by Alex Lempicki, C. Brecher, S. Miller

Introdution

How often do we see that the same type of question pops out in very different and seemingly unrelated fields of Physics? This only attests to the unity and universality of the discipline in which there are no observations or facts that stand alone. Everything is connected to everything.

We will deal with the concept of transparency, whose meaning is clearer, let's say even more transparent, in everyday language than in science. Witness the optics terminology invading language with terms such as "transparent policy", "opaque statement", "obscure statement", "clear statement" even, "photographic memory". We know perfectly well what they mean! However the scientific use is far less clear or transparent. Suppose we encounter objects that are definitely not quite transparent, such as a foggy window, lightly frosted glass or a thin cloud, we are immediately faced with questions, such as what is the degree of transparency, does it have (clear) physical definition, how do you measure it? At this point the scientific language itself becomes fuzzy and terms such as translucency are introduced and we are well on the way towards obscuration. In other words, transparency is a little like pornography, which, as the Supreme Court declared: "We can't define what it is but we know when we see it".

The situation is only slightly improved by consulting Webster's Dictionary, which provides a correct but still qualitative definition of transparency as "having the property of transmitting light without appreciable scattering, so that bodies lying beyond (our emphasis) are entirely visible". The concept of beyond, not very clear from this dictionary definition, is one of the subjects of this paper.

The issue of quantifying transparency has recently surfaced in several areas, as apparently unrelated as heat-seeking missile domes [1] , ceramic lasers [2] and medical scintillators [3] , [4] used as detectors for gamma or x-ray diagnostic imaging. It so happens that these diverse applications have to some extent turned to the use of ceramics as materials capable of better fulfilling their needs: in the case of missile domes, for superior resistance to thermal shock; in medical scintillators, because of the difficulty and cost of growing single crystals of the desired composition. In both of these diverse applications, transparency plays a very important role and therefore the subject has taken on some urgency.

There appear to be two effects that limit the transparency of an object, namely absorption and scattering. While absorption attenuates the transmitted image it does not affect its quality (resolution), while scattering contributes to the blurring of the image, it does not reduce the total light intensity passing through a scattering object. We want primarily to deal with the question how scattering limits transparency, since the role of absorptive attenuation is well understood and quantitatively expressed by optical density.

Fig 1. Back-lighted text pattern, viewed through glass disk with abraded surface at various distances from pattern. Note progressive decrease in legibility as function of separation.

Transparency Introduced

The literature of scattering in materials such as ceramics is full of qualitative, loose and plain wrong statements using terms like transparency, translucency and in-line transmission. It is a common test for distinguishing between translucency and transparency of a ceramic plate, by placing it in contact with a pattern (maybe simply print) on top of a light box. If the sample is sufficiently translucent, then it is quite possible to read the print. However if one lifts the sample a few millimeters above the surface, the pattern becomes totally obliterated. This effect has been recognized by Peelen [5] and is illustrated in Figure 1, using clear glass disks that have been mechanically abraded on one side. Here we see that the legibility of the print is strongly related to the distance separating the plate from the pattern. A specimen can be considered fully transparent only if this distance can become arbitrarily large. This clarifies the statement in the Webster definition of transparency, quoted above.

In order to understand how scattering is responsible for the gradual obliteration of the pattern at sufficiently large distance between source and scatterer, we must examine the pattern of multiple scattering in more detail. For this purpose we shall use the optical layout of Figure 2 in which a parallel laser beam of light enters the scatterer of thickness t at point P. In passing through the sample it acquires a divergence due to scattering, which we may characterize by an angle α. Exiting from the plate, refraction causes an additional spreading of the cone. The concept of a cone is of course an approximation due to the forward peaked nature of the scattered intensity. In reality the emergent scattered light fills up a solid angle of 2π (or 180 degrees in planar view). The diameter of the cone may be arbitrarily defined by the amount of light contained in the cone, such as its FWHM, its 1/e or 90% value. Independent of the particular definition we arrive at cross-sections of the cone, schematically represented by the shaded circles or halos in Figure 2. At successive distances from the scattering plate (0, d, 2d and 4d), the halo grows in area. Thus the scattering is shown to transform the original image, being the cross-section of the laser beam at the entrance to the scatterer, into an ever-expanding halo.

