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We measured the optical properties of drying wood with the moisture contents ranging from 10% to 200%. By using time-resolved near-infrared spectroscopy, the reduced scattering coefficient μs′ and absorption coefficient μa were determined independent of each other, providing information on the chemical and structural changes, respectively, of wood on the nanometer scale. Scattering from dry pores dominated, which allowed us to determine the drying process of large pores during the period of constant drying rate, and the drying process of smaller pores during the period of decreasing drying rate. The surface layer and interior of the wood exhibit different moisture states, which affect the scattering properties of the wood.
© 2016 Optical Society of America
1. Introduction
Understanding the optical properties of biological tissue requires determining which chemical components or pigments are present in the tissue. Many studies have investigated light absorption in and scattering from tissue, such as human skin, fruit, leaves, and wood. Typically, light scattering can be described by a scattering coefficient μs and a reduced scattering coefficient μs′, which depend strongly on the physical structure of the tissue. In addition, the optical absorption, which is linear in the concentration of chemical components, can be described by the absorption coefficient μa.
Light scattering from large scatters (i.e., when the scattering diameter is much larger than the wavelength λ of the light) is described by geometric optics, which describes light propagation in terms of rays. In contrast, for small scatters (i.e., when the scattering diameter is the same as or less than λ) Mie theory (≈λ) [1,2] or Rayleigh theory (<λ) [3] is applicable. These two theories have been widely used in fields involving aerosols and suspensions. An alternative method to study non-spherical scattering centers was developed in fields concerned with interstellar dust and aerosol science. In such approaches, non-spherical particles are approximated as aggregates of spherical particles [4] or as spheroids [5]. Mie-related scattering theories also describe cylindrical scattering by analytically solving Maxwell’s equations for an infinitely long cylinder [6]. In the field of cell biology, cell nuclei [6,7], cell structures [8], and cell states (e.g., cancerous) [9] were investigated, revealing that the structure and state of a cell affects the scattering pattern and intensity.
The optical properties of the plants are also important. Because most plants survive by photosynthesis, they must adapt their morphological structures and the optical properties of their tissue to the demands of their environment. For example, consider the interior surface of leaves, which scatter more strongly than the opposing surface, thereby contributing to the high efficiency of leaf photosynthesis [10]. Or consider the outer-layer cells of flower petals, which reflect sunlight and help attract insects. For many shade plants, which cannot live in direct sunlight, not only is the photoenvironment important but also the photoresponse and the localization of chlorophyll in tissues [11]. Thus, knowledge of the optical properties of plant tissue is important for understanding the physiological mechanisms of plants.
The optical properties of wood differ somewhat from those of other plant tissue. The significant structural characteristic of wood is the hollow cellulosic, yet thick, cell walls. The anatomical structure combining “cell lumens” and a “cell wall” are the most important factor determining the optical properties of wood. Furthermore, water that is retained within a cell wall or cell lumens also significantly affects the optical properties of wood [12].
The variation in light absorption by wood during drying has been studied using near-infrared spectroscopy (NIRS) [12–14]. However, no thorough investigation exists on the light absorption and scattering characteristics of wood that accounts for the anatomical structure of wood, despite the importance of such knowledge having long been recognized. One reason for this deficiency is that light absorption in near-infrared spectroscopy (TR-NIRS) allows the light absorption and scattering coefficients to be calculated separately. Kienle et al. and D’Andrea et al. used this method to derive the reduced scattering coefficient μs′ for dry and wet wood [15,16]. D’Andrea et al. furthermore determined μs′ for wood during drying and with an initial moisture content below 30% (which only included water absorbed in cell walls) and reported that moisture content and scattering are not related [13].
In this study, we consider the variation in the optical properties of wood, which range from the fully water-saturated condition (including free water) to the air-dried condition. By combining TR-NIRS with the diffuse approximation of the transport equation, we determine μa and μs′. To improve our understanding of the optical properties of wood, we analyze the mechanisms underlying the variations in the two optical coefficients. The scattering coefficient μs′ is decomposed into two components: in the regime where only single scattering events need be considered, these two components consist of (i) the number of scattering events per unit path length, and (ii) the angular distribution of scattered intensity. Furthermore, to relate μs′ to the state of water in wood and to the wood microstructure, we relate μs′ to each drying stage. In addition, the coefficient μa is consistent with its value calculated based on water absorption and wavelength. Thus, this work demonstrates the potential of μs′ and μa for determining the state of moisture of wood, such as whether adsorbed water or free water exists at the surface or inside the wood. The dynamic change of scattering properties well corresponds to the microstructure and water states. Although, we targeted on the wood drying, our method can be applicable for monitoring other physical dominant phenomena, such as water absorption-desorption and coagulation process from protein denaturation.
