Loss-induced super scattering and gain-induced absorption

Simin Feng

doi:10.1364/OE.24.001291

1. Introduction

In their pioneering work, Bender and Boettcher [1] proved that non-Hermitian Hamiltonian with parity-time $(P T)$ symmetry may exhibit entirely real spectrum below a phase transition point. Inspired by this emerging concept, in the past decade there has been a growing interest in studying $P T$ -symmetric Hamiltonian in the framework of optics [2–32] where the $P T$ complex potential in quantum mechanics is mapped into a complex permittivity satisfying ε(r) = ε(−r). In optical systems, most $P T$ -symmetric structures are constructed by parallel waveguides or media with alternating gain and loss either along or across the propagation direction. The periodic gain-loss spatial modulation can lead to many intriguing phenomena such as asymmetric light propagation [9–11] and invisibility [12–14], power oscillation [15, 16], coherent perfect absorption [17–21], loss-induced transparency [22], nonreciprocal Bloch oscillation [23], unidirectional tunneling [25], optical switch [26,27], and laser-absorber [28–30]. The $P T$ -symmetric systems are a subset of open quantum systems for which Hamiltonian is non-Hermitian with complex eigenvalues [33]. The unique properties associated with non-Hermitian Hamiltonian are exceptional points and spectral singularities. An exceptional point is a branch point singularity related to level repulsion [34–36] and symmetry breaking [22, 30]. The existence of the exceptional point has been observed in optical [45] and microwave [37, 38] experiments. Spectral singularity is related to scattering resonance of non-Hermitian Hamiltonian [39–43] and manifests itself as giant transmission and reflection with vanishing bandwidth, often corresponding to resonance narrowing and laser oscillation [43]. Exceptional point and spectral singularity have given rise to many interesting electromagnetic properties [44–47] and attracted enormous attention lately.

Parity-time synthetic materials represent a new class of metamaterials with intriguing electromagnetic properties arising from a delicate balance between loss and gain. Global $P T$ symmetry is a demanding condition. Systems with local $P T$ -symmetry are easier to implement. Array of $P T$ -symmetric dimers where each gain-loss pair in itself possesses local $P T$ -symmetry with respect to its own center allows for real spectrum in the right parameter region [3]. Except for loss compensation, active plasmonic materials offer an ideal platform for studying non-Hermitian Hamiltonian in the electromagnetic domain at the subwavelength scale. Most studies on $P T$ -symmetric structures use analytical models based on either one dimensional scalar Helmholtz equation or two dimensional scalar paraxial wave equation. For plasmonic metamaterials, above analytical methods are not applicable due to the resonant nature of subwavelength ”meta-atom” and the coupling between the components of the electric and magnetic fields. Nevertheless, the plasmonic metafilms are complex quantum systems and can be described by non-Hermitian Hamiltonian in strong coupling regime. Recently, based on the spaser concept [48], Zheludev et. al. [49] suggested “lasing spaser” where dark modes are changed into radiative modes via structural symmetry breaking. Follow-on experiments have also been reported [50, 51].

In the past decades, the investigation of composite structured materials are primarily based on the manipulations of the real part of the permittivity of the materials. Recently, it was discovered that the manipulations of the imaginary part can often result in counter-intuitive electromagnetic phenomena [18, 52–54]. The imaginary part of permittivity is associated with the distributed loss and gain in the media. In this paper, we investigate electromagnetic properties of a $P T$ -synthetic plasmonic metafilm composed of a planar array of coupled $P T$ -symmetric dimers. This study will shed light on the effects of the imaginary part of permittivity and the interaction between loss and gain on electromagnetic properties of the metafilms. We have found that this structure can display super scattering by controlling the loss of the system. When the metafilm is steered to the vicinity of an exceptional point, tuning either the substrate dissipation or the loss of the dimers can lead to substantially amplified waves radiated from both sides of the metafilm, acting as a $P T$ -plasmonic thinfilm laser. In addition, we also found that increasing the gain of the dimers can increase the absorption near the exceptional point.

