1. CHAPTER 2 Dielectric Waveguides and Optical Fibers

2. 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)

3. 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 (in Shanghai, China)
b (in Amherst, NS, Canada)
b. 1930

4. 2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE

5. 2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE A. Waveguide Condition

6. 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.

7. 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.

8. 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

10. 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

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

12. 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)

13. 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

14. 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

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

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 2. 1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE B
2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE B. Single and Multimode Waveguide

23. 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

25. 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.

26. 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.

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

28. 2.1 SYMMETRIC PLANAR DIELECTRIC SLAB WAVEGUIDE C. TE and TM Modes

29. 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// .

30. 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.

31. 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.

32. EXAMPLE 2.1.1 Waveguide modes
Solution (1/4)

33. EXAMPLE 2.1.1 Waveguide modes
Solution (2/4)

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

35. EXAMPLE 2.1.1 Waveguide modes
Solution (4/4)

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

37. EXAMPLE 2.1.2 V-number and the number of modes
Solution

38. 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

39. EXAMPLE 2.1.3 Mode field distance (MFD), 2w0

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

41. 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

42. 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.

43. Group velocity

44. Dispersion diagram

45. 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

46. 2. 2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE B
2.2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE B. Intermodal Dispersion

47. 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.

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

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

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

51. 2. 2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE C
2.2 MODAL AND WAVEGUIDE DISPERSION IN THE PLANAR WAVEGUIDE C. Intramodal Dispersion

52. 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.

53. 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.

54. 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.

55. 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.

56. 2.3 STEP INDEX FIBER

57. Reeled optical fibers on a drum

58. 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

59. 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

60. 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

61.  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.

62. 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

63. 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.

64. 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.

65. 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

66. V-number (normalized frequency)

67. Single mode fiber
Jl-1(V)= 0 for l  1
 J0(V)= 0 V =
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.

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

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

70. 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)

71. 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

72. 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.

73. 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.

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

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

76. EXAMPLE 2.3.2 A single mode fiber
Solution

77. 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.

78. 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) = ?

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

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

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

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

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

84. EXAMPLE 2.3.4 Group velocity and delay
Solution

85. EXAMPLE 2.3.4 Group velocity and delay
Solution

86. EXAMPLE 2.3.4 Group velocity and delay
Solution

87. 2.4 NUMERICAL APERTURE

88. 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.

89. 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.

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

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

92. V-number and NA
V-number
Numerical aperture 

93. 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 = ?

94. EXAMPLE 2.4.1 A multimode fiber and total acceptance angle
Solution

95. 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 = ?

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

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

98. 2.5 DISPERSION IN SINGLE MODE FIBERS

99. 2.5 DISPERSION IN SINGLE MODE FIBERS A. Material Dispersion

100. 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

101. 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”.

102. 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.

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

104. 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

105. 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.

106. 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

107. 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

108. 2.5 DISPERSION IN SINGLE MODE FIBERS B. Waveguide Dispersion

109. 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.

110. 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

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

112. 2. 5 DISPERSION IN SINGLE MODE FIBERS C
2.5 DISPERSION IN SINGLE MODE FIBERS C. Chromatic Dispersion or Total Dispersion

113.  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 

114. Chromatic dispersion

115.  1300 nm

116. 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

117. 2. 5 DISPERSION IN SINGLE MODE FIBERS D
2.5 DISPERSION IN SINGLE MODE FIBERS D. Profile and Polarization Dispersion Effects

118. 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

119. 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.

120. 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

121. 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

122. 2.5 DISPERSION IN SINGLE MODE FIBERS D. Dispersion Flattened Fibers

123. 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

124. 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.

125. 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  = 1.55 m with 1/2 = 100 nm ?
What is the material dispersion effect per km of silica fiber operated from a laser diode  = 1.55 m with 1/2 = 2 nm ?

126. 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!).

127. EXAMPLE 2.5.2 Material, Waveguide, and chromatic dispersion
Single mode optical fiber
Core: SiO % 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  = 1.5 m ?

128. 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

129. 2.6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH

130. 2. 6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH A
2.6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH A. Bit Rate and Dispersion

131. 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 

132. 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.

133. 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

134. 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 

135. 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 

136. 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

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

138. 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

139. 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   = 1/2
Dch : chromatic dispersion coefficient
The rms dispersion :
The BL product:
Maximum bit rate  distance

140. 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  = 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.

141. 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

142. 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

143. 2. 6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH B
2.6 BIT RATE, DISPERSION, ELECTRICAL, AND OPTICAL BANDWIDTH B. Optical and Electrical Bandwidth

144. 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.

145. 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.

146. 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

147. 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

148. 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

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

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

151. 2.7 THE GRADED INDEX (GRIN) OPTICAL FIBER

152. Single mode step index fiber
Drawback: relatively small numerical aperture NA (see EXAMPLE 2.4.2) EX
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

153. 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.

154. 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.

155. 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

156. 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

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

158. Refractive index profile
Power law

159. Optimal profile index
The intermodal dispersion is minimum when

160. 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

161. 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 = ?

162. 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

163. 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

164. 2.8 LIGHT ABSORPTION AND SCATTERING

165. 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

166. 2.8 LIGHT ABSORPTION AND SCATTERING A. Absorption

167. 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.

168. 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.

169. 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).

170. 2.8 LIGHT ABSORPTION AND SCATTERING B. Scattering

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

172. 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.

173. 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.

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

175. 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.

176. 2.9 ATTENUATION IN OPTICAL FIBERS

177. 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

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

179. Attenuation in dB/length

180. 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.

181. 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 1310 nm
~1.55 m
The lowest attenuation
for long-haul communications

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

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

184. 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.

185. 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.

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

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

188. EXAMPLE 2.9.1 Rayleigh scattering limit
Solution
Absolute Temp.

189. 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

190. 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)?

191. 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.