2. Material and method
2.1. Samples
From a single log, samples of Douglas fir (Pseudotsuga meziesii) were cut along the radial or tangential planes. To examine how light propagation depends on wood thickness, the samples were cut into 50 mm × 50 mm wafers with thicknesses of 3, 5, and 7 mm. Five samples were cut and measured for each thickness and two independent experiments were conducted on each sample. The samples had an anatomically simple structure, as shown in Fig. 1 (cross-sectional view). Tracheary elements are an especially important scatterer in softwood because they are mainly composed of tracheids (90 vol%). We prepared radial- or tangential-plane samples because the transverse section does not significantly scatter the light in transmittance, as discussed by Fujimoto et al. [17].
The samples were immersed in water under reduced pressure (about 20 hPa) for at least 100 h, resulting in complete water saturation and a maximum moisture content (MC) of about 200%. After reaching saturation, the samples were removed from the water and exposed to the ambient atmosphere. While air drying, the sample-weight, surface temperature, and scattering dynamics were measured at regular time intervals. The measurements were repeated several times for each sample as they went from water saturation to air dried (MC ~10%). The subsequent drying process was then monitored for 33, 38, and 72 h for the 3-, 5-, and 7-mm-thick sample, respectively. The air temperature and relative humidity (RH) were recorded by using a laboratory thermometer and hygrometer. The average room temperature and relative humidity were 25°C ( ± 0.4 °C) and 55% ( ± 6%), respectively.
To investigate how the moisture state at the sample surface affects light scattering and absorption, the drying rate and surface temperature of the wood samples were also monitored, which allowed us to calculate the generalized drying curve in which the rate of drying as a percent of the constant rate is plotted as a function of moisture content [18]. “Constant-rate drying” is when drying proceeds as though the water were at a free surface. The temperature at the wood surface was detected and recorded by using a thermal-imaging camera (TS-9230, Nippon Avionics Co., Tokyo, Japan).
2.2 TR-NIR measurement
Figure 2 shows the TR-NIRS setup. The system is mainly composed of a picosecond pulsed laser, with a wavelength of 846 nm, and a pulse width of 70 ps (PLP-10, Hamamtsu Photonics Co., Hamamatsu, Japan). Time-resolved transmission (TRT) of the diffusely transmitted laser pulse through the samples was recorded with a streak camera with a time interval of 10.8 ps (C5680, Hamamtsu Photonics Co., Hamamatsu, Japan). The pulsed laser and the streak camera were separated by 370 mm and the wood sample was 140 mm from the laser source. The instrumental response function (IRF) was measured as a reference signal without sample using two neutral-density filters, which were placed directly in front of the camera slit. The TRT was acquired over a range of 5 ns and with an acquisition time of 60 s to improve the signal-to-noise ratio.
Fig. 2 Schematic diagram of measuring system. The streak camera (10.8 ps time interval) recourd 70 ps NIR pulses after their transmission through a wood sample.
2.3 Analysis
The coefficients μa and μs′ are calculated from the TRT by applying the diffuse approximation of the transport equation with the boundary conditions of a laterally infinite slab, as proposed by Patterson et al. [19]. The boundary conditions used in this analysis assume no reflection from internal media and that no light enters the medium from the outside except for the initial incident light (Dirichlet condition). The analytical expression for the time-of-flight distribution (TOFD) T(d, t) of photons through the sample at time t and for sample thickness d, which is the number of photons reaching the detector surface per unit area per unit time, is
where c is the speed of the light in the sample, (i.e., speed of light in vacuum divided by the index of refraction n of the sample). The index of refraction nwood of the wood is the volume-weighted mean of the indices of refraction for air, water, and wood:where fair, fwater, and fcw are the fractions of a unit volume occupied by air, water, and cell wall, respectively, and ncw = 1.55 is the index of refraction of the cell wall of the wood [15], which is expected to fall in the range of the index of refraction of cellulose (1.46), hemicellulose (1.53), and lignin (1.61). The index of refraction nwater = 1.33 is for pure water [20], and nair = 1.00 is for air. The quantity u is moisture content (%), and ρ0 is the oven-dry density (g/cm3) of the wood samples. In Eq. (1), we use the diffusion coefficient D = [3(μs′ + μa)]−1 obtained from the original approximation. The results for μa and μs’ calculated with D = [3(μs′ + μa)]−1 and with D = (3μs′)−1 (obtained by Furutsu et al. [21]) are almost the same at 846 nm, where μs′ is much larger than μa in woody material.To account for the finite instrument response, the calculated curve is convolved with the IRF and normalized by the maximum of the experimental curve. The coefficients μa and μs′ are obtained by fitting the TRT to the convolution of the IRF and the analytical solution of the diffusion approximation of the transport equation. The fitting was done by using the lsqcurvefit function in the nonlinear least-squares package in MATLAB ver. R2013a (MathWorks Co., Massachusetts, USA).
