Post on 14-Apr-2022
EE485 Introduction to PhotonicsIntroduction
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Nature of Light
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“They could but make the best of it and went around with woebegone faces, sadly complaining that on Mondays, Wednesdays, and Fridays, they must look on light as a wave; on Tuesdays, Thursdays, and Saturdays, as a particle. On Sundays they simply prayed.”
The Strange Story of the QuantumBanesh Hoffmann, 1947
Geometrical (ray) optics
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History of Optics• Geometrical optics (Ray optics)
– Enunciated by Euclid in Catoptrics, 300 B.C.• Early 1600: First telescope by Galileo Galilei• End of the 17th century: Light as wave to explain reflection and refraction, by
Christian Huygens• 1704: Corpuscular nature of light (light as moving particles) to explain refraction,
dispersion, diffraction, and polarization, by Issac Newton• Early 1800: Interference experiment by Thomas Young – light is wave• Maxwell equation (1864) ― Light as electromagnetic waves, by James Clerk
Maxwell• How about emission and absorption?• Quantum theory ― Light as photons
– 1900: Max Plank – quantum theory of light– 1905: Albert Einstein – photoelectric effect experiment, light behaves as
particles with energies E = h– 1925-1935: de Broglie –quantum mechanics explaining the wave-particle
duality of light• 1950s: Communication and information theory• 1960: First laser
Quantum optics
E-M wave opticsRay optics
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Topics we plan to cover
• Light as electromagnetic waves• Polarized light• Superposition of waves and interference• Diffraction• Photon and laser basics• Laser operation• Nonlinear optics and light modulation
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Electromagnetic Spectrum
Optical frequencies
EE485 Introduction to PhotonicsLight as Electromagnetic Waves1. Wave equations2. Harmonic waves3. Electromagnetic waves4. Energy flow and absorption5. Fiber optics
Reading: Pedrotti3, Sec. 4.1-4.8, Sec. 10.1-10.6
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Historic Young’s Double-slit Experiment (1802)
Light is wave
Water waves from two point sources
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What does an optical wave look like
Direct measurement of light waves
(Goulielmakis, et al., Science, V. 305, p. 1267-1269, August 2004)
Water waves
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1-D Wave Equation
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1-D traveling wave function: ( )y f x vt
They satisfy 1-D differential wave equation: 2 2
2 2 2
1y yx v t
Quiz: Which one(s) of the following wave functions represent traveling waves? What is the magnitude and direction of the wave velocity?
2( , ) cos [ ( )]y z t A t z 2 2( , ) ( 4 4 )y x t A x xt t
2( , ) ( )y x t A Bx t
0t t2
0 2 20
( , )( )by x t t
a x x
Exercise: Consider a pulse propagating in the –x direcion with speed v. The shape of the pulse at is given by
Such a pulse is known as a Lorentzian pulse. Determine the shape of the pulse at an arbitrary time t.
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Harmonic Waves
0 0sin sin
[ ( ) ] or [( ) ]cos cos
y A k x vt A kx t
Harmonic waves with different A, 0, k and vor k and form a complete set of functions. Any periodic wave form can be decomposed into linear combination of harmonic wave functions. → Fourier Optics.
A snapshot in time
2 : Propagation constant k
+
+
+…
=
2 : Angular frequencyf
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ExerciseA red diode laser, with = 650 nm in free space, incidents from air to a medium with refractive index n equal to 1.5, as shown below. Derive its harmonic wave functions in the air and in the medium. The speed of light in the medium is ⁄ .
Note: Light speed in free c = 3 x 108 m/s. Assume amplitude A remains constant as the wave enters the medium. Amplitude displacement at the interface = / 2A
x
y(Snapshot at t = 0)
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Plane Waves and Spherical WavesPlane wave in +x-direction Plane wave in any direction Spherical wave
( cos )i kr tAe Define k to represent the propagation constant and direction.
