Bogdan Ksiezopolski

Adam Wierzbicki

Damian Rusinek

On the modelling of the computer security impact on the reputation systems

Reputation systems are an important factor for building trust in virtual communities. In the article we introduce reputation module for Quality of Protection Modelling Language which allows to represent the reputation system as a part of the protocol, where all operations and communication steps can be consistently modelled. Owing to the proposed approach the reputation systems can be formally specified and computer security impact can be considered as a factor of the metrics in the reputation systems. Finally, we model and analyse the case study of the eBay reputation system with modification which will refer to the computer security impact.

Introduction

Reputation systems are one of the most successful and important forms of trust management in open communities. Sophisticated algorithms and system designs can be applied in order to increase the resilience of reputation systems to various forms of attacks, such as coalition attacks, whitewashing, Sybil attacks and many others. Attacks against reputation systems depend on the capabilities of the attackers and defence mechanisms that protect reputation systems which significant part are based on technical measures [START_REF] Hoffman | A survey of attack and defense techniques for reputation systems[END_REF]. The complete analysis of the computer security impact for the distributed reputation systems is a very difficult task, which should nevertheless be performed, because only then we can be sure that a reputation system has been properly tested. The completeness of analysis can be achieved by means of a formal approach, where reputation system can be represented as a part of the protocol, where all operations and communication steps can be consistently modelled. In the article [START_REF] Ksiezopolski | Quality of Protection modelling language for cryptographic protocols[END_REF] B.Ksiezopolski introduced the Quality of Protection Modelling Language (QoP-ML) which provides the modelling language for making abstraction of cryptographic protocols that put emphasis on the details concerning quality of protection. The intended use of QoP-ML is to represent the series of steps which are described as a cryptographic protocol. Additionally, in the QoP-ML the security economics analysis can be performed which is named adaptable security in literature [START_REF] Ksiezopolski | Adaptable security mechanism for the dynamic environments[END_REF][START_REF] Ksiezopolski | Adaptive approach to network security[END_REF].

In this article we would like to present the syntax and semantics of a new structure for the QoP-ML which is required for analysing a reputation system.

A major contribution of this work is introducing the ability of formally specifying distributed reputation systems together with other protocols and functions of information security that support the reputation systems. Owing to the introduced reputation module the reputation systems can be analysed from the technical and information security perspectives. The reputation values of agents are calculated according to the defined algorithms which are abstracted as a process in the operating system which is realized by means of the host. This host is defined as a part of the whole IT architecture by means of which distributed communities can be abstracted. In this infrastructure one can model defence mechanisms and analyse their impact on the reputation system. The reputation module proposed in this paper, which is a part of the QoP-ML, is the first modelling language which allows abstracting and analysing reputation systems from the technical and information security perspectives in a formal way.

Reputation in QoP-ML

In this article we introduce the new reputation module owing to which the reputation analysis can be prepared simultaneously with standard quality of protection analysis of used security mechanisms. The reputation analysis will be will be performed simultaneously with the QoP evaluation according to the methodology presented in the article [START_REF] Ksiezopolski | Quality of Protection modelling language for cryptographic protocols[END_REF].

For reputation modelling in the QoP-ML one has to use two structures which were introduced in the QoP-ML: functions and security metrics. Additionally, we would like to introduce a new structure which will be used for reputation modelling, the modules structure. The semantics of all structures which are required during the reputation modelling in the QoP-ML is presented in the next sections.

Functions

The function modifies the states of the variables and pass the objects by communication channels. The function with the reputation qop parameters is declared as follows:

fun post(id)[Reputation: par1, par2, par3, par4, par5]
This function is named post and includes two types of factors. The functional parameters, which are written in round brackets, they are necessary for the execution of the function. The additional parameters, which are written in square brackets, influence the system reputation.

Security metrics

In the case of representing the reputation metrics, the structure data* will be used because the reputation can not be measured but can be modelled [START_REF] Ksiezopolski | Quality of Protection modelling language for cryptographic protocols[END_REF]. Below we present the example of the metrics structure used for reputation modelling.

metrics { } data*() { primhead[function][reputation:algorithm] primitive[post][alg1] } }
The body of the data* structures contains two operators: primhead and primitive. The primhead operator defines the required parameters for agent actions which influence its reputation. In the presented example for data* the two parameters were defined, the first one is the function name which describes the operation influencing the modelled reputation. The second one defines the name of the additional module (reputation) for the previously defined function with the name of the algorithm which calculates the value of the reputation of this function. Then, the primitive operator is used which defines the details about previously defined functions. In our example the data* operator defines that the post function will be calculated according to the algorithm defined in the module reputation and the name of this algorithm is alg1.

Modules

In this article we introduce a new structure which will be used for defining details for different analysis modules according to the base QoP-ML analysis. This structure is named modules. In the presented approach, the structure for the reputation analysis will be presented. The security metrics structure in the QoP-ML approach is based only on static values which are defined as the primitive structure. One of the main features of the modules structure is enabling the representation of the results in the dynamic way. It means that the results defined in this structure can be estimated by means of the algorithms define mathematics operations.

In the next part of this section the exemplary declaration of this structure will be presented. Afterwards, this structure will be described.

modules { reputation { # rep=0 alg1(par1, par2, par3, par4, par5){ if(rep<=100){ extra = (par4 * par5)/2; rep = rep + (par1 * par2 * par3) + extra; } if(rep>100 || rep <200){ extra = (par4 * par5)/2; rep = rep + par1 + par2 + par3 + extra; } else{ rep = rep + (par1 + par2 + par3 + par4 + par5)/5 } } } }
The modules structure is started by the operator modules. Inside the body of the modules structure one can define different types of modules. In the presented example the reputation module is described. In the specific modules, the initial values of variables can be defined, they are precoded by the # operator. After this, the algorithm which estimates the reputation value is defined alg1(par1, par2, par3, par4, par5){}. The name of the algorithm is not restricted and in the presented example it is alg1. In the round brackets the parameters of the algorithm are defined. The values of these parameters are defined during the QoP-ML protocol modelling in the specific function which are taken into consideration during reputation modelling. The body of the alg1 algorithm includes the arithmetic operations which define the algorithm of reputation value calculation. In the modules structure one can use condition statements. In this example, three possible calculations can be prepared and they are changed depending on the current value of rep variable. When the rep variable will be lower or equal to 100 (rep<=100), then the first conditional statement will be true and the relevant algorithm will be executed. When the rep variable will be higher than 100 and lower than 200 (rep>100 || rep <200), then the second conditional statement will be true. In other cases the else structure will be executed.

In the modules structure one can use other operators which are the same as in the language C.

Case Study -reputation in the eBay

In the article we would like to model the eBay reputation system [START_REF] Schlosser | On the Simulation of Global Reputation Systems[END_REF] with modification which will refer to the computer security impact. In the QoP-ML one can model the reputation system where the mark will be modified depending on the security measures used. One can imagine the scenario where the rates of seller transaction will be modified by the attacker as the part of Men in the Middle Attack. That kind of attack can be easily performed when the rate will be submitted by means of non-encrypted channel. As the defence from this attack can be usage of TLS protocol, which first of all authenticates the eBay server and encrypts transmitted data. In this case study we would like to present the possibility of modelling the reputation systems with computer security impact and in the basic way the reputation of an agent (seller) a can be computed as:

r(a) = n m=1 (m • s) (1)
where: r(a) -the overall reputation value of the agent a; m -the single rate for the transaction from the set {-1, 0, 1}; s -the security impact value, s ∈ R; n -the number of transactions.

In the case study we presented three scenarios which differ in the type of security technology used for rating the eBay transactions. We modelled the agents reputation system based on formula 1. The security impact s will be equal to 1.3 when the transaction rating is sent by the security channel (with the TLS protocol) and will be equal to 0.7 when the the security channel is not used. It means that the transaction rate will be increased by 30% when the rate is secured against technical attacks and will be decreased by 30% when it is not secured.

