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Terahertz anisotropy in fascia and lean meat tissues

Open Access Open Access

Abstract

Terahertz (THz) spectroscopy provides multifaceted capabilities for observing low-energy responses of macromolecules, cells and tissues, understanding THz biophysical effects, and expecting to realize the application of THz technology in biomedicine. However, its high-frequency characteristics of limited penetration depth and strong absorption of water in the body comparable to microwaves are impeding the proliferation of THz spectroscopy. Here we show that THz spectroscopy makes possible the observation of THz anisotropy phenomena for the first time in fascia and lean tissue. Through optical microscopy, we infer that the microscopic mechanism of THz anisotropy comes from the periodic stripe structure of the biological tissue. The above related experimental findings may be expected to promote the application of THz technology in biomedicine.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Terahertz (THz) electromagnetic radiation with the frequency range of 0.1-10 THz and wavelengths between 30 µm and 3 mm is located between millimeter waves and near-infrared light. With the rapid development of THz technology, the power of THz sources and the sensitivity of detectors have been significantly improved [16], further promoting the application of THz wireless communication [7,8], nondestructive testing [9,10], environmental monitoring and other fields [11,12]. Due to the low photon energy and high penetration capability of THz radiation, THz technology can also be used in biomedical applications [1315]. Biological macromolecules interactions are the key drivers of major vital phenomena and lesion generation, and THz photon energy covers the energy level range of the spatial conformation of biological macromolecules. Using THz waves one can obtain important information such as spatial conformation that directly represents the function of biological macromolecules, which cannot be detected by other electromagnetic frequency bands [1618]. Therefore, THz technology has the potential to provide advanced technical means for future disease diagnosis and effective intervention.

The skin covers the entire surface of the body and is one of the largest organs in the body, accounting for approximately 16% of body weight. It is the tissue that is in direct contact with the external environment and has protective, sensory, secretory, excretory, and respiratory functions. The skin area of adults is about 1.2-2.0 m2. The thickness of the skin varies throughout the body with an average thickness of 0.5-4.0 mm, with the back, collar, palms, and soles of the feet being the thickest, and the armpits and face being the thinnest [19]. Although the thickness of the skin varies, it can be divided into two layers: the epidermis and the dermis, which are connected to the deeper layers by the subcutaneous tissue [20]. THz spectroscopic imaging has been applied in the fields of skin cancer detection, burn degree discrimination, and transdermal drug delivery detection [2123]. The high sensitivity of THz waves to water has been mainly utilized in these works as a basis for disease diagnosis. Cancerous tissues have a higher water content than normal tissues. During burn detection, burn depth is discerned by quantifying the wound edema and its spatial-temporal distribution. The process of drug diffusion is usually accompanied by changes in the water content inside the biological skin. THz imaging can be used as a label-free method to assess the efficiency of different transdermal drug delivery. The decisive factor is that the tissue water content establishes a hydration model while ignoring the structural characteristics of different tissues. In addition, skin aging detection has also become a key research component. Skin aging is a complex multifactorial process whose baseline rate is genetically determined, but environmental factors such as light exposure can accelerate skin aging [24]. Skin aging is more than just producing aesthetic adverse effects such as wrinkles and hyperpigmentation. More importantly, skin aging leads to variations in the internal structure of the skin, increasing its fragility, reducing its ability to heal itself, greatly increasing the risk of toxicological skin damage, and increasing the risk of developing various diseases. Therefore, we should pay more attention to the structural changes of the skin in addition to the effects of skin hydration. Taking full advantage of the non-invasive and non-ionizing characteristics of weak-field THz waves, it enables nondestructive skin inspection.

In this work, we exploit the polarization detection characteristics of the THz pulse technique to investigate the various anisotropic phenomena in different biological tissues [25]. We prepare biological samples with uniform thickness and flat surface by using the paraffin sectioning technique, while excluding the effect of water content in biological samples on the experimental results. So that the results of the transmission experiments are only related to the cell morphology and arrangement. We use a THz time-domain (THz-TDS) system to measure the transmission signals of pig dorsal skin, pig hoof skin, lean meat longitudinal cut, lean meat transverse cut, fascia, and fat meat with the thickness of 60 µm, and conclude that the pig dorsal skin has the lowest transmission efficiency, while the fat has the highest transmission capability. Rotating the sample azimuthal angles with 10° intervals, we find that the striped samples possess THz anisotropic phenomena, such as fascia and lean longitudinal cuts. A skin model is developed using the Bruggeman effective medium approximation (EMA) model to qualitatively explain the cause of the anisotropy behavior.

