2.1.

Principle of Device

A Fabry–Pérot interferometer consists of a resonating cavity formed between two mirrors and an optically transparent medium between them, as shown in Fig. 1(a). Part of the light incident on the mirror gets reflected repeatedly between the two mirrors, with a periodic resonance condition occurring at every wavelength, for which the thickness of the cavity is equal to an integral number of effective half wavelengths (absolute wavelength divided by the refractive index). To detect the ultrasound wave, the intensity of reflected light over a range of interrogation wavelengths is studied, and the value corresponding to the highest slope is chosen for maximum sensitivity. Under interrogation using this optimal wavelength, when the acoustic pressure wave hits the cavity, it changes the thickness of the cavity and consequently the wavelengths, in which optical resonances occur. This results in a change in the intensity of the received light at the tuned wavelength, and thus the ultrasound wave can be detected.

Fig. 1

Fabry–Pérot ultrasound detector: (a) structure of the device showing NIR wavelength interrogation and (b) shift in resonance upon ultrasound detection.

2.2.

Fabry–Pérot Cavities: Effect of Waveguide on Q-Factor

For an ideal resonator, both resonator mirrors are unconfined and perfectly reflecting, and no power is lost during the round trip. However, for a practical resonator with small mirror areas, lateral losses due to multiple reflections at nonnormal incidence become significant, as illustrated by Li et al.¹⁹ These losses reduce the overall energy of the interferometer, which is equivalent to reduced reflectivity of the mirrors, and also cause phase dispersion due to increased path lengths. Therefore, the $Q$-factor of the resonator is significantly reduced. This loss can be compensated for in two ways by: increasing the reflectivity of the mirrors and restricting the diffraction loss in the lateral direction by introducing a waveguide in the resonator cavity. A waveguide can be created in the cavity by inducing a small refractive index change in the resonator medium to create a step-index profile. The following sections describe simulation studies carried out to study the effect of waveguides on $Q$-factor in cavities of different thicknesses to decide the optimal thickness, and the fabrication and testing of a prototype waveguide device. Thereafter, further refined simulation models have been created to study the effect of various physical parameters on the optical performance with a view to use the results from these simulations to guide further development and optimization of the fabricated device.

2.3.

Optical Resonator Q-factor and Acoustic Sensitivity

An important metric to evaluate the performance of an ultrasound detector is its NEP, the smallest detectable pressure signal for the device. For a Fabry–Pérot resonator, this minimal detectable pressure is inversely proportional to the resonator finesse $F$²⁵

Resonator $Q$ factor can be defined as

Therefore, using Eqs. (4) and (5), we can rewrite Eq. (3) as

This demonstrates that a higher $Q$-factor for the optical resonator implies a lower NEP and, consequently, a greater acoustic sensitivity. In this paper, therefore, the resonator $Q$-factor has been used as the performance metric to optimize the sensor design.

3.1.

Waveguide to Address Diffraction Losses

In order to visually demonstrate the efficacy of a waveguide in restricting beam walk off upon transmission over a distance corresponding to several round trips, a simulation study was set up. This simulation has been performed over a thickness of $1000\mu \mathrm{m}$ even though the envisaged final device thickness is only $\sim 20\mu \mathrm{m}$ in order to understand and compare the beam divergence for the two cases that would result after multiple round trips within the resonator. The refractive index of the material within the cavity was set to 1.5 (comparable to glass). The optical field in the polymer was studied to understand the distribution of energy. As seen in Fig. 2(a), the near-infrared (NIR) field spreads in the polymer to a diameter of around $100\mu \mathrm{m}$ at an axial distance of $800\mu \mathrm{m}$ from the source.

Fig. 2

Simulation results showing power distribution in dB for Fabry–Pérot interferometer devices (a) without a waveguide and (b) with waveguide.

