CHAPTER 2 Dielectric Waveguides and Optical Fibers

Charles K. Kao (高錕) The inventor of fiber optics “The introduction of optical fiber systems will revolutionize the communications network. The low-transmission loss and the large bandwidth capability of the fiber systems allow signals to be transmitted for establishing communications contacts over large distances with few or no provisions of intermediate amplification.” Charles K. Kao, Optical Fiber Systems: Technology, Design, and Applications (McGraw-Hill, New York, 1982), p.1. Professor Charles Kao receiving an IEE prize from Professor John Midwinter (1998 at IEE Savoy Place, London, UK)

The Nobel Prize in Physics 2009 “for groundbreaking achievements concerning the transmission of light in fibers for optical communication” "for the invention of an imaging semiconductor circuit – the CCD sensor" Willard S. Boyle George E. Smith Charles K. Kao b. 1933 (in Shanghai, China) b. 1924 (in Amherst, NS, Canada) b. 1930

2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE

2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE A. Waveguide Condition

Planar dielectric slab waveguide core cladding A slab of dielectric of refractive index n1 is sandwiched between two semi-infinite regions both of refractive index n2 (n2 < n1). Core- the region of higher refractive index (n1) Cladding- the region of lower refractive index (n2) sandwiching the core. A light ray can readily propagate along such a waveguide, in a zigzag fashion, provided it can undergo total internal reflection (TIR) at the dielectric boundaries.

A light ray traveling in the slab waveguide Electric field (TE wave) Consider a plane wave type of light ray propagating in the waveguide. The ray is guided in a zigzag fashion along the guide axis by reflections from the boundaries.  Effective propagation along z The wave interferes with itself.  Only certain reflection angles  give rise to the constructive interference and hence only certain wave can exist in the guide.

constructive interference TE mode, under TIR  0 <  < 180(see Fig. 1.12) k1 = n1k = n1(2 / ), k, : the free space wavevector and wavelength Phase difference between A and C constructive interference

Constructive interference For wave propagation along the guide, we need Only certain  and  values can satisfy this equation for a given integer m. But,  depends on , i.e.  (). For each m, there will be one allowed angle m and corresponding m. Constructive interference

m indicates that  is a function of incidence angle m Waveguide condition Waveguide condition m indicates that  is a function of incidence angle m

Ray 1 at B, just after two reflection, has a phase k1AB2 We can derive the same waveguide condition if we took a wider angle for . Ray 1 at B, just after two reflection, has a phase k1AB2 Ray 2 at B has a phase k1(AB) For constructive interference, phase difference Waveguide condition Problem 2.1(a)

Higher m values yields lower m. Waveguide condition We can resolve the wavevector k1 into two propagation constants,  and  Only certain m are allowed within the guide corresponding to m = 0, 1,  Higher m values yields lower m. Each choice of m leads to a different reflection angle m. Each different m value leads different propagation constant  m and m. Propagation constant along the guide Transverse propagation constant

Ray 1 and ray 2 meet at C, distance y above the guide center. /2 Consider the resultant of two parallel rays that have m satisfying the waveguide condition. Ray 1 and ray 2 meet at C, distance y above the guide center.  phase difference m between ray 1 and ray 2

The phase difference between ray 1 and ray 2 at C m is a function of y for a given m

A traveling wave along z, due to the cos(tmz) term, whose amplitude along y is modulated by the cos(my + m /2) term.

Possible waves in the guide  Field pattern for m = 0 mode Field pattern for m = 0, 1, 2 modes field distribution along y for a given m The distribution Em(y) across the guide is traveling down the guide along z.

The lowest mode, m  0 The field pattern has a maximum intensity at the center. The whole field distribution is moving along z with a propagation vector 0. The field penetrates into the cladding, which is due to a propagating evanescent wave in the cladding near the boundary. The field pattern in the core exhibits a harmonic variation across the guide.

Mode of propagation Each m leads to an allowed m value that corresponds to a particular traveling wave in the z-direction with particular wavevector m. Each of these traveling waves with Em(y) constitutes a mode of propagation. m : mode number m is smaller for larger m  higher modes exhibit more reflection and penetrate much more into the cladding.

Mode of propagation The lowest mode, m = 0 0  90 wave is said to travel axially Light that is launched into the core can travel only along the guide in the allowed modes specified by Eq. (3) These modes will travel down the guide at different group velocities.  Light pulse spreads as it travels along the guide.

