Four-Wave Mixing techniques (FWM)
Recent work has demonstrated the ability to measure the E-field using nonlinear four wave mix-ing (FWM) techniques, in particular, a technique which is very similar to coherent anti-Stokes Raman spectroscopy8, a well-known diagnostic for measurement of gas temperatures, species con-centrations, and vibrational distribution functions . This basic technique was first described and disassembled by Gavrilenko et al. in 1992  and is described in detail in Refs [102, 103].
According to Simeni et al. , the FWM method has many practical applications using various nonlinear media in the visible-to-infrared region of the electromagnetic spectrum (e.g., dispersion compensation, cavity resonators, and image processing), viable nonlinear media for 8 also called Coherent anti-Stokes Raman scattering spectroscopy (CARS), is a nonlinear Raman spectroscopy tech-nique that uses two very strong collinear lasers to irradiate a sample. The frequency is usually kept constant, with the second laser tuned so that the frequency difference between the two lasers equals to the frequency of a Raman-active mode of interest wavelengths longer than 10 µm have not been fully developed. Using four-wave mixing plasma diagnostics, important plasma parameters such as wave resonances, plasma susceptibility, electron temperature, plasma velocity, plasma density, magnetic field strength, etc., may be measured non intrusively by electromagnetic waves. The FWM beam intensity is proportional to the squared E-field integrated along the length of the discharge electrodes. Figure 1.11 shows a schematic of the FWM experimental made by Muller et al. .
Electric-Field Induced Second Harmonic generation (E-FISH) method
In the presence of an external E-field, the symmetry can be broken, allowing the molecules to radiate light at the second harmonic frequency normaly forbiden in centro-symetric media like gas. Electric-field-induced second harmonic generation (E-FISH), a non-linear optical phe-nomenon, arises from the interaction between the E-field of an external bias and that of two inci-dent photons. The technique was first discovered in the 1960s and 1970s, and has since been used largely to measure the hyperpolarizabilities of different species . It has recently demonstrated significant potential as a method for making absolute E-field measurements in non-equilibrium plasmas and gas discharges. Research at the Laboratory of Plasma Physics (LPP9) has focused on both the application and development of the E-FISH method (see Figure 1.12). For more informa-tion about this technique we can refer to [91, 106, 107].
Electric field measurement methods
Figure 1.12: Schematic of the E-FISH optical setup from . BT: beam trap; DM: dichroic mirror; DP: dispersive prism; FL: plano-convex spherical lens; HWP: half-wave plate; LP: long pass filter; PB: 532 nm polarizer; PD: photodiode; PMT: photomultiplier tube with attached iris and 532 nm band pass filter.
Electro-Optical method (EO)
The goal of this research was to develop and evaluate a diagnostic technique to perform the real time analysis of the E-field vector associated to plasma discharges. The technique is based on electric field measurement via the Pockels’ effect and will be used for the measurement of vectorial electric field in DBDs and plasma jets and will be detailed later in this manuscript (Chapter 3).
Figure 1.13: Schematic of an EO setup.
The use of this effect for high voltage measurement was considered in 1999 by  and . Pigtailed EO sensors are naturally becoming a reliable and consistent mean of characterization for many applications, e.g. high power microwaves (HPM), electromagnetic interference (EMI), on-chip diagnostic, bio-electromagnetism (e.g. influence of mobile phones on the human body) [110, 111]. An example of a schematic of this EO technique exploiting a fibered sensor is represented in Figure 1.13.
Advantages and limitations of different techniques
In this section, we will discuss some advantages and limitations of all the techniques above-mentioned and summarized in Table 1.2.
For non-invasive E-field measurement, techniques that monitor the field-induced Stark shift, such as Stark spectroscopy and LIF-dip Stark spectroscopy, have been used with great success and allow the measurement of E-fields in neutral gases and low ionization plasmas. But the main limitation is the pressure of the gas of the discharge (the maximum pressure has been estimated to be around 104 Pa) due to effects such as the collision quenching of the intermediate level reduces the fluorescence intensity at higher pressure.
The FWM technique has been used for several years to measure discharge fields containing hy-drogen and nitrogen. While being a useful measuring tool, it suffers from the drawback that it relies on molecular resonances and has therefore only been used in the gases above-mentioned. Additionally, due to phase matching requirements, a collinear geometry must be used, limiting the spatial resolution of the technique. The temporal resolution of these measurements is also limited by laser pulse duration and discharge time jitter .
E-FISH, which has lately been redeveloped and successfully employed as a diagnostic for E-field measurements in plasma, presents the key advantage to be applied in virtually any gas, without constraints on the gas composition . The signal production is governed by the duration of the laser excitation, and in principle permits sub-ps temporal resolution with ultrashort laser pulses. But it also remains some disadvantages that rely on the high intensity of the laser (some experi-ments done in our laboratory verify the initiation of plasma by a femtosecond laser). Indeed, the measurement uncertainty of the absolute E-field and the Gouy phase-shift10.
EO sensors have been the object of growing attention for more than four decades . They present a frequency bandwidth spreading from 30 Hz up to several GHz, a dynamic range higher than 130 dB, a sensitivity much better than 1 V/m, a 1 dB compression point exceeding several 100 kV/m, and a spatial resolution lower than 1 mm3. Those pigtailed EO sensors constitute handful tools suitable for many applications, especially for absolute and vectorial measurements of high E-fields in harsh environments [113–117]. But it also presents some limitations inherent to the sensor used. These limitations mainly concern the disturbance induced by the sensor in case of measurement within the plasma and the potenial presence of space charges onto the sensor itself.
Generalities and applications of THz radiations
Electromagnetic (EM) waves are one of the most powerful tools for studying the world around us. Among the various domains of the electromagnetic spectrum, the visible domain corresponds to a small window through which we see the world every day (spectral interval between 7.1014 Hz and 4.1014 Hz). On one side of this frequency range are the ultraviolet (UV) rays, responsible for our summer tanned complexion, and on the other side is the infrared (IR) domain discovered by William Herschel in 1800. In recent years, the scientific community has paid increasing attention to EM waves whose frequencies are in the terahertz range. This spectral range located between 100 GHz and 10 THz, corresponds to the far infrared of opticians or the sub-millimeter range of electronics. In this chapter, we are interested in studying the terahertz « THz » domain. Section 2.1 is devoted to the presentation of THz sources and applications related to the THz range while section 2.2 presents their different method of generation and detection using femtosecond laser pulses.
Generalities and applications of THz radiations
Among the different wavelength ranges of the electromagnetic spectrum, the scientific com-munity studies very intensely the waves of TeraHertz frequencies (THz). It defines the area of the spectrum of electromagnetic waves whose frequencies spread from 100 GHz to 10 THz, which corresponds to wavelengths between 30 µm and 3 mm . The electromagnetic spectrum is represented as a function of frequency and wavelength in Figure 2.1.
Figure 2.1: Diagram of the electromagnetic radiation spectrum, showing the location of terahertz (THz) radiation .
Table of contents :
I Plasma as intense field source and electro-optic detection
1 Overview of plasmas and focus on low frequency plasmas
1.1 Generalities and sources of plasmas
1.2 Classification of plasmas
1.2.1 Physical parameters of plasmas
1.2.2 Types of plasma
1.2.3 Plasma sources and applications
1.2.4 Plasma diagnostics
1.3 Electric field measurement methods
1.3.1 Stark polarization spectroscopy
1.3.2 Four-Wave Mixing techniques (FWM)
1.3.3 Electric-Field Induced Second Harmonic generation (E-FISH) method
1.3.4 Electro-Optical method (EO)
1.3.5 Advantages and limitations of different techniques
2 Plasma induced by femtosecond laser as THz emitter
2.1 Generalities and applications of THz radiations
2.2 Generation and detection of THz radiations
2.2.1 THz generation techniques
2.2.2 THz radiations Detectors
3 Electro-optics techniques from quasi DC to THz range
3.1 Electro-optics in isotropic crystals (vectorial calculation)
3.1.1 Electro-optic effect
3.1.2 Isotropic crystal
3.1.3 Theoretical calculation
3.1.4 Several crystal cut
3.2 Real time electro-optic measurement techniques at « low » frequency (< 100 GHz)
3.3 Equivalent time electro-optic measurement techniques at « high » frequency (THz)
II Vectorial and real time characterization of plasmas by EO method
4 Filamentary discharges: Vectorial characterization of the electric field induced by dielectric barrier discharges
4.1 Experimental setup
4.2 Experimental results: transient measurement and analysis of the field surrounding the DBD
4.2.1 Voltage threshold & simulation analysis of the DBD configuration
4.2.2 1D mapping of the radial field
4.2.3 Additional experiment campaigns involving two EO probes
4.3 Influence of the probe on the DBD behavior
5 Vectorial analysis of the electric field induced by a cold atmospheric pressure plasma jet
5.1 Experimental setup
5.2 Specific calibration of the EO probe
5.3 Experimental results and discussions
5.3.1 Spatial evolution of the Laplacian field along the tube & Electrostatic simulation
5.3.2 Electric field measurements: Mapping of radial and longitudinal E-field components associated to the APPJ
5.3.3 Spatio temporal evolution of both components of the field
5.3.4 Polarimetric mapping of the E-field
5.3.5 Determination of ionization wave front velocity
5.3.6 Influence of an organic target on the field behavior
5.4 Perturbation of the field induced by the EO probe
III Non-linear optics for THz generation (different aspects involving optics and plasmas)
6 Impact of gases and plasmas on THz generation
6.1 Experimental setup and THz pulse
6.2 Experimental measurements and results
6.2.1 Influence of plasma jet length on the orientation of the THz pulse
6.2.2 Influence of helium gas on the modulus and orientation of the THz pulse
6.3 Model description
6.3.1 Linear distribution of THz sources : z-scan
6.3.2 Study of the efficiency of THz generation
6.3.3 Results and interpretations
General conclusion & Perspectives
Perspectives and future work