1 Introduction

An intense femtosecond (fs) laser pulse excites a molecule through the rotational Raman transitions, and creates a rotational wave packet in the molecule, which exhibits transient alignment and anti-alignment periodically with interval of its revival time [1,2,3]. Utilizing the field-free alignment induced by an intense fs laser pulse, we demonstrated isotope-selective ionization of 14N2 and 15N2 isotopologues [4]. In the demonstration, we shined an intense fs pulse with linear polarization (alignment pulse) to create rotational wave packets in these two isotopologues. At specific time delays after the alignment pulse irradiation, one isotopologue was aligned and the other one was anti-aligned due to difference between the revival times T i rev (i = 14 or 15). At one of the time delays, we shined another linearly-polarized intense fs pulse (ionization pulse) to ionize the isotopologues. Because of angular dependence of the N2 ionization probability [5,6,7,8,9,10], ionization of the aligned isotopologue is enhanced, whereas that of the anti-aligned one is suppressed. Our experiment showed that the ion yield ratio R [= I(15N2 +) / I(14N2 +)] could be controlled in the range of R = 0.85–1.22, by changing the time delay t delay between the alignment and ionization pulses [4].

The molecular laser isotope separation (MLIS) method based on the field-free alignment is a strong candidate for an isotope separation method of heavy isotopes, because the isotope selectivity does not decrease significantly as isotope mass of isotopologues increases [11]. Our simulations demonstrated that the selectivity has only a small dependence on the isotope mass [11]. This feature is an advantage over the conventional MLIS methods, based on vibrational excitation induced by an infrared laser light with a narrow linewidth [12,13,14]. The selectivity in the vibrational excitation depends on vibrational frequency difference between isotopologues, which decreases as the isotope mass increases [15]. Thus, in the conventional MLIS methods, the selectivity decreases as the mass of isotopologues increases. Additionally, some groups have realized an MLIS method utilizing vibrational wave packet propagation [16,17,18,19]. In the method, the isotope selectivity originates from difference in time-evolution between vibrational wave packets. Time evolution makes the vibrational wave packets spatially separated in the vibrational coordinate due to difference in vibrational period ∆T vib. However, the difference ∆T vib also decreases as the isotope mass increases. For a binary mixture of homonuclear diatomic isotopologues (αX2 and βX2), as an example, ∆T vib is proportional to ∆m/(m β)1/2, where m β is the atomic mass of βX, and ∆m is the difference between m β and m α (∆m = m β − m α), as described in Ref [11]. This relation indicates that the time difference decreases as m β increases, resulting in lowering of the selectivity.

To clearly separate the vibrational wave packets in the spatial coordinate, we need to wait for a number of vibrational recurrences until the wave packet of the heavier isotopologue delayed by a half of the vibrational period T vib [16]. Additionally, the vibrational wave packets collapse and revive periodically due to the vibrational anharmonicity. The vibrational revival does not always coincide with the timing when the delay between the vibrational wave packets is equal to T vib/2 [16]. On the other hand, at the first revival of the rotational wave packets, the influence of the centrifugal distortion is small and the isotope selectivity is achieved at a reasonable level [11].

In the MLIS based on the field-free alignment, the isotope selectivity depends strongly on the degree of alignment of isotopologues. The degree of alignment can be improved by increasing the intensity of the fs alignment pulse [20,21,22]. However, the alignment improvement is restricted by non-resonant ionization [23,24,25,26]. An intense alignment pulse also causes the ionization, which degrades the isotope selectivity. Hence, the isotope selectivity cannot be improved significantly as the pulse intensity is increased. The restriction on the alignment improvement due to ionization is serious for isotopologues including heavy isotopes, because of its low ionization potential. Multiple pulse irradiation, that is, a train of fs alignment pulses also can improve the degree of alignment without increasing the pulse intensity [25, 27,28,29,30]. Actually, our experiments for 14N2 and 15N2 isotopologues demonstrated that a train of four fs pulses with the pulse interval of 125.7 ps (= 15 T 14 rev = 14 T 15 rev) creates rotational wave packets in the isotopologues exhibiting higher degree of alignment [31]. In this case, we obtained the higher isotope selectivity with R ranging from 0.49 to 2.00. Our numerical simulations showed that selectivity can be further improved by increasing a number of the fs alignment pulses to 12, which leads to the wider R range of 0.31–3.0 [31]. However, the selectivity can no longer be improved, when the number of pulses increases to more than 12. This is because the centrifugal distortion induces time-mismatch between the pulse train and the rotational revivals [28, 32, 33]. The other strategies using more elaborated pulse-shaping techniques can make molecules well-aligned at a reasonable laser intensity, as demonstrated theoretically [23, 24, 27, 30, 34,35,36,37,38] and experimentally [39, 40]. Some of them can align multiple isotopologues simultaneously [23, 30], possibly leading to the higher selectivity.

