25 Optics of Metals

25
Optics of Metals

25.1

INTRODUCTION

We have been concerned with the propagation of light in nonconducting media. We
now turn our attention to describing the interaction of light with conducting materials,
namely, metals and semiconductors. Metals and semiconductors, absorbing media, are
crystalline aggregates consisting of small crystals of random orientation. Unlike true
crystals they do not have repetitive structures throughout their entire forms.
The phenomenon of conductivity is associated with the appearance of heat; it is
very often called Joule heat. It is a thermodynamically irreversible process in which
electromagnetic energy is transformed to heat. As a result, the optical field within a
conductor is attenuated. The very high conductivity exhibited by metals and semiconductors causes them to be practically opaque. The phenomenon of conduction
and strong absorption corresponds to high reflectivity so that metallic surfaces act as
excellent mirrors. In fact, up to the latter part of the nineteenth century most large
reflecting astronomical telescope mirrors were metallic. Eventually, metal mirrors
were replaced with parabolic glass surfaces overcoated with silver, a material with a
very high reflectivity. Unfortunately, silver oxidizes in a relatively short time with
oxygen and sulfur compounds in the atmosphere and turns black. Consequently,
silver-coated mirrors must be recoated nearly every other year or so, a difficult, timeconsuming, expensive process. This problem was finally solved by Strong in the
1930s with his method of evaporating aluminum on to the surface of optical glass.
In the following sections we shall not deal with the theory of metals. Rather, we
shall concentrate on the phenomenological description of the interaction of polarized
light with metallic surfaces. Therefore, in Section 25.2 we develop Maxwell’s equations for conducting media. We discover that for conducting media the refractive
index becomes complex and has the form n ¼ n(1Ài) where n is the real refractive
index and  is the extinction coefficient. Furthermore, Fresnel’s equations for reflection and transmission continue to be valid for conducting (absorbing) media.
However, because of the rapid attenuation of the optical field within an absorbing
medium, Fresnel’s equations for transmission are inapplicable. Using the complex

refractive index, we develop Fresnel’s equations for reflection at normal incidence

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


and describe them in terms of a quantity called the reflectivity. It is possible to
develop Fresnel’s reflection equations for non-normal incidence. However, the
forms are very complicated and so approximate forms are derived for the s and p
polarizations. It is rather remarkable that the phenomenon of conductivity may be
taken into account simply by introducing a complex index of refraction. A complete
understanding of the significance of n and  can only be understood on the basis of
the dispersion theory of metals. However, experience does show that large values of
reflectivity correspond to large values of .
In Sections 25.3 and 25.4 we discuss the measurement of the optical constants n
and . A number of methods have been developed over the past 100 years, nearly all
of which are null-intensity methods. That is, n and  are obtained from the null
condition on the reflected intensity. The best-known null method is the principle
angle of incidence/principle azimuthal angle method (Section 25.3). In this method
a beam of light is incident on the sample and the incidence angle is varied until an
incidence angle is reached where a phase shift of /2 occurs. The incidence angle
where this takes place is known as the principle angle of incidence. An additional
phase shift of /2 is now introduced into the reflected light with a quarter-wave
retarder. The condition of the principal angle of incidence and the quarter-wave
shift and the introduction of the quarter-wave retarder, as we shall see, creates
linearly polarized light. Analyzing the phase-shifted reflected light with a polarizer
that is rotated around its azimuthal angle leads to a null intensity (at the principal
azimuthal angle) from which n and  can be determined.
Classical null methods were developed long before the advent of quantitative
detectors, digital voltmeters, and digital computers. Nulling methods are very valuable, but they have a serious drawback: the method requires a mechanical arm that
must be rotated along with the azimuthal rotation of a Babinet–Soleil compensator

and analyzer until a null intensity is found. In addition, a mechanical arm that yields
scientifically useful readings is quite expensive. It is possible to overcome these
drawbacks by reconsidering Fresnel’s equations for reflection at an incidence
angle of 45 . It is well known that Fresnel’s equations for reflection simplify at
normal incidence and at the Brewster angle for nonabsorbing (dielectric) materials.
Less well known is that Fresnel’s equations also simplify at an incidence angle of 45 .
All of these simplifications were discussed in Chapter 8 assuming dielectric media.
The simplifications at the incidence angle of 45 hold even for absorbing media.
Therefore, in Section 25.4 we describe the measurement of an optically absorbing
surface at an incidence angle of 45 . This method, called digital refractometry, overcomes the nulling problems and leads to equations to determine n and  that can be
solved on a digital computer by iteration.
25.2

