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We theoretically investigate resonant absorption in a multiple-port surface-plasmon polaritons (SPP) resonator near the condition of critical coupling at which internal loss is comparable to radiation coupling. We show that total absorption is obtainable in a multiple-port system by properly configuring multiple coherent lightwaves at the condition of critical coupling. We further derive analytic expressions for the partial absorbance at each port, the total absorbance, and their sum rule, which provide a non-perturbing method to probe coupling characteristics of highly localized optical modes. Rigorous simulation results modeling a surface-plasmon resonance grating in the multiple-order diffraction regime show excellent agreements with the analytic expressions.
©2010 Optical Society of America
1. Introduction
Coupling of optical power between waveguides and resonators is of use in many photonic devices and systems [1–4]. On coupling a free-space input wave to a periodic waveguide, a leaky-mode resonance can localize light thereby enhancing light-matter interaction [5,6]. Effective confinement of light at a resonance site can yield high absorption if the site possesses dissipative loss [7]. Many means providing strong absorption of light have been suggested utilizing, for example, attenuated total reflection prisms [8,9], diffraction gratings [10,11], metallic mesoporous surfaces [12], or metamaterials [13]. These resonance structures can be mapped onto the same category of resonators, namely dissipative open resonators [14].
Substantial light absorption can be induced by the excitation of surface-plasmon polaritons (SPP) on metallic nanostructures. Whereas Wood’s anomalies were observed in reflectance spectra of corrugated metal surfaces [10], enormous interest in SPP resonance phenomena has arisen owing to their considerable role in contemporary nanoplasmonics [15]. Although the various nanoplasmonic systems may exhibit diverse embodiments, each possesses features that are inherent in dissipative resonators: a metal-dielectric interface forms a resonator; heat dissipation in the metal represents an internal resonator loss; SPPs trapped at the interface have well-defined eigenfrequencies and eigenmodes.
Several analytic approaches treat such resonance systems including use of the Fresnel reflection formula [8,16], multiple interference model [17], scattering matrix formalisms [7], or temporal coupled-mode theory [2,18]. A common conclusion is that total absorption of electromagnetic waves, thus maximum enhancement of localized intensity, occurs only at the condition of critical coupling [1,2,12,14,19], when the internal loss rate (γ int) equals the radiation coupling rate (γ rad). In particular, the critical coupling condition for SPP resonators was derived by analogically considering the excitation of SPP modes as a problem of surface wave localization on a one-dimensional dissipative oscillator with semitransparent walls [7]. This oscillator model also suggested that the transmission coefficients of the two walls, which play the same role as the coupling coefficients of a resonator with two coupling ports, can be estimated in a phenomenological way by measuring the transformation ratio of incoming waves into evanescent waves. It is formidable, however, to apply this oscillator model to a general SPP system because of its particular resonance geometry with two coupling ports. Therefore, it is of interest to develop an analytic model to derive the critical coupling condition of γ rad = γ int in a universal way, especially for SPP resonators with multiple coupling ports.
In this paper, we use a multiple-port resonator model grounded in lossless temporal coupled-mode theory [2,18]. Upon introducing a small internal loss, we find that the model provides a thorough understanding of resonant absorption phenomena of multiple-port, dissipative SPP resonators. We also present rigorous simulation results that show an excellent agreement with our analytic estimation of resonant absorption at a multiple-port, grating-coupled SPP resonator.
2. Temporal coupled-mode theory for a dissipative resonator
Figure 1 schematically illustrates a resonance mode, g(t), coupled with N pairs of incoming (fm+) and outgoing (fm –) radiation modes. We assume that |g|2 and |fm ± |2 are normalized to represent energy content of the resonance mode and power transported by the ports’ radiation modes, respectively, and that g(t) dissipates its energy via both internal loss and radiation coupling with a total damping rate of γ tot = γ rad + γ int. Temporal behavior of g(t) resonant at a frequency ω 0 can be generally described by the coupled-mode equations using a vector notation of |X ± 〉 = (X 1 ± … Xm ± … XN ± )T in the limit that γ tot/ω 0 << 1 [2,18]:
Fig. 1 Schematic of a general SPP resonator with N coupling ports. N pairs of incoming ( + ) and outgoing (-) radiation modes expressed by fm ± (t) are coupled to a single resonance mode with amplitude g(t). Cnm and κm ± represent direct scattering coefficients from port m to port n and coupling coefficients between g(t) and fm ± (t) at port m, respectively.
