1 Introduction

Conjugated organic polymers have emerged as potential materials for photonics due to their inherently large ultrafast nonresonant third-order optical nonlinearity and scope for the improvement by way of molecular engineering. Among the variety of conjugated polymers, polydiacetylenes possess one of the largest nonlinear responses [1]. However, none of the existing conjugated polymers so far possesses the requisite figure of merit for the practical realization of all optical signal processing devices. The device grade nonlinear optical polymers still remain a distant dream and new avenues are needed to be explored.

It has been observed that in conjugated molecules the cubic hyperpolarizability increases very rapidly with the molecular dimensions according to a power law as against that in saturated molecules with the same number of valence electrons [2]. According to the free electron model of Rustagi and Ducuing [3], the cubic hyperpolarizability of π conjugated molecules scales as the fifth power of the chain length. Agrawal et al. [4] calculated γ for polydiacetylenes in the tight binding approximation and predicted the sixth power dependence on the chain length. These scaling laws when casted in solid-state terminology predict the inverse sixth power dependence on the bandgap [4, 5]. Thus, a natural prescription for enhancing the nonlinearity of conjugated polymers is to engineer new low bandgap oligomers or polymers. Control of the bandgap of a polymer by way of molecular engineering not only yields high nonlinearity, but also allows control of its electronic and opto-electronic properties as well. With this regard, several strategies suggested from quantum chemical calculations have been successfully implemented by organic chemists [6]. May et al. [7–9] have used donor–acceptor substitution to increase the third-order polarizability of small cyanoethynylethene molecules by lowering the energy corresponding to the first order optical transition. Bredas et al. [10, 11] have shown that the bandgap of polyheteroarylene, such as polythiophene, is a strong function of molecular geometry. Their quantum chemical calculations predicted 2 eV bandgap of polythiophene with aromatic geometry in close agreement with the experimental value and a dramatically low bandgap of 0.47 eV for its quinoid isomer. Theoretical calculations of Hoogmartens et al. [12] confirmed very small bandgap of quinoid polythiophene. It was Jenekhe who first proposed poly(heteroarylene methines) which directly incorporated quinoid moieties in the conjugated polymer backbone (Fig. 1) as a class of small bandgap conducting polymers [13]. He suggested that by incorporating both aromatic and quinoid isomers in the same polymer chain, materials with bandgap values intermediate between the theoretical limits of 0.47 and 2 eV for the quinoid and aromatic geometries would be obtained. Theoretical calculation of the electronic structure of the poly(heteroarylene methines) has confirmed the small bandgap ranging from 0.75 to 1.21 eV for these polymers [14–19]. Thus, one may expect that the alternating aromatic and quinoid segments in this superlattice like 1-D molecular structure would produce spatially modulated potential barrier for electrons which may enhance the anharmonicity and hence the nonlinearity [20]. Degree of anharmonicity, bandgap and hence their optical properties can be controlled by synthetic manipulation of the molecular parameters x and y or by controlling the quinoid fraction Q = y/(x + y) of the chain (Fig. 1). Additional structural parameters affecting their electronic and optical properties can be the side group R, the heteroatom X and the number of repeat units m in the structure. Variety of these organic molecular superlattices and structure enabled control of their bandgap have been demonstrated [21–25].

Fig. 1
figure 1

Ground state structure of novel tunable, narrow bandgap, 1-D molecular superlattice polymer containing alternating aromatic and quinoid segments

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While the nonlinear optical response of polythiophenes has been extensively investigated [26–32], similar studies on heteroarylene methines are sparse. Jenekhe et al. [20, 33] have reported large resonant second hyperpolarizability values of heteroarylene methine derivatives. They found that the electron donating side group in one copolymer had significant effect on both the linear and nonlinear optical properties as well as the dynamics of the nonlinear response of the material. D’Amore et al. [34] have reported a three photon resonantly enhanced value of χ(3) = 3.0 × 10−11 esu at 1.5 μm for a quinoid molecule. Yi et al. [35] studied the resonant third-order optical nonlinearities of a series of poly(heteroarylene methines) containing one pyrrole or thiophene ring at 532 nm using picosecond degenerate four-wave mixing and showed that the resulting polymers exhibited χ(3) as large as 10−8 esu.