MAXWELL’S EQUATIONS FOR ABSORBING MEDIA

We now solve Maxwell’s equations for a homogeneous isotropic medium described
by a dielectric constant ", a permeability , and a conductivity . Using material
equations (also called the constitutive relations):
D ¼ "E
B ¼ H
j ¼ E

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ð25-1aÞ
ð25-1bÞ
ð25-1cÞ


Maxwell’s equations become, in MKSA units,

@E
¼ E
@t
@H
=ÂEþ
¼0
@t

=ÁE¼
"
=ÁH¼0
=ÂHÀ"

ð25-2aÞ
ð25-2bÞ
ð25-2cÞ
ð25-2dÞ

These equations describe the propagation of the optical field within and at the
boundary of a conducting medium. To find the equation for the propagation of
the field E we eliminate H between (25-2a) and (25-2b). We take the curl of
(25-2b) and substitute (25-2a) into the resulting equation to obtain
= Â ð= Â EÞ þ ð"Þ

@2 E
@E
¼0
þ 
2
@t

@t

ð25-3Þ

Expanding the = Â ð=ÂÞ operator, we find that (25-3) becomes
=2 E ¼ "

@2 E
@E
þ 
@t
@t2

ð25-4Þ

Equation (25-4) is the familiar wave equation modified by an additional term. From
our knowledge of differential equations the additional term described by @E=@t
corresponds to damping or attenuation of a wave. Thus, (25-4) can be considered
the damped or attenuated wave equation.
We proceed now with the solution of (25-4). If the field is strictly monochromatic and of angular frequency ! so that E  Eðr, tÞ ¼ EðrÞ expði!tÞ, then substituting this form into (25-4) yields
=2 EðrÞ ¼ ðÀ"!2 ÞEðrÞ þ ði!ÞEðrÞ
which can be written as
h
  i
=2 EðrÞ ¼ ðÀ!2 Þ " À i
EðrÞ
!

ð25-5Þ

ð25-6Þ

In this form, (25-6) is identical to the wave equation except that now the dielectric
constant is complex; thus,

"¼"Ài
ð25-7Þ
!
where " is the real dielectric constant.
The correspondence with nonconducting media is readily seen if " is defined in
terms of a complex refractive index n (we set  ¼ 1 since we are not dealing with
magnetic materials):
" ¼ n2

ð25-8Þ

We now express n in terms of the refractive index and the absorption of the medium.
To find the form of n which describes both the refractive and absorbing behavior
of a propagating field, we first consider the intensity I(z) of the field after it has

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


propagated a distance z. We know that the intensity is attenuated after a distance z
has been traveled, so the intensity can be described by
IðzÞ ¼ I0 expðÀzÞ

ð25-9Þ

where  is the attenuation or absorption coefficient. We wish to relate  to , the

extinction coefficient or attenuation index. We first note that n is a dimensionless
quantity, whereas from (25-9)  has the dimensions of inverse length. We can express
z as a dimensionless parameter by assuming that after a distance equal to a wavelength  the intensity has been reduced to
IðÞ ¼ I0 expðÀ4Þ
Equating the arguments of the exponents in (25-9) and (25-10), we have
 
4
¼
 ¼ 2k


ð25-10Þ

ð25-11Þ

where k ¼ 2/ is the wavenumber. Equation (25-9) can then be written as
  !
4
IðzÞ ¼ I0 exp À
z
ð25-12Þ

From this result we can write the corresponding field E(z) as
  !
2
EðzÞ ¼ E0 exp À
z


ð25-13Þ

or
EðzÞ ¼ E0 expðÀkzÞ

ð25-14Þ

Thus, the field propagating in the z direction can be described by
EðzÞ ¼ E0 expðÀkzÞ exp½ið!t À kzފ
The argument of (25-15) can be written as
 
  !
k
k
i! t À
zþi
z
!
!
!
k
¼ i! t À f1 À igz
!
But k ¼ !/v ¼ !n/c, so (25-16b) becomes
h
i
n
i! t À f1 À igz
c
h
 n i

¼ i! t À
z
c

ð25-15Þ

ð25-16aÞ
ð25-16bÞ

ð25-17aÞ
ð25-17bÞ

where
n ¼ nð1 À iÞ
Thus, the propagating field (25-15) can be written in the form:
h 
 n  i
EðzÞ ¼ E0 exp i! t À
z
c