For all scenarios we assume that agents (sellers) have 100 transactions and these transactions are rated by other agents (buyers or reviewers). We assume that all of these transactions are good and only honest buyers can asses the transactions. In the first scenario (version 1) the buyers will send 100 reviews by means of the secured channel (with the TLS protocol). In the second scenario (version 2) the buyers will send 100 reviews by means of not secured channel (without the TLS protocol). In the third scenario (version 3) the buyers will send 50 reviews by means of the secured channel (with the TLS protocol) and 50 reviews by means of the not secured channel (without the TLS protocol).

QoP and reputation evaluation

The QoP evaluation investigates the influences of the security mechanisms for the ensuring security attributes. That kind of evaluation can be found in the articles [START_REF] Ksiezopolski | Quality of Protection modelling language for cryptographic protocols[END_REF][START_REF] Ksiezopolski | On the efficiency modelling of cryptographic protocols by means of the Quality of Protection Modelling Language (QoP-ML)[END_REF]. In this article we would like to focus on the reputation evaluation which is a new type of evaluation introduced here. The QoP and reputation evaluation algorithms are implemented in the Automated Quality of Protection Analysis tool (AQoPA). The AQoPA tool can be downloaded from the web page of the QoP-ML Project [START_REF]The web page of QoP-ML project[END_REF]. In the QoP-ML models library (included in the AQoPA tool) one can find presented in this article eBay reputation model. In the article we analysed a simple reputation algorithm based on the eBay reputation system where we added modification referring to the computer se-curity impact. In Fig. 1 the results, which refer to the three analysed versions (scenarios) are presented. One can notice significant difference between the overall reputation level of the seller for versions 1 and 2. Version 1 refers to the rates, given by the secured channel, while in version 2 rates are not encrypted. The advantage of transaction rates which are submitted by secure channel is that we are sure that these rates were not modified as the results of one of technical attacks. In the second version we can not be sure that the transactions rates were not modified. This lower level of credibility of the submitted transactions reviews causes lower overall reputation level of the seller. The third version shows the combination of two earlier analysed behaviours of the transactions reviewer. In Fig. 1 one can notice that the last 50 rates submitted by means of not secure channel have less contribution to the overall reputation level of the seller.

Conclusions

The aim of this study was to present the new structure for the Quality of Protection Modelling Language (QoP-ML) which is required for analysing a reputation system. Owing to this module one can model and analyse the reputation systems from the technical and information security perspectives. In the article we modelled the case study in the QoP-ML where the eBay reputation systems were modified. In that system the transactions rates were modified depending on the fact of securing the transmission channel between the transaction reviewer and the eBay portal. We have shown that the reputation systems, taking into account factors related to information security, can provide protection against technical attacks on them.

Fig. 1 .

 1 Fig. 1. The reputation evaluation for the modelled system

Acknowledgements

This work is supported by Polish National Science Centre grant 2012/05/B/ST6/03364


Do It Yourself: make your own thermoacoustic engine with steel wool or rice

Keywords: numbers: 43.35.Ud Thermoacoustics, Self-sustained oscillator, DIY

The construction of two demonstrators of thermoacoustic engines is proposed in this paper, using common material from hardware stores. A brief discussion about the thermoacoustic effect highlights the main parameters controlling the generation of self-sustained acoustic oscillations from heat. The construction of the two tabletop demonstrators is then described in detail, so that they can be reproduced with minimum equipment (rice, steel wool, and other hardware easy to find in hardware stores). Experimental results are also presented for both engines and a discussion about their use for teaching purpose is also provided.

Introduction

There is a large number of problems in acoustics in which considering the propagation of sound as an adiabatic process is a good assumption. It is usual to characterise the acoustic process by means of pressure and velocity fluctuations, but a gas parcel submitted to an acoustic wave also experiences temperature (and density) fluctuations. The temperature of a gas parcel compressed adiabatically rises (as can be experienced when operating a bicycle hand pump), while adiabatic expansion leads to a temperature decrease (think of the cold gas coming out of a spray can). Still considering an adiabatic process, the temperature fluctuations τ (t) of a gas parcel can be easily calculated from its pressure fluctuations p(t), using the relation [START_REF] Pierce | Acoustics: an introduction to its physical principles and applications[END_REF][START_REF] Temkin | Elements of acoustics[END_REF] p = ρ 0 C p τ , where ρ 0 and C p respectively stand for the mean (non-oscillating) density and for the isobaric heat capacity of the fluid. Having in mind an order of magnitude of the amplitude of these fluctuations is important: if one considers for instance a plane wave propagating through air under standard conditions (static pressure P 0 = 10 5 Pa, temperature T 0 = 300 K) with frequency f = 200 Hz, and if the associated pressure level is fixed to the threshold of pain for human hearing 20 log 10 (p rms /p 0 ) = 120 dB SPL where p 0 = 2 × 10 -5 Pa stands for the pressure at the human hearing threshold, then the peak amplitude of pressure fluctuations is equal to p p = 20 √ 2 Pa, the peak amplitude of gas displacement equals ξ p = p p /(2πρ 0 c 0 f ) ≈ 50 µm where c 0 stands for the adiabatic sound speed, while the peak amplitude of temperature fluctuation, equals τ p ≈ 23 mK. From these values, one can conclude that temperature (as well as pressure) fluctuations are very small even at large acoustic levels, but it is also noticeable that the temperature gradient experienced by a gas parcel during half an acoustic cycle, namely τ p /ξ p ≈ 460 K/m, is not small. However, the assumption of an adiabatic process fails as soon as the propagation of an acoustic wave through a porous medium is considered. In such situation, the thermal coupling between the solid frame and the oscillating fluid leads to sound attenuation due to irreversible heat exchanges between the fluid and the solid, taking place within the so-called thermal boundary layers. Actually, and although the heat conductivity of a gas is generally very small (typically 26 mW/m/K for air), the losses due to heat diffusion have the same order of magnitude than the viscous losses in porous materials. The mechanism by which acoustic energy is dissipated is a relaxation process, which itself is due to the existence of a phase shift between pressure and temperature fluctuations in the vicinity of the solid walls. Due to this phase shift, the small amount of heat that a gas parcel gives to the solid during its phase of greatest condensation is larger than the amount of heat given by the solid to the gas parcel during its phase of greatest rarefaction.

There is nonetheless a way to transform attenuation into amplification of sound if one finds a mean to provide heat to the gas parcel during its phase of greatest condensation, and (partly) take it away during its phase of greatest rarefaction. This amplification process was identified more than one century ago by Lord Rayleigh in his famous textbook [START_REF] Strutt | The theory of sound[END_REF] when trying to explain qualitatively the operation of devices giving rise to sound generation from heat, such as the Sondhauss tube [START_REF] Sondhauss | Über die Schallschwingungen der Luft in erhitzten Glasröhren und in gedeckten Pfeifen von ungleicher Weite[END_REF] and the Rijke tube [START_REF] Pl Rijke | Notiz über eine neue Art, die in einer an beiden Enden offenen Röhre enthaltene Luft in Schwingungen zu versetzen[END_REF]. More generally, the phase relationship between pressure oscillations and the rate of unsteady heat released to the gas is a key control parameter (now referred to as the Rayleigh criterion) which governs the generation of sound from heat. This Rayleigh criterion prevails for the description of combustion instabilities in gas turbines and rocket engines [START_REF] Tc Lieuwen | Unsteady combustor physics[END_REF], as it also prevails for thermoacoustic engines which are within the scope of this article.

A sketch of a simple thermoacoustic engine is provided in figure 1. The basic principle of such a device involves the application of a steep temperature gradient along a porous medium (the so-called stack) inserted inside an acoustic resonator: if the temperature has the appropriate sign and amplitude (typically larger than τ p /ξ p ), then the thermal coupling between the fluid and the gas may give rise to the reversal of natural heat exchange processes, or in other words (and accordingly with the Rayleigh criterion) to the promotion of acoustic oscillations from the provision of heat during the phase of greatest condensation of a gas parcel. This will lead to the onset of selfsustained acoustic oscillations at the frequency of the most unstable acoustic mode of the device.

Though nowadays the thermoacoustic effect (more precisely the one involved in thermoacoustic engines) is studied by an international scientific community, it has long been considered as a curiosity. It was probably first observed by glass-blowers that their cold blowing cane would sometimes emit a low pitch continuous sound as a molten glass gob was mounted on it. The oldest evidence of a thermoacoustic engine strictly speaking comes from 16th century Japan, with the diary of a Buddhist monk referring to an object called "Kibitsunokama", which translates as "Kettle of Kibitsu". A collection of tales published in 1776 [START_REF] Ueda | Ugetsu Monogatari (Tales of Moonlight and Rain)[END_REF] gives a better description of it: this instrument, that is still used today during ceremonies in the Shintoist temple of Kibitsu, consists of a barrel mounted on an iron bowl containing water. A mesh screen in the barrel is loaded with rice. Put on a fire, the Kibitsunokama will emit a sound much like the bellowing of an ox [START_REF] Ueda | Ugetsu Monogatari (Tales of Moonlight and Rain)[END_REF]. The objective of this paper is neither to provide a detailed description of the thermoacoustic effect (which can be found elsewhere [START_REF] Rott | [END_REF][START_REF] Swift | Thermoacoustic engines[END_REF][START_REF] Garrett | Resource letter: Ta-1: thermoacoustic engines and refrigerators[END_REF]) nor to present new research results on the topic, but it is rather to explain how to build two simple versions of the devices mentioned above with minimum equipment.

Several tabletop build-ups of thermoacoustic engines (and refrigerators) have been proposed in the literature [START_REF] Garrett | The power of sound[END_REF][START_REF] Garrett | Acoustic Laser" Kit Instructions[END_REF][START_REF] Jeromen | A simplified thermoacoustic engine demonstration[END_REF][START_REF] Kj Bastyr | High-frequency thermoacoustic-Stirling heat engine demonstration device[END_REF][START_REF] Ki Matveev | Small-scale thermoacoustic engine demonstrator[END_REF], but they usually require either materials that are not commonly available (such as square pore ceramics used in catalytic exhaust systems, or machined parts of complex geometry) or minimum scientific hardware (e.g. a DC power supply). The two devices described in this paper are portable (they are compact and do not necessary require an electrical input) and very easy to build using widely available materials. We believe that these two devices could be used easily for teaching purpose in order to illustrate basic concepts of acoustics, thermodynamics, or self-sustained oscillations. Section 2 presents the most important design guidelines for building a simple thermoacoustic oscillator. The construction of a test tube thermoacoustic engine (a modern version of the Sondhauss tube) is presented in section 3, and a Kibitsunokama rice-based thermoacoustic engine is described in section 4. Finally, a conclusion is provided in section 5, where the possible use of these two devices for teaching purposes is highlighted.

Basic principles and design guidelines

Two operating principles are to be distinguished for the thermoacoustic effect, according to the conversion process. In both cases, the review paper [START_REF] Swift | Thermoacoustic engines[END_REF] is a good starting point for more complete information.

The first case consists in sustaining a large amplitude acoustic wave through a porous material initially at uniform temperature. The porous material is usually called a stack, because first realisations were made of a pile of thin plates (but any open cell porous media could be used instead). The properly phased displacement and variations of pressure of a small volume of gas allows for it to give away a bit of its heat to the surrounding colder walls when the gas is compressed and "hot", and further take away a bit of heat from the slightly warmer walls when the gas parcel is dilated and cold. By a cumulative effect, heat is advected from an extremity of the stack to the other and a temperature gradient gradually emerges. This use of an acoustic work to produce a thermal gradient is the definition of a thermoacoustic heat-pump or refrigerator, whether the hot or cold side of the system is used. These systems will not be treated further here. Instructions to build a tabletop demonstrator of thermoacoustic refrigerator may be found in reference [START_REF] Da Russell | Tabletop thermoacoustic refrigerator for demonstrations[END_REF].

The second case, the one we are interested in, is the reciprocate effect. Consider a gas parcel oscillating in a porous material subject to a temperature gradient ∆T /l s along the stack of length l s . For a sufficiently large temperature difference ∆T along the stack, and with the proper phase-shift between its displacement and the pressure oscillation, this parcel receives a small amount of heat from the stack when it is compressed and hot, and further gives away a bit of heat to the colder stack when it is dilated and cold. The gas oscillation is thus amplified by the presence of the temperature gradient in the stack. If this thermoacoustic amplification is large enough to compensate all losses in the system, self-sustained oscillations are generated at the frequency of the most unstable mode of the acoustic network. This system is then called a thermoacoustic prime-mover or engine.

At the threshold of the autonomous oscillations, the behaviour of thermoacoustic engines with a known geometry (in particular that of the pores of the stack) can be described thanks to the linear theory of thermoacoustics [START_REF] Rott | [END_REF][START_REF] Swift | Thermoacoustic engines[END_REF]. This theory however fails to predict accurately the behaviour once the self-sustained instability is initiated. Indeed, the saturation of the oscillation in the singing engine is piloted by different nonlinear effects due to the high acoustic amplitudes (amongst them: acoustically enhanced heat conduction, acoustically induced streaming, wave steepening or vortex generation at the ends of the stack). Therefore, the development of an analytical model able to predict accurately the operation of a thermoacoustic engine is a hard task [START_REF] Guedra | Experimental and theoretical study of the dynamics of self-sustained oscillations in a standing wave thermoacoustic engine[END_REF], even for the simple demonstrators presented hereafter.

Nevertheless, some general principles for the configurations presented here can be considered. In this paper, we will limit ourselves to the simple configuration of a stack placed into a resonator open at one end and closed at the other. A schematic view of such a quarter-wavelength thermoacoustic engine is given in figure 1, showing the main elements and dimensions for its characterisation. In this configuration, the working frequency f of the engine is given by the length of the open/close resonator, and is approximated as f c 0 /4L, where c 0 346 m s -1 is the speed of sound in air at room temperature.

The stack is where the thermoacoustic conversion takes place. It must allow the passage of the acoustic wave and therefore have open pores with a low resistance to fluid flow. The pore size, which is evaluated by the average hydraulic radius r h (ratio of the cross sectional area to the wet perimeter), must be such that the heat exchanges between the gas subject to a standing wave and the solid walls are quasi-adiabatic. This condition is fulfilled for a pore size of a few times the thermal boundary layer thickness r h ∼ 2.5δ κ , the latter being frequency dependent such that δ κ ∼ 2.6 × 10 -3 / √ f for air at ambient pressure. For the thermoacoustic amplification to take place, the temperature gradient has to be imposed appropriately. For the case of a quarter wavelength thermoacoustic prime-mover considered in this paper, the temperature gradient must be oriented accordingly with the axial gradient of pressure amplitude of the acoustic mode (otherwise damping would occur instead of amplification). Since thermoacoustic amplification depends on both the relative amplitude of pressure and velocity fluctuations, there exists an optimum position of the stack along the waveguide to minimise the threshold or to maximise the efficiency of power conversion. In the examples described below, the optimum position of the stack is around one quarter to one third of the length of the resonator, h opt ≈ L/4 or L/3, starting from the rigid plug (see figure 1).

Test tube thermoacoustic engine

Following the principles given above, one of the simplest thermoacoustic engine can be built with very few materials, namely steel wool, a test tube and a burner.

