#  Polarization-dependent Intensity Ratios in Double Resonance Spectroscopy

Kevin K. Lehmann  Departments of Chemistry and Physics, University of
Virginia, Charlottesville VA, 22904-4319

###### Abstract

Double Resonance is a powerful method spectroscopic method that provides
unambiguous assignment of the rigorous quantum numbers of one state of a
transition. However, there is often ambiguity as to the branch ($\Delta J$) of
the transition. The dependence of the intensity of the double resonance signal
on the relative polarization of pump and probe radiation can be used to
resolve this ambiguity and has used for this in the past. However, the
published theoretical predictions for this ratio are based upon a weak (i.e.
non-saturating) field approximation. In this paper, we present theoretical
predictions for these intensity ratios for cases where the pump field is
strongly saturating, in the two limits of transitions dominated by homogeneous
and inhomogeneous broadening. While saturation, as can be expected, reduces
the magnitude of the polarization effect (driving the intensity ratio closer
to unity), polarization anisotropy remains even with a strongly saturating
probe field in most cases. For the case of an inhomogeneous broadened line, as
when Doppler broaden linewidth dominates over even the power broadened
homogeneous line width, a large fraction of the low pump power anisotropy
remains. Results are presented for both the case of linear and circular pump
and probe field polarizations. The present predictions are compared with
experimental measurements on CH4 ground state $\rightarrow\nu_{3}\rightarrow
3\nu_{3}$ transitions recently reported by de Oliveira et al and found to be
in better agreement than the weak field predictions.

Double resonance (DR) has long been one of the most powerful methods in the
spectroscopist’s toolkit.carroll66  This is an intrinsically nonlinear
spectroscopy using two coherent light sources, at least one (the pump) that
creates a nonequilibrium population distribution in a sample, and another (the
probe) that measures an absorption, emission, scattering, or action spectrum
of the resulting nonequilibrium sample. There are thus three states linked by
two transitions. We neglect in this paper “4-level” DR transitionsPursell90 ;
Mandal93  where a collision transfers population from a state of one
transition to a state of the other. DR results in greatly simplified spectra
Teets76 ; Coy86a ; Coy86b ; Lehmann88 ; Coy89 ; Lehmann86 ; Gambogi94 ;
Gambogi96 ; Kabir03 ; Zhao04  and allows the unambiguous assignment of the
starting state of probe transitions, which is often required when spectra are
highly perturbed and do not follow regular patterns due to the breakdown of
the separation of degrees of freedom, such as vibration and rotation.Park15
It is also useful for observation of spectra from states without significant
thermal population Martinez04 ; Settersten03 ; Foltynowicz21a ; Foltynowicz21b
and cases where homogeneous or inhomogeneous broadening results in substantial
overlap of individual transitions, which can result, for example, in a broad
rotational contour without resolvable features.Henck95a ; Henck95b ;
Callegari97 ; Callegari00 ; Callegari03  DR allows the selective population of
states that have negligible thermal populations under available experimental
conditions, allowing novel spectroscopic transitions to be observed. Barnum20
; Yang20  In many cases, at least some of the transitions from these states
reach final states that are weak or forbidden from the thermally well-
populated states due to symmetry or propensity selection rules.Chevalier87 ;
Frye87 ; Kasahara94 ; Kasahara99 ; Srivastava20b ; Dia21  When DR is performed
using narrow bandwidth lasers, one can largely eliminate inhomogeneous
broadening, as the pump laser will produce a Bennet holeElbel90  in the
velocity distribution of the initial state and corresponding Bennet hill in
the upper state of the pump transition. Probe spectra will display sub-Doppler
features whose widths are on the order of the homogeneous widths, which can be
orders of magnitude below Doppler broadened widths. Kaminsky76 ; Wieman76  Due
to elastic collisions, the narrow feature will often ride on top of a Doppler
broadened feature, but the former usually has a much higher peak strength. DR,
especially with a pulsed pump source and continuous wav (CW) probe field, has
been widely used to study elastic, reorientation, and inelastic collision
rates and kernels. Greene83 ; Coy92 ; Klaassen94 ; Coy95 ; Shin92 ; Soriano98
; Abel99 ; Bayram06 ; Forthomme15  Resonant 3-wave mixing, which is another
form of DR spectroscopy, can be used to both measure enantiomeric excess of
chiral molecules Patterson13a ; Patterson13b ; Lobsigner15 ; Lehmann18a  and
to create enantiomeric excess in single rotational states. Eibenberger17

If the common level in the DR scheme is the lowest energy level (called V-type
DR), the DR signal will be a narrow depletion of the background Doppler
Broadened probe transition. If the common level is not the lowest energy
state, the pump will create a narrow absorption or emission depending upon
whether the final state is higher or lower in energy than the pumped
(intermediate) state. These are known as ladder- and $\Lambda$-type DR
respectively. If there is a negligible thermal population in the intermediate
state, the probe transitions will be a new narrow transition. The very narrow
width of DR transitions can be a drawback when one needs to sample the probe
transition spectrum over a broad spectral range (say 30 THz). The time
required to search such a spectral range will on the order of 10 times the
detection time constant times the ratio of the scan range divided by the width
of the probe DR transitions. In such cases, detection of probe transitions of
a few MHz or less width requires either very long search times or a very short
time constant, which reduces the signal to noise ratio (SNR) of the probe
spectrum. The probe spectrum scan needs to be repeated for each pump
transition studied. The recent demonstration of DR using a stabilized
frequency comb for the probe allows one to simultaneously sample the probe
spectrum at the frequencies of tens of thousands of comb teeth has
dramatically reduced this limitation. Nishiyama16 ; Foltynowicz21a ;
Foltynowicz21b

Selection rules greatly reduced the final states that are observed in double
resonance using a particular pump transition. However, one often retains an
ambiguity as to the total angular momentum quantum number $J$ of the final
state of a probe transition due to the $\Delta J=0,\pm 1$ selection rule for
dipole transitions. The pump transition produces a nonequilibrium alignment of
the angular momentum projection quantum number $M$ of the initial and final
states, and this results in the probe absorption strength depending upon the
relative polarization state of the pump and probe fields, which was first
reported by Frankel and Steinfeld (1975).Frankel75  By comparing the strength
of the DR signal with parallel and perpendicular linear polarizations, one
obtains a polarization ratio that can be used to assign the value of $\Delta
J$ for the probe transition. One sensitive implementation of DR is
polarization spectroscopy which uses nearly crossed polarizers for the probe
wave, placed before and after the sample.Li04 ; Steeves05  The pump laser-
induced birefringence of the sample results in a change in the transmission of
the probe that is observed on a greatly reduced background intensity which
will result in an increased signal-to-noise ratio if the probe field is
dominated by technical intensity noise. Another sensitive variation is
polarization modulation where a change in probe transmission is produced by a
polarization modulation of the pump field.Shin92 ; Soriano98  In the limit
that the pump transition is not saturated, the predicted polarization
dependences of the DR signals are easily derived from the dependence of pump
and probe field transition intensity on $M$. Expressions have been given in
the literature Chevalier87 ; Jonas92 ; Ferber97 ; Kabir03 ; Chen08 ;
Petrovic08  though often not in a form most useful for DR spectroscopy, and
almost always neglecting the effects of optical saturation. Significantly, one
optimizes the strength of DR transitions by working with sufficient pump power
to have substantial saturation of the pump transitions as that produces a
larger pump-induced disequilibrium of the sample yet quantitative discussion
of these cases is largely absent. One of the few exceptions was the study of
Spano and Lehmann Spano92  who examined the case of polarization spectroscopy
of a sample that is optically thick for the pump transition. It was found that
for excitation of a dipole transition with a pulse duration much shorter than
the relaxation time, a strong pump pulse evolves into an area-preserving pulse
similar to the self-induced transparency of a two-level system.McCall69  This
pulse produces an even greater fractional alignment of the sample than that
produced by excitation with negligible saturation. For DR with continuous wave
pump fields, which produce a steady-state response of the sample, the analysis
of Spano and LehmannSpano92  is not applicable. In this paper, we will present
an analysis of the polarization dependence of DR signal strength for cases of
a steady-state response of the sample. First, the weak field case will be
considered, which results in simple analytical expressions. This is followed
by results for a saturating pump field both for the cases of homogeneous and
inhomogeneous broadened transitions.

