Hairpin probes are used to determine electron densities via measuring the shift of the resonant frequency of the probe structure when immersed in a plasma. This manuscript presents new developments in hairpin probe hardware and theory that have enabled measurements in a high electron density plasma, up to approximately 10^{12} cm^{−3}, corresponding to a plasma frequency of about 9 GHz. Hardware developments include the use of both quarter-wavelength and three-quarter-wavelength partially covered hairpin probes in a transmission mode together with an easily reproducible implementation of the associated microwave electronics using commercial off-the-shelf components. The three-quarter-wavelength structure is operated at its second harmonic with the purpose of measuring higher electron densities. New theory developments for interpreting the probe measurements include the use of a transmission line model to find an accurate relationship between the resonant frequency of the probe and the electron density, including effects of partially covering the probes with epoxy. Measurements are taken in an inductively coupled plasma sustained in argon at pressures below 50 mTorr. Results are compared with Langmuir probe and interferometry measurements.

## I. INTRODUCTION

The hairpin resonator probe was first introduced by Stenzel^{1} in 1976 as a technique to determine the plasma density via measurements of the resonant frequency of a quarter-wavelength resonant transmission line immersed in a plasma. The plasma density is determined by measuring the difference between the resonant frequency of the transmission line in vacuum and that in a plasma. The hairpin probe has been utilized in a number of laboratory settings, typically in unmagnetized plasmas with lower densities (below ∼10^{11} cm^{−3}).^{2–9} Hairpin probes have also been utilized to determine other plasma parameters that influence the plasma dielectric, including the electron-neutral collision frequency,^{10} and also to determine the electron density in magnetized plasmas.^{11,12}

In this paper, we present a more accurate model for the resonant frequency of a hairpin probe in a plasma in addition to a novel and more robust approach to electronics for measuring the resonant frequency. These contributions, along with utilizing both quarter-wavelength and three-quarter-wavelength resonators, expand the range of measurable densities (in principle, up to 10^{13} cm^{−3}), simplify measurement, and provide a more rigorous basis for density determination. In the current work, results obtained from the hairpin probe are compared with Langmuir probe and microwave interferometer measurements. Langmuir probe measurements are known to be greatly dependent on the analysis method used, which relies on assumptions about probe geometry and surface contamination conditions.^{13} Hairpin probes on the other hand suffer from less ambiguity during analysis and are less affected by plasma conditions (i.e., sheath, surface contamination, and rf fluctuation potential), although as we discuss, the smaller probes necessary to measure high density are hard to construct reliably. We find, in general, that accurate density measurements require cross-calibration with a microwave interferometer.

Hairpin probes are typically made from a wire bent into a quarter-wavelength transmission line shape (the hairpin wire). The hairpin wire is coupled electrically to measurement electronics via coupling loops, which provide an indirect (capacitive/inductive) connection. The hairpin wire can be supported mechanically using epoxy resin to secure it between the coupling loops of the probe. An accurate model of the resonant frequency of the structure requires accounting for the effects of the epoxy. Previous attempts to account for the effect of the epoxy have utilized a heuristic approach, using a weighted mean of the resonant frequency for a fully bare and a fully covered hairpin.^{14} We introduce a more rigorous transmission line model, representing the hairpin as two connected transmission lines (one bare and one epoxy coated) and demonstrate that it results in densities that are consistent with interferometry.

The measurement of the probe resonance at microwave frequencies is typically done using a network analyzer. We present a microwave circuit using commercial off-the-shelf components that are relatively inexpensive (under $900). The circuit allows time resolution as low as 10 *µ*s, limited by the time response of the mixer circuit. The noise introduced by the circuit was only significant at the low end of its frequency spectrum (8 GHz), which did not affect measurements at the density of interest. The circuitry’s setup for data acquisition is easily implemented and the data analysis involved to obtain the resonant frequency of the hairpin in the plasma with such a circuit is easily performed.

As the density increases, the quarter-wavelength resonator must be made smaller to operate at the relevant frequency. We found that a 7 mm probe with a resonant frequency of approximately 10 GHz is the smallest we could reliably make. To overcome this limitation, we employed a three-quarter-wavelength resonator,^{15} allowing a larger (and more manageable) probe to be operated at higher frequency. The three-quarter-wavelength hairpin doubled the quality factor (Q ∼ 40) of its counterpart. However, the signal to noise ratio when using such a hairpin limited the maximum measurable density to 9.0 × 10^{11} cm^{−3}.