A second simulation was done to evaluate the effectiveness of including the waveguide in containing the field distribution. This was done by selectively modifying the refractive index of the waveguide to mimic the core and cladding of the fiber (1.506 and 1.500, respectively). The results of this simulation are seen in Fig. 2(b). Without a waveguide, there is significant lateral loss in power over a transmission of $1000\mu \mathrm{m}$. In the case of the device with waveguide, for a distance of $1000\mu \mathrm{m}$ (25 round trips for a $20\text{-}\mu \mathrm{m}$-thick device), the power is still restricted within the $9\text{-}\mu \mathrm{m}$ core, proving the efficacy of the waveguide in reducing lateral losses.

A subsequent simulation of the full device with real dimensions is presented in Fig. 3, which shows a quantitative confirmation of the same effect via improvement in $Q$-factor. The values of the refractive indices assigned to the waveguide and surrounding portions of the polymer were 1.506 and 1.500, respectively. A $45\text{-}\mu \mathrm{m}$ cavity was simulated between two gold mirrors of 30-nm thickness with and without a waveguide. The corresponding optical resonances confirm that the $Q$-factor was seen to improve significantly—from 1853.33 without a waveguide to 3173.04 after including a waveguide.

Fig. 3

Improvement in $Q$-factor of the resonator upon introduction of waveguide.

Since beam divergence increases with higher thickness, it was expected that the improvement with including a waveguide would be more drastic for thicker devices. To confirm this, a study was done comparing the improvement with the introduction of a waveguide for three different Fabry–Pérot interferometer thicknesses, 20, 45, and $80\mu \mathrm{m}$. These thicknesses correspond to FSR of 40, 17, and 10 nm, respectively. As expected, the improvement in $Q$-Factor with the presence of a waveguide is seen to increase with cavity thickness (Fig. 4).

Fig. 4

The improvement of $Q$-Factor with the presence of a waveguide for different cavity thicknesses.

The improvement in $Q$-factor upon the introduction of the waveguide was highest for the $80\text{-}\mu \mathrm{m}$ device, from 3034.01 to 4779.94. In the case of the $45\text{-}\mu \mathrm{m}$ device, the improvement was reduced, and the $Q$-factor changed from 1853.33 to 3173.04. The least improvement, from 1301.04 to 1403.46, was observed in the case of the $20\text{-}\mu \mathrm{m}$ device. These simulations served as a confirmation that a self-aligned waveguide would provide significant improvement to the $Q$-factor of the device (Table 1).

Table 1

Improvement in Q-factor upon the introduction of the waveguide for different thicknesses.

3.2.

Effect of Mirror Reflectivity on Q-Factor

In the work discussed in the previous section, the mirrors for the resonating cavity were 30-nm gold layers. Gold was chosen as a mirror for the preliminary design because of the ease of fabrication, sufficient reflectivity at NIR wavelengths ($>95\%$) used for interrogation of the cavity, and poor reflectivity at UV wavelengths, which need to penetrate the gold layer to cure the epoxy—the optical medium of the resonator. However, since there are multiple successive reflections, there is significant energy lost from the cavity even with 95% reflectivity offered by the gold mirrors. Since the finesse, and therefore, the $Q$-factor of the resonator depend on mirror reflectivity,²⁵ the $Q$-factor for devices with gold mirrors was not very high. This can be improved using dielectric Bragg mirrors instead.

Dielectric mirrors consist of multiple thin layers of (usually two) different transparent optical materials and offer very high reflectivity (above 99%) because the reflections from the multiple interfaces constructively interfere. Bragg mirrors comprise several dielectric layers, each of which has a thickness equal to quarter of the design wavelength. This design leads to the highest possible reflectivity for a given number of layer pairs and given materials. Tadayon et al.²¹ used similar Bragg mirrors in their Fabry–Pérot interferometer in large size multimode devices, consisting of eight quarter-wavelength layers of titanium dioxide ($n=2.19$) and seven quarter-wavelength layers of silicon dioxide ($n=1.46$), designed for high reflectivity at 1550 nm.

A $45\text{-}\mu \mathrm{m}$ thick cavity simulated using this design showed an improved $Q$-factor of 7855.96 as compared to 2095.32 with gold mirrors, as seen in Fig. 5. Including a waveguide structure in addition to the dielectric mirrors further improved the $Q$-factor of the device to 10827.66. This is especially significant as it implies that high $Q$-factors may be achieved even at lower thicknesses, which allows for greater acoustic bandwidths.

