2.1.

Experimental Setup

Several goniometer setups are described in the literature. Those involving cameras as detection systems^{13, 22} have a limited solid angle of detection and a small dynamic range, but work with short acquisition times. To increase solid angle and dynamic range, mechanical scanning setups are used, at the price of increased acquisition times. ^{8, 9, 10, 11, 12, 13, 14, 15} These systems mechanically scan the detector around one axis. In this study, a two-axis goniometer was designed to measure the phase function with its full elevation and azimuthal angle dependence at high angular resolution combined with a high-dynamic-range detection system.

The mechanical design (see Fig. 1 ) enables the detector fiber to scan the full surface of a sphere, while the sample is positioned in the sphere’s origin. The scanning unit was designed such that the sample and the detection fiber can be lowered into a tank for index-matching purposes. The tank walls were covered with black cloth to absorb scattered light and minimize stray light. Alignment of the probe and the laser to the goniometer arm is critical and can lead to asymmetric artifacts of the measured scattering pattern.

Fig. 1

Design of the goniometric system. The mechanical part has two axes of freedom to scan the detector fiber on the whole scattering sphere ( $\theta =[0\dots 180\mathrm{deg}]$ , $\varphi =[0\dots 360\mathrm{deg}]$ ). The direction of the incident laser beam is $\theta =90\mathrm{deg}$ , $\varphi =0\mathrm{deg}$ .

The mechanical movement was operated by two stepper motors (TECO, S0302) with a belt transmission for the $\varphi$ axis (3072 steps/rev., $\Delta \varphi =0.12\mathrm{deg}$ ) and a cogwheel transmission for the $\theta$ axis (1600 steps/rev., $\Delta \theta =0.23\mathrm{deg}$ ) in half-step mode. The radial distance between sample and fiber can be varied between 1 and $10\mathrm{cm}$ , so that a $600\text{-}\mu \mathrm{m}$ fiber covers an opening angle between 3.4 and 0.34 deg. Due to the mechanical movement, the measuring speed is about 8 data points/s. The shortening of the circumference of the latitude toward the poles decreases the required number of data points with increasing latitude. Still, there are about 1 600 000 data points at the highest angular resolution, which takes about $2\text{days}$ to acquire.

The arm of the detection fiber is able to scan the full solid angle. But when the arm is close to the backward direction, it shields the incident beam. So there is a blind zone of about $3\mathrm{deg}$ around the backward direction, at a $10\text{-}\mathrm{cm}$ radial distance, that cannot be measured.

The light source was a five-color polarized helium-neon laser (Laser2000, LSTP-1010). For the detection a black-coated $600\text{-}\mu \mathrm{m}$ fiber was used to guide the light either onto a photodiode (Hamamatsu, S2386-18L) or a photomultiplier (Hamamatsu, H5700-02). The photodiode was used for a relative high scattered intensity, whereas the photomultiplier was applied when the scattered intensity was low. In this study, no polarizer was used in front of the detector fiber.

To evaluate the photodiode currents, whose dynamic range spans more than 8 orders of magnitude (10 pA to 3 mA), a monolithic logarithmic amplifier (Texas Instruments, LOG101) was used. The transfer function of the logarithmic amplifier is

The photomultiplier was coupled via a preamplifier (Stanford Research Systems, SR440) to a $300\text{-}\mathrm{MHz}$ single-photon counter (Stanford Research Systems, SR400). The dynamic range of the single-photon-counting system is about six orders of magnitude, while the sensitivity is considerably enhanced compared to the photodiode. Both detector systems were validated with neutral density filters and their combination provides 12 orders of magnitude dynamic range (240 dB or 40-bit resolution).

Since single scatterers have small cross sections, most of the incident light passes the sample unscattered. Due to the geometrical overlap of laser beam and detection fiber, unscattered light will be detected up to scattering angles of $1\mathrm{deg}$ . In theory, it should be possible to remove the incident beam via subtraction of a background measurement. In practice, due to mechanical tolerances, we found that this is not a reliable method.

2.2.

