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

Optical Bistability 309

16.2 The Carrier Equation

To obtain a dynamic equation for the total carrier density, we sum the

equation for the carrier distribution, see Eq. (12.19)

= −2Im













E(t)+



q=k

|k−q|

(t)



∗

(t)







, (16.15)

where we used the explicit expression (12.18) for the generalized Rabi fre-

quency. We can convince ourselves that the Coulomb term in Eq. (16.15)

does not contribute. To show this, we write

2Im



k,q

|k−q|

∗

=Im



k,q

|k−q|

∗

+ V

|q−k|

∗

, (16.16)

where we interchanged q ←→ k in the second half of the sum. The RHS of

this equation has the structure

Im(A + A

∗

)=Im(2ReA) ≡ 0 . (16.17)

Hence, Eq. (16.15) simpliﬁes to

= −





E(t)P

∗

(t) . (16.18)

Now, we express the time-dependent polarization through its Fourier trans-

form, which we write as product of susceptibility and ﬁeld,

= −





E(t)



dω

2π

∗

(ω)E

∗

(ω)e

iωt

, (16.19)

where we should remember, that the susceptibility function contains only

the resonant term, since P

= P

(k), Eq. (12.9). For a monochromatic

ﬁeld of the form

E(t)=E

−iω

, (16.20)

E(ω)=2πE

δ(ω − ω

) , (16.21)

Eq. (16.19) yields

= −



Im E(t) E



∗

(ω

) e

iω

. (16.22)

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

310 Quantum Theory of the Optical and Electronic Properties of Semiconductors

To take the imaginary part, we write

Im E(t) E

∗

(t)



∗

(ω

)=|E(t)|



∗

(ω

) . (16.23)

Hence, Eq. (16.22) becomes



|E(t)|



(ω

) , (16.24)

whereweevaluatedthek-summation. Expressing the imaginary part of

the susceptibility in terms of the intensity absorption coeﬃcient, we get

= |E(t)|

4π

α(ω

)

ω

α(ω

)

ω

I, (16.25)

where we introduced the intensity

I = |E(t)|

4π

. (16.26)

Note, that in the spirit of the quasi-equilibrium approximation, the ﬁeld

amplitude and therefore also the intensity can be slow functions of time.

Replacing ω

by ω and phenomenologically adding carrier recombination

and diﬀusion, we write the rate equation for the carrier density N as

∂N

∂t

= −

α(ω, N)

ω

I + ∇D∇N . (16.27)

electron–hole–pair rate equation

Here, τ is the carrier life time and D is the electron–hole–pair diﬀusion

coeﬃcient, also called the ambi-polar diﬀusion coeﬃcient.

Eqs. (16.12) and (16.27), together with an expression for the nonlinear

susceptibility determine the quasi-equilibrium nonlinear optical response of

semiconductors. Note, that this approach involves the adiabatic elimination

of the polarization dynamics, which is often justiﬁed if the ﬁeld variations

are slow compared to the fast phase destroying processes.

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

Optical Bistability 311

16.3 Bistability in Semiconductor Resonators

Dispersive optical bistability or bistability through decreasing absorption

may be obtained if a semiconductor is brought into an optical resonator,

which introduces the required feedback for the light ﬁeld. In the following,

we consider a Fabry–Perot resonator (Fig. 16.1) consisting of two lossless

mirrors of reﬂectivity R and transmissivity T =1−R. The nonlinear semi-

conductor material ﬁlls the space between the mirrors. In many practical

applications, the mirrors are actually the end faces of the semiconductor

crystal itself, and R is just the natural reﬂectivity, or R is increased through

additional high reﬂectivity coatings evaporated onto the semiconductor sur-

faces.

