Pulse broadening and beam spread of polarized laser pulse beam on slant path in turbulence atmospheric

Optik 126 (2015) 4651–4657

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Pulse broadening and beam spread of polarized laser pulse beam on slant path in turbulence atmospheric Ming Gao, Yan Li ∗ , Lei Gong, Hong Lv School of Optoelectronic Engineering, Xi’an Technological University, Xi’an 710021, China

a r t i c l e

i n f o

Article history: Received 28 July 2014 Accepted 18 August 2015 Keywords: Elliptically polarization Gaussian pulses Atmospheric turbulence Pulse broadening Beam spread

a b s t r a c t This paper is based on the extended Huygens – Fresnel principle, the two-frequency mutual coherence function and the Rytov approximation method. We have derived the novel expressions for the pulse broadening and beam effective spread radius of the elliptically polarized Gaussian pulse beam propagating through atmospheric turbulence along a slant path. Meanwhile the effect of weak optical turbulence on the pulse broadening and the beam spread in a slant path are analyzed. The results show that the pulse broadening of the elliptically polarized Gaussian pulse beam on slant path turbulence atmospheric depends on the zenith angle  and receiver height H. The inner scale factor ˇ appears to have an insignificant effect on the pulse broadening. Beam spread is mainly depends on the initial waist radius w0 , wavelength  and polarization angle ϕ. With the increasing of propagation distance L, beam effective spread radius w(L) decreases as w0 increases, and it increases with , in addition, w(L) increases as ϕ increases. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction In recent years, more and more laser pulse beams are put into wide applications of laser systems. It has become the carriers of information in the fields of laser radar, laser guidance, optical communication system, laser imaging systems et al. and obtained important means of additional information [1–3]. It is noted that when the laser beams propagate through the atmosphere, the atmospheric refractive index occurs ups and downs randomly for the influence of the turbulence atmospheric, resulting in the intensity fluctuations, beam spot wandering, beam spread, phase fluctuations and angle-of-arrival fluctuations etc [4–6]. The turbulence-induced spatial spreading and temporal broadening of laser pulse beams are limiting factors in most applications, such as the remote sensing and atmospheric optical communication etc. Reducing these effects will provide a theoretical basis for improving the performance of the laser systems. At present, lots of researchers pay much attention to the propagation problems of laser pulse beams through the random media. J.J. Laserna et al. [7] investigated the propagation of nanosecond pulse beams on the laser induced breakdown spectroscopy measurements through atmospheric turbulence. Lü’s team studied the propagation properties of ultrashort chirped pulsed

∗ Corresponding author. Tel.: +86 13630287903. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.ijleo.2015.08.061 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

Gaussian beams with constant diffraction length, then taken the pulsed Gaussian beam as an example, the way of elimination of the spatial singularity of ultrashort pulsed beams in dispersive media are derived; and taken the ultrashort pulsed Laguerre – Gaussian (LG) beam as a typical example of multimode pulsed beams, the spectral and temporal properties of ultrashort pulsed LG beams in dispersive media are studied by using the Fourier transform [8–10]. In recent years, many researchers have focused on the propagation of polarized laser beam for the advantages of the polarized beam in laser communication systems, remote sensing and laser imaging systems. For example, it can accurately identify man-made objects in complex background and simultaneously obtains the radiation intensity and polarization information. F. Gori et al. [11] proposed the partially polarized Gaussian Schell-model beams, which provides a theoretical source model for the study of the polarization characteristics of laser beam in free space. Furthermore, Yang Lihong et al. [12] analyzed the depolarization properties of polarized laser beam in turbulence atmospheric based on the Monte Carlo method, in the condition of different fog particle radius, mist and weak fog. So far, the studies on the propagation issues of laser beam are mostly restricted to the polarized laser beams or laser pulse beams, and less analysis on the pulse beams with polarization characteristics. Moreover, only very few papers have dealt with the relevant questions about propagation of polarized laser pulse beam in turbulent medium. In this paper, we take the elliptically polarized Gaussian pulses beam as a typical example. The pulse broadening and the beam

