8 Mueller Matrices for Reflection and Transmission

Brewster, who was led to enunciate his famous law relating the polarization ofthe reflected light and the refractive index of the glass to the incident anglenow known as the Brewster angl

of polarized light One of the major reasons for discussing the Stokes parametersand the Mueller matrices in these earlier chapters is that they provide us with anexcellent tool for treating many physical problems in a much simpler way than isusually done in optical textbooks In fact, one quickly discovers that many of theseproblems are sufficiently complex that they preclude any but the simplest to beconsidered without the application of the Stokes parameters and the Muellermatrix formalism.

One of the earliest problems encountered in the study of optics is the behavior

of light that is reflected and transmitted at an air–glass interface Around 1808,

E Malus discovered, quite by accident, that unpolarized light became polarizedwhen it was reflected from glass Further investigations were made shortly afterward

by D Brewster, who was led to enunciate his famous law relating the polarization ofthe reflected light and the refractive index of the glass to the incident anglenow known as the Brewster angle; the practical importance of this discovery wasimmediately recognized by Brewster’s contemporaries The study of the interaction

of light with material media and its reflection and transmission as well as itspolarization is a topic of great importance

The interaction of light beams with dielectric surfaces and its subsequentreflection and transmission is expressed mathematically by a set of equationsknown as Fresnel’s equations for reflection and transmission Fresnel’s equations

can be derived from Maxwell’s equations We shall derive Fresnel’s equations in thenext Section.

In practice, if one attempts to apply Fresnel’s equations to any but thesimplest problems, one quickly finds that the algebraic manipulation is veryinvolved This complexity accounts for the omission of many important derivations

in numerous textbooks Furthermore, the cases that are treated are usually restricted

to, say, incident linearly polarized light If one is dealing with a different state ofpolarized light, e.g., circularly polarized or unpolarized light, one must usually beginthe problem anew We see that the Stokes parameters and the Mueller matrixare ideal to handle this task

The problems of complexity and polarization can be readily treated byexpressing Fresnel’s equations in the form of Stokes vectors and Mueller matrices.This formulation of Fresnel’s equations and its application to a number of interest-ing problems is the basic aim of the present chapter As we shall see, both reflectionand refraction (transmission) lead to Mueller matrices that correspond to polarizersfor materials characterized by a real refractive index n Furthermore, for totalinternal reflection (TIR) at the critical angle the Mueller matrix for refractionreduces to a null Mueller matrix, whereas the Mueller matrix for reflection becomesthe Mueller matrix for a phase shifter (retarder)

The Mueller matrices for reflection and refraction are quite complicated.However, there are three angles for which the Mueller matrices reduce to verysimple forms These are for (1) normal incidence, (2) the Brewster angle, and (3)

an incident angle of 45
All three reduced matrix forms suggest interesting ways tomeasure the refractive index  of the dielectric material These methods will bediscussed in detail

In practice, however, we must deal not only with a single air–dielectricinterface but also with a dielectric medium of finite thickness, that is, dielectricplates Thus, we must consider the reflection and transmission of light at multiplesurfaces In order to treat these more complicated problems, we must multiply theMueller matrices We quickly discover, however, that the matrix multiplicationrequires a considerable amount of effort because of the presence of the off-diagonalterms in the Mueller matrices This suggests that we first transform the Muellermatrices to a diagonal representation; matrix multiplication of diagonal matricesleads to another diagonal matrix Therefore, in the final chapters of this part ofthe book, we introduce the diagonalized Mueller matrices and treat the problem

of transmission through a single dielectric plate and through several dielectricplates This last problem is of particular importance, because at present it is one

of the major ways to create polarized light in the infrared spectrum

TRANSMISSION

In this section we derive Fresnel’s equations Although this material can be found

in many texts, it is useful and instructive to reproduce it here because it is

so intimately tied to the polarization of light Understanding the behavior ofboth the amplitude and phase of the components of light is essential to designingpolarization components or analyzing optical system performance We start with areview of concepts from electromagnetism

is the electric displacement

H*is the magnetic field

"0 is the permittivity of free space

8.2.2 Boundary Conditions

In order to complete our review of concepts from electromagnetism, we mustrecall the boundary conditions for the electric and magnetic field components Theintegral form of Maxwell’s first equation, (8.2a), is

