Modified biological spectrum and SNR of Laguerre-Gaussian pulsed beams with orbital angular momentum in turbulent tissue

Ye Li; Yixin Zhang; Yun Zhu; Lin Yu

doi:10.1364/OE.27.009749

1. Introduction

The relationship between optical and biological properties of tissue is the underlying mechanism for optical biomedical technology used in optical imaging, diagnostics and therapeutics [1–4]. The scattering of light propagation in biological tissue is a principal component which affects the quality of optical imaging and the accuracy of diagnostic.

Recently, the nature of light scattering in biological tissue has been actively studied by experimental measurements [4,5] and theoretical simulations [6–9]. Xu and Alfano [6] presented a fractal continuous model of the biological spectrum and correlation function, and showed that the simulation scattering results are in good agreement with experimental results for liver and other tissues. Sheppard [7] further simplified the Xu and Alfano’s model of light scattering and extended the range of allowable power laws into the subfractal regime. Glaser [8] presented a 3D model of light propagation through fractal medium using the fractal propagation method, and revealed the beam steering and aberration of focused Gaussian and Bessel beams in fractal media agreeing with experimental measurements of these effects in fresh mouse esophageal tissue. However, the limits of the sensitivity of light scattering to different structural length-scales in a biological tissue have effects on detecting and quantifying subdiffractional structures. To study the short length-scale or long length-scale of “particles”, a new fractal model of light scattering in biological tissues and cells containing only short length-scale and long length-scale was proposed by Radosevich [9]. But the simulation results of the power spectral in the region of high spatial wave number or low spatial wave number are inconsistent with experimental results.

In recent years, optical beams with orbital angular momentum (OAM) propagating in random media have attracted much attention. The main motivation is that beams carrying OAM can realize arbitrary base-N quantum digits in principle [10–12], which offers the additional degree of freedom for information coding and increases information capacity of optical links. The propagation of the optical signal through a biological medium suffers attenuation and wavefront distortion because of the presence of various tissue constituents and the fluctuations of the refractive index of biological tissue [13]. Turbulent biological tissue distorts OAM modes of optical beams, which introduces intermodal crosstalk between OAM channels. In addition, since optical pulse propagation promises broad bandwidth and high data rate at high frequencies that has potential advantages to improve the quality of optical imaging and diagnostic accuracy [14–18]. Considering the merits of the vortex pulsed beam, the applications of the pulsed beam carrying OAM in wireless optical communications have been investigated [19–22]. Li et al. [19] investigated the influence of oceanic turbulence on Laguerre-Gaussian (LG) pulsed beam, and showed that the vortex pulsed beams can improve the communication quality and capacity of the optical link. Nie et al. [20] studied the spatiotemporal coupling characteristics and the intensity distribution variations of ultrashort Gaussian vortex pulse and found that the topological charge number of the beam affects the intensity and the spatiotemporal coupling. Liu and Gao [21] investigated the time-varying of the LG pulsed beam, and found that a higher OAM quantum number can cause the larger spread of pulse width in short propagating range. Martínez-Matos et al. [22] analyzed the spatial pulse shaping of ultrashort LG beams, and revealed that the topological charge drastically jumps with evolving the pulse in time. The signal to noise ratio (SNR) of signal light plays a key role in biophotonics research which provides quantitative guidance for disease diagnosis or screening. To the best of our knowledge, the SNR of LG pulsed beam with OAM in turbulent biological tissue with short length-scales and long length-scales has not been investigated.

In this paper, we propose two-scale modified biological power spectrum of the refractive index fluctuations for biological tissues including both short length-scales and long length-scales, and develop a two-frequency mutual coherence function (MCF) of LG pulsed beams with OAM. Then, we derive the analytic expression for the coherent function of LG pulsed beams with OAM propagating through turbulent biological tissue based on the modified biological spectrum and the two-frequency MCF. We establish the spatiotemporal model of the SNR of OAM states for LG pulsed beams propagating in turbulent biological tissue.

2. Modified biological spectrum

2.1. Two-scale modified biological spectrum model

The conventional power spectrum of the refractive index fluctuations of biological tissues is represented by [9]:

ϕ_{n} (κ) = \frac{A_{n} l_{c}^{3} Γ (D / 2)}{π^{3 / 2} 2^{(5 - D) / 2} {(1 + κ^{2} l_{c}^{2})}^{D / 2}},

where

A_{n}

is the fluctuation strength,

l_{c}

is the characteristic length of heterogeneity,

D

is the fractal dimension and determines the shape of the distribution,

κ

is the spatial frequency of the tissue fluctuation, and

Γ (\cdot)

represents the gamma function.

