Abstract

The thermal blooming effect of laser beams propagating through seawater is studied in detail by using the numerical simulation method. It is found that an increase of the salinity in the seawater causes the more severe thermal blooming. As compared with the wavelength, the absorption coefficient is the main factor that dominates the thermal blooming because the absorption coefficient is very high in the seawater. In the seawater the thermal blooming becomes more severe for the wavelength corresponding to the higher absorption coefficient. Furthermore, both the behavior of the thermal blooming effect and the main factor dominating the thermal blooming effect in the shallow sea region are different from those in the deep sea region. In the shallow sea region, the dependence of the thermal blooming on the depth is not monotonic as the time increases. However, in the deep sea region, the thermal blooming effect becomes more severe monotonously as the depth increases. The physical explanations for the main results obtained in this paper are presented.

© 2017 Optical Society of America

1. Introduction

When a laser beam propagates through a medium, a fraction of the beam energy is absorbed by the medium. This absorbed energy heats the medium and alters the index of refraction of the path, and leads to a distortion of the beam itself, which is called the thermal blooming effect [1]. Until now, many works have been carried out concerning the thermal blooming effect of laser beams propagating through the atmosphere. For example, Fleck et al. studied the thermal blooming by using the four-dimensional (4D) computer code of the time-dependent propagation of high power laser beams through the atmosphere [2]. Fortes et al. estimated the thermal blooming degradation in adaptive-optics [3]. Gebhardt presented an overview of thermal blooming until 1990 [4]. Barchers et al. provided the linear analysis of thermal blooming compensation instabilities in laser propagation [5]. Zandt et al. presented an enhanced and fast-running scaling law model of thermal blooming and turbulence effects on high energy laser propagation [6]. Recently, we studied the effect of thermal blooming of an Airy beam propagating through the atmosphere, and found that an Airy beam can’t retain its shape and the structure due to thermal blooming effect [7].

Since recently the interest in active optical underwater communications, imaging and sensing appeared [8, 9], it has become more important to deeply understand how the seawater affects light beam propagation. Numerous works have been carried out concerning the oceanic turbulence effects on the light beam characteristics [10–15] and the imaging [16–19]. However, the thermal blooming is one of the main factors that affect the characteristics of laser beams propagating through the seawater. Until now, studies were only restricted to the thermal blooming effect of laser beams propagating through a piece of liquid sample [20, 21], but the thermal blooming effect of laser beams propagating through the seawater hasn’t been studied.

There are some differences between characteristics of the seawater and the atmosphere. Firstly, the salinity exists in the seawater. Thus, an interesting question arises: How does the salinity of the seawater affect the thermal blooming? Secondly, in the seawater the absorption coefficient is much higher than that in the atmosphere, and the absorption coefficient varies rapidly with the wavelength. If all parameters are constant except the wavelength, the thermal blooming effect becomes more severe as the wavelength decreases [22]. Thus, a question arises: Is the law the same when the absorption coefficient changes with the wavelength in the seawater? Finally, at different depth of the seawater, the values of the temperature, the salinity, the pressure, the density and the current velocity are different. Thus, another interesting question arises: How does the thermal blooming change with the depth of seawater? In this paper, the influence of the salinity and the depth of seawater and the wavelength on the thermal blooming is studied in detail by using the 4D computer code of the time-dependent propagation of laser beams through the seawater. In addition, the physical explanations for the main results obtained in this paper are also given.

2. Theoretical model

Under the paraxial approximation, Maxwell’s wave equation is expressed as [2]

2ikEz=2E+k2(n2n021)E,
where2=2/x2+2/y2is the transverse Laplace operator, E is the envelope of the electric field, k is the wave-number related to the wave lengthλbyk=2π/λ, n0 is the liquid refractive index before perturbation, n is the liquid refraction index with perturbation. When a laser beam propagates through the liquid, it can be expressed as [23]
n2n021=(n021)(n02+2)3n02ρ1ρ0,
whereρ0is the liquid density before perturbation and its corresponding first-order small quantity isρ1. According to [24], a simpler expression can be approximately written as
(n021)(n02+2)3n022nTβ,
where β is the thermal expansion coefficient, nT is the thermo-optic coefficient, and nT=dn/dT(T is the temperature of liquid).

The hydrodynamic equation under the isobaric approximation is expressed as [25]

ρ1t+vρ1=βαCpI,
whereCpis the isobaric specific heat capacity, α is absorption coefficient, v is the current velocity, and I is the intensity. It is noted that the thermal conduction term in Eq. (4) is ignored because the thermal conduction and the viscosity may be neglected in water.

Assume the incident field at the plane z = 0 has the Gaussian beam profile which can be written as

E=Pπw02exp(x2+y22w02),
where P is the input beam power, w0 is the beam radius.

Considering beam propagation along z axis, letEnbe the complete solution to Eq. (1) at the plane z = zn, the solution to Eq. (1) at the plane zn+1 = zn + Δz can be obtained by using the second-order accuracy of the symmetrically split operator (see appendix A in [2]), i.e.,

En+1=exp(i4kΔz2)exp(is)exp(i4kΔz2)En,
wheres=(k/2)znzn+1(n2/n021)dzis the phase modulation due to thermal blooming. Equation (6) indicates that the propagating field from zn to zn+1 can be divided into three steps: a vacuum propagation of the field over a distanceΔz/2, an incrementing of the phase due to thermal blooming over a distanceΔz, a vacuum propagation of the field over a distanceΔz/2. In this paper, we design a 4D computer code of the time-dependent propagation of laser beams through the seawater by using the multi-phase screen method [2].

The dimensionless quantity called the distortion parameter N of Gaussian beams propagating through the liquid is derived by Gebhardt et al., i.e., [20]

N=(2nTI0zn0ρ0Cpvw0)[1(1eαz)αz],
where I0 is the peak intensity of the incident laser beam. The distortion parameter N can describe the thermal distortion effect of a moving medium on a collimated Gaussian laser beam. The higher distortion parameter N means the more severe thermal blooming.

In this paper, several parameters are adopted to characterize the thermal blooming effect of laser beams propagating through the seawater, which are given as follows:

(i) Intensity Strehl ratio (SR)

The intensity Strehl ratio is defined as [26]

SR=ImaxI0max,
where Imax and I0max are the peak intensity with and without thermal blooming, respectively. It is clear thatSR1, and the lower value of SR is, the more the peak intensity decreases.

