cal1 n(x,y>. is
given in radians and has B
1/~
15
dependence on
the
It
VBve-
2.1.1
16
Table 1. Often-occurring Fourier Transform Relationships
Function
Fourier transform
Pupil field E(x)
Focal plane field
Detector field El ex)
Field in FP aperture q(u)
Detector record l(x)
FP field estimate
Pupil field ACF(x)
FP field squared
~(u)
~(u) I~I·
(u)
2.1.1
17
~ength
of the radiation used to probe
~
general, waves
we
to
specimen.
require an infinite ensemble
describe the scattered
assumption a
would
the
waves.
of
However,
In
plane on
the
that nearly all the scattering of interest is in
small cone about the forward
direction,
we
approximate
kW(x,y) by a £inite series sin
UM,
=
UM
2nmx/L,
o S m S L/2 where
we have suppressed the transverse y dimension in
notation with no essential cost to the argument. L
should be indefinitely large to allow
scattering
angles;
scattering,
the
in
the digital
vanishingly
approximation
% (A •
-
iB)
e-
1U •
e-
I U
small to
the
specimen
We find that a more econo-
notation emerges if we replace 2 cos u by e 1U
and 2i sin u by e 1u
our
The length
length L is apt to represent the
width or a small multiple of it. mical
(2.1>
e-
•
1u
The mth term o£ kW(x) becomes •
% (A -
iB)
e
(2.2)
l U
To recover a plane wave interpretation we may set
where
eM
k,x
= kx
sin
eM
=
UM
k3Z
= kz
cos
eM
~
kz
(2.3)
is an angle measured from the (forward) z axis.
On
comparing the two plane waves a exp (1Ck a z - k,x» a
to
(2.1)
=
(A
and
+
i8)/2,
(2.2),
or a· exp (i(k 3 z
and a e
=
+
k,x»,
(A - i8)/2
we need only a factor
(2.4)
of
exp(ik 3 z)
2.1.1
18
multiplying each term of kW(x) to exhibit plane former
plane
wave
has a surface normal with
along negative x when sin e is taken positive. plane
wave
a
A
complex number.
The
component
We refer to a
of this form as a "positive tilt"
coefficient
because
its
is corresponds to the usual notation for
~
a
The corresponding "negative tilt" plane wave
s-.
has the complex conjugate amplitude (2.3)
In
waves.
we noted that the z-component kJ
of
the
wavenumber
k is a function of the scattering angle e..
Our
assumption
of
a "small cone about the
forward
direction"
becomes more definite if we now require angles small so that cos e. multiply
is
sensibly constant.
In that case we could
all the terms of interest in (2.1) by
factor exp(ikz) to exhibit plane waves. of no interest and would be dropped.
enough
a
constant
A constant factor is
However,
if we were to
take an interest in the wave field distribution at a "small" distance
Z
approximation cos
eM)
from the object or its image, would be too crude.
1
The phase factor exp(ikZ
in the plane waves of index m should
somehow into E(x),
~
the cos e.
e.g., the phasors aM
be
inserted
and aM- should each
be altered by multiplying by exp(ikZ) and by exp(-ikZe.
8
/2).
This approximation ignores inverse-square falloff of amplitudes scattered by the object in the above -small" distance. The quadratic phase factor can be expressed more usefully in terms of UM using (2.3) for sin eM
= uM/k,
whence the
phase
19
2.1.1
becomes -inZm 8 /Z o ,
change distance
where the
characteristic
axial
Za for a grating of period L diffracting radiation
of wavelength
is
~
L8/~.
"Small" Z means that Z/Zo is a very
sma.ll ratio. Following
a phase object the fie.ld distribution may
be taken proportional to exp(in(x». To first order we have E(x;z) = e' At
a
II.
(1
+
1: aM
exp( -iuM)
+
large but finite distance from
aM - exp(iuM» the
object,
(2.5) we
may
consider that the description of the field by p.lane waves as in (2.5) vi.l.l fai.l and that these beams wi.l.l separate enough so that a square-lav detector of convenient size wi.l.l have a response proportiona.l to aM-aM = laM that
notion
think
of
equal.ly axis.
18
in beam number m.
is an experimenta.l fiction,
samp.ling spaced
we can
the power f.low out through
angular regions centered on the
at
.least
finite
and
forward
Divided by the incident pover f.low and changed to
amp.litude
by
a square root operation,
we have the
an
of
fie.ld distribution E(x) separates E(x) into p.lane waves
or beams at a .large distance and assigns aM than
the above model,
the
however,
modulus and the phase,
which that P1
z
ingre-
dients of understanding that a digita.l Fourier transform the
If
i.e.,
va.lue.e.
Better
the transform returns both a comp.lex number for
serves as a phasor having 2 components.
Thus we
(2.5) describes the fie.ld distribution near the in
Figure 1 and that a set of aM va.lues
describes
aM say
pupi.l the
2. 1. 1
20
field distribution in the focal plane FP of Figure 1. Behind the stops in the focal plane ve should set to zero those values that correspond to blocked rays. the At
In particular,
aM all
amplitudes corresponding to negative tilts are removed. the
(infocus)
image of the object we
have
the
field
distribution (to first order in n(x) for nov)
E,
=1
(x~z=O)
... E aM exp(-iuM)
(2.6)
and ve obtain the detector record I
(x)
= E,
e
El
(2.7)
Suppose that in the absence of any object we call the plex)
field distribution at the detector R(x),
(com-
so that
in
general with a weak object O(x) present E, (x) = R(x) ... o(x)
(2.8)
Then the detector record becomes lex)
= ReR
... ReO ... RO· ... 0·0
(2.9)
For any recording geometry of interest the function R(x) may be
made
system. side
purely real by a trivial In
rotation
of
coordinate
microscopy the illumination can extend far out-
of the region of interest without e sudden
change
in
magnitude, so the function R(x) can be taken to be 1 with no appreciable loss in
generality~
lex) = 1 ... which
The detector record becomes
a ...
O· ... 0·0
(2.10)
exhibits terms linear in the object amplitudes and
a
second order imaging function O·(x)O(x) that causes concern. A Fourier transform of I(x),
which we shall denote by
~(u),
2.1.1
21
recovers (if noise vanishes) the set of field amplitudes
at
positive
tilts
They are separated from the
~(u).
complex
conjugate set of amplitudes q- (u) by placement of the latter in
negative tilt pOSitions in the
both which
positions is
computer
array.
distribution
q(u)
~
are compromised by a function q(u)
the autocorrelation function (ACF) of
Sadly,
the
field
that was transmitted through the positive tilt
region of the focal plane [5). To
outline
a theory accurate to second
order,
we
e.g.,
we
need at least two spatial frequencies from (2.1), can
call them u and v in abbreviation for
UK
and
where
UN,
the indices m and n are distinct. Then let us write n(x) where
=a
by
e-
LU
~
b e-
"negative
obtainable
from
conjugates,
terms
symmetrically
1Y
~
tilts"
negative tilts, we
mean
those written out by that
correspond
v > u,
additional
~orming
to
placed about the forward z
(2.11)
the
terms complex
scattering axis.
angles
The
field
distribution that corresponds to the phase distribution n(x) to second order is (apart from the z and t E(x) = exp(ln(x»
=
1 • in(x) -
dependence) ~
ne(x)
(2.12)
If we write out
• negative tilts and
remember that the radiation corresponding to
(2.13) "negative
2.1.1
tilt
22
plane waves" will be blocked,
we can set down £or the
£ields in the detector plane E, (x) = Eo Eo
E,
(x)
... i
E, (x) - Ea (x)
=G
(1 - P)
=
e-
a
I U
...
b e-
Ea (x) = a- b e-' c v ...
e-
(aa
lau
U I
l
...
ba
...
v
ab e-' C v + U e-
18V
I
)/2
(2. 14)
where the zero-£requency (dc) amplitudes were attenuated
by
a transmission £actor G £or generality and the "power" P, (2.15)
represents positive
a sum over the absolute squares o£ amplitudes at tilts
approximation
within
the £ocal
plane
aperture
that cubic and higher terms
are
in
tne
negligible.
Then to quadratic order in the £ields the detector record is lex)
= GR
(1-2P) ... G(iE, (x) - Ea (x» ... G(-iE, -
(x)
-
Ea-
(x»
(2. 16)
where the by
the last term,
arising £rom mutual inter£erences
object-scattered £ields in the £ocal plane not the kni£e or other stops,
o£
blocked
contributes P to the dc level
and a "cross term" CT(x) plus its complex conjugate CT- (x) E,-(x}E, (x) = P
...
CT(x}
CT(x) = a·b e- , From spatial
the
linear
C
v-
...
CT- (x), (2.17)
U I ,
theory [3] we
expect
that
£requency o£ the object is shi£ted by a quarter
each o£
2.1.1 its we
23
corresponding grating period in the image. need to consider only,
say,
To see this
£requency u in the sum
o£
iE. (x) and -iE.· (x) in equation (2. 16) : i a eComparing
lU
-
i
with (2.1),
ae
e lU
=-
B cos u
+
we £ind that A has been replaced
(2.18) by -B and that B has been replaced by A, responds phasor
( 2. 18)
A sin u
to a phasor advance by 90 degrees.
which
in cor-
The choice
advance or retardation depends on the choice o£ i
o£ or
-i in (2.12), which is not important £or a consistent set o£ digital Fourier trans£orms. Thus the present £ormalism gives the
expected position shi£ts o£ each Fourier
the
object
in
the linear terms
representing
component the
o£
image.
Higher order e££ects will modi£y these shi£ts. Next choices
we
want to notice what
happens
o£ the transmission £actor G.
£or
various
The mutual inter£er-
ence (last) term in (2.16) is not at all attenuated when the unde£lected edge
(dc) beam is attenuated by inserting the
into the central Airy disk.
£erence
is distinctly un£avorable;
But strong mutual we wanted all
second order terms to be as small as possible.
kni£e inter-
the
0·0
There£ore we
would like G to be o£ unit value, or at least as near to one as cuts
i t can be. the
central Airy disk.
negative tilts, as we can.
We reject a geometry in which the kni£e edge Yet we want to cut
away
all
i.e., as much o£ one side o£ the £irst ring
An adequate tradeo££ is achieved by placing
the
2.1.1
24
kni£e that
edge tangent to the £irst dark ring.
Note,
this choice leaves a line of object 1ields
knife
edge in,
say,
however, along
the +y direction which have
the
matching
complex conjugate £ields on the opposite side 01 the central Airy disk (at -y) 10r which DSB interferences may cause loss of
modulation
plane
in the detector record
I(x,y).
fields would have to be discarded during
Such
£ocal
reconstruc-
tion unless the worker finds it 1easible to indent the knife edge with a half-circle that encircles the central Airy disk
during
recording,
[81.
Actually,
tron
microscopy,
as has been done in electron
microscopy
the Airy disk is inacceSSibly small in elecbut the half-circle has a radius
smaller
than that for the lowest Fourier components of a crystalline specimen. An
even more compelling reason to prefer the
G
=1
geometry is a cancellation to second order of the cross term CT(x) of (2.17) with the first term in Ea (x) 01 latter
(2.14)~
term has a coe1ficient -G to second order in (2.16).
A more complete discussion of the cross term CT(x) is
given
Here we emphasize that the cross term CT(x) involves
below.
the difference frequency v - u, nents
the
of
unwanted
the term.
so that low £requency compo-
object are especially at peril due By comparison,
to
all the other second
terms in E.(x) are Bum frequencies.
this order
If the spatial frequency
corresponding to the outer edge of the focal plane
aperture
2. 1. 1
is
25
then those ti1ts u and v £or which 2u,
~
are
greater o£
£ields
Remember at a11,
than
~
give no inter£erence;
or u + v
the £oca1
these harmonic terms are blocked
by
plane
the
stop.
that what we would pre£er is no second order terms so we should seize any opportunity to be rid o£ one.
1£ we p1aced detectors in the £ocal plane, in
2v,
section
important
3,
we could not avoid the unwanted
di££erence £requencies.
detector
and
as we shall show
a
By using a
particular recording
£ields pupil
geometry
to the di££erence frequency
plane
we
mutual
at
become
nearly
blind
inter£e-
rences
o£ a pure phase object because the squaring o£
n(x)
in (2.12) and the absolute squaring o£ E. (x) at the detector caused the cancellation o£ CT(x). Taking then G I
(x)
=
= (1 - P)
1 in (2.16) and (2.14), +
iE.
