Classes of Balanced Gain-and-Loss Waveguides as Non-Hermtian Potential Hierarchies

Abstract

In this work, we report the construction of different classes of complex-valued refractive index landscapes, with real spectra, in the framework of the factorization method. The particular case of guiding hyperbolic-type profiles is considered in the $\mathcal{PT}$- and non-$\mathcal{PT}$-symmetric configurations. In both schemes, the imaginary part of the refractive index satisfies the zero-total-area condition indicating that the total transverse optical power is preserved, allowing stable propagating modes to be obtained. The spectra and the guided modal field amplitudes are obtained and their orthogonality relations are established.

1. Introduction

The task of processing and manipulating light states has been one of the most promising achievements in optical sciences. The possibility of designing and implementing control protocols, such that an optical signal behaves in a very specific fashion, has enabled the emergence of a number of remarkable applications. These, in turn, have stimulated the development of new theoretical models and experimental methodologies, addressed to describe and engineer light–matter interactions, leading to further results and novel applications. In this context, gradient index (GRIN) optics has played an important part. The study of light propagation phenomena in non-homogeneous media, as well as the continuous improvements in the design and manufacture techniques of GRIN materials, has assisted, for instance, image enhancing, trajectory control of light rays, focusing and collimation of light beams and manipulation and processing of optical signals in guiding devices [1,2]. In particular, the quadratic and hyperbolic secant index distributions have deserved a lot of attention due to their focusing and collimating properties and also because the corresponding dynamical equations are reduced to exactly solvable differential problems [2,3,4,5,6,7,8,9,10,11,12]. Furthermore, the hyperbolic secant profile has also proven to be efficient in shortening pulse broadening [3] and in producing diffraction-limited beams and aberration-free meridional rays [4,5].

Concurrently, the development of synthetic optical media such as metamaterials, photonic structures and non-Hermitian systems unveiled the possibility to control light propagation in non-conventional ways [13,14,15,16]. Of special interest are non-Hermitian materials, as they allow one to include non-conservative effects in theoretical models. These structures are, in general, characterized by complex, GRIN profiles, in which the real part represents the guiding refractive index distribution of the material, while the imaginary one models gain/loss events in the propagation processes [2]. It is worth mentioning that, in the last decades, optical systems exhibiting balanced gain-and-loss properties have been exhaustively investigated, from both theoretical and experimental points of view. The interest has been mainly concentrated in the study of $\mathcal{PT}$-symmetric (parity–time-invariant) configurations [15,16,17], for which the refractive index fulfills the condition $n\left(x\right)=n^{*}(-x)$. This is essentially because optical media with complex index distributions in the paraxial regime may be conceived as optical analogues of complex potentials within the framework of non-Hermitian quantum mechanics. The reason lies in the fact that, in such approximation, the electric field envelope satisfies a Schrödinger-like equation (the paraxial Helmholtz equation) [18,19], where the variation in the refractive index with respect to some reference value $n_B$ plays the role of the quantum potential (see Section 2). Yet, in quantum theory, non-Hermitian systems have been a subject of intense discussion after the demonstration that, under some circumstances, complex, but $\mathcal{PT}$-symmetric potentials may posses all-real spectra [20,21]. A plethora of results comprising the construction of $\mathcal{PT}$-symmetric Hamiltonians followed immediately [22,23,24,25,26]. In typical quantum mechanical models, the $\mathcal{PT}$-symmetry condition demands the real and imaginary parts of the complex potentials to be, respectively, even and odd functions of the position. Quite notably, it was soon established that $\mathcal{PT}$-symmetry is not a necessary condition to grant the reality of the spectrum [27], so that more versatile complex potentials may be considered leading to a balanced gain and loss of the probability. The options include, among others, those potentials generated by pseudo-Hermiticity [27,28,29], group-theoretical approaches [22,23,24] and supersymmetric techniques [25,29,30,31,32,33,34,35,36,37,38,39,40,41].

Regarding the equivalence between the Schrödinger and paraxial Helmholtz equations, it was proposed and experimentally demonstrated that $\mathcal{PT}$-symmetry may be realized in the scheme of paraxial optics [42,43]. The results, while offering a reliable platform to verify, in practice, some quantum mechanical phenomena in the non-Hermitian regime, also enable new forms of processing and harnessing light states with high perspectives of further technological applications [13,15,16,17,43,44,45,46,47,48,49,50]. Of particular importance are those phenomena related to the presence of exceptional points (EPs), as it is well known that light experiences unprecedented behavior as the parameter modulating the gain-and-loss properties of the material crosses this point [51]. Among the variety of new processes that may be implemented by the presence of EPs are Bloch oscillations [13], unidirectional invisibility [52,53,54], loss-induced lasing [55], jamming anomaly [56], induced transparency [57] and merge-mode phenomena [53,58], to mention just a few, finding applications in many areas of photonics such as optical sensing [53,58], design of semiconductor lasers [15] and telemetry [59]. From the wave-guiding point of view, it is clear that non-Hermitian architectures allow one to obtain the suppressing and/or controlling loss effects supporting the propagation of non-decaying optical guided modes [60], modulational instability [61] and multi-mode interference [62] in the GRIN playground.

Still, one of the most important challenges in both quantum mechanics and GRIN optics is the construction of analytical solutions to the corresponding dynamical equations. In this regard, the factorization method has proven to be a very efficient technique in quantum mechanics [63,64,65]. The method consists in expressing a hierarchy of Hamiltonians $H_{\ell }$, up to an additive constant, as the product of two first-order operators. The factorizability condition leads to a set of intertwining relations from which the discrete spectrum and the complete set of eigenfunctions can be determined for the entire hierarchy. This technique and its connection to the Darboux transformations encompass one of the most powerful tools in quantum mechanics for the generation of exactly solvable models in the supersymmetric and shape-invariant schemes [22,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79]. In the non-Hermitian regime, supersymmetry has been widely used to construct $\mathcal{PT}$- as well as non-$\mathcal{PT}$-symmetric exactly solvable Hamiltonians [30,33,34,36]. In addition, in recent years, factorization and supersymmetric approaches have been applied in the optical context to describe the propagation of light beams in guiding media [9,10,80] and to the design of optical structures with Hermitian [11,12,81] and non-Hermitian [44,47,48,49,50] configurations.

In this work, we consider the factorization method for the generation of new classes of exactly solvable, non-homogeneous, balanced gain-and-loss optical structures, by taking full advantage of the equivalence between the Schrödinger and Helmholtz equations in the paraxial approximation. All our classes satisfy the zero-total-area condition so that their members support the propagation of constant intensity-guided modes. In our approach, the parameter ℓ that characterizes each member of a particular class is associated to the confining properties of the material and also defines the number of guided modes hosted by the waveguide. We restrict ourselves to the description of transversal electric (TE) modes in the guiding regime, for $\mathcal{PT}$- as well as non-$\mathcal{PT}$-symmetric refractive index landscapes and their limits to the Hermitian scheme. In the non-$\mathcal{PT}$-symmetric configuration, the imaginary part of the refractive index is not an odd function of the position as the distribution of gain-and-loss regions is not symmetric any longer. This fact enables the onset of asymmetric modes that might assist the reduction in the gain value that is necessary to balance the losses in the waveguide [60].

