Download Relationship of the total integrated scattering from

Relationship of the total integrated scattering from
multilayer-coated optics to angle of incidence,
polarization, correlation length, and roughness
cross-correlation properties
J. M. Elson, J. P. Rahn, and J. M. Bennett
Previously published vector equations describing angle-resolved scattering from single-layer- and multilayer-coated optics have been integrated numerically and analytically over all angles in the reflecting hemi-
sphere to obtain numerical results and analytical expressions for total integrated scattering (TIS). The effects of correlation length, polarization, angle of incidence, roughness height distribution, scattered light
missed by the collecting hemisphere, and roughness cross-correlation properties of the multilayer stack on
the TIS expression are considered.
Background material on TIS from optics coated with single opaque re-
flecting layers is given for completeness and comparison to corresponding multilayer TIS results. It is
shown that errors can occur in calculating the true rms surface roughness from actual TIS measurements;
ways to correct these errors are discussed.
1.
Introduction
Total integrated scattering (TIS) measurements have
been used for many years to estimate the rms roughness
of optical surfaces covered with single opaque reflecting
coatings. Davies1 was an early contributor to a theoretical relationship between rms roughness and TIS.
His theory assumed that (1) the surface roughness was
small compared with the wavelength of light illuminating the surface, (2) the surface was perfectly conducting (100%reflectance), and (3) the distribution of
surface heights was Gaussian. Later Bennett and
Porteus 2 and Porteus3 generalized Davies' work. All
these authors based their theories on the scalar Fresnel-Kirchhoff diffraction formula and considered only
opaque reflecting surfaces. Later Eastman4 applied
Airy summations and matrix methods to TIS and other
problems involving multilayer optics. His work was
also scalar, assumed normal incidence illumination, and
invoked the Kirchhoff boundary conditions.5 In all of
the above theoretical treatments, it is assumed that the
lateral separation of surface irregularities is for the most
part greater than a wavelength, so that the scattered
light is approximately confined to a diffuse cone centered about the specular beam. With this assumption,
small angle approximations can be made, although the
assumption is not valid for some common types of surfaces. Additionally, most scalar theories such as those
mentioned above do not consider illumination of the
surface at a non-normal angle of incidence, the degree
of polarization of the incident beam, and the effects of
different correlation functions to describe the statistics
of the rough surface.
In this paper we use angle-resolved scattering (ARS)
formulas6 7 to investigate TIS. These formulas have
been derived by perturbation methods and assume only
that the rms surface roughness is much less than the
wavelength of the incident light beam. The angle of
incidence, polarization, optical constants, and correlation function for the surface roughness are parameters
that may be chosen to fit a specific situation. In the
case of multilayer optics, the autocovariance function
of each interface and the cross-correlation functions
between interfaces may be selected based on available
evidence or reasonable assumptions. The TIS is calculated from the ARS formulas by integrating them
over the hemisphere containing the incident, reflected,
and scattered beams. This approach also gives us the
opportunity to study the effects of scattered light which
may be lost through the aperture in the collecting
hemisphere that admits the incident and specularly
reflected beams as well as any large angle scattering that
The authors are with U.S. Naval Weapons Center, Physics Division,
Michelson Laboratory, China Lake, California 93555.
Received 10 November 1982.
also misses the collecting hemisphere (Fig. 1).
The experimental arrangement for a TIS measurement is shown in Fig. 1. The laser beam is normally
15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS
3207
mirror used to collect the scattered light (Fig. 1), thus
explaining why the roughness of diamond-turned mirrors measured on such an instrument may be much less
than a profile roughness measurement on the same
surface. In Sec. V the relation between TIS and correlation length is discussed for one example of a dielectric multilayer stack, both for normal and nonnormal incidence illumination, and for two types of
roughness cross-correlation. All the above results are
BEAM
INCIDENT BEAM
summarized in Sec. VI.
Sections II-IV consist primarily of background material designed to establish a base for comparison with
TIS from multilayer optics and to discuss some basic
concepts of TIS. The material in these sections, and
more, has been previously discussed in detail by Church
et al.9
Fig. 1. Schematic diagram of apparatus to measure TIS. Most of
the normally incident laser light scattered by sample S is collected
by the aluminized hemisphere (Coblentz sphere) C and focused onto
detector D. Some scattered light (-* ) is lost through the hole along
with the specular beam and beyond the large angle limits.
incident on the sample through an aperture in the collecting hemispherical mirror (Coblentz sphere). The
specular beam also exits through this aperture. Light
scattered by the sample is focused by the sphere onto
a detector, yielding a measure of the diffuse reflectance.
Note that although most of the scattered light is collected, that light scattered close to the incident and
specularly reflected beams is lost as well as that scattered nearly parallel to the surface. TIS is defined according to the relations
TIS =
diffuse reflectance
specular + diffuse reflectance
(la)
where the diffuse reflectance is the fraction of the incident beam which is collected by the Coblentz sphere,
and the specular reflectance is the fraction of the incident beam that is reflected into the specular direction.
Thus the numerator is the light scattered out of the
specular direction, and the denominator is the total
reflected light.
In Sec. II we discuss the concept of spatial wavelengths of the surface roughness and show how these
relate to the autocovariancefunction for the surface and
to the so-called correlation length. Then we integrate
the ARS expressionsto obtain the TIS for the casewhen
the surface correlation length is much longer than the
illuminating wavelength (the usual assumption) and,
second, when the correlation length is much shorter
than the wavelength. A Gaussian autocorrelation
function is assumed in the derivations. All the above
derivations are for normal incidence illumination on the
surface. In Sec. II.B is shown how the results are
modified for non-normal incidence illumination. In
Sec. III we show that it is not necessary to have a
Gaussian distribution of surface heights for the TIS
expressions to hold, and, in fact, the surface height
distribution need not even be symmetric about the
mean surface level. Section IV is important in that it
considers the effect on measured surface roughness of
light lost through the aperture in the hemispherical
3208
APPLIED OPTICS/ Vol. 22, No. 20 / 15 October 1983
11. TIS from an Opaque Reflecting
Coating as a
Function of Correlation Length
In this section we will (1) introduce the concepts of
surface spatial wavelength, autocorrelation function,
correlation length, and power spectral density and (2)
show how the ARS expressions can be integrated
to
obtain a relation for the TIS. The case where the correlation length is long compared to the illuminating
wavelength is considered first, followed by the other
extreme of a correlation length much shorter than the
wavelength. Normal incidence illumination on the
surface is the most common situation (Sec. II.A), but
non-normal incidence illumination (Sec. II.B) is also
considered.
