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4. Optical Fibers. Anatomy of an Optical Fiber. Light confined to core with higher index of refraction Two analysis approaches Ray tracing Field propagation using Maxwell’s equations. Optical Fiber Analysis. Calculation of modes supported by an optical fiber Intensity profile

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4 optical fibers

4. Optical Fibers

Fiber Optics

Fall 2005

anatomy of an optical fiber
Anatomy of an Optical Fiber
  • Light confined to core with higher index of refraction
  • Two analysis approaches
    • Ray tracing
    • Field propagation using Maxwell’s equations

Fiber Optics

Fall 2005

optical fiber analysis
Optical Fiber Analysis
  • Calculation of modes supported by an optical fiber
    • Intensity profile
    • Phase propagation constant
  • Effect of fiber on signal propagation
    • Signal attenuation
    • Pulse spreading through dispersion

Fiber Optics

Fall 2005

critical angle
Critical Angle
  • Ray bends at boundary between materials
    • Snell’s law
  • Light confined to core if propagation angle is greater than the critical angle
    • Total internal reflection (TIR)

Fiber Optics

Fall 2005

constructive interference
Constructive Interference
  • Propagation requires constructive interference
    • Wave stays in phase after multiple reflections
    • Only discrete angles greater than the critical angle are allowed to propagate

Fiber Optics

Fall 2005

numerical aperture
Numerical Aperture
  • The acceptance angle for a fiber defines its numerical aperture (NA)
  • The NA is related to the critical angle of the waveguide and is defined as:
  • Telecommunications optical fiber n1~n2,

Fiber Optics

Fall 2005

modes
Modes
  • The optical fiber support a set of discrete modes
  • Qualitatively these modes can be thought of as different propagation angles
  • A mode is characterized by its propagation constant in the z-direction bz
  • With geometrical optics this is given by
  • The goal is to calculate the value of βz
  • Remember that the range of βz is

Fiber Optics

Fall 2005

optical fiber modes
Optical Fiber Modes
  • The optical fiber has a circular waveguide instead of planar
  • The solutions to Maxwell’s equations
    • Fields in core are non-decaying
      • J, Y Bessel functions of first and second kind
    • Fields in cladding are decaying
      • K modified Bessel functions of second kind
  • Solutions vary with radius r and angle q
  • There are two mode number to specify the mode
    • m is the radial mode number
    • n is the angular mode number

Fiber Optics

Fall 2005

bessel functions
Bessel Functions

Fiber Optics

Fall 2005

transcendental equation
Transcendental Equation
  • Under the weakly guiding approximation (n1-n2)<<1
    • Valid for standard telecommunications fibers
  • Substitute to eliminate the derivatives

HE Modes

EH Modes

Fiber Optics

Fall 2005

bessel function relationships
Bessel Function Relationships
  • Bessel function recursive relationships
  • Small angle approximations

Fiber Optics

Fall 2005

lowest order modes
Lowest Order Modes
  • Look at the l=-1, 0, 1 modes
  • Use bessel function properties to get positive order and highest order on top
  • l=-1
  • l=0

Fiber Optics

Fall 2005

lowest order modes cont
Lowest Order Modes cont.
  • l=+1
  • So the 6 equations collapse down to 2 equations

lowest modes

Fiber Optics

Fall 2005

modes14
Modes

Fiber Optics

Fall 2005

fiber modes
Fiber Modes

Fiber Optics

Fall 2005

hybrid fiber modes
Hybrid Fiber Modes
  • The refractive index difference between the core and cladding is very small
  • There is degeneracy between modes
    • Groups of modes travel with the same velocity (bz equal)
  • These hybrid modes are approximated with nearly linearly polarized modes called LP modes
    • LP01 from HE11
    • LP0m from HE1m
    • LP1m sum of TE0m, TM0m, and HE2m
    • LPnm sum of HEn+1,m and EHn-1,m

Fiber Optics

Fall 2005

first mode cut off
First Mode Cut-Off
  • First mode
    • What is the smallest allowable V
    • Let y  0 and the corresponding x V
    • So V=0, no cut-off for lowest order mode
    • Same as a symmetric slab waveguide

Fiber Optics

Fall 2005

second mode cut off
Second Mode Cut-Off
  • Second mode

Fiber Optics

Fall 2005

number of modes
Number of Modes
  • The number of modes can be characterized by the normalized frequency
  • Most standard optical fibers are characterized by their numerical aperture
  • Normalized frequency is related to numerical aperture
  • The optical fiber is single mode if V<2.405
  • For large normalized frequency the number of modes is approximately

