Effect of annual rings on transmission of 100 GHz millimeter waves through wood

Abstract

For the application of millimeter wave (MMW) technique to nondestructive evaluation of wood, the effect of annual rings on the behavior of a 100 GHz MMW transmitted through wood was examined. The complex amplitude was measured for 2 mm thick flat- (LT) and quarter-sawn (LR) specimens of several species with different annual ring structures at 11 % moisture content, of which the density distribution was measured using X-ray radiography. For the LT specimens of all species and the LR specimens with small density fluctuation or with earlywood width smaller than the wavelength of the MMW (=3 mm), the amplitude and phase of the transmitted wave were similar to those of the wave without a specimen. For the LR specimen with large density fluctuation and with earlywood width close to or larger than the wavelength, the amplitude and phase were different from those of the wave without a specimen. All the measured complex amplitudes were well expressed using a diffraction model. It was concluded that the MMW is deformed by the density distribution, and then its components with periods shorter than the wavelength by diffraction are attenuated.

Introduction

Wood is widely used in industrial products such as furniture and structural members. The constant use of wood in industry, however, has been sometimes disturbed by the fluctuation of physical properties in wood because this fluctuation may cause defects in the wood products. For example, the abrupt change of wood density causes the structural defects such as raised and loosened grains. Thus, the physical properties should be evaluated nondestructively to control the quality of the wood products [1].

A millimeter wave (MMW) technique, which deals with electromagnetic waves in the 30–300 GHz frequency range, has considerable promise as a tool for the nondestructive evaluation (NDE) of the wood products, because it is inexpensive, non-contact, non-invasive, and compact. Recent studies have shown that this technique can obtain the transmission image of wood [2, 3] with higher spatial resolution than the conventional microwave technique in the 3–30 GHz frequency range [4–13]. However, the image presented no quantitative physical property of wood, because the effect of wood structure on the MMW transmission through wood was obscure.

The transmitted waves are probably affected by micro- and macrostructures [14]. The microstructure can be regarded as homogeneous because wood cells, such as the tracheids, fibers, and ray cells, have cross-sectional dimensions much smaller than the wavelength of the MMWs, 1–10 mm. The macrostructure such as annual rings can be regarded as inhomogeneous because annual ring widths are close to or longer than the wavelength.

The effect of the microstructure on the MMW transmission has been explained by dielectric properties of wood [15]. On the other hand, little attention has been paid to the effect of the macrostructure such as annual rings. The understanding of this effect may improve the measurement accuracy of the physical properties of wood.

The purpose of this study was to examine the effect of annual rings on the behavior of the MMWs transmitted through 2 mm thick flat- and quarter-sawn specimens of softwood and ring- and diffuse-porous hardwood species. The complex amplitude of the transmitted wave was measured using linear scanning and discussed based on a model in consideration of both the micro- and macrostructures.

Theory

Macrostructure such as the annual rings is formed by the aggregation of microstructure such as the wood cells. Thus, in this study, the effect of the macrostructure on the MMWs is treated as the distribution of the effect of the microstructure on them.

When an MMW is transmitted through a wood specimen, it will be locally affected by the complex permittivity due to the microstructure and be diffracted in the specimen simultaneously. However, it is difficult to build a model for such a phenomenon. Let us consider that the diffraction starts at the plane which is perpendicular to the wave propagation direction and includes the center of the specimen in the direction, while the wave is affected by the local complex permittivity in all the position in the specimen. Thus, it can be assumed in a right-handed Cartesian coordinate system (x, y, z) that a wave is rectilinearly propagated through a specimen in the z direction, while being locally affected by its complex permittivity and is subsequently diffracted in free space at a distance of half the optical length of the specimen along the z-axis in the specimen and at a distance of l ₀ (=1 mm) in air (Fig. 1) if the double counting of the phase was corrected. The wave was assumed to be polarized constantly and linearly in the y direction, to illuminate perpendicularly the specimen whose longitudinal direction is in the y direction (Fig. 1) and to be expressed as a complex amplitude. The complex permittivity of the specimen in the longitudinal direction was assumed to be constant along the z-axis and represented by ε(x, y).

