Lecture 2: Electromagnetic Waves in Isotropic Media 1
Outline
1
2
3
4
Electromagnetic Waves
Quasi-Monochromatic Light
Electromagnetic Waves Across Interfaces
Snell’s law
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
1
Fundamentals of Polarized Light
Electromagnetic Waves in Matter
Maxwell’s equations ⇒ electromagnetic waves
optics: interaction of electromagnetic waves with matter as
described by material equations
polarization of electromagnetic waves are integral part of optics
Maxwell’s Equations in Matter
~ = 4πρ
∇·D
~
~ − 1 ∂ D = 4π~j
∇×H
c ∂t
c
~
∂
B
1
~ +
∇×E
=0
c ∂t
~ =0
∇·B
Christoph U. Keller, C.U.Keller@astro-uu.nl
Symbols
~
D
ρ
~
H
c
~j
~
E
~
B
t
electric displacement
electric charge density
magnetic field
speed of light in vacuum
electric current density
electric field
magnetic induction
time
Lecture 2: Electromagnetic Waves in Isotropic Media 1
2
Fundamentals of Polarized Light
Electromagnetic Waves in Matter
Maxwell’s equations ⇒ electromagnetic waves
optics: interaction of electromagnetic waves with matter as
described by material equations
polarization of electromagnetic waves are integral part of optics
Maxwell’s Equations in Matter
~ = 4πρ
∇·D
~
~ − 1 ∂ D = 4π~j
∇×H
c ∂t
c
~
∂
B
1
~ +
∇×E
=0
c ∂t
~ =0
∇·B
Christoph U. Keller, C.U.Keller@astro-uu.nl
Symbols
~
D
ρ
~
H
c
~j
~
E
~
B
t
electric displacement
electric charge density
magnetic field
speed of light in vacuum
electric current density
electric field
magnetic induction
time
Lecture 2: Electromagnetic Waves in Isotropic Media 1
2
Fundamentals of Polarized Light
Electromagnetic Waves in Matter
Maxwell’s equations ⇒ electromagnetic waves
optics: interaction of electromagnetic waves with matter as
described by material equations
polarization of electromagnetic waves are integral part of optics
Maxwell’s Equations in Matter
~ = 4πρ
∇·D
~
~ − 1 ∂ D = 4π~j
∇×H
c ∂t
c
~
∂
B
1
~ +
∇×E
=0
c ∂t
~ =0
∇·B
Christoph U. Keller, C.U.Keller@astro-uu.nl
Symbols
~
D
ρ
~
H
c
~j
~
E
~
B
t
electric displacement
electric charge density
magnetic field
speed of light in vacuum
electric current density
electric field
magnetic induction
time
Lecture 2: Electromagnetic Waves in Isotropic Media 1
2
Fundamentals of Polarized Light
Electromagnetic Waves in Matter
Maxwell’s equations ⇒ electromagnetic waves
optics: interaction of electromagnetic waves with matter as
described by material equations
polarization of electromagnetic waves are integral part of optics
Maxwell’s Equations in Matter
~ = 4πρ
∇·D
~
~ − 1 ∂ D = 4π~j
∇×H
c ∂t
c
~
∂
B
1
~ +
∇×E
=0
c ∂t
~ =0
∇·B
Christoph U. Keller, C.U.Keller@astro-uu.nl
Symbols
~
D
ρ
~
H
c
~j
~
E
~
B
t
electric displacement
electric charge density
magnetic field
speed of light in vacuum
electric current density
electric field
magnetic induction
time
Lecture 2: Electromagnetic Waves in Isotropic Media 1
2
Linear Material Equations
Symbols
dielectric constant
~ = E
~
D
~ = µH
~
B
µ magnetic permeability
σ electrical conductivity
~j = σ E
~
Isotropic and Anisotropic Media
isotropic media: and µ are scalars
anisotropic media: and µ are tensors of rank 2
isotropy of medium broken by
anisotropy of material itself (e.g. crystals)
external fields (e.g. Kerr effect)
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
3
Linear Material Equations
Symbols
dielectric constant
~ = E
~
D
~ = µH
~
B
µ magnetic permeability
σ electrical conductivity
~j = σ E
~
Isotropic and Anisotropic Media
isotropic media: and µ are scalars
anisotropic media: and µ are tensors of rank 2
isotropy of medium broken by
anisotropy of material itself (e.g. crystals)
external fields (e.g. Kerr effect)
