Semiconductor laser fundamentals
Major disadvantages of LEDs:
Too broad output beam
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0.8
1
1.2
1.4 1.42 1.44 1.46 1.48
Energy, eV
r/rm
ax
Too broad spectrum
Popt
time
Too slow pulse response
Semiconductor laser fundamentals
Laser definition involves two fundamental processes:light amplification and light emission
In lasers, light amplification is achieved viastimulated emission.
Stimulated emission had been proposed by Albert Einstein in 1915but has been forgotten until 1950s
Absorption, Emission and Stimulated Emission
In atoms or in semiconductor crystals, absorption of a photon will occur only when the quantum energy of the photon precisely matches the energy gap between the initial and final states.
If there is no pair of energy states such that the photon energy can excite the electron from the lower to the upper state, then the matter will be transparent to that radiation.
Stimulated EmissionAs formulated by A. Einstein,
if an electron is already in an excited state (an upper energy level, in contrast to its lowest possible level or "ground state"), then an incoming photon can "stimulate" a
transition to that lower level, producing a second photon of the same energy.
Absorption – Stimulated emission competition
According to Einstein, WABS= WSE
The rate of absorption = Number of electrons in the state 1 × the probability to absorb the photon:AR = N1 WABS
The rate of Stimulated emission = Number of electrons in the state 2 × the probability to emit the photon:SER = N2 WSE
Normally, in solid states and in semiconductors, in equilibrium, N1 > N2because E1 < E2
It follows that in equilibrium or close to equilibrium,AR > SER; i.e. absorption dominates.
Population Inversion (PI): N2 > N1
is the key condition for the stimulated emission and the laser action
Electrons will normally reside in the lowest available energy state. They can be elevated to excited states by absorption, but no significant
collection of electrons can be accumulated by absorption alone since both spontaneous emission and stimulated emission will bring them back down.
E2, N2
E1, N1
ω ω ωω
ωRate b 12
Absorption Spontaneous Emission Stimulated Emission
Rate a 12
E2 - E 1 = ω
The total emission rate is always greater than the absorption rate.Hence, the population inversion can never be achieved by absorbing the photons(with the same energy that the one to be emitted).
Population Inversion (PI) in Semiconductors
Conduction band
Valence band
E
EC
EVMajority of electrons
Very few electrons
No electrons (forbidden band)
n(EC) << n(EV)
No PI
Electron energy distribution in semiconductors.
The probability to find an electron with the energy E:
⎟⎠⎞
⎜⎝⎛ −
+=
kTEE
EfFexp1
1)(
f(EF) = ½Example:
EC – EF = 1 eV; kT = 0.026 eV; f(EC) = 2×10-17
Under normal (quasi-equilibrium conditions) the free electron concentration must be increased by around 17 orders of magnitude to reach
the PI condition
f(E)
Ev EF Ec
1/2
Fermi Energy
Electron and hole concentrations requirements for population inversion in semiconductors
Ev EF Ec
1/2
The PI condition:n(EC) > n(EV)
NC fn(EC) > NV fn(EV)NC,, NV are the numbers of energy
positions available.Assume NC = NV.
Then the PI condition is:fn(EC) > fn(EV)
In the valence band, the absence of electron means the presence of the hole:fn(EV) + fp(EV) =1;fn(EV) =1 - fp(EV);
Then the PI condition is:
fn(EC) > 1 - fp(EV), or:fn(EC ) + fp(EV) > 1
fn(EC)fp(E)
fn(E)
fp(EC)
The Fermi energies for both electrons and holes must be positioned inside the conductance and valence bands correspondingly.
In other words, both electron and hole concentrations must be very high simultaneously
Electron and hole concentrations requirements for population inversion in semiconductors
Ev Ec
fn=1/2EFn
fp=1/2EFp
fn(EC)fp(EV)
The PI condition can be reformulated asEFn – EFp > (EC-EV)
1
1n
Fnf E
E EkT
( )exp
=−⎛ ⎞
+ ⎜ ⎟⎝ ⎠
1
1p
Fpf E
E EkT
( )exp
=−⎛ ⎞
+ ⎜ ⎟⎜ ⎟⎝ ⎠
Forward biased p-n junction is one way to approach the PI condition
In the first semiconductor lasers, the PI has been achieved in a heavily doped forward biased p-n junctions. The pumping current was too high to operate at
room temperature in CW-mode
P-n junction in equilibrium,n×p = ni
2; fn(EC ) + fp(EV) <1
Forward biased p-n junction,n×p >> ni
2; fn(EC ) + fp(EV) <1
Forward biased heavily doped p-n junction,n×p >> ni
2; fn(EC ) + fp(EV) >1
EF
EF
p n
EFEFnEFp
EFn
EFp
Zero bias
Forward bias
Zero bias
Forward bias
Quantum well heterostructure laser allows to achieving the PI at much lower pumping currents
Quantum well heterostructure laser
Laser gain
0 xI x I( ) ( ) eγ=
If the PI condition is met, the intensity of stimulated emission I(x) increasesthe the optical beam propagates along the p-n junction plane:
x=0 is the coordinate corresponding to one of the sample facets and x is the positioninside the sample along the junction plane.
