- Email: [email protected]
Average intensity and spreading of partially coherent model beams propagating in a turbulent biological tissue
Author’s Accepted Manuscript Average intensity and spreading of partially coherent model beams propagating in a turbulent biological tissue Yuqian Wu, Yixin Zhang, Qiu Wang, Zhengda Hu www.elsevier.com/locate/jqsrt
PII: DOI: Reference:
S0022-4073(16)30269-2 http://dx.doi.org/10.1016/j.jqsrt.2016.08.001 JQSRT5416
To appear in: Journal of Quantitative Spectroscopy and Radiative Transfer Received date: 8 May 2016 Revised date: 1 August 2016 Accepted date: 1 August 2016 Cite this article as: Yuqian Wu, Yixin Zhang, Qiu Wang and Zhengda Hu, Average intensity and spreading of partially coherent model beams propagating in a turbulent biological tissue, Journal of Quantitative Spectroscopy and Radiative Transfer, http://dx.doi.org/10.1016/j.jqsrt.2016.08.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Average intensity and spreading of partially coherent model beams propagating in a turbulent biological tissue Yuqian Wua , Yixin Zhanga,b,∗, Qiu Wanga , Zhengda Hua,b b Jiangsu
a School of Science, Jiangnan University, Wuxi 214122, China Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, Wuxi 214122, China
Abstract For Gaussian beams with three different partially coherent models, including Gaussian-Schell model (GSM), LaguerreGaussian Schell-model (LGSM) and Bessel-Gaussian Schell-model (BGSM) beams propagating through a biological turbulent tissue, the expression of the spatial coherence radius of a spherical wave propagating in a turbulent biological tissue, and the average intensity and beam spreading for GSM, LGSM and BGSM beams are derived based on the fractal model of power spectrum of refractive-index variations in biological tissue. Effects of partially coherent model and parameters of biological turbulence on such beams are studied in numerical simulations. Our results reveal that the spreading of GSM beams is smaller than LGSM and BGSM beams on the same conditions, and the beam with larger source coherence width has smaller beam spreading than that with smaller coherence width. The results are useful for any applications involved light beam propagation through tissues, especially the cases where the average intensity and spreading properties of the light should be taken into account to evaluate the system performance and investigations in the structures of biological tissue. Keywords: Biological tissue, Average intensity, Beam spreading, Turbulence, Partially coherent beam,
1. Introduction With the development of imaging technology in the biological tissue including optical coherence tomography (OCT) [1–3], the propagation of optical beams through a biological tissue [4–14] is a subject of considerable importance which has received more and more attention from researchers. As an important approach to provide quantitative guidance for disease diagnosis or screening, the average intensity and spreading of signal light play key roles in biophotonics research [4]. Based on the experimental results of intensity measurement, Wu et al. [5] indicated that the unified Mie and fractal model of cell light scattering provides an accurate interpretation of the scattering spectra measured from intact cells in suspension and culture and quantitative information about cellular structural features and organization can be deduced from this unified model from wavelength- and angular-resolved light scattering spectroscopy. Biological tissue is a very complex system in ∗ Corresponding
author Email address: [email protected] (Yixin Zhang)
Preprint submitted to Journal of Quantitative Spectroscopy & Radiative Transfer
August 4, 2016
which the light is strongly scattered in propagation due to the spatial fluctuation of its refractive index. Schmitt and Kumar have measured and analyzed statistical properties of the refractive-index variations in specimens of biological tissue and they found that spatial correlations quantitatively similar to those produced by atmospheric turbulence [6]. Based on this model of power spectrum of refractive-index variations in biological tissue, Gao made lots of efforts to explore the changes of coherence and the state of polarization of optical beams propagating through a biological tissue [7–11]. The unified theory of coherence and polarization for random electromagnetic beams was employed to analyze the change of coherence due to the fluctuations of the index of refraction of tissue [12]. Xu and Alfano utilized the fractal continuous random media to model visible and near-infrared light scattering by biological tissue and cell suspensions [13]. For an improvement of the model based on a continuous refractive index variation presented in [6], Sheppard proposed a fractal model of light scattering in biological tissue and cells based on K-distribution, the range of allowable power laws was extended into the subfractal regime [14]. As an optical beam propagates through the turbulent biological tissue, it will experience random deflections due to refractive turbulence. As a result, the average intensity of beam will decrease and the beam width will increase as a result of the effect of diffraction [15]. For stochastic electromagnetic vortex beams propagation through the turbulent biological tissues, the spectral density, the spectral degree of coherence and the spectral degree of polarization were investigated in detail [16]. The analytical formulae of anomalous hollow beams (AHBs) propagating through the turbulent biological tissues based on the extended Huygens-Fresnel integral formula and the irradiance and spreading properties of AHBs in turbulent biological tissues were studied numerically in [17]. Some published computational papers [18, 19] have suggested that the intensity and spreading of beams with partially coherent sources are less sensitive to the effects of turbulence than fully coherent ones. Evidence shows that choosing the appropriate partially coherent model beam is one of the most effective methods to reduce the influence of turbulence on the laser beams propagating through the random medium [20, 21]. However, to the best of our knowledge, the intensity and spreading of partially coherent model beams propagating through a turbulent biological tissue have not been studied. In this paper, we put forward a model of average intensity for GSM, Laguerre-Gaussian Schell-model (LGSM) and Bessel-Gaussian Schell-model (BGSM) beams, spreading of LGSM and BGSM beams propagating in the biological tissue turbulence on the basis of the fractal model the of the refractive index power spectrum of fluctuations in biological turbulent tissue. The paper is organized as follows, in section 2, we derive the expressions of average intensity for GSM, LGSM and BGSM beams in turbulent biological tissue. The spreading of partially coherent model beams through turbulent biological tissue is given in section 3. The numerical simulations and analysis are given in section 4. Finally, conclusions are given in section 5.
2
Fig. 1: Notation for a partially coherent beam propagating through a turbulent biological tissue.