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ð25-18Þ

ð25-19Þ


Equation (25-19) shows that conducting (i.e., absorbing) media lead to the same
solutions as nonconducting media except that the real refractive index n is replaced

by a complex refractive index n. Equation (25-18) relates the complex refractive
index to the real refractive index and the absorption behavior of the medium and
will be used throughout the text.
From (25-7), (25-8), and (25-18) we can relate n and  to . We have

" ¼ n2 ¼ n2 ð1 À iÞ2 ¼ " À i
ð25-20Þ
!
which leads immediately to
n2 ð1À2 Þ ¼"

ð25-21aÞ



¼
n2  ¼
2! 4

ð25-21bÞ

where  ¼ !=2
We solve these equations to obtain
"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#
  2
1
n2 ¼
þ"
"2 þ

2
4
"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#
  2
1
2 2
2
À"
n  ¼
" þ
2
4

ð25-22aÞ

ð25-22bÞ

Equation (25-22) is important because it enables us to relate the (measured) values of
n and  to the constants " and  of a metal or semiconductor. Because metals are
opaque, it is not possible to measure these constants optically.
Since the wave equation for conducting media is identical to the wave equation
for dielectrics, except for the appearance of a complex refractive index, we would
expect the boundary conditions and all of its consequences to remain unchanged.
This is indeed the case. Thus, Snell’s law of refraction becomes
sin i ¼ n sin r

ð25-23Þ

where the refractive index is now complex. Similarly, Fresnel’s law of reflection and

refraction continue to be valid. Since optical measurements cannot be made with
Fresnel’s refraction equations, only Fresnel’s reflection equations are of practical
interest. We recall these equations are given by
Rs ¼ À

Rp ¼

sinði À r Þ
E
sinði þ r Þ s

tanði À r Þ
E
tanði þ r Þ p

ð25-24aÞ

ð25-24bÞ

In (25-24) i is the angle of incidence and r is the angle of refraction, and Rs, Rp, Es,
and Ep have their usual meanings.
We now derive the equations for the reflected intensity, using (25-24).
We consider (25-24a) first. We expand the trigonometric sum and difference terms,

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


substitute sin r ¼ n sin r into the result, and find that
!
Rs

cos i À n cos r
¼
Es
cos i þ n cos r

ð25-25Þ

We first use (25-25) to obtain the reflectivity, that is, the normalized intensity at
normal incidence. The reflectivity for the s polarization, (25-25) is defined to be
 2
R 
Rs   s 
ð25-26Þ
Es
At normal incidence i ¼ 0, so from Snell’s law, (25-23), r ¼ 0 and (25-25) reduces to
!
Rs
1Àn
¼
ð25-27Þ
1þn
Es
Replacing n with the explicit form given by (25-18) yields
!
Rs
ð1 À nÞ þ in
¼
ð1 þ nÞ À in
Es

ð25-28Þ

From the definition of the reflectivity (25-26) we then see that (25-28) yields
"
#
ðn À 1Þ2 þ ðnÞ2
Rs ¼
ð25-29Þ
ðn þ 1Þ2 þ ðnÞ2
We observe that for nonabsorbing media ( ¼ 0), (25-29) reduces to the well-known
results for dielectrics. We also note that for this condition and for n ¼ 1 the reflectivity is zero, as we would expect. In Fig. 25-1 a plot of (25-29) as a function of 

Figure 25-1

Plot of the reflectivity as a function of . The refractive indices are n ¼ 1.0, 1.5,
and 2.0, respectively.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


is shown. We see that for absorbing media with increasing  the reflectivity
approaches unity. Thus, highly reflecting absorbing media (e.g., metals) are characterized by high values of .
In a similar manner we can find the reflectivity for normal incidence for the p
polarization, (25-24b). Equation (25-24b) can be written as
Rp sinði À r Þ cosði þ r Þ
¼
Ep sinði þ r Þ cosði À r Þ

ð25-30Þ

At normal incidence the cosine factor in (25-30) is unity, and we are left with the
same equation for the s polarization, (25-24a). Hence,
Rp ¼ Rs

