Appendix 5 Spontaneous Emission Term and Factors

Appendix 5 Spontaneous Emission Term and Factors Consider the spontaneous emission term describing the contribution of the spontaneous emission to the laser oscillation, which appears as

Appendix 5 Spontaneous Emission Term and Factors

Consider the spontaneous emission term describing the contribution of the spontaneous emission to the laser oscillation, which appears as the final term on the right-hand side of the rate equation for the photon density (Eq (6.21)) Since this term represents the component of the spontaneous emission belonging to the same mode as the laser oscillation (angular frequency !m), it is closely related to the gain for the oscillation mode Consider a laser of index-guiding type, and let E ¼ E(r) ¼ E(x, y)E(z) be the complex electric field of the oscillation mode The field is normalized in the similar manner as Eq (2.6), so that the optical energy in the resonator corresponds to the energy of a photon:

Z

c

"0nrng

2 jEðrÞj

Then, from Eqs (2.50a) and (2.57), the transition probability relevant to the photons of this mode can be written as

wabs

wstm n

¼wspt

¼ p 2h EðrÞE h 1jerj 2i

where n is the number of the photons in the resonator Using the direct-transition model, the net number of the stimulated emission direct-transition per unit volume of the active region per unit time can be calculated by integrating wstm multiplied by (1/2p3)( f2
f1) dk The average value of jE(r)j2in the active region is given by

hjEj2ia¼Gð2hh!m="0nrngÞ

Va

ðA5:3Þ

G ¼

R

ajEðrÞj2dV R

where Vais the volume of the active region,  is the confinement factor, and use has been made of Eq (A5.1) Replacing E in Eq (A5.2) by the average given by Eq (A5.3), and calculating the relative time variation in the mode photon number in the resonator of volume Va, we obtain an expression for the mode gain:

GGð!mÞ ¼G pe

2

nrng"0m2!m

jMj2ðf2
f1Þrðhh!mÞ ðA5:5Þ

In the derivation of the above expression, use has been made of Eqs (3.14)– (3.17) This result is consistent with that obtained by rewriting the material gain g given by Eq (3.16) in G ¼ vgg, and then in the mode gain G Let Rsp(!m) be the number of photons spontaneously emitted per unit time in the active region of a volume Va Then Rsp can be calculated by integrating wstm given by Eq (A5.2) multiplied by Va(1/2p3) f2(1
f1) dk

in a similar manner as above, to yield

Rspð!mÞ ¼G pe

2

nrng"0m2!m

jMj2f2ð1
f1Þrðhh!mÞ ðA5:6Þ

Accordingly, from Eqs (A5.5) and (A5.6), the spontaneous emission term and the mode gain are correlated by

nsp¼f2ð1
f1Þ

ðf2
f1Þ ¼ 1
exp hh!m

F

kBT

ðA5:8Þ

where
F ¼ Fc
Fv is the difference between the quasi-Fermi levels The parameter nspin Eq (A5.8) is referred to as the population inversion factor The optical waves in an ordinary laser structure include many radiation modes with the propagation vector not parallel to the waveguide axis The majority of the spontaneous emissions belong to such radiation modes, and therefore the guided mode component is negligibly small Therefore, the expression given by Eq (3.20) for a homogeneous semiconductor can also be used to describe approximately the spontaneous emission spectrum in a laser structure By integrating it, the total spontaneous emission Rsp can be calculated Approximating the spontaneous emission spectrum by a Lorentzian distribution with a half-width
! at half-maximum and using

300 Appendix 5

an approximation that spontaneous emission peak frequency oscillation frequency, we obtain an expression for Rsp:

Rsp¼ nre2!

pm2c3"0

jMj2f2ð1
f1Þrðhh!mÞ



Z

ð
!=2Þ2 ð!
!mÞ2þ ð
!=2Þ2 d!

¼nre2!
!

2m2c3"0 jMj

The spontaneous emission term CsN sat the end of the right-hand side of the rate equation given by Eq (6.21) was introduced, by representing the total spontaneous emission per unit volume in the active region approximately as Rsp¼N s, and by representing the component belonging

to the oscillation mode per unit volume in the active region Rsp(!m)/Vaas

CsN s¼CsRsp Therefore, we see from Eqs (A5.6) and (A5.9) that the spontaneous emission factor Csis given by

3

n2

rngVa!2
!¼

G4

4p2n2

It should be noted that the above result does not apply for lasers of gain-guiding type, where the guided mode cannot be described independently to the carrier injection Since the guided mode in gain-guiding lasers has curved wavefronts, the coupling of the spontaneous emission to the guided mode is stronger, and the value of Csis several times the value given by Eq (A5.10)

Spontaneous Emission Term and Factors 301

Spontaneous Emission Term and Factors 301...

The spontaneous emission term CsN sat the end of the right-hand side of the rate equation given by Eq (6.21) was introduced, by representing the total spontaneous emission. .. s¼CsRsp Therefore, we see from Eqs (A5.6) and (A5.9) that the spontaneous emission factor Csis given by

3

n2