by Alex Lempicki, C. Brecher, S. Miller

Introdution

The present report is a continuation of the description of our work on the transparency and light scattering in polycrystalline ceramics. The first report [1] dealt primarily with the concept of transparency and examined ways of quantifying this property. The driving force for continuing this investigation was therefore greatly influenced by interest in optical properties of ceramics (this time truly single phase, polycrystalline materials) because of their application as windows, armor, scintillators or lasers, making the subject of light scattering in these media of immediate importance. While ceramics made from isotropic materials are reasonably easy to produce with a high degree of transparency, the anisotropic ceramics represent the real challenge, often requir­ing consideration of nanograined materials and development of new consolidation technologies.

Continuing the study of transparency we made extensive use of a scattering goniometer, manufactured by TMA Technologies [2] , originally developed by J. Stover [3] and described more fully in Section 2. It soon became quite evident that a number of translucent and transpar­ent materials exhibit laser speckle when a laser beam traverses the specimen. This was an obvi­ous sequel to the work done first at Boston University in the 1990’s.

Speckle patterns are mostly associated with interference of laser light reflected from rough surfaces and have been described in voluminous literature as well as several books [4] , [5] . Speckle has been applied extensively to the study of surfaces and to interferometry, and is an important factor in astronomy, where atmospheric speckle degrades photographic quality. However, fluctua­tions in whatever bulk property of solid materials that can contribute to speckle have attracted very little attention. The subject is disposed by a three-page section of the book by Goodman, which provides only two references [6] to work describing the use of speckle for detecting a highly scattering region artificially embedded in a less scattering medium (white Plexiglas within a clear plastic). One other related observation is on diffraction of light by twinned domains in ortho­rhombic crystals [7] . An extensive discussion of the use of scattering data for the study of sur­faces is contained in the book by Stover [3] . References 3-7 appear to exhaust the main literature on the contribution of bulk effects to the formation of laser speckle.

It is somewhat surprising that the work performed at Boston University on bulk speckle pro­duced by glasses and glass ceramics in the early nineties [8] , [9] , [10] , [11] , [12] is not referenced in books and articles published at later times. Yet this work reported a detailed experimental and theoretical treatment of bulk speckle in glasses caused by distributed random fluctuation in refractive index. One would expect this work to have stimulated further exploration of the glass transition in glasses and amorphous semiconductors [13] by means of light scattering, but this did not happen.

Here we report a series of experiments that reveal the presence of bulk speckle in a much larger class of materials than had previously been thought. Indeed, our results suggest that any material that transmits light to some degree, scattering or nonscattering, will show some degree of speckle. In glassy materials the origin of this effect has been ascribed to frozen density fluctuations that are an inevitable consequence of the bulk disorder. This effect is exacerbated in glass ceramics, where the presence of both amorphous and crystalline phases introduces yet more disorder. But neither of these is likely to be present in single crystals, so that the root cause of speckle remains unknown.

We shall review preliminary results on laser speckle obtained on materials which to our belief have not been subject to such studies. The sequence will be as follows:

Glasses: This category of amorphous materials, (specifically CVD-made silica glass) has been the subject of work described in Refs. [8] - [12] . The source of speckle effects was attributed to density fluctuations frozen in the solid glass. These “defects” arise on a scale much small­ler than the wavelength of light and densities on the order of 1017 cm-3 [9] . No new results are reported here and the only purpose of repeating some experiments was to ascertain that measurements made using the Stover scatterometer are reasonably close to our older results.

Anisotropic (polycrystalline) ceramcs (APC): The difference between these materials and the glass ceramics dealt with in Ref. [12] is that there is no amorphous medium in which crys­tallites, of dimensions comparable to the wavelength of light, are embedded. In polycrys­talline ceramics, individual crystalline grains fill all the volume and their dimensions are usually considerably larger than visible light wavelength (single and double digit microns). Light scattering in APCs is predominantly caused by discontinuities of refractive index between differently oriented grains. It is quite conceivable that this could be the source of interference, leading to speckle. However the characteristics of such a coarse speckle could be quite different than that found in glasses. Also, it may be expected that speckle size or coherence areas would be related to grain size. This investigation carries a certain promise of leading to further insights towards improving the transparency of ceramics. To our knowledge no work has been done on this potential source of speckle.

Isotropic ceramics: It is well known that polycrystalline ceramics made of cubic materials can be prepared in a highly transparent state [14] , because the refractive index on both sides of a grain boundary is the same. True, the interface region itself may be a source of scattering, but, being much smaller than the wavelength of light, may contribute very little. Neverthe­less a question on ultimate transparency of isotropic ceramics is important for laser tech­nology: How transparent can a body resembling extremely fine fiberglass wool actually be?