Spectrum model of Eq. (1) is relatively tractable model and commonly used in theoretical studies of optical wave propagation, which doesn’t include the structural length-scale. However, the practical applications for medical diagnosis and imaging in biological tissue have indicated that the influence of structural length-scale sensitivities on optical scattering cannot be ignored [9]. To make the power spectrum of biological tissues agreeing well with the experimental data in the high and low wave number region, we need to introduce a function which includes the structural length-scales. Thus, we define the excess refractive index which contributes to scattering as $n_{Δ} (\vec{r}) = n (\vec{r}) / n_{0} - 1$ , where $n (\vec{r})$ represents a continuous distribution of fluctuating refractive index, $\vec{r}$ is the spatial position and $n_{0}$ is the mean refractive index. We know that the turbulent fluctuations can be divided into the regions of low frequencies and high frequencies. In the low frequency region, the refractive index fluctuation is caused by large scale eddies. The refractive index fluctuation of large scale eddies includes the difference of refractive index between different eddies and the fluctuation of refractive index at different positions within a certain eddies. However, in high frequencies region, the refractive index fluctuation caused by small scale eddies. Since the eddies are small enough, the refractive index of each turbulence is approximately constant. Therefore, the refractive index fluctuation of small scale eddies only includes the difference of refractive index between different eddies. From the above discussion and the discussion in [9], we can express the the refractive index fluctuation $n_{Δ} (\vec{r})$ as

n_{Δ}^{S} (r) = n_{Δ} (\vec{r}) * G_{S} (\vec{r}),

n_{Δ}^{L} (r) = n_{Δ} (\vec{r}) * [δ (\vec{r}) + G_{L} (\vec{r})],

where

G_{S} (\vec{r}) = a_{S} \exp (- b_{S} {\vec{r}}^{2} / w_{S}^{2})

represents the modified short length-scale Gaussian,

w_{S}

is the short length-scale parameter,

a_{S}

and

b_{S}

are constants,

G_{L} (\vec{r}) = a_{L} \exp (- b_{L} {\vec{r}}^{2} / w_{L}^{2})

represents the modified long length-scale Gaussian,

w_{L}

is the long length-scale parameter,

a_{L}

and

b_{L}

are constants,

δ (\vec{r})

represents unit impulse function.

Since $n_{Δ} (\vec{r})$ is a random variable, the statistical description of refractive index fluctuations through the structure function $D_{n}^{c} (r)$ can be described as [9]

D_{n}^{c} (r) = \int n_{Δ}^{S} (\vec{r}) n_{Δ}^{L} (r_{d} - \vec{r}) d \vec{r} = n_{Δ}^{S} (r) * n_{Δ}^{L} (r) .

Using the convolution theorem of Fourier transform, the structure function in Eq. (3) can then be found as

D_{n}^{c} (r) = F^{- 1} [{| F [n_{Δ} (\vec{r})] |}^{2} | F [G_{S} (\vec{r})] (1 + F [G_{L} (\vec{r})]) |],

where

F (\cdot)

represents the Fourier transform and

F^{- 1} (\cdot)

represents the inverse Fourier transform.

Under the Born approximation, $ϕ_{n}^{c} (κ)$ is the Fourier sine transform of $D_{n}^{c} (r)$ [9]. The structure function $D_{n}^{c} (r)$ can be described as

D_{n}^{c} (r) = 4 π \int_{0}^{\infty} κ ϕ_{n}^{c} (κ) \frac{\sin (κ r)}{r} d κ .

where

ϕ_{n}^{c} (κ)

is the modified biological spectrum including both short length-scale and long length-scale parameters and can be calculated by Eqs. (4) and (5) as

ϕ_{n}^{c} (κ) = ϕ_{n} (κ) | F [G_{S} (\vec{r})] (1 + F [G_{L} (\vec{r})]) | .

where

ϕ_{n} (κ) = {| F [n_{Δ} (\vec{r})] |}^{2}

.

With the help of the integral expressions [23,24]

\int_{- \infty}^{+ \infty} \exp (- a x^{2} + i b x) d x = \frac{\sqrt{π}}{\sqrt{a}} exp (- \frac{b^{2}}{4 a}),

we obtain a two-scale modified biological spectrum which is given as

ϕ_{n}^{c} (κ) = \frac{A_{n} l_{c}^{3} Γ (D / 2)}{π^{3 / 2} 2^{(5 - D) / 2} {(1 + κ^{2} l_{c}^{2})}^{D / 2}} \frac{a_{S} a_{L} π w_{S} w_{L}}{2 \sqrt{b_{S} b_{L}}} \exp (- \frac{w_{S}^{2} κ^{2}}{4 b_{S}}) [1 + \exp (- \frac{w_{L}^{2} κ^{2}}{4 b_{L}})] .

