Light Propagation in Media
Boundary problems
Absorption coefficient, a,
and
refractive index, n.
Reflected and refracted
beams from surfaces – the
Snell’s law
Incident, transmitted, and
reflected beams determined
by boundary conditions
The "Fresnel Equations"
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Light is an Electromagnetic Wave

~ i ( kx t )
E y (r , t )  E y e

~ i ( kx t )
Bz ( r , t )  Bz e
2

E
2
 E   2  0
t
2

B
2
 B   2  0
dt
1. The electric field, the magnetic field, and the k-vector are all perpendicular:
EB  k
2. The electric and magnetic fields are in phase.
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c
1
 0 0
 3 *108 m / s
Wave Fronts
At a given time, a wave's "wave-fronts" are the planes where the
wave has its maxima.
A plane wave's wave-fronts are
equally spaced, a wavelength
apart.
They're perpendicular to the
propagation direction, and they
propagate with time.
Wave-fronts are helpful for drawing
pictures of interfering waves.
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Wave Propagation in Media
Typically, the speed of light, the wavelength, and the amplitude decrease.
Vacuum (or air)
Medium
n=1
n=2

Absorption depth = 1/
Absorptive
nk
k
n
E(x,t) = E0 exp[i(kx – t)]
Wavelength decreases
Dispersive
E(x,t) = E0exp[(–/2)x]exp[i(nkx– t)]
where  is the "absorption coefficient" and n is the "refractive index."
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I(z) = I(0) exp(- x)
Polarization Notation
Parallel ("p")
polarization
Perpendicular("s")
polarization
Note the little lines and circles.
“P” polarization is the parallel
polarization, and it lies parallel to the
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plane of incidence.
y
z
x
“S” polarization is the perpendicular
polarization, and it sticks up out of the
plane of incidence
Light Encounters A Surface
Constructive interference occurs
for a reflected beam if the angle
of incidence = the angle of reflection.
Constructive interference occurs
for a transmitted beam if the sine of
the angle of incidence = sine of the
angle of "refraction." (Snell's Law)
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Refraction and Snell's Law
AD = BD/sin(qi)
AD = AE/sin(qt)
qi
So:
BD/sin(qi) = AE/sin(qt)
But:
BD = vi Dt = (c0/ni) Dt
&
AE = vt Dt = (c0/nt ) Dt
So:
(c0/ni) Dt / sin(qi)
= (c0/nt) Dt / sin(qt)
Or:
ni sin(qi) = nt sin(qt)
qt
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Snell's Law Explains Many Effects
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Reflection and Transmission of Waves
y A  yB
y A y B

x
x
y1 t  x c A  y3 t  x cB 
apply continuity conditions
for separate components
y2 t  x c A 
y


x
x
y A  y1  t    y2  t  
 cA 
 cA 

x
y B  y3  t  
 cB 
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A
Reflection
Coefficient
Transmission
Coefficient
combine forward and
reflected waves to give total
fields for each region
B
x
hence derive fractional
transmission and reflection
y2 t  cB  c A
r12 

