## Critical coupling in dissipative surface-plasmon resonators with multiple ports |

Optics Express, Vol. 18, Issue 25, pp. 25702-25711 (2010)

http://dx.doi.org/10.1364/OE.18.025702

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### Abstract

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 OSA

## 1. Introduction

1. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**(4), 321–322 (2000). [CrossRef]

4. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **62**(55 Pt B), 7389–7404 (2000). [CrossRef] [PubMed]

5. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express **12**(9), 1885–1891 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-9-1885. [CrossRef] [PubMed]

6. K. J. Lee, R. LaComb, B. Britton, M. Shokooh-Saremi, H. Silva, E. Donkor, Y. Ding, and R. Magnusson, “Silicon-layer guided-mode resonance polarizer with 40-nm bandwidth,” IEEE Photon. Technol. Lett. **20**(22), 1857–1859 (2008). [CrossRef]

7. K. Yu. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. **97**(24), 243904 (2006). [CrossRef]

9. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. **216**(4), 398–410 (1968). [CrossRef]

11. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. **100**(6), 066408 (2008). [CrossRef] [PubMed]

12. T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlet, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics **2**(5), 299–301 (2008). [CrossRef]

13. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. **100**(20), 207402 (2008). [CrossRef] [PubMed]

14. K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media,” Rev. Mod. Phys. **80**(4), 1201–1213 (2008). [CrossRef]

15. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**(3-4), 131–314 (2005). [CrossRef]

16. K. Kurihara and K. Suzuki, “Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretchmann’s theory,” Anal. Chem. **74**(3), 696–701 (2002). [CrossRef] [PubMed]

17. A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A **14**(3), 588–595 (1997). [CrossRef]

7. K. Yu. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. **97**(24), 243904 (2006). [CrossRef]

18. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**(3), 569–572 (2003). [CrossRef]

*critical coupling*[1

1. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**(4), 321–322 (2000). [CrossRef]

12. T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlet, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics **2**(5), 299–301 (2008). [CrossRef]

14. K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media,” Rev. Mod. Phys. **80**(4), 1201–1213 (2008). [CrossRef]

19. Y. P. Bliokh, J. Felsteiner, and Y. Z. Slutsker, “Total absorption of an electromagnetic wave by an overdense plasma,” Phys. Rev. Lett. **95**(16), 165003 (2005). [CrossRef] [PubMed]

*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

7. K. Yu. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. **97**(24), 243904 (2006). [CrossRef]

*γ*

_{rad}=

*γ*

_{int}in a universal way, especially for SPP resonators with multiple coupling ports.

18. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**(3), 569–572 (2003). [CrossRef]

## 2. Temporal coupled-mode theory for a dissipative resonator

*g*(

*t*), coupled with

*N*pairs of incoming (

*f*) and outgoing (

_{m+}*f*

_{m}_{–}) radiation modes. We assume that |

*g*|

^{2}and |

*f*

_{m}_{±}|

^{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 ±}…

*X*

_{m}_{±}…

*X*

_{N}_{±})

*in the limit that*

^{T}*γ*

_{tot}/

*ω*

_{0}<< 1 [2,18

18. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**(3), 569–572 (2003). [CrossRef]

**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

**20**(3), 569–572 (2003). [CrossRef]

*g*(

*t*) describing the harmonic mode.

**C**and the mode coupling coefficients |

*κ*

_{±}〉:

**CC**

^{†}=

**I**, |

*κ*

_{+}〉 = |

*κ*

_{–}〉 ≡ |

*κ*〉, 〈

*κ*|

*κ*〉 = 2

*γ*

_{rad}, and

**C**|

*κ*〉* = –|

*κ*〉 as shown by Fan et al. [18

**20**(3), 569–572 (2003). [CrossRef]

*κ*|

_{m}^{2}/2 =

*γ*as a partial radiation coupling rate at port

_{m}*m*and

*γ*

_{rad}= (

*γ*

_{1}+ … +

*γ*+ … +

_{m}*γ*) 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

_{N}**20**(3), 569–572 (2003). [CrossRef]

*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

*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

**20**(3), 569–572 (2003). [CrossRef]

*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

16. K. Kurihara and K. Suzuki, “Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretchmann’s theory,” Anal. Chem. **74**(3), 696–701 (2002). [CrossRef] [PubMed]

17. A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A **14**(3), 588–595 (1997). [CrossRef]

19. Y. P. Bliokh, J. Felsteiner, and Y. Z. Slutsker, “Total absorption of an electromagnetic wave by an overdense plasma,” Phys. Rev. Lett. **95**(16), 165003 (2005). [CrossRef] [PubMed]

*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.

*γ*

_{rad}=

*γ*

_{int}), they yieldwhere

*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}.

