Lecture 3

ExampleA common optical component is the equal-power splitter

which splits the incoming optical power evenly among M outputs. By reversing this component, we can make a combiner, which can be made to deliver to a single output the sum of the input powers if multimode fiber is used, but which splits the power incoming to each port by a factor of M if single-mode fiber is used.(a) Compare the loss in dB between the worst-case pair of

nodes for the 3 topologies if the number of nodes is N = 128 and multimode is used. Assume that, for the tree, there are 32 nodes in each of the top two clusters and 64 nodes in the bottom one.

(b) What would these numbers become if single-mode fiber were used?

(c) How would you go about reducing the very large accumulated splitting loss for the bus?

ExampleA light wave communication link, operating at a

wavelength of 1500 nm and a bit rate of 1 Gbps, has a receiver consisting of a cascaded optical amplifier, narrow optical filter, and a photodetector. It ideally takes a t least 130 photons/bit to achieve 10-15 bit error rate.

(a) How many photons/bit would it take to achieve the same error rate at 10 Gbps?

(b) At this wavelength, 1 mW of power is carried by 7.5 x 1015 photons/s, what is the received power level for 10-15 bit error rate at 1 Gbps?(c) Same as (b) but at 10 Gbps?

Four-port optical couplersBy definition: 1 2

12

1

1 2

2

1 11

1

out out in

outout

in inout

out out

inout

P P LP

PP

rLP rLP

Pr

rP rP

LPP

r

Four-port optical couplersPower division matrix

1 111 12

2 221 22

11 22

12 21

11 12

1

1

out in

out in

P PC C

P PC C

rLC C

rL

C Cr

C C L

Four-port optical couplersr = 0 C11 = 0, C12 = L

All input power crossover to output 2.

r = ∞ C11 = L, C12 = 0

All input power goes straight through.

0 1

1 0ijC L

1 0

0 1ijC L

Four-port optical couplersr = 1 C11 = C12 = L/2

3-dB coupler or ‘50-50’ coupler.

0.5 0.5

0.5 0.5ijC L

Example For the 4-port fiber optic directional coupler, the network below

uses 8 of these couplers in a unidirectional bus. Assume that the excess loss of each coupler is 1 dB.

(a) If the splitting ratio is 1 for all of the couplers, what is the worst case loss between any Tx and Rx combination in dB?(b) What is the least loss between any Tx and Rx?

Multimode fiber

Rays incident at angle still travel farther but also low average faster ( )

Length bandwidth product is lot greater than one of SI multimode fiber. Useful for telecommunications.

1n n

Multimode fiber

Rays incident at an angle to axis travel further than rays incident parallel to an axis.

Low-length bandwidth product (<100 MHz-km) not widely used in telecommunications.

Single-mode fiber

Only one mode propagates: neglecting dispulsion all incident light arrives at fiber end at the same time.

Length bandwidth product > 100 GHz-km.Much greater bandwidth than any multimode

fiber. suitable for long live intercity applications.

Modes in fibers

Modes in fibersIt begins with Maxwell’s equations to define a

wave equation.

In an isotropic medium:2 2

22 2

2 2 22

2 2 2

22

02,

n EE

c t

x y z

nn

c

Modes in fibersWe have 3 equations with solution of Ei for

each axis which is not generally independent.Assume that wave travels in z-direction:

Substitute these into a wave equation, it yields

( )( , , , ) ( , ) e

propagation constant

2

i z tE x y z t E x y

Modes in fibers2 2

2 2

0

2 2

2 2

( , ) ( , )

2

2We know .

( , ) ( , )0

E x y E x y

x y

n n

c c

k

E x y E x y

x y

Modes in fibersFor guided mode: n2 < neff < n1

For radiation mode: neff < n2

Modes in FibersIf we rewrite a wave equation in scalar, we

get

2 22 2 2

02 2

2 2 2 2

2 2 2 2 2

2 2

0

( , , ) or (propagation in z-direction)

( , ) ( )

1 1From

cos , sin , ,

x y

u un k u

x y

u x y z E E

n x y n r

r rx y r r

x r y r r x y

Modes in FibersSolutions for the last equation are

st

stn

( ) 0, 1, 2,...

Solutions to scalar wave equation for step index fiber:

( ) ;

( , )

( ) Bessel Function of 1 kind of order m.

K ( ) Modified Bessel Function of 1 kind of order

il

illm

m

u r e for l

AJ pr e r a

u r

J x

x

m.

Modes in Fibers

Modes in FibersIt is convenient to define a useful parameter

called ‘V-number’ as

V is dimensionless.V determines

Number of modes.Strength of guiding of guided modes.

2 2V a p s

V

Modes in Fibers

Modes in FibersMode designation LPlm

l = angular dependence of field amplitude eil (l = 0,1,..)

m = number of zeroes in radial function u(r)

Fundamental LP01 mode: no cutoff. It can guide no matter how small r is.u(r,) = u01(r) ….circular symmetric maximum at r

= 0.

Modes in FibersTwo mode fiber guide LP01 and LP11

modes:

Modes in FibersMode Cutoff condition V at cutoff @ m =1,2,3

LP0m l=0 J-1(r)=0 0 3.832 7.016

LP1m l=1 J0(r)=0

LP2m l=2 J1(r)=0

LP01 HE11

LP11 TE01, TM01, HE21

For large V, number of guide modes = V2/2

ExampleFind a core diameter for a single-mode fiber

with λ=1330 nm.