2. 1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE B 2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE B. Single and Multimode Waveguide

Maximum value of m Waveguide condition The allowed incident angle m must satisfy TIR, i.e., sinm > sinc  We can only have up to a certain maximum number of modes being allowed in the waveguide The mode number m must satisfy in which V, called V-number, is defined by Maximum value of m

V-number Other names V-parameter normalized thickness normalized frequency For a given free space wavelength , V depends on the waveguide geometry (2a) and waveguide properties, n1 and n2.  V is a characteristic parameter of the waveguide.

Single mode planar waveguide Suppose there is only lowest mode (m = 0), and suppose the propagation is due to m 90 then   When V < /2 , m = 0 is the only possibility and only the fundamental mode (m=0) propagates along the dielectric slab waveguide, which is then termed a single mode planar waveguide.

Cut-off wavelength Above this wavelength, only one-mode, the fundamental mode, will propagate.  Cut-off wavelength

2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE C. TE and TM Modes

TE and TM modes (a) E field  plan of incidence (indicated by E). E = Ex (b) B field  plan of incidence (indicated by B). E field  E// Any other field direction can be resolved to have electric field components along E// and E E// and Eexperience different phase changes, // and , and require different angle m to propagate along the guide TE modes (transverse electric field modes) (denoted by TEm ) : The modes associated with E . TM modes (transverse magnetic field modes) (denoted by TMm ) : The modes associated with E// .

Propagating longitudinal fields For TM modes  Ez : a propagating longitudinal electric field For TE modes  Bz : a propagating longitudinal magnetic field In free space, it is impossible for such longitudinal fields, but within an optical guide, due to interference, it is possible to have a longitudinal field.

EXAMPLE 2.1.1 Waveguide modes Planer dielectric waveguide core thickness d = 2a = 20m n1 = 1.455, n2 = 1.440  = 900 nm Waveguide condition  in TIR for the TE mode (Ch. 1) : Find angles m for all the modes.

EXAMPLE 2.1.1 Waveguide modes Solution (1/4)

EXAMPLE 2.1.1 Waveguide modes Solution (2/4)

EXAMPLE 2.1.1 Waveguide modes Solution (3/4) m = 0 m = 1 m = 2 m = 3 81.77

EXAMPLE 2.1.1 Waveguide modes Solution (4/4)

EXAMPLE 2.1.2 V-number and the number of modes Planar dielectric waveguide d = 2a = 100 m n1 = 1.490, n2 = 1.470 @  = 1 m Estimate the number of modes Compare the estimate with the formula Number of modes Integer function

EXAMPLE 2.1.2 V-number and the number of modes Solution

EXAMPLE 2.1.3 Mode field distance (MFD), 2w0 The field distribution along y penetrates into the cladding. Within the core, the field distribution is harmonic. From the boundary into the cladding, the field is due to the evanescent wave and it decays exponentially.  y is measured from the boundary  decay constant for the evanescent wave in medium 2

EXAMPLE 2.1.3 Mode field distance (MFD), 2w0

EXAMPLE 2.1.3 Mode field distance (MFD), 2w0 In cylindrical dielectric waveguide (i.e. optical fiber) MFD is called mode field diameter.

2. 2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE A 2.2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE A. Waveguide Dispersion Diagram

Light propagation in a slab dielectric waveguide Each mode propagates with a different propagation constant even if illumination is by monochromatic radiation. Group velocity Vg along the guide is the velocity at which the energy or information is transported. The higher modes penetrate more into the cladding where the refractive index is smaller and the waves travel faster.

Group velocity

Dispersion diagram

Cut-off frequency cut-off cut-off corresponds to the cut-off condition   c when V  /2 For  > cut-off  there are more than one mode Two immediate consequences The group velocity at one frequency changes from one mode to another. For a given mode, the group velocity changes with the frequency.   c when V  /2

2. 2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE B 2.2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE B. Intermodal Dispersion

Intermodal dispersion Multimode operation  > cut-off m = 0 mode  the slowest group velocity vg  c/n1 the highest mode  the highest group velocity The modes take different times to travel the length of the waveguide  modal dispersion (or intermodal dispersion)  a broadened signal Single mode waveguide only one mode can propagate (m = 0) There will be no modal dispersion.

Modal dispersion Vgmin : the minimum group velocity of the slowest mode Vgmax : the maximum group velocity of the fastest mode Modal dispersion

Modal dispersion The lowest order mode (m = 0) when  > cut-off Vgmin  c/n1 The highest order mode Vgmax  c/n2 

Modal dispersion n1 = 1.48 (core) n2 = 1.46 (cladding)

2. 2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE C 2.2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE C. Intramodal Dispersion

In single mode operation Even if we use the guide in single mode operation, as long as the excitation source has a finite spectrum, it will contain various frequencies. These frequencies will then travel with different group velocities and hence arrive at different time.