It is to be noted that all the above studies were performed at one single resonant wavelength and hence did not offer any useful insight into the nonlinear response of heteroarylene methines. On the other hand, spectral dispersion characteristics of γ are the key to understand the electronic structure–nonlinearity relationship in a given class of molecule. Jeffrey S. Meth et al. [36] studied the spectral dispersion of χ(3) in heteroarylene methines, though with very coarse spectral resolution (~100 nm). They probed the spectral response of χ(3) for poly(heteroarylene methines) with x = y = 1, 2 and 3, and accounted it in terms of three-essential states model but did not correlate it to the chemical structure of the samples. In this article, we present results of the spectral dispersion studies of γ conducted on three heteroarylene methines with different R groups. The observed spectral response of their hyperpolarizability has been analyzed in the framework of three-essential states model and a correlation with their electronic and chemical structures has been discussed. The energy of the two-photon state, transition dipole moments and the line width of transitions have also been estimated. The estimated γ values have been compared to the fundamental quantum mechanical limit.

2 Experiment

The heteroarylene methine derivatives having three different substituents, henceforth represented by abbreviation PTBTQ (R = o–H), PTCTQ (R = o-Cl) and PTBBTQ (R = o-OBu), were synthesized following a reported methodology [6]. The scheme is shown in Fig. 2. The chemical name of these compounds is as follows: poly[(2,5-thiophenediyl) benzylidene (2,5-thiophene-qinodimethanediyl)] [PTBTQ], poly[(2,5-thiophenediyl)-2-chlorobenzylidene(2,5-thiophene-quinodimethanediyl)] [PTCTQ], and poly[(2,5-thiophenediyl)-2-butoxybenzylidene(2,5-thiophene-qinodimethanediyl)] [PTBBTQ]. The samples were characterized by FTIR, 1H NMR, 13C NMR and UV–visible absorption spectroscopic techniques. The molecular weight of these samples was determined by gel permeation chromatography (GPC). A detailed report of the synthesis and characterization of these three derivatives of heteroarylene methine will be published elsewhere.

Fig. 2
figure 2

Scheme for the synthesis of heteroarylene methines

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The UV–Vis absorption spectra of the three samples in toluene were taken with JASCO V-530 UV/Vis spectrophotometer and are shown in Fig. 3. The absorption spectra reflect three overlapping peaks in all the three samples. In the literature, the peak in UV region is attributed to the monomer and another peak at ~380 nm is ascribed to the terthiophene [26]. The π – π* optical bandgap in these materials has been assigned to the lowest lying energy peak within 460–540 nm region [6]. The absorption spectra show that the peaks are less resolved and appear as broad humps at the above-mentioned locations. To ascertain their locations with better certainty, we fitted the observed absorption spectra with the mixture of three spectral lines with Gaussian lineshape. This analysis assigns the π − π* transition peaks in PTBTQ, PTCTQ and PTBBTQ samples at 479 nm (2.59 eV), 470 nm (2.64 eV) and 537 nm (2.31 eV), respectively.

