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Keywords:

  • optical rotation;
  • circular dichroism;
  • linear birefringence;
  • linear dichroism;
  • spherulite;
  • sorbitol;
  • Mueller matrix

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MUELLER MATRIX MICROSCOPY
  5. RESULTS
  6. DISCUSSION
  7. EXPERIMENTAL
  8. LITERATURE CITED

It is well-known that circular dichroism (CD) measurements of anisotropic media may contain artifacts that result from mixed linear anisotropies. Such artifacts are generally considered a nuisance. However, systematic artifacts, carefully measured, may contain valuable information. Herein, polycrystalline spherulites of D-sorbitol grown from the melt were analyzed with a Mueller matrix microscope, among other differential polarization images devices. As spherulites grew into one another they developed strong apparent optical rotation and CD signals at the boundaries between spherulites. These signals are shown not to have a chiroptical origin but rather resultfrom the interactions of linear anisotropies in polycrystalline bodies. Such chiroptical artifacts should not be dismissed reflexively. Rather, they are comprehensible crystal-optical effects that serve to define mesoscale structure. Chirality 21:E20–E27, 2009. © 2009 Wiley-Liss, Inc.


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MUELLER MATRIX MICROSCOPY
  5. RESULTS
  6. DISCUSSION
  7. EXPERIMENTAL
  8. LITERATURE CITED

For generations, measurements of chiroptical phenomena [circular birefringence (CB) and circular dichroism (CD)] have foundered on much larger linear anisotropies that otherwise disappear in isotropic solutions.1 For instance, in 1982, Maestre and Katz adapted a Cary spectropolarimeter to a microscope2 for single point measurements of the CD spectra of chromatin. They encountered instrumental artifacts3, 4 arising from electronic polarization modulators in commercial instruments that typically generate sinusoidally varying polarization states,5 thereby introducing a small admixture of linearly polarized light into the circularly polarized output. Residual ellipticity, when coupled with the linear birefringence (LB) and linear dichroism (LD) of ordered media, generated artifactual CD signals.6, 7 Strain in photoelastic modulators (PEMs) compounds these artifacts.8 Attempts have been made to skirt these problems by adding additional compensatory modulators,9 rotating the sample,10, 11 or carefully selecting optical components with the smallest polarization biases.12

In some cases, irrespective of the ideality of the optical components in the systems, CB and CD artifacts will persist due to heterogeneities in the sample encountered along the light propagation direction. One such sample, analyzed herein, is composed by overlapping polycrystalline spherulites of D-sorbitol. We recently used such samples to reckon a confusion in the literature regarding whether absorbance (A) or transmittance (T) in thin films shows a cos2θ angular dependence with respect to incident linearly polarized light.13 An ideal sample for making this distinction is a thin section of a sphere in which chromophores are arranged uniaxially with respect to the radii. D-sorbitol spherulites14 doped with the azo dye amaranth (acid red 27, C.I. 16185) met this criterion. In such media, every dye orientation can be evaluated in normal incidence eliminating the need for moving samples, rotating optical components, and intensity corrections. To further characterize polycrystalline D-sorbitol spherulites as well as other polycrystalline patterns, we turned to Mueller matrix microscopy [(MMM) the same acronym is here adopted for Mueller matrix microscope].15

MMM is a form of differential polarization imaging,16 the evaluation of transmitted intensities made with variable polarization state generators (PSGs) and polarization state analyzers (PSAs). We have pursued various aspects of differential polarization imaging during the past 8 years.17 Differential polarization imaging is naturally suited to problems of polycrystalline pattern formation, and MMM is particular instructive in this regard because it permits the simultaneous analysis of linear and circular anisotropies.

Individual doped D-sorbitol spherulites have vanishingly small CB and CD. However, when sorbitol spherulites grow into one another, substantial apparent CB and CD signals develop between adjacent spherulites. The analysis of these signals using MMM is developed herein. To date, MMM has been applied to problems in biomedical optics,18–25 turbid media,26–32 liquid crystal display components,33–38 and even to the analysis of a magnetic fluid,39 but there are no applications in crystallography to the best of our knowledge, a yawning gap and a terrific opportunity.

The Stokes-Mueller calculus that underlies the operation of the MMM, and that will be discussed in the following section, should be most famililar to readers of Chirality through its extensive coverage of vibrational circular dichroism (VCD), a technique that has revolutionized the assignment of absolute configuration.40, 41 The Stokes-Mueller calculus was used to account for VCD artifacts in solid samples.42, 43 However, while in previous work, artifacts were obviated or disposed of, herein we choose to exploit them.