Fiber Optics

Fall 2005

intensity profiles
Intensity Profiles

Fiber Optics

Fall 2005

standard single mode optical fibers
Standard Single Mode Optical Fibers
  • Most common single mode optical fiber: SMF28 from Corning
    • Core diameter dcore=8.2 mm
    • Outer cladding diameter: dclad=125mm
    • Step index
    • Numerical Aperture NA=0.14
      • NA=sin(q)
      • Dq=8°
      • lcutoff = 1260nm (single mode for l>lcutoff)
      • Single mode for both l=1300nm and l=1550nm standard telecommunications wavelengths

Fiber Optics

Fall 2005

standard multimode optical fibers
Standard Multimode Optical Fibers
  • Most common multimode optical fiber: 62.5/125 from Corning
    • Core diameter dcore= 62.5 mm
    • Outer cladding diameter: dclad=125mm
    • Graded index
    • Numerical Aperture NA=0.275
      • NA=sin(q)
      • Dq=16°
      • Many modes

Fiber Optics

Fall 2005

5 optical fibers attenuation

5. Optical Fibers Attenuation

Fiber Optics

Fall 2005

fiber attenuation
Fiber Attenuation
  • Loss or attenuation is a limiting parameter in fiber optic systems
  • Fiber optic transmission systems became competitive with electrical transmission lines only when losses were reduced to allow signal transmission over distances greater than 10 km
  • Fiber attenuation can be described by the general relation:

where a is the power attenuation coefficient per unit length

  • If Pin power is launched into the fiber, the power remaining after propagating a length L within the fiber Pout is

Fiber Optics

Fall 2005

fiber attenuation27
Fiber Attenuation
  • Attenuation is conveniently expressed in terms of dB/km
  • Power is often expressed in dBm (dBm is dB from 1mW)

Fiber Optics

Fall 2005

fiber attenuation28
Fiber Attenuation
  • Example: 10mW of power is launched into an optical fiber that has an attenuation of a=0.6 dB/km. What is the received power after traveling a distance of 100 km?
    • Initial power is: Pin = 10 dBm
    • Received power is: Pout= Pin– a L=10 dBm – (0.6)(100) = -50 dBm
  • Example: 8mW of power is launched into an optical fiber that has an attenuation of a=0.6 dB/km. The received power needs to be -22dBm. What is the maximum transmission distance?
    • Initial power is: Pin = 10log10(8) = 9 dBm
    • Received power is: Pout = 1mW 10-2.2 = 6.3 mW
    • Pout - Pin = 9dBm - (-22dBm) = 31dB = 0.6 L
    • L=51.7 km

Fiber Optics

Fall 2005

material absorption
Material Absorption
  • Material absorption
    • Intrinsic: caused by atomic resonance of the fiber material
      • Ultra-violet
      • Infra-red: primary intrinsic absorption for optical communications
    • Extrinsic: caused by atomic absorptions of external particles in the fiber
      • Primarily caused by the O-H bond in water that has absorption peaks at l=2.8, 1.4, 0.93, 0.7 mm
      • Interaction between O-H bond and SiO2 glass at l=1.24 mm
      • The most important absorption peaks are at l=1.4 mm and 1.24 mm

Fiber Optics

Fall 2005

scattering loss
Scattering Loss
  • There are four primary kinds of scattering loss
    • Rayleigh scattering is the most important

where cR is the Rayleigh scattering coefficient and is the range from 0.8 to 1.0 (dB/km)·(mm)4

  • Mie scattering is caused by inhomogeneity in the surface of the waveguide
    • Mie scattering is typically very small in optical fibers
  • Brillouin and Raman scattering depend on the intensity of the power in the optical fiber
    • Insignificant unless the power is greater than 100mW

Fiber Optics

Fall 2005

absorption and scattering loss
Absorption and Scattering Loss

Fiber Optics

Fall 2005

absorption and scattering loss32
Absorption and Scattering Loss

Fiber Optics

Fall 2005

loss on standard optical fiber
Loss on Standard Optical Fiber

Fiber Optics

Fall 2005

external losses
External Losses
  • Bending loss
    • Radiation loss at bends in the optical fiber
    • Insignificant unless R<1mm
    • Larger radius of curvature becomes more significant if there are accumulated bending losses over a long distance
  • Coupling and splicing loss
    • Misalignment of core centers
    • Tilt
    • Air gaps
    • End face reflections
    • Mode mismatches

