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Abstract
We proposed a novel perfect absorber with an asymmetric single resonator supporting two degenerate resonant modes, whose operation concept is mimicking a one-port system by making only one of the modes experience loss while using the other for an internal 100% reflector in conjunction with the background scattering. We confirmed the operation principle and the design requirement from a theoretical study using the temporal coupled-mode theory. We also designed an example device based on the guided-mode resonances (GMRs) in a slab-waveguide grating and numerically demonstrated a high absorption of ∼ 99.95% in monolayer graphene with greatly enhanced fabrication error tolerance in comparison to the previously proposed scheme. Our proposed scheme will find various useful applications due to the intuitive design process and relatively easier fabrication, which is attributed to the one-port mimicking operation concept with a single GMR-based broadband flat-top reflector.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Perfect absorption of an incident wave under single-sided illumination is of great importance in many photonic device applications. Especially, a perfect absorber plays an important role to implement a high-efficiency photodetector when an extremely thin layer of an absorbing medium such as graphene is used [1–5].
There have been many studies on resonant structure-based perfect absorbers [6–23]. The perfect absorber, by definition, should have zero reflection and transmission under single-sided illumination. The most straightforward approach to realize this is to use a one-port resonant system which has a 100% mirror on the backside [6–15]. The 100% backside mirror guarantees zero transmission, and in the front (incident) side, zero reflection for the perfect absorption is achieved by balancing the internal loss rate and the external decay (or coupling) rate of the resonator, which is dubbed as the ‘critical coupling’ condition. For the 100% mirror, a distributed Bragg reflector (DBR) or a metal mirror can be used. However, DBR requires a sophisticate growth technique limited to a certain material system or a complicated fabrication process, and metal mirror inevitably induces ohmic loss which is undesirable for the application such as photodetectors. There has been another approach to realize the one-port resonant system based on a prism coupling scheme, in which the total internal reflection (TIR) is employed for zero transmission [16–19]. While the prism coupling scheme can provide an ultra-broad absorption bandwidth owing to its relatively weak resonance feature, its drawback is that it works only for an oblique incidence angle.
In order to avoid the usage of the mirror or the TIR for zero transmission, a perfect absorber based on a single resonator with two modes of opposite symmetry has been proposed, in which the incoming wave from one side excites two degenerate modes of even and odd symmetries simultaneously, and the sum of their absorptions brings about the perfect absorption without the mirror [20]. In this scheme, each mode should satisfy its own critical coupling condition, which is a rather strict requirement and thus, its performance is quite sensitive to the design parameter variation. Moreover, the usage of the two modes of opposite symmetry requires that the resonator should have mirror-symmetry along the wave propagation direction, which will limit practicality or cause design and fabrication complexity. There has been another type of perfect absorber scheme based on a single resonator of broken Cz2 symmetry, in which two opposite-direction propagating modes with nonzero in-plane momentum (${\vec{k}_\parallel } \ne 0$) were used [21]. This scheme works only for an oblique incidence angle, and the design and fabrication of the structure of broken Cz2 symmetry is rather complicated. As an alternative to avoid the specific structural symmetry requirement, a scheme based on two coupled resonators has been proposed, the concept of which is to mimic the one-port system in that only one of the two resonators is lossy and the other plays a role of a resonant 100% reflector [22]. In this scheme, its design process is straightforward due to the simple and intuitive operation concept, and more importantly, its performance shows enhanced tolerance for fabrication error. Another mirror-less perfect absorber based on the all-pass filter composed of two coupled resonators has been studied [23]. In this scheme, the two resonators should have the same losses to satisfy the critical coupling condition, which is quite difficult to implement in practice.
