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Multiphoton nonlinear interactions in conjugated organic polymers
Synthetic Metals, 37 (1990) 231 - 247
231
M U L T I P H O T O N N O N L I N E A R I N T E R A C T I O N S IN C O N J U G A T E D ORGANIC POLYMERS H. J. BYRNE* and W. BLAU Department of Physics, Trinity College, Dublin (Ireland)
Abstract The optical nonlinearity at a wavelength of 1.064 #m has been studied in a range of soluble conjugated organic polymers using the technique of degenerate forward four-wave mixing. This technique enables determination of the third-order nonlinear susceptibility IX¢3)I of the material. In all cases, determination of the response time of the material nonlinearity is limited by the experimental resolution ( ~ 70 ps). Measurements of I~~3)] as a function of polymer solution concentration were performed and contributions from both real and imaginary components were observed and estimated. The origin of the imaginary contribution in these 'off-resonant' conditions was investigated by intensity-dependent measurements of the four-wave mixing signal. In solutions of poly-3 and -4(butoxycarbonylmethylurethane diacetylene), (BCMU) in both chloroform and chlorobenzene, a dependence characteristic of a fifth-order nonlinear interaction is observed. Furthermore, in chloroform solutions of poly(3-butylthiophene) a seventh-order nonlinearity is observed. These high-order nonlinearities are associated with two-photon and threephoton resonant enhancement of the four-photon process and are characterized by fifth- and seventh-order susceptibilities, X¢5) and ~(7) respectively. The presence of imaginary contributions to the optical nonlinearity in all materials investigated implies that this nonlinearity is dominated in all cases by the nearest resonance. This conclusion renders redundant such simplistic models as the free-electron model.
Introduction Organic polymers have been predicted to display large ultrafast Kerrlike nonlinearities because of their highly polarizable, delocalized ~-electron backbone. However, a quantitative understanding of the relationship between these large nonlinearities and the electron delocalization and polymer backbone structure is as yet lacking. Notably, the role of 'resonant'
*Author to whom correspondence should be addressed. 0379-6779/90/$3.50
© Elsevier Sequoia/Printed in The Netherlands
232
contributions to the nonlinearity at a wavelength which is far from the primary absorption and, indeed, the nature of this off-resonant nonlinearity are unclear. In this study, the optical nonlinearity has been measured in the transparency region of a selection of conjugated organic polymers in an attempt to address matters. Concentration dependences of the nonlinearities have been studied to establish the contributions, if any, of imaginary components, and intensity-dependent measurements have been made in order to establish the nature of influential resonances.
Experimental
The experimental method used in these studies was that of forced light scattering from laser-induced gratings, a technique which may be viewed as a degenerate four-wave mixing process in the forward direction [1]. The set-up is depicted schematically in Fig. 1. The light source is an amplified, passively mode-locked Nda+:YAG laser emitting linearly polarized pulses of 50 + 25 ps duration and of wavelength ,~ = 1.064 gm at a frequency of 3 Hz. Peak powers of up to 50 MW were readily available. The experimental method, which is described in detail elsewhere [2], is based on the interference at the sample of two spatially and temporarily overlapped beams, producing a spatial modulation of the intensity-dependent refractive index of the material. This modulation acts as a diffraction grating from which the pulses may self-diffract. Under thin grating conditions [1], satisfied experimentally by keeping the angle between the two beams small ( < 1°), an expression relating the diffraction efficiency (~/) in the first order to the third-order material nonlinearity may be derived
Iz3>I=
(1)
4~°cn 22(~I) 1/2 3~dlo
where c is the speed of light, eo is the permittivity of free space, n is the refractive index of the sample, d is the sample thickness and I0 is the input pulse intensity. In the experiments reported here, d was 1 mm and n was Beam Splitter
Delay Line
Photodiode
Fig. 1. Experimental set-up for the self-diffraction technique.
233
taken to be the refractive index of the solvent, because of the low fractional volume of the solute. Equation (1) holds for materials which are transparent at the operating wavelength. It can be seen from eqn. (1) that verification of the presence of a true third-order nonlinear process may be obtained by monitoring the intensity dependence of the diffraction efficiency. For a true third-order process (2)
= (I1/Io) ~ 12
where 11 is the intensity diffracted into the first order. Such a verification is important as fifth- and seventh-order processes, originating in two- and three-photon resonant enhancement of the material nonlinearities, have been observed in organic conjugated materials [2, 3]. IX(3)I may have both real and imaginary components originating from (3) . the polymer as well as a contribution from the solvent, X~o~v. which is purely real and positive in the case of most organic solvents, including n-hexane [4]. For the concentration range used in this work, the oligomer fractional volume is negligible. Thus = L~ol-~ +
Re~pol)
+ "~--~po1~ '
where ReX(p3o ), and Im~pol -"(~) are the real and imaginary components of the material nonlinearity. By monitoringthe concentration dependenceoflx(3)l, the contribution X.(3) solv due to the solvent may be extracted and the magnitude
of ReX~pao ),and ImX~p~o)1may be determined. Furthermore, the sign of Zex~o),may be determined from the concentration dependence of the real part of IX(a)].
R e s u l t s and d i s c u s s i o n Polydiacetylenes
The chemical structures of the polymeric materials chosen for this study are depicted in Fig. 2. A particular series of polydiacetylenes which has been extensively studied because of their ease of synthesis and good environmental stability, is the n-BCMU series [5] which has the backbone structure of Fig. 2(a) with side groups R = R' = --(CH2) n - O C - O N H C H z C O O ( CH2)3CH3 The materials employed in this study have n = 3 and 4. Both poly-3- and -4BCMU are soluble in organic solvents such as chloroform and dimethylformamide, forming strongly coloured yellow solutions. Solutions of poly-3and -4BCMU in chlorobenzene are, however, blue and red in colour respectively. Maximum concentrations of poly-3- and -4BCMU available were, however, limited to 1.3 and 2.0 g/1 respectively. A range of concentrations, for the purpose of concentration-dependent measurements, was generated by dilution of poly-3- and -4BCMU gels, made by the method of ref. 3, at 100 °C. With
234
II
n
(a)
(b) 0C18H37
"['C=C
C=C n
OC18H37 (c) Fig. 2. Some one-dimensional conjugated polymers: (a) polydiaeetylene; (b) polythiophene; (c) polymer P28.
decreasing concentration, the gels became fluid and approached solutions at low concentrations. Yellow solutions of poly-4BCMU in chloroform were made up with a maximum concentration of 1.1 g/1. In order to examine the nature of the process giving rise to the nonlinear diffraction, the intensity dependence of the diffraction efficiency was examined. In the case of neat solvent, the intensity dependence of the diffraction efficiency was found to vary as the square of the input intensity for both chloroform and chlorobenzene, as is expected for a third-order nonlinear process. Similarly for polymer solutions at low concentrations, the intensity dependence of the diffraction efficiency is characteristic of a true third-order process, as is shown in Fig. 3. However, at concentrations greater than ~ 0.2 g/l, the polymer solutions exhibit an intensity dependence of the diffraction efficiency greater than squared. Figure 4 shows the intensity dependences for the range of poly-4BCMU solutions in chloroform. Clearly, an intensity-dependent process of higher order is contributing to the grating formation. Such high-order processes have, indeed, been observed previously in polydiacetylenes [3, 6]. A relationship between the intensity-dependent change in the refractive index of the sample, in the presence of a periodically modulated intensity distribution, may be derived and is given by Ax -
AZ 2(1 + Z) ~/2
(4)
In the case where the nonlinear behaviour is purely third order, Ax is given by a .(a) Ak A l A Z i j = 4~ijkl
(5)
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Fig. 3. Intensity dependences of the diffraction efficiency: (A) chloroform; (O) 0.1 g/1 poly4BCMU in chloroform; (-) chlorobenzene; ((3) 0.01 g/1 poly-3BCMU in ehlorobenzene; ( + ) 0.2 g/1 poly-4BCMU in chlorobenzene.
