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The influence of oceanic turbulence on the spectral properties of chirped Gaussian pulsed beam
Optics & Laser Technology 82 (2016) 76–81
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
The influence of oceanic turbulence on the spectral properties of chirped Gaussian pulsed beam Dajun Liu a,n, Yaochuan Wang a, Guiqiu Wang a, Hongming Yin a, Jinren Wang b a b
Department of Physics, Dalian Maritime University, Dalian 116026, China Sifang College, Shijiazhuang Tiedao University, Shijiazhuang 051132, China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 October 2015 Received in revised form 22 January 2016 Accepted 25 February 2016
Based on the extended Huygens-Fresnel principle, the spectral behaviors of a chirped Gaussian pulsed beam propagating in oceanic turbulence are illustrated. The influence of the parameters of oceanic turbulence (the rate of dissipation of turbulent kinetic energy per unit mass of fluid, rate of dissipation of mean-square temperature, relative strength of temperature and salinity fluctuations), relative position parameter and propagation distance on the spectra shift is analysed and given by numerical examples. The research results have the potential application in underwater wireless laser communication and remote sensing. & 2016 Published by Elsevier Ltd.
Keywords: Chirped Gaussian pulse beam Oceanic turbulence Laser propagation
1. Introduction In the recent years, the spectral properties of optical beams propagating in the different channels have been widely studied [1–5]. For examples, the polarization properties of partially coherent pulsed propagating in fibers are studied [1]. The propagation properties of partially coherent electromagnetic pulsed beams propagation in free space are analysed by Ding et al. [2,3]. The spectral properties of chirped Gaussian pulsed beams propagating through turbulent atmosphere have been studied, and the influence of structure constant of the relative index Cn2, chirp parameter C and pulse duration T on the structure are analysed by Ji et al. [4]. The anomalous spectral behaviors of chirped Gaussian pulses beam diffracted by an annular aperture propagating in turbulent atmosphere are studied by Yang and Duan [5]. And with the development of laser technology, laser beams propagation in oceanic turbulence has obtained much attention. The Influences of temperature and salinity fluctuations propagation behavior of partially coherent beams in oceanic turbulence are firstly given by Lu et al. [6]. The propagation properties of a short light in oceanic flow is studied based on the Monte Carlo simulation by Bogucki et al. [7]. The polarization evolution properties of stochastic beams propagating in turbulent clear-water ocean based on the extended Huygens-Fresnel principle and the unified theory of coherence and polarization for light are investigate by Korotkova et al. [8], the parameters w = − 2.5 , xT = 10−6 , ε = 10−7 of oceanic turbulence are used in the numerical analysis. The radiative transfer in ocean n
Corresponding author. E-mail address: [email protected] (D. Liu).
http://dx.doi.org/10.1016/j.optlastec.2016.02.019 0030-3992/& 2016 Published by Elsevier Ltd.
turbulence is investigated by Xu et al. [9]. The influences of oceanic turbulence on the evolution properties of Gaussian Schell-model vortex beams in oceanic turbulence are given by Huang et al. [10]. The propagation properties of stochastic electromagnetic vortex beam propagation in oceanic turbulence based on the extended Huygens-Fresnel principle are investigated by Xu and Zhao [11]. The regions of spreading properties for Gaussian array beam propagating in oceanic turbulence are studied by Tang and Zhao [12]. And, Baykal has studied the influence of oceanic turbulence on the intensity fluctuations of multimode laser beams [13]. Lu et al. have studied the average intensity of Gaussian array beams propagation in oceanic turbulence [14]. However, to the best of our knowledge, there has been no analysis about the spectral properties of chirped Gaussian pulsed beam propagation in oceanic turbulence. In this work, we mainly investigate the spectral properties of chirped Gaussian pulsed beam propagation in oceanic turbulence.
2. Theory analysis In the Cartesian coordinate system, the Gaussian pulsed beam in the space-time domain propagating in the half space z 40 at the plane z ¼0 can be expressed as [15]:
⎛ x 2 + y2 ⎞ ⎡ (1 + iC ) t 2 ⎤ 0 0⎟ ⎥ exp ( −iω 0 t ) E0 ( r0, 0, t ) = exp ⎜⎜ − ⎟ exp ⎢ − 2 ⎣ ⎦ w 2T 2 ⎝ ⎠
(1)
where ω0 is the central frequency, T is the pulse duration and C is the chirp parameter, w is the waist width which is independent of the frequency.
