AIAA 2010-4881
41st Plasmadynamics and Lasers Conference
28 June - 1 July 2010, Chicago, Illinois
Measurement of Rubidium Number Density under
Optically Thick Conditions
Matthew D. Rotondaro ∗
Charles F. Wisniewski † and Martiqua L. Post
‡
Department of Aeronautical Engineering, US Air Force Academy, CO, 80840, USA
Gordon D. Hager
§
Department of Engineering Physics, AFIT, Wright Paterson AFB, OH, 45433, USA
A measurement of rubidium number density under optically thick conditions has been
demonstrated by measuring the wings of the D1 absorption spectra using a laser with a
0.16 nm (75 GHz) fine tuning range. This technique can measure the absolute concentration
in rubidium under conditions where the absorption coefficient and path length product
yield conditions where the central region of the line is opaque. The laser was tuned to a
region sufficiently far into the short wavelength wing of the absorption where transmission
through the cell was possible. The laser was then scanned through the central opaque
region of the line to the adjacent long wavelength wing. The wavelength of the scan was
calibrated by using a 1.5 GHz etalon and a cell containing only naturally occurring rubidium
as a frequency reference. The measured absorption spectra for various cell conditions of
temperature and pressure were then fit to a pressure broadened Voigt profile thereby
allowing the determination of the rubidium number density.
Nomenclature
λ
ν
ν0
σ(ν)
τ
AF it
c
fF
fiso
I
k
L
M
N
P
R
SF F ′
T
Wavelength
Frequency
Frequency at line center
Absorption cross section
Radiative lifetime (s)
Amplitude of the Voigt profile derived from curve fitting the data
Speed of light (299, 792, 458 m/s)
The statistical distribution of the F state
The isotopic abundance
Intensity
2
kg
)
Boltzmann constant (1.3806503x10−23 ms2 K
Cell Length (m)
Mass of rubidium (kg)
Number Density (/m3 )
Pressure (T orr)
Hz
Pressure broadening rate ( M
T orr )
The relative hyperfine transition strength factor
Temperature (K)
∗ Senior
Scientist, USAF/DFAN, 2410 Faculty Dr, USAF Academy 80840, and AIAA Member.
Professor, USAF/DFAN, 2410 Faculty Dr, USAF Academy 80840, and AIAA Member.
‡ Associate Professor, USAF/DFAN, 2410 Faculty Dr, USAF Academy 80840, and AIAA Senior Member.
§ Professor, AFIT, Wright Paterson AFB, OH, 45433.
† Assistant
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Institute
of Aeronautics and Astronautics
Copyright © 2010 by the American Institute of Aeronautics and
Astronautics,
Inc.
The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.
All other rights are reserved by the copyright owner.
Subscripts
F
The hyperfine quantum number of the starting level
The hyperfine quantum number of the ending level
F′
I.
Background
A new class of laser, the optically pumped alkali-metal vapor laser (OPAL), has gained increasing attention over the past few years as a means to develop efficient laser systems. These three-level lasers are
pumped on the D2 line 2 S1/2 → 2 P3/2 and support lasing on the D1 line 2 P1/2 → 2 S1/2 of the alkali metal.
The 2 P1/2 level is populated by rapid spin orbit relaxation from the excited 2 P3/2 level. This relaxation is
induced by collisions with an additive relaxant species. The small energy defect in these three-level systems,
554 cm−1 , 237 cm−1 , and 57.7 cm−1 for Caesium, Rubidium, and Potassium respectively lead to intrinsic
quantum efficiencies of greater than 95% and to the possibility of very efficient electric laser systems. While
these alkali metals offer great promise, there are several issues which need to be resolved.1 One issue is
the challenge of holding the alkali metal number density constant during laser operation. The ability to
maintain constant alkali metal concentration is dependent on the ability to measure the concentration. This
measurement, while in principle is not difficult, is extremely difficult under the optically thick conditions at
which an alkali metal laser operates with typical alkali metal number densities of 1013 cm−3 to 1014 cm−3 .
The measurement of the concentration is performed by scanning a laser across the D1 or D2 line and measuring the absorption.2 Once this absorption spectra is measured then equation 1 can be applied to relate
the observed spectra I(ν)
I0 to the number density N.
(
)
I(ν)
ln
= −N σ(ν)L
(1)
I0
Under conditions favorable for the operation of an alkali laser the number density of the alkali metal is
sufficiently high to render the cell opaque at line center. This issue, can be circumvented by starting a laser
scan in a wing, scanning through the opaque region and then capturing the wing on the other side of the peak.
These peak fragments can then be fit using a highly constrained fitting function thereby reconstructing the
spectra. While this is difficult at low pressures, at high pressure where the line becomes significantly broad
this becomes even more difficult. An alkali metal laser requires high pressure of buffer gas to induce spinorbit relaxation of the D2 to D1 line. The buffer gas also has the effect of broadening the lines. Therefore,
to accomplish a wing to wing laser scan, the laser must have a tuning range on the order of 75 GHz with a
step size in the range of 500 MHz. Fortunately, a laser is available that can scan such a broad region and
will be employed to make these concentration measurements.
