Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors

Подождите немного. Документ загружается.

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

Semiconductor Laser 339

(ω)=



∞

−∞

dτ e

−iωτ

E

∗

(τ)E(0)

= E

∆ω

(ω − ω

)

+(∆ω/2)

, (17.75)

where

∆ω =

4ω

(1 + α

) . (17.76)

Eq. (17.76) is the Schawlow–Townes line-width formula, except for the ad-

ditional term α

.Sinceα

is often larger than one in semiconductor gain

media, the laser line width usually exceeds the Schawlow–Townes limit.

Hence, the expression line-width enhancement factor for α. The expres-

sion (17.76) for the semiconductor laser line width has been derived ﬁrst

by Haug and Haken (1967), but only later has it been recognized that the

line width enhancement factor α in semiconductors can be much larger

than unity, because generally the density-dependent refractive changes of

the complex gain function are quite large in semiconductors as compared

to atomic systems.

Fourier transformation of the full correlation function (17.73) is not

possible analytically. In order to show the basic features, we ignore the

term proportional B

and approximate exp(−|τ|/τ

)  1, β  ω

, assum-

ing weakly damped relaxation oscillations. We use the associated series

(Abramowitz and Stegun, 1972) for the modiﬁed Bessel functions I

write

[cos(ω

τ )−1]

= I

)+2

∞



n=1

)cos(nω

τ) . (17.77)

Inserting our approximations and Eq. (17.77) into Eq. (17.73), and evalu-

ating Eq. (17.75) yields

(ω)=E

∆ω

∞



n=−∞

−B

)

(ω − ω

− nω

)

+(∆ω/2)

, (17.78)

showing that the laser spectrum consists of a series of lines at ω = ω

+nω

An example of the spectrum is shown in Fig. 17.2. The main peak is at

and the sidebands occur at multiples of the carrier relaxation oscillation

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

340 Quantum Theory of the Optical and Electronic Properties of Semiconductors

-4 -2 0 2 4

Normalized Detuning

Normalized Spectrum

Fig. 17.2 Plot of the main mode and the ﬁrst two sidemodes of the normalized laser

spectrum, Eq. (17.75). For illustration, we choose the frequency of the relaxation oscil-

lations ω

=2, B

=1, ∆ω/ω

=0.25.

frequency. Such spectra, usually with stronger damped sidemodes, are in-

deed observed in semiconductor lasers above threshold, see, e.g., Thompson

(1980) or Agrawal and Dutta (1986).

17.5 Nonlinear Dynamics with Delayed Feedback

A single-mode semiconductor laser diode displays a rich scenario of nonlin-

ear dynamical eﬀects with bistable, quasi-periodic and chaotic behavior, if

one provides an optical feedback which couples a part of the emitted light

ﬁeld E(t −τ) with an appropriate delay τ back into the resonator. In order

to get these instabilities, the delay-time has to be of the order of an in-

verse relaxation oscillation frequency, i.e. τω

 1. Because the relaxation

oscillation frequencies are typically in the MHZ-region, experimental real-

izations typically use the ﬁeld from a distant optical reﬂector.

In the presence of delayed feedback, the appropriate ﬁeld equation in

the slowly varying amplitude approximation is



− iω



(t)=



G(N) −



(t)+ηE

(t − τ) , (17.79)

where η is the strength of the feedback. The intensity gain G diﬀers by a

factor of 2 from the amplitude gain g deﬁned in (17.1). It is considered here

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

Semiconductor Laser 341

as a complex gain function, i.e. G



=2g, while the imaginary part describes

the dispersive inﬂuence. In terms of the complex dielectric function, the

complex gain is G =8πiωχ/(n

c). Similarly, the inverse photon lifetime

is related to the cavity loss κ by T

−1

=2κ. The material equation is

according to (17.2)

= r

− r

= r

− G



I(t) −

. (17.80)

Writing the ﬁeld E

√

i(ω

t+Φ)

in terms of the intensity I(t) and a

phase Φ(t), the ﬁeld equation yields





−



I(t)+2η



I(t)I(t − τ)cos



τ +Φ(t) − Φ(t − τ)



(17.81)

and

dΦ



(N) −η



I(t − τ)

I(t)

sin



τ +Φ(t) − Φ(t − τ)



. (17.82)

Next, we simplify the gain by linearizing it around its value at threshold.