##  I  Polarization Dependence of Pump Transitions

Consider a DR signal that results from a pump transition between a pair of
levels $1$ and $2$ and a probe transition between levels $2$ and $3$, and
label the total angular momentum quantum numbers for the three levels as
$J_{1},J_{2}$, and $J_{3}$ respectively. Let the pump (probe) transition be
driven by waves $a(b)$ with angular frequency and wavevector $\omega_{a,b}$
and $\vec{k}_{a,b}$ respectively. Each $J_{2},M$ state will contribute to the
DR signal proportional to the population change in that state produced by the
pump laser, $\Delta\rho_{22}(M,\Delta\omega_{12})$ where $\Delta\omega_{12}$
is the detuning of the pump from resonance, times the absorption coefficient
of the probe laser by that state, $S(M)$, both of which will be dependent on
$M$ and the polarization directions of pump and probe fields respectively.
Both pump and probe strengths depend upon the respective transition dipole
moment matrix element for the respective transition which has the form
$\left<i,M|\vec{\mu}|j,M^{\prime}\right>\cdot\hat{G}=\left<i|\mu_{g}|j\right>\left<i,M|\phi_{gG}|j,M^{\prime}\right>$.
where $g$ gives the direction of the transition dipole moment, $\mu$, between
states $i$ and $j$ in the molecular frame, $\hat{G}$ gives the direction of
the optical electric field, $E$ in the laboratory fixed frame, and
$\left<i,M|\phi_{gG}|j,M^{\prime}\right>$ is the direction cosine matrix
element which is the matrix element of $\hat{g}\cdot\hat{G}$. The transitions
direction cosine matrix elements are given in Table 4.4 of Microwave
Spectroscopy by Townes and SchawlowTownes&Schawlow  and reproduced in Table 1
for completeness. The direction cosine matrix elements consist of three
factors but only the one that depends on $J$ and $M$ for each state,
$\phi_{G}(J,M,J^{\prime}.M^{\prime})$, of a transition is needed for
predicting the polarization dependence – the other two factors are independent
of $M$ and the polarization state of the radiation field.

We begin by considering linear polarization for the pump and probe fields. The
total signals are independent of how we align the laboratory axes; we will
take the $Z$ axis as along the polarization direction of the pump wave and $Y$
axis along the propagation direction of both pump and probe fields. With this
axis assignments, we use $\phi_{Z}(J_{1},M,J_{2},M)$ for the pump matrix
element and for the probe $\phi_{Z}(J_{2},M,J_{3},M)$ for parallel and
$\phi_{X}(J_{2},M,J_{3},M\pm 1)$ for perpendicular relative polarizations.

Treating the pump transition as a two-level system for each $M$ value, the
steady-state change in population in each $M$ state of level $2$ can be
written in terms of the equilibrium population density difference between
levels $1$ and $2$, $\rho_{11}^{e}-\rho_{22}^{e}$; the population and
coherence relaxation rates of the pump transition, $\gamma_{1},\gamma_{2}$;
the pump Rabi frequency $\Omega_{12}(M)=\mu_{12}(M)E/\hbar$ where $E$ is the
amplitude of the field driving the $1\leftrightarrow 2$ pump transition; and
the detuning from resonance,
$\Delta\omega_{12}=\omega_{a}-\vec{k}_{a}\cdot\vec{v}-\omega_{12}$ with
$\vec{v}$ the velocity of the absorber:Karplus48 ; Townes&Schawlow

$\Delta\rho_{22}(M,\Delta\omega_{12})=\frac{(\rho_{11}^{e}-\rho_{22}^{e})}{2(2J_{2}+1)}\frac{(\Omega_{12}(M)^{2}\gamma_{2}/\gamma_{1})}{\Delta\omega_{12}^{2}+\gamma_{2}^{2}\left(1+(\Omega_{12}(M)^{2}/\gamma_{1}\gamma_{2})\right)}$
(1)

This is a Lorentzian with half width half maximum (HWHM) of
$\gamma_{2}\sqrt{1+(\Omega_{12}(M)^{2}/\gamma_{1}\gamma_{2})}$. We will treat
the multistate transition with different values of $M$ as a sum of two-level
systems with different transition dipole moments (hence the $M$ labels in Eq.
1). SchwendemanSchwendeman80 ; Schwendeman99  has pointed out this is an
approximation, but concluded it holds if one neglects pure $M$ changing
collisions (elastic $J$-reorientation). We will assume that the probe
transition is unsaturated and thus each $M$ value has an absorption strength
$S(M)$ proportional to $\phi_{Z}(J_{2},M,J_{3},M)^{2}$ for $\parallel$
alignment and
$\phi_{X}(J_{2},M,J_{3},M+1)^{2}$+$\phi_{X}(J_{2},M,J_{3},M-1)^{2}$ for
$\perp$ alignment.

The total DR signal ratio for a given probe polarization, $G$, can be written
as:

$R_{\rm
lin}=\frac{I_{\parallel}}{I_{\perp}}=\frac{\sum_{M}(\int\Delta\rho_{22}(M,\Delta\omega_{12})d\omega_{12})\phi_{Z}(J_{2},M,J_{3},M)^{2}}{\sum_{M}(\int\Delta\rho_{22}(M,\Delta\omega_{12})d\omega_{12})(\phi_{X}(J_{2},M,J_{3},M+1)^{2}+\phi_{X}(J_{2},M,J_{3},M-1)^{2})}$
(2)

If the transition is homogeneously broadened, there is only a single value of
$\Delta\omega_{12}$ and thus not integral of this detuning.

###  I.1 Unsaturated pump transitions

In most CW gas-phase DR experiments, the pump Rabi frequencies,
$\Omega_{12}(M)$ are far below the Doppler width of the pump transition, in
which case the pump will burn a Bennet hole in the velocity distribution of
the lower energy state and a Bennet hill in the upper energy state. If we
assume that the pump transition is inhomogeneously Doppler broadened with
lineshape function $g_{\rm D}$ that has width $\Delta\omega_{\rm
D}>>\gamma_{1},\Omega_{12}(M)$, we can integrate Eq. 1 over the Doppler
detuning to give an integrated steady-state population change for level $2$

$\Delta\rho_{22}(M)=\frac{\pi}{2(2J_{2}+1)}(\rho_{11}^{e}-\rho_{22}^{e})g_{\rm
D}(\omega-\omega_{12})\frac{(\Omega_{12}(M)^{2}}{\gamma_{1}\sqrt{1+(\Omega_{12}(M)^{2}/\gamma_{1}\gamma_{2})}}$
(3)

In the limit of low saturation, $\Omega_{12}^{2}(M)<<\gamma_{1}\gamma_{2}$,
get

$\Delta\rho_{22}(M)\rightarrow\frac{\pi}{2(2J_{2}+1)}(\rho_{11}^{e}-\rho_{22}^{e})g_{\rm
D}(\omega_{a}-\omega_{12})|\Omega_{12}(M)|^{2}/\gamma_{1}$ (4)

In this limit, the fraction pumped of each $M$ is proportional to
$\Omega_{12}^{2}$ and thus proportional to the intensity and the square of the
transition matrix element.

For non-saturated probe transitions, $\Omega_{23}<<\gamma_{1}\gamma_{2}$, the
absorption coefficient of the probe is given by

$\alpha_{23}(\Delta\omega_{23})=\frac{\omega_{23}}{\epsilon_{0}c\hbar}\frac{\gamma_{2}}{\gamma_{2}^{2}+\Delta\omega_{23}^{2}}\cdot\sum_{M=-J_{2}}^{J_{2}}\mu_{23}(M)^{2}\Delta\rho_{22}(M)$
(5)

Integrating over the probe detuning, we get an integrated absorption
coefficient we get for the unsaturated case:

$I_{G}=\int\alpha_{23}d\omega_{23}=\frac{\pi^{2}\omega_{23}}{(J_{2}+1)(\epsilon_{0}c)^{2}\hbar^{3}\gamma_{1}}(\rho_{11}^{e}-\rho_{22}^{e})g_{\rm
D}(\omega-\omega_{12})I_{p}\cdot\sum_{M=-J_{2}}^{J_{2}}\mu_{23}(M)^{2}\mu_{12}(M)^{2}$
(6)

Using the axes assignments given above, for symmetric top transitions
$J_{1},K_{1},M\rightarrow J_{2},K_{2},M\rightarrow J_{3},K_{3},M^{\prime}$ we
have

$\displaystyle\sum_{M=-J_{2}}^{J_{2}}\mu_{23}(M)^{2}\mu_{12}(M)^{2}$
$\displaystyle=\mu_{12}^{2}\,\mu_{23}^{2}\,\phi_{J}(J_{1},J_{2})^{2}\phi_{g}(J_{1},K_{1},J_{2},K_{2})^{2}\phi_{J}(J_{2},J_{3})^{2}\phi_{g^{\prime}}(J_{2},K_{2},J_{3},K_{3})^{2}\times$
(7)
$\displaystyle\sum_{M=-J_{2}}^{J_{2}}\phi_{Z}(J_{1},M,J_{2},M)^{2}\sum_{M^{\prime}}\phi_{G}(J_{2},M,J_{3},M^{\prime})^{2}$

In this equation, $\mu_{12}$ and $\mu_{23}$ are the transition dipole moment
matrix elements in the molecular frame for the pump and probe transitions, and
$G=Z$ or $X$ depending on whether the probe is polarized parallel to the pump,
giving a signal $I_{\parallel}$ or perpendicular to the pump, giving a signal
$I_{\perp}$. When $G=Z$, the selection rule gives $M^{\prime}=M$, and when
$G=X$, the selection rule $M^{\prime}=M\pm 1$.

