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Abstract
Parity-time (PT)-symmetric optical sensors operating around exceptional points have recently gained much attraction due to their improved sensitivity for measuring small perturbations. Previously, most of the PT-symmetric sensors have been based on detecting the mode splitting that arises due to the perturbation-induced change in coupling strength between two sub-cavities of the PT-symmetric system. Here, we present a model of a fiber Fabry-Pérot linear cavities sensor tailored to operate in the broken parity-time symmetric region. We propose, what we believe to be, a new sensing metric based on the mode’s linewidth change as a function of perturbation-induced loss changes in one of the cavities. The coupling strength between the two sub-cavities remains unchanged in our proposed sensor. We derive a mathematical expression that describes a square root dependence of the full-width-half-maximum change as a function of the refractive index change-induced loss, which is introduced via a tapered fiber in one of the fiber cavities. The proposed fiber Fabry-Pérot refractive index sensor has a maximum sensitivity of 2.26 × 107 GHz/RIU and the lowest detection limit of 10−9 RIU, widely outperforming comparable cavity sensors subject to the same refractive index change, gain, and loss settings.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Can we use coupled resonant systems in optical fibers to develop sensitive and viable sensing systems? The paper aims to address this question by proposing a fiber-based, coupled Fabry-Pérot sensing platform. Fiber Fabry-Pérot etalons (FFPE) have been used widely in various domains, such as lasers [1], wavelength filters [2], and optomechanical devices [3]. The simple structure, low cost, and immunity to electromagnetic radiation make FFPE a desirable platform for sensing applications [4,5].
For a wide-scale adoption across industries, FFPEs remain to be developed with improved sensitivity and a lower detection limit to meet the relevant performance requirements. Previously, in a single-cavity FFPE platform, resonant frequency and full-width-half-maximum (FWHM) changes have been employed as sensing metrics [6]. Researchers have also used active cavities to enhance the sensitivity of FFPE sensors [7]. To improve the sensing sensitivity further, a plausible platform consists of coupled gain and loss cavities that are arranged in a non-Hermitian parity-time symmetry (PTS) configuration [8]. This enables the resulting FFPE to perform exceptional point sensing, which is not explored in detail for the coupled fiber cavities.
The exceptional point phenomenon has recently generated increased interest in optical sensing within small footprint configurations [9]. Researchers have designed various optical systems such as microcavities [10], ring resonators [11], gyroscopes [12], and resonant optical tunneling effect resonators [13] for refractive index, temperature, and pressure sensing. Although in most proposed schemes, the sensing was performed by relating the amount of the mode-splitting to the change in the coupling strength between coupled resonators. However, there were some exceptions. For example, in one of the gyroscope-based works [14], researchers looked over the mode’s FWHM changes as a function of perturbations-induced coupling strength changes in coupled ring resonators.
The exceptional point sensing in optical fiber-based cavities remains largely unexplored. In one of the fiber loop resonator works, researchers sensed the mode-splitting as a function of coupling strength changes induced by temperature changes [15]. The linear fiber cavities are more sensitive as the propagating mode samples perturbing regions twice than loop cavities [7]. However, in linear fiber cavities, perturbations-introduced coupling strength changes are challenging to exploit for sensing purposes.
In the present work, we provide the first proposal, to our knowledge, for using coupled and linear fiber cavities in the broken PT-symmetric region and introduce a new sensing metric of FWHM change as a function of induced loss in one of the cavities. Notably, the coupling strength remains constant in our designed sensor. The sensor consists of a tapered fiber in one of the cavities (loss-cavity) as a sensing head that responds to external refractive index perturbations. Consequently, changes are induced in the gain-loss balance within the PT-symmetric cavity without affecting the joint coupling strength of the system. We demonstrate that our FFPE sensor exhibits a square-root dependency of FWHM on the loss change in the cavity due to refractive index change. We also show that, compared to a single-cavity fiber sensor, the FWHM enhancement is significantly larger in the proposed PT-symmetric system. We determine that our sensor has a maximum sensitivity of $2.26\times 10^7$ GHz/RIU and the lowest detection limit of $10^{-9}$ RIU, offering a promising platform for physical and chemical sensing in various applications.
