Dispersion of the third-order nonlinear-optical properties ofpoly(p-phenylene benzobisthiazole)
and its molecular composites with polyamides
Herman Vanherzeele and Jeffrey S. Meth
Central Research and Development, Du Pont, RO. Box 80356, Wilmington, Delaware 19880-0356
Sampson A. Jenekhe and Michael F. Roberts
Department of Chemical Engineering and Center for Photoinduced Charge Transfer, University of Rochester,Rochester, New York 14627-0166
Received May 8, 1991; revised manuscript received July 18, 1991
With third-harmonic generation I 3 k(-3c; co, o, co)I has been measured in the range 0.8-2.4 Am for thin films ofpoly(p-phenylene benzobisthiazole) (PBZT) and its molecular composites with nylon-66 and poly(trimethylhexa-methylene terephthalamide). For pure PBZT, I 3)(-3w; co, co, co) = (8.3 ± 1.72 x 10-" esu near the peak of thethree-photon resonance at 1.3 Am, and I( 3 (-3.; c., , c)I = (6.0 + 1.5) x 10 2 esu off resonance at 2.4 Jim. Anew theoretical treatment that includes the effects of inhomogeneous broadening and vibronic transitions onX(3) and permits accurate analytic fits of the dispersion data is presented. By preparation of molecular com-posites it has been discovered that these composites induce macroscopic ordering, or orientational anisotropy, ofthe PBZT, which can lead to an apparent enhancement ofy(3). The implications of the dispersion of the nonlin-ear properties of these materials are discussed. We have also measured the damage threshold of PBZT andfound it to be -50 GW/cm 2 for 30-50-ps pulses at 1.9 ,um.
1. INTRODUCTION
Currently there is a strong interest in exploring the third-order nonlinear-optical (NLO) properties of organic mate-rials and conjugated polymers in particular in order tounderstand the relationship between structure and non-linearity.` 5 Conjugated heterocyclic rigid-rod polymersare of special interest as NLO materials because of theirrobust physical properties, including their excellent me-chanical strength and thermal stability. In the caseof highly oriented poly(p-phenylene benzobisthiazole)(PBZT) fibers, a tensile strength of 3 GPa and a modulusof 330 GPa were reported.7 Further improvements in themechanical properties are achieved in molecular com-posites of PBZT with flexible-chain polymers such asnylon 66.3 Early reports3 4 revealed that PBZT has a rela-tively large nonresonant third-order susceptibility [X(3)]and a fast response time. However, detailed studies ofthe NLO properties of PBZT and related rigid-rod poly-mers were hindered by the difficulty of processing thesematerials into thin films with good optical quality becauseof the insolubility of the materials in organic solvents.Rao et al.3 studied a 33-,m-thick free-standing PBZT filmby degenerate four-wave mixing (DFWM) near 0.6 mand found noticeable effects of in-plane anisotropy inX(3) (- ); ), -t, Ct). This result was due most probably tothe liquid-crystalline nature of the pure material. Thevalue reported for the nonlinearity X(3)(-cv; cv -(,( c),6 x 10-12 to 7 x 10-12 esu, was surprisingly low for a near-resonant measurement. Garito and Teng4 performedthird-harmonic generation (THG) at 1.9 ,m on a 50-pAm-thick PBZT film. Their value for X 3 (-3) ; c, co, v), (50-
100) X 10-12 esu has a large error, again because of thepoor quality of the film. More recently Lee et al.5 remea-sured PBZT by DFWM, again at 0.6 m, and foundX(3) (-o; , -t, C) = 4.5 x 10'` esu. They also fabricatedcomposite films of PBZT/Zytel and reported improved op-tical properties. These widely disparate results warranta more detailed and accurate study of the PBZT X(3) andits dispersion.
Recent advances 9-" in the solubilization and processingof heterocyclic rigid-rod polymers in organic solvents nowpermit the preparation of optical-quality thin films forNLO measurements. This development allows us toevaluate the third-order nonlinear properties of PBZT andits composites with polyamides away from resonances.' 2
Here we refer to a molecular composite as a compositematerial in which a rigid-rod polymer is molecularly dis-persed in the matrix of a flexible-coil polymer.' 3 Al-though the processing and mechanical properties ofmolecular composites of PBZT and other conjugated poly-mers have been widely studied,7'8 3 their linear and NLOproperties have not been fully investigated. We arespecifically interested in the potentials offered by molecu-lar composites for exploring the effects of structure, com-position, and morphology on NLO properties of organicmaterials, including the figure of merit Re[x(3 )(-w;to, -o, c)]/a, where a is the linear loss.