This paper is organized as follows: In Sec. II, we describe the design and construction of hairpin probes as well as the novel microwave circuit utilized to measure the resonant frequency of the probes. Section III describes a theoretical model of the hairpin probe, including the correction necessary to accurately describe a partially covered hairpin. Section IV presents measurements, both bench testing (vacuum measurements) and in-plasma measurements using a series of 1/4 and 3/4 wavelength resonators. Plasma measurements are performed in an inductively coupled plasma (ICP) sustained in argon at pressures below 50 mTorr. The density measurements are compared with Langmuir probe^{16} and interferometry^{17} measurements under the same conditions.

## II. EXPERIMENTAL SETUP

### A. Hairpin design

The hairpin resonator is excited using microwave signals that are coupled using a small loop placed in the vicinity of the hairpin. Either the same loop or a second loop detects the resonant response of the hairpin. There are two common connections for these loops—directly and indirectly coupled. A directly coupled resonator requires the loop to be electrically attached to the resonator, while the indirectly coupled resonator floats without any direct contact to the loop. Directly coupled resonators have been reported to produce a stronger signal. However, because the probe is grounded via the microwave circuit, a thicker sheath will form around the wires, which interferes more with the measurement, particularly, in RF plasmas.^{5} For this work, we also constructed and tested a directly coupled hairpin probe (not shown in this manuscript); however, it was not able to produce a good signal at 9 GHz frequency. Therefore, we used the indirectly coupled resonator method where the hairpin does not contact the feed or return coaxes and electrically sits at the local plasma floating potential.

There are also two common techniques for acquiring signals from hairpin resonator probes in low electron density plasmas: reflection and transmission. Previous researchers have used the reflection mode because it only requires one coupling loop that is easier to construct and slightly smaller. In this case, the scattering parameter^{18} S11 is used to measure the ratio of reflected to incident power on a single coax. However, for higher frequency operation associated with higher target densities, the transmission mode provides superior performance; here, we measure S21, the ratio of transmitted to incident power. We found that, in reflection measurements, standing wave modes in the coax obscured the target peak, as can be seen in Figs. 10 and 11. Details of the testing are in Sec. IV A. This agrees with the transmission measurement of Piejak *et al.*,^{4} where the authors report that the measurable density is as high as 1.3 × 10^{12} cm^{−3}.

Semi-rigid coaxial cables carry the microwave signals to and from the hairpin. These are type UT-034M from Amawave^{19} with OD = 0.034 in. (0.87 mm), as shown in Fig. 1, emerging from a 2-hole ceramic support tube. The cables are approximately 110 cm long with a section of 5 cm–10 cm sitting inside the ceramic. The rest of the cable sits in a stainless probe shaft, which enables the translation and rotation of the probe inside the chamber. Attenuation in these cables is about 3 dB/m, among the lowest for these types of cables at relevant frequencies. The input and pick-up loops are at the end of the coaxial cables and made by soldering the inner conductor of the coax to the outer conductor. The connection is made using Sn5Pb93.5Ag1.5 solder with a melting temperature of ∼300 °C. Before fixing the hairpin between the two loops, the axes of the loops are mostly parallel, but the final alignment is adjusted to give the best transmission and to minimize spurious resonances arising from the loops themselves. The location of the hairpin with respect to the loops is also adjusted to give the sharpest resonance. Note that the coaxial outer shields can interfere with the resonant frequency of the system and should be located as out of the way as possible. Finally, we use epoxy^{20} to keep them in position. The construction process of the hairpin is shown in Fig. 1.

### B. Microwave circuit

An inexpensive microwave circuit has been designed and constructed to realize the hairpin measurement. A block diagram showing the circuit connection to the hairpin is presented in Fig. 2. X-Microwave^{21} provides a convenient modular scheme to assemble all the needed microwave components. The voltage controlled oscillator (VCO) is HMC588LC4B from Analog Devices with the frequency range from 8.0 GHz to 12.5 GHz and an output power of 5 dBm. We use an FPC06074 directional coupler from Knowles Dielectric Labs, which has a coupling coefficient from 10 dB to 12 dB for the frequency range 8 GHz–12 GHz. The pivotal part of the system is the mixer, which we choose to use SIM-153LH+ from Mini-Circuits. It is a Level 10 (local oscillator Power +10 dBm) mixer for 3.2 GHz–15 GHz signals. In order to make the local oscillator (LO) port receive enough power to output a signal through the intermediate frequency (IF) port, we use the amplifier CMD157P3 from Custom MMIC to give a +24 dB boost to the signal between 6 GHz and 18 GHz. The whole system fits into an X-Microwave housing, 1.34 × 2.83 in., as shown in Fig. 3.