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
3
Wave Equation in Matter
for most materials: ρ = 0, µ = 1
combine Maxwell, material equations ⇒ differential equations for
damped (vector) wave
~ −
∇2 E
~
~
∂2E
4πσ ∂ E
−
=0
c 2 ∂t 2
c 2 ∂t
~ −
∇2 H
~
~
∂2H
4πσ ∂ H
− 2
=0
2
2
c ∂t
c ∂t
damping controlled by conductivity σ
~ and H
~ are equivalent ⇒ sufficient to consider E
~
E
~
interaction with matter almost always through E
~ are crucial
but: at interfaces, boundary conditions for H
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
4
Wave Equation in Matter
for most materials: ρ = 0, µ = 1
combine Maxwell, material equations ⇒ differential equations for
damped (vector) wave
~ −
∇2 E
~
~
∂2E
4πσ ∂ E
−
=0
c 2 ∂t 2
c 2 ∂t
~ −
∇2 H
~
~
∂2H
4πσ ∂ H
− 2
=0
2
2
c ∂t
c ∂t
damping controlled by conductivity σ
~ and H
~ are equivalent ⇒ sufficient to consider E
~
E
~
interaction with matter almost always through E
~ are crucial
but: at interfaces, boundary conditions for H
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
4
Wave Equation in Matter
for most materials: ρ = 0, µ = 1
combine Maxwell, material equations ⇒ differential equations for
damped (vector) wave
~ −
∇2 E
~
~
∂2E
4πσ ∂ E
−
=0
c 2 ∂t 2
c 2 ∂t
~ −
∇2 H
~
~
∂2H
4πσ ∂ H
− 2
=0
2
2
c ∂t
c ∂t
damping controlled by conductivity σ
~ and H
~ are equivalent ⇒ sufficient to consider E
~
E
~
interaction with matter almost always through E
~ are crucial
but: at interfaces, boundary conditions for H
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
4
Wave Equation in Matter
for most materials: ρ = 0, µ = 1
combine Maxwell, material equations ⇒ differential equations for
damped (vector) wave
~ −
∇2 E
~
~
∂2E
4πσ ∂ E
−
=0
c 2 ∂t 2
c 2 ∂t
~ −
∇2 H
~
~
∂2H
4πσ ∂ H
− 2
=0
2
2
c ∂t
c ∂t
damping controlled by conductivity σ
~ and H
~ are equivalent ⇒ sufficient to consider E
~
E
~
interaction with matter almost always through E
~ are crucial
but: at interfaces, boundary conditions for H
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
4
Plane-Wave Solutions
~~
~ =E
~ 0 ei (k ·x −ωt )
Plane Vector Wave ansatz E
~k
~k
~
|k |
~x
ω
t
~0
E
spatially and temporally constant wave vector
normal to surfaces of constant phase
wave number
spatial location
angular frequency (2π× frequency)
time
(generally complex) vector independent of time and space
~ =E
~ 0 e−i (~k ·~x −ωt )
could also use E
damping if ~k is complex
~
real electric field vector given by real part of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
5
Plane-Wave Solutions
~~
~ =E
~ 0 ei (k ·x −ωt )
Plane Vector Wave ansatz E
~k
~k
~
|k |
~x
ω
t
~0
E
spatially and temporally constant wave vector
normal to surfaces of constant phase
wave number
spatial location
angular frequency (2π× frequency)
time
(generally complex) vector independent of time and space
~ =E
~ 0 e−i (~k ·~x −ωt )
could also use E
damping if ~k is complex
~
real electric field vector given by real part of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
5
Plane-Wave Solutions
~~
~ =E
~ 0 ei (k ·x −ωt )
Plane Vector Wave ansatz E
~k
~k
~
|k |
~x
ω
t
~0
E
spatially and temporally constant wave vector
normal to surfaces of constant phase
wave number
spatial location
angular frequency (2π× frequency)
time
(generally complex) vector independent of time and space
~ =E
~ 0 e−i (~k ·~x −ωt )
could also use E
damping if ~k is complex
~
real electric field vector given by real part of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
5
Plane-Wave Solutions
~~
~ =E
~ 0 ei (k ·x −ωt )
Plane Vector Wave ansatz E
~k
~k
~
|k |
~x
ω
t
~0
E
spatially and temporally constant wave vector
normal to surfaces of constant phase