*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 by

16. K. Kurihara and K. Suzuki, “Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretchmann’s theory,” Anal. Chem. **74**(3), 696–701 (2002). [CrossRef] [PubMed]

17. A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A **14**(3), 588–595 (1997). [CrossRef]

12. T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlet, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics **2**(5), 299–301 (2008). [CrossRef]

*P*

_{abs}= 2

*γ*

_{int}|

*g*|

^{2}and the absorbance is a ratio of

*P*to incident power

_{abs}*P*, i.e.,

_{inc}*A*=

*P*/

_{abs}*P*. For a single incoming radiation at port

_{inc}*q*with amplitude

*F*and frequency

_{q}*ω*, the partial absorbance is

*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

20. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. **9**(3), 193–204 (2010). [CrossRef] [PubMed]

*A*/

_{n}*A*=

_{m}*γ*/

_{n}*γ*= |

_{m}*κ*/

_{n}*κ*|

_{m}^{2}and Σ

*A*(

_{q}*ω*

_{0}) = 4

*η*

_{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.

*ψ*〉 ≡

*U*

^{1/2}|

*κ*〉*, where

*U*is a normalization factor. First, its relative power at port

*m*, |

*ψ*|

_{m}^{2}, can be simply given by the partial absorbance

*A*(

_{m}*ω*

_{0}) since |

*ψ*|

_{m}^{2}=

*U*|

*κ*|

_{m}^{2}=

*U*·

*γ*and

_{m}*A*= (

_{m}*γ*

_{int}/

*γ*

_{tot}

^{2})·

*γ*. If we normalize |

_{m}*ψ*〉 so that 〈

*ψ*|

*ψ*〉 = 1, then

*U*=

*γ*

_{rad}

^{–1}and

*ψ*), 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

_{m}*f*

_{1+}=

*ψ*

_{1}exp(

*iϕ*

_{1}) and

*f*

_{2+}=

*ψ*

_{2}exp(

*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

*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 sin

^{2}[…] 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

*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.

*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(

*i*2

*α*)], 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(

*i*2

*α*). By increasing

*ω*from lower to upper far off-resonance limits,

*σ*(

*ω*) rotates counterclockwise from the lower (2

*α*= –

*π*) limit to the upper limit (2

*α*= +

*π*) via

*σ*(

*ω*) = 2

*η*

_{rad}at

*ω*

_{0}(blue dot at 2

*α*= 0) while the directly scattered amplitude (red arrow) remains constant.

*ρ*

_{tot}(

*ω*) is now represented by a blue arrow directing a point at 2

*α*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.

*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

21. M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. **19**(3), 431–436 (1976). [CrossRef]

22. R. A. Depine, V. L. Brudny, and J. M. Simon, “Phase behavior near total absorption by a metallic grating,” Opt. Lett. **12**(3), 143–145 (1987). [CrossRef] [PubMed]

*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

*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

*k*= (

_{x}*ω/c*)sin

*θ*=

_{m}*k*

_{SPP}–2

*πm*/Λ, where

*m*denotes the diffraction order. The incoming mode

*f*incident at an angle

_{m+}*θ*excites the SPP mode via +

_{m}*m*order diffraction, and its zero-th order reflection corresponds to

*f*

_{m}_{–}. The excited SPP mode also loses its energy toward

*f*

_{1–}and

*f*

_{2–}as leakage radiation modes.

23. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. **72**(7), 839–846 (1982). [CrossRef]

_{0}) frequencies are 6.35 × 2

*πc*/

*μm*and 0.125 × 2

*πc*/

*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.

*A*

_{tot}(

*ω*

_{0}) and

*A*(

_{q}*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

*A*= 1−Σ

_{q}*|W*

_{p}*|*

_{pq}^{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 (Γ

_{0}) 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

25. J. Yoon, S. H. Song, and J.-H. Kim, “Extraction efficiency of highly confined surface plasmon-polaritons to far-field radiation: an upper limit,” Opt. Express **16**, 1269 (2008), http://www.opticsinfobase.org/oe/abstra ct.cfm?URI=oe-16-2-1269.

*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/Γ

_{0}= 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

20. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. **9**(3), 193–204 (2010). [CrossRef] [PubMed]

## Appendix: The fundamental constraints of coupling constants in a dissipative system

**E**,

**H**} → {

**E***,

**H***} and {

*ε*,

*μ*} → {–

*ε**, –

*μ**} [26

26. A. Lakhtakia, “Conjugation symmetry in linear electromagnetism in extension of materials with negative real permittivity and permeability scalars,” Microw. Opt. Technol. Lett. **40**(2), 160–161 (2004). [CrossRef]

27. J. Yoon, S. H. Song, C. H. Oh, and P. S. Kim, “Backpropagating modes of surface polaritons on a cross-negative interface,” Opt. Express **13**(2), 417–427 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-2-417. [CrossRef] [PubMed]

**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.

**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).

*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*〉 = 2

*γ*

_{rad}, and

**C**|

*κ*〉* = –|

*κ*〉, which are identical to those for a lossless resonance system.

