Wave and complex issue, Lecture notes of Electromagnetic Engineering

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Bablu K. Ghosh

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, where v is

the velocity of the wave.

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f ( x )

f ( x- 3)

f ( x- 2)

f ( x- 1)

x 0 1 2 3

f ( x - v t )

f ( x + v t )

For an EM wave, we could have E = f( x ± v t )

The Phase

Velocity

How fast is the wave

traveling?

Velocity is a reference

distance

divided by a reference time.

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The phase velocity is the wavelength / period: up =  / 

Since f = 1/ :

In terms of k, k = 2/ , and

the angular frequency,  = 2/ , this is:

up =  f

up =  / k

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)

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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.

…or sometimes j = √(-1)

The argument of the cosine function represents the phase of the wave, ϕ, or the

fraction of a complete cycle of the wave.

Using complex numbers, we can write the harmonic wave equation as:

i.e., E = E 0

cos() + i E 0

sin(), where the ‘real’ part of the expression actually

represents the wave.

We also need to specify the displacement E at x = 0 and t = 0, i.e., the ‘initial’

displacement.

Waves using complex

numbers

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E  E

0

cos k(x  ct);   k(x  ct)

E  E

0

e

ik( x ct )

 E

0

e

i( kx  t )

Amplitude and Absolute phase

E ( x,t ) = A cos[( k x –  t ) –  ]

A = Amplitude

 = Absolute phase (or initial, constant phase) at x = 0, t

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kx

Waves using complex

amplitudes

We can let the amplitude be complex:

Where the constant stuf is separated from the rapidly changing stuf.

The resulting "complex amplitude”:

is constant in this case (as E 0 and θ are constant), which implies that

the medium in which the wave is propagating is no absorbing.

What happens to the wave amplitude upon interaction with matter?

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

E(x,t)  E 0

exp[i(kx  t  )]

E(x,t)  E 0

 exp(i^ )exp[i(kx^ ^ t)]



E 0

 exp(i^ )

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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 ( )

tot

E 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 E 1

, E

2

, and E 3

Adding waves of the same frequency, but different initial phase,

yields a wave of the same frequency.

EM propagation is space

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Wave Model of Light

• (^) To better understand the wave model of light…
  • (^) Think of waves of light as being transverse waves on a rope
  • (^) If you shake a rope through a fence with vertical slats, only

waves that vibrate up and down will pass through

• (^) If you shake the rope side to side, the waves will be blocked
• (^) A polarizing light filter acts like the slats in a fence.
  • (^) It only allows waves that vibrate in one direction to pass through

Particle Model of Light

• (^) Sometimes light can even cause an electron

to move so much that it is knocked out of the

substance

• (^) This process is called the photoelectric effect

Particle and wave!!

Absorption of EM radiation

Recall the expression for the flux density of an EM wave (Poynting

vector):

When absorption occurs, the flux density of the absorbed frequencies is

reduced.

F 

1

2

c  0

E

2

Energy in Electromagnetic Waves

• For a wave, intensity = energy flow through unit area per

unit time = P/A

• In the case of an EM wave, intensity corresponds to the

“brightness” of the radiation

• Studies of the PHOTOELECTRIC EFFECT by Lennard

(~1900) gave results that could not be explained by the

classical wave picture of light………

I 

c 

2

E