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Fig. 4. Intensity dependence of the diffraction efficiency of solutions of poly-4BCMU in chloroform.
If, h o w e v e r , h i g h e r o r d e r t e r m s a r e significant, t h e y m u s t be t a k e n into a c c o u n t , h e n c e A~, in the case w h e r e all fields a r e d e g e n e r a t e , t a k e s t h e form, A ~ i j = ' 4 } ~ i j .(S) kl
A k A l ±~'(5) T8Aijklmn
A k AA l
m
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(6)
T r e a t m e n t of t h e r e l a t i o n s h i p b e t w e e n t h e d i f f r a c t i o n efficiency of a l a s e r - i n d u c e d g r a t i n g a n d the n o n l i n e a r susceptibilities, in this case, m a y be p e r f o r m e d by c h a r a c t e r i z i n g t h e p r o c e s s by a n effective t h i r d - o r d e r
236
susceptibility, XefP - (3) given by Zeff(3)__--•(3)
+ 5)((S)AnAm
= X(a) + ~
51
. . . .
X(s) . . . .
(7)
This effective susceptibility is measured experimentally in the selfdiffraction experiments and may be related to the diffraction efficiency into the first order by the relationship (3) Zee
4~oCn2,~07) 1/2 -
(8)
3udI
as for the case of a pure third-order susceptibility. In this case, however, the material absorption must be taken into consideration. In the presence of a two-photon absorption, the material absorption coefficient becomes intensity dependent and has the form = ~0 + ~21
(9)
where a o is the linear absorption coefficient and a2 is the two-photon absorption coefficient. In the present situation, a0 is small and may be assumed to be equal to zero [3]. Similarly, the sample transmission becomes intensity dependent and may be written [7] as T = [1 -
a2Id] -1
(10)
Inclusion of material absorption in the expression for the effective material nonlinearity, assuming that the term o~2Id ~ 1, yields (3) 4~oCn2'~(t/)~/2 ( 1 :¢2) Zea -3, I-d + 2
(11)
~(3) eff is therefore composed of two factors. The first is identical to the nonlinearity in the case of a pure third-order process and may be written 1~(3){ -- 4eocn2~0/)1/2
(12)
3udI
The second term is intensity dependent and arises from the additional fifth-order contribution due to the two-photon resonant enhancement of the nonlinearity. Thus 2 a 2 n,~(t/) 1/2
Iz(5)l = -
5.I 3Z(3)eocnd~2
10
(13)
This relationship between X(5) and ~(3) contributes to the understanding of the nature of the fifth-order process. The microscopic hyperpolarizability 5 is determined by the third-order hyperpolarizability 7 and by the change in
237
Fig. 5. Degenerate six-wave mixing process.
the first-order hyperpolarizability ~z 6 = "tu2
(14)
The fifth-order nonlinearity is, therefore, a cascade process. ~(5) is not a true fifth-order polarizability, but is rather a two-photon resonantly enhanced ~(~), and the process is illustrated schematically in Fig. 5. It should be noted that this process is essentially a degenerate six-wave mixing process. Equation (11) accomodates the intensity dependence of the diffraction efficiency, in the proximity of a two-photon resonance, of intensity up to the power of four. This accounts qualitatively for the high-order intensity dependences of the diffraction efficiency of poly-4BCMU solutions in chloroform, as shown in Fig. 4. Similar high-order intensity dependences are exhibited by poly-4BCMU/chlorobenzene gels and, to a lesser degree, poly-3BCMU/ chlorobenzene gels. The smaller contribution of the fifth-order process in the poly-3BCMU gels is consistent with a larger detuning from the two-photon resonance, as observed by Kajzar and Messier [6]. However, the resolution is of critical importance and also the quantitative evaluation of the contributing components to the effective nonlinear susceptibility and their relative magnitudes. According to eqn. (3), this susceptibility is given by =
X~o,v+Re
X~)ol+~-~-k)Xpo,)) +
(3) Im Xpo,+
(5) X,ol
(15)
The diffraction efficiency is then given by
~ = /~ ) 3~d
\I2 (-~2Id)21;~(3)1o~ 1 2
(16)
Figure 6 shows the variation with concentration of the diffraction efficiency for the range of solutions of poly-4BCMU in chloroform at a constant intensity of 7 x 1013W/m 2. The curve shows a concentration dependence which is markedly different from that of the nonlinear susceptibility, which increases linearly with concentration. Although the diffraction efficiency increases initially from that due to the solvent alone, indicating a positive real component of the polymer nonlinearity, the increase is not parabolic. In this concentration range and at this intensity, the diffraction efficiency appears to increase initially approximately linearly with concentration.
238 xlO -3 |
I
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I
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[] D I
,
0.0
I
0.2
,
[
0.4
,
[
,
0.6
I
0.8
,
I
,
1.0
Concentration (g/l) Fig. 6. Concentration dependence of the diffraction efficiency of solutions of poly-4BCMU in chloroform at 7 x 1013W/m 2.
At the maximum concentration, however, this increase appears to be saturated. Similar behaviour is seen in the concentration dependence of the diffraction efficiency of both poly-3- and -4BCMU gels. Equation (16) shows the predicted behaviour of the diffraction efficiency as a function of concentration, in the presence of a two-photon absorption. The influence of the two-photon term (afld) is simply to reduce, by absorption, the observed diffracted signal. In the development of this equation, the assumption that ~2Id ~ 1 was made. Measured values of the two-photon absorption coefficient of poly-BCMUs are reproduced in Table 1, from ref. 3. Also listed in Table 1 are the values of ~2Id predicted for the present cases. In all cases, therefore, the above approximation holds. However, the
TABLE 1 Two-photon absorption coefficients of poly-BCMU solutions a ~2(m/W) (for 1 g/l)
a2Id
Poly-3BCMU in chlorobenzene
2.26 x 10 -13
0.011
Poly-4BCMU in chlorobenzene
4.47 x 10-13
0.022
Poly-4BCMU in chloroform
< 1 x I0-14
< i x 10 - 3
Polymer
aTaken from ref. 3.