The paper is organized in the following way: In Section 2, we consider the propagation of TE, linearly polarized modes in a planar GRIN waveguide. It is shown that, in the weakly guiding regime, the Helmholtz equation can be reduced to a one-dimensional eigenvalue problem of the Schrödinger type for the electric field envelope. In Section 3, different classes of complex index distributions are determined by means of the factorization method. Particular classes of the hyperbolic type are presented including $\mathcal{PT}$- and non-$\mathcal{PT}$-symmetric structures. In the Hermitian limit, the $\mathcal{PT}$-symmetric profiles are reduced to the well-known hyperbolic secant refractive index family. In the non-$\mathcal{PT}$-symmetric case, the same limit leads to the family of Scarf II potentials in quantum mechanics. In Section 4, we present the determination of the field amplitudes in the guiding regime and, in Section 5, we establish orthogonality relations in the context of the biorthogonal approach. Finally, in Section 6, we present a summary of our results and some conclusions and perspectives.

2. The Paraxial Helmholtz Eigenvalue Problem in One Dimension

The propagation of an electromagnetic (EM) wave linearly polarized, for instance, in the $\widehat{\mathit{y}}$-direction is governed by the Helmholtz equation

$$\frac{{\partial }^2}{\partial x^2}\mathcal{E}(x,z)+\frac{{\partial }^2}{\partial z^2}\mathcal{E}(x,z)+k_0^2{n}^2\left(x\right)\mathcal{E}(x,z)=0,$$

(1)

where $\mathbf{E}(x,z,t)=\mathcal{E}(x,z)e^{-i\omega t}\widehat{\mathit{y}}$ is the corresponding electric field, $k_0=\frac{\omega }{c}$ is the wave number in free space and $n\left(x\right)$ is the refractive index depending only on the x-coordinate. In the weakly guiding approximation, where the position-dependence of the refractive index corresponds to small fluctuations $\Delta n\left(x\right)$ around a constant real reference value $n_B$ (called bulk refractive index), we may write

$$n\left(x\right)=n_B+\Delta n\left(x\right),\mathrm{with}|\Delta n\left(x\right)|\ll n_B.$$

(2)

If this is the case, the paraxial approximation is valid and the electric field amplitude $\mathcal{E}$ can be expressed as a plane wave, propagating in the z-direction, modulated by an enveloping function $E(x,z)$, as follows:

$$\mathcal{E}(x,z)=E(x,z)e^{i{k}_0{n}_{B}z}.$$

(3)

In this approximation, the second derivative of E with respect to the longitudinal coordinate z can be neglected and the finite transversal power condition must be fulfilled. Under these circumstances, the modulating amplitude E satisfies the paraxial Helmholtz equation

$$\left\{-\frac{1}{2k_0^2{n}_B}\frac{{\partial }^2}{\partial x^2}+n_B-n\left(x\right)\right\}E(x,z)=\frac{i}{k_0}\frac{\partial }{\partial z}E(x,z),$$

(4)

together with the square integrability condition in the transversal coordinate x. Note also that, as the refractive index is independent of z, this equation admits stationary solutions of the form

$$E(x,z)=E_0\varphi \left(x\right)e^{-i{k}_0\epsilon z},$$

(5)

where $E_0$ is a constant with units of electric field representing the field strength of the EM wave, the parameter $\epsilon$ defines the spectrum of propagation constants and the function $\varphi \left(x\right)$ is a solution of the eigenvalue problem [50,82,83]

$$\left\{-\frac{1}{2k_0^2{n}_B}\frac{d^2}{d{x}^2}+n_B-n\left(x\right)\right\}\varphi =\epsilon \varphi .$$

(6)

For the sake of simplicity, it is convenient to introduce the dimensionless variable $r=r\left(x\right)=\sqrt{2k_0^2{n}_B}x$, in such a way that Equation (6) can be transformed into

$$H\psi \left(r\right)=\epsilon \psi \left(r\right),$$

(7)

with

$$H=-\frac{d^2}{d{r}^2}+v\left(r\right),$$

(8)

and $\psi$ and v being two functions of the variable r defined, respectively, by $\varphi \left(x\right)=\psi \left(r\right(x\left)\right)$ and

$$n\left(x\right)=n_B-v\left(r\left(x\right)\right).$$

(9)

Remark that the eigenvalue problem (7) and (8) have the form of a one-dimensional stationary Schrödinger equation with H being the Hamiltonian operator and $v\left(r\right)$ being the quantum potential. In this work, we take full advantage of this resemblance in order to generate new exactly solvable refractive index profiles in the non-Hermitian regime.

3. Balanced Gain-and-Loss Waveguides

3.1. Optical Potential Hierarchies

Let us consider a class of complex refractive index profiles $n_{\ell }\left(x\right)$, labeled by the parameter $\ell ={\ell }_0+s,{\ell }_0>0,s\in \mathbb{N}$, associated to a hierarchy of operators $H_{\ell }$ through (8) and (9), where

$$H_{\ell }=-\frac{d^2}{d{r}^2}+v_{\ell }\left(r\right).$$

(10)

The function $v_{\ell }\left(r\right)$, called optical potential, may be, in general, a complex-valued function. We are interested in solving the eigenvalue problem

$$H_{\ell }{\psi }_{\ell }(r;\epsilon )=\epsilon {\psi }_{\ell }(r;\epsilon ),$$

(11)

with $\epsilon$ being a real constant, in the regime of guided modes where the field amplitudes satisfy the finite transversal optical power condition.

We assume that the hierarchy elements $H_{\ell }$ are factorizable in the following sense [63,66,67,68]:

$$H_{\ell }=a_{\ell }{b}_{\ell }+{ϵ}_{\ell }=b_{\ell -1}{a}_{\ell -1}+{ϵ}_{\ell -1},$$

(12)

where $a_{\ell }$ and $b_{\ell }$ are first-order operators of the form

$$a_{\ell }=-\frac{d}{dr}+w_{\ell }\left(r\right),b_{\ell }=\frac{d}{dr}+w_{\ell }\left(r\right),$$

(13)

with $w_{\ell }$ being a complex-valued function to be determined and ${ϵ}_{\ell }$ being a real constant to be fixed. The substitution of (13) into (12) leads to the pair of Riccati equations

$$-\frac{d}{dr}{w}_{\ell }\left(r\right)+w_{\ell }^2\left(r\right)=v_{\ell }\left(r\right)-{ϵ}_{\ell },$$

(14)

$$\frac{d}{dr}{w}_{\ell -1}\left(r\right)+w_{\ell -1}^2\left(r\right)=v_{\ell }\left(r\right)-{ϵ}_{\ell -1}\left(r\right),$$

(15)

that must be simultaneously satisfied. This set of equations can be solved for $v_{\ell }$ providing a condition to construct the function $w_{\ell }$.

