## Abstract

We provide a modified Fano resonance formula applicable to dissipative two-port resonance systems. Based on a generic coupled-resonator model, the formula embodies loss-related correction terms and fundamental resonance parameters that can be determined by an analytic method or experimentally as opposed to finding phenomenological parameters by fitting to numerical results. The theory applies physically meaningful resonance parameters including resonance frequency, total decay rates, and partial radiation probabilities. For example, it shows that the classic Fano shape parameter *q* is given directly in terms of the phase difference between the resonant and non-resonant transmission pathways. Our new resonance formula quantitatively expresses the resonance spectra pertaining to modal nanophotonic and surface-plasmonic thin-film structures as verified by comparing with exact numerical models.

©2013 Optical Society of America

## 1. Introduction

Fano resonance is a fundamental expression of wave interaction with a system supporting localized modes. The characteristically asymmetric Fano spectral profile is explained by the interference of light following two distinct scattering pathways. These pathways are the resonant pathway associated with a discrete state pertaining to a localized mode and the non-resonant pathway contributed by the continuum [1,2]. This interference picture, originally suggested to explain quantum-mechanical resonance effects, also applies to classical resonance phenomena. Currently, nanophotonic and nanoplasmonic systems are of high global interest stimulated by discoveries such as extraordinary optical transmission through metallic nano apertures [3,4], plasmonic metamaterials [5–8], and guided-mode resonance effects [9–11] in periodic thin films. In these systems, and many others, Fano resonance signatures appear.

In dissipative resonant systems, the conventional Fano resonance formula is not strictly valid because the probability flux is taken as conserved in the original formulation [1]. Clearly, material absorption and scattering loss can play significant roles in nanostructured optical systems as widely noted in the literature [12,13]. Recently, the effect of dissipation on asymmetric Fano profiles has been studied to precisely describe resonance spectra in nanophotonic and nanoplasmonic systems [6,14–16]. Coupled-mode theory of light scattering by subwavelength particles [14] and generalized electromagnetic formulation of plasmonic metamaterials [6,15] showed loss-induced degradation of resonance contrast. A linear systems approach was used to classify Fano resonance profiles in the presence of dissipation [16].

In this paper, we provide a Fano resonance formula applicable to lossy two-port systems. Included in this formula are explicit analytic interpretations of fundamental interference parameters and formal corrections to the conventional lossless Fano resonance expression. In our approach, time reversibility dictates the phase difference between the two interfering pathways to be determined by the relative strength of the non-resonant pathway to the resonant pathway. Using this general phase relation, the Fano resonance expression in the presence of dissipation is reformulated with correction terms including a linear decrease of the contribution arising from the lossless Fano term as a function of the absorption probability and peak efficiency degradation due to resonance-enhanced absorption. On this basis, we provide a comprehensive picture of the resonant transmission and reflection in lossy two-port systems where the resonance-enhanced absorption induces transmission peak suppression and nonreciprocal behavior in reflection [17,18]. We also present rigorous numerical calculations pertaining to guided-mode resonance effects in a semiconductor grating structure as well as surface-plasmonic resonance effects in a metallic nanocavity. The modified Fano resonance formula shows quantitative agreement with the numerical calculations for both of these representative device classes.

## 2. Fano resonance formula for dissipative two-port systems

The classical Fano resonance is described by the conventional formula [1,3]

where*t*

_{D}is the non-resonant transfer amplitude via the continuum,

*δ*= (

*ω*–

*ω*

_{0})/

*γ*is the reduced frequency of a discrete state at

*ω*

_{0}–

*iγ*, and

*q*is the shape factor describing the interference between the resonant and non-resonant scattering pathways. Equation (1) has been widely used to describe transmission spectra associated with classical resonances with

*ω*

_{0},

*γ*,

*q*, and

*t*

_{D}determined by curve fitting to numerical or experimental results.

Fano resonance can be visualized using the coupling configuration schematically depicted in Fig. 1. The resonant scattering pathway is governed by the radiation probability *η _{n}* quantifying the fraction of radiation that the localized mode emits to the reflection (

*n*= 1) and transmission (

*n*= 2) sides. The non-resonant pathway is described by the amplitude

*r*

_{D}and

*t*

_{D}. The absorption probability

*η*

_{abs}of the localized mode parameterizes dissipative loss due to material absorption and random scattering. Using this model, the transmittance is expressed as [19]

*δ*= 0). In Eq. (2), the first and second terms separately represent the non-resonant and resonant contributions, respectively, and the last term corresponds to the interference between them.

