The electrical engineering handbook CH042

The electrical engineering handbook

Agbo, S.O., Cherin, A.H, Tariyal, B.K “Lightwave”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

Lightwave

42.1 Lightwave WaveguidesRay Theory • Wave Equation for Dielectric Materials • Modes in Slab Waveguides • Fields in Cylindrical Fibers • Modes in Step-Index Fibers • Modes in Graded-Index Fibers • Attenuation • Dispersion and Pulse Spreading

42.2 Optical Fibers and CablesIntroduction • Classification of OpticalFibers and Attractive Features • Fiber Transmission Characteristics • Optical Fiber Cable Manufacturing

42.1 Lightwave Waveguides

Samuel O Agbo

Lightwave waveguides fall into two broad categories: dielectric slab waveguides and optical fibers As illustrated

in Fig 42.1, slab waveguides generally consist of a middle layer (the film) of refractive indexn1 and lower andupper layers of refractive indices n2 and n3, respectively

Optical fibers are slender glass or plastic cylinders with annular cross sections The core has a refractiveindex, n1, which is greater than the refractive index, n2, of the annular region (the cladding) Light propagation

is confined to the core by total internal reflection, even when the fiber is bent into curves and loops Opticalfibers fall into two main categories: step-index and graded-index (GRIN) fibers For step-index fibers, therefractive index is constant within the core For GRIN fibers, the refractive index is a function of radius r given by

(42.1)

In Eq (42.1), D is the relative refractive index difference, a is the core radius, and a defines the type ofgraded-index profile For triangular, parabolic, and step-index profiles, a is, respectively, 1, 2, and ¥ Figure 42.2shows the raypaths in step-index and graded-index fibers and the cylindrical coordinate system used in theanalysis of lightwave propagation through fibers Because rays propagating within the core in a GRIN fiberundergo progressive refraction, the raypaths are curved (sinusoidal in the case of parabolic profile)

Ray Theory

Consider Fig 42.3, which shows possible raypaths for light coupled from air (refractive index n0) into the film

of a slab waveguide or the core of a step-index fiber At each interface, the transmitted raypath is governed bySnell’s law As q0 (the acceptance angle from air into the waveguide) decreases, the angle of incidence qi increases

ö ø

é ë

ê ê

ù û

ú

ì í

ï ï

î

ï ï

1

1 2

1

1 2 2

D D

until it equals the critical angle, qc, making q0 equal to the maximum acceptance angle, qa According to raytheory, all rays with acceptance angles less than qa propagate in the waveguide by total internal reflections.Hence, the numerical aperture (NA) for the waveguide, a measure of its light-gathering ability, is given by

waveguide; (c) light guiding in a slab waveguide.

graded-index profiles; (c) raypaths in step-graded-index fiber; (d) raypaths in graded-graded-index fiber.

è

ö ø

=

where NA is the numerical aperture for meridional rays and g is the angle between the core radius and theprojection of the ray onto a plane normal to the fiber axis.

Wave Equation for Dielectric Materials

Only certain discrete angles, instead of all acceptance angles less than the maximum acceptance angle, lead toguided propagation in lightwave waveguides Hence, ray theory is inadequate, and wave theory is necessary,for analysis of light propagation in optical waveguides

For lightwave propagation in an unbounded dielectric medium, the assumption of a linear, homogeneous,charge-free, and nonconducting medium is appropriate Assuming also sinusoidal time dependence of thefields, the applicable Maxwell’s equations are

Modes in Slab Waveguides

Consider a plane wave polarized in the y direction and propagating in z direction in an unbounded dielectric

medium in the Cartesian coordinates The vector wave equations (42.6) lead to the scalar equations:

(42.9a)

(42.9b)The solutions are

(42.10b)

where A is a constant and h = is the intrinsic impedance of the medium

Because the film is bounded by the upper and lower layers, the rays follow the zigzag paths as shown in

Fig 42.3 The upward and downward traveling waves interfere to create a standing wave pattern Within the film,

the fields transverse to the z axis, which have even and odd symmetry about the x axis, are given, respectively, by

where band h are the components of k parallel to and normal to the z axis, respectively The fields in the upper

and lower layers are evanescent fields decaying rapidly with attenuation factors a3 and a2, respectively, and are

given by

(42.12a)

(42.12b)

Only waves with raypaths for which the total phase change for a complete (up and down) zigzag path is an

integral multiple of 2p undergo constructive interference, resulting in guided modes Waves with raypaths not

satisfying this mode condition interfere destructively and die out rapidly In terms of a raypath with an angle

of incidence qi = q in Fig 42.3, the mode conditions [Haus, 1984] for fields transverse to the z axis and with

even and odd symmetry about the x axis are given, respectively, by

ö ø

-ö ø

p

/

where h = k cos q = (2pn1/l) cos q and l is the free space wavelength.

