Determination of 3 D molecular orientation by concurrent polarization analysis of multiple Raman modes in broadband CARS spectroscopy

A theoretical description is presented about a new analysis method to determine three-dimensional (3D) molecular orientation by concurrently analyzing multiple Raman polarization profiles. Conventional approaches to polarization Raman spectroscopy are based on single peaks, and their 2D-projected polarization profiles are limited in providing 3D orientational information. Our new method analyzes multiple Raman profiles acquired by a single polarization scanning measurement of broadband coherent anti-Stokes Raman scattering (BCARS). Because the analysis uses only dimensionless quantities, such as intensity ratios and phase difference between multiple profiles, the results are not affected by sample concentration and the system response function. We describe how to determine the 3D molecular orientation with the dimensionless observables by using two simplified model cases. In addition, we discuss the effect of orientational broadening on the polarization profiles in the two model cases. We find that in the presence of broadening we can still determine the mean 3D orientation angles and, furthermore, the degree of orientational broadening. Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States. OCIS codes: (300.6230) Spectroscopy, coherent anti-Stokes Raman scattering; (300.6450) Spectroscopy, Raman; (260.5430) Polarization. References and links 1. M. Tanaka and R. J. Young, “Review polarised Raman spectroscopy for the study of molecular orientation distributions in polymers,” J. Mater. Sci. 41(3), 963–991 (2006). 2. S. H. Parekh and K. F. Domke, “Watching orientational ordering at the nanoscale with coherent anti-stokes Raman microscopy,” Chemistry 19(36), 11822–11830 (2013). 3. M. Pigeon, R. E. Prud’homme, and M. Pézolet, “Characterization of molecular orientation in polyethylene by Raman spectroscopy,” Macromolecules 24(20), 5687–5694 (1991). 4. M. Richard-Lacroix and C. Pellerin, “Molecular orientation in electrospun fibers: from mats to single fibers,” Macromolecules 46(24), 9473–9493 (2013). 5. F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105(12), 123903 (2010). 6. B. G. Saar, H.-S. Park, X. S. Xie, and O. D. Lavrentovich, “Three-dimensional imaging of chemical bond orientation in liquid crystals by coherent antiStokes Raman scattering microscopy,” Opt. Express 15(21), 13585–13596 (2007). 7. J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015). 8. E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006). 9. N. Sandeau, L. Le Xuan, D. Chauvat, C. Zhou, J.-F. Roch, and S. Brasselet, “Defocused imaging of second harmonic generation from a single nanocrystal,” Opt. Express 15(24), 16051–16060 (2007). #248369 Received 21 Aug 2015; revised 24 Sep 2015; accepted 18 Oct 2015; published 30 Oct 2015 © 2015 OSA 2 Nov 2015 | Vol. 23, No. 22 | DOI:10.1364/OE.23.029279 | OPTICS EXPRESS 29279 10. E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008). 11. A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007). 12. Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012). 13. R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008). 14. C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014). 15. D. I. Bower, “Investigation of molecular orientation distributions by polarized Raman scattering and polarized fluorescence,” J. Polym. Sci., Polym. Phys. Ed. 10(11), 2135–2153 (1972). 16. M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain KramersKronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012). 17. C. Khatri and K. Mardia, “The von Mises-Fisher matrix distribution in orientation statistics,” J. R. Stat. Soc. B 39(1), 95–106 (1977). 18. G. W. H. Wurpel, H. A. Rinia, and M. Müller, “Imaging orientational order and lipid density in multilamellar vesicles with multiplex CARS microscopy,” J. Microsc. 218(1), 37–45 (2005). 19. G. Turrell, “Analysis of polarization measurements in Raman microspectroscopy,” J. Raman Spectrosc. 15(2), 103–108 (1984).