In this paper we report our X(3) dispersion characteriza-tion of thin films of PBZT, PBZT/nylon 66, and PBZT/poly(trimethylhexamethylene terephthalamide) (PTMHT)molecular composites by THG. The structure of PBZT,nylon 66, and PTMHT are shown in Fig. 1. We develop atheory that includes the effects of inhomogeneous broad-
PTMHTFig. 1. Molecular structure of PBZT, nylon 66, and PTMHT.
ening and vibronic transitions on X(3)- The effect ofcomposite composition on (3) is explained in terms of ori-entational anisotropy or, equivalently, macroscopic order-ing of the PBZT in the composite. The dispersive effectsare described in terms of the sum-over-states perturba-tion expression for X(3)- We find that the nonlinearity in
essary to include the effects of inhomogeneous broadeningand vibronic transitions in the expression for the secondhyperpolarizability in order to account for the dispersionof the susceptibility of the system. The inhomogeneousbroadening arises from the fact that different chro-mophores experience interactions with different localenvironments inside the media. The overall linewidth re-sulting from these different environments can be muchbroader than the homogeneous broadening caused by life-time (Ti) effects or dephasing (T2) phenomena. For in-stance, in PBZT we measured a fluorescence lifetime T, of-1.3 ns, corresponding to a broadening of 109 Hz, whereasthe linear absorption profile possesses a line shape of_1013 Hz, 4 orders of magnitude greater. Dephasing pro-
cesses can result in linewidths of 1012 Hz, which are stillsecondary to inhomogeneities. Thus various local fields,such as the strain field and the dipole field, shift the tran-sition frequency of the chromophore and may have a pro-found effect on the dispersion of the nonlinearity. Thevibronic transitions arise from mechanical degrees offreedom in the system and also contribute to the disper-sion of the susceptibility.
The perturbation expression for the second hyperpolar-izability, y(-3w; w, w), including homogeneous broaden-ing, is14"5
the region studied can be adequately expressed by consid-ering only two states of the system, the ground state andthe lowest-lying dipole-allowed transition. The differentcomposites are seen to affect the broadening of this stateto different extents, indicating that the microscopic uni-formity in the samples vary. The model is used to gaininformation about the orientational anistropy and to pre-dict the X(3)(-@; to, -o, w) for PBZT. This analysis allowsus to evaluate PBZT as a NLO polymer and to suggestother promising structures.
2. THEORY
From conventional perturbation theory an expression forthe second hyperpolarizability of a material was previ-ously derived.' 4 "5 This expression includes only the
fdgif dwogm f d(gn
homogeneous broadening i]alistic materials this expr(
In this expression g represents the ground state; 1, m, andn represent excited states of the system; cij and Fjj repre-sent the transition energy and the damping, respectively,between states i and j; and the matrix elements are thetransition dipole moments between the states. The sum-mations are carried out over all states, excluding theground state.
To account for a distribution of resonance frequenciesof the excited states, it is necessary to convolute over theo variables with some weighting function. The hyper-bolic secant has been chosen as the weighting function fortwo reasons: It provides a reasonable description of theline shapes observed, and it admits of an analytical solu-tion to the problem. Since the integral of a sum is thesum of the integrals, the problem reduces to the evaluationof triple integrals of the form
nherent in the system. For re- where the subscript c refers to a central frequency and sjession is incomplete. It is nec- refers to the width of the hyperbolic secant. Several ap-
Vanherzeele et al.
526 J. Opt. Soc. Am. B/Vol. 9, No. 4/April 1992
proximations are necessary. First, it is assumed that thetransition dipole moments between states are indepen-dent of their energy separation. This assumption per-mits the matrix elements to be removed from the integral.Although this approximation does not truly model thetransition, it results in an effective dipole moment be-tween the two states, which is sufficient for our purposesfor this model.
The second approximation is the assumption that theintegration over the product of three variables can bebroken up into the product of three integrals. If thetransition frequencies were independent variables, thisassumption would not be an approximation but a mathe-matical truism. However, in a conjugated polymer theenergies of the excited states are related to one anotherfor a particular chromophore. From the many-bodyground state g there is a distribution of excited-state en-ergies coupled to the ground state, corresponding to ex-cited states in different local environments and to theFranck-Condon overlap between the ground state and theexcited state. Once a particular excited state is created,then the transition from that state, 1, to another state, m,is not inhomogeneously broadened, because I and m arelocalized in the same spatial region of the solid. It is theFranck-Condon factor that permits oscillator strength be-tween state I and states with energies distributed aroundthe energy of state m. This distribution is also modeledas a hyperbolic secant. The peak energy of the distri-bution does not rigorously correspond to the energy ofstate m; it corresponds to the maximum oscillatorstrength out of state 1. Thus it is feasible to write theconvolution of products as the product of convolutions forthis case, and one is left with only one term to convolvewith the hyperbolic secant. Because of this approxima-tion the theory accounts for not only inhomogeneousbroadening but vibronic transitions as well. The theoryin essence approximates the absorption spectrum and theexcited-state-excited-state spectrum of the material bybroad hyperbolic secant line shapes.
