Transcript ECE 245

L8 Lasers UConn ECE 4211 03/10/2015 F. Jain
•Conditions of Lasing
•Threshold Current Density Jth
•Reduction of Jth: Heterostructure Lasers
•Optical Power-Current Behavior
•Carrier confinement in a double Heterostructure (DH) laser
•Laser Design Exercise
•Quantum Well/Wire/dot Lasers
•Distributed Feedback Lasers
Operating parameters:
Operating wavelength: green, red, blue, fiber optic wavelength 1.55 microns
Optical power output, expected external and wall conversion efficiency,
Operating structure
Cavity or Distributed Feedback type
Edge emitting or surface emitting.
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General Conditions of Lasing:
Rate of emission = Rate of absorption
Rate of spontaneous emission + rate of stimulated emission =Rate of absorption
A21N2
+
B21 r(hn12) N2
=
B12 r(hn12) N1
(1) Rate of stimulated emission >> rate of absorption gives
B21 r(hn12) N2
>>
B12 r(hn12) N1
Or, ( N2 /N1) >> 1 ………Condition known as population inversion.
(Using Planck’s distribution law, we can show that B12=B21).
(2) Rate of stimulated emission >> rate of spontaneous emission gives
B21 r(hn12) N2 >>
Or, r(hn12)
>>
A21N2
A21/B21
Photon density higher than a value.
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Conditions of Lasing:
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General Conditions of Lasing:
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General Conditions of Lasing:
(a) Representing the amplitude/magnitude
(b) Phase condition
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Resonant Cavity: Condition I for Lasing
Figure2. Cavity with parallel end faces
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The emission spectrum high lighting cavity
modes
Figure3 shows the emission spectrum highlighting cavity modes
(also known as the longitudinal or axial modes) for the GaAs laser diode.
Conditions and Calculations: GaAs
λ = 0.85μm
nr = 3.59
L = 1000μm
Δλ=2.01 Å
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Equivalent of population inversion in semiconductor lasers: Condition II for Lasing
This condition is based on the fact that the rate of stimulated emission has to be greater than the rate of absorption.
B21 n2 r (hn 12 ) > B12 n1 r (hn 12 )
(23)
(27)
Strictly speaking, the rate of stimulated emission is proportional to:
(i)
the probability per unit time that a stimulated transition takes place (B21)
(ii)
probability that the upper level E2 or Ec in the conduction band is occupied


1

f e=
( E c - E fn )


 1 + e kT 
(iii)
(iv)
(v)
, Efn = quasi-fermi level for electrons.
(24)
joint density of states Nj(E=hv12)
density of photons with energy hv12, ρ(hv12)
probability that a level E1 or Ev in the valence band is empty (i.e. a hole is there)
The rate of stimulated emission:


1


f h = 1E
E
v
fp 

 1+ e kT 




1
1
= B21 * 
*
(E
=
h
)
*
r
(h
)
*
1



N
n
n
j
12
12
E c - E fn
E v - E fp
 1 + e kT 
 1 + e kT 
(25)
(26)
Similarly, the rate of absorption:




1
1
= B12 * 
*
(E
=
h
)
*
r
(h
)
*
1



N
n
n
j
12
12
E v E fp
E c E fn
 1 + e kT 
 1 + e kT 
(27)
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[1]M.G.A.
Bernard and G.Duraffourg, Physica Status Solidi, vol. 1, pp.699-703, July 1961
Using the condition that the rate of stimulated emission > rate of absorption; (assuming B 21=B12), simplifying
Equation (26) and Equation (27)




1
1
1
* 1 * 1 >

E c - E fn
E v - E fp
E v - E fp
E c - E fn
1 + e kT
 1 + e kT  1 + e kT
 1 + e kT 
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Further mathematical simplification yields if we use
E c - E v  hn 12 = hn
(28)
(29)
E fn - E fp > E c - E v
(30)
Bernard - Douraffourg Condition[1]
E fn - E fp > hn
(31)
Equation (31) is the equivalent of population inversion in a semiconductor laser. For band to band transitions hn  E g
E fn - E fp  hn  E g
(32)
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Definition of quasi Fermi-levels
E fn
E fn- E i
n = nie
kT
; n  N C e kT
E i- E fp
p = nie
kT
; p  NV e
E fp  Eg
kT
Gain coefficient g and Threshold Current Density Jth
The gain coefficient g is a function of operating current density and the operating
wavelength λ. It can be expressed in terms of absorption coefficient α(hv12) involving, for
example, band-to-band transition.
g = - o (1 - f e - f h )
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Derivation of JTH
Rate of stimulated emission
Rate of absorption
= B21feNj(E=hn) r(hn12) fh (Where r(hn12) = P vn hn ns)
= B21fefh (vg) nn Nn ns
= (1-fh)(1-fe) vg nn Nn ns B12
Net rate = Stimulated – absorption
= [fe fh – (1-fe)(1-fh)] vg nn Nn ns B21
= -[1-fe-fh]  vg nn Nn ns B21 and also note that B12 = B21)
The gain coefficient is
g = -  (1- f e - f h )
Rate of spontaneous emission
Rn  rnn s = f e f h(  ov g )Nnn s
(34)
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where:
αovg = probability of absorbing a photon
Nv = number of modes for photon per unit frequency interval
Δvs = width of the spontaneous emission line
Equation (34) gives
o =
rn n s
f e f h v g Nn n s
(35)
The total rate of spontaneous emission
(36)
Rc =  rn dn  rn n s
Rc =
(37)
J
1
I
1
  =
 
q
d A d q
where:
Rc = Rate per unit volume
η = quantum efficiency of photon (spontaneous) emission
d = active layer width
J  1
A = junction cross-section
   ( f e + f h - 1)
Equations (33), (35), (36) and (37) give
 q d 
g=
f e f h v g Nn n s
(38)
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Nn =
(39)
8 n2rn 2
2
c vg
hn -( E fn - E fp )
f e+ f h -1
= 1 - e kT
= z(t)
fe fh
(40)
Substituting for Nv and f e + f h - 1 , we get
fe fh
2
hn - 
J
c vg 
g=
1
e kT 
2 2

qd v g n s 8 nr n 
(41)
 = E fn - E fp
The condition of oscillation, Equation (15), gives
g = +
1
1
ln
2L R1 R 2
(15)
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Threshold current density
Using Equation (15) and Equation (42)
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hn - 
1
1
c vg 
J th
1 - e kT  =  + ln
2 2

qd v g n s 8 nr n 
2L R1 R2
(43)
8 n2rn 2 qdn s 
1
1 
  + ln
J th =

2
 c z(t)
2L
R1 R2 

(44)
When the emitted stimulated emission is not confined in the active
layer thickness d, Equation (44) gets modified by Γ, the confinement
factor (which goes in the denominator).
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