## Abstract

Small amounts of ellipticity in the nominally linearly polarized light used in magnetic rotation spectroscopy play an important role in determining the character of the signals developed in these experiments. For example, ellipticity introduced by stress-induced birefringence can easily influence such signals more than does a nonzero polarizer extinction ratio. In addition, for nearly-crossed polarizers, an initial ellipticity allows one to probe magnetic circular dichroism instead of the more commonly investigated magnetic circular birefringence. A general expression for the magnetic rotation spectroscopy signal is derived and compared to experimental results. An expression for the detection sensitivity is developed by taking shot noise and rms laser power fluctuations to be the dominant noise sources.

©1999 Optical Society of America

## 1. Introduction

Magnetic rotation spectroscopy (MRS) is a powerful zero-background spectroscopic
technique useful for reducing signal noise due to a laser source [1]. In addition, MRS can be helpful in sorting out complicated
spectra [2] since it is mainly sensitive to transitions between states
with low angular momentum quantum numbers. Additionally, MRS is sensitive only to
paramagnetic species, which can be either an advantage or a disadvantage. Several
molecular species have been investigated using the MRS technique, with an
experimental focus placed on NO and NO_{2} because of the important role
these molecules play as trace atmospheric constituents. One of the earliest reviews
of magnetic rotation spectroscopy, that of Buckingham and Stephens [3], still provides a good theoretical overview of the
technique. As somewhat of a side issue, those authors speculate on the possibility
of using a slight variation, in which elliptically polarized light replaces linearly
polarized light, of the usual MRS experimental configuration to examine various
types of signals. We pursue this point by considering the effect on MRS signals of
imperfect polarizers and birefringent cell windows.

A second focus of this paper is on the detection sensitivity of the magnetic rotation technique. We investigate how absorption and dispersion terms interact when one uses elliptically polarized light and an analyzing polarizer slightly offset from the null transmission position. It is found that one can obtain a signal-to-noise ratio (SNR)using elliptically polarized light which is comparable to that obtained with linearly polarized light incident on the sample. We discuss signal-to-noise ratios in MRS considering shot noise and rms laser power fluctuations to be the dominant noise sources.

## 2. Background

MRS is based on the change in the state of the polarization of light due to passage
through a gas cell immersed in a longitudinal magnetic field. Normally, this change
is monitored by placing the gas cell between two polarizers and detecting the light
transmitted by the system. There are two contributions to the change in polarization
state, magnetic circular birefringence (MCB) and magnetic circular dichroism (MCD).
MCB (sometimes referred to as magnetic optical rotation) arises from the differing
indices of refraction *n*_{L}
and
*n*_{R}
for left- and right-circular polarized light. In the
simplest case, relevant to this work, linearly polarized light, made up of RCP and
LCP components, will experience a rotation of the plane of polarization during
passage through the cell due to the relative phase shifts experienced by RCP and LCP
light, and thus the signal is proportional to the difference between
*n*_{R}
, and *n*_{L}
. The MCB
signal is then given by the difference between two dispersion curves offset in
frequency by a small amount, which in turn gives rise to a curve with two
zero-crossings. The maximum rotation of polarization occurs at the original
unperturbed transition frequency.

Magnetic circular dichroism (MCD) can be qualitatively explained by similar
arguments. The absorption coefficients *κ*_{L}
,
and *κ*_{R}
for zero applied magnetic field are
identical and the MCD signal, dependent on the difference between the two, vanishes
for zero applied field. With an applied *B*-field, the absorption
peaks shift in opposite directions and the difference of the absorption coefficients
gives a curve with a single zero-crossing. Expressed another way, after passing
through the sample, light which is initially linearly polarized will in general be
elliptically polarized.

A characteristic of most investigations known to these authors is the assumption made
at the outset of theoretical analyses that the species was very weakly absorbing. Of
the two components comprising the magnetic rotation signal, MCB and MCD, the
explicit assumption is made that the MCD contribution is negligible. One purpose of
this paper is to clarify exactly the extent to which one may make this
approximation. In fact, the statement that one could have a signal due to large
index of refraction changes *without* a corresponding absorption
contribution should probably seem surprising, given the fact that absorption and
dispersion are so closely coupled through the Kramers-Kronig relations [4]. For example, a great deal of work has been carried out over
the past several years to investigate coherence effects in atomic systems which may
lead to large dispersion at frequencies for which absorption is small [5]. In fact, our analysis show that the crucial point for
assumptions made in previous analyses to be valid is not the strength of the
absorption at all, but that the analyzing polarizer must be offset to a large enough
angle such that the dominant MRS contribution comes from the difference in indices
of refraction for LCP and RCP. Although some of the points to be made in this paper
have been noticed previously, we wish to unite and extend several different analyses
and to present experimental results in support of our analysis.

