Resonant nonlinear magneto-optical effects in atoms∗ - The Budker ...

B. Polarimetry
For a 1-mW visible-light beam and χ ≃ 0.5, this corresponds
to ∼2 × 10 −8 rad Hz −1/2 . Similar shot-noiselimited
sensitivity can be achieved with a balanced po-
Successful application of the NMOE-based experimental
methods (Sec. XII) depends on one’s ability to perform
precision polarimetry or, more specifically, laser Various modifications of the spectropolarimetry techlarimeter
(Birich et al., 1994).
spectropolarimetry.
niques have been developed to meet specific experimental
In a “balanced polarimeter” (Huard, 1997), a sample is requirements, with the common challenge of realizing the
placed between a polarizer and a polarizing beam splitter
(analyzer) whose transmission axes are oriented at and polarimetry of condensed matter was given by Za-
shot-noise limit of Eq. (31). A review of magneto-optics
45 ◦ to one another. The optical rotation ϕ due to the passkii and Feofilov (1975). Birich et al. (1994) considered
in detail experimental approaches to measurement
optical activity of the sample can be found from a simple
expression valid for ϕ ≪ 1
of small optical rotation and their limiting factors.
ϕ = I 1 − I 2
2(I 1 + I 2 ) , (29)
C. Nonlinear magneto-optical rotation with
where I 1 and I 1 are the light intensities detected in the
frequency-modulated light
two output channels of the analyzer. In order to measure
the ellipticity ɛ of the transmitted light, a λ/4-plate is
placed in front of the analyzer, oriented at 45 ◦ Budker et al. (2000b) showed that NMOR can be used
to its
for measurements of sub-microgauss magnetic fields with
axis, to form a circular analyzer (Huard, 1997, see also
sensitivity potentially exceeding 10 −11 G Hz −1/2 (Sec.
Sec. XI.E).
VIII.E). However, for many applications (such as geophysics,
magnetic prospecting, and navigation), it is nec-
Another method of polarimetry employs a crossed polarizer
and analyzer (Fig. I and Sec. II.B) and polarization
modulation. A polarization modulator using a
essary to have a magnetometer with dynamic range ∼1
G. Budker et al. (2002b) recently demonstrated that if
Faraday-glass element is shown in Fig. XI.A. The signal
frequency-modulated light is used to induce and detect
in the “dark” (crossed) channel of the analyzer is given
nonlinear magneto-optical rotation (FM NMOR), the ultranarrow
features in the magnetic-field dependence of
by
(
I s (t) ≃ χI 0 r e + 1 )
2 α2 m+ ϕ 2 optical rotation normally centered at B = 0 can be translated
to much larger magnetic fields. In this setup, light
+ 2χI 0 α m ϕ sin ω m t − 1 (30)
polarization modulation (Fig. XI.A) is replaced by frequency
modulation of the laser, and the time-dependent
2 χI 0αmcos 2 2ω m t.
optical rotation is measured at a harmonic of the light
Here χ is the coefficient defined by absorption and scattering
of light by the atomic vapor cell, I 0 is the intenfects
both optical pumping and probing of atomic polar-
modulation frequency Ω m . The frequency modulation afsity
of the linearly polarized light (in photons per unit ization. This technique should enable the dynamic range
time) transmitted through the polarizer, r e is polarizer of an NMOR-based magnetometer to extend beyond the
(analyzer) extinction ratio, and α m and ω m are polarization
modulation amplitude and frequency, respectively.
Earth-field range.
For sufficiently low light intensities (so that the optical
The amplitude of the first harmonic of the signal Eq.
pumping saturation parameter does not exceed unity),
(30) is a measure of the polarization rotation caused by
FM NMOR can be understood as a three-stage (pump,
the sample. The first-harmonic amplitude is also proportional
to the modulation amplitude α m . Typically, with
precession, probe) process (Sec. V.B). Due to the frequency
modulation of the laser light, the optical pumping
a Faraday modulator with a high-Verdet-constant glass
(such as Hoya FR-5), the modulation α m is ∼10 −2 and probing acquire a periodic time dependence. When
rad
the pumping rate is synchronized with the precession of
(Wolfenden et al., 1990, 1991) at modulation frequencies
atomic polarization, a resonance occurs and the atomic
∼1 kHz. (High frequency modulation is often limited
medium is pumped into an aligned state whose axis rotates
at the Larmor frequency Ω L . The optical proper-
by the inductance of the Faraday modulator solenoid.)
Resonant photoelastic modulators can also be used for
ties of the medium are modulated at 2Ω L , due to the
polarimetry (as in the work of, for example, Oakberg,
symmetry of atomic alignment. This periodic change of
1995; Wang and Oakberg, 1999), allowing for larger polarization
modulation and higher modulation frequencies
the optical properties of the atomic vapor modulates the
angle of the light polarization, leading to the high-field
(several tens of kilohertz).
FM NMOR resonances. If the time-dependent optical
From Eq. (30), one can obtain the sensitivity of the polarimeter
with data accumulation time T for shot-noise-
rotation is measured at the first harmonic of Ω m , a resonance
occurs when Ω m coincides with 2Ω L . Additional
limited detection of the first-harmonic signal in the case
of an “ideal polarimeter” (αm 2 ≫ r e + ϕ 2 resonances can be observed at higher harmonics.
):
It should be noted that this technique is closely related
to the work of Bell and Bloom (1961), in which the
1
δϕ s ≃
2 √ χI 0 T . (31) intensity of pump light was modulated in order to op-
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