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1 Stephen SchultzFiber Optics Fall 2005 4. Optical Fibers

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2 Stephen SchultzFiber 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

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3 Stephen SchultzFiber 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

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4 Stephen SchultzFiber 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)

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5 Stephen SchultzFiber 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

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6 Stephen SchultzFiber 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 n 1 ~n 2,

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7 Stephen SchultzFiber 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 z With geometrical optics this is given by The goal is to calculate the value of β z Remember that the range of β z is

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8 Stephen SchultzFiber 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 There are two mode number to specify the mode – m is the radial mode number – is the angular mode number

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9 Stephen SchultzFiber Optics Fall 2005 Bessel Functions

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10 Stephen SchultzFiber Optics Fall 2005 Transcendental Equation Under the weakly guiding approximation (n 1 -n 2 )<<1 –Valid for standard telecommunications fibers Substitute to eliminate the derivatives HE Modes EH Modes

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11 Stephen SchultzFiber Optics Fall 2005 Bessel Function Relationships Bessel function recursive relationships Small angle approximations

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12 Stephen SchultzFiber 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

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13 Stephen SchultzFiber Optics Fall 2005 Lowest Order Modes cont. l=+1 So the 6 equations collapse down to 2 equations lowest modes

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14 Stephen SchultzFiber Optics Fall 2005 Modes

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15 Stephen SchultzFiber Optics Fall 2005 Fiber Modes

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16 Stephen SchultzFiber 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 ( z equal) These hybrid modes are approximated with nearly linearly polarized modes called LP modes –LP 01 from HE 11 –LP 0m from HE 1m –LP 1m sum of TE 0m, TM 0m, and HE 2m –LP m sum of HE +1,m and EH -1,m

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17 Stephen SchultzFiber 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

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18 Stephen SchultzFiber Optics Fall 2005 Second Mode Cut-Off Second mode

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19 Stephen SchultzFiber Optics Fall 2005 Cut-off V-parameter for low-order LP lm modes m=1m=2m=3 l=003.8327.016 l=12.4055.5208.654

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20 Stephen SchultzFiber 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

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21 Stephen SchultzFiber Optics Fall 2005 Intensity Profiles

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22 Stephen SchultzFiber Optics Fall 2005 Standard Single Mode Optical Fibers Most common single mode optical fiber: SMF28 from Corning –Core diameter d core =8.2 m –Outer cladding diameter: d clad =125 m –Step index –Numerical Aperture NA=0.14 NA=sin( ) =8° cutoff = 1260nm (single mode for cutoff ) Single mode for both =1300nm and =1550nm standard telecommunications wavelengths

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23 Stephen SchultzFiber Optics Fall 2005 Standard Multimode Optical Fibers Most common multimode optical fiber: 62.5/125 from Corning –Core diameter d core = 62.5 m –Outer cladding diameter: d clad =125 m –Graded index –Numerical Aperture NA=0.275 NA=sin( ) =16° Many modes

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24 Stephen SchultzFiber Optics Fall 2005 5. Optical Fibers Attenuation

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25 Stephen SchultzFiber Optics Fall 2005 Coaxial Vs. Optical Fiber Attenuation

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26 Stephen SchultzFiber 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 is the power attenuation coefficient per unit length If P in power is launched into the fiber, the power remaining after propagating a length L within the fiber P out is

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27 Stephen SchultzFiber 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)

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28 Stephen SchultzFiber Optics Fall 2005 Fiber Attenuation Example: 10mW of power is launched into an optical fiber that has an attenuation of =0.6 dB/km. What is the received power after traveling a distance of 100 km? –Initial power is: P in = 10 dBm –Received power is: P out = P in – L=10 dBm – (0.6)(100) = -50 dBm Example: 8mW of power is launched into an optical fiber that has an attenuation of =0.6 dB/km. The received power needs to be -22dBm. What is the maximum transmission distance? –Initial power is: P in = 10log 10 (8) = 9 dBm –Received power is: P out = 1mW 10 -2.2 = 6.3 W –P out - P in = 9dBm - (-22dBm) = 31dB = 0.6 L –L=51.7 km

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29 Stephen SchultzFiber 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 =2.8, 1.4, 0.93, 0.7 m Interaction between O-H bond and SiO 2 glass at =1.24 m The most important absorption peaks are at =1.4 m and 1.24 m

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30 Stephen SchultzFiber Optics Fall 2005 Scattering Loss There are four primary kinds of scattering loss –Rayleigh scattering is the most important where c R is the Rayleigh scattering coefficient and is the range from 0.8 to 1.0 (dB/km)·( m) 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

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31 Stephen SchultzFiber Optics Fall 2005 Absorption and Scattering Loss

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32 Stephen SchultzFiber Optics Fall 2005 Absorption and Scattering Loss

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33 Stephen SchultzFiber Optics Fall 2005 Loss on Standard Optical Fiber WavelengthSMF2862.5/125 850 nm1.8 dB/km2.72 dB/km 1300 nm0.35 dB/km0.52 dB/km 1380 nm0.50 dB/km0.92 dB/km 1550 nm0.19 dB/km0.29 dB/km

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34 Stephen SchultzFiber 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

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35 Stephen SchultzFiber Optics Fall 2005 6. Optical Fiber Dispersion

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36 Stephen SchultzFiber 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

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37 Stephen SchultzFiber Optics Fall 2005 Intermodal Dispersion Higher order modes have a longer path length –Longer path length has a longer propagation time –Temporal pulse separation –v g is used as the propagation speed for the rays to take into account the material dispersion

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38 Stephen SchultzFiber Optics Fall 2005 Group Velocity Remember that group velocity is defined as For a plane wave traveling in glass of index n 1 Resulting in

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39 Stephen SchultzFiber 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

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40 Stephen SchultzFiber 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 ( ~ c )Lowest order mode ( ~90°) Dispersion becomes The modes are not equally excited –The overall dispersed pulse has an rms pulse spread of approximately

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41 Stephen SchultzFiber 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

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42 Stephen SchultzFiber Optics Fall 2005 Graded Index Multimode Fiber Refractive index profile The intermodal dispersion is smaller than for step index multimode fiber

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43 Stephen SchultzFiber 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 is

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44 Stephen SchultzFiber 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

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45 Stephen SchultzFiber Optics Fall 2005 Material Dispersion Material dispersion depends on the material

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46 Stephen SchultzFiber 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 (n 1 -n 2 <<1) resulting in the simplification

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47 Stephen SchultzFiber 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 =1.3 m Dispersion shift fiber has the zero dispersion shifted to around =1.55 m

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48 Stephen SchultzFiber Optics Fall 2005 Standard Optical Fiber Dispersion Standard optical fiber –Step index ≈0.0036 –Graded index ≈0.02 Dispersion –Step index multi-mode optical fiber (D tot ~10ns/km) –Graded index multi-mode optical fiber (D tot ~0.5ns/km) –Single mode optical fiber (D intra ~18ps/km nm)

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49 Stephen SchultzFiber 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 laser ~2nm –Multimode FP laser or LED laser ~30nm –DFB laser laser ~0.01nm Laser linewidth due to modulation – f~2B

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