To determine the accuracies of μa and μs′, error bars are calculated based on the residual sum of squares (RSS) for the fits to the data (Figs. 4, 5, 8). After the RSS is normalized by the value at the solution, the ranges of μa and μs′ are computed, along with the RSS, up to unity. The error bars or estimation error (EE) serve to compare the accuracies of μa and μs′.
3. Light scattering properties of drying wood
We estimated μs′ over a wide range of MC in the wood. Figures 3(a)–3(c) show the 846 nm laser-pulse time-of-flight for wood samples with an MC of about 12% (air-dried), 30% (fiber saturation point), and 120% (water saturated), respectively. The coefficients μa and μs′ were estimated by fitting the TRT, which is convolved with the IRF and the analytical solution of the diffusion approximation of the transport equation (black solid line). Figure 4 shows plots of μs′ as a function of MC, acquired while the wood was drying. The coefficient μs′ represents the average number of scattering events per unit length, where a series of isotropic scatterings is considered a single scattering. The results show that the coefficient μs′ increases as the wood dries. We also calculate the transport mean free path l* (i.e., 1/μs′; see Fig. 5), which represents the mean distance traveled by photons before the scattering becomes isotropic. The mean free path l* decreases as MC decreases and reaches an equilibrium at approximately 0.1 mm. This result means that the scattering becomes isotropic only after photons travel 2 to 5 times the distance of the diameter of a tracheid, which is 20 to 60 μm in air-dried softwood.
Fig. 3 Measured time-of-flight for 70 ps, 846 nm pulses transmitted through wood samples with moisture contents of about 12% (air dried), 30% (fiber-saturation point), and 120% (water saturated). The dots show the normalized number of photons arriving after transmission through the wood sample, and the solid lines are fit by using the diffusion model (see text). The sample thickness d is marked in each panel.
Fig. 4 Reduced scattering coefficient μs′ as a function of moisture content for wood. Low moisture content indicates dry wood. The sample thickness d is marked in each panel.
Fig. 5 Transport mean free path l* as a function of moisture content in wood. The sample thickness d is marked in each panel.
4. Light-scattering properties at each stage of wood drying
Figure 6 shows the generalized drying curve, in which the rate of drying as a percentage of the constant rate is plotted as a function of MC (blue open squares, right axis). In addition, the results for μs′ are also shown in Fig. 6 (black solid circles, left axis). The generalized drying curve reveals four drying stages (labeled A–D). The variation in the drying rate reflects the moisture state in the wood. The drying of wood as though the water were at a free surface is called “constant-rate drying” (stage A in Fig. 6) [22]. During stage A, the water on the wood surface evaporates. The constant drying rate changes to a slower rate at MC ~150%, which is the onset of the “first decreasing-rate period” (stage B in Fig. 6). During this stage, the free water inside large pores in wood is transferred to the wood surface so that the surface MC remains constant, resulting in a gradual decline in the drying rate as the bulk MC decreases. At the later part of the first decreasing-rate period (stage C in Fig. 6), condensed capillary water (exists with adsorbed water and strongly retained by capillary forces) at the surface evaporates [23]. Below the fiber-saturation point, when only water adsorbed in cell walls remains, the MC at the wood surface equilibrates with the ambient air and the MC inside the wood gradually reaches a constant value [24]. This period is called the second decreasing drying rate period (stage D in Fig. 6). Interestingly, the change in μs′ corresponds well to the change in drying rate.