## Acknowledgments

## References and links

1. | A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. |

2. | H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984). |

3. | C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. |

4. | Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

5. | Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express |

6. | K. J. Lee, R. LaComb, B. Britton, M. Shokooh-Saremi, H. Silva, E. Donkor, Y. Ding, and R. Magnusson, “Silicon-layer guided-mode resonance polarizer with 40-nm bandwidth,” IEEE Photon. Technol. Lett. |

7. | K. Yu. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. |

8. | E. Kretchmann and H. Reather, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A |

9. | A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. |

10. | R. W. Wood, “On the remarkable case of uneven distribution of a light in a diffracted grating spectrum,” Philos. Mag. |

11. | J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. |

12. | T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlet, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics |

13. | N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. |

14. | K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media,” Rev. Mod. Phys. |

15. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. |

16. | K. Kurihara and K. Suzuki, “Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretchmann’s theory,” Anal. Chem. |

17. | A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A |

18. | S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A |

19. | Y. P. Bliokh, J. Felsteiner, and Y. Z. Slutsker, “Total absorption of an electromagnetic wave by an overdense plasma,” Phys. Rev. Lett. |

20. | J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. |

21. | M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. |

22. | R. A. Depine, V. L. Brudny, and J. M. Simon, “Phase behavior near total absorption by a metallic grating,” Opt. Lett. |

23. | J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. |

24. | E. D. Palik, |

25. | J. Yoon, S. H. Song, and J.-H. Kim, “Extraction efficiency of highly confined surface plasmon-polaritons to far-field radiation: an upper limit,” Opt. Express |

26. | A. Lakhtakia, “Conjugation symmetry in linear electromagnetism in extension of materials with negative real permittivity and permeability scalars,” Microw. Opt. Technol. Lett. |

27. | J. Yoon, S. H. Song, C. H. Oh, and P. S. Kim, “Backpropagating modes of surface polaritons on a cross-negative interface,” Opt. Express |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(300.1030) Spectroscopy : Absorption

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: September 27, 2010

Revised Manuscript: November 8, 2010

Manuscript Accepted: November 16, 2010

Published: November 23, 2010

**Citation**

Jaewoong Yoon, Kang Hee Seol, Seok Ho Song, and Robert Magnusson, "Critical coupling in dissipative surface-plasmon resonators with multiple ports," Opt. Express **18**, 25702-25711 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25702

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### References

- A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
- C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999). [CrossRef]
- Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000). [CrossRef] [PubMed]
- Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express 12(9), 1885–1891 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-9-1885 . [CrossRef] [PubMed]
- K. J. Lee, R. LaComb, B. Britton, M. Shokooh-Saremi, H. Silva, E. Donkor, Y. Ding, and R. Magnusson, “Silicon-layer guided-mode resonance polarizer with 40-nm bandwidth,” IEEE Photon. Technol. Lett. 20(22), 1857–1859 (2008). [CrossRef]
- K. Yu. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. 97(24), 243904 (2006). [CrossRef]
- E. Kretchmann and H. Reather, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).
- A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216(4), 398–410 (1968). [CrossRef]
- R. W. Wood, “On the remarkable case of uneven distribution of a light in a diffracted grating spectrum,” Philos. Mag. 4, 396–402 (1902).
- J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100(6), 066408 (2008). [CrossRef] [PubMed]
- T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlet, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008). [CrossRef]
- N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef] [PubMed]
- K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media,” Rev. Mod. Phys. 80(4), 1201–1213 (2008). [CrossRef]
- A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]
- K. Kurihara and K. Suzuki, “Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretchmann’s theory,” Anal. Chem. 74(3), 696–701 (2002). [CrossRef] [PubMed]
- A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14(3), 588–595 (1997). [CrossRef]
- S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]
- Y. P. Bliokh, J. Felsteiner, and Y. Z. Slutsker, “Total absorption of an electromagnetic wave by an overdense plasma,” Phys. Rev. Lett. 95(16), 165003 (2005). [CrossRef] [PubMed]
- J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010). [CrossRef] [PubMed]
- M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19(3), 431–436 (1976). [CrossRef]
- R. A. Depine, V. L. Brudny, and J. M. Simon, “Phase behavior near total absorption by a metallic grating,” Opt. Lett. 12(3), 143–145 (1987). [CrossRef] [PubMed]
- J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72(7), 839–846 (1982). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids II (Academic Press, San Diego, 1998).
- J. Yoon, S. H. Song, and J.-H. Kim, “Extraction efficiency of highly confined surface plasmon-polaritons to far-field radiation: an upper limit,” Opt. Express 16, 1269 (2008), http://www.opticsinfobase.org/oe/abstra ct.cfm ?URI=oe-16-2-1269.
- A. Lakhtakia, “Conjugation symmetry in linear electromagnetism in extension of materials with negative real permittivity and permeability scalars,” Microw. Opt. Technol. Lett. 40(2), 160–161 (2004). [CrossRef]
- J. Yoon, S. H. Song, C. H. Oh, and P. S. Kim, “Backpropagating modes of surface polaritons on a cross-negative interface,” Opt. Express 13(2), 417–427 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-2-417 . [CrossRef] [PubMed]

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