If we add a second laser beam entering the sample at a distance r from the first beam we obtain a superposition of two partially overlapping Gaussian beams, shown in Figure 3. The resulting profile becomes very similar to the intensity profile at the Rayleigh limit of resolution imposed by diffraction, treated in most optics books [7] . In both cases there is a certain arbitrariness as to how small r can get before it becomes impossible to resolve the two entrance points of the two beams. It is clear, however, that at some distance d between the scatterer and the detector, the two cones will completely merge, with essentially no minimum between them. We can already sense that this is what is behind the obliteration of the print pattern when the scatterer is lifted above the light box. To complete the picture we must show that moving the detector away from the scatterer is essentially equivalent to moving the scatterer above the light box.

Let us consider Figure 4. A point source is located at P and the detector (observer's eye) on the other side of the plate or scatterer S. If the plate is non-scattering the eye receives light from one direction, along the line joining them. If the plate is scattering then the direction from which the eye can receive the light remains unique only when d = 0. For d ≠ 0 receiving from other directions is possible. Consider the light emanating from P and striking the scatterer at P'. It will emerge from the scatterer over a range defined by a cone having an apex angle α. The light propagating along the edge of the cone can still reach the eye, which when focused on the plane of P will appear to the eye as coming from the "perimeter" of a fuzzy halo having a radius r. The size of the halo will of course grow with the distance d. If a second point source is also located on the P plane there will come a distance at which the respective halos overlap too much for the eye to discern the presence of two point sources. Using this reasoning the pattern of print on the plane P will no longer be visible.

Fig 2. Schematic diagram of scattering. A narrow parallel beam of light enters a scattering material at P, is scattered within the bulk, is refracted upon exit and its profile scanned by a detector placed at three distances d, 2d and 3d from sample. The cross-sections of the 'cone' of light are indicated by the shaded circles.

Fig 3. Scanning profile (similar to those of Figure 2), but obtained when two laser beams separated by a distance r, enter the scattering plate. Note resemblance to well known Rayleigh resolution criterion due to diffraction.

Fig 4. Layout of scattering corresponding to 'nude in the shower' effect (see Ref. [6]).

We see that this reasoning does not require anything more than what has been described previously, where the variable distance was that between the scatterer and the detector. The phenomenon of loss of resolution with distance in the presence of a scattering medium is sometimes referred to as "nude in the shower" and beautifully described by Bohren [6] . If the shower has scattering walls then the hand of a nude placed against the wall is perfectly imaged, while the body being further from the wall is only visible in outline.

Since tests such as the one illustrated in Figure 1 are constantly used in the literature on the transparency of ceramics, it is important to understand their background. Another frequently quoted test is the measurement of "in-line transmission". For this purpose the sample, again illuminated by a laser beam, is placed near the entrance slit of a monochromator and a detector at the exit. If the monochromator has a small acceptance angle, the measured light intensity corresponds to the predominantly forward-scattered light. It is generally assumed that the larger that signal, the more "transparent" is the sample. Although this is presumably true, we feel that a more detailed understanding of the scattered light distribution should be reached.

Scattering of Light

As we have said, because of scattering a linear laser beam directed at a ceramic plate emerges with components distributed over the full solid angle of 2α. The intensity distribution, however, is not uniform, being more concentrated in the forward direction. This is true regardless of the nature of the individual grains that make up the ceramic, even if the grains themselves are anisotropic. Since the grains are randomly oriented, the average optical properties are identical no matter which direction the light travels. The random orientation of the grains prevents any particular direction from having unique characteristics. This has an important consequence for the study of light scattering by ceramics: because of the isotropic nature of the bulk ceramic, it must have a circular symmetry. We can therefore obtain all the information about the scattered distribution by angularly scanning the intensity in any plane containing the axis of the incoming beam. The resulting curve P(Θ), which we shall call a scattering scan, is proportional to the so called "phase function", a term sometimes used in the literature [8] , but having nothing to do with the phases of the light.