With ∆N = N −N

thr

, we get for the real part of the gain



(∆N)=G



thr

∂G



(N)

∂N

∆N =

+ A∆N, (17.83)

where we used the fact that at threshold the gain equals the losses

.For

the imaginary part, we introduce the line-width enhancement factor in the

form

α = −

∂G



/∂N

∂G



/∂N

, (17.84)

compare (17.72) and write



(∆N)=G



thr

∂G



∂N

∆N = G



thr

) − αA∆N. (17.85)

For suﬃciently strong laser ﬁelds, it may happen that the gain is spectrally

deformed since the stimulated emission predominantly removes carriers that

are in states energetically close to the laser resonance. This spectral hole

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

342 Quantum Theory of the Optical and Electronic Properties of Semiconductors

burning is often treated phenomenologically by introducing an intensity-

dependent gain in the form



(∆N,I)=



(∆N)

1+I(t)

 G



(∆N)



1 − I(t)



, (17.86)

where the expanded form can be used for weak saturation. This approxima-

tion can indeed be motivated by detailed studies of the scattering kinetics

(see Schuster and Haug (1996)). Ignoring a constant phase shift, we obtain

the following equations for the intensity, phase, and the deviations of the

total carrier number from its threshold value





(∆N,I)−



I(t)+η



I(t)I(t−τ )cos



τ +Φ(t) − Φ(t−τ)



(17.87)

dΦ

= −

A∆N(t) − η



I(t − τ)

I(t)

sin



τ +Φ(t) − Φ(t − τ)



, (17.88)

and

d∆N

=∆r

− G



(∆N,I)I(t) −

∆N

. (17.89)

These equations are know as the Lang–Kobayashi equations.

As an application, we now analyze under which conditions a weak level

of feedback can transform the relaxation oscillations of a semiconductor

laser into self-sustained oscillations. In the relaxation oscillations, the car-

rier and photon numbers oscillate out of phase about their mean value.

Because the oscillations are damped, they are usually present only after

fast switching processes or as a consequence of noise. The delayed external

feedback, however, can counteract the damping leading to stable oscilla-

tions. In terms of the parameters of the Lang-Kobayashi, (17.87) – (17.89),

the relaxation oscillation frequency (17.66) becomes

∂G



∂N

∆r

. (17.90)

This form shows that the relaxation oscillation frequency varies as the

square root of the pump rate above threshold. Decomposing the phase

into a part linear in time and an oscillating one, we get

Φ(t)=∆ω

t + φ(t) . (17.91)

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

Semiconductor Laser 343

The time average <dΦ(t)/dt >=∆ω

t describes the shift of the laser fre-

quency by the feedback. For external laser modes, which obey the condition

= ω

+∆ω

(2M − 1/2)π

, (17.92)

where M is a positive integer, the trigonometric functions simplify to

cos



τ +Φ(t) − Φ(t − τ)



=sin



φ(t) − φ(t − τ )



, (17.93)

and

sin



τ +Φ(t) − Φ(t − τ )



= −cos



φ(t) − φ(t − τ )



. (17.94)

The cos term can be replaced by its temporal average. Without this replace-

ment, one would get additionally the contributions of the higher harmonics

2mω with m =1, 2,... . A detailed analysis shows further that under the

chosen conditions one can use the following simpliﬁcations



I(t)I(t − τ)  I

and



I(t − τ)/I(t)  1 , (17.95)

where I

is the stationary mean intensity value. As only oscillations with

small amplitudes will be considered, one can linearize the product G



(t)I(t)

around its stationary value. Denoting the stationary values as I

and N

we deﬁne the oscillatory deviations p(t) and n(t) via

I(t)=I



1+p(t)



and N(t)=N

n(t)