Fig. 3
figure 3

Linear absorption spectra of PTBTQ, PTCTQ and PTBBTQ

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The third-order nonlinear susceptibility χ(3) of the samples was measured using antiresonant ring interferometric nonlinear spectroscopic (ARINS) technique [37]. The configuration used for the ARINS setup is based on the Sagnac interferometer [38]. This technique utilizes the dressing of two unequal-intensity counter-propagating pulsed beams with differential nonlinear phases, which occur upon traversing the sample. This difference in phase manifests itself in the intensity dependent transmission. Photodetection of the transmission of the ARINS yields spatially and temporally integrated response. In this technique, a 50–50 beam splitter divides the incoming pulsed beam into two counter-propagating pulses having a π phase difference. The pulses propagating in the clockwise (CW) direction are reflected by an uncoated flat mirror with wedged rear surface, while that propagating in the counterclockwise (CCW) direction are reflected by a high reflectivity mirror. Thus, this technique is based on the dressing of two unequal-intensity counter-propagating pulses. These two pulses again recombine at the beam splitter to yield ARINS transmission, \( | {E_{\rm out} } |^{2} \propto {E_{\rm cw} + E_{\rm ccw} }|^{2} \), where E cw and E ccw are the optical fields traveling in clockwise and counterclockwise directions, respectively. Inside the ring there is a unit magnification telescope comprising a pair of identical lenses in 2f configuration. One of the lenses focuses the pulses into the sample placed at its focus while the other one re-collimates them. The two counter-propagating fields traversing the ring acquire linear as well as intensity dependent nonlinear phase shifts (if the sample exhibits nonlinear response). Since both the fields traverse the same optical path through the ring and encounter the same interactions with the optical elements, linear interactions would affect their amplitude and phase in the identical manner. For an exactly 50 % beam splitter, in the absence of any nonlinear interactions, the two returning fields will be having the same amplitude and phase difference between them and consequently they will interfere destructively at the beam splitter to yield zero transmission. In this ‘balanced’ condition, all the input power is reflected back to the incident direction. Measurement against this dark background provides the basis for the improved sensitivity essential for measuring relatively weak signals. Any small deviation (δ) from the ideal splitting ratio results in a leakage from the ARINS and is responsible for the background that limits the sensitivity of the measurement. If the sample under investigation exhibits nonlinear response, then the two counter-propagating pulses having unequal intensity will undergo different phase changes after passing through the sample. Their superposition on the beam splitter will result in the intensity dependent transmission of the ARINS, which is related to the nonlinear response of the sample. However, the differential dressing of the two counter-propagating pulses with nonlinear phases is possible only when the two pulses do not interact simultaneously in the sample. Otherwise, the same nonlinear phase will be impressed on both by the cross-action phenomenon. Therefore, we prevent the temporal overlap of the two pulses in the sample by spatially offsetting it with respect to the center of the ARINS in such a way that the intense beam (i.e. CCW beam in our geometry) should reach the sample first in order to initiate nonlinear processes. The time difference between the arrivals of the two pulses (Δτ arr) determines the nature of the nonlinear optical process that can be studied depending on its response time. Nonlinear processes with decay time longer than (Δτ arr) do not contribute to the intensity dependent transmission of the ARINS as both the pulses will be affected simultaneously in that case. The delay window thus acts as an ultrafast gate. Hence this technique has the unique ability to filter the nonresonant electronic contributions from integrating (or slow, e.g. thermal) nonlinearities or those arising from long lived (resonant) states and makes it ideal for time-resolved studies and ultrafast gating. If the arrival times of the two pulses in the sample were reversed, one would measure the unfiltered response.

The typical ARINS experimental setup is shown in Fig. 4. Transform limited, 80 fs pulses (repetition rate 100 MHz and energy per pulse 5 nJ) from a self-modelocked Ti:sapphire laser were used for the experiment. The incoming pulse beam was split by a 50–50 beam splitter into two counter-propagating pulses. The two counter-propagating laser pulses inside the ring were focussed in the sample by two identical lenses of 10 cm focal length each. In order to avoid the contribution of integrating slow nonlinearities and for the measurement of fast electronic nonlinearity only, the solution sample in 1 mm path length quartz cuvette was placed inside the ring such that the intense pulse arrived earlier at the sample in order to initiate nonlinear processes. The delay between the intense and weak counter-propagating pulses was kept ≤3 ps. Energy of the incident and the transmitted pulses was measured using precalibrated photodetectors. Laser pulse duration was determined at each wavelength. In order to avoid any misalignment due to the fluctuating air current pressure during the course of intensity scan, the experimental setup was kept inside an enclosure made of perspex. The experimental setup was standardized using toluene which yielded n 2 = 6.2 × 10−27 cm2/GW and a negligibly small value of β = 2.5 × 10−3 cm/GW corresponding to Re{χ (3)} = (3.5 ± 0.5) × 10−14 and Im{χ (3)} = (8.8 ± 1.3) × 10−16 esu, respectively. This measurement of Re{χ (3)} value of toluene is in excellent agreement with its earlier reported value (3.8 × 10−14 esu) determined by another nonlinear interferometric technique [39]. Earlier self-focussing studies on toluene have determined sign of its nonlinearity to be positive [40, 41].

Fig. 4
figure 4

ARINS experimental setup. M Mirror, L 1, L 2 lenses, HR high reflectivity mirror, S sample, VA variable attenuator, D 1, D 2 detectors, Osc digital oscilloscope, PC computer

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The {χ (3)} dispersion of the solution and the solvent was studied at various wavelengths in 725-820 nm range and the second molecular hyperpolarizability γ of the solute, based on a pairwise additive model for noninteracting molecules, was evaluated using [42]

$$ \gamma = \frac{{\chi_{\rm solution}^{(3)} - \chi_{\rm solvent}^{(3)} }}{{L^{4} N_{\rm solute} }} $$
(1)

Here \( \chi_{\rm solution}^{(3)} \) and \( \chi_{\rm solvent}^{(3)} \) are the third-order susceptibility of solution and solvent, respectively, N solute is the number density of the solute, \( L = {{( {n_{0}^{2} + 2} )} \mathord{/ {\vphantom {{( {n_{0}^{2} + 2} )} 3}} \kern-0pt} 3} \) is the local field factor and n 0 (≈1.49 for toluene) is the refractive index of the solvent.

