25 Optics of Metals

We discover that for conducting media the refractiveindex becomes complex and has the form n ¼ n1i where n is the real refractiveindex and is the extinction coefficient.. Using the comple

The phenomenon of conductivity is associated with the appearance of heat; it isvery often called Joule heat It is a thermodynamically irreversible process in whichelectromagnetic energy is transformed to heat As a result, the optical field within aconductor is attenuated The very high conductivity exhibited by metals and semi-conductors causes them to be practically opaque The phenomenon of conductionand strong absorption corresponds to high reflectivity so that metallic surfaces act asexcellent mirrors In fact, up to the latter part of the nineteenth century most largereflecting astronomical telescope mirrors were metallic Eventually, metal mirrorswere replaced with parabolic glass surfaces overcoated with silver, a material with avery high reflectivity Unfortunately, silver oxidizes in a relatively short time withoxygen and sulfur compounds in the atmosphere and turns black Consequently,silver-coated mirrors must be recoated nearly every other year or so, a difficult, time-consuming, expensive process This problem was finally solved by Strong in the1930s 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, weshall concentrate on the phenomenological description of the interaction of polarizedlight with metallic surfaces Therefore, in Section 25.2 we develop Maxwell’s equa-tions for conducting media We discover that for conducting media the refractiveindex becomes complex and has the form n ¼ n(1i) where n is the real refractiveindex and is the extinction coefficient Furthermore, Fresnel’s equations for reflec-tion and transmission continue to be valid for conducting (absorbing) media.However, because of the rapid attenuation of the optical field within an absorbingmedium, Fresnel’s equations for transmission are inapplicable Using the complexrefractive index, we develop Fresnel’s equations for reflection at normal incidence

and describe them in terms of a quantity called the reflectivity It is possible todevelop Fresnel’s reflection equations for non-normal incidence However, theforms are very complicated and so approximate forms are derived for the s and ppolarizations It is rather remarkable that the phenomenon of conductivity may betaken into account simply by introducing a complex index of refraction A completeunderstanding of the significance of n and can only be understood on the basis ofthe dispersion theory of metals However, experience does show that large values ofreflectivity correspond to large values of

In Sections 25.3 and 25.4 we discuss the measurement of the optical constants nand 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 nullcondition on the reflected intensity The best-known null method is the principleangle 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 anincidence angle is reached where a phase shift of /2 occurs The incidence anglewhere this takes place is known as the principle angle of incidence An additionalphase shift of /2 is now introduced into the reflected light with a quarter-waveretarder The condition of the principal angle of incidence and the quarter-waveshift and the introduction of the quarter-wave retarder, as we shall see, createslinearly polarized light Analyzing the phase-shifted reflected light with a polarizerthat is rotated around its azimuthal angle leads to a null intensity (at the principalazimuthal angle) from which n and can be determined

Classical null methods were developed long before the advent of quantitativedetectors, digital voltmeters, and digital computers Nulling methods are very valu-able, but they have a serious drawback: the method requires a mechanical arm thatmust be rotated along with the azimuthal rotation of a Babinet–Soleil compensatorand analyzer until a null intensity is found In addition, a mechanical arm that yieldsscientifically useful readings is quite expensive It is possible to overcome thesedrawbacks by reconsidering Fresnel’s equations for reflection at an incidenceangle of 45
It is well known that Fresnel’s equations for reflection simplify atnormal 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 absorbingsurface at an incidence angle of 45
This method, called digital refractometry, over-comes the nulling problems and leads to equations to determine n and that can besolved on a digital computer by iteration

We now solve Maxwell’s equations for a homogeneous isotropic medium described

by a dielectric constant ", a permeability , and a conductivity Using materialequations (also called the constitutive relations):

Maxwell’s equations become, in MKSA units,

We proceed now with the solution of (25-4) If the field is strictly matic and of angular frequency ! so that E  Eðr, tÞ ¼ EðrÞ expði!tÞ, then substitut-ing this form into (25-4) yields

which can be written as

The correspondence with nonconducting media is readily seen if " is defined interms of a complex refractive index n (we set  ¼ 1 since we are not dealing withmagnetic materials):