2. Theoretical approach

The structure we investigated is depicted in Fig. 1 showing a planar subwavelength square array composed of gain-loss elements embedded in a metallic thinfilm. The gain-loss dimer repeats in the x-y plane. The plasmonic metafilm satisfies the $P T$ symmetry with respect to the x directions, i.e. ε(x,y,z) = ε^* (−x,y,z) for Δx/2 < |x| < b + Δx/2. This structure cannot be described by the paraxial wave equation due to the abrupt change of electromagnetic (EM) field at the metal-dielectric interfaces. For the infrared $P T$ -synthetic materials, the dispersion of metal, which is aluminum (Al) in our case, cannot be neglected. Assume a harmonic time dependence exp(−iωt) for the electromagnetic field, the permittivity of Al was obtained by curve-fitting experimental data [55] with a Drude model,

ε_{m} = 1 - \frac{ω_{p}^{2}}{ω^{2} + i γ ω},

where the “plasma frequency” ω_p = 9.38 μm⁻¹ and the damping constant γ = 0.048 μm⁻¹. Schrödinger equation is scalar whereas Maxwell’s equations are vectorial, and in many cases the components of the electric and magnetic fields are coupled. To investigate this type of media from a quantum perspective, it is desirable to recast Maxwell’s equations into a vector-Schrödinger-type form [56]:

i \frac{\partial}{\partial z} (\begin{matrix} E_{t} \\ \hat{z} \times H_{t} \end{matrix}) = \tilde{H} \cdot (\begin{matrix} E_{t} \\ \hat{z} \times H_{t} \end{matrix}),

where the subscript ‘t’ refers to the transverse (x,y) components of the EM field on the meta-surface; and ẑ is the unit vector along the z direction. The Hamiltonian is given by

\tilde{H} = (\begin{matrix} 0 & k_{0} μ_{t} {\hat{I}}_{t} + \frac{1}{k_{0}} \nabla_{t} \frac{1}{ε_{z}} \nabla_{t} \\ k_{0} ε_{t} {\hat{I}}_{t} + \frac{1}{k_{0}} \hat{z} \times \nabla_{t} \frac{1}{μ_{z}} \hat{z} \times \nabla_{t} & 0 \end{matrix}),

where k₀ = ω/c, and the c is the speed of light in vacuum. The subscript ‘z’ refers to the component in the z direction. Î_t = Î−ẑẑ is the two-dimensional unit dyadic; and

\nabla_{t} \equiv \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} .

Fig. 1 A schematic showing (a) a unit cell composed of $P T$ -symmetric loss (blue) and gain (red) subwavelength elements embedded in an aluminum substrate and (b) a two-dimensional array of the unit cells in the x-y plane with the same period in both directions. The dimers and the aluminum film have the same thickness, i.e. the metallic mesh is filled with the gain-loss elements. The real part of the relative permittivity of the loss and gain elements is fixed at 3.6 through out this work. The imaginary part varies, but satisfies $ε_{g a i n}^{″} = - ε_{l o s s}^{″}$ to ensure the $P T$ symmetry. The permeability is unit for all the materials. Period p = 3.5 μm, the dimer length a = 2.5 μm, the width b = 1.0 μm, and the separation between the loss and gain Δx = 0.5 μm are fixed throughout the paper. The incidence wave is p-polarized with the electric field parallel to the x-z plane.

Download Full Size | PDF

The form of Eq. (3) can handle uniaxial anisotropic materials with the optical axis along the z direction. The Hamiltonian in Eq. (3) is non-Hermitian. Equations (2) and (3) are accurate without any approximation. They are obtained by separating the longitudinal and transverse components in Maxwell’s equations and expressing the longitudinal components in terms of the transverse components. We numerically solve above vector equations based on rigorous coupled-wave analysis [57]. Numerical approaches can handle more complicated structures for practical implementation of the extraordinary properties predicted by analytical theory. In this work, scattering and transfer matrices, as well as transmittance and reflectance are calculated numerically based on above equations. The transfer matrix which connects the field at the output and the input surfaces is defined as

| Ψ_{i} 〉 = (\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) | Ψ_{o} 〉,

where the subscripts ’o’ and ’i’ refer to the EM field at the output and input surfaces, respectively; and

| Ψ 〉 \equiv (\begin{matrix} E_{t} \\ \hat{z} \times H_{t} \end{matrix}) .