Making it

The configuration pictured on figure 2 uses a 140 mm long pyrex test tube of 14 mm in diameter. The stack is a 20 × 40 mm 2 fine steel wool rectangle, cut in a common unrolled steel wool pad. The rectangle is rolled along its width and inserted to three quarters of the length of the tube (i.e. at 4.5 cm of the end). The tube is heated below the end of the stack on the closed side of the tube with a flame, either a lighter or a small spirit lamp. The other side of the stack is maintained at ambient temperature by a soaked paper towel strip rolled around the pipe. A short video clip, showing the fabrication and a run of the engine, has been made available at https://youtu.be/owbjLWrC86g. Built with a 14 cm tube, this engine sings at a frequency close to f c 0 /4L = 630 Hz. The average pore size for which the onset temperature difference will be minimised is then r h ∼ 250 µm. Several trials might be required to adjust the quantity of steel wool to obtain the ideal size of pores in the stack.

Its performances

Figure 3(a) shows a recording of the acoustic pressure at 10 cm of the opening of the tube. The flame is lit at t = 10 s, the engine starts to sing a few tens of seconds later with a maximum sound level around L ac = 98 dB SPL (corresponding to peak pressure variations of 2.3 Pa). It keeps singing as long as the temperature gradient is maintained by the flame and the soaked paper, in this case until t = 200 s when the flame is blown out. The temperature distribution in the engine is hard to evaluate. If available, thermocouples can be integrated in the engine to give a coarse insight of the evolution of the temperature, as shown in figure 3(b), even if the incertitude on the position of the thermocouples and their weak coupling with the oscillating fluid do not allow for absolute measurements. The colour of the iron oxide layer at the surface of the steel wool (see insert in figure 2) is however a good indicator of the maximum temperature reached at the hot side of the stack: depending on the flame used, T h goes from 300 • C to >400 • C, thus covering a range from purple-blue to gray-blue [START_REF]Tempering (metallurgy[END_REF]. Figure 4 shows another view of the same measurements in which a zoom has been performed around the onset of the auto-oscillation, with a focus on the temperature at the cold extremity of the stack. The behaviour of this engine appears simple enough during this period so that different theoretical stages can be qualitatively isolated in accordance with the linear theory of thermoacoustics [START_REF] Rott | [END_REF][START_REF] Swift | Thermoacoustic engines[END_REF]. At first, the temperature gradient (T h -T c )/l s is not high enough for the thermoacoustic amplification to compensate the losses in the whole set-up. Acoustic perturbations are damped (state labelled 1, until t = 44.5 s). The evolution of the cold temperature T c is at this point controlled mainly by the heat diffusion through the stack. When a sufficient temperature gradient has been reached, any small acoustic perturbation is amplified exponentially at the frequency of the least stable mode of the engine (here the quarterwavelength mode), as is predicted by the linear theory of thermoacoustics [START_REF] Rott | [END_REF][START_REF] Swift | Thermoacoustic engines[END_REF] (state 2, from 44.5 s to 45.6 s). The black curves are an exponential fit of the acoustic amplitude. Thermoacoustic amplification then moves away from an exponential growth due to several saturation processes (state 3). One of these saturation processes consists of thermoacoustic streaming, namely the advective heat transport from the hot to the cold side of the stack induced by acoustic oscillations. As the amplitude of the acoustic oscillation rises, this acoustic heat pumping adds to the conduction. Thus T c rises at an increased rate and T h grows more slowly: the amplitude growth of the acoustic oscillation slows down and saturates when a thermal equilibrium is reached (i.e. after ∼100 s, see figure 3).

The construction of this thermoacoustic engine requires just a few elements, that are common in a scientific laboratory and inexpensive. It can therefore easily be built by students in many copies for further experimental studies, as will be discussed in section 5.

Kibitsunokama: a rice-based thermoacoustic engine

An equivalent construction of the traditional Kibitsunokama is proposed here, which itself is inspired from the demonstrator presented by Pr. Ueda during the 2nd International Workshop on Thermoacoustics in Sendai, Japan on May 2014. This configuration, in addition to the use of rice as the stack material, has the advantage of requiring a limited temperature difference for its operation: the cold side of the stack remains almost at room temperature, when the hot side only reaches 100 • C.

Building it

Figure 5(a) shows a photography of different elements that can be used to build a ricebased thermoacoustic engine for less than 30e: long grain rice (1), grids to support the rice (2), a worm gear clamp (3), a fireplace pipe clean-out cap with flat bottom (4), a ventilation duct connector (5), a ventilation duct reducer (6), a flexible foil ventilation ducting [START_REF] Ueda | Ugetsu Monogatari (Tales of Moonlight and Rain)[END_REF], or a single wall chimney pipe (8) that can replace elements (5-7) and a 20 cL cup [START_REF] Swift | Thermoacoustic engines[END_REF]. Elements (3) through ( 8) have a diameter of 125 mm. They can be found in the ventilation or heating area of a local hardware store. The different elements are put together so that the engine total length is L = 46 cm and the stack is positioned at h = 5.5 cm of the closed end. One of the grid is rigid enough to support the rice but its perforations are too large and let the rice through. It is stacked with a second grid, finer but pliable, that has been cut out of a frying pan spatter screen. The cup ( 9) is used to measure both the water quantity and the rice quantity. A schematic view of the assembled system is given on figure 5(b). The grids supporting the rice are simply jammed in the bulge of the reducer. The position of the stack can easily be modified by adjusting the height of the clamp (3) on the reducer [START_REF] Tc Lieuwen | Unsteady combustor physics[END_REF]. The flexible duct can be extended to adjust the overall length L of the engine. The engine is driven by an electric hotplate, which power is limited to a few hundreds watts. A higher heating power allows to reach the boiling point faster, but the large amount of vapour emitted rapidly warms the stack, thus impeding the thermoacoustic amplification to take place. A video clip, showing the fabrication and a run of two configurations of this engine, has been made available https://youtu.be/dBqa5t8nefo.

The onset condition being defined as the compensation of all losses, working on the tightness of the different seals is primordial to get the longest and loudest sound. Electrical tape between the pieces or duct tape over the seams greatly helps with reducing the acoustical losses.

Its operation

Figure 6 presents measurements of the acoustic pressure variations and the temperature on each side of the layer of rice. The time t = 0 is the instant at which the engine is placed on the (cold) hotplate, and the power is switched on. Since the engine starts to sing only a few minutes later, the first 10 min are compressed in the representation. The engine starts to sing when a sufficient temperature difference T h -T c is reached between both sides of the layer of rice, at t = 625 s in this measurement. The onset of the auto-oscillation takes place after 5 min to 10 min, depending on the heating power. The working frequency is close to f 180 Hz.

The emitted acoustic level is unstable but can exceed L ac 116 dB SPL (corresponding to 18 Pa) at its maximum value measured at 10 cm from the open end. The sound maintains for 30 s to several minutes. For the results presented in figure 6, the singing last for almost 90 seconds, time after which the temperature difference is not enough anymore, due to the homogenisation of the temperature of the rice layer by the steam passing through it. For the same reason, too much heat brings the water to a violent boiling that releases an excessive amount of vapour escaping through the Basmati rice is the stack material that offered the best experimental results. Other types of kernels with different size and shapes (cereals or beans) will provide stacks with various porosity and pore size. For example, a green lentils stack works, but the engine sings with a lower acoustic level L ac 95 dB SPL for only a few tens of seconds.

The low temperature operation of this engine is allowed by the presence of water vapour that facilitates the heat exchanges by condensing and evaporating at the interface on the stack during each acoustic cycle. Though the phenomena involved are not yet fully understood, experimental studies [START_REF] Noda | A thermoacoustic oscillator powered by vaporized water and ethanol[END_REF][START_REF] Tsuda | Abrupt reduction of the critical temperature difference of a thermoacoustic engine by adding water[END_REF] have shown that the presence of the vapour can reduce the onset temperature gradient by ∼200 • C for water, and even more for ethanol. The overall behaviour exhibited by this engine is more complex than for the test tube engine. As clearly shown in figure 6, the device rapidly reaches very high sound levels, but the dynamics of evolution of the amplitude exhibits a complex behaviour with a succession of pulses before the final damping. The reason why such a complicated dynamics is observed is not clear, but it might be related to a competition between different high amplitude effects impacting the temperature distribution in the device. Such kind of dynamics have already been reported in other simple thermoacoustic devices [START_REF] Guedra | Experimental and theoretical study of the dynamics of self-sustained oscillations in a standing wave thermoacoustic engine[END_REF][START_REF] Yu | Fishbone-like instability in a looped-tube thermoacoustic engine[END_REF] and research works are still in progress to get a deeper understanding of the processes saturating wave amplitude in thermoacoustic oscillators.

Conclusion

Two demonstrators of thermoacoustic engine have been presented. Based on very simple configurations built from largely available materials, they allow to demonstrate the principle of self-sustained oscillations by thermoacoustic amplification. The construction of such devices is well-suited for under-graduate student's project, and it provides a very demonstrative way to prove that sound can be generated from heat. They are also a basis for further experimental studies. First of all, parameters which influence the thermoacoustic amplification can be investigated empirically in order to improve the acoustic pressure generated (a basic modelling of the device can even be made using the freely available software DELTA-EC [START_REF] Ward | Design Environment for Low-amplitude Thermoacoustic Energy Conversion (DeltaEC Version 6.2) Users Guide[END_REF] developed at Los Alamos National Laboratory). For example, the influences of the stack position along the resonator, of the geometry or material of the stack can be studied. The sound frequency emitted can also be explored by changing the resonator length. The study of the impact of these two parameters can be the starting point to design a thermoacoustic organ pipe [START_REF] Stodola | Development of a thermoacoustic organ[END_REF]. Such a musical instrument can be built by using several resonators of different length, each resonator corresponding to one note.

Secondly, these demonstrators can serve to highlight the interest of thermoacoustic engine in an industrial point of view. Hence, the acoustic power generated by the engine can be converted into electric energy by using an alternator (like a loudspeaker functioning as a linear alternator [START_REF] Yu | A method of characterising performance of audio loudspeakers for linear alternator applications in low-cost thermoacoustic electricity generators[END_REF], a bi-directional turbines [START_REF] Owczarek | Bi-directional turbines for converting acoustic wave power into electricity[END_REF], or a piezoelectric transducer [START_REF] Jensen | Thermoacoustic power conversion using a piezoelectric transducer[END_REF]) or into heat energy by using a thermoacoustic heat-pump [START_REF] Gardner | Waste-heat-driven thermoacoustic engine and refrigerator[END_REF]. The heat energy source provided by the flame can also be replaced by a parabolic solar dish concentrating the sun's rays on the hot side of the stack.

Finally, more academic experiments for pedagogical demonstrations can also be imagined to explore the use of a thermoacoustic oscillator as a specific kind of autonomous, dynamical system. For instance, the coupling of several oscillators [START_REF] Ps Spoor | The Huygens entrainment phenomenon and thermoacoustic engines[END_REF][START_REF] Biwa | Amplitude death in coupled thermoacoustic oscillators[END_REF], or the forcing of a thermoacoustic oscillator by an external sound source [START_REF] Penelet | Synchronization of a thermoacoustic oscillator by an external sound source[END_REF] leads to synchronisation phenomena and other nonlinear processes (quenching, chaos [START_REF] Yazaki | Experimental observation of thermoacoustic turbulence and universal properties at the quasiperiodic transition to chaos[END_REF]). Therefore, thermoacoustic oscillators could be used advantageously for pedagogical purposes in a class devoted to dynamical systems.

Figure 1 .

 1 Figure 1. (a) Schematic representation of quarter-wave length thermoacoustic engine with main dimensions. (b) (Arbitrary) temperature distribution in the fluid, imposed by a couple of heat exchangers, varying between Tc and T h . (c) Schematic representation of the amplitude distribution of acoustic pressure p (green, maximum at closed end and null at open end) and particular velocity v (blue, null at closed end and maximum at open end), normalised by their maximum value.

Figure 2 .

 2 Figure 2. Thermoacoustic engine built from a test tube and a small piece of fine or medium steel wool (enlarged at bottom left). A strip of wet paper towel is wrapped around the cold extremity of the stack to maintain its temperature. The heat source is a small spirit lamp, or a lighter. The color of the steel is a good indicator of the temperature reached at the hot extremity of the stack: straw yellow indicates a temperature of ∼210 • C, brown indicates ∼260 • C and deep blue ∼310 • C [18].

Figure 3 .

 3 Figure 3. Temporal evolution of (a) the acoustic pressure p at 10 cm of the opening of the tube and (b) the temperatures Tc and T h close to respectively the cold (blue) and hot (red) extremities of the stack. The vertical dashed lines symbolise the instants the flame was lit and blown out.

Figure 4 .

 4 Figure 4. Zoom on the onset of the acoustic instability. Temporal evolution of (a) the acoustic pressure variation at 10 cm of the opening of the tube and (b) the temperature Tc close to the cold extremity of the stack. Three regimes can be distinguished. (1): Stability. (2): Exponential growth of amplitude, fitted with the black curve. (3): Progressive saturation by thermoacoustic heat transport.

Figure 5 .

 5 Figure 5. Different elements for the build of a rice-based thermoacoustic engine. (1): long grain rice, (2): grids to support the rice, (3): worm gear clamp, (4): fireplace pipe clean-out cap with flat bottom, (5): ventilation duct connector (6): ventilation duct reducer, (7): flexible foil ventilation ducting, (8): single wall chimney pipe, (9): 20 cL cup. The final assembly is schematically presented on the left.

  rice and thus impeding the set-up of the temperature gradient. When the engine stops to sing, taking it off the hotplate sometimes makes it start singing again abruptly, as shown by the measurements on figure 6 at t = 715 s.

  660 680 700 720 740 760 Time t [s]

Figure 6 .

 6 Figure 6. Temporal evolution of (a) the acoustic pressure p at 10 cm of the opening of the tube and (b) the temperature Tc and T h at both extremities of the stack. The vertical gray line shows the moment just after the extinction of the sound, when the engine has been moved from the hotplate to the workbench. After a instant, the short engine starts to sing again. The first ten minutes of heating have been compressed in the representation, the change of slope at t = 600 s in the hot temperature is due to the change of temporal scale.


Jérôme Degert
email: jerome.degert@ubordeaux.fr

Marion Cornet

Emmanuel Abraham

Eric Freysz

Simple and distortion-free optical sampling of terahertz pulses via heterodyne detection schemes

Keywords: Ultrafast nonlinear optics, (320.7100) Ultrafast measurements, (300.6495) Spectroscopy, terahertz

We present and demonstrate experimentally two simple and distortion-free methods to sample terahertz pulse in zinc-blende-type crystals like ZnTe or GaP. They are based on an optical heterodyne detection scheme which makes it possible to measure the terahertz field by detecting the optical probe beam or its second harmonic. Both are compared to conventional electro-optic sampling and give results in good agreement with the latter.

