General phenomenology of ionization from aligned molecular ensembles

The one-photon case has been extensively treated in the literature [28, 30, 39, 41]. We recount here the salient details, with a specific focus on the coupling of the observable to the A_K,Q, then proceed to determine the properties of specific types of measurement and extend the formalism to the N-photon case.

The full photoelectron angular distribution can be most generally expressed as a multipole expansion (analogous to $P(\theta ,\phi ,t)$ discussed above, see equation (2)):

$$\begin{eqnarray}&&I(\theta ,\phi ,t)=\mathop{\sum }\limits_{L,M}{{\beta }_{L,M}}(t){{Y}_{L,M}}(\theta ,\phi ).\end{eqnarray} \tag{ 12 }$$

Here the polar coordinates reference the LF, as defined by the probe pulse (see figure 1(d)), in which the photoelectron flux as a function of angle and time is measured⁶ . The LF ${{\beta }_{L,M}}(t)$ can be written in terms of the coherent square of the dipole matrix elements: for the ionization of an aligned ensemble, in the perturbative and dipole approximations, and assuming that all time-dependence is contained in the axis distribution, the ${{\beta }_{L,M}}(t)$ can be written as [28, 39]:

$$\begin{eqnarray}\begin{array}{rcl} {{\beta }_{L,M}}(t) & = & {{(2L+1)}^{1/2}}\mathop{\sum }\limits_{P}{{(-1)}^{P}}\left( \begin{array}{ccccccccccccccc} 1 & 1 & P \\ p & -p & R \\ \end{array} \right){{e}_{-p}}e_{-p}^{*} \\ {} & {} & \times \mathop{\sum }\limits_{K}\mathop{\sum }\limits_{Q}{{(2K+1)}^{1/2}}\left( \begin{array}{ccccccccccccccc} P & K & L \\ Q-M & -Q & M \\ \end{array} \right){{A}_{K,-Q}}(t) \\ {} & {} & \times \mathop{\sum }\limits_{q,q^{\prime} }{{(-1)}_{{{q}^{\prime }}}}\left( \begin{array}{ccccccccccccccc} 1 & 1 & P \\ q & -q^{\prime} & q^{\prime} -q \\ \end{array} \right)\left( \begin{array}{ccccccccccccccc} P & K & L \\ q-q^{\prime} & q^{\prime} -q & 0 \\ \end{array} \right) \\ {} & {} & \times \mathop{\sum }\limits_{l,l^{\prime} }\mathop{\sum }\limits_{\lambda ,\lambda ^{\prime} }{{(-1)}^{\lambda ^{\prime} }}{{(2l+1)}^{1/2}}{{(2l^{\prime} +1)}^{1/2}}\left( \begin{array}{ccccccccccccccc} l & l^{\prime} & L \\ \lambda & -\lambda ^{\prime} & M \\ \end{array} \right)\left( \begin{array}{ccccccccccccccc} l & l^{\prime} & L \\ 0 & 0 & 0 \\ \end{array} \right) \\ {} & {} & \times \;{{(-i)}^{l^{\prime} -l}}\mathop{\sum }\limits_{\Gamma ,\Gamma ^{\prime} }\mathop{\sum }\limits_{\mu ,\mu ^{\prime} }\mathop{\sum }\limits_{h,h^{\prime} }b_{hl\lambda }^{\Gamma \mu *}b_{h^{\prime} l^{\prime} \lambda ^{\prime} }^{\Gamma ^{\prime} \mu ^{\prime} }{\boldsymbol{D}} _{hl}^{\Gamma \mu *}(q){\boldsymbol{D}} _{h^{\prime} l^{\prime} }^{\Gamma ^{\prime} \mu ^{\prime} }(q^{\prime} ). \\ \end{array}\end{eqnarray} \tag{ 13 }$$

The first line of equation (13) describes the polarization state of the ionizing radiation; the photon carries 1 unit of angular momentum with projection p onto the lab frame z-axis. For linearly polarized light aligned with the LF z-axis p = 0, hence from the 3-j symbol P = 0, 2 and R = 0. The spherical tensor components ${{e}_{-p}}$ describe the polarization and amplitude of the ionizing radiation, for the case of linearly polarized light along the z-axis ${{e}_{-p}}={{e}_{0}}={{e}_{z}}$ and the term ${{e}_{z}}e_{z}^{*}$ can be set to equal unity.

The second and third lines of equation (13) describe the convolution of the MF with the aligned axis distribution, P(θ, t), expressed as ADMs. The light field has MF projection terms q. Terms in q = 0 thus represent ionizing light polarized along the MF axis, while $q=\pm 1$ terms represent light polarized perpendicular to the MF axis. If the LF and MF are coincident then a single value of q = p is selected, while an arbitrary rotation serves to mix terms in q as the LF polarization axis is projected onto different MF axis. This mixing (and averaging), due to the ADMs, is described by the coupling of P and K into the final multipole moments L.

The remaining lines of equation (13) deal with the photoelectron and 'molecular' terms. Here $(l,\lambda )$ represent the photoelectron partial wave components [38, 47], with (orbital) angular momentum l, and MF projection λ. The terms ${\boldsymbol{D}} _{hl}^{\Gamma \mu }(q)$ represent the symmetrized radial components, with symmetrization coefficients $b_{hl\lambda }^{\Gamma \mu }$ (see appendix B), of the (radial) dipole matrix elements for each symmetry-allowed continuum Γ [39, 48, 49],

$$\begin{eqnarray}&&{\boldsymbol{D}} _{hl}^{\Gamma \mu }(q)=\langle {{\Psi }^{+}};\psi _{hl}^{\Gamma \mu ,e}|\mathop{\sum }\limits_{s}{{r}_{s}}{{Y}_{1,q}}({{{\bf \hat{r}}}_{s}})|{{\Psi }^{i}}\rangle ,\end{eqnarray} \tag{ 14 }$$

