ECEN 5645 Introduc0on to Optoelectronics Class Mee0ng 2 Refrac0ve Index And Wave Velocity Note On Problem Sets •  Problem sets are due on Tuesdays •  Problem sets consist of 4 problems •  Problems will be assigned to individual students to solve, outline their solu0on of a power point, submit to me before class on the assigned day and to discuss in class (5 minutes max). •  The assignments begin next Tuesday (9/1/15) Today’s Topics •  Refrac0ve Index –  Phase velocity and index –  Index as a func0on of wavelength –  Sellmeier and Cauchy expansions •  Group Velocity –  Wave packets –  Group index Refractive Index
When an EM wave is traveling in a dielectric medium, the oscilla0ng electric field polarizes the molecules of the medium at the frequency of the wave The stronger is the interac0on between the field and the dipoles, the slower is the propaga0on of the wave Refractive Index
Maxwell’s Wave Equation in an isotropic medium
∂2E ∂2E ∂2E
∂2E
+ 2 + 2 − ε oε r µ o 2 = 0
2
∂x
∂y
∂z
∂t
A plane wave is a solution of Maxwell’s wave equation
Ex = Eo cos(ωt - kz + φο)
The phase velocity of this plane wave in the medium is given by
v=
ω
k
=
1
ε oε r µo
The phase velocity in vacuum is
c=
ω
ko
=
1
ε o µo
Phase Velocity and εr
The relative permittivity εr measures the ease with which the
medium becomes polarized and hence it indicates the extent of
interaction between the field and the induced dipoles.
For an EM wave traveling in a nonmagnetic dielectric medium of
relative permittivity εr, the phase velocity v is given by
ν=
1
ε rε o µo
Refractive Index n
1
Phase Velocity and εr
ν=
Refractive index n
definition
c
n = = εr
v
ε rε o µo
Optical frequencies
Typical frequencies that are involved in optoelectronic devices are in the infrared (including far infrared), visible, and UV, and we generically refer to these frequencies as op0cal frequencies Somewhat arbitrary range: Roughly 1012 Hz to 1016 Hz Frequency Wavelength Low frequency (LF) rela0ve permi[vity εr(LF) and refrac0ve index n.
Energy Bond Energies •  An EM wave excites QM resonances •  Electronic resonances absorb in the visible and ionize in the UV – require no mo0on •  Vibra0onal resonances absorb in the N&m IR •  Rota0onal resonances absorb in the FIR (THz) •  Larger organic molecules may have collec0ve resonances that extend THz/mmwave Refractive Index and Propagation Constant
ko
ko
λo
k
λ
Free-space propagation constant (wave vector)
2π/λο
Free-space wavelength
Propagation constant (wave vector) in the medium
Wavelength in the medium
k
n=
ko
In noncrystalline materials such as glasses and liquids, the material structure is the same in all direcAons and n does not depend on the direcAon. The refracAve index is then isotropic Refractive Index and Wavelength
It is customary to drop the subscript o on k and λ
kmedium = nk
In free space
λmedium = λ /n
n depends on the wavelength λ
Dispersion relation: n = n(λ)
The simplest electronic polarization gives
⎛ N at Ze
n = 1 + ⎜⎜
⎝ ε ome
2
2
2
2
⎞⎛ λo ⎞ λ
⎟⎟⎜⎜
⎟⎟ 2 2
⎠⎝ 2πc ⎠ λ − λo
Nat =Number of atoms per unit
volume
Z = Number of electrons in the
atom (atomic number)
λo = A “resonant
frequency”
Sellmeier Equation
2
2
2
A3λ
A1λ
A2 λ
n = 1+ 2
+ 2
+ 2
2
2
2
λ − λ1 λ − λ2 λ − λ3
2
Polarizability at a Resonance •  Un-­‐broadened polarizability takes the form of a Lorentzian near a resonance n depends on the wavelength λ
Cauchy dispersion rela0on n = n(υ) n = n-­‐2(hυ)-­‐2 + n0 + n2(hυ)2 + n4(hυ)4 n depends on the wavelength λ
Some Sellmeier Plots (Kasap Table) GeO2 SiO2 SiGe Group Velocity and Group Index
•  There are no perfect
monochromatic waves
•  We have to consider the way in
which a group of waves differing
slightly in wavelength travel
along the z-direction
Group Velocity
Two slightly different wavelength waves traveling in the same direction result in a wave
packet that has an amplitude variation that travels at the group velocity.
dω
vg =
dk
Group Velocity
Consider two sinusoidal waves that are close in frequency, that is, they have frequencies ω -­‐ δω and ω + δω. Their wavevectors will be k -­‐ δk and k + δk. The resultant wave is Ex(z,t) = Eocos[(ω -­‐ δω)t -­‐ (k -­‐ δk)z] + Eocos[(ω + δω)t -­‐ (k + δk)z] By using the trigonometric iden0ty cosA + cosB = 2cos[1/2(A -­‐ B)]cos[1/2(A + B)] we arrive at Ex(z,t) = 2Eocos[(δω)t -­‐ (δk)z][cos(ωt -­‐ kz)] Ex(z,t) = 2Eocos[(δω)t - (δk)z][cos(ωt - kz)]
This represents a sinusoidal wave of frequency ω . This is
amplitude modulated by a very slowly varying sinusoidal of
frequency δω. This system of waves, i.e. the modulation,
travels along z at a speed determined by the modulating
term, cos[(δω)t - (δk)z]. The maximum in the field occurs
when [(δω)t - (δk)z] = 2mπ = constant (m is an integer),
which travels with a velocity
dz δω
=
dt δk
or dω
vg =
dk
This is the group velocity of the waves because it determines the speed of propagaAon of the maximum electric field along z. The group velocity therefore defines the speed with which energy or informaAon is propagated. dω
vg =
dk
ω = 2πc/λo and k = 2πn/λo, λo is the free space wavelength.
Differentiate the above equations in red
dω = -(2πc/λo2)dλo
⎛ dn ⎞
⎟⎟dλo
dk = 2πn( −1 / λ )dλo + (2π / λo )⎜⎜
⎝ dλo ⎠
2
o
⎛
dn ⎞
⎟⎟dλo
dk = −(2π / λ )⎜⎜ n − λo
dλo ⎠
⎝
2
o
∴
dω
vg =
=
dk
− (2πc / λ2o )dλo
c
=
dn
dn ⎞
2 ⎛
⎟⎟dλo n − λo
− (2π / λo )⎜⎜ n − λo
dλo
dλo ⎠
⎝
Group Velocity and Group Index where n = n(λ) is a function of the wavelength. The group
velocity vg in a medium is given by,
dω
c
vg (medium) =
=
dk n − λ dn
dλ
This can be wriEen as c
vg (medium) =
Ng
Group Index
dn
Ng = n − λ
dλ
is defined as the group index of the medium In general, for many materials the refracAve index n and hence the group index Ng depend on the wavelength of light. Such materials are called dispersive Refractive Index and Group Index
Refractive index n and the group index Ng of pure SiO2 (silica) glass as a function of
wavelength.