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4. Optical Fibers Stephen Schultz Fiber Optics Fall 2005 1 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 Stephen Schultz Fiber Optics Fall 2005 2 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 Stephen Schultz Fiber Optics Fall 2005 3 Critical Angle • Ray bends at boundary between materials n2 sin 2 n1 sin 1 – Snell’s law 2 sin 1 n2 n sin 1 • 1 Light confined to core if propagation angle is greater than the critical angle – Total internal reflection (TIR) 1 c sin 1 n2 n Stephen Schultz Fiber Optics Fall 2005 1 4 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 Stephen Schultz Fiber Optics Fall 2005 5 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: NA sin i n12 n22 • Telecommunications optical fiber n1~n2, NA n12 n22 n1 n2 n1 n2 NA n1 2 Stephen Schultz Fiber Optics Fall 2005 2 n12 n1 n2 n1 n1 n2 n1 6 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 Stephen Schultz b zi n1 k0 sin i n2 ko b z n1 ko Fiber Optics Fall 2005 7 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 – n is the angular mode number Stephen Schultz Fiber Optics Fall 2005 8 Bessel Functions Stephen Schultz Fiber Optics Fall 2005 9 Transcendental Equation • Under the weakly guiding approximation (n1-n2)<<1 – Valid for standard telecommunications fibers J l' kT a K l' γ a kT a γa J l kT a K l γ a kT2 n12 ko2 b 2 • 2 b 2 n22 ko2 Substitute to eliminate the derivatives J l x J x J l 1 x l ' l kT a K l x K x K l 1 x l ' l x J l 1 kT a K a a l 1 or J l κ a K l a kT a HE Modes Stephen Schultz x J l 1 kT a K a a l 1 J l kT a K l a EH Modes Fiber Optics Fall 2005 10 Bessel Function Relationships • Bessel function recursive relationships K n x K n x J n x 1 J n x n 2n J n 1 x J n x J n 1 x x 2n K n 1 x K n x K n 1 x x 2 K 0 x K1 x K 2 x x 2 J 0 x J1 x J 2 x x • Small angle approximations x ln 0.5772 n 0 2 K n x n n 1 ! 2 n0 2 x Stephen Schultz Fiber Optics Fall 2005 11 Lowest Order Modes • Look at the l=-1, 0, 1 modes x kT a • • y a V x2 y2 a n12 n22 Use bessel function properties to get positive order and highest order on top l=-1 J x K y x 2 y 2 J 1 x K 1 y x l=0 x J 1 x K y y 1 J 0 x K0 y x x J1 x K y y 1 J 0 x K0 y Stephen Schultz J 0 x K y y 0 J1 x K1 y J 2 x K2 y x y J1 x K1 y J x K y x 2 y 2 J1 x K1 y • 2 Fiber Optics Fall 2005 J1 x K y y 1 J 0 x K0 y 12 Lowest Order Modes cont. • l=+1 x J 0 x K y y 0 J1 x K1 y x • x J 2 x K y y 2 J1 x K1 y J 2 x K y y 2 J1 x K1 y So the 6 equations collapse down to 2 equations x J1 x K1 y x y J 0 x K0 y J 2 x K y y 2 J1 x K1 y lowest modes Stephen Schultz Fiber Optics Fall 2005 13 Modes 20 LHS, RHS 15 10 5 0 0 2 4 6 8 10 x (kT a) Stephen Schultz Fiber Optics Fall 2005 14 Fiber Modes Stephen Schultz Fiber Optics Fall 2005 15 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 Stephen Schultz Fiber Optics Fall 2005 16 First Mode Cut-Off • First mode – What is the smallest allowable V – Let y 0 and the corresponding x V 1 2 y J V K y 2 y V 1 lim y 1 lim 0 y 0 y 0 y J 0 V K0 y ln 0.5772 2 J1 V 0 – So V=0, no cut-off for lowest order mode – Same as a symmetric slab waveguide Stephen Schultz Fiber Optics Fall 2005 17 Second Mode Cut-Off • Second mode 2 1 2 y J V K y 2 y V 2 lim y 2 lim 2 y 0 y 0 J1 V K1 y 1 2 2 y J 2 V 2 J1 V V 2n J n 1 x J n x J n 1 x x 2 J 2 x J1 x J 0 x x 2 2 J1 V J o V J1 V V V J o V 0 V 2.405 Stephen Schultz Fiber Optics Fall 2005 18 Cut-off V-parameter for low-order LPlm modes m=1 m=2 m=3 l=0 0 3.832 7.016 l=1 2.405 5.520 8.654 Stephen Schultz Fiber Optics Fall 2005 19 Number of Modes • The number of modes can be characterized by the normalized frequency V • 2 a n12 n22 Most standard optical fibers are characterized by their numerical aperture NA n12 n22 • Normalized frequency is related to numerical aperture V • • 2 a NA The optical fiber is single mode if V<2.405 For large normalized frequency the number of modes is approximately # Modes Stephen Schultz 4 2 V 2 Fiber Optics Fall 2005 V 1 20 Intensity Profiles Stephen Schultz Fiber Optics Fall 2005 21 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() • =8° • cutoff = 1260nm (single mode for cutoff) • Single mode for both =1300nm and =1550nm standard telecommunications wavelengths Stephen Schultz Fiber Optics Fall 2005 22 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() • =16° • Many modes Stephen Schultz Fiber Optics Fall 2005 23 5. Optical Fibers Attenuation Stephen Schultz Fiber Optics Fall 2005 24 Coaxial Vs. Optical Fiber Attenuation Stephen Schultz Fiber Optics Fall 2005 25 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: a P dz where a is the power attenuation coefficient per unit length dP • If Pin power is launched into the fiber, the power remaining after propagating a length L within the fiber Pout is Pout Pin exp a L Stephen Schultz Fiber Optics Fall 2005 26 Fiber Attenuation • Attenuation is conveniently expressed in terms of dB/km a dB km P 10 log 10 out L Pin Pine aL 10 log 10 L P in 10 