wave number
spatial location
angular frequency (2π× frequency)
time
(generally complex) vector independent of time and space
~ =E
~ 0 e−i (~k ·~x −ωt )
could also use E
damping if ~k is complex
~
real electric field vector given by real part of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
5
Complex Index of Refraction
temporal derivatives ⇒ Helmholtz equation
ω2
4πσ ~
2~
E =0
∇ E + 2 +i
ω
c
dispersion relation between ~k and ω
2
~k · ~k = ω
c2
complex index of refraction
4πσ
+i
ω
4πσ ~ ~
ω2
, k · k = 2 ñ2
ω
c
split into real (n: index of refraction) and imaginary parts (k :
extinction coefficient)
ñ = n + ik
ñ2 = + i
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
6
Complex Index of Refraction
temporal derivatives ⇒ Helmholtz equation
ω2
4πσ ~
2~
E =0
∇ E + 2 +i
ω
c
dispersion relation between ~k and ω
2
~k · ~k = ω
c2
complex index of refraction
4πσ
+i
ω
4πσ ~ ~
ω2
, k · k = 2 ñ2
ω
c
split into real (n: index of refraction) and imaginary parts (k :
extinction coefficient)
ñ = n + ik
ñ2 = + i
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
6
Complex Index of Refraction
temporal derivatives ⇒ Helmholtz equation
ω2
4πσ ~
2~
E =0
∇ E + 2 +i
ω
c
dispersion relation between ~k and ω
2
~k · ~k = ω
c2
complex index of refraction
4πσ
+i
ω
4πσ ~ ~
ω2
, k · k = 2 ñ2
ω
c
split into real (n: index of refraction) and imaginary parts (k :
extinction coefficient)
ñ = n + ik
ñ2 = + i
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
6
Complex Index of Refraction
temporal derivatives ⇒ Helmholtz equation
ω2
4πσ ~
2~
E =0
∇ E + 2 +i
ω
c
dispersion relation between ~k and ω
2
~k · ~k = ω
c2
complex index of refraction
4πσ
+i
ω
4πσ ~ ~
ω2
, k · k = 2 ñ2
ω
c
split into real (n: index of refraction) and imaginary parts (k :
extinction coefficient)
ñ = n + ik
ñ2 = + i
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
6
Transverse Waves
plane-wave solution must also fulfill Maxwell’s equations
~
~ 0 · ~k = 0, H
~ 0 · ~k = 0, H
~ 0 = ñ k × E
~0
E
|~k |
isotropic media: electric, magnetic field vectors normal to wave
vector ⇒ transverse waves
~ 0, H
~ 0 , and ~k orthogonal to each other, right-handed vector-triple
E
~ 0 and H
~ 0 out of phase
conductive medium ⇒ complex ñ, E
~
~
~
E0 and H0 have constant relationship ⇒ consider only E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
7
Transverse Waves
plane-wave solution must also fulfill Maxwell’s equations
~
~ 0 · ~k = 0, H
~ 0 · ~k = 0, H
~ 0 = ñ k × E
~0
E
|~k |
isotropic media: electric, magnetic field vectors normal to wave
vector ⇒ transverse waves
~ 0, H
~ 0 , and ~k orthogonal to each other, right-handed vector-triple
E
~ 0 and H
~ 0 out of phase
conductive medium ⇒ complex ñ, E
~
~
~
E0 and H0 have constant relationship ⇒ consider only E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
7
Transverse Waves
plane-wave solution must also fulfill Maxwell’s equations
~
~ 0 · ~k = 0, H
~ 0 · ~k = 0, H
~ 0 = ñ k × E
~0
E
|~k |
isotropic media: electric, magnetic field vectors normal to wave
vector ⇒ transverse waves
~ 0, H
~ 0 , and ~k orthogonal to each other, right-handed vector-triple
E
~ 0 and H
~ 0 out of phase
conductive medium ⇒ complex ñ, E
~
~
~
E0 and H0 have constant relationship ⇒ consider only E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
7
Transverse Waves
plane-wave solution must also fulfill Maxwell’s equations
~
~ 0 · ~k = 0, H
~ 0 · ~k = 0, H
~ 0 = ñ k × E
~0
E
|~k |
isotropic media: electric, magnetic field vectors normal to wave
vector ⇒ transverse waves
~ 0, H
~ 0 , and ~k orthogonal to each other, right-handed vector-triple
E
~ 0 and H
~ 0 out of phase
