Optical beam propagation in soft anisotropic biological tissues

Xi Chen; Olga Korotkova

doi:10.1364/OSAC.1.001055

1. Introduction

During the last several decades optical methods of biological tissue analysis have become the significant part of medical diagnostics and treatment [1–3]. Quasi-elastic light scattering [4], optical coherence tomography [5], Mueller matrix polarimetry [6], two-photon fluorescence microscopy [7] are only a few in the variety of well-established methods developed for efficient sensing and imaging of biological structures.

Although the average optical properties of the multitude of tissue types are readily measured by modern optical instrumentation and well documented [2,3] the higher-order spatial, spectral and polarimetric properties, such as correlation functions, structure functions, power spectra, etc. are still unknown for the majority of tissues. These properties are fundamental for accurate theoretical predictions of light statistics [8]. The substantial advance in this direction was made in Ref. [9] where for relatively homogeneous tissues like human and mouse dermis and liver the two-dimensional spatial power spectra have been obtained from the microscopic images of tissue slices. It was discovered that the power spectra of several bio-tissues are of the negative power-law form (i.e., decreasing lines in logarithmic scale) with slopes depending on the tissue type and differing slightly from the classic Kolmogorov type of turbulence [10]. Hence, like other natural random media composed by a large number of differently sized inhomogeneities, such as atmospheric turbulence and oceanic salty water turbulence, the biological tissues have been shown to be the random process with the Richardson’s energy distribution [11]. Moreover, the malignant portions of tissues generally have power spectra deviations which can be employed for cancer diagnostics [12]. Alternatively, the shape of the two-point spatial correlation function of the refractive index, being the Fourier transform of the power spectrum, can also provide the information about the cancer genesis [13]. Very recent experimental studies suggest that the biological tissues might also have multi-fractal structure [14]. The measurements in Ref. [9] have also shown that while the outer scale of tissue inhomogeneities is on the order of several microns, the inner scale takes values below optical wavelengths. This general property of bio-tissues sets them in striking difference with other turbulent media since the dominance of small scales present in the power spectrum is responsible for scattering rather than refraction of light. Hence certain effects based on light interaction with large-scale inhomogeneites, such as beam wander [15] are not pertinent to bio-tissue propagation.

The three-dimensional power spectrum of the refractive index fluctuations of bio-tissues has been developed in Refs. [16,17] and was later extensively used in light propagation and scattering calculations [18–25]. However, all these studies were pertaining to isotropic biological tissues. In reality, some biological tissues may exhibit anisotropy due to their intrinsic cell composition, for instance, tissues containing fibers, due to external mechanical stresses, or both [26–29]. One should not confuse the geometrical anisotropy, with the electromagnetic anisotropy, in which the random medium perturbs the electric field components differently and may couple then, causing the loss in the degree of polarization [30–33]. The geometrical anisotropy treated in this paper causes the spectral density of the beam to acquire elliptical shape for sufficiently large propagation distances but does not change its polarimetric content.

In other natural random media, e.g. turbulent atmosphere and oceans, the treatment of geometrical anisotropy and its effects on propagating light has been recently suggested by introducing the anisotropy factors to the isotropic power spectrum [34–36]. It was assumed so far that out of three factors only one is different from the other two, and, in this connection, two different scenarios have been considered: (I) the anisotropy factor different from two others is along the direction of light propagation and (II) the two different anisotropy factors are along the two directions transverse to light propagation. While in the former case the anisotropy was shown to introduce only scaling effects for the beam statistics, acting as an effective refractive index variance (structure parameter) [34], in the latter case it manifests itself in different spreading rates of the beam intensity along the two transverse directions, causing an initially circularly symmetric (say, Gaussian) beam to acquire an elliptic shape [35].