Introduction
Information on molecular orientation is critical to understanding of the chemical and physical properties of materials and improving their performances for various applications [1,2].Various spectroscopic techniques have been used to investigate the molecular orientation on different length scales.X-ray scattering and NMR can provide inter-and intra-molecular structural information on the atomic scale, but due to ensemble-averaged measurement over a large volume, they are not applicable for spatially heterogeneous materials on the micrometer scale.Polarization fluorescence microscopy can measure the orientation of fluorophores embedded in an oriented matrix with a very high sensitivity and a sub-micrometer spatial resolution.However, it requires either fluorescing groups or exogenous fluorophores and only measures the orientation of the specific fluorophores.On the other hand, Raman spectroscopy is a label-free technique with high chemical sensitivity, and its polarization dependence specific to vibrational modes has been used to characterize the molecular orientation [1,3].Additionally microscopy approaches based on polarization Raman have characterized spatially-resolved molecular orientations of heterogeneous materials [4][5][6][7].However, conventional polarization Raman spectroscopy measures only the azimuthal projection of a 3D orientated Raman mode to the 2D polarization plane and so it misses the axial component of the 3D orientation.One can know the 3D orientation information only when the exact number density of a specific Raman mode is known at the interrogated position and its corresponding Raman signal intensity is calibrated for all 3D orientation angles.Both prerequisites are not trivial to acquire, in particular, from spatially heterogeneous samples.
Recently, new methods have been demonstrated for 3D orientation measurement of various material systems.They analyze a series of slightly defocused images and reconstruct the 3D orientated image of molecules or particles of interest.These methods were applied to various samples by various spectroscopic techniques, including single molecule spectroscopy of fluorophores [8], second harmonic generation of single nanocrystals [9], confocal Raman [10] and coherent anti-Stokes Raman scattering (CARS) of liquid crystals [11].However, these methods are only applicable for isolated molecules or defects that can be defocused, and not applicable for measuring molecular orientations of continuous samples.
In this paper, we propose a new approach to determine the 3D molecular orientation by concurrently analyzing multiple polarization Raman profiles.Our analysis is based on the broadband CARS (BCARS) microscopy technique [12], which can acquire polarization profiles of multiple Raman bands simultaneously.The coherent, multiphoton signal generation also allows rapid acquisition sufficient for hyperspectral imaging with the diffraction-limited spatial resolution.Here, we present a theoretical description about how to find a unique solution for the 3D molecular orientation by using two simplified model cases.We use only dimensionless, or system-independent, quantities to determine 3D molecular orientational angles.After we discuss the effect of orientational broadening on polarization profiles, we demonstrate that we can still determine the mean orientation angles and moreover the degree of broadening.

