Dynamics of the moments of angular momentum distribution during generation of laser-induced polarization spectroscopy (LIPS) signal

Sukesh Roy; Robert P. Lucht; James R. Gord

doi:10.1364/JOSAB.28.000208

1. INTRODUCTION

Laser-induced polarization spectroscopy (LIPS) has emerged as a valuable spectroscopic tool for measuring the concentration of minor species in reacting flows [1, 2, 3, 4, 5, 6, 7, 8, 9] and plasmas [10]. LIPS is a pump-probe arrangement, where the pump and probe beams are crossed at the measurement location, as shown in Fig. 1. The probe beam is linearly polarized before entering the medium of interest. In LIPS, either a circularly or a linearly polarized pump beam (polarization of the pump beam is $45 °$ relative to that of the probe beam) is employed for selective pumping of the population from the ground state to the excited Zeeman states. Because of the anisotropy of the medium induced by the pump beam, the probe-beam polarization becomes slightly elliptical or rotates slightly while passing through the medium. As a consequence, a portion of the probe-beam intensity leaks through a polarization analyzer whose transmission axis is orthogonal to the original probe-beam polarization. This leakage through the polarizer is the LIPS signal [11].

Analysis of the experimental LIPS signals to determine the concentration of minor species in reacting flows remains a challenge. Recent efforts by Reichardt and Lucht [12] and Roy et al. [13] toward the theoretical calculation of LIPS signals and understanding the physics of LIPS signal generation by solving the time-dependent density matrix equations represent significant steps toward establishing a theoretical framework for the interpretation of experimental LIPS signals. Roy et al. [13] and Reichardt et al. [4] showed that the short-pulse laser (laser pulse width $τ_{L}$ less than characteristic collision time $τ_{C}$ ) significantly decreases the collision-rate dependence of the LIPS signal as compared to that in the long-pulse laser case ( $τ_{L} > τ_{C}$ ).

The objective of this study was to investigate the effects of the pump-induced anisotropy on the generation of the short-pulse LIPS signal. The influence of the saturation and pre existing ground-state anisotropy on the LIPS signal was also investigated. The use of picosecond lasers allows experimental investigation of the rates at which these anisotropies are destroyed in collisional environments. Recently, short-pulse LIPS and degenerate-four-wave-mixing (DFWM) experiments have been performed to investigate the relaxation rates of the anisotropy for OH and NH [14, 15] and to study the rotational-level dependence of ground-state recovery rates for OH [16]. Using two-color polarization spectroscopy, the rotational energy transfer and the relaxation of orientation and alignment in the ground-electronic state of OH was investigated in an atmospheric-pressure methane-air flame by Chen et al. [17]. The evolution of the orientation and alignment of rotational angular momentum in the course of inelastic collisions has been studied by Costen et al. [18] using a perturbative theoretical approach. These investigators concluded that the rate of collisional depolarization of alignment is approximately twice that of population transfer in atmospheric-pressure flames. An understanding of the effects of anisotropies along with their relaxation rates will allow modeling of the LIPS signal-generation process for interpretation of experimental LIPS signals.

The anisotropy induced in the medium due to the selective pumping of population from the ground state to the excited Zeeman states can best be described in terms of the various moments of the angular momentum distribution of the ensemble [19, 20, 21]. Wasserman et al. employed DFWM to probe the anisotropy induced in a medium by a pump beam with different polarizations [22]. The contributions of the population, orientation, and alignment gratings in the generation of the resonant-four-wave-mixing (RFWM) signal in the perturbative limit were discussed by Williams et al. [23, 24] and Wasserman et al. [25]. In their theoretical and experimental analysis, they showed that the absolute DFWM signal intensity strongly depends on the population, orientation, and alignment relaxation rates. The role of orientation and alignment in generating the LIPS signal was also briefly discussed by Teets et al. [26].

2. CALCULATION OF THE LIPS SIGNAL FOR A MULTISTATE SYSTEM

In LIPS, a third-order polarization is induced in the medium because of the interaction of the molecules with the pump and probe beams. These molecules then radiate along the phase-matched direction to produce a coherent LIPS signal at the probe frequency. The induced third-order polarization is calculated by solving the density matrix equations using direct numerical integration (DNI). The time-dependent density matrix equations for a multistate system irradiated by laser beams are [27]

\frac{\partial ρ_{k k} (\vec{r}, t)}{\partial t} = - \frac{i}{ℏ} \sum_{m} (V_{k m} (\vec{r}, t) ρ_{m k} (\vec{r}, t) - ρ_{k m} (\vec{r}, t) V_{m k} (\vec{r}, t)) - Γ_{k} ρ_{k k} (\vec{r}, t) + \sum_{m} Γ_{m k} ρ_{m m} (\vec{r}, t),

\frac{\partial ρ_{k j} (\vec{r}, t)}{\partial t} = - ρ_{k j} (\vec{r}, t) (i ω_{k j} + γ_{k j}) - \frac{i}{ℏ} \sum_{m} (V_{k m} (\vec{r}, t) ρ_{m j} (\vec{r}, t) - ρ_{k m} (\vec{r}, t) V_{m j} (\vec{r}, t)),

where the diagonal matrix elements

ρ_{k k}

are proportional to the population of state k; the off-diagonal matrix elements

ρ_{k j}

describe the coherence between states k and j; ℏ is Planck’s constant

(J s)

