Abstract:
In a polarimeter for analyzing a state of polarization of a light beam incident thereon, the polarimeter including first and second variable retarders configured to exhibit first and second retardance values, respectively, variable over an overall retardance range, and a detector arrangement, a method includes the steps of directing the light beam through the first and second variable retarders and sweeping a selected one of the first and second retardance values progressively and unidirectionally through at least a part of the overall retardance range to produce a plurality of retardance values. The method further includes the steps of, for the plurality of retardance values, detecting at the detector arrangement at least a spatial portion of the beam and extracting the state of polarization based on the spatial portion of the light beam detected at the detector arrangement corresponding to the plurality of retardance values.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to polarimeters and, more specifically, to polarimeters based on liquid crystal variable retarders for determining the state of polarization of light incident thereon. 
     In applications ranging from astronomy to telecommunications, it is often desired to have knowledge of the state of polarization (SOP) of light. For example, astronomical applications include the utilization of polarization information of light received at a telescope as a tool for mapping solar magnetic fields. Chemical and pharmaceutical industries exploit the effect of enantiomerically enriched chiral compounds on the state of polarization for light passed through such compounds, i.e. optical activity. The state of polarization plays a significant role in telecommunications since polarization mode dispersion and polarization-dependent loss present considerable impediments to increased optical bandwidth. Furthermore, polarimetric measurements are used in a wide array of materials characterization, such as in quantification of thin film thickness and index and as a tool for mapping internal material strain via stress-induced birefringence. 
     The art and science of polarimetry is vast with a history that extends well over a century, and, accordingly, various mathematical descriptions of polarized light have long been established. For example, in the Stokes vector representation, the full SOP is characterized as a four element Stokes vector {overscore (S)}, which is defined as                S   _     ≡     (           S   0               S   1               S   2               S   3           )             (   1   )                                
     where 
     S 0 =Total light intensity, 
     S 1 =Intensity difference between horizontal and vertical linearly polarized components, 
     S 2 =Intensity difference between ±45° linearly polarized components and 
     S 3 =Intensity difference between right and left circularly polarized components 
     Other important and often utilized polarization parameters, such as the degree of polarization (DOP), degree of linear polarization (DOLP), degree of circular polarization (DOCP), ellipticity and orientation of major axis, are directly obtainable from the Stokes vector components. For example,              DOP   =           S   1   2     +     S   2   2     -     S   3   2           S   0               (   2   )               DOLP   =           S   1   2     +     S   2   2           S   0               (   3   )               DOCP   =       S   3       S   0               (   4   )                                
     Turning now to the drawings, wherein like components are indicated by like reference numbers throughout the various figures where possible, attention is immediately directed to FIG. 1, which illustrates a Poincaré sphere  10 . The Pioncaré sphere is a commonly used graphical visualization aid for the SOP. As shown in FIG. 1, a Poincaré sphere  10  represents a mapping of all possible SOPs onto the surface of a sphere. A north pole  12  and a south pole  14  of Poincaré sphere  10  correspond to right and left circularly polarized light, respectively. An equator  15  corresponds to linearly polarized light. Arbitrarily chosen opposing points  16  and  18  along the equator represent horizontal and vertical linear polarizations, and opposing points  20  and  22 , which define a line orthogonal to the line defined by points  16  and  18 , represent +45° and −45° linear polarizations, respectively. In terms of the Stokes vector of Eq. (1), the bottom three components of the Stokes vector define a three-dimensional vector that points from the center of the Poincaré sphere to a point on the surface of the sphere. 
     A Stokes polarimeter is a device for determining the SOP of light incident thereon by measuring the components of the Stokes vector of Eq. (1). In terms of the Poincaré sphere, the Stokes polarimeter determines the components of the Stokes vector by measuring the projections along the orthogonal axes of the Poincaré sphere. For example, passing the light through a horizontal linear polarizer is equivalent to measuring the projection of the Stokes vector along the horizontal axis. As another example, for measurements of circular polarization components, a quarterwave plate can be utilized to convert circular polarization components into a linear polarization, from which a linear polarizer may then be used to determine the projection. In general, multiple measurements must be made in order to obtain all four components of the Stokes vector. 
     Currently available polarimeter technologies use polarization optics to extract the polarization information of input light, which is received at one or more detectors and converted to electrical signals. There are mainly four types of existing polarimeters, the basic configurations of which are illustrated in FIGS. 2A-2D. 
     FIG. 2A illustrates a manually operated polarimeter  30  including an optical assembly  32  and a detector  39 . Optical assembly  32  includes a casing  33 , which contains passive optical elements (not shown) such as a polarizing element and an optical retarder. Casing  33  includes an opening  35  for accepting an input light  37  such that input light  37  is acted upon by the polarizing element and the optical retarder within optical assembly  32 , and at least a portion of input light  37  is transmitted through optical assembly  32  to be detected by a detector  38 . During normal operation, the user of polarimeter  30  manually rotates and flips optical assembly  32  to obtain data at detector  38 . Optical assembly  32  is usually configured to have at least four measurement positions, and the data obtained at detector  38  is analyzed by a computer  39  to convert the four measurements into Stokes parameters. A device based on the design as shown in FIG. 2A is available from Optics for Research, for example, and such a design has been described in the literature. 1  A manually operated polarimeter such as polarimeter  30  is limited in that a relatively long time (i.e., several seconds) is required to take the full set of measurements. As a result, the calculated parameters are susceptible to inaccuracies due to power and polarization fluctuations in the input light. Also, since passive, static optical elements are used in the optical assembly, the wavelength range is limited due to the effective range of the optical elements. 
