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Transmission Mueller matrix ellipsometry of chirality switching phenomena
Thin Solid Films 519 (2011) 2617–2623
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Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Transmission Mueller matrix ellipsometry of chirality switching phenomena Oriol Arteaga a,⁎, Zoubir El-Hachemi b, Adolf Canillas a, Josep Maria Ribó b a b
Física Aplicada i Òptica Department and IN2UB, C/Martí i Franquès 1, Universitat de Barcelona, Catalonia, Spain Química Orgànica Department and Institute of Cosmos Science, C/Martí i Franquès 1, Universitat de Barcelona, Catalonia, Spain
a r t i c l e
i n f o
Available online 10 December 2010 Keywords: Polarimetry Mueller matrix Ellipsometry Optical activity Anisotropy Chirality
a b s t r a c t Mechanisms of molecular chirality induction are fundamental to many questions in chemistry. Interest in these mechanisms is shifting toward media of increasing complexity that simultaneously exhibit linear birefringence and dichroism and where the common assumption that optical activity is the only optical effect that affects light polarization is no longer valid. Light propagation through several of these anisotropic media can be appropriately studied with transmission Mueller matrix ellipsometry. The applications presented herein include the measurement of optical activity in stirred solutions of soft-matter nanophases and the determination of chiral domains in solid-state samples. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical activity was first observed in quartz crystals in 1811 by the French-Catalan astronomer François Arago. Nowadays, nearly two hundred years after this discovery, scientists know lots of substances and materials showing optical activity and its measurement has become of essential importance in many fields: chemistry, biology, material science, optics, medicine, etc. The measurement of the azimuthal rotation of linearly polarized light passing through certain media was the first experimental approach to the measurement of optical activity. Soon it was discovered that such rotation could be explained in the basis of a difference in the refractive indices for left- and right-circularly polarized light, which is known as circular birefringence (CB). The measurement of the unequal absorption for left and right circularly polarized light, or circular dichroism (CD), was described by Aimé Cotton only in 1895. Despite the broad knowledge of chiroptical phenomena, chiroptical measurement is still challenging because the magnitude of CD and CB effects is usually small. Moreover, in systems where linear birefringence and/or linear dichroism are also present they usually hinder the effect of CB and CD as they are orders of magnitude larger. A clear example is the measurement of optical rotation in crystals for directions off of the optic axes, in which the minor perturbation in the ellipticity of polarization state caused by the optical activity of the crystal is difficult to detect from an experimental point of view. Standard commercial chiroptical methods cannot be used to investigate optical activity in anisotropic samples, because they can only
⁎ Corresponding author. E-mail address: [email protected] (O. Arteaga). 0040-6090/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.11.083
take accurate measurements of CB and/or CD when no linear linear birefringence or linear dichroism are present. Recently, transmission Mueller matrix ellipsometry, sometimes also referred so as to Mueller polarimetry, has been applied to the study of anisotropic samples with the intent of studying their optical activity. Interesting systems to study with this methodology correspond to samples containing oriented molecules [1] or to crystals [2–4]. In a similar way that, years ago, generalized ellipsometry and Mueller matrix ellipsometry started being used to measure anisotropic samples that could not be studied with standard ellipsometry, it seems reasonable to think that the future of polarimetric measurements of optical activity in anisotropic samples will rely on characteristics of the Mueller matrix. In this paper we present the approach that we use to study optical activity in anisotropic samples from measurements of its Mueller matrix in transmission. We focus on heterogeneous samples that have undergone symmetry breaking in some way. Methods for the analysis of the measured Mueller matrices are also briefly discussed, and, with some examples, we illustrate how the information contained in the Mueller matrix becomes relevant in the study of optical activity in anisotropic media.
2. Mueller matrix measurements To measure the transmission Mueller matrix of a sample, the light from the source passes through the polarization state generator (PSG), transmits through the sample and then goes through a polarization state analyzer (PSA). In order to have access to all the elements of the Mueller matrix both PSG and PSA must contain some kind of compensator element. For example, it can be a quarter-wave plate, a Fresnel rhomb, a liquid crystal modulator or a photoelastic modulator.
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We employ a two modulator generalized ellipsometer (2-MGE) [5,6] to obtain spectroscopic and space resolved measurements of the transmission Mueller matrix of transparent samples. Such an instrument uses two polarizer–photoelastic modulator pairs, one as PSG and the other as PSA. Photoelastic modulators (PEMs) are resonant devices that are cut to oscillate at different high frequencies (~50 and ~ 60 kHz in our instrument). Since measurements of optical activity require a very high sensitivity, the use of PEMs as a source of polarization modulation in this kind of experiments is usually the best option, since they provide a high modulation purity, efficiency and stability. All the elements of the Mueller matrix can be obtained in a 2-MGE with four measurements made at different azimuthal orientations of the PSG and PSA. To allow this change of azimuth both the PSG and PSA are mounted on precision mechanical rotators. However, if the rotation axis does not coincide with the direction of the light beam and, there is some beam translation even for quite well aligned systems associated to the mechanical rotation. The amount of beam translation depends, of course, on the quality of the alignment of the mechanical rotator element with respect to the light beam and is also proportional to the thickness of the optical elements that are rotated. Because our PSG and PSA are several tens of millimeters thick the beam translation occurring when rotating the PSG and PSA is appreciable. This fact is not a major problem for samples with homogeneous in-plane characteristics but, for system that do not maintain their optical properties at different points of their surface, their Mueller matrix cannot be measured with accuracy with a 2-MGE. To skirt this problem we don't rotate the PSG and PSA. Instead we introduce a calibrated optical rotator that acts as a substitute of the mechanical rotation. Z-cut quartz plates (with the optical axis of the crystal perpendicular to the plate surface) are used as optical rotators. Two sets of quartz plates comprising plates of thicknesses 1 mm and 0.25 mm are mounted on filter wheels and placed between the PSG and the sample (quartz0), and between the sample and the PSA (quartz1). The thickness of the quartz plates is proportional to the amount of optical rotation introduced as shown in the CB definition of Table 1 (CB is twice the optical rotation). Fig. 1 shows the approximate dispersive behavior of the optical rotation for the two types of quartz plates as calculated from [2]. The ideal optical rotations for a 2-MGE are 45∘ (it would correspond to the situation in which the axes of the PEM are at 45∘ one to the other) which approximately occurs at 240 nm for the 0.25 mm plate and at 430 nm for the 1 mm plate. However, it is also possible to make measurements for arbitrary wavelengths and associated rotations if properly calibrated. For example we can infer from Fig. 1 the wavelengths for which the plates provide optical rotations between 20∘ and 70∘ that may be adequate wavelengths for space-resolved measurements. With this criterion the 1 mm rotator covers a wide range of wavelengths from 350 nm to 630 nm while the
Table 1 Symbols used and definitions. Effect
Symbol
Definitiona
Isotropic phase retardation Isotropic amplitude absorption
η κ
Horizontal linear dichroism projection
LD
Horizontal linear birefringence projection
LB
45∘ linear dichroism projection
LD′
45∘ linear birefringence projection
LB′
Circular dichroism
CD
Circular birefringence
CB
2πnl/λ0 2πkl/λ0 2π kx −ky l λ0 2π nx −ny l λ0 2π ðk45 −k135 Þl λ0 2π ðn45 −n135 Þl λ0 2π ðk− −kþ Þl λ0 2π ðn− −nþ Þl λ0
a
n stands for refractive index, k for the extinction coefficient, l for path length through the medium, and λ0 for the vacuum wavelength of light. Subscripts specify the polarization of light as, x, y, 45∘ to the x axis, 135∘ to the x axis, circular right +, or left −.
