Download EM fields

Chapter 7 - Review of EM Theory
Gabriel Popescu
University of Illinois at Urbana-Champaign
Beckman Institute
Quantitative
Q
tit ti Li
Light
ht Imaging
I
i Laboratory
L b t
http://light.ece.uiuc.edu
Principles of Optical Imaging
Electrical and Computer Engineering, UIUC
ECE 460 – Optical Imaging
7.1 Maxwell’ss Equations
7.1 Maxwell
Equations
 B(r , t )
  E (r , t )  
t
(7.1)
E (r , t )  instantaneous electric field intensity (V/m)
B(r , t )  instantaneous magnetic flux density (Wb/m)
Chapter 7: Review of EM Theory
2
ECE 460 – Optical Imaging
7.1 Maxwell’ss Equations
7.1 Maxwell
Equations
 D(r , t )
  H (r , t )  j s (r , t )  j c (r , t ) 
t
Material
source
(7.2)
(induced)
H (r , t )  instantaneous magnetic field intensity (A/m)
D(r , t )  iinstantaneous electric
l i flux
fl density
d i (C/m
(C/ 2 )
j s (r , t )  instantaneous conduction current density (A/m 2 )
j c (r , t )  instantaneous source current density (A/m 2 )
Chapter 7: Review of EM Theory
3
ECE 460 – Optical Imaging
7.1 Maxwell’ss Equations
7.1 Maxwell
Equations
  B(r , t )  0
(7.3)
  D(r , t )   (r , t )
( )
(7.4)
source
B (r , t )  instantaneous magnetic flux density (Wb/m)
D(r , t )  instantaneous electric flux density (C/m 2 )
 (r , t )  instantaneous volume charge density (C/m3 )
Chapter 7: Review of EM Theory
4
ECE 460 – Optical Imaging
7.2 Constitutive Relations
7.2 Constitutive Relations
D   E ,  = 0 r
((7.5))
B   H ,  =0 r
(7.6)
jc   E
(7.7)
The permittivity ( ), permeability ( ), and the condunctivity ( ), are
spatially dependent for inhomogeneous media, orientation dependent (tensor)
for anisotropic media, and field dependent for nonlinear media. They are
simple scalar constants for linear homogeneous isotropic (LHI) media.
Chapter 7: Review of EM Theory
5
ECE 460 – Optical Imaging
7.2 Constitutive Relations
7.2 Constitutive Relations
 = 0 r
 =0 r
 r andd r are ddefined
fi d as the
th relative
l ti permittivity
itti it andd relative
l ti permeability
bilit
respectively.  0 and 0 are the permittivity and permeability of free space.
 0  8.854 1012
109
F /m