In the present work, we focus our attention on another field-free alignment method, in which we consider an intense nanosecond (ns) laser field with rapid turn off at the maximum field (Fig. 1a) [41, 42]. Hereafter, we call the shaped pulse the “switched” ns pulse. In the slowly turned-on laser field, a molecule exhibits a high degree of alignment along the laser electric field in an adiabatic manner. At the rapid turn off, a rotational wave packet is created in a nonadiabatic manner, exhibiting transient alignment and anti-alignment periodically in the temporal propagation (Fig. 1b). The switched ns pulse makes molecules well-aligned at a lower intensity [I 0 ~ 1 TW/cm2 (= 1 × 1012 W/cm2)] [43,44,45] than a typical intensity (I 0 = 10–100 TW/cm2) of the fs alignment pulse. The feature of the alignment with the switched ns pulse is useful to reduce the ionization probability, and especially suitable for alignment of isotopologues including heavy isotopes.

Fig. 1
figure 1

Conceptual diagrams of the field-free alignment utilizing a ns laser field with rapid turn off; a Temporal profile of the laser field, and b time dependence of anisotropy parameter β with sketches of the molecular alignment

Full size image

We demonstrate theoretically isotope separation of I79Br and I81Br isotopologues aligned with a switched ns pulse, as an example. At a certain delay from the switched pulse, the isotopologues are irradiated with an intense fs pulse for the non-resonant ionization. We simulate the field-free alignment of the IBr isotopologues induced by the switched ns pulse that a laser pulse with 6-ns duration in full width at half maximum (FWHM) is truncated with the falling time of ~ 200 fs. The degree of IBr alignment is compared with two results simulated for unshaped pulses: the IBr alignment induced by a fs pulse with 60-fs duration in FWHM, and that induced by a ps pulse with 3-ps duration. To evaluate the isotope selectivity and the required laser condition, ion yield ratio of the isotopologues, R = I(I81Br+)/I(I79Br+), is calculated as a function of t delay of the fs ionization pulse from the timing of the rapid turn off.

2 Computational details

We solve the time-dependent Schrödinger equation (TDSE) for time evolution of rotational wave packets in IBr isotopologues [31, 46]. The Hamiltonian is expressed by the field-free rotational one \(^{i}{\widehat {H}_0}\) for IiBr isotopologue (i = 79 or 81), and the interaction potential with the alignment laser field \({\widehat {H}_{{\text{ind}}}}(t)\) [1,2,3], as follows:

$$^{i}\widehat {H}(t)~{=^i}{\widehat {H}_0}~+{\widehat {H}_{{\text{ind}}}}(t).$$
(1)

An eigenenergy of the first term \(^{i}{\widehat {H}_0}~\) at the rotational quantum number J is calculated by utilizing a set of Dunham-type parameters Y k,l , including the quadratic and higher-order centrifugal distortion terms [47], as follows:

$$^{i}{\widehat {H}_0}=\sum\limits_{{k,l}} {{Y_{k,l}}} {\left( {{\text{1}}/{\text{2}}} \right)^k}{\left[ {J\left( {J+{\text{ 1}}} \right)} \right]^l}.$$
(2)

