A method for determining the spatially resolved acoustic field inside a water-filled microchannel is presented. The acoustic field, both amplitude and phase, is determined by measuring the change of the index of refraction of the water due to local pressure using stroboscopic illumination. Pressure distributions are measured for the fundamental pressure resonance in the water and two higher harmonic modes. By combining measurement at a range of excitation frequencies, a frequency map of modes is made, from which the spectral line width and -factor of individual resonances can be obtained.
© 2015 Optical Society of America
Lab-on-a-chip (LoC) microfluidics is a developing field and has grown tremendously over the past decades [1,2]. Various laboratory functions, such as particle separation , cell sorting , particle trapping , droplet manipulation , biological assays , and mixing , are integrated into a single microchip using very small fluid volumes compared with traditional methods.
Following the trend toward the microscale, various acoustic techniques such as surface acoustic waves [9–11] (SAWs), acoustic bubbles , or acoustic actuation  have been developed for use in LoC technology. Likewise, the use of sound-based particle separation and trapping was explored [14–17].
Ultrasonic standing waves for particle manipulation have been integrated onto silicon/glass microchips by precise microfabrication techniques and various configurations have emerged on the silicon/glass platform [18–22]. To aid design and improve performance, physical models have been developed to describe the acoustic resonances [23–28], which predict the resonance frequency and pressure amplitude distributions well for thin chips .
Due to the high sensitivity of the acoustic resonances to resonator geometry, chip geometry, ultrasonic transducer or SAW generator, acoustic coupling, and other experimental factors, it is generally hard to predict and measure the exact acoustic mode and resonance strength, hindering direct comparison among different experiments and configurations. For measuring the acoustic amplitude, particle-based methods [30–32] provide high sensitivity but require sufficiently large particles (typically 5 μm in diameter or larger) to be present in the region of interest. As such the acoustic amplitude cannot be measured in experiments without large particles or in regions where no particles can be placed. As the acoustic force exerted on the particles depends on the pressure squared, only the amplitude of the pressure can be measured and full information of the modes, including the acoustic phase, cannot be obtained.
In this work, we present a novel and particle-less method for measuring the local acoustic field inside a microchannel. The method is based on optically measuring the local change in the index of refraction of water due to acoustic pressure. Stroboscopic, incoherent illumination allows measuring both the local acoustic amplitude and phase of the pressure field. By combining measurements from a range of excitation frequencies, mode spectrograms can be made giving an overview of the modes present. The resonance strength and frequency of a mode can be determined from the spectral line obtained from the spectrogram.
We study pressure resonances in a silicon/glass chip containing a microchannel filled with water, which is used as a platform for acoustophoresis [25,30]. Our method is based on light traveling through the glass and water, reflecting at the bottom of the channel, and returning along the same path, thereby probing the pressure-induced variation of index of refraction of the water (see Fig. 1). Any acoustic density variations in the glass layer will also contribute to the measured variation. This contribution from the glass is determined separately by measuring the light reflected from the glass–water interface and is found to be relatively small (up to 20%) compared with the acoustic variation due to the pressure of the water.
We spatially resolve the acoustic pressure field, both amplitude and phase, of the fundamental resonance and two higher harmonic modes using a stroboscopic illumination scheme. The acoustic amplitudes measured are , , and . The measured pressure distributions match the theory well.
In the first section, we give the theoretical basis of the pressure modes of the chip and the theory of acousto-optical interaction. After that the experimental setup and measurement method are discussed, followed by results and concluding remarks.
2. THEORETICAL BACKGROUND
A. Acoustic Resonances
The silicon/glass walls surrounding the water are assumed to be infinitely hard. The acoustic pressure in water in a lossless rectangular, hard-walled channel at resonance is given by26]. The corresponding resonance frequencies are 1.
The acoustic energy density at resonance follows a Lorentzian line shape as function of frequency around the resonance frequency :
B. Acousto-optic Interaction
The interaction strength between the acoustic pressure and index of refraction is expressed by the adiabatic acousto-optic coefficient . The starting point of derivation is the Lorentz–Lorenz (LL) (or Clausius–Mossotti) equation, which relates the index of refraction to the molecular polarizability :34].
