Patent Publication Number: US-2018045889-A1

Title: Method and system for polarization state generation

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     This application claims the benefit of U.S. Provisional Patent Application 62/117,014 filed Feb. 17, 2015 to She et al., titled “Method and System for Polarization State Generation,” the contents of which are incorporated herein by reference in their entirety. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with Government support under Grant No. N66001-13-1-2007, awarded by the U.S. Navy, Space and Naval Warfare Systems Center Pacific. The Government has certain rights in the invention. 
    
    
     FIELD OF THE DISCLOSURE 
     The present disclosure, in general, relates to apparatus and methods for polarization state generation and phase control. 
     BACKGROUND 
     A polarization state generator transforms incoming light into a beam of a desired polarization state. If the generator can produce a beam of any desired polarization state, it is called a complete polarization state generator. It is desirable for a polarization state generator to produce a beam of light with a well-defined state of polarization and phase on demand. Current polarization state generators depend on technologies such as control of optical path length for phase control, indices of refraction and birefringence of optical materials. However, due to the use of moving parts, such as rotating wave plates or bending fibers, current polarization state generators lack stability and repeatability of generated polarization states. Therefore, current techniques for polarization state generation often require additional control systems to compensate for the lack of stability and repeatability of the generated polarization states. In addition, the use of moving parts limits the speed of polarization state generators in changing polarization state. Furthermore, birefringent materials and phase control devices may not easily act on a broad spectrum of wavelengths simultaneously, and moreover may not be readily available in some desired wavelength spectra, such as for an X-ray spectrum or other high-energy electromagnetic radiation spectrum. Cost effective and easy-to-use polarization state generators with improved speed and stability for a broad spectrum of electromagnetic radiation are desired. 
     SUMMARY 
     In an aspect, an apparatus for polarization state generation and phase control includes a Stokes Basis generator to generate multiple Stokes Bases from an input beam, an intensity modulator to modulate an intensity of each of the Stokes Bases, and a beam combiner to combine the modulated Stokes Bases into an output beam. 
     In an aspect, a method of polarization state generation and phase control includes generating multiple Stokes Bases from an input beam; modulating an intensity of each of Stokes Bases; and combining the modulated Stokes Bases into an output beam. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  illustrates a Poincaré sphere. 
         FIG. 2  illustrates an example of a tetrahedron inscribed on a Poincaré sphere with four vertices representing Stokes Bases, according to an embodiment of the present disclosure. 
         FIG. 3  illustrates a high-level schematic of an example of a polarization state generation and phase control device according to an embodiment of the present disclosure. 
         FIG. 4  illustrates internal functional modules of an example of a polarization state generation and phase control device according to an embodiment of the present disclosure. 
         FIG. 5  illustrates an example of a device configured to perform intensity modulation of Stokes Bases according to an embodiment of the present disclosure. 
         FIG. 6  illustrates an example of a device configured to perform intensity modulation of Stokes Bases according to an embodiment of the present disclosure. 
         FIG. 7A  illustrates an example of a device configured to perform intensity modulation of Stokes Bases according to an embodiment of the present disclosure. 
         FIG. 7B  illustrates an example of a digital micromirror device with four regions, each acting as an intensity modulator for one Stokes Basis according to an embodiment of the present disclosure. 
         FIG. 8  illustrates an example of a coherent Stokes Basis generator according to an embodiment of the present disclosure. 
         FIG. 9  illustrates an example of a coherent Stokes Basis generator according to an embodiment of the present disclosure. 
         FIG. 10  illustrates an example of a coherent Stokes Basis generator according to an embodiment of the present disclosure. 
         FIG. 11  illustrates an example of a beam combiner according to an embodiment of the present disclosure. 
         FIG. 12  illustrates an example of a beam combiner according to an embodiment of the present disclosure. 
         FIG. 13  illustrates an example of a polarization state generator according to an embodiment of the present disclosure. 
         FIG. 14  illustrates an example of a polarization state generator according to an embodiment of the present disclosure. 
         FIG. 15A  illustrates an example of a polarization state generator according to an embodiment of the present disclosure. 
         FIG. 15B  illustrates an example of quadrant modulation displayed on a digital micromirror device surface according to an embodiment of the present disclosure. 
         FIG. 16A  illustrates examples of polarization states generated by the polarization state generator illustrated in  FIG. 15A , traced on a Poincaré sphere as the intensity modulator varies the linear combination of pairs of Stokes Bases, according to an embodiment of the present disclosure. 
         FIG. 16B  illustrates an eye diagram of a random bit stream using horizontal and vertical polarization modulation by the polarization state generator illustrated in  FIG. 15A . 
         FIG. 17  illustrates an example of a polarization state generator architecture according to an embodiment of the present disclosure. 
         FIG. 18  illustrates, by way of a Poincaré sphere, an example of a system with four Stokes Bases having a degenerate state of polarization. 
         FIG. 19  illustrates, by way of a Poincaré sphere, an example of a system with four Stokes Bases C 1 , C 2 , C 3 , C 4  optimized for improved uniformity of state of polarization coverage. 
         FIG. 20A ,  FIG. 20B ,  FIG. 20C  and  FIG. 20D  illustrate experimental data for the Stokes Bases of  FIG. 19 . 
         FIG. 21  illustrates an example of a computing device. 
     
