Effects of Linewidth Broadening Method on Recoil of Sodium Laser Guide Star

: The linewidth broadening of the circular-polarized continuous wave laser mitigates the recoil effects of the sodium laser guide star very well. By choice of the optimal laser linewidth, the relations between the laser intensity and average spontaneous emission rates are obtained. The numerically simulated results show that the 1–100 MHz linewidth broadening effectively weakens recoil and enhances the average spontaneous emission rates. For laser powers from 10 W to 60 W, considering the intensity distribution with random at the mesospheric sodium layer, when the laser linewidth is broadened to be 1–100 MHz from 0 MHz, increments of the return photons go up to 110% from 50% and do not have an effect on the spot sizes of the sodium laser guide star. Several cases have proven that the linewidth broadening method is correct. Further calculations show that the linewidth broadening method similarly applies to the case of the multi-mode laser. Furthermore, the linewidth broadening of re-pumping should be taken into account. and investigation, and and


Introduction
There is 600 kg of sodium in the atmosphere all around the earth, which provides a fundamental condition for the form of the sodium laser guide star [1]. It is well known that the sodium laser guide star is applied in the wave-front detection of adaptive optics and helps to improve high resolution imaging through the atmosphere. The high return photons are beneficial to enhancing the signal to noise ratio in the wave-front detection. When the laser interacts with mesospheric sodium atoms, the three evils named by Holzlöhner [2] cause the drop in the brightness of the sodium laser guide star. The geomagnetic field effects weaken the polarization of the circular-polarized laser. The stimulated emission caused by a high intensity reduces the spontaneous radiation of sodium fluorescence. Recoil makes sodium atoms in the range illuminated by the laser drift to a higher and higher frequency as time goes on [3]. The recoil phenomenon leads to a decrease in the total number of excited sodium atoms and a reduction in spontaneous emission rates. Fortunately, the laser with 1.713 GHz sidebands has been employed to pump sodium atoms from F = 1 to F = 2 ground states [4]. The continuous wave laser with limited powers enables the stimulated emission to be effectively dropped [5]. Not long ago, Bustos et al. [6] proposed a blue-drift method to drop recoil effects for the 0 MHz linewidth laser. This work shows a 60% increase in return photons and a 50% decrease in re-pumping power for the sodium laser guide star excited by a continuous wave laser. In this article, we focus on the study of the linewidth broadening of the circular-polarized laser to alleviate recoil effects of the sodium laser guide star. In fact, the linewidth broadening method is used to modulate the intensity distribution of the laser in the spectrum to weaken recoil effects. The linewidth of the continuous wave laser is always less than 1 MHz, which is theoretically regarded as 0 MHz. We anticipate that there is a laser modulation technology that will be applied to broaden the laser linewidth from 0 MHz or tens of kHz to above 1 MHz. Up to now, Chamoun and Digonnet [7] have proposed a Gaussian white noise (GWN) phase modulation to broaden the laser linewidth which uses the linear Pockels effect of the electro-optic crystal by an external electric field. Its advantages lie in that the laser wavelength can remain stable, and the modulated laser linewidth is independent of the natural linewidth of the laser. By the theoretical models and the numerical simulations, the return photons from the sodium laser guide star excited by the continuous wave (CW) laser are calculated. The study results show that the linewidth broadening method markedly increases the return photons. In the process of the research, theoretical models are presented in Section 2. The steady state solution of the two-level Bloch equation is applied in the excitation probability of sodium atoms. The average spontaneous emission rate of excited sodium atoms is introduced. The expressions of return photons and spot sizes of the sodium laser guide star are described. In Section 3, the numerical method is described in detail and the parameters are listed. In Section 4, results of the numerical simulations are obtained. Powerful evidence indicates that linewidth broadening can effectively weaken the recoil effects. Then, the choice of the optimal linewidth broadening is analyzed, and relational expressions between laser intensity and average spontaneous emission rates are obtained. In Section 5, we discuss the effects of linewidth broadening on the return photons and spot sizes of the sodium laser guide star and consider the linewidth broadening for the re-pumping laser. In particular, we further simulate the influence of linewidth broadening on the recoil effects of the sodium laser guide star in the case of the multi-mode laser. Finally, we summarize our work, including the optimal range of laser linewidth broadening and the impact on the returned photons from the sodium laser guide star.