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spread of the beam propagating through the atmospheric turbulence along a slant path have are discussed. Based on the extended Huygens-Fresnel principle, the two-frequency mutual coherence function and Rytov approximation method, the analytic expressions for average intensity, pulse broadening and beam effective spread radius of elliptically polarized Gaussian pulse beam in receiver through atmospheric turbulence in a slant path are in turn derived. Finally, we make a numerical calculation and analysis. 2. Basic principles Considering a random, wide-sense stationary and electromagnetic field, we assume that an elliptically polarized Gaussian pulse beam is generated in the source plane r(z = 0) and propagates into the half-space (z > 0) close to the z-axis in the turbulence atmospheric. The initial electromagnetic field distribution function in the source plane is given by [13]: U (r, 0, t) = U (r, 0)fi (t)

(1)

where, U (r, 0) is the complex amplitude of TEM00 mode Gaussian beam,



1 U (r, 0) = U0 exp − ˛0 kr2 2



(2)

where, U0 = Ax cos  ex + Ay sin  exp(iϕ)ey , ˛0 = i/F0 + 2/kw02 , Ax and Ay denote the amplitudes of optical field components in i and j directions respectively,  is the direction angle of elliptically polarized Gaussian pulse beam, ϕ is the phase difference, ϕ = / n (n = 1,2,3. . .). w0 is the initial waist width of the beam, F0 denotes the phase front radius of curvature of the beam wave in the transmitter, k = ω/c is wave number. In the free space, i.e., ignoring the turbulence, the diffraction characteristics of the Gaussian beam in the transmitter and the receiver can be described by Entrance pupil parameters 0 and 0 and the exit pupil parameters  and : 0 = 1 − =

L , F0

0 20

+ 20

2L

0 = =

,

(3)

kw02 0 20

(4)

+ 20

fi (t) is the time function of the input Gaussian pulse signal, that can be written as [14]



 2

fi (t) = exp −t 2 /T0 exp (−iω0 t)

(5)

where, ω0 is center angular frequency, T0 denotes the initial halfpulse. From which we deduce the complex envelope of the output pulse f0 ( , L, t): f0 ( , L, t) =

1 2



+∞

−∞

fi (ω)U ( , L, ω) exp(−iωt)dω

= U0



2

ik exp (ikL) 2z

+ 2



 

+∞

U(r, 0) exp −∞

ik exp ikL + (1 −  + i ) 2 2L



 ik

2L





I(␳, L, t) =

T02

 

4



(7)

According to Eqs. (9)–(12) in Ref. [14], we can derive the temporal average intensity in the receiver by setting the two-time,







1 1 exp − ωc2 T02 exp − ωd2 T02 2 8



(8)

×2 ( , , L, ω1 + ω0 , ω2 + ω0 ) exp(−iωd t)dωc dωd where,  2 ( 1 , 2 , L, ω1 + ω0 , ω2 + ω0 ) is the two frequency Mutual Coherence Function(MCF)of the elliptically polarized Gaussian pulse beam propagating in turbulence atmospheric. Let k1 = (ω1 + ω0 )/c, k2 = (ω2 + ω0 )/c, the two frequency MCF defined by [14]: 2 ( 1 , 2 , L, ω1 + ω0 , ω2 + ω0 ) = 20 ( 1 , 2 , L, k1 , k2 )M2 ( 1 , 2 , L, k1 , k2 )

(9)

20 ( 1 , 2 , L, k1 , k2 )

where, represents the two frequency MCF of the elliptically polarized Gaussian pulse beam propagating in the free-space. M2 ( 1 , 2 , L, k1 , k2 )caused by atmospheric turbulence. According to the extended Huygens – Fresnel principle, the two frequencies MCF of the elliptically polarized Gaussian pulse beam propagating in free space can be described by [15]: 20 ( 1 , 2 , L, k1 , k2 ) = U( 1 , L, k1 )U∗ ( 2 , L, k2 ) = U0 · U∗0 × exp