2. Method

2.1 Experimental setup

The THz-TDS system is driven by a femtosecond fiber laser oscillator (model: FemtoFErb FD 6.5, Topica) with a wavelength of 1560 nm, a repetition frequency of 100 MHz, a pulse width of 60 fs, and an effective frequency range of 0.2-2 THz. The femtosecond laser excites the InGaAs semiconductor to produce free electrons, which are accelerated by the bias voltage to radiate THz waves. The detection process is the inverse of THz generation, converting THz waves into photocurrents. The diagrammatic sketch is shown in Fig. 1(a). The radiated THz pulses are first collimated by a TPX lens (L1, focal length = 50.0 mm), and focused onto the sample holder via another TPX lens (L2, focal length = 100.0 mm). The focused spot is about 4 mm in diameter. The THz waves carrying the sample information are refocused onto the detector through the third (L3, focal length = 100.0 mm) and the fourth TPX lenses (L4, focal length = 50.0 mm).

 figure: Fig. 1.

Fig. 1. Schematic diagram of the experimental setup and the sample preparation. (a) The THz time-domain spectrometer. L1-L4: TPX lens. (b) Paraffin sections preparision process and the measurement scene.

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Biological samples have a high water content, and water is highly absorbent to THz radiation, which can affect the experimental results, so paraffin sections are used to prepare the samples. It is possible to exclude the influence of water which in biological samples on THz radiation, and to ensure a flat surface and uniform thickness. Before the measurement, it is found that the slide has a strong attenuation effect on the THz waves, so we design a new sample substrate using an acrylic plate, as drawn in Fig. s1a-b. Fig. s1c verifies that there is no anisotropy phenomenon in the acrylic plate. The sample thickness is only 60 µm could not be removed completely from the acrylic plate, so the reference signal is a single layer of acrylic plate, and the sample signal is a combination of the bio-paraffin section and the acrylic plate, as illustrated in Fig. 1(b). All these measurements are carried out in ambient air.

2.2 Sample preparation process

The process of preparing paraffin sections is mainly divided into three steps: tissue fixation, dehydration and transparency, and embedding sections [26]. Paraformaldehyde is used to soak the biological samples overnight to ensure the structural integrity of the cells. After cutting and trimming the surface of the sample using a surgical blade, the excess paraformaldehyde solution is rinsed with running water. Ethanol solutions with concentration gradients of 70%, 80%, 90%, and 100% are used to soak the samples one at a time for 1.5 h, where the 100% ethanol solution needed to be soaked twice to displace the entire paraformaldehyde solution. Xylene I and Xylene II are used to replace the ethanol solution, and the length of each soaking is 15 min. Melting paraffin during the rehydration and transparency process, the melting point of paraffin I is 54-56°C, and the melting point of paraffin II and paraffin III is 56-58°C. The samples are soaked in paraffin for 1 hour each in turn. After the soak is completed, the samples are placed in the embedding box and refrigerated. Biological tissues are completely embedded in melted paraffin wax. After waiting for the paraffin to solidify, the bio-samples are sectioned using a microtome, and the section thickness is set at 60 µm. The biological tissue is located in the center of the sample, surrounded by paraffin, with no paraffin on the top and bottom surfaces.

2.3 Data processing

The experimental data are not acquired on the same day and the power of the lasers varied slightly, and we pre-processed the data according to Eq. (1) when comparing the transmission efficiency of different biological samples.

$${E_{nor\_sam}} = \frac{{{E_{sam}}}}{{{E_{\max \_ref}}}}$$
where, Esam is the THz time-domain signal of different biological samples, and Emax_ref is the reference signal of the corresponding sample,.

The refractive index of the sample is obtained from Eq. (2).

$$n(w) = \frac{{c\phi (w)}}{{wd}} + 1$$
where, ϕ (w) is the phase difference between the reference signal and the sample signal; d is the sample thickness, and c is the speed of light in free space.

We also calculate the equivalent refractive index using Eq. (3).

$$n = \frac{{\Delta t \cdot c}}{d} + 1$$
where, $\Delta t$ is the THz temporal waveform peak value time difference between the sample and the reference signals.