Fig. 6 Drying rate ratio (blue open squares, right axis) and reduced scattering coefficient μs′ (solid circles, left axis) as a function of moisture content in wood. The sample thickness d is marked in each panel. Four drying stages are identified based on the slope of the drying-rate ratio: constant-rate period (stage A), the initial part of the first decreasing-rate period (stage B), the later part (stage C), and the second decreasing-rate period (stage D).
To investigate the surface water state on shorter time scales, we also measured the wood surface temperature, with the results shown in Fig. 7(a) (red open squares, right axis). The surface temperature is a decreasing linear function of the drying-rate ratio and reflects moisture states at the surface, which contributes to decreasing the drying rate of the wood. In particular, the surface temperature occasionally rises, indicating low MC at the surface [Fig. 7(b), upward arrow]. Interestingly, μs′ also increases at the same point as the surface temperature. This result indicates that μs′ increases when the surface MC is maintained low for a short time.
Fig. 7 Surface temperature (open squares, right axis) and (a) drying rate ratio (solid squares, left axis) and (b) reduced scattering coefficient μs′ (solid circles, left axis), all as a function of moisture content in wood.
5. Characteristics of light absorption during drying
Figure 8 shows the optical absorption coefficient μa as a function of MC (i.e., as the wood dries). The results show that μa tends to decrease as the wood dries. This is consistent with the results of D’Andrea et al. [13] and Bargigia et al. [14], who also reported a decrease in μa as wood dries. They measured μa with an MC below 30%, including only water adsorbed in cell walls. In the present study, we examine the variation in μa as the MC ranged from 200% to 10% and include both adsorbed water and free water. Thus, the same tendency is not obtained, which indicates that free water and adsorbed water have similar effects on μa. By definition, μa equals the absorption ratio per unit path length of light travel and depends on the volume concentration of the chemical composition. Thus, theoretically, μa must be an increasing linear function of the volume fraction of water, where the slope would correspond to the absorption coefficient of water. The result we obtain is similar to that reported by Hale et al. (4.33 × 10−3 mm−1) [20]. This suggests that the decrease in μa might be due to a decrease in the absorption assigned to (2ν1 + ν3) in the water molecule, which peaks at 970 nm.
Fig. 8 Optical absorption coefficient μa as a function of moisture content during wood drying. The sample thickness d is marked in each panel.
However, note that the EE (see experimental section) in Fig. 8 is much larger than that shown in Fig. 4, which is because μa is much more sensitive than μs′ to the RSS between the measured TRT and the simulated TOFD. This implies that not only the standard deviation but also the EE calculated from the RSS is effeective for accurately determining μa and μs′. Moreover, the RSS, EE, and SD depend on sample thickness, which suggests that a sample with more appropriate thickness will improve the accuracy with which the optical coefficients are determined. Specifically, thicker and lower MC sample has the lower EE, which imply such sample well corresponds the analysis model by detecting completely uniform angular distribution of photons after passing through the sample, under our measuremnt with the same angular selective detection for every sample using lens (omitted in Fig. 2). Hence, thicker sample with enough transmittance for photon counting will give us more accurate optical coefficients.
6. Light scattering patterns of drying wood
The results show that the scattering coefficient μs′ increases as the wood dries. Several authors also report that the air-dried wood has a larger scattering coefficient or attenuation due to scattering in comparison with water-saturated wood [15,16,25]. Andersson et al. explained that the larger scattering observed in air-dried wood is due to reduced index-matching upon drying (which translates into an increase in the difference in the index of refraction between scatterers and medium) [12]. The large difference Δn in the index of refraction between scatterers and medium reduces the forward-scattering intensity g. Conversely, a small mismatch in the index of refraction at the water–cell-wall interface (nwater = 1.33, ncw = 1.55, Δn = 0.22) results in low reflectance and high transmittance T = 99.4% at the surface (i.e., large forward-scattering intensity). With the small MC of wood, however, scattering from air–cell-wall or air–water interfaces are relatively large because of the large difference Δn = 0.33 and 0.55, respectively, and results in large reflectance and low transmittance (T = 98.0% and 95.3%, respectively). The end result is small forward-scattering intensity. Thus, an increase in Δn leads to a decrease in the forward-scattering intensity.