A 5 V DC source powers the oscillator. A 0 V–10 V ramp signal sweeping in 10 ms serves as the control voltage, tuning the output signal frequency from 7.5 GHz to 12.9 GHz. The curve of frequency vs tuning voltage is shown in Fig. 4. As diagrammed in Fig. 2, the signal from the oscillator directly goes into the coupler, and a −7 dBm coupling signal is sent into the RF port of the mixer, while the major part of the signal is sent into the input loop that couples the signal onto the hairpin. Any signals picked by the pick-up loop from the hairpin are sent into the LO port of the mixer through the amplifier.

Signal attenuation is always an issue for hairpin probes. In our case, we find a fairly abrupt reduction in the measured signal above a resonant frequency of 11.5 GHz. A combination of attenuators and amplifiers (6 GHz–18 GHz) is used as a variable gain stage for the hairpin probe to enhance the signal to noise ratio. The amount of attenuation was adjusted to give the best signal for given hairpin probes. These elements are placed outside the microwave circuit, usually at the end of the high frequency coax cable. In the case of a 3-quarter-wavelength copper hairpin, all the resonance signals reduce to below the noise level once the frequency surpasses 11.5 GHz. The resonant peak is no longer distinguishable even with the auxiliary amplifier.

The signal output from the single-ended mixer has phase information encoded in it as the frequency is swept. This is seen in the oscillating waveforms, as shown in Fig. 5. Figure 5 is a composite of the waveforms taken at different RF powers and consequently different densities. To determine the resonant frequency at a given density, we select the peaks of the absolute value of the waveform and construct the profile of these over the frequency range. We fit an offset Gaussian function to this envelope and locate the center of the Gaussian as the resonant frequency. Figure 6 shows such a fit for an RF power of 230 W.

### C. Plasma chamber

The hairpin probe is tested in an industrial plasma etch tool modified for experimental access.^{16,22} Figure 7 is the schematic of the experimental chamber. A double-wound three-turn circular rf antenna mounts above an alumina window that forms the top of the chamber. The rf generator operates at 2 ± 0.1 MHz. Probes are inserted into the chamber from a side mounted flange utilizing a ball valve feedthrough.^{23} The probe shaft can translate in and out through the ball valve as well as rotate with the ball as a pivot point. The wafer is considered to be located at *z* = 0, with its center at *x* = 0, *y* = 0. Positive $x^$ points toward the probe insert location. In this manuscript, the plasma operates at steady state and, for this work, the bias voltage is not applied on the wafer.

## III. HAIRPIN THEORY

As discussed above, the hairpin probe is modeled as a transmission line shorted at one end. The lowest frequency resonance occurs when a quarter wavelength fits approximately in the length of its legs. When embedded in a dielectric medium, the resonant frequency is given approximately by the formula

where *f*_{0} is the resonant frequency in vacuum, *c* is the speed of light, *l* is the length of one leg of the hairpin, and *ϵ*_{r} is the relative dielectric constant of the surrounding medium.^{4} Without any external magnetic field, the real component of the dielectric constant of the plasma in the cold plasma approximation is

where $\omega p=2\pi fp=(ne2\u03f50m)1/2$ is the plasma frequency with *e* and *m* being the charge and mass of an electron respectively, *ϵ*_{0} being the vacuum permittivity, and *n* being the electron density.^{1,4,24,25} Operating at frequencies f > *f*_{p}, Eq. (2) is always valid. Plugging Eq. (2) into Eq. (1), we can derive the relation between the frequencies under the condition that *f* = *f*_{r},

A sheath will form around the legs of the hairpin when placed in the plasma. The sheath is estimated as concentric cylindrical around each wire that depletes electrons, which leads to an underestimation of the electron density between two legs of the hairpin. To accurately measure the density, a correction is introduced, as discussed in the work of Sands *et al.*^{25} Then, Eq. (3) becomes

where *f*_{p}′ is the corrected plasma frequency and *ζ*_{s} is the correction factor. In a lumped element sense, the sheath can be treated as a capacitance in series with the resonator; therefore, the corrected frequency has a new relation with the hairpin dimension from which the correction factor is eventually derived. *ζ*_{s} is given by the formula^{25}

where *r* is the radius of the hairpin wire, $w$ is the separation between the two hairpin wires, and *b* is the radius of the cylindrical sheath around the wire. Here, we assume that the sheath extends one electron Debye length, $\lambda D=(\u03f50Tenee)1/2$, out from the surface of the hairpin wire. The wire we use to construct the hairpin is of radius *r* = 0.25 mm, and in these plasmas, the Debye length is about 13 *μ*m–40 *μ*m. We, further, assume that in these low-pressure plasmas, the collision correction is negligible. The formula for the corrected electron density is then

The hairpin probes used in this work are operated in a transmission mode to measure the resonant response as described above. The floating hairpin is fixed between the two signal loops using epoxy.^{20} Furthermore, the hairpin wire utilized has a dielectric coating, although we ignored this due to its negligible effect, as explained in Subsection III A. However, a correction due to the epoxy covered portion of the hairpin cannot be introduced in the same manner as Sands *et al.*^{25} since the resonator’s frequency becomes dependent on how much of its length *l* is covered. A number of authors have introduced an averaging scheme between the covered and uncovered portions of the transmission line, introducing an effective permittivity^{14} for the lumped element model. In contrast, we believe the following is a more accurate treatment.