For asymmetric top molecules, we can expand the rotational wavefunction for
each state with quantum numbers $i,J_{i},\tau_{i},M$ as
$\phi(i,J_{i},\tau_{i},M)=\sum_{K}A(i,J_{i},\tau_{i},K)\phi_{J_{i},K,M}$ where
$\phi_{J_{1},K,M}$ are symmetric top wavefunctions. In Eq. 7, we replace the
terms $\phi_{g}(J_{i},K_{i},J_{j},K_{j})^{2}$ by
$|\sum_{K_{i},K_{j}}A(i,J_{i},\tau_{i},K_{i},)A(j,J_{j},\tau_{j},K_{j})|^{2}\phi_{g}(J_{i},K_{i},J_{j},K_{j})|^{2}$,

The DR signal strength for an arbitrary angle, $\theta$, between pump and
probe polarizations can be written as
$I(\theta)=I_{\parallel}\cos^{2}\theta+I_{\perp}\sin^{2}(\theta)$. For
$\theta_{m}=\cos^{-1}(1/\sqrt{3})$ (known as the magic angle)
$I(\theta_{m})=(I_{\parallel}+2I_{\perp})/3$. The sum
$\phi_{Z}(J,M,J^{\prime},M)^{2}+2\phi_{X}(J,M,J^{\prime},M+1)^{2}+2\phi_{X}(J,M,J^{\prime},M-1)^{2}$
is $M$ independent and so, at the magic angle of relative polarization, the
probe absorption strength is proportional to the total population in the
intermediate energy level $J_{2},K_{2}$.

The ratio of $I_{\parallel}$ and $I_{\perp}$ depends only on the sum over $M$
values as all other factors are independent of pump or probe polarization,
thus the polarization ratio in the unsaturated pump case is

$R^{\rm us}_{\rm
lin}=\frac{I_{\parallel}}{I_{\perp}}=\frac{\sum_{M=-J_{2}}^{J_{2}}\phi_{Z}(J_{1},M,J_{2}.M)^{2}\phi_{Z}(J_{2},M,J_{3},M)^{2}}{\sum_{M=-J_{2}}^{J_{2}}\phi_{Z}(J_{1},M,J_{2}.M)^{2}(\phi_{X}(J_{2},M,J_{3},M-1)^{2}+\phi_{X}(J_{2},M,J_{3},M+1)^{2})}$
(8)

Given the $\Delta J=0,\pm 1$ selection rule for both pump and probe
transition, there are 9 possible cases and the ratio given in Eq. 8 can be
evaluated using the expressions for $\phi_{G}$ given in table 1. The resulting
analytical expressions (with sums over $M$ evaluated using Mathematica) are
presented in Table 2, both in symbolic form and numerical values for
$J_{2}=0-10$. It is traditional to label molecular transitions with $R,Q,P$
for transitions when $J$ for the upper state minus $J$ for the lower states
$=+1,0,-1$ respectively. These labels change for the three different DR
schemes: (ladder-type with $E_{1}<E_{2}<E_{4}$, V-type with
$E_{2}<E_{1},E_{2}$, and $\Lambda$-type with $E_{2}>E_{1},E_{3}$. Missing
entries in the table correspond to dipole forbidden transitions which require
at least one of the two $J$ values for each transition to be greater than $0$.

Table 1: Table of nonzero direction cosine matrix elements taken from Townes & Schawlow. The symmetric top matrix element of $<J,K,M|\mu_{g}E_{h}|J^{\prime},K^{\prime},M^{\prime}>=\phi_{(}J,J^{\prime})\phi_{g}(J,K,J^{\prime},K^{\prime})^{*}\phi_{h}(J,M,J^{\prime},M^{\prime})$ where $g$ is the direction of the transition moment in the molecular axis system and $h$ is the direction of the electric field in the laboratory fixed axis system. The root mean square values $\phi_{\rm rms}^{2}=\sum_{M=-J}^{J}\phi_{Z}(J,M,J^{\prime},M)^{2}/(2J+1)=2\sum_{M=-J}^{J}\phi_{X}(J,M,J^{\prime},M\pm 1)^{2}/(2J+1)=2\sum_{M=-J}^{J}|\phi_{Y}(J,M,J^{\prime},M\pm 1)|^{2}/(2J+1)$ | $J^{\prime}=J+1$ | $J^{\prime}=J$ | $J^{\prime}=J-1$
---|---|---|---
$\phi(J,J^{\prime})$ | $\left[4(J+1)\sqrt{(2J+1)(2J+3)}\right]^{-1}$ | $\left[4J(J+1)\right]^{-1}$ | $\left[4J\sqrt{(2J-1)(2J+1)}\right]^{-1}$
$\phi_{Z}(J,M,J^{\prime},M)$ | $2\sqrt{(J+1)^{2}-M^{2}}$ | $2M$ | $2\sqrt{J^{2}-M^{2}}$
$(\phi_{X}{\rm\,\,or\pm\phi_{Y}})(J,M,J^{\prime},M\pm 1)$ | $\mp\sqrt{(J\pm M+1)(J\pm M+2)}$ | $\sqrt{(J\mp M)(J\pm M+1)}$ | $\pm\sqrt{(J\mp M+1)(J\mp M-1)}$
$\phi_{\rm rms}(J,J^{\prime})^{2}$ | $\frac{4}{3}(J+1)(2J+3)$ | $\frac{4}{3}J(J+1)$ | $\frac{4}{3}J(2J-1)$
Table 2: Ratio of double resonance signals for parallel over perpendicular relative linear polarization for unsaturated double resonance. J is the rotational total angular momentum quantum number for the state common to the two transitions  ladder-type | R then R | R then Q | R then P | Q then R | Q then Q | Q then P | P then R | P then Q | P then P
---|---|---|---|---|---|---|---|---|---
V-type | P then R | P then Q | P then P | Q then R | Q then Q | Q then P | R then R | R then Q | R then P
$\Lambda$-type | R then P | R then Q | R then R | Q then P | Q then Q | Q then R | P then P | P then Q | P then R
$J_{1}=$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}$ | $J_{2}$ | $J_{2}$ | $J_{2}+1$ | $J_{2}+1$ | $J_{2}+1$
$J_{3}=$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$
$J_{2}$ | $\frac{4}{3}$ | $\frac{2(J_{2}-1)}{4J_{2}+1}$ | $\frac{8J_{2}^{2}+2}{(J_{2}-1)(6J_{2}+1)}$ | $\frac{2J_{2}+4}{4J_{2}+3}$ | $\frac{6J_{2}^{2}+6J_{2}-2}{2J_{2}^{2}+2J_{2}+1}$ | $\frac{2J_{2}-2}{4J_{2}+1}$ | $\frac{8J_{2}^{2}+16J_{2}+10}{(J_{2}+2)(6J_{2}+5)}$ | $\frac{2J_{2}+4}{4J_{2}+3}$ | $\frac{4}{3}$
0 |  |  |  |  |  |  | 1. |  |
1 | 1.3333 | 0. | $\infty$ | 0.8571 | 2. | 0. | 1.0303 | 0.8571 | 1.3333
2 | 1.3333 | 0.2222 | 2.6154 | 0.7273 | 2.6154 | 0.2222 | 1.0882 | 0.7273 | 1.3333
3 | 1.3333 | 0.3077 | 1.9474 | 0.6667 | 2.8000 | 0.3077 | 1.1304 | 0.6667 | 1.3333
4 | 1.3333 | 0.3529 | 1.7333 | 0.6316 | 2.8780 | 0.3529 | 1.1609 | 0.6316 | 1.3333
5 | 1.3333 | 0.3810 | 1.6290 | 0.6087 | 2.9180 | 0.3810 | 1.1837 | 0.6087 | 1.3333
6 | 1.3333 | 0.4000 | 1.5676 | 0.5926 | 2.9412 | 0.4000 | 1.2012 | 0.5926 | 1.3333
7 | 1.3333 | 0.4138 | 1.5271 | 0.5806 | 2.9558 | 0.4138 | 1.2151 | 0.5806 | 1.3333
8 | 1.3333 | 0.4242 | 1.4985 | 0.5714 | 2.9655 | 0.4242 | 1.2264 | 0.5714 | 1.3333
9 | 1.3333 | 0.4324 | 1.4773 | 0.5641 | 2.9724 | 0.4324 | 1.2357 | 0.5641 | 1.3333
10 | 1.3333 | 0.4390 | 1.4608 | 0.5581 | 2.9774 | 0.4390 | 1.2436 | 0.5581 | 1.3333

