Introduction. Optical resonators have emerged as powerful optical devices for next-generation, high-performance pressure sensors for photoacoustic imaging [1]. Among the possible implementations, Fabry–Perot (FP)-based pressure sensors have proven especially promising and have been successfully used for bio-imaging applications [2,3]. However, a comprehensive picture on the performance of FP pressure sensors is still missing. Recent work in the field focused on the effects that surface roughness [4], mirror parallelism [5], as well as optical aberrations [6] have on the shape and sensitivity of the interferometer transfer function (ITF) of an FP pressure sensor. Here, we extend this work by investigating the often present asymmetry in the ITF of FP pressure sensors as well as explaining the origins of this phenomenon and methods for tackling it’s effects under experimental conditions.

This property, characteristic for FPs illuminated with focused beams, has not been extensively described or studied in the literature, yet has important implications for accurately quantifying the sensitivity of such sensors which would allow for more standardized comparisons between FP sensors with different cavity parameters and application regimes. Also, this becomes especially crucial for recently described approaches for improving FP sensitivity by Adaptive Optics (AO) enhancement of the interrogation beam properties [6] where the experimentally quantified sensitivity is used as a feedback metric for the optimization. Here, we discuss the origin of ITF asymmetry in FP pressure sensors, the impact of common FP imperfections on this asymmetry, as well as propose a model for fitting asymmetric ITFs that improves the accuracy of the sensitivity measurement.

Origins of transfer function asymmetry. In this work, we focus on free-space detection as this mode is more applicable to be combined with AO enhancement. The ITF for free-space detection of an FP pressure sensor can be defined as

As the Gouy phase accumulation is dependent on the beam divergence/diameter [Fig. 1(b)], a larger divergence leads to an increased asymmetry which can be seen in Fig. 1(c). Additionally, as the effects of divergence can be partially overshadowed by the limited finesse of the cavity, the asymmetry is also dependent on the mirror reflectivity with stronger reflecting mirrors displaying enhanced asymmetry. This asymmetry has interesting properties when it comes to the optical sensitivity of the FP pressure sensor. We observe that the normalized optical sensitivity ($S_{norm}^\pm$) [10] is

Effects of mirror misalignment. The simplest FP cavity imperfection to consider is mirror misalignment, which results in the cavity becoming wedged, because of the non-parallel orientation of the mirrors. As the FP pressure sensor is a solid cavity, these imperfections, which originate from the manufacturing process, are very difficult to correct. Thus, it is important to evaluate the effects a cavity wedge angle has on the ITF asymmetry.

 

Fig. 1. (a) Comparison of the ITF sensitivity $S(\lambda )$ between a normal Airy ITF and an Airy ITF modified with a Gouy phase term (Airy*). (b) Gouy phase accumulation across different reflections. (c) Changes of the ITF asymmetry with beam radius ($w_0$) and mirror reflectivity ($R$). (d) Relation of the normalized optical sensitivity ($S_{norm}^\pm$) to beam diameter. (e) ITF asymmetry dependence on FPI wedge angle and mirror reflectivity ($R$) for a $w_0=25\,\mu$m beam. (f) Effects of optical aberration magnitude on the ITF asymmetry. For all panels: cavity thickness is $20 \ \mu$m; refractive index inside the cavity is 1.639; simulated wavelength range is 1560–1562 nm.

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To do so, we adopted a simple model based on a tilting frame of Ref. [7]:

Effects of optical aberrations. It was previously described that optical aberrations lead to a generation of higher-order Laguerre–Gauss modes in Gaussian beams [11]. These higher-order modes are more likely to cause ITF asymmetry in the FP system because the Gouy phase is dependent on the beam order [Eq. (5)]. To test this, we used the theoretical framework developed in our previous work (see Ref. [6]) to evaluate the effects of beam aberrations on ITF asymmetry. An ITF for an arbitrary electric field $E(x,y,\lambda )$ can be calculated by decomposing it in a Laguerre–Gauss basis:

In this case, $E(x,y,\lambda )=G(x,y,z,\lambda )\exp (ik\sum _i \alpha _i Z_i)$ and the total aberration $Z_{tot}=\sum _{i=3}^{15} |\alpha _i|$, where $G(x,y,z,\lambda )$ and $LG_i(x,y,z,\lambda )$ are a Gaussian beam and a Laguerre–Gauss beam, respectively, and $Z_i$ is a Zernike polynomial (see Supplement 1 for details). We found that indeed stronger aberrations cause the ITF to become more asymmetric [Fig. 1(f)] which is in line with the intuitive prediction from analyzing the Gouy phase term. This sheds additional light on the possible mechanism of the previously described ITF response to optical beam aberrations [6], which also behaves asymmetrically, with one edge being more resistant to the effects of Zernike-type aberrations.

