In situ millimeter wave spectroscopy of microplasma within a photonic crystal Open Access

Microplasmas have been a growing area of interest due to their attractive features, such as high electron density and discharge stability at gas pressures up to at least 1atm.^1,2 Microwave driven microplasma can be ignited and sustained by using a power of less than 1 W.^3–6 High electron density (>10¹⁹ m⁻³) is attributed to distinct heating mechanisms for electrons at higher excitation frequencies. Hoskinson et al. experimentally showed the fundamental dependence of argon plasma on excitation frequency from 0.5 to 12 GHz: a higher frequency resulted in a higher density and lower electrode voltage due to electron confinement.⁷ The plasma dependence on excitation frequency up to 400 GHz is also investigated by several theoretical and simulation papers.^8–10 An increase in the electron density is commonly predicted up to 10 GHz. Uncertainty in the model assumptions, however, results in contradictory predictions at frequencies above 40 GHz due to complex heating mechanisms for electrons and a lack of experimental data for validation. The experimental results of the present work may be used to benchmark future high frequency plasma models and theories.

Advances in high power and high frequency electronics have allowed microplasma generation at higher excitation frequencies. For example, formation of stable plasma at 27 and 43 GHz has been demonstrated by using a photonic crystal.^11,12 The photonic crystal (PhC) is typically configured as a periodic lattice of dielectrics or conductors. Originally, the PhC was designed to manipulate high frequency electromagnetic waves within a specific bandgap frequency.^13,14 If a vacancy defect is introduced into the lattice, the resulting PhC cavity possesses a defect-induced resonance. The electric field of incoming waves resonates in the PhC vacancy defect such that microplasma can be formed within the PhC when the resonating electric field exceeds the gas breakdown voltage.¹⁵ This work focuses on the properties of a plasma that is ignited and sustained by millimeter wave radiation at 43.66 GHz.

Using plasma to manipulate the electromagnetic transmission properties of a photonic crystal has been the subject of research for more than a decade. As plasma properties such as electron density and collision frequency are readily adjustable, the plasma-loaded PhCs possess tunable characteristics with a fixed mechanical structure.^16–20 In this literature, however, the plasmas are all conventional DC or low-frequency AC discharges, typically confined within cylindrical discharge tubes. The plasmas are externally ignited by conventional power supplies. The physical behavior of these low frequency discharges is readily measured and well known, with an electron density limited to 10¹⁹ m⁻³ as shown in Ref. 17. In the present work, we seek to understand a MMW-driven plasma which produces an electron density on the order of 10²⁰ m⁻³. The MMW plasma is distinct because it is sustained within the PhC by the incident waves, rather than an external low-frequency power source. In addition, the plasma is not confined to a discharge tube but is freely floating within the PhC. A detailed investigation of the MMW plasma has been restricted due to difficulties in plasma diagnostics within the obstructed spaces and small volume of the PhC. Even the size and shape of the plasma are neither visible nor known, but these questions along with electron density, gas temperature, and MMW spectroscopy transmission are addressed here.

Previously, a simple Drude model has been used to estimate electron density based on the simulated wave transmission spectra of the PhC.¹¹ This model estimates the volume-averaged electron density within a uniform ellipsoid, but with substantial uncertainty in the approximation of the plasma dimensions, the electron density gradient, and the gas temperature. Our recent work performed an analysis of the emission spectrum of the microplasma formed in a 43 GHz photonic crystal and showed the validity of the simple Drude model at a high argon pressure.²¹ In that work, the rotational gas temperature was measured and used to better calculate the plasma collision frequency. In addition, the central electron density has been measured by Stark broadening at a continuous wave (cw) MMW power of 1.5 W. While electron densities (∼10²⁰ m⁻³) from both the Stark and transmission methods are relatively consistent at high pressures (above 200 Torr), a discrepancy has been reported at low pressures. Specifically, the volume-averaged density is an order of magnitude less than the measured electron density at 25 Torr. In this work, we directly measure the PhC transmission spectrum and compare the experimental results with the simulated transmission. The discrepancy previously observed at low gas pressures is found to be caused by a very large density gradient in the microplasma that alters the scattering properties of the discharge at low pressures. In particular, one only directly observes the weak periphery of the plasma within the PhC, but we demonstrate that the MMW plasma has an intense central core region of less than 1 mm in diameter and that the only readily observed plasma emissions are from the weak boundary regions of the plasma. The intense core region scatters the radiation, but weak boundary regions have very little effect on altering the transmission properties of the PhC.