Fig 1. Optical layout of the Stover scatterometer.

Single crystals: We define single crystals as ordered bodies that show no identifiable defects re­vealed by x-ray diffraction (twins, abnormal line broadening, presence of multiple phases). It is difficult to see what exactly is the source of laser speckle in these materials, and why has it not been reported. Some indication of the subtlety of matter is the observation of speckle in such optically perfect material as CVD silica glass, mentioned above. The conclusion may be that the scatterers may be so small and dispersed that they become difficult to observe by any other means.

Experimental

The basic optical layout of the Stover scatterometer is shown in Fig. 1. In this configuration the laser beam is focused on the aperture of the detector, not on the sample as in other instruments; con­sequently, alignment of the Stover in­cludes focusing the laser at the θ = 0 po­sition. The rotation of the goniometer arm allows scanning in transmission, from nearly -90 to +90 degrees. The forward (θ = 0) beam is not dumped, but is allowed to propagate straight through to the detector. The scattering volume is defined by the thickness of the speci­men and the cross-section of the beam at that point. The Stover instrument is not designed for measurement of such quantitative scattering values as Rayleigh ratios, especially in liquids.

Fig 2. Details of the detector assembly of the Stover scatterometer.

In the current work we used the Stover scat­terometer in this normal mode, with only small modifications. The most important of these is a provision to enable measurement of polariza­tion effects. The He-Ne laser is itself polarized in the vertical (s) plane, perpendicular to the plane (p) of the angular scan. This polarization direction, however, can be readily rotated to the horizontal plane by simply inserting an appropriately oriented half-wave plate between the laser and the specimen. Then, by placing a sheet analyzer directly in front of the detector, as indicated in Fig. 2, we can obtain scans under four different types of polarization conditions: ss, pp, sp, and ps, where the first letter indicates the polarization of the beam and the second the position of the analyzer. The first two, where the analyzer is aligned with the beam polarization, are termed parallel scans; the other two, crossed.

The dynamic range of the instrument is illustrated by its so-called signature, an angular scan without any sample present, shown in Fig. 3. The signal for θ ¹ 0 is thus due only to the wings of the (approximately Gaussian) beam, augmented by scattering from the instrument’s own optics, stray reflections from the sample holder and atmospheric dust. Note that the shape of the signature scan is essentially independent of the presence or orientation of the half-wave plate, and that where differences are observed (mostly eleven orders of magnitude down from the peak), they are random and not reproducible; this demonstrates that there is no inherent instrumental bias associated with the direction of polarization, and that any polarization-related differences that we observe (greater than the background level) originate in the specimen alone.

Fig 3. Two signature scans (no sample) from -10 to +10 degrees. Each scan has three distinct regions, with ver­tical arrows indicating angular width of pedestal. Note that the shapes are essentially independent of polarization.

We can distinguish three regions of the scan. The dominant feature is the forward peak, which contains some xx% of the total light crammed into a width of only about 0.03 degree, an arc length of about 0.25 mm. This spike is perched atop a broader feature spanning about ± 1 degree, which we call the pedestal; this is the region most sensitive to instrumental design [15] , because it is impossible to entirely elimi­nate scattering from the optics of the instru­ment itself [16] . Finally the wide angle re­gion, constituting the rest of the scan, repre­sents the averaged, wide angle, angular de­pendence of the scattering. Both the opti­cal aperture and the angular steps are auto­matically adjusted to cope with the wide dynamic range of the signal and to provide high angular resolution at very small angles without requiring an unacceptably long time to cover the full angular range.

We should note at this point that while the shape of the signature scan is essentially independent of the polarization of the inci­dent laser beam, it is most decidedly not in­dependent of the orientation of the analyzer. This is because in the crossed configuration the intensities of both forward peak and pedestal are reduced by more than three or­ders of magnitude. While this still leaves the former as the dominant feature, it re­duces the pedestal to the point where underlying instrument-related fine struc­ture can emerge.