2. Average intensity of partially coherent beams propagating in a turbulent biological tissue The cross-spectral density function (CSDF) of GSM, LGSM and BGSM random fields at the planar source surface can be written as [20]:
2 ρ + ρ2 (1) W (0) ρ1 , ρ2 ; ω = exp − 1 2 2 µ(0) ρ2 − ρ1 ; ω , 4σ s where ρ1 and ρ2 are two-dimensional position vectors at the source plane (the notation for a partially coherent beam propagating through a turbulent biological tissue is shown in Fig. 1), σ s is the rms width of the source, ω is the angular frequency, µ(0) is the spectral degree of coherence of the source for GSM, LGSM, BGSM beams and takes the following form [20, 21]:
|ρ2 − ρ1 |2 , ρ2 − ρ1 ; ω = exp − (2) 2σ2µ |ρ2 − ρ1 |2 |ρ2 − ρ1 |2 , L (3) µ(0) ρ − ρ ; ω = exp − n 2 1 L 2σ2µ 2σ2µ |ρ2 − ρ1 |2 |ρ2 − ρ1 | (0) µB ρ2 − ρ1 ; ω = exp − (4) J0 β , 2σ2µ 2σ2µ where β and σµ are real constants, J0 (·) is the zeroth-order Bessel function of the first kind, and Ln (·) is the n-order µG(0)
Laguerre polynomial and n is the order of Laguerre polynomial. Eq. (1) corresponds to the collimated beam. The condition on the LGSM and BGSM sources for beamlike field generation is the same as that for the classic GSM source [20]: 1 1 2π2 + 2 << 2 , 2 4σ s σµ λ
(5)
with λ being the wavelength of the source. According to the extended Huygens-Fresnel principle, upon propagation from the source plane to any plane with z > 0 and by the paraxial approximation, the CSDF takes the form [15] 0
0
W(ρ1 , ρ2 , z; ω) 0 0 ZZ ZZ (ρ1 − ρ1 )2 − (ρ2 − ρ2 )2 k2 (0) = W ρ , ρ ; ω exp −ik 1 2 2z (2πz)2 D h iE 0 0 × exp Ψ∗ ρ1 , ρ1 , z; ω + Ψ ρ2 , ρ2 , z; ω d2 ρ1 d2 ρ2 a
3
(6)
0
0
where ρ1 and ρ2 are two-dimensional position vectors at the output plane, k = 2πn0 /λ is the wave number of light, 0 with n0 being the background refractive index, Ψ∗ ρ1 , ρ1 , z; ω represents the random part of the complex phase of a spherical wave propagating in biological tissue turbulence, the asterisk stands for the complex conjugate, and h·ia denotes the ensemble average of in biological tissue turbulence. The last term in above integrand can be given by the expression [15]: D h iE 0 0 exp Ψ∗ ρ1 , ρ1 , z; ω + Ψ ρ2 , ρ2 , z; ω
a
o iZ ∞ π k zh 0 0 0 0 = exp − (ρ1 − ρ2 )2 + (ρ1 − ρ2 )(ρ1 − ρ2 ) + (ρ1 − ρ2 )2 κ3 Φ(κ)dκ 3 0 h i 0 0 0 0 1 − , = exp (ρ1 − ρ2 )2 + (ρ1 − ρ2 )(ρ1 − ρ2 ) + (ρ1 − ρ2 )2 ρ2 2 2
n
(7)
0
where Φ(κ) is the power spectrum of fluctuations in the refractive index of the biological tissues, with κ << 2k being spatial frequency. κ is defined as 2π/(eddy size), ρ0 is the coherence length of a spherical wave propagating in the turbulent biological tissue and Φ(κ) is given by [14] Φ(κ) = σ
2
2 L
!n−3/2
Γ(n) in , h Γ(n − 3/2) 1 + (κL)2
(8)
where L is the outer scale of the refractive-index inhomogeneity size, and σ2 is defined as 7−D
σ2 =
2 η0 lmax f , 7 − D f π2
(9)
where 2 η0 stands for a constant, D f represents fractal dimension, with lmax being the cutoff correlation length. Note that n ≥ 1.5 is required in Eq. (8). According to Eqs. (7) and (8) and considering the value of lmax is small enough within an order of µm magnitude, thus we can use 1/lmax instead of infinity [13] ρ−2 0 =
π2 k 2 z 3
Z
1/lmax
κ3 Φ(κ)dκ.
(10)
0
Then, we use the integral relation obtained by the software Mathematica 9.0, i.e., Z 0
b
x3
h in dx = 1 + (ax)2
1−n h i 1 − 1 + a2 b2 1 + a2 b2 (n − 1) 2a4 (n2 − 3n + 2)
,
! 1 Re ,0 , ab
(11)
Thus, we can obtain the spatial coherence radius of a spherical wave propagating in the turbulent biological tissue 1−n 2 2 (n − 1) π2 k2 σ2 ηz 1 − 1 + L2 /lmax 1 + L2 /lmax (12) , ρ−2 0 = 6L4 n2 − 3n + 2 where η=
2 L
!n−3/2
4
Γ(n) , Γ (n − 3/2)
(13)
Considering a case of Kolmogorov fractal in [14], for n = 11/6 , D f = 10/3 , Eq. (12) can be reduced to: ( " !#)−1/2 −5/6 5 11/3 2 2 |ρ0 (z)| = 0.145k2 2 η0 zL−13/3 lmax 1 − 1 + L2 /lmax 1 + L2 /lmax . 6
(14)
Submitting Eqs. (14) and (7) into Eq. (6), for GSM beams we have 0
0
W(ρ1 , ρ2 , z; ω) 0 0 ZZ ZZ (ρ1 − ρ1 )2 − (ρ2 − ρ2 )2 k2 −ik exp = 2z (2πz)2 i 0 0 0 0 1 h 2 2 − × exp (ρ1 − ρ2 ) + (ρ1 − ρ2 )(ρ1 − ρ2 ) + (ρ1 − ρ2 ) ρ2 0 2 ρ + ρ2 |ρ − ρ |2 × exp − 1 2 2 − 2 2 1 d2 ρ1 d2 ρ2 , 4σ s 2σµ Then, we utilize the integral relation [22]: ! √ Z +∞ b2 π 2 exp −ax + bx dx = exp , a 4a2 −∞
Re(a2 ) > 0,
(15)
(16)
and performing the calculation, the spectral density function of a GSM beam is reduced to 0
0
W(ρ1 , ρ2 , z; ω) 0 0 0 0 (ρ22 − ρ12 ) (ρ1 + ρ2 )2 1 = exp 2 − ik A1 (z) 2A2 (z) 8σ s A1 (z) 0 1 1 + σ2s z2 , × exp (ρ1 − ρ2 )2 + − 4 2 2A1 (z) ρ0 2k2 σ2s ρ0 A1 (z)
(17)
where z2 A1 (z) = 1 + 2 2 k σs
1 2 + 2 , B1 ρ0
k2 σ2s zA1 (z) , 2 2 k2 σ2s A1 (z) + z2 ρ−2 0 − k σs −1 1 1 B1 = 2 + 2 , 4σ s σµ
A2 (z) =
0
(18) (19) (20)
0
From Eq. (17), by setting ρ1 = ρ2 = ρ , we can derive the spectral density at any point at z plane [23] " # ρ2 1 S (ρ, z; ω) = exp − 2 , A1 (z) 2σ s A1 (z)
(21)
Similarly, utilizing of the method in [21] and by square approximation [24], we can obtain the average intensity of LGSM and BGSM beams propagating through a turbulent biological tissue, 2 ! Z r 4π2 σ2s ∞ krρ 2 S L (ρ, z; ω) = 2 2 r exp(−B2 r )Ln 2 J0 dr, z λz 2σµ 0 4π2 σ2 S B (ρ, z; ω) = 2 2 s λz
Z
∞ 2
r exp(−B2 r )J0 0
5
! ! βr krρ J0 dr, σµ z
(22)
(23)
where
k2 σ2s 1 1 z 1 + + 2+ 1− 2 2 F0 8σ s 2σµ ρ0 2z2 where F0 denotes the wave-front radius of curvature of the initial beam. B2 =
!2 ,
(24)
Using the integral formula [22] Z ∞
! ! 1 1 x exp − αx2 Ln βx2 J0 (xy)dx 2 2 0 (25) ! " # 2 n y (α − β) βy2 exp − = L , y > 0, Re(α) > 0 n 2α 2α(β − α) αn+1 ! ! Z ∞ 1 2βγ β2 + γ2 (26) x exp(−αx2 )Jν (2βx)Jν (2γx)dx = Iν exp , Re(ν) > −1, 2α α α 0 where Iν is the modified Bessel functions of the first kind with ν being the order and then Eqs. (22) and (23) can be reduced to S L (ρ, z; ω) =