A fundamental property of Fano interference is inferred by applying time reversibility. Time reversibility constrains the interference to occur in a specific way as discussed by Yoon *et al.* [19], i.e., the phase difference Δ is not a free, independent parameter. It is given in terms of |*t*_{D}| and *η _{n}* by

*β*= 4

*η*

_{1}

*η*

_{2}/(

*η*

_{1}+

*η*

_{2})

^{2}is the coupling-symmetry factor and the parity factor

*p*takes

*p*= + 1 for the even-like mode or

*p*= –1 for the odd-like mode [19]. Obviously from Eq. (2), the coupling-symmetry factor

*β*corresponds to the intensity of the resonant transmission at

*δ*= 0 for lossless cases where

*η*

_{1}+

*η*

_{2}= 1. Therefore, the phase difference is determined by the relative strength of the non-resonant pathway to the resonant pathway. The shape factor

*q*in Eq. (1) is then simply expressed in terms of the phase difference Δ byWe note here that for given

*η*and |

_{n}*t*

_{D}|, the interference is completely deterministic, not requiring any further knowledge of the system’s geometrical and parametric details.

Using Eqs. (3) and (4), Eq. (2) can be re-written in terms of the conventional Fano resonance formula and correction terms as

*η*

_{rad}=

*η*

_{1}+

*η*

_{2}= 1–

*η*

_{abs}is the total radiation probability of the localized mode. The reflectance is then given bywhere the last term on the right-hand side represents resonance-enhanced absorbance [19,20].

For lossless systems (*η*_{rad} = 1 and *η*_{abs} = 0), *T*(*δ*) = *T*_{Fano}(*δ*). As the absorption probability *η*_{abs} increases, the contribution from *T*_{Fano} decreases linearly with the factor *η*_{rad} = 1–*η*_{abs}. The second term on the right-hand side of Eq. (5) further degrades the resonance peak intensity. In particular, this Lorentzian correction term is maximized to *β*/4 at the critical coupling condition *η*_{rad} = *η*_{abs} = 1/2 where the resonance-enhanced absorption becomes maximal [20]. Note that *β* corresponds to the peak value of *T*_{Fano}, i.e., [*T*_{Fano}]_{max} = *T*_{Fano}(1/*q*) = *β*. These dependences are summarized in Figs. 2(a) and 2(b) for *β* = 0.7 and |*t*_{D}|^{2} = 0.1. Figure 2(c) shows the coupling-symmetry factor *β* as a function of the normalized difference |*η*_{1}–*η*_{2}|/*η*_{rad} in the radiation probabilities.

## 3. Consistency of the lossy Fano resonance formula with rigorous calculation

We present the following numerical results to demonstrate the validity and general applicability of our model. We treat two completely different Fano resonance systems: the first system pertains to guided-mode resonance (GMR) in a periodic semiconductor thin film, and the second refers to surface-plasmon (SP) resonance in a metallic nanocavity. Consistent with the formulation, we limit our study to a two-port system supporting a single resonance mode. Accordingly, the two exemplary systems are chosen so that the resonance bandwidth is relatively narrow and the resonance center wavelength is longer than the first-order Rayleigh wavelength where additional radiation ports emerge with higher diffraction orders.

We treat a silicon-on-insulator structure with the top Si layer partially etched to form a periodically modulated surface as shown in the inset of Fig. 3 (see figure caption for geometrical parameters). The GMR occurs in this structure under phase-matching of incident light to a leaky waveguide mode [9–11]. We assume Si refractive index *n*_{Si} = 3.5 and use rigorous coupled-wave analysis (RCWA) [21] for numerical calculations.

Figure 3 shows the calculated zero-order transmission (*T*_{0}) and reflection (*R*_{0}) spectra for TE-polarized light at normal incidence. Sharp resonance features are evident in the spectra. This GMR is induced by a TE_{1} mode at 0.912 μm as confirmed by the associated electric field distribution in the inset.

