1 October 1999

Ž .Optics Communications 169 1999 177–181www.elsevier.comrlocateroptcom

Bistable behavior of transmittivity of an ultrathin film comprisedof linear molecular aggregates

V.A. Malyshev 1, H. Glaeske, K.-H. Feller )

Fachhochschule Jena, Fachbereich MedizintechnikrPhysikalische Technik, Tatzendpromenade 1b, D-07745 Jena, Germany

Received 5 May 1999; accepted 12 July 1999

Abstract

Ž .A theoretical study of transmittivity of an ultrathin film of thickness smaller than an optical wavelength consisting oflinear molecular aggregates is carried out. We show that within a certain domain of the incident field amplitude, opticaltransitions of a single disordered aggregate, the eigenstates of which are represented by Frenkel exciton states, can be treatedin the frame of two- level approximation. This allows us to claim that an ensemble of linear aggregates is analogous to thatcomprised of inhomogeneously broadened two-level systems. Based on this analogy, we report that transmittivity of the film

Ž .can manifest a bistable behavior originating from coupling of aggregates via the emission far-zone field. Estimates ofdriving parameters show that such well studied objects as J-aggregates of polymethine dyes provide a real basis to realize it.q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 42.65.Pc; 78.66.-wKeywords: Ensemble of molecular aggregates; Optical bistability; Disorder

Since the last time, the problem of bistable behav-ior in optical response of a single linear molecular

w x w xaggregate 1–4 and even of a dimer 1,5–7 cur-rently created much interest. The effect consists of asudden switching of the population from a low levelto a higher one as the pump intensity rises andoriginates from the dynamical resonance frequencyshift depending on the population of the system. Theshift mentioned reflects, in fact, the fermionic char-

) Corresponding author. Tel: q49-3641-205621; fax: q49-3641-205601; e-mail: feller@mt.fh-jena.de

1 Permanent address: All-Russian Research Center ‘‘VavilovState Optical Institute’’, Birzhevaya Liniya 12, 199034 Saint-Petersburg, Russia.

acter of elementary excitations of an aggregate beingw xFrenkel excitons 8 . An experimental realization of

the bistable behavior of an aggregate would give us areal chance to create an all-optical switching unit,which being integrated in an array could serve as amacro-unit of a logical all-optical device.

w xAs has been shown in Refs. 2,4 , an aggregate ofsize less than the emission wavelength does notdisplay a bistable behavior. This finding makes theabove mechanism of the aggregate bistability to behardly realizable, at least, for such well studied untilnow objects as J-aggregates, in spite of the fact thatthey normally incorporate thousands of molecules,having subsequently a size larger than an opticalwavelength. However, the presence of disorder of

0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 99 00390-9

( )V.A. MalysheÕ et al.rOptics Communications 169 1999 177–181178

the surroundings leads to the exciton eigenfunctionsbeing spread over the whole aggregate when disorderis absent, reducing their extension to a localizationsegment of mean size N ) depending on the degree

w xof disorder 9 . From the experimental data related tow xJ-aggregates 10–14 , it follows that even at low

temperatures N ) does not exceed one hundred,meaning that a typical size of a localization segmentis less than the emission wavelength. In view of theabove, these objects could manifest the population-dependent-resonance-frequency-shift bistability pro-vided the increase of size of the localization domainN ).

Note that the above mechanism of bistability isattributed to an indiÕidual homogeneous aggregateor a localization segment. In this context, the ques-tion can arise whether an ensemble of localizationsegments, just being shorter than the emission wave-length, may display a bistable behavior. As was

w xshown in Refs. 15,16 , a model system of homoge-neously broadened two-level molecules elaborated inthe form of a thin slab of a thickness smaller than anemission wavelength manifests a bistable behavior of

Ž .transmittivity or reflectivity resulting from the cou-Ž .pling of molecules via the emission far-zone field.

Note that the previous mechanism of bistability, infact, originated from the near-zone intermolecular

w xinteraction 2,3 .In this paper, we show that under the conditions

of low-temperature experiments, J-aggregated molec-ular systems can be modeled by an ensemble ofinhomogeneously broadened two-level systems. Sincethe typical concentrations of molecules needed for

Ž y 4 y 3the aggregation is rather high 10 –103.molrdm , such that a square of linear size of the

order of the emission wavelength incorporates manymolecules, a thin film comprized of J-aggregatesmay manifest a bistable behavior of transmittivitydue to the coupling of localization segments via theemission field. Critical parameters for the occurrenceof such a behavior seems to be achievable for J-ag-gregates of polymethine dyes.

