3.46 PHOTONIC MATERIALS AND DEVICES
Lecture 4: Ray Optics, Electromagnetic Optics, Guided Wave Optics
Lecture
Notes
Light
photon
™ exchanges energy with medium
¾ Emission
¾ absorption
¾ scattering
electromagnetic wave
™ nondissipative medium
¾ Propagation
¾ Interference
¾ Diffraction
ray optics
™ small λ approx.
¾ Geometric optics
Photon
E = hν
h = 6.626 x 10-34 J⋅s
c
λ=
ν
mass = 0; charge = 0; spin = 1
Ray Optics
“Optical” properties
Complex index of refraction
ncomplex = n + iK
n = refractive index
K = extinction coefficient
Complex dielectric function
ε = ε1 + iε 2
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 1 of 13
Lecture
Notes
Kramers-Kronig relations
Relate ε1(ω) and ε2(ω)
α ≡ absorption coefficient
α=
2ωK
c
Reflectivity (normal incidence)
2
(n − 1) + K
R=
2
(n + 1) + K
•
in transparent range of ω:
2
⎛ n − 1⎞⎟ K→0; R → ⎜⎜
⎜⎝n + 1⎠⎟⎟
Snell’s Law
sin θ1 n2
=
sin θ 2
n1
Total internal reflection
⎛n ⎞
θ1 > θ ext = sin−1 ⎜⎜⎜ 2 ⎟⎟⎟
⎝ n1 ⎠⎟
Reflection (materials n1, n2)
2
⎛ n − 1⎞⎟
R = ⎜⎜
normal incidence
⎜⎝n + 1⎠⎟⎟
Diamond: n ≈ 2.4
n = 2.6
TiO2:
ZrSiO4: n = 1.9
Material
Water
Glass
Crystal glass
diamond
θc
48.6°
41.8°
31.8°
24.4°
n
1.33
1.50
1.90
2.42
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
R
0.02
0.04
0.10
0.17
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 2 of 13
Notes
Lecture
Index matching
n(medium) = n(material) ⇒ no reflection
Anti-reflection coating
R=
n1 (air)
n2 (coating)
n3 (material)
n22 − n1
n3
n2
2 + n1n3
↓
t
↑
= 0 when n2 = n1n3
Example
for solar cell: n3 (silicon)
n2 t =
λ
4
quarter wave film
for glass:
n3 = 1.5; air : n1 = 1.0 ⇒ n2 = 1.22
n2 = 1.384
⇒ R = 0.12
MgF2
Example
AR coating for silicon
nSi = 3.5 ⇒ nAR = 1.87 nSiO = 1.51 2
λ = 550 nm
→ t = 91 nm
R
400
550
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
700 nm
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 3 of 13
Notes
Lecture
Electromagnetic optics
K K
K K
E(r,t) , H(r,t)
Electromagnetic Field
Maxwell’s Equations
K
K
K K
K K
∂H
∂E
∇× H = ε0
∇× E = −μ0
∂t
∂t
K K
∇⋅ E = 0
K K
∇⋅ H = 0
Monochromatic EM Wave
JJK K
JK K
E r,t = Re E′ r exp ( jωt)
( )
{ ()
}
Each of the six scalar components of
K
K
E & H must satisfy the Helmholtz Equation
∇2u + k 2u = 0
wave vector:
k=
c=
ω
k
nω
2π
ω
1/ 2
= ω (εμ0 ) = nk 0 =
=
λ
c
c0
: phase velocity; velocity v g =
dω
= group
dk
The carrier propagates with the phase velocity c. The slowly varying envelop propagates at the group velocity, vg. 3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 4 of 13
Lecture
Notes
Transverse EM Plane Waves (TEM)
K K
K K
• E(r,t) , H(r,t) are plane waves with