     Another prior art polarimeter is shown in FIG. 2B. A polarimeter  40  of FIG. 2B is a “division of aperture” or “division of amplitude” type polarimeter. Polarimeter  40  includes a beam expander  42 , a collimator  43 , a Stokes filter array  45 , which includes at least four filters  46 , and a detector array  47 , in which a plurality of detectors  48  are aranged to detect light transmitted through each of filters  46 . Input light  37  is expanded by beam expander  42  then collimated by collimator  43  to be incident on Stokes filter array  45 . Each filter  46  is configured to be preferentially sensitive to different polarizations such that at least four simultaneous measurements may be taken to obtain the complete Stokes vector. Polarimeters based on the design shown in FIG. 2B are commercially available from companies such as A flash Corporation, Gaertner Scientific, Santec and General Photonics. Various modifications of polarimeter  40  are disclosed in the literature, such as the “photopolarimeter” which uses non-normal illumination of four detectors arranged in a non-planar configuration. 2  Polarimeter  40  is advantageous in comparison to polarimeter  30  of FIG. 2A due to the high speed in which data may be acquired, limited only by the detector speed. However, polarimeter  40  is still limited in the useful wavelength range due to the wavelength-dependence of the Stokes filters, and the need for a plurality of balanced detectors adds to the total cost of the system. Also, polarimeter  40  is extremely sensitive to the angle of incidence of the input light. In order to overcome this incidence angle sensitivity, other researchers have suggested various configurations in which light propagation into the polarimeter is confined by the use of optical fibers. 3-9  Yet, the use of fibers adds to the complexity and cost of the polarimeter while further limiting useful optical bandwidth, and therefore is not desirable in many applications. 
     A third type of polarimeter is shown in FIG. 2C. A polarimeter  50  includes first and second spinning retarders  52 A and  52 B, respectively, which spin in a direction indicated by curved arrows  53 . Polarimeter  50  further includes an analyzer  54  and a detector  56 . First and second spinning retarders  52 A and  52 B are configured to spin at different rate such that they modulate input light  37  at different rates, and detector  56  is configured to cooperate with first and second spinning retarders  52 A and  52 B to detect and demodulate the light received thereon by lock-in detection. The specific rate of spin, as well as the retardance values of the spinning retarders, are flexible, although certain retardance values will not work in the system (e.g., a one-wave retarder). Such polarimeters are commercially available from, for example, Thor Labs and are disclosed in a number of patents. 10-11  Tremendous accuracy (&lt;0.001% error) has been achieved using the design shown in FIG. 2C. 12  This approach can be advantageous over the polarimeter design shown in FIG. 2A because the duty cycle for measurements is 100% (i.e., data is not just taken at four discrete steps but continuously while the retarders are spinning) and advantageous over both implementations shown in FIGS. 2A and 2B because rapid spin rates, coupled with lock-in detection, limit measurement bandwidth and transfer detection frequencies well above 1/f and other low frequency noise sources. Furthermore, polarimeter  50  requires only a single detector, which can lead to both cost savings and decreased angle of incidence sensitivity over multiple detector designs. However, the polarimeter design of FIG. 2C requires motors to spin the retarders, and the utilizable optical bandwidth is still limited due to the fixed retardance values of the spinning retarders. 
     A fourth type of polarimeter, as shown in FIG. 2D, is similar in configuration to polarimeter  50  as shown in FIG. 2C but the spinning passive retarders are replaced by stationary active or variable retarders. A polarimeter  60  includes first and second variable retarders  62 A and  62 B, respectively, with the orientation of optic axes of the two retarders being held fixed (thereby eliminating the need for motors or moving parts) while the retardance values of the two variable retarders are either switched between sets of specific, predetermined values, or rapidly modulated at different rates for the two different retarders. 
     In the case where the variable retarders are switched between specific, predetermined values, the retardance values of first and second variable retarders  62 A and  62 B are set to the predetermined values in a discrete, stepwise fashion, and an intensity measurement is taken at each of the predetermined values. The Stokes vector components are calculated based on the measurements taken at detector  56  at the predetermined values. One example of such a polarimeter based on the configuration shown in FIG.  2 D and the step-wise measurement scheme is the LC Stokes polarimeter of Meadowlark Optics. 13  The LC Stokes polarimeter is configured to take a set of discrete measurements with the variable retarders set at discrete, predetermined retardance values in order to calculate the Stokes vector. 
     Continuing to refer to FIG. 2D in conjunction with FIG. 2C, polarimeter  60  has the added advantage of wavelength versatility since the retardance values of the variable retarders are not fixed and no moving parts are required, thereby simplifying the system design. The orientations of optical axes, represented by arrows  63 A and  63 B, of first and second variable retarders  62 A and  62 B, respectively, are typically chosen such that, for instance, optical axis  63 A is vertical while optical axis  63 B is positioned at an angle  65  away from the vertical. Small variations in angle  65  are usually accounted for in calibration process. 