In order to eliminate the effects of coefficients on the two-scale modified biological spectrum, we set

a_{S} = \sqrt{2 b_{S}} / (\sqrt{π} w_{S}), a_{L} = \sqrt{2 b_{L}} / (\sqrt{π} w_{L}),

and the two-scale modified biological spectrum is rewritten as

ϕ_{n}^{c} (κ) = \frac{A_{n} l_{c}^{3} Γ (D / 2)}{π^{3 / 2} 2^{(5 - D) / 2} {(1 + κ^{2} l_{c}^{2})}^{D / 2}} \exp (- \frac{w_{S}^{2} κ^{2}}{4 b_{S}}) [1 + \exp (- \frac{w_{L}^{2} κ^{2}}{4 b_{L}})] .

2.2. Feasibility analysis of modified biological spectrum

In order to verify the rationality of the two-scale modified biological spectrum derived in the paper, we compare the results of different theoretical spectral models with experimental data, and analyze the consistency with the two-scale modified biological spectrum, the common spectrum and the experimental data.

With the values and the parameters of the mouse liver tissue shown in Table 1 obtained by Schmitt et al. [25], we simulate the two-scale modified biological spectrum described by Eq. (10) in Fig. 1. Compared with the Sheppard’s model (S’s model) [7] and Radosevich’s model (R’s model) [9], our model has a more accurate match with experimental data from low to high spatial wave number region, when the parameters are set as $D = 3$ , $l_{c} = 10.4 μ m$ , $b_{S} = - 2.77$ , $b_{L} = 2.77$ , $w_{s} = 0.6 μ m$ and $w_{L} = 120 μ m$ for mouse liver tissue. This is the first result in the paper.

Fig. 1 Power spectrum of refractive index variations in mouse liver tissue fits to the theoretical power spectrum of Eq. (10). The round solid points represent experimental data reported in Ref [25]. The chain-dotted line with circles is theoretical curve in Ref [7]. The two dotted curves are theoretical curves in Ref [9], and the solid line is our theoretical fit. Fitting yields $D = 3$ , $l_{c} = 10.4 μ m$ , $w_{s} = 0.6 μ m$ and $w_{L} = 120 μ m$ .

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We know that Xu and Alfano [6] fit their theoretical model to the reduced scattering coefficient with the changing wavelength from $600 nm$ to $1400 nm$ in rat liver tissue reported by Parsa et al. [26]. Evidently, the reduced scattering coefficient within the range $600 nm$ to $1400 nm$ is not in good agreement with experimental data. This wavelength range is important for optical communications in biological tissue. In Fig. 2, we fit the reduced scattering coefficients of Eq. (11) within the range $600 nm$ to $1400 nm$ to the experimental data [27]. Compared with the Xu and Alfano’s model (X’s model) [6], our model based on the modified biological spectrum has better agree with those extracted from the light scattering data for rat liver tissue [26].

Fig. 2 Wavelength dependence of the reduced scattering coefficient ${μ^{'}}_{s}$ of rat liver tissue fitted to the power law. The round solid points represent experimental data reported in Ref [26]. The dotted curve is derived from the model in Ref [6], and the chain-dotted line is the fitted curve of our model. Fitting yields $D = 3.94$ , $l_{c} = 2.3 μ m$ , $w_{S} = 0.26 μ m$ and $w_{L} = 12 μ m$ .

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The reduced scattering coefficient of beam propagating in a multiple scattering medium is expressed by [27]

{u^{'}}_{s} = 4 π^{2} k^{4} \int_{- 1}^{1} ϕ_{n}^{c} (κ) (1 - \cos θ) (1 + \cos^{2} θ) d \cos θ,

where

κ = k \sqrt{2 - 2 \cos θ}

,

k = 2 π / λ

is the wavenumber of light and

θ

is the scattering angle.

The Euclidean distances between the experimental data points of the power spectrum and our new model, the S’s model, the R’s model 1, the R’s model 2 are 0.8928, 1.1539, 1.5098, 1.4395, respectively. In addition, the Euclidean distances between the experimental data points of reduced scattering coefficient and our new model, the X’s model are 2.4390, 3.3470, respectively. Thus, the new biological spectrum derived from the theory is more reliable.

3. Vortex pulsed beam propagation in biological tissue

3.1. The complex envelope of the output LG pulsed beams in biological tissues

Now, we consider the propagation of an input ultrashort LG pulsed beams in turbulent biological tissues at $z$ plane of a cylindrical coordinate system $(r, φ, z)$ . Based on the developed two-scale modified biological spectrum in the second part, the complex envelope of the output LG pulsed beams propagation in biological tissue can be obtained by [28,29]

v_{0} (r, φ, z; t) = \int_{- \infty}^{+ \infty} V_{j} (ω) \frac{\exp (- i ω t)}{2 π} U_{m}^{L G} (r, φ, z; ω + ω_{0}) d ω,

where

V_{j} (ω)

represents the Fourier transform of the amplitude of the input pulse, and

ω_{0}

is the carrier frequency,

r

denotes the radial coordinate at the plane

z

,

φ

is the azimuthally angle,

z

is the transmission distance,

m

is the OAM quantum number or topological charge which describes the vortex strength,

ω = k c

is the angular frequency of light, and

c

is the speed of light.