239
influence of such values of a2Id is negligible, producing an attenuation of the diffracted signal of order of 1 to 2%. The concentration dependence of the modulus of X(3) must, therefore, be considered. Because of the cascade nature of the fifth-order nonlinear process (eqn. (13)), the concentration dependence of 1~(3)1may be written in the form 1
1~(3)1= {(A + C(1 +
Sc)c) 2 + (D(1 + Be))2} 5
(17)
where A, B, C and D are constants and c is the polymer concentration. In this case, to account for the behaviour shown in Fig. 6, B is the relevant parameter. Combination of eqns. (13) and (15) yields an expression for B given by
IBI \ 2c ]Id
(18)
The magnitude of this component is similarly of the order of a few percent and its influence cannot account for the concentration dependences of the diffraction efficiencies. In relation, therefore, to eqn. (15), the dependence of the nonlinear susceptibilities on the polymer concentration appears not to be a well-behaved function. Extraction of a quantitative understanding of the nonlinear susceptibilities in these solutions is prevented by what appears to be the effects of some form of concentration-dependent interchain effects. However, some information may be extracted. Considering first the solutions of poly-4BCMU in chloroform, a first observation is that the real component of the nonlinear susceptibility of the polymer is of positive sign. Observation of a positive real component of the polymer nonlinearity is in agreement with the findings of Nunzi and Grec [3]. In all cases, however, the nonlinearity appears to be dominated by imaginary components and, therefore, by the Kramers-Kronig relationship: the sign of the real component of the nonlinearity at the laser wavelength is determined by the position of that wavelength with respect to the influencing resonance. Kajzar has shown the sensitivity of the sign of the real component of the nonlinearity to this positioning [6] and the differing sign of the yellow solutions from that measured for enine oligomers [8] may be accounted for by the fact that the nonlinearity in each case is dominated by a different resonance. Although the magnitude of the susceptibility has not been calculated, an estimate of the contributing component may be made, considering the low concentration regime. A best-fit of a well-behaved third-order nonlinearity to the low concentration regime of Fig. 6 yields a value of the magnitude of the molecular hyperpolarizability calculated per monomer unit of 171 = 1.5 _ 0.5 × 10 t2 mS/V 2. Comparison of this to the power-law dependence observed for the oligomeric enine series [8] predicts a value of the chain length of the order of 70/~. Since the polymer is of considerably longer chain length, a deviation from this power-law dependence may be inferred. This deviation is not, however, surprising because of the differing nature of the nonlinearities. In both cases the nonlinearity is dominated by imaginary components,
240 since the polymer is influenced by the proximity of a two-photon resonance. As has been illustrated by Su et al. [9], the electronic transitions in one-dimensional conjugated polymers are localized because of lattice coupling and give rise to lattice deformations in the form of solitons or bipolarons. The spatial extent of such lattice deformations is predicted to be of the order of 15 repeat units [9], which may be estimated to be 70/k. If, therefore, polarization of the u-electron system occurs via electronic transitions resulting in soliton-type lattice deformations, then this polarization itself limits the u-electron delocalization. Notably, the chain length of the polymer estimated from a fit of the nonlinearity to the power-law dependence of the oligomers compares rather favourably with the estimated extent of deformation. Deviation from the power-law dependence is then interpreted by a limitation of the effective chain length, or conjugation length, by the nature of the polymer response to the polarization. In the concentration dependence of the diffraction efficiency of both poly-3- and -4BCMU, a negative real component of the nonlinearity is observed. The process is again dominated by a strong imaginary component, as may be seen from the non-zero minimum in both cases. The magnitude of these components may be estimated, again by fitting a well-behaved nonlinearity to the low concentration regions of the curve. Table 2 lists the estimated values for all three polymer solutions. Although the increase in the nonlinearity from the yellow to blue forms is consistent with the proposed increase in the degree of conjugation in the red and blue forms, the dominance again of imaginary components implies that the material nonlinearity may not be considered within the simplistic formalisms of free electrons. It has been pointed out that the nature of the visible absorptions in both the blue and red forms are excitonic in nature, as opposed to the interband transition of the yellow solution. It may be said, therefore, that this excitonic nature gives rise to a higher material nonlinearity at 1.064 #m. The magnitude of this nonlinearity appears to be determined by the proximity of the two-photon resonance. However, the absorption cross
TABLE 2 Estimated polymer nonlinearities (in m~/V2) Polymer
Re~
IImTI
171
P o l y - 3 B C M U in chlorobenzene
-4
× 10 -42
1 × 10 -41
1.1 x 1 0 - 4 1
Poly-4BCMUin chlorobenzene
-5
× 10 - 4 2
6 x 10 42
8 × 10 -42
Poly-4BCMU in chloroform
2 × 10-42
241
section at the wavelength of operation is larger for the red gel than for the blue and, yet, the blue form yields a higher nonlinearity. In view of the above discussion, however, the nonlinearity must be considered to be determined by the excited state polarization which appears to be largest for the blue excitonic species.
Poly(3-butylthiophene) Another material of considerable interest is polythiophene (Fig. 2(b)). It has a heterocyclic backbone in which the sulfur stabilizes the cis-acetyleniclike structure. There is considerable evidence that the sulfur interacts only weakly with the ~-electron structure [10] and so the increased backbone rigidity should enhance the ~-electron delocalization. Polythiophene itself is insoluble and therfore the polymer used in this study was the substituted poly(3-butylthiophene) which was studied in chloroform solution. The intensity dependence of the diffraction efficiency was monitored for polymer solutions in chloroform in the range 0- 8.6 g/1. The results for (a) neat solvent, and solutions (b) 0.052 g/l, (c) 0.207 g/1 and (d) 3.32 g/l are shown in Fig. 7. In the case of neat solvent, the diffraction efficiency was found to vary as the square of the input intensity, as is expected for a third-order process. The intensity dependences of the polymer solutions, however, were found to be strongly concentration dependent, with an dependence of up to the sixth power of intensity being observed, indicating the presence of a seventh-order nonlinearity. This may be associated with a three-photon resonant enhancement of the nonlinearity, giving rise to an eight-wave mixing process. As in the case of the two-photon resonant enhancement of the nonlinearity observed in polydiacetylenes, the process may be characterized by an effective third-order susceptibility given by I
~(3} = ~(3> _~_7 (
i
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10-5
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.
(19)
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50
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60
. . . .
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Fig. 7. Intensity dependence of the diffraction efficiency of solutions of poly(3-butylthiophene) in chloroform.
242 10-4
10"5.
x
x
x x x
10-6. x 10-7 10"3
10"2
x
10-1
10 0
101
Concentration (g/l) Fig. 8. Concentration dependence of the diffraction efficiency of solutions of poly(3-butylehiophene) in chloroform.
Again, the process may be interpreted as a cascade process, and the nonlinearity is considered a resonant-enhanced Z~3~ rather than a true X~7~. Although a correct treatment of this process relies on knowledge of the positioning and oscillator strength of the three-photon resonance, it may be dealt with, as in the case of the two-photon enhanced process, in the form of an effective third-order susceptibility, Xe~-¢a~ The variation of the diffraction efficiency as a function of polymer concentration at constant intensity (I = 3 × 1012 W/m 2) is shown in Fig. 8. The curve initially decreases and goes through a minimum at a concentration of 0.052 g/1. This is a result of the cancellation of the positive real contribution to the nonlinearity from the solvent by a negative real polymer contribution. The non-zero minimum of the curve is indicative of an imaginary component of the polymer nonlinearity. However, any attempt to fit to this curve a linear dependence of the nonlinearity on the concentration fails. Indeed, this nonlinearity with respect to concentration is self-evident from the variation of intensity dependences with concentration; as outlined above, for concentrations in the midrange, the nonlinearity is dominated by the proximity of a three-photon resonance, but at higher and lower concentrations the influence of this resonance appears to diminish. Variation of imaginary linear susceptibilities with concentration is common in organic dye systems and is attributed to aggregation phenomena [11]. The shift of oscillator strength from the primary to a secondary dimer resonance is the result of the change in the local field experienced by the monomer on dimerization, which gives rise to Stark effects. The deviation from linearity of the nonlinear susceptibility in these materials may be understood in terms of concentration-dependent interchain effects which give rise to a variation of the local fields experienced by the polymer. In the case of a seventh-order nonlinearity, the macroscopic susceptibility depends on the local field correction factor to the eighth power and, hence, is highly sensitive to variation of it. These variations can give rise to a shift of the three-photon resonance by the Stark effect and, indeed, such a
243
shift is observed in the linear absorption spectrum on going from dilute solution to a thin film [2]. Similar aggregation processes have been observed in polydiacetylenes [12]. An estimate of the material nonlinearity may, however, be made by fitting a well-behaved nonlinear susceptibility to the low concentration region of Fig. 8. Such a fit gives values of 7R = - - 5 . 7 X 10-43mS/V 2 and 17~1= 1.2 x 10-42 mS/V 2, resulting in 171= 1.35 x 10-42 mS/V 2. Since the influence of the three-photon resonance is considerable even at low concentrations, these values are representative of an effective nonlinearity.
Polymer P28 The third polymer investigated in this study is the 'ridigized backbone' diacetylene-type polymer shown in Fig. 2(c). This material was synthesized by R. Giesa of Mainz and will be referred to as P28. Similar to the case of polythiophene, the cyclic structures of the anthracene and phenol rings in the backbone should increase the rigidity of the backbone and, in so doing, increase the n-electron delocalization. A range of solutions of the polymer P28 was made up in toluene with a maximum concentration of 3 g/1. To investigate the possible contribution of multiphoton absorptions, the intensity dependence of the diffraction efficiency was examined. Figure 9 shows the dependences observed for a range of concentrations. For all concentrations, the behaviour of the nonlinear signal is characteristic of a pure third-order process with no indication of multiphoton resonant enhancement. Figure 10 shows the concentration dependence of the diffraction efficiency at a constant intensity of 3 x 1013W/m 2. No decrease of the efficiency from the initial solvent value is observed, indicating a positive value of the real component of the nonlinearity. However, although the
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Intensity (GW/m2) Fig. 9. Intensity dependence of the diffraction efficiency of P28 solutions in toluene.
