Homogeneity testing of optical glass byholographic interferometry
Diana Tentori
A holographic interferometry technique for the measurement of optical glass homogeneity of plate samples ispresented. It is shown that this immersion technique is more accurate than methods used for this purposebased on classical interferometry without the need of quality optics.
1. Introduction
The degree of spatial variations of refractive indexwithin a melt or blank is evaluated considering twoparameters: optical homogeneity and striae. Thevariations called striae correspond to strong inhomo-geneities of a very local nature. They produce achange on the propagation direction of light that isdetected using schlieren techniques. Optical homoge-neity corresponds to smooth spatial variations of therefractive index that produce negligible variations onthe direction of propagation of glass through the glasssample.
Optical homogeneity is tested by transmission usingclassical interferometry.1-3 The wavefront transmit-ted through the sample is compared interferometrical-ly with a reference wavefront. Hence conditions im-posed on the quality of the optics used in theseinstruments are strong. The sample itself must bewell polished or conveniently tested1' 2 to separate itssurface deformations from the contributions due tooptical homogeneity. Another alternative is to use amatching liquid3 to lower the requirements on thepolishing of the sample. Nevertheless, the flatnesscontribution of the mirror surfaces cannot be neglect-ed. In this case the point-to-point thickness variationbetween the interferometer mirrors must be known.Furthermore, because of the use of the matching liq-uid, the reference surfaces must be conveniently coat-ed to improve the visibility of the fringe pattern. In
The author is with CICESE, Apdo. Postal 2732, Ensenada, B. C.,Mexico.
this case as well as in Ref. 1, the use of multiple beaminterferometry has been proposed, but as the separa-tion between the mirrors increases, the collimationtolerances on the light beam are more strict, requiringa higher quality collimator lens. Classical techniquesare not suitable for visual inspection of a nonprocessedinterference pattern.
In this work an holographic interferometry tech-nique is presented that uses an immersion device toavoid the need for separating surface contributions ofthe sample. For visual inspection, Young carrierfringes can be introduced by laterally displacing a dif-fuser plate located just after the immersion tank. Asusual, in hologram interferometry, quality optics is notrequired. The wavefront transmitted through theglass plate is tested against the wavefront transmittedthrough the same test plate rotated with respect to itsoriginal position.
II. Optical Homogeneity Evaluation Using ClassicalInterferometry: Immersion Technique
In the optical arrangement presented in Ref. 3, theoptical homogeneity of glass plates is measured using aFizeau interferometer. The sample is immersed in amatching liquid between the two reference mirrors.Classical interferometry uses known reference wave-fronts to compare the state of interference on a point inthe interference pattern, with the state of interferencein another point of the same interferogram. To calcu-late the contribution of the flatness tolerance of eachsurface, Fig. 1 is used. It shows the two referencemirrors, the immersion liquid, and the plate sample.The trajectory followed by ray 1 is considered as thereference, and the trajectory followed by ray 2 containsthe optical inhomogeneity to be detected. The fabri-cation tolerances on each surface are represented bydifferent variables. Flatness tolerances of the frontand back mirrors are written as T1 and 6T2, respec-
Fig. 1. Parameters used for ray tracing within the cavity formed by
two reference mirrors in a Fizeau interferometer. There is a match-
ing fluid between the mirrors and sample.
tively. Flatness tolerances of the first and secondsurfaces of the glass plate are represented by 3T1 and5T2 .
The reference optical path of the ray travelingthrough the sample can be written as
OP1 = 2(N'T + Nt,), (1)
where N' is the refractive index of the matching liquid,T7 is the sum of the liquid thicknesses before and afterthe sample, seen by ray 1, N is the refractive index ofthe sample, and t1 is its thickness.
The optical path for ray 2 is given by
OP2 = 2(N'T2 + Nt2 + aN6 ), (2)
where T12 is the thickness of the matching liquid seenby ray 2. Considering the thickness variations, it canbe written as
r2=TdlT+ 6T T 2 6tl±bat 2 . (3)
The parameter t2 in Eq. (2) represents the thickness ofthe sample seen by ray 2. Introducing the thicknessvariations,
T2 = T t, F at 2. (4)
The variation of the optical homogeneity of the sampleis given by bNbT, where AN is the change in the value ofthe refractive index and 6r is the thickness of theinhomogeneity.
In Eq. (5) the matching has been considered betweenthe refractive index of the sample and the immersionliquid using N' = N + AN. Canceling terms and usingthe same tolerance for both faces of the sample,
bt = U2= t, as well as for both reference mirrors,6T1 = T2 = T, we obtain
(6)OP2-OP1 = 2(2ANbt + 2N'6T + Nsbr).
The wavefront emerging from the mirror cavity willcontain only information about the optical inhomoge-neity, if the first two terms can be neglected. Consid-ering that both faces of the sample are polished to +Xand a measurement limit of X/200 (commercial phaseshifting systems), a simple calculation shows that aperfect matching of the indices is not required. If itsatisfies AN < 10-3, this contribution can be neglected.Here it is not considered the thickness variation due tolack of parallelism that is used to produce Fizeau carri-er fringes.
Regarding the flatness tolerance of the mirrors, us-ing 3T = X/20, this term contribution is always largerthan X/4 without considering the influence of parallel-ism between the mirrors. Hence, even for visual in-spection, the form of the mirrors and the angle betweenthem cannot be neglected. The cavity contribution tothe interference pattern must be measured for eachpoint, as the authors of Refs. 1 and 2 report. Testing itwith the same arrangement produces an additionalerror contribution that doubles the final error mea-surement, if it is considered that there is no contribu-tion to error due to the identification of each point onthe interferogram; otherwise the error contribution iseven larger. With this method the spacing and orien-tation of carrier fringes cannot be varied.
