Abstract:
A method of measuring fiber twist in a multi-core optical fiber bearing an FBG with polarization dependent reflectivity. The state of polarization of the launched light is adjusted until the reflected FBG wavelength is maximal, indicating that light reaching the FBG is linearly polarized, and the polarization axis of the light reaching the FBG is aligned with the slow birefringent axis of the FBG; the SOP of launched light is now measured. Bending experienced by the fiber is measured conventionally, and birefringence produced by bending of the multi-core optical fiber is calculated. A candidate amount of twist between the launch location and the FBG is proposed, and the corresponding twist-induced birefringence is calculated. When calculations show that light with the launched SOP becomes linearly polarized and aligned with the FBG after traversing a fiber section with the calculated birefringences and proposed rotation, the amount of twist has been properly identified.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims priority to U.S. Provisional Patent. Application Ser. No. 61/112,227, entitled “MEASUREMENT OF FIBER TWIST BY POLARIZATION TRACKING,” to Askins. This application is a divisional of U.S. patent application Ser. No. 12/614,309 to Askins. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention relates in general to a method and apparatus for measuring twist in an optical fiber which may concurrently experience bending, and more particularly to a method and apparatus for measuring twist by tracking polarization of light in the optical fiber. 
       BACKGROUND OF THE INVENTION 
       [0003]    For a range, of applications, it is useful to deduce the shape that a flexible optical fiber occupies in space using only measurements of light traveling within the fiber; this shape generally includes both bending and twisting deflections. Traditional shape estimation failed to address torsion. Such a failure is permissible in a torsionally rigid optical fiber, wherein bend states of direction and curvature fully and accurately describe the three-dimensional geometry of the optical fiber. However, torsional deflection of the optical fiber can make for entirely different geometries as compared to the geometry of a torsionally rigid optical fiber. 
         [0004]    As shown in  FIG. 1 , a prior art method and apparatus  10  use fiber Bragg gratings  20  (“FBGs”) in a multi-core optical fiber  30  having multiple cores and a helical bias of some peripheral cores. See, e.g., C. G. Askins, et al., Proc. Bragg Gratings, Photosensitivity and Poling in Glass Waveguides 2007, JWA39, Quebec Canada, incorporated herein by reference, and C. G. Askins et al., OFC/NFOEC 2008. Tech. Digest, OMT3, San Diego Calif. (“Askins OFC 2008”), incorporated herein by reference. In that approach, local uniaxial strains sensed, by the variously placed and oriented FBGs are used to deduce both bending and twisting. The prior art apparatus  10  includes a polarization controller  40 , a charge-coupled device (“CCD”) spectrometer  50 , a processor  60 , and a broadband light source  70 . The polarization controller  40  is computer-driven, and is capable of outputting any state of polarization (“SOP”) from any input SOP. The spectrometer  5 . 0  permits high-speed, determination of the center wavelength of light reflected from the multiple FBGs. The processor  60  adjusts the controller in response to the wavelength measured by the spectrometer  50 . 
         [0005]    In addition to the physical bases of that approach, polarization-dependent reflectivity (“PDR”) is another optical effect in optical fibers bearing Bragg gratings which may be exploited. With PDR, the spectrum of light reflected by a FBG is seen to shift in wavelength depending on the SOP of the light illuminating it. In particular, when light is linearly polarized and aligned with the “slow” PDR axis, the longest Bragg reflection wavelength is observed; light polarized normal to this direction produces the shortest Bragg reflection wavelength (“Bragg wavelength” is used as shorthand for the midpoint or centroid of a FBG reflection spectrum having finite width).  FIG. 2  shows an experimental graph of PDR, where two distinct reflection spectra are seen from the same FBG illuminated with orthogonal, polarizations. PDR results both from intrinsic fiber birefringence and from the grating inscription process; the orientation of the two birefringence components may not necessarily align. The rotational orientation of the net birefringence is fixed with respect to the fiber, and is nearly unperturbed by significant bending or twisting for gratings in cores at the centerline of the fiber. While large lateral stresses can disturb PDR orientation and magnitude, such stresses may be avoided in many applications of interest. 