The roughness on an optical surface can be considered to be composed of a Fourier series of roughness
components of various amplitudes and periods. For an
isotropic polished surface, these components will be
oriented in random directions; but for a diamondturned or lapped surface, most of the roughness components willbe aligned parallel to the cutting or lapping
direction. Each of the Fourier roughness components
will have a single periodicity or spatial wavelength,
which is the reciprocal of the spatial frequency, and will
diffract light in a direction that is determined by the
well-known diffraction grating equation, X= d Isin 0o ±
sinOI. Here X is the illuminating wavelength, d the
spatial wavelength (grating spacing), 00 the angle of
incidence (measured from the surface normal), and 0 the
diffraction angle (also measured from the surface normal). In this equation it is assumed that the plane of
incidence is perpendicular to the direction of the spatial
wavelengths (grating grooves). From this equation we
can obtain two very important facts about optical
scattering: (1) surface spatial wavelengths which are
large compared with the wavelength of the illuminating
light produce scattering very close to the specular direction; (2) large angle scattering (away from the specular direction) is produced by progressively shorter
spatial wavelengths. In the limit when d = X, the
scattered light is at a grazing angle along the surface for
normal incidence illumination. Surface spatial wavelengths that are shorter than the incident wavelength
cannot scatter into the reflecting hemisphere except by
other mechanisms such as surface plasmon absorption
and subsequent reemission.
Most optical surfaces do not have a single spatial
wavelength roughness component but contain a large
range of roughness spatial wavelength components.
The roughness of these surfaces can be conveniently
described by a so-called autocovariance function 8 which
takes into account both the amplitudes and periods of
the various roughness components. This function is a
measure of the average correlation between two points
separated by a distance x, called the correlation or lag
distance. For calculational purposes, it is convenient
to assume a Gaussian autocovariance function, which
has a value equal to the mean square surface roughness
for zero correlation distance.
tropic so that G(r) = G(r) and g(k) = g(k), where
result
J'
J0
ticularly for lag distances slightly larger than zero. The
distance at which the autocovariance function drops to
I/e of its initial value is frequently called the correlation
length and will be so defined in this paper. When this
correlation length is large compared with the illuminating wavelength, most of the light will be scattered
dkkg(k)
=
2ir
2
(3)
.
It should be emphasized that this equation is useful for
calculating the mean square roughness 62 only when
g(k) is known over the full range of surface wave numbers k.
The autocovariance function G (r) is a measure of the
average correlation between two points separated by a
distance '. A common example of a G(T) function is
the Gaussian autocovariance function:
Many real surfaces have
autocovariance functions closer to exponentials, par-
=
Jr and k = I k . Since d 2k = kdkd4, we can integrate
Eq. (2a) over 0 for the case when r = 0 and obtain the
G(T) = 62exp(-r 2 /uy2 ),
(4a)
where u is the correlation length. From Eq. (2b) we find
that a Gaussian autocovariance function also has a
Gaussian power spectral density function:
g(k =
r52U2exp(-k2u2/4).
(4b)
hemisphere to yield the customaryrelation between TIS
The power spectral density function plays a significant
role in determining the angular distribution of scattered
light. This is the part of the angular scattering expression [Eqs. (5)] which contains the statistical properties of the surface roughness.
An important assumption made in deriving Eq. (lb),
and rms roughness for a surface covered with an opaque
reflecting coating1 0 :
long compared with X. This is an assumption that is
into a cone about the specular direction; when it is equal
to or smaller than the illuminating wavelength, there
will be appreciable large angle scattering. The ARS
expressions 6 can be integrated over the scattering
147b
62
TIS-,
(lb)
where 6 is the rms roughness and X the illuminating
wavelength. The
designation indicates that the
correlation length u [defined in Eq. (4a) and discussed
later in this section] is long compared with the illuminating wavelength. Equation (lb) is for normal incidence illumination. In practice, 6 would be calculated
from Eq. (lb) using the definition of TIS in Eq. (la).
To establish the notation we will be using, we will now
discuss autocovariance functions, power spectral density
functions, and their respective parameters: rms
roughness and correlation length. The autocovariance
function 8 and power spectral density function form a
Fourier transform pair where
G(=
(12
, fd kg(k)
2
exp-ik
*r)
(2a)
is the autocovariance function of the surface roughness
and
2
g(k) = Sd rG(r) exp(ik
T)
(2b)
is the power spectral density of the surface roughness.
The autocovariance function may be defined by G(r)
= (z (p)z (p + r)), where ( ... ) denotes ensemble average
and z (p) describes the height of the surface roughness
above and below the mean surface level at point p =
(x,y). From this definition, it can be shown that G(0)
= 62,11 where 2 is the mean square of the surface
roughness. G(T) and g(k) are 2-D functions. In this
paper, we will assume that the optical surfaces are iso-
as we shall see, is that the autocorrelation
length a is
not necessarily realistic; in what follows we will see how
this assumption can lead to serious errors. We reproduce the ARS expression from Ref. 6:
1 dP
((J/C)4
-=
P dQ
72
XI
cOO cos 2O1ll- EI2 g (k-ko)
IX012
+
1x04'
ljq + ql 2 jq + q2}
I
(5a)
where
X =
(q'q' cosk - kkoe) costk' (w/c)q' sino sino'
qo + q0
q0 + qoe
() [q' sino
C
coso'
q0 +qoE
(w/c) cosk sinl/]
qc+q0
(5b)
(5c)
The left-hand side of Eq. (5a) represents the differential
power per unit solid angle dQ = sinOdOdk scattered into
direction (0,k) (Fig. 2). The scattered power is normalized to the power P incident on the surface. As
shown in Fig. 2, Oois the polar angle of incidence where
the plane of incidence is the x - y plane. The angles
0 and 0 are the polar and azimuthal angles of scattering,
respectively. The angle ' is the angle of the incident
electric field vector relative to the plane of incidence.
If O = 0 (r/2), the incident beam is p-polarized (spolarized); i.e., the incident electric field vector is parallel (perpendicular) to the plane of incidence. The
dielectric constant of the scattering surface is , which
can be a complex quantity.
The free-space wave
number is /c = 27r/X, where X is the incident wavelength. Other definitions include q = (/c) cosO, q =
(WO/c)cos00 k = (/c)
sinO, ko = (co/c) sinGo, q' =
15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS
[E(W/c)
2
3209
Alternately, Eq. (9a) can be written in terms of k-space
with dQ = d 2k/[(cw/c)2 cosO], where d 2k = kdkd5. Integrating over 0 yields
DENT BEAM
P
SCATTERED BEAM
TIS = -=
(w/c)2
2]
[2- (kX/27)
0 2,r/X
7d
d
O
g
)X
(9b)
Po
7 f,
~[I
- X201/
where g(k) may be modeled based on experimental
OPTICAL SURFACE
evidence. Recall that Eqs. (9) are for normal incidence
illumination, Ro = 1, and isotropic surface roughness.
If the correlation length a >> , g(k) in Eq. (4b) may
be approximated by
Fig. 2. Schematic diagram showing notation for the ARS formulas.
Light is incident at angle 0o; the polar and azimuthal scattering angles
are 0 and 0, respectively.
k2]1 2 , q' = [(w/c)
- ko]l/2 . For the special case of
normal incidence (00 = 0, ko = 0), Eqs. (5) reduce to
4
1 dP (wO/C)
- 12
= (
cos20 g(k)
P dQ
712
I 1+ ~-J2
X [I q'I2 CoS20+ (W/C)2 in 2 k.
(5d)
X~j q -,I2 + q
2Sd)
-
2
.'
According to the definition of TIS in Eq. (la), we may
relate Eqs. (5) to TIS by the expression
TIS=
P/P
P
Ro +P/Po
(6a)
RPo
d
(6b)
is the scattered light integrated over the entire scattering hemisphere (SH) and Ro is the specular reflectance. The approximation in Eq. (6a) is consistent with
first-order theory in that the diffuse reflectance (scattering) is much less than the specular reflectance.
