Terahertz (THz) spectroscopy has made great progress in numerous fields owing to its attractive and exclusive attributes[
Chinese Optics Letters, Volume. 20, Issue 6, 063701(2022)
Revisiting the relationship between composite multiscale entropy and THz optical parameters with exterior product
With the framework of exterior product, we investigate the relationship between composite multiscale entropy (CMSE) and refractive index and absorption coefficient by reanalyzing six concentrations of bovine serum albumin aqueous solutions from the published work. Two bivectors are constructed by CMSE and its square by the refractive index and absorption coefficient under vectorization. The desirable linear behaviors can be captured, not only between the defined two bivectors in normalized magnitudes, but also between the normalized magnitude of bivectors pertinent to CMSE and the magnitude of a single vector on the refractive index or absorption coefficient, with the processing of optimum selection. Besides that, the relationship between the coefficients of two bivectors is also considered. The results reveal that plenty of sound linear behaviors can be found and also suggest the scale of 15, 16 and frequency of 0.2, 0.21 THz are prominent for those linear behaviors. This work provides a new insight into the correlation between terahertz (THz) time and frequency domain information.
1. Introduction
Terahertz (THz) spectroscopy has made great progress in numerous fields owing to its attractive and exclusive attributes[
Geometry algebra (GA)[
The exterior product, an indispensable component of the geometric product, is involved in some applications. Li et al.[
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In this study, we investigate the characteristic of exterior product constructed by the vectorized refractive index and absorption coefficient as a function of frequency and the further relationship between CMSE and absorption coefficient, with the refractive index resorting to the exterior product by reusing the experiment data in the previous work[
2. Method
It is acknowledged that cross product between vectors is restricted to operate in three dimensions in linear algebra[
3. Results and Discussion
The bandwidth for absorption coefficients and refractive indices of six concentrations of bovine serum albumin (BSA) solutions in Ref. [9] is from 0.1 to 0.5 THz with the resolution of 0.01 THz. From Eq. (3), taking 5000 µg/mL as an example, the mean profile of can be seen in Fig. 1. It should be noted that the vertical axis represents the coefficients in Eq. (3), and the horizontal axis is the exterior product of basis vectors for all the possible pairs (i.e., basis bivectors), in which the numbers represent () in order. The profile of the first several basis bivectors is very similar to that of absorption coefficient, and the vector of the absorption coefficient is seemingly associated with . It is accessible that the last several indices represent high frequency components interactions between the refractive index and absorption coefficient, and the profile corresponding to low frequency parts diminishes with the increasing index of the basis bivector. Additionally, for further insight into , we investigated the relationship between and as well as between and . It can be seen from Fig. 2 that both of the R-squared reach above 0.9 and the R-squared of linear fitting between and is higher than those of and .
Figure 1.Mean profile of
Figure 2.Linear fitting between A0 and modulus of the (a) absorption coefficient and (b) refractive index.
In order to further explore the relationship between CMSE and the refractive index or absorption coefficient in the framework of exterior product, similar to the bivector , the bivector on CMSE is defined as , which is composed of CMSE and its square. The quantity , normalized magnitude of bivector , is given as follows:
In compliance with the method of our previous work, VMD was performed for that relationship investigation. In terms of the previous search results[
Figure 3.Distribution on standard deviation of CMSE over 20 scales at α = 210.
Figure 4.Linear fitting between B0 at (a) α = 200 and (b) α = 210 from scales 15 to 20 and A0 from 0.2 to 0.38 THz.
Figure 5.Linear fitting between B0 from scales 15 to 20 and modulus of (a) absorption coefficient and (b) refractive index from 0.2 to 0.38 THz.
Additionally, the correlation on the coefficients of bivectors between and should be further studied because they, respectively, reveal the interrelations between the two correspondingly constituted vectors. The bivector coefficient profiles of from 0.2 to 0.38 THz and from scales 15 to 20 are shown in Figs. 6(a) and 6(b), which separately depict the interactions between the refractive index and absorption coefficient and between CMSE and with the increasing index of the basis bivector. The behavior in Fig. 6(a) is similar to that in Fig. 1, and some different disparities exist on the coefficients of among these concentrations of samples including index 1 to 9. The linear fitting between them was performed by the exhaustive search that implemented linear fittings for all possible coefficients of (from 0.2 to 0.38 THz) and (from 15 to 20). Some representative cases were chosen to be exhibited in Figs. 7(a) to 7(e), which indicates some specific coefficients of bivectors can successfully describe the linear fitting for some subspaces between and . Furthermore, according to the combinations of basis bivectors , Figs. 8(a) and 8(b) show the weight distribution of the representative frequency pairs and scale pairs in determination linear fitting. The index of the basis bivector was transformed into the upper triangular matrix that is shown as Eq. (6) in order to get a better interpretability. The subscripts above correspond to the frequencies 0.2, 0.21, 0.22,…, 0.38 THz in turn for Fig. 8(a) and the scales 15, 16,…, 20 in turn for Fig. 8(b). The intensity of the color bar denotes the proportions of highly linear behaviors (). It should be noted that the parts in are shown with zeros. It can be seen that the first two lines are noticeable, which demonstrates that the scales 15, 16 and frequencies 0.2, 0.21 THz coupling with the remaining scales or some frequencies can acquire much more sound linear behaviors, thereby implying the importance of the scales 15, 16 and frequencies 0.2, 0.21 THz:
Figure 6.Profile of the coefficients of (a)
Figure 7.Linear fitting between coefficients of
Figure 8.Weight distribution on coefficients of bivector (a)
4. Conclusions
In this work, by reanalyzing the data of previous work, we further study the relationship between CMSE and refractive index or absorption coefficient under the framework of exterior product, which presents a novel insight into the relationship between THz time and frequency domain information. The quantity bivectors and were established by vectorizing the corresponding parameters, and their linear behaviors in normalized magnitude and bivector coefficients were given with optimization processing, which provides their correlation on the second graded information (subspaces) and further validates the results of previous work as well. It is feasible to introduce the exterior product to characterize the properties of CMSE, refractive index, and absorption coefficient. Besides that, the coefficients of bivectors have large potential in sifting important variables depending on the performance of corresponding constructed vectors. The magnitude and coefficients of bivectors established in this work may also fertilize the methodology in substance identification of THz spectroscopy.
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Haishun Liu, Zhenwei Zhang, Meiyan Liang, Cunlin Zhang. Revisiting the relationship between composite multiscale entropy and THz optical parameters with exterior product[J]. Chinese Optics Letters, 2022, 20(6): 063701
Category: Infrared and Terahertz Photonics
Received: Jan. 25, 2022
Accepted: Mar. 18, 2022
Posted: Mar. 21, 2022
Published Online: May. 6, 2022
The Author Email: Haishun Liu (phscdream@163.com), Zhenwei Zhang (zhangzw@cnu.edu.cn)