Theoretically, light propagation can be determined from the scattering coefficient μs and the anisotropy factor g (0 < g < 1), which describes the number of scattering events per unit length, and light-scattering intensity as a function of scattering angle; that is, g = 1 represents forward scattering and g = 0 represents isotropic scattering. As explained above, light scattering by large scatterers (when the scattering diameter is much larger than the wavelength λ of the light) can be described by geometric optics, whereas for small scatterers (when the scattering diameter is the same as or less than λ), Mie theory (scattering diameter ≈λ) [1, 2] or Rayleigh theory (scattering diameter < λ) [3] is applicable.
For scattering in wood samples of NIR light (λ = 800 to 2500 nm), the angular distribution of scattered intensity determined by the structure of the cell wall (for softwood, this is the tracheid structure, with a diameter of 10–50 μm) may be described by geometric optics. Kienle et al. [15] calculated μs and g due to tracheid by solving Maxwell’s equation for wet and dry wood. They assumed that cell lumens fill with water (n = 1.33) for wet wood and fill with air (n = 1.00) for dry wood, whereas the index of refraction of the cell wall remains at 1.55 regardless of whether the cell is filled with air or with water. Although they used the differing values of g = 0.959 and g = 0.850 for wet and dry wood, respectively, they obtained almost the same result for μs: for water-saturated wood, μs = 43.8 mm−1 and, for dry wood, μs = 44.6 mm−1 [15]. This result implies that the variation in the angular distribution of scattered intensity (variation in g) is the dominant factor governing the difference μs′ = μs(1 − g) between water-saturated and air-dried wood. The softwood used by Kienle et al. consists mainly of the tracheids (90 vol%), which provide the main contribution to scattering in the geometric-optic regime. Kienle et al. also describe the scattering coefficient μs, iso due to all other scattering media (pits, ray cells, rough border between lumen and wood cell-wall substance) [15].
In the present study, we divide small particles into two groups that can be explained as Mie and Rayleigh scatterers. For the Mie-scattering range, the scattering properties of the wood are defined by scattering from small parts of the wood with the micrometer-size structure, such as the latewood tracheids or pits. We calculated the scattering intensity of the smaller scatters by Mie theory [1,2]. The calculated scattering angles were determined by g = 0.92 and 0.84 for wet and dry wood, respectively.
We compare the intensity pattern of light scattered from small particles by using Mie theory [Fig. 9(a), blue line] and from the tracheid structure as calculated by Henyey–Greenstein function using the values obtained by Kienle et al. [15] [Fig. 9(a), red line]. The qualitative agreement between the two models shows that the change in the angular distribution of the scattered intensity for small particle and tracheid is similar.
Fig. 9 Angle distributions of scattered intensity for wet (full lines) and dry (broken lines) wood. The scattering patterns from the cell-wall structure (red lines) and from small particles in the cell wall (blue lines) are calculated based on the Henyey–Greenstein function using the values given by Kienle et al. and by Mie theory, respectively.
Figure 10 shows a schematic diagram of light scattering from the scale of the tracheid to that of the wood tissue under different MCs. Light scattering at the tracheid scale can be understood by considering the mean free path l* traveled by light, which for dried wood is 2 to 5 times larger than the tracheid diameter [Fig. 10(b-2)]. We conclude from this that the scattering at the tracheid scale still produces forward scattering [see inside of arcs in Fig. 10(b)]. However, this forward scattering clearly decreases when we include multiple scattering from multiple cell walls. The calculated mean free path l* shows that the distance at which light begins to undergo isotropic scattering is 10 to 25 times greater than the diameter of the tracheids [for water-saturated conditions; see Fig. 10(b-1)]. Thus, these results demonstrate that a decrease in MC modifies the index of refraction of the scatterers, which in turn modifies the scattering pattern due to single scattering.
Fig. 10 Schematic diagram of optical scattering on the scale (a) of tracheids and (b) of wood tissue for (1) water-saturated conditions and of (2) low moisture content.
We also investigated how tissue anisotropy and density affect optical scattering. For these experiments, samples were cut in two different directions (radial and tangential planes). However, measurements on samples cut in different directions gave almost the same result forμs′. Although the results may change by using the simulated solutions which take account of the tissue anisotropy, this indicates that, compared with MC, the impact of wood-tissue anisotropy on scattering is negligible, at least for the wood with its axis perpendicular to the tracheids. Similarly, we find that density also has negligible impact on scattering compared with MC, at least when considering a single, given species in the density range 0.41–0.53 g/cm3.