A.
Normal Incidence Illumination
For the moment we will confine ourselves to normal
incidence illumination and consider only Eq. (5d). To
calculate TIS as defined in Eq. (6a), we must integrate
(1/Po)dP/d over 0 and 0, where 0 ranges over the angular region 0 - 7r/2 and 0 ranges over 0 - 27r, and divide by Ro. We first assume that the surface has a reflectance of 100% (Ro = 1), so that e
I
1 and 0' = 0
(electric vector defining the 1 = 0 direction).
(5d) then becomes
1 d=
PodQ
d
Po PO SH dQ
where the integration is performed over the entire
scattering hemisphere.
Also, since g[(w/c sinO] is assumed to be independent of 1,the integration over 0 can
be performed with the result that
TIS =
3210
J
(11)
The subscript in Eq. (11) again refers to u/X being
large compared with unity. Although a Gaussian form
of autocovariance function [Eq. (4a)] was used in the
derivation, any other form of the autocovariance function would also be valid as long as /X >> 1. The point
we are making here is that when a TIS measurement is
made on a surface for which /X >> 1 and the reflectance
is high, the resulting value of 62obtained from Eqs. (lb)
It is also possible to omit the assumption of 100%
specular reflectance. To do this, we again assume that
/X >> 1 but impose no restrictions on I 1l. We let N 2
= , where N is the complex refractive index. Since a/X
>> 1, most of the scattered light will be concentrated in
a cone about the specular direction and the significant
range of integration of Eq. (5d) will occur for 0 << 7r/2.
We can then approximate q' n (co/c)-v'Tand q
(co/c),
since Ie >> sin 20 for 0 << r/2. Equation (5d) then becomes
1 dP
dO sin0(1 + cos 2 O)g[(co/c) sinO].
(W/C)4 1
Po dQ
71.2
-
1
N12
+N
Thus, when /X >> 1, the ARS is proportional to the
normal incidence specular reflectance Ro = (1 - N)/(1
+ N) 2. Using Eq. (4b) forg(k) and integrating Eq. (12)
over the reflectance hemisphere yields
1
dP
P
167r262
PO
0
(13a)
X
so that
(7)
where k = IkI has been replaced by (co/c) sinO. Since
Ro = 1, the TIS in Eq. (6a) is given by
1
TIS. = (161r262)/X2
POf dQ
712
P
where 6(k)is the Dirac 6-function,not to be confused
with the rms roughness 6. Using this result in Eq. (9b)
yields
Equation
2
(COS20+ sin 2o COS
0)g[(W/C) sinO],
()
(10)
or (11) is accurate.
where
JSH d
27rb2 [6(k)/k],
g(k) -
(9a)
APPLIED OPTICS / Vol. 22, No. 20 / 15 October 1983
TIS = P
RoPo
167r6
A2
(13b)
in agreement with Eq. (11). Equation (13b) is valid for
surfaces of any reflectance and /X >> 1.
The approximation aIX- - for g(k) in Eq. (10) need
not be the only approach to deriving Eq. (lb). An alternate method is to recognizethat the argument of the
exponential term of Eq. (4b) is 7r2 (af/X)2 sin 2 G [since k
= (co/c)sinG],and for a>>X the major contribution to
the integral over 0 in Eq. (9a) is for 0 << 7r/2, as mentioned above. Thus we replace sinG - 0, cosO 1 -
02/2, and extend the range of integration over 0 to 0
-. The result is again Eq. (11). At this point, we call
attention to the fact that the TIS in Eq. (11) varies as
X- 2 , since for /X << 1, considered in the next paragraph,
the wavelength dependence will be as X-4 .
We now consider the other extreme case when the
correlation length is much smaller than the illuminating
wavelength. Now the argument of the exponential term
in Eq. (4b) is small for k values from 0
-
27r/X (or 0
values from 0 - 7r/2), and the exponential term in this
equation may be approximated by unity:
2 2
7r6
t U.
g(k)
(14)
From Eqs. (9) we now obtain the result that
64 r4 62o-2
TISo =-
3
(15)
4
X
where the subscript 0 refers to the case a/X <<1. The
TIS in Eq. (15) varies as \- 4 and also depends ol the
correlation length . Since the exponential term in Eq.
(4b) was approximated by unity, any power spectral
density function which is nearly constant over the range
with Eqs. (11) and (15). However, the quadratic region
is only in effect for a/X $S 0.1, which is a shorter corre-
lation length than can be measured with a surface profiling instrument
(for X = 0.6328 ,m, a
0.06
Am),
12
although correlation lengths considerably smaller than
0.1 um have been calculated from surface plasmon
measurements on rough silver films.1 3 On the other
hand, for /X ' 0.6, the plot levels off at unity and remains there for larger values of /X. This yields a- >
0.38 Am for X = 0.6328 Am. Thus, it would appear that
a/Xfrequently lies in the transition region between the
quadratic range and the constant value of unity. This
transition region can be approximated by a straight line,
as TIS/TIS. = 2.58(a/X) - 0.161 for 0.1 < a/X < 0.4.
For this range of /X, 6 _ (X/47r)(TIS)l/2 [2.58(a/X) -
0.161]1/2. Note also that for /X > 0.6, 6 _ (X/4r)
(TIS) 1/2 ; for /X < 0.1, 6 (/8)(X2/72af)(TIS)l/2.
In
summary, whenever a-/X< 0.6, 6 calculated from TIS_
[(Eq. (11)] will be too small.
of integration can be used in the deviation. The result
B.
as in Eq. (15), however, will depend on the value of the
In all the preceding derivations it was assumed that
the surface was illuminated at normal incidence. For
non-normal incidence illumination, it will be shown in
Sec. III that for 6 <<X and >>X
power spectral density function at the origin. The results obtained from the two limiting cases in Eqs. (11)
and (15) do not depend on the overall shape of the power
spectral density function.
Obviously, a value of a/X not
within the range of approximations used in Eqs. (11)
and (15) would mean that we would need to know the
detailed shape of g(k) over the range of wave numbers
k applicable to the TIS scattering hemisphere.
When
2
is calculated from Eq. (11) or (15), the results
will generally be different. As an example, let =
0.6328 m and a = 0.1 um, typical values for many
polished glass surfaces. For a given TIS measurement,
calculation of the rms roughness from either Eq. (11) or
(15) yields the ratio
(bo2
I'DJ
3
4wr2 (cr/X)2 '(16)
which is 3.02 for these parameters.
The 60and 65. are
the rms roughnesses calculated from Eqs. (15) and (11),
respectively, and their ratio is thus 1.74. The question
arises as to which formula is more appropriate to use
with state-of-the-art optical components. Since correlation lengths are typically 0.1-0.2 m for supersmooth polished fused quartz surfaces coated with
silver or aluminum, it would seem that Eq. (15) would
be more appropriate for such surfaces in light of the a/X
<< 1 approximation.
Non-normal Incidence Illumination
TIS
show TIS/TIS
0
vs /X for
= 30 and 45°, respectively.
