High-order exceptional point based optical sensor

Yulin Wu; Yulin Wu; Yulin Wu; Yulin Wu; Peiji Zhou; Peiji Zhou; Ting Li; Ting Li; Ting Li; Weishi Wan; Yi Zou

doi:10.1364/OE.418644

1. Introduction

In quantum mechanics or optical systems involving gain or loss, the Hamiltonian operator is usually not Hermitian [1]. The systematic research of non-Hermitian system can be traced back to 1998, Bender and Boettcher showed that a non-Hermitian Hamiltonian could still possess a real spectrum if it preserves parity-time (PT) -symmetry [1]. Due to the isomorphic equivalence between the Schrödinger equation and the classical wave equation, optics has become an ideal platform for studying the concepts of non-Hermitian systems both theoretically and experimentally [2]. A simplest non-Hermitian optical system, which is also a PT-symmetric system, consists of two coupled waveguides or resonators with gain and loss media, where the loss is achieved by absorption or radiation whereas the gain is realized by optical or electrical pumping [3,4]. The eigenvalues show different characteristics for the PT-symmetric phase, where eigenvalues are real, and PT broken phase, where eigenvalues are complex, separated by an abrupt phase transition point, the exceptional point (EP). In more detail, EPs collapse two or more eigenvalues and their corresponding eigenstates in the parameter space [5]. These counter-intuitive characters lead to extensive research in non-Hermitian systems [6–8], such as unidirectional light transport [3,9–11], loss-induced transparency [12], perfect absorption [13], selective single-mode lasing [14,15], topological chirality [16,17], Bloch oscillations [18–20] and high-order exceptional points [21,22].

Recently, it has been shown that the bifurcation properties of non-Hermitian degeneracies can provide a means of enhancing the sensitivity of resonant optical structures to external perturbations [22–25]. In particular, [22] showed that for a non-Hermitian system designed around the EP, the eigenfrequency splitting Δω has an ɛ^1/N-dependency (referred as “response order”) with the external perturbation ɛ, where N is the order of the EP (referred as “EP order”), indicating the coalescent of multiple eigenvectors. This characteristic is extremely important for designing a highly sensitive sensor, since an N^th root of the small perturbation ɛ, i.e. |ɛ|<<1, actually boosts the response compared to the linear case, of which N=1. This statement provides a basic rule for designing a high-order non-Hermitian optical sensor. However, it is not always right. In some cases, the “response order” does not match with the “EP order”, especially when the system is highly symmetric, which is referred to as the degeneracy of response order.

In this paper, we will firstly discuss the response order of high-order non-Hermitian systems and derive the requirements for achieving a nondegenerated response order, which will be a useful criterion for the following high-order non-Hermitian sensor design. After that, a fourth-order non-Hermitian optical sensor will be designed based on the proposed criterion, of which the coalescence of four eigenvalues would form a fourth-order EP, featuring a fourth root of the perturbation related wavelength splitting that could significantly amplify the response of small perturbations, leading to higher sensitivity.

2. Analysis of N^th-order non-Hermitian system

A microresonator, which can convert the variation of the surrounding refractive index to a resonant wavelength shift, widely used in sensing applications, is chosen as the basic building block for our device. We firstly consider a coupled microresonators based non-Hermitian system, as shown in Fig. 1, of which three resonators with the same isolated eigenvalue β₀ and different coupling coefficients κ and iκ line up. The Hamiltonian for this system is:

(1)$$H = \left[ {\begin{array}{{cc}} {{\beta_0}}&{\begin{array}{{cc}} \kappa &0 \end{array}}\\ {\begin{array}{{c}} \kappa \\ 0 \end{array}}&{\begin{array}{{cc}} {{\beta_0}}&{i\kappa }\\ {i\kappa }&{{\beta_0}} \end{array}} \end{array}} \right]$$

where the diagonal elements represent the eigenvalues of each resonator, and the off-diagonal elements represent the coupling between them. The Hamiltonian has a triple degenerated eigenvalue β_1,2,3=${\beta _0}$ as well as a triple degenerated eigenvector (1, 0, i)^T. In this case, the point where the eigenvalue β_1,2,3=${\beta _0}$ is a typical third-order EP.

