Soliton Propagation in Optical Fibers
Russell Herman
UNC Wilmington
March 21, 2003
Outline
• History
– Optical Fibers
– Transmission
– Communications
•
•
•
•
Linear Wave Propagation
Nonlinear Schrödinger Equation
Solitons
Other Fiber Characteristics
Geometric Optics
• Reflection
• Refraction
• Total Internal Reflection
n1 sin 1  n2 sin 2
Internal Reflection in Water
• Daniel Colladon
– 1826 velocity of sound in water
– Introduced Compressed air
– 1841 Beam in jet of water
• John Tyndall
– 1853 Royal Institute talks
– 1854 needed demo
• Faraday suggested demo
• Sir Francis Bolton
– 1884 Illuminated Fountains, London
Internal Reflection in Glass
• Glass – Egypt 1600 BCE
• Medievel glass blowers
• 1842 Jacques Babinet
– Light Guided in Glass Rods
• 1880s William Wheeler
– Patent for Light Pipes in Homes
Most glass is a mixture of silica obtained
from beds of fine sand or from
pulverized sandstone; an alkali to lower
the melting point, usually a form of soda
or, for finer glass, potash; lime as a
stabilizer; and cullet (waste glass) to assist
in melting the mixture. The properties of
glass are varied by adding other
substances, commonly in the form of
oxides, e.g., lead, for brilliance and weight;
boron, for thermal and electrical resistance;
barium, to increase the refractive index, as
in optical glass; cerium, to absorb infrared
rays; metallic oxides, to impart color; and
manganese, for decolorizing.
-http://www.infoplease.com/ce6/society/A0858420.html
Spun Glass Fibers
• Rene de Reamur – First in 18th Century
• Charles Vernon Boys
– Measurement of Delicate Forces – Mass on thread
– 1887 First quartz fibers
– Radiomicrometer – measured candle heat over 2 mi
• Herman Hammesfahr
– Glass Blower, American Patent for glass fibers
– Glass Fabric - Dresses for 1892 World’s Fair - $30,000
– Not Practical – scratched, fibers easily broke
• Owens-Illinois Glass Company
– 1931 Mass Production – glass wool
• Joint venture with Corning Glass Works => Owens-Corning Fiberglass
– 1935 Woven into Clothing – without breaking!
Image Transmission
• First Facsimile – 1840’s
• Alexander Graham Bell – 1875 Telautograph
• Henry C. Saint-Rene’
– 1895 – First Bundle of glass rods
• John Logie Baird
– Mechanical TV inventor, London
– 1925 First Public Demo of TV
– Bundle of Fibers, 8 lines/frame
• Clarence W. Hansell
– GE, RCA – 300 Patents
– 1930 Bundling of fibers to transmit images
• Heinrich Lamm
– Medical Student - Munich
– First transmitted fiber optic image - 1930
Light Leakage
• Brian O’Brien,
– Opt. Soc. Am., Rochester
• Abraham Van Heel
– Netherlands, Periscopes, Scramblers
– Metal Coating, Lacquer, …
• Cladding Hard – clean, smooth, no touching
– 1952
• Holger Moller Hansen