The rough scattering surface is assumed to be silver with
E = (-16.4, 0.53) at X = 0.6328 gm. In both cases, the
~I
C
:
w
To answer this question more
origin and levels off to unity as a increases in agreement
(17)
where 00is the angle of incidence of the illuminating
beam. There is no restriction on the reflectance of the
surface. Equation (17) is thus analogous to Eqs. (lb)
and (11) for normal incidence illumination. By comparing Eq. (17) to Eq. (11),it is seen that the TIS values
are in the ratio of cos 2 00. Thus 6, calculated from Eq.
(11) but using non-normal incidence illumination, will
be too small. The error will increase as 0 increases.
When a-/Xis no longer large compared to unity, it is
most convenient to integrate numerically Eq. (5a) to
obtain the dependence of TIS on /X. Figures 4 and 5
I
I
I
I
I
1.0
I
I
I
/
0.8
/
t 0.6-
precisely, we have numerically integrated Eq. (5d) over
the reflectance hemisphere and plotted the results in
Fig. 3, using Eq. (4b) for g(k). The y axis is the TIS
normalized with respect to TISc-, Eq. (11), while the x
axis is the ratio of correlation length to wavelength.
The scattering surface is assumed to be silver with =
(-16.4, 0.53) at X = 0.6328 m [ = El + iE2]. In fact,
very nearly the same curve is obtained for uncoated
glass as long as the scattered light is divided by the
specular reflectance [see Eq. (la) for the definition of
TIS]. As seen in Fig. 3 the plot is quadratic near the
= (47rbcosOo)2
o
0.4-
n
0.2-
0
Z
0
0
0.2
0.4
0.6
0.8
1.0
CORRELATION
LENGTH/WAVELENGTH,
a/X
Fig. 3. Normalized TIS/TIS_ vs o-/Xfor an opaque highly reflecting
surface. The light is assumed to be normally incident on the surface,
and the scattered light is collected for all angles between 0 and 90°.
TIS_ is for the same surface when
>> X.
15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS
3211
4
Although in most analytical treatments of scattering
1.0
the heights of surface irregularities are assumed to have
C-)
a Gaussian distribution about the mean surface level,
this is not necessary for the TIS relations to hold. In
fact, the height distribution function need not even be
@ 0.8
ZT
t 0.6
symmetric, as will be shown in this section.
Since en-
ergy is conserved, the TIS for an opaque highly reflecting surface must be equal to 1 - R 0 , where Ro is the
04
I-_
0
N
4
specular or coherent reflectance. We assume that the
ratio of rms roughness to wavelength is <<1and that
0.2
there are no correlation length effects. Thus we assume
0z
n 1,s
-0
0.5
1.0
1.5
2.0
CORRELATION LENGTH/WAVELENGTH,a/X
Fig. 4. Same as Fig. 3 except that the angle of incidence o is 30°.
TIS_ is for normal incidence and a >>X. The solid (dashed) curve
is for p-polarized (s-polarized) incident light. For aIX > 2, the
TIS/TIS.
curve is nearly constant at cos 230' = 0.75.
Qir
us
that the exiting wave fronts have height variations that
are in 2:1 correspondence to the respective surface
height variations independent of the transverse separation of the scatterers. With these assumptions, we
use the Kirchhoff diffraction integral to compute the
specular fields ER and ES for a rough and smooth surface, respectively. The ratio Ro = (ER/ES)2 , which for
6/X << 1 will be slightly less than unity, and, therefore,
TIS = 1 - (ER/Es) 2 .
The form of the Kirchhoff diffraction integral for the
specular field produced by the reflection of a plane wave
0
from a slightly rough surface A' at a distance r >>
8
0
is
V)
~_
T
E(r) = cosoo
da' E(x'y) exp 27- 1 rO+ 2(x',y') cosOol
(18)
N
where 0 0 is the angle of incidence.
.:
To
Z0
0 IC
I
0.5
CORRELATION'
1.0
1.5
LENGTH/WAVELENGTH,
2.0
a/\
Fig. 5. Same as Figs. 3 and 4, except that the angle of incidence 0 0
is 45°. The solid (dashed) curve is for p-polarized (s-polarized) in-
cident light. For a/ > 2, the TIS/TIS- curve is nearly constant at
cos245
0.50.
results are significantly different than for 00= 0, and
also the curves are different for s- or p-polarized incident light. The asymptotic values of the curves in Figs.
3-5 are proportional to
cos2 00 ,
as predicted in Eq.
(17).
All the theoretical calculations in Figs. 3-5 have assumed a Gaussian autocovariance function for the rough
surface. This is not necessary since the formulas could
be numerically integrated with any analytical autocovariance function that fits the experimental surface.
Also, the assumption of a Gaussian distribution of
surface heights is not necessary, as has been discussed
by Porteus 3 and is further considered in the following
section.
Ill. TIS Independent of the Form of the Surface
Roughness Height DistributionFunction
In this section wepresent a development based on the
Kirchhoff diffraction integral. The primary reason for
this change in formalism is that the surface roughness
height distribution function is relatively simple to incorporate. Conclusions similar to those of this section
have been obtained by Church et al.9 by verbal and
analytical arguments.
3212
APPLIED OPTICS / Vol. 22, No. 20 / 15 October 1983
A point on the sur-
face is defined by its position (x',y') and height (in the
z direction) ¢(x',y'). The parameter r = {(x - XI)2 +
(y - y') 2 + [ - (x',y')]2}1 /2 is the distance from the
point on the surface [x',y',(x',y')] to the observation
y') 2 + z 2] 1/2 is the
distance from the point on the mean surface plane (x',
point (x,y,z); ro = [(x
- x') 2 + (y
-
y',O) to the observation point. Since the observation
point is many wavelengths removed from the surface,
we may write r n ro, and then the only dependence on
the roughness t(x',y') is in the exponential term. The
average value of the electric field for an ensemble of
surfaces may be computed by using the height probability density function D(P). This function has the
propertiesthat
SJD(O)dt = 1,
(19a)
or that the area under the probability density is unity,
and
S
D(g)tdv
= A
(19b)
which is taken to be zero ( = 0) by definition of the
mean surface level. Also
X D()? 2d!=
a,
(19c)
which is the mean square roughness value. More generally,
E.
D()f()d
= (f()),
(19d)
wql
where (f (g)) is the ensemble averaged value of a function f(A). Note that wehave made no assumptions that
D (P)is symmetric about the origin D= 0 (mean surface
level). Since the surface roughness is nondeterministic,
we cannot calculate Eq. (18) exactly. Instead we must
use ensemble averaging techniques as in Eq. (19d). The
ensemble average of E(r) is
(E(r)) =
dA' E
Ai
2Xi
A'
i
X exp
o
IV. Surface Autocovariance Functions and TIS
Angular Measurement Limitations
In previous sections we have considered only the TIS
C
d D
E
[ro + 2t(x',y')
The primary conclusion from this section is that the
standard TIS formula does not depend on the surface
height distribution function being Gaussian or even
symmetric about the mean surface level if 6/X <<1.
arising from the spatial wavelengths on a rough surface
coso]
(20a)
and have not considered the limitations introduced by
the system that measures the TIS. A hemispherical
collector of the type shown in Fig. 1 must have a hole to
which may be rewritten as
(E(r)) = El(r)
dgD(P) exp
X
cosOo),
Ar
(20b)
where
El(r)
=
E
cos~o
r dAE(x',y')
10 dA'
2Xi fA'
ro
(27riro(
exp t
(21)
For a perfectly smooth surface t(x',y) = 0 and D(P) =
W(r),which is the Dirac -function. This yields the
simple result that
(E(r)) = E1 = Es.