∂G



/∂N

. (17.96)

The mean photon number is

=∆r



, where T



1+/(AT

)

 T

. (17.97)

The mean shift of the laser frequency is



Φ = ηcos



φ(t) − φ(t − τ)



 , (17.98)

and the stationary carrier density is

= N

thr

−

∂G



/∂I

∂G



/∂N

. (17.99)

The equations of motion for p(t),φ(t), and m(t) are

= −2Γ

p(t)+

n(t)

+2η sin



φ(t) − φ(t − τ)



, (17.100)

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

344 Quantum Theory of the Optical and Electronic Properties of Semiconductors

dφ

n(t) , (17.101)

and

= −2Γ

n(t) −

Ω

p(t) . (17.102)

Here,

Ω is the relaxation oscillation frequency of the solitary laser

Ω

− (Γ

+Γ

)

 ω

. The damping rates Γ

and Γ

are given by

2Γ

= −

∂G/∂I

∂G/∂N

and 2Γ

+ ω

. (17.103)

An approximative solution of these equations is obtained in the form of

simple harmonic oscillations

φ(t)=

cos(ωt) , (17.104)

p(t)=a

cos(ωt − µ

),a

(ω/ω

)

α cos(µ

)

,µ

=arctan(

2Γ

) , (17.105)

and

n(t)=−a

sin(ωt) with a

ωT

. (17.106)

The amplitude and period of the limit cycle is obtained from the equations

of motion using the expansion sin(b sin(x))  2J

(b)sin(x),whereJ

(b) is

the ﬁrst order Bessel function. With b = a

sin(

ωτ

) one gets the implicit

equation

(b)

, (17.107)

where the critical feedback strength η

for the formation of the limit cycle

with the period ω is given by





+Γ

α sin

(ωτ/2)

. (17.108)

Using a quadratic expansion of the Bessel function 2J

(b)/b  1 −b

/8 one

gets a square root law of the form

b =2

√



1 −

, (17.109)

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

Semiconductor Laser 345

which is typical for a so-called Hopf bifurcation. The frequency of the limit

cycle is found from

−

2(Γ

+Γ

)

tan(ωτ/2)

. (17.110)

This equation has in principle an inﬁnite series of solutions. As an example,

we consider feedback levels η ≥ η

, where only two solutions ω

with i =1, 2

are possible. In Fig. 17.3, we compare the analytical results to numerical

solutions of the full equations. Fig. 17.3a shows the dominant characteristic

frequencies of the two solutions and Fig. 17.3b displays the amplitudes of

the intensity oscillations which are marked by the arrows in Fig. 17.3a. In

Fig. 17.3c, we plot the parameter ranges in which the diﬀerent stationary,

single frequency (monostable) and bistable solutions appear.

(b)

0.2

0.0

-0.2

3.05

3.10

3.15

(c)

monostable

bistable

stationary

()w

(,)ww

1 2

1.00

1.05

1.10

0.95

0.90

3.05

3.10

3.15

W(J) [GHZ rad]

(a)

3.05

3.10

3.15

3.4

3.2

3.0

2.8

i=1

i=2

[Ghz rad]

a( )

hh/

c,x

Fig. 17.3 Two bistable limit cycles as a function of the relaxation frequency Ω(J)=

),whereJ = r

is the pump current. (a) Frequencies of the two limit cycles. (b)

Amplitudes of the oscillating photon density. The circles give the results of the numerical

integration. (c) Critical feedback rates for the various types of solutions. The parameters

are α =6,T

=2ps,T

=2ns,τ=1ns, (∂G



/∂I)/(∂G



/∂N) = −4.6. [According to

Ritter and Haug (1993).]