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
77
Fig. 1. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence (a) On axis (0, 0), (b) off axis (6 cm, 6 cm).
The spectral component with the angular frequency ω of the Gaussian pulsed beam can be obtained by Fourier transform.
E ( r0, 0, ω) =
1 2π
∞
∫−∞ E ( r0, 0, t ) exp (−iωt ) dt
(2)
(
)
M=
k2 4π 2z 2
+∞
∭ ∫−∞
(3)
W0 ( r10, r20, 0)
×
π 2k 2z 3
∫0
∞
dκκΦ (κ )
Φ (κ ) = 0.388 × 10−8ε−11/3 ⎡⎣ 1 + 2.35 (κη)2/3⎤⎦ f ( κ , ς , χT )
(6)
(7)
where ε is the rate of dissipation of dissipation of turbulent kinetic energy per unit mass of fluid which may vary in rage from 10−1m2s−3 to 10−10m2s−3. η = 10−3 is the Kolmogorov micro scale (inner scale), and
f ( κ , ς , χT ) =
⎡ ik ⎤ ik 2 × exp ⎢ − ( r−r10) + ( r−r20)2⎥⎦ ⎣ 2z 2z exp ⎡⎣ ψ ( r, r10) + ψ * ( r, r20) ⎤⎦ dr10dr20
(5)
and
According to the extended Huygens-Fresnel principle, the spectral density function of laser beams propagating in oceanic turbulence can be expressed as [12–14]:
W (r, r, z ) =
exp ⎡⎣ ψ ( r, r10) + ψ * ( r, r20) ⎤⎦ = exp ⎡⎣ −M ( r10 − r20)2⎤⎦ with
By substituting Eq. (1) into Eq. (2), we can obtain.
⎛ x 2 + y2 ⎞ T2 (1 − iC ) 1 0 0⎟ E0 ( r0, 0, ω) = exp ⎜⎜ − ⎟ 2π w2 ⎠ 1 + C2 ⎝ ⎤ ⎡ T2 (1 − iC ) 2 ω − ω0 ) ⎥ × exp ⎢ − ( ⎥ ⎢ 2 1 + C2 ⎦ ⎣
the source plane. The ensemble average in Eq. (4) can be expressed as [12–14]:
χT ⎡ 2 ⎤ ⎣ ς exp ( −AT δ ) + exp ( −AS δ ) − 2ς exp ( −ATS δ ) ⎦
ς2
(8)
With χT is the rate of dissipation of mean square temperature
(4)
where k = 2π /λ is the wave number; ψ is the solution to the Rytov method that represents the random part of the complex phase; the asterisk denotes the complex conjugation; r = (x, y ) and r0 = (x0, y0 ) are the position vectors at the output plane z and the input plane z¼ 0, respectively; W0 (r10, r20, 0) = E0 (r0, 0, ω) E0* (r0, 0, ω) is the cross-spectral density function of chirped Gaussian pulsed beam at
taking value in the range from 10−4K2s−1 to 10−10K2s−1, − 2 − 4 AT = 1.863 × 10 , AS = 1.9 × 10 , ATS = 9.41 × 10−3, 2 4/3 δ = 8.284 (κη) + 12.978 (κη) , ς is the relative strength of temperature and salinity fluctuations, which in the ocean waters can vary in the interval [−5, 0]. On substituting Eq. (3) into Eq. (4), we can obtain the spectrum of a chirped Gaussian pulsed beam propagating in oceanic turbulence.
78
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
Fig. 2. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence for the different C (a) On axis (0, 0), (b) off axis (3 cm, 3 cm).