MUELLER MATRIX MICROSCOPY

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MUELLER MATRIX MICROSCOPY
  5. RESULTS
  6. DISCUSSION
  7. EXPERIMENTAL
  8. LITERATURE CITED

Mueller matrix microscopy (MMM) analyzes the polarization properties of a sample in terms of the transformation of an input Stokes vector (Sin, eq. 1) via the Mueller matrix (M): Sout = MSinxliv.44 Here, Ex and Ey are orthogonal electric field components. ℜ is the real part and ℑ is the imaginary part. Sout is the output Stokes vector. A MMM is a crossed polarizer microscope with two rotating quarter wave plates added above and below the sample that act as a complete PSG/PSA pair. By collecting

  • equation image(1)
  • equation image(2)

images as a function of rotations of the PSA (θ) and PSG, (θ′), eq. 2 can be applied to the recorded intensities (I) to solve the M of a sample at each pixel through pseudo-inversion.45 A MMM generates 16 images, each representing one of the elements of the 4 × 4 Mueller matrix (Fig. 3), expressed in terms of measurable input and output polarization state intensities.16 Our instrument is based on designs found elsewhere.46–48

Unfortunately, the 16 images are not simply related to the fundamental optical constants of interest: absorbance (A), LB, LD, CB, and CD.6 The contributions of the fundamental optical quantities to each matrix element was first derived by Jensen et al.49 They showed that a given matrix element, for instance M03 (in the top row and right most column), contains the convolution of contributions from CD, LB, and LD, as well as LB′ and LD′, the LB and LD along the directions ±45° with respect to an arbitrary fixed sample coordinate system that defines LB and LD. Other matrix elements are likewise complex. Separating linear and circular anisotropies from one another is often our aim and it is a matter of ongoing research in other laboratories.50–52

To achieve this separation, we have implemented the differential analysis pioneered by Azzam:53

  • equation image
  • thumbnail image
  • equation image(3)
  • equation image(4)

Here, z is the distance of propagation and m is the differential Mueller matrix describing the transformation of the Stokes vector for some infinitesimally small path Δz. Equation 3 can be integrated to give the following:

  • equation image(5)

The matrix m can be obtained numerically54 by a approximating the logarithm of matrices. The process of deriving the differential matrix m unfolds the convolution of linear and circular anisotropies so that each fundamental quantity has a unique place in the matrix as in eq. 4. This decomposition55 of the resulting Mueller matrices in Figure 3 subverts ambiguities56 resulting from decomposition methods that parse M into noncommuting components.57, 51

RESULTS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MUELLER MATRIX MICROSCOPY
  5. RESULTS
  6. DISCUSSION
  7. EXPERIMENTAL
  8. LITERATURE CITED

Images representing the “raw” Mueller matrices (Ms) for each pixel of sorbitol and amaranth-doped sorbitol spherulites are shown in Figure 3. The differential matrix elements m12 and m03 are shown in Figure 4 after data reduction. These matrix elements correspond, in principle, to CB and CD. D-sorbitol is optically active and we would expect CB, although the contribution from its crystals is likely to be too small for the sensitivity of our instrument. In the case of dyed sorbitol, it is possible that we would also observe an induced CD, however, likewise, we expect that effect to be too small to measure in the present configuration. As can be seen in Figure 4, the bulk of the material shows little natural optical activity. Analysis of the transmittance reveals that the regions that display the apparent chiroptical signatures correspond to the boundaries between adjacent spherulites. When m12 and m03 are extracted from the raw data, we discover a web-like pattern of positive and negative “CB” and “CD” (Fig. 4A, 4B, and 4D). The Mueller matrix decomposition is, strictly speaking, only applicable to homogeneous anisotropic materials. Given the fact that anisotropic fibrils cross at the boundaries, we sought an alternative interpretation of these signals.

To confirm that our MMM and data analysis were working properly (in addition to measuring Ms for standard optical components, see “Experimental” section), we analyzed the samples with other differential polarization imaging devices.17 Figures 5A and 5B show images of the LB and “CD” made with a LB imaging system and a circular extinction (CE) imaging microscope, respectively. In the former case, we use a rotating linear polarizer as the PSG and a fixed circular analyzer as the PSA.58 In the latter case, we use linear polarizer that oscillates between two positions only and a fixed quarter wave plate (to generate right and left circularly polarized light) as the PSG.59 There is no PSA. The computation of the resulting output is comparatively simple in these cases.17

