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

InterferenceDiffraction

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 22

2 2

2 2 2

2 2 2 2

2 2

2 2 2

v

v

v v

1v

1v

y y

xt

y y x f x ft x t x t x

y y y x f x ft t t

x t

x t t x x t xf y y

x x t

′∂= −

∂′ ′∂ ∂ ∂ ∂ ∂ ∂

= = = −′ ′ ′∂ ∂ ∂ ∂ ∂ ∂

′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠∂ ∂ ∂

=

⇒

∂ ∂

=′∂ ∂ ∂

=∂

⇒∂

1-D differential wave equation

2 2

2 2

v :

1

Now develop the general one D wave ex x t

xx

y y x f x fx x x x x x

y y y x f x fx 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 v22

,

y A k x t or y A k x t

Harmonic wave repeats after one wavelength changes phaseof function by for fixed t

y A kx k t A k x k t

A kx k t

k k

Harmonic wavealso repeats after one period T

π

λ

ππλ πλ

= ± = ±⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= + = + +⎡ ⎤⎣ ⎦= + +

= =

( ) ( )[ ]

2 :

sin v sin v

sin v 21 v vvT 2

2v

changes phaseof functionby for fixed x

y A kx k t A kx k t T

A kx k tkk

T

π

π

π νπ λ

νλ

= + = + +⎡ ⎤⎣ ⎦= + +

= = = =

∴ =

Propagation constant (전파상수)

v cn 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 tA e ρ ωψρ

±=

Spot size : w(z)

Beam waist : wo

4-8. Electromagnetic Waves

0 sin( )E E k r tω= ⋅ −

0 sin( )B B k r tω= ⋅ −

E c B=

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 30 8.854 10 / / : permittivity of vacuumC J m C s kg mε − ⎡ ⎤= × ⋅ = ⋅ ⋅⎣ ⎦

Physical meaning of the Magnetic flux density

HM

For most of materials in optical frequency, M = 0

7 2 2 24 10 / /kg m C kg m A sπ − ⎡ ⎤= × ⋅ = ⋅⎣ ⎦

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 2 ( ) 0, onk r k nkc

ψ ω∇ + = = =

Helmholtz Equation

Energy density (energy per unit volume)

• Energy density stored in an electric field

• Energy density stored in a magnetic field

32 ,

21

mJEu oE ε=

32 ,

21

mJBu

oB μ=

cEB =

22

2

1 12 2B o E

o

Eu E uc

εμ

= = =

2 21E B o

o

u u u E Bεμ

⎛ ⎞= + = = ⎜ ⎟

⎝ ⎠Total energy density

E B ou u u cEBε= =

PowerIn free space, wave propagates with speed c

Power passing through A :Energy u V u Ac tP ucA

t t t⋅Δ ⋅ Δ

= = = =Δ Δ Δ

Power per unit area : 20

PS uc c EBA

ε= = =

2 1o

o

S c E B E B E Hεμ

⎛ ⎞= × = × = ×⎜ ⎟

⎝ ⎠Poynting vector

Irradiance (Intensity)

Irradiance (or, Intensity): time average of the power per unit area

2 20 0 0

2 20

sin ( )

1 1 12 2 2

e

o o o o oo

S E I

I c E B k r t

cI cE B cE B

ε ω

ε εμ

≡ =

= ⋅ ±

⎛ ⎞= = = ⎜ ⎟

⎝ ⎠

4-9. Light polarization

0 sin( )B B kz t yω= −

x

y

z

0 sin( )E E kz t xω= − 2 20 0 sin ( )S cE kz t zε ω= −

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.

There is one relative motion between them.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,