Methods and apparatus of determining a stress-related characteristic of a substrate

Abstract

Apparatus includes a first beam source, a prism, a polarization filter array, and a first detector. The first detector is configured to detect at least a portion of the first beam emitted from the first beam source after the first beam impinges the prism and contains information about transverse electric and transverse magnetic mode spectra associated with the substrate. The polarization filter array is positioned between the prism and the first detector along a detection light axis. The polarization filter array comprises at least two portions of a second type of polarization filter sandwiching a first portion of the first type of polarization filter in a direction perpendicular to the light detection axis. Methods involve passing a light beam that has impinged a substrate and the prism through the polarization filter array before it impinges the first detector.

Description

ATTORNEY DOCKET NO. SP24-280METHODS AND APPARATUS OF DETERMINING A STRESS-RELATED CHARACTERISTIC OF A SUBSTRATE

[0001] This application claims the benefit of priority to U.S. Provisional Patent Application No. 63 / 715,305 filed November 01, 2024, the content of which is relied upon and incorporated herein by reference in its entirety.FIELD

[0002] The present disclosure relates generally to methods and apparatus for determining a stress-related characteristic of a substrate and, more particularly, to methods and apparatus of determining a stress-related characteristic of a substrate using a polarization filter.BACKGROUND

[0003] The evanescent prism-coupling spectroscopy (EPCS) method measures refractive index and birefringence profiles (and thus stress-related characteristics) of samples. The EPCS method passes input light through a coupling prism (e.g., reference block) and the sample being measured. The coupling prism is also used to couple light out of the sample to filter components corresponding to a transverse electric (TE) mode spectrum and a transverse magnetic (TM) spectrum that are measured by an image sensor. The TE and TM modes spectrum are analyzed to extract stress-related characteristics, including a stress profile.

[0004] Light-scattered polarimetry (LSP) uses scattered polarized light to determine stress- related characteristics of samples capable of scattering light from within the sample material. The sample is irradiated with input light at a relatively shallow angle. The light polarization is varied continuously between different polarization states using an optical compensator. The scattered light is detected by an image sensor. Stress in the sample causes optical retardation along the light path, with the amount of stress being proportional to the derivative of the optical retardation. The amount of optical retardation can be determined from the detected scattered light intensity distribution, which varies due to the constructive and destructive interference for the different effective path lengths of the detected light. The stress-related properties that can be measured include stress profiles, central tension (CT) and depth of compression (DOC). However, measurements can be subject to noise and measurements for the region in compression can be unreliable.ATTORNEY DOCKET NO. SP24-280

[0005] The determination of stress-related characteristics often relies on a combination of LSP and EPCS measurements. Consequently, there is a need for apparatus and methods to produce increased accuracy and precision in the estimate of stress-based characteristics.SUMMARY

[0006] There are set forth herein apparatus and methods for determining a stress-related characteristic of a substrate using a prism-coupling apparatus having a polarization filter array in accordance with the present disclosure. Providing multiple portions of a first type and / or a second type of polarizer in the polarization filter array enables higher accuracy and higher precision measurements of at least one stress-related characteristic to be performed (even in the presence of tilt in the mode spectra - including inconsistent tilt - and / or optical aberrations from optical elements). As discussed herein, the polarization filter array can have portions of different types of polarization filters arranged in a line and / or a two-dimensional array. Further, the Examples discussed herein demonstrate the increased accuracy and precision enabled by the polarization filter arrays in accordance with aspects of the present disclosure.

[0007] In aspects, the polarization filter array can be part of an EPCS apparatus in the prism-coupling apparatus. Further, the prism-coupling apparatus can include another measurement apparatus (e.g., LSP apparatus, refractive near field (RNF) apparatus). For example, a combined apparatus for measurements using LSP and EPCS can simplify and speed up the measurement process. Also, the combined apparatus reduces the risk of sample breakage because less handling is required to load the sample into the combined apparatus as compared to two separate apparatus. Methods of using the combined apparatus can additionally produce more reliable measurements for overall stress profiles.

[0008] Some example aspects of the disclosure are described below with the understanding that any of the features of the various aspects may be used alone or in combination with one another.

[0009] Aspect 1. A prism-coupling apparatus for determining at least one stress-related characteristic of a substrate, the prism-coupling apparatus comprising: a sample holder comprising a cavity configured to receive the substrate and defining a viewing aperture; a prism configured to allow a first beam of a measurement beam to travel between the prism and the cavity configured to receive the substrate; a first beam source configured to transmit the first beam that impinges the prism and the viewing aperture;ATTORNEY DOCKET NO. SP24-280 a first detector configured to detect at least a portion of the first beam after the first beam impinges the prism and contains information about a transverse electric (TE) mode spectrum and a transverse magnetic (TM) mode spectrum associated with the substrate; and a polarization filter array positioned between the prism and the first detector along a detection light axis, wherein the polarization filter array comprises a first type of polarization filter configured to permit light with a first polarization therethrough and a second type of polarization filter configured to permit light with a second polarization therethrough, the first polarization is perpendicular to the second polarization, and the polarization filter array comprises at least two portions of the second type of polarization filter sandwiching a first portion of the first type of polarization filter in a first direction perpendicular to the detection light axis.

[0010] Aspect 2. The prism -coupling apparatus of aspect 1, wherein the first direction is substantially perpendicular to a principal axis of the TE mode spectrum, the TM mode spectrum, or both.

[0011] Aspect 3. The prism -coupling apparatus of any one of aspects 1-2, wherein the prism-coupling apparatus is configured to produce a signal at the first detector corresponding to a portion of the TE mode spectrum sandwiched between portions of the TM mode spectrum.

[0012] Aspect 4. The prism -coupling apparatus of any one of aspects 1-2, wherein the prism-coupling apparatus is configured to produce a signal at the first detector corresponding to a portion of the TM mode spectrum sandwiched between portions of the TE mode spectrum.

[0013] Aspect 5. The prism -coupling apparatus of any one of aspects 1-4, wherein the polarization filter array further comprises a second portion of the first type of polarization filter, the first portion and the second portion of the first type of polarization filter sandwiches the second portion of the second type of polarization filter in the first direction.

[0014] Aspect 6. The prism-coupling apparatus of any one of aspects 1-5, wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a line along the first direction with the portions of the first type of polarization filter alternating with the portions of the second type of polarization filter along the line.

[0015] Aspect 7. The prism-coupling apparatus of any one of aspects 1-5, wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a two-dimensional array perpendicular to the detection light axis, wherein a portion of the first type of polarization filter is positioned between adjacent pairs of portions of the second type of polarization filter.ATTORNEY DOCKET NO. SP24-280

[0016] Aspect 8. The prism-coupling apparatus of aspect 7, wherein the polarization filter array further comprises a third type of polarization filter and fourth type of polarization filter each configured to permit portions of light with the first polarization and light with the second polarization therethrough, a polarization of the third type of polarization filter is different than a polarization of the fourth type of polarization filter, and the two-dimensional array further includes portions of the third type of polarization filter and the fourth type of polarization filter.

[0017] Aspect 9. The prism-coupling apparatus of aspect 8, wherein the two-dimensional array of polarization filters is arranged as a two-dimensional array of pixels, where each pixel corresponds to a two-by-two arrangement of a portion of the first type of polarization filter, a portion of the second type of polarization filter, a portion of the third type of polarization filter, and a portion of the fourth type of polarization filter.

[0018] Aspect 10. The prism-coupling apparatus of any one of aspects 1-9, wherein the first beam source is configured to produce the first beam that is monochromatic and uncollimated.

[0019] Aspect 11. The prism -coupling apparatus of any one of aspects 1-10, wherein the first beam source, the first detector, and the polarization filter array is an evanescent prismcoupling spectroscopy sub-system.

[0020] Aspect 12. The prism -coupling apparatus of any one of aspects 1-11, further comprising a light-scattering polarimetry sub-system comprising: a second beam source configured to transmit a second beam of the measurement beam that impinges the prism and the viewing aperture; and a second detector of the light-scattering polarimetry sub-system is configured to detect at least a portion of the second beam after the second beam impinges the prism.

[0021] Aspect 13. The prism-coupling apparatus of any one of aspects 1-12, wherein a projection of the polarization filter array onto an imaging plane of the first detector along the detection light axis covers substantially an entire field-of-view of the first detector.

[0022] Aspect 14. The prism -coupling apparatus of any one of aspects 1-13, wherein the polarization filter array further comprises a light-blocking portion between adjacent polarization filters of the polarization filter array.

[0023] Aspect 15. The prism -coupling apparatus of any one of aspects 1-14, wherein the at least one stress-related characteristic comprises a stress profile, a knee stress, a central tension, a tension-strain energy, a birefringence, a spike depth, a depth of layer, a surface compressive stress, a refractive index profile, or combinations thereof.ATTORNEY DOCKET NO. SP24-280

[0024] Aspect 16. The prism -coupling apparatus of any one of aspects 1-15, wherein the prism-coupling apparatus is configured to work with the substrate comprising a glass-based material.

[0025] Aspect 17. The prism -coupling apparatus of any one of aspects 1-15, wherein the prism-coupling apparatus is configured to work with the substrate comprising a glass-ceramic material.

[0026] Aspect 18. A method of determining at least one stress-related characteristic of a substrate comprising: disposing the substrate in a cavity of a sample holder; disposing a coupling prism over a first surface of the substrate, a coupling liquid positioned between a coupling surface of a prism and the first surface of the substrate in a viewing aperture of the sample holder; transmitting a first beam from a first beam source that impinges the prism, the coupling liquid, and the first surface of the substrate at a measurement location; passing at least a portion of the first beam through a polarization filter array after the first light beam impinges the prism and contains information about a transverse electric (TE) mode spectrum and a transverse magnetic (TM) mode spectrum to generate a filtered first beam; detecting at least a portion of the first filtered beam with a first detector to generate a first signal; and determining the at least one stress-related characteristic based on at least the first signal, wherein the at least the portion of the first beam passing through the polarization filter array is travelling along a detection light axis, the polarization filter array comprises a first type of polarization filter permitting light containing information corresponding to the TE mode spectrum to travel therethrough and a second type of polarization filter permitting light containing information corresponding to the TM mode spectrum to travel therethrough, the polarization filter array comprises at least two portions of the second type of polarization filter sandwiching a first portion of the first type of polarization filter in a first direction perpendicular to the detection light axis.

[0027] Aspect 19. The method of aspect 18, wherein the first beam source, the polarization filter array, and the first detector are part of an evanescent prism-coupling spectroscopy subsystem, and the determining comprises processing the first signal to form an optical retardation versus depth curve based on information in the TE mode spectrum and the TM mode spectrum.ATTORNEY DOCKET NO. SP24-280

[0028] Aspect 20. The method of any one of aspects 18-19, wherein a light-scattering polarimetry sub-system comprises the second beam source, the second prism, and the second detector, the method further comprises: emitting a second light beam from the second beam source; impinging the second light beam on the coupling prism and the substrate; detecting the second light beam after the impinging by the second detector to generate a second signal; and the determining comprises processing the second signal is processed to determine an optical retardance.

[0029] Aspect 21. The method of any one of aspects 18-20, wherein a position of a TM mode line in the TM mode spectrum is based on an average TM position calculated based on information corresponding to the TM mode spectrum that passed through each portion of the second type of polarizer.

[0030] Aspect 22. The method of aspect 21, wherein a position of a TE mode line in the TE mode spectrum is based on an average TE position calculated based on information corresponding to the TM mode spectrum that passed through each portion of the first type of polarizer.

[0031] Aspect 23. The method of aspect 22, wherein an optical retardance is calculated as a difference between the average TM position and the average TE position.

[0032] Aspect 24. The method of any one of claims 18-20, wherein a fitted TM mode line in the TM mode spectrum is based on a linear regression based on information corresponding to the TM mode spectrum that passed through each portion of the first type of polarizer.

