Residual stresses are a major factor in the quality ofglass products because these stresses determine themechanical strength and failure mode,1 which arekey for many applications, especially the high-volume applications of automotive and architecturalwindows. Low internal stresses are required to en-able fabrication of a glass object to a desired size andshape by means of cutting or grinding. The com-pressive surface stresses created when a glass win-dow is tempered can increase its breaking strengthby factors from 3 to 5 and double the maximum ve-locity with which a stone can hit a tempered glasswindow and have the window survive without dam-age. The balancing internal tensile stresses will en-sure that, upon breaking, the resulting pieces will besmall and safe rather than result in large sharpshards that can badly cut an accident victim. How-ever, excessive stresses, even at a single point, cangreatly weaken a window and even lead to spontane-ous breakage. Thus knowledge of the residualstresses is important to ensuring the quality of glassproducts.
The side and rear windows in cars and trucks are
common examples of tempered glass windows. Atempered car window should have compressive sur-face stresses of approximately 100 MPa �15,000pounds per square inch, psi� at both surfaces. Half-way between these two surfaces the glass shouldhave a tensile stress that is half of the magnitude ofthe surface stress. Safety regulations �Federal Mo-tor Vehicle Safety Standard �FMVSS 205�, U.S. De-partment of Transportation� require that thetempered automotive glass be tested with the four-point break test on a sample of the production runand that the distribution of sizes of the resultingpieces be measured. Other tests for tempered glassinclude the dropping of steel balls of a specifiedweight from a specified height onto the temperedglass window. These procedures are costly becauseof the amount of substandard products that are pro-duced before a defective product is first detected andbecause the results do not provide much useful infor-mation that would help identify the causes of thedefective product. Once a tempered piece of glass isbroken, all that is left is a pile of small pieces. Thereare substantial national and global economic impactsof these costs because glass manufacture is extremelyenergy intensive. In fact, glass is the most energy-intensive component in car and truck manufacturing,which means that automotive glass manufacturingby itself is a major energy consumer in the UnitedStates and the world. A method for rapid, nonde-structive, and spatially resolved measurement of re-sidual stress in glass could help improve the energyefficiency and reduce costs of tempered glass produc-tion.
Stress is a tensor quantity that we can specify by
B. D. Cannon �[email protected]�, C. Shepard, and M.Khaleel are with the Pacific Northwest National Laboratory, 902Battelle Boulevard, P.O. Box 999, Richland, Washington 99352-0999.
Received 10 October 2000; revised manuscript received 12 June2001.
giving the components along the three principal axesand the orientation of these axes relative to the objectat each point in the object. At any surface, one of theprincipal axes is normal to the surface, and that prin-cipal stress component is zero. For uniform temper-ing, the two principal stresses in the plane of thesurface are degenerate and compressive, and thestress in the direction normal to the surface is zerothrough the thickness of the glass.2 In a temperedplate, the stress along a given direction in the planeof the surface is compressive on the surface, becomestensile in the middle, and then compressive on theother surface. For uniformly tempered glass, thisstress profile is symmetric about the midplane of theplate, and the integral of this stress through thethickness of the plate is zero. For thin glass plates,this stress profile is parabolic, and the magnitude ofthe compression on the surfaces is twice the magni-tude of the tension at the midplane. Edges and anytransverse spatial variations in the cooling ratesbreak the degeneracy of the two in-plane principalstress axes and create deviations from this idealstress pattern.
The current nondestructive methods to measuresurface stress are optical measurements3 based onthe equation
��na � �nb��n � B�Sa � Sb�,
where �a, b, c� is a Cartesian coordinate system andthe light propagates along the c axis, ��na � �nb��n isthe difference in the fractional change in the refrac-tive index for the a and b components of the electricfield of a light wave that is due to the differencebetween the a and b components of the stress, �Sa �Sb� and B is the stress optical coefficient. For lightpropagating in the plane of a tempered glass surfaceand the b axis normal to that plane, then Sb is zeroand the birefringence ��na � �nb��n is proportional toSa. That Sb is zero does not mean that �nb is zerobecause a stress in a given direction changes therefractive index both parallel and perpendicular tothe direction of stress.4 A typical compressive sur-face stress is 15,000 psi for tempered soda-lime floatglass and B for soda-lime glass5 is 2.6 � 10�12 Pa�1
or 1.8 � 10�8 psi�1, so the birefringence would be2.7 � 10�4 or 2.3 mm of travel for one wave of retar-dation at 633 nm.
In contrast, the most common optical stress mea-surement method is to measure the change in polar-ization state of light after it passes through thesample traveling normal to the surface. Thismethod is fast, nondestructive, and noncontact, butprovides no information on the surface stresses. Inthis case, the c axis is normal to the surface, and thebirefringence is proportional to the difference in theprincipal stresses in the planes parallel to the sur-face. Small differences in stresses can be measuredthis way but the effect is integrated over the full paththrough the glass and so is not sensitive to surfacestress. For a tempered glass sheet, Sa and Sb arenearly the same except near the edges, and the inte-
grals of Sa and of Sb through the thickness of theglass are nearly zero. Thus, except for near an edge,this measurement gives no information about thesurface stresses that strengthen or weaken the glassnor information about tensile stresses in the middleof the thickness that are responsible for breaking intosmall pieces. At an edge, the stress component nor-mal to the edge surface is zero, so if the b axis isnormal to the edge surface then Sa � Sb reduces toSa, and this technique can measure the surface stressof the edge. However, this measurement cannot bemade in many cases, such as automotive glass wherethe edges are beveled and ground to reduce the oc-currence of flaws originating at an edge. It is thesurface compression and midplane tension in the di-rections parallel to the surfaces that are most impor-tant in the determination of the performance oftempered glass, but this common measurement doesnot measure any of these stresses. Only techniquesin which light travels parallel to the surface can mea-sure these stresses.
The grazing angle surface polarimeter �StrainopticTechnologies, Inc.� is an instrument that can mea-sure surface stress. This instrument works best onthe tinned surface of float glass; that is, the surface ofthe glass plate that touches the molten tin bath whenthe sheet of float glass is formed. High-index prismsand index-matching fluid are used to couple polarizedlight into and out of the waveguide formed by the tinthat diffused into the glass near the surface. Theneed to align the instrument for each piece of glassand the need to clean off the index-matching fluidfrom the sample make these measurements slow andrequire a highly skilled operator. However, this in-strument can measure the stress in surfaces that donot have too much curvature, which makes it themost general surface stress measurement technique.
The Rayleigh fringe technique is a method of mea-suring the in-plane stress under the surface.6,7 Thismethod requires the injection of polarized light inthrough the edge of the sample or at a shallow angleto the surface by use of index-matching fluid or acoupling prism. This method measures the birefrin-gence in the stressed glass by means of observing theRayleigh-scattered light. The angular distributionof Rayleigh scattering from linearly polarized light iszero along the polarization axis and a maximum nor-mal to that axis. Thus fringes in the Rayleigh-scattered light can be observed as light travelsthrough a birefringent medium and the fringe periodis the distance for one wave of retardation. Thesefringes have the largest contrast when the observa-tion direction and the axis of the initial linear polar-ization are both at 45° to the principal stress axes inthe plane normal to the propagation direction.However, for tempered glass one of the principal axesis close to normal to the surface, and so the lightscattered at 45° to this principal axis will be trappedby total internal reflection unless index matching orcoupling prisms are used. Another option is to ob-serve with an external angle of 45° to the surfacenormal, rotate the direction of the initial linear po-
larization axis to match the internal observation an-gle of 27°, and accept the loss of approximately 25% ofthe peak signal.