     Referring now to FIG. 2E in conjunction with FIG. 2D, an example of a retardance value schematic for a stepwise polarimeter  50  is illustrated. This example is based on a prior publication regarding the LC Stokes polarimeter published by Meadowlark Optics. 13  A graph  70  of FIG. 2E includes a vertical axis  71  representing retardance (in units of waves λ) and a horizontal axis  72  representing time (in arbitrary units). A first, solid line  73  shows the retardance value settings of first variable retarder  62 A, and a second, dashed line  75  shows the retardance value settings of second variable retarder  62 B. Times T 1 -T 6  are times at which measurements are taken in order to generate the data for extraction of the Stokes parameters. As shown in graph  70 , first variable retarder  62 A is first set to a retardance value of zero waves for the measurements taken at times T 1 -T 4 , then later held at a retardance value of λ/4 for the measurements taken at times T 5  and T 6 . Second variable retarder  62 B begins at zero wave retardance for time T 1 , then is switched to a retardance value of λ/2 for a measurement at time T 2 . Second variable retarder  62 B is reset to a retardance value of λ/4 for a measurement at time T 3 , and the second variable retarder is twice switched between retardance values of λ/4 and 3λ/4 for measurements at times T 3 -T 6 . In this way, the first Stokes parameter S 0  of Eq. (1) is determined from any one set of two measurements (T1 and T2, T3 and T4, or T5 and T6), the S 1  component is found by comparing the detector readings obtained at times T 1  and T 2 , the S 3 -component is calculated by using the detector readings obtained at times T 3  and T 4 , and the S 2  component is obtained by using the detector readings at times T 5  and T 6 . In other words, the retardance values of the first and second variable retarders are set to predetermined values in a stepwise fashion, with discrete measurements being taken when the first and second variable retarders are fixed at predetermined retardance values for extraction of the Stokes parameters. 
     Although the Meadowlark LC Stokes polarimeter is sufficient for most applications, this device does have one disadvantage by a relatively slow acquisition time (&gt;hundreds of milliseconds) due to the response times of the LC material and, since only single, discrete measurement are made, also a small measurement duty cycle. Also, it is not trivial to set the retardance values of the LC variable retarder at the exact, discrete values required for the step-wise Stokes parameter measurements. 
     As an alternative to the scheme shown in FIG. 2E, the retardance values of the two retarders of FIG. 2D may be rapidly oscillated such that the portion of input light  37  transmitted through first and second variable retarders  62 A and  62 B and analyzer  54  is detected at detector  56  by lock-in detection. The first and second variable retarders are modulated at first and second known frequencies, respectively, such that the polarization information of the input light can be obtained by analysis of the signal detected at the lock-in detector at the first and second known frequencies and their harmonics (usually twice the first and second known frequencies). The first and second variable retarders generally must be driven at different frequencies such that the resulting signals detected at the lock-in detector can be distinguished as being the result of the modulation by the first variable retarder or the second variable retarder. This scheme is analogous to the spinning waveplate technique, except with stationary variable retarders, the retardance of which are oscillated at predetermined frequencies. Various electro-optic and photoelastic materials may be used as the variable retarders.  14-16  For example, KDP or other electro-optic crystalline material is appropriate for use as the variable retarder in this scheme. Alternatively, piezo-electric elements can be used to stress optical fiber or a variety of bulk materials with sufficiently large stress-optical coefficients, such as calcium fluoride, lithium fluoride, or fused silica to induce birefringence. When driven at a resonant frequency, such photo-elastic modulators can obtain sufficient stroke for polarimetry applications. While both crystalline electro-optic and photo-elastic variable retarders are attractive due to their high modulation frequencies, they are expensive and require high voltage driving electronics as well as complex lock-in amplification and/or detection schemes. Liquid crystal materials may provide cost savings and simplified, low-voltage drive electronics, but are traditionally slow in comparison to KDP and electro-optic materials, therefore liquid crystals are generally not suited for use in the retardance oscillation polarimeters. 
     A sample measuring polarimeter generally measures the SOP of light that has been transmitted through or reflected from a sample. Numerous examples of sample measuring polarimeters exist in the literature. For instance, Oldenbourg et al. disclose a step-wise algorithm for obtaining polarization information from a sample using a polarized microscope including LC variable retarders. 17  The algorithm of Oldenbourg et al. sets the LC variable retarders at predetermined, discrete settings by applying a set of predetermined voltage values to the LC variable retarders. Intensity measurements of light transmitted through the LC variable retarders and the sample are taken at a detector, such as a CCD array. Then, the intensity values measured at the predetermined, discrete retardance values are inserted into an algorithm to calculate the retardance values at different portions of the sample. For example, the LC variable retarder of Oldenbourg et al. must be set to four different sets of applied voltages, and thereby retardance value, combinations in order to measure the retardance values of the sample specimen. It is noted that Oldenbourg et al. does not disclose or suggest in any way the measurement or calculation of Stokes parameters. Also, the disclosure of Oldenbourg et al. is limited to applications in a polarized light microscope. 
     The present invention provides a polarimeter and associated method which serves to reduce or eliminate the foregoing problems in a highly advantageous and heretofore unseen way and which provides still further advantages. 
     SUMMARY OF THE INVENTION 
     As will be disclosed in more detail hereinafter, there is disclosed herein a method for use in a polarimeter for analyzing a state of polarization of a light beam incident thereon. The polarimeter includes first and second variable retarders and a detector arrangement, wherein the first and second variable retarders are configured to exhibit first and second retardance values, respectively, which first and second retardance values are variable over an overall retardance range. The method includes the steps of directing the light beam through the first and second variable retarders, and sweeping a selected one of the first and second retardance values progressively and unidirectionally through at least a part of the overall retardance range. The method further includes the steps of, for a plurality of retardance values that are produced as the selected one of the first and second retardance values is progressively and unidirectionally swept through the part of the overall retardance range, detecting at the detector arrangement at least a spatial portion of the light beam, and extracting the state of polarization based on the spatial portion of the light beam detected at the detector arrangement corresponding to the plurality of retardance values. 