(x)
-
(E.
(x)
we find
CT (x)
-
)
(2.16')
+ negative tilts In E.
(2.16') I(x) corresponds closely to the
detector
including the second order sum frequencies u+v,
(x),
and v+v surviving in the last bracket of (2.16'), ding
field
the negative tilt terms.
conjugate surface)
waves are
caused
but exclu-
The negative tilts (complex
by the squaring
no problem [3];
u+u,
they
at
should
the be
detector considered
redundant numbers required by a purely real detector record. In the Fourier trans£orm of lex), negative
u
contain
!.(u),
the same information
the components as
the
at
desired
2.1.1
26
waves, at
those at positive u. If we set to zero those phasors
negative u,
equation (2.16')
revea~s
that
positive
th~
tilt components are the same as the original fields, for
CT(u),
stops
placed
~ays
in the portion of the
plane passed by the
This happy circumstance suggests
two
to process the detector record lex) so as to find
the
fields
there.
foca~
except
at the detector surface when the wavefront
tions are not too large,
deforma-
i.e., when scattering is weak.
A first approach is to ignore all second order terms in
(2.16'),
to depend on the
cancel~ation
dominant effect so that other effects are result
cou~d
I(x);
its Fourier transform
u;
the
be a
simp~e
of CT(x) to be a unimportant.
The
linear filtering of the data record ~(u)
is set to zero at negative
inverse transform approximates E, (x) so that en arc
tangent routine gives the phases of the fields at the detector
plane.
Those phases are proportional to the
wavefront
deformation after the radiation exited from the specimen. We do
not assert yet that ve vill not be harmed by the loss of
the
radiation (information) having negative
~eft
the back surface of the object. A reasonable simUlation
Yi~l
be needed to assess that loss.
ti~ts
after
it
One small complication is posed by the fact that the dc What
level of I(x) should be 1 - P when we do not yet know P. we can do about that is to set the mean value
detector record to 1:
of
the
2. 1. 1
27
I. (x) = 1
The
(iE I (x) - E. (x)
+
transform of I.
Fourier
CT (x) ) / ( 1-P)
+
!"t
(x),
(2.19)
has
(u),
a
one-sided
"power" estimate satisfying
EST = P/(1-P)8 which
has
(2.20)
one unique solution for P that can
numerically.
Then
be
obtained
every term of !..I (u) can be multiplied by
1-P to restore a normalization corresponding to (2.16').
procedure can be carried out on lex) before
equivalent
An the
Fourier transform. The
(2.16')
second approach is to note the form of
and make an attempt to reconstruct the "fields" in the focal plane
as
second
accurately as possible;
transform of the detector record,
attempt to put it back in. we
can
know
that
only the cross term CT(u) is absent from
order
Fourier
since we
zero
!..(u),
we
to the
could
To estimate the cross term CT(u)
all the tilts of !..(u)
not
corresponding
to
radiation passed by the focal plane aperture of Figure 1 and also
zero the dc position.
After transforming back to
the
detector plane we obtain an estimate of the function E. whos~
f
(x)
= i
EI
absolute square is, 1.
Upon
0
0
f
(x)
transforming
again, positive
we
see tilts
= P
(x)
-
Ell
+
(x)
CT ( x
(2.21)
)
to quadratic quantities, +
CT (x)
+
CT-
(2.22)
(x)
the latter function to the
from (2.22) that the set of within the focal plane
focal
plane
amplitudes
aperture
gives
at an
2.1.1
28
estimate
ox CT(u).
In that case we can estimate
£ield
~he
distribution in the £ocal plane at positive tilts using Q,; (u) = !. (u) - CT ( u ) Axter ~
( 2. 23)
putting zeroes at all negative tilts,
(u)
to
Os (x).
the detector plane allows a
a transxorm o£
phase
estimate
This prescription was that o£ Misell [5]. Whether
CT(u) is to be subtracted £rom !.(u) or not,
we might try to £ind out by additional calculations Os (x)
is
consistent
started.
The
with
the lex)
data
with
phases ox Os (x) just £ound,
which
we
I£ we use the
replace the modulus by II/·(X),
trans£orm to the £ooal plane,
we usually £ind that the
amplitudes
outside ox the £ocal plane aperture
small
not
but
whether
£unction II/·(X) is proportional to the total
wave modulus at each point x ox the detector.
and
£or
zero.
Trimming outside the
region
are
aperture
with
zeroes and inside the aperture with (2.23), we can trans£orm to
the
detector plane and update the
phases
there.
This
process can be iterated until no changes occur in Q,;(u>. The iterations
o£
this paragraph correspond
those suggested by Gerchberg & Saxton [16]. tion
approximately
Again, a simula-
is desirable to assess when any o£ these second
complications are worth doing.
to
order
29 2.1.2
Application to Other Problems The foregoing scalar diffraction theory for a
tering the
scat-
problem is suited to small angle scattering in which
polarization
observed.
The
of the scattered particle
is
not
being
same theory is aptly suited for high voltage
electron microscopy of thin specimens. A definition of -thin specimen-
will be given later.
In this section we want
to
note some expressions related to the theory already outlined that
may be of interest when measurement of a
components spatial object is
of
few
some object might be aided by
frequency
shifting
the
We want to include
an
such as the imperfect figure of a telescope when
it
being
higher or lower.
Fourier
used to image an isolated star in
companion
objects
very much fainter,
a
search
such as
for
planets
in
orbit around it. In
electron microscopy a
spe~imen
of interest
must
usually be supported on another material. A common choice is amorphous support the
object.
though
amorphous
is certain to compromise every Fourier component of
Fourier
the
carbon even though scattering from the
A crystalline support would affect only a
components.
The specimen is often a grating
we need not insist on that situation.
product of two transmission functions
following
also,
In any
ari ••s.
few
case, In
the
Fourier transform plane we shall be interested in
sums and differences of spatial frequencies.
2.1.2
30
In measuring
the planet search there can be speci£ic
di££iculties
spatial £requency ranges,
in
notably
the
lowest £requencies o£ the telescope £igure errors, where the bright star and a £ew lowest £requencies must be blocked
to
control edge-di££racted light (see section 3.5.8).
we
give
Thus
here a second order calculation o£ the detector record
£or a weak object next to a £airly strong phase grating. Let the weak object be represented by (2.24) For the grating we will need second order terms 1 but
ve
i
~
vill
b
e-~v
i b-
~
e~v
- b-b
harmonics
~
assume v > 2u and that the harmonics
phase grating will be blocked in the £ocal plane. plying
these expressions and a£ter dropping
(2.25) o£
the
On multi-
negative
tilt
terms, we £ind £or the £ields at the (in £ocus) detector El
(x)
= 1
b e-
~
i
-
a- b e-'
l
i
a
e-'
I V - U I
-
ab e-
v
~
U
~ I
v.
II I
- b-b
(2.26)
Calling b-b by the name P in analogy to (2.15),
we £ind £or
the detector record I(x)
=
(l-P)·
~
(l-P)(ib e- lV
~
-ab e-·I C v •
ia e- lU - a-b e-IIV-UI II I
~
negative tilts)
~
iba- a e-' v - iab- b e-' U
~
a-a
~
b-b
~
~
negative tilts
a- b e- I I V- U
I
2.1.2
31
• Paa b e-
l
, \I - U)
-
(1-P) ab e-
I , \I. U )
• negative tilts About
(2.27)
these terms we note 1) a dc term 1 - P;
term
slightly
I-Pi
3)
increased in
cancellation if
an
4) a di£ference term that was a near
and is of minor importance if P is not
the
frequency
p~
v
addition of a-a to
a tilt u term slightly decreased in modulus by
extra subtraction of
i. e. ,
modulus by
2) a tilt
phase grating is not too strong;
large,
5)
a
sum
The
term that survives with full contrast.
last
term could be of interest when the tilt u term is not accessible
at
passes
low
spatial £requencies (we
the dc beam).
last term) are e-
i. e. ,
I II ,
presume
All the object waves (ia e-
multiplied by the
fixed
a
pinhole
1U
in
quantity
i(1-P)b
the low-frequency object information has
shifted
by
the phase grating to frequencies around
course,
at
these same frequencies there is also high
quency
object information.
options. frequency.
Option The
a)
The experimenter then
components
has
from v upward
v/2
Of fretwo
Ignore the weak object waves at
Fourier
can be used.
been v.
high
can
shifted down by v and the previously given knife edge rithm
the
be
algo-
Those Fourier components from v down to
are more affected by the ignored object waves but the
they
carry
information about the negative tilts of
object,
which
we vill find are nearly redundant information about a
,.. 32
2.1.2
'
weak phase object. [17-19)
but
A similar suggestion was made by
without noting that the positive and
tilts should be processed separately. tional low
Option b)
negative Make addi-
observations that allow separation ox the
xrequency object waves near £requency v.
quency
object
third
term
waves are o:f the :form
o:f
(2.27),
high
and
The high :fre-
ia(I-2P)e-
Since the third term
others
aU
in
contains
the no
dependence upon the position o:f the phase grating while
the
sum
the
xrequency
(£i:fth) term o:f (2.27) does depend upon
orientation o:f the phasor b ,
there might be the possibility
ox shixting the phase grating by hal:f its period. second detector record corresponding to (2.27), rence
ox b is replaced by -b.
Then in a each occur-
The records can be added
to
estimate term 3 and subtracted to estimate term 5.
2.2
Second Order Theory :for an Amplitude Object As before we begin with a plane wave o:f unit modulus along the z direction.
propagating
We take a
transmission
:function t(x) ox the :form t(x)
=1
-
2P
+
a e-&U
+ b
e-
lV
+
negative tilts, (2.28)
v > u,
where as be£ore P is the one-sided sum o:f squared amplitudes over
all
positive tilts.
The choice :for dc
bias
assures
constant total power in the xooal plane to :first order in P, because
2.2
33
It (x) III = (1 - 2P) II
= and
(1
(1 -
+
negative tilts
2P) (1
-
2P) E. (x) ... 2 P
+
+
4p8 1 (1 - 2P)
+
2 Ea (x)
E. (x) ... • •• ) ( 2. 29)
+
as ve saw in (2.22) that the E. (x) terms
total
power
negative
in the £ocal plane of P.
tilts corresponding to
frequency, £ields,
contribute
a
With another P
£rom
E,- (x) and 1 - 2P at
zero
ve have a total pover of 1 to second order in the
i. e. ,
to £irst power in P.
Inserting a kni£e edge
that does not touch the dc beam, the detector £ields become E,
= 1
(x)
- P +
a
e-' U
b
+
e-'
(2.30)
Y
The detector record is I(x)
=
(1 - P)8
=
(1 - P)(1 ... p a /(1-P) + E, (x)
+
(1 - PlEa (x) ... P
CT(x)
+
+
negative tilts
+
CT(x)/(l-P)
+
••• )
(2.31) The
expressions (2.29) and (2.31) give the DSB
SSB images of an trast
of
provided
object,
respectively.
first order E, (x) terms
the 4pa
amp~itude
«
1
-
2P.
exhibit
For example,
for
a
The no
and con-
change
sinusoidal
grating object we have a or b
=
much less than 1 - 2P = 1/2.
For such a deep modulation the
1/2,
P = 1/4, and 4pa is not
SSB image haa better first order contrast. no
overtone terms for the SSB image;
(2.29)
for
difference
the
DSB image.
In (2.31) we see
they are
Both image types
frequency term CT(x),
but in
that have been normalized to unit mean,
present contain
detector
in the
records
the SSB coefficient
2.2
34
for CT(x) is 1/(1 - P); the DSB coefficient of 2/(1 - 2P) is more
than
objects
twice as large.
We conclude that for
amplitude
we should always prefer the SSB image over the
eSB
image, other things being equal.
2.3
Understanding the Second Order Terms The second order product O·(x)O(x) in (2.10) emerged
in
our
treatment o£ two frequencies as the
e-
1 Cy - u I •
This
product
aab·
second order term arose from mutual
inter-
ferences between the focal plane fields resulting from scattering by the object into a range of positive tilts. there
are many frequency pairs v and u.
For a
Usually
given
dif-
ference frequency v - u a more complete formalism would show that
we
could
are to £orm a sum of products of phasors
appreciate
the nature of the sum better if
moment we would express the phasors in terms of ~
= U
~
iV and
~
a-b.
