Biological tissues have complex structures and display abundant spectral behavior in the optical field. The light can be scattered and absorbed when it is transmitted to subcutaneous tissues and organs through the skin. Consequently, the received light carries physiological and pathological information about biological tissues. Some promising technologies have been made for optical noninvasive early diagnosis of biological tissue diseases, such as laser-induced fluorescence, polarized light imaging and elastic-scattering measurement. Schmitt et al. explored spatial variation of the refractive index of various mammal tissues and found that their structure functions were analogous with the classical Kolmogorov model in atmospheric turbulence, where the model of the power spectrum refractive-index in biological tissue was also presented^[1]. According to this power spectrum model, the changes of coherence and polarization for an electromagnetic Gaussian Schell-model beam in biological tissues were studied by Gao et al^[2], and then their coherences and polarizations for anisotropic electromagnetic Gaussian Schell-model beams were further analyzed by Zhao et al. ^[3]. In 2021, Zhang et al. studied the average intensity and beam quality of Hermite-Gaussian correlated Schell-model beams in biological tissue, and found that it is less sensitive to tissue turbulence than that of Gaussian Schell-model beams^[4].

On the other hand, Optical Coherent Lattices (OCLs) have inspired prominent research interests due to their special periodic reciprocities^[5-7]. New properties have been found that a Gaussian profile at the source plane gradually becomes a periodic array in the far-field zone. Much effort has been devoted to exploring their propagation properties in atmospheric and oceanic turbulence^[8-9]. In addition, the effect of the embedded optical vortex or rotating elliptical Gaussian factor on OCLs in a turbulent atmosphere was also studied^[10-11]. It has been found that the lattice periodicity reciprocity can be preserved over a certain distance even in the atmosphere or oceanic turbulence, and its scintillation index is lower than that of Gaussian beams. The optical coherence lattices can preclude Talbot self-imaging in free space, which is beneficial to robust optical imaging or the transfer of information in biological tissue^[6]. However their researches are mainly focused on the propagation dynamics of OCLs in the field of monochromatic light, and the spectral behavior of a polychromatic field in biological tissue is not addressed. What happens in the polychromatic light fields of a chirped Gaussian pulse and vortex function if it is brought into OCLs?

The motivation of this paper is to investigate the averaged intensity and spectral shift of Partially Coherent Chirped Optical Coherence Vortex Lattices (PCCOCVLs) in biological tissue turbulence. The results show that optical lattice structures are modulated by lattice parameters and topological charge, and that rapid spectral transitions vanish over a longer distance, which provides valueable applications in developing image recognition technology, medical devices and noninvasive optical diagnoses in biological tissue.

Assume that the propagation direction of PCCOCVLs in biological tissue turbulence propagates along the z-axis, whose cross-spectral density at the source plane can be expressed by^[12-14]

$W({{\boldsymbol{\rho}'}_1},{{\boldsymbol{\rho}'}_2},0,\omega ) = {S_{00}}(\omega ){\left( {{{{\boldsymbol{\rho}'}}_1}{{{\boldsymbol{\rho}'}}_2}} \right)^{\left| m \right|}}\exp \left[ {im\left( {{\theta _1} - {\theta _2}} \right)} \right] \prod \limits_{s = x,y} \sum\limits_{{n_s} = 0}^N \frac{{{V_{{n_s}}}}}{{\sqrt {\text{π}} }}\exp \left[ { - \frac{{2i{\text{π}} {n_s}\left( {{{s'}_1} - {{s'}_2}} \right)}}{{a{w_0}}} - \frac{{{s'}_1^2 + {s'}_2^2}}{{w_0^2}} - \frac{{{{\left( {{{s'}_1} - {{s'}_2}} \right)}^2}}}{{{\sigma _0}^2}}} \right], $ (1)

where ${x'_j} = {\rho '_j}\cos {\theta _j}$ and ${y'_j} = {\rho '_j}\sin {\theta _j}$ (j=1, 2) are positions of two points at z=0 in the Cartesian coordinate, w₀ is the radius of the beam waist, ${V_{{n_s}}}$ is the power distribution of the pseudo-modes constituting the lattice, a is the lattice constant, m is the topological charge of the vortex, σ is the correlation length and N indicates the number of lattice lobes, respectively. In Eq. (1) the incident pulsed beam is assumed to be a chirped Gaussian given by ^[15-16]

