1

Anisotropic media with geometric álgebra

Marlene Lucete Matias Rocha nº 57531

Instituto Superior Técnico

Av. Rovisco Pais, 1049-001 Lisboa, Portugal

E-mail: m_lucete@hotmail.com

Abstract

This paper presents a new mathematical

formalism, designated geometric algebra or

Clifford. It begins by addressing the bases of

this algebra concept, introducing the basic

definition, the product of vectors: the

geometric product. Based on this product, two

new geometric objects appear the bivector

trivector and important concepts such as rotors

and contractions. We introduce the concept of

anisotropy and demonstrate the application of

geometric algebra to uniaxial and biaxial

anisotropic crystals in a less complex way

using a coordinate free system to analyze the

propagation of electromagnetic waves in

anisotropic media, obtaining characteristic

wave and the constitutive relationships of the

crystals, and finally an application in

anisotropic media are analyzed half wave

retarder plates, quarter-wave and full wave.

Keywords: Geometric algebra, bivector,

trivector, geometric product, anisotropic

media, uniaxial crystal and biaxial crystal.

1. Introduction

The history of geometric algebra has its

beginnings in Ancient Greece ([1] and [2])

with the writing of geometric relations in an

algebraic form, however, the formalism

inherent to geometric algebra as a

mathematical tool with uses on the most

diverse areas as we know it today has its

beginning only in nineteenth century.

In 1843, Sir William Hamilton Rowam (1805-

1865) in an attempt to solve the problem of the

three-dimensional rotations generalized

complex numbers for all three dimensions,

giving rise to the quaternions of Hamilton.

A year later, in 1844 Herman Gunter

Grassmann (1809-1877), develops algebra

itself, origining the exterior product, proving

that the relation in geometric algebra is not

restricted to three dimensions, i.e. the exterior

product is definable in n dimensions. This

product is characterized by the properties of

associativity and anti-commutativity and also

to be a product that does not depend on any

metric.

William Kingdon Clifford (1845-1879), an

English mathematician who in 1978 unified

algebraic structure the algebras of Hamilton

and Grassmann, yielding geometric algebra.

This algebra is characterized by a product

between the vectors so-called geometric

product. This product is associative as the

2

cross product of Grassmann but is invertible as

quaternions of Hamilton. At the end of the

century comes the cross product of Josiah

Willard Gibbs (1839-1903), which is the usual

three-dimensional space that is defined in three

dimensions.

Only at the beginning of the XX century,

Einstein published the theory of relativity in

1905, and was replaced by the need to work

with four dimensions, one begins to question

the outer product of Gibbs and consequently

begins to think about the utility of studies,

Grassmann and particularly by Clifford, but

however only in 1920 geometric algebra

reappears in the form of spin matrices

indispensable to quantum mechanics.

In 1970, when David Orlin Hestenes (1833 - )

deepened his knowledge in quantum

mechanics, it was realized that geometric

algebra could be a unifying mathematical

language of many of the same areas.

2. Basic Concepts of geometric

algebra

The geometric product or Clifford’s product is

the key definition of geometric algebra.

Considering the linear space 3 with

Euclidean metric and an orthonormal basis3

1 2 3{ , , }e e e ,

1,    j = k= =

0,   j kj k jk

e e

(1)

where , {1,2,3}j k so that 1 2 3= = =1e e e

Given the vector 31 2 3r =     e + zex e + y it’s

respective length is 2 2 2= = + +x y zr r r .

In geometric algebra 3 we introduce a new

product between vectors the Clifford or

geometric product, which we denote by 2=rr r

By requiring that 2 2  =  r r and using the

distributive rule, without assuming

commutative, we get

22 2 2 2 2 2

1 2 2 1 1 3 3 1

2 3 3 2

2 1 1 2

2 2

3 1 1 3

3 2 2 3

x + y + z x + y + z

xy + + xz +

+ yz + 0

=

     =        =

=

=

e e e e e e e e

e e e e =

e e e e

r r e e e e

e e e e

(2)

In the general case the geometric product is

not be commutative.

2 2 2

1 2 1 2= = 1 e e e e (3)

So when the square of 1 2e e is negative, it is not

a vector neither a scalar so it is a bivector. A

bivector represents a targeted area, which has

clockwise or countersclockwise orientation.

1 2 12ˆ = =F e e e represents a unit bivector as

shown in figure 1.

