Point Group Symmetry Determination via Observables Revealed by

Article pubs.acs.org/ac

Point Group Symmetry Determination via Observables Revealed by Polarized Second-Harmonic Generation Microscopy: (1) Theory Monique A. van der Veen,*,†,‡ Frederik Vermoortele,† Dirk E. De Vos,† and Thierry, Verbiest‡ †

Centre for Surface Chemistry and Catalysis, KU Leuven, University of Leuven, 3001-Leuven, Belgium Molecular Electronics and Photonics, KU Leuven, University of Leuven, 3001-Leuven

‡

S Supporting Information *

ABSTRACT: We present a methodology based on polarization-controlled second-harmonic generation microscopy that allows one to determine the point group symmetry of noncentrosymmetric structures in situ and in vivo in complex systems regardless of the occurrence of periodicity. Small, randomly oriented structures suffice for the analysis, which is based on simple recognition of observables in four tests. These can be performed in any standard SHG-microscope that allows polarization control of the incident and detected light. The method is resilient to birefringence and light dispersion.

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ing11−14 and more recently in materials science6,8,15−19 and nanophotonics20−22 as well. Indeed, noncentrosymmetric materials, i.e., materials lacking an inversion center, are important functional materials as they include piezoelectric, ferroelectric, pyroelectric, and frequency doubling materials that can be used in a myriad of applications.23,24 Also biological structures and biomimetic materials often lack a center of inversion, as these supramolecular structures generally are chiral and polar.25 In second-harmonic generation, coherent beams of a specific polarization state are generated. Combined with its sensitivity to noncentrosymmetric systems, this makes second-harmonic generation microscopy a technique which is extremely sensitive to the organization of a system.26,27 SHG is sensitive to local structure, the degree of disorder, symmetry, and orientation. This is why an increasing effort is invested in deducing structural information from SHG-images. Examples can be found both in biological and materials science and include investigation of the helical pitch angle,28,29 polarity,30−32 orientation, disorder, and agglomeration of nanoparticles,15−17 domain structures,8,18 host/guest systems,6,19 and structural transitions.25 Yet the extraction of quantitative structural information, such as 3-D orientation maps, needs the use of a model that

tructural information is important to unravel the structure− activity relationship of functional compounds and materials. The point group or symmetry relations of a material are directly linked to its properties, or as stated in Neumann’s principle, the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal.1,2 Typically diffraction based methods are used to infer this information. However, not for all structures with a certain degree of organization, the point group symmetry can be determined with these methods. Examples are materials that are ordered yet lack periodicity,3−5 host/guest systems in which the periodicity of the guest is incommensurate with the periodicity of the host6,7 and systems where a certain amount of structural information can be inferred from XRD, yet a correct point group symmetry determination seems out of reach.8,9 We present here, in the case of noncentrosymmetric structures, a methodology that still allows a point group symmetry determination. This methodology is based on polarized second-harmonic imaging. Second-harmonic generation (SHG) is a second-order nonlinear effect that doubles the frequency of light. Within the electrical-dipole approximation, only noncentrosymmetric structures can generate SHG.10 As SHG-microscopy (SHGM) generally specifically visualizes these noncentrosymmetric structures, the technique provides enhanced contrast. SHGmicroscopy has thus emerged during the last 2 decades as an increasingly popular tool in biomedical and biological imag© 2012 American Chemical Society

Received: April 30, 2012 Accepted: July 9, 2012 Published: July 9, 2012 6378

dx.doi.org/10.1021/ac300936q | Anal. Chem. 2012, 84, 6378−6385

Analytical Chemistry

Article

developed for such single-beam experiments. We did not assume Kleinman symmetry as the routine failure of Kleinman symmetry, even under nonresonance conditions, has been reported frequently for various systems (see the Supporting Information, section SI.5).39−42 To track the observables theoretically, it is necessary that we define for each point group symmetry a set of nonzero tensor components that reflects the symmetry properties of the point group exactly (see the Supporting Information, section SI.1). The choice of these nonzero tensor components determines the natural coordinate system (xyz). To describe the orientation of the structure, we use a transformation matrix to connect the natural coordinate system (xyz) with the laboratory frame (XYZ):