= K
a-b
~
iY,
whence
(UK
=
dot product • i cross product
VY)
~
i
(UY - VX) (2.32)
When the phasors entering the aab products are numerous have
a
the
components,
=
~
for
We
wide variety of orientations,
cross products tend to have both signs.
their vector dot
and or
The algebraic sum of
a large number of such vector products is then a random walk in two dimensions and tends to yield a small phasor sum. The phasor
~
in the focal plane will have components U
~
i~
when
2.3
it
35
describes a wavefront deformation in the pupil
cosine-like
at
i.e.,
it describes a bump with maximum at
when
that
we
the
spatial frequency associated
happen to have chosen.
The phasors
~
that
is
with
~
an or
origin in
~
the
focal plane will have random orientation whenever the object in
the
pupil plane Pl of Figure 1 has the character
diffuser Random
--- no
matter how weak that
orientation
scene
diffuser
of b means that nothing in
defines a highly special origin.
of
might the
~
be.
object
The sharp edge of a
lens defines a special origin so that the Airy rings of edge diffraction have a unique pattern of focal plane phases. surface the
But
errors have no special position in the aperture
resultant
consequence
~
the
phasors are random in
orientation.
ACF of focal plane fields caused
As
by
roughness tends to vanish for any v not equal to u.
so a
lens
A simi-
lar result should be expected for the wavefront deformations caused by passage of electrons through biological molecules, even
if the molecules adopt a crystalline pattern of
larity.
When the crystal unit is repeated N times across the
image width L, is
regu-
each Fourier component of that crystal
unit
necessarily folloved by N-l phasors of zero length.
fastening often
attention on the non-zero components,
we
should
expect random orientation of the phasors because
tangled
molecules
of
biology often
have
special
origin within the crystal unit.
no
But
the
discernible
Smallness of
the
2.3
ACF
36
is very helpful in case of appreciable
the transfer optics. lation
of
in
They would seriously complicate calcu-
CT(u) since the image amplitudes
attenuated
aberrations
would
be
by an envelope function whose effects should
be
~(u)
known and corrected (201.
Super-Resolution and the Knife Edge
2.4
The
phases
at
the exit pupil plane
a
band-
limited version of those at the entrance pupil plane,
since
the focal plane aperture is finite.
are
The resolution would be
twice higher if we automatically knew the fields at negative tilts where they were blocked by the knife.
We offer a
few
remarks about when that is or is not possible. The
expansion of exp(in(x»
order amplitudes (-B +
iA)/2.
+
iA)/2;
gave the
~(u)
first-
at negative tilts they are (B
Filling in negative tilts with this symmetry using
data
for positive tilts in the focal aperture is valid
very
weak phase objects.
The even-order phase harmonics do
not have this symmetry but if na(x) could be estimated the
for
from
measures on a weak phase object (evidently n(x) must be
of order unity or less),
the appropriate additional
ampli-
tudes could be filled in at negative tilts. Such an activity is a pursuit of super-resolution by a factor of 2, which has little chance in the presence of A more costly procedure,
appr~ciable
in general,
detector noise.
is a reversing of the
2.4
37
knife edge to measure the fields at negative tilts.
The new
adjustments are usually not trivial.
reports
that
Section 3.5.2
the filling in of negative tilts with the first
approximation
symmetry
gains no advantage over
order
using
the
exit
pupil phases estimated from the usual band pass as the
best
estimate
report
for the phases at
the study,
algorithm
however,
the
entrance
decoder
of
We
because the performance of the
is close to that of a perfect
electronic
pupil.
(noiseless)
the detector record
lex).
optoSuch
a
device also makes errors of second order. When that
the wavefront deformations are large enough so
terms like nJ (x) are important in the
description
the radiation fields as they exit the speCimen, could
tion
those terms
be calculated in prinCiple from convolutions
Fourier
transform
of
a
function, bution
~(u)
function
with itself j with
itself
times. results
of
of
the
Since convoluin
a
broader
we should expect a broadening of the power distri-
in the focal plane as the deformations
magnitude. Such high order terms,
increase
in
when they are appreciable,
result in a decrease of the central spike until it cannot be discerned
and a loss of symmetry because the center of sym-
metry vanishes. The electron number
use
of half-plane apertures also
microscopy of
papers
(EM) and has
been
concern the use of a
appeared
reviewed Hilbert
[21].
in A
transform
2.4
38
integral
and
exp (in (x) ) detector.
In
apertures
was
without [22] in
(expansion
1 + in(x) only) to 1'ind the
to
instrument
:first Born approximation
the
evel'y
case
the need :for measures No suggestion is
assumed.
phases
known
aberrations could be sought :from one
special symmetries in the object,
at
the
with
two
that
the
micrograph
although Lohmann
has remarked on the use o£ a priori in1'ormation in general,
and in SSB images
super-resolution EM:
atomic
3.5.7,
aa
phase
multiple
in
are large according
di:f1'raction
e££ects are
EM
Attempting
particular.
outlined above is not appropriate shi:fts
01'
to
1'or
section
appreciable
£or
realizable specimen thicknesses according to several simulationa
[23-28],
damage-induced phases, the
and
during shi£ting
observation thermal o£ atoms will
blur
vibration the
observed
di£:ferent planes of re1'lection are studied
specimen is crystalline.
Rather,
the object
must
observed at various tilts o:f beam and specimen [29,30]; data must be combined in a computer [31,321.
and
i1' be the
SIMULATION OF A ROUGH SURFACE
3.0
The next three sections vill reveal that the concept o£
a
ftrough sur£ace" haa caused enough debate so that
proposal to achieve
realism is apt to
anything approaching
provoke
considerable
de£ense
o£
skepticism.
For
any
some
realism will not be important:
readers, the
regime
the o£
applicability o£ the algorithm might be central instead. Any simulation
that
which
scattering becomes weaker at
the
corresponds to scattering by an object
angles is apt to be use£ul.
larger
scattering
Certainly those concerned about
electron microscopy are apt to £all in this
category.
readers
it
may
want
coincidence approximates solid
to skip to section 3.4;
that the
matter
in
the scattering lav we
use
happens also
scattering properties o£ thin
by
closely
slices
in the angular region important to
Such
o£
improving
the limiting resolution o£ their microscopes. Because
o£ a single
interaction plane instead o£ a
sequence o£ interactions that occur in a volume, the sur£ace problem But
is the simplest possible choice £or
the
simulation.
notion o£ a slightly rough sur£ace should
underestimated. tools
a
We
shall
have to invoke
rather
not
be
power£ul
in our argument and make connections to subjects that
may seem £ar a£ield,
e.g.,
electrical
The
systems.
the annoying 1/£ noise in
many
point ox view espoused here
vill
39
3.0
40
doubtless be cause o£ some controversy. £urther research,
It should stimulate
some of vhich viII be suggested explicit-
ly. But de£inite specifications £or the computer viII emerge and the reader may be confident that the basis in reality. from
Not only so, .~
simulations have a
but the reflection of light
a slightly rough surface viII take its place alongside
many closely related problems of imaging in microscopy.
Experiments on and Theories o£ Light
3.1
Scattered from Rough Surfaces Random figure errors from the grinding and polishing process remain on any lens. In astronomy the resultant point spread
function (PSF) has extended vings that may influence
photometry.
Therefore
studied [13], the
a
variety of telescopes
have
and the PSF declines as the inverse square of
angle measured from a point light source for angles
excess light
of 100 as
itself
scattering
~rad.
A decline of the amount
slowly as a pover law declines is
and
been
viII be the subject of the next property
surfaces (333,
is fully general
for
of
scattered
remarkable section. clean
in
in This
polished
since all experimental studies show that the
scattered light declines inversely as a pover of the sine of the
scattering angle when the incident light was normal
the
surface.
The exponent is 2 for a clean
dust or patchy films make a contribution,
surface.
to When
the light tends to
3.1 ~all
41
a little less rapidly at large angles '[34).
o~f
scattering scratches
is
symmetric about the
on
the
surface
normal
surface are so prominent and
The
unless
so
nearly
regular that the surface is like a grating. For
very
general reasons that we will
discuss
section 3. 3,
the inverse square decline at moderate
must
an inverse cube law at small angles.
become
writing
the author knows of one
visible
light,
the
data
At
case
Kormendy (35) on
angles this
involving
the
48-inch
reproduced in Figure 2.
The in-
o~
Palomar Schmidt telescope,
con~irming
in
verse cube decline seen below 100 prad (1 arc second = 4.848 prad) by a
is so rapid that it is process whose
pretation
exponent
overwhelmed at larger
angles
becomes -1.7.
inter-
However,
of the inverse cube relationship is made
compli-
cated by the fact that edge diffraction by a circular ture
also declines as an inverse cube law.
that
the
Kormendy data are not just edge
aper-
Can we be
sure
diffraction?
We
review edge diffraction next to help in the decision. The familiar Airy ring pattern has an irradiance due to edge diffraction alone of the form B(r) The
asymptotic
=
(2J,
(3.1)
(r)/r)IJ
form of the Bessel function J, (r) at
large
argument r
J,B (r)
=
(2/nr) cos s r',
r'
=
r
(3.2)
- 3n/4
shows that the overall angular decline of B(r) is
r-~
multi-
42 10
\ II
12
13 14
15 16
fLB 17 18 '\
~
'\
19
\
\
\
20
\ \
\ \
21
\
\ -2 \
22
\ \
\
23
\
\
24~--~--~----~--~---L--~----~--~
-1.'2
-.8
-.4
o
log
.4
r' (arc
.8
1.2
1.6
2.0
mIn)
Figure 2. Data of Kormendy [35] on scattered light at the Palomar 48 inch Schmidt telescope, blue stellar magnitudes per square arc second referred to a zero magnitude star in the range of scattering angles from 6 arc seconds to 1 degree. Humps on the curve near 1 arc minute and 5 arc minutes are artifacts due to internal reflections in the corrector plate.
3.1
43 by a
these
evenly spaced Airy rings in the radial direction with
the
sinusoid
whose period in r
is n.
plied
=0
integer n and start with n
1£ we
at the central
count
maximum,
we have rN
= n
(n
(3.3)
3/4)
...
The angular units counted by the integer n are
eo for wavelength
~
>. 1 D
=
(3.4)
and aperture diameter D.
If we average the
irrediance from one dark ring to the next, we find B (r .. )
This
approximate
central 10~
=4
n-
4
(n
...
relation gives an
3/4) -
(3.5)
3
estimate for the n
Airy disk that is 10 times too low.
at n = 1,
number
n.
surprisingly,
to
it becomes exact at large ring
When a central circular obscuration
there are two waveforms of the above form. involves
It is good
=0
the diameter tD instead of D,
is
present
If the second one
the beat pattern of
the two waveforms [36] increases the mean irradiance (due to the sum of squares) by a factor (l-t)/(l-t R
).,
which is only
2.7 for t as large as 0.5. In
Figure 2 the Kormendy star profile is given as a
radiance, referred to a star of zero magnitude, whereas B(r) is an irradiance. At 10.5
6
We will need a relationship between them.
arcseconds we reed off a relative radiance stellar magnitudes per arc-second
magnitudes
is
8
factor of 6.31 x lo-a.
squared.
of The
Changing to
about 10.5 micro-
3.1
44
radians
have a radiance of 2.68 x 10- 5 prad- a
we
rings of blue ( A = 0.45 pm) light at D 0.369
prado
= 1.22
To change (3.5) to a radiance,
m,
at
78.9
aD
i.e.,
=
we need to use
the fraction of the light inside the first dark ring at 1.22
aD,
which
0.838.
has the closed and exact form 1
This needs to be divided by B(ro)
solid angle of the first dark ring
- J o ·(3.833)
=
= 0.0973
and by the
aD)·.
The resul-
n(1.220
ting radiance unit £or the unobstructed telescope not having any figure errors becomes (3.6) and
the
mean
radiance centered on bright ring
n
due
to
di£fraction by an aperture of diameter D becomes bB(rN) = 7.56 x 10- 8
(n
~
3/4)-~
ao - 8
(3.7)
for rings greater than zero. At 78.9 rings the mean radiance for the above values is 1.10 x 10- 5 pradonly.
edge
a
due to the
outer
The central obscuration for the Palomar Schmidt
camera is a plateholder which is a square about 17 inches on a side. Diffraction by a square of side § gives an amplitude proportional to sine u sine v, and
=
u
declines edges; each
The square of
as slowly as narrow
a-a
the
amplitude
perpendicular to the
spikes in the form of a
~
are 4
therefore plateholder
sign are seen
bright star on the Palomar Schmidt plate
spikes from
a.51 A.
where sinc x = (sin nx)/(nx)
increased a little in strength
by
prints.
