once a signal is acquired, it can be recirculated and hence processed in any number of different ways with the TAD.
LITERATURE CITED
(4) Gary Horlick, A p ~ l Spectrosc., . 30, 113 (1976). (5) K. R. Betty and Gary Horlick, Appl. Spectrosc., 30, 23 (1976).
RECEIVEDfor review July 20,1976. Accepted August 25,1976. Financial support by the University of Alberta and the National Research Council of Canada is gratefully acknowledged.
(1) H. E. Kallmann, Roc. IRE, 28, 302 (1940). (2) Gary Horlick, Anal. Chem., 48, 783A (1976). (3) A. Savltzky and M. J. E. Golay, Anal. Chem., 38, 1627 (1964).
Holographic Interferometric Microscopy R. F. Cournoyer,' M. B. Rhodes," and Sidney Siggia Chemistry Deparfment, University of Massachusetts, Amherst, Mass. 0 1002
Holographic interferometric microscopy (HIM) is described. Holographlc Interferograms are presented that reveal both the qualitative and quantitative aspects of HIM. The usefulness of this technique is demonstrated by considering a varlety of static and dynamic chemical systems. HIM offers the analyst a new approach to old problems and now makes It experimentally feasible to conveniently perform previously lmpossible or complicated interferometric investigations.
Holographic interferometric microscopy (HIM) has been discussed by R. F. van Ligten and H. Osterberg ( I ) , G. W. Ellis ( Z ) , R. H. McFee (3),M. B. Rhodes ( 4 ) ,and W. H. Carter et al. (5), but there are few applications of this technique described in the chemical literature. Microscopy is an important analytical tool; therefore, developments that extend the capabilities of this technique are of interest to the analytical chemist. This report describes HIM and its use in the investigation of a variety of chemical systems. The holographic process allows one to encode, on photographic film, a radiation wavefront emanating from a subject so that, at some future time, this wavefront may be reconstructed. The hologram is produced by exposing a photographic emulsion with two coherent off-axis wavefronts, one being termed the subject or information containing beam and the other the reference beam. The reconstructed wavefront is produced by illuminating the processed hologram and will contain the phase and amplitude information found in the original subject wavefront. Introductory descriptions and applications of the holographic process may be found in the article by E. N. Leith and J. Upatnieks (6) and in the monographs by H. M. Smith (7) and G. W. Stroke (8). The studies to be described use a subject wavefront magnified with microscope optics prior to the recording of the hologram. It has been demonstrated that it is possible to do darkfield, phase contrast, and polarization microscopy with the reconstructed wavefront (1-5). It was also demonstrated that the reconstructed wavefront can be focused in various optical planes. The ability to process and focus the reconstructed wavefront a t one's leisure are useful advantages of the holographic technique but these advantages are overshadowed by the ability to do interferometry. The holographic interferograms to be considered in this work are produced by the spatial superposition of two reconstructed wavefronts. These wavefronts are reconstructions of those transmitted through systems of interest to the chemist. Holographic interferometry Present address, Industrial Laboratories, Eastman Kodak Co., Rochester, N.Y. 14650.
generally produces results more conveniently than conventional interferometric techniques because exact alignment and high quality optical components are not required. An interferogram unique to holography can be produced by superimposing reconstructions of two wavefronts that existed at different times. When the two superimposed wavefronts are of different subject fields, the interferogram is termed a time differential holographic interferogram. The interference phenomena that occur from the superposition of wavefronts results in fringes which can be used to indicate differences in the optical paths between the two wavefronts. The following expression describes the optical thickness or optical path
(W, OP = ( n ) ( d )
(1)
where ( d ) is the distance that light travels in a medium of refractive index (n).Introductory material on holographic interferometry is offered by A. E. Ennos (9) in a review article. Interferograms are characterized by fringes whose appearance is dependent on differences in the phase and amplitude and the spatial superposition of the two superimposed wavefronts. The reconstructed wavefronts are said to be exactly superimposed when they appear to originate from the same point. Any observed fringes are due only to differences in optical paths inherent in the superimposed subject systems. When the reconstructed wavefronts appear to be originating from different points, then the wavefronts are said to be nearly superimposed. The near superposition of wavefronts gives a series of equidistant parallel fringes extending across the interferogram in addition to the fringes due to differences in the superimposed wavefronts. The spatial frequency of these parallel fringes decreases as exact superposition of the two reconstructed wavefronts is approached. A continuum exists as the wavefront superposition goes from near to exact, in which the spatial fringe frequency decreases and finally becomes so small that no fringes caused by differences between the spatial superimposed wavefronts are observed. (These effects are all illustrated in Figures 2 and 3.) The fringes in the interferograms are used t o determine the phase shifts in the sample. The phase shift is correlated with the system of interest by the following expression
r = nldl - nodo
(2)
where r is the observed phase shift and is measured in units of A, the length of one optical path. The length of one optical path X is equal to the wavelength of the illuminating radiation. The optical thickness, nldl, is the product of the refractive index (nl)of the subject field and its thickness ( d l )and nodo is the product of the refractive index of the reference field and
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T h e observation of the fringe displacement allows the determination of phase shifts ranging from a few A’S down to small fractions of a h. Under favorable circumstances and with the use of appropriate techniques, phase shifts of Ah00 or even less may be determined. The use of both the fringe counting and fringe displacement measurements allows the determination of four orders of magnitude of phase shifts ranging from Ah00 to 100A. This means that optical path differences ranging from about a 100 8, up t o a fraction of a millimeter may be quantitatively determined with holographic interferometric microscopy.