S (x, y , z, ω) = W (x, y , z ) =
⎡ ⎤ k2 T2 T2 2 exp ⎢ − ( ω − ω0 ) ⎥ 2 ⎣ 1 + C2 ⎦ 8abπz 1 + C2 2 ⎤ ⎡ 1⎛ k ⎞ × exp ⎢ − ⎜ ⎟ x2 + y2 ⎥ ⎥⎦ ⎢⎣ a ⎝ 2z ⎠
(
)
⎤ ⎡ k2 ⎛ M⎞ ⎜ 1 − ⎟ x 2 + y2 ⎥ × exp ⎢ − 2⎝ ⎠ a ⎦ ⎣ 8bz
(
)
(9)
with.
a=
1 ik − +M 2z w2
(10a)
b=
ik 1 M2 + +M+ 2 2z a w
(10b)
By using Eqs. (9) and (10), we can directly obtain the spectral properties of chirped Gaussian pulsed beam propagating in oceanic turbulence.
3. Numerical examples and analysis In this section, numerical calculations are made to illustrate the influence of oceanic turbulence on the spectral behaviors of chirped Gaussian pulsed beams. The following calculation parameters are used in the calculations unless the other values of parameters are specified in the caption: λ 0 = 532nm, w = 5mm,
C = 2, T = 5fs , ς = − 2.5, χT = 10−9K2/s , ε = 10−6m2s−1, ω0 = 2πc /λ 0 and c = 3 × 108m /s . Fig. 1 gives the normalized spectra of chirped Gaussian pulsed beam propagating in oceanic turbulence for the different ς at the propagation distance z = 700m. As can be seen, the spectra of chirped Gaussian pulsed beam at the different position has the different evolution properties, the on-axis spectra is blue-shift, while the off-axis spectra is red-shift with the decreasing of ς . Figs. 2 and 3 show the normalized spectra of chirped Gaussian pulsed beam propagating in oceanic turbulence at the propagation distance z = 700m for the different C and T. It is found that the onaxis spectra of chirped Gaussian pulsed beam is blue-shift with the increasing of C or the decreasing of T; the off-axis spectra of Gaussian beam is red-shift with the increasing of C or the decreasing of T. Fig. 4 gives the relative spectra shift of chirped Gaussian pulsed beams versus the position parameter r = x 2 + y2 at the propagation distance z = 500m for the different C and T. It is shown that the on-axis spectra is blue shift, and the blue shift become bigger with the increasing of C or the decreasing of T; it is also found that blue shift will gradually become red shift with the increasing of position parameter r . Fig. 5 gives the on-axis and off-axis relative spectral shift of chirped Gaussian pulsed beams propagating in oceanic turbulence for the different propagation distance. Form Fig. 5, we can see that the on-axis spectral shift will firstly increase, and then gradually decrease with the propagation distance increasing; while the offaxis spectral shift appear a spectral switch at the certain propagation distance. But the on-axis and off-axis spectral shifts are
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
Fig. 3. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence for the different T (a) On axis (0, 0), (b) off axis (3 cm, 3 cm).
Fig. 4. The relative spectral shift of chirped Gaussian pulsed beams in oceanic turbulence versus the position parameter r .
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80
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
Fig. 5. The relative spectral shift of chirped Gaussian pulsed beams in oceanic turbulence for the different propagation distance (a) on-axis (0, 0), (b) off-axis ( r = 1cm ).
Fig. 6. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence for the different xT and ε .
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
much smaller than the spectral shift due to the influence of position parameter r (Fig. 6), and the larger influence of the position parameter which makes no sense. Fig. 6 gives the normalized spectra of chirped Gaussian pulsed beam propagating in oceanic turbulence at the propagation distance z = 500m and r = 8cm for the different xT and ε . It can be seen that the off-axis spectra of Gaussian pulsed beam is the blue shift with the xT increasing or the ε decreasing.
4. Conclusions In this paper, the spectral properties of chirped Gaussian pulsed beam propagating in oceanic turbulence is analysed based on the extended Huygens-Fresnel principle. It is found that on-axis spectra of the Gaussian pulsed beam is blue-shift with the increasing of the chirp parameter C or the decreasing of the pulse duration T; while the off-axis spectra of Gaussian pulsed beam is red-shift with the increasing of the chirp parameter C or the decreasing of the pulse duration T, and the off-axis spectra is the blue-shift with the xT (the rate of dissipation of mean square temperature taking value) increasing, and with the ς (the relative strength of temperature and salinity fluctuations) or ε (the rate of dissipation of dissipation of turbulent kinetic energy per unit mass of fluid) decreasing; the relative on-axis spectra blue shift becomes larger with the increasing of C or the decreasing of T, but the blue shift will gradually become red shift with the increasing of position parameter r , and the on-axis and off-axis spectral shifts affected by the propagation distance are much smaller than the spectral shift introduced by position parameter.
Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (Grants nos. 3132015152, 3132015233, 3132014337 and 3132014327), National Natural Science Foundation
81
of China (11404048 and 11375034), the program for Liaoning Educational Committee (L2015071).
References [1] W. Huang, S. Ponomarenko, M. Cada, G. Agrawal, Polarization changes of partially coherent pulses propagating in optical fibers, J. Opt. Soc. Am. A 24 (2007) 3063–3068. [2] C. Ding, L. Pan, B. Lü, Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams, New J. Phys. 11 (2009) 083001. [3] C. Ding, L. Pan, B. Lü, Changes in the spectrum, polarization, and coherence of stochastic spatially and spectrally partially coherent electromagnetic J0-correlated Schell-model pulsed beams propagating in free space, Eur. Phys. J. D 59 (2010) 329–336. [4] X. Ji, E. Zhang, B. Lü, Spectral properties of chirped Gaussian pulsed beams propagating through the turbulent atmosphere, J. Mod. Opt. 54 (2007) 541–553. [5] Y. Yang, K. Duan, Spectral anomalies of chirped Gaussian pulses from An annular aperture in the turbulent atmosphere, Opt. Eng. 48 (2009) 058001. [6] W. Lu, L. Liu, J. Sun, Influence of temperature and salinity fluctuations propagation behavior of partially coherent beams in oceanic turbulence, J. Opt. A 8 (2006) 1052–1058. [7] D. Bogucki, M. Piskozub, G. Spiers, Monte Carlo simulation of propagation of a short light beam through turbulent oceanic flow, Opt. Express 15 (13) (2007) 988–13,996. [8] O. Korotkova, N. Farwell, Effect of oceanic turbulence on polarization of stochastic beams, Opt. Commun. 284 (2011) 1740–1746. [9] Z. Xu, X. Guo, L. Shen, D. Yue, Radiative transfer in ocean turbulence and its effect on underwater light field, J. Geophys. Res. 117 (2012) C00H18, http://dx. doi.org/10.1029/2011JC007351. [10] Y. Huang, B. Zhang, Z. Gao, G. Zhao, Z. Duan, Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence, Opt. Express 22 (2014) 17723–17734. [11] J. Xu, D. Zhao, Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence, Opt. Laser Technol. 57 (2014) 189–193. [12] M. Tang, D. Zhao, Regions of spreading of Gaussian array beams propagating through oceanic turbulence, Appl. Opt. 54 (2015) 3407–3411. [13] Y. Baykal, Intensity fluctuations of multimode laser beam in underwater medium, J. Opt. Soc. Am. A 32 (2015) 593598. [14] L. Lu, Z. Wang, J. Zhang, P. Zhang, C. Qiao, C. Fan, X. Ji, Average intensity of M N Gaussian array beams in oceanic turbulence, Appl. Opt. 54 (2015) 7500–7507. [15] G. Agrawal, Spectrum-induced changes in diffraction of pulsed optical beams, Opt. Commun. 157 (1998) 52–56.
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
The influence of oceanic turbulence on the spectral properties of chirped Gaussian pulsed beam Dajun Liu a,n, Yaochuan Wang a, Guiqiu Wang a, Hongming Yin a, Jinren Wang b a b
Department of Physics, Dalian Maritime University, Dalian 116026, China Sifang College, Shijiazhuang Tiedao University, Shijiazhuang 051132, China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 October 2015 Received in revised form 22 January 2016 Accepted 25 February 2016
Based on the extended Huygens-Fresnel principle, the spectral behaviors of a chirped Gaussian pulsed beam propagating in oceanic turbulence are illustrated. The influence of the parameters of oceanic turbulence (the rate of dissipation of turbulent kinetic energy per unit mass of fluid, rate of dissipation of mean-square temperature, relative strength of temperature and salinity fluctuations), relative position parameter and propagation distance on the spectra shift is analysed and given by numerical examples. The research results have the potential application in underwater wireless laser communication and remote sensing. & 2016 Published by Elsevier Ltd.