In this paper we first introduce the anisotropic power spectrum for the soft biological tissues in scenario (II), i.e. for the case when two different anisotropic factors are along the two mutually orthogonal directions, transverse to direction of propagation. Then we apply the extended Huygens-Fresnel integral for a wide-sense stationary light beam propagation in a tissue with such a power spectrum [15,37]. In particular, we assume that the beam is generated by a scalar isotropic Gaussian Schell-Model (GSM) source [37] and examine the effects of typical anisotropic tissue for two source properties: the spectral density and the spectral degree of coherence.

2. GSM beam propagation in anisotropic biological tissues

We begin by assuming that a scalar GSM light beam is incident on a soft anisotropic biological tissue with anisotropy factors μ_x = μ_z ≠ μ_y, in the plane z = 0, termed source plane or plane of incidence and propagates through it in the positive half-space z > 0 (see Fig. 1). The cross-spectral density of the GSM beam with a unit on-axis spectral density has form [37]:

W^{(0)} (r_{1}, r_{2}; ω) = exp [- \frac{r_{1}^{2} + r_{2}^{2}}{4 σ_{0}^{2}}] exp [- \frac{{(r_{2} - r_{1})}^{2}}{2 δ_{0}^{2}}],

where r₁ ≡ (x₁, y₁) and r₂ ≡ (x₂, y₂) are two position vectors in the source plane and ω is the angular frequency of light; σ₀ and δ₀ denote the transverse source r.m.s. beam width and r.m.s. coherence width, respectively. On assuming that the GSM source is quasi-monochromatic we omit the possible dependence of σ₀ and δ₀ on angular frequency ω throughout the paper.

Fig. 1 Illustration of beam propagation in biological tissues.

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For biological tissues with geometrical anisotropy the three-dimensional power spectrum can be written in the following form, in similarity with other anisotropic random media [35]:

Φ_{n} (κ_{x}, κ_{y}, κ_{z}) = \frac{{(2 π)}^{3} σ_{n}^{2} μ_{x} μ_{y} μ_{z} exp (- κ_{x}^{2} / κ_{m x}^{2} - κ_{y}^{2} / κ_{m y}^{2} - κ_{z}^{2} / κ_{m z}^{2})}{κ_{0}^{3 - α} {[κ_{0}^{2} + 4 π^{2} (μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2} + μ_{z}^{2} κ_{z}^{2})]}^{α / 2}}, 3 < α < 4 .

Here

σ_{n}^{2}

is the variance of the refractive index of the bio-tissue, μ_x, μ_y and μ_z are the anisotropic strength coefficients in each direction. Further, κ₀ is the large-scale cut-off frequency vector with magnitude

κ_{0} = \sqrt{κ_{x 0}^{2} + κ_{y 0}^{2} + κ_{z 0}^{2}}

and components

κ_{x 0} = 2 π / L_{x}

,

κ_{y 0} = 2 π / L_{y}

,

κ_{z 0} = 2 π / L_{z}

, with L_x, L_y and L_z being the outer scales along x, z and y directions, respectively: L_x = μ_xL₀, L_y = μ_yL₀ and L_z = μ_z L₀. Small-scale cut-off frequency vector κ_m has magnitude

κ_{m} = \sqrt{κ_{x m}^{2} + κ_{y m}^{2} + κ_{z m}^{2}} = 2

and components κ_mx = 2π/l_x, κ_my = 2π/l_y and κ_mz = 2π/l_z with l_x = l₀μ_x, l_y = l₀μ_y and l_z = l₀μ_z, being components of the inner scale vector l₀ of the tissue. We also note that for our anisotropy case l_x = l_z ≠ l_y and L_x = L_z ≠ L_y. For μ_x = μ_y = μ_z = 1 power spectrum in Eq. (2) reduces to 3D isotropic spectrum of Ref. [17].

As mentioned above, the inner scale of a typical bio-tissue is smaller than the wavelength but finite. Its values were not reported in [9] but are crucial for convergence of the integrals of interest. Since we employ the concept of the turbulent cascade of scales for the treatment of light propagation it appears plausible to choose the inner scale, i.e. the smallest scale affecting the beam to be comparable with the smallest inhomogeneity in the tissue cell. It is well known that the smallest micro-structures participating in the scattering process from the soft bio-tissues are organelles, being on the order of 0.2–0.5 μm, i.e. roughly comparable with the half of a typical optical wavelength [38]. Hence we will select the values of the inner scale in this range.