The outline of this paper is as follows: We experimentally demonstrate the manipulation of MMWs by directly measuring the wave transmission spectra (S₂₁) in the presence of microplasma in a photonic crystal. Variations of S₂₁ are observed to follow the electron density within the vacancy, which is simultaneously measured by emission spectroscopy. The measured S₂₁ is compared to the simulated S₂₁ using an improved plasma model that employs an electron density gradient. This improved plasma model reconciles the differences between previously reported volume-averaged models and the electron density determined by Stark measurements.

Figure 1 shows a schematic of the experiment. The PhC consists of a two-dimensional array (11 × 7) of alumina rods that are 1 mm in diameter, 2.85 mm long, and placed 2.8 mm apart. The central rod is removed from the array, creating a vacancy defect. Additional details are described in the modeling section.

The plasma's cw drive frequency (43.66 GHz) is produced by using a cw drive signal generator (Keysight, N5183A) and a 4× active frequency multiplier (Millitech, AMC-22) similar to the configuration in a previous work.²¹ The transmission spectrum is found using sweep frequencies (43.5–44.1 GHz) produced by another oscillator (HP, 8350B and 83550A plug-in) and a 3× passive frequency multiplier (SAGE Millimeter Inc., SFP-223SF-S1). The drive and the sweep signals are connected to variable attenuators (VAs) that independently adjust the amplitudes. The outputs from each attenuator are connected to a combiner (SAGE Millimeter Inc., SWP-40350302-22-S1). Then, the combined MMWs signals are amplified by a power amplifier (Millitech, AMP-22-40060) and transferred to PhC using standard waveguides (WR-22) and a custom waveguide vacuum port. A directional coupler (D/C, Millitech, CL4-22) samples the forward power (P_in) and the reflected power (P_ref), these are measured by power sensors (Keysight, N8487A) and a power meter (Keysight, E4417A). The power of the sweep signal is set to ∼10 mW, which is much less than the cw drive power (0.1–1.5 W) and therefore has negligible effect on the plasma.

For the determination of the transmission spectra, MMWs transmitted through the PhC are connected to a horn antenna. This horn antenna transfers MMWs to another identical horn that is outside the vacuum chamber. Thus providing a second mechanically flexible MMW feedthrough port. Finally, the wave transmission spectrum (S₂₁) is measured by a spectrum analyzer (Agilent, E4407B) with a harmonic mixer (Agilent, 11970U, 40–60 GHz). Calibration of the system including the spectrum analyzer was performed by measuring S₂₁ without the presence of the PhC, and this transmitted power defined the 0 dB reference level. In this work, we report variations of S₂₁(ω) with respect to plasma parameters in a narrow sweep frequency band (0.6 GHz) near the resonant frequency of PhC (43.66 GHz). The full wave transmission spectrum of this PhC without plasma can be found in a previous work.²² 

For optical plasma diagnostics, the PhC has a 600 μm optical access hole located at the center of the vacancy. The spectrometer (Princeton Instruments, SP-2500) collects plasma emission from the center of the microplasma through an optical fiber (Thorlabs, M29L, Ø600 μm) and a 20 μm wide entrance slit at the output of fiber. A grating (1800 grooves/mm) disperses light on the CCD detector (Horiba Jobin-Yvon, Synapse 2048 × 512 BIUV) with 13.5 × 13.5 μm pixels. Plasma photographs are taken by a camera (Nikon, D90) using a macrolens (Nikon, AF-S DX Micro-NIKKOR 85 mm F/3.5) through a glass window, as shown in Fig. 1. Luminous plasma light intensity is measured to estimate the plasma dimensions. In order to avoid a non-linear response of optical intensity, the shutter speed of the camera is adjusted with a minimum aperture to guarantee that the camera sensor remains in the linear region (<150 out of 255 counts).

The PhC is placed in a vacuum chamber and evacuated by an oil-lubricated vacuum pump, maintaining a base pressure of 10 mTorr with a leak rate of 10⁻⁴ Torr min⁻¹. Argon and 500 ppm hydrogen are fed into the chamber, where the pressure is measured by a piezo transducer gauge (MKS, 902B). All experiments are performed using a static gas fill to maintain consistent levels of hydrogen for Stark measurements. Plasma is initially ignited at 40 Torr using a MMW drive power of 2 W, then gas pressure and drive power are adjusted to the desired conditions.

Plasma behaves as a frequency-dependent dielectric having a complex permittivity. The complex plasma permittivity (ɛ_p = ɛ_re + jɛ_im) is a function of the plasma frequency (ω_pe) and electron-neutral collision frequency (ν_m) given as

where ω is the angular frequency of the excitation electromagnetic field, n_e is the electron density, ɛ₀ is the permittivity of free space, and e and m_e are electron charge and mass. In the collisionless regime (ν_m << ω), the real part of the plasma permittivity is much greater than the imaginary part. Plasma permittivity decreases from approximately 1 to negative values as the electron density increases and ω_pe> ω. When plasma has a negative permittivity, waves cannot propagate and merely evanesce. In the high-pressure collisional regime (ω << ν_m), plasma permittivity is positive but less than 1. Waves may propagate through the plasma medium even with a high electron density, but losses within the collisional plasma significantly attenuate the incoming waves.

Figure 2 shows the measured plasma parameters of the peak electron density (n₀), the rotational gas temperature (T_g), the complex plasma permittivity of the real part (ɛ_re) and the imaginary part (ɛ_im), and the ratio of collision frequency and angular excitation frequency (ν_m/ω) vs MMW drive power in a pressure range of 3–400 Torr. The optical measurement collects light from the central core region of the plasma and is primarily sensitive to the maximum electron density. Details regarding the spectroscopic measurement and analysis of n₀ and T_g are omitted, but are fully described in previous works.^21,23 Complex plasma permittivity is obtained from Eq. (1) using the measured peak density (n₀) and the gas temperature (T_g). Collision frequency at a discharge pressure (p) is approximated based on a numerically obtained value at p = 10 Torr and T_g = 300 K¹⁵ given by ν_m = 5.5 × 10⁹× p × 300 × T_g⁻¹ s⁻¹. The data point for lowest drive power of 0.1 W at 3 Torr is absent since these conditions failed to sustain plasma. While the peak electron density was relatively unchanged, gas temperature was increased by a higher drive power. The plasma is essentially collisionless below 10 Torr (ν_m/ω ≤ 0.2) and has a negative value of real permittivity. Collision frequency and angular excitation frequency are nearly equal at 100 Torr, where the plasma still has a negative real permittivity. Plasma becomes collisional at 400 Torr and has positive real permittivity. The microplasma's core, however, is overdense (ω_pe> ω) as the plasma frequency (f_pe ≥ 50 GHz) is above the excitation frequency (43.66 GHz) for all measured conditions (n_o > 2.3 × 10¹⁹ m⁻³).

The plasma properties are coupled with the photonic crystal such that the propagation of incident MMWs is altered by incident power and gas pressure. Figure 3 shows the measured transmission spectra (S₂₁) at the same discharge conditions reported in Fig. 2. The large peak displayed at 43.66 GHz is due to the plasma drive power; it appears because no drive power was applied during the calibration of S₂₁. For low pressures (ν_m << ω), the observed plasma permittivity shifts the initial resonant frequency of the PhC (43.66 GHz) to a higher frequency. The largest shift of PhC resonance is observed at 10 Torr where we also observe the minimum in the real plasma permittivity. Although the plasma has a negative permittivity at the plasma core, waves are observed to propagate through the plasma medium. This is because the plasma has a density gradient within its volume such that the plasma volume has a positive average permittivity. More details about spatial density distribution will be presented in Sec. III B.

Unlike the low pressure observations, the collisional plasma (>100 Torr) has no PhC resonance because the high density plasma scatters the waves. The microplasma scattering is similar to that of the dielectric rods in the PhC but with plasma-induced losses and wave absorption. In the presence of high pressure microplasma, the photonic crystal becomes “defectless” but lossy. As a result, the PhC no longer has a resonating cavity and the wave transmission throughout the entire frequency range was decreased.

In this section, we determine the spatially resolved electron density using a diffusion model that is validated by comparing the measured and computed microplasma emission intensity. Since observation of the plasma core is obscured by dielectric rods (see Fig. 4) as well as top and bottom conducting boundaries, only the radial variation of the optical intensity near the periphery can be measured. This measured intensity profile will be numerically fitted to a line-integrated model prediction with the assumption that emission intensity is proportional to electron density. This assumption is reasonable for low pressure plasmas with limited optical self-absorption. In addition, the region being measured optically should have a uniform electron temperature such that the excitation frequency is primarily a function of electron density only. These conditions are met in the low pressure glow outside the region of power absorption (as shown at the end of this section).

Figure 4 shows a typical plasma image and the measured intensity profiles along the central plane of the PhC (see the white dotted line). From the profiles, one notes that the peak intensity and overall size of plasma were increased by a higher power. While a symmetric intensity profile is seen at low powers, it became asymmetric for higher powers. The plasma core shifts slightly to the left, where the MMWs are incident. The maximum plasma radius (R_max) is 2.19 mm for the maximum power of 1.5 W. The distance from the center to the edge of a next dielectric rod (d) is 2.28 mm, thus plasma does not contact the dielectric rods at this pressure (10 Torr). It is difficult to estimate the height of the plasma due to the obstructed view, but it is less than the height of inner structure (h = 2.85 mm).

The measured intensity is actually the line-integrated intensity along the optical path of the camera lens (y axis). The Abel inversion is sometimes used to derive the local intensity, but this method requires a more complete dataset. Since viewing plasma in other dimensions is blocked, we approach the Abel inversion problem from the opposite direction by using a modeled spatial electron density distribution. Once the approximate electron density distribution is computed, it is then numerically line-integrated to obtain the computed plasma emission intensity profile. The computed intensity profiles are then fitted to the experimental intensity profiles in Fig. 4.

We use a simple diffusion model to approximate the electron density distribution in the low pressure regime. The spherically symmetric plasma was modeled with assumptions that there is no three-body recombination (L = 0) or ionization (G = 0) within the volume of diffuse plasma except for ionization in the central plasma core (r < R₀). The diffusion equation for r ≥ R₀ becomes

where ∂n/∂t = 0 in the steady state, and D_a is the ambipolar diffusion coefficient. From Eq. (5), the electron density as a function of radius r is found to be

where k and c are constants. Equation (6) is solved by using boundary conditions from the measurements. The peak electron density (n₀) in the plasma core was measured by Stark broadening. The maximum plasma radius (R_max) was obtained from the plasma contours in Fig. 4, such that

where R₀ is the radius of the central plasma core, n(R₀) = n₀ and R₀ << R_max. The term R₀/R_max is necessary to preserve the condition that n(R_max) ≈ 0. This theoretical electron density distribution of a spherical microplasma is numerically integrated along the y-coordinate axes with the known parameters of n₀ and R_max to create a computed plasma image. Figure 5 shows the computed plasma image along with the computed plasma intensity profiles in one dimension that confirm the measured intensity profiles at 10 Torr. The black rectangular area at the center of the image represents occlusion by the dielectric rods. The computed intensity profile shows that the peak intensity in the plasma core is much higher than intensities of the visible region. For the purpose of curve fitting, the center of the plasma core was shifted left by 0.08 mm at the maximum power of 1.5 W. The error in the estimated maximum plasma radius (R_max) is ∼0.04 mm, which corresponds to the interval between pixels in the photograph.

While the peripheral electron density distribution n(r) is accurately approximated by the diffusion model, the model has uncertainty in determining the size of the central core R₀. Specifically, the computed intensity profile outside the plasma core (r >> R₀) is not sensitive to R₀. Thus, another method is required to more precisely determine the dimension of the electron-rich core. We used a 3D electromagnetic simulation (ANSYS EM19.2, HFSS) that computes the wave transmission spectra (S₂₁) of the PhC based on the measured plasma properties and theoretical gradient. Because the simulated wave transmission spectra is highly sensitive to R₀, it is possible to derive R₀ by matching the measured S₂₁ (Fig. 3) to the simulated S₂₁. The computed intensity profiles, as shown in Fig. 5, were obtained using this method to determine R₀. Details regarding the electromagnetic simulation will be discussed in Sec. III C.

Figure 6 shows a summary of the plasma dimensions for the radius of the plasma core (R₀) and the maximum plasma radius (R_max) at pressures of 3 and 10 Torr. Higher pressure data could not be accurately fit by the diffusion model due to the emergence of three-body recombination. The core volume of plasma expands at higher power. At lower pressures, more rapid diffusion results in a larger R_max but with a smaller R₀. The volume-averaged density, however, is almost constant regardless of R₀, and the estimated effective real permittivity varied between 0.97 and 1 for all the conditions investigated. This average permittivity is consistent with the estimated permittivity reported by previous models using a volume-averaged plasma.

Next, we examine the validity of the model assumption that ionization is restricted to the central core (r < R₀). Using a predefined R₀, the power density absorbed by the microplasma can be found from P_abs = ¹/₂σE² [W m⁻³]. The plasma conductivity σ is proportional to electron density. The electric field is primarily in the z-direction with the form E_z(y) = E_ocos(πy/h_y), where h_y is the PhC periodicity. Using the diffusion profile of Eq. (7), the power absorption can be approximated as

where δ is a constant and n(0) = n₀. Figure 7 shows the normalized plasma power absorption along the y axis (1.5 W at 10 Torr). This power absorption profile confirms that most of the power is absorbed within the small volume of the plasma core and is consistent with the approximation used in the diffusion model: little or no ionization occurs except for the microplasma core (r < R₀).

