of Light from a Random Amplifying with Disorder in the ...shodhganga.inflibnet.ac.in/bitstream/10603/180/1/10_chapter3.pdf · pllotonh 111 rh~, case of an amplifying or absorbing

Reflection Medium

of Light from a Random Amplifying with Disorder in the Complex

Refractive Index: Statistics of Fluctuations T I j t ~ii.s/rihution ofthe reflection coefficient for light reflected

,/rorn (I ~~ni,-tiiwz~~n.sional random amplzfjiing medium with cross-correlated

disorder rrr the reui and imaginaryparts of the refractive index is derived

using tllt rnc'thoii o f ' invariant imbedding. The statistics of jluctuations

have bc'1.11 oh~oined ,for both the correlated telegraph noise and the

G'trllssiu~~ i*,/iitt noise rnodels for the disorder. In both cases an enhanced

huckscu~~~~rrrrg i.suj)c,r-reflection with reflection coefficient grater than

liniry) ~ ~ ' . Y L I / / . S ~ C ' C ~ U U S ~ of coherent feed back due to Anderson localization

itnd colic~~t~rrr urw/)/~/ii~cr~ion in the medium. The results show that the effects

o f r n n r i o r ~ ~ ~ ~ r , ~ . 111 l i l ~ imaginary part of the refractive index on localization

o~zd suj~or - Y L J / ~ E ( , I I O I I ix quulitatively dzfferent [1,2].

A n d c r s o ~ ~ locali/atiot~ (31 is a phenomenon that is entirely based upon the

i~~tcrfercricc ~:tlcct i t 1 random medium. Since interference is a common

proper[! t1I all nave phenomenon, , i t is natural to extend electron

locali~anoii to photon localization in disordered dielectric media. There are

svtnc inlportant differences between electron transport and photon

transpor-I. 1 he ilumber of electrons is always conserved while it is not for

pllotonh 1 1 1 r h ~ , case of an amplifying or absorbing random medium.

Ilo\vevci ttrc irlrportant phenomenon which happens only in the case of

photons I S laser action in a disordered gain medium. In case of strong

scattering and gun, recurrent scattering events could provide coherent

feedback and las~ng results [4,5,6]. Such a laser is called random laser.

I t reprc,hcnrh ti ~ohercnt effect in an effectively amplifying random

mcdiu~ii

An important thcorctical study in this field was conducted by Pradhan and

Kumar / 71. I t predicted enhancement in reflected light intensity, as a

result ot the conlinement of the wave by Anderson localization and

coherent amplification by the active medium. Related works were reported

18,9]. In all tl>esc lus~ng action was predicted.

f h r random alnpl~Sying medium can be readily realized as a colloidal

suspensio~i ot dielectric microspheres in the solution of a laser active dye,

opticallj 1~~1lnpcci b! an appropriate pulsed laser. The experiment

conductetl h! I.n\\andy et a1.[4], were performed on colloidal solutions

containirip 1.hot1anlinc 640 perchlorate laser dye in methanol as active

medium. I llc I.I( J llano particles were coated with a layer of A1?O3 and

thih colloidal surl>crlsion was used as the optical scattering random

n~ediun~. I t ic, clnishion from the system exhibited spectral and temporal

propcrt~c\ il~aracrci-~sric of a multimode laser oscillator; even though the

systrni iitd no1 have any external resonant cavity. Interestingly there was

no lastils ;ictiori when there was no Ti02 in the active solution. Figure 3.2

slio\\s llleil- experimental results. It is found that the threshold excitation

energ! i t l t - 1;1ser action is very low as indicated by the line width collapse.

ire 3.1 I:spcritner~ls of'lawandy et al. 111 the tit-st picture the container shown contains rhodamine 640 dissolved in methanol as a laser active medium. When a green light \ \a\ puniped into the medium, no lasing action was b ~hscrv~.d i r i ~ h c hccond figure the same experiment repeated with added iolloid;~l ritania particles which is basically white paint. A bright ~ ~ x ~ r st1~1\\ 5 the glow due to lasing action.