K
wave vector k
K K K
• E, H, k are mutually orthogonal
KK
KK
K K
K K
E(r ) = E0e-jkr , H(r ) = H0 e-jkr
Phenomenology of Properties
Absorption
χ = χ ′− iχ ′′ ; ε = ε 0 (1+
1
2
k = ω (εμ 0 ) = (1+
)
1
2
)
1
k 0 = (1+ χ ′ + i ′′)2 k 0
1
= β−i α
2
U( x) = Ae−ikx = Ae
−αx
−iβx
2
e
2
I( x) ∝ U( x) ∝ e−αx
Resonant atoms in host medium
n (ν) ≈ n0 +
χ ′ ( ν)
2n0
⎛ 2πν ⎞⎟
⎟ χ ′′ (ν)
, α (ν) ≈ −⎜⎜⎜
⎜⎝ n0 0 ⎠⎟⎟
Fiber materials for transmission
• Electronic polarizability not important
for IR fibers
• Heavy atom → weaker bond
→ long λ0
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 5 of 13
Notes
Lecture
Frequency dependence of the several contributions to polarizability. Dispersion
≡
dn
dλ
normal
dispersion
Normal dispersive
medium →
group index
ng = n − λ 0
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
dn
dλ 0
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 6 of 13
Lecture
Notes
group velocity
−1
⎛
c
dn ⎞⎟
⎟
v g = 0 = c 0 ⎜⎜⎜n − λ 0
⎜⎝
ng
dλ 0 ⎠⎟⎟
Dispersion coefficient
λ 0 d2n
d ⎛⎜ 1 ⎞⎟⎟
⎜
Dλ =
⎟=−
dλ ⎜⎜⎝ v g ⎠⎟⎟
c 0 dλ 0 2
Dλ =
temporal spread
ps
=
length ⋅ spectral width km ⋅ nm
D λ σλ =
σλ
seconds of pulse broadening
distance travel
: spectral width
pulse delay: τ d =
z
v
pulse spreading: Dν =
d ⎛⎜ 1 ⎞⎟⎟
⎜ ⎟
dν ⎜⎜⎝ v g ⎠⎟⎟
σ τ = D ν σ ν z temporal width
Gaussian pulse
⎛ t 2 ⎞⎟
A(0,t) = exp −⎜⎜⎜ 2 ⎟⎟
⎜⎝ τ ⎠⎟
0
τ0
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
FWHM = τ0
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 7 of 13
Lecture
Notes
1
⎡ ⎛ ⎞2 ⎤ 2
z
τ 2 = τ0 ⎢⎢1+ ⎜⎜⎜ ⎟⎟⎟ ⎥⎥
⎢⎣ ⎝⎜ z0 ⎠⎟ ⎦⎥
Dr
0
z
πτ0
for z z0
t
Polarization
JK K
The time course of direction of E r,t
( )
Helical rotation of circular polarization
1. Plane Polarization
JK
K
E at fixed direction of k
JK
K i kz−ωt)
E (z,t) = a y ye (
; ω = kc
monochromatic light
⎧⎪ JK
JK K
⎡
E r,t = Re ⎪⎨A exp ⎢i2π
⎪⎩⎪
⎣⎢
( )
⎛ z ⎞⎟⎤ ⎫⎪⎪
⎜⎜t − ⎟⎥ ⎬
⎜⎝ c ⎠⎟⎥ ⎪
⎦⎪
⎭
ν = frequency of photons z = direction of propagation c = phase velocity K
K
Amplitude has x and y component:
JK
K
K
A = Ax x + Ay y
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 8 of 13
Lecture
Notes
JK
K
K
E (z,t) = E x x + E y y
↓
⎡
⎤
⎛ z⎞
axcos ⎢ 2πν ⎜⎜t − ⎟⎟⎟ + φ x ⎥
⎢⎣
⎥⎦
⎝⎜ c ⎠
J
K
⇒ at fixed z, E rotates periodically in x-y
plane
2. General Solution: elliptical
polarization
2
E xE y
E2x E y
+
− 2cosφ
= sin2 φ
2
2
ax ay
axay
Matrix Representation
Matrix representation is a simplified way
to perform first order calculations where
small angles can be assumed. It can be
used for order of magnitude calculations
to obtain general values for a broad range
of optical devices.
