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Lecture 6:Electromagnetic Theory of Light
ELG 4117
Optoelectronics and Optical Components
September 23, 2013
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Electromagnetic Optics
ray
beam
wave
EM Electromagnetic optics accounts for the
polarization (vector nature) of theelectric and magnetic fields in the lightwaves
Wave optics is scalar approximation toelectromagnetic optics
Ray optics is the approximation to waveoptics when the objects that the lightinteracts with are much larger than thewavelength
Vector nature of light determines the amount of light reflected
and refracted at the boundaries; governs the light propagation
in waveguides and laser resonators.
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Electromagnetic Wave
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Electromagnetic Wave
http://electronicsgurukulam.blogspot.ca/2012/04/how-electro-magnetic-wave-propagates.html
E(r, t)
H(r, t)
Described by coupled electric and magnetic fieldvectors, changing in time and space.
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Maxwell's Equations
In free space
H=0E t
E=0H
t
E=0
H=0
c0=1
00
Ex, Ey , Ez
Hx , Hy, Hz
satisfy 2u1
c02
2 u
t2=0
Wave equation stems from Maxwell'sequations: Take to prove.(E)
0(1/36)109 F/m
0(4)107
H /m
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Maxwell's Equations
In a mediumIn a medium with no free electric charges or currents, twoadditional vector fields are required:
D(r, t)
B (r, t)
- electric flux density (electric displacement);
- magnetic flux density
H=D t
E=
B
t
D=0
B=0
These two extra vectors include mediumresponse to the electromagnetic field.
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Constitutive Relationships
D=0E+P;B=0H+0M
P - polarization density;
M- magnetization density
H=D
t
E=B t
D=0
B=0
In free space:
P=0 ;
M=0 ;
D=0E;
B=0H;
Maxwell's equations reduce tothose in free space.
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Boundary Conditions
In a homogeneous medium:
At the Interfaces:
E,H ,D,B are continuous.
Et ,1=Et ,2
Ht ,1=Ht ,2
Dn ,1=Dn ,2Bn ,1=Bn , 2
Tangential components of electric and magnetic fields, and normalcomponents of electric and magnetic flux densities should becontinuous accross an interface between two media.
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Intensity, Power, Energy
The flow of electromagnetic power is governed by Poynting vector:
S=EH
The magnitude of time-averaged Poynting vectoris optical intensity:
E
H
S
I(r, t)=S
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Poynting Theorem
Energy Conservation Law
(EH)=(E)H(H)E
Applying the vector product identity
and Maxwell's equations, we arrive at
S= t(
1
20E
2+1
20H
2)+EP t +0HM t
Energy density storedin electric and magnetic field
Power densities deliveredto electric and magnetic
dipolesThe power flow escaping from the surface of
small volume equals the time rate of change of
the energy stored inside the volume.
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Electromagnetic Waves in Dielectrics
The medium is linear, ifP(r,t) depends on E(r,t) linearly. The medium is nondispersive, if response is instantaneous.
The medium is homogeneous, if the relation between Pdepends on E does not depend on position r.
The medium is isotropic, if the relation between P depends on Edoes not depend on the direction of vectorE.
E(r, t) P(r, t)optical medium
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Linear, Nondispersive, Homogeneous, Isotropic
P=0E
- susceptibility of the medium.
E P
- electric permittivity - magnetic permeability
D=E,B=H ,
=0(1+)
=0
H= E t
E=H t
E=0
H=0
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Linear, Nondispersive, Homogeneous, Isotropic
H= E t
E=H t
E=0
H=0
Similar to free-space Maxwell's equations.
Hence, each component of electric andmagnetic fields satisfies the wave equation:
2u1
c2
2 u
t2=0, c=
1
n= c0c= 0 0
=0,Nonmagnetic medium: n=0=1+
Poynting Theorem:
S=W t
W=1
2E2+
1
2H2 - energy density
stored in the medium
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Monochromatic Electromagnetic Waves
E
(r, t
)=Re [E
(r
)exp(i
t
)]H(r, t)=Re [H(r)exp(i t)]
H=iD
E=iBD=0B=0
H=iE
E=iHE=0H=0
in linear isotropichomogeneous
nondispersive medium:
=2
k=n k0=
2U+k2U=0
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Electromagnetic Waves
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