Sample Preparation

For the measurements on the polystyrene spheres, a drop of a diluted emulsion of spheres having a diameter of $5.58\mu \mathrm{m}$ was dried on a coverslip. The spheres remained attached to the surface even when a drop of water was added again later. Then, a second coverslip, fastened by a thin double-adhesive tape, capped the spheres for protection. For the goniometric experiment an area was chosen with the aid of a microscope, where the spheres were not aggregated and the surface density was so low that about three spheres remained within the laser spot. The index-matching tank was filled with distilled water.

The cylinder phantoms are glass cylinders (E-glass, specimen EC 12 60tex, Institut für Polymerforschung, Dresden, Germany) of various radii with a refractive index of $n=1.547$ . Besides the advantage that the cylinders are easier to handle mechanically compared to polystyrene spheres, they also illustrate the importance of measuring the full scattering pattern as it is not rotationally symmetric.

Dentin is a bonelike material forming the main mass of the tooth. It has a channel system, the tubules, which proceed from the dentin-enamel-junction (DEJ) to the dentin-pulp-junction in S-shaped curves. Figure 2 shows three microscopic images along the tubules’ route of a dentin slab having a thickness of $20\mu \mathrm{m}$ . At the DEJ the tubules are relatively thin (radius: $0.7\pm 0.1\mu \mathrm{m}$ ) and far apart (mean distance $7.5\pm 1.5\mu \mathrm{m}$ ). About half way to the pulp, the tubules’ radius increases $(0.8\pm 0.1\mu \mathrm{m})$ and their mean distance decreases $(6.0\pm 1.0\mu \mathrm{m})$ . At the dentin-pulp-junction the tubules are relatively thick (radius: $1.0\pm 0.15\mu \mathrm{m}$ ) and close together $(5.0\pm 1.5\mu \mathrm{m})$ .

Fig. 2

Schematic image of a tooth. The shape and arrangement of dentinal tubules are documented with three microscopic pictures on the right hand side. Experiments are carried out as depicted in the central region of the tooth between the pulp and DEJ.

Extracted human molars were embedded in cold-curing resin (SpeziFix20) and about $300\text{-}\mu \mathrm{m}$ slabs were cut using a diamond saw. The slabs were thinned to the desired thickness using silicon-carbide grinding papers of decreasing grit sizes $\left(25\text{to}5\mu \mathrm{m}\right)$ . Finally, the surfaces were polished with diamond suspensions. The slabs were stored in a NaCl solution.

Fried ⁹ pointed out the importance of index matching the samples in goniometric experiments because surface scattering can significantly contribute to the measured signal. In our setup, we found that using glycerine $(n=1.45)$ instead of water $(n=1.33)$ did not significantly affect the results for the measurements of dentin slabs $(n=1.52)$ . We conclude that the surface scattering is negligible for our samples due to the careful grinding and polishing.

3.1.

Scattering by Polystyrene Spheres

The goniometer data of the scattering by polystyrene spheres are shown in Fig. 3 in a 3-D representation with a logarith mic colored intensity scale. The goniometer data are in complete for scattering angles in the $\varphi =[175\dots 185\mathrm{deg}]$ , $\theta =[85\dots 95\mathrm{deg}]$ sector, where the detector arm blocks the incident laser. Due to the small cross section of a single polystyrene sphere, the detected intensity of the scattering pattern is about 8 to 12 orders of magnitude lower than the incident laser intensity.

Fig. 3

Scattering pattern from polystyrene spheres with a diameter of $5.58\mu \mathrm{m}$ diluted in water using a $543\text{-}\mathrm{nm}$ HeNe laser. The display of the scattered light is logarithmically scaled. Only a part of all data are shown for better visualization.

The scattering plane is defined by the unit vectors of the incident $s_i$ and scattered $s_s$ direction. The parallel and perpendicular component of the incident $E^i$ and scattered electric field $E^s$ is referenced to the scattering plane. At large distances $kR⪢1$ , the scattered field can be calculated through the complex valued amplitude scattering matrix $\mathsf{S}$ and the incident field:

Fig. 4

Scattering profiles along the pole and equator from Fig. 3. The profile along the pole corresponds to perpendicular polarization, while the profile along the equator is parallel polarized with respect to the incident beam. The curves for parallel polarized light were divided by 100 to improve their visibility.

