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Chapter 2

Introduction to Optical Networks. Chapter 2. Propagation of Signals in Optical Fiber. 2.Propagation of Signals in Optical Fiber. Advantages Low loss ~0.2dB/km at 1550nm Enormous bandwidth at least 25THz Light weight Flexible Immunity to interferences Low cost

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Chapter 2

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  1. Introduction to Optical Networks Chapter 2 Propagation of Signals in Optical Fiber

  2. 2.Propagation of Signals in Optical Fiber Advantages • Low loss ~0.2dB/km at 1550nm • Enormous bandwidth at least 25THz • Light weight • Flexible • Immunity to interferences • Low cost Disadvantages and Impairments • Difficult to handle • Chromatic dispersion • Nonlinear Effects

  3. 2.1LightPropagation in Optical Fiber Cladding refractive index ≈1.45 core 8~10μm, 50μm, 62.5μm doped

  4. 2.1.1Geometrical Optical Approach (Ray Theory) This approach is only applicable to multimode fibers. incident angle (入射角) refraction angle (折射角) reflection angle (反射角) Snell’s Law

  5. =>Critical angle When total internal reflection occurs. let = air refractive index = acceptance angle (total reflection will occur at core/cladding interface)

  6. (2.2)

  7. If Δ is small (less than 0.01) For (multimode) Numerical Aperture NA= Because different modes have different lengths of paths, intermodal dispersion occurs.

  8. Infermode dispersion will cause digital pulse spreading Let L be the length of the fiber The ray travels along the center of the core The ray is incident at (slow ray)

  9. Assume that the bit rate = Bb/s Bit duration The capacity is measured by BL (ignore loss) Foe example, if

  10. For optimum graded-index fibers, δT is shorter than that in the step-index fibers, because the ray travels along the center slows down (n is larger) and the ray traveling longer paths travels faster (n is small)

  11. The time difference is given by (For Optical graded-index profile) and (single mode ) If Long haul systems use single-mode fibers

  12. 2.1.2 Wave Theory Approach Maxwell’s equations D.1 D.2 D.3 D.4 : the charge density, : the current density : the electric flux density, : the magnetic flux density : the electric field, : the magnetic field ρ

  13. Because the field are function of time and location in the space, we denote them by and , where and t are position vector and time. Assume the space is linear and time-invariant the Fourier transform of is 2.4 let be the induced electric polarization 2.5 : the permittivity of vacuum 2.6 : the magnetic polarization : the permeability of vacuum 注意有些書Fourier transform定義為

  14. Locality of Response: and related to dispersion and nonlinearities If the response to the applied electric field is local depends only on not on other values of This property holds in the 0.5~2μm wavelength Isotropy: The electromagnetic properties are the same for all directions in the medium Birefringence: The refraction indexes along two different directions are different (lithium niobate, LiNbO , modulator, isolator, tunable filter)

  15. Linearity： (Convolution Integral) 2.7 : linear susceptibility The Fourier transform of is 2.8 Where is the Fourier transform of ( is similar to the impulse response) is function of frequency => Chromatic dispersion

  16. Homogeneity: A homogeneous medium has the same electromagnetic properties at all points The core of a graded-index fiber is inhomogeneous Losslessness： No loss in the medium At first we will only consider the core and cladding regions of the fiber are locally responsive, isotropic, linear, homogeneous, and lossless. The refractive index is defined as 2.9 For silica fibers def

  17. From Appendix D For (zero charge) (zero conductivity, dielectric material) For nonmagnetic material

  18. Assume linear and homogenence

  19. Take Fourier transform Recall 2.8 Denote c: speed of light (Locally response, isotropic, linear, homogeneous, lossless) 2.9

  20. palacian operation 2.10 (free space wave number)

  21. For Cartesian coordinates For Cylindrical coordinatesρ. φ and z n:{ a: radius of the core Similarly 2.11 Boundary conditions is finite and continuity of field at ρ=a References: G.P. Agrawal “Fiber-Optical Communication System” Chapter 2 John Senior “Optical Fiber Communications, Principles and practice” John Gowar “Optical Communication Systems” 注意有些書在time domain運算 有些書在frequency domain運算