INTRODUCTION

Electro optic (EO) sampling, hereafter reported as conven tional EO sampling (CEOS), is a widely used technique for the measurement of the electric field of a terahertz (THz) pulse with subpicosecond time resolution [1 3]: in a zinc blende crystal, the polarization state of a linearly polarized ultrashort optical pulse is modulated by the THz pulse through the Pockels effect. The corresponding phase retardation Γ, propor tional to THz electric field amplitude E THz , is then measured by ellipsometry for different time delays τ between the two pulses. The ellipsometer, consisting of a λ∕4 plate followed by a suitably oriented Wollaston polarizer, converts the phase modulation into an amplitude modulation of the optical probe pulse measured by a balanced photodetector. Different alterna tives to this technique have been proposed, but they suffer from some detection nonlinearities [START_REF] Jiang | Electro optic sampling near zero optical transmission point[END_REF]5]. Even though these nonli nearities can be corrected, they increase the complexity of the measurement process with respect to conventional EO. Very re cently, new methods based on a technique commonly used in optically heterodyne detected optical Kerr effect spectroscopy have been introduced [START_REF] Brunner | Distortion free enhancement of terahertz signals measured by electrooptic sampling. I. Theory[END_REF][START_REF] Johnson | Distortion free enhancement of terahertz signals measured by electrooptic sampling. II. Experiment[END_REF]. These distortion free techniques are based on two measurements at opposite optical biases near the zero transmission point in a crossed polarizers detection geometry. They lead to an increase, by 1 order of magnitude, of the signal enhancement factor, defined as the ratio of the normalized detected balanced difference signal to Γ.

Hereafter, we propose two novel and simple schemes, based on a special geometry of the polarization states of the optical and THz beams. The first one leads to an amplitude modulation of the optical pulse at the output of the zinc blende crystal and makes it possible to separate the optical pulse generated by the Pockels effect, hereafter called the signal pulse, from the in cident pulse [START_REF] Cornet | Frequency resolved cross correlation between optical and terahertz pulses: application to ultrashort laser pulse characterization[END_REF]. Thanks to an optical heterodyne detection of these signal pulses, one can perform a distortion free sampling of THz pulses within a single measurement. In the second one, we take advantage of this geometry where the Pockels nonlinearity couples to the intrinsic second order nonlinearity of the EO crys tal, leading to the generation of a second harmonic signal [the Pockels induced second harmonic (PISH) effect], which can be used, through an optical heterodyne detection, to sample tera hertz pulses [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF]. Since the latter detection scheme, which shows some similarities with the first one, offers interesting perspectives for the sampling of THz pulses generated by near infrared pulses, we will present it here with more details than in [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF]. This paper is organized as follows. In Section 2, we present the experimental setup of our optically heterodyne detected EO sampling and derive the expression of the detected signal. The latter expression is then analyzed and confronted with the experiment. Next, in Section 3, we give a comprehensive analysis of the optically heterodyne detected PISH signal and compare it, experimentally, with conventional EO sampling. Finally, in Section 4, we compare both detection schemes.

OPTICALLY HETERODYNE-DETECTED ELECTRO-OPTIC SAMPLING A. Experimental Setup

The experiment is the following [Fig. 1(a)]: a linearly polarized THz pulse is emitted from air ionized by a two color, namely 400 and 800 nm, femtosecond laser field [START_REF] Thomson | Broadband THz emission from gas plasmas induced by femtosecond optical pulses: from fundamentals to applications[END_REF]. This THz pump field is then collimated and focused onto a h110i EO crystal (ZnTe or GaP), with a thickness L (L 300 μm for ZnTe and 200 μm for GaP), by two off axis paraboloidal mirrors with a 150 mm focal length. Inside the crystal, the ŷ polarized THz beam propagates collinearly with a weak 800 nm probe beam, initially ẑ polarized by a half wave plate and a polarizer, the ẑ axis coinciding with the h001i axis of the crystal [Fig. 1(b)]. After the crystal, the transmitted optical probe beam is split by a 50/50 nonpolarizing beam splitter. Both beams are then sent through a quarter wave plate and a polarizer, and their intensity is recorded by photodiodes. The fast axis of the quarter wave plate in front of the photodiode PD1 (resp. PD2) is oriented at an angle β (resp. β) with respect to the ẑ axis. The axes of the polarizers are set along the ŷ axis, so that the incident probe beam is blocked in the absence of THz and β 0°. However, when the probe and THz pulses temporally overlap in the crystal, their interactions within the crystal generates a weak optical signal polarized along the ŷ axis [START_REF] Cornet | Frequency resolved cross correlation between optical and terahertz pulses: application to ultrashort laser pulse characterization[END_REF]. The latter is therefore transmitted by the polarizers. The THz beam is chopped and the signal, ΔI Γ; β I 1 Γ; β I 2 Γ; β, mea sured by the two photodiodes versus the time delay τ between the probe and THz pulses, is recorded by a lock in amplifier.

B. Electro-Optic Sampling by Optical Heterodyne Detection

Let us see what is the expression of the signal ΔI Γ; β provided by the setup of Fig. 1. In the ŷ; ẑ basis, the electric field of the incident probe beam writes

E E 0 0 1 : (1)
At the output of the EO crystal, after interaction with the THz pulse, this field is given by [START_REF] Cornet | Frequency resolved cross correlation between optical and terahertz pulses: application to ultrashort laser pulse characterization[END_REF]

E E 0 i sinΓ∕2 cosΓ∕2 ; (2)
with

Γ 2πL λ n 3 ω r 41 E THz (3)
the phase retardation induced by the THz field, where λ is the wavelength of the probe beam, n ω is the refractive index of the crystal in the optical range, and r 41 its electro optic coefficient. Thus, the electric field of the signal pulse is given by E sig iE 0 sinΓ∕2 ≃ iE 0 Γ∕2 in the weak THz field limit Γ ≪ 1.

Using the Jones matrix formalism [START_REF] Fowles | Introduction to Modern Optics[END_REF], one can derive the expression of the electric fields E 1 and E 1 transmitted by polar izers P 1 and P 2 , respectively, and their intensities I 1 and I 2 detected by photodiodes PD1 and PD2, respectively. All cal culations done, one has

I 1 Γ;β I 0 2 sin 2 Γ 2 cos 2 2β 1 2 sin 2 2β 1 2 sin2βsin Γ (4)
and I 2 Γ; β I 1 Γ; β. The second term in Eq. ( 4) corre sponds to the intensity of the local oscillator,

I LO I 1 0; β I 0 4 sin 2 2β; (5)
and the intensity of the homodyne signal on PD1, I sig , is given by

I sig I 1 Γ; 0 I 0 2 sin 2 Γ 2 : (6)
Now, if one considers the modulation depth M Γ; β I 1 Γ; β I 1 0; β∕I 1 0; β of the signal detected by PD1 [START_REF] Brunner | Distortion free enhancement of terahertz signals measured by electrooptic sampling. I. Theory[END_REF], it is then possible to express the enhancement factor g and the distortion parameter d associated with our method. They are related to the modulation depth, in the weak field limit, by

M Γ; β gΓ d gΓ 2 OΓ 3 : (7)
From Eqs. ( 4) and ( 5), one has

g 1 sin2β ; (8)
d 1 2 cos 2 2β: (9)
Equation ( 8) agrees with the expression of the enhancement factor found for EO sampling expressed in the terminology of optical heterodyne detection [START_REF] Brunner | Distortion free enhancement of terahertz signals measured by electrooptic sampling. I. Theory[END_REF], where the heterodyning of the signal is performed in front of the EO crystal. On the other hand, the distortion parameter we found is multiplied by a fac tor cos 2 2β compared to [START_REF] Brunner | Distortion free enhancement of terahertz signals measured by electrooptic sampling. I. Theory[END_REF], meaning that our detection scheme is less sensitive to distortion as β increases. However, since the enhancement factor is better for small values of β, d ≃ 1∕2 over the optimal operating range of our detection scheme.