[0033] Aspect 25. The method of aspect 24, wherein a fitted TE mode line in the TE mode spectrum is based on a linear regression based on information corresponding to the TE mode spectrum that passed through each portion of the first type of polarizer.

[0034] Aspect 26. The method of aspect 25, wherein an optical retardance is calculated as a difference between the fitted TE mode line and the fitted TM mode line at a location where the fitted TE mode line and the fitted TM mode line are interpolated.

[0035] Aspect 27. The method of any one of aspects 18-26, wherein the first signal generated at the first detector includes information corresponding to a portion of the TE mode spectrum sandwiched between portions of the TM mode spectrum detected by the first detector.

[0036] Aspect 28. The method of any one of aspects 18-27, wherein the first signal generated at the first detector includes information corresponding to a portion of the TM mode spectrum sandwiched between portions of the TE mode spectrum detected by the first detector.ATTORNEY DOCKET NO. SP24-280

[0037] Aspect 29. The method of any one of aspects 18-28, wherein the polarization filter array further comprises a second portion of the first type of polarization filter, the first portion and the second portion of the first type of polarization filter sandwiches the second portion of the second type of polarization filter in the first direction.

[0038] Aspect 30. The method of any one of claims 18-29, wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a line along the first direction with the portions first type of polarization filter alternating with the portions of the second type of polarization filter along the line.

[0039] Aspect 31. The method of any one of aspects 18-29, wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a two-dimensional array perpendicular to the detection light axis, wherein a portion of the first type of polarization filter is positioned between adjacent pairs of portions of the second type of polarization filters.

[0040] Aspect 32. The method of aspect 31, wherein the polarization filter array further comprises a third type of polarization filter and fourth type of polarization filter each permitting portions of light with light containing information about both the TE mode spectrum and the TM mode spectrum therethrough, a polarization of the third type of polarization filter is different than a polarization of the fourth type of polarization filter, and the two-dimensional array further includes portions of the third type of polarization filter and the fourth type of polarization filter.

[0041] Aspect 33. The method of aspect 32, wherein the two-dimensional array of polarization filters is arranged as a two-dimensional array of pixels, where each pixel corresponds to a two-by-two arrangement of a portion of the first type of polarization filter, a portion of the second type of polarization filter, a portion of the third type of polarization filter, and a portion of the fourth type of polarization filter.

[0042] Aspect 34. The method of any one of aspects 18-33, wherein the first beam is monochromatic and uncollimated.

[0043] Aspect 35. The method of any one of aspects 18-34, wherein a projection of the polarization filter array onto an imaging plane of the first detector along the detection light axis covers substantially an entire field-of-view of the first detector.

[0044] Aspect 36. The method of any one of aspects 18-35, wherein the polarization filter array further comprises a light-blocking portion between adjacent polarization filters of the polarization filter array.ATTORNEY DOCKET NO. SP24-280

[0045] Aspect 37. The method of any one of aspects 18-36, wherein the stress-related characteristic comprises a stress profde, a knee stress, a central tension, a tension-strain energy, a birefringence, a spike depth, a depth of layer, a surface compressive stress, a refractive index profile, or combinations thereof.

[0046] Aspect 38. The method of any one of aspects 18-37, wherein the prism-coupling apparatus is configured to work with the substrate comprising a glass-based material.

[0047] Aspect 39. The method of any one of aspects 18-37, wherein the prism-coupling apparatus is configured to work with the substrate comprising a glass-based material.BRIEF DESCRIPTION OF THE DRAWINGS

[0048] The above and other features and advantages of aspects of the present disclosure are better understood when the following detailed description is read with reference to the accompanying drawings, in which:

[0049] FIG. 1 is a schematic block diagram of an example prism-coupling according to some aspects;

[0050] FIG. 2 is a schematic view of an example prism-coupling apparatus according to some aspects;

[0051] FIG. 3 is a schematic view of another prism-coupling apparatus according to some aspects;

[0052] FIG. 4 schematically shows a view along line 4 — 4 of any of FIGS. 2-3 of an exemplary polarization filter array in accordance with some aspects;

[0053] FIG. 5 schematically shows a view along line 4 — 4 of any of FIGS. 2-3 or corresponding to a view of polarization filter array 329 of FIG. 3 representing another exemplary polarization filter in accordance with some aspects;

[0054] FIG. 6 schematically shows TE and TM mode spectra produced by a two-portion polarization filter (polarization filter array AA);

[0055] FIG. 7 schematically shows multiple portions of TE and TM mode spectra produced by a four-portion polarization filter array (polarization filter array 1) arranged in a line in accordance with some aspects;

[0056] FIG. 8 schematic shows measured data corresponding to fitted mode lines, where the fitted mode lines have the same tilt;

[0057] FIG. 9 schematically shows measured data corresponding to fitted mode lines, where the fitted mode lines are not parallel to one another;

[0058] FIG. 10 presents results showing the consistency of measured distance between TE and TM mode lines in the absence of tilt using a two-portion polarization filter;ATTORNEY DOCKET NO. SP24-280

[0059] FIG. 11 presents results showing the inconsistency of measured distance between TE and TM mode lines in the presence of tilt using a two-portion polarization filter;

[0060] FIG. 12 presents results showing the consistency of measured distance between TE and TM mode lines in the absence of tilt using a polarization fdter array in accordance with the present disclosure;

[0061] FIG. 13 presents results showing the consistency of measured distance between TE and TM mode lines in the presence of tilt using a polarization fdter array in accordance with the present disclosure; and

[0062] FIG. 14 schematically shows an exemplary polarization fdter array configured to cover an entire aperture of the first detector in accordance with aspects.

[0063] Throughout the disclosure, the drawings are used to emphasize certain aspects. As such, it should not be assumed that the relative size of different regions, portions, and substrates shown in the drawings are proportional to its actual relative size, unless explicitly indicated otherwise.DETAILED DESCRIPTION

[0064] Aspects will now be described more fully hereinafter with reference to the accompanying drawings in which example aspects are shown. Whenever possible, the same reference numerals are used throughout the drawings to refer to the same or like parts.

[0065] FIGS. 1-3 illustrate views of a prism-coupling apparatus 101 and / or 121 having a polarization fdter array 129 and / or 329 in accordance with aspects of the present disclosure. FIGS. 4-5 present exemplary polarization fdter arrays 401 and / or 501 in accordance with the present disclosure comprising a first type of polarizer sandwiched between portions of a second type of polarizer. In aspects, the first type of polarizer can be configured to permit light associated with a transverse electric (TE) mode spectrum while the second type of polarizer can be configured to permit light associated with a transverse magnetic (TM) mode spectrum. Alternatively, the first type of polarizer can be configured to permit light associated with a TM mode spectrum while the second type of polarizer can be configured to permit light associated with a TE mode spectrum. Providing multiple portions of the second type of polarizer enables higher accuracy and higher precision measurements of at least one stress-related characteristic to be performed (even in the presence of tilt in the mode spectra - including inconsistent tilt - and / or optical aberrations from optical elements). Unless otherwise noted, a discussion of features of aspects of one apparatus can apply equally to corresponding features of any of the aspects of the disclosure. For example, identical part numbers throughout the disclosure can indicate that, in aspects, the identified features are identical to one another and that the discussion of the identified feature of one aspect, unlessATTORNEY DOCKET NO. SP24-280 otherwise noted, can apply equally to the identified feature of any of the other aspects of the disclosure.

[0066] FIGS. 1-3 illustrate views of a prism-coupling apparatus 101 and / or 121 in accordance with the aspects of the disclosure. The apparatus 101 shown in FIGS. 1-2 will now be discussed with an understanding that the description of components of the apparatus 101 are equally applicable to the prism-coupling apparatus 301 shown in FIG. 3, unless indicated otherwise. As shown in FIGS. 1-2, the apparatus can be an apparatus 101 comprising a lightscattering polarimeter (LSP) apparatus 131 and an evanescent prism-coupling system (EPCS) apparatus 121. FIG. 3 shows a different view of the EPCS apparatus 121 (and / or a standalone EPCS system as the prism-coupling apparatus 301), and FIG. 2 shows a different view of the LSP apparatus 131.

[0067] FIGS. 1-2 schematically illustrate the apparatus 101 comprising a housing 107 enclosing the LSP apparatus 131 and the EPCS apparatus 121. In aspects, the housing 107 or other area of the apparatus 101 can comprise first dimension LI and a second dimension L2 where components of the apparatus 101 are confined within an area defined by the first dimension LI and the second dimension L2. In aspects, LI and / or L2 can be in a range from 200 mm to 1 meter, from 200 mm to 500 mm, from 200 mm to 300 mm, or any range or subrange therebetween. In aspects, although not shown, the controller 141 can be positioned outside of the housing 107.

[0068] As shown in FIGS. 1-2, the LSP apparatus 131 comprises a first polarizationswitching light source 133. In aspects, as shown in FIG. 2, the first polarization-switching light source 133 comprises a first light source 201 and a first optical compensator 203. The first light source 201 can comprise a laser, a light-emitting diode (LED), and / or an organic light emitting diode. In aspects, the first light source 201 can be configured to emit a first light beam comprising a first optical wavelength. In further aspects, the first optical wavelength can be in a range from 300 nanometers (nm) to 1,000 nm, from 350 nm to 900 nm, from 400 to 800 nm, or any range or subrange therebetween. In even further aspects, the first wavelength can be (and / or be centered on) 365 nm, 405 nm, 415 nm, 450 nm, 510 nm, 590 nm, 650 nm, or 780 nm. In aspects, the first light source 201 can be configured to selectively emit the first light beam comprising one of plurality of optical wavelengths, for example, by sequentially emitting different optical wavelengths from different component light sources of a plurality of light sources in the first light source. In aspects, the first polarization-switching light source 133 is configured to emit a first polarization-switched light beam along a first path 205 (e.g., portions 205F, 205S), as shown in FIG. 2.

[0069] In aspects, the first optical compensator 203 can comprise a polarizing beam splitter. In further aspects, the first optical compensator 203 can comprise a half-wave plate and aATTORNEY DOCKET NO. SP24-280 quarter-wave plate. In even further aspects, one of the half-wave plate or the quarter-wave plate can be rotatable relative to the other, which can change the polarization of a light beam passing through it. In further aspects, the first optical compensator 203 can comprise an electronically controlled polarization modulator, for example, a liquid-crystal-based modulator or a ferroelectric liquid-crystal-based modulator. In further aspects, the first optical compensator 203 can be controlled by the controller 141 (discussed below).

[0070] As used herein, the first polarization-switching light source 133 (e.g., including the first optical compensator 203) is configured to cycle between two or more polarization states (polarizations). In aspects, the first polarization-switching light source 133 can be configured to switch between (e.g., cycle through) up to eight different polarizations that combine the linear, elliptical, and / or circular polarizations. In further aspects, the first polarization-switching light source 133 can be configured to go through a full polarization cycle (e.g., change between two or more polarizations) in a range from less than 1 second to 10 seconds.

[0071] In aspects, as shown in FIG. 2, the LSP apparatus 131 can comprise a first focusing lens 135. As shown, the first focusing lens 135 can be positioned along the first path 205 of the first polarization-switched light beam. After passing through the first focusing lens 135, the first polarization-switched light beam can be focused along portion 205F of the first path 205. The first focusing lens 135 can comprise a convex lens and / or an adjustable focal length lens. In aspects, the first focusing lens 135 can be configured to collimate the first polarization-switched light beam along portion 205F of the first path 205. In aspects, although not shown, a bandpass filter, additional focusing lenses, a light diffuser, a beam splitter, and / or an attenuator can be positioned along portions 205F and / or 205S of the first path 205. In further aspects, one or more of these additional elements can be controlled by the controller 141.