In case of the profiles along the poles, the electric fields are still polarized in the equator plane. But now the scattering plane has changed, so that incident and scattered fields are perpendicularly polarized with respect to the pole scattering plane. The measured phase function, also shown in Fig. 4, is proportional to the perpendicular polarization ${\mid S_1\left(\varphi \right)\mid }^2$ .

The scattering pattern is altered by several effects due to experimental conditions. There is 4% reflection of the unscattered beam at the second cuvette interface, which passes through the polystyrene spheres again. The scattered light of that backward beam changes the scattering pattern predominantly for $\varphi$ greater than $90\mathrm{deg}$ . For the quantitative comparison with theory we therefore omitted the backward region. Furthermore, the presence of the cuvette in the near field of the polystyrene spheres changes the scattering pattern in an analytically unknown way. Fresnel reflection at the cuvette interfaces also alters the intensity of the scattered light. Therefore, these experimental conditions are not ideally suited for accurate comparison with theory. Instead we use a glass cylinder, where these complications do not occur.

3.2.

Scattering by a Perpendicularly Illuminated Cylinder

Figure 5 shows the 3-D scattering pattern of a $4.030\text{-}\mu \mathrm{m}$ cylinder in air illuminated perpendicularly to the cylinder axis. The reference planes for the definition of the beam polarization in the cylinder case are the incident plane, defined by the cylinder axis and the incident direction, and the scattering plane, defined by the cylinder axis and the scattered direction. The incident polarization is again in the equator plane, which is now defined as perpendicular polarized. In case of perpendicular illumination, the scattered light is confined only to the equator plane. The amplitude scattering matrix of this cylinder is also diagonal, so the electric fields are polarized in the equator plane. The phase function along the equator profile and the corresponding theoretical phase function $\left({\mid S_1\left(\varphi \right)\mid }^2\right)$ are shown in Fig. 6 . The goniometer data agree accurately with the analytical cylinder theory.²³ At a scattering angle of $\approx 0\mathrm{deg}$ , the unscattered incident beam, which is more than 2 orders of magnitude stronger than the scattered intensity, is visible. The scattering profiles were used to determine the radius of the glass cylinders within a few nanometers, knowing the refractive indices of the cylinders and the surrounding medium. Experiments with cylinders in liquids were also carried out (data not shown). With these measurements, the refractive index of liquids could be determined with high accuracy.

Fig. 5

Scattering pattern of a cylinder with a radius of $4.030\mu \mathrm{m}$ at perpendicular illumination. The display of the scattered light is logarithmically scaled.

Fig. 6

Quantitative comparison of the goniometric profile taken from the scattering pattern of the perpendicularly illuminated cylinder in Fig. 5 with the analytical data obtained from the scattering theory. At small angles of the experimental data, the unscattered incident beam, polarized parallel to the equator, is measured.

3.3.

Scattering by an Obliquely Illuminated Cylinder

If a cylinder is tilted with respect to the incident beam, the scattered light is confined to a cone, which arises from rotating the unscattered beam around the cylinder axis, as shown²³ in Fig. 7 . Perpendicularly incident light, which was described in Sec. 3.2, is a special case, where the rotation of the incident beam around the cylinder axis gives the equator plane.

Fig. 7

Typical cone-shaped scattering pattern by a perfect infinite cylinder that is incident at $\xi =65\mathrm{deg}$ with respect to the cylinder axis. The cone arises from rotation of the unscattered beam around the cylinder axis.

