8. Fiber optics

Principal Types of Optical Fiber

8.1 Step-index fibers

• Cladding typically pure silicaC d d ith i t i i d• Core doped with germanium to increase index

• Index difference referred to as “delta” in units of percent (typically 0.3-1.0%)( yp y )

• Tradeoff between coupling and bending losses• Index discontinuity at core clad boundary• Index discontinuity at core-clad boundary

8.1 Step-index fibers

Meridional Rays

Skew Rays

8.1 Step-index fibers

For Corning SMF-28 optical fiber, nco=1.4504, nCL=1.4447 at 1550 nm

NA = 0.13, Acceptance angle = 7.35 degrees

8.1 Step-index fibers

2 22 21 1 0U U U n k U∂ ∂ ∂

+ + + =

Helmholtz equation :

02 2 2 0n k Ur r r r φ

+ + + =∂ ∂ ∂

With a propagation constant β { }n k n kβ> >With a propagation constant β { }1 0 2 0n k n kβ> >

n2

n1

a

0zE ≈

8.1 Step-index fibers

Tk : transverse propagation constant

γ : decay constant

8.1 Step-index fibersSpatial (radial) distribution function

l 0l = 0

l = 0

l = 3l = 1

l = 2

8.1 Step-index fibers

V parameter of fibers (Normalized frequency of fibers)

V parameter or Normalized frequency2 2V k k NA V-parameter or Normalized frequency2 20 1 2 0V k a n n k a NA= − =

8.1 Step-index fibers

: Characteristic Equation

8.1 Step-index fibers

: Characteristic Equation

2 2V k

01X X

2 20 1 2V k a n n= −

0102X

03X

Fiber modes Linearly polarized modes modeslmLP

8.1 Step-index fibers

Linearly polarized modes in fibers modeslmLP

2( )u r( )u r

01LP

2( )u r

34LP

8.1 Step-index fibers

modeslmLP

8.1 Step-index fibers

(for low V)

8.1 Step-index fibers

8.1 Step-index fibers

If V >> 1;;

8.1 Step-index fibers

8.2 Graded-index fibers

8.2 Graded-index fibers

Ray Propagation in Graded-Index Fibers

8.2 Graded-index fibers

If 2 ( b li i d fil )If p = 2 (parabolic index-profile)

Comparison : the number of modesd

0

2 dMλ

=1-d Mirror Guide For the mirror guide the number of modes is just the number of ½ wavelengths that can fit.

0

2 dM NAλ

≈1-d Dielectric GuideFor dielectric guides it is the number that can fitbut now limited by the angular

22

4dM πλ

⎛ ⎞⎟⎜ ⎟≈ ⎜ ⎟⎜ ⎟⎜2-d Mirror Guide

cutoff characterized by the NA of the guide

The V parameter04 λ ⎟⎜⎝ ⎠

22dπ ⎛ ⎞⎟⎜

characterizes the number ofwavelengths that can fit acrossthe core guiding region in a fiber.

0

24

dM NAπλ

⎟⎜ ⎟≈ ⎜ ⎟⎜ ⎟⎜⎝ ⎠2-d Dielectric Guide

22

20

4 16 aM V NAπ λ

⎛ ⎞⎟⎜ ⎟≈ = ⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠2-d Dielectric fiber (step-index) ( / 2)a d=

2-d dielectric fiber (graded-index) 21 for p = 24

M V≈

8.3 Attenuation and dispersion

Dispersion

DefinitionsDefinitions

Group velocity1−

⎟⎠⎞

⎜⎝⎛∂∂

=ωβ

gv

Group delay

⎠⎝ ∂ω

∂∂ ββτ 1Group delaykc ∂

=∂

=ω

τ

Dispersionλτ∂∂

=D

Types of dispersionDispersion

Types of dispersion∂λτ∂∂

=D

• Modal dispersion: different modes propagate at different

λ∂

• Modal dispersion: different modes propagate at different group velocities

• Material dispersion: the index of refraction of the mediumMaterial dispersion: the index of refraction of the medium changes with wavelength

• Waveguide dispersion: different wavelengths in a modeWaveguide dispersion: different wavelengths in a mode have different delays

• Polarization mode dispersion: if waveguide is birefringentp g g

Dispersion

Dispersion

Dispersion

Dispersion

Dispersion