 1

21 + 21

− (k1 1 12 2L



22 + 22 exp[i(k1 − k2 )L]



(10)

i ˜ 1 2 + k2  ˜ 2 2 ) (k1  1 2 2L

+ k2 2 22 ) +

˜ = 1 − . where,  Meanwhile, according to the Rytov approximation method, M2 ( 1 , 2 , L, k1 , k2 ) can be written by [15]:



M2 ( 1 , 2 , L, k1 , k2 ) = exp





∗ ( , L, k ) 2 2

( 1 , L, k1 ) +

= exp [2E1 (0, 0, L, k1 , k2 ) + E2 ( 1 , 2 , L, k1 , k2 )] where:



H



E1 (0, 0, L, k1 , k2 ) = −2 (k12 + k22 ) sec 

 H × exp −

i 2 L 2





∗

1 − 2 k1 k2



∞

0

˚n ( , h)J0 ( | 1 1 − 2∗ 2 |) 0

(12)

∞

E2 ( 1 , 2 , L, k1 , k2 ) = 42 k1 k2 sec 



(11)

˚n ( , h)d dh 0

0

(13)

(h) d dh

˜ + i )(h), where, J0 (x) is zero-order Bessel function, = 1 − ( (h) = 1 − h/H, is spatial radial frequency, ˚n ( , h) is the refractive index power spectral density function. Using the modified von Karman spectral model, ˚n ( , h) can be given by [15]: ˚n ( , h) = 0.033Cn2 (h)

( − r)2 dr

+∞

−∞

(6)

where, U ( , L, ω) is the complex amplitude of the elliptically polarized Gaussian pulse beam in the space-frequency domain propagating from transmitter plane r(z = 0) to receiver plane

(z = L). Assuming that this beam propagates into the receiver plane after a distance z, and then using the extended Huygens – Fresnel principle, the complex amplitude of the optical field distribution can be derived: U( , L, k) = −

two-point correlation function at 1 = 2 = and t1 = t2 = t. This leads to

2) exp(− 2 / m

( 2 + 02 )

11/6

(0 ≤ ≤ ∞)

(14)

where, Cn2 (h) is the refractive-index structure function on slant path which is a measure of the strength of turbulence, 0 = 1/L0 , m = 5.92/l0 , L0 and l0 are the outer scale and the inner scale of turbulence, respectively. In Fig. 1, assuming the transmitter placed on level ground, H is the vertical height of the receiver,  is the path zenith angle, L is the distance on slant path from the receiver to the transmitter. h and z (h ≤ H, z ≤ L) are the vertical propagation height and the slant propagation distance of the beam on slant path, respectively, and both of which are variables. Using ITU-R P.1621 recommend atmospheric refractive index structure constant model, whose description of the atmospheric distribution of the refractive index structure constant is changing with height, and the wind speed and turbulence are taken into

M. Gao et al. / Optik 126 (2015) 4651–4657

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From which we deduce the two-frequency MCF as

 2 ( , , L, ω0 + ωc , ωd ) ≈ U





2 L + c 2Lc

exp −ωd2 + i

w02

2 (ω0 + ωc )2

2Lc



 ωd −2

w02



2

2Lc

(ω0 + ωc )2

(22)

Upon substituting Eq. (22) into Eq. (8) and evaluating the expression we find that the temporal average intensity can be written as Fig. 1. Diagram of the polarized laser pulse beams propagation through atmospheric turbulence on slant path.



account, it can be described [16]:

I( , L, t) =

Cn2 (h) = 8.148 × 10−56 v2rms h10 e−h/1000 + 2.7 × 10−16 e−h/1500 + C0 e−h/100 (m−2/3 )

