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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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ECSE-6660Introduction to Optical Networking & Relevant Optics Fundamentals
http://www.pde.rpi.edu/Or
http://www.ecse.rpi.edu/Homepages/shivkuma/
Shivkumar KalyanaramanRensselaer Polytechnic Institute
[email protected] in part on textbooks of S.V.Kartalopoulos (DWDM) and H. Dutton (Understanding Optical communications), and slides of Partha Dutta
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Overview
Quick History
Relevant Properties of Light
Components of Fiber Optic Transmission and Switching Systems
Chapter 2 of Ramaswami/Sivarajan
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Quick History of Optical Networking1958: Laser discoveredMid-60s: Guided wave optics demonstrated1970: Production of low-loss fibers
Made long-distance optical transmission possible!1970: invention of semiconductor laser diode
Made optical transceivers highly refined!70s-80s: Use of fiber in telephony: SONETMid-80s: LANs/MANs: broadcast-and-selectarchitectures1988: First trans-atlantic optical fiber laidLate-80s: EDFA (optical amplifier) developed
Greatly alleviated distance limitations!Mid/late-90s: DWDM systems explodeLate-90s: Intelligent Optical networks
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Big Picture: Optical Transmission System Pieces
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Big Picture: DWDM Optical components
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Evolution of Fiber Transmission Systems
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Bigger Picture: Key Features of Photonics
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Electromagnetic Spectrum
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What is Light? Theories of Light
Historical Development
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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What is Light? Wave nature:
Reflection, refraction, diffraction, interference, polarization, fading, loss …
Transverse EM (TEM) wave:Interacts with any charges in nearby space… Characterized by frequency, wavelength, phase and propagation speedSimplified Maxwell’s equations-analysis for monochromatic, planar wavesPhotometric terms: luminous flux, candle intensity, illuminance, Luminance…
Particle nature:Number of photons, min energy: E = hu“Free” space => no matter OR EM fieldsTrajectory affected by strong EM fields
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Light Attributes of InterestDual Nature: EM wave and particleMany λs: wide & continuous spectrumPolarization: circular, elliptic, linear: affected by fields and matterOptical Power: wide range; affected by matterPropagation:
Straight path in free spaceIn matter it is affected variously (absorbed, scattered, through); In waveguides, it follows bends
Propagation speed: diff λs travel at diff speeds in matterPhase: affected by variations in fields and matter
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Interaction of Light with Matter
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Goal: Light Transmission on Optical Fiber
Need to understand basic ideas of λ interacts with λs and with matter
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Light interaction with other λs and interaction with matter
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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• Light rays travel in straight lines
Interaction with Matter: Ray Optics
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Reflection of Light
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Reflection Applications: Mirrors & MEMS
Plane
Paraboloidal
Spherical
Elliptical
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Refraction of Light
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Ray Deflection by Prism
• Newton’s Rainbow: Deflection angle dependent on the wavelength;
• Used in optical multiplexers and de-multiplexers !
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Optical Multiplexer & DeMultiplexer
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Internal & External Reflections
• Critical Angle for Total Internal Reflection:
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Total Internal Reflection
• Total internal reflection forms the back-bone for fiber optical communication
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Light (Wave) Guides: Reflection vs Total Internal Reflection
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Light Guiding: Concept of Optical Fiber
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Geometrical Optics: Fiber StructureFiber Made of Silica: SiO2 (primarily)Refractive Index, n = cvacuum/cmaterialncore > ncladding
Numerical Aperture:Measures light-gatheringcapability
n~1.45n~1.43
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Light Coupling into a fiber
Effect of numerical aperture…
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Light Coupling is Polarization Dependent
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Geometrical Optics Applied to Fiber
Light propagates by total internal reflectionModal Dispersion: Different path lengths cause energy in narrow pulse to spread outδT = time difference between fastest and slowest ray
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Total Internal Reflection & ModesImpacts how much a fiber can be bent!
Micro-bends can eat up energy, kill some modes!
Modes are standing wave patterns in wave- or EM-optics!
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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EM Optics: Optical Electromagnetic Wave
Linear polarization assumed …
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Amplitude Fluctuations of TEM Waves
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Speed of Light in a MediumAs a monochromatic wave propagates through media of different refractive indices, its frequency remains same, but its velocity, wavelength and wavenumber are altered.