For more general cases where
(22)
/X << 1 and
62 =
( t2(x',y') ), we may expand the exponential term in the
integrand in Eq. (20a) to yield
ER = (E(r) )
X [1 + 4-i
E1
pass the incident and specularly reflected beams; it also
cannot collect all the light scattered at grazing angles
to the surface. Thus very near angle scattering and
grazing angle scattering will be missed, and the roughnesses of the surface spatial wavelengths that produce
this scattering will not be measured. [Any roughness
features with surface spatial wavelengths that are
shorter than the illuminating wavelength (for normal
incidence illumination) cannot be measured even with
a perfect collecting hemisphere since they would produce scattering at larger than grazing angles to the
surface.] In this section we will introduce the concept
of an effective roughness 6e which is that value that
would be measured if the polar angle limits on the TIS
measurements
and show how
d gD()
cosSo + I (4ri COSo)
(23)
2].
By the conditions given in Eqs. (19), the first two integrals are unity and zero, respectively.
Since TIS = 1 -
(ER/ES)2 , we obtain from Eqs. (23) and (22)
were >0 and <7r/2.
We will derive
general relations with variable collection angle limits
6
e
differs from
as a function of these
limits and of the correlation length of the surface.
Finally, we will give examples of surfaces having different correlation lengths including one that has both
long and short correlation lengths to show how the effective roughness measured by TIS differs from the true
roughness.
We will assume that g(k), the power spectral density
of the true surface roughness, is known. Then, if the
surface is isotropic so that there is azimuthal symmetry,
TIS = 1 - [1 -
2
2
87r252 cos 00/X2]
16r22 cos2
0
4)
Note the dependence of the TIS on angle of incidence
00 and also that, for normal incidence (
0
Eq. (3) yields the true mean square roughness
surface is less than a wavelength. In terms of the spatial
wavelength components on the surface, no wave front
height variation may occur when the spatial wavelengths are less than the illuminating wavelength and
the surface is being illuminated at normal incidence. In
other words, the light is not scattered by more than 90°
from the mean surface normal.
6
e
by
the relation
= 0), the result
reduces to that of Eqs. (lb) and (11). Note also that
this result does not depend on D(t) being Gaussian or
even symmetrical. This is by virtue of the assumption
b/X <<1, which means that the light is not sensitive to
the roughness peak-to-valley ratio because it cannot
resolve this ratio.
The result for TIS given in Eq. (24) assumed that all
the roughness frequency spectrum contributed to the
scattering, which is equivalent to the assumption o/X
>> 1. It does not take into account the reduction in
exiting wave front height variation which occurs when
the spacing between adjacent peaks and valleys on the
By
62.
analogy we can define an effective rms roughness
52=
f'dkkg(k),
e
27r
(25)
where the limits of integration are variable to include
the nonideal collection limits of the hemispherical collecting mirror. In k-space, these limits are ca= (27r/X)
sinG1 , ( > 0), and 3 = (2ir/X) sinG2 , ( < 27r/X), where
G1 and 02 are the polar angles subtended by the limits
of the specular exit hole and the rim of the hemisphere.
In other words, some scattered light is lost through the
specular exit hole and also beyond the large angle limit
of the Coblentz sphere, as shown in Fig. 1. These losses
can affect the accuracy of 62 obtained from actual TIS
measurements.
Equation (25) can be related to the autocovariance
function using Eq. (2b) for (ko = 0), which yields
eS=
J
dk k f
d T G()Jo(k,),
(26)
where G(r) = G (T) (azimuthally symmetric), and the
15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS
3213
wI
azimuth integration of r has been done yielding the
zero-order Bessel function JO. If we choose a Gaussian
autocovariance function G(T) = 62 exp(-r 2 /o 2 ) and an
exponential autocovariance function G (r) = 62
exp(-IT I/o), we may derive from Eq. (26)
(1e)2
1
22d
1 P2a2\
2 - exp (-
l(^)2= exp (-
)
(27a)
Equation (32a) agrees with Eq. (11). However, Eq. (15)
is larger than Eq. (32b) by a factor of 4/3. The difference is attributed to the lack of information in the development of this section of the angular dependence of
dipole scattering, which is inherent in the ARS formulas. To a first approximation and for normally incident
light, the dipole scattering currents on a slightly rough
surface are oriented in the same direction as the incident
for the Gaussian case, and
electric field vector. Thus the dipole scattering cur(27b)
rents are parallel to the mean surface plane. A Rayleigh
for the exponential case. Equations (27) are in agreement with Church et al.9 In the indicated asymptotic
angles. Hence it is clear why Eq. (32b) (which does not
lf
=
1
1+
I
cases we find
(Se2
I
)
fJr
2
(
2
-
ae2)/4
_ exp(-a2a2/4)
a <<A,
(28a)
a >> A,
(28b)
a <<A,
a»> A,
(28c)
(28d)
for Eq. (27a) and
re2
0a2(02-
a 2 )/2
l1/ar
for Eq. (27b). The result for >>Xindicates that most
of the scattered light is passing through the specular exit
hole, and thus the amount of light collected approaches
zero. In this case, the effective roughness be obviously
approaches zero. Note, however, that the results of
Eqs. (28b) and (28d) are very different in their asymptotic behavior. Because of this, it is possible to realize
large numerical differences with these asymptotic formulas. In this case, the choice of the autocovariance
function becomes very important. On the other extreme, when /X << 1, the TIS also approaches zero.
This is because the surface spatial wavelengths are, for
the most part, shorter than the incident wavelengthand
thus cannot produce direct scattering into the hemisphere (for normal incidence illumination).
In the perfect case when a = 0 and 1 = 27r/X,we
have
(5e)2
)l
72oa2 /X2
I
A,
a >> ,
a
(29a)
(29b)
for the Gaussian case and
lbeU
(b )/
f 2ir2 a 2 /A2
1
a << A,
(29c)
a >> A,
(29d)
for the exponential case.
To relate Eqs. (29a) and (29b)to previous TIS results
[Eqs. (11) and (15) for Gaussian autocovariance functions, a = 0 and 13 27r/A]in the same limiting cases, we
write
TIS = (167r2 52)/X 2,
(30)
where e has replaced 6 in Eq. (11). From Eq. (27a) for
a = 0 and 13= 2r/X and Eq. (30), we have
dipole exhibits a cos2 0 dependence [seeEq. (12)], and
thus the scattering intensity falls off for larger scattering
take into account dipole scattering properties) is slightly
larger than Eq. (15) (which does contain the dipole angular scattering properties). The result of Porteus3 was
limited to small angle approximations and thus did not
contain the larger angle dipole scattering properties.