Fig. 5

$Q$-factor enhancement with waveguides and dielectric mirrors.

3.3.

Planar Versus Plano Concave Simulation Models

The planar structure described in Secs. 3.1 and 3.2 provides a simplified model to test the improvement in $Q$-factor upon the inclusion of a waveguide. However, this does not accurately describe the structure fabricated on the tip of a single-mode fiber. Dip-coating of the epoxy would result in a plano-concave structure due to surface tension holding the epoxy on the fiber tip surface. In order to accurately model this and understand the effect of cavity shape on the device performance, the planar simulation model was studied in comparison to plano-concave structures. The largest contact angle possible for a plano-concave device would be 90 deg, resulting in a hemispherical drop. For lower contact angles, a gentler curvature on the droplet can be expected. Figure 6 shows a comparison of the optical characteristics of planar and hemispherical devices plotted alongside the focused plano-concave cavity created with the dimensions of the device presented by Guggenheim et al.²⁰ The total height of the planar and hemispherical devices was kept the same for a fair comparison.

Fig. 6

Comparison of characteristic curves of (a) planar, (b) hemispherical, and (c) focusing resonator structures.

The hemispherical device showed a negligible $Q$-factor of 387.18 (finesse of 3.22) in comparison to the planar cavity with $Q$-factor of 3123.17 and finesse of 26.23 for the planar device. This difference is because the hemispherical cavity is unable to focus light effectively back into the fiber cavity. The focused plano-concave model, on the other hand, can do so most effectively as its curvature exactly matches the emerging wave front. This approach has been used to improve finesse by a factor of 4 as compared to planar devices.²⁷ An important difference in our approach, however, is the presence of a waveguide—implying a flat wave front as opposed to the divergent beam that would be present in a device without a waveguide. This dictates that we require the second mirror to be as flat and parallel to the first mirror as possible for optimal performance. In order to understand the effect of the device curvature in the case of waveguided devices, a second set of simulations was performed, comparing waveguided hemispherical and planar cavities, presented in Fig. 7.

Fig. 7

Characteristic curves of hemispherical and planar waveguided cavities illustrate the importance of having a flat mirror for waveguided cavities.

As expected, the performance of the planar cavity surpassed that of the hemispherical cavity as the flat second mirror is better suited to reflect the plane wave front that emerges from the waveguide.

3.4.

Plano-Concave Resonators with Gentle Curvature Approaching Flat Mirrors: the Fiber-Ferrule Device

Although it is possible to control the curvature of the concave second mirror to an extent using epoxies with different contact angles, it is difficult to achieve the flat parallel resonator structure required for optimal performance of the waveguided device on a bare fiber. One way to achieve this is to insert the fiber into a ferrule and use the larger surface area to deposit a flat parallel mirror. This approach was evaluated through simulations using a model based on dimensions resulting from experimental measurements of dip-coating using a fiber-ferrule assembly (described in Sec. 4.3). The distance between the tip of the pillar and the second mirror can be varied depending on how far beyond the surface of the ferrule the fiber is placed and secured. To determine to what extent this positioning might affect the device $Q$-factor, and therefore, the tolerance on the position of the fiber with respect to the ferrule, a set of simulation models were designed to scale (for the portion of the device demarcated in Fig. 13) using the contact angle of 37 deg from the experiment described in Sec. 4.3, and the ferrule surface of diameter 1.8 mm.

As shown in Fig. 8, it is observed that the distance between the tip of the fiber and the second mirror significantly affects the $Q$-factor. The $Q$-factor for the device without a waveguide was 1558.30 (finesse 14.11), and those of the devices with waveguides that were 75%, 85%, 95%, and 100% of the total height of the device were 3113.59 (finesse 26.20), 3916.51 (finesse 32.22), 5255.45 (finesse of 39.79), and 5205.40 (finesse 40.40), respectively. From this study, it was inferred that the fiber must be positioned above the ferrule at a position such that the waveguide structure is at least 90% of the total device height. Running this test also provides us with the best estimate for the $Q$-factor of a device fabricated using this approach. As discussed in Sec. 3.2, dielectric mirrors in place of the gold mirrors used would result in a significantly higher $Q$-factor.

