## Abstract

We show numerically that both coherent perfect absorption and transparency can be realized in a monolayer graphene. The graphene film, doped and patterned with a periodical array of holes, can support plasmonic resonances in the Mid-infrared range. Under the illumination of two counter-propagating coherent optical beams, resonant optical absorption may be tuned continuously from 99.93% to less than 0.01% by controlling their relative phase which gives a modulation contrast of 40 dB (about 30 dB for transmission). The phenomenon provides a versatile platform for manipulating the interaction between light and graphene and may serve applications in optical modulators, transducers, sensors and coherent detectors.

© 2014 Optical Society of America

Optical absorption plays an important role in a variety of applications such as photodetectors and photovoltaics. As a result, there has recently been lots of interest in realizing total light absorption and two different routes have generally been employed. Firstly, one can take advantage of diffusion or multi-scattering of light on very disordered lossy surfaces such as carbon nanotube arrays [1, 2]. The second method exploits the critical coupling effect [3]. When the coupling rate is equal to the dissipative rate of the structure and other energy loss channels such as diffraction and polarization conversion are blocked, critical coupling will lead to perfect absorption [4, 5]. Such kind of perfect absorption has been realized in various micro- and nano-structures such as plasmonic surfaces, metallic particle arrays and metamaterials [6–11].

Most perfect absorbers have only one port and critical coupling happens under the illumination of one beam. A generation to the two port situation leads to the so-called ’Coherent Perfect Absorption (CPA)’ [12]. Similar to the critical coupling effect, CPA relys on the coherent interference of light where two coherent beams are incident on the absorber from opposite sides and the reflected light of one beam interfere with the transmitted light of the other and vice versa [13, 14]. CPA was first realized using a silicon slab cavity [15], and has recently been reported in composite absorptive films, planar metamaterials, waveguides and so on [16–23]. Coherent absorption provides an additional flexibility to tune the absorption by changing the relative phase of two beams and shows promising potentials for applications ranging from nanoscale light manipulations to data processing [19, 21, 24, 25].

Graphene, a single layer of carbon atoms arranged in plane with a honey comb lattice, shows promising potentials in optics and optoelectronics [26] and has recently attracted great attentions in many applications, such as photodetectors, optical modulators, tunable filters and polarizers [27–30]. Even though the optical absorption of monolayer graphene is quite weak (about 2.3% in the visible range), enhanced absorption by unstructured graphene can be realized by placing it in an optical cavity or exploiting the critical coupling effect [31–34]. Moreover, doped and patterned graphene can support localized plasmonic resonances in the infrared and THz ranges which significantly enhance the absorption [35–37]. Specifically, complete light absorption has been reported in an array of doped graphene disks backed by a metallic mirror or a dielectric with high optical permittivity [38]. Graphene plasmonics provide an effective route to enhance light-graphene interactions.

Here we show CPA in a nanostructured graphene film. The graphene film, patterned with periodical arrays of holes, displays a strong plasmonic resonance in the Mid-infrared wavelength range. By controlling the relative phase of the two coherent beams incident on the graphene film, we are able to either enhance or suppress the resonant absorption in the graphene. Besides CPA, its opposite effect, coherent perfect transparency (CPT), can also be realized due to the extremely thin thickness of the graphene sheet which is less than one ten-thousandth of a wavelength thick.

It has been known that there is a universal limit to absorption by a thin layer with its thickness much smaller than the wavelength of light [38, 39]. We consider a thin film at the interface between two media, medium 1 and medium 2, with different refractive indices, *n*_{1} and *n*_{2}. If light impinges on the film from medium 1 at normal incidence, the combined reflection and transmission coefficients are

*η*is the self-consistent amplitude by the thin film, while

*r*and

*t*are the Fresnel coefficients of the bare

*n*

_{1}|

*n*

_{2}interface

The absorption is

The condition of maximum absorption is *∂A/∂η* = 0 (*∂*^{2}*A/∂*^{2}*η* is real and negative) and we get

*A*= 1/(1 +

_{max}*χ*), where

*χ*=

*n*

_{2}/

*n*

_{1}. The corresponding reflection and transmission coefficients are

For any thin layer with symmetric environments, i.e., *n*_{1} = *n*_{2}, the limit to maximum absorption is *A _{max}* = 0.5. The maximum absorption can be increased in asymmetric environments and most perfect absorbers employ a back mirror to block the transmission channel.