where $\psi _{hl}^{\Gamma \mu ,e}$ are the partial wave components of the photoelectron wavefunction ${{\Psi }^{e}}$ and the summation is over all electrons s. These matrix elements are complex, and may also be written in the form ${\boldsymbol{D}} _{hl}^{\Gamma \mu }=|{\boldsymbol{D}} _{hl}^{\Gamma \mu }|{{{\rm e}}^{-{\rm i}\eta _{hl}^{\Gamma \mu }}}$, where η is the total phase of the matrix element, often called the scattering phase. The radial matrix elements and phases are the only part of equation (13) which are not analytic functions and, in general, must be determined numerically [50, 51] or from experiment [52–54] for quantitative understanding of a given system. Symmetry-based arguments can, however, provide a means of determining which integrals are non-zero, hence which $(l,\lambda )$ can appear in ${{\Psi }^{e}}$. Such considerations therefore allow for phenomenological, qualitative, or possibly semi-quantitative, treatments of photoionization for a given molecule, and are discussed in appendix B.

The effect of the averaging over a distribution of molecular axis directions is to lose sensitivity in the PADs. In particular, the observed anisotropy in the LFPAD cannot be more than that arising from the coupling of the probe photon to the aligned distribution of molecules, as can be seen from the 3-j term linking terms $P,K,L$. This limits L to the range $|P-K|...P+K$ in integer steps. For instance, if the alignment is prepared by a single pump photon then a ${{{\rm cos} }^{2}}\theta$ axis distribution is created, and the only non-zero alignment parameters are ${{A}_{0,0}}$ and ${{A}_{2,0}}$. Because P = 0, 2 only, the alignment in this case would restrict ${{\beta }_{L,M}}(t)$ to terms with L = 0, 2, 4 (additionally, for cylindrically symmetric cases, $M=-Q=0$). As the degree of alignment increases higher-order K terms are required to describe the axis distribution and the LF ensemble result approaches the true MF [28]. Higher order terms in equation (13) can be observed, hence more information is present in the LFPAD and a greater sensitivity to any property which affects the PADs, e.g. the evolution of the axis distribution itself, intermediate state dynamics in a pump–probe experiment, and so on, may be obtained.

For an angle-integrated measurement (photoelectron yield), integration over $\{\theta ,\phi \}$ leaves only the leading term ${{\beta }_{0,0}}(t)$, and angular coherences between partial waves are integrated out of the measurement. In this case the terms remaining in (13) are significantly restricted by the 3-j terms. Allowed terms have l = l'and P = K. For instance, for one-photon ionization of a distribution with P = 0, 2, only ${{A}_{K,Q}}(t)$ with K = 0, 2 will be coupled to the ionization yield. Depending on the magnitudes of the parallel and perpendicular ionization matrix elements, this can result in the one-photon yield mapping the ${{A}_{2,0}}(t)$ somewhat directly [31].

To make the geometric convolution of the axis distribution and MF photoionization more explicit, equation (13) can also be written in the form [28]:

$$\begin{eqnarray}&&{{\beta }_{L,M}}(t)=\mathop{\sum }\limits_{K,Q}{{A}_{K,-Q}}(t){{a}_{KLM}},\end{eqnarray} \tag{ 15 }$$

where the a_KLM contain all the other terms of equation (13). As per equation (13), the allowed values of M and Q are coupled; for cylindrically symmetric geometries, i.e. probe polarization parallel to the pump polarization, Q = M = 0. In this case the photoelectron yield is given by:

$$\begin{eqnarray}&&{{\beta }_{0,0}}(t)=\mathop{\sum }\limits_{K=0,2}{{A}_{K,0}}(t){{a}_{K00}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&=\;{{A}_{0,0}}(t){{a}_{000}}+{{A}_{2,0}}(t){{a}_{200}}.\end{eqnarray} \tag{ 17 }$$

Since the zero-order term ${{A}_{0,0}}(t)$ is just the total population of the rotational states forming the wavepacket (often normalized to unity), it is a constant in the absence of any population dynamics, and any time-dependence observed in the ionization yield is due to the second-order term, as asserted above.

The effect of frame rotations on the ${{\beta }_{L,M}}(t)$ can also be simply expressed using this form, and employing the Wigner rotation matrix:

$$\begin{eqnarray}&&\beta _{L,M^{\prime} }^{\prime }(t;\Omega )=\mathop{\sum }\limits_{K}\mathop{\sum }\limits_{Q,Q^{\prime} }D_{Q^{\prime} ,Q}^{K}(\Omega ){{A}_{K,-Q}}(t){{a}_{KLM}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&=\;\mathop{\sum }\limits_{K}\mathop{\sum }\limits_{{{Q}^{\prime }}}A_{K,-Q^{\prime} }^{\prime }(t){{a}_{KL{{M}^{\prime }}}},\end{eqnarray} \tag{ 19 }$$

where the second form follows from the assumption that Q = 0. Here $\Omega =\{\Phi ,\Theta ,\chi \}$ is the set of Euler angles describing the frame rotation between the alignment field and the probe field, and the properties in the rotated frame are denoted by primes. The rotation mixes multipole components Q within a given rank K. Because of the coupling between Q and M (the second 3-j term in equation (13)) different terms, M', are allowed in the rotated frame. As discussed above, and illustrated in figures 1 and 2, such a frame rotation can break the symmetry of an initially cylindrically symmetric distribution, hence M' may be non-zero even if M = 0.

