Mechanism of propagating graphene plasmons ... - OSA Publishing

Vol. 26, No. 3 | 5 Feb 2018 | OPTICS EXPRESS 3709

Mechanism of propagating graphene plasmons excitation for tunable infrared photonic devices LINLONG TANG,1 WEI WEI,2 XINGZHAN WEI,1,3 JINPENG NONG,1,2 CHUNLEI DU,1 AND HAOFEI SHI1,4 1Chongqing

Key Laboratory of Multi-scale Manufacturing Technology, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, 400714, China 2Key Laboratory of Optoelectronic Technology & Systems, Ministry of Education of China, College of Optoelectronic Engineering, Chongqing University, Chongqing 400044, China [email protected] [email protected]

Abstract: The mechanism of propagating graphene plasmons excitation using a nano-grating and a Fabry-Pérot cavity as the optical coupling components is studied. It is demonstrated that the system could be well described within the temporal coupled mode theory using two phenomenological parameters, namely, the intrinsic loss rate and the coupling rate of a graphene plasmonic mode, and their analytical expressions are derived. It is found that the intrinsic loss rate is solely determined by the electron relaxation time of graphene, while independent of the field distributions of the modes. Such result originates from the negligible magnetic field energy of the graphene plasmonic mode. The coupling rate is governed by the optical coupling components parameters, and varies periodically with the Fabry-Pérot cavity length. By modulating the two rates, quality factors and absorption rates can be adjusted. Furthermore, it is revealed that low refractive index of the Fabry-Pérot cavity material is vital to the enlargement of tunable band, and the underlying physics is discussed. Such plasmon excitation configuration is insensitive to light incident angle and could serve as a platform for many tunable infrared photonic device, such as surface-enhanced infrared absorption spectroscopies, infrared detectors and modulators. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (240.6680) Surface plasmons; (050.6624) Subwavelength structures; (260.5740) Resonance.

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#318899 Journal © 2018

https://doi.org/10.1364/OE.26.003709 Received 3 Jan 2018; revised 27 Jan 2018; accepted 27 Jan 2018; published 2 Feb 2018


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1. Introduction Graphene, a two dimensional material composed of carbon atoms arranged in a honeycomb lattice, has being investigated intensively due to its unique properties and potential applications [1–4]. One of the most striking properties of graphene is its capability of supporting low loss and highly confined surface plasmons at the infrared and terahertz range, whose dispersions can be dynamically tuned by varying Fermi energy via gating voltage, thanks to the low electron density of states near Dirac points [5–14]. These advantages make graphene plasmons promising in a wide range of tunable photonic applications, such as surface-enhanced infrared absorption spectroscopy (SEIRAS) [15–19], infrared and terahertz photodetectors [20–23], infrared sources [24], absorbers [25–29] and modulators [6,30,31]. In various applications, graphene plasmons are anticipated to have proper quality factors and band tunabilities. For example, in SEIRAS high quality factor is desired so as to increase the stored energy in a graphene plasmonic mode [18], and hence is crucial to the improvement of the enhancement factor. Wideband tunability is also required in SEIRAS so that the whole fingerprint of molecules (~6-16 μm) can be covered [16]. Currently, graphene are usually patterned into nano-structures to excite localized graphene plasmons, such as nano-disks [27,32], nano-ribbons [29,33], and many kinds of metamaterial cells [25,26,28]. In theory, such method could enable high quality factor by assuming a large mobility of graphene. But in practice, the patterning of graphene unavoidably introduces a great amount of boundaries and defects into graphene, resulting in severe decrease of the mobility and the quality factor. Therefore, graphene patterning may not be ideal for plasmon excitation from a practical point of view. Another effective but less studied method is to use nano-gratings to excite propagating graphene plasmons, which do not require the patterning of graphene and hence its high mobility can be preserved [34–37]. Because of this merit, nano-gratings based graphene excitation configurations could be more suitable platforms for infrared photonic devices like SEIRAS, infrared photodetectors and graphene plasmonic infrared sources, where high mobility of graphene (or high quality factor of the plasmonic mode) is crucial to their performances. Previous theoretical studies mainly focus on the dispersion relations of propagating graphene plasmons for the prediction of resonant frequencies, but it remains challenge to analytically predict the quality factors, bandwidths, absorption rates and tunable ranges of propagating graphene plasmons. Therefore, it is of fundamental importance to reveal the underlying physics of propagating graphene plasmon excitations, and build an intuitive picture for designing the devices to achieve proper quality factors and band tunabilities. In this paper, a propagating graphene plasmon excitation configuration employing a nanograting and a Fabry-Pérot (FP) cavity as the optical coupling components is proposed, and a general framework based on the temporal coupled mode theory is established to study its work mechanisms. We show that the phenomenological parameters in the temporal coupled mode theory, namely, the intrinsic loss rate and the coupling rate, could be derived analytically for propagating graphene plasmonic modes. The intrinsic loss rate is solely determined by the electron relaxation time (or electron mobility) of graphene, while independent of the field distributions of the modes. It is demonstrated that such result is the consequence of the negligible magnetic field energy of graphene plasmonic modes. The coupling rate relates to parameters of the optical coupling components, and can be tuned periodically by the FP cavity length. By varying the two phenomenological parameters, the quality factor and the absorption rate of the plasmonic mode can be adjusted. Meanwhile, the refractive index of FP cavity material is identified to be the crucial factor that influence the band tunability, and a low refractive index is beneficial to enlarge the tunable band. Finally,


