3.1 Experimental setup
Feasibility of the proposed method was experimentally checked. Schematic layout of the experimental setup is presented in Fig. 2
. A coherent laser light at the wavelength of 633 nm and power of 30 mW was provided by a HeNe laser. The laser beam was collimated and its diameter was equal to 1.5 mm. Incidence angle of the beam was equal to approximately 30 degrees. The test object was situated on the rotation stage. Experimental setup was arranged so that the rotation axis of the stage is at the object’s surface and it crosses the center of the illumination spot. Light scattered by the surface was collected by the CMOS camera (Edmund Optics EO-1312M) located at the distance of 10 cm from the object surface. The camera position and illumination angle were chosen so that only diffusely scattered light participates in the speckle-pattern formation at the matrix surface. Moreover, chosen distance ensures that the initial speckle pattern would not totally go out of the camera’s field of view during the measurements, and that the camera would be able to resolve the speckle pattern structure (speckle size on the CMOS surface would be several times bigger than the pixel size).
Fig. 2 Schematic layout of the experimental setup.
Connection of the rotation stage with the computer controlled micrometric translation stage (Newport MFA-CC) through the lever has allowed for very fine changes of the illumination angle. In our experiments the length of the lever was equal to 127 ± 1 mm, and the translation stage was displacing with the step of 11 ± 1µm. Therefore, dependence of the correlation function maximum on the illumination angle was inspected with the angular step of (8.7 ± 0.8) ∙10−5 radians. The setup was assembled so that the backlash was minimal, and the vector of the translation stage movement was perpendicular to the lever at the initial moment of time.
Both the camera and translation stage were controlled by the computer with help of the custom-made software. Experiments were carried out for different materials. For each material 41 speckle pattern snapshots were recorded: for initial and 40 altered illumination angles. These snapshots were processed according to the following algorithm. First, the auto-correlation and cross-correlations with all subsequent snapshots are calculated for the first snapshot of the set. Then this operation is repeated for the second snapshot, while the first one is excluded. All consequent snapshots are processed in such a manner until auto-correlation of the last snapshot is calculated. Since initial illumination angle is chosen arbitrary, each of the snapshots from the set can be considered as an initial one in respect to the consequent snapshots. Thus, for each material described data processing algorithm provides us with 41 repetitions of the experiment with no illumination angle changes, 40 - with the illumination angle changed by 8.7 ∙10−5 radians, 39 - with the illumination angle changed by 17.4 ∙10−5 radians and so on. In other words, dependence of the correlation function maximum from the illumination angle is calculated with better statistics in the region of small angle changes, where dependence is the strongest.
3.2 Results and discussion
Dependencies of the correlation function amplitude on the illumination angle for different materials are presented in Fig. 3
. We have abstained from using error-bar representation in Fig. 3
, since the maximal standard error in this set of measurements was only 0.12% due to proper data processing, which makes non-informative this kind of data representation. In our experiments semitransparent materials with both low (white plastic and Teflon) and high (cardboard and pine wood) absorption were studied. In order to show difference between semitransparent and non-transparent (with surface-only scattering) materials, the measurements of the metal surfaces were carried out, as well. There is noticeable difference between the curves corresponding to the metal and to the semitransparent materials. As it was mentioned above, source of this difference is in the process of the speckle pattern formation. Since for the metal there is only surface scattering, due to the properties of the experimental setup in this case structure of the speckle pattern formed by the scattered light will be almost independent from the illumination angle. Therefore, the linear decay of the graph corresponding to the metal is caused solely by the shift of this speckle pattern on the surface of the CMOS matrix. In contrast for semitransparent materials, the resulting speckle pattern is formed by both light scattered at the surface and backscattered from inside of the material. Thus, this speckle pattern should have components both dependent and independent on the illumination angle. As a result, the correlation function amplitude should have bell-shaped decay in the area of small angle changes and linear decay in the area of large angle changes.
Fig. 3 Correlation peak amplitude versus illumination angle measured for different materials.
However, as one can notice, only one semitransparent material, namely pine wood, complies with this description. In particular, the shape of the graph corresponding to the cardboard is pretty close to the linear, while graphs corresponding to the Teflon and white plastic are almost constant after some angle. But, at least for cardboard, this discrepancy between the expected and real shape of dependencies is only apparent. In the cardboard, the optical penetration depth is very small due to the high absorption, and the speckle pattern on the CMOS matrix surface is formed mainly by the light scattered from the surface. Consequently, the dependence of this pattern on the illumination angle is manifested weakly, which is displayed in the graph. In contrast, both samples of Teflon and white plastic studied in our laboratory have very low absorption and large LPD. For this reason decorrelation of speckle pattern component produced by inner scattering is quite noticeable after approximately ten changes of the illumination angle. Thus after ten object rotations influence of the inner scattering on the amplitude of the correlation function maximum becomes negligible, and one can expect to see linear decline of the graphs due to the shift of the speckle pattern formed by the surface scattering on the matrix surface. Absence of the linear decline cannot be explained in the context of combination of inner and surface scattering, and additional studies are required to explain behavior of the graphs. Nevertheless, this fact does not affect applicability of our method, since region of big angle changes, where the discrepancy takes place, is not relevant for LPD estimation.
