1. UNIT-1

2. An optical fiber is a long cylindrical dielectric waveguide, usually of circular cross-section, transparent to light over the operating wavelength. A single solid dielectric of two concentric layers. Inner layer - Core of radius ‘a’ and refractive index n 1 Outer layer – Cladding refractive index n 2. n 2 < n 1 condition necessary for TIR OPTICAL FIBER

3.  For light propagation through the fiber, the conditions for total internal reflection (TIR) should be met at the core-cladding interface Light Propagation through Optical Fiber

4. To understand transmission mechanisms of optical fibers with dimensions approximating to those of a human hair;  Necessary to consider the optical waveguiding of a cylindrical glass fiber. Fiber acts as an open optical waveguide – may be analyzed using simple ray theory – Geometric Optics  Not sufficient when considering all types of optical fibers Electromagnetic Mode Theory for Complete Picture Optical Fiber Wave guiding

5. Light entering from glass-air interface (n 1 >n 2 ) - Refraction At Φ 2 = 90 o, refracted ray moves parallel to interface between dielectrics and Φ1<90° - Limiting case of refraction Angle of incidence, Φ 1 ⇨Φ C ; critical angle Total Internal Reflection

6. Value of critical angle ( Φ C ); sin Φ C = n 2 /n 1 Meridional ray Transmission of light ray in a perfect optical fiber meridional  At angle of incidence greater than critical angle, the light is reflected back into the originating dielectric medium (TIR) with high efficiency (≈99.9%)

7. Not all rays entering the fiber core will continue to be propagated down its length Only rays with sufficiently shallow grazing angle ( i.e. angle to the normal > Φ C ) at the core-cladding interface are transmitted by TIR.  Any ray incident into fiber core at angle > Φ a will be transmitted to core-cladding interface at an angle < Φ C and will not follow TIR. ⇨ Lost (case B) ACCEPTANCE ANGLE

8.  For rays to be transmitted by TIR within the fiber core, they must be incident on the fiber core within an acceptance cone defined by the conical half angle “Φ a ”. ⇨ ‘Φ a ’ is the maximum angle to the axis at which light may enter the fiber in order to be propagated ⇨ Acceptance angle for the fiber Acceptance Cone

9.  A very useful parameter : measure of light collecting ability of fiber. Larger the magnitude of NA, greater the amount of light accepted by the fiber from the external source Acceptance /Emission Cone sin Ө a n 2 core - n 2 cladding NA== NA varies from 0.12- 0.20 for SMFs and 0.20- 0.50 for MMFs Numerical Aperture (NA)

10. NA and Δ (Relative R.I Difference)  In terms of relative R.I. difference  between core and cladding, n 1  n 2 n1 n 2n1 n 2    (for   1)(for   1) n1n1 2n12n1 NA = n 1 (2  ) ½ NA ; independent of core and cladding diameters Holds for diameters as small as 8  m 2

11.  To obtain an detailed understanding of propagation of light in an optical fiber  Necessary to solve Maxwell‟s Equations Very complex analyses - Qualitative aspects only Electromagnetic Theory Light as a variety of EM vibrations E and H fields at right angle to each other and perpendicular to direction of propagation.

12. Maxwell’s Equations  Assuming a linear isotropic dielectric material having no currents and free charges Where D = ϵE and B = µH

13. Maxwell’s Equations Substituting for D and Bandandtaking curloffirstequation Using vectoridentity We get Similarly  Wave equationsfor eachcomponent of the field vectors E& H.& H.

14. n2n1n2n2n1n2 Planar optical waveguide (a) Plane wave propagating in the guide (b) Interference of plane waves in the guide (forming lowest order mode m=0) Wavelength = λ/n 1 Propagation constant β = n 1 k Components of β in z and x directions β z = n 1 k cosӨ β x = n 1 k sinӨ Constructive interference occurs and standing wave obtained in x-direction Concept of Mode A plane monochromatic wave propagating in direction of ray path within the guide of refractive index n 1 sandwiched between two regions of lower refractive index n 2

15.  Components of plane wave in x-direction reflected at core- cladding interface and interfere  Constructive: Total phase change after two reflection is equal to 2mπ radians; m an integer - Standing wave in x-direction  The optical wave is confined within the guide and the electric field distribution in the x-direction does not change as the wave propagate in the z-direction ⇨ Sinusoidally varying in z-direction  The stable field distribution in the x-direction with only a periodic z-dependence is known as a MODE. Specific mode is obtained only when the angle between the propagation vectors or rays and interface have a particular value – Discrete modes typified by a distinct value of Ө Have periodic z-dependence of exp(-jβ z z) or commonly exp(-j βz) Have time dependence with angular frequency β, i.e. exp (jωt)

