Background correction of birefringence measurements

Abstract

One example provides a computer-implemented method for reading data stored as birefringence values ​​in a storage medium, the method including: acquiring an image of voxels of the storage medium, applying a first low-pass filter having a first cutoff frequency to the image of voxels to obtain a first background image, applying a second low-pass filter having a second cutoff frequency to the image of voxels to obtain a second background image, the second cutoff frequency being different from the first cutoff frequency, determining an enhanced background image from the first background image and the second background image, determining a birefringence value for the enhanced background image, determining a birefringence value for the image of voxels, and correcting the birefringence value for the image of voxels based on the birefringence value for the enhanced background image.

Description

[Background technology]

[0001] background

[0001] Over the past decade, much of the world's data has moved to the cloud. To meet the growing demand, cloud providers rely on a variety of data storage technologies. These storage technologies include non-volatile memory (NVM), flash, hard disk drives (HDDs), magnetic tape, and optical disks. These storage technologies differ from each other in terms of cost, latency, throughput, storage density, failure rate, and media lifespan. Summary of the Invention [Problem to be solved by the invention]

[0002] overview

[0002] This Summary is provided to introduce various concepts in a simplified form, further description of which is provided below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter. Moreover, the claimed subject matter is not limited to implementations that solve any or all of the disadvantages noted in any part of this disclosure.

[0003] A promising technology for storing data is encoding data as localized birefringent voxels in a dielectric storage medium. Such data can be stored at high density, and the storage medium can have a long lifetime compared to magnetic and other storage media. Reading a birefringent voxel involves probing the voxel with polarized light and detecting polarization changes in the light resulting from the voxel. However, polarization changes can also arise from background sources both external and internal to the storage medium. While external background effects can be corrected by taking measurements in the absence of the storage medium, such background correction methods do not correct for background polarization changes arising from the storage medium itself. Furthermore, taking background measurements in the absence of the storage medium requires additional measurements, thereby increasing the amount of time utilized to read the data. Accordingly, aspects of the technology disclosed herein use the same image of the storage medium to correct for background arising from background sources external to the storage medium and background arising from the storage medium itself. [Means for solving the problem]

[0004] One aspect provides a computer-implemented method for reading data stored as birefringence values in a storage medium. The method includes acquiring an image of voxels of the storage medium, applying a first low-pass filter having a first cutoff frequency to the image of the voxels to obtain a first background image, and applying a second low-pass filter having a second cutoff frequency to the image of the voxels to obtain a second background image, the second cutoff frequency being different from the first cutoff frequency. From the first and second background images, determining an enhanced background image, determining a birefringence value for the enhanced background image, determining a birefringence value for the image of the voxels, and correcting the birefringence value for the image of the voxels based on the birefringence value for the enhanced background image. The first background image may represent more local background characteristics (such as birefringence noise arising from other voxels of the storage medium), and the second background image may represent more global background characteristics. The effect of the first bandpass filter is to remove high frequency data representing more detailed features of in-focus voxels during the readout process, thereby leaving less detailed features from other out-of-focus voxels. Using this embodiment, a single image of the birefringent sample can be used to correct for background arising from both the storage medium itself and external background sources, without having to take a separate background measurement in the absence of the storage medium. [Brief explanation of the drawings]

[0005] BRIEF DESCRIPTION OF THE DRAWINGS [Figure 1] 5 illustrates a schematic representation of the reading of birefringent voxels of a storage medium. [Figure 2]

[0006] 1 shows a schematic representation of a storage medium containing data encoded as birefringent voxels. [Figure 3]

[0007] The Poincaré sphere is shown, which represents polarization states as locations on the sphere. [Figure 4]

[0008] 1 shows a flow diagram depicting an exemplary method for determining a birefringence value based on a maximum value determined for a likelihood function. [Figure 5]

[0009] 1 shows examples of polarization states for measuring birefringence values. [Figure 6]

[0010] 10 shows exemplary solutions for birefringence values based on two measurements and based on one measurement. [Figure 7]

[0011] 1 shows a flow diagram depicting an exemplary method for obtaining birefringence measurements using wavelength multiplexing. [Figure 8]

[0012] FIG. 1 shows a block diagram of an exemplary system for reading birefringence data using wavelength multiplexing. [Figure 9]

[0013] 1 illustrates an exemplary system for wavelength multiplexing light of different polarization states using light from the same light source. [Figure 10A]

[0014] FIG. 1 shows a flow diagram depicting an exemplary method for measuring birefringent voxels via wavelength multiplexing. [Figure 10B]

[0014] A flow diagram depicting an exemplary method for measuring birefringent voxels via wavelength multiplexing is shown. [Figure 11]

[0015] 1 shows a flow diagram depicting an exemplary method for performing background correction on birefringence values. [Figure 12]

[0016] FIG. 1 illustrates a block diagram of an exemplary computing system. DETAILED DESCRIPTION OF THE INVENTION

[0006] Detailed Description

[0017] As mentioned above, one promising technology for cloud data storage involves the use of high-power, short-pulse laser irradiation to optically write data into dielectric solid substrates such as glass. The irradiation induces localized birefringence at its focal point that can later be read using polarized light imaging. The term "voxel" is used herein to refer to any discrete volume of the substrate that can store individual data values (i.e., symbols). The data stored in a voxel can take a variety of forms. In principle, any of the Mueller matrix coefficients of the substrate lattice can be manipulated to encode data. In an example using a fused silica substrate, the lattice perturbation from focused polarized light irradiation takes the form of non-native birefringence localized at the focal point. Accordingly, each voxel in the substrate can be modeled as a very small waveplate with a retardance magnitude and azimuthal angle. These model parameters can be independently manipulated to write desired symbols into a given voxel. Here, the polarization angle of the beam determines the azimuthal angle of the voxel, and various other factors (pulse amplitude, duration, energy, number and / or spacing between pulses) determine the retardance of the voxel.

[0007]

[0018] By dividing the continuous space of achievable azimuthal angles and / or retardance magnitudes into discrete intervals, a multi-bit data value can be encoded and embedded in each voxel by writing the birefringence of that voxel into one of the discrete intervals. Furthermore, in some examples, multiple parallel layers of voxel structures can be written to the same substrate by focusing laser irradiation to a specified depth below the irradiated surface of the substrate. These features, individually or in combination, enable large amounts of data to be written to a single medium. In some examples, the storage medium comprises a solid, plate-like configuration. In other examples, the storage medium comprises a thin layer formed on another substrate. In further examples, the storage medium may have any other suitable configuration, such as a prism or a cylinder.

