Coherent perfect absorber and laser in purely imaginary conjugate metamaterials

Transcription

1 Coherent perfect absorber and laser in purel imaginar conjugate metamaterials Yangang Fu 1,, Yanan Cao 1, Steven A. Cummer 3, Yadong Xu 1, and Huanang Chen1, 1.College of Phsics, Optoelectronics and Energ, Soochow Universit, No.1 Shizi Street, Suzhou 156, China..Institute of Electromagnetics and Acoustics and Department of Electronic Science, Xiamen Universit, Xiamen 3615, China. 3.Department of Electrical and Computer Engineering, Duke Universit, Durham, North Carolina 778, USA Abstract: Conjugate metamaterials, in which the permittivit and the permeabilit are comple conjugates of each other, possess the elements of loss and gain simultaneousl. B emploing a conjugate metamaterial with a purel imaginar form, we propose a mechanism for realizing both coherent perfect absorber (CPA) and laser modes, which have been widel investigated in parit-time smmetric sstems. Moreover, the general conditions for obtaining CPA and laser modes, including obtaining them simultaneousl, are revealed b analzing the wave scattering properties of a slab made of purel imaginar conjugate metamaterials. Specificall, in a purel imaginar conjugate metamaterial slab with a sub-unit effective refractive inde, perfect absorption can be realized for the incident wave from air. I. Introduction Metamaterials [1, ] provide unprecedented new approaches to manipulate the electromagnetic (EM) wave propagation. With them, a number of amazing optical phenomena and devices have been proposed and well demonstrated in eperiments, such as the invisibilit cloak [3, 4], the perfect lens [5, 6], field rotators [7, 8], and illusion optics [9, 1]. All these cases rel on control of the real parts of both permittivit and permeabilit of metamaterials. On the other hand, the imaginar parts (i.e., the material gain or loss) also ehibit significant impact on the propagation behavior and characteristics of EM waves. For eample, b constructing balanced loss and gain in materials to meet the parit-time (PT) smmetr condition, i.e., n( ) n( ), man significant propagation effects of light have been revealed, such as unidirectional invisibilit [11, 1], coherent perfect absorption (CPA) [13-15], lasing [16, 17], and etraordinar nonlinear effects [18, 19]. Furthermore, b introducing the gain or loss acted as new freedoms to classif all possible metamaterials, the conventional two-dimensional plane could be epanded to three-dimensional space. Such operation greatl enriches the tpes of metamaterials, and most of their EM properties are unclear, deserving to further investigate. One group of interest are the so-called conjugate metamaterials (CMs) [] whose relative permittivit and permeabilit are comple conjugate with each other, i.e., i e i and e, where and are positive numbers and is the phase factor of the materials. Although the possess the elements of loss and gain simultaneousl, their refractive indees are strictl real, and in such media unattenuated propagation of EM waves can be obtained [1, ]. Moreover, the EM properties of CMs are largel dependent of the phase factor. It has been demonstrated in Ref. [] that for /, CMs have positive refractive inde; while for /, CMs have negative one. Such CMs might be used to serve as subwavelength-resolution lens with a perfect lens as the limiting case []. du@suda.edu.cn kenon@mu.edu.cn

