lecture 5. tunable filters Δf defines as the frequency difference between the lowest- and the...

Lecture 5

Tunable Filters

Tunable FiltersΔf defines as the frequency difference between the

lowest- and the highest-frequency channels and f as the spacing between channels.

The maximum number of equally spaced channels can be calculated as Nmax = Δf / f .

The access time is the speed which a tunable filter can be reset from one frequency to another. This should be in the unit of microsecond.

The filter’s transfer function T(f) is not generally equal to unity due to its internal losses.

Tunable FiltersGood filters should be independent to light

polarizations. Tunable-filters have an advantage over the coherent-detection due to that.

With the help of lithography, the low cost filter can be fabricated, but the fiber loss attachment exists. However, with other methods, high-cost processes are involved and that is the great barrier to develop such a technology.

Tunable FiltersWavelength selective filters might be categorized

into:2-port filter

1 N WDM filter

Tunable Filters4-port add-drop filter

Crosstalk

Filters for WDMRequirements:

Center wavelength near 1.55 μm or 1.3 μm. Some local network might have a center wavelength of 0.8 μm.

Frequency Spacing: about 100 GHz.

Number of channels: having been increased to more than 256.

Tuning speed: less than 1 μs.

Some of tunable filtersFabry-perot filtersMach-Zehnder chainGratingAcousto-optic tunable filter (AOTF)Electro-optic tunable filter (EOTF)

Fiber Fabry-Perot Filter

Fiber Fabry-Perot FilterConsider a single mirror

2

2

2

2 2

Reflectance,

Transmittance, 1

1 excess loss

r

i

t

i

t r

i i

Power E

ER

E

ET R A

E

E EA L

E E

Fiber Fabry-Perot FilterAfter a round trip

1 1

4 4round trip phase shift 4

transit time for cavity after light propagating for distance L.

ii iE E R A R Re E R A

L L

c

Fiber Fabry-Perot FilterAfter two round trips

After N round trips

2 2

0

1 1 e

1

e

i ii

Nn

in

i

E E R A R R e

E E R A a

a R

Fiber Fabry-Perot FilterAt steady state (N ∞)

0

2 2

2 2 2 2

1 since < 1

1

1

1

1

1

1

1 (1 cos ) ( sin ) 1 2 cos

Nn

n

i

t

i

a R

aa

E R AE

a

E R AT

E a

a R R R R

Fiber Fabry-Perot FilterAt steady state (N ∞)

2

2

2

max

2

min

2

max

min

1

1 2 cos

max T for cos 1 or 2 ; N = 1,2,...

min T for cos 1 or (2 1) ; N = 1,2,...

1

1

1

1

1

1

R AT

R R

N

N

R AT

R

R AT

R

T R

T R

Fiber Fabry-Perot FilterPlot T vs

Fiber Fabry-Perot Filter

2

2

42

Free Spectral Range (FSR) in frequency24

2

Free Spectral Range (FSR) in wavelength2

L

cc

LL

L

Fiber Fabry-Perot FilterWe can find the bandwidth of the peak by looking at

the denominator expression for T.

2

2

2

2 2

1 2 cos

2 1

cos 12

1 2 2 2(1 )2

1Therefore

12 2 = Full Width at Half Maximum

D R R

N

D R R R R

R

RR

FWHMR

Fiber Fabry-Perot FilterAnother important parameter to characterize a FP

filter is the finesse, F.

This can determine the maximum number of channels in WDM system.

1

FSR RF

FWHM R

ExampleIf we have 10 channels with 100 GHz spacing

for each channel. What shoud the length of FPI filter be?

ExampleConsider a Fabry Perot filter with an air

cavity of length L and R = 0.99 for each mirror, with a free spectral range of 3.2 THz.(a) What is L in μm?(b) For the value of L determined in (a), what is

the wavelength λ0 nearest 1.53 m for which the transmittance is a maximum.

Mach-Zehnder Chain

Mach-Zehnder Chain

2 2 2

12 4

a ba b ii ii it

E EP e e e

Mach-Zehnder Chain

2 2 2

12 4

1 cos2

2

2 2Max for 2

22

a ba b ii ii it

it

a ba b

m a b

m mt

m

m

E EP e e e

PP

n L L

L L L

nL n LP N

cnL

FSRc

c

nL

Mach-Zehnder Chain

11

2

1

Let for MZ chain with M interferometers.2

( ) cos

determines FSR.

m M

Mm

m

m

LL

n LT

c

L

ExampleConsider a 7-stage MZ chain with a FSR of

3.2 THz produced in single mode fiber with n = 1.46 for the fundamental mode and a transmittance maximum for λ1 = 1.53 m. What is the shortest and longest path difference for any interferometer in the chain?

GratingsSpatial period d

diffracted waves interface

(sin sin ); 0, 1, 2,...

diffraction angle

I incidence angle

m d I m

ExampleFind the allowed mode and angles for each

mode for I =30, = 1.53 μm, and d = 1.61 μm.

Gratings

focal length of the lens

From previous exmple, 30 26.7 3.3 ,

suppose we want h = 2 mm. What does f need to be?

h f f I

f

GratingsCalculating wavelength dependence of focused spot

position.

cos

cos

m d

m

d

ExampleFrom previous example, if channel spacing is

100 GHz. What should be a value of h?

Spectral resolutionSpectral resolution is an ability to separate light into

wavelength components.

2

2

0

0

11 2

sin ( )( )

sin

number of illuminated rulings in grating

sin sin

diffraction angle at center of pattern

=

A

NP const

N

d

m N

Gratings

Gratings