4π2 σ2s (2B2 − 1/σ2µ )n λ2 z2 (2B2 )n+1
S B (ρ, z; ω) =
2π2 σ2s I0 B2 λ2 z2
! " # k2 ρ2 k2 ρ2 exp − Ln , 4B2 z2 ) 4B2 σ2s z2 (1/σ2s − 2B2 ) ! β2 k2 ρ2 βkρ . exp − + 2B2 σµ 4B2 σ2µ 4B2 z2
(27) (28)
3. Average beam spreading in a turbulent biological tissue Laser beam spreading induced by the turbulence has been concerned mainly with the determination of the longterm average beam spreading, i.e., the effects of beam spreading are included in [15]. The average beam spreading (normalized rms beam width) of LGSM and BGSM beams [21] through turbulence is given by 1/2 !2 2 2 2 1 2z 1 4 nz z 2 3 wL (z) = 2σ s 1 − + 2 2 + 2 + π z T + 2 2 2 , F0 3 k 4σ s σµ k σµ 1/2 !2 √ 2 z z2 1 1 2 2 3 β2 z2 wB (z) = 2 σ s 1 − + 2 2 + 2 + π z T + 2 2 , F0 3 k 4σ s σµ 2k σµ
(29)
(30)
and the expression of T is given by [25] T=
Z
∞
κ3 Φ(κ)dκ =
0
3 π2 k2 zρ20
.
(31)
Submitting Eq. (31) into Eqs. (29) and (30), we obtain the expressions of the average beam spreading of LGSM and BGSM propagating through biological tissue turbulence 1/2 !2 2 2 2 2z 1 2 z 1 nz wL (z) = 2σ s 1 − + 2 2 + 2 + 2 + 2 2 2 , F0 k 4σ s σµ ρ0 k σµ 1/2 !2 √ 2 z z2 1 1 2 β2 z2 wB (z) = 2 σ s 1 − + 2 2 + 2 + 2 + 2 2 , F0 k 4σ s σµ ρ0 2k σµ
(32)
(33)
For n = 0 and β = 0, the average spreading for LGSM and BGSM beams can be reduced to the case for GSM beams
1/2 !2 2 2 2z z 1 1 2 + 2 2 + 2 + 2 . wG (z) = 2σ s 1 − F0 k 4σ s σµ ρ0 6
(34)
4. Numerical simulations and analysis In this section, we present some numerical simulations and discussions about the average intensity and spreading properties of partially coherent beams including GSM, LGSM, BGSM beams propagating through the turbulent biological tissue. The mouse liver tissue is taken for example and the correlated data in Ref. [13], n0 = 1.53, lmax = 2.3µm, L = 1.4µm, 2 η0 = 9.854 × 10−4 , λ = 0.83µm , σ s = 50µm, σµ = 30µm , F0 = ∞ , n = 5 and β = 5 are employed in our simulations, unless other values are indicated in the figures and illustrations. Firstly, we focus on the average intensity for GSM, LGSM and BGSM beams propagating through the turbulent biological tissue. The numerical simulations in Fig. 2 to Fig. 5 are obtained based on Eqs. (21), (27) and (28) exhibiting on Cartesian coordinates, so that x and y in Fig. 2 to Fig. 5 are related to radial coordinates as ρ = (x, y).
Fig. 2: Average intensity of GSM beams propagating through a turbulent biological tissue versus different propagation distance at z = 0 (a), z = 10µm (b) z = 30µm (c), z = 50µm (d), z = 70µm (e), z = 90µm (f), z = 110µm (g), z = 130µm (h), z = 150µm (i) plane.
To investigate the degradation of average intensity and effects of biological tissue parameters on partially coherent beams. In Figs. 2 to 3, we take GSM beam for an example, Fig. 2 plots the average intensity for GSM beams propagating through biological tissue versus different transmission distance, with |ρ| changing from 0 to 0.3mm. It is clear that from Fig. 2, the average intensity decreases and the Gaussian-like intensity distribution profile spreading outside with increasing propagation distance. Fig. 3 plots the average intensity versus different the outer scale of the fractal behavior at the same propagation distance plane. As indicated in Fig. 3, when the propagation distance 7
Fig. 3: Average intensity of GSM beams propagating through a turbulent biological tissue versus different the outer scale of the fractal behavior for L = 0.8µm (a), L = 1.2µm (b) L = 1.6µm (c), L = 2.0µm (d), L = 2.2µm (e), L = 2.4µm (f) at z = 40µm plane.
z is fixed, the average intensity of GSM beams increases as L increasing. This result indicates that larger value of L will lead to a stronger effect of refraction, the smaller value of L will lead to an effect of stronger scattering, thus the average intensity GSM beams increases as L increasing and the spreading of GSM beams decrease as the increasing outer scale of the fractal behavior. Figs. 4 and 5 display the normalized average intensity for GSM, LGSM and BGSM beams propagating through turbulent biological tissue at different propagation distance z, with |ρ| changing from 0 to 0.03mm. As indicated by Figs. 4 and 5, as the beam propagates, the central peak submerges (Gaussian-distribution like), the GSM beams keeps the Gaussian-distribution profile all the time and the beam width expands as the propagation distance increasing. For LGSM and BGSM beams, as the propagation distance increases long enough, the outer ring is lifted higher simultaneously, finally both of the two beams transform into a well-like profile and spread excessively. This result is in accordance with Fig. 3 in Ref. [26] for the beams propagating through atmospheric turbulence. It is needed to mention that comparing the propagation profile of LGSM and BGSM beams through turbulent biological tissue, LGSM beams needs longer propagation distance to transform into a well-like profile of average intensity than BGSM beams. As indicated in Fig. 6(a) and (b), the collimated LGSM and BGSM beams with different index n and β of the source have different spreading when propagating through a turbulent biological tissue. The larger the value of n and β are, the more spreading the LGSM and BGSM beams are. The average spreading of GSM beams is smaller than LGSM and BGSM beams on the same conditions, this phenomenon can also be seen from Fig. 4. This result is in accordance with [21] in atmospheric turbulence case. Fig. 7(a) and (b) plot the beam width of the LGSM and BGSM 8
Fig. 4: Normalized average intensity of GSM, LGSM and BGSM beams propagating through turbulent biological tissue at z = 30µm in (a), (b) and (c), z = 120µm in (d), (e) and (f), z = 230µm in (g), (h) and (i) plane.