conductive medium ⇒ complex ñ, E
~
~
~
E0 and H0 have constant relationship ⇒ consider only E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
7
Transverse Waves
plane-wave solution must also fulfill Maxwell’s equations
~
~ 0 · ~k = 0, H
~ 0 · ~k = 0, H
~ 0 = ñ k × E
~0
E
|~k |
isotropic media: electric, magnetic field vectors normal to wave
vector ⇒ transverse waves
~ 0, H
~ 0 , and ~k orthogonal to each other, right-handed vector-triple
E
~ 0 and H
~ 0 out of phase
conductive medium ⇒ complex ñ, E
~
~
~
E0 and H0 have constant relationship ⇒ consider only E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
7
Energy Propagation in Isotropic Media
Poynting vector
~ = c E
~ ×H
~
S
4π
~ energy through unit area perpendicular to S
~ per unit time
|S|:
~
direction of S is direction of energy flow
time-averaged Poynting vector given by
D E
~ = c Re E
~0 × H
~∗
S
0
8π
Re real part of complex expression
∗
complex conjugate
h.i time average
energy flow parallel to wave vector (in isotropic media)
D E
~ = c |ñ| |E0 |2
S
8π
Christoph U. Keller, C.U.Keller@astro-uu.nl
~k
|~k |
Lecture 2: Electromagnetic Waves in Isotropic Media 1
8
Energy Propagation in Isotropic Media
Poynting vector
~ = c E
~ ×H
~
S
4π
~ energy through unit area perpendicular to S
~ per unit time
|S|:
~
direction of S is direction of energy flow
time-averaged Poynting vector given by
D E
~ = c Re E
~0 × H
~∗
S
0
8π
Re real part of complex expression
∗
complex conjugate
h.i time average
energy flow parallel to wave vector (in isotropic media)
D E
~ = c |ñ| |E0 |2
S
8π
Christoph U. Keller, C.U.Keller@astro-uu.nl
~k
|~k |
Lecture 2: Electromagnetic Waves in Isotropic Media 1
8
Energy Propagation in Isotropic Media
Poynting vector
~ = c E
~ ×H
~
S
4π
~ energy through unit area perpendicular to S
~ per unit time
|S|:
~
direction of S is direction of energy flow
time-averaged Poynting vector given by
D E
~ = c Re E
~0 × H
~∗
S
0
8π
Re real part of complex expression
∗
complex conjugate
h.i time average
energy flow parallel to wave vector (in isotropic media)
D E
~ = c |ñ| |E0 |2
S
8π
Christoph U. Keller, C.U.Keller@astro-uu.nl
~k
|~k |
Lecture 2: Electromagnetic Waves in Isotropic Media 1
8
Energy Propagation in Isotropic Media
Poynting vector
~ = c E
~ ×H
~
S
4π
~ energy through unit area perpendicular to S
~ per unit time
|S|:
~
direction of S is direction of energy flow
time-averaged Poynting vector given by
D E
~ = c Re E
~0 × H
~∗
S
0
8π
Re real part of complex expression
∗
complex conjugate
h.i time average
energy flow parallel to wave vector (in isotropic media)
D E
~ = c |ñ| |E0 |2
S
8π
Christoph U. Keller, C.U.Keller@astro-uu.nl
~k
|~k |
Lecture 2: Electromagnetic Waves in Isotropic Media 1
8
Quasi-Monochromatic Light
monochromatic light: purely theoretical concept
monochromatic light wave always fully polarized
real life: light includes range of wavelengths ⇒
quasi-monochromatic light
quasi-monochromatic: superposition of mutually incoherent
monochromatic light beams whose wavelengths vary in narrow
range δλ around central wavelength λ0
δλ
1
λ
measurement of quasi-monochromatic light: integral over
measurement time tm
amplitude, phase (slow) functions of time for given spatial location
slow: variations occur on time scales much longer than the mean
period of the wave
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
9
Quasi-Monochromatic Light
monochromatic light: purely theoretical concept
monochromatic light wave always fully polarized
real life: light includes range of wavelengths ⇒
quasi-monochromatic light
quasi-monochromatic: superposition of mutually incoherent
monochromatic light beams whose wavelengths vary in narrow