In another expcrlnlcnt conducted in Rhodamine 590 perchlorate dye in an

aqucouh suspcllsion of polystyrene microspheres 1141 similar lowering of

lasing threshold and gain narrowing was observed. Addition of scatterers

indicated enhanced ernission and reduction in threshold. The important

rcsult obtained here was the observation of lasing action even in the

ballistic Ilrnlt i' , I,. where 1' is the transport mean frce path and L the

system sue. I h ~ s result was explained in terms of statistically sub-mean

frcepath scattering

Wavelength (nm)

Figure 3.2 Results ot cxperiments by Lawandy et nl., ( a ) i-,in~~sion spectrum of a 2 . 5 ~ 1 0 ' ~ M solution of ~ I I O ~ ~ I I I I I I L ~ 640 pcrchlorate in methanol pumped by 3-mJ l)ulses i32nm. (b) and (c) emission spectrum for the Ti02 nanopar~~cles colloidal dye solution pumped by 2.2 pJ and 3ln.l pulses respectively. Scaling of factors 10 and 20 were done f\)r the amplitudes of (b) and (c).

liccentl\ iaslng 'icllon was observed for the first time with coherent feed

back 111 ~ c ~ i i ~ c o i i t i ~ i ~ ~ t o r powder [lo]. They used a mixture of zinc oxide

( % n o ) I I I I ~ gall~unl nitride (GaN) powder and observed lasing emission in

a d ~ e c i ~ o n . 111 order to avoid absorption the frequency doubled output of

a 111od~ i c e I ):sapphire laser was used as the probe light. The

obser\ aiio~i 0 1 riiiitlom laser action in this semiconductor powder provided

direct c\ ~dcncc t c i ~ tlic recurrent scattering of light. The recurrent scattering

of light I , tllc kc\ Ingredient in Anderson model of photon localization.

I'hc csl>~,sI1I1crll also showed that recurrent light scattering can provide

resonani Icc~lha~~L lor lasers.

I'he expcrrnrcllt\ 17) Wiersma et a1.[5], shows enhanced amplification from

a coherc~i~l! ali~plit!ing optical disordered medium due to coherent back

scatteriil I he11 \,iniples consisted of Ti2O3 doped Tisapphire powders.

111 the ~ i s c 01 i~rlc-dimensional models considered previously [11,12,13]

the raridotii \caticl-lngs were modelled by a real Gaussian random refractive

index \\ 111) ;r c\,lislant imaginary part acting as an amplifying term. Thus

the anrpl~ ~icatrol L \ \as assumed to be present homogeneously throughout

the s\s~ciri i 1 1 r - could be valid if the localization length of light was

s~llallcl- riia~i tlic ;i~rrpliiication gain length i.e. a spatially uniform density of

lasing ~ r ~ o l c c u l c ~ 111 units of localization length. For disorder scattering

having iocali/a~lo~l length small compared to the mean free path of lasing

niolecul~~. 0 1 ~ ' I I I I length, the effective amplification term will no longer be

Ilornogcir~~,~li~ \~il l iul~~ted emission will be absent in regions where no

lasine ~ ~ i ~ ~ i c i u l ~ - , iirc present. But since strongly localized regions relatively

h. C I L . k . I ., irrol-c. i~ i~ i i ig tlireshold could be more easily attained compared

to the lack 01 ;ril~plitication in localized regions where there are no lasing

n~o lec~~ i i .~ , I t i ~ ilet effect can be modelled by assuming a spatially

depci~ti~...i~i du1l)lilication. Since the region of amplification could be

rando111 i \ c . '11-c. led to a random imaginary part for the refractive index. We

had as\~ii~ii.d [hill hi~ckground randomness produces the localization. But if

the I t i : ~ iength is larger than the sample size then instead of

dil'fi~si\ I. ir.anspirr[ \vc have ballistic regime for the photons. Lasing action

has bccr~ S C ~ I I 111 such systems [14]. AS a plausible theoretical explanation