Although the two coupled resonator based perfect absorber scheme mimicking the one-port system has many advantages [22], in general, a single resonator based structure is preferred in terms of fabrication simplicity if no specific structural symmetry is required. Therefore, in this work, we propose a novel asymmetric single resonator based perfect absorber which adopts the one-port mimicking concept. In the proposed structure, the asymmetric resonator has two degenerate resonant modes, one of which functions as a 100% reflector in conjunction with the Fabry-Perot (F-P) like background scattering and the other mode is responsible for absorption. Previously, we have shown that a flat-top broadband reflector can be realized by combining a single guided-mode resonance (GMR) and the F-P like background scattering in a slab waveguide grating, and 100% reflection occurs when the GMR coincides with the resonance of the F-P like background scattering ($r = 0$ or $t = 1$) [24]. So, our proposed structure can be understood as adding a lossy resonant mode into the 100% reflector based on the single GMR with the F-P like background scattering. In order to prove this intuitive and simple approach, the operation principle and the conditions for perfect absorption of the proposed structure has been described with the temporal coupled-mode theory (TCMT) [25]. Based on this, we have designed a perfect absorber with undoped monolayer graphene placed on a slab-waveguide grating (SWG) [26], whose performance has been analyzed with the rigorous coupled-wave analysis (RCWA) simulation (a commercial software program, DiffractMOD) [27].
2. Theoretical description of the proposed perfect absorption
In this section, we formulate the temporal coupled mode theory (TCMT) to deal with our proposed perfect absorber depicted in Fig. 1(a), in which undoped monolayer graphene is placed on a SWG without external backside mirror. The proposed SWG structure is vertically asymmetric, and thus can be modeled as an asymmetric single resonator coupled to two external ports (U- and D-port) as shown in Fig. 1(b), where it is assumed that two resonant modes $({{a_1},{a_2}} )$ are supported. The waveguide-like structure in Fig. 1(b) represents the wave propagation channel through which the two resonant modes are indirectly coupled due to the partial reflections, resulting in the identical mutual coupling coefficient (${\mu _{12}} = {\mu _{21}} = \mu $) and the modified resonance frequencies of the modes (${\omega _i} + j\frac{{{\gamma _i} - \gamma _i^\ast }}{2}$) [28]. The whole structure is just the same as the one considered for the unidirectional emitter in our previous work except that one of the modes experiences loss rather than optical gain [28]. So, considering the mutual coupling between the modes, the dynamic equation of mode amplitudes $({{a_1},{a_2}} )$ can be described as [25,28]
A is a $2 \times 1$ vector of the resonant mode amplitudes, where ai is normalized such that |ai|2 represents the energy stored in i-th mode, and ${s_{U + }}\; $ and ${s_{U - }}$ (${s_{D + }}\; $ and ${s_{D - }}$) are the amplitudes of incoming and outgoing waves at U-port (D-port), which are normalized such that |s|2 represents the power carried by the wave. ${\mathrm{\Omega }_m}$ represents the modified resonant frequencies, and ${\mathrm{\Gamma }_m}$ describes the decay and mutual coupling property of the resonant modes, where ${\gamma _i}$ is the decay coefficient of the i-th mode and $\mu $ is the mutual coupling coefficient between the two modes. Due to the mode coupling through the partial reflections in the propagation channel, ${\gamma _i}$ and $\mu $ are complex values in general, which results in the resonant frequency modification [28]. ${{\cal L}}$ represents non-radiative loss, and it is assumed that only one of the modes (${a_1}$) experiences the non-radiative loss in this work, which will be justified later. K is the coupling coefficient matrix between resonant modes and input $({{s_{U + }},\; {s_{D + }}} )$ where ${\kappa _{pq}}$ (p: U, D, q: 1, 2) indicates the coupling coefficient of p-port and q-th mode.
Fig. 1. (a) Schematic of the proposed perfect absorber composed of undoped monolayer graphene and a SWG, where nSWG = 3.40, nSub = 1.45. FF is defined as the ratio of grating width (wGrat) to Period. The red thin layer indicates monolayer graphene of tG = 0.34 nm thickness as an absorbing medium. (b) Schematic of a theoretical model for a two-port system composed of an asymmetric single resonator with two resonant modes and a wave propagation channel with partial reflections.
