1

4. Optical Fibers

2

Anatomy of an Optical Fiber

• Light confined to core with higher index of refraction

• Two analysis approaches

– Ray tracing

– Field propagation using Maxwell’s equations

3

Optical Fiber Analysis

• Calculation of modes supported by an optical fiber

– Intensity profile

– Phase propagation constant

• Effect of fiber on signal propagation

– Signal attenuation

– Pulse spreading through dispersion

4

Critical Angle

• Ray bends at boundary between materials

– Snell’s law

• Light confined to core if propagation angle is greater than the critical angle

– Total internal reflection (TIR)

11

212 sinsin n

n1122 sinsin nn

1

211 sin n

nc

5

Constructive Interference

• Propagation requires constructive interference

– Wave stays in phase after multiple reflections

– Only discrete angles greater than the critical angle are allowed to propagate

6

Numerical Aperture

• The acceptance angle for a fiber defines its numerical aperture (NA)

• The NA is related to the critical angle of the waveguide and is defined as:

• Telecommunications optical fiber n1~n2,

22

21sin nnNA i

1

21212121

22

21 2

n

nnnnnnnnnNA

21nNA1

21

n

nn

7

Modes

• The optical fiber support a set of discrete modes

• Qualitatively these modes can be thought of as different propagation angles

• A mode is characterized by its propagation constant in the z-direction z

• With geometrical optics this is given by

• The goal is to calculate the value of βz

• Remember that the range of βz is

izi kn sin01

ozo knkn 12

8

Optical Fiber Modes

• The optical fiber has a circular waveguide instead of planar

• The solutions to Maxwell’s equations

– Fields in core are non-decaying

• J, Y Bessel functions of first and second kind

– Fields in cladding are decaying

• K modified Bessel functions of second kind

• Solutions vary with radius r and angle • There are two mode number to specify the mode

– m is the radial mode number

– is the angular mode number

9

Bessel Functions

10

Transcendental Equation

• Under the weakly guiding approximation (n1-n2)<<1

– Valid for standard telecommunications fibers

• Substitute to eliminate the derivatives

aγK

aγKγa

akJ

akJak

l

'l

Tl

T'l

T

2221

2 oT knk 222

22okn

x

xJlxJxJ l

ll 1'

x

xKlxKxK l

ll 1'

aK

aKa

akJ

akJakor

aK

aKa

aκJ

akJak

l

l

Tl

TlT

l

l

l

TlT

1111

HE ModesEH Modes

11

Bessel Function Relationships• Bessel function recursive relationships

• Small angle approximations

xJxJ nn

n 1

xKxK nn

xJxJx

nxJ nnn 11

2

xJxJx

xJ 210

2

xKxKx

nxK nnn 11

2

xKxKx

xK 210

2

0

2

2

!1

00.57722

ln

nx

n

nx

xK nn

12

Lowest Order Modes• Look at the l=-1, 0, 1 modes

• Use bessel function properties to get positive order and highest order on top

• l=-1

• l=0

yK

yKy

xJ

xJx

1

2

1

2

yK

yKy

xJ

xJx

1

2

1

2

yK

yKy

xJ

xJx

1

0

1

0

yK

yKy

xJ

xJx

0

1

0

1

yK

yKy

xJ

xJx

0

1

0

1

yK

yKy

xJ

xJx

0

1

0

1

yK

yKy

xJ

xJx

1

2

1

2

akx T ay 22

21

22 2nnayxV

13

Lowest Order Modes cont.• l=+1

• So the 6 equations collapse down to 2 equations

yK

yKy

xJ

xJx

1

0

1

0

yK

yKy

xJ

xJx

1

2

1

2

yK

yKy

xJ

xJx

1

2

1

2

yK

yKy

xJ

xJx

1

2

1

2

yK

yKy

xJ

xJx

0

1

0

1

lowest modes

14

Modes

0 2 4 6 8 100

5

10

15

20

x (kT a)

LHS

, RH

S

15

Fiber Modes

16

Hybrid Fiber Modes

• The refractive index difference between the core and cladding is very small

• There is degeneracy between modes

– Groups of modes travel with the same velocity (z equal)

• These hybrid modes are approximated with nearly linearly polarized modes called LP modes