Equations (42.13a) and (42.13b) are transcendental, have multiple solutions, and are better solved graphically

Let (d/l)0 denote the smallest value of d/l, the film thickness normalized with respect to the wavelength,satisfying Eqs (42.13a) and (42.13b) Other solutions for both even and odd modes are given by

(42.14)

where m is a nonnegative integer denoting the order of the mode.

Figure 42.4 [Palais, 1992] shows a mode chart for a symmetrical slab waveguide obtained by solving

Eqs (42.13a) and (42.13b) For the TE m modes, the E field is transverse to the direction (z) of propagation, while the H field lies in a plane parallel to the z axis For the TM m modes, the reverse is the case The highest-

order mode that can propagate has a value m given by the integer part of

(42.15)

To obtain a single-mode waveguide, d/l should be

smaller than the value required for m = 1, so that only the

m = 0 mode is supported To obtain a multimode

waveguide, d/l should be large enough to support many

modes

Shown in Fig 42.5 are transverse mode patterns for the

electric field in a symmetrical slab waveguide These are

graphical illustrations of the fields given by Eqs (42.11)

and (42.12) Note that, for TE m , the field has m zeros in

the film, and the evanescent field penetrates more deeply

into the upper and lower layers for high-order modes

For asymmetric slab waveguides, the equations and their

solutions are more complex than those for symmetric slab waveguides Shown in Fig 42.6 [Palais, 1992] is the

mode chart for the asymmetric slab waveguide Note that the TE m and TM m modes in this case have different

nm

æ è ç

ö ø

æ è ç

ö ø

÷ +

1 2 2

2 1 2

l

/

symmetric slab waveguide.

propagation constants and do not overlap By contrast, for the symmetric case, TE m and TM m modes are

degenerate, having the same propagation constant and forming effectively one mode for each value of m.

Figure 42.7 shows typical mode patterns in the asymmetric

slab waveguide Note that the asymmetry causes the

evanes-cent fields to have unequal amplitudes at the two boundaries

and to decay at different rates in the two outer layers

The preceding analysis of slab waveguides is in many ways

similar to, and constitutes a good introduction to, the more

complex analysis of cylindrical (optical) fibers Unlike slab

waveguides, cylindrical waveguides are bounded in two

dimensions rather than one Consequently, skew rays exist in

optical fibers, in addition to the meridional rays found in

slab waveguides In addition to transverse modes similar to

those found in slab waveguides, the skew rays give rise to

hybrid modes in optical fibers

Fields in Cylindrical Fibers

Let y represent Ez or H z and b be the component of k in z direction In the cylindrical coordinates of Fig 42.2,

with wave propagation along the z axis, the wave equations (42.6) correspond to the scalar equation

(42.16)The general solution to the preceding equation is

y(r) = C1Jl(hr) + C2Yl (hr) ; k2 > b2 (42.17a)

y(r) = C1Il(qr) + C2Kl (qr) ; k2 < b2 (42.17b)

In Eqs (42.17) and (42.17b), Jl and Yl are Bessel functions of the first kind and second kind, respectively,

of order l; Il and Kl are modified Bessel functions of the first kind and second kind, respectively, of order l;C1 and C2 are constants; h2 = k2 – b2 and q2 = b2 – k2

E z and H z in a fiber core are given by Eq (42.17a) or (42.17b), depending on the sign of k2 – b2 For guidedpropagation in the core, this sign is negative to ensure that the field is evanescent in the cladding One of the

asymmetric slab waveguide.

0

coefficients vanishes because of asymptotic behavior of the respective Bessel functions in the core or cladding.