Theoretical Models
The excitation of the sodium laser guide star needs to launch the sodium laser to the mesospheric sodium layer. Under the usual conditions, the spectrum center of the laser is required to aim at the center frequency of the Doppler shifts of the sodium atom under the thermal equilibrium states. The single-mode mono-mode CW laser is applied in the sodium laser guide star and has the highest intensity in the spectrum center. A great deal of sodium atoms move to a higher Doppler shift after absorbing laser photons, and the Doppler shift increases 50 kHz. This process can be described as follows [8]: where v D is the Doppler shift after a sodium atom absorbs a photon, v D is the former Doppler shift, h is the Plank constant, λ is the 589.159 nm wavelengths, and m Na is the mass of a sodium atom. "+" denotes the motion of a sodium atom along with the direction of laser propagation, and "−" denotes the opposite one. We consider the situation of two energy levels for interactions between the circularly polarized laser and sodium atoms. This process is described by the optical Bloch equation. The relevant theory and experiments [9,10] have proven that when the circular-polarized light with wavelength 589.159 nm and sodium atoms with the D 2 structure interplay, in a very short time, all energy level transitions will come into the cycles between ground states F = 2, m = 2 and excited states F' = 3, m' = 3. The following figure represents the D 2 hyperfine structure of the sodium atom and the two-level cycle [11]. In Figure 1, D 2 denotes the energy levels between 3P 3/2 and 3S 1/2 of sodium atoms, D 2a denotes the level transitions excited by the light with wavelength 589.159 nm, D 2b denotes the level transitions excited by the light with wavelength 589.157 nm, σ + denotes the right-handed circularly polarized light, and the solid double arrows denote the twolevel cycle between 3P 3/2 (3, 3) and 3S 1/2 (2,2). The left and second column data denote the frequency intervals between energy levels. The two-level cycle obtains a stable state in a microsecond order time for the continuous wave laser [12]. The probability of an excited sodium atom is written by [4] where v L is the center frequency of laser, v p is the transition frequency of the two energylevel atoms, I denotes laser intensity, I sat represents the saturated intensity, I sat = πhv/ 3λ 2 τ , τ is the lifetime of sodium atoms in the excited states, and v is the radiative frequency of a photon. At the outset, the distribution of sodium atoms in the mesosphere satisfies the Maxwell velocity distribution law. The normalized distribution of sodium atoms is given by [13] N v D = (4 ln 2/π) 1/2 δv D e −4 ln 2v 2 where N v D is the percent of sodium atoms corresponding to a Doppler shift, δv D denotes the linewidth of sodium atom distributions in the mesosphere, and +∞ −∞ N v D dv D = 1. If ∆v D is regarded as a velocity interval, then the percent of sodium atoms in the small enough velocity interval is For a single-mode laser, the intensity distribution with the Doppler shift, v D , is written as [13] where δv L D is the laser linewidth, and I is the total intensity in a certain area illuminated by laser. The center of the laser intensity is coincident with that of the sodium atomic distributions. Thus, the peak probability of the excited sodium atoms can be obtained when the laser and sodium atoms satisfy the resonant conditions. The probability of excited sodium atoms at every v D is where those sodium atoms away from the resonant conditions are taken into account. These atoms can be excited by near resonance. For a fixed value I(v D ), the excitation probability of sodium atoms including near resonance is the following form [4]: where v D is the Doppler frequency shift relative to v D .
To explicitly indicate the excited effectiveness of the ground sodium atoms, the concept of the average spontaneous emission rate is introduced. At the same time, the stimulated emission is ignored. For the CW laser, without regard to the geomagnetic field, the average spontaneous emission rate is where ∆T is the cycle time of sodium atomic collisions, n denotes the times of excited sodium atomic upward transitions, N v D denotes the sum of sodium atoms of decay from the excited states and the remainder in the ground states after the partial sodium atoms are excited every time, and 5/8 refers to the proportion of sodium atoms in the ground states corresponding to m = 0, ±1, ±2 [14]. Sodium atomic collisions, including velocity exchange, spin damping, and beam exchange, make numerous excited sodium atoms return the F = 2 ground states. Milonni [15] has estimated the velocity exchange time to be 100 µs. This implies that the motional states of all sodium atoms will make a return after 100 µs. However, Holzlöhner [2] has calculated the time to be 35 µs. In this article, 35 µs is regarded as the cycle time.
In adaptive optics, enough return photons from the laser guide star are important for the wave-front detection [16]. For the continuous wave laser, the return photons in the unit area and the unit time on the telescope plane are written by [17] where T 0 is the atmospheric transmissivity, β is the backscattering coefficient of excited sodium atoms, C Na is the column density of sodium atoms in the mesosphere, L is the vertical distance from the telescope plane to the center of the mesospheric sodium layer, ζ is the zenith between the laser beam and the vertical direction, s is the area illuminated by the laser, and f m is the scale factor of depolarization since the geomagnetic field cuts down on the number of sodium atoms in the F = 2 and m = 2 ground states [18]. Values of f m depend on the angles between the circular-polarized laser beam and the direction of the geomagnetic field and the period of Larmor precession. According to an experimental study [19], this factor can be reduced to f m = 1 − 0.6552B/B 0 sin θ , where B and B 0 (B 0 = 0.51 Gs) are the magnitude of the geomagnetic field, and θ is the angle between the directions of the laser beam and the geomagnetic field vector. According to Equations (7) and (8),R relates to laser intensity. Since laser propagation in the atmosphere is easily affected by atmospheric turbulence, laser intensity distributions present random states in the mesosphere. Laser field propagation accords to the following parabolic Equation [20]: where k 1 stands for the wave number, z is the path of laser propagation, E is the amplitude of the light field, and n 1 denotes the fluctuation of the atmospheric refractive index around 1. By solving Equation (10) , the light field at z is achieved. Then, the laser intensity distributions are calculated.
In addition to the return photons, the spot sizes of the sodium laser guide star are required to be small for the wave-front detection. The effective radius of spot size is exploited to characterize the energy focusability of the sodium laser guide star at the mesospheric sodium layer. This concept is defined as [21] R e f f = where I b (x, y) is the fluorescent intensity of the sodium laser guide star at the sodium layer, observed from the orthogonal direction with two-dimensional coordinates (x, y), and r is the distance from I b (x, y) to the centroid of the light spot. I b (x, y) is calculated by the following expression: where T sec(ζ) 0 β C NaR f m denotes the backscattering photons of the sodium laser guide star in unit time, ∆ s is the very small area of radiative fluorescence which meets ∆s = ∆x∆y, and hv stands for photon energy in J.
Based on the above analysis, we conclude that the recoil effects cause the red shifts of sodium atoms. Thus, a mass of sodium atoms miss excitation so that the spontaneous emission rate reduces when recoil occurs. In order to mitigate these effects, we propose that the laser linewidth should be broadened to weaken these recoil effects.

