1. 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 [4–7]. 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.

2. Theory

The third-order nonlinear polarization for CARS is expressed as $P^{(3)}={\chi }^{(3)}{E}^{\text{pu}}{E}^{\text{S*}}{E}^{\text{pr}}$, 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

 

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 ω₁ and ω₂. The Z-axis in the laboratory frame is set to be parallel to the light propagation direction. ${\widehat{e}}_{\text{E}}(\eta )$ denotes the polarization vector of incident lights with the angle η from the X-axis.

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Figure 1 shows an orientation scheme of a molecule whose two orthogonal Raman modes are resonant at different frequencies, ω₁ and ω₂. For simplicity, we assume that their corresponding molecular polarizability tensors, ${\alpha }_{\omega }^{(3)}$, can be represented by a single non-zero tensor component. Then, a tensor can be expressed as a product of a scalar amplitude, ${\alpha }_{\omega }^{(3)}$, and a directional unit vector, ${\widehat{e}}_{\omega }$, 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 CH₂ 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 ${\alpha }^{(3)}={\alpha }_{{\omega }_1}^{(3)}+{\alpha }_{{\omega }_2}^{(3)}={\alpha }_{{\omega }_1}^{(3)}{\widehat{e}}_{{\omega }_1}+{\alpha }_{{\omega }_2}^{(3)}{\widehat{e}}_{{\omega }_2}$, where ω₁ and ω₂ 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.

Back to nonlinear susceptibilities, in fact, the observed CARS signal is contributed by both resonant and nonresonant susceptibilities, ${\displaystyle \sum {\chi }_{{\omega }_i}^{(3)}}$ and ${\chi }_{\text{NR}}^{(3)}$, 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 ${\displaystyle \sum {\chi }_{{\omega }_i}^{(3)}}$, 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

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.

3. Results

3.1 Case I: a primary mode and an orthogonal secondary mode

We begin our discussion with a configuration, where a collection of parallel oriented molecules with two orthogonal Raman modes. Examples of such collection of parallel-oriented molecules will include most of molecular single crystals, biaxial nematic liquid crystals, and crystalline (orthorhombic) polyethylene. For simplicity, we assume that the primary mode (ω₁) is parallel to the z axis and the secondary mode (ω₂) is parallel to the x axis. Then, ${\widehat{e}}_{{\omega }_1}={\widehat{e}}_z=(0,0,1)$ and ${\widehat{e}}_{{\omega }_2}={\widehat{e}}_x=(1,0,0)$, as illustrated in Fig. 1. The effective nonlinear susceptibility for each mode can be calculated from Eq. (7) as:

Figure 2(b) shows plots of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ calculated with Eq. (8) for a constant ψ but various θ, as represented in Fig. 2(a), where the polarization profiles are the same in shape but different in amplitude. Because the phase of the undulation is determined solely by ψ in the cosine term of Eq. (8), we can easily determine ψ from the η value corresponding to the maximum value of a measured polarization profile, as $\psi ={\eta }_{{\omega }_1}^{\max }$. Unlike the phase, however, the amplitude of the undulation is not straightforwardly applicable to determine θ, because the other coefficients in ${\chi }_{{\omega }_1}^{(3)}(=N{\alpha }_{{\omega }_1}^{(3)})$ need to be independently known. When a polarization profile of the secondary mode is analyzed separately, as shown in Figs. 2(c) and 2(d), its interpretation becomes more complicated.

 

Fig. 2 (a) Schematic presentation of nonlinear susceptibilities of the primary mode, ${\chi }_{{\omega }_1}^{(3)}$, in the spherical coordinate of the laboratory frame. (b) Plots of the polarization profiles of effective nonlinear susceptibilities, ${\chi }_{\text{eff},{\omega }_1}^{(3)}$, calculated for the orientations presented in (a), where ψ = 60 °. (c) Schematic presentation of nonlinear susceptibilities of the secondary mode, ${\chi }_{{\omega }_2}^{(3)}$, for various ϕ. The secondary mode is perpendicular to the primary mode. (d) Plots of the polarization profiles of effective nonlinear susceptibilities, ${\chi }_{\text{eff},{\omega }_2}^{(3)}$, calculated for the orientations presented in (c), where ψ = 60 ° and θ = 30 °.

