1. The document discusses optical fibers, specifically step index fibers. It describes step index fibers as having a core with a constant refractive index n1 surrounded by a cladding with a slightly lower refractive index n2. 2. It discusses several factors that determine the number of propagating modes in a step index fiber, including the V-number which is a function of the core radius, wavelengths, and refractive index differences. Fibers with V<2.405 support only one mode. 3. Dispersion effects in step index fibers include intermodal dispersion from different propagation speeds of fiber modes, and material dispersion from the wavelength dependence of the core refractive index.

1. Cylindrical Dielectric Waveguide:
Optical Fibre
 Two types of fibre
1. Step index fibre
2. Graded index fibre (GRIN)

2. Step Index Fibre
• This is essentially a cylindrical dielectric waveguide with inner
core dielectric having refractive index n1 greater than n2 of the
outer dielectric, cladding.
• The normalized index difference
D = (n1 – n2)/n1
• For all practical fibres, n1 & n2 differ only by a small amount (less
than a few percent): D << 1
n
y
n2 n1
Cladding
Core z
y
r
f
Fiber axis2a

3. Step index fibre
• Optical fibre with a core of constant refractive index n1
and a cladding of a slightly lower refractive index n2.
• Could be multimode or single mode.
arn
arn

<
,
,
2
1
n(r) =
n1
n2
2
1
3
n
O

4. Transmission of Light in Optical Fibre
• Two methods for theoretical studying the
propagating characteristics of light in an optical
fibre:
1. Ray optics to study the light acceptance angle &
guiding properties of optical fibres.
– Using ray tracing approach in the case of the ratio of fibre
radius to the wavelength is large.
2. Mode theory to study the field distribution of
individual modes
– Using electromagnetic theory

5. Ray Optics
• Since the core size of multimode fibre is much larger
than the wavelength of the light (~1mm), an intuitive
picture of the propagation mechanism is most easily
seen by a simple geometrical optics representation.
• Two type of rays can propagate in a fibre
1. Meridional rays in the planes that contain the axis of
symmetry of the fibre (core axis)
2. Skew rays that is not confined to a single plane, but follow
a helical-type path along the fibre.

6. Fibre axis
1
2
3
4
5
Skew ray
1
3
2
4
5
Fibre axis
1
2
3
Meridional ray
1, 3
2
(a) A meridional
ray always
crosses the fiber
axis.
(b) A skew ray
does not have
to cross the
fiber axis. It
zigzags around
the fiber axis.
Along the fibre
Ray path projected
on to a plane normal
to fiber axis
Ray path along the fiber
Fig.8: Illustration of the difference
between a meridional ray and skew ray
in step index fibre.

7. Numerical Aperture
• Not all source radiation can be guided along an
optical fibre.
– Only rays falling within a certain cone at the input of the
fibre can normally be propagated through the fibre.
• Fig. 9 shows the path of a light ray launched from
the outside medium of refractive index no into the
fibre core.
– Suppose that the incidence angle at the end of the fibre
core is a & inside the waveguide the ray makes an angle q
with the normal to the fibre axis.
– For total internal reflection within the fibre, there is a
maximum value of a to result in q =qc

8. Numerical aperture &
acceptance angle
• At the no/n1 interface, Snell’s Law gives
sin amax/sin(90° – qc)=n1/no
sin amax = (n1
2 –n2
2)½/no, where sinqc= n2/n1
• The numerical aperture, NA = (n1
2–n2
2)½
• The max acceptance angle, sin amax = NA/no
• Total acceptance angle is 2amax

9. Fig. 9
Cladding
Corea < am ax
A
B
q < qc
A
B
q > qc
a > am ax
n0 n1
n2
Lost
Propagates
Maximum acceptance angle
amax is that which just gives
total internal reflection at the
core-cladding interface, i.e.
when a = amax then q = qc.
Rays with a > amax (e.g. ray
B) become refracted and
penetrate the cladding and are
eventually lost.
Fiber axis
© 1999 S.O. Kasap,Optoelectronics (Prentice Hall)

10. V-number
• Number of propagating modes in a step-index optical
fibre can be determined from the V-number.
• Definition of V-number or normalized frequency
V= (2pa/l) (n1
2 – n2
2)½ = (2pa/l) (2n1
2D)½
– For V < 2.405, only one mode (LP01) can propagate
through the fibre core.
– For V >> 2.405, Number of modes M  V 2/2
– Note: D= (n1 – n2)/n1  (n1
2 – n2
2)/2n1
2

11. Example: A multimode fibre
• Calculate the number of allowed modes in
a multimode step index fibre that has a
core of refractive index of 1.468 and
diameter 100 mm, and a cladding of
refractive index of 1.447 if the source
wavelength is 850 nm.