[0072] In aspects, as shown in FIGS. 1-2, the LSP apparatus 131 comprises a coupling prism 113. In further aspects, as shown in FIGS. 1-3, the prism-coupling apparatus 101 and / or 301 can comprise a unitary (e.g., single) coupling prism 113, although the prism-coupling apparatus can comprise two prisms (e.g., a first coupling prism associated with the EPCS apparatus and a second coupling prism associated with the LSP apparatus) or more in other aspects. In further aspects, as shown in FIG. 2, the coupling prism 113 comprises an input surface 227 and an output surface 223. In further aspects, as shown in FIG. 2, the input surface 227 can comprise an end face of the prism. Although not shown, the portion 205F of the first path 205 can impinge the input surface 227 at an inclination that is substantially normal to the input surface 227. In further aspects, as shown in FIG. 2, the coupling prism 113 comprises an input surface 225 opposite the output surface 223. In further aspects, as shown in FIG. 3, the coupling prism 113 comprises a couplingATTORNEY DOCKET NO. SP24-280 surface 229. In even further aspects, as shown, the coupling surface 229 of the coupling prism 113 can face a coupling liquid 215 and / or a first major surface 105 of the substrate 103. In still further aspects, the coupling surface 229 of the coupling prism 113 can contact the coupling liquid 215. In further aspects, as shown in FIG. 3, the output surface 223, the input surface 225, and the coupling surface 229 can defined a triangular (e.g., equilateral triangle) cross-section of the coupling prism 113, although other arrangements can be provided in further aspects. Also, as shown, the EPCS apparatus 121 and the LSP apparatus 131 can share the same output surface 223 of the coupling prism, although light beams correspond to the EPCS apparatus 121 and the LSP apparatus 131 can exit from different surfaces of the same prism (or even different coupling prisms) in other aspects.

[0073] In aspects, the coupling prism 113 can comprise a first refractive index. Throughout the disclosure, with reference to the coupling prism(s) and the coupling liquid, a refractive index is measured in accordance with ASTM El 967- 19 using light comprising an optical wavelength of 589 nm. In aspects, the first refractive index of the coupling prism 113 may be 1.3 or more, 1.4 or more, 1.45 or more, 1.5 or more, 1.7 or less, 1.6 or less, 1.55 or less, or 1.5 or less. In aspects, the first refractive index of the coupling prism 113 can be in a range from 1.3 to 1.7, from 1.4 to 1.6 from 1.45 to 1.55, from 1.5 to 1.55, or any range or subrange therebetween.

[0074] In aspects, as schematically shown in FIGS. 2-3, the prism-coupling apparatus 101 and / or 301 (e.g., LSP apparatus 131 and / or EPCS apparatus 121) can comprise the coupling liquid 215. In further aspects, as shown, the coupling liquid 215 can contact the coupling prism 113 and comprise a third refractive index that can be greater than, less than, or equal to the first refractive index. In further aspects, a differential equal to the absolute value between the third refractive index and the first refractive index can be in a range from 0.05 or more, 0.06 or more, 0.08 or more, 0.10 or more, 0.12 or more, or 0. 14 or more. In further aspects, a differential equal to the absolute value between the third refractive index and the first refractive index can be in a range from 0.05 to 0.20, from 0.04 to 0.18, from 0.06 to 0.15, from 0.08 to 0.10, or any range or subrange therebetween. In further aspects, as shown, the coupling liquid 215 can be positioned between the coupling prism 113 and the cavity 319 configured to the receive the substrate 103. In further aspects, as shown, the coupling liquid 215 can contact the sample holder 251, be positioned within the viewing aperture 315, and / or contact the first major surface 105 of the substrate 103. In aspects, although not shown, the coupling liquid 215 may not extend past the coupling prism 113 in the x-direction. In aspects, although not shown, the coupling liquid 215 may not extend past the coupling prism 113 in the z-direction and / or the y-direction.ATTORNEY DOCKET NO. SP24-280

[0075] As shown in FIGS. 1-2, the LSP apparatus 131 can comprise a first detector 137. In aspects, the first detector 137 can comprise a digital camera, a CCD, and / or an array of photodetectors. In aspects, the first detector may comprise one or more focusing lenses, although an attenuator and / or a beam splitter can be provided in addition or alternatively in other aspects. In aspects, as shown in FIG. 2, the first detector 137 can comprise an image sensor 217. In further aspects, the image sensor 217 can comprise an array of imaging pixels, which can be arranged in a two-dimensional array. In even further aspects, a maximum dimension of a pixel of the array of imaging pixels can be in a range from 1 micrometer (pm) to 15 pm, from 2 pm to 10 pm, from 5 pm to 8 pm, or any range or subrange therebetween. In further aspects, as shown in FIG. 2, the first detector 137 can face the output surface 223 of the coupling prism 113. In further aspects, the first detector 137 can be positioned along the portion 205S of the first path 205 to detect a signal from the scattered first polarization-switched light beam that traveled through the coupling prism 113. As shown in FIGS. 1-2, the first detector 137 can be connected to the controller 141 by a communication path configured to transmit a signal SB. It is to be understood that another detector can face the second surface of the coupling prism to detect another portion of the scattered first polarization-switched light beam that traveled through the coupling prism, and / or an angle between the first detector, the coupling prism, and the second detector can be substantially a right angle (e.g., in a range from 85° to 95°), which can enable multiple measurements to be captured in the same polarization that can be combined (e.g., averaged) to decrease noise in the measurement and / or reduce the time needed to take the corresponding measurement.

[0076] In aspects, as shown in FIGS. 2-3, the LSP apparatus 131 and / or the EPCS apparatus 121 can be configured to interface with a sample (e.g., substrate) positioned in a cavity 319 of a sample holder 251. As discussed below, the sample holder 251 can at least partially define the cavity 319 configured to receive a substrate 103 (e.g., sample). In further aspects, the sample holder 251 can comprise a viewing aperture 315 configured to enable the first polarization- switched light beam (first path 205) and / or the second light beam (second path 207) to pass therethrough between the coupling prism 113 (and / or the coupling liquid 215) and the substrate 103 (e.g., measurement location ML, cavity 319). As shown, the viewing aperture 315 can be defined by a platform 313 of the sample holder 251, although the viewing aperture can be defined by a free surface of substrate (e.g., first major surface) while the substrate is in the sample holder. In further aspects, although not shown, the sample holder can be translatable, for example to enable measurements to be taken at multiple locations on the substrate (e.g., averaged along a path). In further aspects, the sample holder 251 can be configured to receive a planar substrate and / or aATTORNEY DOCKET NO. SP24-280 curved substrate, where the sample holder can comprise supports (e.g., adjustable supports) to restrain movement of the substrate in the sample holder.

[0077] In aspects, as shown in FIG. 2, the LSP apparatus 131 can be configured so that the first path 205 of the first polarization-switched light beam impinges on the input surface 227 of the coupling prism 113. In further aspects, the coupling prism 113 can be positioned between the first polarization-switching light source 133 and the cavity 319 (of the sample holder 251) and / or the substrate 103. In further aspects, the first path 205 can be configured to impinge the input surface 227 of the coupling prism 113 at a substantially normal angle of incidence (e.g., in a range from 85° to 95° relative to the input surface 227). In further aspects, as discussed above, the first path 205 can impinge the viewing aperture 315 of the sample holder 251 or the substrate 103 before impinging one or more of the output surface 223 of the prism. In even further aspects, as shown, the LSP apparatus 131 can comprise one or more detectors configured to detect a signal from the first polarization-switched light beam.

[0078] As shown in FIGS. 1-3, the EPCS apparatus 121 can comprise a second light source 123. In aspects, as shown in FIGS. 2-3, the second light source 123 can comprise a second light source 221 in combination with focusing optics (e.g., converging lens 125). In further aspects, the second light source 221 can comprise one or more of the light sources discussed above with regards to the first light source 201. In further aspects, the second light source 123 can be configured to emit a second light beam along a second path 207 (e.g., first axis Al). In aspects, the second light source 123 can comprise a second optical compensator, which can comprise one or more of the optical compensators discussed above with regards to the first optical compensator 203. In aspects, as shown in FIGS. 2-3, the EPCS apparatus 121 can comprise a converging lens 125. In further aspects, as shown in FIG. 3, the converging lens 125 can be configured to focus the second light beam passing through the converging lens 125 to form a focal point at an interface between the first major surface 105 of the substrate 103 and coupling liquid 215 (e.g., measurement location ML) and / or within the viewing aperture 315. Consequently, the second light beam can be an uncollimated light beam. In aspects, although not shown, one or more of the optical elements discussed with regards to the first focusing lens 135 (e.g., bandpass filter, additional focusing lenses, a light diffuser, a beam splitter, an attenuator) can be positioned along a portion of the second path 207. In further aspects, one or more of these additional elements can be controlled by the controller 141. For example, in even further aspects, although not shown, a mask 303 (or diverging optics) may be used in combination with the converging lens 125 so that a portion of a cone of light can pass therethrough and focus at the measurement location ML. In even further aspects, the mask 303 may be configured to allow a portion of a circular cross-section of an outerATTORNEY DOCKET NO. SP24-280 periphery of the second light beam to pass through the mask 303 to focus at the measurement location ML. In even further aspects, the mask 303 can be configured to allow a few rays of light to pass through the mask to focus at the measurement location ML.

[0079] In aspects, as shown in FIGS. 1-3, the EPCS apparatus 121 comprises a coupling prism 113. As discussed above, the coupling prism 113 can be shared between multiple subsystems (e.g., EPCS, LSP), although each sub-system can be associated with its own coupling prism in other aspects. As discussed above with reference to FIGS. 2-3, the coupling prism 113 comprises an input surface 225 and an output surface 223 that can be opposite the input surface 225. As shown in FIG. 3, the second path 207 can impinge the input surface 225 at an inclination that is substantially normal to the input surface 225. In further aspects, as shown in FIG. 3, the coupling prism 113 comprises a coupling surface 229 that can face the coupling liquid 215, the cavity 319 of the sample holder 251, and / or the first major surface 105 of the substrate 103. In still further aspects, the coupling surface 229 of the coupling prism 113 can contact the coupling liquid 215. In aspects, the coupling prism 113 for the EPCS apparatus 121 can omit the input surface 227 for the LSP apparatus (e.g., if the prism -coupling system does not include the LSP apparatus or uses a separate coupling prism). In aspects, the input surface 225 can face the second light source 221, and the second path 207 therebetween can extend along a first axis Al. In aspects, the output surface 223 can face the second detector 127, and the portions 207R and / or 207P of the second path therebetween can extend along a second axis A2. In aspects, the second path 207 and / or the first axis Al can impinge the input surface 225 at an inclination that is substantially normal to the input surface 225. In aspects, the portion 207R and / or 207P of the second path and / or the second axis A2 can impinge the output surface 223 at an inclination that is substantially normal to the output surface 223.

[0080] As shown in FIGS. 1-3, the EPCS apparatus 121 can comprise a second detector 127. In aspects, the second detector 127 can comprise a digital camera, a CCD, and / or an array of photodetectors. In aspects, the second detector 127 may comprise one or more focusing lenses 305, an attenuator, and / or a beam splitter. In aspects, as shown in FIGS. 2-3, the second detector 127 can comprise an image sensor 219. In further aspects, although not shown, image sensor 219 can comprise similar or the same attributes as image sensor 217 discussed above. In further aspects, as shown in FIGS. 2-3, the second detector 127 can face the output surface 223 of the coupling prism 113. In further aspects, the second detector 127 can be positioned along the portion 207R of the second path 207 and / or the second axis A2 (e.g., detection light axis) to detect a signal from the refracted second light beam that traveled through the coupling prism 113. As shown in FIGS. 1-3,ATTORNEY DOCKET NO. SP24-280 the second detector 127 can be connected to the controller 141 by a communication path configured to transmit a signal SA.

