1. Introduction

The development of non-invasive and non-destructive imaging methods is in high-demand both in the medical field and in the electronics industry. Ultrasound imaging is one emerging solution. While there are many variations of acoustic microscopes and imaging systems, one approach leverages the relative strengths of ultrasound excitation with optical detection of the ultrasound wave. Using this approach, it is possible to image blood vessels in tissue in 3D [1], image single capillaries in vivo [2], and identify tumors in breast tissue [3]. All of these experiments rely on having a sensitive (low noise and high resolution) method of detecting the ultrasound wave.

One ultrasound sensing technique is based on whispering gallery mode optical resonant cavities. These devices are able to confine light at specific optical frequencies that are defined, in part, by the device geometry and the refractive index [4]. Therefore, when an ultrasound wave hits the surface of the cavity, this resonant frequency changes. A schematic highlighting the general principles of this device is contained in Fig. 1. In previous work using microring cavities, the focus has been on optimizing the quality factor (Q) of the cavity, as a higher Q device has improved sensitivity in response to an impulse [5–7]. This previous work focused on using the effect of material deformation for detection. In this approach, an incoming ultrasound wave changes the device radius. As a result, several other parameters also contribute to the signal. For example, the material constants of the device and its surroundings as well as the intensity of the initial ultrasound pulse. While polymer microring devices have a strong response due to their low bulk modulus, they have moderate quality factors that place a fundamental limit on their ultimate performance.

 

Fig. 1 (a) The resonant wavelength increases and decreases in response to the ultrasound pulse. (b) If the transmission values at a single wavelength (λ_o) are plotted, the characteristic damped oscillator curve is clear. The positive values indicate high pressure and the negative values indicate low pressure.

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Therefore, to develop the ideal ultrasound sensor, it is necessary to explore alternative detection mechanisms which include both optical and mechanical changes. However, within the broader field of resonant cavity ultrasound imaging, there is limited analysis enabling direct comparison of different material systems and device geometries. Therefore, to move the field forward, a more rigorous understanding of these inter-dependencies is needed.

In the present work, we develop and perform COMSOL Multiphysics simulations, using the transient pressure acoustics and RF modules. Subsequently, we verify the model experimentally using a 40MHz ultrasound transducer and silica ultra-high-Q spherical cavities. Given the geometry and material change as well as the three order of magnitude increase in Q, these devices have significantly improved sensitivity as compared to the previous work with microrings. Moreover, as a result of the change in device size and material, a distinct detection mechanism from that observed previously is responsible for the signal generation.

2. Theory

The two key parameters that need to be considered when designing an ultrasound sensor are the sensor sensitivity and the sensor response. The sensitivity is determined by cavity Q whereas the response is determined by the cavity material.

Sensitivity describes the smallest detectable signal change. In these measurements, it is more common to measure transmission than wavelength changes. However, as shown in Fig. 1, as long as the change in wavelength is less than half of the linewidth of the resonance, there is a linear relationship between transmission and wavelength, and, on the linear portion of the spectra, the Q and the slope are linearly proportional. As such, an increase in the Q results in an increase in resolution.

As mentioned, the cavity’s resonance wavelength is defined by the refractive index and the device geometry. Therefore, a change to either parameter will shift the resonant wavelength. The magnitude of both effects are determined by fundamental material constants; therefore, it is possible to predict which mechanism will be the dominant one in a given device system.

The refractive index change is dominated by the photoelastic effect. In the photoelastic effect, the acoustic wave causes a strain in the cavity, which in turn alters the refractive index. This behavior can be defined by [8]:

The cavity can also mechanically deform in response to an ultrasound pulse. In this case, the density changes due to molecular vibrations caused by the sound wave. In regions under compression, the density increases. This change leads to a localized decrease in the device size and an increase in the refractive index, simultaneously. The following equation relates the changing density in a material with an input pressure:

Since the resonance wavelength is also dependent on the geometry of the device, a change in radius will likewise produce a shift. Radius change effects are more apparent in devices with a size similar to that of the ultrasound pulse wavelength and with low K values. As the pulse hits the device, contraction and expansion will occur and will dominate the resonance wavelength change in devices with a size similar to the pulse wavelength. In previous work with polymer microrings, this effect was the dominant detection mechanism [5].

While there are analytical expressions for all of the key physical effects, because the ultrasound wave is time-dependent, the expressions become increasingly complex. For example, the pressure (P) is time-dependent. Therefore, in previous work, researchers have relied on proportionalities or used best fits. However, this approach does not allow for intelligent design of the ideal sensor device.