Previous publications use a Drude plasma model with abrupt boundaries to compute the PhC transmission spectrum. Those models, however, do not predict electron density accurately for an unconfined low-pressure microplasma. Using the diffusion model from the previous section, we now discuss the details of an improved electromagnetic model using a more realistic plasma profile. Figure 8 shows the 3D structure of the photonic crystal and the plasma model. The ellipsoidal shape of Drude plasma was approximated from the plasma image, as shown in Fig. 4. A stepwise diffuse plasma model having 11 shells is placed at the center of photonic crystal. Each ellipsoidal plasma shell has the same aspect ratio (∼1.6:1) but the decaying electron densities (i.e., complex plasma permittivities) follow n(r) in Eq. (6). The radius of the nth shell is selected to be r_n = R₀ × (R_max/R₀)^0.1n, where the radius of the core shell (S₀) and the most external shell (S₁₀) are R₀ and R_max, respectively. The model computes S₂₁, the transmission from wave port 1 to wave port 2 (see Fig. 8), using ∼280 000 finite elements with a convergence accuracy of ΔS = 0.1%.

In these simulations, S₂₁ depends on the radius of the plasma core (R₀). Using experimental results, the simulation's variable parameters, such as n₀ and R_max, were assigned based on the measured value at each power and pressure. Then, R₀ was varied incrementally until the simulated transmission maximum agreed with the experimentally observed maximum. The simulated S₂₁ using the best-fit R₀ is shown in Fig. 9. The resonances of the model match those of the experiments, but there are two issues regarding this comparison. The original resonant frequency of PhC in the simulation (n₀ = 0) is higher than the measured resonance by ∼0.04 GHz. This discrepancy is due to errors in permittivity and dimensions of dielectric rods; it is corrected by using a lower permittivity for the dielectric rods (at 44 GHz) compared to the manufacturer's permittivity specification at 1 GHz. The second issue is that while resonant frequencies in both measured and simulated S₂₁ are the same, transmittance at the resonant frequency is somewhat higher in the simulated S₂₁. Collision frequency affects the wave attenuation rather than resonant frequency, and the approximated collision frequency probably causes this inconsistency in transmittance. Since our collision frequency expression was obtained at T_e = 3.3 eV and 10 Torr, the inconsistency is relatively small at 10 Torr (∼3 dB). However, it becomes larger at 3 Torr (∼8 dB) due to the dependence of collision frequency on the unknown electron temperature.

We have experimentally investigated microplasma formed in a millimeter wave photonic crystal. A method is introduced for in situ measurement of the transmission spectra while simultaneously sustaining a microplasma at 43.66 GHz. In parallel with the wave spectra, an electron density of 0.3–1.5 × 10²⁰ m⁻³ is measured over an argon pressure range of 3–400 Torr. In the center of the microplasma, negative plasma permittivity is found at pressures below 100 Torr. Wave transmission spectra show that the low-pressure plasma core acts as a negative dielectric and shifts the transmission maximum of the PhC. On the other hand, collisional plasma at higher pressures (ω ≤ ν_m) causes a higher power absorption, more wave scattering, and a vanishing resonance of the PhC cavity. In practice, the reduced gas pressures needed for a tunable PhC could be achieved by enclosing the device in a sealed glass envelope similar to a vacuum tube or plasma display.

A major discrepancy in electron densities between Stark broadening and Drude plasma modeling has been reported in previous work. This discrepancy is resolved by using an improved plasma model which is derived from a simple diffusion solution. The diffusion model suggests that the central core of the unconfined microplasma is very small and dense. The dimension of the electron-rich plasma core is found to be much smaller than the overall plasma size, as R₀/R_max ≤ 0.03. This core is the plasma volume measured by the Stark method, hence the Stark density is much greater than the average density derived from simpler Drude models. At low pressures, a low density plasma diffuses beyond this central core. The low density plasma is visible but has little effect on wave scattering compared with the central core and must be modeled carefully. Drude models for unconfined plasma that use abrupt discontinuous plasma boundaries may not be accurate, so a diffuse plasma model should be used within the PhC at low gas pressures (<100 Torr).

All authors contributed equally to this work.

This work was supported by the Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-14-10317 through a Multi-University Research Initiative (MURI) Grant titled “Plasma-Based Reconfigurable Photonic Crystals and Metamaterials” with Dr. Mitat Birkan as the program manager.

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