Now let’s consider two counter-propagating coherent beams, beam 1 and beam 2, impinge on the thin film at normal incidence from opposite sides, medium 1 and medium 2, respectively. We assume the electric field amplitude of beam 1 is 1 and that of beam 2 is *α*. The reflected part of one beam will interfere with the transmitted part of the other one and vice versa. Therefore, the amplitude of scattered light from each side of the thin film will be

*φ*is the phase difference of beam 1 and beam 2.

*R*

_{1}and

*T*

_{1},

*R*

_{2}and

*T*

_{2}are the reflection and transmission coefficients of beam 1 and beam 2 respectively, which are given by Eq. (5)

Apparently, *A _{coh}* gets its maximum of 1 only if

*α*= 1 and

*φ*= 2

*Nπ*(

*N*is an integer). Therefore if a thin film can reach the incoherent absorption limit, coherent perfect absorption can be achieved under the illumination of two counter-propagating coherent beams with equal field amplitudes and a relative phase of zero. Once the amplitude of two coherent beams are fixed to be the same, the coherent absorption depends on their phase difference and can be tuned continuously between ${A}_{\mathit{min}}^{\mathit{coh}}={\left[(1-\chi )/(1+\chi )\right]}^{2}$ (at

*φ*= (2

*N*+ 1)

*π*) and ${A}_{\mathit{coh}}^{\mathit{max}}=1$.

In this part, we study numerically the coherent absorption in a nanostructured graphene. The numerical simulations are conducted using a fully three-dimensional finite element technique (in Comsol MultiPhysics). In the simulation, the graphene is modelled as a conductive surface without thickness [38, 40, 41]. The optical conductivity of graphene can be derived within the random-phase approximation (RPA) in the local limit [42, 43]

Here *k _{B}* is the Boltzmann constant,

*T*is the temperature,

*ω*is the frequency of light,

*τ*is the carrier relaxation lifetime, and

*E*is the Fermi level. The first term in Eq. (11) corresponds to intra-band transitions and the second term is related to inter-band transitions. We restrict our calculations to energies below 0.2 eV in order to avoid the contribution of inter-band contribution and to the Fermi level

_{F}*E*≫ 2

_{F}*k*. Equation (11) reduces to the Drude model if we neglect both inter-band transitions and the effect of temperature (

_{B}T*T*= 0) [44, 45]

*E*depends on the concentration of charged doping and $\tau =\mu {E}_{F}/e{v}_{F}^{2}$, where

_{F}*v*≈ 1 × 10

_{F}^{6}

*m/s*is the Fermi velocity and

*μ*is the dc mobility. Here we use a moderate measured mobility

*μ*= 10000

*cm*

^{2}

*·V*

^{−1}·

*s*

^{−1}[46]. The Fermi level of graphene is assumed to be

*E*= 0.6 eV which corresponds to an doping density of about 2.6 × 10

_{F}^{13}

*cm*

^{−2}and may be realized by electrostatic or chemical doping [46, 47].

Figure 1(a) shows the schematic of coherent absorption in a freestanding graphene film patterned with a periodical array of square holes. Two coherent beams are incident on the graphene film from opposite sides and the coherent absorption is controlled by their relative phase. The geometric parameters of the patterned graphene are shown in Fig. 1(b). The width of hole is *L* = 220 nm and the period of unit cell is *P* = 400 nm in both x- and y- directions.

Figure 2 is the simulated spectra under incoherent as well as coherent illuminations. When a single beam of light is illuminated on the patterned graphene film at normal incidence, part of the energy will be absorbed while others will either be reflected or transmitted, as shown in Fig. 2(a). There is a plasmonic resonance centered at *λ* = 8.476 *μ*m in the studied spectral range with a maximum absorption of *A* = 49.97%. When two coherent beams with equal intensities are incident on the graphene from opposite sides, they will interfere with each other, leading to coherent absorption. Now the absorption depends on the relative phase of the two counter-propagating beams.

