Microscopic Determination of Second-Order Nonlinear Optical

Article pubs.acs.org/JPCC

Microscopic Determination of Second-Order Nonlinear Optical Susceptibility Tensors Liisa Naskali,† Mikko J. Huttunen,*,†,§ Matti Virkki,† Godofredo Bautista,† András Dér,‡ and Martti Kauranen† †

Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland Institute of Biophysics, Biological Research Centre of the Hungarian Academy of Sciences, P.O. Box 521, 6701 Szeged, Hungary

‡

S Supporting Information *

ABSTRACT: We demonstrate a microscopy technique that extracts tensorial information about the second-order nonlinear optical susceptibility and hyperpolarizability of molecular materials. Our technique is based on polarization-dependent second-harmonic generation and a genetic algorithm, using which the best possible match with the measured data, and the possible susceptibility tensor components are found. In contrast to existing techniques, which access only the magnitude of the nonlinear response, our technique also provides information about the phase of the tensor components, which is associated with molecular resonances. After verifying the technique using simulated model structures with wellknown symmetries, we demonstrate its capabilities using model surface samples consisting of single purple membrane (PM) fragments of bacteriorhodopsin (bR) chromoproteins. Since the supramolecular structures of PM, bR, and photoactive retinal molecules are known, complex-valued tensorial information on the molecular hyperpolarizabilities can also be extracted. Our technique opens new possibilities for obtaining detailed structural information on biomolecular samples with microscopic resolution.

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INTRODUCTION Nonlinear optical (NLO) phenomena arise from multiphoton interactions between light and matter.1 Such phenomena are useful for several applications, as they provide different information than linear phenomena about the sample under investigation. Coherent optical spectroscopies, for example, provide information on the order and dynamics of molecular structures.2,3 Coherent NLO techniques are also attractive for imaging applications, since they are ideally label-free, reduce the photodamage of the sample, and can provide deeper penetration into biological tissue.4,5 Four-wave mixing and sum-frequency generation can also provide chemical selectivity.6 Recently, polarization-based NLO microscopy modalities have attracted considerable attention by providing novel structural contrast mechanisms.7 The NLO phenomena are described by respective tensors, whose forms are dictated by the structure and symmetry of the sample.1,8−10 The information on individual molecules is carried by molecular hyperpolarizability tensors, whereas information on macroscopic samples is in susceptibility tensors. For second-harmonic generation (SHG), these tensors are conventionally labeled as β and χ(2), respectively. In addition, the tensor components are complex-valued, which is important for understanding the resonant behavior of the sample. Complex-valued information can be extracted with phasesensitive NLO spectroscopies,11,12 but in general these techniques are difficult to combine with polarization-dependent measurements required for retrieving the tensorial information. © XXXX American Chemical Society

The inversion problem of solving an unknown nonlinear tensor from measurement data is therefore challenging and generally requires the use of a priori information and approximations. In principle, hyper-Rayleigh scattering (HRS) can provide information on the molecular hyperpolarizability but is mainly limited to solution samples.13 In addition, the observables for HRS are real-valued and depend on complicated quadratic combinations of the hyperpolarizability components. It is relatively straightforward to access complex-valued tensorial information on macroscopic susceptibilities but only if the excitation field is assumed to be planar. Such techniques are relatively well-developed for thin films14−17 and have also been used to characterize individual nano-objects using HRS and SHG microscopy.18 This approach, however, has limited applicability, since the excitation fields in high-numerical aperture (NA) confocal microscopy cannot be approximated as plane waves.19,20 In consequence, the measured SHG signals can deviate by 5-fold factors from the predictions of the traditional approach.21 Recently, a technique to determine χ(2) using SHG microscopy combined with interferometry was proposed, but the approach was restricted to real-valued tensor components.22 In this article, we demonstrate that polarization-based SHG microscopy and subsequent data analysis based on a genetic Received: September 18, 2014 Revised: October 16, 2014