9
Fig. 5: 3D view for Fig. 4 (Normalized average intensity of GSM, LGSM and BGSM beams propagating through turbulent biological tissue at z = 30µm in (a), (b) and (c), z = 120µm in (d), (e) and (f), z = 230µm in (g), (h) and (i) plane.)
Fig. 6: Spreading of LGSM (a) and BGSM (b) beams versus the different index n and β of the source propagating through a turbulent biological tissue.
10
Fig. 7: Spreading of LGSM (a) and BGSM (b) beams versus the different coherence parameter σµ of the source propagating through a turbulent biological tissue.
beam versus the propagation distance z and correlation width of the source σµ . As shown in Fig. 7(a) and (b), the LGSM and BGSM beams with different σµ have obviously different spreading when propagating through a turbulent biological tissue. When the transmission distance z is fixed, the larger the σµ , the smaller the beam spreading will be. From Fig. 7(a) with Fig. 7(b), it is clear that beams with larger source coherence width has smaller beam spreading than the beams with smaller σµ .
5. Conclusions In summary, we have developed a model for partially coherent (GSM, LGSM, and BGSM) beams, propagating through a turbulent biological tissue. The expression of the spatial coherence radius of a spherical wave propagating in the turbulent biological tissue based on the fractal model is developed. Formulations of average intensity and beam spreading for these beams are derived based on the extended Huygens-Fresnel principle and shown in numerical simulations. Effects of partially coherent model and outer scale of the fractal behavior in biological tissue turbulence on such beams are studied. Numerical simulations indicate that the average intensity of partially coherent beams decreases with the increasing propagation distance and increases with the increasing outer scale of tissue turbulence when the propagation distance is fixed. Comparing the properties between LGSM and BGSM beams propagating in the turbulent biological tissue, LGSM beams needs longer propagation distance to transform into a well-like profile of average intensity than BGSM beams. The average spreading of GSM beams is smaller than LGSM and BGSM beams on the same conditions. In addition, beams with larger source coherence width have smaller beam spreading than that with smaller coherence width σµ . These findings are useful for the study of quantitative information about cellular structural features and organization which are deduced from the measure of scattering intensity [4, 5]. 11
Acknowledgments This work is supported by the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174) and the Fundamental Research Funds for the Central Universities (JUSRP51517).
References [1] D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, Optical coherence tomography, Science 254 (1991) 1178–1181. [2] A. F. Ferchera, C. Hitzenbergera, G. Kampa, S. Y. El-Zaiat, Measurement of intraocular distances by backscattering spectral interferometry, Opt. Commun. 117 (1995) 43–48. [3] G. Husler, M. W. Lindner, Coherence radar and spectral radarnew tools for dermatological diagnosis, J. Biomed. Opt. 3 (1998) 21–31. [4] J. Yi, V. Backman, Imaging a full set of optical scattering properties of biological tissue by inverse spectroscopic optical coherence tomography, Opt. Lett. 37 (2014) 4443–4445. [5] T. T. Wu, J. Y. Qu, M. Xu, Unified mie and fractal scattering by biological cells and subcellular structures, Opt. Lett. 32 (2007) 2324–2326. [6] J. M. Schmitt, G. Kumar, Turbulent nature of refractive-index variations in biological tissue, Opt. Lett. 21 (1996) 1310–1312. [7] W. Gao, Spectral changes of the light produced by scattering from tissue, Opt. Lett. 35 (2010) 862–864. [8] W. Gao, Changes of polarization of light beams on propagation through tissue, Opt. Commun. 260 (2006) 749–754. [9] W. Gao, O. Korotkova, Changes in the state of polarization of a random electromagnetic beam propagating through tissue, Opt. Commun. 7 (2007) 474–478. [10] W. Gao, Effects of different correlations of partially coherent electromagnetic beam on three-dimensional spectral intensity distribution in the focal region, Opt. Commun. 283 (2010) 4572–4581. [11] W. Gao, Effects of the coherence and the vector properties of the light on the resolution limit in the stimulated emission depletion fluorescent microscopy, J. Opt. Soc. Am. A 25 (2008) 1378–1382. [12] W. Gao, Change of coherence of light produced by tissue turbulence, Journal of Quantitative Spectroscopy & Radiative Transfer 131 (2013) 52–58. [13] M. Xu, R. R. Alfano, Fractal mechanisms of light scattering in biological tissue and cells, Opt. Lett. 30 (2005) 3051–3053. [14] C. J. R. Sheppard, Fractal model of light scattering in biological tissue and cells, Opt. Lett. 32 (2007) 142–144. [15] L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media, SPIE, Bellingham, 2005. [16] M. Luo, Q. Chen, L. Hua, D. Zhao, Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues, Phys. Lett. A 378 (2014) 308–314. [17] X. Lu, X. Zhu, K. Wang, C. Zhao, Y. Cai, Effects of biological tissues on the propagation properties of anomalous hollow beams, Optik 127 (2016) 1842–1842. [18] J. Wu, Propagation of a gaussian-schell beam through turbulent media, J. Mod. Opt. 37 (1990) 671–684. [19] G. Gbur, E. Wolf, Spreading of partially coherent beams in random media, J. Opt. Soc. Am. A 19 (2002) 1592–1598. [20] Z. Mei, O. Korotkova, Random sources generating ring-shaped beams, Opt. Lett. 38 (2013) 91–93. [21] J. Cang, P. Xiu, X. Liu, Propagation of laguerre-gaussian and bessel-gaussian schell-model beams through paraxial optical systems in turbulent atmosphere, Opt. & Laser Technol. 54 (2013) 35–41. [22] O. Korotkova, S. Sahin, E. Shchepakina, Multi-gaussian schell-model beams, J. Opt. Soc. Am. A 29 (2012) 2159–2164. [23] V. L. Minorov, V. V. Nosov, On the theory of spatially limited light beam displacement in a randomly inhomogeneous medium, J. Opt. Soc. Am. 67 (1977) 1073–1080. [24] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals-Series-and Products, Academic Press, Burlington, 2007. [25] Y. Wu, Y. Zhang, Y. Li, Z. Hu, Beam wander of gaussian-schell model beams propagating through oceanic turbulence, Opt. Commun. 371 (2016) 59–66.
