NISTIR 88-3808

Narrow-Angle Laser

Scanning Microscope

System for Linewidth

Measurements on Wafers

D. Nyyssonen

CD Metrology, Inc. Germantown, MD 20874

June 1988

Issued April 1989

Prepared for: U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

National Engineering Laboratory

Center for Manufacturing Engineering

Precision Engineering Division

Gaithersburg, MD 20899

N1STIR 88-3808

Narrow-Angle Laser

Scanning Microscope

System for Linewidth

Measurements on Wafers

D. Nyyssonen

CD Metrology, Inc. Germantown, MD 20874

June 1988

Issued April 1989

National Bureau of Standards became the

National Institute of Standards and Technology

on August 23, 1988. when the Omnibus Trade and

Competitiveness Act was signed. NIST retains

all NBS functions. Its new programs will encourage

improved use of technology by U.S. industry.

Prepared for: U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

National Engineering Laboratory

Center for Manufacturing Engineering

Precision Engineering Division

Gaithersburg, MD 20899

U.S. DEPARTMENT OF COMMERCE

Robert Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

Raymond G. Kammer, Acting Director
'

'

.
Table of Contents

Page

Abstract 1

Introduction 1

Characteristics of Patterned Thin Filins 2

Optical Design of a Metrological Microscope 6

Spatial Coherence and Angle of Incidence 7

Aberrations 11

Alignment 11

Alignment Procedure 13

Measurement of Linewidth 15

Radiometric Signal/Noise Ratio 17

Scalar Theory for Thin-Layer Imaging 17

Vector Theory for Thick-Layer Imaging 20

Accuracy and Precision 24

Design of Linewidth Calibration Standards 26

Summary 27

Acknowledgments 28

References 28

Figures 31

Appendix I 46

Appendix II 55
'
NARROW-ANGLE LASER SCANNING MICROSCOPE SYSTEM

FOR LINEWIDTH MEASUREMENT ON WAFERS*

Diana Nyyssonen**

CD Metrology, Inc.

Germantown, MD 20874

ABSTRACT: The integrated-circuit industry in its push to finer

and finer line geometries approaching submicrometer dimensions

has created a need for ever more accurate and precise featuresize measurements to establish tighter control of fabrication

processes. Under the auspices of the NBS Semiconductor Linewidth Metrology Program, a unique narrow-angle laser measurement

system was developed. This report describes the theory, optical

design, and operation of this system and includes computer software useful for characterizing the pertinent optical parameters

and images for patterned thin layers. For thick layers, the

physics is more complex, and only elements of the theory are

included here. However, for more detail the reader is referred

to several related reports listed in the references.

KEY WORDS: metrology, coherence, critical dimensions, linewidth

measurements, micrometrology, scanning microscopy

INTRODUCTION

The push to submicrometer feature sizes on integrated-circuit (IC)

wafers has resulted in a need for more accurate and precise dimensional measurements in order to establish tighter control of fabrication processes, improve yield, and ensure that lithographic

* Contribution of the National Bureau of Standards; not subject

to copyright.

Formerly with the National Bureau of Standards; collaboration

continues under a Guest Worker Agreement.

* *
- 2 -

and linewidth measurement systems meet specifications. Measurement systems used in the fabrication process must have accuracy

and precision much better than the variation in the parts being

measured and, in turn, the accuracy and precision of calibration

standards must be still better than the instruments being

calibrated.

Under the Semiconductor Metrology Program within the Center for

Electronics and Electrical Engineering at NBS a project was

initiated to develop improved instrumentation, calibration procedures and standard reference materials for linewidth measurement on IC wafers. The result was the development of the NBS

narrow-angle laser linewidth measurement system. This system was

first described in 1978 [1] and discussions of various aspects of

this system have appeared in the literature since then [2-7].

However, there is no single report which adequately describes

the details of its theory, design and operation. This report

attempts to rectify this situation. In order to understand the

motivation behind the development of this system, it is necessary

to understand the optical characteristics of the patterned features on integrated circuits which this system was designed to

measure

.

CHARACTERISTICS OF PATTERNED THIN FILMS

The optical properties of patterned integrated circuit wafers are

best described using the language of ellipsometry . Wafers are

typically made up of layers of insulators and conductors with one

or more of these layers patterned. These layers may vary in

thickness from approximately 0.1 urn to 0.5 urn or more. In this

report discussion is limited to thin layers, those less than onequarter of the illuminating wavelength. Thicker layers cannot be

described by scalar theory and a vector treatment must be used.

See ref. 19. For the moment consider a region on the wafer where

only the top layer is patterned as shown in Fig. 1. A single
-3-

plane wave of wavelength A is incident at an angle ©. This plane

wave is refracted and reflected at each interface. To determine

the complex reflectance, that is, both the amplitude and phase of

the reflected wave, the Fresnel equations [8] are used. In matrix form [9], each layer is characterized by a matrix of the form

M. =

COS Vj

- i u

j

sm Vj

^

sin v

j

COS Vj

( 1 )

where u^ = cos ©.

Tlj cos 0^

parallel polarization

perpendicular polarization

and ri = is the complex index of refraction of the jth

layer. ©^ is found from Snell’s law:

fi sin © = f) . sin ©. = a constant for all j o o 3

and Vj , the effective optical thickness, is given by

v . = ( n . t .cos© .

)

J A J j j

A. UNPATTERNED LAYERS

The characteristic matrix for the composite of N unpatterned

layers is then given by the product of the characteristic

matrices of the individual layers

M

1 N

= M

1

‘ M

2

‘ ‘ #M

j

’ ' <M

N

(2)

with
-4-

m

ll m12

m

21 m

22

The amplitude and phase of the reflected and transmitted waves

are found from the characteristic matrix M. Let

X = mn + Ti sm

i2

Y = m

2 i

+ fi m

22

then the complex reflectance is given by

X-Y

r = (3)

X+Y

If r = x+i y, then the phase of the reflected wave is given by

and

R = | r | 2

B. PATTERNED LAYERS

For the case of a patterned wafer (Fig. 1) where the relative re

flectance and phase difference at an edge are needed, first the

characteristic matrix

N for layers 1 through N is calculated

and then the matrix M

2 N for layers 2 through N. A relative reflectance, R, to be used later, is then defined by
<

R

R

1,N

R

2,N

R

2,N

R

1,N

•1,N

R

2,N

(5)

where the convention is chosen so that R < 1. The corresponding

relative phase difference is defined by

0 = ^2,N - *1,N + 2k

o

t

l

COS 9 ( 6 )

where k = and t. is the thickness of the top layer. At

normal incidence, there is no difference between incident waves

with parallel or perpendicular polarization and these equations

simplify. Appendix I includes a short computer program written

in FORTRAN 77 which calculates R and 0 at normal incidence. The

calculation of R and 0 represents the first step in modeling the

image of a patterned wafer.

C. BEHAVIOR OF R AND 0

R and 0 vary with index and thickness of the layers, angle of

incidence, polarization and wavelength. In addition, dielectrics

behave differently from metals. Figures 2-5 illustrate

variations of R and 0 as a function of these parameters. In

Figs. 2(a) and (b), the curves for R and 0 as a function of t^

are relatively simple for a single patterned layer on a silicon

substrate. These same curves may also be plotted parametrically

as R(t^) vs 0 ( t^

)

as shown in Figs. 2(c) and 2(d).

In Fig. 3, the situation is more complex. With an oxide layer

under the metal, one has the option of varying either the

thickness of the metal layer or the thickness of the oxide layer
- 6 -

underneath. If the metal layer is held constant and the oxide is

varied, the resulting curve is an ellipse.

Figures 2 and 3 are for the case of normal incidence and a single

wavelength. Figures 4 and 5 illustrate the variations with wavelength and angle of incidence for a silicon dioxide layer on

silicon. In real cases, the index of refraction also varies with

wavelength, changing the behavior of the curves for silicon

dioxide shown in Fig. 4.

For accurate linewidth measurements, R and 0 must be constant over

the solid angle of illumination. It is possible to determine the

maximum allowable angle for a given material (or combination of

materials) using eqs. (l)-(6). For example, for Si0

2

on Si, if

a 2% variation in R with © is allowed, © can be determined as

max

a function of thickness of the oxide. See Fig. 5. © is then

max

the maximum allowable illumination angle for the measurement system. For thick layers, this requirement rather than the coherence

requirement will determine the illumination cone angle for the

system.

OPTICAL DESIGN OF A METROLOGICAL MICROSCOPE

The variations in R and 0 with wavelength and angle of incidence

are the driving force behind the design of the narrow-angle, laser

linewidth microscope. As in ellipsometry , accurate dimensional

measurements become exceedingly difficult and the data analysis

time consuming if all of the experimental parameters are allowed

to vary. Ideally then, the optimal solution to dimensional metrology would be a single wavelength, single-angle-of -incidence

system analogous to that used in ellipsometry [10]. Single wavelength is readily achieved using a laser source. However, in

optical microscopy, it is neither desirable nor necessary to use

a single angle of incidence; a narrow illuminating cone angle

over which the resulting R and 0 of the specimen are essentially
-7-

constant is sufficient. The cone angle numerical aperture

(N.A.), however, must be chosen with care for the particular

index of refraction and thickness of the material to be measured.

A schematic of the layout of such a system built at NBS is shown

in Fig. 6. The combination of the rotating ground glass disc and

the illumination optics is used to control the illumination cone

angle and the coherence at the back focal plane of the objective.

With these parameters very tightly prescribed the microscope behaves like a modified bright-field microscope, that is, it operates as an effectively coherent imaging system. Although the

system uses a one-dimensional piezo electric scanning stage [11]

with interferometric readout of distance and a stationary slit in

the image plane, it is also possible to scan the image plane with

a moving slit arrangement. However, the system requirements for

these two modes of operation are different.

The chief disadvantage of the single wavelength, narrow angle

system is the low throughput. A 1-W laser is used on the NBS

system and approximately 1 nW reaches the detector. The principal losses come from use of the rotating ground glass disc, overfilling of apertures to get uniform illumination, the small aperture limiting the illumination cone angle, the oversize illuminated area at the wafer, and the small slit in the image plane.

The requirement on this slit width is that it be 1/6 or less of

the Airy disc diameter of the objective when projected back to

the wafer in order not to degrade the image waveform [12].

SPATIAL COHERENCE AND ANGLE OF INCIDENCE

In order to discuss the coherence aspects of the system, it is

necessary to introduce some concepts related to coherence. It

is conventional in microscopy to describe spatial coherence in a

bright-field microscope in terms of a coherence parameter defined

by the ratio of the numerical aperture of the condenser with

respect to the N.A. of the imaging objective:
- 8 -

_ N.A. condenser

b

B N. A. objective (7)

The limit S —*0 is associated with coherent illumination as in

the case of a collimated .laser beam and the limit S — « with incoherent illumination. In conventional imaging in a microscope,

neither of these limits is ever realized. However, it is useful

to introduce the ideas of effective coherence and incoherence

here. Effective coherence or incoherence means that the images

of lines on wafers would be essentially the same as that seen in

a fully coherent or a fully incoherent system.

For effective coherence:

S_j < 2/3, for thin, high contrast objects such as opaque

For effective incoherence:

S n may have any value for thin low contrast objects with no

For practical purposes, the images formed for these values of S

B

would be indistinguishable from those corresponding to completely

coherent or incoherent images as illustrated in Fig. 7. One consequence of these considerations is that a high numerical aperture conventional bright-field imaging system with 0.9-N.A. dry

microscope objective can never be effectively incoherent since

can never exceed 2; one must therefore deal with partially

coherent or effectively coherent imaging with such a microscope.

photomasks, and

S

B

< 1/5 for thick, low contrast objects with phase

variations such as most wafer features.

phase variations present (rare), and

for all other cases
-9-

For focused-beam scanning systems with the roles of the illumination and detection systems interchanged, the coherence parameter

is defined analogously as the inverse of that given above, i.e.,

c - N,A, collector t o\

* ‘focused beam

(See Ref. 13). For such systems, effective coherence and incoherence are similarly defined for this Sp. Therefore, it is possible to produce a system with an effectively coherent or incoherent response using either a laser or thermal source such as a

tungsten-halogen lamp. By analogy, the narrow-angle, effectivelycoherent NBS system could have been configured with a focused

laser beam and narrow angle collector. However, because of the

response of thin films to angle of incidence, these two configurations would not have produced the same response.

One major difference between coherent and incoherent imaging is

sensitivity to phase changes in the object or feature being measured. This aspect will be discussed further with respect to the

imaging characteristics of the system. The concepts of effective

coherence and incoherence discussed here refer to the plane of

the wafer and assume that the back focal plane of the objective

is incoherently illuminated (which is not fully realizable in

practice )

.

It is useful, therefore to discuss here the reciprocal relationship between coherence and size of the illuminated area at the

back focal plane with respect to the same parameters at the

wafer. This relationship has not yet made its way into textbooks

but was introduced by Wolf [14]. As applied to the microscope,

it states that the coherence at the back focal plane of the objective determines the illuminated area (and intensity distribution) at the wafer, while the size and distribution of the illuminated area at the back focal plane determines the coherence at

the wafer, providing that: 1) the illuminated area is large
- 10-

compared to the coherence area, 2) the intensity is constant or

slowly varying across the back focal plane, and 3) the region

near the edges of the illuminated area is not being considered.

Because of throughput it is not desirable to require a high degree

of incoherence at the back focal plane of the objective. In fact,

if only small line objects are viewed a relatively small circular

field of view is required and the diameter D

c

of the coherence

area at the back focal plane can be found from

0 . 6Af

D

W

(9)

W

where is the desired diameter of the illuminated area on the

wafer, A the wavelength, and f the focal length of the objective.

It is also required that

D®F

<< Df (10)

BF where = 2 (N.A.

conc} ense v ^ * Therefore, for a 4 mm focal length

objective at a wavelength of 0.5 um and = 50 urn these parameters

would be Dj - 1 mm and D

c

> 25 um for an effectively coherent

system. If the coherence area at the back focal plane is reduced,

a larger area on the sample will be illuminated and lower throughput will result. On the other hand, if the coherence area is increased, a smaller area will be illuminated at the sample making

it difficult to find the patterns which need to be measured and

W W possibly violating the requirement that >> .

Only spatial coherence, that is, coherence in the plane of the

wafer or lens aperture, has been discussed. The concept of
-licoherence volume (coherence area times coherence length) is useful

here as well. Because there are no differences in optical path

length in an ideal imaging system, coherence length is not usually

of concern. However, in coherent imaging systems, coherent noise

or speckle becomes a problem due to the extremely long coherence

length of the laser source. Normally, dust particles and scratches on optical surfaces produce diffraction patterns which may be

observed in the image plane due to the long coherence length. As

shown in Fig. 8(a), a system using unfiltered tungsten illumination (white light) has a very short coherence length and small

volume compared to a laser source (Fig. 8(b)) and therefore will

not exhibit these effects. The present narrow-angle laser system

has a peculiar pencil-shaped coherence volume as shown in Fig.

8(c), which eliminates most of the coherence effects normally

associated with coherent imaging.

ABERRATIONS

In a system with a stationary slit and moving wafer where only

the axial image is used, only spherical aberration is of concern.

When the image is scanned with a moving slit, the off-axis aberrations are also of concern. Because it is desirable to eliminate as many variables which effect the measurement as possible,

diffraction-limited performance and aberration tolerances of A/4

or less are desirable. The ultimate test, however, is whether

the system produces a diffraction-limited image waveform. Although aberrations may be taken into account in edge detection

formulas [4], this approach is not recommended.

ALIGNMENT

A bright-field microscope operating in an effectively coherent

mode has much more severe requirements on alignment than a conventional microscope. Because of sensitivity to phase variation,

the illumination must be not only uniform in intensity across the

line pattern but also uniform in phase across the illuminated
area. If this requirement is not met, the resulting nonsymmetric

images invalidate the algorithms used for accurate edge detection. Because poor quality lines will also produce nonsymmetric

images, it is necessary to distinguish between these two sources

of nonsymmetry. One method is to rotate the specimen 180° and

compare image waveforms. System asymmetry will stay the same

while line irregularity will rotate. However, this test is

difficult to interpret when both sources of nonsymmetry are

present.

A special test wafer is therefore used for testing system performance. The ideal wafer is a patterned layer with R = 1, 0 - ti.

The waveform for a 180 nm thick layer of Si02 (R = 1, 0 = 0.6n)

is shown in Fig. 9. This line pattern is ideal because of the

symmetric waveform produced at each edge. Three sources of error

readily show up in the image waveform: 1) misalignment of the

illumination system including decentering of the aperture stop

and tilt errors in the optical elements, 2) tilt of the wafer

with respect to the focal plane of the objective and 3) tilt of

the scanning plane with respect to the image plane when a moving

slit is used.

In order to interpret the waveforms, it is necessary to understand

the diffraction-limited behavior of this waveform with defocus.

Figure 10(a) illustrates the nonsymmetry produced with defocus.

On one side of focus, one of the maxima at the image edge is

enhanced while the other is reduced. On the other side of focus,

the opposite occurs. In addition, as defocus increases, the distance between these peaks increases as shown in Fig. 10(b).

When the only error is the tilt of the wafer with respect to the

focal plane, one edge of the line will lead (or lag) the changing

image waveform as a series of scans are made through focus. A

similar effect occurs when the image plane is tilted with respect

to the scanning plane. This effect is illustrated in Fig. 11.
- 13 -

Misalignment can produce a large variation in waveforms. An example is illustrated in Fig. 12(a). In general, nonsymmetry is

present as focus is varied but is different from that shown in

Fig. 10(a). There is also a loss in resolution, so that the

distance between peaks is larger than that shown in Fig. 10(b).

With both wafer tilt and misalignment present, it is possible to

get waveforms like that shown in Fig. 13 where there is symmetry,

but of the wrong kind.

