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Admittance and Circle Diagrams

Angus Macleod
Thin Film Center, Inc, Tucson, AZ Contributed Original Article

Introduction expression may be complex, but, as usual, we will consider y0, the
Nowadays, few people would dream of calculating the performance of admittance of the incident medium, as always real.
an optical coating without using a computer. Powerful computer pro- If we consider y0 to be constant, then the expression in equation
grams that will run well on even a modest laptop are readily available. 1 can be thought of as transforming Y into ρ, or vice versa, and is a
Even modern programmable calculators can handle straightforward form of a well known expression first studied in the early 19th Century
coating performance calculations. by August Ferdinand Möbius. Therefore, it is sometimes, known
It was not always so. The early pioneers, faced with the immensity as a Möbius transformation, but perhaps more often as a bilinear
of multiple-beam calculations at series of interfaces had to find ways transformation.
of dealing with the problem and developed an array of inventive cal- We can write the transformation as:
culation techniques. Graphical methods were particularly popular and 1  Y y0
effective. Some of these were approximate but others, some of which U (2)
1  Y y0
form the subject of this note, were perfectly accurate in the sense that
the theory contained no approximations and the precision of the result When also
was determined by the quality of the drawing. 1 U
Y y0 (3)
There is now little interest in these techniques from the point of 1 U
view of calculation, but their usefulness as tools for understanding has, Y/y0 is sometimes known as the reduced admittance.
if anything, increased with time. Computers are wonderfully fast in the As written, there is complete symmetry between equations 2 and
accurate manipulation of huge volumes of data in complicated calcula- 3. Unfortunately, there is a complication in the way that we use the
tions, but are of much less help in the explanation of the results. Flawed expressions. We like to think of the complex admittance, Y, in Cartesian
data are handled and presented just as efficiently as reliable data. It is coordinates but the complex amplitude reflection coefficient, ρ, in polar
up to the user of the computer to assess the integrity of the results. It coordinates. Often, particularly when dealing with a completed coating,
is here that graphical techniques are particularly useful. Sketching can we even prefer reflectance, R, over the modulus of the amplitude reflec-
be very quick and great precision is rarely required. Often the problem tion coefficient.
is that the digits of the result presented by the computer are correct, Although equations 2 and 3 are valid for a complex value of y0, the
but the order of magnitude is wrong. A phase shift may be quoted to following techniques and analysis require real y0.
ten significant figures, except that it is exactly 180° in error. There are Let us write
so many different conventions in operation that a problem may simply U U eiM D  iE (4)
turn out to be an incorrect assumption on our part. The knowledge
of an expected magnitude for a result is usually enough to avoid and
difficulties. Y x  iz (5)
The particular techniques we deal with here all involve the drawing
Then
of circles. There are many techniques that include circles, but most are y0  x  iz
U eiM (6)
no longer being used. The early literature contains many references to
y0  x  iz
them. Here we concentrate on two that are still being used and further
developed. The Admittance Diagram concentrates on surface admit- and
tance while the Circle Diagram deals with surface amplitude reflection x  iz 1  D  iE (7)
coefficient. To these two we add the Smith Chart, a calculation tech- y0 1  D  iE
nique closely related to the Circle Diagram. The terminology can be Equations 6 and 7 are the two basic equations for what follows. Equa-
confusing because it is not consistent. For example, all the techniques tion 6 permits us to place contours of constant R and ϕ on a complex
dealt with here, along with some others, are sometimes known by the plane representing the surface admittance. Equation 7 accomplishes the
collective term of Circle Diagrams. opposite: the placing of contours of constant real and imaginary parts
We again use the normal thin-film conventions that have been con- of the surface admittance over a complex plane representing amplitude
sistently employed, and explained earlier, in this series of tutorials. reflection coefficient.
We start with equation 6. We can eliminate ϕ by multiplying each
Fundamentals side by its complex conjugate. On simplifying, we find
The diagrams we are going to discuss are all based on the expression
for the amplitude reflection coefficient of a specular surface. At normal ª1  R º
x 2  z 2  2 y0 « 2
» x  y0 0 (8)
incidence this is: ¬1  R ¼
y0  Y (1)
U where we have replaced |ρ|2 by R, the reflectance. We can treat x and z,
y0  Y the real and imaginary parts of the surface admittance, as coordinates
where ρ is the amplitude reflection coefficient, y0 is the characteristic in the complex plane. Then, if R is constant, equation 8 is the locus of
admittance of the incident medium and Y is the optical admittance points of constant R in the plane, and clearly it is the equation of a circle
of the surface. For a simple surface, Y becomes y1, the characteris- centered on the real axis. The complete set of these circles of constant R,
tic admittance of the material of the surface. All the variables in the forms a nested set, nested about the point, y0. The contours of constant