    
    
     Some or all of the figures are schematic representations by way of example; hence, they do not necessarily depict the actual relative size or locations of the components or devices shown. The figures are presented for the purpose of illustrating one or more embodiments with the explicit understanding that they will not be used to limit the scope or the meaning of the claims that follow below. 
     DETAILED DESCRIPTION 
     The following examples and embodiments serve to illustrate the present disclosure. These examples are in no way intended to limit the scope of the disclosure. 
     The present disclosure provides techniques for polarization state generation and phase control by controlling optical intensity of spatially separated polarization components. Examples of devices using the techniques are further provided. The techniques described allow for polarization state generators with performance at or close to theoretical limits in speed and stability using current technologies of optical intensity modulation, and hence do not depend on a development of other technologies, such as control of optical path length, and indices of refraction or birefringence of optical materials. 
     As described in the present disclosure, a desired state of polarization (degree of polarization) and phase control are achieved by modulating separated polarization components of one or more input light sources, and combining the modulated and separated polarization components into an output light beam with a desired degree of polarization and phase. The input light source may be, for example, a laser beam, which can be monocoherent, multispectral, or broadband (e.g., a super-continuum laser, or a beam composed of an array of spectrally diverse light sources). The output light beam may be a laser beam. The modulating and/or combining may be performed in an analog or a digital manner. 
     The modulated separated polarization components may be modulated and prepared for combining in their polarization states in any order (e.g., first by intensity modulation and then by polarization state preparation, or vice versa). 
     A beam of light with a specific polarization state and phase can be described as in equation (1), where the polarization state is specified by 
     
       
         
           
             
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             , 
           
         
       
     
     E x  and E y  represent the horizontal and vertical polarization components of the electric field of the light wave, and the phase is specified by e iφ . 
     
       
         
           
             
               
                 
                   
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     If the phase shift θ between E x  and E y  is 0, the light wave is linearly polarized. A relative amplitude of E x  and E y  determines an angle of the linear polarization. If there is a phase shift between E x  and E y , the light wave is generally elliptically polarized, or circularly polarized if the phase shift is exactly 90°. 
     A light wave can be described in terms of its total intensity I, degree of polarization p, and shape parameters of the polarization ellipse. An alternative and mathematically convenient description is given by Stokes parameters S 0 , S 1 , S 2  and S 3 . The Stokes parameters S 0 , S 1 , S 2  and S 3  may also be denoted as I, Q, U and V. Neglecting the first Stokes parameter S 0  (intensity I), the three other Stokes parameters can be plotted directly in three-dimensional Cartesian coordinates. The relationship between the Stokes parameters and the intensity and polarization ellipse parameters can be described as in equations (2), (3), (4) and (5), where I 2 , 2ψ and 2χ are spherical coordinates of the polarization state in a three-dimensional space of the Stokes parameters S 1 , S 2 , and S 3 , the three-dimensional space being known as a Poincaré sphere. 
         S   0   =I   (2)
 
         S   1   =Ip  cos 2ψ cos 2χ  (3)
 
         S   2   =Ip  sin 2ψ cos 2χ  (4)
 
         S   3   =Ip  sin 2χ  (5)
 
     Equation (6) describes a power P related to the Stokes parameters S 1 , S 2  and S 3 . 
         P =√{square root over ( S   1   2   +S   2   2   +S   3   2 )}  (6)
 
       FIG. 1  illustrates a Poincaré sphere of radius P. For a polarized wave with a power given by equation (6), the set of all polarization states can be mapped to points on the surface of the Poincaré sphere of radius P, which then graphically represents the Stokes parameters. The factors of two before ψ and χ in equations (3), (4) and (5) reflect that any polarization ellipse is indistinguishable from another one rotated by 180°, or another one with the semi-axis lengths swapped, accompanied by a 90° rotation. 
       FIG. 2  illustrates a tetrahedron inscribed in the Poincaré sphere, wherein the four vertices represent four particular polarization states A 1 , A 2 , A 3  and A 4 , where A x  is referred to herein as a Stokes Basis. A minimal number of Stokes Bases for full coverage of polarization states on the Poincaré sphere is four. Full coverage of all polarization states on the Poincaré sphere can also be achieved with more than four Stokes Bases, such as five or more, six or more, seven or more, and so forth. There is a constraint that a volume of the tetrahedron inscribed in the Poincaré sphere is greater than zero for full coverage of polarization states on the Poincaré sphere. If full coverage of polarization states in the Poincaré sphere is not desired, a polarization state can be generated with less than four Stokes Bases, such as three or less. 
     The selected Stokes Bases can be prepared using a optical elements, such as linear polarizers and rotatable retarders, or variable circular polarizers. Some non-limiting examples are provided in the present disclosure. 
     Four coherent input source beams corresponding to A 1 , A 2 , A 3  and A 4  of  FIG. 2  can each represent a Stokes Basis having a specified polarization state and global phase given equation (7), where n ranges from 1 to 4. 
     
       
         
           
             
               
                 
                   
                     
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     According to equation (7), a light wave with any arbitrary state of polarization and phase can be generated by modulating the intensity of each of the four source beams, which corresponds to modulating a square of the amplitudes of the four source beams, and combining the four modulated coherent source beams such that they are spatially overlapped. Intensity modulation can be achieved as a result of control over one or more optical processes, such as optical absorption, emission (e.g., directly modulated lasers), reflection, diffusion, scattering, deflection, directional coupling, diffraction, and dispersion. Some non-limiting examples are provided in the present disclosure. 
     Equation (7) can be represented more compactly using complex numbers as shown in equation (8), where the global phase and phase shift e iθ  have been included in the complex notation. 
     