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

propagated a distance z We know that the intensity is attenuated after a distance zhas been traveled, so the intensity can be described by

extinction coefficient or attenuation index We first note that n is a dimensionless

as a dimensionless parameter by assuming that after a distance equal to a

ð25-12ÞFrom this result we can write the corresponding field E(z) as

EðzÞ ¼ E0exp  2

z

ð25-13Þor

Thus, the field propagating in the z direction can be described by

The argument of (25-15) can be written as

ð25-17bÞwhere

ð25-19Þ

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

by a complex refractive index n Equation (25-18) relates the complex refractiveindex to the real refractive index and the absorption behavior of the medium andwill be used throughout the text

From (25-7), (25-8), and (25-18) we can relate n and to We have

" ¼ n2¼n2ð1  iÞ2¼"  i

!

ð25-20Þwhich leads immediately to

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 areopaque, it is not possible to measure these constants optically

Since the wave equation for conducting media is identical to the wave equationfor dielectrics, except for the appearance of a complex refractive index, we wouldexpect the boundary conditions and all of its consequences to remain unchanged.This is indeed the case Thus, Snell’s law of refraction becomes

where the refractive index is now complex Similarly, Fresnel’s law of reflection andrefraction continue to be valid Since optical measurements cannot be made withFresnel’s refraction equations, only Fresnel’s reflection equations are of practicalinterest We recall these equations are given by

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,

substitute sin
r¼n sin
rinto the result, and find that

Rs

Es¼

cos
in cos
rcos
iþn cos
r

Rs

Es¼

ð1  nÞ þ inð1 þ nÞ  in

ð25-28ÞFrom the definition of the reflectivity (25-26) we then see that (25-28) yields

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

is shown We see that for absorbing media with increasing the reflectivityapproaches unity Thus, highly reflecting absorbing media (e.g., metals) are charac-terized by high values of

In a similar manner we can find the reflectivity for normal incidence for the ppolarization, (25-24b) Equation (25-24b) can be written as

Rp

Ep¼

sinð
i
rÞsinð
iþ
rÞ

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 againbegin with (25-24a) or, more conveniently, its expanded form, (25-25)

Rs

Es¼

cos
in cos
rcos
iþn cos
r

ð25-25ÞEquation (25-25) is, of course, exact and can be used to obtain an exact expressionfor the reflectivity Rs However, the result is quite complicated Therefore, we derive

an approximate equation, much quoted in the literature, for Rs which is sufficientlyclose to the exact result We replace the factor cos
r by ð1  sin2
rÞ1=2 and usesin
i¼n sin
r Then, (25-25) becomes

cos
iþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2sin2
iq

26

ð25-34ÞThe reflectivity Rs is then

cosð
iþ
rÞ

The first factor is identical to (25-24a), so it can be replaced by its expanded form(25-25):

sinð
i
rÞsinð
iþ
rÞ¼

cos
in cos
rcos
iþn cos
r

Figure 25-2 Reflectance of gold (Au) as a function of incidence angle The refractive indexand the extinction coefficient are 0.36 and 7.70 respectively The normal reflectance value is0.849.

Figure 25-3 Reflectance of silver (Ag) as a function of incidence angle The refractive indexand the extinction coefficient are 0.18 and 20.2, respectively The normal reflectance value is0.951

Figure 25-4 Reflectance of copper (Cu) as a function of incidence angle The refractiveindex and the extinction coefficient are 0.64 and 4.08, respectively The normal reflectancevalue is 0.731.

Figure 25-5 Reflectance of platinum (Pt) as a function of incidence angle The refractiveindex and the extinction coefficient are 2.06 and 2.06, respectively The normal reflectancevalue is 0.699

InFigs 25-2 through 25-5 we observe that the p reflectivity has a minimumvalue This minimum is called the pseudo-Brewster angle minimum because, unlikethe 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 nand 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 classicaltheory of the electron and the dispersion theory The theory provides a theoreticalbasis for the behavior of n and Further details on the nature of metals and, inparticular, the refractive index and the extinction coefficient (n and ) as it appears inthe dispersion theory of metals can be found in the reference texts by Born and Wolfand by Mott and Jones