Fig. 1. Schematic of a third-order non-Hermitian system

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When applying a small perturbation ɛ on Cavity One, the response of the whole device is:

(2)$$\Delta \beta = {\varepsilon ^{\frac{1}{3}}}{\kappa ^{\frac{2}{3}}} \propto {\varepsilon ^{\frac{1}{3}}}$$

It is a third-order response. The same conclusion can be drawn when applying perturbation ɛ on Cavity Three.

However, when perturbation ɛ is applied to Cavity Two, the device response becomes:

(3)$$\Delta \beta = \varepsilon \propto \varepsilon $$

It is a first-order response, even though the EP is a third-order EP. Therefore, high-order EPs do not guarantee high-order responses to the applied perturbations.

To give a guideline for designing a device with a high-order response, we start from a universal case of an N^th-order system. We assume that $\beta = {\beta _0}$ is the multiple-root with multiplicity j for the characteristic equation at EP, which indicates j eigenvalues degenerate to ${\beta _0}$ at EP. Especially when the corresponding j eigenvectors collapse to one, this N^th-order system has a j^th-order EP. Hence, the characteristic equation for an N^th-order device can be expressed as:

(4)$${a_N}{({\beta - {\beta_0}} )^N} + {a_{N - 1}}{(\beta - {\beta _0})^{N - 1}} + \cdots + {a_j}{({\beta - {\beta_0}} )^j} = 0$$

where ${a_\textrm{j}}, \cdots ,{a_{N - 1}},{a_N}$ are constants determined by the Hamiltonian matrix. When a disturbance is applied to a resonator in the device, the perturbed characteristic equation becomes:

(5)$$\begin{array}{c}({{b_N}\varepsilon + {a_N}} ){({\beta - {\beta_0}} )^N} + ({{b_{N - 1}}\varepsilon + {a_{N - 1}}} ){({\beta - {\beta_0}} )^{N - 1}} + \cdots + ({{b_j}\varepsilon + {a_j}} ){({\beta - {\beta_0}} )^j} +\\ {b_{j - 1}}\varepsilon {({\beta - {\beta_0}} )^{j - 1}} + \cdots + {b_k}\varepsilon {({\beta - {\beta_0}} )^k} = 0\end{array}$$

where ${b_k}, \cdots ,{b_N}$ are constants, ${b_k} \ne 0$ and here we assume $\beta = {\beta _0}\; $ is the multiple-root with multiplicity k of the new characteristic equation and k < j. Note that when k=0, $\beta = {\beta _0}$ is no longer the solution of this characteristic equation.

From Eq. (5), we can get an expression of ɛ:

(6)$$\begin{aligned}\varepsilon &={-} \frac{{{a_N}{{(\beta - {\beta _0})}^N} + {a_{N - 1}}{{({\beta - {\beta_0}} )}^{N - 1}} + \cdots + {a_j}{{(\beta - {\beta _0})}^j}}}{{{b_N}{{({\beta - {\beta_0}} )}^N} + \cdots + {b_\textrm{k}}{{({\beta - {\beta_0}} )}^k}}}\\ &={-} {({\beta - {\beta_0}} )^{j - k}}\frac{{{a_N}{{({\beta - {\beta_0}} )}^{N - j}} + {a_{N - 1}}{{({\beta - {\beta_0}} )}^{N - j - 1}} + \cdots + {a_j}}}{{{b_N}{{({\beta - {\beta_0}} )}^{N - k}} + \cdots + {b_\textrm{k}}}}\end{aligned}$$

Note that when β approaches ${\beta _0}$, corresponding to small perturbation condition, Eq. (6) becomes:

(7)$$\varepsilon \approx{-} \frac{{{a_j}}}{{{b_k}}}{({\beta - {\beta_0}} )^{j - k}}$$

Here the coefficient -a_j/b_k is a finite non-zero number, therefore we get the relation between β and ɛ:

(8)$$\beta \approx {\left( { - \frac{{{b_k}\varepsilon }}{{{a_j}}}} \right)^{\frac{1}{{j - k}}}} + {\beta _0}$$

In the complex field, this equation has j-k complex solutions, which are

(9)$$\beta \approx {e^{i\frac{{2m\pi }}{{j - k}}}}{\left( { - \frac{{{b_k}\varepsilon }}{{{a_j}}}} \right)^{\frac{1}{{j - k}}}} + {\beta _0}$$

where m is an integer, and 0≤m < j-k, corresponding to j-k different roots. It is noted that $\beta $-${\beta _0}$ is the perturbation induced eigenvalue variation, in a microcavity-based system, it corresponds to the resonant wavelength shift. The eigenvalue changes and the applied perturbation $\varepsilon $, are related by the response order j-k as described by Eqs. (8) and (9).

Next, we will briefly discuss the relationship between the response order j-k and the EP order, which is related to the number of eigenvectors. For simplicity, here we only consider the case that all the N eigenvalues degenerate to the same value ${\beta _0}$, i.e., ${\beta _1} = {\beta _2} = \ldots = {\beta _N} = {\beta _0}$ and j = N. In this system, if the Hamiltonian H has $k^{\prime} + 1$ eigenvectors, it would have an ($N - k^{\prime}$)^th-order EP, and the rank of ${H_0} = H - {\beta _0}\ast {I_N}$ is $r({{H_0}} )= N - k^{\prime} - 1$, where ${I_N}$ is the N^th-order identity matrix.

When applying perturbations on the i^th cavity, the difference between the perturbed and the unperturbed characteristic polynomial, which corresponds to the difference between the left sides of Eq. (5) and Eq. (4), can be described as:

(10)$$\begin{array}{c}f(\beta )= \varepsilon \ast {\textrm {det}}({{A_{ii}} - \beta \ast {I_{N - 1}}} )\\ = {b_{N - 1}}\varepsilon {({\beta - {\beta_0}} )^{N - 1}} + \cdots + {b_j}\varepsilon {({\beta - {\beta_0}} )^j} + {b_{j - 1}}\varepsilon {({\beta - {\beta_0}} )^{j - 1}} + \cdots + {b_k}\varepsilon {({\beta - {\beta_0}} )^k}\end{array}$$

where ${A_{ii}}$ is a new (N-1)^th-order matrix formed by deleting the i^th row and the i^th column from H, and ${I_{N - 1}}$ is the (N-1)^th-order identity matrix. It is noted that due to the nature of ${A_{ii}}$, we have ${b_N} = 0$ in Eq. (10). We then expand Eq. (10) and get:

(11)$$\begin{array}{c}\varepsilon \ast \textrm{det}({{A_{ii}} - \beta \ast {I_{N - 1}}} )\\ = \mathop \sum \limits_{l = 0}^{N - 1} \mathop \sum \limits_{1 \le {e_1} < \ldots < {e_l} \le N - 1} \varepsilon \ast det\left( {{B_{ii}}\left( {\begin{array}{{ccc}} {{e_1}}& \ldots &{{e_l}}\\ {{e_1}}& \ldots &{{e_l}} \end{array}} \right)} \right)\; {({{\beta_0} - \beta } )^{({N - 1} )- l}}\end{array}$$