L0 (h) =

5

1+

 h−7500 2 , l0 (h) = ˇL0 (h)

where, ˇ is the inner scale factor, 0 < ˇ < 1. When the initial input beam is a spherical wave, the beam parameters can be expressed as  = = 0 [17]. Now, let 1 = 2 = , and under the narrowband assumption ωd « ωc . We can then rewrite Eqs. (10) and (11) as: 20 ( , , L, ω0

 exp i

+ ωc , ωd ) ≈ U

L

2 + c 2Lc



M2 ( , , L, ω0 + ωc , ωd )



≈ exp

−

22 ωd2 c2



w0

(ω0 + ωc )

2Lc

 ωd − 2

2 2

w02

H



sec 

(ω0 + ωc )

2

(17)

 ˚n ( , h)d dh

0

=

exp(−ωd2 )

0

(23) where, T is half-pulse width of the received beam:

 T=

T02 + 8

(24)

As we see from Eq. (24), the pulse broadening is depended on the initial half-pulse width T0 and the turbulence factor . According to Eq. (23) and the Ref. [18], pulse statistics concerning the random arrival time can be described in terms of the average temporal moments







2

0.391 1 + 0.171ˇ2 − 0.287ˇ

 5/3



H

Cn2 (h)[L0 (h)]

= 0

5/3

dh

n = 0, 1, 2, . . .

(25)

Each moment corresponds to a physical characteristic: n = 0 relates to the total energy of the pulse, whereas the first moment relates to the mean arrival time of the pulse and pulse width. Similarly, pulse spread caused by the random medium can be inferred from the second moment. They are described, respectively:





M (0) ( , L)



+∞

=



−∞



4 2  Uw0 T0 2 4L2 c 2





I( , L, t) dt





ω02 T02 + 1 T02 + w0 /Lc





T02 + w0 /Lc

2

2 5/2

⎧ ⎨

 2 ⎫ ⎬ ω0 w0 T0 /Lc exp −     ⎩ 2 T 2 + w0 /Lc 2 ⎭ 0

(26)







+∞

M (1) ( , L) =



(20)

(21)



t I( , L, t) dt −∞

 sec 

c2 × (Turbulence Factor)



t n I( , L, t) dt,

(19)

= I1 + I2 cos ϕ





+∞

M (n) ( , L) =

U0 U∗0

= [Ax cos  ex + Ay sin  exp(iϕ)ey ] · [Ax cos  ex + Ay sin  exp(−iϕ)ey ]

=

0

=

= A2x cos2  + A2y sin  + 2Ax Ay cos  sin  cos ϕ

2 5/2

   2 ⎫ 2  ⎬ 2 t − L/c − 2 /2Lc ω0 w0 T0 /Lc  × exp − exp −  T2 ⎩ 2 T 2 + w0 /Lc2 ⎭

(18)

where: U=



T02 + w0 /Lc

2

−∞



∞



4L2 c 2 T

2

2

2Lc



(16)

2500





Uw04 T02 ω02 T02 + 1 T02 + w0 /Lc

⎧ ⎨

(15)

where, h is the height above the ground, vrms = v2g + 30.69vg + 348.91(m/s) is the wind speed on vertical path, vg the wind speed near the ground, usually taken vg = 2.8 m/s, vrms = 21(m/s); C0 is refractive-index structure constant near the ground(typical values is C0 = 1.7 × 10−14 m−2/3 ). Inner and outer scale models as a function of altitude are taken as [15]:





 Uw4 T 2 L/c + 2 /2Lc ω2 T 2 + 1 T 2 + w /Lc2 0  0 0 0 0 0 · =  5/2   2 4L2 c 2 (27) 2 T02 + w0 /Lc ⎧ ⎫  2 ⎬ ⎨ ω0 w0 T0 /Lc × exp −   2  ⎭ ⎩ 2 2 T0 + w0 /Lc

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M. Gao et al. / Optik 126 (2015) 4651–4657



M (2) ( , L)

 =





t 2 I( , L, t) dt

  2



+∞

= −∞



Uw04 T02 T 2 /4 + L/c + 2 /2Lc

2  





ω02 T02 + 1 T02 + w0 /Lc

4L2 c 2





T02 + w0 /Lc

⎧ ⎨

 2 ⎫ ⎬ ω0 w0 T0 /Lc × exp −     ⎩ 2 T 2 + w0 /Lc 2 ⎭

2

2 5/2

(28)

0

Assuming the pulse signal propagating from a transmitter to a receiver at time t = 0, the mean square pulse width of the pulse signal at transverse position is given by:



2 BW



(2)   (1)  2 M ( , L) M ( , L) T2 −

 = T02 /4 + 2 (29) =

= M (0) ( , L)

4

M (0) ( , L)

Fig. 2. The impact of zenith angle on pulse broadening.

