Abstract
We have developed a Lamb-dip laser-induced fluorescence (LIF) system to precisely measure the local magnetic field strength in a plasma. Utilizing the hole burning effect, we made sharp dips on the LIF spectrum as the frequency markers and accurately determined the Zeeman splitting by reading the frequency interval of the dips. The method is valid even in the conditions where the Doppler broadening of the LIF spectrum is larger than the Zeeman shift. The newly developed LIF system is capable of determining a magnetic field strength on the order of 10 À4 T. It has been demonstrated that the Lamb-dip LIF system can successfully reproduce a very small field inhomogeneity in the HYPER-I linear device.
Introduction
Magnetic field is one of the most fundamental parameters of plasmas and has been measured by various methods. The Zeeman splitting measurement has often been carried out to determine the magnetic field strength, since the frequency measurement is a well established diagnostic technique. [1] [2] [3] [4] [5] From the viewpoint of sensitivity and spatial resolution, laser-aided optical measurements such as laser-induced fluorescence (LIF) spectroscopy are very promising, [3] [4] [5] because the LIF spectroscopy has a good spatial resolution and is a form of non perturbative diagnostics. 6) However, there is still a problem of frequency resolution in weak magnetic field cases, in which the Zeeman shift is smaller than the Doppler broadening of the LIF spectrum. This situation takes place in most conditions of laboratory experiments using a low-temperature plasma. It is necessary to develop a precise and concise method for measuring the magnetic field.
Introduction
To accurately determine the LIF spectrum as a function of frequency, it is necessary to have a high-resolution frequency reference. For example, the absorption line of the gas used for the discharge or iodine vapor is utilized as the reference. 7, 8) However, in the experimental conditions where the spectral shift is smaller than the width of the Dopplerbroadened absorption line, an accurate frequency measurement is difficult even when using peak fitting software. In our previous paper, we proposed a combined system of the LIF spectroscopy and saturated absorption spectroscopy (SAS), in which the saturated absorption spectrum is simultaneously measured as the frequency reference and the flow velocity of the argon neutral atoms has been measured within the error of AE2 m/s.
Introduction
9) The Lamb dip 10) is Dopplerfree and appears at the frequency corresponding to zero velocity in the laboratory frame, which is very convenient for flow velocity measurements.
Introduction
Here, we propose an improved system, where the Lamb dips appear on the LIF spectrum, and the experimental procedure remains almost the same as the previous system. The improved system is called Lamb-dip LIF spectroscopy hereafter, in which the transmitted beam is reflected back into the plasma without damping exactly along the same optical path as the incident beam. When the laser frequency is tuned to the resonant frequency of one of the transitions, the Lamb dip appears on the LIF spectrum by the hole burning effect.
Introduction
11) The width of the Lamb dip is on the order of natural broadening and the position of the Lamb dip is used as the ''frequency marker'' on the LIF spectrum. The Zeeman splitting is determined by reading the frequency interval of the Lamb dips of the AE-transitions of the argon neutral particles.
Introduction
We have demonstrated that the Lamb-dip LIF spectroscopy is capable of distinguishing a very small field inhomogeneity on the order of 10 À4 T, which is one order of magnitude smaller than the results obtained by using SAS 12) or intermodulated fluorescence (IMF) spectroscopy 13, 14) because of the improvement of the measurement system. In addition, the Lamb-dip LIF spectroscopy has high spatial resolution.
Introduction
In x2, the experimental setup of the Lamb-dip LIF system is described. The experimental results are presented in x3, in which a very weak radial inhomogeneity of the magnetic field strength is successfully measured by the Lamb-dip LIF system. The concluding remarks are given in x4.
Experimental Methods
A schematic diagram of the Lamb-dip LIF system is shown in Fig. 1 . An extended cavity diode laser (ECDL) with 3 fm (2 MHz) spectral width is used in the present experiment. The wavelength of the ECDL is tuned to 696.735 nm (vacuum) to excite the argon metastable atoms in the 4s½3=2 2 state into the upper energy state (4p
Experimental Methods
15) The laser power is 5 mW at the entrance window of the chamber, so that the excitation is attained to saturation condition. In addition, the attenuation of laser power by absorption is negligible in the experiment. The excited atoms emit fluorescence photons of 826.679 nm through the de-excitation process into the 4s
Experimental Methods
state. An energy level diagram of the present experiments is shown in Fig. 2 . There are three transitions between the magnetic sublevels, and the Lamb dip appears on each transition.