3. Finite element method simulations

To develop a generalizable model, we leveraged the Multiphysics capabilities of COMSOL Multiphysics 4.3a, specifically the acoustics and RF modules.

3.1 Optical Simulation Design and Parameters

The RF module is used to model the mode volume of the resonance under specific conditions. This model establishes the distribution of the optical mode within the silica and the surrounding medium (water). The 2D simulation geometry is set as the equatorial cross section of a silica microsphere shown in Fig. 2(a). The refractive index of the water was set to 1.332. The mesh size of the simulation was 0.021μm². Due to the symmetry of the device, the computation is greatly simplified. Figure 2(b) shows the results of the simulation with the mode intensity plotted. From this model, mode volume and mode distributions can be calculated [10]. For the present geometry, approximately, 1.5% of the mode is within the water surrounding the resonator, while the rest is inside the silica. Therefore, all values in Eq. (1) and Eq. (2) must be calculated for both water and silica.

 

Fig. 2 (a) Microscope image of a silica microsphere. (b) RF simulation results showing a small part of the sphere in a water environment with the mode distribution plotted at V/m².

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3.2 Acoustic Simulation Design and Parameters

The simulation geometry is shown in Fig. 3(a). All parameters are specified to match subsequent experimental values. A few of the material properties are modified to account for ultrasound effects. The density of all materials is changed to the relation shown in Eq. (2). The speed of sound (ν_s) in a material was defined in terms of density (ρ) and Bulk modulus (K) as follows:

 

Fig. 3 (a) Geometry of 2D FEM simulation in Comsol. (b) Pulse shape used in the model.

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The 1mm wide ultrasound source enters the system, defined as liquid water, from the upper boundary. Similar to the experimental conditions, the silica microsphere resonator is placed adjacent to the transducer and the steel sphere is directly in front of the transducer. However, to reduce memory requirements, the separation distance between the steel sphere and the resonant cavity is reduced. The boundary matching layer (BML) absorbs the ultrasound waves, eliminating reflections and pressure build-up within the simulation area, which would result in a non-physical result.

The two red points on the left of the silica microsphere in Fig. 3(a) indicate the location of the pressure monitors which are located on either side of the water-silica interface. Because the optical cavity is an evanescent field sensor, only changes to the refractive index which occur within the optical mode volume are detected.

The transducer used in the experimental component of the present work exhibited a time-dependent intensity profile which was a composite of an exponential growth and an exponential decay. Therefore, to more accurately mimic experimental conditions, a shaped ultrasonic pulse is defined with this envelope on top of a sine wave of 40MHz, as indicated in Fig. 3(b). The amplitude and width was adjusted to match the experimental data. The boundary was set to a plane wave radiation condition with a defined incident pressure field.

The mesh size is determined by the ultrasound wavelength. Therefore, to maintain 12 degrees of freedom per wavelength, the maximum mesh size is 3 microns. The minimum mesh size was set as the maximum divided by 20. A free triangular geometry was used to construct the mesh. The size of the time steps used in the simulation was determined by the Courant-Friedrichs-Lewy (CFL) condition to be 0.2ns [11].

3.3 Simulation results

A summary of the entire simulation can be viewed in Media 1, which is composed of 70 frames about 17ns apart. As seen in Fig. 4(a), 4(b) and Media 1, the ultrasound waves first interact with the silica microsphere cavity, and then the steel imaging object. The ultrasound echo from the steel object is then reflected onto the silica cavity. It is important to note that the BML is behaving as expected, and there are negligible reflections.

 

Fig. 4 Simulation results. (a) FEM result showing the initial pulse as it passes by the silica microsphere. (b) FEM result for the echo coming from the steel sphere as it reaches the silica surface. (c) Pressure variations recorded with a 0.225μs pulse at the point just outside the silica surface boundary in the water. (d) Pressure variation within the silica material showing the effects of the initial pulse and the secondary echo.

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To quantify this behavior, we evaluate the pressure using the previously mentioned power monitors. Figures 4(c) and 4(d) show the initial and reflected pulses in water and silica resulting from an initial pulse width of 0.225μs. Over large separation distances, the amplitude of the echo should decrease due to scattering and absorption losses in the water. However, to reduce memory requirements, our simulation area is decreased, and these effects ware not observed. The time between the two signals is determined by the distance between the cavity and the steel sphere and the ultrasound speed in water (1480m/s), thus allowing a precise measurement of this distance. Additionally, the initial pulse and the echo mirror each other in relative amplitude and frequency.