£or The
diffraction
vanes holding the plateholder and aligned with
its
3.1
45
edges.
But at 45 degrees to the spikes the di££racted light
arising from the plateholder declines in proportion to which
is
so rapid in
diffraction that
comparison to the
e-~
from the outer circular edge of
8-
4 ,
decline of the the
telescope
the effects of the central obscuration are not percep-
tible away from the • sign spikes.
However,
the light
ab-
sorbed by the plateholder decreases our estimate of the mean radiance
The
at
6 arc seconds by 0.84 to
Kormendy
curve gave 2.7 x 10- 6
Thus his measurements, lute
factor
photometry, of
2.
They
~rad-8
10-~
at
~rad-g.
that
angle.
which should be accurate in an abso-
sense to at least the
graphic
0.92 x
10~
exceed
commonly achieved in
edge diffraction
by
imply that a radiance due
photoabout
to
a
surface
scattering is present and that it is comparable in amount or slightly exceeding that caused by edge di!£raction. We
will next argue that the excess radiation
found
by
our analysis o! the Kormendy curve is just the amount to
be
expected
wave
for a large telescope that passed
criterion" when it was accepted by the
fabricated
it.
a
opticians
Especially in times before the last
final
acceptance tests were done "by eye" and were
tive.
Those
from
the
tests
require that deviations of a
From this it seems not
who
decade subjec-
wavefront
required figure attain a phase value of 2 lY/4
1.57 radians only rarely. able
"quarter
=
unreason-
that their criterion corresponds at least crudely to a
3.1
46
distribution radian.
of
Then
phase errors whose rms average is
a "quarter wave figure error" amounts to
standard deviations. only
12X
The
=
0
1
1.6
Errors larger than this would occur in
oL -the lens for a normal distribution of
notion o£ few errors as large as a
quarter
constrains the rms phase error from above.
errors.
wavelength
Another observa-
tion constrains 0 £rom below. The dark rings o£ edge di££raction are obviously the place
to
e££ects. at
evidence o£ fields
In fact,
discernible
irregular
even tor a simulation o£
the
region
telescope In
to
edge
studied.
At
that to
£inal acceptance o£
it is rare to discern even one
what
£ollows we shall interchangeably
One result of a
o~e
=1
the
most
a
large
Airy
ring
refer
to
dimensional simulation (di£frac-
was generated from a random walk a£ L tilt
extends
radian random sur£Bce ft •
tion by a slit) is given in Figure 3.
remaining
observation
lens,
average
surface
The rings are replaced by an
base of the central maximum
-quarter wave errors" and "a 0
and
(or
pattern (usually called "speckle")
£rom
distant
[371.
due
the bright rings themselves then become
with) light of one wavelength.
right
not
1 the dark rings completely fill in due to
scattering. not
for
The simulations to be reported next will show that
=
0
look
A phase £unction n(x)
= 256
o£ n(x) was removed
and
steps. the
The mean amplitude
was scaled so that the rms deviation o£ the phase
47
~~----~--------~----~------r-------~------~----~--------'
-2
a
o +
~~----~--------~----~l~O------~------~T1~L~T--~l+'OO~----~------~
Figure 3. Simulation of light scattering from the edge of a lens having a) low roughness ¢ = 0.2 radians, b) higher roughness ¢ = 1 radian which fills in minima between "rings" zero to 9. In this plot of log(irradiance) ~ log(scattering angle) a straight line decline signifies a power-law relation which has exponent 2 in this one dimensional simulation.
3.1
48
function vas set to 0. The array vas loaded with L values of exp(in(x»
followed by 3L values of (complex) zero. The fast
Fourier transform (FFT) was calculated and squared. Diffraction maxima 0 to 9 are clearly seen for 0 of
log power
= 0.2
in the
log spatial frequency (or RtiltR).
~
plot
A refe-
rence curve of slope -2 passes close to the tops of maxima 1 to 9. The plot represents the average power from 16 independent
phase functions,
so the results are 4
times
quieter
The results for 0
=1
show
that the bright "ringsR 1 and 2 are barely defined
in
this
average
than anyone power spectrum would be.
remark
of 16 phase functions,
having 0
that
the
equal;
=
=
=
0
Note
0.2
focal
curve is shifted down
by
so
plane
are
1/10
for
1 than for 0
=
0.2.
This is in good agreement
with
prediction by Porteus [38J that the central peak should
times.
This
constraint.
exp(-0a)~
which predicts a decrease by 2.61
relation amounts to a conservation The
of
energy
energy missing from the central peak would
in the dark rings of the Airy ring pattern in the
dimensional display. at
rough-
These curves are normalized
in the pupil and the
our
that the central maximum is 2.75 times lower
be diminished by
fill
1 or more.
variance
the
clarity.
the
with
above that the "rings" fill in for a surface
ness
for 0
in keeping
The mean irradiance required to do
two so
ring n is just equal to that irradiance that would be in
the undisturbed ring.
This is just the excess radiance that
3.1
49
we found for the Kormendy curve,
which completes'our
argu-
ment that he used a -quarter wave" telescope. The above remarks about indiscernible rings when 0 = 1
enables a powerful reductio ad absurdem argument for
necessity light
of an inverse cube fall-off of
ring and going outward.
surface
scattering
a-a
off as
that
from a telescope mirror begins to
fall
starting at the first Airy ring or so instead
more usual angle of a minute of arc or more, validity of all of the
region of the Kormendy data in Figure 2.
scattering then
a
surface emerge
dark
If one tried to require
would have to deny the a-~
surface-scattered
in a finite angular range starting in the first
Airy
the
the
fell off more rapidly than
a- 3
scattering
then
one
data in
the
And if surface
at small
few lowest numbered rings would be
angles,
obliterated
but the opticians would see
far from the central maximum.
of
the
They do not.
by
rings We
can
reject both hypotheses. Therefore we are allowed to conclude that the Kormendy circular
e-~
region is due to edge diffraction by a
aperture plus some admixture of light scattered by
random surface errors. some
The surface errors, moreover,
statistical sense isotropic in the circular
since the
a-~
These
are in
aperture,
region is circularly symmetric. indirect
arguments for the amount
of
light
scattered by surface errors are compelling but still not the sort
of approach to bring joy to
an
experimentalist.
The
3.1
50
cause
of the unsatisfactory experimenta1 situation is quick
to say.
The
9-~
region in Figure 2 involves angles less than
100 prad and therefore grating periods in the mirror surface
of
5 mm or longer. Church
well.
That grating spacing is
never
studied
[39] notes the industry tendency to draw
the
boundary betveen figure errors that will be recorded individually at
and finish errors that vill be treated statistically
a spatial frequency that is about 10 times
the
higher
lowest spatial £requency defined by the lens about 20 samp1es across the diameter,
i. e. ,
ples
in the circular aperture.
With data
t~
diameter,
about 300 sam10 rings there
is little or no motivation to make a Kormendy plot: the
log
square of the Fourier transform of the measured
errors
~
log of the spatial frequency.
tude more rings requires tvo orders points.
o~
than
of
figure
An order of magni-
magnitude more sample
The data handling becomes a big project vith little
motivation smaller
unless
sample
involving
one is going to do
areas.
The
something
portion o£ the
a deformable mirror to correct the
in
those
current
study
small
sample
areas provides that motivation. We
shall review in section 3.5.4 another example of
inverse cube scattering lav, moon
and
developed,
Venus.
that of radar returns from the
Because we vill need
we must vait.
an
concepts yet
to be
51
Power Lavs and Fourier Transxorms
3.2 For
our
central purpose ox demonstrating that
the
knixe edge test vill work xor some "rough" surxaces, ve have learned above that ve must deal vith wavefronts which,
axter
dexormation
light
that
by that surxace,
give rise to scattered
xalls ofx as a power law of a
general
a
obtained
power-lav
spatial
decline of a spectrum
frequency. ~(u)
that
to a theorem stated by Bracewell [40].
spatial
frequency
U-CJ+",
then j
least
was
as the Fourier transform of another function
points to discontinuities in G(x) or its derivatives ding
one
surfaces
u the spectrum
~(u)
In
G(x)
accor-
When at
large
decays as slowly
as
is the lowest derivative of G(x) that has at
discontinuity.
We mentioned the
many
polished
that have a scattered light profile like Figure 2:
the focal plane field function squared proximately as
u-" •
the
the
ACF
of
But
1~1"(u)
fields
E(x)
1~1"(u)
declines
ap-
is the Fourier transform of in
the
preceding
pupil,
in
by a Fourier transform.
is
the
uniquely
the
preceding
Dropping unneeded
constants
and unit conversion factors,
~(u,v) In
= (2n)·- JdX JdY ACF(x,y) exp i(ux
appendix
A
we show that in the case that
~
vy>
(3. 10 >
the
ACF
is
3.3
58
cylindrically symmetric about the origin and declining as an exponential e- cr ,
where r·
= x.
!t.(u,v;c) = (c/2n)(c· The
iactors oi 2"
are
tion oi Church et al.
y.,
T T
UO
+
(3. 11 )
VO )-:1/.
chosen to agree with the normaliza[46] ,
who gave (3.11) as their equa-
tion (29). The Fourier transiorm is linear and i t has a inverse. When we iind irom data like Figure 2 that light is
scatte~ed
can be represented by a sum oi iunctions oi which one
an
more
unique
inverse cube iunction like (3.11) and the or less an inverse square iunction,
other
we must
is
conclude
that the ACF oi the pupil iields has a component with cylindrical symmetry and a radial decay that is exponential. portion
oi
~(9)
proportional to
only at small scattering angles,
9-:1
is apt to
oi course,
be
The
detected
because oi its
rapid decline. Beiore responding
commenting on the component oi the ACF
cor-
to the scattered light decline with (u·
T V·)-I,
the reader may need to be reminded that (3.11) is a
statis-
tical iigure
relation.
For
any particular lens with
its
unique
and iinish errors there is an enormously complicated
(speckle) pattern oi diiiracted light in the iocal plane ior a given wavelength. change
The change with wavelength is mostly
a
in magniiication and a gradual decrease in scattered
pover in proportion to
1/~;
such effects are ignored here.
3.3
59
1£ the lens has the character o£ a grating, anisotropic;
we .ignore that possibility.
Figure 1 has £ocal length F., speckle pattern is F. rings;
we
want
~/D,
that pattern is I£ the lens Ll in
then the linear scale o£
the
the same as the scale o£ the Airy
to pay no attention to that scale
o£
the
pattern. With these limitations we lose nothing i£ we con£ine our attention to the v = 0 line in the £ocal plane.
Next we
notice that i£
= exp(-Icxl)
ACF(x,y) then
(3. 12)
calculating the integral in (3.10) becomes trivial £or
an aperture o£ constant width Y ~.
(u;c)
=
(Y I
2nllc)
which becomes the o£t-observed uc.
The
large
integral cR,
i. e. ,
is
(1 II
+
(3. 13)
(U/C)1I )-&
curve £or u greater
also has this seme value in the limit decay o£ the ACF well inside the needed to de£ine
WI (u;c)
work
Already
the £orm o£ (3.13) strongly suggests that a
component o£ the ACF must be exponential, in the plane o£ observation.
x
o£
radius.
Further
tions
than
more
closely.
this time
second mostly
The wave £ield E(x,y) corruga-
causing this behavior must be rapidly changing in the
direction but relatively slow to change in the
y
direc-
tion. The obvious and only candidate is the set o£ scratches elongated
along the y direction.
The plane o£
observation
3.3
60
can
be
rection
rotated
at will so that set changes as the
y
di-
is rotated. The papers that I have seen give the impression that
a
great
mystery resides in the ubiquity of an ACF that
is
better suited to a one dimensional problem when a two dimensional
surface was the scattering object.