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ϕ are found almost as readily as

y0 (9)
x2  z 2  2 z  y02 0
tan M
Provided ϕ is constant, equation 9 is also the equation of a circle, but,
this time, centered on the imaginary axis and passing through the point
y0 on the real axis. There is a slight difficulty in that tanϕ is ambiguous
and cannot identify the third quadrant from the first, nor the second
from the fourth. We need, therefore, to return to equation 6 to find that
the first and second quadrants correspond to those parts of the locus
below the real axis and the third and fourth to those parts above.
We turn now to equation 7. Eliminating z we find:
x y0 x y0  1 (10)
D2  E2  2 D 0
 x y0   1  x y0   1
which, if x is constant is the equation of a circle centered on the real
axis, the circles for the range of x all passing through (-1,0). Eliminating
x we find
1 (11)
D 2  E 2  2D  2 E 1 0
z y0
With z constant, this is the equation of a circle centered on a line paral-
Figure 1. Circles of constant reflectance in red, and constant phase shift on
lel to the imaginary axis and passing through the point (-1,0).
reflection in blue, over a diagram of complex admittance in Cartesian coordi-
Figure 1 uses equations 8 and 9 to place contours of constant
nates. y0 is assumed to be unity. Zero reflectance corresponds to the point
reflectance, R, and phase shift on reflection, ϕ, over a Cartesian grid of
(1,0) and in order outwards the red circles are 10%, 20%, 30% and so on to
surface admittance, Y. A reflectance of 100% lies over the imaginary
90%. Zero phase corresponds to the blue circle intersecting the origin (a
axis. Any points to the left of the imaginary axis correspond to a
straight line along the real axis) and other circles are spaced 10° apart with
reflectance greater than 100% and in the normal way are excluded. Of
phase increasing counterclockwise so that quadrants 1 and 2 are below the
course, if there is gain in the system then reflectance can exceed 100%.
real axis and 3 and 4 above.
The diagram, therefore, is truncated at a real part of zero so that only
continued on page 24
the first and fourth quadrants of the diagram are present.

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Admittance and Circle Diagrams retardation due to the layer will be twice its phase thickness, so that
continued from page 23 a quarterwave, for example, will be a semicircle. If the characteristic
admittance of the layer is y1, then the reduced admittance is Y/y1, and
Equations 10 and 11 are used in Figure 2. The diagram is bounded
this can be used instead of Y/y0, in equation 2. Suppose we know the
by a unit circle, corresponding to a modulus of ρ of unity and a reflec-
starting admittance, and, hence the starting reduced admittance. When
tance of 100%. Only positive real parts of admittance are within this
we reach the front boundary of the layer, the surface admittance Y will
circle. The first and second quadrants of ρ are above the axis, and so
have changed, but we can find the new value for Y/y1 by reading it
this diagram is inverted with respect to Figure 1, and vice versa.
off the curves on the diagram and can calculate Y by multiplying the
Figure 1 and Figure 2, as they stand, can be used to calculate ampli-
result by y1. Now suppose the next layer has characteristic admittance
tude reflection coefficient from surface admittance, or the reverse. But,
y2. The new reduced admittance will need to be Y/y2. The operation of
although useful, this is not particularly exciting. The theory of high
changing from one layer to the next is simply recognition of the current
frequency transmission lines shares the same structure of fundamental
reduced admittance by reading it from the chart, multiplying by y1/y2
equations as optical coatings, and results from that field were imported
and finding the point corresponding to the new reduced admittance.
into early optical coatings. One of these imports was a calculation
The subsequent layer then rotates from that point around the center,
technique known as the Smith Chart.
through a negative angle of twice its phase thickness, and so on. The
whole process begins with the substrate admittance and ends at the
front surface.
A quite simple chart for a three-layer coating consisting of quarter-
waves, is shown in Figure 3. While obviously it can help in calculation
in the absence of any computer, the understanding that it conveys is
minimal. Any changes in layer thickness imply recalculation of the
transitions, and a termination off the real axis with its complex admit-
tance, is much more difficult to visualize than one on the axis.

