1. Chapter 2 Optical Fibers: Structures and Waveguiding
2.1 The Nature of Light
Linear Polarization
Elliptical and Circular Polarization
The Quantum Nature of Light
2.2 Basic Optical Laws
2.3 Optical Fiber Modes
Fiber Types
Rays and Modes
Step-Index Fiber Structure
2.3.4 Ray Optics Representation

2. Chapter 2 Optical Fibers: Structures and Waveguiding
2.4 Mode Theory for Circular Waveguides
Overview of Modes
Summary of Key Modal Concepts
Modes in Step-Index Fibers
2.5 Single-Mode Fibers
Propagation Modes in SMFs
2.6 Graded-Index Fiber Structure

3. 2.1 The Nature of Light
Light wave is electromagnetic in nature. Polarization effects indicated that light waves are transverse (i.e., the wave motion is perpendicular to the direction in which the wave travels).
The EM waves can be represented by a train of spherical wave fronts, as shown in Fig. 2-1.
A wave front is the locus of all points in the wave train having the same phase. The wave fronts (also called phase fronts) are separated by one wavelength.

4. Figure 2-1. Spherical and plane wave fronts and
2.1 The Nature of Light
Light wave can be represented as a plane wave; its direction of travel can be indicated by a light ray drawn perpendicular to the phase front.
Optical effects of reflection and refraction can be analyzed by the geometrical process of ray tracing. This is referred to as ray or geometrical optics.
Figure Spherical and plane wave fronts and
their associated rays.

5. Linear Polarization
The E or H field of plane linearly polarized waves traveling in direction k can be represented in
A(x, t) = eiAo.exp[j(wt – k.x)] (2-1)
with x = xex + yey + zez representing a general position vector and k = kxex + kyey + kzez representing the wave propagation vector.
Ao is the maximum amplitude of the wave; w=2pn, with n the light frequency; the magnitude of the wave vector k is the wave propagation constant k=2p/l, with l being the wavelength; and ei is a unit vector lying parallel to i-axis.

6. Linear Polarization
If k = kez and A denotes the electric field E with the coordinate axes ei = ex then the real electric field is given by
Ex(z, t) = Re(E) = exEox.cos(wt - kz), (2-2)
This plane wave is traveling in the z direction and is linearly polarized with polarization vector ex.
Figure 2-2. Electric and magnetic fields in a train of EM plane waves.

7. Linear Polarization
Consider the linearly polarized wave Ey(z, t) independent of and orthogonal to Ex(z, t):
Ey(z, t) = eyEoy.cos(wt - kz + d) , (2-3)
where d is the relative phase difference between the waves.
The resultant wave is simply
E(z, t) = Ex(z, t) + Ey(z, t) (2-4) If d = m(2p), m = 0, 1, 2, …, then the waves are in phase.

8. Linear Polarization
Equ. (2-4) is a linearly polarized wave with a polarization vector making an angle
q = tan-1 (Eoy/Eox) (2-5)
with respect to ex and having a magnitude
E = (Eox2 + Eoy2)1/ (2-6)
This case is shown schematically in Fig. 2-3.
Figure 2-3.
Addition of two linearly polarized waves with d = 0
relative phase difference.

9. 2.1.2 Ellipitical and Circular Polarization
The wave given by Eq. (2-4) is elliptically polarized for general values of d. The resultant field vector E will both rotate and change its magnitude as a function of the angular frequency w.
From Eqs. (2-2) and (2-3) we can show that (Ex/E0x)2 + (Ey/E0y)2 - 2(Ex/E0x)(Ey/E0y)cosd
= sin2d (2-7)
which is the general equation of an ellipse.
The axis of the ellipse makes an angle a relative to the x-axis given by
tan(2a) = 2EoxEoycosd/(Eox2-Eoy2) (2-8)

10. 2.1.2 Ellipitical and Circular Polarization
Align the principal axis of the ellipse with the x-axis. Then a = 0, or equivalently, d = ±p/2, ±3p/2, …, so that Eq. (2-7) becomes (Ex/Eox)2 + (Ey/Eoy)2 = (2-9) This is an ellipse centered at the origin with semi-axes Eox and Eoy.
When Eox = Eoy = Eo and d = ±p/2 + 2mp, we have circularly polarized light.
In this case, Eq. (2-9) reduces to
Ex2 + Ey2 = Eo (2-10) which defines a circle.

