Mueller matrix holographic method for small particle characterization: theory and numerical studies

Transcription

1 Mueller matrix holographic method for small particle characterization: theory and numerical studies Meng Gao, 1, * Ping Yang, 2 David McKee, 3 and George W. Kattawar 1 1 Institute for Quantum Science and Engineering, Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA 2 Department of Atmospheric Sciences, Texas A&M University, College Station, Texas 77843, USA 3 Physics Department, University of Strathclyde, 107 Rottenrow, Glasgow G4 ONG, UK *Corresponding author: mgao@tamu.edu Received 3 April 2013; revised 17 June 2013; accepted 18 June 2013; posted 19 June 2013 (Doc. ID ); published 19 July 2013 Holographic imaging has proved to be useful for spherical particle characterization, including the retrieval of particle size, refractive index, and 3D location. In this method, the interference pattern of the incident and scattered light fields is recorded by a camera and compared with the relevant Lorenz Mie solutions. However, the method is limited to spherical particles, and the complete polarized scattering components have not been studied. This work extends the Mueller matrix formalism for the scattered light to describe the interference light field, and proposes a Mueller matrix holography method, through which complete polarization information can be obtained. The mathematical formalism of the holographic Mueller matrix is derived, and numerical examples of birefringent spheres are provided. The Mueller matrix holography method may provide a better opportunity than conventional methods to study anisotropic particles Optical Society of America OCIS codes: ( ) Holography; ( ) Scattering, polarization; ( ) Anisotropic optical materials Introduction Holographic imaging has been widely used to characterize particle size, shape, and position [1 10]. In this method, an incident beam of light impinges on the particles, and the interference pattern of the incident and scattered light fields is recorded and analyzed. The 3D position, refractive index, and radius can be retrieved accurately for isotropic spherical particles by comparing the recorded interference patterns with the Lorenz Mie solutions [11 14]. The light source is normally chosen with one specific polarization state, and the total intensity of the interference pattern is recorded without separating different polarization components. However, the scattered light field can X/13/ $15.00/ Optical Society of America have both linear and circular polarization components. The polarization state of the scattered light may depend on the incident counterpart. The complete polarization information from holography may provide an extra dimension to retrieve particle shape and composition. In this work, a holographic method based on the Mueller matrix formalism is proposed. The transformation of the incident light field into the interference light field, both with arbitrary polarization states, is described by a 4 4 holographic Mueller matrix. The matrix contains all the polarization information for the elastic scattering of small particles, and an analytical expression is derived. The matrix has seven independent elements, and, therefore, provides six additional pieces of information for particle characterization compared with conventional methods. The method is different from polarization holography 20 July 2013 / Vol. 52, No. 21 / APPLIED OPTICS 5289