U_{m}^{L G} (\cdot)

is the complex amplitude of the monochrome scalar field of the continuous LG beam at

z

plane in biological tissues.

Under the Rytov approximation [28], when the LG monochrome beams propagates in turbulence, the complex amplitude of the LG monochrome beams can be written as

U_{m}^{L G} (r, φ, z; ω) = L G_{P}^{m_{0}} (r, φ, z; ω) \exp [ψ (r, φ, z; ω)],

where

m_{0}

is the initial OAM quantum number in the absence of turbulence,

ψ (r, φ, z; ω)

is the complex phase distortion caused by turbulent biological tissues,

L G_{P}^{m_{0}} (\cdot)

is the complex amplitude of the LG monochrome beams in the absence of biological tissues and given by [17,19]

\begin{matrix} L G_{P}^{m_{0}} (r, φ, z; ω) = \exp [- \frac{(1 - i z_{ζ}) r^{2}}{w_{0}^{2} (1 + z_{ζ}^{2})}] \frac{r^{m_{0}}}{w {(z_{ζ})}^{m_{0} + 1}} L_{P}^{m_{0}} [\frac{r^{2}}{w {(z_{ζ})}^{2}}], \\ \times \exp [i k z - i (m_{0} + 1) arc \tan (z_{ζ}) - i m_{0} φ] \end{matrix}

where

w (z_{ζ}) = w_{0} \sqrt{1 + z_{ζ}}

and

z_{ζ} = 2 z / (k w_{0}^{2}) = z / z_{R}

,

z_{R}

represents Rayleigh range,

w_{0}

is the initial radius. In the far-field region

z ≫ z_{R}

, we have the factor

\arctan (z_{ζ}) \approx π / 2

and

w (z_{ζ}) \approx 2 z / (k w_{0})

.

3.2. Two-frequency MCF of OAM states in biological tissues

To investigate the coherence bandwidth of LG pulsed beams propagating through turbulent biological tissues, we should study the two-frequency MCF to explain the pulse bandwidth. The two-point, two-time correlation function of the complex envelope of the output LG pulsed beams is defined by the ensemble average [28]

\begin{matrix} R (r_{1}, r_{2}, φ, φ^{'}, z; t_{1}, t_{2}) = 〈 v_{0} (r_{1}, φ, z; t_{1}) v_{0}^{*} (r_{2}, φ^{'}, z; t_{2}) 〉 = \frac{1}{{(2 π)}^{2}} \int \int_{- \infty}^{+ \infty} V_{j} (ω_{1}) V_{j}^{*} (ω_{2}) \\ \times \exp (- i ω_{1} t + i ω_{2} t_{2}) Γ_{2} (r_{1}, r_{2}, φ, φ^{'}, z; ω_{1} + ω_{0}, ω_{2} + ω_{0}) d ω_{1} d ω_{2}, \end{matrix}

where

Γ_{2} (r_{1}, r_{2}, φ, φ^{'}, z; ω_{1} + ω_{0}, ω_{2} + ω_{0})

is the two-frequency MCF of LG beams with OAM, and

〈 \dots 〉

denotes the ensemble average of turbulent biological tissues.

To simplify notation in the following analysis, let $k_{j} = (ω_{j} + ω_{0}) / c$ ( $j = 1, 2$ ), and then the two-frequency MCF can be expressed as

\begin{matrix} Γ_{2} (r_{1}, r_{2}, φ, φ^{'}, z; k_{1}, k_{2}) = L G_{P}^{m_{0}} (r_{1}, φ, z; k_{1}) L G_{P}^{m_{0} *} (r_{2}, φ^{'}, z; k_{2}) \\ \times 〈 \exp [ψ (r_{1}, φ, k_{1}) + ψ^{*} (r_{2}, φ^{'}, k_{2})] 〉, \end{matrix}

where

〈 \exp [ψ (r_{1}, φ, k_{1}) + ψ^{*} (r_{2}, φ^{'}, k_{2})] 〉

is the coherence function of the complex phase and can be given by [10,28]

〈 \exp [ψ (r, φ, ω_{1} + ω_{0}) + ψ^{*} (r, φ^{'}, ω_{2} + ω_{0})] 〉 = \exp {- 2 [1 - \cos (φ - φ^{'})] r^{2} / ρ_{b}^{2}},

where

ρ_{b}

represents the lateral coherence radius of a spherical wave propagating in turbulent biological tissues, and is given by

ρ_{b} = {[π^{2} {(ω_{1} - ω_{2})}^{2} z \int_{0}^{\infty} κ^{3} ϕ_{n}^{c} (κ) d κ / (3 c^{2})]}^{- 1 / 2} .