The second term of Eq. (1) can be expressed using the temporal amplitude (envelop) of the laser electric field E(t), and the angle between the molecular axis and the laser electric field direction θ, as follows:

$${\widehat {H}_{ind}}(t)~=~ - \frac{1}{4}E{(t)^2}({\alpha _\parallel }{\cos ^2}\theta ~+~{\alpha _ \bot }{\sin ^2}\theta ).$$
(3)

Here \({\alpha _\parallel }\) and \({\alpha _ \bot }\) are the polarizabilities in the parallel and perpendicular directions to the IBr molecular axis, respectively. The laser electric field E(t) of the switched ns pulse is defined as,

$$E\left( t \right){\text{ }}={E_0}\exp [ - 2\left( {\ln \,2} \right){(t/{\tau _r})^2}]\quad ~\left( {t \leqslant 0} \right),$$
(4a)
$$E\left( t \right){\text{ }}={E_0}\exp [ - 2{\text{}}\left( {\ln 2}\right){(t/{\tau _f})^2}]\quad \left( {t>0} \right),$$
(4b)

where the maximum electric field E 0 is set to be the experimentally realized value of 3.88 × 109 V m− 1 (I 0 = 2 TW/cm2) [45], and τ r and τ f represent the rising and falling times of the electric field. Here, we use τ r = 6 ns, which is a typical value of a Nd:YAG laser output, and τ f = 200 fs, which is almost the same as the experimental values [43,44,45]. The polarizabilities \({\alpha _\parallel }\) and \({\alpha _ \bot }\) of IBr are calculated to be \({\alpha _\parallel }\) = 1.17 × 10− 39 C m2 V− 1 and \({\alpha _ \bot }\) = 7.8 × 10− 40 C m2 V− 1, by solving time-dependent Kohn–Sham (TD-KS) equation [48], which is the fundamental equation of the time-dependent density-functional theory (TD-DFT), with KLI-SIC exchange–correlation potential [49, 50]. The TDSE is solved numerically per time step of 2 fs, using the fourth-order Runge–Kutta method [51]. For comparison, the simulation using a 60-fs alignment pulse (τ r = τ f = 60 fs, 0.5-fs time step) with an order-of-magnitude higher intensity of I 0 = 20 TW/cm2, and that using a 3-ps alignment pulse (τ r = τ f = 3 ps, 0.5-fs time step) with I 0 = 20 TW/cm2 are performed in the same manner.

The degree of alignment is quantified using anisotropy parameter β = 3\(\left\langle {\cos ^{2} \theta } \right\rangle\) − 1, where \(\left\langle {\cos ^{2} \theta } \right\rangle\) is the expectation value of cos2 θ. Here, the perfectly aligned, anti-aligned, and isotropic distributions have β values of 2, − 1, and 0, respectively.

To obtain the ion yield ratio R [= I(I81Br+)/I(I79Br+)] as a function of time delay t delay, the ion yield I(IiBr+) is calculated for each IBr isotopologue, by utilizing the time-dependent angular distribution i P(θ, t delay) and the angular-dependent ionization probability Γ(θ), as follows:

$$I({{\text{I}}^i}{\text{B}}{{\text{r}}^+})=\int\limits_{0}^{\pi } {^{i}P\left( {\theta ,~{t_{{\text{delay}}}}} \right)\, \cdot \,\Gamma \left( \theta \right)~{\text{d}}\theta } ,$$
(5)

where i P(θ, t delay) is obtained as the square of the time-dependent rotational wavefunction. The ionization probability Γ(θ) of IBr caused by a fs laser field (16.2-fs duration in FWHM, λ = 800 nm, I 0 = 15 TW/cm2) is simulated as a function of the angle θ between the molecular axis and the laser field direction. The simulation procedure has been reported in detail [52, 53]. Briefly, we solve the TD-KS equation [48] with KLI-SIC exchange–correlation potential [49, 50]. The out-going wave function, which corresponds to the ionization flux, is mimicked by the absorbing boundary condition [52]. To calculate the ion yield I(IiBr+), we assume the ionization pulse with 60-fs duration. However, the simulation of the ionization probability is so time-consuming that Γ(θ) is calculated for the shorter pulse with 16.2-fs duration and is simply multiplied by 3.7 (= 60/16.2) to calculate I(IiBr+).