y1 t  c A  cB
y3 t 
2c B
t12 

y1 t  cB  c A 9
Fresnel's Equations for
Reflection and Refraction
Wave equations
Boundary conditions
ki
Er
Ei
Bi
kr
ni
Br
qi qr
Interface
qt
Et
Bt
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nt
kt
Practical Applications of Fresnel’s
Equations
Lasers use Brewster’s angle components to avoid reflective losses:
R = 100%
0% reflection!
Laser medium
R = 90%
0% reflection!
Optical fibers use total internal reflection. Hollow fibers use highincidence-angle near-unity reflections.
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Fresnel's Equations for
Reflection and Refraction
Wave equations
Boundary conditions: tangential fields
are continuous
ki
Er
Ei
Bi
kr
ni
Br
qi qr
Interface
qt
Et
Bt
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nt
kt
Fresnel Equations
We would like to compute the fraction of a light wave reflected and
transmitted by a flat interface between two media with different refractive indices. Fresnel was the first to do this calculation.
ki
Er
Ei
Bi
kr
Br
qi qr
Interface
Beam geometry
for light with its
electric field perpendicular to the
plane of incidence
(i.e., out of the page)
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y
ni
z
qt
Et
Bt
nt
kt
x
Boundary Condition for the Electric
Field at an Interface
The Tangential Electric Field is Continuous
ki
The total E-field in
the plane of the
interface is
continuous.
Er
Ei
Bi
qi qr
y
x
z
kr
ni
Br
Interface
qt
Et
Here, all E-fields are
in the z-direction,
Bt
kt
which is in the plane
of the interface (xz),
so:
Ei(x, y = 0, z, t) + Er(x, y = 0, z, t) = Et(x, y = 0, z, t)
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nt
Boundary Condition for the Electric
Field at an Interface
The Tangential Magnetic Field* is Continuous
ki
The total B-field in
the plane of the
interface is
continuous.
Here, all B-fields are
in the xy-plane, so we
take the x-components:
Er
Ei
Bi
Interface
x
z
kr
ni
Br
qi qr
qt
Et
Bt
nt
kt
–Bi(x, y = 0, z, t) cos(qi) + Br(x, y = 0, z, t) cos(qr) = –Bt(x, y = 0, z, t)
cos(qt)
*It's really the tangential B/, but we're using   0
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y
Reflection and Transmission for
Perpendicularly Polarized Light
Ignoring the rapidly varying parts of the light wave and keeping
only the complex amplitudes:
E0i  E0 r  E0t
 B0i cos(qi )  B0 r cos(q r )   B0t cos(qt )
But B  E /(c0 / n)  nE / c0 and qi  q r :
ni ( E0 r  E0i ) cos(qi )  nt E0t cos(qt )
Substituting for E0t using E0i  E0 r  E0t :
ni ( E0 r  E0i ) cos(qi )  nt ( E0 r  E0i ) cos(qt )
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Reflection & Transmission Coefficients
for Perpendicularly Polarized Light
Rearranging ni ( E0 r  E0i ) cos(qi )  nt ( E0 r  E0i ) cos(qt ) yields:
E0 r  ni cos(qi )  nt cos(qt )   E0i  ni cos(qi )  nt cos(qt ) 
Solving for E0 r / E0i yields the reflection coefficient :
r  E0 r / E0i   ni cos(q i )  nt cos(q t ) /  ni cos(q i )  nt cos(q t )
Analogously, the transmission coefficient, E0t / E0i , is
t  E0t / E0i  2ni cos(q i ) /  ni cos(q i )  nt cos(q t ) 
These equations are called the Fresnel Equations for
perpendicularly polarized light.
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Fresnel Equations—Parallel E Field
y
ki
kr
Ei
Bi
Br
qi qr
ni
×
z
Er
Interface
Beam geometry
for light with its
electric field
parallel to the
plane of incidence
(i.e., in the page)
qt
Et
Bt
nt
kt
Note that the reflected magnetic field must point into the screen to
achieve E  B  k . The x means “into the screen.”
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x
Reflection & Transmission Coefficients
for Parallel Polarized Light
For parallel polarized light,
and
B0i - B0r = B0t
E0icos(qi) + E0rcos(qr) = E0tcos(qt)
Solving for E0r / E0i yields the reflection coefficient, r||:
r||  E0 r / E0i   ni cos(qt )  nt cos(qi ) /  ni cos(qt )  nt cos(qi )
Analogously, the transmission coefficient, t|| = E0t / E0i, is
t||  E0t / E0i  2ni cos(qi ) /  ni cos(qt )  nt cos(qi )
These equations are called the Fresnel Equations for parallel
polarized light.
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Reflection & Transmission Coefficients
for an Air-to-Glass Interface
nair  1 < nglass  1.5
Total reflection at q = 90°
for both polarizations
Zero reflection for parallel
polarization at 56.3°
“Brewster's angle”
(For different refractive
indices, Brewster’s angle
will be different.)
Reflection coefficient, r
Note:
1.0
Brewster’s angle
.5
r||=0!
0
r
┴
-.5
-1.0
0°
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r||
30°
60°
90°
Incidence angle, qi
Reflection Coefficients for a Glassto-Air Interface
1.0
Note:
Total internal reflection
above the "critical angle"
qcrit  arcsin(nt /ni)
(The sine in Snell's Law
can't be > 1!)
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Reflection coefficient, r
nglass  1.5 > nair  1
Critical
angle
r
┴
.5
Total internal
reflection
0
Brewster’s
angle
-.5
Critical
angle
r||
-1.0
0°
30°
60°
90°
Incidence angle, qi