This explains the agreement of Eq. (31) with the Porteus results.3
If Eq. (31) is plotted as TIS/TIS., vs the ratio of correlation length to wavelength o/X, the curve is essentially identical to that in Fig. 3. Thus, it follows that
in principle Eqs. (27) may be used to correct 6e for
general values of a and for lost scattered light provided
that the autocovariance function is Gaussian. The
values of a and will be known from the geometry of
the collecting hemisphere, but the correlation length a
may not be accurately known for a given sample. If
there is a reason to believe that much of the shape of the
autocovariance function is exponential, perhaps Eqs.
(28) can be used to correct the TIS obtained value 62to
the true value 62. Strictly speaking, exponential autocovariance functions are not physically realistic when
the origin ( = 0) is included. However, many experimentally determined autocovariance functions have an
exponential shape away from the origin.
We will now show how the angular limits on the col-
lecting hemisphere affect the measured TIS and thus
be for surfaces having short ( = 0.2-,m), medium (a
= 2m), and long ( = 10-,m) correlation lengths. The
short correlation length surface might be a conventionally polished glass surface, while the longer correlation lengths might be associated with chemically
polished, electropolished, or diamond-turned surfaces.
All surfaces are assumed to have true rms roughnesses
6 of 10 A. We assume that the hole in the collecting
hemisphere that passes the incident and specularly
reflected beams subtends
a polar angle
1
= 2.85°
(normal incidence illumination) and that the large angle
limit 0 2 is 79.1° (the values for the China Lake instrument). If the measuring wavelengthX= 0.6328,um, /
= 0.316, 3.16, and 15.8, respectively, for the three surfaces. Figure 3 shows that for a/X = 0.316, TIS/TIS_
(31)
= 0.66; for the larger values of o/X, the ratio is unity.
Thus, for a perfect collectinghemisphere, the measured
which is in agreement with the result of Porteus. 3 From
TIS for the two longer correlation length surfaces would
Eq. (31) we find that for the indicated limiting cases
give the correct rms roughness, but the measured TIS
for the 0.2-,m correlation length surface would be too
TIS =
2 [1 - exp(-7r 2a2 /X2 )],
),2
TIS = 167r22/A2
167r462a 2 /X4
3214
a>> ,
a << .
APPLIED OPTICS/ Vol. 22, No. 20
(32a)
(32b)
/
15 October 1983
small, and the value of 6 calculated from Eq. (11) would
be \
X 10 A, or 8.1 A rms.
1.00
0.
Z 061
4
0.2
0
5.00
10.00
15.00 20.00 25.00
CORRELATION LENGTH/WAVELENGTH,
U/\
Fig. 6. TIS/TIS- vs u1Xfor a surface having a two-Gaussian autocorrelation function and measured using an apparatus similar to that shown
in Fig. 1. The short-range correlation length ars is held constant at 0.35 Aum,while the long-range correlation length varies from to 16.0 m.
The short- and long-range rms roughness values are 33.6 and 47.5 A, respectively. For this type of surface (similar to an actual diamond-turned
surface), significant light is lost through the specular beam exit hole.
Now we will calculate what the restricted collection
angles of the hemisphere do to the TIS measurements.
We will assume that the autocovariance functions for
all three surfaces are Gaussian so that Eq. (27a) is applicable. The collection angles 1 = 2.85° and 2 =
79.1° yield a = 0.494 gm- 1 and 13= 9.75 gm- 1 , respectively. Equation (27a) predicts be/6 values of 0.78,0.88,
and 0.047 for the o-values of 0.2, 2, and 10 gm, respec-
tively. Physically, we see that for the o-= 0.2-gm surface the limiting inner aperture introduces essentially
no error, but that appreciable light is scattered beyond
79.10. For the o = 2-gm surface, the inner aperture is
cutting out some of the scattered light, but the large
angle limit has no adverse effect. For the o-= 10-gm
surface, most of the light is scattered into angles of
<2.85° so that only 0.0472 or 0.0022 of the correct TIS
amount is collected by the hemisphere. This last result
is somewhat surprising since if only a single spatial
wavelength of 10 gm were present on the surface it
would be scattered (diffracted) at an angle of 3.60 from
the specular direction and would thus easily be collected
by the hemisphere. This result is particularly relevant
to diamond-turned surfaces and explains why TIS
measurements on some diamond-turned surfaces give
much smaller roughness values than those measured by
stylus instruments or interferometry.1 4 To complete
the calculation that was proposed, by combining the
roughnesses are isotropic so that the previously obtained expressions are valid. As shown above, when a
surface has a long correlation length, most of the light
will be scattered at angles close to the specular direction
and thus will be lost through the specular exit hole in
the collecting hemisphere. Experimentally, it has been
found that for six diamond-turned samples tested the
light lost through the central aperture ranged from 63
to 97%of the total measured scattered light.14 Thus the
TIS measures primarily the short-range roughness of
these surfaces.
To show how the TIS from short- and long-range
roughnesses combine, we can model the surface with a
two-Gaussian autocovariance function. Comparisons
between theory and experiment have shown that generally two-part autocovariance functions are needed.1 5
We can write the two-Gaussian autocovariance function
as 15
G(T)= 6' exp(-T 2 /U2) + t2 exp(-- 2 /aj2).
(33a)
The parameters 6s, L, s, and L are the short-range
rms roughness, long-range rms roughness, short-range
correlation length, and long-range correlation length,
respectively. We note that the long-range parameters
direct scattered light in the near-specular direction,
whereas the short-range parameters yield wide-angle
scattering. The corresponding g(k) is
effects of small aiX with the restricted collection angles
g(k) = 7r sas exp(-k 2 aS/4) + a
of the hemisphere, the three original 10-A rms rough-
exp(-k
2
L/4)].
(33b)
ness surfaces would have roughnesses of 6.3, 8.8, and 0.5
A for o-values of 0.2, 2, and 10 gm, respectively, as cal-
In Eq. (33a) the long-range
culated from uncorrected measured values of TIS.
We carry the TIS and effective surface roughness
calculations one step further by considering a surface
somewhat similar to a diamond-turned surface which
contains both short-range and long-range roughness
components. Diamond-turned surfaces can have
short-range roughness caused by tool chatter, interactions between the chip and the surface, material im-
diamond tool. The short-range as is intended to include the effects of residual random roughness, which
typically has a much shorter correlation length. We
L
is intended to simulate
the effects of the long-range correlation produced by the
realize that a diamond machined surface has anisotropic
the form of grooves cut by the diamond tool. For this
surface topography and that this is contrary to Eqs. (33).
However, similar effects will be seen in TIS measurements from diamond turned surfaces as are predicted
from Eqs. (33). The important condition is that both
long and short range parameters are used, which is
consistent with the situation for diamond turned sur-
example, we will assume that both short- and long-range
faces.
perfections, etc., as well as the long-range roughness in
15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS
3215
W
1.4
r
In
1.2
a
<
1.0
cr
0
0.8
E 0.6
4-1 T
0a
0.4
0.2
z0
Fig. 7.