For arbitrary vectors 3, a b the geometric

product, ab , is the graduate sum of a scalar,

given by the inner (or dot) product between

them, and a bivector which is the exterior (or

outer) product also between.

dot product outer product

= + ab a b a b (4)

12 1 2ˆ = =F e e e

2e

1e

Figure 1: Representation of the unit bivector F̂

3

where

1 1 2 2 3 3= + +a b a b a ba b (5)

and

23 31 12

1 2 3

1 2 3

= =   a a a

b b b

e e e

F a b (6)

So, we have a multivector, u , 2

3= = + = +u ab α F a b a b (7)

A trivector is the outer of three vectors which

can be written as

1 2 3

1 2 3 123 123

1 2 3

ˆ= = = =

a a a

b b b

c c c

V a b c e e V (8)

where and V̂ is the unit trivector such

that 2

123 = 1e .

An arbitrary element 3u , wich we call a

multivector, is a (graded) sum of scalar, a

vector, a bivector and a trivector:

= + + +u a F V , 0

= u , 1

= ua , 2

= uF ,

3= uV , denoting the operation of projecting

onto the terms of a chosen grade k by k.

The geometric algebra of Euclidean 3D space,

3 in the direct sum

2 3

3 3 3

3

scalars vectores trivectors

=bivectors

(9)

where

3

1 2 3

23

23 31 12

33

123

1 1

, , 3

, , 3

1

Basis Subspace Dimension

e e e

e e e

e

so that

3

3dim =1+ 3 + 3 +1= 2 = 8 (10)

The subalgebra of scalar and trivectors is the

center of the algebra, i.e., it consists of those

elements of 3 which commute with every

element in 3 :

3

3

3Cen = (11)

This algebra has an even part 3

and odd

part 3

. The even part, 2

3

3 = ,

constitutes a subalgebra where the odd part, 3

3 3

3 = , is not a subalgebra. The even

subalgebra is isomorphic to the division

algebra of quaternion: 3

and de center

is isomorphic to the division algebra over

the reals: 3Cen .

2.1 Rotors and contractions

Rotors are defined as the geometric product of

two unitary vectors. Considering the linear

space 3 and 3, n m , a rotor is,

=R nm (12) (1.1)

2e

1e

3e

1 2 3= i e e e

Figure 2: Representation of the unit trivector ˆ =V i

4

From this definition (12) we can observe that

R is in fact a multivector defined as,

= = +

ˆ= cos + sin

ˆ = exp

R nm n m n m

B

B

(13)

The rotor defined in (13) can handle a rotation

of 2 in the plane of the corresponding

bivector. So, if we would like to have a

rotation of in the corresponding bivector

plane, we proceed as in

There are two types of contraction, the left

contraction and the right contraction.

Considering the vectors 3a,b,c and the

bivector = B b c , the left contraction is by

definition,

1

= -2

a B aB Ba (14)

We obtain the fundamental rule of left

contraction as

  = a b c a b c a c b (15)

In analogous way we have the right

contraction as

1

  =2

B a Ba aB (16)

Those two contractions can be related by

= a B B  a (17)

We can also write

+

=   +

aB = a B a B

Ba B a B a

(18)

So, we conclude that the left contraction of a

vector with a bivector will step down the

degree of the bivector into the degree of the

vector. The right contraction has the same

consequence, but the new vector is

diametrically opposed to the one obtained in

left contraction.

3. Anisotropy Media

3.1 Anisotropy in geometric algebra

Anisotropy, means that the magnitude of a

property can only be defined along a given

direction. Particularly, if we are speaking of

electrical anisotropy, this means that there is

an angle between the electric field vector, E ,

and the electric displacement vector, D , that

depends on the direction of the Euclidean

space 3 , along which, vector E is applied.

This means that it isn’t possible to write

0= D E , where 0 is the permittivity of

vacuum and is a scalar called the relative

dielectric permittivity of the medium. The

usual solution, with the dyadic analysis,

consists in introducing a permittivity or

dielectric tensor that, in a given coordinate

system, may be written as a 3 3 matrix.

123u = Be

r

r

r

r

B

Figure 3: Rotation the vector r to r

5

With geometric algebra 3 , we simply write

0= D E (19)

where E is a dielectric linear function

3 3 ε: that maps vectors to vectors and it

characterizes the medium only along a certain

direction.