( )i tAe k r ( )i tA er
k r
Intensity (W/m2)2A
r
→ Energy conservation obeyed3-D wave equation: 2
22 2
1v t
( , ) ( )exp( )t i t r r
Helmholtz equation: 2 2( ) ( ) 0k r
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Useful Formulas in Vector Calculus
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Light as Electromagnetic Waves
(Goulielmakis, et al., Science, V. 305, p. 1267-1269, August 2004)
From Maxwell’s equation to Wave equation:
0
0
90
70
00
(1/36 ) 10 ( / ) : Electric permittivity
4 10 ( / ) : Magnetic permeability
t
t
F m
H m
EH
HE
EH
Maxwell’s equations in free space2
22 2
, , , ,
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0 0
1 0
or
1 3 10 ( / )
: Speed of light in free space
x y z x y z
uuc t
u E H
c m s
Wave equationNecessary condition
ˆ ˆ ˆ
: Electric field (V/m)ˆ ˆ ˆ
: Magnetic field (A/m)
x y z
x y z
E E E
H H H
E x y z
H x y z
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E
Maxwell’s Equations in a Medium, : Electric displacement
, : Magnetic flux density
00
t
t
DH D
BE B
DB
Assume a non-magnetic medium with no free electric charges or currents
Physical meaning of the electric displacement:
- - - --
+ + + ++P
0
( if the mediumhas a charge density )
D E PD
0 B HBoundary conditions:• Tangential components of E and H are continuous.• Normal components of D and B are continuous.
E H
B
DPower flow per unit area:2Re{ } Re{ } (W/m ): Poynting vector S E H
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Linear, Nondispersive, Homogeneous, and Isotropic Media
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2 2
In free space
1 0uuc t
0
: Electric susceptibility
P E0
0
(1 )/ : Dielectric constant
D E
Maxwell’s equations:
0
00
t
t
EH
HE
EH
00
t
t
DH
BE
DB
Identical to Maxwell’s equations in free space with replaced by 0.
0 0
1c
Wave equation:2
22 2
In a medium
1 0v
uut
Speed of light: 0
0
1v
/ 1 : Refractive index
cn
n
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Monochromatic Electromagnetic Waves
( )
( )
i t
i t
e
e
E E r
H H r
Let’s relate harmonic waves to electromagnetic waves
( , ) ( )exp( )t i t r r
Maxwell’s equations:
0
00
t
t
EH
HE
EH
0
( ) ( )( ) ( )
( ) 0( ) 0
ii
H r E rE r H r
E rH r
2 2( ) ( ) 0k r
Helmholtz equation:2 2
, , , ,
0 0
( ) ( ) 0( ) ( ) or ( ) x y z x y z
k uu E H
k nk
rr r r
(E-M wave represented by complex numbers)
Optical intensity:
1| | Re ( ) ( )*2
I
S E r H r
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Plane Electromagnetic Wave (I)( )i tAe k r
( )0
( )0
( , ) ( )
( , ) ( )
i t i t
i t i t
t e e
t e e
k r
k r
E r E E r
H r H H r
0
( ) ( )( ) ( )
ii
H r E rE r H r
Substituting into Maxwell’s equations:
0 0
0 0 0
k H Ek E H
→ E, H, and k are mutually orthogonal ― Transverse electromagnetic (TEM) wave.
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Plane Electromagnetic Wave (II)Relationship between the amplitude of the electric field and the magnetic field:
0
0
0
00
0
: Impedance of the medium
120 377 : Impedance of free space
EH
n
Optical intensity:2
0*0 0
1 1| | Re ( ) ( )*2 2 2
EI E H
S E r H r
Example: Let’s describe an optical wave mathematically. A laser beam of radius 1 mm carries a power of 5 mW. (a) Determine its average intensity and the amplitude of its electric and magnetic fields. (b) Assume the laser beam is a TEM wave (actually not a completely correct assumption) with = 650 nm, propagating in x-direction, and the electric field is along y-direction (Slide 11). Determine the complex wave functions for the electric and magnetic fields. (c) Determine the wave functions after entering a medium with n = 1.5.
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ExerciseDetermine the power of a 10-mW laser beam,
(a) With = 440 nm, after traveling 1 km of water at various locations.
(b) With = 1550 nm, after traveling 100 km of optical fiber.
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Absorption Bands of Optical Materials
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Applications of Optical Fibers
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“Wires” for light
Optical fiber structure
Optical fiber transmission system
Inter-continental optical fiber network
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Total Internal Reflection (TIR)
1 2
1sinc
nn
c 1 1
c: Critical angle
Example 1: Diamond
Most of the rays entering the top of the diamond will exit from the top due to total internal reflection.
Example 2: Optical Fiber
ncore
125 m
ncladding
ncore > ncladding
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Numerical Aperture of an Optical FiberLet’s do an exercise:
Show that the maximum incident angle m for TIR in the optical fiber is related to the refractive indices by:
2 20 1 2. . sin mN A n n n
N.A.: Numerical Aperture m: Acceptance angle
Calculate NA and m for a silica glass fiber in air with n1 = 1.475 and n2 = 1.460.