12. Solution
• The V-number for this fibre:
V = (2pa/l) (n1
2 – n2
2)½
= (2p 50/0.850) (1.4682 – 1.4472)½
= 91.44
Since V >> 2.405, the number of modes is
M  V 2/2 = 91.442/2 = 4181

13. Single Mode Fibre
• A fibre that is designed to allow only the
fundamental mode (V < 2.405) to propagate at the
required wavelength is called a single mode fibre.
• Typically, single mode fibre have a much smaller core
radius & a smaller D.
– If the wavelength l of the source is further reduced, a
single mode fibre will become multimode as V >2.405.
– The cut-off wavelength lc above which the fibre becomes
single mode is given by
Vcut-off = 2pa/lc (n1
2 – n2
2)½ = 2.405

14. Multimode fibre
• In multimode fibre, light propagates through
many modes & these are mainly confined to the
core.
– Increasing the core radius or core refractive index n1
increases the number of modes
– Increasing the wavelength or cladding refractive
index n2 decreases the number of modes

15. Mode theory for circular waveguides
• To attain the field distribution pattern (the order of a
mode) in circular waveguide, it is necessary to solve
Maxwell equations.
• Planar waveguide modes
– Transverse Electric (TE)
– Transverse Magnetic (TM)
• Hybrid modes
– Transverse electric field larger than transverse magnetic
field (EH)
– Transverse magnetic field larger than transverse electric
field (HE)

16. Field Distribution in Step Index Fibre
• In a slab-dielectric waveguide, the guided rays zigzag
down the guide & all the rays pass through the axial
plane of the guide.
– So all waves were either TE or TM mode.
• In the step index fibre, the rays that zigzag down the
fibre might not crossing the fibre axis
• Two types of rays in step index fibre
1. Meridional ray that crosses the fibre axis results either TE
or TM mode as in the case of slab-dielectric waveguide
2. Skew ray that zigzags down the fibre without crossing the
axis give rise to modes that have both Ez and Bz, which are
called HE or EH hybrid modes.

17. Linearly Polarized (LP) Modes
• Guided modes in a step index fibre with D<<1 is called weakly
guiding fibres.
– It is generally visualized by travelling waves that are almost plane
polarized.
• LPlm mode can be described by a travelling wave along z of the
form
ELP = Elm (r,  ) exp j(w t – blmz)
in which blm is its propagation constant along z & (r,  , z) is a cylindrical
coordinate system
– l & m are related to the intensity pattern in a LPlm mode
– l represents the extent of helical propagation (skew ray contribution)
– m is directly associated with the reflection angle q of the rays as in the
planar guide.

18. Electric Field Distribution Pattern
of LP modes
• Fig.10(a) shows the electric field pattern (E01) in the
fundamental mode of the step index fibre (LP01 mode)
– The field is maximum at the centre of the core & penetrate
somewhat into the cladding
– Light intensity  E2: Fig 10(b) shows intensity distribution
• Fig.10(c) & (d) show the intensity distribution in the
LP11 and LP21 modes
– l is no of maxima around a circumference divided by 2
– m is no of maxima along r starting from the core centre

19. E
r
E01
Core
Cladding
The electric field distributionof the fundamental mode
inthe transverse plane to the fiber axis z. The light
intensity is greatest at the center of the fiber. Intensity
patterns in LP 01, LP11 and LP21 modes.
(a) The electric field
of the fundamental
mode
(b) The intensity in
the fundamental
mode LP01
(c) The intensity
inLP 11
(d) The intensity
inLP 21
© 1999 S.O. Kasap,Optoelectronics(Prentice Hall)
Fig. 10

20. Dispersion
• Fibre links are limited in path length by attenuation
and pulse distortion.
– When attenuation is the major problem, the system is said to
be power limited.
– If the power is sufficient but the signal shape is distorted due
to the dispersion effect; such systems are said to be
bandwidth limited.
• Dispersion can be divided into two groups
– Modal dispersion (intermodal dispersion)
– Chromatic dispersion (intramodal dispersion)
• Material dispersion
• Waveguide dispersion
• Profile dispersion

21. Dispersion effects in step-index fibre
1. Intermodal dispersion
2. Material dispersion
3. Waveguide dispersion
4. Chromatic dispersion
5. Profile dispersion
6. Polarization dispersion
7. Dispersion flattened fibres

22. 1. Intermodal dispersion
• When a light pulse is fed into the fibre, it travels along
the fibre through various modes of propagation
– Each mode having its own propagation vector blm & its own
electric field pattern Elm.
– Each mode has its own group velocity vg(l,m) that depends on
the w vs blm dispersion behaviour.
• Therefore, these modes emerge at the end of the fibre
with a spread of arrival times
– The output pulse is a broadened version of the input pulse.
– It is an intermodal dispersion phenomenon.