$$\frac{d}{dr}\left(w_{\ell }+w_{\ell -1}\right)-\left(w_{\ell }+w_{\ell -1}\right)\left(w_{\ell }-w_{\ell -1}\right)={ϵ}_{\ell }-{ϵ}_{\ell -1}.$$

(16)

As a simple case, let us assume that $w_{\ell }$ is linear in the parameter ℓ, i.e.,

$$w_{\ell }\left(r\right)=\ell \beta \left(r\right)+\phi \left(r\right),$$

(17)

with $\beta$ and $\phi$ being two functions to be determined in such a way that Equation (16) is satisfied. Then, the parameter ${ϵ}_{\ell }$ fulfills the first-order finite difference equation

$${ϵ}_{\ell }-{ϵ}_{\ell -1}=(2\ell -1)\left[\frac{d}{dr}\beta \left(r\right)-{\beta }^2\left(r\right)\right]+2\left[\frac{d}{dr}\phi \left(r\right)-\beta \left(r\right)\phi \left(r\right)\right],$$

(18)

for which a particular solution can be cast as ${ϵ}_{\ell }=k_2{\ell }^2$, together with the following set of equations for the functions $\beta$ and $\phi$ (see [63]):

$$k_2=\frac{d}{dr}\beta \left(r\right)-{\beta }^2\left(r\right),$$

(19)

$$\frac{d}{dr}\phi \left(r\right)-\beta \left(r\right)\phi \left(r\right)=0.$$

(20)

As ${ϵ}_{\ell }$ is independent of the variable r, we assume that $k_2$ is a real constant. In the sequel, we fix our attention on the case $k_2<0$ (results for more general solutions of Equation (18) are to be reported elsewhere). Thus, we write $k_2=-{\kappa }^2$, $\kappa \in \mathbb{R}$, $\kappa \ne 0$, so that

$${ϵ}_{\ell }=-{\kappa }^2{\ell }^2.$$

(21)

Once the function $w_{\ell }$ is expressed in the form (17), the optical potential can be obtained in terms of the functions $\beta$, $\phi$ and the parameter $\kappa$ by using (14), (19) and (20).

$$v_{\ell }\left(r\right)=\ell (\ell -1)\left[{\beta }^2\left(r\right)-{\kappa }^2\right]+(2\ell -1)\beta \left(r\right)\phi \left(r\right)+{\phi }^2\left(r\right),$$

(22)

defining the class of refractive index profiles

$$n_{\ell }\left(x\right)=n_B-\ell (\ell -1)\left[{\beta }^2\left(r\left(x\right)\right)-{\kappa }^2\right]-(2\ell -1)\beta \left(r\left(x\right)\right)\phi \left(r\left(x\right)\right)-{\phi }^2\left(r\left(x\right)\right).$$

(23)

The problem of constructing optical potentials $v_{\ell }$ that lead to operators $H_{\ell }$ admitting the factorized form (12) has been, so far, reduced to the solution of Equations (19) and (20), with a fixed value of $\kappa$, for $\beta$ and $\phi$. Yet, it is well known that the Riccati Equation (19) can be connected to the linear Schrödinger-type equation for the free particle

$$\frac{d^2}{d{r}^2}u\left(r\right)-{\kappa }^{2}u\left(r\right)=0$$

(24)

through the change in the variable

$$\beta \left(r\right)=-\frac{d}{dr}lnu\left(r\right).$$

(25)

According to (24), different types of real-valued functions $\beta$ can be readily found [84] (see also [22]). Nevertheless, in this work, we are rather interested in complex-valued $\beta$ functions. In order to address the problem, regard that the real and imaginary parts of $\beta$ can be written as [34]

$${\beta }_{\mathrm{R}}\left(r\right)=-\frac{d}{dr}ln\alpha \left(r\right),{\beta }_{\mathrm{I}}=\frac{\lambda }{{\alpha }^2\left(r\right)},$$

(26)

where $\lambda$ is a real constant and $\alpha$ can be either a real or a pure imaginary function fulfilling the equation

$$\frac{d^2}{d{r}^2}\alpha \left(r\right)-{\kappa }^2\alpha \left(r\right)=\frac{{\lambda }^2}{{\alpha }^3\left(r\right)},$$

(27)

named after Ermakov [85]. Furthermore, according to (20) and (26), the function $\phi$ assumes the form

$$\phi \left(r\right)=\frac{{\phi }_0}{\alpha \left(r\right)}exp\left[i\lambda {\int }^r\frac{1}{{\alpha }^2\left(\tau \right)}d\tau \right]=\frac{\delta }{u\left(r\right)},$$

(28)

with $\delta$ and ${\phi }_0$ being integration constants, assumed to be real, that are related by $\delta =\sqrt{a}{\phi }_0$. Quite remarkably, the function $\phi$ corresponds to the missing state in the supersymmetric approach in which the optical potential is constructed as the Darboux deformation of an initial exactly solvable real-valued potential (see [34,50]). On the other hand, the connection between the Ermakov Equation (27) and the linear Schrödinger-type Equation (24) allows us to express the general solution of (27) in terms of a quadratic combination of two linearly independent solutions $u_1\left(r\right)$ and $u_2\left(r\right)$ of (24) (see [34,86] for further details).

$$\alpha \left(r\right)={\left\{a{u}_1^2\left(r\right)+b{u}_1\left(r\right)u_2\left(r\right)+c{u}_2^2\left(r\right)\right\}}^{1/2},$$

(29)

with $a,b$ and c being three constants connected by the condition

$$b^2-4ac=-4\frac{{\lambda }^2}{W_0^2},$$

(30)

and $W_0$ being the Wronskian of $u_1$ and $u_2$. Moreover, the set of Equations (26) and (29) yields an expression for u as a linear combination of $u_1$ and $u_2$ [39], namely,

$$u\left(r\right)=a{u}_1\left(r\right)+\left(\frac{b}{2}-i\frac{\lambda }{W_0}\right)u_2\left(r\right).$$

(31)

On the other hand, the substitution of (25), (26) and (28) into (17) produces

$$w_{\ell }\left(r\right)=\ell \left(-\frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}+i\frac{\lambda }{{\alpha }^2\left(r\right)}\right)+\frac{{\phi }_0}{\alpha \left(r\right)}exp\left[i\lambda {\int }^r\frac{1}{{\alpha }^2\left(\tau \right)}d\tau \right]=-\ell \frac{u^{\prime }\left(r\right)}{u\left(r\right)}+\frac{\delta }{u\left(r\right)},$$

(32)

where the prime stands for derivative with respect to the argument. Then, Equation (14) permits us to express $v_{\ell }$ in terms of either $\alpha$ or u to yield

$$\begin{array}{ccc}\hfill v_{\ell }\left(r\right)& =& \ell (\ell -1)\left[{\left(-\frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}+i\frac{\lambda }{{\alpha }^2\left(r\right)}\right)}^2-{\kappa }^2\right]+\frac{{\phi }_0^2}{{\alpha }^2\left(r\right)}exp\left[2i\lambda {\int }^r\frac{1}{{\alpha }^2\left(\tau \right)}d\tau \right]\hfill \\ \hfill & +& (2\ell -1)\frac{{\phi }_0}{\alpha \left(r\right)}\left(-\frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}+i\frac{\lambda }{{\alpha }^2\left(r\right)}\right)exp\left[i\lambda {\int }^r\frac{1}{{\alpha }^2\left(\tau \right)}d\tau \right]\hfill \\ & =& \ell (\ell -1)\left[{\left(-\frac{u^{\prime }\left(r\right)}{u\left(r\right)}\right)}^2-{\kappa }^2\right]+\frac{{\delta }^2}{u^2\left(r\right)}-(2\ell -1)\delta \frac{u^{\prime }\left(r\right)}{u^2\left(r\right)}.\hfill \end{array}$$