We assume modal amplitudes to describe slowly varying envelope of time-domain electromagnetic fields such that
for the radiation modes and for the resonance mode, e.g., SPP. {e m(r), h m(r)} and {e R(r), h R(r)} are normalized field solutions of frequency-domain Maxwell equations for incoming radiation mode at port m and that for the resonance mode, respectively. Thus, {e m*(r), –h m*(r)} is a field solution for outgoing radiation mode at port m. (C.C) represents the complex conjugate of the left-side term. Accordingly, Eq. (1) is slightly different from the formulation used by Refs [2]. and [18] in the factor g(t) describing the harmonic mode.Energy conservation and time-reversal symmetry in mode coupling enforce fundamental constraints on the relation of the direct scattering matrix C and the mode coupling coefficients |κ ± 〉: CC † = I, |κ +〉 = |κ –〉 ≡ |κ〉, 〈κ|κ〉 = 2γ rad, and C|κ〉* = –|κ〉 as shown by Fan et al. [18]. For later convenience, we define |κm|2/2 = γm as a partial radiation coupling rate at port m and γ rad = (γ 1 + … + γm + … + γN) as a total radiation coupling rate. Although an internal loss is introduced in Eq. (1), the constraints between the coupling coefficients are identical to those of the lossless resonator discussed in [2,18] as a result of two assumptions: time-reversal symmetry of coupled-mode Eqs. (1) and (2) under a loss-gain interchange (γ int → –γ int) and lossless direct scattering, i.e., C is unitary (see Appendix for detailed discussions).
3. Critical coupling, total absorption, and sum rule in partial absorbances
Consider a case of total resonance absorption when |f +〉 ≠ 0 and |f –〉 = 0. In this instance, with the help of the fundamental constraints, the coupled mode Eqs. (1) and (2) are reduced to
at ω = ω 0. In Eq. (8), it is evident that the outgoing radiation mode is suppressed by destructive interference between the directly scattered part C|f +(t)〉 and the leakage radiation part g(t)|κ〉 [2,18]. In the absence of the outgoing radiation mode, Eq. (7) gives a steady state solution for g(t) only if γ rad = γ int (critical coupling); otherwise g(t) is exponentially growing (γ rad > γ int) or become evanescent (γ rad < γ int) with time. By temporally growing g(t) for γ rad > γ int, the leakage radiation part g(t)|κ〉 also temporally increases with g(t). Thus, to satisfy Eq. (8) we should grow |f +(t)〉 at the same rate as g(t) so that the directly scattered part C|f +(t)〉 cancels an excess of the leakage radiation part. On the other hand, in case of γ rad > γ int, keeping |f –〉 = 0 necessarily requires the exponential decrease of |f +(t)〉. Finally, the following results show that the fundamental physics of total resonance absorption in a multi-port system is identical to that of a single-port system [2,16,17,19]. (i) The incoming radiation mode is strongly confined to the resonance mode due to destructive interference in the outgoing radiation mode. (ii) The destructive interference becomes complete when γ rad = γ int because amplitudes of the two interfering parts balance so that they exactly cancel each other; otherwise the outgoing radiation mode survives with excessive leakage radiation (γ rad > γ int) or direct scattering (γ rad < γ int). (iii) The resonance mode grows until its internal loss dissipates the same power as that coupled from the incoming radiation mode; i.e., incoming radiation is totally absorbed by the internal dissipation of the resonance mode.In a multi-port system, however, the destructive interference must occur simultaneously at all outgoing ports to achieve total absorption. In other words, magnitudes and phases of incoming radiation modes at all ports should be orchestrated properly. The incoming mode configuration for total absorption is found by solving Eqs. (7) and (8) with the fundamental constraints of the coupling constants. At the critical coupling condition (γ rad = γ int), they yield
where F 0 is an arbitrary constant. Note that |κ〉* can be interpreted as time reversal of the leakage radiation mode, i.e., phase-conjugated leakage radiation, as |κ〉 represents leakage radiation for the unit excitation of g (refer to the second term on the right-hand side of Eqs. (2) and (8)). Thus, outgoing radiation modes at all ports are suppressed simultaneously due to destructive interference if magnitudes and phases of the incoming radiation modes at all ports are given by the time-reversal form of the leakage radiation mode. Finally, we can conclude that total resonance absorption is obtainable for a multiple-port system by having the incoming radiation mode given by Eq. (9) at the critical coupling condition of γ rad = γ int.With the incoming radiation mode given by Eq. (9) at an arbitrary frequency ω, we obtain spectral responses of the outgoing radiation mode and absorbance as