11. 2.1.2 Ellipitical and Circular Polarization
Choosing the + sign for d, Eqs. (2-2) and (2-3) become
Ex(z, t) = exEocos(wt - kz) (2-11)
Ey(z, t) = -eyEosin(wt - kz) (2-12)
The endpoint of E will trace out a circle, as Fig. 2-5 illustrates.
Figure 2-5.
Addition of two linearly
polarized waves with relative phase d = p/2+2mp results in
a circularly polarized wave.

12. 2.1.2 Ellipitical and Circular Polarization
Pick the endpoint of E field at z = p/k at t = 0.
From Eqs. (2-11) and (2-12), we have
Ex(z, t) = -exEo and Ey(z, t) = 0
so that E lies along the negative x-axis as Fig. 2-5 shows.
At a later time t = p/2w, the E field vector has rotated through 90° and lies along the positive y-axis at zref.
The resultant E field vector rotates clockwise at an angular frequency w. It makes one complete rotation as the wave advances through one wavelength. Such a light wave is right circularly polarized.

13. 2.1.2 Ellipitical and Circular Polarization
If we choose the negative sign for d, then Eqs. (2-2) and (2-3) become the expressions Ex(z, t) = exEocos(wt - kz) Ey(z, t) = eyEosin(wt - kz).
The E field vector then will be
E = Eo[excos(wt - kz) + eysin(wt - kz)] (2-13)
This E field rotates counterclockwise and the wave is left circularly polarized.

14. 2.1.3 The Quantum Nature of Light
In dealing with the interaction of light and matter, such as occurs in dispersion and in the emission and absorption of light, neither the particle theory nor the wave theory of light is appropriate.
Instead, we must turn to quantum theory, which indicates that optical radiation has particle as well as wave properties.
The particle nature arises from the observation that light energy is always emitted or absorbed in discrete units called quanta or photons.
The photon energy is dependent on the frequency n. The energy E and the frequency n of a photon is related with
E = hv (2-14)
where h = x j.s is Planck's constant.

15. 2.1.3 The Quantum Nature of Light
When light is incident on an atom, a photon can transfer its energy to an electron within this atom, thereby exciting it to a higher energy level.
The energy absorbed by the electron must be exactly equal to that required to excite the electron to a higher energy level.
Conversely, an electron in an excited state can drop to a lower state separated from it by an energy hn by emitting a photon of exactly this energy.

16. 2.2 BASIC OPTICAL LAWS
The speed of light is related to the frequency n and the wavelength l by c = nl. Upon entering a dielectric medium the wave travels at a speed n.
The ratio of the light speed in a vacuum to that in matter is the refractive index n of the material :
n = c/n (2-15) Typical values of n are 1.00 for air, 1.33 for water, 1.50 for glass, and 2.42 for diamond.
When a light ray encounters a boundary separating two different media, reflection and refraction will occur. This is shown in Fig. 2-6, where n2 < n1.
The refraction of the light ray at the interface is a result of the difference in the speed of light in two materials that have different refractive indices.

17. 2.2 BASIC OPTICAL LAWS
Fig Refraction and reflection of a light ray at a material boundary.

18. 2.2 BASIC OPTICAL LAWS The Snell's law at the interface is given by
n1 sin f1 = n2 sin f (2-16)
or, equivalently, as
n1 cos q1 = n2 cos q (2-17)
where the angles are defined in Fig. 2-6.
The angle f1 between the incident ray and the normal to the surface is known as the angle of incidence.

19. 2.2 BASIC OPTICAL LAWS
When light traveling in a certain medium is reflected off an optically denser material (one with a higher refractive index), the process is referred to as external reflection.
Conversely, the reflection of light off of less optically dense material (such as light traveling in glass being reflected at a glass-air interface) is called internal reflection.
As the angle of incidence fl in an optically denser material becomes larger, the refracted angle f2 approaches p/2. Beyond this point the light rays become totally internally reflected.

20. 2.2 BASIC OPTICAL LAWS
Consider Fig. 2-7, which shows a glass surface in air. A light ray gets bent toward the glass surface as it leaves the glass in accordance with Snell's law.
If the angle of incidence f1 is increased, a critical angle of incidence fc will eventually be reached where the light ray in air is parallel to the glass surface.
When the incidence angle f1 is greater than the critical angle, the condition for total internal reflection is satisfied; that is, the light is totally reflected back into the glass with no light escaping from the glass surface.

21. 2.2 BASIC OPTICAL LAWS
Consider the glass-air interface shown in Fig When the light ray in air is parallel to the glass surface, then f2 = 90° so that sin f2 = 1.
The critical angle in the glass is thus
sinfc = n2/n (2-18)
Example 2-1:
Using n1 = 1.50 for glass and n2 = 1.00 for air, fc is about 52°.
Any light in the glass incident on the interface at an angle f1 > 52° is totally reflected back into the glass.

