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## 4. Optical Fibers

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**4. Optical Fibers**Fiber Optics Fall 2005**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**Bessel Functions**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**Modes**Fiber Optics Fall 2005**Fiber 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**Second Mode Cut-Off**• Second mode Fiber Optics Fall 2005**Cut-off V-parameter for low-order LPlm modes**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**Intensity Profiles**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**5. Optical Fibers Attenuation**Fiber Optics Fall 2005**Coaxial Vs. Optical Fiber Attenuation**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**Absorption and Scattering Loss**Fiber Optics Fall 2005**Absorption and Scattering Loss**Fiber Optics Fall 2005**Loss on Standard Optical Fiber**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**6. Optical Fiber Dispersion**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**Material Dispersion**• Material dispersion depends on the material 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