       
         
           
             
               
                 
                   
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     Amplitude modulation of the Stokes Bases A 1 , A 2 , A 3  and A 4  can be described with positive coefficients α 1 , α 2 , α 3  and α 4 ; thus, a resultant electric field after combination can be described as shown in equation (9). 
         Ē=α   1     A   1   +α 2     A   2   +α 3     A   3   +α 4     A   4     (9)
 
     Four equations, shown as equations (10), (11), (12) and (13), can be derived by separating the real and imaginary parts of Equation (9). 
     
       
         
           
             
               
                 
                   
                     
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     The four equations (10), (11), (12) and (13) can be rewritten in a matrix form, as shown in equation (14). 
     
       
         
           
             
               
                 
                   
                     
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     By solving matrix equation (14) to find the desired value of α 1 , α 2 , α 3  and α 4 , and modulating each input source beam accordingly, a beam of light with a specific polarization state θ and phase φ can be generated. 
     One way to solve matrix equation (14) such that α 1 , α 2 , α 3  and α 4  are positive is by treating it as a non-negative least square curve fitting problem. Equation (14) can be rewritten as shown in equation (15). 
         A·α=E   (15)
 
     Then, α 1 , α 2 , α 3  and α 4  can be solved by minimizing equation (16), where α 1 , α 2 , α 3  and α 4  are each greater than zero. 
       min∥ A·α−E∥   (16)
 