the el'i'ec~s 01 sub slatistically rare sub localization length scatterings were

in\ohcu 1 i ' 1 . i3i1t a possibility that the randomness of amplifying

sca~terii~y ~ r c l l i.ould play a role is not ruled out. An inhomogeneous

arnplilic,r~ioi~ to1111 \vill thus lnodel super-reflection and random lasing in

such n o t i - d ~ t t u s ~ ~ c systems when it is made active by introducing lasing

i n o l c c ~ ~ l ~

As wc /la\<, hec.11 in all these studies the active random medium is

consider~:d 1 0 sci~l~cr the propagating wave due to the fluctuations in the

rcal par1 b,! the tellactive index (rlr) (real potentials) while the coherent

anll7lilic~ric11r 15 n~odelled by a phenolnenological spatially constant

inlaginar, part ot the ref'ractive index ( r l , ) . However it would be interesting

to examiilc. I I I C cltcci ot'a spatially varying part of the refractive index as

~vcll. 1 . , I > the scattering micro particles (e.g., polystyrelle

n~icrosplr~.tc, i I ( 1 . rutile particles) used in the experiments are not active,

a corre~pil~~tl~il: til~>lilatch in the imaginary part of the refractive index is

l;)ui~d io & : S I ~ I l i i i ; ~ hcen pointed out by Rubio and Kurnar [16] that a

nlis~na~cl I I I i l ~ c 1111;lginary part of the refractive index (imaginary

poteuu'il) c\ould dlways cause a concomitant reflection (scattering) in

a d d i t ~ o ~ ~ Lo the ahsorptioniamplification. Mismatch in 7, alone in an

a n ~ p l ~ t > ~ n g mcd~um (negative imaginary potential) with no mismatch in

7 , can L L I U L ~ rewllant enhancement of the scattering coefficients. In fact the

refleciior~ and ~t.ansniission coefficients can even diverge as can be seen

from tl ic slmplc example of a single imaginary 6 potential in one

diinens~on l'h15 \vould correspond to the experimental situation where the

scattercr-5 (pol) styrene microspheres, say) are suspended in a fluid with the

samc 1, (index matching fluid) in which a laser dye is dissolved and

opticall! punipcd !'he scattering caused by the fluctuations in rl, would,

theretore be cxj~~,cted to have non-trivial effects on the wave propagation

in the ~ I C ~ I L I I I I

The transmittance across a randomly amplifying and absorbing chain was

recentl) considered by Sen [17] numerically and was shown to decay

exponen~~all! wit11 the length of the chain, presumably due to localization.

But the e t l c t of lluctuation in the imaginary part of the refractive index on

lasing ill such raridom media has not been studied so far.

Wc h a w alread~ sccn the importance of the imaginary part of the refractive

index. A spatiall\ dependent imaginary refractive index will give rise to

intrinsic tluctuat~r,ils. llandomness of the imaginary part can thus give rise

to randoni n~ul t~pic scattering, which on the average does attenuation or

amplilicairot~ 1'Ii~,sc backscatterings can act as a feedback for light to

achieve I;~slng tl~rcs~lold.

We consider a one dimensional active disordered medium of length L with

a random coinplex refractive index 7 , 0 < x < L . We have neglected

polarization effects and henct: light is assumed to be a scalar wave. A

physical situation matching our model would be an ~ r ~ * doped and

pumped, polarization maintaining optical fibre intentionally disordered

along its length. We have considered only linear case of gain (absorption)

which is independent of the wave amplitude. Our treatment is for the

possibility for super-reflection ( r ) 1) for an amplifier and not an oscillator

(18,191.

3.2 The Gaussian White Ploise Disorder

We consider a plane wave with wave number ko propagating in a one

dimensional medium of length L, refractive index 17(x) where

17b) = V " ( X ) + i17,b) (3.1)

The real and imaginary p a r s are separately taken as Gaussian random

functions with mean zero and variance given by

This model is very appropl.iate in describing the case of a continuous

random medium such as a laser-dye doped gel or intralipid suspension [20]

where the fluctuations in real and imaginary parts are uncorrelated.