16.  For monochromatic light fields of angular frequency , a mode traveling in positive z-direction hasa timea timeand z-dependence given by exp j(  t-  z) Ray propagation and corresponding TE field patterns of three lower order modes in planar guide. Dominant modes propagating in z- direction with electric field distribution in x-direction formed by rays with m=1,2,3 m denotes number of zeros in this transverse pattern. Also signifies order of the mode and is known as mode number. Modes in Planar Waveguides

17.  Transverse Electric mode (TE): Electric field perpendicular to direction of propagation; E z =0, but a corresponding component of the magnetic field H in the direction of propagation.  Transverse Magnetic (TM) mode: A component of E field in the direction of propagation, but H z =0.  Modes with mode numbers; nomenclature by TE m and TM m  Transverse ElectroMagnetic (TEM) : Total field lies in the transverse plane in that case both E z and H z are zero. Low-order TE or TM mode fields TE and TM modes

18. Wave picture of waveguides

19.  Phase Velocity: For plane wave, there are points of constant phase, these constant phase points forms a surface, referred to as a wavefront.  As light wave propagate along a waveguide in the z-direction, wavefront phase velocity ; v p = // travel at a  Formation of wave packet from combination of two waves of nearly equal frequencies Non-monochromaticity leads to group of waves with closely similar frequencies  Wave Packet Wave packet observed to move at a group velocity, v g =  /    Group Velocity  V g is of great importance in study of TCs of optical fibers:- relates to the propagation characteristics of observable wave groups Phase and Group Velocity

20.  Considering propagation in an infinite medium of R.I. n 1, 2  cc   n k  n  n k  n  n n Propagation constant : 111  c c v Phase velocity : p n 1  c v Group velocity : g dn   N  n  1 g  1 d   Parameter N g is known as the group index of the guide Group Velocity

21.  Another phenomenon of interest under conditions of TIR is the form of the electric field in the cladding of the guide. The transmitted wave field in the cladding is of the form B = B 0 exp(-  2 x) exp j(  t-  z) The amplitude of the field in the cladding is observed to decay exponentially in the x-direction  EvanescentField Exponentially decaying evanescent field in the cladding A field of this type stores energy and transports it in the direction of propagation (z) but does not transport energy in the transverse direction (x). Indicates that optical energy is transmitted into the cladding. Evanescent Field

22.  The evanescent field gives rise to the following requirements for the choice of cladding material  Cladding should be transparent to light at the wavelengths over which the guide is to operate.  Should consist of a solid material in order to avoid both damage to the guide and the accumulation of foreign matter on the guide walls.  Cladding thickness must be sufficient to allow the evanescent field to decay to a low value or losses from the penetrating energy may be encountered.  Most widely used optical fibers consist of a core and cladding, both made of glass.  Although, it give a lower NA for fiber, but provides a far more practical solution. Cladding Material

23.  Exact solution of Maxwell’s Eqns. for a cylindrical dielectric waveguide- very complicated & complex results.  In common with planar waveguide, TE and TM modes are obtained within dielectric cylinder. A cylindrical waveguide is bounded in two dimensions, therefore, two integers, l and m to specify the modes. TE lm and TM lm modes These modes from meridional rays propagation within guide  Hybrid modes where E z and H z are nonzero – results from skew ray propagation within the fiber. Designated as HE lm and EH lm depending upon whether the components of H or E make the larger contribution to transverse field Cylindrical Fiber

24.  Analysis simplified by considering fibers for communication purposes.  Satisfy, weakly guided approximation,  <<1, small grazing angles   Approximate solutions for full set of HE, EH, TE and TM modes may be given by two linearly polarized (LP) components Not exact modes of fiber except for fundamental mode, however, as  is very small, HE-EH modes pairs occur with almost identical propagation constants  Degenerate modes The superposition of these degenerating modes characterized by a common propagation constant corresponds to particular LP modes regardless of their HE, EH, TE or TM configurations. This linear combination of degenerate modes  a useful simplification in the analysis of weakly guiding fibers. Modes in Cylindrical Fibers

25. Correspondence between modes and the traditional formed. the lower order in linearly polarized exactmodes fromwhichtheyare LinearlypolarizedExactExact LP 01 LP 11 LP 21 LP 02 LP 31 LP 12 LP lm HE 11 HE 21, HE 31, HE 12 HE 41, HE 22, TE 01, TM 01 EH 11 EH 21 TE 02, TM 02 HE 2m, TE 0m, TM 0m

26. Intensity Profiles  Electric field configuration for the three lowest LP modes in terms of their constituent exact modes: (a) LP mode designations; (b) exact mode designations; (c) electric field distribution of the exact modes; (d) intensity distribution of E x for exact modes indicating the electric field intensity profile for the corresponding LP modes.  Field strength in the transverse direction is identical for the modes which belong to the same LP mode.