Theory
The third-order nonlinear polarization for CARS is expressed as (3)   (3) pu S* pr = P χ E E E , where χ (3)  is the nonlinear susceptibility tensor, and E pu , E S , and E pr are the electric fields for the pump, Stokes, and probe lights, respectively.χ (3) , defined in the laboratory frame, is the sum of individual molecular polarizability tensors, α (3) .Although detailed description of the connection between the nonlinear susceptibility tensor and molecular polarizability tensors can be found in previous reports [13,14], a more generalized relation between χ (3) and α (3) can be expressed as (3) (3) ( , , ) where N is the number density; and θ, ψ, and φ are the polar, azimuthal, and rotational angles, respectively [15], which are illustrated in Fig. 1.The orientation tensor, R(θ,ψ,φ), can be expressed as cos sin 0 cos 0 sin cos sin 0 ( , , ) sin cos 0 0 1 0 sin cos 0 Fig. 1.Schematic diagram of rotation of a nonlinear molecular polarizability tensor, α (3) , from the molecule frame (xyz) into the laboratory frame (XYZ), where it becomes a nonlinear susceptibility χ (3) .The rotation is described with R(θ,ψ,φ), where θ, ψ, and φ are Euler angles.
The net polarizability tensor, α (3) , is represented as the sum of two linear tensor ellipsoids with resonant frequencies at ω 1 and ω 2 .The Z-axis in the laboratory frame is set to be parallel to the light propagation direction.E ˆ( ) η e denotes the polarization vector of incident lights with the angle η from the X-axis.
Figure 1 shows an orientation scheme of a molecule whose two orthogonal Raman modes are resonant at different frequencies, ω 1 and ω 2 .For simplicity, we assume that their corresponding molecular polarizability tensors, (3)   ω α , can be represented by a single non-zero tensor component.Then, a tensor can be expressed as a product of a scalar amplitude, (3)   ω α , and a directional unit vector, ˆω e , in the molecule frame.Polarizability tensors of localized stretching vibrations (e. g., -C≡N, -N=N-) or skeletal stretching vibrations in linear alkanes or polyethylene (PE), can be considered as their examples.For example, the Raman peak at 1130 cm −1 observed from crystalline PE represents the C-C symmetric stretching mode, which is parallel to the main chain orientation [3,12].The PE Raman peak at 2850 cm −1 represents the CH 2 symmetric stretching, which is perpendicular to the main chain orientation.The net molecular polarizability tensor consisting of those two Raman modes can be expressed as where ω 1 and ω 2 represent the primary and secondary modes, respectively.Similarly, the net nonlinear susceptibility of a molecule can be expressed as the sum of two corresponding nonlinear susceptibilities in the laboratory frame, as shown in Fig. 1.
ˆ( , , ) ( , , ) where (3)   (3) If we assume parallel linear polarization for all incident beams, the thirdorder nonlinear polarization, (3)   ω P , generated from the Raman polarizability axis in the laboratory frame will be expressed as where E ˆ( ) (cos ,sin ,0) is the polarization vector of the incident electric fields; and η denotes the polarization angle from the X-axis.The term in {} represents the amplitude of (3) ω P , and ( , , ) ω θ ψ φ R e represents the direction of the ω mode in the laboratory frame.When a polarizer analyzer is located in the signal collection path, the observed CARS intensity can be expressed as where C represents the response function of the detection system; and ξ is the polarization angle of the analyzing polarizer from the X-axis.Controlling the analyzer polarization could provide complementary information on molecular orientation with more parameters in mathematical descriptions.In this paper, however, we will consider only the mathematically simplest configuration, where both incident and analyzer polarization angles are kept parallel (ξ = η) while the polarization angles are rotated.Back to nonlinear susceptibilities, in fact, the observed CARS signal is contributed by both resonant and nonresonant susceptibilities, , respectively.The nonresonant component is frequency insensitive and can be computationally removed by the time-domain Kramers-Kronig (KK) method or the maximum entropy (ME) method, which yields the summation  , equivalent to the spontaneous Raman spectrum [16].However, these methods, developed to retrieve a spectrally accurate Raman spectrum, do not describe the effect of molecular orientation in non-isotropic samples.In the KK (and ME) method, all the quantities are in the form of scalars as where (3)   eff,ω χ is the effective nonlinear susceptibility that results from the retrieval process.
Such determined (3)   eff,ω χ is expressed with molecular orientation from Eqs. ( 4)-( 6), as follows ( ) While (3)   ω χ is a material property, the quadratic term describes the light polarization angle dependence of (3)   eff,ω χ as a result of the three-photon interaction of the all parallel incident electric fields and the analyzing polarizer.From Eq. ( 7), we can analytically express (3)   eff,ω χ of each Raman mode as a function of η for a given molecular orientation (θ, ψ, and φ).
In the following sections, we demonstrate how to determine the 3D molecular orientation in two types of molecular geometries that consist of two orthogonal Raman modes.We will also discuss the orientational broadening effect on the polarization profiles and their analysis results.We begin our discussion with a configuration, where a collection of parallel oriented molecules with two orthogonal Raman modes.Examples of such collection of paralleloriented molecules will include most of molecular single crystals, biaxial nematic liquid crystals, and crystalline (orthorhombic) polyethylene.For simplicity, we assume that the primary mode (ω 1 ) is parallel to the z axis and the secondary mode (ω 2 ) is parallel to the x axis.Then,

