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L8 Lasers UConn ECE 4211 03/10/2015 F. Jain

Operating parameters:Operating wavelength: green, red, blue, fiber optic wavelength 1.55 microns

Optical power output, expected external and wall conversion efficiency,

Operating structureCavity or Distributed Feedback typeEdge emitting or surface emitting.

•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

General Conditions of Lasing:

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Rate of emission = Rate of absorption Rate of spontaneous emission + rate of stimulated emission =Rate of absorption  A21N2 + B21 r(h12) N2 = B12 (h12) N1

(1) Rate of stimulated emission >> rate of absorption gives 

B21 (h12) N2 >> B12 (h12) 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 (h12) N2 >> A21N2

 Or, (h12) >> A21/B21 Photon density higher than a value.

 

Conditions of Lasing:

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General Conditions of Lasing:

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General Conditions of Lasing:

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(a) Representing the amplitude/magnitude

(b) Phase condition

Resonant Cavity: Condition I for Lasing

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Figure2. Cavity with parallel end faces

The emission spectrum high lighting cavity modes

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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

Equivalent of population inversion in semiconductor lasers: Condition II for Lasing

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)(hnB > )(hnB 1211212221

e+1

1=f

kT

)E-E(e fnc

This condition is based on the fact that the rate of stimulated emission has to be greater than the rate of absorption.

(23)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(iii) joint density of states Nj(E=hv12)

(iv) density of photons with energy hv12, ρ(hv12)

(v) probability that a level E1 or Ev in the valence band is empty (i.e. a hole is there)

, Efn = quasi-fermi level for electrons. (24)

e+1

1-1=f

kT

E-Eh fpv(25)

e+1

1-1*)(h*)h=(EN*

e+1

1*B=

kTE-E1212j

kTE-E21

fpvfnc

e+1

1-1*)(h*)h=(EN*

e+1

1*B=

kTE-E1212j

kTE-E12

fncfpv

The rate of stimulated emission:

(26)

Similarly, the rate of absorption:

(27)

(27)

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Using the condition that the rate of stimulated emission > rate of absorption; (assuming B 21=B12), simplifying

Equation (26) and Equation (27)

e+1

1-1 *

e+1

1 >

e+1

1-1 *

e+1

1

kTE-E

kTE-E

kTE-E

kTE-E fncfpvfpvfnc

(28)

h=hE-E 12vc

E-E > E-E vcfpfn

Further mathematical simplification yields if we use

(29)

(30)

Bernard - Douraffourg Condition[1]

    [1]M.G.A. Bernard and G.Duraffourg, Physica Status Solidi, vol. 1, pp.699-703, July 1961

h > E-E fpfn(31)

Equation (31) is the equivalent of population inversion in a semiconductor laser. For band to band transitions Eh g

fn fp g- hE E E (32)

Definition of quasi Fermi-levels

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;

;

-E Efn i fnkT

-E Ei fp fp gkT

E

kTi C

E E

kTi V

n = n N en e

p = p N en e

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.

)f-f-(1-=g heo

Derivation of JTH

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Rate of stimulated emission

= B21feNj(E=h) (h12) fh (Where (h12) = P v h s)

= B21fefh (vg) n N s

Rate of absorption

= (1-fh)(1-fe) vg n N s B12

Net rate = Stimulated – absorption

= [fe fh – (1-fe)(1-fh)] vg n N s B21

= -[1-fe-fh] vg n N s B21 and also note that B12 = B21)

The gain coefficient is

s o g se hR = ( )f f v Nr

(34)

)f-f-(1-=g he

Rate of spontaneous emission

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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

sghe

so

Nvffr=

(35)

The total rate of spontaneous emission

sc rdr=R

q

1

d A

I=

d

1

q

J=Rc

where:Rc = Rate per unit volume

η = quantum efficiency of photon (spontaneous) emissiond = active layer widthA = junction cross-sectionEquations (33), (35), (36) and (37) give

(36)

(37)

sghe

he

Nvff

1)-f+f(d1

qJ

=g

(38)

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vc

n8=N

g2

22r

z(t)=e-1=ff

1-f+fkT

)E-E(-h

he

hefpfn

(39)

(40)

e-1n8vc

vqd

J=g kT

-h

22r

g2

sg

Substituting for Nv and ff

1-f+f

he

he, we get

(41)

E-E= fpfnThe condition of oscillation, Equation (15), gives

RR

1

2L

1+=g

21

ln

(15)

Threshold current density

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RR

1

2L

1+=e-1

n8vc

vqdJ

21

kT-h

22r

g2

sg

th ln

RR

1

2L

1+

z(t)c

qdn8=J

212

s22

rth ln

Using Equation (15) and Equation (42)

(43)

(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).