Considering the background scattering due to the partial reflectivity in the channel coupled to the external ports, the outgoing waves are related to the incoming waves and the mode amplitudes as
The system described by Eq. (1) and Eq. (2) should satisfy the energy conservation and time-reversal symmetry when the non-radiative loss is not considered (${{\cal L}} = 0$). From the energy conservation, ${M_{decay}}$ and ${\mathrm{\Gamma }_m}$ are related as $M_{decay}^\dagger {M_{decay}} = \mathrm{\Gamma }_m^\dagger + {\mathrm{\Gamma }_m}$. From the time reversal symmetry, we can derive the relations between $\textrm{K},{M_{decay}}$, and ${M_{BS}}$ as ${M_{BS}}M_{decay}^\ast{=}{-} {M_{decay}}$ and $K = {M_{decay}}$. From these relations and the assumption of $r = 0$, we can derive the relations among the coupling coefficient, the decay coefficients, and the mutual coefficient as
The phases of the coupling coefficients are related as
where ${\varphi _{D1,2}}$ and ${\varphi _{U1,2}}$ are the phases of ${\textrm{d}_{D1,2}}$, and ${\textrm{d}_{U1,2}}$, respectively.Assuming single-sided illumination from U-port $({{s_{U + }}\; \ne 0,\; {s_{D + }} = 0} )$, from Eq. (1), at the steady state, the mode amplitudes (${a_1}$ and ${a_2}$) are related to ${s_{U + }}$ at an operating frequency of $\omega $ as follows:
Substituting Eq. (6) into Eq. (2), we have the transmission and the reflection coefficients as
Since we are assuming that only mode 1 (a1) experiences loss, it is reasonable to assume that perfect absorption occurs at the resonant frequency of mode 1, that is, $T({{\omega_o}} )= R({{\omega_o}} )= 0$, where ${\omega _o} = {\omega _1} + j\frac{{{\gamma _1} - \gamma _1^\ast }}{2}$. Substituting Eq. (3) and Eq. (4) into Eq. (7) with the assumption that the resonance of the F-P like background scattering coincides with the perfect absorption frequency ($r = 0$ at $\omega = {\omega _o}$), zero transmission ($T({{\omega_o}} )= 0$) requires
Since non-radiative loss (${\gamma _{loss}}$) is a real value, Eq. (9) is satisfied only when the two modes are degenerated, so that we have
Then, Eq. (9) becomes
Next, with the same assumption and procedure, from Eq. (8), zero reflection ($R({{\omega_o}} )= 0$) condition is found as
Finally, from Eq. (11), Eq. (12), and Eq. (4), we can determine the non-radiative loss of mode 1 (${\gamma _{loss}}$) and the mutual coupling coefficient ($\mu $) for perfect absorption as
Equation (13) can be interpreted as the critical coupling condition for perfect absorption in our system, where one can note that only the decay rate of mode 1 is involved, not the decay rate of mode 2. This implies that the role of mode 2 is a reflector, proving that our approach of mimicking one-port system works.
Another important thing to note is that the zero-transmission condition given by Eq. (11) is not related to the non-radiative loss (${\gamma _{loss}}$). This implies that even for the lossless (${\gamma _{loss}} = 0$) system, the condition for T = 0 will be the same as Eq. (11). This can be easily confirmed from Eq. (9). Note that in the lossless system, T = 0 automatically gives us |R| = 1, resulting in 100% reflector function. From this, we can get an idea that the perfect absorber can be designed in two steps. The first step is to design a passive (lossless) resonator of two degenerate resonant modes having 100% reflection at the target frequency. Then, the second step is to fine-tune the structural parameters to satisfy the critical condition for a given loss value.