Keywords: Chirped Gaussian pulse beam Oceanic turbulence Laser propagation
1. Introduction In the recent years, the spectral properties of optical beams propagating in the different channels have been widely studied [1–5]. For examples, the polarization properties of partially coherent pulsed propagating in fibers are studied [1]. The propagation properties of partially coherent electromagnetic pulsed beams propagation in free space are analysed by Ding et al. [2,3]. The spectral properties of chirped Gaussian pulsed beams propagating through turbulent atmosphere have been studied, and the influence of structure constant of the relative index Cn2, chirp parameter C and pulse duration T on the structure are analysed by Ji et al. [4]. The anomalous spectral behaviors of chirped Gaussian pulses beam diffracted by an annular aperture propagating in turbulent atmosphere are studied by Yang and Duan [5]. And with the development of laser technology, laser beams propagation in oceanic turbulence has obtained much attention. The Influences of temperature and salinity fluctuations propagation behavior of partially coherent beams in oceanic turbulence are firstly given by Lu et al. [6]. The propagation properties of a short light in oceanic flow is studied based on the Monte Carlo simulation by Bogucki et al. [7]. The polarization evolution properties of stochastic beams propagating in turbulent clear-water ocean based on the extended Huygens-Fresnel principle and the unified theory of coherence and polarization for light are investigate by Korotkova et al. [8], the parameters w = − 2.5 , xT = 10−6 , ε = 10−7 of oceanic turbulence are used in the numerical analysis. The radiative transfer in ocean n
Corresponding author. E-mail address: [email protected] (D. Liu).
http://dx.doi.org/10.1016/j.optlastec.2016.02.019 0030-3992/& 2016 Published by Elsevier Ltd.
turbulence is investigated by Xu et al. [9]. The influences of oceanic turbulence on the evolution properties of Gaussian Schell-model vortex beams in oceanic turbulence are given by Huang et al. [10]. The propagation properties of stochastic electromagnetic vortex beam propagation in oceanic turbulence based on the extended Huygens-Fresnel principle are investigated by Xu and Zhao [11]. The regions of spreading properties for Gaussian array beam propagating in oceanic turbulence are studied by Tang and Zhao [12]. And, Baykal has studied the influence of oceanic turbulence on the intensity fluctuations of multimode laser beams [13]. Lu et al. have studied the average intensity of Gaussian array beams propagation in oceanic turbulence [14]. However, to the best of our knowledge, there has been no analysis about the spectral properties of chirped Gaussian pulsed beam propagation in oceanic turbulence. In this work, we mainly investigate the spectral properties of chirped Gaussian pulsed beam propagation in oceanic turbulence.
2. Theory analysis In the Cartesian coordinate system, the Gaussian pulsed beam in the space-time domain propagating in the half space z 40 at the plane z ¼0 can be expressed as [15]:
⎛ x 2 + y2 ⎞ ⎡ (1 + iC ) t 2 ⎤ 0 0⎟ ⎥ exp ( −iω 0 t ) E0 ( r0, 0, t ) = exp ⎜⎜ − ⎟ exp ⎢ − 2 ⎣ ⎦ w 2T 2 ⎝ ⎠
(1)
where ω0 is the central frequency, T is the pulse duration and C is the chirp parameter, w is the waist width which is independent of the frequency.
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
77
Fig. 1. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence (a) On axis (0, 0), (b) off axis (6 cm, 6 cm).
The spectral component with the angular frequency ω of the Gaussian pulsed beam can be obtained by Fourier transform.