The present description is not rigorously correct.There are several factors that are not accounted for prop-erly, and these factors ensure that the integral of theproduct cannot be represented as the product of theintegrals. The first factor is the correlation betweenexcited-state energies for molecules in a certain local en-vironment. The second is the fact that, when a state isused twice in the path for the sum over states, it shouldnot be broadened again. Actually the inhomogeneousbroadening is better represented by a single convolutionthan by a triple convolution. However, the vibronicbroadening is well represented by a triple convolution, ascan be seen when one considers the Franck-Condon fac-tors between states. Even then it is still necessary toplace an energy-conservation constraint on the energiesused in the convolution. The theory presented here is asimple approximation with regard to all these difficulties.We do not claim it to be the ideal, but merely suggest it asa starting point for a more detailed and complete descrip-tion of the phenomena of inhomogeneous broadening andvibronic transitions on X
The third approximation is that the distribution oftransition frequencies is much greater than the homoge-
neous broadening. For systems such as dye molecules inglasses this is a good approximation, and to the extentthat PBZT can be modeled as a collection of chromo-phores dispersed in an amorphous fashion, it would be validfor this system as well. One can then write the reso-nance term as a real, dispersive part plus an imaginary,5-functional part. After convolution the imaginary partbecomes a hyperbolic secant function centered at gl withwidth sgl. The normalization factor is chosen such thatthe integrated area of the imaginary part of the responsefunction is conserved. The real part is obtained by per-forming the Kramers-Kronig transform of the imagin-ary part:
1 f (1/si)sech[(cO' - cog)/Sg1] do!-'r 6_,Oc' - (
(2/T)(og - o)((Ogl - )2 + (Sgi/2) 2 (3)
Thus each term in Eq. (1) is transformed into a convolutedterm in the following manner:
1 -L Ng ( l_ now)(tog, ± no) ± il) isgi Sg
+ (2
/7r)(cogl now)
((g, ± nc) 2 + (7rsgl/2)2(4)
These terms are then substituted into Eq. (1), and amodified expression for y is obtained. Operationally thetheory is replacing the Lorentzian line shapes that areassociated with homogeneous broadening with hyperbolicsecant line shapes.
The microscopic y is related to the macroscopic X(3) bythe expression
Here we are assuming that the nonlinearity is dominatedby the tensor component along the chain. X is a dimen-sionality factor related to the spatial average of cos4
0;
X = 1/5 when this spatial averaging is performed overthree dimensions, and X = 3/8 for two dimensions. Toobtain the Lorentz-Lorenz local-field factors f, and f,one calculates the linear polarizability in the same fashionas for Eq. (1) and uses these calculated values for thedielectric constant. The number density N is the numberof repeat units per unit volume and is calculated from anatomic contribution theory for physical densities, which isgenerally accurate to within 5%.
For modeling the nonlinearity the sum-over-states per-turbation theory is preferred over band theories for sev-eral reasons. Primarily, polymeric materials are notinorganic, crystalline structures and as such do not pos-sess translational symmetry, the foundation of bandstructures. The sum-over-states method allows one toconsider excitations in organic materials as being associ-ated with a certain spatial region in the solid. Also, re-cent developments in quantum-mechanical calculationshave led to a new picture of the mechanism of nonlineari-ties in organics, namely the essential-states mecha-
Vanherzeele et al.
Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. B 527
2.5
2.0
0)Cu
0'I,
1.5
1.0
0.5
0.0200
Fig. 2. AbsorptioiPTMHT.
nism. 16- In thisparticipates in tlnative mechanisierns the nonlineeto the excitonicresonant responEhigher-lying excilnonlinearity (seeone is.21
-23 The
develop some acwhich is more difiapproach.
3. EXPERIM
A. Sample PrepfThe PBZT sample30'C in methanesPolymer Branch Dayton, Ohio. T12,000-18,000 N(Warrington, Pa.)mer Products, Intemperature of 15solutions of PBZILewis acid (GaClas previously desits molecular conprepared in a drycally flat fused-swater and vacuuiness from 0.13 toevidenced by weeTHG response ovsorption spectrunis displayed in Fmaterials show siterized by a stro:and 437 nm and,visible in Fig. 2absorption, sinceshow this tail.that the films arn
but anisotropic in the direction perpendicular to it. Thisdistinction occurs because the PBZT polymer, being arigid-rod structure, tends to orient its long axis parallel tothe plane of the substrate.
B. Laser SystemFor investigation of the dispersion of the third-order sus-ceptibility a high-peak-power and widely tunable lasersource is required, preferably one capable of being con-tinuously tunable. An example of such a laser system is afree-election laser, which in the past was successfullyused to characterize the complex dispersion behavior ofthe X 3)(-3c; w, o, s) of polyacetylene. 24 At Du Pont we
300 400 500 600 700 have built a tabletop continuously tunable picosecondlaser system, specifically developed for our NLO pro-
Wavelength [nm] grams. This versatile system, which has been describedi spectrum of a 230-nm film of 1:1 PBZT/ in complete detail elsewhere,25 operates in two modes,
cw (100 MHz) and pulsed (10 Hz). In either mode the tun-ing range is the same, from 600 nm to 4 ,um. The pulse
s picture a higher-lying excited state also duration can also be selected independently of the otherie nonlinearity. This provides an alter- operating parameters: Either short (typically 5 ps) ora to the two-photon mechanism that gov- long (typically 50 ps) pulses are available. The averageir response of inorganic materialsi" and output power over the entire tuning range is at the milli-bleaching mechanism that governs the watt level for all modes of operation. The computer-se of organic materials.2 0 For PBZT a controlled system is based on a high-power mode-lockedted state is not necessary to explain the Nd:YLF laser,26 which synchronously pumps a dye laserSection 5), but for many other materials and seeds a Nd.YLF regenerative amplifier. Frequencysum-over-states theory also allows one to mixing2 7 and parametric generation-amplification 2 '2 9 incount of inhomogeneous broadening, KTP crystals3 0 are used to obtain the large tunability.ricult to incorporate into the band-theory For X(3) studies the 10-Hz operation is used, and, depend-
ing on whether the geometry used is THG or DFWM, thelong or short pulse duration, respectively, is selected.