To convert changes in pressure to changes in refractive index, we first calculated the pressure-dependent density. The characteristic impedance was computed next, followed by the particle velocity (ν = P/Z_o). From there, the sound intensity could be calculated. Finally, the strain in the medium was computed using the relation:$S=\sqrt{2P\nu /\rho v_S^3}$. Each value was calculated for both silica and water, taking into account the different ultrasound propagation velocity. To preserve the sinusoid shape resulting from the ultrasound pulse, we multiplied the strain by +/−1, depending on whether the pressure at that time point had a negative or a positive sign. The changes in refractive index were then computed using Eq. (1). Based on these simulations and calculations, it becomes evident that there are multiple parameters, in addition to Q, which should be considered when designing a resonant cavity-based ultrasound sensor. For example, when using the photoelastic effect for detection, the pulse length and the bulk modulus plays a dominant role in the fidelity and magnitude of the signal. Specifically, if the Bulk modulus is increased, the refractive index change will be smaller. The density also contributes and has a similar effect to the Bulk modulus, in that its increase will cause a decrease in the overall refractive index changes.

4. Experiment

To verify the simulations, we performed a series of experiments using an ultra-high-Q silica microsphere resonant cavity.

4.1 Experimental setup

The overall testing setup is shown in Fig. 5(a) and 5(b). A narrow linewidth tunable laser centered at 775nm is coupled into a tapered optical fiber, which is used to evanescently couple the laser light into the silica microcavity (Fig. 2(a)) [12, 13]. The taper is aligned with the sphere using high precision nanopositioning stages, and the alignment is monitored using a top view machine vision system (Navitar). The transmission through the optical fiber is received by a photodetector (PD) and sent to a high speed O-scope/digitizer PCI card.

 

Fig. 5 Experimental testing of ultrasound response. (a) Schematic of the testing setup. (b) Rendering of the sensor setup. (c) Resonance peak used in the experiment with a Q of 9.5x10⁷. The red dashed line shows the fit line used to convert simulation data. (d) Response from microsphere to a pulse from the transducer when it is placed in front of the sphere. (e) Transducer response to an echo coming from the silica microsphere.

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During these experiments, the laser is operated in two different modes: 1) frequency modulated and 2) fixed wavelength. To modulate the frequency, a function generator PCI card sends a 100Hz/1VPP signal to the laser controller. This modulation allows for precise scanning across a narrow wavelength range (~0.03nm), which is necessary to measure the quality factor of the cavity (Q = λ/δλ, where δλ is the linewidth). The Q spectrum of the 192 μm diameter device used in the present work is shown in Fig. 5(c).

To perform the ultrasound measurements, the resonator-taper system must be immersed in liquid to maximize the propagation of the ultrasound. In the present work, distilled water was used. However, if the chamber material is not impedance-matched, strong reflections from the boundaries can be generated, significantly interfering with the primary signal and complicating signal analysis.

To meet these requirements, a sample chamber constructed entirely from PDMS was designed (Fig. 5(b)). It was connected to the nano-positioner using a PEEK rod. These materials are impedance-matched to water. Specifically, the testing chamber is formed from a pair of thin PDMS sheets separated by a PDMS slab. In addition to meeting the impedance requirements, PDMS is transparent and allows for imaging of the system during the experiment using the top view camera. The stem of the silica microsphere, shown in Fig. 2(a), is embedded in the PDMS slab. In this configuration, the sphere is suspended in the middle of the chamber.

The transducer is aligned parallel to the microsphere at the same height and slightly behind. The parallel alignment is important for echo detection since the waves return to the same place and a slight angle might result in non-uniform data collection. The transducer used in the experiment is fabricated in the Ultrasound Transducer Resource Center at the University of Southern California and generates 40MHz pulses.

To make sure the system responds as expected, a control experiment is performed. The transducer is placed directly in front of the silica microsphere, instead of behind it. The transducer is pulsed and the response in the microsphere is recorded. Figure 5(d) shows three sample pulses recorded without changing parameters. As can be seen, the signal is stable over all the recordings. The transducer is able to operate in a transmit and receive mode in which it records any ultrasound pulses it receives after sending the initial pulse. Therefore, it can behave as a reference or control sensor. Figure 5(e) shows the signal recorded multiple times by the transducer as an echo returning from the silica microsphere. The signal is once again consistent over the separate recordings and is similar to that detected by the microcavity. Both signals in Fig. 5(d) and 5(e) show 40MHz response, consistent with the transducer output. It is also of interest to note that the microsphere response shows a longer ringing, or more sensitive output, than the transducer.