;

Γ_{m k}

is the population-transfer rate (

s^{- 1}

) from level m to level k;

Γ_{k}

is the population-decay rate of state k (

s^{- 1}

) where

Γ_{k} = \sum_{}^{} Γ_{k m}

;

ω_{k j}

is the angular frequency (

s^{- 1}

) of the resonance between levels k and j; and

γ_{k j}

is the dephasing rate (

s^{- 1}

). The interaction term

V_{k m}

is

V_{k m} (\vec{r}, t) = - μ_{k m} . E (r, t) = - μ_{k m} . [E_{pump} (\vec{r}, t) + E_{pr} (\vec{r}, t)],

where

μ_{k m}

is the electric-dipole matrix element

(C m)

, and

E_{pump} (r, t)

and

E_{pr} (r, t)

are the electric field vectors

(J / C m)

for the probe and pump fields, respectively.

The manipulation of the density matrix equations for DNI calculations is described in detail by Reichardt et al. [27]. The space- and time-dependent polarization $P (y, t)$ along the signal path that oscillates near the input laser frequency is

P (y, t) = \frac{1}{2} (Tr [ρ (y, t) μ] + Tr [μ ρ (y, t)]) = \frac{1}{2} \sum_{p} \sum_{k} [μ_{p k} ρ_{k p} (y, t) + μ_{k p} ρ_{p k} (y, t)] = \frac{1}{2} \sum_{p} \sum_{k} [μ_{p k} σ_{k p} (y, t) \exp (i ω t) + μ_{k p} σ_{p k} (y, t) \exp (- i ω t)],

where

σ_{p k}

is a slowly varying quantity, and

ρ_{p k} = σ_{p k} \exp (- i ω t)

. This polarization term contains both the third-order contribution and contributions from other processes such as absorption. The polarization is multiplied by the phase factor

\exp [i (k_{y} y - ω t)]

and integrated along the

+ y

axis to calculate the LIPS signal. Along the phase-matched direction (

+ y

axis), other polarization components will not be in phase and will average to zero upon spatial integration. The LIPS signal is determined by summing the contribution of

P (y, t)

at M discrete points along the

+ y

axis. The amplitude of the third-order polarization is

A_{3} (t) = \sum_{m = 1}^{m = M} P (y = m Δ y, t) \exp [i (k_{y} y - ω t)] .

The LIPS signal strength

S_{3}

is the integral of the sum of the squares of the real and imaginary parts of the amplitude

A_{3}

over the laser pulse,

S_{3} = \int_{0}^{4 t_{pulse}} {[A_{3 r} (t)]^{2} + [A_{3 i} (t)]^{2}} d t,

where

t_{pulse}

is the full-width-at-half-maximum (FWHM) of the laser pulse electric field amplitude envelope, and the pulses are centered at

t = 2 t_{pulse}

. For the numerical calculation, top-hat spatial profiles were assumed for the laser beams; the temporal profiles of the laser beams were assumed to be Gaussian.

The density matrix element $μ_{e g}$ between a ground state $(J_{g}, M_{g})$ and an excited state $(J_{e}, M_{e})$ is

{\vec{μ}}_{e g} = (α_{e}, J_{e}, M_{e} | \vec{μ} | α_{g}, J_{g}, M_{g}) = μ_{R} (α_{e} J_{e}, α_{g} J_{g}) {\vec{μ}}_{G} (J_{e} M_{e}, J_{g} M_{g}),

where

μ_{R}

is the reduced matrix element and does not depend on the projection quantum numbers

M_{e}

and

M_{g}

of the Zeeman states;

μ_{R}

is given by

μ_{R} (α_{e} J_{e}, α_{g} J_{g}) = \sqrt{\frac{A_{e g}}{Γ (J_{e}, J_{g})} \frac{3 ε_{0} ℏ λ_{e g}^{3}}{8 π^{2}}},

where

A_{e g}

is the spontaneous emission rate [

s^{- 1}

]. Here

α_{e}

and

α_{g}

represent quantum numbers other than the rotational and projection quantum numbers J and M. The function Γ

(J_{e}, J_{g})

is given by

Γ (J_{e}, J_{e} + 1) = (J_{e} + 1) (2 J_{e} + 3) Γ (J_{e}, J_{e}) = J_{e} (J_{e} + 1) Γ (J_{e}, J_{e} + 1) = J_{e} (2 J_{e} - 1) .

However,

{\vec{μ}}_{G}

is dependent on the quantum numbers

M_{e}

and

M_{g}

of the Zeeman states and describes the coupling strength between the upper and lower states. The expressions of

{\vec{μ}}_{G}

for a P-branch transition

(Δ J = J_{e} - J_{g} = - 1)

are [28]

{\vec{μ}}_{G} (J_{e} M_{e}, J_{e} + 1 M_{e} + 1) = - \frac{1}{2} \sqrt{(J_{e} + M_{e} + 1) (J_{e} + M_{e} + 2)} (\hat{x} + i \hat{y}) {\vec{μ}}_{G} (J_{e} M_{e}, J_{e} + 1 M_{e} - 1) = \frac{1}{2} \sqrt{(J_{e} - M_{e} + 1) (J_{e} - M_{e} + 2)} (\hat{x} - i \hat{y}) {\vec{μ}}_{G} (J_{e} M_{e}, J_{e} + 1 M_{e}) = \frac{1}{2} \sqrt{(J_{e} + 1)^{2} - (M_{e})^{2}} \hat{z} .