respectively. An important observation from these solutions is that the outgoing radiation mode can be expressed by a scalar multiple of the time-reversed incoming radiation mode aswhere ϕ = arg(F 0). Note in Eq. (12) that a set of operations that t → –t and the complex conjugation of a modal amplitude produces its exact time reversal (see Appendix for details). Thus, |f +(–t)〉* on the right-hand side of Eq. (12) represents time reversal of the incoming radiation mode. This means that scattering from this particular configuration of incoming radiation modes to all available ports acts like reflection in a single-port resonance system with its reflection coefficient given byThus, total absorbance in Eq. (11) is identical to absorbance obtained for a single-port system by previous theories such as Kretchmann theory [8,16], multiple interference model [17], and quantum Hamiltonian mapping [12].
In many practical cases, however, the incoming radiation mode is not given by a phase-conjugated leakage radiation stated in Eq. (9) but is made incident at one particular port, producing outgoing radiation modes at all available ports. With a single-port incidence, the interferences between direct scattering and leakage radiation parts at all ports cannot be identically destructive and, thus, the incoming energy also cannot be totally absorbed even at the critical coupling condition. The partial absorbance in this case can be derived by considering the excitation strength of the resonance mode. As resonance absorption is described by internal decay of the resonance mode, absorbing power must be P abs = 2γ int|g|2 and the absorbance is a ratio of Pabs to incident power Pinc, i.e., A = Pabs/Pinc. For a single incoming radiation at port q with amplitude Fq and frequency ω, the partial absorbance is
It is worth noting that Eqs. (11) and (14) allow an absorbance sum rule such that
where radiative decay probability η rad is the ratio of γ rad/γ tot. Note again that the accumulated peak absorbance is unity only when η rad = 0.5, that is, γ rad = γ int. We may therefore conclude that the critical coupling condition is a universal constraint for achieving total absorption at a dissipative open resonator and is not limited to a specific geometry or number of coupling ports. Another noteworthy result gathered from the sum rule in Eq. (15) relates to light-emitting applications [20]. By externally measuring peak absorbance at every single port, we can determine relative coupling rates (or relative magnitudes of the coupling coefficients) of all ports by the relations of An/Am = γn/γm = |κn/κm|2 and Σ Aq(ω 0) = 4η rad(1–η rad). Therefore, external measurement of absorption peaks is a practical way to investigate the details of coupling at each port and those of multiple-port plasmonic resonators.Based on partial absorbance measurements, one can easily find the coherent configuration of multiple incoming radiation modes for total absorption, i.e., Eq. (9). We refer this particular form of incoming radiation mode to |ψ〉 ≡ U 1/2|κ〉*, where U is a normalization factor. First, its relative power at port m, |ψm|2, can be simply given by the partial absorbance Am(ω 0) since |ψm|2 = U|κm|2 = U·γm and Am = (γ int/γ tot 2)·γm. If we normalize |ψ〉 so that 〈ψ|ψ〉 = 1, then U = γ rad –1 and
Second, the phase, arg(ψm), can be found by the consecutive maximization of accumulated absorption. Suppose that double radiation modes are incoming at port 1 and 2 with their powers properly given according to Eq. (16) but their relative phases are unknown. In this case, incoming mode amplitudes at two ports can be written by f 1+ = ψ 1exp(iϕ 1) and f 2+ = ψ 2exp(iϕ 2),where ϕ 1 and ϕ 2 is an arbitrary initial phase at port 1 and port 2, respectively. The absorbance in this case of a double port incidence is
The absorbance in this case is phase sensitive and maximized to A 1 + A 2 at the phase matching that ϕ 1 = ϕ 2 as a result of maximization in the resonance mode amplitude. Thus, with port 1 as a reference port, consecutive maximizations of the absorption over all remaining ports finally yield an incoming mode configuration |f +〉 = exp(iϕ 1)|ψ〉 whose magnitudes and phases at all ports are orchestrated so that the outgoing modes at all ports are suppressed simultaneously due to destructive interference while the resonance mode is maximally excited. It is worth noting in Eq. (17) that if two incoming radiation modes are incoherent relative to each other, the term sin2[…] is time averaged to a value 1/2 and A (2) = (A 1 2 + A 2 2)/(A 1 + A 2), which is always less than A 1 + A 2 in the coherent case.