In this work, a steel sphere is used as a proof of concept imaging object. It is placed directly in front of the transducer and microsphere cavity. The focal length of the transducer is between 1.5 and 3mm. Therefore, the object is placed within this distance. To keep the steel sphere in place, the sphere is embedded into a thin strip of PDMS.

To perform an ultrasound measurement, a resonance is first identified by modulating the laser using the function generator. Subsequently, the laser wavelength is set to the value at the steepest part of the left side of the resonance. Using a function generation, the transducer and the oscilloscope are simultaneously triggered. This approach allows the entire response to be recorded on the oscilloscope. For the ultrasound experiments, a high speed detector (150MHz) is used. To adjust the power to fall within the operational range of the detector, an attenuator is placed in line, after the cavity and before the detector.

Given the total system performance, the limiting factor in the data acquisition rate is the oscilloscope PCI card, which can record at 250 MSamples/second (250x10⁶ samples/second) for several hours (limited by the total RAM of the computer). Compared to other data acquisition methods, which rely on either high speed cables or on-board memory on external equipment, by directly acquiring and recording onto the computer, significantly faster and/or longer acquisition times are possible.

4.2 Results and analysis

Figure 6 shows the results from the ultrasound experiments using the peak shown in Fig. 5(c). In Fig. 6(a), three different signals are clearly identifiable. The initial signal is from the pulse leaving the transducer and passing by the microsphere. The second set of oscillations is due to the echo from the steel sphere. The last signal is the echo returning from the air-water interface behind the steel sphere. Based on the time between the initial pulse and the two subsequent pulses and the speed of ultrasound in water, we can calculate the distances. Specifically, the front face of the steel sphere is approximately 2.2mm away and the air-water interface is approximately 3.9mm. These values agree with our experimental design.

 

Fig. 6 Experimental data. (a) The entire signal received by the silica microsphere. (b) Zoomed-in graph of the steel echo. (c) Zoomed-in echo from the water-air interface.

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One interesting point is that the second steel-water interface is not detected. However, if one looks at the simulations, it is clear that the amount of energy which enters the steel is small and does not leave because of the large loss or attenuation of the wave as it propagates through the steel. Therefore, any generated signal is too small to detect.

Figures 6(b) and 6(c) show the echoes from the steel sphere and the air interface. The echo from the steel sphere is not as clean as the theoretically expected signal; however, the echo from the air interface is very clear. It can be noted that the baseline signal between the initial pulse and the steel echo in Fig. 6(a) is much noisier than the rest of the baseline. This is due to the long low-amplitude ringing of the ultrasound pulse that lasts longer than a microsecond. Because of this ringing, the microsphere response during the time the echo returns from the steel sphere is a combination of the continued ringing of the initial pulse and the returning echo. The combination of the two signals produces the non-ideal shape seen in Fig. 6(b). By the time the echo from the air interface returns, on the other hand, the ringing has stopped and the microsphere has only one response resulting from the echo alone.

Additionally, and to a smaller degree, some interference occurs from the echoes coming from the boundaries that exist within our chamber. Even with careful attention to impedance matching, we could still produce extra echoes that interfere and change the total wave as it returns to the resonator.

Additionally, we considered the bandwidth (BW) of our microsphere resonator. The maximum theoretical optical bandwidth is determined by the Q (BW = ν/Q). Based on this expression, the device in Fig. 5(c) has a theoretical bandwidth of 4.06 MHz. To verify this value, the pulse-echo response in the transducer is evaluated. Specifically, the signal is converted to the frequency domain by taking the Fast Fourier Transform, normalizing the result, and converting to dB. The frequency response is the difference between the response of the transducer and the sphere. From this analysis, there is a clear response at 40 MHz, as expected. Defining the BW as the −3dB drop from this value, the bandwidth is approximately 5 MHz, which is close to the predicted value of 4.06 MHz.

4.3 Converting Simulation Data to Match Experiment

In order to directly compare the simulation results with the experimental results it is necessary to convert both data sets into a common unit, the refractive index change.

The conversion of the simulation data relies on the relationship between wavelength change and refractive index discussed previously (Δλ/λ = Δn_eff/n_eff + ΔR/R). In direct contrast to previous work, because the silica sphere is larger than the ultrasound wavelength and the bulk modulus is very high, the change in the radius is negligible in comparison to the refractive index change. Therefore, the expression is simplified to Δλ/λ = Δn_eff/n_eff.