2 [15], where the interference polarization pattern is recorded in polarization sensitive materials. The holographic Mueller matrix can be measured using a series of retarders and polarizers between the light source and particles, and between the particles and the camera [16,17]. As an example of the application of this Mueller matrix holographic method, the scattering properties of birefringent spheres are studied. In previous holographic studies of spherical particles, all particles were assumed to be optically isotropic with no birefringence, and a scalar refractive index could be used with the Lorenz Mie theory to calculate the holographic patterns. However, the materials from which spherical particles are composed can be birefringent with anisotropic refractive indices [18,19]. We will apply the Mueller matrix holography method to birefringent spheres and demonstrate how the polarization information can be used to discriminate the birefringence and orientation of small anisotropic spherical particles. We will compare the results with isotropic particles having no birefringence. Other nonspherical and anisotropic particles, such as biological plates, called iridosomes [20,21], can be studied in a similar way, but are not considered in this paper. To facilitate the retrieval of the shape and composition of specific particles, a database of the polarization scattering properties can be established and used to compare with measured results. 2. Theory of Holographic Mueller Matrix We define the electric field vector of a light field as E E l ;E r T, where E l and E r, respectively, represent the electric field components parallel and perpendicular to the scattering plane. The scattering plane is determined by the incident direction and the scattering direction. Both electric field components are complex quantities. To conveniently represent the intensity and polarized state of a beam of light, a real vector, the Stokes vector I I; Q; U; V T [22] is defined as I E l E l E re r ; (1) Q E l E l E re r ; (2) U E l E r E r E l ; (3) V i E l E r E r E l ; (4) where each of the four components is real. I is the irradiance of the scattered light and Q, U, and V represent the polarization. When light is scattered by a small particle located in a particular plane (plane 1 in Fig. 1), the incident field vector will be transformed into a scattered field vector for each different scattering direction. In the farfieldregion(kr 1),thetransformationcanberepresented using the scattering amplitude matrix S [22]: Fig. 1. Particle is located in plane 1 with a beam of light incident on the particle. Interference pattern is recorded in plane 2. Scattering angle at point P is denoted by θ; ϕ. E s r eik r z ikr SE 0 z ; (5) where E 0 is the incident field vector, E s is the scattered field vector, and S is a 2 2 complex matrix. The ijth component of the scattering amplitude matrix is denoted as S ij. The wavenumber k is defined as k 2πn m λ, where n m is the refractive index of the medium (both the particle and the scattered field are in the same medium). The scattering amplitude matrix is generally a complex matrix and would be more conveniently represented using a real matrix. The transformation of the incident Stokes vector into the scattering Stokes vector can be found by using the scattering Mueller matrix M s [22], I s 1 k 2 r 2 M si 0 ; (6) where I 0 and I s, respectively, correspond to the Stokes vector of the incident and scattered light fields. A video camera (plane 2 in Fig. 1) will record the intensity of the total electric field, which is the interference of the scattered field and the incident field. For the scattering direction near the incident direction, the total electric field vector E t r can be given as the summation of the scattered and incident field vectors E t r E s r E 0 z [12]. The amplitude matrix S 0, which transfers the incident electric field vector into the total electric field vector as E t r S 0 E 0 z, can be obtained by S 0 eik r z S 1; (7) ikr where 1 represents a unitary 2 2 matrix for the incident field. Therefore, the Mueller matrix for the total light field M 0, namely the holographic Mueller matrix, can be given as 5290 APPLIED OPTICS / Vol. 52, No. 21 / 20 July 2013