We model material dissipation by introducing the extinction coefficient *k* in the Si layer. We apply complex refractive index of Si *ñ*_{Si} = 3.5 + *ik* with *k* varying from 0 to 5 × 10^{−3} in the RCWA calculation. To estimate basic resonance parameters such as decay rate *γ* and radiation probabilities to the cover (air) *η*_{cov} and substrate (SiO_{2}) *η*_{sub} for use in the analytic theory calculation by Eqs. (5) and (6), we use the absorbance analysis method suggested in [19,20,22] as opposed to parametric determination by fitting to numerical data. We obtain the resonance parameters listed in Table 1. In the absorbance analysis method, the resonance frequency *ω*_{0} and total decay rate *γ* are directly obtained from the maximum position and width at half maximum of the absorption peak, respectively. Using peak absorbances *A*_{cov} and *A*_{sub} for light incidence from the cover and substrate, the radiation probabilities are determined by *η*_{cov} = *η*_{rad}*A*_{cov}/(*A*_{cov} + *A*_{sub}), *η*_{sub} = *η*_{rad}*A*_{sub}/(*A*_{cov} + *A*_{sub}), and *η*_{rad} = 0.5[1 ± (1–*A*_{cov}–*A*_{sub})^{1/2}], where we take the ( + ) sign for the under-damped case or the (–) sign for the over-damped case. The method fully determines the resonance parameters from the peak value, position, and bandwidth of two absorption peaks, which can readily be implemented in an experiment.

A recent paper by Gallinet and Martin [6] provides a generalized Fano formula describing electromagnetic resonance properties of dispersive and lossy media. Both the final expression and the model parameters differ markedly from our results presented above. Gallinet and Martin extract their model parameters by fitting to results generated by full numerical simulations. In our approach, direct numerical fitting is not necessary.

Figure 4 shows the RCWA calculation results (symbols) and the corresponding analytic-theory curves for the transmittance *T*_{0} and reflectance under the light incidence from cover *R*_{0}^{(cov)} and substrate *R*_{0}^{(sub)}. Note that *η*_{1} = *η*_{cov} and *η*_{2} = *η*_{sub} for *R*_{0}^{(cov)} or *η*_{1} = *η*_{sub} and *η*_{2} = *η*_{cov} for *R*_{0}^{(sub)}. The amplitude |*t*_{D}| is obtained by computing the reflection and transmission coefficients of an equivalent unpatterned structure with the top grating layer replaced by an effective planar film with refractive index *n*_{eff} = [F*ñ*_{Si}^{2} + (1–F)]^{1/2}, where F is the Si fill factor. We confirm excellent agreement of our theory with the numerical calculation.

The effect of dissipation on resonance spectra is exemplified well in Fig. 4. In Fig. 4(a), material loss simply degrades the resonance character of the transmission spectrum. Both the resonance peak and dip approach the non-resonant spectral background level. In contrast, the reflection spectrum exhibits more drastic changes. For example, *R*_{0}^{(cov)} and *R*_{0}^{(sub)} exhibit highly non-reciprocal behavior with increasing *k*. The non-reciprocal reflection is induced by asymmetric radiation probabilities as confirmed in Table 1. In Eq. (6), the last term representing resonance-enhanced absorbance is associated with the reflection dip and it scales with the radiation probability of the reflection side *η*_{1}. Thus, the reflection dip becomes more prominent for light incidence on the side with higher radiation probability as in Fig. 4(c). The theory quantitatively captures these particular properties of resonance in the presence of material dissipation.

We now consider a surface-plasmon (SP) resonance in a metallic nanocavity structure. The nanocavity consists of an SiO_{2} thin film sandwiched by 15-nm-thick Au films as shown in the inset of Fig. 5. The top Au film is modulated by a sinusoidal profile with period 450 nm and depth 100 nm. The thickness of the SiO_{2} thin film is 70 nm on the average and 20 nm at the minimum. For this example, we use a coordinate transformation method, the Chandezon method (C-method), for the numerical calculations [23]. The Au film is modeled as a Drude metal [22,24] with plasma and collision frequencies 1.37 × 10^{16} rad/s and *f*_{abs} × 0.407 × 10^{14} rad/s, respectively. The absorption factor *f*_{abs} adjusts the ohmic damping rate directly proportional to the material absorption of the structure.

Figure 5 shows the zero-order transmittance (*T*_{0}) for the lossless case (*f*_{abs} = 0). A resonance feature at 1.124 μm is caused by SP excitation inside the cavity. The SP mode is strongly localized at the deep subwavelength features, mostly in a 20-nm-thick gap area, as shown in the inset. This tightly localized excitation of the metal-gap SP mode is of special interest in nanoplasmonics on account of its nanofocusing properties and associated ultrahigh nonlinear effects [25,26].