2. Fabry-Pérot sensor model
A linear and coupled FFPE is a non-Hermitian system with a complex-valued spectral response [16]. In the complex spectral response, the imaginary part of the complex eigenvalue (resonant frequency) represents the decay constant related to finite spectral FWHM at the resonance frequency, whereas the real part represents the eigenenergy. Similarly, the eigenstates are the resonant modes of the optical non-Hermitian system. The fundamental work by Bender et al. [17] shows that non-Hermitian systems can also have real spectra if they follow a unique symmetry called PT-symmetry [18]. In the PT-symmetric FFPE, when all the eigenvalues and eigenstates of the system coalesce for a particular parameter space, the system is at an exceptional point. Eigenvalues in the broken symmetry region respond strongly to the external perturbation near an exceptional point.
We consider a fiber Fabry-Pérot sensor having two coupled and linear cavities with end reflection coefficients $R_1$ = $R_3$ = $0.8$ and equal lengths $l_1$ = $l_2$ = $5$ cm, as shown in Fig. 1. We design one of the cavities as a gain medium by incorporating optical amplification and the other cavity as a loss medium using a tapered fiber of diameter 1 $\mu$m. Based on the coupled mode equations, the Hamiltonian of an FFPE sensor comprising two coupled cavities (one representing net gain $B_1$ and the other with net loss $B_2$) of equal length, with resonance frequencies $\omega _1$ and $\omega _2$ can be expressed as $H=\left (\begin {smallmatrix}-\omega _1-i\frac {B_1}{2} & J \\J & -\omega _2-i\frac {B_2}{2} \end {smallmatrix}\right )$, where $J$ is the joint coupling strength between the two cavities [19]. The net gain $B_1$ is the sum of losses due to finite reflectivity $R_1$ of $FBG_1$ and gain due to amplification in the gain cavity. Similarly, the net loss $B_2$ is the cumulative loss due to $FBG_3$’s finite reflectivity $R_3$ and loss due to a tapered fiber segment in the loss cavity, as shown in Fig. 1.
Fig. 1. Proposed FFPE sensor with coupled gain and loss cavities. The amplifier in the gain cavity compensates for all the system losses. The tapered fiber immersed in the liquid adds loss in the loss-cavity through the evanescent field leakage. The FFPE sensor has fixed reflectors, i.e., fiber Bragg gratings (FBGs) with reflection coefficients $R_1$, $R_2$, and $R_3$.
The tapered fiber introduces losses in the cavity through its evanescent field. The refractive index changes in the liquid surrounding the tapered fiber impact the evanescent field and, consequently, loss in the cavity. We quantify the evanescent field using a finite element model of tapered fiber immersed in water. After the base value evanescent field determination due to pure water, we change the refractive index of the water solution from $10^{-9}$ to $10^{-4}$ RIU. Due to the changing refractive index, the fractional evanescent field, $\alpha$, increases from 0.02 to 0.16, as shown in Fig. 2. The increase in the evanescent field also makes physical sense as the propagating mode pulls out when the external medium refractive moves towards the tapered fiber refractive index.
Fig. 2. The fraction of evanescent field leakage: the calculations are performed using the finite element method (FEM). (a) A tapered fiber segment with a length of 5 cm and diameter of 1 $\mu$m immersed in the liquid.
We calculate the absorption loss, $A_{l2}$, due to the fractional evanescent field leakage using $A_{l2} =\alpha C_w$, where $C_w$ is the water absorption coefficient whose value is 7.9959 $cm^{-1}$ at 1550 nm [20]. We determine the overall loss in the loss-cavity using the following expressions [21]:
where $B_1$ and $B_2$ are the intensity decay rates, and $A_{l1}$ and $A_{l2}$ represent the gain and loss in the coupled gain and loss cavities. The sub-cavities lengths and refractive indices are represented by $l_1$, $l_2$ and $n_1$, $n_2$, respectively. The speed of light is represented as $c$, and $T_1$, and $T_3$ are the transmission coefficients of $FBG_1$ and $FBG_3$.We determine $B_2$ = 14.742 GHz using Eq. (2). To achieve an exceptional point in the FFPE sensor, we set $B_1$ = −14.742 GHz to balance gain and loss in the cavities to satisfy the PT-symmetry condition ($B_1$ = -$B_2$) [15]. We then select joint coupling strength [15] $J_{th}=\frac {1}{2}|B_2|=\frac {1}{2}|B_1|$ = 7.371 GHz, to achieve the exceptional point where the system has the same resonance frequencies for both the modes, i.e., $\omega _1$ = $\omega _2$ = $\omega _0$. We determine the $R_2$ of the middle reflector using [22]. The process of achieving an exceptional point is illustrated in Fig. 3.