⎛E x ⎞⎟
⎜⎜ ⎟
⎟
E = ⎜⎜⎜E y ⎟⎟
⎜⎜ ⎟⎟⎟
⎜⎝Ez ⎠⎟
A x = ax
iφx
A y = a y eiφy
K
⎡ A x ⎤
JK
“Jones” vector: J = ⎢⎢ ⎥⎥ = operator on E
⎢⎣ A y ⎦⎥
K
⎡1 ⎤
⎢ ⎥ linear poliarized in x
⎢⎣0⎥⎦
K
⎡cos θ⎤
⎢
⎥ linear poliarized at θ to x
⎢⎣ sin θ ⎥⎦
1 ⎡1⎤
⎢ ⎥ right circular
2 ⎢⎣i ⎥⎦
1 ⎡1 ⎤
⎢ ⎥ left corner
2 ⎢⎣−i⎥⎦
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 9 of 13
Notes
Lecture
Linear polarization ≡ Σ (right + left
circular)
⎡cos θ ⎤
1 −iθ
1 iθ
⎢
⎥=
e
e +
⎢⎣ sin θ ⎥⎦
2
2
Jones Transformation Matrix
optical
system
K
JK JK
J2 = T 1
⎛ A 2x ⎞
⎛T T
⎟⎟⎟ = ⎜⎜ 11 12
⎜⎜⎜
⎝⎜ A 2y ⎠⎟ ⎝⎜T21 T22
⎞⎟⎛⎜ A1x ⎞⎟
⎟⎟⎜⎜ ⎟⎟⎟
⎠⎟⎝⎜A
1y ⎠
Linear Polarizer
⎛1 0 ⎞⎟
⎟ (polarizes wave in x-direction)
T = ⎜⎜
⎜⎝0 0⎠⎟⎟
A1x, A1y → A1x, 0
JK
JK JK
Eout = T in
Guided Wave Optics – Introduction
• Free space
• Guided by confinement in high
refractive index medium
Optical wave guide n2 > n1
n2
n1
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 10 of 13
Lecture
Notes
Planar Mirrors
TEM plane waves
λ
λ= 0
n
k = nk0
k = nk 0
c0
n
polarized in x-direction
K
k in y-z plane at θ to z-axis
c=
JK
1. E ||mirror plane
JK K
2. each reflection → Δφ = π with A, k
unchanged
3. self-consistency: after two reflections, wave reproduces itself ≡ eigenmode of wave ⇒ “bounce angles” θ are discrete (quantized)
mλ = 2dsin θ
m
G
Em (y,z) = Um (y)exp(−iβm z)
β = k z = k cos θ propagation constant = βm (quantized) = kcosθm
Um(y) = transverse distribution
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 11 of 13
Lecture
Notes
(a) Condition of self-consistency: as a wave reflects twice it duplicates itself (b) At angles for which self-consistency is satisfied, the two waves interfere and create a wave that does not change with t.
Optical power
Number of Modes
M≥
2
∝ E ∝ am2
M
2d
λ
M ↑ with d λ max = 2d : cut off λ
c
: cut off ν
νmin =
2d
d ≤ λ ≤ 2d
single mode
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 12 of 13
Lecture
Notes
Field distributions of the modes of a planar-mirror waveguide Group velocity of pulse
vg =
dω
dβ
2
⎛ ω ⎞ m2 π2 β = ⎜⎜ ⎟⎟⎟ − 2 dispersion relation
⎜⎝ c ⎠
d
2
m
v mod e =
β
dω
= c2 m
ω
dβm
= c2
k cos θm
= c ⋅ cos θm
ω
• longer zigzag path → slower group velocity • different modes → different vg →
different transverse u(y) as wave propagates. 3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 4: Ray Optics, EM Optics, Guided Wave Optics
Page 13 of 13