There are two major problems with this technique:�1� the light must be coupled into the glass so that ittravels parallel to the surface at the point where thestress is to be measured, and �2� the stress must beconstant in a volume defined by the cross section ofthe laser beam and a length of at least one fringeperiod. This technique is usually restricted to smallpieces of glass with polished edges. The most gen-eral solution to couple the light into the glass is im-mersion in a tank of index-matching fluid. Clearlythis is not a noncontact method in this case and islimited to glass with low absorption at the laserwavelength and with curvatures within a limitedrange. If the glass is too flat, there would have to bea long path in the glass, which would require lowabsorption. If the glass is too curved, the secondproblem prevents this technique from working.Having the stress be constant over the requisite vol-ume can usually be achieved in the region of maxi-mum tension for tempered glass. However, this canbe much more difficult to achieve for the measure-ment of surface stress or for the measurement ofcurved annealed glass in which the fringe periodwould be long compared to the maximum straightpath of constant stress.
We developed a new method of measuring stressthat is based on use of thermal gratings to couple aprobe beam of light into and out of the glass sample.Figure 1 is a schematic diagram of our approach. Athermal grating is a volume grating formed by a pe-riodic temperature variation, which we form usingnear-infrared �NIR� light at 1064-nm from aninjection-seeded, TEM00 pulsed neodymium–yttriumaluminum garnet laser, Nd:YAG. Polarized green
light at 532 nm from this same laser was incident onthe thermal grating at the Bragg angle, and some ofit is diffracted into a beam traveling parallel to thesurface of the glass. This singly deflected beamprobes the stress birefringence along its path, and asecond thermal grating diffracts a fraction of thisbeam to form a doubly deflected beam. This doublydeflected beam exits the glass sample, and its polar-ization state is measured. The variation of the po-larization state as a function of distance between thetwo thermal gratings is then used to measure thestress in the interior of the glass sample.
Figure 2 shows the vector representation of theBragg condition for the scattering from the two ther-mal gratings kscattered � kincident kG, where thethree terms are the wave vectors of, respectively,the scattered light beam, the incident light beam, andthe thermal grating. At the first thermal grating,kincident is the wave vector of the probe beam kprobe,and kscattered is the wave vector of the singly deflectedbeam kD1. At the second thermal grating, kincident iskD1, and kscattered is the wave vector of the doubledeflected beam kD2. The wave vectors of the inci-dent and scattered light have magnitudes of 2n532�532 nm, and their directions are the direction ofpropagation. The period of the thermal grating is1064 nm�2nNIR, so the wave vector has a magnitudeof 4nNIR�1064 nm and a direction of either orienta-tion along the thermal grating, which corresponds tothe 1 and �1 orders of diffraction. In the expres-sion for the magnitudes, n532 and nNIR are the refrac-tive indices of the glass at 532 and 1064 nm,respectively. Except for the small change in refrac-tive index between the two wavelengths, all threewave vectors have equal magnitudes and they forman equilateral triangle. To make the singly de-flected beam travel parallel to the surface, the probebeam and thermal grating must be at �30° and 30°from the surface normal, respectively. For a refrac-tive index of 1.5, this requires that, in the air, the1064- and 532-nm beams must be at �50° and 50°from the surface normal, respectively. For the ac-tual refractive indices of nNIR equal to 1.507 and n532
Fig. 1. Beams of 1064 nm at an incident angle of �50° from thenormal are retroreflected to form standing waves in the glasssample with a period of 1064 nm��2nNIR�. Within the glass the1064-nm beams are at 30° to the normal. A 532-nm probe beamat an external incident angle of 50° to the glass normal will be at30° within the glass and will be Bragg scattered to give a beamtraveling parallel to the glass surface. A second thermal gratingwill diffract part of this singly deflected beam to form a doublydeflected beam that will exit the glass traveling parallel to butdisplaced from the original probe beam.
Fig. 2. Bragg condition for coherent scattering can be expressedas a conservation of momentum vector diagram with the wavevector of the scattered light being equal to the vector sum of theincident wave vector plus the grating wave vector. �a� At the firstthermal grating the probe wave vector kprobe plus the grating wavevector kG yields the wave vector of the singly deflected beam kD1,which is parallel to the surface of the glass plate. �b� At thesecond thermal grating, kD1 plus kG yields kD2, the wave vector ofthe doubly deflected beam, which is parallel to kprobe.
equal to 1.524, the actual exterior angles are 48.2°for the NIR beam and �51.2° for the green beam.
A comparison of the path taken by light that exitsthe glass in the probe beam with that taken by thedoubly deflected beam in Fig. 1 shows that there arethree elements that could change the polarizationbetween these two beams. The first element is thetwo deflections by the thermal gratings, the second isbirefringence along the path between the two ther-mal gratings, and the last is the difference in thebirefringence in the two paths between the thermalgratings and the exit surface of the glass. In thecase of uniform tempering, the last element wouldnot cause any difference in polarization, and ideally apair of constant Mueller or Jones matrices shouldrepresent the effects of the thermal gratings. In thiscase, a single polarization measurement for the probeand doubly deflected beams would suffice to deter-mine the stress to within an integer number of wavesof retardation. In practice this approach did notwork. However, we found that by measuring thepolarization state of the doubly deflected beam atseveral different separations between the thermalgratings, we could measure the stress accurately bylooking at the changes in polarization with separa-tion. This is the differential double thermal gratingstress measurement technique that we present be-low.
In Section 2 we describe our experimental setups,in Section 3 we present our model of thermal gratingsin soda-lime glass, and in Section 4 we report on ourmeasurements of thermal grating properties and lim-itations on the deflection efficiency of thermal grat-ings. We discuss our calibration of the doublethermal grating method in Section 5, and in Section6 we discuss the measurement of stress in temperedautomotive glass. Finally in Section 7 we concludewith comments on the potential for this techniqueand areas for further research.
2. Experiment
Figure 3 is a schematic diagram of the experimentand shows the laser beams, which are all aligned inthe same horizontal plane and are incident on theglass sample that is mounted vertically. We modi-fied a commercial injection-seeded pulsed Nd:YAGlaser �Continuum Model YG661� with a 10-Hz repe-tition rate for this study by inserting a small aperturein the oscillator cavity and inserting an absorptiveglass filter in the beam between the oscillator and theamplifier to prevent saturation in the amplifier rodfrom degrading the transverse-mode quality. Inearly experiments we found that, after the doublingcrystal, the beam quality of the fundamental beamwas severely degraded. Such an aberrated beamforms a thermal grating with distorted phase frontsand amplitudes and consequently poor diffraction ef-ficiency. To avoid this problem, a 50% beam splittersplits the 1064-nm TEM00 beam from this laser intoa writing beam that creates the two thermal gratingsand a beam that is frequency doubled to provide theprobe beam. The pulse energies in these two writing
beams incident on the sample are approximately 6and 9 mJ and the pulse length �full width at half-height� is 7 ns.
The 532-nm probe beam with typically 3 mJ�pulseis focused to a 0.20-mm-diameter waist at the firstthermal grating. A Berek’s polarization compensa-tor �New Focus Model 5540� after the final turningmirror is used to adjust the elliptical polarization tolinear polarization at 45° from vertical. An absorp-tive neutral-density filter �optical density of 4� ab-sorbs most of the transmitted probe beam and thefirst three of the set of parallel beams generated byFresnel reflections from the back and then front sur-faces of the glass sample. The singly deflected beamoriginates at the intersection of the probe beam withthe first thermal grating and travels in the horizontalplane and parallel to the glass surfaces until itreaches the edge of the glass sample. For studies ofthe singly deflected beam, we used samples that hadpolished edges rather than the roughly ground edgesof the tempered automotive glass samples. The dou-
Fig. 3. TEM00 beam of 1064-nm is split by a 50% beam splitter�BS� into a beam that is frequency doubled �2�� to provide theprobe beam and a writing beam that creates the two thermalgratings. After passing through a Faraday isolator �FI� and ahalf-wave plate �HW� that rotates its polarization to horizontal, thewriting beam passes through a Galilean telescope �T� that forms a0.35-mm-radius waist at the glass sample �GS�. A second beamsplitter and mirror mounted on a translation stage �MM1� split thewriting beam into a pair of parallel beams whose separation ischanged by translation of MM1. The writing beams are retrore-flected by a mirror �M� to form a standing wave. The 532-nmprobe beam passes through a 750-mm focal-length lens �L�, reflectsfrom the final turning mirror, passes through a polarization com-pensator �PC�, is focused at the first thermal grating, and most ofthe transmitted light is absorbed by a neutral-density filter �ND�.The singly deflected beam �SD� travels parallel to the glass sur-faces until it reaches the edge of the glass sample. The doublydeflected �DD� beam is picked off by a mirror mounted on a trans-lation stage �MM2� and sent to the Stokes meter �SM� or otherdetector. P, power meter.
bly deflected beam originates at the intersection ofthe singly deflected beam and the second thermalgrating and travels parallel to the probe beam. Amirror mounted on a translation stage picks off thedoubly deflected beam and sends it to the Stokesmeter or other detectors. The origin of the doublydeflected beam changes with changes in the separa-tion between the two thermal gratings. Translationof the pickoff mirror can compensate for this motionand allows the doubly deflected beam to always enterthe Stokes meter along the same optic axis.