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 constantsdescribe the behavior and performance of optical materials such as metals andsemiconductors, it is very important to know these ‘‘constants’’ over the entireoptical spectrum

Methods have been developed to measure the optical constants One of the bestknown is the principal angle of incidence method The basic idea is as follows.Incident þ45
linearly polarized light is reflected from an optically absorbing mate-rial In general, the reflected light is elliptically polarized; the corresponding polar-ization ellipse is in nonstandard form The angle of incidence of the incident beam isnow changed until a phase shift of 90
is observed in the reflected beam The incidentangle where this takes place is called the principal angle of incidence Its significance isthat, at this angle, the polarization ellipse for the reflected beam is now in its stan-dard form From this condition relatively simple equations can then be found for nand Because the polarization ellipse is now in its standard form, the orthogonalfield components are parallel and perpendicular to the plane of incidence Thereflected beam is now passed through a quarter-wave retarder The beam of lightthat emerges is linearly polarized with its azimuth angle at an unknown angle Thebeam 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 azimuthangle the optical constants n and can then be determined InFig 25-6we show themeasurement configuration

To derive the equations for n and , we begin with Fresnel’s reflection tions for absorbing media:

equa-Rs¼ sinð
i
rÞ

Rp¼tanð
i
rÞ

The angle
ris now complex, so the ratios Rp/Epand Rs/Esare 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 reflectionfrom an optically absorbing medium We now let pand s be the phase changesand pand sthe absolute values of the reflection coefficients rpand rs Then, we canwrite

Substituting (25-43) into (25-45) and using (25-44) yields

12

1CC

@

1C

where
¼ sp

We now allow the incident light to be þ45olinearly polarized so that Ep¼Es

reflected light, which is defined by

1 For normal incidence (
i¼0); then from (25-47) we see that P ¼ 1 and
¼ 

2 For grazing incidence (
i ¼/2); then from (25-47) we see that P ¼ 1 and

¼ 0

Between these two extreme values there exists an angle 

ii called the principalangle of incidencefor which
¼ /2 Let us now see the consequences of obtainingthis condition We first write (25-48b) as

2p2

@

1CCA

@

1CCAð25-50ÞFor incident þ45
linearly polarized light, the Stokes vector is

A¼I0

1010

0B

@

1C

Substituting (25-51) into (25-50), we find the Stokes vector of the reflected light to be

1C

C¼2pI02

1 þ P2

 1  P22P cos

2P sin

0BB

1C

@

1CC

2pI02

1 þ P2

 1  P20

2P

0BB

@

1CC



We must now transform the elliptically polarized light described by the Stokesvector (25-54) to linearly polarized light A quarter-wave retarder can be used totransform elliptically polarized light to linearly polarized light The Mueller matrixfor a quarter-wave retarder oriented at 0
is

@

1C

@

1CC

2pI02

1 þ P2

 1  P22P0

0BB

@

1CC

which is the Stokes vector for linearly polarized light The Mueller matrix for a linearpolarizer at an angle  is

@

1CC

so that (25-61a) can now be written as

We can relate (25-63b) to the principal azimuthal angle  as follows We recall from(25-48a) that

@

1C

A¼I0

1010

0B

@

1C

@

1C

A¼I0

1001

0B

@

1C

@

1C

2pI02

1 þ P2

ð1  P2Þ2P sin
2P cos

0B

@

1C

@

1C

2pI02

1 þ P2

 1  P22P0

0B

@

1C

which is identical to the Stokes vector found in (25-57) Thus, the quarter-waveretarder can be inserted into either the generating or analyzing arm, because the

25- 28ịFrom the denition of the reflectivity (25- 26) we then see that (25- 28) yields

Figure 25- 1 Plot of the reflectivity as a function of The refractive indices...

cos
in cos
rcos
iỵn cos
r

25- 25ịEquation (25- 25) is, of course, exact and can be used to obtain an exact expressionfor the reflectivity... the behavior of n and Further details on the nature of metals and, inparticular, the refractive index and the extinction coefficient (n and ) as it appears inthe dispersion theory of metals can