Fig. 8

Simulations run to estimate the tolerance on distance of the fiber tip from the second mirror and the associated characteristic curves.

4.1.

Bare Fiber Device Fabrication

The first iteration of devices fabricated consisted of plano-concave Fabry–Pérot interferometers constructed on single-mode optical fibers $125\text{-}\mu \mathrm{m}$ diameter. The following steps were followed for fabrication, as presented in our earlier work:²⁸

Fig. 9

Fabrication steps for fiber device: (a) first mirror deposited on fiber, (b) dip coating in waveguide epoxy, (c) pillar obtained by selective curing, (d) dip coating in cladding epoxy, and (e) deposition of second mirror.

Fig. 10

Microscope images of the device (a) after fabrication of the wave-guiding pillar and (b) after the dip-coating of the surrounding epoxy.

As seen in Fig. 10(a), the wave-guiding pillar was not straight and flat as desired but rather had an irregularly shaped tip, which would adversely affect the performance of the device by distorting the emerging wave front. Experiments conducted to further understand the cause for this are presented in subsequent sections. The completed device was then tested for optical resonance and the results are presented below.

4.2.

Results from Testing

4.2.1.

Testing for optical resonance

The completed device was then tested for optical resonance on a tunable NIR laser of wavelength range 1510 to 1640 nm via a fiber circulator. Results are as shown in Fig. 11. The top graph shows results obtained on a device manufactured without a waveguide. In this case, the FSR is 2386 GHz. This corresponds to a droplet height of $40.29\mu \mathrm{m}$. The $Q$-factor and finesse of this device were found to be 5139.80 and 62.79, respectively. The bottom graph shows the improved sensitivity in a device fabricated including a waveguide. In this case, the $Q$-factor and finesse are calculated to be 7728.50 and 93.94, respectively. Although these data do show a slight increase in the case of the device with waveguide, the sensitivity can be vastly improved if the nonideal shape of the pillar is addressed. Methods to do so are described in the following sections.

Fig. 11

Characteristic curves of the device (top) without waveguide and (bottom) with waveguide. Encircled points represent the resonant peaks.

4.3.

Fiber-Ferrule Device Fabrication

The $Q$-factor, and therefore sensitivity, of the interferometer increases with greater thicknesses as seen in Fig. 4. However, it is also desirable to have a thin device in order to improve the sensing bandwidth, leading to a trade-off between these parameters for an optimal design of the ultrasound detector. To fabricate resonator mirrors that are as flat and parallel to each other as possible, a method is proposed that employs a glass ferrule to hold the fiber. This increases the surface area available for dip coating of the cladding and thus offers control over the curvature of the cladding and the second mirror. This process was developed to obtain the dimensions for the model presented in Fig. 8, as well as to confirm the feasibility of depositing mirrors with gentler curvature than a hemispherical droplet.

Figure 12 shows the fabrication method proposed. First the $125\text{-}\mu \mathrm{m}$ bare fiber is inserted into a glass ferrule with inner diameter $125\mu \mathrm{m}$ and outer diameter 1.8 mm and secured using UV-cured epoxy. The fiber surface is above the ferrule surface as shown in Fig. 12(a). Thereafter, the first mirror is deposited using e-beam evaporation. The gold-coated fiber tip is then dipped into the high-refractive index epoxy used to manufacture the waveguiding pillar resulting in a droplet as shown in Fig. 12(b). The dipped fiber is then exposed to 405-nm UV light through the back connector, and then washed in acetone to remove the uncured epoxy. This results in the waveguiding pillar as depicted in Fig. 12(c). To form the cladding, the combination is dipped into the cladding epoxy to the level of the ferrule, as shown in Fig. 12(d). This results in a droplet with a certain contact angle, depending on the epoxy. As an example, the ferrule shown in Fig. 13(b) resulted in a contact angle of 37 deg when dipped in epoxy stereolithography (SLA) 3-D printer resin. Since this curve is spread over a relatively large area of diameter 1.8 mm as compared to the $125\text{-}\mu \mathrm{m}$ diameter of the fiber, the portion of the mirror above the fiber is almost parallel to the first mirror—meeting this important requirement for the waveguide device as described above.