Fig 5. Schematic diagram of the Stover scatterometer.

A particularly useful tool for this kind of study is the Stover scatterometer [9] . The essential functions of the instrument are illustrated in Figure 5. A laser is used as the source of light, which can be focused at any point along the horizontal axis. The sample S is located at the center of a precision goniometor. A solid-state diode detector D is located at the end of a rotating arm. The optics and detection electronics of the instrument allow an extraordinarily large dynamic range of the measured signal, spanning some 14 orders of magnitude. This is accomplished by synchronous detection of the signal and automatic control of an aperture placed in front of the diode. The angular step of rotation can be set constant, or can be automatically controlled by the changes in the signal.

Figure 6 gives the results of an angular scan without any sample in the path of the laser beam, plotted on both linear (right) and logarithmic (left) scales. This is called the signature and gives information about the measurement limits of the instrument. The linear scale lets us quantify the angular resolution of the scatterometer (FWHM 0.03 degrees), but provides no information on the background at larger angles. For this we need the logarithmic scale, which also illustrates the dynamic range. Note that the resolution of the scatterometer is only about half that of the human eye, which is 0.016 degree [10] . The finite resolution and the spreading of the signature curve at large angles, visible only on the logarithmic scale, are due to the characteristics of the laser beam as well as presence of dust in the room and possibly some stray reflections in the instrument.

Fig 6. Signature scans plotted on linear (right) and logarithmic (left) intensity scales. Notice dynamic range of nearly 8 orders of magnitude and resolution of 0.03 degree.

Fig 7. Comparison of signature and fused quartz scans. Note nearly identical forward peaks but difference at larger angles.

Fig 8. Comparison of scattering in various specimens. Note presence of narrow peak in the forward direction, in all except frosted glass. (The x-axis is scaled as the square root of angle.)

Figure 7 gives a superposition of signature and a scan using a plate of optical quality fused quartz. Note that in the forward direction the curves are practically identical as far as amplitude, width and overall shape. At larger angles, however, the signal intensity from the quartz is many orders of magnitude higher; thus scattering is found even in optical quality material.

Figure 8 shows a superposition of several scans obtained using different materials and illustrating the variety of shapes that are encountered. Here we have scaled the horizontal axis as the square root of the angle, to allow display of a larger angular range without losing the small-angle details. Note that only one material, a sample of frosted glass, does not show a prominent peak in the forward (zero degree) direction. All the others, including a high quality plate of fused quartz and three birefringent ceramic samples of various degree of transparency (GOS, hafnate, and alumina ceramics), show forward peaks which, despite widely ranging intensities, have nearly the same width (on the order of +/- 1 deg). As we remarked in the case of Figure 3 the FWHM of the forward peaks are all instrument-limited and individual differences, if indeed there are any, are not observable with this instrument. Note also that while there is substantial variation in the angular profile of the scattered light (i.e., at angles greater than about one degree), the range of intensities is much less we see in the forward peaks.

Fig 9. Photograph of a transparent ceramic alumina disk ~2 cm in diameter and 0.8 mm thick.

Fig 10. Angular scattering scan P(Θ) of the alumina ceramic shown in Figure 9, and its spatial integral I(Θ). (As in Figure 8, the x-axis is scaled as the square root of angle.)

We now wish to examine what immediate information we can obtain from the scattering scans. As a first example we choose a plate of alumina ceramic made by CeraNova Corp., about 2 cm in diameter and 0.8 mm thick. Its appearance to the naked eye is slightly foggy with a reddish cast. There is no question of its transparency as Figure 9 shows. The scattering scan P(Θ) of this sample is shown in Figure 10. Included with the scan, the instrument provides a numerical figure for the so-called Specular Transmission (ST = I_t/I₀) and Specular Reflection (SR = I_r/I₀). From the ST and thickness we obtain the absorption coefficient α = 1.26 cm^-1. The ST of the signature scan is defined as unity.