[0008]

[0019] Birefringent voxels behave anisotropically when light passes through them because different polarization states of light travel at different speeds through the sample. As light travels through or reflects off a birefringent voxel, its polarization state changes in a specific way that depends on the azimuthal angle and retardance of the sample. Therefore, information about the azimuthal angle and retardance of a voxel can be obtained by measuring the polarization state of polarized light that has interacted with the birefringent voxel.

[0009]

[0020] Some methods for measuring the polarization state (and thus the angle and retardance of a voxel) involve performing a series of measurements using different input or output (or both) polarizations. Figure 1 shows a schematic diagram of reading birefringence data stored in a storage medium 105. Light from a light source 102 passes through a polarization state generator (PSG) 104, which outputs polarized light with a polarization angle determined by the PSG. The light source may include an LED, laser, or other light source.

[0010]

[0021] After passing through storage medium 105, light from light source 102 and PSG 104 passes through one or more voxels of storage medium 105, passes through polarization state analyzer (PSA) 106, and then reaches detector 108. The settings of PSA 106 and PSG 104 determine the intensity m of the measurement, as described in more detail below. k defines a polarization state k for . Detector 108 may include, for example, a CMOS image sensor (e.g., a high-resolution / high-frame-rate sensor) or other suitable photodetector array capable of imaging an entire focal plane positioned within storage medium 105, thereby imaging a large number of arranged voxels in the same image. In other examples, point detectors or small detector arrays, such as photodiodes, phototransistors, or SPADs (single-photon avalanche diodes), may be used to construct an image point-by-point. While FIG. 1 shows light rays passing through the storage medium to reach the image sensor, in other examples, light rays may reflect off the storage medium to reach the image sensor.

[0011]

[0022] In examples where data is read from multiple layers of storage medium 105, variable focus optics 110 can be used to adjust the focal plane of detector 108 so that voxels in the focal plane are read while other voxels are out of focus. In other examples, varying the focus can be achieved by moving the sample. Figure 2 shows a schematic diagram of an exemplary storage medium 200 containing multiple layers of voxels, two of which are indicated at 202 and 204.

[0012]

[0023] To describe the voxel measurements that are made, a commonly used structure is the Poincaré sphere, shown at 300 in Figure 3. The Poincaré sphere 300 is used to represent the polarization state of light by mapping the last three components of the 4D Stokes vector to a 3D Cartesian coordinate system. For fully polarized light, a specific measurement state is described by a specific location on the surface of the sphere, which represents a specific polarization state of light. Partial polarization is represented by points inside the surface of the Poincaré sphere. The north pole represents RCP (right-handed circular polarization). The south pole represents LCP (left-handed circular polarization). A state at the equator is linearly polarized, with an angle defined by the azimuth angle on the sphere. A general point on the sphere is elliptically polarized, with an ellipticity determined by the angle between that state and the pole and an azimuth angle defined by the angle around the sphere. An exemplary polarization state is shown at 302 on the sphere. The black dashed curves are circles of constant latitude and therefore constant retardance. The term "swing" refers to the angle 304 of the measurement state relative to the pole of the sphere. The horizontal angle is the azimuthal angle 306. Because the azimuthal angle and retardance of a voxel result in the output state of polarization of the probe light, the azimuthal angle and retardance of a voxel can also be represented as a point on the surface of a sphere.

[0013]

[0024] Current methods for determining birefringence values for voxels of a storage medium involve three or more (typically four) measurements of the voxel taken with different probe light polarization states. Multiple measurements are used because the determination requires four degrees of freedom: voxel retardance, voxel azimuth angle, measurement scale, and measurement offset. The four-measurement method determines all four of these values for each read process. Shribak and Oldenbourg (M. Shribak and R. Oldenbourg, “Techniques for Fast and Sensitive Measurements of Two-Dimensional Birefringence Distributions,” in Applied Optics, Volume 42, Issue 16, 2003, published by the Optical Society of America (https: / / www.osapublishing.org / ao / abstract.cfm?uri=ao-42-16-3009; see also https: / / doi.org / 10.1364 / AO.42.003009) describe a three-measurement method in which the offset is assumed to be zero and the other three parameters are solved for via three measurements. However, the use of three or four measurements impacts the speed at which data can be retrieved from storage due to the number of measurements made.

[0014]

[0025] Accordingly, disclosed herein is an exemplary measurement process that can be used to determine azimuthal and retardance values for birefringent voxels using fewer than three measurements. Briefly, the disclosed method utilizes constraints on retardance values based on prior knowledge of retardance, such that retardance and azimuthal values can be determined with two or even one measurement. The disclosed method can also be used with three measurements, offering advantages over previous three-measurement methods, such as not having to assume offsets to be zero as long as they are known prior to the measurement, and being able to determine relative retardance and angle. Because scale and offset do not vary significantly spatially or temporally in the context of reading data encoded as birefringence on a storage medium, scale and offset can be determined once or periodically using four (or more) measurement techniques, and the determined scale and offset values can then be used for subsequent retardance and angle determinations using fewer than four measurements. Furthermore, as described below, if scale and offset are unknown, they can be determined via numerical optimization techniques.

[0015]

[0026] 4 shows a flow diagram illustrating an exemplary method 400 for determining birefringence values. Method 400 is an example of a technique that can be used to determine scale and offset values for use in subsequent determinations using a smaller number of measurements, and offers the advantage over current methods of being able to make determinations using any probe light polarization state, rather than a predetermined state that is difficult to implement with physical equipment. Method 400 first involves obtaining, at 402, four or more measurements of a voxel at different polarization states, each measurement having a swing χ k and angle θ k The observed intensity m at measurement state k is defined by the measurement polarization state with kThe method 400 further includes determining a likelihood function for the voxel measurement at 404, where the likelihood function represents, for each birefringence value set of the multiple birefringence value sets, the likelihood of the voxel measurement being generated by the birefringence value set. The likelihood is determined as a function of angle and retardance using noise and data model assumptions. Any suitable data and noise model can be used, including Gaussian, Poisson, and a combination of Gaussian and Poisson. As an example, a Gaussian noise model (shown at 406) is used to determine the likelihood of the intensity m given a measurement state k. k The likelihood of measuring is given by equation (1).