Recently we considered the magnetic rotation signals in the molecular oxygen A band [6] (*b*
^{1}${\sum}_{g}^{-}$-*X*
^{3}${\sum}_{g}^{-}$
, transition frequencies ~13120 cm^{-1}, wavelengths ~762 nm),
which is a relatively strongly absorbing transition, even considering the
spin-forbidden nature of the magnetic dipole transition. We have, through a careful
analysis of the MRS signals, including effects of imperfect polarizers and a
possible initial ellipticity of the incident field (such as could arise from
stress-induced birefringence in cell windows), determined that in the limit of
nearly crossed polarizers and a small initial ellipticity of the incident field, the
MCD contribution to the MRS signal is dominant. This behavior has been noted
previously by Yamamoto and coworkers [7–9] for the Voigt configuration (applied magnetic field
transverse to the laser propagation direction direction).

We now wish to address the role of polarizer imperfections and elliptical polarization in a magnetic rotation experiment, along with the question of detection sensitivity and signal-to-noise ratios in light of the results to be presented.

## 3. Theory

#### 3.1 Imperfect Polarizers and Ellipticity

We start by assuming an essentially *x*-polarized electric field
incident on the sample, but allow for the possibility that there is some small
amount of ellipticity. Thus the incident field can be written

$\overrightarrow{E}={E}_{0}\left(\begin{array}{c}1\\ A{e}^{i\Delta}\end{array}\right)\mathrm{exp}\left(i\left(kz-\omega t\right)\right)$

where *A* and Δ are taken to be small numbers and the
normalization is lumped in with the factor *E*
_{0}. By
making a suitable re-definition of axes we can write the above as

$\overrightarrow{E}={E}_{0}\left(\begin{array}{c}1\\ i\delta \end{array}\right)\mathrm{exp}\left(i\left(kz-\omega t\right)\right)$

which we now take to be the incident field. Here
*δ*≡*A* sin
Δ. We may rewrite this as

where we have defined the unit vectors *ê*_{R}
and *ê*_{L}
to be, respectively,

${\hat{e}}_{R}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)$ and ${\hat{e}}_{L}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)$

These represent right- and left-circularly polarized light respectively. Implicit in the above is that the real part of the complex field is the quantity of interest.

Once the incident field begins propagating in the medium, the left- and
right-circularly polarized components propagate with different wavevectors,
*i*.*e*. *k* becomes either
*k*_{R}
or *k*_{L}
respectively. These complex wavevectors are defined by

${k}_{R,L}=\left({n}_{R,L}+i{k}_{R,L}\right)\frac{\omega}{c}.$

The expressions for the field in the medium can be written

$\overrightarrow{E}=\frac{{E}_{0}}{\sqrt{2}}{e}^{-i\omega t}\left({\hat{e}}_{R}\left(1+\delta \right){e}^{i{k}_{R}z}+{\hat{e}}_{L}\left(1-\delta \right){e}^{i{k}_{L}z}\right)$

This is essentially the expression used in our previous work [6], with slight changes in the definition of the unit
vectors. It is not a surprising result in the sense that we know that linearly
polarized light can be considered as a superposition of equal amounts of RCP and
LCP light. A small amount of ellipticity is therefore the result of an imbalance
in these two components; for *δ*>0 we have
here slightly more RCP than LCP light incident on the sample.