0
0
0.5
1.0
1.5
2.0
CORRELATION LENGTH/WAVELENGTH,a/X
TIS/TIS_ vs a/A for light normally incident on a 23-layer dielectric stack, with the scattered light collected from all angles between
0and 90°. The thin films are quarterwave optical thickness at normal incidence. The solid (dashed) curve is for a correlated (uncorrelated)
multilayer stack with Gaussian autocorrelation
functions assumed for the film interfaces.
TIS_ is for an opaque highly reflecting surface
with a >>A. For a/M > 2, the solid curve remains nearly constant at unity.
In the illustrative example, the 6s and
6
L
values are
chosen to be 33.6 and 47.5 A, respectively, which yields
6 = vAJsTWL = 58.2 A. These values are consistent
with experimentally measured quantities. We have
integrated Eq. (5d) over the scattering hemisphere while
limiting the collection angles to the 2.85-79.1° range,
as discussed above. The short-range correlation length
us = 0.35 gm is held constant, while aoLvaries from 0 to
16 gim. Since as is held constant, the TIS for this example will never vanish in small or large limits of aL
because there will always be a background (in this case,
a constant background) of scattered light from the
short-range roughness. The results of the calculations
are plotted in Fig. 6 as TIS/TIS_ vs CL/X. We can learn
several interesting points from Fig. 6. As the long-range
uL approaches the small and large limits, the TIS/TISvalues approach the residual value produced by the
short-range roughness background scattering. When
cYL/X- 0, the long-range roughness scattering ceases
where 62 = 6L + 6S. A plot of this equation for the same
parameters associated with Fig. 6 again yields a plot
which is very nearly identical to Fig. 6.
Thus it is seen that for surfaces having significant
long-range spatial wavelength components, there can
be large errors made in calculating the total rms
roughness because of scattered light lost through the
specular exit hole.
It should be emphasized that the discussion in this
section has been based on the assumption of a known
power spectral density function g(k). Thus the usefulness and validity of the formulas in this section depend on the degree to which g (k) for the unknown surface is Gaussian or Lorentzian. In reality, only a portion of g(k) or G(T) is known or can be measured, and
making assumptions about g(k) beyond the measurable
region can lead to errors. However, it is felt that the
methods outlined in this section can provide reasonable
estimates for corrections to the mean square roughness
to exist because when aL/X << 1 very little scattering can
occur within the hemisphere; also, that which does occur
is not collected because of the large angle limits. When
cJL/X >> 0, the long-range roughness is not collected
for most state-of-the-art optical components, excluding
because the scattered light escapes through the specular
The previous sections have provided background
material for scattering from a surface covered with a
single opaque reflecting coating. The interpretation
of the TIS from such a surface is much simpler than for
a surface covered with a multilayer dielectric stack.
There are two primary reasons for this: (1) thin films
exit hole. The maximum value of TIS/TIS.. occurs for
cJL/X= 0.83, which corresponds to CL = 0.53 gm. The
TIS/TIS., curve passes through a maximum when the
optimum L is reached so that losses through the
specular exit hole and beyond the large angle limits are
minimized.
We can obtain the results of Fig. 6 in another way by
using a two-Gaussian autocovariance function in an
equation analogous to Eq. (27a):
(t)2 = ()2
+
3216
[exp(-a 2 2L/4)- exp(_02ai/4)
2 [exp(-a 2a2/4) - exp(-2 2a /4),
(
APPLIED OPTICS/ Vol. 22, No. 20 / 15 October 1983
diamond-turned optics.
V.
TIS from Multilayer-Coated
Optics
with parallel rough boundaries can produce interference
effects in the scattered light and (2) the statistical relationships between the roughness at a given interface
relative to the other interfaces can significantly affect
the TIS. This latter effect is caused by roughnessinduced phase relationships in the light scattered from
each interface. The angle-resolved vector scattering
theory8 used to predict scattering from multilayer stacks
is not reproduced here. However, its validity is the
same as for the ARS theory presented in previous sections that applies to a single opaque metal coating. In
X
1.4
<
1.2
F
1.0
w
0.8
0
t
a:
0
I-
CI
-
- t
0
5-
0.4
tI
0R
J1
0 d
0.5
1.0
1.5
2.0
CORRELATIONLENGTH/WAVELENGTH,a/X
Fig. 8. TIS/TIS vs a/X for p-polarized light incident at 0o =
I
I
I
l
10
O
0o
30°
I
I
I
I
I
I
I
I
I
2.
3 E0 4.0
LENGTH/WAVELENGTH,
CORRELATION
on the same dielectric stack as in Fig. 7, except that the thin films have
a quarterwave optical thickness at 300 incidence. The solid (dashed)
curve is for a correlated (uncorrelated) multilayer stack with Gaussian
I
0.1
Z0
MI
I
I
E
o
I
- I
O.,
rzj
4
0.2
I
I
0.6
N
I
//
-
t
0
z
I
2-
U/X
Fig. 10. Same as Fig. 8 (p-polarized incident light), except that the
angle of incidence is 450, and the films are quarterwave optical
thickness at 450 incidence. For a/X > 4.5, the TIS/TIS_ curve for
the correlated case is nearly constant at cos 2 450 = 0.50.
autocorrelation functions assumed for the film interfaces. TIS, is
for an opaque highly reflecting surface at 00 = 00 and a >>X. For a/X
> 2, the TIS/TIS_ curve for the correlated case is nearly constant at
cos2 30' = 0.75.
.. .
1.4
4
I
I
I
I
I
I
I
I
I
1.2 -
X
r
0
1.4
4
_
(J
2 Sc0.8 -
1.0
a:
-
4
0
<
1.0
cc
1.2
In
ziG
0.6 0
Z g0.8
0.4
0
D
_
0.6
N
0
E
Or
-
<
n
0
0.2
0
Z
0.5
1.0
1.5
2.0
CORRELATION LENGTH/WAVELENGTH, X
Fig. 9.
0.2
2
0.4
L4
Fig. 11.
I
I
I
I
I
I
I I
I
1.0
2.0
3.0
4.0
CORRELATION LENGTH/WAVELENGTH.
Same as Fig. 10, except that the incident
polarized.
light is s-
Same as Fig. 8, except that the incident light is s-polarized.
this section, we will give examples of scattering from
surfaces covered with multilayer dielectric films and
compare them to previous results for scattering from a
surface covered with a single opaque metal layer.
In this section, we consider a 23-layer dielectric stack
of a SH(LH)1 1A design, where S, H, L, and A represent
the substrate (s = 2.25, 0.0), high index film (H = 5.29,
0.0), low index film (EL = 1.9, 0.0), and air (A = 1.0, 0.0),
respectively, at a wavelength X = 0.6328 gim. The op-
tical thicknesses of the films are assumed to be X/4 at
the angle at which the stack is illuminated. Each interface is assumed to be rough and to have the same
statistical properties of the roughness. However, two
limiting cases are considered:
(1) correlated roughness
and (2) uncorrelated roughness. In the case of correlated roughness, each interface in the stack is identical
in shape so that the roughnesses at all interfaces are
correlated. Naturally, each interface has the same
autocovariance function. Also, all interface pairs have
however, that all interfaces have the same autocovariance function (which is the same as that of the correlated case). However, because of the assumption of
independence between interfaces, all cross-correlation
functions vanish.