The relationship between the transfer and scattering matrices in our case is given by

\begin{array}{l} S_{11} = M_{21} M_{11}^{- 1}, & S_{21} = M_{11}^{- 1}, \\ S_{22} = - M_{11}^{- 1} M_{12}, & S_{12} = M_{22} - M_{21} M_{11}^{- 1} M_{12}, \end{array}

where the S₁₁ and S₂₁ are, respectively, the reflection and transmission coefficients of the electric field. In general, the transfer and scattering matrices are multidimensional due to multiple scattering channels. We have conducted extensive numerical studies and confirmed that for our geometry the magnitudes of the higher order and cross-polarization scatterings are much smaller than that of the first order event (see Appendix) due to the subwavelength period of the structure. Thus, the transfer and scattering matrices reduce to 2 × 2 matrices. The eigenvalues of the transfer matrix is given by

η_{\pm}^{t} = \frac{M_{11} + M_{22}}{2} \pm \sqrt{{(\frac{M_{11} + M_{22}}{2})}^{2} - 1} .

Here the identity det(M) = 1 (which has been numerically validated, see Appendix) has been used. It is easy to show the eigenvalues of the transfer matrix satisfy the condition $| η_{+}^{t} η_{-}^{t} | = 1$ with one magnitude greater than one and the other less than one, corresponding to amplifying and decaying modes, respectively. Also, the symmetry of the structure requires the relationship S₂₁ = S₁₂ and M₁₂ = −M₂₁. These conditions have been confirmed numerically for various parameters (see Appendix). Using these conditions, the eigenvalues of the scattering matrix can be derived as

η_{\pm}^{s} = \frac{M_{21} \pm 1}{M_{11}} .

The eigenfunctions of the non-Hermitian Hamiltonian satisfy bi-orthogonal relationship. The right eigenvectors of the transfer matrix are given by

| Ψ_{\pm}^{r} 〉 = (\begin{matrix} \frac{η_{\pm}^{t} - M_{22}}{M_{21}} \\ 1 \end{matrix}),

and the left eigenvectors

| Ψ_{\pm}^{l} 〉 = (\begin{matrix} \frac{η_{\pm}^{t} - M_{22}}{M_{12}} & 1 \end{matrix})

with the property

〈 ψ_{+}^{l} | ψ_{-}^{r} 〉 = 〈 ψ_{-}^{l} | ψ_{+}^{r} 〉 = 0.

Above bi-orthogonality has also been numerically verified (see Appendix). The transfer and scattering matrices and their eigenvalues and eigenvectors are useful tools for the analysis of exceptional points and spectral singularities.

3. Dissipation-induced super scattering

Many parameters can be changed in Fig. 1. In this work, the period (p = 3.5 μm) of the square array, the distance (Δx = 0.5 μm) between the loss and gain elements, the size (a × b = 2.5 × 1.0 μm²) of the dimers, and the real part of the relative permittivity $(ε_{r}^{'} = 3.6)$ of the dimers are fixed throughout the paper. In the absence of material loss, the metafilm behaves as a Fabry-Pérot cavity filled with high refractive index material, as illustrated in Fig. 2 showing transmission and reflection versus wavelength (upper panel) and thickness of the film (lower panel). The frequency dependent permittivity of aluminum is taken from the real part of Drude model given by Eq. (1). In the lossless case we have T + R = 1 (energy conservation) which is clearly demonstrated in Fig. 2. An increase of the transmission is accompanied by a decrease of the reflection and vice versa as the result of constructive and destructive interferences. The peaks and valleys of the transmittance and the reflectance repeat periodically with the variation of the thickness of the film. The metafilm acts as a low-Q Fabry-Pérot cavity below 9 μm. Above 12 μm, it moves toward to a perfect electric conductor (PEC).

Fig. 2 Transmittance (blue solid curves) and reflectance (red dashed curves) of the normal incident wave on a lossless metafilm versus wavelength (upper panel) and thickness (lower panel) with the electric field parallel to the shorter edge of the dimers. The thickness of the metafilm d = 1.5 μm for the upper panel and the wavelength λ = 6 μm for the lower panel. The relative permittivity of the dimers is real and given by ε_r = 3.6.

