## Abstract

Sodium beacon adaptive optics (AO) system has been proved to be a highly productive tool for improving the resolving power of large-aperture ground-based telescope imaging. The performance of the AO system is mainly limited by photon return of the sodium beacon, which is determined by the coupling efficiency that characterizes the interaction rate between sodium laser and sodium atoms. The interaction processing is strictly influenced by the collisions of sodium atoms with other molecules (N_{2}, O_{2}). Most of the existing collision kernels are assumed as the “memoryless” hard collision, which is completely velocity reset in a Maxwellian distribution of the sodium atoms after scattering. To be more realistic, we adopt a more practical “memory” Cusp weak collision kernel, considering the velocity distribution of sodium atoms after collisions are correlated with the velocity before collision. By solving the Bloch equations, the processing for the interaction between sodium laser and sodium atom with Cusp kernel is established, and the coupling efficiency of sodium beacon with different collision kernel by analyzing the population is obtained. The researching results show that, for “memoryless” kernel, comparing to Cusp kernel with shaping parameter (s) of 100, the coupling efficiency is larger than 56% at best case; for sodium laser with 12% power detuned to D_{2b} line and at a power density ranges from 10 to 100 W/m^{2}, the coupling efficiency of “memoryless” kernel is nearly the same as “memory” Cusp kernel with s of 10, 100 and 3 Cusp kernel.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Adaptive Optics (AO) system originally proposed for astronomical large-aperture telescopes by Babcock in 1953, which compensating turbulent distortions in real time to achieve near diffraction-limited resolution images [1]. However, the scarcity of having a bright natural beacon within the target object isoplanatic area limited the use of AO for astronomical observations. To overcome this limitation, artificial beacons generated by Rayleigh (Rayleigh beacon) or resonance scattering (sodium beacon) were proposed [2]. Comparing to Rayleigh beacon, sodium beacon offers more sufficient sampling of atmospheric turbulence due to higher altitude, thus becomes an essential component of modern AO system for most existing large-aperture ground-based telescopes, like the twin Gemini telescopes, the Keck I/II and the Subaru telescope; and next generation large telescopes, such as the Thirty Meter Telescope, the Giant Magellan Telescope and the European Extremely Large Telescope [3]. Recently, a 12 m diameter Large Optical-infrared Telescope (LOT) is planning constructed in China [4], using a sodium beacon AO system can be a key role to improve the imaging quality significantly.

For a single sodium beacon AO system, the wavefront sensing error induced by the limited photon return of the beacon is a mainly factor of the correction performance of the post-correction AO system, which account for nearly 70% of the entire error [1]. The effective sodium photon return depends on the coupling efficiency which is determined by the action processing between sodium laser and sodium atoms [5]. Sodium atoms pumped by the laser irradiation from ground state into an excited state, after 16 ns of the natural lifetime, atoms in the excited state de-excited back to the ground state by spontaneous decay. The processing is mainly subjected to the following factors: Doppler broadened absorption line of sodium atoms, the recoil of the sodium atoms upon absorption and emission, sodium atoms into and out the laser field and collisions of the sodium atoms with other molecules (N_{2}, O_{2}) [6]. According to Milonni’s research, collisions become important for very long pulses or CW (continuous wave) irradiation, it can limit the degree of optical pumping and can thereby diminish the advantage of using circularly polarized lasers [7]. X. Liu et al. analyzed the influence of collision on the photon return of sodium beacon, and shown that for the linear polarized light, at a power density exceed 0.15W/m^{2}, collision has a large influence on the photon return [8]. Meanwhile, the output power of a sodium laser has reached 50W, with a spot size of FWHM (full width at half maximum) is about 1 arcsec [9], power density of the sodium beacon is more than 10 W/m^{2}. Under this circumstances, the collisions of the sodium atoms may have huge impact on the photon return of the sodium beacon. So, it is very necessary to analyze the impact of the collision on photon return of sodium beacon. Sodium atom D_{2} absorption line profile is well fitted by Gaussians, the frequency spectrum of the pump laser is broadened as well. The amount of the sodium atoms in a frequency (velocity) decreased which is caused by absorbing incident photons from the sodium laser. However, due to the existence of the collision effect, the atoms in other velocity will spread into this velocity, causing diffusion of the pumped atoms in velocity space, and further influence the photon return of sodium beacon. At the same time, the Larmor precession frequency can be measured more accurately by analyzing of the collision mechanism, which can be used to enhance the photon return. In 2018, Bustos et al. conducted an experiment to measure the Larmor precession frequency, and suggested that the collisional processes can be inferred from their approach [10]. So, defining the collision processes can also provide guiding significance for the design of sodium laser.