3 M k 2 r 2 M s 1 Q; (8) kr where 1 is a 4 4 unitary matrix, M s is the scattering Mueller matrix defined in Eq. (6), and Q is the interference matrix representing the interference between the scattered field and the incident field. In the far-field region, kr 1, the term M s k 2 r 2 in Eq. (8) can be neglected. Furthermore, we can obtain the contrast holographic Mueller matrix (M h )by subtracting the intensity of the incident field with no particles present, M h M Q: (9) kr Since the incident plane wave may be experimentally imperfect, in this way the recorded data quality can be enhanced [9]. Hereafter, the holographic Mueller matrix will be referred to according to the definition in Eq. (9). The holographic Mueller matrix imaging will require less illumination intensity than other Mueller matrix imaging methods, such as the backscattering Mueller matrix [23,24], because the holographic Mueller matrix depends on 1 kr, whereas, the scattering Mueller matrix depends on 1 kr 2 and decreases much faster than the holographic signals. Since seven independent elements are required to determine the scattering amplitude matrix S [25], the interference matrix Q also contains seven independent elements and can be denoted as 0 1 Q 11 Q 12 Q 13 Q 14 Q B Q 12 Q 11 Q 23 Q 24 Q 13 Q 23 Q 11 Q 34 A : (10) Q 14 Q 24 Q 34 Q 11 The seven elements can be calculated as follows: Q 11 cos αim S 11 S 22 sin αre S 11 S 22 ; (11) Q 12 cos αim S 11 S 22 sin αre S 11 S 22 ; (12) Q 34 cos αre S 11 S 22 sin αim S 11 S 22 ; (13) Q 13 cos αim S 12 S 21 sin αre S 12 S 21 ; (14) Q 14 cos αre S 12 S 21 sin αim S 12 S 21 ; (15) Q 23 cos αim S 12 S 21 sin αre S 12 S 21 ; (16) Q 24 cos αre S 12 S 21 sin αim S 12 S 21 ; (17) where the phase factor α is defined as α k r z. The holographic Mueller matrix contains all the particle scattering information including both intensity and polarization. Experimentally, the matrix can be obtained by measuring the polarization state of the interference pattern with different incident polarization states [16,17]. Since small particles are in constant Brownian motion, the Mueller matrix elements should be measured simultaneously and efficiently. This may be achieved using electro-optic modulators [26,27] to control the incident and measured polarization states within the digital holographic imaging techniques. The seven independent elements of the holographic Mueller matrix can be classified into two sets: Q 11, Q 12, and Q 34 depend on the diagonal elements of the scattering amplitude matrix; and, Q 13, Q 14, Q 23,andQ 24 depend on the nondiagonal elements. For isotropic spherical particles, the scattering amplitude matrix is diagonal, and the holographic Mueller matrix only has Q 11, Q 12, and Q 34 elements. To the best of the authors knowledge, the analytical expressions for the holographic Mueller matrix have not been reported in the literature. Furthermore, a direct connection exists between the holographic Mueller matrix and the extinction matrix K, which characterizes the extinction cross sections with different incident polarization states for nonspherical particles. The extinction matrix can be calculated by the integration of Q kr over the solid angles around the forward direction [28]. The angular pattern of the holographic Mueller matrix is represented by using the scattering zenith angle θ and scattering azimuthal angle ϕ. As shown in Fig. 1, θ is defined as the angle between the incident direction and the scattering direction. The distance from the particle to camera in plane 2 is z, the distance from the particle to a point P in the screen is r, and the distance between O and P is r p. We can use z and θ to represent both r and r p as r z cos θ and r p z tan θ. Therefore, the holographic Mueller matrix is a function of z and θ. For isotropic spherical particles, both S 11 and S 22 can be calculated using the Lorenz Mie theory. For nonspherical particles, each of the four elements of the scattering amplitude matrix must be calculated using numerical methods, such as the discrete-dipole approximation (DDA) method [29 31] or the finite-difference-timedomain (FDTD) method [32]. 3. Numerical Results for Birefringent Spheres The holographic imaging method can accurately retrieve the size and refractive index of spherical particles [12], but the potential effects of anisotropic refractive index have not been investigated. For anisotropic spheres in our study, the dielectric tensor is defined [33] as D 1 0 x D y ɛ 10 xx ɛ yy 0 A@ E 1 x E y A; (18) D z 0 0 ɛ zz E z where the refractive p index along the x, y, z axes are defined as n x p, ny p, and nz. For a ɛ xx ɛ yy ɛ zz 20 July 2013 / Vol. 52, No. 21 / APPLIED OPTICS 5291