According to the integral relations [24]

\int_{0}^{x} \frac{t^{μ - 1}}{{(1 + β t)}^{v}} d t = \frac{x^{μ}}{μ} {}_{2}F_{1} (v, μ; 1 + μ; - β x), μ > 0,

\int_{0}^{\infty} κ^{2 μ} \frac{\exp (- κ^{2} / κ_{l}^{2})}{{(κ^{2} + κ_{0}^{2})}^{D / 2}} d κ = \frac{1}{2} κ_{0}^{2 μ + 1 - D} Γ (μ + \frac{1}{2}) U (μ + \frac{1}{2}; μ - \frac{1}{3}; \frac{κ_{0}^{2}}{κ_{l}^{2}}), μ > - \frac{1}{2},

where

{}_{2}F_{1} (\cdot)

represents the hypergeometric function,

U (\cdot)

represents the confluent hypergeometric function of the second kind, we can obtain the analytical expression of the lateral coherence length in turbulent biological tissues as follows

ρ_{b}^{- 2} = {(ω_{1} - ω_{2})}^{2} ℧,

where

℧ = \frac{0.59 Γ (D / 2) A_{n} l_{c}^{3} z}{2^{(7 - D) / 2} c^{2}} {\frac{Γ (2)}{l_{c}^{4}} U (2; \frac{7}{6}; \frac{w_{L}^{2} - w_{S}^{2}}{11 l_{c}^{2}}) + \sum_{n = 0}^{N} \frac{{}_{2}F_{1} (D / 2, n + 2; n + 3; - l_{c}^{2} / w_{S}^{2})}{11^{n} n! (n + 2) w_{S}^{4}}}

.

By substituting the Eq. (20) into Eq. (17), the Eq. (17) can be expressed as

〈 \exp [ψ (r, φ, ω_{1} + ω_{0}) + ψ^{*} (r, φ^{'}, ω_{2} + ω_{0})] 〉 = \exp {- Ω ω_{d}^{2}},

where

Ω = 4 \sin^{2} [(φ - φ^{'}) / 2] r^{2} ℧

,

ω_{d} = ω_{1} - ω_{2}

represents the difference frequency.

Under the narrow-band assumption, the two frequency MCF of a collimated LG vortex beam in the far field can be simplified as

\begin{matrix} Γ_{2} (r_{1}, r_{2}, φ, φ^{'}, z; k_{1}, k_{2}) = {(\frac{w_{0}}{2 z c})}^{2 m_{0} + 2} \frac{{(ω_{0} + ω_{c})}^{2 m_{0} + 2}}{{(2 π)}^{2}} r^{2 m_{0}} \exp [i m_{0} (φ - φ^{'})] \\ \times \exp [- \frac{r^{2} w_{0}^{2}}{2 z^{2} c^{2}} {(ω_{0} + ω_{c})}^{2}] \exp (- Ω ω_{d}^{2} - i ω_{d} \frac{z}{c} + i \frac{r^{2}}{2 z c} ω_{d}), \end{matrix}

where

ω_{c} = (ω_{1} + ω_{2}) / 2

represents the sum frequency. In terms of these frequencies the narrow-band assumption can be expressed as

ω_{c}^{2} ≫ ω_{d}^{2}

.

3.3 SNR of LG pulsed beams with OAM in biological tissues

For Gaussian pulse $v_{j} (t) = \exp (- t^{2} / T_{0}^{2})$ , the Fourier transform $V_{j} (ω)$ of the Gaussian pulse is given by [28,29]

V_{j} (ω) = \sqrt{π} T_{0} \exp (- \frac{1}{4} ω^{2} T_{0}^{2}),

where

T_{0}

is the half-pulse width of the input pulse.

We know that the temporal broadening of the pulse is deduced from the temporal mean on-axis intensity. To obtain the temporal broadening of the pulse, we need to obtain the coherence function of LG pulsed beams in turbulent biological tissues. The coherence function is easily obtained by setting $r_{1} = r_{2} = r$ , $t_{1} = t_{2} = t$ in Eq. (15), that is

\begin{matrix} R (r, φ, φ^{'}, z; t) = \frac{1}{{(2 π)}^{2}} \int \int_{- \infty}^{+ \infty} Γ_{2} (r, φ, φ^{'}, z; ω_{c} + ω_{0} + \frac{ω_{d}}{2}, ω_{c} + ω_{0} - \frac{ω_{d}}{2}) \\ \times V_{j} (ω_{1}) V_{j}^{*} (ω_{2}) \exp (- i ω_{d} t) d ω_{c} d ω_{d} . \end{matrix}

With the help of the integral expressions [24]

\int_{- \infty}^{+ \infty} \exp (- a x^{2} + i b x) d x = \frac{\sqrt{π}}{\sqrt{a}} exp (- \frac{b^{2}}{4 a}),