(33)

Observe that, in general, the real and imaginary parts of $v_{\ell }$ are not, respectively, even and odd functions of r. This means that the corresponding landscapes $n_{\ell }$ given by (23) are not necessarily $\mathcal{PT}$-symmetric. Note, also, that the parameter $\lambda$ modulates the imaginary part of the potential, in such a way that the Hermitian case is recovered by taking the limit $\lambda \to 0$. We obtain

$$w_{\ell }\left(r\right)=-\ell \frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}+\frac{{\phi }_0}{\alpha \left(r\right)},$$

$$v_{\ell }\left(r\right)=\ell (\ell -1)\left[{\left(-\frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}\right)}^2-{\kappa }^2\right]+\frac{{\phi }_0^2}{{\alpha }^2\left(r\right)}-(2\ell -1)\frac{{\phi }_0}{\alpha \left(r\right)}\frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}.$$

In this limit, the functions $\alpha$ and u become linearly dependent and

$$u\left(r\right)=\sqrt{a}\alpha \left(r\right)=\sqrt{a}\left(\sqrt{a}{u}_1\left(r\right)+\sqrt{c}{u}_2\left(r\right)\right).$$

On another note, taking $\delta \to 0$$({\phi }_0\to 0)$, we arrive at

$$w_{\ell }\left(r\right)=\ell \left(-\frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}+i\frac{\lambda }{{\alpha }^2\left(r\right)}\right)=-\ell \frac{u^{\prime }\left(r\right)}{u\left(r\right)},$$

$$v_{\ell }\left(r\right)=\ell (\ell -1)\left[{\left(-\frac{{\alpha }^{\prime }\left(r\right)}{\alpha \left(r\right)}+i\frac{\lambda }{{\alpha }^2\left(r\right)}\right)}^2-{\kappa }^2\right]=\ell (\ell -1)\left[{\left(-\frac{u^{\prime }\left(r\right)}{u\left(r\right)}\right)}^2-{\kappa }^2\right].$$

We see, in the next subsections, that this limit corresponds to the $\mathcal{PT}$-symmetric regime.

Hence, regarding (23), it is clear that the expression (33) for $v_{\ell }$ supplies with multi-parametric classes of complex-valued refractive index profiles $n_{\ell }\left(x\right)=n_B+n_R\left(x\right)+i{n}_I\left(x\right)$. The real part $n_B+n_R\left(x\right)=n_B-\mathrm{Re}\left[v_{\ell }\left(r\left(x\right)\right)\right]$ represents the guiding refractive index of the optical medium, while the imaginary part $n_I\left(x\right)=-\mathrm{Im}\left[v_{\ell }\left(r\left(x\right)\right)\right]$, according to its sign, models losses or gains of transversal optical power as the electromagnetic signal propagates throughout the material. Each specific class $n_{\ell }$, $\ell ={\ell }_0+1,{\ell }_0+2.\dots$ is defined by a particular choice of the pair of linearly independent solutions $(u_1,u_2)$ of (24) together with the setting of parameters $\kappa ,\lambda ,\delta ,a$ and b.

3.2. Complex Profiles of the Hyperbolic Type

In order to determine particular classes of optical potentials $v_{\ell }$, we have to choose two appropriate real-valued, linearly independent functions $u_1$ and $u_2$ from the reservoir given by Equation (24) (see, for instance, [84]) and a proper set of parameters $\left\{\kappa ,\lambda ,\delta ,a,b\right\}$. In this context, it is worthwhile to mention that the restriction (30) is sufficient to grant that the function $\alpha$ is free of zeros in $\mathrm{Dom}{v}_{\ell }$ [34]. On the other hand, as $u_1$ and $u_2$ are linearly independent, they do not possess common nodes, so that $u\left(r\right)\ne 0$ in the same domain. Under these conditions, the expressions (25) and (26) lead to the construction of the functions $\beta$ and $v_{\ell }$ free of singularities. With this in mind, let us take, for instance, $u_1\left(r\right)=cosh\left(\kappa r\right)$ and $u_2\left(r\right)=sinh\left(\kappa r\right)$. For the sake of simplicity, let us also set $b=0$. Then, the functions $\alpha$ and u acquire the simple forms

$$\alpha \left(r\right)={\left\{a{cosh}^2\left(\kappa r\right)+\frac{{\lambda }^2}{{\kappa }^{2}a}{sinh}^2\left(\kappa r\right)\right\}}^{1/2},$$

(34)

$$u\left(r\right)=acosh\left(\kappa r\right)-i\frac{\lambda }{\kappa }sinh\left(\kappa r\right),$$

(35)

with $\kappa$, $\lambda$ and a being free parameters. In turn, the functions $\beta$ and $\phi$, from (31) and (28), are given by

$$\beta \left(r\right)=-\frac{d}{dr}ln\left\{acosh\left(\kappa r\right)-i\frac{\lambda }{\kappa }sinh\left(\kappa r\right)\right\},\phi \left(r\right)=\frac{\delta }{acosh\left(\kappa r\right)-i\frac{\lambda }{\kappa }sinh\left(\kappa r\right)}.$$

(36)

Finally, the introduction of these expressions into Equation (23) gives us the class of balanced gain-and-loss refractive index profiles (compare to [22,24]) as follows:

$$\begin{array}{ccc}\hfill n_{\ell }\left(x\right)=n_B& -& \frac{{\kappa }^2}{{\left[a\kappa cosh\left(\kappa r\right(x\left)\right)-i\lambda sinh\left(\kappa r\right(x\left)\right)\right]}^2}\{{\delta }^2-\ell (\ell -1)\left(a^2{\kappa }^2+{\lambda }^2\right)\hfill \\ & +& i\delta (2\ell -1)\left[\lambda cosh\left(\kappa r\right(x\left)\right)+ia\kappa sinh\left(\kappa r\right(x\left)\right)\right]\}.\hfill \end{array}$$

(37)

As stated before, the presence of the constant $\lambda$ in this expression enables us to tune the intensity of its imaginary part, thus allowing us to modulate the gains and losses in the material. Additionally, the parameter $\delta$ defines the parity properties of the real and imaginary parts of $n_{\ell }\left(x\right)$, allowing $\mathcal{PT}$ as well as non-$\mathcal{PT}$ symmetric profiles to be generated. In both cases, it is possible to show that the imaginary part of $n_{\ell }$ fulfills the zero-total-area condition [34,35,36]

$${\int }_{\mathcal{D}}\mathrm{Im}\left[n_{\ell }\left(x\right)\right]dx=0,$$

(38)

where $\mathcal{D}=\mathrm{Dom}{n}_{\ell }$, which means that the total transversal optical power is preserved under propagation processes along the waveguide.