As it leads to substantial tractability of light propagation problems in random media the Markov approximation [15] will be employed, implying that the fluctuations in the refractive index at any pair of points along the direction of propagation are delta-correlated. Hence power spectrum in Eq. (2) becomes:

Φ_{n} (κ_{x}, κ_{y}, 0) = \frac{{(2 π)}^{3} σ_{n}^{2} μ_{x} μ_{y} μ_{z} exp (- κ_{x}^{2} / κ_{m x}^{2} - κ_{y}^{2} / κ_{m y}^{2})}{κ_{0}^{3 - α} {[κ_{0}^{2} + 4 π^{2} (μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2}]}^{α / 2}}, 3 < α < 4 .

In Fig. 2 the power spectrum of a typical anisotropic biological tissue is plotted from Eq. (3) along the x– and y directions. The parameters of the bio-tissue have been selected as follows: σ_n = 2 × 10⁻², L₀ = 5μm, l₀ = 0.2μm, μ_x = 1, μ_y = 3, μ_z = 1, α = 3.5. The discrepancy in the power spectrum along the two orthogonal directions is clearly seen in the inertial range of scales, κ₀ < κ < κ_m, while the two curves merge at both cut-off frequencies, i.e., κ₀ ≈ 0.1μm ⁻¹ and κ_m ≈ 100μm⁻¹.

Fig. 2 Anisotropic power spectrum of the bio-tissue refractive index along the x− and y− directions, plotted from Eq. (3).

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The extended Huygens-Fresnel integral relating the transverse cross-spectral densityW(ρ₁, ρ₂, z) of the beam at distance z in the tissue with that in the source plane Eq. (1) has form [15,37]

W (ρ_{1}, ρ_{2}, z) = \iint W (r_{1}, r_{2}) K (r_{1}, r_{2}; ρ_{1}, ρ_{2}, z) d^{2} r_{1} d^{2} r_{2},

where the integration is performed for all source points r₁ and r₂ and propagator K is given by expression

\begin{matrix} K (r_{1}, r_{2}; ρ_{1}, ρ_{2}, z) = {(\frac{k}{2 π z})}^{2} exp [- i k \frac{{(r_{1} - ρ_{1})}^{2} - {(r_{2} - ρ_{2})}^{2}}{2 z}] \\ \times {〈 exp [Ψ^{*} (ρ_{1}, r_{1}; ω) + Ψ (ρ_{2}, r_{2}; ω)] 〉}_{M} . \end{matrix}

Here the term in the angular brackets represents the correlation function of the complex phase perturbation Ψ(ρ, r; ω) due to the tissue inhomogeneities, and subscript M denotes the average taken over the ensemble of tissue realizations.

Following the calculations of Ref. [35], in anisotropic turbulent medium the complex phase correlation term can be represented as a product of two Cartesian parts:

\begin{array}{l} 〈 exp [Ψ^{*} (ρ_{1}, r_{1}; ω) + Ψ (ρ_{2}, r_{2}; ω)] 〉 = exp [- \frac{π^{2} k^{2} z T (ξ_{d}^{2} + ξ_{d} x_{d} + x_{d}^{2}}{3 μ_{x}^{2}}] \\ \begin{array}{l} \times exp [- \frac{π^{2} k^{2} z T (η_{d}^{2} + η_{d} y_{d} + y_{d}^{2})}{3 μ_{y}^{2}}] \end{array}, \end{array}

where x_d and y_d are Cartesian coordinates of source plane vector r_d = r₂ − r₁ while ξ_d and η_d are Cartesian coordinates of transverse field vector ρ_d = ρ₂ − ρ₁,