we show that the device is not sensitive to the incident angle of infrared light, and can work well in a large angle range. The proposed excitation configuration could serve as platforms for various infrared photonic devices, and the theoretical framework could provide guidance for designing and optimizing these devices. 2. The excitation configuration and its theoretical description The excitation configuration under study is schematically shown in Fig. 1(a), which is composed of six functional layers. And from top to bottom the layers are graphene layer I, the nano-grating, the insulating layer, the graphene layer II, the spacer layer, and the metal mirror. The four top layers form a capacitor structure where the two graphene layers server as the two electrodes. Using such capacitor structure, the Fermi energy of graphene can be dynamically tuned by applying a voltage between the two graphene layers.

Fig. 1. (a) Schematic of the propagating graphene plasmon excitation configuration. (b) The absorption curves of the excitation configuration. Inset is the electric field distribution of the graphene plasmonic mode in a grating period.

Graphene layer I can support the formation of propagating plasmonic waves by using the nano-grating to excite them. The nano-grating is required because the wavevector between an incident light and a graphene plasmonic wave are in huge mismatch, they cannot couple to each other unless the grating is introduced to compensate the mismatch. To be specific, as a light with angular frequency ω impinges on the grating, it will be scattered into many orders of evanescent diffraction waves and the wavevector of the nth order is 2nπ / l + k0sinθ, with l being the period of the grating, k0 the wavevector of free space wave and θ the incident angle. Then, the nth order graphene plasmonic mode can be excited by the diffraction order provided the wavevector match condition Re(q(ω)) = 2nπ / l + k0sinθ is fulfilled, where q(ω) is the dispersion relation of the graphene plasmonic wave with q being its wavevector. Typically, the graphene plasmonic wavevector q is near two orders higher than the free space wavevector k0, meaning that the grating period l should be very small so that 2nπ / l is large


enough to compensate their mismatch. Also, it means the wavevector match condition could be approximated by Re(q(ω)) = 2nπ / l. Graphene layer II is designed to only act as an infrared transparent conductive electrode to dope graphene layer I electrically. It is relatively far away from the nano-grating and the highly localized evanescent diffraction waves of the nano-grating cannot couple to its plasmonic modes, and hence no plasmonic modes could be excited in graphene layer II. The bottom metal mirror is used to reflect the incident light back, combined with the spacer layer to adjust the distance between the mirror and the graphene, the light intensity at the position of graphene layer I can be controlled. In other words, the bottom metal mirror and the spacer layer act as a FP cavity to tune the interaction strength between incident light and graphene plasmons. The interaction process between a graphene plasmonic mode and the incident light is characterized by the light absorption curve of the plasmonic mode, and such process can be described by the temporal coupled mode theory which shows the absorption curve A is determined by A=

4γ 0γ 1

( ω − ωa )

2

+ (γ 0 + γ 1 )

2

(1)

.