22. 2.2 BASIC OPTICAL LAWS Fig. 2-7. The critical angle and total internal
reflection at a glass-air interface.

23. 2.2 BASIC OPTICAL LAWS
When light is totally internally reflected, a phase change d occurs in the reflected wave.
This phase change depends on the angle q1 < p/2 - fc according to the relationships

24. 2.2 BASIC OPTICAL LAWS
Here, dN and dP are the phase shifts of the electric-field wave components normal and parallel to the plane of incidence, respectively, and n = n1/n2.
These phase shifts are shown in Fig. 2-8 for a glass-air interface (n = 1.5 and fc = 52°). The values range from zero at the critical angle to p/2 - fc, when fc=90°.

25. 2.2 BASIC OPTICAL LAWS
Fig Phase shifts from the reflection of wave
components normal (dN) and parallel (dP) to the plane of incidence.

26. 2.3 OPTICAL FIBER MODES 2.3.1 Fiber Types
The propagation of light along a fiber waveguide can be described in terms of a set of guided EM waves called the modes of the waveguide.
Only a certain discrete number of modes are capable of propagating along the guide. These modes are those EM waves that satisfy the homogeneous wave equation in the fiber and the boundary condition at the waveguide surfaces.
In Fig. 2-9, the solid dielectric cylinder of radius a and index of refraction n1 is the core of the fiber. The core is surrounded by a solid dielectric cladding which has a refractive index n2 that is less than n1.

27. Fiber Types
The cladding reduces scattering loss that results from dielectric discontinuities at the core surface, it adds mechanical strength to the fiber, and it protects the core from absorbing surface contaminants.  
Fig Schematic of a single-fiber structure. A solid core of refractive index n1 is surrounded by a cladding having a refractive index n2 < n1. An elastic plastic buffer encapsulates the fiber.

28. Fiber Types
In step-index fiber, the refractive index of the core is uniform throughout and undergoes an abrupt change (or step) at the cladding boundary.
In graded-index fiber, the core refractive index is made to vary as a function of the radial distance from the center of the fiber.
Both the step- and the graded-index fibers can be further divided into single mode and multimode classes. A single-mode fiber sustains only one mode of propagation, whereas multimode fibers contain many hundreds of modes.
A few typical sizes of single- and multimode fibers are given in Fig

29. Fiber Types
Fig Comparison of single-mode and multi-mode step-index and graded-index optical fibers.

30. 2.3.1 Fiber Types Advantages of Multimode Fibers:
The larger core radii of MMFs make it easier to launch optical power into the fiber and facilitate the connecting together of similar fibers.
Light can be launched into a MMF using a LED source, whereas SMFs must generally be excited with laser diodes.  
Although LEDs have less optical output power than laser diodes, they are easier to make, are less expensive, require less complex circuitry, and have longer lifetimes than laser diodes.

31. 2.3.1 Fiber Types Disadvantage of MMFs: Intermodal Dispersion.
Each mode in a MMF travels at a slightly different velocity. The modes in a given optical pulse arrive at the fiber end at slightly different times, thus causing the pulse to spread out in time.  
The effect of intermodal dispersion can be reduced by using a graded-index profile in a fiber core. Graded-index fibers have much larger bandwidths (data rate transmission capabilities) than step-index fibers.
Even higher bandwidths are possible in SMFs, where intermodal dispersion effects are not present.

32. Rays and Modes
For monochromatic light fields of radian frequency w, a mode traveling in the positive z direction is given by
exp[j(wt - bz)]
The factor b is the z component of the wave propagation constant k = 2p/l.
A guided mode traveling in the z direction can be decomposed into a family of superimposed plane waves that collectively form a standing-wave pattern in the direction transverse to the fiber axis.

33. Rays and Modes
The family of plane waves corresponding to a particular mode forms a ray congruence.
Each ray of this particular set travels in the fiber at the same angle relative to the fiber axis.
Only a certain number M of discrete guided modes exist in a fiber, the possible angles of the ray congruences corresponding to these modes are limited to the same number M.

34. 2.3.3 Step-Index Fiber Structure
In the step-index fibers the core of radius a has a refractive index n1 which is typically equal to 1.48.
This is surrounded by a cladding of lightly lower index n2,
n2 = n1(1-D) (2-20)
The parameter D is called the index difference.  

35. 2.3.3 Step-Index Fiber Structure
Figure Ray optics representation of skew rays traveling
in a step-index optical fiber core.