Fig. 1

A diffraction model of MMW during propagation through a specimen. V _i(x, y), f(x, y), and g(x, y), complex amplitudes of an incident, a rectilinearly propagated, and a diffracted waves, respectively; $d\text{Re} \sqrt {\bar{\varepsilon }} /2$ and l ₀, distances of the diffraction areas in the specimen and air, respectively; ε(x, y), complex permittivity distribution in the specimen; d and $\bar{\varepsilon }$, the thickness and averaged complex permittivity of the specimen, respectively

The complex amplitude of the wave propagated rectilinearly through the specimen is formulated as follows [16, 17]:

$$ f(x,y) = S_{21} (x,y)V_{\text{i}} (x,y), $$

(1)

where V _i(x, y) represents the complex amplitude of the incident wave and S ₂₁(x, y), the S-parameter which is formulated as follows:

$$ \begin{gathered} S_{21} (x,y) = \frac{{1 - \left\{ {R(x,y)} \right\}^{2} }}{1 - R(x,y)\Uplambda (x,y)}\Uplambda (x,y), \hfill \\ \left( \begin{gathered} R(x,y) = \frac{{1 - \sqrt {\varepsilon (x,y)} }}{{1 + \sqrt {\varepsilon (x,y)} }}, \hfill \\ \Uplambda (x,y) = \exp \left\{ {{\text{i}}2{{\uppi}} \left (\sqrt {\varepsilon (x,y)} - {1} \right)d/\lambda } \right\} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

(2)

where d represents the specimen thickness (=2 mm) and λ the wavelength of the MMW in air (=3 mm).

The diffraction of the waves in free space is explained by Fourier optics [18], in which the complex amplitude of a diffracted wave at output xy-plane is related to that of a wave at input xy-plane using a transfer function. When the complex amplitude of the wave at input xy-plane is equal to f(x, y) (Fig. 1), that of the diffracted wave, g(x, y), can be formulated as follows:

$$ g(x,y) = \exp ( - {\text{i}}2{{\uppi}}l /\lambda )\int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {H(v_{x} ,v_{y} )F(v_{x} ,v_{y} )\exp \left\{ {{\text{i}}2{{\uppi}}(v_{x} x + v_{y} y)} \right\}} } {\text{d}}v_{x} {\text{d}}v_{y} , $$

(3)

where v _x and v _y represent the spatial frequencies in the x and y directions, respectively, exp(−i2πl/λ) represents the factor which prevents the double counting of the phase, and F(v _x , v _y ), H(v _x , v _y ), and l represent the Fourier transformation, transfer function of f(x, y), and the optical length between input and output planes, respectively, which are formulated as follows:

$$ F(v_{x} ,v_{y} ) = \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {f(x,y)\exp \left\{ { - {\text{i}}\,2\pi (v_{x} x + v_{y} y)} \right\}{\text{d}}x{\text{d}}y} } , $$

(4)

$$ H(v_{x} ,v_{y} ) = \exp \left\{ {{\text{i}}\,2\pi \left( {\frac{1}{{\lambda^{2} }} - v_{x}^{2} - v_{y}^{2} } \right)^{\frac{1}{2}} l} \right\}, $$

(5)

$$ l = l_{0} + \frac{{d\text{Re} \sqrt {\bar{\varepsilon }} }}{2}, $$

(6)

where $\bar{\varepsilon }$ represents the average complex permittivity of the specimen,

$$ \bar{\varepsilon } = \frac{{\int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {\varepsilon (x,y){\text{d}}x{\text{d}}y} } }}{{\int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {{\text{d}}x{\text{d}}y} } }}. $$

(7)

Equations (1)–(7) define a diffraction model in this study and are summarized as follows:

• 1. The incident wave is deformed by the complex permittivity distribution (Eqs. 1, 2).

• 2. The diffraction attenuates the deformed wave components with periods shorter than the wavelength, λ (Eqs. 3–7), because the transfer function H(v _x , v _y ) acts as an attenuator if the spatial period of f(x, y), $(v_{x}^{2} + v_{y}^{2} )^{ - 1/2}$, is less than λ.

Experimental set-up

Specimens

Flat- (large surface parallel to LT plane, LT) and quarter-sawn (LR) specimens, 100 mm square and 2 mm thick, were prepared from sound wood of softwoods, hinoki (Chamaecyparis obtusa), sugi (Cryptomeria japonica), and akamatsu (Pinus densiflora); ring-porous hardwoods, kiri (Paulownia tomentosa) and keyaki (Zelkova serrata); and diffuse-porous hardwood, tochinoki (Aesculus turbinata). The specimens were stored over a saturated NaBr solution in desiccators at 27.5 °C until their moisture content reached 11 %. A Cartesian coordinate system (x, y, z) was defined on the specimen with its larger surface parallel to the xy-plane and its center on the z-axis.

