Modern Optics I – wave properties of light

Special topics course in IAMS

Lecture speaker: Wang-Yau Cheng

2006/4

Outline

• Wave properties of light

• Polarization of light

• Coherence of light

• Special issues on quantum optics

• Properties of wavePropagation

– Phase

– Wave equation

– Phase velocity

– Group velocity

– Refraction of wave

– Interference of wave

– Electro-magnetic (EM) wave

– Spectrum of EM wave

Waves, the Wave Equation, and Phase Velocity

What is a wave?

Forward [f(x-vt)] vs. backward [f(x+vt)] propagating waves

The one-dimensional wave equation

Phase velocity

Complex numbers

What is a wave?A wave is anything that moves.

To displace any function f(x)to the right, just change itsargument from x to x-a,where a is a positive number.

If we let a = v t, where v is positive and t is time, then the displacement will increase with time.

So f(x-vt) represents a rightward, or forward,propagating wave.

Similarly, f(x+vt) represents a leftward, or backward,propagating wave.

v will be the velocity of the wave.

The one-dimensional wave equation

2 2

2 2 2

10

v

f f

x t

We’ll derive the wave equation from Maxwell’s equations. Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f:

Light waves (actually the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light.

Electromagnetism is linear: The principle of “Superposition” holds.

If f1(x,t) and f2(x,t) are solutions to the wave equation,

then f1(x,t) + f2(x,t) is also a solution.

 

Proof: and

   

This means that light beams can pass through each other. It also means that waves can constructively or destructively interfere.

2 2 21 2 1 2

2 2 2

( )f f f f

x x x

2 2 2

1 2 1 22 2 2

( )f f f f

t t t

2 2 2 2 2 21 2 1 2 1 1 2 2

2 2 2 2 2 2 2 2 2

( ) 1 ( ) 1 10

v v v

f f f f f f f f

x t x t x t

The solution to the one-dimensional wave equation

where f (u) can be any twice-differentiable function.

( , ) ( v )f x t f x t

The wave equation has the simple solution:

The 1D wave equation for light waves

We’ll use cosine- and sine-wave solutions: 

or

where:

2 2

2 20

E E

x t

( , ) cos[ ( v )] sin[ ( v )]E x t B k x t C k x t

( , ) cos( ) sin( )E x t B kx t C kx t

1v

k

( v)kx k t

where E is the light electric field

Waves using complex numbers

The electric field of a light wave can be written:

E(x,t) = A cos(kx – t – )

Since exp(i) = cos() + i sin(), E(x,t) can also be written:

E(x,t) = Re { A exp[i(kx – t – )] }

or E(x,t) = 1/2 A exp[i(kx – t – )] + c.c.

where "+ c.c." means "plus the complex conjugate of everything before the plus sign."

We often write these

expressions without the

½, Re, or +c.c.

Waves using complex amplitudes

We can let the amplitude be complex:

where we've separated the constant stuff from the rapidly changing stuff.

The resulting "complex amplitude" is:

So:

, exp

ex, ex( pp )

E x t A i kx t

E x t i k ti xA

0 exp( ) E A i

(note the " ~ ")

0, expE x t E i kx t

How do you know if E0 is real or complex?

Sometimes people use the "~", but not always.So always assume it's complex.

• Properties of wave– Propagation

Phase

Wave equation

Phase velocity

Group velocity

– Refraction of wave

– Interference of wave

– Electro-magnetic (EM) wave

– Spectrum of EM wave

Definitions: Amplitude and Absolute phase

E(x,t) = A cos[(k x – t ) – ]

A = Amplitude = Absolute phase (or initial phase)

Definitions

Spatial quantities:

Temporal quantities:

The Phase Velocity

How to measure the velocity of the moving wave?

First of all, measures the wavelength, secondly, count for how many wave peaks go through per second.

The phase velocity is the wavelength / period: v = /

In terms of the k-vector, k = 2/ , and the angular frequency, = 2/ , this is: v = / k

The Phase of a Wave

The phase is everything inside the cosine.

E(t) = A cos(), where = kx – t –

In terms of the phase,

= – /t

k = /x

and

– /t

v = –––––––

/xThis formula is useful when the wave is really complicated.

When two waves of different frequency interfere, they produce "beats."

Indiv-idual waves

Sum

Envel-ope

Irrad-iance:

When two light waves of different frequency interfere, they produce beats.