3 Results and discussion

A typical ARINS scan of PTBTQ at λ = 725 nm is shown in Fig. 5. Solid curve is the theoretical fit to the experimental data, using the theoretical formalism given in Ref. [37]. The fitted values of nonlinear absorption coefficient and nonlinear refractive index are found to be β = 0.04 cm/GW and n 2 = 6.26 × 10−6 cm2/GW, respectively. From these values of n 2 and β, we calculated the magnitude of real and imaginary part of χ (3) using the relations given in [43] which are respectively given by Re{χ (3)} = 3.76 × 10−13 esu and Im{χ (3)} = 8.77 × 10−14 esu. The second molecular hyperpolarizability per unit monomer unit (γ/N) was calculated at each wavelength using Eq. (1). The measured γ/N values of the three derivatives of heteroarylene methine, PTBTQ, PTCTQ and PTBBTQ, at various wavelengths in the investigated spectral range are shown in Figs. 6, 7 and 8, respectively. A trend of increasing values of both real and imaginary parts of γ/N can be seen for all the three samples as the wavelength is tuned from 814 to 720 nm. Evidently, these enhancements are due to the frequency of the incident radiation approaching that of optical transitions between the electronic states of the material.

Fig. 5
figure 5

ARINS signal for PTBTQ at 725 nm. Solid curve is the theoretical fit using the formulation given in Ref. [37]. Fit parameters are as follows: δ = 0.004, β = 0.04 cm/GW, n 2 = 6.26 × 10−6 cm2/GW

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Fig. 6
figure 6

a Real and b imaginary parts of γ/N of PTBTQ. Solid line is the three-level fit. Open circles are experimental points. The error in γ values is ±15 %

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Fig. 7
figure 7

a Real and b imaginary parts of γ/N of PTCTQ. Solid line is the three-level fit. Open circles are experimental points. The error in γ values is ±15 %

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Fig. 8
figure 8

a Real and b imaginary parts of γ/N of PTBBTQ. Solid line is the three-level fit. Open circles are experimental points. The error in \( \gamma \) values is ±15 %

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For one dimensional conjugated chains having C 2h symmetry, the π-electron states can be classified as even (A g ) and odd (B u ) parity states [44]. One-photon transition dipole moment vanishes between states of the same parity while a two-photon transition is allowed. For chains possessing C 2v symmetry, the transition from the ground state \( 1\,{}^{1}A_{1} \) to \( 2\,{}^{1}A_{1} \) is one-photon allowed for light polarized perpendicular to the chain and to \( 1\,{}^{1}B_{2} \) for light polarized parallel to the chain. Since for quasi 1-D chains, the component of transition dipole moment perpendicular to the chain is vanishingly small, these states can be treated analogous to \( {}^{1}A_{g} \) and \( {}^{1}B_{u} \) states of the C 2h group. The electronic nonlinearity of a molecule arises from the mixing of its ground and the excited states and can be calculated within the framework of perturbation theory by summing over the response from all the states of the system. As we said earlier, it has been shown that, in organic molecules, nonlinear optical response could be adequately explained by considering only the low lying three or four states [45, 46]. In the three-level model, these have been identified as the ground state \( | 0 \rangle \) and the lowest one- and two-photon allowed states \( | 1\rangle \) and \( | 2 \rangle \), respectively. According to the perturbation expansion given by Orr and Ward [47], the second molecular hyperpolarizability in this case is given by

$$ \gamma ( {\omega_{\sigma } ;\,\omega_{1} ,\,\omega_{2} ,\,\omega_{3} } ) = a\,K\,\hbar^{ - 3} \,\mu_{01}^{4} \left[ {\left( {\frac{{\mu_{12} }}{{\mu_{01} }}} \right)^{2} T_{121} + \left( {\frac{\Updelta \mu }{{\mu_{01} }}} \right)^{2} T_{111} - D_{11} } \right] $$
(2)