12
[26] H. T. Eyyuboglu, Propagation of higher order bessel-gaussian beams in turbulence, Appl. Phys. B 88 (2007) 259–265.
13
PII: DOI: Reference:
S0022-4073(16)30269-2 http://dx.doi.org/10.1016/j.jqsrt.2016.08.001 JQSRT5416
To appear in: Journal of Quantitative Spectroscopy and Radiative Transfer Received date: 8 May 2016 Revised date: 1 August 2016 Accepted date: 1 August 2016 Cite this article as: Yuqian Wu, Yixin Zhang, Qiu Wang and Zhengda Hu, Average intensity and spreading of partially coherent model beams propagating in a turbulent biological tissue, Journal of Quantitative Spectroscopy and Radiative Transfer, http://dx.doi.org/10.1016/j.jqsrt.2016.08.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Average intensity and spreading of partially coherent model beams propagating in a turbulent biological tissue Yuqian Wua , Yixin Zhanga,b,∗, Qiu Wanga , Zhengda Hua,b b Jiangsu
a School of Science, Jiangnan University, Wuxi 214122, China Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, Wuxi 214122, China
Abstract For Gaussian beams with three different partially coherent models, including Gaussian-Schell model (GSM), LaguerreGaussian Schell-model (LGSM) and Bessel-Gaussian Schell-model (BGSM) beams propagating through a biological turbulent tissue, the expression of the spatial coherence radius of a spherical wave propagating in a turbulent biological tissue, and the average intensity and beam spreading for GSM, LGSM and BGSM beams are derived based on the fractal model of power spectrum of refractive-index variations in biological tissue. Effects of partially coherent model and parameters of biological turbulence on such beams are studied in numerical simulations. Our results reveal that the spreading of GSM beams is smaller than LGSM and BGSM beams on the same conditions, and the beam with larger source coherence width has smaller beam spreading than that with smaller coherence width. The results are useful for any applications involved light beam propagation through tissues, especially the cases where the average intensity and spreading properties of the light should be taken into account to evaluate the system performance and investigations in the structures of biological tissue. Keywords: Biological tissue, Average intensity, Beam spreading, Turbulence, Partially coherent beam,
1. Introduction With the development of imaging technology in the biological tissue including optical coherence tomography (OCT) [1–3], the propagation of optical beams through a biological tissue [4–14] is a subject of considerable importance which has received more and more attention from researchers. As an important approach to provide quantitative guidance for disease diagnosis or screening, the average intensity and spreading of signal light play key roles in biophotonics research [4]. Based on the experimental results of intensity measurement, Wu et al. [5] indicated that the unified Mie and fractal model of cell light scattering provides an accurate interpretation of the scattering spectra measured from intact cells in suspension and culture and quantitative information about cellular structural features and organization can be deduced from this unified model from wavelength- and angular-resolved light scattering spectroscopy. Biological tissue is a very complex system in ∗ Corresponding
author Email address: [email protected] (Yixin Zhang)
Preprint submitted to Journal of Quantitative Spectroscopy & Radiative Transfer
August 4, 2016
which the light is strongly scattered in propagation due to the spatial fluctuation of its refractive index. Schmitt and Kumar have measured and analyzed statistical properties of the refractive-index variations in specimens of biological tissue and they found that spatial correlations quantitatively similar to those produced by atmospheric turbulence [6]. Based on this model of power spectrum of refractive-index variations in biological tissue, Gao made lots of efforts to explore the changes of coherence and the state of polarization of optical beams propagating through a biological tissue [7–11]. The unified theory of coherence and polarization for random electromagnetic beams was employed to analyze the change of coherence due to the fluctuations of the index of refraction of tissue [12]. Xu and Alfano utilized the fractal continuous random media to model visible and near-infrared light scattering by biological tissue and cell suspensions [13]. For an improvement of the model based on a continuous refractive index variation presented in [6], Sheppard proposed a fractal model of light scattering in biological tissue and cells based on K-distribution, the range of allowable power laws was extended into the subfractal regime [14]. As an optical beam propagates through the turbulent biological tissue, it will experience random deflections due to refractive turbulence. As a result, the average intensity of beam will decrease and the beam width will increase as a result of the effect of diffraction [15]. For stochastic electromagnetic vortex beams propagation through the turbulent biological tissues, the spectral density, the spectral degree of coherence and the spectral degree of polarization were investigated in detail [16]. The analytical formulae of anomalous hollow beams (AHBs) propagating through the turbulent biological tissues based on the extended Huygens-Fresnel integral formula and the irradiance and spreading properties of AHBs in turbulent biological tissues were studied numerically in [17]. Some published computational papers [18, 19] have suggested that the intensity and spreading of beams with partially coherent sources are less sensitive to the effects of turbulence than fully coherent ones. Evidence shows that choosing the appropriate partially coherent model beam is one of the most effective methods to reduce the influence of turbulence on the laser beams propagating through the random medium [20, 21]. However, to the best of our knowledge, the intensity and spreading of partially coherent model beams propagating through a turbulent biological tissue have not been studied. In this paper, we put forward a model of average intensity for GSM, Laguerre-Gaussian Schell-model (LGSM) and Bessel-Gaussian Schell-model (BGSM) beams, spreading of LGSM and BGSM beams propagating in the biological tissue turbulence on the basis of the fractal model the of the refractive index power spectrum of fluctuations in biological turbulent tissue. The paper is organized as follows, in section 2, we derive the expressions of average intensity for GSM, LGSM and BGSM beams in turbulent biological tissue. The spreading of partially coherent model beams through turbulent biological tissue is given in section 3. The numerical simulations and analysis are given in section 4. Finally, conclusions are given in section 5.
2
Fig. 1: Notation for a partially coherent beam propagating through a turbulent biological tissue.