range δλ around central wavelength λ0
δλ
1
λ
measurement of quasi-monochromatic light: integral over
measurement time tm
amplitude, phase (slow) functions of time for given spatial location
slow: variations occur on time scales much longer than the mean
period of the wave
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
9
Quasi-Monochromatic Light
monochromatic light: purely theoretical concept
monochromatic light wave always fully polarized
real life: light includes range of wavelengths ⇒
quasi-monochromatic light
quasi-monochromatic: superposition of mutually incoherent
monochromatic light beams whose wavelengths vary in narrow
range δλ around central wavelength λ0
δλ
1
λ
measurement of quasi-monochromatic light: integral over
measurement time tm
amplitude, phase (slow) functions of time for given spatial location
slow: variations occur on time scales much longer than the mean
period of the wave
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
9
Quasi-Monochromatic Light
monochromatic light: purely theoretical concept
monochromatic light wave always fully polarized
real life: light includes range of wavelengths ⇒
quasi-monochromatic light
quasi-monochromatic: superposition of mutually incoherent
monochromatic light beams whose wavelengths vary in narrow
range δλ around central wavelength λ0
δλ
1
λ
measurement of quasi-monochromatic light: integral over
measurement time tm
amplitude, phase (slow) functions of time for given spatial location
slow: variations occur on time scales much longer than the mean
period of the wave
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
9
Polarization of Quasi-Monochromatic Light
electric field vector for quasi-monochromatic plane wave is sum of
electric field vectors of all monochromatic beams
~ (t) = E
~ 0 (t) ei (~k ·~x −ωt )
E
can write this way because δλ λ0
measured intensity of quasi-monochromatic beam
D
1
tm −>∞ tm
E D
E
~∗ + E
~ ∗ = lim
~xE
~yE
E
x
y
Z
tm /2
−tm /2
~ x (t)E
~ ∗ (t) + E
~ y (t)E
~ ∗ (t)dt
E
x
y
h· · · i: averaging over measurement time tm
measured intensity independent of time
quasi-monochromatic: frequency-dependent material properties
(e.g. index of refraction) are constant within ∆λ
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
10
Polarization of Quasi-Monochromatic Light
electric field vector for quasi-monochromatic plane wave is sum of
electric field vectors of all monochromatic beams
~ (t) = E
~ 0 (t) ei (~k ·~x −ωt )
E
can write this way because δλ λ0
measured intensity of quasi-monochromatic beam
D
1
tm −>∞ tm
E D
E
~∗ + E
~ ∗ = lim
~xE
~yE
E
x
y
Z
tm /2
−tm /2
~ x (t)E
~ ∗ (t) + E
~ y (t)E
~ ∗ (t)dt
E
x
y
h· · · i: averaging over measurement time tm
measured intensity independent of time
quasi-monochromatic: frequency-dependent material properties
(e.g. index of refraction) are constant within ∆λ
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
10
Polychromatic Light or White Light
wavelength range comparable wavelength ( δλ
λ ∼ 1)
incoherent sum of quasi-monochromatic beams that have large
variations in wavelength
cannot write electric field vector in a plane-wave form
must take into account frequency-dependent material
characteristics
intensity of polychromatic light is given by sum of intensities of
constituting quasi-monochromatic beams
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
11
Polychromatic Light or White Light
wavelength range comparable wavelength ( δλ
λ ∼ 1)
incoherent sum of quasi-monochromatic beams that have large
variations in wavelength
cannot write electric field vector in a plane-wave form
must take into account frequency-dependent material