Figure 2(b) shows the normalized total output intensities *S _{tot}* = 1 −

*A*(total input energy minus absorption) when their relative phase equals 0 and

_{tot}*π*, respectively. As the two beams have no phase difference, they are symmetric for the graphene film and form an parity-even mode. In this situation, the interference of the two beams suppresses their scattering and the absorption is enhanced (graphene is at the anti-node of the interference pattern formed by the two coherent beams). Particularly, the total output intensity drops more than 30 dB (

*A*= 99.93%) at the plasmonic resonance of the nanostructured graphene film. On the contrary, they form an parity-odd mode when their phase difference is

*π*and show reduced absorption due to constructive interference of scattered light at the two sides of the graphene film (graphene is at the node of the interference pattern formed by the two coherent beams). As a result, the normalized total output intensity is nearly unitary (

*A*< 0.01%) at the whole studied spectral range. Thus, coherent absorption allows control of resonant absorption in the nanostructured graphene film, either an increase to nearly 100% (corresponding to CPA) or a suppression to almost zero (corresponding to CPT).

Figure 3 shows the normalized coherent absorption in the nanostructured graphene as a function of the relative phase. At the resonance wavelength of 8.476 *μ*m, the coherent absorption varies continuously from 99.93% to less than 0.01% as the relative phase changes from 0 to *π* (see Fig. 3(a)), giving a modulation contrast of 40 dB. At the off-resonance wavelength, coherent perfect absorption cannot be realized and the maximum coherent absorption is about two times of the incoherent absorption (see Fig. 3(b)). Besides the variation of absorption, we also see transfer of energy between scattering at the two sides of the graphene film (*S*_{1} and *S*_{2}) as the relative phase changes because of the interference between the reflected part of each beam with the transmitted part of the other.

Coherent absorption in nanostructured graphene is a quite robust phenomenon and can also be implemented for graphene with asymmetric environments, i.e., on a substrate. We consider the same nanostructured graphene on a semi-infinite substrate with a refractive index of *n* = 1.5. To make our modeling more practical, instead of Eq. (12), here we use Eq. (11) for the conductivity of graphene in which both the intra-band contribution and temperature effect are considered. As in previous sections, we assume *E _{F}* = 0.6

*eV*and

*μ*= 10000

*cm*

^{2}·

*V*

^{−1}·

*s*

^{−1}. The temperature is assumed to be 300 K. Because of the substrate, the resonance wavelength redshifts from 8.476

*μ*m to 10.61

*μ*m. As the dielectric environment now becomes asymmetric for graphene, the incoherent absorption of patterned graphene film is 38.16% for light incident on the graphene from the air side and 57.23 % for light incident on the graphene from the substrate side at the resonance, respectively.

Figure 4 shows the interferometric control of coherent absorption in the nanostructured graphene on a substrate at the resonance wavelength of 10.61 *μ*m. Two coherent beams with equal electric field amplitudes (relative intensity *I*_{1}/*I*_{2} = 1/1.5) impinge on the graphene film from the air and the substrate side, respectively. The normalized coherent absorption varies from 95.4% to 3.8% as the relative phase changes from 0 to *π*. Here the minimum coherent absorption is very close to the ideal lower limit of
${A}_{\mathit{coh}}^{\mathit{min}}={\left[(1-\chi )/(1+\chi )\right]}^{2}=4\%$ while maximum absorption is slightly below coherent perfect absorption. This is because the maximum incoherent absorption of our nanostructured graphene at the resonance is a bit lower than the limit, i.e. 40% and 60% for incidence from the air and substrate side, respectively. Coherent perfect absorption may be realized by optimizing the doping level of graphene and the design of the nanostructures. We can also reduce the minimum absorption with a trade off of decreasing the maximum absorption by changing the relative intensities of input beams, which may increase the modulation contrast. For example, the coherent absorption here can be modulated between about 94.4% and 0.96% for two coherent beams with equal intensities (*I*_{1}/*I*_{2} = 1/1).