Download Full Size | PDF

Now we proceed to investigate the EM scattering from a plasmonic thinfilm with coupled gain-loss elements. The $P T$ -synthetic metafilm no longer behaves as a Fabry-Pérot cavity. Figure 3 shows the effect of the substrate dissipation on transmittance and reflectance of the film with the balanced gain-loss dimers without and with the metallic loss. The dispersion of the metallic substrate is taken from the Drude model in Eq. (1). The relative permittivity of the dimers is given by ε = 3.6(1 ± i0.06). In the absence of the substrate dissipation the sum of the transmittance and the reflectance is close to one, i.e. T + R = 1. The metafilm behaves as a conventional lossless medium where the maximum transmission is accompanied by the minimum reflection, and vice versa. This behavior is consistent with the fact that the film contains the balanced gain-loss subwavelength dimers embedded in the lossless substrate. This situation changes dramatically when the substrate dissipation is taken into account as shown in Fig. 3(b) where the giant transmittance and reflectance occur at the same wavelength, unlike conventional media where the increase of one at the expense of the other. This peculiar property can be analyzed through the scattering parameters S₁₁ and S₂₁ that have a common denominator M₁₁ as shown by Eq. (7). When the denominator of S₁₁ and S₂₁ vanishes, both transmittance and reflectance approach infinity as long as M₂₁ is finite. The lower panels in Fig. 3 show the magnitude of M₁₁ and M₂₁. Without the substrate dissipation, both magnitude of M₁₁ and M₂₁ are large at the wavelength about 8.88 μm [see Fig. 3(c)] which explains the null in the transmittance and the peak in the reflectance [see Fig. 3(a)]. When turn on the substrate dissipation, the |M₁₁| vanishes at the wavelength about 8.92 μm whereas the |M₂₁| is finite [see Fig. 3(d)]. Thus, both transmission and reflection approach infinity. Without the substrate dissipation, the energy generation and absorption is balanced. The substrate dissipation tips the balance in favor of gain, leading to the super scattering in both forward and backward directions. The spectral singularity is a character of non-Hermitian Hamiltonian of the gain medium. From Eq. (9), when M₁₁ → 0, one eigenvalue diverges and the other one is finite, i.e.

η_{+}^{s} \to \infty, η_{-}^{s} \to - \frac{M_{22}}{2} .

Fig. 3 Simulation without (left panels) and with (right panels) metallic substrate dissipation. Upper panels: Transmittance (blue solid curves) and reflectance (red dashed curves) of the normal incident beam with the electric field parallel to the shorter edge of the dimers. Lower panels: The magnitude of M₁₁ (blue solid curves) and M₂₁ (red dashed curves). The thickness of the mesh d = 1.5 μm. The relative permittivity of the gain/loss elements is given by ε = 3.6(1 ± i0.06).

Download Full Size | PDF

The coexistence of the giant transmission and reflection is the manifestation of a resonance in the complex scattering potential, an essential feature of non-Hermitian Hamiltonian with gain. The vanishing M₁₁, a necessary condition for the super scattering to occur, is in fact the condition of lasing threshold [28]. The spectral singularity in the transmission and reflection is inherently associated with a laser. The top and bottom surfaces of the metafilm provide the required feedback mechanism for lasing. This feedback mechanism is observable from the Fabry Pérot effect shown in Fig. 2. The wavelength at which the lasing occurs is associated with the exceptional point that will be discussed in the next section. Here, the spectral singularity is manifested as dissipation-induced lasing or super scattering. Unlike the zero-bandwidth resonance in the longitudinal $P T$ -symmetric structures [39–41], the super scattering in our geometry with the transverse $P T$ -symmetry has a finite bandwidth which is practical and opens up various potential applications.

4. Loss-induced lasing and gain-induced absorption

Our numerical studies indicate that the super scattering occurs in the vicinity of an exceptional point (EP). In the absence of substrate dissipation, the metafilm satisfies $P T$ symmetry with balanced gain and loss arranged in the anti-symmetric distribution of the imaginary part of the permittivity. The two eigenstates are forward and backward waves. Their constructive and destructive interferences modify transmission and reflection, as well as lasing and absorption. An exceptional point is characterized by coalescence of both eigenvalues and associated eigen-vectors, and thus the system reduces to one dimension [36]. Figure 4 depicts eigenvalues and associated eigenfunctions versus wavelength in the neighborhood of the EP indicated by the bifurcation point λ = 8.868 μm where both eigenvalue and eigenfunction coalescence.

Fig. 4 Eigenvalues and associated eigenfunctions versus wavelength near the exceptional point. (a) Magnitude of the eigenvalues of the transfer matrix and (b) the corresponding phase. (c) Real and (d) imaginary parts of the left eigenfunctions. The mesh thickness d = 1.5 μm. The relative permittivity of the dimers is given by ε = 3.6(1 ± i0.06).