3.3. $\mathcal{PT}$-Symmetric Profiles

If parameter $\delta =0$, the refractive index (37), is reduced to

$$\begin{array}{ccc}\hfill n_{\ell }\left(x\right)=n_B& +& \frac{\ell (\ell -1){\kappa }^2(a^2{\kappa }^2+{\lambda }^2)}{{\left[a^2{\kappa }^2{cosh}^2\left(\kappa r\left(x\right)\right)+{\lambda }^2{sinh}^2\left(\kappa r\left(x\right)\right)\right]}^2}\{a^2{\kappa }^2{cosh}^2\left(\kappa r\left(x\right)\right)-{\lambda }^2{sinh}^2\left(\kappa r\left(x\right)\right)\hfill \\ \hfill & +& 2i\lambda a\kappa sinh\left(\kappa r\right(x\left)\right)cosh\left(\kappa r\right(x\left)\right)\}.\hfill \end{array}$$

(39)

Check that the real and imaginary parts of this expression are, respectively, even and odd under parity transformations. This implies that the refractive index fulfills the condition $n_{\ell }\left(x\right)=n_{\ell }^{*}(-x)$ defining a $\mathcal{PT}$-symmetric profile. It is worthwhile to note that, for $\ell <2$, (39) is reduced to the single mode $\mathcal{PT}$-symmetric refractive index constructed as first-order Darboux deformations of the homogeneous medium, reported in [50]. Larger values of ℓ in this expression also allow us to consider multimodal profiles. In Figure 1, it is depicted the behavior of $n_{\ell }$ for some specific values of the parameters. The graphics in the upper panel show $n_R$ (solid blue line) and $n_I$ (dashed red line) for (Figure 1a) the Hermitian ($\lambda =0$) and (Figure 1b) the non-Hermitian ($\lambda \ne 0$) $\mathcal{PT}$-symmetric regimes. In these plots, we used $\kappa =0.2$, $a=5$, $\ell =3.3$ and $n_B=1.5$. More generally, Figure 1c,d shows, respectively, the real and imaginary parts of $n_{\ell }$ as functions of the complexification parameter $\lambda$. Note that both graphics become more intense as $\lambda$ grows, modeling waveguides with increasing confining as well as balanced gain-and-loss properties.

3.4. Non-$\mathcal{PT}$-Symmetric Profiles

If parameter $\delta \ne 0$, the real and imaginary parts of $n_{\ell }$ suffer a parity loss and the profile is no longer $\mathcal{PT}$-symmetric. Yet, it can be shown that its imaginary part still fulfills the zero-total-area condition leading to an all-real spectrum of propagation constants and the global preservation of the transverse optical power. In the Figure 2, we present some plots of the refractive index (37) in the non-$\mathcal{PT}$-symmetric case, for $\delta =1$. The upper row shows $n_R$ (solid blue line) and $n_I$ (dashed red line) for the Hermitian (Figure 2a) and the non-Hermitian (Figure 2b) cases. Observe the loss of the parity in both distributions due to the non-trivial value of $\delta$. This fact is more evident in Figure 2c,d, where the real and imaginary parts of $n_{\ell }$ are presented as functions of the parameter $\delta$. There, it is clear that, as $\delta$ grows from zero to a specific value, these functions become more intense but their symmetry is gradually lost. Additionally, a small valley is formed on one side of the structure that might be relevant in the reduction in multimode dispersion [1].

3.5. Real-Valued Profiles

If we take the limit $\lambda \to 0$ in Equation (37), we obtain a real-valued expression for $n_{\ell }$ as follows:

$$n_{\ell }\left(x\right)=n_B+\left[\ell (\ell -1){\kappa }^2-\frac{{\delta }^2}{a^2}\right]{\mathrm{sech}}^2\left(\kappa r\left(x\right)\right)+(2\ell -1)\kappa \frac{\delta }{a}\mathrm{sech}\left(\kappa r\left(x\right)\right)tanh\left(\kappa r\left(x\right)\right),$$

(40)

which is analogous to the Scarf II potential [68] in quantum mechanics. If, additionally, we take the limit $\delta \to 0$, we obtain the well-known hyperbolic secant refractive index (see the plot in the Figure 1 for the $\lambda =0$ case)

$$n_{\ell }\left(x\right)=n_B+\ell (\ell -1){\kappa }^2{\mathrm{sech}}^2\left(\kappa r\left(x\right)\right),$$

(41)

which is analogous to the hyperbolic Pöschl–Teller potential in quantum mechanics [87].

4. Modal Fields and Spectrum of Propagation Constants

One of the benefits of using the factorization method is that it leads us directly to the solution of the eigenvalue problem by taking full advantage of the algebraic properties of the operator set $\left\{H_{\ell },a_{\ell },b_{\ell }\right\}$ for all the allowed values of ℓ. In fact, the factorized form (12) implies the intertwining of $H_{\ell }$ with their consecutive partners $H_{\ell \pm 1}$ by $b_{\ell }$ and $a_{\ell -1}$ [63,65,69], i.e.,

$$H_{\ell +1}{b}_{\ell }=b_{\ell }{H}_{\ell },H_{\ell -1}{a}_{\ell -1}=a_{\ell -1}{H}_{\ell }.$$

(42)

As a consequence, the eigenfunctions ${\psi }_{\ell \pm 1}$ of $H_{\ell \pm 1}$ can be generated by applying, correspondingly, $b_{\ell }$ and $a_{\ell -1}$ on the eigenfunctions of $H_{\ell }$. This can be easily checked if we consider an eigenfunction ${\psi }_{\ell }(r;\epsilon )$ of $H_{\ell }$ associated to the eigenvalue $\epsilon$. The relations (42) indicate that the functions $a_{\ell -1}{\psi }_{\ell }(r;\epsilon )$ and $b_{\ell }{\psi }_{\ell }(r;\epsilon )$ are, respectively, eigenfunctions of $H_{\ell -1}$ and $H_{\ell +1}$, both with the same eigenvalue $\epsilon$. These facts can be stated as

$$a_{\ell -1}{\psi }_{\ell }(r;\epsilon )\propto {\psi }_{\ell -1}(r;\epsilon ),b_{\ell }{\psi }_{\ell }(r;\epsilon )\propto {\psi }_{\ell +1}(r;\epsilon ).$$

(43)