4. Phasor representation of absorption response
As discussed in Sec. 3, scattering in a multi-port system reduces to a simple reflection response when the incoming radiation modes at all ports are coherent and are given by a scalar multiple of a phase-conjugated leakage radiation mode. The total reflection coefficient in Eq. (13) can be rewritten as
where α(ω) = tan–1[(ω–ω 0)/γ tot]. By representing ρ tot on a complex plane [2], we can intuitively explain resonance behavior of a multiple-port, dissipative resonator in the vicinity of critical coupling.Figure 2(a) shows a phasor representation of ρ tot(ω) as a vectorial superposition of σ(ω), leakage radiation amplitude (black arrow), and –1, directly scattered amplitude (red arrow). Note that σ(ω) = η rad[1 + exp(i2α)], the first term on the right-hand side of Eq. (18). σ(ω) traces a circle with radius η rad as explicitly shown by the term exp(i2α). By increasing ω from lower to upper far off-resonance limits, σ(ω) rotates counterclockwise from the lower (2α = –π) limit to the upper limit (2α = + π) via σ(ω) = 2η rad at ω 0 (blue dot at 2α = 0) while the directly scattered amplitude (red arrow) remains constant. ρ tot(ω) is now represented by a blue arrow directing a point at 2α on the blue circle that crosses the real axis at ω = ω 0. Therefore, the absorption spectrum defined by A tot(ω) = 1−|ρ tot(ω)|2 in Eq. (11) can be geometrically obtained by the square of the blue segment, A tot 1/2, which is stretched perpendicularly from the ρ tot(ω) to the rim of the outer unit circle.
Fig. 2 Resonance behavior of a dissipative SPP resonator for the three cases of under coupling, critical coupling, and over coupling. (a) Phasor representation (blue arrow) of the outgoing response, ρ tot(ω). (b) Circular traces of ρ tot(ω) for three different cases: red, green, and blue circles correspond to under coupling (γ rad = 0.5γ int), critical coupling (γ int = γ rad), and over coupling (γ rad = 4γ int), respectively. Absorption and phase spectra are depicted in (c) and (d), respectively, for the three cases with the same colours as in (b).
Figure 2(b) schematically shows the spectral traces of ρ tot(ω) for three typical cases of under coupling (red circle when γ int > γ rad), critical coupling (green circle when γ int = γ rad), and over coupling (blue circle when γ int < γ rad). Grey unit circle represents the lossless case (γ int = 0). Corresponding absorbance and reflection phase spectra shown in Figs. 2(c) and 2(d) intuitively explain all essential features in the Brewster absorption phenomenon [21,22]. The absorption peak is unity only at critical coupling because the green circle can cross the origin at resonance. The phase spectra in Fig. 2(d) also show quite different resonance behaviors for the three cases. An abrupt π-phase jump occurs at resonance frequency as the system satisfies the critical coupling condition. In the case of under coupling, the phase behavior reveals a peak/dip profile with a phase difference of Δβ max. It is easy to show geometrically that sin(Δβ max/2) = η rad/(1−η rad) as an alternative way to estimate η rad.
