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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Engineering Optics and Optical Techniques
LN#1. Electromagnetic Basics and Maxwell’s Equations (Sections 3-1, 3-2, Appendix 1)
Propagation of Waves (Chapter 2, Sections 3-3, 3-4)
Experimental evidence shows that light propagates as a form of waves consisting
transverse, time-varying electric and magnetic fields. The two amplitude-varying transverse
vectors, electric field strength E and magnetic field strength H, oscillate at the right angles to
each other in phase and to the direction of propagation. They can be expressed in the form of
four fundamental equations known as Maxwell’s Equations.
“It is true that nature begins by reasoning and ends by experience. Nevertheless,
we must begin with experiments and try through it to discover the reason.”
- Leonardo da Vinci –
Read of Chapter 1 for “Brief History” of Optical Science
Homework #1-1
(Clerk) Maxwell’s Equations (1865) – Light is most certainly electromagnetic nature
(Classical electrodynamics)
For vacuum, air, water or glass (no space charge or ion density, ~ zero electric conductivity):
E  0
B  0
Key: Interdependence of E and B
 E  
B
t
  B   o o
E
t
 2 E   o o
2E
t 2
 2 B   o o
2B
t 2
Or their combined and reduced forms,
Where E: Electric field [Force/Charge, N/C]
B ( H ) : Magnetic induction [Force/Charge/Velocity, Ns/Cm]
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Fluid Dynamics vs. Electromagnetics
Continuity Analogy:   V vs.   j
V: flow velocity
j: electric current flux
Extended Analogy
Fluid Dynamics
Electromagnetics
Field
E-Field: FE = qE
g-Field: Fg = mg
M-Field: FM = qVB
Flow current flux V
Continuity
Electric current flux j
= volume flow/area/time
=
= electric charge flow/area/time
Q  L
m 2 s  T 
=
C
m2s
Volume flow rate q:
Electric charge flow rate i:
q   V  dA
i   j  dA : Ampere = C/s
Volume continuity:
Electric charge continuity:

   V  0 , : fluid density
t
E
   j  0 , E: charge density
t
Steady state
 V  0
 j  0
Incompressible flow
Zero or constant space charge condition
 V  0
 j  0
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Arc Angle vs. Solid Angle
Arc angle: defined as ds  Rd
R
Total arc length: s  2R   ds  R  d  R total
ds
d
 total  2
Solid angle: defined as da  Rd (normally outward vector definition)
Total surface area: a  4R 2   da  R 2  d  R 2  total
 total  4
R
d
3
da
Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Coulombic or Gauss Law
F
+q2
F
-q1
r
F  q1q 2 
Experimental finding by Gauss,
F 
1
r2
q1q 2
4o r 2
1
 o vacuum   8.854  10 12 C 2 s 2 / m 3 kg
*Electric permittivity  : measure of the degree to which the material is permeated by the Efield, i.e., the permittivity is higher for more electrically conducting material.
For example,  /  o for water at 20 C is approximately 80, and goes to infinity for a
perfect conductor if exists. When  /  o goes to infinity, charges spread out uniformly in
no time to result in “0” Coulombic force.
**”o” indicates free surface, vacuum or air.
Note: Gauss Law is an inverse square law for the force between charges, which is the central
nature of the force and allows the linear superposition of the effects of different charges.
Electric Field
E
q
E
1
r
E 
4
1
q
4 o r 2
Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
 E  0
Maxwell’s 1st Equation:
… Electric Field Conservation in Free Space
The total force acting on da by q is given as:
E  nda 
1
q
1 q 2
cosda 
r d
2
4 o r
4 o r 2

total
  d  4

For the entire surrounding surface,
q
  E  nda  

o
1
o

E
Divergence theorem:
Therefore,   E 
dV … Gauss Theorem
  E  nda       EdV
E
o
For a vacuum or free space*  E  0 ,
 E  0
[*Most optically thin materials like glass or water can be treated electric charge free.]
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Maxwell’s 2nd Equation:
 B  0
… Steady Magnetic Field Conservation
*Experimental evidence by Biot & Savart (1820) shows that wires carrying electric currents
produce deflections of permanent magnetic dipoles placed around it. This inspires that an electric
current creates the magnetic induction (or equivalently magnetic field B or dB).
Maxwell’s 3rd Equation:
 E  
B
t
… Mutually perpendicular E & B
Farady’s Induction Law (1831) states that a time-varying magnetic flux passing through a closed
conducting loop results in the generation of a current around the loop, i.e.,
The notion is that a time-varying magnetic field will have an electric field associate with
it. This also shows that E and B must be perpendicular each other.
Maxwell’s 4th Equation:
  B   o o
E
t
(Strictly speaking, the above equation is valid for nearly nonconducting or di-electric medium.)
Ampere’s Circutal Law states that a time-varying E-field and j induces a B-field.
The notion is that a time-varying E-field will be accompanied by a B-field.
*  o ~ 4  10 7 m  kg / C 2 : magnetic permeability in vacuum…degree of measure of
magnetic induction (B) for a given magnetic field strength (H), i.e., B  H .
**Ferro-magnetic materials have high values of permeability.
[If interested in detailed analysis for the derivations of the above Maxwell’s equations,
refer to Section 3.1 of the textbook, and for more great details refer to “Classical
Electrodynamics (3rd ed.)” by J. D. Jackson, Wiley, 1999.]
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Maxwell’s Equation for Free Space, Vacuum and ~Air:
 E  0
 B  0
 E  
B
t
  B   o o
E
t
E = E (x,t) and B = B (x,t)
Using a vector identity,     B     B    2 B , the 3rd and 4th equations are expressed as:
2 B 
2B 2B 2B
2B





o o
x2 y2 z2
t 2
2E 2E 2E
2E
 E


  o o
x2 y2 z2
t 2
2
1
 c = 3  108 m/s: speed of light in vacuum
 o o
(This ensures the wave nature of light.)
(Maximum in vacuum and slower in a denser medium)
**Analogy to potential field in acoustics:
1  2
Acoustic pressure wave equations:    2 2 with a being the speed of sound.
a t
2
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
SOUND WAVES - Pressure Waves (Longitudinal) – Needs medium where it travels.
*The speed of sound a: A primary means of information traveling in a media by
propagation of locally pressurized compression, i.e., pressure waves. Thus, a denser
material can transfer the information on the local pressurization more efficiently and
faster. The speed of sound is faster in a denser medium.
a
p