In the guided modes regime, the field amplitudes satisfy the finite transverse power condition entailing the discretization of the parameter $\epsilon$. Thus, for each hierarchy element $H_{\ell }$, we expect a sequence of propagation constants ${\epsilon }_{\ell }^n$, $n=0,1,2,\dots$, for which the functions ${\psi }_{\ell }(r;{\epsilon }_{\ell }^n)$, denoted as ${\psi }_{\ell }^n$ for simplicity, are interpreted as the field amplitudes of localized modes. In the parameter space, this means that each pair $(\ell ,n)$ represents a guided mode of amplitude ${\psi }_{\ell }^n\left(r\right)$ fulfilling appropriate boundary conditions [63]. In this scheme, the fundamental field associated to each hierarchy element $H_{\ell }$ is determined as the function ${\psi }_{\ell }^0$ that is annihilated either by $a_{\ell -1}$ or by $b_{\ell }$ and that satisfies the square integrability condition. Of course, in accordance to (12), the corresponding eigenvalue of this mode can be either ${\epsilon }_{\ell }^0={ϵ}_{\ell -1}$ or ${\epsilon }_{\ell }^0={ϵ}_{\ell }$. In the case of the complexified profiles of the hyperbolic type, the fundamental field amplitude turns out to be annihilated by the operator $a_{\ell -1}$ and corresponds to the eigenvalue ${\epsilon }_{\ell }^0={ϵ}_{\ell -1}$. The differential equation defining ${\psi }_{\ell }^0$, i.e.,

$$a_{\ell -1}{\psi }_{\ell }^0=\left[-\frac{d}{dr}+(\ell -1)\beta +\phi \right]{\psi }_{\ell }^0=0,$$

(44)

can be immediately integrated if we use the expressions of $\beta$ and $\phi$ in terms of u and if we observe that, for u given in (35), it follows that

$$\frac{1}{u\left(r\right)}=\frac{i}{2\sqrt{a^2{\kappa }^2+{\lambda }^2}}ln\left(\frac{1-\xi \left(r\right)}{1+\xi \left(r\right)}\right),$$

(45)

with

$$\xi \left(r\right)=\frac{1}{\sqrt{a^2{\kappa }^2+{\lambda }^2}}\left[\lambda cosh\left(\kappa r\right)+ia\kappa sinh\left(\kappa r\right)\right].$$

(46)

We obtain

$${\psi }_{\ell }^0\left(r\right)=N_{\ell }^0{\left(u\left(r\right)\right)}^{-\ell +1}{\left(\frac{1-\xi \left(r\right)}{1+\xi \left(r\right)}\right)}^{i\frac{\delta }{2\sqrt{a^2{\kappa }^2+{\lambda }^2}}},$$

(47)

with $N_{\ell }^0$ being a normalization factor to be determined.

Higher-order field amplitudes can be constructed by using the intertwining relations (42). Indeed, we may write

$$a_{\ell -1}{\psi }_{\ell }^n\left(r\right)={\theta }_{\ell }^n{\psi }_{\ell -1}^{n-1}\left(r\right),b_{\ell }{\psi }_{\ell }^n\left(r\right)={\vartheta }_{\ell }^n{\psi }_{\ell +1}^{n+1}\left(r\right),$$

(48)

where ${\theta }_{\ell }^n$ and ${\vartheta }_{\ell }^n$ are proportionality constants to be fixed and ${\epsilon }_{\ell +1}^{n+1}={\epsilon }_{\ell }^n={\epsilon }_{\ell -1}^{n-1}$, as stated by (43). Moreover, following (48), the n-th eigenmode ${\psi }_{\ell }^n\left(r\right)$ of $H_{\ell }$ can be generated from the fundamental mode ${\psi }_{\ell -n}^0$ of $H_{\ell -n}$ through the appropriate consecutive application of b-operators as

$${\psi }_{\ell }^n\left(r\right)={\left(\prod _{s=1}^n{\vartheta }_{\ell -s}^{n-s}\right)}^{-1}{b}_{\ell -1}{b}_{\ell -2}\cdots b_{\ell -n}{\psi }_{\ell -n}^0\left(r\right),$$

(49)

with the complementary condition that ${\epsilon }_{\ell }^n={\epsilon }_{\ell -1}^{n-1}=\cdots ={\epsilon }_{\ell -n}^0={ϵ}_{\ell -n-1}=-{\kappa }^2{(\ell -n-1)}^2$, fixing the spectrum of propagation constants. Evidently, the quantity $\ell -n$ must satisfy the condition $\ell -n>1$ to be a valid label for a hierarchy element as stated by (10). This means that, for a fixed value of ℓ, the profile $n_{\ell }$ hosts a finite number of guided modes corresponding to $n=0,1,2,\dots ,\lfloor \ell -1\rfloor$. Note also that

$$b_{\ell }=\frac{d}{dr}+\ell \beta +\phi =u^{\ell }{\left(\frac{1-\xi }{1+\xi }\right)}^{-i\frac{\delta }{2\sqrt{a^2{\kappa }^2+{\lambda }^2}}}\frac{d}{dr}{u}^{-\ell }{\left(\frac{1-\xi }{1+\xi }\right)}^{i\frac{\delta }{2\sqrt{a^2{\kappa }^2+{\lambda }^2}}},$$

(50)

so that the product of b-operators in (49) acquires the compact Rodrigues form

$$b_{\ell -1}{b}_{\ell -2}\cdots b_{\ell -n}=u^{\ell }{\left(\frac{1-\xi }{1+\xi }\right)}^{-i\frac{\delta }{2\sqrt{a^2{\kappa }^2+{\lambda }^2}}}{\left(\frac{1}{u}\frac{d}{dr}\right)}^n{\left(\frac{1-\xi }{1+\xi }\right)}^{i\frac{\delta }{2\sqrt{a^2{\kappa }^2+{\lambda }^2}}}{u}^{-\ell +n}.$$

(51)

This allows us to identify the explicit form of the field amplitudes as

$${\psi }_{\ell }^n\left(r\right)=N_{\ell }^n{\left[u\left(r\right)\right]}^{-\ell +1}{\left(\frac{1-\xi \left(r\right)}{1+\xi \left(r\right)}\right)}^{i\frac{\delta }{2\sqrt{a^2{\kappa }^2+{\lambda }^2}}}{P}_n^{\left(\eta ,{\eta }^{*}\right)}\left(\xi \left(r\right)\right),$$

(52)

where the parameter $\eta$ is given by

$$\eta =-\ell +\frac{1}{2}+i\frac{\delta }{\sqrt{a^2{\kappa }^2+{\lambda }^2}},$$

the symbols $P_n^{\left(\mu ,\nu \right)}\left(z\right)$ stand for the Jacobi polynomials and $N_{\ell }^n$ is the normalization constant given by

$$N_{\ell }^n={\left(\prod _{s=1}^n{\vartheta }_{\ell -s}^{n-s}\right)}^{-1}{N}_{\ell -n}^{0}n!{\left(-2i\sqrt{a^2{\kappa }^2+{\lambda }^2}\right)}^n.$$

(53)

5. Orthogonality and Normalization of the Guided Modes: The Bi-Orthogonal Approach

Since $\epsilon$ is a real parameter, the complex conjugation of Equation (11) implies that

$$H_{\ell }^{†}{\psi }_{\ell }^{*}(r;\epsilon )=\epsilon {\psi }_{\ell }^{*}(r;\epsilon ),$$