By combining the information from the mode volume simulations with the results from the acoustic modeling, the effective refractive index is calculated. Specifically, the time-dependent refractive index of the material (silica or water) is multiplied by the fraction of the optical mode in that material. These values are then combined to form the n_eff, and the resonance shift is determined.

The last step is to convert the resonance shift into a transmission change. Although a spectrum is typically fit to a Lorentzian, in the present work, we operated on the linear portion of the peak. Therefore, it is more appropriate to use a linear relationship between transmission and wavelength. As indicated in Fig. 5(c), we fixed the laser to the left side of the peak; therefore, this side was fit with a linear equation with >98% agreement. The linear fit is shown in Fig. 5(c) as the red dashed line. Additionally, the center wavelength of the resonance is known from the laser controller. To correctly attribute a transmission value for a corresponding wavelength shift, the shift is subtracted from the starting wavelength, and the transmission is calculated at that point.

Based on the simulation results, we can calculate the sensor response, which is related to the sensitivity, and the noise equivalent pressure. For a pulse width of 0.25 μs, the highest pressure to reach the surface of the sphere is 0.35 MPa. This pressure differential results in a voltage change of 1.74V. Defining the response as the pressure per signal (Pa/V), the demonstrated response of the silica cavity is 4911 mV/kPa. In comparison, the highest attained response for microrings with a Q of 4x10⁵ was 66.7 mV/kPa [14]. This significant increase is due to a combination of factors, including the increase in Q and increase in output power (5 mW compared to 240 μW [15]). The noise equivalent pressure (NEP) is related to the device response and the background system noise (NEP = noise/response). The noise in the experiments is primarily generated by coupling instabilities and is 2.63 mV. Therefore, the NEP is 0.535 Pa. For comparison, the NEP for microrings was over an order of magnitude higher (21.4 Pa) [14].

4.4 Comparing simulation and experimental data

Figure 7(a) shows an overlay of experimental and simulation results. In this specific simulation, the pulse length was 0.25μs. While there is general agreement, a more quantitative analysis of the predictive accuracy is important. There are several ways to quantitatively characterize the predictive nature of a model. One approach, based on computer science analysis methods, is to calculate the Accuracy. In this method, the true positives (TP), true negatives (TN), false positives (FP) and false negatives (FN) are determined, and then the Accuracy is calculated using the standard formula:

 

Fig. 7 Comparing experimental and simulation results. (a) Overlapping the experimental data (black) with simulation data (red). (b) Accuracy as a function of pulse width specified in the simulation.

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In the present work, we define all quantities with respect to the experimental results. Specifically, a true positive (TP) is if the simulation correctly predicts the presence of an experimental peak while a false positive (FP) is if the simulation predicts a peak that did not occur in the experiment. Both the peaks and the troughs are considered “peaks”. A true negative (TN) occurs if both the simulation and experimental values are on the zero axis. Similarly, a false negative (FN) is if only the simulation data is zero. For example, the accuracy of the simulation shown in Fig. 7(a) is 58%.

While there are several parameters which control the accuracy, one key value is the pulse width in the simulation. Keeping the pulse shape the same, the pulse duration used in the simulations is changed from 0.1μs to 1μs, and the accuracy analysis detailed above is performed (Fig. 7(b)). Based on this analysis, there are clearly additional ways to improve the predictive. For example, additional parameters that contribute to the microcavity response are: the shape of the ultrasound wave, the wave propagation distance, and boundary reflections. In the current simulation, BMLs are used in conjunction with plane wave radiation with scaled distances. Therefore, some aspects of the experiment are not fully captured.

5. Conclusion

In conclusion, we have demonstrated ultrasound detection based on the photoelastic effect using ultra-high-Q silica microcavities. A COMSOL Multiphysics model of ultrasound propagation and interaction with the optical field of the cavity is developed and experimentally verified. When compared to previous work, the Q factors are nearly 3 orders of magnitude higher [16], improving both the NEP and device response. However, the bulk modulus of silica is also significantly higher, resulting in previously unobserved and unanticipated interference effects, which could be reduced or eliminated if a transducer with a shorter pulse output is used. These results demonstrate the delicate balance between the improved sensitivity enabled by ultra-high Q factors and the deleterious effects of silica. These findings significantly expand the field of optical microcavity-based ultrasound detection, both in terms of material selection for sensor design and approaches for performing detection, and they will enable advances in the field of ultrasound imaging for medical applications [17–19].