     Equation (16) can be solved efficiently using a computing device. 
     Once the intensity (or amplitude) modulation of each Stokes Basis is known, the required intensity modulation can be provided to a device that performs intensity modulation of the input source beams. 
       FIG. 3  illustrates a high-level structure of a polarization state generator  30 . As illustrated, an input light beam S in  is provided to the polarization state generator  30 , which is controlled by a control signal S′ to generate an output light beam S out  of desired state of polarization and phase. The control signal S′ in some embodiments is one or more signals provided by a computing device. 
       FIG. 4  illustrates internal functional modules of an embodiment of the polarization state generator  30  of  FIG. 3 . The device comprises three modules: (1) a beam splitter/preparer  40  (BM 1 ), (2) an intensity modulator  45  (BM 2 ), and (3) a beam combiner  50  (BM 3 ). In some embodiments, a computing device controls configuration of one or more of BM 1 , BM 2  or BM 3 . 
     As shown by way of example in  FIG. 4 , in some embodiments, input light beam S in  is split into four coherent beams in the beam splitter/preparer BM 1 , each being prepared as one of four Stokes Bases. These four beams are individually modulated in intensity by the control signal S′ in the intensity modulator BM 2 , and then combined in the beam combiner BM 3  to form the output light wave S out  with the desired polarization state. As noted above, the input light wave S in  may be split and prepared into 5 or more coherent Stokes Bases. In some embodiments, the input light wave S in  may be split and prepared into 3 or less Stokes Bases. 
     In some embodiments, the Stokes Bases are prepared with substantially equal intensity before entering the intensity modulator BM 2  such that the intensity modulator BM 2  can be omitted, or such that the intensity modulator BM 2  can perform independently of the beam splitter/preparer BM 1 . In some embodiments, intensity values (or other values) can be deemed to be substantially equal if a difference between a largest one of the values and a smallest one of the values is less than or equal to ±10% of the smallest value, such as less than or equal to ±5%, less than or equal to ±4%, less than or equal to ±3%, less than or equal to ±2%, less than or equal to ±1%, less than or equal to ±0.5%, less than or equal to ±0.1%, or less than or equal to ±0.05%. 
     Intensity modulation of Stokes Bases can be performed by a variety of techniques, including but not limited to techniques described in the present disclosure. 
       FIG. 5  illustrates an embodiment of a device  500  configured to perform intensity modulation of Stokes Bases by reflection. Device  500  includes four quadrants. Each quadrant modulates the intensity of one of the four Stokes Bases A 1 , A 2 , A 3  and A 4  by reflective coefficients of α 1 , α 2 , α 3  and α 4 , respectively. Reflected output beams B 1 , B 2 , B 3  and B 4  each have a desired amplitude, phase and state of polarization. 
       FIG. 6  illustrates an embodiment of a device  600  configured to perform intensity modulation of Stokes Bases through transmission. Device  600  includes four quadrants. Each quadrant modulates the intensity of one of the four Stokes Bases A 1 , A 2 , A 3  and A 4  by transmissive coefficients of α 1 , α 2 , α 3  and α 4 , respectively. Transmitted output beams B 1 , B 2 , B 3  and B 4  each have a desired amplitude, phase and state of polarization. 
       FIG. 7A  illustrates an embodiment of a device  700  capable of performing intensity modulation of Stokes Bases by reflection, in the form of a digital micromirror device (DMD) chip  710 . As illustrated, the DMD chip  710  includes many (e.g., hundreds or thousands) of microscopic mirrors  720  arranged in an array (e.g., a rectangular array as shown in  FIG. 7A ) on a surface of the DMD chip  710 . The microscopic mirrors  720  can be individually rotated by certain angle, such as, for example, within about ±10° to about ±12° of a desired angle. In an “on” state, incident light is reflected at the desired angle. In the “off” state, incident light is directed elsewhere, for example, to a heat sink. In some embodiments, an intensity of a total reflected beam (a combined output of the microscopic mirrors  720 ) can be modulated by turning “on” different numbers of microscopic mirrors  720 . In some embodiments, the intensity of the reflected beam can be modulated by turning “on” and “off” different numbers of microscopic mirrors  720  with different “on” time to “off” time ratios. Each microscopic mirror  720  can be controlled individually, and/or a group of microscopic mirrors  720  may be controlled as a group. Control of the microscopic mirrors  720  may be by a computing device. 
       FIG. 7B  illustrates a DMD chip  700  organized into four regions by design or by control, each region acting as an intensity modulator for one Stokes Basis. Each of the four regions can include many (e.g., hundreds or thousands) of microscopic mirrors  720 . Although shown in  FIG. 7A  as approximately equal-area rectangles, in other embodiments, the regions may have a different shape than a rectangle, and each region may have a shape different than other regions. 
     The DMD chip  700  (e.g., as shown in  FIG. 7B ) may be used to implement the reflective intensity modulation of device  500  of  FIG. 5 . It should be noted that other types of arrays may be used to implement the transmissive intensity modulation of device  600  of  FIG. 6  by controlling individual components or groups of components in an array of transmissive components. 
     Referring again to  FIG. 4 , with respect to the beam splitter/preparer BM 1  prior to intensity modulation in the intensity modulator BM 2 , coherent Stokes Bases may be generated by a variety of techniques, including but not limited to the techniques described in the present disclosure. 
       FIG. 8  illustrates an embodiment of the beam splitter/preparer BM 1  described in  FIG. 4 , for generation of coherent Stokes Bases using multiple knife-edge reflectors  810  of substantially equal size. The knife-edge reflectors  810  are offset from each other horizontally and vertically (in the orientation shown). Due to the offsets, the input beam S in  is split into multiple output beams  820 , the number of which is determined by the number of knife-edge reflectors  810  being illuminated by the input beam S in . In the embodiment illustrate in  FIG. 8 , each knife-edge reflector  820  reflects (as output bean  820 ) a substantially equal portion of the input beam S in . The offsets of the knife-edge reflectors  810  also create different delays, and therefore different phases, of the output beams  820  due to different propagation lengths of the paths. A diversion angle ‘a’ between the input light beam S in  and the output beams  820  depends on an incident angle of the input light beam S in  on the reflecting surfaces of the knife-edge reflectors  810 . In the embodiment of  FIG. 8 , the diversion angle ‘a’ is about 90°. In other embodiments, other diversion angles are implemented. Each output beam  820  can be converted to a Stokes Basis beam with defined polarization using a polarizer  830  (e.g., a linear polarizer or a circular polarizer). Each polarizer  830  can be an absorptive polarizer, beam-splitting polarizer, reflective polarizer, birefringent polarizer, thin film polarizer, or other type of polarizer, and a different type of polarizer  830  may be used for each output beam  820 . 