From Eq. (10), it is evident that the coupling between the ground and excited Zeeman states depends on the polarization of the laser beams. The energy-level diagram of the $P_{1} (2)$ transition is shown in Fig. 2, where the allowed $Δ M = 0$ transitions are indicated by dashed arrows, and the $Δ M = \pm 1$ transitions are indicated by solid arrows. The strengths and phases of the transitions are indicated by the numerical value of the x or z components of the geometry-dependent part of the dipole matrix element ${\vec{μ}}_{G}$ .

3. EFFECTS OF ORIENTATION AND ALIGNMENT ON THE LIPS SIGNAL

The moments of the angular momentum distribution such as orientation and alignment are used to describe the anisotropy of the system that is interacting with photons under different collisional environments. The orientation describes the net helicity or spin of the system; whereas the alignment describes the spatial distribution of angular momentum. Consider an ensemble of particles in various angular momentum states $| J M 〉$ characterized by a density matrix ρ with elements $〈 J^{'} M^{'} | ρ | J M 〉 | J^{'} M^{'} 〉〈 J M |$ . The density operator in the ${| J M 〉}$ representation can then be written in the form

ρ = \sum_{J^{'} J M^{'} M} 〈 J^{'} M^{'} | ρ | J M 〉 | J^{'} M^{'} 〉 〈 J M | .

Equation (11) can be rewritten as

ρ = \sum_{J^{'} J K Q} [\sum_{M^{'} M} 〈 J^{'} M^{'} | ρ | J M 〉 (- 1)^{J^{'} - M^{'}} (\begin{matrix} \begin{matrix} J^{'} \\ M^{'} \end{matrix} & \begin{matrix} J \\ - M \end{matrix} & \begin{matrix} K \\ - Q \end{matrix} \end{matrix}) \times (2 K + 1)^{1 / 2}] T (J^{'} J)_{K Q},

where K, known as the rank of the tensor

T (J^{'} J)

, ranges from 0 to

2 J

, and Q ranges in unit steps from

- K

to K [29]. The state multipoles or statistical tensor are defined as

〈 T (J^{'} J)_{K Q}^{†} 〉 = \sum_{M^{'} M} (- 1)^{J^{'} - M^{'}} (2 K + 1)^{1 / 2} (\begin{matrix} \begin{matrix} J^{'} \\ M^{'} \end{matrix} & \begin{matrix} J \\ - M \end{matrix} & \begin{matrix} K \\ - Q \end{matrix} \end{matrix}) \times 〈 J^{'} M^{'} | ρ | J M 〉 .

The multiple moments of the system described by the density matrix can then be defined as [30]

ρ_{K Q} = 〈 T_{K Q}^{†} 〉 T r [ρ] .

The

3 - j

symbol

(\begin{matrix} \begin{matrix} J^{'} \\ M^{'} \end{matrix} & \begin{matrix} J \\ - M \end{matrix} & \begin{matrix} K \\ - Q \end{matrix} \end{matrix})

was evaluated using the computer code of Zare [31]. Here the moments for the ground and excited levels, with angular momentum quantum number

J_{g}

and

J_{e}

, will be evaluated individually for each level

J^{'} = J

. Multipoles with

J^{'} \neq J

describe the coherence between states of different angular momentum J. In the numerical calculation, it is assumed that no population transfer takes place among the ground Zeeman states or among the excited Zeeman states. Population transfer is allowed to occur from the excited to the ground Zeeman states, but the primary population transfer occurs between Zeeman states and the bath levels via rotational-energy-transfer collisions. Recently, more sophisticated treatment of the collisional processes of OH with rare gases has been discussed by Paterson et al. [32], Marinakis et al. [33], Costen et al. [34], and by Dagdigian and Alexander [35, 36, 37]. It was observed that in some cases (particularly, at low J) pure elastic depolarization can contribute significantly to the rotational-energy-transfer rate constant. In this paper, the modeling is carried out for a reacting flow environment with the presence of a significant amount of

H_{2} O

where negligible elastic depolarization collisions may be more appropriate. Furthermore, the assumption of negligible elastic depolarization collisions will not have a significant effect on the generation of the orientation, alignment, and higher order moments during the interaction of the pump and probe beams.

The ensemble of interest in our modeling scheme is an incoherent superposition of states with different quantum numbers M for which Eq. (13) shows that all multipoles with $Q \neq 0$ vanish. The procedure to calculate the multipole moments of a system with $J = 2$ and $M^{'} = M$ is discussed by Greene and Zare [30].