(54)

where $H_{\ell }^{†}\ne H_{\ell }$ is the adjoint operator of $H_{\ell }$ and $z^{*}$ stands for the complex conjugate of z. For the sake of notation it is convenient to denote $H_{\ell }^{†}={\overline{H}}_{\ell }$. Then, the eigenvalue Equation (54) can be rewritten in the form

$${\overline{H}}_{\ell }{\overline{\psi }}_{\ell }(r;\epsilon )=\epsilon {\overline{\psi }}_{\ell }(r;\epsilon ),$$

(55)

where the eigenfunctions ${\overline{\psi }}_{\ell }$ of ${\overline{H}}_{\ell }$ can be immediately identified as the complex conjugates of the eigenfunctions ${\psi }_{\ell }$ of $H_{\ell }$

$${\overline{\psi }}_{\ell }(r;\epsilon )={\psi }_{\ell }^{*}(r;\epsilon ).$$

(56)

Additionally, due to the fact that $H_{\ell }$ is non-Hermitian, neither the orthogonality nor the completeness of its eigenvectors is granted. Yet, the bi-orthogonal approach [31,32,34,36,88] allows us to establish general orthogonality relations if we consider the bi-orthogonal set formed by the eigenvectors of $H_{\ell }$, $\left\{{\psi }_{\ell }^n\left(r\right),n=0,1,2,\dots \right\}$ together with the eigenvectors of ${\overline{H}}_{\ell }$, $\left\{{\overline{\psi }}_{\ell }^m\left(r\right),m=0,1,2,\dots \right\}$ [34,36]. In this context, check that the bi-product

$$\left({\overline{\psi }}_{\ell }^m,{\psi }_{\ell }^n\right)\equiv {\int }_D{\left({\overline{\psi }}_{\ell }^m\left(r\right)\right)}^{*}{\psi }_{\ell }^n\left(r\right)dr,$$

(57)

with $D=\mathrm{Dom}{v}_{\ell }$, vanishes for $m\ne n$, $m,n=0,1,2,\dots$ For $n=m$, it gives the squared of the bi-norm

$$\parallel {\psi }_{\ell }^n{\parallel }_B^2={\int }_D{\left({\psi }_{\ell }^n\left(r\right)\right)}^{2}dr.$$

(58)

Next, observe that the adjoint conjugation of (12) provides the factorized form of ${\overline{H}}_{\ell }$,

$${\overline{H}}_{\ell }=b_{\ell }^{†}{a}_{\ell }^{†}+{ϵ}_{\ell }=a_{\ell -1}^{†}{b}_{\ell -1}^{†}+{ϵ}_{\ell -1},$$

(59)

from which the following intertwining relations are obtained

$${\overline{H}}_{\ell +1}{a}_{\ell }^{†}=a_{\ell }^{†}{\overline{H}}_{\ell },{\overline{H}}_{\ell -1}{b}_{\ell -1}^{†}=b_{\ell -1}^{†}{\overline{H}}_{\ell }.$$

(60)

Accordingly, it follows that the functions ${\overline{\psi }}_{\ell +1}$ and ${\overline{\psi }}_{\ell -1}$ are connected to ${\overline{\psi }}_{\ell }$ through the operators $a_{\ell }^{†}$ and $b_{\ell -1}^{†}$, respectively,

$$a_{\ell }^{†}{\overline{\psi }}_{\ell }(r;\epsilon )\propto {\overline{\psi }}_{\ell +1}(r;\epsilon ),b_{\ell -1}^{†}{\overline{\psi }}_{\ell }(r;\epsilon )\propto {\overline{\psi }}_{\ell -1}(r;\epsilon ),$$

(61)

in complete correspondence to their concomitants (43). Indeed, the complex conjugation of the expressions (48) yields directly the action of the operators $a_{\ell }^{†}$ and $b_{\ell }^{†}$ on the basis vectors ${\overline{\psi }}_{\ell }^n$.

$$a_{\ell }^{†}{\overline{\psi }}_{\ell }^n\left(r\right)={\left({\vartheta }_{\ell }^n\right)}^{*}{\overline{\psi }}_{\ell +1}^{n+1}\left(r\right),b_{\ell -1}^{†}{\overline{\psi }}_{\ell }^n\left(r\right)={\left({\theta }_{\ell }^n\right)}^{*}{\overline{\psi }}_{\ell -1}^{n-1}\left(r\right).$$

(62)

In order to fix the normalization constants ${\vartheta }_{\ell }^n$ and ${\theta }_{\ell }^n$, let us assume that the function ${\psi }_{\ell }^n\left(r\right)$ represents a guided mode. In general, its bi-norm ${∥{\psi }_{\ell }^n∥}_B$ is a complex number such that its modulus is finite and different from zero. We may assume that ${\psi }_{\ell }^n$ is normalized in such a way that $\parallel {\psi }_{\ell }^n{\parallel }_B=1$. Therefore,

$${∥{\psi }_{\ell -1}^{n-1}∥}_B^2=\left({\overline{\psi }}_{\ell -1}^{n-1},{\psi }_{\ell -1}^{n-1}\right)={\left({\theta }_{\ell }^n\right)}^{-2}\left(b_{\ell -1}^{†}{\overline{\psi }}_{\ell }^n,a_{\ell -1}{\psi }_{\ell }^n\right)={\left({\theta }_{\ell }^n\right)}^{-2}({\epsilon }_{\ell }^n-{ϵ}_{\ell -1}),$$

$${∥{\psi }_{\ell +1}^{n+1}∥}_B^2=\left({\overline{\psi }}_{\ell +1}^{n+1},{\psi }_{\ell +1}^{n+1}\right)={\left({\vartheta }_{\ell }^n\right)}^{-2}\left(a_{\ell }^{†}{\overline{\psi }}_{\ell }^n,b_{\ell }{\psi }_{\ell }^n\right)={\left({\vartheta }_{\ell }^n\right)}^{-2}({\epsilon }_{\ell }^n-{ϵ}_{\ell }).$$

Hence, by choosing

$${\theta }_{\ell }^n=\sqrt{{\epsilon }_{\ell }^n-{ϵ}_{\ell -1}}=\kappa \sqrt{n(2\ell -n-2)}{\vartheta }_{\ell }^n=\sqrt{{\epsilon }_{\ell }^n-{ϵ}_{\ell }}=\kappa \sqrt{(n+1)(2\ell -n-1)},$$

(63)

the expressions (48) and (62) take the form

$$\begin{array}{ccc}\hfill {\psi }_{\ell -1}^{n-1}\left(r\right)& =& \frac{1}{\kappa \sqrt{n(2\ell -n-2)}}{a}_{\ell -1}{\psi }_{\ell }^n\left(r\right),\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill {\psi }_{\ell +1}^{n+1}\left(r\right)& =& \frac{1}{\kappa \sqrt{(n+1)(2\ell -n-1)}}{b}_{\ell }{\psi }_{\ell }^n\left(r\right),\hfill \end{array}$$