The glass samples are mounted in a fixture thatallows three orthogonal translations: vertically,horizontally in the plane of the glass sample, andalong the surface normal of the sample. The rangeof the vertical and horizontal motion is large enoughthat any point in a 30 cm � 30 cm plate can be movedinto the laser beam. Dial indicators measure thetranslation of the sample along the surface normal,which allows measurement of the relative position ofthe intersection of the probe beam and the first ther-mal grating from the surface to better than 0.1 mm.This fixture also allows for rotations of the sampleabout a vertical axis and a horizontal axis in theplane of the sample that both pass through the firstthermal grating.
The apparatus is originally aligned starting withthe two NIR beams that form the two thermal grat-ings parallel and in the same horizontal plane. Thesecond beam is then blocked and the telescope T inFig. 3 adjusted to place the waist of the NIR beam onthe first surface of a flat glass plate. This plate andthe NIR beam are adjusted to intersect at a point onthe vertical rotation axis of the sample mount, suchthat the NIR beam is horizontal. The glass plate isaligned to be normal to the incident NIR beam andthen rotated about the vertical axis by �48.2° 51.2°��2 so that the reflection of the NIR beam fromthe first surface of the sample traces the probe beampath in the reverse direction. The 532-nm beam isthen aligned along this reflected NIR beam and thelens L adjusted to place the waist of the green beamat the sample. The glass plate can now be rotatedback 1.5° to have the NIR beam incident at 48.2°.The retroreflecting mirror is adjusted to retroreflectthe NIR beam, and the glass plate is moved approx-imately one fourth its thickness toward the incidentbeams, which places the intersection of the probebeam and the first thermal grating near the middle ofthe plate because the change in the position of thisintersection relative to the surface of the plate is 2.05times the distance the plate is moved. Fine adjust-ments of the angle of the probe beam in the horizontalplane are made to find the singly deflected beam.Adjustments of the vertical and horizontal probebeam angles, the position of lens L, and the focus oftelescope T are used to optimize the singly deflectedsignal. Once the singly deflected beam is optimized,the second NIR beam is unblocked to form the secondthermal grating and generate a doubly deflectedbeam.
The doubly deflected beam is most easily found
after we completely realign the apparatus by trans-lating the probe beam until it intersects the secondthermal grating in the middle of the glass plate tofind the path that the doubly deflected beam willtake. We accomplished translating the probe beamby translating the mirror after lens L in Fig. 3, whichmaintains the angle of the probe beam relative to thethermal gratings. The transmitted probe beam, af-ter suitable attenuation, is then aligned into theStokes meter or other detectors for the doubly de-flected beam. The probe beam is then translatedback to its original position so that it again intersectsthe first thermal grating.
Figure 4 is a schematic diagram of our homemadeStokes meter that uses ferroelectric liquid-crystal�FLC� wave plates �Display Tech Model LV1300-OEM�, a pair of ��8 wave plates �Karl LambrechtModel MWPQC8-12-V532�, a crystal polarizer, and aphotodiode �Hamamatsu Model S1223�. It followsthe design of Gandorfer8 but with the simplificationthat we are not trying to image an object but simplymeasure the polarization of a small monochromaticpulsed light beam. For each laser pulse, the pho-tocurrent from the photodiode was integrated by acharge-sensitive preamplifier �EG&G Ortec Model142A�, amplified �EG&G Ortec Model 575A�, and dig-itized to 12 bits by a card �National InstrumentsModel PCI 6111E� in the data collection computer.The FLC wave plates are ��2 wave plates that canswitch the orientation of their fast axes from 0° to 45°in less than a millisecond. With each FLC havingtwo possible states, there are four possible states forthis optical setup. From the measured signal corre-sponding to each state, all four Stokes parameters ofthe signal beam can be calculated as linear combina-tions of these four measurements. The four Stokesparameters �I, Q, U, V� fully characterize the polar-ization state of a light beam,9,10 and the phase shift �between the horizontal and the vertical componentsof the electric field is given by tan��� � V�U; whichquadrant � lies in is determined by the signs of U andV. Because the FLCs can switch states much fasterthan the 100 ms between laser pulses, it is possible to
Fig. 4. Schematic diagram of the Stokes meter showing the twoFLC ��2 wave plates �FLC A and B�, the two ��8 wave plates, andthe crystal linear polarizer that is oriented for maximum trans-mission of light that is linearly polarized at 22.5° from vertical.The fast axes of the two ��8 wave plates are vertical, and the fastaxis of the FLC A and B switch between vertical and 45° fromvertical depending on the polarity of the applied voltage. The two1-mm-diameter apertures �A� and the 150-mm focal-length lens �L�act to reduce the stray scattered light. PD, photodiode.
make a complete measurement every four laserpulses or every 0.4 s. The computer changes theFLC states with every pulse to obtain a complete setof data every four pulses and typically averagesequivalent data from 30 cycles before it calculates theStokes parameters. This computer also controls thestates of the two FLC wave plates, controls the mo-torized translation stages that move the two mirrorsMM1 and MM2 in Fig. 3, calculates the Stokes vec-tors and phase shifts at four thermal grating separa-tions, and finally calculates the stress based on aleast-squares fit of the phase shifts versus separa-tion.
Gandorfer8 gives the analytical equations relatingthe measured signals to the Stokes parameters forthe case of ideal components and exact alignment.Because the FLC wave plates are not ideal, theseequations are only approximate and so calibration isrequired. We used a fixed crystal polarizer followedby a Babinet–Soleil compensator to generate knownpolarizations that were measured with our Stokesmeter. We used four linear polarizations: horizon-tal, vertical, 45°, and �45°, and plus and minuscircular polarization, which correspond to the Stokesvectors �1,1,0,0�, �1,�1,0,0�, �1,0,1,0�, �1,0,�1,0�,�1,0,0,1�, and �1,0,0,�1�, respectively. We used theRegression macro from Microsoft Excel’s Data Anal-ysis Tool Pack to find the four row vectors of the 4 �4 matrix that gives the best fit of the measured signalwhen we use the Q, U, and V components of theStokes vectors describing the input polarizations asthe independent variables. The matrix inversionfunction in Excel was used to calculate the inverse ofthis 4 � 4 matrix to obtain the calibration matrix.The product of this calibration matrix and the fourvector of measured signals then gives the measuredStokes vector for light of an unknown polarization.This approach assumes that the I component of theStokes vector, which is proportional to the pulse en-ergy, is constant during the measurement of the cal-ibration data set. It appears that drifts in the532-nm pulse energy from the laser over the approx-imate 1 h it takes to collect a full set of calibrationdata are the main limitation in the accuracy of thecalibration. Another error source is the nonideal be-havior of the ferroelectric crystal half-wave plates.The calibration procedure for the Stokes meter cor-rects for the retardation not being exactly half-waveand for the misalignments, but it does not correct fordifferent retardations for the two states of the FLC.The short time required to measure a set of Stokesparameters makes it insensitive to slow laser drifts,but because it takes four laser pulses for each mea-surement of a Stokes vector, it is sensitive to pulse-to-pulse variations in the laser power. Theuncertainties in the accuracy of the phase shifts � is�5° based on the range of deviations from linear fitsof phase shifts versus thermal grating separations.