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Diffraction or Fresnel Phenomenon
Cannot be explained by ray optics!
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Diffraction Pattern from a Circular Aperture
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Diffraction Patterns at Different Axial Positions
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Diffraction Grating• Periodic thickness or refractive index variation (“grooves”)
* Diffraction also occurs w/ pin hole of size of ~λ* In polychromatic light, different wavelengths diffracted differently
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Diffraction Grating as a Spectrum Analyzer
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Interference: Young’s Experiment
Interference is simple superposition, and a wave-phenomenon
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Interference of Two Spherical Waves
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Interference of Two Waves
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Multiple Waves Interference (Equal Amplitude, Equal Phase Differences)
Sinc-squared function
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Application: Bragg Reflection & Interference
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High Intensity, Narrow Pulses from Interference between M Monochromatic Waves
• Used in Phase locked lasers
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Propagation of a Polychromatic Wave
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Optical Splicing Issues: Speckle Patterns
Speckle patterns are time-varying andarise from solution of Maxwell’s equations(> geometric optics)
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Recall: Interaction of Light with Matter
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Optical Transmission: More Light-Matter Interaction Effects
AttenuationDispersion
Nonlinearity
Waveform after 1000 kmTransmitted data waveform
Reflectance
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Absorption vs Scattering
Both are linear effects that lead to “attenuation”. Rayleighscattering effects dominate much more than absorption (in lowerWavelengths, but decreases with wavelength)
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Absorption and Attenuation: Absorption Spectrum
Material absorption (Silica)
0.2 dB/km
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Fiber:Transmission Windows
Lucent’s new AllWave Fiber (1998) eliminates absorption peaksdue to watervapor in the 1400nm area!
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Transmission Bands
Bandwidth: over 35000 Ghz, but limited by bandwidth of EDFAs (optical amplifiers): studied later…
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Optical Amplifier: Limitations on Practical Bandwidths
EDFAs popular in C-bandRaman: proposed for S-bandGain-shifted EFDA for L-band
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Fiber Attenuation
Two windows: 1310 & 1550 nm
1550 window is preferred for long-haul applications
Less attenuationWider windowOptical amplifiers
1190.5 0.486417271 Lucent SMF28 clone fiber spool1190.999 0.4872906771191.499 0.488428184 fiber Spool ID : 021V14645A1CVC1191.998 0.4892913091192.498 0.4891969191192.998 0.4876170791193.497 0.4845246381193.997 0.4805207331194.496 0.4765939481194.996 0.4736238981195.496 0.4719367051195.995 0.4712003141196.495 0.4707222891196.994 0.4699232941197.494 0.468636021197.994 0.4670563051198.493 0.4654864741198.993 0.46414381199.492 0.4631587041199.992 0.4626560831200.492 0.4627433351200.991 0.4633780131201.491 0.4642815531201.99 0.4650492861202.49 0.4653624921202.99 0.465060732
1203.489 0.4640320791203.989 0.462164558
0.2
0.3
0 .4
0 .5
0 .6
1310window
λ
1550window
1190 0.486098221 obtained by measuring about 50 km of
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Fiber Anatomy
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Fiber ManufacturingDopants are added to control RI profile of the fiber (discussed later)Fiber: stronger than glass
A fiber route may have several cablesEach cable may have upto 1000 fibersEach fiber may have upto 160 wavelengthsEach wavelength may operate at 2.5Gbps or 10 Gbps
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
56
Single vs. Multimode FiberSilica-Based Fiber Supports 3 Low-Loss “Windows”: 0.8, 1.3 , 1.55 µm wavelengthMultimode Fibers Propagate Multiple Modes of Light
core diameters from 50 to 85 µmmodal dispersion limitations
Single-mode Fibers Propagate One Mode Onlycore diameters from 8 to 10 µmchromatic dispersion limitations
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
57
Summary: Single-mode vs Multi-mode
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Multimode vs Single mode: Energy distributions
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Single Mode Characteristics (contd)It (almost) eliminates delay spreadMore difficult to splice than multimode due to critical core requirementsMore difficult to couple all photonic energy from a source into it; light propagates both in core and cladding!Difficult to study propagation w/ ray theory; requires Maxwell’s equationsSuitable for transmitting modulated signals at 40 Gb/s and upto 200 km w/o amplificationLong lengths and bit rates >= 10 Gbps bring forth a number of issues due to residual nonlinearity/birefringence of the fiberFiber temperature for long lengths and bit rates > 10 Gbps becomes significant.