(3)
(3) eff , sin cos( ) (3) eff , cos cos cos( ) sin sin( ) (3),max Both Δη and r χ can be expressed analytically as functions of θ and φ, because (3) Eqs. ( 8) and ( 9). 1 ( , ) tan (tan sec ), (3) sin ( , ) cos cos cos( ) sin sin( ) The ratio, r χ , contains the Raman polarizability ratio of the two modes, ω ω χ χ , which can be easily measured from a BCARS spectrum of a randomly oriented region or from an ensemble-averaged BCARS spectrum over a large area that covers all molecular orientations without modifying the measurement configuration.If separate experimental measurement is not possible at all, alternatively quantum calculation can be used to estimate the Raman polarizability ratio of the two modes.From Eqs. ( 12) and ( 13), the contour plots of Δη and r χ can be calculated as functions of θ and φ, as shown in Figs.To demonstrate how this method can determine all the 3D orientation angles, we generated polarization profiles of the two examples with pre-determined 3D orientation angles, as shown in Figs.4(a  orientation angles.By assuming uncertainties of 2 ° in relative angle measurement and 5% in intensity ratio measurement, the uncertainties are estimated as ± 2 ° for ψ, ± 1 ° for θ, and ± 2 ° for φ for the example A; and ± 2 ° for ψ, ± 0.5 ° for θ, and ± 2 ° for φ for the example B.

Case II: a parallel primary mode and a uniformly distributed secondary mode
The next case we consider is a collection of molecules whose primary modes are all parallel but whose secondary modes are randomly distributed around the primary mode axis.Figure 5(a) presents a schematic example of this uniaxial orientational distribution, which includes uniaxial nematic liquid crystals, and pulled semi-crystalline high-density polyethylene.In this case, only two orientational angles, θ and ψ, are to be determined.The polarization profile of 1 (3) eff ,ω χ is identical to that of Case I, as in Eq. ( 8).However, the polarization profile of 2 (3) eff ,ω χ is different because Eq. ( 7) needs to be averaged over φ, as follows: When the 2 ˆω e is perpendicular to 1 ˆω e as in Eq. ( 9), Eq. ( 14) can be simplified as     Alternatively, we can use r χ , defined in Eq. ( 11), to determine θ because r χ can be expressed as an analytical function of θ from Eqs. ( 8) and ( 15), ( ) It is noted that the reliable measurement range for θ with Eq. ( 17) is between 30 ° and 80 °.The difference in the reliable θ range between by m χ and r χ makes the two methods complementary, and furthermore, these two methods can be used to confirm that the molecular orientation of the interrogated sample corresponds to the Case II.   (in the molecule frame, followed by rotation into χ (3) in the laboratory frame.Orientational broadening or disorder can arise for various thermodynamic and kinetic origins.Orientational broadening in the interrogated volume will directly affect the measured polarization profiles of BCARS and can alter the determined orientation angles.We investigate the broadening effect on the measured polarization profiles of BCARS spectra and find methods to determine the mean orientation angles and furthermore to quantify the degree of broadening.To characterize the broadening effect, we use a simple form of a uniaxial broadening model, where the angular probability density function, ρ(β), is a sole function of the axial (or polar) angle, β, from the mean orientation and is not affected by the other (azimuthal and rotational) angles.The direction unit vector of a Raman mode is multiplied by the broadening tensor, R(β,γ,0), in the molecule frame; then, it is transformed into the laboratory frame by the orientation tensor, R(θ,ψ,φ), as represented in Fig. 7(a).For the ω 1 mode, 1 ˆω e in Eq. ( 7) is replaced with