To calibrate the double thermal grating method ofmeasuring stress, we replaced the large glass sam-ples with a small sample of annealed tint glass thatwas mounted in a rectangular steel frame. The
glass was 48.90 mm wide by 41.28 mm high by 3.302mm thick and was mounted between the bottom ofthe steel frame and a piece that slid in the grooves inthe sides of the frame. A 0.75-in.- �1.9-cm-� diameterlead screw with an Acme thread �10 threads�in.� thatwas threaded through the top side of the framepushed on a load cell �Omega Model LC304-5k� thatpushed on this slider. The glass was mounted inadapters made from 0.25-in. �0.63-cm� square steelstock that had slots milled in them that allowed ap-proximately a 0.254-mm clearance for the glass.These slots were filled with Wood’s alloy to fill gapsand allow a uniform pressure to be applied to the topand bottom edges of the glass. These adapters fit inmatching slots in the bottom of the frame and theslider. The stress in the glass was measured by useof a Babinet–Soleil compensator to measure the bi-refringence seen by light traveling through the glassnormal to the faces.
3. Thermal Grating Model
The efficiency with which a thermal grating in glassdeflects light is key to our approach because the sig-nal level is proportional to the square of this effi-ciency. To guide us in optimizing this efficiency, wedeveloped a model for the deflection efficiency. Thethermal grating is formed by a NIR laser pulse andhas a spatial period of 1064 nm�2nNIR � 353 nm.Energy absorbed from this spatially and temporallyvarying light field is the heat source that creates theperiodic temperature variation in the glass thatcauses a periodic variation in the refractive index.This periodic variation in the refractive index is thethermal grating, and it has the same period as thestanding wave.
In developing our model, we first identify thoseprocesses that are fast enough to contribute tochanges in the refractive index during the laserpulse. Second we model the temperature distribu-tion in the thermal grating as a function of time.Third we convert this temperature distribution into arefractive-index distribution that can be inserted intoan expression for the diffraction efficiency of a volumegrating. Finally we integrate this efficiency over theduration of the laser pulse to calculate the net deflec-tion efficiency. Subsections 3.A–3.D describe thesesteps.
A. Time Scales
There are three lengths that are important in thedetermination of the time scales on which the ther-mal grating changes: the period of the standingwave, the radius of the writing beam, and the pathlength through the glass, which is the glass thicknessdivided by cos �30°� and is approximately 3.8 mm forthe samples of automotive glass we used. The ther-mal relaxation rate of a one-dimensional sinusoidaltemperature variation kthermal is given by11
where Dth is the thermal diffusivity of the glass and is the period. For a thermal diffusivity for soda-lime glass of 4.5 � 10�3 cm2 s�1,12–14 and the 353-nmperiod of the NIR standing wave in the glass, thisgives a rate of 1.4 � 108 s�1, which is significant onthe time scale of the NIR laser pulse. In addition,the amplitudes of the sinusoidal temperature varia-tion can vary significantly over the 6–7-ns duration ofthe green pulse depending on the delay between theNIR and green pulses arriving at the sample. Thusthis thermal relaxation must be included in ourmodel. In contrast, the characteristic time scale forradial thermal diffusion is approximately �0
2�4Dth,15,16 which for a beam radius of 0.35 mm is 0.07 sand is not significant during the laser pulse. Simi-larly thermal diffusion from the front to the back ofthe glass is too slow to matter during the laser pulse.
In addition to changes in refractive index that aredue to temperature changes, the index can changebecause of density changes, which propagate at thespeed of sound. An acoustic longitudinal compres-sion wave has a velocity in glass of approximately5000 m�s, which gives time scales of 70 ps, 50 ns, and760 ns for changes in the density on the length scalesof, respectively, the grating period, the beam radius,and the glass thickness. Thus density changes withthe period of the grating will be in steady state withthe temperature changes, and the other densitychanges can be neglected as being much slower thanthe laser pulse.
B. Temperature Distribution
To model the temperature distribution of a thermalgrating, we choose the origin of the coordinate systemas the point where the center of the 1064-nm beamfrom the laser intersects the first surface of the glass�the top surface in Fig. 1� and the direction of thez-unit vector to match the propagation vector of thisbeam. We model the electric fields of the 1064-nmbeam in the glass traveling forward, Ef �r, t, z�, andtraveling backward, Er�r, t, z�, as
Ef�r, t, z� � E�r, t�exp��A2
z � ikz� ,
Er�r, t, z� � E�r, t�exp��A2
d�� �1 � R�exp��
A2
�d � z� � ikz� ,
where A is the power absorption coefficient; d is theglass thickness along the z axis, which is the thermalgrating axis; R is the power reflection coefficient atthe back face; k is the wave vector; and E�r, t� con-tains the transverse and temporal dependences of thelaser pulse. We assume that the Rayleigh range ismuch longer than the distance from the glass to theretroreflecting mirror, so the divergence of the beamcan be neglected. Summing these two fields and cal-culating the square of the magnitude, we obtain theirradiance, which is composed of two terms. One is
a sinusoidal variation of constant amplitude throughthe glass,
which produces the thermal grating; and the other isa decaying term,
IDC�r, t, z� � exp��Az� � �1 � R�2
� exp���Ad � z�� I�r, t�, (1b)
which we neglect in the rest of this model. The ab-sorbed power that forms the thermal grating is Ig�r, t,z� A, which has units of watts per cubic centimeter.We model the transverse distribution as Gaussianand the temporal shape of the pulse as the differencebetween two exponentials by using
I�r, t� �2ENIR
�2 exp��2� r��
2�� abb � a�
� �exp��at��exp��bt��, (2)
where ENIR is the pulse energy and � is the beamradius. The parameters a and b define the shape ofthe laser pulse and are 4 � 108 s�1 and 3 � 108 s�1,respectively. This is not the exact time dependenceof the pulse, but it does have the rapid rise, slowerdecay, and the same full width at half-maximum thatare characteristic of the real pulse shape.
Figure 5 shows an example of a calculated temper-ature distribution that neglects thermal diffusion forVisteon tint glass with a pulse energy of 6 mJ, a0.7-mm-diameter beam at the sample, and with thepolarization of the NIR beam chosen for minimumreflection at the surfaces. To show the sinusoidaltemperature distribution on the same scale as theglass thickness, the period of the standing wave wasincreased by a factor of 300. With neglect of thermal
Fig. 5. Calculated temperature distribution on the axis of theNIR laser in a sample of Visteon tint glass where thermal diffusionis neglected. This temperature distribution is for a 4-mm paththrough the glass and a 6-mJ NIR pulse energy in a 0.7-mm-diameter beam. To show the sinusoidal temperature variation inthis plot, the period of the standing wave was increased by a factorof 300.
diffusion, the on-axis amplitude of the sinusoidalterm is given by
�T � �2ENIR
�2 �� ACp�
��2�1 � R�exp��Ad��,
where Cp is the specific heat capacity of the glass and� is the density. This amplitude is reduced by ther-mal diffusion.
We include the time variation of the heat deposi-tion from the NIR laser pulse, decay of the thermalgrating by thermal diffusion, and arrival time andduration of the green pulse by using the one-dimensional heat equation:
�T�t, z�
�t� Dth
�2T�t, z�
� z2 � H�t, z�,
where T�t, z� is the temperature difference from am-bient as a function of time t and distance through theglass z. H�t, z� is the heat-source term, which isgiven by
H�t, z� � AIg�r � 0, z, t��Cp�,
where Ig�r � 0, z, t� is given by Eq. �1� and that wewrite as
H�t � 0, z� � 0,
H�t � 0, z� � H0 cos�2kz��exp��bt� � exp��at��,
where H0 is a constant that includes the NIR pulseenergy, absorption coefficient, and the heat capacityand has units of degrees Kelvin per second. Withthis heat-source term, the heat equation can besolved analytically to obtain
where c � 4k2Dth. We must convert Eq. �3� into anexpression for the time variation of the refractive-index profile.