where K is a constant that depends on the frequency and degeneracy of a given optical process, μ lm is the electronic transition dipole moment between states \( | l \rangle \) and \( | m \rangle \), and Δμ = μ 11 – μ 00 is the dipole difference, ω 1, ω 2, ω 3 are the frequencies of the perturbing radiation fields, ω σ  = ω 1 + ω 2 + ω 3 is the polarization response frequency while a indicates the orientational average which is 1/5 for an isotropic liquid. For self-action process presented here, ω 1 = ω 2 = −ω 3 = ω. The expressions T 121 and D 11 represent the contributions from the two-photon and one-photon transition channels, respectively, while the remaining part T 111 is nonzero only for non-centrosymmetric structures [48].

Our observation revealed that the two-essential states model comprising the ground state and a one-photon channel cannot describe the observed dispersion of γ/N of heteroarylene methine derivatives. However, the three-essential states model involving the ground state, a one-photon excited state and a two-photon excited state describes the nature of observed dispersion satisfactorily. We also observed that the three-essential states model described the experimental data satisfactorily only when both n 2 and β are assumed to have positive sign. Other possible combinations of the sign of n 2 and β could not be accounted in this theoretical framework. Resulting theoretical fits to the experimental data of real and imaginary parts of {γ/N} for the three samples are shown in Figs. 6, 7 and 8, respectively. We observe that the three-essential states model provides an excellent fit to the experimental data.

Values of the best fit parameters for all the samples are summarized in Table 1. It can be noticed that the substitution of chlorine (Cl) and butoxy (OBu) group on the pendent ring in PTCTQ and PTBBTQ, respectively, results in the reduction of the transition moment to the first excited state as compared to that in PTBTQ. It should also be noted that these substitutions reduce the dipole difference between the first one-photon state and the ground state very substantially. Both one- and two-photon state energies are very nearly the same for PTBTQ and PTCTQ. In case of PTBBTQ, our analysis predicts the two-photon state energy to be higher (~0.1 eV) than that in the other two cases. Also, the observed one-photon state energy for this molecule is lower than the other two by nearly 0.2 eV. One- and two-photon channel contributions to the spectral dispersion of the real and the imaginary components of γ/N in the wavelength range of study were calculated from the parameters given in Table 1. We found that in all the three samples, the two-photon channel contribution is the dominant one and essentially accounts for the observed dispersion. For example, we have plotted the one-photon channel contribution, two-photon channel contribution and the overall contribution to the real and imaginary parts of γ/N for the sample PTCTQ in Fig. 9, which clearly shows that the overall contribution is dominated by that of the two-photon channel.

Table 1 Best-fit parameters (location of excited states, their widths and transition dipole moments between them) used in the three-essential states model for the three samples of heteroarylene methines
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Fig. 9
figure 9

One- and two-photon channel contributions to the (a) real and (b) imaginary components of γ/N of PTCTQ. Dotted line is the one-photon channel contribution while dashed line represents contribution from the two-photon channel. Solid line represents the overall contribution

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The molecular second hyperpolarizability of the oligomers can be calculated from the \( | {{\gamma \mathord{/ {\vphantom {\gamma N}} \kern-0pt} N}} | \) value by multiplying it with the degree of polymerization, D p. Also, the actual transition dipole moment \( \mu_{ij}^{ * } \) of the oligomer molecule can be obtained from the fitted value of \( \mu_{ij} \) given in the Table 1 using:

$$ \mu_{ij}^{ * } \simeq \root{4} \of {{D_{\rm p} }}\,\mu_{ij} $$
(3)

Using Eq. (3), one obtains actual values of transition dipole moments \( \mu_{01}^{ * } \simeq 76.60 \) D and \( \mu_{12}^{ * } \simeq 147.97 \) D for PTBTQ and \( \mu_{01}^{ * } \simeq 69.93 \) D and \( \mu_{12}^{ * } \simeq 116.04 \) D for PTCTQ. Using the parameters given in Table 1, we obtain the nonresonant (zero frequency limit) values of γ for PTBTQ and PTCTQ to be (4.6 ± 0.7) × 10−30 and (3.6 ± 0.5) × 10−30 esu, respectively. One- and two-photon channel contributions to the nonresonant γ value of PTBTQ are −1.2 × 10−30 and 5.8 × 10−30 esu, respectively. The corresponding values for PTCTQ are −3.0 × 10−30 and 6.6 × 10−30 esu, respectively. It can thus be seen that the nonresonant γ value for PTCTQ is somewhat less than that for PTBTQ due to the large extent of destructive interference of one- and two-photon channel contribution.