Waveguide dispersion The higher the wavelength (lower the frequency), the greater the penetration into the cladding. Greater portion of light is carried by the cladding in which the phase velocity is higher. Longer wavelengths propagate faster, even though by the same mode. This is called waveguide dispersion. Waveguide dispersion results from the guiding properties of the dielectric structure.

Material dispersion Refractive index n = n() m behavior depends on n() The change in the group velocity of a given mode due to the n dependence also gives rise to the broadening of a propagation light pulse. This is called material dispersion.

Intramode dispersion Both waveguide dispersion and material dispersion act together to broaden a light pulse propagating within a given mode. Combined dispersion is called intramode dispersion.

2.3 STEP INDEX FIBER

Reeled optical fibers on a drum

Step index optical fiber Cylindrical dielectric waveguide  Optical fiber Inner core index n1 > outer cladding index n2 Normalized index difference For all practical fibers used in optical communications, n1  n2 (n1 > n2)  << 1

Waveguide mode numbers Planar waveguide is bounded only in one-dimension reflections occur only in the y-direction only one mode number m Cylindrical waveguide is bounded in two dimensions reflections occur from all the surfaces  two mode numbers: l , m

Meridional and skew rays Meridional ray enters the fiber through the fiber axis crosses the fiber axis on each reflection as it zigzags down the fiber Skew ray enters the fiber off the fiber axis zigzags down the fiber without crossing the axis has a helical path around the fiber axis

 Guide modes Meridional ray Skew ray hybrid modes: TE modes: Ez = 0 TM modes: Bz = 0 Skew ray HE modes EH modes  hybrid modes: both electric and magnetic field can have components along z.

Modes in weakly guiding fibers   (n1 – n2)/n1 << 1 Guided modes Waves are linearly polarized (LP) and have transverse electric and magnetic field characteristics. LPlm mode For a given l and m, Elm (r,) represents a particular field pattern at z that is propagated along the fiber with an effective wave vector lm

LPlm modes Mode numbers l, m l  There are 2l number of maxima around a circumference. m  There are m number of maxima along r staring from the core center.

LPlm modes Fundamental mode l = 0, m = 1  LP01 mode The field is maximum at the center of the core and penetrates somewhat into the cladding. The intensity distribution has a maximum along the fiber axis.

LPlm modes Each mode has its own propagation vector lm and its own electric filed pattern Elm(r,). Each mode has its own group velocity Vg(l, m) that depends on the  vs. lm dispersion behavior. When a pulse is fed into the fiber, it travels down the fiber through various modes. Output pulse is broadened. Intermodal dispersion

V-number (normalized frequency)

Single mode fiber Jl-1(V)= 0 for l  1  J0(V)= 0 V = 2.4048 If V < 2.405, there is only one mode, the fundamental mode (LP01) can propagate through the fiber core. A fiber that is designed (by the choice of a and ) to allow only the fundamental mode to propagate at the required wavelength is called single mode fiber.

Cut-off wavelength The cut-off wavelength above which the fiber becomes single mode.

Number of modes When V >> 2.405  a  or n1   M 

Intensity loss In a step index fiber, the field of the fundamental mode penetrates into the cladding as an evanescent wave traveling along the boundary. If the cladding is not sufficiently thick, this field will reach the end of the cladding and escape. Intensity loss Typically, cladding diameter > 10 core diameter (in a single-mode step-index fiber)

Normalized propagation constant Given k = 2/ and indices n1 and n2, the normalized propagation constant b is related to  = lm by for   k n1 propagation in the core material for   k n2 propagation in the cladding material

Normalized propagation constant b vs. V-number There is a cut-off V and a corresponding cut-off wavelength for each particular LP mode higher than the fundamental mode. Given the V-parameter of the fiber we can find b, and hence , for the allowed LP modes.

EXAMPLE 2.3.1 A multimode fiber Multimode step index fiber n1 = 1.468 2a = 100 m  a = 50 m n2 = 1.477 Source wavelength = 850 nm Calculate the number of allowed modes.

EXAMPLE 2.3.1 A multimode fiber Solution Since V >> 2.405 M  V2/2 = (91.44)2/22 =4180.6  4181 The number of modes

EXAMPLE 2.3.2 A single mode fiber Source wavelength  = 1.3 m What should be the core radius a ?