Download Full Size | PDF

The presence of such an EP can have a dramatic effect on the EM property of the system. Around this point, lasing can be induced (1) by adding the substrate dissipation as discussed in Sec. 3 (see Fig. 3), or (2) by increasing the loss of the dimers without adding the substrate dissipation. The later effect is illustrated in Fig. 5 for the transmission and reflection coefficients at different losses (δ_l) of the dimers with a fixed gain. The transmission and reflection increase when increasing the loss and reach a maximum at δ_l = 0.43 where the lasing occurs. Further increasing the loss reduces the transmission and reflection. Hence, near the EP increasing the loss of dimers can give rise to an enhanced scattering. A similar phenomenon of loss-induced lasing in a different system was recently demonstrated by using coupled ring resonators operating in the vicinity of an exceptional point [44].

Fig. 5 Transmission (solid curves) and reflection (dashed curves) vs. wavelength near the exceptional point for different losses of the dimers with a fixed gain ε_g = 3.6(1 − i0.06) in the absence of the substrate dissipation. The relative permittivity of the loss elements is ε_l = 3.6(1 + i0.06 + iδ_l). The blue curves δ_l = 0 correspond to the balanced loss and gain where the lasing does not occur. The transmission and reflection increase with the increase of δ_l, reach a maximum at δ_l = 0.43, and then reduce. The mesh thickness d = 1.5 μm.

Download Full Size | PDF

The corresponding poles of scattering matrix in a complex wavelength plane are shown in Fig. 6 for different losses of the dimers. Increasing the loss moves the pole up toward the real axis. Lasing occurs when the pole reaches the real axis ℑ(λ) = 0. In a similar vein, we expect gain-induced absorption to occur in our system. Shown in Fig. 7 is the control of infrared absorption around the EP by varying gain of the dimers with a fixed loss. When the gain (δ_g) increases, both absorption strength and bandwidth increase. Numerical study shows that the maximum absorption can be achieved is about 50% when δ_g = 0.5. The exceptional point gives rise to counterintuitive electromagnetic properties. Away from the EP, negative absorption regions (amplification) are observed in Fig. 7 as the result of the net gain.

Fig. 6 Movement of the pole of the scattering matrix in a complex wavelength plane (x-axis: real λ; y-axis: imaginary λ) when increasing the loss of the dimers with a fixed gain. The arrow indicates the direction of the increase: δ_l = 0.1,0.15,0.2,0.25,0.3,0.35,0.4, and 0.43. The mesh thickness d = 1.5 μm. When δ_l = 0.43, ℑ(λ) = 0 (lasing occurs), corresponding to the highest peak in Fig. 5.

Download Full Size | PDF

Fig. 7 Gain controlled infrared absorption in the vicinity of the exceptional point with a fixed loss ε_l = 3.6(1 + i0.06) of the dimers in the absence of the substrate dissipation. The permittivity of the gain is given by ε_g = 3.6(1 − i0.06 − iδ_g). The mesh thickness d = 1.5 μm. Both absorption peak and bandwidth increase when increasing the gain. The maximum absorption is about 50% at the δ_g = 0.5. Further increasing δ_g, the absorption peak reduces but the bandwidth continues to increase. The negative absorption represents the regions of amplification.

Download Full Size | PDF

5. Radiative plasmonic mode

In Sec. 3, we have shown that without the substrate dissipation, the sum of transmittance and reflectance remains one and the net amplification is zero in the output signals. The periodic dimers provides a array of twin resonators. Incident wave with the electric field parallel to the shorter edge of the rectangular holes can excite the localized cavity mode inside the holes. The $P T$ -synthetic design sustains a localized gain-type plasmon and loss-type plasmon within the unit cell. In this section, we will show that the substrate dissipation can modify the waveguide mode of the substrate and transform the dark mode of the substrate into the radiative mode. Hence, the dissipation provides a mean to couple light out of the system, and the $P T$ -synthetic metafilm behaves as a planar source of coherent radiation.