       FIG. 9  illustrates an embodiment of the beam splitter/preparer BM 1  described in  FIG. 4 , for generation of coherent Stokes Bases using a fiber optic beam splitter  910 . The fiber optic beam splitter  910  can be, for example, a fused biconical taper splitter, a waveguide planar lightwave circuit splitter, or other beam splitters coupled and integrated with optical fibers. An input beam S in  is coupled to an input  920  of the beam splitter  910 . Each output of the beam splitter is provided by an output fiber (e.g., output fibers  911 ,  912 ,  913 ,  914 ). The propagation paths of the outputs may be of different lengths, thus creating different delays and phases in each output beam. The output beams of the beam splitter can each be converted to a Stokes Basis beam with defined polarization using a polarizer  930 , similarly as described above with respect to polarizers  830 . In some embodiments, the polarizer  930  may be a fiber optic polarizer or other integrated optical polarizer. 
       FIG. 10  illustrates an embodiment of the beam splitter/preparer BM 1  described in  FIG. 4 , for generation of coherent Stokes Bases using bulk beam splitters (e.g., BS 1 , BS 2  and BS 3 ) and mirrors (e.g., mirrors M 1  and M 2 ). In some embodiments, a bulk beam splitter is a cube made from two triangular glass prisms glued or otherwise combined together. An interface of the two prisms is designed such that a portion of an incident light beam is reflected and another portion of the incident light beam is transmitted, such as by frustrated total internal reflection or birefringent polarization beam splitting. In some embodiments, the reflected light and the transmitted light are of substantially equal intensity. 
     As illustrated in the embodiment of  FIG. 10 , the input beam S in  is split into a reflected beam  1006  and a transmitted beam  1007  by a beam splitter  1005  (BS 1 ). The transmitted beam  1007  is split into a reflected beam  1011  and a transmitted beam  1012  by a beam splitter  1010  (BS 2 ). The reflected beam  1011  is then reflected as a reflected beam  1021  by a mirror  1020  (M 1 ). The reflected beam  1006  from the beam splitter BS 1  is split into a reflected beam  1031  and a transmitted beam  1032  by a beam splitter  1030  (BS 3 ). The transmitted beam  1032  is reflected as a reflected beam  1041  by a mirror  1040  (M 2 ). As a result, input beam S in  is split into four beams, the transmitted beam  1012 , the reflected beam  1021 , the transmitted beam  1032 , and the reflected beam  1041  by the beam splitter/preparer of  FIG. 10 . Each of these four beams can then be converted to a Stokes Basis beam with defined polarization using a polarizer  1050 , similarly as described above with respect to polarizers  830 . 
     In some embodiments, the bulk beam splitter may incorporate one or more half-silvered mirrors for reflection and transmission (e.g., as the beam splitter BS 1 , BS 2  or BS 3 ). 
     Referring again to  FIG. 4 , with respect to the beam combiner BM 3 , a combination beam may be generated by a variety of techniques, including but not limited to the techniques described in the present disclosure. The beam combiner BM 3  combines output beams of the intensity modulator BM 2  (e.g., the output beams B 1 , B 2 , B 3  and B 4  illustrated in  FIG. 5  or  FIG. 6 ) such that the output beams are spatially overlapped. 
       FIG. 11  illustrates an embodiment of the beam combiner BM 3  of  FIG. 4 . In the embodiment illustrated in  FIG. 11 , fiber optic power splitters are used in a reverse direction to form a beam combiner  1110 . Output beams B 1 , B 2 , B 3  and B 4  from an intensity modulator (e.g., the output beams B 1 , B 2 , B 3  and B 4  of the intensity modulators  500  or  600  respectively illustrated in  FIG. 5  or  FIG. 6 ) are provided as inputs to the beam combiner  1110  of  FIG. 11 . The fiber optic power splitter  1110  illustrated in  FIG. 11  can be the same as or similar to the fiber optic power splitter  910  illustrated in  FIG. 9 . A combined beam S out  is output by the beam combiner  1110 . 
       FIG. 12  illustrates an embodiment of the beam combiner BM 3  of  FIG. 4 . In the embodiment illustrated in  FIG. 12 , non-polarizing beam splitters (e.g., BS 1 ′, BS 2 ′ and BS 3 ′) are used with mirrors (e.g., M 1 ′ and M 2 ′) to form a beam combiner. Output beams B 1 , B 2 , B 3  and B 4  from an intensity modulator (e.g., the output beams B 1 , B 2 , B 3  and B 4  of the intensity modulators  500  or  600  respectively illustrated in  FIG. 5  or  FIG. 6 ) are provided as inputs to the beam combiner of  FIG. 12 . 
     As illustrated in the embodiment of  FIG. 12 , beam B 3  is reflected as a reflected beam  1221  by a mirror  1220  (M 1 ′), and the reflected beam  1221  is partially reflected by a beam splitter  1205  (BS 1 ′) and the partially reflected portion of the reflected beam  1221  is combined with a portion of the beam B 4  transmitted by the beam splitter BS 1 ′ into a transmitted beam  1207 . The reflected beam  1221  is partially transmitted by the beam splitter BS 1 ′, and the partially transmitted portion of the reflected beam  1221  is combined with a portion of the beam B 4  reflected by the beam splitter BS 1 ′ into a reflected beam  1206 , which is directed to an absorber ABS 1 . 
     The beam B 1  is reflected as a reflected beam  1241  by a mirror  1240  (M 2 ′), and the reflected beam  1241  is partially reflected by a beam splitter  1230  (BS 3 ′) and the partially reflected portion of the reflected beam  1241  is combined with a portion of the beam B 2  transmitted by the beam splitter BS 3 ′ into a transmitted beam  1232 . The reflected beam  1241  is partially transmitted by the beam splitter BS 3 ′, and the partially transmitted portion of the reflected beam  1241  is combined with a portion of the beam B 2  reflected by the beam splitter BS 3 ′ into a reflected beam  1231 , which is directed to a beam splitter  1210  (BS 2 ′). 