C. Refractive-Index Variations
At steady state the periodic stresses along the ther-mal grating that are due to temperature and densityvariations are zero, thus
��Sz
�T ��
�T� z� � ���Sz
�� �T
��� z�,
where Sz is the stress along the grating axis z, T is thetemperature, � is the density, �T�z� is the variation oftemperature along the grating, and ���z� is the vari-
ation of density along the grating. This can be re-arranged to
��� z� � �� ��
�Sz�
T��Sz
�T ��
�T� z� � � ��
�T�Sz
�T� z�
� ����T� z�,
where � is the linear coefficient of thermal expansion.For soda-lime glass, � is 9.2 � 10�6 K�1 and � is 2.55g�cm3.12 Thus the variation in the refractive indexalong the thermal grating �n�z� can be expressed interms of �T�z� by
�n� z� � ��n�T�
�
�T� z� � ��n���T
������T� z�, (4)
where the first partial derivative is the change inrefractive index with temperature at constant den-sity and the second is the change in n with density atconstant temperature.
The quantities ���n����T and ��n��T�� can be deter-mined from the pressure dependence of the refractiveindex dn�dp and the temperature dependence at zeropressure dn�dT with the equations
���n���T
�1� �dn
dp� ,
dndT
� ��n�T�
�
� ����n���T
,
where � is the compressibility and � is the volumecoefficient of expansion.17 This last equation showsthat dn�dT is the difference between two effects:the first reflecting the increase in n with temperature
at constant density and the second reflecting thechange in n that is due to thermal expansion and theresulting change in density. Thus dn�dT can be pos-itive or negative depending on which effect domi-nates. This equation for dn�dT differs from that for�n�z���T�z� from Eq. �4� by a factor of only 3 differ-ence between the linear and the volumetric coeffi-cients of expansion.
The values of ��n��T�� and � ��n����T for five com-mercial laser glasses for the wavelength 643.8 nmhave been reported as ranging between 0.64–1.03 �10�5 K�1 and 0.30–0.36, respectively.18 The valuesof ��n��T�� and � ��n����T at 587.6 nm for fused silicaare 0.91 � 10�5 K�1 and 0.32, respectively.17 Al-though these values are all for silicate glasses, noneof these glasses is a close match in composition to theautomotive float glass samples from Visteon thatwere used in this study. In fact these laser glasses
have negative values of dn�dT because the thermal-expansion term is larger than the ��n��T�� term, incontrast to the positive values for soda-lime floatglass in which the ��n��T�� term dominates. How-ever, these are the only values that were found forsilicate glasses. No systematic studies of the effectof composition on ��n��T�� and � ��n����T were found,but composition can have a pronounced effect on dn�dT. In soda-lime glasses with weight percent com-positions of �25-x�Na2O, �x�CaO, and �75�SiO2, dn�dTchanges from �3.95 for x � 0 to 2.87 for x � 10 inunits of 10�6 K�1.19 In a study that compared asoda-lime glass without iron with glass with 2%Fe2O3, the change in optical path length with tem-perature ds�dT was measured by a thermal lensingtechnique.14 The effect of the iron was to increaseds�dT by more than a factor of 2. Because ds�dTdepends on the thermal-expansion coefficient dn�dTand the stress optical coefficients, this result raisesthe question of the effect of the iron content in differ-ent types of automotive glass on the deflection effi-ciency of thermal gratings. We use values of 0.8 �10�5 K�1 for ��n��T�� and 0.34 for ���n����T, whichyield 5 � 10�6 K�1 for �n�z���T�z�. This estimatehas a substantial uncertainty associated with itwhose effect is magnified because the deflection effi-ciency depends on the square of �n�z���T�z�.
D. Deflection Efficiency
Siegman has published an expression for the diffrac-tion efficiency of a Gaussian beam from a volumegrating created by a pair of crossed Gaussianbeams,20 but it is not applicable to this case. Thatexpression assumes that the pair of beams that formthe grating are nearly copropagating and cross at asmall angle, which forms fringes parallel to the axisthat bisects the angle between these two beams.This is in contrast to the counterpropagating casethat we use in which the fringes are perpendicular tothe propagation axis. In addition, Siegman’s treat-ment assumes that the beam to be diffracted is alsotraveling at a small angle to this bisecting axis. Inour case, the equivalent angle is 60°. Thus Sieg-man’s formula is not applicable, and we use a simpli-fied geometry to calculate the diffraction efficiency ofthe thermal grating.
We use the equation derived by Kogelnik21 for aplane holographic grating with a finite thickness Leffthat has a sinusoidal variation in the refractive indexwith an amplitude n1 and is in a medium with aver-age refractive index n0. For the geometry of ourexperiment, this model gives the deflection efficiency��t� at the optimum angle �the Bragg angle� of
��t� �2n1�t�
2Leff2
�2 sin2���, (5)
where � �� 1, � is the wavelength of the green beam�532 nm�, and � is the angle between the probe beamand the thermal grating axis �60.4°�. Note that Leff�sin��� is the length over which the probe beam inter-sects the holographic grating. For Leff we use the
beam radius to 1�e2 of the NIR beam, and for n1�t� weuse �n�z� from Eq. �4� with �T�z� given by T�t� fromEq. �3�.
The product of the green pulse shape, delayed by anamount �t from the NIR pulse, and the deflectionefficiency as a function of time were integrated toobtain the net deflection efficiency. Figure 6 showsthe time-dependent deflection efficiency calculatedfrom this model for a NIR pulse energy of 6 mJ, a4-mm path through the glass, an absorption coeffi-cient matching that of Tint glass �273 m�1, see Table1�, and �n��T of 5 � 10�6 K�1. The pulse shape ofthe green beam with the experimental 2-ns delaywith respect to the NIR beam is also shown where weassumed the same temporal dependence as for theNIR pulse except for a time delay. We numericallyintegrated the product of the time-dependent diffrac-tion efficiency ��t� times the green laser pulse shapeto determine the net deflection efficiency. From themodel, the optimum delay is 3.8 ns, which increasesthe calculated net efficiency from 1.4 � 10�4 to 1.6 �10�4. When we include the dynamics in the model,the calculated deflection efficiency is reduced by a
Fig. 6. Time-dependent deflection efficiency calculated from ourmodel including the time variation of the NIR pulse and decay ofthe thermal grating by thermal diffusion. This calculation is fora NIR pulse energy of 6 mJ, a 4-mm path through the glass, anabsorption coefficient of 0.273 mm�1, and �n��T of 5 � 10�6 K�1.The pulse shape of the green beam with the experimental 2-nsdelay with respect to the NIR beam is also shown. From themodel, the optimum delay is 3.8 ns, which increases the calculatednet efficiency from 1.4 � 10�4 to 1.6 � 10�4.
Table 1. Predicted and Measured Relative Deflection Efficiencies
factor of approximately 6 from the simple model thatneglects thermal diffusion and the time variation ofthe laser pulses.
4. Thermal Grating Characterization
A. Efficiency and Power Scaling
We can obtain the scaling of the efficiency with NIRpulse energy and beam diameter �2Leff� from relation�5� by noting that the refractive-index grating ampli-tude n1�t� is proportional to the amplitude of the tem-perature grating, which in turn is proportional to theNIR pulse energy divided by Leff
2. Thus the effi-ciency scales as
� � �ENIR
Leff2�2
Leff2 �
ENIR2
Leff2 (6)
for Leff much larger than the probe beam diameterand where none of the effects discussed below in Sub-section 4.B are significant. Figure 7 shows the de-flection efficiency versus NIR power. The deflectionefficiency scales as the 2.04 � 0.04 power of the NIRaverage power for pulse energies up to 14 mJ�pulse,in agreement with relation �6�. These data weretaken with the beam splitter that splits the NIR intotwo beams removed. We also confirmed that thedeflection efficiency is independent of green pulse en-ergies up to 4 mJ�pulse. We did not observe devia-tions from the scaling with pulse energy predicted inrelation �6� in these experiments, although, as de-scribed in Subsection 4.B, there are effects that willlimit the deflection efficiency at higher pulse energiesor smaller NIR beam diameters.