Taking the derivative of Eq. (5) with respect to density yields35]. For most liquids is very close to 0 because the intermolecular distance is large such that compression of the medium has no influence on the molecular charge distribution and, thus, also not on the molecular polarizability. High-pressure measurements of water show a modest change in of about 4% at 5.92 GPa and 673 K compared with that under standard conditions , showing that for our experimental conditions we may reasonably take equal to 0.
The adiabatic equation of state relates the pressure and density fluctuations in a fluid :8) at gives the acousto-optic coefficient , while taking the derivative with respect to and subsequent evaluation at gives the quadratic acousto-optic coefficient . Together we get the change in the index of refraction due to acoustic pressure :
Using the positive root of Eq. (7) for condensation as function of acoustic pressure and equilibrium parameters, , wavelength of light , water temperature of 30°C, , , , and , we find the acousto-optic coefficients and . These values will be used in the remainder of the paper.
Other investigators derived a value of assuming a linear dependence of electric susceptibility on density .
3. EXPERIMENTAL SETUP
The setup is based on a white-light phase-shifting Michelson interferometer design as used by Shavrin et al.  (see Fig. 2). A light-emitting diode (LED) with a center wavelength of and a spectral FWHM of is used.
The emission from the LED surface is highly multimode so that only limited coherence exists among different points on the surface. The emitting surface is imaged through the beam splitter onto both the reference mirror in the reference arm and the sample in the sample arm using the illumination lens. Because the beam splitter splits the field coherently, the two images are mutually pointwise coherent . The images are re-imaged though the beam splitter and projected onto the EMCCD detector by the detection lens. The sample is aligned such that the images overlap on the sensor.
Interference of light returning from the sample and reference arms only occurs if the two fields are both spatially and temporally coherent. The first requirement is fulfilled by overlapping the projected sample and reference images onto the detector by rotating the sample. The second condition is satisfied by changing the axial sample position such that the optical path length difference (OPD) between the sample and reference light paths is within the temporal coherence length . The temporal coherence length of the light source is approximately 15 μm.
A. Microchannel Chip
The microchannel was fabricated using photolithography and deep reactive-ion etching on a silicon wafer. Access holes of 1 mm diameter were made from the back in the same way as the channels. The channel wafer is anodically bonded to a 500 μm thick borofloat glass wafer and then diced into glass–silicon microchannel chips measuring 6 cm by 1.5 cm. The length , width , and depth of the microchannel are 4 cm, 380 μm, and 155 μm, respectively. See Fig. 3 for the measurement region in the microchannel. The image is obtained by the EMCCD detector by placing the reference mirror such that no interference is visible, i.e., the OPD between the two arms is larger than the temporal coherence length for all pixels. The length scale is calibrated using the channel width as reference, which was measured during fabrication using an optical inspection microscope with an accuracy of approximately 1 μm.
The channel was completely filled with demineralized water (Milli-Q) through the access holes, which were then sealed with scotch tape. A piezo-element for ultrasonic excitation was attached on the silicon side of the chip using a thin layer of cyano-acrylate glue. The width is chosen such that the fundamental acoustic resonance is present around 2 MHz, which is close to the resonance of the excitation piezo-element. The chip with the piezo-element attached was mounted into the sample arm of the setup.
B. Optical Alignment
The sample is aligned by changing its axial position and orientation using a six-axis sample holder, such that after alignment interference fringes can be seen with the camera at the interface of interest. Two interfaces are measured: the water–silicon interface at the bottom of the channel and the interface at the top of the channel, which is a glass–water interface in the channel and a glass–silicon interface outside. Two possible optical paths are illustrated in Fig. 1. To measure at the bottom of the channel, the sample position is manually adjusted such that the total optical path from the beam splitter to the interface and back () is equal (within the coherence length) to the optical path from the beam splitter to the reference mirror and back. To measure at the top of the channel, the sample is first aligned to the bottom and then moved , with the refractive index of water, in the direction until interference fringes are again observed. The reference mirror is kept at its center position (see Section 4) during the alignment procedure. This method gives a positioning precision better than the coherence length .
