Broadband near total light absorption in non-PT-symmetric waveguide-cavity systems

We introduce broadband waveguide absorbers with near unity absorption. More specifically, we propose a compact non-parity-time-symmetric perfect absorber unit cell, consisting of two metal-dielectric-metal (MDM) stub resonators with unbalanced gain and loss side-coupled to a MDM waveguide, based on unidirectional reflectionlessness at exceptional points. With proper design, light can transport through the perfect absorber unit cell with reflection close to zero in a broad wavelength range. By cascading multiple unit cell structures, the overall absorption spectra are essentially the superposition of the absorption spectra of the individual perfect absorber unit cells, and absorption of ∼ 100% is supported in a wide range of frequencies. c © 2016 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (130.2790) Guided waves; (230.7370) Waveguides; (230.4555) Coupled

Broadband near-perfect absorption can be realized when broadband reflectionless light propagation is achieved in the absorber structures.In the past few years, it has been proposed using parity-time (PT) symmetric optical structures to attain broadband unidirectional light reflectionlessness at exceptional points by careful modulation of the refractive index profile [15][16][17][18][19][20][21][22].However, because gain and loss compensate each other, the transmission in the classical gain/loss balanced PT-symmetric grating structures is unity when light is reflectionless from one side at the exceptional points based on the generalized conservation relation [15,17], and thus, the absorption in general is zero [22].Unidirectional light reflectionlessness can also be attained in non-PT-symmetric structures with unbalanced gain and loss (or even without gain) [23][24][25][26][27][28][29].This is due to the fact that exceptional points exist in a larger family of non-Hermitian Hamiltonians [23].Nevertheless, most of these non-PT-symmetric structures attain unidirectional reflectionlessness only at a specific frequency [24][25][26][27].Although Yang et al. demonstrated broadband unidirectional reflectionless light transport at the exceptional point in a periodic ternary layered structure consisting of lossy and lossless dielectrics [28], large-size periodic structures are not easy to implement in dense integrated optical chips.Most recently, Horsley et al. showed theoretically that broadband unidirectional invisibility can be realized in planar inhomogeneous dielectric media, in which the spatial distributions of the real and imaginary parts of the dielectric permittivity are related by Kramers-Kronig relations [29].In practice, realizing such a complicated permittivity profile is challenging.
In this paper, we introduce a non-PT-symmetric waveguide-cavity system, which consists of two resonators with gain and loss side-coupled to a waveguide.Plasmonic waveguides have shown the potential to guide and manipulate light at deep subwavelength scales [30].We propose a compact non-periodic perfect absorber unit cell, based on the unidirectional reflectionless propagation at exceptional points.Several previously proposed non-PTsymmetric optical systems support unidirectional reflectionless propagation only at a specific frequency [24][25][26][27].Here we present an analysis of the broadband unidirectional near-zero reflection propagation in the proposed absorber unit cell structure.We find that, with proper design, the perfect absorption condition coincides with the broadband near-zero reflection condition, and show that light can transport through the proposed compact non-PT-symmetric perfect absorber unit cells with near-zero reflection in a broad wavelength range.We also show that, by properly cascading the unit cell structures, broadband near-total absorption can be realized.
The remainder of the paper is organized as follows.In Section 2, we employ temporal coupled mode theory (CMT) to account for the behavior of the proposed waveguide-cavity system.In Subsection 3.1, we use this theory to design a perfect absorber unit cell structure based on a non-PT-symmetric plasmonic waveguide system.The realized broadband near-zero reflection is then investigated in Subsection 3.2.In Subsection 3.3, we design a broadband waveguide absorber with near-unity absorption by cascading the perfect absorber unit cell structures designed in Subsection 3.1.Finally, our conclusions are summarized in Section 4. The schematic of a perfect absorber unit cell, which consists of two single mode optical resonators side coupled to waveguide with a single propagating mode, is shown in Fig. 1.Recently, waveguide-cavity structures were used for the realization of on-chip optical diodes in PT-symmetric optical systems [31,32].Light transport in the structure of Fig. 1 can be described using CMT [33,34].For harmonic time dependence of e − jωt , the time evolution of the mode amplitudes a and b of resonators A (left) and B (right), respectively, can be described by the following equations:

Theory and design method
where ω 01 and ω 02 are the resonance frequencies of resonators A and B, respectively.In addition, κ 1 , κ 2 are the input coupling coefficients into resonator A associated with the forward and backward propagating modes in the waveguide, respectively, and κ 3 , κ 4 are similarly defined for resonator B. 1/τ i , i = 1, 2, are the decay rates of the resonator mode amplitudes due to the power escape through the waveguide, 1/τ 0i , i = 1, 2, are the decay (growth) rates due to the internal loss (gain) in the resonators, and L is the distance between the two resonators.S + 1 and S + 2 are the amplitudes of the incoming waves to resonator A from the forward and backward directions, respectively, while S − 1 and S − 2 are the amplitudes of the outgoing waves from resonator A. S + 3 , S + 4 and S − 3 , S − 4 are similarly defined for resonator B. Based on power conservation, the rate of change of the energy in the resonator mode must be equal to the difference between the incoming and outgoing power, and the outgoing waves of the resonators are [33,34] where S +3 = S −2 e −γ L , S +2 = S −3 e −γ L , and γ = α + j β is the complex propagation constant of the propagating mode in the waveguide.Due to the perpendicular symmetry plane of the resonators, we have κ 1 = κ 2 , and κ 3 = κ 4 .Note that κ 1 = 1 τ 1 e jθ 1 , and κ 3 = 1 τ 2 e jθ 2 , where θ i , i = 1, 2, are the phases of the coupling coefficients [33,34].Solving the above coupled-mode equations, the reflection and transmission spectra can be expressed as where , 2, t is the complex transmission coefficient, while r f and r b are the complex reflection coefficients for light incident from the left (forward direction) and right (backward direction), respectively.Thus, when the system is lossless ( 1 τ 01 = 1 τ 02 = 0 and α = 0), the reflection coefficients for light incident in the forward and backward directions are the same (R f = R b ).On the contrary, when the system is lossy, while the transmission coefficients in the forward and backward directions are the same, the reflection coefficients in the forward and backward directions are in general different.
Here we assume that the resonant frequencies ω 01 , ω 02 , and decay rates 1 τ 1 , 1 τ 2 are the same for the two resonators, that is ω 01 = ω 02 = ω 0 , and 1 The reason that we make this assumption will become clear in Subsection 3.2.In order to achieve complete absorption in the structure of Fig. 1, both the transmission and reflection have to be zero.Based on Eq. ( 7), when light is incident on the structure in the forward direction, the transmission T is zero, if the following condition is satisfied In addition, based on Eq. ( 5), the on resonance reflection Here, we choose 1 τ 02 = 0.If the waveguide is lossless (γ = j β), using the above two equations we obtain cos If the waveguide is lossy (γ = α + j β), combining Eqs. ( 8) and ( 9) leads to In both Eqs. ( 10) and ( 11), the decay rate 1 τ 01 due to internal loss in resonator A is positive, which requires that the left resonator (resonator A) be filled with an absorptive material.The absorption for light incident from the forward direction for the system of Fig. 1 (10) [Eq.(11)] is satisfied in the lossless (lossy) waveguide case, we obtain perfect absorption for light incident from the forward direction for the system of Fig. 1.We also note that the condition cos(2 βL) = −1 is satisfied when the distance between the two resonators L is equal to odd multiples of a quarter of the guide wavelength [L = (2n + 1)λ g /4].

Results
In this section, we use a non-PT-symmetric plasmonic-waveguide cavity system, consisting of two MDM stub resonators side coupled to a MDM waveguide [Fig.2(a)], to realize perfect absorption based on the theory discussed in the previous section.Among different plasmonic waveguiding structures, MDM plasmonic waveguides, which are the optical analogue of microwave two conductor transmission lines [13], are of particular interest [35][36][37][38][39][40], because they support modes with deep subwavelength scale over a very wide range of frequencies extending from DC to visible [41], and are relatively easy to fabricate [42].
We use a two-dimensional finite-difference frequency-domain (FDFD) method [43] to numerically calculate the transmission and reflection coefficients in the plasmonic waveguide structure.This method allows us to directly use experimental data for the frequency-dependent dielectric constant of metals such as silver [44], including both the real and imaginary parts, with no approximation.Perfectly matched layer (PML) absorbing boundary conditions are used at all boundaries of the simulation domain [45].