The numerical solutions of the Lang-Kobayashi equations and the ex-

perimental ﬁndings of Mørk et al. (1990) conﬁrm the calculated scenario

for small feedback levels quite well. At higher feedback levels frequency

doubling of the oscillations and quasi-periodic solutions arise, which ﬁnally

go over into chaotic solutions. These chaotic solutions are characterized by

strange attractors, which can be generated, e.g., by plotting the successive

intensity maxima I

max

n+1

versus I

max

of the oscillations.

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

346 Quantum Theory of the Optical and Electronic Properties of Semiconductors

REFERENCES

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions

(Dover Publ., 1972)

For the general laser theory see:

H. Haken, Laser Theory, Handbuch der Physik Vol. XXV/2c, Springer,

Berlin 1970

M. Lax, in 1966 Brandeis University Summer Institute of Theoretical

Physics Vol. II, eds. M. Chretien, E.P. Gross, and S. Deser, Gordon and

Breach, New York 1968

M. Sargent, M.O. Scully, and W. Lamb, Laser Physics, Addison-Wesley,

Reading M. A. 1974

For the description of semiconductor lasers see:

A.Yariv,Optical Electronics,3rded., Holt, Rinehart and Winston, New

York 1985

G.H.B. Thompson, Physics of Semiconductor Laser Devices, Wiley, Chich-

ester 1980

G.P. Agrawal and N.K. Dutta, Long-Wavelength Semiconductor Lasers,

Van Norstrand Reinhold, New York 1986

W.W. Chow and S.W. Koch, Semiconductor-Laser Fundamentals, Springer,

Berlin, 1999

For semiconductor laser noise theory see:

H. Haug, Z. Physik 200, 57 (1967); Phys. Rev. 184, 338 (1969)

H. Haug and H. Haken, Z. Physik 204, 262 (1967)

K. Vahala and A. Yariv, IEEE Journ. Quantum Electron. QE-19, 1096

and 1102 (1983)

Y. Yamamoto and S. Machida, Phys. Rev. A35, 5114 (1987)

H. Haug and S.W. Koch, Phys. Rev. A39, 1887 (1989)

For an analysis of the gain saturation see e.g.:

S. Schuster and H. Haug, JOSA B 13, 1605 (1996)

For a treatment of the nonlinear dynamics of semiconductor lasers with

weak optical feedback see e.g.:

January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2

Semiconductor Laser 347

R. Lang and K. Kobayashi, IEEE JQE-16, 347 (1980)

A. Ritter and H. Haug, JOSA B 10, 130 and 145 (1993); and IEEE JQE-29,

1064 (1993)

J. Mørk, J. Mark, and B. Tromborg, Phys. Rev. Lett. 65, 1999 (1990)

PROBLEMS

Problem 17.1: Show that the commutator relation of the harmonic oscil-

lator is violated if one includes only dissipation in the Heisenberg equation

for b and b

†

Problem 17.2: Derive Eq. (17.18) for the ﬁeld amplitudes from the wave

equation (17.14). Use the eigenmode expansion (17.15) and χ(N =0)=0,

since the background susceptibility has been included in 

Problem 17.3: Use the general ﬂuctuation–dissipation theorem,

Eq. (17.47), to derive the diﬀusion coeﬃcients, Eqs. (17.50) – (17.52).

Problem 17.4: Use the expansion (17.59) to derive Eq. (17.60) from

Maxwell’s wave equation (17.18). Hint: Neglect all second-order derivatives

of the perturbations, all products of perturbations, terms proportional to

(ω

− ω

)E, σE, σφ, and use



4πχ(N

)











Problem 17.5: Solve Eqs. (17.63) – (17.65) using Laplace transformations.

Problem 17.6: Use the solutions of problem 17.5 to compute φ(t+τ)φ(t).

Hint: Keep decaying terms ∝ exp(−ω|τ|), but neglect all terms propor-

tional exp(−ωt), for and ω>0.

This page intentionally left blank