2. Average intensity of partially coherent beams propagating in a turbulent biological tissue The cross-spectral density function (CSDF) of GSM, LGSM and BGSM random fields at the planar source surface can be written as [20]:
2 ρ + ρ2 (1) W (0) ρ1 , ρ2 ; ω = exp − 1 2 2 µ(0) ρ2 − ρ1 ; ω , 4σ s where ρ1 and ρ2 are two-dimensional position vectors at the source plane (the notation for a partially coherent beam propagating through a turbulent biological tissue is shown in Fig. 1), σ s is the rms width of the source, ω is the angular frequency, µ(0) is the spectral degree of coherence of the source for GSM, LGSM, BGSM beams and takes the following form [20, 21]:
|ρ2 − ρ1 |2 , ρ2 − ρ1 ; ω = exp − (2) 2σ2µ |ρ2 − ρ1 |2 |ρ2 − ρ1 |2 , L (3) µ(0) ρ − ρ ; ω = exp − n 2 1 L 2σ2µ 2σ2µ |ρ2 − ρ1 |2 |ρ2 − ρ1 | (0) µB ρ2 − ρ1 ; ω = exp − (4) J0 β , 2σ2µ 2σ2µ where β and σµ are real constants, J0 (·) is the zeroth-order Bessel function of the first kind, and Ln (·) is the n-order µG(0)
Laguerre polynomial and n is the order of Laguerre polynomial. Eq. (1) corresponds to the collimated beam. The condition on the LGSM and BGSM sources for beamlike field generation is the same as that for the classic GSM source [20]: 1 1 2π2 + 2 << 2 , 2 4σ s σµ λ
(5)
with λ being the wavelength of the source. According to the extended Huygens-Fresnel principle, upon propagation from the source plane to any plane with z > 0 and by the paraxial approximation, the CSDF takes the form [15] 0
0
W(ρ1 , ρ2 , z; ω) 0 0 ZZ ZZ (ρ1 − ρ1 )2 − (ρ2 − ρ2 )2 k2 (0) = W ρ , ρ ; ω exp −ik 1 2 2z (2πz)2 D h iE 0 0 × exp Ψ∗ ρ1 , ρ1 , z; ω + Ψ ρ2 , ρ2 , z; ω d2 ρ1 d2 ρ2 a
3
(6)
0
0
where ρ1 and ρ2 are two-dimensional position vectors at the output plane, k = 2πn0 /λ is the wave number of light, 0 with n0 being the background refractive index, Ψ∗ ρ1 , ρ1 , z; ω represents the random part of the complex phase of a spherical wave propagating in biological tissue turbulence, the asterisk stands for the complex conjugate, and h·ia denotes the ensemble average of in biological tissue turbulence. The last term in above integrand can be given by the expression [15]: D h iE 0 0 exp Ψ∗ ρ1 , ρ1 , z; ω + Ψ ρ2 , ρ2 , z; ω
a
o iZ ∞ π k zh 0 0 0 0 = exp − (ρ1 − ρ2 )2 + (ρ1 − ρ2 )(ρ1 − ρ2 ) + (ρ1 − ρ2 )2 κ3 Φ(κ)dκ 3 0 h i 0 0 0 0 1 − , = exp (ρ1 − ρ2 )2 + (ρ1 − ρ2 )(ρ1 − ρ2 ) + (ρ1 − ρ2 )2 ρ2 2 2
n
(7)
0
where Φ(κ) is the power spectrum of fluctuations in the refractive index of the biological tissues, with κ << 2k being spatial frequency. κ is defined as 2π/(eddy size), ρ0 is the coherence length of a spherical wave propagating in the turbulent biological tissue and Φ(κ) is given by [14] Φ(κ) = σ
2
2 L
!n−3/2
Γ(n) in , h Γ(n − 3/2) 1 + (κL)2
(8)
where L is the outer scale of the refractive-index inhomogeneity size, and σ2 is defined as 7−D
σ2 =
2 η0 lmax f , 7 − D f π2
(9)
where 2 η0 stands for a constant, D f represents fractal dimension, with lmax being the cutoff correlation length. Note that n ≥ 1.5 is required in Eq. (8). According to Eqs. (7) and (8) and considering the value of lmax is small enough within an order of µm magnitude, thus we can use 1/lmax instead of infinity [13] ρ−2 0 =
π2 k 2 z 3
Z
1/lmax
κ3 Φ(κ)dκ.
(10)
0
Then, we use the integral relation obtained by the software Mathematica 9.0, i.e., Z 0
b
x3
h in dx = 1 + (ax)2
1−n h i 1 − 1 + a2 b2 1 + a2 b2 (n − 1) 2a4 (n2 − 3n + 2)
,
! 1 Re ,0 , ab
(11)
Thus, we can obtain the spatial coherence radius of a spherical wave propagating in the turbulent biological tissue 1−n 2 2 (n − 1) π2 k2 σ2 ηz 1 − 1 + L2 /lmax 1 + L2 /lmax (12) , ρ−2 0 = 6L4 n2 − 3n + 2 where η=
2 L
!n−3/2
4
Γ(n) , Γ (n − 3/2)
(13)
Considering a case of Kolmogorov fractal in [14], for n = 11/6 , D f = 10/3 , Eq. (12) can be reduced to: ( " !#)−1/2 −5/6 5 11/3 2 2 |ρ0 (z)| = 0.145k2 2 η0 zL−13/3 lmax 1 − 1 + L2 /lmax 1 + L2 /lmax . 6
(14)
Submitting Eqs. (14) and (7) into Eq. (6), for GSM beams we have 0
0
W(ρ1 , ρ2 , z; ω) 0 0 ZZ ZZ (ρ1 − ρ1 )2 − (ρ2 − ρ2 )2 k2 −ik exp = 2z (2πz)2 i 0 0 0 0 1 h 2 2 − × exp (ρ1 − ρ2 ) + (ρ1 − ρ2 )(ρ1 − ρ2 ) + (ρ1 − ρ2 ) ρ2 0 2 ρ + ρ2 |ρ − ρ |2 × exp − 1 2 2 − 2 2 1 d2 ρ1 d2 ρ2 , 4σ s 2σµ Then, we utilize the integral relation [22]: ! √ Z +∞ b2 π 2 exp −ax + bx dx = exp , a 4a2 −∞
Re(a2 ) > 0,
(15)
(16)
and performing the calculation, the spectral density function of a GSM beam is reduced to 0
0
W(ρ1 , ρ2 , z; ω) 0 0 0 0 (ρ22 − ρ12 ) (ρ1 + ρ2 )2 1 = exp 2 − ik A1 (z) 2A2 (z) 8σ s A1 (z) 0 1 1 + σ2s z2 , × exp (ρ1 − ρ2 )2 + − 4 2 2A1 (z) ρ0 2k2 σ2s ρ0 A1 (z)
(17)
where z2 A1 (z) = 1 + 2 2 k σs
1 2 + 2 , B1 ρ0
k2 σ2s zA1 (z) , 2 2 k2 σ2s A1 (z) + z2 ρ−2 0 − k σs −1 1 1 B1 = 2 + 2 , 4σ s σµ
A2 (z) =
0
(18) (19) (20)
0
From Eq. (17), by setting ρ1 = ρ2 = ρ , we can derive the spectral density at any point at z plane [23] " # ρ2 1 S (ρ, z; ω) = exp − 2 , A1 (z) 2σ s A1 (z)
(21)
Similarly, utilizing of the method in [21] and by square approximation [24], we can obtain the average intensity of LGSM and BGSM beams propagating through a turbulent biological tissue, 2 ! Z r 4π2 σ2s ∞ krρ 2 S L (ρ, z; ω) = 2 2 r exp(−B2 r )Ln 2 J0 dr, z λz 2σµ 0 4π2 σ2 S B (ρ, z; ω) = 2 2 s λz
Z
∞ 2
r exp(−B2 r )J0 0
5
! ! βr krρ J0 dr, σµ z
(22)
(23)
where
k2 σ2s 1 1 z 1 + + 2+ 1− 2 2 F0 8σ s 2σµ ρ0 2z2 where F0 denotes the wave-front radius of curvature of the initial beam. B2 =
!2 ,
(24)
Using the integral formula [22] Z ∞
! ! 1 1 x exp − αx2 Ln βx2 J0 (xy)dx 2 2 0 (25) ! " # 2 n y (α − β) βy2 exp − = L , y > 0, Re(α) > 0 n 2α 2α(β − α) αn+1 ! ! Z ∞ 1 2βγ β2 + γ2 (26) x exp(−αx2 )Jν (2βx)Jν (2γx)dx = Iν exp , Re(ν) > −1, 2α α α 0 where Iν is the modified Bessel functions of the first kind with ν being the order and then Eqs. (22) and (23) can be reduced to S L (ρ, z; ω) =