Applying again the stipulations above, the ionization yield for a cylindrically symmetric distribution (Q = 0) under a rotation of Θ between the alignment and ionization fields, is then given by:

$$\begin{eqnarray}&&\beta _{0,0}^{\prime }(t;\Theta )=\mathop{\sum }\limits_{K=0,2}\mathop{\sum }\limits_{{{Q}^{\prime }}}D_{Q^{\prime} ,0}^{K}(0,\Theta ,0){{A}_{K,0}}(t){{a}_{K00}},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&=\mathop{\sum }\limits_{K=0,2}\mathop{\sum }\limits_{{{Q}^{\prime }}}d_{Q^{\prime} ,0}^{K}(\Theta ){{A}_{K,0}}(t){{a}_{K00}},\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}\begin{array}{cll} {} & = & {{A}_{0,0}}(t){{a}_{000}}+d_{0,0}^{2}(\Theta ){{A}_{2,0}}(t){{a}_{200}} \\ {} & {} & +2d_{2,0}^{2}(\Theta ){{A}_{2,0}}(t){{a}_{200}}, \\ \end{array}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&=\;{{A}_{0,0}}(t){{a}_{000}}+(A_{2,0}^{\prime }(t)+2A_{2,2}^{\prime }(t)){{a}_{200}},\end{eqnarray} \tag{ 23 }$$

where $d_{Q^{\prime} ,Q}^{K}(\Theta )$ is the reduced Wigner rotation matrix element, and we have used the identities $d_{0,0}^{0}=1$, $d_{2,0}^{2}=d_{-2,0}^{2}$ [55]. This form makes explicit the fact that the frame rotation mixes additional Q terms into $\beta _{0,0}^{\prime }(t;\Theta )$ as compared to ${{\beta }_{0,0}}(t;\Theta =0)$ (equation (17), see also figure 2). Therefore, in general, a measurement of $\beta _{0,0}^{\prime }(t;\Theta \ne 0)$ will not map ${{A}_{2,0}}(t)$ as directly as $\beta _{0,0}^{\prime }(t;\Theta =0)$.

A measurement of the photoelectron yield as a function of Θ will therefore have the general form (using equation (3.93) from Zare [55]):

$$\begin{eqnarray}\begin{array}{rcl} I(\Theta ;t) & = & \beta _{0,0}^{\prime }(\Theta ;t)={{A}_{0,0}}(t){{a}_{000}}+{{\left( \frac{4\pi }{5} \right)}^{\frac{1}{2}}}{{A}_{2,0}}(t){{a}_{200}}Y_{2,0}^{*}(\Theta ,0) \\ {} & {} & +2{{\left( \frac{4\pi }{5} \right)}^{\frac{1}{2}}}{{A}_{2,0}}(t){{a}_{200}}Y_{2,2}^{*}(\Theta ,0). \\ \end{array}\end{eqnarray} \tag{ 24 }$$

Experimentally, this corresponds to a measurement of the photoionization yield as a function of polarization geometry (defined by Θ, as illustrated in figure 1(d)). Measurements of this form may be made for all Θ, yielding the polarization-angle-resolved ionization yield as a quasi-continuous function, or compared at selected Θ to provide 'transient anisotropy' measurements, for example the standard formulation (which compares yields at Θ = 0 and Θ = π/2) has been explored with respect to ionization [56]. Clearly, a polarization-resolved measurement of this form may be expected to display relatively complex angular structure, with up to four lobes on the interval $0\leqslant \Theta \leqslant 2\pi$ due to the summation of ${{Y}_{2,0}}$ and ${{Y}_{2,2}}$ terms, despite the fact that only the second-order moment of the axis distribution is invoked. This serves to illustrate how even an apparently simple experimental measurement may respond in a more complex fashion than anticipated to an aligned ensemble.

Formally, a direct multi-photon ionization process constitutes a ladder of transitions through virtual states. In terms of the decomposition of this ladder of transitions into dipole matrix elements for each successive photon absorption—a direct extension of the one-photon case discussed above—the complexity rapidly grows, and may be further complicated by near-resonances with real bound states. A full treatment of such cases for atomic ionization has been given by Bebb and Gold [57], including the derivation of an effective N-photon matrix element to reduce the complexity of the problem⁷ . Here we take a similar approach and consider treating the ionization as an effective one-photon transition in which the photon angular momentum is large. The ionization would then have a form essentially as equation (13), but with the photon carrying N units of angular momentum. Such a treatment should allow for at least a qualitative picture of the directionality of the ionization, and in particular the polarization-angle-resolved ionization yields I(Θ,t), with a clear link to the multipoles involved in the molecular axis alignment and ionization process.

The immediate result is that P can take many more values than in the one-photon case. For a cylindrically symmetric distribution equation (20) becomes:

$$\begin{eqnarray}&&\beta _{0,0}^{\prime }(t;\Theta )=\mathop{\sum }\limits_{K=0...2N}\mathop{\sum }\limits_{{{Q}^{\prime }}}D_{Q^{\prime} ,0}^{K}(0,\Theta ,0){{A}_{K,0}}(t){{a}_{K00}}.\end{eqnarray} \tag{ 25 }$$

Hence higher-order ADMs may be coupled to the observed signal, according to the value of N. In practice one would expect, therefore, to observe more structure in the alignment trace (temporal signal) $\beta _{0,0}^{\prime }(t;\Theta )$ for a given Θ, due to the coupling of larger K into the signal. Similarly, more angular structure may be observed in the angle-resolved yield I(Θ,t) and the PADs, $I(\theta ,t;\Theta )$, recorded for a given polarization geometry. Because l_max, the maximum photoelectron angular momentum, will also grow with N, it is likely that high-order ADMs are always coupled to such high-order processes, regardless of the exact details of the photoionization matrix elements.

Physically, one can understand the higher-order angular momenta as signifying a more directional ionization event. For example, in the limit of tunnel ionization, the outgoing electron is confined to a narrow angular spread by the shape of the tunnel, and one can envisage a jet of electron flux centred on the laser polarization axis⁸ . In the framework of angular momentum theory, this is exactly equivalent to a process with high angular momentum. This is the same concept as discussed in section 2.1, where it was seen that contributions from larger angular momenta K give rise to a sharper, or more directionally localized, axis distribution. Ultimately, the angular-dependence of any observable, expressed in terms of an expansion in angular functions, can contain very high-order terms for processes which are highly directional.