244
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4.6 4.4 4.2 4.0 3.8 3.6 3.4
D
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0.0
0.5
1.0
1.5
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I
2.0
2.5
3.0
Concentration (g/l) Fig. 10. Concentration dependence of the diffraction efficiency of P28 solutions in toluene.
intensity dependence of the nonlinearity is well behaved, the concentration dependence of the diffraction efficiency, as with the other polymers investigated, is not well behaved and a linear dependence of the material nonlinearity on the concentration may not be applied. As in the case of polythiophene, concentration-dependent interchain effects are proposed as the driving mechanism of this behaviour. For the purpose of comparison with the previously characterized polymeric solutions, an estimate of the molecular hyperpolarizability of the polymer may be made from the initial rate of rise of the observed signal from the solvent value. The low concentration behaviour is best characterized by I~1 = 1 × 10 41 mS/V 2. Notably, although the diffraction efficiency is small at a fixed concentration, compared to the other polymers, because of the bulkiness of the monomer unit the molecular hyperpolarizability per unit is large. Although in both polymers an increase in the diffracted signal as a function of concentration is observed, this increase is notably less than parabolic. As is observed for polydiacetylene solutions, a saturation-like behaviour of the susceptibility is observed. This behaviour may be attributed to aggregation phenomena. In such an aggregation process, variation of local fields give rise to a shifting of oscillator strength as a result of Stark effects. Although for a dimerization process the shift may be quantitatively described, no simplistic concentration dependence may be fitted to the present situation. Despite the complexities introduced by such aggregation processes, estimates of the material nonlinearity may be made. Of particular interest is a comparison of these with those estimated for the polydiacetylene solutions. In poly(3-butylthiophene) solutions, the third-order nonlinearity is enhanced by the proximity of a three-photon resonance. However, the nonlinearity is lower than the lowest nonlinearity of the three polydiacetylene solutions,
245 that of the yellow, chloroform solutions. In the polydiacetylene solution, the nonlinearity is enhanced by the proximity of a two-photon resonance. Obviously the degree of enhancement experienced will influence the magnitude of the nonlinearity. However, as was stated previously, the dominance of imaginary contributions to the nonlinearity implies that the nature of the excited state ultimately determines the magnitude of the material nonlinearity. In both polymer solutions, the linear optical properties are characteristic of an interband transition. Also, the number of conjugated multiple bonds per monomer unit is the same for polydiacetylene and polythiophene. In both materials, a bipolaronic excited species has been observed [13, 14]. The displacement of the bipolaronic photoinduced absorption from the primary, ground-state absorption indicates a greater degree of localization of the excitation in the polydiacetylene solution. This greater localization should give rise to a lower polarization of the species. However, although the number of conjugated multiple bonds is the same for the two monomers, the number of conjugated electrons is larger in the polydiacetylene monomer unit. This increased charge density may account for the larger nonlinearity. In the P28 polymer solutions, a microscopic polarizability of order 10 41 mS/V 2 is estimated. This nonlinearity is comparable to that of the largest value measured for the polydiacetylene solutions, that of the blue solutions. No multiphoton enhancement of the nonlinearity is observed. This large nonlinearity is, however, deceptive, as the P28 monomeric unit is large and although the backbone has the alternating structure of polydiacetylene, the monomeric unit is the equivalent to two polydiacetylene units. However, comparison of the nonlinearity with that of the yellow polydiacetylene solution is more appropriate, since it exhibits similar linear optical properties to that of P28. There is a sizeable enhancement of the nonlinearity over that of the polydiacetylenes, which may be attributed to increased backbone rigidity. The observation of a three-photon resonant enhancement of the thirdorder nonlinearity in polythiophene solutions highlights the importance of multiphoton nonlinear interactions in organic polymers. A similar multiphoton resonant enhancement of the nonlinearity has been observed in polydiacetylene solutions. Some consideration must, therefore, be directed towards these multiphoton contributions. In conventional solids, the electronic system is extended in three dimensions and may be considered as a Fermi gas of weakly interacting particles. However, under one-dimensional confinement, the motion of the electrons is restricted and the effects of Coulomb interactions are considerably enhanced. It has been shown [15, 16] that in order to account correctly for the ordering of the llBu ~-electron state and the lower lying, two photon 2lAg state in polyenes [17], a proper treatment of the electron-electron interactions must be introduced, and that independent particle theories such as the Huckel theory are inadequate. The importance of these electron-electron interactions is a direct result of the dimensionality of these systems and the result is that the electron
246
motion becomes strongly correlated. The relative contributions of these interactions must be considered also as a function of the electron density within a specific monomer unit. In the case of polythiophene, the monomer unit contains only double bonds and so the electron density is comparatively low compared with other polymers. This results in a positioning of the three-photon level at an energy sufficiently high to influence the four-wave mixing process at 1.064/Lm, indicating an even higher two-photon level. Polydiacetylene has a very much higher electron density per monomer unit and this leads to a two-photon level which is positioned close to the 1.064/~m. Although an imaginary contribution to the nonlinearity of P28 may not be resolved explicitly because of the complexity of the concentration dependence of the nonlinearity, it may be inferred by extrapolation from the series of oligomeric enines. Any imaginary contribution is due to linear absorption, as may be seen from the intensity dependence of the diffraction efficiency (Fig. 10). The absence of a multiphoton resonant contribution to the nonlinearity is consistent with the large electron density of the monomer unit resulting from its complex aromatic structure. This gives rise to a strong electron correlation which shifts the multiphoton resonances far below the laser wavelength.
Conclusions
In all polymeric solutions investigated here, a complex concentration dependence of the material nonlinearity is observed. This behaviour is attributed to concentration-dependent interchain effects. These effects make an exact evaluation of the polymer nonlinearity difficult and a complete study would require a tunable near-infrared source which could track any shifting resonance. However, an estimate of the nonlinearities may be made and a comparison between the polymers is informative. The nonlinearity in the blue form of polydiacetylene proves to be the largest although this value is approached by the 'rigidized' backbone polymer P28. The observation that in all cases the polymer nonlinearity is dominated by imaginary contributions even in their transparency region implies the need for a reassessment of 'non-resonant' nonlinearities. Also, as the fundamental excitations of these materials are inherently localized by virtue of their one dimensionality, the notion of unlimited ~-electron delocalization and subsequently unlimited nonlinearities must also be revised. If the nonlinearity is dominated by resonant effects, then it is limited by the nature of these resonances. The importance is highlighted, therefore, of an understanding of the linear and nonlinear resonant behaviour of these materials. A direct relationship between the monomer structure and the resonance structure has been implied and such a relationship enables the resonance structure of these materials to be chemically tailored. Further development of the nonlinear optical properties of polymers relies, therefore, on expertise in molecular engineering.