At 60 mW of NIR power, corresponding to 6 mJ�pulse, the measured deflection efficiency is 8 � 10�5
compared with the prediction from the model for thesame conditions of 1.4 � 10�4. Several factors maycontribute to this disagreement. The largest uncer-tainty is the value for �n�z���T�z� as described above.
Another possible source is the approximation of aGaussian distribution as a square pulse that is in-herent when relation �5� is used. Finally, the phasefronts of the thermal grating or the green beam maynot have been flat as a result of aberrations in thelaser beams, the beam waists not positioned exactlyat the sample, or inhomogeneity in the refractive in-dex of the glass sample. The scaling with beam di-ameter was not tested.
From Eq. �1� and relation �5�, the relative deflectionefficiency for different types of float glass should varyas A2 exp��2A d� if the heat capacity, density, and�n�z���T�z� are constant among the compositions.Table 1 shows the predicted deflection efficienciesrelative to tint glass based on absorption coefficientsprovided by Visteon for their glasses and our mea-sured results for the relative efficiency. We cor-rected the measured values for absorption of thedeflected green beam using the absorption coeffi-cients for 530 nm. All the glass types in Table 1 aresamples from Visteon, except the clear glass samplethat is a sample of 6-mm-thick window glass forbuildings for which we measured the NIR absorptioncoefficient. The predictions of our model agree quitewell with the measured values, which provideevidence that the variation of �n�z���T�z� with com-position among these glasses is minor. The mea-surement for batch privacy glass has the largestpotential error because the large absorption coeffi-cient for green light magnifies the effect of errorswhen the path length is measured from the thermalgrating to the edge of the sample.
Implicit in our model is the assumption that energyabsorbed from the NIR standing wave creates achange in refractive index on a time scale much fasterthan the laser pulse. We assume that thisrefractive-index change occurs by a thermal mecha-nism. We tested this assumption with a thermallens experiment. We crossed a focused red helium–neon laser beam with the NIR beam at a small anglein the glass. The part of the red beam transmittedthrough a small aperture was detected by a fast pho-todiode and recorded on a digital oscilloscope. Thethermal lens generated by the NIR beam in the glassdeflected the path of the red beam and so changed theamount of light reaching the photodiode. Thechange in the photodiode signal had a rise time of lessthan 10 ns, the sampling rate of the digital oscillo-scope, which confirmed that the change in refractiveindex happens on a time scale at least as fast as theNIR laser pulse.
B. Limitations on Achievable Deflection Efficiency
Relation �6� leads one to naively think that, withsmaller NIR beams and higher pulse energies, thedeflection efficiency can be increased to near 100%.Especially with the NIR beam size, there appears tobe a win–win situation because a smaller NIR beamat the sample �that is, smaller Leff�, gives higher de-flection efficiency and better depth resolution. How-ever, there are at least four effects that start causingproblems as Leff is reduced. The first is the difficulty
Fig. 7. Measured dependence of the deflection efficiency versusNIR power showing a quadratic dependence of the efficiency on theNIR power.
in alignment of the green beam to intersect the ther-mal grating as the diameter of the thermal gratinggets smaller. The second is the damage threshold ofthe glass; at fluences �pulse energy per unit area�above a threshold, a plasma forms on the surfacewhere the laser beam enters the glass that damagesthe surface. This threshold depends on wavelength,pulse duration, glass type, and surface preparation.We measured the damage threshold to be 60 J�cm2 �30% for clear float glass with our NIR laser by focus-ing the NIR laser beam down to a 0.12-mm diameterwith a 250-mm focal-length lens and measuring howclose to the focus we could place the glass beforeobserving a plasma. The peak fluence was calcu-lated as 2 ENIR��2, where � is the beam radius andthe beam profile is Gaussian. For the thermal grat-ing experiments, in which we used a maximum of 14mJ in a 0.7-mm-diameter beam, the maximum flu-ence used was 7 J�cm2, which is well below the dam-age threshold.
The third effect that causes problems as the NIRbeam size is reduced is thermal lensing of the NIRbeam. We discovered that this is a problem whilewe were trying to measure the damage threshold oftint glass. We observed plasma formation on theexit surface of the glass sample and not the entrancesurface. Calculations of the self-focusing of laserbeams22 for 2 mJ in a 0.12-mm-diameter beam andfor the absorption coefficient of tint glass at 1064 nmpredict that, even with absorption losses, focusingthat is due to thermal lensing can double the maxi-mum laser beam intensity. The heating and subse-quent refractive-index change from absorption of theearly part of the NIR laser pulse create a focusinglens in the glass, with a power proportional to ENIR��4, that can focus the latter part of the pulse. In theextreme case, this focusing can increase the intensi-ties to the point where they cause damage on the exitsurface or even in the middle of the glass. Evenwhen thermal lensing does not damage the glass, thefocusing can greatly reduce the deflection efficiencyby two mechanisms. Thermal lensing bends thewave fronts of the NIR beam during the pulse, whichreduces the amplitude of the thermal grating; andwith bent wave fronts, only a part of the thermalgrating is at the Bragg angle to the green beam forstrong constructive interference. However, with a0.7-mm-diameter NIR beam rather than the0.12-mm beam used in the damage threshold mea-surements, thermal lensing is reduced by a factor of840 for the same pulse energy. The calculated de-crease in beam diameter after traversing the glass is0.05%, which is too small to cause an increase inirradiance because of the absorption losses.
The fourth effect that limits the deflection effi-ciency is thermally induced phase shifts in the stand-ing wave. The temperature change during the firstpart of the laser pulse and the resulting change in therefractive index will change the NIR wavelength inthe glass. If this temperature change is too large,then the positions of the maxima and minima of thestanding wave will switch, thus reducing the ampli-
tude of the thermal grating and the deflection effi-ciency. Integrating the product of the totaltemperature change on axis �IDC�r � 0, t � �, z�A��Cp� times ��n��T�� through the thickness of theglass, we obtain the net change in optical path lengthduring the laser pulse. For 3.3-mm-thick tint glass,this optical path-length change is 0.14 wave for a6-mJ pulse with a 0.7-mm diameter. This opticalpath-length change should increase as the inverse ofthe beam area and linearly in the pulse energy andreach one-half wave for pulse energies of 22 mJ forthis beam diameter. The fact that there is no ob-servable reduction in efficiency in Fig. 7 is a littlesurprising, but probably reflects the fact that this is aworst-case estimate.
C. Polarization Behavior
We tested the dependence of the polarization of thedeflected beam on the polarization of the incidentbeam in pieces of annealed glass with a smooth edge.The thermal grating was placed approximately 7 mmfrom the edge to minimize any polarization changesthat are due to residual stresses. For a linearly po-larized incident green beam, the diffracted beam ex-iting through the edge was also linearly polarizedwith extinction ratios Imax�Imin of 200–1000. If theincident green beam was horizontally polarized, thatis, in the plane of the green and NIR beams, or ver-tically polarized, then the deflected beam was polar-ized along the same direction. For an incidentpolarization at 45° from vertical, the deflected beamwas linearly polarized at 36° � 1° from vertical. Thelack of ellipticity in the polarization shows that thediffraction by the thermal grating does not introduceany phase shift between the vertical and the horizon-tal components of the light. The rotation of the po-larization angle was consistent with the largerdeflection efficiency for vertical polarization ratherthan horizontal polarization that we observed. Ko-gelnik’s model21 predicts that light polarized parallelto the plane of incidence �the horizontal plane� willhave a lower scattering efficiency than light polarizedperpendicular to the incident plane by the factorcos2���, where � is the angle between the polarizationvectors of the incident and deflected beams. Thisfactor reflects the smaller projection of the electricfield vector of the incident beam on that of the de-flected beam for the case of horizontal polarization.For our geometry, � is 120°, and the predicted effi-ciency for horizontal polarization is only 25% of thatfor vertical polarization. This prediction is inconsis-tent with our direct measurements that the efficiencyfor horizontally polarized light is 40–60% of that forvertical polarization and the 80% value based on therotation of the polarization angle. We do not havean explanation for this discrepancy.