ENT Thus, specifically, for this study we used the output of theparametric generator-amplifier, consisting of a pair of
tration single 35-45-ps pulses (signal and idler) with energye has an intrinsic viscosity of 18 dL/g at >0.5 mJ each anywhere in the tuning range. Pulse-to-sulfonic acid and was obtained from the pulse energy fluctuations do not exceed 5%. Clearly, with)f the Air Force Materials Laboratories, this kind of energy per pulse and tunability, our system'he nylon 66 sample of molecular weight rivals the capabilities of the free-electron laser, with thevas obtained from Polysciences Inc. added advantages of being more compact and user friendly.. The PTMHT sample (Scientific Poly-c., Ontario, NY.) has a glass transition C. Third-Harmonic-Generation;3C and no observed melting point. The The THG setup is shown schematically in Fig. 3. A'or PBZT/nylon 66 or PBZT/PTMHT in Glan-Taylor polarizer near the output of the laser system3 or AlCl3)-nitromethane were prepared rejects one of the orthogonally polarized output pulsescribed.9'" Thin films of pure PBZT or (signal or idler). The remaining output beam is attenu-iposites with nylon 66 or PTMHT were ated to 100 ,uJ/pulse, and its polarization direction is,box by solution casting onto 1-mm opti- adjusted to vertical by means of a Soleil-Babinet compen-ilica substrates followed by washing in sator followed by a second Glan-Taylor polarizer. Then drying. The films, ranging in thick- energy per pulse is monitored by a calibrated (cooled) Ge
1.6 ,um, were of fair optical quality, as detector (D) that samples a small (4%) fraction of the beam.ik optical scattering and by a uniform The remaining beam is split (90:10) into two beams. Theer a large area of each sample. The ab- weaker beam is used in a reference path, the stronger onei of a 230-nm film of 1:1 PBZT/PTMHT in the sample path. The focusing and collection optics asig. 2. Both pure PBZT and composite well as the THG detection equipment (filters, monochro-milar spectra. The spectra are charac- mator, photomultiplier tubes, and boxcar integrators) inng PBZT absorption with peaks at 468 both paths are identical except for a vacuum cell used onlya shoulder at 410 nm. The spectral tail in the sample path. In the vacuum cell the sample (fusedis caused by scattering rather than by silica as the reference material or the polymer film underspectra of the parent solution do not study on a silica substrate) is rotated about a vertical axis
Polarized optical spectroscopy reveals to generate a THG Maker-fringe pattern. The vacuuma isotropic in the plane of the substrate cell removes from the sample undesirable contributions
Vanherzeele et al.
528 J. Opt. Soc. Am. B/Vol. 9, No. 4/April 1992
SBC
GT - GT
FHt le laser
sample invacuum cell
A f h
V I MA inP
A'I ;7L J ii-
-IZE-LHE boxcar
Fig. 3. Experimental setup. GT's, Glan-Taylor polarizers;SBC, Soleil-Babinet compensator; M's, monochromators;PMT's, photomultiplier tubes; D, IR detector; P polymer film;PC, computer.
where Ak = 17re and al and a3 are standard Naperianabsorption coefficients at the fundamental and thethird-harmonic wavelengths. Obviously, absorption atthe fundamental wavelength will cause film damage. Inpractice, therefore, one has to stay away from this regimeand thus set al = 0 in Eq. (7). In Eq. (6) we neglect dif-ferences between the refractive indices of the polymerfilm and the substrate for both wavelengths. The third-harmonic signals I and I are obtained in the followingway. For a sample that gives a much stronger THG signalfrom the film than for the substrate, no Maker fringes canbe observed. In this case I is the THG signal near normalincidence, and I is inferred from constructing the envel-ope of the Maker fringes of a blank fused-silica plate (alsoplaced in the sample arm). The X(3) value obtained fromEq. (6) then is the final value. In the opposite case, i.e., ifthe signal for the film THG is comparable with that for thesubstrate THG, envelopes for the minima and the maximaof the Maker fringes are constructed, and both I and I, canbe inferred from the minimum (m) and maximum (M)envelopes near normal incidence:
by air to the THG signal. In the reference path a (sta-tionary) highly nonlinear polymer film (P) on a silicasubstrate is used to generate the third harmonic. Thethickness of this film is chosen to be smaller than the co-herence length to avoid Maker fringes in the THG signal,yet it is thick enough to give a THG signal that is 2 ormore orders of magnitude stronger than the one from thesubstrate (to avoid interference). To improve the signal-to-noise ratio, we reject all laser shots for which either theenergy of the fundamental or the third harmonic in thereference arm falls outside predetermined windows. Inthis way the instabilities (in both energy per pulse andpulse width) of the laser source are effectively reduced to±1%. By the same token, any possible degradation of thesample in the reference arm can be monitored in realtime. By taking the ratio of the THG signals in botharms, the effect of fluctuations in both the power and thepulse width of the fundamental beam are eliminated.Finally, averaging the ratio of some 250 laser shots yieldsa precision typically of ±0.5%.
