3.46 PHOTONIC MATERIALS AND DEVICES
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Lecture
Fiber Optics
Optical fiber ≡ core + cladding
guided if n2 > n1
power loss to cladding if n2
< n1
⎛ n
⎞⎟
1⎟
⎟
⎜⎝ n2 ⎠⎟⎟
Notes
n1 < n2
n2
n1
θc = sin−1 ⎜⎜⎜
JK K
each mode travels with β, v g , U (x,y), P, k
single mode (small core)
multi mode (large core)
modal dispersion: modes have different v g
graded index fiber: gradual reduction of n2 ↓
step index fiber: n2 → n1 step change @ boundary
modal dispersion reduced for graded index: v g ↑
(a)
as n ↓ i.e. large θ rays travel farther but faster.
Step Index Fiber
2a
50 μm
=
2b 125 μm
typically:
multi-mode fiber
2a ∼8-10 μm single-mode fiber
Δ
= fractional index change
=
n2 − n1
1
n2
b
Typical dopants to SiO2: Ti, Ge, B
a
n2 : (1.44 −1.46)
Δ : (0.001− 0.02)
numerical aperture: NA = light gathering power guiding of ray incident from air θc = θa for air/core
interface
sin θa = n2 sin θc
1.
n22
nn11 n
1
sinθa
= n2 (1−cos2 θc )2
1
⎡ ⎛ n ⎞2 ⎤ 2
= n2 ⎢⎢1−⎜⎜⎜ 1 ⎟⎟⎟ ⎥⎥
⎢⎣ ⎜⎝ n2 ⎠⎟ ⎥⎦
1
= (n22 − n12 )2
θa = sin−1(NA)
1
1
NA = (n22 − n12 ) 2 ≈ n2 (2Δ) 2
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Page 1 of 5
Notes
Lecture
θa = acceptance ∠ for fiber
≡ exit angle for fiber
θc
θc = complementary critical ∠
e.g. SiO2 fiber
n2 = 1.46, Δ = 0.01
⎛ n ⎞⎟
∴ θc = cos−1 ⎜⎜⎜ 1 ⎟⎟ = 8.1°
⎜⎝ n2 ⎠⎟⎟
θc
θa = 11.9°
NA = 0.206
Unclad fiber
n2 = 1.46 , n1 = 1 , Δ=0.96
θc = 46.8° , θa = 90°
NA = 1
(all rays are guided)
u(r,φ,z) = u(r ) e−jφe− jβz
Guided Waves
Helmholtz equation
∇2u + n2k 02u = 0
(b → ∞)
n = n2 , r < a ; n = n1 , r > a
k0 =
2π
λ0
Condition for guiding
n1k 0 < β < n2
0
kT = rate of change of u(r) in core
rate of decay high ⇒ low penetration
γ = rate of U(r) in cladding
k T 2 = n22k 02 − β2
γ 2 = β2 −n12k 02
k 2T + γ 2
(n22 −n12 )k02 = (NA )
2
k 02
k ↑, γ ↓⇒ penetration into cladding
T
k T > NA ⋅
0
⇒ γ imaginary, wave escape core
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Page 2 of 5
Lecture
Notes
V-parameter
Single mode fiber design
Define:
X = k T = normalized transverse
phase constant in core
Y = γ T = normalized transverse
attenuation constant in clad
2
2
X + Y = V2
a
NA
λ0
V = 2π
= normalized frequency ≤ 2.405 for single mode core radius requirement for single mode
a<
1.2λ 0
1
π(n22 −n12 )2
if Δ = 0.003, a = 8 − 10 μm
most single mode fiber designed @ V = 2.8 for
better confinement of fundamental mode.
Weakly guiding fiber
n2 n1, Δ 1
guided waves are TEM
guided waves are paraxial
linear polarization (x, y) orthogonal
LPlm = linear polarization mode
l = propogation constant
m = spatial distribution
M, number of modes
e.g. SiO2 fiber
JJK
E⊥z
( & fiber axis)
(X + Y polarization travel equally ω no
coupling)
V >> 1
n2 = 1.452, Δ = 0.01, NA = 0.205
λ 0 = 0.85 μm (GaAs)
a (core) = 25 μ m
⇒ V = 37.9, M = 585
remove cladding
⇒ n1 = 1, NA = 1
⇒ V = 184.8,
M > 13,800
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Page 3 of 5
Lecture
vg>>1
Group Velocity
v lm =
Notes
dω
dβlm
⎡
⎢
⎢
⎢⎣
v lm c 2 ⎢1−
2
(l + 2m)
M
⎤
⎥
⎥
⎥⎦
Δ⎥
z < l + 2m <
M
⎛
n ⎞⎟
c 2 > v lm > c 2 ⎜⎜⎜ 1 ⎟⎟
⎜⎝ n2 ⎠⎟
phase velocity > vlm > high order modes
∴
fractional charge in v g Δ
large Δ → large NA
→ large M
(high order modes slower)
high modal dispersion
Single Mode Fibers
small core diameter
small NA
long λ 0
u(r) ~ Gaussian
e.g. SiO2
n2 = 1.447,
Δ
λ 0 = 1.3 μm
= 0.01, NA = 0.205
single mode ⇒ 2a < 4.86 μm
if Δ = 0.0025
single mode ⇒ 2a < 9.72 μm
Graded Index Fiber
reduce modal dispersion
c0 minimum @ center
→shortest travel, slowest velocity
⇒ power low profile
p
⎡
⎛r ⎞ ⎤
n2 (r ) = n22 ⎢⎢1− 2 ⎜⎜ ⎟⎟⎟ Δ⎥⎥
⎝⎜ a ⎠ ⎥
⎢⎣
⎦
n = n2 @ r = 0
= n1 @ r = a
Δ=
O
n1
r≤a
p=1
p=2
p →∞
n2
n
n2(r) linear
n2(r) quadratic
n2(r) step function
n22 − n12
2n22
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Page 4 of 5
Lecture
Notes
Number of Modes
(step index)
p→∞
M ≈
for all other modes (least modal dispersion for
multimode)
V 2
2
⎛a⎞
v = 2π ⎜⎜⎜ ⎟⎟⎟NA
⎟
⎜⎝ λ0 ⎠
Optimal profile
p=2
⇒ v g = c 2
M=
V2
4
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Page 5 of 5