Consider now an imperfect analyzer which transmits a fraction
*αE*_{incident}
along its polarization
axis (with *α*≃1), while also transmitting
a small component *βE*_{incident}
of the field
oriented orthogonally to the polarization axis. If this analyzer is oriented
such that it is nearly crossed with respect to the initial polarizer, but offset
at some angle *θ*, the Jones matrix for the analyzer
becomes

This represents the polarizer operating on the field which has traversed the
magnetically active medium of length *I*. Putting the above
results together to find the field transmitted by the analyzer yields

$$+\frac{{E}_{0}}{2}{e}^{-i\omega t}\left(1-\delta \right){e}^{i{k}_{L}l}\left(\begin{array}{c}\beta {\mathrm{cos}}^{2}\theta +\alpha \phantom{\rule{.2em}{0ex}}{\mathrm{sin}}^{2}\theta \phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}i\phantom{\rule{.2em}{0ex}}\left(\alpha -\beta \right)\phantom{\rule{.2em}{0ex}}\mathrm{sin}\phantom{\rule{.2em}{0ex}}\theta \phantom{\rule{.2em}{0ex}}\mathrm{cos}\phantom{\rule{.2em}{0ex}}\theta \\ \left(\alpha -\beta \right)\phantom{\rule{.2em}{0ex}}\mathrm{cos}\phantom{\rule{.2em}{0ex}}\theta \phantom{\rule{.2em}{0ex}}\mathrm{sin}\phantom{\rule{.2em}{0ex}}\theta \phantom{\rule{.2em}{0ex}}-\phantom{\rule{.2em}{0ex}}i\phantom{\rule{.2em}{0ex}}\left(\beta {\phantom{\rule{.2em}{0ex}}\mathrm{sin}}^{2}\theta \phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}\alpha {\phantom{\rule{.2em}{0ex}}\mathrm{cos}}^{2}\theta \right)\end{array}\right)$$

This can be rewritten again by combining the
“*R*” and
“*L*” terms in one vector. To simplify our
notation, we define the variables

$\Psi =\frac{1}{2}\left({n}_{R}+{n}_{L}\right)\phantom{\rule{.2em}{0ex}}\frac{\omega l}{c}$ $\Xi =\frac{1}{2}\phantom{\rule{.2em}{0ex}}\left({\kappa}_{R}+{\kappa}_{L}\right)\phantom{\rule{.2em}{0ex}}\frac{\omega l}{c}$ $\Theta =\frac{1}{2}\phantom{\rule{.2em}{0ex}}\left({n}_{R}-{n}_{L}\right)\frac{\omega l}{c}$ $\Phi =\frac{1}{2}\left({\kappa}_{R}-{\kappa}_{L}\right)\frac{\omega l}{c}\phantom{\rule{.2em}{0ex}}$

along with the complex angle
Ω̃=Θ+*i*Φ.
Looking back at the definitions of Φ and Θ we see that the
former gives information about magnetic circular dichroism (differential
absorption), whereas the latter involves circular birefringence.

In the most general case, in which all of the above effects are considered together, we can find an analytic solution without making any approximations as to the size of various contributions. After some algebra we find the transmitted intensity to be given by

$$+\left({\beta}^{2}+{\delta}^{2}{\alpha}^{2}\right){e}^{-2\Xi}\left\{\frac{1}{2}\mathrm{cosh}2\Phi +\frac{1}{2}\mathrm{cos}\left(2\theta -2\Theta \right)\right\}$$

$$+{\delta}^{2}{\beta}^{2}{e}^{-2\Xi}\left\{\frac{1}{2}\mathrm{cosh}2\Phi -\frac{1}{2}\mathrm{cos}\left(2\theta -2\Theta \right)\right\}$$

An expansion in the limit of small MRS signals, ellipticity, polarizer
imperfection and polarizer offset shows that the last term in Eq. 4 will always be negligible. In addition we can
simplify Eq. 4 by realizing that
*α*
^{2}+*β*
^{2}≃*α*
^{2}≃1
and
*δ*
^{2}
*α*
^{2}≃*δ*
^{2}.
This leaves

$$+\frac{1}{2}{e}^{-2\Xi}\left({\beta}^{2}+{\delta}^{2}\right)\left\{\mathrm{cosh}2\Phi +\mathrm{cos}\left(2\theta -2\Theta \right)\right\}.$$

This expression takes on a particularly simple form and is the main result of
this paper. The first term in brackets is the usual MRS signal, while the third
term (note that two MRS pieces are added) represents the intensity
“leaked” through the analyzer. The fact that the sinh
2Φ term is proportional to *δ* is of
special note.

