Abstract
In this paper, an analytical expression for describing propagation properties of twisted Gaussian Schell model array (TGSMA) beams through turbulent biological tissues is derived based on the extended Huygens Fresnel integral. With the help of the formulae, properties of the rotation and mergence for the TGSMA beams in turbulent biological tissues are researched in detail. It is found that the TGSMA beams go through the distinct mergence period in the far field besides phenomena of abruption and rotation in the near field, and turbulent biological tissues play a dominated role in mergence of the TGSMA beams. These novel results may be helpful in optical trapping.
1. Introduction
In medicine and other relative fields, image reconstruction technology including optical coherence tomography [1–6] become more and more important and it provided a good insight into the physiological and pathological changes in animal tissues [7, 8]. In the process of the analysis and diagnosis, the optical properties of the biological tissues can be displayed by controlling parameters of incident light beams. Some studies showed that properties of light are directly related to the optical parameters of biological tissues, and phase-contrast microscopy showed that the structure of the refractive index in a variety of mammalian tissues resembles that of frozen turbulence.
An early result concerning structure function is the fundamental result due to Schmitt and Kumar [9], which stated that the observed structure function fits the classical Kolmogorov model turbulence and presented a model of power spectrum of refractive-index variations in biological tissues. By virtue of the power spectrum, many studies related to the stochastic properties of various beams propagating through biological tissues are presented [10–12]. The effect of the upper dermis of human on the state of polarization and coherence of a random electromagnetic beam also have been investigated [13, 14]. In [15], the study provides a result for spectral shift of electromagnetic Gaussian Schell model (GSM) beams propagating through tissues. The statistical properties of anisotropic electromagnetic beams passing through the biological tissues have been studied [16].
On the other hand, due to the wide applications in multiple fields such as holographic optical tweezers [17], particles trapping [18], photonic lithography [19–21], and various array beams have always been a subject of great concern. Compared with the single beam, the coherent array beams have source size which diverges less and can produce higher output power and lattice-like intensity distribution [22]; researchers have made lots of efforts [23–28]. Wan and Zhao [29] presented a new kind of array beams with twist phase termed as twisted Gaussian Schell model array (TGSMA) sources whose spectral density and spectral degree of coherence can gradually rotate along its propagation directions; properties of electromagnetic twisted Gaussian Schell model array beams in free space have been investigated [22]and the effect of anisotropic oceanic turbulence on propagation properties of the TGSMA beams has been reported [30, 31]. Yet, despite, properties of twisted Gaussian Schell model array beams propagating through a turbulent biological tissue have not been studied so far. Hence, it is meaningful to investigate the topic that the TGSMA beams propagate in turbulent biological tissues.
Here, we will explore the propagation characteristics of the TGSMA beams under the action of turbulent biological tissues. In Section 2, an analytical expression for cross-spectral density (CSD) function [29] of TAGSM beams propagating in turbulent biological tissues is derived based on the extended Huygens Fresnel integral. In Section 3, the effect of turbulent biological tissues on propagation characteristics including intensity distribution and degree of coherence (DOC) is discussed in detail. Finally, we present a conclusion in Section 4.
2. Analytical Formula for the CSD Function of TGSMA Beams Propagating in Turbulent Biological Tissues
In the Cartesian coordinate system, the CSD function of the TGSMA beams at source plane z = 0 can be expressed as [29]where r1 = (x1, y1) and r2 = (x2, y2) are arbitrary two-dimensional position vectors; σx and σy denote the beam widths along the x direction and y direction, respectively; u is the twisted strength; and μ(r1, r2) is the degree of coherence and it can be written aswhere P = (Nx − 1)/2 and Q = (Ny − 1)/2, and Nx and Ny are positive integrals which determine the number of lobes of array beams. Cj = 2πRj/δj, δj and Rj (j = x, y) are coherence width of the source and correlations coefficient, respectively. In the presence of the turbulence medium, employing the extended Huygens Fresnel principle, the CSD function of the TGSMA beams between the two arbitrary points r1 = (ρ1, z) and r2 = (ρ2, z) obeys the lawwhere denotes wave number (where λ is the wavelength), ψ stands for complex phase perturbation caused by the medium, and implies averaging over the ensemble of statistical realizations of the turbulence. Under quadratic phase approximations, can be written as [31]where is the spatial power spectrum of refractive-index fluctuations representing the turbulent biological tissue characteristics and it can be written as [10, 11]where is the spatial frequency, η0 is the small-scale factor, lc is the characteristics length of heterogeneity, S is the strength coefficient of the refractive-index fluctuations, Df is the fraction dimension, and Г(.) is the gamma function.