ZZD

*

This equation implies that, at the interface, the normal components on either side

of the interface are equal, i.e.,

The integral form of Maxwell’s second equation, (8.2b), is

ZZB

*

d I*¼

Z Z

r

which implies

i.e., the tangential component of E is continuous across the interface

8.2.3 Derivation of the Fresnel Equations

We now have all the tools we need derive Fresnel’s equations Suppose we have

a light beam intersecting an interface between two linear isotropic media Part ofthe incident beam is reflected and part is refracted The plane in which thisinteraction takes place is called the plane of incidence, and the polarization oflight is defined by the direction of the electric field vector There are two situationsthat can occur The electric field vector can either be perpendicular to the plane

of incidence or parallel to the plane of incidence We consider the perpendicularcase first

Case 1: Eis Perpendicular to the Plane of Incidence

This is the ‘‘s’’ polarization (from the German ‘‘senkrecht’’ for perpendicular) or polarization This is also known as transverse electric, or TE, polarization (refer

to Fig 8-1) Light travels from a medium with (real) index n1 and encounters aninterface with a linear isotropic medium that has index n2 The angles of incidence(or reflection) and refraction are
iand
r, respectively

In Fig 8-1, the y axis points into the plane of the paper consistent with theusual Cartesian coordinate system, and the electric field vectors point out ofthe plane of the paper, consistent with the requirements of the cross product andthe direction of energy flow The electric field vector for the incident field is repre-sented using the symbol E*, whereas the fields for the reflected and transmittedcomponents are represented by R* and T*, respectively Using Maxwell’s thirdequation (8.2c) we can write

where ^an is a unit vector in the direction of the wave vector

Now we can write

0="

Figure 8-1 The plane of incidence for the transverse electric case

"rp

and now Fresnel’s equation for the reflection amplitude is

This last equation can be written, using Snell’s law, n1sin
i¼n2sin
r, to eliminatethe dependence on the index:

Case 2: *Eis Parallel to the Plane of Incidence

This is the ‘‘p’’ polarization (from the German ‘‘parallel’’ for parallel) or polarization This is also known as transverse magnetic, or TM, polarization(refer toFig 8-2).The derivation for the parallel reflection amplitude and transmis-sion amplitude proceeds in a manner similar to the perpendicular case, and Fresnel’sequations for the TM case are

Figures 8-1and8-2have been drawn as if light goes from a lower index medium to

a higher index medium This reflection condition is called an external reflection

Fresnel’s equations also apply if the light is in a higher index mediumand encounters an interface with a lower index medium, a condition known as

an internal reflection

Before we show graphs of the reflection coefficients, there are two specialangles we should consider These are Brewster’s angle and the critical angle.First, consider what happens to the amplitude reflection coefficient in (8-24b)when
i þ
r sums to 90
The amplitude reflection coefficient vanishes forlight polarized parallel to the plane of incidence The incidence angle for whichthis occurs is called Brewster’s angle From Snell’s law, we can relate Brewster’sangle to the refractive indices of the media by a very simple expression, i.e.,

iB¼tan1n2

The other angle of importance is the critical angle When we have an internalreflection, we can see from Snell’s law that the transmitted light bends toever larger angles as the incidence angle increases, and at some point the transmittedlight leaves the higher index medium at a grazing angle This is shown in Fig 8-3

The incidence angle at which this occurs is the critical angle From Snell’s law,

n2 sin
i ¼ n1 sin
r [writing the indices in reverse order to emphasize the lightprogression from high (n2) to low (n1) index], when
r¼90
,

The amplitude reflection coefficients, i.e.,

Fig 8.5 The amplitude reflection coefficients and their absolute values for thesame indices for internal reflection are plotted in Fig 8-6.The phase changes forinternal reflection are plotted inFig 8-7.An important observation to make here isthat the reflection remains total beyond the critical angle, but the phase change is

a continuously changing function of incidence angle The phase changes beyondthe critical angle, i.e., when the incidence angle is greater than the critical angle,are given by

tan’s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2
rsin2
cq

Figure 8-3 The critical angle where the refracted light exists the surface at grazing incidence

where ’s and ’p are the phase changes for the TE and TM cases, respectively.The reflected intensities, i.e., the square of the absolute value of the amplitudereflection coefficients, R ¼ jr2j, for external and internal reflection are plotted in

Figs 8-8 and8-9,respectively

The results in this section have assumed real indices of refraction for linear,isotropic materials This may not always be the case, i.e., the materials may beanisotropic and have complex indices of refraction and, in this case, the expressionsfor the reflection coefficients are not so simple For example, the amplitude reflectioncoefficients for internal reflection at an isotropic to anisotropic interface [as would

be the case for some applications, e.g., attenuated total reflection (see Deibler)], are

