## 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]

*E*is the envelope of the electric field,

*k*is the wave-number related to the wave length$\lambda $by$k=2\pi /\lambda $,

*n*

_{0}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]

*β*is the thermal expansion coefficient,

*n*is the thermo-optic coefficient, and ${n}_{T}=dn/dT$(

_{T}*T*is the temperature of liquid).

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

where${C}_{p}$is 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

*P*is the input beam power,

*w*

_{0}is the beam radius.

Considering beam propagation along *z* axis, let${E}^{n}$be the complete solution to Eq. (1) at the plane *z* = *z ^{n}*, the solution to Eq. (1) at the plane

*z*

^{n+}^{1}=

*z*+ Δ

^{n}*z*can be obtained by using the second-order accuracy of the symmetrically split operator (see appendix A in [2]), i.e.,

*z*to

^{n}*z*

^{n+}^{1}can be divided into three steps: a vacuum propagation of the field over a distance$\Delta z/2$, an incrementing of the phase due to thermal blooming over a distance$\Delta z$, a vacuum propagation of the field over a distance$\Delta z/2$. 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]

*I*

_{0}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 (S*_{R})

_{R})

The intensity Strehl ratio is defined as [26]

where*I*

_{max}and

*I*

_{0max}are the peak intensity with and without thermal blooming, respectively. It is clear that${S}_{R}\le 1$, and the lower value of

*S*is, the more the peak intensity decreases.

_{R}*(ii) Energy Strehl ratio (SR*_{E})

_{E})

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

*w*is the bucket half width chosen as the mean-squared beam width in vacuum,

*I*

_{vacuum}is the intensity in vacuum. It is obvious that$S{R}_{E}\le 1$, and the lower value of

*SR*means the energy focusability is more affected by the thermal blooming.

_{E}*(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],

*σ*is bucket area chosen (e.g., target area). In general,

*σ*is taken as 100 cm

^{2}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*=

*x*and

*y*,$\overline{x}$and$\overline{y}$denote the centroid position of a laser beam in the

*x*direction and

*y*direction respectively,${w}_{x}$and${w}_{y}$denote the mean-squared beam width of a laser beam in the

*x*direction and

*y*direction respectively. The higher the absolute value$\left|\overline{j}\right|$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 *w*_{0} = 0.03m, $T=20\xb0\text{C}$, *v* = 0.1m/s along *x*-axis, the seawater salinity$S=35\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$00$}\right.$, and$\lambda =450\text{nm}$related to$\alpha =0.0145/\text{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 *n _{T}*,

*ρ*,${C}_{p}$and

*β*at the sea surface are adopted, e.g.,

*n*is obtained by using Eq. (3) in [32],

_{T}*ρ*is obtained by using Eq. (10) in [33], and${C}_{p}$and

*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

*n*,

_{T}*ρ*,${C}_{p}$and

*β*change with the seawater depth

*h*, e.g.,

*n*can be obtained by using Eq. (16) in [35],${C}_{p}$is obtained by using Eq. (26) in [36],

_{T}*ρ*can be obtained from Fig. 1.12 in [37], and$\beta =(\text{1/}V){(\partial V/\partial T)}_{P}$can 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 $\lambda $ = 380nm, 400nm, 450nm, 490nm and 510nm respectively, i.e., the dependence of *α* on$\lambda $is not monotonic, and the minimum value of *α* is 0.0145/m corresponding to$\lambda $ = 450nm.

For different values of wavelength, the three-dimensional (3D) intensity distribution and its contour lines, the intensity Strehl ratio *S _{R}*, the energy Strehl ratio

*SR*, the propagation efficiency

_{E}*η*and the centroid position$\overline{x}$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. 1–4. From Figs. 1–4 it can be seen that the dependence of the thermal blooming on the wavelength is not monotonic. When$\lambda =450\text{nm}$corresponding 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

*S*,

_{R}*SR*,

_{E}*η*are largest (see Figs. 2 and 3), and the absolute value$\left|\overline{x}\right|$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.

#### 3.2 Influence of the seawater salinity on the thermal blooming effect

For different values of salinity, the intensity Strehl ratio *S _{R}* and the energy Strehl ratio

*SR*, the propagation efficiency

_{E}*η*and the centroid position$\overline{x}$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,

*SR*and

_{E}*η*decrease (see Figs. 5(b) and 6(a)), and the absolute value$\left|x\right|$increases (see Fig. 6(b)). The physical reason is that

*n*and${C}_{p}$decrease as the salinity increases. When the thermal blooming effect appears, the value of

_{T}*n*is negative. The higher absolute value$\left|{n}_{T}\right|$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,${C}_{p}$is defined as the heat required raising the temperature by one unit (1K) per unit mass (1kg) of medium under the isobaric condition. The lower${C}_{p}$means 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

_{T}*S*on the salinity is not monotonic as the time increases, i.e., at the beginning the

_{R}*S*decreases as the salinity

_{R}*S*increases, and then the situation is opposite (see Fig. 5(a)). The physical reasons are given as follows: at the beginning the

*S*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

_{R}*S*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

_{R}*S*. In particular, from Fig. 5 and Fig. 6 it can be seen that

_{R}*S*,

_{R}*SR*,

_{E}*η*and$\overline{x}$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.

#### 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 *S _{R}* and the energy Strehl ratio

*SR*, the propagation efficiency

_{E}*η*, the mean-squared beam width

*w*,

_{x}*w*and the astigmatism parameter

_{y}*w*/

_{x}*w*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

_{y}*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.,

*SR*and

_{E}*η*increase,

*w*and

_{x}*w*decrease) as the

_{y}*h*increases, and then the thermal blooming effect becomes more severe (e.g.,

*SR*and

_{E}*η*decrease,

*w*and

_{x}*w*increase) as the

_{y}*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

*w*is smaller than

_{x}*w*because of the velocity along

_{y}*x*-axis. The change of

*w*versus

_{x}*t*is not monotonic due to the velocity along

*x*-axis (see Fig. 11(a)), but

*w*increases monotonically as the time

_{y}*t*increases (see Fig. 11(b)). The astigmatism parameter${w}_{x}/{w}_{y}$denotes the beam symmetry, and${w}_{x}/{w}_{y}\le 1$. The higher value of ${w}_{x}/{w}_{y}$is, 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

*S*,

_{R}*SR*,

_{E}*η*,

*w*,

_{x}*w*and${w}_{x}/{w}_{y}$are nearly unchanged when the time

_{y}*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.

### (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 $100\text{dbar}=1\text{MPa}\approx 100\text{m}$is adopted. The energy Strehl ratio *SR _{E}*, the propagation efficiency

*η*and the centroid position$\overline{x}$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.,

*S*and

_{RE}*η*decrease, and$\left|\overline{x}\right|$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 $\left|{n}_{T}\right|$increases and${C}_{p}$decreases 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

*SR*,

_{E}*η*and$\overline{x}$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.

## 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.

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