appropriately describes the system. More specifically, knowledge of the point group symmetry is required. So far, ex situ measurements of the point group symmetry16,17 or assumption of a point group symmetry28−30,32 were used. Additionally, often assumptions concerning the orientation of the structure, e.g., planar in the sample plane, are made.29,32−34 Recently, methods based on SHGM to distinguish chiral and achiral symmetries were proposed. However, these methods are limited to surfaces. It is, moreover, required that the surface is oriented perpendicularly with respect to the direction of the incident light.35,36 Recently it was shown theoretically that the second-order susceptibility tensor describing the SHG-response can be determined from SHGM, from which the point group symmetry may be inferred. The method is limited to structures sufficiently smaller than the diffraction limit (nanosized object). The approach moreover needs a complex microscope design in which the SHG generated by the sample interferes with the SHG generated by a reference and the pulse of the incident light is shaped according to a particular functional. Moreover, determination of a set of complex mathematical expressions is necessary, and Kleinman symmetry of the sample is assumed.37 Kleinman symmetry, however, has been shown to be valid only in very restricted cases.38 In contrast with this method, we introduce here a method of high practicality that can be applied in a standard secondharmonic microscope, both in wide-field and scanning microscopes, both in reflection and transmission geometry. In this method, instead of large crystals with known orientation, small structures of arbitrary orientation suffice. The methodology is based on four simple tests in which the planar polarization of the incident and detected light is controlled. The determination is simply done by visually identifying observables in the obtained polarization patterns. The methodology, moreover, does not assume Kleinman symmetry and is shown to be surprisingly resilient toward birefringence and light dispersion. As the method is based on imaging and is generally only sensitive to noncentrosymmetric structures, it allows one to deduce the symmetry in situ in complex systems and in vivo in biological systems.

⎡X⎤ ⎡x⎤ ⎢ y ⎥ = a⎢ Y ⎥ ⎢ ⎥ ⎢ ⎥ ⎣z⎦ ⎣Z⎦

(3)

The transformation matrix a contains the Euler angles θ, ψ, and φ as defined in Scheme 1. Scheme 1. Schematic Representation of System Frame (xyz) and the Laboratory Frame (XYZ) with the Corresponding Euler Angles (θ, ψ, φ), the Electric Field Vector of the Incident Light Is Also Shown Together with the Angle α Defining the Plane of Polarization

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THEORY SHG or frequency doubling is a second-order nonlinear optical process which is forbidden in a medium with inversion symmetry. This arises from the fact that within the electric dipole approximation, SHG is described by the second-order nonlinear polarization P(2)(2ω) as in following equation: P(2)(2ω) = χ (2) : E(ω)E(ω)

In the laboratory coordinates, the XY plane corresponds with the sample plane. This means that at normal incidence, as is standard in a microscope, the fundamental light can be described by the two electric field components EX and EY solely. Note that this excludes the use of high N.A. objectives (N.A. > 0.60), as with these a significant electric field contribution along the optical axis (Z) is present and the planar wave approximation no longer holds.43,44 The following model is thus applicable to widefield microscopes and scanning microscopes with low N.A. objectives and can be used due to normal incidence in both transmission and reflection geometry. To apply eq 2, we need to transform the electric field components EX and EY from the laboratory frame to the natural frame using the rotation matrix a:

(1)

with χ the second-order susceptibility and E(ω) the electric field of the incoming light at frequency ω. As the second-order susceptibility χ(2) is in fact a tensor, we can rewrite eq 1 as (2)

Pi(2) =

∑ χijk(2) EjEk i,j

(2)

in which the indices ijk refer to the Cartesian coordinates. Equation 2 connects the components Pi(2) of the polarization vector with the components Ei of the electric field vector via the susceptibility tensor components χijk(2).10 Of all 32 crystallographic point groups, 21 are noncentrosymmetric. Of these 21, 20 are SHG-active in single beam SHG-experiments such as in standard SHG-microscopes. The proposed methodology is 6379

Ex = a11EX + a12EY

(4)

Ey = a 21EX + a 22EY

(5)

Ez = a31EX + a32EY

(6)



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Scheme 2. (Left) Schematic Representation of the Four Different Polarization Tests;a (Right) Example of a Polarization Plot with As Observables: 4 Zeros (4z), 4 Symmetry Axes (4sa)

a

L = laser, S = sample plane, F = filter (transmission of SHG-light, absorption of incident light), A = analyzer, D = detector.