Subwavelength hole array in the metallic substrate supports surface plasmon resonances. Figure 8 shows the real and the imaginary parts of the normalized mode propagation constant (β/k₀, k₀ = 2π/λ) versus wavelength with and without the substrate dissipation. The dark mode in the absence of the dissipation has a complex propagation constant (the blue dashed curves in Fig. 8) with the effective mode index greater than one, and thus cannot be accessed from the free space (FS). Therefore, the photons are trapped in the dark mode. When adding the substrate dissipation, the mode propagation constant becomes real and is less than one (the red solid curves in Fig. 8). This mode can be excited by a FS plane wave. Consequently, when this mode is lased it will radiate into the far field. Hence, the substrate dissipation modifies the mode property of the metafilm. The dark mode of the substrate in the absence of the dissipation becomes the radiative mode, which can couple to the external field, in the presence of the dissipation. This radiative branch can be excited by a FS plane wave as further illustrated in Fig. 9 showing the transmission versus wavelength for the different (polar) angles of incidence. The reflectance peaks (not shown) coincide with the transmission peaks. The wavelength of the transmission peaks in Fig. 9 and the corresponding propagation constant of the excited modes are marked by the golden stars in Fig. 8(a). The fact that these golden stars coincide with the radiative branch indicates that the super scattering is the result of lasing to the radiative plasmonic mode by the FS excitation beam. Thus, the substrate dissipation provides a mean to couple the radiation of the $P T$ -synthetic metafilm into the far field. Upon the supper scattering, the transmitted (reflected) electric field experiences a phase shift as shown in Fig. 9(b), which is a typical behavior at the resonance. In the absence of the dissipation, the photons are trapped in the dark modes that cannot radiate without a proper coupling mechanism. The substrate dissipation breaks the $P T$ symmetry and transforms the dark mode into the radiative mode which can couple the light out of the metafilm and releases the trapped photons. The EP-related phenomena, such as resonant frequency, are sensitive to the parameters of the systems. Therefore, the exceptional point offers a preferred mechanism for low-threshold or fast switch, which can be applied to ultra-sensitive sensor to detect small variation of the parameter. Shown in Fig. 10 is the transmission and reflection of the normally incident wave at different thicknesses of the metafilm. Thus, a small variation in the film thickness can tune the resonant scattering frequency or the lasing frequency.

Fig. 8 Mode property of the metafilm. (a) Real and (b) imaginary parts of the normalized propagation constant versus wavelength without (blue dashed curves) and with (red solid curves) the substrate dissipation. The golden starts in (a) correlate to the free space excitation in Fig. 9. The corresponding imaginary part is zero. The arrow indicates the direction of increasing (polar) angle. The mesh thickness d = 1.5 μm. The relative permittivity of the dimers ε = 3.6(1 ± i0.06).

Download Full Size | PDF

Fig. 9 (a) Transmittance of a p-polarized wave and (b) the corresponding phase of the electric field for different (polar) angles of incidence from 0° to 40°. The parameters are the same as those in Fig. 3. The transmission maximums correspond to the excitation of the radiative modes, which are correlated to the golden stars in Fig. 8(a).

Download Full Size | PDF

Fig. 10 Transmittance (blue solid curves) and reflectance (red dashed curves) vs. wavelength when the beam is normally incident on the metafilm of thickness (a) d = 1.91 μm, (b) d = 1.96 μm. The lasing frequency or the resonant scattering frequency can be tuned by varying the thickness of the metafilm. The relative permittivity of the dimers is given by ε = 3.6(1 ± i0.056).

Download Full Size | PDF

6. Summary

In summary we have shown that manipulating the imaginary part of permittivity can lead to novel physical phenomena such as the loss-induced lasing and the gain-induced absorption in the $P T$ -synthetic plasmonic metafilm near the exceptional point. The effect of introducing gain into materials is more than loss compensation. The interplay between loss and gain can transform the $P T$ -synthetic metafilm into a laser, absorber, or laser absorber switch. The gain medium can be fluorescent materials, quantum dots/wells, and semiconductors. The loss-induced lasing and the gain-induced absorption appear to be general characters for structures with separate loss and gain domains where the lasing (or absorbing) mode can avoid to occupy the regions with the largest loss (or gain), reducing the effective loss (or gain) experienced by the mode [58]. The loss-induced lasing can be observed in traditional distributed feedback structures (fiber gratings) without resorting to exceptional points where the loss provides a feedback mechanism for lasing [58]. The super scattering shown here has a finite bandwidth that increases the possibility of detecting the spectral singularity resonance. Our result implies that the scattering characteristics of an optical system can be controlled by background dissipation. Adding substrate dissipation is equivalent to introducing loss into a cavity. It provides a new strategy to control the phase of $P T$ symmetry by using background loss and to control the radiation of a cavity by using cavity dissipation. Our numerical effort is one step closer to practical implementations of $P T$ -synthetic materials. The π-phase jump of the electric field at the super scattering can be used to design a two-state system and light modulation using the concept of $P T$ symmetry. The narrow band feature may find applications in notch filter, sensitive detector, and optical switch.