The N-photon case discussed above assumed that all intermediate states were virtual and, consequently, did not provide any restrictions on the ionization. A distinctly different case arises when there is a single, or multiple, resonant intermediate state(s), because bound–bound transitions carry strict selection rules. In the context of this discussion we are concerned with the directionality, or polarization, of the bound–bound transition, as determined by the electronic dipole selection rules. Here we discuss the simplest case of a single bound–bound transition at the one-photon level prior to ionization—a 1+N resonantly enhanced multiphoton ionization (REMPI) process. The geometric formalism presented could readily be extended to more complex processes.

In the case of a one-photon bound–bound transition the initially prepared P(θ, t) will be raised by an additional power of ${{{\rm cos} }^{2}}(\theta )$ for a parallel transition, or ${{{\rm sin} }^{2}}(\theta )$ for a perpendicular transition, where the transition direction is defined by the symmetry of the bound states involved and the dipole operator. This can be considered purely geometrically, and the transition amplitudes are not required to model this process assuming only a single transition is allowed (or multiple transitions are resolved and can be considered independently). Strictly, such a transition will change the composition of the rotational wavepacket; however, under the assumption that there are no dynamics on the excited state (ionization is effectively instantaneous/rapid following excitation as compared to rotations as discussed above)—hence no rotational wavepacket propagation need be taken into account—then a geometric treatment is valid. Furthermore, the one-photon absorption contains coupling terms which change J by $0,\pm 1$, so will only minimally affect the envelope of a broad wavepacket.

In a purely geometric treatment, the axis distribution following a parallel transition is given by:

$$\begin{eqnarray}&&P_{\parallel }^{\prime }(\theta ,t)={{{\rm cos} }^{2}}(\theta )P(\theta ,t)\end{eqnarray} \tag{ 26 }$$

and for the perpendicular case by:

$$\begin{eqnarray}&&P_{\bot }^{\prime }(\theta ,t)={{{\rm sin} }^{2}}(\theta )P(\theta ,t)\end{eqnarray} \tag{ 27 }$$

The equation above can be written more explicitly in terms of the initial ADMs; for the parallel case we have:

$$\begin{eqnarray}&&P_{\parallel }^{\prime }(\theta ,t)={{{\rm cos} }^{2}}(\theta )\mathop{\sum }\limits_{K,Q}{{A}_{K,Q}}(t){{Y}_{K,Q}}(\theta ,\phi )\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&\propto \frac{1}{3}({{Y}_{2,0}}(\theta ,0)+1)\mathop{\sum }\limits_{K,Q}{{A}_{K,Q}}(t){{Y}_{K,Q}}(\theta ,\phi ),\end{eqnarray} \tag{ 29 }$$

where the ∝ arises because some normalization factors have been neglected. Similarly, for the perpendicular case we have:

$$\begin{eqnarray}&&P_{\bot }^{\prime }(\theta ,t)={{{\rm sin} }^{2}}(\theta )\mathop{\sum }\limits_{K,Q}{{A}_{K,Q}}(t){{Y}_{K,Q}}(\theta ,\phi )\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\propto {{Y}_{2,2}}(\theta ,0)\mathop{\sum }\limits_{K,Q}{{A}_{K,Q}}(t){{Y}_{K,Q}}(\theta ,\phi ).\end{eqnarray} \tag{ 31 }$$

For cylindrically symmetric cases (Q = 0) and single photon ionization (K = 0,2) these equations simplify further to:

$$\begin{eqnarray}&&P_{\parallel }^{\prime }(\theta ,t)\propto {{A}_{0,0}}(t)(1+{{Y}_{2,0}}(\theta ,0))+{{A}_{2,0}}(t)({{Y}_{2,0}}(\theta ,0)+{{Y}_{2,0}}{{(\theta ,0)}^{2}})\end{eqnarray} \tag{ 32 }$$

and

$$\begin{eqnarray}&&P_{\bot }^{\prime }(\theta ,t)\propto {{A}_{0,0}}(t){{Y}_{2,2}}(\theta ,0)+{{A}_{2,0}}(t){{Y}_{2,0}}(\theta ,0){{Y}_{2,2}}(\theta ,0),\end{eqnarray} \tag{ 33 }$$

where ${{Y}_{0,0}}(\theta ,0)$ has been assumed to be normalized to unity.

Hence, equation (17) becomes:

$$\begin{eqnarray}&&\beta _{0,0}^{\parallel }(t)={{A}_{0,0}}(t){{a}_{000}}(1+{{Y}_{2,0}}(\theta ,0))+{{A}_{2,0}}(t){{a}_{200}}({{Y}_{2,0}}(\theta ,0)+{{Y}_{2,0}}{{(\theta ,0)}^{2}})\end{eqnarray} \tag{ 34 }$$

for the parallel case and, for the perpendicular case:

$$\begin{eqnarray}&&\beta _{0,0}^{\bot }(t)={{A}_{0,0}}(t){{a}_{000}}{{Y}_{2,2}}(\theta ,0)+{{A}_{2,0}}(t){{a}_{200}}{{Y}_{2,0}}(\theta ,0){{Y}_{2,2}}(\theta ,0).\end{eqnarray} \tag{ 35 }$$

As expected, in both cases higher-order angular moments appear in the observable, although only the second-order moment of the originally prepared P(θ, t) appears in the final equations. These equations show explicitly how resonant multi-photon ionization processes may respond to aligned ensembles in relatively complex ways, due both to the additional photon angular momentum coupled into the ionization and the angle-dependence of the resonant step. As was the case for frame rotations, such measurements respond to the P(θ, t) but in a less-than-direct manner, and this response is highly dependent on the nature of the probe process.

