Effects of biological tissues on the propagation properties of anomalous hollow beams

Optik 127 (2016) 1842–1847

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Effects of biological tissues on the propagation properties of anomalous hollow beams Xingyuan Lu a , Xinlei Zhu a , Kuilong Wang b , Chengliang Zhao a,∗ , Yangjian Cai a a College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China b Department of Physics, Hangzhou Normal University, Hangzhou 310036, China

a r t i c l e

i n f o

Article history: Received 8 April 2015 Accepted 9 November 2015 Keywords: Anomalous hollow beam Biological tissues Statistical properties

a b s t r a c t We derived analytical formulae of anomalous hollow beams (AHBs) passing through the turbulent biological tissues based on the extended Huygens–Fresnel integral formula. With the help of these formulae, we investigate the propagation properties of AHBs in turbulent biological tissues, the irradiance and spreading properties of AHBs in turbulent biological tissues are studied numerically. It is found that the circular and elliptical AHBs eventually become Gaussian beams in the far field and the central irradiance of the AHB rises more rapidly as the value of Cn2 grows. We also calculate the formulae of the effective beam size of AHB and find that finally Wxz becomes equal to Wyz in turbulent biological tissues which can be used to explain the beam spot eventually becomes circular under the influence of turbulence of biological tissues. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction Dark hollow beams with zero central intensity have attracted much attentions due to its wide applications in atomic optics, free space optical communications, binary optics, optical trapping of particles and medical sciences [1–3]. There are many kinds of hollow beams, such as Hermite–Gauss beams, Laguerre–Gauss beams, Hollow Gaussian beams, Bessel–Gauss beams, Bessel-like beams, and so on, while one of them is very different from others which is named as anomalous hollow beam (AHB). The main difference between conventional dark hollow beam and AHB is that there is an elliptical solid core at the beam center of AHB. In 2005, Wu et al. first observed an anomalous hollow electron beam in an experiment [4], which can be used for studying the transverse instability and provide a powerful tool for studying the linear and nonlinear particle dynamics in the storage ring. In 2007, Cai proposed a theoretical model to describe an AHB [5]. After that, the propagation properties of AHB in free space [6] and uniaxial crystals [7] are studied. Although there are a lot of researches about AHBs [8–12], the exploration is still far from complete. Recently, with the development of molecular biology and medicine, there is an urgent need to understand the living

∗ Corresponding author. Tel.: +86 512 69153532. E-mail address: [email protected] (C. Zhao). http://dx.doi.org/10.1016/j.ijleo.2015.11.039 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

tissues, such as the changes during the process of development of embryos and the physiological and pathological changing processes of animal tissues. Due to the light of the abroad important applications, great efforts are addicted to the beams propagating through tissues [13–16]. In 1996, Schmitt and Kumar found that the structure function of refractive-index inhomogeneities in mammalian tissues fits the classical Kolmogorov model of turbulence [17]. In this paper, we study the propagation of an AHB passing through turbulent biological tissues. We derive analytical formulae of AHB in turbulent biological tissues. Some numerical examples are given to illustrate the propagation properties of an AHB in biological tissues.

2. Theory The electric field of an AHB of elliptical symmetry at z = 0 can be expressed as superposition of astigmatic Gaussian modes and astigmatic doughnut modes as follows [5]:

 E(x, y, 0) =

−2 +

8x2 2 w0x

+

8y2 2 w0y



 exp

−

x2 2 w0x

−

y2 2 w0y

 (1)

where w0x and w0y are the beam waist widths of an astigmatic Gaussian mode in x and y directions, respectively. When w0x = w0y , Eq. (1) reduces to a circular AHB.

X. Lu et al. / Optik 127 (2016) 1842–1847

Based on the extended Huygens–Fresnel integral formula, the propagation of a laser beam in the turbulent biological tissues can be described as [18–21]. k2  (1 , 2 , z) = 42 z 2



∞



−∞

 ik

× exp −

2z

∞



−∞



∞

−∞

∞

E(r 1 , 0) E ∗ (r 2 , 0)

−∞

(r 1 − 1 )2 + ∗

× exp[ (r 1 , 1 ) +

ik (r 2 − 2 )2 2z

 5 =

6 =



(r 2 , 2 )]dr 1 dr 2

(2)

1843

7 =

k2 ∗5/2

2 z 2 w0y by



a∗y ax bx

k2 4 a 5/2 z 2 w0x x

k2 4 a 5/2 z 2 w0y y



ay by



ax bx

k2 − 2 z

2b∗y



1y

2y −

2

+ H3 (y )

a∗y 02

(10)

(A1x + A2x + A3x )

(11)

(A1y + A2y + A3y )

(12)

with ∗

exp[ (r 1 , 1 ) +

(r 2 , 2 )]

