Basic Theoretical Physics: A Concise Overview P19 ppt

Similar wave equations occur in other cases, e.g., for transverse sound waves shear waves, → transverse phonons in solid state physics it is only necessary to replaceE and cB by the tran

where n (= √ ε · μ) is the refractive index2 In optically dense

light-trans-mitting media (e.g., glass or water) n is significantly > 1, which gives rise to the well-known phenomenon of refraction described by Snell’s law :

sin α1

sin α2 =

n2

n1 .

This expression describes the refraction of light rays at a plane boundary

between a vacuum with refractive index n = 1 (or in practice: air) and an (optically) denser medium with n2 > 1 The angle of incidence from the vacuum is α1

The refractive index, n, is thus related both to the (relative) dielectric constant of the material, εr, and also to the (relative) permeability μr This

also means that generally n depends on the wavenumber k and frequency ω.

The above relation for dielectric and/or permeable matter follows from Maxwell’s equations in the absence of charges and currents (i.e., for

≡ j ≡ 0):

divεr ε0

curlE = − ∂ B

∂t ; curl

B

μ r μ0 =j + ε r ε0 ∂ E

Applying the operator curl to the third and fourth equations, together with the identity

curl curlv = grad divv − ∇2v

we obtain the following wave equations:



∇2− 1

c2

m (∂ t)2



E = 0 and



∇2− 1

c2

m (∂ t)2



B = 0 ,

where cm(see below) is the velocity of a stationary electromagnetic wave in the considered medium

Similar wave equations occur in other cases, e.g., for transverse sound

waves (shear waves, → transverse phonons in solid state physics) it is only

necessary to replaceE and cB by the transverse displacements of the atoms

from their rest positions and the velocity of light, cm, by the transverse sound velocity c (s) ⊥ (which in metals is of the order of 103 m/s) With these

re-placements one obtains the same wave equation in totally different contexts

(There are also longitudinal sound waves (compression waves4,→ longitudi-nal phonons) with a significantly higher sound velocity c (s) long..)

2

In some (mainly artificial) materials both ε and μ are negative, i.e., also n These so-called left-handed materials have unusual optical properties.

3

To avoid any misunderstanding, here we explicitly use the lower indexr, although hitherto we did not use this convention

4

In liquids and gases only compression waves exist, and not shear waves.

182 21 Applications of Electrodynamics in the Field of Optics

The relationship

ω = ω( k)

between frequency ω and wavenumber k5 is referred to as the dispersion relation.

For light waves in vacuo the dispersion relation is simply ω = c0· |k|.

In polarizable matter, light waves have only apparently almost the same

dispersion behavior It is true that often ω = cm · |k|, but even then the light

velocity in matter,

c m = c0/

ε r (ω) · μ r (ω) , generally depends on the same frequency ω that one wishes to calculate6, so

that it makes sense to distinguish between the so-called phase velocity

vphase=

ω

k



e k

and the group velocity

vgroup= gradk ω( k)

As a consequence, the components of the group velocityvgroupare calcu-lated as follows:

(vgroup)i= ∂ω( k)

∂k i

The group velocity can thus have a different direction from the wavenumber

k (see below, e.g., the term ray velocity in the subsection on birefringence in

crystals)

As an example for the difference between phase velocity and group veloc-ity consider transverse sound waves propagating in an ideal crystal of cubic symmetry with lattice constant a Let k be parallel to an edge direction.

One then obtains transverse elastic plane waves with the following dispersion relation:

ω( k) = c (s)

⊥

sin k ·a 2 a 2

Thus, if the wavelength λ is much larger than the lattice constant a, i.e., for k · a  1, we have, as expected:

ω = c (s) ⊥ k , and the group velocity,

|vgroup| ,

5 k = 2π

λ e k , where λ is the wavelength and e k the propagation direction of the

plane wave, i.e., according to the usual ansatz Ψ ∝ Re exp(i(k · r − ωt)).

