FUNDAMENTALS OF
`PHOTONICS
`
`BAHAA E. A. SALEH
`
`Department of Electrical and Computer Engineering
`University of Wisconsin —— Madison
`Madison, Wisconsin
`
`MALVIN CARL TEICH
`
`Department of Electrical Engineering
`Columbia University
`New York, New York
`
`A WILEY-INTERSCIENCE PUBLICATION
`
`JOHN WILEY & SONS, INC.
`
`NEW YORK / CHICHESTER / BRISBANE / TORONTO / SINGAPORE
`Page 1
`ILLUMINA, INC. EXHIBIT 1033
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`Page 1
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`Copyright ©1991 by John Wiley & Sons, Inc.
`
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`
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`
`Library of Congress Cataloging in Publication Data:
`
`Saleh, Bahaa E. A., 1944-
`Fundamentals of photonics/Bahaa E. A. Saleh, Malvin Carl Teich.
`p. cm.——(Wiley series in pure and applied optics)
`“A Wiley-Interscience publication.”
`Includes bibliographical references and index.
`ISBN O-471—83965—5
`1. Photonics. I. Teich, Malvin Carl. 11. Title. III. Series.
`
`TA1520.S24 1991
`621.36——-dc2O
`
`90-44694
`CIP
`
`Printed in the United States of America
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`10 9 8 7 6 5 4 3 2 1
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`Page 2
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`

`
`CHAPTER
`
`
`FIBER OPTICS
`
`8.1
`
`STEP-INDEXFIBERS
`
`A. Guided Rays
`B. Guided Waves
`
`C.
`
`Single—Mode Fibers
`
`8.2 GRADED-INDEX FiBERS
`
`A. Guided Waves
`
`B. Propagation Constants and Velocities
`
`8.3 ATTENUATION AND D|SPERSlON
`A. Attenuation
`
`B. Dispersion
`
`C. Pulse Propagation
`
`facture of ultra—low—loss glass fibers.
`
`Dramatic improvements in the development of
`low-loss materials
`for optical
`fibers
`are
`responsible
`for
`the commercial viability of
`fiber-optic communications. Corning Incorpo~
`rated pioneered the development and manu-
`
`272
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`An optical fiber is a cylindrical dielectric waveguide made of low-loss materials such as
`silica glass. It has a central core in which the light is guided, embedded in an outer
`cladding of slightly lower refractive index (Fig. 8.0-1). Light rays incident on the
`core-cladding boundary at angles greater than the critical angle undergo total internal
`reflection and are guided through the core without refraction. Rays of greater inclina~
`tion to the fiber axis lose part of their power into the cladding at each reflection and
`are not guided.
`light can be guided
`As a result of recent technological advances in fabrication,
`through 1 km of glass fiber with a loss as low as = 0.16 dB (= 3.6 %). Optical fibers
`are replacing copper coaxial cables as the preferred transmission medium for electro-
`magnetic waves, thereby revolutionizing terrestrial communications. Applications range
`from long-distance telephone and data communications to computer communications
`in a local area network.
`
`In this chapter we introduce the principles of light transmission in optical fibers.
`These principles are essentially the same as those that apply in planar dielectric
`waveguides (Chap. 7), except for the cylindrical geometry. In both types of waveguide
`light propagates in the form of modes. Each mode travels along the axis of the
`waveguide with a distinct propagation constant and group velocity, maintaining its
`transverse spatial distribution and its polarization. In planar waveguides, we found that
`each mode was the sum of the multiple reflections of a TEM wave bouncing within the
`slab in the direction of an optical ray at a certain bounce angle. This approach is
`approximately applicable to cylindrical waveguides as well. When the core diameter is
`small, only a single mode is permitted and the fiber is said to be a single-mode fiber.
`Fibers with large core diameters are multimode fibers.
`One of the difiiculties associated with light propagation in multimode fibers arises
`from the differences among the group velocities of the modes. This results in a variety
`of travel times so that light pulses are broadened as they travel through the fiber. This
`effect, called modal dispersion, limits the speed at which adjacent pulses can be sent
`without overlapping and therefore the speed at which a fiber—optic communication
`system can operate.
`Modal dispersion can be reduced by grading the refractive index of the fiber core
`from a maximum value at
`its center to a minimum value at
`the core—cladding
`boundary. The fiber is then called a graded-index fiber, whereas conventional fibers
`
`
`
`Figure 8.0-1 An optical fiber is a cylindrical dielectric waveguide.
`
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`972
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`

`
`274
`
`FIBER OPTICS
`
`Figure 8.0-2 Geometry, refractive-index profile, and typical rays in: (a) a multimode step-index
`fiber, (b) a single—mode step-index fiber, and (c) a multimode graded—index fiber.