     In another aspect of the invention, there is disclosed a method for use in a polarimeter for analyzing a state of polarization of a light beam incident thereon. The polarimeter includes first and second liquid crystal variable retarders, a detector arrangement and a control arrangement, wherein the first and second liquid crystal variable retarders are configured to exhibit first and second retardance values, respectively, which first and second retardance values are variable over an overall retardance range. The control arrangement is configured to supply first and second voltage signals to the first and second liquid crystal variable retarders, respectively, so as to vary at least one of the first and second retardance values. The method includes the steps of directing the light beam through the first and second variable retarders, for a selected one of the first and second voltage signals, using the control arrangement, applying an initial voltage value so as to produce a particular condition at a corresponding one of the first and second liquid crystal variable retarders, and, once the particular condition is achieved, applying a different voltage value as the selected one of the first and second voltage signals for a given time period such that the corresponding one of the first and second retardance values varies progressively and unidirectionally during the given time period. The method further includes the steps of, during the given time period, detecting at the detector arrangement at least a spatial portion of the light beam responsive to the changing retardance, and extracting the state of polarization based on the spatial portion of the light beam detected at the detector arrangement. 
     In yet another aspect of the invention, there is disclosed a polarimeter for analyzing a state of polarization of a light beam incident thereon. The polarimeter of the present invention includes first and second variable retarders, wherein at least a selected one of the first and second variable retarders is configured to be progressively and unidirectionally variable through an overall retardance range so as to exhibit a plurality of retardance values. The polarimeter also includes a detector arrangement for detecting at least a spatial portion of the light beam for the plurality of retardance values as the selected one of the first and second variable retarders is progressively and unidirectionally varied through at least a part of the overall retardance range. The polarimeter further includes a control arrangement for causing the selected one of the first and second variable retarders to progressively and unidirectionally vary through the part of the overall retardance range and for extracting the state of polarization based on the spatial portion of the light beam detected at the detector arrangement. 
     In still another aspect of the invention, another polarimeter for analyzing a state of polarization of a light beam incident thereon is disclosed. The polarimeter of this aspect of the invention includes first and second liquid crystal variable retarders and a detector arrangement for detecting at least a portion of the light beam during a given time period. The polarimeter also includes a control arrangement configured to initially apply a first voltage signal then to apply, for the given time period, a second voltage signal to at least a selected one of the first and second liquid crystal variable retarders. The control arrangement is further configured to extract the state of polarization based on the spatial portion of the light beam detected at the detector arrangement. 
     In a further aspect of the invention, there is disclosed another method for use in a polarimeter for analyzing a state of polarization of a light beam incident thereon. The polarimeter includes first and second variable retarders and a detector arrangement for taking a measurement of at least a spatial portion of the light beam, wherein the first and second variable retarders are configured to exhibit first and second retardance values, respectively, which first and second retardance values are variable over an overall retardance range. The method includes the steps of directing the light beam through the first and second variable retarders, and varying a selected one of the first and second retardance values over a selected retardance interval. The method further includes the steps of using the detector arrangement to produce a plurality of measurements corresponding to a plurality of measurement points, which plurality of measurement points are incrementally spaced apart across the selected retardance interval, and extracting the state of polarization based on the plurality of measurements. 
     In a still further aspect of the invention, there is disclosed still another method for use in a polarimeter for analyzing a state of polarization of a light beam incident thereon. The polarimeter includes first and second variable retarders and a detector arrangement, wherein the first and second variable retarders are configured to exhibit first and second retardance values, respectively, which first and second retardance values are variable over an overall retardance range. The method includes the steps of calibrating the polarimeter using a plurality of test input light beams of known polarization states to derive a plurality of basis functions, directing the light beam through the first and second variable retarders, and sweeping a selected one of the first and second retardance values progressively and unidirectionally through at least a part of the overall retardance range. The method further includes the steps of, for a plurality of retardance values that are produced as the selected one of the first and second retardance values is progressively and unidirectionally swept through the part of the overall retardance range, detecting at the detector arrangement at least a portion of the input beam, and extracting the state of polarization by fitting a continuous function to the spatial portion of the light beam detected at the detector arrangement using the plurality of basis functions. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     The present invention may be understood by reference to the following detailed description taken in conjunction with the drawings briefly described below. 
     FIG. 1 is a diagrammatic illustration of a Poincaré sphere, shown here as a foundation for understanding the theoretical formalism of Stokes vector SOP characterization. 
     FIGS. 2A-2D are diagrammatic illustrations of some prior art polarimeters. 
     FIG. 2E is a graph of the retardance values versus time for one example of the prior art polarimeter including variable retarders as shown in FIG.  2 D. 
     FIG. 3A is a diagrammatic illustration of a retardance sweep polarimeter of the present invention. 
     FIG. 3B is a graph of the retardance values versus time for one example of the retardance sweep polarimeter of the present invention. 
     FIGS. 4A and 4B are graphs of the time-dependent basis functions calculated in a calibration process in accordance with the present invention. 
