This document discusses optical waveguides and fiber optic modes. It begins by describing the mode patterns seen in the end faces of small diameter fibers. It then discusses multimode propagation and explains that many modes are excited, resulting in complex field and intensity patterns. Finally, it summarizes the key parameters and solutions used to determine the modes in cylindrical optical fibers.

1. UNIT-2
Mohammad Asif Iqbal
Assistant Professor,
Deptt of ECE,
JETGI, Barabanki

2. Optical Waveguide mode patterns
Optical Waveguide mode
patterns seen in the
end faces of small
diameter fibers
Optics-Hecht & Zajac Photo by Narinder Kapany

3. Multimode Propagation
In general many modes
are excited in the guide
resulting in complicated
field and intensity
patterns that evolve in a
complex way as the light
propagates down the guide

4. Planar Mirror Waveguide
The planar mirror waveguide can be
solved by starting with Maxwells Equations
and the boundary condition that the
parallel component of the E field vanish
at the mirror or by considering that plane
waves already satisfy Maxwell’s equations
and they can be combined at an angle so that
the resulting wave duplicates itself

5. Mode Components Number and Fields

6. Mode Velocity and Polarization Degeneracy
Group Velocity derived
by considering the mode
from the view of rays and
geometrical optics
TE and TM mode polarizations

7. Planar Dielectric guide
Characteristic
Equation and Self-Consistency
Condition
Propagation Constants
Number of
modes vs
frequency
Geometry of Planar Dielectric Guide
The bm all lie between that
expected for a plane wave in the
core and for a plane wave in the
cladding
For a sufficiently low
frequency only 1 mode
can propagate
Fundamentals of Photonics - Saleh and Teich

8. Planar Dielectric Guide
Field components have transverse variation
across the guide, with more nodes for
higher order modes. The changed
boundary conditions for the dielectric
interface result in some evanescent
penetration into the cladding
The ray model can be used for dielectric guides
if the additional phase shift due to the
evanescent wave is accounted for.

9. Two Dimensional Rectangular Planar Guide
In two dimensions the transverse field depends on both kx and ky and
the number of modes goes as the square of d/l The number of modes is
limited by the maximum angle that can propagate qc

10. Modes in cylindrical optical fiber
• Determined by solving Maxwell’s equations in cylindrical
coordinates
0
11 2
2
2
22
2









z
zzz
Eq
E
rr
E
rr
E

0
11 2
2
2
22
2









z
zzz
Hq
H
rr
H
rr
H


11. Key parameters
• q2 is equal to ω2εμ-β2 = k2 – β2. It is sometimes
called u2.
• β is the z component of the wave propagation
constant k, which is also equal to 2π/λ. The
equations can be solved only for certain values
of β, so only certain modes may exist. A mode
may be guided if β lies between nCLk and nCOk.
• V = ka(NA) where a is the radius of the fiber
core. This “normalized frequency” determines
how many different guided modes a fiber can
support.

12. Solutions to Wave Equations
• The solutions are separable in r, φ, and z. The φ
and z functions are exponentials of the form eiθ.
The z function oscillates in space, while the φ
function must have the same value at (φ+2π)
that it does at φ.
• The r function is a combination of Bessel
functions of the first and second kinds. The
separate solutions for the core and cladding
regions must match at the boundary.

13. Resulting types of modes
• Either the electric field component (E) or the magnetic
field component (H) can be completely aligned in the
transverse direction: TE and TM modes.
• The two fields can both have components in the
transverse direction: HE and EH modes.
• For weakly guiding fibers (small delta), the types of
modes listed above become degenerate, and can be
combined into linearly polarized LP modes.
• Each mode has a subscript of two numbers, where the
first is the order of the Bessel function and the second
identifies which of the various roots meets the
boundary condition. If the first subscript is 0, the mode
is meridional. Otherwise, it is skew.

14. Mode characteristics
Each mode has a specific
• Propagation constant β
• Spatial field distribution
• Polarization

15. w-b Mode Diagram
Straight lines of dw/db correspond to the group velocity of the different modes
The group velocities of the guided modes all lie between the phase velocities for
plane waves in the core or cladding c/n1 and c/n2

16. Step Index Cylindrical Guide

17. High Order Fiber modes

18. High Order Fiber Modes 2

19. The Cutoff
• For each mode, there is some value of V below which
it will not be guided because the cladding part of the
solution does not go to zero with increasing r.
• Below V=2.405, only one mode (HE11) can be guided;
fiber is “single-mode.”
• Based on the definition of V, the number of modes is
reduced by decreasing the core radius and by
decreasing ∆.