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[0016]

[0027] The determined scale and offset can be used in determinations using three or fewer measurements. It will be understood that the term "maximum determined value" and similar terms used herein are not intended to indicate the actual global maximum of the likelihood function, but rather to represent the largest observed value for all parameter sets utilized. Furthermore, it will be understood that in other examples, any other suitable method can be used to determine birefringence values, including scale and offset. For example, according to Bayes' theorem, prior knowledge about the angle or retardance distribution can be included, and instead of calculating the likelihood of the parameters, posterior probabilities can be calculated. In this case, the values of the parameters that maximize the posterior probabilities (this is often referred to as MAP (maximum a posteriori)) can be found. In such examples, the parameter values can be determined, for example, by finding the expected values of the posterior probabilities or by using other statistical measures.

[0017]

[0028] As mentioned above, an advantage provided by the use of maximum likelihood to determine birefringence values for a voxel is that the polarization states used for the measurements are arbitrary, rather than predetermined. Nevertheless, in some instances, some configurations of polarization states may provide a more effective use of the available signal than other configurations of states. Two examples of such configurations are as follows. Referring again to the Poincaré sphere of FIG. 3 , a first example of a set of polarization states for determining birefringence values for a voxel includes a circularly polarized input state (as established by a PSG) with one handedness and an elliptically polarized output state (as established by a PSA) with the opposite handedness to the input state and equal ellipticities to each other, but with equally spaced varying azimuth angles across half or the entire circle at equal latitudes on the Poincaré sphere, such as {0, 45, 90, 135} degrees or {0, 22.5, 45, 67.5} degrees. A second example of a polarization state set includes a circularly polarized output state of one handedness and an elliptically polarized input state of opposite handedness to the output state, with equal ellipticities but varying azimuth angles, the azimuth angles equally spaced across half or a full circle on the surface of the Poincaré sphere. Figure 5 depicts an example of three such polarization state configurations, each configuration indicated by a dot with a different symbol than the other configurations. In Figure 5, the diagram of the Poincaré sphere is along the polar axis of the sphere, the dashed circles correspond to selected latitudes containing the polarization states, and the outer boundary corresponds to the sphere's equator.

[0018]

[0029] With the measurements configured in this way, the azimuth angle of a voxel can be determined as follows:

number

[0019]

[0030] where φ is the measurement angle, the subscript k indicates the measurement number, and m k is the intensity of the kth measurement, and 2θ kis the angle of the kth measurement state on the Poincaré sphere. To find the retardance, one suitable method involves numerically optimizing the likelihood of the data given the measurements, as described above with respect to FIG.

[0020]

[0031] As mentioned above, prior knowledge of the scale and offset parameters can be used to determine the azimuth angle and retardance of a voxel with a reduced number of measurements. For example, the azimuth angle of a voxel can be determined by using equation (4) using three measurements plus a known offset value.

number

[0021]

[0032] If some prior information about the retardance is known, but the retardance value itself is unknown, then the birefringence value of the voxel can be determined using a two-measurement method. In some instances, the two-measurement method also utilizes prior knowledge of the scale and offset, while in other instances, the scale and offset are determined by numerical optimization, as mentioned above.

[0022]

[0033] When using two measurements, even with knowledge of the offset and scale, there are two possible solutions for the sample's retardance and angle. This is because each measurement constrains the solution to lie on a 2D plane in the 3D space of the Poincaré sphere. Thus, the two measurements define a line in 3D. Figure 6 shows an example of a two-measurement determination for a hypothetical voxel on the Poincaré sphere 602. In this example, the line 604 defined by the two measurements of the voxel intersects the surface of the sphere at two points 606 and 608.

[0023]

[0034] To determine which of two points corresponds to the actual birefringence value of the measured voxel, a constraint is applied that selects the point with the lower retardance as corresponding to the voxel state because it can be mathematically shown that the other solution is at least approximately as large as the effective retardance of the measured state. For example, considering two measurements on Poincaré sphere 602, point 606 would be selected because it represents a lower retardance value than point 608. This a priori constraint can be used in any system where the effective measured retardance is known to be different (larger or smaller) than the measured retardance. This constraint narrows the solution to a single point on the sphere, from which the sample retardance and azimuth angle can be determined.

[0024]

[0035] A more detailed mathematical description of an exemplary two-measurement method follows. Note that the method can be implemented by using scale and offset values that are first determined using more (e.g., four) measurements and then applying them to the two-measurement determinations, or by using a numerical optimization method to determine the scale and offset values. If the scale and offset values are determined first, any suitable method can be used to determine these parameters, including the maximum likelihood example described above with respect to FIG. 4.

[0025]

[0036] After obtaining measurements, an exemplary two-measurement method involves first solving the following set of simultaneous equations (5, 6, 7) for retardance δ: m1=a(1-cosχ1cosδ+sinχ1sinδsin(2φ-2θ1))+b (5) m2=a(1-cosχ2cosδ+sinχ2sinδsin(2φ-2θ2))+b (6) (sinδsin(2φ-2θ1)) 2 +(sinδsin(2φ-2θ2)) 2 +cos 2 δ=1 (7) Since there are two possible solutions for δ, the smaller value is chosen, as explained above. Then, using equation (8),

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[0026]

[0037] If known values of scale and offset were used in this determination, the determination is complete at this stage. On the other hand, if assumed scale and offset values were used, a and b can be adjusted using prior information. For example, if the distribution of angles is uniform (as is the case for birefringence data when the data writing process is known), the distribution of angles can be calculated and equations (5)-(7) and (8) can be determined iteratively, adjusting a and b until the measured angle histogram is sufficiently similar to that expected.

[0027]

[0038] If the retardance of a voxel is known (and, optionally, if the birefringence write characteristics are known), the birefringence value can be determined via a single measurement, again using known or assumed values for the scale and offset parameters. If assumed values are used, the values can be numerically optimized. Based on the known or assumed scale and offset values, the single measurement 612 defines a two-dimensional plane in three-dimensional space, with reference to the Poincaré sphere 610 in FIG. 6. This two-dimensional plane intersects the surface of the sphere along a circle 614, which represents a continuous range of possible sample angles and retardance values. If the sample retardance (represented by angle 616) is known exactly, the angle can be determined to be within one of two values (represented by points 620, 622, i.e., where the retardance plane 618 intersects the measurement circle 614). By comparison, the two-measurement example described above utilizes less detailed prior information (i.e., retardance is smaller than swing). If the angle is known to lie in a range that spans only half of the total angle available (e.g., 0-90 degrees, 45-135 degrees, or any other 90-degree range), then these two points are constrained to one, thereby determining the retardance and angle of the sample. Mathematically, the operations for performing the disclosed exemplary one-measurement determination are expressed by equations (9) and (10).

number

[0028]

[0039] FIG. 7 shows a flow diagram depicting an exemplary method 700 for determining the birefringence value of a voxel using no more than two measurements. As described above, the method of FIG. 7 can utilize scale and offset parameters determined via the method of FIG. 4, or can first assume these parameters and then numerically optimize them. Method 700 includes, at 702, acquiring measurement data for a birefringent voxel by directing probe light having one or more predetermined polarization states through the birefringent voxel and receiving the light at an image sensor. In some examples, the measurement data may include measurement data acquired using light of a first polarization state and measurement data acquired using light of a second, different polarization state, as shown at 704. In other examples, the measurement data may include data from a single measurement, as shown at 706.