The excitation signal for the sample piezo-element (Pz26 26302, Ferroperm) and the pulse signal for the LED (ELD-870f-515-2, Roithner Laser) are generated by a double function generator (DG 4162, Rigol). The piezo-excitation signal amplitude is set to 30 mV for frequencies from 1.7 to 2.1 MHz and 50 mV otherwise before amplification by an amplifier (2200L, E&I) with a fixed power gain of 53 dB. This results in a peak amplitude at the piezo of approximately 50 Vpp around the piezo-resonance at 2 MHz. The pulse signal is amplified and added to a bias using a homebuilt amplifier. The axial translation stage for the reference mirror is based on a piezo-actuator (PXY80D12, Piezosystem Jena), controlled by a homebuilt position sensor and feedback electronics to ensure linear dependence of position on control voltage. The function generator, control voltage, triggering, and readout of the camera (Ixon DV887, Andor) are controlled by a PC using a data acquisition card (USB-6212, National Instruments) and a homebuilt control program (Labview 2011, National Instruments).
4. ACOUSTO-OPTICAL MEASUREMENT METHOD
An acoustic resonance as described by Eq. (1) induces an instantaneous change in the index of refraction, which results in an instantaneous change in the optical path length of light traversing the water. Using stroboscopic illumination and an interferometric reference it is possible to selectively measure this periodic change, analogous to lock-in detection .
In this experiment, we use short pulses (18 ns FWHM) of light with a repetition rate equal to the acoustic excitation frequency . During every acoustic period, one short snapshot of the instantaneous optical path length is recorded, which is integrated over for many periods (see Fig. 4). By setting the time delay between the acoustic excitation signal and the pulse, the moment of measurement during the acoustic period can be chosen. In our experiments, we measure at four delays spaced a quarter period, , .
The axial position of the reference mirror can be changed using a piezo-actuator. By scanning the mirror position, an interferogram (light intensity as a function of mirror position) can be recorded per pixel. The interferogram peaks at its maximum value when the OPD between the light returning from the sample and the reference arms is 0. An acoustically induced variation will result in a shift of the interferogram equal to the amount of change in the optical path length in the sample. By optically probing four times per acoustic period, both amplitude and phase of the pressure field can be measured. The measured amplitude is the average over the height of the channel and is therefore unable to measure resonances in the -direction ().
At each mirror position four frames are recorded, one for each delay using an integration time of 30 ms each. The mirror is moved in steps of 20 nm over a range of 40 μm for a full scan. To ensure that the mirror is stationary after moving, a stabilization period of 100 ms is used after each step before recording the frames. For the frequency mapping scans, the excitation frequency is varied with steps of 2.5 kHz.
A. Data Post-processing
For each pixel, four interferograms are recorded, one for each time delay. To find the relative shifts among them, each interferogram is first cross correlated with a digital reference signal using a fast Fourier transform-based cross correlation. The cross correlation allows the full trace to contribute to the shift calculation instead of only the region around the maximum of the interferogram, thereby improving the signal-to-noise ratio. The reference is chosen differently for each pixel to accommodate local variations, but it is the same for each of the four interferograms.
The reference signal is described as10) to the 100 interferograms with the strongest signal per scan and using an average value for the reference. The other coefficients are chosen such that the reference approximately matches the signal in terms of amplitude and shift. The shifts are then found from the position of the maximum in the resulting cross correlation, which is extracted with substep accuracy using Fourier interpolation followed by sine fit of the peak.
The amplitudes of the acoustically induced variations in the index of refraction and phase can be calculated using the quadratures and :
The variation of the OPD is assumed to be smaller than and all results larger than this value are rejected to suppress wrongly detected peak positions.
Finally, the instantaneous acoustic pressure is calculated using as a variation in the index of refraction due to pressure and the inverse of Eq. (9):
Measurements are done at two interfaces: subscripts and indicate measurements done at the bottom and top of the channel, respectively (see Fig. 1). The first contains acoustic effects from the water as well as from the glass layer, and the latter only contains effects from the glass. The measured change in the index of refraction of both measurements is expressed in acoustic pressure to be able to directly compare the two measurements and should thus be regarded as the in-water equivalent pressure.