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Here, we propose to concurrently analyze both polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ and to use only dimensionless quantities, which are independent of the system response, the local concentration, and the absolute Raman cross section.

First, we define the phase difference, Δη, and the amplitude ratio, r_χ, from the peak maxima of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$, as described in Fig. 3(a),

 

Fig. 3 (a) Polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$, which are simulated from Eqs. (8) and (9) with ψ = 40 °; θ = 50 °; ϕ = 30 °; and ${\chi }_{{\omega }_1}^{(3)}={\chi }_{{\omega }_2}^{(3)}=1$. Representation of two dimensionless observables, Δη and r_χ, from the polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$, (b) A contour plot of Δη as a function of θ and ϕ. (c) A contour plot of r_χ as a function of θ and ϕ.

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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) and 4(d). First, we can easily determine the azimuthal angle, ψ, from the ${\eta }_{{\omega }_1}^{\max }$ measured from the ${\chi }_{\text{eff},{\omega }_1}^{(3)}$curve. Next, we measure Δη and r_χ from the ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ curves from Eqs. (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. Figures 4(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. 4 Determination of the 3D orientation angles (ψ, θ, and ϕ) from of polarization profiles calculated for two examples A (top row) and B (bottom row). (a) Plots of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ where ψ is determined from ${\eta }_{{\omega }_1}^{\max }$. (b) Contour lines of the Δη and r_χ values determined from the plots of (a). (c) θ and ϕ are determined from the crossing point of the two contour lines of Δη and r_χ. (d)–(f). Similarly, plots of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ are used to determine the 3D 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.

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3.2 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 ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ is identical to that of Case I, as in Eq. (8). However, the polarization profile of ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ is different because Eq. (7) needs to be averaged over ϕ, as follows:

 

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 ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ are calculated from Eqs. (8) and (15) for the examples N and L.

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Figures 5(b) and 5(c) show polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ computed for two orientational examples for Case II with Eqs. (8) and (15). As mentioned above, the primary mode ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ show the same behavior as Case I: the minimum is zero; and the maximum increases with θ. However, the secondary mode ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ behaves differently: the maximum value, ${\chi }_{\text{eff},{\omega }_2}^{(3),\max }$, remains unaffected by θ; and the minimum value, ${\chi }_{\text{eff},{\omega }_2}^{(3),\min }$, becomes non-zero and varies with θ. In Case II, we use the θ dependence of ${\chi }_{\text{eff},{\omega }_2}^{(3),\min }$ to determine θ from a ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ curve. We define its related dimensionless quantity, m_χ, with the ratio of the maximum and minimum values of a ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ curve.

 

Fig. 6 (a) Plots of polarization profiles of ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ calculated for four examples of Case II. Example K: θ = 10 °; example L: θ = 30 °; example M: θ = 40 °; and example N: θ = 70 °. For all examples, ψ = 60 °. (b) m_χ is plotted as a function of θ from Eq. (16). The m_χ values observed from (a) are used to determine the θ values. Polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ are calculated (c) for the example N and (d) the example M. (e) r_χ is plotted as a function of θ from Eq. (17), demonstrating an alternative method to determine the θ values for the examples M and N.

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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),

3.3 Case III: Case I with broadening

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 ω₁ mode, ${\widehat{e}}_{{\omega }_1}$ in Eq. (7) is replaced with ${{\widehat{e}}^{\prime }}_{{\omega }_1}(\beta ,\gamma )=\rho (\beta )R(\beta ,\gamma ,0){\widehat{e}}_{{\omega }_1}$, followed by integration over β and γ. The effective nonlinear susceptibility in Eq. (7) becomes

 

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.