[0081] The polarization filter array 129, 329, 401, 501, and / or 1401 is positioned between the coupling prism 113 and the second detector 127 (e.g., image sensor 219) along the detection light axis (e.g., second axis A2). In aspects, as shown in FIGS. 4-5, the polarization filter array 129, 401, and / or 501 comprises a first type 403a-403f, 503, 523, 533, and / or 543 polarization filter and a second type 405a-405g, 505, 525, 535, and / or 545 of polarization filter. In further aspects, as shown, the polarization filter array 129, 329, 401, and / or 501 comprises a plurality of portions (i.e., at least two portions) of the second type of polarization filter, where a pair (e.g., adjacent pair) of portions of the first type 403c and 403d (or 503 and 523) of polarization filter can sandwich (i.e., be positioned on opposing sides of) a portion of the second type 405d or 505 of polarization filter. Alternatively or additionally, as shown, the polarization filter array 129, 329, 401, and / or 501 comprises a plurality of portions (i.e., at least two or more portions) of the first type of polarization filter, where a pair (e.g., adjacent pair) of portions of the second type 405d and 405e (or 505 and 525) can sandwich (i.e., be positioned on opposing sides of) a portion of the first type 403d or 503 of polarization filter. The polarization of the light passing through the first type of polarization filter can be perpendicular to the polarization of light passing through the second type of polarization filter. In further aspects, the first type of polarizer can be configured to permit light associated with a transverse electric (TE) mode spectrum while the second type of polarizer can be configured to permit light associated with a transverse magnetic (TM) mode spectrum. Alternatively, the first type of polarizer can be configured to permit light associated with a TM mode spectrum while the second type of polarizer can be configured to permit light associated with a TE mode spectrum. Consequently, the detector can generate a signal corresponding to a portion of the TE mode spectrum sandwiched between a pair of portions of the TM mode spectrum (or vice versa). As such, the information about a mode line in a portion of one mode spectrum can be compared with information about a corresponding mode line in a pair of adjacent portions of the other mode spectrum. Providing multiple portions of the second type of polarizer enables higher accuracy and higher precision measurements of at least one stress-related characteristic to be performed (even in the presence of tilt in the mode spectra - including inconsistent tilt - and / or optical aberrations from optical elements).

[0082] In contrast to a polarization filter array having just two portions - a single first type of polarizer and a single second type of polarizer (e.g., see the mode spectra in FIG. 6), polarization filter arrays (in accordance with the present disclosure) having more than two portions allow for increased accuracy and / or precision of the resulting calculation of the at least one stress-relatedATTORNEY DOCKET NO. SP24-280 characteristic. Until now, separate measurements of the TE and TM mode spectra has been deemed sufficient. However, the inventors of the present disclosure have observed limitations of this approach. For example, as shown in FIG. 8, the TE and TM mode lines 813 and 815 calculated from TE and TM points 803 and 805 can be tilted (relative to vertical lines). It had been believed that tilt was due to a misalignment (e.g., rotation) between the substrate and the detector that can be compensated for through careful alignment and calibration procedures. However, as shown in FIG. 9, it has been observed that the TE and TM mode lines 913 and 915 (calculated from TE and TM points 903 and 905) are not always tilted to the same extent (or even in the same direction). In this situation, the distance between the mode lines is not consistent across the plotted area - distance 909 is larger than distance 929. Further, a calculated distance between the actual TE and the TM mode lines differs based on the calculation method from the TE and TM points 903 and 905 measured - for example, distance 919 calculated between the average TE position 904 and the average TM position 906 is greater than either distance 909 or 919. Further, if methods attempted to account for tilt, these methods would be expected to determine an average tilt for both the TE and TM mode lines together, which does not accurately reflect the actual relationship between the TE and TM mode lines. Indeed, the wide variability of the calculated distance (on the vertical axis 1103 - y-axis - in FIG. 11) using either the average positions (curve 1115) and average tilt (curve 1117) for repeated measurements (along the horizontal axis 1101 - x-axis) of the same sample; in fact, the averages (dashed lines 1125 and 1127) do not agree within 10 pixels between these methods. In contrast, using a polarization filter array in accordance with the present disclosure, FIG. 13 shows that the calculated distance (on the vertical axis 1303 - y-axis) using the average tilt (curve 1307) is extremely consistent over repeated measurements (along the horizontal axis 1301 - x-axis) in the presence of tilt, where the calculated distances vary by about 1 pixel or less from the average (dashed line 1317). This represents an unprecedented increase in consistency (e.g., repeatability, precision) and accuracy in calculation of the difference between these mode lines, which is used in the calculation of many stress-related properties.

[0083] In aspects, returning to FIG. 4, the polarization filter array 129 and / or 401 can be oriented such that a direction 402 in which one type of polarization filter is sandwiched by a pair of portions of another type of polarization filter can be substantially perpendicular to a principal axis of the TE mode spectrum, the TM mode spectrum, or both. With reference to FIG. 7, the principal direction of the mode spectra is the direction along which the mode spectra (e.g., mode line 703a, 703b, 705a, and 705b and critical angle transitions 713a, 713b, 715a, and 715b) is distributed - extending left to right in FIG. 7. As used herein, “substantially perpendicular” means within 10° of perpendicular (e.g., from 80° to 100°), although the angle between the principal axisATTORNEY DOCKET NO. SP24-280 and the direction of the line can be from 85° to 95°, from 87° to 93°, from 88° to 92°, from 89° to 91°, or any range or subrange therebetween. Consequently, when the direction 402 is substantially perpendicular to a principal axis of one or more of the mode spectra, signals corresponding to the mode spectra shown in FIG. 7 can be obtained (for the polarization fdter in FIG. 14 having two portions of the first type of polarization filter and two portion of the second type of polarization filter), which allows for direct comparison between the portions of TE mode lines 703a and 703b and the portions of the TM mode lines 705a and 705b. Further, a fitted mode line can be interpolated between corresponding portions (e.g., interpolated portion 703c ofthe fitted mode line based on observed portions of the TE mode lines 703a and 703b) such that a distance 707 between the TE and TM mode lines can be directly calculated in a region where there is actual data for a portion of the TM mode line 703b (or vice versa).

[0084] In further aspects, as shown in FIG. 4, the polarization filter array 129 and / or 401 comprises a plurality of portions of the first type 403a-403f of polarization filter and a plurality of portions of the second type of the second type 405a-405g of polarization filter. In even further aspects, as shown, the plurality of portions of the first type 403a-403f of polarization filter and a plurality of portions of the second type of the second type 405a-405g of polarization filter can be arranged along a line (see direction 402), where the portions of the first type 403a-403f of polarization filter can alternate with the portions of the second type 405a-405g of polarization filter along the line. In even further aspects, a number of alternating pairs (of the first type and the second type of polarization filters) can be 2 or more, 4 or more, 5 or more, 6 or more, 7 or more, 8 or more, 9, or more, or 10 or more (e.g., from 2 to 1024, from 4 to 256, from 5 to 64, from 6 to 32, from 7 to 16, from 8 to 14, from 10 to 12, or any range or subrange therebetween). For example, two alternating pairs are shown in FIG. 14, with the polarization filter array 1401 extending to the edge of a detector aperture 1411 (e.g., mounting ring 1413 that can be attached to an existing detector) for the corresponding detector has two portions 1403a and 1403b of the first type of polarization filter alternating with two portions 1405a and 1405b of the second type of polarization filter with light-blocking regions 1409a-1409c positioned therebetween. Alternatively, for example, 6 full alternating pairs are shown in FIG. 4 with the same type of polarization filter at each end of the line (direction 402). In even further aspects, as shown in FIG. 4, the arrangement of the types of polarization filters can be consistent in a second direction 404 perpendicular to the line (direction 402) that the types of polarization filter alternates along.

[0085] In further aspects, as shown in FIG. 5, the polarization filter array 129 and / or 501 can comprise a plurality of portions of the first type 503, 523, 533, and 543 and a plurality of portions of the second type 505, 525, 535, and 545 arranged in a two-dimensional array (extendingATTORNEY DOCKET NO. SP24-280 in direction 502 and 504). In even further aspects, the polarization filter array 129, 329, and / or 501 can comprise a plurality of pixels (e.g., pixel 511), where each pixel contains portions of each type of polarization fdter. Consequently, a signal generated by the second detector 127 (e.g., image sensor 219) can contain information about substantially the entire TE mode spectrum and substantially the entire TM mode spectrum over the second detector’s field of view, which can be separated (e.g., by a controller) for analysis of each mode spectrum; further, the data or fitted mode lines can be superimposed for an extremely accurate and precise measurement of the difference between corresponding mode lines. In even further aspects, as shown in FIG. 5, the polarization filter array 129 and / or 501 can comprise a third type 507 of polarization filter and a fourth type 509 of polarization filter that are both different from the first type of polarization filter and the second type of polarization filter. In still further aspects, each pixel 511 of a plurality of pixels of the polarization filter array 129 and / or 501 can comprise a first type 503 of polarization filter, a second type 505 of polarization filter, a third type 507 of polarization filter, and a fourth type 509 of polarization filter. Providing the third type of polarization filter and the fourth type of polarization filter can facilitate alignment (e.g., calibration) of the prism-coupling apparatus and / or measurement of the at least one stress-related characteristic even when the prism-coupling apparatus is grossly mis-aligned (e.g., the principal axis of one or more of the mode spectra is close to a 45 ° angle relative to the direction between the first type and second type of polarization filters) . While the polarization filter array 129 is shown as a spatially distinct element from the second detector 127, the polarization filter array (e.g., having a two-dimensional array of portions of different types of polarization filters, a plurality of pixels each having multiple types of polarization filters) can instead be integrated into the second detector, for example as indicated by the dashed polarization filter array 329 (e.g., with the polarization filter array combined with microlenses focusing portions of the second light beam onto pixels and / or sub-pixels of the image sensor 219 therein).

[0086] In further aspects, as shown in FIG. 4, the polarization filter array 129 and / or 401 can further comprise a light-blocking portion 407a-4071. A light-blocking portion 407a-4071 can be positioned between an adjacent pair of polarization filters (e.g., light-blocking portion 407c is between a portion of the first type 403c of polarization filter and a portion of the second type 405c of polarization filter). The light-blocking portion 407a-4071 can avoid interference between the mode spectra in a portion where the light associated with the adjacent pair of polarization filters intersect and / or if there would otherwise be a gap between the adjacent pair of polarization filters.

[0087] In aspects, the polarization filter array can have an optical flatness of less than or equal to 2 / 20, where 2 is the optical wavelength of the measurement beam (light beam(s)). Also,ATTORNEY DOCKET NO. SP24-280 the polarization filter array can have an extinction ratio greater than or equal to 50: 1, greater than or equal to 80: 1 greater than or equal to 100: 1 (e.g., from 50: 1 to 1000: 1, from 80: 1 to 500: 1, from 100: 1 to 250: 1, or any range or subrange therebetween. Providing a sufficiently high extinction ratio can enable sufficient contrast (e.g., signal to noise ratio) to readily identify mode lines in the measured mode spectra. In aspects, the different types of polarization filters can comprise the same material that is oriented relative to one another. For example, a first type of polarization filter (e.g., associated with a TE mode spectrum) can be rotated 90° to obtain a second type of polarization filter (e.g., associated with a TM mode spectrum). Consequently, the relative position and / or orientation of the portions of polarization filters is important in identifying what type of polarization filter a particular polarization filter is functioning as. Also, the polarization filter array 129 can be positioned at a detection aperture (see the polarization filter array 1401 extending to the edge of a detector aperture 1411 and / or the inner periphery of a mounting ring 1413 in FIG. 14) and configured to cover an entire surface area of the detector aperture 1411. Additionally, the polarization filter array 129 and / or 329 can be positioned along the detection axis (second axis A2) such that light travelling from the measurement location ML towards the second detector 127 (e.g., image sensor 219) including reasonable dispersion (spatial broadening) passing through the polarization filter array 129 and / or 329 forms a projection on the second detector 127 (e.g., image sensor 219) that covers substantially an entire field-of-view thereof.