$ \mathrm{S}_{00}(\omega)=|f(\omega)|^{2}\quad, $ (2)

and

$ f\left( \omega \right) = \frac{1}{{\sqrt {2{\text{π}} } }}\sqrt {\frac{{{T^2}}}{{1 + iC}}} \exp \left[ { - \frac{{{T^2}{{\left( {\omega - {\omega _0}} \right)}^2}}}{{2\left( {1 + iC} \right)}}} \right]\quad, $ (3)

with central frequency ω₀, pulse durationT and chirp parameter C.

According to the extended Huygens-Fresnel principle, the cross-spectral density function of PCCOCVLs propagating through biological tissue turbulence can be expressed as

$\begin{aligned} &W({{\boldsymbol{\rho}}_1},{{\boldsymbol{\rho}}_2},{\textit{z}},\omega ) = \frac{k}{{2\text{π} {\textit{z}}}}\iint {{d^2}{{{\boldsymbol{\rho}'}}_1}}{d^2}{{\boldsymbol{\rho}'}_2}W\left( {{{{\boldsymbol{\rho}'}}_1},{{{\boldsymbol{\rho}'}}_2},0,\omega } \right)\\ &\exp \left\{ {\frac{{ik}}{{2{\textit{z}}}}\left[ {\left( {\boldsymbol{\rho}_1^2 - {\boldsymbol{\rho}'}_1^2} \right) - \left( {{\boldsymbol{\rho}}_2^2 - {\boldsymbol{\rho}'}_2^2} \right)} \right]} \right\} \times \\ &{\left\langle {\exp \left[ {\psi \left( {{{\boldsymbol{\rho}}_1} - {{{\boldsymbol{\rho}'}}_1}} \right) + {\psi ^ * }\left( {{{\boldsymbol{\rho}}_2} - {{{\boldsymbol{\rho}'}}_2}} \right)} \right]} \right\rangle _m}\quad, \end{aligned}$ (4)

where k=ω/c represents the wavenumber related to frequency ω and the speed of light in a vacuum c, ψ is the phase function in the refractive-index fluctuations, and ρ₁=(x₁, y₁) and ρ₂=(x₂, y₂) are position vectors of two points at receiver plane, respectively. The ensemble average of the biological tissue turbulent in Eq. (4) is given by

$\begin{aligned} &{\left\langle {\exp \left[ {\psi \left( {{{\boldsymbol{\rho}}_1} - {{{{\boldsymbol{\rho}}'}}_1}} \right) + {\psi ^ * }\left( {{{\boldsymbol{\rho}}_2} - {{{\boldsymbol{\rho}'}}_2}} \right)} \right]} \right\rangle _m} \approx \\ &\exp \left[ { - \frac{{{{\left( {{{{\boldsymbol{\rho}'}}_1} - {{{\boldsymbol{\rho}'}}_2}} \right)}^2} + \left( {{{{\boldsymbol{\rho}'}}_1} - {{{\boldsymbol{\rho}'}}_2}} \right)\left( {{{\boldsymbol{\rho}}_1} - {{\boldsymbol{\rho}}_2}} \right) + {{\left( {{{\boldsymbol{\rho}}_1} - {{\boldsymbol{\rho}}_2}} \right)}^2}}}{{{\rho _0}^2}}} \right] , \end{aligned}$ (5)

where ρ₀ is the spatial coherence length of a spherical wave in biological tissue turbulence, which takes the form of^[17-19]

$ {\rho _0} = {\left( {\frac{2}{3}{\text{π} ^3}C_n^2{k^2}{\textit{z}}} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\quad, $ (6)

where C_n² is the structure constant of the refractive-index of the biological tissues.