a
EXPERIMENTAL
condenser 1 beam
1 splitter
HOLOGRAM
PRODUCTION b
L A 5 E
R
T I
I I
WAVEFRONT
RECONSTRUCTION
Figure 1. (a) Schematic diagram of the holographic microscope; (b) Schematic diagram of the wavefront reconstruction unit
its thickness. Any identical regions in the subject and reference subtract out and do not contribute to the observed phase shift. Therefore, the observed phase shifts are due only to differences between the superimposed wavefronts. Generally do is equal to d l and Expression 2 may be written as,
r = (nl - n o ) ( d )
(3)
The refractive index no of the immersion medium (air, water, oil, etc.) often is known or easily determined. The magnitude of the phase shift for a given subject can be changed by employing immersion media with various refractive indices. If either the subject thickness or its refractive index is known, i t is then possible to determine the remaining unknown physical parameter from the observed phase shift. Alternatively, the use of the phase shifts that are observed when the subject is immersed in two media with differing refractive indices allows the determination of both the subject thickness and refractive index without any prior knowledge of either of these values (IO). When the wavefronts are exactly superimposed, the phase shift is determined by counting the fringes. Phase shifts of several A’s up to over one hundred A’s can be used for quantitative analysis. If the variation in the optical thickness differences between the superimposed wavefronts is less than a few A’s, discrete fringes are not observed. The interferograms in this case exhibit regions of differing contrast. (For example see Figure 3b.) When the near superposition of wavefronts is used to produce the interferograms, the phase shift is determined by observing the displacement of a fringe in the region of interest. The phase shift attributable to a fringe displacement is determined by comparing the magnitude of the displacement ( y )with the fringe spacing (x) using the following
r =Y/x 2254
(4)
Apparatus. An American Optical holographic microscope and reconstruction unit were used to encode and reconstruct the radiation wavefronts of interest. Block diagrams of the instrumentation are included in Figure 1. Helium-neon lasers (632.8nm) with outputs of approximately 3 mW were used. The holograms were taken with either a 20 or a 40 power objective and were recorded on 35 mm AGFA ScientificaF 14-C 75 with exposure times ranging from 1/250 to 1/125 of a second. Kodak D-19 developer was used, and the manufacturers recommendations were followed in the processing of the photographic emulsions. A Kofler hot stage was employed for the studies involving elevated temperatures. Zeiss and Baker interference microscopes and other microscopical techniques verified the conclusions drawn from the holographic interferograms. Procedure. The procedure was to make holograms with wavefronts transmitted by both the subject and reference fields. Several types of reference fields were used. One type was produced by removing the subject from the field of interest. This was accomplished by translating the subject out of field of view. Alternatively,an appropriate reference field could be substituted in the optical path. Another type of reference field was simply the subject field itself prior to some perturbation. Wavefronts reconstructed from these holograms were then superimposed to produce interferograms. The superposition of the reconstructed wavefronts can be accomplished by illuminating two holograms that have been placed together, by illuminating holograms that have been combined, or by doubly exposing a single photographic emulsion. The approach of combining single holograms is comparable to that reported by N. Abramson, ( I I ) , and G. Havener and R. Radley (22) and is a simplification of that described by J. Gates ( 1 3 ) .The positioning of the two separate holograms with respect to one another determines whether the reconstructed wavefronts are to be exactly or nearly superimposed. The doubly exposed holograms are exactly aligned, thus producing exactly superimposed reconstructed wavefronts. The double exposure technique can save time in hologram alignment but at the sacrifice of flexibility since the choice and orientation of the reference wavefront is no longer variable. Identical appearing interferograms have been found to occur when either the doubly exposed holograms or the exactly aligned combined single holograms are used. Once the reconstructed wavefronts are aligned and focused, the resulting interferogram is viewed or photographed.In some instances, the interpretation and measurements are made on enlargements of the interferograms ihcluded in this manuscript.