Perfect absorber unit cell
Figure 2(b) shows the transmission and reflection spectra for the structure of Fig. 2(a) calculated for light incident from both the forward and backward directions using full-wave FDFD simulations (solid lines) and CMT (circles) for w = 50 nm, w 1 = 10 nm, w 2 = 25 nm, h 1 = 67.5 nm, and h 2 = 53 nm.We observe that there is very good agreement between the CMT results and the exact results.The distance between the two stubs is chosen to be L = 285 nm, so that cos(2 βL) = −1 [Eq.(11)].The dielectric is air, and the metal is silver.The left and right stubs are filled with silicon dioxide doped with CdSe quantum dots ( A = 4.0804 − j0.6) [13,46] and InGaAsP ( B = 11.38 + j0.41) [35,47], respectively.The stub dimensions are chosen so that their resonant frequencies are equal ω 01 = ω 02 = ω 0 = 2π193.4THz (λ 0 = 1.55μm), and their decay rates due to power escape through the waveguide are also equal 1 In the presence of pumping, optical gain and loss can be achieved using InGaAsP and silicon dioxide doped with CdSe quantum dots, respectively [13,35].The imaginary part of the dielectric constant of the gain material filling the right stub (0.41) is chosen to compensate the material loss in the metal ( 1 τ 02 = 0), so that the right stub behaves essentially as a lossless stub [35].In addition, the imaginary part of the dielectric constant of the absorptive material filling the left stub (−0.6) is chosen to satisfy the condition 1  τ 01 = e 2α L +1 τ [Eq.(11)].Such gain and loss coefficients are within the limits of currently experimentally achievable values [13,35,46,47].The total absorption in the forward direction can be observed in the magnetic field distributions.When the waveguide mode is incident from the right (backward direction), there is no transmission, and the incident and reflected fields form a strong interference pattern [Figs.2(d) and 2(f)].On the other hand, when the waveguide mode is incident from the left (forward direction), there is hardly any reflection and transmission [Figs.2(c) and 2(e)].
In fact, at the resonant frequency ( f 0 = 193.4THz), the right stub is lossless ( 1 τ 02 = 0), so that it behaves as a perfect reflector.It has a role similar to the one of perfect mirrors or metallic ground planes in classical layered absorbers, such as Salisbury screens, which are based on critical coupling [3,[8][9][10].For light incident from the left on resonance, the wave directly reflected from the left stub, and the decaying amplitude into the backward direction of the resonant cavity fields of the left stub resonator, as well as of the resonator formed between the two stubs interfere destructively, so that the reflection R f reduces to zero, and the system becomes a perfect absorber.
We note here that the optical properties of our proposed two-port system can also be described by the optical scattering matrix of the system which relates the amplitudes of the incoming and the outgoing waves [27].In the presence of loss, such a system is analogous to open quantum systems which are characterized by complex non-Hermitian Hamiltonians [18,23,24,27], and there is a close analogy between optical scattering matrices and Hamiltonian matrices [15,16,18,24,27].When the optical scattering matrix eigenvalues coalesce into a single eigenvalue, and the eigenstates coalesce into a single eigenstate, the system exhibits an exceptional point [27].This leads to unidirectional reflectionless propagation in either the forward or the backward direction.Thus, in the plasmonic system of Fig. 2 the on resonance transmission is zero, and the system therefore exhibits unidirectional total absorption Fig. 3. Phase spectra of the reflection coefficients in the forward (r f , black) and backward (r b , red) directions for the structure of Fig. 2(a).All parameters are as in Fig. 2

(b). (b)
Contrast ratio spectra for the structure of Fig. 2(a).All parameters are as in Fig. 2(b).
In Fig. 3(a), we observe that the phase of the reflection coefficient in the forward direction r f undergoes an abrupt π jump, when the frequency is crossing over the exceptional point on resonance, which actually resembles the phase transition from the PT-symmetric phase to the PT broken phase in optical PT-symmetric systems [17,18,24,26,27].In contrast, the phase of the reflection coefficient r b in the backward direction in general varies smoothly with frequency [26,27].However, Fig. 3(a) shows that the phase of the reflection coefficient in the backward direction r b undergoes an abrupt jump.This is due to the fact that the right stub acts as a perfect reflector for light coming from the right on resonance ( f = f 0 = 193.4THz) [Figs.2(d) and 2(f)].When resonance trapping occurs for reflection, the corresponding delay time, which is proportional to the time that the light is trapped in the stub, diverges.Here the delay time is defined as τ r = dθ r dω , where θ r is the argument of the reflection coefficient [48].