     FIG. 5 is a graph of the data collected by the retardance sweep polarimeter of the present invention and the waveform consisting of a linear combination of the basis functions of FIGS.  4 A and  4 B. 
    
    
     DETAILED DESCRIPTION 
     The following description is presented to enable one of ordinary skill in the art to make and use the invention and is provided in the context of a patent application and its requirements. Various modifications to the described embodiments will be readily apparent to those skilled in the art and the generic principles herein may be applied to other embodiments. Thus, the present invention is not intended to be limited to the embodiment shown but is to be accorded the wildest scope consistent with the principles and features described herein. 
     The configuration of a polarimeter manufactured in accordance with the present invention is shown in FIG. 3A. A polarimeter  100  includes first and second LC variable retarders  102 A and  102 B, respectively, and a detector arrangement (indicated by a box  103 ), which in the embodiment shown includes an analyzer  104  and a detector  106 . Analyzer  104  may be, for example, a vertical linear polarizer such that any light that falls onto detector  106  is always of vertical linear polarization. For instance, first LC variable retarder  102 A may be aligned with its optic axis (not shown) aligned with the optic axis (not shown) of analyzer  104 , and second LC variable retarder  102 B may, for example, be aligned with its optic axis (not shown) at a 45° angle with respect to the optic axes (not shown) of the first LC variable retarder and the analyzer. Polarimeter  100  also includes a controller  108 , which controls first and second LC variable retarders  102 A and  102 B as well as detector  106 . First and second LC variable retarders  102 A and  102 B may be based, for example, on a nematic liquid crystal material. Controller  108  is configured to apply voltage signals  110 A and  110 B to first and second LC variable retarders  102 A and  102 B, respectively, so as to control the retardance values of the LC variable retarders. Voltage signals  110 A and  110 B may be, for example, square wave AC voltage signals or DC signals, depending on the LC material used in the LC variable retarders. For example, AC voltage signals are appropriate for use with nematic LC materials. Controller  108  is also arranged such that, as input light  37  enters polarimeter  100 , controller  108  synchronizes the signal detection process at detector  106  with the control of the retardance values at first and second LC variable retarders  102 A and  102 B in a predetermined way, as will be described in detail immediately hereinafter. 
     Controller  108  is configured to apply a set of predetermined voltage values to first and second LC variable retarders  102 A and  102 B such that LC variable retarders progressively and unidirectionally “sweep” through a range of retardance values. In other words, unlike previously known liquid crystal based polarimeters in which the retarders are stepped between specific, discrete retardance values as data are taken at each of these discrete retardance values (as shown in FIG.  2 E), the retardance values of first and second LC variable retarders  102 A and  102 B of polarimeter  100  are progressively and unidirectionally varied while sampling of the signal at detector  106  is synchronized with the retardance sweeps to take a large number of measurements throughout the retardance sweeps. Therefore, polarimeter  100  manufactured in accordance with the present invention is referred to herein as a retardance sweep polarimeter. The details of such retardance “sweeping” is described in further detail immediately hereinafter. 
     Referring now to FIG. 3B in conjunction with FIG. 3A, an exemplary scheme of retardance values versus time for the LC variable retarders of the retardance sweep polarimeter of the present invention is illustrated. A graph  120  includes a vertical axis  121  representing retardance (in units of waves λ) and a horizontal axis  122  representing time (in arbitrary units). A first, solid line  123  shows the retardance value settings of first LC variable retarder  102 A, and a second, dashed line  125  shows the retardance value settings of second LC variable retarder  102 B. As can be seen in FIG. 3B, the first LC variable retarder is initially set to a retardance value of approximately λ, then, at a time T 1 , the retardance value of the first LC variable retarder is made to fall (i.e., “sweep”) until the retardance value reaches approximately λ/4 at a time T 2 , from which time the retardance value of the first LC variable retarder is held at approximately λ/4. The reduction in retardance may be effected, for example, by the application of an increased voltage to first LC variable retarder  102 A by controller  108 . Similarly, the retardance value of the second LC variable retarder is held at an initial value of approximately λ until time T 2 , at which time the retardance value is made to fall past λ/4 at a time T 3 . In other words, the first LC variable retarder sweeps the retardance range of λ to λ/4 between times T 1  and T 2  while the second LC variable retarder is held at a retardance value of λ. Then, while the first LC variable retarder is held at a retardance value of λ/4, the second LC variable retarder is swept in retardance value over the range λ to λ/4 between times T 2  and T 3 . Therefore, a time period between times T 1  and T 2 , indicated by a double headed arrow  127 A, may be considered to be the time period for a retardance sweep # 1 , and a time period between times T 2  and T 3 , indicated by a double headed arrow  127 B, may be considered to be the time period for a retardance sweep # 2 . In the example shown in FIG. 3B, sweep # 1  predominantly provides polarization information on the Stokes components S 1  and S 3 , while sweep # 2  predominantly provides information on component S 2 . 
     It is noted that the retardance values shown in graph  120  are only approximate. In other words, the specific retardance values terminating these retardance value ranges for the first and second LC variable retarders need not be set with great accuracy in the polarimeter of the present invention in order to obtain the desired polarization information of input light because, as will be described hereinafter, the Stokes parameter information is extracted from a series of data sets including a plurality of measurements made during each sweep. It is not necessary to set the start and end retardance values to the approximate values of λ and λ/4. In fact, an infinite variety of retardance value ranges are suitable for use in the present invention. The retardance sweeps are generally from higher to lower retardance values in the case of nematic LC material-based variable retarders since this direction corresponds to increasing applied voltage signals, to which the nematic LC material responds more quickly. Intensity profiles at detector  106  are recorded synchronously with the retardance sweeps then analyzed to extract the desired polarization information. The light intensity data collected at detector  106  is dependent on the state of polarization of input light  37 ; that is, each SOP will have a unique detector signature. 