247
References 1 H. J. Eichler, P. Gfinter and D. W. Pohl, in Laser Induced Dynamic Gratings, Springer Series in Optical Sciences, Springer, Berlin, 1986. 2 H. J. Byrne, W. Blau and K.-Y. Jen, Synth. Met., 32 (1989) 229. 3 J. M. Nunzi and D. Grec, J. Appl. Phys., 62 (1987) 2198. 4 P. D. Maker, R. W. Terhune and C. M. Savage, Phys. Rev. Left., 12 (1964) 507. 5 G. N. Patel, Polym. Prepr., Am. Chem. Soc., Div. Polym. Chem., 19 (1978) 154. 6 F. Kajzar and J. Messier, in D. Bloor and R. R. Chance (eds.), Polydiacetylenes, NATO A S I Series, Martinus Nijhoff, Dordrecht, 1985. 7 J. A. Giordamaine and J. A. Howe, Phys. Rev. Lett., 11 (1963) 207. 8 H. J. Byrne, W. Blau, R. Giesa and R. C. Schulz, Chem. Phys. Lett., (1989) to be published. 9 A. J. Heeger, S. Kivelson, J. R. Schrieffer and W.-P. Su, Rev. Mod. Phys., 60 (1988) 781. 10 T. C. Chung, J. H. Kaufman, A. J. Heeger and F. Wudl, Phys. Rev. B, 30 (1984) 702. 11 A. Penzkofer and W. Blau, Opt. Electr. Eng., 15 (1983) 325. 12 G. Wenz, M. A. Mfiller, M. Schmidt and G. Wegner, Macromolecules, 17 (1984) 837. 13 F. L. Pratt, K. S. Wong, W. Hayes and D. Bloor, J. Phys. C: Solid State Phys., 20(1987) L41. 14 Z. Vardeny, E. Ehrenfreund, O. Brafinan, A. J. Heeger and F. Wudl, Synth. Met. 13 (1986) 183. 15 B. S. Hudsen and B. E. Kohler, J. Chem. Phys., 64 (1976) 4422. 16 Z. G. Soos and S. Ramasesha, Phys. Rev. B, 29 (1984) 5410. 17 P. Tavan and K. Schulten, Phys. Rev. B, 36 (1987) 4337.
231
M U L T I P H O T O N N O N L I N E A R I N T E R A C T I O N S IN C O N J U G A T E D ORGANIC POLYMERS H. J. BYRNE* and W. BLAU Department of Physics, Trinity College, Dublin (Ireland)
Abstract The optical nonlinearity at a wavelength of 1.064 #m has been studied in a range of soluble conjugated organic polymers using the technique of degenerate forward four-wave mixing. This technique enables determination of the third-order nonlinear susceptibility IX¢3)I of the material. In all cases, determination of the response time of the material nonlinearity is limited by the experimental resolution ( ~ 70 ps). Measurements of I~~3)] as a function of polymer solution concentration were performed and contributions from both real and imaginary components were observed and estimated. The origin of the imaginary contribution in these 'off-resonant' conditions was investigated by intensity-dependent measurements of the four-wave mixing signal. In solutions of poly-3 and -4(butoxycarbonylmethylurethane diacetylene), (BCMU) in both chloroform and chlorobenzene, a dependence characteristic of a fifth-order nonlinear interaction is observed. Furthermore, in chloroform solutions of poly(3-butylthiophene) a seventh-order nonlinearity is observed. These high-order nonlinearities are associated with two-photon and threephoton resonant enhancement of the four-photon process and are characterized by fifth- and seventh-order susceptibilities, X¢5) and ~(7) respectively. The presence of imaginary contributions to the optical nonlinearity in all materials investigated implies that this nonlinearity is dominated in all cases by the nearest resonance. This conclusion renders redundant such simplistic models as the free-electron model.
Introduction Organic polymers have been predicted to display large ultrafast Kerrlike nonlinearities because of their highly polarizable, delocalized ~-electron backbone. However, a quantitative understanding of the relationship between these large nonlinearities and the electron delocalization and polymer backbone structure is as yet lacking. Notably, the role of 'resonant'
*Author to whom correspondence should be addressed. 0379-6779/90/$3.50
© Elsevier Sequoia/Printed in The Netherlands
232
contributions to the nonlinearity at a wavelength which is far from the primary absorption and, indeed, the nature of this off-resonant nonlinearity are unclear. In this study, the optical nonlinearity has been measured in the transparency region of a selection of conjugated organic polymers in an attempt to address matters. Concentration dependences of the nonlinearities have been studied to establish the contributions, if any, of imaginary components, and intensity-dependent measurements have been made in order to establish the nature of influential resonances.
Experimental
The experimental method used in these studies was that of forced light scattering from laser-induced gratings, a technique which may be viewed as a degenerate four-wave mixing process in the forward direction [1]. The set-up is depicted schematically in Fig. 1. The light source is an amplified, passively mode-locked Nda+:YAG laser emitting linearly polarized pulses of 50 + 25 ps duration and of wavelength ,~ = 1.064 gm at a frequency of 3 Hz. Peak powers of up to 50 MW were readily available. The experimental method, which is described in detail elsewhere [2], is based on the interference at the sample of two spatially and temporarily overlapped beams, producing a spatial modulation of the intensity-dependent refractive index of the material. This modulation acts as a diffraction grating from which the pulses may self-diffract. Under thin grating conditions [1], satisfied experimentally by keeping the angle between the two beams small ( < 1°), an expression relating the diffraction efficiency (~/) in the first order to the third-order material nonlinearity may be derived
Iz3>I=
(1)
4~°cn 22(~I) 1/2 3~dlo
where c is the speed of light, eo is the permittivity of free space, n is the refractive index of the sample, d is the sample thickness and I0 is the input pulse intensity. In the experiments reported here, d was 1 mm and n was Beam Splitter
Delay Line
Photodiode
Fig. 1. Experimental set-up for the self-diffraction technique.
233
taken to be the refractive index of the solvent, because of the low fractional volume of the solute. Equation (1) holds for materials which are transparent at the operating wavelength. It can be seen from eqn. (1) that verification of the presence of a true third-order nonlinear process may be obtained by monitoring the intensity dependence of the diffraction efficiency. For a true third-order process (2)
= (I1/Io) ~ 12
where 11 is the intensity diffracted into the first order. Such a verification is important as fifth- and seventh-order processes, originating in two- and three-photon resonant enhancement of the material nonlinearities, have been observed in organic conjugated materials [2, 3]. IX(3)I may have both real and imaginary components originating from (3) . the polymer as well as a contribution from the solvent, X~o~v. which is purely real and positive in the case of most organic solvents, including n-hexane [4]. For the concentration range used in this work, the oligomer fractional volume is negligible. Thus = L~ol-~ +
Re~pol)
+ "~--~po1~ '
where ReX(p3o ), and Im~pol -"(~) are the real and imaginary components of the material nonlinearity. By monitoringthe concentration dependenceoflx(3)l, the contribution X.(3) solv due to the solvent may be extracted and the magnitude
of ReX~pao ),and ImX~p~o)1may be determined. Furthermore, the sign of Zex~o),may be determined from the concentration dependence of the real part of IX(a)].
R e s u l t s and d i s c u s s i o n Polydiacetylenes
The chemical structures of the polymeric materials chosen for this study are depicted in Fig. 2. A particular series of polydiacetylenes which has been extensively studied because of their ease of synthesis and good environmental stability, is the n-BCMU series [5] which has the backbone structure of Fig. 2(a) with side groups R = R' = --(CH2) n - O C - O N H C H z C O O ( CH2)3CH3 The materials employed in this study have n = 3 and 4. Both poly-3- and -4BCMU are soluble in organic solvents such as chloroform and dimethylformamide, forming strongly coloured yellow solutions. Solutions of poly-3and -4BCMU in chlorobenzene are, however, blue and red in colour respectively. Maximum concentrations of poly-3- and -4BCMU available were, however, limited to 1.3 and 2.0 g/1 respectively. A range of concentrations, for the purpose of concentration-dependent measurements, was generated by dilution of poly-3- and -4BCMU gels, made by the method of ref. 3, at 100 °C. With
234
II
n
(a)
(b) 0C18H37
"['C=C
C=C n
OC18H37 (c) Fig. 2. Some one-dimensional conjugated polymers: (a) polydiaeetylene; (b) polythiophene; (c) polymer P28.
decreasing concentration, the gels became fluid and approached solutions at low concentrations. Yellow solutions of poly-4BCMU in chloroform were made up with a maximum concentration of 1.1 g/1. In order to examine the nature of the process giving rise to the nonlinear diffraction, the intensity dependence of the diffraction efficiency was examined. In the case of neat solvent, the intensity dependence of the diffraction efficiency was found to vary as the square of the input intensity for both chloroform and chlorobenzene, as is expected for a third-order nonlinear process. Similarly for polymer solutions at low concentrations, the intensity dependence of the diffraction efficiency is characteristic of a true third-order process, as is shown in Fig. 3. However, at concentrations greater than ~ 0.2 g/l, the polymer solutions exhibit an intensity dependence of the diffraction efficiency greater than squared. Figure 4 shows the intensity dependences for the range of poly-4BCMU solutions in chloroform. Clearly, an intensity-dependent process of higher order is contributing to the grating formation. Such high-order processes have, indeed, been observed previously in polydiacetylenes [3, 6]. A relationship between the intensity-dependent change in the refractive index of the sample, in the presence of a periodically modulated intensity distribution, may be derived and is given by Ax -
AZ 2(1 + Z) ~/2
(4)
In the case where the nonlinear behaviour is purely third order, Ax is given by a .(a) Ak A l A Z i j = 4~ijkl
(5)
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60
80
100
Fig. 3. Intensity dependences of the diffraction efficiency: (A) chloroform; (O) 0.1 g/1 poly4BCMU in chloroform; (-) chlorobenzene; ((3) 0.01 g/1 poly-3BCMU in ehlorobenzene; ( + ) 0.2 g/1 poly-4BCMU in chlorobenzene.