Case III: Case I with broadening
β γ ρ β β γ ′ = e R e , followed by integration over β and γ.The effective nonlinear susceptibility in Eq. ( 7) becomes ( ) For a uniaxial probability distribution function, we use the von Mises−Fisher distribution, which is also known as the spherical normal distribution and widely used in directional statistics [17].For example, the orientation distribution function of electric dipoles in a parallel electric field can be expressed by the von Mises-Fisher distribution.The von Mises−Fisher distribution function with the mean orientation parallel to the z axis is given by  where κ is the concentration parameter.High κ implies a narrow angular distribution, and κ = 0 means random or isotropic distribution.Similarly, Wurpel et al. used a Gaussian distribution function for analysis of lipid chain orientation measured by polarization CARS [18], where a normal distribution on a circle is used instead of one on a sphere.values are zero, with broadening (Case III), the minimum values rise from zero.So we examine if the minimum values can be useful to calculate the mean orientation in the presence of orientational broadening.For the ω 2 mode, the ratio of the minimum to the maximum was already defined as m χ by Eq. ( 16).For the ω 1 mode, similarly, we define a ratio theoretical description can be easily modified and expanded for other types of molecular geometries, broadening functions, and polarization configurations.For example, if the analyzer polarization is rotated to the perpendicular angle to the incident beam, Eq. ( 7) will become ( ) The rest of analysis can be performed in the same way described in this paper, providing a complementary result to the parallel analyzer measurements.Similarly, various complementary measurements can be performed in other polarization configuration without significant modification in experimental schemes.We can also consider an additional nonparallel polarizability tensor in Eq. ( 3), and its analysis can be useful as another complementary measurement for determining the orientational angles or the broadening parameters.Independently obtained knowledge of not only individual Raman tensors (e.g., symmetry, cross section, direction,) of molecules but also structure of molecular assembly can make this analysis method more reliable.Moreover, we note that this concurrent analysis of multiple vibrational modes can be applied to other coherent Raman spectroscopies (e.g.stimulated Raman scattering) as well as linear vibrational spectroscopies (e.g.spontaneous Raman and FTIR).
In actual experiments, several practical factors need to be considered.For example, polarization mixing due to tight focusing [19] can reduce the undulation amplitude of a measured polarization profile; and polarization dependency of the detection system (e.g., grating in a spectrometer) can also distort the relative Raman intensities.However, most of distortion caused by these factors can be corrected by numerical simulation or by separate control experiments with an isotropic sample.In this paper, we have used only "vector-like" polarizabilities for analysis.However, "tensor" polarizabilities can also be used but their analysis will be more complicated including multiple tensor components, whose number depends on the symmetry group and the possible index permutations.
We note that an orientation distribution function can also be described with the molecular orientation distribution coefficients, P lmn , in the generalized Legendre polynomials [1,15].P lmn are widely used in conventional polarization spectroscopy, in particular, for specimens with an axial orientation symmetry.We are currently developing a way to directly determine P lmn from this concurrent analysis of multiple Raman polarization profiles without assuming a model distribution function.

Conclusion
We have presented a theoretical description about the new analysis method for determination of 3D molecular orientation by concurrently analyzing polarization profiles of multiple Raman modes measured by broadband coherent anti-Stokes Raman scattering (BCARS) spectroscopy.We have used only dimensionless quantities obtained from multiple polarization Raman profiles measured by a single polarization scanning.The mathematical relations between the dimensionless observables and orientational angles are used to determine 3D molecular orientation for the two simplified model Cases.We have found that in the presence of broadening, this analysis method can measure not only the mean orientation angles but also the degree of orientational broadening.This polarization analysis method can become more powerful when coupled with high throughput 3D imaging techniques, such as BCARS microscopy.

Fig. 2 .
Fig. 2. (a) Schematic presentation of nonlinear susceptibilities of the primary mode, 1 (3) ω χ , in the spherical coordinate of the laboratory frame.(b) Plots of the polarization profiles of effective nonlinear susceptibilities,

Figure 2
Figure 2(b) shows plots of

Fig. 3 .
Fig. 3. (a) Polarization profiles of 3(b) and 3(c), respectively.The contour plots of Δη and r χ are used to determine the remaining two orientation angles (θ and φ).
) and 4(d).First, we can easily determine the azimuthal angle, ψ, , we measure Δη and r χ from the 1 .(10) and(11).Then, we calculate a line corresponding to the measured Δη as a function of θ and φ with Eq. (12) in the contour plot of Fig.4(b).Likewise, we calculate another line corresponding to the measured r χ with Eq. (13) in the same contour plot of Fig.4(b).The crossing point of the two contour lines in Figs.4(b) and 4(c) yields the "unique" solution of the θ and φ angles.Figures4(d)-4(f) shows another example of determining all the 3D orientation angles of the molecule in the laboratory frame.It is worth reiterating that this measurement is based on dimensionless or relative observables from a single polarization scanning of BCARS measurement.