But all that
we
need to understand is that the set of surface scratches must be
a large number for any selected y direction because
symmetry about the optical axis for,
say,
the
e = 1 mrad scat-
The requirement of large
numbers
of
tering
is excellent.
linear
scratches is certainly not a problem because what we
mean
by
process that
a polished surface is the result of
a
scratching
whose intent is to make the final grooves so
they cannot be seen in a microscope,
i.e.,
having
transverse dimension smaller than the wavelength of light. that
On
this
small
visible
crucial microscopic scale we require
there is a SUbstantial fraction of the grooves
a
only appre-
ciably longer than the width. Alternatively, the grooves may be
more apparent than real:
tion tends to select an -antenna",
a
perhaps the method of observa-
interacti~n
with what seems to be an
series of height fluctuations that act like a
groove or a ridge perpendicular to the plane of observation. If such a projection phenomenon is operating, then we should expect
polarization effects to accompany the ascendence
the one dimensional ACF over the two dimensional ACF.
of
3.3
61
An requires in
experimental way to check the
above 'suggestion
a slight modi£ication o£ the £ocal plane
Figure 1.
Provide a pair o£ holes,
aperture
one on or near
the
optical axis that subtends about 8 0 and a second hole placed o££
axis and
mrad
or
more,
contributing scale
extending £rom 8 to 2 8 o££ axis. the detector would
record
those
to inverse-square-lay scattering.
o£ those £eatures at the lens was at most
=
For 8
1
£eatures
The
linear
~/9
and the
detector needs to £urnish at least 3 samples in that length. The
£eatures are decreased in linear scale at the
by
the ratio Fe I
capability Smartt
Fl
,
vhich may £all below
o£ the detector.
[47J
detector
the
resolving
One attempt o£ this nature
used a central stop instead;
it passed
by
light
scattered through several degrees at all azimuths around the optical axis. linear -the
Needless to say,
resolution integrated
he did not have the
in his £ilm and
signi£icantly
uniform background constitutes
needed remarks,
the
major
source o£ scattering·. Direct methods have established that linear scans on a
polished
sur£ace
yield height distributions
having
an
exponential ACF as in (3.12). Eastman & Baumeister [48] used an
interferometer
Bennett lus.
looking
at a
sur£ece
behind
a
[49] and Elson & Bennett [50] used a scanning
slit. sty-
Rezette [51] studied sanded glass using a tvo-exposure
holographic
method.
The
method is recommendable
£or
the
3.3
62
present tered
argument because it does not omit any o£ the radiation
at the smallest angles where most
scato£
the
scattered radiation occurs. We have asserted that the ACF o£ the
heights
in
any
plane containing the
normal
to
the
sur£ace is exponential because the sur£ace has a vast amount o£
variability in slope at the length scale used
it:
to
probe
the wavelength o£ light. There£ore the polished sur£ace
takes its place with other natural sur£aces whose details we regularly see.
3.4
A Rough Sur£ace in the Computer In the preceding two sections we proved that one way
to
represent a "rough sur£ace" in the one dimensional
is to generate a set o£ L random real numbers o£ mean H (x),
and
case zero,
£orm a height £luctuation £unction z(x) by
per-
£orming the sum (£or integers x) z(x) = H(x) ... z(x-l), z(1) = 0,
(3. 14)
which can also be written as a (one-sided) convolution z(x) = 1: ldx-j) H(j), where
j
S
x
(3.15)
6(x-j) is the Kronecker delta £unction o(x-j) = 1 i£ x
= j,
Such a sum amounts to a random walk, sizable
value £or z(L) in general.
Fourier
trans£orm
straight
o£ zex),
(3.16)
0 otherwise.
which wanders o££ to a In order to calculate a
one usually
subtracts
line passing through z(1) and z(L);
the
o££
a
sequence
3.4
63
then
becomes a constrained random walk and the spectrum
is
altered slightly at the lowest spatial £requency. But at the higher
spatial
£requencies
decreases as a power law,
the
power
in
the
spectrum
in this case as an inverse square
o£ the spatial £requency. The physioal interpretation o£ the numbers H(x) is a height change in one sample spacing,
i.e.,
they describe the slope o£ the sur£ace. Here we examine what constraints we should place on the numbers H(x), out
an
alternate way to generate z(x),
one
then point
that
readily
gives height £unctions z(x,y) £or a two dimensional sur£ace. Our review o£ a variety o£ systems having spectra spectrum ACF
power-law
suggested that one requirement on H(x) is that o£ its ACF should be white,
itsel£
or in turn,
should be proportional to the
the
that the
Kronecker
delta
£unction
c
'"~
"'~ W
t7l
C
...J
d
e
~=-------------------~~~~ 10
TILT
100
Figure 8. Multiple loops of correction of a rougher surface, 0.1%. a) focal plane scattered "light" before correction b) to e) 1 to 4 loops of correction without use of the second order imaging term ~(u). After 2 loops the amount of light on the knife edge at negative tilts is shown by f). After the third loop light at negative tilts shown by g) improved much less than the positive tilts in curve d). The power spectrum of the "true" residual phases at h) was intermediate.
¢ = 1 radian with low detector noise,
3.5.1
87
factor of 100 by dropping Loop
~a(u)
to the detector noise floor.
4 did so by using half the usual value for the
plier
a value that an experimenter could learn
Q,
Associated
knife,
Figure 8, was
which
did
decline
~a
not
~a(-u).
(-u) decline by a factor of 4 decrease it further.
by about a factor of 10,
1000 times. ~8
on
As shown
in
symmetrical on the average until after
3 made
loops
we shall here call
and
~a(u)
the incident paver at positive and negative tilts
quite
Loop
easily.
with the onset of hole burning is a marked asym-
metry of the power in the focal plane aperture the
multi-
Loop
while
or 3
~a(u)
loop
later
so~
made
3.
n.
8
(u)
dropped about
Loops 4 and later did not lower n.a(u),
though
(u) dropped to the noise floor. In this section we have seen simulation results that
suggest
that
corrected focal
so
errors
in a slightly rough
sur£ace
can
that the Wscattered lightW passing through
plane aperture (and its complex conjugate image)
be a can
be decreased 3 to 7 orders of magnitude for attainable noise in
signal at the detector array.
associated
In
summary,
the
with ray paths through the focal plane
errors aperture
rapidly vanished relative to those for ray paths outside the aperture.
In
each
such
data set all the
larger
Fourier
amplitudes were estimated to better than a percent. One may think of any plot like Figure 8 as a sorting of the pupil errors by spatial frequency or wavefront tilts.
3.5.1
88
Inside the aperture the errors become negligible while side
the aperture the errors are untouched.
bined
all
pupil errors,
pupil
errors,
still
But if we com-
as we would in a rms
even though some
This is just what we see in Figure 9,
residual
roughness Loop
corrections,
zero in
(RRR) is plotted as
of
pupil
progress
being made for ray paths through the focal
aperture.
number.
average
we should expect an early drop in the
errors followed by a saturation, is
out-
plane
where an rms
10g(RRR)
loop
~
gives the initial roughness before this
case
0.2,
0.4,
Saturation is evident after 2 or 3 loops,
or
1.0
any
radians.
especially in the
lower curves. Plots the
of RRR furnish an economical way of comparing
effects of various Try iterations for a given
record
before resetting the deformable lens and starting
new loop. For the 0
=1
curves we see that Try
the roughness more rapidly than did Try slightly
detector
less well than Try
=2
=
=2
a
decreased
1. But Try
=
3 did
in the earlier loops.
This
is confirmed for the next lower trio of curves starting with
o
= 0.4 radians.
In a linear processing environment it
clearly important to know when to stop. Thus Try
=2
is
is used
in the reAt of Figure 9. Plots of RRR are also an economical way to study the effects directly
of detector noise.
The detector noise
masquerades
as a signal in the SSB holography detector
record
89
h
.Oll---~---+-----+----...j..----+---~6----'
o
2 LOOP
4
Figure 9. Rms residual roughness after multiple l?ops of correction under varied high noise conditions, us~ng the second order subtraction of ~(u). a) 50% detector noise and initial roughness of 1 radian; al) one Try, i.e., no iteration of focal plane filtering or restoration of detector plane modulus to consistency with the detector record lex); a2) one such iteration; a3) two such iterations; b) to e) initial roughness of 0.4 radians; b) 20% noise, Try numbers as in a), c) 10% noise, d) 5% noise, e) 2% nOise; f) to h) initial roughness of 0.2 radian; f) 5% noise, g) 2% noise, h) 1% noise.
3.5.1
90
lex), which has the £orm (see (2.10» lex) As
=1
+
O(x)
+
+
Noise(x)
a £raction o£ the re£erence wave R(x)
signal the
o(x)
noise
=
(3.23)
1,
the
is proportional to 0 (£or the £irst loop) is proportional to the
£ractional
noise
(FNL).
So
expect
lcg(RRR) p1cts to have the same shape £or
initial
object
i£ we decrease 0
roughness.
and FNL similarly,
we
level should
di££ering
This is just what is seen in the
curves o£ Figure 9 where trios o£ curves starting at 0 and
0.2
FNL/2,
respective1y.
loop 1, dent
are given £or FNL
= 0.1,
lower
= 0.4
and 0.02 and
£or
The correspondence vas only £air
£or
but improved thereafter.
0.05,
and
The same tendency is evi-
£or the relative1y high noise runs (FNL = 0.5 or
0.2)
starting at 0 = 1 or 0.4, but the £latter curve £or 0 = 1 is a
little un£air.
A £ew o£ the £iles out o£ the ensemble c£
files were extraordinari1y slow in progress but they contributed large increases in RRR. Exactly the same set o£ random numbers went into the sur£ace de£ormations,
so we can blame
the onset o£ strong non1inear ef£ect~ £or the slower decline at large roughness. The numerical value at which RRR should saturate can be
estimated £rom the shape of the power spectrum using
elementary ca1culation. the
an
Let the Nyquist frequency be Nand
focal aperture from zero to N/2.
The tota1 variance at
the beginning of the numerical experiment is,
apart from
B
3.5.1
91
normalization not needed here, pover spectrum
I
v. =
equal to the integral o£ the
rJ dx (c'
+
x')-
a
=
(2/c)
tan-
a
(N/c)
(3.24)
-~
while
in
the
£ocal aperture and
its
·complex
aperture-
the
amount o£ variance that vill be
conjugate essentially
decreased to zero is Va
The
= (2/c) tan- a (N/2c) = (2/c) (n/2 - tan- a (2c/N»
angles involving the ratios H/c or N/2c are so
degrees,
(3.25) nearly
90
it is best to work with the complement angles,
as in (3.25).
Then the expected decrease in rms
quantities
(ED) is given by ( ED ) •
= ( V. - V 1
)
I V.
= 2c I
In the case that c = 1 and N = 128,
( Nn -
(3.26)
2c)
we £ind ED
= 7.1~.
largest declines in RRR in Figure 9 were to about initial larger not
roughness.
Note
8~
The
o£
the
that the ED values would be
i£ the variance associated with negative
much
tilts
go dovn as much as the positive tilts within the
did £ocal
plane aperture. In Similar
Figure
vas
subtracted.
runs without the subtraction o£ CT(u) are presented
in Figure 10. environment First,
9 the cross term CT(u)
The results are so similar in this high noise that I need to point out the main
di££erences.
the best choice £or no CT(u) subtraction is Try
This is especially true £or high initial roughness, lower
curves
= 1.
but the
£1 and £2 can hardly be distinguished by
the
92
.01 0
LOOP
6
Figure 10. Rms residual roughness after multiple loops of correction under varied high noise conditions, not using the subtraction of ~(u) except for curve c). a) 50% detector noise, Try number as in Figure 9, b) 20% noise, Try = 1; c) 10% noise, Try = 2 and ~(u) subtracted, d) same noise, Try = 2 and ~(u) not subtracted, e) same noise, Try = 1; f) 5% noise. Curve g) shows the changes of another indicator of residual roughness based on the Fourier transform of the detector record.
3.5.1
93
parameter focal
Try.
This means that the first estimate for
plane fields gave the best estimate when
the
the
object
phase bumps and their locations were estimated by the detector plane fields.
=2
What is not shown here is that Try
and
3 gave slightly better estimates of the phases of the fields
in the focal plane and in the detector plane. values are not our goal. to
=
Try
was
not
But the latter
The worsening of Try
=2
relative
1 as noise increased tells us that our
an optimum treatment of the
procedure
information
about
noise.