4 Fig. 2. (a) Birefringent sphere with a radius a and an optical axis along the y direction. The refractive indices along the x, y, and z axes are n x, n y, and n z, where n x n z n y. The incident direction of the light is along z. (b) Sphere is rotated around the x axis with an orientation angle β, and the optical axis is along the y direction. (c) Angular distribution diagram for the holographic Mueller matrix. Zenith angles θ from 0 to 40 and azimuthal angles ϕ from 0 to 360 are shown. uniaxial birefringent system, only one optical axis is present, and the refractive indices along the directions perpendicular to the optical axis are the same. As shown in Fig. 2(a), the optical axis is along the y direction, and therefore n x n z n y. The incident light is along the z direction, and the optical axis of a particle can have arbitrary orientation relative to the incident direction. As shown in Fig. 2(b), an orientation angle β represents the current orientation of the optical axis after the particle is rotated around the x axis. We will study the holography of two uniaxial birefringent systems: TiO 2 (rutile) and CaCO 3 (calcite) spheres. The TiO 2 sphere is strongly birefringent, and the scattering properties have been modeled using the FDTD method [34]. In our study, TiO 2 spheres are embedded in oil with a refractive index of 1.55, and the relative refractive indices are n x n z 1.7 and n y 1.9 (at wavelength 0.56 μm) with a birefringence of 0.2 [35]. CaCO 3 spheres are chosen as an example with less birefringence. The refractive indices relative to water of 1.33 are n x n z 1.25 and n y 1.12 (at wavelength 0.56 μm) with a birefringence of 0.13 [36]. The angular patterns of the holographic Mueller matrix for TiO 2 and CaCO 3 will be discussed with spheres of different orientations and size parameters. The scattering amplitude matrices are calculated using the Amsterdam DDA (ADDA) code developed by Yurkin and Hoekstra [31], and the results are inserted into Eqs. (11) (17) to calculate the holographic Mueller matrix. The range of the size parameters (ka, where a is sphere radius) is from 1 to 10, which corresponds to a radius of approximately 0.1 to 0.6 μm. The conventional holographic method has been used to characterize the size and refractive index of leucosomes, a type of biological particle, which correspond to this size range [14]. If anisotropy is present in such structures, it may be distinguished by the Mueller matrix holographic method. The simulation of large size particles using the DDA method can be done in the same way but will require more computational resources. In order to improve the simulation efficiency for large particles, geometric optics based methods, such as the physical-geometric optics hybrid method [37,38], can be used. For other anisotropic and nonspherical systems, the holographic patterns can be calculated in a similar manner. The Mueller matrix elements depend on particlecamera distance kz through cos α kr and sin α kr with α k r z and r z cos θ as shown in Eqs. (11) (17). Larger values of kz will result in more variations of the Mueller matrix versus scattering zenith angles. These variations of the Mueller matrix can be used to determine the distance from the particle to the camera [12]. As an example, kz is chosen to be 200, which is much larger than 1, and therefore the contribution of the scattering Mueller matrix can be neglected according to Eq. (9) and only the interference matrix is evaluated. The angular resolution of the scattering amplitude matrix used in this study is Δϕ 6 andδθ 0.5. All 4 4 elements of the holographic Mueller matrix for TiO 2 spheres with radius size parameter ka 5 at orientation β 0 [Fig. 2(a)] are plotted in Fig. 3. Both M h 12 and Mh 21 are quite sensitive to the birefringence of the spheres. The element M h 21 corresponds to the measured difference between the horizontal and vertical components of the interference pattern with an unpolarized incident light field. Since M h 12 and Mh 21 are equal, it is sufficient to measure M h 12, which corresponds to the difference between the total field in the interference pattern with horizontal and vertical polarized incident light fields [16]. Hereafter, we will focus on a discussion of M h 12 for birefringent spheres and compare with the behavior of M h 11. Different particle shapes and compositions may have different prominent holographic Mueller matrix elements. If the sphere is isotropic, e.g., n x n y n z 1.7, the scattering amplitude matrix will be diagonal, and the M h 11, Mh 12, and Mh 34 elements are nonvanishing; therefore, only the two diagonal 2 2 blocks in the Mueller matrix are nonzero, as shown in Fig. 4, and the patterns are all circular. From a comparison of Fig. 3 with Fig. 4, both the noncircular angular patterns and the nonzero terms in the nondiagonal 2 2 blocks show features that are caused by particle anisotropy. When nonspherical particles are considered, these patterns could potentially also be a sign of particle nonsphericity APPLIED OPTICS / Vol. 52, No. 21 / 20 July 2013

5 Fig. 3. Angular distribution [according to Fig. 2(c)] ofm h ij for TiO 2 with size parameter ka 5. The scattering properties of an anisotropic sphere also depend on the incident direction relative to its optical axis. For TiO 2 spheres, we varied its optical axis with the orientation angle β 0, 30, and 90 at size parameter ka 5. The interference pattern for M h 11 and Mh 12 are plotted in Fig. 5. When the optical axis is along the z direction (β 90 ), the interference patterns of both M h 11 and Mh 12 become circular. When β 0, the pattern is symmetric with respect to both the x and y axes, but when β 30, the pattern is only symmetric with respect to the y axis. The symmetry property can be used to determine the particle orientation. More specifically, the data of M h 11 and Mh 12 at both ϕ 0 ( x) andϕ 90 ( y) are plotted in Fig. 6. ForM h 11, only a small variation is present between the values at these two Fig. 4. Same as Fig. 3 but for an isotropic sphere with n x n y n z July 2013 / Vol. 52, No. 21 / APPLIED OPTICS 5293