[0088] While the concept of plurality of pixels (e.g., discussed with reference to the polarization filter array 129 and / or 501 shown in FIG. 5) can also mirror the arrangement of pixels in a color camera, the function of the second detector 127 (e.g., image sensor 219) with the polarization filter array 129 and / or 501 is completely different. Typically color cameras are used to record the actual appearance of an object, and a polarization-filtered version of said object is generally insufficient to determine the sort of stress-based characteristics that can be determined using the prism-coupling apparatus of the present disclosure. Further, even if a birefringent material were directly imaged using the polarization filter array of the present disclosure in a manner akin to a color camera, any information about stress in the birefringent material throughout the optical path is mixed together such that a stress profile, refractive index profile, etc. cannot be determined. In contrast, the function of the polarization filter array in the prism-coupling apparatus in accordance with aspects of the present disclosure separate (a portion of) the TE mode spectrum from (a portion of) the TM mode spectrum that have been extracted from the substrate using the particular coupling arrangement of the prism-coupling apparatus.

[0089] In aspects, as shown in FIG. 3, the EPCS apparatus 121 and / or 301 can be configured to interface with a sample (e.g., substrate) positioned in a sample holder 251. AsATTORNEY DOCKET NO. SP24-280 discussed below, the sample holder 251 can at least partially define the cavity configured to receive a substrate 103 (e.g., sample). In further aspects, the sample holder 251 can comprise a viewing aperture 315 configured to enable the second light beam to pass therethrough between the coupling prism 113 (and / or the coupling liquid 215) and the substrate 103 (e.g., measurement location ML). In further aspects, the sample holder 251 can be translatable in a direction 119 (e.g., x-direction).

[0090] As used herein, the term “controller” can encompass all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The processor can include, in addition to hardware, code that creates an execution environment for the computer program in question (e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or combinations thereof). In aspects, the controller can comprise and / or be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Aspects of controllers described herein can be implemented as one or more computer program products (e.g., one or more modules of computer program instructions encoded on a tangible program carrier for execution by, or to control the operation of, data processing apparatus). The tangible program carrier can be a computer-readable medium. The computer-readable medium can be a machine-readable storage device, a machine -readable storage substrate, a memory device, or a combination of one or more of them. A computer program can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network. The processes described herein can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry (e.g., a field programmable gate array) or an application specific integrated circuit). Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors. Generally,ATTORNEY DOCKET NO. SP24-280 a processor will receive instructions and data from a read only memory, a random-access memory, or both. The essential elements of a computer are a processor for performing instructions and one or more data memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data (e.g., magnetic, magneto optical disks, or optical disks); however, a computer need not have such devices. Moreover, a computer can be embedded in another device (e.g., a mobile telephone). Computer-readable media suitable for storing computer program instructions and data include all forms of data memory including nonvolatile memory, media and memory devices, including by way of example semiconductor memory devices (e.g., EPROM, EEPROM, and flash memory devices), magnetic disks (e.g., internal hard disks or removable disks), magneto-optical disks (e.g., CD ROM, DVD-ROM disks). The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry. To provide for interaction with a user, aspects described herein can be implemented on a computer having a display device for displaying information to the user and a keyboard and a pointing device (or a touch screen) by which the user can provide input to the computer. Other kinds of devices can be used to provide for interaction with a user as well; for example, input from the user can be received in any form, including acoustic, speech, or tactile input. Aspects described herein can be implemented in a computing system that includes a back-end component, e.g., as a data server, or that includes a middleware component, e.g., an application server, or that includes a front-end component, e.g., a client computer having a graphical user interface or a Web browser through which a user can interact with implementations of the subject matter described herein, or any combination of one or more such back-end, middleware, or front-end components. The components of the system can be interconnected by any form or medium of digital data communication, e.g., a communication network. Aspects of communication networks include a local area network (“LAN”) and a wide area network (“WAN”) (e.g., the Internet). The computing system can include clients and servers. A client and server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises from computer programs running on the respective computers or between processes on a computer, and having a client-server relationship to each other.

[0091] The controller 141 can be configured to determine at least one stress-related characteristic based on the signals SA and / or SB received from the LSP apparatus 131 and / or the EPCS apparatus 121. For example, the stress-related characteristic can include a stress profile, a knee stress, a central tension, a tension-strain energy, a birefringence, a spike depth, a depth of layer, a surface compressive stress, a refractive index profile, or combinations thereof. In aspects,ATTORNEY DOCKET NO. SP24-280 the substrate 103 can comprise a glass-based material that can be chemically strengthened. As used herein, “glass-based” includes both glasses and glass-ceramics, wherein glass-ceramics have one or more crystalline phases and an amorphous, residual glass phase. Glass-based material cools or has already cooled into a glass, glass-ceramic, and / or that upon further processing becomes a glassceramic material. A glass-based material (e.g., glass-based substrate) may comprise an amorphous material (e.g., glass) and optionally one or more crystalline materials (e.g., ceramic). Amorphous materials and glass-based materials may be strengthened. As used herein, the term “strengthened” may refer to a material that has been chemically strengthened, for example, through ion-exchange of larger ions for smaller ions in the surface of the substrate. Exemplary glass-based materials, which may be free of lithia or not, comprise soda lime glass, alkali aluminosilicate glass, alkali- containing borosilicate glass, alkali-containing aluminoborosilicate glass, alkali-containing phosphosilicate glass, and alkali-containing aluminophosphosilicate glass. In one or more aspects, a glass-based material may comprise, in mole percent (mol %): SiCh in a range from 40 mol % to 80%, AI2O3 in a range from 10 mol % to 30 mol %, B2O3 in a range from 0 mol % to 10 mol %, ZrCh in a range from 0 mol% to 5 mol %, P2O5 in a range from 0 mol % to 15 mol %, TiCh in a range from 0 mol % to 2 mol %, R2O in a range from 0 mol % to 20 mol %, and RO in a range from 0 mol % to 15 mol %. As used herein, R2O can refer to a total amount of an alkali metal oxide, for example, Li2O, Na2O, K2O, Rb2O, and CS2O. As used herein, RO refers a total amount of MgO, CaO, SrO, BaO, and ZnO. In aspects, a glass-based substrate may optionally further comprise in a range from 0 mol % to 2 mol % of each of Na2SO4, NaCl, NaF, NaBr, K2SO4, KC1, KF, KBr, AS2O3, Sb2O3, SnO2, Fe20s, MnO, Mn02, MnOs, M1T2O3. MnsO4, Mn20?. “Glassceramics” include materials produced through controlled crystallization of glass. In aspects, glassceramics have 1% to 99% crystallinity. Examples of suitable glass-ceramics may include Li2O- AhO3-SiO2 system (i.e. LAS-System) glass-ceramics, MgO-AhO3-SiO2 system (i.e. MAS- System) glass-ceramics, ZnO x AI2O3xnSiO2 (i.e. ZAS system), and / or glass-ceramics that include a predominant crystal phase including P-quartz solid solution, -spodumene, cordierite, petalite, and / or lithium disilicate. The glass-ceramic substrates may be strengthened. For example, MAS-System glass-ceramic substrates may be strengthened in Li2SO4 molten salt, whereby an exchange of 2Li+for Mg2+can occur. In aspects, the substrate comprising the glass-based substrate can be optically transparent. As used herein, “optically transparent” or “optically clear” means an average transmittance of 70% or more in the wavelength range of 400 nm to 700 nm through a 1.0 mm thick piece of a material. In aspects, an “optically transparent material” or an “optically clear material” may have an average transmittance of 75% or more, 80% or more, 85% or more, 90% or more, 92% or more, 94% or more, 96% or more in the wavelength range of 400 nm to 700 nmATTORNEY DOCKET NO. SP24-280 through a 1.0 mm thick piece of the material. The average transmittance in the wavelength range of 400 nm to 700 nm is calculated by measuring the transmittance of whole number wavelengths from 400 nm to 700 nm and averaging the measurements.

[0092] The substrate 103 can comprise dimensions that can correspond to dimensions of a consumer electronic product. In aspects, the substrate 103 can comprise a consumer electronic product. The consumer electronic product can comprise a glass-based portion and further comprise electrical components at least partially within a housing. The electrical components can comprise a controller, a memory, and a display. A display can be at or adjacent the front surface of the housing. The consumer electronic product can comprise a cover substrate disposed over the display. In aspects, a thickness of the substrate defined as an average distance between the first major surface and a second major surface (opposite the first major surface) of the substrate can be 20 pm or more, 40 pm or more, 100 pm or more, 150 pm or more, 200 pm or more, 250 pm or more, 500 pm or more, 800 pm or more, 1 mm or more, 2 mm or more, 5 mm or less, 3 mm or less, 2 mm or less, 1 mm or less, 500 pm or less, 300 pm or less, 200 pm or less, or 100 pm or less. In aspects, the thickness can be in a range from 20 pm to 5 mm, from 40 pm to 3 mm, from 100 pm to 1 mm, from 150 pm to 500 pm, or any range or subrange therebetween. In aspects, the thickness can be substantially constant across the first major surface.

[0093] Aspects of methods of determining a stress-related characteristic (for example using the prism-coupling apparatus 101, 121, and / or 301 in FIGS. 1-3 and / or the polarization filter arrays 401 and / or 501 in FIG. 4-5) will now be discussed in accordance with the aspects of the disclosure.

[0094] In aspects, as shown in FIG. 1-3, methods can comprise disposing the substrate 103 on the sample holder 251 (e.g., in the cavity 319), for example, with the coupling liquid 215 disposed on the first major surface 105 of the substrate 103 (facing the viewing aperture 315). In further aspects, methods can comprise disposing the coupling prism 113 over the first major surface 105 of the substrate 103, where the coupling surface 229 of the coupling prism 113 and the first major surface 105 of the substrate face one another. The coupling liquid 215 (if present) can be positioned between the first major surface 105 of the substrate 103 and the coupling surface 229 of the coupling prism 113. In further aspects, as shown, the sample holder 251 is positioned such that the viewing aperture 315 is aligned with the coupling surface 229 of the coupling prism 113 (e.g., configured to perform a measurement using the EPCS apparatus 121 and / or using the LSP apparatus 131). In aspects, as shown, a plane tangent to the first major surface 105 of the substrate 103 at the measurement location ML can be parallel to the coupling surface 229 of the coupling prism 113.ATTORNEY DOCKET NO. SP24-280

[0095] Then, as shown in FIGS. 2-3, aspects of methods can proceed to transmitting a second beam (e.g., along the second path 207, second axis A2) from a second beam source (e.g., second light source 123 that impinges the input surface 225 of the coupling prism 113 and the measurement location ML (e.g., substrate 103) (as well as the coupling liquid 215, if present). In further aspects, as shown, the second beam (e.g., travelling along the second path 207) can be focused by the focusing optics (e.g., converging lens 125) before impinging the coupling prism 113 and the measurement location ML. After the second beam interacts with the substrate 103 (e.g., at the measurement location ML), (at least a portion of) the second beam travelling along a reflected beam path 207R (e.g., along second axis A2) contains information about a transverse electric (TE) mode spectrum and a transverse magnetic (TM) mode spectrum associated with the substrate 103. Then, (the at least a portion of) the second beam can impinge and / or pass through the polarization filter array 129 and / or 329 to form a filtered second beam that can travel along the filtered beam path 207P towards the second detector 127 (e.g., image sensor 219). As discussed above, the polarization filter array 129 and / or 329 can selectively filter portions of (the at least a portion of) the second beam to permit information associated with the TE mode spectrum to pass in region(s) associated with one type of polarization filter and to permit information associated with the TM mode spectrum to pass in region(s) associated with another type of polarization filter. The polarization filter array 129 and / or 329 can comprise any of the aspects discussed above with reference to polarization filter arrays 129, 401, and / or 501 shown in FIGS. 4-5. For example, a portion of a first type of polarization filter can be sandwiched between a pair of portions of a second type of polarization filter, where these portions are arranged along the direction 402 perpendicular to the light detections axis (second axis A2, beam paths 207R and / or 207P).