The TIS from dielectric stacks having correlated and
uncorrelated roughness is quite different, as we will illustrate by the following examples. In Fig. 7 we show
a plot similar to that in Fig. 3 except that the TIS is for
the multilayer stack (the optical thickness of the layers
is X/4 for normal incidence illumination), and TIS is
the scattering that would be obtained from a highly
reflecting opaque surface of the same roughness but
having a large correlation length/wavelength ratio.
Note that even though the illumination is at normal
incidence, the correlation properties of the surface
greatly affect the amount of TIS. The curve for the
correlated
case is very much like that in Fig. 3 for an
opaque highly reflecting surface, but the TIS for the
the same cross-correlation functions which are identical
uncorrelated
to the autocovariance function. In the case of uncorrelated roughness, the roughness shapes between different interfaces are independent. It is assumed,
Since the illumination is at normal incidence, there is
no difference between p-polarized and s-polarized incidence.
case is approximately a factor of 2 lower.
15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS
3217
xxxx
a
0
z~KzS
4LI
:f
2
2
<t
11
2
Z
Fig. 12. Plot of the electric field intensity for p-polarized (solid curves) and s-polarized (dashed curves) light incident on a 23-layer dielectric
stack. Angles of incidence are (a) O (b) 30, and (c) 45°. The optical thicknesses of the layers are quarterwave at the given angle of incidence.
At normal incidence there is no difference between s- and p-polarized incident light.
Figures 8-11 show curves similar to those in Figs. 4
and 5 for multilayer stacks illuminated at 30 and 450
angles of incidence, respectively. The optical thicknesses of the layers have been adjusted to be X/4 at the
appropriate illumination angle. Also TIS_ is again the
scattering that would be obtained from a highly reflecting opaque surface of the same roughness, with a
large a/X ratio, and illuminated at normal incidence.
With regard to Figs. 8-11, first consider the curves for
correlated roughness and s- and p-polarized incident
light (solid curves in Figs. 8-11). These curves are quite
similar to the corresponding curves in Figs. 4 and 5 in
that the long-correlation length limits for the TISMtIS_
ratio tend toward the ratios that would be calculated
from the cos20 value for the illumination angle:
0.75
(0.50) for 30° (45°) incident illumination. The interpretation is straightforward, since the roughness at each
interface is identical, and, therefore, the scattering
currents are correlated in phase throughout the multilayer stack. Also the layer pairs are halfwave optical
thickness. Thus considering the optical thicknesses
and phase relationships of the scattering currents, it
follows that for scattering reasonably near the specular
direction the scattered light will behave much like the
specular beam from a single opaque surface. Since
ratios oaI > 1 confine scattered light reasonably near
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APPLIED OPTICS/ Vol. 22, No. 20 / 15 October 1983
the specular beam, this explains why the solid curves of
Figs. 8-11 approach the corresponding curves of Figs.
4 and 5.
Next consider the curves for uncorrelated roughness
and s- and p-polarized incident light (dashed curves in
Figs. 8-11). In the case of uncorrelated roughness,
there is no correlated phase relationship of the roughness-induced scattering currents between interfaces.
This lack of correlation can produce significant differences in the ARS1 5,1 6 and, consequently, also the TIS.
Note that in Figs.7-11 for s-polarized incident light, the
TIS for the uncorrelated case generally decreases with
increasing angle of illumination. On the other hand,
for p-polarized incident light and uncorrelated roughness, the TIS generally increases with increasing angle
of illumination. This is somewhat easier to understand
by looking at the electric field distribution in the 23layer stack for angles of incidence of 0, 30, and 45°
shown in Fig. 12. In the case of s-polarized incident
light, the field intensity consistsof nodes and antinodes.
The intensity of the antinodes decreases with increasing
angle of incidence.
This is consistent with the decrease
in TIS, for s-polarized incident light, as the angle of
incidence increases. In other words, the dipole scattering currents generated by the fields at the antinodes
decrease with increasing angle of incidence. Further-
more, for uncorrelated stacks, each interface is an independent sheet of currents, and thus phase relationships due to roughness shape do not yield the interference effects seen in the correlated stacks. It follows that
when the electric fields decrease, the TIS will also decrease.
In the case of p-polarized incident light, there are
nodes and antinodes in the electric field for normal incidence. However, for the 30 and 450 cases, there are
no nodes because of the normal component of the incident field. The original antinodes actually decrease
very little when going from normal incidence to nonnormal incidence illumination. Thus, considering this
fact along with the disappearance of the nodes, the
overall distribution of the electric field in the stack increases with increasing angle of incidence for p-polarized incident light. This is consistent with the increase
of TIS for increasing angle of incidence with p-polarized
incident light and uncorrelated roughness.
VI.
have been given of surfaces having different roughness
correlation lengths to show the extent of the decrease
of their TIS from the ideal TIS. Also an example is
given of a surface that has both short-range and longrange correlation length roughnesses to simulate a diamond-turned surface. The results for this surface
clearly show that the scattering from the long-range
roughness component passes through the hole in the
collecting hemisphere, so that only the scattering from
the short-range roughness is actually collected.
Finally (Sec. V), the TIS has been calculated for a
23-layer multilayer stack assuming that the roughness
between adjacent layers in the stack is either correlated
or uncorrelated. Normal incidence, 300, and 450 incident illumination are considered. The magnitude of
the TIS is quite different depending on whether the
roughness between adjacent layers is correlated or uncorrelated. It is also shownthat the relative amount of
TIS for the correlated and uncorrelated cases depends
on the incident polarization.
Conclusions
We have shown that the previously published vector
equations 6 for ARS from an opaque reflecting surface
(normal incidence illumination) can be integrated over
all angles in the reflecting hemisphere to yield an expression for the TIS (Sec. II). When the correlation
length a of the surface roughness is long compared with
the illuminating wavelength X, the TIS expression is
identical to the previously published expression and
shows that TIS is directly proportional to the square of
the rms roughness 6 and inversely proportional to the
square of the wavelength. When a is small compared
to X, the TIS is directly proportional to the product 62a-2
and inversely proportional to 4. The ARS expression
has also been numerically integrated to yield TIS for
intermediate values of a relative to Xand is plotted in
Fig. 3.
The ARS expressions for non-normal incidence illumination have also been numerically integrated for in-
cident illumination angles of 30 and 450, respectively
(Sec. IIB), and the results are plotted in Figs. 4 and 5.
The curves show that when /X > 1, the TIS value is
identical to that predicted by the customary TIS
equation for non-normal incidence [Eq. (17)].
A proof based on the Kirchhoff diffraction integral
(Sec. III) shows that when 6 << X the TIS is independent
of the form of the function for the distribution of surface
heights. Specifically, the height distribution function
does not need to be Gaussian or even symmetric about
the mean surface level.
One type of instrument used to measure TIS (Fig. 1)
contains an aluminized hemisphere with a central hole
through which the normally incident light beam passes
and the specularly reflected beam exits. The hemisphere on one such instrument collects scattered light
for angles from 3 to 790 from the specular beam and
thus misses the very near angle and very large angle
scattered light. In Sec. IV the ideal TIS expressions
have been modified to take into account the limited
collection angles for the scattered light, and examples
The authors would like to thank the referees for
helpful comments and suggestions.
References
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