where ${B_{ii}}$ is a new (N-1)^th-order matrix formed by deleting the i^th row and the i^th column from ${H_0}$, and ${B_{ii}}\left( {\begin{array}{{ccc}} {{e_1}}& \ldots &{{e_l}}\\ {{e_1}}& \ldots &{{e_l}} \end{array}} \right)$ is the $l$^th-order principal submatrix of ${B_{ii}}$. It is clear that the rank of ${B_{ii}}$ satisfies that $r({{B_{ii}}} )\le r({{H_0}} )$. Also, the ranks of all the principal submatrices should be no more than $r({{B_{ii}}} )$. Therefore, if $\textrm{N} - 1 \ge l \ge r({{B_{ii}}} )+ 1$, then the $l$^th-order principal submatrix is not full rank, which results in the corresponding determinant equals to 0. In this case, in Eq. (11), the coefficients of the terms ${({\beta _0} - \beta )^0}$∼${({\beta _0} - \beta )^{({N - 1} )- ({r({{B_{ii}}} )+ 1} )}}$ are 0, corresponding to ${b_{({N - 1} )- ({r({{B_{ii}}} )+ 1} )}} = \ldots {b_0} = 0$. One should note that, for $0 \le l \le r({{B_{ii}}} )$, it only determines the high-order coefficients, i.e., ${b_{({N - 1} )- r({{B_{ii}}} )}}$∼${b_{N - 1}}$. Compared with Eq. (10), since ${b_k} \ne 0$, we get $N - 1 - ({r({{B_{ii}}} )+ 1} )\le k - 1$. Since $r({{B_{ii}}} )\le r({{H_0}} )$ and $r({{H_0}} )= N - k^{\prime} - 1$, we have $k^{\prime} \le k$, which indicates the response order $N - k$ is no more than the EP order $N - k^{\prime}$. Therefore, to achieve the maximum response, it requires $N - k = N - k^{\prime} = N$, which is $k^{\prime} = k = 0$.

From the derivation above, two specific cases are noted:

(1) For an N^th-order system with N different eigenvectors, such as a typical Hermitian system, according to the definition of EP, this is a first-order EP (usually we don’t call it “EP”). As discussed above, the system processes a first-order response order, which means the eigenvalues vary linearly with the applied perturbations.

For better understanding this concept, let’s consider the simplest third-order system,

(12)$$H = \left[ {\begin{array}{{ccc}} {{\beta_0}}&0&0\\ 0&{{\beta_0}}&0\\ 0&0&{{\beta_0}} \end{array}} \right]$$

It has three degenerated eigenvalues of ${\beta _0}$ and three different eigenvectors, (1, 0, 0)^T, (0, 1, 0)^T, and (0, 0, 1)^T. The perturbation response is:

(13)$$\Delta \beta = \varepsilon \propto \varepsilon $$

Therefore, the N^th-order system with N different eigenvectors can only have a first-order response.

(2) For an N^th-order system, the N^th-order response requires j-k = N, which translates to j = N and k=0. Here the j = N means a_N-1=…=a₀=0, indicating the Hamiltonian has N degenerated eigenvalues under no disturbance condition. The k=0 leads to b_k=b₀≠0, meaning the $\; \beta = {\beta _0}$ is not the solution of the perturbed characteristic equation Eq. (5). In other words, all the eigenvalues will deviate from the original value, which is ${\beta _0}$, when a disturbance is applied. It’s worth noting that, from the conclusion above, since $k^{\prime} \le k$, we have $k^{\prime} = 0$, and the Hamiltonian has only $k^{\prime} + 1 = 1$ eigenvector from the definition of $k^{\prime}$. Therefore, in this case, all the eigenvectors collapse to one point, leading to an N^th-order EP.

When applying perturbations on the i^th cavity, from Eqs. (10) and (11), we have

(14)$${b_0} = det\left( {{B_{ii}}\left( {\begin{array}{{ccc}} {{e_1}}& \ldots &{{e_{N - 1}}}\\ {{e_1}}& \ldots &{{e_{N - 1}}} \end{array}} \right)} \right) = det({{B_{ii}}} )$$

The system shows an N^th-order response only when b₀=$\; det({{B_{ii}}} )$≠0. Let’s take the system mentioned in Fig. 1 as an example. When perturbation ɛ is applied on Cavity One, we get