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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

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 comparison of the polarization profiles with broadening (κ = 10) and without broadening (κ = ∞). First, the phase of the polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ is not affected by broadening. Therefore, ${\eta }_{{\omega }_1}^{\max }$ and ${\eta }_{{\omega }_{\text{}2}}^{\max }$ 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 ${\chi }_{\text{eff},{\omega }_1}^{(3),\max }$ and ${\chi }_{\text{eff},{\omega }_2}^{(3),\max }$ in the presence 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. 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 ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ calculated with broadening (Case III, κ = 10, dashed lines) are compared with those with no broadening (Case I, κ = ∞, solid lines). (b) ${\chi }_{\text{eff},{\omega }_1}^{(3),\max }$ and ${\chi }_{\text{eff},{\omega }_2}^{(3),\max }$ are plotted as a function of κ. For the random distribution (κ = 0), both values converge to 0.2. (c) The contour lines corresponding to the r_χ (or r_χ') and Δη values determined from the plots in (a). The r_χ' value, shifted by broadening (κ = 10), and shifts, results in false orientation angles (θ' = 49.8 ° and ϕ' = 30.1 °) while the unshifted r_χ value yields the true mean orientation angles (θ = 50 ° and ϕ = 30 °). (d)‑(f) Similar to the example A, broadening effect on the example B on determined orientation values: false angles for κ = 10 are θ' = 36.1 ° and ϕ' = –18.8 °, compared with the true values of θ = 30 ° and ϕ = –20 °.

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As mentioned above, orientation broadening affects not only the maximum but the minimum values of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$. While without broadening (Case I), the minimum 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 ω₂ mode, the ratio of the minimum to the maximum was already defined as m_χ by Eq. (16). For the ω₁ mode, similarly, we define a ratio

Because the ω₁ mode is not related with ϕ, n_χ can be expressed as a function of θ and κ, as shown in Fig. 9(a). In contrast, m_χ, which defined for the ω₂ mode, is a function of all three variables, θ, κ, and ϕ. Figures 9(b)–9(c) shows the contour plots of m_χ as a function of θ and κ for three different ϕ. Once ϕ is known, contour lines of both n_χ and m_χ can be plotted on the same contour plot for θ and κ so that the crossing point of the two contour lines can be used to determine θ and κ.

 

Fig. 9 (a) A contour plot of n_χ as a function of θ and κ for the ω₁ mode of Case III. (b)–(d) Contour plots of m_χ for the ω₂ mode of Case III for various ϕ.

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Figure 10 shows how the dimensionless values (${\eta }_{{\omega }_1}^{\max }$, Δη, r_χ', n_χ and m_χ) observed from the polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ are used to determine the mean orientation angles (ψ, θ and ϕ) as well as the degree of broadening (κ). First, the ψ can be easily determined by ${\eta }_{{\omega }_1}^{\max }$, which is not affected by broadening, in Fig. 10(a). Next, the observed r_χ', which is shifted by broadening, is plotted together with Δη in a contour plot, in Fig. 10(b). Their crossing point yields θ' and ϕ', which are deviated from the true mean orientational angles (θ and ϕ), as shown in Figs. 8(c) and 8(f). Then, we use the crossing point of the contour lines of measured m_χ and n_χ to determine θ and κ. Although the true contour line of m_χ requires the true ϕ, we find that the curve of m_χ with ϕ' is sufficiently close to the curve of m_χ with true ϕ for determining θ and κ. In Fig. 10(c), the crossing point of the contour lines of n_χ and m_χ (approximated by ϕ') returns θ and κ. Next, the newly determined θ is used to determine ϕ from the contour line of Δη in Fig. 10(d), which is insensitive to broadening. This process can be performed iteratively between the last two steps in Figs. 10(c) and 10(d) until ϕ, θ and κ converge the mean orientation angles, if necessary. In this way, we can determine the mean orientation angles even in the presence of orientational broadening, and additionally, the degree of broadening can also be quantitatively estimated.

 

Fig. 10 (a) Plots of polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ calculated for the example B with broadening (Case III, κ = 10). (b) Contour lines of Δη and r_χ' determined from the ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ curves in (a) are plotted for θ and κ. Their crossing point yields a preliminary (false) value of ϕ'. (c) The preliminary ϕ' is used to the contour line of m_χ corresponding to the m_χ value determined from the ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ curve in (a). Another line corresponding to the n_χ value determined from the ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ curve is plotted together with the contour line of m_χ. Their crossing point yields θ and κ. (d) Such determined θ is used to determine the true value of ϕ from the contour line of Δη.