6 The frequency dependence of the light velocity may involve not only the ampli-tude, but also the direction (see below)

and phase velocity, c (s) ⊥ , are identical; but for k · a → π, i.e., with decreasing

λ → 2a, the group velocity converges to zero,

∂ω

∂k → 0

The meaning of “group velocity” can be illustrated by considering a wave packet generated by the superposition of monochromatic plane waves with slightly differentk-vectors in the interval Δk:

Ψ (x, t) =



Δk dka(k)e i(k ·x−ω(k)·t) . (21.5)

One obtains beats, which can be treated by a Taylor expansion, as follows:

If K0is the center of the interval Δk, i.e.,

k = K0 + k  (with k  ∈(− K0

2 , + K0

2

)

) then we have a(k) ≈ a(K0) and

ω(k) · t ≈ ω(K0)· t + k  dω

dk · t +

Thus

Ψ (x, t) ≈ a(K0)ei(K0x −ω(K0 )·t)

K0

2



− K0

2

dk eik  ·(x−vgroup·t) . (21.6)

The factor immediately in front of the integral, which can also be written

as eiK0·(x−vphaset) (i.e., with the phase velocity vphase) describes the rapid os-cillation of the amplitude of the wave In contrast, the integral itself describes

the (much slower) wave motion of an envelope function, which is propagated

at the group velocity vgroup; in particular we have

|Ψ(x, t)|2∝ |a(K0)|2·









K0

2



− K0

2

dk eik  ·(x−vgroup·t)









2

The transport of energy in a wave packet propagates with the group veloc-ity, not the phase velocveloc-ity, and according to Einstein’s theory of relativity we have the constraint that |vgroup| (and not |vphase|) must always be ≤ c (see below).

The difference between phase velocity and group velocity becomes clear

if we consider an electromagnetic wave reflected “back and forth” between

two metal plates perfectly parallel to the plane z ≡ 0 (These parallel plates

184 21 Applications of Electrodynamics in the Field of Optics

form the simplest version of a waveguide.) We assume that this guided elec-tromagnetic wave propagates in the x-direction The phase velocity is given

by

(vphase)x= c

cos θ , which is > c

In contrast, the group velocity is

(vgroup)x= c · cos θ , i.e , ≤ c ,

as expected Here θ is the grazing angle at which the electromagnetic wave

meets the metal plates

In the context of our treatment of phase and group velocity we should

mention not only Planck’s radiation formula (1900, see above) but also Ein-stein’s “photon” hypothesis (1905, see below), which essentially resolved the problem of whether light should be considered as a particle phenomenon

(Newton) or a wave phenomenon (Huygens) The answer is “both” ( →

wave-particle duality, see Part III) According to Einstein (1905, → Nobel prize

1921) electromagnetic waves result from the emission of individual relativistic

quanta with velocity c ( → “photons”, see below), possessing energy E = hν,

momentum

p = E

c =

hν

c (=k) , and vanishing rest mass The internal energy U (T ) of the electromagnetic field in a large cavity of volume V at Kelvin temperature T is then (see above, and Parts III and IV) given according to Planck’s formula:

U (T ) = V ·

∞



ν=0

8πν2dν

exp



hν

k B T



where the factor

V · 8πν2dν

c3

is the number of wave modes with frequencies ν ∈ dν; hν is the energy of

a single photon of wavelength ν c, and the thermal expectation valuen ν of the number of these photons is

n ν=

∞

n=0

n · e −n kB T hν

∞

n=0

e−n hν

kB T

exp



hν

k B T



− 1 . (kB is Boltzmann’s constant.)

If in this formula the integration variable

8πν2dν

c3

is replaced by the more general expression

e f · d3k (2π)3 , where for photons the degeneracy factor e f has the value 2 and the function

ω( k) = c · k

has to be used, then Planck’s formula can be generalized

For example, for the excitations of so-called quasi-particles resembling

photons, e.g for “phonons” or “magnons” (the quanta of sound waves and

of spin waves, respectively) or for “plasmons” (quanta corresponding to os-cillations of the charge density in solids), one has similar properties as for

photons, but different dispersion relations (e.g ω(k) = D · k2for magnons in ferromagnets and

ω(k) = ω p + b · k2

for plasmons, where ωp is the plasma frequency and b a positive factor.)