[0029]

[0040] Continuing, method 700 includes, at 708, determining, based on the measurement data, two points on the surface of the Poincaré sphere that correspond to two possible birefringence states of the birefringent voxel, each state including one birefringence value set, and applying a constraint to determine the birefringence value of the voxel. For example, if measurements are made at two polarization states, then there are two azimuthal solutions to equations (5)-(7) above. In this example, method 700 includes, at 710, applying a constraint specifying that the birefringence value set having the lower retardance value is the correct set. By selecting the point on the Poincaré sphere that represents the lower retardance, the azimuthal angle can be solved using equation 8 above, thereby providing a determined birefringence value set for the voxel.

[0030]

[0041] If a single measurement is used, method 700 includes, at 712, determining a circle on the surface of the Poincaré sphere based on the measurement, the circle including two locations where the circle intersects with a plane representing the known retardance of the voxel. A point can then be selected at 714 that represents the birefringence value of the voxel based on the azimuth angle of the point that falls within the expected range of angles, and the birefringence value can be determined using equations (9) and (10) above. Method 700 optionally further includes, at 716, numerically optimizing a likelihood function based on the determined azimuth angle and retardance value to determine the scale and offset values in instances where these values are not initially known. Method 700 further includes, at 718, outputting the birefringence value of the voxel.

[0031]

[0042] The examples described above can help reduce the time and computational resources consumed when reading birefringent voxels of a storage medium compared to methods that use four or more measurements. Alternatively or additionally, other processes can be used to provide efficient reading of birefringent voxels. For example, wavelength multiplexing can be used to reduce the number of individual images acquired during the reading process.

[0032]

[0043] FIG. 8 shows a schematic illustration of an exemplary system 800 for reading birefringent storage media. System 800 utilizes wavelength multiplexing, in which N different wavelength bands of light with different polarization states are multiplexed to obtain N measurements in a time-overlapping manner. System 800 includes N light sources, shown as first light source 802, second light source 804, and Nth light source 806, each configured to output light in a different wavelength band (e.g., red, green, and blue). Each light source directs light to a corresponding PSG (shown as PSG 808, PSG 810, and PSG 812 for light sources 802, 804, and 806, respectively) so that a different polarization state can be set for each wavelength band. In another example, a system for performing two time-overlapping measurements may have two light sources and corresponding PSGs.

[0033]

[0044] Light from each PSG is directed towards a storage medium 814 located in a sample region of the system to read the storage medium. The term "sample region" is used herein to denote where the storage medium is located for reading. In the depicted example, N-1 beam combiners (shown as beam combiner 1 816 and beam combiner N-1 818) are used to combine the light from each PSG into a beam to probe the sample medium.

[0034]

[0045] In the depicted embodiment, optics in the form of a collection lens 820 directs light through the storage medium, and an objective lens 817 focuses the light onto a detector in the form of an image sensor 822 that images the entire data layer in the storage medium 814 for each image frame. An achromatic PSA 824 is positioned between the storage medium and the image sensor 822. The image sensor 822 includes multiple integrated wavelength-selective bandpass filters such that light of different wavelength bands passes through different filters and reaches different areas of the pixels of the image sensor 822. In this manner, the intensity of each wavelength band of light (each of which has a different polarization setting) can be measured in the same image frame.

[0035]

[0046] In some examples, one or more physical masks can be used in the pupil design to help improve the quality of the signal used to read the birefringence-encoded data of the storage medium. Examples of masks that can be used are shown schematically as intensity mask 825 and phase mask 826. The pupil design can depend on the voxel layout (e.g., x, y, z spatial distribution) of the storage medium. For example, a ring-shaped intensity mask added to the lens pupil generates a Bessel beam rather than a traditional Gaussian beam. Also, a phase mask can be applied to design the polarization field at the sample plane to optimize the pupil profile for the intended type of birefringence distribution of the sample. Thus, if a specific measurement light probe is desired, the shape of the input light can be designed to tailor the signal available as input to these methods with the goal of improving the quality of the measurement.

[0036]

[0047] In some examples, the adjustable focus optics can be moved to selectively focus on voxels of specific layers within the volume of the storage medium 814, thereby enabling reading of different layers. In other examples, the storage medium can be moved to focus on voxels of different layers. While the image sensor depicted includes an integrated bandpass filter, in other examples, the bandpass filter can be included elsewhere in the system. For example, the system can utilize multiple PSGs and wavelength multiplexing in combination with a rotating bandpass filter (e.g., a color wheel) to enable sequential acquisition of images of different polarization states.

[0037]

[0048] As mentioned above, in some examples, more wavelength bands with different polarization states can be generated using fewer light sources. Figure 9 shows an exemplary light source configuration in which light from a single light source 902 is split into two beams 906, 908 of different wavelength bands via a dichroic beam splitter 904. The beams 906, 908 are directed through respective PSGs 912, 914 using any suitable optics (such as mirrors 916, 918 in the depicted example), which set different polarization states for the beams 906, 908. After passing through the PSGs, the beams 906, 908 are combined in a dichroic beam combiner 920 to probe the storage medium. In other examples, light from a suitable light source can be split into three or more different wavelength bands.

[0038]

[0049] 10A and 10B show a flow diagram depicting an exemplary method 1000 for obtaining multiple temporally overlapping measurements via wavelength multiplexing. Referring initially to FIG. 10A , method 1000 includes, at 1002, generating a first polarization of a first wavelength band, the first polarization comprising a first polarization state, and generating a second polarization of a second wavelength band different from the first wavelength band, the second polarization comprising a second polarization state different from the first polarization state. In some examples, as shown at 1004, light of the first wavelength band is output via a first light source, and light of the second wavelength band is output by a second light source. In other examples, light can be output from fewer light sources and then split into more beams of different wavelength bands, as shown at 1006. It will be appreciated that additional wavelength bands greater than two can be used. Thus, at 1008, the method 1000 may include outputting light of three or more different wavelength bands with different polarization states to generate a third polarization and potentially additional other polarization beams.