23. 2. Material Dispersion
• In a single mode, step-index fibre, there is no intermodal
dispersion of an input light pulse.
– But there will still be dispersion due to the variation of the core
refractive index, n1 , with wavelength of light coupled into the
fibre.
• Therefore, the propagation velocity of the guided wave
depends on the wavelength.
– The dispersion due to the wavelength dependence of the
material properties of the guide is called material dispersion.
• No practical light source is perfectly monochromatic

24. t
t
Spread,D t
t
0
l
Spectrum,D l
l1
l2
lo
Intensity Intensity Intensity
Cladding
Core
Emitter
Very short
light pulse
vg
(l2
)
vg
(l1
)
Input
Output
All excitation sources are inherently non-monochromatic and emit within a
spectrum, Dl of wavelengths. Waves in the guide with different free space
wavelengths travel at different group velocities due to the wavelength dependence
of n1. The waves arrive at the end of the fiber at different times and hence result in
a broadened output pulse.
Fig. 11

25. Broaden output pulse
due to material dispersion
• The output is a pulse that is broadened by Dt due to
the spread in the arrival time t of the waves.
• Material dispersion is expressed as







D=
D
2
2
index,refractivetheofderivativesecondby the
givenelyapproximatisitt,coefficiendispersionmaterialisin which
l
l
l
t
d
nd
c
D
D
D
L
m
m
m

26. Broadening and attenuation of two adjacent pulses as they travel along a
fibre. The overlapping of pulses limits the information capacity of a fibre.

27. 3. Waveguide Dispersion
• It is due to the dependence of the group velocity
vg(0,1) of the fundamental mode on the V-number,
which depends on the source wavelength l.
– Even if n1 and n2 were constant
– Even if no material dispersion, we would still have
waveguide dispersion that is due to the guiding properties
of the waveguide.
• A spectrum of source wavelengths will result in
different V-Number for each wavelength & hence
different propagation velocities.
– There will be a spread in the group delay times of the
fundamental-mode waves with different l.

28. Explanation Waveguide Dispersion
• The profile of the fibre has a very significant effect on the
group velocity.
– the E and M fields of light extend into the cladding.
• The amount that the fields overlap between core and cladding
depends strongly on the wavelength.
– The longer the wavelength the further the EM wave extends into the
cladding.
• n experienced by the wave is an average of the n of core and
cladding depending on the relative proportion of the wave that
travels there.
– Since a greater proportion of the wave at shorter wavelengths is
confined within the core, the shorter wavelengths “see” a higher RI
than do longer wavelengths.
– Therefore shorter wavelengths tend to travel more slowly than longer
ones and signals are dispersed.

29. Broaden output pulse
due to waveguide dispersion
• If we use a light pulse of very short duration as input with a
wavelength spectrum of Dl, the broadening per unit length in
output light pulse is
( )
2).(mediumcladdingthe
ofindicesrefractiveandgrouptheareandin which
22
984.1
byedapproximatnmkmpsingivenisit2.4,1.5
rangeover theand,propertieswaveguideon thedependsIt
t.coefficiendispersionwaveguidetheisin which
22
2
2
2
2
11
nN
cna
N
D
V
D
D
L
w
--
w
w
g
g
p
l
t
=
<<
D=
D

30. 4. Chromatic Dispersion
or Total Dispersion
• In single mode fibres the dispersion of a propagating
pulse arises because of finite width Dl of the source
spectrum.
– It is not perfectly monochromatic source
• The dispersion caused by a range of source
wavelengths is generally termed as chromatic
dispersion.
– It includes both material & waveguide dispersion.
– As a first approximation, the two dispersion effect can be
simply added as
Dt/L = Dm + DwDl
in which Dch = Dm + Dw is the chromatic dispersion

31. 0
1.2 1.3 1.4 1.5 1.61.1
-30
20
30
10
-20
-10
l (mm)
Dm
Dm + Dw
Dw
l0
Dispersion coefficient (ps km -1 nm-1)
Material dispersion coefficient (Dm) for the core material (taken as
SiO2), waveguide dispersion coefficient (Dw) (a = 4.2 mm) and the
total or chromatic dispersion coefficient Dch (= Dm + Dw) as a
function of free space wavelength, l.
© 1999 S.O. Kasap,Optoelectronics(Prentice Hall)