0 1 1 0 2 2

1 2 1 2

1 2 1 2

0 0

( , ) exp ( ) exp ( )

Let and 2 2

Similiarly, and 2 2

So:

( , ) exp ( ) exp (

tot

ave

ave

tot ave ave ave

E x t E i k x t E i k x t

k k k kk k

E x t E i k x kx t t E i k x kx

0

0

0

)

exp ( ) exp ( ) exp{ ( )}

2 exp ( )cos( )

Real part: 2 cos( ) cos( )

ave

ave ave

ave ave

ave ave

t t

E i k x t i kx t i kx t

E i k x t kx t

E k x t kx t

Group velocity

vg d /dk

Light wave beats (continued):

Etot(x,t) = 2E0 cos(kavex–avet) cos(kx–t)

This is a rapidly oscillating wave [cos(kavex–avet)]

with a slowly varying amplitude [2E0 cos(kx–t)]

The phase velocity comes from the rapidly varying part: v = ave / kave

What about the other velocity?

Define the "group velocity:" vg /k

In general, we define the group velocity as:

Group velocity is not equal to phase velocityif the medium is dispersive (i.e., n varies).

0 1 0 2

1 1 2 2

1 2

0 01 21 2

1 2

For our example, v

where and are the k-vectors in vacuum.

If , v phase velocity

g

g

kc k c k

n k n k

k k

c ck kn n n

n k k n

1 2

If , vgn n c

vg d /dk

Now, is the same in or out of the medium, but k = k0n, where k0 is the k-vector in vacuum, and n is what depends on the medium. So it's easier to think of as the independent variable:

Using k = n() / c0, calculate: dk /d = ( n + dn/d ) / c0

vg = c0 / ( n + dn/d) = (c0/n) / (1 + /n dn/d )

Finally:

So the group velocity equals the phase velocity when dn/d = 0,such as in vacuum. Otherwise, since n increases with , dn/d > 0, and: vg < vphase.

Calculating the Group velocity

1v /g dk d

v v / 1g phase

dn

n d

0

0

20 0 0 0

0 0 2 20 0 0

0

00

2 22 /

(2 / ) 2

v / 1

2v / 1

g

g

ddn dn

d d d

d c cc

d c c

c dn

n n d

cc

n

Use the chain rule :

Now, , so :

Recalling that :

we have : 20

0 0 02

dn

n d c

or :

Calculating Group Velocity vs. WavelengthWe more often think of the refractive index in terms of wavelength,so let's write the group velocity in terms of the vacuum wavelength 0.

0 0

0

v / 1g

c dn

n n d

The group velocity is the velocity of the envelope or irradiance: the math.

0( ) ( v ) exp[ ( v )]gE t E z t ik z t

( ) ( v ) exp[ ( v )]gE t I z t ik z t

And the envelope propagates at the group velocity:

Or, equivalently, the irradiance propagates at the group velocity:

The carrier wave propagates at the phase velocity.

The group velocity can exceed c0 whendispersion is anomalous.

vg = c0 / (n + dn/d )

dn/d is negative in regions of anomalous dispersion, that is, near a resonance. So vg can exceed c0 for these frequencies!

One problem is that absorption is strong in these regions. Also, dn/d is only steep when the resonance is narrow, so only a narrow range of frequencies has vg > c0. Frequencies outside this range have vg < c0.

Pulses of light (which are broadband) therefore break up into a mess.

Beating the speed of lightTo exceed c, we need a region of negative dn/d over a fairly large range of frequencies. And the slope should not vary much—to avoidpulse break-up. And absorption should be minimal.

One trick is to excite the medium in advance with a laser pulse, whichcreates gain (instead of absorption), which inverts the curve.Then two nearby resonances have a region in between with minimalabsorption and near-linear negative slope:

Negative dispersion (vg = c0 / (n + dn/d) and dn/d)

1. Naturally

2. Artificially1. Grating pair

2. Optical fiber (or, some special designed waveguide)

3. Photonic crystal

4. Prisms in mode-locked lasers

5. EIT

• Properties of wave– Propagation

– Phase

– Wave equation

– Phase velocity

– Group velocity

Refraction of wave

Interference of wave

– Electro-magnetic (EM) wave

– Spectrum of EM wave

An interesting question is what happensto wave when it encounters a surface.

At an oblique angle, light can be completely transmitted or completely reflected.

"Total internal reflection" is the basis of optical fibers, a billion dollar industry.

9. Standing Waves, Beats, and Group Velocity

Group velocity: the speed of information

Going faster than light...

Superposition again

Standing waves: the sum of two oppositely

traveling waves

Beats: the sum of two different frequencies

Superposition allows waves to pass through each other.

Otherwise they'd get screwed up while overlapping

Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency.

This isn't so obvious using trigonometric functions, but it's easywith complex exponentials:

1 2 3

1 2 3

( , ) exp ( ) exp ( ) exp ( )

( ) exp ( )totE x t E i kx t E i kx t E i kx t

E E E i kx t

where all initial phases are lumped into E1, E2, and E3.