First, analyze the spectroscopic data related toJ-aggregates from the viewpoint of the concept of

w xhidden structure 17,18 , according to which anylocalization segment can be considered as a homoge-

Žneous aggregate with the corresponding to the size.of segment exciton energy structure of few lowest

states. The J-band is then formed by the transitionsfrom the ground state to the lowest states of the localŽ .related to a certain segment of an aggregate one-exciton bands having dominating oscillator strengthsw x17–19 . The transition from a one- to two-excitonlocal band is blue-shifted with respect to that fromthe ground state to a one-exciton local band. Inparticular, for J-aggregates of pseudo-isocyanine-

Ž .bromide PIC-Br this shift has an order of severaly1 Ž y1 . w xtens cm ;30 cm 12–14 , while the radiative

damping constant of the lowest one-exciton statescoupled to the light is at least one order of magni-

w xtude smaller 10,11 , i.e., does not overlap the blueshift. From the above mentioned, we come to theconclusion that the transition from the ground stateto bottom of a local one-exciton band can be consid-ered as an isolated two-level transition provided theRabi frequency of the external field is smaller thanthe blue shift mentioned. Due to the fluctuations ofsizes of localization segments, the edge energies ofthe ‘local’ exciton bands also undergo fluctuations.This implies, under the conditions outlined, to modelan aggregate as a meso-ensemble of inhomoge-neously broadened two-level systems and a J-aggre-gated sample as a macro-ensemble of those, onlywith characteristics attributed to localized excitons.

The mathematical basis of the above conceptimplies the use of the standard set of semiclassical

Ž .equations for the slowly varying in time amplitudesE and R, respectively, of the electric field andoff-diagonal density matrix element as well as forthe population difference between the one-excitonand ground states Z. For a sample in the form of athin film of thickness L smaller than wavelengthŽ .considered in this paper it has the form

R sy iD qG R qm VZ , 1aŽ . Ž .N N N N N N

1 ) )Z sy m V R qV R yg Z q1 ,Ž .Ž .N N N N N N2

1bŽ .

VsV qg p N m R . 1cŽ . Ž .Ýi R N NN

Here the subscript N denotes the size of localizationsegment; D sv yv is the resonance detuningN N i

Ž . Ž .between the incident v and transition v fre-i N

quencies; g is the spontaneous emission constant ofN

the optically-active one-exciton state: g fg N withN 0

( )V.A. MalysheÕ et al.rOptics Communications 169 1999 177–181 179

g being the analogous constant for an isolated0

molecule; G sg r2qG is the dephasing con-N N 2Ž .stant including that G not connected with the2

spontaneous emission; V s d Er" and V sN iŽ .d E r" are the whole also transmitted and incidentN i

Ž .electric fields in the frequency units , respectively,with d being the transition dipole moment for aN 'segment of mean size N scaled as d fd N , whereN

d is the transition dipole moment of an isolatedŽ .molecule; m sd rd ; p N is the distributionN N N

function of localization segments over sizes; g sR2 2 Ž2p d nk Lr"s2p d n kLr", where ksv rc cN i 0 N

.is speed of light and n sNn is the original concen-0Ž .tration of molecules in the film before aggregation .

The quantity g describes the superradiant radiativeRw xdamping of an ensemble of two-level molecules 20 .

Being expressed through the spontaneous emissionconstant of an isolated molecule g s4d2 k 3r3", it0

Ž . 2reads g s 3r8p g n l L, where l is the transi-R 0 0

tion wavelength. Introducing the surface density ofmolecules n sn L, one can interpret the factorS 0

n l2Lsn l2 as the number of molecules within a0 S

l2-square.