(ii) Energy Strehl ratio (SRE)

The energy Strehl ratio is defined as the ratio of real beam power to ideal beam power within a given bucket, i.e., [27]

SRE=x2+y2w2I(x,y,z)dxdyx2+y2w2Ivacuum(x,y,z)dxdy,
where w is the bucket half width chosen as the mean-squared beam width in vacuum, Ivacuum is the intensity in vacuum. It is obvious thatSRE1, and the lower value of SRE means the energy focusability is more affected by the thermal blooming.

(iii) Propagation efficiency

The propagation efficiency indicates what fraction of the total beam power is within a given bucket at a propagation distance z, which is defined as [28],

η=σI(x,y,z)dxdyI(x,y,0)dxdy,
where σ is bucket area chosen (e.g., target area). In general, σ is taken as 100 cm2 and the corresponding radius is 5.6 cm [28].

(iv) Centroid position and Mean-squared beam width

The centroid position and the mean-squared beam width are defined as [29]

j¯=jI(x,y,z)dxdyI(x,y,z)dxdy,
and
wj2=4(jj¯)2I(x,y,z)dxdyI(x,y,z)dxdy,
respectively, where j = x and y,x¯andy¯denote the centroid position of a laser beam in the x direction and y direction respectively,wxandwydenote the mean-squared beam width of a laser beam in the x direction and y direction respectively. The higher the absolute value|j¯|is, the further away the centroid position is from propagation z-axis.

3. Numerical calculation results and analysis

In this paper, thermal blooming effect of laser beams propagating through the seawater is studied by using the numerical simulation method. Unless specified, the calculation parameters are taken as w0 = 0.03m, T=20°C, v = 0.1m/s along x-axis, the seawater salinityS=35000, andλ=450nmrelated toα=0.0145/m [30]. It is noted that the salinity adopted in this paper is the absolute salinity, which is defined as the mass fraction of dissolved material in seawater [31]. In sections 3.1 and 3.2, the influence of wavelength and the seawater salinity on the thermal blooming effect is investigated respectively, where the values of parameters nT, ρ,Cpand β at the sea surface are adopted, e.g., nT is obtained by using Eq. (3) in [32], ρ is obtained by using Eq. (10) in [33], andCpand β are obtained by using Tab. 17 and Tab. 22 in [34] respectively. In section 3.3, the influence of the seawater depth on the thermal blooming effect is examined, where nT, ρ,Cpand β change with the seawater depth h, e.g., nT can be obtained by using Eq. (16) in [35],Cpis obtained by using Eq. (26) in [36], ρ can be obtained from Fig. 1.12 in [37], andβ=(1/V)(V/T)Pcan be obtained from [38], with V being the liquid volume.

3.1 Influence of the wavelength on the thermal blooming effect

It is known that in the seawater the absorption coefficient is very high, and it changes rapidly with the wavelength. In this paper, the blue-green light is considered because its absorption coefficient is relatively low and it usually used in practice. Smith and Baker used measured values of the diffuse attenuation function from very clear water (e.g., Grater Lake, Oregon, USA and the Sargasso Sea) to estimate the absorption coefficient of pure water [30]. There is strong evidence that there is no significant (<10%) difference between absorption coefficient for freshwater and saltwater for wavelengths longer than ~375 nm [39,40]. In addition, according to [41], the influence of temperature on the absorption coefficient is also small. In this paper, the absorption coefficient of saltwater approximately adopts that of pure water given by Smith and Baker. From Table 1 in [30], we have the values of the absorption coefficient α = 0.022/m, 0.017/m, 0.0145/m, 0.0196/m and 0.0357/m when the wavelength λ = 380nm, 400nm, 450nm, 490nm and 510nm respectively, i.e., the dependence of α onλis not monotonic, and the minimum value of α is 0.0145/m corresponding toλ = 450nm.

For different values of wavelength, the three-dimensional (3D) intensity distribution and its contour lines, the intensity Strehl ratio SR, the energy Strehl ratio SRE, the propagation efficiency η and the centroid positionx¯are shown in Fig. 1, Fig. 2, Fig. 3 and Fig. 4, respectively. Meanwhile, the change of the absorption coefficient versus the wavelength is considered in Figs. 14. From Figs. 1–4 it can be seen that the dependence of the thermal blooming on the wavelength is not monotonic. Whenλ=450nmcorresponding to the minimum value of the absorption coefficient, the thermal blooming effect is smallest, e.g., the intensity distortion is smallest (see Fig. 1(c)), the values of SR, SRE, η are largest (see Figs. 2 and 3), and the absolute value|x¯|is smallest (see Fig. 4). In the seawater, the thermal blooming becomes more severe for the wavelength which corresponds to the higher absorption coefficient. The thermal blooming effect becomes more severe as the wavelength decreases if all parameters are constant except the wavelength [22], while the thermal blooming effect becomes more severe as the absorption coefficient increases regardless of the wavelength. Furthermore, the absorption coefficient is very high in the seawater. This is the physical reason why in the seawater the absorption coefficient is the main factor that dominates the thermal blooming as compared with the wavelength. In addition, the propagation efficiency η decreases rapidly as the propagation distance increases because the absorption coefficient is very high in the seawater (see Fig. 3(a)). It is mentioned that the absorption coefficient in the seawater is much higher than that in the atmosphere, which results in the thermal blooming occurring at lower power levels in the seawater than that in the atmosphere.

 

Fig. 1 3D intensity distribution and contour lines for different values of the wavelength λ. t = 1 s, P = 3kW, z = 100m.

Download Full Size | PPT Slide | PDF

 

Fig. 2 (a) SR and (b) SRE versus time t for different values of the wavelength λ. P = 3kW, z = 100m.

Download Full Size | PPT Slide | PDF

 

Fig. 3 η versus (a) the propagation distance z and (b) time t for different values of the wavelength λ. P = 3kW.

Download Full Size | PPT Slide | PDF

 

Fig. 4 x¯versus time t for different values of the wavelength λ. P = 3kW, z = 100m.