To compare more closely with expressions used by other authors we rewrite Eq. 5 with *δ*=0 and
*β*≠0, and find for the intensity

$$=\frac{1}{2}{e}^{-2\Xi}\left[\mathrm{cosh}2\Phi -\mathrm{cos}\left(2\theta -2\Theta \right)+\xi \right]$$

$$\simeq \frac{1}{2}\left[\left(1-\mathrm{cos}2\theta \right)-2\Theta \mathrm{sin}2\theta +\xi \right]$$

To arrive at Eq. 6, we make the approximations that
*θ*≫Φ,
*Xi*≪1, and that Θ≪1. Note that
we do not compare Θ and Φ since, as we shall see, they are
of similar size. Compared to Eq. 5 the “noise” term ξ
involves more than just simply the polarizer offset angle; there are
contributions due to the magnetic rotation signals as well. However, those other
contributions are of higher order in the small quantities
*β*, Φ and Θ. This expression
agrees with the results used by previous authors, in which the polarizer term
was added as a phenomenological constant; here we have justified in more detail
the inclusion of that term. Setting *β*=0 leads to an
expression identical to Eq. 1 in both Ref. [1] and Ref. [11], as well as to the working relations used by Pfeiffer,
*et at* [10] and McCarthy, *et al*. [12].

Following the development Yamamoto and co-workers used in analyzing the Voigt
effect [7–9] we expand the expression for the transmitted intensity
in the general case to second order in terms involving
Φ,Θ,*β*,
*δ* and *θ*, which leads
to

which agrees with their results for the Faraday effect signal
(*e*.*g*. Eq. 2 in Ref. [8]) except for a discrepancy in sign of the final two
terms, presumably resulting from a definition difference in Φ and
Θ. This form is particularly useful in determining the parametric
dependence of MRS signals. To proceed further we must first determine the
relative sizes of the various terms. Analyzer offsets will be chosen in the
range 0<*θ*<0.1 (radians), while
*β*
^{2}, which represents the polarizer
intensity extinction ratio is ~10^{-4} for our experimental setup. For
purposes of comparison to theory we arrange the experiment so as to be able to
choose *δ* as well by simply inserting a
λ/4 plate before the interaction region but after the initial
polarizer. For the experimental results shown *δ* will
be ~10^{-2}. Thus we see the crucial points: Since
Φ~Θ it is actually the ratio of
*δ* to *θ* which is
important. As long as
*θ*>*δ* or
*θ*>Φ, the MCB contribution
will dominate.

The magnetic rotation terms can be estimated by modeling the linewidth as a
Lorentzian (approximately valid in the limit of an
atmospheric-pressure-broadened line). For the *PP*(1, 1)
transition studied here the on-resonance absorption for the 48 cm interaction
region pathlength is such that
1-exp(-*αl*)≃3.4×10^{-3}.
Taking the MRS signals to be the result of the difference between identical
curves for the index of refraction and absorption we find maximum values of
Θ≃1.2×10^{-4} and Φ≃7.5×10^{-5}.
The maximum value for Θ occurs at the frequency of the unperturbed
transition, whereas that for Φ is at a detuning of approximately
Γ/2, where Γ is the pressure-broadened linewidth.

With these values it is clear that in the expression above the terms in
Φ^{2} and Θ^{2} are negligible. The
*β*
^{2} and
*δ*
^{2} terms are roughly of the same size
and represent intensity “leaked” through the analyzer, or
background. The relative magnitude of the remaining terms must be approached
with some care. Clearly, in the limit *θ*=0, the only
term which contributes to the MRS signal is
2*δ*Φ, which is the magnetic circular
dichroism contribution usually neglected at the outset. For analyzer offset
angles of *θ*≥5° the terms in
*θ*
^{2} and
2*θ*Θ dominate the background
“noise” and the MRS signal, respectively. The latter case
is that assumed for essentially all MRS investigations to date.

A set of theoretical curves is shown in Fig.1 for parameters similar to those of the experiment discussed in Sec. 4. The curves were generated using Eq. 5. A calculation of the Voigt profile was implemented using FORTRAN to find the linewidths for the atmospheric-pressure oxygen, although they actually differ from a Lorentz profile of the same width by very little. The second-harmonic component of the MRS signal is shown in Fig. 1 to allow comparison to the experimental results presented later.