Given the difficulty of determining and correcting these errors

when all of these effects (misalignment, wafer tilt, poor line

edge quality) are present, a procedure which will now be discussed has been worked out for laser alignment of the microscope.

ALIGNMENT PROCEDURE

We have found that in a high quality microscope the individual

microscope optics are usually aligned adequately in their own

mounts. However, these components when assembled to form the

microscope are generally improperly aligned. This is probably

because manufacturing tolerances of conventional microscopes are

not adequate for this highly demanding mode of operation. It is

common practice, for example, to correct errors in one component

by an offsetting adjustment in another. One can expect, therefore, to have to shim and in some cases redesign mounts and

adjustments to achieve the desired alignment accuracy.

The required tools for this job are a small HeNe laser (1 mW or

less), neutral density filters to reduce the power to a comfortable level for visual viewing through the microscope, a laser

beam steerer or other method for controlling the position and

tilt of the laser beam, a polished silicon wafer or other highly

reflective, flat surface, and a wafer holder with tilt adjustment.
The first step is to disassemble the microscope. If possible,

remove all optics except the beam splitter in the head and the

wafer on the stage. Then select a small aperture on the illumination side and one on the viewing side for reference. The

center of these apertures together with the requirement that the

return beam fall on the exit aperture of the laser determines the

optical axis of the system. All components will be aligned to

this axis as illustrated in Fig. 14.

The principle of this procedure is to add components one at a

time and make sure that each is aligned to this axis. This is

achieved by making sure that the return beam goes back to the

laser aperture and the forward beam remains centered on the

chosen reference aperture for each added optical element. In

general, centering is done first and if the forward and return

beams cannot both be returned to their reference points, then

tilt must be adjusted by shimming or other means. In general,

the tilt must be under- or over-corrected and the element recentered and these steps repeated until the desired alignment

accuracy is achieved.

One difficulty with this method is that the laser beam changes

diameter at the reference aperture as elements are added. In

some cases, it may be necessary to temporarily remove an element

already aligned if it can be replaced exactly in order to keep

the beam size small and maintain the desired accuracy. No rule

of thumb can be given for "tolerable errors." Because of the

large number of components (9 lenses, a beam splitter and 2

apertures in the NBS system) and the variations in microscope

design, it is best to align every element as accurately as

possible, that is, within a small fraction of the laser beam

diameter. Because of its high magnification, the microscope

objective is left for last. The aperture pinhole, which determines the illumination cone angle, will be next to last. The

ultimate test of the accuracy achieved is the symmetry of the

resulting waveform in a series of profiles with increasing

amounts of defocus.
- 15 -

When all of the microscope optics have been aligned adequately, the argon or other high-power laser source is brought into

coincidence with the alignment laser as shown in Fig. 15 and the

beam expander and ground glass are aligned. Through these last

steps, the aperture pinhole as viewed through the microscope with

an auxiliary alignment telescope (such as used for centering the

disc in phase contrast microscopy) or other device must remain

stationary and uniformly illuminated.

One element that needs special attention is the rotating ground

glass disc. If the normal to the surface precesses about the

optical axis, fluctuations in intensity in the image plane will

be observed. For this reason, a flexible coupling and a precision

bearing at the drive motor are recommended.

After the alignment is completed, the alignment wafer is replaced

with the SiC^ test wafer. A scan of a line is made and the tilt

of the wafer adjusted if indicated. If nonsymmetry due to misalignment is detectable, the alignment procedure was not performed

accurately enough. Once alignment is deemed satisfactory, other

adjustments of optical elements should be required or made thereafter. With each new wafer, only wafer tilt and focus are

adjusted.

MEASUREMENT OF LINEWIDTH

The system must produce and maintain an ideal image waveform for

the test line object because of the demands of accurate edge detection. An equally important requirement is the accurate measurement of distance (i.e. linewidth) . To scan the image, the

NBS laser system moves the wafer and measures the motion with a

laser interferometer, thus providing traceability to fundamental

standards of length. Aside from the usual demands of accurate

laser interferometry, this system has some unique aspects principally involving alignment of the elements of the scanning system

including scanning slit, line object, axis of stage motion and

interferometer axis. The principal difficulty stems from the
extremely short distance of motion of the piezo electric stage,

typically less than 50 urn.

The easiest method of alignment is to use the crosshair in the

viewing eyepiece as a fiducial mark. The slit and line object

can be centered and aligned to the vertical axis visually. In

addition, the axis of stage motion can be aligned to the horizontal axis visually by inspecting the motion of a horizontal line

object as it traverses the field of view. In order to align the

interferometer axis parallel to the piezo electric stage axis, an

auxiliary mirror has been used. This mirror has a mount that fits

into the holes at the pivot points on the stage (See Ref. 11.)

and is constructed so that the mirror face is accurately perpendicular to the direction of motion. Without this auxiliary

mirror, the method is one of trial and error; minimizing a

measured linespacing to eliminate the possible cosine error.

Fortunately, because of the short distances scanned, the angle

accuracy required for a given tolerance on a one micrometer linewidth is not very demanding. The final check, however, is measurement of a known linespacing traceable to national standards

of length.

The major sources of distance measurement errors are vibration

(the system should be mounted on a massive vibration isolation

system) , the least count of the interferometer, and temperature

effects on the system. Because of the short distances, temperature effects on the wafers being measured are negligible.

In the NBS system, the precision of the interferometry is a fundamental limit on the precision of linewidth measurements. One

cannot measure the size of an object to better precision than

that of the distance measurement. This limitation is to some

extent due to the basic design of the microscope, which is sensitive to both acoustic and mechanical sources of vibration and

temperature

.
-17-

RADIOMETRIC SIGNAL/NOISE RATIO

In most microscope linewidth measurement systems employing tungsten sources, the radiometric precision is limited by photon

noise. Here the laser power has been increased in order to maintain the single wavelength, narrow angle mode of operation. Thus

photon noise has been traded for laser output fluctuations. However, the specifications on the laser (< 0.5% variation) are adequate in this case. In addition to increasing the power, the

system uses a variable speed chopper and lock-in amplifier

operating at approximately 350 Hz with high and low-pass filters.

The resulting signal/noise ratio is better than 200/1.

SCALAR THEORY FOR THIN-LAYER IMAGING

Scalar theory of partially coherent imaging has been developed

using several different approaches including convolution integrals

and Fourier analysis. The most efficient approach for computer

calculations is the use of the transmission crosscoefficient of

the optical system [15] as applied to the imaging of line objects

by Kintner [16]. Based on the methods of Fourier analysis, the

complex amplitude transmittance of the patterned line object is

described by

(1 0 < x < W/2

t ( x ) = J (11)

f VRexp(i0) W/2 < x < P

which is Expanded in the Fourier series

t|x) = I Vos (12)

m

where the line object is repeated at a period P which may be

chosen arbitrarily large to describe isolated line objects. R

and 0 are the relative reflectance and phase difference at the

line edge as introduced earlier. (See Eqs. 5 and 6.) The image

is calculated from the Fourier series equation
I<y, . ^ V». (T^

n=-»

where, for a symmetric line object.

oo

and

b = b

n -n

where A

n

are the Fourier coefficients for the line object as given

in Eg. 12.

The function is called the transmission crosscoefficient [14]

and characterizes the optical system including the state of partial coherence of the illumination. For a one-dimensional line

object, following Ref. 15, the transmission crosscoefficient is

given by

(15)

• FU

X + I", n" )F*U 2 + t", n")dn"dr

where >e/(^ ", n") is the two-dimensional intensity

distribution in the condenser aperture, F is the two-dimensional

equivalent of the pupil function, and * denotes the complex conjugate (wide-field Kohler or critical illumination assumed). The

function^/ is sometimes called the effective source function [17]

(assumed to be incoherent, i.e., the coherence interval is small

compared with the effective source size).

Because of the dependence of R and 0 on angle of incidence the

image structure will also vary with angle of incidence. This
formulation assumes that R and 0 are constant over the illuminating solid angle. If they are not, it is still possible to calculate the image from these equations. However, R and 0 become

functions of and n and the integrals can no longer be separated

as shown in Eqs. 14 and 15.

In this scalar theory approach, the line object is described by

the planar function t(x). Hence, there is no ambiguity about

focus; the object thickness is much thinner than the depth of

field of the optics. If the plane of the object is not coincident with the focal plane of the lens, the defocus aberration

term is included in the pupil function,

Fdefocus (u) = exp(ika

2 u

2 )Rec u | f

(16)

where a£ is the constant that indicates the amount of defocus in

number of waves (units of A ) , k is the wave number, and Rec is

the rectangular function of width M/P, which defines the aperture

diameter

.

With lines patterned in thin layers (less than approximately A/4

thick) and vertical edges, the images can be accurately described

by these scalar equations and the coherent optical edge detection

threshold T can be used for linewidth measurement [4].

c

T = 0.25 ( 1 + R + 2 JR cos 0), (17)

However, in order to use T , R and 0 must be known. R (the ratio

c

of reflectances on either side of the edge) is best determined

from the image waveform. If 0 (the phase change at the edge from

Eq. 6) is unknown, the dual threshold method [6] illustrated in

Fig. 16(a) may be used to determine linewidth.

Because of the complex waveforms which result from the coherent

imaging of line features of varying contrast with phase discont-
- 20-

inuities present, best focus is difficult to determine. One objective criterion currently in use is minimization of as defined

in Fig. 16(b). This distance can also be toleranced to ensure

that measurements made with inaccurate focus are rejected. That

is, at best focus is a minimum but by plotting versus the

change in linewidth with defocus, an acceptable range for may

be specified for a desired measurement accuracy.

In Fig. 17, calculated image waveforms are given for lines

patterned in silicon dioxide on silicon and for chromium on

glass. The computer software used to calculate the theoretical

images is given in Appendix II. The reproducibility of these

waveforms for linewidth measurement is determined principally by

the accuracy of alignment of the line to be measured to the

reference crosshair, accuracy of leveling of the wafer, and

accuracy of focus. All of these operations should be automated

so that they become operator independent and the required

accuracy can be specified and maintained.

VECTOR THEORY FOR THICK-LAYER IMAGING

Scalar theory is unable to accurately predict the image profiles

for line objects which violate the initial assumptions of infinitesimally thin (planar) objects and vertical edges characterized

by abrupt discontinuities in R and 0. For patterned layers thicker than approximately one-quarter of the illumination wavelength,

the multiple reflections which occur within semi-transparent

layers result in constructive and destructive interference, which

affects R and 0 and the scattering patterns as well. In addition,

both metals and dielectrics exhibit waveguide effects near edges

which also influence the nature of the image waveforms. (See

Fig. 18.) The major differences for thick layers as compared

to thin layers are (1) the broadening of the minimum at the line

edge, (2) enhanced maxima on either side of the edge particularly

with sloping edge geometry, and (3) edge ringing which extends

farther from the line edge in some cases.
Imaging of lines patterned in thick layers may be modeled using

vector theory. For lines patterned in a thick layer with vertical edges, the complex dielectric constant of the material rather

than the complex reflectance function is expanded in a Fourier

series

<«) s? - E “

(¥*)

m

The appropriate wave equation

V

2 E„ + k

2 eE„ = 0 (TE-mode)* (19)

y o y

and

v2h

y -Hi-5£ + k

o® H

y

= 0 (TM-mode) (20)

where k = 2 ti/A is solved for the E- and H-fields within the

o

layer. In this case, when the Fourier series expansion for

e(x) is substituted into Eq. (19), the resulting equation is

Hill's equation [18], which has solutions of the form

vx ' z > = m

• z

exp (a z) + A' exp (-a z) m m m m

B. _ exp( 2ni jx/P

)

3 /it\

where the a 's are the eigenvalues and the B. 's are the eigen

m 3 ,m

vector solutions to Hill's equation. The Ammand A' are weighting

constants which must be determined from the boundary conditions.

Each of these terns represents an inhomogeneous eigenfunction or

waveguide mode which is supported by the line structure.

* Defined as that polarization component with its electric

field parallel to the line being measured.
With a single plane wave incident, the boundary condition equations allow for solution for the Fourier series coefficients in

the plane wave expansion for the reflected (scattered) field

e

r

(x,z) ( 22 )

For normal or nearly normal incidence, the magnitudes of the Eand H- fields are equal and no polarization effects are present.

R Therefore, the E

^ coefficients in Eq. (22) can be substituted for

the A

n

’s in the scalar imaging equation (Eq.(14)). In this case,

Eq. (22) may be regarded as representing the equivalent planar

object which would produce the same image as the thick object of

Eq. (18). The equivalent planar object is taken as located in

the plane of the top surface of the thick layer. This concept

is important to understanding the problem of focusing for thick

layers

.

For nonvertical edges, the single thick layer may be subdivided

into a set of sublayers each of which may vary in linewidth, complex index of refraction and offset (to allow for asymmetric line

objects). Such a representation is shown in Fig. 19. When boundary conditions are applied at each sublayer interface, the solution of the resulting equations yields the scattered field in the

same form as Eq. (22). Thus any nonplanar structure can be represented by an equivalent planar structure and its image determined. For details of the method, see Refs. 5 and 19.

This method of computing both the reflected field and the corresponding microscope image requires no approximations of the type

usually found in calculations of the scattered field, such as

limits on the conductivity or slope of the surface. Limitations

may be imposed, however, by the computation capability available.

First, increasing the number of layers used to approximate the

structure increases the computing time linearly. In most cases
- 23 -

of interest seven to nine layers were sufficient to produce significant results.

The second limitation is in the truncation of the series, i.e.,

the matrix sizes used in the computations. In this case, as for

a single layer [4], all of the reflected plane waves which have

diffraction angles less than + ti2 in the air are included. With

P = 12 urn and A =0.53 urn, 22 diffracted orders are included.

This requires a 45 x 45 complex eigenvalue matrix and a 90 x 90

complex matrix for solution of the boundary condition equations.

This choice necessarily truncates the series which represents the

- in the layers with higher refractive index. This truncation

does not appear to significantly affect the results except for

very small linewidths and near resonances, that is, either where

the thickness of the layer or the linewidth is approximately equal

to the wavelength of the illumination.

Also, for grating objects with P < 12 urn, the assumed periodicity

is true. However, for isolated line objects near resonances, P = 12 urn is not large enough to eliminate the effect of the assumed

adjacent lines on the calculated image. In order to calculate

images of isolated lines, a larger period and, therefore, larger

matrix sizes would have to be used.

As discussed earlier, this approach to the imaging of lines patterned in thick layers involves replacement of the thick line object by its equivalent planar object located in the plane coinciding with the top surface of the patterned layer. It can also be

shown that displacement of the top surface of the thick layer (or

equivalent planar object) along the optical axis of the imaging

system introduces a focus error as in conventional scalar imaging

equations with the accompanying loss of resolution and distortion

of the image profiles.

At this time, no universal, simple, and accurate edge detection

methods have been found for thick layer imaging. The complex
image structure in the vicinity of the line edge depends upon

thickness and edge geometry as well as wavelength and other parameters of the imaging system. It is difficult to see how, in

such a complex relationship of these parameters, a single number

will be adequate to characterize line geometry.

Calculated images have been compared with experimentally measured

images from the narrow-angle laser linewidth system. Some results are shown in Figs. 20 and 21. One of the major difficulties

in getting agreement between theory and experiment is finding line

objects with well characterized edge geometries.

ACCURACY AND PRECISION

Both the accuracy and precision of any metrology system needs to

be established. Precision can be determined by repeated measurements on a control specimen. In the present case, the quality of

the line specimens may limit precision. That is, specimens with

rough edges cannot be placed in exactly the same position each

time, and, therefore, the precision of the measurements is a

function of edge roughness as well as system parameters. Unfortunately, the quality of most available processed wafers is not

suitable for standard reference materials. Indications are that

for the best thin layer materials available, the precision of the

measurements when the coherent edge detection threshold is used

is comparable to that of the photomask system used for calibration of SRM 475, which is approximately + 0.05 uni (three standard

deviations). The narrow-angle laser system at NBS has not yet

reached the operational level of the photomask system. Until such

time, there will be operator dependence due to focus, alignment

and leveling errors. The difference here is that at the operational level all image scans which do not meet specified tolerance criteria are rejected automatically with an accompanying

improvement in the long term precision of the system. The laser

system has not been in routine use at this level long enough to

get an accurate number for measurement precision.
- 25-

Accuracy is customarily assessed based on fundamental physics considerations and/or comparison with other measurement techniques.

In this case, at the level of accuracy being considered, comparison with other techniques cannot provide comparison numbers.

There are no other fundamentally accurate optical techniques for

dimensional measurements on thick objects with accuracies at the

0.05 um (A/10) level available at this time. A common recourse

is to compare optical with scanning electron microscope (SEM)

measurements. However, it has been shown that at this level SEM

measurements are suspect [20] due to both electron beam interactions with the specimen and instrument errors. Therefore, comparison with SEM can only be expected to indicate gross errors at

best. Even edge slope (or geometry) for thin layers (< 200 nm)

can only be determined crudely in the SEM. (Magnifications of

more than 100,000X are required.)

Therefore, accuracy needs to be assessed in terms of fundamental

physics of the measurement process. For the photomask calibration

system, edge slopes greater than approximately 70° produce images

which are indistinguishable from vertical (90°) edges. This is

supported by the fact that structure or variations which occur

within a distance less than approximately 1/6 the Airy disc diameter of the imaging objective do not affect the image. Hence,

for lines with edge slopes greater than approximately 70°, there

is an uncertainty in the measurement (for lines patterned in a

150 nm thick layer and 0.9 N.A.) of approximately + 0.05 um (worst

case) if the measurement is taken to be the mean width. See Fig.

22. For the photomask case, SEM measurements on Cr-CrO masks

corroborated this value.

