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The favorable downscaling behavior of photoacoustic spectroscopy has provoked in recent years a growing interest in the miniaturization of photoacoustic sensors. The individual components of the sensor, namely widely tunable quantum cascade lasers, low loss mid infrared (midIR) waveguides, and efficient microelectromechanical systems (MEMS) microphones are becoming available in complementary metal–oxide–semiconductor (CMOS) compatible technologies. This paves the way for the joint processes of miniaturization and full integration. Recently, a prototype microsensor has been designed by the means of a specifically designed coupled opticalacoustic model. This paper discusses the new, or more intense, challenges faced if downscaling is continued. The first limitation in miniaturization is physical: the light source modulation, which matches the increasing cell acoustic resonance frequency, must be kept much slower than the collisional relaxation process. Secondly, from the acoustic modeling point of view, one faces the limit of validity of the continuum hypothesis. Namely, at some point, velocity slip and temperature jump boundary conditions must be used, instead of the continuous boundary conditions, which are valid at the macroscale. Finally, on the technological side, solutions exist to realize a complete labonachip, even if it remains a demanding integration problem.
Photoacoustic (PA) spectroscopy is a wellestablished technique and numerous gas sensors designs, relying on a single cavity, based on Helmholtz resonance, or enhanced by quartz tuning forks have been imagined and implemented [
In PA spectroscopy, the modulation of a light source creates a timeperiodic variation of temperature due to nonradiative relaxation of excited molecules, and thus an acoustic wave in the chamber. The use of laser sources in the mid infrared (midIR) wavelength range, from 3 to 12 μm, maximizes the light absorption by a number of molecules of interest for a wide variety of applications. The combination of PA detection with midIR pumping is particularly adapted to the detection of chemicals in trace amounts.
The acoustic signal is inversely proportional to the volume of the resonant cell [
All the individual components of the PA sensor, such as widely tunable quantum cascade laser (QCL) sources [
Among the various PA sensors principles available, the differential Helmholtz resonator (DHR) [
In a previous article [
A significant downsizing step has thus been achieved. However, the question of the dimensions at which the full potential of miniaturization will be obtained still remains open. In this paper the consequences of further miniaturization on the physics, models and microfabrication technology are discussed from a theoretical point of view. In the first part of the paper, two aspects of the consequences of downsizing on the models are investigated. Namely: (i) the dynamics of gas molecule excitation by modulated IR light and relaxation is studied; and (ii) the applicability of the continuum model hypothesis to ever shrinking gas sensors is reevaluated. In the last part of the paper, an overview of the MEMS fabrication challenges and the resulting constraints on the microdevice design is presented. Even though the focus is placed here on the μPA cell currently under fabrication, our expectation is to provide as general insights as possible.
It is not economically viable to build and test a large number of different DHR cells since the technological steps required for their fabrication are expensive and time consuming. Resorting to modeling and numerical simulation to optimize the performance of the sensor is a way to partially overcome these limitations. However, one should note that a global design methodology, iteratively adapting the simulation tools with the fabrication process flow, is necessary. In fact, at the microscale, even more than at the macroscale, the choices of fabrication methods and the constraints imposed on the device dimensions drive new model developments, while modeling results help exploring novel routes for the technological implementation.
Due to the very diverse domains of physics involved, the model should address: (i) the midIR electromagnetic mode propagation in the waveguide; (ii) the light illumination of the chamber; (iii) the interaction of light with the molecules of interest and the relaxation process; (iv) the creation and propagation of acoustic waves in the cell; and finally (v) the acoustic wave sensing by the microphone. Many of these physics are intimately coupled but, for the problem to remain tractable, advantage should be taken of all the uncoupling possibilities, such as that occurring because of the linear dependence of the signal on the deposited energy.
The optical model generally used for PA sensors assimilates the laser illumination geometry to a straight beam for which the flux follows a Gaussian distribution within a cross section [
A new optical model, briefly recalled here, has been devised to cope with the above mentioned problems [
In the μPA device, a QCL operating in transverse magnetic polarization is coupled to a SiGe/Si waveguide, designed to be monomodal at the midIR wavelength of interest [
The length scale ratio between the size of the illuminated chamber (several millimeters long) and the midIR wavelength (∼3–12 μm) is too large for practical solution of the Maxwell's equations by exact fullwave methods, such as the finite difference time domain method or the finite element method. Nevertheless, in this regime, the propagation of light can be modeled by geometric optic tools, which are less computationally demanding, while providing almost as accurate results. The raytracing method relies on the plane wave decomposition of the electromagnetic source (
An illustration of raytracing, involving only a few rays for clarity, is presented in
In photoacoustic spectroscopy of gases, intensity or frequency modulated light from the laser enters the measurement cell filled with the gaseous sample. A portion of the incident radiation is absorbed by the gas resulting in a pressure disturbance. When a gas molecule absorbs a photon, it goes from its ground state to an excited state, the energy difference between the states being the energy of the absorbed photon. In a subsequent step, the molecule loses this excess energy and returns to the ground state in one among several ways: radiative deexcitation, intersystem energy transfer, and energy transfer by collision with other molecules [
In the models used at the macroscale, it is assumed that: (i) the absorbing molecular transition is not saturated; and (ii) the relaxation time is much smaller than the period of the source modulation [
The rate equation approach is adequate to study the joint effects of absorption and excited molecules relaxation, and therefore delineate the different operating regimes of the heat source. The analysis made by Kreuzer [
In this equation,
The discussion of Kreuzer [
The characteristic scales used for the density of excited molecule
The results of the solution of
The typical working point of the μPA cell (
The previous computations have been performed assuming a uniform illumination of the chamber. However, in the μPA case, due to beam divergence, the latter is highly variable throughout the illuminated chamber. Thus, it would be useful to get the more precise picture obtained with a local rate equation formulation. The local conservation equation can be derived from
If the generally adopted assumption that the absorption of the midIR flux by the gas molecules is low enough, is valid, the local flux density
It should also be noted that the relaxation time vary on several orders of magnitude (see for instance
The real picture is thus more complex than the simple case of the intensity modulated twolevel system with uniform illumination computed above but the broad lines of the reasoning are still valid. It appears that (i) the heat source modulation loss is significant for a cell resonating at 20 kHz; and that (ii) the modulation will further decrease with miniaturization, as resonance frequency will increase. In fact, set apart quartz enhanced photoacoustic spectroscopy [
The pressure acoustic model is commonly used to study PA device behavior. Assuming adiabatic propagation in an ideal lossless gas, it consists of a single inhomogeneous Helmholtz equation for the unknown pressure [
However, various volume and surface dissipation processes are at work in the gas, in the bulk of the propagation medium and close to the cell walls, respectively [
Approximate models can be adapted from the pressure acoustic model to take into account the dissipation effects, for instance with eigenmode expansion and introduction of quality factors [
A viscothermal model [
The necessity to use the sophisticated, but computationally demanding, viscothermal model is illustrated in
It has been assumed up to this point, that the gas contained in the μPA chamber, although composed of a myriad of rapidly moving and colliding molecules, is a continuous medium, represented by locally averaged macroscopic quantities, such as density, pressure, and temperature. These macroscopic quantities are defined as local averages on fluid elements that, ideally, are large enough to contain a considerable number of molecules, typically 10^{6}, but small enough to permit the use of differential calculus. In these conditions, statistical fluctuations of the macroscopic quantities are not significant (<0.1%) and molecular chaos conditions are achieved due to frequent collisions. The macroscopic quantities vary continuously in space and time, and are governed by the mass, energy, and momentum conservation laws. These equations, referred to as the NavierStokes equations in the following text, can be handled by numerical methods, such as the FEM or the finite volume method. However, as downscaling is carried on, it is natural to investigate the domain of validity of the continuum framework. The brief discussion that follows is inspired from the microfluidics [
