4. Wave Equations
Last Lecture
• Stops, Pupils, Windows
• Optical Magnifiers
• Microscopes
• Telescopes
This Lecture
• Equations of traveling waves
• Harmonic Waves
• Plane Waves
• EM Wave Equations
Interference Diffraction
When do we use Wave Optics?
Lih Y. Lin, http://www.ee.washington.edu/people/faculty/lin_lih/EE485/
4-1. One-Dimensional Wave Equation
' ( ') y = f x
( )
y = f x vt− ' ( ') y = f x
'
x = −x vt
Traveling waves
1-D wave pulse of arbitrary shape
( )
y = f x vt+
Move toward +x direction
Move toward -x direction
One-Dimensional Wave Equation
v O’(x’, y’)
O(x, y)
2 2
2
2 2
2 2 2
2 2 2 2
2 2
2 2 2
v v
v v
1 v
1 v
y y
x t
y y x f x f
t x t x t x
y y y x f x f
t t t
x t
x t t x x t x
f y y
x x t
∂ ′ = −
∂
′ ′
∂ ∂ ∂ ∂ ∂ ∂
= = = −
′ ′ ′
∂ ∂ ∂ ∂ ∂ ∂
′ ′
∂∂ = ∂ ∂∂ ∂⎛⎜⎝ ⎞⎟⎠ = ∂∂ ∂′⎝⎛⎜ ∂ ⎞⎟⎠∂∂ = ∂∂′⎛⎜⎝− ∂∂ ′⎞⎟⎠∂∂ = ∂∂ ′
∂ ∂ ∂
=
⇒
∂ ∂
′ =
∂ ∂ ∂
= ∂
⇒ ∂
1-D differential wave equation
2 2
2 2
v :
1 Now develop the general one D wave e
x x t
x x
y y x f x f
x x x x x x
y y y x f x f
x x x x x x x x
quation
x x
′ = −
∂ ′ =
∂
′ ′
∂ = ∂ ∂ = ∂ ∂ = ∂
′ ′ ′
∂ ∂ ∂ ∂ ∂ ∂
′ ′
∂∂ = ∂∂ ∂⎛⎜⎝ ∂ ⎞⎟⎠ = ∂∂ ∂′⎛⎜⎝∂ ⎞⎟⎠∂∂ = ∂∂ ∂′⎛⎜⎝∂ ′⎞⎟⎠∂∂ = ∂∂ ′
−
⇒
( )
y= f x±vt
One-dimensional Wave Equation
v = 1 m/s, -z v = 2 m/s, +x
4-2. Harmonic Waves
– Wavelength and Propagation Constant -
Harmonic Waves
( ) ( )
( ) ( )
[ ]
sin v cos v
, 2 :
sin v sin v
sin 2 v
2 2
,
y A k x t or y A k x t
Harmonic wave repeats after one wavelength changes phase of function by for fixed t
y A kx k t A k x k t
A kx k t
k k
Harmonic wave also repeats after one period T
π λ
π λ π π
λ
= ⎡⎣ ± ⎤⎦ = ⎡⎣ ± ⎤⎦
= + = ⎡⎣ + + ⎤⎦
= + +
= =
( ) ( )
[ ]
2 :
sin v sin v
sin v 2
1 v v
vT 2
2 v
changes phase of function by for fixed x
y A kx k t A kx k t T
A kx k t k k
T
π
π
π ν
π λ νλ
= + = ⎡⎣ + + ⎤⎦
= + +
= = = =
∴ =
Propagation constant (전파상수)
v c
n k
= = ω
k ω
n1 n2
Called light line
Propagation velocity (전파속도)
Harmonic Waves - Period and Frequency -
Harmonic Waves as Complex Numbers
Plane Waves and Spherical Waves
3-D Wave Equation and Helmholtz Equation
4-7. Other Harmonic Waves
Cylindrical waves
Gaussian beams
( )
i k t
A e ρ ω
ψ ρ
= ±
Spot size : w(z) Beam waist : wo
4-8. Electromagnetic Waves
0 sin( )
EG = EG k rG ⋅ −G ωt
0 sin( )
BG = BG k rG ⋅ −G ωt EG = c BG
Let’s derive the EM Wave Equations from Maxwell’s Equations
Physical meaning of the Electric flux density (or, Electric displacement)
For a linear, homogeneous, isotropic, and nondispersive media,
12 2 2 2 3
0 8.854 10 C /J m C s /kg m : permittivity of vacuum
ε = × − ⎡⎣ ⋅ = ⋅ ⋅ ⎤⎦
Physical meaning of the Magnetic flux density
M H
For most of materials in optical frequency, M = 0
7 2 2 2
4π 10− ⎡kg m C/ kg m A s/ ⎤
= × ⎣ ⋅ = ⋅ ⎦
Boundary conditions of EM waves
At the boundary surface :
Tangential components of E and H are continuous Normal components of D and B are continuous.
Ampere-Maxwell’s Law
Stokes theorem (very general)
Faraday’s Law (Faraday 1775-1836)
(dielectric insulator)
Linear, homogeneous, and isotropic media
In 3 dimension,
Cartesian
Cylindrical
Spherical
( 2 k2) ( )r 0, k n nko
ψ c ω
∇ + = = =
Helmholtz Equation
Energy density (energy per unit volume)
• Energy density stored in an electric field
• Energy density stored in a magnetic field
3
2 ,
2 1
m E J
uE = εo
3
2 ,
2 1
m B J
u
o
B = μ
c B = E
2
2 2
1 1
2 2
B o E
o
u E E u
c ε
= μ = =
2 1 2
E B o
o
u u u ε E B
μ
⎛ ⎞
= + = = ⎜ ⎟
⎝ ⎠
Total energy density
E B o
u = u u = ε cEB
Power
In free space, wave propagates with speed c
Power passing through A : Energy u V u Ac t
P ucA
t t t
⋅ Δ ⋅ Δ
= = = =
Δ Δ Δ
Power per unit area : 2
0
S P uc c EB
A ε
= = =
2 1
o
o
S ε c E B E B E H μ
⎛ ⎞
= × = ×⎜ ⎟ = ×
⎝ ⎠
G G G G G G G
Poynting vector
Irradiance (Intensity)
Irradiance (or, Intensity): time average of the power per unit area
2 2
0 0 0
2 2
0
sin ( )
1 1 1
2 2 2
e
o o o o o
o
S E I
I c E B k r t
I cE B cE c B
ε ω
ε ε
μ
≡ =
= ⋅ ±
⎛ ⎞
= = = ⎜ ⎟
⎝ ⎠ G
4-9. Light polarization
0 sin( )
BG = B kz −ωt y
x
y
z
0 sin( )
EG = E kz −ωt x
2 2
0 0 sin ( ) SG = ε cE kz −ωt z
Direction of the electric field = polarization
Linear, circular, and elliptical polarizations
Circular pol. : π phase different between x and y components
Circular polarization
Circular polarization
stems from the intrinsic angular momentum
(“spin”) of the photons that make up the beam.
Doppler Effect in light waves
A source of light waves moving to the right with velocity 0.7c.
The frequency is higher on the right, and lower on the left.
Since light waves can propagate in vacuum,
there is no longer physical basis for the distinction between moving observer and moving source.
ÆThere is one relative motion between them.
Æ Relativistic Doppler effect
Assume the observer and the source are moving away from each other with a relative velocity, v.
is the original frequency of the wave emitted from the source.
If they are approaching each other.
For sound waves,