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3.4 Case IV: Case II with broadening

Now we discuss the broadening effect on the polarization profiles of Case II, where the secondary mode is randomly distributed over ϕ. The orientation distribution function of the ω₁ mode is the von Mises–Fisher function, the same in Eq. (18) for Case II. The broadening effect on the ω₂ mode is numerically calculated by integration over ϕ.

In Figs. 11(a) and 11(b), the polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ calculated with Eqs. (18) and (21), respectively, are compared between without broadening (Case II) and with broadening (Case IV; κ = 10). First, broadening does not alter the phase of undulating polarization profiles, which means that ψ can be easily determined from ${\eta }_{{\omega }_1}^{\max }$, like all other previous cases. However, broadening reduces the undulation amplitude of both polarization profiles and, therefore, changes the observed m_χ and r_χ' values. In other words, m_χ and r_χ' are functions of not only θ but κ. The method of using m_χ and r_χ with Eqs. (16) and (17) to determine θ works only for no broadening (κ = ∞). In the presence of broadening, the observed m_χ and r_χ' values will generate false θ' values, which are different from the true θ values as shown in Figs. 11(c) and 11(d).

 

Fig. 11 Broadening effect on polarization profiles of ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ for the two examples (a) M and (b) N with broadening (Case IV, κ = 10, dashed lines) are compared with those with no broadening (Case II, κ = ∞, solid lines). (c) The θ values are determined from the m_χ values from the curves of (a) and (b) by using the m_χ – θ plot, presented in Fig. 6(b). Broadening (κ = 10) makes the θ' values deviate from the true θ values. (d) Similarly, θ values are determined from the r_χ (or r_χ') values from the curves of (a) and (b) by using the r_χ – θ plot, presented in Fig. 6(e).

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Instead, we numerically calculate m_χ and r_χ' for various θ and κ and create their contour plots, as shown in Figs. 12(a) and 12(b). We draw contour lines corresponding to m_χ and r_χ' measured from the curves in Figs. 11(a) and 11(b). The crossing point yields the true mean orientation angle θ not a false θ'. The concentration parameter, κ, can also be determined from the crossing point. In addition, another dimensionless observable, n_χ, can be plotted in the same contour plot for θ and κ. In this theoretical model, crossing points of any pair or all three ratios (m_χ, n_χ, and r_χ') can be used to determine the mean θ and κ. Practically, however, one pair of observables may become more reliable than other pairs depending on orientation angles and degree of broadening. Figure 12 shows two examples of simultaneously determining the mean θ and κ from the simulated curves of the ω₁ and ω₂ modes for Case IV. This method of using the crossing point of m_χ, r_χ (or r_χ') curves works good for large broadening (κ <100). When the broadening is narrow (κ >100), the ratio n_χ becomes uncertain because the minimum value of the ω₁ mode becomes close to zero; hence, increased uncertainty of κ. However, the mean θ can be determined simply from the curve of m_χ and r_χ as functions of θ, as demonstrated in Figs. 5 and 6.

 

Fig. 12 Contour plots of (a) m_χ and (b) r_χ' as a function of θ and κ for Case IV. (c) θ and κ are determined from the contour lines of m_χ and r_χ' acquired from ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ of the example M in Fig. 11(a). The contour line of n_χ is plotted together for confirmation. (d) Similarly, θ and κ are determined from the contour lines of m_χ, r_χ', and n_χ acquired from ${\chi }_{\text{eff},{\omega }_1}^{(3)}$ and ${\chi }_{\text{eff},{\omega }_2}^{(3)}$ of the example N in Fig. 11(b).

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4. Discussion

So far our theoretical descriptions are based on two simplified molecular geometries by using polarization profiles acquired by parallel, linearly-polarized BCARS measurements. This 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

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.

5. 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.