This brings up another important point: Note that fluctuations in the signal appear to be greater and more frequent in the range of ± 4 degrees than in the rest of the scan. This is a direct consequence of the auto­matic adjustments in both the optical aper­ture and the angular steps that had been programmed into the full scan, as indicat­ed earlier. Thus it is only in this narrow angular range that both optical aperture and angular steps are small enough to enable resolution of the fine details. Over the rest of the scan, the pro­grammed measurement parameters essentially average out any closely spaced spatial fluctuations of intensity, as would be caused by speckle. Historically, speckle has always been viewed as an impediment to the primary purpose of the instrument, which is to measure the geometric depend­ence of light scattering [17] , and the Stover was deliberately designed with that in mind. Yet, as mentioned in Ref. [5] , it can still be used for observation of speckle, which is particularly useful for the study of surface roughness, by maintaining small apertures and angular steps over the entire range of interest. Consequently, we have chosen a standard set of conditions (which we call High Resolution or HR) under which transmission speckle is regularly seen. These condi­tions include a fixed aperture of 287 µm, a field stop of 800 µm and a constant angular step of 0.01 degree. And, having already seen that the pedestal region (±1 degree) is dominated by instru­mental effects, we have arbitrarily chosen to avoid that region entirely.

Fig 4. Signature scans with polarizer and analyzer in place, in both parallel (ss) and crossed (sp) alignments.

Most of our samples were rather thin (0.5 to 3 mm) disks, hence the scattering volume was represented by a cylindrical region of about 1 mm diameter (of the beam) and length given by the thickness of the sample. This, it must be emphasized, is quite different from the optics of the Brook­haven instrument [8] , which uses a laser beam focused inside the sample and a different set of pin­hole and slit to limit the size of the spot on the photocathode. Furthermore according to Ref. [18] thin disks do not provide the ideal geometry from the point of view of maximizing the S/N ratio.

Because of the similarity between speckle and noise traces it is important to prove that we are dealing with a genuine phenomenon. Thus we must examine the reproducibility of repeated meas­urements made under identical conditions. If the beam were moved to a different position be­tween scans, one should not expect any correlation between the two. The same lack of correla­tion would be expected if the observed fluctuations were strictly a matter of random noise. But speckle, being in our case a stationary process, requires that scans taken in succession, without moving the position of the beam, should be exactly reproducible. And indeed, as shown in Fig. 5, this is just what we observe. This test was performed on all the samples investigated in this study and they all behaved in the same manner. (An interesting experiment would be to calculate the correlation coefficient between scans taken at progressively greater displacement of the sample; this may be a method for estimating the size of the coherence area).

Fig 5. Two consecutive scans of a thin disk of trans­parent alumina, with beam at the same spot. Note exact reproducibility of detail in superimposed traces, demon­strating that the fluctuations are speckle, not noise.

It is not presently known if the theory of a fully developed speckle, outlined in Sec. 3 of Ref. [xx] is applicable to our case. If so, it would be useful to know the number (n) of speckle areas that are imaged on the photocathode. This can be estimated in two different ways. One is from the geometry, which leads to the approximate expression

n = AΩ/λ2 , (1)

where A is the spot size on the photocathode and Ω is the solid angle

Ω = A/R2 (2)

(R being the scatterometer arm length of 50 cm). We do not know the spot size on the photocathode because of the presence of the lens and field stop (Fig. 2) and will therefore assume that A is given by the area of the aperture, which amounts to 3.5 x 10-3 cm2. This gives a solid angle of Ω = 1.4 x 10-6, which in turn gives a value for n of A2/λ2R2 = 1.36 .

Another way to estimate n is from the contrast C of the speckle pattern, defined as the ratio of the standard deviation σ to the average mean amplitude of the speckle:

C = σ/<I> . (3)

The statistics of independent speckle intensities predicts that

n2 = 1/C , (4)

with a signal-to-noise ratio of <I>/σ = n1/2 .

Equations (1) through (4) are only approximate but they certainly support the identification of the rapid intensity fluctuations as evidence of laser speckle. To obtain the same value of n as above, the speckle contrast would have to be 0.54 .

For the initial experiments we chose a material in which speckle is expected to be present, namely a clear microscope glass. True, the specimens used in [8] - [12] were generally much thicker, typically 6 or 7 mm, while most of our present samples are 0.5-3 mm disks. Following a deri­vation of the speckle intensity [18] emitted into the angle Ω, the geometry of the scattering vol­ume of the Stover scatterometer (being a short but rather fat cylinder) is less propitious for the study of speckle than that of the Brookhaven. The dimensions of the scattering volume are de­fined by the thickness of the samples and the 0.5 mm diameter of the unfocused laser beam.

In what follows, all scans designated as high resolution (HR) were performed with a constant aperture of 287 µm diameter and a step size of 0.01 degree. Moreover, all the scans reported here were performed over the range of 2 to 5 degrees, which was a sufficient range to provide the numbers needed to characterize the speckle. In further work the relationship between the speckle and the wide-angle scans should also be investigated because the latter carry information about the nature of the scatterers.

