1. Nonlinear Optics Lab. Hanyang Univ. Chapter 3. Propagation of Optical Beams in Fibers 3.0 Introduction Optical fibers  Optical communication - Minimal loss - Minimal spread - Minimal contamination by noise - High-data-rate In this chapter, - Optical guided modes in fibers - Pulse spreading due to group velocity dispersion - Compensation for group velocity dispersion

2. Nonlinear Optics Lab. Hanyang Univ. 3.1 Wave Equations in Cylindrical Coordinates Refractive index profiles of most fibers are cylindrical symmetric  Cylindrical coordinate system The wave equation for z component of the field vectors : where, and Since we are concerned with the propagation along the waveguide, we assume that every component of the field vector has the same z- and t-dependence of exp[i(  t-  z)]  # Solve forfirst and then expressing in terms of

3. Nonlinear Optics Lab. Hanyang Univ. From Maxwell’s curl equations :  in terms of We can solve for

4. Nonlinear Optics Lab. Hanyang Univ.  (3.1-1) Now, let’s determine The solution takes the form : where, 1) 2) where, : Bessel functions of the 1 st and 2 nd kind order of l : Modified Bessel functions of the 1 st and 2 nd kind of order l

5. Nonlinear Optics Lab. Hanyang Univ. Asymptotic forms of Bessel functions :

6. Nonlinear Optics Lab. Hanyang Univ. 3.2 The Step-Index Circular Waveguide 1)(cladding region) : The field of confined modes : * : evanescent (decay) wave * : virtually zero at is not proper for the solution where,

7. Nonlinear Optics Lab. Hanyang Univ. 2)(core region) : * : finite at * : propagating wave is not proper for the solution where, * Necessary condition for confined modes to exist :

8. Nonlinear Optics Lab. Hanyang Univ. Other field components

9. Nonlinear Optics Lab. Hanyang Univ. Boundary condition : tangential components of field are continuous at (3.2-10)

10. Nonlinear Optics Lab. Hanyang Univ. Amplitude ratios : [ from (3.2-10) with determined eigenvalue  Report] : the relative amount of E z and H z in a mode Condition for nontrivial solution to exist : (Report) is to be determined for each l (3.2-11)

11. Nonlinear Optics Lab. Hanyang Univ. Mode characteristics and Cutoff conditions (3.2-11) is quadratic in  Two classes in solutions can be obtained, and designated as the EH and HE modes. (Hybrid modes) (3.2-11)  By using the Bessel function relations : where, : EH modes : HE modes : Can be solved graphically(3.2-15)

12. Nonlinear Optics Lab. Hanyang Univ. Special case (l=0) 1) HE modes (3.2-15b) & From (3.2-10), (Report) Therefore, from (3.2-6)~(3.2-9), nonvanishing components are(TE modes) (3.2-15a) & From (3.2-10), (Report) Therefore, from (3.2-6)~(3.2-9), nonvanishing components are(TM modes) 2) EH modes

13. Nonlinear Optics Lab. Hanyang Univ. Graphical Solution for the confined TE modes (l=0) should be real to achieve the exponential decay of the field in the cladding * Roots of J 0 (ha)=0

14. Nonlinear Optics Lab. Hanyang Univ. * If the max value of ha, V is smaller than the first root of J 0 (x), 2.405 => no TE mode * Cutoff value (a/ ) for TE 0m (or TM 0m ) waves : where, : mth zero of J 0 (x) * Asymtotic formula for higher zeros :

15. Nonlinear Optics Lab. Hanyang Univ. Special case (l=1) * HE mode does not have a cutoff. * All other HE 1m, EH 1m modes have cutoff value of a/  * Asymptotic formula for higher zero :

16. Nonlinear Optics Lab. Hanyang Univ. The cutoff value for a/ (l>1) where, z lm is the mth root of

17. Nonlinear Optics Lab. Hanyang Univ. Propagation constant,  : (effective) mode index # : poorly confined # : tightly confined # V<2.405  Only the fundamental HE 11 mode can propagate (single mode fiber)

18. Nonlinear Optics Lab. Hanyang Univ. 3.3 Linearly Polarized Modes The exact expression for the hybrid modes (EH lm, HE lm ) are very complicated. If we assume n 1 -n 2 <<1 (reasonable in most fibers) a good approximation of the field components and mode condition can be obtained. (D. Gloge, 1971)  Cartesian components of the field vectors may be used. 1) y-polarized waves (2.4-1), (3.1-2) & assume E z <<E y 