The next useful piece of information is the spatial distribution of the scattered light. Since the experimentally measured P(Θ) function provides the angular distribution only in the horizontal plane, we must perform an integration over a full circle (=360�) around the laser beam itself, which is defined as the z-axis. This converts the planar angle ? into the vertex angle of a cone centered on the laser beam. Such an integration is permissible because of the aforementioned average isotropic nature of the ceramic bulk:

where r is the radius of the detector circle and the angles correspond to the standard definitions in spherical coordinates. The integral I can then be numerically calculated as a function of the upper limit of Θ, (neglecting the proportionality constant 2πr²), as also shown in Figure 10. The consecutive figures along the line indicate the value of I(Θ) up to that point. Hence the area under the forward peak is 15.57 - 2.58 = 12.99. This value divided by the total area from -60 to +60 degrees (18.10) indicates that the forward peak corresponds to 71.7% of the total light transmitted by the sample into a cone with an apex angle of 120 degrees, or practically the entire forward hemisphere.

Some "Measure" of Transparency.

We have noted above (see Figure 7) that materials that are perfectly transparent, such as quartz, and partially transparent, such as alumina ceramics, exhibit a narrow forward peak in the scattering scans. We also noted that the width of the peak is limited by the resolution of the scatterometer. As long as this is the case, we can make the conjecture that the amplitude of the forward peak can be taken as a relative measure of transparency. Again referring to Figure 7 we can state that the transparency of fused quartz is greater than that of the alumina window, which in turn is greater than that of the hafnate. As has been stated in the literature [5] , it is not unreasonable to suggest that the fraction of the light contained in the forward peak goes through the material without being scattered and without a random shift in phase. Thus a progressively decreasing fraction of light traverses the specimens without scattering, the loss probably being compensated by an increased scattering at larger angles. All of this is of course valid only when there is no absorption.

A convenient measure of transparency may be the fraction of light contained in the forward peak. Aside from the practical value of such a measure the existence of a pronounced forward peak in evidently not fully transparent ceramics (such as illustrated in Figures 8 and 10) raises several interesting points. .How this is possible in anisotropic alumina remains to be answered by theory. If this were true, then one would expect that in the forward direction there would be no attenuation of light with increasing thickness, again assuming no absorption.

A further curious possibility is that the existence of the forward peak is a necessary condition for the existence of "snake light", or light transmitted through a scattering medium on the time scale of picosecond pulses, investigated by Alfano et al. [11] . This would mean that frosted glass samples, which show no forward peak, should not exhibit any snake light.

Separation of Transparency and Absorption.

Both scattering and absorptive loss result in the attenuation of a light beam in the forward direction, absorption through conversion to heat, scattering through redistribution to large angles. When both are present, separation of the two losses is a perennial and rather difficult problem. There does not seem to be simple and completely trustworthy experimental procedures for this task.

Fig 11. Scattering profiles of four neutral density filters, differing by an order of magnitude in transmission.

Fig 12. Scattering profiles of four frosted glass samples, abraded on one side. Note that here the scattering wings do not scale with the forward peaks, which in one case (ED-8) is totally absent.

To illustrate the effect of absorption on scattering scans, we have chosen four neutral density filters whose absorption at the He-Ne wavelength increases by an order of magnitude (NG3, 1 mm; NG3, 2 mm, NG3, 3 mm and NG1, 1 mm). The scans shown in Figure 11 indicate that the amplitude of the forward peak decreases stepwise by an order of magnitude, as may be expected. In fact the entire curve is proportionately depressed each time, although the shapes of the scattering "wings" (beyond +/-0.1 degree) are distorted by instrumental effects. We surmise that in this series of filters the only change is in the absorption.

To illustrate the effect of changes of scattering only, a number of glass microscope plates were abraded on one surface for different amounts of time, and therefore increased scattering. The progression is indicated by the numbers 1 through 7. We also measured a commercial plate of ground glass obtained from Edmunds Scientific (designated as ED-8). Figure 12 gives small angle scans of plates 1, 4, 6 and ED-8. It is interesting that (unlike what we see in Figure 11) the intensities of the scattered light do not scale with the forward peaks, so that at only 1 degree away from the normal, specimen ED-8, which has absolutely no forward peak and no transparency, shows the same light intensity as the most transparent plate F1. These examples illustrate how difficult it is to disentangle the separate effects of absorption and scattering when changes are occurring in both properties.