In summary, we have studied coherent absorption in a single layer of nanostructured graphene. The thickness of monolayer graphene is about 3.4 *A ^{o}* which is less than one ten-thousandth of a wavelength. The coherent absorption can be tuned continuously from nearly 99.93% (corresponding to CPA) to less than 0.01% (corresponding to CPT) depending on the relative phase of the two counter-propagating coherent beams. This phenomenon relies on interplay of plasmonic resonances and the interference of optical beams on the patterned graphene and provides functionality that can be implemented freely across a broad Mid-infrared to THz range by varying the structural design and graphene doping level. Coherent absorption provides a very versatile platform for manipulating the interaction between graphene and light. It may serve applications in optical modulators, transducers, sensors or coherent detectors.

## Acknowledgments

This work was supported by National Natural Science Foundation of China [Grant Nos. 11304389, 61177051 and 61205087] and the State Key Program for Basic Research of China [Grant No. 2012CB933501].

## References and links

**1. **Z.-P. Yang, L. Ci, J. A. Bur, S.-Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. **8**, 446–451 (2008). [CrossRef] [PubMed]

**2. **K. Mizuno, J. Ishii, H. Kishida, Y. Hayamizu, S. Yasuda, D. N. Futaba, M. Yumura, and K. Hata, “A black body absorber from vertically aligned single-walled carbon nanotubes,” Proc. Natl. Acad. Sci. USA **106**, 6044–6047 (2009). [CrossRef] [PubMed]

**3. **M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. **85**, 74 (2000). [CrossRef] [PubMed]

**4. **K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. SavelŁv, and F. Nori, “Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media,” Rev. Mod. Phys. **80**, 1201 (2008). [CrossRef]

**5. **J. Zhang, J. Ou, K. MacDonald, and N. Zheludev, “Optical response of plasmonic relief meta-surfaces,” Journal of Optics **14**, 114002 (2012). [CrossRef]

**6. **T. V. Teperik, F. G. De Abajo, A. Borisov, M. Abdelsalam, P. Bartlett, Y. Sugawara, and J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics **2**, 299–301 (2008). [CrossRef]

**7. **N. Landy, S. Sajuyigbe, J. Mock, D. Smith, and W. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. **100**, 207402 (2008). [CrossRef] [PubMed]

**8. **N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. **10**, 2342–2348 (2010). [CrossRef] [PubMed]

**9. **X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. **104**, 207403 (2010). [CrossRef] [PubMed]

**10. **J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. **96**, 251104 (2010). [CrossRef]

**11. **K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Comm. **2**, 517 (2011). [CrossRef]

**12. **Y. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. **105**, 053901 (2010). [CrossRef] [PubMed]

**13. **A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Elect. Lett. **36**, 321–322 (2000). [CrossRef]

**14. **L. Ken, “Phase effect on guided resonance in photonic crystal slabs,” Chin. Phys. Lett. **22**, 2294 (2005). [CrossRef]

**15. **W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science **331**, 889–892 (2011). [CrossRef] [PubMed]

**16. **S. Longhi, “Coherent perfect absorption in a homogeneously broadened two-level medium,” Phys. Rev. A **83**, 055804 (2011). [CrossRef]

**17. **S. Dutta-Gupta, O. J. Martin, S. Dutta Gupta, and G. Agarwal, “Controllable coherent perfect absorption in a composite film,” Opt. Express **20**, 1330–1336 (2012). [CrossRef] [PubMed]

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

**19. **H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. **108**, 186805 (2012). [CrossRef] [PubMed]

**20. **M. Pu, Q. Feng, M. Wang, C. Hu, C. Huang, X. Ma, Z. Zhao, C. Wang, and X. Luo, “Ultrathin broadband nearly perfect absorber with symmetrical coherent illumination,” Opt. Express **20**, 2246–2254 (2012). [CrossRef] [PubMed]

**21. **J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light: Sci. Appl. **1**, e18 (2012); doi: [CrossRef]

**22. **N. Gutman, A. A. Sukhorukov, Y. Chong, and C. M. de Sterke, “Coherent perfect absorption and reflection in slow-light waveguides,” Opt. Lett. **38**, 4970–4973 (2013). [CrossRef] [PubMed]

**23. **R. Bruck and O. L. Muskens, “Plasmonic nanoantennas as integrated coherent perfect absorbers on soi waveguides for modulators and all-optical switches,” Opt. Express **21**, 27652–27661 (2013). [CrossRef]

**24. **S. A. Mousavi, E. Plum, J. Shi, and N. I. Zheludev, “Coherent control of optical activity and optical anisotropy of thin metamaterials,” arXiv preprint arXiv:1312.0414 (2013).