32. 5. Profile and Polarization Effects
• Profile dispersion arises because the group velocity,
vg(01), of the fundamental mode also depends on the
refractive index difference D=D(l).
– If D changes with wavelength, different wavelengths would
have different group velocities & experience different group
delays leading to pulse broadening
• It is part of chromatic dispersion,
Dt/L = DpDl
in which Dp is the profile dispersion coefficient
• The overall chromatic dispersion coefficient becomes
Dch = Dm + Dw + Dp
– Dp < 1 ps nm–1 km–1 ,negligible compared with Dw

33. 6. Polarization dispersion
• It arises when the fibre is not perfectly symmetric and
homogenous
– that is the refractive index is not isotropic.
– It is due to various variations in the fabrication process such as small
changes in the glass composition, geometry & induced local strains
(either during fibre drawing or cabling)
• The refractive index depends on the direction of the electric
field
– the propagation constant of a given mode depends its polarization
• Suppose n1 has the values n1x and n1y when the electric field is
parallel to the x & y axes respectively.
– The propagation constant for fields along x and y would be different,
bx(01) and by(01).
– It leads to different group delays and dispersion

34. Core
z
n1x
// x
n1y
// y
Ey
Ex
Ex
Ey
E
Dt = Pulse spread
Input light pulse
Output light pulse
t
t
Dt
Intensity
Suppose that the core refractive index has different values along two orthogonal
directions corresponding to electric field oscillation direction (polarizations). We can
take x and y axes along these directions. An input light willtravel along the fiber with Ex
and Ey polarizations having different group velocities and hence arrive at the output at
different times
© 1999 S.O. Kasap, Optoelectronics(Prentice Hall)
Fig. 12

35. 7.Dispersion Flattened Fibres
• The doping of the core material to shift material dispersion
(Dm) and hence overall dispersion to longer wavelength results
in an increased attenuation of the signal.
– It is desirable to have minimal dispersion over a range of wavelength
• Waveguide dispersion (Dw) can be adjusted by changing the
waveguide geometry
– We can alter the waveguide geometry (refractive index profile) to result
a total chromatic dispersion that is flattened between two wavelengths
l1 & l2.
• Fig.13 shows the dispersion flattened fibre.
– The refractive index profile of such a fibre looks like a W (doubly clad)

36. 20
-10
-20
-30
10
1.1 1.2 1.3 1.4 1.5 1.6 1.7
0
30
l (mm)
Dm
Dw
Dch =D m + Dw
l1
Dispersion coefficient (ps km -1 nm-1)
l2
n
r
Thin layer of cladding
with a depressed index
Dispersion flattened fiber example. The materialdispersioncoefficient ( Dm) for the
core material and waveguide dispersioncoefficient ( Dw) for the doublyclad fiber
result ina flattened smallchromatic dispersion between l1 and l2.
© 1999 S.O. Kasap,Optoelectronics(Prentice Hall)
Fig. 13:

37. Announcement
- Submission date for Group A Lab report is on 22 Feb 2013
- Lab session for Group B is on 20 Feb 2013
- Assignment 2 will be held on 21 Feb 2013

38. Bit rate and Dispersion
• In digital communication, signal are generally sent as
light pulses along an optical fibre
– Information is first converted to an electrical signal in the form
of pulses
– The pulses represent bits of information in digital form
– The pulses is very short with a well defined duration
• The electrical signal drives a light emitter whose light is
coupled into a fibre for transmission
– At the destination end of the fibre, the light is coupled to a
photodetector to convert optical signal back to electric signal
– The information is then decoded from this electrical signal

39. Bit rate capacity
• Digital communications engineers are
interested in the maximum rate at which
the digital data can be transmitted along
the fibre.
– The rate is called the bit rate capacity, B (bits
per second) of the fibre
– It is directly related to the dispersion
characteristics.

40. Full width at half power (FWHP)
• If a light pulse is fed into the fibre, the output pulse will be
delayed by the transit time t.
• Due to various dispersion mechanism, there will be a spread Dt
in the arrival times of different guided waves.
– The dispersion is measured between half-power points & is called full
width at half-power (FWHP) or full width at half-maximum (FWHM), Dt
= Dt½.
• To clearly distinguish between two consecutive output pulses (no
intersymbol interference), the time-separate from peak to peak
is at least 2Dt½
– So we can only feed in pulses at every 2Dt½ seconds
– Thus the maximum bit rate B is roughly 1/(2Dt½).
B  0.5/(Dt½). [10]