Numerical Simulation Methods
To explore the linewidth broadening mitigating recoil effects of sodium laser guide star, numerical simulations are carried out. A fundamental assumption is that the two-energy level cycle of sodium atoms is able to be very well maintained due to enough re-pumping. Since the re-pumping power is about 10%, even less than 10%, in the total laser power [22], this power is ignored in the numerical simulations. The average spontaneous emission rates and return photons with respect to this power are attributed to the total values of the cycles between ground states F = 2, m = 2 and excited states F' = 3, m' = 3.
According to the theoretical models, Equations (3)-(10) are discretized. A numerically simulated method is employed to solve Equation (8). Its discrete formation is written as where nτ = ∆T, τ = 2τ, τ represents the time of decay and once again the excitation of a sodium atom, i is defined as the number of velocity groups, N v D (i) denotes the number of sodium atoms in the i-th velocity group, and p 2 (i) denotes the excitation probability of sodium atoms in Equation (7). For the purpose of obtaining enough return photons, from Equations (7) and (8),R is required to be maximum under the same other parameters. We set 200001 velocity groups with the adjacent interval ∆v D = 1.0 × 10 4 Hz. The range of Doppler shifts is taken from −1.0 GHz to 1.0 GHz.
To solve Equation (10), multi-phase screen method [23] is employed. Moreover, the atmospheric turbulence model of Greenwood [24] and power spectrum of Kolmogorov [25] are used in simulations of laser atmospheric propagation. Laser intensity distributions are discretized as 512 × 512 grids. Laser intensity is thought as concentrating on a plane through the whole sodium layer. Then, the return photons are calculated according to Equation (9). Similarly, Equation (11) is discretized as the following form [21]:  (14) where I b (m, n) is intensity of sodium laser guide star in the m-th row and n-th column, and m and n are, respectively, the row and column ordinals of 512 × 512 grids. Due to the effects of atmospheric turbulence, the distribution of laser intensity is randomized in the mesospheric sodium layer. To simulate laser intensity, the multi-phase screen method is used to solve Equation (10) [23]. The power spectrum of Kolmogorov turbulence is taken into account, and its expression is [24] Φ(k) = 0.033r −5/3 where r 0 is atmospheric coherent length, k is spatial frequency, r 0 = 0.185 , C 2 n is refractive index structure constant for atmosphere, and h is the atmospheric vertical height from the ground in m. The atmospheric turbulence model of Greenwood is [25] On the thin layer perpendicular to the laser transmission direction, the power spectrum of atmospheric phase is written as [26] Φ n (k) = 2π(2π/λ) 2 0.033k −11/3 z+∆z z C 2 n (ξ)dξ.
Then, Equation (17) is filtered by a complex Gaussian random matrix a (m, n) and through the inverse Fourier transform to the discretized phase screen as follows [27]: where L x and L y are side lengths, and N x and N y are the number of grids. Furthermore, the third harmonic method is used to compensate for the low frequency inadequacy. Finally, the total phase S(r, z), including the low and high frequency components, modulates the light field. Thus, the solution of Equation (10) is expressed as [28] E(r, z + ∆z) = exp i 2k where exp i 2k z+∆z z ∇ 2 ⊥ dξ is caused by vacuum diffraction.

Simulation Parameters
This simulation study involves laser characteristics, atmospheric properties, and sodium layer features. All relevant parameters are listed in Table 1 [2].
When θ = 30 • and B = 0.228 Gs, the scale factor of depolarization f m = 0.8466. Specifically, a laser with TEM00 mode is launched at collimation.

Recoil and Linewidth Broadening
The continuous wave laser is single-mode with a 0 or 2.0 MHz linewidth. For the 2.0 MHz linewidth laser, its intensity distribution is expressed as Equation (5). The total intensity of the laser is taken as I = 150 W/m 2 . It is assumed that sodium atoms are excited every 32 ns due to the cycle time of excited states. The tens of nanoseconds in the ascending stage are ignored before steady states. For the 0 MHz laser, the normalized distributions of sodium atoms after recoil are simulated at t = 10 µs, 20 µs, and 35 µs as in Figure 2. In order to study the effects of linewidth broadening on the mitigation of recoil, the linewidth of the continuous wave laser is taken to be 2.0 MHz in Equation (5). After t = 10 µs, 20 µs, and 35 µs, the normalized distributions of the sodium atoms are presented in Figure 3.  From Figure 2, one can see that recoil results in the accumulation of sodium atoms at higher and higher Doppler shifts as time goes on. Compared with Figure 2, after linewidth broadening is employed, the peaks of recoil greatly drop in Figure 4, and the corresponding three sodium atomic distributions are coincident. In addition to this, the laser intensity also influences recoil, as is shown in Figure 4. With the same linewidth broadening method as the above, after t = 35 µs for I = 50 W/m 2 , 100 W/m 2 , and 150 W/m 2 , the situations of mitigated recoil are shown in Figure 5.   Figure 4 shows that high intensity causes more drastic recoil and aggravates the adverse situations. Simultaneously, the higher intensity makes sodium atoms drift to the higher Doppler frequency shifts. Figure 5 reveals that the linewidth broadening method can effectively alleviate the recoil effects for different laser intensities.

Choice of Optimal Laser Linewidth
In practice, if the recoil effects have to be dropped, and the laser is required to modulate the intensity distribution in Equation (5). The linewidth broadening of the laser intensity distribution aims at achieving the maximal excitation probability of mesospheric sodium atoms. The maximal average spontaneous emission rate is necessary. Therefore, we simulate the average spontaneous emission rates by the linewidth broadening from 0 to 1.0 GHz. In light of Equations (2)-(9), the average spontaneous emission rates with the intensity from 0 to 1500 W/m 2 are simulated in Figures 6 and 7.  When the laser is broadened to a larger linewidth, the average spontaneous emission rates instead drop. In the case of lower intensity, the laser linewidth broadening finitely gains the average spontaneous emission rates in the range of l-100 MHz. However, it is not that the wider linewidth can obtain the best effect, but that the average spontaneous emission rates have a maximum for the linewidth from 1 MHz to 100 MHz. However, the average spontaneous emission rate at δv L D = 0 MHz is lower than the peak values. In Figures 6 and 7, the peak values of average spontaneous emission rates are the same in terms of linewidth.
We hope that the linewidth broadening of laser intensity distributions makes the average spontaneous emission rate maximal. Figures 8 and 9 simulate the average spontaneous emission rates for laser linewidth from 1 to 10 3 MHz and laser intensity from 5 to 1500 W/m 2 .

Effects of Linewidth Broadening on the Return Photons and Spot Sizes
Generally, the higher laser power forms higher peak values of intensity in the mesospheric sodium layer. In accordance with the simulation method and parameters in Section 3, the return photons are calculated for a laser power from 10 W to 60 W at δv L D = 0 and 10 MHz. Meanwhile, the spot sizes of the sodium laser guide star are computed. These data are listed in Table 2. Based on the above data, we can summarize three results. First, linewidth broadening is able to achieve the most return photons. We find that the laser linewidth at δv L D = 10 MHz produces more return photons than that at = 0 MHz. Compared with that at δv L D = 0, the return photons at δv L D = 10 MHz increase 50.4%. Second, increments of the return photons increase with laser power. For laser power from 10 W to 60 W, calculations show that the increment of return photons goes up to 110% from 50.4%. Thirdly, in general, the effective radii at δv L D = 10 MHz are equal to those at δv L D = 0 MHz. Therefore, the linewidth broadening method is useful to improve the signal-to-noise ratio of wave-front detection in adaptive optics.
It is well known that the strength of atmospheric turbulence can affect the intensity distribution of the laser. For the atmospheric turbulence model of Greenwood, the whole atmospheric coherent length is 15.6 cm (perpendicular to ground upward for wavelength 500 nm). When the whole atmospheric coherent length is 21.6 cm, such as the atmospheric turbulence model of Mod-HV [29], the intensity distribution of the laser will trend to the higher values in space. However, because linewidth broadening leads to the decrease of peak intensity in the spectrum, effects of recoil will be weakened .

Considerations of Linewidth Broadening Method
The two energy-level cycles of sodium atoms are influenced by several factors. Among them, depolarizations of the geomagnetic field enable a number of sodium atoms detuning to F = 2 m = 2 ↔ F = 3 m = 3 cycles. To maintain these two energy-level cycles, repumping ( D 2b in Figure 1) is used to excite the sodium atoms, trapping in F = 1 ground states [2]. The laser is generally modulated with a sideband by 1.713 GHz. This sideband linewidth should be broadened at δv L D = 1-100 MHz. Otherwise, the recoil will weaken the efficiency of re-pumping.
The single-frequency CW laser was once used to excite the sodium laser guide star [30]. This laser's power is 4-5 W, but the efficiency is low because of the 0 MHz linewidth. A modeless CW laser with a 3 GHz linewidth is thought to be more efficient due to the wide spectrum, but its efficiency is not very high according to the above analysis [31]. The most famous CW laser is the 50-watt FASOR at the Starfire Optical Range [32]. On 30 May 2006, the experimental average value of the return photons was about 1.8 × 10 7 ph/m 2 /s for the 40-watt FASOR, and the zenith of the laser launch was 0 degrees. Furthermore, the angle of the laser beam and geomagnetic field vector was about 28 degrees, similar to the 30 degrees in Table 1. Moreover, the column density of the sodium layer is nearly as much as 4 × 10 13 cm −2 , and the atmospheric transmission is 0.91. In particular, the laser linewidth is 1 × 10 4 Hz instead of 0 MHz. Thus, the average spontaneous emission rates are more than that of the 0 MHz linewidth laser.Therefore, in Table 2, the calculated value of the return photons for the 0 MHz linewidth laser with 40 W is basically in line with the actual situation. Figure 11 depicts the average spontaneous emission rates for the laser linewidth at δv L D = 0, 0.01, 1 MHz. From Figure 11, one can see that the average spontaneous emission rates at δv L D = 0.01 MHz are the same as half at δv L D = 1 MHz for a laser intensity more than 25 W/m 2 . If the linewidth of the CW laser at the Starfire Optical Range is broadened to be 1 MHz, the return photons will increase about one times. At Gemini North (Mauna Kea), the CW laser, which is mono-mode, has a linewidth at 10 MHz, and has a zenith of 45 degrees, the return photons are simulated to be 2.22 × 10 6 ph/m 2 /s by an approximately 10 W laser [33].
The column density of the sodium layer is merely 2 × 10 13 cm −2 . If this parameter is taken as 4 × 10 13 cm −2 in Table 1, the calculated value is approximately 4.44 × 10 6 ph/m 2 /s. This result is relatively close to 5.36 × 10 6 ph/m 2 /s in Table 2.

Effects of Linewidth Broadening on Recoil of the Multi-Mode Laser
The multi-mode CW laser which is utilized to excite the sodium laser guide star has the advantage of a wide bandwidth. However, its efficiency is not seemingly as good as a single-mode laser [33].This laser has several centers of frequency. The expression of the laser intensity is [8] where j stands for the number of modes, k is an integer, v 1 is an interval of adjacent modes, v D is the Doppler shift, I(v D ) = I(0) exp −4 ln 2(v D /δv b ) 2 is the envelop curve, I(0) is the peak power, and δv b is the linewidth of envelop curve. δv 0 is the mode linewidth.
Taking v 1 = 150 MHz, k = 1, δv 0 = 10 4 Hz, and δv b = 1 GHz results in a laser spectrum with three modes as in Figure 12. When the laser mode linewidth is broadened to be 60 MHz, the laser spectrum with three modes is shown as Figure 13.
In light of the above simulated method, at I = 150 W/m 2 , we simulate linewidth broadening influences on recoil in Figure 14. Results indicate that the linewidth broadening of three modes effectively weakens the recoil peaks of the distribution of sodium atoms.   According to the results in Figure 15, when the linewidth of the three-mode CW laser is broadened to 60 MHz, the average spontaneous emission rates increase over 10 times at I = 150 W/m 2 and exceed those values of the mono-mode laser at linewidth 0 MHz. Such a case greatly enhances the return photons of the sodium laser guide star. Thus, we produce a result that the linewidth broadening method is capable of improving the recoil effects of the sodium laser guide star in the case of the multi-mode laser.

Conclusions
This article carefully investigates the effects of the linewidth broadening method on the recoil of the sodium laser guide star. The numerical results show that the linewidth broadening method increases the return photons of the sodium laser guide star. Moreover, for a laser over 50 W, the return photons increase more than one time. Furthermore, this method will not make a difference in the spot sizes of the sodium laser guide star.
The optimal values of linewidth broadening are 1-100 MHz according to our simulations. This linewidth is capable of achieving wholly optimal average spontaneous emission rates. Our simulations reveal that the mono-mode CW laser at a linewidth less than 1 MHz and more than 100 MHz cannot obtain high average spontaneous emission rates. In order to obtain the best results, the laser linewidth should be modulated to be 1 MHz or between 1 and 100 MHz. In addition, the sideband linewidth of re-pumping should be correspondingly broadened. Further analysis shows that the linewidth broadening method is capable of weakening the recoil effects of the sodium laser guide star for the multimode laser. A simulated result of the three-mode laser indicates that the average spontaneous emission rates of the sodium laser guide star significantly improve.
Several cases have proven that the linewidth broadening method is consistent with the actual situation. The current laser linewidth modulation method can provide a reference for laser linewidth broadening. We hope that our results are proven by experiments.