Density measurement with X-ray radiography

The density of specimens was measured using X-ray radiography. Figure 2 shows an experimental set-up for X-ray radiography using an X-ray television inspection equipment (FI-30, Shimadzu). The X-rays were generated by an X-ray tube at a tube voltage and current of 32 kV and 1.0 mA, respectively. The rays that passed through the specimen were converted into a visible image by an image intensifier (I.I.). The image was captured by a CCD camera (640 × 480 pixel, 256 gray levels) and sent to a computer to obtain the intensity of the rays, I(X, Y), in a Cartesian coordinate system (X, Y), where X and Y are in the width and height directions of the image, respectively (Fig. 2). The specimen was set so that its center was at (X, Y) = (320, 240) and its longitudinal direction was parallel to the Y-axis.

Fig. 2

Experimental set-up for X-ray radiography of wood specimens

The intensity of the rays that passed through the specimen at (X, Y) is formulated as follows:

$$ I(X,Y) = I_{0} (X,Y)\exp \left\{ { - \mu \rho (X,Y)d} \right\} , $$

(8)

where I ₀(X, Y) represents intensity of the rays at (X, Y) without a specimen, ρ(X, Y) the air-dry density of the specimen averaged along the z-axis, and μ the mass attenuation coefficient. Assuming that μ is constant for all the specimens, Eq. (8) can be transformed into:

$$ \Upgamma = - \mu \rho_{\text{a}} d , $$

(9)

where Г and ρ _a represent the averages of the logarithmic intensity attenuation and air-dry density over the coverage area of the image, respectively, and they are expressed using discrete values for each pixel as follows:

$$ \Upgamma = \frac{{\sum\nolimits_{X = 1}^{640} {\sum\nolimits_{Y = 1}^{480} {\ln \frac{I(X,Y)}{{I_{0} (X,Y)}}} } }}{640 \times 480} , $$

(10)

$$ \rho_{\text{a}} = \frac{{\sum\nolimits_{X = 1}^{640} {\sum\nolimits_{Y = 1}^{480} {\rho (X,Y)} } }}{640 \times 480} . $$

(11)

The relation of the average attenuation, Г, to the product of the average air-dry density and the thickness, ρ _a d, was obtained for all specimens in a preliminary experiment (Fig. 3). There was a good linear relation between them, and the mass attenuation, μ, was determined to be 0.4640 cm²/g using Eq. (9) from the slope of the regression line.

Fig. 3

Relation of average attenuation, Г, to product of average air-dry density, ρ _a, and thickness, d, for all specimens

The density of a specimen, ρ(X, Y), was estimated using Eq. (8) for 640 × 480 pixel points and resampled to ρ(x, y) with 1024 × 1024 pixels over an area of −24 mm ≤ x ≤ 24 mm and −20 mm ≤ y ≤ 20 mm by Shepard’s method [19] using the X-ray image of a square mesh of 0.35 mm thick steel wires with 5 mm spacing to perform a fast Fourier transform (FFT) and an inverse FFT (IFFT) in the discussion described below. Furthermore, an oven-dry density, ρ ₀(x, y), was obtained from the air-dry one, ρ(x, y), using the equation, ρ = 1.11ρ ₀/(1 + 0.094ρ ₀), for wood at 11 % MC [20].

Oven-dry density, ρ ₀(x, y), obtained in this way is shown in Figs. 4 and 5 as grayscale images for the LT and LR specimens, respectively. The LT specimens showed smooth tones of gray (Fig. 4), while the LR specimens showed remarkable stripped patterns due to the annual rings (Fig. 5).

Fig. 4

Images of oven-dry density distributions in xy-plane for LT specimens

Fig. 5

Images of oven-dry density distributions in xy-plane for LR specimens

The complex permittivity in the longitudinal direction of wood at 11 % MC, ε(x, y), was determined as follows:

$$ \varepsilon (x,y) = 1 + \frac{{0.760(\varepsilon_{\text{WS}} - 1)\rho_{0} (x,y)}}{{1 + 0.094\rho_{0} (x,y)}}. $$

(12)

This equation was derived from a parallel capacitor formula, ε = 1 − v _WS + v _WS ε _WS [21, 22], where ε _WS represents complex permittivity of wood substance and v _WS, volume fraction of wood substance which was expressed as 0.760ρ ₀/(1 + 0.094ρ ₀) at 11 % MC [23]. In an experiment described in another article submitted to this journal, Eq. (12) was confirmed to be well fitted to the complex permittivity in the longitudinal direction for LT specimens of nine wood species at 11 % MC measured using a free space method in the range of ρ ₀ = 0.2–1.0 g/cm³, and the complex value of ε _WS was estimated to be 4.1 (1 + 0.12i).