`
`with constant refractive indices in the core and the cladding are called step-index
`fibers. In a graded—index fiber the velocity increases with distance from the core axis
`(since the refractive index decreases). Although rays of greater inclination to the fiber
`axis must travel farther, they travel faster, so that the travel times of the different rays
`are equalized. Optical fibers are therefore classified as step—index or graded-index, and
`multimode or single-mode, as illustrated in Fig. 8.02.
`This chapter emphasizes the nature of optical modes and their group velocities in
`step-index and graded—index fibers. These topics are presented in Secs. 8.1 and 8.2,
`respectively. The optical properties of the fiber material (which is usually fused silica),
`including its attenuation and the effects of material, modal, and waveguide dispersion
`on the transmission of light pulses, are discussed in Sec. 8.3. Optical fibers are revisited
`in Chap. 22, which is devoted to their use in lightwave communication systems.
`
`8.1
`
`STEP-INDEX FIBERS
`
`A step-index fiber is a cylindrical dielectric waveguide specified by its core and cladding
`refractive indices, n, and n2, and the radii a and b (see Fig. 8.0-1). Examples of
`standard core and cladding diameters 2a /2b are 8/125, 50/125, 62.5 / 125, 85/125,
`100/140 (units of ptm). The refractive indices differ only slightly, so that the fractional
`refractive—index change
`
`"1 — "2
`A = ——-—-
`"1
`
`8.1-1
`
`(
`
`)
`
`is small (A < 1).
`
`Almost all fibers currently used in optical communication systems are made of fused
`silica glass (SiO2) of high chemical purity. Slight changes in the refractive index are
`
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`
`lex
`
`ex
`
`(is
`er
`
`LYS
`ad
`
`made by the addition of low concentrations of doping materials (titanium, germanium,
`or boron, for example). The refractive index rzl
`is in the range from 1.44 to 1.46,
`depending on the wavelength, and A typically lies between 0.001 and 0.02.
`
`STEP-INDEX FlBERS
`
`275
`
`A. Guided Rays
`
`An optical ray is guided by total internal refiections within the fiber core if its angle of
`incidence on the core-cladding boundary is greater than the critical angle 66 =
`sin”1(n2/nl), and remains so as the ray bounces.
`
`.
`Meridional Rays
`The guiding condition is simple to see for meridional rays (rays in planes passing
`through the fiber axis), as illustrated in Fig. 8.1-1. These rays intersect the fiber axis
`and reflect in the same plane without changing their angle of incidence, as if they were
`in a planar waveguide. Meridional rays are guided if their angle 0 with the fiber axis is
`smaller than the complement of the critical angle BC = 72'/2 ~ 06 = cos”1(n2/n1).
`Since rt, = I12, 5c is usually small and the guided rays are approximately paraxial.
`
`
`
`Meridional plane
`
`Figure 8.1-1 The trajec_tory of a meridional ray lies in a plane passing through the fiber axis.
`The ray is guided if 0 < BC = cos‘1(n1/n2).
`
`Skewed Rays
`An arbitrary ray is identified by its plane of incidence, a plane parallel to the fiber axis
`and passing through the ray, and by the angle with that axis, as illustrated in Fig. 8.1-2.
`The plane of incidence intersects the core—cladding cylindrical boundary at an angle gb
`with the normal to the boundary and lies at a distance R from the fiber axis. The ray is
`identified by its angle 0 with the fiber axis and by the angle 45 of its plane. When (is a6 0
`(R #- 0) the ray is said to be skewed. For meridional rays <1) = O and R = 0.
`A skewed ray reflects repeatedly into planes that make the same angle (12 with the
`core—cladding boundary, and follows a helical trajectory confined within a cylindrical
`shell of radii R and a, as illustrated in Fig. 81-2. The projection "of the trajectory onto
`the transverse (x—y) plane is a regular polygon, not necessarily closed. It can be shown
`that the condition for a skewed ray to always undergo total internal reflection is that its
`angle 6 with the z axis be smaller than 06.
`
`Numerical Aperture
`A ray incident from air into the fiber becomes a guided ray _if upon refraction into the
`core it makes an angle 9 with the fiber axis smaller than 06. Applying Snell’s law at
`the air—core boundary, the angle 0,, in air corresponding to BC in the core is given by
`the relation 1-sin Ba =n1sin 0C,
`from which (see Fig. 8.1-3 and Exercise 1.2-5)
`sin 0,, = n1(1 — cos25C)1/2 = n1[1 —— (n2/n,)2]1/2 = (11:12 -- 21%)”. Therefore
`
`0a == sin‘1NA,
`
`(8.1-2)
`
`Page 6