T = \frac{1}{μ_{x} μ_{y}} \int_{0}^{\infty} κ^{' 3} Φ_{n}^{'} (κ^{'}) d κ^{'},

and the adjusted spectrum Φ′_n(κ′) is obtained from Φ_n(κ) given in Eq. (3) by changing spatial frequency vector κ = (κ_x, κ_y) to κ′ = (κ′_x, κ′_y), where κ′_x = μ_xκ_x, κ′_y = μ_yκ_y to take form:

Φ_{n}^{'} (κ^{'}) = {(2 π)}^{3} σ_{n}^{2} μ_{x} μ_{y} μ_{z} \frac{exp [- {| κ^{'} |}^{2} / κ_{m}^{2}]}{κ_{0}^{3} {(1 + 4 π^{2} {| κ^{'} |}^{2} / κ_{0}^{2})}^{α / 2}} .

Then term T becomes

\begin{array}{l} T = \frac{μ_{z} σ_{n}^{2} κ_{0}}{4 π (α - 2)} [(2 + 4 π^{2} (α - 2) {(κ_{m} / κ_{0})}^{2}) {(2 π)}^{2 - α} {(κ_{m} / κ_{0})}^{2 - α} \\ \begin{array}{l} \times exp [κ_{0}^{2} / 4 π^{2} κ_{m}^{2}] Γ (2 - \frac{α}{2}, {(2 π κ_{m} / κ_{0})}^{- 2}) - 2] \end{array}, \end{array}

where Γ (·, ·) is the incomplete Gamma function. Finally, on substituting from Eqs. (5), (6), and (9) into Eq. (4) we find that the cross-spectral density of the beam factorizes as:

W (ρ_{1}, ρ_{2}, z) = W_{x} (ξ_{1}, ξ_{2}) W_{y} (η_{1}, η_{2}),

where ρ₁ = (ξ₁, η₁), ρ₂ = (ξ₂, η₂) and

\begin{array}{l} W_{x} (ξ_{1}, ξ_{2}) = \frac{1}{\sqrt{Δ_{x} (z)}} exp [- \frac{ξ_{1}^{2} + ξ_{2}^{2}}{4 σ_{0}^{2} Δ_{x} (z)}] exp [- \frac{i k (ξ_{1}^{2} - ξ_{2}^{2}}{2 R_{x} (z)}] \\ \begin{array}{l} \times exp [- (\frac{1}{2 δ_{0}^{2} Δ_{x} (z)} + \frac{π^{2} k^{2} T_{z}}{3 μ_{x}^{2}} (1 + \frac{2}{Δ_{x} (z)} - \frac{π^{4} k^{2} T^{2} z^{4}}{18 μ_{x}^{4} Δ_{x} (z) σ_{0}^{2}}) {(ξ_{1} - ξ_{2})}^{2}] \end{array} \end{array}

with

\begin{array}{l} Δ_{x} (z) = 1 + [\frac{1}{4 k^{2} σ_{0}^{4}} + \frac{1}{k^{2} σ_{0}^{2}} (\frac{1}{δ_{0}^{2}} + \frac{2 π^{2} k^{2} T z}{3 μ_{x}^{2}})] z^{2}, \\ R_{x} (z) = z + \frac{σ_{0}^{2} z - π^{2} T z^{4} / 3 μ_{x}^{2}}{(Δ_{x} (z) - 1) σ_{0}^{2} + π^{2} T z^{3} / 3 μ_{x}^{2}} . \end{array}

Factor W_y(η₁, η₂) in Eq. (10) has the same form as W_x(ξ₁, ξ₂). The spectral density and the degree of coherence of a beam-like field are defined by expressions [37]

\begin{matrix} S (ρ, z) = W (ρ, ρ, z), \\ μ (ρ_{1}, ρ_{2}, z) = \frac{W (ρ_{1}, ρ_{2}, z)}{\sqrt{S (ρ_{1}, z) S (ρ_{2}, z)}} . \end{matrix}