     The reflected beam  1241  is partially reflected by the beam splitter BS 3 ′ and the partially reflected portion of the reflected beam  1241  is combined with a portion of the beam B 4  transmitted by the beam splitter BS 3 ′ into a transmitted beam  1232 , which is directed to an absorber ABS 3 . 
     The transmitted beam  1207  is partially reflected by the beam splitter BS 2 ′, and is combined with a portion of the reflected beam  1231  transmitted by the beam splitter BS 2 ′ into a reflected beam  1211 , which is directed to an absorber ABS 2 . The transmitted beam  1207  is partially transmitted by the beam splitter BS 2 ′, and is combined with a portion of the reflected beam  1231  reflected by the beam splitter BS 2 ′ into a transmitted beam  1212 , which is an output S out  of the beam combiner. As a result, the four beams B 1 , B 2 , B 3  and B 4  are combined into a single output S out . 
     A portion of beam B 4  is transmitted through a beam splitter BS 1 ′, and another portion of the output beam B 4  is reflected by the beam splitter BS 1 ′ to a light absorber ABS 1 . A portion of an output beam B 3  from the intensity modulator (BM 2 ) is reflected by a mirror M 1 ′ and also by the beam splitter BS 1 ′, and is combined with the transmitted portion of the output beam B 4 . Another portion of the output beam B 3  is transmitted through the beam splitter BS 1 ′ to the light absorber ABS 1 . The combined beam of output beams B 3  and B 4  after the beam splitter BS 1 ′ is partially transmitted through a beam splitter BS 2 ′ to form a part of the output beam S out , and is partially reflected to a light absorber ABS 2 . A portion of an output beam B 2  from the intensity modulator (BM 2 ) is transmitted through a beam splitter BS 3 ′ and is absorbed by a light absorber ABS 3 ; another portion of the output beam B 2  is reflected by the beam splitter BS 3 ′ towards the beam splitter BS 2 ′. A portion of an output beam B 1  from the intensity modulator (BM 2 ) is reflected by a mirror M 2 ′, is further reflected by the beam splitter BS 3 ′ and is absorbed by the light absorber ABS 3 ; another portion of the output beam B 1  is transmitted through the beam splitter BS 3 ′ and is combined with the reflected portion of the output beam B 2 . The combined beam of output beams B 1  and B 2  after the beam splitter BS 3 ′ is partially reflected by the beam splitter BS 2 ′ to form another part of the output beam S out , and is partially transmitted through the beam splitter BS 2 ′ towards the light absorber ABS 2 . The output beam S out  thus comprises a portion of each of the output beams B 1 , B 2 , B 3  and B 4  from the intensity modulator (BM 2 ). 
     In some embodiments, the polarization state generator disclosed herein has an accuracy of polarization generation of better than about 1°, better than about 0.5°, better than about 0.2°, better than about 0.1°, better than about 0.05°, or better than about 0.01°. In some embodiments, the polarization state generator disclosed herein has a repeatability of polarization generation of better than about 1°, better than about 0.5°, better than about 0.2°, better than about 0.1°, better than about 0.05°, or better than about 0.01°. 
     The polarization state generator disclosed herein can be used for polarization modulation in optical communication. For example, polarization state generators may be used for polarization modulation in polarization-division multiplexing (PDM), which can be used together with phase modulation, optical quadrature amplitude modulation (QAM) or other advanced coding techniques, allowing transmission speeds of about 100 Gigabit per second or more using a single wavelength. Sets of polarization-division multiplexed wavelength signals can then be carried over a wavelength-division multiplexing (WDM) infrastructure to substantially expand its capacity. 
     The polarization state generator disclosed herein can also be used in polarization analysis, spectropolarimetry, spectral ellipsometry, swept-wavelength measurement, and monitoring of polarization-related parameters and signal-to-noise ratios of optical networks. The polarization state generator disclosed herein can also be used in other technology areas, such as chemistry, biology or astronomy. 
     Having described the individual techniques of the present disclosure, some examples are next provided of polarization state generation and phase control devices. 
     EXAMPLES 
     Example 1 
       FIG. 13  illustrates an example of an embodiment of a polarization state generation and phase control device according to the structure illustrated in  FIG. 4  (reproduced in  FIG. 13  as structure  1300 ). In the embodiment illustrated in  FIG. 13 , the beam splitter/preparer BM 1  for Stokes Bases generation is implemented using bulk beam splitters and mirrors as illustrated and described with respect to  FIG. 10  (represented as beam splitter/preparer  1310  in  FIG. 13 ). The intensity modulator BM 2  is implemented as a reflective DMD as illustrated and described with respect to  FIG. 5  (represented as intensity modulator  1320  in  FIG. 13 ). The beam combiner BM 3  is implemented using bulk beam splitters, mirrors and light absorbers as illustrated and described with respect to  FIG. 12  (represented as beam combiner  1330  in  FIG. 13 ). 
     Example 2 
       FIG. 14  illustrates an example of an embodiment of a polarization state generation and phase control device according to the structure illustrated in  FIG. 4  (reproduced in  FIG. 13  as structure  1400 ). In the embodiment illustrated in  FIG. 14 , the beam splitter/preparer BM 1  for Stokes Bases generation is a fiber or waveguide power splitter as illustrated and described with respect to  FIG. 9  (represented as beam splitter/preparer  1410  in  FIG. 14 ). The intensity modulator BM 2  is a transmissive intensity modulator as illustrated and described with respect to  FIG. 6  (represented as intensity modulator  1420  in  FIG. 14 ). The beam combiner BM 3  is a fiber or waveguide power splitter used in reverse direction as illustrated and described with respect to  FIG. 11  (represented as beam combiner  1340  in  FIG. 14 ). 
     Example 3 
       FIG. 15A  illustrates an example of an embodiment of a polarization state generation and phase control device according to the structure illustrated in  FIG. 4 . In the embodiment illustrated in  FIG. 15A , a light beam  1502  generated by a light source  1501  (e.g., a He—Ne laser) is reflected by a mirror  1505 , and the reflected beam  1506  is passed through a 45° linear polarizer  1507  to set an initial polarization state of beam  1508  to a known state. The linearly polarized beam  1508  may be the input S in  illustrated in  FIG. 4 . 