7. Water state and wood microstructure: effect on scattering
We have seen that the variation in μs′ differs for the various stages of drying. Table 1 suggests scattering pores in the wood for each stage of drying. In the constant-rate period (stage A), μs′ changes radically. Next, in the initial part of the first decreasing-rate period (stage B), μs′ changes gradually. In the later part (stage C), μs′ changes somewhat more rapidly. In the second decreasing-rate period (stage D), μs′ again changes radically. Andersson et al. reported similar results [12]; they demonstrated a radical increase in scattering from wood at the end of the drying period.
Table 1. Characteristic of reduced scattering coefficient μs' and the scattering characteristic of pores in wood for each drying stage. The radical change should be related to the moisture condition and to the wood microstructure, and these characteristics are presented in the same row with the corresponding drying stage.
In the wood-drying process, the spatial distribution of moisture affects the angular distribution of scattered intensity because it affects the nonuniformity of the index of refraction. Specifically, the dried sections in the wood structure most strongly affect the scattering (Table 1). During the constant-rate period (stage A), the free water in the subsurface layer evaporates and the dried sections play a leading role in scattering. During the initial part of the first decreasing-rate period (stage B), inside the wood, large pores such as dry tracheids or partly dry tracheids with small air bubbles start to appear and contribute to scattering. After this period, the subsurface layer releases the condensed water from the capillaries, and so the interior of the wood also starts to dry. On the subsurface layer, the small pores such as pits, ray cells, or the edge of the tracheids all begin to contribute to scattering. In the later part of the period (stage C), the surface layer releases any adsorbed water. Wood substrates also contribute to scattering as minute scatters. Similarly, the interior sections of the wood release the relatively strained water from inside small pores, rather than tracheids, leaving the small pores to scatter the light. In the second decreasing-rate period (stage D), the subsurface layer is already dry and only the interior of the wood continues to dry. The interior of the wood also releases adsorbed water, allowing all the pores of the wood to dry up and more strongly contribute to scattering.
The angular distribution of scattered intensity is related to the average size of scatterers in the wood (Table 1). Rayleigh scatterers produce isotropic scattering, whereas Mie scatterers and geometric-optic scatterers produce anisotropic scattering. As the stage of drying evolves, the anisotropy factor g changes drastically, which indicates a decrease in the average size of scatters. In going from drying stage A to D, the spatially averaged size of scatterers in wood decreases, which implies that the spatially averaged scattering pores in wood transform from geometric-optic scatters for high MC to Mie scatterers and Rayleigh scatterers for low MC. The scattering becomes more isotropic as drying evolves from stage A to D. However, the drastic change in μs′ during the constant-rate period (stage A) is not explained only by the strongly nonisotropic scattering of the scatters. It might also be affected by the surface roughness at the onset of drying and the isotropic patterns of the dry tracheid [15].
8. Conclusions
We used time-resolved near-infrared spectroscopy to investigate the optical properties of wood material over a wide range of moisture content from fully water saturated to air dried. Although we targeted on the wood sample, this paper may be the first study to reveal the time course of optical coefficients during the drying of polymeric material. Hence, our results would also helpful for drying process of other products containing water, such as solid food, pharmaceutical tablets and organic materials made of rubber or prastic. Also, to our knowledge, this may be the first detailed study to show the usefulness of optical scattering properties for other physical dominant phenomena relating microstructure and water, such as water absorption-desorption and coagulation process from protein denaturation, each of which is related to quality degradation, rotting and bacteria growth and texture and taste, respectively. Our results showed that knowledge of the reduced scattering and absorption coefficients should help in understanding the optical properties of polymeric material with a non-uniform moisture distribution because they are useful for simulating the propagation of light through polymeric material. Knowledge of the scattering coefficient and anisotropy factor should also help to determine microstructure of polymeric material by inspecting the angular distribution of scattered intensity and comparing them with known patterns from given structural scales. We demonstrate the potential for detecting changes in the structure of the fine pores that are two orders of magnitude smaller than the scattering wavelength (Rayleigh-scattering regime). Finally, although the diffusion equation only accounts for uniform polymeric material, it provides accurate results for inhomogeneous porous media.
Acknowledgments
This work is supported by Grant- in-Aids for Scientific Research from Japan Society for the Promotion of Science (Grant Nos. 25660135). The authors thank to Prof. Eiji Okada (Keio University) for valuable suggestions on the measurement geometry and the data analysis. We also thank to Prof. Hiroyuki Yamamoto (Nagoya University) for technical assistance.
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