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dx.doi.org/10.1021/jp509453b | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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using a GA to find viable χ(2) solution candidates (see Supporting Information, Figure S1). Provided that the data consist of a sufficient number of independent measurements, the problem is transformed to achieve good convergence of the search algorithm and to find a unique solution. The amount of needed information about the measurements depends on the sample and how much a priori information on the sample is known. The allowed solution space can be restricted by using a priori information and is in practice essential for ensuring that a unique solution is found. For example, the molecular structure or the overall symmetry of the sample can be used to restrict the solution space. For the case of a highly symmetric sample, i.e., an isotropic surface, only a few of the χ(2) components are nonzero. In addition, if the excitation and emission frequencies are far away from any resonances, the tensor components are also real-valued. For such a sample both transmitted and reflected SH intensities can be measured and a converged solution is found relatively quickly. For complicated samples, however, additional restrictions or more diverse measurement data are necessary to ensure a unique solution. Here we measure the total SHG intensities with only seven excitation polarizations to demonstrate how only a few measurements can be used to ensure that a unique solution is found. In fact, for certain sample types, such as isotropic surfaces, even fewer input polarizations would be adequate. For improving the accuracy, the amount of data could be further increased, for example, by performing Stokes polarimetry on the SH emission30 or by holographic SHG measurements.22

algorithm (GA) can be utilized to determine the relative complex-valued components of the χ(2) tensor. In addition, if the supramolecular structure and overall symmetry of the sample are known, it is possible to relate the macroscopic measurements to the molecular β, providing information not accessible otherwise. First, we demonstrate our technique by simulating the second-order responses of two isotropic surfaces and determining their effective χ(2). Second, we use the technique to determine the macroscopic and molecular level SHG responses of single cell membrane fragments from Halobacterium salinarum consisting of light harvesting bacteriorhodopsin (bR) proteins.23−26 Before the actual experiments, the technique is validated by simulations also for the C3 symmetry of the bR surface sample. We find that the technique can provide in situ complex-valued tensorial information at the spatial resolution of confocal microscopy, and therefore, it opens new possibilities for more quantitative NLO microscopy and spectroscopy.

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THEORETICAL BASIS The second-order polarization at the second-harmonic (SH) frequency 2ω, induced by the fundamental field at frequency ω, at point r is Pi(2ω ,r) =

∑ χijk(2) Ej(r)Ek(r) (1)

jk

where i,j,k refer to Cartesian components (x, y ,z) in the frame of reference of the macroscopic sample, χ(2) is the second-order susceptibility tensor of the sample for SHG, and E(r) is the local electric field at the fundamental frequency in the focal volume.27 Note that this approach takes into account the variations of the electric field vector in the focal volume. The SHG signal is proportional to the total emitted SHG power, I(2ω) ∝

∫A ∫V G̅ (R,r)χ (2) : E(r) E(r) dV dA

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MATERIALS AND METHODS To demonstrate the performance of the algorithm, we simulated SHG responses from two isotropic surfaces that exhibit achiral (C∞v point group) and chiral (C∞ point group) symmetries. A vast majority of the surface-specific second-order spectroscopic techniques have been developed for measuring surface samples with these symmetries,31 justifying their use to study the performance of the algorithm. The target χ(2) components (zzz, xxz, zxx) for the achiral isotropic surface were randomly generated real numbers, normalized to zzz. Similarly, the achiral target components (zzz, xxz, zxx) for the chiral isotropic surface were randomly generated real numbers, but the chiral component (xyz) was taken as imaginary to study whether complex-valued components could also be extracted. Next, the suitability of the algorithm to complex structures was investigated by applying the technique to a cell membrane fragment of Halobacterium salinarum. The membrane consists of bR proteins organized into crystalline patches, also known as purple membranes.32 The bR protein is the simplest proton pump found in biological systems.33 Proton pumps are key players in biological energy transduction,34 and bR is often considered as a model molecule of the others. Moreover, bR is the best-characterized member of the diverse rhodopsin family, hence playing a paradigmatic role also in rhodopsin research and in the investigation of the seven-helix receptor family, in general.33 On the other hand, because of its favorable physical properties (high stability and photoelectric and optical properties), bR has become the primary target protein of recent opto- and bioelectronic applications.35−37 The structures of bR and PMs are relatively simple and well-known.23,32,38,39 The repeating unit in the PM is a bR trimer consisting of three