Fig. 12

Fabrication process proposed for fiber-ferrule devices: (a) first mirror deposited on fiber affixed to ferrule, (b) dip coating in waveguide epoxy, (c) pillar obtained by selective curing, (d) dip coating in cladding epoxy, and (e) deposition of second mirror.

Fig. 13

Fiber-ferrule assembly highlighting (a) the simulated section and (b) a photograph of the same used for contact angle estimation.

As seen in Fig. 12(d), the distance between the tip of the pillar and the second mirror can be varied depending on how far beyond the surface of the ferrule the fiber is placed and secured. As shown in Fig. 8, it is observed that the distance between the tip of the fiber and the second mirror significantly affects the $Q$-factor.

Since the goal is to fabricate this device at the smallest possible size scale, it would be desirable to separate the fiber from the ferrule once the device has been completed. There are several ways this can be done:

4.4.

Nonideal Pillar Shape and Refractive Index Tuning

The greatest challenge in our work so far has been to control the shape of the epoxy pillar produced, which serves as the waveguide. A perfectly flat waveguide is essential in order to preserve the flat wave front emerging from the fiber core. To understand the process of UV-curing within the polymer, two sets of experiments were conducted. First, the hemispherical drop of epoxy on the fiber tip resulting from dip coating was exposed to UV light through the fiber connector while varying the duration of exposure. The fibers were then washed in acetone, and the resulting pillars imaged under the microscope. As seen in Fig. 14, the pillars obtained from shorter duration exposures are seen to have a sharp, pointed tip, rather than the desirable cylindrical shape, and a larger mass appears to cure on top of this pointed pillar when exposure time is increased to 60 s. To better understand the shapes of the pillars produced, the study was repeated while keeping the fiber tip immersed in a bulk volume of the epoxy during curing and washing in acetone thereafter. As expected, the shapes obtained reflected complex geometries.

Fig. 14

Comparison of epoxy pillars resulting from (top row) droplet cure and (bottom row) bulk cure for cure times of 30 s, 60 s, 2 min, and 5 min.

A possible reason for this could be because the UV source used to cure the epoxy was broadband, containing wavelengths from 320 to 500 nm, and the fiber is not a single-mode conductor for UV wavelengths. In addition, the curing of the epoxy is a dynamic process, in which portions of the epoxy that get cured first decide the path for UV light reaching the volume beyond. Therefore, it was desirable to use longer wavelengths so that their penetration depth is greater than the device thickness, eliminating the complications resulting from progressive curing of epoxies at different depths. Another possibility was to change the epoxy being used since the photochemistry of the epoxy and the photoinitiator also affect the curing process. To test this, fibers were dip-coated in different epoxies and cured using a 405-nm laser through the back connector, and then dipped in acetone to wash away uncured epoxy. It was seen that epoxy SLA 3-D printer resin, designed to cure at 405 nm UV produced pillars that were consistently seen to be straight, perpendicular to the fiber surface, and with flat tops, satisfying our most important requirement. This is seen in Fig. 15.

Fig. 15

Photographs of straight pillars obtained using 405-nm curing and multiple samples demonstrate repeatability of the process.

This experiment confirmed that it was possible to fabricate epoxy waveguides with flat top surfaces by selective curing of UV sensitive epoxies, thus overcoming our biggest challenge. This waveguide structure can be incorporated into either bare fiber devices or fiber-ferrule assemblies as described above. Further, since the waveguide is self-aligned, this fabrication method can easily be extended to fiber bundles as well. Future work shall include complete characterization—optical and acoustic performance—of the devices fabricated with the waveguide and a cladding surrounding the waveguide with the requisite refractive index difference between the two materials for single-mode operation ($\sim 0.006$, calculated theoretically and confirmed via simulations), as well as dielectric mirrors for improved reflectivity.