– LP01 from HE11

– LP0m from HE1m

– LP1m sum of TE0m, TM0m, and HE2m

– LPm sum of HE+1,m and EH-1,m

17

First Mode Cut-Off• First mode

– What is the smallest allowable V

– Let y 0 and the corresponding x V

– So V=0, no cut-off for lowest order mode

– Same as a symmetric slab waveguide

0

5772.02

ln

221

limlim0

0

1

00

1

y

yy

yK

yKy

VJ

VJV

yy

01 VJ

18

Second Mode Cut-Off• Second mode

VJV

VJ 12

2

2

221

221

limlim

2

01

2

01

2

y

yy

yK

yKy

VJ

VJV

yy

xJxJx

nxJ nnn 11

2

xJxJx

xJ 012

2

VJV

VJVJV o 11

22

0VJo

405.2V

19

Cut-off V-parameter for low-order LPlm modes

m=1 m=2 m=3

l=0 0 3.832 7.016

l=1 2.405 5.520 8.654

20

Number of Modes

• The number of modes can be characterized by the normalized frequency

• Most standard optical fibers are characterized by their numerical aperture

• Normalized frequency is related to numerical aperture

• The optical fiber is single mode if V<2.405

• For large normalized frequency the number of modes is approximately

22

21

2nnaV

22

21 nnNA

NAaV2

14

Modes# 22

VV

21

Intensity Profiles

22

Standard Single Mode Optical Fibers

• Most common single mode optical fiber: SMF28 from Corning

– Core diameter dcore=8.2 m

– Outer cladding diameter: dclad=125m

– Step index

– Numerical Aperture NA=0.14

• NA=sin()

• =8°

• cutoff = 1260nm (single mode for cutoff)

• Single mode for both =1300nm and =1550nm standard telecommunications wavelengths

23

Standard Multimode Optical Fibers

• Most common multimode optical fiber: 62.5/125 from Corning

– Core diameter dcore= 62.5 m

– Outer cladding diameter: dclad=125m

– Graded index

– Numerical Aperture NA=0.275

• NA=sin()

• =16°

• Many modes

24

5. Optical Fibers Attenuation

25

Coaxial Vs. Optical Fiber Attenuation

26

Fiber Attenuation

• Loss or attenuation is a limiting parameter in fiber optic systems

• Fiber optic transmission systems became competitive with electrical transmission lines only when losses were reduced to allow signal transmission over distances greater than 10 km

• Fiber attenuation can be described by the general relation:

where is the power attenuation coefficient per unit length

• If Pin power is launched into the fiber, the power remaining after propagating a length L within the fiber Pout is

PdzdP

LPP inout exp

27

Fiber Attenuation

• Attenuation is conveniently expressed in terms of dB/km

• Power is often expressed in dBm (dBm is dB from 1mW)

34.4

log10

log10

log10

10

10

10

eLL

P

eP

L

P

P

LkmdB

in

Lin

in

out

dBmmW

mWmWP 10

1

10log1010 10

mWmWdBmP 50110127 10

27

28

Fiber Attenuation

• Example: 10mW of power is launched into an optical fiber that has an attenuation of =0.6 dB/km. What is the received power after traveling a distance of 100 km?

– Initial power is: Pin = 10 dBm

– Received power is: Pout= Pin– L=10 dBm – (0.6)(100) = -50 dBm

• Example: 8mW of power is launched into an optical fiber that has an attenuation of =0.6 dB/km. The received power needs to be -22dBm. What is the maximum transmission distance?

– Initial power is: Pin = 10log10(8) = 9 dBm

– Received power is: Pout = 1mW 10-2.2 = 6.3 W

– Pout - Pin = 9dBm - (-22dBm) = 31dB = 0.6 L

– L=51.7 km

nWmWPout 10110 1050

29

Material Absorption

• Material absorption

– Intrinsic: caused by atomic resonance of the fiber material

• Ultra-violet

• Infra-red: primary intrinsic absorption for optical communications

– Extrinsic: caused by atomic absorptions of external particles in the fiber

• Primarily caused by the O-H bond in water that has absorption peaks at =2.8, 1.4, 0.93, 0.7 m

• Interaction between O-H bond and SiO2 glass at =1.24 m

• The most important absorption peaks are at =1.4 m and 1.24 m

30

Scattering Loss

• There are four primary kinds of scattering loss

– Rayleigh scattering is the most important

where cR is the Rayleigh scattering coefficient and is the range from 0.8 to 1.0 (dB/km)·(m)4

• Mie scattering is caused by inhomogeneity in the surface of the waveguide

– Mie scattering is typically very small in optical fibers

• Brillouin and Raman scattering depend on the intensity of the power in the optical fiber

– Insignificant unless the power is greater than 100mW

kmdBcRR /1

4

31

Absorption and Scattering Loss

32

Absorption and Scattering Loss

33

Loss on Standard Optical Fiber

Wavelength SMF28 62.5/125

850 nm 1.8 dB/km 2.72 dB/km

1300 nm 0.35 dB/km 0.52 dB/km

1380 nm 0.50 dB/km 0.92 dB/km

1550 nm 0.19 dB/km 0.29 dB/km

34

External Losses

• Bending loss

– Radiation loss at bends in the optical fiber

– Insignificant unless R<1mm

– Larger radius of curvature becomes more significant if there are accumulated bending losses over a long distance