Fortunately other methods exist that may be helpful in this respect. A common method is to use an integrating sphere in a manner illustrated in Figure 13, Case A. First a laser beam is allowed to enter the sphere and a quantity I₀, proportional to the beam intensity, is measured by the PMT. This is sometimes referred to as calibration of the sphere. Next the plate is placed at the entrance to the sphere, and if the scatterometer scans of the plate indicate the presence of a forward peak, an opening is provided in the sphere to let that portion of the light out of the sphere. The remaining signal Is provides a measure of the total light scattered by the plate into a 2π solid angle. Finally, to measure the absorption, we place the plate in the path of the beam in a closed sphere. This measurement provides the quantity I_s - P, where P is the light absorbed.

The problem with this procedure is that it lumps together the absorption of the forward component, which has traveled directly through the plate, with the absorption of the scattered light, which may have traveled a much longer and circuitous path before emerging. As a result it does not lead to a unique value of an absorption coefficient. Nevertheless the method can be used, with the omission of step (3) if the absorption is known from other experiments.

A second method (Case B) is based on the absorption of diffused light within an integrating sphere. The basic concept appears to have been applied for the first time by Rhower and Martin [12] , in connection with an absolute measurement of quantum efficiency in luminescent materials. We shall describe here a simplified version contained in a bulletin of Labsphere Corp. [13] and directly adaptable to the measurement of absorption in a scattering sample.

Fig 13. Schematic diagram of an arrangement for measuring absorption of a scattering plate by the use of an integrating sphere.

As before, the first step in Case B is a calibration of the sphere, which yields I₀ The next step, however, is to introduce into the sphere a sample whose absorption is to be measured, but not in the path of the beam. The signal measured will be I₁= I₀ - uI₀, where u is the reduction of the diffused radiation due to the presence of the sample. This procedure is slightly idealized because in the presence of the beam, the radiation density in the sphere is not necessarily ideally uniform. Finally, we move the sample into the beam and again measure the signal. Here we will obtain an ordinary exponential attenuation of the beam, by a factor (1 - A), where A is the attenuation in the sample. Since we do not know the average path length of the photons traversing the sample, which in general will be larger than its thickness d, we can only relate A to an effective optical density U, by using the relation A = exp(-U), or an effective absorption constant a = U/d. All the experiments described so far are of course limited to the wavelength of the laser source.

One more method, used by Electro-Optics Technology, Inc. [14] , consists of a measurement of scattering, using a low power laser. The measurement of absorption is performed using a higher power laser and observing a lensing effect caused by a radial temperature gradient due to the heating of the medium. This ingenious method is based on work of Andrade and Innocenzi et al. [15] . In principle it also suffers from the same criticism that absorption of the scattered light is not taken into account. From these examples it is seen that the disentanglement of scattering and absorption is not a trivial problem if we want to relate it to basic material quantities, such as cross-sections for absorption and scattering.

Transparency and Optical Transfer Function.

Image formation is an important and venerable part of optics. Some books state wrongly that it has been inherited from electromagnetic communication theory, whereas maybe the converse is closer to the truth. It is not surprising that it all started with Rayleigh giving a mathematical formulation to Abbe's theories of image formation in a microscope. It became clear to these gentlemen, over a century ago, that all the elements of an object, contribute to one single element of the image. Because of the wave nature of light, there is no strict one-to-one correspondence between elements of object and image. This is due to diffraction, which in principle extends the effects of one image element to infinity. There are simply no optical elements (lenses, apertures, etc.) that are free of diffraction and aberration, which invariably contribute to the blurring of the image.

It appears that adding scattering trivializes the problem and overshadows all the other effects. This is undoubtedly the case, but the methods of taking into account the image degradation process remain the same. The analogy between an optical system and a filter in electrical engineering becomes sometimes useful, while both are treated as linear systems.