Finally, this distortion is removed by measuring the subtrac tion of the signals detected by both photodiodes,

ΔI Γ; β I 0 2 sin2β sin Γ; (10)
whose enhancement factor is 2g. For comparison, all these fields are nor malized to one and superimposed to the normalized THz fields measured with a CEOS setup, as described in [START_REF] Wu | Ultrafast electro optical field sensors[END_REF], using the same EO crystal. At first sight, there is a very good agreement between both detection schemes, especially in terms of signal to noise ratio, which is about 10 3 . Discrepancies between the signals, if any, might be due to some small mismatch between the heterodyning angles of the quarter wave plates, leading to a nonvanishing contribution from the homodyne signal, which would distort the measured signal shape. We kept such a source of distortion as small as possible by tuning the angle of the quarter wave plates so that the intensities of the local oscillator on each photodiode, I 1 0; β and I 2 0; β, respectively, were equal, the signals delivered by PD1 and PD2 being measured by means of a voltmeter. However, a closer look at the detected waveforms, especially their dependence with respect to the heterodyning angle β [Figs. 3(a) and 3(c)], reveals some discrepancies. This is clearly evidenced by Figs. 3(b) and 3(c), which display the evolution of the peak value of the nonnormalized signal ΔI with respect to the angle β. According to Eq. ( 10), this amplitude is expected to scale like sin2β. For β ≤ 20°, the experimental results are in agreement with this angular dependence, as shown by the fits on Figs. 3(b) and 3(d). But, for higher values of β, the peak value of ΔI saturates and no longer obeys to Eq. [START_REF] Thomson | Broadband THz emission from gas plasmas induced by femtosecond optical pulses: from fundamentals to applications[END_REF]. We attribute this behavior, and the subsequent distortion of the detected THz waveforms, to the saturation of PD1 and PD2 by the local oscillator signal, I LO , once β > 20°. Nevertheless, this saturation of the photodetectors is not detrimental since, according to Eq. ( 8), it is better to work at small values of β.

Indeed, the smaller the heterodyning angle β, the better is the enhancement factor 2g of EO sampling by optical heterodyne detection. For instance, for β 5°, the enhancement factor is 2g ≃ 1∕β ∼ 10, a value identical to the enhancement factor obtained in [START_REF] Johnson | Distortion free enhancement of terahertz signals measured by electrooptic sampling. II. Experiment[END_REF].

TERAHERTZ PULSES SAMPLING THROUGH OPTICALLY HETERODYNE-DETECTED PISH SIGNAL

In a recent paper [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF], we have shown that it was possible to detect THz pulses via the detection of the second harmonic (SH) of an optical pulse induced through its interaction with an intense THz pulse in a zinc blende crystal (the so called PISH effect). Briefly, with the geometry of Fig. 1, an optical signal pulse is induced by the THz pulse through the Pockels effect. For the sake of simplicity, assuming monochro matic plane waves at angular frequencies ω in the optical range and Ω in the THz range, the electric field of the signal is

E sig E ωΩ y ≃ E ω y , since Ω ≪ ω.
This field then couples to the fundamental electric field, E ω z , through the second order nonlinearity of the crystal, χ 2 X Y Z , to generate a second harmonic signal with an electric field given by [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF]:

E PISH ∝ χ 2 X Y Z r 41 E ω z 2 E THz L∕λ.
Since the PISH signal is proportional to the THz electric field, it has been used to detect the latter. However, such a detection requires to heterodyne the PISH signal in order to avoid losing the phase of the THz waveform. This was done by inserting a quarter wave plate on the optical beam pathway in front of the crystal.

Recently, to circumvent some limitations of lithium niobate or ZnTe (two or three photon absorption in the spectral range of the pump, absorption in the THz range, phase mismatch), new materials have been developed to generate intense THz pulses by optical rectification: orientation patterned GaAs [START_REF] Vodopyanov | Terahertz wave gener ation in orientation patterned GaAs using resonantly enhanced scheme[END_REF] or stilbazolium salt single crystals [START_REF] Yang | Large size bulk and thin film stilbazolium salt single crystals for nonlinear optics and THz genera tion[END_REF], for example. They share the common characteristic of being pumped in the near infrared, and may require some frequency doubling of the pump pulse for further characterization of the emitted THz wave by EO sampling [START_REF] Yang | Large size bulk and thin film stilbazolium salt single crystals for nonlinear optics and THz genera tion[END_REF]. One can avoid the insertion of a nonlinear crystal for SH generation of the pump beam prior to EO sampling, or the use of a THz Michelson interferometer to retrieve the THz waveform by first order autocorrelation [START_REF] Vicario | High efficiency THz gen eration in DSTMS, DAST and OH1 pumped by Cr:forsterite laser[END_REF], by resorting to the heterodyne detection of the THz wave via THz field induced second harmonic generation in a plasma [START_REF] Karpowicz | Coherent heterodyne time domain spectrometry covering the entire "terahertz gap[END_REF] or in a suitable EO crystal as we did.

Consequently, since THz detection through optically heterodyne detected PISH signal deserves some attention, we are going to present it in detail compared to the elementary treatment of [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF] where, for instance, no derivation of the ana lytical expression for the measured signal or evaluation of the enhancement factor was given.

A. Experimental Setup and Analytical Expression of the Optically Heterodyne-Detected PISH Signal

The experimental setup is the same as in [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF]. THz pulses are generated in LiNbO 3 by optical rectification of tilted pulse front, 800 nm, femtosecond pulses, and the EO crystal is a 300 μm thick ZnTe. A quarter wave plate, with its fast axis oriented at an angle β with respect to the ẑ axis, is inserted on the optical probe beam pathway in front of the EO crystal (Fig. 4). The THz beam is modulated by a mechanical chopper. The THz induced SH signal, filtered by a 400 nm bandpass filter and measured by a photomultiplier tube with a monochromator, is recorded by a lock in amplifier as a function of the time delay τ between the probe and THz pulses. Note that, contrary to usual heterodyne detection schemes, this method does not require any polarizer prior to the detector.

Without any loss of generality, we model the probe and THz beams by monochromatic nondepleted waves [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF]:

E l t; x 1 2 E ω l x exp iωt k ω x c:c: l y; z (11
) and

E THz t; x 1 2 E THz exp iΩt k Ω xŷ c:c:; ( 12
)
where k ω ωn ω ∕c, k Ω Ωn Ω ∕c, and n ω (resp. n Ω ) is the re fractive index of the crystal at the probe (resp. THz) frequency. At the entrance of the crystal, for a positive heterodyning angle β, the probe beam amplitude writes

E ω y β; 0 E ω z β; 0 E 0 1i 2 sin2β 1 isin 2 β i : (13)
The h110i cut ZnTe crystal, due to the applied THz field, is birefringent with the following principal axes [START_REF] Yariv | Quantum Electronics[END_REF]: ê1 x, ê2 ŷ ẑ∕ 2 p , and ê3 ŷ ẑ∕ 2 p , whose refractive indices are n 1 n ω , n 2 n ω 1 2 n 3 ω r 41 E THz , and n 3 n ω 1 2 n 3 ω r 41 E THz , respectively. As a consequence, in the frame of the crystal [see Fig. 1(b)], the probe field amplitude at each point x of the crystal is given by

E ω X β; x E ω Y β; x; (14)
E ω Y β; x E 0 2 2 p f β e iΓ x ∕2 g β e -iΓ x ∕2 ; (15)
E ω Z β; x E 0 2 f β e iΓ x ∕2 g β e -iΓ x ∕2 ; (16)
where

f β 1 i 2 sin2β 1 isin 2 β i; (17)
g β 1 i 2 sin2β i 1 isin 2 β; (18)
and Γ x 2πxn 3 ω r 41 E THz ∕λ. These equations make it possible to compute the second order polarization radiating the THz induced field at frequency 2ω. Its amplitude is given by [17]

P 2ω i 1 2 ε 0 χ 2 ijk E ω j E ω k ; (19)
where i ≠ j ≠ k X ; Y ; Z and χ 2 X Y Z χ 2 Y Z X χ 2 Z X Y χ 2 .
In the frame of the laboratory, it writes

P 2ω x β; x 0; (20)
P 2ω y β; x ε 0 χ 2 E 2 0 8 f 2 β e iΓ x g 2 β e -iΓ x ; (21)
P 2ω z β; x ε 0 χ 2 E 2 0 16 f β e iΓ x ∕2 g β e -iΓ x ∕2 2 ; (22)
and acts as a source term for the nonlinear wave equation of the SH field amplitude. Within the slowly varying amplitude approximation, this equation is