ð25-31Þ

and for normal incidence the reflectivity is the same for the s and p polarizations.
We now derive the reflectivity equations for non-normal incidence. We again
begin with (25-24a) or, more conveniently, its expanded form, (25-25)
!
Rs
cos i À n cos r
¼
ð25-25Þ
Es
cos i þ n cos r
Equation (25-25) is, of course, exact and can be used to obtain an exact expression
for the reflectivity Rs . However, the result is quite complicated. Therefore, we derive
an approximate equation, much quoted in the literature, for Rs which is sufficiently
close to the exact result. We replace the factor cos r by ð1 À sin2 r Þ1=2 and use
sin i ¼ n sin r. Then, (25-25) becomes
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
2
2
Rs 6cos i À n À sin i 7
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5
¼4
ð25-32Þ
Es

cos i þ n2 À sin2 i
Equation (25-32) can be approximated by noting that n2 ) sin2i, so (25-32) can be
written as
Rs cos i À n
¼
Es cos i þ n

ð25-33Þ

We now substitute (25-18) into (25-33) and group the terms into real and imaginary
parts:
!
Rs
ðcos i À nÞ þ in
¼
ð25-34Þ
ðcos i þ nÞ À in
Ep
The reflectivity Rs is then
"
#
ðn À cos i Þ2 þ ðnÞ2
Rs ¼
ðn þ cos i Þ2 þ ðnÞ2

ð25-35Þ

We now develop a similar, approximate, equation for Rp . We first write
(25-24b) as
Rs sinði À r Þ cosði þ r Þ

¼
Ep sinði þ r Þ cosði À r Þ

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð25-30Þ


The first factor is identical to (25-24a), so it can be replaced by its expanded form
(25-25):
!
sinði À r Þ
cos i À n cos r
¼
ð25-36Þ
sinði þ r Þ
cos i þ n cos r
The second factor in (25-30) is now expanded, and again we use cos r ¼
ð1 À sin2 r Þ1=2 and sin i ¼ n sin r:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
cosði þ r Þ ðcos i Þ n À sin i À sin i
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
ð25-37Þ
cosði À r Þ
ðcos i Þ n2 À sin2 i þ sin2 i
Because n2 ) sin2i, (25-37) can approximated as

cosði þ r Þ n cos i À sin2 i
ffi
cosði À r Þ n cos i þ sin2 i

ð25-38Þ

We now multiply (25-36) by (25-38) to obtain
!


Rp
cos i À n n cos i À sin2 i
¼
cos i þ n n cos i þ sin2 i
Ep

ð25-39Þ

Carrying out the multiplication in (25-39), we find that there is a sin2 i cos i term.
This term is always much smaller than the remaining terms and can be dropped. The
remaining terms then lead to
Rp Àn cos i þ 1
ð25-40aÞ
¼
n cos i þ 1
Ep
or
Rp
cos i À 1=n
¼À

cos i þ 1=n
Ep

ð25-40bÞ

Replacing n by n(1 À i), grouping terms into real and imaginary parts, and ignoring
the negative sign because it will vanish when we determine the reflectivity, gives
Rp ðn À 1= cos i Þ À in
ð25-41Þ
¼
Ep ðn þ 1= cos i Þ þ in
Multiplying (25-41) by its complex conjugate, we obtain the reflectivity Rp :
Rp ¼

ðn À 1= cos i Þ2 þ ðnÞ2
ðn þ 1= cos i Þ2 þ ðnÞ2

For convenience we write the equation for Rs , (25-35), here also:
"
#
ðn À cos i Þ2 þ ðnÞ2
Rs ¼
ðn þ cos i Þ2 þ ðnÞ2

ð25-42Þ

ð25-35Þ

In Figs. 25-2 through 25-5 plots are shown for the reflectivity as a function of
the incidence angle i of gold (Au), silver (Ag), copper (Cu), and platinum (Pt), using

(25-35) and (25-39). The values for n and  are taken from Wood’s classic text
Physical Optics.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Figure 25-2 Reflectance of gold (Au) as a function of incidence angle. The refractive index
and the extinction coefficient are 0.36 and 7.70 respectively. The normal reflectance value is
0.849.

Figure 25-3 Reflectance of silver (Ag) as a function of incidence angle. The refractive index
and the extinction coefficient are 0.18 and 20.2, respectively. The normal reflectance value is
0.951.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Figure 25-4

Reflectance of copper (Cu) as a function of incidence angle. The refractive
index and the extinction coefficient are 0.64 and 4.08, respectively. The normal reflectance
value is 0.731.