As we have seen in Section 2, an optical image tends to be blurred by scattering, which is quite analogous to diffraction but on a much stronger scale. Thus the blurring of a point light source will be described by a point spread function (PSF); a blurred slit by a line spread function (LSF) and a blurred edge by an edge spread function (ESF), each time the spreading being determined by the scattering properties.

Fig 14. Dependence of modulation of output signal upon blurring and frequency. (a) Ideal system, no blurring: Mod=1. (b) Same frequency as (a), but system introduces some blurring: Mod less 1. (c) Increased frequency causes further reduction in modulation: Mod less less 1.

The quantity that provides a measure of the degradation, or loss of resolution, of a pattern is called the Modulation Transfer Function (MTF). Suppose the object is a pattern consisting of a sequence of black and white lines as shown in Figure 14a. The frequency of the pattern is defined by the number of line pairs (one black, one white) per unit length, usually a millimeter. Suppose now we want to observe the pattern, forming its image on the retina or on photographic film. If a system existed that would not introduce any blurring at any frequency of the pattern, then its contrast, defined as the modulation (amplitude in / amplitude out), would be unity as shown in Figure 14a. As soon as there is some blurring the contrast is reduced to that of Figure 14b. An increase of frequency further reduces the contrast as shown in Figure 14c. The Modulation Transfer Function (MTF) quantifies this dependence of modulation on pattern frequency.

The MTF gives a full description of the resolution of the system, and tells us how small the details of the object can be and still be observed. Methods of obtaining MTFs are described in a vast literature and do not have to be repeated here in detail. The principle is first to obtain information on the degree of blurring of a sharp feature of the object (such as a point source, a slit or a sharp edge), on the basis of their respective Spread Functions. Slit or Edge Spread Functions are experimentally preferred, being related in that the former is the derivative of the latter:

SSF(x) = d/dx(ESF(x)), or conversely, ESF(x) = _-∞ ∫^∞ SSF(x')dx' .

Suppose ESF is obtained experimentally by scanning an edge with a narrow slit or using an array of sensors (such as a CCD camera). The derivative is then computed to obtain the SSF, upon which a Fourier Transform is performed. The transform reveals all the frequencies (and their amplitudes) that are responsible for the blur. The assumed linearity of the system assures that each sinusoidal term in the input is reproduced in the output without distortion, but different frequencies are attenuated by different factors. In general an overall waveform, which represents the output, will be degraded (blurred) from that which represents the input. The curve representing the attenuation of the various frequency components is again the Modulation Transfer Function, described above.

Fig 15. MTFs of frosted glass plates 1 mm thick, sandwiched between the razor edge and the CCD detector. On the left (a) the abraded surface was in contact with the blade and therefore 1 mm from the detector, while on the right the abraded surface was in direct contact with the detector.

It is now becoming evident that angular scattering scans and MTFs provide complementary information on the process of scattering. To illustrate both methods, we wish therefore to describe the MTFs of some of the same objects that have been subjected to angular scans. We chose the frosted plates of Figure 12, assuming again that the only difference between them was the degree of scattering. To obtain MTFs of the ground plates we placed each specimen in turn onto a CCD array with the abraded surface up, upon which we then placed a razor blade. A yellow LED light source illuminated the assembly for 200 ms and shut off during CCD readout. The details of the measurement are described in a publication by Samei, Flynn and Reimann [16] . Figure 15a shows the MTFs of the five slides, one clear and four frosted. We see a rapid degradation of the MTF with more pronounced grinding. It should be remembered that the ground surface was 1 mm above the edge (thickness of the slide). If we now reverse the slides so that the ground surface is in contact with the detector we obtain a set of essentially superimposed curves, as shown in Figure 15b, where we see that the MTF is now independent of the amount of grinding, and indeed virtually the same as that of the fully transparent (clear) slide. This holds true even for the ED-8 slide, where the complete absence of any forward peak (see Figure 14) would appear to rule out any imaging capability whatsoever. The reason for this somewhat puzzling behavior is that in the ground glasses the scattering layer is very thin, being just confined to the surface, so that the halo of the exiting light (see Figure 2) must indeed be of very small diameter. The conclusion must therefore be that a scattering layer, even if it completely eliminates transparency, can still maintain high resolution, provided that the distance traversed by the light, both within the scattering medium itself and between the scatterer and the detector, is small. This observation may have a bearing on some medical diagnostic imaging, where the detector is placed in contact with the organ being investigated (e.g., dental or mammographic imaging).