In order to study the possibility for getting abistable behavior of the system we are dealing with,

Ž .one should look for a solution to Eqs. 1 under˙ ˙Ž .steady-state conditions R sZ s0 . It is the mat-N N

ter of simple algebra to arrive at the followingequation for the transmitted field intensity

2GN2 2< < Ž .V 1qg p N mÝR N 22 2 2ž /< <D q G qm G V rgN N N N NN

2DN2 2Ž .q g p N m s V .ÝR N i22 2 2ž /< <D q G qm G V rgN N N N NN

2Ž .

Usually, J-bands at low temperatures are inhomoge-neously broadened, which means that the main effectof the size dispersion results from the fluctuation ofthe transition energies. Since we have no informationrelative to the size distribution, it seems to be reason-able to replace the average over sizes by that withrespect to detunings with a suitable distribution func-tion allowing us to carry out the average analytically.

Ž . Ž .Fig. 1. Effect of the inhomogeneous broadening or size dispersion on the appearance of a three-valued solution to Eq. 3 . The dashedŽregion represents the domain of parameters GrG and g rG where the bistable behavior can appear same in the insert, only for a widerR

.domain of GrG and g rG .R

( )V.A. MalysheÕ et al.rOptics Communications 169 1999 177–181180

A Lorentz-shaped distribution just provides such aŽ .possibility. Thus, we make in Eq. 2 the substitution

`

p N ™ dDp DŽ . Ž .Ý Hy`N

` G 1s dD ,H 2 2py` DyD qGŽ .0

where G plays the role of the J-band width andD sv yv is the deviation of the external fre-0 0 i

quency v on the central frequency of the Lorentzian,i

and replace all other quantities by their mean values.Ž .It is straightforward to evaluate integrals in Eq. 2

evidently. For the sake of simplicity, we will presenthere the result of integration only for a particular

Ž .case D s0 holding also at D <G . It reads0 02

g rGR2< <V 1q1r2 1r22 2< < < <GrG q 1q V rgG 1q V rgGŽ . Ž .

s V 2 . 3Ž .i

Ž .In Fig. 1, the results of numerical solution of Eq. 2are depicted showing the domain of parameters g rGR

and Grg for which the bistable behavior can beŽ .realized shadowed region . As is seen from this

Ž . Žfigure, the threshold of bistability is g rG s8 atR c. w xGs0 15 . The dispersion of sizes acts in such a

manner that the higher ratio GrG is the value ofg rG is required to get the bistability effect.R

Fig. 2 represents the numerical solution of Eqs.Ž .1 at adiabatic up-and-down scanning of the input

Fig. 2. Above-threshold optical hysteresis loop of the transmission2 2 Ž . Ž .T s V rV solid line calculated by means of Eqs. 1 ati

adiabatic up-and-down scan of the input intensity xs V 2rgGiŽ .g s10G , the dispersion of sizes is neglected . The dotted curveR

Ž .represents the stationary solution obtained on the basis of Eq. 2at Gs0.

intensity V 2 displaying the possibility of getting thei

hysteresis and high-contrast switching of transmittiv-ity of the film.

In conclusion, we discuss parameters of real sys-tems. In this sense, J-aggregates of PIC-Br presentone of the most studied objects of such a kind. Asalready mentioned above, the blue shift of the transi-tion from one-to-two exciton bands with respect tothat from the ground state to the one exciton band

Ž .for the red J-band ls576.1 nm is of the order ofy1 Ž y1 .30 cm or 1 ps in frequency units . The radia-

tive damping constant of the optically active one-ex-Ž . w xciton state g has an order of 1r70 ps 10,11 and,

as already seen, does not overlap the energy separa-tion between the above mentioned transitions. On theother hand, switching of the transmittivity occurs atsaturating intensities of the field inside the film,V 2 ;gG , so that this field will never exceed the

Ž . 2blue shift fG due to the inequality gGrG <1Ž .we have here in mind the case G)G . Thus, thetwo-level model we are dealing with seems to beapplicable to the problem under study.

Estimate now the superradiant damping constantŽ . 2 Ž .g s 3r8p g n l L. Taking g s 1r3.7 ns , nR 0 0 0 0

s2=1018 cmy3 and Lflf5.73=10y5 cm, wethen obtain g f13 psy1, i.e., a magnitude that isR

remarkably larger than the width of the J-band Gf1psy1. Manipulating then the temperature, the re-quired ratios of g rG and GrG can be achieved toR

Žbe within the bistable region shown in Fig. 1 as.shadowed .