D. Beam Profiles
Figure 8 shows the profile of the singly deflectedbeam in the horizontal plane as a function of depththrough the glass from a thoroughly annealed piece oftint glass that has surface stresses of 0 � 100 psi as
measured with a laser-based grazing angle surfacepolarimeter �Laser-GASP, Strainoptic Technologies,Inc.�. We measured these profiles by directing thebeam exiting the polished edge of this sample onto alinear diode array �EG&G Reticon Model RC1000,pixel dimensions 2.5 mm high by 25 �m wide� andconverting the displacements on the array to anglesin the glass by dividing by the distance and the re-fractive index. The angles are approximate becausethe polished edge has a slight curvature and the zeroangles are only approximately equal to being parallelto the surface of the glass. The corresponding diver-gence in the vertical plane was a fraction of a milli-radian. This qualitative behavior is seen with allglass samples that have an edge that is smoothenough to allow the light to escape as a beam, al-though measurements on an annealed clear glasssample showed much less distortion of the beam.Even well away from the surfaces, these beams havea complicated structure in the horizontal plane. Ex-cept for near the surfaces, these curves all have ap-proximately the same area. The amount ofdivergence in the horizontal plane in even the worstcase does not significantly increase the size of thesingly deflected beam over the typical 15-mm dis-tances between thermal gratings. However, thesedivergences do exceed the full with at half-maximumacceptance angle for Bragg scattering from thesethermal gratings, ��, that is given by21
�� � �2��nLeff
and equals 0.9 mrad for Leff of 0.35 mm.We do not understand the cause of these beam
shapes, but it appears to be related to the distance orthe direction the beam travels through the glass sam-ple. The profiles of the doubly deflected beams havemuch smaller divergences and usually only a singlemaximum as shown in Fig. 9. These doubly de-flected beams were recorded immediately beforethose in Fig. 8 and with no change in sample or
alignment of the NIR and probe beams. It is possi-ble that this beam distortion is due to fluctuations inthe refractive index with depth that is analogous tobeam breakup because of atmospheric turbulence.The process of drawing the molten glass from the tinbath into a sheet would tend to make compositionmore uniform in planes parallel to the surfaces, butnot through the thickness. This is only a supposi-tion, and we have not studied these beam shapes inenough detail to come to any conclusions.
Figure 10 shows the dependence of the singly anddoubly deflected signals versus depth by use of theareas from Fig. 9 for the doubly deflected signals andpower measurements taken at the same time on thesingly deflected signals. The maximum doubly de-flected signal occurs at a depth of 0.6 mm from themiddle of the glass, which corresponds to the cleanestsingly deflected beam profile in Fig. 8. This is con-sistent with the narrow angular acceptance for Bragg
Fig. 8. Horizontal beam profiles of singly deflected beams as afunction of depth in annealed tint glass. Each profile is labeled bythe distance of the intersection of the thermal grating and theprobe beam from the midplane of the glass. The positions in theglass are the same as those used in Fig. 9.
Fig. 9. Horizontal beam profiles of doubly deflected beams as afunction of depth in annealed tint glass. Each profile is labeled bythe distance of the intersection of the thermal grating and theprobe beam from the midplane of the glass. The positions in theglass are the same as those used in Fig. 8.
Fig. 10. Signal strength versus depth in annealed tint glass forboth the singly deflected beam �open diamonds� and the doublydeflected beam �filled squares�. The doubly deflected beam dataare the areas of the peaks in Fig. 9 plus two replicate measure-ments and a background measurement at 2 mm. The verticaldashed lines indicate the surfaces of the glass as determined by themeasured thickness and centered on the signal.
scattering from the second thermal grating. At theoptimum depth and with careful alignment, we foundthat the efficiency of converting the probe beam intoa doubly deflected signal approaches 7 � 10�8 in tintglass with our experiment. This is comparable tothe product of the two efficiencies predicted by ourmodel for the 6- and 9-mJ pulse energies, 1.4 � 10�4
and 3.2 � 10�4, but almost a factor of 5 higher thanthe expected efficiency based on the observed deflec-tion efficiency from a single thermal grating.
In tempered glass samples, the power in the singlydeflected beam is approximately constant with depthin the glass except for near the surfaces, just as inannealed glass as shown in Fig. 10. However, thedeflection efficiency is lower in tempered glass thanin annealed glass. The depth dependence of thedoubly deflected signal in tempered glass is muchsharper than in annealed glass and often shows twomaxima of different heights. These variations indoubly deflected signal with depth vary with positionin the sample.
E. Scattered Light
Figure 11 shows a measurement of the background ofscattered 532-nm light above which we detect ourdoubly deflected signal. We collected this data usingjust a photodiode as the detector and by translatingmirror MM2 in Fig. 3 to vary the separation of theoptical axis of the detection from the axis of the trans-mitted probe beam. Neutral-density filters wereused to reduce the signal levels to within the dynamicrange of the detection system, and the plotted signallevels were corrected for their transmissions. Thepeak at zero is the transmitted probe beam, and thesubsequent peaks are beams that are parallel to theprobe beam and that are created by a pair of Fresnelreflections at air–glass interfaces from the previousbeam. For example, the Fresnel reflection of the
probe beam at the exit surface of the glass creates abeam that undergoes Fresnel reflection at the en-trance surface of the sample that creates the beamthat causes the peak at approximately 2 mm. It isthese reflections that determine the minimum usableseparation between thermal gratings. With thickerglass samples, it should be possible to make measure-ments at smaller separations, that is, between thesepeaks. Figure 11 also shows the size of a doublydeflected signal on the same scale.
5. Stress Measurement Calibration
Given the unexplained beam shapes of the singlydeflected beams and the variation in signal strengthof the doubly deflected beam with depth in the glass,it was essential to confirm that the double thermalgrating technique was in fact measuring stress and tocalibrate those measurements. Figure 12 shows thestress determined when we measured the birefrin-gence in the test glass samples in our calibrationframe versus nominal applied stress. The nominalapplied stress is the applied force �measured by theload cell� divided by the cross-sectional area of theglass sample. These results are for light travelingnormal to the face and through the middle of the facesof the sample and use a stress optic coefficient of2.68 � 10�12 Pa�1. The downward curvature of theplot indicates that the force is not uniformly distrib-uted over the glass and is concentrated in the center.The curve is a quadratic fit to the data. Here in thecenter of the sample, the stress was uniform as mea-sured by the birefringence, but even approximately 1cm from the edge the stress was much less uniformand weaker. Thus the stress was not uniform acrossthe width of the glass.
Figure 13 shows the results of when we look fornonuniformities in the stress through the thickness
Fig. 11. Scattered green light and a doubly deflected signal as afunction of distance from the transmitted probe beam. Thesesignals are in millivolts from a photodiode terminated in 1 M� andscaled by the transmission of the neutral-density filters needed tokeep the signals linear but still above the noise floor.
Fig. 12. Stress in the middle of the calibration sample measuredthrough the thickness as a function of the applied nominal stress�applied force divided by the cross-sectional area of the sample�.This stress is determined from the birefringence seen by lighttraveling through the glass normal to the faces. The curve is aquadratic fit of the data, and the curvature indicates that the forceis not uniformly distributed.
of the sample over a range of applied force using thedouble thermal grating technique. For these mea-surements the singly deflected beam was positionedat the center of the thickness of the sample and at�0.5 mm from the center, and the pair of thermalgratings were approximately centered on the middleof the width of the sample. The error bars are the 1standard deviation error estimates from the linearleast-squares fit of the phase retardation versus dou-ble grating separation. There is no systematic vari-ation in these measured stresses with positionthrough the thickness. Thus the through-thicknessbirefringence measurements should give an accuratemeasurement of the compressive stresses.
