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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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Order Number 8726811

The knife edge test as a wavefront sensor

KenKnight, Charles Elman, Ph.D. The University of Arizona, 1987

U·M·I

300 N. Zeeb Rd. Ann Arbor. MI48106

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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



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



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



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



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



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



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



£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



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,



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



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



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



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



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



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



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



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



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



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



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) +



(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,



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 =