The techniques in Refs. 1 and 2 require the evalua-tion of four interferograms; hence their accuracy iseven lower.
111. Holographic Interferometry Technique for Optical
Homogeneity Evaluation
A diagram of the hologram interferometer discussedhere is shown in Fig. 2. The reference beam is pro-duced by a point source. The collimated object beamilluminates the glass sample S located within an im-mersion tank. Immediately after the tank, there is adiffuser plate D. The hologram plate H is aligned withthe light source L, the sample, and the optical viewingsystem V.
The procedure used to test optical homogeneity is asfollows. A hologram of the object beam with the sam-ple within the immersion fluid is recorded. Whendeveloped, the orientation of the sample plate ischanged to look for homogeneity variations. If nofringes are observed, a lateral displacement of the dif-fuser can be introduced to produce carrier fringes forvisual inspection.4
IV. Theory
The optical path up to the diffuser is changed byvarying the orientation of the sample plate. If thischange is a lateral displacement or a rotation, a regionof the sample is compared with another region of thesame sample. In the following it is demonstrated thatthis technique allows direct detection of the presenceof optical inhomogeneities in the glass sample.
Fig. 2. Diagram of the holo-graphic interferometer used foroptical homogeneity testing. Theglass sample S, diffuser plate D,hologram H, light source L, andoptical viewing system V are
aligned.
In the object beam, the optical path from the lightsource to the diffuser can be written as
L1 = L + N'T + Nt, (7)
where Lo is the optical path from the light source to apoint P on the diffuser plate without considering thetrajectory through the immersion tank. The opticalpath through the immersion fluid is N'T, where N' isthe refractive index of the fluid and T is the sum of thethicknesses of the fluid before and after the glass sam-ple. The optical path through the sample is Nt, whereN is the glass sample refractive index and t is the glasssample thickness (Fig. 3).
After varying the orientation of the sample plate, theoptical path relation with the same point P on thediffuser is
L2 = Lo + N'(T - t) + N(t + t) + 6N6r. (8)
In this equation it has been considered that the samplethickness in the new region has a different value givenby t + t and that the ray travels through a regionwhere there is an inhomogeneity. The change in therefractive index value is represented by N 6N. Thethickness of the inhomogeneity is given by &r. Sincethe sample thickness is different, the immersion fluidthickness seen by the ray changes to T F U.
Using Eqs. (7) and (8), the optical path difference isgiven by
DCO = ANU + Nr. (9)
Here again, as in Eq. (6), the first term can be canceledusing the proper values for the refractive index of thematching liquid and the tolerances for sample thick-ness. In this case the interferogram will contain justthe information about the homogeneity variation inthe glass sample.
The tolerances for sample thickness correspond toflatness and parallelism, since different regions of theglass plate are compared. To determine the propervalues for flatness and parallelism, it is necessary toconsider the difference between the refractive index ofthe matching fluid and that of the sample as well as theaccuracy of the measurement. For this techniquesample fabrication or index matching must be accom-
plished to a higher precision to be able to perform thehomogeneity evaluation of the sample from a singleinterferogram. Some simple calculations show thatfor AN < 10-3 the thickness tolerance is higher than 4X;this value must include parallelism as well as flatnesscontributions.
To avoid a difference in the optical thickness seen bythe light traversing the sample, due to tilt, using adynamic mount of the type utilized to hold holographicplates in real-time holographic interferometry is rec-ommended.
V. Experimental Results
Using the technique presented here, an optical glassplate with a 10-mm thickness was tested in air andwithin an immersion tank. Figures 4 and 5 show theinterferograms obtained for the BK7 grade A sample.In Fig. 4 the test plate is immersed in an immersionfluid (ND = 1.5150 2 X 10-4 at 230C) used for opticalmicroscopy. The matching is -10-3. Since no fringes
(N+SN)S-
V t-
, + T = T
Fig. 3. Parameters used for ray tracing within the immersion tankin the holographic interferometer. The glass sample is immersed in
Fig. 4. Interferogram for a BK7 grade A sample after a rotation of
200. The sample is within an immersion fluid; refractive index
matching is -10-3. Carrier fringes were introduced.
were observed after changing the orientation of thesample, Young carrier fringes were introduced.
Using the same changes of orientation, it can be seenin Fig. 5 how the strong contributions of flatness andparallelism (a dynamic mount was used for position-ing) obscure the contribution of optical inhomogenei-ty.
VI. Conclusions
The measurement procedure presented here can bemade sensitive just to optical homogeneity variationsof the sample. Since a single hologram must be evalu-ated, this technique has an accuracy higher than thatobtained with classical interferometry. It does notrequire high quality optics, as is usual in holographicinterferometry.
For visual inspection Young carrier fringes can beproduced by laterally displacing the diffuser plate.The orientation and spacing of these carrier fringes can
Fig. 5. Interferogram for a BK7 grade A sample after a rotation of
200. The sample is in air. Thickness contributions mainly due to a
lack of parallelism obscure the optical homogeneity contribution.No carrier fringes were introduced.
be varied to increase the sensitivity to optical homoge-neity variations of the optical glass samples.
I wish to thank Adriana Lopez for her help with theexperimental work.
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4. D. Tentori and D. Salazar, "Hologram Interferometry: Carrier