36. 2.3.4 Ray Optics Representation
Meridional rays can be divided into: bound rays that are trapped in the core and propagate along the fiber axis, and unbound rays that are refracted out of the fiber core.
Skew rays are not confined to a single plane, but instead tend to follow a helical-type path along the fiber as shown in Fig  
The meridional ray is shown in Fig for a step-index fiber. The light ray enters the fiber core from a medium of refractive index n at an angle qo with respect to the fiber axis and strikes the core-cladding interface at a normal angle f.  

37. 2.3.3 Step-Index Fiber Structure
Fig Meridional ray optics representation of the propagation mechanism in an ideal step-index optical waveguide.

38. 2.3.4 Ray Optics Representation
From Snell's law, the minimum angle fmin that supports total internal reflection for the meridional ray is sin fmin = n2 / n (2-21) Rays striking the core-cladding interface at angles < fmin will refract out of the core.
By applying Snell's law to the air-fiber face boundary, the condition of Eq. (2-21) can be related to the max. entrance angle qo,max :
n sin qo,max = n1 sin qc = (n12 - n22)1/2 (2-22)
where qc = p/2 - fc.
Those rays having entrance angles qo < qo,max will be totally internally reflected at the core-cladding interface.

39. 2.3.4 Ray Optics Representation
Equation (2-22) defines the numerical aperture (NA) of a step-index fiber for meridional rays:  
NA = n sin(qo,max) = (n12-n22)1/2 = n1(2D)1/ (2-23)
The approximation on the right-hand side is valid for the typical case where D, as defined by Eq. (2-20), is much less than 1.
The NA is commonly used to describe the light acceptance or gathering capability of a fiber and to calculate source-to-fiber optical power coupling efficiencies. The NA is a dimensionless quantity which is < 1.0, with values ranging from 0.14 to 0.50.

40. 2.3.4 Ray Optics Representation
Example 2.2 : Preferred sizes of multimode glass optical fibers and their corresponding numerical apertures are as follows:
Core diameter Clad diameter Numerical Aperture (mm) (mm)

41. 2.4 MODE THEORY FOR CIRCULAR WAVEGUIDES
In optical fibers the coupling between E and H fields results in HE or EH hybrid modes.
The two lowest-order modes are HE11 and TE01, the subscripts refer to possible propagation modes of the optical field.
Fibers are constructed so that n1-n2 << 1. The field components are called linearly polarized (LP) modes and are labeled LPjm where j and m designate mode solutions.

42. 2.4 MODE THEORY FOR CIRCULAR WAVEGUIDES
For the lowest-order modes, each LP0m mode is derived from HE1m mode and each LP1m mode comes from TE0m, TM0m and HE0m modes.
The fundamental LP01 mode corresponds to an HE11 mode.
Figure Electric field distributions for several of the lower-order guided modes in a symmetrical-slab waveguide.

43. Overview of Modes
Figure 2-14 shows the field patterns of several of the lower order TE modes in the planar dielectric slab waveguide. The order of a mode is equal to the number of field zeros across the guide.  
The order of the mode is related to the angle that the ray congruence corresponding to this mode makes with the plane of the waveguide. The steeper the angle, the higher the order of the mode.
The leaky modes are only partially confined to the core region, and attenuate by continuously radiating their power out of the core as they propagate along the fiber. This power radiation out of the waveguide results from a tunnel effect.

44. Overview of Modes
A mode remains guided as long as b satisfies the condition
n2k < b < n1k
where n1 and n2 are the refractive indices of the core and cladding, respectively, and k = 2p/l.
The boundary between truly guided modes and leaky modes is defined by the cutoff condition b = n2k. As soon as b < n2k, power leaks out of the core into the cladding region.

45. 2.4.2 Summary of Key Modal Concepts
The V-number defined by
V = (2pa/l)(n12-n22)1/2
= (2pa/l)NA (2-27)
determines how many modes a fiber can support.
Except for the lowest-order HE11 mode, each mode can exist only for values of V that exceed a certain limiting value.
The modes are cut off when b = n2k. This occurs when V < The HE11 mode has not cutoff and ceases to exist only when the core diameter is zero.

46. 2.4.2 Summary of Key Modal Concepts
The V-number can be used to express the number of modes M in a multimode fiber when V is large.
An estimate of the total number of modes supported in a fiber is
(2-28)

47. 2.4.2 Summary of Key Modal Concepts
When the V-number approaches cutoff for any particular mode, more of the mode power is in the cladding.
Far from cutoff -- for large values of V -- the fraction of the average optical power residing in the cladding can be estimated by Pclad/P = (4/3)M-1/ (2-29) where P is the total optical power in the fiber.
Since M is proportional to V2, the power flow in the cladding decreases as V increases. This increases the number of modes in the fiber, which is not desirable for a high-bandwidth capability.