The sodium layer locate from 85 to 100 kilometers with a peak near 90 km, and stated at a temperature around 185 K. The collisions, triggered by the thermal motion of the sodium atoms and other molecules, could result in an energy and momentum exchanging. Happer believes that there are two main aspects of the impact of collisions [11]. First, the velocity exchanging collision resulted from the velocity distribution changing of sodium atoms. Second, the spin exchanging collision originated from the exchanging of spin angular momentum of electron between sodium atoms and O_{2} or O. Morgan’s researches show that the two collisions are uncorrelated, so the influence of the two collisions on the process of laser and sodium atoms can be analyzed separately [12]. In 1999, Milonni et al. analyzed the collisions of the sodium atoms with other mesospheric species, the value of the collision rate has a significant influence on sodium beacon photon return for CW or long pulse sodium laser [7]. In 2010, Holzlöhner et al. conducted a numerical simulation using Bloch-equation with velocity exchanging collisions, however, their simulation assumes that the velocity exchanging collisions are “memoryless” (completely velocity reset to Maxwell-Boltzmann distribution after scattering) which need to be refined [6]. In the course of his follow-up research, the velocity-changing were considered, but the model didn’t state [13]. Similarity, in 2016, Hackett et al. realized that “memoryless” collision model can’t describe the process of interaction between Narrow linewidth continuous laser and sodium atoms accurately, and then a new Non Boltzmann velocity-changing collision model was established [14], which modified the collision rate based on experience, but no quantitative analysis results were given for the accuracy of the empirical model [15]. In 2013, McGuyer et al. aimed at the defect of the general Kernel-Storer (KS) collision model, which can’t describe the collision process at low pressure buffer gases accurately, Cusp Kernels was proposed and is perfect for modeling the effect of velocity exchanging collisions of sodium atoms [16]. At the same year, Bhamre et al. experimentally demonstrated Cusp Kernels is the best describing of the collision processing between potassium atoms and low pressure helium [17]; Marsland et al. used a 3 Cusp Kernel to describe velocity exchanging collisions between rubidium and helium, which is in excellent agreement with the experiments in laboratory [18].

All these previous works focus on establishing a velocity exchanging collision, and there are few researches applied Cusp Kernel to sodium beacon photon return simulation or experimental verification. However, the processing of velocity exchanging collision makes the absorbing frequency of the atoms (velocity of atoms) not match the pumping laser, then effects the performance of sodium beacon. So, establishing a velocity exchanging collision model more accurately, like “memory” Cusp weak collision kernel, can simulate the photon return of sodium beacon more accurately.

In section 2, the law of thermal motion of atoms and molecules with Cusp and KS collision kernel are compared. In section 3, the Cusp kernel is applied to the simulation of interaction processing of sodium atoms and other molecules by solving Bloch equations. In section 4, the coupling efficiency of sodium beacon is obtained. In section 5, the population distribution at each sub-levels at different power densities are discussed. In section 6, conclusions and future works are presented.

## 2. Cusp Kernel for collisions

The thermal movement causes collisions between molecules all the time, thus changing the velocity distribution of molecules. In the state of thermal equilibrium, even the collision between molecules exist, the overall velocity distribution of molecules does not change, namely, the velocity distribution still obeys Maxwell-Boltzmann distribution. For laser interacting with atoms, KS collision kernel is usually used to describe the velocity variation of atoms in the direction of laser propagation, which has an analytical form [16]:

*y*to

*x*after collision along the direction of pump laser transmission.

*m*is the atomic mass.

*k*is Boltzmann’s constant.