Quantifying FPI sensitivity from experimental data. Practical use of FP pressure sensors requires characterization prior to measurement to determine the optimal bias wavelength. At the same time, comparison between different sensors requires quantification of the optical sensitivity. The recently described AO methods [10] for enhancing FP sensitivity require even more robust sensitivity quantification as it is used as a feedback readout for the AO optimization. All of these applications require information to be extracted from an experimentally measured ITF. One of the most commonly used approaches is to fit the experimental ITF data with a generic line shape function, such as a pseudo-Voigt function [10,12]:

 

Fig. 2. (a) Exemplary fit to simulated data showing the superiority of the asymmetric pseudo-Voigt function ($pVoigt_{asym}$) in fitting highly asymmetric ITFs. The functions are fitted to simulated data based on Eq. (1) that includes shot noise to recreate experimental SNR conditions as well as provide the ground truth necessary to estimate the fitting error. (b) Dependence of the log-fold error ($\sigma _{fold}$) on the ground truth asymmetry of the ITF. (c) (left) Exemplary fit of an experimentally acquired ITF comparing the symmetric and asymmetric pseudo-Voigt functions as well as their fitting residuals showing a clear improvement in the fit quality of the peak region (gray box). (right) Quantification of fit quality by root mean square (rms) error and maximum deviation (MaxDev) in the peak region. (d) ITF asymmetry quantification of a realistic FP pressure sensor ($R\sim 0.98$, $l_0\sim 20\ \mu$m) using $pVoigt_{asym}$ fitting. (e) Exemplary $S_{raw}$ improvement over the AO optimization procedure for both $pVoigt_{sym}$ and $pVoigt_{asym}$. (f) AO optimization validation by quantifying the US sensitivity of the FPI for optimization shown in panel (e). (g) Effective sensitivity improvement quantification over several points on the FPI. The Wilcoxon signed-rank test was used to compare the distributions of the data.

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Fundamentally, the lower bound of the mean fold error ($\overline {\sigma }_{fold}$) when fitting an asymmetric ITF using the symmetric pseudo-Voigt function is limited by the degree of symmetry [$s$, Fig. 2(b)]:

This can be corrected by allowing an asymmetrical line shape [see Figs. 2(a) and 2(b)], which can be realized in multiple ways. One of the simplest and most robust approaches is to allow the linewidth to vary as a function of the wavelength using a sigmoid function [13]:

We have used the $pVoigt_{asym}$ model to fit experimentally measured ITFs [Fig. 2(c), see Supplement 1 for details] and experimentally quantified the ITF asymmetry dependence on the beam radius based on the fit parameters. The experimental results qualitatively follow the predictions from our simulations where a smaller beam radius (and consequently higher divergence) leads to stronger asymmetry [Fig. 2(d)], corroborating the usefulness of our theoretical findings to experimental work.

Effects of ITF asymmetry on AO optimization. The ITF asymmetry can have detrimental effects on the robustness of quantifying the FPI sensitivity. This capability, however, is vital in situations that depend on their proper quantification, e.g., when used as a feedback mechanism to increase sensitivity via AO-optimization of the interrogation beam [10]. To further highlight the importance of proper ITF modeling/fitting, we have therefore compared the performance of symmetric and asymmetric pseudo-Voigt fitting schemes on AO optimization in an FPI displaying significant ITF asymmetry (see Supplement 1 for details).

Here we observed clear differences in behavior between the two schemes [Fig. 2(e)]. In particular, the $pVoigt_{sym}$ underestimated the actual sensitivity of the FPI which led to an erratic optimization trajectory that saturated quickly. However, the $pVoigt_{asym}$ was able to properly determine the FPI sensitivity, leading to a more gradual AO optimization trajectory as expected for indirect AO schemes.

To further validate these findings, we measured the effective increase in sensitivity by comparing the response of the FPI to synthetic ultrasound pulses generated with a ultrasound transducer [Fig. 2(f), Supplement 1 Figs. S1(b) and S1(c)]. Also in this case, we observed that $pVoigt_{asym}$ optimization led to a higher increase in sensitivity corroborating our previous observations. We have repeated these measurements, also normalizing for the AO-induced shifts in the photodiode working point, on several points across the FPI to ascertain that the improvements are not a result of statistical fluctuations [Fig. 2(g)].

Discussion. Fabry–Perot pressure sensors are a rapidly developing type of optical ultrasound detectors for photoacoustic imaging. Accurately quantifying their sensitivity is important not only for optimizing their design, but is also key for using advanced optical techniques (such as AO) to further improve their performance. Here, we discussed why ITF asymmetry arises in FP pressure sensors and how can it be affected by system imperfections (e.g., a wedged cavity or beam aberrations). Moreover, we showed that implementing an asymmetric line shape model allows for more accurate estimation of the optical sensitivity in theory, as well as showed experimentally that the fitted line asymmetry follows qualitative predictions from our simulations.

We believe that taking into account the ITF asymmetry will allow to better quantify the performance of FP pressure sensors which will aid in pushing both AO based sensitivity enhancement further in future realizations as well as allow more standardized comparisons between FP sensors with different cavity parameters and application regimes.