Here, ω is the angular frequency of the incident light, ωa is the resonant frequency of the plasmonic mode, γ0 is the intrinsic loss rate of the plasmonic mode, and γ1 is the coupling rate between the plasmonic mode and the incident light. Equation (1) suggests that at resonant frequencies (i.e. ω = ωa), the absorption rate A could reach 100% if γ0 = γ1 is fulfilled, and this is the critical coupling condition. Since the temporal coupled mode theory is a general model, it gives no information about the specific expressions of γ0, γ1. Hence, the main purpose of this paper is to establish the relations between γ0, γ1 and the excitation configuration parameters, and to analyze how to achieve desired quality factors, absorption rates and band tunabilities. In the follows, we will calculate the absorption curves of graphene plasmonic modes numerically by finite element method using Comsol Multiphysics. Graphene is modeled as a surface current boundary condition in the software with the intraband surface conductivity σ given by a Drude-like model [38], σ ( ω= )

2e

2

π

2







 2k B T

k B T ⋅ ln  2 ⋅ cos h 

Ef

i    ⋅ ω + iτ .  −1

(2)

The interband contribution of σ is negligible in what our study concerns. In Eq. (2), e is the elementary charge,  is the reduced Plank constant, kB is the Boltzmann constant, T = 300 K is the temperature, Ef is the Fermi energy, and τ is the electron relaxation time. We first numerically calculate the absorption curve of a graphene plasmonic mode, and compare it with the analytical one given by Eq. (1) to demonstrate the validity of the temporal couple mode theory. In the calculations, the parameters are set as follows: The metal mirror is assumed to be gold of 500 nm thick, whose conductivity is described by the standard Drude model with plasma frequency of 8.55 eV and damping rate of 18.4 meV [39]. The spacer layer is ZnSe with thickness of 1.5 μm and refractive index of 2.4. The insulating layer is KBr with thickness of 300 nm and refractive index of 1.5. The grating material is 100 nm thick ZnSe, and the grating period is 200 nm. The Fermi energy and relaxation time of graphene layer I are 0.4 eV and 0.4 ps, and those of graphene layer II are - 0.4 eV and 0.05 ps. The relaxation time of graphene layer II is assumed to be lower than graphene layer I by considering that the quality of layer II may deteriorate in experiments because it is capsulated by the spacer and insulating layer which would introduce many electron scatter centers. The numerical results calculated by Comsol Multiphysics are presented in Fig. 1(b), where a sharp resonant peak of the total absorption rate at about 10 μm is observed. Such peak


absorption rate composed of three parts: graphene plasmon absorptions in graphene layer I, Ohmic absorptions in graphene layer II and gold mirror, and obviously the peak is mainly caused by graphene plasmons absorptions though the Ohmic absorptions also contributes a little. Moreover, the electromagnetic field distribution at 10 μm is displayed in the inset of Fig. 1(b), confirming that a graphene plasmonic mode in graphene layer I is excited and no plasmonic mode is excited in graphene layer II. At other wavelengths that out of the graphene plasmon resonance, the total absorption rates is small and dominated by Ohmic absorptions of the two graphene layers and the metal mirror. We fitted the above simulation results with Eq. (1) by setting γ0 = 1.25 THz, γ1 = 0.26 THz, and compared them in Fig. 1(b). It is clear that the analytical curve matches well with the numerical one, indicating the behavior of the excitation configuration can be well described by temporal coupled mode theory. 3. Results and discussions The electron relaxation time τ in Eq. (2) characterize the loss of graphene, and hence it is expected to connect with the intrinsic loss rate γ0 of a graphene plasmon mode. To verify this idea, we changed the relaxation time from 0.05 ps to 5 ps to numerically calculate the total absorption spectra of the excitation configuration, and kept other parameters the same as in calculating Fig. 1(b). The result spectra are presented in Fig. 2(a), which shows all the resonant peaks are around 10 μm, signifying the relaxation time does not affect resonant wavelengths. Another feature in Fig. 2(a) is that the resonant bandwidth decreases monotonously as the relaxation time rises. Such trend could also be clearly seem from Fig. 2(b), where three spectra of different relaxation time are extracted from Fig. 2(a). More importantly, it is found that the resonant absorption rate increases as the relaxation time rise from 0.05 ps to 0.3 ps, and reach a maximum value of 100% at about 0.3 ps, and then decreases as the relaxation time further rises. Figure 2(c) is the extracted resonant absorption peaks from Fig. 2(a), which clearly shows such variation trend.

Fig. 2. (a) The evolution of absorption spectra with graphene electron relaxation time. (b) Three extracted absorption spectra from the position marked by white dashed lines in (a). (c) The extracted absorption peaks from (a), and the analytical results.