The wave is incident on .he left at x=O and partially reflected with

reflection amplitude R(LJ and transmitted through with transmission

amplitude T (L). Ignoring polarization effects the electric vector of the

wave satisfies the Helmholtz equation.

l 'hc 1,angevin equation for the co~nplex amplitude reflection coefficient

R(L) follows from the invariant imbedding method [6] froin the Eq.3.3. as

3.2.1 The Fokker-Planck Equation

Taking R(L)=&" the Langevin Eq.3.4 reduces to two coupled

differential equations.

According to the van-Kampcn lemma [21] these two stochastic coupled

differential equations will produce a flow of the density Q(r ,8 ) in the ( r , 8 )

space according to the stochastic Liouville equation with increasing length

of the sample L. Q ( ~ , B ) is the solution of the stochastic Liouville equation.

ar ao . We substitute for a n d - In Eq.3.6 to get a~ a~

a x cos 6' k0[$(r a.,(nR a .Q)]

sin 8 + ko [?(Ix ar + ,rx jcos8 + 2r)q, Q - - a8 a ( r % ) Q (3.8)

'1.0 get the Fokker-Planck (:quation for the probability distribution of

r and8 parameterised by L, P ( ~ , B ; L ) , Eq.3.8 has to be averaged over the

stochastic aspect which is crier all realizations of the random potentials

vR and 7, . While doing this vie encounter terms (?,Q) . To evaluate such OR

terms we use Novikov's theoi.em [22], which for the Gaussian white noise

disorder is

a ((Ix - - rr"sinB((Q)))

( ( Q ~ R ) , , ~ ) , , , = - ~ R ~ Q

+r-']7((Q)))+$((Q)) cos 8 I

After averaging out the disorder aspect in Eq.3.8 and writing

( ( ~ ( r . 6 ' ) ) ) = R ' ( ~ , B ; L ) we get the Fokker-Planck equation for the '11r

probability distribution W(r, 6 ; L),

The Fokker-Planck equation we obtained above in the ( r , ~ ) space is

difficult to solve both analytically and numerically due to the presence of

the phase factor. To remove the contribution of the phase factor from this

we do a random phase appraximation (-A) which is valid for weak

disorder. The physical meaning of random phase approximation is that the

incoming wave has to undergo multiple reflections before escaping a

localization length and in the process the wave randomises its phase. In the

work by Stone et a l [ 2 3 ] it was showed that the distribution of the phase is

insensitive to the disorder strcngth in the limit of weak as well as strong

disorder. For the weak disorder case the localization length is large and

hence the phase of the wave gets randomised. Thus one can write

P ( ~ , B ) = - ~ ( r ) . Here P(r , e ) factorises and 6' is uniformly distributed (2:) over 271

The three length scales L,, L, and L, are

L, and L, are the localization lengths associated with the disorder of the

real and imaginary parts of the refractive index respectively. L, is the

gainlattenuation length assoc ated with the average gainlattenuation. Here

a negative gives amplificatio 1 and a positive gives attenuation.

3.2.2 Results and Discussio~~

In order to elucidate the lasing action due to the randomness in the

imaginary part (finite gain l:ngth L,) and randomness in the real part

(localization length L , ) we firs1 consider the special case gr = 0 (L, =a).

In this case the probability dktribution depends only on the scaled sample

L size 1 = - and the ratio of tht: disorder and gain lengths a ,

LX

For sample length much larger than the localization length L, the

asymptotic solution of the Fokker- Planck equation is valid and is

Wm, is a normalization constalt. For gain length L,, much larger than the

amplification disorder localizzlion length L , , a + 0 and ~ , ( r ) tends to a

universal one w"(Y) , in the sense that it becomes independent of the sample

parameters. -I

W" ( r ) = - 1

Now we consider the case in which all three length scales Lr , L, and L, are

finite. The asymptotic soluticn to the full Fokker-Planck equation in this

case (corresponding to the sample size much larger than the length scales);

for L, ) 2Lr

For L, ( 2L,

where

W, is a normnalization constant.

We have plotted the asymptotic probability for the various length scales.

The two regimes L, ) 2Lr ant1 L, ( 2L, can be understood as those regimes

where the localization effects due to randomness in the real part of

refractive index are stronger than the amplification and vice versa.

