8. Wave Equations 8. Wave Equations

Last Lecture• Eye• Optical Magnifiers• Microscopes• Telescopes

This Lecture• Wave Equations• Harmonic Waves• Plane Waves• EM Waves

Light is a Ray (Geometrical Optics)

1. Nature of light2. Production and measurement of light3. Geometrical optics4. Matrix methods in paraxial optics5. Aberration theory6. Optical instrumentation

Light is a Wave (Physical Optics)

25. Fourier optics16. Fraunhofer diffraction17. The diffraction grating18. Fresnel diffraction19. Theory of multilayer films20. Fresnel equations* Evanescent waves

26. Nonlinear optics

Light is a Wave (Physical Optics)

8. Wave equations27. optical properties of materials9. Superposition of waves10. Interference of light11. Optical interferometry12. Coherence13. Holography14. Matrix treatment of polarization15. Production of polarized light

Light is a Photon (Quantum Optics)

21. Laser basics22. Characteristics of laser beams23. Laser applications24. Fiber optics

So far …Wave Optics

InterferenceDiffraction

When do we use Wave Optics?

Lih Y. Lin, http://www.ee.washington.edu/people/faculty/lin_lih/EE485/

Regimes of (EM) Wave Optics

d > λ d < λd ~ λ

Micro lensDOE lensHybrid lensBLULED lightingBeam shaping…

Flexible BLUBeam shapingLED lightingResonance gratingWDM filtersDFB, DBR, …PhC deviceSilicon device…

Super lens CDEWMetal wireSPP waveguideNano-photonics…

Far-field Near-field Evan.-field

Science, Vol. 297, pp. 820-822, 2 August 2002.

Ag film, hole diameter=250nm, groove periodicity=500nm, groove depth=60nm, film thickness=300nm

Light transmission through a metallic subwavelength holed < λ

One-dimensional Traveling Wave

One-Dimensional Wave Equation: The Traveling Wave

One-Dimensional Wave Equation: The Traveling Wave

( )

( ) ( )

2

2

:v

v

:

1 v

In coordinate system Oy f x

In coordinate system Ox x t

y y f x f x t

Now develop general one D wave equationx xx t

y y x f x fx x x x x x

y y y x fx x x x x x x x

′

′ ′=

′ = −′ ′= = = −

−′ ′∂ ∂= = −

∂ ∂

′ ′∂ ∂ ∂ ∂ ∂ ∂= = =

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

2

2x fx x′∂ ∂⎞ =⎜ ⎟ ′∂ ∂⎝ ⎠

v

O’(x’, y’)

O(x, y)

1-D wave pulse of arbitrary shape

One-Dimensional Wave Equation: The Traveling Wave

One-Dimensional Wave Equation: The Traveling Wave

2 22

2 2

2 2 2

2 2 2 2

2 2

2 2 2

v

v

v v

1v

1v

xt

y y x f x ft x t x t xy y y x f x f

t t t x t t x x t x

f y yx x t

y yx t

′∂= −

∂

′ ′∂ ∂ ∂ ∂ ∂ ∂= = = −

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

∂ ∂ ∂= =

′∂ ∂ ∂

∂ ∂⇒ =

∂ ∂1-D differential wave equation

One-dimensional Wave Equation

v = 1 m/s, -z

v = 2 m/s, +x

Harmonic Waves – Wavelength and Propagation Constant -

Harmonic Waves - Period and Frequency -

Harmonic Waves 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 wave also 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

Harmonic Waves as Complex Numbers

Complex representationComplex representation

[ ]1 cos(2 ) cos( )2

A B tω α β α β= + + + −

( , ) cos( ) cos( )oE z t E t kz E t kzω ω= − = − ( ) ( )( , ) j t kz j t kzoE z t E e E eω ω− −= =

Consider the time-averaged values which are meaningful, rather than the instantaneous values of many physical quantities.(Since the field vectors are rapidly varying function of time; for example λ = 1 μm has 0.33 x 10-14 sec time-varying period!)

{ }Re ( ) ( ) cos(2 )a t b t A B tω α β= + +

Identicalrepresentation !![ ] [ ] *1 1( ) ( ) Re ( ) Re ( ) Re cos( )

2 2a t b t a t b t AB AB α β⎡ ⎤= = = −⎣ ⎦

(real form)

(complex form)

Complex representation of real quantities : ExamplesComplex representation of real quantities : Examples

Plane Waves and Spherical Waves

Plane WavesPlane Waves

A plane wave

3-D Wave Equation and Helmholtz Equation

In 3 dimension,

Cartesian

Cylindrical

Spherical

Helmholtz equationHelmholtz equation

Helmholtz, Hermann von (1821-1894)

Helmholtz sought to synthesize Maxwell's electromagnetic theory of light with the central force theorem. To accomplish this, he formulated an electrodynamic theory of action at a distance in which electric and magnetic forces were propagated instantaneously.