As shown schematically in Fig. 16.1, it is useful to decompose the com-

plex transverse ﬁeld amplitude E into the forward and backward propa-

gating parts to treat the feedback introduced by the mirrors. We write

E = E

+ E

= ξ

−iφ

+ ξ

iφ

, (16.28)

where the amplitudes ξ and the phases φ are real quantities. Depending

on the phase relation between E

and E

, there can be either constructive

or destructive interference, leading to a maximum or minimum of the light

intensity transmitted through the etalon.

R,T

Fig. 16.1 Schematic drawing of a Fabry-Perot etalon. The mirrors have reﬂectivity R,

transmissivity T =1− R,andE

are incident, forward traveling, backward

traveling, and transmitted ﬁeld, respectively.

Dominantly dispersive nonlinearities can be observed in semiconductors,

since the refractive index of the medium changes via the carrier density

with the light intensity, causing the optical path between the mirrors of

the etalon to change with intensity. These intensity-induced changes tune

the etalon in or out of resonance with light of a ﬁxed frequency. In reality,

absorptive and dispersive changes occur simultaneously, but either one may

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

312 Quantum Theory of the Optical and Electronic Properties of Semiconductors

be dominant in a particular frequency regime.

The boundary conditions of the Fabry-Perot resonator can be written

(z =0)=

√

T E

√

R E

(z =0) (16.29)

(z =0)=

√

R E

(z = L)e

iβ/2−α

tot

L/2

(16.30)

(z = L)=

√

= E

(z =0)e

iβ/2−α

tot

L/2

, (16.31)

yielding



tot

L/2

− Re

−α

tot

L/2

)

+4Rsin

(β/2)

, (16.32)

whereweused

cos(β)=1− 2sin

(β/2) .

The eﬀective absorption α

tot

, as well as the phase shift β still have to be

computed. The total phase shift of the light after passing through the

resonator can be written as

β = φ

(z = L) − φ

(z = L) − 2δ, (16.33)

where 2δ contains all carrier-density-independent phase shifts of the linear

medium and of the mirrors. If we ignore transverse variations, and insert

Eq. (16.28) into Eq. (16.12), we obtain for the ﬁeld amplitudes



∂

∂t

∂

∂t

α(ω,N)



F,B

=0 , (16.34)

and for the phases



∂

∂z

∂

∂t



F,B

= ∓

ω∆n(ω, N)

, (16.35)

where the minus sign is for φ

Most semiconductor bistability experiments are done for resonator

lengths L  0.5 − 2µm. Under these conditions, the round-trip time for

the light in the resonator is substantially shorter than the carrier relaxation

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

Optical Bistability 313

time τ and it is justiﬁed to adiabatically eliminate the dynamics of the light

ﬁeld (bad cavity limit),

∂ξ

∂t

 0 and

∂φ

∂t

 0 . (16.36)

Solving Eqs. (16.34) and (16.35) for these conditions yields

F,B

(z)=ξ

F,B

(0)exp

−







N(z



)



(16.37)

and

F,B

(z)=∓





∆n



N(z



)



, (16.38)

showing that α

tot

and β in Eq. (16.32) are given by

tot



dz α[N (z)] (16.39)

and

= −

δ +



dz ∆n



N(z)



. (16.40)

The spatial carrier distribution N (z) has to be computed from Eq. (16.27).

To describe typical semiconductor etalons, it is often a good approxi-

mation to neglect all spatial density variations (diﬀusion dominated case).

Then one can trivially evaluate the integrals in Eqs. (16.39) and (16.40),

and Eq. (16.32) becomes



α(ω,N)

− Re

−α(ω,N)



+4R sin



δ + ω∆n(ω, N)



(16.41)

transmision through a resonator

Eq. (16.41) is the well-known equation for the transmission of a Fabry-

Perot etalon. This transmission exhibits peaks whenever the argument of

the sin

-term equals integer multiples of π.