On substituting from Eqs. (10)–(12) into Eq. (13) we can examine the evolution of the GSM beams in anositropic tissues. Further, the 1/e values of S and μ of the GSM beams are conventionally used for calculations of the r.m.s. widths of these distributions. In particular, along the x and y directions we obtain the formulas (see also [35]):

σ_{i} (z) = σ_{0} \sqrt{Δ_{i} (z)} = \sqrt{σ_{0}^{2} + \frac{z^{2}}{4 k^{2} σ_{0}^{2}} + \frac{z^{2}}{k^{2} δ_{0}^{2}} + \frac{2 π^{2} T z^{3}}{3 μ_{i}^{2}}}, (i = x, y)

δ_{i} (z) = {[\frac{1}{δ_{0}^{2} Δ_{i} (z)} + \frac{2 π^{2} k^{2} T z}{3 μ_{i}^{2}} + \frac{2 π^{2} k^{2} T z}{3 μ_{i}^{2} Δ_{i} (z)} (2 - \frac{π^{2} T z^{3}}{6 μ_{i}^{2} σ_{0}^{2}})]}^{- \frac{1}{2}}, (i = x, y)

3. Numerical examples

We will now discuss two aspects of the beam-tissue interaction problem based on a number of numerical examples. On the one hand, we study how the statistics of light change on passing through the bio-tissues that can be characterized by various values of the power-spectrum parameters: the variance of the refractive index $σ_{n}^{2}$ , the ratio of the anisotropic factors along the x and y axes, μ_x/μ_y, the slope α in the inertial range, and the inner and outer scales, l₀ and L₀. The other facet of the problem is to explore how light beams radiated by sources with different properties, such as the initial r.m.s. beam width, σ₀, and the initial coherence width, δ₀, evolve on passing through the same tissue. For all the examples below the numerical values of parameters are set to: λ = 0.6328μm, σ₀ = 3mm, $σ_{n}^{2} = 4 \times 10^{- 4}$ , L₀ = 5μm, l₀ = 0.2μm, μ_x = 1, μ_y = 3, μ_z = 1, α = 3.5, unless different values are specified in figure captions. Figures 3–7 examine the evolution of the beam radiated by a coherent source (δ₀ → ∞) while Fig. 8 illustrates the effects of partial source coherence (0 < δ₀ < ∞). The horizontal and vertical axes in Figs. 3–6 are the Cartesian coordinates of vector ρ = (ξ, η).

Fig. 3 Spectral density of the coherent GSM beam (δ₀ → ∞) propagating in the bio-tissue. (A) z = 0; (B) z = 1.2mm; (C) z = 1.8mm, (D) z = 1cm.

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Fig. 4 Spectral density of the coherent GSM beam (δ₀ → ∞) propagating in the bio-tissue at z = 1cm with different anisotropy ratios: (A) μ_x/μ_y = 1 : 1; (B) μ_x/μ_y = 1 : 1.5, (C) μ_x/μ_y = 1 : 3, (D) μ_x/μ_y = 1 : 5.

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Fig. 5 Spectral density of the coherent GSM beam (δ₀ → ∞) at z = 1mm with different α (A) α = 3.1, (B) α = 3.3, (C) α = 3.6, (D) α = 3.9.

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Fig. 6 Spectral density of the coherent GSM beam (δ₀ → ∞) at z = 1mm with different initial beam width (A) σ₀ = 1mm, (B) σ₀ = 2mm, (C) σ₀ = 3mm, (D) σ₀ = 5mm.

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Fig. 7 The r.m.s. beam widths σ_i (i = x, y) of the coherent GSM beam (δ₀ → ∞) vs. propagation distance z.

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Fig. 8 (A), (C) σ_i, (i = x, y) and (B), (D) δ_i (i = x, y) of the GSM beam with (A), (B) δ₀ = σ₀ = 3mm ; (C), (D) δ₀ = λ = 0.6328μm, as a function of propagation distance z.