3. Perfect absorber design based on a slab-waveguide grating
Based on the TCMT analysis, a mirror-less perfect absorber has been designed by mimicking one-port resonant system, whose schematic is shown in Fig. 1(a). The designed perfect absorber consists of a SWG of Si stacked on a glass substrate, and monolayer graphene is placed just above the SWG. Although an identical material is assumed for ridge and slab region for design simplicity in this work, different materials can also be used. Refractive indices of Si and glass are 3.40 and 1.45, respectively. For the permittivity of graphene, Kubo formulation [7,29] was used with a graphene thickness of 0.34nm, Fermi velocity of 106 m/s, and mobility of 0.5 m2/Vs. In this work, undoped monolayer graphene is assumed. The light of transverse electric (TE) polarization, whose electric field is perpendicular to the incident plane, is incident normally from the air. Optimal design parameters for perfect absorption (A > 99.9%) are as follows: Period = 1 μm, dGrat = 0.498 μm, dSlab = 0.318 μm, FF = 0.535, λ = 1.53519 μm, where FF is the fill factor defined as wGrat/Period. In the optimization of the design parameters, we investigated the reflection (|R|2) and absorption (A) spectra and the field profiles of various resonant modes in the SWG by using the RCWA simulation [27].
Figure 2(a) shows the reflection map as a function of FF when only the graphene layer is removed in the designed optimal absorber, where GMR modes with a diffraction order (m) and a slab waveguide modal order (n) are indexed as GMRmn. As seen in the field profiles in Fig. 2(a), the diffraction order (m) is related to the number of the null points in the field profile within one period in the horizontal direction, which is 2 m, and the waveguide modal order (n) is the same as the number of the peak points in the field profile in the vertical direction. To clearly show the related GMR mode of each resonant reflection branch, the field (|Ey|) profiles for FF = 0.99 are shown, where the branches are distinctly separate due to high quality factors stemming from extremely weak scattering strength of the grating. For moderate values of the fill factor of our interest (∼ 0.5 < FF < ∼ 0.6), the GMR modes of the first diffraction order (in particular, GMR13) show relatively lower quality factors and bring about broadband flat-top reflection spectrum in conjunction with the F-P like background scattering [24]. Whereas, the second diffraction order mode, GMR21 shows much higher quality factor even for the moderate fill factors due to the strong field confinement in the slab region of the SWG as seen in the field profile (FF = 0.6) of Fig. 2(a), which is the appropriate feature for absorption enhancement. So, it is quite natural to choose the second order diffraction GMR mode for absorption function and the first diffraction order GMR mode for an internal reflector function. Moreover, when monolayer graphene is placed on the slab region of the SWG as depicted in Fig. 1(a), the loss experienced by the first diffraction order GMR mode is quite negligible since most of field is confined in the ridge region of the SWG, which is another suitable feature for mimicking one-port system, resulting in enhanced tolerance for fabrication error.
Fig. 2. (a) Reflection map as a function of FF when only the graphene is removed for optimized graphene perfect absorber. The inset shows the normalized electric field distribution (|Ey|) at seven different points marked by open circles: FF = 0.99 (top), FF = 0.6 (middle), FF = 0.535 (bottom). (b) Absorption map as a function of FF for optimized graphene perfect absorber. The inset on the right indicates the close-up view of dashed box area. In all the calculations, Period = 1 μm, dGrat = 0.498 μm, dSlab = 0.318 μm.
For perfect absorption, the degeneracy of those two modes is needed as discussed in the previous section. The sharp reflection peak branch of GMR21 and the broad reflection branch of GMR13 intersect near the point of FF = 0.535, λ = 1.53519 μm, which corresponds to the optimal design point. One can see that almost perfect absorption (A > 99.95%) is achieved at the point in the absorption map plotted in Fig. 2(b), where the situation is the same as considered in Fig. 2(a) except that monolayer graphene is included. At the optimal point, the field profile in the absorber is very close to GMR21 mode, proving that the GMR21 mode is dominantly responsible for the absorption. So, the main absorption peak locus in Fig. 2(b) follows the sharp reflection peak branch of GMR21 mode in Fig. 2(a). Even when there is no internal reflection function, that is, the GMR21 mode peak does not overlap with the high internal reflection peak branch of GMR13 mode, GMR21 mode alone shows somewhat enhanced absorption due to the enhance graphene-guided-mode interaction at the GMR, which, however, never reach 100%. Much more enhanced absorption close to 100% occurs only with help of the broad internal reflection function of GMR13 mode.