For $J_{g} = 2.5$ and $J_{e} = 1.5$ , K will range from 0 to 5 and from 0 to 3, respectively. The tensors $T_{K 0}$ for the ground Zeeman states with $J_{g} = 2.5$ are

T_{00} = (\begin{matrix} 0.41 \\ \begin{array}{l} 0.41 \\ 0.41 \\ 0.41 \\ 0.41 \end{array} \\ 0.41 \end{matrix}), T_{10} = (\begin{matrix} - 0.60 \\ \begin{array}{l} - 0.36 \\ - 0.12 \\ 0.12 \\ 0.36 \end{array} \\ 0.60 \end{matrix}) {, T}_{20} = (\begin{matrix} 0.55 \\ \begin{array}{l} - 0.11 \\ - 0.44 \\ - 0.44 \\ - 0.11 \end{array} \\ 0.55 \end{matrix}), T_{30} = (\begin{matrix} - 0.37 \\ \begin{array}{l} 0.52 \\ 0.30 \\ - 0.30 \\ - 0.52 \end{array} \\ 0.37 \end{matrix}), T_{40} = (\begin{matrix} 0.19 \\ \begin{array}{l} - 0.57 \\ 0.38 \\ 0.38 \\ - 0.57 \end{array} \\ 0.19 \end{matrix}), and T_{50} = (\begin{matrix} - 0.06 \\ \begin{array}{l} 0.32 \\ - 0.63 \\ 0.63 \\ - 0.32 \end{array} \\ 0.06 \end{matrix}) .

The tensors

T_{K 0}

for the excited Zeeman states with

J_{e} = 1.5

are

T_{00} = (\begin{matrix} 0.5 \\ \begin{array}{l} 0.5 \\ 0.5 \end{array} \\ 0.5 \end{matrix}), T_{10} = (\begin{matrix} - 0.67 \\ \begin{array}{l} - 0.22 \\ 0.22 \end{array} \\ 0.67 \end{matrix}), T_{20} = (\begin{matrix} 0.5 \\ \begin{array}{l} - 0.5 \\ - 0.5 \end{array} \\ 0.5 \end{matrix}), and T_{30} = (\begin{array}{l} - 0.22 \\ \begin{matrix} 0.67 \\ - 0.67 \end{matrix} \\ 0.22 \end{array}) .

The orientation (proportional to the dipole moment of the ensemble with

K = 1

), alignment (proportional to the quadrupole moment of the ensemble with

K = 2

), the octopole moment (with

K = 3

) for

J_{g} = 2.5

and

J_{e} = 1.5

are calculated as follows [38]:

{Orientation}_{J = 2.5} = 0.6 * (- ρ_{M = - 2.5} + ρ_{M = 2.5}) + 0.36 * (- ρ_{M = - 1.5} + ρ_{M = 1.5}) + 0.12 * (- ρ_{M = - 0.5} + ρ_{M = 0.5}), {Orientation}_{J = 1.5} = 0.67 * (- ρ_{M = - 1.5} + ρ_{M = 1.5}) + 0.22 * (- ρ_{M = - 0.5} + ρ_{M = 0.5}), {Alignment}_{J = 2.5} = 0.55 * (ρ_{M = - 2.5} + ρ_{M = 2.5}) - 0.11 * (ρ_{M = - 1.5} + ρ_{M = 1.5}) - 0.44 * (ρ_{M = - 0.5} + ρ_{M = 0.5}), {Alignment}_{J = 1.5} = 0.50 * (ρ_{M = - 1.5} + ρ_{M = 1.5}) - 0.50 * (ρ_{M = - 0.5} + ρ_{M = 0.5}), {Octopole}_{J = 2.5} = 0.37 * (- ρ_{M = - 2.5} + ρ_{M = 2.5}) + 0.52 * (ρ_{M = - 1.5} - ρ_{M = 1.5}) + 0.30 * (ρ_{M = - 0.5} - ρ_{M = 0.5}), {Octopole}_{J = 1.5} = 0.22 * (- ρ_{M = - 1.5} + ρ_{M = 1.5}) + 0.67 * (ρ_{M = - 0.5} - ρ_{M = 0.5}) .

Here,

ρ_{M}

refers to the population at a particular Zeeman state with magnetic quantum number M, which is available from the solution of Eq. (1). The population simply reflects the population density in the Zeeman states; the orientation, alignment, and the higher order moments describe the degree of anisotropy introduced into the medium that is due to the interaction of the medium with the laser beams in the presence of different collisional and Doppler-broadened environments.