= exp[−0.5 D (r 1 − r 2 )]



= exp

−

8 =



1

[(1 − 2 )2 + (1 − 2 )(r 1 − r 2 ) + (r 1 − r 2 )2 ] 2

0

k2 2 w 2 b∗5/2 b 5/2 z 2 w0x y 0y x



(3) ×

where |0 | = 0.22(Cn2 k2 z)

−1/2

, Cn2 =



ın2  L0 (2 − ς)

(1x − 2x )2 + (1y − 2y ) ik 2 2 2 2 − 2x − 2y )− (1x + 1y 2z 2



× exp −

k2 4bx z 2

 1x −



2x

2 −

ax 02

k2 4by z 2

0



1y −



2y





k2 × 2bx − 2 z

ay =

ay 02

1 2 w0x

1 2 w0y

bx = a∗x −

(5)



a∗y ax

(6)

ax ay bx by k2

2 z 2 w0x bx

5/2

k2 2bx − 2 z

ax ay by

k2

∗5/2

2 z 2 w0x bx

a∗x ay by

2b∗x

k2 − 2 z

 k 2 z 2 w0y by

2

5/2



+ H3 (y )

a∗y 02



2

2x

(13)

2

1y

2y −

1x −

+ H2 (y )

ay 02



2

2y

+ H2 (x )

ax 02

by = a∗y −

−

ik 1 + 2, 2z 0

a∗x =

−

ik 1 + 2, 2z 0

a∗y =

1

, b∗x = ax −

ax 04 1 ay 04

, b∗y = ay −

1 2 w0x

ay ax bx

2by −

2

k z2

 1x −

 2x −

 1y −

2x

+ H2 (x )

ax 02 1x

2

ay 02

2

9/2

04 bx

A2x = −

(14)

1 a∗y 04



(9)

1x −

×

 1x −



ax 02

1x −

ax 02

− H0 (x )

ax 02

− H0 (x )

1x −

2x

2

2x

4

2x



z 2 04 bx

k2 6bx − 2 z

ik 1 + 2 2z 0

1

7/2



+

a∗x 04

2 −k2 02 2x + ikzx

ik 1 + 2 2z 0

1

k2 − 12bx 2 z



(8)

+ H2 (y )

12bx

2

k4 + 4 z

(7)

+ H3 (x )

a∗x 02 2y

2

A1x =

1

+

2 w0y



k2



4 =

×

k2 2b∗y − 2 z

+ H3 (x )

a∗x 02

x = 2x − 1x , y = 2y − 1y



3 =



ax =

where

2 =

k2 2 w 2 b∗5/2 b 5/2 z 2 w0x x 0y y

2

× (1 − 2 − 3 − 4 − 5 + 6 + 7 + 8 + 9 )]

z2

1y −

2

1x

2

k2 k2 × exp − 2 − 2 exp[H(x ) + H(y )] 4ax z 2 2x 4ay z 2 2y

1 =

2x −

a∗x ay

with

 (1x , 2x , 1y , 2y , z) = exp −

9 =

 

k2 × 2by − 2 z

(4)

In Eq. (3), r = (x,y) and  = (x , y ) are the position vectors at the input plane (z = 0) and output plane (z), respectively  (1 , 2 , z) is the second-order correlation at the output plane (z), E(r1 ,0) is the electric field of the laser beam at the source plane (z = 0), and dr1 dr2 = dx1 dy1 dx2 dy2 . In Eq. (4), Cn2 is the structure constant of the refractive-index of the biological tissues. L0 is the outer scale of the refractive-index size. ς is related to the fractal dimension of the tissue which is an indication of the classical turbulent behavior of biological tissue. ın2  is the ensemble-averaged variance of the refractive index. Substituting Eq. (1) into Eq. (2), after tedious but straightforward integration, we obtain the following expression for the secondorder correlation of AHB passing through the biological tissues:



k2 2b∗x − 2 z



2x ax 02

 − H0 (x )

2

− H0 (x )

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A3x =

X. Lu et al. / Optik 127 (2016) 1842–1847



1

2 k2 2x

2ax −

5/2 bx

z2



×

2bx −

z 2 x2 + 2ikz02 2x x

+

z 2 04



k2 z2

2x

1x −

A1y =

9/2

04 by

k4 + 4 z

A2y = −

×

A3y =

1y −



1

2ay −

5/2

by

 1y −

2 k2 2y

z2



H(x ) k2 =− 2 4z bx

+

kax 04



izx

2 −

k02

 +

k2 H(y ) = − 2 4z by



2y ay 02

+

−

02

a∗x 2x

kay 04

izx



k02

02

 + a∗x

+ 21x ax

2y −

ay k02

izy



k02

 + a∗y 2y

−2





izy

−2 02

 + a∗y

 + 21y ay





z 2 (1 − ax 02 ) x2 + 2ikz02 2x + ax 2 04 1x − ax 02 (1x + 2x ) x

 2

z 2 ax 2 08



z 2 (1 − ay 02 ) y2 + 2ikz0 2y + a2y 04 1y − ay 02 (1y + 2y ) y z 2 a2y 08



2



4 ∗ 2 z 2 (1 − a∗x 02 ) x2 + 2ikz02 1x + a∗2 x 0 2x − ax 0 (1x + 2x ) x

z (1 − 2

H3 (y ) =

iz(−1 + ay 02 )y



2

2

H3 (x ) =

− H0 (y )