Substituting Eqs. (1)−(3) into Eqs. (4)−(6), and lettingx₁=x₂=x and y₁=y₂=y in the Cartesian coordinate, the analytical spectral intensity of PCCOCVLs propagating through biological tissue turbulence at the receiver plane is expressed as

$ S(x,y,{\textit{z}},\omega ) = {S_{00}}(\omega )W\left( {x,y,{\textit{z}},\omega } \right)\quad, $ (7)

where

$ \begin{split} &W\left( {x,y,{\textit{z}},\omega } \right) = {\left( {\frac{1}{{w_0^2}}} \right)^{2N}}\frac{{{\omega ^2}}}{{4{\text{π}} {c^2}{{\textit{z}}^2}}}\sum\limits_{{{n'}_x} = 0}^N {\sum\limits_{{{n'}_y} = 0}^N {\prod\limits_{t = 1}^4 {\frac{1}{{\sqrt {{A_t}} }}} } } \\ &\exp \left( {\frac{{B_t^2}}{{4{A_t}}}} \right)\sum\limits_{{l_1} = 0}^M {\frac{{M!{i^{{l_1}}}}}{{{l_1}!\left( {M - {l_1}} \right)!}}} \sum\limits_{{l_2} = 0}^M {\frac{{M!{{\left( { - i} \right)}^{{l_2}}}}}{{{l_2}!\left( {M - {l_2}} \right)!}}}\times\\ &\sum\limits_{{k_1} = 0}^{\left[ {\frac{{{D_1}}}{2}} \right]} {\frac{{{2^{ - {D_1}}}{D_1}!A_1^{{k_1} - {D_1}}}}{{{k_1}!{m_1}!}}} \sum\limits_{{k_2} = 0}^{{m_1}} {\frac{{{m_1}!B_1^{{m_1} - {k_2}}{{C'}^{{k_2}}}}}{{{k_2}!\left( {{m_1} - {k_2}} \right)!}}}\\ &\sum\limits_{{k_3} = 0}^{\left[ {\frac{{{D_2}}}{2}} \right]} {\frac{{{2^{ - {D_2}}}{D_2}!A_2^{{k_3} - {D_2}}B_2^{{m_2}}}}{{{k_3}!{m_2}!}}} \sum\limits_{{j_1} = 0}^{\left[ {\frac{{{D_3}}}{2}} \right]} {\frac{{{2^{ - {D_3}}}{D_3}!}}{{{j_1}!{m_3}!}}} A_3^{{j_1} - {D_3}}\times\\ &\sum\limits_{{j_2} = 0}^{{m_1}} {\frac{{{m_3}!B_3^{{m_3} - {j_2}}{{C'}^{{j_2}}}}}{{{j_2}!\left( {{m_3} - {j_2}} \right)!}}} \sum\limits_{{j_3} = 0}^{\left[ {\frac{{{D_4}}}{2}} \right]} {\frac{{{2^{ - {D_4}}}{D_4}!A_4^{{j_3} - {D_4}}B_4^{{m_4}}}}{{{j_3}!{m_4}!}}}\quad,\end{split}$ (8)

with

$\begin{aligned} &{A_1}={A_3}=\frac{1}{{w_0^2}} - \frac{{ik}}{{2{\textit{z}}}} + \frac{{C'}}{2}, {A_2}={A_4}=\frac{1}{{w_0^2}} + \frac{{ik}}{{2{\textit{z}}}} + \frac{{C'}}{2} - \frac{{{{C'}^2}}}{{4{A_1}}}, \\ &{B_1}=\frac{{2\text{π} iV{{n'}_y}}}{{a{w_0}}} - \frac{{iky}}{{\textit{z}}} \quad, \end{aligned}$ ()

$\begin{aligned} &{B_2}{\text{ = }}\frac{{ - 2\text{π} iV{{n'}_y}}}{{a{w_0}}} + \frac{{iky}}{{\textit{z}}} + \frac{{{B_1}C'}}{{2{A_1}}} , {B_3}{\text{ = }}\frac{{2\text{π} iV{{n'}_x}}}{{a{w_0}}} - \frac{{ikx}}{{\textit{z}}}\quad, \\ &{B_4}{\text{ = }}\frac{{ - 2\text{π} iV{{n'}_x}}}{{a{w_0}}} + \frac{{ikx}}{{\textit{z}}} + \frac{{{B_3}C'}}{{2{A_1}}}\quad, \end{aligned}$ ()

$ \begin{aligned}&{D_1} = 2N + {l_2}, {D_2} = 2N + {l_1} + {k_2}, {D_3} = 2N + m - {l_2}\quad, \\ &{D_4} = 2N + m - {l_1} + {j_2}\quad, \end{aligned}\;$ ()