Figure 25-5

Reflectance of platinum (Pt) as a function of incidence angle. The refractive
index and the extinction coefficient are 2.06 and 2.06, respectively. The normal reflectance
value is 0.699.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

In Figs. 25-2 through 25-5 we observe that the p reflectivity has a minimum
value. This minimum is called the pseudo-Brewster angle minimum because, unlike
the Brewster angle for dielectrics, the intensity does not go to zero for metals.
Nevertheless, a technique based on this minimum has been used to determine n
and . The interested reader is referred to the article by Potter.
Finally, we see that the refractive index can be less than unity for many metals.
Born and Wolf have shown that this is a natural consequence of the simple classical
theory of the electron and the dispersion theory. The theory provides a theoretical
basis for the behavior of n and . Further details on the nature of metals and, in
particular, the refractive index and the extinction coefficient (n and ) as it appears in
the dispersion theory of metals can be found in the reference texts by Born and Wolf
and by Mott and Jones.

25.3

PRINCIPAL ANGLE OF INCIDENCE MEASUREMENT OF
REFRACTIVE INDEX AND EXTINCTION COEFFICIENT OF
OPTICALLY ABSORBING MATERIALS

In the previous section we saw that optically absorbing materials are characterized
by a real refractive index n and an extinction coefficient . Because these constants
describe the behavior and performance of optical materials such as metals and
semiconductors, it is very important to know these ‘‘constants’’ over the entire
optical spectrum.
Methods have been developed to measure the optical constants. One of the best
known is the principal angle of incidence method. The basic idea is as follows.
Incident þ45 linearly polarized light is reflected from an optically absorbing material. In general, the reflected light is elliptically polarized; the corresponding polarization ellipse is in nonstandard form. The angle of incidence of the incident beam is
now changed until a phase shift of 90 is observed in the reflected beam. The incident

angle where this takes place is called the principal angle of incidence. Its significance is
that, at this angle, the polarization ellipse for the reflected beam is now in its standard form. From this condition relatively simple equations can then be found for n
and . Because the polarization ellipse is now in its standard form, the orthogonal
field components are parallel and perpendicular to the plane of incidence. The
reflected beam is now passed through a quarter-wave retarder. The beam of light
that emerges is linearly polarized with its azimuth angle at an unknown angle. The
beam then passes through an analyzing polarizer that is rotated until a null intensity
is found. The angle at which this null takes place is called the principal azimuth angle.
From the measurement of the principal angle of incidence and the principal azimuth
angle the optical constants n and  can then be determined. In Fig. 25-6 we show the
measurement configuration.
To derive the equations for n and , we begin with Fresnel’s reflection equations for absorbing media:
Rs ¼ À
Rp ¼

sinði À r Þ
E
sinði þ r Þ s

tanði À r Þ
E
tanði þ r Þ p

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð25-24aÞ
ð25-24bÞ


Figure 25-6

Measurement of the principal angle of incidence and the principal azimuth

angle.

The angle r is now complex, so the ratios Rp/Ep and Rs/Es are also complex. Thus,
the amplitude and phase change on being reflected from optically absorbing media.
Incident polarized light will, in general, become elliptically polarized on reflection
from an optically absorbing medium. We now let p and s be the phase changes
and p and s the absolute values of the reflection coefficients rp and rs. Then, we can
write
rp ¼

Rp
¼ p expðip Þ
Ep

ð25-43aÞ

rs ¼

Rs
¼ s expðis Þ
Es

ð25-43bÞ

Equation (25-43) can be transformed to the Stokes parameters. The Stokes parameters for the incident beam are
S0 ¼ cos i ðEs Esà þ Ep Epà Þ

ð25-44aÞ

S1 ¼ cos i ðEs Esà À Ep Epà Þ

ð25-44bÞ

S2 ¼ cos i ðEs Epà þ Ep Esà Þ

ð25-44cÞ

S3 ¼ i cos i ðEs Epà À Ep Esà Þ

ð25-44dÞ

Similarly, the Stokes parameters for the reflected beam are defined as
S00 ¼ cos i ðRs RÃs þ Rp RÃp Þ

ð25-45aÞ

S01 ¼ cos i ðRs RÃs À Rp RÃp Þ

ð25-45bÞ

S02 ¼ cos i ðRs RÃp þ Rp RÃs Þ

ð25-45cÞ

S03 ¼ i cos i ðRs RÃp À Rp RÃs Þ

ð25-45dÞ

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Substituting (25-43) into (25-45) and using (25-44) yields
0 2
10 1
0 01
s þ 2p 2s À 2p
0
0
S0
S0
CB S C
2
2
2
2
B S01 C 1 B
À


þ

0
0
B
C

1

p
s
p
B 0 C¼ B s
CB C
@S A 2@ 0
0
2s p cos Á 2s p sin Á A@ S2 A
2
S3
S03
0
0
À2s p sin Á 2s p cos Á