20. Number of Modes
Graphical Construction
to estimate the
total number of Modes
Propagation constant of the lowest
mode vs. V number

21. Number of Modes—Step Index Fiber
• At low V, M4V2/π2+2
• At higher V, MV2/2

22. Graded-index Fiber
For r between 0 and a.
Number of modes is
 








a
r
nrn 211
  


2
1
2
aknM



23. Comparison of the number of modes
0
2
d
M NA
l
»
0
2
d
M
l
=
2
0
2
4
d
M
p
l
æ ö÷ç ÷» ç ÷ç ÷çè ø
2
2
2
0
4
16
d
M V NA
p l
æ ö÷ç ÷» = ç ÷ç ÷çè ø
2
0
2
4
d
M NA
p
l
æ ö÷ç ÷» ç ÷ç ÷çè ø
1-d Mirror Guide
1-d Dielectric Guide
2-d Mirror Guide
2-d Dielectric Guide
2-d Cylindrical Dielectric Guide 0
a
V=2
λ
NAp
The V parameter
characterizes the number of
wavelengths that can fit across
the core guiding region in a fiber.
For the mirror guide the number of
modes is just the number of ½
wavelengths that can fit.
For dielectric guides it is the numbe
that can fit but now limited by the
angular cutoff characterized by the
NA of the guide

24. Power propagating through core
• For each mode, the shape of the Bessel
functions determines how much of the optical
power propagates along the core, with the rest
going down the cladding.
• The effective index of the fiber is the weighted
average of the core and cladding indices, based
on how much power propagates in each area.
• For multimode fiber, each mode has a different
effective index. This is another way of
understanding the different speed that optical
signals have in different modes.

25. Total energy in cladding
The total average power propagating in the
cladding is approximately equal to
MP
Pclad
3
4


26. Power Confinement vs V-Number
This shows the fraction of the power
that is propagating in the cladding
vs the V number for different modes.
V, for constant wavelength, and
material
indices of refraction is proportional to
the core diameter a
As the core diameter is dereased more
and more of each mode propagates in
the cladding. Eventually it all
propagates
in the cladding and the mode is no
longer
guided
Note: misleading ordinate lable

27. Macro bending Loss
One thing that the geometrical ray view point cannot calculate is the amount of bending loss
encountered by low order modes. Loss goes approximately exponentially with decreasing radius
until a discontinuity is reached….when the fiber breaks!

28. Signal Attenuation & Distortion in
Optical Fibers
• What are the loss or signal attenuation mechanism in a
fiber?
• Why & to what degree do optical signals get distorted as
they propagate down a fiber?
• Signal attenuation (fiber loss) largely determines the
maximum repeaterless separation between optical
transmitter & receiver.
• Signal distortion cause that optical pulses to broaden as
they travel along a fiber, the overlap between neighboring
pulses, creating errors in the receiver output, resulting in
the limitation of information-carrying capacity of a fiber.

29. Attenuation (fiber loss)
• Power loss along a fiber:
• The parameter is called fiber attenuation coefficient in a units of for
example [1/km] or [nepers/km]. A more common unit is [dB/km] that
is defined by:
Z=0
P(0) mW
Z= l
lp
ePlP

 )0()( mw
zp
ePzP

 )0()( [3-1]
p
]km/1[343.4
)(
)0(
log
10
]dB/km[ p
lP
P
l
 





 [3-2]

30. Fiber loss in dB/km
• Where [dBm] or dB milliwat is 10log(P [mW]).
z=0 Z=l
]dBm)[0(P
]km[]dB/km[]dBm)[0(]dBm)[( lPlP   [3-3]

31. Optical fiber attenuation vs. wavelength

32. Absorption
• Absorption is caused by three different mechanisms:
1- Impurities in fiber material: from transition metal ions
(must be in order of ppb) & particularly from OH ions with
absorption peaks at wavelengths 2700 nm, 400 nm, 950 nm
& 725nm.
2- Intrinsic absorption (fundamental lower limit): electronic
absorption band (UV region) & atomic bond vibration band
(IR region) in basic SiO2.
3- Radiation defects