In the unsaturated limit, the polarization ratios are the same for
homogeneously and inhomogeneously broadened cases. We are not aware in the
literature of explicit expressions for the predicted DR polarization intensity
ratio, $I_{\parallel}/I_{\perp}$, for all cases. These expressions in Table 2
can be derived from those previously given in Table 5.1 of the text Angular
Momentum by Richard Zare,ZareAM  which gives the degree of polarization for
fluorescence, which is analogous to the $\Lambda$-type DR scheme. Zare gave
expressions for the polarization anisotropy defined as
$P=(I_{\parallel}-I_{\perp})/(I_{\parallel}+I_{\perp})$. The expressions for
$P$ were converted by using $I_{\parallel}/I_{\perp}=(P+1)/(P-1)$ and found to
agree with those given in Table 2, after correcting for the fact that Zare
used what we have written as $J_{1}$ for his expressions and we have used
$J_{2}$, as that allows all three DR schemes to be combined. If the sample is
optically thick for the pump, the polarization ratio, $R_{\rm lin}$, will not
change. If the probe is optically thick, then $R_{\rm lin}$ will give the
ratio of the change in the probe field absorbance induced by the pump field.

It is also possible to do the polarization measurement using circular
polarization for the pump and probe fields.Schwendeman99  In this case, we
assign the $Z$ axis to the direction of propagation of the pump field and that
(or it’s inverse) of the probe field. The pump and probe fields will be
assumed to be circularly polarized with positive or negative helicity and the
propagation direction, $\vec{k}$. The nonzero direction cosine matrix elements
are $\phi_{\pm}(J,M,J^{\prime},M\pm 1)=(\phi_{X}(J,M,J^{\prime},M\pm 1)\pm
i\phi_{Y}(J,M,J^{\prime},M\pm
1))/\sqrt{2}=\sqrt{2}\phi_{X}(J,M,J^{\prime},M\pm 1)$. The positive sign is
used for absorption from a wave of positive helicity or stimulated emission
from a wave of negative helicity. If the wave propagates with $\vec{k}$
antiparallel to the $Z$ axis, the signs are switched. We compare the predicted
DR signal strength for the cases where the pump and probe co-rotate together
or counter-rotate, with signal strengths $I_{\rm same}$ and $I_{\rm opposite}$
respectively. Note that if the pump and probe waves have parallel $\vec{k}$,
the waves have the same or opposite helicities for co-rotating and counter-
rotating respectively. However, if they have antiparallel $\vec{k}$, then co-
and counter rotation corresponds to the helicities being opposite or the same
respectively. The helicity of a wave is reversed upon reflection at normal
incidence from a mirror, so one can use a double pass or even a linear
enhancement cavity to increase the DR signal strength assuming the mirrors
have negligible birefringence and dichroism. Using the direction cosine matrix
elements, one can evaluate the polarization ratios for ladder- and V-type DR
as

$R^{\rm us}_{\rm cir}=\frac{I_{\rm same}}{I_{\rm
opposite}}=\frac{\sum_{M=-J_{2}}^{J_{2}}\phi_{X}(J_{1},M-1,J_{2}.M)^{2}\phi_{X}(J_{2},M,J_{3},M+1)^{2}}{\sum_{M=-J_{2}}^{J_{2}}\phi_{X}(J_{1},M-1,J_{2},M)^{2}\phi_{X}(J_{2},M,J_{3},M-1)^{2}}$
(9)

As the change in the $M$ quantum number for stimulate emission is opposite
that for absorption for fixed helicity, $R^{\rm us}_{\rm cir}$ for
$\Lambda$-type DR will be the inverse of that given in Eq. 9. Table 3 gives
the predicted non-saturated DR circular polarization ratios. It is evident
that the polarization effects are generally larger for the comparison of
circular vs linear polarization DR signals. In particular, for linear
polarization, the predicted ratios for probe transitions with $\Delta J=\pm 1$
approach each other as $J$ grows, making discrimination difficult. For
circular polarization, the two $\Delta J=\pm 1$ probe transitions are most
easily distinguished.

Zare’s text gives the degree of circular polarization, $C(J_{1})=(I_{\rm
same}-I_{\rm opposite})/I_{\rm same}+I_{\rm opposite})$ for fluorescence in
Table 5.2. Correcting for the inversion for emission vs absorption (
equivalently, changing the sign of $C(J_{1})$) his expressions can be
converted to those given in Table 3 and are found to agree with the exception
of the case he labels $(R\uparrow\,Q\downarrow)$. However, recalculating that
$C(J_{1})$ value using his eq. 5.124 shows that there was a sign error in his
printed table for that entry. With this correction, the two expressions are in
agreement in that case as well.

Table 3: Double resonance signal polarization ratios using circularly polarized radiation for unsaturated transitions. For ladder- and V-type double resonance, what is tabulated is same helicity / opposite helicity. For $\Lambda$-type DR, what is tabulated is opposite helicity / same helicity. J is the rotational total angular momentum quantum number for the state common to the two transitions  ladder-type | R then R | R then Q | R then P | Q then R | Q then Q | Q then P | P then R | P then Q | P then P
---|---|---|---|---|---|---|---|---|---
V-type | P then R | P then Q | P then P | Q then R | Q then Q | Q then P | R then R | R then Q | R then P
$\Lambda$-type | R then P | R then Q | R then R | Q then P | Q then Q | Q then R | P then P | P then Q | P then R
$J_{1}=$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}$ | $J_{2}$ | $J_{2}$ | $J_{2}+1$ | $J_{2}+1$ | $J_{2}+1$
$J_{3}=$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$
$J_{2}$ | 6 | $\frac{3J_{2}-3}{3J_{2}+2}$ | $\frac{(J_{2}-1)(2J_{2}-3)}{12J_{2}^{2}-2}$ | $\frac{3J_{2}+6}{3J_{2}+1}$ | $\frac{(2J_{2}+3)(2J_{2}-1)}{4J_{2}^{2}+4J_{2}+2}$ | $\frac{3J_{2}-3}{3J_{2}+2}$ | $\frac{(J_{2}+2)(2J_{2}+5)}{12J_{2}^{2}+24J_{2}+10}$ | $\frac{3J_{2}+6}{3J_{2}+1}$ | 6.
0 |  |  |  |  |  |  | 1. |  |
1 | 6 | 0. | 0. | 2.2500 | 0.5 | 0. | 0.4565 | 2.2500 | 6
2 | 6 | 0.3750 | 0.0217 | 1.7143 | 0.8077 | 0.3750 | 0.3396 | 1.7143 | 6
3 | 6 | 0.5455 | 0.0566 | 1.5000 | 0.9000 | 0.5455 | 0.2895 | 1.5000 | 6
4 | 6 | 0.6429 | 0.0789 | 1.3846 | 0.9390 | 0.6429 | 0.2617 | 1.3846 | 6
5 | 6 | 0.7059 | 0.0940 | 1.3125 | 0.9590 | 0.7059 | 0.2442 | 1.3125 | 6
6 | 6 | 0.7500 | 0.1047 | 1.2632 | 0.9706 | 0.7500 | 0.2321 | 1.2632 | 6
7 | 6 | 0.7826 | 0.1126 | 1.2273 | 0.9779 | 0.7826 | 0.2232 | 1.2273 | 6
8 | 6 | 0.8077 | 0.1188 | 1.2000 | 0.9828 | 0.8077 | 0.2165 | 1.2000 | 6
9 | 6 | 0.8276 | 0.1237 | 1.1786 | 0.9862 | 0.8276 | 0.2112 | 1.1786 | 6
10 | 6 | 0.8438 | 0.1277 | 1.1613 | 0.9887 | 0.8438 | 0.2069 | 1.1613 | 6