First, the Stokes parameters must be defined appropriately for the field within andexternal to the dielectric medium The first Stokes parameter represents the totalintensity of the radiation and must correspond to a quantity known as the Poyntingvector This vector describes the flow of power of the propagating field components

of the electromagnetic field The Poynting vector is defined to be

In an isotropic dielectric medium, the time-averaged Poynting vector is

With these considerations, we will arrive at the correct Mueller matrices forreflection and transmission at a dielectric interface, as we will now show

AT AN AIR–DIELECTRIC INTERFACE

The Stokes parameters for the incident field in air (n ¼ 1) are defined to be

1

p.Similarly, the Stokes parameters for the reflected field are

(8-24a) into (8-33) and using (8-32), the Stokes vector for the reflected beam SR isfound to be related to the Stokes vector of the incident beam S by

1CC

C¼

12

tan
sin
þ



0BBB

1CCC

1CC

1CC

Comparing (8-34) with (8-35) we see that the 4  4 matrix in (8-34) corresponds

to a Mueller matrix of a polarizer; this is to be expected from the form ofFresnel’s equations, (8-22) and (8-24), in Section 8.2

The Stokes parameters for the transmitted field are defined to be

where the subscript T indicates the Stokes parameters of the transmitted beam, and

Ts and Tp are the transmitted field components perpendicular and parallel to the

plane of incidence Substituting the values of Tsand Tpfrom Eqs (8-23) and (8-25)into (8-36) and using (8-32), the Stokes vector ST is found to be

C¼ sin 2
isin 2
r2ðsin
þcos
Þ2

1CC

1C

The Stokes vector for unpolarized light is

S ¼ I0

1000

0B

@

1C

@

1C

12

tan
sin
þ



cos2
þcos2
þcos2
cos2
þ00

0B

@

1C

In general, because the numerator in (8-41) is less than the denominator, thedegree of polarization is less than 1 However, a closer inspection of (8-41) showsthat if cos
þis zero, then P ¼ 1; that is, the degree of polarization is 100% Thiscondition occurs at

is confirmed by setting cos
þ¼0 in (8-40), which then reduces to

@

1C

0B

@

1C

The Stokes vector in (8-43) shows that the reflected light is linearly horizontallypolarized Because the degree of polarization is 1 (100%) at the angle of incidencewhich satisfies (8-42b), we have labeled
ias
iB, Brewster’s angle

In Fig 8-10 we have plotted (8-41), the degree of polarization P versusthe incident angle
i, for a material with a refractive index of 1.50 Figure 8-10shows that as the incident angle is increased P increases, reaches a maximum, andthen returns to zero at
i¼90
Thus, P is always less than 1 everywhere except at the

Figure 8-10 Plot of the degree of polarization P versus the incident angle
ifor incidentunpolarized light which is reflected from glass with a refractive index of 1.5

maximum The angle at which the maximum takes place is 56.7 (this will be shownshortly) and P is 0.9998 or 1.000 to three significant places At this particular angleincident unpolarized light becomes completely polarized on being reflected Thisangle is known as the polarization or Brewster angle (written
iB) We shallsee shortly that at the Brewster angle the Mueller matrix for reflection (8-34)simplifies significantly This discovery by Brewster is very important because itallows one not only to create completely polarized light but partially polarizedlight as well This latter fact is very often overlooked Thus, if we have a perfectunpolarized light source, we can by a single reflection obtain partially polarizedlight to any degree we wish In addition to this behavior of unpolarized light anextraordinarily simple mathematical relation emerges between the Brewster angleand the refractive indices of the dielectric materials, i.e., (8-26): this relation wasused to obtain the value 56.7
.

With respect to creating partially polarized light, it is of interest to determinethe intensity of the reflected light From (8-40) we see that the intensity IR of thereflected beam is

IR¼1

2

tan
sin
þ

In Fig 8-11 we have plotted the magnitude of the reflected intensity IR as

a function of incident angle
i for a dielectric (glass) with a refractive index of1.5 Figure 8-11 shows that as the incidence angle increases, the reflected intensityincreases, particularly at the larger incidence angles This explains why when thesun is low in the sky the light reflected from the surface of water appears to bequite strong In fact, at these ‘‘low’’ angles polarizing sunglasses are only partially

Figure 8-11 Plot of the intensity of a beam reflected by a dielectric of refractive index of 1.5.The incident beam is unpolarized