Appendix

In this appendix we provide numerical results to verify the claims in Sec. 2. Figure 11 shows the magnitudes of the different diffraction orders and polarizations in the transmittance and reflectance of the normal incident p-polarized wave. In the simulation, each direction (x or y) has ±3 Fourier components, resulted in a total of 49 diffraction orders in both transmission and reflection directions and each diffraction order has two polarizations. Among all the diffraction beams, the most significant component is the zero-order diffraction as shown in Fig. 11(a) for p-polarization (p₀₀-component) and Fig. 11(b) for s-polarization (s₀₀-component). The power in the main cross polarization (s₀₀) is three-order magnitude smaller than that of the main p₀₀₋component. In Fig. 11(a) the transmission and reflection powers in the p₀₀-component almost satisfies energy conservation, indicating that the higher-order scattering and cross polarization can be ignored. The similarity between Fig. 11(b) and Fig. 11(d) further proves that the energy scattered out of the main diffraction is negligible. It is worth mentioning that although the higher order and cross-polarization terms can be neglected, this structure cannot be described by the effective medium theory because of spatial dispersion, i.e. the effective refractive index changes with the angle of incidence. This effect is implied in Fig. 9. Therefore, the structure cannot be characterized by a unique effective refractive index. The effective medium theory usually applies to non-resonant structures with the period much smaller than the operating wavelength. Figure 12 shows numerical verifications of the various conditions given in Sec. 2. These numerical confirmations are necessary to ensure self-consistency of the algorithm when neglecting the higher-order diffraction and cross polarization terms.

Fig. 11 Transmittance (blue solid curves) and reflectance (red dashed curves) vs. wavelength for different diffraction orders. (a) Zero-order of p-polarized component (main diffraction). (b) Zero-order of s-polarized component (main cross-polarization). (c) The (11)-order scattering including both polarizations. (d) All the scattering orders including both polarizations minus the main diffraction (p₀₀-component). Simulation parameters are the same as those in Fig. 2.

Download Full Size | PDF

Fig. 12 Numerical validations of the mathematical conditions. (a) Bi-orthogonality Eq. 12. (b) Identity det(M) = 1 (blue solid line) and $| η_{+}^{t} η_{-}^{t} | = 1$ (red dashed line). Relationships (c) S₂₁ = S₁₂ or |S₂₁ − S₁₂| = 0, and (d) M₁₂ = −M₂₁ or |M₁₂ + M₂₁| = 0. Simulation parameters are the same as those in Fig. 4.

Download Full Size | PDF

Acknowledgments

The author acknowledges the support from K. Boulais and J. Solka. This project is funded by In-House Applied Research program at NSWC Dahlgren and Office of Naval Research.

References and links

1. C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 805243 (1998). [CrossRef]

2. S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008). [CrossRef] [PubMed]

3. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43, 265305 (2010). [CrossRef]

4. H. F. Jones, “Analytic results for a PT-symmetric optical structure,” J. Phys. A 45, 135306 (2012). [CrossRef]

5. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT Symmetry Breaking in Polarization Space with Terahertz Metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014). [CrossRef] [PubMed]

6. H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014). [CrossRef]

7. H. Benisty, A. Degiron, A. Lupu, A. D. Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Léronde, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19, 18004 (2011). [CrossRef] [PubMed]

8. J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015). [CrossRef]

9. S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013). [CrossRef]

10. X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nature Mater. 12, 175 (2013). [CrossRef]

11. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333729 (2011). [CrossRef] [PubMed]

12. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108 (2013). [CrossRef]

13. S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A 44, 485302 (2011). [CrossRef]

14. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-Symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011). [CrossRef] [PubMed]

15. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192 (2010). [CrossRef]

16. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Paritytime synthetic photonic lattices,” Nature 488, 167 (2012). [CrossRef] [PubMed]

17. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010). [CrossRef] [PubMed]

18. S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B 86, 165103 (2012). [CrossRef]

19. Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112, 143903 (2014). [CrossRef] [PubMed]