###  I.2 Saturated Inhomogeneously broadened Pump Transitions

In steady-state, saturation of the pump and/or probe transitions will
generally reduce the polarization effects as it will reduce the degree of
alignment produced by the pump beam and/or reduce the impact of alignment on
the probe absorption. We will only consider the case with saturation of the
pump transition. We start by considering the case where the inhomogeneous
Doppler width is the dominant broadening. This is the common situation in
continuous wave excitation experiments due to the fact it is difficult to
realize a Rabi excitation frequency greater than the Doppler width of the
transition, at least for thermal samples at near ambient temperatures or
above. In the limit of a strongly saturating pump wave,
$\Omega_{12}^{2}(M)>>\gamma_{1}\gamma_{2}$, for all nonzero values of
$\mu_{12}(M)$. We again will assume that the inhomogeneous Doppler width
dominates over $\Omega_{12},\gamma_{1}$ and $\gamma_{2}$. Returning to Eq. 3
and taking this limit, we have for the total change in population in state 2:

$\Delta\rho_{22}(M)\rightarrow\frac{\pi}{2(2J_{2}+1)}(\rho_{11}^{e}-\rho_{22}^{e})g_{\rm
D}(\omega_{a}-\omega_{12})|\Omega_{12}(M)|\sqrt{\gamma_{2}/\gamma_{1})}$ (10)

and the population transferred to level $2$ is proportional to the
$|\Omega_{12}(M)|$ and thus the square root of the pump intensity. We thus get
the following sums for DR transitions $J_{1},K_{1}\rightarrow
J_{2},K_{2},\rightarrow J_{3},K_{3}$ of symmetric tops

$\displaystyle\sum_{M=-J_{2}}^{J_{2}}\mu_{23}(M)^{2}|\mu_{12}(M)|$
$\displaystyle=|\mu_{12}|\,\mu_{23}^{2}\,\phi_{J}(J_{1},J_{2})|\phi_{g}(J_{1},K_{1},J_{2},K_{2})|\phi_{J}(J_{2},J_{3})^{2}\phi_{g^{\prime}}(J_{2},K_{2},J_{3},K_{3})^{2}\times$
(11)
$\displaystyle\sum_{M=-J_{2}}^{J_{2}}|\phi_{Z}(J_{1},M,J_{2},M)|\sum_{M^{\prime}}\phi_{G}(J_{2},M,J_{3},M^{\prime})^{2}$

As above, the asymmetric top case can be found by replacing the terms
$|\phi_{g}(J_{i},K_{i},J_{j},K_{j})|$ by
$|\sum_{K_{i},K_{j}}A(i,J_{i},\tau_{i},K_{i},)A(j,J_{j},\tau_{j},K_{j})|^{2}\phi_{g}(J_{i},K_{i},J_{j},K_{j})|$.
These results leads to the expressions, in the limit of strong pump
saturation,

$I^{\rm
sat}_{G}=\frac{\pi^{2}\omega_{23}}{2(J_{2}+1)(\epsilon_{0}c)^{3/2}\hbar^{2}}(\rho_{11}^{e}-\rho_{22}^{e})g_{\rm
D}(\omega-\omega_{12})\sqrt{\gamma_{2}/\gamma_{1})}\sqrt{2I_{p}}\cdot\sum_{M=-J_{2}}^{J_{2}}\mu_{23}(M)^{2}|\mu_{12}(M)|$
(12)

and

$R^{\rm sat}_{\rm
lin}=\frac{I_{\parallel}}{I_{\perp}}=\frac{\sum_{M=-J_{2}}^{J_{2}}\phi_{Z}(J_{1},M,J_{2}.M)\phi_{Z}(J_{2},M,J_{3},M)^{2}}{\sum_{M=-J_{2}}^{J_{2}}\phi_{Z}(J_{1},M,J_{2}.M)(\phi_{X}(J_{2},M,J_{3},M-1)^{2}+\phi_{X}(J_{2},M,J_{3},M+1)^{2})}$
(13) $R^{\rm sat}_{\rm cir}=\frac{I_{\rm same}}{I_{\rm
opposite}}=\frac{\sum_{M=-J_{2}}^{J_{2}}\phi_{X}(J_{1},M-1,J_{2}.M)\phi_{X}(J_{2},M,J_{3},M+1)^{2}}{\sum_{M=-J_{2}}^{J_{2}}\phi_{X}(J_{1},M-1,J_{2},M)\phi_{X}(J_{2},M,J_{3},M-1)^{2}}$
(14)

Due the fact that most of the factors $|\phi_{Z}(J_{1},M,J_{2},M)|$ and
$|\phi_{X}(J_{1},M,J_{2},M+1)|$ are square roots of polynomials in $J_{2}$ and
$M$, the sums over $M$ values do not lead to compact expressions in the
saturated case. The exceptions are for Q pump ratios for linear polarization.
However, it is straightforward to numerically calculate the relevant sums for
any particular values of $J_{1},J_{2}$, and $J_{3}$. This has been done and
the results for linear and circular polarization are presented in tables 4 and
5. In the comparison of the tables for unsaturated and saturated pump
conditions, it is evident that, while pump saturation reduces the polarization
effects, this reduction is modest and polarization can be used to
unambiguously assign transitions. The polarization ratios are independent of
the optical depth of the pump transition as long as the steady-state
excitation remains strongly saturated through the sample.

Table 4: Ratio of double resonance signals for parallel over perpendicular relative linear polarization for strongly saturated inhomogeneously broadened pump transitions but unsaturated probe transitions. J is the rotational total angular momentum quantum number for the state common to the two transitions. The first transition listed is the pump’ and the 2nd the probe.  ladder-type | R then R | R then Q | R then P | Q then R | Q then Q | Q then P | P then R | P then Q | P then P
---|---|---|---|---|---|---|---|---|---
V-type | P then R | P then Q | P then P | Q then R | Q then Q | Q then P | R then R | R then Q | R then P
$\Lambda$-type | R then P | R then Q | R then R | Q then P | Q then Q | Q then R | P then P | P then Q | P then R
$J_{1}=$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}$ | $J_{2}$ | $J_{2}$ | $J_{2}+1$ | $J_{2}+1$ | $J_{2}+1$
$J_{3}=$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$
$J_{2}$ |  |  |  | $\frac{2J_{2}+4}{3J_{2}+4}$ | 2. | $\frac{2J_{2}-2}{3J_{2}-1}$ |  |  |
0 |  |  |  |  |  |  | 1. |  |
1 | 1.3333 | 0. | $\infty$ | 0.8571 | 2. | 0. | 1.0148 | 0.9282 | 1.1547
2 | 1.3244 | 0.2363 | 2.5559 | 0.8000 | 2. | 0.4 | 1.0449 | 0.8527 | 1.1634
3 | 1.3038 | 0.3514 | 1.8494 | 0.7692 | 2. | 0.5 | 1.0682 | 0.8125 | 1.1690
4 | 1.2868 | 0.4194 | 1.6207 | 0.7500 | 2. | 0.5455 | 1.0856 | 0.7873 | 1.1703
5 | 1.2739 | 0.4640 | 1.5089 | 0.7368 | 2. | 0.5714 | 1.0990 | 0.7698 | 1.1759
6 | 1.2640 | 0.4954 | 1.4431 | 0.7273 | 2. | 0.5882 | 1.1096 | 0.7570 | 1.1783
7 | 1.2562 | 0.5187 | 1.3998 | 0.7200 | 2. | 0.6000 | 1.1182 | 0.7471 | 1.1802
8 | 1.2500 | 0.5365 | 1.3694 | 0.7143 | 2. | 0.6087 | 1.1253 | 0.7392 | 1.1817
9 | 1.2450 | 0.5507 | 1.3468 | 0.7097 | 2. | 0.6154 | 1.1312 | 0.7328 | 1.1830
10 | 1.2408 | 0.5621 | 1.3294 | 0.7059 | 2. | 0.6207 | 1.1362 | 0.7275 | 1.1842
Table 5: Double resonance signal polarization ratios using circularly polarized radiation for strongly saturated inhomogeneously broadened pump transitions but unsaturated probe transitions. For ladder- and V-type Double resonance, what is tabulated is the same helicity / opposite helicity. For $\Lambda$-type DR, what is tabulated is opposite helicity / same helicity. J is the rotational total angular momentum quantum number for the state common to the two transitions. The first transition listed is the pump; the 2nd the probe.  ladder-type | R then R | R then Q | R then P | Q then R | Q then Q | Q then P | P then R | P then Q | P then P
---|---|---|---|---|---|---|---|---|---
V-type | P then R | P then Q | P then P | Q then R | Q then Q | Q then P | R then R | R then Q | R then P
$\Lambda$-type | R then P | R then Q | R then R | Q then P | Q then Q | Q then R | P then P | P then Q | P then R
$J_{1}=$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}$ | $J_{2}$ | $J_{2}$ | $J_{2}+1$ | $J_{2}+1$ | $J_{2}+1$
$J_{3}=$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$
$J_{2}=0$ |  |  |  |  |  |  | 1. |  |
1 | 6. | 0. | 0. | 2.25 | 0.5 | 0. | 0.6531 | 1.5306 | 2.4495
2 | 4.4016 | 0.4936 | 0.0479 | 1.7071 | 0.8040 | 0.3876 | 0.5459 | 1.3439 | 2.5777
3 | 3.8908 | 0.6628 | 0.1234 | 1.4914 | 0.8967 | 0.5582 | 0.4930 | 1.2549 | 2.6562
4 | 3.6454 | 0.7477 | 0.1701 | 1.3761 | 0.9365 | 0.6542 | 0.4614 | 1.2028 | 2.7095
5 | 3.5030 | 0.7985 | 0.2005 | 1.3046 | 0.9571 | 0.7159 | 0.4403 | 1.1684 | 2.7482
6 | 3.4106 | 0.8323 | 0.2216 | 1.2559 | 0.9691 | 0.7589 | 0.4252 | 1.1440 | 2.7776
7 | 3.3461 | 0.8564 | 0.2371 | 1.2206 | 0.9767 | 0.7906 | 0.4139 | 1.1258 | 2.8008
8 | 3.2986 | 0.8745 | 0.2489 | 1.1938 | 0.9818 | 0.8149 | 0.4051 | 1.1117 | 2.8195
9 | 3.2623 | 0.8885 | 0.2581 | 1.1729 | 0.9854 | 0.8341 | 0.3980 | 1.1004 | 2.8350
10 | 3.2337 | 0.8997 | 0.2656 | 1.1560 | 0.9880 | 0.8497 | 0.3922 | 1.0912 | 2.8480