4π2 σ2s (2B2 − 1/σ2µ )n λ2 z2 (2B2 )n+1
S B (ρ, z; ω) =
2π2 σ2s I0 B2 λ2 z2
! " # k2 ρ2 k2 ρ2 exp − Ln , 4B2 z2 ) 4B2 σ2s z2 (1/σ2s − 2B2 ) ! β2 k2 ρ2 βkρ . exp − + 2B2 σµ 4B2 σ2µ 4B2 z2
(27) (28)
3. Average beam spreading in a turbulent biological tissue Laser beam spreading induced by the turbulence has been concerned mainly with the determination of the longterm average beam spreading, i.e., the effects of beam spreading are included in [15]. The average beam spreading (normalized rms beam width) of LGSM and BGSM beams [21] through turbulence is given by 1/2 !2 2 2 2 1 2z 1 4 nz z 2 3 wL (z) = 2σ s 1 − + 2 2 + 2 + π z T + 2 2 2 , F0 3 k 4σ s σµ k σµ 1/2 !2 √ 2 z z2 1 1 2 2 3 β2 z2 wB (z) = 2 σ s 1 − + 2 2 + 2 + π z T + 2 2 , F0 3 k 4σ s σµ 2k σµ
(29)
(30)
and the expression of T is given by [25] T=
Z
∞
κ3 Φ(κ)dκ =
0
3 π2 k2 zρ20
.
(31)
Submitting Eq. (31) into Eqs. (29) and (30), we obtain the expressions of the average beam spreading of LGSM and BGSM propagating through biological tissue turbulence 1/2 !2 2 2 2 2z 1 2 z 1 nz wL (z) = 2σ s 1 − + 2 2 + 2 + 2 + 2 2 2 , F0 k 4σ s σµ ρ0 k σµ 1/2 !2 √ 2 z z2 1 1 2 β2 z2 wB (z) = 2 σ s 1 − + 2 2 + 2 + 2 + 2 2 , F0 k 4σ s σµ ρ0 2k σµ
(32)
(33)
For n = 0 and β = 0, the average spreading for LGSM and BGSM beams can be reduced to the case for GSM beams
1/2 !2 2 2 2z z 1 1 2 + 2 2 + 2 + 2 . wG (z) = 2σ s 1 − F0 k 4σ s σµ ρ0 6
(34)
4. Numerical simulations and analysis In this section, we present some numerical simulations and discussions about the average intensity and spreading properties of partially coherent beams including GSM, LGSM, BGSM beams propagating through the turbulent biological tissue. The mouse liver tissue is taken for example and the correlated data in Ref. [13], n0 = 1.53, lmax = 2.3µm, L = 1.4µm, 2 η0 = 9.854 × 10−4 , λ = 0.83µm , σ s = 50µm, σµ = 30µm , F0 = ∞ , n = 5 and β = 5 are employed in our simulations, unless other values are indicated in the figures and illustrations. Firstly, we focus on the average intensity for GSM, LGSM and BGSM beams propagating through the turbulent biological tissue. The numerical simulations in Fig. 2 to Fig. 5 are obtained based on Eqs. (21), (27) and (28) exhibiting on Cartesian coordinates, so that x and y in Fig. 2 to Fig. 5 are related to radial coordinates as ρ = (x, y).
Fig. 2: Average intensity of GSM beams propagating through a turbulent biological tissue versus different propagation distance at z = 0 (a), z = 10µm (b) z = 30µm (c), z = 50µm (d), z = 70µm (e), z = 90µm (f), z = 110µm (g), z = 130µm (h), z = 150µm (i) plane.
To investigate the degradation of average intensity and effects of biological tissue parameters on partially coherent beams. In Figs. 2 to 3, we take GSM beam for an example, Fig. 2 plots the average intensity for GSM beams propagating through biological tissue versus different transmission distance, with |ρ| changing from 0 to 0.3mm. It is clear that from Fig. 2, the average intensity decreases and the Gaussian-like intensity distribution profile spreading outside with increasing propagation distance. Fig. 3 plots the average intensity versus different the outer scale of the fractal behavior at the same propagation distance plane. As indicated in Fig. 3, when the propagation distance 7
Fig. 3: Average intensity of GSM beams propagating through a turbulent biological tissue versus different the outer scale of the fractal behavior for L = 0.8µm (a), L = 1.2µm (b) L = 1.6µm (c), L = 2.0µm (d), L = 2.2µm (e), L = 2.4µm (f) at z = 40µm plane.
z is fixed, the average intensity of GSM beams increases as L increasing. This result indicates that larger value of L will lead to a stronger effect of refraction, the smaller value of L will lead to an effect of stronger scattering, thus the average intensity GSM beams increases as L increasing and the spreading of GSM beams decrease as the increasing outer scale of the fractal behavior. Figs. 4 and 5 display the normalized average intensity for GSM, LGSM and BGSM beams propagating through turbulent biological tissue at different propagation distance z, with |ρ| changing from 0 to 0.03mm. As indicated by Figs. 4 and 5, as the beam propagates, the central peak submerges (Gaussian-distribution like), the GSM beams keeps the Gaussian-distribution profile all the time and the beam width expands as the propagation distance increasing. For LGSM and BGSM beams, as the propagation distance increases long enough, the outer ring is lifted higher simultaneously, finally both of the two beams transform into a well-like profile and spread excessively. This result is in accordance with Fig. 3 in Ref. [26] for the beams propagating through atmospheric turbulence. It is needed to mention that comparing the propagation profile of LGSM and BGSM beams through turbulent biological tissue, LGSM beams needs longer propagation distance to transform into a well-like profile of average intensity than BGSM beams. As indicated in Fig. 6(a) and (b), the collimated LGSM and BGSM beams with different index n and β of the source have different spreading when propagating through a turbulent biological tissue. The larger the value of n and β are, the more spreading the LGSM and BGSM beams are. The average spreading of GSM beams is smaller than LGSM and BGSM beams on the same conditions, this phenomenon can also be seen from Fig. 4. This result is in accordance with [21] in atmospheric turbulence case. Fig. 7(a) and (b) plot the beam width of the LGSM and BGSM 8
Fig. 4: Normalized average intensity of GSM, LGSM and BGSM beams propagating through turbulent biological tissue at z = 30µm in (a), (b) and (c), z = 120µm in (d), (e) and (f), z = 230µm in (g), (h) and (i) plane.