When L, ( 2L, we have the probability distribution starting from the

maximum and then decreases to zero with a long tail. As gain length

increases the initial peak valrle also increases for fixed L, and L,. As gain

length decreases the tail broadens out for larger values for r making large

values of r more probable. As L, decreases the distribution gets more

sharply peaked at origin. Tkus increasing L, increases the probability of

amplification.

Figure 3.4 The probability distribution of reflectivity ~ ( r ; c o ) in the case of white noise disorder. Here the real part disorder is dominating (g, = 1 .0,g, = 0.1). The amplification parameter is a = -0.25.

For L7 ) 2L, we have the distribution starting from a finite value to a

maximum and then falling off to zero slowly. As gain length L, is

increased for fixed L, and L, the initial value and the peak decreases and

the distribution broadens o ~ t with more probability in the tail. By

increasing L, (for fixed L, and L,) the tail can be substantially reduced

and distribution to lie most around its peak. Increasing gain length does not

change the distribution much. Thus for large Lr it is the amplification

disorder that plays the dominant role. When L, gets comparable to 2L, the

peak gets closer to origin and the behaviour shifts to L, ( 2.4 case.

r

Figure 3.5 The probi~bility distribution ~ ( r ; m ) in the case of white noise disorder for a pure imaginary mismatch ( g , = 0). 'The an~plification parameter is a = -1.0

Finally we see that the Fokker-Planck equation reduces to that of Pradhan

and Kurnar [7] when g, = 0 . The full equation describes the interplay of

localization and amplificatior /attenuation due to separate randomness in

real and imaginary parts of the refractive index. It can be seen that the

disorder in a~nplification plays a nontrivial part in the amplification of

reflected light.

3.3 Correlated Telegraph Disorder

We classify the two types of random media as discrete and continuous. A

discrete random medium can be modelled as a suspension of micro-

particles and a continuous medium as gels and biological tissue. In case of

most experiments a random amplifying medium is realized as a colloidal

suspension of dielectric microspheres in the solution of a laser active dye

[4]. In such discrete random media the real and imaginary parts of the

refractive index fluctuate spatially in a correlated manner and hence can be

described by the same stoch;lstic process. A telegraph noise with a finite

correlation length is most appropriate to describe such a situation. The

gainlabsorption coefficient is always physically bounded from above and

hence it is better to describe the fluctuations in the imaginary part of the

refractive index by this dichoiornic Markov process.

We have taken 77,< = a q ( ~ ) and 7, = br7(L) where 7 ( ~ ) is the dichotomic

Markovian process which can take on the values ? A such that ( r l ( ~ ) ) = 0 ,

( 7 ( ~ , ) I ~ ( L , )) = A' exp(- r ~ , - l., l)and, a and Pare constants.

Following the invariant imbedding method the Langevin equation for the

reflection amplitude R ( L ) = ele gives two coupled stochastic differential

equations:

According to the van-Kampen lemma [21] these two stochastic coupled

differential equations will produce a flow of the density Q(r,@) in the ( r , ~ )

space according to the stochastic Liouville equation with increasing length

of the samples, i.e., Q(r ,8) is the solution of the stochastic Liouville

equation

ar a8 . We substitute for -and - In Eq. 3.1 8 and then proceeding as in the case a~ a~ of Gaussian noise we get

where

, = k ( I - r)sin 0

Let (Q) = P. and (Q?) = 4, . ' I

To average over the dichotorr.ous configurations of A we use the formulae

of differentiation of Shapiro and Loginov [24].

d 4(a(rb, dr [a) = (.C),A 14) - v(akb. b b .