We wish also to be able to make comparisons to other analyses of the
signal-to-noise ratio (SNR) for MRS. However, we should first define more
carefully what we mean by this term. We follow the discussion of McCarthy,
*et at* [12] as their approach seems to be the most careful and
complete in the literature to date. It is possible to define a laser excess
noise term, which can be determined experimentally, as
*γ*≡
〈*δP*
^{2}〉^{1/2}/〈*P*〉;
this term represents the rms fractional fluctuations in laser power. We consider
laser excess noise along with photon shot noise to be the dominant sources of
noise when considering MRS signal-to-noise ratios. One should distinguish as
well between the detection sensitivity, which can be defined as the maximum
signal size compared to the noise when the laser is far from resonance, and the
signal to noise ratio (SNR). The latter is more properly defined as the ratio of
the signal to the noise on the signal at the same frequency [12].

#### 3.2 Magnetic Circular Birefringence

In this section we examine the MCB signal and the corresponding signal-to-noise ratio. The difference in absorption coefficients for the left- and right-circularly polarized components of the light vanishes on resonance, so the index difference part of the signal can be separated from the absorption difference part by looking at the intensity at the frequency of the unperturbed transition. In this case, we can write Eq. 5 as

*I*_{t}
=*I*
_{0}
*e*
^{-2Ξ}[sin^{2}(*θ*-Θ)+(*β*
^{2}+*δ*
^{2})cos^{2}(*θ*-Θ)]

where we take *α*=1 and ignore terms of order
*δ*
^{2}
*β*
^{2}.
For the experimental case of interest
Θ(*ω*=*ω*
_{0})=1.2×10^{-4}.
For discussions of the shot noise limit it is convenient to translate to the
number of detected photons. The conversion is

${N}_{0}=\frac{{P}_{0}{\tau}_{\mathrm{det}}}{h\nu}\approx 4\times {10}^{11}\mathrm{photons}$

for a power of 1mW incident on the sample and a detection bandwidth of 10 kHz. Since the overall absorption is small and independent of the parameters to be varied, the expression for the number of detected photons can be taken to be

In our experimental setup, *β*=0.01,
*α*≈1, and
*δ*=0.02. If we take the noise in a measurement of
Θ to be entirely due to intensity noise, then the noise is given as

Here,

=*N*=

*γ*

^{2}

*N*

^{2}

where *N* is given by Eq. 8. A typical number for the power fluctuations for the
diode laser used in the experiment is
*γ*=4×10^{-4}; for the
theoretical curves to be presented we chose a value of
*γ*=1×10^{-4} to emphasize the
relative contributions of the excess laser power and shot noise fluctuations.
Other intensity-independent noise sources such as amplifier noise could be
included as well. The net effect of these is to decrease the SNR and to shift
the maximum of the curves to be presented below to larger values of
*δ*(or *γ*). The
signal-to-noise ratio (SNR) is simply

If shot noise is ignored, the expression for the SNR is

$SNR=\frac{\Theta \left(1-{\beta}^{2}\right)\left(1-{\delta}^{2}\right)\mathrm{sin}\left(2\theta -2\Theta \right)}{\gamma \left[\left(1+{\delta}^{2}{\beta}^{2}\right){\mathrm{sin}}^{2}\left(\theta -\Theta \right)+\left({\beta}^{2}+{\delta}^{2}\right){\mathrm{cos}}^{2}\left(\theta -\Theta \right)\right]}$

which, to lowest order becomes

This expression has a maximum at $\theta =\Theta \sqrt{{\delta}^{2}+{\beta}^{2}}$,while the maximum value of the SNR in this limit is

This result would give an infinite maximum SNR for perfect polarizers and no ellipticity. We show below that this unphysical result is not obtained when shot noise is included.

The fundamental noise limit is the shot-noise limit and is formally obtained by
ignoring the amplitude fluctuation noise represented by
*γ* in Eq. 10. In that limit the SNR becomes

$SNR=\frac{\Theta \sqrt{{N}_{0}}\left(1-{\beta}^{2}\right)\left(1-{\delta}^{2}\right)\mathrm{sin}\left(2\theta -2\Theta \right)}{\sqrt{\left(1+{\delta}^{2}{\beta}^{2}\right){\mathrm{sin}}^{2}\left(\theta -\Theta \right)+\left({\delta}^{2}+{\beta}^{2}\right){\mathrm{cos}}^{2}\left(\theta -\Theta \right)}}$

For small *β*, *δ* and
(*θ*-Θ), this function is a maximum for

*θ*=Θ±(*β*
^{2}+*δ*
^{2})^{1/4}

and at the maximum has the value

Thus, if shot noise is dominant, the maximum SNR occurs at a somewhat larger offset angle than when excess laser noise dominates. In the latter case the maximum value is determined by the polarizer imperfections and ellipticity while in the former case it is determined by the intensity of the laser source.