3 Results

3.1 Field-free alignment

Figure 2a shows time dependence of anisotropy parameters β for IBr isotopologues (the rotational temperature of T rot = 5 K) when a switched 6-ns pulse with I 0 = 2 TW/cm2 is used for the alignment. The two isotopologues behave in the same manner, excepting the revival times T i rev (T 79 rev = 294.0 ps, T 81 rev = 298.6 ps). For each isotopologue, β gradually increases in the time range of t < 0, and after the sudden decrease in the laser intensity at t = 0, it shows a spike-like dip with two small side peaks at every half revival of t = (n + 1/2)T i rev (n is an integer), and a spike-like peak with two small side dips at every full revival of t = n T i rev. These behaviors are the typical features of the field-free alignment induced by a switched ns pulse [41, 42]. The anisotropy parameter β varies in the range of − 0.72 to 1.40 for each isotopologue. The variation range caused by the switched 6-ns pulse is twice as wide as that by a 60-fs pulse irradiation with an order-of-magnitude higher intensity of I 0 = 20 TW/cm2 (β = − 0.49 ~ 0.70), as shown in Fig. 2a, b.

Fig. 2
figure 2

a Time dependence of anisotropy parameters β for I79Br (red dashed curve) and I81Br (blue solid curve) isotopologues (T rot = 5 K) when a switched ns pulse (τ r = 6 ns, τ f = 200 fs) with I 0 = 2 TW/cm2 is used for the alignment. The down arrows with dotted stem indicate the revival times of I79Br, and the up arrows indicate those of I81Br. b, c Same as a for a 60-fs pulse (τ r = τ f = 60 fs) with I 0 = 20 TW/cm2, and for a 3-ps pulse (τ r = τ f = 3 ps) with I 0 = 20 TW/cm2, respectively

Full size image

It is known that the degree of alignment in the non-adiabatic regime (when the pulse duration ∆t is much shorter than the rotational period T i rev) is maximized by the pulse energy. Quantum mechanical calculations have demonstrated that the non-adiabatic regime would remain unchanged if the laser pulse is chirped up to ∆t ~ T i rev/100 [54]. For T i rev ~ 300 ps of IBr isotopologues, we can extend the duration up to ~ 3 ps, 50 times longer than the 60-fs pulse. Figure 2c shows the result for a 3-ps pulse with I 0 = 20 TW/cm2. The anisotropy parameter β ranges from − 0.30 to 1.49, which is wider than that for the 60-fs pulse (Fig. 2b) due to its 50 times larger pulse energy. The maximum of 1.49 is larger than that for the switched 6-ns pulse (β = 1.40). However, the maximum appears at a short time delay of t ~ 0.45 ps, when the β curve of I79Br almost overlaps with that of I81Br. The negligible difference will give us only a small isotope selectivity. Excepting the short delay range of t < 10 ps, β ranges from − 0.30 to 1.10 for each isotopologue. The β range is narrower than that for the switched 6-ns pulse (β = − 0.72 to 1.40) because, in the non-adiabatic regime, a single-pulse irradiation imposes a restriction on the achievable alignment [40, 55, 56]. These results indicate that the switched 6-ns pulse can induce a better IBr alignment than those induced by the unshaped pulses, in spite of its lower laser intensity.