Figure 3 shows the population distribution of the ground Zeeman states at time $t = 0 \sec$ and of the excited Zeeman states at time $t = 200 ps$ , $325 ps$ , and $450 ps$ for the $P_{1} (2)$ transition of an OH molecule. The populations shown here are calculated for a fixed $+ y$ location. In the numerical calculation the interaction length of the pump and probe beams is divided into 1000 discrete locations. The LIPS signal is determined by summing the contribution of polarization at those discrete points along the $+ y$ axis. For these calculations the pump and probe beams are centered at $200 ps$ with FWHM of $100 ps$ . The characteristic time for population transfer between the ground and excited states was assumed to be $2500 ps$ , and the dephasing rates were assumed to be $2 \times 10^{9} s^{- 1}$ [3, 4, 13]. The pump beam is circularly polarized with an intensity of $5 \times 10^{8} W / m^{2}$ , whereas the probe beam is linearly polarized. Initially, 1% of the total population is assumed to be isotropically distributed among the ground Zeeman states with no population at the excited states. The remaining 99% population is assumed to be in the ground bath level. The pumping rate of population from the ground to the excited Zeeman states depends on the dipole-moment strength between the coupled Zeeman states, as is evident from the anisotropic distribution of population among the excited Zeeman states shown in Fig. 3b. It is clear that, for pumping by a circularly polarized light, the anisotropy is mostly oriented in nature. The temporal profile of the orientation and alignment of the ground ( $J_{g} = 2.5$ ) and excited ( $J_{e} = 1.5$ ) levels along with the LIPS signal are shown in Figs. 4a, 4b, respectively. The orientation and alignment are calculated using Eq. (17). The LIPS signal is predominantly due to the orientation of the angular momentum distribution. The magnitude of the orientation is $\sim 3$ times greater for the ground level and $\sim 7$ times greater for the excited level than that of the alignment for the $P_{1} (2)$ transition pumped by a circularly polarized pump beam.

The value of the orientation and alignment in the excited level is significantly lower than unity. This is to be expected since $\sum_{Excited (J_{e} = 1.5)} M_{J} = 1$ was not imposed during the calculation. For example, at $t \sim 300 ps$ , $\sum_{Excited (J_{e} = 1.5)} M_{J} = 2.1282 \times 10^{- 4}$ ; the value of the orientation and the alignment were not normalized by the state-level population in order to highlight the dynamic evolution of the anisotropies with respect to the laser pulse.

Figure 5a shows the population distribution among the excited Zeeman states for the $P_{1} (2)$ transition pumped by a linearly polarized beam. The polarization of the pump beam was set at $45 °$ with respect to the polarization of the probe beam. The pump beam intensity was $5 \times 10^{8} W / m^{2}$ . As is evident in Fig. 5a, the distribution is primarily aligned. The temporal evolution of the orientation and alignment along with the normalized LIPS signal for the ground and excited levels are shown in Figs. 5b, 5c, respectively. The magnitude of the alignment is $\sim 15$ times greater for the ground level and $\sim 7$ times greater for the excited level than the magnitude of the orientation for the $P_{1} (2)$ transition pumped by a linearly polarized pump beam. A comparison of Figs. 4, 5 reveals that the maximum magnitude of the excited-level orientation for the $P_{1} (2)$ transition pumped by a circularly polarized beam is $\sim 6.6$ times higher than that of the excited-level alignment for the same transition pumped by a linearly polarized beam. For the ground level, the maximum magnitude of the orientation is $\sim 3$ times higher than that of the alignment. The LIPS signal is $\sim 20$ times stronger for the circularly polarized pump beam than for the linearly polarized pump beam.

The population distributions of the excited Zeeman states pumped by a linearly and a right circularly polarized beam for the $Q_{1} (2)$ transition at three different times are shown in Figs. 6a, 6b, respectively. The initial population distribution of the ground Zeeman states is assumed to be isotropic. From Fig. 6, it is apparent that the distribution in the excited Zeeman states is almost completely aligned when the pump beam is linearly polarized. When the pump beam is right circularly polarized, the distribution displays both aligned and oriented anisotropy. When the $Q_{1} (2)$ transition is pumped by a right circularly polarized beam, the $M_{e} = - 2.5$ state is not coupled with any of the ground Zeeman states; the absence of population in the $M_{e} = - 2.5$ state creates a net helicity for the ensemble. The temporal profiles of the LIPS signal, orientation, and alignment for the $Q_{1} (2)$ transition pumped by linearly and circularly polarized beams are shown in Figs. 7a, 7b, respectively. For the circularly polarized pump beam, the orientation and alignment are comparable. The LIPS signal is $\sim 50$ times stronger when pumped by a linearly polarized beam rather than a circularly polarized beam.

Figure 8 shows the temporal profiles of the LIPS signal for the $P_{1} (8)$ transition pumped by circularly and linearly polarized pump beams. The pump beam intensity is set at $5 \times 10^{8} W / m^{2}$ . The LIPS signal is $\sim 46$ times stronger for the circularly polarized pump beam than for the linearly polarized pump beam. For the $P_{1} (2)$ transition, the LIPS signal was $\sim 20$ times stronger when pumped by the circularly polarized beam compared to the linearly polarized beam, as discussed above. The population distributions for the excited Zeeman states pumped by right circularly polarized and linearly polarized beams for the $P_{1} (8)$ transition at three instants are shown in Figs. 9a, 9b, respectively. The temporal profiles of the orientation and alignment for the $P_{1} (8)$ transition pumped by circularly polarized and linearly polarized beams are shown in Figs. 10a, 10b, respectively. The magnitude of the alignment is $\sim 5$ times weaker than that of the orientation when the transition is pumped by a circularly polarized light. However, when the transition is pumped by a linearly polarized light, the contribution of the orientation toward LIPS signal generation is significantly less than that of the alignment, unlike the transition with a low J number. Therefore, the numerical code employed was able to simulate quantum energy levels with large rotational quantum numbers. For the $P_{1} (8)$ transition, 34 quantum states were included in the calculations.