characteristics
intensity of polychromatic light is given by sum of intensities of
constituting quasi-monochromatic beams
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
11
Polychromatic Light or White Light
wavelength range comparable wavelength ( δλ
λ ∼ 1)
incoherent sum of quasi-monochromatic beams that have large
variations in wavelength
cannot write electric field vector in a plane-wave form
must take into account frequency-dependent material
characteristics
intensity of polychromatic light is given by sum of intensities of
constituting quasi-monochromatic beams
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
11
Polychromatic Light or White Light
wavelength range comparable wavelength ( δλ
λ ∼ 1)
incoherent sum of quasi-monochromatic beams that have large
variations in wavelength
cannot write electric field vector in a plane-wave form
must take into account frequency-dependent material
characteristics
intensity of polychromatic light is given by sum of intensities of
constituting quasi-monochromatic beams
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
11
Electromagnetic Waves Across Interfaces
Fields at Interfaces
classical optics due to interfaces between 2 different media
from Maxwell’s equations in integral form at interface from
medium 1 to medium 2
~2 − D
~ 1 · ~n = 0
D
~2 − B
~ 1 · ~n = 0
B
~2 − E
~ 1 × ~n = 0
E
~2 − H
~ 1 × ~n = 0
H
~n normal on interface, points from medium 1 to medium 2
~ and B
~ are continuous across interface
normal components of D
~ and H
~ are continuous across interface
tangential components of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
12
Electromagnetic Waves Across Interfaces
Fields at Interfaces
classical optics due to interfaces between 2 different media
from Maxwell’s equations in integral form at interface from
medium 1 to medium 2
~2 − D
~ 1 · ~n = 0
D
~2 − B
~ 1 · ~n = 0
B
~2 − E
~ 1 × ~n = 0
E
~2 − H
~ 1 × ~n = 0
H
~n normal on interface, points from medium 1 to medium 2
~ and B
~ are continuous across interface
normal components of D
~ and H
~ are continuous across interface
tangential components of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
12
Electromagnetic Waves Across Interfaces
Fields at Interfaces
classical optics due to interfaces between 2 different media
from Maxwell’s equations in integral form at interface from
medium 1 to medium 2
~2 − D
~ 1 · ~n = 0
D
~2 − B
~ 1 · ~n = 0
B
~2 − E
~ 1 × ~n = 0
E
~2 − H
~ 1 × ~n = 0
H
~n normal on interface, points from medium 1 to medium 2
~ and B
~ are continuous across interface
normal components of D
~ and H
~ are continuous across interface
tangential components of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
12
Plane of Incidence
plane wave onto interface
incident (i ), reflected (r ), and
transmitted (t ) waves
~ i,r ,t
E
~ i,r ,t
H
~ i,r ,t ei (~k i,r ,t ·~x −ωt )
= E
0
c ~ i,r ,t ~ i,r ,t
k
×E
=
ω
interface normal ~n k z-axis
spatial, temporal behavior at interface the same for all 3 waves
(~k i · ~x )z=0 = (~k r · ~x )z=0 = (~k t · ~x )z=0
valid for all ~x in interface ⇒ all 3 wave vectors in one plane, plane
of incidence
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
13
Plane of Incidence
plane wave onto interface
incident (i ), reflected (r ), and
transmitted (t ) waves
~ i,r ,t
E
~ i,r ,t
H
~ i,r ,t ei (~k i,r ,t ·~x −ωt )
= E
0
c ~ i,r ,t ~ i,r ,t
k
×E
=
ω
interface normal ~n k z-axis
spatial, temporal behavior at interface the same for all 3 waves
(~k i · ~x )z=0 = (~k r · ~x )z=0 = (~k t · ~x )z=0
valid for all ~x in interface ⇒ all 3 wave vectors in one plane, plane
of incidence