For thin layers on wafers, the same argument may be applied with

similar results. However, the agreement between theoretical and

experimental image profiles must also be considered. In the wafer

case, there is much more variation in materials and image profiles

as well as greater system sensitivity to optical alignment, leveling, and focus errors. Therefore, while the narrow-angle laser
system has the capacity for accuracies on thin layers comparable

to that of the photomask system, the accuracy should be assessed

on a case-by-case basis with the above considerations taken into

account

.

For thick layers, assessment of accuracy is more complex. First,

theory is based on a model which has yet to be fully evaluated.

There are inappropriate assumptions in the modeling near resonances. Although the calculations are known to be in agreement

for thin layers, the accuracy of the calculations for other cases

has not been established. When discrepancies occur between calculated and experimentally measured image profiles, it is not

known whether these differences are due to inaccuracies in the

calculations, deviations of the system response from the ideal,

poorly characterized line geometry, poorly known optical constants, or all of these. More work needs to be done on comparisons with line objects of known geometry, perhaps preferentially

etched silicon samples, and on testing the accuracy of the calculated image profiles.

DESIGN OF LINEWIDTH CALIBRATION STANDARDS

Photomask materials have relatively little variation in optical

constants. Hence, choice of a standard reference material was

simplified. The most commonly used mask materials were chosen,

CrO on Cr on glass, and an appropriate warning given about calibration of systems used to measure other materials was also given

[21]. For wafers, there is an enormous variation in index of refraction and thickness of the layers found on wafers and, therefore, in R and 0 values as well. In fact, different combinations

of materials may produce the same R, 0 values as well. Rather

than sample all variables (R, 0, and linewidth w) over the ranges

of interest (0 < R < 1, 0 < 0 < ti, 0.5 urn < w < 5 urn) which is

impractical, it is possible to determine the nature of the expected errors, that is, their dependence on the variables R, 0, and w

and apply experimental design methodology to the design of a
- 27 -

standard. For linewidth measurements on wafers, the expected

errors are known to be of low order [6]. Work by Dr. James

Lechner, Carol Croarkin, and Ruth N. Varner at NBS resulted in

the proposed optimal six-point design illustrated in Fig. 23.

The expected error surface shown in Fig. 7(a) of Ref. 6 was found

to fit a polynomial of the form

E(R, 0) = ( 1-R) (A + BR + CR2

+ E cos 0 + F cos

2

0) (23)

A search for a D-optimal design [22] was made, based on the

polynomial model of the expected error surface. In combination

with other factors such as ease of fabrication, the six-point

design of Fig. 23 was selected as optimal. It is also a good

design for polynomial surfaces of lower order such as those of

Fig. 7(b) and (c) of Ref. 6. The design points (R^, cos 0^) can

be fabricated from two materials with a silicon substrate using

the combinations shown in Table 2. In each case, only the top

layer is patterned. This design is also relatively insensitive

to small changes in R and 0 such as would occur in normal fabrication of the standard. Thin-layer standards could thus be provided and calibrated with the present narrow-angle laser microscope. However, these standards would not be directly applicable

for applications involving thick layers on silicon.

SUMMARY

This report has described the development of the narrow-angle

laser linewidth measurement system at NBS and its application to

calibration of linewidth measurement standard reference materials

for the IC industry. This system represents a major move toward

optical systems with well characterized waveforms suitable for

accurate linewidth measurements at dimensions on the order of the

illumination wavelength. In the course of its development, major

theoretical advances have also been made in the theory of optical

scattering from and imaging of objects with dimensions on the

order of a micrometer.
- 28- ACKNOWLEDGMENTS

A number of people have contributed at various times to the developments described here. The original computer software for

partially coherent imaging of line objects patterned in thin

layers was written by Dr. Eric Kintner and later expanded and

documented by Clinton R. Gable. Thanks are given to Ruth N.

Varner for help in preparing the software for publication.

Credit for the application of experimental design methodology to

the design of a reference standard goes principally to Dr. James

Lechner, Carol C. Croarkin, and Ruth N. Varner.

REFERENCES

1. Nyyssonen, D., "Optical linewidth measurements on wafers,"

Proc. SPIE Vol. 135, Developments in Semiconductor Microlithography III, 115-119 (1978).

2. Nyyssonen, D. , "Calibration of optical systems for linewidth

measurements on wafers," Proc. SPIE Vol. 221, Semiconductor

Microlithography V, 119-126 (1980).

3. Nyyssonen, D., "Design of an optical linewidth standard

reference material for wafers," Proc. SPIE Vol. 342,

Integrated Circuit Metrology, 27-34 (1982).

4. Nyyssonen, D., "Theory of optical edge detection and imaging

of thick layers," J. Opt. Soc. Am. 72^, 1425-1436 (1982).

5. Nyyssonen, D., and Kirk, C., "Modeling the optical microscope

images of thick layers for the purpose of linewidth measurement," Proc. SPIE Vol. 538, Optical Microlithography IV, 179-

187 (1985).
-29-

6. Nyyssonen, D. , "A practical method for edge detection and

focusing for linewidth measurements on wafers," Opt. Eng. 26 , 81-85 (1987).

7. Nyyssonen, D., "Focused-beam vs. conventional bright-field

scanning microscopy for integrated circuit metrology," Proc.

SPIE Vol. 565, Micron and Submicron Integrated Circuit

Metrology, 180-186 (1985).

8. Stone, J. M. , Radiation and Optics , (McGraw-Hill, New York,

1963) Chapter 16.

9. Driscoll, W. G., ed. , Handbook of Optics , (McGraw-Hill, New

York, 1978) pp. 8-42 to 8-43.

10. Chandler-Horowitz , D. , "Ellipsometric accuracy and the principal angle of incidence," Proc. SPIE Vol. 342, Integrated

Circuit Metrology, 121-130 (1982).

11. Scire, F. E. and E. C. Teague, "Piezodriven 50-um range state

with subnanometer resolution". Rev. Sci. Instrum. _£9, 1735-

1740 (1978).

12. Kinzly, R. E. "Partially coherent imaging in a microdensitometer," J. Opt. Soc. Am. 6_2, 386-394 (1972).

13. Sheppard, C. J. R. and T. Wilson, "Image formation in scanning

microscopes with partially coherent source and detector,"

Optica Acta 25_, 315-325 (1978).

14. Wolf, E., "The radiant intensity from planar sources of any

state of coherence," J. Opt. Soc. Am. 68_, 1597-1605 (1978).

15. Born, M. And E. Wolf, Principles of Optics, 5th ed. (Pergamon,

New York, 1975) pp. 530-532.
16. Kintner, E. C., "Method for the calculation of partially coherent imagery," Appl. Opt. 17 , 2747-2753 (1978).

17. Hopkins, H. H. , "Applications of coherence theory in microscopy and interferometry," J. Opt. Soc. Am. 47, 508-526

(1957)

.

18. Magnus, W. and S. Winkler, Hill’s Equation (John Wiley and

Sons, New York, 1966 and Dover, New York, 1979).

19. Nyyssonen, D. , "Computer software for the computation of the

scattered field and the optical microscope images of line

objects patterned in thick layers," NBSIR 87-3618 (December

1987)

.

20. Postek, M. T. and D. C. Joy, "Microelectronics dimensional

metrology in the scanning electron microscope," Solid State

Tech. Part I 2_9, 145-150 (Nov. 1986) and Part II 2_9, 77-85

( Dec . 1986 )

.

21. Nyyssonen, "Linewidth calibration for bright-chromium photomasks," NBSIR 86-3357 (May 1986).

22. Raghavarao, D. , Construction and Combinatorial Problems in

Design of Experiments (John Wiley and Sons, New York, 1971).
Fig. 1 Schematic of multi-layer structure on a wafer with definition of parameters used for calculation of R and 0.
- 32 -

33NV133U3M 3AI1V13JJ

uo

4-1

4-J a0ox0

ft CM0G

Cn

•H

Pm • G

CO

O

ft 0

•H

C rH E -H

ft

W

cn

Co

CM -H

CPH

•H

Cl,

O

O

33NVJL33133U 3A!lV13d

C-t o

4-1

PJft00X0 • G - 0 — u

X5 -H '— rH

CM -HW 0U G G 0

0>

•H £

ft

GG

W -H

ft EG O pH E ft ft cn 'OG

ft

CM -HE • O

c*

u

•h .c

ft o -6- Ec

coV)

LLI z

acoz

3DNV103-133H 3AI1 V~13b co 0 *H 0£ PJ W pJ 4-1 ft

ft

OG

-G E -H

ftO co E

>h CO

G

T) 4-1 0 H C G rH

ft T3

M -H 0 0

05 4J -H 0 ftGH 0 rH pJ pJ U G ft G 0 C7> E • ft pH C 0 0

•P ft

H

M

U 0 U >iG G 0 U 0 0

<H

G ft

O T> 4h O > G -H 0 0 0 0 U -H H E G

rH 0 *H 0 -H 4-1 T3 > CO C rH

•H CO ft ft pJ P G E

ft o o 0 u

rH 4-1 -H 'O

O 0 4-> -H G

05 St ft

X GOP 0 tr-H ft - 0 0T3

ft

G

— -0H C E

CM

k (t

O 2 0 G 0 • 4-1 WHO

O' 4-1 0 H

n

•H *H P *H lO

ft T3 ft

M 30NVi331J3ii 3AI1V13H i TJ CO 4H 0 G 0 o G ft P h us: W 0 o 0 0 TJ 0 ft 0 -H 0 X 0 0 OG O'

G5 *H 4->

C

p 'd o

4H rH • G O ft W O > 0 0t

•h E 0 0 rH O 4-» 0 -H CO ft GW 0

ft *H

4-> P • T3 0 O G G CP OH

rH 0

4-1 0 mH. 0 C UH k k G -H ft 0 W 0 k W > 0 G 0

•H 4-1 o W

pJ 4H

W

ft *H 0 0 H'd'O c 0 -H

05 0 U 0 W PH • ft -H ^ CQ CP 0 0ft > — c o> u

CM 0 o G

G

G 0H 0 • 4-> -H *d H C 0

•H 4-1 -H

O G5

ft 0 W ft pJ
-33-

Fig. 3 Same as Fig. 2(b) for chromium with the addition of a

silicon-dioxide layer of varying thickness between the chromium

and silicon. The tangent point of each ellipse indicates the

thickness of the chromium patterned layer. Varying the silicon

dioxide between zero and 180 nm produces the ellipse.

A (nm)

Fig. 4 Variation of R and 0 with \ for a 600 nm. thick

layer of SiO~ on Si.
-34-

TE TM

Fig. 5 Variation of R and 0 with angle of incidence © for thicknesses of SiC^ on Si shown.

Collector

*- aser

r lens

beam 3 = ^1 x

^

Rotating

ground glass

Fig. 6 Ray path for ref lected-light laser-scanning microscope system.
-35-

Fig. 7 Comparison of edge profiles calculated for fully coherent

imaging (dashed curve) and for a coherence parameter of 0.2

(solid curve). Edge is located at 5 urn.

Fig. 8 Relative sizes of coherence volumes for microscopes using

the sources indicated: (a) white light thermal source, (b) laser

source, (c) narrow-angle laser system with rotating ground glass

and laser source.
_> u~

OC.I- — OC J Z •- LLl Z !/> — - >-

Fig.

10

Same

as

Figure

9

with

defocus
OPTICAL

INTENSITY

s

•

OPTICAL

INTENSITY

DISTANCE IN MICROMETERS

11 Image waveform with tilt present in wafer

nonsymmetry.

DISTANCE IN MICROMETERS

Fig. 12 Image waveform with misaslignment present
OPTICAL

INTENSITY

OPTICAL

INTENSITY

-38-

(a)

1

DISTANCE IN MICROMETERS

rf

DISTANCE IN MICROMETERS

Fig. 13 Image waveform with slight misalignment and

tilt (b) with correct alignment (a) shown for comparison.
Fig. 14(a) Defining the optical axis of the system.

Fig. 14(b) Alignment errors with addition of lens.
-f 

Fig. 15 Alignment laser beam and high power laser beam

must be brought into coincidence at both A and B.
Fig. It i Proposed dual-threshold edge detection criteria where

T

2 = RT. In this paper, T. is taken to be 0.95 times the reflectance of either the line material or surround, whichever is

higher

Fig. 16(b) Dual-threshold focus criterion for wafers. The edge

width 6 is a minimum at focus. For wafers, the threshold 7. and

T n are taken the same as in Fig. 16(a).
-42-

Fig • 17(a). Comparison. of experimental ( — ) and theoretical (•) ima ge profiles for a window etched in a 150—nm~ thick layer of silicon dioxide on silicon (0.85 objective N.A., 0.2 condenser

N.A., and 530-nm wavelength).

Fig. 17(b) Calculated image profiles from thick (vector, soli

line) and thin (scalar, dotted line) models for a

chromium on glass line.
-43-

0

p

T

ICA

L

IN

T£N

S

!TY

DISTANCE IN m!CRCTE t ERS

Fig. 18 Calculated image waveforms at edges of a single,

vertical edge patterned layer for (a) Cr on SiO- on Si and

(b) SiC^ on Si. 1

X

Air (n-j)

Silicon substrate (ns)

(a)

ni

x-0

zr I M. l X

"2 \

\

V

"3

Nr

\

1

r

( /

JL

n4

"5 , z

(b)

Fig. 19 Cross section of a typical thick line object (a)

and the corresponding multilayer representation (b).
- 44 -

Fig. 20 (a) Comparison of experimental (solid curve) and

theoretical ( ) (based on waveguide model) image profiles for a

window etched in a 616-nm-thick layer of silicon dioxide in

silicon. The calculated curve is based on rj = 1.46,tj 8 = 4.1 +

i(0.06), 0.85 objective N . A. 0.14 condenser N.A., a wavelength of

514 nm, and a linewidth of 4.85 urn (b) SEM image of oxide line

for wafer samples used in (a).

Fig. 21 Comparison of theoretical and experimental image

profiles for 1 urn thick resist on Si for (a) vertical and

(b) non-vertical.
-45-

Fig. 22 Schematic of the profile for an opaque line on

SRM 474 showing uncertainty U.

Fig. 23 Values of R and cos 0 (o) for six-point design

super imposed on error surface. Corresponding to using

the minima at the line edge as the edge detection threshold.
-46-

Appendix I - Software for Calculation of R and 0

from the Fresnel Equations
-47-

1 PROGRAM TH2LR

2 C

3 C THIS PROGRAM COMPUTES THE RELATIVE REFLECTANCE AND PHASE

4 C DIFFERENCE FOR A PATTERNED LAYER WITH A SUBLAYER AND

5 C SUBSTRATE BOTH OF WHICH MAY HAVE COMPLEX INDICES.

6 C

7 C THIS PROGRAM WAS WRITTEN BY D. NYYSSONEN , CD METROLOGY , INC.

8 C

£ Q ************************************************************

10 c

11 C INTERMEDIATE LAYER THICKNESS IS HELD CONSTANT , WHILE THE

12 C PATTERNED LAYER THICKNESS VARIES FROM 0 TO A MAXIMUM VALUE

13 C (TO BE INPUT). THE PATTERNED LAYER THICKNESSES ARE

14 C INCREMENTED IN STEPS OF 0.002 MICROMETERS (20 ANGSTROMS).

15 C

q ************************************************************

17 C

18 C OUTPUT IS ON LOGICAL UNIT IOUT

• ; - ^ 2u DIMENSION T(2)

21 COMPLEX CM ( 2 , 2 , 2 ) , DEL ( 2 )

, CN ( 2 ) , CBASE , CML (2,2), WAVEL

,

22 * X,Y,R

23 DATA I OUT/10/

24 OPEN ( UNIT=IOUT , FILE= ' TAPE10 '

)

25 C

26 C WAVELENGTH OF LIGHT USED=0.53 MICROMETERS.

27 C

28 WAVEL=0 .3

29 C

30 C THICKNESS ENTERED FOR PATTERNED LAYER, T(l), IS MAXIMUM

31 C FOR WHICH CALCULATIONS ARE MADE, INCREMENT IS 0.002 UM

32 C

33 PRINT*, 'INPUT THICKNESS AND COMPLEX INDICES OF LAYERS WITH’,

34 * PATTERNED LAYER FIRST'

35 C

36 C ENTER MAX. THICKNESS OF PATTERNED LAYER IN MICROMETERS

37 C FOLLOWED BY THE COMPLEX INDEX OF THE PATTERNED LAYER

38 C

39 READ*, T( 1)

,

CN ( 1

)

40 PRINT*, 'INPUT THICKNESS AND COMPLEX INDEX OF',

41 * INTERMEDIATE LAYER'

42 C

43 C ENTER THICKNESS OF INTERMEDIATE LAYER IN MICROMETERS

44 C FOLLOWED BY THE COMPLEX INDEX OF THE INTERMEDIATE LAYER,

45 C

46 READ*, T ( 2 )

,

CN(2)

47 PRINT*, 'INPUT COMPLEX INDEX OF SUBSTRATE’

48 C

49 C ENTER COMPLEX INDEX OF SUBSTRATE ( CBASE

)

50 C

51 READ*, CBASE

52 WRITE ( IOUT, *

)

'FOR PATTERNED LAYER, MAX THICKNESS IS ', T(l),

53 * ' AND INDEX IS ', CN ( 1

)

54 WRITE( IOUT, * ) 'FOR INTERMEDIATE LAYER, THICKNESS IS ', T(2),

55 * ' AND INDEX IS ', CN ( 2

)
56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

-48-

WRITE ( IOUT, *

)

WRITE! IOUT, * WRITE ( IOUT , *

)

WRITE! IOUT,*)

WRITE! IOUT,*)

' RNO RM

WRITE! IOUT,*)

TMAX=T( 1)

’INDEX OF SUBSTRATE IS ’, CBASE

’THIS PROGRAM VARIES THICKNESS OF PATTERNED LAYER'

' THICK R-PAT R-INT P-PAT P-INT DELAY',

PNORM COS-P’

T ( 1

) =0 .