So we have

(go) = a(Qn)" + r(nn). 0

aL

ap. a a a - =--[4lp, .]--[Pf,p, .]--[f ,P.] a~ ar ar ar

a a. a ---[d3(:.1-,bf4p,. a$ l - - [ . f 6 p * ] ar

In order to integrate out the phase factor from this as done in the white

noise case we proceed with a random phase approximation (RPA) which is

valid for weak disorder. Tkus one can write f ( r ,@)= - ~ ( r ) . Here

P ( ~ , B ) factorises and 6' is uniforlnly distributed over 2~

We define ( E ) , - P and = w ( ~ , Q ; L ) , to get

(3.23)

and

ap1 - + T P , = - d 2 (3.24) aL

Proceeding further and puttin;: (P,), = ~ ( r , 8 ) and ( P , , ) ~ = w ( ~ , Q ; L )

a p a p - = -2k, - + aL,P + ( a ~ , + ,?L2)w (3.25) a~ as aw an. - = A ~ ( & , + P L , ) P - ~ ~ , - - + u L , w - T w , (3.26) a~ a t3

where the two linear operators L, and L, are

C

2

A, In the limit of A2 + m, r + m and keeping - constant, the Eq.3.25 and r Eq.3.26 goes to the corresporlding Eq.3.11 in the white noise case.

The equation in the asymptotic limit L + m using the random phase

approximation is

and

2aT where A = -

'

The form of the telegraph no se equation for ~ ( r , m ) is identical to that for

the white noise case in the caq;e of pure real part disorder ( P = 0 ) . The only

2I-u change is in the coefficient A = -- In the case of pure imaginary part

k,A2 '

disorder (a = 0) again the fi~rm of the telegraphic noise equation is the

2r'a same as that in the white noise case, but with A =

'

The

imaginary part of the refractive index is always positive (absorbing) or

negative (amplifying) for ( la1 . The solutions which we obtained in the

case of white noise disorder are valid here also but in the interval 0 ( r ( 1

for the absorbing medium and 1 ( r ( m for the amplifying medium.

The complele solution for tke Eqns.2.27 and 2.28 is given by

A A+ =

- ai(r t1-rk2)

The solution can also be written in the forln

1 1 2 ~ ( ~ + ( r ) + l - ( r ) ) h

where

for + ) 2 and

j l o j p ' k $ ) - z a ' ] pa where = a* + P 2 i-, A ck =

l*,@+a.

3.3.1 Results and Discussion

The solutions for one-sided disorder in the imaginary part

The solutions exhibit three qu;llitatively different behaviour corresponding

to the choice of the paranleters a , p and a .

Figure 3.6 The probability distribution ~ ( r ; c o ) in case of correlatec telegraph noise 1, ) l K for one sided

disorder with disorder in both the real and imaginar:~ parts.

2 2 Pa Let us examine the case a + p - - = 0 , here we notice A

that

(a2 + P 2 ) - l , - ' is the localization length and pa = l C - ' , the effective

an~pl i f~ca~~tr l i Icligth. So we notice that there are two regimes the one in

\\.hick1 rllc , i~~~l ) l~f ica t ion dominates the localization and the second one

whcrc tllc iocal~/ation dominates. In the first region the disorder in the real

part ib >lnaiI and does not affect the statistics much. Further amplification

domir~a~i , . t11c localization ( a 2 + P2 - @ (0), and the solutions exhibit

monoto1iic drcrca5ing behaviour in this region (1 5 r ( w ) (Figure 3.6).

Figure 3.' I hc p~ohabllity distribution ~(r;w) in case of correlated \clcgrcipll noise for 1< ( I g for one sided disorder with disorder

111 borl1 tcal and imaginary parts.

I.or I ! . ,; I U 2 0 . localization dominates and the solutions of the

cquatll~i, t l i \ c ~ p ~ at i.-"'. The solutions for Eq.28 are valid in the range

0 1 r - I t i i t11 P ( ~ , o ) = 0 outside. If 2 A ) 1, localization

d I . i\ C. have a broad distribution with peak at r,, ) r (*) and

I,(,. [ L ) . . , ! I I L I I S I C ~ C . (Figure 3.7). The value of rmXis large for small disorder

i n the 1-1,,11 p.11.1 I ( I + /f2 - fllal 2 O), and decreases as rxincreases. The real

part ~ I ' I I I L . r ~ t i . ~ i i ~ r \ ~ index is dominating the behaviour in this region.