The third-order susceptibility xt3 o(-3w; wcoco) of thesample (thin polymer film on a fused-silica substrate) isobtained relative to fused silica in the following way.First, an approximate value for Ix)3 I(-3co;cvcvco)j of thepolymer film is inferred by a procedure similar to the oneoutlined by Kanetake et al.3 The Maker-fringe patternof the sample is compared to the Maker-fringe pattern of ablank substrate that is also placed in the sample arm.Since the thickness e of our polymer films is much lessthan their coherence length, one can use the followingapproximation3 1 :
In Eq. (6) the subscript s refers to the substrate (silica), Iis the third-harmonic signal, (,,, represents the coherencelength of the substrate, and f(al, a 3, e) is defined by32
I = AM, (8a)
4s= (M + m - 2_eM)exp(a 3 ). (8b)
For this case an exact technique, based on the transfer-matrix theory of Bethune3 3 or, equivalently, Neher et al.,3 4
was also developed. The approximate X(3) value, obtainedfrom Eq. (6), then is used as a starting value for the exactfitting procedure. However, the large uncertainty in thethickness of the films (as great as ±20%) and the largenonlinearity of the films studied in this work made theemployment of the exact-fitting technique impractical forsome samples. Therefore only the magnitude, and notthe phase, of x(3)(-3wo; a, cv, c) is reported here.
For the silica reference we used X(3)(-3co;c, cv, c) =2.8 x 10-1' esu at 1.9 ,um.35 In order to correct this valuefor dispersion, we used, as a simple approximation,Miller's rule, where X(3)(-3@); °, °o, &) c [xl')(3o)] [X(1)(&v)] 3.
D. Index of Refraction and Film Thickness MeasurementTo derive X(3) from the Maker-fringe patterns as explainedabove, it is necessary to know the film thickness and thelinear index of refraction of the polymer at the fundamen-tal and the third-harmonic wavelengths. Film thicknessmeasurements were carried out on a mechanical pro-filometer (Dec-Tac IIA). To measure refractive indices,we used the method described by Swaenepoel.36 Etalonfringes of a thin film were measured in transmission andcombined with the independently measured film thick-ness to produce a modified Sellmeier equation. Theresults were confirmed at 633 nm by using the m-linetechnique for relatively thick samples (Metricon PC-2000).
4. RESULTS
A. Linear Index of RefractionUsing the measured maxima and minima for the interfer-ence fringes and using the film thickness, we have derived
N
N
Vanherzeele et al.
Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. B 529
C)
0)
0
0
2.6-
2.4
2.2
2.0
1.8
0.4 0.8 1.2 1.6
Wavelength (,um)
2.0 2.4
Fig. 4. Dispersion of the refractive index of PBZT. The solidcurve represents Eq. (6); the filled circles are the experimen-tal data.
2
U)
7
-7
sa)
s
s~
0 A-.0.0 0.2 0.4 0.6 0.8 1.0
Mole Fraction PBZT
Fig. 5. Composition dependence of Ix 3)(- 3w; w), w, w)I at 1.9 pumof PBZT/nylon 66 and PBZT/PTMHT molecular composites.Filled squares refer to PBZT/nylon 66 and filled triangles toPBZT/PTMHT. The solid line is a least-squares fit for thePBZT/nylon 66 data.
the following modified Sellmeier equation for the refrac-tive index of PBZT thin films:
2 10.12
(A - 490 1)0.42 , (9)
where A is expressed in nanometers. In Fig. 4 we presentthe observed and the calculated results. The data pointsrepresent the position of the transmission extremum asthe abscissa, and the ordinate is the index at that wave-length as calculated by the expression n = mA/2d, wherem is the order number and d is the thickness. This calcu-lated expression reproduces the observed values to within2% over the investigated wavelength range. The majorsource of error is the inability to locate the precise posi-tion of the transmission extremum.
B. Composition Dependence of X(3)
Figure 5 shows the composition dependence of theIX13)(-3w; w, w, w)I of thin films of PBZT/nylon 66 andPBZT/PTMHT molecular composites measured at thefundamental wavelength A = 1.9 pum. As for the purematerial, no in-plane anisotropy in X(3) was detected in themolecular composites by polarized light microscopy. Inthe case of the PBZT/nylon 66 composite system X(3) showsa linear behavior that is proportional to the mole fraction
of the conjugated polymer PBZT. However, the X(3) of thePBZT/PTMHT composites versus composition deviatesfrom linear behavior. For instance, at 50 mol. %, X(3) =
(1.07 ± 0.21) x 10-"1 esu, which is more than 50% largerthan the corresponding PBZT/nylon 66 composite.
C. Dispersion of X(3)In Fig. 6 we present the dispersion data for pure PBZT, a1:1 composite of PBZT/PTMHT, and a 1:1 composite ofPBZT/nylon 66. All three materials show a resonant en-hancement of X(3) near 1 eV(1.3 pum). This energy isa three-photon resonance with the first excited state,whose band edge is at 2.65 eV(473 nm) and whose peak isat 2.9 eV(437 nm), as is shown in Fig. 2. It can be seenthat throughout the entire measurement range there is noenergy region in which X(3) is constant. This result indi-cates that for these materials there is no region that canbe considered to be totally resonance free for THG. Be-cause of errors in the film thickness measurements andthe index of refraction measurements, the errors in theX(3) values are ±20%. However, relative (repeatability)errors amount to only ± 5%.