Calculation results for P(θ, t), in the region of the half-revival of a rotational wavepacket calculated for butadiene under typical experimental conditions, have already been illustrated in figure 3. The axis distributions were mapped in both Cartesian and polar space, and also expressed in terms of the associated second-order alignment metrics, $\langle {{{\rm cos} }^{2}}(\theta ,t)\rangle$ and ${{A}_{2,0}}(t)$, in figure 4. As discussed above, and clear from the complex angular structure visible in figure 3, a full description of P(θ, t) requires higher-order moments of the axis distribution to be taken into account. Figure 5 illustrates this point with the temporal evolution of all ADMs up to K = 40 for this axis distribution. As already noted w.r.t. the P(θ, t) distributions, in this case it is clear that the degree of alignment is high, and many terms in K are significant in the ${{A}_{K,Q}}(t)$ expansion around the half-revival. Away from the peaks of the revival the degree of alignment is still high, due to an effective DC contribution to P(θ, t), although there are also high-frequency temporal modulations, particularly visible at the poles of the P(θ, t) distribution.

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The ${{A}_{K,Q}}(t)$ which parametrize the full P(θ, t) distribution also show complex behaviour, as expected from the form of P(θ, t). Since we are concerned with the coupling of the aligned distribution—parametrized as a set of ${{A}_{K,Q}}(t)$—into an observable, we discuss here the evolution of the axis distribution in terms of the ${{A}_{K,Q}}(t)$ rather than the underlying rotational wavepacket. We also focus on the comparison of the higher-order terms with K = 2, since this corresponds to the most often used metric of alignment.

At the main peak of the alignment, $t\sim 59$ ps, all of the terms up to $K\simeq 24$ peak, reflecting the maximal axis alignment obtained at the half-revival. The widths of the features reveal a more complex temporal dependence of different order terms, with a narrowing of the revival peak at higher K. Immediately before and after this feature, at around $t\sim 58$ and $t\sim 60$ ps respectively, the traces show additional satellite features appearing at higher K. The satellite features around the half-revival have the effect of blurring out the anti-alignment features, relative to K = 2, hence would potentially reduce the observable contrast of the revival for an observable sensitive to higher-order terms as compared to a low-order observable. In general, it is clear that one would expect observables which couple to different order terms to exhibit different revival contrast and markedly different temporal evolution. Finally, it is worth noting that the frequency content of the distribution also scales with K (this is a direct consequence of terms with high $\Delta J$ preferentially coupling into higher-order moments of the distribution). A high-order measurement away from the revival peak may, therefore, be difficult to interpret in terms of the expected (typically low-order) response and the complex temporal response may even be dismissed as experimental noise⁹ .

As compared with $\langle {{{\rm cos} }^{2}}(\theta ,t)\rangle$, there is significantly more temporal structure in the higher-order K terms which is not coupled into $\langle {{{\rm cos} }^{2}}(\theta ,t)\rangle$. This is clear from comparison of figure 4, which shows the direct correspondence between $\langle {{{\rm cos} }^{2}}(\theta ,t)\rangle$ and ${{A}_{2,0}}(t)$. This follows from the overlap integral of equation (11) which, for n = 2, will be most significant for K = 2. Furthermore, the first few terms in K have similar temporal response to K = 2, and smaller magnitudes. The net result is that any contributions to the $\langle {{{\rm cos} }^{2}}(\theta ,t)\rangle$ line-shape from these ADMs will have only a minimal effect on the overall temporal profile, and the $\langle {{{\rm cos} }^{2}}(\theta ,t)\rangle$ metric can be considered as essentially equivalent to ${{A}_{2,0}}(t)$ in this case, as stated earlier (section 2.3) without explanation.

It may be concluded that, in general, a highly-aligned ensemble will contain high-order ${{A}_{K,Q}}(t)$ (effectively by definition), regardless of the precise details of the molecule under study. An appreciation for the coupling of this ensemble into the measurement is, therefore, essential for an understanding of experimental results. In particular, analysis of results based solely on $\langle {{{\rm cos} }^{2}}(\theta ,t)\rangle$ will generally not be sufficient for interpretation of experimental data, even at a phenomenological level, beyond the simplest single-photon ionization yield probe scheme.

In order to treat the case of 1 + 1' REMPI, we consider distributions $P_{\parallel }^{\prime }(\theta ,t)$ and $P_{\bot }^{\prime }(\theta ,t)$ following parallel and perpendicular bound–bound transitions respectively, obtained via equations (26) and (27). These distributions are shown in figure 6. The parallel transition enhances the alignment, with a slight sharpening of the distribution at the poles (see figure 3), as might be intuitively expected from the form of equation (26). However, the perpendicular transition significantly changes the axis alignment. This is also an intuitive result: essentially the perpendicular transition couples preferentially to axes aligned in the $(x,y)$ plane, thus addressing only population away from the poles (z-axis) of the initial distribution. Consequently the peaks in the distribution are shifted both spatially (into the ($x,y$) plane) and temporally, with the peak in the (x, y) plane of $P_{\bot }^{\prime }(\theta ,t)$ corresponding with the anti-alignment feature in the initial P(θ, t). As a consequence of the coupling, the high-order oscillations in the $(x,y)$ plane are, in effect, amplified.

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Figure 7 illustrates these effects in terms of the ADMs and shows the ${{A}_{K,Q}}(t)$, up to K = 10, for both cases. In particular the effect of a parallel transition is to raise the second-order moment (K = 2) of the distribution, while a perpendicular transition will decrease the second-order moment. The temporal profiles are also affected, with the additional modulations appearing leading to significant temporal asymmetry in the K = 2 traces, particularly in the case of a perpendicular transition. Essentially the effect of the transition is to mix higher-order terms into a given K; for example the ${{A}_{2,0}}(t)$ trace for $P_{\parallel }^{\prime }(\theta ,t)$ has characteristics of both ${{A}_{2,0}}(t)$ and ${{A}_{4,0}}(t)$ of the initial P(θ, t)—this can be seen by comparison of figures 5 and 7 (top panel). Because single photon ionization is only sensitive to K = 0, 2 terms of the aligned ensemble, as discussed above, the resonant excitation step has the effect of allowing K = 4 terms from the initial P(θ, t) to contribute to this observable since these are now mixed into the ${{A}_{2,0}}(t)$ ADMs for the excited state distribution P'(θ, t).