2 Particularl, when, the corresponding CMs are purel imaginar conjugate metamaterials (PICMs), which are given b and. As the transition case [] from positive to negative refractive inde, PICMs offer interesting underling phsics to eplore. In this work, owing to comple conjugate propert, is set for PICMs to uncover / i i = significant new behavior of waves. For simplicit, is assumed. While if the are different ( ), similar wave propagations will still happen. We thoroughl stud EM wave behavior in PICMs, and find that both CPA and laser modes can be supported in a slab composed of PICMs. B analzing the wave scattering properties of a PICM slab, we deduce the general conditions for realizing CPA and laser modes, which are verified b numerical simulations. We also derive the conditions for obtaining CPA and laser modes simultaneousl. Furthermore, when the PICMs are provided with a sub-unit effective refractive inde, there is critical angle for wave impinging from air to them. In contrast to the traditional case, where total reflection happens for the incident angle beond the critical angle, a PICM slab can function as a perfect absorber beond the critical angle, absorbing the incident wave without an reflection and transmission. n II. Eigen modes analsis for CPA and laser in a PICM slab We first investigate wave scattering from a PICM slab in air, which is schematicall shown in Fig. 1(a). The blue and red arrows are the incoming and outgoing waves respectivel, and the black arrows are the forward and backward waves inside the slab. The EM parameters of PICMs are given as and ( ) for the transverse electric (TE) polarization (the electric field is along z direction). In this work, we will focus on the case of TE polarization. The similar results can be obtained for the transverse magnetic (TM) polarization (the magnetic field is along direction). Here the related parameters and i i z n are real numbers, as a result the effective refractive inde of PICMs is n, which is a real number. The EM parameters of PICMs contain gain and loss simultaneousl. Therefore, depending on which dominates in our PICM slab, an incident wave might induce quite intense transmission and reflection defined as laser modes, and as a result the injected signal (blue arrows) can be ignored (see schematic diagram in Fig. 1(b)). Or, if the loss dominates, the slab ma support coherent perfect absorber (CPA) modes, which might absorb incoming coherent waves without an outgoing waves (or reflections). The schematic diagram for obtaining CPA mode is displaed in Fig. 1(c). FIG. 1. (Color Online) (a) The general schematic diagram of wave scattering for the PICM slab in air. (b) and (c) The corresponding schematic diagrams of wave scattering for laser modes and coherent perfect absorber (CPA) modes, respectivel. (d) The schematic diagram of wave scattering for perfect absorber (PA) modes in semi-infinite spaces consisting of air and PICM medium, which are treated as the simplified cases of CPA modes. B analzing the wave scattering of laser modes and CPA modes (for details, see in Appendi A), the dispersion relationships of laser modes are epressed as,

3 icot( kd / ) (odd modes), (1) i tan( kd / ) (even modes), () and the corresponding dispersion relationships of CPA modes are written as, where k / k, k k icot( kd / ) (odd modes), (3) i tan( kd / ) (even modes), (4), k n k, and, is the propagation n = constant along direction, and and are the wavevectors along direction in air and PICMs, respectivel. Even modes are defined as smmetric modes, i.e., the field distribution in direction is smmetric, and odd modes are anti-smmetric modes, i.e., the field distribution in direction is anti-smmetric. In fact, for a single slab composed of PICMs ( ) or their analogs ( ), the above dispersion relationships are satisfied. Accordingl, CPA and laser modes can be perfectl ecited owing to the real solutions for in Eqs. (1)-(4). = / k k While for CMs with As a result, perfect ecitation for CPA and laser modes will not happen and the performance of CPA and laser modes will decrease (for more detailed discussions, please see in Appendi A). Furthermore, under some circumstances (we will eplain in the following part of, i.e., CMs with real parts, there are comple solutions for. n 1), the problem of two coherent incoming waves for CPA modes in a PICM slab could be treated as the case of a single incoming wave for perfect absorber (PA) modes in the semi-infinite spaces consisting of air and PICMs (see the schematic in Fig. 1(d), which is simplified from Fig. 1(c) b seeing the black dashed frames). The dispersion relationship for PA modes in the semi-infinite spaces is calculated as (for details, see in Appendi A), where k k, k i n k, and n n=1 k 1, k (5). From the above equations, we find that is a special value, as k k and the effective refractive inde of PICMs is 1 and thus equal to that in the air background medium. Therefore, in the following we will eplore the case of a PICM slab with in air, and then investigate the case of a PICM slab with n 1 in air. n 1 III. CPA and laser in PICMs with n 1 n 1 In this section, we will investigate the PICM slab with. Based on Eqs. (1)-(4), the corresponding dispersion relationships between the effective refractive inde n of PICM slab and the propagation constant are displaed in Figs. (a) and (b), which are the cases of laser modes and CPA modes respectivel. The red and blue curves are the corresponding even modes and odd modes. For n =1.5 as an eample (see Figs. (a) and (b)), we choose the propagation constants =.67k and =.577k to obtain laser mode and CPA mode respectivel. To verif the analtical results, we carr out numerical simulations b using COMSOL Multiphsics based finite element method. For instance, to match the tangential momentum of the laser mode, i.e., =.67k, the wave with incident angle 38.8 is striking from the left side on the PICM slab (see the white arrow in Fig. (c)), then the incoming wave oscillates inside the slab to accumulate energ, similar to the mechanism of laser modes in the cavit sstems with gain media. As a result, the amplified electric field