In the present example the function
noise
values
from tilt 65 to 128.
Rather than
we
have has
~(u)
use
those
numbers in any way, ve set them to zero and transformed back to
the
noise
detector plane,
whereas we could have
obtained
a
level estimate for a Wiener filtering that would have
de-emphasized
those Fourier amplitudes corrupted by
In a nonlinear procedure we could hope to use those
noise. numbers
even more effectively. The second small improvement of Figure 10 over Figure 9
vas
e larger drop in RRR as a result of the first
loop,
the only loop in most applications. Curves c and d of Figure 10
are given for this comparison at medium
curves 9a
in Figure lOa are enough lover than those in
to be seen readily on
saturation prevents
roughness.
of
tvo
panels.
Figure
Unfortunately,
RRR after some roughness had been
The
the
corrected
Figure 10 from conveying to us the dramatic
drops
3.5.1 in
~.
94
(u) later than loop 1.
I also tried another
indicator
of residual roughness but experienced the same problem. indicator
was
the sum over positive tilts of Ia(u)
the focal plane aperture.
RI we
within
Defin~ng
(2 L
(3.27)
~a(u»·/R
in Figure lag that the RI curve is similar to
find
RRR
=
The
curve.
All the same conclusions about Figure 10
follow using the RI indicator. requests
were
indicator
the would
The curves for different Try
RI
as
the
so RRR vas retained
for
have established the advantage of not using
the
of
even less distinguishable using residual roughness,
illustration purposes. We correction
term
CT(u).
Let
conSidering a simpler example.
us try to understand
why
by
For Figure 11 we deformed the
entering wavefront by the sum of two sinusoidal waves,
just
as
sine
in the theoretical treatment of section 2.1.
wave
had amplitude equal to 0.7
radians.
The
Each
frequencies
were 29 and 32, which are relatively prime to each other, their
difference 3,
and to their sum 61.
The result is
line spectrum featuring the difference frequency 3. line
to
is shown the incident power at the focal
plane
a
At each ~R
(u)
and the pover spectra of the detector records from the first and second loops (after no correction and one correction). As might be expected,
the strongest lines in Figure
11 are the fundamental frequencies 29 and 32. Next strongest
95
a
" +
.... o ."
N
o
I 20
40
TILT
60
Figure 11. Power spectra for a phase screen having 2 superimposed sine waves with spatial frequencies 29 and 32; the amplitudes were both 0.7 radian. Three responses plotted at each frequency are, left to right, incident light, squared modulus of the Fourier transform of the first detector record, same for a detector record after one loop of correction of the phase screen using no subtraction of ~(u) •
3.5.1 are
96 the sum and difference
frequencies.
However,
at
the
difference frequency for the spectrum of the detector record 1~1-(u=3)
the power is low by 4.5 orders of
agreement
with
magnitude,
our theoretical prediction in (2.16)
of
second order cancellation of the difference frequency.
in a
There
are other strong lines that differ from 3, 29, 32, and 61 by integer multiples of the difference frequency 3.
In many of
these
~Q(u)
latter
different loop CT(u)
cases the magnitudes of
by an order of magnitude.
of
correction
~(u)
and
The main effect of one
of the "roughness" without
doing
subtraction was a decrease of the lines of
frequencies
29
and 32,
magnitude.
The
satellite
lines
lines
are
the
I-(u)
at
a decrease by 2 or more orders
of
were decreased to the level
around them.
Later cycles
of
the
of
correction
decreased all lines more or less democratically,
once the 2
biggest phase perturbations were decreased. A concern might be raised that the ACF of the
focal
plane fields was estimated from a mere estimate of the focal plane
fields
and that the poor results of subtracting
ACF estimate was caused by a poor estimate. excellent facility
to deny or affirm such a question. to
example
A simulation is I provided
compare the directly calculated ACF
indirect estimate.
The results are in Figure 12,
of Figure 11.
the
with
a the
using the
The lines that are prominent in
the
97
+
"
;::' u
o .....'"
):.
o
20
III
'It
-
-
-
+
40
TILT
60
Figure 12. Testing the estimated ACF of the focal plane fields against the ACF calculated directly from the "true" fields. At each of the frequencies are: at left, the estimated modulus squared; at right, directly calculated modulus squared; at bottom is shown the ACF phase, + for zero, _ for 180 degrees, x for a disagreement.
r··
3.5.1
98
enough in
magnitude to deny that the indirect
estimate
is
causing trouble. ACF are 29,
32,
and 3,
as one should expect £rom the many
lines in Figure 11 that di££ered by these values. that
several
di££erent
by
lines
It is true
o£ the indirectly calculated
hal£ an order o£ magnitude or more
directly calculated values.
ACF
are
£rom
the
But the main lines are accurate
A next objection if the magnitudes were close enough is
that some phases of the ACF estimate were
wrong.
Since
the ACF was estimated by Fourier trans£orming a real and (by arrangement) quency
symmetric
must be zero or
respectively,
in
the
n,
freor
+
In 4 weak lines marked with X
So the important phases were correct.
examples reveal why one should
not
subtract
focal plane ACF (or any multiple of it) when estimating
the detector plane fields. order
of
tilts.
The ACF at tilts 29 and 32 was an
magnitude smaller in modulus than
Thus
correction
~(u)
at either u = 29 or 32.
tiny
But since
those
appreciable ~(3)
is
very
the subtraction of even
fraction of CT(3) causes a large
apparent field at that frequency.
at
~(u)
- CT(u) did not amount to an
much smaller in modulus than CT(3), a
the phases at each
which are marked with
Figure 12.
the phases disagreed. These
£unction,
increase
in
the
Such a correction hinders
the decrease in the largest phase perturbations, those at 29 and
32 in this example.
Since our aim vas to lover all the
3.5.1
99
lines
in
Figure 11 to the noise £loor caused
by
detector
noise in the smallest number o£ corrective loops, correction £or CT(u) is not desirable. The results presented ved rather low noise levels.
BS
power spectra so £ar invol-
In Figure 13 we study FNL = 0.1
£or
large to small initial roughness and £or a single loop.
The
white noise £loor (FNL)A/L intersects the
£or
0
= 0.5
at a high tilt number,
about 45.
about number 50 made some progress in one loop. were
light
All tilts to Higher tilts
made worse by the phase estimates made £rom
noisy detector record,
level
the
very
i.e., the guesses £or bump size were
worthless. At initial roughness 0 = 0.1 the amount o£ "scattered
light"
vas 25 times lower than at 0
= 0.5,
so
tilt
number 10 was at the white noise £loor. Again there was some progress to tilt 14; the highest tilts got worse by up to 20 times in
~A
(u).
A perhaps surprising result occurred when the ting
roughness
vas lowered to 0
= 0.01
radians.
For
case all ox the "scattered light" was below the white £loor. tilts
The
e££ect o£ one loop made
1 and 2 (by a percent or so).
became worse only by 3 times. the
estimated
~a(u)
smaller only
But the highest
starthat noise £or tilts
As we shall see in Figure 15,
phases (bump locations) at the detector
and
the £ocal plane were garbage, as should be expected when the signal is smaller than the noise in Signal.
As noted at the
100
-
-
+
'"
.....
'"
---
Il'I
C'I
....o
10
100
Figure 13. High noise conditions, 10% due to detector, showing effects of 1 loop of optimum correction at varied initial roughness: a) and b) 0.5 radian; c) and d) 0.1 radian; e) and f) 0.01 radian. Solid curves, incident "light" in the focal plane; dashed curves, light after one loop of correction using a ~(u) correction. The arrow marks the level of power due to the detector noise.
3.5.1
101
end of appendix C, phases
the optimum procedure when the estimated
are uncorrelated with the true phases is to set
the
multiplier Q in (3.21) to zero, i.e., make no correction. An extremely experienced experimenter might have enough
infor-
mation about signal and noise-in-signal to adopt this procedure.
Most
of us would make some correction and sooner
or
later ruin the 0 = 0.01 radian surface. A survey of the amount of improvement in the surface roughness in
after analysis of one data set I(x) is
Figure
roughness
14.
If
nl (x)
01
is the rms average of
after
optimum
correction
the for
phases as in (3.21) and if 0 was the initial rms then Figure 14 plots 10g(0 1 /0) lOX,
lX,
and
O.lX.
Each
~
presented residual estimated roughness,
10g(0) for 3 noise levels:
curve has a broad minimum
with
respect to initial roughness where noise in signal is balanced
against
increasing
difficulty
imaging terms at high roughness. and
0
that
to
second-order
In the curve for lOX noise
less than 0.5 radians we already saw from Figure the
making
floor of vhite noise was giving
rapid
whether not.
due
progress.
The results depend
difficulties slightly
the correction for the cross term CT(u) is done
For detector noise as low as 0.1Y. in Figure
14f,
13 in upon or the
improvement in one loop was nearly constant for a wide range of
initial roughness because one loop was enough to correct
the roughness as well as the detector noise would allow. The
102
1.r-"==::c:::::::=---r-----,.----...----r---i
.....
'"'E"
'">
0
"-
Co
~
....., ...
;: u
'"
"-
c
.1
0.1
.01
1.
Initial Roughness (radian)
1.r----===::::::::~--------------1
f
.1L-----~--------~------~----~--------~----~ .01 0.1 1.
Figure 14. Progress in one loop of correction measured by rros residual roughness divided by initial roughness as a function of initial roughness and for 3 levels of detector noise: a) and d) 10%; b) and e) 1%; c) and f) 0.1%. The upper curves used a subtraction of ~(u); the lower did not.
3.5.1
103
generally
poorer
progress at high initial
GT(u) was subtracted,
roughness
when
as in Figure 14c, means that the sub-
traction worsens the effect of second order imaging terms in The steep rise in the ratio 0 1 /0 near 0 = 1 radian
general. signals
collapse
=
of the algorithm at or just above 0
1
radian. The tions. phase
interpretation
complica-
The detector deals with a band-limited image of object
estimating the
of Figure 14 has its
whose Wtrue R roughness we know
is
0.
the After
a band-limited version of residual phases we did
optimum
subtraction of
using
(3.21),
a
correlation
between Rtrue W object and estimated object forever denied to an
experimenter.
Then we calculated an rms average of
the
residual,
which again included high frequency detail hidden
from
the
observer.
more
like that in Figure 15,
We would prefer a measure of
progress
which at least compares
numbers available from experiment with true values,
only
in this
case phases in the focal plane. Suppose
that
we
average
the
squared
difference
between an estimated quantity S and its actual value
T.
We
obtain (3.28)
or
in terms of standard errOrS and a normalized correlation
coefficient
Y.
T ,
; : a.· .
aT·
-
2
a.
aT
't. T
(3.29)
104
1.
..... c:
o
l-
e:>.
E
....o I-
o ..... u
.....'" .1. 01
0.1
1.
Initial Roughness (radian)
Figure 15. Correlation of estimates of focal plane phases with "true" phases, calculated as an unweighted average over all frequencies passed by the focal plane aperture and expressed as a "factor of improvement" (see text) so that low values mean a small difference between estimated and "true" values. a) detector noise 10%; b) noise 1%: c) noise 0.1%. The higher values on a) show a complete uncertainty about the "true" values based on the estimated values.
3.5.1
105
When the differences are arnall,
we can estimate a £actor of
improvement from (3.30)
Applied
to the difference between estimated
phases
at the detector plane and true phases at the object,
I find
that
the average o£ FI values for one loop can not be
tinguished
from
the
ratio
plotted
0./0
especially when the 1atter is sma11, rapid.
f.,
goes to zero,
14,
i.e., when progress is
which happens for any phase comparison of
when the multi-loop process is close to
stopping.
it is a small inconvenience of (3.30) that the
of improvement has the limit 2 l / 8 In at
Figure
Slov progress means that the correlation coefficient
interest Then
in
dis-
the
,
rather than 1.
Figure 15 we plot the FI for focal plane
subtraction
starting
of the ACF. in Figure 14.
phases
=
end o£ one loop for which we selected Try
presented
factor
2
The runs are the same survey
runs
Again.we find the
that
situation
with either very high or very low roughness is not
favorable:
high
second-order
roughness
has plenty of
signal
but
The curve for lOY. detector noise
might lead us to the conclusion that SSB holography is The
correlation
the
the other
imaging terms are poorly estimated,
loses signal in the noise.
hopeless.
and
of a11 the focal plane
then
phases,
without any weighting for the associated modulus, was always worse than 63X and vas much vorse at 10w initial
roughness.