Fig. 1. Angle of rotation as a function of the wavelength for quartz rotator elements of different thicknesses. The horizontal lines indicate the angles of rotation of 20∘ and 70∘. They delimit, for each rotator element, the interval of wavelengths suitable for Mueller matrix measurements using the quartz method.
0.25 mm rotator is mostly suitable for wavelengths between 205 nm and 330 nm. Arbitrary optical rotations between 0∘ and 90∘ induced by a quartz plates, yield combinations of elements of the Mueller matrix rather than pure single elements. Nevertheless, as the 2-MGE is already able to measure 8 elements of the Mueller matrix simultaneously in a configuration not assisted by the quartz plates we can use these known Mueller matrix elements to solve for the others. It is possible to measure all the elements of the Mueller matrix in a 2MGE with the assistance of quartz plates if the following configurations are used: ● ● ● ●
Configuration I. PSG/sample/PSA. Configuration II. PSG/quartz 0/sample/PSA. Configuration III. PSG/sample/quartz 1/PSA. Configuration IV. PSG/quartz 0/sample/quartz 1/PSA.
The Mueller matrix elements measured in each configuration are given in the following equation: 0
1 B m10;III B M=@ m20;I m30;I
m01;II m11;IV m21;II m31;II
m02;I m12;III m22;I m32;I
1 m03;I m13;III C C: m23;I A m33;I
ð1Þ
Note that we use configuration I to measure 8 Mueller matrix elements, configurations II and III to measure 3 elements and configuration IV to measure only one. The most important feature is that for all the configurations the PSG and the PSA remain at the same position (in our case with the two PEMs oriented parallel). Filter wheels allow rapidly transition from one configuration to another. Apart from the optical rotation effect, the quartz plates can also produce some perturbations on the measurement if their optic axis is not perfectly parallel to the light beam because, in this case, some linear birefringence emerges. In order to calibrate and possibly correct this small effect we can follow the same strategies that are used in ellipsometry to considerate the influence of strained windows [7]. In the case of a 2-MGE it can be easily done if the windows are calibrated with anticipation to the measurement. 3. Light transmission through anisotropic optically active media We can distinguish two different main approaches to study light propagation through anisotropic optically active media. The first consists in solving Maxwell equations with the adequate tensorial constitutive relations characteristic of the medium under study. The calculations corresponding to this method are rather complicated, but diverse elegant matrix formalisms have been developed to systematize and simplify the calculations, most notably the 4 × 4 matrix
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method by Berreman [8]. Even with these tools, to our knowledge analytic solutions are only available either for anisotropic but nongyrotropic (i. e. without optical activity) media with certain geometries [9] or for gyrotropic but isotropic media [10]. The remaining cases cannot be handled analytically. The second approach is semiphenomenological and it was introduced by Jones [11] in the framework of his formalism to represent polarized light. It is based on an infinitesimal representation of the medium using a differential matrix calculus that does not involve the introduction of the constitutive equations of the medium. Instead, it uses the eight basic optical effects given in Table 1 as a basis set for constructing the properties of any complex optical element. In this work we use the Jones matrix method. This method, at least in the form it will be presented here, is only described for normal incidence and it does not take into account multiple reflections at the layer interfaces. Despite these limitations, it is especially useful to study light transmission through a homogeneous anisotropic media with optical activity, because it involves simple algebra and yields analytic results that can be easily correlated to the experiment. The Jones matrix J that describes light transmission through any kind of homogeneous non-depolarizing medium can be obtained from the exponential[12]: Jðω; zÞ = exp½zNðωÞ;
ð2Þ
incident medium (usually air) to the material and the other one between the material and the exiting medium (also usually air). If multiple reflections are excluded, the transmission Mueller matrix of the complete system is then given by: M = MI1 MS MI0 ;
ΝðωÞ = limz→0
Jðω; zÞ−I : z
ð3Þ
Once the infinitesimal matrix has been built from an infinitesimal version of the basis in Table 1, the Jones matrix in Eq. (2) is calculated [11,13]: 0
T iL T cos − sin 2 T 2 −iχ = 2 B B J=e @ ðC + iL′ Þ T sin − T 2
1 ðC−iL′ Þ T sin T 2 C C; A T iL T cos + sin 2 T 2
ð4Þ
ð9Þ
where MI0 and MI1 are, respectively, the Mueller matrices corresponding to the incident and exiting interfaces. MS is the Mueller– Jones matrix of the “bulk” material calculated from in Eq. (4), and constructed from the thickness-dependent parameters given in Table 1. At normal-incidence, MI0 and MI1 can only have a certain contribution if the measured sample is anisotropic, otherwise the matrices MI0 and MI1 are the identity matrix. If the sample under study is nonabsorbent (i.e. if LD, LD′ and CD in MS vanish) then MI0 and MI1 describe the difference polarization transmission at the interface, i.e. a diattenuation, and, in this case, the interface matrices do not include any retardation. In the case where MS contains any anisotropic absorption then there appears a phase shift at the interface that translates in Mueller matrices MI0 and MI1 including also retardation terms. The interface diattenuation for two orthogonal polarization states, e.g. x and y, as it appears in the Mueller matrix, can be calculated as: I
where ω is the frequency and z is the distance into the medium (z = 0 denotes the origin of the scattering medium). The matrix N is the infinitesimal generator of J, and satisfies:
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Dyx =
ty ty −tx tx ; ty ty + tx tx
ð10Þ
where the superscript I refers to the interface and ty and tx are the complex transmission coefficients, that, at normal incidence, are: tk =
2n0 ; k = x; y n0 + nk −ikk
ð11Þ
where n0 is the refractive index in the incident medium and nk − ikk is the complex refractive index of the sample for light polarized along the k direction. A subtlety that needs to be noted is that, for an interface diattenuation, the real-valued terms tktk used in Eq. (10) are not directly the intensity transmission coefficients Tx and Ty for light polarized along the x or y, used in other definitions of the global diattenuation [16]. These transmission coefficients are
nk = 1−Rk :ðnormal incidenceÞ n0
where we have defined a complex retardation for each birefringence– dichroism pairs:
Tk = tk tk
χ = η−iκ;
ð5Þ
L = LB−iLD′
ð6Þ
L′ = LB′ −iLD′ ;
ð7Þ
where the factor nk/n0 is due to the change of medium. The reason why this transmittance coefficient, Tk, should not be used here is that the diattenuation occurring at one interface depends only on the amplitude of the transmitted fields as a function of the incident polarization, and the index of refraction is not included in the definitions of the Jones vectors or in the Stokes parameters. In case we were looking at the diattenuation of the reflected light, the reflection coefficients Rk could be used for the calculation of the diattenuation [17]. The retardation for the transmitted beam upon the interface for two orthogonal polarization states, e.g. x and y, is
C = CB−iCD; ð8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 and T = L2 + L′ + C2 . The Mueller–Jones matrix corresponding to the Jones matrix in Eq. (4) can also be analytically calculated as shown in [13,14]. Given the non-depolarizing Mueller–Jones matrix estimate of an experimental normalized Mueller matrix it is also possible to obtain the CD, CB, LD, LD′, LB and LB′ following the procedure described in [15].