9
Fig. 5: 3D view for Fig. 4 (Normalized average intensity of GSM, LGSM and BGSM beams propagating through turbulent biological tissue at z = 30µm in (a), (b) and (c), z = 120µm in (d), (e) and (f), z = 230µm in (g), (h) and (i) plane.)
Fig. 6: Spreading of LGSM (a) and BGSM (b) beams versus the different index n and β of the source propagating through a turbulent biological tissue.
10
Fig. 7: Spreading of LGSM (a) and BGSM (b) beams versus the different coherence parameter σµ of the source propagating through a turbulent biological tissue.
beam versus the propagation distance z and correlation width of the source σµ . As shown in Fig. 7(a) and (b), the LGSM and BGSM beams with different σµ have obviously different spreading when propagating through a turbulent biological tissue. When the transmission distance z is fixed, the larger the σµ , the smaller the beam spreading will be. From Fig. 7(a) with Fig. 7(b), it is clear that beams with larger source coherence width has smaller beam spreading than the beams with smaller σµ .
5. Conclusions In summary, we have developed a model for partially coherent (GSM, LGSM, and BGSM) beams, propagating through a turbulent biological tissue. The expression of the spatial coherence radius of a spherical wave propagating in the turbulent biological tissue based on the fractal model is developed. Formulations of average intensity and beam spreading for these beams are derived based on the extended Huygens-Fresnel principle and shown in numerical simulations. Effects of partially coherent model and outer scale of the fractal behavior in biological tissue turbulence on such beams are studied. Numerical simulations indicate that the average intensity of partially coherent beams decreases with the increasing propagation distance and increases with the increasing outer scale of tissue turbulence when the propagation distance is fixed. Comparing the properties between LGSM and BGSM beams propagating in the turbulent biological tissue, LGSM beams needs longer propagation distance to transform into a well-like profile of average intensity than BGSM beams. The average spreading of GSM beams is smaller than LGSM and BGSM beams on the same conditions. In addition, beams with larger source coherence width have smaller beam spreading than that with smaller coherence width σµ . These findings are useful for the study of quantitative information about cellular structural features and organization which are deduced from the measure of scattering intensity [4, 5]. 11
Acknowledgments This work is supported by the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174) and the Fundamental Research Funds for the Central Universities (JUSRP51517).
References [1] D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, Optical coherence tomography, Science 254 (1991) 1178–1181. [2] A. F. Ferchera, C. Hitzenbergera, G. Kampa, S. Y. El-Zaiat, Measurement of intraocular distances by backscattering spectral interferometry, Opt. Commun. 117 (1995) 43–48. [3] G. Husler, M. W. Lindner, Coherence radar and spectral radarnew tools for dermatological diagnosis, J. Biomed. Opt. 3 (1998) 21–31. [4] J. Yi, V. Backman, Imaging a full set of optical scattering properties of biological tissue by inverse spectroscopic optical coherence tomography, Opt. Lett. 37 (2014) 4443–4445. [5] T. T. Wu, J. Y. Qu, M. Xu, Unified mie and fractal scattering by biological cells and subcellular structures, Opt. Lett. 32 (2007) 2324–2326. [6] J. M. Schmitt, G. Kumar, Turbulent nature of refractive-index variations in biological tissue, Opt. Lett. 21 (1996) 1310–1312. [7] W. Gao, Spectral changes of the light produced by scattering from tissue, Opt. Lett. 35 (2010) 862–864. [8] W. Gao, Changes of polarization of light beams on propagation through tissue, Opt. Commun. 260 (2006) 749–754. [9] W. Gao, O. Korotkova, Changes in the state of polarization of a random electromagnetic beam propagating through tissue, Opt. Commun. 7 (2007) 474–478. [10] W. Gao, Effects of different correlations of partially coherent electromagnetic beam on three-dimensional spectral intensity distribution in the focal region, Opt. Commun. 283 (2010) 4572–4581. [11] W. Gao, Effects of the coherence and the vector properties of the light on the resolution limit in the stimulated emission depletion fluorescent microscopy, J. Opt. Soc. Am. A 25 (2008) 1378–1382. [12] W. Gao, Change of coherence of light produced by tissue turbulence, Journal of Quantitative Spectroscopy & Radiative Transfer 131 (2013) 52–58. [13] M. Xu, R. R. Alfano, Fractal mechanisms of light scattering in biological tissue and cells, Opt. Lett. 30 (2005) 3051–3053. [14] C. J. R. Sheppard, Fractal model of light scattering in biological tissue and cells, Opt. Lett. 32 (2007) 142–144. [15] L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media, SPIE, Bellingham, 2005. [16] M. Luo, Q. Chen, L. Hua, D. Zhao, Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues, Phys. Lett. A 378 (2014) 308–314. [17] X. Lu, X. Zhu, K. Wang, C. Zhao, Y. Cai, Effects of biological tissues on the propagation properties of anomalous hollow beams, Optik 127 (2016) 1842–1842. [18] J. Wu, Propagation of a gaussian-schell beam through turbulent media, J. Mod. Opt. 37 (1990) 671–684. [19] G. Gbur, E. Wolf, Spreading of partially coherent beams in random media, J. Opt. Soc. Am. A 19 (2002) 1592–1598. [20] Z. Mei, O. Korotkova, Random sources generating ring-shaped beams, Opt. Lett. 38 (2013) 91–93. [21] J. Cang, P. Xiu, X. Liu, Propagation of laguerre-gaussian and bessel-gaussian schell-model beams through paraxial optical systems in turbulent atmosphere, Opt. & Laser Technol. 54 (2013) 35–41. [22] O. Korotkova, S. Sahin, E. Shchepakina, Multi-gaussian schell-model beams, J. Opt. Soc. Am. A 29 (2012) 2159–2164. [23] V. L. Minorov, V. V. Nosov, On the theory of spatially limited light beam displacement in a randomly inhomogeneous medium, J. Opt. Soc. Am. 67 (1977) 1073–1080. [24] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals-Series-and Products, Academic Press, Burlington, 2007. [25] Y. Wu, Y. Zhang, Y. Li, Z. Hu, Beam wander of gaussian-schell model beams propagating through oceanic turbulence, Opt. Commun. 371 (2016) 59–66.
12
[26] H. T. Eyyuboglu, Propagation of higher order bessel-gaussian beams in turbulence, Appl. Phys. B 88 (2007) 259–265.
13