J. Phys. Chem. 1992, 96, 2843-2848
’1 100 PURE POLYMER
50
COPOLYMER (5050)
100 PURE POLYMER
Figure 10. Resonant x ( ~ as ) a function of copolymer composition for PSPQ/PBAPQ,PBPQIPBAPQ,and PSPQIPBPQ.
perimental studies on oligomers and model compounds of conjugated molecules where the third-order susceptibility x(’) or second hyperpolarizability y has been shown to strongly depend on the A,- or optical bandgap.’4-2’ Therefore, it is essential that systematic studies aimed at elucidating the structured’) property relationships in polymers include high molecular weight polymer structures. Effects of Copolymerization. In Figure 10, the resonant x ( j ) is represented as a function of copolymer composition for all the three copolymers. The x ( j ) of copolymer PSPQ/PBAPQ (5050) lies on a straight line joining the x(’)values of the two constituent homopolymers, while in the case of copolymers PBAPQ/PBPQ (5050) and PSPQ/PBPQ (5050) a marginal enhancement in resonant x(’) values is observed compared to the respective molar averages of the x(’) values of the constituent homopolymers. In PBAPQ/PBPQ, an increase of 15% in resonant x(’) was observed esu calculated from compared to the molar average of 22.9 X the x(’) values of constituent homopolymers PBAPQ and PBPQ. Similarly, an incrtase of 38% was observed in copolymer PSPQ/PBPQ compared to the calculated molar average of 13.7 X esu. The THG spectroscopy results, both in the resonance and the nonresonance regions of the copolymers, suggest that the introduction of disorder in the copolymer backbone does not have a significant effect on the third-order optical nonlinearities. Conclusions The third-order nonlinear optical properties of a series of systematically designed nine homopolymers and three random
2843
copolymers in the class of conjugated rigid-rod polyquinolines and polyanthrazolines have been studied using third harmonic generation (THG) spectroscopy. Three-photon resonance peaks at 3A- were observed in the x(’) spectra of the polymers. Although off-resonance a t 2.38 pm, the x(’) values were found to be very close to each other for all the polymers within the experimental errors, the peak value of the three-photon enhanced x(’)varied in the range 0.8 X 10-”-3.3 X 10-l’ esu with the backbone structure of the polymers. The results suggested that there was no significant dependence of the nonresonant ~ (values 9 on the molecular structure or degree of *-electron delocalization of the polymers. The resonant x(’) values, which varied significantly on the other hand, did not correlate well with the degree of a-electron delocalization of the polymers. A theoretical three-level model based on essential states mechanism was used to obtain a fit to the x(’)dispersion of the polymers. The superior fit obtained using a three-level model in constrast to a two-level model suggests that more than one excited state is responsible for the observed third-order optical nonlinearity of the materials. On the basis of the experimental and theoretical x(’) spectra of the series of systematically derived polymer structures, we can say that the magnitude of x(’) does not always scale with the *-electron delocalization of the conjugated polymers and that knowledge of the overall electronic excited-state structure is essential in determining the third-order nonlinear optical response of the materials. The results of the x(’)spectra of the three random copolymers showed that the x ( j ) values of the copolymers were the molar averages of the x ( ~values ) of the respective constituent homopolymers, suggesting the absence of significant enhancement or reduction in the third-order optical nonlinearities with the introduction of disorder in the copolymer backbone. Thus, random copolymerization provides one approach to tailoring the value of the third-order NLO susceptibility from known nonlinear optical polymers.
Acknowledgment. Work at University of Rochester was sup ported by the Amoco Foundation and the National Science Foundation (Grant CHE-88 1-0024). The nonlinear optical characterization was carried out at Du Pont. H.V. acknowledges the valuable technical assistance of J. Kelly. Registry No. 1, 139102-46-8; 2a, 75460-97-8;2b, 135614-64-1;2c, 86527-06-2; 2 4 76996-76-4; 3a, 59827-44-0 3b, 137091-74-8; 3c, 139102-47-9; 3d, 137091-72-6.
Theory of Anomalous Photon Echo Decays in Confined Excitonic Systems Frank C . Spano Department of Chemistry, Temple University. Philadelphia, Pennsylvania 191 22 (Received: September 3, 1991)
An analytical expression for the photon echo decay in a linear aggregate consisting of N coupled two-level molecules is derived
in the weak field limit. The expression has two terms: one which involves transitions between the ground state and the one-exciton state, and a second which involves two-photon absorption to the doubly excited states or two-excitons. For interpulse separation times Idwhich obey Y 1
wo - excitons]
e = e,
e = e*
I
I
IK=O,q=l>'
I
Figure 2. Photon echo pulse sequence consisting of two excitation pulses separated by a delay time td. The polarization is refocused at 2 t d , producing the photon echo pulse.
Ik=O>
Figure 1. Optically allowed energy levels for a small aggregate where kern is approximately independent of n (k is the exciting pulse wavevector and r, is the position of the nth molecule in the aggregate). Not shown are the three- and higher-exciton states. The states shown are all that are needed to calculate the third-order (weak field) photon echo signal exactly.
number of excited molecules. The ground state is denoted as 10) and represents the state with no excitations. The one- and twoexciton eigenstates are Ik) = ( 1 / f i f e 2 T i k " / N l n )
k = 0, 1, ..., N - 1
(2.3a)
n= 1
IKq) = (2/N)xe2"iKR/Nsin (rqs/N)IR - s/2, R R.s
K=0,1,
+ s/2)
..., N - 1 (2.3b)
..., N - 2 ( N - 3)
K even
= 0, 2, ..., N - 3 ( N - 2)
K odd
q = 1, 3,
respectively. Here, the upper limit for q is for N odd (even).15 In eq 2.3, In) denotes the state with only molecule n excited, whereas Imn) denotes the state with both molecule m and molecule n excited. R is the center of mass coordinate between sites 1 and m, R = (1 m)/2, while s is the relative distance, s = I/- ml. The eigenenergies are
+
= WM
+ 2V COS ( 2 ~ k / N )
(2.4a)
for the one-exciton state and
Q& = 2wy
+ 4V COS (Ak/N)
cos (?rq/N)
(2.4b)
for the two-exciton state on the Mth aggregate. The one-exciton eigenstates in eq 2.3a are the familiar Frenkel excitons states. The two-exciton states cannot be expressed as the product of two Frenkel exciton states because Frenkel excitons are not bosons. The difference arises from the Pauli exclusion principle or the fact that two excitations cannot reside on the same site (see eq 2.2). When the secular contributions of H'are included, the first-order eigenenergies for the one- and two-exciton states become w: + A,,, and Q& + 2AM, respectively, where A M is defined as
In Figure 1 the optically allowed energy levels are shown for a small molecular aggregate (whose length is much smaller than an optical wavelength) up to and including the two-exciton levels. Note that only the k = 0 exciton in the one-exciton manifold can be optically excited, while ( N - 1)/2 states (when N is odd) in the two-exciton manifold carry oscillator strength from the k = 0 one-exciton state. There are thus a total of N ( N + 1)/2 levels which contribute to the photon echo in the weak-field limit. This is a substantially reduced number compared to the total number of 2N levels which include the three and higher exciton states. 111. Photon Echo A photon echo experiment consists of directing two laser pulses (usually at a slight angle with respect to each other) into the sample and measuring the sample polarization at a time 2td, where td is the time delay between the first and second pulse. The second pulse serves to reverse the inhomogeneous dephasing of the polarization following the first pulse; at 2td the polarization is again in phase emitting a burst of radiation known as the photon echo. The pulse sequence is shown in Figure 2. The decay of the echo intensity as a function of td provides a direct measure of the homogeneous relaxation times. The maximum polarization will result from the application of a a / 2 pulse followed by a A pulse, where ~ / and 2 A refer to the pulse area. However, echo formation with reduced intensity will result from the application of two pulses with arbitrary area (as long as they avoid the values of nr, where n = 0, 1, ...). In particular, echo experiments can be carried out with weak pulses with areas much smaller than unity. In this weak-field regime the echo polarization scales as 0(81822),where 8, is the pulse area of the nth pulse. The echo phenomenon can understood in terms of a third-order nonlinear optical process, and only the energy levels shown in Figure 1 and none higher are needed to exactly describe its beha~i0r.l~ For larger area pulses, three-, four-, and higher-excitons contribute to the echo polarization, greatly complicating the analysis. In the strong-field limit all 2N states are needed for a general description of the photon echo. Not surprisingly, analytical or numerical solutions to the general problem are not available for N larger than about 12. Fortunately, in the weak field regime only ( N 1)/2 wave functions are required for exact solutions; in addition it is in this regime where Skinner et al. predicted the unusual oscillations in the echo intensity vs 22, for dimers. The natural question to ask is "How does the oscillatory behavior depend on the aggregate size N?" When the excitation pulse areas 8, are small and equal to 8, the photon echo signal may be calculated using third-order perturbation theory. From nonlinear response theory16 the total third-order refocused aggregate polarization at 2td is
+
N
AM = (1/N)C6, n= 1
(2.5)
Note that A,,, is the average of N normal random variables with standard deviation Umm. Therefore A M is also normally distributed The but with a reduced standard deviation equal to aintra/dN. first-order approximation therefore holds as long as P ~ ~is ~ much smaller than min[wf - W E ] . Using eq 2.4a, this becomes qntra
O
Figure 4. Normalized echo intensity as a function of dimensionless interpulse time delay l u t d for much larger aggregate sizes than in Figure
3. The solid curves are calculated from eq 3.3 and the dashed curves from eq 3.5. The behavior is linear up to a value t d = N/2V; therefore as N approaches infinity, the response is perfectly linear (for td >> VI), Note that in the linear regime the echo intensity scales as O(N).
increases, resulting in the oscillatory behavior with a peak value which is approximately 15P.Since the second term is dominated by the K = 0, q = 1 term, the period of oscillation is easily shown to be M/uV, which agrees well with the curves in Figure 3. This behavior is quite different from what one expects from higher intensity echo sequences where dipoledipole dephasing leads to an initial decrease in echo amplitude with time. This anomalous behavior was shown previously for the case of a dimergJOand is extended here for a one-dimensional exciton. In Figure 4 1(2fd) vs t d is shown for much larger aggregates than in Figure 3. Note the approach to linear behavior for times less than about N/ZVas N increases. For later times (not shown) the oscillatory behavior is recovered however. (The linear behavior is just discernible in Figure 3 for N = 40). To understand the origin of the linear behavior, eq 3.3 was analyzed in the limit that N approaches infinity. The asymptotic result is P(2td) = 4NibB3e-4i"dll/zJo(4Vfd)- 4vtd[J](4vtd) - iJo(4vtd)]l (3.4)
-
When 4Vtd >> 1, the Bessel functions appearing in eq 3.4 can be replaced by the large-argument asymptotic form, Le., Jo(z) ( 2 / ~ z ) Icos / ~ (z - 7r/4) and J l ( z ) (2/rz)l/* cos ( z - 3?r/4). Then for times t d >> V 1we get
-
Z(2fd) = (128/T)@Vfd
(3.5)
The dashed curve in Figure 4 shows eq 3.5, which is in excellent agreement with the numerical results. Thus for infinite aggregates the linear behavior persists for all time (eq 3.5) but for finite size aggregates is persists for intermediate times N P > td >> VI, where the upper bound was estimated from the numerical results. For longer times the oscillatory behavior of Figure 3 is recovered.
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2847
Confined Excitonic Systems
i
Nw=20 75
Nw=20
i
_i
751
Nw40
00 00
15000
3000 0
45000
6000 0
lvltd
0.0
Figure 5. Normalized echo intensity as a function of dimensionless interpulse time delay I ut,,for several aggregate size distributions. The polarization is calculated from eq 3.3 and averaged according to eq 4.1. The initial anomalous increase arises from the linear behavior (see eq 3.5), and the longtime convergence results from a cancellation of the second term in eq 3.3 in the individual aggregate responses.
IV. Distribution of Aggregate Sizes It is highly unlikely that any real sample of molecular aggregates consists of a single-size aggregate, as was assumed up to this point. In reality there exists a distribution of sizes governed by some distribution function g(N). The calculation of the echo signal therefore requires two averages; one over the site transition frequency and one over the aggregate size. For the remainder of the calculations a Gaussian distribution of aggregate sizes is assumed: (4.1) Calculating the echo polarization from eq 3.3 and averaging according to eq 4.1 lead to the echo decays shown in Figure 5 for several values of N,. The most obvious characteristic is the convergenceof the echo intensity to a constant value for long times. This arises because the oscillations with period p / a V within the distribution effectively cancel out. The only term which survives the destructive interference is the first term in eq 3. With only this term present the echo intensity simply reduces to Z(2fd) = 4(N2)2 (4.2) which, for the Gaussian distribution of eq 4.1 is Z(2td) = Nw4,a value which is in excellent agreement with the numerical results of Figure 5 . A second noteworthy characteristic of the curves in Figure 5 is the approximate linear behavior for short time. This can also be accurately estimated by simply averaging the polarization from eq 3.4 in the large-N limit (assuming that the signal is dominated by aggregate sizes where the linear behavior of eq 3.5 is applicable), giving Z(2td) = (128 / a ) (N)2vf,j
(4.3)
which, for the Gaussian distribution of eq 4.1, reduces to Z(2td)
00
1500.0
30000
45000
60000
lVlk Figure 6. Normalized echo intensity as a function of dimensionless interpulse time delay lutd for the same size distributions of Figure 5 except that superradiant damping is included as discussed in section V with y = 2 X 10-5V. The anomalous behavior is still present but with a reduced peak time t p as compared to the undamped case in Figure 5. The long-time behavior approaches the superradiant fluorescence of eq 5.2.
= (128/az)NW2vtd.This also agrees well with the initial slope measured from Figure 5. Using the results of the short- and long-time limits the time a t which the signal peaks in Figure 5 is approximately given by
(4.4) which gives t p = a2NW2/128V for the Gaussian distribution.
V. Superradiant Damping Thus far damping and dephasing mechanisms have been neglected. It is by now well established that damping in some low-temperature molecular aggregates is dominated by excitonic radiative decay or ~uperradiance."~'~J*Since the radiative decay rate is proportional to the square of the transition dipole moment connecting the excited and ground states and because the transition dipole moments for one- and two-excitons scale as the square root of the number of molecules comprising the aggregate N, we get from eq 3.2a the kth one-exciton coherence decay rate: Yk = (NY/2)6&,0
(5.la)
and the K = 0, qth two-exciton coherence decay rate:
r4 = WN)
cot2 ( w w
(5.lb)
(17) Fidder, H.; Wiersma, D. A. Phys. Rev. Left. 1991,66, 1501. Fidder, H.; Terpstra, J.; Wiersma, D. A. J. Chem. Phys. 1991, 94, 6895. Fidder, H.; Knoester, J.; Wiersma, D. A,, to be published. (18) Spano, F. C.; Mukamel, S. J . Chem. Phys. 1989,91,683. Spano, F. C.; Kukliniski, J. R.; Mukamel, S. Phys. Rev. Lerr. 1990, 65, 211.
2848 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 These quantities are introduced into eq 3.3 by simply adding the factor exp[-2yotd] to the first term and exp[-(2yo + rq)td] to the second term. When this is done, the echo decays from Figure 5 are modified to those shown in Figure 6. Note that the initial linear rise is still present. Since the long-time decay is dominated by the first term in eq 3.3, it becomes a size-averaged superradiant exponential decay: (5.2) which is identical to the form of the superradiant fluorescence decay. VI. Discussion and Summary An exact solution for the echo polarization as a function of td for a linear array of N coupled two-level molecules is given in eq 3.3 and simplified for an infinite aggregate in eq 3.4. The echo decay consists of two basic terms: the first one in eq 3.3 involves three single-photon exchanges between the ground state and the k = 0 exciton state. The first photon is supplied by the first pulse and the remainder from the second pulse. The second term in eq 3.3 involves two-photon absorption to the two-exciton states. The first photon from the first pulse creates an exciton coherence and the next two photons from the second pulse directly excite a two-exciton state. A similar decomposition has recently been shown to exist for the general third-order aggregate hyperpolarizability.l33l9 Since the second term has an opposite sign compared to the first term at td = 0, the two destructively interfere yielding a result which depends linearly on aggregate size, P(td=O) = 2NipB3. For times shorter than NVF'but longer than V-I the echo intensity increases linearly with delay time with a slope equal to 128VMlr. As td increases well beyond NV' the echo intensity oscillates with period N/rV; when the two terms are in-phase the polarization scales as O(IV2). This is one example of the giant nonlinearities originally predicted by Hanamura.20 The enhanced behavior can exist only for small aggregates which is assumed throughout. As the aggregate length approaches an optical wavelength the polarization will change over to a linear scaling with N.19 The oscillatory behavior is a direct consequence of the Pauli exclusion principle, which states that two excitations cannot reside on the same site. A two-exciton wave function is not a simple product of two one-exciton wave functions as it would be if excitons were bosons. Accordingly the two-exciton frequency no is not simply twice the k = 0 exciton frequency 2wk=0. The difference is the oscillation beat frequency as is evident from eq 3.3. For a distribution of aggregate sizes the oscillations from different size aggregates largely cancel out leaving the first term in eq 3.3 as the main contributor to the longtime behavior. Interestingly enough, for long times the effect of dipole-dipole coupling is absent from the echo decay. This is demonstrated clearly in Figure 6 (and eq 5.2), which shows the echo decay approaching the fluorescence decay for long times. In effect the weak-field echo sequence has refocused the dipole-dipole influence on the echo decay at long times. For short times though the linear behavior persists creating an anomalous echo decay or one which initially increases with time. Estimates of the time for peak echo intensity can be made from eq 4.4 when superradiance is neglected; when superradiance is included the peak time is shorter, as is evident from Figure 6. Thus far the anomalous behavior has not been detected experimentally. This could be because the 2-3-ps pulses used in (19) Spano, F. C.;Mukamel, S. Phys. Reu. Lett. 1991, 66, 1197. (20) Hanamura, E.Phys. Reu. B 1988, 37, 1273.
Spano the J-aggregate echo experiments4J7are much longer than the echo peak time, t,. For J aggregates with V = -600 cm-l, the upper bound is t, = 4 X 10-'5Nw2s, which is of the order of the experimental resolution for N, < 50. However, it is probably more likely that the motional narrowing assumption is too severe for accurate modeling of the J aggregate, which is known to have significant inhomogeneous broadening outside this regime.17 The present results cannot explain the fact that the low-temperature echo decay time is about 6 times shorter than the fluorescence decay time. It is likely that this observation is a direct result of strongly localized Frenkel excitonsI7 and could adequately be modeled by allowing uinrrato take on values of the order of the exciton bandwidth. As discussed earlier, Warren and &wail7 have shown that in this regime the echo decay may be enhanced because of imperfect refocusing of the inhomogeneous dephasing. At the same time, increasing disorder distributes oscillator strength over the entire exciton manifold leading to a corresponding decrease in the fluorescence decay rate.I7-I8 The opposite behaviors of the two observables may very well provide the explanation for the measured low temperature disparity. The details of the description of the echo decay behavior just presented depend of the several assumptions introduced in section 11, Le., nearest-neighbor coupling, periodic boundary conditions, and weak intraaggregate inhomogeneous broadening. However, the basic two-term structure of the echo decay is independent of these assumptions and quite general features of echo decays in excitonic systems can be inferred from eq 3.1. The most important observation is that the sums in eq 3.1 for td = 0 must give a polarization that scales as N. This can be shown directly from eq 3.1 where the sum can be made independently of the details of the eigenstates in analogy with the oscillator sum rule. Of course the subsequent evolution depends on the eigenstates and eigenvalues of the one and two excited-state manifolds and these will vary depending on the dimension of the exciton, the range of the intermolecular interactions, and the extent of charge transfer. When the two terms are in phase at a later time the polarization will be enhanced over the td = 0 value as long as the oscillator strengths for some subset of eigenstates scale as N , 1 / 2 with N , >> 1. N, is defined as the coherence length or number and is related to the number of molecules in the aggregate whose polarizations remains phase coherent; when exciton-phonon scattering and other homogeneous broadening mechanisms are neglected and the disorder remains within the motional narrowing regime, N, = N (as we have assumed in our detailed calculations); otherwise, N, < N. Therefore, when the in-phase time is reached the echo intensity will peak to values of order O[N,Z], much greater than the td = 0 value. Hence, the anomalous echo behavior is a basic feature of excitonic systems. The form if the intensity increases will however depend on the eigenstate details and the nature of the broadening; the linear form derived here may be restricted to one-dimensional excitons with N, = N. Finally, when the &function pulse assumption is relaxed, the anomalous increase will be a strong function of detuning of the laser frequency wL from the center molecular frequency to wo. This is easy to see because molecules excited far from resonance (wL - oo>> V) will not have sufficient time to interact with each other via dipole-dipole coupling. They will behave like independent monomers and not excitons, and a monotonic decrease in echo intensity with interpulse separation time will result. Acknowledgment. Acknowledgments are made to Temple University and the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research.