EXAMPLE 2.3.2 A single mode fiber Solution

EXAMPLE 2.3.2 A single mode fiber Solution Single mode fiber core radius a  2.01 m This is rather thin for easy coupling of the fiber to a light source or to another fiber and special coupling techniques must be used.  = 1.3 m  a ~  This means that the geometric ray picture cannot be used to describe light propagation.

EXAMPLE 2.3.3 Single mode cut-off wavelength Single mode operation A fiber has 2a = 7 m a = 3.5 m n1 = 1.458 n2 = 1.452 Cut-off wavelength c = ? If  = 1.3 m V-number V = ? Mode field diameter (MFD) = ?

EXAMPLE 2.3.3 Single mode cut-off wavelength Solution For single mode operation Wavelength shorter than 1.208 m will result in a multimode propagation

EXAMPLE 2.3.3 Single mode cut-off wavelength Solution At  = 1.3 m Single mode  Mode field diameter MFD

EXAMPLE 2.3.4 Group velocity and delay Single mode fiber a = 3 m n1 = 1.448 n2 = 1.440 operating @  = 1.5 m If the fundamental mode normalized propagation constant propagation constant   ?

EXAMPLE 2.3.4 Group velocity and delay If    (changed 0.01%)  =  (1 + 0.01%) = 1.50015 m new propagation constant   ? Group velocity Vg @ 1.5 m = ? Group delay g over 1 km = ?

EXAMPLE 2.3.4 Group velocity and delay Solution For weakly guiding fiber (n1  n2)  = (n1 - n2) / n1

EXAMPLE 2.3.4 Group velocity and delay Solution

EXAMPLE 2.3.4 Group velocity and delay Solution

EXAMPLE 2.3.4 Group velocity and delay Solution

2.4 NUMERICAL APERTURE

Not all source radiation can be guided along an optical fiber. Only rays falling within a certain cone at the input of the fiber can normally be propagated through the fiber.

Maximum acceptance angle max max gives total internal reflection at the core-cladding interface When  = max then   c Rays with  > max become refracted and penetrate the cladding and are eventually lost.

Maximum acceptance angle max At the n0 /n1 interface Snell’s law

Numerical Aperture (NA) and Maximum acceptance angle max Define numerical aperture Maximum acceptance angle max Total acceptance angle 2max

V-number and NA V-number Numerical aperture 

EXAMPLE 2.4.1 A multimode fiber and total acceptance angle Step index fiber 2a = 100 m n1 = 1.480 n2 = 1.460 Source wavelength  = 850 nm NA = ? max = ? The number of modes = ?

EXAMPLE 2.4.1 A multimode fiber and total acceptance angle Solution

EXAMPLE 2.4.2 A single mode fiber normalized index difference  = (n2 – n1)/n1 = 0.3% cladding diameter = 125 m NA = ? max = ? Single mode cut-off wavelength c = ?

EXAMPLE 2.4.2 A single mode fiber Solution Drawback of single mode step index fiber

EXAMPLE 2.4.2 A single mode fiber Solution Illumination wavelength shorter than 1.18 m will result in multimode operation.

2.5 DISPERSION IN SINGLE MODE FIBERS

2.5 DISPERSION IN SINGLE MODE FIBERS A. Material Dispersion

The advantage of the single mode fiber There is no intermodal dispersion. There will still be dispersion due to the variation of refractive index n1 of the core glass with the wavelength of light coupled into the fiber. The advantage of the single mode fiber

Material dispersion n1 = n1 () vg depends on n  vg depends on  This type of dispersion that results from the wavelength dependence of material properties is called “material dispersion”.

No practical light source is perfectly monochromatic. There are waves in the guide with various space wavelengths. Each radiation with a different  will propagate at a different group velocity vg The waves will arrive at different time at the end of fiber Broadened output light pulse.

Group index Ng (for silica glass)  Zero material dispersion around   1300 nm.

Material dispersion Input: very short-duration light pulse Output: a pulse that is broadened by     L Material dispersion L Material dispersion coefficient   Dm may be negative or positive

Material dispersion coefficient (Dm) For silica (SiO2), Dm vs.  curve passes through zero at   1.27m Silica is doped with GeO2 (germania) to increase the refractive index  Dm vs.  curve shifts slightly to higher wavelength.