In order to understand the observed results, we analyzed in detail the relations between the intrinsic loss rate γ0 of a graphene plasmon mode and the relaxation time τ of the graphene


[see Appendix A], and find that due to the negligible magnetic field energy of graphene plasmonic modes, a simple but universal analytical relation exists,

γ 0 = 1 / (2τ ).

(3)

To test this relation, we fitted the total absorption spectra at τ = 0.3 ps by setting γ0 = 1/(2τ) = 1.67THz and γ1 = γ0 in Eq. (1). The result is shown in Fig. 2(b), it is clear that not only the peak absorption rate can be recovered, but also the bandwidth can be well fitted by the analytical curve, indicating Eq. (3) is valid. Furthermore, the peak absorption rate can also be well fitted by using Eq. (3) together with a constant coupling rate (γ1 = 1.67THz) as shown in Fig. 2(c), which means the variation of relaxation time has no influence on the coupling rate γ1. It is noted that there are a little discrepancies between the numerical and analytical results in Fig. 2(c). Such discrepancies is reasonable because Eq. (3) is the intrinsic loss rate of a plasmonic mode, while the Ohmic losses of the metal reflector and graphene layer II which also contribute the total absorption rate are not incorporated in Eq. (3). The bandwidth and absorption peak variations in Fig. 2(a) could also be intuitively understood from Eq. (3). Note that the resonant bandwidth is 2(γ0 + γ1) according to Eq. (1). As the relaxation time τ increases, γ0 decreases while γ1 keeps unchanged, and therefore the bandwidth γ0 + γ1 decreases monotonously. On the other hand, Eq. (1) shows the resonant absorption peak is given by 4 γ0γ1/(γ0 + γ1)2 which is not a monotonous function with respect to γ0. It first increases as γ0 increases, and reach its maximum value of 100% when γ0 = γ1, and then decrease as γ0 further increases. Such results are in accordance with the observations in Fig. 2(a).

Fig. 3. (a) The evolution of absorption spectra with spacer thickness. (b) Three extracted absorption spectra from the position marked in white dashed lines in (a). (c) The extracted absorption peaks of every spectrum in (a), and the analytical results calculated by Eq. (1) and (4).

Next, we investigate the influence of the configuration geometric parameters on the absorption rates. It is found that many parameters could have an impact on the absorption spectra, such as the height and the fill factor of the grating. But we find the thickness of the


spacer layer (i.e., the length of the FP cavity) could influence the absorption rate significantly, hence in the next we focus on this parameter. We calculate the absorption spectra versus spacer thickness using Comsol Multiphysics. In the simulations, we fixed the relaxation time τ of graphene to 0.4 ps so that the intrinsic loss rate γ0 is unchanged, and varied the spacer thickness from 50 nm to 10 μm to calculate the absorption spectra as presented in Fig. 3(a). From the figure, we can see the resonant peaks of every spectrum is around 10 μm, indicating the spacer thickness has little effect on the dispersions of graphene plasmons. In contrast, the resonant absorption peaks and bandwidths vary periodically with the spacer thickness, and the period is about 2 μm. In every period, the bandwidths first becomes wider, reaches a maximum value, and then becomes narrower. Similarly, the resonant peak first rises till reaching the maximum value of 100%, and then decreases to about 0%. These results signify that by adjusting the spacer thickness, the resonant bandwidth can be controlled and the absorption peak can be tuned from 0% to 100%. To better quantify such trend, we extract three spectra from Fig. 3(a) and plotted them in Fig. 3(b). It shows the resonant wavelengths have small shifts at different spacer thickness, which is caused by the ‘avoid crossing’ behavior as two modes (here are the graphene plasmon mode and the FP cavity mode) couple together. It is also evident that the bandwidth and resonant peaks indeed first increased and then decreased. Figure 3(c) is the extracted absorption peaks from Fig. 3(a), we can clearly see the periodically variation trend. In order to understand the above simulation results, we should know how the coupling rate γ1 relates to the spacer thickness h. It is noted that the absorption peak variation period h0 has a simple relation with the resonant free space wavelength λa as h0 = λa / 2ns, with ns being the refractive index of the spacer layer. To be specific, Fig. 3(c) shows the period h0 = 2 μm, and since the resonant wavelength λa = 10μm and ns = 2.4, therefore λa / 2ns = 2.1 μm ≈h0. This result implies the phenomenon may originate from the FP cavity, where the interference between the incident wave and the reflected wave cause the light intensity to vary periodically. Inspired by this idea, we speculate that the coupling rate γ1 is proportional to the light intensity at the position of graphene layer I. By employing the transfer matrix method, we calculated the light intensity [see Appendix B] and obtained the following relation, γ1 (h) =

