Spectral singularities of odd-$\mathcal{P}\mathcal{T}$-symmetric potentials

We describe one-dimensional stationary scattering of a two-component wave field by a non-Hermitian matrix potential which features odd-$\mathcal{P}\mathcal{T}$ symmetry, i.e., symmetry with ${\left(\mathcal{P}\mathcal{T}\right)}^2=-1$. The scattering is characterized by a $4\times 4$ transfer matrix. The main attention is focused on spectral singularities which are classified into two types. Weak spectral singularities are characterized by the zero determinant of a diagonal $2\times 2$ block of the transfer matrix. This situation corresponds to the lasing or coherent perfect absorption of pairs of oppositely polarized modes. Strong spectral singularities are characterized by a zero diagonal block of the transfer matrix. We show that in odd-$\mathcal{P}\mathcal{T}$-symmetric systems any spectral singularity is self-dual, i.e., lasing and coherent perfect absorption occur simultaneously. A detailed analysis is performed for a case example of a $\mathcal{P}\mathcal{T}$-symmetric coupler consisting of two waveguides, one with localized gain and another with localized absorption, which are coupled by a localized antisymmetric medium. For this coupler, we discuss weak self-dual spectral singularities and their splitting into complex-conjugate eigenvalues which represent bound states characterized by propagation constants with real parts lying in the continuum. A rather counterintuitive restoration of the unbroken odd-$\mathcal{P}\mathcal{T}$-symmetric phase subject to the increase of the gain-and-loss strength is also revealed. A comparison between odd- and even-$\mathcal{P}\mathcal{T}$-symmetric couplers, the latter characterized by ${\left(\mathcal{P}\mathcal{T}\right)}^2=1$, is also presented.

Figure 1

Schematic of two waveguides which are locally coupled through an antisymmetric medium characterized by $V_2\left(x\right)=-V_2^{*}(-x)$. In the domain of coupling, the waveguides are doped by active impurities resulting in gain $+i{V}_1\left(x\right)$ and loss $-i{V}_1\left(x\right)$. A paraxial beam propagates along $z$ axis directed toward the reader.

Figure 2

Scattering by the odd-$\mathcal{P}\mathcal{T}$-symmetric potential barrier with $v_0=8$ and $\ell =1$. (a) Values of $v_1$ and $v_2$ that correspond to weak SSs are plotted as curves in the $(v_1,v_2)$ plane. The lower (red and blue) solid lines separate the domains of unbroken (shaded area) and broken (white area) $\mathcal{P}\mathcal{T}$-symmetric phases. The black dots labeled a–k correspond to Figs. 3–3, respectively. (b) Weak self-dual SSs $k_{★}$ as functions of the coupling strength $v_2$. (c) Ratio $|a_4/a_3|$ between the amplitudes of the two polarizations for the laser solution. (d) Magnitudes of reflection and transition coefficients $r_{\alpha \beta }^{L,R}$ and $t_{\alpha \beta }^{L,R}$ defined in Appendix pp1 as functions of the wave number $k$ driven through the self-dual SS for $v_1\approx 10.685$ and $v_2=15$.

Figure 3

Real and imaginary parts of eigenvalues for different combinations of parameters labeled with black dots in Fig. 2. The thick line shows the continuous spectrum and circles correspond to isolated eigenvalues.

Figure 4

Scattering on the odd-$\mathcal{P}\mathcal{T}$-symmetric potential well with $v_0=-3$ and $\ell =1$. (a) Values of $v_1$ and $v_2$ that correspond to weak SSs are plotted as curves in the $(v_1,v_2)$ plane. (b) Dependence of $k_{★}$ on the coupling strength $v_2$. (c) Ratio $|a_4/a_3|$ between amplitudes of the two polarizations of the lasing solution. Black dots labeled a–c correspond to Figs. 5–5, respectively. Here $\mathcal{P}\mathcal{T}$ symmetry is broken for any $v_1>0$ with one or more complex-conjugate pairs in the spectrum.

Figure 5

Real and imaginary parts of eigenvalues for different combinations of parameters labeled with black dots in Fig. 4. The thick line shows the continuous spectrum and circles correspond to isolated eigenvalues.

Figure 6

Scattering by the even-$\mathcal{P}\mathcal{T}$-symmetric potential barrier with $v_0=8$ and $\ell =1$. (a) Values of $v_1$ and $v_2$ that correspond to weak SSs are plotted as curves in the $(v_1,v_2)$ plane and separate the unbroken $\mathcal{P}\mathcal{T}$-symmetric phase (shaded domain) and the broken one (white domain). The black dots labeled a–d correspond to Figs. 7–7, respectively. (b) Weak SS vs coupling strength. (c) Ratio $|a_4/a_3|$ between amplitudes of the two polarizations of the laser solution.

Figure 7

Real and imaginary parts of eigenvalues for different combinations of parameters labeled with black dots in Fig. 6. The thick line shows the continuous spectrum and circles correspond to isolated eigenvalues.

Figure 8

Scattering by the even-$\mathcal{P}\mathcal{T}$-symmetric potential well with $v_0=-3$ and $\ell =1$. (a) Values of $v_1$ and $v_2$ that correspond to weak SSs are plotted as curves in the $(v_1,v_2)$ plane. Shaded and white domains correspond to the unbroken and broken $\mathcal{P}\mathcal{T}$-symmetric phase, respectively. The black dots labeled a–c correspond to Figs. 9–9, respectively. (b) Weak SS vs the coupling strength $v_2$. (c) Ratio $|a_4/a_3|$ between amplitudes of the two polarizations of the laser solution.

Figure 9

Real and imaginary parts of eigenvalues for different combinations of parameters labeled with black dots in Fig. 8. The thick line shows the continuous spectrum and circles correspond to isolated eigenvalues.