     The linearly polarized beam  1508  is split into four beams A 1 , A 2 , A 3  and A 4  through a beam splitter (BS)  1510  and four variable circular polarizers (VCPs)  1511 ,  1512 ,  1513 ,  1514 . The BS  1510  and the VCPs  1511 ,  1512 ,  1513 ,  1514  together form a beam splitter/preparer (BM 1  in  FIG. 4 ). 
     The intensities of the four beams A 1 , A 2 , A 3  and A 4  are equalized by four variable neutral density filters (VNDFs)  1515 ,  1516 ,  1517 ,  1518 , respectively, and output as four spatially-separated beams which are then reflected by a respective mirror  1520 ,  1521 ,  1522 ,  1523  labeled ‘M’ (e.g., in an array of mirrors) to a modulator  1525 . The four spatially-separated beams are modulated by the modulator  1525  and output as beams B 1 , B 2 , B 3  and B 4 . In an embodiment, a DMD device (e.g., a Texas Instruments DMD device DLP3000) is used as the modulator  1525 , and the four spatially-separated beams strike four quadrants on the surface of the DMD and are reflected off the DMD after modulation. The VNDFs and the DMD together from an intensity modulator (BM 2  in  FIG. 4 ). 
     The four beams A 1 , A 2 , A 3  and A 4  in this embodiment are the Stokes Bases prior to modulation, and the four output beams B 1 , B 2 , B 3  and B 4  are the modulated Stokes Bases. An iris  1530  selects the strongest diffraction order of each of the four output beams B 1 , B 2 , B 3  and B 4 . 
     The four beams B 1 , B 2 , B 3  and B 4  are recombined using mirrors  1535 ,  1536 ,  1537 ,  1538  and  1540  and beams splitters  1545 ,  1546 ,  1547 , to generate a single polarized, modulated light beam output S out  with a desired state of polarization or degree of polarization. The mirrors and beam splitters together form a beam combiner (BM 3  in  FIG. 4 ). 
       FIG. 15B  illustrates an example of four-quadrant modulation displayed on a DMD surface. 
       FIG. 16A  illustrates polarization states generated by the polarization state generator illustrated in  FIG. 15A , traced on a Poincaré sphere as the intensity modulator varies a linear combination of pairs of Stokes Bases. A trajectory is measured by a polarimeter, and the data are plotted. The Stokes Bases are labeled C 1 -C 4 . Polarization trajectories are generated by coherent combination and incoherent combination as indicated. A polarization trajectory is generated by coherent combination by maintaining an optical path length between the two Stokes Bases C 2  and C 3  well below the coherence length of the laser (approximately 20 cm). An incoherent polarization trajectory following the geodesic path between Stokes Bases C 2  and C 4  is generated by making the optical path length between Stokes Bases much longer than the coherence length of the laser (hence reducing the mutual coherence). 
       FIG. 16B  illustrates an eye diagram of a bit stream using horizontal and vertical polarization modulation by the polarization state generator illustrated in  FIG. 15A , where the bit stream is a random bit stream at 4 kilohertz. 
     Example 4 
       FIG. 17  illustrates an example of a polarization state generator architecture, in which a number N of light sources (A 1 , A 2 , A 3  . . . A N ) with well-defined states of polarization and relative phase (representing a number N of Stokes Bases) are provided to respective modulators  1711 ,  1712 ,  1713  . . .  171 N. Output beams B 1 , B 2 , B 3  . . . B N  from respective modulators  1711 ,  1712 ,  1713  . . .  171 N are combined by weighted linear superposition at a beam combiner  1720  to produce a desired output signal S out . 
     Example 5 
       FIG. 18  illustrates, by way of a Poincaré sphere, an example of a system with four Stokes Bases C 1 , C 2 , C 3 , C 4  having a degenerate state of polarization. The four states of polarization in the system are linear horizontal (C 1 ), vertical (C 2 ), +45° with a 180° phase shift (C 3 ), and right circular polarization (C 4 ). A Monte Carlo simulation was performed (results shown by the dots) by randomly varying intensity modulation parameters. As can be seen, the results show relatively complete, yet non-uniform coverage of states of polarization over the Poincaré sphere. A polarization trajectory between Stokes Bases C 3  to C 4  is shown for coherent combination (line  1810 ) and incoherent combination (line  1820 ). Incoherent trajectories are geodesics. 
     Example 6 
       FIG. 19  illustrates, by way of a Poincaré sphere, an example of a system with four Stokes Bases C 1 , C 2 , C 3 , C 4  optimized for improved uniformity of state of polarization coverage. With comparison to the degenerate Stokes Bases system described by  FIG. 18 , the Stokes Bases C 1 , C 2 , C 3 , C 4  of  FIG. 19  are vertices of a regular tetrahedron inscribed in the Poincaré sphere. In Jones vector notation, the Stokes Bases C 1 , C 2 , C 3 , C 4  or  FIG. 19  are respectively [0.7071, 0.7071i], [−9.856, 0.1691i], [0.5141, 0.7941−0.3242i], and [0.5141, −0.7941−0.3242 i].    
       FIGS. 20A-20D  illustrate experimental data for the Stokes Bases C 1 , C 2 , C 3 , C 4  of  FIG. 19  using the polarization state generation and phase control device of  FIG. 15A . The Stokes Bases are set to states of polarization approximating (within the error of tuning the VCPs) a regular tetrahedron on the Poincaré sphere. The Stokes Bases C 1 , C 2 , C 3 , and C 4  were measured and the resulting tetrahedron drawn ( FIG. 20A ). Coherent polarization trajectories from each Stokes Basis to every other Stokes Basis were generated by modulating Stokes Bases intensities in twenty discrete increments spanning twenty seconds, and the raw data as measured by a polarimeter are shown ( FIG. 20A ). A Monte Carlo experiment was performed in which 200 random intensity modulation parameters α were used. The results are shown on a Poincaré sphere ( FIG. 20B ), indicating good uniformity of coverage of the states of polarization. 
       FIG. 20C  plots time series data (dots on the plot) of a coherent polarization trajectory between Stokes Bases C 2  and C 4  (of  FIG. 20A ) against theoretical calculations (dotted lines on the plot), and show good agreement, where S 1 , S 2  and S 3  are elements of the Stokes vector. 
       FIG. 20D  is an eye pattern generated for a polarization signal that switches between linear horizontal and vertical polarizations using a DLP3000 DMD device. The data are shown for a pseudorandom bitstream modulated at 1 kHz. The inset  2010  is an enlarged view of the rectangle  2020 , showing a measured settling time (eye rise and fall time) to be 3.5 microseconds (μs), following an exponential path. 
       FIG. 21  illustrates an example of a computing device  200 , such as may be used to control components in a polarization signal generation system according to an embodiment of the present disclosure. Computing device  200  that includes a processor  210 , a memory  220 , an input/output interface  230 , and a communication interface  240 . A bus  250  provides a communication path between two or more of the components of computing device  200 . The components shown are provided by way of illustration and are not limiting. Computing device  200  may have additional or fewer components, or multiple of the same component. 