### A. Partially covered hairpin correction model

To obtain the plasma frequency from the resonant frequency of the hairpin in the plasma, we model the hairpin as two transmission lines in series. The first transmission line includes the part of the hairpin that is covered by epoxy and is shorted at the end. The second transmission line includes the hairpin wires in the plasma. We assume that the dielectric constant for the covered part is constant and unaffected by the presence of plasma, i.e., the electric field in this region is wholly confined to the epoxy. This arrangement necessarily makes the hairpin less sensitive to the plasma since only the exposed portion of the wires responds to changes in the plasma dielectric.

The capacitance and inductance per length for a lossless parallel wire transmission line are given by

where $w$ is the width from the center axis of one wire to the other, *r* is the radius of the wire, *ϵ* is the permittivity, and *μ* is the permeability of the media. The impedance at a distance *l* from the load *Z*_{L} for a lossless transmission line is given by

where $Z0=L/C$ is the characteristic impedance, *β* = 2*π*/*λ* is the wave number, and $i=\u22121$.

The first transmission line is the portion of the hairpin that is covered by epoxy, see Fig. 8. The capacitance *C*_{1} and inductance per length *L* for this transmission line are given by Eqs. (7) and (8), where *ϵ* = *ϵ*_{0}*ϵ*_{e} and *μ* = *μ*_{0}, respectively. The relative permittivity of epoxy *ϵ*_{e} = 3.17 was determined by comparing a hairpin’s resonant frequency *f*_{s} fully covered with epoxy to its resonant frequency *f*_{0} in vacuum and using $\u03f5e=(f0/fs)2$. *f*_{s} and *f*_{0} were measured by an Agilent N5230C vector network analyzer (VNA). Using Eq. (9) with *Z*_{L} = 0, the impedance at a distance *l*_{1} from the short is then

where $Z01=L/C1$ is the characteristic impedance and $\beta 1=2\pi \u03f5efr/c$ is the wave number of the first transmission line with *f*_{r} being the (to be determined) resonant frequency of the hairpin and *c* being the speed of light.

The second transmission line is made up of the portion of the hairpin wire in the plasma, see Fig. 8. For this transmission line, the load impedance is *Z*_{1}, the impedance at the end of the first transmission line. The inductance per length of this transmission line is the same as the first. The capacitance per length *C*_{2} is given by Eq. (7), where *ϵ* = *ϵ*_{0}*ϵ*_{p}. The impedance, given by Eq. (9) with *Z*_{L} ≡ *Z*_{1}, at a distance *l*_{2} from the epoxy is given by

where $Z02=L/C2$ is the characteristic impedance and $\beta 2=\u03f5p/\u03f5e\beta 1$ is the wave number of the second transmission line. For the quarter-wavelength resonance, we expect the impedance looking into the second transmission to be infinite. Setting the denominator to zero, we obtain

Using the dispersion relation for cold, unmagnetized plasma given by Eq. (2), we can rewrite Eq. (12) as

where the only unknowns are *f*_{p} and *f*_{r}. A summary of all the parameters of interest in Eq. (13) is given in Table I. Being able to effectively measure the resonant frequency of the hairpin in the presence of plasma allows the calculation of *f*_{p} and, consequently, the electron density $n=\u03f50m(2\pi fp)2/e2$.