3.2 Evaluation of the isotope selectivity

To evaluate the isotope selectivity, we simulate the ion yield for each IiBr isotopologue (T rot = 5 K) irradiated with the switched 6-ns pulse (I 0 = 2 TW/cm2), followed by the 60-fs ionization pulse with I 0 = 15 TW/cm2 as a function of time delay t delay of the ionization pulse from the timing of the rapid turn off of the switched ns pulse. Here, angular dependence of ionization probability Γ(θ) shown in Fig. 3 is used in the simulation of the ion yield I(IiBr+) represented by Eq. (5). The angular dependence shows a butterfly-like pattern, which implies that the ion yield decreases when IBr is aligned to the laser electric field, and the yield increases when IBr is anti-aligned. Figure 4a shows the ion yields I(I79Br+) and I(I81Br+) in the two ranges of t delay = − 0.1 ~  0.7 ns, and 9.3 ~ 10.0 ns. In the former range, the first couple of full revivals of the two isotopologues shift slightly. In the latter range, a full revival of I79Br coincides with a half revival of I81Br, and vice versa. Because of the alignment dependence of the ionization, I(IiBr+) shows a peak at every half revival, and a dip at every full revival. In the delay range of 9.3 to 10.0 ns, the dips at the full revivals are smaller, and the peaks at the half revivals are slightly smaller than those in the delay range of − 0.1 to 0.7 ns. These results are attributed to the centrifugal distortion effect, which leads to dephasing of the rotational wave packet. In addition to the half and full revivals, a small peak of I(IiBr+) appears at every quarter revival of t = (n + 1/2)T i rev /2, whereas no peak appears in β (Fig. 2a). At the quarter revivals, the wave packet shows a butterfly-like angular distribution i P(θ), being similar with the ionization probability Γ(θ) shown in Fig. 3. The butterfly-like shape of i P(θ) causes the increase of I(IiBr+) at the quarter revival due to the large overlap with Γ(θ).

Fig. 3
figure 3

Angular dependence of the ionization probability Γ(θ) of IBr obtained by solving the TD-KS equation (λ = 800 nm, I 0 = 15 TW/cm2) [52, 53]

Full size image
Fig. 4
figure 4

a The simulated ion yields of I79Br+ (red dashed curve) and I81Br+ (blue solid curve), as a function of time delay t delay in the two ranges of − 0.1 to 0.7, and 9.3–10.0 ns. b The ratio R [= I(I81Br+)/I(I79Br+)] as a function of t delay

Full size image

Figure 4b shows the ion yield ratio R [= I(I81Br+)/I(I79Br +)], in which several peaks and dips appear, demonstrating that the present approach works well as an MLIS method. The maximum is obtained to be R max = 2.01 at t delay = 294.0 ps, where the main dip of I(I79Br+) at the first full revival coincides with the side peak of I(I81Br+) preceding the main dip, as shown in left inset in Fig. 4a. The minimum is R min = 0.49 at t delay = 298.4 ps, where the main dip of I(I81Br+) at the first full revival coincides with the side peak of I(I79Br+) following the main dip (left inset in Fig. 4a).

A wide range of R is expected when a main peak of one isotopologue coincides with a main dip of the other one. This out-of-phase condition is realized when a half revival of one isotopologue coincides with a full revival of the other one [11, 57], that is, (n + 1/2) T i rev ≈ n′ T j rev. The time delays for the out-of-phase condition are derived as:

$${t_{{\text{out-of-phase}}}}\left( k \right) \approx ({\text{2}}k - {\text{1}}){T_{{\text{79}}}}^{{{\text{rev}}}}{T_{{\text{81}}}}^{{{\text{rev}}}}/[{\text{2}}({T_{{\text{81}}}}^{{{\text{rev}}}} - {T_{{\text{79}}}}^{{{\text{rev}}}})],$$
(6)

where k = n′ – n [11]. The ratio R around the shortest t out-of-phase(k = 1) ≈ 9.5 ns is shown in the right panel of Fig. 4b. Although the temporally varying I(I79Br+) and I(I81Br+) are out-of-phase around t delay = 9.5 ns (right panel of Fig. 4a), the selectivity is lower than that around the first revival; R at the peak is smaller than R max = 2.01 at t delay = 294.0 ps, and R at the dip is larger than R min = 0.49 at t delay = 298.4 ps. This situation is different from the N2 isotope selective ionization [4, 31]. In the N2 case, the higher selectivity is obtained at t delay ≈ t out-of-phase(k = 1).