     Processor  210  represents a programmable processor, which may be, for example, a general-purpose processor, digital signal processor, microprocessor, microcontroller, application specific integrated circuit (ASIC), field programmable gate array (FPGA), other circuitry effecting processor functionality, or multiple ones or combinations of the foregoing, along with associated logic and interface circuitry. Processor  210  may be incorporated in a system on a chip. 
     Computing device  200  may include code that creates an execution environment for a computer program, such as code that constitutes processor firmware, a protocol stack, a database management system, an operating system, a cross-platform runtime environment, a virtual machine, or a combination of one or more of the foregoing. 
     A computer program (also known as a program, software, software application, script, instructions or code) can be written in any form of programming language, including compiled or interpreted languages, declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, object, or other unit suitable for use in a computing environment. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub-programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a network. 
     Memory  220  represents one or both of volatile and non-volatile memory for storing information (e.g., instructions and data). Examples of memory include semiconductor memory devices such as EPROM, EEPROM, flash memory, RAM, or ROM devices, magnetic media such as internal hard disks or removable disks or magnetic tape, magneto-optical disks, CD-ROM and DVD-ROM disks, holographic disks, and the like. 
     Portions of controlling a polarization signal generation system may be implemented as computer-readable instructions in memory  220  of computing device  200 , executed by processor  210 . 
     An embodiment of the disclosure relates to a non-transitory computer-readable storage medium (e.g., memory  220 ) having computer code thereon for performing various computer-implemented operations. The term “computer-readable storage medium” is used herein to include any medium that is capable of storing or encoding a sequence of instructions or computer codes for performing the operations, methodologies, and techniques described herein. The media and computer code may be those specially designed and constructed for the purposes of the embodiments of the disclosure, or they may be of the kind well known and available to those having skill in the computer software arts. 
     Examples of computer code include machine code, such as produced by a compiler, and files containing higher-level code that are executed by a computer using an interpreter or a compiler. For example, an embodiment of the disclosure may be implemented using Java, C++, or other object-oriented programming language and development tools. Additional examples of computer code include encrypted code and compressed code. Moreover, an embodiment of the disclosure may be downloaded as a computer program product, which may be transferred from a remote computer (e.g., a server computer) to a requesting computer (e.g., a client computer or a different server computer) via a transmission channel. Another embodiment of the disclosure may be implemented in hardwired circuitry in place of, or in combination with, machine-executable software instructions. 
     Input/output interface  230  represents electrical components and optional code that together provide an interface from the internal components of computing device  200  to external components. Examples include a driver integrated circuit with associated programming. 
     Communication interface  240  represents electrical components and optional code that together provides an interface from the internal components of computing device  200  to external networks. Communication interface  240  may be bi-directional, such that, for example, data may be sent from computing device  200 , and instructions and updates may be received by computing device  200 . 
     Bus  250  represents one or more interfaces between components within computing device  200 . For example, bus  250  may include a dedicated connection between processor  210  and memory  220  as well as a shared connection between processor  210  and multiple other components of computing device  200 . 
     As used herein, the terms “substantially,” “substantial,” “approximately” and “about” are used to describe and account for small variations. When used in conjunction with an event or circumstance, the terms can encompass instances in which the event or circumstance occurs precisely as well as instances in which the event or circumstance occurs to a close approximation. For example, when used in conjunction with a numerical value, the terms can encompass a range of variation of less than or equal to ±10% of that numerical value, such as less than or equal to ±5%, less than or equal to ±4%, less than or equal to ±3%, less than or equal to ±2%, less than or equal to ±1%, less than or equal to ±0.5%, less than or equal to ±0.1%, or less than or equal to ±0.05%. 
     While the present disclosure has been described and illustrated with reference to specific embodiments thereof, these descriptions and illustrations are not limiting. It should be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the true spirit and scope of the present disclosure as defined by the appended claims. The illustrations may not necessarily be drawn to scale. There may be distinctions between the artistic renditions in the present disclosure and the actual apparatus due to manufacturing processes and tolerances. There may be other embodiments of the present disclosure which are not specifically illustrated. The specification and the drawings are to be regarded as illustrative rather than restrictive. Modifications may be made to adapt a particular situation, material, composition of matter, method, or process to the objective, spirit and scope of the present disclosure. All such modifications are intended to be within the scope of the claims appended hereto. While the methods disclosed herein have been described with reference to particular operations performed in a particular order, it will be understood that these operations may be combined, sub-divided, or re-ordered to form an equivalent method without departing from the teachings of the present disclosure. Accordingly, unless specifically indicated herein, the order and grouping of the operations are not limitations.