ax k02

−21y



2

2x −

2

H2 (y ) =

2

− H0 (y )



izx

k02



− H0 (y )

ay 02

H0 (y ) =

,

−21x

izy



2y

z 2 04

02

+

H2 (x ) =

ay 02



iz −1 + ax 02 x

− H0 (y )

z 2 y2 + 2ikz02 2y y

1y −



H0 (x ) =

ay 02



k2 2by − 2 z

×

2y

2

2y

− H0 (y )

ay 02



7/2

2

4

2y

1y −

z 2 04 by

k2 6by − 2 z

1y −



2(−k2 02 2y + ikzy )





k2 z2

2

12by − 12by

3. Numerical examples

− H0 (x )

ax 02

 1



2 a∗y 02 ) y2

+

2ikz02



8 z 2 a∗2 x 0

1y +

4 a∗2 y 0 2y

−

a∗y 02 (1y



+ 2y ) y

8 z 2 a∗2 y 0

(15) The irradiance of AHB at the output plane is given by I(1x , 1y , z) =  (1x , 1y , 1x , 1y , z). Eqs. (5)–(15) are the main analytical results of this paper, which provide a convenient way to analyze the propagation properties of AHBs passing through the biological tissues (Cn2 > 0). When Cn2 = 0, Eq. (5) reduce to the propagation formular of AHBs in free space.

Based on the propagation formulae obtained in Section 2, we study the propagation properties of circular and elliptical AHBs passing through the biological tissues. In this section, we mainly study the influence of propagation distance (z), beam width and structure constant of the refractive-index (Cn2 ) on propagation properties of AHBs. Furthermore, we study the influence of beam width, wavelength and structure constant of the refractive-index (Cn2 ) on effective beam size. Fig. 1 shows the cross line (y = 0) of normalized irradiance distribution of a circular AHB at several propagation distances in biological tissues with  = 632.8 nm, w0x = w0y = 10 ␮m and Cn2 = 5 × 10−7 ␮m−2/3 . From Fig. 1, we can see that the circular AHB eventually becomes a Gaussian beam in the far field under the influence of the turbulence biological tissues. At certain propagation distance (see Fig. 1(e)), we can observe a flat-topped beam profile. For the convenience of observation, in Fig. 2, we simulated the contour graph of the irradiance of the AHB in the biological tissues at the same propagation distance with Fig. 1. Also, the other parameters of Figs. 1 and 2 are the same. Furthermore, in order to study the influence of initial beam width of AHB on propagation properties, we change the initial beam width and then get another set of results in Fig. 3 with  = 632.8 nm, w0x = 25 ␮m, w0y = 10 ␮m and Cn2 = 5 × 10−7 ␮m−2/3 . Similarly, elliptical AHB eventually becomes a Gaussian beam in the far field and also we can observe a flat-topped beam profile at certain propagation distance. Interestingly, in the far field, both of circular and elliptical AHBs become into a circular Gaussian beam. These phenomena are similar to the propagation properties of AHB in the turbulent atmosphere, only the propagation distance in tissues is much less than that in atmosphere for stronger turbulence. Fig. 4 shows the cross line (y = 0) of normalized irradiance distribution of a circular AHB with several values of Cn2 in biological tissues with  = 632.8 nm, w0x = w0y = 10 ␮m and z = 100 ␮m. From Fig. 4, we can see that the AHB is more like a Gaussian beam as the value of Cn2 grows at the same propagation distance. In the other word, the central irradiance of the AHB rises more rapidly as the value of Cn2 grows. For example, as we can see in Fig. 4(a), the beam profile of AHB remains almost invariant, while it changes into a flattopped beam when the value of Cn2 changes from 5 × 10−8 ␮m−2/3 into 1.3 × 10−6 ␮m−2/3 . Based on this property, we can determine the disturbance of the biological tissue. In order to learn the spreading properties of AHBs, we study the evolution properties of the effective beam spot of an AHB. The effective beam spot can be defined as [22]

 ∞ ∞ Wsz =

2

−∞

−∞





s2 I(x, y, z) dxdy

−∞ −∞ ∞ ∞ 



I(x, y, z) dxdy

(s = x, y)