27.  The scalar wave equation for homogeneous core waveguide under weak guidanceconditions 1 d  is   n 2 k 2   2    0 22 d d  1 d1 d  1 dr 2dr 2 r 2r 2 d2d2 rdr  is the field (E or H).  The propagation constant for the guided modes  lie in the range n 2 k<  <n 1 k  Solution of wave equation for cylindrical fiber have the form Here,  Represents the dominant transverse electric field component. The periodic dependence on  gives a mode of radial order l. Solutions of Wave Equation

28. Introducing the equation d E solutionto waveequationresults in a differential 22  1 dE r dr ll  n 2 k 2  2  n 2 k 2  2   E  0E  0 1 dr 2dr 2 r 2r 2   For a SI fiber with constant RI core, it is a Bessel differential equation and the solutions are cylinder functions. In the core region the solutions are Bessel functions denoted by J l (Gradually damped oscillatory functions w.r.t. r)  The field is finite at r =0 and is represented by the Zero order Bessel function J 0.  However, the field vanishes as r goes to infinity and the solutions in the cladding are therefore modified Bessel functions denoted by K l – These modified functions decay exponentially w.r.t. r.

29. Figures Showing (a) Variation of the Bessel function J l (r) for l = 0, 1,2,3 ( first four orders), plotted against r. (b) Graph of the modified Bessel function K l (r) against r for l = 0, 1.

30.  The electric field is given by E(r) = GJ l (UR) = GJ l (U) K l (WR)/K l (W) for R<1 (core) for R>1(cladding) where G; amplitude coefficient, R=r/a; normalized radial coordinate, U eigen values in the core and cladding respectively & W are  The sum of squares of U & W defines a very useful quantity usually referred to as normalized frequency V V = (U 2 +W 2 ) ½ = ka(n 1 2 -n 2 2 ) ½ U; radial phase parameter or radial propagation constant W; cladding decay parameter U = a(n 1 2 k 2 -  2 ) ½ and W= a(  2 -n 2 2 k 2 ) ½ Bessel Function Solutions

31.  Normalized Frequency, V may be expressed in terms of NA and , as 22 22 1 V V  a (NA) a (NA)  a n (2) 2a n (2) 2 1  Normalized frequency is a dimensionless parameter and simply called V-number or value of the fiber.  It combines in a very useful manner the information about three parameters, a,  and.  Limiting parameter for single and multimode propagation in optical fiber.  V  2.405 for SM operation V-Number

32.  Lower order modes obtained in a cylindrical homogeneous corewaveguide Allowed regions for the LP modes of order l = 0,1 against normalized frequency (V) for a circular optical waveguide with a constant refractive index core (SI) Value of V, where J 0 and J 1 cross the zero gives the cutoff point for various modes. V = V c ; V c is different for different modes = 0 for LP 01 mode = 2.405 for LP 11 = 3.83 for LP 02 Allowed LP modes

33.  Limit of mode propagation i.e. n 2 k<  <n 1 k  Cut OFF: When,  = n 2 k ; the mode phase velocity is equal to the velocity of light in the cladding and mode is no longer properly guided.  Mode is said to be cut off and eigenvalue W=0  Unguided (Radiation, Leaky) modes have frequencies below cutoff, where  <n 2 k and hence W is imaginary. Nevertheless, wave propagation does not cease abruptly below cutoff. Modes exist near the core-cladding interface.  Solns of wave equation giving these states are called leaky modes, and often behaves as very lousy guided modes rather than radiation modes.  Guided Modes: For  > n 2 k, less power is propagated in the cladding until at  = n 1 k - all the power is confined to the fiber core.  This range of values for  signifies guided modes of the fiber. Leaky & Guided Modes

34. Step Index / Graded Index

35. Fiber with a core of constant refractive index n 1 and a cladding of slightly lower refractive index n 2.  Refractive index profile makesastep change at the core-cladding interface Refractive index profile n1n2n1n2 ;r<a (core) n(r) = r  a (cladding) ; Multimode Step Index Single mode Step Index The refractive index profile and ray transmission in step index fibers: (a) multimode step index fiber. (b) single-mode step index fiber. Step Index / Graded Index

36.  MM SI fibers allow the propagation of a finite number of guided modes along the channel.  Number of guided modes is dependent upon the physical parameters ; a,  of fibers and wavelength of the transmitted light – included in V-number  For example: A MM SI fiber of core diameter 80  m, core refractive index 1.48, relative index difference of 1.5% and operating at 850nm  supports 2873 guided modes. The total number of guided modes or mode volume M s for SI fiber is related to V-number for the fiber by approximate expression M s  V 2 /2  Allows an estimate of number of guided modes propagating in a particular MM SI fiber. Modes in SI Fibers