[0096] As shown, methods can further comprise detecting at least a portion of the second beam (e.g., the second filtered beam traveling along the filter beam path 207P of the second path) with the second detector 127 to generate signal SA. For example, the signal can correspond to the composite mode spectra 701 shown in FIG. 7. Then, methods can determine the at least one stress- related characteristic based on at least the signal SA generated by the second detector 127, which can occur at the controller or using another processing device (e.g., processor, program, computer). Various calculation methods that can be used in determining the at least one stress-related characteristic.

[0097] In a first aspect of calculation methods, a position of a mode line (including a critical angle transition) in a mode spectrum can be calculated as an average position (along the principal axis of the mode spectrum - e.g., average TE position of a TE mode line, average TM position of a TM mode line) from the information (e.g., data) associated with the corresponding mode line. InATTORNEY DOCKET NO. SP24-280 further aspects, the calculated average position can use information from multiple portions (or even all portions) of the corresponding mode spectrum (e.g., mode line of interest) associated with the multiple portions of the corresponding type of polarization fdter. In further aspects, a difference between mode lines (in the same mode spectrum or between the TE and TM mode spectra - e.g., a difference between the average TE position and the average TM position of corresponding mode lines) can be calculated as the difference between the average positions calculated.

[0098] In a second aspect of calculation methods, a fitted mode line can be calculated based on a linear regression of data corresponding to that mode line. In further aspects, the linear regression can use the information (e.g., data) associated with the corresponding mode line from multiple portions (or even all portions) of the corresponding mode spectrum (e.g., mode line of interest) associated with the multiple portions of the corresponding type of polarization filter. In further aspects, a difference between fitted mode lines corresponding to different mode lines (in the same mode spectrum or between the TE and TM mode spectra - e.g., a difference between the fitted TE mode line and the fitted TM mode line of corresponding mode lines). In even further aspects, as discussed herein with reference to FIG. 7 for difference 707, the difference between the mode lines can be calculated at a location where one of the fitted mode lines is interpolated (interpolated portion 705c is within the range of sampled data - having portions 705a and 705b of the mode line on either side of the location) and the other mode was fit with data at (or immediately adjacent) to the location (portion 703b). Alternatively or additionally, the fitted mode line (e.g., slope) can be used to rotate the data (or fitted mode line) to remove any tilt present. For example, data from multiple (or even all) portions of the mode line can be used to calculate the fitted mode line; the data in at least one portion of the mode line can be rotated to remove any tilt present (based on the slope of the fitted mode line); and the position of the mode line can be taken as the average position of this corrected (e.g., rotated) data, which can be in just one of the regions of the mode line, although multiple regions can be used for the final average in other aspects, where the difference between the average positions of the corresponding corrected mode lines as the calculated difference.

[0099] Additionally or alternatively, when using polarization filter array 329 and / or 501 shown in FIGS. 3 and 5, information from the third type 507 and fourth type 509 of polarization filter can be used to further improve the accuracy and / or precision of location of mode lines and / or distances therebetween. The shown arrangement of the sub-pixels (e.g., types 503, 505, 507, and 509 of polarization filters) in each pixel 511 of the plurality of pixels in FIG. 5 has the first type 503 of polarization filter (horizontally and / or diagonally) offset (in at least one of directions 504 and / or 502) relative to the second type 505 of polarization filter by half of the pixel 511 size. ThisATTORNEY DOCKET NO. SP24-280 offset can be accounted for in processing the signal. However, increased precision (e.g., at the subpixel level of resolution) can be obtained using the third type 507 and fourth type 509 of polarization fdter. For example, when the portions of the third type 507 and fourth type 509 of polarization fdter are configured to filter light at a polarization rotated 45° relative to the first type 503 and second type 505 of polarization filter. In this situation, the portion(s) of the signal detected from light passing through third type 507 and fourth type 509 of polarization filter corresponding to the mode spectra (or mode line(s) of interest) will be alternating pixels (e.g., extending) along the same line (e.g., diagonally; if the mode lines were essentially vertical in locations corresponding to the first type 503 and second type 505 of polarization filter). Consequently, the distance between the corresponding diagonal (sub-)pixels is 70.7% (l / sqrt(2) as a percent) of the distance between the vertical (sub-)pixels. This allows for refinement of the position of the mode lines by combining these measurements. Of course, calculation methods can fit the slope (e.g., apparent tilt) of the (diagonal) mode lines that can be accounted for (e.g., rotating the data or fitted line) before comparison and / or combination with data from the (vertical) mode lines corresponding to the other types of polarization filters.

[0100] In aspects, methods can optionally comprise making a measurement using the LSP apparatus (or another apparatus combined with the EPCS apparatus in the prism-coupling apparatus). As shown in FIG. 2, aspects of methods can proceed to transmitting a second beam (e.g., along the first path 205) from a second beam source (e.g., first polarization-switching light source 133 that impinges the input surface 227 of the coupling prism 113, the coupling liquid 215, and the measurement location ML. As shown, methods can further comprise detecting at least a portion of the second beam (e.g., traveling along portion 205S of the first path) with the first detector 137 to generate signal SB, which can be transmitted as signal SB to the controller 141. The LSP apparatus 131 can measure a retardation profile of the substrate 103, which can be transmitted as signal SA to the controller 141. As used herein, the retardation profile means an amount of optical retardation of the signal as a function of the depth that the first polarization- switched light beam traveled into the substrate 103. As used herein, the optical retardation means a phase shift between two orthogonal light polarizations, which can be measured in radians (rad) or nanometers (nm). Without wishing to be bound by theory, the amount of optical retardation can be determined from the detected signal, which varies due to the constructive and destructive interference for the different effective path lengths of the detected signal through the sample. Without wishing to be bound by theory, stress in the sample can cause optical retardation along the first path of the first polarization-switched light beam, with the amount of stress encountered being proportional to the derivative of the optical retardation.ATTORNEY DOCKET NO. SP24-280

[0101] Based on the signal SA and / or SB (e.g., at least signal SA) received by the controller 141, the controller can calculate at least one stress-related characteristic of the substrate 103. As discussed above, the stress-related characteristic can include a stress profile, a knee stress, a central tension, a tension-strain energy, a birefringence, a spike depth, a depth of layer, a surface compressive stress, a refractive index profile, or combinations thereof. For example, the signal(s) SA and / or SB can contain information about a refractive index profile for orthogonal polarization states (e.g., TE and TM), and a stress profile can be calculated by taking the difference between the two measured refractive index profiles (and dividing the difference by a stress optical coefficient, which can be measured using any means known to those skilled in the art).EXAMPLES

[0102] Various aspects will be further clarified by the following examples a prismcoupling apparatus with various polarization filter arrays under various conditions. FIGS. 8-9 will be used to generally discuss difficulties while FIGS. 6-7 and 12-13 present data for polarization filter arrays and conditions. Polarization filter array AA contained just two portion: only a single portion of a first type of polarization filter associated with the TE mode spectrum and a single portion of a second type of polarization filter associated with the TM mode spectrum. Using polarization filter array AA produced the mode spectra shown in FIG. 6. Polarization filter array 1 is shown in FIG. 14 as polarization filter array 1401 containing two pairs of alternating portions arranged in a line: two portions of a first type of polarization filter associated with the TE mode spectrum and two portions of the second type of polarization filter associated with the TM mode spectrum. Using polarization filter array 1 produced the mode spectra shown in FIG. 7. Polarization filter array 2 resembles the polarization filter array 501 shown in FIG. 5 (e.g., corresponding to polarization filter array 329 in FIG. 3) comprising a plurality of pixels, where each pixel has four types of polarizers. The raw data from the recorded mode spectra is complicated and cannot be reproduced in a way that is readily understood to the naked eye. Instead, information corresponding to “sub-pixels” associated with each type of polarization filter were separated into their own image, which looks each look like a complete mode spectrum (spanning the entire field of view of the detector), which can be analyzed and the fitted mode lines to superimposed to produce mode lines 813, 815, 913, and 915 shown in FIGS. 8-9 where the underlying data would extend for the full extent of each mode line.

[0103] Data shown in FIGS. 6 and 10-11 were obtained using polarization filter array AA (two portion polarization filter array having only a single portion of a first type of polarization filter associated with the TE mode spectrum and a single portion of a second type of polarization filter associated with the TM mode spectrum). FIG. 6 shows a composite mode spectra 601 havingATTORNEY DOCKET NO. SP24-280 a portion of TE mode spectrum on the top and a portion of the TM mode spectrum on the bottom with a blocked region 609 in between (corresponding to a light-blocking region in the corresponding polarization fdter array AA). In FIG. 6, a TE mode line 603, a TM mode line 605, a TE critical angle transition 613, and a TM critical angle transition 615 are visible. Under the ideal conditions used to obtain FIG. 6, the mode lines 603 and 605 are well resolved and parallel to one another with no tilt. In such ideal conditions, a mode line difference 607 can be reliably estimated based on the positions of the mode lines 603 and 605 and / or a critical angle transition difference 617 can be reliably estimated based on the positions of the corresponding critical angle transitions 613 and 615.

[0104] FIG. 10 presents the calculated difference (d) in pixels (px) on the vertical axis 1003 (e.g., y-axis) over repeated measurements (n = 12) along the horizontal axis 1001 (e.g., x- axis), where the measurements are all taking on the same sample using polarization fdter array AA. The measurements underlying the data in FIG. 10 all used mode spectra that were substantially free of tilt. Curve 1005 was generated using the average position of the data associated with each mode line to calculate the difference therebetween while curve 1007 was generated using linear fits to the data associated with mode line that were then compared to calculate the difference therebetween. Dashed line 1015 represents the average distance calculated for curve 1005 (average position), and dashed line 1017 represents the average distance calculated for curve 1007 (linear fits). As shown in FIG. 10, both curves 1005 and 1007 tended towards larger distances as the measurement number (n) increased, which represents a systematic error (drift). While this drift amounts to less than a single pixel over these 12 measurements, the drift could accumulate over time. This drift would not be apparent if different samples were analyzed, as is typically the case. Further, the method of calculating the difference leads to significantly different calculated differences (about 4% difference with the range of the points between the methods not overlapping), even under ideal conditions - the linear fit method (curve 1007 and dashed line 1017) is closer to the actual value than the average position method.

[0105] As discussed above, tilt (and inconsistent tilt) can occur for various reasons, which can impact the reliability and / or accuracy of measuring a difference between mode lines (and / or critical angle transitions). FIG. 11 presents the calculated difference (d) in pixels (px) on the vertical axes 1103 and 1107 (e.g., y-axis - left and right) over repeated measurements (n = 24) along the horizontal axis 1101 (e.g., x-axis), where the measurements are all taking on the same sample using polarization filter array AA. Here, the underlying mode spectra had noticeable tilt, which was achieved by rotating the detector including the polarization filter array 30° (relative to the substrate) between each measurement. This is intended corresponds to a situation where thereATTORNEY DOCKET NO. SP24-280 are optical aberrations in the optical elements and / or there is not proper alignment (or calibration) for the sample being measured relative to the detector. Curve 1115 uses vertical axis 1103 (left) and corresponds to the calculated difference using the average position of the data associated with each mode line, and dashed line 1125 represents the average of these calculated differences. Curve 1117 uses vertical axis 1107 (right) and corresponds to the calculated difference using linear fits associated with each mode line, and dashed line 1127 represents the average of these calculated differences. As shown, the measurements for the average method (curve 1115) are wildly inconsistent with a range of greater than 100 pixels (compare measurement 1 with measurement 12). Using the linear fit method (curve 1117) has a smaller range (about 2 pixels - representing 4% or more of the average value), but it is still highly variable (e.g., in comparison to curve 1007 or 1017 in FIG. 10) Further, the methods produce vastly different estimates for the difference (and even the average of these calculated differences still differing by more than 10 pixels - 20% or more of the underlying difference). Indeed, the actual difference is about 50.5 pixels, and neither method accurately estimates this value (averages of 48.2 pixels for the linear fit method).