γ is the laser gain
Laser gain and loss
0 xI x I( ) ( ) e α−=
In practical lasers, the are regions with the gain (PI) and with absorption.We can say that the gain can be positive (the actual gain) or negative (the absorption)
The net laser gain is the difference between the gain in the PI region and the absorption in the rest of the laser
Regular semiconductor material α is the absorption coefficient
0 xI x I( ) ( ) eγ=Laser γ is the gain
I(x)
I(x)
Self-sustainable laser emission
The stimulated emission must be initiated by the incoming photon.After all the photons have passed through the semiconductor sample the emission is over.
outputFeedback
Amplifier
Laser
Lasing can be achieved by redirecting a portion of the out coming photons back to the input
Cleaved facets as a Fabri-Perot etalon in heterolasers
The power reflection coefficient for the mirror is RWave amplitude reflection coefficient is R1/2
Waveguide structure of hetero-lasers
Laser Fabri-Perot resonator
Laser Fabri-Perot resonator –amplitude balance
Light output
Light output
P1 RP1 RP1 P1
L
The mirror power reflection factor R nn
=−+
⎛⎝⎜
⎞⎠⎟
11
2 where n is the refraction index. In GaAs, n ≈ 3.5 and R ≈ 0.31.
Optical power change after a full roundtrip is R1R2 exp(2γL)(γ is the net gain of the entire laser structure).
The condition for continuous emission is: R1R2 exp(2γL)= 1, or:
1 2
1 12
L lnR R
γ⎛ ⎞
= ⎜ ⎟⎝ ⎠
lasing condition (Fabri-Perot laser equation)
(2): R1P1
R1 R2
(1): P1
(3:) (R1P1)eγL
(4): R2 R1P1eγL
(5):R2 R1P1e2γL
1 2
1 12m ln
L R Rα
⎛ ⎞= ⎜ ⎟
⎝ ⎠is called the “mirror loss”. Then, the lasing condition: mγ α=
Under the equilibrium lasing condition, the electromagnetic wave phase should remain unaltered after a round trip (the path is 2L). Otherwise the wave superposition will decrease the beam intensity.
2r
n Lnλ
= nr is the semiconductor refractive index;n is any integer
Laser Fabri-Perot resonator –phase balance
Find the value of the integer n for operation at1.4 µm, assuming nr =3.5, L=250 µm.
Example
n ≈ 1250
Find ∆λ for the above example.
Laser mode separation:
∆λ ~ 1 nm.
Carrier and light intensity distribution in the transverse direction
Loss LossGain
0.5 1 1.5 20
2
1
0.5 1 1.5 20
0.5 1 1.5 20
3
2
1
6
-6
Distance (µm)
0.5 1 1.5 20
0.3
0.2
0.1
Light confinement
n-type p-type
ActiveGaAs region
Al 0.3 Ga0.7 As Al 0.3 Ga0.7 As
0.5 1 1.5 20
2
1
0.5 1 1.5 20
0.5 1 1.5 20
3
2
1
6
-6
Distance (µm)
0.5 1 1.5 20
0.3
0.2
0.1
Light confinement
n-type p-type
ActiveGaAs region
Al 0.3 Ga0.7 As Al 0.3 Ga0.7 As
Electrical and optical confinement in heterostructure lasers
GaAs/AlGaAs QWsGaAs/AlGaAs QWs
Optical gain as a function of injected carrier concentration
Loss
• The Gain must be greater or equal to the total loss in the cavity for the lasing.• The loss comes from the cavity loss outside the QW and from the leak through
the mirrors
Pumping current
Gain < Loss; no lasing
Gain > Loss; lasing
Laser threshold current
Loss
• d - the active layer thickness,
• L and W - the resonator length and width
• nth - the threshold electron-hole density
corresponding to Gain = Loss
• τe - the electron-hole recombination time.
th th ej q×d×n /τ=
th th eI q×L×W×d×n /τ=
Threshold current density:
Threshold current:
Steady-state Photon Number
The net photon population is controlled by three processes : (a) the generation by stimulated emission, (b) the absorption through various other processes, e.g., free-carrier absorption, interface scattering, inter-valence band absorption and (d) the spontaneous emission.
The laser rate equation shows the photon balance
The steady-state number of photons PhspR
NG α
=−
PhPh Ph sp
NG N N R
tα
∂= − +
∂where Nph is the total number of photons in the cavity.
• GNph is the rate of photon generation through stimulated emission• αNph is the rate of photon decay through absorption and end-surface loss, • Rsp is the rate of spontaneous emission.
In the absence of spontaneous recombination (Rsp = 0), the lasing will not initiate: for NPh(0)=0 YdN/dt = 0
Output Optical Power
where αr is the loss in the resonator and αm is the mirror loss; for R1 = R2:
thint i ph
I IP E
qη
−= ⋅ ⋅
1 1m ln
L Rα ⎛ ⎞= ⎜ ⎟
⎝ ⎠
When the laser current exceeds the threshold, all the additionally supplied e-h pairs convert their energy into stimulated emission.
Therefore, the “internal” optical power (no light is leaking through the mirrors) is:
where ηi is the internal quantum efficiency and Eph is the photon energy
A fraction of the power is coupled out through the mirrors (cleaved facets). The OUTPUT power for identical mirrors:
th mout i ph
r m
I IP E
qα
ηα α
−= ⋅ ⋅ ⋅
+
The photon lifetime in lasers τp, is the time the photon spends in the resonator before being emitted or absorbed.
Typically, τp ≅ 4…10 single-path travel times (2..5 roundtrip times).Photon life time estimate:ttr1 = L / v0; L = 100 µm = 1E-4 m; v0 = 3E8/nr m/s; nr≅ 2.5Single path travel time: ttr1 = 0.75 E-12 s = 0.75 ps;tph = 4 … 10 ttr1 = 3 … 7.5 ps => very fast modulation is possible
Photon life time and laser modulation speed
Laser emission spectrum
Just below the threshold
Above the threshold (@ 5mW)
Laser –LED spectrum comparison
Above the threshold (@ 5mW)
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Wavelength, um
Pow
er d
ensi
ty
1.52 um LED