36
0  4 107 H / m
Chapter 7: Review of EM Theory
6
ECE 460 – Optical Imaging
7.3 Boundary Conditions
7.3 Boundary Conditions
nˆ  ( E1  E 2 )  0, tangential E is continuous
(7.8)
nˆ  ( H 1  H 2 )  j s , j s is the surface current
(7.9)
nˆ  ( B1  B 2 )  0, Normal B is continuous
(7.10)
nˆ  ( D1  D 2 )   s ,  s is the surface charge
(7.11)
nˆ is the normal to the interface between the two regions
For a perfect conductor, E and js are continuous
Etan
Bnorm
Chapter 7: Review of EM Theory
7
ECE 460 – Optical Imaging
7.4 Maxwell’s Equations
i f
in freq. domain
d
i
  E (r ,  )  i B(r ,  )
  H (r ,  )  js (r ,  )  jc (r ,  )  i D
  B(r ,  )  0
  D(r ,  )   (r ,  )
((7.12))
(7.13)
(7.14)
(7.15)
E, H, D, B, js , jc , and  are time independent complex quantities.
Chapter 7: Review of EM Theory
8
ECE 460 – Optical Imaging
7.5 Complex Constitutive Relations and Boundary Conditions
d
di i
D(r ,  )   E (r ,  ),  = 0 r
(7.16)
B (r ,  )   H (r ,  ),  =0 r
(7.17)
j c (r ,  )   E (r ,  )
(7.18)
Bounday conditions are the same as before:
nˆ  ( E1  E2 )  0,
0
nˆ  ( B1  B2 )  0,
Chapter 7: Review of EM Theory
nˆ  ( H1  H 2 )  js
nˆ  ( D1  D2 )   s
9
ECE 460 – Optical Imaging
7.6 The Wave Equation
7.6 The Wave Equation
For linear, isotropic, and source free region:
  E  i H ,
D  0
  H  (  i ) E ,
B  0
(7.19; 7.20)
(7.21; 7.22)
    E  i   H
(7.23)
(  E )   2 E  i (  i ) E for constant 
(7.24)
  D    E  0    E  0
Chapter 7: Review of EM Theory
for constant 
(7.25)
10
ECE 460 – Optical Imaging
7.6 The Wave Equation
7.6 The Wave Equation
 2 E  k 2 E  0,
k 2   2   i
(7.26)
A nd sim m ilarly w e can show that:
 H k H 0
2
2
(7.27)
The two differential vector wave equations above are mathematically identical.
Th i solutions
Their
l ti
are similar
i il in
i the
th functional
f ti l behavior
b h i but
b t off course they
th differ
diff in
i their
th i
amplitudes since they represent two different physical quantities.
Chapter 7: Review of EM Theory
11
ECE 460 – Optical Imaging
7.7 The Solution to the Wave Equation in Rectangular Coordinates
l
di
2 F  k 2 F  0
2
2
2
ˆ
ˆ
 Fx x  Fy y  Fz zˆ  k (Fx xˆ  Fy yˆ  Fz zˆ)  0
2
2 Fx  k 2 Fx  0,, 2 Fy  k 2 Fy  0,, 2 Fz  k 2 Fz  0
All three equations are of the form:
2 f  k 2 f  0
Chapter 7: Review of EM Theory
12
ECE 460 – Optical Imaging
7.7 The Solution to the Wave Equation in Rectangular Coordinates
l
di
Using the seperation of variable method, we assume solution of the form:
f ( x, y, z ,  )  X ( x,  )Y ( y,  ) Z ( z ,  )
 X
Y
 Z
YZ 2  XZ 2  YX 2  k 2 XYZ  0
x
y
z
2
2
2
1  X 1Y 1 Z
2


k 0
2
2
2
X x
Y y
Z z
2
Chapter 7: Review of EM Theory
2
2
(7.28)
13
ECE 460 – Optical Imaging
7.7 The Solution to the Wave Equation in Rectangular Coordinates
l
di
Each term must be constant
Expect f ( x, y, z ) to be
exponential e
i ( kx x  k y y  kz z )
1 2 X
 ik x x
2


k
,
X

e
x
2
X x
1  2Y
 ik y y
2
 k y , Y  e
2
Y y
1 2Z
2


k
z ,
2
Z z
Z  e  ik z z
((7.29))
(
(7.30)
)
(7.31)
where k x 2  k y 2  k z 2  k 2   2   i  is the dispersion relation
of the medium. Any three quantities k x 2 , k y 2 , and k z 2 including complex
quantities are possible solutions if they satisfy the dispersion relation.
Chapter 7: Review of EM Theory
14
ECE 460 – Optical Imaging
7.8 EM Theory


Each valid set of kx,, ky,, and kz is associated with a mode.
The electromagnetic field associated with a mode is:
i ( kx x  k y y  kz z )
 Ae  i k r
 f ( x, y, z ,  )  Ae
 The total electromagnetic field is the sum of all possible modes:
 i kv r
f
(
r
,

)