1 1 a 1 \ \ L i o ~ ~ \ ~ c ~ c r the region where 2A

-I j ( 1. Here amplification 14- (<. - 4 ) ~

d o n l ~ n ~ i ~ ~ \ ' I I I ~ rhi l y 28 diverges at r."'. This divergence implies that

( I , I I ~~cdhetl 'it that point. This behaviour is basically a condition on

the corrc.I,itlo~~ ler~gth and can be understood from the second condition

rem rittel, I -

, " ; , i ( i i i /iXr*' + 2/?4ai - P)]. The correlation length is (I,, = T-') , I

and the c,rlld~i~on I i i d i d for small r , where the correlation length is large.

l h e n the ~ ~ t l c c t ~ o l ~ 1s basically from a single potential barrier and thus has

a sl~arpl! ~ !c I i~ l t : t l \ a l ~ i c

The solutions for the case of a two side disorder for the imaginary part:

( / ; ~i J are s ~ r ~ l ~ l a r to the solutions for the white noise case. It should be

~lotcd t h , ~ ~ tllerc tloes not exist real L('' which falls into the physical region

of interest ( 0 <- r ;o ). Here large disorder in the imaginary part (P) causes

Figure 3.8 I he probability distribution ~ ( r ; m ) in case of correlated telegraph noise for I , ) 1, for two sided

J ~ w r d e r and pure imaginary mismatch.

tllc e l I localization to dominate. However in all cases of

1 anl~)lific~itioi~ lo1 a tin~le A and u2 + p2 -/la1 z- 0, there is a universal r

tail lor the /'(/.,a). For the case of pure imaginary disorder ( a = 0), we

see .I rnor~ot~~i~ically decreasing behaviour of ~ ( r ; o o ) with r for one-sided

disorclc~ ( ~ . I ~ I I C 3.8) and a ~ ( r ; m ) with a peak for two sided disorder

(1,'lglirc- 3 9 1.

I'hc d c ~ m a ~ l ~ Q I I validity of our treatment and the results therefrom, for the

s u p e l - : c i l c c t ~ ~ ~ ~ , lrom a random amplifying medium is restricted to

opcraiiirg, coll t l~t~o~is corresponding to below the threshold of lasing i .e . , to

the parameter regilne IL ( l r . The random ampliQing medium operating in

the rellcct~orl irlodc acts as a one sided cavity of size 1,essentially open

(11eni.c Icahlng I I I thc direction of the incident beam (of course deep inside

the med~unl. a p11oko11 i~~jected, for example, through spontaneous emission

will underg,~ intiefinitc amplification in an effectively closed cavity of size

1, Sucti an ainp1il'ic.d spontaneous emission will lead to large storage of

photon. \vhicl1 \\ 1 1 1 eventually be limited by non-linear effects in real

systenih,

As I L app~oa~lic , 11om below (lC 21,) the statistical weight of reflection

coeft i~ic~i i I ~ I O \ L \ c i l h19he1 values of reflectivity as indeed can be seen

iron1 1 igure i i , r i ~ t I l*igure 3.9 and finally at I, ) l g , we would expect the

randon1 ;~mpl~l i~. i t i) bccolne a random oscillator with self-sustaining

oscillatio~~, a1 t i r i clgc.11 111odes of the system. Thus one may suspect the

rcsults to! . I 11surc 3.6 and Figure 3.8) to lie outside the validity of

our treatment i t has been pointed out that [19] the Time Independent Wave

Iquatii>~r ! ' i ' iU'I-) and the associated stationary state scattering does not

descr~bc thc situation above the threshold of lasing when the gain-length

produci cxceccis criticality. Their numerical results based on a Time

llepe~ldcnt U.:I\C !:quation give a transmission, which grows exponentially

in timi

Figure 3.9 I Ilc l'robab~llty distribution P ( I ; c ~ ) in case of correlated t~~lrgrapti noise for lL ( I, for two sided disorder and pure

iiildgindrk inismatch.