19. Nonlinear Optics Lab. Hanyang Univ. Expressions for the field components in core (r<a) After tedious calculations, (3.3-6)~(3.3-17), … (x, y)  Expressions for the field components in cladding (r>a) Continuity condition :

20. Nonlinear Optics Lab. Hanyang Univ. 2) x-polarized waves (similar procedure to the case y-polarized waves) In core (r<a) In cladding (r>a) Continuity condition  Mode condition : and/or simpler than (3.2-11) : This results also can be obtained from the y-polarized wave solution.  x- and y-modes are degenerated.

21. Nonlinear Optics Lab. Hanyang Univ. Graphical Solution for the confined modes (l=0)

22. Nonlinear Optics Lab. Hanyang Univ. Mode cutoff value of a/ where, (3.3-27)  Ex) l=0, Ref : Table 3-1Cutoff value of V for some low-order LP Asymptotic formula for higher modes :

23. Nonlinear Optics Lab. Hanyang Univ. Power flow and power density The time-averaged Poynting vector along the waveguide : (3.3-18), (2.3-19) 

24. Nonlinear Optics Lab. Hanyang Univ. The ratio of cladding power to the total power,  2 :

25. Nonlinear Optics Lab. Hanyang Univ. 3.4 Optical Pulse Propagation and Pulse Spreading in Fibers One bit of information = digital pulse Limit ability to reduce the pulse width : Group velocity dispersion Group velocity dispersion Considering a Single mode / Gaussian pulse, temporal envelope at z=0 (input plane of fiber) : where,: transverse modal profile of the mode Fourier transformation : where,

26. Nonlinear Optics Lab. Hanyang Univ. Propagation delay factor for wave with the frequency of Let’s take complex expression and omit the (are not invloved in the analysis and can be restored when needed) Taylor series expansion : where, (3.4-5) : Field envelope

27. Nonlinear Optics Lab. Hanyang Univ. The pulse spreading is caused by the group velocity dispersion characterized by the parameter, (3.4-3)  (3.4-5) :

28. Nonlinear Optics Lab. Hanyang Univ. If we use the definition of factor a, # Pulse duration  at z (FWHM) initial pulse width # |aL|>>  0 (large distance) : Practical Expression : where, : pulse transmission time through length L of the fiber

29. Nonlinear Optics Lab. Hanyang Univ. Group velocity dispersion 1) Material dispersion : n(  ) depends on   Waveguide dispersion :  lm depends on  (& geometry of fiber) i) : modal dispersion ii) Single mode fiber, material dispersion waveguide dispersion (3.4-18)

30. Nonlinear Optics Lab. Hanyang Univ. From the uniform dielectric perturbation theory, where, : Fractions of power flowing in the core and cladding (3.4-18) 

31. Nonlinear Optics Lab. Hanyang Univ. In weakly guiding fiber : n 1 ~n 2 Group velocity dispersion : ex) GeO 2 -doped silica : # depends on core diameter, n1, n2  control the waveguide shape

32. Nonlinear Optics Lab. Hanyang Univ. Group velocity dispersion & dispersion-flattened and dispersion-shifted fibers

33. Nonlinear Optics Lab. Hanyang Univ. Frequency chirping : modification of the optical frequency due to the dispersion (3.4-6)  where, Total optical phase : Optical frequency :

34. Nonlinear Optics Lab. Hanyang Univ. 3.5 Compensation for Group Velocity Dispersion (3.4-5)  Fiber transfer function By convolution theorem, (1.6-2), : envelop impulse response of a fiber of length z

35. Nonlinear Optics Lab. Hanyang Univ. Compensation for pulse broadening 1) By optical fiber with opposite dispersion (a 1 L=-a 2 L)

36. Nonlinear Optics Lab. Hanyang Univ. 2) By phase conjugation (a 1 L=a 2 L)

37. Nonlinear Optics Lab. Hanyang Univ. Where are (b) and (c) ??  Refer to the text

38. Nonlinear Optics Lab. Hanyang Univ. 3.7 Attenuation in Silica Fibers Recently, 400 Mb/s, 100 km @ 1.55  m Residual OH contamination of the glass 1.55  m is favored for long-distance optical communication