0

20 IF(T( 1 ) .GE. (TMAX-0 . 001 ) ) GO TO 60

DO 40 J=1 ,

2

DEL! J)=6. 28318* CN(J)*T(J) /WAVEL

CM ( J , 1 , 1

) =CCOS ( DEL ( J )

)

CM ( J , 1 , 2

) =CSIN ( DEL ( J ) ) /CN ( J

) *CMPLX ( 0 . 0

, -1 . 0

)

CM ( J , 2 , 1

) =CSIN ( DEL ( J )

) * CN ( J

) * CMPLX ( 0 . 0, - 1 . 0

)

CM ( J , 2 , 2

) =CM ( J , 1 , 1

)

40 CONTINUE

CML (1,1) =CM (1,1,1)* CM (2, 1,1) +CM (1,1, 2)* CM (2, 2,1)

CML (1,2) =CM (1,1,1)* CM (2,1,2) +CM (1,1,2)* CM (2,2,2)

CML (2,1) =CM (1,2,1)* CM (2,1,1) +CM (1,2,2)* CM (2,2,1)

CML (2,2) =CM (1,2,1)* CM (2, 1,2) +CM (1,2, 2)* CM (2, 2, 2)

X=CML (1,1) +CML (1,2)* CBASE

Y=CML (2,1) +CML (2,2) *CBASE

R= ( X-Y ) / ( X+Y

)

S=CABS ( R

)

RL=S* * 2

EL=ATAN2 ( AIMAG ( R) , REAL ( R )

)

EL=EL/3. 14159

X=CM (2,1,1) +CM (2,1,2) *CBASE

Y=CM (2,2,1) +CM (2,2,2)* CBASE

R= ( X-Y ) / ( X+Y

)

S=CABS ( R

)

RS=S* * 2

ES=ATAN2 ( AIMAG ( R) , REAL ( R)

)

ES=ES/3. 14159

C=4.0*T(1) /WAVEL

IF(RS.LT.RL) THEN

RO=RS/RL

ELSE

RO=RL/RS

END IF

EO=ES-EL+C

CSE=COS ( 3 . 14159*EO)

WRITE ( IOUT, 80 )T( 1 ) , RL , RS , EL , ES , C , RO , EO , CSE

C

C INCREMENT THE PATTERNED LAYER THICKNESS BY 0.002 UM

C

T( 1

) =T( 1 ) +0 . 002

GO TO 20

60 CONTINUE

CLOSE (UNIT=IOUT)

STOP

80 FORMAT ( 9F8 . 3

)

END

(EOF)
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-49-

TEST CASE FOR TH2LR

FOR PATTERNED LAYER, MAX THICKNESS IS .2 AND INDEX IS (1.46,0.)

FOR INTERMEDIATE LAYER, THICKNESS IS 0. AND INDEX IS (1.46,0.)

INDEX OF SUBSTRATE IS (4.1,.l)

THIS PROGRAM VARIES THICKNESS OF PATTERNED LAYER

THICK R-PAT R-INT P-PAT P-INT DELAY RNORM PNORM COS-P

.000 .370 .370 -.996 -.996 .000 1.000 .000 1.000

.002 .369 .370 -.982 -.996 .015 .999 .001 1.000

.004 .369 .370 -.968 -.996 .030 .997 .002 1.000

.006 .367 .370 -.954 -.996 .045 .993 .003 1.000

.008 .366 .370 -.940 -.996 .060 .989 .004 1.000

.010 .363 .370 -.926 -.996 .075 .983 .005 1.000

.012 .361 .370 -.911 -.996 .091 .976 .006 1.000

.014 .358 .370 -.897 -.996 .106 .968 .007 1.000

.016 .354 .370 -.882 -.996 .121 .958 . 007 1.000

.018 .350 .370 -.868 -.996 .136 .947 .008 1.000

.020 .346 . 370 -.853 -.996 .151 .936 .008 1.000

.022 .341 .370 - . 838 -.996 .166 .923 . 008 1.000

.024 .336 . 370 -.823 -.996 .181 .908 . 008 1.000

.026 .330 .370 -.807 -.996 .196 .893 .008 1.000

.028 .324 .370 -.792 -.996 .211 .877 . 007 1.000

.030 .318 .370 -.776 -.996 .226 .859 . 006 1.000

.032 .311 . 370 -.760 -.996 .242 .840 . 005 1.000

.034 .303 . 370 -.743 -.996 .257 .821 .004 1.000

.036 .296 .370 -.726 -.996 .272 .800 .002 1.000

.038 .288 .370 - .709 -.996 .287 .779 .000 1.000

.040 .280 . 370 -.692 -.996 .302 .756 - . 003 1.000

.042 .271 . 370 -.674 -.996 .317 .733 - . 005 1.000

.044 .252 .370 -.655 -.996 .332 .709 -.009 1 . 000

.046 .253 .370 -.636 -.996 .347 .685 -.013 .999

.048 . 244 . 370 -.617 -.996 .362 .660 -.017 .999

.050 .234 .370 -.597 -.996 . 377 .634 -.022 .998

.052 .225 .370 - . 576 -.996 .392 .608 -.028 .996

.054 .215 .370 -.554 -.996 . 408 .582 -.034 .994

.056 .206 . 370 -.532 -.996 .423 .556 -.041 .992

.058 .196 . 370 -.510 -.996 .438 .530 - .049 .988

.060 .187 . 370 -.486 -.996 .453 .505 -.057 .984

.062 . 177 . 370 -.461 -.996 . 468 . 480 -.067 . 978

.064 .168 .370 -.436 -.996 .483 .455 - . 077 . 971

.066 .159 .3 70 - .409 -.996 .498 .431 - .089 .961

.068 .151 . 370 -.381 -.996 . 513 .408 -.101 .950

.070 .143 . 370 -.352 -.996 .528 .387 -.115 .935

.072 .135 .370 - .322 -.996 . 543 .366 -.130 .918

.074 .128 .370 -.291 -.996 .558 .347 -.146 .896

.076 .122 .370 -.259 - . 996 .574 .330 - .164 . 871

. 078 .117 . 370 -.225 - . 996 .589 .315 -.182 .841

.080 .112 .370 -.190 -.996 .604 .302 - .202 .806

.082 .108 . 370 - . 154 -.996 . 619 .291 -.223 .765

.084 .104 . 370 -.118 - .996 .634 .282 - .244 .719

.086 .102 .370 - . 080 -.996 . 649 .276 -.267 .669

.088 .101 . 370 -.042 -.996 .664 .272 - .290 .613

.090 .100 . 370 - . 003 - .996 . 679 . 270 -.313 .554
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.092 .100 .370 .035 -.996 .694 .271 -.337 .491

.094 .102 . 370 .073 -.996 .709 .275 -.360 .426

.096 .104 .370 .111 -.996 .725 .281 -.382 .361

.098 .107 .370 .148 -.996 .740 .289 -.404 .296

.100 .111 .370 .184 -.996 .755 .300 -.425 .233

.102 .116 .370 .219 -.996 .770 .313 -.445 .171

.104 .121 .370 .253 -.996 .785 .328 -.464 .113

.106 .127 .370 .286 -.996 .800 .344 -.482 .058

.108 .134 .370 .317 -.996 .815 .363 -.498 .007

.110 .142 .370 .347 -.996 .830 .383 -.513 -.041

.112 .149 .370 .376 -.996 .845 .404 -.527 -.084

.114 .158 .370 .404 -.996 .860 .427 -.540 -.124

.116 .167 .370 .431 -.996 .875 .451 -.551 -.161

.118 .176 .370 .457 -.996 .891 .475 -.562 -.194

.120 .185 .370 .481 -.996 .906 .500 -.572 -.224

.122 .194 .370 .505 -.996 .921 .526 -.581 -.250

.124 .204 .370 .528 -.996 .936 .552 -.588 -.274

.126 .214 .370 .551 -.996 .951 .578 -.596 -.296

.128 .223 .370 .572 -.996 .966 .604 -.602 -.315

.130 .233 .370 .593 -.996 .981 .630 -.608 -.332

.132 .242 .370 .613 -.996 .996 .655 -.613 -.347

.134 .252 .370 .633 -.996 1.011 .680 -.617 -.360

.136 .261 .370 .652 -.996 1.026 .705 -.621 -.372

.138 .270 . 370 .670 -.996 1.042 .729 -.625 -.382

.140 .278 .370 .688 -.996 1.057 .752 -.628 -.391

.142 .286 .370 .706 -.996 1.072 .775 -.630 -.398

.144 .294 .370 .723 -.996 1.087 .796 -.632 -.404

.146 .302 .370 .740 -.996 1.102 .817 -.634 -.409

.148 .309 .370 .757 -.996 1.117 .837 -.636 -.414

.150 .316 .370 .773 -.996 1.132 .856 -.637 -.417

.152 .323 .370 .789 -.996 1.147 .874 -.638 -.419

.154 .329 .370 .805 -.996 1.162 .890 -.638 -.421

.156 .335 .370 .820 -.996 1.177 .906 -.639 -.422

.158 .340 .370 .835 -.996 1.192 .920 -.639 -.422 . 160 .345 .370 .850 -.996 1.208 .933 -.639 -.422

.162 .350 .370 .865 -.996 1.223 . 945 - .638 -.421

.164 .354 .370 . 880 -.996 1.238 .956 -.638 -.420

.166 .357 . 370 .894 -.996 1.253 .966 -.637 -.418

.168 .360 . 370 .909 -.996 1.268 .974 -.637 - .416 . 170 .363 . 370 .923 -.996 1.283 .982 -.636 -.414

.172 .365 .370 .937 -.996 1.298 .988 -.635 -.412 . 174 .367 .370 .951 -.996 1.313 .993 -.634 -.409

.176 .368 .370 .965 -.996 1.328 .996 -.633 -.406

.178 .369 .370 .979 -.996 1.343 .999 -.632 -.403

.180 .370 .370 .993 -.996 1.358 1.000 -.631 -.400

.182 . 370 .370 -.993 -.996 1.374 1 . 000 1.370 -.397

.184 .369 .370 -.978 -.996 1.389 .999 1.371 -.394

.186 .368 . 370 -.964 -.996 1.404 .996 1.372 -.391

.188 .367 .370 - .950 -.996 1.419 .992 1.373 -.388

.190 .365 . 370 -.936 -.996 1.434 .987 1.374 -.385

.192 .363 . 370 - .922 -.996 1 .449 .981 1.375 -.382

.194 .360 . 370 -.908 -.996 1.464 .974 1.376 - .380

.196 .357 . 370 - . 893 -.996 1.479 .965 1.377 -.378

.198 .353 . 370 - . 879 -.996 1 . 494 .956 1.377 -.377
-51-

1

2 C

3 C

4 C

5 C

6 C

7 C

8 C

9 C

10 C

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12 C

13 C

14 C

15 C

16 C

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53

54

PROGRAM TH2SV

THIS PROGRAM COMPUTES THE RELATIVE REFLECTANCE AND PHASE DIFFERENCE FOR A PATTERNED LAYER WITH A SUBLAYER AND SUBSTRATE BOTH OF

WHICH MAY HAVE COMPLEX INDICES.

THIS PROGRAM WAS WRITTEN BY D. NYYSSONEN , CD METROLOGY , INC. ******************************************************************

PATTERNED LAYER THICKNESS IS HELD CONSTANT, WHILE THE

INTERMEDIATE LAYER THICKNESS VARIES FROM 0 TO A MAXIMUM

VALUE (TO BE INPUT). THE INTERMEDIATE LAYER THICKNESSES

ARE INCREMENTED IN STEPS OF 0.002 MICROMETERS 20 ANGSTROMS). ******************************************************************

OUTPUT IS ON LOGICAL UNIT IOUT

DIMENSION T ( 2

)

COMPLEX CM( 2, 2,2) , DEL ( 2 ) , CN ( 2 ) , CBASE , CML (2,2) ,WAVEL,

* X,Y,R

DATA I OUT/10/

OPEN ( IOUT,FILE= ' TAPE10 '

)

WAVELENGTH OF LIGHT USED=0.53 MICROMETERS.

WAVEL=0 .53

THICKNESS ENTERED FOR INTERMEDIATE LAYER IS MAXIMUM THICKNESS FOR

WHICH CALCULATIONS ARE MADE, INCREMENT IS 0.002 UM

PRINT*, 'ENTER THE NUMBER OF DIFFERENT PATTERN LAYER'

PRINT*, 'THICKNESSES TO BE EXPLORED'

READ* ,K

DO 60 L=1,K

PRINT*, 'INPUT THICKNESS AND COMPLEX INDICES OF LAYERS WITH',

* ' PATTERNED LAYER FIRST'

READ *, T ( 1 ) , CN ( 1

)

PRINT*, ’ INPUT MAX. THICKNESS AND COMPLEX INDICES OF',

* ' INTERMEDIATE LAYER'

READ*, T ( 2 )

,

CN ( 2

)

PRINT*, 'INPUT COMPLEX INDEX OF SUBSTRATE'

READ*, CBASE

WRITE( IOUT, *

)

'FOR PATTERNED LAYER, THICKNESS IS ', T(l),

* ' AND INDEX IS ' , CN ( 1

)

WRITE ( IOUT, *

)

'FOR INTERMEDIATE LAYER, THICKNESS IS ', T(2),

* ’ AND INDEX OF ' , CN ( 2

)

WRITE! IOUT, *

)

'INDEX OF SUBSTRATE IS CBASE
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-52-

WRITE ( IOUT , *

)

WRITE ( I OUT , * ) 'THIS PROGRAM VARIES THICKNESS OF INTERMEDIATE'

,

* ' LAYER'

WRITE ( IOUT, * )

' THICK R-PAT R-INT P-PAT P-INT DELAY',

* ' RNO RM PNORM COS-P'

TMAX=T( 2)

T( 2

) =0 .

0

20 DO 40 J=1 ,

2

DEL (J)=6. 28318* CN(J)*T(J) /WAVEL

CM( J,l,l)=COS(DEL( J)

)

CM ( J , 1 , 2

) =S IN ( DEL ( J ) ) /CN( J

) * CMPLX ( 0 . 0

, - 1 . 0

)

CM ( J , 2 , 1

) =SIN ( DEL ( J )

) *CN ( J) *CMPLX ( 0 . 0

, - 1 . 0

)

CM ( J , 2 , 2 ) =CM ( J , 1 , 1

)

40 CONTINUE

CML (1,1) =CM (1,1,1)* CM (2, 1,1) +CM (1,1, 2)* CM (2, 2,1)

CML( 1,2) =CM( 1,1,1 )*CM( 2,1,2 )+CM( 1,1,2 )*CM( 2,2,2)

CML (2,1) =CM (1,2,1)* CM (2, 1,1) +CM (1,2,2)*CM(2,2,1)

CML( 2,2) =CM( 1,2,1 )*CM( 2,1,2 )+CM( 1,2,2 )*CM( 2,2,2)

X=CML (1,1) +CML (1,2) *CBASE

Y=CML (2,1) +CML (2,2) *CBASE

R= ( X-Y ) / ( X+Y

)

S=CABS ( R

)

RL=S* * 2

EL=ATAN2 ( AIMAG ( R ) , REAL ( R)

)

EL=EL/3 . 14159

X=CM (2,1,1) +CM (2,1,2)* CBASE

Y=CM (2,2,1) +CM (2,2,2) *CBASE

R= ( X-Y ) / ( X+Y

)

S=CABS ( R

)

RS=S* * 2

ES=ATAN2 ( AIMAG ( R ) , REAL (R)

)

ES=ES/3 . 14159

C=4.0*T(1) /WAVEL

I F ( RS . LT . RL ) THEN

RO=RS/RL

ELSE

RO=RL/RS

END IF

EO=ES-EL+C

CSE=COS ( 3 . 14159*EO)

WRITE ( IOUT , 8 0 ) T ( 2 ) , RL , RS , EL , ES , C , RO , EO , CSE

IF ( T ( 2 ) . GE . TMAX ) THEN

GO TO 60

ENDIF

T( 2

) =T( 2 ) +0 . 002

GO TO 20

60 CONTINUE

CLOSE (UNIT=IOUT)

STOP

80 FORMAT ( 9F8 . 3

)

END

( EOR

)
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54

-53-

test CASE FOR TH2SV

FOR PATTERNED LAYER , THICKNESS IS .08 AND INDEX IS (2. 9, 4. 2)

FOR INTERMEDIATE LAYER , THICKNESS IS .2 AND INDEX OF (1.46,0.)