  22. Fiber Modes cladding core x z y

  23. must satisfy 2.10, 2.11 and the boundary conditions. let Where are unit vectors For the fundamental mode, the longitudinal component is the propagation constant

  24. : Bessel functions The transverse components For cylindrical symmetry of the fiber In general, we can write (Appendix E)

  25. Where The multimode fiber can support many modes. A single mode fiber only supports the fundamental mode. Different modes have different β, such that they propagate at different speeds.=>mode dispersion (We can think of a “mode” as one possible path that a guided ray can take)

  26. For a fiber with core and cladding , if a wave propagating purely in the core, then the propagation constant is λ: free space wavelength The wave number Similarly if the wave propagating purely in the cladding, then The fiber modes propagate partly in the cladding and partly in the core, so Define the effective index The speed of the wave in the fiber=

  27. For a fiber with core radius a , the cutoff condition is : normalized wave number Recall V↓ when a↓ and △ ↓ For a single mode fiber, the typical values are a=4μm and △=0.003

  28. The light energy is distributed in the core and the cladding.

  29. Since Δ is small, a significant portion of the light energy can propagate in the cladding, the modes are weakly guided. The energy distribution of the core and the cladding depends on wavelength. It causes waveguide dispersion (different from material dispersion) ( Appendix E ) For longer wave, it has more energy in the cladding and vice versa.

  30. mode A multimode fiber has a large value of V The number of modes For example a=25μm, Δ=0.005 V=28 at 0.8μm Define the normalized propagation constant (or normalized effective index)

  31. Polarization Two fundamental modes exist for all λ. Others only exist for λ< λcutoff, Linearly polarized field : Itsdirection is constant. For the fundamental mode in a single-mode fiber

  32. Fibers are not perfectly circularly symmetric. The two orthogonally polarized fundamental modes have different β =>Polarization-mode dispersion (PMD) Differential group delay (DGD) Δτ=Δβ/w～typical value Δτ=0.5 100 km => 50 ps Practically PMD varies randomly along the fiber and may be cancelled from an segment to another segment. Empirically, Δτ ～0.1-1 Some elements such as isolators, circulators, filters may have polarization-dependent loss (PDL).

  33. 2.2 Loss and Bandwidth

  34. Recall Take the bandwidth over which the loss in dB/km is with a factor of 2 of its minimum. 80nm at 1.3μm, 180nm at 1.55μm =>BW=35 THz

  35. Erbium-Doped Fiber Amplifiers (EDFA) operate in the c and L bands, Fiber Raman Amplifiers (FRA) operate in the S band. All Wave fiber eliminates the absorption peaks due to water.

  36. 2.2.1 Bending loss A bend with r = 4cm, loss < 0.01dB r↓ loss↑ 2.2.3 Chromatic Dispersion Different spectral components travel at different velocities. • Material dispersion n(w) • Waveguide dispersion, different wavelengths have different energy distributions in core and cladding =>different β, kn2< β < kn1

  37. 2.3.1 Chirped Gaussian Pulses Chirped: frequency of the pulse changes with time. Cause of chirp: direct modulation, nonlinear effects, generated on purpose. (soliton)

  38. Appendix E, or Govind P. Agrawal “ Fiber- Optic Communication Systems” 2nd Edition, John Wiley & Sons. Inc. PP47~51 A chirped Gaussian pulse at z=0 is given by The instantaneous angular frequency

  39. k = The chirp factor Define: The linearly chirped pulse: the instantaneous angular frequency increases or decreases with time, (k=constant) Note Solve with the initial condition (E.7) We get (E.8) A(z,t) is also Gaussian pulse

  40. Broadening of chirped Gaussian pulses They have the same of broadening length. Note

  41. In Fig 2.9, , it is true for standard fibers at 1.55μm Let be the dispersion length If , dispersion can be neglected

  42. ↓decreases increases ↑ for certain z For kβ2 < 0 For β2 > 0, high frequency travels faster => the tail travels fasters => compression => make βk < 0, LD increases (Fig 2.10)

  43. Note β2> 0 , k< 0 β2k<0

  44. 2.3.2 Controlling the Dispersion Profile Def: Chromatic dispersion parameter D = D = DM + Dw The standard single mode fiber has small chromatic dispersion at 1.3 μm but large at 1.55 μm

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