By substituting equations (1) and (2) and (4) and (5) into equation (3) and performing mathematical calculation, we finally obtain the following expression for the CSD function of the TGSMA beams through turbulent biological tissueswith
Based on equation (6), the average intensity and the DOC for the TGSMA beams propagating in turbulent biological tissue can be expressed as [30]
3. Numerical Example and Analysis
Due to complex structure of power spectrum model of biological turbulence, in the following section, we examine numerically behavior of the TGSMA beams propagating in turbulent biological tissues. Unless it is specified in the figure captions, otherwise, we assume the following parameter values of the source: λ = 0.632 μm, Nx = Ny = 3, σx = δx = 1 μm, σy = δy = 0.3 μm, Rx = 2Ry = 3 μm, and u = 3 μm−1 and the biological tissue types and its parameters are chosen as [9] Df = 2.60, lc = 10.24 μm, for liver parenchyma (mouse), Df = 2.67, lc = 11.48 μm, for intestinal epithelium (mouse), Df = 2.67, lc = 5.24 μm, for upper dermis (human), and Df = 2.72, lc = 5.29 μm, for deep dermis (mouse). These comparative values in the four biological tissues denote that they possess different values for the pairs of fractal dimension and characteristic length of heterogeneity.
Figure 1 illustrates evolution of the spectral intensity and its rotating angle of the TGSMA beams propagating through turbulent biological tissues at several selected distances. It can be seen from Figure 1(a) that the intensity distribution of the array beams goes through two pronounced periods: one is the abruption and rotation and another is mergence of the array beams. Firstly, in the near field, owing to effect of turbulent medium on the array beams is comparatively weaker. Propagating behavior of the array beams depends on the initial parameters, when transmission distance increases, the initial Gaussian ellipse spots progressively split into a rotating lattice-like field upon propagation, and all lobes rotate around its respective center in a synchronous motion, and the rotation angle of the array beams increases gradually from 0° to 90°. Figure 1(b) shows the behavior of rotation for different values of twisted strength, one can find that, with the increase of propagation distance, rotation angle increases monotonically, and the rotating behavior of the array beams relies on choice of twisted parameters. A larger value of twist strength leads to more obvious rotation of the TGSMA beam. When propagation distance is at approximately z = 20 μm, rotation angle approaches 90o degrees, which is similar to the case described in [29]. Secondly, in far field, with the increase of propagation distance, the accumulated effect of turbulent biological tissues on the array beams becomes gradually obvious, which are shown in Figure 2. As can been seen, mergence of the array beams occurs. Intensity of one lobe (we defined as main beam lobe) increases while intensity of the other beam lobes (we defined as secondary beam lobes) degenerates and gradually vanishes. Meanwhile, the main beam lobe moves its position as the propagation distance changes, and this implies the fact that, when the array beams propagate through sufficiently large distances, the turbulent biological tissues play a dominate role for determining intensity of main beam lobes and its position. This novel result of the array beams in turbulent biological tissue provides a further insight into the TGSMA beams and may be applied in optical trapping.