From Maxwell’s Equations to Wave Equations

12 2 2 2 30

7 2 2 20

8.854 10 / / : permittivity of vacuum

4 10 / / : permeability of vacuum

C J m C s kg m

kg m C kg m A s

ε

μ π

−

−

⎡ ⎤= × ⋅ = ⋅ ⋅⎣ ⎦⎡ ⎤= × ⋅ = ⋅⎣ ⎦

Energy density (energy per unit volume)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 = 3

22

2 ,21

21

mJE

cEu o

oB ε

μ==

Energy density Energy density

2Euuu oBE ε=+=

Now if Now if E = EE = Eoosin(sin(ωωt+t+φφ)) and and ωω is very largeis very large

We will see only a time average of EWe will see only a time average of E

( ) ( )21sin1sin 22 =+=+ ∫

+

dttT

tTt

t

ϕωϕω

2

21

ooEu ε=

Intensity or IrradianceIntensity or Irradiance

kr

In free space, wave propagates with speed cIn free space, wave propagates with speed c

c c ΔΔtt

AA

In time In time ΔΔt, all energy in this volume passes through A.t, all energy in this volume passes through A.Thus, the total energy passing through A is,Thus, the total energy passing through A is,

JoulestAcu Δ=Ξ

Intensity or IrradianceIntensity or Irradiance

cAut

P =ΔΞ

=Power passing through A is,Power passing through A is,

Define: Define: IntensityIntensity or or IrradianceIrradiance as the as the power per unit areapower per unit area

2

21

ooEcI

cuI

ε=

=

Intensity in a dielectric mediumIntensity in a dielectric medium

In a dielectric medium,In a dielectric medium,

Consequently, the irradiance or intensity is,Consequently, the irradiance or intensity is,

2

21

oEvI ε=

ncvandn o == εε 2

Poynting vectorPoynting vector

1

o

S E H Poynting Vector

S E Bμ

= × ≡

= ×

r r r

r r r

For an isotropic media energy flows in the direction of propagatFor an isotropic media energy flows in the direction of propagation, soion, soboth the magnitude and direction of this flow is given by,both the magnitude and direction of this flow is given by,

( ) HEStIIrr

×===

The corresponding intensity or irradiance is then,The corresponding intensity or irradiance is then,

Poynting vectorPoynting vector

( )( )HEcS

cBEcSEcS

oo

o

o

με

εε

2

2

=

==

9. Superposition of Waves 9. Superposition of Waves

1 2

2 2 2 22

2 2 2 2 2

1 2

1 2

:

1v

. ,,

Suppose that and are both solutionsof the wave equation

x y z t

Then any linear combination of and isalso a solution of the wave equation For exampleif a and b are constants then

a b

is a

ψ ψ

ψ ψ ψ ψψ

ψ ψ

ψ ψ ψ

∂ ∂ ∂ ∂+ + = ∇ =

∂ ∂ ∂ ∂

= +

.lso a solution of the wave equation

E. Hecht, Optics, Chapter 7.

Superposition of Waves of the Same Frequency Superposition of Waves of the Same Frequency

In-phase case of superposing 2 waves,

Superposition principle :

Superposition of Waves of the Same Frequency Superposition of Waves of the Same Frequency Out-of-phase case,

The sum is again a harmonic wave of the same frequency

Superposition of Multiple Waves of the Same Frequency

Superposition of Multiple Waves of the Same Frequency

( ) ( ) ( )

( )

0 01

02 2 10 0 0 0

1 10

1

:

, sin sin

sin2 cos tan

cos

N

i i ii

N

i iN N Ni

i i j i j Ni j i i

i ii

The relations just developed can be extended to the addition of an arbitrary number N of waves

E x t E t E t

EE E E E

E

ω α ω α

αα α α

α

=

=

= > =

=

= + = +

= + − =

∑

∑∑ ∑∑

∑

E. Hecht, Optics, Chapter 7.

Superposition of Waves with Different Frequency Superposition of Waves with Different Frequency

: beat frequency

kp =

kg =

Phase velocity and Group velocity Phase velocity and Group velocity

When the waves have also a time dependence,

1 2

1 2

2

2

p

g

ω ωω

ω ωω

+=

−=

1 2

1 2

2

2

p

g

k kk

k kk

+=

−=

1 1 1

2 2 2

( , ) cos( )( , ) cos( )x t A k x tx t A k x t

ψ ωψ ω

= −= −

higher frequency wave

lower frequency wave (envelope)

phase velocity : 1 2

1 2

pp

p

vk k k kω ω ω ω+

= = ≈+

1 2

1 2

gg

g

dvk k k dkω ω ω ω−

= = ≈−

group velocity :

( )

( )

2 1

1 2 /

g

pp p

p p p

p

dvdk

dvd kv v kdk dk

d c c dn k dnv k v k vdk n n dk n dk

dnv kn d

ω

λ π λλ

=

⎛ ⎞= = + ⎜ ⎟

⎝ ⎠− ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + = + = +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞= + ← =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