(15)$${b_0} = {B_{11}} = \det \left( {\left[ {\begin{array}{{cc}} {{\beta_0}}&{i\kappa }\\ {i\kappa }&{{\beta_0}} \end{array}} \right] - {\beta_0}\ast {I_{N - 1}}} \right) = \left|{\begin{array}{{cc}} 0&{i\kappa }\\ {i\kappa }&0 \end{array}} \right|= {\kappa ^2} \ne 0$$

A third-order response is observed as mentioned before. While applying perturbation ɛ on Cavity Two, we get

(16)$${b_0} = {B_{22}} = \det \left( {\left[ {\begin{array}{{cc}} {{\beta_0}}&0\\ 0&{{\beta_0}} \end{array}} \right] - {\beta_0}\ast {I_{N - 1}}} \right) = \left|{\begin{array}{{cc}} 0&0\\ 0&0 \end{array}} \right|= 0$$

In this case, the response order degeneracy occurs, and a first-order response order is obtained.

As a result, to have a system with an N^th-order response, the easiest way is to build an N^th-order system with all the eigenvalues and eigenvectors degenerate to one point and make sure that the cofactor of the i^th diagonal elements of ${H_0} = H - {\beta _0}\ast {I_N}$, $det({{B_{ii}}} )$, is non-zero, indicating the $\; \beta = {\beta _0}$ is not the solution of the perturbed characteristic equation.

3. Design of a fourth-order non-Hermitian sensor

Based on the criterion derived above, we design a high-order response non-Hermitian sensor. We choose a fourth-order non-Hermitian optical sensor as an example, due to its symmetric configuration and high order nature [see Fig. 2(a)]. The corresponding general Hamiltonian matrix for this system could be written as:

(17)$${H_s} = \left[ {\begin{array}{{cc}} {\begin{array}{{cc}} {iA\gamma }&{\; C\kappa }\\ {\; C\kappa }&{iB\gamma } \end{array}}&{\begin{array}{{cc}} 0&{D\kappa }\\ {D\kappa }&0 \end{array}}\\ {\begin{array}{{cc}} 0&{D\kappa }\\ {\; D\kappa }&0 \end{array}}&{\begin{array}{{cc}} { - iB\gamma }&{\; C\kappa }\\ {\; \; C\kappa }&{ - iA\gamma } \end{array}} \end{array}} \right]$$

where + iγ (-iγ) represents the gain (loss), and κ accounts for the coupling between microresonators. Besides, A, B, C, and D are positive numbers indicating the strength of gain/loss and coupling. It is worth noting that the parameters on the diagonal stand for the overall effect, including loss from scattering, material, and coupling between input/output strip waveguides, as well as gain from the material. Besides, to make sure that all eigenvalues coalesce at one point, we assume the above device satisfies the following condition:

(18)$$\left( {\beta - \rho \sqrt {{\kappa^2} - {\sigma^2}{\gamma^2}} } \right)\left( {\beta + \rho \sqrt {{\kappa^2} - {\sigma^2}{\gamma^2}} } \right)\left( {\beta - \sqrt {{\kappa^2} - {\sigma^2}{\gamma^2}} } \right)\left( {\beta + \sqrt {{\kappa^2} - {\sigma^2}{\gamma^2}} } \right) = 0$$

where β is the eigenvalue, of which the real part and the imaginary part represent the resonant wavelength shift and the gain/loss respectively, $\rho $ and $\sigma $ are the undetermined coefficients. Here we set |$\; \rho $ |≠1 to avoid multiple roots outside the EP. Therefore, the position of EP in the parameter space is determined by σ only. For the sake of simplicity, we set σ = 1. A solution for this fourth-order system is achieved when $A = 3 + 2\sqrt 2 $, $B = 1$, $C = 2 + \sqrt 2 $, and $D = 1 + \sqrt 2 ,$ indicating the special optical device consisting of four resonators with low loss, low gain, high loss, and high gain respectively. The system possesses a quadruple degenerated eigenvalue ${\beta _{1,2,3,4}} = 0$ as well as a quadruple degenerated eigenvector ${\left( {1, - i\sqrt 2 - i, - 1 - \sqrt 2 ,i} \right)^T}$, which are the signatures of a fourth-order EP. It’s easy to confirm that the corresponding perturbed characteristic equation satisfies the criterion that a_n-1=…=a₀=0 and b_k=b₀≠0. Therefore, the device will feature a fourth-order response no matter on which resonator the perturbation is applied. The eigenvalues of Eq. (17) can thus be expressed as:

(19)$$\beta ={\pm} \sqrt {p({\; {\kappa^2} - {\gamma^2}} )\pm q({\; {\kappa^2} - {\gamma^2}} )} $$

where $p = 9 + 6\sqrt 2 $, $q = 8 + 6\sqrt 2 $. From Eq. (19), it is obvious that the coalescence of all four states would occur when γ/κ=1. The corresponding eigenvalues of our resonator-based device are plotted in Fig. 2(b) as a function of γ/κ. When γ<κ, there are four real eigenvalues branches and only one imaginary eigenvalue branch, which equals to zero, indicating the lossless condition for all these four eigenstates. Subsequently, a phase transition, an EP, occurs at γ=κ, where all eigenvalues collapse. When γ>κ, all eigenvalues become purely imaginary. In this case, all states share the same real part of eigenvalues, which is the resonant wavelength of the isolated resonators, and only experience gain or loss.

Fig. 2. (a) Schematic of the proposed system, where only two different coupling coefficients (Cκ and Dκ) and two pairs of gain and loss coefficients (Aγ, -Aγ, Bγ and -Bγ) are considered. (b) Real and imaginary parts of the eigenvalues in the proposed system as functions of γ/κ.

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When applying perturbations to the device works around the EP (γ/κ=1), here we choose the resonator with low loss, the perturbed Hamiltonian becomes:

(20)$$\left[ {\begin{array}{{cc}} {\begin{array}{{cc}} {iA\kappa }&{C\kappa }\\ {\; C\kappa }&{iB\kappa } \end{array}}&{\begin{array}{{cc}} 0&{D\kappa }\\ {D\kappa }&0 \end{array}}\\ {\begin{array}{{cc}} 0&{D\kappa }\\ {D\kappa }&0 \end{array}}&{\begin{array}{{cc}} { - iB\kappa + \varepsilon }&{\; C\kappa }\\ {\; \; C\kappa }&{ - iA\kappa } \end{array}} \end{array}} \right]$$

As a result, the corresponding characteristic equation around the fourth-order EP is

(21)$${\beta ^4} - \varepsilon {\beta ^3} + i\varepsilon \kappa {\beta ^2} - l\varepsilon {\kappa ^2}\beta + is\varepsilon {\kappa ^3} = 0$$

where $l = 8 + 6\sqrt 2 $, $s = 48 + 34\sqrt 2 $. By numerically solving Eq. (21), we get the real part and imaginary part of the eigenwavelength shift, λ_n, and plot them in Figs. 3(a) and 3(b). The two cross-sections of Figs. 3(a) and (b), when γ/κ=1 (the EP), are highlighted in Figs. 3(c) and 3(d) (solid cures), showing the responses to perturbation in both the real and imaginary domains. We also plot the splitting between two eigenvalues in wavelength form (λ₁ and λ₄) as a function of Δn_eff, which corresponds to the effective index variations of the low loss resonator shown in Fig. 3(e) (solid curve). The slope of the perturbation response in the double logarithmic form is about 1/4, thus confirming the fourth root relation.

Fig. 3. The real part (a) and imaginary part (b) of the eigenwavelengths of the fourth-order non-Hermitian optical device as functions of γ/κ and the effective index variation of the low loss resonator. All eigenwavelengths collapse to one point when the device is operating at the EP without any disturbance. The variation of the real part (c) and imaginary part (d) of the eigenwavelength as functions of the effective index variation using the analytic (dotted) and the numerical (solid) methods. Note that the analytic and numerical results are from Eqs. (23) and (21), respectively. The calculated eigenwavelength splitting as a function of the external perturbation (the analytic (dotted) and the numerical (solid) methods) in the linear (e) and the double logarithmic forms (inset).