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

314 Quantum Theory of the Optical and Electronic Properties of Semiconductors

Through α and ∆n, Eq. (16.41) is coupled to the spatially averaged rate

equation (16.27)

= −

α(ω,N)

ω

I, (16.42)

where I has to be taken as the average intensity inside the resonator

I  I

1+R

. (16.43)

Note, that the incident intensity I

in Eq. (16.41) may still be time-

dependent, but all variations have to be slow on the time scale of the

resonator round-trip time (adiabatic approximation).

To explicitly solve Eqs. (16.41) – (16.43) for an example of practical

interest, we use the band-edge nonlinearities of room-temperature GaAs

(Fig. 15.8) as obtained from the theory in Sec. 15.3. Inserting the computed

α(N) and ∆n(N ) into Eqs. (16.41) – (16.43) allows us to directly study

nonlinear optical device performance. We obtain bistable hysteresis curves

by plotting transmitted intensity versus input intensity. In Fig. 16.2, we

show some typical results for slightly diﬀerent resonator lengths L, i.e., for

diﬀerent detunings of the excitation frequency ω with respect to the nearest

resonator eigenfrequency ω

<ω,where

= m

πc

,m=0, 1, 2, ··· . (16.44)

0 10 20 30

Transmitted Intensity

Incident Intensity (kW/cm )

Fig. 16.2 Optical bistability for GaAs at T =300K, as computed from Eq. (16.41)

using the absorption and refractive index data shown in Fig. 15.8. The mirror reﬂectivity

R =0.9, the resonator lengths are 2.0168 µm (1), 2.0188 µm (2), and 2.0233 µm(3),

respectively, and the operating frequency ω is chosen such that (ω − E

)/E

= −4.

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

Optical Bistability 315

Curve 2 in Fig. 16.2 shows marginal and curve 3 shows well developed opti-

cal bistability for some range of input intensities, whereas curve 1 exhibits

only nonlinear transmission.

For a situation similar to the one in curve 3 of Fig. 16.2, a stability ana-

lysis shows that the intermediate branch with the negative slope is unstable

and therefore not realized under usual experimental conditions. Hence,

increasing the incident intensity I

from 0 leads to a transmitted intensity

that follows the lower bistable branch until I

reaches the switch-up

intensity, denoted by B in Fig. 16.3, then I

follows the upper branch.

On the other hand, lowering I

from an original value I

>Bleads to a

transmission following the upper branch, until I

reaches the switch-down

intensity, denoted by A in Fig. 16.3.

Optically nonlinear or bistable semiconductor etalons may be used as

all-optical logic or switching devices, see Gibbs (1985). Since the bistable

etalon maintains one of two discrete output states for some range of input

intensity, it is possible to use it in the so-called latched mode.Inthatcase,

one divides the total input beam into several beams, which may serve as

holding beam or switching beam(s). The holding beam has an intensity

positioned between the switch points of the bistable loop (A and B in

Fig. 16.3) and is used to bias the device. The switching beam — which

actually is the input logic beam — then only has to be suﬃciently large

for the total incident intensity to exceed the switch-up threshold B.The

device, once switched, would remain in a high transmission state even if

the switching beam is removed. It has to be turned oﬀ by interrupting its

bias power or by some other mechanism.

Fig. 16.3 Optical bistability for GaAs at T =300K, where B, A denote the switch-up

and switch-down intensities.

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

316 Quantum Theory of the Optical and Electronic Properties of Semiconductors

In this latched mode of operation, one can obtain signal ampliﬁcation.

When a nonlinear etalon is operated as a passive device, it obviously has

no overall gain relative to the total incident power. However, it is possible

to achieve diﬀerential gain, in which case the device is able to transmit a

larger signal than the signal used to switch it. Diﬀerential gain makes it

possible to use the output of one device as input (switching beam) for one

or more other devices. This process is called cascading and the number of

devices that can be switched with the output of a single device is usually

referred to as fan-out. Using two beams as input in addition to the proper

bias beam makes it possible to realize all-optical gates which perform logic

functions, such as AND, OR, or NOR.