Start
End

Figure 2. Circles of constant real part of surface admittance, Y, in red and

constant imaginary part in blue, over a plot of amplitude reflection coeffi-
cient in polar coordinates. y0 is assumed to be unity. A real part of zero
coincides with the bounding unit circle and the other circles in decreasing
size are 0.5, 1.0, 1.5 and so on to 7.0. The blue circles run from a value of
-5.0 at intervals of 0.5 to +5.0, zero corresponding to the real axis and nega-
tive value being above the real axis in the first and second quadrants and
positive values below.
Figure 3. A Smith Chart showing the calculation of the reflectance of a
The Smith Chart simple three quarterwave coating design, consisting of Air | HLH | Glass with
The Smith Chart [1] dates back to the early 1930’s and is hardly yglass=1.52, yair=1.0, yH=2.45 and yL=1.45. The final amplitude reflection coef-
ever used now for calculation, but it still appears occasionally in the ficient has modulus 0.84 and the phase shift on reflection is 180°. High-
literature. It is based on the diagram of Figure 2. As a result, Figure 2 index circles are black, low-index cyan, and the transitions at the interfaces
is sometimes called a Smith Chart, but the term refers to a calculation orange.
method, and Figure 2 is not really a Smith Chart until it is used as
such. Circle Diagrams
The Smith Chart method records the amplitude reflection coef- The major problem with the use of the Smith chart in understand-
ficient calculated at a reference plane, parallel to the substrate surface, ing rather than in calculation, is the jump from one layer to the next.
as it moves smoothly through a coating from back to front. When the However, it can be shown that the amplitude coefficient of a dielec-
plane is sliding through a single layer, the effective incident medium tric layer growing in thickness describes a circle even if the incident
has the same characteristic admittance as the layer, so it is only the medium has a different characteristic admittance. The circle is still
phase of the amplitude reflection coefficient that changes. In the thin- described clockwise, but its center is displaced along the real axis.
film convention, the phase shift becomes increasingly retarded. The Calculation of the circle is rather more difficult than in the Smith
locus of this change in reflection coefficient, plotted in Figure 2, will Chart, and so the Smith Chart was the more important calculation tool,
be a circle centered on the point (0,0). The magnitude of the phase but with the modern ease of calculation, such complications are rela-

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tively unimportant. The circle diagram assumes that the optical coating
is growing within a constant incident medium and records the com-
plete record of amplitude reflection coefficient as the entire thin-film
structure is notionally constructed. This record is clearly a continuous
one. Thus we have a continuous locus in the diagram consisting of a
series of arcs of circles, each arc corresponding to a particular dielectric
layer. A change in wavelength will cause the arcs to grow or shrink
in length and change their position but, with practice, the qualitative
details of the change can be readily understood. Mostly, this is all that
is required. Accurate calculation will be performed by the computer. It
is helpful to know that all the circles that can correspond to a particular
layer material form a nested set, nested about the amplitude reflection
coefficient of the layer material in the given incident medium. No two
circles corresponding to a single material can possibly intersect. The
circles for two different materials, do intersect and permit the desired
movement over the diagram.
Frank C. Rock has been credited with the early use of this technique
in explaining the functioning of a coating design [2]. A 1969 patent
describes the construction of a four-layer two-material antireflection
coating for the visible region. A circle diagram, called a polar coordi-
nate phase diagram in the specification, is included to explain the con-
struction. The design is based on the well-known quarter-half-quarter
coating [3] but the layer of intermediate characteristic admittance
next to the substrate is replaced by a pair of rather thinner layers of
high and then low index. This coating actually dates back to the 1940’s
and Walter H Geffcken [4], and has appeared again and again in the
literature.
Im

Low

Intermediate
High Re

Low High

Figure 4. The four-layer locus plotted in orange over the original three-layer
quarter-half-quarter design in blue. The loci are made up of arcs of circles.
The black dots mark the terminations of the layers. The locus ends at the
origin where the reflectance is zero. The halfwave layer of high index forms
the complete circle on the left. As the wavelength changes and the loci
change in length, the halfwave layer tends to compensate for the changes in
the other layers, keeping the terminal point close to the origin and zero
reflectance.