Fig. 3. Exceptional point: (a) Tuning of gain in a cavity to achieve the PT-symmetric state in the coupled cavities. (b) The joint coupling strength ($J_{th}$) between gain and loss cavities to achieve the exceptional point (EP).
With a fixed joint coupling strength ($J_{th}$), when the loss in the cavity changes via tapered fiber, the system can experience three regimes [22]: $(i)$ PT-symmetry broken regime with $|B_1-B_2|>J_{th}$ where the change in the eigenfrequency of the system has equal real parts (no mode-splitting) but has equal and opposite imaginary parts leading to FWHM broadening, $(ii)$ PT-symmetric unbroken regime with $|B_1-B_2|<J_{th}$ where eigenfrequency of the system is real, leading to mode-splitting, and $(iii)$ The exceptional point (EP) where eigenfrequencies and eigenvectors coalesce. These three regimes are shown in Fig. 4.
Fig. 4. The evolution of real and imaginary parts of resonant frequency in an FFPE sensor when a loss in the loss cavity is varied. The exceptional point is located at $J_{th}$ = 7.371 GHz, having balanced gain and loss, i.e., $B_1$ = -$B_2$ = 14.742 GHz. (a) the real part of eigenfrequencies in the broken PT-symmetric region. (b) The imaginary part of the change in the resonant frequency of the system in the broken PT-symmetric region.
In a fiber Fabry-Pérot sensor at the exceptional point (EP), where $B_1$ = -$B_2$ and $B_2$ = 2$J_{th}$, a perturbation due to varying loss shifts the system into the broken PT-symmetric region, i.e., the gain and loss in the coupled cavities are no more balanced. There is a change in the natural frequency of the cavity corresponding to the loss change. In a fiber Fabry-Pérot sensor, leakage through the tapered fiber increases and adds an absorption loss to the cavity due to the changing refractive index in the tapered fiber surrounding medium. There is a change in the system’s resonance frequency ($\Delta \omega$) due to a loss change in a coupled cavity. The eigenfrequencies of the resonant modes in a coupled cavities system are given in Eq. (3):
The change in the eigenfrequency of the coupled cavity system due to a slight change in the refractive index of the loss-cavity in the broken PT-symmetric region is given as:
Separating the real and imaginary parts, we get:
The $\Delta \omega _{PTS\_b}$ represents the eigenfrequencies difference in the broken PT-symmetric region, where $\Delta \omega = 4\pi c\Delta nl/K\lambda ^2_o$ [23] and K is an integer whose value is 2237419 at 1550 nm. The origins of K lie in a standard cavity equation, which states that the integral multiple of wavelengths can fit in a cavity of length $l$, i.e., $K \lambda _o=2nl$. The imaginary part of $\Delta \omega _{PTS\_b}$ gives information about FWHM of the system.
In the broken PT-symmetric region, near the exceptional point, where $\Delta \omega ^2\ll 16J_{th}^2$, we can simplify Eq. (5) to determine the change in FWHM of the sensor as:
In the broken PT-symmetric region, near the exceptional point, we can write the equation of FWHM ($\Delta \omega _{FWHM}$) in terms of the refractive index change ($\Delta n$) as:
As previously stated, we assume the sensing head is a tapered silica fiber immersed in a liquid within the loss cavity, as illustrated in Fig. 1. When the surrounding refractive index ($\Delta n$) increases, the electromagnetic field will gradually leak from the tapered fiber, as depicted in the simulation results shown in Fig. 2. Consequently, this phenomenon induces losses within the cavity, ultimately resulting in variations in $\Delta \omega _{FWHM}$, as expressed in Eq. (7), with respect to $\Delta n$.
Equation (7) shows a square-root dependency of FWHM on the refractive index change ($\Delta n^{'}$) near the exceptional point in a broken PT-symmetric region. There is no mode splitting as the real parts of the eigenfrequencies are the same. However, there is a change in the imaginary parts of $\Delta \omega$ of the same magnitude but in the opposite direction leading to FWHM enhancement. The FWHM enhancement is more profound in the region near the exceptional point where the refractive index change is small.