aL log 10 e L 4.34a • Power is often expressed in dBm (dBm is dB from 1mW) 10 mW 10 dBm P 10 mW 10 log 10 1 mW 27 10 P 27 dBm 1 mW 10 501mW Stephen Schultz Fiber Optics Fall 2005 27 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 Pout 1050 10 1 mW 10 nW • 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 Stephen Schultz Fiber Optics Fall 2005 28 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 mm • Interaction between O-H bond and SiO2 glass at =1.24 mm • The most important absorption peaks are at =1.4 mm and 1.24 mm Stephen Schultz Fiber Optics Fall 2005 29 Scattering Loss • There are four primary kinds of scattering loss – Rayleigh scattering is the most important a R cR • • 1 dB / km 4 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 Stephen Schultz Fiber Optics Fall 2005 30 Absorption and Scattering Loss Stephen Schultz Fiber Optics Fall 2005 31 Absorption and Scattering Loss Stephen Schultz Fiber Optics Fall 2005 32 Loss on Standard Optical Fiber Wavelength SMF28 62.5/125 850 nm 1.8 dB/km 2.72 dB/km 1300 nm 0.35 dB/km 0.52 dB/km 1380 nm 0.50 dB/km 0.92 dB/km 1550 nm 0.19 dB/km 0.29 dB/km Stephen Schultz Fiber Optics Fall 2005 33 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 Stephen Schultz Fiber Optics Fall 2005 34 6. Optical Fiber Dispersion Stephen Schultz Fiber Optics Fall 2005 35 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 Stephen Schultz Fiber Optics Fall 2005 36 Intermodal Dispersion • Higher order modes have a longer path length – Longer path length has a longer propagation time – Temporal pulse separation L vg – vg is used as the propagation speed for the rays to take into account the material dispersion Stephen Schultz Fiber Optics Fall 2005 37 Group Velocity • Remember that group velocity is defined as • For a plane wave traveling in glass of index n1 • Resulting in b n1 n1 c c 1 n1 n1 c b vg b n1 1 c n1g c 1 c b vg n1g Stephen Schultz Fiber Optics Fall 2005 n1g n1 n1 38 Intermodal Dispersion • • • • Path length PL depends on the propagation angle L PL sin 1 The travel time for a longitudinal distance of L is Temporal pulse separation PL L vg vg sin 1 1 1 L v sin v sin 2 g 1 g The dispersion is time delay per unit length or Stephen Schultz Fiber Optics Fall 2005 1 1 D v sin v sin 2 g 1 g 39 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°) n1g 1 1 v g1 v g sin 90 c n1g n1 1 1 v g 2 vg sin c c n2 • Dispersion becomes n1g n1 n1g 1 D c n2 c • n1 n2 n1g n2 c The modes are not equally excited – The overall dispersed pulse has an rms pulse spread of approximately n1g D c 2 Stephen Schultz Fiber Optics Fall 2005 40 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 Stephen Schultz Fiber Optics Fall 2005 41 Graded Index Multimode Fiber • Refractive index profile 2 r n1 1 2 a nr n 1 2 n 2 1 • ra ra The intermodal dispersion is smaller than for step index multimode fiber n1g 2 Dinter c 4 Stephen Schultz Fiber Optics Fall 2005 42 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 2 g 1 2 g g g o o o 2 2 • Dispersion is defined as Dintra • g Propagation delay becomes 1 b z v g g g o o Dintra • 1 o 2 Dintra 2 Keeping the first two terms, the pulse width increase for a laser linewidth of is g Dintra Stephen Schultz Fiber Optics Fall 2005 43 Intramodal Dispersion • Intramodal dispersion is Dintra • There are two components to intramodal dispersion Dintra • • b z b z b1 b1 1 n1g b z n1g b z Dmaterial Dwaveguide c b1 c b1 Material dispersion is related to the dependence of index of refraction on wavelength Waveguide dispersion is related to dimensions of the waveguide Stephen Schultz Fiber Optics Fall 2005 44 Material Dispersion • Material dispersion depends on the material Dmaterial Stephen Schultz 1 n1g b z c b1 Fiber Optics Fall 2005 45 Waveguide Dispersion • Waveguide dispersion depends on the dimensions of the waveguide Dwaveguide • Expanded to give n1g b z c b1 n1g n1g 2 2 b z b z Dwaveguide V 2 V 2 c n1 V b1 V b1 where V is the normalized frequency V k a NA • Practical optical fibers are weekly guiding (n1-n2 <<1) resulting in the simplification Dwaveguide Stephen Schultz n 1g n2 g 2 V b V 2 c V Fiber Optics Fall 2005 46 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 mm Dispersion shift fiber has the zero dispersion shifted to around =1.55 mm Stephen Schultz Fiber Optics Fall 2005 47 Standard Optical Fiber Dispersion • Standard optical fiber – Step index ≈0.0036 – Graded index ≈0.02 • Dispersion – Step index multi-mode optical fiber (Dtot~10ns/km) n1g Dtot Dinter c 2 – Graded index multi-mode optical fiber (Dtot~0.5ns/km) n1g 2 Dtot Dinter c 4 – Single mode optical fiber (Dintra~18ps/km nm) Dtot Dintra Stephen Schultz Fiber Optics Fall 2005 48 What is the laser linewidth? • Wavelength linewidth is a combination of inherent laser linewidth and linewidth change caused by modulation 2laser 2mod • – 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 c f f c 2 Stephen Schultz 2 c 2 c f 2B Fiber Optics Fall 2005 49