4 emits toward - and + directions b observing the field pattern and energ flu (black arrows) in Fig. (c), in which quite intense reflection and transmission take place in the left and right sides of the PICM slab. To ecite a CPA mode, the two incoming waves must be coherent with a specific amplitude and phase relationship at the boundaries (for details, see in Appendi A). With A and D as the comple amplitudes of the two incoming coherent waves, CPA requires either (smmetric mode) or A D (anti-smmetric mode). When the in-phase coherent waves ( A D ) with are incident from the left and right sides (see a pair of white arrows in Fig. (d)), the match one of the tangential wave numbers of CPA modes, i.e.,. As such, CPA is produced b the combination of destructive interference and dissipation in the PICM slab, which is well demonstrated b observing the field pattern and energ flues (black arrows) in Fig. (d). Therefore, both laser modes and CPA modes are supported in a PICM slab, if the tangential momenta are matched. A D =.577k 35. FIG.. (Color Online) (a) and (b) The corresponding dispersion relationships ( n vs ) for laser modes and CPA modes, respectivel, where the red and blue curves are the corresponding even modes and odd modes. (c) and (d) The simulated electric field patterns for a laser mode and a CPA mode, respectivel (for ). In all the cases, n 1, d and. n=1.5 =1 IV. CPA and laser in PICMs with n 1 As we know, total internal reflection should occur when light is incident from a medium with a higher refractive inde to a medium with a lower one. It turns out that the phsics are substantiall different for PICMs with lower refractive indees. We consider a PICM slab with n 1 placed in air schematicall shown in Fig. 3(a). In the following, we take n.5 for the PICM slab as an eample to reveal the underling phsics. Based on the general coefficients of transmission and reflection of a PICM slab (see the Eq. (S18) and Eq. (S19) in Appendi B), the corresponding relationships between transmission/reflection of the PICM slab and the incident angle are shown in a logarithmic scale in Fig. 3(b), where the red curve is the reflection ( R r ) and the blue dashed curve is the transmission ( T t ). From Fig. 3(b), we find that when the incident angle is near 18. or 7.6, the transmission and reflection are etremel large. Moreover, when the incident angle is near =5.6 or =39.3, the reflection is zero. In particular, for the incident angle with =5.6, the transmission is unit, which means that total transmission occurs. However, for the incident angle with 39.3, the transmission is almost zero, impling that "no transmission and no reflection" occurs.

5 FIG. 3. (Color Online) (a) The schematic diagram of a PICM slab placed in air. (b) The relationships between transmission\reflection of the PICM slab and the incident angle in a logarithmic scale. The working wavelength is and the thickness of slab is. For the slab, we set and..5i.5i 1 d To verif the above intriguing results, numerical simulations are performed b using COMSOL Multiphsics. For eample, when the wave with incident angle of impinges on the slab, the corresponding field pattern is shown in Fig. 4(a), with strong transmission and reflection. Likewise, for the wave with an incident angle of, the predicted strong transmission and reflection are seen in the field pattern in Fig. 4(b). The simulated field pattern for the incident wave with is shown in Fig. 4(c), where the incident wave passes through the slab without an reflection and with unit transmission. For the incident wave with, which is beond the critical angle c arcsin n 3, the corresponding field pattern is shown in Fig. 4(d), and it seems that the incident wave is bounded at the interface between air and PICM just like a surface wave, without an reflection and transmission FIG. 4. (Color Online) The simulated electric field patterns for the TE plane wave incident on the PICM slab with different incident angles. (a) 18., (b) 7.6, (c) 5.6, (d) To eplain the above intense and vanishing transmission and reflection, we emplo Eqs.(1)-(5) to displa the corresponding dispersion relationships ( n vs ) for laser modes and CPA modes in Figs. 5(a) and 5(b), where the red curve and blue curve are the corresponding even modes and odd modes and the green one is related to PA modes. For the case of n.5 shown in Figs. 5(a) and 5(b), the corresponding tangential propagation constants marked b the black dashed lines are.31k,.46k, and.63k, which are consistent with the incident angles 18., 7.6 and Therefore, the unusual transmission and reflection in Figs. 4(a) and 4(b) and Fig. 4(d) are related to the ecited laser modes and CPA mode, respectivel. In fact, from Figs. 5(a) and 5(b), laser and CPA modes can be ecited in the PICM slab with n 1, if the incident angle is less than. However, for larger than, the transmission will be tremendousl reduced, as c c