3. Results

3.1 THz transmission response

We measure the time-domain signals of pig dorsal skin, pig hoof skin, fat, lean longitudinal cut, lean transverse cut and fascia, respectively. The reference signal is measured before the sample signal. Figure 2(a) shows the attenuation effect of THz waves for different bio-samples calculated by dividing the sample signal by the peak-to-peak value of the reference signal. By comparing the maximum value of the THz electric field, we find that the THz attenuation of the fat is the smallest, while the pig dorsal skin is the largest, and the attenuation for the fascia and lean tissues are almost the same. It is noteworthy that the attenuation of the THz waves differs significantly between the pig hoof skin and the pig dorsal skin, despite the fact that they are both skin samples. Compared with their fresh tissue, it can be found that the structure of pig hoof skin is looser than that of pig dorsal skin. Therefore, for the same thickness of the same biological samples, the THz field amplitude can, to some extent, reflect the structure of the biological sample.

 figure: Fig. 2.

Fig. 2. THz transmission response for different tissues. (a) THz temporal waveforms for pig dorsal skin, pig hoof skin, fat, lean longitudinal cut, lean transverse cut and fascia, respectively. (b) Corresponding equivalent refractive indices.

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Using the sample data in Fig. 2(a) combined with the reference signal, we can calculate the refractive indices corresponding to these tissues. As displayed in Fig. 2(b), the THz refractive indices of our samples can be obviously divided into two groups in the 0.2-2.0 THz frequency band. The maximum refractive index of the pig dorsal skin is ∼2.3, while that of other tissues is ∼1.4-1.8. Paraffin slices form a thin film layer on the acrylic plate, and THz wave irradiation on the sample produces a Fabry-Perot effect [2729]. The change on the surface of the biological sample leads to a change in the interference pattern. The refractive index is oscillations relative to the reference spectrum of the acrylic plate. Even so, we can obtain the macroscopic THz electromagnetic response of these tissues by THz time-domain spectroscopy. This may be very helpful for the early diagnosis and treatment of psoriatic arthritis and other related skin diseases.

3.2 THz anisotropic phenomena in biological samples

Striated structures are present on the surface of paraffin sections of the lean and fascial tissues. Figure 3(a) exhibits s the azimuthal relationship between the THz wave and the striated structure on the bio-sample section surface. We define the azimuth angle as 0 degrees when the THz wave polarization direction is perpendicular to the sample striated structure. The ellipticity of the generated line polarization is ∼81.2. A set of time-domain signals is recorded at 10° azimuthal intervals for each biological sample. We plot their peak-to-peak values with the red line and the equivalent refractive index with the blue line in Fig. 3(b)-(g). We find that the azimuthal angle dependence patterns for different samples are various [30]. Some have no dependence on the angle, while others show various anisotropy phenomena. In comparing the anisotropic phenomena of different biological samples, we ensure that the THz electric field $\Delta PK$ is spaced at about 1 a.u., allowing different samples to be compared at a uniform scale. Figure 3(b) and Fig. 3(c) are for pigskin tissues, and they have no obvious anisotropy phenomena, where the $\Delta PK$ value for the pig dorsal skin is smaller than that of the pig hoof skin. Their refractive indices are also basically circular in shape. Figure 3(d) is lean longitudinal tissue, while Fig. 3(e) is lean transverse cut tissue. When the lean meat is cut longitudinally, there is a clear anisotropy, while for the transverse cut case, the THz anisotropy phenomenon is not obvious. When the lean meat is transected, there is almost no anisotropy phenomenon in the azimuthal angles from 0° to 180°, whereas there is an insignificant diminution of the peak value at 240°. When the lean meat is cut longitudinally, the azimuthal angle of 90-270 degrees refractive index is significantly greater than that of 0-180 degrees. However, when the lean meat is cut transversely, its refractive index shows an almost perfect circle. Figure 3(f) gives the fascial tissue, whose anisotropy is similar to that of the lean longitudinal tissue, both of which are shaped like “peanuts”. The refractive index at 150-330 degrees azimuth is slightly greater than that at 60-240 degrees azimuth. The trend in refractive index is less pronounced than in the longitudinal cut of lean meat, and we speculate that the possible reason is due to the laminar structure of the fascia. Figure 3 g shows the fat tissue, which has no anisotropy phenomenon, but the transmitted THz peak amplitude is the largest among all the biological samples, again proving that the fat tissue has the least attenuation effect on the THz radiation. The refractive index of the fat also shows an approximate perfectly round shape.

 figure: Fig. 3.