Information about the LPD can be obtained from the data shown in Fig. 3
through the least squares fitting. To do the fitting a proper model is required. Behavior of the decorrelation of speckle patterns from inner scattering has been already discussed in the section 2.2, while influence of surface scattering is described by the linear equation in a slope–intercept form. Therefore, the dependence of the correlation function maximum on the illumination angle can be represented as:
is the autocorrelation function amplitude, Z
is the effective LPD, ζ
is a term accounting the alteration of inner speckles caused by the illumination angle change, δθ
is the angular step of the measurements, A
is the slope of the linear decline corresponding to the surface speckles, and B
is the constant. Fitting of this model to the experimental data becomes much easier task if following variable transformations are done:
Substitution of Eq. (10)
into Eq. (9)
gives us following model:
are the model parameters, and δθ
is the independent variable.
The most interesting parameters for us are A
, and α
. First two parameters are representing influence of the surface speckles. Using these parameters one can exclude their influence on the experimental data and analyze separately the behavior of inner speckles as a function of the illumination angle. In its turn, the parameter α
is needed for estimation of the LPD. With known refractive index of the studied material one can calculate penetration depth using the following equation:
But even without knowledge of the refractive index the reciprocal value of the dimensionless parameter α
allows for qualitative comparison of the LPD in semitransparent materials. Note that, according to Eq. (12)
, coefficient 1/α
have the same order of magnitude as the ratio of effective penetration depth to the wavelength of the illuminating beam. Dependence of the correlation function amplitude from the illumination angle for speckle pattern formed by the light scattered inside the material and the reciprocal of α
parameter for semitransparent materials are presented in Fig. 4
and Table 1
Fig. 4 Correlation peak amplitude versus illumination angle only for the light scattered inside the material. Dots are representing the experimental data, while solid lines – least squares fitting.
Table 1 Reciprocal Value of the Parameter α for Different Materials
Due to our assumptions, width of the graphs in Fig. 4
is inversely proportional to the LPD. Consequently, the peak width of the graphs corresponding to the materials with surface-only scattering should be infinite. For this reason, the least-squares fitting of the bell-shaped function into the metal data provide incorrect results. Although there is no reciprocal α
for the metal, one can see that the graph representing this material is a straight line. Thus, experimental data confirms that there is no LPD in metal. As for semitransparent materials, it is clear from both Fig. 4
and Table 1
that the largest LPD is for the Teflon and the smallest one is for the cardboard. It is interesting to note that at constant angular resolution increasing of the LPD diminishes the peak amplitude of the graphs in Fig. 4
. This fact is easy to explain, since the magnitude of the speckle pattern variation is directly proportional to the penetration depth, and less angular steps are required for complete decorrelation. This means that chosen angular resolution can be insufficient for some LPD values, at which decorrelation occurs even after one rotational step. Thus, the system resolution should be optimized with respect to the optical properties of the material. There is possibility that at the very large LPD values the speckle pattern formed on the CMOS matrix surface by the light experienced multiple scatterings inside the material will be no longer dependent on the illumination angle. In other words, it is possible that there is maximal LPD value evaluable by our method. However, this speculation should be checked experimentally.
Experimental data has clearly shown feasibility of the proposed method for LPD estimation. To our mind, the main advantage of this method is its versatility. The main requirements to the medium are stable inner structure during the measurements and capability to form a speckle pattern. Although the former requirement cannot be usually fulfilled for gases and liquids, it is always fulfilled for the solid materials. The latter requirement is fulfilled a priori because the laser speckles can be observed from virtually any material. Moreover, we have shown that the method can be applied to both absorbing and non-absorbing scattering media, as well as to both isotropic and anisotropic media, though the later with some reservations. Therefore, our method is applicable to wide range of materials.
Unlike experimental methods mentioned in the introduction, presented method requires knowledge of the refractive index, which could be considered as a limitation. However, in this report we have shown that there is no need to know any optical properties of the material for qualitative estimation of the LPD. Moreover, the refractive index in turbid media can be measured by variety of methods, see for example [7
S. Xie, H. Li, and B. Li, “Measurement of optical penetration depth and refractive index of human tissue,” Chin. Opt. Lett. 1, 44–46 (2003).
F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28(12), 2297–2303 (1989). [CrossRef] [PubMed]
Unfortunately, optical properties of turbid media are not widely reported. Among the tested materials, there is trustworthy information only about Teflon PTFE: its refractive index was reported to be approximately 1.35 at the wavelength of 633 nm. Consequently, we estimate the LPD of our Teflon sample as 1.39 mm by using Eq. (12)
and the fourth line of the Table 1
. However, we are not able to compare this value with the published data because the search for LPD value of the Teflon in the visible range was unsuccessful.
Another thing that we need to discuss is a maximal LPD value measurable by our method. On the one hand, this value is directly related to the angular resolution of the measuring system. As we see it, the resolution can be adjusted in quite a large range and it is not a limiting factor. On the other hand, we can only speculate about behavior of the light after multiple scatterings inside the material. We believe that there is possibility that at some LPD value the speckle pattern formed by this light will lose connection with the inner structure of the studied material. In any case this issue should be thoroughly studied. It is also necessary to perform additional experiments and compare results of different method. However, summing together all advantages of proposed method (i.e., simplicity of the measuring system, non-contact and non-destructive usage, versatility, original data processing algorithm) we believe that the proposed method is very promising.