Fig. 5 .
Fig. 5. (a) Schematic presentation of Case II, where the secondary mode is uniformly distributed over φ about the primary mode and the secondary mode is kept to be perpendicular to the primary mode.Polarization profiles of

Figures 2 (
Figures 5(b) and 5(c) show polarization profiles of The m χ value can be visualized as the minimum value when the (constant) maximum value corresponds to one as shown in Figs.6(a) and 6(b).The m χ value can be used to determine θ K-N with four different θ.In Fig.6(b), the dimensionless m χ values are used to determine θ values without knowledge(15).It is noted that we can determine θ reliably for the range between 10 ° and 60 ° from the curve (e.g., Examples K-M); however, when θ approaches 90 ° and the #248369 corresponding m χ value decreases close to zero (Example N), such determined θ value becomes unreliable.

Fig. 7 .
Fig. 7. (a) Schematic representation of a uniaxial orientational broadening of α(3) in the molecule frame, followed by rotation into χ(3) in the laboratory frame.(b) and (c) Plots and graphical presentation of the von Mises−Fisher probability distribution function ρ(β) for two different concentration parameters, κ.
Fig. 7. (a) Schematic representation of a uniaxial orientational broadening of α(3) in the molecule frame, followed by rotation into χ(3) in the laboratory frame.(b) and (c) Plots and graphical presentation of the von Mises−Fisher probability distribution function ρ(β) for two different concentration parameters, κ.

Fig. 8 . 1 (
Fig. 8. Broadening effect on the two examples A (Left column) and B (right column), which were presented for Case I in Fig. 4. (a) Polarization profiles of

η 1 (
Figures 7(b) and 7(c) show a few examples of the von Mises−Fisher distribution functions for different κ values.Using the von Mises-Fisher function, we computed the broadening effect on the polarization profiles of the examples used in Case I. Figures 8(a) and 8(b) show comparisonof the polarization profiles with broadening (κ = 10) and without broadening (κ = ∞).First, the phase of the polarization profiles of 1 are unchanged, and so is Δη.In contrast, the minimum values and the maximum are affected apparently, which means that r χ changes with κ.For comparison with r χ in the absence of broadening, we call r χ ' measured from of broadening.The κ dependence of apparent r χ ' is also very different for different orientations.As in Case I, we use Δη and r χ ' to determine θ and φ and to compare such determined angles with the mean angles used for the examples.The contour line corresponding to apparent r χ ' is plotted along with Δη, and the orientational angles θ and φ are determined from the crossing points.When apparent r χ ' does not vary much with κ, as in Fig.8(b) for the example A, the broadening effect on the apparent values of θ and φ are not significantly different from the true mean θ and φ, as shown in Fig.8(c).In contrast, when r χ ' varies rapidly with κ, as in Fig.8(e) for the example B, the corresponding contour line shifts noticeably, and such determined values of θ and φ also deviate from the mean values farther with greater broadening (smaller κ), as shown in Fig.8(f).This shows that in the presence of orientational broadening, the method used for Case I cannot determine the mean orientation angles θ and φ, reliably.

Fig. 9 ..
Fig. 9. (a) A contour plot of n χ as a function of θ and κ for the ω 1 mode of Case III.(b)-(d) Contour plots of m χ for the ω 2 mode of Case III for various φ.As mentioned above, orientation broadening affects not only the maximum but the minimum values of 1(3) eff ,ω χ