Fig. 3. Biological sample anisotropy phenomenon. (a) Schematic diagram of THz wave polarization direction and streak direction of biological samples. The peak-to-peak transmitted THz electric field amplitude as a function of the azimuthal angle for different biological samples. (b) Pig dorsal skin. (c) Pig hoof skin. (d) Lean longitudinal cut. (e) Lean transverse cut. (f) Fascia. (g) Fat meat.

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3.3 Optical microstructures of biological samples

To further investigate the cause of the THz anisotropy phenomenon in biological samples, we observe the microstructure of biological paraffin sections using an optical microscope (Model: SDPTOP MX4R). Figure 4(a)-(f) shows the microscopic images for pig dorsal skin, pig hoof skin, lean longitudinal cut, lean transverse cut, fascia and fat meat, respectively. The microscopic images of pig dorsal skin have no obvious structural features, while the observation of the pig hoof skin reveals the presence of porous structure, which explains why the transmission amplitude of the pig hoof skin is higher than that for the pig dorsal skin from the microstructural level. The microstructure of the lean longitudinal cuts and fascia have a distinct striated structure, being consistent with the anisotropy phenomenon. The striated structure of the fascia (Fig. 4(e)) is more pronounced and the anisotropy phenomenon is more centrally symmetrical than the behavior in the lean longitudinal cuts (Fig. 4(c)). Two structures exist in the microscopic images of the lean transverse samples. A cross-section of muscle cells is shown in Fig. 4(d) with a circular and uniform arrangement, while a streak-like structure in Fig. 4(c), which exhibits an introduced artifact during the sample cutting process. This results in the presence of an insignificant anisotropy in the lean transected tissue. If the lean transected sample is prepared to contain only the cross-sectional structure of the severed muscle cells, there would be no anisotropy in the lean transected tissue. It is very difficult to prepare such samples, so we need to further study the preparation methods of samples. The microscopic images of the fat tissue sections well verify our speculation, and we observe from Fig. 4(f) that the fat cells are basically round and uniformly arranged. There is no anisotropy phenomenon. Since we have excluded the effects of the water and thickness of the samples, the fat cells are the most loosely arranged and therefore have the largest transmission amplitude of the THz waves. Consequently, we can safely claim, to some extent, that THz spectroscopy can be used to investigate cell structure and its arrangement.

 figure: Fig. 4.

Fig. 4. Optical microscopic images for (a) pig dorsal skin, (b) pig hoof skin, (c) lean longitudinal cut, (d) lean transverse cut, (e) fascia, and (f) fat meat, respectively.

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3.4 THz bio-anisotropy model

To further understand the origin of the THz anisotropy, we borrow a birefringence model stem from ordered condensed matter [31,32]. Crystals are a special condensed state of matter, generally presenting a solid phase, characterized by a certain regularity in appearance and an orderly and periodic arrangement of internal atoms, which are mutually exclusive. Periodicity or symmetry in the microstructure of a crystal leads to anisotropy phenomena in its macroscopic physical properties. Numerous experiments have demonstrated that a beam of light incident onto a crystal is usually divided into two beams propagating through the crystal, known as birefringence [33]. The light that obeys the law of refraction in homogeneous media is called ordinary light with its light speed defined as of vo, while that does not obey the law is extra-ordinary light using ve for its speed. The optical wavefront of e-light is a rotating ellipsoidal surface whose axis is the optical axis z. The propagation speed ve(ξ) of e-light is related to the azimuth angle ξ, as shown in Eq. (4).

$${v_e}(\xi ) = \left\{ {\begin{array}{{c}} {{v_o},\textrm{ (}\xi \textrm{ = 0)}}\\ {{v_e},\textrm{ (}\xi \textrm{ = }\frac{\pi }{2}\textrm{)}} \end{array}} \right.$$
When the length scale of the optically homogeneous material of particles is smaller than the wavelength of light in the medium, the optical properties of the composite can usually be approximated by a uniform effective medium. The effective medium can be simulation using the generalized anisotropic Bruggeman EMA model [3436]. We can derive the model from the generalized form of Maxwell-Garnett's effective medium approximation employing an expression for the ellipsoidal particle polarizability [37].
$$\frac{{\varepsilon - {\varepsilon _0}}}{{{\varepsilon _0} + L(\varepsilon - {\varepsilon _0})}} = \sum\limits_{i = 1}^N {{f_i}\frac{{{\varepsilon _i} - {\varepsilon _0}}}{{{\varepsilon _0} + L({\varepsilon _i} - {\varepsilon _0})}}} $$
where ɛ is the effective medium permittivity, ɛ0 is the permittivity of the intercellular matrix, and ɛi and fi are the permittivities and volume fractions of each of the N cells. L is the depolarization factor, depending on the shape of the cell and the direction of the electric field.