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Again, in general one can conclude from this discussion that for more complex ionization processes higher-order terms become necessary to understand and interpret experimental results. In particular the perpendicular case illustrates how a process with strong selection rules (i.e. a highly directional response to the molecular axis alignment) acts as a strong filter on the initially prepared axis distribution. In the most general case of multiphoton ionization, where multiple resonances may be accessed sequentially or via different competing ionization pathways, it is clear that a very complex P'(θ, t) may be created which has little obvious correspondence to the initially prepared P(θ, t) and, without some appreciation of the underlying probe process, the temporal response of the observable may be inexplicable.

We first consider some limiting cases in order to investigate the response of the time-resolved photoelectron yield, ${{\beta }_{0,0}}(t)$, to P(θ, t). We consider the case of (a) one-photon ionization from the rotationally-excited ground state, (b) one-photon ionization following a parallel (${{{\rm cos} }^{2}}(\theta )$) resonant excitation and (c) one-photon ionization following a perpendicular (${{{\rm sin} }^{2}}(\theta )$) resonant excitation. The calculated P(θ, t) and the extracted ${{A}_{K,Q}}(t)$ for these cases have already been shown in figures 3, 5, 6 and 7, and discussed above. In all cases we assume that the probe (ionization) pulse polarization is parallel to the pump (alignment) pulse polarization, i.e. Θ = 0. For the excitation step the laser pulse polarization is assumed to be either parallel or perpendicular to the alignment pulse polarization, as appropriate for the excitation, and its effect is treated geometrically (no temporal evolution of the wavepacket in the excited state) as discussed in section 2.4.3

The dipole matrix elements in equation (13) typically represent unknown quantities. In order to explore limitations on the ionization yield we consider the yield as a function of the parallel versus perpendicular ionization cross-sections (${{\sigma }_{\parallel }}$ and ${{\sigma }_{\bot }}$). Specifically, in the case of butadiene—which is used here as an exemplar system—ionization of the S₂ excited state in ${{C}_{2h}}$ symmetry leads to allowed continuum symmetries of A_g and B_g character for parallel and perpendicular ionization events respectively (see appendix B for details). The effect of varying the ratio of the ionization matrix elements is shown in figure 8. Here the matrix elements were set as

$$\begin{eqnarray*}&&{\boldsymbol{D}} _{hl}^{\Gamma \mu }\equiv \mathop{\sum }\limits_{q}{\boldsymbol{D}} _{|m|l}^{\Gamma }(q)=r{\boldsymbol{D}} _{|m|l}^{{{A}_{g}}}(0)+(1-r){\boldsymbol{D}} _{|m|l}^{{{B}_{g}}}(\pm 1)\end{eqnarray*}$$

and all symmetry-allowed terms, up to ${{l}_{\,{\rm max} }}=4$, were included and set to unity. The matrix elements were further re-normalized such that $|{{\sum }_{l,|m|}}b_{|m|l|m|}^{{{A}_{g}}}{\boldsymbol{D}} _{|m|l}^{{{A}_{g}}}(0){{|}^{2}}=1$ and $|{{\sum }_{l,|m|}}b_{|m|l|m|}^{{{B}_{g}}}{\boldsymbol{D}} _{|m|l}^{{{B}_{g}}}(\pm 1){{|}^{2}}=1$. Hence r = 1 corresponds to the A_g continuum only, and a purely parallel ionization event in the MF, while r = 0 corresponds to the B_g continuum and a purely perpendicular ionization. Interestingly, the same type of calculation can also be used to extract the ratio of the matrix elements from experimental data by fitting of the matrix elements [30], although, due to the structure of equation (13), and as emphasized in equation (17), the ${{\beta }_{0,0}}(t)$ are sensitive only to the amplitudes of each l (although can contain cross-terms in λ), so phase information between different l is not defined, and cannot be obtained, from this observable alone.

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These results—figure 8(a)—show quantitatively a number of features which might intuitively be expected. The ionization yield is sensitive only to ${{A}_{2,0}}(t)$, as shown in equation (17). For the purely parallel case (r = 1), the ${{A}_{2,0}}(t)$ is mapped faithfully (this is shown in detail in figure 9(a)), while the purely perpendicular case (r = 0) shows an inverted trace, essentially mapping $\langle {{{\rm sin} }^{2}}(\theta ,t)\rangle$. Additionally, the total yield is decreased in the perpendicular case due to the geometry—P(θ, t) is heavily peaked along the z-axis as shown in figure 3, so the axis distribution geometrically favours q = 0 transitions which are parallel to the molecular axis, hence aligned near to the LF z-axis. Between these limits, $0\lt r\lt 1$, the ionization yield is less sensitive to the axis distribution, and even shows regions of no sensitivity where the balance of parallel and perpendicular components is such that there is no observable modulation in the yield. In other words, these are regions where there is no angular dependence for a second-order observable due to the equal magnitudes of parallel and perpendicular ionization cross-sections. The import of this is that, for a given molecule, it is possible that an observable sensitive to only the lowest order ADM shows no alignment dependence, despite the presence of an aligned distribution (see measuring at magic angle in order to cancel out ${{{\rm cos} }^{2}}(\theta )$ terms). Since this depends on a molecular property—the details of the ionization matrix elements—it is generally out of the control of the experimentalist, and may not be anticipated a priori unless the molecule under study is already well-characterized. Such a situation becomes less likely as higher-order moments are coupled to the observable.

