Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects

A fully automated Mueller-matrix ellipsometer with a division-of-amplitude photopolarimeter as the polarization-state detector is described. This device achieves Mueller-matrix ellipsometry by measuring the Stokes parameters of reflected light as a function of the fast axis C of a quarter-wave retarder, which, in combination with a fixed linear polarizer, determines the polarization state of incident light. The reflected Stokes parameters were Fourier analyzed to give the 16 elements of the Mueller matrix. We investigated depolarization of polarized light on reflection from rough, heterogeneous, and anisotropic surfaces by obtaining measurements on rolled aluminum and plant leaves. The results demonstrate (1) a variation of degree of polarization of reflected light with the input polarization state, (2) the precision with which the measured matrices describe the depolarization results, (3) effects of surface anisotropy (rolling direction) on depolarization and cross polarization by reflection from aluminum surfaces, and (4) large values and differences in the depolarization effects from conifer and deciduous leaves. Depolarization of light reflected by the aluminum surfaces was most sensitive to the angle between the plane of incidence and the rolling direction when the incident Stokes parameters S(1), S(2), and S(3) were equal.     

Polarization contrast imaging of biological tissues by polarization-sensitive Fourier-domain optical coherence tomography

Jones matrix imaging of biological samples by a polarization-sensitive Fourier-domain optical coherence tomography has been demonstrated using a two-dimensional CCD camera to obtain two spectra corresponding to the orthogonal polarization components simultaneously. The measurement results of a quarter-wave plate are compared between the two incident polarization sets, H-V linear and R-L circular polarization. Jones matrix imaging of the bovine tendon is demonstrated. Measured Jones matrix images are converted to equivalent Müller matrix images. Local polarization properties are obtained by longitudinal differentiation of Jones matrix components. The layered structure of the bovine tendon and birefringence are revealed.     

Fiber-based polarization-sensitive Mueller matrix optical coherence tomography with continuous source polarization modulation

We report on a new configuration of fiber-based polarization-sensitive Mueller matrix optical coherence tomography that permits the acquisition of the round-trip Jones matrix of a biological sample using only one light source and a single depth scan. In this new configuration, a polarization modulator is used in the source arm to continuously modulate the incident polarization state for both the reference and the sample arms. The Jones matrix of the sample can be calculated from the two frequency terms in the two detection channels. The first term is modulated by the carrier frequency, which is determined by the longitudinal scanning mechanism, whereas the other term is modulated by the beat frequency between the carrier frequency and the second harmonic of the modulation frequency of the polarization modulator. One important feature of this system is that, for the first time to our knowledge, the Jones matrix of the sample can be calculated with a single detection channel and a single measurement when diattenuation is negligible. The system was successfully tested by imaging both standard polarization elements and biological samples.     

Theoretical model of the polarization properties of the retinal nerve fiber layer in reflection

In several optical technologies for glaucoma diagnosis, polarized light is used to assess the retinal nerve fiber layer (RNFL) of the eye. For better understanding of the polarization properties of the RNFL, it was modeled as a thick birefringent slab containing parallel light-scattering cylinders, and the Mueller matrix for reflectance was derived. The model predicts that (1) the RNFL reflectance has weak intrinsic diattenuation; (2) the diattenuation spectrum depends strongly on the relative refractive indices of the cylinders; (3) both scattering and birefringence contribute to retardation; and (4) the RNFL reflectance generally preserves polarization, but depolarization may be detectable for thick RNFL at short wavelengths.     

Normal-incidence generalized ellipsometry using the two-modulator generalized ellipsometry microscope

A new microscope is described that is capable of measuring the polarization characteristics of materials in normal-incidence reflection with a demonstrated lateral resolution of 4 microm. The instrument measures eight parameters of the sample Mueller matrix, which can be related to the diattenuation, retardation, circular diattenuation, direction of the principal axis, and the polarization factor. With proper calibration, the eight elements of the sample Mueller matrix can be determined to better than 0.001-0.002 for small values. Examples are given for aluminum, rutile (TiO2), and calcite (CaCO3).     

减小无源OCT中初始双折射影响的方法

针对无源光学电流互感器的线性双折射问题,用琼斯矩阵分析了双折射对系统输出的影响.分析了输出光退化成圆偏振情况下光学电流互感器仍能传感电流的现象,以及输出光保持线偏振条件下系统输出电压仍受线性双折射影响的现象.分析了入射光偏转角、入射光湘移与初始线性双折射对光学电流互感器输出的联合影响.结果表明,当初始双折射存在的情况下,随着入射光相移的变化,系统输出相对偏差存在极小值,且极小值对应的入射光相移随着初始双折射值而单调递增.由此提出增加入射光相移来减小初始线性双折射对系统输出影响的方法.用计算机仿真了理论计算结果,实验验证了这种方法的可行性.对改进无源光学电流互感器的性能、开发实用光学电流互感器具有参考意义.     