48. 2.4.7 Modes in Step-Index Fibers
Schematics of the transverse electric field patterns for the four lowest-order modes are shown in Fig
An important parameter connected with the cutoff condition is the normalized frequency V (also called
the V-number or V-parameter) defined by
V2 = (2pa/l)2(n12-n22)
= (2pa/l)2NA (2-57)
V-parameter is a dimensionless number that determines how many modes a fiber can support.

49. 2.4.7 Modes in Step-Index Fibers
The number of modes that can exist in a waveguide,
as a function of V, may be represented in terms of a normalized propagation constant b defined by
b = [(b/k)2-n22]/(n12-n22)
Figure Cross-sectional
views of the transverse electrical
field vectors for the four lowest-order modes in a step-index fiber.

50. 2.4.7 Modes in Step-Index Fibers
The plot of b (in terms of b/k) as a function of V is shown in Fig for a few of the low-order modes. The figure shows that the modes are cut off when b/k = n2.
The HE11 mode has no cutoff and ceases to exist only when the core diameter is zero. By appropriately choosing a, n1, and n2, so that
V = (2pa/l)(n12-n22)1/2 < (2-58)
which is the value at which all modes except the HE11 mode are cut off.

51. 2.4.7 Modes in Step-Index Fibers
Fig Plots of the propagation constant (in terms of b/k ) as a function of V for a few of the lowest-order modes.

52. 2.5 SINGLE-MODE FIBERS
SMFs are constructed by letting the dimensions of the core diameter be a few wavelengths (usually 8-12) and by having small index differences between the core and the cladding.
On Eq. (2-27) or (2-58) with V = 2.405, single-mode propagation is possible for fairly large variations in values of the physical core size a and the core-cladding index differences D.  

53. 2.5 SINGLE-MODE FIBERS
In practical SMFs, the core-cladding index difference varies between 0.2 and 1.0 %, and the core diameter is just below the cutoff of the first higher-order mode; that is, for V slightly < 2.4.
For example, a typical SMF may have a core radius of 3-mm and an NA of 0.1 at a wavelength of 0.8-mm.
From Eqs. (2-23) and (2-27), this yields V =

54. 2.5.2 Propagation Modes in SMFs
In any SMFs, two propagation modes may be chosen as the horizontal (H) and the vertical (V) polarizations as shown in Fig
The two modes propagate with different phase velocities (kx =/= ky). The difference between their effective refractive indices is called the fiber birefringence,
Bf = ny - nx or b = ko(ny - nx) (2-76)
where ko = 2p/l is the free-space propagation constant.

55. 2.5.2 Propagation Modes in SMFs
One polarization mode will be delayed in phase relative to the other as they propagate.
When the phase difference is an integral multiple of 2p, the two modes will beat and the input polarization state will be reproduced.
The length over which the beating occurs is the fiber beat length, Lp = 2p / b (2-77)
Figure 2-24.
Two polarizations of the
fundamental HE11 mode in a single-mode fiber.

56. 2.6 GRADED-INDEX FIBER STRUCTURE
In the graded-index fiber design the most commonly used refractive-index variation in the core is the power law relationship
(2-78)
Here, r is the radial distance from the fiber axis, a is the core radius, n1 and n2 are the refractive indices at the core and the cladding, and the parameter a defines the shape of the index profile.

57. 2.6 GRADED-INDEX FIBER STRUCTURE
The index difference D for the graded-index fiber is given by
D = [n12-n22]/2n12 = [n1-n2]/n (2-79)
The approximation on the right-hand side of this equation reduces the expression for D to that of the step-index fiber given by Eq. (2-20).
For a = ∞, Eq. (2-78) reduces to the step-index profile n(r) = n1.

58. 2.6 GRADED-INDEX FIBER STRUCTURE
Light incident on the core at position r will propagate as a guided mode if it is within the local numerical aperture NA(r) at that point.
The local numerical aperture is defined as
(2-80a)
where the axial numerical aperture is defined as NA(0) = [n2(0)-n22]1/2 = [n12 - n22]1/2 = n1(2D)1/2 (2-80b)
The NA of a graded-index fiber decreases from NA(0) to zero as r moves from the fiber axis to the core-cladding boundary.

59. 2.6 GRADED-INDEX FIBER STRUCTURE
A comparison of the numerical apertures for fibers having various a profiles is shown is shown in Fig The number of bound modes in a graded-index fiber is
M = [a/(a+2)]a2k2n12D (2-81)
Figure Comparison of NAs for fibers having various a profiles.