Thus in analogous manner to that for photons one obtains different results

for the internal energy U (T ) of the quasi-particle gas considered One sees here that the dispersion relation ω( k) of a wave plays an important role not

only in optics but also in many other branches of physics

21.2 From Wave Optics to Geometrical Optics;

Fermat’s Principle

By avoiding abrupt changes (e.g., by neglecting such quantities as

|λ · gradA(r)| ,

see below) one can start with wave optics and arrive at the field of geometrical

or ray optics as follows.

Commencing with the wave equation

∇2−

 1

c m

2

where Ψ is one of the Cartesian components of the electromagnetic fields

or one of the equivalent wave quantities considered7, we try to solve this

7

Neglecting the vectorial character of the electromagnetic field is already an

ap-proximation, the so-called scalar approximation This can already be essential,

which should not be forgotten

186 21 Applications of Electrodynamics in the Field of Optics

equation with the usual stationary ansatz Thus we are led to the following wave equation:

∇2+ k2(r) ψ( r) !

with

k( r) = ω

c m = k0· n(r) ,

where k0= ω

c0 and n( r) is the refractive index, which is a real quantity (see

above) For cases involving dielectric losses or for Ohmic behavior, one should

replace n( r) by the complex quantity

˜

n := n + iκ , where κ −1

is the absorption length

The solution is now

ψ( r) := A(r) · exp ik0· S(r) ,

with the so-called eikonal S( r) In spite of its strange name, which has mainly

historical reasons, this function S is most important for geometrical optics;

it has the dimensionality of an effective length, viz the minimum distance

between two equivalent wave fronts.8

One now assumes that with the formation of the second derivatives the terms∝ k2 dominate, e.g.,

∂2ψ

∂x2 ≈ −k2(∂S

∂x)

2ψ + , where the dots represent neglected terms, which are not proportional to k2,

but only to the first or zeroth power of k0.9 In this way the wave equation

(21.10) is systematically replaced by the so-called eikonal equation10

(gradS( r))2

Here the surfaces S( r) = constant describe the wave fronts, and their

gradients describes the ray directions.

The eikonal approximation of a scalar wave equation, derived from Maxwell’s theory, is the basis of geometrical (or ray) optics In particular

one can derive from it Fermat’s principle of the “shortest optical path”:

r2



r1

dln( r) !

8

The eikonal function S( r) should of course not be confused with the Poynting

vector S or the entropy S(T ) of statistical physics (Part IV) In each of these

cases the same letter S is used.

9 Note that k0 is not small, but large (k0= 2π

λ0, where λ0is the wavelength)

10 A similar approximation leads from Schr¨odinger’s wave equation of quantum mechanics to the Hamilton-Jacobi equations of classical mechanics (see below)

for the real path of a ray of light Due to lack of space, further details on Fermat’s principle will not be described here We only mention that not only

Snell’s law of refraction (see above), but essentially the whole of lense optics,

including optical microscopy, can be derived from it.11

The eikonal approximation is significant because it contains the transition from (i) the wave picture of light, explored by Huygens and Young (with the basic issue of the ability to interfere; → holography, see below) to the (ii) particle representation.