We now verify Eqs. (5) and (6) with numerical calculations for the SP excitation at 1.124 μm. Again, the basic resonance parameters with increasing *f*_{abs} are obtained by the absorbance analysis method and listed in Table 2. We estimate |*t*_{D}|^{2} by interpolating the numerical *T*_{0} spectrum from off-resonant wavelength ranges. Figure 6 shows the numerical results compared with the corresponding analytic curves. Degraded resonance characteristics in transmission as well as absorption-induced nonreciprocity in reflection are clearly observed. Again we confirm excellent quantitative agreement of the analytic theory with full numerical modeling. Interestingly, the minimum of *R*_{0}^{(sub)} in Fig. 6(c) is almost zero for the particular level of absorption (*f*_{abs} = 0.2). This property can be understood as a general aspect of a system with small non-resonant transmission (|*t*_{D}|^{2} << 1). It is readily derived from Eq. (6) under light incidence from the substrate that *R*(*δ* = 0) ≈1–4*η*_{sub} + 4*η*_{sub}^{2} for |*t*_{D}|^{2} << 1, implying *R*(*δ* = 0) ≈0 when *η*_{sub} = 1/2. We confirm *η*_{sub} = 0.489 ≈1/2 in Table 2. Therefore, the reflection dip is strongest if the critical coupling condition is satisfied at the incident port; i.e., the radiation probability of the resonance mode to the incident port is close to 1/2.

## 3. Conclusion

In conclusion, we developed a general theory of Fano resonance in dissipative two-port systems. Using a coupled-resonator model with minimal assumptions, the Fano resonance formula is expressed in terms of the resonance frequency, total decay rate, and radiation probabilities that are experimentally measurable parameters. This expression provides formal corrections to the lossless Fano resonance formula. We show that the corrected lossy Fano resonance formula quantitatively captures the effects of material dissipation on the resonant transmission and reflection spectra pertaining to guided-mode resonance in a thin-film semiconductor grating and deep subwavelength surface-plasmon excitation in a metallic nanocavity structure.

In the previous literature, Fano profiles in lossy systems are described by the general form

Avrusky*et al.*recently obtained this expression with a linear systems approach that treats a resonant scattering amplitude as a function of simple pole and zero [16]. Using Eq. (7), they built a parametric classification scheme for various resonance profiles; Eq. (5) for the transmission and Eq. (6) for the reflection basically take this form. As revealed in Figs. 4 and 6, Eqs. (5) and (6) generally describe various resonance profiles such as symmetric peaks, highly asymmetric profiles, and symmetric dips with more physically transparent definitions of the profile amplitude

*a*, shape factor

*q*, and damping parameter

*b*. Therefore, our formula supports intuitive physical interpretations to the parametric classification scheme in [16]. Gallinet and Martin also derived Eq. (7) by adopting the original quantum-mechanical formalism to classical electromagnetic equations [6,15]. They found analytic expressions for the shape factor

*q*and damping parameter

*b*. In contrast to parametric definitions based on radiation decay probabilities, their expressions are based on coupling constants, center frequencies, and bandwidths of the two interfering states. We believe that the two different interpretations must be consistent with each other because our approach is based on time-reversibility that is also a fundamental property of electromagnetic fields. Our theory is based on the phenomenological coupled-mode formalism as opposed to the more rigorous electromagnetic formalism by Gallinet and Martin. Therefore, the general model parameters in our approach must be found by additional mathematical treatment for any given materials and resonance system geometry. However, the approach by Gallinet and Martin also includes phenomenological aspects in defining fields associated with discrete and continuum states. To the best of our knowledge, there is no known analytic method that defines exact field configurations associated with leaky, quasi-bound states. We emphasize that the phenomenological coupled-mode formalism has been successfully used to describe various electromagnetic resonance effects [14,19,20], and our model parameters all have straightforward physical interpretation.

We apply absorption analysis for parametric extraction and thereby avoid the tedium and possible ambiguity of direct numerical fitting. The theory reveals the essential physics of Fano resonance in lossy systems in a general manner and is not limited to particular materials and geometries. We find that the spectral contrast of a Fano-resonance feature in transmission simply degrades with increasing material dissipation while the resonance contrast in the reflection spectrum is maximized when the critical coupling condition is satisfied at the incident port. The coupling symmetry of a localized mode determines the lossless peak transmittance and the loss-induced resonance degradation. The classic Fano shape parameter *q* is given directly in terms of the phase difference between the resonant and non-resonant transmission pathways. The theory is generally applicable to various classical resonance systems such as narrowband filters based on thin-film gratings and photonic crystals, high-quality-factor ring and disc microresonators, and devices containing plasmonic metamaterials.

## Acknowledgments

This work is supported by the Texas Instruments Distinguished University Chair in Nanoelectronics endowment.

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