Ae


v
Chapter 7: Review of EM Theory
15
ECE 460 – Optical Imaging
7.8 EM Theory

Recall that:
k 2      i
define:
ik    Fi 
f (r ,  )  Ae  r e  i r
f (r , t   Re[ Ae r e  i r eit ]
f (r , t )  A e  r cos(t   r   
where A  A ei
Chapter 7: Review of EM Theory
16
ECE 460 – Optical Imaging
7.8 EM Theory
 The field amplitude is determined by
p
y
A e  r
 And the plane wave is being attenuated along α direction
define :   attenuation constant  Im[k ]
 The phase is determined by
t   r  
 And it is propagating along the β
And it is propagating along the β direction
define :   phase propagation constant  Re[k ]
Chapter 7: Review of EM Theory
17
ECE 460 – Optical Imaging
7.8 EM Theory
 Definitions:
 Wavelength λ : distance between two points of 2π phase difference.
 
2

 Phase velocity vp : velocity of points of constant phase.
vp 
Chapter 7: Review of EM Theory


18
ECE 460 – Optical Imaging
7.8 EM Theory
 In lossless media α=0, β=k, and
k           ) ( r  r )
1
2


c
1
2
1
2
(  r  r   k0 (  r  r 
1
2
1
2
 In dielectric media at optical frequencies μr=1 and εr1/2 is defined as the refractive index n, and k=k0 n is called the wave number of the region.
Chapter 7: Review of EM Theory
19
ECE 460 – Optical Imaging
7.9 The Electric and the Magnetic Fields
A
Associated
i t d with
ith a Plane
Pl
Wave
W
 The electric field associated with a plane wave may be written as
E (r ,    E0eikr
 Where E0 is the complex constant vector amplitude
x E  i

Chapter 7: Review of EM Theory
k  E0

e
 ikr

kE

20
ECE 460 – Optical Imaging
7.9 The Electric and the Magnetic Fields
A
Associated
i t d with
ith a Plane
Pl
Wave
W
 The characteristic impedance is defined as
1
2

   




k
  i 


 One can show that
k    ikr
k 
E
e 
  i
  i
Chapter 7: Review of EM Theory
21
ECE 460 – Optical Imaging
7.9 The Electric and the Magnetic Fields
A
Associated
i t d with
ith a Plane
Pl
Wave
W

Note that
1
2

  
k

 


(  i )
  i 


 Since H=0 and E=0 for linear homogenous isotropic media it follows that:
media, it follows that:
kE  0
Chapter 7: Review of EM Theory
and
k  
22
ECE 460 – Optical Imaging
7.9 The Electric and the Magnetic Fields
A
Associated
i t d with
ith a Plane
Pl
Wave
W
 Both the electric field and the magnetic field are normal to the g
direction of propagation and normal to each other. That is, the three vectors are mutually orthagonal. The electric and magnetic fields may not have a component along the direction
magnetic fields may not have a component along the direction of propagation.
Chapter 7: Review of EM Theory
23
ECE 460 – Optical Imaging
7.10 Power Flow Density

Poynting Theorem: Intantaneous power flow density (W/m
y
g
p
y ( / 2)
P (r , t )  E (r , t )   (r , t )
 Re[
R [ E (r ,  eit ] 
 r ,  eit ]
1
1
it
 [ E (r ,  e  c.c]  [ r ,  eit  c.c]
2
2
1
1
 [ E    ]  [ E  ei 2t  c.c]
4
4
1
1
 Re[ E    ]  Re[ E  ei 2t ]
2
2
Chapter 7: Review of EM Theory
24
ECE 460 – Optical Imaging
7.10 Power Flow Density

Time average power flow density
g p
y
1
 P (r )  Re[ E  * ]
2

For a plane wave
1  E x (k * x E * ) 
 P (r )  Re 

2 


1  k * ( E  E * )  E * (k *  E ) 
 Re 

2 



Chapter 7: Review of EM Theory
1 2  k 
E Re 
 , kE  0
2



25
ECE 460 – Optical Imaging
7.10 Power Flow Density

Similarlyy
1 2  k 
 P (r ) 

 Re
R  
2
  
Chapter 7: Review of EM Theory
26