The scattering pattern of a tilted cylinder $(\xi =65\mathrm{deg})$ having a radius of $4.916\mu \mathrm{m}$ is shown in Fig. 8 . The scattering profile along the scattering cone and the corresponding theoretical curve are shown in Fig. 9 . Again, the experimental measurements are accurately confirmed by the theory.²⁴ As the incident light is perpendicularly polarized (in the equator plane) and there is no polarizer used in front of the detection fiber, the theoretical curve is proportional to ${\mid S_3\left(\varphi \right)\mid }^2+{\mid S_1\left(\varphi \right)\mid }^2$ . At about $\varphi =90\mathrm{deg}$ (see Fig. 8), the shadow of the fiber holder can be seen. This holder also causes the dent at $90\mathrm{deg}$ in the scattering profile in Fig. 9 and was not used in the measurement shown in subsection 3.2.

Fig. 8

Measurement of the scattering pattern of a cylinder in air tilted at $\xi =65\mathrm{deg}$ with respect to the cylinder axis. The radius of the cylinder is $4.916\mu \mathrm{m}$ . The shadow of the fiber holder can be seen at $\varphi =90\mathrm{deg}$ . The display of the scattered light is logarithmically scaled.

Fig. 9

Quantitative comparison of the goniometric profile taken from the scattering pattern of the tilted cylinder in Fig. 8 with cylinder theory.

3.4.

Scattering by Two Parallel Cylinders

In this subsection, we investigate the angular resolution of the goniometer. In the evaluation so far, only scattering patterns of single scatterers were discussed. Now the effect of a second scatterer is investigated. In principle, this is the well-known double-slit experiment, except that instead of slits, two parallel glass cylinders are used. The cylinders are held perpendicular to the incident beam, so that their scattered light overlaps in the equator plane and interference effects can be observed.

If independent scattering is assumed, the measured phase function $p^{*}\left(\varphi \right)$ could be described by the multiplication of the single cylinder phase function $p\left(\varphi \right)$ with the double-slit interference term:

Figure 10 shows the goniometer data together with the double-cylinder theory at a separation of the cylinders of $97.0\mu \mathrm{m}$ . The radius of both cylinders is $6.935\mu \mathrm{m}$ . The goniometric data show the unscattered light at small scattering angles $[0\dots 1\mathrm{deg}]$ . As in the previous experiments, the goniometrical measurements are in good agreement with the theoretical data.

Fig. 10

Quantitative comparison of the goniometric profile with double-cylinder theory at a $97.0\text{-}\mu \mathrm{m}$ cylinder separation. The radii of both cylinders are $6.935\mu \mathrm{m}$ .

For larger cylinder separations, the interference term gives rise to so rapid oscillations that the angular resolution of the goniometer was not high enough to follow them. In fact, this was the case with the polystyrene spheres (see Fig. 3). There, the large separation between the three spheres caused interference oscillations so rapidly that the goniometer could not resolve them and only the average scattering pattern of a single polystyrene sphere was measured.

3.5.

Measurements on Dentin

Depending on the location where the dentin slabs were cut from the whole tooth, the tubules have different orientation within the slab. If the slab section is cut vertically through the tooth center, the tubules lie within the slab, otherwise they intersect with the slab’s surface. Figure 11 shows the scattering pattern of a sample with the tubules lying within the slab. The geometry of the tubules in this sample is similar to the microscope images in Fig. 2. Thus, the scattering pattern is similar to the perpendicularly illuminated cylinder in Fig. 5. Note that the scattering data in Fig. 11 (and Fig. 12 ) are presented in an equal-area projection for better quantitative comparison.

Fig. 11

Scattering pattern of a dentin sample with tubules perpendicularly illuminated. The data are displayed in an equal-area projection using a logarithmic scale. The thickness of the slab is $20\mu \mathrm{m}$ .

Fig. 12

Scattering pattern of a dentin sample with tubules tilted within the slab. The data are displayed in an equal-area projection using a logarithmic scale. The thickness of the slab is $20\mu \mathrm{m}$ .

The scattering pattern of tubules intersecting the slab is shown in Fig. 12. The typical cone-shaped pattern of a tilted cylinder is clearly visible. At the position $\theta =90\mathrm{deg}$ and $\varphi =160\mathrm{deg}$ is a brighter area, which arises from the reflection of the incident laser beam at the slab surface.