6 total internal reflection happens (e.g., see the transmission curve in Fig. 3(b)). As a result, the laser modes do not eist for >, as wave deca will take place in the PICM slab. c However, CPA modes do eist in the case of. For eample, when the two incoming waves are incident on the slab, the will be bounded at the left and right interfaces without an reflection. For PICMs possessing higher refractive indices, the bounded waves inside the slab with a fied thickness (here d ) do not interact with each other as their deca lengths are short. As a result (see Figs. 1(c) and 1(d)), the problem of two coherent incoming waves for CPA modes in a PICM slab will be treated as the case of a single incoming wave for perfect absorber (PA) modes in the semi-infinite spaces consisting of air and PICMs. Consequentl, PA modes, which can perfectl absorb incident wave, could be realized in the PICM slab b replacing the coherent waves with a single incoming wave, which has been demonstrated in Fig. 4(d). This phenomena is shown clearl in the dispersion relationships in Fig. 5(b), where the lowest blue curve of CPA modes in the case of is c coincident with the green curve of PA modes in the region of modes can be replaced b PA modes. For CPA modes. In such case, CPA modes can be realized in the PICM slab b the incoming coherent waves. For the single incident wave shining on the PICM slab, although there is no reflection, the transmitted wave will increase for the lower refractive indices of PICMs. Therefore, PA modes are not obtained in the PICM slab in this parameter regime. This is because for PICM slab with the lower refractive indices, the bound waves, which are from the two incoming coherent waves for realizing CPA mode, interact with each other inside due to longer decaed lengths. Furthermore, to clarif the behavior of the bounded wave inside the PICM slab, we analze the time-averaged energ flues along and directions inside the PICM slab, i.e., S Re[ H E ]/ and S Re[ H E ]/, which can effectivel reveal the energ flu n.5, impling that CPA n.5, marked in the gra region in Fig. 5(b), PA modes graduall deviate from z z distribution inside the slab. From the fields inside the PICM slab (for details, please see in the Appendi C), we find that S in the PICM slab. In particular, S k k = sin( ) / ( ) is sinusoidal distribution inside the PICM slab for c S= ap e / ( ) is decaed distribution for c c, and. We also displa the simulated time-averaged energ flues shown in Figs. 5(c) and 5(d), where the incident angles are =7.6 and c =39.3 c for laser mode and PA mode, respectivel. From the information in Figs. 5(c) and 5(d), inside the PICM slab, S ehibits the epected sinusoidal distribution for the ecited laser mode and decaed distribution for PA mode, but S in the PICM slab for both cases. The numerical results are thus consistent with the analtical results. Therefore, the bounded wave at the interface of PICM slab and air in Fig. 4(d) is absorbed, rather than a surface wave propagating along direction. Furthermore, total transmission in Fig. 4(c) results from the Fabr-Pérot resonances, i.e., kd m.

7 FIG. 5. (Color Online) (a) and (b) The corresponding dispersion relationships ( n vs ) for laser modes and CPA modes, respectivel, where the red and blue curves are the corresponding even modes and odd modes. In (b), the green curve is related to the perfect absorber (PA) modes in the semi-infinite spaces. In all the cases, 1, and. (c) and (d) The simulated time-averaged energ flues along and directions inside the PICM slab for the cases of laser mode and PA mode, respectivel. n d =1 V. The condition for realizing CPA and laser modes simultaneousl B observing these dispersion relationships of (Figs. (a) and (b)) and n 1(see Figs. 5(a) and 5(b)), generall, CPA modes and laser modes take place in different conditions. In other word, for a fied refractive inde of PICM slab, CPA modes and laser modes share different propagation constants, which have been demonstrated for the n 1 cases of n=1.5 and n =.5. However, in some cases where CPA and laser modes share opposite smmetric modes, CPA modes and laser modes could be realized simultaneousl. To be eact, the odd laser modes (see Eq. (1)) can have a solution identical to the even CPA modes (see Eq. (4)), or the even laser modes (see Eq. ()) can have the same solution as the odd CPA modes (see Eq. (3)). In both cases, we can get a condition for realizing CPA modes and laser modes simultaneousl with opposite smmetric modes, i.e., k / k= i. After some simplification, such condition for d can be further given as, n= m/ 1/8 ( m 1,, 3... ) or n=1. For, i.e., in the case of normal incidence [3, 4], the required effective inde of the PICM is n=1/8, 3/8, 5/8,... (m 1) / 8, which can clearl be found in Figs. (a) and (b) and Figs. 5(a) and 5(b), where CPA modes and laser modes happen simultaneousl et with opposite smmetric modes (see the colors of dispersion curves). In addition, for n=1, there are several transverse propagation constants to realize CPA modes and laser modes simultaneousl, which are given as =.485 k,.78 k,.97 k,.993k. VI. Discussion and Conclusion = In conclusion, we have analzed the interaction of incident plane waves with uniform PICM slabs. We find, both analticall and numericall, that CPA modes and laser modes are supported in the PICM slab, and it is a mechanism different from that which occurs in PT smmetric sstems, in which loss and gain are separated in different regions. In addition, the general conditions for realizing CPA and laser modes have been derived. We also identif the conditions for realizing CPA mode and laser mode simultaneousl, with the case of normal = with