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
13
Plane of Incidence
plane wave onto interface
incident (i ), reflected (r ), and
transmitted (t ) waves
~ i,r ,t
E
~ i,r ,t
H
~ i,r ,t ei (~k i,r ,t ·~x −ωt )
= E
0
c ~ i,r ,t ~ i,r ,t
k
×E
=
ω
interface normal ~n k z-axis
spatial, temporal behavior at interface the same for all 3 waves
(~k i · ~x )z=0 = (~k r · ~x )z=0 = (~k t · ~x )z=0
valid for all ~x in interface ⇒ all 3 wave vectors in one plane, plane
of incidence
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
13
Snell’s Law
spatial, temporal behavior the
same for all three waves
(~k i ·~x )z=0 = (~k r ·~x )z=0 = (~k t ·~x )z=0
~ ω
k = c ñ
ω, c the same for all 3 waves
Snell’s law
ñ1 sin θi = ñ1 sin θr = ñ2 sin θt
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
14
Snell’s Law
spatial, temporal behavior the
same for all three waves
(~k i ·~x )z=0 = (~k r ·~x )z=0 = (~k t ·~x )z=0
~ ω
k = c ñ
ω, c the same for all 3 waves
Snell’s law
ñ1 sin θi = ñ1 sin θr = ñ2 sin θt
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
14
Snell’s Law
spatial, temporal behavior the
same for all three waves
(~k i ·~x )z=0 = (~k r ·~x )z=0 = (~k t ·~x )z=0
~ ω
k = c ñ
ω, c the same for all 3 waves
Snell’s law
ñ1 sin θi = ñ1 sin θr = ñ2 sin θt
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
14
Monochromatic Wave at Interface
~ i,r ,t , B
~ i,r ,t = c ~k i,r ,t × E
~ i,r ,t
~ i,r ,t = c ~k i,r ,t × E
H
0
0
0
0
ω
ω
boundary conditions for monochromatic plane wave:
4 equations are not independent
only need to consider last two equations (tangential components
~ 0 and H
~ 0 are continuous)
of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
15
Monochromatic Wave at Interface
~ i,r ,t , B
~ i,r ,t = c ~k i,r ,t × E
~ i,r ,t
~ i,r ,t = c ~k i,r ,t × E
H
0
0
0
0
ω
ω
boundary conditions for monochromatic plane wave:
~ i + ñ2 E
~ r − ñ2 E
~ t · ~n = 0
ñ12 E
0
1 0
2 0
~k i × E
~ i + ~k r × E
~ r − ~k t × E
~ t · ~n = 0
0
0
0
~i +E
~r −E
~ t × ~n = 0
E
0
0
0
~k i × E
~ i + ~k r × E
~ r − ~k t × E
~ t × ~n = 0
0
0
0
4 equations are not independent
only need to consider last two equations (tangential components
~ 0 and H
~ 0 are continuous)
of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
15
Monochromatic Wave at Interface
~ i,r ,t , B
~ i,r ,t = c ~k i,r ,t × E
~ i,r ,t
~ i,r ,t = c ~k i,r ,t × E
H
0
0
0
0
ω
ω
boundary conditions for monochromatic plane wave:
~i +E
~r −E
~ t × ~n = 0
E
0
0
0
~k i × E
~ i + ~k r × E
~ r − ~k t × E
~ t × ~n = 0
0
0
0
4 equations are not independent
only need to consider last two equations (tangential components
~ 0 and H
~ 0 are continuous)
of E
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
15
Two Special Cases
TM, p
TE, s
1
electric field parallel to plane of incidence ⇒ magnetic field is
transverse to plane of incidence (TM)
2
electric field particular (German: senkrecht) or transverse to plane
of incidence (TE)
general solution as (coherent) superposition of two cases
choose direction of magnetic field vector such that Poynting vector
parallel, same direction as corresponding wave vector
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
16
Two Special Cases
TM, p
TE, s
1
electric field parallel to plane of incidence ⇒ magnetic field is
transverse to plane of incidence (TM)
2
electric field particular (German: senkrecht) or transverse to plane
of incidence (TE)
general solution as (coherent) superposition of two cases
choose direction of magnetic field vector such that Poynting vector
parallel, same direction as corresponding wave vector
Christoph U. Keller, C.U.Keller@astro-uu.nl
Lecture 2: Electromagnetic Waves in Isotropic Media 1
16