5. Consistency with rigorous simulation
To numerically confirm the critical coupling condition of total absorption and the absorbance sum rule in a multiple-port SPP resonator, we present in Fig. 3 an example of grating induced SPP resonance. Figure 3(a) shows a single SPP mode propagating along the + x direction on an Ag surface (the x-z plane) corrugated by a periodic array of 25-nm-deep and 175-nm-wide (FWHM) Gaussian grooves with a period (Λ) of 700 nm. The SPP mode is coupled with multiple ports; for example, it is coupled with two ports, Port 1 and Port 2, carrying two incoming modes, f 1 + and f 2 +, and two outgoing modes, f 1– and f 2– as depicted in Fig. 3(a). Each of the incoming modes excites an SPP mode via diffraction under phase-matching condition that kx = (ω/c)sinθm = k SPP–2πm/Λ, where m denotes the diffraction order. The incoming mode fm+ incident at an angle θm excites the SPP mode via + m order diffraction, and its zero-th order reflection corresponds to fm –. The excited SPP mode also loses its energy toward f 1– and f 2– as leakage radiation modes.
Fig. 3 Numerical simulation results of strong absorption induced by a grating coupled SPP resonator. (a) Schematic of the SPP resonator with a metal grating of 700-nm-period Gaussian grooves. (b) Zero-th order reflectance (R 0) showing SPP dispersion relations of which curves are marked by diffraction order of m. (c), (d), and (e) show the peak absorbances at three different resonance wavelengths. The square symbols and the continuous curves respectively represent the peak values obtained numerically and analytically.
The numerical calculation used here is a coordinate transformation algorithm known as Chandezon method [23]. Ag is modeled as a Drude metal whose plasma and collision (Γ0) frequencies are 6.35 × 2πc/μm and 0.125 × 2πc/μm (c is speed of light in vacuum) at room temperature (300 K), respectively [24]. The dark bands on the R 0 spectrum in Fig. 3(b) clearly show periodic series of SPP dispersion curves. At three different wavelengths of λ 0 = 825 nm, 580 nm, and 440 nm marked by the horizontal dot-lines, the grating configuration corresponds to an SPP resonator with the number of coupling ports, N = 1, 2, and 3, respectively. The two-port case depicted in Fig. 3(a) is thus fit for λ 0 = 580 nm, and the circular dots on the dispersion curves are for the two pairs of incoming and outgoing plane waves forming two coupled ports.
Dependences of A tot(ω 0) and Aq(ω 0) on η rad are indicated in Figs. 3(c), 3(d), and 3(e). The numerical results are represented by the red, blue, green, and black squares for A 1, A 2, A 3, and A tot, respectively. In the numerical calculation, we first find a scattering matrix W describing the system response such that |f –〉 = W|f +〉 and then the absorbance at each port is obtained by the relation of Aq = 1−Σp |Wpq|2. By obtaining six different values of η rad starting from the left to the right shown by square symbols in all three plots, the collision frequency Γ of Ag is varied by an order of 10%, 20%, 40%, 60%, 80%, and 100% of that (Γ0) at room temperature. In the low-collision frequency limit, γ int is linearly proportional to Γ while γ rad remains constant; therefore, the exact values of η rad are obtained by a linear extrapolation of the absorption bandwidth at Γ = 0. The analytical results obtained from Eqs. (14) and (15) are also plotted by the solid curves [25]. Note that the difference between the square symbols and the analytic curves in the whole η rad range is less than 0.02; this excellent agreement strongly supports our analytic theory. We also confirmed that det(W) = 0, which means A tot = 1, at Γ/Γ0 = 0.4837 and 0.4396 for λ 0 = 580 and 440 nm, respectively, of which η rad values are exactly equivalent to 0.5 at those collision frequency ratios. Consequently, we can say that the results in Figs. 3(c)-3(e) confirm the universality of the critical coupling condition for total absorption at a dissipative open resonator with multiple coupling ports.