         [0006]    According to this prior art method, a fiber-squeezing polarization scrambler is used to vary the SOP of light illuminating the FBG. Alternatively, any of several designs of polarization controller using various means of optical phase modulation (e.g., electro-optic modulation and adjustable bulk optics), are also capable of this function. The initial purpose of the scrambler is to rapidly vary the SOP to remove polarization sensitivity. However, an alternative to polarization scrambling is systematic modulation of SOP in response to rapid measurements of Bragg wavelength to permit tracking of the maximum (or minimum) PDR wavelength. The signals are monitored by CCD spectrometer to obtain a real-time feedback signal to adjust the force on three fiber-squeezing actuators, constituting an electro-mechanical polarization controller. Representative shifts in measured wavelength (vertical axis) are plotted versus time (horizontal axis) in  FIG. 3 . The signal represents how the measured Bragg wavelength is seen to vary as a representative sampling of all possible states of polarization (SOP) sequentially illuminate the FBG by use of a rapidly adjustable polarization controller. For the example, the strain state of the FBG is not varied during the measurement interval. The graph shows the smoothly-evolving nature of the signal, and its suitability for an algorithm which automatically “tracks” the maximum or minimum Bragg wavelength by applying adjustments to the polarization controller. Dashed line “A” is the maximum measured Bragg wavelength for a constant strain state: this occurs when light is linearly polarized and aligned with the slow birefringent axis at the location of the fiber Bragg grating (FBG). Dashed line “B” is the minimum measured Bragg wavelength: this occurs when light is linearly polarized and aligned with the fast birefringent axis at the location of the fiber Bragg grating (FBG). 
         [0007]    Polarization controllers are commonly available and apply birefringence to an optical path (which may constitute, or be connected to, an optical fiber) and monitor the resulting SOP with a polarization analyzer. By way of example, Applicant&#39;s polarization tracking algorithm is described as follows. The polarization control is modified in response to a spectral measurement and finds and tracks maxima or minima of Bragg wavelengths, thereby maintaining polarization alignment with the FBG. Light from a broadband source is guided by a single-moded optical fiber which is subjected to lateral compression by three or more actuators. Each actuator modulates a compressive force along a direction perpendicular to the fiber&#39;s axis, and substantially differing from each of the other actuators. Each actuator is capable of applying sufficient stress to produce birefringence exceeding a full wave of phase difference. This configuration is sufficient to convert any input SOP to any other. The measured Bragg wavelength of an FBG is recorded as all squeezers are rapidly modulated. When the largest observed Bragg wavelength is identified, the squeezer drive signals are noted. From this point forward, the squeezers are only “dithered” over a reduced range and the Bragg wavelength monitored to guide small, frequent offsets to the average setting of each squeezer. Occasionally, the end of the adjustment range of a squeezer is approached, and a large step adjustment corresponding to a one-wave phase shift is applied toward the center of the squeezer&#39;s operational range. In the example, fiber squeezers are used to produce adjustable birefringence. Alternatively, electro-optic modulators, bulk optical elements, or other suitable phase modulation devices may be used for the same purpose. Examples of suitable commercial products include Adaptif model A3000 and Thorlabs model IPM5300. 
         [0008]    Although the FBG is not disturbed (i.e., no changes in temperature or strain), when the birefringence-inducing fiber squeezers are modulated, the indicated strain also modulates. The computer or processor rapidly executes the above-mentioned algorithm to adjust fiber squeezer drive signals to converge on the maximum (or minimum) PDR wavelength, which amounts to establishing linearly polarized light aligned with the “slow” (or fast) PDR axis of the FBG. As a new strain state occurs in the fiber due to incremental bending or twisting, the algorithm is applied to re-establish polarization alignment with the FBG. 
       BRIEF SUMMARY OF THE INVENTION 
       [0009]    [NOTE TO SELF: ADD SUMMARY OF INVENTION PARAGRAPHS ONCE CLAIMS HAVE BEEN APPROVED]. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]      FIG. 1  is a graph of illustrative spectra showing PDR in a FBG. 
           [0011]      FIG. 2  is a schematic illustration of a prior art apparatus for tracking PDR alignment in a FBG. 
           [0012]      FIG. 3  is a graph of illustrative data showing variation in the Bragg reflected wavelength as the SOP is modified by a polarization controller. 
           [0013]      FIG. 4  is a schematic illustration of an embodiment of the inventive apparatus to measure twist and bending in a multi-core optical fiber by tracking PDR axes of FBGs in the center core. 
           [0014]      FIG. 5  is a schematic illustration of another embodiment of the inventive apparatus to measure twist and bending in a multi-core optical fiber by tracking PDR axes of FBGs in the center core. 