33. Scattering Loss
• Small (compared to wavelength) variation in material density, chemical
composition, and structural inhomogeneity scatter light in other directions
and absorb energy from guided optical wave.
• The essential mechanism is the Rayleigh scattering. Since the black body
radiation classically is proportional to (this is true for wavelength
typically greater than 5 micrometer), the attenuation coefficient due to
Rayleigh scattering is approximately proportional to . This seems to me
not precise, where the attenuation of fibers at 1.3 & 1.55 micrometer can be
exactly predicted with Planck’s formula & can not be described with
Rayleigh-Jeans law. Therefore I believe that the more accurate formula for
scattering loss is
4

4

1
5
)exp(









Tk
hc
B
scat


eTemperatur:,JK103806.1Js,10626.6 -12334
Tkh B



34. Absorption & scattering losses in fibers

35. Typical spectral absorption & scattering
attenuations for a single mode-fiber

36. Bending Loss (Macrobending & Microbending)
• Macrobending Loss: The
curvature of the bend is much
larger than fiber diameter.
Lightwave suffers sever loss due
to radiation of the evanescent
field in the cladding region. As
the radius of the curvature
decreases, the loss increases
exponentially until it reaches at a
certain critical radius. For any
radius a bit smaller than this
point, the losses suddenly
becomes extremely large. Higher
order modes radiate away faster
than lower order modes.

37. Microbending Loss
• Microbending Loss:
microscopic bends of the fiber
axis that can arise when the
fibers are incorporated into
cables. The power is dissipated
through the microbended fiber,
because of the repetitive
coupling of energy between
guided modes & the leaky or
radiation modes in the fiber.

38. Dispersion in Optical Fibers
• Dispersion: Any phenomenon in which the velocity of propagation of any
electromagnetic wave is wavelength dependent.
• In communication, dispersion is used to describe any process by which any
electromagnetic signal propagating in a physical medium is degraded
because the various wave characteristics (i.e., frequencies) of the signal
have different propagation velocities within the physical medium.
• There are 3 dispersion types in the optical fibers, in general:
1- Material Dispersion
2- Waveguide Dispersion
3- Polarization-Mode Dispersion
Material & waveguide dispersions are main causes of Intramodal
Dispersion.

39. Group Velocity
• Wave Velocities:
• 1- Plane wave velocity: For a plane wave propagating along z-axis in an
unbounded homogeneous region of refractive index , which is
represented by , the velocity of constant phase plane is:
• 2- Modal wave phase velocity: For a modal wave propagating along z-axis
represented by , the velocity of constant phase plane is:
3- For transmission system operation the most important & useful type of
velocity is the group velocity, . This is the actual velocity which the
signal information & energy is traveling down the fiber. It is always less
than the speed of light in the medium. The observable delay experiences by
the optical signal waveform & energy, when traveling a length of l along the
fiber is commonly referred to as group delay.
1n
)ωexp( 1zjktj 
11 n
c
k
v 

)ωexp( zjtj 

ω
pv
[3-4]
[3-5]
gV

40. Group Velocity & Group Delay
• The group velocity is given by:
• The group delay is given by:
• It is important to note that all above quantities depend both on frequency &
the propagation mode. In order to see the effect of these parameters on
group velocity and delay, the following analysis would be helpful.
d
d
Vg
ω
 [3-6]
ωd
d
l
V
l
g
g

  [3-7]

41. Input/Output signals in Fiber Transmission
System
• The optical signal (complex) waveform at the input of fiber of length l is
f(t). The propagation constant of a particular modal wave carrying the signal
is . Let us find the output signal waveform g(t).)ω(
z-=0 Z=l








c
c
deftf tj
)(
~
)( [3-8]









c
c
deftg ljtj )(
)(
~
)( [3-9]
bandwidth.signalopticaltheis

42. ...)(
2
1
)()()(
If
2
2
2



ccc
c
cc
d
d
d
d









)()(
)(
~
)(
~
)(
~
)(
)()(
)(2/
2/
)(
)]()([2/
2/
)(
2/
2/
g
ljlj
d
d
ltj
lj
l
d
d
jtj
ljtj
tfe
d
d
ltfe
defe
defdeftg
c
c
c
c
c
c
c
c
c
cc
c
c
c









































g
g
V
l
d
d
l
c




[3-10]
[3-11]
[3-14]