RESULTS AND DISCUSSION Several representative inorganic crystalline systems will be discussed in order to establish the interpretive background for the additional HIM studies that will be presented. Figure 2a is a photograph of the reconstructed image of a sodium chloride crystal and is identical to the image observed when the system is examined with a helium-neon laser illuminated microscope. T h e interferogram that results from the exact superposition of the reconstructed subject and reference wavefronts is presented in Figure 2b. The exact superposition of the wavefronts was assured in this case by reconstructing. the wavefronts from a doubly exposed hologram. The observed fringes are fringes of constant optical path and indicate that the crystal is pyramid shaped with attenuated apex planar faces or possibly stepped faces. T h e ambiguity in the face geometry is resolved when Figures 2c and d are considered. Figures 2c and d are interferograms produced by the near superposition of the reconstructed subject and reference wavefronts. Each wavefront was reconstructed from a separate
ANALYTICAL CHEMISTRY, VOL. 48, NO. 14, DECEMBER 1976
Figure 2. (a)Reconstructed image of sodium chloride crystal (X400); (b) Interferogram of a sodium chloride crystal produced by the exact superposition of wavefronts; (c, d) Interferograms of sodium chloride crystal produced by the near superposition of wavefronts
hologram. The variable spatial fringe frequency at the experimenter’s disposal is demonstrated in these interferograms. The tracing of a fringe as it proceeds from the empty part of the field into the crystal confirms the pyramid shape with planar faces suggested from consideration of Figure 2b. The fact that the faces are planar is discerned by observing that the fringes undergoing displacement are approximately linear, indicating a continuous increase in optical path. The jitter in the fringes around the subject base is due to crystal decoration a t the edges. Figure 2b may be interpreted quantitatively to give the thickness of the sodium chloride crystal. The nine fringes observed between the edge and center of the crystal indicate a phase shift a t the crystal center of nine optical paths. The maximum error in the fringe counting procedure is observed to be less than one-half fringe. The sodium chloride crystal is cubic and has a single refractive index of 1.54. The refractive index of the air immersion medium is 1.00. Using this information and Expression 3, the crystal is determined to be 16.6 f 0.9 gm thick. The application of Expressions 3 and 4 to the computation of the phase shift from the fringe displacement in Figures 2c and d produces quantitative results that do not appear to agree with those calculated from Figure 2b. This is because the crystal optical thickness is dependent upon the observation direction. Figure 3a is the reconstructed image of a planar lead iodide crystal. Figure 3b was produced by the exact superposition
of the subject and reference wavefronts. The planar crystal in Figure 3b is not expected to produce a series of contour fringes because the phase shift introduced by the subject is nearly uniform. This uniform phase shift manifests itself in this case as a darkening of the field in the region of the crystal. The crystal decoration observed in Figure 3a appear as the lighted regions in the interferogram in Figure 3b. This latter interferogram illustrates how the exact superposition of wavefronts can result in interference contrast. Although some qualitative information results from consideration of Figure 3b, the interferogram in Figure 3c is used for quantitative investigation. This interferogram resulted from the near superposition of wavefronts reconstructed from individual holograms of the crystal and reference field. The displacement of the lower fringe in Figure 3c at the point where the fringe intersects the crystal will be used to demonstrate the determination of crystal thickness. A phase shift of 0.29 f 0.10 X is observed indicating that the crystal is 0.29 f 0.10 optical paths thick. Using this information, the refractive index of the crystal, and the relationships in expressions 3 and 4, the edge thickness is determined to be 0.08 f 0.03 gm. The reported error in the determination of the fringe displacement can be decreased to less than 0.01 X by the application of vernier ( 1 4 ) , densitometric procedures, or the Sabbatier effect ( 1 5 ) . More points become available for measurement purposes if one increases the spatial fringe frequency as is demonstrated
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,