INDEX OF SUBSTRATE IS (4.1,.l)

THIS PROGRAM VARIES THICKNESS OF INTERMEDIATE LAYER

THICK R-PAT R-INT P-PAT P-INT DELAY RNORM PNORM COS-P

.000 .647 .370 -.897 -.996 .604 .572 . 505 -.015

.002 .647 .369 -.897 -.982 .604 .571 .519 -.059

.004 .647 .369 -.897 -.968 .604 .570 .533 -.103

.006 .647 .367 -.897 -.954 .604 .568 .547 -.147

.008 .647 .366 -.897 -.940 .604 .565 .561 -.191

.010 .647 .363 -.897 -.926 .604 .562 .575 -.234

.012 .647 .361 -.897 -.911 .604 .558 .590 -.278

.014 .647 .358 -.897 -.897 .604 .553 .604 -.321

.016 .647 .354 -.897 -.882 .604 .548 .618 -.364

.018 .647 .350 -.897 -.868 .604 .542 .633 -.406

.020 .647 .346 -.897 -.853 .604 .535 .648 -.448

.022 .647 .341 -.897 -.838 .604 .527 .663 -.490

.024 . 647 .336 -.897 -.823 .604 .519 .678 -.531

.026 .647 .330 -.897 -.807 .604 .510 .693 -.571

.028 .647 .324 -.897 -.792 .604 .501 .709 -.611

.030 . 647 .318 -.897 -.776 .604 .491 .725 -.649

.032 . 647 .311 -.897 -.760 .604 .480 .741 -.687

.034 .647 .303 -.897 -.743 .604 .469 .758 -.724

.036 .647 .296 -.897 -.726 .604 .457 .774 -.759

.038 .647 .288 -.897 -.709 .604 .445 .792 -.793

.040 . 647 .280 -.897 -.692 .604 .432 . 809 -.826

.042 .647 .271 -.897 -.674 .604 .419 .827 -.856

.044 .647 .262 -.897 -.655 .604 .405 .846 -.885

.046 .647 .253 -.897 -.636 .604 .391 .865 -.911

.048 .647 .244 -.897 -.617 . 604 .377 . 884 -.934

.050 .647 .234 -.897 -.597 .604 .362 .904 -.955

.052 .647 .225 -.897 -.576 .604 .348 .925 -.972

.054 .647 .215 -.897 -.554 .604 .333 .946 -.986

.056 . 647 .206 -.897 -.532 .604 .318 .968 -.995

.058 .647 .196 -.897 -.510 . 604 .303 .991 -1.000

.060 .647 .187 -.897 -.486 . 604 .288 1.015 -.999

.062 .647 . 177 -.897 -.461 . 604 .274 1.040 -.992

. 064 .647 .168 -.897 -.436 .604 . 260 1.065 -.979

.066 . 647 .159 -.897 -.409 . 604 .246 1 . 092 - .959

. 068 .647 .151 -.897 -.381 .604 .233 1.119 -.930

. 070 .647 . 143 -.897 -.352 . 604 .221 1.148 -.894

. 072 .647 .135 -.897 -.322 .604 .209 1 . 178 -.847

.074 .647 . 128 - . 897 -.291 . 604 .199 1.209 - .791

.076 .647 .122 -.897 -.259 .604 .189 1.242 - .725

.078 .647 .117 -.897 -.225 .604 .180 1.276 -.648

.080 .647 .112 -.897 -.190 .604 . 173 1.310 -.561

.082 .647 . 108 -.897 -.154 . 604 .166 1.346 -.464

.084 .647 .104 -.897 -.118 . 604 .161 1.383 -.359

.086 . 647 .102 - .897 -.080 .604 .158 1.421 -.24 7 .088 . 647 .101 - .897 -.042 . 604 .155 1.459 - . 129

. 090 . 647 .100 -.897 - . 003 .604 .155 1.497 -.009

.092 .647 .100 - .897 .035 .604 .155 1 .536 .112

. 094 . 647 . 102 -.897 . 073 . 604 .157 1.574 . 230
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096 .647 .104 -.897 .111 .604 .161 1.612 .344

098 .647 .107 -.897 .148 .604 .165 1.649 .450

100 .647 .111 -.897 .184 .604 .171 1.685 .548

102 .647 .116 -.897 .219 .604 .179 1.720 .637

104 .647 .121 -.897 .253 .604 .187 1.754 .715

106 .647 .127 -.897 .286 .604 .197 1.786 .783

108 .647 .134 -.897 .317 .604 .207 1.818 . 840

110 .647 .142 -.897 .347 .604 .219 1.848 .888

112 .647 .149 -.897 .376 .604 .231 1.877 .926

114 .647 .158 -.897 .404 .604 .244 1.905 .956

116 .647 .167 -.897 .431 .604 .258 1.932 .977

118 .647 .176 -.897 .457 .604 .272 1.957 .991

120 .647 .185 -.897 .481 .604 ,286 1.982 .998

122 .647 .194 -.897 .505 .604 .301 2.006 1.000

124 .647 .204 -.897 .528 .604 .315 2.029 .996

126 .647 .214 -.897 .551 .604 .330 2.051 .987

128 .647 .223 -.897 .572 .604 .345 2.073 .974

130 .647 .233 -.897 .593 .604 .360 2.094 .957

132 .647 .242 -.897 .613 .604 .375 2.114 .937

134 .647 .252 -.897 .633 .604 .389 2.133 .913

136 .647 .261 -.897 .652 .604 .403 2.152 .888

138 . 647 .270 -.897 .670 .604 .417 2.171 .859

140 .647 .278 -.897 .688 .604 .430 2.189 .829

142 .647 .286 -.897 .706 .604 .443 2.207 .796

144 .647 .294 -.897 .723 .604 .455 2.224 .762

146 .647 .302 -.897 .740 .604 .467 2.241 .727

148 .647 .309 -.897 .757 .604 .479 2.257 .690

150 .647 .316 -.897 .773 .604 .489 2.274 .653

152 .647 .323 -.897 .789 .604 .499 2.290 .614

154 .647 .329 -.897 .805 . 604 . 509 2.305 .574

156 .647 .335 -.897 .820 .604 .518 2.321 .534

158 .647 .340 -.897 .835 .604 .526 2.336 .493

160 .647 .345 -.897 .850 .604 .534 2.351 .451

162 .647 .350 -.897 .865 .604 .541 2.366 .409

164 .647 .354 -.897 .880 .604 .547 2.380 .367

166 .647 .357 -.897 .894 .604 .552 2.395 .324

168 .647 .360 -.897 .909 .604 .557 2.409 .281

170 . 647 . 363 -.897 .923 .604 . 561 2.424 .237

172 . 647 .365 -.897 .937 . 604 .565 2 .438 . 194

174 .647 .367 -.897 .951 .604 .568 2.452 .150

176 .647 .368 -.897 .965 .604 . 570 2.466 .106

178 .647 .369 -.897 .979 . 604 . 571 2.480 .062

180 .647 . 370 -.897 .993 .604 . 572 2.494 .018

182 .647 .370 -.897 -.993 .604 .572 .508 -.026

184 .647 .369 -.897 -.978 .604 .571 .522 -.070

186 .647 .368 -.897 -.964 .604 .569 .536 -.114

188 .647 . 367 -.897 -.950 .604 . 567 .550 -.158

190 . 647 .365 -.897 -.936 .604 .564 . 565 -.201

192 .647 .363 -.897 -.922 . 604 . 561 . 579 -.245

194 .647 .360 -.897 -.908 . 604 .557 . 593 -.288

196 .647 .357 -.897 -.893 . 604 .552 .607 -.331

198 . 647 .353 -.897 -.879 .604 .546 .622 -.374

200 .647 .349 -.897 -.864 .604 .540 .637 -.416

202 .647 .345 -.857 -.849 . 604 .533 .652 -.458
- 55 -

Appendix II - Software for Calculation of Partially Coherent

Imaging of Planar Line Objects
-56-

1 PROGRAM PCIMAG2

2 C PROFPLAY . MAIN

3 C

4 (2

* ********************************************************************* * 5 C

6 C THIS IS THE MAIN PROGRAM FOR COMPUTING PARTIALLY COHERENT

7 C IMAGERY OF 1-D PERIODIC OBJECTS.

8 C

9 C THE COMPUTATIONS EMPLOY THE METHODS OF FOURIER OPTICS.

10 C

11 C THIS PROGRAM CONTROLS THE FLOW OF THE CALCULATIONS BY TESTING

12 C CERTAIN INPUT PARAMETERS. IT ALLOWS FOR THE CHOICE OF 1-D OR 2-D

13 C OPTICS, AND THE INPUT PARAMETERS CAN BE CHANGED WITHOUT HALTING

14 C THE EXECUTION OF THE PROGRAM.

15 C

Q* ********************************************************************* * 17 C

18 C PARAMETER DEFINITIONS - 19 C

20 C WIDTH - LINEWIDTH OF OBJECT IN MICROMETERS

21 C WT - HALF-WIDTH OF FOREGROUND OBJECT

22 C PER - PERIOD

23 C XIR - FUNDAMENTAL FREQUENCY

24 C

25 q* ********************************************************************* * 26 C

27 CHARACTER* 2 ANSWER

28 CHARACTER* 5 TYPE

29 CHARACTER* 1 2 SIZE

30 COMMON WIDTH, WAVE, OBJ , SSO, SAO , AB2 , AB4 , AP , TBO , PB , TTO , PT, SLIT,

31 * DATAS ( 13

)

32 COMMON/PAR/PI ,TWOPI

33 COMMON/MN/XIR, WT, PER, NX

34 COMMON/ IM/DUM1 ( 2000 ) ,NX1 ,DUM2,DUM3 ,DUM4

35 COMMON/ 10/ INA, IOUTA, IOUTB

36 DATA INA, IOUTA, IOUTB/ 10, 16, 40/

37 PI=4.0*ATAN( 1.0)

38 TWOPI=2 . 0*PI

39 OPEN ( UNIT=INA, FILE= ' INDATA '

)

40 OPEN ( UNIT=IOUTA , FILE= ' PRTDATA '

)

41 OPEN ( UNIT=IOUTB , FI LE= ' PLOTDAT '

)

42 C

43 c* ********************************************************************* * 44 C

45 C SUMMARY OF THE INPUT/ OUTPUT STRUCTURE.

46 c 47 C

48

49

C

C

FILE VARIABLE LOGICAL UNIT DEFINITION

50 C INDATA INA 10 INPUT DATA FILE

51 C PRTDATA IOUTA 16 OUTPUT DATA FILE - PRINTER

52 C PLOTDAT IOUTB 40 OJTPUT DATA FILE - PLOTTER

53 C

54 q* **********************************************************************

55 C

56 PRINT*

57 READ ( INA, *) ANSWER

58 C

59 q************************************************************************

60 C
-57-

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C READ THE INPUT PARAMETERS THAT CHARACTERIZE THE OPTICAL SYSTEM.

C CONVERT INPUT DATA FOR INTENSITY TRANSMITTANCE (OR REFLECTANCE)

C AND PHASE OF THE OBJECT TO COMPLEX AMPLITUDE TRANSMITTANCE

C (OR REFLECTANCE).

C

q* ***************** *************************** * ************* *********** * C

100 CALL RDATA

C

0 ***********************************************************************

C

C TEST 'WIDTH' INPUT PARAMETER.

C THE WIDTH CAN VARY FROM 0.01 TO 10.00 MICROMETERS.

C

C NOTE: WIDTH IS MULTIPLIED BY 2 WHEN CALCULATING 'NX' IN ORDER

C TO VIEW THE BEHAVIOR OF THE INTENSITY PROFILE OF LINES

C OR SPACES FURTHER FROM THE EDGE.

C

C AN ERROR MESSAGE OCCURS IF THE WIDTH LIES OUTSIDE THIS RANGE

C THE PROGRAM THEN DEMANDS A NEW SET OF INPUT DATA

C

0 ************************************************************************

c 200 IF ( WIDTH . GT .2.5. OR . WIDTH . LE .10.0) THEN

WT=0 .

PER=5 . 0*WIDTH

SIZE=' GREATER THAN'

NX=100 . 0*WIDTH+0 . 0001

ELSEIF ( WIDTH . LE . 2 . 5 . AND . WIDTH . GT . 0 . 0 ) THEN

WT=WIDTH/ 10 .

PER=5 .

SIZE= ' LESS THAN’

NX=1 0 0 . 0 * WIDTH* 2 .0 + 0 .0 001

ELSE

PRINT* ERROR IN WIDTH. PROGRAM STOPS.'

STOP

ENDIF

XIR=WAVE/ ( PER*OBJ

)

C

0***********************************************************************

c

C COMPUTE THE COMPLEX FOURIER COEFFICIENTS, A ( N )

,

OF A PERIODIC

C OBJECT WITH A SYMMETRIC RECTANGULAR WAVEFORM.

C

0***********************************************************************

c CALL OBJECT

C

0***********************************************************************

C

C COMPUTE THE REAL FOURIER COEFFICIENTS, C(N), WHICH CHARACTERIZE

C THE INTENSITY DISTRIBUTION IN THE IMAGE PLANE.

C

C (' CROS1D ' IS CALLED FOR 1-D OPTICS, WHILE ' CROS2D ' IS CALLED FOR

C 2-D OPTICS.

)

C

0***********************************************************************

c IF ( ANSWER . EQ

.

’ ID ' ) THEN

CALL CROS1D
-58-

121 ELSE

122 CALL CR0S2D

123 ENDIF

124 C

125 ^,* ******************************************************* It * * It It It It 1r h 1c It It ic 1r t 126 C

127 C INCLUDE THE EFFECT OF A SCANNING SLIT IN THE IMAGE PLANE BY

128 C ADJUSTING THE VALUES OF THE REAL FOURIER COEFFICIENTS , C(N).

129 C (THIS IS NOT NECESSARY IF THE SLIT IS LESS THAN 0.01 MICROMETERS

130 C IN WIDTH.

)

131 C

132 q* ********************************************************************* * 133 C

134 IF ( SLIT . GE .0.01) CALL CSLIT

135 C

135 C* ********************************************************************* * 137 C

138 C CALCULATE THE INTENSITY DISTRIBUTION OF A PLANAR OBJECT USING

139 C PARTIALLY COHERENT IMAGING FORMULAS.

140 C

141 Q* ********************************************************************* * 142 C

143 CALL IMAGE

144 C

145 q* ********************************************************************* * 146 C

147 C TEST FOREGROUND TRANSMITTANCE IN ORDER TO SET CERTAIN PARAMETERS

148 C PRIOR TO NORMALIZATION.

149 C

150 q* ********************************************************************* * 151 C

152 I F ( TTO . EQ .1.0) THEN

153 NUM=1

154 TYPE=' SPACE’

155 ELSE

156 NUM=NX1

157 TYPE= ' LINE

158 ENDIF

159 C

160 c* **********************************************************************

161 C

162 CALL YNORM(NUM)

163 C

164 C

* ********************************************************************* * 165 C

166 C PRINT A TABLE OF NORMALIZED INTENSITY VERSUS DISTANCE ONTO

167 C AN OUTPUT DEVICE. ALSO PRINT TWO COLUMNS OF INTENSITY AND

168 C DISTANCE OUT TO FILE PLOTDAT FOR PLOTTING PURPOSES.

169 C

170 c***********************************************************************

171 C

172 CALL PRINT ( TYPE , SI ZE , ANSWER

)

173 C

174 c***********************************************************************

175 C

176 CLOSE (UNIT=INA)

177 CLOSE (UNIT=IOUTA)

178 CLOSE (UNIT=IOUTB)

179 STOP

180 END
-59-

181 C

182 C* ********************************************************************* * 183 C

184 C THIS SUBROUTINE COMPUTES THE COMPLEX FOURIER COEFFICIENTS, A(N), OF

185 C A PERIODIC OBJECT WITH A SYMMETRIC RECTANGULAR WAVEFORM.

186 C

187 C THE COMPUTED COEFFICIENTS ARE STORED IN LABLED COMMON/ ACOEF/

188 C

189 Q^* ************************ ********************'************************ * 190 C

191 SUBROUTINE OBJECT

192 C

193 q* ********************************************************************* * 194 C

195 COMPLEX TB, TT, A(100), AO, CC

196 COMMON/ ACOEF/ A, AO

197 COMMON/MN/DUM1 ,WT,DUM2 ,DUM3

198 COMMON/RD1/TB ,TT

199 COMMON/ PAR/ PI, TWOPI

200 C

201 q* ********************************************************************* * 202 C

203 C SEE EQUATION #21A, 'METHOD FOR THE CALCULATION OF PARTIALLY COHERENT

204 C IMAGERY', ERIC KINTNER, J. APPLIED OPTICS, VOL. 17, NO. 17, PAGE 2747

205 C

206 A0=TB+2 . 0* (TT-TB) *WT

207 C

208 C COMPUTE CONSTANT COMMON TO ALL COEFFICIENTS.

209 C

210 CC= ( TT-TB ) /PI

211 C

212 C NOTE THE USE OF THE IDENTITY - 213 C SIN ( N*X ) = 2 . 0*COS ( X

) *SIN ( (N-1)*X) - SIN ( ( N- 2) *X )

214 C

215 COSN=COS( TWOPI *WT)

216 S=0 .

217 S1=-SIN ( TWOPI *WT

)

218 DO 300 N=l, 100

219 S2=S1

220 S1=S

221 S=2 . 0*COSN*S1-S2

222 A(N)=CC*S/N

223 300 CONTINUE

224 RETURN

225 END

226 C

227 C* ********************************************************************* * 228 C

229 C THIS SUBROUTINE COMPUTES THE REAL FOURIER COEFFICIENTS, C(N), WHEN

230 C THE 1-D OPTION OF THE PROGRAM IS SELECTED.

231 C

232 q* ********************************************************************* * 233 C

234 SUBROUTINE CROS1D

235 C

236 q* ********************************************************************* * 237 C

238 COMMON WIDTH , WAVE , OBJ , SSO , SAO , AB2 , AB4 , AP

2 3 9 C OMMON/ MN / XI R , DUM1 , DUM2 , DUK 3

240 COMMON /RD 2 / SS , SA
-60-

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COMMON / CROS / C ( 3 2 5 ) ,LJ,LJ1,C0

COMPLEX CCF1D, A

C

C******************************************************* ****************

cC SEE EQUATION #18, 'METHOD FOR THE CALCULATION OF PARTIALLY COHERENT

C IMAGERY', ERIC KINTNER, J. APPLIED OPTICS, VOL. 17, NO. 17, PAGE 2747

C FOR THE MATHEMATICAL FORMULATION OF THE FOURIER COEFFICIENTS, C(N).