In statistical mechanics, the starting point is the Liouville equation, which expresses the conservation of the particle distribution function in the phase space, consisting in all the possible values of the position, momentum and internal states of all the molecules. Due to the large number of particles (
Additionally, if the fluid is close enough to thermodynamic equilibrium, the NavierStokes equations can be used. The frequency of collisions in the gas bulk, a measure of the proximity to thermodynamic equilibrium, is governed by the Knudsen number,
Special attention must be taken when approaching the fluidsolid interface. If the frequency of collisions at the chamber wall is high enough, the thermodynamic equilibrium is established between the gas and solid particles. Continuity of particles velocity and temperature at the fluidsolid interface respectively lead to the no velocity slip and no temperature jump boundary conditions. However, if the collision frequency is not high enough, the thermodynamical equilibrium is not established. Tangential velocity slip and temperature jump, ruled by tangential momentum and thermal accommodation coefficients (σ
For simplicity, in
A summary of the molecular and continuum flow models and the major hypotheses allowing passing from a model to the next one is presented on
The domain of validity of the continuum approximation, with continuous boundary conditions, is represented, in
A typical working point of the μPA cell is placed in the diagram (
The working point is far above the insignificant statistical fluctuations limit (
The working point is located close to the limit where the use of velocity slip and temperature jump boundary conditions becomes compulsory.
The reduction of the device size, going along with the increase of the resonance frequency, and thus the decrease of the boundary layer thickness would bring the limit even closer. The same effect would be reached by the reduction of the working pressure.
Modeling seems to validate the approach of using PA cells of submillimeter dimensions but shrinking the size of the cell at this level remains a technological challenge. Although precision machining has been successfully used to realize absorption cells several millimeters long [
Regarding the realization of the PA cavity itself, silicon manufacturing technologies come with design rules which impact directly the shape and size of the acoustic cavity. For example, lithography and deep reactive ion etching processes lead to vertical walls geometry, with constrained aspect ratio, restricting the shape of the cavities to parallelepipedic volumes. It must also be noted that it is not straightforward to realize two connected volumes of different depths in a single wafer without creating nondesired over etched zones. This can be detrimental to realize a DHR cell, for which the capillaries need to be much smaller in depth than the chambers. Thus, integrating the chambers and the capillaries in a single silicon wafer is challenging. To a certain extent however, it is possible to stack two or more wafers together by using eutectic sealing or wafer bonding, using a metallic or polymer layer. This technique allows processing separately the integrated MEMS microphones, whose complex fabrication can be developed and stabilized independently. One possible resulting configuration is depicted in
Several MEMS microphones types are nowadays available for sensing the acoustic pressure within the cavity. Generally they rely on the realization of a selfstanding membrane, able to oscillate under the effect of the acoustic pressure wave. The amplitude of vibration is converted into an electric signal via optical, capacitive, or piezoelectric transducing. Of course, reducing the size of the chambers directly impacts the size of the membrane, and thus its performances, in terms of resonance frequency, detectivity, and signal to noise ratio. Recently, cantilever microphones have been proposed [
Quite serious challenges occur in the choice and fabrication of the optical source. One possibility relies on hot filaments. These blackbody sources present a very large bandwidth, at the expense of a low wavelength power density, and must be combined with a high rejection filter in order to address a single gas absorption transition to keep the benefit of the optical sensor selectivity. Moreover, they have a slow operation dynamics. Relying on QCL constitutes an appealing alternative because they are monochromatic and powerful sources, matching closely the needs of the μPA sensor. Multigas sensing can be addressed, due to the recent development of widely tunable sources constituted by a monolithic array of several tens of distributed feedback QCL [
