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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
Constitutive relations
use the four-dimensional
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
+
++
+
−−−−
Constitutive relations
+
++
+
−−−−
obtained in a similar fashion
• We will assume isotropic materials and ignore spatial
dispersion (k-dependence).
• We will discuss the frequency dependence later on.
use the four-dimensional
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)
Solution in the frequency domain
time domain frequency domain
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)
Solution in the frequency domain
time domain frequency domain
(any field with – convention) = (the same field with + convention)*
two equally valid conventions for
expressing the time dependence
(the complex conjugate of the field)
Solution in the frequency domain
time domain frequency domain
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
• Maxwell’s equations
• Plane waves
• Pulses and group velocity
• Polarization of light
• Transmission and reflection at an interface
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.
Homogeneous medium: plane wave
Define
refractive index
free-space wavenumber
(alternative forms
of dispersion Eq.)electric
field
magnetic
field
raywavefronts
wavelength
phase velocity
(when n is real)
Homogeneous medium: plane wave
(vectors are generally complex)
?
Homogeneous medium: plane wave
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
• Maxwell’s equations
• Plane waves
• Pulses and group velocity
• Polarization of light
• Transmission and reflection at an interface
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”
Optical pulse (wave packet)
(pulse = superposition of
monochromatic plane waves)
The longer the pulse in time, the narrower the
spread of the spectra in the frequency domain.
Optical pulse (wave packet)
Example: an idealized pulse with duration Δ𝑡 and frequency 𝜔0
(pulse = superposition of
monochromatic plane waves)
sharply peaked around
Propagation of optical pulse
(pulse = superposition of
monochromatic plane waves)
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
(pulse = superposition of
monochromatic plane waves)
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
• Maxwell’s equations
• Plane waves
• Pulses and group velocity
• Polarization of light
• Transmission and reflection at an interface
Polarization of light
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
Polarization of light
(polarizer)
(analyzer)
when 𝜃 = 900 (crossed polarizers)
Polarization of light
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
Polarization of light
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
• Maxwell’s equations
• Plane waves
• Pulses and group velocity
• Polarization of light
• Transmission and reflection at an interface
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.
(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.
and ...
1) Laws of reflection and refraction
... the longitudinal components (x) of the k-vectors
can be calculated as
(dispersion Eq.)
(for the chosen coordinate system)
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