\int_{- \infty}^{+ \infty} x^{n} e^{- p x^{2} + 2 q x} d x = n! e^{q^{2} / p} \sqrt{\frac{π}{p}} {(\frac{q}{p})}^{n} \sum_{H = 1}^{⌊ n / 2 ⌋} \frac{1}{(n - 2 H)! H!} {(\frac{p}{4 q^{2}})}^{H} [p > 0],

where

H

represents any positive integer between

1

and

n / 2

, the analytic expression of the coherence function by substituting of Eq. (22) and Eq. (23) into Eq. (24) can be given by

\begin{array}{l} R (r, φ, φ^{'}, z; t) = {(\frac{w_{0}}{2 z c})}^{2 m_{0} + 2} \exp (- \frac{ω_{0}^{2} T_{0}^{2} {[r w_{0} / (z c)]}^{2}}{2 {[r w_{0} / (z c)]}^{2} + 2 T_{0}^{2}}) \exp [i m_{0} (φ - φ^{'})] \frac{T_{0}^{2} r^{2 m_{0}}}{\sqrt{(T_{0}^{2} + 8 Ω)}} \\ \times \sum_{H = 1}^{m_{0} + 1} \frac{(2 m_{0} + 2)! {({[r w_{0} / (z c)]}^{2} + T_{0}^{2})}^{H - 2 m_{0} - 5 / 2}}{(2 m_{0} + 2 - 2 H)! H! 2^{H} {(ω_{0} T_{0}^{2})}^{2 H - 2 m_{0} - 2}} exp {- 2 {[\frac{r^{2}}{2 z c} - (t - \frac{z}{c})]}^{2} / (T_{0}^{2} + 8 Ω)}, \end{array}

where

⌊ n ⌋

represents the integral part of the real number

n

. This is the second result of our paper.

Consider a single cycled pulse, the pulse resulting wave $v_{0} (r, φ, z; ω_{0})$ as a superposition of eigenstates is given as [19]

v_{0} (r, φ, z; ω) = \frac{\sqrt{T_{0}}}{2 π} \sum_{P} a_{P,}_{m, t} (z, t) R_{P} (r) \exp [i (ω t_{0} + m φ)],

where

a_{P,}_{m, t} (z, t)

is the superposition coefficients,

t

represents a continuous time within the half-pulse width. Utilizing the basis projections in Eq. (27), we obtain the coefficients

a_{P,}_{m, t} (z, t)

as

a_{P,}_{m, t} (z, t) = \frac{\sqrt{T_{0}}}{2 π} ∭ R_{P}^{*} (r) \exp (- i m φ) \exp (- i ω t) v_{0} (r, φ, z; ω) r d r d φ d ω,

where the superscript

*

represents the complex conjugate.

By summing the probabilities associated with eigenvalues, the conditional probability distribution of the monochromatic OAM states of LG pulsed beam is given by

P (m | m_{0}, t) = \sum_{P} {| a_{P,}_{m, t} (z, t) |}^{2},

where

P (m | m_{0}, t)

represents the received probability for given the initial OAM quantum number

m_{0}

and the time

t

with the change of the OAM quantum number in turbulent biological tissues

m

.

Substituting Eq. (28) into Eq. (29), and using the completeness of the radial basis [11],

\sum_{P} R_{P} (r) R_{P}^{*} (r^{'}) = δ (r, r^{'}) / r,

and integrating over

r^{'}

, we can obtain the conditional probability distribution of the OAM states of the pulsed beam

\begin{matrix} P (m | m_{0}, t) = \frac{T_{0}}{{(2 π)}^{2}} ∭ \iint \exp [- i m (φ - φ^{'})] v_{0} (r, φ, z; ω) v_{0}^{*} (r, φ^{'}, z; ω^{'}) \\ \times \exp [- i (ω - ω^{'}) t] r d r d φ d φ^{'} d ω d ω^{'} . \end{matrix}

Since the aberrations caused by turbulent biological tissue are random, the received probability distribution of monochromatic OAM states of LG pulsed beams by taking the ensemble average can be obtained. Based on the received probability of OAM states, the SNR of the OAM state of LG pulsed beams in the receiver plane can be expressed as [13]

S N R = \frac{〈 P (m_{0} | m_{0}, t) 〉}{\sum_{h = - \infty}^{\infty} 〈 P (h | m_{0}, t) 〉 - 〈 P (m_{0} | m_{0}, t) 〉} = \frac{P (m = m_{0}, t)}{\sum_{h = - \infty}^{\infty} P (m = h, t) - P (m = m_{0}, t)},

where

P (m = m_{0}, t)

represents the received probability of signal OAM states where the OAM quantum number in turbulent biological tissues

m

is equal to the initial OAM quantum number

m_{0}

,

\sum_{h = - \infty}^{\infty} P (m = h, t)

represents all the received probability of OAM states for a given initial OAM quantum number

m_{0}

, the expression of

\sum_{h = - \infty}^{\infty} P (m = h, t) - P (m = m_{0}, t)

