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We present novel experimental method for estimation of the light penetration depth (LPD) in turbid media based on the analysis of the speckle pattern structure. Under the certain illumination conditions this structure is strongly dependent on the penetration depth. Presented theoretical model based on the Bragg diffraction from the thick holograms allows LPD estimation if only one parameter of the material, namely refractive index, of the material is known. Feasibility of the method was checked experimentally. Experimental results obtained for variety of the materials are in good agreement with the theoretical assumptions. It was shown that qualitative LPD comparison does not require knowledge of the material properties.
©2012 Optical Society of America
1. Introduction
Coherent illumination has found multiple applications in modern medicine, biology, industry, and many other fields [1]. In many of these applications physical parameters of a material are either estimated or altered by means of the laser light. Consequently, it is important to know how deep the laser light does penetrate inside the material. In a medium with negligible internal scattering, the light penetration depth (LPD) can be easily found using Beer-Lambert law. However, this law is not applicable in case of turbid media. For that reason quite a few methods for LPD estimation were developed. Mainly, these methods are computational and analytical, each possessing unique advantages and limitations. The most widely used computational method is the Monte Carlo simulation [2,3]. While this method can predict behavior of the light in the medium with very high precision, its implementation requires significant computational resources. Analytical methods are usually simpler, but these methods provide less precise predictions, and in some cases accurate only under certain conditions [4–6]. It is important to note, that both computational and analytical methods require prior knowledge of the optical properties of the studied material. In many cases it is difficult to define these parameters, while rough estimation of the LPD would be enough. In this case implementation of simple experimental technique would be preferable.
However, there are only a few experimental methods suitable for estimation of LPD in turbid media. One of the methods, proposed by Xie et al., is based on direct measurements of the fluence rate inside the material [7]. Fluence rate is measured by the optical-fiber probe inserted in the needle and introduced into the material from the side opposite to the illuminated one. Position of the probe inside the material can be adjusted by means of the micrometric translation stage, allowing recovery of the fluence rate depth profile. Information concerning LPD is obtained by least squares fitting of the exponential function to the fluence rate data. However, operating principle, which makes this method so simple and straightforward, is a source of method’s drawbacks: destructiveness and inapplicability to the hard materials.
Another simple but nondestructive experimental method was described in [8]. It is based on the analysis of the light introduced into the sample and backscattered by medium. One normally oriented fiber situated right against the sample surface is used as a point light source. Similar fiber is used as a detecting probe for measurements of the backscattered light intensity at a distance from the source. Using simple analytical formula one can estimate the maximal depth from which light has come to the detector. Thus, the light penetration depth can be estimated by finding such a source-detector distance at which the intensity of detected light is e times smaller than that at the source. However, the formula is valid only at the assumption of homogenous medium with low absorption, which greatly limits applicability area of the method.
In this work we propose a novel experimental method for LPD estimation exploiting the laser speckle effect. This method is based on the analysis of the spatial structure of the speckle patterns formed by the light scattered at the surface and inside the material. Information concerning the optical penetration depth is provided by the fine variations of the speckle pattern structure caused by the change of the illumination conditions. Since laser speckles can be obtained from any scattering material, proposed method is very versatile. Taking into account that the proposed method is also non-destructive and non-contact, one can consider it as a good alternative for abovementioned techniques.
2. Theory and methods
2.1 Method description
As known, a speckle pattern inevitably appears when a coherent light illuminates an optically rough surface. In many experimental configurations, including surface illumination by the collimated laser beam, either displacement of the illuminating beam or its tilting in respect to the object surface leads to a spatial shift of the speckle pattern without significant change of its structure. The later, however, is true only for the materials with surface-only scattering. For turbid media even small change of illumination angle causes considerable change of the speckle pattern structure besides its spatial displacement.
This occurs because of the difference in processes of the speckle pattern formation. In contrast to materials with small LPD or with surface-only scattering, formation of the speckle pattern in turbid media is strongly affected by light scattered from inner layers of the material. In the general case the turbid medium can be represented as a random set of the scattering centers. Tilt of the illumination beam in respect to the object surface will result in the change of phase differences between waves reflected from a large number of scatterers. The speckle pattern formed by the interference of these waves will change accordingly. Formation of the speckle pattern by the interference of a large number of waves scattered from the volume of the media resembles the reconstruction of a thick hologram recorded by the incident beam as reference wave and that speckle pattern as an object. As the angle selectivity of the volume hologram during its reconstruction can give the information about the hologram thickness, the incident beam tilt angle range within which the speckle pattern still maintains its fine structure can provide the information about mean free photon path in turbid media. Therefore, it is possible to estimate LPD by tracking changes of the speckle pattern caused by the illumination angle variation.
Besides the volume scattering, light is also diffusely reflected by the surface of the sample which can be considered as a thin hologram which does not have strong angular selectivity. Therefore, part of the speckle pattern will maintain its fine structure even when tilt angle of the incident beam is changed considerably.
2.2 Theoretical model
One of the common ways to monitor changes speckle pattern structure is to make snapshots of the pattern before and right after it was changed. The difference between these snapshots is typically estimated by calculation of the cross-correlation function. Position and amplitude of the function maximum provides information concerning the nature and magnitude of the pattern transformation. However, interpretation of this information requires application of at least rough model of the processes which occur in turbid media. Below we present such a model.