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Ionization yields following one-photon excitation, figures 8(b) and (c), show similar behaviour, with the main difference appearing due to changes in the temporal line-shapes of $P^{\prime} (\theta ,t)$ as discussed in section 3.1.2. The parallel yield following a parallel excitation, i.e. ionization of the distribution $P_{\parallel }^{\prime }(\theta ,t)$ with r = 1, is larger for all t (relative to case (a)), and is not modulated as strongly around the revival feature; conversely the perpendicular yield is reduced relative to case (a). The opposite is seen following a perpendicular excitation, i.e. ionization of the distribution $P_{\bot }^{\prime }(\theta ,t)$ (case (c)), which enhances the yield in the case of perpendicular ionization, and reduces it for parallel ionization. In both cases (b) and (c) this effect is due to the coupling of K = 4 terms from the initial axis distribution into the observable, as discussed above (section 2.4.3), and consequently also results in more complex temporal evolution of the signal, as would be expected from the form of the ${{A}_{K,Q}}(t)$ (figure 5). Despite this additional term, there are still values of r where the yield shows very little sensitivity to the ADMs, although the width of this region of r-space appears significantly reduced in the perpendicular case.

In the multiphoton case, as detailed in section 2.4.2, the expectation is for higher-order ${{A}_{K,Q}}(t)$ terms to become significant as the photon order of the process increases. This behaviour is illustrated in figure 9 for N = 1 − 3, and compared directly with the contributing ${{A}_{K,Q}}(t)$. In these calculations r = 1, hence the ionization is purely parallel and, as for the one-photon case above, the probe and alignment pulse polarization are parallel (Θ = 0). The line-outs show how the total ionization yield drops as a function of N, and also how the line-shapes change as couplings with larger K become allowed. Comparison of ${{\beta }_{0,0}}(t)$ with ${{A}_{K,Q}}(t)$ clearly shows that the line-shapes contain contributions from ${{A}_{K,Q}}(t)$ up to ${{K}_{{\rm max} }}=2N$ (see also equation (25)): there is a narrowing of the features observed over the revival, and an increased complexity and temporal asymmetry to the line-shape as a function of N (equivalently as a function of K_max), as already observed in the ADMs (section 3.1.1). The oscillations away from the revival also increase, again as expected from the ${{A}_{K,Q}}(t)$ traces. Naturally, the exact coupling will depend on both r (as illustrated for the one-photon case in the preceding section) and Θ (see below), but these general features will likely be apparent to some degree in an N-photon observable.

Angle-resolved measurements, based on the response of the ionization yield to the probe polarization, are given by ${{\beta }_{0,0}}(\Theta ,t)$ as defined in equation (25). This is essentially the same observable as discussed above—the total ionization yield, angle-integrated w.r.t. the photoelectron—except that measurements are made as a function of pump–probe polarization geometry, i.e. the angle Θ, as illustrated in figure 1(d).

Figure 10 shows surface plots for of ${{\beta }_{0,0}}(t;\Theta )$ for N = 1 − 8 and Θ = 0, 0.45, 0.95 rad (0°, 26°, 54°) respectively (all other parameters were the same as those used in section 3.2.2); the data is re-normalized for each N to emphasize the changes in the line-shapes independently of the total yield. For ease of comparison, panel (a) shows the same results as figure 9 for N = 1 − 3, and the narrowing of the features with N, as discussed above, is again very clear. In cases (b) and (c) very different surfaces are observed, as expected from equation (25). The line-shapes are more complex, and this is particularly apparent in the splitting of the main feature observed for $N\gt 3$ in figure 10(b). As would also be expected, the 54° case shows reduced contrast as the aligned distribution is now rotated by close to $\pi /4$ from the probe pulse, hence the probe now maps a combination of $\langle {{{\rm cos} }^{K}}(\theta )\rangle$ and $\langle {{{\rm sin} }^{K}}(\theta )\rangle$. For N = 3 in figure 9(b) and N = 4 in figure 9(c) a decreased sensitivity to alignment is observed. This is, presumably, due to the most geometrically significant ${{A}_{K,Q}}(t)$ term (most strongly coupled according to equation (25)) coming close to its magic angle, but may also be due to the distinct Θ-dependence of different order terms leading to regions where contrast is washed out. This is essentially the same effect discussed above, and shown in figure 9(c) for Θ = 0 and N = 3, in which the t-dependence, rather than Θ-dependence, of different terms led to a decrease in revival contrast.

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To further visualize this behaviour, figure 11 shows full surfaces of $I(\Theta ,t)$ for N = 1 − 3 in polar form. These plots show more clearly the narrowing of the observed distributions at the poles as N increases. Comparison with figure 3(b) shows that the narrowing of the equator, as well as the increased spatial and temporal complexity, of higher N cases approach the full P(θ, t) surface—a direct result of higher ${{A}_{K,Q}}(t)$ terms becoming coupled into the observable, hence an increase in information content or fidelity w.r.t. the axis distribution. In this context, the most direct and detailed experimental measure for mapping an aligned distribution should be a multi-photon probe of high order, with terms up to ${{K}_{{\rm max} }}=2N$ present in the observable in a—relatively—transparent manner, provided that the Θ-dependence of the probe process (i.e. the ionization matrix elements) is well-defined as in this illustration.

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As was the case for the N-photon ionization yields discussed above, PADs can be considered as high-order observables with many ${{\beta }_{L,M}}(t)$ contributing to the observable. Hence, as indicated in equation (13), many ${{A}_{K,Q}}(t)$ may be mapped by the PAD. In terms of phenomenology, the extra sensitivity of the observed PAD to both the amplitudes and phases of the ionization matrix elements makes it difficult to choose realistic representative values for use in limiting case calculations. In this case we simply use the same limiting cases as above, i.e. ${{A}_{g}}+{{B}_{g}}$ continuum functions with amplitudes set to unity and phases set to zero, and explore the observables as a function of r. Generally, one could hope to obtain the matrix elements from experimental measurements of PADs if the alignment is known [8, 30, 31]; conversely, one could also use experimentally-measured PADs to map alignment in the case where the ionization matrix elements are known [27, 58].