###  I.3  Comparison with Experimental Data

The author and collaborators have performed IR-IR double resonance experiments
in the spectral range of the CH4 ground $\rightarrow\nu_{3}\rightarrow
3\nu_{3}$. A CW 3.3 $\mu$m Optical Parametric Oscillator was used for the
pump, and a 1.65 $\mu$m centered frequency comb as the probe. The pump was
strongly saturated with the pump Rabi frequency about an order of magnitude
larger than the collisional dephasing rate. A preliminary experiment using a
single pass, liquid N2 cooled cell has been published.Foltynowicz21a ;
Foltynowicz21b  The method has greatly improved sensitivity by using a finesse
optical cavity for the probe radiation.deoliveira2023subdoppler  Table 6
presents a set of measured double resonance transitions from this latter work,
each measured with both parallel and perpendicular relative polarization of
the pump and probe waves. The probe $\Delta J$ values were assigned based on
combination differences and also by comparison with highly accurate
theoretical predictions.Rey18  The observed and calculated polarization ratios
have significant quantitative differences, with observed values systematically
closer to unity. Such a bias towards unity can be expected from errors in the
relative polarization state of the pump and probe lasers; as that experimental
setup has been replaced, we will not now speculate on the specific reason(s)
for the deviation of the experimental polarization ratios from the predicated
values. Despite this, the saturated wave predictions are generally closer to
the observed values. This demonstrates that even with strong saturation of the
pump transitions, which optimizes detection sensitivity, polarization ratios
can still be used to unambiguously assign the $\Delta J$ for the observed DR
transitions.

Table 6: Comparison of observed and predicted polarization intensity ratios for Methane ground state $\rightarrow\,\,\nu_{3}\,\,\rightarrow 3\nu_{3}$ double resonance intensity ratios. Experimental values taken from de Oliveria et al.deoliveira2023subdoppler  The probe intensity is the integrated probe absorption of the sub-Doppler feature in units of 10-9 cm-2. Pump | Probe | Wavenumber | Final Term Value | Probe | Polarization | Saturation | Unsaturated
---|---|---|---|---|---|---|---
trans | trans | cm-1 | cm-1 | Intensity | Ratio | Prediction | Prediction
P(2F2) | R(1) | 5948.267590(3) | 8978.704010(3) | 2.26(5) | 0.91(4) | 1.0148 | 1.0303
P(2F2) | R(1) | 5964.06227(2) | 8994.49869(2) | 0.081(3) | 1.01(7) | 1.0148 | 1.0303
P(2F2) | R(1) | 5979.042972(3) | 9009.479392(3) | 0.56(1) | 0.98(4) | 1.0148 | 1.0303
Q(2F2) | Q(2F1) | 5928.61142(2) | 8978.70401(2) | 0.56(3) | 1.50(10) | 2.0000 | 2.6154
Q(2F2) | Q(2F1) | 5944.40608(2) | 8994.49868(2) | 0.103(3) | 1.55(6) | 2.0000 | 2.6154
Q(2F2) | R(2F1) | 5958.673574(6) | 9008.766169(6) | 2.38(6) | 0.83(5) | 0.8000 | 0.7273
Q(2F2) | Q(2F1) | 5959.386797(5) | 9009.479392(5) | 0.89(2) | 1.84(7) | 2.0000 | 2.6154
R(2F2) | R(3F1) | 5913.18732(2) | 8992.78303(2) | 0.062(4) | 1.1(1) | 1.3038 | 1.3333
R(2F2) | R(3F1) | 5918.14141(1) | 8997.73712(1) | 0.096(4) | 1.05(7) | 1.3038 | 1.3333
R(2F2) | R(3F1) | 5923.94848(1) | 9003.54418(1) | 0.44(2) | 1.19(6) | 1.3038 | 1.3333
R(2F2) | R(3F1) | 5924.26536(2) | 9003.86107(2) | 0.175(5) | 1.10(8) | 1.3038 | 1.3333
R(2F2) | Q(3F1) | 5929.170466(3) | 9008.766172(3) | 4.7(1) | 0.44(7) | 0.3514 | 0.3077
R(2F2) | P(3F1) | 5929.883687(4) | 9009.479393(4) | 0.85(2) | 1.61(4) | 1.8494 | 1.7333
R(2F2) | R(3F1) | 5932.279186(9) | 9011.874892(9) | 0.100(3) | 1.21(9) | 1.3038 | 1.3333
R(2F2) | R(3F1) | 5935.245195(3) | 9014.840901(3) | 0.74(2) | 1.36(5) | 1.3038 | 1.3333

###  I.4  Double Resonance Polarization Ratios for Saturated and Homogeneously
Broadened pump transitions

For the sake of completeness, we now consider the saturation of saturated and
homogeneously dominated broadened pump transitions. As mentioned above, for
unsaturated transitions, the same polarization ratios apply to the homogeneous
and inhomogeneously broadened cases. In the homogeneously broadened case, the
steady-state $\Delta\rho_{22}(M)$ is proportional to $x(M)/2(1+x(M))$ with
$x(M)=S\,\phi_{Z}(J_{1},M,J_{2},M)^{2}/\phi_{\rm rms}(J_{1},J_{2})^{2}$ in the
linear pump polarization case and
$x=2\,S\,\phi_{X}(J_{1},M,J_{2},M)^{2}/\phi_{\rm rms}(J_{1},J_{2})^{2}$ for
circular pump polarization. $S$ is the saturation parameter which equals the
ratio of the pump rate neglecting saturation divided by the population
relaxation rate and $\phi_{\rm
rms}(J_{1},J_{2})^{2}=\sum_{M=-J_{2}}^{J_{2}}\phi_{Z}(J_{1},M,J_{2},M)^{2}/(2J_{2}+1)=2\sum_{M=-J_{2}}^{J_{2}}\phi_{X}(J_{1},M-1,J_{2},M)^{2}/(2J_{2}+1)$.
Expressions for $\phi_{\rm rms}$ are given in Table 1.