_{B}*T*is temperature. $\alpha$ is the “memory parameter” with 0≤$\alpha$<1 and a corresponding width $\beta = \sqrt {1 - {\alpha ^2}}$: the processing will be hard collision with $\alpha$ approaching to 0, and the velocity distribution before collision and after collision are uncorrelated, which is adopted by Holzlöhner; the processing will be a “weak collision” with $\alpha$ approaching to 1, the velocity distribution of sodium atoms after and before collisions are correlated. In reality, there will be a sharp peak near the initial velocity distribution after collision, so the KS kernel should be refined [19,20]. McGuyer et al. revised the KS kernel to have the value of $\alpha$ approaching to 1, a probability density is defined [16]: Where

*s*is the nonzero “sharpness parameter”, which describes the model matches a hard or a weak collision kernel. For large positive sharpness, the kernel is closer to weak collision, the smaller the value of s is, the kernel is closer to hard collision. The α(0≤$\alpha$<1) is the “memory parameter”. Then, the analytical form for Cusp collision model can be obtained:

*y*to

*x*after collision along the direction of pump laser transmission with Cusp collision kernel. ${W_{cusp}}(x,y)$ satisfied two conditions as below:

- (1). $\int {{W_{cusp}}(x,y)dx = 1}$
- (2). In the state of thermal equilibrium, the velocity distribution after collision is still Maxwell-Boltzmann distribution.

Based from Eq. (1) and Eq. (3), the initial velocity of sodium atom is 500 m/s at the temperature of 185 K, the velocity distribution after collision with KS kernel and Cusp kernel were shown in Fig. 1. Figure 1(a) shows that for KS kernel, with the increasing of the “memory parameter “ $\alpha$, especially approaching to 1, the more concentrated the velocity distribution of sodium atoms after collision is near the initial velocity; for $\alpha$ is 0, the velocity distribution of sodium atoms after collision is independent of the initial velocity. Figure 1(b) shows that for Cusp kernel and the “sharpness parameter” *s*>>1, in contrast to KS kernel, the velocity distribution display a sharp peak near the initial velocity (500 m/s). To better approximate a real collision kernel, a 3 Cusp kernel which is established by superposing three Cusp kernels is compared, the superposition parameters are [s_{1}, s_{2}, s_{3}] = [7.8, 27.2, 500] and corresponding weights are [f_{1}, f_{2}, f_{3}] = [0.13, 0.37,0.5], which is labeled 3 Cusp.

## 3. Methodology overview

Semiclassical theory is often used to simulate the physical process of laser interacted with sodium atoms, the laser is treated as a classical electromagnetic wave and expressed by Maxwell equation, the atom is treated as a quantized system expressed by the Bloch equation, the evolution of the state of the sodium atoms which interacted with sodium laser can be obtained by solving Bloch equations.

In order to calculate the photon return of sodium beacon more accurately, the influence of external environment on the interaction process of sodium laser and sodium atoms should be taken into account, such as intensity and zenith angle of the geomagnetic field, the Doppler broadening spectrum of sodium atom, rate of atom into and out the laser field, recoil, spin exchanging collision and velocity exchanging collision.

#### 3.1 D_{2} line transition process of sodium atom

When a pulse of sodium laser pass through sodium layer, some atoms in 3S_{1/2} state will be excited to 3P_{3/2} state with a photon absorbed, then, some atoms in 3P_{3/2} state may de-excite back to 3S_{1/2} state by spontaneous decay, and a photon in the same wavelength will be emitted isotropically. Thus, a sodium beacon was generated. Hyperfine energy level structure of sodium atom in 3S_{1/2} and 3P_{3/2} is shown in Fig. 2 [21]. The ground energy level 3S_{1/2} is split into two hyperfine energy levels F=1 and F=2 with frequency separation 1.772 GHz; the excited energy level is split into four hyperfine energy levels which are labeled F’=0,1,2,3, with frequency separation 15.8 MHz, 34.4 MHz and 58.3 MHz respectively. An atom excited form F=2 to 3P_{3/2} with absorbing a photon in specific frequency which called D_{2a} line, there are three hyperfine transition proceedings for D_{2a} line: |F=2>→|F’=3>, |F=2>→|F’=2>, |F=2>→|F’=1 > . An atom excited from F=1 to 3P_{3/2}, with absorbing a photon in specific frequency which called D_{2b} line, the hyperfine transition proceedings for D_{2b} line are |F=1>→|F’=2>, |F=1>→|F’=1> and |F=1>→|F’=0 > .

#### 3.2 Establishment of Bloch equations

The evolution of laser interacted with sodium atoms can be described by Liouville equation [22]:

^{−34}Js. $\boldsymbol{H}$ is a total Hamiltonian, in this paper, $\boldsymbol{H} = {\boldsymbol{H_0}} + {\boldsymbol{H_E}} + {\boldsymbol{H_B}}$, where ${\boldsymbol{H}_0}$ is the unperturbed energy Hamiltonian; ${\boldsymbol{H_E}}$ is the atom interact with external fields, ${\boldsymbol{H_E}} = d \cdot E$, where

**is electric dipole operator,**

*d***is the electric field intensity of the laser; ${\boldsymbol{H_B}}$ is the Hamiltonian of magnetic dipole interaction, ${\boldsymbol{H_B}} ={-} \boldsymbol{\mu} \cdot \boldsymbol{B}$, where $\boldsymbol{\mu}$ is the atomic magnetic moment,**

*E***is the geomagnetic intensity. $\hat{\mathbf\Gamma }$ is the relaxation matrix, which is a diagonal matrix, the element on the diagonal is the decay rate of each energy level, the relaxation is caused by spontaneous emission, atom into and out the laser field and velocity exchanging collision. $\mathbf{\Lambda} $ is the repopulation matrix, which describes the repopulation of the ground state levels due to the spontaneous decay of the excited states, atom into and out the laser field, velocity exchanging collisions, spin exchanging collisions and recoil.**

*B*#### 3.3 Spontaneous emission

When sodium atoms spontaneous emission photons, it caused the relaxation of each Zeeman sub-level, the relaxation is equal to the spontaneous emission rate of the atom from the sub-level decay to lower state. In the population regeneration matrix ${\bf \Lambda }$, the transformation from upper state density matrix ${{\boldsymbol \rho }_{rs}}$ to lower state ${{\boldsymbol \rho }_{mn}}$ which caused by spontaneous emission can be expressed as [22]:

**d**is the electric dipole operator.

#### 3.4 Doppler broadening

In reality, the sodium atoms are normally in a low temperature, around 185K, the thermal motion of the atoms causes Doppler broaden of absorption line when laser interacted with these atoms. For the convenience of analyzing the effect of Doppler broaden, the velocity distribution of the sodium atoms are divided into several velocity groups, the more groups, the analyzing is more accurate. In this paper, the division method adopted is used by Holzlöhner [6] and in LGSBloch software package [23], which divided the entire frequency range into 500 velocity groups. For a given velocity group labeled *i*, the frequency shift Δ can be expressed as $\Delta = k\upsilon$, where *k* is wave number, $\upsilon$ is the velocity of sodium atoms in this group, [a,b] is the range of frequency shift which corresponding to the velocity group. The probability of the sodium atom in the velocity group *i* can be expressed as ${P_i}$ :

*i*.

#### 3.5 Atom into and out the laser field

Due to the factors such as thermal motion of the atoms, wind speed and laser beam wander and so on, an action field existed where sodium atoms into or out to interacted with the laser field. The sodium atoms are all in the ground state without laser pumping, that means, the atoms are all in ground state when they into the action field, and the atoms may in excited state when they exit the action field and called regenerated population. So, in the processing of establishing the equation, treating the regenerated population as in ground state, will be equally distributed into each Zeeman sub-levels of ground state, and the relaxation caused by into and out action field will exists in each energy level.

#### 3.6 Recoil

The energy of a sodium atom will increase $h\nu$ with a photon absorbed, the momentum exchanges by $\hbar k$, the velocity will increase ${\upsilon _r} = \hbar k/m$ in the direction of laser propagation, this phenomenon called recoil. The excited sodium atom decay back to the ground state by spontaneous, the emitted photon is incoherently and isotropically, the averaged velocity change with spontaneous is 0. For a given velocity group *i*, the range for the group is ($\varDelta {\upsilon _i}$), after the sodium atoms absorbing photons, the part ${\upsilon _r}/\varDelta {\upsilon _i}$ of the sodium atoms will move to the group called *i*+1, the remaining part ($1 - {\upsilon _r}/\varDelta {\upsilon _i}$) still stay in group *i*.

#### 3.7 Spin exchanging collision

The spin exchanging collision occurs when spin angular momentum of electron changes, which is induced by the colliding between sodium atom and O_{2} or O. Scholtes et al. analyzed the physical mechanism of spin exchanging collision [24], the variation of the population in ground state which effected by spin exchanging collision can be expressed as:

**S**is the electron spin operator.

#### 3.8 Velocity exchanging collision

The Cusp velocity exchanging model is used to analyze the velocity exchanging. After the sodium atoms in velocity group *i* colliding with sodium atoms or other molecules (N_{2}, O_{2}), the probability of the atoms still stay in the same group *i* is ${P_{i,i}}^{\prime}$, the probability of the atoms move out to other groups is $1 - {P_{i,i}}^{\prime}$, ${P_{i,i}}^{\prime}$ can be expressed as:

For the sodium atom in velocity group *j* before collision, the probability of moving into group *i* after velocity exchanging collision can be expressed as:

The rate of the velocity exchanging collision occurrence is assumed as ${\gamma _{V - \textrm{collision}}}$, thus, for the given group *i*, the rate of moving out from group *i* which caused by collision can be expressed as:${\gamma _i} = (1 - {P_{i,i}}^{\prime}){\gamma _{V - \textrm{collision}}}$. The rate of group *i* moving into group *j* caused by collision can be expressed as: ${\gamma _{j,i}}^{\prime} = {P_{j,i}}^{\prime}{\gamma _{V - \textrm{collision}}}$. At the same time, the relaxation of group *i* caused by collision is ${\gamma _i}$, the population regeneration from group *j* to group *i* caused by collision is ${\gamma _{j,i}}^{\prime}{{\boldsymbol \rho }_j}$.

Based on the analyzation above, considering the influence of spontaneous emission, intensity and zenith angle of the geomagnetic field, the Doppler broadening spectrum of sodium atom, rate of atom into and out the laser field, recoil, spin exchanging collision and velocity exchanging collision, a Bloch equation can be established which is used to describe the processing of the sodium atoms pumped by laser, by solving the equation, the state evolution of the atoms can be obtained. The method above is based on continuous laser, it also can be applied to long pulse laser, because according to the research of Holzlöhner [6] and Kane [25], the pulse laser can be translated to continuous laser for analyzing.

## 4. Coupling efficiency numerical simulation

A numerical simulation is performed using ADM platform [26] and LGSBloch simulation platform [23], and then the simulation program of Cusp kernel can be established. T. Bhamre’s research results shown that shaping parameter (s) is on the order of 10 [17]. R. Marsland simulated a Cusp kernel superposed by 3 values of s (s=7.8, 27.3 and 500) with different weight factors (0.13, 0.37 and 0.5), which is up to the order of 100 [18]. In our simulation, s is adopted 10 and 100, and the 3 Cusp kernel is also compared.

The coupling efficiency for circular and linear polarized pumped laser under four conditions are simulated: 1) Cusp kernel with shaping parameter of 10; 2) Cusp kernel with shaping parameter of 100; 3) 3 Cusp kernel; 4) “memoryless” LGSBloch hard collision model. The numerical simulation parameters are listed in Table 1. The energy level parameters for sodium atom come from information collected by Steck [21], the Geographical environment parameters are for Lijiang gaomeigu Astronomical observatory [27], some properties of the sodium layer come from the research of Holzlöhner [6].

The coupling efficiency defined by Holzlöhner indicates the photon return of the sodium beacon and the process of laser interacted with sodium atoms. The coupling efficiency with different polarized pump laser and different shaping parameters are compared. On the whole, the coupling efficiency of the laser with circular polarization is nearly 40% higher than linear at best case, which has nothing to do with the collision model, this phenomenon is in good agreement with the experimental [28,9]. Secondly, for Cusp collision kernel with s of 10 and 100, the coupling efficiency is lower than the LGSBloch collision kernel whether circular or linear polarized. Thirdly, for Cusp collision kernel with s of 100, the coupling efficiency is lower than s of 10 which is no matter what the polarization is.

Figure 3(a) shows that for q = 0, the max difference value between Cusp kernel with s=10 and LGSBloch is 13 photons/s/sr/atom/(W/m^{2}) at power density of near 7W/m^{2}. Secondly, the max difference value between Cusp kernel with s=100 and LGSBloch is 32 photons/s/sr/atom/(W/m^{2}) at power density of near 4W/m^{2}, which reached to 56% compared to LGSBloch kernel. The coupling efficiency of 3 Cusp kernel is between the Cusp kernel with 10 and 100. For q=0.12, the coupling efficiency is nearly the same at the power density of less than 100W/m^{2} for LGSBloch, Cusp kernel with s=10, Cusp kernel with s=100 and 3 Cusp kernel these four models; at the power density of exceeding 100W/m^{2}, the coupling efficiency for Cusp kernel with s=100 is significantly lower than Cusp kernel with s=10 and LGSBloch kernel, but is nearly the same with 3 Cusp kernel. Figure 3(b) is the coupling efficiency that sodium atom is pumped by circular polarized laser, and shows similar conclusions with a).

## 5. Discussion

#### 5.1 Population distribution

The populations redistribution for sub-levels can effect the optical pumping [6], hence the photon return is also as a function of atomic-velocity group. To study the dynamics of the atomic system, it is helpful to plot atomic populations as a function of velocity group. S. M. Rochester et al. analyzed the influence of the laser modes on the population [29]. In this paper, in order to analyze the influence of these three collision models, include “memoryless” LGSBloch hard collision kernel, “memory” Cusp weak collision kernel for s=100 and “memory” 3 Cusp kernel, the circular and linear polarized laser at a power density of 1W/m^{2}, 10W/m^{2} and 100W/m^{2} are used to pump the sodium atoms, the populations in velocity space at different energy state are obtained, as shown in Fig. 4, Fig. 5, Fig. 6 and Fig. 7.

Figure 4 shows that the population distribution at ground state with these three collision kernels. The recession is due to hole burning and the bulge is due to recoil. When the atoms are pumped at these three power densities, the population of “memoryless” kernel is larger than Cusp weak collision kernel which is due to hole burning at ground state, the 3 Cusp kernel is between of them. But at a power density of 1W/m^{2}, q=0 and linear polarized laser is applied, the 3 Cusp kernel is significantly less than the other two.

The extent of hole burning for these three kernels nearly the same if repumping is applied; the effect of the hole burning for circular pump laser is evident than linear pump laser at a power density of 1W/m^{2} and 10W/m^{2}.

Figure 5 shows the population distribution of the hyperfine sub-level F=1 in ground state with these three collision kernels: “memoryless” LGSBloch hard collision kernel, “memory” Cusp weak collision kernel for s=100 and 3 Cusp kernel. The two bulges correspond to the increased populations of hyperfine sub-level F=1 at ground state, which is caused by downpumping after the transition of |F=2>→|F’=1> and |F=2>→|F’=2> respectively when the atoms is pumped by the laser in D_{2a} line; the two recessions on the dashed lines due to the hole burning of the transition of |F=1>→|F’=1> and |F=1>→|F’=2> by D_{2b} laser pumping.

Comparing these figures, we can observe that the increased population at the hyperfine sub-levels at ground state F=1 are caused by the downpumping of D_{2a}, the “memory” Cusp weak collision kernel for s=100 is higher than “memoryless” LGSBloch hard collision kernel, the 3 Cusp kernel is between of them. At a power density of 10W/m^{2} and 100W/m^{2}, the curves for 3 Cusp kernel is nearly the same with the Cusp kernel of s=100; but at a power density of 1W/m^{2}, the difference between these two kernels is more obvious. For laser detuned 0.12 power to D_{2b}, at a high power density (10W/m^{2}, 100W/m^{2}), s has little effect on the populations. But at a power density of 1W/m^{2}, the effect of s is more obvious.

Figure 6 shows the population distribution of the hyperfine sub-level F=2 in ground state with these three collision kernels: “memoryless” LGSBloch hard collision kernel, “memory” Cusp weak collision kernel for s=100 and 3 Cusp kernel. The hole burning of the transition: |F=2>→|F’=1>, |F=2>→|F’=2> and |F=2>→|F’=3>, are marked in Fig. 6(a). We can observe that the hole burning of |F=2>→|F’=1> and |F=2>→|F’=2> is more obvious than |F=2>→|F’=3>, that means these two hyperfine sub-levels have a greater influence on the population distribution. For the pump laser tuned on D_{2a} only, for three power density: 1W/m^{2}, 10W/m^{2} and 100W/m^{2}, the population at sub-level F=2 with Cusp weak collision (s=100) kernel is smaller than that with LGSBloch hard collision kernel, the factor up to a maximum of 0.5, and the 3 Cusp kernel is between of them. The same thing will happen like Fig. 5, at a power density of 10W/m^{2} and 100W/m^{2}, the curves for 3 Cusp kernel is nearly the same with the kernel s=100; but at a power density of 1W/m^{2}, the difference between these two kernels is more obvious. For the laser detuned 0.12 power to D_{2b} and at a power density of 10W/m^{2} and 100W/m^{2}, the population of these three kernels are nearly the same; but at a power density of 1W/m^{2}, the greater the s is, the larger the population is.

Figure 7 shows the population distribution at excited state with these three collision kernels: “memoryless” LGSBloch hard collision kernel, “memory” Cusp weak collision kernel for s=100 and 3 Cusp kernel. We can observe that the population at excited state with Cusp weak collision kernel is smaller than that with LGSBloch hard collision kernel; the population at excited state which pumped by circular laser is larger than pumped by linear laser; at the condition of higher power density, like 100W/m^{2}, the population distribution at excited state with D_{2a}+D_{2b} repumping higher than only D_{2a} pumping by a factor of 2.9 and 4.5 for hard and weak collision kernel of s=100 separately. At a power density of 10W/m^{2} and 100W/m^{2} and without repumping, the population at excited state for the 3 Cusp kernel is between the other two kernels; if the laser power detuned q=0.12 to D_{2b}, the population at excited state for these three kernels are nearly the same. But the population at excited state with these three kernels are nearly the same at a power density of 1W/m^{2}, that means the population have nothing to do with the fraction detuned to D_{2b} and polarization.

## 6. Conclusions

In this paper, a Cusp kernel collision modeling method using Bloch equation for analyzing the photon return of sodium beacon has been stated. The coupling efficiency was analyzed in “memoryless” LGSBloch hard collision kernel, “memory” Cusp weak collision kernel for s=10, s=100 and 3 Cusp kernel these four conditions. At the same time, the population distribution in velocity space with LGSBloch collision kernel or Cusp kernel were discussed. All of the simulations and quantitative results are based on LGSBloch package. Some optimization conclusions are summarized as follows:

- 1) When sodium atoms are pumped with narrow D
_{2a}line, the coupling efficiency of Cusp weak collision kernel (s=100) is the lowest while LGSBloch hard collision kernel is the highest whether the laser is linear polarized or circular polarized. In some case, the coupling efficiency of LGSBloch hard collision kernel is 56% larger than the Cusp weak collision kernel (s=100). - 2) When sodium atoms are pumped with narrow D
_{2a}line, the variation of the population at the ground state sub-level (F=1, F=2) which caused by hole burning and downpumping of |F=2>→|F’=1> and |F=2>→|F’=2> is much bigger than the influence of transition |F=2>→|F’=3> whether for hard or weak collision kernel. However, the transition |F=2>→|F’=1> and |F=2>→|F’=2> have little influence on population distribution at excited and ground state, that means it has little influence on photon return. For Cusp weak collision kernel with s=100, the phenomenon above is more obvious, and for 3 Cusp kernel, the phenomenon is between Cusp weak collision kernel with s=100 and LGSBloch hard collision kernel. - 3) When 12% of the laser power detuned to D
_{2b}, the power density ranges from 10W/m^{2}to 100W/m^{2}, the influence of shaping parameter can be ignored. The population distribution at excited state are nearly the same for LGSBloch kernel, Cusp kernel of s=100 and 3 Cusp kernel.

In the next step, a more accurate kernel which the collision rate is not a fixed rate constant (1/(35µs)), but as a function of gas number density will be established, and the simulations will be run for at least two different altitudes. The source code can be retrieved from GitHub (https://github.com/huangjian3058/Cusp-kernel-for-sodium-LGS).

## Funding

Research Startup Funding Project of Chongqing Technology and Business University (1956022); Chongqing Municipal Education Commission Foundation (KJQN201900805).

## Acknowledgments

We are thankful for the advice, guidance and support given by Doctor Kai Wei, Doctor Kai Jin and Doctor Keran Deng from the Key Laboratory on Adaptive Optics, Institute of Optics and Electronics, Chinese Academy of Sciences. Last but not least, we thank all the other members of our group for their help.

## Disclosures

The authors declare no conflicts of interest.

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