In the laser intensity range that the laser supported detonation (LSD) wave can be maintained, dissociation, ionization and radiation take a substantial part of the incidence laser energy. There is little treatment on the phenomenon in the existing models, which brings obvious discrepancies between their predictions and the experiment results. Taking into account the impact of dissociation, ionization and radiation in the conservations of mass, momentum and energy, a modified LSD wave model is developed which fits the experimental data more effectively rather than the existing models. Taking into consideration the pressure decay of the normal and the radial rarefaction, the laser induced impulse that is delivered to the target surface is calculated in the air; and the dependencies of impulse performance on laser intensity, pulse width, ambient pressure and spot size are indicated. The results confirm that the dissociation is the pivotal factor of the appearance of the momentum coupling coefficient extremum. This study focuses on a more thorough understanding of LSD and the interaction between laser and matter.

## I. INTRODUCTION

When a pulsed high-powered laser irradiates a target in an atmosphere, the target will be ablated and reduce the breakdown threshold of the ambient gas. Thus a strong absorbing region will form above the target surface and propagate against the incident direction supersonically. The incident laser energy is used to heat, decompose and ionize the gas along the path. The high pressure behind the LSD wave delivers an impulse to the target, which is employed in many domains, such as, laser propulsion, pulsed laser deposition of thin film, laser machining and so on.^{1} The fundamentals and applications of the LSD wave have attracted the attentions of a large number of researchers.^{2,3}

Raizer had taken the LSD wave as a hydrodynamic discontinuity, and assumed that the laser energy was completely absorbed at the discontinuity surface.^{4} Through counting the conservations of mass, momentum and energy, and using the Jouguet condition, he deduced that the relationship between the LSD wave parameters and the laser parameters. Based on Raizer's theory, Pirri^{5} and Reilly^{6} analyzed the performance of the laser induced impulse with one-dimensional and two-dimensional simplifications. The most apparent deficiency of these works lies in the fact that they do not consider the energy cost of dissociation, ionization and radiation, which makes their results much larger than the experimental data. Shimamura *et al.*^{7} measured the internal structure and the electron density distribution with the two-wavelength Mach–Zehnder interferometer. Shimamura's results show that, when the laser intensity is 5.0 × 10^{7} W/cm^{2}, the electron density can rise up to 2 × 10^{24} m^{−3} in 1 atm atmosphere and the degree of ionization is about 4%. If the LSD wave velocity is 3 × 10^{3} m/s, dissociation and ionization energy account for 36% of the total incident energy of the laser pulse. It is evident that dissociation, ionization and radiation play an important role in the conversion of laser energy. Initiating and maintaining a LSD wave need an intensity much higher than 10^{7} W/cm^{2},^{6} in this range the energy cost by dissociation, ionization and radiation has a profound influence on the LSD wave propagation, the flow field parameters and the efficiency of impulse. It is necessary to modify the classical model to obtain a more accurate understanding of the mechanism of the LSD wave.

Taking into consideration the effect of dissociation, ionization and radiation in mass, momentum and energy conservation, a modified model of the LSD wave is developed in this paper, the laser intensity range that we focus on is 3.0 × 10^{7}–1.0 × 10^{9} W/cm^{2}. Grounded on the flow configuration of the LSD wave (Fig. 1), taking into account the pressure decay by normal and radial rarefaction, the impulse coupling efficiencies are calculated under different laser intensity, pulse width, ambient pressure and spot size. The optimum laser parameters are also discussed.

## II. MODIFIED MODEL OF LSD WAVE

The hypothesis of the strong hydrodynamic discontinuity is employed. The incident laser is completely absorbed in the small region behind the discontinuity. In the inertial coordinate system fixed at the LSD wave front, an infinitely thin control volume is established to deduce the relationship between the parameters of the LSD wave front and the parameters of the wave rear. The mass, momentum and energy conservations are as follows

Where, *D* is the LSD wave velocity; *I*_{0} is the incident laser intensity; *ρ* is the density, *p* is the pressure, *e* is the internal energy and *u* is the gas velocity, the subscripts 1 and 2 denote the front and back of the LSD wave respectively. Δ*L* is the energy cost by dissociation and ionization, determined by the degree of ionization α_{i}

*L*_{d} and *L*_{i} are dissociation energy and ionization energy, respectively. *Q*_{rad} is the radiation energy consumption. With optical thin model,^{8} the formula for high-temperature air^{9} is in the case by

Where σ_{b} is the Stefan-Boltzmann constant and the unit of *Q*_{rad} is W/m^{2}.

In order to illustrate the effect of dissociation, ionization and radiation, it is appropriate to define a parameter *R*_{Ec} as

which denotes the proportion of energy cost by dissociation, ionization and radiation to the incident laser energy per unit mass along the path of the LSD wave moving.

In the paper, we take the gas besides the LSD wave as polytropic gas, and the state parameters meet the relationship *p*_{i} = *n*_{i}*kT*_{i} and *e*_{i} = *p*_{i}/ρ_{i}(γ_{i} − 1), where *i* = 1, 2. The particle number densities of the wave front and the wave rear are related by

where, *N* is the number of atoms contained in the gas molecule.

For now, the analysis suggests that the linchpin to solve the equations is the plasma degree of ionization. Assuming the ionization is in equilibrium and satisfies the Saha equation

So, the relationship among the degree of ionization, the pressure and the temperature is established. Thus, in case the parameters before the LSD wave and the incident laser intensity are given, only one added condition is needed to determine the parameters behind the LSD wave, i.e. Jouguet conditions

Where *c*_{2} is the speed of sound behind the LSD wave given by

## III. LASER INDUCED IMPULSE

We take into consideration that the LSD wave is maintained by a pulsed laser with a flat-top spatial and temporal beam profiles. The flow configuration induced by the LSD wave in the air is shown in Fig. 1. The spot radius is *R*_{p}, and the target radius is *R*_{T}. As the LSD is initiated, a set of radial rarefaction waves are propagating towards the spot center and a blast wave outwards from the edge of the laser spot. When the pulse is turned off, the radial rarefaction waves keep travelling while a set of normal rarefaction waves start moving from the position of the detonation wave front towards the target. The interaction of the normal and radial rarefaction wave finally determines the pressure distribution on the target surface.

### A. Normal flow field above the ablation surface

Vertically, the wave structure and the characteristic lines of the flow behind the LSD wave are shown in Fig. 2, where *τ*_{p} is the laser pulse width and *τ*_{S} is the normal relaxation time when the normal rarefaction wave reaches the target surface. The laser pulse is extinguished at the point of A, and the LSD wave is terminated simultaneously. AB is the intersection line of the rarefaction wave regions issued from point O and point A respectively. The state parameters on AB meet both the characteristic relations of the two regions, that is:

Hence we get the expressions of AB which is of the form

OB can be expressed as

The time *τ*_{B}, corresponding to B is

Therefore, the normal relaxation time *τ*_{S} is evaluated as follows

When t < τ_{S}, the pressure on the target is constant, according to Taylor wave's relationship,^{6} the pressure *p*_{S} is as follows

### B. Radial rarefaction

The wave structure and the characteristic lines of the radial rarefaction are shown in Fig. 3. The continuity equation for cylindrical flow is of the form

For convenience to solve this equation, a quasi-one-dimensional hypothesis is employed by neglecting the axisymmetric term ρ*v*/*r*.^{6} In the range of *r* > 0, the Riemann's invariant is *u* + 2*c*/(γ_{2} − 1) = *const*. The radial relaxation time τ_{O} when the radial rarefaction wave gets the spot center is

The pressure distribution on the target surface covered by the radial rarefaction wave is as follows

With the radial rarefaction, a blast wave is propagated outwards correspondingly. Employing the one-dimension simplification and according to the relationship of rarefaction wave and blast wave, when *p*_{S} is much larger than *p*_{1} the location of the blast wave can be written as

For γ_{2} = 1.2,^{5} the average pressure in the laser spot is depended on the time and can be expressed as

When *t* = τ_{O}, we can obtain |$\bar p_{S_O } = \bar p_S (\tau _O) = 0.728p_S$|$p\xafSO=p\xafS(\tau O)=0.728pS$ and location of the blast wave is *R*_{2} = 12*R*_{p}.

### C. Impulse calculation

The characteristics of the pressure distribution are closely dependent on the time sequence of the arrival of either the radial rarefaction at the center or the normal rarefaction at the surface. The sequence determines the pressure decay model directly. The pressure decay models used in this paper are summarized in Table I.

1) *τ*_{p} < *τ*_{S} < *τ*_{O}

According to the findings of Reilly^{6} and Ferriter,^{11} the pressure on the surface is |$\bar p_S$|$p\xafS$ when *t* < *τ*_{S}; for *τ*_{S} < *t* < *τ*_{O}, the pressure follows the Planar decay model; and the Spherical decay model is appropriate when *t* >*τ*_{O}. The impulse delivered to the target can be expressed as

Where |$\tau _\infty = \tau _S ^{5/9} \tau _O ^{4/9} ( {{{\bar p_{S_O } } / {p_1 }}} )^{5/6}$|$\tau \u221e=\tau S5/9\tau O4/9(p\xafSO/p1)5/6$ is the time when the pressure in the spot has decayed to the ambient pressure *p*_{1}. *R*_{t} = min (*R*_{p}(*t*/τ_{O})^{2/5}, *R*_{T}) is the effective radius in which pressure delivers impulse actually. *A*_{p} is the spot area.

2) *τ*_{p} < *τ*_{O} < *τ*_{S}

In this region, the radial rarefaction wave arrives at the center earlier than the normal rarefaction wave, when the laser is switched off. For t < *τ*_{O}, pressure is |$\bar p_S$|$p\xafS$. When *t* > *τ*_{O}, the pressure follows the spherical decay model. The total impulse is given by

where |$\tau _\infty = \tau _O ( {{{\bar p_{S_O } } / {p_1 }}} )^{5/6}$|$\tau \u221e=\tau O(p\xafSO/p1)5/6$.

3) *τ*_{O} < *τ*_{p} < *τ*_{S}

In this case, the radial rarefaction wave gets the spot center when the laser is still on. The untimely rarefaction contributes directly to the low performance because of disturbing the active state of the follow-up pulse.^{11} The method mentioned in the case of *τ*_{p} < *τ*_{O} < *τ*_{S} is used in the calculation.

When the impulse delivered to the target is obtained, we can obtain another important parameter. That is the momentum coupling coefficient, which is defined as the impulse per unit of the incident laser energy with the form as follows

## IV. RESULTS AND DISCUSSION

In the environment of air, the LSD wave parameters and the impulse delivered to the target are calculated with the modified model. The air parameters involved in the calculations are shown in Table II.

### A. LSD wave parameters

The velocity of the LSD wave is described in Fig. 4. The results of Raizer's model and the experimental data^{13} are also shown in the figure. According to literature 4, Raizer evaluates the LSD speed *D*_{R} and surface pressure *p*_{SR} as follows

It is evident in Fig. 4 that our model fits the experimental data much better by considering the impact of dissociation, ionization and radiation. In order to illuminate the discrepancies between our model and Raizer's model, the energy cost rate *R*_{Ec} are displayed in Fig. 4 with dotted lines. As the laser intensity increases, *R*_{Ec} decreases first and then increases, which corresponds to the discrepancy trend. When the laser intensity is weaker (below 1.0 × 10^{8} W/cm^{2}), the temperature and the degree of ionization of the wave rear are quite small, dissociation is the main means of energy consumption and the *R*_{Ec} is reduced in this region (circle symbol line). As the laser intensity is larger than 1.0 × 10^{8} W/cm^{2}, the energy cost rate of ionization and radiation rise rapidly, and *R*_{Ec} increases consequently.

According to Raizer's model, the LSD wave velocity is directly proportional to the cube root of the quotients of laser intensity divided by the ambient density. However, the simple dependence may be changed when the impacts of dissociation, ionization and radiation are taken into account. As shown in Fig. 5, the LSD wave velocity increases with the growth of the ambient density when the quotients are larger than 1.2 × 10^{12} W·m/kg. The prime reason lays in the value of *R*_{Ec} which falls off correspondingly, and it means more energy can be used to support the detonation propagation.

The variation of the pressure on the target surface with the laser intensity for *p*_{1} = 1 atm is shown in Fig. 6. Comparisons with the experimental data^{14–17} and the prediction of Raizer's model are also presented in Fig. 6. It can be seen that the divergent of the experimental data is visible, and Raizer's model fits the upper of the experimental results well, while our result approximates to the average value better. The deviation between the two models turns upon the value of *R*_{Ec}, the larger *R*_{Ec} is, the more noticeable the deviation shows.

### B. Laser induced impulse

The relationship between the momentum coupling coefficient and the laser intensity is shown in Fig. 7. The calculation program is designed by FORTRAN with special interfaces for Raizer's model and our modified model. In the figure, the solid line and the dotted line denote the results of our model and Raizer's model respectively, and the experimental data^{18} are expressed by the discrete points. It is evident that the experimental results rely on the target properties closely when the laser intensity is less than 5 × 10^{7} W/cm^{2}; for the target properties, such as, the ionization energies, determine the electron number densities of the region where the LSD wave is initiated. Our result fits the experimental data of C best, because the ionization energy of C is larger than both Ti and Al, which makes the circumstance more approximate to the condition of our model. Compared with Raizer's model, the modified model has an obvious advantage. Particularly, the trend of the variation of the momentum coupling coefficient with laser intensity has been described accurately, i.e. increasing at first and then decreasing. The result indicates that dissociation, ionization and radiation are important reasons for the appearance of *C*_{m} extremum.^{19} In light of Fig. 4, dissociation takes the largest part of the energy cost at the optimum laser intensity which is corresponding to *C*_{m} extremum (about 5 × 10^{7} W/cm^{2}). So dissociation is the pivotal factor to determine the variation tendency of *C*_{m}.

The momentum coupling coefficient vs laser intensity under different ambient pressure is plotted in Fig. 8. On the whole, all lines change with the same tendency as the laser intensity increases. The optimum laser intensity rises with the growth of the ambient pressure, while the *C*_{m} extremum remains nearly constant. According to Fig. 5, the flow states behind the LSD wave are determined by the quotient of laser intensity divided by the ambient density when the quotient is less than 1.2 × 10^{12} W·m/kg. As illustrated in Fig. 8, all the optimum laser intensities are below 1.0 × 10^{8} W/cm^{2} and the corresponding quotients are less than 5.0 × 10^{11} W·m/kg. This means that the *C*_{m} extremum and the quotient of the ambient density into optimum intensity are almost invariable when the ambient pressures are under 2 atm. The optimum laser intensity and the *C*_{m} maximum vs ambient pressure is presented in Fig. 9, for *R*_{p} = 1.0 cm, *R*_{T} = 4.0 cm and *τ*_{p} = 1.0 *μ*s. As the increasing of the ambient pressure, the *C*_{m} maximum rises slowly and the optimum laser intensity is directly proportional to the ambient pressure which can be expressed as

Fig. 10 illustrates the impact of spot area on *C*_{m}. It can be seen that as the spot radius increases the coupling coefficient increases at first and then decreases. In Fig. 10, the line marked with delta symbol denotes the ratio of radial relaxation time to normal relaxation time, i.e. *τ*_{2D}/*τ*_{S}. The best performance of *C*_{m} is obtained as *R*_{p} = 2 cm, which is corresponding to the condition of *τ*_{2D}/*τ*_{S} = 1.0. This phenomenon can be observed when the ambient pressure or the pulse width is changed. Therefore it is rational that *τ*_{2D}/*τ*_{S} = 1.0 is an important requirement to achieve the optimum coupling coefficient. In addition, the optimum laser intensities for different spot radiuses are approximately the same. This suggests that the flow states behind the LSD wave is the main factor to decide the tendency of *C*_{m}, and *τ*_{2D}/*τ*_{S} denoting the effect of the two-dimensional decay determines the optimal laser spot radius.

*C*_{m} for different pulse widths are shown in Fig. 11. The coupling coefficient increases as the pulse width shortens, because the two-dimensional decay is tending towards one-dimension decay, and the two-dimensional effect is less evident. Fig. 12 shows the optimum momentum coupling coefficient and the single pulse impulse delivered to the target as a function of the pulse width. The results in the figure are obtained as follows. First, the optimum laser intensity is ascertained by substituting the ambient pressure in Eq. (26). Then the flow field is obtained using the modified model. Thirdly, according to the relationship of *τ*_{2D}/*τ*_{S} = 1.0, the optimum spot radius is determined. Finally, the total impulse and *C*_{m} are calculated through the process presented in Section III C. As shown in Fig. 12, the single pulse impulse increases with the growth of pulse width, and the increasing scale is slackening off step by step. The *C*_{m} extremum keeps nearly the same when the pulse width is below 1μs, and then drops swiftly. We can hold the opinion that 1μs is the optimum pulse duration.

## V. CONCLUSIONS

A modified LSD wave model with the impacts of dissociation, ionization and radiation has been developed. Based on the modified model and considering the pressure decay by normal and radial rarefaction, the laser induced impulse delivered to the target surface is calculated in air; and the dependence of the impulse performance on laser intensity, pulse width, ambient pressure and spot size have been indicated. The modified model fits the experimental data quite well, and the following conclusions can be made:

Dissociation, ionization and radiation will cost a remarkable part of the incident laser energy. When the laser intensity is below 1.0 × 10

^{8}W/cm^{2}, dissociation is the main means of energy consumption, then ionization and radiation take the place.The energy cost of dissociation, ionization and radiation induced the appearance of

*C*_{m}extremum. As the increasing of the ambient pressure, the*C*_{m}extremum rises slowly and the optimum laser intensity increases linearly. The optimum laser intensities are below 1.0 × 10^{8}W/cm^{2}when the ambient pressures are under 2 atm, and dissociation is the pivotal factor to determine the variation tendency of*C*_{m}.The laser intensity and ambient density are the main factors to determine the variation trend of

*C*_{m}, while*τ*_{2D}/*τ*_{S}= 1.0 decides the optimal laser spot radius. The best impulse performance is obtained when the pulse duration is 1μs.

The continuation of this work should be carried on from two aspects. Firstly, the actual profile of the laser pulse need to take into account, for the spatial and temporal distribution of the laser intensity will affect the target ablation and of the LSD wave initiation. Secondly, the meticulous process of plasma absorption during the laser propagation is complex, while it is necessary to understand the interior configuration of LSD wave.

## ACKNOWLEDGMENTS

This work has been supported by the National Natural Science Foundation of China under Grant No.51306203.