Henceforth, it will be considered that the

media in study are not magnetic, lossless,

unbounded and linear. In non magnetic media

we have = 0M , so the relationship between B

and H is same of vacuum, i.e.,

0= B H (20)

Formally, considering figure 4, when an

electric field =E E s

in the direction

characterized by unit vector s is applied, the

media responds with an electric displacement

field =D D t . The relation between s and t is

such that = coss t and = sins t F . Now,

we are in conditions of characterizing the

media correctly. So it follows, in such way:

= = cos

= = sin

ˆ = =sin

ˆ= = = sin

D s D D

D r D D

s tF sr

F E D E D s t F

(21)

s

s

= s = s = s cos

D =

D

E

(22)

The following considerations can also be made

in accordance to Figure 4,

=

= + = = +

ˆ= cos + sin

ˆ= exp

E D

ED E D E D st s t s t

F

ED F

(23)

(1.2)

In expression (22), s = s is the

permittivity along s and s is the dielectric

function. Specifically, in an anisotropic

medium, to each direction s of the space

corresponds a scalar s = s s .

In order to exemplify better this concept of

relative dielectric constant let us consider an

example. Admitting that the linear operator 3 3ε : has three real distinct eigenvalues

1 2 3> > . For convenience of future

calculus, consider the following:

2

3 3 3 1

2 2 2

3 1 2 1 2 1

2 2

1 1 3 2 3

= + 2 = 2

+ =1       =         = 2

= 2 = 2

 

 

   

(24)

Figure 4: Electrical anisotropy characterized by bivector

= F E D .

6

Considering the orthonormal basis

3

1 2 3= e ,e ,e , formed by the

eigenvectors of considered operator, i.e., such

that

1 1 1 1

2 2 2 2

3 3 3 3

eixo X    =

eixo X    =

eixo X    =

e e

e e

e e

(25)

.

According to Figure 5, we define,

1 1 1 3 3

2 1 1 3 3

= +

= +

d e e

d e e (26)

The relation between 1d and

2d is the

following,

2 1= R R d d (27)

where R is a rotor given by

31 31= exp = cos + sin2 2 2

R

e e (28)

This equation means that 2d is the rotation of

1d in 3 . We are now in condition of writing

the linear operator in terms of the vectors

defined in expression (26), so considering the

generic vector 3a it follows,

1 2 2 1= + + a a a d d a d d (29)

where 2= and 3 1

1=

2 .

A linear operator 3 3ε : with three real

positive and distinct eigenvalues, can be

written in terms of two unit vectors 1d and

2d

non parallel. For this reason it is said that this

is biaxial operator.

A uniaxial operator is a particular case of the

biaxial operator. Continuing to consider the

same example as before and considering now

that 1 2= =

and 3 = we have the

following,

1 2

1 2

= cos = = 1  

    = 0  

  = =

d d

d d c

(30)

= + a a c a c (31)

In the uniaxial case, is considered the non

degenerate eigenvalue and corresponds to the

eigenvector c (axis of the crystal). The is

the doubly degenerated eigenvalue

corresponding to the other two eigenvectors

which defining the orthogonal plane to the axis

of the crystal.

The inverse of the dielectric function is

the impermeability function 1= , such that

0

=

DE .

If 1 1 2 2 3 3= + +E E e E e E e and

1e , 2e and

3e

are the principal dielectric axes corresponding

to the eigenvalues 1 , 2 and

3 (respectively),

with 3 2 1> > , then

1 1 1 2 2 2 3 3 3+ + D = E e E e E e and

1 1 1 2 2 2 3 3 3= D + D + D E e e e where 1=i i

with =1,2,3i . One can readily show that, is 1d

and 2d are the two unit vectors that

2d

3e

3X

1d

1

1e

1

1X

2X

3

Figure 5: Representation of the vectors 1d and

2d , which

define the linear operator as an biaxial operator.

7

characterize , then 1c and

2c , are the two unit

vectors that characterize .

The process of obtaining the inverse of the

dielectric function is similar to the process for

the case of the dielectric function. So

considering Figure 6 we can write the

following,

1 1 1 3 3

2 1 1 3 3

= +

= +

c e e

c e e

(32)

where,

3

1 1

2

1

3 3

2

=

=

(33)

with 1 = sin2

and 3 = sin2

.

Similarly to Equation (30), we can now write,

2 1= R R c c (34)

where 2c is the rotation of

1c in 3 originated

by the rotor

31

31= exp = cos + sin2 2 2

R

ee (35)

Whence,

1 1

2 2

=

c d

c d

(36)

where,

3 1

3 1

= , +

2

1=

1

and

1 3

2

=

.