$ \begin{aligned} &C' = \frac{2}{{\rho _0^2}} + \frac{1}{{\sigma _0^2}} , {m_1} = {D_1} - 2{k_1}, {m_2} = {D_2} - 2{k_3}\quad, \\ &{m_3} = {D_3} - 2{j_1}, {m_4} = D{}_4 - 2{j_3}\quad. \end{aligned}\quad\quad\;\;$ ()

In monochromatic lightof ω=ω₀=2πc/λ₀, the Eq. (8) denotes the averaged intensity of the PCCOCVLs in biological tissue. For the polychromatic light fields, the Eq. (8) describes the spectral intensity of PCCOCVLs, which depends on lattice constant a, observation point (x, y, z), pulse durationT, chirp parameter C and biological tissue turbulence parameters.

The frequency ω_max of the maximum spectral intensities for the PCCOCVLs are determined by

$ \frac{{\partial S(x,y,{\textit{z}},\omega )}}{{\partial \omega }} = 0\quad. $ (13)

The relative spectral shift is described by

$ \frac{\delta \omega}{\omega_{0}}=\frac{\omega_{\max }-\omega_{0}}{\omega_{0}} \quad. $ (14)

If δω>0, the spectrum is blue-shifted, whereas it is red-shifted forδω<0.

We choose a human upper dermis, a mouse′s deep dermis and a mouse′s intestinal epithelium as the specimens for numerical calculation. The refractive indices of the biological tissues are C_n²=0.06×10⁻³ μm⁻¹, C_n²=0.22×10⁻³ μm⁻¹, and C_n²=0.44×10⁻³ μm^−1[17], respectively. Numerical calculations are performed to illustrate the influence of pulse duration T and chirped parameter C, observation point (x, y, z) and the biological tissue turbulence parameters (i.e. C_n²) on the spectral behaviors of PCCOCVLs. The calculation parameters are fixed by λ₀=0.83 μm, σ=2 μm, z=1.5 cm, ω₀=2 πc/λ₀, w₀=5 mm, C_n²=0.22×10⁻³ μm⁻¹, T=2fs, C=2, N=2, m=2 and c=3×10⁸ m/s unless otherwise specified.

Fig.1 (Color online) gives the intensity evolution of PCCOCVLs with monochromatic light in biological tissue for different lattices parameter a. It is found that the beam at the source plane of z=0 presents an annular structure with a vortex core and that it gradually evolves into a periodic array of lobes with a dark zone at its origin as the distance increases. As the distance further increases, the turbulence effect in biological tissue continues to accumulate, the profile of the annular structure and the array eventually disappear and become a Gaussian pattern. Although the patterns are the same at the source plane for different lattice parameters, a larger lattice parameter changes circular array to a rectangular structure in propagation. It should especially be noted that the dark lines in the last line of Figs. 1 (b)−(d) are not edge dislocations.

Fig. 2 (Color online) describes the intensity profiles of PCCOCVLs with monochromatic light in biological tissue for different topological charges m with canonical (a=1) and noncanonical lattices parameters (a=1+i). Where there is a vortex, it can be seen that noncanonical lattices parameters lead to an annular structure with periodic lobes, and the dark zone at the origin increases with an increasing topological charge. For the case of non-vortex, the noncanonical lattices parameter of a=1+i exhibit a periodic array (e.g. 3×3 spot array) rather than that of canonical lattices of a=1. The results show that lattice parameters and topological charge change dramatically with changes in the structures of periodic arrays.

To further investigate the spectral behavior of polychromatic light, Fig. 3 shows a relative spectral shift of the PCCOCVL beams over the transverse coordinate x for different lattices parameters. One can see that their spectrums are not influenced by lattices parameters, and the spectrum becomes red-shifted with an increase in the transverse coordinate x. When the propagation distance in biological tissue is small (e.g z=1 or 2 cm) it is found that there is a rapid spectral transition at the critical value of x_c=1.06 cm, and the changes of Δ(δω/ω₀) are 0.9 and 0.6 for z=1 and 2 cm, respectively. As the beam further propagates in biological tissue, the spectrum becomes more flat.