Group delay time Transit time  : Group delay time g a delay time of information between the output and the input. Group delay time g the signal delay time per unit distance g   /L

Group delay time and dispersion If 01 is the propagation constant,  g = g () will be a function of wavelength Material dispersion is the spread in g due to the dependence of 01 on wavelength through Ng. Fundamental mode group delay time

2.5 DISPERSION IN SINGLE MODE FIBERS B. Waveguide Dispersion

Waveguide dispersion Waveguid dispersion is due to the dependence of the group velocity vg(01) on the V-number, which depends on the source , even if n1 and n2 were constants.  A spectrum of source wavelength will result in different V-number for each source wavelength and hence different propagation velocities.

Waveguide dispersion If we use a light pulse of very short duration with a wavelength spectrum of , the broadening or dispersion per unit length /L in the output light pulse due to waveguide dispersion is Waveguide dispersion coefficient Ng2 and n2: the group and refractive indices of the cladding Dw depends on the guide geometry through the core radius a

Waveguide dispersion coefficient (Dw) The dependence of Dw on  for a core radius a = 4.2 m . Dw and Dm have opposite tendencies.

2. 5 DISPERSION IN SINGLE MODE FIBERS C 2.5 DISPERSION IN SINGLE MODE FIBERS C. Chromatic Dispersion or Total Dispersion

 Chromatic dispersion In single mode fibers the dispersion of a propagating pulse arises because of the finite width  of the source spectrum. The source spectrum is not perfectly monochromatic. Chromatic dispersion: The type of dispersion causes by a range of source wavelength. Chromatic dispersion = Material dispersion + Waveguide dispersion 

Chromatic dispersion

 1300 nm

Dispersion shifted fibers Since Dw depends on the guide geometry, it is possible to shift the zero dispersion wavelength 0 by suitably designing the guide. Such fibers are called dispersion shifted fibers. Ex. core radius a , increasing the core doping 0 = 1300 nm  0 =1550 nm where light attenuation in the fiber is minimal

2. 5 DISPERSION IN SINGLE MODE FIBERS D 2.5 DISPERSION IN SINGLE MODE FIBERS D. Profile and Polarization Dispersion Effects

Profile dispersion coefficient Chief sources of broadening Profile dispersion additional dispersion that arises because the group velocity, vg(01), of the fundamental mode also depends on the refractive index difference  Profile dispersion coefficient Dp is negligible compared with Dw

Overall chromatic dispersion coefficient Dch = Dm + Dw + Dp Dm : material dispersion coefficient Dw : waveguide dispersion coefficient Dp : profile dispersion coefficient The reason  exhibits a wavelength dependence is due to material dispersion characteristics, i.e. n1 vs.  and n2 vs.  Profile dispersion originates from material dispersion.

Polarization dispersion Polarization dispersion arise when the fiber is not perfectly symmetric and homogeneous. the refractive index is not isotropic the refractive index depends on the direction of the electric field the propagation constant depends on its polarization

Polarization dispersion The refractive indices n1 and n2 may not be isotropic There are different group delays and hence dispersion even if the source wavelength is monochromatic Typically,  / L < ps/km The propagation constant for fields along x and y would be different

2.5 DISPERSION IN SINGLE MODE FIBERS D. Dispersion Flattened Fibers

Dispersion Flattened Fibers It may be desirable to have minimal dispersion over a range of wavelength not just at zero-crossing wavelength 0. We can alter the waveguide geometry to yield a total chromatic dispersion that is flattened between 1 and 2  Dispersion flattened fiber

Doubly clad fiber Singly clad fiber Doubly clad fiber simple step fiber Doubly clad fiber Such fibers can exhibit excellent chromatic dispersion 1-3 ps km-1 nm-1 over the wavelength 1.3 – 1.6 m Doubly clad fiber allows wavelength multiplexing , i.e., using a number of wavelengths (e.g.1.3, 1.55 m) as communication channels.

EXAMPLE 2.5.1 Material dispersion 1/2 (linewidth) the width of intensity vs. wavelength spectrum between the half-intensity points. 1/2 the width of the output light intensity vs. time between the half-intensity points. What is the material dispersion effect per km of silica fiber operated from a LED emitting @  = 1.55 m with 1/2 = 100 nm ? What is the material dispersion effect per km of silica fiber operated from a laser diode emitting @  = 1.55 m with 1/2 = 2 nm ?

EXAMPLE 2.5.1 Material dispersion Solution Material dispersion coefficient Dm( = 1.55m) = 22 ps km-1 nm-1 For the LED, 1/2 = 100 nm For the laser diode, 1/2 = 2 nm 22 1.55  There is a big difference between the dispersion effect of the two sources.  If the fiber is properly dispersion shifted so that Dm+ Dw = 0 at  = 1.55m  1/2 / L ~ ps/km for a typical laser diode (but not zero!).