20. T. S. Luk, S. Campione, I. Kim, S. Feng, Y. C. Jun, S. Liu, J. B. Wright, I. Brener, P. B. Catrysse, S. Fan, and M. B. Sinclair, “Directional perfect absorption using deep subwavelength low-permittivity films,” Phys. Rev. B 90, 085411 (2014). [CrossRef]

21. S. Yu, X. Piao, J. Hong, and N. Park, “Progress toward high-Q perfect absorption: A Fano antilaser,” Phys. Rev. A 92, 011802 (2015). [CrossRef]

22. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef] [PubMed]

23. S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009). [CrossRef] [PubMed]

24. R. Fleury, D. L. Sounas, and A. Alú, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014). [CrossRef] [PubMed]

25. S. Savoia, G. Castaldi, Vincenzo Galdi, Andrea Alú, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014). [CrossRef]

26. A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010). [CrossRef]

27. A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 21651 (2013). [CrossRef] [PubMed]

28. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010). [CrossRef]

29. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nature Commun. 54034 (2014). [CrossRef]

30. Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011). [CrossRef] [PubMed]

31. G. Castaldi, S. Savoia, V. Galdi, A. Alú, and N. Engheta, “PT Metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013). [CrossRef] [PubMed]

32. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760 (2014). [CrossRef] [PubMed]

33. I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009). [CrossRef]

34. I. Rotter, “Branch points in the complex plane and geometric phases,” Phys. Rev. E 65, 026217 (2002). [CrossRef]

35. W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929 (2000). [CrossRef]

36. W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012). [CrossRef]

37. C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental Observation of the Topological Structure of Exceptional Points,” Phys. Rev. Lett. 86, 787 (2001). [CrossRef] [PubMed]

38. S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89032909 (2014). [CrossRef]

39. A. Mostafazadeh, “Optical spectral singularities as threshold resonances,” Phys. Rev. A 83, 045801 (2011). [CrossRef]

40. A. Mostafazadeh, “Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies,” Phys. Rev. Lett. 102, 220402 (2009). [CrossRef] [PubMed]

41. R. Aalipour, “Optical spectral singularities as zero-width resonance frequencies of a Fabry-Perot resonator,” Phys. Rev. A 90, 013820 (2014). [CrossRef]

42. S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010). [CrossRef]

43. S. Longhi, “Optical Realization of Relativistic Non-Hermitian Quantum Mechanics,” Phys. Rev. Lett. 105, 013903 (2010). [CrossRef] [PubMed]

44. B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014). [CrossRef] [PubMed]

45. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014). [CrossRef] [PubMed]

46. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time symmetric microring lasers,” Science 346, 975 (2014). [CrossRef] [PubMed]

47. M. Kang, H.-X. Cui, T.-F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89, 065801 (2014). [CrossRef]

48. D. J. Bergman and M.I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett. 90, 027402 (2003). [CrossRef] [PubMed]

49. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nature Photon. 2, 351 (2008). [CrossRef]

50. E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17, 8548 (2009). [CrossRef] [PubMed]

51. F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. Hooft, and M. P. van Exter, “Surface Plasmon Lasing Observed in Metal Hole Arrays,” Phys. Rev. Lett. 110, 206802 (2013). [CrossRef] [PubMed]

52. S. Feng, “Loss-Induced Omnidirectional Bending to the Normal in ε-Near-Zero Metamaterials,” Phys. Rev. Lett. 108, 193904 (2012). [CrossRef]

53. L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 101, 241101 (2012). [CrossRef]

54. V. Y. Fedorov and T. Nakajima, “All-angle collimation of incident light in μ-near-zero metamaterials,” Opt. Express 21, 27789 (2013). [CrossRef]

55. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

56. S. Feng, “Dissipation-Induced Super Scattering and Lasing PT-Spaser,” http://arxiv.org/abs/1503.00188.

57. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811 (1981). [CrossRef]

58. S. Longhi and G. D. Valle, “Loss-induced lasing: new findings in laser theory?” http://arxiv.org/abs/1505.03028.

Loss-induced super scattering and gain-induced absorption

Abstract

1. Introduction

2. Theoretical approach

3. Dissipation-induced super scattering

4. Loss-induced lasing and gain-induced absorption

5. Radiative plasmonic mode

6. Summary

Appendix

Acknowledgments

References and links

Cited By

Figures (12)

Equations (13)

Optics Express