43. Intramodal Dispersion
• As we have seen from Input/output signal relationship in optical fiber, the
output is proportional to the delayed version of the input signal, and the
delay is inversely proportional to the group velocity of the wave. Since the
propagation constant, , is frequency dependent over band width
sitting at the center frequency , at each frequency, we have one
propagation constant resulting in a specific delay time. As the output signal
is collectively represented by group velocity & group delay this
phenomenon is called intramodal dispersion or Group Velocity
Dispersion (GVD). This phenomenon arises due to a finite bandwidth
of the optical source, dependency of refractive index on the wavelength
and the modal dependency of the group velocity.
• In the case of optical pulse propagation down the fiber, GVD causes pulse
broadening, leading to Inter Symbol Interference (ISI).
)ω( ω
cω

44. Dispersion & ISI
A measure of information capacity of
an optical fiber for digital
transmission is usually specified by
the bandwidth distance product
in GHz.km.
For multi-mode step index fiber this
quantity is about 20 MHz.km, for
graded index fiber is about 2.5
GHz.km & for single mode fibers are
higher than 10 GHz.km.
LBW 

45. How to characterize dispersion?
• Group delay per unit length can be defined as:
• If the spectral width of the optical source is not too wide, then the delay
difference per unit wavelength along the propagation path is approximately
For spectral components which are apart, symmetrical around center
wavelength, the total delay difference over a distance L is:




d
d
cdk
d
cd
d
L
g
2
1
ω
2
 [3-15]


d
d g













































2
2
2
2
2
2
2
d
d
L
V
L
d
d
d
d
d
d
d
d
c
L
d
d
g
g
[3-16]

46. • is called GVD parameter, and shows how much a light pulse
broadens as it travels along an optical fiber. The more common parameter is
called Dispersion, and can be defined as the delay difference per unit length
per unit wavelength as follows:
• In the case of optical pulse, if the spectral width of the optical source is
characterized by its rms value of the Gaussian pulse , the pulse
spreading over the length of L, can be well approximated by:
• D has a typical unit of [ps/(nm.km)].
2
2
2



d
d

22
211




 c
Vd
d
d
d
L
D
g
g









 [3-17]

g
 


 DL
d
d g
g  [3-18]

47. 
t
Spread, ² 
t
0

Spectrum, ² 
1
2o
Intensity Intensity Intensity
Cladding
Core
Emitter
Very short
light pulse
vg(2)
vg(1)
Input
Output
All excitation sources are inherentlynon-monochromatic and emit within a
spectrum, ² , of wavelengths. Waves in the guide with different free space
wavelengths travelat 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.
© 1999 S.O. Kasap,Optoelectronics(Prentice Hall)
Material Dispersion

48. Material Dispersion
• The refractive index of the material varies as a function of wavelength,
• Material-induced dispersion for a plane wave propagation in homogeneous
medium of refractive index n:
• The pulse spread due to material dispersion is therefore:
)(n
























d
dn
n
c
L
n
d
d
L
cd
d
L
cd
d
Lmat )(
2
22ω
22
[3-19]
)(2
2







 

 mat
mat
g DL
d
nd
c
L
d
d
 [3-20]
)(matD is material dispersion

49. Material Dispersion Diagrams

50. Waveguide Dispersion
• Waveguide dispersion is due to the dependency of the group velocity of the
fundamental mode as well as other modes on the V number, (see Fig 2-18 of
the textbook). In order to calculate waveguide dispersion, we consider that n
is not dependent on wavelength. Defining the normalized propagation
constant b as:
• solving for propagation constant:
• Using V number:
21
2
2
2
2
1
2
2
22
//
nn
nk
nn
nk
b






 [3-21]
)1(2  bkn [3-22]
 2)( 2
2/12
2
2
1 kannnkaV [3-23]

51. Waveguide Dispersion
• Delay time due to waveguide dispersion can then be expressed as:





dV
Vbd
nn
c
L
wg
)(
22 [3-24]

52. Waveguide dispersion in single mode fibers
• For single mode fibers, waveguide dispersion is in the same order of
material dispersion. The pulse spread can be well approximated as:
2
2
2 )(
)(
dV
Vbd
V
c
Ln
DL
d
d
wg
wg
wg