ð25-46Þ

where Á ¼ s À p.
We now allow the incident light to be þ45o linearly polarized so that Ep ¼ Es.
Furthermore, we introduce an azimuthal angle  (generally complex) for the
reflected light, which is defined by
tan  ¼

Rs
cosði À r Þ
¼ P expðiÁÞ
¼À
cosði þ r Þ
Rp

ð25-47Þ

where we have used (25-24), and P is real and we write it as
P ¼ tan
where

ð25-48aÞ

is called the azimuthal angle. From (25-43) we also see that

P¼

s
p

Á ¼ s À p

ð25-48bÞ

We note that  is real in the following two cases:
1. For normal incidence (i ¼ 0); then from (25-47) we see that P ¼ 1 and Á ¼ .
2. For grazing incidence (i ¼ /2); then from (25-47) we see that P ¼ 1 and
Á ¼ 0.
Between these two extreme values there exists an angle "i called the principal
angle of incidence for which Á ¼ /2. Let us now see the consequences of obtaining
this condition. We first write (25-48b) as
s ¼ Pp

ð25-49Þ

Substituting (25-49) into (25-46), we obtain the Stokes vector of the reflected light
to be
À
Á
0 01
0
10 1
1 þ P2
À 1 À P2
S0
0
0
S0
Á
À
Á
B 0C
B
C
C
2B À
B S1 C p B À 1 À P2
CB S1 C
1 þ P2
0
0
B C¼ B
CB C
B S0 C
CB C

2B
@ 2A
@
0
0
2P cos Á 2P sin Á A@ S2 A
S3
S03
0
0
À2P sin Á 2P cos Á
ð25-50Þ
For incident þ45 linearly polarized light, the Stokes vector is
0 1
0 1
S0
1
B S1 C
B0C
B C ¼ I0 B C
@ S2 A
@1A
0
S3

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð25-51Þ

Substituting (25-51) into (25-50), we find the Stokes vector of the reflected light to be
1
0
0 01
S0
1 þ P2
Á
B S0 C 2 I B À
2 C
p 0BÀ 1 À P C
B 1C
ð25-52Þ
¼
C
B
B 0C
@ S2 A
2 @ 2P cos Á A
S03
À2P sin Á
The ellipticity angle  is
 


1 À1 S03
1 À1 À2P sin Á
 ¼ sin
¼ sin
2
2

S00
1 þ P2
Similarly, the orientation angle


1
À2P cos Á
¼ tanÀ1
2
1 À P2

ð25-53aÞ

is
ð25-53bÞ

We see that  is greatest when Á ¼ /2 but then ¼ 0; i.e., the polarization ellipse
corresponding to (25-52) is in its standard, nonrotated, form.
For Á ¼ /2 the Stokes vector, (25-52), becomes
0
1
0 01
S0
1 þ P2
ÁC
B À
B 0C
B S1 C 2p I0 B À 1 À P2 C
B
C

B C¼
ð25-54Þ
C
B S0 C
2 B
0
@
A
@ 2A
S03
À2P
and  and

of the polarization ellipse corresponding to (25-54) are,
 


1 À1 S03
1 À1 À2P
 ¼ sin
¼ sin
2
2
S00
1 þ P2
 0
1
À1 S2
¼ tan
¼0

2
S01

ð25-55aÞ

ð25-55bÞ

We must now transform the elliptically polarized light described by the Stokes
vector (25-54) to linearly polarized light. A quarter-wave retarder can be used to
transform elliptically polarized light to linearly polarized light. The Mueller matrix
for a quarter-wave retarder oriented at 0 is
0
1
1 0 0 0
B0 1 0 0 C
C
M¼B
ð25-56Þ
@ 0 0 0 À1 A
0 0 1 0
Multiplying (25-54) by (25-56) yields
0
1
0 01
S0
1 þ P2
ÁC
B À
B 0C
B S1 C 2p I0 B À 1 À P2 C

B
C
B C¼
C
B S0 C
2 B
2P
@
A
@ 2A
S03
0

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð25-57Þ


which is the Stokes vector for linearly polarized light. The Mueller matrix for a linear
polarizer at an angle
is
0
1
1
cos 2

sin 2

0
B

C
cos2 2

sin 2
cos 2
0 C
1 B cos 2

C
M¼ B
ð25-58Þ
2B
sin2 2

0C
@ sin 2
sin 2
cos 2