We see the mean square pulse width in Eq. (29) is proportional to the half-pulse width in Eq. (24). In the limit T0 → ∞ and regardless of the turbulence impacted on transmission frequency, by using the quadratic approximation of Rytov’s phase structure function, we can derive [19]:



M2 (r1 , r2 , , L; k) = exp[ (r1 , , L; k) +



(r1 − r2 ) ∼ = exp − r02

2



∗ (r , , L; k)] 2 m



(31)



where, r0 = 0.423k2 sec 

Z H

−3/5

Cn2 (h)dh

is the spatial coher-

ence radius of a spherical wave propagating in the turbulent medium. Therefore we can express the average intensity in the receiver as: I( , L) = I1 W1 + I2 cos ϕW2

(32)

where: I1 = A2x cos2  + A2y sin2 , I2 = 2Ax Ay cos sin ; Wi = 1+

w2 0

2 i

+

ω2

0 w2 Ki 0 2w2 0 r2 0

0 = 1 −



exp

−2 2 w2 Ki 0



(i = 1, 2), Ki = 20 + ςi 20 (i = 1, 2); ςi =

(i = 1, 2); L , F0

0 =

2L . kw2 0

Combined with the second moment of the optical field, the effective spread radius of the beam can be written as:

# ! ∞ ∞ 2 ! 4 −∞ −∞ I( , L)d2

I1 w02 K1 + I2 cos ϕw02 K2 " w(L) = 2 = ∞ ∞ 2 −∞

−∞

I( , L)d

I1 + I2 cos ϕ

(33)

3. Numerical calculation and analysis We assume that an elliptically polarized Gaussian pulse beam with the parameters as follows: wavelength  = 1.06 ␮m, initial waist width w0 = 1 cm, the amplitudes of electric field components in i and j directions respectively are Ax = 2 and Ay = 1. Different elliptically polarized light will be obtained by different phase ϕ. In Section 2, we have defined the incident beam is a spherical wave, and whose beam parameters are  = = 0. Therefore we can ignore the impact of the diffraction characteristics of the beam on the pulse broadening and the beam spread.

3.1. The impact of zenith angle, receiver height and inner scale factor on the pulse broadening From Eq. (24), we can see that the pulse broadening depended on the initial half-pulse width T0 and turbulence factor. But the zenith angle, receiver height H and inner scale factor ˇ are all the function of the turbulence factor. Therefore, we mainly analyze the impact of zenith angle, receiver height and inner scale factor on the pulse broadening. Fig. 2 shows that propagating under the slant path, while the receiver height is 100 m, a detailed analysis of the impact of zenith angle on the pulse broadening is made. The values of all the parameters used in calculations are illustrated in Table 1. Setting the zenith angle:  = 0,  = 0.5 and  = 1.0. As shown by Fig. 2 and Table 1, when the initial half-pulse width is greater than 15 fs, the variation of the pulse broadening with changing zenith angles is not obvious, i.e., the zenith angle has a small impact on the pulse broadening. However, when the initial half-pulse width is less than 10 fs, the range of pulse broadening is comparatively larger. When the initial halfpulse width of pulse is fix, pulse broadening changes with different zenith angle, and it increases with an increase of the zenith angle. Fig. 3 shows that propagating under the slant path, while the zenith angle denote  = 0.5, a detailed analysis of the impact of receiver height on the pulse broadening is made. The values of all the parameters used in calculations are illustrated in Table 2. Now we set the receiver height: H = 10 m, 100 m, 500 m and 1000 m. As illustrated by Fig. 3 and Table 2, when initial half-pulse width is greater than 20 fs, the variation of the pulse broadening along with the receiver heights change is similar. However, when the initial half-pulse width is less than 10 fs, and the initial half-pulse widths are equal, the pulse broadening increases as receiver height increases. When the receiver heights are 10 m and 100 m, respectively, the extent of the pulse broadening are much larger. When Table 1 Simulation parameters with respect to the impact of zenith angel on pulse broadening. Wavelength Refractive index Initial waist width Inner scale factor Receiver height T0 = 5.0505 fs T0 = 10.1010 fs T0 = 15.1515 fs T0 = 30.3030 fs