(16)

Wxz and Wyz are the effective beam sizes of AHBs in the x and y directions, respectively. For an elliptical AHB, Wxz =





17/10 w0x

and Wyz = 17/10 w0y [5]. Substituting Eq. (5) into Eq. (16), we can calculate the effective beam size of AHB in the biological tissues. Fig. 5 shows the dependence of effective beam size Wxz of circular AHB on propagation distance z in biological tissues for different values of wavelength with Cn2 = 5 × 10−6 ␮m−2/3 and w0x = w0y = 10 ␮m. Fig. 6 shows the dependence of effective beam size Wxz of elliptical AHB on propagation distance z in biological tissues for different values of wavelength with Cn2 = 5 × 10−6 ␮m−2/3 and w0x = 25 ␮m, w0y = 10 ␮m. From Figs. 5 and 6, we can conclude that the beam spot of AHB spread more rapidly with lager wavelength. However, in the case of paraxial transmission, the difference is very subtle at visible wavelengths. Fig. 7 shows the dependence of effective beam size Wxz and Wyz

X. Lu et al. / Optik 127 (2016) 1842–1847

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Fig. 1. Cross line (y = 0) of normalized irradiance distribution of a circular AHB at several propagation distances in biological tissues with  = 632.8 nm, w0x = w0y = 10 ␮m and Cn2 = 5 × 10−7 ␮m−2/3 .

Fig. 2. The contour graph of the irradiance of the circular AHB at several propagation distances in biological tissues with  = 632.8 nm, w0x = w0y = 10 ␮m and Cn2 = 5 × 10−7 ␮m−2/3 .

Fig. 3. The contour graph of the irradiance of the elliptical AHB at several propagation distances in biological tissues with  = 632.8 nm, w0x = 25 ␮m, w0y = 10 ␮m and Cn2 = 5 × 10−7 ␮m−2/3 .

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X. Lu et al. / Optik 127 (2016) 1842–1847

Fig. 4. Cross line (y = 0) of normalized irradiance distribution of a circular AHB with several values of Cn2 in biological tissues with  = 632.8 nm, w0x = w0y = 10 ␮m and z = 100 ␮m. (a) Cn2 = 5 × 10−8 ␮m−2/3 , (b) Cn2 = 3 × 10−7 ␮m−2/3 , (c) Cn2 = 1.3 × 10−6 ␮m−2/3 , (d) Cn2 = 5 × 10−6 ␮m−2/3 .

Fig. 7. Dependence of effective beam size Wxz and Wyz of elliptical AHB on propagation distance z in biological tissues for different values of structure constant Cn2 (␮m−2/3 ). The solid line represents Wxz and the dotted line represents Wyz .

4. Conclusion Above all, we have derived the analytical formulae which provide a convenient way to analyze the propagation properties of AHBs passing through the biological tissues. Based on these formulae, we investigate the irradiance and spreading properties of AHBs numerically. We conclude that the circular and elliptical AHBs eventually become Gaussian beams in the far field. The AHB is more like a Gaussian beam as the value of structure constant Cn2 grows at the same propagation distance. In the other word, the central irradiance of the AHB rises more rapidly as the value of Cn2 grows. Also, we calculate the effective beam size of AHB which provide a convenient way to learn the spreading properties of AHBs. We find that finally Wxz becomes equal to Wyz in biological tissues which can be used to explain the beam spot eventually becomes circular under the influence of turbulence of biological tissues.

Fig. 5. Dependence of effective beam size Wxz of circular AHB on propagation distance z in biological tissues for different values of wavelength with Cn2 = 5 × 10−6 ␮m−2/3 and wox = woy = 10 ␮m.

Acknowledgments This research is supported by the National Natural Science Foundation of China under Grant Nos. 11274005, 11374222, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Universities Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the Qing Lan Project of Jiangsu Province, the 2013 High Educational Reform Project of Soochow University under Grant No. 5731501713, the Zhejiang Provincial Natural Science Foundation of China under Grant No. LY12A04012. References

Fig. 6. Dependence of effective beam size Wxz of elliptical AHB on propagation distance z in biological tissues for different values of wavelength with Cn2 = 5 × 10−6 ␮m−2/3 and wox = 25 ␮m, wox = 10 ␮m.

of elliptical AHB on propagation distance z in biological tissues for different values of structure constant Cn2 with  = 632.8 nm and w0x = 25 ␮m, w0y = 10 ␮m. The solid line represents Wxz and the dotted line represents Wyz . The solid black line (Cn2 = 0) reduces to the propagation of AHBs in free space. From Fig. 7, we can find that the effective beam sizes of AHB gradually increase on propagation and finally Wxz becomes equal to Wyz in biological tissues which can be used to explain the beam spot eventually becomes circular under the influence of the turbulence of biological tissues.

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