[0106] As discussed above, even the linear fit method can have problems when there is inconsistent tilt between the mode lines (the mode lines are not parallel across mode spectra). If it was believed that tilt applied to all mode lines equally (as expected for a misalignment - rotation - between the substrate and the detector), the situation shown in FIG. 8 would be expected to occur. For the polarization filter array AA, the data would be limited to TE and TM points 803 and 805 (not extending beyond divider 811 corresponding to a light blocking member in the polarization filter array; and likewise, with reference to FIG. 9, limited to TE and TM points 903 and 905 that do not extend beyond divider 911). As shown in FIG. 8, the TE and TM mode lines 813 and 815 calculated as liner fits to TE and TM points 803 and 805, respectively. In this situation, an average difference 809 between the TE and TM average positions 804 and 806 is expected to be the same as the linear difference 819 calculated between fitted TM and TE model lines 813 and 815, although the absolute position of the mode lines is different due to the tilt.

[0107] However, as shown in FIG. 9, it has been observed that the TE and TM mode lines 913 and 915 (calculated from TE and TM points 903 and 905) are not always tilted to the same extent (or even in the same direction). In this situation, the distance between the mode lines is not consistent across the plotted area - distance 909 is larger than distance 929. Further, a calculated distance between the actual TE and the TM mode lines differs based on the calculation method from the TE and TM points 903 and 905 measured - for example, distance 919 calculated between the average TE position 904 and the average TM position 906 is greater than either distance 909 or 919. Further, if methods attempted to account for tilt, these methods would beATTORNEY DOCKET NO. SP24-280 expected to determine an average tilt for both the TE and TM mode lines together, which does not accurately reflect the actual relationship between the TE and TM mode lines. In this situation, the wild variability seen in FIG. 11 can occur, which is a problem not recognized until now.

[0108] Polarization fdter array 1 was used to produce the composite mode spectra 701 shown in FIG. 7. As shown, portions of the TE mode spectra (TE1, TE2) alternate with portions of the TM mode spectra (TM1, TM2). Consequently, two portions of the TE mode line 703a and 703b are visible, and two portions of the TM mode line 705a and 705b are visible. While making use of the same field-of-view as polarization filter array AA (see FIG. 6), polarization filter array 1 provides information on each mode spectrum over a wider range by alternating (e.g., interleaving) portions of different types of polarization filters corresponding to different mode spectra. Consequently, a slope (e.g., tilt) of each mode line can more accurately be determined since the slope is more pronounced relative to noise due to the portions of the mode line being further apart (in polarization filter array 1) than in polarization filter array AA. Additionally or alternatively, as shown, the mode lines can be interpolated (see interpolated portion 703c or 705c between portions of the mode lines 703a-703b or 705a-705b) between the observed portions. Consequently, a distance 707 between the TE and TM mode lines can be directly calculated in a region where there is actual data for a portion of the TM mode line 703b (or vice versa), although average locations of each mode line can be used to calculate the difference therebetween in other methods. In contrast, calculation of differences using the composite mode spectra 601 shown in FIG. 6 would rely on extrapolation of the fitted mode lines, which are believed to be less accurate than the interpolation of fitted mode lines using the composite mode spectra 701 shown in FIG. 7. Likewise, the critical angle transition can be interpolated (interpolated portions 713c and 715c) between portions of the TE and TM critical angle transitions 713a-713b and 715a-715b, respectively; then, the critical angle transition difference 717 can be calculated using an interpolated portion 715c (e.g., TM) in a region where there is observed data on corresponding feature (critical angle transition 713b) in the other polarization (TE).

[0109] FIGS. 12-13 presents the calculated difference (d) in pixels (px) on the vertical axis 1203 or 1303 (e.g., y-axis) over repeated measurements (n = 12 for FIG. 12, n = 24 for FIG. 13) along the horizontal axis 1201 or 1301 (e.g., x-axis), where the measurements are all taking on the same sample using polarization filter array 2 (see FIG. 5). These measurements used linear fits to each mode line, and the difference is calculated therebetween at a central location in the field of view. The measurements underlying the data in FIGS. 12 all used mode spectra that were substantially free of tilt (similar to FIG. 10). Curve 1207 represents the calculated difference (using this linear fit method), and dashed line 1217 represents the average of these measurements. AsATTORNEY DOCKET NO. SP24-280 shown, all of the measurements (curve 1207) are within less than 1 pixel of each other (less than 0.5 pixels and about 0.2 pixels or less) and the average (dashed line 1217) - the range of these measurements is about 0.2 pixels or less and there is no discernable drift. In comparison, as discussed above, even the linear fit method (curve 1005 in FIG. 10) has a range of about 1 pixel (5 times more than for curve 1207) and exhibits drift. Consequently, using the polarization filter arrays in accordance with the present disclosure can provide improved precision and / or repeatability over other polarization filter arrays (AA), even under ideal conditions.

[0110] For FIG. 13, the measurements underlying the data had noticeable tilt (similar to FIG. 11). Curve 1307 represents the calculated difference (using this linear fit method), and dashed line 1317 represents the average of these measurements. As shown, all of the measurements (curve 1307) are within about 1 pixel of each other and the average (dashed line 1317) - the range of these measurements is about 1 pixels or less and there is no discernable drift. For comparison, these measurements in the presence of tilt (FIG. 13) has comparable precision (or better) than using polarization filter array AA under ideal conditions (FIG. 10). Also, in comparison, the range is about 2 pixels or more (for curve 1117 in FIG. 11 - 100% more than for curve 1307 in FIG. 13). Additionally, the accuracy is noticeably improved from 48.2 (4.6% error) for the average difference (dashed line 1127) calculated in FIG. 11 to about 50.5 (essentially 0% error - perhaps 0.1% error) for the average difference (dashed line 1317).

[0111] Table 1 presents the performance of the different polarization filter arrays in the absence of tilt and in the presence to tilt as a summary of the data presented in FIGS. 10-13. Most notably, the high variability using polarization filter array AA in the presence of tilt with the average position method had a standard deviation nearly as large as the average measurement itself (and two orders of magnitude greater than for polarization filter array 2 under any condition). Overall, using polarization filter array 2 (or other polarization filter arrays in accordance with aspects of the present disclosure) provide improved accuracy and precision in the calculated difference between mode lines.Table 1 : Average and Standard Deviation for the Calculated Difference (in pixels) between TE and TM Mode Lines using Polarization Filter Arrays AA and 2 under Different Conditions and with Different Calculation MethodsATTORNEY DOCKET NO. SP24-280

[0112] The above can be combined to provide apparatus and methods for determining a stress-related characteristic of a substrate using a prism-coupling apparatus having a polarization filter array in accordance with the present disclosure. Providing multiple portions of a first type and / or a second type of polarizer in the polarization filter array enables higher accuracy and higher precision measurements of at least one stress-related characteristic to be performed (even in the presence of tilt in the mode spectra - including inconsistent tilt - and / or optical aberrations from optical elements). As discussed herein, the polarization filter array can have portions of different types of polarization filters arranged in a line and / or a two-dimensional array. Further, the Examples discussed herein demonstrate the increased accuracy and precision enabled by the polarization filter arrays in accordance with aspects of the present disclosure.

[0113] In aspects, the polarization filter array can be part of an EPCS apparatus in the prism-coupling apparatus. Further, the prism-coupling apparatus can include another measurement apparatus (e.g., LSP apparatus, refractive near field (RNF) apparatus). For example, a combined apparatus for measurements using LSP and EPCS can simplify and speed up the measurement process. Also, the combined apparatus reduces the risk of sample breakage because less handling is required to load the sample into the combined apparatus as compared to two separate apparatus. Methods of using the combined apparatus can additionally produce more reliable measurements for overall stress profiles.

[0114] In contrast to a polarization filter array having just two portions - a single first type of polarizer and a single second type of polarizer (e.g., see the mode spectra in FIG. 6), polarization filter arrays (in accordance with the present disclosure) having more than two portions allow for increased accuracy and / or precision of the resulting calculation of the at least one stress- related characteristic. Until now, separate measurements of the TE and TM mode spectra has been deemed sufficient. However, the inventors of the present disclosure have observed limitations of this approach. For example, as shown in FIG. 8, the TE and TM mode lines 813 and 815 calculated from TE and TM points 803 and 805 can be tilted (relative to vertical lines). It had been believed that tilt was due to a misalignment (e.g., rotation) between the substrate and the detector that can be compensated for through careful alignment and calibration procedures. However, as shown in FIG. 9, it has been observed that the TE and TM mode lines 913 and 915 (calculated from TE andATTORNEY DOCKET NO. SP24-280TM points 903 and 905) are not always tilted to the same extent (or even in the same direction). In this situation, the distance between the mode lines is not consistent across the plotted area - distance 909 is larger than distance 929. Further, a calculated distance between the actual TE and the TM mode lines differs based on the calculation method from the TE and TM points 903 and 905 measured - for example, distance 919 calculated between the average TE position 904 and the average TM position 906 is greater than either distance 909 or 919. Further, if methods attempted to account for tilt, these methods would be expected to determine an average tilt for both the TE and TM mode lines together, which does not accurately reflect the actual relationship between the TE and TM mode lines. Indeed, the wide variability of the calculated distance (on the vertical axis 1103 - y-axis - in FIG. 11) using either the average positions (curve 1115) and average tilt (curve 1117) for repeated measurements (along the horizontal axis 1101 - x-axis) of the same sample; in fact, the averages (dashed lines 1125 and 1127) do not agree within 10 pixels between these methods. In contrast, using a polarization filter array in accordance with the present disclosure, FIG. 13 shows that the calculated distance (on the vertical axis 1303 - y-axis) using the average tilt (curve 1307) is extremely consistent over repeated measurements (along the horizontal axis 1301 - x-axis) in the presence of tilt, where the calculated distances vary by about 1 pixel or less from the average (dashed line 1317). This represents an unprecedented increase in consistency (e.g., repeatability, precision) and accuracy in calculation of the difference between these mode lines, which is used in the calculation of many stress-related properties.

[0115] While the concept of plurality of pixels (e.g., discussed with reference to the polarization filter array 129 and / or 501 shown in FIG. 5) can also mirror the arrangement of pixels in a color camera, the function of the second detector 127 (e.g., image sensor 219) with the polarization filter array 129 and / or 501 is completely different. Typically color cameras are used to record the actual appearance of an object, and a polarization-filtered version of said object is generally insufficient to determine the sort of stress-based characteristics that can be determined using the prism-coupling apparatus of the present disclosure. Further, even if a birefringent material were directly imaged using the polarization filter array of the present disclosure in a manner akin to a color camera, any information about stress in the birefringent material throughout the optical path is mixed together such that a stress profile, refractive index profile, etc. cannot be determined. In contrast, the function of the polarization filter array in the prism-coupling apparatus in accordance with aspects of the present disclosure separate (a portion of) the TE mode spectrum from (a portion of) the TM mode spectrum that have been extracted from the substrate using the particular coupling arrangement of the prism-coupling apparatus.ATTORNEY DOCKET NO. SP24-280

[0116] Directional terms as used herein — for example, up, down, right, left, front, back, top, bottom — are made only with reference to the figures as drawn and are not intended to imply absolute orientation.