. | Parameter summary . |
---|---|

ϵ_{e} | Relative permittivity of epoxy |

$w$ | Separation between hairpin wires |

r | Radius of the hairpin wire |

l_{1} | Transmission line length in epoxy |

l_{2} | Transmission line length in the plasma |

f_{r} | Hairpin resonant frequency in the plasma |

f_{p} | Plasma frequency |

c | Speed of light |

. | Parameter summary . |
---|---|

ϵ_{e} | Relative permittivity of epoxy |

$w$ | Separation between hairpin wires |

r | Radius of the hairpin wire |

l_{1} | Transmission line length in epoxy |

l_{2} | Transmission line length in the plasma |

f_{r} | Hairpin resonant frequency in the plasma |

f_{p} | Plasma frequency |

c | Speed of light |

If desired, a correction due to the plasma sheath and the dielectric cover can also be included in *C*_{2} by adding it as a capacitance in series, as described, for instance, by Piejack.^{4} Introducing a correction for a plasma sheath of 40 *μ*m or less, which is typical for a floating hairpin in our plasma conditions, and the dielectric cover of 30 *μ*m made of polyvinyl formal (*ϵ*_{c} = 2.76^{26}) resulted in a plasma density difference of 2.2%, decreasing with increasing density. This complicates the algebraic expressions in this work without adding new insight, and we ignore it here.

Letting *l*_{1} → 0 with the total length *l* = *l*_{1} + *l*_{2} held constant in Eq. (12) gives the simple formula for the resonant frequency of a quarter-wavelength transmission line *f*_{r} = *c*/4*l*. In practice, the shorted end of the transmission line is not a perfect short and has some inductance. Furthermore, the vacuum resonant frequency of the hairpin is influenced by its proximity to the coax cables whose center conductors form the coupling loops. The total length *l* is measured before we apply the epoxy to stabilize the assembly. We use an adjustable parameter Δ*l* such that *l*_{eff} = *l*_{1} + *l*_{2} + Δ*l* to account for the observed shift in the resonant frequency from its nominal value. Δ*l* can be found by setting *f*_{p} = 0 in Eq. (13) and measuring the vacuum resonant frequency *f*_{r} and *l*_{2} of the finished hairpin. A summary of these measurements for three different hairpins is given in Table II. In these cases, the values of Δ*l* are small such that the effective lengths *l*_{eff} are within 2.2% of the measured value. However, the boundary between the epoxy covered portion and the portion in the plasma is not abrupt, and it is difficult to measure an exact value for *l*_{2}. We have found that if an independent measure of the density is available, the estimate for Δ*l* can be improved by cross-calibration. This is discussed, further, in Sec. IV. On the other hand, if an independent measurement is unavailable, simply setting Δ*l* = 0 gives an adequate approximation. For the hairpins in this study that can be cross-calibrated, we found an average Δ*l* to be about 6.3% of the total length. The corrected length provides a measurably more accurate value for the resonant frequency of the hairpin without any epoxy,

l_{1} (mm)
. | l_{2} (mm)
. | $w$ (mm) . | f_{0} (GHz)
. | Δl (mm)
. |
---|---|---|---|---|

4.32 | 13.40 | 2.07 | 4.038 | 0.29 |

2.64 | 6.39 | 1.98 | 7.710 | 0.20 |

2.56 | 4.14 | 1.64 | 10.169 | 1 × 10^{−3} |

l_{1} (mm)
. | l_{2} (mm)
. | $w$ (mm) . | f_{0} (GHz)
. | Δl (mm)
. |
---|---|---|---|---|

4.32 | 13.40 | 2.07 | 4.038 | 0.29 |

2.64 | 6.39 | 1.98 | 7.710 | 0.20 |

2.56 | 4.14 | 1.64 | 10.169 | 1 × 10^{−3} |

Gogna^{14} described an alternate model for a partially covered hairpin using a weighted average of the effective permittivity in the epoxy and plasma. The plasma frequency *f*_{p} is obtained by solving the equation

where $f0*$ is the resonance frequency of the hairpin without external dielectric calculated from the dimensions of the hairpin. In Gogna’s work, *f*_{0m} is the vacuum resonance frequency of the hairpin partially covered measured experimentally, corresponding roughly to our quantity designated *f*_{0}. *f*_{r} is the measured resonance frequency of the probe in the plasma.

We compare the predictions of the transmission line model to Gogna’s weighted average scheme using a theoretical hairpin of *l* = 7 mm and arbitrarily choosing different *f*_{0m}. *l*_{1} and *l*_{2} were obtained from Eq. (13), where *f*_{p} = 0 and *f*_{r} = *f*_{0m}. As shown in Fig. 9, Gogna’s model always gives a higher plasma frequency than the transmission line model. At *f*_{r} = *f*_{p} (i.e., *ϵ*_{p} = 0), Eq. (13) is undefined, resulting in a maximum measurable *f*_{p}, so we constrain the comparison to *f*_{p}/*f*_{r} < 1. Decreasing the vacuum resonant frequency of the hairpin—for instance, by covering a greater portion of the hairpin with epoxy—leads to a lower maximum measurable electron density. This highlights the importance of constructing hairpins with vacuum resonant frequencies as high as possible to measure high electron densities. We confirm the accuracy of the transmission line model by comparing hairpin measurements with Langmuir probe and interferometry measurements in Sec. IV B.