Apparatus and measurement

Figure 6 shows an experimental set-up of an MMW measurement system arranged in a right-handed Cartesian coordinate system (x, y, z). An MMW signal of 100 GHz with an average output power of 10 mW was oscillated in a Gunn diode and divided into two signals, measurement and reference, by a directional coupler. The reference signal was sent to an in-phase/quadrature mixer (IQ-mixer) through a dielectric waveguide. The measurement signal was modulated using a PIN diode at a switching frequency of 100 kHz and passed through a phase shifter. The wave linearly polarized in the y direction was transmitted by a conical horn antenna toward the z direction and collimated using a Teflon plano-parabolic lens with a diameter of 80 mm and a focal length of 33 mm. The wave that passed through the specimen was received by an open-ended rectangular waveguide with an inner cross-section of 2.54 mm in the x direction and 1.27 mm in the y direction and sent to the IQ-mixer, where the in-phase (I) and quadrature (Q) components were discriminated by comparing the measurement and reference signals. The signals from the IQ-mixer were processed into the voltages, V _I and V _Q, in a lock-in amplifier. The front face of the waveguide, excluding the aperture, was covered with MMW absorbent material to prevent reflections from the waveguide and surrounding apparatuses.

Fig. 6

Experimental set-up of an MMW measurement system arranged in a right-handed Cartesian coordinate system (x, y, z). 1 Gunn diode (100 GHz), 2 Isolator, 3 Directional coupler, 4 PIN diode, 5 Phase shifter, 6 Conical horn antenna, 7. Teflon plano-parabolic lens, 8 Specimen, 9 Open-ended rectangular waveguide (inner cross-section: x × y = 2.54 × 1.27 mm) covered with millimeter wave absorbent except open end, 10 Quadrature phase shifter, 11 Switching frequency generator (100 kHz), 12 Mixer, 13 Lock-in amplifier, 14 IQ-mixer, and 15 Dielectric waveguide. V _I and V _Q, output voltages of in-phase (I) and quadrature (Q) components of an MMW signal

The complex amplitude of the MMW was determined as:

$$ V = V_{\text{I}} + {\text{i }}\left( {\frac{{V_{\text{I}} }}{{\tan (\xi_{\text{Q}} - \xi_{\text{I}} )}} - \frac{{V_{\text{Q}} }}{{(A_{\text{Q}} /A_{\text{I}} ){ \sin }(\xi_{\text{Q}} - \xi_{\text{I}} )}}} \right) , $$

(13)

where A _I, A _Q, and ξ _I, ξ _Q are the amplitude and phase of the sinusoidal functions obtained as the regression lines of the relation between V _I, V _Q and the phase controlled by the phase shifter and were determined in a preliminary experiment.

The waveguide was set at (y, z) = (0, 113 mm) and moved along the x-axis in steps of 1 mm within a range of −20 mm ≤ x ≤ 20 mm for line scanning with and without a specimen (Fig. 6). Furthermore, it was set at z = 110 mm for area scanning of an incident wave (−24 mm ≤ x ≤ 24 mm and −20 mm ≤ y ≤ 20 mm, 1 mm step along the x- and y-axes) (Fig. 6).

Figure 7a and b show the distributions of the amplitude, |V _i(x, y)|, and phase, argV _i(x, y), of the incident MMW at z = 110 mm. Figure 7a shows a bright region in the center and concentric bright rings around it, so called main lobe and side lobes, respectively [24–27]. In Fig. 7b, there are three concentric regions, gray, dark gray, and black, corresponding to one main and two side lobes. These distributions were obtained over the area of −24 mm ≤ x ≤ 24 mm and −20 mm ≤ y ≤ 20 mm with 49 × 41 pixels. To perform the FFT and IFFT in the discussion described below, they were resampled to the distributions with 1024 × 1024 pixels using Shepard’s method [19] and the complex amplitude, V _i(x, y), was obtained.