4. EFFECTS OF SATURATION ON THE LASER-INDUCED ANISOTROPY

In this section we will discuss how orientation, alignment, and the higher order moments evolve at saturation and their relationship with the generation of the LIPS signal. Temporal profiles of the LIPS signal for the $P_{1} (2)$ transition at four pump-beam intensities are shown in Fig. 11. In this figure, the normalized pump-beam intensity profile is shown by the solid black line. At the onset of saturation, the signal intensity first decreased without affecting the temporal shape of the LIPS signal when the pump-beam intensity was increased from $5 \times 10^{9} W / m^{2}$ to $10^{10} W / m^{2}$ . At $5 \times 10^{10} W / m^{2}$ , the temporal envelope of the LIPS signal was drastically reduced with a FWHM of $\sim 45 ps$ . No LIPS signal generation occurred after $\sim 240 ps$ , even though the pump pulse lasted until $\sim 340 ps$ . At this intensity no Rabi-flopping behavior was observed. The oscillation of the LIPS signal due to Rabi flopping at saturation was observed when the intensity of the pump beam was further increased to $10^{11} W / m^{2}$ . Saturation intensities for $P_{1} (2)$ transition of an OH molecule was estimated based on the measurements discussed in [4].

The temporal evolution of the orientation, alignment, and octopole moments and their relative phase difference during saturation will now be discussed in an attempt to explain the behavior observed in Fig. 11. The temporal profiles of the excited-level orientation, alignment, and octopole moments of the angular momentum distribution for pump-beam intensities of $5 \times 10^{9} W / m^{2}$ and $10^{10} W / m^{2}$ are shown in Fig. 12. Clearly, the phase relationship between the moments for $5 \times 10^{9} W / m^{2}$ and for $10^{10} W / m^{2}$ is similar. The magnitude of the orientation at $10^{10} W / m^{2}$ is only $\sim 4 %$ lower than that at $5 \times 10^{9} W / m^{2}$ , resulting in a reduction of LIPS signal by $\sim 55 %$ . From the relative magnitudes of the alignment and octopole moments, it is clear that the anisotropy is still due predominantly to orientation. Even though the temporal profile of the LIPS signal at $5 \times 10^{9} W / m^{2}$ appears to be similar to the profiles in the perturbative regime, it defines the onset of saturation, as evidenced in Fig. 12b by the significant pumping of the ground-level population to the excited states. For P transitions pumped by circularly polarized light in the perturbative regime, LIPS signal generation occurs when the sign of the excited-level orientation is negative, as evident in Figs. 4, 10. In the saturated regime when the P transition is pumped by circularly polarized light, the magnitudes of all of the moments become comparable; signal generation occurs when the net anisotropy is oriented with a negative sign, which will now be discussed in detail.

Figure 13 displays the temporal evolution of the excited- level moments for the $P_{1} (2)$ transition for a pump-beam intensity of $5 \times 10^{10} W / m^{2}$ . The LIPS signal is shown here for reference. At saturation the magnitudes of the orientation and alignment become comparable, and the population distribution no longer maintains a preferred anisotropy. In addition to the orientation and alignment, higher order moments such as octopole also begin to play a significant role in the saturated regime. For $J_{e} = 1.5$ there are no moments higher than the octopole moment as K varies from 0 to 3; for this reason we have shown up to octopole moments in Fig. 13. As discussed earlier, at this intensity no LIPS signal generation occurs after $\sim 240 ps$ , even though the magnitudes of the orientation, alignments, and octopole moments are nonzero beyond $240 ps$ . Shown in Figs. 14a, 14b, respectively, are the population distribution of the excited Zeeman states and the temporal profile of the normalized LIPS signal and various moments of the angular momentum distribution for the excited level in the saturated regime for a pump-beam intensity of $10^{11} W / m^{2}$ . The saturated LIPS signal was normalized to have a maximum value of unity. The oscillations in the temporal profiles shown in Fig. 14b are due to the Rabi-flopping behavior exhibited at saturation [13]. Again Figs. 14a, 14b illustrate that the population distribution cannot be described as being nearly oriented or aligned in nature. During saturation, the signs of the orientation and alignment become reversed, as shown in Figs. 13, 14b, unlike the behavior observed in the unsaturated regime. This implies that the population distribution dictated by the Zeeman-states coupling scheme shown in Fig. 2 no longer holds because of the pumping of population out of the excited states that is exhibited by the Rabi-flopping behavior. Moreover, these reversals of the anisotropies do not necessarily contribute to the generation of the LIPS signal, as shown in Figs. 13, 14b.