One should note that = cosh , = sinh

thus leading to = tanh ,

3 1= ln 1+ 1 2 = ln 4 ∕ ∕ ∕ and

3 1tan = tan 2 ∕ ∕ ∕ .

Accordingly, in comparison with Equation

(29), one has

2 3 1 1 2 2 1= + 2 + D D D c c D c c∕

(37)

The expression (37) is the inverse of the

dielectric function for the biaxial crystals.

Now doing the same that was made for

equation (31), we can find the inverse of the

dielectric function for uniaxial crystals.

= + D D D c c (38)

3.2 Monochromatic plane waves

For electromagnetic field variation of the form

0exp = expi t ik ct k r n r (39)

with

1

1

2X

1 1

1e

1X

2c

2d

1d

1c

3X

3

3e

Figure 6: Representation of the vectors 1d and

2d , which

define the linear operator and the vectors 1c and

2c

which define in a biaxial medium.

8

0 0= =

ˆ=

k kc  

n

k n

n k

(40)

Maxwell equation in 3 may be simply

written, for source-free regions, as

123

123

=

=

= 0

= 0

c

c

n E Be

n H De

n D

n H

(41)

3.3 Uniaxial crystals

As we saw previously, for the case of a

uniaxial medium we have the following

dielectric function:

= + E   E c E c (42)

Normally, an uniaxial crystals it is common to

consider 2= no and 2= ne . The crystals are

considered positive when > and negative

if < .

Remembering the constitutive relations of the

media in equation (19) and also remembering

that the media is non magnetic, 0= B H , the

following comes,

2 2

123 123 0 123

0

= =

=

1 = =

n n n

cc

n   n E E E n E

n   Be n B e = n H e

D E

(43)

Analyzing the expression (43), we can infer

the following,

2 2

2

=          =

=         =

n n

n

E E E E

E E E E E E

(44)

The parallel component of electric field, E , is

defined as its component as k̂ , i.e., such that

ˆ = 0k E .

2

2 2 2

1 1= =

= =

n n

n n n

E n E       E n E n

   E E E E n E n

(45)

Considering the wave equation,

2

ˆ = 0

ˆ ˆ ˆ= = 0n

k E

k E k E k E

(46)

Figure 7: Positive uniaxial crystal

Figure 8: Negative uniaxial crystal

9

One should note that,

0 2

0

0 2 2

0 0

0 0

1= =

1 1

= +

ˆ ˆ= +

e o

n   

= =n n

E E c E c

k E k E c E k c

(47)

Finally we get the wave equation in a uniaxial

crystal.

2 2

0 0ˆ ˆ1 + = 0 n n k E c E k c (48)

Applying the left contraction with ˆ k c to

equation (48) we obtain the two eigenwaves or

isonormal wave (extraordinary wave and

ordinary wave) of the uniaxial crystal

2

2 2

0 0

onda ordináriaonda extraordinária

ˆ1 + = 0n n

k c c E

(49)

Finally we can obtain from (49) the equation

of the ordinary wave,

2 2

0

0

1= =n n

(50)

And the equation of the extraordinary wave,

2 2

2

2 22 2ˆ ˆ

o e

e o

n nn =

n n k c k c

(51)

4.2 Biaxial crystals

Considering the equations (29), (37), (40), (41)

and monochromatic plane wave propagation in

a biaxial media, it comes immediately, like we

saw for a uniaxial crystal,

2

ˆ ˆ=

n

E = E

E E E = E E k k

(52)

Accordingly, in terms of permeability function

of equation (39) we may also write

2

ˆ = 0

= n

k w

w E E

(53)

Using a similar procedure to that used for the

case of uniaxial crystals the wave equation for

the biaxial crystals, i.e.,

2

0

2

0 1 2 2 1

ˆ1 +

ˆ ˆ+

n

n

k E

c E k c c E k c (54)

Where 0 2= and 3 1

0 =2

.

Introducing the following vectors,

1 123

2 123

ˆ=

ˆ=

u k c e

v k c e (55)

and applying the left contraction of bivectors

1ˆ k c and 2

ˆ k c with (38), with equation (54),

we obtain,

2 2

0

1 22

0 0

2 2

0

2 12

0 0

=1 +

=1 +

n

n

n

n

uc E c E

u v

vc E c E

u v

(56)

The eingenwaves corresponding to the

direction of propagation k̂ (the normal wave)

are characterized by two distinct refractive

indexes (birefringence), such that n and n

,

such that

2 2

0 02

1= +

n

u v u v (57)

10

Analyzing the equation (57), we can conclude

that for waves propagating along 1c or

2c we

have, according to equation (54), = 0 u or

= 0v (respectively) and hence 2

2

0

1= =n

,

i.e., the two refractive indexes are equal. But

then, according to the definition of optic axes,

we conclude that the two unit vectors 1c and

2c

that characterize the impermeability function

are, in fact, the two optic axes of the biaxial

crystal. These two waves, contrarily to what

happens for uniaxial crystals, cannot be

divided into ordinary and extraordinary waves,

because both of them have simultaneously

characteristics of both ordinary and

extraordinary waves as we can see in Figure 6.