– Gastroscope, 1951 Patent, rejected
• Avram Hirsch Goldbogen
– Mike Todd, 1950
– Cinerama – 3 cameras
Clad Optical Fibers
• Hopkins and Kapany
• Basil Hirshowitz
– Gastroentologist
– 1956 First endoscope at U. Michigan
• Lawrence E. Curtiss
– Undergraduate
– 1956 First glass-clad fiber, tube+rod
– $5500
• J. Wilbur Hicks
– Image Scramblers at AO => CIA
Wireless Communication
• Optical Telegraphs
– Semaphores
• Bell’s Photophone 1880
– Used Selenium, 700 ft
• “Wireless” – Marconi 1898
• Communication Satellites
– Arthur C. Clarke 1945
– John R. Pierce 1950s
• Optical Communication Concerns
– Radio Competition
– Bandwidth
– Transparency
• Pipes and Switches - Telephones
Wireless World, October 1945, pages 305-308
Bell’s Photophone
On Bell's Photophone...
http://www.alecbell.org/Invent-Photophone.html
"The ordinary man...will find a little difficulty in comprehending how sunbeams are to be used. Does Prof.
Bell intend to connect Boston and Cambridge...with a line of sunbeams hung on telegraph posts, and, if so,
what diameter are the sunbeams to be...?...will it be necessary to insulate them against the weather...?...until
(the public) sees a man going through the streets with a coil of No. 12 sunbeams on his shoulder, and
suspending them from pole to pole, there will be a general feeling that there is something about Prof. Bell's
photophone which places a tremendous strain on human credulity."
New York Times Editorial, 30 August 1880
Source: International Fiber Optics & Communications, June, 1986, p.29
Bandwidth
• C.W. Hansell – RCA
– 1920s transatlantic 57 kHz, 5.26 km
– 1925 – 20 MHz, 15 m – Vacuum Tubes
• South America in Daytime – lower cost
• Telephone Engineers
– Higher frequency & multiplexing (24-phone channels)
• 1939 – 500 MHz – C.W. Hansell
– Aimed for TV demands
• WWII – microwaves passed 1 GHz
• Relay Towers – 50 mi apart vs Coaxial Cables in 50s
• Next?
– Alec Harvey Reeves, – 1937 ITT Paris/ 1950s STL
– digital signals to lessen noise problems
– Telepathy?
– Shorter Wavelengths – Weather problems
Waveguides
• Hollow Pipes
–
–
–
–
–
BCs
Cutoff Wavelength
100 MHz – Wavelength = 3 m => 1.5 waveguide
GHz – 10 cm
Bell Circular, hollow, D=5 cm for 60 GHz/5 m – 1950
– Stewart E Miller
• 1956 – Holmdel – 3.2 km – leakage from
bends/kinks
• 1958 – 50.8 mm, 80,000 conversations, 35-75
GHz, digitized => 160 million bits/s
Maxwell’s Equations
B
E  
t
D
H  J f 
t
D  f
B  0
D   0E  P
B  0 H  M
Wave Equation
E
P
    E   0 0 2  0 2
t
t
2
Vaccum -
2
   E  (  E)   2E
1  2E
 E 2 2
c t
c
2
Linear and Homogeneous
Medium -
1
 0 0
P(r, t )   0  (1) E
v
1
0
  (1   (1) ) 0  n 2 0
Waveguides – add BCs => modes and cutoff frequency
Fiber Modes
E(r,  ) 