Δyx = arg ty −argðtx Þ;
ð13Þ
which, at normal incidence, is 3 ky ðn0 + nx Þ−kx n0 + ny 5; = atan4 ðn0 + nx Þ n0 + ny + kx ky 2
3.1. The effect of the interfaces Δyx The infinitesimal matrix method introduced by Jones does not take into account the effect of the interfaces. When studying light transmission through a certain homogeneous material the most common situation is having two well defined interfaces: one delimiting the
ð12Þ
ð14Þ
demonstrating that for kx = ky = 0 no retardation appears in the interface.
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At normal incidence the effect of the interfaces will be, in general, very small and only will be noticeable for samples with large anisotropy. For example, in a crystal with huge birefringence such as calcite, the transmission Mueller matrices at the interfaces are: 2
3 1 0:0668 0 0 6 0:0668 7 1 0 0 7; MI0 = 6 4 0 0 0:9978 0 5 0 0 0:9978 3 2 0 1 −0:0426 0 0 6 −0:0426 1 0 0 7 7; MI1 = 6 4 0 0 0:9991 0 5 0 0 0 0:9991
ð15Þ
where we have assumed that the optic axis is parallel to the y axis of the laboratory frame (ny = ne = 1.486, nx = no = 1.658). Only when studying samples with large anisotropy does the effect of the interfaces need to be considered in detail. Otherwise the Mueller interface matrices may be taken as the identity matrix without introducing significant errors. In the examples that follow the effect of the interfaces has not been considered. 4. Examples 4.1. Stirred solutions Transmission Mueller matrix ellipsometry has been applied to the study of a surprising phenomenon that has attracted attention in chemistry: the induction and switching of chirality in solutions of certain nanophases under the effect of a stirring vortex. In these experiments the handedness of the induced optical activity depends on direction of stirring in a completely reversible process. This effect has been reported in a number of recent publications but various interpretations are not supported by experiments [18–20]. Our approach to the problem has been directed toward obtaining as much information as possible from polarimetric experiments on stirred solutions. Experiments were performed in ~ 10 mm pathlength cuvettes that were stirred using small magnetic bars. Cuvettes containing Jaggregates of different types of porphyrins have been investigated, ones showing an intense induction and switch of optical activity (usually easily traceable using CD and CB signals) and others showing no appreciable change in the values of these spectra with respect to the stagnant configuration. For example J-aggregates H2TPPS3 porphyrin do not show any change in the optical activity upon stirring [21], while for another similar porphyrin, H2TPPS4, the one used in this work, there are dramatic changes. All these J-aggregates are ordered soft-matter structures which tend to be thin (few nanometers) and long (hundreds of nanometers), and for which flow gradients can sculpture their shapes [22], and, possibly, modify the order patterns on the electronic interactions that sustain the aggregate. Mueller matrix measurements performed in-situ in cuvettes stirred in CW and CCW directions reveal several aspects about the interrelation between the nanosized H2TPPS4 porphyrin aggregates and the macroscopic flows. With spectroscopic measurements [1] we confirmed that the effect was completely traceable from both the CD and CB values obtained from the Mueller matrix and thus, it was clear that a process of induction and switch of optical activity was responsible for the observed effect. Space resolved measurements in square section cuvettes have revealed that the induction of optical activity takes place in the central part of the cuvette (see Fig. 2), in coincidence with the central chiral descending vortex. Fig. 2 shows that at 485 nm the switch of optical activity is clearly visible in the CD maps, because this wavelength corresponds to the high-energy peak of a bisignated CD absorption band. For CB the change of sign can also be noted, but is less appreciable because this wavelength does not
Fig. 2. Scanning of CD and CB inside a 10 mm pathlength cuvette for clockwise (CW), counter-clockwise stirring (CCW) and stagnant (no stirring). CD and CB have been calculated from space-resolved measurements of the Mueller matrix performed in situ at 485 nm, that corresponds to a peak at high energy of the CD bisignated CD band.
correspond to a peak of CB. The cuvette stirring increases the errors of stochastic nature of the measurements, but, interestingly, not all the elements of the Mueller matrix are affected in the same degree. For example, the stochastic errors of elements m03 and m30, mostly contributed by CD, are barely not incremented when stirring, but, in other elements, such as m01 and m10, that mostly depict the linear dichroism, the stochastic errors greatly increase when stirring. This difference is understandable if we attend to the fact that linear dichroism (but not CD) is very sensitive to the size and orientation dispersions of the moving particles of the fluid that traverse the light beam. Fig. 3 shows a vectorial representation of the measured projections of linear dichroism. The length of the vectors, l, and their orientation with respect to the horizontal laboratory axis, θ have been calculated according to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l∝ LD2 + LD′ ; θ=
1 LD′ arctan : 2 LD
ð16Þ ð17Þ
This figure shows how the measured values for linear anisotropic effects can be correlated with the average orientation of the nanophases in the solution. For both CW and CCW stirring there appears to be a small bend of the vortex axis at the top of cuvette, that allows areas with weaker flows (top-left corner for CW stirring and top-right corner for CCW stirring), where the particles can maintain more stable orientation, yielding greater values of LD and LD′.
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4.2. Chiral domains in benzil crystallizations
Fig. 3. Superposition of the CD color map (already presented in Fig. 2) with a vectorial representation of in-plane projection of the linear dichroism, also calculated from the Mueller matrix.