8 incidence [-4] as a special case. More interestingl, as a simplification of CPA modes, single-sided PA modes can be obtained in a PICM slab with a sub-unit effective inde for the incident angle beond the critical angle. Overall, we find that PICMs produce et more unusual wave behavior that can be realized through metamaterials. We acknowledge that PICMs are ver hard to realize in practice as loss and gain parameters are included in the same medium. However, recentl two schemes [3, 4] to realize materials with properties similar to PICMs have been proposed. B emploing PT-smmetric materials in laered structures [3], PICMs can be effectivel mimicked, with the similar CPA-laser effect realized. The other method is that, based on the effective medium theor [5], PICMs could be well designed b using a photonic crstal composed of core-shell rods [4], in which loss and gain media are distributed in either the cores or the shells. Therefore, considering the recent eperimental progress in optical gain [17], PICMs might be realized eperimentall in the coming future. As we focus on the CMs in this work, thus the equal amplitudes of EM parameters are considered for PICMs. In fact, in a more general case, i.e., and with, the similar phenomena including CPA, laser and PA can still be obtained. If we consider TM polarization in our proposed PICMs, the left sides of Eqs. (1)-(4) will change the signs as TM k / k k / k. As a result, the dispersion relationships of laser (CPA) modes for TE polarization will be transformed into these of CPA (laser) modes for TM polarization. To be eact, for a PICM slab of interest, when it is used to realize a laser (CPA) for TE polarization, accordingl, a CPA (laser) will be achieved for TM polarization. Therefore, our proposed PICMs can be emploed to realize CPA and laser modes without limit of polarizations, which is available for obtaining CPA and laser modes simultaneousl. As CPA and laser modes can be effectivel realized in our PICMs, PICMs also can be used to achieve negative refraction [6] and planar imaging [7], in which CPA (as the timereversed laser [8]) and laser are respectivel obtained in the loss and gain media in PT smmetric sstems. i i ACKNOWLEDGMENTS This work was supported b the National Natural Science Foundation of China (grant No ), the National Science Foundation of China for Ecellent Young Scientists (grant no ), the Postdoctoral Science Foundation of China (grant no. 15M58456), and the Fundamental Research Funds for the Central Universities (Grant No ). We thank Prof. Hong Chen for the helpful discussions. REFERENCE 1. W. Cai and V. M. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 9).. Y. Xu, Y. Fu, and H. Chen, Nat. Rev. Mater. 1, 1667 (16). 3. D. Schurig, et al. Science 314, 977(6). 4. T. Ergin,, N. Stenger, P. Brenner, J. B. Pendr, and M. Wegener, Science 38, (1). 5. J. B. Pendr, Phs. Rev. Lett. 85, (). 6. R. A. Shelb, D. R. Smith, and S. Schultz, Science 9, (1). 7. H. Chen, and C. T. Chan, Appl. Phs. Lett. 9, 4115 (7). 8. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phs. Rev. Lett. 1, 18393(9). 9. Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Zhang, and C. T. Chan, Phs. Rev. Lett. 1, 539 (9). 1. Y. Xu, S. Du, Lei Gao, and H. Chen, New J. Phs. 13, 31 (11). 11. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phs. Rev. Lett. 16, 1391 (11). 1. Y. Fu, Y. Xu, and H. Y. Chen, Opt. Epress 4, 1648 (16).