represents the received probability of crosstalk OAM states which is the noise. The received probability of signal OAM states can be given as [11]

\begin{array}{l} P (m = m_{0}, t) = \int \iint {(\frac{w_{0}}{2 z c})}^{2 m_{0} + 2} \frac{r^{2 m_{0} + 1} T_{0}^{2}}{2 π \sqrt{(T_{0}^{2} + 8 Ω)}} \sum_{H = 1}^{m_{0} + 1} \frac{(2 m_{0} + 2)! {({[r w_{0} / (z c)]}^{2} + T_{0}^{2})}^{H - 2 m_{0} - 5 / 2}}{(2 m_{0} + 2 - 2 H)! H! 2^{H} {(ω_{0} T_{0}^{2})}^{2 H - 2 m_{0} - 2}} \\ \times \exp (- \frac{ω_{0}^{2} T_{0}^{2} {[r w_{0} / (z c)]}^{2}}{2 {[r w_{0} / (z c)]}^{2} + 2 T_{0}^{2}}) exp {- 2 {[\frac{r^{2}}{2 z c} - (t - \frac{z}{c})]}^{2} / (T_{0}^{2} + 8 Ω)} d r d φ d φ^{'} . \end{array}

4. Numerical simulation

We investigate the statistical properties of the LG pulsed beam with OAM states propagating through turbulent biological tissue by numerical simulation. In the following analysis, the calculation parameters are $λ = 0.633 μ m$ [6,29,33], $A_{n} = 4.2 \times 10^{- 5}$ [7], $z = 0.1 m$ [13], $w_{S} = 0.6 μ m$ [9], $m_{0} = 1$ , $m = 1$ , $l_{c} = 10.4 μ m$ [7,9], $w_{L} = 60 μ m$ [9], $D = 3$ [32], unless other variable parameters are specified in calculation.

The SNR of OAM states are depicted in Fig. 3 as a function of the co-moving coordinate $ζ = t - z / c$ . As can be seen, the largest SNR of the OAM states appears at the axis $ζ = 0$ . In other words, the LG pulsed beam propagating through turbulent biological tissue has hardly pulse delay. This is caused by the short transmission distance in turbulent biological tissue. In order to investigate the effects of OAM states on the pulsed beam propagation properties, we focus our attention on the SNR of the OAM state with different OAM quantum number $m_{0}$ . Figure 3(a) shows the SNR of the OAM states for various values of the OAM quantum numbers $m_{0}$ . As would be expected, the SNR of the OAM state becomes smaller with the increasing OAM quantum number $m_{0}$ .

Fig. 3 (a) SNR and (b) NSNR of OAM states versus co-moving coordinate $ζ = t - z / c$ for different OAM quantum number $m_{0}$ .

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To obtain high data rates in optical communication and avoid pulse–pulse interference for LG pulsed beams, one must know how much of a time buffer to be allowed between pulses. We know that the pulse broadening can be described by the full width half maximum value (FWHM) of the pulse profile [30]. In order to objectively describe the pulse width broadening during pulse transmission in different turbulent media, we use the peak SNR of each OAM mode to normalize the SNR profile of the corresponding OAM mode. In following discussion, we use the absolute difference between two co-moving coordinates where the normalized signal to noise ratio (NSNR) of the OAM mode is the half of the maximum NSNR, as the pulse width.

Figure 3(b) shows the NSNR of the OAM state as a function of the OAM quantum number $m_{0}$ . In Fig. 3(b), as the OAM quantum number increases, the pulse width generates a spread slightly. Such as, the OAM quantum number $m_{0}$ increases from 1 to 5, the pulse broadening $Δ τ$ increase from $24.8 fs$ to $26.1 fs$ . That is, the influence of OAM quantum number $m_{0}$ on the NSNR is very tiny. This result reveals that the pulsed vortex beam carrying a large quantum number $m_{0}$ can realize a higher information capacity than the continuous vortex beam.

For practical applications for medical diagnosis and imaging in biological tissue, the effects of structural length-scales of biological tissue on the SNR of the OAM state should be considered. Figures 4 and 5 show that the SNR of the OAM state for LG pulsed beams as functions of the co-moving coordinate $ζ$ for different short length-scales $w_{S}$ and long length-scales $w_{L}$ , respectively. We can see from Fig. 4 that, as the short length-scale $w_{S}$ increases, the SNR of the OAM state of the LG pulsed beams increases. This result can be explained that, as the short length-scale increases, the lateral coherence length in Eq. (18) decreases. The smaller lateral coherence length has a better resistance to turbulence. Figure 4 obviously exhibits that the smaller value of short length-scale will cause the larger pulse broadening $Δ τ$ . This implies that the LG pulsed beam propagating through the biological tissue of a larger short length-scale has a large bandwidth ( $Δ ω = 2 π / Δ τ$ ), which increases the precision in optical imaging and diagnostic.