Analogous experiments have been repeated in cylindrical flasks yielding differences in the sign of the induced signals with respect to the case of square section cuvettes. We attribute the differences to the fact that for a cylindrical cuvette an ascending flow is dominant. A more detailed account of these experiments is available in Ref. [23].
One of the most useful applications of Mueller matrix polarimetry is to investigate optical activity in transparent solid samples which, in many cases, show some degree of optical anisotropy. When the structure or composition of the sample varies from point to point, imaging or mapping techniques are the only way to discern the different optical effects that may be present in the sample. Benzil is a molecular crystal; its molecules do not exhibit any optical activity when they are in solution [24], but they crystallize in a chiral enantiomorphous group of symmetry D43 or D63 (the same group of symmetry than α-quartz). The crystal of benzil is built upon a hexagonal lattice and the unit cell accommodates three molecules disposed spirally around the trigonal axis. Therefore the benzil crystal is uniaxial and shows optical activity, which at visible regions of the spectra manifest not only in CB, but also in CD [25] because benzil molecules have an absorption band centered near 400 nm. As it is common for anisotropic crystals, optical activity in benzil can be easily measured for light propagating along the optical axis, but in other directions is more difficult because linear birefringence and linear dichroism effects become much greater. Benzil (1,2-diphenyl-1,2-ethanedione, Aldrich) was recrystallized twice from toluene, as means of purification, yielding six-sided prisms of about 5 mm in length. Later they were manually ground in an agate
Fig. 4. Mapping of the normalized Mueller matrix of a thin polycristaline film of benzil measured at 400 nm. It covers an area of 10×10 mm with an step size of 60 μm.
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mortar to get a fine powder. 2–3 mg of the powder were sandwiched between a blank microscope slide and a glass cover slips. The slide was raised to a temperature above the melting point of benzil (378 K) in a hot stage, held for 15 min, and left to cool at room temperature. A polycrystalline film is eventually developed from the metastable melt. Most of the surface of the sample was scanned by the 2-MGE maintaining a high lateral resolution during the measurement. Imaging representations of the transmission Mueller matrix of the sample in regions around 10 × 10 mm were obtained for several different samples (see one of them in Fig. 4). The pixel size used for generating these mapping plots is 60 μm, which is slightly below the optical resolution of the measurement (85–90 μm). In our case, this resolution is determined by the size of the light spot and the width and height of the monochromator slits placed before the detector. Fig. 5 shows the CD, CB and fraction β of polarized light (1 for completely polarized and 0 for completely depolarized) calculated from the Mueller matrix in Fig. 4. Well delimited zones with CD values of opposite signs are observed. A similar separation is obtained for CB, but, in this case, it is worse defined because the average error of this CB measurement (0.004) is greater than the average error associated to CD (0.001). In Fig. 6 the small area delimited by a black square in Fig. 5 is presented with more detail. In this figure the vectorial representation of
Fig. 6. Superposition of the CD color map (squared area in Fig. 5) with a vectorial representation of in-plane projection of the linear birefringence, also calculated from the Mueller matrix. The vectors of this figure correspond to linear birefringence projections with magnitudes around 0.6 rad.
the projections of linear birefringence, calculated with Eqs. (16) and (17) but now using LB and LB′ instead of LD and LD′, has been superposed to the color map of CD. This figure confirms that the boundaries between areas of positive and negative CD (mostly delimited by strait lines) do not correspond with changes on the orientation of the projected directions of the optic axis. Thus, zones in distinct false color correspond to enantiomorphs otherwise in comparable optical orientations.
5. Summary In this paper, we have shown that transmission Mueller matrix ellipsometry can be applied to study anisotropic samples that exhibit optical activity. From an experimental point of view the 2-MGE operated in transmission mode is an ideal instrument for this purpose because it allows the measurement of the complete Mueller matrix and retains the high sensitivity and wide spectral range characteristics of phase modulated ellipsometry. We have also shown that the same instrument can be used to obtain mappings of the Mueller matrix with a certain spatial resolution if the mechanical rotators of the instruments are substituted by optical rotators. For normal incidence transmission in bulk transparent samples the effect of the interfaces can be in most cases omitted and the differential matrix method pioneered by Jones can be used to interpret the measured matrices. We showed how space resolved Mueller matrix measurements can be applied in solution and in the solid state. We illustrated the induction of optical activity by effect of flows in solutions of some soft matter nanophases. We also detected chiral domains in polycrystalline samples. In summary, this work shows how instruments and techniques developed mainly in the framework of ellipsometry can be directly applied to work on the old problem of measuring optical activity in anisotropic media.
Acknowledgements
Fig. 5. Values of CD and CB obtained from the Mueller matrix in Fig. 4 using the inversion process [15] of the Mueller matrix based on the Jones differential representation of a medium. The black square indicates the area that is studied with more detail in Fig. 6.
We thank Bart Kahr for revising the manuscript. This work was funded by the Spanish Government (AYA2009-13920-C02-01 and AYA2009-13920-C02-02). This work was funded by the Spanish Government (AYA2009-13920-C02-01 and AYA2009-13920-C02-02) and is part of the COST Action CM0703 Systems Chemistry. O.A. acknowledges the financial support from FPU AP2006-00193.