9 13. S. Longhi, Phs. Rev. A 8, 3181 (1). 14. Y. D. Chong, L. Ge, and A. D. Stone, Phs. Rev. Lett. 16, 939 (11). 15. Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, Phs. Rev. Lett. 11, (14). 16. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Science 346, 61 (14). 17. Z. Wong, Y. Xu, J. Kim, K. OBrien, Y. Wang, L. Feng, and X. Zhang, Nat. Photon. 1, (16). 18. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebski, and T. Kottos, Phs. Rev. Lett. 19, 339 (1). 19. N. Lazarides and G. P. Tsironis, Phs. Rev. Lett. 11, 5391 (13).. Y. D. Xu, Y. Y. Fu, and H. Y. Chen, Opt. Epress 5, 495 (17). 1. D. Dragoman, Optics Communications 84, (11).. A. Basiri, I. Vitebski, and T. Kottos, Phs. Rev. A 91, (15). 3. S. Xiao, J. Gear, S. Rotter and J. Li, New J. Phs. 18, 854 (16). 4. P. Bai, K. Ding, G. Wang, J. Luo, Z. Zhang, C. T. Chan, Y. Wu, and Y. Lai, Phs. Rev. A 94, (16). 5. Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, Phs. Rev. B 74, (6). 6. R. Fleur, D. L. Sounas, and A. Alù, Phs. Rev. Lett. 113, 393 (14). 7. F. Monticone, C. A. Valagiannopoulos, and A. Alù, Phs. Rev. X 6, 4118 (16). 8. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, H. Cao, Science 331, (11). Appendi A: CPA and laser modes of PICM slab Laser modes: For the general wave scattering of a PICM slab ( i and i, and are positive real numbers), it includes incoming waves (the blue arrows) and outgoing waves (the red arrows), which is schematicall shown in Fig. 1(a). For laser modes, the injected signals (incoming waves) can be ignored, as the outgoing waves are etremel powerful compared with the incoming waves, and the schematic of wave scattering is plotted in Fig. 1(b). Therefore, the corresponding electric field distributions in different regions can be written as, ik i E1 Ae e zˆ,, (S1) E ( Be Ce ) e z, d, (S) ik ik i ˆ ik d i E3 De e zˆ, d, (S3) and the corresponding magnetic field distributions can be obtained b the Mawell's Equation H E / i. B matching the boundar conditions at the interfaces and d, the dispersion relationships for laser modes are, icot( k d / ) (odd modes), (S4) i tan( kd / ) (even modes), (S5) where k / k, k k, k n k, and n =. Consider that the ecited wave is plane wave incident from air with k, then k k. If the slab is composed of PICMs ( = ) or their analogs ( ), and are purel imaginar numbers: a) For nk, k/ k is purel imaginar number, and the right sides of the Eq. (S4) and Eq. (S5) are also purel imaginar numbers.

10 b) For nk, k / k is real number. As k i n k is purel imaginar number, the right sides of the Eq. (S4) and Eq. (S5) are real numbers. Therefore, Eq. (S4) and Eq. (S5) can be satisfied with real solutions for. As wave from air possesses a real tangential wavevector, perfect ecitation for laser modes will happen owing to momentum matching. As a result, PICMs or their analogs can support perfect laser modes. However, if the slab is made of CMs with real parts, i.e.,, and are comple numbers: nk / c): For, is comple number, while the right sides of the Eq. (S4) and Eq. (S5) are purel imaginar numbers. d): For nk k / k, k / k is comple number; as k i n k is imaginar number, the right sides of the Eq. (S4) and Eq. (S5) are real numbers. Therefore, Eq. (S4) and Eq. (S5) are not satisfied with real solutions for. As wave from air possesses a real tangential wavevector, perfect ecitation for laser modes will not happen due to momentum mismatching. As a result, the performance of laser modes will decrease, and CMs with do not support perfect laser modes. / Coherent perfect absorber (CPA) modes: For obtaining CPA modes, the outgoing signals can be ignored, i.e., the outgoing waves are zero, and the schematic of wave scattering is plotted in Fig. 1(c). Therefore, the corresponding electric field distributions in different regions can be written as, ik i E1 Ae e zˆ,, (S6) E ( Be Ce ) e z, d, (S7) ik ik i ˆ -ik d i E3 De e zˆ, d, (S8) and the corresponding magnetic field distributions can be figured out b the Mawell's Equation H E / i. B matching the boundar conditions at the interfaces and, the dispersion relationships for CPA modes are, icot( k d / ) (odd modes), (S9) i tan( kd / ) (even modes), (S1) d where k / k, k k, k n k, and n =. Similar to the = discussion of laser modes, perfect ecitation of CPA modes can onl eist in PICMs ( ) or their analogs ( ). For realizing CPA modes, the two incident signals should be coherent with a specific amplitude and phase relationship at the boundaries, i.e., ik d A (1 ) / (1 ) e D. Perfect absorber (PA) modes: In our PICM slab with the effective refractive inde less than 1, when the incident angle is beond, the incident wave could be perfectl bounded at the interface between air and PICM without an reflection and transmission, defined as perfect absorber (PA) modes. When the two incoming waves for CPA are incident on the slab, the will be bounded at the left and right interfaces. For PICM slab with a fied thickness, when PICMs possess a higher effective refractive indices, the bounded waves inside the slab are difficult to interact with each other, which can effectivel throttle the outgoing waves, On this occasion, the problem of CPA modes will become the case of PA modes in the semi- c