     It is emphasized that the polarimeter of the present invention yields complete polarization information without the need for high frequency modulation of the variable retarders, high voltage driver electronics or lock-in amplifiers and/or detectors. Especially in contrast to the variable retarder oscillation and lock-in detection scheme discussed in the Background section, the polarimeter of the present invention requires only a progressive, unidirectional retardance sweep rather than repeated retardance oscillation. That is, the polarimeter of the present invention achieves fast, SOP measurements using low cost LC variable retarders and detection system in place of expensive electro-optic or photoelastic modulators with lock-in detection. 
     In order to minimize noise and error due to time-dependent shifts in input light polarization, it is desired to obtain this detector signature in as short a time as possible. For instance, it is possible to improve the response time of a variable retarder based on a nematic liquid crystal material by taking advantage of the transient nematic effect (TNE). TNE is a technique of overshooting a voltage change in order to increase the speed of the LC response. For example, to change a LC retarder from one retardance to a second retardance , rather than just changing the voltage to the steady-state value for the new retardance, the voltage change is temporarily greatly increased. 18  The polarimeter of the present invention takes advantage of the high speed obtainable with TNE and the versatility and low cost of LC materials. By taking advantage of TNE, it is possible to apply predetermined voltage signals from controller  108  so as to cause first and second LC variable retarders  102 A and  102 B to sweep through a range of retardance values very rapidly. 
     In order to understand the operation of polarimeter  100  as shown in FIG. 3A, the theory behind polarimeter  100  is described using Mueller matrix formulation. 19.20  As is well known, each optical component in an optical system may be represented as a 4×4 matrix called a Mueller matrix. Since the initial SOP of input light may be represented by an input Stokes vector, the SOP of input light after passing through the optical component is conveniently provided by an output Stokes vector resulting from multiplying the input Stokes vector by the Mueller matrix of the optical component. The Mueller matrices for some polarization optical components of interest are listed below:              Retarder   ,                optic                 axis                 0      °     ,               retardance                 of                 δ        :            (         1       0       0       0           0       1       0       0           0       0         Cos        (   δ   )             Sin        (   δ   )               0       0         -     Sin        (   δ   )                            Cos        (   δ   )               )               (   5   )               Retarder   ,                optic                 axis                 45      °     ,               retardance                 of                 δ        :            (         1       0       0       0           0         Cos        (   δ   )           0         -     Sin        (   δ   )                 0       0       1       0           0         Sin        (   δ   )           0         Cos        (   δ   )             )               (   6   )                 Linear                 Polarizer     ,               optic                 axis                 0      °        :            (         1       1       0       0           1       1       0       0           0       0       0       0           0       0       0       0         )               (   7   )                                
     Furthermore, an optical arrangement consisting of several optical components in series may itself be represented by a single Mueller matrix, which is constructed by multiplying the Mueller matrices for the optical components in the order in which input light encounters the components. 
     For an ideal, retardance sweep polarimeter (i.e., no loss, no noise, no misalignment, etc.) of FIG. 3A, the resulting Mueller matrix for the combination of first and second LC variable retarders  102 A and  102 B and analyzer  104  is                Perfect                 LC                   Polarimeter   :                
            1   /   2          (         1         Cos        (   δ1   )               Sin        (   δ1   )            Sin        (   δ2   )                 Cos        (   δ2   )            Sin        (   δ1   )                 1         Cos        (   δ1   )               Sin        (   δ1   )            Sin        (   δ2   )                 Cos        (   δ2   )            Sin        (   δ1   )                 0       0       0       0           0       0       0       0         )           ,           (   8   )                                
     which is obtained by matrix multiplication of the matrices in Eqs. (5)-(7). However, an ideal polarimeter is generally a mathematical abstract. For instance, the alignment of the optic axes of the LC variable retarders is rarely exactly 45° in actual devices, which would provide additional non-zero elements on the Mueller matrix of Eqs. (6). Furthermore, losses due to effects such as polarization scattering and retardance-dependent etalon effects (e.g., diattenuation) would require additional matrices not represented in Eqs. (5)-(6). Such non-ideal effects add to the complexity of the Mueller matrix for the polarimeter, but nevertheless a Mueller matrix for each real-life polarimeter does exist. 
     As will be further described, it is recognized in the context of the present invention that determination of the full, exact Mueller matrix for the non-ideal, real life polarimeter is not necessary to achieve a useful and accurate polarimeter. In the retardance sweep polarimeter of the present invention as shown in FIG. 3A, the retardance values of first and second LC variable retarders  102 A and  102 B are rapidly swept by application of the appropriate voltage from controller  108 . It is further recognized that the actual retardance values of the first and second LC variable retarders are not required to accurately obtain polarization information regarding input light incident on the retardance sweep polarimeter, as will be discussed in further detail immediately hereinafter. 