Materials and Results

With the exception of “frosted glass”, all of the samples to be discussed here were in the form of disks or plates, polished on both sides by using a y grit. While this does not represent an ultimate optical polish, any surface contribution to speckle should be similar in magnitude from specimen to spe­cimen, and consequently too small to ac­count for the substantial differences in scat­tered intensity that are actually observed.

Amorphous: Glass

There is plenty of literature citing evi­dence that speckle is primarily caused by surface irregularities [3] , [4] . The problem of unambiguously distinguishing between surface- and bulk-generated speckle is rather difficult, as discussed in Ref. [5] . We decided therefore to start this investigation by comparing a clear microscope slide with one that had been ground on one side to be­come “frosted” (#F4 described in Ref. [1] , ground by hand with z grit). Although work on this particular type of glass was not in­cluded in the earlier work [8] - [11] , there is no reason to believe that it would have proper­ties drastically different from CVD glass. The four polarized scans are shown in Fig. 6 and the results summarized in Table 1.

First we note that the scattered intensity is much higher than in all other samples. The scattering is only weakly depolarized (sp/ss » 0.004; ps/pp » 0.0014), there being roughly three orders of magnitude differ­ence between the parallel and crossed scans. Also we note little correlation be­tween ss and pp traces. These two features are compatible and both indicate, as one would expect, an anisotropic character of the “defect” giving rise to speckle. In this case the damaged surface is clearly the dominant factor in the speckle.

Fig 6. Polarized scans of frosted glass plate (F4). Note major separation of parallel and crossed scans, indicating very little depolarization. There appears to be little cor­relation between parallel (ss and pp) scans.

Fig 7. Polarized scans of clear glass slide. For parallel scans (ss and pp), the signal is at least three orders of mag­nitude weaker than in Figure 6. Here again polarization is preserved, but now the crossed scans (sp and ps) are too weak to separate from the background and are not shown. Note the definite correlation of the two parallel scans.

A comparison with the scans obtained from a clear (undamaged) slide, presented in Fig. 7 and Table 2, tends to confirm this conclusion. As expected the intensity of the scattered light is much lower than in the previous case. The depolarization ra­tios (sp/ss = 0.0579; ps/pp = 0.0381) are still small, but not as small as for the frost­ed slide. Moreover, here the ss and pp do show definite correlation, indicating a change in the character of the defect toward lower anisotropy. This, as we shall see, appears to be a characteristic of a more ordered structure. It is interesting that the contrast ratio and the value of n are practically the same. The meaning of this observation is presently not understood.

Anisotropic Ceramics: Alumina, LSO

We now turn our attention to polycrystalline ceramics. One would expect these materials to have all the attributes to generate speckle in the bulk, yet this has not been investigated previously. We start with the archetypal ceramic of alumina which has consumed more research and develop­ment dollars than any other. The reason for this is that for many years it has been a material of choice for the lighting industry, as the envelope of arc lamps. In addition it is a favorite ceramic of the military for use as heat-resistant nose cones. In both thermal and mechanical strength it is superior to its crystalline form of sapphire. Currently, extensive work is being carried out in several laboratories, to further improve the transparency of alumina.

Fig 8. Polarized scans of thick alumina ceramic disk. All scans overlap, indicating complete depolarization. No correlation is seen.

Fig 9. Polarized scans of highly scattering LSO ce­ramic. As in alumina all scans overlap, indicating high depolarization. While the fluctuations are fully repro­ducible, there is little correlation between them.

Among optically important ceramics, alumina is very well positioned for achieving transpar­ency because its crystal structure is uniaxial, whose two refractive indices differ by only 0.008 in the visible. This is an exceptionally low number, which allowed its light scattering properties to be mathematically modeled according to the classical theories of the Rayleigh-Gans type, totally disregarding multiple scattering [19] . From our point of view alumina is of great interest since it is the first polycrystalline ceramic in which speckle has been investigated.

The scanning curves (Fig. 8) and ac­companying Table 3 show a radical differ­ence from the glass. The contrast is now much lower and the depolarization almost complete, (sp/ss =0.745; ps/pp = 0.616). This we claim provides the most compel­ling proof that the speckles we observe are due to the bulk of the material. The sur­face of the alumina sample is certainly no better than that of the clear glass slide, and yet the speckle patterns are very different. Moreover, the virtually complete depolar­ization would indicate a “defect” quite different and far more isotropic than that encountered in glass.