EXAMPLE 2.5.2 Material, Waveguide, and chromatic dispersion Single mode optical fiber Core: SiO2 - 13.5% GeO2 for diameter = 2a Laser source  = 1.5 m 1/2 = 2 nm What is the dispersion per km of fiber if 2a = 8 m ? What should be 2a for zero chromatic dispersion @  = 1.5 m ?

EXAMPLE 2.5.2 Material, Waveguide, and chromatic dispersion Solution   = 1.5 m Dm = +10 ps km-1 nm-1 a = 4 m  Dw = 6 ps km-1 nm-1 Chromatic dispersion coefficient Dch = Dm + Dw = 10  6 = 4 ps km-1 nm-1 Total dispersion or chromatic dispersion per km Dispersion will be zero at 1.5m when Dm Dw  Dch = Dm + Dw = 0  Dw Dm  10 ps km-1 nm-1 at  = 1.5m From Fig. 2.21, at  = 1.5, Dw 10 ps km-1 nm-1  a = 3.0m

2.6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH

2. 6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH A 2.6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH A. Bit Rate and Dispersion

Digital communications The pulses represent bits of information that are in digital form. In digital communications Signals  Light pulses along an optical fiber  Information  Electrical digital signal (pulses)  Light emitter (laser diode or LED)  Light output is coupled into a fiber  Photodetector  Electrical signal 1/2 

Bit rate capacity Bit rate capacity B (bits per second) of the fiber The maximum rate at which the digital data can be transmitted along the fiber. Bit rate capacity is directly related to the dispersion characteristics.

FWHM or FWHM Suppose that we feed a light pulse into the fiber. The output pulse will be delayed by the transit time . There will be a spread  in the arrival times. This dispersion is typically measured between half-power points and is called full width at half power (FWHP) or full width at half maximum (FWHM)  1/2  1/2

Maximum bit rate If there is no intersymbol interference Clear distinguishability between two consecutive output pulses. They be time-sparated from peak to peak at least 21/2 We can only feed in pulses at the input, at best, at every 21/2 seconds. The period of the input pulses T = 21/2 The maximum bit rate (or simply the bit rate) Intuitive RZ bit rate and dispersion 1/2 

Return-to-zero and Nonreturn-to-zero bit rate Return-to-zero (RZ) bit rate or data rate Two consecutive pulses for two consecutives 1’s must have a zero in between. Nonreturn-to-zero (NRZ) bit rate Two consecutive binary 1 pulses without having to return to zero at the end of each 1-pulse. The two pulses can be brought closer until T  1/2 Such a maximum data rate is called nonreturn-to-zero bit rate. Bit rate B(NRZ) = 2 B(RZ) 1/2 

Maximum RZ bit rate : root-mean-square dispersion rms: full-width rms time spread  rms = 2 The bit rate in terms of  requires that two consecutive light output pulse are separated by 4 between their peaks Maximum RZ bit rate and dispersion

Maximum RZ bit rate for Gaussian pulse For a Gaussian pulse B is ~ 18% greater than the intuitive estimate (B  0.5/1/2)

Bit rate B and BL product Dispersion increases with fiber length L and also with the range of source wavelength 1/2  Dispersion   as L  and 1/2   Bit rate B = 0.25/  as L  and 1/2  It is customary to specify the product of the bit rate B with the fiber length L at the operating wavelength for a given emitter. The BL product, called the bit-rate  distance product, is given by

Maximum bit rate  distance BL product  : the rms spread of wavelengths in the light output spectrum of the emitter 1/2: wavelength spread for Gaussian output   = 0.425 1/2 Dch : chromatic dispersion coefficient The rms dispersion : The BL product: Maximum bit rate  distance

BL product BL is a characteristic of the fiber, through Dch and also of the range of source wavelengths. In specifications, the fiber length is taken as 1km. Ex., step-index single mode fiber @  = 1300 nm and excited by a laser diode source  BL ~ Gb s-1 km Dividing this by the actual length of the fiber gives the operating bit rate for the length.