∂ ∂x α 2ω E 2ω l iω cn 2ω ε 0 P 2ω l expixΔk l y;z; (23)
where α 2ω is the absorption coefficient at 2ω and Δk 2ωn ω n 2ω ∕c is the phase mismatch between the fundamen tal and SH fields. Since, in our experiment, α 2ω x ≫ 1 and α 2ω ; Δk ≫ Γ x ∕x, one gets, at the output of the crystal:

E 2ω y β; L i A 4 f 2 β e iΓ g 2 β e -iΓ ; (24)
E 2ω z β; L i A 8 f β e iΓ∕2
As expected, the heterodyned PISH signal is proportional to the THz electric field and makes it possible to measure the latter. Note that, in the equation given in [START_REF] Cornet | Terahertz field induced second harmonic generation through Pockels effect in zinc telluride crystal[END_REF] for ΔI 2ω Γ; β, the square power law dependence regarding I ω , χ 2 , and r 41 is missing. sig . On the other hand, their sub straction gives a signal directly proportional to the THz electric field, as evidenced by Fig. 5(b) where ΔI 2ω Γ; β I 2ω Γ; β I 2ω Γ; β is compared to the THz waveform measured by CEOS. As can be seen, the agreement between our technique and CEOS is excellent. This is also evidenced by the corre sponding spectra displayed in Fig. 5(c). The signal to noise ratio was found to be ∼10 3 in both cases. Note that, contrary to Section 2, these experiments were not performed in dry air, explaining why the measured THz waveforms and spectra are less clean than those of Fig. 2.

B. Results and Discussion

COMPARISON OF THE TWO DETECTION SCHEMES

First, the two detection schemes presented above offer the same performances in terms of enhancement factor and distortion parameter. So, the choice of one technique rather than the other will depend on the user's aim. If one intends to measure THz pulses generated by amplified Ti:sapphire laser systems, the recommended scheme is the optically heterodyne detected electro optic sampling (Section 2) for the following reasons: first of all, this setup is very close to the widespread CEOS setup usually dedicated to this task; it makes it possible to acquire the THz waveforms in one scan, limiting the distortions that may arise from any drift of the experimental setup between two consecutive scans for two different heterodyning angles; finally, it does not require the use of a photomultiplier tube, which is much more expensive and fragile than a photodiode. However, this scheme, like the second one, requires good polarizing optics (polarizers, wave plates). Furthermore, these optics must be identical (i.e., from the same manufacturer) on both arms after the EO crystal in order to reduce as much as possible the influence of any intrinsic birefringence. As already mentioned in Section 3, the second detection scheme is interesting if one intends to characterize THz pulses from a setup pumped by near infrared laser pulses. It is a good alternative to much more complex detection techniques, like THz field induced second harmonic generation in a plasma or the use of a Michelson interferometer, since it is very close to the setup of CEOS and requires very few modifications of the latter to be imple mented in the laboratory.

CONCLUSION

Two simple and distortion free methods to sample THz pulses in zinc blende type crystals, based on an optical heterodyne detection scheme, have been presented and demonstrated experimentally. The first one makes it possible to measure, in a single measurement and without distortion, a THz waveform by detecting an optical probe beam. Regarding the second one, its relies on the so called PISH effect and makes it possible to sample a THz wave through THz induced second harmonic generation. It might be useful in characterizing THz pulses generated by optical rectification of near infrared pulsed sources other than Ti:sapphire lasers. Both methods are as good as the more conven tional, and widespread, electro optic sampling technique.

Fig. 1 .

 1 Fig. 1. (a) Experimental setup. BS, beam splitter; nBS, nonpolariz ing beam splitter; P, polarizer; PD, photodiode. (b) Geometry of the experiment. X; Ŷ; Ẑ, Cartesian frame of the crystal; x; ŷ; ẑ, Cartesian frame of the laboratory. k THz (resp. k pr ) is the wave vector of the THz (resp. probe) pulse.

Fig. 2 .

 2 Fig. 2. EO sampling by optical heterodyne detection performed in ZnTe [(a) and (b)] and GaP [(c) and (d)]. (a),(c) Normalized THz electric field, for different β angles, versus the time delay τ compared to the field measured with a CEOS setup (dotted line). (c),(d) Corresponding spectra.

Fig. 3 .

 3 Fig. 3. EO sampling by optical heterodyne detection performed in ZnTe and (b)] and GaP [(c) and (d)]. (a),(c) THz electric field, for different β angles, versus the time delay τ. (c),(d) Evolution of the amplitude of the signal displayed in (a) and (c) with respect to the heterodyning angle β. The solid line curve corresponds to a fit of the data with the function sin2β.

Fig.

  Fig. Experimental setup. PMT, photomultiplier tube; x; ŷ; ẑ, Cartesian frame of the laboratory.

Figure 5 (

 5 Figure 5(a) displays the PISH intensities, I 2ω Γ; β, measured for β 10°. Both signals oscillate around zero thanks to the mechanical chopper and the lock in amplifier, which make it possible to get rid of the contribution of the local oscillator, I 2ω LO . Note also that, by adding these two signals, one would get the homodyne signal, I 2ωsig . On the other hand, their sub straction gives a signal directly proportional to the THz electric field, as evidenced by Fig.5(b) where ΔI 2ω Γ; β I 2ω Γ; β I 2ω Γ; β is compared to the THz waveform measured by CEOS. As can be seen, the agreement between our technique and CEOS is excellent. This is also evidenced by the corre sponding spectra displayed in Fig.5(c). The signal to noise ratio was found to be ∼10 3 in both cases. Note that, contrary to Section 2, these experiments were not performed in dry air, explaining why the measured THz waveforms and spectra are less clean than those of Fig.2.

Funding.

  Conseil Régional d'Aquitaine (2011 1603 004).

  g β e -iΓ∕2 2 ;

					The first term in Eq. (26) corresponds to the intensity of the
					SH local oscillator,
					I 2ω LO I 2ω 0; β ∝	jAj 2 2	sin 2 2β;	(27)
					whereas the intensity of the SH homodyne signal, I 2ω sig , is
					given by
					I 2ω sig I 2ω Γ; 0 ∝	jAj 2 4	sin 2 Γ:	(28)
					In the terminology of optical heterodyne detection, one has
					to consider the modulation depth, M Γ; β I 2ω Γ; β I 2ω LO ∕I 2ω LO , of the PISH signal detected for a positive heterodyn ing angle β. As Γ; β → 0, we found that
					M Γ; β gΓ d gΓ 2 OΓ 3 ;	(29)
					with g 1∕ sin2β and d 1∕2, results identical to the one
					found for EO sampling in this context [6].
					Finally, the heterodyned PISH signal is
					ΔI 2ω Γ; β I 2ω Γ; β I 2ω Γ; β;
					∝	jAj 2 2	sin2β sin2Γ;
					∝ 2	L λ	βχ 2 r 41 2 I 2 ω E THz :
					(25)
	where A ωχ 2 E 2 0 e iΔkL ∕2cn 2ω iΔk α 2ω . The expansion of f 2 β , g 2 z β; L, assuming β , and E 2ω Γ; β ≪ 1, makes it possible to calculate the SH intensity
	for an heterodyning angle β, I 2ω Γ; β ∝ jE 2ω y β; Lj 2 jE 2ω z β; Lj 2 . Neglecting irrelevant terms of order 3 and more
	in Γ and β, one has
	I 2ω Γ; β ∝	jAj 2 4	2 sin 2 2β sin 2 Γ sin2β sin2Γ
	3 2	sin 2 2βsin 2 Γ:	(26)

Acknowledgment. M. C. is grateful for thesis funding from the Direction Générale de l'Armement (DGA) and the Région Aquitaine.