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Fig. 4. Intensity dependence of the diffraction efficiency of solutions of poly-4BCMU in chloroform.
If, h o w e v e r , h i g h e r o r d e r t e r m s a r e significant, t h e y m u s t be t a k e n into a c c o u n t , h e n c e A~, in the case w h e r e all fields a r e d e g e n e r a t e , t a k e s t h e form, A ~ i j = ' 4 } ~ i j .(S) kl
A k A l ±~'(5) T8Aijklmn
A k AA l
m
A n .-.
(6)
T r e a t m e n t of t h e r e l a t i o n s h i p b e t w e e n t h e d i f f r a c t i o n efficiency of a l a s e r - i n d u c e d g r a t i n g a n d the n o n l i n e a r susceptibilities, in this case, m a y be p e r f o r m e d by c h a r a c t e r i z i n g t h e p r o c e s s by a n effective t h i r d - o r d e r
236
susceptibility, XefP - (3) given by Zeff(3)__--•(3)
+ 5)((S)AnAm
= X(a) + ~
51
. . . .
X(s) . . . .
(7)
This effective susceptibility is measured experimentally in the selfdiffraction experiments and may be related to the diffraction efficiency into the first order by the relationship (3) Zee
4~oCn2,~07) 1/2 -
(8)
3udI
as for the case of a pure third-order susceptibility. In this case, however, the material absorption must be taken into consideration. In the presence of a two-photon absorption, the material absorption coefficient becomes intensity dependent and has the form = ~0 + ~21
(9)
where a o is the linear absorption coefficient and a2 is the two-photon absorption coefficient. In the present situation, a0 is small and may be assumed to be equal to zero [3]. Similarly, the sample transmission becomes intensity dependent and may be written [7] as T = [1 -
a2Id] -1
(10)
Inclusion of material absorption in the expression for the effective material nonlinearity, assuming that the term o~2Id ~ 1, yields (3) 4~oCn2'~(t/)~/2 ( 1 :¢2) Zea -3, I-d + 2
(11)
~(3) eff is therefore composed of two factors. The first is identical to the nonlinearity in the case of a pure third-order process and may be written 1~(3){ -- 4eocn2~0/)1/2
(12)
3udI
The second term is intensity dependent and arises from the additional fifth-order contribution due to the two-photon resonant enhancement of the nonlinearity. Thus 2 a 2 n,~(t/) 1/2
Iz(5)l = -
5.I 3Z(3)eocnd~2
10
(13)
This relationship between X(5) and ~(3) contributes to the understanding of the nature of the fifth-order process. The microscopic hyperpolarizability 5 is determined by the third-order hyperpolarizability 7 and by the change in
237
Fig. 5. Degenerate six-wave mixing process.
the first-order hyperpolarizability ~z 6 = "tu2
(14)
The fifth-order nonlinearity is, therefore, a cascade process. ~(5) is not a true fifth-order polarizability, but is rather a two-photon resonantly enhanced ~(~), and the process is illustrated schematically in Fig. 5. It should be noted that this process is essentially a degenerate six-wave mixing process. Equation (11) accomodates the intensity dependence of the diffraction efficiency, in the proximity of a two-photon resonance, of intensity up to the power of four. This accounts qualitatively for the high-order intensity dependences of the diffraction efficiency of poly-4BCMU solutions in chloroform, as shown in Fig. 4. Similar high-order intensity dependences are exhibited by poly-4BCMU/chlorobenzene gels and, to a lesser degree, poly-3BCMU/ chlorobenzene gels. The smaller contribution of the fifth-order process in the poly-3BCMU gels is consistent with a larger detuning from the two-photon resonance, as observed by Kajzar and Messier [6]. However, the resolution is of critical importance and also the quantitative evaluation of the contributing components to the effective nonlinear susceptibility and their relative magnitudes. According to eqn. (3), this susceptibility is given by =
X~o,v+Re
X~)ol+~-~-k)Xpo,)) +
(3) Im Xpo,+
(5) X,ol
(15)
The diffraction efficiency is then given by
~ = /~ ) 3~d
\I2 (-~2Id)21;~(3)1o~ 1 2
(16)
Figure 6 shows the variation with concentration of the diffraction efficiency for the range of solutions of poly-4BCMU in chloroform at a constant intensity of 7 x 1013W/m 2. The curve shows a concentration dependence which is markedly different from that of the nonlinear susceptibility, which increases linearly with concentration. Although the diffraction efficiency increases initially from that due to the solvent alone, indicating a positive real component of the polymer nonlinearity, the increase is not parabolic. In this concentration range and at this intensity, the diffraction efficiency appears to increase initially approximately linearly with concentration.
238 xlO -3 |
I
I
I
I
I Q
8
6
4
[] D I
,
0.0
I
0.2
,
[
0.4
,
[
,
0.6
I
0.8
,
I
,
1.0
Concentration (g/l) Fig. 6. Concentration dependence of the diffraction efficiency of solutions of poly-4BCMU in chloroform at 7 x 1013W/m 2.
At the maximum concentration, however, this increase appears to be saturated. Similar behaviour is seen in the concentration dependence of the diffraction efficiency of both poly-3- and -4BCMU gels. Equation (16) shows the predicted behaviour of the diffraction efficiency as a function of concentration, in the presence of a two-photon absorption. The influence of the two-photon term (afld) is simply to reduce, by absorption, the observed diffracted signal. In the development of this equation, the assumption that ~2Id ~ 1 was made. Measured values of the two-photon absorption coefficient of poly-BCMUs are reproduced in Table 1, from ref. 3. Also listed in Table 1 are the values of ~2Id predicted for the present cases. In all cases, therefore, the above approximation holds. However, the
TABLE 1 Two-photon absorption coefficients of poly-BCMU solutions a ~2(m/W) (for 1 g/l)
a2Id
Poly-3BCMU in chlorobenzene
2.26 x 10 -13
0.011
Poly-4BCMU in chlorobenzene
4.47 x 10-13
0.022
Poly-4BCMU in chloroform
< 1 x I0-14
< i x 10 - 3
Polymer
aTaken from ref. 3.