will evidently show the same observables. These are the point groups that differ from each other only by having either a 4-, 6-, or ∞-fold rotation axis, for example, C6v, C4v, and C∞v. For the documentation of the number of observables for each test per point group symmetry, we started for structures for which none of the Euler angles have values of jπ/2 with j = ±0, 1, 2, .... These are further on denoted as “arbitrarily oriented structures”. The result is tabulated in Table 1. Similarly we documented the number of observables for structures that are aligned in part or completely with the laboratory frame and thus have an Euler angle θ and/or ψ with values of jπ/2 (j = ±0, 1, 2, ...) (for φ values of jπ/2 vide infra). The cases for which the observables are different from the arbitrarily oriented structures are documented in Table 1. The latter always show a higher number of observables compared to the arbitrarily oriented structures. An example of a crystal with D3h symmetry is shown in Figure 1: the polarization plots for tests 1 and 3 are given for different values of the out-of-plane angle θ. As can be seen, when θ equals 0 or π/2, a higher number of observables are present with respect to arbitrary orientations of θ. Concretely this means if we have an array of crystals with different orientation then the measured SHG polarization plots for individual crystallites can be divided into two groups. One group with the lower number of observables, which are the same for each of these structures, and another group with a higher amount of observables. On the basis of the former, arbitrarily oriented group and the data in Table 1, we formulated a flowchart to make a first division into groups of point group symmetries. The result is depicted in Scheme 3. On the basis of this flowchart we can directly assign the point group symmetries C6/C4/C∞, C6v/C4v/C∞v, D6/D4/D∞, and D2d. The other point group symmetries can be divided into four different subgroups (I−IV). For subgroups I−IV, further determination of the exact point group can be established based on the flowcharts presented in Scheme 4. The determination now relies on the observables present in the polarization plots of the group of structures with θ and/or ψ corresponding to jπ/2 with j = 0, 1, 2, ... (thus a higher number of observables). For symmetry groups I, II, and III (see Scheme 3) these observables will unambiguously point toward the point group symmetry as different observables are present for each point group symmetry. However, in symmetry group IV, for some cases the discrimination only depends on the lack of certain

Applying these relations to eq 2 provides the nonlinear polarization components Px, Py, Pz in the natural coordinate frame. Transformation of Px, Py, Pz to the laboratory frame delivers the polarization components PX and PY, which are directly related to the measured quantities IX and IY by IX ∼ |PX |2

(7)

IY ∼ |PY |2

(8)

In order to be able to determine the point group symmetry, we defined four different polarization tests (see Scheme 2). Test 1 is rotation of the plane of linearly polarized incident light over 360° along the direction of light propagation while detecting all the generated second-harmonic light. Test 2 is rotation of the plane of linearly polarized light over 360° along the direction of light propagation while detecting only the generated secondharmonic light polarized along an arbitrarily chosen direction (thus via the use of an analyzer). Test 3 is rotating the sample over 360° while the plane of polarization of the incident light is parallel with the plane of polarization of the detected secondharmonic light (alternatively the sample is kept fixed while the plane of polarization of both incident and detected light is rotated over 360° while keeping a parallel direction). Test 4 is rotating the sample over 360° while the plane of polarization of the incident light is perpendicular with the plane of polarization of the detected second-harmonic light. When plotting the SHGintensity over 360° for each of these tests, polarization patterns such as that in Scheme 2 (right) are obtained. We defined two types of easily distinguishable observables in these polarization patterns. One type is the number of polarization positions at which the SHG-intensity becomes zero during the test (abbreviated “z”, Scheme 2b) over 360°. The other type is the number of times over 360° the function of the SHG-intensity behaves as an even function around a minimum or maximum value M, thus the number of times following relation is true: f(M − α) = f(M + α) with α the independent variable. The latter is designated the number of symmetry axes (abbreviated “sa”, see Scheme 2b). Note also that it necessarily follows that the distance must be equal between all symmetry axes (so 2 sa are 180° apart, 4sa are 90° apart, and 6 sa are each 60° apart). We grouped the point group symmetries that are described by the same set of independent tensor components χijk(2) as these 6380



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Table 1. Overview of the Observables Present in the SHG Polarization Dependency Tests for the Different Point Group Symmetriesa

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Table 1. continued

id. = identical, z = zero, sa = symmetry axis, j = ± 0, 1, 2,... If the polarization response corresponds to a very simple function, this function is also given (the influence of birefringence and light dispersion is not taken into account in these functions). C and D are fitting parameters.

a

Figure 1. Simulated SHG intensity for (a) test 1 and (b) test 3 for a crystal with D3h symmetry, orientation: ψ = 0.3π and different values for θ. In part a φ = 0, while in part b φ is the angle defining the sample rotation.

Euler angle φ has thus been discarded. This angle only has an effect on test 2. When φ makes an angle of jπ/2 (j = ±0, 1, 2, ...) with the direction of the polarization of the detected secondharmonic light in test 2, the graphs can contain extra observables. These are tabulated in the Supporting Information, section SI.2 and are not taken into account in the methodology. By measuring test 2 for two different orientations of the polarization direction of the detected light in which the angle between these two directions is different from jπ/2 (j = ±0, 1, 2, ...) (e.g., a 45° difference), and subsequently disregarding the graphs that contain extra observables for only one of the polarization

observables. More specifically the determination of Cs instead of C3v is based on the absence of the 6 z, 12 sa observables. Also the distinction between C1 and the other members in symmetry group IV is based on the absence of any observables. Obviously the determination of Cs and C1 should be performed carefully: a large number of structures need to be measured. For final assignment of the observables, it is recommendable to check whether all the results conform with Table 1 for all four tests. Several notes must be made for the use of the methodology. To determine the observables used in the flowcharts in Scheme 4, only the orientations with Euler angles θ and ψ corresponding to jπ/2 (j = 0, 1, 2, ...) were taken into account (see Table 1). The 6382



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Scheme 3. Flow Chart for Arbitrarily Oriented Structuresa

a

The analysis starts with the use of this flowchart.