The Admittance Diagram

The greatest difficulty in sketching a circle diagram is the calculation of
the circle and particularly the intersections of the loci with the real axis.
This problem is largely avoided in the Admittance Diagram, which has
some other advantages.
We can insert a notional plane, parallel to the substrate surface, at
any point in a system of layers and calculate the surface admittance
at that plane. This admittance is a function of the structure behind
the plane only. It takes no account of what is in front. Thus a record
continued on page 26

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Admittance and Circle Diagrams It is very easy to add lines of constant electric field amplitude to
continued from page 25 an admittance diagram. Provided the materials are all dielectric, the
contours of constant electric field amplitude are lines drawn normal
of surface admittance as this notional plane is slid through the entire
to the real axis. The field amplitude is proportional to the reciprocal
coating is a continuous record that is indistinguishable from the record
of the square root of the value of real admittance. This permits ready
that would have been achieved by a continuous measurement of outer
identification of those points in the coating where the field is at its
surface admittance during the deposition of the coating. The resulting
highest or at its lowest. Point A in Figure 5, the outer surface, is clearly
locus of admittance when plotted in the complex plane can therefore be
where the field is highest, and point B, the mid-point of the high admit-
considered to represent the progress of layer monitoring, or a graphical
tance halfwave layer, where it is lowest. The interaction of the light with
representation of the calculation of the coating properties. Once again,
the material of the coating is proportional to the square of the electric
dielectric layers are represented by circles centered on the real axis and
field amplitude and so we can recognize that the outer surface of this
described clockwise but here the two points of intersection with the
antireflection coating is particularly vulnerable. In fact, this is true for
real axis are governed by the quarterwave rule [3]. If the two points of
virtually all antireflection coatings.
intersection have admittances a and b then we can write
We can also deduce that the extrema of reflectance during the depo-
ab y 2f (12) sition of the layers coincide with the intersection of the loci with the
where yf is the characteristic admittance of the layer material. The real axis. This is important in the study of monitoring errors and their
circles corresponding to any particular material then form a nested set, compensation, particularly in narrowband filters.
nested about the point yf on the real axis. Each semicircle above and Contours of constant electric field in the circle diagram are circles
below the real axis represents a quarterwave and a circle, centered on centered on the real axis and similar to the red circles of constant real
the origin and passing through the point yf, cuts each quarterwave into admittance. They are a little more difficult to interpret but the conclu-
eighth waves. These simple relationships make sketching the locus par- sions derived from them are identical.
ticularly straightforward, because both estimates of thickness and calcu- There is much more to the admittance diagram and we will return to
lation of the points of intersection with the real axis are so very simple. it in a future note.
The designs of Figure 4 yield a similar diagram, shown in Figure 5.
Figure 5. The admittance diagram corresponding to the circle diagram of
Conclusion
As optical coatings become more and more complex and force us to
Intermediate rely completely on computers for accurate calculation, so it becomes
more and more important to have some simple techniques to help us in
Low assessing the validity of the results. Circle and admittance diagrams are
High
High especially useful in this.

A Start
References
B 1. Smith, Phillip H., Electronic applications of the Smith Chart. 1969: McGraw-Hill Book
Company.
2. Rock, F.C. Optical Coating Laboratory Inc., “Antireflection coating and assembly having
synthesized layer of index of refraction.” 1969. USA Patent 3,432,225.
3. Macleod, Angus, “Fundamentals of optical coatings.” SVC News Bulletin, 2005 (Winter):
p. 28-29, 31-32, 42.
Low 4. Thelen, Alfred, “The pioneering contributions of W Geffcken,” in Thin Films on Glass, H.
Bach and D. Krause, Editors. 1997, Springer-Verlag: Berlin, Heidelberg. p. 227-239.

Figure 4. Note that the imaginary axis is drawn at a real admittance value
of 0.5. Further Reading
For more information see:
Macleod, H A, Thin-Film Optical Filters. Third ed. 2001, Institute of Physics Publishing

24/7
(Now Taylor and Francis).
Willey, Ronald R., Practical Design and Production of Optical Thin Films. Second ed. 2002,
Register on-line at Marcel Dekker Inc.

www.svc.org
Angus Macleod is currently Vice President of

SVC Conference Registration the Society of Vacuum Coaters. He was born and
educated in Scotland. Then in 1979 he moved to

and Housing is Now Open! Tucson, where he is President of Thin Film Center,
Inc. and Professor Emeritus of Optical Sciences at
the University of Arizona. His best-known publica-
We’ve made it easy to register for the Conference and reserve tion is Thin-Film Optical Filters, now in its third
your hotel room with our integrated Web Site. edition. In 2002 he received the Society’s Nathaniel
H Sugerman Memorial Award.
Note: After registering on-line, you will receive acknowledgements For further information, contact
from SVC and the hotel. The information in these E-mail confirma-
Angus Macleod, Thin Film Center, Inc. at
tions is very important. For this reason, we would like to remind
angus@thinfilmcenter.com.
you to add several E-mail addresses to your whitelist so they are not Angus Macleod
sent to your SPAM box.
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acknowledgement@pkghlrss.com

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