Fig. 7

Images of a amplitude, |V _i|, and b phase, argV _i, of incident wave at z = 110 mm, and c distributions of amplitude, |V ₀|, and phase, argV ₀, of wave without a specimen along x-axis at (y, z) = (0, 113 mm)

Figure 7c shows the distributions of the amplitude, |V ₀|, and phase, argV ₀, of the MMW without a specimen along x-axis at (y, z) = (0, 113 mm). The distribution, |V ₀|, has a large peak in center and two small peaks near either end, and argV ₀ has three plateau regions in the range of −12 mm ≤ x ≤ 12 mm,−20 mm ≤ x ≤ −15 mm, and 15 mm ≤ x ≤ 20 mm, corresponding to one main and two side lobes.

Results and discussion

The amplitude, |V|, and phase, argV, of the transmitted MMWs measured along x-axis at (y, z) = (0, 113 mm) are shown in Figs. 8 and 9 for the LT and LR specimens, respectively. For the LT specimens of all species (Fig. 8) and LR specimens of hinoki, keyaki, and tochinoki (Fig. 9a, e, f), the variation patterns of |V| and argV were similar to those of |V ₀| and argV ₀ in Fig. 7c, though |V| was slightly attenuated in comparison with |V ₀| and argV was shifted away from argV ₀ toward the positive direction. For the LR specimens of sugi, akamatsu, and kiri (Fig. 9b–d), on the other hand, the patterns of |V| and argV were different from those in |V ₀| and argV ₀ (Fig. 7c).

Fig. 8

Distributions of amplitude and phase of transmitted MMWs for LT specimens measured along x-axis at (y, z) = (0, 113 mm). V, measured complex amplitude; g _s(x), complex amplitude estimated from a diffraction model

Fig. 9

Distributions of amplitude and phase of transmitted MMWs for LR specimens measured along x-axis at (y, z) = (0, 113 mm). V, measured complex amplitude; g _s(x), complex amplitude estimated from a diffraction model

The complex amplitude of a diffraction model, g(x, y), was estimated for z = 113 mm by substituting the interpolated complexes, V _i(x, y) and ρ ₀(x, y), for Eqs. (1)–(7) and (12), of which Eqs. (3), (4), and (7) were computed using the IFFT, FFT, and numerical integration, respectively, in a discrete-finite form. In consideration of the aperture area of the receiving waveguide in Fig. 6, the simple moving averaged complex amplitude, g _s(x, y), was calculated as simple moving averages of g(x, y) over 2.54 mm in the x direction and 1.27 mm in the y direction. The amplitude, |g _s(x)|, and phase, argg _s(x), estimated from the diffraction model are shown as solid and dotted lines, respectively, in Figs. 8 and 9. The plots, |V| and argV, for all the specimens matched well the lines estimated from the model (Figs. 8, 9). This indicates that the diffraction model is valid for MMWs transmitted through wood specimen and that the waves may be deformed by the complex permittivity distribution due to the density distribution as shown in Figs. 4 and 5, and their components with periods shorter than the wavelength of the MMWs, λ (=3 mm), must be attenuated.

The LT specimens of all species and LR specimens of hinoki, keyaki, and tochinoki had small density fluctuations in the xy-plane or earlywood widths smaller than λ (Figs. 4, 5a, e, f), while the LR specimens of sugi, akamatsu, and kiri had large density fluctuations in the xy-plane and earlywood widths close to or larger than λ (Fig. 5b–d). This is why the amplitude and phase of the transmitted wave for LR specimens of sugi, akamatsu, and kiri were different from those of the wave without a specimen on the basis of the diffraction model.

Conclusion

The complex amplitude was measured for 2 mm thick flat- (LT) and quarter-sawn (LR) wood specimens with different annual ring structures, of which the density distribution was measured using X-ray radiography. For the LT specimens of all species and the LR specimens with small density fluctuation or with earlywood width smaller than the wavelength of the MMWs, λ (=3 mm), the amplitude and phase of the transmitted wave were similar to those of the wave without a specimen. For the LR specimen with large density fluctuation along its large surface and with earlywood width close to or larger than λ, the amplitude and phase were different from those of the wave without a specimen. All the measured complex amplitudes were well expressed using a diffraction model. It was concluded from these findings that the incident wave is deformed by the density distribution and then its components with periods shorter than λ by the diffraction are attenuated. Furthermore, these findings will not only improve the measurement accuracy of the physical properties of wood, but also provide a new technique such as the detection of the annual rings without scanning. Further study is necessary to examine these effects for the specimens whose large surfaces are not parallel to LT and LR planes.