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