2

(2)

where G̅ (R,r) is the dyadic Green’s function for signal propagation from the source point r to the far-field point R. The integrations are performed over the sample volume V (position r) and the detection plane of the far-field SHG emission A(R). The forward problem is then to solve I(2ω) when E(r) and χ(2) are known, whereas in the inverse problem I(2ω) and E(r) are known and χ(2) is the unknown to be solved. If the SHG field would be measured using holographic methods, the problem could be turned to be linear.22 Instead, we measure the SHG intensity which is not linearly proportional to the polarization or electric field and depends on quadratic combinations of the χ(2) components. This leads to a problem that is not solvable linearly. The main idea of our approach is to collect information from the SH measurements of the sample, to model these measurements for a sample with trial χ(2), and to investigate what values of χ(2) components give rise to similar responses as observed experimentally. To find the effective χ(2) of a sample, we use numerical methods to solve the inversion problem determined by eq 2. GAs are a class of evolutionary algorithms that can be used to find a global solution.28 We chose to use GAs because of their popularity and flexibility and because the forward problem of calculating I(2ω) using the known focused excitation E(r) and a trial χ(2) is straightforward and relatively fast to solve.21,29 The data set is generated by measuring the SH emission as a function of polarization of the excitation field and B



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indicate a perfectly matched solution but is in practice unreachable because of measurement error. The measurement technique is based on a polarized SHG microscopy setup, described in detail elsewhere.21,44 Briefly, a femtosecond (fs) laser beam is used for excitation (wavelength 1060 nm, pulse duration 200 fs, repetition rate 82 MHz) and focused with an objective (NA 0.8) onto the sample. SHG in the focal volume occurs, and the reflected SHG signal is collected by the same objective. The signal is separated from the fundamental wavelength by dichroic mirrors and filters and focused onto a photomultiplier tube. The samples were prepared by drop-casting PM fragment solution (10 mg/mL wild-type bR in water) onto glass substrates, and the sample exhibiting the lowest concentration was used. C3 is the highest rotational symmetry group that can be resolved by SHG from even higher symmetries, which behave as isotropic surfaces. The susceptibility tensor for C3 has all the tensor components of an isotropic chiral surface (xyz = −yxz = xzy = −yzx, xzx = yzy = xxz = yyz, zxx = zyy, zzz) complemented by the components specific to C3 (xxx = −xyy = −yyx = −yxy, yyy = −yxx = −xxy = −xyx), where the latter components have strictly in-plane character. These in-plane components are peculiar in the sense that, for unpolarized detection, they give rise to a SHG response that is independent of the direction of linear polarization; i.e., the response is isotropic.9,10 In addition, the response is isotropic for circular polarizations. Any possible deviations from isotropic behavior can occur for elliptical polarizations, but this can be considered only as a small higher-order correction to the predominant isotropic response. Indeed, we have verified through extensive simulations that in our experiments, where the focused beams are also properly treated, the small deviations from isotropy can only lead to uncertainties that are much smaller than any other uncertainties in the experiment. The excitation polarizations were chosen to be at 22.5° quarter-wave plate rotation angle intervals consisting of leftand right-handed circular polarizations, linear polarization (LP), and four elliptical polarizations between them. As already mentioned, the direction of LP can be arbitrary, since the PM fragment possesses C3 symmetry and the total SHG signal does not depend on the direction of input LP.9,10 We verified this by measuring SHG from several different PM fragments, for which only a few showed dependence on the direction of LP. We attributed this behavior to folded PM fragments and excluded them in the data analysis. SHG intensity can be measured in transmission and reflection geometries. For simulated isotropic samples, both of these collection geometries are used in calculations. For the simulated sample possessing a symmetry that is similar to bR, as well as for the real bR sample, only the reflected SHG is taken into account.

photoactive retinal molecules in a known geometry possessing C3 symmetry (see Figure 1a and Figure 1b) as observed previously using X-ray diffraction analysis40 and atomic force microscopy.41

Figure 1. (a) Structure of the bR trimer with a photoactive retinal molecule (PDB code 2NTU) in each subunit.39 Scale bar = 2 nm. (b) Schematics of the bR monomer and the crystalline PM fragment consisting of bR trimers with the retinal molecules highlighted in red.

