Thermal blooming effect of laser beams propagating through seawater

Yuqiu Zhang; Xiaoling Ji; Xiaoqing Li; Hong Yu

doi:10.1364/OE.25.005861

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]

2 i k \frac{\partial E}{\partial z} = \nabla_{⊥}^{2} E + k^{2} (\frac{n^{2}}{n_{0}^{2}} - 1) E,

where

\nabla_{⊥}^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

is the transverse Laplace operator, E is the envelope of the electric field, k is the wave-number related to the wave length

λ

by

k = 2 π / 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]

\frac{n^{2}}{n_{0}^{2}} - 1 = \frac{(n_{0}^{2} - 1) (n_{0}^{2} + 2)}{3 n_{0}^{2}} \frac{ρ_{1}}{ρ_{0}},

where

ρ_{0}

is the liquid density before perturbation and its corresponding first-order small quantity is

ρ_{1}

. According to [24], a simpler expression can be approximately written as

\frac{(n_{0}^{2} - 1) (n_{0}^{2} + 2)}{3 n_{0}^{2}} \approx - 2 \frac{n_{T}}{β},

where β is the thermal expansion coefficient, n_T is the thermo-optic coefficient, and

n_{T} = d n / d T

(T is the temperature of liquid).

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

\frac{\partial ρ_{1}}{\partial t} + v \cdot \nabla ρ_{1} = - \frac{β α}{C_{p}} I,

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

E = \sqrt{\frac{P}{π w_{0}^{2}}} \exp (- \frac{x^{2} + y^{2}}{2 w_{0}^{2}}),

where P is the input beam power, w₀ is the beam radius.

Considering beam propagation along z axis, let $E^{n}$ be 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.,

E^{n + 1} = \exp (- \frac{i}{4 k} Δ z \nabla_{⊥}^{2}) \exp (- i s) \exp (- \frac{i}{4 k} Δ z \nabla_{⊥}^{2}) E^{n},

where

s = (k / 2) \int_{z^{n}}^{z^{n + 1}} (n^{2} / n_{0}^{2} - 1) d z

is 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 distance

Δ z / 2

, an incrementing of the phase due to thermal blooming over a distance

Δ z

, a vacuum propagation of the field over a distance

Δ 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]

N = (\frac{- 2 n_{T} I_{0} z}{n_{0} ρ_{0} C_{p} v w_{0}}) [1 - \frac{(1 - e^{- α z})}{α z}],

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 (S_R)

The intensity Strehl ratio is defined as [26]

S_{R} = \frac{I_{max}}{I_{0 max}},

where I_max and I_0max are the peak intensity with and without thermal blooming, respectively. It is clear that

S_{R} \leq 1

, and the lower value of S_R is, the more the peak intensity decreases.

(ii) Energy Strehl ratio (SR_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]

S R_{E} = \frac{\iint_{x^{2} + y^{2} \leq w^{2}} I (x, y, z) d x d y}{\iint_{x^{2} + y^{2} \leq w^{2}} I_{vacuum} (x, y, z) d x d y},

where 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} \leq 1

, and the lower value of SR_E 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 = \frac{\iint_{σ} I (x, y, z) d x d y}{\iint I (x, y, 0) d x d y},

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]

\bar{j} = \frac{\iint j I (x, y, z) d x d y}{\iint I (x, y, z) d x d y},

and

w_{j}^{2} = \frac{4 \iint (j - \bar{j})^{2} I (x, y, z) d x d y}{\iint I (x, y, z) d x d y},

respectively, where j = x and y,

\bar{x}

and

\bar{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

| \bar{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 salinity $S = 35 \frac{0}{00}$ , and $λ = 450 nm$ related to $α = 0.0145 / m$ [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 n_T, ρ, $C_{p}$ and β at the sea surface are adopted, e.g., n_T is obtained by using Eq. (3) in [32], 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_T can be obtained by using Eq. (16) in [35], $C_{p}$ is obtained by using Eq. (26) in [36], ρ can be obtained from Fig. 1.12 in [37], and $β = (1/ V) {(\partial V / \partial T)}_{P}$ 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 $λ$ = 380nm, 400nm, 450nm, 490nm and 510nm respectively, i.e., the dependence of α on $λ$ is not monotonic, and the minimum value of α is 0.0145/m corresponding to $λ$ = 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_E, the propagation efficiency η and the centroid position $\bar{x}$ 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 $λ = 450 nm$ 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 $| \bar{x} |$ 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.

Abstract

1. Introduction

2. Theoretical model

(i) Intensity Strehl ratio (SR)

(ii) Energy Strehl ratio (SRE)

(iii) Propagation efficiency

(iv) Centroid position and Mean-squared beam width

3. Numerical calculation results and analysis

3.1 Influence of the wavelength on the thermal blooming effect

3.2 Influence of the seawater salinity on the thermal blooming effect

3.3 Influence of the seawater depth on the thermal blooming effect

(i) Shallow sea region

(ii) Deep sea region

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (16)

Equations (12)

Optics Express

(i) Intensity Strehl ratio (S_R)

(ii) Energy Strehl ratio (SR_E)