Total rms dispersion When both chromatic (or intramodal ) and intermodal dispersion are present, Overall dispersion in terms of an rms dispersion :  can be used in this equation to approximately find B. Total rms dispersion

Bit rate and pulse shape Bit rate B  0.25/ To determine B from 1/2 we need to know the pulse shape For a rectangular pulse For an ideal Gaussian pulse T t

2. 6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH B 2.6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH B. Optical and Electrical Bandwidth

Transmitting analog signals The emitter can also be driven, or modulated, by an analog signal that varies continuously with time. Ex., a sinusoidal signal. We can determine the transfer characteristics of the fiber by feeding in sinusoidal light intensity signals, which have the same intensity but different modulation frequencies f.

Optical transfer characteristic of fiber Pi = Input light power Po = Output light power Po/Pi = The output light power per unit input light power, which is the observed optical transfer characteristic of the fiber. Po/Pi is flat and then falls with frequency. Reason: the frequency becomes too fast so that dispersion effects smear out the light at the output.

for Gaussian dispersion Optical bandwidth Optical bandwidth fop The frequency in which the output intensity is 50% below the flat region. The useful frequency range in which modulated optical signals can be transferred along the fiber. If the fiber dispersion characteristics are Gaussian, then Optical bandwidth for Gaussian dispersion

Electrical bandwidth Electrical bandwidth fel for electrical signals is measured where the signals is 70.7% of its low frequency value.  fel < fop The relationship between fel and fop depends on the dispersion through the fiber.  For Gaussian dispersion fel  0.71 fop

EXAMPLE 2.6.1 Bit rate and dispersion Optical fiber: chromatic dispersion coefficient Dch = 8 ps km-1 nm-1 at  = 1.5m L = 10 km Laser diode source: FWHP linewidth 1/2= 2 nm Calculate Bit rate  distance product (BL) Optical bandwidth fop Electrical bandwidth fel

EXAMPLE 2.6.1 Bit rate and dispersion Solution For chromatic dispersion Assuming a Gaussian light pulse shape  RZ bit rate  distance product (BL)

EXAMPLE 2.6.1 Bit rate and dispersion Solution For a Gaussian light pulse shape  Optical bandwidth  Electrical bandwidth

2.7 THE GRADED INDEX (GRIN) OPTICAL FIBER

Single mode step index fiber Drawback: relatively small numerical aperture NA (see EXAMPLE 2.4.2) EX. 4.2.2 The difficulty in the amount of light that can be coupled into the fiber. Increasing the NA means only increasing the V-number, which must be less than 2.405.

Multimode fiber step index fiber Have a relatively larger NA Accept more light from broader angle. Easier light coupling and more optical power to be launched into the fiber Intermodal dispersion Ray paths are different so that rays arrive at different times.

Graded index (GRIN) fiber The refractive index is not constant within the core but decreases from n1 at the center, as a power law, to n2 at the cladding. Refractive index profile across the core ~ parabolic Ray paths are different but so are the velocities along the paths so that all the rays a arrive at the same time. Absence of modal dispersion.

GRIN fiber as many concentric layers na > nb > nc>  Ray 1: A = c(ab) = sin1 (nb/na) TIR at A and arrives at O Ray 2 B < c(ab) = sin1 (nb/na)  Refracted and enters layer b where its angle is B' . If we choose nb appropriately  we can direct B'  ray 2 impinges layer c at B' OB' = B'O'. If we choose nc appropriately  TIR at B', i.e. B' > c(bc) = sin1 (nc/nb) By appropriate choice of na, nb and nc, we can ensure ray 1 and ray 2 pass through O', and both arrive nearly at the same time. The appropriate choice for the refractive indices follow an approximately parabolic decrease of n from the core axis. Critical angle

Continuous change of n (a) n decreases step by step There is an immediate sequence of refractions (b) n decreases continuously The ray path becomes bent continuously until the ray suffers TIR Ray paths  curved trajectories

Graded index (GRIN) fiber Intetrmodal dispersion is not totally absent though it is reduced by orders of magnitude from the multimode step index fiber.

Refractive index profile Power law

Optimal profile index The intermodal dispersion is minimum when

Dispersion in graded index fiber With the optimal profile index, the rms dispersion intermodal (2 = rms) per unit length is Dispersion in graded index fiber

EXAMPLE 2.7.1 Dispersion in a graded-index fiber and bit rate What will be the bit rate  distance product ? (i.e. BL = ?) Multimode step index fiber light output is nearly rectangular    0.29  ( : full spread) BL = ?