16.4 Intrinsic Optical Bistability

From a conceptual point of view, the simplest example of optical bistability

is obtained, if one considers a medium whose absorption increases with in-

creasing excitation density. Bistability in such a system may occur without

any external feedback since the system provides its own internal feedback.

Increasing the carrier density leads to an increasing absorption that causes

the generation of even more carriers, etc.. There are numerous mechanisms

which may cause such an induced absorption in semiconductors and other

systems. Here, we concentrate on the induced absorption which is observed

in semiconductors like CdS at low temperatures, as a consequence of the

band-gap reduction (Koch et al., 1985). However, most of the macroscopic

features discussed below are quite general and may very well also occur in

other systems.

In Fig. 16.4, we show the absorption spectra for CdS, which have been

computed using the theory of Sec. 15.3. For some frequencies below the

exciton resonance, we see that the absorption increases with increasing car-

rier density. Assuming now that the semiconductor is excited at such a

frequency below the exciton, one has only weak absorption for low intensi-

ties. Nevertheless, if the exciting laser is suﬃciently strong, even this weak

absorption generates a density of electron–hole pairs which causes a reduc-

tion of the semiconductor band gap. Eventually, the band edge shifts below

the frequency of the exciting laser giving rise to a substantially increased

one-photon absorption coeﬃcient. Consequently, the absorption increases

with increasing carrier density.

Since we want to emphasize here the general aspects of increasing ab-

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

Optical Bistability 317

0.5

-2 -1.5 -1 -0.5 0

(h - E )/Ew

aa/

N=1x10

N=4x10

N=2x10

N=8x10

Fig. 16.4 Computed absorption spectra for CdS at T =30K. These results have been

obtained using the theory of Sec. 15.3, with the parameters: m

=0.235, m

=1, 35,



=8.87, a

=30.1 Å, E

=27meV, Γ=0.04E

,andα

=10

/cm.

sorption optical bistability, we do not use the original CdS data, but a

simple generic model:

α(N)=











,N<N

+(α

− α

)sin



π(N−N

)

2(N

−N

)



<N<N

(16.45)

Here, α

and α

denote the low and high absorption values.

The coupling between carrier density N and light intensity I is described

by Eq. (16.27), which in the stationary, spatially homogeneous case leads

α(N)=

ω

. (16.46)

This relation can be bistable, as indicated in Fig. 16.5, where we plot α(N),

Eq. (16.45), and the straight line, which is the RHS of Eq. (16.46). The

slope of this straight line is inversely proportional to I. The intersection

points with the curve α(N ) are the graphical solutions of Eq. (16.46), clearly

showing the occurrence of three simultaneous solutions indicating intrinsic

optical bistability without resonator feedback (see Fig. 16.6). It is worth-

while to stress at this point that for such an induced absorption bistability

only a single pass of the light beam through the medium is required. Hence,

one has no superposition of forward and backward traveling waves as, e.g.,

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

318 Quantum Theory of the Optical and Electronic Properties of Semiconductors

a(N)

Nh / Iwt

Fig. 16.5 Graphical solution of Eq. (16.46) using α(N) given by Eq. (16.45).

in the Fabry-Perot resonator. Therefore, one may neglect nonlinear disper-

sive eﬀects as long as one is only interested in the characteristic variation

of the transmitted intensity and not in transverse beam-proﬁle variations

or in diﬀraction eﬀects.

To obtain an equation for the light intensity, we multiply Eq. (16.12)

by E

∗

and add the complex conjugate equation



∂

∂t

∂

∂z

α(N)



∗

= −

2kn

E∇

∗

− c.c.

. (16.47)

The RHS of Eq. (16.47) describes beam diﬀraction. This eﬀect can be

neglected if the length over which the light propagates is much smaller

Fig. 16.6 Bistability of the carrier density N as function of intensity I, obtained as

solution of Eq. (16.46). The lines with the arrows indicate switch-up of carrier density

N when I is increased from 0, and switch-down when I is decreased from a suﬃciently

high value.