Flgure 3. (a) Reconstructed image of a lead iodide crystal (X200); (b) Interferogram of the lead iodide crystal produced by exact superposition of wavefronts; (c, d) Interferograms of the lead iodide crystal produced by the near superposition of wavefronts
Flgure 5. Time differential holographic interferogram of a thymol crystal crystallizing from the melt (X200)
(h)
Figure 4. (a, b) Interferograms of a KH2P04crystal dissolving in water (X200)
in Figure 3d. The near linearity of the fringes in this interferogram in the region of the subject verifies the uniformity of the crystal’s optical thickness. The displacement of the fringes in the decorated region of the crystal is in the same 2256
direction as that a t the crystal edge indicating an additional increase in optical thickness due to the decoration. In this manner, fringe displacements can be correlated with increases or decreases in the subject’s optical thickness. The analysis of the interferogram in Figures 2 and 3 demonstrates that the appropriate selection of the quantitative measurement technique (fringe counting or fringe displacement) is dependent on the optical thickness of the subject field. At times, both measurement techniques are necessary to give a complete qualitative and quantitative picture of the system under investigation. Figure 4 shows two interferograms produced by the near superposition of a reconstructed wavefront of a KHzP04 crystal dissolving in aqueous media with a reference wavefront. The reference wavefront is a reconstruction of a wavefront transmitted by a saturated aqueous solution of KHzP04.
ANALYTICAL CHEMISTRY, VOL. 48, NO. 14, DECEMBER 1976
a
(bl
b
Flgure 7. (a) Reconstructed image of a melt crystallizing polyethylene oxide spherulite (X300); (b) Interferogram of the melt crystallizing
Figure 6. (a) Reconstructed image of two thymol crystals growing from the melt (X200); (b) Interferogram of the crystallizing thymol produced by near superposition of wavefronts
polyethylene oxide spherulite produced by the near superposition of wavefronts
The fringes profile regions of constant optical thickness. The variation in optical thickness is due to concentration gradients in the solution produced by the dissolving crystal. A knowledge of the relationship between concentration and optical thickness would allow the quantitative determination of the concentration gradients in this system. The alignment of the two holograms was varied to produce the two interferograms found in Figures 4a and b. Each interferogram provides different information although the same two holograms were used in the interferogram construction. Details concerning the determination of concentration gradients with interferometry have been reported elsewhere (16). Figure 5 is a time differential holographic interferogram produced by the double exposure technique. This technique produces interferograms in which there is exact superposition of the reconstructed wavefronts. The subject is a thick preparation of a super-cooled thymol crystal growing from the melt. The observed fringes indicate the optical path changes accompanying crystal growth that occurred during the time interval between exposures. The observed fringe pattern is due to differences in the thermal or density gradients as crystallization proceeds. The gradients were accentuated by the choice of these experimental conditions. Melt crystallization has been studied previously by both classical (17) and holographic interferometric methods ( 3 ) . A single reconstructed image of thymol crystals growing toward one another with a melt zone between them is shown in Figure 6a. The interferogram in Figure 6b was produced by illuminating the combination of the hologram used to reconstruct the image shown in Figure 6a and a reference hologram of molten thymol. In Figure 6b, the wavefronts were nearly
superimposed producing the observed fringes. The parallel equidistant fringes are sharply displaced a t the crystal-melt boundary. The geometrical thickness of the entire field is constant; therefore the fringe displacement is due only to differences in the melt and crystal refractive indices. The reconstructed image of a melt crystallizing polyethylene oxide spherulite is shown in Figure 7a. Consideration of the fringes in Figure 7b, produced by near superposition of the reconstructed melt wavefront and the spherulite wavefront, shows evenly spaced parallel fringes in the melt and jittery fringes in the region of the spherulite. The jitter in the fringes is due to the optical inhomogenieties in the spherulite. A gradual