=0 1.2170 1.0584 1.0264 1.0067

 = 1.06 ␮m C0 = 1.7 × 10−14 m−2/3 w0 = 0.01 m ˇ = 0.005 H = 100 m  = 0.5  = 1.0 1.2442 1.3748 1.0663 1.1057 1.0300 1.0483 1.0076 1.0123

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Table 3 Simulation parameters with respect to the impact of inner scale factor on pulse broadening.  = 1.06 ␮m C0 = 1.7 × 10−14 m−2/3 w0 = 0.01 m  = 0.5 H = 100 m

Wavelength Refractive index Initial waist width Zenith angle Receiver height

T0 = 5.0505 fs T0 = 10.1010 fs T0 = 15.1515 fs T0 = 30.3030 fs

ˇ = 0.005 1.24421 1.06631 1.03000 1.00758

ˇ = 0.01 1.244191.3748 1.066301.1057 1.030001.0483 1.007581.0123

half-pulse width is less than 10 fs, the range of the pulse broadening is very intense. Therefore, it is investigated that the impact of the inner scale factor on the pulse broadening is little. Fig. 3. The impact of receiver height on pulse broadening. Table 2 Simulation parameters with respect to the impact of receiver height on pulse broadening.  = 1.06 ␮m C0 = 1.7 × 10−14 m−2/3 w0 = 0.01 m ˇ = 0.005  = 0.5

Wavelength Refractive index Initial waist width Inner scale factor Zenith angle

T0 = 5.0505 fs T0 = 10.1010 fs T0 = 15.1515 fs T0 = 30.3030 fs

H = 10 m 1.0396 1.0100 1.0045 1.0011

H = 100 m 1.2442 1.0663 1.0300 1.0076

H = 500 m 1.3868 1.1094 1.0500 1.0127

H = 1000 m 1.4097 1.1166 1.0534 1.0136

the receiver heights are 500 m and 1000 m, respectively, the extent of the pulse broadening are smaller. In addition, when the pulse broadening is fixed, the receiver height decreases with the increase of the initial half-pulse width. Fig. 4 shows that propagating under the slant path, while the zenith angle denote  = 0.5, and the receiver height is 100 m, a detailed analysis of the impact of inner scale factor on the pulse broadening is made. The values of all the parameters used in calculations are illustrated in Table 3. Now we let the inner scale factor: ˇ = 0.005 and ˇ = 0.01. As shown by Fig. 4 and Table 3, the two curves are nearly coincide. When the initial half-pulse width is greater than 20 fs, the variation of the pulse broadening along with the inner scale factor change is smaller. However, when the initial