(65)

so that the operators ${\left({\theta }_{\ell }^n\right)}^{-1}{a}_{\ell -1}$ and ${\left({\vartheta }_{\ell }^n\right)}^{-1}{b}_{\ell }$ and their adjoints transform normalized functions into normalized ones. This fact, according to (49), enables to construct a bi-orthogonal set of functions $\left\{{\psi }_{\ell }^n,{\overline{\psi }}_{\ell }^n,n=0,1,2\dots \right\}$, fulfilling the bi-orthonormality relation

$$\left({\overline{\psi }}_{\ell }^m,{\psi }_{\ell }^n\right)={\delta }_{m,n},m,n=0,1,2,\dots$$

(66)

through the appropriate subsequent application of b-($a^{†}$-)operators on the normalized fundamental mode amplitude ${\psi }_{\ell -n}^0$ (${\overline{\psi }}_{\ell -n}^0$). The key point, thus, is to choose the constant $N_{\ell }^0$ in (47) such that

$$\left({\overline{\psi }}_{\ell }^0,{\psi }_{\ell }^0\right)={\int }_D{\left({\psi }_{\ell }^0\left(r\right)\right)}^{2}dr=1.$$

For the particular case of the field amplitude (47), after some calculations, we obtain

$$N_{\ell }^0=\sqrt{\frac{\kappa {(a^2{\kappa }^2+{\lambda }^2)}^{\ell -1}}{\sqrt{\pi }\mathsf{\Gamma }\left(\ell -\frac{1}{2}\right)\mathsf{\Gamma }\left(\ell -1\right)}}{\kappa }^{-\ell +1}\left|\mathsf{\Gamma }\left(-{\eta }^{*}\right)\right|.$$

(67)

Additionally, from (63), we arrive at

$$\prod _{s=1}^n{\vartheta }_{\ell -s}^{n-s}=2^{-\ell +n+1}{\kappa }^n\sqrt{\frac{\sqrt{\pi }n!\mathsf{\Gamma }(2\ell -n-1)}{\mathsf{\Gamma }(\ell -n-\frac{1}{2})\mathsf{\Gamma }(\ell -n)}},$$

so that the normalization constant $N_{\ell }^n$ of the field (52) can be finally expressed as

$$N_{\ell }^n={(-i)}^n{\left(\frac{2}{\kappa }\right)}^{\ell -1}\sqrt{\frac{\kappa n!(\ell -n-1){\left(a^2{\kappa }^2+{\lambda }^2\right)}^{\ell -1}}{\pi \mathsf{\Gamma }(2\ell -n-1)}}|\mathsf{\Gamma }\left(-{\eta }^{*}-n\right)|.$$

(68)

In Figure 3, we depict the behavior of the three guided modes hosted by the generalized hyperbolic-type waveguide characterized by $\kappa =0.2$, $\lambda =0.4$, $a=5$, $\ell =3.3$ and $n_B=1.5$ (same parameters as in Figure 1b) in the $\mathcal{PT}$-symmetric regime ($\delta =0$). In the upper row, the real (solid blue curve) and imaginary (dashed red) parts of each field amplitude are presented. We can see that, as n grows, the imaginary part of the wave function becomes as pronounced as the real one, implying that the balanced gain-and-loss processes become more important for higher-order modes. Additionally, for these particular sets of parameters, we can observe that the real and imaginary parts of the field amplitudes exhibit some interlacing properties of their nodal points. This fact can be also noted in the Argand–Wessel diagrams presented in the central panel of this figure. The lower row contains the optical intensity distributions $|{\varphi }_{\ell }^n\left(x\right){|}^2=|{\psi }_{\ell }^n\left(r\left(x\right)\right){|}^2$, $n=0,1,2$. Check the absence of nodal points in these plots due to the interlacing properties of the zeroes of the corresponding amplitudes [35]. In Figure 4, we present similar results in the non-$\mathcal{PT}$-symmetric case ($\delta \ne 0$) with the same values of the parameters $\kappa$, $\lambda$, a, ℓ and $n_B$. Additionally, observe, in all diagrams, the symmetry loss displayed by the real and imaginary parts of ${\varphi }_{\ell }^n$. This asymmetry becomes more evident for higher-order modes as it is shown in Figure 4c, where we can see that almost all intensity is concentrated in a region of small values of loss in the waveguide. In a similar way to the $\mathcal{PT}$-symmetric case, for the chosen values of the parameters, the real and imaginary parts exhibit interlacing properties of their nodal points, so that the associated intensity distributions are nodal-free.

6. Conclusions

We present a method to construct different classes of exactly solvable guiding refractive index distributions in the GRIN non-Hermitian regime. This was accomplished in the framework given by the equivalence between the Helmholtz equation in the paraxial approximation and the time-dependent Schrödinger equation. For planar waveguides, the paraxial Helmholtz equation was reduced to a one-dimensional Schrödinger-type eigenvalue problem for the electric field envelope. At such point, it is possible to apply all the techniques addressed to generate exactly solvable models in quantum mechanics. Some previous results have been obtained in the supersymmetric context [50] (see also [34]) where the optical potential was determined as the n-th order Darboux deformation of an initial exactly solvable potential. In this work, alternatively, we employed the factorization method to generate new families of complex-valued refractive index landscapes associated to real spectra. As an example, the complex hyperbolic class was explicitly determined. We included $\mathcal{PT}$- as well as non-$\mathcal{PT}$-symmetric profiles (compare with [24,25,89,90]). In any case, the integral in the complete domain of the imaginary part of the refractive indices thus obtained is null. This property, called the zero-total-area condition, implies the conservation of the total transverse optical power as the optical signal propagates through the waveguide. In the Hermitian limit, we obtained the optical analogues to the well-known Pöschl–Teller [34,50,87] and Scarf [68] potentials of type II. The intertwining relations, defined by the factorizability condition, allowed us to obtain the point spectra and the set of modal field amplitudes in the guiding regime. Due to the non-Hermiticity of the system, it is not possible to grant the orthogonality of the set of modal fields. Yet, using the bi-orthogonal approach, we established orthogonality relations and normalization properties in the sense of the bi-product of modal fields and their concomitants.

The profiles here obtained may be useful in managing loss compensation or control in guiding devices and in enhancing transmission in non-Hermitian optical structures due to their GRIN nature and the multiple parameters available to adjust their properties. Additionally, the quadratic behavior of the spectra of the propagation constants suggests the appearance of multi-mode interference in the class members supporting higher-order guided modes [62]. Jamming anomalies may also be expected for the highest-order mode of class elements with large values of the parameter ℓ, leading to the concentration of transversal optical power in the lossy regions (see, for instance Figure 3c). Although the scope of this paper only includes the study of guided modes, scattering states may be also analyzed in order to evaluate the transversal transmission and reflection properties of our structures, as well as their possible application in the implementation of unidirectional invisibility processes [53,54] (results on this matter are to be presented elsewhere).

Finally, in regards to the overall technique, it is worthwhile to stress that it offers a versatile route to design non-Hermitian optical structures, as it places at our disposal different choices of the pair $(u_1,u_2)$ in combination with the set of constants $\left\{\kappa ,\lambda ,\delta ,a,b\right\}$ in order to obtain a wide variety of $n_{\ell }$-classes with diverse confining and gain–loss properties that may be tuned by properly adjusting the relevant parameters.