Figure 5A shows the |sin δ| micrograph where δ = 2πΔnLλ−1n is the LB, L is the thickness, and λ is the wavelength of light). Since the retardance δ is the first-order, the micrograph corresponds to sin δ which is qualitatively similar to Figure 1B where the LB is expressed in terms of (LB2+ LB′2)1/2. The circular eyes reveal spherulite nuclei where the fibrils are growing in the direction of the wave vector. These uniaxial ensembles show no birefringence at their centers. A close-up of adjacent nuclei in Figure 5B, overlaid with radial grids, show that where fibrils cross at ±90°, there are additional nulls where the retardance of adjacent spherulites cancels out one another due to orthogonal contributions. The corresponding CE micrographs for the same areas are shown in Figures 5C and 5D. The progression of the CE signal, measured directly as the differential transmission of left and right circularly polarized light ((IRIL)/Io), is only evident at the border between spherulites. Thus, the apparent chiroptical signals are a consequence of the overlap of linearly anisotropic elements. The sign of the signal changes every 90° with angular overlap of the elements, and is null at 0° and ±90°.2

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Figure 1. (A) D-sorbitol spherulite grown from the melt shown under cross polarizers. (B) The absolute retardance (δ = 2πΔnLλ−1) is in radians. 532 nm illumination.

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Figure 2. Schematic Mueller matrix microscope.

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Figure 3. Mueller matrix element images of D-sorbitol spherulites. 532 nm illumination (A) Dye doped with 1% by weight amaranth (acid red 27, C.I. No. 16185). (B) Un-dyed D-sorbitol spherulites. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Figure 4. (A,C) m03 for dyed and pure D-sorbitol spherulites, respectively. (B,D) m12 for dyed and pure D-sorbitol spherulites, respectively. If the sample were homogeneous, A and C would correspond to CB, and B and D would correspond to CD. 532 nm illumination. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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DISCUSSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MUELLER MATRIX MICROSCOPY
  5. RESULTS
  6. DISCUSSION
  7. EXPERIMENTAL
  8. LITERATURE CITED

From earlier studies we know that D-sorbitol spherulites grow as fibrils that are averaged to give an optical uniaxial body where the radii are effective optical axes.13 As the CB and CD of dyed sorbitol fibrils are expected to be small compared with their LB and LD, the infinitesimal Mueller matrices for each fibril can be approximated as:

  • equation image(6)

In any one spherulite, the values of LB and LD vary as does the radial growth angle with respect to the wave vector of the light. Because the vibration directions of the fibrils are similarly projected onto the plane perpendicular to the wave vector, the differential ms commute. M can be expressed as the sum below where m1, m2, m3….represent successive infinitesimal matrices sampled by the light in normal incidence:

  • equation image(7)

The LB at any point in a spherulite is defined by the distance of propagation z in the image plane, the height of nucleation h, the birefringence of the spherulite Δn, and the distance to the nucleus, d.

  • equation image(8)

LB can then be expressed in integral form from z = 0 to z = 1 at the top of the sample where Δn is the maximum birefringence that is evident when the fibrils are perpendicular to the wave vector of the light.

  • equation image(9)
  • equation image
  • equation image

B(z,h,d) is the value of the integrand at any point z. Equations 8 and 9 show that the measured LB at a point in the image plane is a function of the nuclei's distance from the center of the sample and the distance to that nucleus. To simulate the apparent “CB” and “CD” in doped sorbitol, the micrographs were divided into discrete spherulites, numbers 1–6 in Figure 7. To define each spherulite basin, micrographs were inspected to exclude the overlapping regions. Then, for each spherulite, every point within a basin was assigned a distance from its nucleus in pixels, and the LB was averaged for all positions of a given radial distance. Linear regression was used to find the maximum LB and LD, as well as z and h.

The results from the fitting are summarized in Figure 7B however, the absolute location of the nucleus can not be found in transmission since both positive and negative sloping fibrils evidence the same optical properties.

The “CB” and “CD” data contain this information. As shown in Figure 5, the borders between spherulites show strong signals for CB and CD with a characteristic sign progression: [+,−,+,−] or [−,+,−,+]. The absolute alternation of the sign is dictated by which spherulite lies above or below the other within the sample. Figure 8 shows a simulation of overlapping spherulites above and below one another. These images were generated from the A, LB, and LD values derived from the fit shown in Figure 7b. These values were used to calculate differential Mueller matrices for each spherulite that were then overlapped via matrix multiplication. Finally, the resulting Mueller matrix images were decomposed using the exponential form to derive the “CD” and “CB” signals shown here.6

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Figure 5. (A,B) Images of |sin δ| for 1 wt % amaranth doped D-sorbitol spherulites made with linear birefringence imaging system. δ = 2πΔnλ−1. (C,D) Corresponding images of CE ((IR-IL)/Io). Nuclei in C and D are overlaid with radial grids that indicate directions of fibril growth. The signals are null where grid lines are parallel or perpendicular. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Figure 6. Schematic of spherulite growing between two glasses at z1 and z0. Nucleus is at height h while d is the distance from the nucleus in the image plane.