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From the analytic point of view, for the fourth-order device proposed here, Eq. (6) can be rewritten as

(22)$$\frac{{{\beta ^4}}}{{{\beta ^3} - i\kappa {\beta ^2} + l{\kappa ^2}\beta - is{\kappa ^3}}} = \varepsilon $$

When β approaches 0, the roots of Eq. (22) are:

(23)$$\left\{ {\begin{array}{{c}} {\begin{array}{{c}} {{\beta_1} = \nu {e^{\frac{{3\pi }}{8}i}}{\kappa^{\frac{3}{4}}}{\varepsilon^{\frac{1}{4}}}\; }\\ {{\beta_2} ={-} \nu {e^{\frac{{3\pi }}{8}i}}{\kappa^{\frac{3}{4}}}{\varepsilon^{\frac{1}{4}}}} \end{array}}\\ {\begin{array}{{c}} {{\beta_3} = i\nu {e^{\frac{{3\pi }}{8}i}}{\kappa^{\frac{3}{4}}}{\varepsilon^{\frac{1}{4}}}}\\ {{\beta_4} ={-} i\nu {e^{\frac{{3\pi }}{8}i}}{\kappa^{\frac{3}{4}}}{\varepsilon^{\frac{1}{4}}}} \end{array}} \end{array}} \right.$$

where $\nu = {\left( {48 + 34\sqrt 2 } \right)^{\frac{1}{4}}} \approx 3.13$. The plotting of Eq. (23) is shown in Figs. 3(c)–3(e) (dotted lines), which is the approximate analytic solution of perturbed characteristic equation Eq. (21) under small perturbation conditions, agrees well with the numerical solutions for the entire spectrum.

4. Device performance

Based on the above model and corresponding parameters, we design a device consisting of four racetrack micro-resonators, shown in Fig. 4(a), and simulate it using Lumerical’s INTERCONNECT software. The device is made of InGaAsP waveguide (400 nm wide and 250 nm tall) sitting on SiO₂ with air as the cladding. In our design, each racetrack resonator consists of four 90° arcs (r=30 µm) and four straight waveguides with a length of 2.88 µm, therefore the perimeter of a resonator is about 200 µm. There are two different gaps (g₁=85 nm corresponding to Cκ and g₂=120 nm corresponding to Dκ) between the adjacent resonators to satisfy the required working condition according to Eq. (18). Signal light around 1550 nm is impinging from the top waveguide and couples into the low loss resonator, and the output is collected from the bottom straight waveguide near the high gain resonator. We set Aγ=0.003, which is the imaginary part of the effective index of the high gain/loss resonator based on the data from [26], and calculate the imaginary part of the effective index of the low gain/loss one, which is Bγ≈0.0005. Optical beam with 840 nm wavelength could be used to provide gain energy. By properly adjusting the pumping power, the device can work in its linear region to avoid the associated optical nonlinearities [26]. The output without any perturbations is shown in Fig. 4(b), and the very sharp peak indicates that the device is at the EP where all eigenwavelengths collapse into one point. The disturbance is induced at the low loss resonator by varying the refractive index of the cladding, and a clear peak splitting is observed in Fig. 4(c) when the effective index increases $5 \times {10^{ - 4}}$. Note that for sensing purposes, we are more interested in the cladding induced effective index variation, and based on our waveguide design, the effective index will increase by the amount of $1.732 \times {10^{ - 4}}$ when the cladding refractive index changes ${10^{ - 3}}$. Here we ignore the coupling variation induced resonant wavelength splitting, originally coming from the top cladding index change, since it is more than two orders smaller than the one induced by the effective index variation. It is also worth noting that, in theory, when a disturbance is applied to the device, the output peak should split into four peaks, which differs from our observation, i.e., only two peaks in the output spectrum. The reason is that the other two unobservable peaks are either with high loss or high gain, featuring a wider output linewidth, which is more pronounced than the separation. Therefore, we calculate the four individual transmissions of each resonance in Fig. 4(c) and the summation of them fits well to the overall transmission. The evolution of resonant wavelength is plotted in Fig. 4(d), showing the separation of the two peaks becomes wider as the perturbations increase. We also plot the wavelength splitting as a function of the cladding refractive index variation in Fig. 4(e), together with the double logarithmic form in the inset. The slope of the perturbation response in double logarithmic form is approximately 1/4, indicating a fourth root relation between wavelength splitting and the induced perturbations, showing a good agreement between theory and simulation.