In conclusion, the conditions for the model pro-posed in this paper seems to be indeed realizable foran ensemble of J-aggregates of PIC-Br at low tem-peratures. A certain support of our findings is givenby the recent publications reporting on the coopera-tive emission in p-conjugated polymer thin filmw x21,22 as well as the superradiant lasing from theJ-aggregated BIC molecules adsorbed onto colloidal

w xsilica or silver 23,24 , just meaning a collectiviza-tion of polymers and J-aggregates via the emissionfield we need for the effect under consideration.

Acknowledgements

We greatly acknowledge the support of the Bun-desministerium fur Bildung, Wissenschaft, Forschung¨

( )V.A. MalysheÕ et al.rOptics Communications 169 1999 177–181 181

und Technologie within the Hochschulsonderpro-gramm III. V.A.M. thanks the Deutsche Forschungs-gemeinschaft for a grant as well as acknowledges apartial support from the Russian Foundation for Ba-

Ž .sic Research project 97-03-09221 .

References

w x Ž .1 V.V. Gusev, Adv. Mater. Opt. Electr. 1 1992 235.w x Ž .2 V. Malyshev, P. Moreno, Phys. Rev. A 53 1996 416.w x3 V.A. Malyshev, H. Glaeske, K.-H. Feller, Opt. Commun.

Ž . Ž .140 1997 83; J. Lumin. 76&77 445 1998 .w x4 V.A. Malyshev, H. Glaeske, K.-H. Feller, Phys. Rev. A 58

Ž .1998 670.w x Ž .5 J. Heber, Z. Phys. B 68 1987 115.w x Ž .6 N. Bodenschatz, J. Heber, Phys. Rev. A 54 1996 4428.w x7 V.A. Malyshev, H. Glaeske, K.-H. Feller, Phys. Rev. A 58

Ž .1998 1496.w x Ž .8 D.B. Chesnut, A. Suna, J. Chem. Phys. 39 1963 146.w x Ž .9 E.W. Knapp, Chem. Phys. 85 1984 73.

w x Ž .10 S. de Boer, D.A. Wiersma, Chem. Phys. Lett. 165 1990 45.

w x11 H. Fidder, J. Knoester, D.A. Wiersma, Chem. Phys. Lett. 171Ž .1990 529.

w x12 H. Fidder, J. Knoester, D.A. Wiersma, J. Chem. Phys. 98Ž .1993 6564.

w x13 R. Gagel, R. Gadonas, A. Laubereau, Chem. Phys. Lett. 217Ž .1994 228.

w x14 K. Minoshima, M. Taiji, K. Misawa, T. Kobayashi, Chem.Ž .Phys. Lett. 218 1994 67.

w x Ž .15 S.M. Zakharov, E.A. Manykin, Poverkhnost’ Moscow 2Ž . w x1988 137 in Russian .

w x Ž . w16 A.M. Basharov, Zh. Exp. Teor. Fiz. 94 1988 12 Engl.Ž . xtransl.: JETP 67 1988 1741 .

w x Ž . w17 V. Malyshev, Opt. i Spektrosk. 71 1991 873 Engl. transl.:Ž . x Ž .Opt. Spectrosc. 71 1991 505 ; J. Lumin. 55 1993 225.

w x Ž .18 V. Malyshev, P. Moreno, Phys. Rev. B 51 1995 587.w x19 H. Fidder, J. Knoester, D.A. Wiersma, J. Chem. Phys. 95

Ž .1991 7880.w x Ž .20 R.H. Dicke, Phys. Rev. 93 1954 99.w x21 S.V. Frolov, W. Gellermann, M. Ozaki, K. Yoshino, Z.V.

Ž .Vardeny, Phys. Rev. Lett. 78 1997 729.w x22 X. Long, A. Malinovski, D.D.S. Bradley, M. Inbasekaran,

Ž .E.P. Woo, Chem. Phys. Lett. 272 1997 6.¨w x Ž .23 S. Ozcelik, D.L. Akins, Appl. Phys. Lett. 71 1997 3057.¨ ¨w x24 S. Ozcelik, I. Ozcelik, D.L. Akins, Appl. Phys. Lett. 73

Ž .1998 1949.