## IV. EXPERIMENTAL RESULTS AND DISCUSSION

### A. Vacuum results

We constructed a number of hairpins for the purpose of testing. This manuscript describes a subset of nine. Their dimensions are listed in Tables III and IV. Hairpins 1–3 are only different in their dimensions. They are constructed to determine the best quality factor (Q). Hairpins 4 and 5 are constructed to test the validity of 3/4 wavelength hairpins in measuring higher resonance frequencies. Hairpins 1–5 have never been tested in the plasma. After acquiring solid bench testing results with the above hairpins, we constructed hairpins 6–9 to test in the plasma.

Dimensions . | Hairpin1 . | Hairpin2 . | Hairpin3 . |
---|---|---|---|

Width ($w$) (mm) | 1.78 | 2.90 | 3.90 |

Length (l) (mm) | 9.25 | 9.15 | 8.30 |

Ratio (l/$w$) | 5.20 | 3.16 | 2.13 |

f_{0} (GHz) | 7.5 | 7.2 | 7.2 |

Q factor | 37.5 | 24.0 | 14.4 |

Dimensions . | Hairpin1 . | Hairpin2 . | Hairpin3 . |
---|---|---|---|

Width ($w$) (mm) | 1.78 | 2.90 | 3.90 |

Length (l) (mm) | 9.25 | 9.15 | 8.30 |

Ratio (l/$w$) | 5.20 | 3.16 | 2.13 |

f_{0} (GHz) | 7.5 | 7.2 | 7.2 |

Q factor | 37.5 | 24.0 | 14.4 |

Probe . | l (mm)
. | $w$ (mm) . | f_{0} (GHz)
. | l_{2} (mm)
. | Δl (mm)
. |
---|---|---|---|---|---|

Hairpin 6 (copper) | 7.96 | 1.30 | 9.06 | 4.46 | −0.53 |

Hairpin 7 (copper) | 7.00 | 1.35 | 9.73 | 4.04 | −0.13 |

Hairpin 8 (silver) | 6.90 | 0.92 | 10.33 | 3.45 | −0.62 |

Hairpin 9 (copper) | 22.83 | 1.30 | 9.19 | 17.19 | −1.34 |

Probe . | l (mm)
. | $w$ (mm) . | f_{0} (GHz)
. | l_{2} (mm)
. | Δl (mm)
. |
---|---|---|---|---|---|

Hairpin 6 (copper) | 7.96 | 1.30 | 9.06 | 4.46 | −0.53 |

Hairpin 7 (copper) | 7.00 | 1.35 | 9.73 | 4.04 | −0.13 |

Hairpin 8 (silver) | 6.90 | 0.92 | 10.33 | 3.45 | −0.62 |

Hairpin 9 (copper) | 22.83 | 1.30 | 9.19 | 17.19 | −1.34 |

The quality factor *Q* of a hairpin depends on its dimensions. Although the resonant frequency is primarily determined by the total length of the hairpin, the *Q* factor is affected by the ratio of *l* and $w$ (leg length and width between two legs).^{25} We verified this relation by comparing hairpins 1–3 with almost the same resonant frequency in vacuum but different *l* to $w$ ratios. The results from the network analyzer are shown in Fig. 10, and using these, we can calculate the *Q* factors for three hairpins (see Table III).

In Fig. 10, the blue (S11) and green (S22) curves are the measured reflected power from each of the two loops of the hairpin constructed in Fig. 1. This demonstrates that the use of the hairpin probe in a reflection mode is problematic. When using the reflection mode, the resonant frequency of the hairpin is almost lost in a standing wave signal arising from the impedance mismatch at the coupling to the hairpin. This effect is worse when the frequency is high and the transmission line is much longer than the wavelength. On the other hand, the transmission mode (labeled hairpin 1/2/3 in Fig. 10) only relies on accurate termination at the receiving end, which is easy to accomplish.

Higher aspect ratios give larger *Q* values, which helps both in detectable signal and density resolution. Ideally, in order to build the most sensitive hairpin, this ratio should be as high as possible. Therefore, decreasing the width and increasing the length can optimize *Q* (hairpin 1). However, the width is limited by both the physical characteristics of the material and the tools available to construct the hairpins. On the other hand, as the length increases, the resonant frequency drops and the perturbation caused by the hairpin becomes larger. In practice, using this method, we were able to construct hairpins with adequate *Q* for quarter-wavelength resonant frequencies as high as 11 GHz, which corresponds to a useful density of about 10^{12} cm^{−3}.