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Figure 3 shows the changes in the spectral density of the beam along the propagation path, showing that the beam spreads much sooner in the path along the axis with the smaller anisotropic factor (y–axis) than that with the larger one (x–axis). Hence, we can clearly see the gradual appearance of the elliptical beam profile after propagating at distances on the order of 1mm from the plane of incidence. As the propagation distance increases to values on the order of 1cm, the ratio of the semi-minor axis and the semi-major axis of the ellipse approaches 1/3 being exactly the ratio of the anisotropic factors, μ_x/μ_y.

Figure 4 illustrates how different ratios μ_x/μ_y of anisotropic factors affect the spectral density at a sufficiently large but fixed propagation distance from the source, say, 1 cm. As expected, the smaller ratio μ_x/μ_y, i.e. the stronger anisotropy of the tissue results in the larger ratio of the two semi-axes of the ellipse. At larger distances the ratio of the minor- to major semi-axes of the ellipse is approximately the same as μ_x/μ_y.

Figure 5 demonstrates how different values of the power spectrum slope, α, affect the evolution of the spectral density of the beam. The smaller values of α [Figs. 5(A), 5(B)] put more weight on the smaller inhomogeneities, leading to more small-scale scattering effects and less refraction-like effects, hence, resulting in spectral density expanding sooner along the propagation path and exhibiting the elliptical profiles at smaller distances from the source. On the other hand, larger values of α [Figs. 5(C), 5(D)] produce more of the refraction-like effects on the beam and hence its spectral density preserves its source plane profile for large distances.

Figure 6 shows the effect of the initial beam width on the spectral density distribution of the propagating beam. We can see that, in terms of shape-invariance, the initially smaller beams [Fig. 6(A), 6(B)] are more susceptible to the bio-tissue anisotropy, gradually chaning to the elliptic profiles, while beams radiated by the larger sources [Fig. 6(C), 6(D)] are preserving the initial, circular, profiles much better.

In Fig. 7 we explore the effects of the bio-tissue parameters on the evolution of the r.m.s. beam widths σ_i (i = x, y) calculated from Eq. (14), with increasing propagating distance, for a fixed ratio of anisotropic factors μ_x/μ_y = 1/3, for a beam radiated by a coherent source (δ₀ = ∞). Figure 7(A) shows σ_i (i = x, y) changing along the propagation path for two values of the local strength of fluctuations, $σ_{n}^{2}$ , taken from Ref. [8]. We confirm that the larger values of $σ_{n}^{2}$ correspond to the larger discrepancies between the beam widths along the x− and y directions, and more so at larger distances from the source. Figure 7(B) illustrates the effect of the slope α of the bio-tissue power spectrum on the beam spreading. In agreement with results in Fig. 5, the smaller values of α lead to larger beam expansion rates. Figures 7(C) and 7(D) show the influence of the outer scale L₀ and the inner l₀ of the power spectrum on the beam profiles. The inner scale of the spectrum generally has stronger influence on the expansion rates of the beam than the outer scale. Bio-tissues with smaller inner scales l₀ indicating finer structures give more scattering effects on the beam, and hence the beam has larger expansion rate. Tissues with larger outer scales L₀ result in slightly smaller expansion rate.

We will now demonstrate the effects of the source coherence on the beam spread and on the coherence state of the propagating light. In Fig. 8 we show σ_i [Fig. 8(A) and 8(C)] and δ_i [Fig. 8(B) and 8(D)] (i = x, y), calculated from Eqs. (14) and (15), respectively, for fairly coherent source [Fig. 8(A) and 8(B)] with δ₀ on the order of the source width and for nearly incoherent source [Fig. 8(C) and 8(D)], respectively, with δ₀ being on the order of the wavelength. The beam expansion is the same in both cases, as seen from Figs. 8(A) and 8(C), i.e. the source coherence state does not affect the beam size at all. This is explained by the fact that in both cases the source coherence width decreases rapidly with the propagating distance and attains values smaller than the wavelength of light at distances as little as several μm from the source. Hence the width and the coherence state of the beam are solely determined by the bio-tissues correlations, in particular by the ratio of the anisotropic factors μ_x/μ_y.