Thus, with A1 and A2 as arbitrary constants, the fields in the core and cladding are given, respectively, by

Because of the cylindrical symmetry,

Thus, the usual approach is to solve for E z and H z and then express E r , Ef, H r , and Hf in terms of E z and H z

Modes in Step-Index Fibers

Derivation of the exact modal field relations for optical fibers is complex Fortunately, fibers used in opticalcommunication satisfy the weekly guiding approximation in which the relative index difference, Ñ , is muchless than unity In this approximation, application of the requirement for continuity of transverse and tangential

electric field components at the core-cladding interface (at r = a) to Eqs (42.18a) and (42.18b) results in the

following eigenvalue equation [Snyder, 1969]:

for the mode

As with planar waveguides, TE (E z = 0) and TM (H z = 0) modes corresponding to meridional rays exist in

the fiber They are denoted by EH or HE modes, depending on which component, E or H, is stronger in the

plane transverse to the direction or propagation Because the cylindrical fiber is bounded in two dimensionsrather than one, two integers, l and m, are needed to specify the modes, unlike one integer, m, required forplanar waveguides The exact modes, TElm, TMlm, EHlm, and HElm, may be given by two linearly polarizedmodes, LPlm The subscript l is now such that LPlm corresponds to HEl + 1,m, EHl – 1,m, TEl – 1,m, and TMl – 1,m

In general, there are 2l field maxima around the fiber core circumference and m field maxima along a radiusvector Figure 42.8 illustrates the correspondence between the exact modes and the LP modes and their fieldconfigurations for the three lowest LP modes

Figure 42.9 gives the mode chart for step-index fiber on a plot of the refractive index, b/k0, against the

normalized frequency Note that for a single-mode (LP01 or HE11) fiber, V < 2.405 The number of modes supported as a function of V is given by

(42.22)

J ha

qa qa qa

l l

l l

Modes in Graded-Index Fibers

A rigorous modal analysis for optical fibers based on the solution of Maxwell’s equations is possible only forstep-index fiber For graded-index fibers, approximate methods are used The most widely used approximation

is the WKB (Wenzel, Kramers, and Brillouin) method [Marcuse, 1982] This method gives good modal solutions

step-index fiber: (a) mode designations; (b) electric field patterns; (c) intensity distribution (Source: J M Senior, Optical Fiber

Communications: Principles and Practice, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p 36 With permission.)

D B Keck, Fundamentals of Optical Fiber Communications, M K Barnoski, Ed., New York: Academic Press, 1981, p 13.

With permission.)

for graded-index fiber with arbitrary profiles, when the refractive index does not change appreciably overdistances comparable to the guided wavelength [Yariv, 1991] In this method, the transverse components ofthe fields are expressed as

d

2 2

ê ê

ù û

ú ú

= ( ) y ( )

M INIATURE R ADAR

n inexpensive miniaturized radar system developed at Lawrence Livermore National Labs(LLNL) may become the most successful technology ever privatized by a federal lab, with apotential market for the product estimated at between $100 million and $150 million

The micropower impulse radar was developed by engineer Tom McEwan as part of a device designed

to measure the one billion pulses of light emitted from LLNL’s Nova laser in a single second The system

he developed is the size of a cigarette box and consists of about $10 worth of parts The same measurementhad been made previously using $40,000 worth of equipment

Titan Technologies of Edmonton, AL, Canada, was the first to bring to market a product using thetechnology when they introduced storage-tank fluid sensors incorporating the system The new radarallowed Titan to reduce its devices from the size of an apple crate to the size of a softball, and to sellthem for one-third the cost of a comparable device The Federal Highway Administration is preparing

to use the radar for highway inspections and the Army Corps of Engineers has contracted with LLNL touse the system for search and rescue radar Other applications include a monitoring device to check theheartbeats of infants to guard against Sudden Infant Death Syndrome (SIDS), robot guide sensors,automatic on/off switches for bathroom hand dryers, hand-held tools, automobile back-up warningsystems, and home security

AERES, a San Jose-based company, has developed a new approach to ground-penetrating radar usingimpulse radar The first application of the technology was an airborne system for detecting undergroundbunkers The design can be altered to provide high depth capability for large targets, or high resolutionfor smaller targets near the surface This supports requirements in land mine searches and explosiveordinance disposal for the military AERAS has developed both aircraft and ground-based systemsdesigned for civilian applications as well as military Underground utility mapping, such as locating pipesand cables; highway and bridge under-surface inspection; and geological and archeological surveying are

examples of the possible civilian applications (Reprinted with permission from NASA Tech Briefs, 20(10),

24, 1996.)