4α

n 1+  n

2

1

 2π n cot  cos θ  λ cos θ1 2

a

2

 h cos θ   

2

(4)

2

where α is the coupling rate determined by the grating, n2 represents the effective refractive index of the spacer, dielectric and grating layer. Because the spacer layer is much thicker than the dielectric and grating layer, n2 is mainly contributed from the refractive index of the spacer layer ns, which means n2 ≈ns. n1 is the refractive index of air, and θ1, θ2 are the incident and refraction angle which is zero when light incident perpendicularly (i.e. cosθ1 = cosθ2 = 1 in our case). The term cot (2πn2h / λa) in Eq. (4) indicates that γ1 indeed varies periodically with h, and the period is exactly λa / 2n2, in accordance with the results in Fig. 3. To further check the applicability of Eq. (4), we insert the equation into Eq. (1) and set ω = ωa in Eq. (1) to calculate the resonant peaks. In the calculations, the parameters are as follows: γ0 = 1.25THz in Eq. (1) which corresponds to the electron relaxation time of 0.4 ps of graphene according to Eq. (2), and α = 0.4 THz, n1 = 1, n2 = 2.4 and λa = 10.05 μm in Eq. (4). The obtained analytical results are also plotted in Fig. 3(c), from which we can see the analytical and numerical results agree well, indicating Eq. (4) is a good approximation. Based on the above analysis, it becomes clear that the intrinsic loss rate γ0 is only determined by the electron relaxation time τ of graphene (Eq. (3)), and the coupling rate γ1 relates to the structure parameters of the excitation configuration (Eq. (4)). Complete absorption of the incident light requires γ0 = γ1, which means γ0 and γ1 can be simultaneously larger or smaller to control the resonant bandwidth (given by 2(γ0 + γ1)) while maintaining the


100% absorption. In Fig. 4, two absorption spectra with resonant absorption of 100% but different resonant bandwidth are shown, confirming the bandwidth can be adjusted by controlling γ0 and γ1. Actually, the bandwidth relates directly to the quality factor Q of a graphene plasmonic mode through Q = ωa / 2(γ0 + γ1), indicating that the quality factor can be improved by lower γ0 and γ1 to reduce the bandwidth. Also, it is obvious that the upper limit of the quality factor Qmax is ωa / 2γ0 if we tune γ1 to be zero, and the plasmonic mode can no longer be excited by the incident light and becomes a dark mode. Employing Eq. (3) we can further get Qmax = ωa τ, and since the electron relaxation time τ is proportional to electron mobility [7], hence we can conclude that a high mobility graphene is crucial to the improvement of the plasmonic mode quality factor.

Fig. 4. Two absorption spectra with peak absorption rates of 100% but different bandwidths. The relaxation times, spacer thicknesses for the broad and narrow spectrum are 0.35 ps, 0.85 μm and 1 ps, 1.2 μm, respectively.