fringe displacement is observed for the growing spherulite in contrast to the sharp displacement a t the thymol melt-crystal interface. The gradual displacement in the fringes is thought to be due to a continuous change from disordered melt to the ordered crystalline phase ( 4 ) .An additional explanation might be that impurities are being excluded a t the melt-crystal boundary by the growing crystalline phase ( 1 8 )or that thermal gradients due to the exothermic crystallization are present. Figures 8a and b are time differential holographic interferograms with exact superposition of reconstructed wavefronts of a polyvinylpyrrolidone particle absorbing moisture from a saturated water atmosphere a t room temperature. The double exposure procedure was used to ensure exact superposition of the reconstructed wavefronts. The double exposures in Figure 8a were produced at time zero and after 4 min in the water saturated atmosphere while those in Figure 8b were recorded a t 4 and 11 min. More fringes are found in Figure 8a than 8b even though the time interval in 8b is almost
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a
a
b b Figure 8. Time differential holographic interferograms of a PVP particle absorbing moisture, (a) 0 to 4 min, (b) 4 to 11 min (X200)
twice as long as that in Figure 8a. These interferograms indicate that the rate of moisture uptake by these polymer particles initially is large and decreases as the uptake proceeds. It was possible to determine when the particles had equilibrated with the moist surroundings by continuing to produce interferograms a t longer and longer time intervals. The absence of fringes indicates that the system is no longer absorbing moisture. (The dark bands around the particle circumferences are due to an optical artifact.) A nylon fiber in an oil immersion medium is the subject in the interferograms presented in Figures 9a and b. The interferograms were produced by the near superposition of the reconstructed wavefront of the fiber in immersion oil and a wavefront of the immersion oil alone. The fiber axis orientation is varied 90' with respect to the fringe direction. In Figure 9a, the fringes reveal the cylindrical geometry of the fiber. The jitter in the fringes intersecting the fiber is due to minor optical heterogeneities within the subject. The optical homogeneity along the fiber axis is demonstrated by the nearly straight fringes shown in Figure 9b. The application of classical interferometric techniques to the study of fibers has been discussed in detail by R. C. Faust (19). A variety of additional systems that are of chemical interest have been investigated with promising results. Included in these studies were the quantitative determination of relaxation of a cellulose acetate film after removal of a mechanical
2258
Figure 9. Interferograms of nylon fiber produced by the near superposition of wavefronts (X200)
stress and the study of the diffusion of a penetrant in a amorphous polymer film (20).
LITERATURE CITED R. F. van Ligten and H. Osterberg, Nature (London), 211, 282 (1966). G. W. Ellis, Science, 154, 1195 (1966). R. H. McFee. Appl. Opt., 9, 834 (1970). M. B. Rhodes, Appl. Opt., 13, 2263 (1974). W. H. Carter et ai., J. QuantumElectronics, QE2, 44 (1966). E. N. Leith and J. Upatnieks, J. Opt. SOC.Am., 59, 1545 (1969). H. M. Smith, "Principles of Holography", Wiley-lnterscience, New York, 1969. G. W. Stroke, "An Introduction to Coherent Optics and Holography", Academic Press New York, 1966. A. E. Ennos, in "Advances in Quantum Electronics", D. W. Goodin, Ed., Academic Press, New York, 1970, p 199. F. Waiter, Sci. Techn. Inform., Engi. Ed., 1, 1-28 (1964). N. Abramson, "Sandwich Hologram Interferometry", paper presented at April 1974 meeting of the Optical Society of America. G. Havener and R. Radley, Opto-Nectronics, 4, 349 (1972). J. W. C. Gates, Nature (London), 220, 473 (1968). K. Stetson (G. C. Optronics, Inc.), US. Patent No. 3,612,693 (1971). 0. E. Lau and W. Krug, "Die Aquidensitometrie", Grundlagen, Verfahren and Anwendungsgebiete. Berlin, 1957. C. Knox, R. Sayano, E. Seo, and H. Silverman, J. Phys. Chem.. 71, 3102 (1968). V. S. Doladuginaand E. E. Berezina, in "Growth of Crystals", N. N. Sheftal, Ed., Consultants Bureau, New York, Vol. 55, 1968, p 177. H. D. Keith and F. G. Padden, J. Appl. Phys., 35, 1270 (1964). R. C. Faust, Proc. Roy. SOC.London, Ser. A, 211, 240 (1952). R. F. Cournoyer, M. B. Rhodes, and S. Siggia, J. Polym. Sci., 13, 1023 (1975).
RECEIVEDfor review December 27,1974. Accepted July 19, 1976.
ANALYTICAL CHEMISTRY, VOL. 48, NO. 14, DECEMBER 1976