3.2. The impact of initial waist width of an elliptically polarized Gaussian pulse beam on the beam spread Fig. 5 shows that propagating under the slant path, while the initial waist widths are different, a detailed analysis of the effective spread radius varies with the propagation distance is made. The values of all the parameters used in calculations are illustrated in Table 4. Fig. 5(a) gives the beam spread of the elliptically polarized Gaussian pulsed beam in different initial waist widths, where we set the wavelength of the elliptically polarized Gaussian pulse beam is  = 1.06 ␮m, the receiver height is h = 100 m, the polarization angle is  = /4. The initial waist radius w0 denotes 0.001 m, 0.01 m, 0.1 m and 1 m, respectively. As shown by Fig. 5(a) with the increase of the initial waist radius, their effective spread radius decreases with the increase of propagation distance. The larger extent of the beam spread is, the smaller initial waist radius will be. Meanwhile it shows that the turbulence has an important effect on the beam. In Fig. 5(a), while the initial waist radius is 1 m, the range of the beam spread is very little. The curve is close to a horizontal line. As with the increase of the waist radius, the beam will approximately be a plane wave. Traditionally, the expansion effect of the plane wave is very small. However, when the initial waist radius is 0.001 m, the extent of the beam spread is much larger. Fig. 5(b) shows a comparison of an elliptically polarized Gaussian pulse beam with unpolarized Gaussian pulse beam at different initial waist radius of beam on the beam spread. We assume that the wavelength is  = 1.06 ␮m, receiver height is 100 m, the initial waist radius w0 denotes 0.001 m and 0.01 m, respectively, and the polarization angle is 3/4. From Fig. 5(b) we can see that along with the increase of the propagation distance, the elliptically polarized Gaussian pulse beam with different initial waist radius, whose Table 4 Simulation parameters with respect to the impact of initial waist width on beam spread. Wavelength Refractive index Receiver height Polarized: ϕ = 3/4 L = 200 m L = 1000 m L = 3000 m L = 5000 m

Fig. 4. The impact of inner scale factor on pulse broadening.

L = 200 m L = 1000 m L = 3000 m L = 5000 m

 = 1.06 ␮m C0 = 1.7 × 10−14 m−2/3 H = 100 m w0 = 0.001 m w0 = 0.01 m 0.03008 0.02566 0.14220 0.08282 0.42565 0.24192 0.70929 0.40232 Unpolarized: ϕ = 0 w0 = 0.01 m w0 = 0.001 m 0.02992 0.02547 0.14141 0.08145 0.42402 0.23904 0.70800 0.40004

w0 = 0.1 m w0 = 1.0 m 0.20019 1.00004 0.20482 1.00095 0.23990 1.00854 0.29792 1.02355 Polarized: ϕ = 3/4 w0 = 0.001 m w0 = 0.01 m 0.03089 0.02660 0.14653 0.09005 0.43940 0.26536 0.73358 0.44374

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M. Gao et al. / Optik 126 (2015) 4651–4657

Fig. 6. The impact of wavelength on beam spread: (a) elliptically polarized Gaussian pulse beams and (b) unpolarized Gaussian pulse beams. Fig. 5. The impact of initial waist width on beam spread: (a) elliptically polarized Gaussian pulse beams and (b) a comparison of elliptically polarized Gaussian pulse beams with unpolarized Gaussian pulse beams.

effective spread radius is greater than the unpolarized Gaussian pulse. It is illustrated that the impact of the turbulence atmospheric on the elliptically polarized Gaussian pulse beam is larger than which is on the unpolarized Gaussian pulse beam, but the changes in magnitude are smaller. 3.3. The impact of wavelength and polarization angle of an elliptically polarized Gaussian pulse beam on the beam spread Figs. 6 and 7 give the variation of effective spread radius of an elliptically polarized Gaussian pulse beam propagating through the atmospheric turbulence on slant path with different wavelength and different polarization angles along with the propagation distance. In Fig. 6, assuming that the polarization angle of an elliptically polarized Gaussian pulse beam is /4, the initial waist radius w0 denotes 0.01 m, receiver height is 500 m, and the wavelength  sets: 0.532 ␮m, 1.06 ␮m, 3.8 ␮m and 10.6 ␮m. The values of all the parameters used in calculations are illustrated in Table 5. A comparison of Fig. 6(a) with Fig. 6(b) is made, which shows with

Fig. 7. Beam spread of elliptically polarized Gaussian pulse beams in different polarized angle.