Download Full Size | PPT Slide | PDF

3.2 Influence of the seawater salinity on the thermal blooming effect

For different values of salinity, the intensity Strehl ratio SR and the energy Strehl ratio SRE, the propagation efficiency η and the centroid positionx¯are shown in Figs. 5 and 6, respectively. It can be seen that the thermal blooming effect becomes more severe with increasing salinity, e.g., as the salinity increases, SRE and η decrease (see Figs. 5(b) and 6(a)), and the absolute value|x|increases (see Fig. 6(b)). The physical reason is that nT andCpdecrease as the salinity increases. When the thermal blooming effect appears, the value of nT is negative. The higher absolute value|nT|denotes the larger change of the refractive index for the same temperature change, namely, the thermal blooming effect becomes more severe. On the other hand,Cpis defined as the heat required raising the temperature by one unit (1K) per unit mass (1kg) of medium under the isobaric condition. The lowerCpmeans the larger change of the temperature when the medium absorbs the same energy, namely, the thermal blooming effect becomes more severe. It is noted that the dependence of SR on the salinity is not monotonic as the time increases, i.e., at the beginning the SR decreases as the salinity S increases, and then the situation is opposite (see Fig. 5(a)). The physical reasons are given as follows: at the beginning the SR decreases because of beam spreading due to thermal blooming effect. The more severe the thermal blooming effect is, the larger the beam spread is, and the lower SR is. As the time increases, the intensity distribution takes a crescent-like pattern due to current velocity of the seawater. The more severe the thermal blooming effect is, the more severe the beam distortion is (i.e., the thinner the crescent shape is), which results in an increase of SR. In particular, from Fig. 5 and Fig. 6 it can be seen that SR, SRE, η andx¯are nearly unchanged when the time t is long enough, i.e., the steady state thermal blooming arrives. In addition, from Fig. 7 it can be seen that the value of distortion parameter N increases as the salinity increases, namely, the thermal blooming effect becomes more severe as the salinity increases, which is in agreement with the result obtained from Figs. 5 and 6 for the steady state thermal blooming case.

 

Fig. 5 (a) SR and (b) SRE versus time t for different values of the salinity S. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 6 (a) η and (b)x¯versus time t for different values of the salinity S. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 7 Distortion parameter N versus the salinity S. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

3.3 Influence of the seawater depth on the thermal blooming effect

At different depth of the seawater, the values of the temperature, the salinity, the pressure, the density and the current velocity are different. Furthermore, the situation in the shallow sea region is different from that in the deep sea region. Thus, in this section, the influence of the depth on the thermal blooming effect in the shallow and deep sea regions is investigated, respectively.

(i) Shallow sea region

According to the Fig. 2.4 in [37], we can estimate the values of the temperature, the salinity, the density and the current velocity at different depth in the shallow sea region. 3D intensity distribution and its contour lines, the intensity Strehl ratio SR and the energy Strehl ratio SRE, the propagation efficiency η, the mean-squared beam width wx, wy and the astigmatism parameter wx / wy are shown in Figs. 8–11 respectively, where the different values of the depth in the shallow sea region are considered (in this paper only h = 0~200m is adopted). Figure 8 indicates that the thermal blooming effect becomes more severe as h increases when t = 1 s. However, in the shallow sea region, the dependence of the thermal blooming on the depth h is not monotonic as the time increases (see Fig. 9, Fig. 10, and Fig. 11), namely, at the beginning the thermal blooming effect becomes smaller (e.g., SRE and η increase, wx and wy decrease) as the h increases, and then the thermal blooming effect becomes more severe (e.g., SRE and η decrease, wx and wy increase) as the h increases. The physical reason is that the temperature and the current velocity increase as the depth h decreases in the shallow sea region. It is known that the temperature and the current velocity are two of the main physical factors that dominate the thermal blooming effect. The thermal blooming becomes more severe due to higher temperature (see Fig. 12), while the thermal blooming is suppressed because of current velocity. At the beginning, the temperature dominates the thermal blooming, and the thermal blooming becomes more severe with decreasing h because the temperature increases with decreasing h. And then, the current velocity dominates the thermal blooming, and the thermal blooming becomes smaller with decreasing h because the current velocity increases with decreasing h. From Fig. 11 it can be seen that wx is smaller than wy because of the velocity along x-axis. The change of wx versus t is not monotonic due to the velocity along x-axis (see Fig. 11(a)), but wy increases monotonically as the time t increases (see Fig. 11(b)). The astigmatism parameterwx/wydenotes the beam symmetry, andwx/wy1. The higher value of wx/wyis, the better the beam symmetry is. From Fig. 11(c) it can be seen that the beam symmetry depends on both h and t. At the beginning, the beam symmetry becomes better as h increases, and then the situation is opposite. In particular, Figs. 9–11 indicate that SR, SRE, η, wx, wy andwx/wyare nearly unchanged when the time t is long enough, i.e., the steady state thermal blooming appears. In addition, Fig. 13 shows that the value of beam distortion parameter N decreases as the depth h decreases, which is in agreement with the result obtained from Figs. 9-11 for the steady state thermal blooming case.

 

Fig. 8 3D intensity distribution and contour lines for different values of the depth h within shallow sea region. t = 1 s, P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 9 (a) SR and (b) SRE versus time t for different values of the depth h within shallow sea region. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 10 η versus time t for different values of the depth h within shallow sea region. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 11 (a) wx, (b) wy and (c) wx / wy versus time t for different values of the depth h within shallow sea region. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 12 3D intensity distribution and contour lines for different values of the temperature T. t = 1 s, P = 200W, z = 1km, h = 0.