Fiber Optics

Fall 2005

6 optical fiber dispersion

6. Optical Fiber Dispersion

Fiber Optics

Fall 2005

dispersion
Dispersion
  • Dispersive medium: velocity of propagation depends on frequency
  • Dispersion causes temporal pulse spreading
    • Pulse overlap results in indistinguishable data
    • Inter symbol interference (ISI)
  • Dispersion is related to the velocity of the pulse

Fiber Optics

Fall 2005

intermodal dispersion
Intermodal Dispersion
  • Higher order modes have a longer path length
    • Longer path length has a longer propagation time
    • Temporal pulse separation
    • vg is used as the propagation speed for the rays to take into account the material dispersion

Fiber Optics

Fall 2005

group velocity
Group Velocity
  • Remember that group velocity is defined as
  • For a plane wave traveling in glass of index n1
  • Resulting in

Fiber Optics

Fall 2005

intermodal dispersion39
Intermodal Dispersion
  • Path length PL depends on the propagation angle
  • The travel time for a longitudinal distance of L is
  • Temporal pulse separation
  • The dispersion is time delay per unit length or

Fiber Optics

Fall 2005

step index multimode fiber
Step Index Multimode Fiber
  • Step index multimode fiber has a large number of modes
  • Intermodal dispersion is the maximum delay minus the minimum delay
  • Highest order mode (q~qc) Lowest order mode (q~90°)
  • Dispersion becomes
  • The modes are not equally excited
    • The overall dispersed pulse has an rms pulse spread of approximately

Fiber Optics

Fall 2005

graded index multimode fiber
Graded Index Multimode Fiber
  • Higher order modes
    • Larger propagation length
    • Travel farther into the cladding
    • Speed increases with distance away from the core (decreasing index of refraction)
    • Relative difference in propagation speed is less

Fiber Optics

Fall 2005

graded index multimode fiber42
Graded Index Multimode Fiber
  • Refractive index profile
  • The intermodal dispersion is smaller than for step index multimode fiber

Fiber Optics

Fall 2005

intramodal dispersion
Intramodal Dispersion
  • Single mode optical fibers have zero intermodal dispersion (only one mode)
  • Propagation velocity of the signal depends on the wavelength
  • Expand the propagation delay as a Taylor series
  • Dispersion is defined as
  • Propagation delay becomes
  • Keeping the first two terms, the pulse width increase for a laser linewidth of Dl is

Fiber Optics

Fall 2005

intramodal dispersion44
Intramodal Dispersion
  • Intramodal dispersion is
  • There are two components to intramodal dispersion
  • Material dispersion is related to the dependence of index of refraction on wavelength
  • Waveguide dispersion is related to dimensions of the waveguide

Fiber Optics

Fall 2005

material dispersion
Material Dispersion
  • Material dispersion depends on the material

Fiber Optics

Fall 2005

waveguide dispersion
Waveguide Dispersion
  • Waveguide dispersion depends on the dimensions of the waveguide
  • Expanded to give

where V is the normalized frequency

  • Practical optical fibers are weekly guiding (n1-n2 <<1) resulting in the simplification

Fiber Optics

Fall 2005

total intramodal dispersion
Total Intramodal Dispersion
  • Total dispersion can be designed to be zero at a specific wavelength
  • Standard single mode telecommunications fiber has zero dispersion around l=1.3 mm
  • Dispersion shift fiber has the zero dispersion shifted to around l=1.55 mm

Fiber Optics

Fall 2005

standard optical fiber dispersion
Standard Optical Fiber Dispersion
  • Standard optical fiber
    • Step index D≈0.0036
    • Graded index D≈0.02
  • Dispersion
    • Step index multi-mode optical fiber (Dtot~10ns/km)
    • Graded index multi-mode optical fiber (Dtot~0.5ns/km)
    • Single mode optical fiber (Dintra~18ps/km nm)

Fiber Optics

Fall 2005

what is the laser linewidth
What is the laser linewidth?
  • Wavelength linewidth is a combination of inherent laser linewidth and linewidth change caused by modulation
    • Single mode FP laser Dllaser~2nm
    • Multimode FP laser or LED Dllaser~30nm
    • DFB laser Dllaser~0.01nm
  • Laser linewidth due to modulation
    • Df~2B

Fiber Optics

Fall 2005