Analytical Study on the Steady-State Thermal Blooming of Incoherent Combining Beam

Abstract

The incoherent combined beam of lasers has many important applications due to its simple technology and excellent expansibility. Thermal blooming is one factor that will seriously affect the propagation of the combined beam in the atmosphere. Based on the analytical expression of thermal blooming for a Gaussian beam, the thermal blooming of the incoherent combined beam with a basic arrangement was studied and analyzed. From the evolution of thermal blooming for the incoherent combined beam with a basic arrangement, the thermal blooming for the incoherent combined beam with a complex arrangement can be predicted. As an example, incoherent beams with seven channels were studied. Results show that the thermal lens effect will split the beam located at the central path of heat into two parts.

1. Introduction

In recent years, multi-kW average power level with diffraction-limited beam quality of a single fiber laser has been achieved. It has improved the application for long distance penetration of a laser beam in the atmosphere, such as for clearing space debris [1], optoelectronic countermeasure [2], laser propulsion [3] and laser defense [4]. However, due to limitations including thermal self-focusing, onset of non-linear effects, facet damage, and/or the availability of high brightness pump diodes [5], it is difficult to obtain more high output power from a single fiber.

Combining several lasers into one beam (beam combining) is regarded as the most efficient way to overcome the limitations of the power of a single laser [5,6]. Beam combining includes coherent, incoherent, and spectral beam combining [7]. Coherent beam combining is to combine several laser beams to obtain a single beam with correspondingly higher power and beam quality. The essential question is the control of mutual coherence [8]. Incoherent beam combining is to combine several laser beams to obtain a single beam but with correspondingly higher power [9]. Comparing with coherent beam combining, the technology of incoherent beam combining is simpler [10].

There are many factors that will affect the propagation of a high-power laser beam in the atmosphere for a long distance. Turbulence and thermal blooming are two of the most important factors [11,12]. The effect of turbulence is linear and results from the small-scale, irregular air motions, which will lead to the fluctuation of the refractive index and cause beam spread, angle-of-arrival fluctuation, and scintillation [13,14]. Thermal blooming is nonlinear and result from the change of the refractive index due to the absorption of the atmosphere [15]. The absorption will cause the energy dissipation of the beam and a negative lens effect of the atmosphere [16,17,18]. In the past decades, thermal blooming effects on the beam propagation have attracted much attention [19,20,21,22,23,24,25].

The thermal blooming of a Gaussian beam has been analytically studied and proven in experiments when the thermal blooming effect was not strong. However, it is difficult to analytically study the thermal blooming of other beams due to its nonlinear effect. In the present paper, we analytically studied the thermal blooming of the incoherent combined beam and calculate the effects of one laser beam on the thermal blooming of another nearby beam, which is relevant for incoherent beam combination when the thermal blooming effects are not strong. It is useful both in the propagation of the combined beam and the verification of many simulations on thermal blooming.

2. Theory of the Steady-State Thermal Blooming

In the assumptions of paraxial ray approximation and neglect of index gradients in the propagation direction z, the intensity equation of the laser in the atmosphere can be written as [16].

$$\frac{I\left(x,y,z\right)}{I\left(x,y,0\right)}=\exp \left[-\alpha z-{\displaystyle {\int }_0^z\left({\nabla }_t+\frac{{\nabla }_{t}I}{I}\right)}\cdot {\displaystyle {\int }_0^{z^{\prime }}\frac{{\nabla }_t\mu }{{\mu }_0}d{z}^{″}d{z}^{\prime }}\right]$$

(1)

where $\mu$ is the index of refraction of the air, $I\left(x,y,0\right)$ and $I\left(x,y,z\right)$ are the intensity distribution of the laser in the initial and receive plane, respectively; (x, y, z) are the space coordinates, ${\nabla }_t=\frac{\partial }{\partial x}\widehat{i}+\frac{\partial }{\partial y}\widehat{j}$ is the transverse gradient, $\widehat{i}$ and $\widehat{j}$ are the unit vector along x and y.