We would now analyze the predicted values of transition dipole moments \( \mu_{01}^{ * } \), \( \mu_{12}^{ * } \) and γ (−0; 0, 0, −0) from our analysis in the light of their sum-rule restricted limits of three-level model derived by Kuzyk [49]. These fundamental limits for the position matrix element \( r_{01} = \langle {1| {\,r} |0} \rangle \), \( r_{12} = \langle {2| {\,r} |1} \rangle \) and second molecular hyperpolarizability \( \gamma \) for a noncentrosymmetric molecule are, respectively, given by:

$$ \left. \begin{aligned} r_{01} &\le \hbar \sqrt {\frac{{N_{e} }}{{2mE_{01} }}} \hfill \\ r_{12} &\le \sqrt {\frac{{2\left| {r_{01} } \right|^{2} E}}{(1 - E)}} \hfill \\ \gamma &\le 4\frac{{e^{4} \hbar^{4} }}{{m^{2} }}\left( {\frac{{N_{e}^{2} }}{{E_{01}^{5} }}} \right) \hfill \\ \end{aligned} \right\} $$
(4)

where E = E 01/E 02; E 01 and E 02 being the lowest one-photon state and two-photon state energies, respectively, m is the mass of electron and N e is the effective number of electrons in the molecule. For determining the effective number of electrons N e that participate in the excitation, two electrons are associated with each double bond. N e is given by Kuzyk [50]:

$$ N_{\rm e} = \sqrt {\sum\limits_{i} {N_{i}^{2} } } $$
(5)

where the sum is over all the submolecules. The sum-rule restricted fundamental limiting value of γ has not been realized so far. To the best of our knowledge, nonresonant γ values of even the best known molecules so far have fallen short of this limiting value by more than a factor of 103/2 [51]. More recently, May et al. [7–9] have reported nonlinearities in their donor-substituted cyanoethynylethene molecules within a factor of 50 of the fundamental limit.

Using Eq. (5), we estimate N e ~ 173 for PTBTQ and ~213 for PTCTQ, respectively. This then fixes the fundamental limits \( {\mu_{01}^{ * } } |_{L} = 76.8 \) D, \( {\mu_{12}^{ * } } |_{L} = 180.48 \) D and γ max = 7.7 × 10−30 esu for PTBTQ while \( {\mu_{01}^{ * } } |_{L} = 84 \) D, \( {\mu_{12}^{ * } } |_{L} = 209.28 \) D and γ max = 10.6 × 10−30 esu for PTCTQ, respectively. It is extremely heartening to note that, for the first time, the nonresonant values of γ estimated from our dispersion studies {(4.6 ± 0.7) × 10−30 esu for PTBTQ and (3.6 ± 0.5) × 10−30 esu for PTCTQ} approach the fundamental limit within a factor of 2 and 3.5 for PTBTQ and PTCTQ, respectively. It may also be noted that the transition dipole moment for the lowest one-photon allowed state is at its limiting value in PTBTQ but somewhat lower for PTCTQ. This, precisely, is the reason for lower nonresonant value of γ for PTCTQ than that of PTBTQ. If one assumes oligomer molecular density to be 1020 cm3 and Lorentz field factor L 4 ~ 5.0 in an ordered sample of PTBTQ, its χ (3) value will be 2 × 10−9 esu, highest value for a molecule of this size (17 monomer units only).

4 Conclusions

We have studied the spectral dispersion of second molecular hyperpolarizability γ(−ω; ω, −ω, ω) of three derivatives of conjugated heteroarylene methine. The third-order nonlinear optical susceptibility was measured by ARINS technique using femtosecond modelocked Ti:sapphire laser in the spectral range of 725–820 nm. The observed dispersion of γ has been explained in the framework of three-essential states model involving the ground state, a one-photon excited state and a two-photon excited state. The energy of two-photon state, transition dipole moments and line width of the transitions have been estimated. The dispersion of γ in the spectral range investigated has been correlated with the electronic and chemical structures of the three derivatives of heteroarylene methines. The estimated γ values have been compared to the fundamental quantum mechanical limit, and we have found that the heteroarylene methines approach this limit within a factor of 2. To the best of our knowledge, this is the largest intrinsic second hyperpolarizability reported so far.