     As shown in FIG. 3B, at the onset of each of retardance sweeps # 1  and # 2 , the retardance of one LC variable retarder is caused to begin changing rapidly with applied voltage (and thereby time). Noting that there is a one-to-one correspondence between time and retardance after application of a predetermined voltage signal (e.g., a voltage step), the parameters in the following derivations are considered to be functions of time rather than retardance. It is notable that the following derivation may also be performed in terms of the measurement point index, i.e., the index number of the sampling points at which detector measurements are taken. 
     Considering first the intensity signal as detected at detector  106  of FIG. 3A, the light incident on the detector in this case is always of a known polarization due to the presence of analyzer  104 . Therefore, the detector may be considered to provide an accurate measure of the first Stokes component (i.e., total light intensity) for the Stokes vector of light which has passed through the polarimeter. The relation may be expressed as                (             S   0   ′          (   t   )                 –                 –               –                 –               –                 –           )     =       (           a        (   t   )             b        (   t   )             c        (   t   )             d        (   t   )                 –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –           )          (           S   0               S   1               S   2               S   3           )               (   9   )                                
     where 
     a(t), b(t), c(t) and d(t)=time-dependent elements of the top row of the Mueller matrix for the polarimeter, 
     S 0 ′(t)=a(t)S 0 +b(t)S 1 +c(t)S 2 +d(t)S 3 =intensity signal at detector, and 
     S 0 , S 1 , S 2 , S 3 =Stokes parameters of the incident light. 
     The 4×4 matrix in Eq. (9) is the Mueller matrix for the non-ideal polarimeter, i.e., including manufacturing imperfections, misalignments, etc. The dashes in Eq. (9) represent vector and matrix elements which are not of concern because the retardance sweep polarimeter is not sensitive to those elements. Accordingly, it is desired to extract the elements in the top row of the retardance sweep polarimeter Mueller matrix as a function of time (to be described) for the retardance sweeps, i.e., the a(t), b(t), c(t) and d(t) functions. Then, once these functions are known, the detected waveform S 0 ′(t) may be least squares fit with the unknown Stokes components S 0 , S 1 , S 2  and S 3  as floated parameters. The Stokes parameters S 0 , S 1 , S 2  and S 3  of the incident light are the desired polarization information. 
     It is notable that there exists an infinite number of measurement sets from which the Stokes vector projections may be extracted because a projection along any arbitrary axis of the Poincaré sphere is related to the projection along the desired axis through a transformation. The process of determining the transformation is equivalent to calibrating the polarimeter. Therefore, determining the a(t), b(t), c(t) and d(t) functions is the calibration process. The calibration is accomplished by application of a series of input light beams with known SOP and recording the signal detected at the detector. Determination of the four desired Mueller matrix elements a(t), b(t), c(t) and d(t) in Eq. (9) requires the application of four known polarizations and the subsequent recording of four intensity waveforms, which may be denoted P 1 , P 2 , P 3  and P 4 . A convenient, but certainly not the only way to perform this process is to sequentially apply light of horizontal linear polarization, vertical linear polarization, +45° linear polarization and right circular polarization in succession while each time sweeping through the retardance values as described above in reference to FIG.  3 B. By substituting the corresponding Stokes vectors for the known input polarization states into Eq. (9), the following relationships are obtained:                (             P      1          (   t   )                 –                 –               –                 –               –                 –           )     =           (           a        (   t   )             b        (   t   )             c        (   t   )             d        (   t   )                 –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –           )          (         1           1           0           0         )       →       P      1     (   t   )       =       a   (   t   )     +     b   (   t   )                 (   10   )                 (             P      2          (   t   )                 –                 –               –                 –               –                 –           )     =           (           a        (   t   )             b        (   t   )             c        (   t   )             d        (   t   )                 –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –           )          (         1             -   1             0           0         )       →       P      2     (   t   )       =       a   (   t   )     -     b   (   t   )                 (   11   )                 (             P      3          (   t   )                 –                 –               –                 –               –                 –           )     =           (           a        (   t   )             b        (   t   )             c        (   t   )             d        (   t   )                 –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –           )          (         1           0           1           0         )       →       P      3     (   t   )       =       a   (   t   )     +     c   (   t   )                 (   12   )                 (             P      4          (   t   )                 –                 –               –                 –               –                 –           )     =           (           a        (   t   )             b        (   t   )             c        (   t   )             d        (   t   )                 –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –               –                 –           –                 –           –                 –           –                 –           )          (         1           0           0           1         )       →       P      4     (   t   )       =       a   (   t   )     +     c   (   t   )                 (   13   )                                
     From Eqs. (10)-(13), it is straight forward to extract the desired a(t), b(t), c(t) and d(t) functions: 
     
       
           a ( t )= P   1 ( t )+ P   2 ( t ) 
       
     
     
       
           b ( t )= P   1 ( t )− P   2 ( t ) 
       
     
     
       
           c ( t )= P   3 ( t )− a ( t ) 
       
     
     
       
           d ( t )= P   4 ( t )− a ( t )  (14) 
       
     
     The aforedescribed calibration process results in four 1-D arrays for the functions a(t), b(t), c(t) and d(t), with the number of elements in the array being dependent upon the sweep duration and sampling speed. For example, typical sweep times in an exemplary polarimeter range from a few milliseconds up to tens of milliseconds, with typical array sizes of a hundred elements. 