4 Discussion

4.1 Influence of ionization induced by alignment laser pulses

To achieve a high isotope selectivity, we should take care of not only enhancement of the alignment but also suppression of the ionization induced by the alignment pulse. Although the switched 6-ns pulse with an order-of-magnitude lower intensity will lead to a lower ionization rate than that of the 60-fs pulse (or the 3-ps pulse), we should take the pulse durations into account, because of the switched 6-ns pulse which is five (three) orders-of-magnitude longer than that of the 60-fs pulse (the 3-ps pulse). In the following estimation of the ionization probabilities, we assume a Ti:sapphire laser (λ ~ 800 nm, τ r = τ f = 60 fs) as an example of the unshaped pulses and a Nd:YAG laser (λ ~ 1064 nm, τ r = 6 ns, τ f = 200 fs) for the switched ns pulse. Here, we consider two limiting cases of nonlinear ionization, multiphoton and tunnel ionizations [58]. On the basis of the IBr ionization potential (I p = 9.79 eV [59]), the IBr ionization requires seven photons at λ ~ 800 nm and 9 photons at λ ~ 1064 nm. In the multiphoton ionization limit, where the Keldysh’s adiabaticity parameter γ (\(=\sqrt {{I_{\text{p}}}/2{U_{\text{p}}}}\), where U p is the ponderomotive energy) ≫ 1, the probability can be estimated using the ionization cross section σ(λ), the laser intensity I 0, and the pulse duration ∆t, as:

$${P_{{\text{ion}}}}(\lambda ) \approx \sigma (\lambda ){I_0}^{N}\Delta t,$$
(7)

where N is the required number of photons. As far as we know, σ(λ) of IBr is not available at λ = 800 and 1064 nm. Therefore, in the present estimation, we suppose that σ(1064 nm) = σ(800 nm). As the intensity I 0 of 1064-nm pulse is decreased by one order-of-magnitude, the ionization rate (= σ(λ) I 0 N) is decreased by nine orders-of-magnitude. Considering the switched 6-ns pulse being five orders-of-magnitude longer than the 60-fs pulse, P ion(λ) induced by the switched 6-ns pulse with I 0 = 2 TW/cm2 is four (= 9–5) orders-of-magnitude lower than that induced by the 60-fs pulse with I 0 = 20 TW/cm2. On the basis of this order estimation, it is obvious that P ion(λ) induced by the switched 6-ns pulse with a realistic intensity is lower than that induced by the 60-fs pulse, unless σ(1064 nm) is four or more orders-of-magnitude larger than σ(800 nm). In the tunnel ionization limit, where γ << 1, we use the simple ADK theory [60] to estimate the ionization rate. The ADK rate of IBr ionization at I 0 = 2 TW/cm2 is sixteen orders-of-magnitude lower than that at I 0 = 20 TW/cm2, independently of the wavelength. Taking the pulse durations into account, we estimate that the ionization probability induced by the switched 6-ns pulse is 11 (= 16–5) orders-of-magnitude lower than that induced by the 60-fs pulse. These estimations for the two limiting cases suggest that the ionization induced by the switched 6-ns pulse (γ = 6.8 for 1064-nm pulse with I 0 = 2 TW/cm2) also has a lower probability than that induced by the 60-fs pulse (γ = 2.9 for 800-nm pulse with I 0 = 20 TW/cm2). Therefore, we expect that the present approach using the switched ns pulse works effectively for isotope separation of heavy isotopes.

4.2 Rotational dephasing

The lower selectivity of IBr at t delay ≈ t out-of-phase(k = 1) (Fig. 4b) is due to the small dips and the small peaks in each I(IiBr+). As shown by the dashed curves in insets in Fig. 4a, the main dip shape of I(I79Br+) observed at the first full revival (left inset) is not identically reproduced at the 33rd full revival (right inset), where I(I79Br+) shows an asymmetric “chirped” structure [61]. The chirped structure is also observed at the 33rd half revival of I(I81Br+) (solid curve in the right inset). These chirped structures cause the lower contrast in each I(IiBr+) (right inset in Fig. 4a), resulting in the lower selectivity at t delay = t out-of-phase(k = 1) ≈ 9.5 ns (Fig. 4b).

These chirped structures imply serious dephasing of the rotational wave packet, which is a consequence of the centrifugal distortion effect [61]. However, centrifugal distortion effect of IBr is smaller than that of N2. In Table 1, we compare a ratio D 0/ B 0 of I79Br [47] to that of 14N2 [62], because the ratio is a nice index of the centrifugal distortion effect per revival [32, 33]. Here, D 0 and B 0 are the centrifugal distortion and rotational constants of the vibrational ground state, respectively. Actually, the ratio D 0/ B 0 of I79Br is an order-of-magnitude smaller than that of 14N2, implying a smaller IBr dephasing at t delay ~ T i rev.

