The 3D wave equation
Plane wave
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Spherical wave
Planar and Spherical Wavefronts
Planar wavefront (plane wave):
The wave phase is constant along a
planar surface (the wavefront).
As time evolves, the wavefronts propagate
at the wave speed without changing;
we say that the wavefronts are invariant to
propagation in this case.
Spherical wavefront (spherical wave):
The wave phase is constant along a
spherical surface (the wavefront).
As time evolves, the wavefronts propagate
at the wave speed and expand outwards
while preserving the wave’s energy.
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Wavefronts, rays, and wave vectors
Rays are:
k
1) normals to the wavefront surfaces
2) trajectories of “particles of light”
Wave vectors:
At each point on the wavefront, we may
assign a normal vector k
k
k
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This is known as the wave vector;
it magnitude k is the wave number and it
is defined as
3D wave vector from the wave equation
wa
x
kx
ve
k
ky
y
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nt
tor
ec
ve v
wa
fro
kz
z
3D wave vector and the Descartes sphere
The wave vector represents the
momentum of the wave.
Consistent with Geometrical Optics,
its magnitude is constrained to be
proportional to the refractive index n
(2π/λfree is a normalization factor)
In wave optics, the Descartes sphere
is also known as Ewald sphere
or simply as the k-sphere.
(Ewald sphere may be familiar to you
from solid state physics)
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Spherical wave
“point”
source
Outgoing
wavefronts
(spherical)
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Outgoing
rays
Dispersive waves
Dispersion curves for glass
Fig. 9X,Y in Jenkins, Francis A., and Harvey E. White.
Fundamentals of Optics. 4th ed. New York, NY:
McGraw-Hill, 1976. ISBN: 9780070323308.
(c) McGraw-Hill. All rights reserved. This content
is excluded from our Creative Commons license.
For more information, see http://ocw.mit.edu/fairuse .
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cut-off
frequency
Dispersion diagram for
metal waveguide of width
a=1µm
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Superposition of waves at different frequencies
Fig. 7.16a,b,c in Hecht, Eugene. Optics. Reading, MA: Addison-Wesley,
2001. ISBN: 9780805385663. (c) Addison-Wesley. All rights reserved.
This content is excluded from ourCreative Commons license.
For more information, see http://ocw.mit.edu/fairuse .
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Group and phase velocity
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Spatial frequencies
We now turn to a monochromatic (single color) optical field.
The field is often observed (or measured) at a planar surface along the optical axis z. The wavefront shape at the observation plane is, therefore, of particular interest.
θ
x
observation planes
x
z
Plane wave
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Spherical wave
Spatial frequencies
Plane wave
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Spherical wave
Today
•
Electromagnetics
– Electric (Coulomb) and magnetic forces
– Gauss Law: electrical
– Gauss Law: magnetic
– Faraday’s Law
– Ampère-Maxwell Law
– Maxwell’s equations ⇒ Wave equation
– Energy propagation
• Poynting vector
• average Poynting vector: intensity
– Calculation of the intensity from phasors
• Intensity
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Electric and magnetic forces
I
r
q
+
+
q´
F
dl
free charges
r
F
I´
Coulomb force
(dielectric) permitivity
of free space
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Magnetic force
(magnetic) permeability
of free space
Electric and magnetic fields
Observation
Generation
F
electric E
field
magnetic
induction
⊗
B
E
velocity
v
+
+
q
electric
charge
v
B
Lorentz force
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static charge:
⇒
electric field
+
+
moving charge
(electric current):
⇒
magnetic field
Gauss Law: electric field
E
+
E
da
da
+
V
A
Gauss theorem
ρ: charge density
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Gauss Law: magnetic field
there are no
magnetic
charges
V
A
Gauss theorem
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da
B
Faraday’s Law: electromotive force
E
dl
B(t) (in/de)creasing
A
C
Stokes theorem
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Ampère-Maxwell Law: magnetic induction
B
dl
B
dl
E
I
capacitor
current
A
A
C
C
Stokes theorem
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Maxwell’s Equations (free space)
Integral form
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Differential form
Wave Equation for electromagnetic waves
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2.71 / 2.710 Optics
Spring 2009
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