Even in the limit of highly saturating pump intensity, the polarization ratios
do not go to unity except for a $J_{1}=J_{2}+1$ pump transition as the
selection rules for the $J_{1}=J_{2}$ or $J_{2}-1$ cases prevent pumping all
$M$ values of state $2$. Below, in tables 7 and 8 are listed the
$S\rightarrow\infty$ polarization ratios values for different cases. Note that
the polarizations ratios in the limit of infinite saturation $\rightarrow 1$
as $J\rightarrow\infty$ in all cases, as in that limit, the non-pumped $M$
values are a negligible fraction of the total. Figures [1-3] plot the linear
polarization ratios as a function of $S$ for the three probe transitions when
a homogeneously broadened $R(5),P(5)$, and $Q(5)$ pump is used. Figures [4-6]
plot the circular polarization ratios for the same pump transitions. Table 9
and 10 report the values of the saturation parameter, $S$, that results in a
polarization ratio halfway between the unsaturated and $S\rightarrow\infty$
values for the linear and circular polarization DR experiments respectively,
Note that the $J_{2}=1,J_{1}=J_{3}=0$ entry is empty for the linear
polarization case as the polarization ratio, in that case, is $\infty$
regardless of $S$

Table 7: Polarization DR signal ratios for parallel over perpendicularly polarized waves for homogeneously broadened pump transition in the limit of saturation parameter $\rightarrow\infty$. $J$ is the total angular momentum quantum number of the state common in both transitions.  | $J_{2}-J_{1}=1$ | $J_{2}-J_{1}=0$ | $J_{2}-J_{1}=-1$
---|---|---|---
$J_{3}-J_{2}=1$ | $\frac{(2J_{2}+1)(J_{2}+3)}{2J_{2}^{2}+4J_{2}+3}$ | $\frac{8J_{2}+10}{8J_{2}+13}$ | 1
$J_{3}-J_{2}=0$ | $\frac{J_{2}-1}{J_{2}+2}$ | $\frac{4J_{2}+2}{4J_{2}-1}$ | 1
$J_{3}-J_{2}=-1$ | $\frac{J_{2}+1}{2J_{2}-2}$ | $\frac{2(J_{2}-1)(4J_{2}+1)}{8J_{2}^{2}-3J_{2}+1}$ | 1
Table 8: Polarization DR signal ratios for V- and ladder-type for co- over counter-rotating waves for Homogeneously broadened pump transition in the limit of saturation parameter $\rightarrow\infty$. $J$ is the total angular momentum quantum number of the state common in both transitions. For $\Lambda$-type DR, the ratio should be inverted. | $J_{2}-J_{1}=1$ | $J_{2}-J_{1}=0$ | $J_{2}-J_{1}=-1$
---|---|---|---
$J_{3}-J_{2}=1$ | $\frac{2J_{2}^{2}+7J_{2}+9}{(J_{2}(2J_{2}+1)}$ | $\frac{4J_{2}^{2}+12J_{2}+11}{(2J_{2}+1)(2J_{2}+1)}$ | 1
$J_{3}-J_{2}=0$ | $\frac{(J_{2}-1)(J_{2}+3)}{J_{2}(J_{2}+2)}$ | $\frac{(J_{2}+2)(2J_{2}-1)}{(J_{2}+1)(2J_{2}+1)}$ | 1
$J_{3}-J_{2}=-1$ | $\frac{(J_{2}-1)(2J_{2}-3)}{J_{2}(2J_{2}+1)}$ | $\frac{(2J_{2}-2)}{(2J_{2}+1)}$ | 1
Table 9:  Pump saturation parameter, $S$, required for the linear polarization ratio to be halfway between unsaturated and $S\rightarrow\infty$ limit  ladder-type | R then R | R then Q | R then P | Q then R | Q then Q | Q then P | P then R | P then Q | P then P
---|---|---|---|---|---|---|---|---|---
V-type | P then R | P then Q | P then P | Q then R | Q then Q | Q then P | R then R | R then Q | R then P
$\Lambda$-type | R then P | R then Q | R then R | Q then P | Q then Q | Q then R | P then P | P then Q | P then R
$J_{1}=$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}$ | $J_{2}$ | $J_{2}$ | $J_{2}+1$ | $J_{2}+1$ | $J_{2}+1$
$J_{3}=$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$
$J_{2}=1$ | 2.0000 | 2.0000 |  | 1.9541 | 2.0000 | 2.0000 | 0.5502 | 0.5835 | 1.2506
2 | 0.9213 | 0.9377 | 0.9030 | 1.3278 | 0.9288 | 1.5002 | 0.7924 | 0.8975 | 1.3969
3 | 1.1211 | 1.1877 | 1.0847 | 1.7102 | 1.0027 | 1.9302 | 0.9743 | 1.1458 | 1.5106
4 | 1.2688 | 1.3904 | 1.2250 | 1.9762 | 1.0431 | 2.206 | 1.1173 | 1.3474 | 1.6015
5 | 1.3840 | 1.5583 | 1.3374 | 2.1697 | 1.0703 | 2.3947 | 1.2334 | 1.515 | 1.676
6 | 1.4772 | 1.7001 | 1.4300 | 2.3167 | 1.0904 | 2.5315 | 1.3299 | 1.6568 | 1.7381
7 | 1.5544 | 1.8216 | 1.5076 | 2.4321 | 1.1062 | 2.6353 | 1.4116 | 1.7787 | 1.7908
8 | 1.6197 | 1.9270 | 1.5739 | 2.5253 | 1.1189 | 2.7167 | 1.4817 | 1.8847 | 1.8361
9 | 1.6758 | 2.0195 | 1.6312 | 2.6022 | 1.1294 | 2.7824 | 1.5427 | 1.9779 | 1.8755
10 | 1.7245 | 2.1014 | 1.6812 | 2.6666 | 1.1382 | 2.8366 | 1.5962 | 2.0605 | 1.9100
Table 10:  Pump saturation parameter, $S$, required for the circular polarization ratio to be halfway between unsaturated and $S\rightarrow\infty$ limit  ladder-type | R then R | R then Q | R then P | Q then R | Q then Q | Q then P | P then R | P then Q | P then P
---|---|---|---|---|---|---|---|---|---
V-type | P then R | P then Q | P then P | Q then R | Q then Q | Q then P | R then R | R then Q | R then P
$\Lambda$-type | R then P | R then Q | R then R | Q then P | Q then Q | Q then R | P then P | P then Q | P then R
$J_{1}=$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}-1$ | $J_{2}$ | $J_{2}$ | $J_{2}$ | $J_{2}+1$ | $J_{2}+1$ | $J_{2}+1$
$J_{3}=$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$ | $J_{2}+1$ | $J_{2}$ | $J_{2}-1$
$J_{2}=1$ | 2.0000 | 2.0000 | 2.0000 | 1.9541 | 2.0000 | 2.0000 | 0.5502 | 0.5835 | 1.2506
2 | 0.9213 | 0.9377 | 0.9030 | 1.3278 | 0.9288 | 1.5002 | 0.7924 | 0.8975 | 1.3969
3 | 1.1211 | 1.1877 | 1.0847 | 1.7102 | 1.0027 | 1.9302 | 0.9743 | 1.1458 | 1.5106
4 | 1.2688 | 1.3904 | 1.2250 | 1.9762 | 1.0431 | 2.2060 | 1.1173 | 1.3474 | 1.6015
5 | 1.3840 | 1.5583 | 1.3374 | 2.1697 | 1.0703 | 2.3947 | 1.2334 | 1.5150 | 1.6760
6 | 1.4772 | 1.7001 | 1.4300 | 2.3167 | 1.0904 | 2.5315 | 1.3299 | 1.6568 | 1.7381
7 | 1.5544 | 1.8216 | 1.5076 | 2.4321 | 1.1062 | 2.6353 | 1.4116 | 1.7787 | 1.7908
8 | 1.6197 | 1.927 | 1.5739 | 2.5253 | 1.1189 | 2.7167 | 1.4817 | 1.8847 | 1.8361
9 | 1.6758 | 2.0195 | 1.6312 | 2.6022 | 1.1294 | 2.7824 | 1.5427 | 1.9779 | 1.8755
10 | 1.7245 | 2.1014 | 1.6812 | 2.6666 | 1.1382 | 2.8366 | 1.5962 | 2.0605 | 1.9100

##  II Summary and Conclusions

This work has presented expressions that allow the prediction of changes in
signal strength as a function of relative pump and probe polarization
applicable to a wide range of DR experiments performed with the pump
population transfer in the steady-state limit. It is found that, even in the
case of a strongly saturated pump field, most of the polarization anisotropy
remains in the case of an inhomogeneously broadened pump transition due to the
different power broadened widths of different $M$ projection states. This
allows polarization ratios to be used to unambiguously assign the $\Delta J$
values for the probe transitions. Combined with the assignment of the pump
transition, this allows the final state term value, symmetry, and total
angular momentum quantum numbers to be determined for the terminal state of
each observed probe transition. Even in the homogeneously and strongly
saturated case, polarization effects remain for low to modest $J$ values due
to the fact that not all possible $M$ values can be pumped.