Scheme 4. Flowchart for Further Analysis of Subgroups I−IV That Should Be Used on the Polarization Graphs with Extra Observables

C∞v symmetry with a significant χzzz(2), χzxx(2), and χxzx(2) and is specifically taken into account in the flowchart in Scheme 3. Note that the methodology has been developed for single uniform structures. If this is not the case, e.g., for polycrystalline structures or aggregates, less observables will be present in the polarization tests. Two strategies to identify uniform monocrystalline structures were developed by Brasselet et al. Single particles can be distinguished from aggregates based on size and SHG intensity discrimination.15 Monocrystalline structures can be distinguished from polycrystalline structures via variation of the polarization excitation in two-photon fluorescence imaging as the shape of the polarization response should always be the same regardless of the polarization direction of the detected light.16

directions of the detected light, we very conveniently get rid of these unwanted extra observables. Note also that the found symmetry is a reflection of the occurrence of the different tensor components χijk(2) for each point group symmetry. This means that when certain tensor components χijk(2) are negligible with respect to the other components, the SHGM based methodology may suggest a higher symmetry (but never lower) than the actual point group symmetry. A typical case is that of a dominant component χzzz(2) along the polar z axis. This case will appear as C6v/C4v/C∞v symmetry with a single tensor component χzzz(2). This specific case has more observables associated with it than C6v/C4v/ 6383



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Notes

So far, the influence of birefringence was not included in the model. As it was shown before that birefringence can influence the SHG polarization response,33 we investigated the influence of birefringence on the presence of observables. The theoretical treatment can be found in section SI.3 in the Supporting Information. Moreover, we also included the influence of natural dispersion or the occurrence of a different refractive index at the wavelength of the fundamental and the second-harmonic light (see section SI.4 in the Supporting Information). The qualitative results of this treatment are the same for birefringence with or without light dispersion. Moreover, the qualitative results are the same for scanning and wide-field microscopes and for transmission and reflection geometries. The shape and intensity of the polarization plots in the tests 1−4 are influenced by birefringence and dispersion. Yet the number, type, and position of the observables generally remain the same for all four tests. It can occur for high degrees of birefringence and/or dispersion, roughly a phase shift larger than 20° over the thickness of the sample, that for specific combinations of orientation, thickness, birefringence, and dispersion, a higher, but never lower, number of observables may be found. Thus for a sample with structures with a variety of orientations, this will thus only be observed for some of the analyzed structures. For thick samples showing a large phase shift due to dispersion and birefringence, it is necessary to be cautious of this effect.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.A.v.d.V. thanks FWO Vlaanderen for a postdoctoral fellowship. T.V. and M.A.v.d.V. acknowledge financial support from the KU Leuven (GOA). D.E.D.V. is indebted to the Methusalem CASAS Grant and to IAP 6/27 Functional Supramolecular Systems. We thank Bert Sels and Jasper Van Noyen for providing us with the SAPO-5 sample.

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CONCLUSIONS We developed a methodology to determine the point group symmetry of noncentrosymmetric structures with polarization controlled second-harmonic generation microscopy. For the methodology, homogeneous structures or crystals with a variety of orientations are required, though an a priori knowledge of the orientation is not required. The simple methodology relies on four different polarization tests for which observables can be visually determined. We showed that the method can correctly assign all point group symmetries regardless of the presence of birefringence or light dispersion. The presented methodology would thus be the first method that allows one to determine the point group symmetry in vivo and in situ in complex systems. Moreover, periodicity of the material is not required in contrast with the requirements of X-ray diffraction analysis and other diffraction based methods such as electron diffraction. Also, a correct assignment of point group symmetry will allow one to employ a model that accurately describes the system, which allows one to correctly extract spatially resolved quantitative structural information with SHG-microscopy.

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ASSOCIATED CONTENT

S Supporting Information *

Overview of second-order susceptibility tensor elements for different point group symmetries; overview of observables in test 2 for the Euler angle φ being equal to jπ/2 (with j = 0,1,2,...); theoretical treatment of influence of birefringence and dispersion on the methodology; explanation Kleinman symmetry and its routine failure. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*Address: Kasteelpark Arenberg 23, Box 2461, University of Leuven, 3001-Leuven, Belgium. E-mail: monique.vanderveen@ biw.kuleuven.be. Fax: +32 16 321998. Phone: +32 16 32 7159. 6384



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Point Group Symmetry Determination via Observables Revealed by

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