Now, let us turn to the band tunability of the excitation configuration by varying the Fermi energy of graphene. In our simulations, all parameters are set the same as in Fig. 1(b) except that the refractive index of spacer is 4.2 and the Fermi energy of the graphene are varying from 0.15 eV to 0.7 eV. For every Fermi energy the absorption spectrum are calculated numerically, and the results are presented in Fig. 5(a), which shows the resonant wavelength varies with the Fermi energy and actually their relations follow the graphene plasmon dispersions [35]. We extracted the spectra at Fermi energy from 0.15 eV to 0.7 eV with intervals of 0.05 eV from Fig. 5(a) and plotted them in Fig. 5(c). From Fig. 5(c), we can see that as the Fermi energy increases, the resonant wavelength moves to the shorter region, and the resonant absorption peak first increase, reach to the maximum value of 100%, and then decreased. The red curve in Fig. 5(c) represents the resonant absorption peaks extracted from Fig. 5(a), showing the absorption rate can maintain more than 90% at the Fermi energy range of 0.25 eV to 0.4 eV, but falling significantly when Fermi energy is out of this range. Next, we employed Eq. (1), (3) and (4) to calculate the analytical resonant absorption peaks and compared them with the numerical results in Fig. 5(c). It is clear that the trend of the numerical results can be recovered by the analytical ones. The above results indicate that the resonant wavelength can be tuned dynamically by Fermi energy without the change of the excitation configuration, which is of great importance in practical applications. But the tunable range that can preserve high absorption rate is relatively narrow. Hence, we analyze why the resonant absorption peak varies with the Fermi energy to determine whether it is possible to enlarge the band tunable range. Recall that the absorption peak is determined by 4 γ0γ1/(γ0 + γ1)2 according to Eq. (1), and γ0 is only connected to the electron relaxation time of graphene which is not varied in our simulations. Therefore, the variation is due to the change of γ1. From Eq. (4), we noted that the variation of resonant wavelength could change the term (n2cosθ1/n1cosθ2)cot(2πn2h cosθ2/λa) in the


denominator. In order to reduce the impact of the variation of this term on γ1, we can either increase n1 or decrease n2, and since n1 is the refractive index of air and can hardly be changed, thus a feasible method is to decrease the refractive index of the spacer layer n2. In other words, the tunable range of the excitation configuration can be enlarged by reducing the refractive index of the spacer layer according to Eq. (4).

Fig. 5. The evolution of absorption spectra with Fermi energy as the refractive index of spacer layer is 4.2 (a) and 1.5 (b). (c) and (d) are extracted spectra and absorption peaks from (a) and (b), respectively. In (c) and (d), the analytical absorption peaks versus resonant wavelength are also shown, which is calculated using Eq. (1), (3) and (4) by setting γ0 = 1.25 THz and α = 0.4 THz.

We validate such speculation by numerical simulations. We calculated the absorption spectra by reducing the spacer refractive index from 4.2 to 1.5, and plotted them in Fig. 5(b) to compare with the previous case with spacer refractive index of 4.2. In the figure, the absorption rate at the regions marked by white dashed rectangles are obviously improved, indicating the band tunable range are now enlarged. We also extract the representative spectra at Fermi energies from 0.15 eV to 0.7 eV with intervals of 0.05 eV, and the results as presented in Fig. 5(d) show the tunable range that preserving up to 80% absorption rate span nearly all the spectrum. To further test the speculation, we also calculated the spectra when the refractive index of spacer is 2.4 (not shown here). As expected, the tunable range in such case is larger than the case with spacer refractive index of 4.2, but smaller that the case with spacer refractive index of 1.5. Therefore, these results clearly show the smaller the refractive index of spacer layer, the wilder the band tunable range. Next, we investigate the performance of the excitation configuration when the light incident angle changes. The parameters in the simulations are the same as in Fig. 5(b), except that now the Fermi energy is not varied but fixed at 0.4 eV, and the incident angle are varied from −90 to 90 degrees to calculate the absorption spectra of the excitation configuration. The spectra are illustrated in Fig. 6(a), it shows the resonant wavelengths are nearly independent of the incident angles, and they are all at about 10 μm. This result can be readily understood because the wavevector of a graphene plasmonic wave is much larger than that of a free space wave, and is mainly provided by the reciprocal vector of the grating. As a result, though the part of free space wavevector parallel to graphene varies as incident angle changes, it


contributes little to the variation of wavevector of graphene plasmonic wave and hence the variation of resonant wavelength. Another important result in Fig. 6(a) is that the resonant absorption rate is also nearly independent of the incident angle. To have a more clear view of this, we extracted the resonant absorption rate of every spectrum in Fig. 6(a), and plotted them in Fig. 6(b). This figure clearly shows the absorption rate can maintain nearly 100% from −60 to 60 degree, and varies rapidly only when the angle is smaller than −75 or larger than 75 degree. In order to understand such phenomenon, we analytically calculated the coupling rate γ1 at different incident angles using Eq. (4) by setting α = 0.4 THz, and find that γ1 also varies slowly in a wide angle range, suggesting the light intensity at the position of graphene layer I is insensitive to the angle variation. Using the γ1, we further calculated the resonant absorption rate by Eq. (1) and compared such analytical result with the numerical ones in Fig. 6(b), which shows the two results are in good agreement, further proving our theory is valid. From the above analysis, it becomes clear that the excitation configuration performs well at a large angle range, and such property could benefit many graphene plasmons based photonic devices.