LIGHT WAVES – Electromagnetic Waves (Transverse) – Medium is not necessary.
In vacuum:
In a medium:
*The speed of light c is the fastest for vacuum and slower for a denser medium.
(e.g. mirage or mirror-like road surface on a hot and sunny day)
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
1-D Wave Equation and Plane Waves
Maxwell’s Equation for E- field:  2 E 
2E 2E 2E
2E





o o
x2 y2 z2
t 2
 2  x, t  1  2  x, t 
… Linear* 2nd order PDE
 2
2
2
x
V
t
General solutions are:   x, t   c1 f  x  Vt   c2 g x  Vt 
Since the waves are harmonic (sine or cosine), we choose
  x, t   A sin k  x  Vt 
and also,  x, t   A sin k x  Vt   A sin k x    Vt 
Thus, k  2 or k 

V

2

: wave propagation number
…wave frequency
  2 …angular frequency
1

1

  …wave number
  …wave period


Now more generally,  x, t   A sin kx  kVt  A sin kx  t     A  Im e i kx t   **
*Linearity conditions: 1) If  1 and  2 are solutions,  1   2 is also a solution.
2) If  1 is a solution, C 1 is also a solution.
** e i  cos   i sin 
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
1-D Wave Equation and Plane Waves
§ Def’n of plane: collection of all r vectors.
k  r  ro   0
or k  r  const
(k is the wave propagation direction.)
A set of planes over which  r  varies in space sinusoidally, i.e., a way to express wave
propagation
 r   Asin k  r 
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
*Example: general expressions for 2-D waves
k  r  k x  k y  k z    x  y  z   0  k sin   k cos    x  y  z 
 ky sin   kz cos 
2

 y sin   z cos 
 2

Thus, the wave is expressed as:   y, z , t   A sin   y sin   z cos    t   


The second wave   0,   is expressed as,
 2

z   t   
 

  y, z , t   B sin 
And the resulting superposed wave is given as
 sup erposed       ....
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Poynting Vector and Irradiance
The radiant energy per unit volume, or energy density, u:
u  u E  u B  2u E   o E 2 
B2
o
[energy/volume = force*length/length**3 = force/length**2 = pressure]

E2 
 u E   o
 … Homework: EOC problem 3-8
2 

uE  uB  and E  cB … Homework #2-1
c  1/  o o
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
The radiant energy flux, or radiation power per unit area, i.e., Poynting vector S:
S
Energy total u  totalvolume u  ct  A
1


 uc 
EB  c 2  o EB
Area  Time
At
At
o
S
Power
 Radiation Intensity
Area
Note: The energy density has a pressure dimension.
The Poynting vector has a pressure  velocity dimension.
Energy beam
A
or in a vector form,
S t  
1
o
E  B  c 2  o E t   Bt  … Poynting vector
S t   u(t )  c
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
The irradiance, the time-average energy per unit area per unit time, I:
I  S t 
T

1
T

t T / 2
t T / 2
S t dt 
1
T

t T / 2
t T / 2
c 2  o E t   Bt dt
where E t   E o cosk  r  t  and Bt   Bo cosk  r  t  .
Using,
1
T
I
c o 2
Eo :
2

t T / 2
t T / 2
cos 2 k  r  t dt 
1
2
Irradiance: Time-Averaged Radiation Power per Unit Area
(Intensity)
* I  r 2  Const and E o r   r  Const ' .
**The irradiance is proportional to 1/r2 (Inverse Square Law)
and the amplitude of E-field, E o ,drops off inversely with r.
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Radiation Pressure
Basis for the corpuscular theory: the light is a stream of (weightless, m = 0) particles?
Radiation pressure p ~
F E
S
S
~
 u  , thus F   A
A V
c
c
Time-average radiation pressure is p 
S
c
p 
E-o-C. 3-29
normal?

S
c
[cf. F  ma 
d
mV  ]
dt
I
… perfectly absorbing surface
c
2
I
… perfectly reflecting surface
c
Time-average radiation pressure for an oblique incidence at an angle  with the
Poynting vector S

*Possible applications:
Optical levitation of a small particle
Micro-capsule accelerator (particle gun)
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Engineering Optics and Optical Techniques, LN No. 1
Prof. K. D. Kihm, Spring 2007
Homework Assignment #1
Due by 6:45 p.m. on January 23 (Tuesday), 2007 at the classroom.
Homework #2-1
For a plane wave propagating in vacuum, show that E  c B .
[Hint: section 3.2.1]
Homework # 2-2
For an Nd:YAG laser generate 400 mJ/pulse light wave with its pulse duration of 7 ns. Calculate
the maximum radiation pressure that the laser can exert on a totally reflecting surface. Also
calculate the maximum diameter of a totally reflecting silver particle that the laser can levitate
against the gravity for the pulse duration. Assume the laser illumination diameter of 10-microns
hitting the particle.
Solve End-of-Chapter problems: 3-5, 3-14, 3-15, 3-19, and 3-33.
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