4. Retarder plates

A retarder (or waveplate) alters the

polarization of light in a manner on the

retardance and the angle between the retarder

fast axis and the input plane of polarization.

The most common types of waveplates are

quarter-wave plates (λ/4 plates) and half-wave

plates (λ/2 plates), where the difference of

phase delays between the two linear

polarization directions is π/2 or π, respectively,

corresponding to propagation phase shifts over

a distance of λ / 4 or λ / 2, respectively.

4.1 Quarter-Wave retarder

A quarter-wave retarder is used to convert

linear polarization to circular polarization and

vice-versa. It change linearity polarized light

to circularly polarized light, when the angle

between the input polarization and the retarde

fast axis is 45 . The thickness of the quarter

waveplate is such that the phase difference is

4 1 4, Zero order , or certain multiple of

1 4 -wavelength 2 1 1 4, multiple orderm .

A quarter wave retarder has 90 .

4.2 Half -Wave retarder

When the angle between the retarder fast axis

and the input plane of polarization is 45°,

horizontal polarized light is converted to

vertical. A half-wave retarder rotates a linear

polarized input by twice the angle between the

retarder fast axis and the input plane of

polarization, as shown in figure 7. The

thickness of half waveplate is such that the

phase difference is 2 1 2, Zero order or

3X

1X

2c 1c

1

1

3

2

3

2

+n

n

1

2

Figure 9: Representation of the refractive indexes, which

correspond to the eigenwaves of the biaxial crystal

Figure 10: Conversion between linear and circular polarization

by a quarter-wave plate.

11

certain multiple of 1 2 -wavelength

2 1 1 2, multiple orderm . A half wave

retarder has 180 of retardance.

4.3 Multiple-order and Zero-order plates

The retarder plates may by zero-order i.e.,

m = 0 or multiple order, with m 1 .

The zero-order retarders are almost impossible

to fabricate because the have extremely small

thicknesses, but may be constructed from two

multiple-order retarders with handlable

thicknesses. Regarding multiple order plates

are easy to fabricate but have a disadvantage

of having reduced sensitivity to small

deviations.

4.2 Conclusions and future work

In this thesis were studied the fundamentals of

basic algebra geometry of Euclidean space.

Layed up and studied the product which

characterizes this algebra, the product

geometric or Clifford. This product is the sum

of the inner product with the outer product,

and is generally not commutative and

associative however is invertible. Regarding

the outer product (Grassmann been found that

this is definable in n dimensions and is

independently any metric, unlike the cross

product Gibbs which is definable in three

dimensions and only requires a metric.

Through outer product appears new geometric

objects, the bivector (segment oriented plan)

result of exterior product of two vectors, and

trivector (volume oriented) outcome of

exterior product of three vectors. Stands out

even of extreme importance two operators

such as the rotors which allow the making of

spatial rotations and planar rotations and

contractions of vectors are useful to define the

geometric product of vectors for bivectores

and vice versa. The geometric algebra of

Euclidean space allows complete formulation

of the developments of the classical areas of

physics, mainly from electromagnetism.

In the context of electromagnetism, geometric

algebra stand out showed up tensor algebra for

its independence of any coordinate system,

where the concept of tensor is unnecessary.

With geometric algebra the study of the

propagation of electromagnetic waves in a

uniaxial and biaxial crystal can the simplified,

without resorting the properties of matrices, as

once was done using tensor calculus or dyadic.

It becomes possible to determine the refractive

indices for any direction of the optical axis,

and any direction of propagation. This paper

aims to disclose geometric algebra as a clear

structure, intuitive and geometric as well as

becoming an universal and unifying language

for the study and application in different areas

of physics and engineering, in this case, as an

application to anisotropic media. As an

application in anisotropic media retardant half-

wave plates, quarter-wave and full-wave are

analyses.

.

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df , 03-06-12

Figure 11: Rotation of linear polarization by a half-wave

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12

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