           [0015]      FIG. 6  is an illustrative multi-core optical fiber for use with the instant invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0016]    The basis for sensing twist in a fiber with the inventive apparatus and method may be conveyed by a thought experiment. Consider any of several optical elements which exhibit reflectivity which is dependent upon the polarization of the light illuminating their surfaces. Determination of the optic&#39;s rotational orientation (with 180-degree degeneracy) is readily obtained by measuring the reflected intensity of a linearly polarized, probing light beam as the light&#39;s polarization axis is systematically swept through 360 degrees. Determining the orientation of a FBG with PDR is an extension of this principle. In this case, the probing light travels along an optical path which modifies the SOP of propagating light in the presence of bending and twisting according to well understood physical laws. Further, a fiber Bragg grating may exhibit a sharp dependence of reflectivity on wavelength. 
         [0017]    An embodiment of the invention includes a method comprising the following steps or operations. 
         [0018]    Operation A involves calibrating fiber birefringence. In Operation A, optical measurements are performed to calibrate how much birefringence is produced by bending and twisting in a multi-core optical fiber. Standard methods for making such measurements are applied. 
         [0019]    Operation B involves aligning polarization. In Operation B, the maximum (or minimum) instantaneous Bragg wavelength occurs when light illuminating the FBG is linearly polarized, and when the light&#39;s polarization direction aligns with the FBG&#39;s slow (or fast) birefringent axis. More specifically, Operation B involves three sub-operations. In Sub-operation B 1 , launch light is launched from a source which emits light over a spectral range that includes all Bragg wavelengths reflected by a fiber Bragg grating being interrogated and located in a center core of the multi-core optical fiber. In Sub-operation B 2 , a Bragg wavelength of reflected light is measured. In Sub-operation B 3 , the SOP of launched light is systematically varied (as opposed to randomly varied) until the instantaneous maximum (or minimum) PDR wavelength is found. Standard apparatuses for varying SOP in optical fibers are used. 
         [0020]    Operation C involves measuring the SOP of the launched light which produces the maximum (or minimum) PDR Bragg wavelength, to be referred herein as the SOP at Maximum (or Minimum) PDR Wavelength (“SMPW”). Standard methods and standard apparatuses to measure SOP are used. 
         [0021]    Operation D involves determining local fiber bending. Standard methods to measure the direction and magnitude of bending in a fiber using optical measurements of differential strain in multiple cores displaced from the fiber&#39;s central axis are used. Illustrative standard methods use twisted peripheral cores in the multi-core optical fiber or straight peripheral cores in the multi-core optical fiber. 
         [0022]    Operation E involves calculating birefringence. The calibration obtained in Operation A is used to calculate the birefringence produced by the known bending between the launch end of the fiber and the FBG. 
         [0023]    Operation F involves estimating twist from known quantities. The most probable description of fiber twist which is consistent with the calibrations and measurements is calculated. An iterative illustrative approach is described as follows with the following sub-operations. 
         [0024]    In Sub-operation F 1  an amount and direction of fiber twist is conjectured and then how the SMPW changes by traveling through the fiber (having known bend shape, and the conjectured twist) to the location of the FBG is calculated. The SOP at the location of the FBG is calculated in the standard manner by representing the SMPW and each length of fiber having known birefringence with Stokes matrices. 
         [0025]    In Sub-operation F 2 , an error figure-of-merit is assigned for how well the estimated SOP from step F 1  equals a linear state of polarization at the position of the FBG of interest, and with the polarization axis rotated by the same amount as the conjectured twist. As an example, let error 1  equal 1 divided by the ellipticity of the SOP calculated in Sub-operation F 1  at the FBG of interest. Linear polarization gives a error 1  value of zero. Let error 2  equal the angle in radians between the major axis of the SOP from F 1  and the conjectured twist angle. The error figure-of-merit is, in this example, expressed as the sum of error 1  and error 2 . 
         [0026]    In Sub-operation F 3 , the conjectured fiber twist is adjusted within a range constrained by the physical bounds of the application and guided by trends of results from previous conjectures while noting the associated error figure-of-merit. 
         [0027]    In Sub-operation F 4 , Sub-operations F 1 , F 2  and F 3  are repeated to minimize the error figure-of-merit; and the conjectured twist with the minimum error figure-of-merit is taken as the estimate of twist. 