5. EFFECTS OF GROUND-LEVEL ANISOTROPY ON THE LIPS SIGNAL

In the analysis discussed above, the population distribution of the ground Zeeman states at $t = 0 \sec$ was assumed to be isotropic. Wasserman et al. [19] derived a perturbative model based on equations for the DFWM signal with anisotropic population distribution. We examined the effect of preexisting ground-level anisotropy on the LIPS signal and also on the orientation and alignment of the excited level. Three types of anisotropic distribution (i.e., aligned, LCP oriented, and RCP oriented), as shown in Fig. 15, were considered. The anisotropies displayed in Figs. 15b, 15c are labeled as LCP oriented and RCP oriented since these would be the orientations of the system pumped by left circularly polarized (LCP) and right circularly polarized (RCP) pump beams, respectively. The LIPS signals for the $P_{1} (2)$ transition pumped by a circularly polarized beam, where the initial ground-level population distribution is isotropic, aligned, and oriented, are shown in Fig. 16 for a pump-beam intensity of $10^{9} W / m^{2}$ . The intensity of the LIPS signal is higher for anisotropic distributions of the ground-level population. The intensity of the LIPS signal is $\sim 8$ times stronger when the initial ground-level population distribution is LCP oriented in nature rather than isotropic and $\sim 20$ times stronger when the ground-level population is RCP oriented. These enhancements of the LIPS signals can best be explained by the laser-induced anisotropy of the excited level.

Displayed in Fig. 17 are the orientation and alignment of the excited level for the three cases shown in Fig. 15 with initial ground-level anisotropic population distribution. The magnitudes of the alignments for all anisotropic cases are significantly higher than that for the isotropic ground-level case, as is evident from a comparison of Figs. 4, 17. Moreover, the relative sign of the orientation and alignment also has a significant impact on the strength of the LIPS signal. The signs of the orientation and alignment for the RCP-oriented anisotropy, shown in Fig. 17c, are the same, unlike in all other cases, where the signs of the orientation and alignment are opposite. This similarity results in a significantly stronger LIPS signal for the RCP-oriented anisotropy, even though the magnitude of the orientation and alignment is smaller than that for the other anisotropic cases and the isotropic case, as shown in Figs. 4, 17. In our numerical code, we verified that the preexisting ground-level anisotropy does not produce LIPS signals in the absence of the pump beam.

For the LCP-oriented ground-level population distribution, the maximum LIPS signal occurs before the orientation reaches its maximum magnitude. This is unlike all the other cases where the maximum LIPS signal occurs only after either the orientation or the alignment reaches its maximum magnitude, which could be due to the significant pumping of population to the excited states that is exhibited by the strong coherence before the orientation reaches its maximum value. The pumping of population from the ground Zeeman states depends, not only on the coherences established by the laser beams, but also on the dipole-moment strength for a particular transition. Figure 18 shows the temporal profile of the real and imaginary components of $(μ_{e g} σ_{g e})$ multiplied by the appropriate phase factor $\exp [i (k_{y} y - ω t)]$ along the $+ y$ axis for the $Δ M = + 1$ transitions with LCP-oriented ground-level anisotropy. This is because the third-order polarization induced in the medium is calculated from the product of the dipole- moment matrix element μ and the density matrix element $σ_{e g}$ , as shown in Eq. (4), where the summation is carried out for all possible transitions. As discussed above, the off-diagonal element of the density matrix equation $ρ_{e g}$ is a slowly varying quantity and, as such, is written as $σ_{e g} \exp (- i ω t)$ . As is clear from Fig. 18b, the maximum value of $[μ_{e g} σ_{g e}]_{img}$ occurs earlier than that of $[μ_{e g} σ_{g e}]_{real}$ . The contribution of $[μ_{e g} σ_{g e}]_{img}$ and $[μ_{e g} σ_{g e}]_{real}$ from all the possible transitions yields the LIPS signal.

From the analysis discussed above, it is clear that the orientation and the alignment play a decisive role in the generation of the LIPS signal, depending either on the polarization of the pump beam or on the ground-level anisotropy.

6. CONCLUSIONS

The effects of laser-induced anisotropy on the generation of short-pulse (laser pulse width $τ_{L}$ less than characteristic collision time $τ_{C}$ ) LIPS signals for the P- and Q-branch transitions was investigated. The calculations were performed by solving the time-dependent density matrix equations for a multistate system. In the unsaturated regime the oriented anisotropy is mainly responsible for generation of the LIPS signal for the $P_{1} (2)$ transition with isotropic ground-level population pumped by a circularly polarized light, whereas alignment is responsible for LIPS signal generation when the transition is pumped by a linearly polarized light. The contributions of the orientation and alignment to the generation of the LIPS signal become comparable in the saturated regime. The magnitude of the LIPS signal increases by more than a factor of 20 for an initial oriented anisotropic distribution of ground-state population, as compared to the isotropically distributed population. For the $Q_{1} (2)$ transition pumped by a circularly polarized light, the contributions of the orientation and alignment to the LIPS signal are comparable. An understanding of the effects of anisotropy on the LIPS signal will facilitate correct modeling of the LIPS signal-generation processes for the interpretation of experimental LIPS signals.

ACKNOWLEDGMENTS

Funding for this research was provided by the U.S. Air Force Research Laboratory [Ms. Amy Lunch, Program Manager] under contract FA8650-10-C-2008, by the Air Force Office of Scientific Research [Dr. Tatjana Curcic, Program Manager], and by the U.S. Department of Energy, Division of Chemical Sciences, Geosciences, and Biosciences, under grant FG02-03ER15391.

Fig. 1 Geometry of LIPS signal-generation process.