41. Fig. 14:
t
0
Emitter
Very short
light pulses
Input Output
Fibre
Photodetector
Digital signal
Information Information
t
0
~2Dt1/2
T
t
Output IntensityInput Intensity
Dt1/2
An optical fibre link for transmitting digital information and the effect of
dispersion in the fibre on the output pulses.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

42. Return-to-zero bit rate
• The maximum bit rate B assumes a pulse
representing the binary information 1 must return to
zero for a duration before the next binary
information.
• Two consecutive binary 1 pulses have a zero in
between as in the output pulses shown in Fig. 14
– The bit rate is called the return-to-zero bit rate (RZ)

43. Nonreturn-to-zero bit rate
• It is also possible to send two consecutive binary 1
pulses without having to return to zero at the end
of each 1-pulse
– Two 1-pulses are immediately next to each other
• Two pulses in Fig. 15 can be brought closer until the
repetition period T Dt½
– The signal is nearly uniform over the length of these two
consecutive 1
• Such a maximum data rate is called nonreturn-to-
zero bit rate (NRZ)
• NRZ bit rate is twice the RZ bit rate.

44. Analysis for RZ transmission
• For a more rigorous analysis, the temporal shape of
signal & the criterion for discerning the information
should be known.
• For a Gaussian output light pulse, tolerable
interference between two consecutive light output
pulses is 4s between their peaks
• Thus, the bit rate should be B  0.25/s
• Given s = 0.425Dt½ , B = 0.59/Dt½ .
– This is ~18% greater than the intuitive estimation in eqn[10]

45. t
Output optical power
Dt1/2
T = 4s1
0.5
0.61 2s
A Gaussian output light pulse and some tolerable intersymbol
interference between two consecutive output light pulses ( y-axis in
relative units). At time t = s from the pulse center, the relative
magnitude is e-1/2 = 0.607 and full width root mean square (rms)
spread is Dtrms = 2s.
© 1999 S.O. Kasap,Optoelectronics(Prentice Hall)
Fig. 15:

46. BL product
• RMS spread of wavelengths in the Gaussian
output spectrum, sl = 0.425Dl½
• RMS dispersion of output pulse, s = LDchsl
• The bit rate  distance product,
BL  0.25L/s = 0.25/ (sl Dch)

47. Intramodal & Intermodal Dispersions
• For intramodal dispersion such as material &
waveguide dispersion, the net effect is simply the
linear addition of two dispersion coefficients,
Dch=Dm+ Dw
• For combining intramodal and intermodal
dispersions, overall dispersion must be found
from individual rms dispersion as
s 2 = s 2
intermodal + s 2
intramodal

48. Pulse shape and Bit rate
• To determine B from Dt½, we need to know
the pulse shape.
• For the rectangular pulse,
– the full width DT = Dt½
– B = 0.25/s = 0.87/ DT = 0.87/ Dt½
• For an ideal Gaussian pulse,
– s = 0.425Dt½
– B = 0.25/s = 0.59/Dt½

49. Optical Bandwidth
• The emitter can be driven using a sinusoidal signal as in Fig.16
– The input light intensity will be modulated at the same frequency as the
driving signal.
– The output light intensity should be a sinusoidal with a phase shift due to
the time it takes to travel along the fibre.
• Fig.16 shows the optical transfer characteristic of the fibre
– It is defined as output light power (Po) per unit input light power (Pi) as a
function of modulation frequency (f)
– The response is flat and falls to 50% below the flat region at frequency fop.
– fop is defined as optical bandwidth of the fibre.
– Intuitively fop= B, but it is not true because B can tolerate pulse
overlapping.
– For Gaussian dispersion, Optical bandwidth fop  0.75B  0.19/s

50. Electrical Bandwidth
• Electrical signal from the photodetector does not
exhibit the same bandwidth as optical signal.
– Bandwidth for electrical signal fel is measured where the
signal is 70.7% of its low frequency value.
• The relationship between fel & fop depends on the
dispersion through the fibre
– For Gaussian dispersion, fel  0.71 fop
– For Rectangular dispersion with full width DT,
fel  0.73 fop

51. t
0
Pi = Input light power
Emitter
Optical
Input
Optical
Output
Fiber
Photodetector
Sinusoidal signal
Sinusoidal electrical signalt
t
0
f
1 kHz 1 MHz 1 GHz
Po / Pi
fop
0.1
0.05
f = Modulation frequency
An optical fiber link for transmitting analog signals and the effect of dispersion in the
fiber on the bandwidth, fop.
Po = Output light power
Electrical signal(photocurrent)
fel
1
0.707
f
1 kHz 1 MHz 1 GHz
© 1999 S.O. Kasap, Optoelectronics(Prentice Hall)
Fig. 16: