Collisionless absorption of short laser pulses in a deuterium cluster: dependence of redshift of resonance absorption peak on laser polarization, intensity and wavelength
S. S. Mahalik M. Kundu






 

































Appl. Phys. Lett. 
J. Phys. B: At. Mol. Opt. Phys. 
Plasma Phys. Control. Fusion 
Phys. Plasmas 
Phys. Rev. A 
Phys. Rev. E 
Phys. Rev. Lett. 
et al.


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    Institute for Plasma Research, HBNI, Bhat, Gandhinagar - 382 428, Gujarat, India
    
    We study collisionless absorption of short laser pulses of various intensity, wavelength (λ) and polarization in a deuterium cluster using molecular dynamics (MD) simulation. For a given laser energy and a pulse duration ≈ 5-fs (fwhm), it is found that maximum laser absorption does not happen at the welknown static Mie-resonance  or linear resonance (LR) wavelength of ≈ 263 nm (for deuterium cluster) irrespective of linear polarization (LP) and circular polarization (CP) state of laser.
    As the laser intensity increases, the absorption peak is gradually red-shifted to a higher λ in the marginally over-dense regime of λ≈(1-1.5) from the expected static  owing to gradual outer ionization and cluster expansion; and above an intensity the resonance absorption 
    peak disappears (sometimes followed by  even a growth of absorption) when outer ionization saturates at 100% for both LP and CP.
    This disappearance of the resonance absorption peak should not be misinterpreted as the negligible (or no) role of Mie-resonance. In fact, in this marginally over-dense band of λ≈(1-1.5), some electrons undergo dynamic Mie-resonance (dynamic LR) and others anharmonic resonance when they are freed. 
    
    
    It is also found that before the
    absorption peak, laser absorption due to LP and CP lasers are almost equally efficient (CP case being inappreciably higher than LP) for all intensities and λ. However, after the absorption peak, at lower intensities, absorption due to LP inappreciably dominates
    absorption due to CP with increasing λ which gradually reverses at higher intensities.   
    MD results are also supported by a naive rigid sphere model of cluster. 
    
    
    
    
    
    36.40.Gk, 52.25.Os, 52.50.Jm
    Collisionless absorption of short laser pulses in a deuterium cluster: dependence of redshift of resonance absorption peak on laser polarization, intensity and wavelength
    
    
        S. S. Mahalik and M. Kundu
        December 30, 2023
    =============================================================================================================================================================================
    
    
    
    
    § INTRODUCTION
    
    
    The interaction of intense laser pulses with nanoscale targets, particularly with atomic clusters demonstrates enhanced absorption of laser <cit.> than expected from isolated atoms/molecules irradiated by same laser pulses. The localized solid-like density of a clustered target (deuterium cluster here) allows the laser field to penetrate fully with no attenuation, which helps in such an extraordinary amount of laser energy absorption.
    The rising edge of the laser pulse of intensity >10^15 ionizes the constituent atoms of the cluster (inner ionization) through optical field ionization (OFI) and forms a nano-plasma. These inner ionized electrons are governed by the laser field plus the transient local electrostatic field (due to charge separation) and  escape from the cluster periphery (outer ionization) after absorbing energy from the remaining part of the laser pulse. 
    Subsequent outer ionization of electrons leaves the cluster with a net positive charge which explodes due to inter-ionic Coulomb repulsion. The high energy absorption by escaping electrons  <cit.>, emission of x-rays <cit.> (typically in the KeV range) and finally the explosion of cluster resulting emission of ions with MeV energies <cit.> and MeV neutrals <cit.> in some conditions.
    
    
    Several experimental, theoretical and particle-simulation works on laser-cluster interaction have reported the effect of various parameters of laser and cluster (i.e., peak intensity, wavelength, pulse duration, cluster size and type etc.) on  
    
    average charge per atom, mean electron and ion energies and also total absorbed energy. However, Petrov et al <cit.> by MD simulations claimed that (i) there is “no enhancement of absorbed energy near the plasmon resonance”<cit.> while laser wavelength λ is varied, (ii) absorbed energy depends linearly on λ<cit.> and (iii) the Mie-resonance or linear resonance (LR) plays no role for the enhancement of absorbed energy, without giving any plausible justification. Note that Petrov et al <cit.> considered  only three well separated λ = 100, 248, 800 nm and did not resolve the wavelength-space meticulously while passing through the Mie-resonance, which possibly led to such misleading conclusions. On the contrary, we reported dependence of energy absorption on λ for an argon cluster with linearly polarized (LP) laser by MD simulation <cit.>; and MD results were supported by a rigid sphere model (RSM). We showed that for a given pulse energy, maximum laser absorption in a cluster happens  not at the well-known static Mie-resonance wavelength of λ_M, but at a red-shifted λ which lies in the marginally over-dense band of wavelength Λ_d ≈ (1-1.5)λ_M. Note that linear Mie-theory (as in the case of nano-plasma model <cit.>) is valid only at lower intensities where static Mie-resonance at λ=λ_M is possible which is often identified as a sharp peak in the absorption curve. However, with increasing laser intensity, linear Mie-theory fails and resonance absorption peak is gradually
    red-shifted from λ_M <cit.>. At this shifted Λ_d ≈ (1-1.5)λ_M all the possible resonances, i.e., dynamical linear resonance (LR) and anharmonic resonance (AHR) are unified   <cit.> to yield maximum laser absorption. We termed this combined resonance as the unified dynamical linear resonance (UDLR) and strongly concluded that there is always a redshift of the absorption peak with respect to λ_M in the marginally over-dense band of Λ_d ≈ (1-1.5)λ_M irrespective of the laser intensity, cluster size and laser pulse type <cit.>.
    
    
    
    However, for different types of clusters, it is still unknown how the redshift of the absorption peak changes with laser intensity. This is particularly very important to justify and validate the above mentioned UDLR <cit.> in the wavelength band of Λ_d ≈ (1-1.5)λ_M with a different cluster type (other than the argon cluster) for its universal acceptance. Further, the degree of polarization of the impinging laser pulse is also an important parameter that changes the dynamics of the cluster electrons, hence it  might effect the UDLR and the redshift of the absorption peak which remains to be  explored.  
    An electron under the influence of a circularly polarized (CP) laser moves spirally, whereas in LP laser it executes oscillatory motion perpendicular to the direction of the laser propagation. 
    The spiral motion of electron in CP prevents the electron-rescattering from the cluster boundary, whereas the probability of rescattering is more for LP. 
    
    
    
    
    
    Particle-in-cell (PIC) simulations <cit.> performed for a xenon cluster at different intensities and cluster charge densities but at a  fixedλ = 1056 nm (with immobile ions) concluded that energy absorption and outer ionization in CP and LP laser field are almost equally efficient. However, it is not known (i) how absorbed energies compare with LP and CP laser fields with the variation of λ and (ii) how UDLR (in the marginally overdense regime) helps in the redshift of absorption peaks from the λ_M in LP and CP fields. Note that, even if laser intensity is kept fixed, ponderomotive energy U_p = E_0^2λ^2/4 of electron and its dynamics is affected by varying λ which is expected to contribute to the redshift of the absorption peak. We thus present energy absorption and outer ionization for a deuterium cluster (as a different cluster type other than argon) with CP and LP lasers. Particularly, we show the effect of laser polarization on the redshift of the absorption peak in the collisionless regime.    
    In passing, we also disprove some of the claims of Petrov et al <cit.> by 
    performing additional MD simulations and discuss some deficits there corroborating to plasmon resonance.
     
    
    We consider a deuterium cluster of radius R_0≈ 2.05 nm (charge state Z =1) which is irradiated by LP and CP laser pulses of different peak intensities I_0 = 5 × 10^15 - 5 × 10^17 and λ = 100-800 nm. For a given intensity and polarization we vary λ. Similar to previous studies with argon clusters <cit.>, here we find that, for both LP and CP, redshifts of the absorption peaks from the static Mie-resonance wavelength λ_M = 263 nm (for deuterium cluster) still persist, which also lie in the marginally over-dense band of wavelength Λ_d ≈ (1-1.5)λ_M as long as outer-ionization is below 100%. Redshift of the absorption peak monotonically increases with increasing 0. Additionally, above a certain 0, absorption peak is found to disappear with further increase in I_0 ≥ 10^17 as outer ionization saturates at 100%. 
    It is also found that before the absorption peak in the band of Λ_d ≈ (1-1.5)λ_M, laser absorption due to LP and CP lasers are almost equally efficient
    (CP case being inappreciably higher than LP) for all intensities and λ. However, after the absorption peak, at lower intensities ≤ 10^17, absorption due to LP inappreciably dominates absorption due to CP with increasing λ which gradually reverses at higher intensities ≥ 10^17.
    
    
    Atomic units (i.e., = | -e | =1, 4πϵ_0 = 1, ħ = 1) are used in this work unless mentioned explicitly. Section <ref> discusses the form of the laser pulse.
    Section <ref> illustrates laser absorption and the role of laser polarization on the redshift of the absorption peak in the deuterium cluster by a MD simulation. Section<ref> gives justification of MD results by RSM analysis. Summary and conclusion are given in Sec.<ref>.
    
    
    
    
    § THE LASER PULSE
    
    
    
    As laser wavelengths (λ=100-800 nm) are much longer
    than cluster sizes (2-4 nm), as considered here, 
    the effect of propagation of laser (directed in z) is disregarded and the dipole approximation
    for laser vector potential A⃗(z,t) = A⃗(t)exp(-i 2π z/λ) ≈A⃗(t) is assumed. 
    
    In general, we write
    
    
        A⃗(t) =  E_0/ωsin ^2(ω t/2n)[δcos (ω t) x̂ + √(1-δ^2)sin (ω t) ŷ],
    
     for 0≤ t ≤ n T. Here δ is the degree of ellipticity (0 ≤δ≤ 1); 
     δ = 1,1/√(2) for LP and CP respectively; 
     n is the number of laser period T; 
    τ = n T is the total pulse duration and E_0=√(8π I_0/c)
    is the field strength for the peak intensity I_0. The components of driving laser electric field E⃗_⃗l⃗ (t)=-dA⃗ /dt along x, y and z directions are,
    
        E⃗_⃗l⃗^x (t) = δE_0/ω∑_i=1^3c_iω_isin(ω_i t)    if  0 ≤ t ≤ nT
        
        0    otherwise;
    
    
        E⃗_⃗l⃗^y (t) = √(1-δ^2)E_0/ω∑_i=1^3c_iω_icos(ω_i t)    if  0 ≤ t ≤ nT
        
        0    otherwise;
    
    
        E⃗_⃗l⃗^z (t) = 0.
    
    
    
    
    
    
    
    
    
    
    
    Where c_1=1/2, c_2=c_3= -1/4, ω_1 = ω, ω_2 = (1+1/n)ω, and ω_3 = (1-1/n)ω.
    Note that the ponderomotive energy U_p = E_0^2/4ω^2
    is the same for both LP and CP. For LP, only the x-component of laser electric field E⃗_⃗l⃗^x (t) survives which may vanish at the completion of each laser cycle. Whereas, for CP, electric field components E⃗_⃗l⃗^x (t), E⃗_⃗l⃗^y (t) in x,y do not vanish simultaneously. Therefore, CP laser is expected to yield different
    results than LP laser while interacting with a cluster. Particularly, it is not known how resonance absorption maxima shifts w.r.t. polarization state of laser. 
    
    
    
    
    § RESONANCE ABSORPTION BY MD SIMULATION
    
    In a previous study <cit.> we reported MD simulation results for argon clusters of different sizes (2-4 nm) irradiated by LP laser fields with different peak intensities and wavelengths λ =100-800 nm. We showed that in the short pulse regime (5 fs, fwhm) and at a given laser intensity, linear resonance (LR) and an-harmonic resonance (AHR) can be dynamically unified to yield maximum laser energy absorption at a particular λ, typically in the UV regime. The possible unification of all resonances comprises: (i) LR in the initial time of plasma creation, (ii) LR during Coulomb explosion in the later time and (iii) AHR for electrons in the intermediate time during the laser cluster interaction, leading to maximum energy absorption, maximum outer ionization and also maximum average charge states for the argon cluster. We found the wavelength Λ_d of the absorption maxima is gradually red-shifted from the conventional static Mie-resonance wavelength λ_M in the band of Λ_d ≈ (1-1.5)λ_M for increasing I_0 = 5 × 10^15 - 10^17. We coined this marginally over-dense regime Λ_d ≈ (1-1.5)λ_M as the regime of unified dynamical linear resonance (UDLR), where dynamical LR and AHR are efficiently unified <cit.>. 
    
    Here, we report a comparative study of laser absorption and outer ionization and the dependence of 
    the red-shift of absorption maxima in the UDLR regime with a deuterium cluster for different polarization states of laser pulses.
    
    
    
    
    
    
    
    
     §.§ Details of MD simulation
    
    The main workhorse here is the MD simulation code. More details are given in Refs.<cit.>. For conciseness, we only mention necessary
    points here.
    
    A single deuterium cluster of radius R_0≈ 2.05 nm and N = 1791 number of neutral atoms is considered, 
    unless mentioned explicitly. Initially, atoms are placed according to the Wigner-Seitz radius r_w≈ 0.17 nm, giving R_0=r_w N^1/3≈ 2.05 nm. When all atoms are ionized initially, it gives a charge density ≈ 7× 10^-3 a.u. and =√(4π/3) = 0.1735 a.u.. For 800 nm, it represents an over-dense plasma with /≈ 27.87 and /ω≈ 3.05, where ≈ 1.75× 10^27 m^-3 is the critical density.
    Ionization of deuterium atoms is treated by “over the barrier” ionization (OBI) model Bethe of optical field ionization (OFI) <cit.>, 
    
    
        E_c = I_p^2 (Z)/4Z,
     
    which is valid at higher intensities >10^14. Here  I_p(Z) is the ionization potential for the charge state Z and E_c is the corresponding critical field. The position and velocity of a newly born electron are assumed same as the parent ion conserving the momentum and energy. 
    
    The equation of motion (EOM) of i-th charge particle in the laser field with the electric field E⃗_l(t) and the magnetic field B⃗_l(t) reads,
    
    
    
    
    
    
        dp⃗_⃗i⃗/dt    =  F⃗_⃗i⃗(r_i,v_i,t) + q_i [ E⃗_l(t) + v⃗_⃗i⃗×B⃗_l ],
    
    where F⃗_⃗i⃗ = ∑_j=1, i j^N_pq_i q_jr⃗_⃗i⃗j⃗/r_ij^3 
    is the Coulomb force on i-th particle of charge q_i due to all 
    other N_p-1 particles each of charge q_j, where N_p is the total number of particles including ions and electrons. Components of laser electric fields are chosen from Eqs. eq:laserfieldx,eq:laserfieldyeq:laserfieldz depending upon LP and CP cases. Accordingly, components of laser magnetic fields are oriented. 
    For 0< 10^18 usually B_l(t)≪ 1.
    To mitigate Coulomb singularity of F⃗_⃗i⃗ for r_ij→ 0, an artificial smoothing parameter r_0 is added with r_ij. For a given cluster we choose r_0=r_w, which produces accurate Mie-plasma frequency  (see Ref.<cit.>). 
    The modified Coulomb force on i-th particle and 
    the corresponding potential at its location are, 
    
    
        F⃗_i = ∑_j=1,i j^N_pq_i q_j r⃗_⃗i⃗j⃗/(r_ij^2+r_0 ^2)^3/2,      ϕ_i = ∑_j=1, i j^N_pq_j/(r_ij^2+r_0 ^2)^1/2.
    
    
    
    Eq. eom2 is solved using the velocity verlet method with a time step Δ t = 0.01 a.u. to resolve the . 
    
    The deuterium cluster is irradiated by short laser pulses of duration 13.5 fs (FWHM of 5-fs) at different I_0 = 5 × 10^15 - 5 × 10^17. The pulse energy is kept fixed for a particular I_0 and wavelength is varied in the range of λ = 100-800 nm for both LP and CP laser light.
    
    
    
    
    
    
     §.§ Absorption and outer ionization with LP laser
    
    
    
    
    Figures <ref>(a)-(b) depicts average total absorbed energy ℰ_t = ∑_1^N_p (v_i^2/2 + q_iϕ_i)/N per atom and corresponding fractional outer ionization N = ^out/N of electrons at the end of 13.5 fs LP laser pulses versus λ for different I_0, where ^out is the number of electrons outside the initial radius R_0. At lower 0 ≲ 5 × 10^16, ℰ_t and N initially increase with increasing λ, attain different maximum values at different λ between 263-400 nm, i.e., in the band of λ =Λ_d≈ 330 ± 67 nm which is equivalent to Λ_d≈ (1-1.5)λ_M, then drop as λ is increased further. For the deuterium cluster, the static Mie-resonance (or the static LR) with a sharp absorption maxima is often conventionally expected at λ=≈ 263 nm (marked by vertical dashed line) according to the nano-plasma model. In stead, it is seen that ℰ_t and N attain maximum values in the band of Λ_d≈ (1-1.5)λ_M which are red-shifted from the expected ≈ 263 nm, and the redshift increases with increasing values of ℰ_t and N for increasing 0 ≲ 5 × 10^16. For higher I_0>5 × 10^16, absorption peaks gradually disappear due to faster saturation of outer ionization of electrons to 100%.  
    
    Approaching towards ≈ 263 nm from 100 nm, higher intensity pulse expels more electrons from the cluster at a much faster rate than at a lower intensity [see Figs.<ref>(a)-(b)]. As electrons move far from the cluster, the laser field dominates over the restoring field of background ions acting on those free/quasi-free electrons. Therefore, after the 100% outer ionization for an intensity, the average ℰ_t may also grow [see Fig.<ref>(a) for I_0 ≥ 10^17] with increasing λ since the average energy of a laser-driven free electron scales as ≈ U_p ∝ E_0^2 λ ^2/4. Because electrons (free or bound) do not have same energy and liberated at different times, the growth of ℰ_t is slower than the scaling of ≈ U_p. In this regime of 100% outer ionization, absorption maxima does not show up and the absorption curve does not bend down with increasing λ. So, the survival of the absorption peak (and it's redshift from the static LR wavelength ≈ 263 nm) depends on the population of bound and free electrons during the interaction. As the free population of electrons increases with increasing 0, absorption peak gradually shifts towards higher λ in the band of Λ_d≈ (1-1.5)λ_M from , and finally it disappears with the free population eventually reaching to 100%. 
    
    
    To understand the growth of average energy with increasing I_0 and dis-appearance of the absorption peak when outer ionization is 100% saturated for some higher I_0, we look into the kinetic energy scaling of free/quasi-free electrons which are far from the cluster. We extract the average total kinetic energy of the electrons ℰ_k = ∑_1^Nm_e v_i^2/2/N for the highest intensity of 5× 10^17 corresponding to Fig. <ref> and compared with the ponderomotive energy scaling U_p ∝ E_0^2λ^2/4. Fig. <ref> shows ℰ_k for those electrons which are beyond different radii 3 R_0, 5 R_0, 10 R_020 R_0 for I_0 = 5 × 10^17. The solid (blue) line with circle  represents U_p/5 for different λ. It is found that the growth of ℰ_k is ∝ U_p, but it is slower than U_p scaling as different electrons are liberated from the cluster potential at different times by experiencing different laser fields and restoring forces of background ions. Moreover, the scaling U_p ∝ E_0^2λ^2/4 is an over-estimation for a short pulsed light, i.e., shorter the pulses more is the over-estimation. Thus Fig. <ref> justifies that growth of absorption beyond some
    intensity and wavelength (when outer ionization reaches 100% in Fig.<ref>) is due to increasing population of free electrons.  
    
    
    
     
    The redshifts of absorption peaks in the case of deuterium cluster for lower intensities in the band of Λ_d≈ (1-1.5)λ_M also resembles redshifts in the case of argon cluster in the previous work <cit.>, which justifies red-shifting of the absorption maxima irrespective of the atom type of a cluster. However, for an argon cluster, absorption peak does not disappear due to supply of electrons via inner ionization and unsaturated outer ionization < 100% for the same laser intensity ≤ 5× 10^17 (see Ref.<cit.>). For deuterium cluster, in Fig.<ref>, the red-shifted absorption peaks for I_0 = 5× 10^15 and 5× 10^16 are located at Λ_d = 320, 360 nm respectively and the corresponding free electron's population are 88.8%, 98.6%. With increasing 0, for the deuterium cluster, average charge state is quickly saturated at Z = 1 in the early time of the laser pulse and remaining pulse energy helps in increasing the absorption, outer ionization of electrons (also early Coulomb explosion of cluster) and gradual increase of redshift of the absorption peak from the expected ≈ 263 nm. Note that Coulomb explosion of background ions also contributes to the redshift of the absorption peak in the band of Λ_d≈ (1-1.5)λ_M from the .
    Therefore larger the peak intensity, larger will be the redshift in the absorption peak from the expected =263 nm when Z = 1 is saturated as compared to argon cluster <cit.>. 
    
    The occurrence of distinct maxima in the absorption and outer ionization in the red-shifted band of Λ_d≈ (1-1.5)λ_M for lower intensities and also dis-appearance of absorption maxima after the 100% saturation of outer ionization for some higher intensities clearly shows the effect of λ variation. Our results in Figs.<ref> contradicts earlier MD simulation works by Petrov et al.<cit.> where they considered only three wavelengths λ = 100,248,800 nm at a fixed intensity of I_0 = 5× 10^16. They could not find any maxima in the absorption while passing through Mie-resonance and nullified any role of Mie-resonance for laser absorption. They further argued that “non-uniform electron density” inside the cluster is responsible for the “null effect” of Mie-resonance  <cit.>. Firstly, we point out that, electron density is always non-uniform within the cluster and its surroundings (during the laser pulse driving; also by birth due to the random distribution of parent atoms in the position and velocity space) that does not explain the absence of Mie-resonance absorption peak. Secondly, only three well-separated wavelengths λ = 100,248,800 nm chosen by Petrov et al. <cit.> can not resolve the absorption maxima which is evident from Fig.<ref>. Thirdly, due to high intensity, resonance absorption maxima is  actually red-shifted in the band of Λ_d≈ (1-1.5)λ_M from the commonly expected λ_M of Mie-resonance as shown in Fig.<ref> and linear Mie-resonance theory is invalid here. It is delusive to look for the absorption maxima exactly at the λ_M in presence of non-zero outer ionization at higher intensities. However, the absence of absorption maxima at the expected  does not mean that Mie-resonance has “null effect” in the absorption and outer ionization <cit.>. In stead, dynamic LR (dynamic Mie-resonance) and AHR work in unison
    (i.e., UDLR works) in the band of Λ_d≈ (1-1.5)λ_M very efficiently and the near-the-LR effective field E_eff [as understood from the simple estimate of E_eff = E_0/(ω_M^2 - ω^2)] inside the cluster is  so much enhanced that it forces almost 90% electrons to be outer ionized  even at the near-resonance (under-dense) values of λ≈ 250-260 nm before the λ_M≈ 263 nm and responsible for the resonance peak shift towards a higher λ>λ_M as seen in Fig.<ref>. 
    For elaborate discussion on UDLR see Ref.<cit.> and Sec.<ref>.
    
    
    
    
    
    
      §.§.§ Resonance peak shift with radius variation of cluster
    
    
    For a cluster, the static Mie-resonance can be met by a transition from over-dense to under-dense plasma regime (artificially) by increasing the cluster radius (mimicking cluster expansion) for a fixed number of ions/electrons at a fixed λ, e.g., 800 nm as performed by Petrov  <cit.> by a MD test simulation where “no enhancement of energy absorption” was found “near the plasmon resonance”. Contrarily, we show by our MD simulation that LR indeed plays a major role for pronounced laser absorption during the variation of cluster radius, but red-shift of absorption peak occurs depending upon laser intensity due to the UDLR in presence of outer ionization. 
    
    
    The deuterium cluster (N=1791 atoms) is irradiated by LP laser
    pulses of fixed λ=800 nm and τ=13.5 fs. At a given I_0, cluster radius R  is varied from R = R_0 = 2.05 nm to R = 15R_0 with fixed N and immobile ions. Thus plasma density is successively reduced. For R = 2.05 nm, the density is = 27.87, where ≈ 1.75× 10^21 cm^-3 and /ω≈ 3.05. Since ^2 ∝ 1/R^3, the static LR condition = λ (or = ω) is achieved when R ≈ 2.1 R_0.
    
    Test MD results with above parameters in Figs. <ref>(a)-(b) show the absorbed energy ℰ_t per atom and fractional outer ionization N of electrons versus R/R_0 (top row) for different I_0 at the end of LP laser pulses. Same results are shown in Figs. <ref>(c)-(d) with the corresponding /λ (bottom row). Our results are different than Petrov  <cit.>. It is seen that maximum absorption and outer ionization occur for all three intensities but at red-shifted wavelength ratios of /λ before meeting the static LR condition at /λ≈ 1 (or R/R_0≈ 2.1, vertical dashed lines). The gradual red-shifting of absorption and outer ionization maxima with increasing intensity in Fig. <ref> in the marginally over-dense regime of 0.7≲/λ≤ 1 (or 1.5≲ R/R_0 ≤ 2.1) is very much consistent with the findings in Figs. <ref>(a)-(b) with varying λ and mobile ions.  
    
    It is known that for a over-dense cluster AHR happens <cit.> for different electrons at different times when dynamical frequency Ω[r_i(t)] of i-th electron at the position r_i(t) satisfies Ω[r_i(t)]=ω. If plasma is too much over-dense w.r.t. ω, the absorption via AHR can not be collective and absorption peak is not expected. As we reach to the marginally over-dense regime of Λ_d=(1-1.5)λ_M, where  and ω become very close, many electrons may pass through AHR at the same time at ease, being excited by the near-the-LR enhanced effective field E_eff = E_0/(ω_M^2 - ω^2), depending upon the laser intensity. 
    
    
    In this case, the dynamical LR and AHR are often indistinguishable where they work together (UDLR happens) to maximize the laser absorption <cit.>. The absorption peak in Figs. <ref>,<ref> at various intensities is the clear manifestation of collective effect of 
    UDLR in the marginally over-dense band of Λ_d=(1-1.5)λ_M dominated by the near-LR (near Mie-resonance) enhanced field effects.
    
    
    
    
     §.§ Absorption and outer ionization with CP vs LP light
    
    
    To see the impact of laser polarization on the absorption and outer ionization; and to know how the resonance absorption maxima shifts due to it, we simulate the same deuterium cluster with CP light (Eqn.eq:laserfieldx,eq:laserfieldy with δ = 1/√(2)) with similar laser parameters as in Fig. <ref> of the LP case. 
    
    Laser electric field components being different for LP and CP at a given instant, overall electron dynamics becomes different for a given I_0 and λ which may impact differently (i.e., CP may yield different results than LP) on energy absorption, outer ionization and the redshift of the resonance absorption peak. 
    
    
    
    Figure <ref>(a)-(b) show the comparison of average total absorbed energy ℰ_t = ∑_1^N_p (v_i^2/2 + q_iϕ_i)/N per atom and corresponding fractional outer ionization N = ^out/N of electrons at the end of 13.5 fs LP and CP laser pulses versus λ for different 0. The solid lines correspond to LP results from Figs.<ref>(a)-(b) and the dashed lines correspond to CP. The vertical dashed line represents the wavelength of static Mie-resonance λ_M = 263 nm. 
    
    
    In the case of CP, ℰ_t and N initially increase with increasing λ, attain different maximum values in the range of λ≈ 330 ± 67 nm, then drop as λ is increased further for intensities < 10^17 similar to the case of LP. Surprisingly, for CP, the maxima of ℰ_t and N for different intensities are also red-shifted from the expected λ_M for I_0≤ 10^17 in the  same marginally over-dense band of Λ_d ≈ (1-1.5)λ_M (shaded region) as in the case of LP,  in spite of electron dynamics and laser field configuration are different in CP and LP. In fact, for some intensities (specially for low intensities) CP and LP results for ℰ_t and N have only minor differences in the entire range of λ = 100-800 nm; and up to the static Mie-resonance λ=λ_M, ℰ_t and N have almost no difference, irrespective of the laser polarization.
    
    In the microscopic level, main difference between CP and LP results is that the magnitude of the maximum of ℰ_t and N are little more for CP as compared to LP within the band of Λ_d ≈ (1-1.5)λ_M. For example, the value of max(ℰ_t) for LP (CP) case are 21, 24, and 33 a.u. (23, 26, and 36 a.u.) for 0 = 5× 10^15,  10^16,  5× 10^16 respectively; and corresponding percentages of max(N) are 88.8%, 92.6%, and 98.6% (89%, 94.8%, and 98.9%) respectively. However, after the absorption maximum, for lower intensities, absorption due to LP either remains almost same as CP or dominates absorption due to CP with increasing λ outside the band of Λ_d ≈ (1-1.5)λ_M which gradually reverses its tendency at higher values of 0=10^17, 5× 10^17 (i.e., CP dominates LP); although N remains mostly higher for LP than the CP case except for a very high 0=5× 10^17 where absorption peak disappears (outer ionization saturates to 100%) and absorption grows with increasing λ similar to the LP case as seen after 400 nm (explained in Fig.<ref>). We shall further justify this findings of MD by the RSM in Sec.<ref>. 
    
    However, above microscopic differences may not be captured in real laser-cluster experiments using CP and LP pulses. We conclude that the efficiency of laser energy absorption and outer ionization are almost same for LP and CP laser pulses of moderate 0 < 10^17. We have checked (not presented here) that absorption and outer ionization in Fig.<ref> do not depend on the handedness of the CP laser, i.e., right-handed CP laser and left-handed CP laser yield almost same results, since the components of electric field amplitudes remain unchanged and the cluster is almost spherically symmetric.
    
    
    
    
      §.§.§ Time domain analysis of resonance shift with CP vs LP light
    
    
    By analyzing the results in time domain, we now justify the shifting of absorption maxima from the expected static LR wavelength of λ_M=263 nm in Fig.<ref> at some intensity of 10^16 (red, dashed square line of Fig.<ref>) and various λ with LP and CP light. Similar analysis for the redshift of absorption maxima was previously reported in Ref. <cit.> in detail for a more complicated argon cluster illuminated by LP light.
    
    Figures <ref>(a)-<ref>(c) present comparison of scaled Mie-frequency (t)/ω, total absorbed energy ℰ_t(t) per atom and fractional outer ionization N(t) = ^out(t)/(t) of electrons respectively with LP (solid line) and CP (dashed line) versus normalized time in units of the laser period T of λ =800 nm. The horizontal dashed line in Fig. <ref>(a) indicates the line of static LR condition = ω. The dynamical
    (t) is calculated from the relation ^2(t) = Q_0(t)/R_0^3, where Q_0(t) is the instantaneous total positive charge defined by Q_0(t) = (t) Z(t) inside the initial cluster radius R_0 having (t) number of ions with average charge Z(t)<cit.>. 
    At I_0 = 10^16, the only charge state Z = 1 of the deuterium ions for all λ is created during t/T = 0.9-1.1 by the OFI Bethe and the corresponding dynamical ω_M(t)/ω jumps form zero to its maximum for both LP and CP. After the respective maximum values, ω_M(t)/ω drops due to Coulomb explosion. For a chosen λ, although there are minor differences in ℰ_t(t) and N(t), surprisingly temporal variation of ω_M(t)/ω remains indistinguishable until the end of laser pulses for both LP and CP. 
    
    
    
    
    
    It is known that the efficient transfer of energy from an oscillatory driver (laser fields) to the oscillators (bound electrons) is possible  not only by the frequency matching condition = ω of LR, the time spent by the system near the LR and the strength of the driver at that time are also crucial <cit.>. The longer time spent by the dynamical ω_M(t)/ω near the line of LR (ω_M/ω = 1) and in the marginally over-dense band of Λ_d ≈ (1-1.5)λ_M, the higher is the absorption and outer ionization due to the combined effect of AHR and dynamical LR <cit.>, coined as unified dynamical LR (UDLR) <cit.>. In this marginally over-dense band of Λ_d = (1-1.5)λ_M, some electrons undergo dynamical LR (due to dynamical ω_M(t)) with great enhancement of the near-LR effective field E_eff≈ E_0/(ω_M^2(t) - ω^2) <cit.> and some other electrons undergo AHR by meeting their time dependent frequencies Ω_i(t)=ω in the anharmonic part of the potential at that time  at ease for the close proximity of Ω_i(t) to ω_M(t) and the effect of near-LR enhanced field E_eff≈ E_0/(ω_M^2(t) - ω^2) <cit.>. 
    
    For the deuterium cluster also (in this work) the absorbed energy and outer ionization are enhanced (see Figs. <ref>(b),(c)) for those wavelengths which lie in the marginally over-dense band of λ≈ 330 ± 67 nm (e.g., 260 nm, 320 nm and 360 nm), equivalent to Λ_d ≈ (1-1.5)λ_M, as compared to the other wavelengths which lie outside this band (e.g., 200 nm and 800 nm). Among the three intermediate λ, dynamical ω_M(t)/ω for λ=260 nm (magenta line) is very close to the static LR, ω_M/ω = 1 (where λ_M = 263 nm), but the absorption and outer ionization are not still at their expected maximum. For 260 nm, the dynamical ω_M(t)/ω just meet the line of static LR  only for a very short time during t/T = 0.9-1.2 and rest of the time it remains in the under-dense regime below the line of static LR (where AHR does not work), so can not absorb energy efficiently. Instead, for λ=320,360 nm (green, red lines), the respective dynamical ω_M(t)/ω continues to remain in the marginally over-dense band of Λ_d ≈ (1-1.5)λ_M for a prolonged period up to t/T ≈ 4.1,5.0, where both AHR and dynamical LR with near-LR enhanced effective field E_eff = E_0/(ω_M^2(t) - ω^2) contribute unitedly (i.e., UDLR happens) to maximize absorption and outer ionization for both LP and CP. 
    Thus UDLR explains the red-shift of absorption maxima from the static LR wavelength of λ_M=263 nm in Fig.<ref> for LP and CP light. It is observed that the maximum of ℰ_t(t) is little enhanced for CP in comparison to LP for those λ which lie in the UDLR regime of Λ_d ≈ (1-1.5)λ_M. This enhanced energy is partly due to non-zero rotating electric field vector of the CP laser, although N(t) for CP light is either less (or nearly equal) compared to LP light in Fig. <ref>(b).
    
    The oscillatory nature of ℰ_t(t) and N(t) for LP are due to the oscillating electric field vector while for CP they smoothly increase due to non-zero rotating electric field vector.
    
    
    
    § RIGID SPHERE MODEL : ABSORPTION AND OUTER IONIZATION WITH CP VS LP LASER
    
    We further justify MD results (qualitatively) by a simple rigid sphere model (RSM)<cit.>.
    Here, cluster is assumed to be pre-ionized and consists of uniformly charged spheres of ions (e.g., deuterons) and electrons of equal radii =  =. The ionic sphere is considered immobile (ions are frozen) for the short laser pulse duration < 14-fs and laser magnetic field is also neglected for  0<10^18. 
    The same deuterium cluster with N=1791, R_0=2.05 nm and charge density ≈ 7× 10^-3 a.u. is considered. For λ = 800 nm, it represents an over-dense plasma of /ω≈ 3.05. For simulating multi-electron system by the RSM,
    we assume N=1791 non-interacting electron spheres with their centers uniformly positioned inside the ionic sphere at the initial time. Center of each electron sphere mimics a real point size electron. 
    We simulate the system with LP and CP laser pulses with same parameters as in Sec. <ref>. 
    
    
    
    
    The dynamics of each electron sphere is governed by the laser field pulse the electrostatic restoring field in the background potential of positively charged ion sphere. In the case of LP light (polarized in x), the equation of motion (EOM) of an electron sphere can be written as <cit.>,
    
        r̈⃗̈+r⃗g(r)/r = x̂ (/) E_l(t)/R_0,
    
    where r⃗ = x⃗/R_0 and r = |r⃗|. 
    The potential ϕ(r) and the corresponding electrostatic restoring field g(r) are respectively,
    
    
        ϕ (r) = ^2 R_0^2 ×3/2-r^2/2   if  0≤ r ≤ 1
        1/r   if  r ≥ 1
     -0.5cm
    
    
        g(r) = ^2 R_0 ×
        r    if 0≤ r ≤ 1
        1/r^2   if r ≥ 1.
    
    The electric field component polarized in x-direction for LP is given by Eq. <ref> with δ = 1. Each electron sphere will oscillate in the potential ϕ(r) with a position dependent frequency Ω[r(t)] <cit.>. 
    
    
    To study the dynamics of electron spheres in the potential (<ref>) with CP light we write EOM in two dimension as <cit.>,
    
        r̈⃗̈_̈⃗̈ẍ⃗̈ + r⃗_⃗x⃗ g(r)/r = x̂( /) E_l^x(t)/R_0 
        r̈⃗̈_̈⃗̈ÿ⃗̈ + r⃗_⃗y⃗ g(r)/r = ŷ( /) E_l^y(t)/R_0.
    
    Here r⃗_⃗x⃗ = x⃗/R_0, r⃗_⃗y⃗ = y⃗/R_0, r = √(r_x^2 + r_y^2).
    The electric field components for CP light are given by Eqn. eq:laserfieldx and eq:laserfieldy with δ = 1/√(2).
    
    
    
    
    Similar to MD study in sec. <ref>, here also we have same U_p = E_0^2/4 ω^2 as in the LP case. Each electron sphere moves spirally in the potential ϕ(r) driven by CP light. 
    
    Since initial positions of electron spheres are different, they experience different restoring forces. As laser drives those electron spheres, they will climb up in the potential and some of them will leave the potential at different instant of time experiencing different total fields. Inside the potential their motion is harmonic with a constant frequency Ω[r(t)] = ω_M. As soon as they cross the cluster boundary (r>1) their motion becomes anharmonic with a gradual decrease of Ω[r(t)]<cit.>. When Ω[r(t)]=ω is met for an electron, AHR happens, and the electron leaves the cluster <cit.> by absorbing energy. Thus, RSM can be used to understand both LR and/or AHR processes depending upon the laser frequency and intensity. 
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Figures <ref>(a)-(b) show comparison of average total absorbed energy ℰ_t = ∑_1^N (m_s v_i^2/2 + q_sϕ_i)/N per electron sphere for LP and CP laser pulses (after τ=13.5 fs) versus λ for different I_0. Figure <ref>(c)-(d) are the corresponding fraction of outer ionized electrons N=N_out/N. Results are separated in two regimes of intensities for more clarity. Figures <ref>(a),(c) are for I_0 ≤ 10^17 and Figs. <ref>(b),(d) are for I_0 = 5 × 10^17. 
    
    It is noted that
    RSM results (for both LP and CP) resemble the MD results in Fig. <ref>: (i) growth of absorption and outer ionization with increasing λ for 0≲ 10^17 with distinct maxima located in the band of Λ_d ≈ (1-1.5)λ_M, i.e., in between 263-400 nm in Figs. <ref>(a),(c); (ii) increasing redshift of the absorption and outer ionization maxima from the static LR wavelength of λ_M ≈ 263 nm (vertical dashed line) with increasing 0≲ 10^17 due to increasing outer ionization; and (iii) gradual disappearance of absorption maxima (followed by  even a growth of ℰ_t for λ > 400 nm) due to 100% saturation of N when 0 exceeds 10^17 in Figs. <ref>(b),(d). 
    
    Similar to the MD results in Fig. <ref> the magnitude of the maximum value of ℰ_t and N are more for CP than LP in the marginally overdense band of Λ_d ≈ (1-1.5)λ_M.
    The maxima of ℰ_t for LP (CP) are 19.1, 31.9, 54.9, and 64.5 a.u. (40.3, 49.4, 62.4, and 72.8 a.u.) for I_0 = 5× 10^15,  10^16,  5× 10^16, 10^17 respectively and corresponding N are 33%, 57%, 89%, and 94% (74%, 90%, 98%, and  99%) for respective I_0 in Figs. <ref>(a),(c).
    Also, after the maxima, for lower intensities, both ℰ_t and N with CP light drop below the values with LP light which, on the other hand, gradually reverses at higher intensities (i.e., CP dominates LP as in Figs. <ref>(b),(d)), as λ is increased. 
    At the higher intensity 5 × 10^17, after λ≈ 320 nm, N is saturated at 100% for CP and almost 98.5% for LP but ℰ_t gradually grows with increasing λ>320 nm for both CP and LP. For higher intensities ≥ 5× 10^17, when 100% outer ionization is reached (almost all electrons are freed) in the early cycles of a laser pulse due to efficient UDLR, the average absorbed energy ℰ_t per electron sphere may grow (as for MD in Fig. <ref> at the intensity 5 × 10^17) with increasing λ due to the dominant variation of free electron kinetic energy as U_p∝ E_0^2 λ ^2/4. To support this again for the RSM, we look at the free electrons kinetic energy scaling similar to the MD case presented in Fig. <ref>. Inset Fig. <ref>(e) show the comparison of the average kinetic energy ℰ_k per electron sphere versus higher λ≥ 500 nm for both LP (solid-squared orange line) and CP (dashed-squared orange line) with U_p ∝ E_0^2 λ ^2/4 (dashed-dotted blue line) at 5 × 10^17 corresponding to Fig. <ref>(b). One can see the growth of ℰ_k∝ U_p, but it is slower than the scaling of U_p for both LP and CP since the electrons (free or bound) do not have same kinetic energy and liberated at different times of the laser cycle. Also, growth ℰ_k is more close to U_p for CP than LP due to more number outer electrons with higher kinetic energies at different λ for CP than LP. This justifies the growth of ℰ_t with λ at a higher 0=5 × 10^17 for CP and LP; and also explains why ℰ_t for CP dominates LP as in Fig. <ref> for λ beyond the band of Λ_d ≈ (1-1.5)λ_M.
    
    
    Thus RSM results (in Fig. <ref>) bring out most of the features of MD results (in Fig. <ref>) which justify the dependence of redshift of absorption maxima on laser polarization and intensity and also disappearance of absorption maxima followed by a growth of absorption at higher intensity of 5 × 10^17 for CP and LP, with increasing λ. 
    
    
    
    
    
    
    
    
    
    § SUMMARY AND CONCLUSION
    
    
    
    We demonstrate the effect of laser polarization on the red-shift of the resonance absorption peak for a deuterium cluster irradiated by short 5-fs (fwhm) laser pulses using MD simulation and supported by RSM analysis. For both the polarization cases (LP and CP), we show that for a given intensity < 10^17 the optimized wavelength for maximum laser absorption in deuterium cluster lie in the band of wavelengths λ≈ 330± 67 nm in stead of the commonly expected static LR (Mie-resonance) wavelength of λ_M=263 nm. MD simulation and the RSM show gradual red-shift of the absorption maxima towards higher wavelengths in the marginally over-dense band of Λ_d ≈ (1-1.5)λ_M from λ_M of static LR with increasing laser intensity; and for higher intensities > 10^17 absorption peak disappears as outer ionization saturates at 100% for both LP and CP. This disappearance of the resonance absorption peak should not be misinterpreted as the negligible (or no) role of Mie-resonance. In fact, in this marginally over-dense band of Λ_d≈(1-1.5), both AHR and dynamical LR with near-LR enhanced effective field E_eff = E_0/(ω_M^2(t) - ω^2) contribute in unison very efficiently –  UDLR happens – to maximize absorption and outer ionization for both LP and CP. 
    It is also found that before the absorption peak, laser absorption due to LP and CP lasers are almost equally efficient (CP case being inappreciably higher than LP) for all intensities and λ. However, after the absorption peak, at lower intensities, absorption due to LP inappreciably dominates
    absorption due to CP with increasing λ which gradually reverses at higher intensities. Neglecting marginal differences for LP and CP cases, which may not be differentiated in real experiments, we conclude that laser absorption and outer ionization are almost same irrespective of the states of laser polarization for intensities < 10^17. Analysis (Fig.<ref>) of dynamical ω_M(t) suggests that Coulomb explosion of deuterium cluster is also almost independent of laser polarization. 
    
    Our results agree with the general observation from the nano-plasma model <cit.> that absorption peaks exist but red-shifted in the band of Λ_d≈(1-1.5). It is perhaps misleading to look for the absorption maxima  exactly at the λ_M in presence of non-zero outer ionization at higher intensities since linear Mie-resonance theory is invalid here. Our future work will report the pulse length dependence of the red-shift of the absorption peak.
    
    
    
    
    Authors would like to thank Sudip Sengupta for careful reading of the manuscript.
    
    *