where we see that the MTF is now independent of the amount of grinding, and indeed virtually the same as that of the fully transparent (clear) slide. This holds true even for the ED-8 slide, where the complete absence of any forward peak (see Figure 14) would appear to rule out any imaging capability whatsoever. The reason for this somewhat puzzling behavior is that in the ground glasses the scattering layer is very thin, being just confined to the surface, so that the halo of the exiting light (see Figure 2) must indeed be of very small diameter. The conclusion must therefore be that a scattering layer, even if it completely eliminates transparency, can still maintain high resolution, provided that the distance traversed by the light, both within the scattering medium itself and between the scatterer and the detector, is small. This observation may have a bearing on some medical diagnostic imaging, where the detector is placed in contact with the organ being investigated (e.g., dental or mammographic imaging).

Fig 16. Photographs of razor edge as viewed through frosted glass plate at various distances from the edge.

The effect of distance upon the capability to perceive images through scattering media is a recurring theme. It is important to recognize that while the degree of scattering may be a material property, the imaging performance most decidedly is not. This is seen particularly graphically in Figure 16, where we present photographs of a razor edge as viewed through one of the frosted glass specimens, where the specimen is positioned at various distances from the edge. Thus distance between scatterer and detector exacerbates any image degradation caused by the frosted surface. Consequently, if the MTFs were measured with the plates positioned 1 mm away from the detector, the set on the right, taken with the abraded surfaces facing the detector, would no longer superimpose, but would closely resemble Figure 15a, while the set on the left, with the abraded surfaces now 2 mm away, would show even poorer resolution than before.

Conclusions.

We have attempted to give some meaning to the concept of transparency as distinguishable from translucency. Unfortunately, it is not possible to simply quantify transparency. Comparisons can meaningfully be carried out between objects completely free of absorption, and of the same thickness. The forward peak appears to be a necessary condition for any transparency, but its magnitude is influenced both by absorption and by degree of scattering (Figure 13). The unambiguous separation of absorption and scattering is not possible in the absence of knowledge of the scattering mean free path.

The nature of the forward peak still remains somewhat of a mystery. We have seen from Figure 8 that while its amplitude may vary over some four orders of magnitude, all of the chosen ceramic samples are definitely transparent to some degree. In each case the simple test of observing a distant object by looking through the sample is fulfilled. The differences in amplitude of the forward peak are most reasonably attributable to the redistribution of the scattered light, and it is highly unlikely that absorption is in any way responsible. If indeed the forward peak represents the fraction of the light that passes through the medium without being scattered, it is puzzling how it can traverse few hundred grains of a highly anisotropic ceramic like GOS without spreading.

The deeper problems of the transparency of ceramics have not been discussed in this paper. It is likely that attempts to improve the transparency must involve consideration of grain size and the fact that ceramics represent an upper limit of disorder, since the scatterers are not widely dispersed in an otherwise isotropic medium (which is a commonly-used approximation [17] ), but rather in intimate contact with each other. It is also likely that intensity fluctuations discovered by us some 20 years ago in glasses and glass ceramics [18] , [19] will, if present, have a different cause from the density fluctuations of the frozen liquids. We expect that much will be learned from high-resolution scans of ceramics, as well as from the shapes of the scattering curves at angles larger than those corresponding to the forward peak.

We also believe that coherent back scattering, on which we have not touched at all in this paper, can provide other important information, such as the relation between the photon mean free path and the grain size of the ceramic. Interestingly when coherent back scattering and photon localization emerged as analogs of Anderson electron localization in the last decade of 20th century, no experiments were reported on ceramics, which seemed to be ideal media for such a study. We are aware of only one work on this subject, a chapter in an unpublished doctoral thesis by J.-J. Chen of Boston University in 2000 [20] .

References