Fig. 6. (a) The evolution of absorption spectra with the incident angle of light. (b) The extracted absorption peaks from (a) and the analytical results.

At last, we point out some potential applications of the excitation configuration in infrared photonics. The configuration could serve as a platform for SEIRAS, where high quality factor, high absorption rate and wideband tunabilities are crucial to their performances. It could also act as a band tunable infrared photodetector by exploring the photo-thermoelectric effect and introducing asymmetries to separate the hot electrons [23]. Beside these, the excitation configuration could be the building blocks of many other infrared photonic devices, such as broadband infrared sources [24], tunable absorbers, high speed modulators, and efficient nonlinear components [40]. 4. Conclusion In summary, we proposed a propagating graphene plasmon excitation configuration and studied its working mechanisms. It is demonstrated that the excitation configuration could be well described by the temporal coupled mode theory using the intrinsic loss rate γ0 and the coupling rate γ1. Meanwhile, it is found that γ0 is solely determined by the relaxation time of graphene, and γ1 relates to the grating and FP cavity parameters and can be tuned periodically by the spacer thickness. Complete absorption can be reached as γ0 = γ1, and the quality factor of plasmonic mode can be improved by reducing them. Moreover, it is proved that low refractive index of the FP cavity spacer layer is vital to broaden the tunable band of the excitation configuration, and can be well explained by the analytical expressions of the coupling rate. In addition, the excitation configuration is found to be insensitive to the incident angle of light and can work well even when the angle is as large as 75 degree. Our theoretical results is general and could be applied to other graphene plasmonic systems, and


the proposed excitation configuration can serve as a platform for various infrared photonic applications. Appendix A: The derivation of the relaxation rate γ0 In the special case of the coupling rate γ1 = 0, Eq. (1) shows that the bandwidth of the absorption rate is 2γ0. As the quality factor Q of a mode is defined as the ratio of the resonant center frequency to the bandwidth, therefore Q = ωa / 2γ 0 ,

(5)

which means γ0 can be calculated from the quality factor of the graphene plasmonic mode in this special case. Moreover, since γ0 is independent of γ1, thus γ0 calculated in this case is also applicable to other cases in which γ1 is nonzero. Another equivalent definition of the quality factor is as follows,

Q ≡ ωa

Wstored , Ploss

(6)

where Wstored is the stored energy in the graphene plasmonic mode, and Ploss is the loss power of the plasmonic mode. Because we consider the case of γ1 = 0, so the out coupling loss is zero and Ploss is only caused by the graphene Ohmic loss. Combining Eq. (5) and (6), we get

γ0 =

Ploss , 2Wstored

(7)

In the follows, we will calculate γ0 according to such equation. Normally, the energy of a plasmonic mode Wstored can be divided into three parts, i.e., the electric field energy WE, the magnetic field energy WH and the electron kinetic energy WK, and they are related by [41] W = WH + WK E

(8)

For a graphene plasmonic mode, its electric and magnetic field energy are stored in the space out of graphene, while electron kinetic energy are stored in the free electrons in the graphene layer. In lossless dielectric environments, the electric field and magnetic field do not lose any energy, and therefore the intrinsic loss of a plasmonic mode energy is only caused by the loss of electron kinetic energy. By employing the Drude model of graphene conductivity, Ohmic loss power expression Ploss = Re {∫Ω J*•E dr / 2 } and Ohm’s law J =σ E, where Ploss is the Ohmic loss power of electron kinetic energy, Ω represents the graphene layer, J is the current due to electron motions, and E is the parallel electric field in the graphene layer, we can obtain the following concise expressions for the loss power of electron kinetic energy, 2 Ploss = WK .