For more detailed analysis, the reflection spectrum of the designed structure without graphene (Fig. 3(a)) and the spectra of the reflection and the absorption of the designed structure (Fig. 3(b)) are plotted. One can see that the degeneracy between the two GMR modes is incomplete in the present absorber design, which is mainly due to a finite resolution in the structural parameter variation. In this work, we used a resolution of 1 nm considering practical fabrication errors. However, in spite of this, almost perfect absorption (A > 99.95%) was achieved at the resonant wavelength of the high quality factor GMR mode. This is because the low quality factor GMR mode offers still high enough reflection of ∼ 0.999 at the resonant wavelength of the peak absorption. This shows the importance of the broadband flat-top reflection coming from the interaction between the first diffraction order GMR mode and the background scattering and the possibility that the proposed perfect absorber scheme may be tolerant to the design parameter variation. Another important thing to note in Fig. 3(a) is that the introduction of sharp GMR21 does not distort the broadband reflection property of GMR13, unlike the most of transmission filters based on multiple GMRs [30]. This is attributed to the proper indirect (or mutual) coupling strength between GMR13 and GMR21 through the properly designed partial reflection in the internal wave propagation channel.
Fig. 3. (a) Reflection spectrum (when only the graphene is removed), and (b) reflection and absorption spectra for optimized graphene perfect absorber, where Period = 1 μm, dGrat = 0.498 μm, dSlab = 0.318 μm, FF = 0.535.
The tolerance of the designed absorber for the structural parameter variation was numerically analyzed and compared to the scheme based on two modes of opposite-symmetry [20]. As seen in Fig. 4, our proposed scheme shows approximately an order of magnitude larger tolerance in comparison to the other scheme, confirming that the advantage of the one-port mimicking concept is maintained in our asymmetric single resonator based scheme.
Fig. 4. Fabrication tolerance in the optimal conditions for (a) the proposed SWG perfect absorber (Period = 1 μm, wGrat = 0.535 μm, dGrat = 0.498 μm, dSlab = 0.318 μm, λ = 1.53519 μm) and (b) the HCG perfect absorber as a reference (Period = 1 μm, wGrat = 0.953 μm, dGrat = 0.2914 μm, λ = 1.42007μm). The inset on the top shows a 2D schematic of each perfect absorber.
The quality factor Q is ∼ 1 × 103 (in detail, Q = λo/Δλ = 1535.19nm/1.52nm = 1,010) for the optimized graphene perfect absorber, as estimated in Fig. 3(b). The Q-factor can be improved by applying high order resonant modes, which can be supported in absorber structure of a thicker slab thickness. The additional details are presented in Supplement 1.
4. Conclusion
In this work, we proposed a novel perfect absorber scheme based on an asymmetric single resonator with two degenerate GMR modes adopting the one-port mimicking concept in which only one mode experiences loss and the other functions as a reflector in conjunction with the background scattering. With the TCMT formalism, the operation concept has been confirmed, and the design requirements have been derived, which reveals that a properly chosen mutual coupling strength between two modes is important. Based on the theoretic analysis, a perfect absorber has been designed, where the GMR modes in a SWG are used. An almost perfect absorption (A > 99.95%) was numerically demonstrated, and our designed device showed enhanced fabrication error tolerance which was approximately an order of magnitude larger in comparison to the scheme based on two modes of opposite-symmetry.
Since our proposed perfect absorber structure does not require specific structural symmetry, its design is straightforward and its fabrication will be easier. Our works can be also applied for different kind of absorbing materials beside graphene. Therefore, we believe that our proposed scheme will find various useful applications such as optical sensors, solar cells, thermal emitters, and nonlinear optics, as well as photodetectors.
Funding
National Research Foundation of Korea (2020R1A2B5B01002681, 2021R1A4A1033155).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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