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Fundamental of Optical Engineering
Lecture 4

Recall for the four Maxwell’s equation:
D
 H 
J
t
.D  
B
 E  
t
.B  0
where E  electric field intensity vector
D  electric displacement vector
B  magnetic flux density vector
H  magnetic field intensity vector
  free charge density
J = current density due to free charges

The wave equation is derived from the
assumptions of
◦ Non-magnetic material,
B  0 H
0  magnetic permeability of free space
◦ Uniform dielectric medium
D   E  n 2 0 E
  permittivity
 0  permittivity of free space
n = refractive index
◦ No current or
  0 and J  0
Therefore, the simplified form of Maxwell’s equation
can be written as
E
  H  n 0
t
H
  E   0
t
.E  0
2
.H  0

From (2):


    E   0

 H
t


2

E
  E  n2 0 0 2
t



From (1):

Since

Similarly, we have
   A    A  2 A ,
we end up with

Equation (3) and (4) are wave equations of the
form
1 2 A
 A 2 2
v t
2
where
A is a function of x,y,z, and t.
1 2 A
A( x, y, z, t )  2 2
v t

Generally, we only are interested in electric
field. The wave equation may be written as
E  Ex eˆx  Ey eˆy  Ez eˆz

Assume that the wave propagates in only zdirection
d2
1 d 2 E ( z, t )
E ( z, t )  2
2
dz
v
dt 2

Then assume that E(z,t) = E(z)E(t) and put it
into (5)
d 2 Ez Ez d 2 Et
Et
 2
0
2
2
dz
v dt
or


We clearly see that the left side of (6) is
dependent on ‘z’ only, and the right side of
(6) is on ‘t’ only.
Both sides must be equal to the same
constant, which we arbitrarily denote as -2.
d 2 Ez  2

Ez  0
2
v
dz
d 2 Et
2


Et  0
2
dt

The general solutions of these equations are
Ez  C1ei ( / v ) z  C2 ei ( / v ) z
Et  D1eit  D2 eit

Constants C1, C2, D1, and D2 could be found
by the boundary conditions.

We now can express the general solution
E(z,t) as
E ( z , t )  Ez Et  e  i ( / v ) .e  it

A wave travelling from left to right has a
function of the form
E ( z, t )  E0 ei (t   z )
  propagation constant
  radian frequency  2

From
2
2 E 2 E 2 E

E
2



n


0 0
x 2 y 2 z 2
t 2
2 E 2 E
 2 0
2
x
y
2 E
2



E
2
z
2 E
2



E
2
t
  2 E   2 n 2 0 0 E

Phase velocity: dt=dz
dz 
  v ph
dt 

Write the expression of a plane wave traveling
in z-direction that has maximum amplitude
of unity and a wavelength of 514.4 nm.
S  (E  H * )
S  complex poynting vector


The time average power density:
For plane wave propagation in z-direction,
using Maxwell’s equations and definition of s,
we find that
1 
PA 
Ex2 eˆz
2 0

Let  be Gaussian beam solution and assume
propagation in z-direction
   ( x, y , z ) e i (  t   z )
2 P0 i
 
.e .e
2
w
i 0 r 2

2R

.e
r2
w2
.ei (t   z )
r  x 2  y 2  distance in cylindrical coordinates

2 n

P0  Total power in beam
R = Radius of curvature of a phase front
w = beam half width
2 2

(
z

z
)
 
2
2
0
w  w0 1 

2 2 4
n

w
0


w  half width at beam waist
z 0  location of beam waist along z-axis

 2 n 2 w04 
R  ( z  z0 ) 1  2
2 
  ( z  z0 ) 
1  ( z  z0 ) 
  tan 

2
n

w
0


For large (z-z 0 )
w2
w
2 2

(
z

z
)
 
2
0
w0  2 2 4 
 n  w0 
( z  z0 ) 
n w0
r

Diverging angle, tan  

z  z0 n w0
For small  , tan

2 P0
Power per unit area, PA    
e
2
w
*
(
2r2
w
2
)

Geometrical optics may be employed to
determine the beam waist location in
Gaussian problems.
w
tan  
z  z0

w
n

Consider a HeNe laser with λ = 0.63 μm.
Calculate the radius of curvature for mirrors in the
figure below.

Calculate beam width at mirrors from the
previous example and at a distance of 1m, 1
km, and 1,000 km from center of laser
(assuming that mirrors do not deform beam)

Consider a colliminated Nd:YAG laser beam
(λ=1.06 μm) with a diameter to e-2 relative
power density of 10 cm at the beam waist
with z0 = 0. What is the beam half width to
e-2 relative power density at z = 1m, 100 m,
10 km, and 1,000 km?

From the previous example, what is power
density on beam axis at each distance,
assuming the total power is 5 W? What is the
divergence angle of beam to e-2 and e-4
relative power density?

Two identical thin lenses with f = 15 cm and
D = 5 cm are located in plane z = 0 and z =
L. A Gaussian beam of diameter 0.5 cm to e-2
relavtive power density for λ = 0.63 μm is
incident on the first lens. The value of L is
constained such that the e-2 relative power
density locus is contained within the aperture
of the second lens.
(a) For what value of L will the smallest spot be obtained
for some value of z0 > 0? What is the value of z0
corresponding to the location of that spot? What is the
diameter of that spot?

(b) For what value of L will the smallest spot size be
obtained on the surface of the moon at a distance of
300,000 km? What is the beam diameter on the moon
surface?