         [0028]    In practice, the above-described operations are performed initially on a straight and relaxed fiber to provide a baseline set of optical measurements. Variations from the baseline measurements are then unambiguously attributed to the presence of bending and twisting. In concept, several discrete values for twist can transform a given SMPW to the same SOP at the FBG, in the same manner that a multiple order waveplate can produce the same transmitted SOP as a “zeroth order” waveplate. However, in practice, the magnitude of the twist-induced birefringence is quite small for reasonably expected values of twist. Therefore, the lowest twist-induced birefringence, which is consistent with the other measurements, is the correctly assumed value. 
         [0029]    Referring to  FIGS. 4 and 6 , apparatus embodiments of the invention are described as follows. Multi-core optical fiber  100 , the twist (or torsion) of which is to be measured, has a light-guiding center core  102  placed essentially at the fiber&#39;s centerline, and at least one (though preferably multiple) FBG reflectors  110 ,  120  spaced along its length. Additional, parallel cores  104 ,  106 ,  108  positioned near the multi-core optical fiber&#39;s outside diameter are provided with similar FBGs to quantify bending deflections. 
         [0030]    The polarization of the launched light is determined by a standard polarization vector analyzer  160 . The computer-controlled polarization controller squeezes the single core optical fiber  190  to adjust the polarization of the launched light. When the polarization of the launched light is adjusted such that the longest (or shortest) Bragg wavelength is reflected from the FBG, the light is known to be linearly polarized and in alignment with a fast (or slow) PDR axis when it reaches the location of the grating. The SOP is not likely to be the same at the launch end and at the FBG; rather, SOP evolves smoothly along the intervening path. Transformation of SOP along the fiber occurs deterministically in response to intrinsic birefringence, geometrical phase (i.e., path shape), bend- and twist-induced stress optical birefringence, and grating-inscription birefringence. Intrinsic birefringence is a stable property that can be calibrated for each sample, and inscription birefringence is directly measurable. Length-dependent evolution of SOP amounts to accumulated phase differences in light propagating with different polarizations; the effects of shape-induced waveguide perturbations on optical paths are well-described in the prior art literature. See, e.g., Ulrich et al., “Bend-Induced Birefringence in Single-Mode Fibers, OPTICS LETTERS, Vol. 5, No. 6, June 2008, p. 273-275, incorporated herein by reference, and H. C. LeFevre, “Single-mode Fibre Fractional Wave Devices and Polarization Controllers,” ELECTRONICS LETTERS, Vol. 16, No. 20, 25 Sep. 1980, pp 778-780, incorporated herein by reference. Precise elasto-optical coefficients of particular fibers may be obtained by calibration, and applied to the independent measurements of bending. 
         [0031]    A short segment of well-characterized fiber is assumed, over which twist is posited to be constant, and simple bending is independently resolved. Starting from the measured SOP of light at the launch end, a most-probable combination of known bending and (a single unknown value of) twist will cause the light to reach the FBG in a linear, aligned state. Multiple, consistent solutions for twist may be found by convergent iteration, and further selected in order of least deflection. The approach need not initially consider twist-induced birefringence, because the actual effect is quite small for moderate rates of twist. Further, the first estimate of twist may then be used to reduce error in subsequent iterations which include twist-induced birefringence. 
         [0032]      FIG. 4  shows the optical components needed for twist monitoring: the tested multi-core optical fiber  100  with FBGs  110 ,  120 , a broadband light source  130 , a wavelengthe measuring device for measuring the Bragg wavelength from light reflected by FBG&#39;s  140  (or other instrument) for providing real-time measurements of the reflected Bragg wavelength, and a standard computer-controlled polarization controller  150 , a standard polarization vector analyzer  160 , a single core optical fiber  190 , a processor  200 , and a non-polarizing beam splitter with fiber coupling optics  210 . The wavelength measuring device  140  is any device capable of measuring the wavelength of light reflected by an FBG with an accuracy of a few picometers, and capable of measuring changes in Bragg wavelength as small as about 0.1 picometers, and which changes occur within less than one second. Preferably, wavelength measuring device  140  is for example a fiber-coupled CCD spectrometer (e.g., the spectrometer disclosed in U.S. Pat. No. 6,233,373 to Askins, incorporated herein by reference) or a scanning Mach-Zehnder interferometer (e.g., the interferometer disclosed in Flockhart et al., “Differential phase tracking applied to Bragg gratings in multicore fiber for high accuracy curvature,measurement,” Smart Structures and Materials 2006: Smart Sensor Monitoring Systems and Applications, Proc. of SPIE Vol. 6167, 616701, (2006), incorporated herein by reference). The fiber-coupled CCD spectrometer provides high detection sensitivity and parallel measurement of many FBGs simultaneously, and can be further expanded to address many more FBGs by the inclusion of time-division multiplexing as described in U.S. Pat. No. 6,233,373 to Askins. The Mach-Zehnder interferometer provides very high speed and very high wavelength resolution for a single FBG at a time, or for several FBGs at one time by the inclusion of wavelength multiplexing optics, or by the inclusion of time division multiplexing. Examples of illustrative polarization controllers are found in U.S. Pat. No. 6,480,637 to Yao and U.S. Pat. No. 4,389,090 to LeFevre, both incorporated herein by reference. Examples of illustrative polarization vector analyzers  160  are Adaptif model A1000 and Thorlabs model series PAX 5000. 