Download Full Size | PPT Slide | PDF

 

Fig. 13 Distortion parameter N versus the depth h within shallow sea region. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

(ii) Deep sea region

According to the Fig. 1.12 in [37], we can estimate the values of the temperature, the salinity, density and current velocity at different depth in the deep sea region, where 100dbar=1MPa100mis adopted. The energy Strehl ratio SRE, the propagation efficiency η and the centroid positionx¯are shown in Figs. 14(a), 14(b) and 14(c) respectively, where the different values of the depth h in the deep sea region are considered (in this paper only h = 2000m~5000m is adopted). It is clearly seen that the behavior of the thermal blooming effect in the deep sea region is different from that in the shallow sea region. Figure 14 indicates that, in the deep sea region, the thermal blooming effect becomes more severe (e.g., SRE and η decrease, and|x¯|increases) as the h increases. Actually, in the deep sea region, the pressure is the main physical factor that dominates the thermal blooming effect, namely, the thermal blooming effect becomes more severe with increasing pressure. The physical reason is that |nT|increases andCpdecreases as the depth h increases (see Fig. 15), while the temperature, the salinity, density and current velocity are nearly unchanged as the depth h increases (i.e., as the pressure increases). In particular, Fig. 14 shows that SRE, η andx¯are nearly unchanged when the time t is long enough, i.e., it reaches the steady state thermal blooming. The curve of the distortion parameter N versus the depth h in the deep sea region is plotted in Fig. 16. From Fig. 16 it can be seen that N increases (i.e., the thermal blooming becomes more severe) as the depth h increases, which is in agreement with the result obtained from Fig. 15.

 

Fig. 14 (a) SRE, (b) η and (c)x¯versus time t for different values of the depth h within deep sea region. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 15 (a) nT, (b)Cpversus the depth h within deep sea region. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

 

Fig. 16 Distortion parameter N versus the depth h within deep sea region. P = 200W, z = 1km.

Download Full Size | PPT Slide | PDF

4. Conclusion

In this paper, the effects of the salinity and the depth of seawater and the wavelength on the thermal blooming of laser beams propagating through the seawater are investigated in detail by using the numerical simulation method. It is found that the thermal blooming becomes more severe due to an increase of the salinity in the seawater. Furthermore, the dependence of the peak intensity on the salinity is not monotonic as the time increases, i.e., at the beginning the peak intensity decreases as the salinity increases, and then the situation is opposite. As compared with the wavelength, the absorption coefficient of the seawater is the main factor dominating the thermal blooming effect. In the seawater, the thermal blooming becomes more severe for the wavelength corresponding to the higher absorption coefficient. Comparing the shallow sea region with the deep sea region, both the behavior of the thermal blooming effect and the physical factors dominating the thermal blooming effect are different. In the shallow sea region, the dependence of the thermal blooming on the depth is not monotonic as the time increases, i.e., at the beginning the thermal blooming becomes smaller as the depth increases, which is dominated by the temperature; and then the situation is opposite, which is dominated by the current velocity. However, in deep sea region, the thermal blooming effect becomes more severe monotonously as the depth increases, and the pressure is the main factor dominating the thermal blooming effect. The results obtained in this paper will be very useful for applications of laser beams in the seawater.

Funding

National Natural Science Foundation of China (NSFC) (61475105, 61505130).

Acknowledgments

The authors are very thankful to the reviewers for their very valuable comments.

References and links

1. D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE 65(12), 1679–1714 (1977). [CrossRef]  

2. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976). [CrossRef]  

3. B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE 3706, 361–367 (1999). [CrossRef]  

4. F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE 1221, 2–25 (1990). [CrossRef]  

5. J. D. Barchers, “Linear analysis of thermal blooming compensation instabilities in laser propagation,” J. Opt. Soc. Am. A 26(7), 1638–1653 (2009). [CrossRef]   [PubMed]  

6. N. R. Van Zandt, S. T. Fiorino, and K. J. Keefer, “Enhanced, fast-running scaling law model of thermal blooming and turbulence effects on high energy laser propagation,” Opt. Express 21(12), 14789–14798 (2013). [CrossRef]   [PubMed]  

7. X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013). [CrossRef]   [PubMed]  

8. J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992). [CrossRef]  

9. S. Arnon and D. Kedar, “Non-line-of-sight underwater optical wireless communication network,” J. Opt. Soc. Am. A 26(3), 530–539 (2009). [CrossRef]   [PubMed]  

10. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006). [CrossRef]  

11. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012). [CrossRef]  

12. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012). [CrossRef]  

13. M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013). [CrossRef]  

14. L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22(22), 27112–27122 (2014). [CrossRef]   [PubMed]  

15. Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014). [CrossRef]   [PubMed]  

16. W. Hou, “A simple underwater imaging model,” Opt. Lett. 34(17), 2688–2690 (2009). [CrossRef]   [PubMed]  

17. W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012). [CrossRef]   [PubMed]  

18. W. Hou, E. Jarosz, S. Woods, W. Goode, and A. Weidemann, “Impacts of underwater turbulence on acoustical and optical signals and their linkage,” Opt. Express 21(4), 4367–4375 (2013). [CrossRef]   [PubMed]  

19. H. Pu and X. Ji, “Oceanic turbulence effects on long-exposure and short-exposure imaging,” J. Opt. 18(10), 105704 (2016). [CrossRef]  

20. F. G. Gebhardt and D. C. Smith, “Self-Induced Thermal Distortion in the Near Field for a Laser Beam in a Moving Medium,” IEEE J. Quantum Electron. 7(2), 63–73 (1971). [CrossRef]  

21. E. Wild and M. Maier, “Thermal blooming in liquid N2 during high repetition rate stimulated Raman Scattering,” J. Appl. Phys. 51(6), 3078–3080 (1980). [CrossRef]  

22. W. Liu, T. Liao, and Q. Gao, “Numerical simulation of thermal blooming effect of laser propagation in atmosphere,” Proc. SPIE 8192, 81924 (2011). [CrossRef]  

23. J. N. Hayes, “Thermal Blooming of Laser Beams in Fluids,” Appl. Opt. 11(2), 455–461 (1972). [CrossRef]   [PubMed]  

24. D. Kovsh, D. Hagan, and E. Van Stryland, “Numerical modeling of thermal refraction inliquids in the transient regime,” Opt. Express 4(8), 315–327 (1999). [CrossRef]   [PubMed]  

25. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Vol.6: Fluid Mechanics (Pergamon, 1987).

26. S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008). [CrossRef]  

27. Y. H. Yan, B. Q. Liu, B. Zhou, and D. S. Wu, “Studies of energy Strehl ratio of a collimated Gaussian beam in turbulent atmosphere,” Proc. SPIE 8906, 89062H1 (2013). [CrossRef]  

28. P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009). [CrossRef]  

29. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–S1049 (1992). [CrossRef]  

30. R. C. Smith and K. S. Baker, “Optical properties of the clearest natural waters (200-800 nm),” Appl. Opt. 20(2), 177–184 (1981). [CrossRef]   [PubMed]  

31. F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008). [CrossRef]  

32. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995). [CrossRef]   [PubMed]  

33. A. Poisson, C. Brunet, and J. C. Brun-Cottan, “Density of standard seawater solutions at atmospheric pressure,” Deep-Sea Res. 27(12), 1013–1028 (1980). [CrossRef]  

34. R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. 58(1), 43–114 (2003). [CrossRef]  

35. G. T. McNeil, “Metrical fundamentals of underwater lens system,” Opt. Eng. 16(2), 129–139 (1977). [CrossRef]  

36. N. P. Fofonoff and R. C. Millard, “Algorithms for computation of fundamental properties of seawater,” Paris: UNESCO Tech. Pap. Mar. Sci. 44, 31–37 (1983).

37. S. A. Thorpe, An Introduction to Ocean Turbulence (Cambridge University, 2008).

38. A. Bradshaw and K. E. Schleicher, “Direct measurement of thermal expansion of sea water under pressure,” Deep-Sea Res. 17(4), 691–698 (1970).

39. G. L. Clarke and H. R. James, “Laboratory analysis of the selective absorption of light by sea water,” JOSA 29(2), 43–53 (1939). [CrossRef]  

40. S. A. Sullivan, “Experimental study of the absorption in distilled water, artificial sea water, and heavy water in the visible region of the spectrum,” JOSA 53(8), 962–968 (1963). [CrossRef]  

41. W. S. Pegau and J. R. Zaneveld, “Temperature-dependent absorption of water in the red and near-infrared portions of the spectrum,” Limnol. Oceanogr. 38(1), 188–192 (1993). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE 65(12), 1679–1714 (1977).
    [Crossref]
  2. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
    [Crossref]
  3. B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE 3706, 361–367 (1999).
    [Crossref]
  4. F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE 1221, 2–25 (1990).
    [Crossref]
  5. J. D. Barchers, “Linear analysis of thermal blooming compensation instabilities in laser propagation,” J. Opt. Soc. Am. A 26(7), 1638–1653 (2009).
    [Crossref] [PubMed]
  6. N. R. Van Zandt, S. T. Fiorino, and K. J. Keefer, “Enhanced, fast-running scaling law model of thermal blooming and turbulence effects on high energy laser propagation,” Opt. Express 21(12), 14789–14798 (2013).
    [Crossref] [PubMed]
  7. X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013).
    [Crossref] [PubMed]
  8. J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
    [Crossref]
  9. S. Arnon and D. Kedar, “Non-line-of-sight underwater optical wireless communication network,” J. Opt. Soc. Am. A 26(3), 530–539 (2009).
    [Crossref] [PubMed]
  10. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
    [Crossref]
  11. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
    [Crossref]
  12. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
    [Crossref]
  13. M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
    [Crossref]
  14. L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22(22), 27112–27122 (2014).
    [Crossref] [PubMed]
  15. Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014).
    [Crossref] [PubMed]
  16. W. Hou, “A simple underwater imaging model,” Opt. Lett. 34(17), 2688–2690 (2009).
    [Crossref] [PubMed]
  17. W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
    [Crossref] [PubMed]
  18. W. Hou, E. Jarosz, S. Woods, W. Goode, and A. Weidemann, “Impacts of underwater turbulence on acoustical and optical signals and their linkage,” Opt. Express 21(4), 4367–4375 (2013).
    [Crossref] [PubMed]
  19. H. Pu and X. Ji, “Oceanic turbulence effects on long-exposure and short-exposure imaging,” J. Opt. 18(10), 105704 (2016).
    [Crossref]
  20. F. G. Gebhardt and D. C. Smith, “Self-Induced Thermal Distortion in the Near Field for a Laser Beam in a Moving Medium,” IEEE J. Quantum Electron. 7(2), 63–73 (1971).
    [Crossref]
  21. E. Wild and M. Maier, “Thermal blooming in liquid N2 during high repetition rate stimulated Raman Scattering,” J. Appl. Phys. 51(6), 3078–3080 (1980).
    [Crossref]
  22. W. Liu, T. Liao, and Q. Gao, “Numerical simulation of thermal blooming effect of laser propagation in atmosphere,” Proc. SPIE 8192, 81924 (2011).
    [Crossref]
  23. J. N. Hayes, “Thermal Blooming of Laser Beams in Fluids,” Appl. Opt. 11(2), 455–461 (1972).
    [Crossref] [PubMed]
  24. D. Kovsh, D. Hagan, and E. Van Stryland, “Numerical modeling of thermal refraction inliquids in the transient regime,” Opt. Express 4(8), 315–327 (1999).
    [Crossref] [PubMed]
  25. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Vol.6: Fluid Mechanics (Pergamon, 1987).
  26. S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
    [Crossref]
  27. Y. H. Yan, B. Q. Liu, B. Zhou, and D. S. Wu, “Studies of energy Strehl ratio of a collimated Gaussian beam in turbulent atmosphere,” Proc. SPIE 8906, 89062H1 (2013).
    [Crossref]
  28. P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009).
    [Crossref]
  29. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–S1049 (1992).
    [Crossref]
  30. R. C. Smith and K. S. Baker, “Optical properties of the clearest natural waters (200-800 nm),” Appl. Opt. 20(2), 177–184 (1981).
    [Crossref] [PubMed]
  31. F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008).
    [Crossref]
  32. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995).
    [Crossref] [PubMed]
  33. A. Poisson, C. Brunet, and J. C. Brun-Cottan, “Density of standard seawater solutions at atmospheric pressure,” Deep-Sea Res. 27(12), 1013–1028 (1980).
    [Crossref]
  34. R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. 58(1), 43–114 (2003).
    [Crossref]
  35. G. T. McNeil, “Metrical fundamentals of underwater lens system,” Opt. Eng. 16(2), 129–139 (1977).
    [Crossref]
  36. N. P. Fofonoff and R. C. Millard, “Algorithms for computation of fundamental properties of seawater,” Paris: UNESCO Tech. Pap. Mar. Sci. 44, 31–37 (1983).
  37. S. A. Thorpe, An Introduction to Ocean Turbulence (Cambridge University, 2008).
  38. A. Bradshaw and K. E. Schleicher, “Direct measurement of thermal expansion of sea water under pressure,” Deep-Sea Res. 17(4), 691–698 (1970).
  39. G. L. Clarke and H. R. James, “Laboratory analysis of the selective absorption of light by sea water,” JOSA 29(2), 43–53 (1939).
    [Crossref]
  40. S. A. Sullivan, “Experimental study of the absorption in distilled water, artificial sea water, and heavy water in the visible region of the spectrum,” JOSA 53(8), 962–968 (1963).
    [Crossref]
  41. W. S. Pegau and J. R. Zaneveld, “Temperature-dependent absorption of water in the red and near-infrared portions of the spectrum,” Limnol. Oceanogr. 38(1), 188–192 (1993).
    [Crossref]

2016 (1)

H. Pu and X. Ji, “Oceanic turbulence effects on long-exposure and short-exposure imaging,” J. Opt. 18(10), 105704 (2016).
[Crossref]

2014 (2)

2013 (5)

2012 (3)

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
[Crossref] [PubMed]

2011 (1)

W. Liu, T. Liao, and Q. Gao, “Numerical simulation of thermal blooming effect of laser propagation in atmosphere,” Proc. SPIE 8192, 81924 (2011).
[Crossref]

2009 (4)

2008 (2)

F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008).
[Crossref]

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

2006 (1)

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

2003 (1)

R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. 58(1), 43–114 (2003).
[Crossref]

1999 (2)

D. Kovsh, D. Hagan, and E. Van Stryland, “Numerical modeling of thermal refraction inliquids in the transient regime,” Opt. Express 4(8), 315–327 (1999).
[Crossref] [PubMed]

B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE 3706, 361–367 (1999).
[Crossref]

1995 (1)

1993 (1)

W. S. Pegau and J. R. Zaneveld, “Temperature-dependent absorption of water in the red and near-infrared portions of the spectrum,” Limnol. Oceanogr. 38(1), 188–192 (1993).
[Crossref]

1992 (2)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–S1049 (1992).
[Crossref]

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

1990 (1)

F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE 1221, 2–25 (1990).
[Crossref]

1983 (1)

N. P. Fofonoff and R. C. Millard, “Algorithms for computation of fundamental properties of seawater,” Paris: UNESCO Tech. Pap. Mar. Sci. 44, 31–37 (1983).

1981 (1)

1980 (2)

A. Poisson, C. Brunet, and J. C. Brun-Cottan, “Density of standard seawater solutions at atmospheric pressure,” Deep-Sea Res. 27(12), 1013–1028 (1980).
[Crossref]

E. Wild and M. Maier, “Thermal blooming in liquid N2 during high repetition rate stimulated Raman Scattering,” J. Appl. Phys. 51(6), 3078–3080 (1980).
[Crossref]

1977 (2)

G. T. McNeil, “Metrical fundamentals of underwater lens system,” Opt. Eng. 16(2), 129–139 (1977).
[Crossref]

D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE 65(12), 1679–1714 (1977).
[Crossref]

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

1972 (1)

1971 (1)

F. G. Gebhardt and D. C. Smith, “Self-Induced Thermal Distortion in the Near Field for a Laser Beam in a Moving Medium,” IEEE J. Quantum Electron. 7(2), 63–73 (1971).
[Crossref]

1970 (1)

A. Bradshaw and K. E. Schleicher, “Direct measurement of thermal expansion of sea water under pressure,” Deep-Sea Res. 17(4), 691–698 (1970).

1963 (1)

S. A. Sullivan, “Experimental study of the absorption in distilled water, artificial sea water, and heavy water in the visible region of the spectrum,” JOSA 53(8), 962–968 (1963).
[Crossref]

1939 (1)

G. L. Clarke and H. R. James, “Laboratory analysis of the selective absorption of light by sea water,” JOSA 29(2), 43–53 (1939).
[Crossref]

Arnon, S.

Ata, Y.

Baker, K. S.

Barchers, J. D.

Baykal, Y.

Bradford, L. W.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Bradshaw, A.

A. Bradshaw and K. E. Schleicher, “Direct measurement of thermal expansion of sea water under pressure,” Deep-Sea Res. 17(4), 691–698 (1970).

Brun-Cottan, J. C.

A. Poisson, C. Brunet, and J. C. Brun-Cottan, “Density of standard seawater solutions at atmospheric pressure,” Deep-Sea Res. 27(12), 1013–1028 (1980).
[Crossref]

Brunet, C.

A. Poisson, C. Brunet, and J. C. Brun-Cottan, “Density of standard seawater solutions at atmospheric pressure,” Deep-Sea Res. 27(12), 1013–1028 (1980).
[Crossref]

Christou, J. C.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Clarke, G. L.

G. L. Clarke and H. R. James, “Laboratory analysis of the selective absorption of light by sea water,” JOSA 29(2), 43–53 (1939).
[Crossref]

Eyyuboglu, H. T.

Farwell, N.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

Feistel, R.

F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008).
[Crossref]

R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. 58(1), 43–114 (2003).
[Crossref]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Fiorino, S. T.

Fischer, R.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009).
[Crossref]

Flatley, J. P.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Fofonoff, N. P.

N. P. Fofonoff and R. C. Millard, “Algorithms for computation of fundamental properties of seawater,” Paris: UNESCO Tech. Pap. Mar. Sci. 44, 31–37 (1983).

Fortes, B. V.

B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE 3706, 361–367 (1999).
[Crossref]

Freeman, D. E.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Fry, E. S.

Gao, Q.

W. Liu, T. Liao, and Q. Gao, “Numerical simulation of thermal blooming effect of laser propagation in atmosphere,” Proc. SPIE 8192, 81924 (2011).
[Crossref]

Gebhardt, F. G.

F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE 1221, 2–25 (1990).
[Crossref]

F. G. Gebhardt and D. C. Smith, “Self-Induced Thermal Distortion in the Near Field for a Laser Beam in a Moving Medium,” IEEE J. Quantum Electron. 7(2), 63–73 (1971).
[Crossref]

Gladysz, S.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Goode, W.

Hafizi, B.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009).
[Crossref]

Hagan, D.

Hayes, J. N.

Hou, W.

James, H. R.

G. L. Clarke and H. R. James, “Laboratory analysis of the selective absorption of light by sea water,” JOSA 29(2), 43–53 (1939).
[Crossref]

Jarosz, E.

Ji, G.

Ji, X.

Jia, X.

Kedar, D.

Keefer, K. J.

Korotkova, O.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

Kovsh, D.

Landry, M. A.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Liao, T.

W. Liu, T. Liao, and Q. Gao, “Numerical simulation of thermal blooming effect of laser propagation in atmosphere,” Proc. SPIE 8192, 81924 (2011).
[Crossref]

Lindstrom, C. E.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Liu, B. Q.

Y. H. Yan, B. Q. Liu, B. Zhou, and D. S. Wu, “Studies of energy Strehl ratio of a collimated Gaussian beam in turbulent atmosphere,” Proc. SPIE 8906, 89062H1 (2013).
[Crossref]

Liu, L.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Liu, W.

W. Liu, T. Liao, and Q. Gao, “Numerical simulation of thermal blooming effect of laser propagation in atmosphere,” Proc. SPIE 8192, 81924 (2011).
[Crossref]

Longacre, J. R.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Lu, L.

Lu, W.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Lukin, V. P.

B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE 3706, 361–367 (1999).
[Crossref]

Maier, M.

E. Wild and M. Maier, “Thermal blooming in liquid N2 during high repetition rate stimulated Raman Scattering,” J. Appl. Phys. 51(6), 3078–3080 (1980).
[Crossref]

McDougall, T. J.

F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008).
[Crossref]

McNeil, G. T.

G. T. McNeil, “Metrical fundamentals of underwater lens system,” Opt. Eng. 16(2), 129–139 (1977).
[Crossref]

Millard, R. C.

N. P. Fofonoff and R. C. Millard, “Algorithms for computation of fundamental properties of seawater,” Paris: UNESCO Tech. Pap. Mar. Sci. 44, 31–37 (1983).

Millero, F. J.

F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008).
[Crossref]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Pegau, W. S.

W. S. Pegau and J. R. Zaneveld, “Temperature-dependent absorption of water in the red and near-infrared portions of the spectrum,” Limnol. Oceanogr. 38(1), 188–192 (1993).
[Crossref]

Penano, J.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009).
[Crossref]

Poisson, A.

A. Poisson, C. Brunet, and J. C. Brun-Cottan, “Density of standard seawater solutions at atmospheric pressure,” Deep-Sea Res. 27(12), 1013–1028 (1980).
[Crossref]

Pu, H.

H. Pu and X. Ji, “Oceanic turbulence effects on long-exposure and short-exposure imaging,” J. Opt. 18(10), 105704 (2016).
[Crossref]

Quan, X.

Roberts, L. C.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Schleicher, K. E.

A. Bradshaw and K. E. Schleicher, “Direct measurement of thermal expansion of sea water under pressure,” Deep-Sea Res. 17(4), 691–698 (1970).

Schwartz, J. A.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Shchepakina, E.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

Smith, D. C.

D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE 65(12), 1679–1714 (1977).
[Crossref]

F. G. Gebhardt and D. C. Smith, “Self-Induced Thermal Distortion in the Near Field for a Laser Beam in a Moving Medium,” IEEE J. Quantum Electron. 7(2), 63–73 (1971).
[Crossref]

Smith, R. C.

Snow, J. B.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Sprangle, P.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009).
[Crossref]

Sullivan, S. A.

S. A. Sullivan, “Experimental study of the absorption in distilled water, artificial sea water, and heavy water in the visible region of the spectrum,” JOSA 53(8), 962–968 (1963).
[Crossref]

Sun, J.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Tang, M.

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
[Crossref]

Ting, A.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009).
[Crossref]

Van Stryland, E.

Van Zandt, N. R.

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–S1049 (1992).
[Crossref]

Weidemann, A.

Wild, E.

E. Wild and M. Maier, “Thermal blooming in liquid N2 during high repetition rate stimulated Raman Scattering,” J. Appl. Phys. 51(6), 3078–3080 (1980).
[Crossref]

Woods, S.

Wright, D. G.

F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008).
[Crossref]

Wu, D. S.

Y. H. Yan, B. Q. Liu, B. Zhou, and D. S. Wu, “Studies of energy Strehl ratio of a collimated Gaussian beam in turbulent atmosphere,” Proc. SPIE 8906, 89062H1 (2013).
[Crossref]

Yan, Y. H.

Y. H. Yan, B. Q. Liu, B. Zhou, and D. S. Wu, “Studies of energy Strehl ratio of a collimated Gaussian beam in turbulent atmosphere,” Proc. SPIE 8906, 89062H1 (2013).
[Crossref]

Zaneveld, J. R.

W. S. Pegau and J. R. Zaneveld, “Temperature-dependent absorption of water in the red and near-infrared portions of the spectrum,” Limnol. Oceanogr. 38(1), 188–192 (1993).
[Crossref]

Zhao, D.

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
[Crossref]

Zhou, B.

Y. H. Yan, B. Q. Liu, B. Zhou, and D. S. Wu, “Studies of energy Strehl ratio of a collimated Gaussian beam in turbulent atmosphere,” Proc. SPIE 8906, 89062H1 (2013).
[Crossref]

Appl. Opt. (4)

Appl. Phys. (Berl.) (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Appl. Phys. B (1)

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
[Crossref]

Deep-Sea Res. (3)

A. Poisson, C. Brunet, and J. C. Brun-Cottan, “Density of standard seawater solutions at atmospheric pressure,” Deep-Sea Res. 27(12), 1013–1028 (1980).
[Crossref]

F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep-Sea Res. 55(1), 50–72 (2008).
[Crossref]

A. Bradshaw and K. E. Schleicher, “Direct measurement of thermal expansion of sea water under pressure,” Deep-Sea Res. 17(4), 691–698 (1970).

IEEE J. Quantum Electron. (2)

F. G. Gebhardt and D. C. Smith, “Self-Induced Thermal Distortion in the Near Field for a Laser Beam in a Moving Medium,” IEEE J. Quantum Electron. 7(2), 63–73 (1971).
[Crossref]

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmosphere propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45(2), 138–148 (2009).
[Crossref]

J. Appl. Phys. (1)

E. Wild and M. Maier, “Thermal blooming in liquid N2 during high repetition rate stimulated Raman Scattering,” J. Appl. Phys. 51(6), 3078–3080 (1980).
[Crossref]

J. Opt. (1)

H. Pu and X. Ji, “Oceanic turbulence effects on long-exposure and short-exposure imaging,” J. Opt. 18(10), 105704 (2016).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

J. Opt. Soc. Am. A (3)

JOSA (2)

G. L. Clarke and H. R. James, “Laboratory analysis of the selective absorption of light by sea water,” JOSA 29(2), 43–53 (1939).
[Crossref]

S. A. Sullivan, “Experimental study of the absorption in distilled water, artificial sea water, and heavy water in the visible region of the spectrum,” JOSA 53(8), 962–968 (1963).
[Crossref]

Limnol. Oceanogr. (1)

W. S. Pegau and J. R. Zaneveld, “Temperature-dependent absorption of water in the red and near-infrared portions of the spectrum,” Limnol. Oceanogr. 38(1), 188–192 (1993).
[Crossref]

Opt. Commun. (1)

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

Opt. Eng. (1)

G. T. McNeil, “Metrical fundamentals of underwater lens system,” Opt. Eng. 16(2), 129–139 (1977).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–S1049 (1992).
[Crossref]

Paris: UNESCO Tech. Pap. Mar. Sci. (1)

N. P. Fofonoff and R. C. Millard, “Algorithms for computation of fundamental properties of seawater,” Paris: UNESCO Tech. Pap. Mar. Sci. 44, 31–37 (1983).

Proc. IEEE (1)

D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE 65(12), 1679–1714 (1977).
[Crossref]

Proc. SPIE (5)

B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE 3706, 361–367 (1999).
[Crossref]

F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE 1221, 2–25 (1990).
[Crossref]

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

W. Liu, T. Liao, and Q. Gao, “Numerical simulation of thermal blooming effect of laser propagation in atmosphere,” Proc. SPIE 8192, 81924 (2011).
[Crossref]

Y. H. Yan, B. Q. Liu, B. Zhou, and D. S. Wu, “Studies of energy Strehl ratio of a collimated Gaussian beam in turbulent atmosphere,” Proc. SPIE 8906, 89062H1 (2013).
[Crossref]

Prog. Oceanogr. (1)

R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. 58(1), 43–114 (2003).
[Crossref]

Publ. Astron. Soc. Pac. (1)

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Waves Random Complex Media (1)

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

Other (2)

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Vol.6: Fluid Mechanics (Pergamon, 1987).

S. A. Thorpe, An Introduction to Ocean Turbulence (Cambridge University, 2008).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1 3D intensity distribution and contour lines for different values of the wavelength λ. t = 1 s, P = 3kW, z = 100m.
Fig. 2
Fig. 2 (a) SR and (b) SRE versus time t for different values of the wavelength λ. P = 3kW, z = 100m.
Fig. 3
Fig. 3 η versus (a) the propagation distance z and (b) time t for different values of the wavelength λ. P = 3kW.
Fig. 4
Fig. 4 x ¯ versus time t for different values of the wavelength λ. P = 3kW, z = 100m.
Fig. 5
Fig. 5 (a) SR and (b) SRE versus time t for different values of the salinity S. P = 200W, z = 1km.
Fig. 6
Fig. 6 (a) η and (b) x ¯ versus time t for different values of the salinity S. P = 200W, z = 1km.
Fig. 7
Fig. 7 Distortion parameter N versus the salinity S. P = 200W, z = 1km.
Fig. 8
Fig. 8 3D intensity distribution and contour lines for different values of the depth h within shallow sea region. t = 1 s, P = 200W, z = 1km.
Fig. 9
Fig. 9 (a) SR and (b) SRE versus time t for different values of the depth h within shallow sea region. P = 200W, z = 1km.
Fig. 10
Fig. 10 η versus time t for different values of the depth h within shallow sea region. P = 200W, z = 1km.
Fig. 11
Fig. 11 (a) wx, (b) wy and (c) wx / wy versus time t for different values of the depth h within shallow sea region. P = 200W, z = 1km.
Fig. 12
Fig. 12 3D intensity distribution and contour lines for different values of the temperature T. t = 1 s, P = 200W, z = 1km, h = 0.
Fig. 13
Fig. 13 Distortion parameter N versus the depth h within shallow sea region. P = 200W, z = 1km.
Fig. 14
Fig. 14 (a) SRE, (b) η and (c) x ¯ versus time t for different values of the depth h within deep sea region. P = 200W, z = 1km.
Fig. 15
Fig. 15 (a) nT, (b) C p versus the depth h within deep sea region. P = 200W, z = 1km.
Fig. 16
Fig. 16 Distortion parameter N versus the depth h within deep sea region. P = 200W, z = 1km.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

2ik E z = 2 E+ k 2 ( n 2 n 0 2 1)E,
n 2 n 0 2 1= ( n 0 2 1)( n 0 2 +2) 3 n 0 2 ρ 1 ρ 0 ,
( n 0 2 1)( n 0 2 +2) 3 n 0 2 2 n T β ,
ρ 1 t +v ρ 1 = βα C p I,
E= P π w 0 2 exp( x 2 + y 2 2 w 0 2 ),
E n+1 =exp( i 4k Δz 2 )exp(is)exp( i 4k Δz 2 ) E n ,
N=( 2 n T I 0 z n 0 ρ 0 C p v w 0 )[ 1 (1 e αz ) αz ],
S R = I max I 0max ,
S R E = x 2 + y 2 w 2 I(x,y,z)dxdy x 2 + y 2 w 2 I vacuum (x,y,z)dxdy ,
η= σ I(x,y,z)dxdy I(x,y,0)dxdy ,
j ¯ = jI(x,y,z)dxdy I(x,y,z)dxdy ,
w j 2 = 4 (j j ¯ ) 2 I(x,y,z)dxdy I(x,y,z)dxdy ,

Metrics