6. Conclusion
We have explored the resonance behavior of multiple-port, dissipative surface-plasmon resonators near the critical coupling condition. Generalizing the temporal coupled-mode theory of leaky-mode resonators by incorporating a small internal loss provides an analytic expression of absorption spectra that explicitly reveals the universality of the matching condition, not limited to specific resonance geometries and the number of coupling ports. The phasor representation of the expression intuitively explains all the main features of complex resonance behavior in the vicinity of critical coupling.
Our model depends only on internal loss and radiation coupling rates; therefore, we can investigate the internal characteristics of resonators in depth by externally measuring the peak absorbance at each coupling port. In particular, the sum rule for peak absorbance will be a practical guideline for estimating light extraction efficiency of nanoplasmonic light emitters [20] because efficient emission of light mediated by SPP resonance is analogous to the time reversal of strong light absorption in a dissipative leaky-mode resonator.
Appendix: The fundamental constraints of coupling constants in a dissipative system
The loss-gain interchange requirement for time reversibility corresponds to the conjugation invariance (C-invariance) of linear electromagnetism, which has been often quoted to explain propagation characteristics of electromagnetic waves in negative-index metamaterials. The discussion concerning field characteristics in a negative-index metamaterial is based on the invariance of frequency-domain Maxwell equations under the set of operations that {E, H} → {E*, H*} and {ε, μ} → {–ε*, –μ*} [26,27]. The C-invariance has another interpretation for the time reversal of fields since frequency-domain Maxwell equations are also invariant under a set of operations that {E, H} → {E*, −H*} and {ε, μ} → {ε*, μ*}. Therefore, if {E, H} is a solution of frequency-domain Maxwell equations in a system with {ε, μ}, then {E*, −H*} must be a solution for the conjugated system with {ε*, μ*}. Physical interpretation of this fundamental property is that the field-scattering process in a dissipative system is reversible when the absorbing process is reversed to a gain process in the time-reversal operation. The operation {E, H} → {E*, −H*} reverses phase and group velocity of field propagation while the operation {ε, μ} → {ε*, μ*} turns a material loss into a gain by changing signs of Im(ε) and Im(μ). Note that the loss-gain interchange corresponds to the time reversal of the absorbing process that is physically forbidden by the second law of thermodynamics. It is inevitable for the absorbed energy to be thermally redistributed into other quanta in an irreversible way in the macroscopic level. In other words, the dissipative system is time irreversible, but the electromagnetic scattering process itself is reversible under the allowance of the loss-gain interchange and, thus, the scattering coefficients in a dissipative system are subject to the virtual time-reversal symmetry provided by C-invariance of linear electromagnetism.
Direct scattering in an absorbing system is not lossless in general; i.e., C is not unitary. However, in most cases anomalously strong absorption due to a resonance arises in a system that presumably exhibits negligible absorption in off-resonance condition. Thus, for SPP resonance structures consisting of noble metals, it is acceptable to assume lossless direct scattering in Eq. (2).
Based on the above two arguments considering time-reversal symmetry in a dissipative system and the assumption of lossless direct scattering, we obtained the fundamental constraints of |κ〉 and C. The time reversal of fields is given by the operation {E(r,t), H(r,t)} → {E(r,–t), –H(r, –t)} in the time domain. Thus, in a time-reversed situation,
for port’s radiation modes and for the resonance mode. By comparing Eqs. (A1) ~(A4) to Eqs. (3) ~(6), the time reversal of fields corresponds to the transformation of the modal amplitudes such thatWith an additional operation γ int → –γ int as a loss-gain interchange, the coupled-mode Eqs. (1) and (2) transform into in a time-reversed situation. Requiring Eqs. (A6) and (A7) to be identical to Eqs. (1) and (2) yields the fundamental constraints of CC † = I, |κ +〉 = |κ –〉 ≡ |κ〉, 〈κ|κ〉 = 2γ rad, and C|κ〉* = –|κ〉, which are identical to those for a lossless resonance system.Acknowledgments
This work was supported by the National Research Foundation of Korea grant funded by the Korea Government (MEST) [2010-0000256] and the IT R&D program [2008-F-022-01] of the MKE/IITA, Korea.
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