Finally, it is important to handle, filter or shape the optical beam from the light source to the acoustic resonant cavity. Compared to free space propagation, integrated photonics is very attractive for this purpose. Silicon photonics, based on silicon on insulator or silicon on sapphire technologies, has proven its efficiency at telecommunication wavelength (1.55 μm) and up to about 5 μm but can not convey light in the whole midIR wavelength range of interest (∼3–12 μm) because of silica absorption. Developments targeting a complete CMOS compatible midIR photonic platform have been made recently: a waveguide solution involving a variable composition SiGe core in a silicon cladding has shown its capability to produce low loss single mode waveguides up to a wavelength of 8 μm, as well as more complex functions that could prove useful in a fully integrated μPA LOC [
The favorable scaling behavior of PAbased gas sensors and the availability of the different components of the system in CMOScompatible technologies support the idea of developing a LOC μPA sensor. For this reason, optical and acoustic models, especially adapted to the LOC size and characteristics, have been devised and coupled to assist in the design of a microdevice currently under fabrication [
The question whether further miniaturization would be beneficial is discussed in this paper. The consequences on the physics, models and microfabrication technology are assessed. As chamber downscaling comes with an increase in resonance frequency, the first restriction met when miniaturizing PA sensors is the loss of thermal source modulation due to the competing effects of the periodic excitation of the molecules of interest and their delayed relaxation. The sensibility of the sensor to low concentrations of gases having a long relaxation time would decrease dramatically for microdevices characterized by resonance frequencies above 20–30 kHz. This suggests a limit in miniaturization as the expected signal improvement is, at least partly, cancelled out by the heat source modulation loss.
On the model side, as the working point of the current μPA device is close to the limit of validity of the continuum model, and as any size reduction or pressure decrease would bring it even closer, the implementation of velocity slip and temperature jump boundary conditions in the viscothermal model has to be undertaken to help, at least, estimate the influence of the boundary conditions in the μPA context. Further improvements of the model can also be expected from the inclusion of the rate equation governing the excitationrelaxation process and the MEMS microphones response and resolution.
On the technological side, even if realizing a complete LOC remains a difficult integration problem, many building blocks such as MEMS microphones and midIR waveguides are already available. So far, the main bottleneck is the heterogeneous integration of the midIR source in CMOS compatible technologies, which has not yet been reported.
This research was supported by the French Agence Nationale de la Recherche (project MIRIADE, ANR11ECOT04) and by the Seventh Program Framework of the European Union (project MIRIFISENS, FP7 317884). The authors thank Lars Duggen from Mads Clausen Institute and Damien Barbès from Commissariat à l'Energie Atomique et aux Energies Alternatives, for their assistance in acoustic model implementation. The authors gratefully acknowledge valuable discussions with Jaroslaw Czarny and Salim Boutami from Commissariat à l'Energie Atomique et aux Energies Alternatives.
The authors declare no conflict of interest.
Schematic view of (
Raytracing in the μPA DHR cell. Crosstalk appears when light emitted by the laser source is refracted towards the nonilluminated chamber, where it can interact with the gas.
Plot of the nondimensional excited molecules density
Time evolution of the heat source density in typical μPA conditions. The relaxation rate time constant is 1 μs (blue thin line), 11 μs (red thick line), and 100 μs (black dashed line). (
Frequency response of the μPA cell obtained with four different models: pressure acoustic (dashdotted line), pressure acoustic with correction for dissipation effects by quality factors (dashed line) or boundary conditions (thin line), and viscothermal model (thick line).
Summary of available molecular and continuum flow models.
Domain of validity of the continuum approximation without (dark grey triangle) or with (light grey area) velocity slip and temperature jump boundary conditions, and typical working point of the μPA cell (●). Figure inspired from Karniadakis and Beskok [
Relaxation time of several gas molecules on nitrogen at 20 °C and 101,325 Pa.
Relaxation time (μs)  11 [ 
1.7 [ 
0.1 [ 
30 [ 
Characteristic values for the μPA cell working point at 20 °C and 101,325 Pa.



3.4 nm  66 nm  15 μm 
δ/ 


9.2  4.4 × 10^{−3}  4400 