[0039]

[0050] Continuing, method 1000 includes, at 1010, passing the first polarization and the second polarization through voxels of the storage medium, thereby changing a first polarization state of the first polarization to a first modified polarization state and changing a second polarization state of the second polarization to a second modified polarization state. Passing the first polarization and the second polarization through voxels of the storage medium may include, at 1012, combining the first polarization and the second polarization via a beam combiner before passing the first and second polarizations through the storage medium. Further, process 1010 may also include, at 1014, combining a third polarization and polarization of any additional wavelength bands with the first and second polarizations before passing the light through the storage medium. In some examples, one or more masks can be used to implement a designed pupil. Accordingly, method 1000 may include, at 1016, passing the polarized light through an intensity mask before passing the polarized light through voxels of the storage medium. Alternatively or additionally, at 1018, the method 1000 may include passing the polarized light through a phase mask before passing the polarized light through the voxels of the storage medium.

[0040]

[0051] 10B , at 1022, method 1000 includes passing the polarized light through voxels of the storage medium, followed by passing the first polarized light, the second polarized light, and polarized light of any additional wavelength bands through a polarization state analyzer, where the analyzer attenuates the intensity of the light relative to the analyzer state based on the polarization state of the light. Then, at 1024, method 1000 includes passing the first polarized light through a first bandpass filter to an image sensor and passing the second polarized light through a second bandpass filter to the image sensor, where the first bandpass filter selectively passes the first wavelength band and the second bandpass filter selectively passes the second wavelength band. Furthermore, as shown at 1026, polarized light of the additional wavelength bands having different polarization states can be passed through corresponding additional bandpass filters. In this method, the intensities of the first wavelength band, the second wavelength band, and any additional wavelength bands are imaged separately.

[0041]

[0052] In some examples, the first bandpass filter, the second bandpass filter, and any additional bandpass filters are integrated with the image sensor as spatially separate filters integrated with the pixels of the image sensor. In such examples, method 1000 includes, at 1028, passing a first polarization to a first region of the image sensor, passing a second polarization to a second region of the image sensor, and passing polarization of any additional wavelength bands to the image sensor. In this method, images for each polarization state can be acquired separately in the same image frame. In other examples, different bandpass filters can be applied in a time-multiplexed manner, such as with a color wheel. In such examples, different image frames are acquired for each wavelength band. After acquiring measurements using different wavelength bands for different polarization settings, method 1000 includes, at 1030, determining a birefringence value of the voxel based on the first polarization received at the image sensor through the first bandpass filter, the second polarization received at the image sensor through the second bandpass filter, and any additional wavelength-multiplexed polarizations. The birefringence values can be determined using the exemplary methods disclosed above or in any other suitable manner.

[0042]

[0053] Using any of the methods described above, the system and medium through which the probe light passes when probing a voxel may impart additional rotation and / or retardance to the polarization used to probe the voxel. For example, reading a layer of voxels in a data storage medium containing a three-dimensional array of voxels can result in the polarized probe light being rotated during the reading process by passing through other, out-of-focus layers of voxels. System imperfections can also lead to background noise. Therefore, it is often interesting to measure the birefringence properties of a sample, such as a data storage medium, in relation to system and storage medium imperfections (e.g., the presence of other, unread voxels in the optical path). This is called background subtraction.

[0043]

[0054] Current methods involve the removal of background signals, which is done by subtracting the k} plus image set {b k} and calculate the background angle and retardance. k} is {m k} are captured using the same instrument with the same set of polarization states as {b k} is captured simply by removing the actual sample from the field of view. Also, {m k} to {b k It is also possible to estimate the blurring effect. Blurring can be achieved by a simple low-pass filter. However, such methods cannot adequately compensate for both system imperfections and sample imperfections (e.g., other voxels).

[0044]

[0055] Therefore, based on the two-step process, k} to {b kAn example related to estimating {} is disclosed, in which a two-step process is used to calculate and combine two low-pass versions of the measured intensity using a multiplicative constant to form an enhanced background image. This method estimates both local and global background intensities, thereby representing a more accurate estimate of background intensity at large and small scales, since the large-scale background can compensate for imperfect systems and the small-scale background can compensate for the three-dimensional nature of storage media. The filter parameters and multiplicative constants can be derived, for example, by minimizing the error between the known sample retardance and the calculated retardance distribution. Such a background correction method can be more efficient than separately acquiring a background image for the system (e.g., without the presence of a data storage medium in the sample area) because fewer physical measurement processes are performed, thereby saving time and resources that would be used for separate physical background measurements. As a more specific example, for a sample in which known data has been encoded and embedded in glass and a set of image frames has been acquired, initial filter parameters and multiplicative constants are set, and the encoded data is decoded. This is then repeated using updated filter parameters and multiplicative constants. The parameters and multiplicative constants that provide the determined minimum error in decoding are then selected. The process of updating the filter parameters can be, in various examples, a brute search or gradient method in the parameter space.

[0045]

[0056] 11 shows a flow diagram depicting an example method 1100 for correcting birefringence values using multiple low-pass filters. Method 1100 includes, at 1102, acquiring an intensity image of a voxel of a storage medium. To measure the birefringence of the voxel, multiple images of the voxel are acquired at different polarization states, as shown at 1104. In some examples, the images may be of multiple voxels arranged in an image plane within the storage medium, as shown at 1106. The image of the voxel includes high-frequency image information resulting from the birefringence state of the voxel and low-frequency image information resulting from one or more birefringent regions of the storage medium other than the voxel.

[0046]

[0057] At 1108, method 1100 includes applying a first low-pass filter to the image of the voxel to obtain a first background image and applying a second low-pass filter to the image of the voxel to obtain a second background image. As described above, the first low-pass filter and the second low-pass filter may have different cutoff frequencies, such that the first background image represents more localized background characteristics (birefringence noise arising from other voxels in the storage medium) and the second background image may have a lower cutoff frequency than the first low-pass filter and represent more global background characteristics. The effect of the more localized low-pass filter may be to remove high-frequency data representing more detailed features of in-focus voxels during the readout process, thereby leaving less detailed features from other out-of-focus voxels. As shown at 1110, the first and second low-pass filters are applied to each measurement image acquired for the voxel to form a background image for each measurement image.