M. Gao et al. / Optik 126 (2015) 4651–4657 Table 5 Simulation parameters with respect to the impact of wavelength on beam spread. Initial waist width Refractive index The receiver height Polarized: ϕ = /4 L = 200 m L = 1000 m L = 3000 m L = 5000 m2.39 = 5000 m Unpolarized: ϕ = /4 L = 200 m L = 1000 m L = 3000 m L = 5000 m

w0 = 0.01 m C0 = 1.7 × 10−14 m−2/3 H = 500 m  = 0.532 ␮m 0.02155 0.04488 0.12218 0.20188  = 0.532 ␮m 0.02110 0.03917 0.10300 0.16959

 = 1.06 ␮m 0.02558 0.08223 0.24013 0.39932  = 1.06 ␮m 0.01949 0.06848 0.20805 0.34908

 = 3.8 ␮m 0.06054 0.28641 0.85737 1.42870  = 3.8 ␮m 0.03721 0.18194 0.55296 0.92455

 = 10.6 ␮m 0.16064 0.79721 2.39096 3.98485  = 10.6 ␮m 0.11295 0.61720 1.88076 3.14448

Table 6 Simulation parameters with respect to the impact of polarized angle on beam spread.  = 1.06 ␮m w0 = 0.01 m C0 = 1.7 × 10−14 m−2/3 H = 500 m

Wavelength Initial waist width Refractive index Receiver height

L = 200 m L = 1000 m L = 3000 m L = 5000 m2.39 = 5000 m

ϕ= 0.02703 0.09317 0.27418 0.45704

ϕ = 3/4 0.02660 0.09005 0.26536 0.44374

ϕ = /2 0.25950 0.08509 0.24942 0.41579

ϕ = /4 0.02558 0.08232 0.24170 0.40446

4657

distance, the effective spread radius of the elliptically polarized Gaussian pulse beam decreases as the radius of the initial waist radius increases and which increases as the wavelength increases. In addition, the greater the polarization angle of the beam is, the greater effective spread radius will be. Finally, we made a comparison between the polarized Gaussian pulse beam and the unpolarized Gaussian pulse beam on the beam spread. Acknowledgments This paper is supported by the Natural Science Basic Research Plan in Shaanxi Province (Nos. 2012JM8008, 2013JQ8018), the National Natural Science Foundation of China (61308071) and the funded projects of the Natural Science Special of the Department of Education in Shaanxi Province (2013JK0633). The program is financed by the open foundation of Shaanxi Key Laboratory of Photoelectric Measurement and Instrument Technology. References

ϕ=0 0.02547 0.08145 0.23904 0.40004

the increase of the wavelength, the effective spread radius of the elliptically polarized Gaussian pulse beam increases with increasing of propagation distance. Therefore, it can be anticipated that the wavelength has an important influence on the beam spread, and the elliptically polarized Gaussian pulse beam is weaker than the unpolarized Gaussian pulse beam on the extent of beam spread. Fig. 7 shows the variation of the beam spread of an elliptically polarized Gaussian pulse beam along with different polarization angle in a slant path. The values of all the parameters used in calculations are illustrated in Table 6. By assuming the initial waist radius w0 of the elliptically polarized Gaussian pulse beam is 0.01 m, receiver height denotes 500 m and the wavelength  is 1.06 ␮m. Now setting that the polarization angle ϕ are 0 rad, /4, /2, 3/4 and , respectively. In Fig. 7, with the increasing of the polarization angle, the effective spread radius of the elliptically polarized Gaussian pulse beam increases as the propagation distance increases. Therefore we find that the polarization angle has a significant influence on the beam spread. In this paper, the novel expressions were derived for the pulse broadening and the beam spread of the elliptically polarized Gaussian pulse beam propagating through the atmospheric turbulence along a slant path, by using the two-frequency MCF of a pulse beam. The results showed that: The pulse broadening mainly depends on the zenith angle  and receiver height H, and the inner scale factor appears to have an insignificant effect on the pulse broadening. The beam spread mainly depends on the initial waist radius w0 , wavelength  and the polarization angle of an elliptically polarized Gaussian pulse beam. With the increase of the propagation

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