**25. **X. Fang, M. L. Tseng, J.-Y. Ou, K. F. MacDonald, D. P. Tsai, and N. I. Zheludev, “Ultrafast all-optical switching via coherent modulation of metamaterial absorption,” Appl. Phys. Lett. **104**, 141102 (2014). [CrossRef]

**26. **F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics **4**, 611–622 (2010). [CrossRef]

**27. **F. Xia, T. Mueller, Y.-m. Lin, A. Valdes-Garcia, and P. Avouris, “Ultrafast graphene photodetector,” Nat. Nanotechnol. **4**, 839–843 (2009). [CrossRef] [PubMed]

**28. **M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature **474**, 64–67 (2011). [CrossRef] [PubMed]

**29. **H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. **7**, 330–334 (2012). [CrossRef] [PubMed]

**30. **Z. Zhu, C. Guo, K. Liu, J. Zhang, W. Ye, X. Yuan, and S. Qin, “Electrically tunable polarizer based on anisotropic absorption of graphene ribbons,” Appl. Phys. A **114**, 1017–1021 (2014).

**31. **M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. **12**, 2773–2777 (2012). [CrossRef] [PubMed]

**32. **J.-T. Liu, N.-H. Liu, J. Li, X. J. Li, and J.-H. Huang, “Enhanced absorption of graphene with one-dimensional photonic crystal,” Appl. Phys. Lett. **101**, 052104 (2012). [CrossRef]

**33. **J. Piper and S. Fan, “Total absorption in a graphene monolayer in the optical regime by critical coupling with a photonic crystal guided resonance,” ACS Photonics **1**, 347353 (2014). [CrossRef]

**34. **F. Liu, Y. Chong, S. Adam, and M. Polini, “Gate-tunable coherent perfect absorption of terahertz radiation in graphene,” arXiv preprint arXiv:1402.2368 (2014).

**35. **A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics **6**, 749–758 (2012). [CrossRef]

**36. **R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon meta-material,” Opt. Express **20**, 28017–28024 (2012). [CrossRef] [PubMed]

**37. **N. Papasimakis, S. Thongrattanasiri, N. I. Zheludev, and F. G. de Abajo, “The magnetic response of graphene split-ring metamaterials,” Light: Sci. Appl. **2**, e78 (2013); doi: [CrossRef]

**38. **S. Thongrattanasiri, F. H. Koppens, and F. J. G. de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. **108**, 047401 (2012). [CrossRef] [PubMed]

**39. **L. N. Hadley and D. Dennison, “Reflection and transmission interference filters,” J. Opt. Soc. Am. **37**, 451–453 (1947). [CrossRef] [PubMed]

**40. **A. Vakil and N. Engheta, “Transformation optics using graphene,” Science **332**, 1291–1294 (2011). [CrossRef] [PubMed]

**41. **Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. **13**, 1257–1264 (2013). [CrossRef] [PubMed]

**42. **L. Falkovsky and S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B **76**, 153410 (2007). [CrossRef]

**43. **L. Falkovsky and A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. Jour. B **56**, 281–284 (2007). [CrossRef]

**44. **G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” Jour. Appl. Phys. **104**, 084314 (2008). [CrossRef]

**45. **M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B **80**, 245435 (2009). [CrossRef]

**46. **K. S. Novoselov, A. K. Geim, S. Morozov, D. Jiang, Y. Zhang, S. Dubonos, I. Grigorieva, and A. Firsov, “Electric field effect in atomically thin carbon films,” Science **306**, 666–669 (2004). [CrossRef] [PubMed]

**47. **Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano **7**, 2388–2395 (2013). [CrossRef] [PubMed]