Broadband unidirectional reflectionlessness of perfect absorber unit cell
Broadband total absorption becomes possible only when broadband reflectionless light propagation can be achieved in the absorber structures.However, non-PT-symmetric optical systems typically exhibit unidirectional reflectionless propagation only within a very narrow wavelength range around the exceptional point [24][25][26][27].This is especially true for nonperiodic non-PT-symmetric systems, which are relatively easy to fabricate and more compact.This, however, limits potential applications of compact non-PT-symmetric optical systems as broadband perfect absorbers.In addition, an optical medium that satisfies the PT-symmetry condition can behave as a coherent perfect absorber, fully absorbing incoming coherent waves with appropriate amplitudes and phases injected from both sides of the medium [2].
The underlying mechanism of such a PT-symmetric coherent perfect absorber is not based on the unidirectional reflectionlessness at exceptional points.However, such PT-symmetric coherent perfect absorbers also exhibit total absorption only at specific frequencies, and are not reflectionless in a broad wavelength range [2].
In this subsection, we investigate the broadband unidirectional near-zero reflection in our proposed non-PT-symmetric perfect absorber unit cell structure [Fig.2(a)], which was introduced in the previous section.We consider the general case without making any assumptions, and derive the general conditions for broadband near-zero reflection in the forward direction.From Eq. ( 5), zero reflection in the forward direction (R f = 0) leads to Broadband zero reflection in the forward direction (R f = 0) requires the solution of Eq. ( 12) to be weakly dependent on frequency.To meet this requirement, we set the terms in Eq. ( 12) which include ω equal to zero and obtain sin(2 βL) = 0, and cos(2 βL) Since 1 τ 1 and 1 τ 2 , which are the decay rates of the field in the resonators due to the power escape through the waveguide, are both positive, we must have cos(2 βL) = −1, and we therefore obtain Substituting cos(2 βL) = −1 and Eq. ( 14) into Eq.( 12), we obtain When αL is small, e 2α L ∼ 1, and Eqs. ( 14), ( 15) become These are the general conditions for broadband near-zero reflection in the forward direction (R f = 0) for our proposed waveguide-cavity system.Thus, to achieve broadband near-zero reflection, the conditions 1 τ 1 = 1 τ 2 and ω 01 = ω 02 have to be satisfied.This is the reason that we chose 1 τ 1 = 1 τ 2 = 1 τ and ω 01 = ω 02 = ω 0 in Section 2. We note that for frequencies far away from the resonant frequency ω 0 , the denominator of Eq. ( 5) is proportional to (ω − ω 0 ) 4  (note that ω 01 = ω 02 = ω 0 ), whereas the numerator is only proportional to (ω − ω 0 ) 2 .Thus, the reflection R f remains small even when the frequency is not in the vicinity of the resonant frequency ω 0 , and the system exhibits broadband near-zero reflection.We also note that for our proposed perfect absorber unit cell structure, the decay rate 1 τ 02 was chosen zero [Eqs.( 10) and ( 11)], and thus the broadband unidirectional near-zero reflection condition [Eq.( 16)] coincides with the perfect absorption condition [Eq.(11)].
In Fig. 2(b) we indeed observe that our proposed non-PT-symmetric perfect absorber structure shown in Fig. 2(a) exhibits near-zero reflection in the forward direction over a broad wavelength range (black).For the range of frequencies shown, the maximum reflection is less than 2%.In addition, the contrast ratio between the forward and backward reflection, defined , as a function of frequency for the structure of Fig. 2(b) is shown in Fig. 3(b).The contrast ratio between the forward and backward reflection is higher than 99% from 180.4 THz to 207.4 THz.In contrast to perfect mirrors used in the layer based perfect absorbers [3,9,10], the right stub in our proposed structure is not a perfect reflector off-resonance.This enables our structure to have the broadband near-zero reflection property discussed above.Taking advantage of the broadband near-zero reflection property of the proposed non-PTsymmetric perfect absorber unit cell of Fig. 2(a), we next cascade multiple such structures to obtain a broadband near-total absorber (Fig. 5).In particular, here we consider a structure with four unit cells, each consisting of two stubs.The geometrical parameters and imaginary part of the dielectric constant of the material filling each stub are chosen so that the absorption cell is compact and suitable for ultradense chip-scale integration.Finally, we showed that, by cascading multiple properly designed unit cells, we obtain a structure in which light of different wavelengths can be almost completely absorbed at different positions in the structure.For such a structure the overall absorption spectra are essentially the superposition of the absorption spectra of the individual perfect absorber unit cells, and thus broadband near-total light absorption can be realized.