Experimental Methods
In the Lamb-dip LIF spectroscopy, the incident and reflected laser beams, which counter-propagate each other, are radially introduced into the plasma as the pump beams. When the laser frequency differs from the resonant frequency of the atoms at rest 0 , the velocity selective resonance occurs at different velocities for each beam; 2ð À 0 Þ=k for the incident beam and À2ð À 0 Þ=k for the reflected beam, where k is the wave number of the laser beam. On the other hand, when the laser frequency coincides with the resonant frequency 0 , the resonant excitation takes place at the same particle velocity (zero velocity in the laser propagation direction). Since the resonant particles have been already excited by the incident beam, the reflected laser beam meets the depleted velocity distribution made by the incident beam (hole burning effect). Consequently, the LIF spectrum exhibits the Lamb dip when the laser frequency coincides with the resonant frequency 0 . The spectral width of the Lamb dip is very narrow and is of the order of natural broadening ($10 MHz). By sweeping the laser frequency, the Lamb-dip LIF spectrum as a function of frequency is obtained.
Experimental Methods
To control the polarization of the incident laser beam, a =2 wave plate (WP) and a polarization beam splitter (PBS) are placed in front of the viewing port of the chamber, as shown in Fig. 1 . The polarization of the laser is aligned to the y-direction for the -transition (E ? B) and the z-direction for the -transition (E k B). An electro-optical modulator (EOM) modulates the amplitude of the laser beam at 100 kHz to realize the synchronized detection of the fluorescence photons of the photomultiplier tube (PMT).
Experimental Methods
The PMT with the collection optics is installed on the top viewing port of the chamber and receives the fluorescence photons. The LIF signal from the PMT output is amplified by a current amplifier and is detected by a lock-in amplifier. The radial resolution of the LIF system is set to about 2 mm by a slit in front of the PMT. In addition, the cross-section of the laser beam is 3 mm in width and 1 mm in height. Therefore, the LIF system has 6 mm 3 spatial resolution. The collection optics is horizontally movable to carry out the radial scan.
Experimental Methods
The secondary laser beam is divided from the primary beam and introduced into a Fabry-Pérot interferometer (FPI). The free spectral range of the FPI is 294 MHz and the output fringes are used as the scale of the laser frequency. The FPI signal is simultaneously recorded with the LIF signal. The system is operated as a conventional LIF system by removing the reflection mirror (MR).
Experimental Methods
The experiment has been made in the high-density plasma experiment (HYPER-I) device at National Institute for Fusion Science (NIFS). The details of the experiment using the HYPER-I device have been described elsewhere. 17, 18) The plasma is produced by the electron cyclotron heating of a 2.45 GHz microwave (TE11 mode) introduced from an open end of the cylindrical chamber.
Experimental Methods
19) The microwave power was 40 W and the operation pressure of argon gas was 7.5 mTorr. Figure 3 shows the Lamb-dip LIF spectra for the -transition case [ Figs. 3(a) and 3(b) ] and the -transition case [ Fig. 3(c) ]. It is clearly seen in Fig. 3(a) that the Zeeman splitting with an interval of about 4 GHz takes place in the -transition case. As seen in the figure, there are three Lamb dips around the peak of each Gaussian-shaped LIF spectrum, which come from the transitions between the lower and upper magnetic sublevels denoted by A AE , B AE , and C AE in Fig. 2 . The observed LIF spectrum is the sum of contributions from each magnetic sublevel. By measuring the frequency interval of a pair of Lamb dips [C þ and C À in Fig. 3(b) ], we determined the Zeeman splitting with a very high resolution ($1 MHz).
Experimental Results
An atomic electron in a magnetic field has an energy ÁE associated with the magnetic moment, 20 )
Experimental Results
where the quantity B is the Bohr magneton, M i is the magnetic quantum number, g i is the Landé g-factor of the i-th quantum state, and B is the magnetic field strength. When a magnetic field is present, the energy level splitting (Zeeman splitting) takes place according to eq. (1). By measuring the frequency interval of the Zeeman splitting, we can determine the local magnetic field strength B as
Experimental Results
where Á is the frequency interval of the Zeeman splitting, h is Planck's constant, and i and k denote the i-th and the k-th energy levels, respectively. The LIF spectra obtained with the conventional LIF spectroscopy are also shown in Fig. 3 . It is noted that the amplitude of the Lamb-dip LIF signal is about 76% higher than that of the conventional LIF measurement, indicating an enhancement of the sensitivity. This is because both the laser beams (the incident and reflected beams) contribute to the emission of the fluorescence photons.
Experimental Results
We have confirmed the performance of the Lamb-dip LIF system by measuring a small inhomogeneity of the magnetic field in the HYPER-I device. There is a small radial variation in the field strength of the order of 10 À5 jason93@example.com T in the HYPER-I device. Figure 4(a) shows the observed radial profile of the magnetic field, in which the solid line indicates the calculated value from the coil current. The solid circles and the open circles are the results of the Lamb-dip LIF spectroscopy and the conventional LIF spectroscopy, respectively. In the conventional LIF procedure, the LIF spectrum is decomposed into three Gaussian distributions, whose intensity ratios are specified by the selection rules 21) with the assumption of uniform population densities in the magnetic sublevels in the lower state.
Experimental Results
22) The frequency of Zeeman splitting is determined by reading the frequency interval of the sublevel distributions using peak fitting software. It is noted in the figure that the results of the Lamb-dip LIF spectroscopy well agree with the expected values calculated from the coil current. The results of the conventional LIF spectroscopy deviated by about 2% from the results of the Lamb-dip LIF. This means that the decomposition of the LIF spectrum by Gaussian-shaped subpeaks is not accurate enough compared with the Lambdip LIF spectroscopy. It should be pointed out that there is a very small systematic difference in the field strength between the results obtained by the Lamb-dip LIF spectroscopy and the expected values, which is attributable to the accuracy of the external current source, the geometrical factor of the magnetic field coils, and the geomagnetic field. When we take a correction factor, the observed magnetic field coincides with the calculated ones, which is indicated by the dashed line in the figure. Figure 4(b) shows the spatial difference of the magnetic field strength in the HYPER-I device, indicating a good agreement with the results of the Lamb-dip LIF spectroscopy. It is concluded that the Lambdip LIF spectroscopy provides a very precise way for measuring magnetic field strength.
Discussion and Conclusions
We applied peak fitting software using the sublevel distributions with the Lamb dips, which are expressed as the sum of each contribution composed of a Gaussian distribution and a Lorentzian distribution, 23) (3), the derived magnetic field differs between the two. In the decomposition of the conventional LIF spectrum, the spectral intensity ratios of the sublevel distributions are fixed to the theoretical values (1 : 2 : 6), which is based on the assumption of uniform population density in the magnetic sublevels. We did not have a reasonable decomposition result without specifying the intensity ratios, since the magnitude of the Zeeman shift is smaller than the width of Doppler broadening of the subpeaks. In the present case, the amplitude of the Gaussian profile is nearly flat in the neighborhood of the peak position, and the result of decomposition is not sensitive to a small shift of the subpeaks. Consequently, ambiguity arises in the decomposition into subpeaks. In the case of Lamb-dip LIF spectrum, we need not specify the intensity ratios of the sublevels and treat them as free parameters, since the existence of the Lamb dips helps the peak fitting software to converge on the reasonable value. As seen in Fig. 5 , the intensity ratios obtained from the Lamb-dip LIF spectrum are different from those obtained in the conventional case. This difference comes from the nonuniformity of the population density in the magnetic sublevels. Anisotropy of the plasma causes the nonuniform population, which is called selfalignment. 24, 25) The optical pumping effect of the excitation laser also changes the population density. 26) In our experimental condition, we confirmed that both the mechanisms affect the population density. This means the assumption of equal population density in the lower magnetic sublevels is not justified for the present case. The difference in the assumption of the population density causes the systematic error of the magnetic field derived by conventional LIF.
IðÞ
It should be emphasized that the difference in the field strength determined by the direct reading of the dips on the Lamb-dip LIF spectrum and that determined with the peak fitting procedure using eq. (3) are negligibly small (about 1 Â 10 À4 T). We can conclude that for almost all the cases, the peak fitting procedure is not needed to determine the Zeeman splitting, and the Lamb-dip LIF spectroscopy provides a quick and precise technique to measure the magnetic field in a plasma.
IðÞ
We have developed a Lamb-dip LIF system for precise measurements of the magnetic field strength in a plasma. The incident laser beam is reflected back into the plasma exactly along the same optical path as the incident beam, and we have obtained an LIF spectrum with the Lamb dips, which have been utilized as the frequency reference in determining the Zeeman shift of the magnetic sublevels. Since the width of the Lamb dip is narrower than the Doppler broadening of sublevel distributions, we can precisely determine the Zeeman shift of sublevel distributions, and accordingly determine the magnetic field strength very accurately. The performance of the Lamb-dip LIF spectroscopy has been confirmed by measuring a small radial variation of the magnetic field in the HYPER-I plasma.

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