Another interesting limit is that of perfect polarizers and no ellipticity with both technical noise and shot noise. In this limit the SNR is,

$SNR=\frac{\Theta \sqrt{{N}_{0}}\mathrm{sin}\left(2\theta -2\Theta \right)}{\sqrt{{\gamma}^{2}{N}_{0}{\mathrm{sin}}^{4}\left(\theta -\Theta \right)+{\mathrm{sin}}^{2}\left(\theta -\Theta \right)}}=\frac{2\Theta \sqrt{{N}_{0}}\mathrm{cos}\left(\theta -\Theta \right)}{\sqrt{{\gamma}^{2}{N}_{0}{\mathrm{sin}}^{2}\left(\theta -\Theta \right)+1}}$

The maximum value of this function clearly occurs at
*θ*=±Θ and has the value, ${SNR}_{max}=2\Theta \sqrt{{N}_{0}}$. Thus, in the absence of polarizer imperfections and
ellipticity the maximum signal to noise ratio is not limited by excess laser
noise at all.

In the general case the offset angle that gives the maximum signal to noise ratio satisfies the relation

(*δ*
^{2}+*β*
^{2})^{1/2}≤|*θ*-Θ|≤(*δ*
^{2}+*β*
^{2})^{1/4}

and the maximum SNR is somewhat less than the lesser of the values given in Eqs. 12 and 13. Fig. 2 shows a comparison of the SNR as a function of offset angle for shot noise limited detection, laser excess-noise-limited detection, and for the case appropriate for our experimental setup. Note that both shot noise and excess noise play a role. The relative importance of these terms is determined by the ratio

$\frac{\delta {N}_{\mathit{laser}\phantom{\rule{.2em}{0ex}}\mathit{power}}^{2}}{\delta {N}_{\mathit{shot}\phantom{\rule{.2em}{0ex}}\mathit{noise}}^{2}}={\gamma}^{2}{N}_{0}\left\{\left(1+{\beta}^{2}{\delta}^{2}\right){\mathrm{sin}}^{2}\left(\theta -\Theta \right)+\left({\delta}^{2}+{\beta}^{2}\right){\mathrm{cos}}^{2}\left(\theta -\Theta \right)\right\}\simeq {\gamma}^{2}{N}_{0}.$

#### 3.3 Magnetic Circular Dichroism

For parameters corresponding to our experiment the signal has the maximum size
Φ=7.5×10^{-5}. To separate the absorption
difference signal from the index different signal it is useful to set
*θ*=Θ. This essentially would mean to
set the offset angle to as close to zero as would be experimentally feasible. In
this case the number of detected photons takes the form

*N*=*N*
_{0}[1+*δ*
^{2})sinh^{2}Φ-2*δ*sinhΦcoshΦ+*β*
^{2}+*δ*
^{2}]

For the usual small parameter choices, an excellent approximate form is

*N*=*N*
_{0}[Φ^{2}-2*δ*Φ+*β*
^{2}+*δ*
^{2}]

The SNR is found to be

$SNR=\frac{\Phi \mid \frac{\partial N}{\partial \Phi}\mid}{\sqrt{{\gamma}^{2}{N}^{2}+N}}\approx \frac{2\Phi \left(\Phi -\delta \right)}{\sqrt{{\gamma}^{2}{N}^{2}+N}}$

If shot noise is ignored the SNR is maximized for an ellipticity given by
*δ*=Φ+*β*.
At this value the SNR has the value
SNR_{max}=Φ/*β*_{γ}
.
This indicates that an infinite signal to noise ratio could be obtained by
eliminating the polarizer imperfection. This unphysical result is again a
consequence of ignoring shot noise. In the shot noise limit the SNR for no
ellipticity has the approximate value

${SNR}_{\gamma =0,\delta =0}=\frac{2\Phi \sqrt{{N}_{0}}}{\sqrt{1+{\beta}^{2}\u2044{\Phi}^{2}}}$

which passes through zero at *δ*=Φ and then
rises to a constant value of

when *δ*≫*β*.