239
influence of such values of a2Id is negligible, producing an attenuation of the diffracted signal of order of 1 to 2%. The concentration dependence of the modulus of X(3) must, therefore, be considered. Because of the cascade nature of the fifth-order nonlinear process (eqn. (13)), the concentration dependence of 1~(3)1may be written in the form 1
1~(3)1= {(A + C(1 +
Sc)c) 2 + (D(1 + Be))2} 5
(17)
where A, B, C and D are constants and c is the polymer concentration. In this case, to account for the behaviour shown in Fig. 6, B is the relevant parameter. Combination of eqns. (13) and (15) yields an expression for B given by
IBI \ 2c ]Id
(18)
The magnitude of this component is similarly of the order of a few percent and its influence cannot account for the concentration dependences of the diffraction efficiencies. In relation, therefore, to eqn. (15), the dependence of the nonlinear susceptibilities on the polymer concentration appears not to be a well-behaved function. Extraction of a quantitative understanding of the nonlinear susceptibilities in these solutions is prevented by what appears to be the effects of some form of concentration-dependent interchain effects. However, some information may be extracted. Considering first the solutions of poly-4BCMU in chloroform, a first observation is that the real component of the nonlinear susceptibility of the polymer is of positive sign. Observation of a positive real component of the polymer nonlinearity is in agreement with the findings of Nunzi and Grec [3]. In all cases, however, the nonlinearity appears to be dominated by imaginary components and, therefore, by the Kramers-Kronig relationship: the sign of the real component of the nonlinearity at the laser wavelength is determined by the position of that wavelength with respect to the influencing resonance. Kajzar has shown the sensitivity of the sign of the real component of the nonlinearity to this positioning [6] and the differing sign of the yellow solutions from that measured for enine oligomers [8] may be accounted for by the fact that the nonlinearity in each case is dominated by a different resonance. Although the magnitude of the susceptibility has not been calculated, an estimate of the contributing component may be made, considering the low concentration regime. A best-fit of a well-behaved third-order nonlinearity to the low concentration regime of Fig. 6 yields a value of the magnitude of the molecular hyperpolarizability calculated per monomer unit of 171 = 1.5 _ 0.5 × 10 t2 mS/V 2. Comparison of this to the power-law dependence observed for the oligomeric enine series [8] predicts a value of the chain length of the order of 70/~. Since the polymer is of considerably longer chain length, a deviation from this power-law dependence may be inferred. This deviation is not, however, surprising because of the differing nature of the nonlinearities. In both cases the nonlinearity is dominated by imaginary components,
240 since the polymer is influenced by the proximity of a two-photon resonance. As has been illustrated by Su et al. [9], the electronic transitions in one-dimensional conjugated polymers are localized because of lattice coupling and give rise to lattice deformations in the form of solitons or bipolarons. The spatial extent of such lattice deformations is predicted to be of the order of 15 repeat units [9], which may be estimated to be 70/k. If, therefore, polarization of the u-electron system occurs via electronic transitions resulting in soliton-type lattice deformations, then this polarization itself limits the u-electron delocalization. Notably, the chain length of the polymer estimated from a fit of the nonlinearity to the power-law dependence of the oligomers compares rather favourably with the estimated extent of deformation. Deviation from the power-law dependence is then interpreted by a limitation of the effective chain length, or conjugation length, by the nature of the polymer response to the polarization. In the concentration dependence of the diffraction efficiency of both poly-3- and -4BCMU, a negative real component of the nonlinearity is observed. The process is again dominated by a strong imaginary component, as may be seen from the non-zero minimum in both cases. The magnitude of these components may be estimated, again by fitting a well-behaved nonlinearity to the low concentration regions of the curve. Table 2 lists the estimated values for all three polymer solutions. Although the increase in the nonlinearity from the yellow to blue forms is consistent with the proposed increase in the degree of conjugation in the red and blue forms, the dominance again of imaginary components implies that the material nonlinearity may not be considered within the simplistic formalisms of free electrons. It has been pointed out that the nature of the visible absorptions in both the blue and red forms are excitonic in nature, as opposed to the interband transition of the yellow solution. It may be said, therefore, that this excitonic nature gives rise to a higher material nonlinearity at 1.064 #m. The magnitude of this nonlinearity appears to be determined by the proximity of the two-photon resonance. However, the absorption cross
TABLE 2 Estimated polymer nonlinearities (in m~/V2) Polymer
Re~
IImTI
171
P o l y - 3 B C M U in chlorobenzene
-4
× 10 -42
1 × 10 -41
1.1 x 1 0 - 4 1
Poly-4BCMUin chlorobenzene
-5
× 10 - 4 2
6 x 10 42
8 × 10 -42
Poly-4BCMU in chloroform
2 × 10-42
241
section at the wavelength of operation is larger for the red gel than for the blue and, yet, the blue form yields a higher nonlinearity. In view of the above discussion, however, the nonlinearity must be considered to be determined by the excited state polarization which appears to be largest for the blue excitonic species.
Poly(3-butylthiophene) Another material of considerable interest is polythiophene (Fig. 2(b)). It has a heterocyclic backbone in which the sulfur stabilizes the cis-acetyleniclike structure. There is considerable evidence that the sulfur interacts only weakly with the ~-electron structure [10] and so the increased backbone rigidity should enhance the ~-electron delocalization. Polythiophene itself is insoluble and therfore the polymer used in this study was the substituted poly(3-butylthiophene) which was studied in chloroform solution. The intensity dependence of the diffraction efficiency was monitored for polymer solutions in chloroform in the range 0- 8.6 g/1. The results for (a) neat solvent, and solutions (b) 0.052 g/l, (c) 0.207 g/1 and (d) 3.32 g/l are shown in Fig. 7. In the case of neat solvent, the diffraction efficiency was found to vary as the square of the input intensity, as is expected for a third-order process. The intensity dependences of the polymer solutions, however, were found to be strongly concentration dependent, with an dependence of up to the sixth power of intensity being observed, indicating the presence of a seventh-order nonlinearity. This may be associated with a three-photon resonant enhancement of the nonlinearity, giving rise to an eight-wave mixing process. As in the case of the two-photon resonant enhancement of the nonlinearity observed in polydiacetylenes, the process may be characterized by an effective third-order susceptibility given by I
~(3} = ~(3> _~_7 (
i
.
10-5
.
.
2 ~(7)
.
(19)
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.
.
.
.
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.
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3.32g/1- 14
.
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I
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.
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Fig. 7. Intensity dependence of the diffraction efficiency of solutions of poly(3-butylthiophene) in chloroform.
242 10-4
10"5.
x
x
x x x
10-6. x 10-7 10"3
10"2
x
10-1
10 0
101
Concentration (g/l) Fig. 8. Concentration dependence of the diffraction efficiency of solutions of poly(3-butylehiophene) in chloroform.
Again, the process may be interpreted as a cascade process, and the nonlinearity is considered a resonant-enhanced Z~3~ rather than a true X~7~. Although a correct treatment of this process relies on knowledge of the positioning and oscillator strength of the three-photon resonance, it may be dealt with, as in the case of the two-photon enhanced process, in the form of an effective third-order susceptibility, Xe~-¢a~ The variation of the diffraction efficiency as a function of polymer concentration at constant intensity (I = 3 × 1012 W/m 2) is shown in Fig. 8. The curve initially decreases and goes through a minimum at a concentration of 0.052 g/1. This is a result of the cancellation of the positive real contribution to the nonlinearity from the solvent by a negative real polymer contribution. The non-zero minimum of the curve is indicative of an imaginary component of the polymer nonlinearity. However, any attempt to fit to this curve a linear dependence of the nonlinearity on the concentration fails. Indeed, this nonlinearity with respect to concentration is self-evident from the variation of intensity dependences with concentration; as outlined above, for concentrations in the midrange, the nonlinearity is dominated by the proximity of a three-photon resonance, but at higher and lower concentrations the influence of this resonance appears to diminish. Variation of imaginary linear susceptibilities with concentration is common in organic dye systems and is attributed to aggregation phenomena [11]. The shift of oscillator strength from the primary to a secondary dimer resonance is the result of the change in the local field experienced by the monomer on dimerization, which gives rise to Stark effects. The deviation from linearity of the nonlinear susceptibility in these materials may be understood in terms of concentration-dependent interchain effects which give rise to a variation of the local fields experienced by the polymer. In the case of a seventh-order nonlinearity, the macroscopic susceptibility depends on the local field correction factor to the eighth power and, hence, is highly sensitive to variation of it. These variations can give rise to a shift of the three-photon resonance by the Stark effect and, indeed, such a
243
shift is observed in the linear absorption spectrum on going from dilute solution to a thin film [2]. Similar aggregation processes have been observed in polydiacetylenes [12]. An estimate of the material nonlinearity may, however, be made by fitting a well-behaved nonlinear susceptibility to the low concentration region of Fig. 8. Such a fit gives values of 7R = - - 5 . 7 X 10-43mS/V 2 and 17~1= 1.2 x 10-42 mS/V 2, resulting in 171= 1.35 x 10-42 mS/V 2. Since the influence of the three-photon resonance is considerable even at low concentrations, these values are representative of an effective nonlinearity.