The smallest quarter-wavelength resonators we could make gave a highest measurable density of 10^{12} cm^{−3}, which is insufficient for the higher density plasmas typically produced at the Basic Plasma Science Facility (BaPSF). With the goal of measuring densities up to 10^{13} cm^{−3}, we explored the possible usage of the three-quarter-wavelength resonant frequency.^{15} Thus, a hairpin with lowest resonance at 11 GHz could potentially be over-moded to operate at a much higher frequency. We explore this potential application by comparing hairpins 4 (1/4-wavelength) and 5 (3/4-wavelength) around the resonance frequency of 8.2 GHz. Figure 11 shows the result from a network analyzer. The 1/4-wavelength hairpin has *Q* ≈ 20 at *f*_{0}, while the 3/4-wavelength version has *Q* ≈ 41. Its advantage is that, for a given hairpin, the frequency can be three times higher, and at a given resonant frequency, the *Q* factor is larger than the 1/4 resonator shown for comparison; this is consistent with the higher *l* to $w$ aspect ratio.

No matter what dimensions of the hairpins we choose, our *Q* values are below 100 when the resonant frequencies are higher than 8.0 GHz. The hairpins reported by Sands *et al.*^{25} and Peterson *et al.*^{10} have Q > 100, but their resonant frequencies fall in the range of 2.0 GHz–3.1 GHz. Lower frequency gives a larger and sharper signal. We found the compromised accuracy to be unavoidable when pursuing higher measurable frequencies.

### B. Results in plasma

With experience gained from the bench tests, we constructed hairpin numbers 6–9 and tested them in the ICP. The information about these hairpins is listed in Table IV. Here, the total length (*l*) and width ($w$) are measured using a caliper. The length covered by epoxy is estimated using the transmission line model.

Figure 12 shows the frequency spectra of the received signal from hairpins 6–9 in the argon plasma at 45 mTorr with a generator power of 230 W. Examining the spectrum, we first notice that hairpin 8, made out of silver, has a much smaller signal amplitude than the rest. This will result in a smaller signal to noise ratio, even though hairpin 8 has a higher *Q* factor than hairpins 6 and 7. In principle, a material with higher conductivity would constitute a hairpin with less loss and more sensitivity. In this test, the silver hairpin had a higher *Q* than copper possibly due to its higher conductivity (although at only ∼10% that would not seem to be enough to account for the difference). However, the small signal amplitude when inside the plasma puts it at a disadvantage against copper. The *Q* value of hairpin 9 (3/4-wavelength) is higher than the 1/4-wavelength hairpins, as expected. However, the resonant peak of hairpin 9 drops below the noise when the input plasma power goes higher than 600 W (9 × 10^{11} cm^{−3}). As the RF power increases, the signals at the receiver are dominated by detected oscillations that match the lower resonant frequency of the hairpin.

The lengths *l*_{2} and Δ*l* were found by a least squares fit between hairpin and interferometer^{17} data and requiring that the plasma density is zero when *f*_{r} = *f*_{0}. In Fig. 13, we calculated the density measurements of hairpins in Table IV using this technique, assuming that the plasma density with the probe present is unperturbed from the interferometer measurement (this is unlikely to be a perfect assumption^{27–29}). This enables reproducing the density inferred from the interferometer quite accurately. The interferometer measures a line-average density, but the measurements were scaled by a measured profile peaking factor, as described below. The lengths outside the epoxy *l*_{2} found for the hairpins in Table IV are all reasonable quantities based on how hairpins were generally constructed. The adjustable parameter Δ*l* is small such that the average is about 6.3% of the total length of the hairpins. One source of error in this experiment is the lack of tight control on the gas flow rate, which was later found to influence the density for a given power subsequent to these experiments, even when the pressure was maintained at a constant level.

In order to compare hairpin and interferometer measurements, interferometer measurements are scaled such that they represent the peak density at the center of the chamber. We used hairpin 6 to obtain profile measurements of the chamber at different powers. Profile measurements provided us with a peaking factor (peak density to line average ratio) at these powers. The peaking factor as a function of power is estimated as a linear trendline fitted with the peaking factors we measured and calculated for three density profiles under the corresponding power setting. Using this trendline, we scaled the interferometer data at each power to estimate point measurements for comparison with the hairpin measurements. Those three density profiles were obtained by using the transmission line model with Δ*l* = 0 and *l*_{2} given by Eq. (13) with *f*_{r} = *f*_{0} and *f*_{p} = 0. Presenting the density profiles more precisely, the profiles shown in Fig. 14 are calculated with *l*_{2} and Δ*l* providing the best least squares fit between the hairpin and interferometer measurements.