II.
Experimental Apparatus
The experimental apparatus is depicted in figure 1. Starting at the laser, the broadband finely tunable
laser source is routed through three beam splitters and into an etalon. The first beam splitter directed
the laser beam into the sample cell. The sample cell was enclosed in a heat block allowing good control of
the temperature. The sample cell contained rubidium 87 and 400 Torr of Ethane. The second beam-path
was directed into a second cell which contained only naturally occurring rubidium. This cell was used as
an absolute frequency reference. The third beam-path monitored the laser power throughout each scan.
Recording the laser power was absolutely necessary due to the large changes in laser power (approximately
30%) during a scan. The final beam-path went to an etalon. This etalon was used to linearize the scan
axis and to establish the correct spacing between each data point. The axis was then shifted to the correct
wavelength using the rubidium frequency reference.
The laser has a large tuning region (794-804 nm) which can be scanned, using the stepper motor, in
increments as small as 0.01 nm (4.75 GHz). This step size is too coarse for the purpose of measuring
the pressure broadening and therefore the number density. Subsequently, it was necessary to use the fine
frequency scan mode of the laser which employs a piezoelectric crystal. The piezoelectric crystal voltage
could be set between 0% and 100% of maximum in increments of 0.4%. The fine resolution scan length is
0.16 nm (75G Hz) the minimum fine scan step size is approximately 0.64 pm (300 MHz). By slowly adjusting
the piezoelectric crystal voltage scanning from 100% to 0% the pressure broadened absorption spectra from
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Figure 1. Experimental apparatus
the sample cell, the hyperfine spectra of the reference cell along with the etalon resonance peaks could be
recorded.
III.
Data Acquisition
In preparation for data acquisition, the first step was to ensure the cells were set to appropriate temperatures. A typical scan required approximately ten minutes to complete therefore, it was necessary to ensure
the temperature of the sample cell was fully stabilized before scanning the cell. A temperature change of
only 1 K could have a significant effect on the number density within the cell thereby distorting the spectra
during a scan. An appropriate temperature for the reference cell meant ensuring there was a sufficiently
strong absorption spectra to ensure a good quality fit later in the analysis.
During the warmup period for the heat block, the laser was set to its initial scanning position. This
was accomplished by setting the initial wavelength of the laser to a sufficiently low wavelength to ensure it
was well outside of any of the absorption features. The piezoelectric crystal was set to 50% of its range.
Then, the laser was scanned slowly, using the laser’s stepper motor on its smallest step size, towards shorter
wavelengths until the reference cell absorption spectra began to appear on the data acquisition monitor. At
that point the laser’s course scan was stopped. This placed the laser at the center of the absorption feature.
The laser could then be scanned from 0-100% of the piezoelectric crystal range allowing the acquisition of
the absorption spectra.
While the laser was scanned four channels of data were collected. These channels were the signal from the
sample cell, rubidium reference cell, laser power and the etalon transmission. The laser power fluctuations
were corrected by dividing the sample cell, reference cell and etalon data by the reference power. Once the
laser power correction was accomplished, the etalon data and the reference cell data were fit and used to
establish the wavelength axis which was then applied to the sample cell data. Sample cell data and reference
cell data with the wavelength axis corrected are depicted in figure 2.
IV.
Results
Once the data has been acquired and the wavelength axis has been linearized, scaled and shifted to the
correct wavelength region the data could be used to extract the rubidium number density. The sample cell
contained the isotope rubidium 87 and 400 Torr of ethane. At these pressures, the line shape is predominantly
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Figure 2. Sample cell and reference cell data that has been scaled and shifted using the etalon and reference data
Lorentzian, but to develop a generalized methodology for measuring the alkali metal number density at
various pressures and temperatures a Voigt profile was used to characterize the line shape. A Voigt line
shape is represented by equation 2,


∫ ∞
2
1
∆νL exp(−t )
 dt.

√
gV oigt (ν, νF F ′ ) =
(2)
2π π −∞ (ν − νF F ′ − t √∆νD )2 + ( ∆ν2 L )2
4 ln(2)
Using the Voigt line shape and summing over all of the appropriately weighted allowed transitions equation 3 can be used to construct the line shape for the rubidium sample cell spectra. The resulting line shape
can be curve fit to the data and the Voigt full width at half maximum (FWHM) and the amplitude can be
obtained. This method works well when the entire rubidium spectra is observed but, under highly opaque
conditions the Voigt profile is quite difficult to fit because only the wings of the profile are available.
∑
g(ν) =
fF fiso SF F ′ ∗ gV oigt (ν, νF F ′ )
(3)
F,F ′ ,iso
To improve the quality of the fit and subsequently the quality of the number density measurement, the
Gaussian and the Lorentzian FWHM are calculated using equations 4 and 5, respectively. The isotopic
abundance, hyperfine F state statistical distribution, relative hyperfine transition strength factors and relative
line positions were all obtained from3 and,4
√
8kT ln(2)
∆νD = ν0
,
(4)
M c2
and
∆νL = R(ν)P.
(5)
Hz
5
The value used for the broadening rate R was 27.8 M
T orr measured at 314.15 K and it was obtained from.
These FWHM values were then fixed and the only parameters that were allowed to float during the curve
fitting were the baseline and the amplitude of the Voigt profile. Typical data with their associated fits are
depicted in figure 3.
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Figure 3. Absorption spectra of Rubidium 87 with 400 Torr or Ethane at various temperatures with their associated
curve fits
Once the data has been curve fit, the amplitude of the spectra is determined using Beer’s Law (equation 1)
and the relationship between the absorption cross section and the line shape, equation 6. Equations 6
through can be combined to form equation 9 which is used to calculate the number density. The results
of these calculations can be seen in figure 4 also depicted is the number density calculated using the vapor
pressure curve equation 10 from.4 Note, this equation has been modified to yield number density, N m−3 ,
as a function and temperature, T , in Kelvin.
σ(ν) =
(
− ln
I(ν)
I0 (ν)
λ2
g(ν, νF F ′ , ∆νD , ∆νL )
8πτ
)
= N σ(ν)L = N L
λ2
g(ν, νF F ′ , ∆νD , ∆νL )
8πτ
AF it (ν)g(ν, νF F ′ , ∆νD , ∆νL ) = N L
N = AF it (ν)
N=
λ2
g(ν, νF F ′ , ∆νD , ∆νL )
8πτ
8πτ
Lλ2
−4040
2.077 × 109
· 10 T
kT
V.
(6)
(7)
(8)
(9)
(10)
Conclusion
As can be seen in figure 4 there is very good agreement between the number density measured by directly
probing the heat pipe when compared to the expected value derived from equation 10 using the temperature
measured from the sample cell. The number density measurements at higher temperatures resulted in a
larger uncertainty because the rubidium vapor pressure was sufficiently high to render the cell opaque at
line center. Even under these conditions, there was sufficient transmission through the cell off line center
to uniquely characterize the line shape. The laser’s broad tuning range and its small step size through that
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Figure 4. Measured rubidium number density as a function of temperature with the vapor pressure curve for rubidium.
tuning range made it possible to scan from one wing to the other through the opaque region allowing the
measurement of the wings of the spectra possible.
This methodology demonstrates a robust capability for measuring the number density of rubidium under
conditions that are favorable for rubidium laser operation. Therefore, this technique could be used on a
rubidium laser during laser operation. The ability to determine the alkali metal number density while the
laser is operating would provide insight into the thermal loading and yield valuable information that would
enhance both our theoretical and experimental knowledge of alkali metal laser systems.
Acknowledgments
I would like to acknowledge the United States Air Force academy for its outstanding support of this
research. Specifically, the authors would like to note the contribution of Dr. Tom Mclaughlin. Dr Mclaughlin
went to extraordinary lengths to ensure this work was adequately funded and without him none of this work
would exist. Also of note, Christopher Seaver, he was instrumental in finding adequate facilities and procuring
the necessary equipment to conduct this research. Finally, research can not be done without good quality
technical support and the authors want to single out Ken Ostasiewski for knowing where the indigenous
equipment could be found saving the authors significant time, money and effort.
References
1 Knize, R., Ehrenreich, T., and Zhdanov, B., “Highly Efficient Scalable Cesium Vapor Laser,” Journal of Directed Energy
2 , 2006, pp. 145–150.
2 Rotondaro, M. D. and Perram, G. P., “Collisional Broadening and Shift of the Rubidium D1 and D2 lines (52 S
1/2 −
52 P1/2 , 52 P3/2 ) by Rare Gases, H2, D2, N 2, CH4 and CF 4,” Journal of Quantitative Spectroscopy and Radiative Transfer ,
Vol. 57, 1997, pp. 497.
3 Steck, D. A., Rubidium 85 D Line Data, Vol. 0.2.1, Oregon Center for Optics and Department of Physics, University of
Oregon, Department of Physics, 1274 University of Oregon, Eugene, Oregon 97403-1274, April 2009.
4 Steck, D. A., Rubidium 87 D Line Data, Vol. 2.1.1, Oregon Center for Optics and Department of Physics, University of
Oregon, Department of Physics, 1274 University of Oregon, Eugene, Oregon 97403-1274, April 2009.
5 Zameroski, N. D. and Hager, G. D., “Pressure Broadening and Collisional Shift of Rubidium by CH4, C2H6, C3H8,
N C4H10, He,” Unpublished.
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