 


 [3-25]
)(wgD

53. Polarization Mode dispersion
Core
z
n1x
// x
n1y
// y
Ey
Ex
Ex
Ey
E
 = Pulse spread
Input light pulse
Output light pulse
t
t

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)

54. Polarization Mode dispersion
• The effects of fiber-birefringence on the polarization states of an optical are
another source of pulse broadening. Polarization mode dispersion (PMD)
is due to slightly different velocity for each polarization mode because of
the lack of perfectly symmetric & anisotropicity of the fiber. If the group
velocities of two orthogonal polarization modes are then the
differential time delay between these two polarization over a
distance L is
• The rms value of the differential group delay can be approximated as:
gygx vv and
pol
gygx
pol
v
L
v
L
 [3-26]
LDPMDpol  [3-27]

55. Chromatic & Total Dispersion
• Chromatic dispersion includes the material & waveguide dispersions.
• Total dispersion is the sum of chromatic , polarization dispersion and other
dispersion types and the total rms pulse spreading can be approximately
written as:


LD
DDD
chch
wgmatch
)(
)(


[3-28]
 LD
DDD
totaltotal
polchtotal

 ...
[3-29]

56. Total Dispersion, zero Dispersion
Fact 1) Minimum distortion at wavelength about 1300 nm for single mode silica fiber.
Fact 2) Minimum attenuation is at 1550 nm for sinlge mode silica fiber.
Strategy: shifting the zero-dispersion to longer wavelength for minimum attenuation and dispersion.

57. Optimum single mode fiber &
distortion/attenuation characteristics
Fact 1) Minimum distortion at wavelength about 1300 nm for single mode
silica fiber.
Fact 2) Minimum attenuation is at 1550 nm for sinlge mode silica fiber.
Strategy: shifting the zero-dispersion to longer wavelength for minimum
attenuation and dispersion by Modifying waveguide dispersion by
changing from a simple step-index core profile to more complicated
profiles. There are four major categories to do that:
1- 1300 nm optimized single mode step-fibers: matched cladding (mode
diameter 9.6 micrometer) and depressed-cladding (mode diameter about 9
micrometer)
2- Dispersion shifted fibers.
3- Dispersion-flattened fibers.
4- Large-effective area (LEA) fibers (less nonlinearities for fiber optical
amplifier applications, effective cross section areas are typically greater
than 100 ).2
m

59. Single mode fiber dispersion

60. Single mode fiber dispersion

61. Single mode Cut-off wavelength & Dispersion
• Fundamental mode is with V=2.405 and
• Dispersion:
• For non-dispersion-shifted fibers (1270 nm – 1340 nm)
• For dispersion shifted fibers (1500 nm- 1600 nm)
0111 LPorHE 2
2
2
1
2
nn
V
a
c 


[3-30]





LD
DD
d
d
D wgmat
)(
)()()(

 [3-31]
[3-32]

62. Dispersion for non-dispersion-shifted fibers
(1270 nm – 1340 nm)
• is relative delay minimum at the zero-dispersion wavelength , and
is the value of the dispersion slope in .
2
2
00
0 )(
8
)(


 
S
0 0 0S
.km)ps/(nm2
0
)( 00




d
dD
SS
[3-33]
[3-34]






 400
)(1
4
)(



S
D [3-35]

63. Dispersion for dispersion shifted fibers (1500
nm- 1600 nm)
2
0
0
0 )(
2
)(  
S
00 )()( SD  
[3-36]
[3-37]

64. Example of dispersion
Performance curve for
Set of SM-fiber

65. Example of BW vs wavelength for various optical sources for
SM-fiber.

66. MFD

67. Bending Loss

68. Bending effects on loss vs MFD

69. Bend loss versus bend radius
07.0;1056.3
m60;m6.3
2
233





n
nn
ba 

70. Kerr effect
Innn 20  Kerr nonlinearity in fiber, where I is the intensity of
Optical wave.
Temporal changes in a narrow optical pulse that is subjected to Kerr nonlinearity in
A dispersive medium with positive GVD.
Kerr effect

71. First-order Soliton
Temporal changes in a medium with Kerr nonlinearity and negative GVD. Since dispersion tends to broaden
the pulse, Kerr Nonlinearity tends to squeeze the pulse, resulting in a formation of optical soliton.

72. THANK YOU!