[0047]

[0058] Continuing, method 1100 includes determining an enhanced background image from the first background image and the second background image at 1112. The enhanced background image may be determined in any suitable manner. In some examples, the enhanced background image may be determined by combining the first background image and the second background image using a multiplicative constant, as shown at 1114. A more specific example utilizes the following equation: b k =α(lowpass 1(m k )-Lowpass 2(m k ))+Lowpass2(m k ) (11) In the formula, measured intensity = m k and the local background intensity is the low-pass 1(m k ) and the overall background intensity is expressed as low-pass 2 (m k ), where α is a multiplicative constant used as a scale factor, and b k is the enhanced background image.

[0048]

[0059] The method 1100 further includes determining birefringence values for the enhanced background image at 1116 and determining birefringence values for the image of the voxel at 118. Each set of birefringence values includes a retardance value and an azimuthal value angle and may be determined using the examples described above or in any other suitable manner.

[0049]

[0060] The method 1100 further includes, at 1120, correcting the birefringence values for the image of the voxel based on the birefringence values for the enhanced background image. At 1122, correcting the birefringence values for the image may include, for example, determining a relative angle and relative retardance of the birefringence values for the image of the voxel compared to the birefringence values for the enhanced background image. In one example, the measured retardance δ is determined using, for example, any of the methods described above or other suitable methods.m and angle θ m is {m k} and the background retardance δ b and angle θ b is {b k}. The relative angle θ r and retardance δ r is determined by the following equation:

number

[0050]

[0061] In some embodiments, the methods and processes described herein may relate to the computing systems of one or more computing devices. In particular, such methods and processes may be implemented as computer application programs or services, application programming interfaces (APIs), libraries, and / or other computer program products.

[0051]

[0062] 12 illustrates generally a non-limiting embodiment of a computing system 1200 capable of implementing one or more of the methods and processes described above. The computing system 1200 is illustrated in simplified form. The computing system 1200 may take the form of one or more personal computers, server computers, tablet computers, home entertainment computers, network computing devices, gaming devices, mobile computing devices, mobile communication devices (e.g., smartphones), and / or other computing devices.

[0052]

[0063] Computing system 1200 includes a logic subsystem 1202 and a storage subsystem 1204. Computing system 1200 may optionally include a display subsystem 1206, an input subsystem 1208, a communication subsystem 1210, and / or other components not shown in FIG.

[0053]

[0064] Logical subsystem 1202 includes one or more physical devices configured to execute instructions. For example, a logical subsystem may be configured to execute instructions that are part of one or more applications, services, programs, routines, libraries, objects, components, data structures, or other logical constructs. Such instructions may be implemented to perform a task, implement a data type, transform the state of one or more components, achieve a technical effect, or reach a desired result.

[0054]

[0065] A logical subsystem may include one or more processors configured to execute software instructions. Additionally or alternatively, a logical subsystem may include one or more hardware or firmware logical subsystems configured to execute hardware or firmware instructions. The processors of a logical subsystem may be single-core or multi-core, and the instructions executed on the processors may be configured for sequential, parallel, and / or distributed processing. Individual components of a logical subsystem may optionally be remotely located and / or distributed among two or more separate devices configured for cooperative processing. Aspects of a logical subsystem may be virtualized and executed by remotely accessible networked computing devices arranged in a cloud computing configuration.

[0055]

[0066] Storage subsystem 1204 includes one or more physical devices configured to hold instructions executable by the logical subsystem to implement the methods and processes described herein. When such methods and processes are implemented, the state of storage subsystem 1204 may be transformed, for example, to hold different data.

[0056]

[0067] The storage subsystem 1204 may include removable and / or built-in devices. The storage subsystem 1204 may include optical memory (e.g., CD, DVD, HD-DVD, Blu-ray disc, etc.), semiconductor memory (e.g., RAM, EPROM, EEPROM, etc.), and / or magnetic memory (e.g., hard disk drive, floppy disk drive, tape drive, MRAM, etc.), among others. The storage subsystem 1204 may include volatile, non-volatile, dynamic, static, read / write, read-only, random access, sequential access, location-addressable, file-addressable, and / or content-addressable devices.

[0057]

[0068] It will be appreciated that storage subsystem 1204 may include one or more physical devices, however, aspects of the instructions described herein may alternatively be propagated by a communications medium (e.g., electromagnetic signals, optical signals, etc.) that is not carried by a physical device for a finite period of time.

[0058]

[0069] Aspects of logic subsystem 1202 and storage subsystem 1204 may be integrated together into one or more hardware logic components, which may include, for example, field programmable gate arrays (FPGAs), program and application specific integrated circuits (PASICs / ASICs), program and application specific standard products (PSSPs / ASSPs), systems on a chip (SOCs), and complex programmable logic devices (CPLDs).

[0059]

[0070] The term "program" may be used to describe aspects of computing system 1200 that are implemented to perform a particular function. In some cases, a program may be instantiated through logic subsystem 1202 executing instructions held by storage subsystem 1204. It will be understood that different programs may be instantiated from the same application, service, code block, object, library, routine, API, function, etc. Similarly, the same program may be instantiated from different applications, services, code blocks, objects, routines, APIs, functions, etc. The term "program" may encompass individual or groups of executable files, data files, libraries, drivers, scripts, database records, etc.

[0060]

[0071] A "service," as used herein, is understood to be an application program that can run across multiple user sessions. A service may be available to one or more system components, programs, and / or other services. In some implementations, a service may run on one or more server computing devices.

[0061]

[0072] The display subsystem 1206, if included, can be used to present a visual display of the data maintained by the storage subsystem 1204. This visual display can take the form of a graphical user interface (GUI). Because the methods and processes described herein modify the data maintained by the storage subsystem, and thus transform the state of the storage subsystem, the state of the display subsystem 1206 can likewise be transformed to visually represent the underlying data changes. The display subsystem 1206 can include one or more display devices utilizing virtually any type of technology. Such display devices can be combined with the logic subsystem 1202 and / or the storage subsystem 1204 in a shared enclosure, or such display devices can be peripheral display devices.

[0062]

[0073] If included, the input subsystem 1208 may include or interface with one or more user input devices, such as a keyboard, mouse, touchscreen, or game controller. In some embodiments, the input subsystem may include or interface with selected natural user input (NUI) components. Such components may be integrated or peripheral, and input action introduction and / or processing may be handled on-board or off-board. Exemplary NUI components may include microphones for voice and / or voiceprint authentication; infrared, color, stereo, and / or depth cameras for machine vision and / or gesture recognition; head trackers, eye trackers, accelerometers, and / or gyroscopes for motion detection and / or intent recognition; and electric field sensing components for assessing brain activity.