C

Q***********************************************************************

c

C COMPUTE THE NUMBER OF FOURIER COEFFICIENTS NEEDED IN THE SUMMATION

C WITHOUT EFFECTING THE ACCURACY.

C

LJ=2 . 0/XIR+1 .

0

LJ1=LJ+1

LN0=( 1 . 0+SS ) /XIR+1 .

0

C

C TEMPORARILY SET AB2 AND AB4 TO ZERO BEFORE ENTERING CCF1D FUNCTION

C THROUGH CNORM1

.

C

TEMP1=AB2

TEMP2=AB4

A32=0 .

0

AB4=0 .

0

C

C COMPUTE NORMALIZING FACTOR FOR COEFFICIENTS (CNO). THIS IS DONE

C THROUGH THE FUNCTION CNORM1

.

C

CNO=CNORM1(SA)

C

C SET AB2 AND AB4 BACK TO THEIR ORIGINAL VALUES.

C

AB2=TEMP1

AB4=TEMP2

C

C CALCULATE THE FOURIER COEFFICIENTS CORRESPONDING TO THE PRIMARY

C AXIS FIRST (CO)

.

C

C0=CABS(A( 0) )**2.0*CCF1D( 0.0, 0,0)

DO 300 N=1 , LN0

C0=C0+REAL ( CABS ( A( N )

) **2*CCF1D(XIR,N,N)

1 +CABS ( A

( -N ) ) ** 2*CCF1D ( XIR, -N , -N )

)

300 CONTINUE

C0=C0/CN0

C

C COMPUTE FOURIER COEFFICIENTS, C(N), VIA EQUATION #18 (KINTNER'S REF.)

C

DO 500 J=l, LJ

LN=LN0+J

CT=REAL ( A ( J

) *CONJG ( A ( 0 )) *CCF1D ( XIR, J , 0 )

)

DO 400 N=1 , LN

CT=CT+REAL( A( J+N) *CONJG( A(N) ) *CCF1D ( XIR , J+N , N

)

1 +A( J-N) *CONJG(A( -N) ) *CCF1D( XIR, J-N, -N)

)

400 CONTINUE

C( J)=CT/CN0

500 CONTINUE

RETURN

END

C
- 61 -

301 c* * * ******************************************************************* * 302 C

303 C THIS FUNCTION RETRIEVES THE FOURIER COEFFICIENTS FROM COMMON/A/.

304 C IT IS CALLED FROM THE FUNCTION CCF1D AND CCF2D

305 C

306 C* **************** * ***** * ********************************************** * 307 C

308 COMPLEX FUNCTION A(N)

309 C

32.0 q* ********************************************************************* * 311 C

312 COMPLEX AA(100), A0

313 COMMON / ACOEF/ AA, A0

314 C

22_5 £* ******************** **************** ******** ********* * **************** * 316 C

317 NA=I ABS ( N

)

318 IF!NA .GT. 100) THEN

31' A=(0.0,0.0)

3 2

,

RETURN

321 ELSEIF ( NA .EQ. 0) THEN

322 A=A0

323 RETURN

324 ELSE

325 A=AA( NA)

326 RETURN

327 ENDIF

328 END

329 C

330 q* ********************************************************************* * 331 C

332 C FUNCTION TO NORMALIZE IMAGERY.

333 C FOR BRIGHT-FIELD IMAGERY, FIELD IS UNITY.

334 C FOR DARK-FIELD IMAGERY, FIELD WITHOUT SOURCE STOP IS UNITY.

335 C

336 C THIS FUNCTION IS CALLED FROM THE SUBROUTINE CROS1D (ONLY NEEDED

337 C WHEN THE 1-D OPTION OF THE PROGRAM IS USED).

338 C

339 q

* ********************************************************************* * 340 C

341 FUNCTION CNORMl(SA)

342 C

343 C

* ********************************************************************* * 344 C

345 IF ( SA .GE. 1.0) THEN

346 CNORMl=l .

347 RETURN

348 ELSE

349 CN0RM1=REAL(CCF1D( 0. 0,0,0)

)

350 RETURN

351 ENDIF

352 END

353 C

354 q ***********************************************************************

355 C

356 C FUNCTION TO NORMALIZE IMAGERY.

357 C FOR BRIGHT-FIELD IMAGERY, FIELD IS UNITY.

358 C FOR DARK-FIELD IMAGERY, FIELD WITHOUT SOURCE STOP IS UNITY.

359 C

360 C THIS FUNCTION IS CALLED FROM THE SUBROUTINE CROS2D (ONLY NEEDED
- 62 -

361 C WHEN THE 2-D OPTION OF THE PROGRAM IS USED).

362 C

303 Q* ********************************************************************* * 364 C

365 FUNCTION CNORM2(SA)

366 C

367 c***********************************************************************

368 C

369 IF ( SA .GE. 1.0) THEN

370 CNORM2=l .

0

371 RETURN

372 ELSE

373 CNORM2=REAL( CCF2D( 0 . 0 , 0 , 0 )

)

374 RETURN

375 ENDIF

376 END

377 C

378 q* ********************************************************************* * 379 C

380 C FUNCTION TO COMPUTE THE TRANSMISSION CROSS-COEFFICIENT FOR A

381 C ONE DIMENSIONAL PARTIALLY COHERENT IMAGING SYSTEM WITH

382 C DEFOCUSING, SPHERICAL ABERRATION, AND GAUSSIAN APODIZATION OF

383 C BOTH SOURCE AND PUPIL.

384 C

385 C (USED ONLY WITH THE 1-D OPTION OF THE PROGRAM)

386 C

3g7 q

* ********************************************************************* * 388 C

389 COMPLEX FUNCTION CCF1D( XIR,N1 ,N2

)

390 C

291 q* ********************************************************************* * 392 C

393 C PARAMETER DEFINITIONS - 394 C XIR - FUNDAMENTAL FREQUENCY.

395 C N1 HARMONIC FOR FIRST FREQUENCY.

396 C N2 HARMONIC FOR SECOND FREQUENCY.

397 C SS - SOURCE SIZE.

398 C SA - SOURCE APODIZATION.

399 C AB2 - DEFOCUSING

400 C AB4 - SPHERICAL ABERRATION.

401 C AP - PUPIL APODIZATION.

402 C

403 C ROUTINE ASSUMES THAT (N1 .GE. N2

)

404 C

405 c***********************************************************************

406 C

407 COMPLEX ZERO, SUM, Y

408 COMMON WIDTH , WAVE , OBJ , SSO , SAO , AB2 , AB4 , AP

409 COMMON/RD2/SS , SA

410 COMMON /PAR/ PI ,TWOPI

411 DATA ZERO, EPS/ (0.0 ,0.0), 0.01/

412 DATA N0/10/

413 C

414 c***********************************************************************

415 C

416 X1=N1*XIR

417 X2=N2*XIR

418 CCFlD=ZERO

419 C

420 C IN THE COHERENT LIMIT, USE ALTERNATE ROUTINE.
421 C

422

423 C

424 C

425 C

426

427

428

429 C

430 C

431 C

432 C

433

434

435

436

437 C

438 C

439 C

440 C

441 C

442 C

443

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475 C

476 C

477 C

478

479

480 C

-63-

IF ( SS .LT. EPS) GOTO 100

RETURN WITH ZERO IF SOURCE AND TWO PUPILS DO NOT INTERSECT.

IF ( ( X1-X2 ) .GT. 2.0) RETURN

IF ( ( XI —1.0) .GT. SS) RETURN

IF ( - ( 1 . 0+X2 ) .GT. SS) RETURN

DETERMINE LIMITS OF INTEGRATION. ASSUML LIMITS DETERMINED BY

PUPILS UNLESS SOURCE IS SMALLER.

RL2=1 . 0-X1

RLl=-( 1.0+X2)

IF ( SS .LT. RL2

)

RL2=SS

IF ( -SS .GT. RL1 ) RL1=-SS

SET UP INTEGRATION BY SIMPSON’S RULE.

SCALE THE INTEGRATION INTERVAL TO BE APPROXIMATELY EQUAL FOR

ALL CASES.

(NO IS THE APPROXIMATE NUMBER OF INTERVALS PER UNIT LENGTH.)

NC=1 . 0+ ( RL2 -RL1 ) *N0

DC=(RL2-RL1)/NC

DC2=DC/2.

Q=RL1

Q1=Q+X1

Q2=Q+X2

QQ1=Q1*Q1

QQ2=Q2*Q2

AR=SA*Q*Q+AP* ( QQ1+QQ2

)

AI=TWOPI* ( AB2* ( QQ1-QQ2 ) +AB4* ( QQ1*QQ1-QQ2*QQ2 )

)

SUM=CEXP ( CMPLX ( AR , AI )

)

DO 50 N=1 , NC

QN=RL1+N*DC

Q=QN-DC2

Q1=Q+X1

Q2=Q+X2

QQ1=Q1*Q1

QQ2=Q2*Q2

AR=SA*Q*Q+AP* ( QQ1+QQ2

)

AI=TWOPI* ( AB2* ( QQ1-QQ2 ) +AB4* ( QQ1*QQ1-QQ2*QQ2 )

)

Y=CEXP ( CMPLX ( AR , AI )

)

SUM=SUM+4 . 0*Y

Q=QN

Q1=Q+X1

Q2=Q+X2

QQ1=Q1 *Q1

QQ2=Q2*Q2

AR=SA*Q*Q+AP* ( QQ1+QQ2

)

AI=TWOPI* ( AB2* ( QQ1-QQ2 ) +AB4* ( QQi*QQl-QQ2*QQ2 )

)

Y=CEXP ( CMPLX ( AR , AI )

)

SUM=SUM+2 . 0 *Y

50 CONTINUE

EXTRA FACTOR OF TWO IN DENOMINATOR CORRECTLY NORMALIZES RESULT.

CCF1D= ( SUM-Y ) *DC2/ 6 .

RETURN
-64-

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482

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491

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493

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531

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536

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539

540

C ALTERNATE ROUTINE FOR COHERENT LIMIT.

C

100 IF ( ABS ( XI ) .GT. 1.0) RETURN

IF ( ABS ( X2 ) .GT. 1.0) RETURN

QQ1=X1*X1

QQ2=X2*X2

AR=AP* ( QQ1+QQ2

)

AI=TWOPI * ( AB2 * ( QQ1-QQ2 ) +AB4* ( QQ1*QQ1-QQ2*QQ2 )

)

CCF1D=CEXP ( CMPLX ( AR , AI )

)

RETURN

END

C

cc

c

c

c

cc

cc

c

cc

c

cc

c

c

cc

c

***********************************************************************

FUNCTION TO COMPUTE TRANSMISSION CROSS-COEFFICIENT FOR CIRCULAR

OPTICAL SYSTEM (ANNULAR SOURCE) WITH NO ABERRATIONS OR

APODIZATION

.

(USED ONLY WITH THE 2-D OPTION OF THE PROGRAM) ***********************************************************************

PARAMETER DEFINITIONS - XI R - FUNDAMENTAL FREQUENCY.

N1 - HARMONIC FOR FIRST FREQUENCY.

N2 - HARMONIC FOR SECOND FREQUENCY.

SS - OUTER RADIUS OF SOURCE ANNULUS.

SA - INNER RADIUS OF SOURCE ANNULUS.

ROUTINE ASSUMES THAT (N1 .GE. N2 )

.

**********************************************************************

COMPLEX FUNCTION CCF2D(XIR,N1 ,N2

)

C

0 **********************************************************************

C

COMMON / RD 2 / S S , S

A

COMMON / PAR/PI , TWOP

I

C

0 **********************************************************************

C

X1=N1*XIR

X2=N2*XIR

CCF2D=(0. 0,0.0)

C

C USE ALTERNATE ROUTINE FOR COHERENT SOURCE.

C

IF ( SS .LT. 0.01) THEN

GOTO 2000

ENDIF

C

C RETURN WITH ZERO IF SOURCE AND PUPILS DO NOT INTERSECT.

C

IF ( ( X1-X2 ) .GT. 2.0) RETURN

IF ( (XI— 1.0) .GT. SS) RETURN

IF ( - ( 1 . 0+X2 ) .GT. SS) RETURN

C

C MAXIMUM AND MINIMUM EXTENT OF INTERSECTION OF TWO PUPILS.

C

XMAX=X2+i .

0
-65-

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599

600

XMIN=X1-1.0

IFLAG=0

C

C SET FLAG IF SOURCE IS ANNULAR - TWO SEPARATE PASSES REQUIRED.

C

IF ( SA .GT. 0.01) IFLAG=1

XS=SS

100 XS2=XS*XS

C

C IF SOURCE SIZE IS LESS THAN PUPIL SIZE, USE ALTERNATE ROUTINE.

C

IF ( XS .LT. 1.0) GOTO 400

IF ( XS .LT. XMAX .OR. -XS .GT. XMIN) GOTO 200

C

C IF SOURCE ENVELOPES AREA COMMON TO BOTH PUPILS, CALCULATE THIS

C AREA.

C

Y=AREA( 1.0, 1.0, (XI -X2)

)

GOTO 1100

200 IF ( XS2 .LE. (X1*X2+1.0)) GOTO 300

C

C IF REQUIRED AREA IS BOUNDED BY SOURCE AND BOTH PUPILS, CALCULATE

C THIS AREA.

C

Y=AREA ( 1 . 0 , 1 . 0 , ( X1-X2 )

) +AREA( XS ,1.0, AMIN1 ( ABS ( XI ) ,ABS(X2) )

) -PI

GOTO 1100

C

C IF REQUIRED AREA IS BOUNDED ONLY BY SOURCE AND ONE PUPIL,

C CALCULATE THIS AREA.

C

300 Y=AREA ( XS ,1.0, AMAX1 ( ABS ( XI ) , ABS ( X2 ) )

)

GOTO 1100

C

C ALTERNATE ROUTINE FOR SOURCE SMALLER THAN PUPIL.

C

C FIND WHETHER BOUNDARY OF SOURCE EXTENDS BEYOND THE AREA COMMON

C TO BOTH PUPILS, IN EITHER DIRECTION.

C

400 IFL1=0

IFL2=0

IF (XMIN .LE. -XS) IFL1=1

IF ( XMAX .GE. XS) IFL2=1

IF ( IFL1+IFL2 -1 ) 900, 600, 500

C

C IF SOURCE IS ENVELOPED BY AREA COMMON TO BOTH PUPILS, CALCULATE

C AREA OF SOURCE.

C

500 Y=PI *XS2

GOTO 1100

C

C IF REQUIRED AREA IS BOUNDED BY SOURCE AND EITHER PUPIL (ALONE),

C CALCULATE AREA IN COMMON BETWEEN SOURCE AND THIS PUPIL.

C (ZERO IS IMPOSSIBLE IN IF- STATEMENT AT 600.)

C

600 IF ( IFL1-IFL2 ) 700, 1100, 800

700 Y=AREA ( 1 . 0 , XS , ABS ( XI )

)

GOTO 1100

800 Y=AREA( 1 . 0 , XS , ABS ( X2 )

)

GOTO 1100

900 IF( (XI* X 2+1.0) .LE. XS2

)

GOTO 1000
- 66-

601 C

602 C IF REQUIRED AREA IS BOUNDED BY SOURCE AND BOTH PUPILS , CALCULATE

603 C THIS AREA.

604 C

605 Y=AREA(1.0,XS,ABS(X1) ) +AREA ( 1 . 0 , XS , ABS ( X2 ) )-PI*XS2

606 GOTO 1100

607 C

608 C IF SOURCE ENVELOPES AREA COMMON TO BOTH PUPILS , CALCULATE THIS

609 C AREA.

610 C

611 1000 Y=AREA(1. 0,1.0, (X1-X2)

)

612 GOTO 1100

613 C

614 C CHECK FLAG FOR CIRCULAR SOURCE, OR FIRST OR SECOND PASS WITH

615 C ANNULAR SOURCE.

616 C

617 1100 IF ( I FLAG- 1 ) 1500, 1600, 1700

618 C

619 C FOR CIRCULAR SOURCE, NORMALIZE AND RETURN.

620 C

621 1500 CCF2D=Y/PI

622 RETURN

623 C

624 C FOR FIRST PASS WITH ANNULAR SOURCE, SAVE RESULTS FROM OUTER

625 C BOUNDARY OF ANNULUS, SET INNER BOUNDARY AS BOUNDARY OF INNER

626 C SOURCE.

627 C

628 1600 IFLAG=2

629 CCF2D=Y

630 Y=0 .

0

631 XS=SA

632 IF ( (XI— 1-0) .GT. SA) GOTO 1100

633 IF ( - ( X2+1 . 0

)

.GT. SA) GOTO 1100

634 GOTO 100

635 C

636 C FOR SECOND PASS WITH ANNULAR SOURCE, SUBTRACT RESULTS OF INNER

637 C SOURCE FROM SAVED RESULTS OF OUTER SOURCE. NORMALIZE AND RETURN.

638 C

639 1700 CCF2D=( CCF2D-Y) /PI

640 RETURN

641 C

642 C ALTERNATE ROUTINE FOR COHERENT SOURCE.

643 C

644 2000 IF ( ABS ( XI ) .GT. 1.0) RETURN

645 IF ( ABS ( X2 ) .GT. 1.0) RETURN

646 CCF2D=(1. 0,0.0)

647 RETURN

648 END

649 C

650 c***********************************************************************

651 C

652 C FUNCTION TO COMPUTE THE AREA COMMON TO TWO DISPLACED CIRCLES.

653 C

654 C FOR THE MATHEMATICAL FORMULATION PERTAINING TO THIS FUNCTION,

655 C SEE THE APPENDIX OF 'METHOD FOR THE CALCULATION OF PARTIALLY COHERENT

656 C IMAGERY', ERIC KINTNER, J. APPLIED OPTICS, VOL. 17, NO. 17, PAGE 2750.

657 C

658 q

* ********************************************************************* * 659 C

660 C PARAMETER DEFINITIONS -
-67-

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

704

705

706

707

708

709

710

711

712

713

714

715

716

717

718

719

720

C R1 - RADIUS OF FIRST CIRCLE.