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[14] J. Schellman, H.P. Jensen, Chem. Rev. 87 (6) (1987) 1359, doi:10.1021/ cr00082a0048 URL http://dx.doi.org/10.1021/cr00082a004. [15] O. Arteaga, A. Canillas, Opt. Lett. 35 (4) (2010) 559, doi:10.1364/OL.35.0005598 URL http://dx.doi.org/10.1364/OL.35.000559. [16] S.-Y. Lu, R.A. Chipman, J. Opt. Soc. Am. A 13 (5) (1996) 1106, doi:10.1364/ JOSAA.13.0011068 URL http://dx.doi.org/10.1364/JOSAA.13.001106. [17] J.D. Jellison, C.M. Hunn, Appl. Opt. 45 (22) (2006) 5479, doi:10.1364/ AO.45.0054798 URL http://dx.doi.org/10.1364/AO.45.005479. [18] A. D'Urso, R. Randazzo, L. Lo Faro, R. Purrello, Angew. Chem. Int. Ed. 49 (1) (2010) 108 , d oi :1 0.1 002 /anie .2 0090 354 38 URL http://dx.doi.org/1 0.100 2/ anie.200903543. [19] A. Tsuda, T. Md, T. Harada, N. Yamaguchi, T. Ishii, Aida, Angew. Chem. Int. Ed. 46 (43) (2007) 8198, doi:10.1002/anie.2007030838 URL http://dx.doi.org/10.1002/ anie.200703083. [20] M. Wolffs, S.J. George, Z. Tomović, S.C. Meskers, A.P. Schenning, E.W. Meijer, Angew. Chem. Int. Ed Engl. 46 (43) (2007) 8203, doi:10.1002/anie.2007030758 URL http://dx.doi.org/10.1002/anie.200703075. [21] O. Arteaga, C. Escudero, G. Oncins, Z. El-Hachemi, J. Llorens, J. Crusats, A. Canillas, J.M. Ribó, Chem. Asian J. 4 (11) (2009) 1687, doi:10.1002/asia.2009002548 URL http://dx. doi.org/10.1002/asia.200900254. [22] C. Escudero, J. Crusats, I. Díez-Pérez, Z. El-Hachemi, J.M. Ribó, Angew. Chem. 118 (47) (2006) 8200, doi:10.1002/ange.2006031828 URL http://dx.doi.org/10.1002/ ange.200603182. [23] O. Arteaga, A. Canillas, J. Crusats, Z. El-Hachemi, J. Llorens, E. Sacristan, J.M. Ribó, ChemPhysChem 11 (16) (2010) 3511, doi:10.1002/cphc.201000658. [24] N.K. Chaudhuri, M.A. El-Sayed, J. Chem. Phys. 47 (1967) 1133, doi: 10.1063/1.17119988 URL http://dx.doi.org/10.1063/1.1711998. [25] J. Ríha, I. Vysín, H. Lapanská, Mol. Cryst. Liq. Cryst. 442 (2005) 181, doi: 10.1080/1542140909648898 URL http://dx.doi.org/10.1080/154214090964889.
Contents lists available at ScienceDirect
Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Transmission Mueller matrix ellipsometry of chirality switching phenomena Oriol Arteaga a,⁎, Zoubir El-Hachemi b, Adolf Canillas a, Josep Maria Ribó b a b
Física Aplicada i Òptica Department and IN2UB, C/Martí i Franquès 1, Universitat de Barcelona, Catalonia, Spain Química Orgànica Department and Institute of Cosmos Science, C/Martí i Franquès 1, Universitat de Barcelona, Catalonia, Spain
a r t i c l e
i n f o
Available online 10 December 2010 Keywords: Polarimetry Mueller matrix Ellipsometry Optical activity Anisotropy Chirality
a b s t r a c t Mechanisms of molecular chirality induction are fundamental to many questions in chemistry. Interest in these mechanisms is shifting toward media of increasing complexity that simultaneously exhibit linear birefringence and dichroism and where the common assumption that optical activity is the only optical effect that affects light polarization is no longer valid. Light propagation through several of these anisotropic media can be appropriately studied with transmission Mueller matrix ellipsometry. The applications presented herein include the measurement of optical activity in stirred solutions of soft-matter nanophases and the determination of chiral domains in solid-state samples. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Optical activity was first observed in quartz crystals in 1811 by the French-Catalan astronomer François Arago. Nowadays, nearly two hundred years after this discovery, scientists know lots of substances and materials showing optical activity and its measurement has become of essential importance in many fields: chemistry, biology, material science, optics, medicine, etc. The measurement of the azimuthal rotation of linearly polarized light passing through certain media was the first experimental approach to the measurement of optical activity. Soon it was discovered that such rotation could be explained in the basis of a difference in the refractive indices for left- and right-circularly polarized light, which is known as circular birefringence (CB). The measurement of the unequal absorption for left and right circularly polarized light, or circular dichroism (CD), was described by Aimé Cotton only in 1895. Despite the broad knowledge of chiroptical phenomena, chiroptical measurement is still challenging because the magnitude of CD and CB effects is usually small. Moreover, in systems where linear birefringence and/or linear dichroism are also present they usually hinder the effect of CB and CD as they are orders of magnitude larger. A clear example is the measurement of optical rotation in crystals for directions off of the optic axes, in which the minor perturbation in the ellipticity of polarization state caused by the optical activity of the crystal is difficult to detect from an experimental point of view. Standard commercial chiroptical methods cannot be used to investigate optical activity in anisotropic samples, because they can only
⁎ Corresponding author. E-mail address: [email protected] (O. Arteaga). 0040-6090/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.11.083
take accurate measurements of CB and/or CD when no linear linear birefringence or linear dichroism are present. Recently, transmission Mueller matrix ellipsometry, sometimes also referred so as to Mueller polarimetry, has been applied to the study of anisotropic samples with the intent of studying their optical activity. Interesting systems to study with this methodology correspond to samples containing oriented molecules [1] or to crystals [2–4]. In a similar way that, years ago, generalized ellipsometry and Mueller matrix ellipsometry started being used to measure anisotropic samples that could not be studied with standard ellipsometry, it seems reasonable to think that the future of polarimetric measurements of optical activity in anisotropic samples will rely on characteristics of the Mueller matrix. In this paper we present the approach that we use to study optical activity in anisotropic samples from measurements of its Mueller matrix in transmission. We focus on heterogeneous samples that have undergone symmetry breaking in some way. Methods for the analysis of the measured Mueller matrices are also briefly discussed, and, with some examples, we illustrate how the information contained in the Mueller matrix becomes relevant in the study of optical activity in anisotropic media.
2. Mueller matrix measurements To measure the transmission Mueller matrix of a sample, the light from the source passes through the polarization state generator (PSG), transmits through the sample and then goes through a polarization state analyzer (PSA). In order to have access to all the elements of the Mueller matrix both PSG and PSA must contain some kind of compensator element. For example, it can be a quarter-wave plate, a Fresnel rhomb, a liquid crystal modulator or a photoelastic modulator.