37. The total average cladding power is thus approximated by P  4343  1 M  2M  2 cladclad  Here M is the total number of modes entering the fiber  totaltotal P  Since M is proportional to V 2, the power flow in the cladding decreases as V increases.  For V = 1;  70% of power flow in cladding  For V = 2.405;  20% of power flow in cladding.  Far from the cutoff the average power in the cladding has been derived for the fibers in which many modes can propagate.  Because of the large number of modes, those few modes that are appreciably close to cutoff can be ignored to a reasonable approximation. Power Flow in Step-Index Fibers

38. Fractional power flow in the cladding of a SI function of V. fiber as a Power Flow in Step-Index Fibers

39.  GI fibers do not have a constant refractive index in the core, but a decreasing core index n(r) with radial distance from a maximum value of n 1 at the axis to a constant value n 2 beyond the core Radius a in the cladding. – Inhomogeneous core fibers Index variation is represented as where,  is relative refractive index difference and  is the profile parameter which gives the characteristic RI profile of the fiber core. Graded Index Fiber Structure

40. The refractive index profile and ray transmission in a multimode graded index fiber. Possible fiber refractive index profiles for different values of   =  ; Step index profile  = 2; Parabolic profile  =1 Triangular profile

41. ncnc nfnf ncnc n varies quadratically Graded Index Fiber

42.  The parameters defined for SI fibers ( NA, , V) may be applied to GI fibers and give comparison between two. However, in GI fibers situation is more complicated because of radial variation of RI function of radial distance. of core from the axis, NA is also Local numerical aperture Axial numerical aperture Number of bound modes in graded index fiber is  V2 V2    M g = V /4  half the number (n ka)2  (n ka)2     M    g1    2    2     2  2   2  2 For parabolic profile core (  =2), 2 supported by SI fiber with sane V value Graded Index Fiber Parameters

43. Single mode (mono-mode) Fibers SMFs: Most important for long-haul use (carrier and Internet core). Small core (8 to 10 microns) that forces the light to follow a single path down its length. Lasers are the usual light source. Most expensive and delicate to handle, linear Highest bandwidths (GHz) and distance ratings (more than 100 km).

44. Single Mode fibers Only one mode of propagation Core diameter 8-12 μm and V = 2.4 Δ varies between 0.2 and 1.0 percent Core diameter must be just below the cut off of the first higher order mode LP 01 mode alone exists 0 <V<2.405

45. Relatively large diameter core (50 to 100 microns) Step-index multimode cable has an abrupt change between core and cladding. It is limited to about 50 Mbits/sec Graded-index multimode cables has agradualchangebetween core andcladding.Itislimitedto1Gbit/sec. SI GI Multimode Fibers

46. Numerical Aperture (NA): NA = sin  a = [(n 1 ) 2 -(n 2 ) 2 ] 1/2 0.12-0.20 for SMF, 0.15-0.25 for MMF Relative Refractive Index Difference (  ):  = (n 1 –n 2 )/n ; n- the average refractive index <0.4% for SMF,>1% for MMF Normalized Frequency or V-Number: V = [(2  a)/ ] NA V  2.405 for SMF;  10 for MMF DESIGNER’S PARAMETERS

47. Classification of Optical Fibers Classified on basis of :  Core and Cladding materials  Refractive index profile  Modes of propagation

48. a. Glass core and cladding (SCS: silca-clad silica) Low attenuation & best propagation characteristics Least rugged – delicate to handle b.Glass core with plastic cladding (PCS: plastic cladsilica) More rugged than glass; attractive to military applications Medium attenuation and propagation characteristics c.Plastic core and cladding More flexible and more rugged Easy to install, better withstand stress, less expensive, weigh 60% less than glass High attenuation- limited to short runs. Three Varieties:

49. Refractive IndexProfile: Two types Step Index : Graded Index : Refractive index makes abrupt change Refractive index is made to vary as a function of the radial distance from the centre ofthe fiber Mode of propagations : Two types Single mode : Single path of light Multimode : Multiple paths

50. Application Areas  Single mode fibers: Mostly Step index type  Ideally suited for high bandwidth, very long-haul applications using single-mode ILD sources; Telecommunication, MANs  Multimode fibers : Step index, Graded index Step Index Fibers: Best suited for short-haul, limited bandwidth and relatively low cost applications. Graded Index Fibers: Best suited for medium-haul, medium to high bandwidth applications using incoherent and coherent sources (LEDs and ILDs); LANs

51. Cutoff Wavelength It may be noted that single-mode operation only occurs above a theoretical cutoff wavelength λc given by: where Vc is the cutoff normalized frequency. Hence λc is the wavelength above which a particular fiber becomes single-moded

52. Cutoff Wavelength Thus for step index fiber where Vc = 2.405, the cutoff wavelength is given by:

55. IMPORTANT FORMULAS 1.Refractive index n = c/v 2.Critical angle 3.Numerical aperture 4. Relative refractive index difference Δ,

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