[0117] It will be appreciated that the various disclosed aspects may involve features, elements, or steps that are described in connection with that aspect. It will also be appreciated that a feature, element, or step, although described in relation to one aspect, may be interchanged or combined with alternate aspects in various non-illustrated combinations or permutations.

[0118] It is also to be understood that, as used herein the terms “the,” “a,” or “an,” mean “at least one,” and should not be limited to “only one” unless explicitly indicated to the contrary. For example, reference to “a component” comprises aspects having two or more such components unless the context clearly indicates otherwise. Likewise, a “plurality” is intended to denote “more than one.”

[0119] As used herein, the term “about” means that amounts, sizes, formulations, parameters, and other quantities and characteristics are not and need not be exact, but may be approximate and / or larger or smaller, as desired, reflecting tolerances, conversion factors, rounding off, measurement error and the like, and other factors known to those of skill in the art. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint and independently of the other endpoint.

[0120] The terms “substantial,” “substantially,” and variations thereof as used herein are intended to note that a described feature is equal or approximately equal to a value or description. For example, a “substantially planar” surface is intended to denote a surface that is planar or approximately planar. Moreover, as defined above, “substantially similar” is intended to denote that two values are equal or approximately equal. In aspects, “substantially similar” may denote values within about 10% of each other, for example, within about 5% of each other, or within about 2% of each other.

[0121] Unless otherwise expressly stated, it is in no way intended that any method set forth herein be construed as requiring that its steps be performed in a specific order. Accordingly, where a method claim does not actually recite an order to be followed by its steps or it is not otherwise specifically stated in the claims or descriptions that the steps are to be limited to a specific order, it is in no way intended that any particular order be inferred.

[0122] While various features, elements, or steps of particular aspects may be disclosed using the transitional phrase “comprising,” it is to be understood that alternative aspects, including those that may be described using the transitional phrases “consisting of’ or “consisting essentially of,” are implied. Thus, for example, implied alternative aspects to an apparatus that comprisesATTORNEY DOCKET NO. SP24-280A+B+C include aspects where an apparatus consists of A+B+C and aspects where an apparatus consists essentially of A+B+C. As used herein, the terms “comprising” and “including”, and variations thereof shall be construed as synonymous and open-ended unless otherwise indicated.

[0123] The above aspects, and the features of those aspects, are exemplary and can be provided alone or in any combination with any one or more features of other aspects provided herein without departing from the scope of the disclosure.

[0124] It will be apparent to those skilled in the art that various modifications and variations can be made to the present disclosure without departing from the spirit and scope of the disclosure. Thus, it is intended that the present disclosure cover the modifications and variations of the aspects herein provided they come within the scope of the appended claims and their equivalents.

Claims

ATTORNEY DOCKET NO. SP24-280What is claimed is:

1. A prism-coupling apparatus for determining at least one stress-related characteristic of a substrate, the prism-coupling apparatus comprising: a sample holder comprising a cavity configured to receive the substrate and defining a viewing aperture; a prism configured to allow a first beam of a measurement beam to travel between the prism and the cavity configured to receive the substrate; a first beam source configured to transmit the first beam that impinges the prism and the viewing aperture; a first detector configured to detect at least a portion of the first beam after the first beam impinges the prism and contains information about a transverse electric (TE) mode spectrum and a transverse magnetic (TM) mode spectrum associated with the substrate; and a polarization filter array positioned between the prism and the first detector along a detection light axis, wherein the polarization filter array comprises a first type of polarization filter configured to permit light with a first polarization therethrough and a second type of polarization filter configured to permit light with a second polarization therethrough, the first polarization is perpendicular to the second polarization, and the polarization filter array comprises at least two portions of the second type of polarization filter sandwiching a first portion of the first type of polarization filter in a first direction perpendicular to the detection light axis.

2. The prism-coupling apparatus of claim 1, wherein the first direction is substantially perpendicular to a principal axis of the TE mode spectrum, the TM mode spectrum, or both.

3. The prism-coupling apparatus of any one of claims 1-2, wherein the prism-coupling apparatus is configured to produce a signal at the first detector corresponding to a portion of the TE mode spectrum sandwiched between portions of the TM mode spectrum.

4. The prism-coupling apparatus of any one of claims 1-2, wherein the prism-coupling apparatus is configured to produce a signal at the first detector corresponding to a portion of the TM mode spectrum sandwiched between portions of the TE mode spectrum.ATTORNEY DOCKET NO. SP24-2805. The prism -coupling apparatus of any one of claims 1 -4, wherein the polarization filter array further comprises a second portion of the first type of polarization filter, the first portion and the second portion of the first type of polarization filter sandwiches the second portion of the second type of polarization filter in the first direction.

6. The prism -coupling apparatus of any one of claims 1 -5 , wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a line along the first direction with the portions of the first type of polarization filter alternating with the portions of the second type of polarization filter along the line.

7. The prism -coupling apparatus of any one of claims 1 -5 , wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a two-dimensional array perpendicular to the detection light axis, wherein a portion of the first type of polarization filter is positioned between adjacent pairs of portions of the second type of polarization filter.

8. The prism-coupling apparatus of claim 7, wherein the polarization filter array further comprises a third type of polarization filter and fourth type of polarization filter each configured to permit portions of light with the first polarization and light with the second polarization therethrough, a polarization of the third type of polarization filter is different than a polarization of the fourth type of polarization filter, and the two-dimensional array further includes portions of the third type of polarization filter and the fourth type of polarization filter.

9. The prism-coupling apparatus of claim 8, wherein the two-dimensional array of polarization filters is arranged as a two-dimensional array of pixels, where each pixel corresponds to a two-by-two arrangement of a portion of the first type of polarization filter, a portion of the second type of polarization filter, a portion of the third type of polarization filter, and a portion of the fourth type of polarization filter.

10. The prism -coupling apparatus of any one of claims 1-9, wherein the first beam source is configured to produce the first beam that is monochromatic and uncollimated.ATTORNEY DOCKET NO. SP24-28011. The prism-coupling apparatus of any one of claims 1-10, wherein the first beam source, the first detector, and the polarization filter array is an evanescent prism-coupling spectroscopy subsystem.

12. The prism -coupling apparatus of any one of claims 1-11, further comprising a lightscattering polarimetry sub-system comprising: a second beam source configured to transmit a second beam of the measurement beam that impinges the prism and the viewing aperture; and a second detector of the light-scattering polarimetry sub-system is configured to detect at least a portion of the second beam after the second beam impinges the prism.

13. The prism -coupling apparatus of any one of claims 1-12, wherein the polarization filter array further comprises a light-blocking portion between adjacent polarization filters of the polarization filter array.

14. A method of determining at least one stress-related characteristic of a substrate comprising: disposing the substrate in a cavity of a sample holder; disposing a coupling prism over a first surface of the substrate, a coupling liquid positioned between a coupling surface of a prism and the first surface of the substrate in a viewing aperture of the sample holder; transmitting a first beam from a first beam source that impinges the prism, the coupling liquid, and the first surface of the substrate at a measurement location; passing at least a portion of the first beam through a polarization filter array after the first light beam impinges the prism and contains information about a transverse electric (TE) mode spectrum and a transverse magnetic (TM) mode spectrum to generate a filtered first beam; detecting at least a portion of the first filtered beam with a first detector to generate a first signal; and determining the at least one stress-related characteristic based on at least the first signal, wherein the at least the portion of the first beam passing through the polarization filter array is travelling along a detection light axis, the polarization filter array comprises a first type of polarization filter permitting light containing information corresponding to the TE mode spectrum to travel therethrough and a second type of polarization filter permitting light containing information corresponding to the TM mode spectrum to travel therethrough, the polarization filter array comprises at least two portions of the second type of polarization filter sandwiching a firstATTORNEY DOCKET NO. SP24-280 portion of the first type of polarization filter in a first direction perpendicular to the detection light axis.

15. The method of claim 14, wherein the first beam source, the polarization filter array, and the first detector are part of an evanescent prism-coupling spectroscopy sub-system, and the determining comprises processing the first signal to form an optical retardation versus depth curve based on information in the TE mode spectrum and the TM mode spectrum.

16. The method of any one of claims 14-15, wherein a light-scattering polarimetry sub-system comprises the second beam source, the second prism, and the second detector, the method further comprises: emitting a second light beam from the second beam source; impinging the second light beam on the coupling prism and the substrate; detecting the second light beam after the impinging by the second detector to generate a second signal; and the determining comprises processing the second signal is processed to determine an optical retardance.

17. The method of any one of claims 14-16, wherein a position of a TM mode line in the TM mode spectrum is based on an average TM position calculated based on information corresponding to the TM mode spectrum that passed through each portion of the second type of polarizer.

18. The method of claim 17, wherein a position of a TE mode line in the TE mode spectrum is based on an average TE position calculated based on information corresponding to the TM mode spectrum that passed through each portion of the first type of polarizer.

19. The method of any one of claims 16-18, wherein a fitted TM mode line in the TM mode spectrum is based on a linear regression based on information corresponding to the TM mode spectrum that passed through each portion of the first type of polarizer.

20. The method of claim 19, wherein a fitted TE mode line in the TE mode spectrum is based on a linear regression based on information corresponding to the TE mode spectrum that passed through each portion of the first type of polarizer.ATTORNEY DOCKET NO. SP24-28021. The method of claim 20, wherein an optical retardance is calculated as a difference between the fitted TE mode line and the fitted TM mode line at a location where the fitted TE mode line and the fitted TM mode line are interpolated.

22. The method of any one of claims 16-21, wherein the first signal generated at the first detector includes information corresponding to a portion of the TE mode spectrum sandwiched between portions of the TM mode spectrum detected by the first detector.

23. The method of any one of claims 16-21, wherein the first signal generated at the first detector includes information corresponding to a portion of the TM mode spectrum sandwiched between portions of the TE mode spectrum detected by the first detector.

24. The method of any one of claims 16-23, wherein the polarization filter array further comprises a second portion of the first type of polarization filter, the first portion and the second portion of the first type of polarization filter sandwiches the second portion of the second type of polarization filter in the first direction.

25. The method of any one of claims 16-24, wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a line along the first direction with the portions first type of polarization filter alternating with the portions of the second type of polarization filter along the line.

26. The method of any one of claims 16-24, wherein the polarization filter array comprises a plurality of portions of the first type of polarization filter and a plurality of portions of the second type of polarization filter arranged in a two-dimensional array perpendicular to the detection light axis, wherein a portion of the first type of polarization filter is positioned between adjacent pairs of portions of the second type of polarization filters.

27. The method of claim 26, wherein the polarization filter array further comprises a third type of polarization filter and fourth type of polarization filter each permitting portions of light with light containing information about both the TE mode spectrum and the TM mode spectrum therethrough, a polarization of the third type of polarization filter is different than a polarization ofATTORNEY DOCKET NO. SP24-280 the fourth type of polarization filter, and the two-dimensional array further includes portions of the third type of polarization filter and the fourth type of polarization filter.

28. The method of claim 27, wherein the two-dimensional array of polarization filters is arranged as a two-dimensional array of pixels, where each pixel corresponds to a two-by-two arrangement of a portion of the first type of polarization filter, a portion of the second type of polarization filter, a portion of the third type of polarization filter, and a portion of the fourth type of polarization filter.

29. The method of any one of claims 16-28, wherein the polarization filter array further comprises a light-blocking portion between adjacent polarization filters of the polarization filter array.

30. The method of any one of claims 16-29, wherein the stress-related characteristic comprises a stress profile, a knee stress, a central tension, a tension-strain energy, a birefringence, a spike depth, a depth of layer, a surface compressive stress, a refractive index profile, or combinations thereof.

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