3.5.1
106
Most
ox
small
the
phases estimated in this example
modulus
detector
although
noise
we know that the
will trouble the small
more than the large ones. is
telling
little
£loor
modulus
o£
that
white
The curve £or lOX detector
nothing about most components in a passes
64
independent
a
amplitudes
us that even en optimum procedure can
or
aperture
1nvolved
noise
tell
£ocal
us
plane
components
or
·channels·. For so much detector noise we were too greedy in asking
£or 64 channels ox in£ormation about the
object
in
parallel. Figures
14
and
15 agree about
£act. When the detector noise is large,
another
we vill be £orced to
study only objects with large phase excursions. py dant
unwelcome
In microsco-
that tends to mean thicker specimens with all the attentroubles
with multiple scattering
and,
in
electron
microscopy, an accumulation o£ inelastic events. When important,
reduction Figures
ox
scattered
light
is
16 and 17 are instructional.
centrally
They
plots ox the change in the logarithm o£ the scattered due
to
initial
the £irst resetting o£ the de£ormable lens roughness
0 and 3 noise levels.
Results
lowest available tilt are shown in Figure 16. improved
by an optimum procedure,
gest component in the signal. involves
are light
£or £or
an the
It was always
since tilt 1 is the lar-
Figure 17, on the other hand,
the highest tilts passed through the
£ocal
plane
107
2~----~--------~----~------~------~------;
o~====~~-------------------------------j
b
d
.01
0.1
1.
Initial Roughness (radian)
Figure 16. Decreases in scattered light at the lowest (and strongest) spatial frequency after one loop of roughness correction as a function of initial rms roughness. a) detector noise 10%; b) noise 1%, c) noise 0.1%: d) same noise but no subtraction of the second order imaging term ~(u) during estimation of the needed lens figure changes. For microscopy these curves allow an estimate of the accuracy with which the modulus of this spatial frequency can be measured in a single frame of data.
108
2~------~---------r------~------~~--------~------,
_ - -__ a
O~------------------~~-------------------------------1 b
~
CI> ~
0
c..
-2
c
c::n
0 .oJ
give
We turned therefore cross-correlation
between some known scene 1unction
and
the
measured phases. We did not need the entire function CCF(x); the
value at its origin was enough.
Thereby we reduced the
problem to a scalar function of a few variables.
4.0
172 For
the
defocus
least 3 variables: besides of
problem just stated there p~
are at
a small relative shift in each
2 dimensions is generally needed to align the known phase functions.
~easured
In an example that will follow we
have a parabola of defocus centered on axiS, waves
of
and
defocus at the Nyquist
spatial
uQ
= 0,
and 2
frequency.
That
parabolic trend could also be represented by 2 waves of tilt in the length of the srray in the focal plane plus a quadratic
function whose mean and tilt is zero.
A tilt trend
of
that size causes a translation of the detector function by 2 samples. tary,
A shift by an integer number of samples is elemen-
but a fractional shift thereafter is not trivial as it
requires an interpolation scheme. sines and cosines attractive,
I found interpolation
by
using the Fourier translation
theorem. The
finding of a best shift was a sufficiently non-
linear problem to require a line search procedure [81l. size
of
spacing or
an
interesting
at each harmonic.
fundamental and
of a sample
proportionally more
A procedure to guess a sign for that step
We recall that the correlation of 2 functions f(x)
and g(x) involves the complex product transform
half
which is a phase shift in Fourier space of
less~
n per L samples at the
follows.
step is clear:
The
phaeors
autocorrelation
~(u)
and a(u).
function
we
~·(u)~(u)
of 2 Fourier
In the discussion of
the
(2.32)
this
noted
in
that
173
4.0
product
amounts
to
vector dot and cross products
phasors.
Thus the spectrum o£ CCFCx),
ensemble
o£ phasors whose real (imaginary) part
large
and
o£
namely CCFCu),
the is an
should
positive (close to zero) at each £requency
alignment is close. CCF(x)
=L
be when
In particular, i£ we set Aft cos nu
+
Bft sin nu, u
= 2nx/L,
(4.2)
we shall be interested only in the sum CCF ( 0 ) = 1: Aft
(4.3)
which might increase i£ the phasors are rotated slightly accord
with the
translation
trix Rft with elements C
= cos
Rn
theorem.
in
1£ the rotation ma-
n9 and S = sin nS is given by
= (4.4)
then
(4.5)
The
elements C and S may be updated in Rft using
trigonome-
tric addition £ormulae to gain speed. Note that a£ter CCF(u) is
calculated,
we need only alter the phasor
and per£orm the sum (4.3). to
orientations
Only one other Fourier transform
assure that CCF(x) has its maximum at the origin may
be
needed; experience might avoid even that labor. I£ there were only one harmonic n, needed angle S exactly by maximizing
(B' ) II
[82]:
we could £ind the
(A')- or by
minimizing
4.0
174
tan 2nS
=
2 AB
(4.6)
n
"ore w~,
we have many harmonics,
genera~ly
a weighting' scheme
and we obtain S from a sum over the harmonics tan
~~
:2 1: n w"AB En v" (Aa
The
sign of
e
but correct a large fraction of the
vas time.
implementations of the shifting procedure have been Three
st.udied.
phasor positions are enough to fit the
(4.3) to a quadratic,
short words. maximum
of is
sum
estimate the maximum, and react. accor-
Such an approach is less suited to a
dingly.
which
(4.7)
Ba)
obtained from the use of uniform weights
sometimes wrong, Several
-
vith
comp~ter
Therefore instead of looking for the quadratic CCF(x),
availab~e
we can look for the zero by multiplying each
of
CCF'(x),
positive-frequency
harmonic of (4.2) by n and forming the sum CCF' (0) = [
Again
(4.8)
nB"
3 posit.ions are enough for a
quadrat.ic fit,
but nov
the zero of the fitted curve is sought. We focus
many
along
the
quency.
The
applied the CCF formalism by estimating the times vhen the
true defocus was
ray corresponding to the
Nyquist
de-
2 wavelengths spatia~
spatial frequency for the inside of the
frefocal
plane aperture was half that large, so the amount of defocus was small,
0.5 wavelengths in the aperture.
We divided the
4.0
175
rms
scatter in defocus estimate AZ by the true defocus Z
=
0.5 and plotted the ratio aZ/Z in Figure 38 as a function of detector
noise.
We
used 3 types of input phase
·calibrators· of defocus. radians
That amount of surface
limited the focus search at low
but
differently for the 3 calibrator types.
was
used
to estimate defocus.
1.
detector
noise,
A single
One Try worked as
several Gerchberg-Saxton iterations,
=
for
A background roughness of 0 = 0.01
was common to all the runs.
roughness
data
loop
well
as
so these runs used Try
The results in Figure 38 are for a correlation of the
fields
arising
from the input phase distribution
and
the
detector plane estimation procedure. That is, for a calibrating phase function 9 c (x), of
exp'
(5.4)
8
\J
'
where the chromatic coefficient Cc for the objective lens is typically nearly equal to both C. and the lens focal length, ~
U is the beam energy,
and
be£ore
I have ignored lens current
the specimen.
tions in (5.4) The
BS
U is the rms beam energy spread
a smaller e££ect.
existence o£ a ring o£ high modulation xor TIBF
images o£ amorphous £ilms is known [95,96), papers
[97-100) have attempted to implicate
electrons ignored even
£luctua-
in the high modulation ring.
but a series o£ the
inelastic
Each o£ the
papers
the rings centered on the di1£ractogram axis
though
such centered rings point to
[95],
spherical
waves
that in this case are not coherent with the tilted beam. The centered to
rings are highly variable in strength £rom as
sample,
such
though a second order process is
involved,
as elastic scattering o£ inelastic electrons.
reject
inelastic electrons as the cause o£
components
sample
We
dominant
can image
whose tilt change caused by an amorphous £ilm is
about 10-:a rad,
since the de£lection o£ inelastic electrons
is mostly less than the ratio o£ a plasmon energy (about eV) keV.
to
the beam energy [11,101],
Neither
10
which is 10-- rad at 100
can the beam divergence s be very
important,
since the ring 01 high modulation had a radius equal to
the
5.1
183
beam
while the zeroes of
t~lt
matic
V~
are fixed.
On the ftachro-
ringft [96] around the axis of the microscope and that
includes the electrons udeflected by the specimen, the phase difference due to the aberrations of (5.1) vanish since is
symmetric
about
the optical
axis.
We
can
~(~)
generally
express the object wave field E'S the product of an amplitude £unction A(x) with a phase function exp(in(x».
The Fourier
components of any compact feature in the (thin) object have paths verse
t~rough
the achromatic ring will suffer no trans-
shifts in the transfer optics and vill be
with good contrast by the detector. nents
of
to
cycle
registered
This is true for compo-
either A(x) or n(x) when the achromatic
ring
smear in the detector plane due to
position
shift,
their
quarter
a smearing that increases with
radial width of an ftachromatic ring".
suppose
contrast
inside the ring is consistent with large chromatic
envelope
in
The loss
we
of
effects,
that both contribute.
the
Without more detailed
experiments to distinguish components of A(x) from n(x), must
is
But we saw in section 2 that the components of n(x)
narrow. tend
that
keeping with estimates [84,102,103] that agree
that present 100 keV microscopes are
l~mited
to about 0.3 nm
resolution when used with axial illumination, and that chromatism
is
the
limiting effect if
corrected out [94].
In addition,
X,
would be a waste to place the knife edge at the
axis [29,105].
For example,
optical
i£ we wished to take advantage
o£ the insensitivity to the beam divergence s,
then we could where
Q ')(.
has a zero (see Figure 39) and use object components to
and
put the beam and kni£e edge on the optical axis,
just
beyond
resolution -
011.
the
u· = 0 zero o£ V "X..
if we place the
and work to u
c
+
beam and
011 ••
But
we
double
aperture edge at u
the
=
The ef£ect of the chromatism
envelope is then vorst £or a Fourier component of the object that gives
B
ray parallel to the optical axis,
but no worse
5.2
185
than
£or the choice o£ kni£e edge on axis and a stop at u So we
doub~e
=
the resolution at no cost that we were
not prepared to £ace. Chromatism, like de£ocus, is dependent on
the
square o£ a tilt
re~evant
ti~t
ang~e
change.
Here
the
band
angles extends £rom the beam to the
of
optical
axis only. For the same resolution in a SSB image as the DSB image,
chromatism
phase
dif£erences
geometry. phase
The
and defocus ef£ects that
are
involve
4 times smaller
aberration in
SSB
the
£ourth power of spherical aberration
allows
difference magnitudes that are 16 times smaller.
advantage is decisive when the aberrations must be
The
measured
in the presence of noise. The increased compact
aberration resolution.
search should be greatly The
equivalent
aided
for EM of
by
the
phase object per grating spacing in Figure
one
38c
or
38d would be a heavy atom derivative of a crystalline specimen.
The
"through its quent
user focus
may need to recognize the heavy atom
in
series" of playback (averaged) images
position is not adequately known a priori.
The
aberration search steps may be loosely said
a
when subse-
to
make
those atoms appear round and small. The EM equivalent of the diffuser in Figure 30e would be a £ractional atomic layer of a strongly adhering heavy element on a crystalline Bubatx'ate such
as boron [106).
compounds
The use of crystalline substrates
such as BeD [1071,
"gO [108l,
of
or graphite oxide
5.2
186
(109)
may
surface
be less suited for a high bonding
site.
observation, integer
If
the
heavy atoms do not
energy
at
migrate
during
the a priori knowledge that they are placed at
multiples of the crystal lattice is enough for
aberration
a
search.
But such an attempt is attractive
the only
for aperiodic specimens that have great resistance to radiation damage by the beam. As
a
specific
example let us consider
electrons of 100 keV energy, 1 Sch is (
= 81.6
nm,
~/C.) 1 14
=
1 Gl
pm, C.
= 1.8
for
mm [84):
0.55 nm, and the characteristic tilt
6.73 mrad.
=
= 3.7
A
(5.1)
At the o:ff-axis zero of
'f:/'I.
the
aberration value (in cycles) is X( Da
and
I
a)
= -
~
(5.5)
Da
the aberration has a ring of value zero at u
=
(2D)1/8.