Figure 12 shows examples of ${{\beta }_{L,M}}(t;\Theta =0)$ calculated for these example cases, as a function of r as per the results discussed in section 3.2.1 and shown in figure 8. Similarly to the $I(\Theta ,t)$ for N-photon ionization discussed above, higher-order L terms couple to higher-order K, resulting in more complex line-shapes which map primarily different ${{A}_{K,Q}}(t)$ as a function of L. One immediate result is that, with the exception of r = 0.5 and L = 6, there are no regions totally insensitive to the axis alignment in this case, in contrast to the L = 0 results shown in figure 8(a). Additionally, the higher-order terms are sensitive to the phases of the ionization matrix elements, so will display a much enhanced sensitivity to molecular properties (e.g. vibronic dynamics) as compared to the yields (L = 0) alone. Another interesting observation in this particular case is that the temporal peak in the ${{\beta }_{2,0}}(t)$ (at the half-revival) does not move significantly with r, although it does narrow and shift slightly. This is very different to the ionization yield, where the observable essentially mapped ${{A}_{2,0}}(t)$ for r = 1 (parallel ionization) but was inverted for r = 0 (perpendicular ionization), hence the half-revival feature appeared out of phase for r = 0 as compared to r = 1 (see figure 8(a)). However, for the ${{\beta }_{4,0}}(t)$ very different behaviour is observed, and there is a significant temporal shift in the main feature as a function of r, although the switch is not abrupt and does not occur at r = 0.5 as is the case for the yield (and as one might intuitively expect). Naturally, the specific details of these types of behaviour are highly dependent on the ionization matrix elements, but in general it is clear that a more complex temporal response is expected regardless of the exact details of the ionization matrix elements.

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To further emphasize the sensitivity of the PADs to the ADMs, figures 13 and 14 show selected ${{\beta }_{L,M}}$ (L = 2 and L = 4) for ionization following one-photon absorption, for a parallel (i.e. axis distribution given by $P_{\parallel }^{\prime }(\theta ,t)$) and perpendicular (i.e. axis distribution given by $P_{\bot }^{\prime }(\theta ,t)$) absorption; in the following discussion we denote the ${{\beta }_{L,M}}(t)$ correlated to these cases as $\beta _{L,M}^{\parallel }(t)$ and $\beta _{L,M}^{\bot }(t)$ respectively. As expected, all of the ${{\beta }_{L,M}}$ respond to the change in the ADMs following absorption, and are much more sensitive to the details of the axis distribution than the yields alone (section 3.2.1). In particular:

• Higher-order terms become more strongly coupled. This is apparent from, for instance, the appearance of strong modulations in the ${{\beta }_{2,0}}(t)$ around 40–45 ps, far from the main half-revival feature, and the increase in contrast of such features in the ${{\beta }_{4,0}}$.
• The range and magnitudes of the ${{\beta }_{L,M}}$ change. Interestingly, for the set of matrix elements used here, $\beta _{2,0}^{\parallel }(t)$ has only negative values for all (t, r), while the sign of $\beta _{2,0}^{\bot }(t)$ changes at $r\sim 0.5$ (except at the half-revival feature). In both cases the average value of ${{\beta }_{2,0}}(t)$ for a given r is significantly different from ionization of the initial distribution P(θ, t); the range of ${{\beta }_{L,M}}$ values is similar for ionization of P(θ, t) and $P_{\parallel }^{\prime }(\theta ,t)$, although offset, but much reduced for $P_{\bot }^{\prime }(\theta ,t)$ (this is particularly evident for $\beta _{4,0}^{\bot }(t)$). Experimentally this would mean less significant changes in the time-resolved PADs would be observed in the latter case.
• The position of the temporal maxima, as noted above, move only slightly as a function of r for ${{\beta }_{2,0}}(t)$, but do move significantly for ${{\beta }_{4,0}}(t)$. Additionally, for $\beta _{2,0}^{\parallel }(t)$, the width and magnitude of the temporal maxima at the half-revival is almost constant for all r; this is quite distinct from the other cases.
• For $\beta _{4,0}^{\bot }(t)$ the line-shapes over the half-revival are more complex for all r, this is distinct from the behaviour observed for $\beta _{4,0}^{\parallel }(t)$ and all L = 2 cases, for which the half-revival features remain qualitatively similar in temporal complexity to the unexcited case. This change therefore reflects both the significant change in the ADMs for $P_{\bot }^{\prime }(\theta ,t)$ (see figure 7) and the enhanced coupling to higher-order ADMs for L = 4 as compared to L = 2.

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These specific features indicate how complex the response of the ${{\beta }_{L,M}}(t)$ may be in any given case, with the temporal response reflecting both the ionization matrix elements and the ADMs. However, we again emphasize that one can begin to build some intuition on how these observables may respond phenomenologically to the ADMs and the experimental configuration in general terms, even if the precise details are molecule dependent. In this regard it is clear that one might expect a strong response to molecular alignment in all ${{\beta }_{L,M}}(t)$ as compared to the ionization yield (even in cases where the yield is insensitive to alignment), and that higher-order terms will contain higher-frequency components due to the coupling to higher-order ADMs.

This mapping has been discussed extensively by Seideman and co-workers (e.g. [30, 31]), including the possibility of extracting ionization matrix elements via fitting experimental data in this case. From an experimental perspective, measuring PADs may therefore be useful for characterizing alignment and ionization dynamics, but requires a rather involved analysis due to the high information content and complexity of the coupling [8]. However, the benefit of such measurements is precisely this high information content, so in some cases this effort is worthwhile. Similarly, in experiments where PADs are used as probes for other molecular properties (for a recent review of applications, see [59]), it is clear that the effect of rotational dynamics must be carefully considered precisely due to this complexity.