[0063]

[0074] If included, communications subsystem 1210 may be configured to communicatively couple computing system 1200 with one or more other computing devices. Communications subsystem 1210 may include wired and / or wireless communication devices compatible with one or more different communications protocols. By way of non-limiting example, communications subsystem may be configured to communicate over a wireless telephone network or over a wired or wireless local or wide area network. In some embodiments, communications subsystem may enable computing system 1200 to send messages to and / or receive messages from other devices over a network, such as the Internet.

[0064]

[0075] Another example provides a computer-implemented method for reading data stored as birefringence values in a storage medium, the method including: acquiring an image of voxels of the storage medium, the image of the voxels including high-frequency image information resulting from the birefringence state of the voxels and low-frequency image information resulting from one or more birefringent regions of the storage medium other than the voxels; applying a first low-pass filter having a first cutoff frequency to the image of the voxels to obtain a first background image; applying a second low-pass filter having a second cutoff frequency to the image of the voxels to obtain a second background image, the second cutoff frequency being different from the first cutoff frequency; determining an enhanced background image from the first background image and the second background image; determining a birefringence value for the enhanced background image; determining a birefringence value for the image of the voxels; and correcting the birefringence value for the image of the voxels based on the birefringence value for the enhanced background image. In some such examples, acquiring images of voxels of the storage medium includes acquiring multiple images of the voxels at different polarization states, and applying the first low pass filter and the second low pass filter to each image of the multiple images of the voxels. In some such examples, determining an enhanced background image includes combining the first background image and the second background image using a multiplicative constant. In some such examples, an image m of a voxel acquired at polarization state k includes applying the first low pass filter and the second low pass filter to each image of the multiple images of the voxels. k The improved background image for b k is shown as b k =α(lowpass 1(m k )-Lowpass 2(m k ))+Lowpass2(m k ) where the first low pass filter is Low Pass 1, the second low pass filter is Low Pass 2, and α is a multiplicative constant. In some such examples, the birefringence value for the enhanced background image is determined by b k The background retardance δ determined from b and background angle θ b and the birefringence value for the image of a voxel is m k The measured retardance δ determined from m and measurement angle θ m The method includes:

number

[0065]

[0076] Another example is a computer system including a logic subsystem and a storage subsystem including instructions executable by the logic subsystem to read birefringence data from a storage medium, wherein reading the birefringence data from the storage medium includes receiving an image of voxels of the storage medium, the image including high frequency image information resulting from birefringent states of the voxels and low frequency image information resulting from one or more birefringent regions of the storage medium other than the voxels; applying a first low pass filter having a first cutoff frequency to the image of the voxels to obtain a first background image; and applying a second low pass filter having a first cutoff frequency to the image of the voxels to obtain a second background image. and applying a second low-pass filter having a second cutoff frequency to the image of the voxel to obtain an enhanced background image, the second cutoff frequency being different from the first cutoff frequency; determining an enhanced background image from the first background image and the second background image; determining a birefringence value for the enhanced background image; determining a birefringence value for the image of the voxel; and correcting the birefringence value for the image of the voxel based on the birefringence value for the enhanced background image. In some such examples, the instructions executable to acquire the images of the voxel on the storage medium are executable to acquire multiple images of the voxel at different polarization states and to apply the first low-pass filter and the second low-pass filter to each of the multiple images of the voxel. In some such examples, the instructions executable to determine the enhanced background image include instructions executable to combine the first background image and the second background image using a multiplicative constant. In some such examples, the image m of the voxel acquired at polarization state k is k The improved background image for b k and the instruction is b k =α(lowpass 1(m k )-Lowpass 2(m k ))+Lowpass2(m k ) By bk where the first low pass filter is Low Pass 1, the second low pass filter is Low Pass 2, and α is a multiplicative constant. In some such examples, the birefringence value for the enhanced background image is b k The background retardance δ determined from b and background angle θ b and the birefringence value for the image of a voxel is m k The measured retardance δ determined from m and measurement angle θ m and the instruction includes:

number

[0013] In some such examples, the instructions are executable to correct the birefringence values by determining α. In some such examples, the instructions are further executable to determine α by optimizing the error between the known sample retardance and the calculated retardance distribution. In some such examples, the instructions are executable to obtain images of voxels by acquiring an image of an array of voxels at an image plane, wherein one or more birefringent regions of the storage medium other than the voxels include one or more other voxels outside the image plane. Another example is a computer-readable storage device including instructions executable by a computing system to read birefringence data from a storage medium, wherein reading the birefringence data from the storage medium includes receiving images of voxels of the storage medium, the image including high frequency image information resulting from the birefringent states of the voxels and low frequency image information resulting from the one or more birefringent regions of the storage medium other than the voxels; applying a first low pass filter having a first cutoff frequency to the images of the voxels to obtain a first background image; and applying a second low pass filter to the images of the voxels to obtain a second background image. applying a second low-pass filter having a second cutoff frequency to the image of the voxel, the second cutoff frequency being different from the first cutoff frequency; determining an enhanced background image from the first background image and the second background image; determining a birefringence value for the enhanced background image; determining a birefringence value for the image of the voxel; and correcting the birefringence value for the image of the voxel based on the birefringence value for the enhanced background image. In some such examples, the instructions executable to acquire the image of the voxel in the storage medium are executable to acquire multiple images of the voxel at different polarization states and to apply the first low-pass filter and the second low-pass filter to each image of the multiple images of the voxel.In some such examples, the instructions executable to determine an enhanced background image include instructions executable to combine the first background image and the second background image using a multiplicative constant. In some such examples, the image m of a voxel acquired at polarization state k is k The improved background image for b k and the instruction is b k =α(lowpass 1(m k )-Lowpass 2(m k ))+Lowpass2(m k ) By b k where the first low pass filter is Low Pass 1, the second low pass filter is Low Pass 2, and α is a multiplicative constant. In some such examples, the birefringence value for the enhanced background image is b k The background retardance δ determined from b and background angle θ b and the birefringence value for the image of a voxel is m k The measured retardance δ determined from m and measurement angle θ m and the instruction includes:

number

[0066]

[0077] Another example is a method for reading data stored as birefringence values in a storage medium on a computing device, comprising: measuring the intensity m of an image of a voxel of the storage medium using light of polarization state k;k and determining a likelihood function for an image of a voxel, the likelihood function representing, for each birefringence value set of a number of possible birefringence value sets, a likelihood of the intensity of the voxel being generated by the birefringence value set, the likelihood function being based on a selected data model and a selected noise model; determining a maximum value of the likelihood function; and determining a most probable birefringence value set for the voxel based on the birefringence value set that produces the maximum value of the likelihood function. In some such examples, the noise model includes a Gaussian noise model. In some such examples, determining an intensity m of the image of the voxel for a measurement state k k The likelihood of measuring

number

number

number

[0067]

[0078] Because many variations are possible, it will be understood that the configurations and / or techniques described herein are exemplary in nature, and that these specific embodiments or examples should not be considered limiting. The particular routines or methods described herein may represent one or more of any number of processing strategies. Thus, various acts shown and / or described may be performed in the order shown and / or described, in other orders, in parallel, or omitted. Similarly, the order of the processes described above may be changed.

