We’ll start with optics

Optional optics texts:

Eugene Hecht, Optics, 4th ed.

J.F. James, A Student's Guide to

Fourier Transforms

Optics played a major role in all the

physics revolutions of the 20th

century, so we’ll do some.

And waves and the Fourier

transform play major roles in all of

science, so we’ll do that, too.

Waves, the Wave Equation, and Phase

Velocity

What is a wave?

Forward [f(x-vt)] andbackward [f(x+vt)]propagating waves

The one-dimensional wave equation

Harmonic waves

Wavelength, frequency, period, etc.

Phase velocity

Complex numbers Plane waves and laser beams

f(x)

f(x-3)

f(x-2)

f(x-1)

x0 1 2 3

What is a wave?

A wave is anything that moves.

To displace any function f(x) to the

right, just change its argument 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 represents a rightward,

or forward, propagating wave.

Similarly, represents a

leftward, or backward, propagating

wave.

v will be the velocity of the wave.

f(x)

f(x-3)

f(x-2)

f(x-1)

x0 1 2 3

f(x - v t)

f(x + v t)

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.

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:

Proof that f (x ± vt) solves the wave equation

Write f (x ± vt) as f (u), where u = x ± vt. So and

Now, use the chain rule:

So ! and !

Substituting into the wave equation:

1u

x

!

!= v

u

t

!

!= ±

f f u

x u x

! ! !

! ! !=

f f u

t u t

! ! !

! ! !=

f f

x u

! !

! != v

f f

t u

! !

! != ±

2 2

2

2 2v

f f

t u

! !

! !=

2 2

2 2

f f

x u

! !

! !=

2 2 2 2

2

2 2 2 2 2 2

1 1v 0

v v

f f f f

x t u u

! ! ! !

! ! ! !

" #$ = $ =% &

' (

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

! !µ"

! !# =

( , ) 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

!

µ"= =

( v)kx k t±

where E is the

light electric field

The speed of light in

vacuum, usually called

“c”, is 3 x 1010 cm/s.

A simpler equation for a harmonic wave:

E(x,t) = A cos[(kx – !t) – "]

Use the trigonometric identity:

cos(z–y) = cos(z) cos(y) + sin(z) sin(y)

where z = k x – ! t and y = " to obtain:

E(x,t) = A cos(kx – !t) cos(") + A sin(kx – !t) sin(")

which is the same result as before,

as long as:

A cos(") = B and A sin(") = C

( , ) cos( ) sin( )E x t B kx t C kx t! != " + "

For simplicity, we’ll

just use the forward-

propagating 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 fast is the wave traveling?

Velocity is a reference distance

divided by a reference time.

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

Since % = 1/$ :

In terms of the k-vector, k = 2" / #, and

the angular frequency, ! = 2" / $, this is:

v = # v

v = ! / k

Human wave

A typical human wave has a phase velocity of about 20 seats

per second.

The phase is everything inside the cosine.

E(x,t) = A cos(&), where & = k x – ! t – "

& = &(x,y,z,t) and is not a constant, like " !

In terms of the phase,

! = – $& /$t

k = $& /$x

And – $& /$t

v = ––––––– $& /$x

The Phase of a Wave

This formula is useful

when the wave is

really complicated.

Complex numbers

So, instead of using an ordered pair, (x,y), we write:

P = x + i y

= A cos(#) + i A sin(#)

where i = (-1)1/2

Consider a point,

P = (x,y), on a 2D

Cartesian grid.

Let the x-coordinate be the real part

and the y-coordinate the imaginary part

of a complex number.

Euler's Formula

exp(i&) = cos(&) + i sin(&)

so the point, P = A cos(#) + i A sin(#), can be written:

P = A exp(i&)

where

A = Amplitude

& = Phase

Proof of Euler's Formula

Use Taylor Series:

2 3 4

2 4 3

exp( ) 1 ...1! 2! 3! 4!

1 ... ...2! 4! 1! 3!

cos( ) sin( )

i ii

i

i

! ! ! !!

! ! ! !

! !

= + " " + +

# $ # $= " + + + " +% & % &' ( ' (

= +

2 3

( ) (0) '(0) ''(0) '''(0) ...1! 2! 3!

x x xf x f f f f= + + + +

exp(i&) = cos(&) + i sin(&)

2 3 4

2 4 6 8

3 5 7 9

exp( ) 1 ...1! 2! 3! 4!

cos( ) 1 ...2! 4! 6! 8!

sin( ) ...1! 3! 5! 7! 9!

x x x xx

x x x xx

x x x x xx

= + + + + +

= ! + ! + +

= ! + ! + +

If we substitute x = i&

into exp(x), then:

Complex number theorems

[ ]

[ ]

[ ]

[ ]

1 1 2 2 1 2 1 2

1 1 2 2 1 2 1 2

exp( ) 1

exp( / 2)

exp(- ) cos( ) sin( )

1 cos( ) exp( ) exp( )

2

1 sin( ) exp( ) exp( )

2

exp( ) exp( ) exp ( )

exp( ) / exp( ) / exp ( )

i

i i

i i

i i

i ii

A i A i A A i

A i A i A A i

!

!

" " "

" " "

" " "

" " " "

" " " "

= #

=

= #

= + #

= # #

$ = +

= #

exp( ) cos( ) sin( )i i! ! != +If

More complex number theorems

Any complex number, z, can be written:

z = Re{ z } + i Im{ z }So

Re{ z } = 1/2 ( z + z* )and

Im{ z } = 1/2i ( z – z* )

where z* is the complex conjugate of z ( i % –i )

The "magnitude," | z |, of a complex number is:

| z |2 = z z* = Re{ z }2 + Im{ z }2

To convert z into polar form, A exp(i&):

A2 = Re{ z }2 + Im{ z }2

tan(#) = Im{ z } / Re{ z }

We can also differentiate exp(ikx) as if

the argument were real.

[ ]

[ ]

exp( ) exp( )

cos( ) sin( ) sin( ) cos( )

1 sin( ) cos( )

1/ sin( ) cos( )

dikx ik ikx

dx

dkx i kx k kx ik kx

dx

ik kx kxi

i i ik i kx kx

=

+ = ! +

" #= ! +$ %& '

! = = +

Proof :

But , so :

Waves using complex numbers

Recall that the electric field of a 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 everythingbefore the plus sign."

We often

write these

expressions

without the

!, 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.

As written, this entire

field is complex!

Complex numbers simplify waves!

This isn't so obvious using trigonometric functions, but it's easy

with 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 different initial phase,

yields a wave of the same frequency.

The 3D wave equation for the electric field

and its solution!

or

which has the solution:

where

and

2 2 2 2

2 2 2 20

E E E E

x y z t

! ! ! !µ"

! ! ! !+ + # =

0( , , , ) exp[ ( )]E x y z t E i k r t!= " #

! !

" "

x y zk r k x k y k z! " + +

! !

2 2 2 2

x y zk k k k! + +

2

2

20

EE

tµ!

"# $ =

"

!A wave can propagate in any

direction in space. So we must allow

the space derivative to be 3D:

( ), ,x y zk k k k!

!

( ), ,r x y z!!

is called a plane wave.

A plane wave's wave-fronts are equally

spaced, a wavelength apart.

They're perpendicular to the propagation

direction.

Wave-fronts

are helpful

for drawing

pictures of

interfering

waves.

A wave's wave-

fronts sweep

along at the

speed of light.

A plane wave’s contours of maximum phase, called wave-fronts or

phase-fronts, are planes. They extend over all space.

0 exp[ ( )]E i k r t!" #

! !

"

Usually, we just

draw lines; it’s

easier.

Laser beams vs. Plane waves

A plane wave has flat wave-fronts throughout

all space. It also has infinite energy.

It doesn’t exist in reality.

A laser beam is more localized. We can approximate a laser

beam as a plane wave vs. z times a Gaussian in x and y:

2 2

20( , , , ) exp[ ( )exp ]x y

E x y z t E i kz tw

!=" #+$%

&$'

(! !

Laser beam

spot on wall

w

x

y

Localized wave-fronts

z