When the optical homogeneous materials of the same kind of particles are in an orderly arrangement generated birefringence is called form birefringence [3840]. In these biological sample sections we used, the pig dorsal skin had no obvious structural features (Fig. 4(a)). Although the pig hoof skin has a porous structure, it is disordered (Fig. 4(b)). The fat is arranged in microstructure in an orderly manner (Fig. 4(f)), but they are not oriented in their arrangement, and all angles are arranged in a consistent manner. The microstructure of these three samples does not meet the requirements of form birefringence, so we cannot observe THz anisotropy phenomenon. However, the fascia (Fig. 4(e)) and lean longitudinal (Fig. 4(c)) samples have a very distinct striated structure in their microstructures. We equate these samples with stripe-like structures as rod-shaped ordered arranged particles, and the THz-TDS system produces a line polarization equivalent to a monochromatic plane wave with a central frequency of ∼0.7 THz, providing an idealized model of biological tissue. Figure 5 shows the schematic diagram of the cell model principle.

 figure: Fig. 5.

Fig. 5. Schematic diagram of the form birefringence model.

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Assume that t1 and t2 are the cell thickness and intercellular space width, respectively, and ɛ1 and ɛ2 are the dielectric constants for the cell and the medium in which the cell is immersed, respectively. Both intracellular and intercellular are uniform fields. Suppose that the electric vector is perpendicular to the cell surface and the normal component of the electric displacement should be continuous when passing through the surface of the abrupt change in medium property. Therefore, the electrical displacement D should be the same in intracellular and intercellular. Equation (6) is the average electric field E averaged over the total volume.

$$E = \frac{{{t_1}\frac{D}{{{\varepsilon _1}}} + {t_2}\frac{D}{{{\varepsilon _2}}}}}{{{t_1} + {t_2}}}$$
Thus the effective dielectric constant ${\varepsilon _ \bot }$ is shown in Eq. (7),
$${\varepsilon _ \bot } = \frac{D}{E} = \frac{{({t_1} + {t_2}){\varepsilon _1}{\varepsilon _2}}}{{{t_1}{\varepsilon _2} + {t_2}{\varepsilon _1}}} = \frac{{{\varepsilon _1}{\varepsilon _2}}}{{{f_1}{\varepsilon _2} + {f_2}{\varepsilon _1}}}$$
where, ${f_1} = {t_1}/({t_1} + {t_2}),\textrm{ }{f_2} = {t_2}/({t_1} + {t_2}) = 1 - {f_1}$ are the percentage of the intracellular and surrounding medium in the total volume, respectively.

Similarly, for the parallel case, the electric field E will have the same value in the intracellular and cellular gaps. That is, the average electric displacement D can be expressed as,

$$D = \frac{{{t_1}{\varepsilon _1}E + {t_2}{\varepsilon _2}E}}{{{t_1} + {t_2}}}$$
The effective dielectric constant at this point is given by Eq. (9):
$${\varepsilon _\parallel } = \frac{D}{E} = \frac{{{t_1}{\varepsilon _1} + {t_2}{\varepsilon _2}}}{{{t_1} + {t_2}}} = {f_1}{\varepsilon _1} + {f_2}{\varepsilon _2}$$
Since ${\varepsilon _\parallel }$ and ${\varepsilon _ \bot }$ are the effective permittivity for the vibrational directions parallel and perpendicular to the cell surface, respectively. Therefore, the optical properties for these cell lines are like a uniaxial crystal with its optical axis perpendicular to the cell surface. The constant positive dielectric constant difference ${\varepsilon _\parallel } - {\varepsilon _ \bot }$ means that the optical properties of this plate cell tether set are like a negative uniaxial crystal.
$${\varepsilon _\parallel } - {\varepsilon _ \bot } = \frac{{{f_1}{f_2}{{({\varepsilon _1} - {\varepsilon _2})}^2}}}{{{f_1}{\varepsilon _2} + {f_2}{\varepsilon _1}}} \ge 0$$
Using the refractive index description, Eq. (11) can be rewritten as follows,
$$n_e^2 - n_o^2 ={-} \frac{{{f_1}{f_2}{{(n_1^2 - n_2^2)}^2}}}{{{f_1}n_2^2 + {f_2}n_1^2}}$$
where, $n_o^2 = {\varepsilon _\parallel },\textrm{ }n_e^2 = {\varepsilon _ \bot },\textrm{ }n_1^2 = {\varepsilon _1},\textrm{ }n_2^2 = {\varepsilon _2}$, no, ne, n1, n2 are the equivalent refractive indices for THz waves parallel and perpendicular to the set of cell tethers, intracellular and intercellular immersion solutions, respectively. Therefore, we use the theoretical model of form birefringence to well qualitatively explain the observed experimental phenomenon of THz birefringence in the skin and related biological tissues.