Let us consider illumination of the isotropic turbid medium by a collimated laser beam. For simplicity we will assume that the incident light has uniform intensity distribution at the surface, and its scattering inside the medium is isotropic. Assumption about the intensity distribution is valid when the illuminated area is sufficiently larger than the light penetration depth. Due to absorption and scattering processes, the light intensity of the incident wave inside the medium attenuates exponentially. At the depth of z it will be equal to:
where I0 is the intensity of the incident light at the surface, and Z is the depth at which the light intensity is e times smaller than at the surface. In other words, the parameter Z is an effective LPD.Since the intensity of the light is proportional to its amplitude squared, the light-wave amplitude attenuation in the medium can be described as
The speckle pattern is formed after back scattering of the light from the inner material layers. Each material layer scatters back only some fraction of the incident light. Proportion of backscattered to transmitted light depends on the material properties, but usually it is quite small. On its way to the object surface light is once again attenuated. Thus, the small increment of the amplitude of the observed speckle pattern provided by the light wave backscattered from the depth z can be estimated asAs described above, we assume the turbid medium as a random thick hologram with low diffraction efficiency. If in this case the incident wave has low attenuation or even does not decay, but the hologram is not uniform in z-direction and modulated by the factor e-z/Z, then the increment of the amplitude of the diffracted light wave from the depth z is also defined by the Eq. (3). The angular selectivity of such a hologram can be illustrated by the well-known method of Ewald spheres [9].
Figure 1 presents all light waves and the hologram in a spatial-frequency domain. Spheres with the radiuses of 2π/λ and 2πn/λ are representing Ewald spheres for all possible light waves in the air and medium, respectively. Two vectors pointing up-left are representing incident and refracted waves prior to the object rotation. Angles θ and θn are incidence angles in the air and in the medium, respectively. From the diagram of Fig. 1 one can see that which represents the Snell’s law. The angles δθ and δθn show the change of the incidence angles due to the object rotation. The horizontal vector pointing right is a wave diffracted from the hologram whose vector is denoted as K.
Fig. 1 Schematic representation of the diffraction from the thick hologram in spatial-frequency domain. Spheres with the radiuses of 2π/λ and 2πn/λ are representing Ewald spheres for all possible light waves in the air and medium, respectively. Angles θ and δθ are incidence angle and its change due to rotation in the air, while θn and δθn – in the medium. Two vectors pointing up-left are representing incident and refracted waves. The horizontal vector pointing right is a diffracted wave. Vector denoted as K is a hologram vector.
In this coordinate system, fulfillment of the Bragg’s law for thick holograms requires coincidence of end points of the incident and diffracted wave vectors with the end points of the grating vector K. If the hologram vector length was set strictly, then diffraction would have been possible only at certain angle of incidence. However, there is small uncertainty in the vector length caused by the finite thickness of the hologram. Consequently, there is some set of angles at which incident light can be diffracted. Efficiency of this diffraction depends on proximity of the incident angle to the Bragg angle. As a result of the finite hologram size in z coordinate, the length of the vector K has uncertainty, so that the amplitude of the reconstructed hologram will not be zero even when the end-point of the grating vector K does not touch the corresponding sphere but it is located at a distance ζ. Since the space-frequency domain is nothing else but the Fourier domain, the relative amplitude of the reconstructed hologram can be evaluated as:
Solution of this integral isAnd the intensity of the reconstructed wave is proportional to the square modulus of this equation:Exactly the same dependence should be for the maximum of the correlation function calculated for two speckle pattern snapshots made before and after the object rotation leading to the mismatch from the Bragg condition by the parameter ζ in the spatial frequency domain because these speckle patterns can be considered as an initial hologram and its weakened version reconstructed after the rotation with the lower diffraction efficiency.Now it is necessary to define how the mismatch parameter ζ does depend on the incident angle. It is easy to see that when the object is rotated by the angle of δθ, the refracted beam is also changed by the angle of δθn. In this case the end-point of the corresponding vector follows the arc shown by a thick line in Fig. 1. Unlike wave vectors corresponding to the light waves, vector of the hologram cannot be rotated since hologram is fixed in respect to the object. If the initial point of the hologram vector K is fixed at the end-point of the wave vector corresponding to the refracted incident beam, then the end-point of the hologram vector also follows the arc when the object is rotated. In this case it goes out of the sphere of possible positions for the reconstructed wave, which leads to the Bragg condition mismatch.
The arc representing the track of the right end-point of the vector K is congruent to the arc which shows the track of the left end-point of this vector. The mismatch parameter can be found as the change of the horizontal coordinate of the either right or left end-point since they are the same. It is easier to calculate the coordinate of the left vector’s end. As one can see, the horizontal coordinate of this end-point is defined as:
Shift of the end-point caused by the change of the illumination angle on δθ can be defined through differentiation of the Eq. (7) with respect to the angle:Substitution of Eq. (8) to Eq. (6) provides equation which describes the dependence of the amplitude of the correlation function on the illumination angle.3. Experiment
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).
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.
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:
where C0 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:where α, β, A and B are the model parameters, and δθ is the independent variable.The most interesting parameters for us are A, B, 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 , respectively.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 [7,10].
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.
4. Conclusion
In this work we have presented novel method for estimation of LPD in turbid media based on the laser speckles. Here we proposed simple theoretical model based on the theory of Bragg diffraction from volume holograms. The model allows estimation of LPD if only one parameter of the inspected media, namely the refractive index, is known. Feasibility of the method was shown experimentally. Experimental results are in good agreement with the theoretical model. We have shown that qualitative LPD estimation is possible without knowledge of any optical properties of the material. The method is very versatile, since speckle pattern can be formed by virtually any material. Moreover, the method is applicable to isotropic and anisotropic, absorbing and non-absorbing scattering materials. Although additional studies are required, we believe that simplicity, versatility and other strong points make this method quite promising.
Acknowledgments
The Finnish Funding Agency for Technology and Innovation (TEKES) and the Academy of Finland are acknowledged for financial support of this research (projects No. 70058/09 and 128582, respectively).
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