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We employ a two modulator generalized ellipsometer (2-MGE) [5,6] to obtain spectroscopic and space resolved measurements of the transmission Mueller matrix of transparent samples. Such an instrument uses two polarizer–photoelastic modulator pairs, one as PSG and the other as PSA. Photoelastic modulators (PEMs) are resonant devices that are cut to oscillate at different high frequencies (~50 and ~ 60 kHz in our instrument). Since measurements of optical activity require a very high sensitivity, the use of PEMs as a source of polarization modulation in this kind of experiments is usually the best option, since they provide a high modulation purity, efficiency and stability. All the elements of the Mueller matrix can be obtained in a 2-MGE with four measurements made at different azimuthal orientations of the PSG and PSA. To allow this change of azimuth both the PSG and PSA are mounted on precision mechanical rotators. However, if the rotation axis does not coincide with the direction of the light beam and, there is some beam translation even for quite well aligned systems associated to the mechanical rotation. The amount of beam translation depends, of course, on the quality of the alignment of the mechanical rotator element with respect to the light beam and is also proportional to the thickness of the optical elements that are rotated. Because our PSG and PSA are several tens of millimeters thick the beam translation occurring when rotating the PSG and PSA is appreciable. This fact is not a major problem for samples with homogeneous in-plane characteristics but, for system that do not maintain their optical properties at different points of their surface, their Mueller matrix cannot be measured with accuracy with a 2-MGE. To skirt this problem we don't rotate the PSG and PSA. Instead we introduce a calibrated optical rotator that acts as a substitute of the mechanical rotation. Z-cut quartz plates (with the optical axis of the crystal perpendicular to the plate surface) are used as optical rotators. Two sets of quartz plates comprising plates of thicknesses 1 mm and 0.25 mm are mounted on filter wheels and placed between the PSG and the sample (quartz0), and between the sample and the PSA (quartz1). The thickness of the quartz plates is proportional to the amount of optical rotation introduced as shown in the CB definition of Table 1 (CB is twice the optical rotation). Fig. 1 shows the approximate dispersive behavior of the optical rotation for the two types of quartz plates as calculated from [2]. The ideal optical rotations for a 2-MGE are 45∘ (it would correspond to the situation in which the axes of the PEM are at 45∘ one to the other) which approximately occurs at 240 nm for the 0.25 mm plate and at 430 nm for the 1 mm plate. However, it is also possible to make measurements for arbitrary wavelengths and associated rotations if properly calibrated. For example we can infer from Fig. 1 the wavelengths for which the plates provide optical rotations between 20∘ and 70∘ that may be adequate wavelengths for space-resolved measurements. With this criterion the 1 mm rotator covers a wide range of wavelengths from 350 nm to 630 nm while the
Table 1 Symbols used and definitions. Effect
Symbol
Definitiona
Isotropic phase retardation Isotropic amplitude absorption
η κ
Horizontal linear dichroism projection
LD
Horizontal linear birefringence projection
LB
45∘ linear dichroism projection
LD′
45∘ linear birefringence projection
LB′
Circular dichroism
CD
Circular birefringence
CB
2πnl/λ0 2πkl/λ0 2π kx −ky l λ0 2π nx −ny l λ0 2π ðk45 −k135 Þl λ0 2π ðn45 −n135 Þl λ0 2π ðk− −kþ Þl λ0 2π ðn− −nþ Þl λ0
a
n stands for refractive index, k for the extinction coefficient, l for path length through the medium, and λ0 for the vacuum wavelength of light. Subscripts specify the polarization of light as, x, y, 45∘ to the x axis, 135∘ to the x axis, circular right +, or left −.
Fig. 1. Angle of rotation as a function of the wavelength for quartz rotator elements of different thicknesses. The horizontal lines indicate the angles of rotation of 20∘ and 70∘. They delimit, for each rotator element, the interval of wavelengths suitable for Mueller matrix measurements using the quartz method.
0.25 mm rotator is mostly suitable for wavelengths between 205 nm and 330 nm. Arbitrary optical rotations between 0∘ and 90∘ induced by a quartz plates, yield combinations of elements of the Mueller matrix rather than pure single elements. Nevertheless, as the 2-MGE is already able to measure 8 elements of the Mueller matrix simultaneously in a configuration not assisted by the quartz plates we can use these known Mueller matrix elements to solve for the others. It is possible to measure all the elements of the Mueller matrix in a 2MGE with the assistance of quartz plates if the following configurations are used: ● ● ● ●
Configuration I. PSG/sample/PSA. Configuration II. PSG/quartz 0/sample/PSA. Configuration III. PSG/sample/quartz 1/PSA. Configuration IV. PSG/quartz 0/sample/quartz 1/PSA.
The Mueller matrix elements measured in each configuration are given in the following equation: 0
1 B m10;III B M=@ m20;I m30;I
m01;II m11;IV m21;II m31;II
m02;I m12;III m22;I m32;I
1 m03;I m13;III C C: m23;I A m33;I
ð1Þ
Note that we use configuration I to measure 8 Mueller matrix elements, configurations II and III to measure 3 elements and configuration IV to measure only one. The most important feature is that for all the configurations the PSG and the PSA remain at the same position (in our case with the two PEMs oriented parallel). Filter wheels allow rapidly transition from one configuration to another. Apart from the optical rotation effect, the quartz plates can also produce some perturbations on the measurement if their optic axis is not perfectly parallel to the light beam because, in this case, some linear birefringence emerges. In order to calibrate and possibly correct this small effect we can follow the same strategies that are used in ellipsometry to considerate the influence of strained windows [7]. In the case of a 2-MGE it can be easily done if the windows are calibrated with anticipation to the measurement. 3. Light transmission through anisotropic optically active media We can distinguish two different main approaches to study light propagation through anisotropic optically active media. The first consists in solving Maxwell equations with the adequate tensorial constitutive relations characteristic of the medium under study. The calculations corresponding to this method are rather complicated, but diverse elegant matrix formalisms have been developed to systematize and simplify the calculations, most notably the 4 × 4 matrix
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method by Berreman [8]. Even with these tools, to our knowledge analytic solutions are only available either for anisotropic but nongyrotropic (i. e. without optical activity) media with certain geometries [9] or for gyrotropic but isotropic media [10]. The remaining cases cannot be handled analytically. The second approach is semiphenomenological and it was introduced by Jones [11] in the framework of his formalism to represent polarized light. It is based on an infinitesimal representation of the medium using a differential matrix calculus that does not involve the introduction of the constitutive equations of the medium. Instead, it uses the eight basic optical effects given in Table 1 as a basis set for constructing the properties of any complex optical element. In this work we use the Jones matrix method. This method, at least in the form it will be presented here, is only described for normal incidence and it does not take into account multiple reflections at the layer interfaces. Despite these limitations, it is especially useful to study light transmission through a homogeneous anisotropic media with optical activity, because it involves simple algebra and yields analytic results that can be easily correlated to the experiment. The Jones matrix J that describes light transmission through any kind of homogeneous non-depolarizing medium can be obtained from the exponential[12]: Jðω; zÞ = exp½zNðωÞ;