Pure -modes like or mixed modes such as with low can be regarded to be constant in the -direction within the measurement area. In such cases, the average can be taken over the -direction to show the mode profile, while the standard deviation calculated over the same data is an indication of the lower limit in sensitivity. Averages are indicated as and are calculated by averaging the quadratures and then following the rest of the calculation as above. The standard deviation is calculated over the pressure from Eq. (15).
A. Acoustic Pressure Distributions at Resonance
The measured instantaneous local pressures for the modes , , and are shown in Fig. 5. On the left, the pressure distributions as measured on the bottom of the channel and on the right the -averaged values (in red) and (in gray) are shown. In each case, a phase is added to in Eq. (15) such that the pressure is shown at the peak moment in time.
In the pressure distributions, some pixels are missing values (shown as black). As the bottom of the microchannel is slightly rough (2.1 μm RMS), light scatters strongly and at certain locations insufficient light returns from the sample, resulting in unreliable values. This limits the spatial resolution to approximately 25 μm in highly scattering regions. In sufficiently smooth regions, the resolution is equal to the optical resolution, which is 5 μm. Some pixels show a phase difference of compared with the surrounding pixels, which is especially apparent in Fig. 5(b). This is not physical but an artifact due to faulty phase assignment. This could possibly be corrected by using an improved algorithm with a priori knowledge, such as setting a limit for the spatial variation of the pressure.
The resonance frequencies found are lower than expected from theory, up to 5% for . This could be partially explained by a lower temperature used in experiment resulting in a lower speed of sound and resonance frequency.
The wavelengths of the resonances and the pressure amplitudes are found by fitting a cosine to . Confidence bounds at 95% are included. The results from the fit are , , and , and the corresponding values for are , , and . The acoustic pressures show a clear sinusoidal variation in the -direction as indicated by the small confidence bounds and are almost constant in the -direction.
The uncertainty in the pressure is given by the standard deviation as indicated by the shaded area in the graph. The acoustic wavelengths found match the expected wavelengths within .
The acoustic amplitudes are comparable with values found by other studies. Using a comparable geometry, Barnkob et al. measured amplitudes of 80–660 kPa using excitation voltages of 0.5–1.9 Vpp for the fundamental resonance, which would correspond to over 16 MPa at 50 Vpp, which we used for the mode . The excitation voltage for and is much lower, at approximately 15 Vpp, due to a lower piezo-efficiency at the corresponding frequencies. Dron and Aider obtained a pressure amplitude of 110 kPa with 5 Vpp excitation  and Lakämper et al. measured 53–160 kPa at 10 Vpp , values comparable to the ones we obtained.
The measurements done at the top of the channel are shown as on the right in Fig. 5. Because the top interface is smooth, all pixels show a good signal and the uncertainty is lower than those of the measurements done at the channel bottom.
The acoustic variation in the OPD, shown as in-water equivalent pressure , is much smaller than the acoustic variation in the OPD due to the pressure resonances . The shape of is asymmetric. This is explained by an acoustic resonance in the glass layer with a much larger wavelength than the width of the channel. As such only part of the wavelength is seen and results in an asymmetric pressure distribution without clear nodes or antinodes within the field of view.
An important result is that the acousto-optical signal originating from the glass layer is much smaller than . Light traveling through the glass and water picks up acoustic signals from both media and because the signals add harmonically, one cannot distinguish between the two without knowing the relative phase between the two contributions. However, since is much smaller than , we can be certain that the most significant part of the signal in is in fact from the pressure mode in the water and not from the glass.
The pressure sensitivity of this method is limited by camera noise. Without spatial averaging, the sensitivity is approximately 50 kPa as given by the standard deviation in ; with averaging over the -direction (57 values), the sensitivity is approximately 5 kPa as seen from the fit confidence bounds.
B. Pressure Mode Spectroscopy
By measuring pressure distributions at a range of frequencies and plotting as a function of excitation frequency , spectrograms of acoustic resonances can be made. The result for frequency ranges around the resonances in Section 5.A is shown in Fig. 6. The spectrograms give an overview of the pressure resonances present in the system. Relative peak amplitudes and the spatial distribution over the width of the channel can be compared. The order of the modes can be read from the number of nodes and the resonance quality (-factor) of resonances can be estimated from the spectral width of a resonance line.