4. Conclusion

In summary, we systematically study the macroscopic THz electromagnetic response of skin and related tissues by THz-TDS. We observe that the highest amplitude of the THz pulse is acquired when fat tissue is measured. This once again proves the possibility of using THz technology to study fat-related diseases. More interestingly, by rotating the azimuth of the bio-samples, we find that some bio-samples with periodic stripe structure under the optical microscope can cause an obvious THz birefringence phenomenon. Through the theoretical model of form birefringence, we can qualitatively explain the observed phenomenon. Our results may be valuable for further understanding the interaction mechanism between THz wave and biological tissue and promoting the application of THz technology in biomedicine.

Funding

National Natural Science Foundation of China (11827807, 61905007); National Key R&D Project (2019YFB2203102); the Open Project Program of Wuhan National Laboratory for Optoelectronics (2018WNLOKF001); the Open Fund of Guangdong Provincial Key Laboratory of Information Photonics Technology (Guangdong University of Technology, No. GKPT20).

Acknowledgments

We would like to thank Prof. Lin Feng for providing the solution and instruments required for the preparation of paraffin sections

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Paraffin section substrate selection and preparation process

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. Schematic diagram of the experimental setup and the sample preparation. (a) The THz time-domain spectrometer. L1-L4: TPX lens. (b) Paraffin sections preparision process and the measurement scene.
Fig. 2.
Fig. 2. THz transmission response for different tissues. (a) THz temporal waveforms for pig dorsal skin, pig hoof skin, fat, lean longitudinal cut, lean transverse cut and fascia, respectively. (b) Corresponding equivalent refractive indices.
Fig. 3.
Fig. 3. Biological sample anisotropy phenomenon. (a) Schematic diagram of THz wave polarization direction and streak direction of biological samples. The peak-to-peak transmitted THz electric field amplitude as a function of the azimuthal angle for different biological samples. (b) Pig dorsal skin. (c) Pig hoof skin. (d) Lean longitudinal cut. (e) Lean transverse cut. (f) Fascia. (g) Fat meat.
Fig. 4.
Fig. 4. Optical microscopic images for (a) pig dorsal skin, (b) pig hoof skin, (c) lean longitudinal cut, (d) lean transverse cut, (e) fascia, and (f) fat meat, respectively.
Fig. 5.
Fig. 5. Schematic diagram of the form birefringence model.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

E n o r _ s a m = E s a m E max _ r e f
n ( w ) = c ϕ ( w ) w d + 1
n = Δ t c d + 1
v e ( ξ ) = { v o ,  ( ξ  = 0) v e ,  ( ξ  =  π 2 )
ε ε 0 ε 0 + L ( ε ε 0 ) = i = 1 N f i ε i ε 0 ε 0 + L ( ε i ε 0 )
E = t 1 D ε 1 + t 2 D ε 2 t 1 + t 2
ε = D E = ( t 1 + t 2 ) ε 1 ε 2 t 1 ε 2 + t 2 ε 1 = ε 1 ε 2 f 1 ε 2 + f 2 ε 1
D = t 1 ε 1 E + t 2 ε 2 E t 1 + t 2
ε = D E = t 1 ε 1 + t 2 ε 2 t 1 + t 2 = f 1 ε 1 + f 2 ε 2
ε ε = f 1 f 2 ( ε 1 ε 2 ) 2 f 1 ε 2 + f 2 ε 1 0
n e 2 n o 2 = f 1 f 2 ( n 1 2 n 2 2 ) 2 f 1 n 2 2 + f 2 n 1 2
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