Enhanced Spontaneous Emission at Third-Order Dirac Exceptional Points in Inverse-Designed Photonic Crystals

Abstract

We formulate and exploit a computational inverse-design method based on topology optimization to demonstrate photonic crystal structures supporting complex spectral degeneracies. In particular, we discover photonic crystals exhibiting third-order Dirac points formed by the accidental degeneracy of monopolar, dipolar, and quadrupolar modes. We show that, under suitable conditions, these modes can coalesce and form a third-order exceptional point, leading to strong modifications in the spontaneous emission (SE) of emitters, related to the local density of states. We find that SE can be enhanced by a factor of 8 in passive structures, with larger enhancements ∼ ffiffiffiffiffi n 3 p possible at exceptional points of higher order n. [3,4], and as precursors to nontrivial topological effects [5][6][7]. Recent work also showed that Dirac-point degeneracies can give rise to rings of exceptional points [8]. An exceptional point (EP) is a singularity in a non-Hermitian system where two or more eigenvectors and their corresponding complex eigenvalues coalesce, leading to a nondiagonalizable, defective Hamiltonian [9,10]. EPs have been studied in various physical contexts, most notably lasers and atomic as well as molecular systems [11,12]. In recent decades, interest in EPs has been reignited in connection with non-Hermitian parity-time symmetric systems [13], especially optical media involving carefully designed gain and loss profiles [14][15][16][17][18][19][20], where they can lead to intriguing phenomena such as excess noise [21,22], chiral modes [23], directional transport [24,25], and anomalous lasing behavior [26][27][28]. Also recently, it became possible to directly observe EPs in photonic crystals (PhCs) [8] and optoelectronic microcavities [29]. Thus far, however, the main focus of these works has been the effect of second-order exceptional points (EP2s) realized through photonic radiations, where only two modes coalesce; apart from a few mathematical analyses [30][31][32] or works focused on acoustic systems [33], there has been little or no investigation into the design and consequences of EPs of higher order (where more than two modes collapse).In this Letter, we formulate and exploit a powerful inverse-design method, based on topology optimization (TO), to develop complex photonic crystals supporting Dirac points formed out of the accidental degeneracy [34] of modes belonging to different symmetry representations. We show that such higher-order Dirac points can be exploited to create third-order exceptional points (EP3s) along with complex contours of EP2s. Furthermore, we consider possible enhancements and spectral modifications in the spontaneous emission (SE) rate of emitters, showing that the local density of states (LDOS) at an EP3 (14) can be enhanced eightfold (in passive systems) and can exhibit a cubic Lorentzian spectrum under special conditions. More generally, we find enhancement factors ∼ ffiffiffiffiffi n 3 p with increasing EP order n. Although the area of photonic inverse design is not new ...

General theory of spontaneous emission near exceptional points

“…For example, at third order passive EPs (n = 3), the enhancement is at most 8-fold (as some of us recently confirmed in [76]). …”

Unidirectional reflectionless light propagation at exceptional points

“…In addition, a few recent studies of multidimensional PT-symmetric systems have demonstrated intriguing properties not found in 1D PT-symmetric systems, such as conical diffraction, third-order EPs, continuous rings of EPs, and rotating input by asymmetric coupling between wave vectors [14,[93][94][95][96][97][98]. Most recently, Fan et al considered 2D PT-symmetric PhCs, whose non-Hermitian primitive cell is an integer multiple of the primitive cell of the underlying Hermitian system [15].…”

Electrical tunability due to coalescence of exceptional points in parity-time symmetric waveguides

“…The transition point is called the exceptional point (EP), at which two or more eigenvalues coincide. The existence of EPs have been investigated in various types of PT-symmetric physical systems [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], giving rise to a wide range of counterintuitive wave phenomena, such as loss-induced transparency [3], nonreciprocal Bloch oscillation [4], unidirectional invisibility [5][6][7], coexisting coherent perfect absorption and lasing [8,9], enhanced spontaneous emission [10], and enhanced nano-particle sensing [11].…”

“…Recently, there has been strong interest in more complex phenomena closely related to the occurrence of EPs in multistate systems and high-order EPs [10,[20][21][22][23][24][25][26][27]. Multiple optical waveguide systems [20,21] and photonic crystals [10,22] have been proposed for the realization of high-order EPs.…”

“…The transition point is called the exceptional point (EP), at which two or more eigenvalues coincide. The existence of EPs have been investigated in various types of PT-symmetric physical systems [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], giving rise to a wide range of counterintuitive wave phenomena, such as loss-induced transparency [3], nonreciprocal Bloch oscillation [4], unidirectional invisibility [5][6][7], coexisting coherent perfect absorption and lasing [8,9], enhanced spontaneous emission [10], and enhanced nano-particle sensing [11].Recently, there has been strong interest in more complex phenomena closely related to the occurrence of EPs in multistate systems and high-order EPs [10,[20][21][22][23][24][25][26][27]. Multiple optical waveguide systems [20,21] and photonic crystals [10,22] have been proposed for the realization of high-order EPs.…”