Fig. 4 SNR of OAM states versus co-moving coordinate $ζ = t - z / c$ for different short length-scale $w_{S}$ .

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Fig. 5 SNR of OAM states versus co-moving coordinate $ζ = t - z / c$ for different long length-scale $w_{L}$ .

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Figure 5 also shows that, as long length-scale $w_{L}$ of turbulent biological tissue increases, the SNR of the OAM state minutely increases and the pulse broadening minutely decreases. The phenomenon can be explained that the biological tissue with the long length-scale $w_{L}$ causes a tiny light scattering. As a consequence, the tiny light scattering will cause tiny pulse broadening $Δ τ$ which leads to weak influence on the SNR of OAM state. In other words, the long length-scale $w_{L}$ of turbulent biological tissue causes a tiny influence on the quality of optical imaging and the accuracy of diagnostic.

To comprehend the regularity of the SNR of the OAM state in different turbulent biological tissue, we analyze the influence of the shape of the distribution $D$ on the SNR of the OAM state in Fig. 6. It is evident from Fig. 6, the SNR of the OAM state for the LG pulsed beam increases with the increasing shape of the distribution $D$ from $3$ to $4$ . The reason of this phenomenon is that the large shape of the distribution $D$ induces a small light scattering caused by turbulent biological tissue [31]. As would be expected, the pulse broadening decreases with the increasing shape of the distribution $D$ . That is to say, we can choose the large shape of the distribution to alleviate the effect of turbulent biological tissue and extend bandwidth. The results will be very useful to improve optical imaging and medical diagnosis in turbulent biological tissue.

Fig. 6 SNR of OAM states versus co-moving coordinate $ζ = t - z / c$ for different shape of the distribution $D$ .

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The characteristic length of heterogeneity $l_{c}$ is the essential features of light scattering turbulent biological tissue. In Fig. 7, we analyze the influence of the characteristic length of heterogeneity $l_{c}$ on the SNR of the OAM state and the pulse broadening. Figure 7(a) shows the SNR of the OAM states under different characteristic lengths of heterogeneity $l_{c}$ . The larger characteristic length of heterogeneity $l_{c}$ is, the smaller the SNR of the OAM state becomes. In Fig. 7(b), the pulse broadening increases with the increasing characteristic length of heterogeneity. Thus, we can use the results to realize a more information capacity and improve the quality of optical imaging and the accuracy of diagnostic.

Fig. 7 (a) SNR, (b) NSNR of the OAM states versus co-moving coordinate $ζ = t - z / c$ for different characteristic length of heterogeneity $l_{c}$ .

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Figure 8 shows the SNR of the OAM state for various values of the initial half-pulse width $T_{0}$ . As we expected, Fig. 8 shows that the pulse broadening increases as the initial half-pulse width increases. In addition, from Fig. 8, the SNR of the OAM states increases with the increasing initial half-pulse width. This is because that the larger initial half-pulse width is, the smaller temporal scintillation caused by turbulence becomes [32].

Fig. 8 SNR of the OAM states versus co-moving coordinate $ζ = t - z / c$ for different initial half-pulse width $T_{0}$ .

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5. Conclusion

In this paper, by introducing short length-scale and long length-scale factors of biological tissue, a two-scale modified biological spectrum is proposed, which is more consistent with the experimental data than the existing biological spectrum. Based on the two-scale modified biological spectrum, we derived the two-frequency mutual coherence function of LG pulsed beam and established the model of the SNR of OAM states for LG pulsed beams propagation in the turbulent biological tissues which contains the spatial-temporal characteristics. By numerical simulation with published experimental parameters, we further analyzed the effects of turbulent biological tissues on the SNR and temporal properties of LG pulsed beams carrying OAM. The results demonstrate that the turbulent biological tissues with the longer short length-scale, the greater shape of distribution, and the small characteristic length of heterogeneity can increase the SNR of OAM states and the bandwidth of vortex pulsed beams. Further, the LG pulsed beam with smaller OAM quantum number and larger initial half-pulse width has better resistance to the distraction of turbulent biological tissues. These results indicate that, for medical instruments, the vortex pulsed light is a better choice than the conventional light source.

Funding

National Natural Science Foundation of China (61871202, 61701196, 11847109).

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Modified biological spectrum and SNR of Laguerre-Gaussian pulsed beams with orbital angular momentum in turbulent tissue

Abstract

1. Introduction

2. Modified biological spectrum

2.1. Two-scale modified biological spectrum model

2.2. Feasibility analysis of modified biological spectrum

3. Vortex pulsed beam propagation in biological tissue

3.1. The complex envelope of the output LG pulsed beams in biological tissues

3.2. Two-frequency MCF of OAM states in biological tissues

3.3 SNR of LG pulsed beams with OAM in biological tissues

4. Numerical simulation

5. Conclusion

Funding

References

Cited By

Figures (8)

Equations (36)

Optics Express