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Figure 3 illustrates normalized intensity distribution at certain propagation distances for the selected source parameters. One can find from Figures 3(a)–3(d) that normalized intensity distribution is significantly affected by the initial parameters. More specifically, the width σx and coherence width δx decrease while width σy and coherence width δy increase; intensity of secondary beam lobes and position of the main lobe change more rapidly. Here, it is necessary to mention that, in Figures 3(a) and 3(b), we choose the ratio of Ri/δi (i = x, y) to remain unchanged so as to explore simplistic effect of coherence width on position of the beam lobes.
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The influence of turbulent biological tissues on the intensity profile when propagation distance z = 200 μm is shown in Figure 4. In Figure 4(a), for different strength coefficients of the refractive-index fluctuation S, in Figure 4(b), for different characteristic lengths of heterogeneity lc, in Figure 4(c), for different small length-scale factor η0, in Figure 4(d), for different fractal dimension pairs Df, and in Figure 4(e), for various biological tissue types. One finds from Figures 4(a)–4(e) that biological tissue parameters and biological tissue types play an important role in behavior of mergence for the array beams in the far field. It indicates that when the parameter S increases as seen in Figure 4(a), while the parameters lc,η0, and Df decrease as seen in Figures 4(b)–4(d) and Figures 4(b)–4(d) can all cause larger secondary beam lobes reductions. The larger parameter S and smaller parameters lc,η0, and Df mean larger turbulence strength, thus more obvious mergence of the beam lobes occurs. Figure 4(e) presents intensity variations of the array beams in various biological tissue types, respectively. For all the chosen tissue types, it is observed that at certain transmission distance, the upper dermis (human) has the largest impact on the array optical field among the four tissue types and others are the deep dermis (mouse), parenchyma (mouse), and intestinal epithelium (mouse) from the largest intensity variation to the smallest intensity variation successively.
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We now turn our attention to the DOC of the TGSMA beams. Figures 5(a) and 5(b) present variation of modulus of the DOC and its rotation angle of the optical array beams by the same source parameters in Figure 1. It is found from Figure 5(a) that the DOC rotates counter clockwise around the beam center upon propagation and degenerated gradually a Gaussian profile from initial lattice-like profile. In Figure 5(b), one can find that variation of rotation angle depends on the twist strength u; the behaviors are similar to those in Figure 1(a), except rotating in opposite directions.
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The effect of turbulent biological tissue medium on the DOC of the array beams at propagation distance z = 200 μm is shown in Figure 6 by the same biological tissue parameters as in Figure 4. One can see from Figures 6(a)–6(e) that biological tissue parameters are responsible for broadening of the DOC of the TGSMA beams. In general, no matter when the parameter S increases or the parameters lc,η0, and Df decrease, it can all cause more obvious broadening of the DOC. Especially, it can be found that, among the four biological tissue types, variation of the DOC caused by liver parenchyma (mouse) fits approximately to those by the intestinal epithelium (mouse) and change of the DOC caused by the upper dermis (human) is similar to those by the deep dermis (mouse).
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4. Conclusion
We have obtained an analytical expression for describing polarization properties of the TGSMA beams in turbulent biological tissues based on the extended Huygens Fresnel integral. With the help of these formulae, properties of the rotation and mergence of twisted Gaussian Schell model array beams in turbulent biological tissues are investigated in detail. The results indicate that, when transmission distance is relatively shorter, the beam spots go through the abruption and rotation. The similar phenomena also exist in the DOC of the beam, which is dominated by the source parameters, and with increase of propagation distance, the intensity of main lobe increases rapidly, while intensity of the secondary lobes reduces. Meanwhile, the DOC of the beam degenerated gradually a Gaussian profile from the initial lattice-like profile. With the former observations, we can reach such conclusion that, despite turbulent biological tissues have an important effect on the TGSMA beams, with proper choices of source parameters and propagation distance, it can control propagation behavior of the array beams. These novel features of the array beams in turbulent biological tissues may provide a possible approach for further applying the TGSMA beams.
Data Availability
The authors declare that the researched data are available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first author thanks Prof. W. Fu and Doctor F. Zhang for their kind help. This study was funded by the Science Foundation of Gansu Province in China (Grant no. 21JR7RM188).