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Single Mode Light Propagation
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Dispersion
Interference
Dispersion causes the pulse to spread as it travels along the fiberChromatic dispersion important for single mode fiber
Depends on fiber type and laser usedDegradation scales as (data-rate)2
Was not important for < 2.5Gbps, < 500km SMF fibersModal dispersion limits use of multimode fiber to short distances
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Effects of Dispersion
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Pulse-Widening Effect on ISI & BER
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Combating Modal Dispersion in Multimode Fiber: Refractive Index Profiles
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Graded Index (contd)
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Graded Index MultiModeCharacteristics (contd)
Minimizes delay spread (modal dispersion), but it is still significant at long lengthsOne percent index difference between core/cladding amounts to 1-5ns/km delay spread
Step index has 50 ns/km spreadEasier to splice and couple light into itBit rate is limited (100 Mbps etc) for 40 km. Higher bit rates for shorter distancesFiber span w/o amplification is limited Dispersion effects for long lengths, high bit rates is a limiting factor
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Chromatic DispersionDifferent spectral components of a pulse travel at different velocitiesAlso called group-velocity-dispersion (GVD),
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Chromatic DispersionDifferent spectral components of a pulse travel at different velocitiesAlso called group-velocity-dispersion (GVD), aka β2Sub-components:
Material dispersion: frequency-dependent RIWaveguide dispersion: light energy propagates partially in core and cladding.Effective RI lies between the two (weighted by the power distribution). Power distribution of a mode between core/cladding a function of wavelength!
GVD parameter (β2) > 0 => normal dispersion (1.3µm)GVD parameter (β2) < 0 => anomalous dispersion (1.55µm)
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Pulse Shaping: Chirped Gaussian PulsesSince chromatic dispersion affects pulse shape, we study how pulse shaping may affect the outcomeGaussian: envelope of pulseChirped: frequency of launched pulse changes with timeSemiconductor lasers + modulation, or nonlinear effects also lead to chirpingWith anomalous c-dispersion in normal 1.55 um fibers (β2< 0), and negative chirping (κ < 0, natural for semi-laser outputs), the pulse broadening effects are exacerbated (next slide)Key parameter: dispersion length (LD)
@1.55um, LD = 1800 km for OC-48 and LD = 155 km for OC-192)If d << LD then chromatic dispersion negligible
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Chromatic Dispersion effect on Unchirped/Chirped Pulses
Unchirped
(Negatively) Chirped
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Chirped Pulses May Compress (I.e. not broaden)!
* Depends upon chirpingparameter (κ) and GVDParameter (β2), I.e κ β2<0* Pulse may compress upto aparticular distance andthen expand (disperse)* Corning’s metrocor fiber:positive β2 in 1.55 um band!
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Combating Chromatic Dispersion: Dispersion Shifted Fiber
Though material dispersion cannot be attacked, waveguide dispersion can be reduced (aka “shifted”) => DSF fiber
•Deployed a lot in Japan
•RI profile can also be varied to combat residualC-dispersion
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Dispersion Shifted Fiber (contd)* Waveguide dispersion may be reduced by changing the RI-profile of the single-mode fiber from a step-profile to a trapezoidal profile (see below)* This operation effectively “shifts” the zero-chromatic dispersion point to 1550nm & the average value in the band is 3.3 ps/nm/km* Alternatively a length of “compensating” fiber can be used
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Fiber Dispersion
Wavelengthl
Dis
pers
ion
ps/
nm-k
m 18
01310 nm 1550nm
Normal fiberNon-dispersion shifted fiber (NDSF) >95% of deployed plant
Reduced dispersion fibersDispersion shifted fiber (DSF)Non-zero dispersion shifted fibers (NZDSF)
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Dispersion Compensation Modules
Instead of DSF fibers, use dispersion compensation modulesEg: In-fiber chirped bragg gratings (carefully reflect selected λs and make then travel a longer path segment) to compensate for C-dispersion
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Residual Dispersion after DCMs
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Role of Polarization• Polarization: Time course of the direction of the electric
field vector- Linear, Elliptical, Circular, Non-polar
• Polarization plays an important role in the interaction of light with matter- Amount of light reflected at the boundary between two
materials- Light Absorption, Scattering, Rotation- Refractive index of anisotropic materials depends on
