ECE 228A Fall 2008 Daniel J. Blumenthal! 4.1!Lecture 4 - Propagation in Optical Fibers!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.2!Step Index Fibers!a!n!n1!n2!Fiber core!Fiber cladding!Definition: Fractional refractive index difference !Δ = (n1 - n2)/n1!Typical value for silica (glass) fibers!n1 = 1.48, n2 = 1.46!Δ = .0135 ≈ 1%!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.3!Geometrical Optics Model! Use of total internal refraction for optical field guiding!Light rays that enter the fiber with an angle smaller than an “acceptance angle” θ0 will be guided by total internal reflection within the fiber when:!−=≤−022211max,00sinnnnθθUnguided ray!n1!n2!Fiber core!Fiber cladding!θ1!Air n0 = 1.0!θ0!Unwanted unguided ray!Guided ray!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.4!Numerical Aperture!Definition: The light collecting capacity of the optical fiber is measured by the Numerical Aperature (NA)!NA = n0sinθ0,max= n12− n22≈ n12Δ Example: if we couple light from air into a fiber with Δ = .01 and n1 = 1.5, then the NA ≈ 0.2121 and θ0,max ≈ 12˚!(For small Δ)!The maximum acceptable “angular error” when launching an optical beam into a fiber is consequently of the order of θ0,max ≈ 12˚!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.5!Modes in Step Index Fibers!Definition: Modes are light intensity profiles (patterns) that propagate down the fiber maintaining their transversal field shape!• Multimode fibers can support many thousands of modes. !• Single mode fibers support one mode.!Gaussian first order mode intensity profile!Gaussian secon order mode intensity profile!E(x, y, z, t) = J(x, y)Cos(ω0t −β(ω0)z)In order to accurately study optical modes, the complete Maxwell equations are to be solved.!Anyway, for multimode fibers, the following intuitive explanation can be given:!Each mode corresponds to a light beam traveling inside the fiber core with different angles!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.6!Normalized Frequency Parameter V!( )Δ=−=2212122210annnaVλπκV is a design parameter that takes into account the fiber parameters (n1, n2 and a) and the free space wavelength λ0.!It can be shown that:!In order to have a Single Mode Fibers: !V ≤ 2.405!In order to have a Multimode Fibers: !V > 2.405!Important consequence:!Given the parameters n1, n2 and a fixed wavelength, a fiber is single mode if the core radius a is smaller than a given value (of the order of 10 µm at 1550 nm)!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.7!Multimode Fibers! Each mode will propagate in the fiber at as if it had its own index of refraction n. ! The index of refraction for each mode n lies between n1 and n2 (from the solution of the Maxwell equations)! Intuitive explanation: each mode has different portions of the field overlap with different amounts of the core and cladding! Consequence: each mode will travel along the fiber at slightly different speeds, giving rise to multimode fiber dispersion!N =V22For large V, the number of modes propagating in a multimode fiber is approximately!Example: A multimode fiber with core diameter 2a=50µm, Δ=5x10-3 and λ=1.3µm supports about 160 modes!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.8!Multimode Fiber Dispersion!Since each mode travels at a different velocity on the fiber, an optical bit launched into the fiber will distort as it propagates.!t!Optical intensity!Multimode Fiber!t!Optical intensity!• The resulting distortion is actually a pulse broadening!• The amount of pulse broadening in a multimode fiber is given by:!δT =Ln12cn2ΔL!T!T+δT!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.9!Bit Rate Limit for Multimode Fibers! When dealing with digital transmission, each pulse represent a bit! A pulse spreading leads to intersymbol interference (ISI)! Let’s assume a bit cannot spread by more than half the allocated bit period in order to have an acceptable ISI level!ΔT <12BLcn12n2Δ <12BBL <n2c2n12ΔWe can define the Bandwidth-Distance product (BL) for multimode fibers as:!TB = 1/B!TB = 1/B!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.10!Bit Rate Limit for Multimode Fibers! The previous formula give rise to the ultimate bit-rate limitation of a standard multimode fiber!10010110210-510-410-310-2Distance, L (km)Bit Rate, B (Gbps)BL <n2c2n12ΔΔ = 0.01!n1 = 1.5!On a standard 1-km long step-index multimode fiber, the resulting maximum bit rate is 1 Mbit/s only !!!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.11!Graded index fibers! The multimode dispersion limit can be drastically changed by using a proper index of refraction profile!radius n1 n2 radius n1 n2 Step Index!Graded Index!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.12!Dispersion limit for graded index fibers!Gb/s  The limit for graded index fibers is of the order of (for example) 1 Gbit/s at 2 Km! With particular techniques (misplaced launch) this limit can be somehow increased! Anyway, high bit rates and long haul link are NOT feasible on multimode fibers, even using graded index profiles!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.13!Multimode vs. single mode fiber! Multimode fiber! They have a limit in terms of maximum bit rate of the order of 1Gbit/1Km, due to multimode dispersion! They have a relatively large core! Splicing is easier! Connectors are less expensive! Installation is simpler! They are intrinsically more resilient to mechanical and environmental stress! They are thus mostly used in LAN application! Single mode fibers! We will see that they are not affected by multimode dispersion, and their bandwidth limit is extremely higher! They have a small core! Splicing is more difficult! Connectors are more expensive! Installation is more difficult! They are thus used in all applications where the distance to be covered is significantly higher than 1Km! In the rest of the course, we will mostly focus on single mode fibers!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.14!Step Index Circular Waveguide!(lossless, isotropic)!• Simplest type of fiber!• (Most fiber these days is far more complex)!• Cylindrical symmetry!ECE 228A Fall 2008 Daniel J. Blumenthal! 4.15!Step Index Circular Waveguide!(lossless, isotropic)!• Simplest type of fiber!• (Most fiber these days is far more complex)!• Cylindrical