##  III Acknowledgements

Kevin K. Lehmann recognizes Steven L.Coy for conversations on this topic over
many years and his and Aleksandra Foltynowicz’s encouragement to publish these
results. He also acknowledges support for the US National Science Foundation.

## References

  * (1) T. Carroll and G. Wolga, IEEE Journal of Quantum Electronics 2, 456 (1966).
  * (2) C. J. Pursell, D. P. Weliky, and T. Oka, The Journal of Chemical Physics 93, 7041 (1990).
  * (3) S. Mandal and P. N. Ghosh, Phys. Rev. A 47, 4934 (1993).
  * (4) R. Teets, R. Feinberg, T. W. Hansch, and A. L. Schawlow, Physical Review Letters 37, 683 (1976).
  * (5) S. L. Coy, K. K. Lehmann, and F. C. Delucia, Journal of Chemical Physics 85, 4297 (1986).
  * (6) S. L. Coy and K. K. Lehmann, Journal of Chemical Physics 84, 5239 (1986).
  * (7) K. K. Lehmann and S. L. Coy, Berichte Der Bunsen-Gesellschaft-Physical Chemistry Chemical Physics 92, 306 (1988).
  * (8) S. L. Coy and K. K. Lehmann, Spectrochimica Acta Part a-Molecular and Biomolecular Spectroscopy 45, 47 (1989).
  * (9) K. K. Lehmann and S. L. Coy, Journal of the Chemical Society, Faraday Transactions 2 84, 1389 (1988).
  * (10) J. E. Gambogi, E. R. T. Kerstel, K. K. Lehmann, and G. Scoles, Journal of Chemical Physics 100, 2612 (1994).
  * (11) J. E. Gambogi et al., Journal of Molecular Spectroscopy 175, 198 (1996).
  * (12) M. H. Kabir et al., Journal of Chemical Physics 119, 3691 (2003).
  * (13) R. Zhao, I. M. Konen, and R. N. Zare, The Journal of Chemical Physics 121, 9938 (2004).
  * (14) G. B. Park et al., Journal of Chemical Physics 142, 12 (2015).
  * (15) R. Z. Martinez, K. K. Lehmann, and S. Carter, J Chem Phys 120, 691 (2004).
  * (16) T. B. Settersten, R. L. Farrow, and J. A. Gray, Chemical Physics Letters 370, 204 (2003).
  * (17) A. Foltynowicz et al., Physical Review Letters 126, 063001 (2021).
  * (18) A. Foltynowicz et al., Physical Review A 103, 022810 (2021).
  * (19) S. A. Henck et al., Journal of Chemical Physics 102, 4783 (1995).
  * (20) S. A. Henck et al., Journal of Chemical Physics 102, 4772 (1995).
  * (21) A. Callegari et al., Journal of Chemical Physics 106, 432 (1997).
  * (22) A. Callegari et al., Journal of Chemical Physics 113, 10583 (2000).
  * (23) A. Callegari et al., Molecular Physics 101, 551 (2003).
  * (24) T. Barnum, Thesis, Massachusetts Institute of Technology, 2020.
  * (25) B. D. Yang, Y. Liu, and J. M. Wang, Optics Communications 474, 126102 (2020).
  * (26) M. Chevalier and A. Demartino, Chemical Physics Letters 135, 446 (1987).
  * (27) D. Frye, H. T. Liou, and H. L. Dai, Chemical Physics Letters 133, 249 (1987).
  * (28) S. Kasahara, H. Ikoma, and H. Kato, Journal of Chemical Physics 100, 63 (1994).
  * (29) S. Kasahara et al., Journal of Chemical Physics 111, 8857 (1999).
  * (30) H. K. Srivastava et al., Review of Scientific Instruments 71, 4032 (2000).
  * (31) A. Dia, M. Abboud, P. J. Nacher, and G. Tastevin, European Physical Journal D 75, 223 (2021).
  * (32) M. Elbel, M. Simon, and H. Welp, Quantum Optics: Journal of the European Optical Society Part B 2, 351 (1990).
  * (33) M. E. Kaminsky, R. T. Hawkins, F. V. Kowalski, and A. L. Schawlow, Physical Review Letters 36, 671 (1976).
  * (34) C. Wieman and T. W. Hansch, Physical Review Letters 36, 1170 (1976).
  * (35) C. H. Greene and R. N. Zare, The Journal of Chemical Physics 78, 6741 (1983).
  * (36) S. L. Coy, S. D. Halle, J. L. Kinsey, and R. W. Field, Journal of Molecular Spectroscopy 153, 340 (1992).
  * (37) J. J. Klaassen, S. L. Coy, J. I. Steinfeld, and C. Roche, Journal of Chemical Physics 100, 5519 (1994).
  * (38) S. L. Coy, S. D. Halle, J. L. Kinsey, and R. W. Field, Journal of Molecular Spectroscopy 169, 292 (1995).
  * (39) U. Y. Shin and R. H. Schwendeman, Journal of Chemical Physics 96, 8699 (1992).
  * (40) G. M. Soriano and R. H. Schwendeman, Journal of Molecular Spectroscopy 190, 294 (1998).
  * (41) B. Abel, N. Lange, F. Reiche, and J. Troe, Journal of Chemical Physics 110, 1389 (1999).
  * (42) S. B. Bayram, S. Kin, M. J. Welsh, and J. D. Hinkle, Physical Review A 73, 042713 (2006).
  * (43) D. Forthomme et al., Journal of Physical Chemistry A 119, 7439 (2015).
  * (44) D. Patterson and J. M. Doyle, Physical Review Letters 111, 023008 (2013).
  * (45) D. Patterson and M. Schnell, Physical Chemistry Chemical Physics 16, 11114 (2014).
  * (46) S. Lobsiger et al., Journal of Physical Chemistry Letters 6, 196 (2015).
  * (47) K. K. Lehmann, in Frontiers and Advances in Molecular Spectroscopy, edited by J. Laane (Elsevier, Amsterdam, 2018), book section 21, pp. 713–734.
  * (48) S. Eibenberger, J. Doyle, and D. Patterson, Physical Review Letters 118, 123002 (2017).
  * (49) A. Nishiyama et al., Optics Express 24, 25894 (2016).
  * (50) D. S. Frankel and J. I. Steinfeld, Journal of Chemical Physics 62, 3358 (1975).
  * (51) Z. S. Li et al., Applied Physics B-Lasers and Optics 79, 135 (2004).
  * (52) A. H. Steeves, H. A. Bechtel, S. L. Coy, and R. W. Field, Journal of Chemical Physics 123, 141102 (2005).
  * (53) D. M. Jonas, X. Yang, and A. M. Wodtke, The Journal of Chemical Physics 97, 2284 (1992).
  * (54) R. Ferber, W. Jastrzebski, and P. Kowalczyk, Journal of Quantitative Spectroscopy & Radiative Transfer 58, 53 (1997).
  * (55) K. Chen and E. S. Yeung, The Journal of Chemical Physics 69, 43 (2008).
  * (56) V. S. Petrovic and R. W. Field, The Journal of Chemical Physics 128, 014301 (2008).
  * (57) F. C. Spano and K. K. Lehmann, Physical Review A 45, 7997 (1992).
  * (58) S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1969).
  * (59) C. Townes and A. Schawlow, Microwave Spectroscopy, 2nd ed. (Dover Books, New York, 2012), p. 720.
  * (60) R. Karplus and J. Schwinger, Physical Review 73, 1020 (1948).
  * (61) R. H. Schwendeman, Journal of Chemical Physics 73, 4838 (1980).
  * (62) R. H. Schwendeman, Journal of Molecular Spectroscopy 198, 129 (1999).
  * (63) R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, 1st ed. (Wiley-Interscience, New York, 1991), p. 368.
  * (64) V. S. de Oliveira et al., Sub-Doppler optical-optical double-resonance spectroscopy using a cavity-enhanced frequency comb probe, 2023, preprint available on the Arxiv at webpage https://arxiv.org/abs/2307.03256.
  * (65) M. Rey et al., Icarus 303, 114 (2018).

Figure 1: Probe linear polarization ratios as a function of saturation
parameter for R(5) pump transition

Figure 2: Probe linear polarization ratios as a function of saturation
parameter for P(5) pump transition

Figure 3: Probe linear polarization ratios as a function of saturation
parameter for Q(5) pump transition

Figure 4: Probe circular polarization ratios as a function of saturation
parameter for R(5) pump transition

Figure 5: Probe circular polarization ratios as a function of saturation
parameter for P(5) pump transition

Figure 6: Probe circular polarization ratios as a function of saturation
parameter for Q(5) pump transition