• Coupling and splicing loss

– Misalignment of core centers

– Tilt

– Air gaps

– End face reflections

– Mode mismatches

35

6. Optical Fiber Dispersion

36

Dispersion

• Dispersive medium: velocity of propagation depends on frequency

• Dispersion causes temporal pulse spreading

– Pulse overlap results in indistinguishable data

– Inter symbol interference (ISI)

• Dispersion is related to the velocity of the pulse

37

Intermodal Dispersion

• Higher order modes have a longer path length

– Longer path length has a longer propagation time

– Temporal pulse separation

– vg is used as the propagation speed for the rays to take into account the material dispersion

gv

L

38

Group Velocity

• Remember that group velocity is defined as

• For a plane wave traveling in glass of index n1

• Resulting in

1

gv

cn 1

c

n

nn

c

n

cc

n

g1

11

11

1

gg n

cv

1

1

111

nnn g

39

Intermodal Dispersion

• Path length PL depends on the propagation angle

• The travel time for a longitudinal distance of L is

• Temporal pulse separation

• The dispersion is time delay per unit length or

1sin

gg v

L

v

PL

12 sin

1

sin

1

gg vvL

1sinL

PL

12 sin

1

sin

1

gg vvD

40

Step Index Multimode Fiber

• Step index multimode fiber has a large number of modes

• Intermodal dispersion is the maximum delay minus the minimum delay

• Highest order mode (~c) Lowest order mode (~90°)

• Dispersion becomes

• The modes are not equally excited

– The overall dispersed pulse has an rms pulse spread of approximately

2

11

2 sin

11

n

n

c

n

vvg

cgg

c

n

vvg

gg

1

1 90sin

11

c

n

n

nn

c

n

n

n

c

nD ggg 1

2

211

2

11 1

21

c

nD g

41

Graded Index Multimode Fiber

• Higher order modes

– Larger propagation length

– Travel farther into the cladding

– Speed increases with distance away from the core (decreasing index of refraction)

– Relative difference in propagation speed is less

42

Graded Index Multimode Fiber

• Refractive index profile

• The intermodal dispersion is smaller than for step index multimode fiber

arnn

ara

rn

rn

21

2

1

21

21

4

21

inter

c

nD g

43

Intramodal Dispersion

• Single mode optical fibers have zero intermodal dispersion (only one mode)

• Propagation velocity of the signal depends on the wavelength

• Expand the propagation delay as a Taylor series

• Dispersion is defined as

• Propagation delay becomes

• Keeping the first two terms, the pulse width increase for a laser linewidth of is

2

22

2

1

go

googg

z

g

g

vD

1intra

intra2intra 2

1 DD ooogg

intraDg

44

Intramodal Dispersion

• Intramodal dispersion is

• There are two components to intramodal dispersion

• Material dispersion is related to the dependence of index of refraction on wavelength

• Waveguide dispersion is related to dimensions of the waveguide

1

1intra

zzD

waveguidematerial1

1

1

1intra

1DD

c

nn

cD zgzg

45

Material Dispersion

• Material dispersion depends on the material

1

1material

1

zgn

cD

46

Waveguide Dispersion

• Waveguide dispersion depends on the dimensions of the waveguide

• Expanded to give

where V is the normalized frequency

• Practical optical fibers are weekly guiding (n1-n2 <<1) resulting in the simplification

1

1waveguide

zg

c

nD

11

2

22

1

11waveguide 2

zzgg

VV

VV

n

n

c

nD

NAakV

bVV

Vc

nnD gg

2

221

waveguide

47

Total Intramodal Dispersion

• Total dispersion can be designed to be zero at a specific wavelength

• Standard single mode telecommunications fiber has zero dispersion around =1.3 m

• Dispersion shift fiber has the zero dispersion shifted to around =1.55 m

48

Standard Optical Fiber Dispersion

• Standard optical fiber

– Step index ≈0.0036

– Graded index ≈0.02

• Dispersion

– Step index multi-mode optical fiber (Dtot~10ns/km)

– Graded index multi-mode optical fiber (Dtot~0.5ns/km)

– Single mode optical fiber (Dintra~18ps/km nm)

4

21

intertot

c

nDD g

intratot DD

21

inter

c

nDD g

tot

49

What is the laser linewidth?

• Wavelength linewidth is a combination of inherent laser linewidth and linewidth change caused by modulation

– Single mode FP laser laser~2nm

– Multimode FP laser or LED laser~30nm

– DFB laser laser~0.01nm

• Laser linewidth due to modulation

– f~2B

2mod

2laser

Bc

fc

cf

cf

22

2

2