The absolute value is used instead of , displaying the spatial distribution of the pressure amplitude over the width of the channel and does not include the phase . The phase depends on the total time delay between signal generation at the function generator and the excitation of the acoustic mode. As such it does not only depend on the acoustic resonance alone, but also on resonances in the excitation piezo, the response of the electronic equipment, the mode coupling between the excitation piezo and the acoustic mode, etc. Therefore, using as a function of frequency would result in a very chaotic map, which defeats the purpose of giving an overview of the acoustic modes available in the microchannel.
To estimate the influence of glass modes on , measurements at the top layer have also been done. Within each of the three ranges displayed in Fig. 6, the highest in-water equivalent pressures are 1.7 MPa at 1.93 MHz [as shown in Fig. 5(a)], 56 kPa at 3.54 MHz, and 68 kPa at 5.86 MHz. Glass modes therefore have no significant effect and the pressure distributions shown are due to acousto-optic effects in the water.
The pure modes , , and as studied in Section 5.A are clearly visible at , , and , respectively. Mixed modes are also visible. These tend to have a lower peak amplitude and are lossier (spectrally broader). These can partially be explained by the theory as being mixed higher order modes. In other cases, it might be needed to include acoustic resonances in the surrounding silicon chip .
A notable exception is the relatively strong mode , which cannot be explained, as all modes represented by Eq. (1) satisfy the hard-wall condition (antinodes at the boundary). The pressure distribution of this mode has nodes at the boundary and one antinode in the middle. No notable signal is measured at this frequency at the top of the channel.
1. Determination of Resonance Quality
For experiments and design purposes, it is important to know the resonance quality of an acoustic resonance, as measured by the -factor. As an example, we will calculate the resonance quality of the mode using the measured spectrogram.
The peak amplitude , calculated by averaging over the width of the channel, is expressed as acoustic energy density using Eq. (3) and is plotted as a function of frequency (see Fig. 7). A Lorentzian function defined by Eq. (4) is fitted to the data using the least-square method. The fitted values are , peak acoustic energy density , and a -factor of 139. The corresponding spectral line width equals .
6. CONCLUDING REMARKS
Using a stroboscopic interferometric technique, we have determined the acoustic field, both pressure amplitude and phase, in a water-filled microchannel for various excitation frequencies. The field was determined by measuring the OPD due to change in the index of refraction due to the local acoustic pressure. The mode profiles at resonance for three measured resonances closely resemble the theoretical model. Fields measured at a range of excitation frequencies can be combined into mode spectrograms showing which modes are present at which frequency. From the measurements the local pressure amplitude, acoustic energy density, and -factor of modes can also be determined.
The influence of acoustic modes in the glass top layer could be investigated separately by virtue of the incoherence of the light used. We have shown that the optical path length of light traversing the glass and water is affected most significantly by the acoustic resonances in the water, compared with acoustic effects in the glass. This provides a basis for particle-free optical resonance tracking. In principle, the optical tracking could be done for various frequencies simultaneously, which is of interest for multifrequency acoustophoresis experiments [18,46].
Our method has a sensitivity of approximately 50 kPa without and 5 kPa with spatial averaging over the field of view and can measure pressures well over 9.5 MPa without the use of particles. This opens up the possibility of in situ optical characterization of acoustic modes on the microscale, which is important for a variety of systems such as those used in shaped acoustic fields , acoustophoresis , and even piezo-printheads. Due to the spatial selectivity, the local pressure around micro-objects can be measured and novel configurations in acoustic microfluidics can be investigated.
Foundation for Fundamental Research on Matter (FOM); Wetsus, European Centre of Excellence for Sustainable Water Technology.
This work is part of the research program of the Foundation for Fundamental Research on Matter, which is part of the Netherlands Organization for Scientific Research. This work was performed in the cooperation framework of Wetsus, European Centre of Excellence for Sustainable Water Technology (www.wetsus.nl). Wetsus is funded by the Dutch Ministry of Economic Affairs and Ministry of Infrastructure and Environment. The authors would like to thank the participants of the research theme Sensoring for the fruitful discussions and their financial support.
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