For the general case the best signal to noise ratio is achieved if there are no polarizer imperfections and the ellipticity is very close to zero. In this case the maximum SNR is given by Eq. 14. The SNR for the parameters of our experiment is compared with shot noise and laser excess-noise limits in Fig. 3.

## 4. Experiment

The experimental setup is much the same as that in our previous work [6] and is shown in Fig. 4. The main change for the purposes of the present work is the addition of a quarter-waveplate (QWP) placed after the first polarizer. We use this waveplate to introduce a small amount of ellipticity to the incident laser field. If the QWP is set such that its optic axis is aligned with the transmission axis of the first polarizer, the incident polarization is linear. For a slight offset of the QWP (in the results to be shown below, this offset is ~1°) the incident field is elliptical.

The laser source is a semiconductor diode laser, Mitsubishi 4405-01, held at a
temperature of ~33°C and dc injection current of ~72 mA to reach the
PP(1,1) rotational transition of the
*b*
^{1}${\sum}_{g}^{-}$
(*ν′*=0)←*X*
^{3}${\sum}_{g}^{-}$
(*ν″*=0) transition at 13118.0
cm^{-1} (762.3 nm). The injection current is scanned at a rate of ~0.07
cm^{-1}/sec. and modulated at a rate of 1 kHz. These modulation signals
are suitably added and attenuated and fed to the external input of the commercial
current controller (ILX-Lightwave LDX-3620), which provides 0.3 mA/V transfer
function. The transition frequency is monitored using a wavemeter (Burleigh WA-1000)
with a resolution of 0.01 cm^{-1} and an absolute accuracy of 1 part in
10^{6}.

Ambient oxygen is used for these experiments, and an air-cooled solenoid wrapped on a
5 cm diameter, 48 cm long PVC tube is the source for the magnetic field. At the
typical operating current of 7.1 A the solenoid center magnetic field is 83G (8.3
mT). The polarizers at the entrance an exit to the solenoid provide an extinction
ratio of 7×10^{-5}. The first polarizer is a broadband polarizing
beamsplitter and the analyzer is a Polarcor polarizer mounted in a rotation stage.

Data are collected by first choosing an offset angle for the analyzing polarizer. The
laser frequency is scanned across the oxygen absorption feature and the
2*f* lock-in amplifier output signal recorded on an oscilloscope (H-P
54602A) which can be used in either single trace or averaging mode. The trace is
then saved as a set of voltage vs. time data points (1000 points per sweep).
Analysis of the data consists mainly of subtraction of an identical data trace taken
with “zero” magnetic field. In this case,
“zero” means that no current is flowing in the solenoid, but
the Earth’s magnetic field has not been compensated. Typically the
lock-in amplifier has a 12 dB/octave rolloff and a time constant of 100 ms.

In Fig. 5 we show data which typify the signals observed. The
QWP is offset by 1.0°±0.5°. As the analyzer offset
is increased from the crossed position (*θ*=0°)
the signal changes from one which resembles most the expected MCD contribution to a
more symmetric signal with two zero-crossings, as expected for the MCB contribution.

A similar set of data taken with the QWP set at 0° shows no trace at all of the asymmetric MCD signal. It was also observed experimentally that with the QWP set at this position the transmission minimum position of the analyzer was the same and gave the same leakage as was the case with no QWP present. In contrast to this, for a QWP setting of 1.0° the extinction ratio of the polarizer/analyzer pair appeared to be degraded, a result of the ellipticity introduced by the QWP.

## 5. Conclusions

We have presented a simple, general result describing magnetic rotation spectroscopy
signals in all parameter regimes. Our result takes into account specifically effects
due to imperfect polarizers and of any ellipticity in the incident field
polarization. Comparable signal-to-noise ratios may be obtained for both MCB and MCD
signals with the appropriate choice of parameters, mainly the ellipticity,
*δ*, and the analyzer offset angle
*θ*.

In this paper we hope to have clarified some of the approximations commonly made in magnetic rotation spectroscopy experiments, as well as to have offered an analysis of signal-to-noise ratios that goes somewhat beyond that present in the literature. Particularly, we have shown that the weakness of an absorbing transition is not the determining factor in deriving an expression for MRS signals. Individually, some of the results presented here have been noted in previously; however, many statements made justifying these approximations seem to have been made without a careful consideration of all parameters involved. In the end, our results, theoretical as well as experimental, should serve to clarify some points with regards to MRS signals.

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