it
E
(
r
,
t
)
e
dt


0   E  n ( )
2
Eei ( k r t )
E
or
2
Cylindrical Symmetry
Central Core + Cladding
Normalized Frequency
2
c
2
E
E z (r,  )  A( ) F (  )eim ei z
V  k0 a n12  n22
Radial Equation
d 2 F 1 dF
m2
2

 [  2 ]F  0
2
d
 d

Solutions
 2  n 2 k02   2
 J ( ),   a
F ( )   m
 K m ( ),   a
BCs => Eigenvalue Problem for mj
Single Mode Condition (HE11)
Ex:
 2   2  n 2 k02
 2   2  k02 (n 2  n 2 )
1
2
V  k0 a n12  n22  2.405
n1  n2  0.005, a  4 m    1.2 m
Still Needed: coherent beams, clean fiber material
LASERs
• Charles H. Townes
– Coherent Microwave Oscillator – MASER – 1951
– With Arthur L.Schawlow (Bell Labs) – LASER
• Theodore Maiman 1960
– Hughes Research
– Ruby laser
– PRL rejected paper!
• Ali Javan 1960
– 1.15 micrometer He-Ne Laser
– First gas laser
– First continuous beam laser
– Later: Bell Labs 633 nm version
• Visible, stable, coherent
Other Lasers
• Semiconductor Laser 1962
– Short endurance at -196 C
• Communications problems
– Ruby – 25 mi – could not see
– He-Ne – 1.6 mi – large spread in good weather
• Georg Goubau 1958
– Beam Waveguides
– 15 cm x 970 m with 10 lenses
• Rudolf Kompfner/Stewart E. Miller 1963
– models of waveguides
– Hollow Optical Light Pipes, Fiber Optics
The Transparency Problem
• Light Pipes – Confocal Waveguides
– Impossible tolerances
• Fibers – mode problem
– Multimodes messy
– Pulse Spreading
• Antoni Karbowiak/Len Lewin/Charles K. Kao, STL
– Multimode Calculations 1960s
– Rescaled microwave results by 100,000
– Needed .001 mm – too fine to see or handle
The Transparency Solution
• C.K. Kao and George Hockham – Single mode
fiber
– Rods in air, energy along surface, low absorption loss
– 0.1-0.2 microns thick
– Added Cladding! – 1% index change => O(10)
increased diameter
– Easier to focus light on core
– New Problem – light travels in core => optical losses
– Paper – loss can be < 20 dB/km 1965-6
• Robert Maurer Corning first low loss fibers
Nonlinear Wave Equation
 2 PNL
 2 PL
1  2E
 E  2 2   0 2  0
c t
t
t 2
2
P(r, t )   0 [  (1)  E   (2 )  EE   (3) EEE 
Isotropic –
Nonlinear -
In Silica -
 (1)  n2  1
]
 (2)  0
Third harmonic generation, four wave mixing,
nonlinear refraction
n( ,| E | )  n( )  n2 | E |
2
2
n2 
3 (3)
 xxx
8n
Basic Propagation Equation
2E   ( )k02E  0
 ( )  1   xx (1) ( ) 
 (n  i
Assumptions:
•PNL small
•Polarization along length – scalar
•Quasimonochromatic – small width
•Instantaneous response
•Neglect molecular vibrations
3
 xxxx (3) | E |2
4
c 2
)  n 2  2nn
2
n  n2 | E |2 
i
2k0
E (r,   0 )  F ( x, y) A( z,   0 )ei0 z
2
2 F 2 F
2
 2  [ ( )k0   ]F  0
2
x
y
2
A
2i  0
 (    02 ) A  0
z
Amplitude Equation
2
A
2i  0
 (    02 ) A  0
z
   0  
A
 i[  ( )     0 ] A
z
1
2
 ( )   0  1 (  0 )   2 (   0 ) 2 
 3 d 2n
2 
2 c 2 d  2
GVD – Group Velocity Dispersion
1
1 
vg
= 0 near 1.27 m
>0 Normal dispersion
<0 Anomalous dispersion (Higher f moves slower)
Nonlinear Schrödinger Equation
A
A i
2 A 
 1
  2 2  A  i | A |2 A
z
t 2
t
2
Nonlinear Schrödinger Equation
T  t  1 z
 0
A 1  2 A
i
  2 2   | A |2 A  0
z 2 T
Balance between dispersion and nonlinearity
Optical Solitons
• Hasegawa and Tappert – 1973
• Mollenauer, et. al. – 1980
– 7 ps, 1.2 W, 1.55 mm, single mode – 700 m
Optical Losses
Solitons
• John Scott Russell 1834
– "... I followed it on horseback, and overtook it still
rolling on at a rate of some eight or nine miles per hour,
preserving its original figure some 30 feet long and a
foot to a foot and a half in height." - J.S. Russell
•Airy, 50 yr dispute
•Rayleigh and Bussinesq 1872
•Korteweg and deVries 1895
ut  6uux  uxxx  0
Recreation in 1995 in Glasgow
Inverse Scattering Method
•
•
•
•
•
Kruskal and Zabusky - 1965
Gardner, Greene, Kruskal, Muira – 1967
Zahkarov and Shabat – NLS – 70’s
…. Sine-Gordon, Toda Lattice, Relativity, etc.
AKNS – Ablowitz, Kaup, Newell, Segur 1974
Two Soliton Solution of the NLS
u( x t)  4 e
 i x 



 cosh( 4 t)  4 cosh( 2 t)  3 cos ( 4 x) 
cosh( 3 t)  3 e
 4 i x
 cosh( t)
Other Nonlinear Effects
A
A 1  2 A

2
i
 i 1
 1 2  i | A | A  A
z
t 2
2
t
2
i
3 A


|
A
|
  3 3  ia1 (| A |2 A)  ia2 A
6
t
t
t
•Soliton Perturbation Theory
•Coupled NLS
•Dark Solitons – Normal Dispersion Regime
•Raman Pumping
Summary
• History
– Optical Fibers
– Transmission
– Communications
•
•
•
•
•
Linear Wave Propagation
Nonlinear Schrödinger Equation
Solitons
Perturbations
Other Applications
– Soliton Lasers and Switching
– Coupled Equations