11 infinite spaces consisting of air and PICMs as shown in Fig. 1(d). For this case, the electric field distributions in air ( ) and PICM media ( ik i E1 Ae e zˆ,, (S11) ik i E Be e zˆ,, (S1) and the corresponding magnetic field distributions can be obtained b the Mawell's Equation H E / i. B matching the boundar conditions at the interface =, the dispersion relationship for PA modes is, where k k, k i n k, and = k 1, k n ) can be respectivel written as, (S13). If the slab is composed of PICMs ( ) or their analogs ( ), and are purel imaginar numbers, then Eq. (S13) can be satisfied with real solutions for. While for CMs with real parts, there are comple solutions for. Hence, for our proposed PICMs or their analogs, PA modes can eist. Appendi B: Transmission and reflection of PICM slab in air Now we consider the reflection and refraction for a PICM slab in air (see Fig. 3(a)). Suppose the two interfaces of the CM slab and air are and, where d is the thickness of the slab. For an incident plane wave, ik Eint e zˆ,, (S14) the reflected wave is written as, E ik i re zˆ,, (S15) ref while the transmitted wave is epressed as, ik ( d ) E te i zˆ, trn d d, (S16) where k sin is the wave vector along the slab, cos k k and k k sin are the wave vectors normal to the slab in air and CM slab, respectivel, is the incident angle, r and t are the coefficients of reflection and transmission, respectivel. For the PICM slab, ( ik i ik E a e a e i ) zˆ, d, (S17) where a p and a n PICM p n are the corresponding coefficients of positive and negative components. After matching the boundar conditions at the interfaces of and r t a p a n 1 i sin and 1 epi 1 epi 4 1 epi 1 epi ( ) 1 1 epi ( )epi 1 1 epi where k/ k, kd is the phase change in the slab., d,, we have, (S18) (S19), (S), (S1)

12 Appendi C: Energ flu in PICM slab Based on the results in Appendi B, the corresponding EM wave inside the PICM slab can be written as, ( ik ik E a e a e ) e i zˆ, (S) z p n H ( a e a e ) e zˆ, (S3) ik ik i p n k H ( a e a e ) e zˆ. (S4) ik ik i p n The time-averaged energ flues in and directions are given as and S. Re( HEz ) / a): For the EM wave in the PICM slab wave being propagating wave ( S Re( H Ez ) / sin ), k sin and k k sin. As k/ k is purel imaginar number, based - - on Eq. (S) and Eq. (S1), we can easil deduce a = a, and ( ik ik E a e a e ) e i zˆ, therefore, S 1 Re[ H Ez ] S p n z p n 1 k = Re[ ( ik - ik ap e an e )], 1 Re[ H Ez ] We assume a = i and a = and p 1 = Re[ ( ik - ik ap + an + ap e an e )]., then e i n e 1 k Re[ ( k sin( )] S i k sin( k ), 1 i Re[ ( S (cos( k ) 1))] (S5). (S6) b): For the EM waves in the PICM slab are decaed waves ( k ik i. As i sin = ep =ep( d) sin ), k sin, we have a / 1 p, with k / k= k /, k k cos, and a. As a result, as k / k= k / is a real number and a p is also a real one, we have, 1 i S Re[ a e ] a e p p 1 i Re[ ap e ] S n, (S7). (S8) Therefore, based on the above results, we can conclude that energ flu in the PICM slab onl propagates along direction as S = alwas happens in the PICM slab.