     The a(t), b(t), c(t) and d(t) functions may be thought of as a set of basis functions such that the detected intensity profile S 0 ′ (t)=a(t)S 0 +b(t)S 1 +c(t)S 2 +d(t)S 3 , which is transmitted through the polarimeter, is a linear combination of these basis functions with Stokes components S 0 , S 1 , S 2  and S 3  of the incident light as the weighting factors. Examples of calibration curves for the two retardance sweeps according to FIG. 3B are shown in FIGS. 4A and 4B. A graph  130 , shown in FIG. 4A, includes the calculated values versus time for a(t) (a solid line  132   a ), b(t) (a dotted line  132   b ), c(t) (a dashed line  132   c ) and d(t) (a dot-dash combination line  132   d ) for the aforedescribed retardance sweep # 1 . Similarly, a graph  135  of FIG. 4B includes the calculated values versus time for a(t) (a solid line  137   a ), b(t) (a dotted line  137   b ), c(t) (a dashed line  137   c ) and d(t) (a dot-dash combination line  137   d ) for the aforedescribed retardance sweep # 2 . By using these functions a(t), b(t), c(t) and d(t) as basis functions, the Stokes parameters for any input light may be calculated by curve fitting. In this way, since any non-ideal characteristic of the polarimeter is absorbed in the calculation of the a(t), b(t), c(t) and d(t) functions, imperfections in the polarimeter system are accounted for (i.e., calibrated out) in the calculation of the SOP of incident light. For example, it is noted that the c(t) curve in FIG. 4A should be a straight horizontal line according to the Mueller matrix for the ideal polarimeter, Eq. (6), but the measured values of detected intensity are not constant due to the fact that the actual polarimeter on which these measurements were made is not an ideal device. By accounting for such deviations from theory in the calibration process, the SOP of the input light may be calculated with a high degree of accuracy. 
     Turning now to FIG. 5, an example of the extraction of the full SOP for an input light beam is shown performed using calibrated, polarimeter  100  as shown in FIG. 3A. A graph  140  of FIG. 5 includes a plurality of dots, representing the actual data taken, as well as curves  142 A and  142 B obtained by curve fitting the actual data during retardance sweeps # 1  and # 2 , respectively. The best fit curves  142 A and  142 B are found using a least-squares fitting routine using a linear combination of the basis functions a(t), b(t), c(t) and d(t) as the model function, thereby fitting the two retardance sweeps simultaneously to find the Stokes parameters S 0 , S 1 , S 2  and S 3  for the incident light. The data, from which the Stokes vector is extracted, in the case shown in FIG. 5 is a 200 element long array. The use of the fitting routine effectively includes contributions from all 200 elements. 
     Continuing to refer to FIG. 5, it is emphasized that all of the plurality of data points collected during the retardance sweep procedure are utilized in the calculation of the SOP in the polarimeter of the present invention. This characteristic of the present invention is in contrast to, for example, the aforementioned step-wise approach of Oldenbourg et al., in which only four discrete data points are used to calculate the Stokes parameters. There is a significant signal-to-noise advantage to utilizing all of the data collected in the retardance sweeps rather than only four points. Furthermore, the speed enhancement achieved by the use of TNE allows the full data set from the retardance sweeps to be collected over an extremely short time frame of a few to up to tens of milliseconds. As a result, the retardance sweep polarimeter of the present invention is less susceptible to noise in comparison to prior art polarimeters, and measurement-to-measurement reproducibility error of less than 0.1% has been achieved, even with a largely fluctuating source (e.g., power fluctuations on the order of up to 20% over a time scale of tens of seconds). 
     Although each of the aforedescribed embodiments have been illustrated with various components having particular respective orientations, it should be understood that the present invention may take on a variety of specific configurations with the various components being located in a wide variety of positions and mutual orientations and still remain within the spirit and scope of the present invention. Furthermore, suitable equivalents may be used in place of or in addition to the various components, the function and use of such substitute or additional components being held to be familiar to those skilled in the art and arc therefore regarded as falling within the scope of the present invention. For example, although the aforedescribed example embodiment utilizes two sets of retardance sweeps to obtain the complete Stokes vector, it is also possible to calculate all of the Stokes vector components using a single sweep by using a simultaneous, concerted sweep of both retarders. The retarders may be synchronously swept at different rates in a manner analogous to the aforedescribed spinning waveplate approach, in which two passive waveplates are rotated at different rates. Proper timing of the synchronous sweeps would enable the extraction of full SOP data in a single set of measurements (rather than two sequential sweeps). Alternatively, the retardance sweeps of the two retarders may start and end at different times, with the sweeps partially overlapping during a certain time period, while light intensity data detected at the detector arrangement is recorded and analyzed to extract the polarization information. In other words, the retardance sweeps of the retarders need not be triggered simultaneously, as long as the sweeps are performed in a reproducible fashion so as to enable reproducible measurement and calibration. Another possible modification is the use of faster relaxation time liquid-crystal material. With a standard nematic LC, which can be switched quickly in one retardance sweep direction but not as quickly in the reverse sweep direction, thereby decreasing the duty cycle. Faster relaxation time, and thereby quick retardance sweeps in both directions, would increase the duty-cycle and, if the sweep times are fast enough, data could also be recorded in both sweep directions. Moreover, it would be possible to calibrate the retardance sweep polarimeter of the present invention over a certain range of wavelengths, then interpolate the calibration data to enable the measurement of SOP data at wavelengths away from the calibration range of wavelengths. 
     Therefore, the present examples are to be considered as illustrative and not restrictive, and the invention is not to be limited to the details given herein, but may be modified within the scope of the appended claims. 
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