Adding waves of the same frequency, but opposite direction, yields a "standing wave."

0 0

0

0

( , ) exp ( ) exp ( )

exp( )[exp( ) exp( )]

2 exp( )cos( )

totE x t E i kx t E i kx t

E ikx i t i t

E ikx t

0( , ) 2 cos( ) cos( )totE x t E kx t

Since we must take the real part of the field, this becomes:

(taking E0 to be real)

Standing waves are important inside lasers, where beams areconstantly bouncing back and forth.

Waves propagating in opposite directions:

A Standing Wave

0

( , )

2 cos( )cos( )totE x t

E kx t

A Standing WaveAgain…

0( , ) 2 cos( ) cos( )totE x t E kx t

A Standing Wave: Experiment3.9 GHz microwaves

Note the node at the reflector at left.

Mirror

Input beam

The same effect occurs in lasers.

Interfering spherical waves also yield a standing wave

Antinodes

Two Point Sources

Different separations. Note the different node patterns.

When two waves of different frequency interfere, they produce beats.

0 1 0 2

1 2 1 2

0 0

0

( ) Re{ exp( ) exp( )}

2 2

( ) Re{ exp ( ) exp ( )}

Re{ exp( )[exp( ) exp( )]}

tot

ave

tot ave ave

ave

E t E i t E i t

E t E i t t E i t t

E i t i t i t

Let and

So :

0

0

Re{2 exp( )cos( )}

2 cos( )cos( )

).

ave

ave

ave

E i t t

E t t

Taking the real part yields the product of a rapidly varying

cosine ( ) and a slowly varying cosine (

Take E0 to be real.

What happens when light passes through two slits?

The idea is central to many laser techniques, such as holography, ultrafast photography, and acousto-optic modulators.Tests of quantum mechanics also use it.

Light pattern that emerges“fringes”

Young’s Two-Slit Experiment

Diffraction

Light bends around corners. This is called diffraction.

The diffraction pattern far away is the Fourier transform of the slit transmission vs. position.

Light patterns after passing through rectangular slit(s):

One slit:

Two slits:

Fourier decomposing functions plays a big role in optics.

Here, we write a square wave as a sum of sine waves of different frequency.

It converts a function of time to one of frequency:

The Fourier transform is perhaps one of the most important equation in optics.

12 ( ) ( ) exp( )E t E i t d

and converting back uses almost the same formula:

˜ E () E(t)exp( i t)dt

Often, they do so by themselves.

What do we hope to achieve with theFourier Transform?We desire a measure of the frequencies present in a wave. This will

lead to a definition of the term, the “spectrum.”

Plane waves have only one frequency, .

This light wave has many frequencies. And the frequency increases in time (from red to blue).

It will be nice if our measure also tells us when each frequency occurs.

Ligh

t el

ectr

ic f

ield

Time

• Properties of wave– Propagation

– Phase

– Wave equation

– Phase velocity

– Group velocity

– Refraction of wave

– Interference of wave

Electro-magnetic (EM) wave

Spectrum of EM wave

• “Light is just electromagnetic wave”– Review of Maxwell equations– The solutions which is convenient for optics– EM wave spectrum

The equations of optics are Maxwell’s equations.

where is the electric field, is the magnetic field, is the charge density, is the permittivity, and is the permeability of the medium.

/

0

BE E

t

EB B

t

E

B

Longitudinal vs. Transverse waves

Motion is alongthe direction ofPropagation

Motion is transverse

to the direction of

Propagation

Space has 3 dimensions, of which 2 directions are transverse tothe propagation direction, so there are 2 transverse waves in ad-dition to the potential longitudinal one.

Transverse:

Longitudinal:

Vector fieldsLight is a 3D vector field.

A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space.

( )f r

22

2

2 2 2 2

2 2 2 2

0

0

EE

t

E E E E

x y z t

The 3D vector wave equation for the electric field

which has the vector field solution:

Note the vector symbol over the E.

, expE r t A i k r t ��������������������������������������������������������

0, expE r t E i k r t ������������������������������������������

This is really just three independent wave equations, one each for the x-, y-, and z-components of E.

Waves using complex vector amplitudes

We must now allow the complex field and its amplitude to be vectors:

0, expE r t E i k r t

0 (Re{ } Im{ },Re{ } Im{ },Re{ } Im{ })x x y y z zE E i E E i E E i E

The complex vector amplitude has six numbers that must be

specified to completely determine it!

0E

E

Note the arrows over the E’s!

