Electromagnetic fields and wavesphysics.fme.vutbr.cz/~jirka/FP/L_01_EM_fields_and_waves.pdf · • The concept of fields was introduced to explain “action at distance.” The physical

Electromagnetic fields and waves

Maxwell’s rainbow

Outline

• Maxwell’s equations

• Plane waves

• Pulses and group velocity

• Polarization of light

• Transmission and reflection at an interface

Macroscopic Maxwell’s equations

• The concept of fields was introduced to explain “action at distance.” The

physical observables are forces.

• Macroscopic Maxwell’s equations deal with fields that are local spatial

averages over microscopic fields associated with discrete charges. Charge

and current densities are considered as continuous functions of space.

Continuity equation

Take divergencezero for any

vector field

The conservation of charge is implicitly contained in Maxwell’s equations.

(continuity equation)

Constitutive relations

Maxwell’s equations are incomplete. The

fields are connected to one another by

constitutive relations (material equations)

describing the electromagnetic response

of media.

Polarization – material dependent!

+

++

+

−−−−

Magnetization

Relation between D and E

Assumptions:

• a linear medium (P is proportional to E)

• an isotropic medium

• an instantaneous response (no temporal dispersion)

• a local response (no spatial dispersion)

Response function (tensor)

+

++

+

−−−−

Response

Temporal dispersion: P (or D) at time t

depends on E at all times t′ previous to t

(non-instantaneous response).

Temporal dispersion is widely encountered

phenomenon and it is important to

accurately take it into account.

Spatial dispersion: P (or D) at a point [x,y,z] also depends on the

values of the electric field at neighboring points [x′,y′,z′]. A

spatially dispersive medium is therefore also called a nonlocal

medium. Nonlocal effects can be observed at interfaces between

diffrent media or in metallic objects with sizes comparable with

the mean-free path of electrons. In most cases of interest the

effect is very week and we can safely ignore it.

+

++

+

−−−−

(a bit of math)

use the four-dimensional

Fourier transform

(the Fourier transform ≡ FT)

(the inverse Fourier transform ≡ IFT)

the angular

frequency of

the field

the spatial

frequency of

the field



Fourier transform

the angular

frequency of

the field

the spatial

frequency of

the field

• non-instantaneous response leads to 𝜔-dependence

• non-local response leads to k-dependence

+

++

+

−−−−


+

++

+

−−−−

obtained in a similar fashion

• We will assume isotropic materials and ignore spatial

dispersion (k-dependence).

• We will discuss the frequency dependence later on.


Fourier transform

Solution in the frequency domain

time domain frequency domain

We assume a monochromatic field

i.e.

real

Complex amplitudes

they depend on the angular frequency

(the dependence is not explicitly shown)



time dependent

field (real)

The same Eqs. are

obtained for the spectral

components

its spectrum

(complex)

not shown

(FT)

(IFT)

(a consequence for the

spectral components)



(any field with – convention) = (the same field with + convention)*

two equally valid conventions for

expressing the time dependence

(the complex conjugate of the field)



For simpler notation, textbook authors often drop the argument in the

fields and material parameters. It is the context of the problem which

determines which of the fields is used.

?

Complex dielectric constant

no source current!

we will call it “complex”

dielectric constant and

denote again with

check this!

Complex dielectric constant

no source current!

• Just handy convention – it will be used here.

• The formulation does not distinguish between conduction

currents (free charges) and polarization currents (bound charges).

• Energy dissipation is associated with the imaginary part of the

dielectric constant.

... independent Eqs.

check that they follow from ...

we will solve them

TE and TM solutions (modes)

assume homogeneity in y-direction

TE solution: TM solution:

=> two independent sets of equations:

which have two independent sets of solutions

Symmetry:

Outline


• Plane waves




Homogeneous medium: plane wave

The magnitude of the wavevector

(=wavenumber) is related to the

angular frequency by the dispersion

equation.

monochromatic plane

wave solutions:

wavevector

Significance of plane waves: Any solution of the Eqs.

can be expressed as linear combination of plane waves.


Define

refractive index

free-space wavenumber

(alternative forms

of dispersion Eq.)electric

field

magnetic

field

raywavefronts

wavelength

phase velocity

(when n is real)


(vectors are generally complex)

?


The evanescent wave

• k is complex even for a purely real

refractive index n, i.e., nꞌꞌ = 0

• cannot occur in an infinite

homogenous medium (exponential

growth)

• we will disuse together with total

internal reflection

Evanescent fields play a central role in

nano-optics.

The homogeneous wave –

occurs for complex n

looks complicated

however, 2 important solutions are simple

dispersion Eq.

choose k in z direction =>

- define phase velocity

- acts as a “normal” n loss

(or gain)

• Define all the material properties

• Frequency dependent

• Transparent materials can be

described by a purely real refractive

index n

• Absorbing materials (or materials

with gain): n = nꞌ + inꞌꞌ is complex

valid for the most transparent media

Refractive index

- define phase velocity

- acts as a “normal” n loss

(or gain)

extinction coefficient

Refractive index

Poynting vector and optical intensity

The (instantaneous) Poynting vector

The time-averaged Poynting vector

dependence assumed

• the direction and magnitude of the energy flux

• the energy flux = the power per unit area [W/m2]

carried by the field

Photodetection measurements are slow compared

with the oscillation period of a wave. Therefore the

measured quantity is a time average ⟨… ⟩

power [W] = the

energy flux through

the given surface

Intensity (power flux, irradiance)

a unit vector

normal to the

surface

Poynting vector and optical intensity

For a monochromatic plane wave

often

The time-averaged Poynting vector

dependence assumed

Photodetection measurements are slow compared

with the oscillation period of a wave. Therefore the

measured quantity is a time average ⟨… ⟩

(describe conditions)

Plane wave, beam, ray

A ray is a line drawn in space corresponding to the

direction of flow of radiant energy. In practice we can

produce beams (pencils) of light and we can imagine a ray

as the limit on the narrowness of such a beam. In

homogenous isotropic materials, rays are straight lines

parallel to k-vector.