C R2 - RADIUS OF SECOND CIRCLE.

C D - SEPARATION BETWEEN FOCII OF TWO CIRCLES.

C

C ROUTINE ASSUMES SOME CONTACT EXISTS , AND EXPECTS ALL THREE

C PARAMETERS TO BE POSITIVE.

C

c***********************************************************************

c FUNCTION AREA( R1 ,R2 ,D)

COMMON/ PAR/ PI ,TWOPI

C

£* ****** * ************ ********** ****** ***** * * * * * * * * ****** ******** * ****** * c

C TRAP FOR CONCENTRIC CIRCLES.

C

IF ( D .LT. 0.001) GOTO 100

C

C FIND ANGLE IN EACH CIRCLE SUBTENDED BY COMMON AREA. NOTE TRAPS

C TO SUPPRESS ERRORS IN ARC-COSINE ROUTINES.

C

D2=D*D

RADII=R1*R1-R2*R2

DENOM=2 . 0*D

T= ( D2+RADII ) / ( DENOM*Rl

)

IF ( T .GT. 1.0) T=1 .

THETAl=ACOS ( T

)

T=( D2-RADII ) / ( DENOM*R2

)

IF ( T .GT. 1.0) T=1 .

THETA2=ACOS(T)

C

C COMPUTE HALF-LENGTH OF CHORD COMMON TO THE TWO COMPUTED ANGLES.

C

C=R1 *SIN ( THETA1

)

C

C COMPUTE THE AREA OF THE REGION OF OVERLAP.

C

AREA= ( R1**2*THETA1+R2**2*THETA2 ) -C*D

RETURN

C

C IF CIRCLES ARE CONCENTRIC, CALCULATE AREA OF SMALLER CIRCLE.

C

100 AREA=PI * ( AMIN1 ( Rl , R2 )

) * * 2

RETURN

END

C***********************************************************************

C

C THIS SUBROUTINE CORRECTS THE FOURIER COEFFICIENTS, C(N), FOR THE

C EFFECT OF THE SCANNING SLIT IN THE IMAGE PLANE.

C

Q****************:V ******************************************************

c SUBROUTINE CSLIT

C

Q***********************************************************************

C

COMMON WIDTH

,

WAVE

,

OBJ , SSO , SAO , AB2 , AB4 , AP , TBO , PB , TTO , PT , SLIT

COMMON/MN/XIR , DUM1 , DUM2 , DUM3

COMMON/ CROS/ C ( 3 2 5 ) ,LJ

COMMON/PAR/PI ,TWOPI
- 68 -

721

722

723

724

725

726

727

728

729

730

731

732

733

734

735

736

737

738

739

740

741

742

743

744

745

746

747

748

749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

768

769

770

771

772

773

774

775

776

777

778

779

780

C

0 *********************************************************************^^

c

C SEE EQUATION #19 IN 'METHOD FOR THE CALCULATION OF PARTIALLY COHERENT

C IMAGERY', ERIC KINTNER, J. APPLIED OPTICS, VOL. 17, NO. 17, PAGE 2747

C FOR THE MATHEMATICAL FORMULATIONS RELATING TO THIS SUBROUTINE.

C

0

* *********************** *********** ******** * * * * * * * ******** * * * ***** * * * * * c

ARG=PI *XIR* ( SLIT*OBJ /WAVE

)

COSN2=2 . 0*COS ( ARG

)

S=0.0

S1=-SIN( ARG)

DO 700 J=l, LJ

S2=S1

S1=S

S=C0SN2*S1-S2

C(J)=C(J)*S/( J* ARG

)

700 CONTINUE

RETURN

END

C

0 ***********************************************************************

C

C THIS SUBROUTINE COMPUTES THE REAL FOURIER COEFFICIENTS, C(N), WHEN

C THE 2-D OPTION OF THE PROGRAM IS SELECTED.

C

0 ***********************************************************************

C

SUBROUTINE CROS2D

C

0 ***********************************************************************

C

COMMON/MN/XIR , DUM1 , DUM2 , DUM3

COMMON/RD2/SS , SA

COMMON / CROS / C ( 3 2 5 ) ,LJ,LJ1,C0

COMPLEX CCF2D, A

C

0 ***********************************************************************

cC SEE EQUATION #18 IN 'METHOD FOR THE CALCULATION OF PARTIALLY COHERENT

C IMAGERY', ERIC KINTNER, J. APPIED OPTICS, VOL. 17, NO. 17, PAGE 2747

C FOR THE MATHEMATICAL FORMULATION OF THE FOURIER COEFFICIENTS, C(N).

C

0 ***********************************************************************

c

C COMPUTE THE NUMBER OF FOURIER COEFFICIENTS NEEDED IN THE SUMMATION

C

LJ=2 . 0/XIR+1 .

0

LJ1=LJ+1

LN0=(1.0+SS) /XIR+1 .

0

C

C COMPUTE THE NORMALIZING FACTOR FOR THE COEFFICIENTS (CN0). THIS

C IS DONE THROUGH THE FUNCTION CNORM2

.

C

C.NO=CNORM2 ( SA)

C

C CALCULATE THE FOURIER COEFFICIENTS CORRESPONDING TO THE PRIMARY

C AXIS FIRST ( CO )

C
-69-

781

782

783

784

785

786

787

788

789

790

791

792

793

794

795

796

797

798

799

800

801

802

803

804

805

806

807

808

809

810

811

812

813

814

815

816

817

818

819

820

821

822

823

824

825

826

827

828

829

830

831

832

833

834

835

836

837

838

839

840

C0=CABS(A(0) ) **2*CCF2D ( 0 . 0 , 0 , 0 ) DO 300 N=1 , LNO

C0=C0+REAL ( CABS ( A ( N ) ) **2*CCF2D( XIR,N,N

)

1 +CABS ( A

( -N) )**2*CCF2D(XIR,-N,-N)

)

300 CONTINUE

C0=C0/CN0

C

C COMPUTE FOURIER COEFFICIENTS , C(N), VIA EQUATION #18 (KINTNER'S REF.)

C

DO 500 J=1 , LJ

LN=LN0+J

CT=REAL(A( J)*CONJG(A(0) ) *CCF2D(XIR, J, 0 )

)

DO 400 N=1 / LN

CT=CT+REAL ( A ( J+N ) * CONJG ( A(N) ) * CCF 2D ( XIR, J+N , N

)

1 +A( J-N ) *CONJG ( A( -N) ) *CCF2D(XIR, J-N, -N)

)

400 CONTINUE

C( J)=CT/CN0

500 CONTINUE

RETURN

END

C

Q^* ********************************************************************* * c

C THIS SUBROUTINE NORMALIZES ALL OF THE INTENSITY VALUES STORED IN

C THE ARRAY YLIST(N).

C

q* ********************************************************************* * C

SUBROUTINE YNORM(NUM)

C

q* ********************************************************************* * c COMMON/ IM/YLI ST ( 2000) , NX1 , YMAXBN , DUM1 , YMAXAN

C

Q* **********************************************************************

C

C SINCE SCANNING IS ASSUMED TO START FROM THE CENTER OF THE LINE

C OBJECT ( CORRESPONDING TO X=0 ) , THE INTENSITY VALUES ARE NORMALIZED

C RELATIVE TO THE FOREGOUND INTENSITY WHEN THE INPUT VALUE FOR THE

C INTENSITY TRANSMITTACE ( TTO ) IS 1.0.

C OTHERWISE, THE INTENSITY VALUES ARE NORMALIZED RELATIVE TO THE

C BACKGROUND INTENSITY CORRESPONDING TO AN INPUT OF 1 . 0 FOR THE

C BACKGROUND INTENSITY ( TBO )

C

£***********************************************************************

c

C YMAXAN THE MAXIMUM INTENSITY VALUE AFTER NORMALIZATION

C

YMAXAN=YMAXBN / YLI S T ( NUM

)

CONST=YLI ST ( NUM

)

DO 100 1=1, NX1

YLI ST ( I

) =YLI ST ( I ) / CONST

100 CONTINUE

RETURN

END

C

Q* **********************************************************************

r

C THIS SUBROUTINE READS THE INPUT PARAMETERS THAT CHARACTERIZE THE

C OPTICAL SYSTEM TO BE MODELED. THE COMPLEX AMPLITUDE TRANSMITTANCE
-70-

841

842

843

844

845

846

847

848

849

850

851

852

853

854

855

856

857

858

859

860

861

862

863

864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

C (OR REFLECTANCE) IS THEN CALCULATED FROM THE INPUTTED INTENSITY

C TRANSMITTANCE AND PHASE OF THE OBJECT.

C

q***********************************************************************

C

SUBROUTINE RDATA

C

0 ***********************************************************************

C

C PARAMETER DEFINITIONS - C

C WIDTH - C WAVE

C OBJ

C SSO

C SAO

C

C AB2

C AB4

C AP

C

C TBO

C PB

C TTO

C PT

C SLIT

c ss

C SA

C TB

C TT

C

0 ***********************************************************************

c COMMON WIDTH , WAVE , OBJ , SSO , SAO , AB2 , AB4 , AP , TBO , PB , TTO , PT , SLIT

,

* DATAS ( 13

)

COMPLEX TB , TT

COMMON/IO/INA, IOUTA, IOUTB

COMMON/RD1/TB , TT

COMMON/RD2/SS , SA

COMMON/PAR/PI / TWOPI

C

q***********************************************************************

C

C READ THE INPUT PARAMETERS AND STORE THEM IN THE ARRAY ' DATAS (N) ' C

READ ( INA, * ) ( DATAS ( I ) ,1=1,13)

C

C ASSIGN VARIABLE NAMES TO EACH ELEMENT OF ARRAY ’DATAS':

C

WIDTH=DATAS( 1)

WAVE=DATAS ( 2

)

OBJ=DATAS ( 3

)

SSO=DATAS ( 4

)

SAO=DATAS ( 5

AB2=DATAS ( 6

AB4=DATAS ( 7

)

AP=DATAS ( 8

)

TBO=DATAS ( 9

)

PB=DATAS ( 10

TTO=DATAS( 11)

LINEWIDTH OF OBJECT IN MICROMETERS

WAVELENGTH OF LIGHT IN MICROMETERS

OBJECTIVE NUMERICAL APERTURE

CONDENSER NUMERICAL APERTURE

NUMERICAL APERATURE OF CENTRAL OBSTRUCTION OF ANNULAR

CONDENSER (SET TO ZERO FOR BRIGHT FIELD IMAGE)

AMOUNT OF DEFOCUS IN WAVES

AMOUNT OF SPHERICAL ABERRATION IN WAVES

GAUSSIAN APODIZATION PARAMETER

(MUST BE ZERO UNLESS USING 1-D OPTICS)

INTENSITY TRANSMITTANCE OF BACKGROUND

PHASE IN UNITS OF PI OF BACKGROUND

INTENSITY TRANSMITTANCE OF LINE OBJECT

PHASE OF LINE OBJECT

EFFECTIVE SCANNING SLIT WIDTH IN MICROMETERS

COHERENCE PARAMETER

NORMALIZED RADIUS OF CENTRAL OBSTRUCTION

COMPLEX AMPLITUDE TRANSMITTANCE OF BACKGROUND

COMPLEX AMPLITUDE TRANSMITTANCE OF LINE OBJECT
-71-

901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

951

952

953

954

955

956

957

958

959

960

PT=DATAS (12)

SLIT=DATAS (13)

SS=SSO/OBJ

SA=SAO/OBJ

AMPL=SQRT ( TBO

)

PHASE=PI*PB

TB=CMPLX( AMPL* COS (PHASE) , AMPL* SIN ( PHASE )

)

AMPL=SQRT ( TTO

)

PHASE=PI *PT

TT=CMPLX( AMPL* COS (PHASE) , AMPL*SIN ( PHASE )

)

RETURN

END

C

Q **********************************************************************

c

C THIS SUBROUTINE PRINTS A TABLE OF DISTANCE VERSUS INTENSITY VALUES

C FOR THE PARTIALLY COHERENT IMAGING OF PERIODIC OBJECTS OUT TO

C FILE ' PRTDATA '

C

C TWO COLUMNS CONSISTING OF X-VALUES AND CORRESPONDING INTENSITY

C VALUES ARE OUTPUT TO FILE ' PLOTDAT ' . THESE VALUES MAY BE USED AS

C A PLOT FILE TO PRODUCE A THEORETICAL OPTICAL PROFILE FOR THE LINE

C OBJECT.

C

q* **********************************************************************

C

SUBROUTINE PRINT ( TYPE , SI ZE , ANSWER)

C

Q* ********************************************************************* * c COMMON WIDTH , WAVE , OBJ , SSO , S AO , AB 2 , AB 4 , AP , TBO , PB , TTO , PT, SLIT

COMMON / I O / I NA , I OUTA , I OUTB

COMMON /IM/YLIST( 2000) , NX1 , YMAXBN , XMAX , YMAXAN

CHARACTER* 5 TYPE

CHARACTER* 1 2 SIZE

CHARACTER* 2 ANSWER

C

Q* **********************************************************************

c

C PRINT PERTINENT INFORMATION FOLLOWED BY A TABLE OF INTENSITY VALUES

C FOR THE DISTANCE TRAVERSED:

C

WRITE ( IOUTA, 1 ) TYPE, SIZE

WRITE ( I OUTA , 2 ) NX1

WRITE ( IOUTA , 3 ) YMAXBN

WRITE ( IOUTA , 4 ) YMAXAN

WRITE ( IOUTA , 5 ) XMAX

IF ( ANSWER. EQ. ' 2D' ) THEN

PRINT*,' 2-D OPTION SELECTED'

ELSE

PRINT*,' 1-D OPTION SELECTED'

ENDIF

WRITE (IOUTA,*)

WRITE ( IOUTA,*

)

WRITE ( IOUTA, 6

)

WRITE ( IOUTA, 7 ) WIDTH, WAVE, OBJ, SSO,SAO,AB2 ,AB4 ,AP, TBO, PB , TTO, PT, SLIT

WRITE ( IOUTA, 8

)

WRITE ( IOUTA, 9

)

WRITE ( IOUTA, 10

)

NA=1
-72-

961

962

963

964

965

966

967

968

969

970 C

971 C

972 C

973 C

974 C

975 C

976 C

977 C

978 C

979 C

980 C

981 C

982 C

983 C

984

985

986 C

987 C

988 C

989 C

990

991

992

993

994

995 C

996 C

997 C

998

999

1000

1001

1002

1003

1004 C

1005 C

1006 C

1007 C

1008

1009

1010 C

1011 C

1012 C

1013

1014

1015 C

1016 C

1017 C

1018

1019

1020

NB=10

X=0 .

0

100 WRITE ( IOUTA, 11 ) X, (YLIST(NP), NP=NA,NB)

NA=NA+10

NB=NB+10

X=X+0.1

IF(NB.GT.NXl) NB=NX1

IF(NA.GT.NXl) GOTO 200

GOTO 100

THE FOLLOWING CODE PRINTS TWO COLUMNS OF DISTANCE AND INTENSITY

VALUES OUT TO FILE ' PLOTDAT ' . SINCE YLIST(N) ONLY CONTAINS INTENSITY

VALUES STARTING FROM THE CENTER OF THE LINE OBJECT, A ROUTINE HAS BEEN

DEVELOPED TO REPEAT THE CORRESPONDING INTENSITY VALUES FOR NEGATIVE

X-VALUES (LYING TO THE LEFT OF THE LINE OBJECT'S CENTER).

THIS ROUTINE ALSO COMPARES THE SLOPE OF A SET OF (X,Y) POINTS

AND PRINTS ONLY THOSE POINTS WHICH HAVE INTENSITY VALUE DIFFERENCES

GREATER THAN 0.001. IN THIS WAY, THE PLOT FILE GENERATED MAY

BECOME MORE EFFICIENT BY ELIMINATING UNNECESSARY POINTS.

THIS PORTION OF CODE PRINTS OUT CORRESPONDING NEGATIVE X-VALUES

AND INTENSITY VALUES - STARTS FROM YLIST(NXl) AND GOES TO YLIST(l):

200 N=NX1

J=0

J IS COUNTER FOR NUMBER OF POINTS APART TO COMPARE WITH (UP TO 15)

SUBTRACT J BECAUSE WE ARE HEADING TOWARDS CENTER OF PROFILE ... YLIST ( 1

)

300 N=NJ

J=0

DO 400 1=1,15

J=J+1

DIFF=ABS ( YLIST ( N

) -YLIST ( N-I )

)

IF YLSIT(l) HAS BEEN REACHED, SWITCH TO SECOND PART OF ROUTINE (BELOW)

IF ( (N-I) . LE . 1 ) GOTO 600

IF(DIFF.GE. 0.001) GOTO 500

400 CONTINUE

500 X= ( N- J-l ) *0 . 01 • WRITE ( IOUTB, 12 ) YLIST(N-J), -X

GOTO 300

THIS PORTION OF CODE PRINTS OUT THE POSITIVE X-VALUES AND

INTENSITY VALUES - STARTS FROM YLIST(l) AND GOES TO YLIST(NXl):

600 X=0.0

WRITE( IOUTB, 12) YLIST(l), X

START AT LOWEST INDEX NUMBER OF ARRAY AND WORK UP TO HIGHEST.

N=1

J = 0

ADD J BECAUSE WE START WITH YLIST(l) AND PROCEED TO YLIST (NX1).