5. DISCUSSIONThe reason for the nonlinear dependence of X(3) on com-position for the PBZT/PTMHT composites lies in theorientational anisotropy of the films in the direction per-pendicular to the plane of the film. Although PBZT is athree-dimensional molecule, for this study it is treated asone dimensional, with yim11 being the only nonzero tensorelement. It is well known that, compared with a materialoriented in one dimension, (y) ill is reduced by a factor of3/8 when the material is randomly oriented in a plane andby 1/5 for a truly isotropic material. We hypothesize thatthese materials, being thin films, should be thought of asmaterials whose chromophores (PBZT strands) are par-tially oriented and have an orientation factor correspond-ing to averaging over a dimension between two and three.This makes intuitive sense, since the PBZT molecule isessentially a rigid rod. When that rod is deposited onto asubstrate, it tends to lie in the plane of the substrate.The chromophores are randomly distributed in the planeof the substrate and are also distributed with some non-random weighting function in the direction perpendicularto the substrate. That weighting function depends onseveral factors, most notably the thickness of the film, thecomposite material, and the conditions under which thefilms were deposited. These factors result in an averageprojection of the transition dipole moment onto the planeof the substrate, and the THG measurements detect thisorientational anisotropy. To observe a linear dependenceof X(3) on composition, the anisotropy of the compositemust be independent of composite makeup. This is thebehavior observed in the PBZT/nylon 66 composites. Ifthe orientational anisotropy were to vary as a functionof PBZT percentage, then the resulting dependence of X(3)on composition would deviate from linearity. This isprecisely the behavior observed for the PBZT/PTMHTcomposites. This observation suggests that a propitiouschoice of complementary polymer materials will be influ-ential not only in the processing but also in the resultingNLO properties of the composite.
Vanherzeele et al.
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530 J. Opt. Soc. Am. B/Vol. 9, No. 4/April 1992
10.0- Scanning electron microscopy revealed the materials to0 PBZT be uniform on the micrometer scale, thus eliminating any
possibility that inhomogeneities in the film contributed8.0- to the systematic enhancements observed in the PBZT/
PTMHT composite. We do not see voids or particles in6.0\ our samples, though voids and particles may be seen in
6.0 - I \extruded samples.5 Since the films are cast from truesolutions, aggregation is not likely during the casting
m9 / \process, which lasts only for seconds. Polarized optical9 4.0- microscopy clearly confirms that the films are isotropic in
the plane and anisotropic perpendicular to it, adding fur-ther support to the proposed interpretation of the compo-
x 2.0- sition dependence. Anisotropic waveguides are moredifficult to use and are not desirable for device applica-
0 \ tions, so the enhancement seen in the PBZT/PTMHT0- composites is not necessarily advantageous. However, it
0 0.5 1 1.5 is important to note the effect of the constituent polymersEnergy [eV] on the overall structure of the composite and to under-
(a) stand that the dimensionality can be influenced in thisfashion. The observation also shows that in general thin
6.0- films are not isotropic structures and that this anisotropyo PBZT/PTMHT needs to be considered in the interpretation of all mea-
5.0 0 surements made on thin films in order to arrive at correctconclusions about the microscopic nonlinearity.
In Fig. 6 the dispersion data of PBZT and the two7 4.0 - 1:1 composites are fitted by using a two-level model in the
perturbation expression described in Section 2. The two3.0 - f \ states are the ground state and an odd-parity state cen-3.0 / \tered at 2.9 eV The two-state model provides an accurate
account of the dispersion, with no need to include a3 2.0- higher-lying even-parity state. The fitting parameters
co2s V are collected in Table 1. In the third column are the1.0 o/ \ transition dipole moments derived by assuming that the.0 - \material is isotropically oriented in three dimensions
o 0 (dim). In the second column are the moments derived by0- [ assuming that they are oriented in the plane of the film
0.5 1 1.5 with no projection perpendicular to the plane of the film.Energy [eV] The fourth column shows the energy of the excited state,
(b) and the fifth column its phenomenological width. Anyattempt to fit the data by using the unmodified, homoge-
5.0- neously broadened expression yields large systematic de-o PBZT/Nylon 66 o viations between the fit and the data. This discrepancy
n 4.0 stems from the fact that the wings of the observed reso-a 4.0 - / \nance are not Lorentzian in shape, and the theory would
overestimate the off-resonance value.o 1 \ The parameters in Table 1 indicate some important as-_ 3.0 pects of these materials. First, the energy of the odd-
parity state is the same, within the errors of the fit, for all
2.0-4 Table 1. Fitting Parameters for the
Two-Level ModelX 1.0
01 , _0 0.5 1 1.5
Energy [eV]
(c)
Fig. 6. Dispersion of Ix 3)(-3cot; toco, c)I and two-level fit to thedata, assuming an isotropic transition dipole distribution, for(a) PBZT, (b) PBZT/PTMHT, (c) PBZT/nylon 66. See Table 1 forparameters.
three materials. This is as expected. This energy isclose to the absorption peak at 437 nm (see Fig. 2). Intheory, this energy represents the weighted mean energyof the first absorption band. The fact that this peak isnot at the band edge underscores the importance of con-sidering all the transitions, including vibronic transitions,associated with the first band.