A

Let r1 and r2 be roots of p2(r) = 0 such that r1 < r2 A ray propagating in the core does not necessarily reach

the core-cladding interface or the fiber axis In general, it is confined to an annular cylinder bounded by the

two caustic surfaces defined by r1 and r2 As illustrated in Fig 42.10, the field is oscillatory within this annularcylinder and evanescent outside it The fields obtained as solutions to Eq (42.25) are

In Eq (42.28) l and m are the integers denoting the modes A closed analytical solution of this equation for

b is possible only for a few simple graded-index profiles For other cases, numerical or approximate methods

field that is oscillatory between r1 and r2 and evanescent outside that region.

ù û

ù û

2 2

1 2

21

2

é ë

are used It can be shown [Marcuse, 1982] that for fiber of graded index profile a, the number of modes

supported N g , and the normalized frequency V, (and hence the core radius) for single mode operations are

given, respectively, by

(42.29)

(42.30)

For parabolic (a = 2) index profile Eq (40.29) give N g = , which is half the corresponding number of modes

for step index fiber, and Eq (40.30) gives V £ 2.405 Thus, compared with step index fiber, graded index

fiber will have larger core radios for single mode operation, and for the same core radius, will support a fewernumber of modes

on transmission loss of low-OH content optical fibers,” Electronic Letters, vol 12, no 21, p 550, 1976 With permission.)

+

æ è

ø

÷ æ è

ø

÷

a a

2

is caused by thermal vibration of the hydroxyl ion Extrinsic absorption is strong in the range of normal fiberoperation Thus, it is important that impurity level be limited.

Rayleigh Scattering

Rayleigh scattering is caused by localized variations in refractive index in the dielectric medium, which aresmall relative to the optic wavelength It is strong in the ultraviolet region It increases with decreasing wave-length, being proportional to l– 4 It contributes a loss factor of exp(–aR z) The Rayleigh scattering coefficient,

Dispersion and Pulse Spreading

Dispersion refers to the variation of velocity with frequency or wavelength Dispersion causes pulse spreading,but other nonwavelength-dependent mechanisms also contribute to pulse spreading in optical waveguides Themechanisms responsible for pulse spreading in optical waveguides include material dispersion, waveguidedispersion, and multimode pulse spreading

Material Dispersion

In material dispersion, the velocity variation is caused by some property of the medium In glass, it is caused

by the wavelength dependence of refractive index For a given pulse, the resulting pulse spread per unit length

is the difference between the travel times of the slowest and fastest wavelengths in the pulse It is given by

(42.32)

In Eq 42.32, n² is the second derivative of the refractive index with respect to l, M = ( l/c)n² is the material

dispersion, and Dl is the linewidth of the pulse Figure 42.12 shows the wavelength dependence of materialdispersion [Wemple, 1979] Note that for silica, zero dispersion occurs around 1.3 mm, and material dispersion

is small in the wavelength range of small fiber attenuation

Waveguide Dispersion

The effective refractive index for any mode varies with wavelength for a fixed film thickness, for a slab waveguide,

or a fixed core radius, for an optical fiber This variation causes pulse spreading, which is termed waveguidedispersion The resulting pulse spread is given by

÷

8 3

3 4

Polarization Mode Dispersion

The HE11 propagating in a single mode fiber actually consists of two orthogonally polarized modes, but the

two modes have the same effective refractive index and propagation velocity except in birefringent fibers.Birefringent fibers have asymmetric cores or asymmetric refractive index distribution in the core, which result

in different refractive indices and group velocities for the orthogonally polarized modes The different groupvelocities result in a group delay of one mode relative to the other, known as polarization mode dispersion.Birefringent fibers are polarization preserving and are required for several applications, including coherentoptical detection and fiber optic gyroscopes In high birefringence fibers, polarization dispersion can exceed

1 ns/km However, in low birefringence fibers, polarization mode dispersion is negligible relative to other pulsespreading mechanisms [Payne et al., 1982]

Multimode Pulse Spreading

In a multimode waveguide, different modes travel different path lengths This results in different travel timesand, hence, in pulse spreading Because this pulse spreading is not wavelength dependent, it is not usuallyreferred to as dispersion Multimode pulse spreads are given, respectively, for a slab waveguide, a step-indexfiber, and a parabolic graded-index fiber by the following equations:

dispersion in optical fibers,” Applied Optics, vol 18, no 1, p 33, 1979 With permission.)

8