         [0033]    In addition, optics  170  for achieving multiple-core fan-out or an optical switch allows connection of interrogation instruments  180  to evaluate bending by calculation from strain sensed in the outer cores of the fiber. An example of a high speed switch for selectively coupling multiple cores of an arbitrary geometry multicore fiber to optical instrumentation is reported in C. G. Askins, Presentation MM3, Proceedings of the 21st Annual Meeting of the Lasers and Electro-Optical Society, Newport Beach, Calif., 9-13 Nov. 2008, incorporated herein by reference and Askins OFC 2008. An example of a bend-sensing system  180  is found in G. M. H. Flockhart et al, Opt. Lett. Vol. 28, 387-389 (2003), incorporated herein by reference. 
         [0034]    Optionally, an embodiment of the inventive method includes a tunable narrow-band filter  133  inserted between the broadband light source  130  and the coupler  135 , as shown in  FIG. 4 . An example of a suitable tunable narrow-band filter  133  is a tunable Fiber Fabry-Perot filter made by Micron Optics. The bandwidth of the tunable filter  133  is wider than the spectral width of the FBG of interest so that the polarization analyzer  160  may unambiguously evaluate the SOP of the light which illuminates the FBG of interest. 
         [0035]    An alternative embodiment of the inventive apparatus is shown in  FIG. 5 . A narrow-band filter  220  inserted between the beamsplitter  210  and the polarization vector analyzer  160  so that the SOP of only a narrow portion (preferably less than 1 nm) is measured by the analyzer  160 . Optionally, the wavelength passed by the-narrow-band filter  220  is within the wavelength range emitted by the light source, but not equal to any;Bragg wavelength of the various FBGs so as to avoid reflected light from the polarization analyzer interfering with the light from any of the various FBGs. For a stable arrangement of components  150 ,  210 ,  220  and  170 , and a known SOP of light at each wavelength entering the polarization controller  150 , the SOP of each small wavelength interval launched into the multi-core optical fiber  100  through the beamsplitter  210  is determined relative to the SOP of all other wavelength intervals. By this observation, knowledge of the SOP of the narrow wavelength band transmitted by the tunable narrow-band filter  220  to the polarization analyzer  160  enables calculation of the SOP of the light which illuminates any FBG of known. Bragg wavelength after launch through the fanout or fiber switch  170 . The SOP of light illuminating all FBGs may then be determined simultaneously without reconfiguring any optical elements. Optionally, a high-brightness narrow bandwidth light source  139  (preferably a laser diode) with a wavelength matching the transmission pass band of the narrow-band filter  220  is coupled with the broadband light source  130  by polarization-maintaining coupler  240 . Optionally, the single-core optical fiber  190  and polarization-maintaining couplers  240 ,  242  are made with polarization-maintaining fiber, and linear polarizers  230 ,  232  assure that light from the light sources  130 ,  139  are linearly polarized and aligned with a polarization axis of the fiber  190  so that the SOP of all wavelengths entering the polarization controller  190  is equivalent. Optionally, the beamsplitter and accompanying coupling optics constitute bulk, free-space optics because of their highly stable properties with respect to the SOP of transmitted light. Optionally, the beamsplitter and coupling optics constitute planar waveguide optics or fiber-based optics because of their compactness. 
         [0036]    Optionally, an embodiment of the inventive method starts with providing the multi-core optical fiber  100  in a relaxed state. In the relaxed state, a baseline birefringence map is made for each section of the multi-core optical fiber between successive FBGs in the center core. Once the baseline map is generated, then when the multi-core optical fiber  100  takes on a new geometry, relative changes to the baseline map can be determined. Those relative changes or departures from the baseline map are attributable to bending and twist geometries of the multi-core optical fiber  100 . 