For simplicity, we assume that thermal blooming is in a steady state with the transverse wind, neglect turbulence and thermal diffusion effects, and assume the variation of refractive index is due only to molecular absorption, thus the relationship of temperature change with molecular absorption is [16]

$$\rho c_p\overrightarrow{v}\cdot \nabla T=-\alpha I$$

(2)

where $\rho$ is the gas density, $c_p$, the specified heat at constant pressure, $\overrightarrow{v}$, wind velocity, T, the temperature, I, the intensity of beams, and $\alpha$, the absorption coefficient. $\nabla T$ is the gradient, $\overrightarrow{v}\cdot \nabla T$ denotes dot product of velocity and the temperature gradient. The relationship between the density of the air and the index of refraction is given by the Gladestone–Dale relationship [16]

$$\mu -1=K\rho$$

(3)

where K is the Gladestone–Dale constant and changes with gas and the wavelength of laser.

$$\nabla \mu =\nabla T{\mu }_T$$

(4)

Here ${\mu }_T$ is the rate of change for the index of refraction of the gas with respect to the temperature at a constant pressure. From Equation (1) the intensity distribution can be obtained when the high-power laser travel in the atmosphere as [16]

$$I\left(x,y,z\right)=I\left(x,y,0\right)\exp \left(-\alpha z\right)\exp \left[-\frac{{\mu }_T\alpha I_{\max }}{\mu \rho c_{p}v}Q\left(x,y\right)\right]$$

(5)

where $Q\left(x,y\right)$ is a factor that determines the distribution of the intensity and can be expressed as [16]

$$Q\left(x,y\right)={\displaystyle {\int }_0^z{\displaystyle {\int }_0^{z^{\prime }}\left[\frac{\partial I^{\prime }}{\partial x}+\frac{1}{2}\frac{{\partial }^2}{\partial y^2}{\displaystyle {\int }_{-\infty }^x{I}^{\prime }d{x}^{\prime }+\frac{1}{2I^{\prime }}\frac{\partial I^{\prime }}{\partial y}{\displaystyle {\int }_{-\infty }^x\frac{\partial I^{\prime }}{\partial y}}d{x}^{\prime }}\right]d{z}^{\prime }d{z}^{″}}}$$

(6)

where $I^{\prime }(x,y,0)=I\left(x,y,0\right)/I_{\max }$ is the normalized intensity. It should be noted that Equation (6) is valid only when the blooming is not too strong. For stronger blooming conditions, nonlinear problem is needed to solve, numerically. If we neglect the variation of the intensity along propagation direction, Equation (6) can be written as

$$I\left(x,y,z\right)=I\left(x,y,0\right)\exp \left(-\alpha z\right)\exp \left[-N_{c}G\left(x,y\right)\right]$$

(7)

where G (x, y) can be regarded as the distribution of temperature.

$$G\left(x,y\right)=\left(\frac{\partial I^{\prime }}{\partial x}+\frac{1}{2}\frac{{\partial }^2}{\partial y^2}{\displaystyle {\int }_{-\infty }^x{I}^{\prime }d{x}^{\prime }+\frac{1}{2I^{\prime }}\frac{\partial I^{\prime }}{\partial y}{\displaystyle {\int }_{-\infty }^x\frac{\partial I^{\prime }}{\partial y}}d{x}^{\prime }}\right)R$$

(8)

where $N_c\approx \frac{-{\mu }_T\alpha I_{\max }{z}^2}{\mu \rho c_{p}vR}$ is a distort factor that reflect the decrement of the beam during propagation, R is the radius of the transmitter for the incoherent beams. It should be noted that Equations (7) and (8) is an approximate representation where the diffraction and the variation of intensity along the propagation direction are neglected. Under this condition, the valid of the expression is tested by many experiments [15,16].

3. Thermal Blooming of the Incoherent Combined Beams

We assume that the intensity of a laser has a Gaussian distribution, the center of each sub-aperture is ${\left(a,b\right)}_{tm}$. Thus, the intensity of the sub-aperture is

$$I_{tm}\left(x,y,0\right)=I_{\max }\exp \left\{-\frac{2}{w_0^2}\left[{\left(x-a_{tm}\right)}^2+{\left(y-b_{tm}\right)}^2\right]\right\}$$

(9)

where $I_{\max }$ is the maximum intensity of the laser. Because incoherent beam combining is the superposition of the intensity, the intensity distribution of the incoherent combined beams is [26].

$$I_{IC}\left(x,y,0\right)=I_{\max }{\displaystyle \sum _m^M\exp \left\{-\frac{2}{w_0^2}\left[{\left(x-a_{tm}\right)}^2+{\left(y-b_{tm}\right)}^2\right]\right\}}$$

(10)

where M is the number of channels of the beams. To show the thermal blooming of the incoherent combined beam, a high-power laser with $1.06 \mathsf{\mu }\mathrm{m}$ wavelength is selected as the example. The total absorption of atmosphere near ground is about $6\times {10}^{-5}/\mathrm{m}$, $\rho =1.29 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and $c_{\rho }=1005 \mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}$, the value of ${\mu }_T$ is about 2.92 × 10⁻⁴ in air.