Geometrical optics is an approximate method that describes

light propagation in terms of rays. It is applicable when

dimensions of obstacles (lenses, mirrors) and also widths of

beams >> wavelength.

Outline


• Plane waves




Optical pulse (wave packet)

• Monochromatic waves are idealizations

never strictly realized in practice

• “Any” wave can be expressed as a

superposition of monochromatic waves

(IFT) (see page 13)

found by solving the Maxwell’s Eqs. in the frequency domain

For monochromatic plane

waves propagating in z

“negative” frequencies = positive

frequencies with the minus sign

“pulse”


(pulse = superposition of

monochromatic plane waves)

The longer the pulse in time, the narrower the

spread of the spectra in the frequency domain.


Example: an idealized pulse with duration Δ𝑡 and frequency 𝜔0



sharply peaked around

Propagation of optical pulse



no dispersion weak dispersion

In a dispersive medium, each monochromatic plane propagates at its own phase velocity

=> pulse becomes distorted (=dispersion)

Propagation of optical pulse



assume it is real

for simplicity

(small)

(fast varying) phase factor slowly varying envelope that

travels with a velocity

(group

velocity)

sharply peaked around

Group velocity

Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot

moves with the phase velocity, and the green dots propagate with the group velocity. In this

deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two

green dots when moving from the left to the right of the figure.

http://en.wikipedia.org/wiki/Group_velocity

(fast varying) phase factor slowly varying envelope that

travels with a velocity

(group

velocity)

Three velocities

(group velocity)

(phase velocity)

(energy velocity)

energy density not valid in dispersive media!

more details: [Novotny and Hecht] sect. 2.11;

[J. D. Jackson, Classical Electrodynamics. New York: Wiley, 3rd ed. (1999)] page 263, Eq. (6.126b);

or [L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press (1960)]

The velocities need to be carefully examined to avoid wrong interpretations!

- lies in the direction of the wavevector

- perpendicular to phase fronts

Outline


• Plane waves




Polarization of light


incident ray, intensity 𝐼0

unpolarized light

polarizing sheet

linearly polarized when incident light (with

intensity 𝐼0) is linearly

polarized

when incident light is

unpolarized


(polarizer)

(analyzer)

when 𝜃 = 900 (crossed polarizers)


The polarizer sheet absorbs radiation

polarized in a direction parallel to the

long molecules; radiation perpendicular

to them passes trough.

with polarizerno filter

http://en.wikipedia.org/wiki/File:CircularPolarizer.jpg

http://en.wikipedia.org/wiki/Polarization_%28waves%29


Elliptically polarized wave

http://en.wikipedia.org/wiki/Polarization_%28waves%29

z

y

x

(ellipse)

Elliptically polarized wave

(ellipse)

when 𝑎𝑥 = 𝑎𝑦

𝛿

𝛿

Outline


• Plane waves




Transmission and reflection at an interface

?

?

(for the chosen coordinate system)

The plane of incidence: defined by

the incident ray and a line normal to

the interface - here (x,z) plane.

We use continuity of the tangent E- and H-field

and obtain:

1) the laws of reflection and refraction

2) relations for the amplitudes of the reflected

and transmitted waves (Fresnel reflection

and transmission coefficients)

1) Laws of reflection and refraction

The transverse components of the k-vectors are

conserved (here: y and z component) =>

All three k-vectors lie in the plane of incidence.


The plane of incidence: defined by

the incident ray and a line normal to

the interface - here (x,z) plane.

and ...

1) Laws of reflection and refraction

... the longitudinal components (x) of the k-vectors

can be calculated as

(dispersion Eq.)


transmitted wave

reflected wave

To left

eye

To right

eye

Air

Chromatic dispersion

sunlight water drops

to observer

Rainbow

http://www.atoptics.co.uk/rainbows/primcone.htm

TE polarization, s, perpendicular

(continuity)

2) Fresnel reflection and transmission

coefficients

TM polarization, p, parallel

(Fresnel

coefficients)

Total internal

reflection

Brewster angle

(reflectance and transmittance)

For now:

Brewster angle

Brewster angle

for TM polarization

The reflected light is fully polarized (TE polarization)

Brewster angle

The reflected light is fully polarized (TE polarization)

Total internal reflection

Total internal

reflection

Evanescent wave

Check that this is a pure

imaginary number in the case

of total internal reflection.

Evanescent wave

In the medium with

1) the field propagates along the surface

(z direction)

2) the field does not propagate in x but

rather decays exponentially

3) no time-averaged energy flux in x

Documents

Electromagnetic fields and wavesphysics.fme.vutbr.cz/~jirka/FP/L_01_EM_fields_and_waves.pdf · • The concept of fields was introduced to explain “action at distance.” The physical