Producing optical vortices through forked holographic grating: study of polarization

It is important to know what happens to initial polarization during the formation of optical vortices. We use a computer-generated forked holographic grating to produce optical vortices in the laboratory and study changes in polarization, introduced by the grating, which generates optical vortices in its diffracted orders. The Mueller matrix has been used to quantify changes in the polarization in diffracted orders containing optical vortices. Decomposition of the Mueller matrices estimates the polarizing properties, such as diattenuation, retardance and depolarization of the system. We find that the system is a non-depolarizing system. The study also shows that spin and the orbital angular momentum of photons are not coupled in this process.     

Depth-resolved two-dimensional stokes vectors of backscattered light and mueller matrices of biological tissue measured with optical coherence tomography

Mueller matrices provide a complete characterization of the optical polarization properties of biological tissue. A polarization-sensitive optical coherence tomography (OCT) system was built and used to investigate the optical polarization properties of biological tissues and other turbid media. The apparent degree of polarization (DOP) of the backscattered light was measured with both liquid and solid scattering samples. The DOP maintains the value of unity within the detectable depth for the solid sample, whereas the DOP decreases with the optical depth for the liquid sample. Two-dimensional depth-resolved images of both the Stokes vectors of the backscattered light and the full Mueller matrices of biological tissue were measured with this system. These polarization measurements revealed some tissue structures that are not perceptible with standard OCT.     

Mathematical modeling of retinal birefringence scanning.

Retinal birefringence scanning (RBS) is a new technique that is used to detect the fixation of the eye remotely and noninvasively. The method is based on analysis of polarization changes induced by the retina. In this study, the principles of RBS were mathematically modeled to facilitate a better understanding of the origins of the signals obtained. Stokes vector analysis and Mueller matrix multiplication were augmented with Poincaré sphere representation. The cornea was modeled as a linear retarder. The foveal area was modeled as a radially symmetric birefringent medium. The model accurately predicted the frequency and phase of RBS signals obtained during central and paracentral fixation. The signal that indicates central fixation during RBS likely results from a combination of the radial birefringence of the Henle fibers and the overlying corneal birefringence.     

Scattering matrices for large ice crystal particles

The problem of light scattering by ice crystal particles whose sizes are essentially larger than the incident wavelength is divided into two parts. First, the scattered field is represented as a set of plane-parallel outgoing beams in the near zone of the particle. Then, in the far zone the scattered field is represented as a result of both diffraction and interference of these beams within the framework of physical optics. A proper ray-tracing algorithm for calculation of the amplitude (Jones) scattering matrix is developed and applied. For large particles, a number of reduced Mueller matrices are introduced and discussed, since the pure Mueller matrix obtained from the Jones matrix becomes a rather cumbersome and quickly oscillating value. Backscattering by hexagonal ice crystals, including polarization properties, is considered in detail.     

Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition

A product decomposition of a nondepolarizing Mueller matrix consisting of the sequence of three factors--a first linear retarder, a horizontal or vertical "retarding diattenuator," and a second linear retarder--is proposed. Each matrix factor can be readily identified with one or two basic polarization devices such as partial polarizers and retardation waveplates. The decomposition allows for a straightforward interpretation and parameterization of an experimentally determined Mueller matrix in terms of an arrangement of polarization devices and their characteristic parameters: diattenuations, retardances, and axis azimuths. Its application is illustrated on an experimentally determined Mueller matrix.     

Distorted wave front produced by a high-resolution projection optical system having rotationally symmetric birefringence

The influence of birefringence caused by rotationally symmetric stress distribution in a high-resolution projection optical system is investigated. The general form of the pupil function is derived based on the Jones matrix calculation, expressing the wave front as a combination of the two orthogonal polarization components. Assuming a linearly polarized incident beam, it is found that the main polarization portion of the wave front exiting the projection lens has astigmatic aberration in the Seidel region and shows phase singularity at four pupil points at which the amplitude transmittance becomes zero.     

Structure of a general pure Mueller matrix

Changes in the radiance and state of polarization of a beam of radiation can often be described by means of a pure Mueller matrix. Such a 4 × 4 matrix transforms Stokes parameters and can be expressed in terms of the elements of a 2 × 2 Jones matrix. Relations between the two types of matrix are discussed. Explicit expressions are given for changes of a pure Mueller matrix that are caused by certain elementary changes of its Jones matrix, such as when its transpose, complex conjugate, or Hermitian conjugate are taken. It is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements. All possible interrelations for the elements of a general pure Mueller matrix are derived from this simple structure.     

Transmission Mueller matrix ellipsometry of chirality switching phenomena

Mechanisms of molecular chirality induction are fundamental to many questions in chemistry. Interest in these mechanisms is shifting toward media of increasing complexity that simultaneously exhibit linear birefringence and dichroism and where the common assumption that optical activity is the only optical effect that affects light polarization is no longer valid. Light propagation through several of these anisotropic media can be appropriately studied with transmission Mueller matrix ellipsometry. The applications presented herein include the measurement of optical activity in stirred solutions of soft-matter nanophases and the determination of chiral domains in solid-state samples.     