Figure 14 shows the variation of the stress mea-sured by the double thermal grating technique versusthe through-thickness stress. The quadratic fitshown in Fig. 12 was used to convert the appliedforces measured with the load cell into through-thickness stresses. These data show a good linearcorrelation between the two stress measurement
methods. The linear least-squares fit of these datagives a slope of 0.85 � 0.02 where the error estimateis 1 standard deviation of the fit. There are twoobvious reasons why this slope differs from unity.The first is that the two methods do not measure thestress at exactly the same point in the glass. Thesecond is that the two methods do not measure ex-actly the same quantity. The through-thicknessmeasurement measures Sb � Sa whereas the doublethermal grating technique measures Sb � Sc wherethe vertical b axis is the direction of the compression,the c axis is normal to the faces of the sample, and thehorizontal a axis is in the plane of the glass. Thedata in Fig. 13 indicate that Sc is near zero, thereforeSb � Sc should be close to Sb. In contrast, the non-uniform stress indicated in Fig. 12 would predict thatSa would be a tensile stress near the middle of theglass; therefore the through-thickness method, whichmeasures Sb � Sa, would yield a value larger than Sbby the magnitude of Sa. With no force applied, Sa iszero in this annealed sample but �Sa� to first orderincreases linearly with the applied force as does Sb.Thus, to first order, Sb � Sa should be proportional tobut larger than Sb because the tensile stress Sa isnegative. This is consistent with the slope in Fig. 14that is less than 1.
6. Tempered Glass Measurement
Figure 15 shows a measurement of the tensile stressin a sample of tempered automotive glass made fromVisteon tint glass that has surface stresses in therange of 15,000–16,000 psi as measured with a laser-based grazing angle surface polarimeter. From theslope of the line that was fit to the four data points,the tensile stress is measured to be 8074 � 98 psiwhere the error estimate is the 1 standard deviationerror estimate. This value is in excellent agreementwith the expected value of one half of the surfacecompression that is predicted for a quadratic stressprofile through the thickness. The second thermalgrating was approximately 45 mm from the roundedand ground edge of the sample, and the laser beamswere aligned so that the singly deflected beam struck
Fig. 13. Double thermal grating measurements of the stress atthe center of the thickness �open circles� 0.5 mm from the centertoward the front �filled diamonds� and 0.5 mm toward the back�open triangles� show that there are no large nonuniformities inthe stress through the thickness of the sample. The error barscorrespond to �1 standard deviation from the linear regression ofthe phase angle versus the double grating separation.
Fig. 14. Comparison of through-thickness stress versus doublegrating measured stress for compressive stresses.
Fig. 15. Double thermal grating measurement of the in-planestress at the center of the thickness of a tempered automotive glasssample made of Visteon tint glass.
the middle of the edge of the sample. A more gen-eral method of measuring the depth at which theprobe beam intersects the first thermal grating is tomeasure the separation between the two pointswhere the NIR and green beams intersect the surfaceof the glass sample. For these data, we measuredthe phase retardation of the doubly deflected beam bythe slow process of rotating a piece of polarizing filmto locate the orientation and ellipticity of the polar-ization. This method has different possible system-atic errors from our Stokes meter measurements, andit is gratifying to see that this method of measuringthe retardation also gives data that are well fit by astraight line.
The absolute accuracy of the stress measurement islimited not only by the retardation data but also bythe uncertainty in the value of the stress optic coef-ficient for this glass as well as possible other system-atic errors. Fontana showed that at roomtemperature the stress optic coefficient of annealedfibers of a soda-lime glass �Corning Code 0080� was10% lower than that for unannealed fibers.5 In ad-dition, the effects of glass composition, including theamounts of components such as ferric and ferrousions, on the stress birefringence coefficient would alsoneed to be included to achieve high absolute accuracyin the stress measurement. However, for most ap-plications the requirement is not for high accuracybut for better information about the spatial variationof the stress, which the double thermal grating tech-nique can provide.
The variation in the stresses in tempered glassadds two effects that affect the double thermal grat-ing technique. The first is due to the finite size ofthe singly deflected beam, which is approximately 0.5mm in diameter in this study. Near the surfaces,the stress, and hence the birefringence, varies rapidlyover that distance. This works to depolarize thelight in the singly deflected beam and limits how closeto the surface this technique will work. Another ef-fect of the varying stress versus depth in temperedglass is that it creates a gradient in the refractiveindex that changes the propagation of the singly de-flected beam, and the vertical and horizontal polar-izations see different index gradients. For auniaxial stress S, the changes in the refractive index�n for the electric field parallel and perpendicular tothe direction of stress are
�nparallel � nSE �q
v� 2�
pv� ,
�nperpendicular � nSE �����
qv
� �1 � ��pv� ,
where E is Young’s modulus, � is Poisson’s ratio, andp�v and q�v are the strain optic coefficients that are,respectively, 0.31 and 0.21 for soda-lime glass.4 Forequal in-plane stresses, the refractive-index changeseen by light traveling in the plane and polarized
either in plane or out of plane, the refractive-indexchanges are
�nin plane � �nparallel � �nperpendicular,
�nout of plane � 2�nperpendicular,
and the difference between these two equations givethe strain birefringence. For example, consider a3.5-mm-thick piece of glass with a quadratic stressprofile that gives a surface stress of 16,000 psi. Witha 15-mm path length that is 0.25 mm from the mid-plane of the glass, the refractive-index gradientswould cause deflections of 1.7 and 2.9 mrad for theout-of-plane and in-plane polarizations, respectively.These values are much larger than the predicted an-gular acceptance of these thermal gratings; and justlike a lens, the deflection increases with the distancefrom the center. It appears that it is only the wideangular spread in the horizontal plane of the singlydeflected beam that allows the double thermal grat-ing technique to work in tempered auto glass. Bothof these consequences of the large stress variationswith depth restrict the application of this double ther-mal grating technique to the middle for temperedglass samples.
7. Conclusion
The double thermal grating technique works to non-destructively measure the tensile stresses at the mid-plane of tempered glass plates and should beapplicable to other glass objects. Although we donot have a complete understanding of this technique,the high quality of the calibration data plotted in Fig.14 verifies that our technique does measure stressesin the middle of glass samples. Chief among thethings we do not understand is the mechanisms forthe multiple peaks seen in the singly deflected beamand the relationship to the properties of the glasssamples. We have developed a useful model of thethermal grating that accurately predicts the scalingof the deflections efficiency with changing laser andglass parameters and absolute efficiencies within afactor of approximately 2. The excellent linearity ofthe data in Fig. 15 argues that the potential mea-surement precision of the double thermal gratingtechnique is high and that our Stokes meter needsimprovement to achieve this potential precision.
We have automated the stress measurement partof our technique, but automation of the alignmentand sample positioning still remain to be done for acompletely automated instrument. For the laserbeams aligned in a plane, there are two angles andone distance that control the generation of the singlydeflected beam at the center of the thickness andtraveling parallel to the surface of the glass sheet.These are the angle in air between the 1064- and the532-nm beams ���, the orientation of these two beamsrelative to the normal of the glass ���, and the dis-tance from the surface of the glass to the intersection���. For the efficiency to be greater than 50% of max-imum for our experiment, Snell’s law predicts thatthe tolerance for � is �0.85 mrad and the tolerance
for � must be �29 mrad. Thus the angle betweenthe beams is more critical that the orientation of thesurface normal relative to the beams. To control thedistance � in the glass to �0.2 mm, which is 5% of thethickness of our tempered auto glass samples, re-quires one to control the position to �0.1 mm. Tomeet these tolerances for a sample such as a carwindow would require an active feedback system, butthis is practical.
This research was supported by the Office of In-dustrial Technologies of the U.S. Department of En-ergy �DOE�. We greatly appreciate the continuedsupport of Theodore Johnson, DOE manager of theglass industries of the future. We thank Jim Hay-wood at Visteon Glass Divisions and K. K. Koram atPPG Industries, Inc. for many fruitful technical dis-cussions about stress measurement in glass. Spe-cial thanks go to Vincent Henry for his continuousinterest in our research and his guidance. PacificNorthwest National Laboratory is operated by Bat-telle for the DOE under contract DE-AC06-76RL01830.
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