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Figure 7. The average retardance versus distance was calculated for the six spherulites shown above. The birefringence was then matched by eq. 8 using nonlinear least squares regression to solve for the nucleation height and innate birefringence of the sample. The experimental versus theoretical birefringence for three sorbitol spherulites is shown on the right. The distance of nucleation from center of the sample was calculated to be 0.9, 0.16, and 0.18 mm. Sample thickness = 0.34 mm. Normalized retardance was found to be 0.013 rad/mm with differential absorption of 0.0013 /mm. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Figure 8. Overlapping spherulites result in strong “CD” and “CB” signals. (A) The “CD” signal generated by dyed (top) and undyed (bottom) overlapping spherulites. (B) The “CB” signal resulting from dyed or undyed overlapping spherulites. These signals are artifacts of the form: (LB × LD′−LB′ × LD), and (LB × LB′ + LD × LD′). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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It is useful in the context of the Mueller calculus to define, in addition to the standard crystal optical quantities, A, LD, LB, LD′, LB′, CD, CB, whose relation to experiment is given in Ref.6, the additional birfreingence-dichroism pairs:

  • equation image
  • equation image(10)
  • equation image

In computations and experiments, the components of the generalized pairs (eq. 10) commute with one another. This makes the calculation and measurement of systems containing only one such pair straightforward (eg. CB and CD from an isotropic medium in which L and L′ are zero by definition). The derivation of the infinitesimal Mueller matrix m allows us to subvert the great complexity of the general Mueller matrix M that arises from the simultaneous measurement of L, L′, and C.60 However, this only applies for homogeneous media. When L, L′, and C are combined from heterogeneous components, artifacts persist. The elements m12 and m03 that would correspond to CB and CD in a homogeneous system, contain residues of the following form, respectively:

  • equation image(11)
  • equation image(12)

The implication of these terms in the elements of m is that we will generate artifactual CB and CD signals whenever we have imperfect optical components in our optical train (e.g. a circular polarizer that produces slightly elliptically polarized light. See Ref.59 and discussion therein) or whenever we have crossed, linearly anisotropic elements in an inhomogeneous sample. The artifactual signals disappear when fibrils cross at ±90° and 0°. In the latter case, the absorption and refraction ellipsoids of the two sets of fibrils are aligned. Thus, LB′ and LD′ are effectively zero. At ±90°, the signal changes sign as the overlap angle moves from acute to obtuse; by necessity it must go through zero.

Equations 11 and 12 computed with linear anisotropies derived from the nonoverlapping regions of spherulites agrees to within < |10%| with “CB” and “CD” in the overlapping regions. In summary, we can conclude that the micrographs in Figures 5C and 5D represent in largest measure the quantities in eqs. 10 and 11, respectively. These signals plainly indicate the absolute orientation of anisotropic fibrils with respect to one another, information about mesoscale stereochemistry, the relative heights of sorbitol spherulites. While it will not offend us to discover that the relative heights of polycrystalline spherulites may not be actively sought by legions of investigators, it doesn't take a leap of imagination to recognize that such clear-cut artifacts can be used to establish the texture of liquid crystals, the relative positions of closely packed cells, or to unravel complex pathological structures comprised of overlapping fibrils.61 The exploitation of such signals in biomedical optical applications will be the focus of future research.

EXPERIMENTAL

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MUELLER MATRIX MICROSCOPY
  5. RESULTS
  6. DISCUSSION
  7. EXPERIMENTAL
  8. LITERATURE CITED

The growth of dye-doped sorbitol spherulites can be found in Ref.1.

The Mueller matrix microscope was constructed of a 532 nm Nd:YAG laser, a rotating optical diffuser (glue stick), two linear polarizers (Thorlabs), two achromatic quarter wave plates (Thorlabs) mounted in rotation stages, and a Canon Rebel XTi camera. Images were collected without a sample in the optical train using 10° steps of the retarders. The data inversion algorithm was adopted from Ahmad and Takakura.62 The resulting intensity map was used to find the offset of the top polarizer from crossed position (0.408°) and retardance (88.49° and 89.12°) of the wave plates with linear regression. With standard optical components, a linear polarizer and quarter wave plate, the averaged Mueller matrix images had values within 1% of the expected across the whole image field.

LITERATURE CITED

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MUELLER MATRIX MICROSCOPY
  5. RESULTS
  6. DISCUSSION
  7. EXPERIMENTAL
  8. LITERATURE CITED