ð2Þ
incident medium (usually air) to the material and the other one between the material and the exiting medium (also usually air). If multiple reflections are excluded, the transmission Mueller matrix of the complete system is then given by: M = MI1 MS MI0 ;
ΝðωÞ = limz→0
Jðω; zÞ−I : z
ð3Þ
Once the infinitesimal matrix has been built from an infinitesimal version of the basis in Table 1, the Jones matrix in Eq. (2) is calculated [11,13]: 0
T iL T cos − sin 2 T 2 −iχ = 2 B B J=e @ ðC + iL′ Þ T sin − T 2
1 ðC−iL′ Þ T sin T 2 C C; A T iL T cos + sin 2 T 2
ð4Þ
ð9Þ
where MI0 and MI1 are, respectively, the Mueller matrices corresponding to the incident and exiting interfaces. MS is the Mueller– Jones matrix of the “bulk” material calculated from in Eq. (4), and constructed from the thickness-dependent parameters given in Table 1. At normal-incidence, MI0 and MI1 can only have a certain contribution if the measured sample is anisotropic, otherwise the matrices MI0 and MI1 are the identity matrix. If the sample under study is nonabsorbent (i.e. if LD, LD′ and CD in MS vanish) then MI0 and MI1 describe the difference polarization transmission at the interface, i.e. a diattenuation, and, in this case, the interface matrices do not include any retardation. In the case where MS contains any anisotropic absorption then there appears a phase shift at the interface that translates in Mueller matrices MI0 and MI1 including also retardation terms. The interface diattenuation for two orthogonal polarization states, e.g. x and y, as it appears in the Mueller matrix, can be calculated as: I
where ω is the frequency and z is the distance into the medium (z = 0 denotes the origin of the scattering medium). The matrix N is the infinitesimal generator of J, and satisfies:
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Dyx =
ty ty −tx tx ; ty ty + tx tx
ð10Þ
where the superscript I refers to the interface and ty and tx are the complex transmission coefficients, that, at normal incidence, are: tk =
2n0 ; k = x; y n0 + nk −ikk
ð11Þ
where n0 is the refractive index in the incident medium and nk − ikk is the complex refractive index of the sample for light polarized along the k direction. A subtlety that needs to be noted is that, for an interface diattenuation, the real-valued terms tktk used in Eq. (10) are not directly the intensity transmission coefficients Tx and Ty for light polarized along the x or y, used in other definitions of the global diattenuation [16]. These transmission coefficients are
nk = 1−Rk :ðnormal incidenceÞ n0
where we have defined a complex retardation for each birefringence– dichroism pairs:
Tk = tk tk
χ = η−iκ;
ð5Þ
L = LB−iLD′
ð6Þ
L′ = LB′ −iLD′ ;
ð7Þ
where the factor nk/n0 is due to the change of medium. The reason why this transmittance coefficient, Tk, should not be used here is that the diattenuation occurring at one interface depends only on the amplitude of the transmitted fields as a function of the incident polarization, and the index of refraction is not included in the definitions of the Jones vectors or in the Stokes parameters. In case we were looking at the diattenuation of the reflected light, the reflection coefficients Rk could be used for the calculation of the diattenuation [17]. The retardation for the transmitted beam upon the interface for two orthogonal polarization states, e.g. x and y, is
C = CB−iCD; ð8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 and T = L2 + L′ + C2 . The Mueller–Jones matrix corresponding to the Jones matrix in Eq. (4) can also be analytically calculated as shown in [13,14]. Given the non-depolarizing Mueller–Jones matrix estimate of an experimental normalized Mueller matrix it is also possible to obtain the CD, CB, LD, LD′, LB and LB′ following the procedure described in [15].
Δyx = arg ty −argðtx Þ;
ð13Þ
which, at normal incidence, is 3 ky ðn0 + nx Þ−kx n0 + ny 5; = atan4 ðn0 + nx Þ n0 + ny + kx ky 2
3.1. The effect of the interfaces Δyx The infinitesimal matrix method introduced by Jones does not take into account the effect of the interfaces. When studying light transmission through a certain homogeneous material the most common situation is having two well defined interfaces: one delimiting the
ð12Þ
ð14Þ
demonstrating that for kx = ky = 0 no retardation appears in the interface.
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At normal incidence the effect of the interfaces will be, in general, very small and only will be noticeable for samples with large anisotropy. For example, in a crystal with huge birefringence such as calcite, the transmission Mueller matrices at the interfaces are: 2
3 1 0:0668 0 0 6 0:0668 7 1 0 0 7; MI0 = 6 4 0 0 0:9978 0 5 0 0 0:9978 3 2 0 1 −0:0426 0 0 6 −0:0426 1 0 0 7 7; MI1 = 6 4 0 0 0:9991 0 5 0 0 0 0:9991
ð15Þ
where we have assumed that the optic axis is parallel to the y axis of the laboratory frame (ny = ne = 1.486, nx = no = 1.658). Only when studying samples with large anisotropy does the effect of the interfaces need to be considered in detail. Otherwise the Mueller interface matrices may be taken as the identity matrix without introducing significant errors. In the examples that follow the effect of the interfaces has not been considered. 4. Examples 4.1. Stirred solutions Transmission Mueller matrix ellipsometry has been applied to the study of a surprising phenomenon that has attracted attention in chemistry: the induction and switching of chirality in solutions of certain nanophases under the effect of a stirring vortex. In these experiments the handedness of the induced optical activity depends on direction of stirring in a completely reversible process. This effect has been reported in a number of recent publications but various interpretations are not supported by experiments [18–20]. Our approach to the problem has been directed toward obtaining as much information as possible from polarimetric experiments on stirred solutions. Experiments were performed in ~ 10 mm pathlength cuvettes that were stirred using small magnetic bars. Cuvettes containing Jaggregates of different types of porphyrins have been investigated, ones showing an intense induction and switch of optical activity (usually easily traceable using CD and CB signals) and others showing no appreciable change in the values of these spectra with respect to the stagnant configuration. For example J-aggregates H2TPPS3 porphyrin do not show any change in the optical activity upon stirring [21], while for another similar porphyrin, H2TPPS4, the one used in this work, there are dramatic changes. All these J-aggregates are ordered soft-matter structures which tend to be thin (few nanometers) and long (hundreds of nanometers), and for which flow gradients can sculpture their shapes [22], and, possibly, modify the order patterns on the electronic interactions that sustain the aggregate. Mueller matrix measurements performed in-situ in cuvettes stirred in CW and CCW directions reveal several aspects about the interrelation between the nanosized H2TPPS4 porphyrin aggregates and the macroscopic flows. With spectroscopic measurements [1] we confirmed that the effect was completely traceable from both the CD and CB values obtained from the Mueller matrix and thus, it was clear that a process of induction and switch of optical activity was responsible for the observed effect. Space resolved measurements in square section cuvettes have revealed that the induction of optical activity takes place in the central part of the cuvette (see Fig. 2), in coincidence with the central chiral descending vortex. Fig. 2 shows that at 485 nm the switch of optical activity is clearly visible in the CD maps, because this wavelength corresponds to the high-energy peak of a bisignated CD absorption band. For CB the change of sign can also be noted, but is less appreciable because this wavelength does not
Fig. 2. Scanning of CD and CB inside a 10 mm pathlength cuvette for clockwise (CW), counter-clockwise stirring (CCW) and stagnant (no stirring). CD and CB have been calculated from space-resolved measurements of the Mueller matrix performed in situ at 485 nm, that corresponds to a peak at high energy of the CD bisignated CD band.