After parking the coupled FFPE system at the exceptional point, we vary the refractive index of water by $10^{-9}$ to $10^{-4}$ RIU in which the tapered fiber is assumed to be immersed. As mentioned previously, we use FEM to determine the evanescent field for the tapered fiber for each refractive index change and calculate the corresponding loss $B_2$ experienced by the FFPE system. The perturbations due to varying loss shift the system into the broken PT-symmetric region where the cavities’ gain and loss are no longer balanced.
3. Numerical simulation results
The numerical simulations in Fig. 5 show FWHM enhancement in the broken PT-symmetric region of an FFPE sensor for $10^{-9}$ to $10^{-4}$ RIU range of refractive index change. The FWHM increases sharply with a square-root dependence on the refractive index change in the broken PT-symmetric region. The blue plot represents theoretically predicted FWHM ($\Delta \omega _{FWHM}$) using the imaginary part of Eq. (5). Whereas the red circle plot shows our derived approximate FWHM using Eq. (7). The approximate FWHM overlaps the theoretically predicted FWHM near the exceptional point in the broken PT-symmetric region. However, when we move away from the exceptional point, the approximate FWHM no longer represents the system’s FWHM.
Fig. 5. FWHM change comparison of an FFPE sensor with coupled gain and loss cavities with a single-cavity (1C) sensor having balanced gain and loss and a single-cavity (1C) sensor with loss only. For an FFPE sensor operating in the broken PT-symmetric region, the blue plot shows theoretically predicted FWHM without approximation using the imaginary part of Eq. (5), whereas the red circles show our derived approximate FWHM using Eq. (7). The yellow-filled circles represent the numerical simulations of FWHM using Lorentzian fit of the transmission profiles evaluated (Eq. (8)) for the selected refractive index change $\Delta n^{'}$ values. The numerical simulation results of FWHM shown in green and purple plots are of single cavity (1C) with loss only and single cavity (1C) with balanced gain and loss, respectively, using the Lorentzian fit of their transmission profiles evaluated using Eq. (8). The inset is the log-log plot of the FWHMs for 1C (L), 1C (G+L), and PTS sensor near an exceptional point.
To validate our derived FWHM expression, we evaluate the FWHM of the transmission profiles for selected values of the refractive index change. We determine the transmission profiles for the assumed values of refractive index change using the transmission amplitude expression derived using the electromagnetic model of the multicavity Fabry-Pérot etalon presented in [23] as:
Fig. 6. An analysis of tapered fiber (sensing area) length and refractive index change ($\Delta n^{'}$) limit for which expression Eq. (3) holds.
Now, we compare our proposed FFPE sensor operated in the broken PT-symmetric region with (i) a traditional single-cavity FFPE sensor with loss only and (ii) a single-cavity FFPE sensor with gain and loss. For a fair comparison, we make some assumptions, i.e., $(i)$ the end reflectors’ reflection coefficients will be the same for all the sensors, $(ii)$ the net optical path lengths of the fiber cavities for all the sensors will be equal, and $(iii)$ all the sensors will have identical tapered fiber segments. In Fig. 5, we obtain simulation results for $(i)$ a traditional single-cavity FFPE sensor with loss only and $(ii)$ a single-cavity FFPE sensor with gain and loss shown in green and purple color, respectively, by determining FWHM using the Lorentzian fit of the transmission profiles evaluated using Eq. (8). In these simulations, we assume $R_2$ = 0 to convert the multicavity model into a single cavity system. The results show that the FWHM change is directly proportional to the refractive index change in a traditional single cavity sensor with loss only. However, when we add the same gain as the loss in the single cavity due to the tapered fiber, the slope of the FWHM change increases for the same refractive index change as in the single cavity sensor without gain. The plot in blue color is the imaginary part of Eq. (7) representing FWHM without approximation. The red circles shows our derived approximate FWHM using Eq. (7), and yellow-filled circles represent the numerical simulations of FWHM using Lorentzian fit of the transmission profiles evaluated using Eq. (8) for the selected refractive index change $\Delta n^{'}$ values in our proposed sensor operated in the broken PT-symmetric region. With the square-root dependence of FWHM change on the refractive index change, our proposed FFPE sensor in the broken PT-symmetric region outperforms the traditional single-cavity FFPE sensor with a maximum sensitivity of $2.26\times 10^7$ GHz/RIU. The inset in Fig. 5 is the log-log plot of the FWHM curves of the three sensors. In the log-log plots, the single cavity FFPE sensor in both the cases mentioned in (i) and (ii) shows a linear response. In contrast, the log-log plot of our proposed FFPE sensor in the broken PT-symmetric region shows a linear response with a half-slope as compared to (i) and (ii) due to square root dependence in the refractive index change on a linear scale.