Derivation of the Wave Equation from Maxwell’s Equations

Take of:

 

Change the order of differentiation on the RHS:

B

Et

[ ] [ ]B

Et

[ ] [ ]E Bt

Derivation of the Wave Equation from Maxwell’s Equations (cont’d)

But:

 

Substituting for , we have:

 

 

Or:

EB

t

B

[ ] [ ]E

Et t

2

2[ ]

EE

t

[ ] [ ]E Bt

assuming that and are constant in time.

Derivation of the Wave Equation from Maxwell’s Equations (cont’d)

Using the lemma,

 

becomes:

 

If we now assume zero charge density: = 0, then

 

and we’re left with the Wave Equation!

2

2[ ]

EE

t

22

2( )

EE E

t

0E

22

2

EE

t

Why light waves are transverseSuppose a wave propagates in the x-direction. Then it’s a function of x and t (and not y or z), so all y- and z-derivatives are zero: 

  

Now, in a charge-free medium,  that is,  

Substituting, we have:

0y yz zE BE B

y z y z

0 0E B

0 0y yx xz zE BE BE B

x y z x y z

0 0x xE B

x x

and

The magnetic-field direction in a light wave

Suppose a wave propagates in the x-direction and has its electric field along the y-direction [so Ex = Ez= 0, and Ey = Ey(x,t)].

What is the direction of the magnetic field?

 

Use:

 

So:

 

In other words: 

And the magnetic field points in the z-direction.

0,0, yEB

t x

yzEB

t x

, ,y yx xz zE EE EB E E

Et y z z x x y

Suppose a wave propagates in the x-direction and has its electric field in the y-direction. What is the strength of the magnetic field?

The magnetic-field strength in a light wave

0

( , ) ( ,0)

t

yz z

EB x t B x dt

x

We can integrate:

Take Bz(x,0) = 0

Differentiating Ey with respect to x yields an ik, and integrating with respect to t yields a 1/-i.

0, expyE r t E i kx t

yz

EB

t x

Start with: and

0( , ) exp ( )z

ikB x t E i kx t

i

1( , ) ( , )z yB x t E x t

c

So:

But / k = c:

An Electromagnetic Wave

The electric field, the magnetic field, and the k-vector are

all perpendicular:

The electric and magnetic fields are in phase.

E B k

The Energy Density of a Light Wave

The energy density of an electric field is:

The energy density of a magnetic field is:

Using B = E/c, and , which together imply that

we have:

Total energy density:

So the electrical and magnetic energy densities in light are equal.

2

2

1

21 1

2

E

B

U E

U B

2 21 1 1

2 2B EU E E U

2E BU U U E

1c

B E

Why we neglect the magnetic field

The force on a charge, q, is:

 

 

 

so:

 

 

Since B = E/c:

vF qE q B

vmagnetic

electrical

F q B

F qE

vmagnetic

electrical

F

F c

So as long as a charge’s velocity is much less than the speed of light, we can neglect the light’s magnetic force compared to its electric force.

Felectrical Fmagnetic

where is the charge velocity

v

The Poynting Vector: S = c2 E x BThe power per unit area in a beam.

Justification (but not a proof):

Energy passing through area A in time t:

= U V = U A c t

So the energy per unit time per unit area:

= U V / ( A t ) = U A c t / ( A t ) = U c = c E2

= c2 E B

And the direction is reasonable.E B k

V = A c t

The Irradiance (often called the Intensity)A light wave’s average power per unit area is the “irradiance.”

 

Substituting a light wave into the expression for the Poynting vector,

, yields:

 

The average of cos2 is 1/2:

2S c E B

2 20 0( , ) cos ( )S r t c E B k r t

/ 2

/ 2

1( , ) ( , ') '

t T

Tt T

S r t S r t dtT

real amplitudes

20 0

( , ) ( , )

(1/ 2)

TI r t S r t

c E B

The Irradiance (continued)

Since the electric and magnetic fields are perpendicular and B0 = E0 / c,

 

becomes:

 

 

 

where

 

210 02I c E B

0

2

12 ~

I c E

2 * * *0 0 0 0 0 0 0x x y y z zE E E E E E E

The Electromagnetic Spectrum

infrared X-rayUVvisible

wavelength (nm)

microwave

radio

105106

gamma-ray

The transition wavelengths are a bit arbitrary…

The Electromagnetic Spectrum

The Long-Wavelength

Electro-magnetic Spectrum

Radio & microwave regions (3 kHz – 300 GHz)

• Key points – Why light is just the electromagnetic wave?

– Why light is a transverse EM wave?

– Speed of light is by definition

– How to use Maxwell eq. depends on your conditions

– What’s the so-called “instantaneous frequency”?

– What’s the three ways of the solutions of wave equation?

– What’s the amplitude and phase of EM wave?

– Wave equation