The FWHM of the spectrum in Fig. 2 is approxi-mately 0.6 eV. The FWHM for a hyperbolic secant is2 cosh-'(2)sgl = 1.2 eV for these materials. The increasedwidth in the fitted parameter is a result of the near-IRloss. If one studies the width of this state for the variousmaterials, it is seen that the nylon 66 composite shows anarrower inhomogeneous broadening than either of theother two materials. This improvement is caused by thesharper three-photon resonance seen in Fig. 6(c), indicat-ing that this material produces a more regular micro-scopic environment for the PBZT than the PTMHT or thepure material. This effect is important because, if onewishes to maximize the figure of merit, it is necessary toproduce as narrow a distribution as possible.
By studying the magnitudes of the transition dipole mo-ments, one may obtain information about the macroscopicorientation of the PBZT, which affects the composition de-pendence described above. From Table 1 we see that theeffective transition dipole increases in the order PBZT <PBZT/nylon 66 < PBZT/PTMHT for both the two- andthe three-dimensional fits. We conclude that, accord-ing to this model, the pure PBZT is closest to a three-dimensional dipole orientation and that the PBZT/PTMHTcomposite is closest to a two-dimensional orientation.The fact that in the composition dependence of X(3) thereis a linear relationship between the X(3) of the PBZT/nylon66 composites and the pure PBZT is strictly fortuitous. Acombination of anisotropy and broadening makes the purePBZT point lie along the same line as the PBZT/nylon 66points. Thus the results of the fit reinforce the interpre-tation of macroscopic ordering as the cause for the ob-served composition dependence. Because we have notmeasured the dispersion for all the different compositions,we do not know how the width or the anisotropy of thecomposites varies with composition. The inhomogeneouswidth of the PBZT/nylon 66 composites must vary withcomposition, because its width must approach that of thepure PBZT as the percentage of nylon 66 in the compositeapproaches zero. With this in mind, it is possible that thelinear composition dependence seen for the thin films ofthis material is also coincidental, being the result of inter-play between anisotropy and broadening throughout theentire composition range tested.
The magnitudes of the calculated dipole moments areunrealistically large, corresponding to oscillator strengthsgreater than the 16 electrons present in the PBZT repeatunit. This situation occurs for several reasons. First,we are considering only a two-level model, which ignoresall higher lying o- -> o* transitions. Large dipole mo-ments result from using only this transition to account forthe index of refraction, which is subsequently used in thelocal field. A simple correction to this difficulty is toassume that the higher-lying transitions would add aconstant, real part to the dielectric tensor and then to cal-culate the resulting index. This modification can reduce
the dipole moments by as much as 50%, depending on thecontribution attributed to the higher-lying states, butleaves the ordering of the dipole moments, and hence theorientational anisotropy interpretation of the compositiondependence, unchanged. The second source of error is inthe number density. We are using an atomic contributiontheory to calculate a physical density of 1.55 g/cm', closeto the measured density. However, it may not be properto use the repeat-unit density as the number density. Ifthe repeat unit is smaller, the number density increases,and the transition dipole moment decreases. With thesefactors it is possible to reduce the transition dipole mo-ment to -15 D, corresponding to an oscillator strength of-7 electrons and an effective dipole length of -3 A. Thisexercise simply demonstrates the obvious fact that fittingthe dispersion curves is not an accurate method for deter-mining the transition dipole moments. However, it ispossible to analyze trends in certain families of materials.
It is possible to estimate the proper value for the transi-tion dipole moment from linear spectroscopy. Our life-time studies have shown that the fluorescence lifetime ofPBZT at 77 K is 1.3 ns. In a simple Einstein model thislifetime corresponds to a transition dipole moment of-8 D. It is also possible to estimate the dipole momentfrom the integrated absorption of the first optical band,which yields -10 D. These estimates underscore the in-terpretation of the calculated transition dipole momentsas fitting parameters, not realistic quantities. This esti-mated value for the transition dipole can be comparedwith recent measurements of transition dipoles of poly-acetylene oligomers,3 7 from which we can infer aneffective conjugation length of 7-9 double bonds, or ap-proximately one PBZT repeat unit. The theoretical posi-tion of the band center at 2.9 eV is also consistent withthis interpretation. This means that twists of the phenylrings in the PBZT structure hinder delocalization andlimit the nonlinearity. This result is evidence that it isnecessary to increase the delocalization of the or electronsto increase the nonlinearity. This implication is an in-centive for exploring other structures, such as ladder poly-mers, 13 in which rotations are hindered.
If a third state, possessing even parity, is added to thetheoretical model, the fit improves slightly but not signifi-cantly. The third state, which produces the best fit, hasan energy of 4.5-5.0 eV and a width similar to that of theodd-parity state. The transition dipole from the odd-parity state to the even-parity state is 1.5-2.0 times thetransition dipole from the ground state to the odd-paritystate. These parameters generally agree with predic-tions about the nature of the even-parity state. 6 7 To ob-serve the effects of this even-parity state, we would needto tune the fundamental wavelength to -2.5 eV, which isbeyond our THG detection capabilities. In recent publi-cations concerning other classes of materials, we haveshown how this even-parity state affects the dispersion of
(3) because of interference effects. 21 22
Our theoretical expression allows us to predictx(3)(-co; co, -to, co) using the same parameters that were ob-tained from fitting x (3(-3wo; co, co, c). Displayed in Fig. 7is the spectrum of x(3)(- o; a, -w, co), calculated by usingthe parameters for PBZT in Table 1 and assuming a three-dimensional dipole orientation factor. A positive value
Vanherzeele et al.