[0068]

[0079] The subject matter of this disclosure includes all novel and non-obvious combinations and subcombinations of the various processes, systems and configurations, and other features, functions, acts and / or properties disclosed herein, and any and all equivalents thereof.

Claims

1. 1. A computer-implemented method for reading data stored as birefringence values in a storage medium, comprising: acquiring an image of a voxel of the storage medium, the image of the voxel including high frequency image information resulting from a birefringent state of the voxel and low frequency image information resulting from one or more birefringent regions of the storage medium other than the voxel; applying a first low-pass filter having a first cutoff frequency to the image of the voxels to obtain a first background image, the first low-pass filter removing high frequency data representing more detailed features of in-focus voxels and leaving background characteristics from other out-of-focus voxels; applying a second low pass filter having a second cutoff frequency lower than the first cutoff frequency to the image of voxels to obtain a second background image, the second low pass filter retaining more overall background characteristics; determining an enhanced background image from the first background image and the second background image; determining a birefringence value for the enhanced background image; determining a birefringence value for the image of the voxels; correcting the birefringence values for the image of the voxels based on the birefringence values for the enhanced background image; and A method comprising:

2. 2. The method of claim 1, wherein acquiring the images of the voxels of the storage medium includes acquiring multiple images of the voxels at different polarization states, and applying the first low-pass filter and the second low-pass filter to the images of the voxels includes applying the first low-pass filter and the second low-pass filter to each image of the multiple images of the voxels.

3. The method of claim 1 , wherein determining the enhanced background image comprises combining the first background image and the second background image using a multiplicative constant.

4. Image m of said voxel acquired at polarization state k ｋ The enhanced background image for b ｋ is shown as b ｋ = α (low pass １ (m ｋ ) - Low Pass ２ (m ｋ )) + low pass ２ (m ｋ ) is determined by wherein the first low-pass filter is a low-pass １ and the second low-pass filter is ２ 4. The method of claim 3, wherein α is a multiplicative constant.

5. The birefringence value for the enhanced background image is b ｋ The background retardance δ determined from ｂ and background angle θ ｂ and the birefringence value for the image of the voxel is m ｋ The measured retardance δ determined from ｍ and measurement angle θ ｍ 5. The method of claim 4, comprising: [Equation 1] By this, the relative angle θ ｒ and the relative retardance δ ｒ The method further comprising correcting the birefringence value by determining:

6. The method of claim 4 , further comprising determining α based on optimizing the error between the retardance of a known sample and the calculated retardance distribution.

7. 2. The method of claim 1 , wherein acquiring the image of the voxels comprises acquiring an image of an array of voxels at an image plane, and wherein the one or more birefringent regions of the storage medium other than the voxels comprise one or more other voxels outside the image plane.

8. The method of claim 1 , wherein the substrate comprises fused silica.

9. a logical subsystem; a storage subsystem containing instructions executable by said logic subsystem to read birefringence data from a storage medium; and A computer system comprising: reading the birefringence data from the storage medium; receiving an image of voxels of the storage medium, the image including high frequency image information arising from birefringent states of the voxels and low frequency image information arising from one or more birefringent regions of the storage medium other than the voxels; applying a first low-pass filter having a first cutoff frequency to the image of the voxels to obtain a first background image, the first low-pass filter removing high frequency data representing more detailed features of in-focus voxels and leaving background characteristics from other out-of-focus voxels; applying a second low pass filter having a second cutoff frequency lower than the first cutoff frequency to the image of voxels to obtain a second background image, the second low pass filter retaining more overall background characteristics; determining an enhanced background image from the first background image and the second background image; determining a birefringence value for the enhanced background image; determining a birefringence value for the image of the voxels; correcting the birefringence values for the image of the voxels based on the birefringence values for the enhanced background image; and A computer system.

10. 10. The computer system of claim 9, wherein the instructions executable to acquire the images of the voxels of the storage medium are executable to acquire multiple images of the voxels at different polarization states and to apply the first low-pass filter and the second low-pass filter to each image of the multiple images of the voxels.

11. Image m of said voxel acquired at polarization state k ｋ The enhanced background image for b ｋ and the instruction is b ｋ = α (low pass １ (m ｋ ) - Low Pass ２ (m ｋ )) + low pass ２ (m ｋ ) By b ｋ is feasible to determine wherein the first low-pass filter is a low-pass １ and the second low-pass filter is ２ 10. The computer system of claim 9, wherein α is a multiplicative constant.

12. The birefringence value for the enhanced background image is b ｋ The background retardance δ determined from ｂ and background angle θ ｂ and the birefringence value for the image of the voxel is m ｋ The measured retardance δ determined from ｍ and measurement angle θ ｍ wherein the instructions include: [Equation 2] The relative angle θ given by ｒ and the relative retardance δ ｒ 12. The computer system of claim 11, wherein the computer system is executable to correct the birefringence value by determining:

13. 12. The computer system of claim 11, wherein the instructions are further executable to determine α by optimizing an error between a retardance of a known sample and a calculated retardance distribution.

14. 10. The computer system of claim 9, wherein the instructions are executable to acquire the image of the voxels by acquiring an image of an array of voxels in an image plane, and the one or more birefringent regions of the storage medium other than the voxels include one or more other voxels outside the image plane.

15. 10. The computer system of claim 9, wherein the storage medium comprises fused silica.

16. 8. A computer program stored on a storage medium of a computer system and comprising instructions executable by a processor of said computer system, said instructions, when executed by said processor, causing said computer system to perform a method according to any one of claims 1 to 7.

Images