Polymer P28 The third polymer investigated in this study is the 'ridigized backbone' diacetylene-type polymer shown in Fig. 2(c). This material was synthesized by R. Giesa of Mainz and will be referred to as P28. Similar to the case of polythiophene, the cyclic structures of the anthracene and phenol rings in the backbone should increase the rigidity of the backbone and, in so doing, increase the n-electron delocalization. A range of solutions of the polymer P28 was made up in toluene with a maximum concentration of 3 g/1. To investigate the possible contribution of multiphoton absorptions, the intensity dependence of the diffraction efficiency was examined. Figure 9 shows the dependences observed for a range of concentrations. For all concentrations, the behaviour of the nonlinear signal is characteristic of a pure third-order process with no indication of multiphoton resonant enhancement. Figure 10 shows the concentration dependence of the diffraction efficiency at a constant intensity of 3 x 1013W/m 2. No decrease of the efficiency from the initial solvent value is observed, indicating a positive value of the real component of the nonlinearity. However, although the
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244
xlO -4 I []
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D
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2.5
3.0
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intensity dependence of the nonlinearity is well behaved, the concentration dependence of the diffraction efficiency, as with the other polymers investigated, is not well behaved and a linear dependence of the material nonlinearity on the concentration may not be applied. As in the case of polythiophene, concentration-dependent interchain effects are proposed as the driving mechanism of this behaviour. For the purpose of comparison with the previously characterized polymeric solutions, an estimate of the molecular hyperpolarizability of the polymer may be made from the initial rate of rise of the observed signal from the solvent value. The low concentration behaviour is best characterized by I~1 = 1 × 10 41 mS/V 2. Notably, although the diffraction efficiency is small at a fixed concentration, compared to the other polymers, because of the bulkiness of the monomer unit the molecular hyperpolarizability per unit is large. Although in both polymers an increase in the diffracted signal as a function of concentration is observed, this increase is notably less than parabolic. As is observed for polydiacetylene solutions, a saturation-like behaviour of the susceptibility is observed. This behaviour may be attributed to aggregation phenomena. In such an aggregation process, variation of local fields give rise to a shifting of oscillator strength as a result of Stark effects. Although for a dimerization process the shift may be quantitatively described, no simplistic concentration dependence may be fitted to the present situation. Despite the complexities introduced by such aggregation processes, estimates of the material nonlinearity may be made. Of particular interest is a comparison of these with those estimated for the polydiacetylene solutions. In poly(3-butylthiophene) solutions, the third-order nonlinearity is enhanced by the proximity of a three-photon resonance. However, the nonlinearity is lower than the lowest nonlinearity of the three polydiacetylene solutions,
245 that of the yellow, chloroform solutions. In the polydiacetylene solution, the nonlinearity is enhanced by the proximity of a two-photon resonance. Obviously the degree of enhancement experienced will influence the magnitude of the nonlinearity. However, as was stated previously, the dominance of imaginary contributions to the nonlinearity implies that the nature of the excited state ultimately determines the magnitude of the material nonlinearity. In both polymer solutions, the linear optical properties are characteristic of an interband transition. Also, the number of conjugated multiple bonds per monomer unit is the same for polydiacetylene and polythiophene. In both materials, a bipolaronic excited species has been observed [13, 14]. The displacement of the bipolaronic photoinduced absorption from the primary, ground-state absorption indicates a greater degree of localization of the excitation in the polydiacetylene solution. This greater localization should give rise to a lower polarization of the species. However, although the number of conjugated multiple bonds is the same for the two monomers, the number of conjugated electrons is larger in the polydiacetylene monomer unit. This increased charge density may account for the larger nonlinearity. In the P28 polymer solutions, a microscopic polarizability of order 10 41 mS/V 2 is estimated. This nonlinearity is comparable to that of the largest value measured for the polydiacetylene solutions, that of the blue solutions. No multiphoton enhancement of the nonlinearity is observed. This large nonlinearity is, however, deceptive, as the P28 monomeric unit is large and although the backbone has the alternating structure of polydiacetylene, the monomeric unit is the equivalent to two polydiacetylene units. However, comparison of the nonlinearity with that of the yellow polydiacetylene solution is more appropriate, since it exhibits similar linear optical properties to that of P28. There is a sizeable enhancement of the nonlinearity over that of the polydiacetylenes, which may be attributed to increased backbone rigidity. The observation of a three-photon resonant enhancement of the thirdorder nonlinearity in polythiophene solutions highlights the importance of multiphoton nonlinear interactions in organic polymers. A similar multiphoton resonant enhancement of the nonlinearity has been observed in polydiacetylene solutions. Some consideration must, therefore, be directed towards these multiphoton contributions. In conventional solids, the electronic system is extended in three dimensions and may be considered as a Fermi gas of weakly interacting particles. However, under one-dimensional confinement, the motion of the electrons is restricted and the effects of Coulomb interactions are considerably enhanced. It has been shown [15, 16] that in order to account correctly for the ordering of the llBu ~-electron state and the lower lying, two photon 2lAg state in polyenes [17], a proper treatment of the electron-electron interactions must be introduced, and that independent particle theories such as the Huckel theory are inadequate. The importance of these electron-electron interactions is a direct result of the dimensionality of these systems and the result is that the electron
246
motion becomes strongly correlated. The relative contributions of these interactions must be considered also as a function of the electron density within a specific monomer unit. In the case of polythiophene, the monomer unit contains only double bonds and so the electron density is comparatively low compared with other polymers. This results in a positioning of the three-photon level at an energy sufficiently high to influence the four-wave mixing process at 1.064/Lm, indicating an even higher two-photon level. Polydiacetylene has a very much higher electron density per monomer unit and this leads to a two-photon level which is positioned close to the 1.064/~m. Although an imaginary contribution to the nonlinearity of P28 may not be resolved explicitly because of the complexity of the concentration dependence of the nonlinearity, it may be inferred by extrapolation from the series of oligomeric enines. Any imaginary contribution is due to linear absorption, as may be seen from the intensity dependence of the diffraction efficiency (Fig. 10). The absence of a multiphoton resonant contribution to the nonlinearity is consistent with the large electron density of the monomer unit resulting from its complex aromatic structure. This gives rise to a strong electron correlation which shifts the multiphoton resonances far below the laser wavelength.
Conclusions
In all polymeric solutions investigated here, a complex concentration dependence of the material nonlinearity is observed. This behaviour is attributed to concentration-dependent interchain effects. These effects make an exact evaluation of the polymer nonlinearity difficult and a complete study would require a tunable near-infrared source which could track any shifting resonance. However, an estimate of the nonlinearities may be made and a comparison between the polymers is informative. The nonlinearity in the blue form of polydiacetylene proves to be the largest although this value is approached by the 'rigidized' backbone polymer P28. The observation that in all cases the polymer nonlinearity is dominated by imaginary contributions even in their transparency region implies the need for a reassessment of 'non-resonant' nonlinearities. Also, as the fundamental excitations of these materials are inherently localized by virtue of their one dimensionality, the notion of unlimited ~-electron delocalization and subsequently unlimited nonlinearities must also be revised. If the nonlinearity is dominated by resonant effects, then it is limited by the nature of these resonances. The importance is highlighted, therefore, of an understanding of the linear and nonlinear resonant behaviour of these materials. A direct relationship between the monomer structure and the resonance structure has been implied and such a relationship enables the resonance structure of these materials to be chemically tailored. Further development of the nonlinear optical properties of polymers relies, therefore, on expertise in molecular engineering.
247
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