Wc no\\ clar~l! and interpret our results in the following manner. We will

also coils~der ; I I al-rrj-l'erol set up as treated in [19]. Thus we have a gain

lncdiuiii of lerlgrl~ L between the facets with reflection coefficients rand

transmission cuei'ficients r placed between two distant absorbers. The

retlcction anti rr;~nsniission coefficients at the facets are related to the

compl~,\ \v;I\ C. \ CL tor k = k'+ikU (k" ( 0 for the case of amplification) in the

k i;, ~liediiil~i ab r

2k - Re'$ and T = - where k , is the wave vector in A: . k , , k + k ,

free s p x e out\tcli It can be readily shown that for a wave (e-'") incident

at the t 1 1 \i i'icc~ , 0. the wave amplitude at the second face at time r is

& I \ en b \

\+here , i d ' L , Int denotes the integer value, I = - and v is the

I ? - V

speed oi propagallon in the medium. It is seen that the first part on the right

hand sit l~ i \\-har i \ z bvould get from a scattering treatment based on the

'1'1W1. (Me ilnvc c~~nsidercd here the case of transmission for the ease of

corlipariht~n with 119). but the case of reflection can be treated similarly),

i.c.. as 121 as 011> Lerm is considered the expression obtained below

tl~reshol~! contlmlei analytically in the expression obtained above

tl~rcslxc~lc~ l'hc s c~or l~ l term on the right hand side is what is not contained

in this analytical ~oiitlnuation. It indeed gives the exponential growth of the

t~-ansmlt(cci alnpli~uitc as in 1191. 'This growing terrn (which may eventually

c I ~ n i t ~ c l I 7 ~r~in-lincarities considered here) essentially is a noise

1mpo5ed or1 the rel,~t~ve weak transmission noted above. Further rewriting

the secoird pnn L I ~

4 u c ncrtc ihar t h ~ \ cxponcntially growing part is at an effective frequency - . r

Note that this frequency is nothing but a rate of change of accumulated

phase shift arising ti.on1 multiple reflections at the interfaces, due to the

mismatcl~ 111 thc imaginary part of the refractive index. The growing

ainplitud~, is cxtrciirelq sensitive to the change in parameters (e.g. R , T ) of

the systeni in thc l~mit I 4 m . Indeed, in principle, it is possible to pickup

the siuall t in~te part referred to above, as it is synchronous with the incident

wave. I hus. oui rcsults in the regime above the threshold based on the

'1'1 WF (cg . . I:igurc 3.7 and Figure 3.9) represent just this synchronous part.

This in our \'leu gives an operational meaning to the results given by Eq.28

in the above-the-threshold regime and shown in Figures 3.7 and 3.9.

I'hc tre'itlnent M C h'ice chosen above for the purpose of illustration is of

course i leteiin~n~\t~c Eor the random case the interpretation has to be

probabili\l~c

3.3 Conclusion

We ha\c studied thc statistics of super-reflection from a one-dimensional

disordered s\steili \ \ i t h spatial randomness both in the real and imaginary

pans of ihc conrplc~ reli-active index. We have discussed the models of

disorder I I I I I I ~ I applicable to experimental systems such as

intcr~[~onali! disordered optical fibres with gain ( ~ r ~ ' doped) and obtained

I ~ I I tlistribution function of the reflectivity for the cases of a

\ v l ~ i ~ c - i ~ o ~ i c disorder and a correlated telegraph disorder. In both cases

enhauced rrtlcction results because of coherent feed back due to Anderson

locaii/ation and coherent amplification. In the case of white noise-disorder

thc. s i ; ~ i ~ s t ~ c i t t r i ~lualitatively different in the two regimes of the real part

disordcr dolu~nating ( L, ) 2 4 ) and the imaginary part disorder dominating

, 111 thc case of telegraph disorder, we obtain three qualitatively

differciit hellct~Ioclrs for p j r ; m ) depending on threshold conditions

i n I li)calization length, amplification length and correlation

length I'he lluctuations in the imaginary part of the refractive index is seen

to have. a nun-tr~vial and qualitatively different effect on localization and

lasing I ~ O I I I suc11 r;~ndom media. As the phenomenon considered here is

concer~l~d \ \ i t l r the issue of statistical fluctuations, we propose for it the

acronynl RAMAN (Random Amplifying Media and Noise)

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