To examine this further we chose a dif­ferent polycrystalline ceramic, namely Lu₂SiO₅:Ce, otherwise known as LSO, cur­rently being developed [20] as a scintilla­tor for medical imaging. Unlike the alu­mina, LSO is a biaxial crystal with higher anisotropy (∆nmax = 0.028) and thus far from transparent at this stage of develop­ment. The scattering scan is shown in Fig. 9 and the corresponding data in Table 4.

The scattering is again completely de­polarized (sp/ss =1.003; ps/pp = 0.969), the contrast further reduced and n signifi­cantly increased. These two examples in­dicate that there may be a common thread in anisotropic ceramics, even as different as alumina and LSO.

Isotropic Ceramics: Lu₂O₃

Fully transparent ceramics made from materials having a cubic crystal structure have been known for some time. One such material, lanthana-strengthened yttria, has shown particularly attractive properties as IR-transmitting domes and windows [21] . Other lanthanide oxides doped with Eu, Ce and Pr, have been developed as medical scintillators [22] , the latest being Lu₂O₃:Eu [14] . Scans and relevant data for the latter material are presented in Fig. 10 and Ta­ble 5. Here we see that the speckle is not as depolarized as in the cases of alumina and LSO (Figs. 8 and 9) and the contrast ratio is significantly larger. Correlation between ss and pp scans, while not as strong as in the other transparent specimens (clear glass, Fig. 5; sapphire, Fig. 11), is consider­ably greater than in the other ceramics.

Fig 10. Polarized scans of isotropic (cubic) ceramic Lu2O3. Not as depolarized as alumina or LSO but also showing correlation between ss and pp scans. As in Figure 7, the crossed scans (sp and ps) are too weak to separate from the background and are not shown.

Fig 11. Polarized scans of a single crystal sapphire disk. Very highly correlated parallel scans (ss and pp). Crossed scans too weak to extract from background, in­dicating little or no depolarization.

Single crystals: Sapphire

To cap this survey we finally give an example of speckle in a single crystal of sapphire, which can be viewed as the fully ordered counterpart of alumina ceramic. The results on a plate of sapphire crystal are presented in Fig. 11 and Table 6.

Conclusions

The experiments described in this re­port invite the generalization that any solid that transmits light, whether transparent like single crystals or isotropic ceramics, mildly scattering like the alumina speci­mens, or very highly scattering like LSO ceramic, does produce laser speckle, with the evidence pointing unmistakably to the bulk as the source of the interference. We are led to this conclusion by observing that samples with optically polished surfaces still show pronounced speckle. Furthermore we have shown that even glass whose sur­face has been brutally damaged by coarse grinding continues to show speckle almost as strongly polarized as in unabraded sam­ples, although correlation between parallel scans has been essentially obliterated.

Although the examples described in this survey were chosen arbitrarily simply to illustrate the speckle effect, we still perceive some commonality in behavior. For one thing, the presence of structural correlation between vertical and horizontal parallel scans appears to follow transparency; the more scattering the material, the less correlation is seen. But scatter alone does not destroy polarization, as measurements on the abraded glass specimen demonstrate; it is only when the material is both scattering and optically anisotropic that we see a high degree of depolarization. Of course, these generalities stem from measurements on only a limited and eclectic population of materials, and whether they will hold as definitive rules remains to be seen.

As for the origin of the observed speckle, it is reasonable to assume that it is in some way connected to random variations of the refractive index. Whereas in glasses this has been attributed to frozen density fluctuations and in ceramics to refraction at grain boundaries, these are unlikely to play much of a role in isotropic ceramics or single crystals. In the latter especially, it is difficult to see how even the wide range of point defects that must be present in the lattice on thermodynamic considerations could generate refractive index fluctuations large enough to account for the speckle. However the facts seem to speak otherwise, and the nature and source of the fine structure of the refractive index that gives rise to speckle remains to be determined. Per­haps it is so fine that it can be detectable only by the interference effects of coherent light. What­ever is the nature of this fine structure, it must be on a scale that does not interfere with the opti­cal properties that we collectively call “transparency” or “optical quality”, since speckle is seen even in high quality glasses (CVD silica) and single crystals. It is therefore our opinion that the role of density fluctuations as the primary source of speckle in glass may have to be revised.

Another observation which will merit further attention is that no definitive correlation is evi­dent between speckle and certain prominent scattering features, such as the presence or absence of a forward peak. It seems that if any light gets through, speckle will follow!

Lastly (and perhaps most importantly), we feel that the discovery of bulk speckle from single crystals opens a new area of investigation and that further work on ceramics should not neglect information carried by speckle.

References