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Fig. 2 Energy-level diagram for the Zeeman-state structure of the $P_{1} (2)$ transition. Allowed $Δ M = 0$ transitions are indicated by dashed arrows; $Δ M = \pm 1$ transitions are indicated by solid arrows. Strengths and phases of the transitions are indicated by the numerical value of the x or z components of the geometry-dependent part of the dipole matrix element ${\vec{μ}}_{G}$ .

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Fig. 3 Population distribution in the (a) ground Zeeman states at $t = 0 \sec$ and (b) excited Zeeman states for the $P_{1} (2)$ transition of OH pumped by a circularly polarized beam. $ρ_{g g}^{0}$ is the ground-state population at $t = 0 \sec$ .

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Fig. 4 Temporal profiles of the orientation and alignment of the (a) ground level ( $J_{g} = 2.5$ ) and (b) excited level ( $J_{e} = 1.5$ ) for the $P_{1} (2)$ transition pumped by a circularly polarized beam. LIPS signal is shown for reference. The intensity of the pump beam is $5 \times 10^{8} W / m^{2}$ .

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Fig. 5 (a) Population distribution in the excited Zeeman states, (b) temporal profiles of the various moments for the ground level, and (c) temporal profiles of the various moments at the excited level along with the LIPS signal for the $P_{1} (2)$ transition pumped by a linearly polarized beam. The intensity of the pump beam is $5 \times 10^{8} W / m^{2}$ .

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Fig. 6 Population distribution in the excited Zeeman states for the $Q_{1} (2)$ transition pumped by (a) a linearly polarized beam and (b) a right circularly polarized beam. $ρ_{g g}^{0}$ is the ground-state population at $t = 0 \sec$ .

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Fig. 7 Temporal profiles of the orientation and the alignment of the excited state for the $Q_{1} (2)$ transition for (a) a linearly polarized pump beam and (b) a right circularly polarized pump beam.

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Fig. 8 Temporal profiles of the LIPS signal for the $P_{1} (8)$ transition pumped by a circularly and linearly polarized pump beam. The intensity of the pump beam is $5 \times 10^{9} W / m^{2}$ .

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Fig. 9 Population distribution in the excited Zeeman states for the $P_{1} (8)$ transition pumped by (a) a right circularly polarized beam and (b) a linearly polarized beam. $ρ_{g g}^{0}$ is the ground-state population at $t = 0 \sec$ .

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Fig. 10 Temporal profiles of the orientation and the alignment of the excited state for the $P_{1} (8)$ transition for (a) a right circularly polarized pump beam and (b) a linearly polarized pump beam.

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Fig. 11 Temporal profiles of the LIPS signal for the $P_{1} (2)$ transition pumped by a circularly polarized beam for different intensities.

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Fig. 12 (a) Temporal profiles of the orientation, alignment, and octopole moments of the excited level and (b) population distribution in the excited Zeeman states for the $P_{1} (2)$ transition pumped by a right circularly polarized beam at $5 \times 10^{9} W / m^{2}$ to $10^{10} W / m^{2}$ .

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Fig. 13 Temporal profiles of the orientation, alignment, and octopole moments of the excited state for the $P_{1} (2)$ transition pumped by a right circularly polarized beam at $5 \times 10^{10} W / m^{2}$ . The LIPS signal is shown for reference.

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Fig. 14 (a) Excited-state population distribution and (b) temporal profiles of orientation, alignment, and octopole moments for the $P_{1} (2)$ transition pumped by a circularly polarized light with an intensity of $10^{11} W / m^{2}$ .

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Fig. 15 The (a) aligned, (b) LCP-oriented, and (c) RCP-oriented population distribution among the ground Zeeman states at $t = 0 \sec$ . These three initial population distributions are considered to investigate the effect of preexisting anisotropy on the LIPS signal.

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Fig. 16 LIPS signals for the $P_{1} (2)$ transition pumped by a circularly polarized beam where the initial ground-level population distribution is isotropic, aligned, or oriented.

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Fig. 17 Temporal profiles of the excited-level orientation and alignment for the $P_{1} (2)$ transition pumped by a circularly polarized beam, where the initial ground-level population is (a) aligned, (b) LCP oriented, and (c) RCP oriented. The LIPS signal is shown for reference.

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Fig. 18 Temporal profiles of the (a) real and (b) imaginary components of $(μ_{e g} σ_{g e})$ multiplied by the appropriate phase factor $\exp [i (k_{z} y - ω t)]$ along the $+ y$ axis for the $Δ M = + 1$ transitions with LCP-oriented ground-level anisotropy.

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Dynamics of the moments of angular momentum distribution during generation of laser-induced polarization spectroscopy (LIPS) signal

Abstract

1. INTRODUCTION

2. CALCULATION OF THE LIPS SIGNAL FOR A MULTISTATE SYSTEM

3. EFFECTS OF ORIENTATION AND ALIGNMENT ON THE LIPS SIGNAL

4. EFFECTS OF SATURATION ON THE LASER-INDUCED ANISOTROPY

5. EFFECTS OF GROUND-LEVEL ANISOTROPY ON THE LIPS SIGNAL

6. CONCLUSIONS

ACKNOWLEDGMENTS

Cited By

Figures (18)

Equations (17)

Journal of the Optical Society of America B