(9)

τ

Based on Eq. (7) - (9), and note that Wstored = WE + WH + WK, we obtain 2

1

WK W Ploss 1 τ= τ K , γ0 = = = 2Wstored 2 (WE + WH + WK ) 2 (1 + β )WK 2 (1 + β )τ

(10)

where β ≡ WH WK , a parameter that depend on the details of the mode. Equation (10) shows that in general cases, the intrinsic loss rate depends not only on the electron relaxation time τ, but also on the field distribution of the plasmonic mode (since β is a mode dependent


parameter). However, for a highly confined plasmonic mode, the magnetic field energy can be neglected, meaning that β = 0 in this case. Graphene plasmonic modes should belong to such special case, because the dimensions of their mode volumes are about two orders smaller than that of the incident wavelengths. In fact, we numerically calculated the magnetic field energy of propagating plasmonic modes, and find they are typically 2-3 orders smaller than the electric field energy. Therefore, we can neglect the magnetic field energy of a graphene plasmonic mode, and obtain the following equation from Eq. (10), 1 (11) . 2τ In derivation of Eq. (11), we do not employ the specific forms of the plasmonic mode fields. It means that Eq. (11) should be general applicable to all kinds of highly confined graphene plasmonic modes in lossless dielectric environments, including localized graphene plasmonic modes. In addition, when the magnetic energy is negligible, Eq. (7) becomes WE = WK, indicating that 50% of the total energy of a graphene plasmonic mode is stored as electric field energy out of graphene, and the other 50% is stored as electron kinetic energy in the graphene layer. Equation (11) could also be derived using another method. Specifically, by approximating the grating as a uniform medium, we can analytically express the electromagnetic field of the propagating plasmons in a plane wave like form. After that, we can also calculate the stored energy and loss power of the propagating plasmonic modes to obtain γ0 by employing Eq. (7).

γ0 =

Appendix B: The derivation of the coupling rate γ1 The coupling rate γ1 are assumed to be determined by the grating parameters and the light intensity enhancement factor N at the position of graphene layer I, namely, γ1 = αN with α being the coupling rate due to grating in the absence of the FP cavity. We calculate the light intensity enhancement factor based on the transfer matrix method. As shown in Fig. 7, the layers beneath graphene layer I are approximated as an effective medium with refractive index of n2. The graphene layer I is on the interface between air and the effective medium, and its impact on the light intensity is neglected. As a light with an incident angle of θ1 impinges on the device, it will undergo multiple reflection at the upper and lower interfaces of the effective medium, and we first calculate the total reflection rate r. After that, the light intensity enhancement factor N can be calculated by |1 + r|2.

Fig. 7. Schematic of the transfer matrix method calculation of the light intensity.

Using the transfer matrix method, the following relation can be obtained


iδ 1  1 1 r12   0 e  1 1 r23  t  =      ,  r  t  r 1  − iδ   12  12  e 0  t23  r23 1  0 

(12)

In the equation, r12 and t12 are the reflection and transmission coefficient as light propagate from the air to the effective medium, r23 and t23 are the reflection and transmission coefficient as light propagate from the effective medium to the metal mirror, and t is the total transmission coefficient. δ = 2πn2h cosθ2/λ with h being the effective medium thickness (equals the thickness of the spacer layer), θ2 the refraction angle and relate to the incident angle θ1 through Snell’s law, and λ the incident light wavelength. After some algebraic manipulations, and using r23 = −1, we can obtain the total reflection coefficient from Eq. (12), r=

r12 − ei 2δ . 1 − r12 ei 2δ

Then, we calculate the light intensity enhancement using |1 + r|2, and arrive at, N = 1+ r = 2

1−

2 [1 − cos ( 2δ )] 2r12

(1 − r )

2

[1 + cos ( 2δ )]

.

(13)

12

For TM waves, r12 is given by (n2cosθ1 – n1cosθ2) / (n2cosθ1 + n1cosθ2) according to Fresnel equations. Therefore, Eq. (13) can be further cast into, 4

N = 1+ r = 2

. 2  n2 cos θ1  1−  cot δ   n1 cos θ 2  Finally, recalling that γ1 = αN and δ = 2πn2h cosθ2/λ, we eventually obtain Eq. (4) used in the main text, γ1 =

4α

 n cos θ1  2π n2 h cos θ 2   1−  2 cot   λ θ cos n    1 2

2

.

Funding National Natural Science Foundation of China (11374359, 11574308, 61505207, 61405021, 61675037); the Basic Science and Frontier Technology Research Program of Chongqing (cstc2017jcyjA0442, cstc2015jcyjA50018); Natural Science Foundation Project of Chongqing (cstc2017jcyjB0284); the CAS Western Light Program 2016; National High-tech R&D Program of China (863 Program, 2015AA034801).