Within the latter ring the mean value of / is - D a /6 and the average value o:f its square is D4/30, is
Da 1
(180) I
strikingly
I.,
whence its rms
about 1/27 cycle at DR
only
=
value
1/2.
small variation of the phase differences
The
inside
the (2D)I/. ring suggests that we try to arrange a tilt just outside
this
ring
such that most points inside
the
involve
a quarter cycle phase change (see Figure 39)
ring rela-
tive to the main beam phase
Xo = The
~
(u ll
target tilt then is
period
is
Q.55/2.674
Uo
-
2"
a )u·
= 1.337;
= 0.206
= ... 1/6
(5.6)
the
limiting
grating
nm on the
diameter
of
the
5.2
187
achromatic circle having 9.0 mrad tilt. to
use an under£ocus o£ 57.7 nm,
1/4
1/12 cycle £or
t
circle. 0.82
79~
Thus i£ we happened
the phase shi£t would
o£ the area inside the
the chromatism estimate o£ Chiu & Glaeser
The critical value o£ the beam divergence,
circle
£arthest from the undeviated
gence can be negligible. !
0.1 Sch
changes the
the
magnitude
the e££ects o£ beam
Hissing the de£ocus D target
of
electrons.
since the beam divergence can be an order o£
lower than this critical value,
[84J.
0.99 mrad, would
a loss o£ modulation by exp(-l) at the point
achromatic But
achromatic
The loss o£ modulation on the optical axis would be
£or
cause
be
diver-
= 0.7
Sch by
phase shi£t o£ 1/4 cycle by
!0.09 cycle. This choice £or imaging conditions avoids blurring
a wide range o£ Fourier components o£ the object phase
n(x}. A(x)
1£ we wish to avoid blurring the amplitude modulation in the scene,
we would drop the tilt to the range
4.8 to 6.7 mrad and target D grating
= 0.7
to 1.0 Sch.
of
The limiting
period in a circular aperture is then 0.39 to
0.28
nm,
respectively. Since in this example the achromatic cir-
cle
coincides with the circle that is insensitive
divergence, contrast.
the
second
arrangement
has
to
beam
especially
high
188 The Challenge of Charging Aberrations
5.3
A
small fraction of the beam current flowing
to
good metal aperture would cause Ohm's law voltage drops small
to annoy an electron microscope.
vacuum The
contaminants by fast electrons
film
decreases
too
But films made from are
semiconductors.
conductivity drops exponentially with
temperature,
a
increasing
so heating the aperture edge [110,111) greatly
the perturbing fields in addition
to
decreasing
the rate of film growth.
Quite small fringing fields at the aperture edge are
A phase change of b0 due to passing through
troublesome. voltage
change
V(z)
can
be
estimated
from
a
geometrical
optics:
A
0
= 2".
J dz
= 211' Ih
( ). -, - >'0 - ,
f dz
(p - po)
(5.7)
Using the nonrelativistic relation for momentum p and energy
p8 = 2me (U
+
V(z»
(5.8)
and the smallness of V/U, we find [112)
A0 = A
phase
V(Z)
n
5dz V(z)
change of n/2 will
hav~ng
magnitude
>'oU/2,
1
>-D U
arise for a path which is 0.185
(5.9)
integral ~m-Volt
of
at 100
keY and changing nearly as U"_. In our example above with a focal
length
radius is 18
of about 2 mm and 9 mrad tilt, ~m.
the
aperture
Even a contact potential difference between
5.3
189
the
metal and the £ilm amounting to a £ev tenths o£ a
£ringing over a 1 pm dimension along z can cause e£fects apart For
for
a beam just inside the aperture
volt
observable
[1131,
quite
from Ohm's law voltages across the contaminant
film.
50 years [114J the marked effects of vacuum
impurities
on the apparent photoelectric or thermal work function been
well known.
have
Experiment will be essential to establish
those conditions required £or a small and stable V. Part o£ the solution mat be keeping the beam 1 or
so from the aperture edge.
nique
will
not
The aberration search
mrad tech-
impaired if the observation geometry
b~
is
recorded so that the image Fourier components suf£ering from D5B inter£erences can be excluded £orm the cross-correlation £unction. When
it
should
providing For dc
perturbing
fields
are
arranged
they may still be deemed not small enough.
stable, case
the
be possible to control
them
to
In that
actively
a microcircuit on the back side o£ the
be
by
aperture.
some measurements an ability to adjust the phase of the beam at will would aid in separating amplitude
contrast
from phase contrast. The microscope faced
di£ficulties posed by passing the main electron beam near to
realistically.
possibility
to
S
£ocal plane aperture need
to
be
Balanced against them is the apparent
obtain 5SB images for which
we
here
have
5.3 given
190
an
algorithm that permits direct
interpretation
in
terms o£ wave£ront de£ormations induced by one projection o£ the charge density o£ the (thin) specimen. Under some conditions ration
it should be possible to measure the microscope aberparameters by consideration o£ image data
micrograph
(or
at
least the composite data £rom
exposure to record a £ragile specimen and a longer to locate its pre£erably crystalline support).
£rom a
one brie£
exposure
CONCLUSION
6.
We
showed that thE
correctable
by
successive
limited by noise in should be
roughness on a good lens should
signa~
approximations to
an
due to the detector.
better than linear,
the residual
accuracy
Convergence
roughness
de-
creasing by an improving fraction per correction step, until detector
noise limits progress.
Increasing the density
detector
pixels tends to lower the final roughness
more detailed information is available at each stage.
of
because There
is a real-time optical processing possibility for microscope data,
although no studies of the effects of the
contributed by the
optoelectronic loop were
Defocus of the detector and other the required transfer optics slightly,
but a
coefficients,
calculation was found to be robust: linearly with
an electron the
finite
limiting factor in findbut the cross-correlation parameter error increa-
detector noise.
These
prompted a detailed proposal for microscope at nearly atomic
wavefront
aberrations of
sensitive scheme to find those aberrations
ing the aberration
properties
reported here.
degrade the system performance
was studied. Detector noise is the
ses only
low order
extra noise
deformations
thickness,
but the
are
observing with
resolution.
caused by a
mathematics
encouraging
that
There
specimen
of
estimate the
deformations at the exit face of the specimen are the same. 191
6.
192 The
algorithm discussed here is easily
implemented
on a digital computer. It is £ast when implemented with £ast Fourier trans£orms. number that
It is a £ixed procedure with a de£inite
o£ iterations. would
terion.
Convergence is assured £or
pass the common
When
the
method
quarter-wave is used £or
sur£aces
aberration
acceptance
cri-
optical
testing, the light may come £rom an extended source partially
covered
"White" of
by the kni£e in the geometry
of
Dakin
[115].
light could be used since the act of blocking
the £ocal plane is achromatic.
aberrations
in
The existence of
hal£ £inite
the trans£er optics will restrain both
the
temporal and angular extent o£ the source. Those limits were considered
quantitatively
in the proposed
electron microscopy in section 5,
application
to
where third order spheri-
cal aberration is dominantly important. The limits £or other application should be generalized £rom that section. The comparison to
o£ SSB holography has
tering
su££ers
playback
from £lare light,
neglected
stage.
is
in
Analog
a small angle
which was the £irst motivation for this
inability to control second order
latter
been
to off-axis holography because o£ the hesitation
use digital processing for the
processing
the
use
imaging
scat-
paper,
and
terms.
The
also dominantly a small angle scattering
effect
during playback.
A variety o£ problems should bene£it
SSB
fractional-wavelength optical testing
holography:
from and
6.
193
correction, tvo dimensional optical processing, three dimensional :few.
microscopy
vi th electrol.s and light,
to suggest
a
7.
APPENDICES
7.A
Exponential ACF in a Circular Pupil We
want
to
autocorrelation
perform the Fourier
transform
of
an
function (ACF) that declines radially as an
exponential in a region whose radius R is the diameter of lens.
We
will use normalized coordinates.
The two
a
dimen-
sional power spectral density becomes WCu, v~c) = C2n)-.Sf exp(-c:x. + yl >' • exp i(ux In
the pupil we let x
=p
to
the
optical axis.
We find ux
+
dx dy W = C2n>-.S.1t
• f
= (2n) - IS
dp
y
v
~
p
=
=w
vy =
=P
p
(Al)
sin 9,
sin 'f'
sine of an angle
and in the where w
is
from
the
measured
w cos (9 - '1' )
(A2) (A3)
d9
df
exp(-cp)
1l1r de
exp i (
fll
cos (e - '4'
»
I)
F dp
= (2n)-lcw-I,.
vy) dx dy
cos 9,
focal plane we let u = w cos proportional
+
II)
exp(-cp) 3 0 (Wp>
Jl' pi' •
c- I exp(-cp> 3 0
(lip> (wf) l
0
,.
df (A4)
Now (A4) is in the standard form of a Hankel transform if we can extend the radial integral from R to infinity [116], so W = ( 2n ) -
In section 3.2 we
l
c· (c.
argued that the 194
+
Wi > -
:I , •
spatial frequency
(AS)
c
is
7.A
2"
195
divided by the length of the measurement interval,
which
we may take to be the aperture diameter for the present purpose.
In
expC-2n)
(A4)
= 1.87
x
f) is
the
function
10-~
in the region from R to infinity that
vas included in (AS).
exp(-c
smaller
than
Therefore the finite integral (A4) may
fluctuate about (AS) by a few parts per thousand.
7.8
Normalizina Power Spectra £rom the FFT In
minimizing
the
residual
in
trigonometric
a
approximation S(aO,al,al-,.')
=
one requires the sums
1:(80
y o l,
+
ale'"
+
B,-e-'"
u
= 2n/L,
t yoexp(iu),
I:
+
•••
-y)., (B1>
yoexpCiu), etc.
These sums are just the sequence £urnished by a FFT o£ y(x). In real space the total variance o£ yCx) is given by t(y-dc)· = Lay·
(82)
The FFT sorts the variance into £requency bins, to
but we need
understand whether the normalization is correct.
£ore
we consider sequences y(x) that yield a
spectrum in an ensemble sense.
£lat
There(white)
Consider any FFT bin such as
ty"exp(jiu). Then i£ we can estimate the variance associated with every
this £requency bin,
we have the average variance
such bin and can evaluate the total variance in
£or £re-
quency space. The values
by
trick is to note that we can always group pairs £or which ju haa one value and that
y(x) value
plus n. Let them be y~
= y(x.)
exp(jiu)
y. = yCx.) expCi(ju 196
(B3) +
n»
(84)
7.B
197
We then have L/2 iteme in a sum o£ the £orm = CoE«y. - y_ )1>
= CoE
+
-2
(B5)
where the brackets indicate an ensemble average and C cipates the need £or a normalizing constant. lation
anti-
But the corre-
vanishes £or white noise and the average There£ore (B5) gives CLa y l £or one
quency. IlL
We have L such £requency bins.
o£ £re-
=
We need to take C
to normalize the squares o£ the FFT o£ y(x) when calcu-
lating a power spectrum that conserves variance. O£ten it is more convenient to calculate the FFT y(x)/L
since it yields directly the sequence ao,
al,
etc.
When thi9 sequence o£ coherent imaging signal amplitudes squared to obtain the sequence SJ requires
= aJeaJ,
o£
is
the above result
that LS J be calculated to obtain the set o£
power
spectrum values that conserve variance. Since the resolution o£
such a FFT is the £requency £l
=
IlL,
we note that
the
sequence SJ/£l conserves variance.
With this interpretation
the
power
latter
sequence
is called a
spectral
denSity
£unction. Suppose (FNL)
by
that
we
de£ine a £ractional
dividing (B2) by L,
taking a
square
noise root,
level and
dividing by the dc level
FNL
=AI
dc
= ay/dc
(B6)
7.B
198
where
A
is the rms value for y(x) - dc.
generality
FNL
= Gy
we may scale
With no
quantities such that dc
and in a plot of the sequence SJ'
level caused by FNL is FNLa/L.
loss
= 1.
the white
in Then
noise
7.C
"atching Two Correlated Given
estimates
two
s. and
wave~orms
o~
Wave~orms
mean zero and standard error
£ind the £actor Q that minimizes
S"
the
variance sum
v Dividing
= 1: (Y. (x) - Qy. (x»,
(el>
by the number of samples L and using
brackets
to
denote averages V/L =