Generalized ellipsometry for materials characterization

Ellipsometry experiments normally measure two to four parameters, which are converted to the ellipsometric parameters ψ and Δ. This is usually sufficient for many samples, but more complicated situations (such as anisotropic or depolarizing samples) require more sophisticated measurements. Over the last 7 years, we have developed the two-modulator generalized ellipsometer (2-MGE), which measures eight elements of the sample Mueller matrix simultaneously either in reflection or transmission. In reflection, the 2-MGE totally characterizes light reflection from anisotropic samples, measuring the normal ellipsometry parameters, as well as the cross-polarization and depolarization effects. Applications include the determination of the spectroscopic optical functions of uniaxial materials (such as TiO₂ and ZnO), and the measurement of cross-polarization from diffractive structures. In transmission, the 2-MGE completely characterizes a general linear diattenuator and retarder. Applications include the measurement of the retardation and diattenuation of film polarizers and internal electric fields in LiNbO₃ and CdZnTe under bias.     

Transmission two-modulator generalized ellipsometry measurements

The two-modulator generalized ellipsometer has been used to measure samples in transmission. In this configuration, the instrument can completely characterize a linear diattenuator and retarder, measuring birefringence, diattenuation, the angle of the principal axis, and the sample depolarization simultaneously and accurately. This instrument can be operated in two modes: (1) spectroscopic, in which measurements are made through the entire sample aperture as a function of wavelength, and (2) spatially resolved, in which measurements are made at a single wavelength and a birefringence picture is made of the sample. Current spatially resolved measurements have been made at a resolution of approximately 40 microm. Four samples have been examined with this instrument: (1) a mica plate, (2) a Polaroid polarizer, and (3) two quartz plates.     

琼斯矩阵在分布式光纤传感器偏振态分析中的应用

针对基于Sagnac原理的分布式光纤传感器中光波偏振态在双折射影响下所带来的干涉信号"偏振诱导衰落"问题,运用琼斯矩阵分析法,建立了传输光偏振态影响系统功率传输系数的数学模型;根据仿真分析的结果,发现使用反射镜作为反射元件,只能消除光纤圆双折射的影响,而不能消除线性双折射的影响.因此,提出了使用法拉第旋转镜提高系统抗偏振衰落能力的改进方法,仿真结果表明可以很好地消除传感光纤的线性双折射和圆双折射的影响.     

Depolarization of diffusely reflecting man-made objects

The polarization properties of light scattered or diffusely reflected from seven different man-made samples are studied. For each diffusely reflecting sample an in-plane Mueller matrix bidirectional reflectance distribution function is measured at a fixed bistatic angle using a Mueller matrix imaging polarimeter. The measured profile of depolarization index with changing scattering geometry for most samples is well approximated by an inverted Gaussian function. Depolarization is minimum for specular reflection and increases asymptotically in a Gaussian fashion as the angles of incidence and scatter increase. Parameters of the Gaussian profiles fitted to the depolarization data are used to compare samples. The dependence of depolarization on the incident polarization state is compared for each Stokes basis vector: horizontal, vertical, 45 degrees, 135 degrees, and right- and left-circular polarized light. Linear states exhibit similar depolarization profiles that typically differ in value by less than 0.06 (where 1.0 indicates complete depolarization). Circular polarization states are depolarized more than linear states for all samples tested, with the output degree of polarization reduced from that of linear states by as much as 0.15. The depolarization difference between linear and circular states varies significantly between samples.     

Angular dependence of polarizing beam-splitter cubes

The field-of-view dependence of polarizing beam-splitter cubes has been studied to characterize their behavior in imaging systems such as optical computers and optical correlators and in other applications that involve noncollimated light. Significant polarization aberration is present in polarizing beamsplitter cubes for two reasons: (1) the s- and p-component orientations, which define the polarizing axes, at the beam-splitting interface vary with the direction of propagation, and (2) the performance of the coating is a function of the angle of incidence. We describe the polarization aberration of a polarizing beam-splitter cube in terms of its diattenuation (polarizing efficiency). We use an imaging polarimeter to measure six figures of merit for three polarizing beam-splitter cubes demonstrating typical polarization aberrations. Finally, we derive the Mueller matrix for a polarizing beam-splitter cube in terms of the s and p transmittance and reflectance and the phase retardances, the parameters generally calculated with thin-film analysis programs.     

Degree of polarization as a criterion to obtain the nine bilinear constraints between the Mueller-Jones matrix elements

The degree of polarization is employed as a criterion to find the nine independent relations among the elements of the Mueller-Jones matrix. This procedure is applied by considering a previously determined, physically realizable Mueller matrix. On the other hand, the nine bilinear constrains are obtained by directly measuring the degree of polarization from an outgoing beam of light from an optical system by considering nine incident states of light taken from the Poincaré sphere. For practical purposes, all the incident polarization states must be scanned from the Poincaré sphere in order to satisfy the over-polarization and the over-gain conditions, respectively, for the physical realizability of the Mueller matrix.