In the particle representation, light rays without the ability to interfere

are interpreted as particle trajectories, while the wave fronts S( r) are

inter-preted as purely fictitious mathematical quantities This is analogous to the

transition from quantum mechanics to classical mechanics in semiclassical theories: one tries to solve the Schr¨odinger equation

−2 2m ∇2ψ + V ( r)ψ = Eψ ,

which plays the role of a matter-wave equation, by a kind of eikonal ansatz,

ψ( r) := A(r) · exp i

S( r)

(Here S( r) has the dimensionality of action, i.e., the same dimensionality

as Planck’s constant h (  is Planck’s constant divided by (2π).) An expansion

in powers of 1 (taking into account only the dominant terms analogously

to the eikonal ansatz ) leads to the so-called Hamilton-Jakobi equations of classical mechanics, viz

(gradS)2

2m + V ( r) ≡ E

The Hamilton-Jakobi equation contains the totality of classical

mechan-ics, e.g., the Hamilton function, the canonical equations and also Hamilton’s principle of least action, which ultimately corresponds to Fermat’s princi-ple.12

11See, e.g., Fliessbach’s book on electrodynamics, problem 36.2

12

The equivalence can be rather easily shown: Hamilton’s principle says that

Rt2

t1 dt L !

≡ extremal for variation of all virtual orbits in the space of gen-eralized coordinates q i, for fixed initial and final coordinates (The momenta

p i ≡ ∂L/∂ ˙q i result implicitly) Using a Legendre transformation, one replaces

L(q i , ˙ q i)→Pi p i q˙i − H, with the Hamilton function H(p i , q k), where the gener-alized coordinates and momenta are varied independently Meanwhile one sub-stitutesH by the constant E, i.e., one does not vary all orbits in phase space,

but only those with constantH ≡ E In this way one obtains the so-called

Mau-pertuis principle (→ Landau-Lifshitz I, [13], Chap 44) of classical mechanics In

the field of optics, it corresponds exactly to Fermat’s principle

188 21 Applications of Electrodynamics in the Field of Optics

21.3 Crystal Optics and Birefringence

For (i) fluids (gases and liquids), (ii) polycrystalline solids, (iii) amorphous substances and (iv) cubic crystals, the dielectric displacement D is

propor-tional to E, i.e., as assumed hitherto:

D = ε r ε0 E

In contrast, for non-cubic crystals plus all solid systems under strong uniaxial tension or compression, the behavior is more complicated For such systems

we have (for i = 1, 2, 3)

D i= 3



k=1

where the dielectric constants εi,k now form a tensor.

In the following we shall assume (i) that we are dealing with nonmagnetic material, such that the relative permeability is 1, and (ii) that there are no magnetic fields, such that

ε i,k ≡ ε k,i The tensor ε i,k can be diagonalized by a suitable rotation, i.e., in the new orthogonal basis it has the diagonal representation

ε α,β=

⎛

⎝ε1 0 , ε , 0 , 02, 0

0 , 0 , ε3

⎞

The quadratic form

we=1

2E · D ≡ ε0

2

3



i ,k=1

ε i,k E i E k

is thus diagonal in the eigenvector basis, i.e., there are analogous relations as for the mechanical rotational energy of a rigid body (see Part I)

The relation

we= 1

2E · D

for the energy density of the electric field corresponds in fact to the mechan-ical relation

TRot =1

2ω · L ,

and the mechanical relation

L i= 3



k=1

Θ i,k ω k ,

with the components Θi,k of the inertia tensor, corresponds to the electric

relation

D i /ε0=

k

ε i,k E k;

i.e., ω, L, and the inertia ellipsoid of classical mechanics correspond to the

quantitiesE, D and the ε i,k-ellipsoid of crystal optics.

However, as in mechanics, where in addition to the inertia ellipsoid (

ω-ellipsoid) there is also an equivalent second form, the so-called Binet ellipsoid

(L-ellipsoid), which can be used to express the rotational energy in the two

equivalent ellipsoid forms

2· TRot (i) = Θ1ω12+ Θ2ω22+ Θ23ω32(ii)= L

2

Θ1 +

L2

Θ2 +

L2

Θ3 ,

in the field of crystal optics it is also helpful to use two methods One may prefer either (i) theE-ellipsoid or (ii) the equivalent D-ellipsoid, as follows:

(i) if E and/or the Poynting vector S(= E × H) are preferred, then one

should use the Fresnel ellipsoid ( E-ellipsoid) On the other hand, (ii), if D

and/or the propagation vectork are involved, then one should use the index

ellipsoid ( D-ellipsoid) These two ellipsoids are:

we (i)= ε0

2 · ε1E12+ ε2E22+ ε23E32 (ii)= 1

2ε0 ·



D21 ε1 +

D22 ε2 +

D32 ε3



. (21.15)

For a givenE, the direction of the vector D is obtained in a similar way to

that in the mechanics of rigid bodies In that case the direction of the vectorL

for a givenω must be determined by means of a Poinsot construction: L has

the direction of the normal to the tangential plane belonging to ω, tangential

to the inertia ellipsoid Analogously, in the present case, the direction ofD has

to be determined by a Poinsot construction w.r.t.E, i.e., by constructing the

normal to the tangential plane of theE-ellipsoid, and vice versa In general,

the vectorE is thus rotated with respect to the direction of D (The vector

S is rotated with respect to the vector k by exactly the same amount about

the same rotation axis,H, see below.)

In fact, the three vectors S, E and H form a right-handed trihedron

(S ≡ E × H), similar to the three vectors k, D and H These statements

follow from Maxwell’s equations: the monochromatic ansatz

D ∝ D kei(k·r−ωt) ,

plus analogous assumptions for the other vectors, implies with

divD = 0 that k · D k ≡ 0

Similarly

divB = 0 implies that k · H k ≡ 0

190 21 Applications of Electrodynamics in the Field of Optics

Furthermore the relation

curlH = ∂ D

∂t implies that k × H k=−ωD k; and with

curlE = − ∂ B

∂t it follows that k × E k = μ0ωH k The four vectors D, E, k and S thus all lie in the same plane

perpen-dicular to H The vectors D and k (as well as E and S) are perpendicular

to each other E originates from D (and S from k) by a rotation with the

same angle about the common rotation axis H13.

One therefore distinguishes between (i) the phase velocity vphase, which is immediately related to the wave-propagation vectork, and (ii) the so-called

ray velocity vray, which is instead related to the energy-current density vector

S (These different velocities correspond to the phase and group velocities

discussed above.)

In most cases one of the three principal axes of the dielectric tensor, the c-axis, is crystallographically distinguished, and the other two orthogonal axes are equivalent, i.e., one is dealing with the symmetry of an ellipsoid of revolution.

This is true for tetragonal, hexagonal and trigonal14 crystal symmetry Only for orthorhombic15, monoclinic and triclinic crystal symmetry can all three eigenvalues of the dielectricity tensor be different (This exhausts all

crystal classes) In the first case one is dealing with so-called optically uniaxial crystals, in the latter case with optically biaxial crystals.

We have now arrived at the phenomenon of birefringence, and as a simple

example we shall consider a linearly polarized monochromatic electromag-netic wave with wavenumberk, incident at right angles from a vacuum onto

the surface of a non-cubic crystal In the interior of the crystal,k must also

be perpendicular to the surface For fixed k we should use the D-ellipsoid

(index ellipsoid ) which is assumed (by the symmetry of the crystal) in gen-eral to be inclined to the direction of incidence, i.e., the propagation vector

k is not necessarily parallel to a principal axis of the dielectric tensor.

The values of D k corresponding to the vector k are all located on an

elliptical section of the index ellipsoid with the plane perpendicular to k

through the origin

13

With this (somewhat rough) formulation we want to state that the unit vector

ˆ

E originates from ˆ D by the above-mentioned rotation.

14

This corresponds to the symmetry of a distorted cube stressed along one space

diagonal; the c-axis is this diagonal, and in the plane perpendicular to it one has

trigonal symmetry

15 This corresponds to the symmetry of a cube, which is differently strained in the three edge directions

for... we shall assume (i) that we are dealing with nonmagnetic material, such that the relative permeability is 1, and (ii) that there are no magnetic fields, such that

ε i,k...

The Hamilton-Jakobi equation contains the totality of classical

mechan-ics, e.g., the Hamilton function, the canonical equations and also Hamilton’s principle of least action, which