polarization (Brewster’s law)
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Linearly Polarized Light
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Circularly Polarized Light
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Polarizing Filters
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Rotating Polarizations
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Optical Isolator
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Single Mode Issues: Birefringence, PMD
Even in single mode, there are 2 linearly independent solutions for every λ (to maxwell’s equations)State of polarization (SOP): distribution of light energy between the (two transverse) polarization modes Ex and EyPolarization Vector: The electric dipole moment per unit volumeIn perfectly circular-symmetric fiber, the modes should have the same velocityPractical fibers have a slight difference in these velocities (birefringence): separate un-polarized light into two rays with different polarizationsThis leads to pulse-spreading called Polarization Mode Dispersion (PMD)
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AnIsotropy and BirefringenceSilica used in fiber is isotropic
Birefringence can alsobe understood as differentrefractive indices in different directions
It can be exploited (eg: Lithium niobate) for tunable filters, isolators, modulators etc
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Birefringence
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Polarization Mode Dispersion (PMD)Most severe in older fiberCaused by several sources
Core shapeExternal stressMaterial properties
Note: another issue is polarization-dependent loss (PDL)Both become dominant issue at OC-192 and OC-768
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Polarization Mode Dispersion
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Non-linear EffectsLinearity: a light-matter interaction assumption
Induced dielectric polarization is a convolution of material’s susceptibility (χ) and the electric field (E)
Linearity: low power (few mW) & bit rates (2.4 Gbps)
Non-linearity:↑ bit rates (10 Gbps) and ↑ power => non-linearities↑ channels (eg: DWDM) => more prominent even in moderate bit rates etc
Two categories:A) λ-phonon interaction & scattering (SRS, SBS)B) RI-dependence upon light intensity (SPM, FWM)
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Non-linearity Scattering EffectsStimulated Raman or Brillouin Scattering (SRS or SBS)
Energy transferred from one λ to another at a longer λ(or lower energy)The latter wave is called the “Stokes wave”Former wave is also called the “pump”Pump loses power as it propagates and Stokes wave gains power
SBS: pump is signal wave & Stokes is unwanted waveSRS: pump is high-power wave, and Stokes wave is signal wave that is amplified at the expense of the pumpParameters:
g: gain coefficient (strength of the effect)∆f: Spectral width over which the gain is present
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SRS: Photon Emission Mechanics
Photons interact with atoms: eg: May be absorbed to reach an “excited” state (“meta-stable”, I.e. cant hang around!) In the excited state, certain photons may trigger them to fall back, and release energy in the form of photons/phononsPhoton-Atom vs Photon-Atom-Photon interactionsMost of these effects are “third order” effects
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Stimulated Raman Scattering (SRS)Power transferred from lower-λ to higher-λchannelsCan be used as basis for optical amplification and lasers!Photons of lower-λ have higher energy (aka“pump”) that excite atoms and lead to stimulate emission at higher-λEffect smaller than SBS, but can affect both forward and reverse directionsEffect is also wider: I.e a broadband effect (15 Thz)
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Raman Scattering
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Stimulated Brillouin Scattering (SBS)Triggered by interaction between a photon and an acoustic phonon (I.e. molecular vibrations)Affects a narrowband: 20 Mhz (compare with 15 Thz effect in SRS)
Can combat it by making source linewidth widerThe downshifted wavelength waves propagate in the opposite direction (reverse gain): need isolation at source!Dominant when the spectral power (brightness) of the source is large and abruptly increases beyond a threshold (5-10 mW)Limits launched power per channel, but may be used in amplification
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SBS: Threshold Variation
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Electro-Optic RI EffectsElectro-optic effects:
Refractive index (RI) depends upon amplitude (and hence intensity) of electric field (E)Result: induced birefringence, dispersion
Pockels Effect: ∆n = (a1)EKerr Effect: (second order) ∆n = (λK)E2
The second order magnification in Kerr effect may be used to create ultra high speed modulators (> 10Gbps)