EXAMPLE 2.7.1 Dispersion in a graded-index fiber and bit rate Solution Graded index fiber Normalized refractive index difference Dispersion for 1 km BL product

EXAMPLE 2.7.1 Dispersion in a graded-index fiber and bit rate Solution Multimode step-index fiber Full dispersion (total spread) which is nearly 1,000 times smaller

2.8 LIGHT ABSORPTION AND SCATTERING

Attenuation of light in the direction of propagation Absorption (lattice absorption) Scattering (Rayleigh scattering) Both give rise to a loss of intensity in the regular direction of propagation

2.8 LIGHT ABSORPTION AND SCATTERING A. Absorption

Absorption In absorption, some of the energy from the propagation wave is converted to other forms of energy, for example, to heat by the generation of lattice vibration.

Lattice absorption An EM wave displaces the oppositely charges ions in opposite directions and forces them to vibrate at the frequency of the wave. The medium experiences ionic polarization. As the ions is made to vibrate by the passing EM wave, some energy is coupled into the lattice vibrations of the solid.

Lattice absorption The coupled energy peaks when the frequency of the wave is close to the natural lattice vibrational frequencies.  f ~ infrared region. Most of the energy is absorbed from the EM wave and converted to lattice vibrational energy (heat).

2.8 LIGHT ABSORPTION AND SCATTERING B. Scattering

Scattering A portion of the energy in a light beam is directed away from the original direction of propagation.

Rayleigh scattering Rayleigh scattering involves the polarization of a small dielectric particle or a region that is much smaller than the light wavelength. The field forces dipole oscillations in the particle (by polarizing it) which leads to the emission of EM waves in "many" directions so that a portion of the light energy is directed away from the incident beam.

Small fluctuations in the relative permittivity Rayleigh scattering of waves in a medium aries whenever there are small inhomogeneous regions in which the refractive index different than the medium. The small inhomogeneous region acts a small dielectric particle and scatters the propagating wave in different directions. The glass fiber has small fluctuations in the relative permittivity that lead to Rayleigh scattering.

Net effect of scattering The incident wave becomes partially reradiated in different directions and hence loss intensity in its original direction of propagation.

Rayleigh Scattering Scattering coefficient   1/4 Scattering decreases with increasing wavelength. Blue light is scattered more strongly by air molecules. The sun appears yellow. The sky is blue.

2.9 ATTENUATION IN OPTICAL FIBERS

Attenuation coefficient The fractional decrease in the optical power per unit distance Definition of attenuation coefficient P(x): optical power in the fiber at a distance x from the input

Attenuation coefficient Pin : the input power Pout: the output power L : the length of fiber

Attenuation in dB/length

Attenuation coefficient of a silica based optical fiber The sharp increase at  > 1.6m (infrared region)  the absorption by “lattice vibration” of the constitute ions.  Energy absorption corresponds to the stretching of the Si-O bonds.

Attenuation coefficient of a silica based optical fiber Attenuation peaks: 1.4 m (marked) 1.24 m (minor) due to OH- absorption Two windows: ~1.3 m  for optical communication @ 1310 nm ~1.55 m The lowest attenuation for long-haul communications

Rayleigh scattering in silica at Tf the liquid structure during the cooling of fiber is frozen to become the glass structure.

Attenuation coefficient of a silica based fiber Rayleigh scattering represents the lowest attenuation one can achieve using a glass structure.

Microbending loss Microbending loss is due to a “sharp” local bending of the fiber that changes the guide geometry and refractive index profile locally, which leads to some of the light energy radiating away from the guiding direction.

Microbending loss If ' <   a transmitted wave or a greater cladding penetration. If ' < c, there will be no total internal reflection and substantial light power will be radiated into cladding and eventually to the outside medium.

Microbending loss Microbending loss B with R  (the radius of curvature) R <~ 10 mm can lead to appreciate microbending loss.

EXAMPLE 2.9.1 Rayleigh scattering limit Tf = 1730C (softening temperature) T  710-11 m2N-1 n = 1.4446 @ 1.5m R = ? (Rayleigh scattering attenuation)

EXAMPLE 2.9.1 Rayleigh scattering limit Solution Absolute Temp.

EXAMPLE 2.9.1 Rayleigh scattering limit Solution Attenuation in dB per km The lowest possible attenuation for a silica glass fiber at 1.55 m

EXAMPLE 2.9.2 Attenuation along an optical fiber Optical power into a single-mode fiber Pin  1 mW The photodetector required Pout > 10 nW for a clear signal  = 1.3 m dB = 0.4 dB km-1 What is the maximum length of fiber that can be used without inserting a repeater (to regenerate the single)?

EXAMPLE 2.9.2 Attenuation along an optical fiber Solution For long distance communications, the signal has to be amplified, using an optical amplifier, after a distance about 50 ~ 100 km.