6 Fig. 5. Angular distribution of M h 11 and Mh 12 for a TiO 2 sphere with size parameter ka 5 and orientation β 0, 30, and 90. Angular distribution for each element is shown according to Fig. 2(c). azimuthal angles. But for M h 12, the anisotropic effect is prominent, and the data almost have opposite signs between the results at ϕ 0 and ϕ 90. To quantify the anisotropy of the angular patterns, we define the relative difference between azimuthal angle ϕ 0, and ϕ 90 as R θm hδm h ij i 0 dθ sin θjmh ij ϕ 0 ;θ Mh ij ϕ 90 ;θ j R θm 0 dθ sin θjmh 11 ϕ 0 ;θ Mh 11 ϕ 90 ;θ j; (19) for each ij-th element of the holographic Mueller matrix. The quantity is evaluated around the forward direction with a maximum zenith angle θ m 30. Based on this quantity, we can compare the performance for each element of the holographic Mueller matrix in discriminating the birefringence of small spheres. The results of M h 11 and Mh 12 for TiO 2 spheres are represented by hδm h 11 i and hδmh 12i in Fig. 7. For size parameter ka < 4, hδm h 11i is less than 5.4%, but hδm h 12i is always larger than 12% and its maximum can reach 70%; for ka > 4, there are oscillations of the value of hδm h 11i and its maximum can reach 20%, but hδm h 12i can be as much as 46% to 76%. Therefore, M h 12 provides higher contrast than Mh 11 for discriminating the anisotropy of birefringent spheres, and its performance depends on the particle size. The performance of each element in discriminating birefringent properties of nonspherical particles has still to be tested. CaCO 3 spheres have smaller birefringence than TiO 2 spheres. The angular distributions for M h ij at Fig. 6. Same as the M h 11 and Mh 12 results in Fig. 5, but only for ϕ 0 (solid lines) and ϕ 90 (dashed lines). Fig. 7. Relative differences of M h 11 and Mh 12 between ϕ 0 and ϕ 90 for TiO 2 spheres at β 0 with size parameters from 1 to APPLIED OPTICS / Vol. 52, No. 21 / 20 July 2013

7 Fig. 8. M h ij for a CaCO 3 sphere with ka 5 and β 0. Angular distribution for each element is shown according to Fig. 2(c). Fig. 9. Same as Fig. 7 but for CaCO 3 spheres. orientation β 0 and size parameter ka 5 are plotted in Fig. 8. M h 14, Mh 41, Mh 32, and Mh 23 (M h 14 Mh 41, Mh 32 Mh 23 ) are all almost zero. The M h 11 interference pattern is more circular than that for TiO 2 shown in Fig. 3. In Fig. 9, hδm h 11i is less than 1.4%, but hδm h 12i is above 36% for all size parameters between 1 and 10. Therefore, the interference pattern of the intensity element M h 11 provides little evidence to discriminate the birefringence of the CaCO 3 spheres; however, the polarization elements such as M h 12 are a potentially useful option by which to study the anisotropy of small spheres with relatively small birefringence. 4. Conclusion We have generalized the holographic imaging method to include arbitrary incident and scattered polarization states using the Mueller matrix for particle characterizations. General expressions for the holographic Mueller matrix, including seven independent elements, are analytically derived. The birefringence of anisotropic spheres is studied using the holographic Mueller matrix method with different sizes and orientations. The present numerical results demonstrate that, unlike the intensity element M h 11 usually measured by the conventional holographic method, the interference patterns of the polarized components, such as M h 12, can be used to discriminate the birefringence of small spheres. The specific properties of all seven elements of the holographic Mueller matrix may be sensitive to deviations from sphericity or from material uniformity, thus potentially offering more information about general particle characterization. This research was partially supported by the ONR MURI program N , ONR N , NSF Grant OCE , NSF Grant ATM , and NASA Grant NNX11AK37G. The authors would like to thank Dr. Alex Nimmo Smith and Dr. David G. Grier for helpful discussions, and the Texas A&M Supercomputing Facility for providing computing resources in conducting the research reported in this paper. The authors would also like to thank Sinan Karaveli and Dr. Rashid Zia for introducing the holography imaging method to us, and Dr. Emlyn Davies and three other reviewers for the constructive comments and suggestions on improving the readability of this paper. References 1. B. J. Thompson, Holographic particle sizing techniques, J. Phys. E 7, (1974). 20 July 2013 / Vol. 52, No. 21 / APPLIED OPTICS 5295

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