Fig. 4. (a) Schematic of the proposed device. The output of the device operating at EP when no disturbance applied (b) and disturbance applied (c). The dashed curves represent the four resonant peaks and the dotted curve represents the sum up of the four resonant peaks. (d) The output evolution when the refractive index of cladding gradually increased. (e) Wavelength splitting as a function of the cladding index variation $\Delta \textrm{n}$ in a linear form and log-log form (inset).

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To evaluate the performance of our fourth-order non-Hermitian sensor, we also compare the wavelength splitting of different-order non-Hermitian devices together with the single microresonator (liner device) in Fig. 5, where the real part of effective indices and the coupling constants κ for all devices are kept the same. Defined as the ratio of the wavelength splitting between the non-Hermitian devices and that of the linear device, the sensitivity enhancement of over 100 is observed. Shown in Fig. 5, the perturbation response of the fourth-order device is much more pronounced, compared to its counterparts of the lower-order non-Hermitian devices. The enhancement factor is above 3 for the whole testing range, where Δn is small than $5 \times {10^{ - 3}}$. The dynamic range is limited by the free spectral range (FSR) from two adjacent resonant modes, which in our case is about ${10^{ - 2}}$. Within the entire dynamic range, the sensitivity of our non-Hermitian device is about twofold higher than a linear device. More significantly, the enhancement will be much more obvious for smaller disturbance due to its fourth root nature. It is worth noting that the dynamic range and enhancement factor can be adjusted according to different applications.

Fig. 5. Wavelength splitting and enhancement factor as a function of the cladding index variation Δn, for non-Hermitian devices with different orders and linear devices. All the devices here have the same real part of the effective index (2.316) and coupling constant κ (120 nm gap) between the adjacent resonators. The third-order device consists of three in-line ring resonators with gain, neutral, and loss, respectively, whose imaginary parts of the effective indices are 0.0014, 0, and −0.0014, and only the coupling between the nearest neighbors are considered. The second-order device consists of two ring resonators with imaginary parts of effective indices of 0.001 (gain) and −0.001 (loss). For the liner device, there is only one ring resonator with imaginary part of effective index of −1.2×10⁻⁷.

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5. Conclusion

In conclusion, we investigate the response order of high-order non-Hermitian systems and provide a criterion for designing a high-order non-Hermitian sensor without the response order degeneracy. Based on the criterion, a fourth-order non-Hermitian optical sensor is proposed and its associated characteristics are analyzed. The four eigenvalues of our optical sensor simultaneously collapse at the high-order EP in the parameter space, providing a ɛ^1/4-dependence of the eigenwavelength splitting $\Delta \lambda $ that can translate the surrounding perturbations to resonant wavelength shifts. A clear sensitivity boost, more than 100 times higher, is observed by comparing the wavelength splitting with traditional single ring resonator for small perturbation.

Funding

National Natural Science Foundation of China (61705099); Science and Technology Commission of Shanghai Municipality (Y7360k1D01).

Disclosures

The authors declare no conflicts of interest.

References

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High-order exceptional point based optical sensor

Abstract

1. Introduction

2. Analysis of N^th-order non-Hermitian system

3. Design of a fourth-order non-Hermitian sensor

4. Device performance

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (23)

Optics Express

Abstract

1. Introduction

2. Analysis of Nth-order non-Hermitian system

3. Design of a fourth-order non-Hermitian sensor

4. Device performance

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (23)

Optics Express

2. Analysis of N^th-order non-Hermitian system