532 J. Opt. Soc. Am. B/Vol. 9, No. 4/April 1992
6.0
13CO0)
910
.-3
a.
IF
4.0
2.0
0
-2.0
-4.0
-6.0
-8.00 1 2 3 4 5
Energy [eV]Fig. 7. Dispersion of X(3)(-); (c, c(, -() for PBZT calculated byusing the parameters from Table 1; Imag, imaginary.
for Im[x(3 )(-c;cv,-cv, c)] represents two-photon absorp-tion, whereas a negative value corresponds to bleaching, orsaturation of the linear absorption. An interesting obser-vation about the spectrum is that there necessarily existsan energy at which Im[x(3)(-cv; cv, -co, v)] = 0. At thisfrequency one has a nonsaturating absorption, for whichthe two-photon absorption is balanced by the saturation ofthe linear absorption. Thus for certain wavelengths non-linear index changes can be decoupled from nonlinearlosses, even though they are inextricably related. Thispossibility implies that there exists a frequency for anymaterial at which the two-photon absorption is no longer alimitation to all-optical switching, a topic discussed byother researchers.3 9 This situation occurs in the pres-ence of higher-lying even-parity states as well. Unfortu-nately, there will always be some linear absorption at thisfrequency. In some systems it may be possible to movethe position of this nonsaturating absorption to a placewhere a is extremely small. At this frequency Re[x(3)(-c;co, - , cv)] will be resonantly enhanced by as much as anorder of magnitude and could lead to an increase in thefigure of merit, Re[x(3)(-co; cv, -co, cv)]/a. The spectrum inFig. 7 also shows the relatively larger IX(3)1 values obtainedby DFWM compared with those for THG. To accuratelypredict observed DFWM results, which are typically per-formed closer to resonance, it would be necessary to in-clude higher-lying even-parity states. Our data do notrequire us to consider these other states because of theenergy range over which the data were collected. How-ever, the predicted DFWM X(3) compares favorably withthe values in the literature.3 5 At 602 nm the model pre-dicts ix 3 )(-c; cv, -cvcv)l = 3.7 x 10`0 esu, compared withthe measured 4.5 x 10-1 esu.5
As for the figure of merit, Re[x, 3y(-c ; c, -co, v)]/a, forthese materials, no great improvement was achieved bymaking the composite. The decrease in the nonlinearityis accompanied by an equivalent decrease in the measuredloss. For the PBZT/PTMHT composites the figure ofmerit is improved by 50%, but at the expense of creating amore anisotropic material. Most of the loss in the near
IR is caused by scattering from the material and is-103 cm-'. One of the contributions to this loss is fromsurface roughness, which may be remedied by overcoatingthe film with a buffer layer. There may be a contributionto the loss from crystallinity in the polymer, while some ofthe loss is caused by inherent scattering in the polymer,which may be caused by vacancies created when the Lewisacid is rinsed from the polymer. To alleviate this loss it isnecessary to choose a diluent polymer that has approxi-mately the same refractive index as the NLO active poly-mer. This diluent will serve to index match the activematerial, which will reduce inherent scattering losses.For the PBZT, which has a large index of refraction, thismethod will be difficult. However, it is an avenue that isbeing considered for other materials. Recent investiga-tions by Lee et al.5 have been successful in this regard.They have fabricated a PBZT/Zytel 330 composite thathas a figure of merit two times higher than the purematerial. This result demonstrates the possibility of im-proving the nonlinearity of materials through careful pro-cessing and composite formation.
6. CONCLUSIONS
In summary, we have prepared optical-quality films ofpoly(p-phenylene benzobisthiazole) (PBZT) and its mol-ecular composites with nylon 66 and poly(trimethylhexa-methylene terephthalamide) and characterized theirthird-order nonlinear optical properties by third-harmonic generation. The alterations made to perturba-tion theory to account for inhomogeneous broadening andvibronic transition yield an expression that can be used tofit the data analytically. We have found that the disper-sion of X(3) can be accounted for by a two-level model.The effect of forming a composite is to alter the orienta-tional anisotropy of the PBZT and to change the effectivelinewidth of the nonlinear-optical active conjugated poly-mer. From estimates of the transition dipole moment wecan conclude that the PBZT is no better a chromophorethan a,c-diphenyl-octatetraene, 3 7 although it is environ-mentally more robust. The need to improve the oscillatorstrengths of prospective materials is important, and theconsideration of ladder polymers is the logical next step.While we were not able to improve the figure of meritsignificantly for these materials, the composite approachto optimizing Re[x, 3y(-c ; , -c, c)]/a may still succeedwith other materials.5 Finally, the ability to predict four-wave mixing results from the presented model is a power-ful tool for comparing different experiments performed onthe same material.
ACKNOWLEDGMENTS
The optical characterization was performed at Du Pont.We thank J. Caspar at Du Pont for the lifetime measure-ments. We also are indebted to G. Meredith for stimu-lating discussions. H. Vanherzeele acknowledges thevaluable technical assistance of J. Kelly. Research at theUniversity of Rochester was supported by the NationalScience Foundation (grant CHE-881-0024) and by a grantfrom the Amoco Foundation.
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