correspond to a peak of CB. The cuvette stirring increases the errors of stochastic nature of the measurements, but, interestingly, not all the elements of the Mueller matrix are affected in the same degree. For example, the stochastic errors of elements m03 and m30, mostly contributed by CD, are barely not incremented when stirring, but, in other elements, such as m01 and m10, that mostly depict the linear dichroism, the stochastic errors greatly increase when stirring. This difference is understandable if we attend to the fact that linear dichroism (but not CD) is very sensitive to the size and orientation dispersions of the moving particles of the fluid that traverse the light beam. Fig. 3 shows a vectorial representation of the measured projections of linear dichroism. The length of the vectors, l, and their orientation with respect to the horizontal laboratory axis, θ have been calculated according to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l∝ LD2 + LD′ ; θ=
1 LD′ arctan : 2 LD
ð16Þ ð17Þ
This figure shows how the measured values for linear anisotropic effects can be correlated with the average orientation of the nanophases in the solution. For both CW and CCW stirring there appears to be a small bend of the vortex axis at the top of cuvette, that allows areas with weaker flows (top-left corner for CW stirring and top-right corner for CCW stirring), where the particles can maintain more stable orientation, yielding greater values of LD and LD′.
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4.2. Chiral domains in benzil crystallizations
Fig. 3. Superposition of the CD color map (already presented in Fig. 2) with a vectorial representation of in-plane projection of the linear dichroism, also calculated from the Mueller matrix.
Analogous experiments have been repeated in cylindrical flasks yielding differences in the sign of the induced signals with respect to the case of square section cuvettes. We attribute the differences to the fact that for a cylindrical cuvette an ascending flow is dominant. A more detailed account of these experiments is available in Ref. [23].
One of the most useful applications of Mueller matrix polarimetry is to investigate optical activity in transparent solid samples which, in many cases, show some degree of optical anisotropy. When the structure or composition of the sample varies from point to point, imaging or mapping techniques are the only way to discern the different optical effects that may be present in the sample. Benzil is a molecular crystal; its molecules do not exhibit any optical activity when they are in solution [24], but they crystallize in a chiral enantiomorphous group of symmetry D43 or D63 (the same group of symmetry than α-quartz). The crystal of benzil is built upon a hexagonal lattice and the unit cell accommodates three molecules disposed spirally around the trigonal axis. Therefore the benzil crystal is uniaxial and shows optical activity, which at visible regions of the spectra manifest not only in CB, but also in CD [25] because benzil molecules have an absorption band centered near 400 nm. As it is common for anisotropic crystals, optical activity in benzil can be easily measured for light propagating along the optical axis, but in other directions is more difficult because linear birefringence and linear dichroism effects become much greater. Benzil (1,2-diphenyl-1,2-ethanedione, Aldrich) was recrystallized twice from toluene, as means of purification, yielding six-sided prisms of about 5 mm in length. Later they were manually ground in an agate
Fig. 4. Mapping of the normalized Mueller matrix of a thin polycristaline film of benzil measured at 400 nm. It covers an area of 10×10 mm with an step size of 60 μm.
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mortar to get a fine powder. 2–3 mg of the powder were sandwiched between a blank microscope slide and a glass cover slips. The slide was raised to a temperature above the melting point of benzil (378 K) in a hot stage, held for 15 min, and left to cool at room temperature. A polycrystalline film is eventually developed from the metastable melt. Most of the surface of the sample was scanned by the 2-MGE maintaining a high lateral resolution during the measurement. Imaging representations of the transmission Mueller matrix of the sample in regions around 10 × 10 mm were obtained for several different samples (see one of them in Fig. 4). The pixel size used for generating these mapping plots is 60 μm, which is slightly below the optical resolution of the measurement (85–90 μm). In our case, this resolution is determined by the size of the light spot and the width and height of the monochromator slits placed before the detector. Fig. 5 shows the CD, CB and fraction β of polarized light (1 for completely polarized and 0 for completely depolarized) calculated from the Mueller matrix in Fig. 4. Well delimited zones with CD values of opposite signs are observed. A similar separation is obtained for CB, but, in this case, it is worse defined because the average error of this CB measurement (0.004) is greater than the average error associated to CD (0.001). In Fig. 6 the small area delimited by a black square in Fig. 5 is presented with more detail. In this figure the vectorial representation of
Fig. 6. Superposition of the CD color map (squared area in Fig. 5) with a vectorial representation of in-plane projection of the linear birefringence, also calculated from the Mueller matrix. The vectors of this figure correspond to linear birefringence projections with magnitudes around 0.6 rad.
the projections of linear birefringence, calculated with Eqs. (16) and (17) but now using LB and LB′ instead of LD and LD′, has been superposed to the color map of CD. This figure confirms that the boundaries between areas of positive and negative CD (mostly delimited by strait lines) do not correspond with changes on the orientation of the projected directions of the optic axis. Thus, zones in distinct false color correspond to enantiomorphs otherwise in comparable optical orientations.
5. Summary In this paper, we have shown that transmission Mueller matrix ellipsometry can be applied to study anisotropic samples that exhibit optical activity. From an experimental point of view the 2-MGE operated in transmission mode is an ideal instrument for this purpose because it allows the measurement of the complete Mueller matrix and retains the high sensitivity and wide spectral range characteristics of phase modulated ellipsometry. We have also shown that the same instrument can be used to obtain mappings of the Mueller matrix with a certain spatial resolution if the mechanical rotators of the instruments are substituted by optical rotators. For normal incidence transmission in bulk transparent samples the effect of the interfaces can be in most cases omitted and the differential matrix method pioneered by Jones can be used to interpret the measured matrices. We showed how space resolved Mueller matrix measurements can be applied in solution and in the solid state. We illustrated the induction of optical activity by effect of flows in solutions of some soft matter nanophases. We also detected chiral domains in polycrystalline samples. In summary, this work shows how instruments and techniques developed mainly in the framework of ellipsometry can be directly applied to work on the old problem of measuring optical activity in anisotropic media.
Acknowledgements
Fig. 5. Values of CD and CB obtained from the Mueller matrix in Fig. 4 using the inversion process [15] of the Mueller matrix based on the Jones differential representation of a medium. The black square indicates the area that is studied with more detail in Fig. 6.
We thank Bart Kahr for revising the manuscript. This work was funded by the Spanish Government (AYA2009-13920-C02-01 and AYA2009-13920-C02-02). This work was funded by the Spanish Government (AYA2009-13920-C02-01 and AYA2009-13920-C02-02) and is part of the COST Action CM0703 Systems Chemistry. O.A. acknowledges the financial support from FPU AP2006-00193.
O. Arteaga et al. / Thin Solid Films 519 (2011) 2617–2623
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