System with broadband near-total absorption
Our results could be potentially important for developing a new generation of highly compact unidirectional integrated nanophotonics devices.We also note that our proposal is scalable.In addition, the general design principle in our structures can be extended to other multiple layer systems, such as microwave structures with unidirectional invisibility.
As final remarks, we note that our proposed MDM waveguide based absorber is more compact and exhibits near-total absorption in a wider wavelength range compared to other MDM absorbers based on optical Tamm states [14].In addition, the size of our proposed absorber could be further reduced by decreasing the guide wavelength through decreasing the width of the MDM waveguide [37].We also found that, based on coupled mode theory, decreasing the decay rate of the field in the resonators due to the power escape through the waveguide (by separating the resonators and the waveguide with a small distance instead of directly coupling them [35]) could further increase the bandwidth of unidirectional reflectionlessness of the proposed perfect absorber unit cell.In addition, we note that the proposed broadband near-total absorbers can be realized in similar plasmonic waveguidecavity systems based on other plasmonic two-conductor waveguides, such as three-dimensional plasmonic coaxial waveguides [55].The proposed broadband near-total absorbers can also be realized by other structures such as microring and photonic crystal cavities [56,57].In such lossless structures gain media are not required to implement the proposed waveguide-cavity systems, since there is no internal loss in the resonators.

Fig. 1 .
Fig. 1.Schematic of a waveguide side coupled to two resonators.Both resonators have a symmetry plane perpendicular to the waveguide.S + 1 and S + 2 are the amplitudes of the incoming waves to resonator A from the forward and backward directions, respectively; S − 1 and S − 2 are the amplitudes of the outgoing waves from resonator A. S + 3 , S + 4 and S − 3 , S − 4

Figure 2 (Fig. 2 .
Figure2(b) also shows the absorption spectra in the forward direction calculated using FDFD (green solid line).The numerical results of Fig.2(b)show that the structure of Fig.2(a), which includes loss and gain, is unidirectional reflectionless at f = 193.4THz (R f = 0, R b 0).More interestingly, the results confirm that the on resonance reflection in the backward direction is unity, as predicted by CMT [Eq.(6)].Thus, for light incident from the left the on resonance absorption is unity, while for light incident from the right the on resonance reflection is unity [Fig.2(b)].As expected by reciprocity, the on resonance transmission for light incident from both the forward and backward directions is zero.

Fig. 4 .
Fig. 4. (a) (a) Reflection spectra in the forward direction (R f ) as a function of the life time τ 01 for the structure of Fig. 2(a).All other parameters are as in Fig. 2(b).(b) Reflection spectra in the forward direction (R f ) as a function of the distance L for the structure of Fig. 2(a).All other parameters are as in Fig. 2(b).(c) Absorption spectra in the forward direction as a function of the life time τ 01 for the structure of Fig. 2(a).All other parameters are as in Fig. 2(b).(d) Absorption spectra in the forward direction as a function of the distance L for the structure of Fig. 2(a).All other parameters are as in Fig. 2(b).

Figure 4 (
Figure4(a) shows the reflection spectra in the forward direction as a function of the life time τ 01 for the structure of Fig.2(a).All other parameters are as in Fig.2(b).Note that 1τ 01 is the decay rate due to internal loss in the left stub resonator (resonator A).This decay rate1  τ 01 can be controlled by tuning the absorption coefficient of the material filling the left resonator.At f = f 0 = 193.4THz, when 1 τ 01 → 2 τ , or equivalently Im( A ) → −0.6 [note that this is equivalent to τ 01 → 6 (2/ω 0 ), where ω 0 2π = 193.4THz, and corresponds to the white dashed line in Fig.4(a)], the reflection in the forward direction R f of the proposed unit cell structure of Fig.2(a) goes

Figure 4 (
b) shows the reflection spectra in the forward direction as a function of the distance between the two resonators L for the structure of Fig.2(a).All other parameters are as in Fig.We observe that approximately zero reflection R f can be obtained for different L. However, broadband near-zero reflection can only be realized when L = 285 nm [white dashed line in Fig. 4(b)], which is obtained from cos(2 βL) = −1 at f = 193.4THz, and corresponds to a quarter of the guide wavelength (L = λ g /4).Perfect absorption in the forward direction is also obtained for L = 285 nm [Fig.4(d)].