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Intensity-dependent RI EffectsSelf-phase Modulation (SPM), Cross-Phase Modulation (CPM) & Four-wave mixing (FWM)SPM: Pulses undergo induced chirping at higher power levels due to RI variations that depend upon intensityIn conjunction with chromatic dispersion, this can lead to even more pulse spreading & ISI
But it could be used to advantage depending upon the sign of the GVD parameter
CPM: Multiple channels: induced chirp depends upon variation of RI with intensity in other channels!FWM: A DWDM phenomena: tight channel spacing
Existence of f1, … fn gives rise to new frequencies 2fi – fj and fi + fj – fk etcIn-band and out-of-band crosstalk
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Self-Phase Modulation
Example of (positive) chirp or frequency fluctuations induced by self-phase modulation
Modulation instability or self-modulation: In the frequency domain, we see new sidelobes
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Four-Wave Mixing (FWM)
Creates in-band crosstalk(superposition of uncorrelated data) that can not be filteredSignal power depletionSNR degradationProblem increases geometrically with
Number of λsSpacing between λsOptical power level
Chromatic dispersion minimizes FWM (!!)Need to increase channel spacing and manage power carefully
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Four-Wave Mixing Effects
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Fiber Dispersion (revisited)
Wavelengthl
Dis
pers
ion
ps/
nm-k
m 18
01310 nm 1550nm
Normal fiberNon-dispersion shifted fiber (NDSF) >95% of deployed plant
Reduced dispersion fibersDispersion shifted fiber (DSF)Non-zero dispersion shifted fibers (NZDSF)
* Dispersion-shifted (DSF) is good for chromatic dispersionbut bad for non-linear effects. * NZ-DSF: puts back a small amount of C-dispersion!
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Non-Zero Dispersion Shifted Fiber
• NZ-DSF: puts back a small amount of C-dispersion!• Note: The goal of RI-profile shaping is different here than graded-index in multimode fiber
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Fibers: chromatic dispersion story…
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Latest Fibers & BandsLEAF fibers have largereffective area=> bettertradeoff for non-linearities
Fiber Bands: O-band: (Original) 1260-1360nmE-band: (Extended) 1360-1460nmS-band: (Short) 1460-1530nmC-band: (Conventional): 1530-1565nmL-band: (Long) 1565-1625nmU-band: (Ultra-long): 1625-1675nm
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Terrestrial vs Submarine Fibers
* Positive (chromatic) dispersion fibers (CDF) used in terrestrial, and negative CDF used in submarine apps. * Due to modulation instability (interaction between SPM and chromatic dispersion at high power levels)
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Fiber Dispersion (contd)
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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SolitonsKey idea: SPM induced chirping actually depends upon the time-domain envelope of the pulse!If pulse envelope right, SPM induced chirping will exactly combat the chromatic dispersion (GVD) chirping!
Soliton Regime: input powerdistribution shape, effective area/cross-section of fiber core and fiber type
DWDM with pure solitonsnot practical since solitons may “collide” and exchange energy over a length of fiber
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Solitons (contd)Family of pulse shapes which undergo no change or periodic changesFundamental solitons: no change in shapeHigher-order solitons: periodic changes in shapeSignificance: completely overcome chromatic dispersionWith optical amplifiers, high powers, the properties maintained => long, very high rate, repeaterless transmissionEg: 80 Gb/s for 10,000km demonstrated in lab (1999)!Dispersion-managed solitons:
An approximation of soliton pulse, but can operate on existing fiberThis can be used for DWDM: 25-channel, 40 Gbps, 1500km has been shown in lab (2001)
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Summary: Fiber and Optical Amplifier TrendsBandwidth-span product:
SMF: 1310 nm, 1983 => 2.5Gbps for 640 km w/o amplification or 10 Gbps for 100 kmRecent SMF: 2.5 Gbps for 4400 km; 10 Gbps for 500 kmMultiply these by # of DWDM channels! (eg: 40-160)…
Fiber amplifiers: Erbium doped (EDFA): 1550 nm rangePraseodymium-doped flouride fiber (PDFFA): 1310 nmThorium-doped (ThDFA): 1350-1450nmThulium-doped (TmDFA): 1450-1530 nmTellerium-erbium-doped (Te-EDFA): 1532-1608 nmRaman amplifiers: address an extended spectrum using standard single-mode fiber… (1150 –1675 nm!)
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Optical Amplifier: Limitations on Practical Bandwidths
EDFAs popular in C-bandRaman: proposed for S-bandGain-shifted EFDA for L-band
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Future: Hollow Nano-tube Waveguides
Perhaps carbon nanotubes developed at RPI could be used? ☺
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Summary: Interaction of Light with Matter
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Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Metrics and Parameters in Optics