The range of refractive index change for which our derived FHWM expression holds depends on the length of the tapered fiber, which is also the length of the loss cavity. We analyze the shift in refractive index change limit for different tapered fiber lengths in the 1 cm to 10 cm range, as shown in Fig. 6. We notice that for the given lengths, the deviation in the refractive index change limit for which our expression of FWHM change holds is small. Hence, we can take any value of the tapered fiber length from 1 cm to 10 cm.
4. Discussion
FFPE sensor operated in the broken PT-symmetry region, the change in the real part of the eigenfrequencies becomes zero leading to no mode splitting. In contrast, the FWHM of the sensor enhances due to the square root topology of the complex energy near the exceptional point. For an FFPE sensor, one can achieve a larger sensitivity enhancement for smaller perturbations near the exceptional point in a broken PT-symmetric region. The FWHM change as a function of the refractive index is used as a sensing tool in a single-cavity fiber sensor. However, previously, it has not been used for sensitivity enhancement in an FFPE sensor with coupled cavities. This paper uses FWHM change as a function of the refractive index in the coupled-cavity fiber sensor designed to operate in the broken PT-symmetry region. The novelty in our work is that we have tuned the loss in the sensor cavity by independently changing the refractive index in the tapered fiber without changing the coupling strength of the gain-loss sub-cavities. The coupling strength depends on the reflectivity of the central FBG (Fig. 1), which is fixed and independent of gain and loss tuning. In coupled ring resonators, mode-splitting enhancement is generally used as a sensing technique in which loss is added to the resonator while perturbing the coupling strength [24].
An experimental implementation of the proposed refractive index sensor can be realizable by using off-the-shelf fiber components. The loss-cavity can be built by joining a tapered fiber with the FBGs on single-mode fibers (SMF28) with a central wavelength of 1550 nm. The gain-cavity can be designed using an amplifier [7] or a custom-made erbium-doped fiber amplifier (EDFA) with the specified amplification in dB required to balance loss in the loss-cavity. A tunable laser source and an optical detector can be used as input and output devices for the sensor.
The challenges we may face while implementing the setup are noise sources in the sensor, including the amplifier’s nonlinearity, gain fluctuations, and ambient temperature effects on FBGs and the analyte. One can minimize the FBGs’ reflectivities shift due to the temperature variations by experimenting in a temperature-controlled environment and choosing the FBGs with narrow bandwidth to ensure the wavelength scanning range is small compared to the optical amplifier bandwidth. One can also implement the proposed sensor on a chip with an onboard amplifier and tapered fiber cavity-based sensing head.
5. Conclusion
In summary, we propose an ultra-sensitive FFPE refractive index sensor operated in the broken PT-symmetry region. A significant enhancement in FWHM is predicted for an ultra-small refractive index change in the sensor’s fiber sub-cavity. We validate FWHM enhancement and its dependence on the discretely added loss in the fiber sub-cavity through theoretical analysis and numerical simulations while keeping the joint coupling strength fixed. We compare the FWHM change of our proposed FFPE sensor having coupled gain and loss sub-cavities with the traditional single cavity FFPE sensor having the same gain and loss settings. Due to the square-root dependence of FWHM change on the refractive index change, our FFPE sensor’s sensitivity is much better than that of the single-cavity FFPE sensors with linear FWHM change curves. Our results indicate that an FFPE sensor operating in the broken PT-symmetric region outperforms a traditional single-cavity fiber sensor with a maximum sensitivity enhancement of $2.26\times 10^7$ for the $10^{-9}$ to $10^{-4}$ RIU range of refractive index change. We anticipate that the present work will find various liquid and gas phase sensing applications involving fiber cavities.
Funding
Syed Babar Ali Research Award (GRA-011).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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