Here, our focus is given to rapid spectral transition. Fig. 4 (Color online) plots the relative spectral shift of PCCOCVLs versus transverse coordinate x for different C and T at z=2 cm. It is found that there exist critical values x_c of spectral transition, and these values decrease with the increase of chirp parameter C and the decrease of pulse durationT. For example, the critical values of x_c are 8.6 mm, 14.7 mm and 19.4 mm for C=0, 1 and 3, respectively. In addition, the red-shift spectrum is presented at a smaller transverse coordinate x, and its value decreases and then rapidly increases with an increase in coordinate x. The phenomenon means that the spectrum is sensitive to the transverse coordinate for different C and T, which is significant in the detection and acquisition of spectrum signals in biological tissues.

Fig. 5 (Color online) gives the relative spectral shifts of the PCCOCVL beam versus transverse coordinatex for different biological tissue turbulences (i.e. C_n²). When the propagation distance in biological tissue is small, the turbulence-induced spectral difference is not significant, and their spectral behaviors in red-shift zone are similar to those of Fig. 4. The rapid spectral transition disappears, to be replaced by a smooth curve in the spectrum when the beam travels a longer distance in the biological tissue, e.g. z=10 cm or z=20 cm. At the same transverse coordinate, stronger turbulence leads to a smaller red-shift value in the spectrum, which indicates that the accumulated turbulence effect in a longer distance can suppress not only spectral transition, but also spectral shift.

Fig. 6 (Color  online) gives physical explanations for the rapid spectral transition of PCCOCVL beams where the critical coordinate x_c=14.6 cm and C=1 as shown in Fig. 4(a). One can see that there exists only one spectral maximum S_max1 at (ω_max1−ω₀)/ω₀=−0.16 in Fig. 6(a), but two spectral maximums S_max1 and S_max2 are found at (ω_max1−ω₀)/ω₀=−0.16 and (ω_max2−ω₀)/ω₀=−0.8 at critical transverse coordinate x_c=14.7 cm as shown in Fig. 6(b). The second spectral maximumS_max2 continues to maintain its previous maximal value, but the spectrum of S_max1 is suppressed for the coordinate x=14.8 cm. The behavior is the result of spectral competition in two red-shift spectra. If one of them has a disadvantage, then the other presents superiority in the spectrum.

Intensity evolution and spectral behavior of PCCOCVLs passing through biological tissue turbulence are investigated by using the extended Huygens-Fresnel principle. The analysis of the evolution of its intensity shows that the beam in the monochromatic optical field evolves from an annular structure profile with a vortex core into a periodic array of lobes with a dark zone, and it finally presents a Gaussian-like structure when the distance in biological tissue increases. Moreover, the noncanonical lattices parameter of a=1+i leads to an annular structure with periodic lobes where there is a vortex, while a non-vortex presents a periodic array rather than that of canonical lattices where a=1. These results indicate that lattice parameters and topological charges change dramatically according to the structure of the periodic array, and the cumulative effect of turbulence in a longer distance results in the disappearance of optical lattices and the appearance of a Gaussian-like pattern.

However, in a polychromatic optical field, the effect of lattice parameters on spectral shift is negligible. Although the spectrum in red-shifts smoothens with an increasing distance, it presents a rapid transition for a smaller distance. There are some critical transverse coordinates in spectral rapid transitions, whose decreases are accompanied by an increase of chirp parameter C and a decrease of pulse durationT. The influence of cumulative turbulence in a longer distance on spectral behavior results in the disappearance of the rapid spectral transition which is replaced by a smooth curve in the spectrum. Stronger biological tissue turbulence in a longer distance suppresses the spectrum shift and spectral transition. The appearance of rapid spectral transitions in PCCOCVLs is also physically explained by spectral competitions.

Refs.[4, 8, 9] investigated the lattice profile and spatial degree of coherence in atmospheric or oceanic turbulence and studied the polarization behavior and intensity in biological tissue turbulence. In contrast, where we focus the introduction of a chirped Gaussian pulse and a vortex function to the optical coherence lattices in polychromatic light field, and its sensitivity to spectrum signals such as rapid spectral transitions and spectral shifts in biological tissue. The results obtained here should be useful for the noninvasive optical diagnoses including the detection and acquisition of spectrum signals in biological tissue.