4. Discussion

Regardless from the formal similarities in the power spectra of the biological tissues and those for the turbulent atmosphere and oceans the distributions of energy among the spatial scales for these media are very different. First, the refractive index structure parameter of the biological tissue power spectrum (around 10⁻⁴) is many orders of magnitude stronger than that of the atmosphere (10⁻¹³ − 10⁻¹⁷). Moreover, the inertial range of scales in a soft bio-tissue extends from a fraction of a micron to several microns. This is also in striking difference with the turbulent atmosphere where such range can vary from millimeters to tens of meters. Hence, in bio-tissues the scales that are comparable with the light wavelength lead to predominantly scattering effects on light, while virtually no refractive effects are present. The relatively small inertial range of the bio-tissue and its high refractive index variance lead to the values of parameter T, defined in Eqs. (7) and (9) and accounting for the tissue effects on the beam, to be in the range from 0.4 to 7.2 mm⁻¹ compared with 10⁻¹⁷ to 10⁻¹³mm⁻¹ for the atmosphere. That is why, already after a very small propagation distance within the bio-tissue, the large value of T rapidly drives the coherence state of the beam to zero, regardless of its source value. On the contrary, in the atmospheric and oceanic turbulence there is always a region present close to the source where the degree of coherence of an initially partially coherent or incoherent beam can increase, on the basis of the van Cittert-Zernike theorem, and only after a sufficient propagation distance it starts decreasing, when the accumulated medium fluctuations become dominating. Thus, the refractive-index fluctuations of the bio-tissue play the crucial part for spreading of the beam compared with its source coherence state, and start affecting the beam virtually from the source plane. Analytically this statement can be verified by examining Eq. (14). Here the large values of T make the fourth term much more dominant compared with the third term responsible for the source coherence state, via δ₀. On setting these two terms equal we can estimate a typical propagation distance z at which fluctuations in the medium start dominating those of the source:

\frac{z^{2}}{k^{2} δ_{0}^{2}} = \frac{2 π^{2} T z^{3}}{3 μ_{i}^{2}} .

Solving for z gives:

z_{c} = \frac{3 μ_{i}^{2}}{2 π^{2} T k^{2} δ_{0}^{2}} .

For typical values of source and tissue parameters z_c takes sub-micron values. Moreover, the forth term in Eq. (14) has inverse dependence on the anisotropy parameter. This explains that starting from very small distances from the source the r.m.s. beam widths expand in inverse proportion with the anisotropic factors.

5. Summary

Treating a soft anisotropic biological tissue as a statistically stationary random medium described by the spatial power spectrum of a non-Kolmogorov type with different strengths along the two mutually orthogonal directions we have analyzed propagation in it of coherent and partially coherent, scalar light beams with the optical axis laying in the direction transverse to the tissue anisotropy plane. We have applied the extended Huygens-Fresnel method for analytic evaluation of the propagating beam spectral density and its coherence state, within the validity of the Markov approximation.

We have found that the source coherence state does not play significant part in the beam spreading. Since the bio-tissue is composed of inhomogeneites on the order of the light wavelength decoherence of the beam due to the bio-tissue can occur within a micron-like distance from the source. Further, the effect of the bio-tissue on the beam expansion rate starts occurring at distances on the order of hundreds of microns to millimeters. At these distances the beam acquires the elliptical shape, the ratio of semi-major to semi-minor axes being equal to the bio-tissue anisotropy ratio. We have provided a variety of numerical examples illustrating the effects of the source and tissue parameters on the beam evolution. Also, we have suggested a qualitative comparison of the typical beam behavior in the soft bio-tissue and other random natural media. The results of this study may be of use in medical diagnostics and treatment of anisotropic bio-tissues by means of optical radiation.

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Optical beam propagation in soft anisotropic biological tissues

Abstract

1. Introduction

2. GSM beam propagation in anisotropic biological tissues

3. Numerical examples

4. Discussion

5. Summary

References

Cited By

Figures (8)

Equations (17)

OSA Continuum