700 N=N+

J

J = 0

DO 800 1=1,15
-73-

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

1031

1032

1033

1034

1035

1036

1037

1038

1039

1040

1041

1042

1043

1044

1045

1046

1047

1048

1049

1050

1051

1052

1053

1054

1055

1056

1057

1058

1059

1060

1061

1062

1063

1064

1065

1066

1067

1068

1069

1070

1071

1072

1073

1074

1075

1076

1077

1078

1079

1080

J=J+1

DIFF=ABS ( YLIST ( N

) -YLIST ( N+I )

)

IF( (N+I) .GE.NX1) GOTO 1000

IF(DIFF.GE. 0.001) GOTO 900

800 CONTINUE

900 X= ( N+J-l ) *0 . 01

WRITE (IOUTB, 12) YLIST(N+J ) , X

GOTO 700

1000 X= ( NX1-1 ) *0 . 01

WRITE ( IOUTB, 12) YLIST ( NX1 ) , X

WRITE (IOUTB,*) 'END OF DATA'

1100 ENDFILE 40

RETURN

1 FORMAT ( IX/ /IX, 'THE FOLLOWING DATA CORRESPONDS TO A ' ,A5/1X,

* A12 ,

' 2.5 MICROMETERS IN WIDTH: ’/)

2 FORMAT (IX, 'THE NUMBER OF DATA POINTS = ', 14/)

3 FORMAT ( IX, 'THE MAXIMUM INTENSITY VALUE BEFORE NORMALIZATION =',

* F7.4)

4 FORMAT ( IX, 'THE MAXIMUM INTENSITY VALUE AFTER NORMALIZATION =',

* F7 . 4)

5 FORMAT ( IX, 'THE CORRESPONDING X-VALUE TO THESE MAXIMUM INTENSITIES

*=' , F6.2//)

6 FORMAT ( IX, 5HWIDTH, 6H WAVE,4H OBJ,5H SSO,5H SAO,5H AB2,5H AB4

,

* 5H AP,6H TBO , 6H PB,6H TTO,6H PT,6H SLIT)

7 FORMAT (IX, 8(F5.2), 4(F6.2), F5.2///)

8 FORMAT ( IX, 'LEFT-MOST RELATIVE INTENSITY VALUES CORRESPONDING TO

* STEPS OF'

)

9 FORMAT ( IX, ’ X- VALUES 0.01 MICROMETERS FROM LEFT TO RIGHT')

10 FORMAT ( IX, '

*

'/)

11 FORMAT! IX, F6 . 2, 2X, 10F6 . 3

)

12 FORMAT ( IX , F7 . 4 , 3X , F7 . 2

)

END

C

q***********************************************************************

C

C THIS SUBROUTINE COMPUTES THE IMAGE INTENSITY IN THE OBJECT PLANE

C BY AN INVERSE FOURIER TRANSFORM METHOD.

C

q* **********************************************************************

c SUBROUTINE IMAGE

C

Q* ********************************************************************* * c COMMON WIDTH, WAVE, OBJ

COMMON /MN/DUM1 ,DUM2 , PER, NX

COMMON / CROS / C ( 325) ,LJ,LJ1,C0

COMMON/ IM/YLI ST ( 2000 )

, NX1 , YMAXBN,XMAX,DUM3

COMMON/ PAR/ PI ,TWOPI

C

q* **********************************************************************

C

C THE MATHEMATICAL FORMULATIONS RELATING TO THIS SUBROUTINE CAN

C BE FOUND IN EQUATIONS #14 AND #17 IN 'METHOD FOR THE CALCULATION

C OF PARTIALLY COHERENT IMAGERY' , ERIC KINTNER, J. APPIED OPTICS,

C VOL. 17, NO. 17, PAGE 2747.

C

Q* **********************************************************************
1081

1082

1083

1084

1085

1086

1087

1088

1089

1090

1091

1092

1093

1094

1095

1096

1097

1098

1099

1100

1101

1102

1103

1104

1105

1106

1107

1108

1109

1110

1111

1112

1113

1114

1115

1116

1117

1118

1119

1120

1121

1122

1123

1124

1125

1126

1127

- 74 -

PARAMETER DEFINITIONS -

NX1

YMAXBN

XMAX

X

Y

YLIST(N)

TOTAL NUMBER OF CALCULATED IMAGE POINTS

MAXIMUM INTENSITY VALUE BEFORE NORMALIZATION

X-VALUE CORRESPONDING TO THE MAXIMUM INTENSITY VALUE

TRANSVERSE DISTANCE ACROSS THE OBJECT (STARTING AT X=0

)

INTENSITY VALUE CORRESPONDING TO EACH X-VALUE OF

OBJECT

ARRAY THAT CONTAINS EACH INTENSITY VALUE CORRESPONDING

TO ACCENDING VALUES OF X. *********************************************************************

X0=0.01

NX1=NX+1

YMBN=0 .

0

DO 900 N=1 , NX1

X= ( N-l ) *X0

C

C NOTE THE USE OF THE IDENTITY:

C COSNX = 2*COS(X) *COS(N-l) *X - COS(N-2)*X

C

COSN=COS ( TWOPI *X/PER

)

COSN2=2 . 0*COSN

CK1=0 .

0

CK=0 .

0

DO 800 J=l, LJ

CK2=CK1

CK1=CK

CK=COSN2*CKl-CK2+2 . 0*C ( LJ1- J

)

800 CONTINUE

CK2=CK1

CK1=CK

CK=COSN2*CK1-CK2+CO

Y=CK-COSN*CKl

IF ( Y .LE. YMBN) THEN

GOTO 850

ENDIF

XM=X

YMBN=Y

850 YLIST ( N ) =Y

900 CONTINUE

XMAX=XM

YMAX3N=YMBN

RETURN

END

( EOR)
-75-

TEST CASE FOR PCXMAG2

INPUT DATA

Line

No.

1

2

'ID'

3.00 .53 .85 .17 .00 .00 .00

A A A A A

T3

<D

COP

cou

<1) T3

4-> iH

<1) (I) E *H

fO m-i u

fO M

a u

<d

<D T3

CO

CD U

X o

4-> 4-1

CO3

u

o

M-l0) T3

.00 1.00 0.00 1.00 1.00 0.20

A A A A A

c

oH

P

<GU

u

u

0) si

<d

Pi

T3

c

3

o

J-l O'

u

<d

S <D <u C c

•H •rH C i—

r—

P

0 J-l J-l

J-l O O

O' *4-1 M-l o Pi Si

<d Si

A

fdu

•H

Pi(U x:

cu

CO

Notes

1) phase angles are given in units of ti 2) both defocus and spherical aberrations are given in

"number of waves"

3) all distances are given in um

slit

width

at

object

plane
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

- / bOUTPUT

THE FOLLOWING DATA CORRESPONDS TO A SPACE

GREATER THAN 2.5 MICROMETERS IN WIDTH:

THE NUMBER OF DATA POINTS = 301

THE MAXIMUM INTENSITY VALUEBEFORE NORMALIZATION = 1.2948

THE MAXIMUM INTENSITY VALUE AFTER NORMALIZATION = 1.2989

THE CORRESPONDING X-VALUE TO THESE MAXIMUM INTENSITIES = 1.17

WIDTH WAVE OBJ

3.00 .53 .85

SSO SAO AB2

.17 .00 .00

AB4

.00

AP TBO

.00 1.00

PB TTO PT SLIT

.00 1.00 1.00 .20

LEFT-MOST RELATIVE INTENSITY VALUES CORRESPONDING TO STEPS OF

X-VALUES 0.01 MICROMETERS FROM LEFT TO RIGHT

.00 1.000 1.000 1.000 1.000 .999 .999 .998 .998 .997 .996

.10 .996 .995 .994 .994 .993 .993 .992 .992 .991 .991

.20 .991 .991 .991 .992 .992 .993 .994 .995 .997 .998

.30 1.000 1.002 1.004 1.006 1.009 1.011 1.014 1.017 1.020 1.024

.40 1.027 1.030 1.033 1.037 1.040 1.043 1.046 1.049 1.051 1.053

.50 1.055 1.057 1.058 1.058 1.058 1.058 1.057 1.055 1.053 1.050

.60 1.047 1.043 1.038 1.033 1.027 1.021 1.014 1.007 1.000 .992

.70 .984 .976 .967 .959 .951 .943 .935 .928 .921 .914

.80 .909 .903 .899 .896 .893 .892 .891 .892 .894 .897

.90 . 901 .907 .914 .923 .933 .944 .957 .971 .986 1.002

1.00 1.020 1.039 1.058 1.078 1.099 1.120 1.141 1.162 1.182 1.202

1 .10 1.221 1.239 1.255 1.269 1.280 1.290 1.296 1.299 1.298 1.294

1.20 1.286 1.274 1.258 1.237 1.212 1.183 1.149 1.112 1.070 1.025

1.30 . 977 .926 .872 .817 .759 .701 . 643 .585 . 528 .473

1 .40 .420 . 370 .324 .281 .243 .210 .182 . 161 .145 .136

1.50 . 132 .136 .146 .162 .184 .211 .245 .283 .326 .373

1 .60 .423 .476 .531 .588 .646 .705 .763 .820 . 875 .929

1.70 .980 1.028 1.073 1.114 1.152 1.185 1.213 1.238 1.258 1.274

1.80 1.285 1.293 1.296 1.296 1.292 1.285 1.275 1.263 1.248 1.231

1.90 1.213 1 .194 1.173 1.152 1.131 1.109 1.088 1.067 1.047 1.027

2.00 1.009 .991 .975 .961 .947 .935 .925 .916 .908 .902

2.10 .897 .894 .892 .892 .892 .894 .897 .900 .905 .911

2.20 .917 . 924 .931 .939 .947 .955 .964 .972 .981 .989

2.30 .998 1.005 1.013 1.020 1.027 1.033 1.039 1 .043 1.048 1.051

2.40 1.054 1.057 1.058 1.059 1.059 1.059 1.058 1 . 057 1.055 1.053

2.50 1.050 1.047 1.044 1.040 1 .036 1.033 1.029 1.025 1.021 1.017

2.60 1 .014 1.010 1.007 1.004 1.001 .999 .997 .995 .993 .991

2.70 .990 .989 .988 .988 .988 .987 .987 .988 .988 .988

2.80 . 989 . 990 .990 .991 .992 .993 .994 .994 .995 .996

2.90

3.00

.9 97

1.002

. 998 .998 .999 1.000 1.000 1.001 1.001 1.002 1.002
-77-

List of Program Elements

1. MAIN

2. Subroutine OBJECT - Calculates Fourier series

coefficients for input line object.

3. Subroutine CR0S1D - Calculates Fourier coefficients

of the image for 1-D optics using transmission

cross-coefficients

4. Function A(N) - Retrieves Fourier coefficients for

the line object from calculated array assuming

symmetric object.

5. Function CN0RM1 - Normalizes transmission crosscoefficients for 1-D optics.

6. Function CN0RM2 - Normalizes transmission crosscoefficients for 2-D optics.

7. Function CCF1D - Calculates transmission crosscoefficients for 1-D optics.

8. Function CCF2D - Calculates transmission crosscoefficients for 2-D optics.

9. Function AREA - Calculates the overlapping area

of circular lens apertures; used by CCF2D.

10. Subroutine CSLIT - Multiplies image Fourier

coefficients by scanning slit function.

11. Subroutine CROS2D - Calculates Fourier coefficents

of the image for 2-D optics using transmission

cross -coefficients

12. Subroutine YNORM - Normalizes the image to 1.0 at

either the center or the edge of the image, whichever

has the higher intensity.

13. Subroutine RDATA - Reads input data and sets up

parameters used in calculations.

14. Subroutine PRINT - Creates two print files, one for

printing of a table of image data (optical intensity vs.

distance) and the second for a plot file. The plot file

reduces the number of data points by eliminating values

for which there is less than a 0.1% change in intensiy.

15. Subroutine IMAGE - Calculates the image intensity vs.

distance from the image Fourier coefficients.
1 FLOW DIAGRAM FOR 'MAIN' PROGRAM
- 79 -

NOTES:

PER = PERIOD

WT = NORMALIZED WIDTH

XIR = NORMALIZED FREQUENCY

NX = INCREMENT OF DISTANCE IN IMAGE
-80-

2. FLOW DIAGRAM OF SUBROUTINE 'OBJECT'

^ RETURN ^
-81-

3. FLOW DIAGRAM OF SUBROUTINE ' CROS1D'
- 82 -

4. FLOW DIAGRAM OF THE FUNCTION 'A(N)'
500

J = 1 , LJ
- 84 -

FLOW DIAGRAM OF THE FUNCTION ' CNORM1
FLOW DIAGRAM OF FUNCTION ' CNORM2
- 86 -

7. FLOW DIAGRAM OF FUNCTION CCF1D (XIR,N1,N2)

V
-87-

i

RLl = -SS

NC = 1 . 0+ ( RL2

- RLl ) *N0

DC = ( RL2-RL1

)

/NC

DC2 = DC/ 2.0

0

= RLl

01 = o+xi

22 = 0+X2

001 = 01*01

002 = 02*02

AR = SA*0*0

+AP*( 001+002)

AI = TWOPI * ( AB2 * (001-002)

+AB4 *

( 001*001

-002*002)

)

SUM = CEXP ( CMPLX

( AR , AI )

)

RL1+N*DC

QN=DC2

Q+Xl

Q+X2 = 01*01

= 02*02

SA*0*0+AP*

(Q21+QQ2)

1 TWOOT i n vr i

(AS2M001-Q02)

+A54* ( 001*001

-002*002)

)

CEXP ( CKPLX

( AR , a: )

)

- SUM- 4 . 0*Y

ON

0+xi

0+X2

= 01*01

= 02*02

TWOPI*

( A32* ( 001-002

)

+ A34 *

( 01*001

- 002 * 002 )

)

SA*0*0+AP*

( 001+002

)

CEXP ( CKPLX

( AR , AI )

)

= SUX-r 2 . C * Y

T

T
CCF1D = (SUM-Y)

*DC2/6 .

0

RETURN
-89-

8. FLOW DIAGRAM OF THE FUNCTION ' CCF2D
- 90 -

I

Y = AREA ( 1 . C

,

1.0, (X1-X2)

)
- 91 -

T

- 93-
-94-

9. FLOW DIAGRAM FOR FUNCTION 'AREA'

<EED
— J 

10. FLOW DIAGRAM FOR SUBROUTINE ' CSLIT'
— 7 0 —

11. FLOW DIAGRAM FOR SUBROUTINE ' CROS2D
- 97 -
FLOW DIAGRAM FOR THE SUBROUTINE YNORM

YNORM

( NUM)

i

YMAXAN = YMAXBN/

YLIST(NUM)

CONST = YLIST(NUM)
FLOW DIAGRAM FOR THE SUBROUTINE ' RDATA

^

RDATA ^

1 /DATAS

I = 1,13

WIDTH = DATAS(l)

WAVE = DATAS ( 2

)

OBJ = DATAS ( 3

)

SSO = DATAS ( 4

)

SAO = DATAS ( 5

)

AB2 = DATAS ( 6

)

AB4 = DATAS ( 7

)

A? = DATAS ( 8

)

TBO = DATAS ( 9

)

PB = DATAS (10)

TTO = DATAS ( 11

)

PT = DATAS (12)

SLIT = DATAS (13)

SS = SSO/OBJ

SA = SAO/ OBJ

AMPL = SQRT(TBO)

PHASE = PI *PB

TB = CM? LX (AMPL*

CCS (PHASE) , AMI

*SIN( PHASE)

)

AMPL = SQRT(TTO)

PHASE = PI *PT

TT = CMPLX ( AMPL*

COS (PHASE) , AMI

* SIN (PHASE)

)
14. FLOW DIAGRAM FOR THE SUBROUTINE PRINT
- 101 -
- 102 -
-103-

15. FLOW DIAGRAM FOR SUBROUTINE 'IMAGE'
- 104 -

i f J = J+l

DIFF =

| YLIST ( N ) - YLIST ( N+I )

|
- 105 -

XKAX = XX

YKAX3N = yxbk

(

RETURN ^
Acknowledgments

Support for documentation and publication of this software was

provided by VLSI Standards, Inc. under an NBS Research Associate

Agreement. A PASCAL version of this software has also been developed. In addition, the assistance of Ruth N. Varner and

Carmelo Montanez in converting this software for use on the

CYBER 855 is gratefully acknowledged.
NBS.114A (REV. 2-8C> U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NISTIR 88-3808

2. Performing Organ. Report NoJ 3. Publication Date

APRIL 1989

4. TITLE AND SUBTITLE

Narrow-Angle Laser Scanning Microscope System for Linewidth Measurement

on Wafers

5. AUTHOR(S)

D. Nyyssonen

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Stote, ZIP)

10.

SUPPLEMENTARY notes

Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a si gn if i cant bibliography or literature survey, mention it here)

The integrated-circuit industry in its push to finer and finer line geometries

approaching submicrometer dimensions has created a need for ever more accurate

and precise feature-size measurements to establish tighter control of fabrication processes. In conjunction with the NBS Semiconductor Linewidth Metrology

Program, a unique narrow-angle laser measuremnt system was developed. This

report describes the theory, optical design, and operation of this system and

includes computer software useful for characterizing the pertinent optical

parameters and images for patterned thin layers. For thick layers, the physics

is more detail the reader is referred to several related reports listed in the

references .

12.

KEY WORDS (Six to twelve entries; a Iph a beti cal order; capitalize only proper names; and separate key words by semi colon s) coherence; critical dimensions; linewidth measurements; metrology; micrometrology;

scanning microscopy

13. AVAILABILITY

U n 1 1 mi ted

' For Official Distribution. Do Not Release to NTIS

~ Order From Superintenaent of Documents, U.S. Government Printing Office, Washington, D.C. '

' 20402.

|y y O rder From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

11C

15. Pri ce

$ 19.95

UlSCOMM-DC 6043-P 80
.

.

.