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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

- 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

- 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

- 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

- 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

- 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

- 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

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 recursive relationships
- Small angle approximations

Fiber Optics

Fall 2005

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.

- l=+1
- So the 6 equations collapse down to 2 equations

lowest modes

Fiber Optics

Fall 2005

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
- 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

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

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

- 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

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 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 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
- 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

- 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

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

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

- 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

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

Fiber Optics

Fall 2005

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 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

- 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 Fiber

- Refractive index profile
- The intermodal dispersion is smaller than for step index multimode fiber

Fiber Optics

Fall 2005

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 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

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 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
- 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?

- 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

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