Substituting Equation (9) into Equations (7) and (8), the evolution of the thermal blooming of high-power for the combing beam can be analytically studied. There are many arrangements of beam combining, for more details, see Reference [7]. To understand the mechanism of thermal blooming for the incoherent combined beam, the thermal blooming of two beams arranged along the direction of the wind is given in Figure 1 where the direction of the wind is from the left to right.

In the present paper, we set w₀ = 0.05 m, and the distance between two beams is 0.15 m. Thermal blooming of two-channel incoherent beams arranged along the direction of wind is given in Figure 1. It can be seen that the evolution of the beam on the left has the same thermal blooming of a Gaussian beam, namely, curvature of the beam increases with the power of the beam. This is because the heat of the right beam does not affect the temperature field around the left beam. However, the thermal blooming of the right beam is very different from that of a Gaussian beam because of the left beam. The beam gradually splits into two parts with the increase of the distort factor N_c.

The thermal blooming of three beams arranged along the direction of the wind is given in Figure 2. It can be seen that the beam on the left has the same evolution as that of a single Gaussian beam because the heat of the beams on the right hardly affects the temperature field around the left beam. As the temperature field around the central beam is only affected by the left beam, the thermal blooming of the central beam is the same as that of the right beam in Figure 1. The air surrounded the right beam in Figure 2 is heated by the middle and left beams. Compared with the middle beam, a stronger thermal lens effect will split the right beam into two parts rapidly.

The thermal blooming of seven incoherent beams arranged as a circle is shown in Figure 3. It can be seen that the arrangement can be regarded as superposition of the fundamental combination in Figure 1 and Figure 2. The two beams in the top and bottom lines are the same as in Figure 1. The arrangement of the three beams in the middle line is the same as in the Figure 2.

In practice, the direction of wind varies with time. If the arrangement of incoherent beams is shown as in Figure 4 and the direction of wind from left to right is assumed, the thermal blooming shows a different evolution from that in Figure 1 and Figure 2. The results show that each beam is independent and presents the same property of a single Gaussian beam when N_c = 1. Increasing N_c, the beam at the left has the same evolution of the thermal blooming of a Gaussian beam because the temperature field around it is not affected by the other beams. However, the thermal blooming of the two beams on the right is different from that of a Gaussian beam because the temperature field around the two beams is affected by the left beam.

If we assume that the two beams are on the left and one beam is on the right, the thermal blooming of the three beams is shown in Figure 5. Because the heat of the beam on the right does not affect the two beams on the left, the two beams on the left show the same property of one Gaussian beam. However, the air temperature around the beam on the right changes due to the heat transfer from the beam on the left; the thermal blooming of the beam on the right is different from that of a single Gaussian beam.

When we rotate Figure 3 by 90 degrees, or the direction of wind rotates 90 degrees, the evolution of the thermal blooming for seven incoherent beams is given by Figure 6. It shows great differences from that in Figure 3. Namely, the change of the wind direction will result in the variation of the thermal blooming for the incoherent combined beam. Comparing with Figure 3, we can see that only the two beams on the right gradually split into two parts. The thermal blooming of other five beams have a similar property to that of a single Gaussian beam.

From Equations (7) and (8), we can see that G is the distribution of normalized difference in temperature, or the change in normalized index of refraction. Figure 7 shows that in the line of beam array, the change in the normalized index of refraction is larger, which results in a change of the intensity of the beam. Along the direction of wind, the change in the normalized index of refraction increases. It will cause larger distortion for the beam at the end of the wind direction.

4. Conclusions

In conclusion, we studied analytically the thermal blooming of incoherent laser beams. To see the evolution of the thermal blooming for different arrangements, the thermal blooming of a basic configuration, such as two and three beams along the wind direction, a triangle, and an inverted triangle arrangement, were studied. From the property of the thermal blooming of the basic configuration, the evolution of the thermal blooming of the incoherent combined beam with different arrangements can be predicted. As an example, the thermal blooming of seven channels of incoherent beams was studied. To see the variation of the thermal blooming, the seven channels of incoherent beams were rotated by 90 degrees and studied. The results show that the beam can be split into two parts when heat passes through the beam. It can be explained as due to atmospheric absorption heating the air resulting in a thermal lens effect. The beam will be split into two parts when it passes through the heated air.