Figure 15 shows the corrected density measured by hairpin 6 using the transmission line model plotted against plasma power. Langmuir probe measurements based on ion saturation current^{16} (*I*_{isat}) and scaled interferometer measurements are also shown for comparison. The curve labeled uncorrected density is calculated from Eq. (3), which does not include the effects due to epoxy. The transmission line model shows a good agreement with interferometer measurements. Neither shows good agreement with the Langmuir probe *I*_{isat} data, but we attribute this to defects in the latter. Using the transmission line model increased the density calculations by 53% on average from the uncorrected density for hairpin 6.

## V. CONCLUSION

In this paper, we explored the use of hairpin resonators to measure the electron density up to 10^{12} cm^{−3}. Various aspects about the design of the system, data analysis, plasma frequency corrections, and Q values of the probes have been discussed. Some improvements are put forward according to observation and calculation.

We have been able to construct hairpins with ∼2 cm legs (resonant frequency 3 GHz–4 GHz), which are capable of *a priori* density measurement. That is, the predicted response matched the measured response, as cross-calibrated by the interferometer. We have not been able to construct a hairpin that resonates in the 10 GHz range of frequencies accurate enough to do similar *a priori* measurements. We attribute this to several problems all associated with the small dimensions, i.e., with legs shorter than about 7.5 mm. The first problem is that the bottom of the “U” shape, which should be an electrical short in order to accurately apply the nominal transmission line theory described in the text. On this scale, the length of this leg is a macroscopic fraction of a wavelength, which, consequently, has some inductance. Second, the boundary between the two transmission lines is assumed to be perfect in the theory from two points of view: (1) The transmission line medium extends to infinity in the transverse direction and (2) the boundary is sharp. Neither of these is true. Third, we assume that we know the dielectric constant of the epoxy. While we have characterized the dielectric constant of the medium in the coated portion of the hairpin, as described in the text, this is only an approximation. Finally, the inductive loops positioned close to the hairpin modify its resonant frequency by a small amount, depending on just how the assembly goes together. We are pursuing schemes for more accurately fabricating small hairpins, with the potential of developing such probes into a more accurate diagnostic in the future.

In order to measure higher resonant frequencies, the dimensions of the probe must be reduced. The highest density so far reported for a hairpin probe measurement is about 1.3 × 10^{12} cm^{−3} (plasma frequency *f*_{p} ≈ 10 GHz) corresponding to a vacuum quarter wavelength of 7 mm.^{4} Another scheme for getting to higher frequencies relaxes the size constraint by operating at a 3-quarter-wavelength resonance. A comparison between the quarter-wavelength and three-quarter-wavelength hairpin probes reveals that the latter has a higher *Q* value in general. One drawback is that the resonant signal of the three-quarter-wavelength probe has too high of a signal to noise ratio at a high frequency. This can potentially be overcome with a lower noise microwave circuit. On the other hand, we have taken advantage of this to use a single hairpin over a broad density range by using the higher frequency resonances at a higher density.

The microwave circuit described, herein, offers a viable and much less costly alternative to a network analyzer often used for measurements with transmission mode hairpin probes. All the circuit components are available in modules that are easy to connect using specially designed impedance-matched bridges. The signal response of the circuit can be easily analyzed to determine the resonant frequency of the probe in the plasma. We note that the measurements can be improved by using an IQ mixer. With this modification, the output of the microwave circuit would have two signals with 90° phase difference. The signals are expected to be sinusoidal with increasing amplitude as the frequency gets closer to the resonant frequency of the probe. If these two signals are matched in magnitude and exactly 90° out of phase, adding the square of each signal should give an envelope like graph whose peak is the resonant frequency of the probe. This mixer is expected to mix the signals such that the outputs obtained have less noise.

The hairpin mounting can affect the applicability of a simple model to describe the plasma frequency. In our case, an epoxy support covers part of the resonator, which unavoidably alters the resonant frequency. We describe a transmission line model that accounts for the covered portion of the hairpin resonator structure. The corresponding correction changes the computed electron densities by 53% on average. This correction is more significant than the sheath and coating corrections, especially at a high density. Hairpin measurements agree well with the interferometer measurements.

## ACKNOWLEDGMENTS

This work was supported by the National Science Foundation (Grant Nos. PHY-1500099 and PHY-1500126), the Department of Energy Office of Fusion Energy Science (Grant Nos. DE-SC0001319 and DE-SC0014132), and Lam Research Corp. We thank Lam Research Corp. for the donation of the plasma chamber. We acknowledge the expert technical help of Zoltan Lucky, Tai Ly, and Marvin Drandell.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.