Table 1 Rotational constant B 0 and ratio D 0 /B 0 of the centrifugal distortion constant to B 0
Full size table

The rotational dephasing of IBr at t delay ≈ t out-of-phase(k = 1) results from the following two factors: the wide rotational state distribution, and the large n and n′ satisfying t delay ≈ t out-of-phase(k = 1). We discuss how the dephasing is affected by the two factors, comparing the IBr case with the N2 case.

First, IBr has a two orders-of-magnitude smaller B 0 than N2, as listed in Table 1. As far as T rot is the same, the rotational state distribution of IBr is wider than that of N2. At T rot = 5 K, as an example, ca. 0.9 of 14N2 molecules populate only in the J = 0 and J = 1 rotational levels, whereas ca. 0.9 of I79Br molecules populate in the J = 0 ~ 11 levels. The centrifugal distortion effect quadratically increases as J increases [63]. Hence, the centrifugal distortion at J = 11 is two orders-of-magnitude larger than that at J = 1. Although D 0/ B 0 of IBr is one order-of-magnitude smaller than that of N2, IBr in the J = 11 level has, in total, an order-of-magnitude larger effect than N2 in the J = 1, resulting in the larger dephasing of IBr.

Next, n (= 32) and n′ (= 33) used in IBr are four times as large as those (n = 8, n′ = 7) in N2 [4]. In spite of the smaller dephasing per revival, the larger numbers of the revivals enhance the effect, resulting in the larger dephasing of IBr at t delay ≈ t out-of-phase(k = 1).

Both these two factors should contribute to the rotational dephasing of IBr observed at t delay ≈ t out-of-phase(k = 1). Isotopologues including heavy isotopes commonly have these two factors, the wide rotational state distribution and the large n and n′. When we apply the present MLIS method for heavy isotopes, serious dephasing will be observed at t delay ≈ t out-of-phase(k = 1). Therefore, the delay range around the first revival will provide higher isotope selectivity.

5 Conclusion and outlook

In the present study, we proposed an MLIS method based on field-free alignment of isotopologues induced by a switched ns laser pulse, as a strong candidate for an isotope separation method of heavy isotopes. To demonstrate how the method works effectively for heavy isotopes, we performed the numerical simulations. Our simulations for IBr isotopologues (T rot = 5 K) showed that a switched 6-ns laser pulse with I 0 = 2 TW/cm2 can induce the better alignment with its variation range of β being wider than those obtained by unshaped pulses (∆t = 60 fs or 3 ps) with an order-of-magnitude higher intensity of I 0 = 20 TW/cm2. This simulated result means that we can make isotopologues including heavy isotopes well-aligned with the switched ns pulse. Although the heavy isotopologues are easily ionized due to their low ionization potentials, the switched ns pulse can suppress the ionization probability significantly. Hence, the switched ns pulse has a significant advantage for isotope separation of heavy isotopes.

Additionally, we simulated the ion yield ratio R [= I(I81Br+)/I(I79Br +)] of the IBr isotopologues for the ionization caused by the fs ionization pulse at I 0 = 15 TW/cm2, utilizing the ionization probability Γ(θ) shown in Fig. 3. The ratio varied in the range of 0.49–2.01, as a function of t delay between the switched pulse and the fs ionization pulse. The R variation demonstrates the effectiveness of the present method utilizing the switched ns pulse.

The present MLIS utilizing a switched ns pulse can be applied to the other isotopologues by simply tuning the time delay, because the method does not use any resonance transitions [4, 31]. This fact means that we do not have to tune the laser wavelength, which makes a contrast to conventional laser isotope separation (LIS) methods, including conventional MLIS [12,13,14] and atomic vapor LIS methods [64, 65], where a tunable laser is needed to utilize isotope shift of resonant excitation. This advantage allows us to easily apply the present method to a wider variety of isotopes, including heavy isotopes.