The molecular β tensor is defined in the molecular frame (x′, y′, z′). The macroscopic χ(2) tensor is defined in the laboratory frame (x, y, z) and is the effective response of the molecular species present in the sample volume.1 Recent works suggest that the χ(2) can be approximated as a coherent sum of the molecular β, transformed by direction cosines to the laboratory frame.42 Then, provided that the structure of the sample is known, tensorial information on the β can be accessed by measuring the macroscopic SHG response. SHG responses of bR are due to three retinal molecules, which are usually approximated as rodlike. Here we assume the retinals to be nonplanar.43 We are interested in the relative values of the components; therefore, we normalize with respect to z′z′z′, where z′ is parallel to the long axis of the molecule. As the retinal does not protrude much in y′ direction of the microscopic frame, the β components z′y′y′ and y′y′z′ are assumed negligible [see Figure 1a for the definition of the molecular frame (x′, y′, z′)]. The dominant β components are then taken as z′z′z′, z′x′x′, and x′x′z′ = x′z′x′.43 The macroscopic χ(2) of a PM fragment is then calculated by converting and summing the β of the retinals to the macroscopic frame using direction cosines and finally normalized with respect to zzz. From previous crystallography measurements, the angle between the retinal chain and the sample normal is known to be θ = −69° and the rotation angle is ψ = −20° (Figure 1).38,39 Therefore, only two complexvalued variables (z′x′x′ and x′x′z′) remain to be solved to determine the relative retinal β and the χ(2) of the PM fragment (see Supporting Information). The fitting parameter f describing how closely the solution fits to the measured data was defined as f=

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RESULTS We first verified our algorithm by simulating its performance for the two isotropic surfaces with randomly generated target values of the tensor components and for a hypothetical sample that closely resembles bR. We used the target values to generate polarization-dependent SHG data. In the case of the bR, the target values were chosen to lead to approximately similar polarization-dependent SHG signals as measured in the experiment. The generated SHG data were then fed into our algorithm to yield the solutions for the tensor components, which were compared to the original target values.

∑N |ΔIN | N

(3)

where ΔIN is the difference between the calculated and the measured SHG signal for the given measurement N with particular polarizations. The asymptotic zero value for f would C



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Figure 2. Complex-valued (a) β and (b) χ(2) tensor solutions from calculations based on measured SHG intensities of the bR sample. The β (χ(2)) values are normalized to z′z′z′ (zzz). Note that the actual deviation of the solution candidates is much smaller than what appears in the figure because the solutions clustered close to the average are difficult to distinguish.

Table 1. Relative Nonzero β Components of the Retinal and χ(2) Components of a PM Fragment (Equivalent to a bR Trimer) with Their Standard Deviations, Calculated from the Measured SHG Intensities of a bR Sample β

χ(2)

nonzero component

average value

standard deviation of real part

standard deviation of imaginary part

x′z′x′ = x′x′z′ z′x′x′ z′z′z′ xxx = −xyy = −yyx = −yxy xyx = xxy = yxx = −yyy xzx = xxz = yzy = yyz yzx = yxz = −xzy = −xyz zxx = zyy zzz

0.09 + 0.04i 0.40 + 0.08i 1 −1.22 + 0.60i −0.47 − 0.09i 2.40 − 0.15i 0.64 + 0.03i 3.21 − 0.11i 1

0.06 0.13

0.04 0.12

0.29 0.04 0.17 0.35 0.30

0.28 0.04 0.17 0.29 0.21

emphasize that in the present work, the primary quantity determined is the hyperpolarizability β, which has fewer independent components. The susceptibility χ(2) is calculated by converting and summing the β to the macroscopic frame using direction cosines and finally normalized with respect to zzz. In order to directly determine the susceptibility, the number of independent measurements would have to be increased. Convergence toward certain β and χ(2) values is clear within different measured data sets. Regardless of the dispersion of β and χ(2) values between the sets, all 30 different solutions are relatively similar to each other and the different tensor components are clearly separated from each other. Alongside the simulations, this experimental verification of the method shows that a global and unique solution can be found for the tensor components of the sample by measuring the SHG intensity with only few different polarization states of input light.