The scattering pattern of the dentin slab shown in Fig. 2 was investigated. The slab was placed perpendicular to the laser in the tank filled with water. About 15 measurements were carried out along a line from the DEJ to the pulp in the central region of the tooth, similar to the sites of the microscope images in Fig. 2.

Scattering profiles were extracted from the goniometric data. The profiles are along the pole or the equator (see Fig. 13 ). The typical oscillations obtained with Mie theory in the profiles cannot expected to be observed, because the shape of a single tubule is not perfectly cylindric and the ensemble of tubules in the measured region have a distribution of radii. Both scattering profiles are symmetric with respect to the unscattered beam. The intensity of the pole profile is about two orders of magnitude lower than that of the equator profile. There is a pronounced first-order interference peak at about $5\mathrm{deg}$ in the profile along the scattering cone due to the quasi-periodic tubules arrangement. The peak position shifts in accordance with the tubules spacing (data not shown), which itself decreases from the DEJ to the pulp. The slab geometry of the tooth samples mask all scattering data in the plane perpendicular to the incident beam. Due to bending of the samples, the masking of the profiles occurs at a scattering angle of $\approx 80\mathrm{deg}$ for the profiles along the scattering cone instead at $90\mathrm{deg}$ .

Fig. 13

Profiles extracted parallel and perpendicular to the scattering plane from the dentin scattering pattern (see Fig. 11). The interpolated profiles used for the anisotropy factor calculation are also shown.

The form of the phase function is often characterized by their first Legendre moments $\left(g_i\right)$ , where $g=g_1=⟨\cos \left(\theta \right)⟩$ is the anisotropy factor.

In Fig. 13, 15 profiles and their average are shown. Their shape is described by the multiplication of the single phase function and the interference function, similar to the double cylinder case [see Eq. 2]. The tubule positions have only a periodic order at short distances. Therefore the amplitudes of the oscillation of the interference function diminishes quickly and vanishes after the second minimum. To reconstruct the single tubuli phase function we use two models to fit the averaged profile in the angular region $[0\dots 12\mathrm{deg}]$ . Since the interference function oscillates around unity for more than one period, the fitted function is a reasonable solution to recover the single phase function without the interference effects. The first model function is the infinite cylinder, where we fixed the refractive indices of the tubule $(n=1.33)$ in dentin background $(n=1.52)$ and fitted the radius $R$ and a scaling factor. The second model is the Fraunhofer diffraction formula of a slit [Eq. 7], which is a valid approximation for small-angle scattering of optically soft particles. The slit radius $L$ and the scaling constant $C$ were fitted

In the regions around 90 and $180\mathrm{deg}$ , the light is blocked, so that the measured data cannot be used. There, the reconstruction is carried out by a linear interpolation between beginning and end. The $g$ factor is not much affected because its integral contribution is small.

The moments of the full scattering pattern were calculated from Fig. 11. Equation 3 was used to calculate $g_1$ from the full scattering pattern, whereas Eq. 4 was used for the profiles (analogously for the higher moments). Thus, the calculation from the profiles falsely assumes a rotationally symmetric scattering pattern. Table 1 shows the results for the three first moments of the three investigated phase functions.

Table 1

The three first moments of the phase functions for the full 2-D pattern, along (parallel profile) and perpendicular to the scattering cone (perpendicular profile) shown in Fig. 13.

Surprisingly small differences of the moments are found, although the shape of the phase functions are quite different. However, if the reduced scattering coefficient $[{\mu }_s^{\prime }={\mu }_s(1-g_1)]$ , which is the important quantity if the light propagation at large distances is considered, is calculated from the obtained $g$ factors, we find large differences. For, example, ${\mu }_s^{\prime }$ derived for the perpendicular profile is 80% larger than that derived for the parallel profile, whereas the reduced scattering coefficient calculated with the full 2-D pattern lies between these values. Thus, it is important to measure the phase function in the whole solid angle, if tissue is investigated that has an anisotropic light propagation.