E ( r0, 0, ω) =
1 2π
∞
∫−∞ E ( r0, 0, t ) exp (−iωt ) dt
(2)
(
)
M=
k2 4π 2z 2
+∞
∭ ∫−∞
(3)
W0 ( r10, r20, 0)
×
π 2k 2z 3
∫0
∞
dκκΦ (κ )
Φ (κ ) = 0.388 × 10−8ε−11/3 ⎡⎣ 1 + 2.35 (κη)2/3⎤⎦ f ( κ , ς , χT )
(6)
(7)
where ε is the rate of dissipation of dissipation of turbulent kinetic energy per unit mass of fluid which may vary in rage from 10−1m2s−3 to 10−10m2s−3. η = 10−3 is the Kolmogorov micro scale (inner scale), and
f ( κ , ς , χT ) =
⎡ ik ⎤ ik 2 × exp ⎢ − ( r−r10) + ( r−r20)2⎥⎦ ⎣ 2z 2z exp ⎡⎣ ψ ( r, r10) + ψ * ( r, r20) ⎤⎦ dr10dr20
(5)
and
According to the extended Huygens-Fresnel principle, the spectral density function of laser beams propagating in oceanic turbulence can be expressed as [12–14]:
W (r, r, z ) =
exp ⎡⎣ ψ ( r, r10) + ψ * ( r, r20) ⎤⎦ = exp ⎡⎣ −M ( r10 − r20)2⎤⎦ with
By substituting Eq. (1) into Eq. (2), we can obtain.
⎛ x 2 + y2 ⎞ T2 (1 − iC ) 1 0 0⎟ E0 ( r0, 0, ω) = exp ⎜⎜ − ⎟ 2π w2 ⎠ 1 + C2 ⎝ ⎤ ⎡ T2 (1 − iC ) 2 ω − ω0 ) ⎥ × exp ⎢ − ( ⎥ ⎢ 2 1 + C2 ⎦ ⎣
the source plane. The ensemble average in Eq. (4) can be expressed as [12–14]:
χT ⎡ 2 ⎤ ⎣ ς exp ( −AT δ ) + exp ( −AS δ ) − 2ς exp ( −ATS δ ) ⎦
ς2
(8)
With χT is the rate of dissipation of mean square temperature
(4)
where k = 2π /λ is the wave number; ψ is the solution to the Rytov method that represents the random part of the complex phase; the asterisk denotes the complex conjugation; r = (x, y ) and r0 = (x0, y0 ) are the position vectors at the output plane z and the input plane z¼ 0, respectively; W0 (r10, r20, 0) = E0 (r0, 0, ω) E0* (r0, 0, ω) is the cross-spectral density function of chirped Gaussian pulsed beam at
taking value in the range from 10−4K2s−1 to 10−10K2s−1, − 2 − 4 AT = 1.863 × 10 , AS = 1.9 × 10 , ATS = 9.41 × 10−3, 2 4/3 δ = 8.284 (κη) + 12.978 (κη) , ς is the relative strength of temperature and salinity fluctuations, which in the ocean waters can vary in the interval [−5, 0]. On substituting Eq. (3) into Eq. (4), we can obtain the spectrum of a chirped Gaussian pulsed beam propagating in oceanic turbulence.
78
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
Fig. 2. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence for the different C (a) On axis (0, 0), (b) off axis (3 cm, 3 cm).
S (x, y , z, ω) = W (x, y , z ) =
⎡ ⎤ k2 T2 T2 2 exp ⎢ − ( ω − ω0 ) ⎥ 2 ⎣ 1 + C2 ⎦ 8abπz 1 + C2 2 ⎤ ⎡ 1⎛ k ⎞ × exp ⎢ − ⎜ ⎟ x2 + y2 ⎥ ⎥⎦ ⎢⎣ a ⎝ 2z ⎠
(
)
⎤ ⎡ k2 ⎛ M⎞ ⎜ 1 − ⎟ x 2 + y2 ⎥ × exp ⎢ − 2⎝ ⎠ a ⎦ ⎣ 8bz
(
)
(9)
with.
a=
1 ik − +M 2z w2
(10a)
b=
ik 1 M2 + +M+ 2 2z a w
(10b)
By using Eqs. (9) and (10), we can directly obtain the spectral properties of chirped Gaussian pulsed beam propagating in oceanic turbulence.
