Download The Fabry-Perot Cavity

The Fabry-Perot Cavity
Reflecting
surface
Pf
Ef
2
R2
Reflecting
surface
Pi
Ei
Steady state EM oscillations
L
Optical cavity resonator
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
1
Cavity axis
R1
x
Derivation of the Laser Threshold Condition
In the steady-state, the light (plane wave assumption) should remain unchanged after one
round trip (2L). In other words, the gain = loss at threshold. An energy pumped in above
threshold is converted into photons.
R1 and R2 are the power reflectivities of mirrors 1 and 2, respectively. g is the (intensity) gain
per unit length, αint is the internal scattering loss in the waveguide per unit length, L is the
cavity length of the laser, k is the wave number of the plane wave, E0 is the electric field
amplitude.
E 0 exp(gL) R1R2 exp(−α int L)exp(2ikL) = E 0
By equating magnitude and phase on the two sides of this equation, one obtains:
1 ⎛ 1 ⎞
g = α int +
ln⎜
⎟ = α int + α mir = α cav
2L ⎝ R1R2 ⎠
2kL = 2mπ
or
ν m = mc /2n g L
αmir is the mirror loss
m is an integer
Some typical device parameters for LD’s
L = 200 - 400 µm, R1 = R2 = 0.3 (cleaved mirrors)
αmir = (1/2L)ln(1/R1R2) = 30 - 60 cm-1
Γ ≤ 0.4 (the optical confinement factor)
αint = 2-4 cm-1
g = Γgm = 32 - 64 cm-1 (gm is the material gain)
∆ν = 100-200 GHz (mode spacing in GHz)
Look at fig. (d) for illustration of Γ
(a )
n
p
p
AlGaAs
GaAs
AlGaAs
(a) A double
heterostructure diode has
two junctions which are
between two different
bandgap semiconductors
(GaAs and AlGaAs).
(~0.1 µm)
Electrons in CB
Ec
∆Ec
Ec
1.4 eV
2 eV
2 eV
(b)
Ev
Ev
Holes in VB
Refractive
index
(c )
Photon
density
Active
region
∆n ~ 5%
(d)
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
(b) Simplified energy
band diagram under a
large forward bias.
Lasing recombination
takes place in the pGaAs layer, the
active layer
(c) Higher bandgap
materials have a
lower refractive
index
(d) AlGaAs layers
provide lateral optical
confinement.
Equations for the Light-Current Curve
Optical P ower
Optical Power
Optical P ower
Laser
LED
Stimulated
emission
λ
Optical P ower
Spontaneous
emission
λ
0
Laser
I
Ith
λ
Typical output optical power vs. diode current (I) characteristics and the corresponding
output spectrum of a laser diode.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Equations for Laser Diode Characteristics
I take a slightly different approach here compared to Agrawal to motivate the
power output per mirror facet and the threshold current.
hω ηintα mir
Pe =
(I − Ith )
2q α mir + α int
The “2” in the denominator takes into
account that the power is split between
symmetric mirrors
dPe hω
ηintα mir
=
ηd and ηd =
dI
2q
α mir + α int
Ith =
qn thV
τc
ηint is close to 100% for modern
laser diodes
1.24
dPe
=
ηd
dI 2 λ (µm)
dPe/dI is called the slope efficiency (units of W/A)
ηd is called the differential quantum efficiency (units of %)
The Laser Rate Equations
dP
P
= GP + Rsp −
τp
dt
dN I N
= − − GP
dt q τ c
Any term that subtracts from
N creates damping in the
oscillator
G = Γv g g = GN (N − N 0 )
τp = cavity photon lifetime
τc = carrier (recombination) lifetime
P = photon number
N = electron number, N0 = the transparency electron number
GN = differential gain
G = the threshold gain, net rate of stimulated emission
g = material gain
vg = group velocity = c/ng
Rsp = rate of spontaneous emission coupled into the lasing mode
I = injected current
The results of small-signal analysis on the
laser rate equations
Ω2R + ΓR2
H (ω m ) =
(ΩR + ω m − iΓR )(ΩR + ω m + iΓR )
[
ΩR = GGN Pb − (ΓP − ΓN ) /4
ΓR = (ΓP + ΓN ) /2
ΓP = Rsp /Pb + εNL GPb
ΓN = τ −1
c + GN Pb
2
]
A damped resonator
1/ 2
Normally, Γp is
assumed to be
zero
ΩR is called the resonance frequency
ΓR is called the damping rate
ωm is the modulation frequency
Typical modulation response of the LD
Note the resonance damping at higher current levels
f3db = 1.55fr for the Ib/Ith ≤ 4.6
The 3-dB modulation bandwidth and its dependence
on certain device parameters
f 3dB
[
1/ 2
1
=
Ω2R + ΓR2 + 2(Ω4R + Ω2R ΓR2 + ΓR4 )
2π
]
1/ 2
at low power ΓR << ΩR (not much damping)
f 3dB
12
⎞
⎛
3ΩR
3G P
≈
≈ ⎜⎜ N2 b ⎟⎟
2π
⎝ 4π τ p ⎠
To get a high bandwidth semiconductor laser, you want
to run it at high power, short cavity length (small τp) and
large differential gain.
High-speed laser cavities
Temperature Performance of a 1.3 µm InGaAsP/InP
LD used for uncooled applications
Ith = I0exp(T/T0)
T0 is called the characteristic temperature. Typical values
are from 50 to 100 K.