The convergence of the algorithm clearly depends on the symmetry of the sample. The highest symmetry with the fewest independent tensor components is seen to converge very rapidly, whereas the bR requires more iteration loops for convergence (Supporting Information, Figure S3). For each simulated case, the algorithm is found to yield final solutions that match very closely with the original target values of tensor components (Supporting Information, Tables S1 and S2). We also simulated the case of bR for its robustness to noise in the experimental data. This was performed for two different noise levels corresponding to 1% and 3.7% as normalized to the maximum SHG signal levels. The simulated noise levels correspond to the range of noise levels in the actual experiments. The average results obtained in the presence of noise are again very close to the target values (Supporting Information, Figure S4). However, their uncertainty increases with noise as expected. On the basis of these simulations, we can conclude that the main uncertainty in our method is due to the inherent measurement error, and the accuracy of the calculations could be further increased with longer iteration times and by measuring more data. After verifying our algorithm by simulations, we applied it to the actual bR sample. SHG intensity was measured 30 times with each of the seven polarizations. The algorithm was then driven separately for all 30 intensity patterns, and this was repeated five times with 2000 loops. The resulting nonzero β (χ(2)) components are shown in Figure 2a (Figure 2b) with their standard deviation bars. The bars are centered at the average of the results of each component (numerically shown in Table 1), and the length of each bar in the horizontal (vertical) direction equals the corresponding standard deviation of the real (imaginary) parts of individual results. We

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DISCUSSION Our results agree qualitatively with previous SHG studies from ensembles of PM fragments and confirm that the chiral SHG responses most plausibly arise because of the three-dimensional orientation of the retinals.25,26,45,46 The detailed comparison of our results with previous works, however, is difficult because the excitation wavelengths and the sample preparation are different. To the best of our knowledge, our work provides the first demonstration of microscopic tensor analysis done at the individual to few PM fragment levels whereas the other most recent studies have focused on ensemble samples consisting of several hundreds of PM layers.45,46 Moreover, unlike previous studies, we provide complex-valued information on both the D



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retinal β and χ(2) of a PM fragment, which because of the crystallinity of the membrane is equivalent to χ(2) of the bR trimer. Interestingly, the values of the imaginary parts of the relative β components are relatively small. This implies that the phase differences between the tensor components are small, as the absolute phase remains undetermined. Furthermore, this suggests that all tensor components are associated with the same resonance of the molecules. In the present case, the resonance corresponds to the SHG wavelength being close to the broad absorption peak of PM fragments near 560 nm.45 Nevertheless, the nonzero complex-valued responses elucidate the molecular origins of the reported SHG circular-difference responses of bR.26 In the future, it will be interesting to study the wavelength-dependent nonlinear responses of the bR and related samples47 closer to the resonance and to combine the technique with pump−probe measurements to study the dynamic responses of bR during their photocycle. It is also worth noting that our method is not restricted to this particular application but can be extended to different kinds of optical measurements that warrant full tensorial characterization of single nano-objects at high spatial resolution, e.g., imaging of individual nanoparticles using inhomogeneous focal fields.48

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Grant 134973 from the Academy of Finland. This work is partially supported by TUT’s Strategy Funding 2014 (Grant 84010). G.B. acknowledges funding from the Academy of Finland (Grant 267847). M.V. acknowledges funding from TUT graduate school. A.D. acknowledges the support of Research Grant TÁ MOP-4.2.2.A-11/1/KONV2012-0060. This work was performed in the context of the European COST Action MP1302 Nanospectroscopy.

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CONCLUSION We have demonstrated a microscopic second-harmonic generation technique to determine the complex-valued second-order susceptibility and hyperpolarizability. The technique is based on polarization measurements and iterative data algorithms. The technique can provide complex-valued tensorial information in situ with the spatial resolution of a confocal microscope. We demonstrated the algorithm with three different kinds of simulated samples and used the technique to determine molecular tensorial response of the retinal molecules of light harvesting bacteriorhodopsin proteins as well as the macroscopic second-order responses of purple membrane fragments consisting of bacteriorhodopsin. The present samples belonged to symmetry group C3, which has several beneficial properties for the demonstration of the technique on the basis of a relatively limited set of measurements. However, the basic approach can be extended for more complicated samples by proper choice of the set of experiments.

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ASSOCIATED CONTENT

S Supporting Information *

Description of the used genetic algorithm, information about boundary conditions, and details of simulation results. This material is available free of charge via the Internet at http:// pubs.acs.org.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +358 50 4921 272. Present Address §

M.J.H.: COMP Centre of Excellence and Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland. E



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Microscopic Determination of Second-Order Nonlinear Optical

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