3. Numerical examples and analysis In this section, numerical calculations are made to illustrate the influence of oceanic turbulence on the spectral behaviors of chirped Gaussian pulsed beams. The following calculation parameters are used in the calculations unless the other values of parameters are specified in the caption: λ 0 = 532nm, w = 5mm,
C = 2, T = 5fs , ς = − 2.5, χT = 10−9K2/s , ε = 10−6m2s−1, ω0 = 2πc /λ 0 and c = 3 × 108m /s . Fig. 1 gives the normalized spectra of chirped Gaussian pulsed beam propagating in oceanic turbulence for the different ς at the propagation distance z = 700m. As can be seen, the spectra of chirped Gaussian pulsed beam at the different position has the different evolution properties, the on-axis spectra is blue-shift, while the off-axis spectra is red-shift with the decreasing of ς . Figs. 2 and 3 show the normalized spectra of chirped Gaussian pulsed beam propagating in oceanic turbulence at the propagation distance z = 700m for the different C and T. It is found that the onaxis spectra of chirped Gaussian pulsed beam is blue-shift with the increasing of C or the decreasing of T; the off-axis spectra of Gaussian beam is red-shift with the increasing of C or the decreasing of T. Fig. 4 gives the relative spectra shift of chirped Gaussian pulsed beams versus the position parameter r = x 2 + y2 at the propagation distance z = 500m for the different C and T. It is shown that the on-axis spectra is blue shift, and the blue shift become bigger with the increasing of C or the decreasing of T; it is also found that blue shift will gradually become red shift with the increasing of position parameter r . Fig. 5 gives the on-axis and off-axis relative spectral shift of chirped Gaussian pulsed beams propagating in oceanic turbulence for the different propagation distance. Form Fig. 5, we can see that the on-axis spectral shift will firstly increase, and then gradually decrease with the propagation distance increasing; while the offaxis spectral shift appear a spectral switch at the certain propagation distance. But the on-axis and off-axis spectral shifts are
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
Fig. 3. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence for the different T (a) On axis (0, 0), (b) off axis (3 cm, 3 cm).
Fig. 4. The relative spectral shift of chirped Gaussian pulsed beams in oceanic turbulence versus the position parameter r .
79
80
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
Fig. 5. The relative spectral shift of chirped Gaussian pulsed beams in oceanic turbulence for the different propagation distance (a) on-axis (0, 0), (b) off-axis ( r = 1cm ).
Fig. 6. Normalized spectra of chirped Gaussian pulsed beam in oceanic turbulence for the different xT and ε .
D. Liu et al. / Optics & Laser Technology 82 (2016) 76–81
much smaller than the spectral shift due to the influence of position parameter r (Fig. 6), and the larger influence of the position parameter which makes no sense. Fig. 6 gives the normalized spectra of chirped Gaussian pulsed beam propagating in oceanic turbulence at the propagation distance z = 500m and r = 8cm for the different xT and ε . It can be seen that the off-axis spectra of Gaussian pulsed beam is the blue shift with the xT increasing or the ε decreasing.
4. Conclusions In this paper, the spectral properties of chirped Gaussian pulsed beam propagating in oceanic turbulence is analysed based on the extended Huygens-Fresnel principle. It is found that on-axis spectra of the Gaussian pulsed beam is blue-shift with the increasing of the chirp parameter C or the decreasing of the pulse duration T; while the off-axis spectra of Gaussian pulsed beam is red-shift with the increasing of the chirp parameter C or the decreasing of the pulse duration T, and the off-axis spectra is the blue-shift with the xT (the rate of dissipation of mean square temperature taking value) increasing, and with the ς (the relative strength of temperature and salinity fluctuations) or ε (the rate of dissipation of dissipation of turbulent kinetic energy per unit mass of fluid) decreasing; the relative on-axis spectra blue shift becomes larger with the increasing of C or the decreasing of T, but the blue shift will gradually become red shift with the increasing of position parameter r , and the on-axis and off-axis spectral shifts affected by the propagation distance are much smaller than the spectral shift introduced by position parameter.
Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (Grants nos. 3132015152, 3132015233, 3132014337 and 3132014327), National Natural Science Foundation
81
of China (11404048 and 11375034), the program for Liaoning Educational Committee (L2015071).
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