         [0037]    The method and apparatus are most successful when applied successively to short segments of fiber subjected to levels of bend and twist that vary slowly in time and along the fiber&#39;s length. Such slowly varying conditions are typical for many applications where fiber shape must be evaluated. For the purpose of this discussion, slowly varying bend and twist means that that the shape is effectively static during the time required to apply the polarization tracking method, and that curvature and twist only vary smoothly and slightly between successive locations along the fiber. That is, bend and twist along the fiber&#39;s length must be sufficiently slowly varying that sufficient sampling of bending stresses is measured to accurately infer bending shape. An increased time rate of bending and twisting can be compensated for by commensurately more rapid detection of the described optical signals, and more rapid execution of the described algorithms. More spatially rapid variations along the fiber&#39;s length are accommodated by a reduced spacing between successive FBGs along the multicore fiber&#39;s length. By example, a spacing between bend sensing FBGs of less than 10 cm would permit accommodation of curvatures that vary appreciably only at separations greater than 10 cm. Fiber twist changes occurring at a rate of 10 degrees per second would require completion of the polarization tracking algorithm more than 10 times per second. A radius of curvature greater than 1 cm over 10 cm of fiber length, and twist rates below 2π radians per meter over 10 cm both produce less than ¼ wave birefringence at wavelengths between 500 and 1700 nm. Most shape-sensing applications involve gentler curvatures and slower rates of change of twist are therefore well-serviced by the instant method. Once twist and bend are solved for the segment of fiber between launch and the first FBG, optical and geometrical conditions at the first FBG is essentially as well known as at the launch location. The method is then repeated with each solved FBG location serving as an equivalent launch location. This permits twist (and shape) estimation throughout a length of fiber bearing a series of FBGs. Unambiguous measurement of successive FBGs along a single waveguide is accomplished by wavelength-division-multiplexing (“WDM”) or time-division-multiplexing (“TDM”), or a combination of both. In the case of WDM, calculation of SOP evolution along a waveguide must consider wavelength-dependent optical path differences. Over small ranges of wavelengths, and for moderate inter-grating fiber lengths, length-dependent phase differences (SOP) may be safely scaled from solutions obtained at nearby wavelengths. This permits concatenation of calculated SOP and fiber geometry using a series of gratings at different wavelengths. Since gradients in twist are not easily supported by a nominally free fiber, adjacent estimates of twist can be screened for consistency, and treated statistically to reduce estimation noise. 
         [0038]    In other embodiments of the invention, enhanced PDR would improve the sensitivity of the SOP measurements. Also, the optical effects producing PDR should dominate handling-induced birefringence effects so that the rotational orientation of the PDR is fairly stable. For reference, a ¼ wave retardation may be effected in single mode fiber with ˜20 cm wound at 30 mm diameter (corresponding to ˜10{circumflex over (0)}−6 birefringence.) Many shape sensing applications may not encounter such high curvature. PDR resulting in a 10 picometer polarization-dependence of Bragg wavelength is routinely observable, corresponding to &gt;10{circumflex over (0)}−5 birefringence. Larger PDR should be obtainable with optimization. This might be achieved by optimized optical exposures during grating writing, or by the impression of differential radial stresses during exposure. Differential stress might be effected through transparent compression plates, or optically-induced, transient thermal gradients (non-contact, and best with pulsed exposures). Where handling effects can significantly reorient PDR, the effect can be computationally corrected in proportion to the independently determined local curvature. 
         [0039]    It should be appreciated that various embodiments of the present invention may be implemented as a sequence of computer implemented acts or program modules running on a computing system and/or as interconnected machine logic circuits or circuit modules within the computing system. The implementation is a matter of choice dependent on the performance requirements of the computing system implementing the invention. Accordingly, logical operations including related algorithms can be referred to variously as operations, structural devices, acts or modules. It will be recognized by one skilled in the art that these operations, structural devices, acts and modules may be implemented, in software, firmware, special purpose digital logic, and any combination thereof without deviating from the spirit and scope of the present invention as described herein. 
         [0040]    Because numerous modifications and variations of the above-described invention will occur to those of ordinary skill in the art, it is not desired to limit the invention to the exact construction and operation illustrated and described. Accordingly, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention.