## Abstract

We study the reflectivity spectra of photonic crystal slab cavities using an extension of the scattering matrix method that allows treating finite sizes of the spot of the excitation beam. The details of the implementation of the method are presented and then we show that Fano resonances arise as a consequence of the electromagnetic interference between the discrete contribution of the fundamental cavity mode and the continuum contribution of the light scattered by the photonic crystal pattern. We control the asymmetry lineshape of the Fano resonance through the polarization of the incident field, which determines the relative phase between the two electromagnetic contributions to the interference. We analyse the electric field profile inside and outside of the crystal to help in the understanding of the dependence on polarization of the reflectivity lineshape. We also study with our implementation the dependence of the Fano resonances on the size of the incident radiation spot.

© 2013 Optical Society of America

## 1. Introduction

Fano interference is an important phenomenon that takes place in a large variety of physical systems. The initial formulation was made in 1961 by Ugo Fano [1] to explain asymmetric sharp profiles in the absorption spectra of Rydberg atoms. Nevertheless, the universal character of the Fano theory makes it a very powerful tool to explain and reduce complex phenomena to a few simple parameters in several branches of physics, such as neutron scattering [2], resonant tunneling in heterostructures [3], conductance of quantum dots [4], electronic waveguides [5], microwave metal cavities [6] and photonic crystals [7–10]. The latter has become an important field of intense research during the last two decades, since photonic crystals, which are the photonic counterpart of solid state electronic crystals, are promising systems to create efficient integrated optical devices. The understanding of the communication channels between the fundamental constituents (defect cavities and waveguides) of photonic devices based on photonic crystals is of great importance to achieve the desired fully optical information processing. Since Fano resonances are the product of the interference between discrete and continuum channels, the study of its properties and the conditions in which it can arise in photonic crystal systems becomes very relevant to comprehend the interaction between the confined photonic modes and the electromagnetic environment, thus helping to design efficient optical devices. The characteristic feature of a Fano resonance is an asymmetric sharp lineshape in the response function of the system. The sharp asymmetric lineshape means that there is a very large change of the response function in a range narrower than the full width of the resonance itself. This property has been exploited in photonic crystals to produce optical Fano diodes [11] and all-optical switching and bistability with high contrast between the two stable states [12].

In the context of photonic crystal slabs, which are two-dimensional photonic crystals with finite thickness in the third dimension [13], Fano resonances are the product of the coupling between guided modes in the slab (discrete states) and external radiation (continuum of states), constituting in this way an efficient mechanism to transfer information within the slab to external media [14]. In this work, we make a detailed study of Fano resonances in slab photonic crystal cavities using the scattering matrix formalism; we apply the method to the L3 cavity, which consists in a line of three holes missing from a hexagonal lattice of holes in the slab [15]. In this case, the discrete channel of the Fano resonance is the light that leaks from the cavity mode while the continuum channel is the radiation reflected by the photonic crystal pattern, including the cavity. We extend the scattering matrix method to allow treating finite spot sizes of the incident radiation. We present the details of the implementation of the method and then show that the asymmetry of the Fano resonance in the reflectivity spectra depends on the incident field linear polarization and the size of the spot of the excitation. Good agreement between our theoretical calculations and reported experimental results is obtained.

## 2. Scattering matrix formalism

To investigate the Fano resonances in the photonic crystal cavity we implement the scattering matrix (SM) method, initially formulated by Ko and Ikson [3] to model electronic tunneling in multilayer semiconductor structures, and then modified by Whittaker and Culshaw [16] to apply it to multilayer photonic structures. The principal advantage of this formulation is that the external radiative modes (continuum states) are taken into account in a natural way and are connected to the internal modes of the photonic structure (bounded states) through boundary conditions, which is fundamental to characterize the Fano interference phenomenon. We present in this section the main aspects of the SM method following Whittaker and Culshaw [16], but with a new approach for the treatment of the incident field to describe finite spot sizes.

We consider a sequence of layers, in the *z* direction, with a periodic photonic pattern embedded in the *xy* plane. A convenient zero-divergence expansion of the magnetic field in each layer is:

*ϕ*(

_{x}**G**) and

*ϕ*(

_{y}**G**) are the expansion coefficients,

*k*,

_{x}*k*and

_{y}*q*are the wavenumbers in

*x*,

*y*and

*z*directions, respectively, and

**G**is the reciprocal lattice vector of the pattern. With this magnetic field expansion, the Maxwell curl equations determine the following eigenvalue problem for

*q*: where

*ϕ*= (

*ϕ*,

_{x}*ϕ*)

_{y}*, with*

^{T}*T*denoting the transpose vector. The block matrices

*ℰ*,

*K*,

*𝒦*are defined as:

*ε̂*has elements

*ε̂*

**(**

_{G′G}*z*) =

*ε̃*(

**G′**−

**G**,

*z*), where

*ε̃*(

**G′**−

**G**,

*z*) is the Fourier transform of the dielectric function and

*k̂*and

_{x}*k̂*are diagonal matrices with components (

_{y}*k̂*)

_{x}**=**

_{GG}*k*+

_{x}*G*and (

_{x}*k̂*)

_{y}**=**

_{GG}*k*+

_{y}*G*, and

_{y}*η̂*fulfills

*ε̂η̂*= 1. The quantities

*q*can be complex even if

*ℰ*is real. The planar components

*x*,

*y*of the fields are expanded in terms of forward and backward propagating waves with wavenumbers

*q*, and amplitudes

_{n}*a*and

_{n}*b*respectively, which are determined through the electromagnetic boundary conditions in each interface of the structure. The index

_{n}*n*denotes the

*n*-th eigenvalue of Eq. (2). The sign of

*q*is chosen to make Im{

_{n}*q*} > 0, maximizing the amplitudes

_{n}*a*and

_{n}*b*throughout the structure and preventing exponential growths in the

_{n}*z*dependence which would make the method unstable. In a compact notation the field expansions are written as:

*ϕ*,

_{n}*f̂*(

*z*) and

*q̂*are diagonal matrices with components

*f̂*(

*z*)

*=*

_{nn}*e*

^{iqnz}and

*q̂*=

_{nn}*q*respectively,

_{n}*a*is defined as (

*a*

_{1},

*a*

_{2}, ···)

*,*

^{T}*b*is defined as (

*b*

_{1},

*b*

_{2}, ···)

*and*

^{T}*d*is the thickness of the layer. The vectors

*h*(

_{x}*z*) and

*h*(

_{y}*z*) contain the expansion coefficients of the

*H*(

_{x}**r**

_{||},

*z*) and

*H*(

_{y}**r**

_{||},

*z*) components, respectively, in the plane wave basis

*e*

^{iG· r||}. The equivalent expression for the electric field is:

*M*defined as:

*M*makes the transformation from the forward

*a*

^{(l)}and backward

*b*

^{(l)}waves representation to the Fourier or momentum representation of the in-plane electromagnetic fields, ${h}_{\left|\right|}^{(l)}(z)$ and ${e}_{\left|\right|}^{(l)}(z)$. At this point, we can introduce the scattering matrix

*S*(

*l′*,

*l*), which relates the waves going forward and backward in different layers

*l*and

*l′*of the structure:

*a*

^{(l)}is the vector ${\left({a}_{1}^{(l)},{a}_{2}^{(l)},\cdots \right)}^{T}$ in layer

*l*and

*b*

^{(l′)}is the vector ${\left({b}_{1}^{({l}^{\prime})},{b}_{2}^{({l}^{\prime})},\cdots \right)}^{T}$ in layer

*l′*. The form of

*S*(

*l′*,

*l*) is obtained applying the boundary conditions of the fields in the interfaces of the multilayer structure, and

*S*(

*l′*,

*l*+ 1) can be calculated recursively from

*S*(

*l′*,

*l*) [16].

We are interested in the case where there is an electromagnetic field *a*^{(0)} incident from the *l* = 0 layer into the *l* = 1 layer of the multilayer system, and is then reflected, *b*^{(0)}, and transmitted, *a*^{(N)}, to the *l* = *N* layer on the other side. To obtain the reflectivity and transmission spectra of the structure we need the waves vector coefficients *b*^{(0)} and *a*^{(N)}, which represent the reflected and transmitted waves, respectively. With the scattering matrix *S*(0, *N*) and the vector coefficient *a*^{(0)} (the incident field) it is straightforward to obtain *b*^{(0)} and *a*^{(N)}:

*b*

^{(N)}= 0 since there is no field incident on the system from the

*N*layer. Using the translating equation Eq. (6) and the Maxwell curl equations to obtain

*h*

_{||}(

*z*), we construct the magnetic field expansion in the complex space at

*d*

_{0}for the reflected field and at

*z*for the transmitted field:

_{t}*d*

_{0}and

*z*are the perpendicular distances from the structure at which the reflectivity and transmission are calculated, respectively, and

_{t}*Ref*and

*Tra*denote the reflected and transmitted fields. The same expression stands for the electric field, although with expansion coefficients

*e*

_{||}(

*z*) and

*e*(

_{z}*z*). To obtain the vector

*a*

^{(0)}from the incident fields

**H**and

**E**we put their expansion coefficients (in the basis

*e*

^{iG· r||}) into the corresponding components of

*h*

_{||}and

*e*

_{||}, and we use again the translating equation, Eq. (6).

To describe more realistic phenomena it is essential to consider the shape and size of the incident field. It is possible to extend the scattering matrix method in order to treat finite sizes of the incident radiation spot. If an arbitrary-shape function *P*(**r**_{||}), whose maximum value is 1 and minimum value is 0, is expanded in the same plane wave basis that describes the system,

*P*(

**r**

_{||}). Consequently, the incident vector coefficients in the momentum representation become:

*E*,

_{x}*E*,

_{y}*H*and

_{x}*H*are the complex space components of the in-plane fields.

_{y}Finally, once we expressed the field in the complex space, it is possible to calculate the reflectivity and transmittance:

*is the flux of the Poynting vector in the*

_{z}*z*direction through a desired area

*A*:

*Inc*is for the incident fields, while

*Ref*and

*Tra*stand for the reflected and transmitted fields, as before. Note that area

*A*here can have any arbitrary shape.

When the integration is made in an area of orthogonality (*A _{RO}*) of the plane waves

*e*

^{iG · r||}, the integral of Eq. (14) reduces to:

*r*the integral in Eq. (14) has the analytical solution:

*J*

_{1}being the Bessel function of the first kind of order one. The expressions of Eq. (13) are a generalization to calculate

*R*and

*T*; when there is no pattern in the plane

*xy*, these quantities reduce to the square moduli of the well known Fresnel coefficients.

## 3. Implementation of the method

The method described in section 2 can be applied to any slab photonic crystal structure. In this section, we employ the method to calculate the reflectivity spectra of a L3 photonic crystal slab cavity at normal incidence. The photonic crystal we are interested in consists of a semiconductor membrane with a hexagonal pattern of air holes. The cavity is produced by missing three holes in a row [15]. This air-bridge photonic crystal can be modeled as a three layer structure: a layer of thickness *d* interposed between two semi-infinite layers. In this way, the layer index *l* takes three possible values: 0, 1 and 2. Figure 1(a) shows a scheme of the photonic crystal in *z* direction. The layer *l* = 1 has a dielectric function *ε*(*x*, *y*) which is constant along *z* in the layer, and contains the hexagonal photonic pattern of air holes in the plane; the layers *l* = 0 and *l* = 2 have a dielectric constant associated to air, *ε* = 1; and the quantities *a*^{(0)}, *b*^{(0)} and *a*^{(2)} represent the incident, reflected and transmitted waves, respectively.

A representative scheme of the reflectivity calculation for the L3 cavity is shown in Fig. 1(b). The incident electric and magnetic fields are **E*** _{Inc}* and

**H**

*, respectively, with wave vector*

_{Inc}**k**

*and polarization angle*

_{Inc}*φ*. The red circle with radius

*r*in the figure corresponds to the incident excitation spot, and the green circle with radius

_{s}*r*corresponds to the area over which the reflected Poynting vector is integrated. The reflected flux calculation is performed at the same distance

_{f}*d*

_{0}from the crystal at which the excitation is made.

Currently, it has become usual to measure reflectivity spectra of photonic crystals by cross-polarized spectroscopy [17, 18], which consists in collecting the radiation reflected from the sample at a polarization orthogonal to the polarization of the incident radiation. This filters the scattered radiation with the same polarization of the incident light and ensures a measurement of the sample response only. We adopt a cross-polarized scheme in our SM calculations.

## 4. Results and discussion

We choose a L3 cavity that has an outward displacement of the end lateral holes in order to increase the quality factor of the fundamental mode, Fig. 2(a); this shift has a magnitude of *s* = 0.15*a* [15], where *a* is the lattice parameter of the hexagonal distribution of holes, whose radius are *r _{h}* = 0.29

*a*. The fundamental mode of the cavity has a dominant field polarization in the direction perpendicular to the L3 defect (our

*y*direction), as demonstrated in Fig. 2(b), which shows the electric field vector plot of the fundamental mode calculated with the SM method. Note that the vector field is predominant in the

*y*direction, although the

*x*component has a nonzero contribution [19, 20]. It is important to take into account the mode polarization in order to understand the behavior of the reflectivity lineshapes. Figures 2(c) and 2(d) show, respectively, the dielectric function and the fundamental mode intensity distribution of the system considered in the SM calculations. Here,

*A*= 12

*a*is the supercell parameter. The refractive index of the dielectric membrane is

*n*= 3.5 and its thickness is

_{s}*d*= 0.5

*a*. The base dimension is fixed at 441 plane waves, which determine a well defined field distribution and a low computational cost. The small asymmetry seen in the mode profile of Fig. 2(d) is caused by the choice of a hexagonal lattice in the supercell approximation and the finite dimension of the base. The hexagonal supercell was chosen to preserve the photonic crystal symmetry.

To calculate the reflectivity spectrum we choose a circular incident spot of radius *r _{s}* = 2.15

*a*with the purpose to excite only the cavity. The

*P̃*(

**G**) coefficients of Eq. (11) for this geometrical shape are easily computed using simple integral identities. The vertical distance at which the crystal is excited, which is the same distance at which the Poynting vector flux is computed, is fixed at

*d*

_{0}= 20

*a*. In addition to this, since there are as many supercells as plane waves in the basis, the radius

*r*of the flux area must be appropriately chosen to collect radiation of only one supercell. In the reflection process of the crystal we expect a Fano resonance, which is the product of the electromagnetic interference between the radiation that leaks from the cavity mode (discrete channel) and the radiation reflected by the whole crystal pattern (continuum channel). If the discrete and continuum contributions to the interference are comparable, the resonance peak has a characteristic sharp asymmetrical shape. Nevertheless, when the incident wave is polarized at

_{f}*φ*= 0° or

*φ*= 90°, the reflected wave has the same polarization. The cross-polarized collection thus filters the continuum contribution and only the emission from the cavity is seen, as this has non-zero

*x*and

*y*components (see the vector field plot of the fundamental mode in Fig. 2b). Therefore, we expect symmetrical peaks in the reflectivity at

*φ*= 0° and

*φ*= 90° polarization [21]. In our model, collection is understood as the integration of the Poynting vector given by Eq. (14). With these physical ideas, whose origin comes from the Fano interference phenomenon, it is possible to find the appropriated

*r*to collect radiation of only one supercell:

_{f}*r*is reduced until symmetrical peaks at 0° and 90° are obtained in the reflectivity spectrum. We find that the optimal value of

_{f}*r*to make the polarization-dependent reflectivity calculations at

_{f}*d*

_{0}= 20

*a*is 0.8

*a*.

The polarization-dependent reflectivity spectra are shown in Fig. 3 for several values of *φ*. At *φ* = 0°, as we discussed above, a symmetrical lineshape is obtained. The degree of asymmetry increases with the polarization angle, until the lineshape is inverted around 52°. Above this angle the lines become asymmetrical again, albeit with a reversed shape. Finally, the degree of asymmetry decreases, reaching the symmetric case again at *φ* = 90°. The asymmetrical line-shapes can be understood in the following way. When the angle of polarization of the incident wave has an arbitrary value, the polarization of the reflected wave at the mode frequency is slightly different from the incident one, since the transmitted *y* component is higher than the *x* component at the mode frequency (due to the dominant mode polarization in the *y* direction). Consequently, the reflected wave has an increased *x* component, *i.e.*, the polarization of the reflected wave is rotated. This effect produces a non-zero contribution of the reflected wave in the cross-polarized collection around the mode frequency, and therefore the radiation emitted by the cavity (discrete channel) interferes with the radiation reflected by the photonic crystal (continuum channel), generating the characteristic Fano asymetrical lineshapes. At *φ* = 52.7°, the reflected radiation interferes destructively with the light from the cavity mode.

The spectra in Fig. 3 were fitted with the standard Fano formula [1]. The behavior of the Fano asymmetry factor is as expected, *i.e.*, large absolute values when the lines are symmetrical (0° and 90°), zero for the inverted lineshape (∼ 52°) and around 1 for the intermediate polarization angles. The linewidths obtained from the fit do not change significantly with the polarization angle and correspond to a quality factor of *Q* = 22800 ± 100. Figure 3 evidences that the degree of asymmetry can be controlled through the angle of polarization of the incident wave. These results are in good agreement with a recent experimental work made by Valentim *et al.* [22]. Control of the degree of asymmetry of the Fano resonance has also been reported by Driessen *et al.* [23] and Babic *et al.* [24] in photonic crystals without cavities, using the angle of incidence. In this way, the polarization angle of the incident field is another important parameter which must be taken into account in the Fano interference process [25]. The right side of Fig. 3 shows the imaginary part of the *x* and *y* electric field components, in the middle of the slab, for the polarization angle *φ* = 55°, at the energies corresponding to the minimum and maximum of the reflectivity. There is a clear phase difference in both components of the electric field between the reflectivity minimum and maximum.

The electromagnetic interference producing the characteristic sharp peaks seen in Fig. 3 depends on the relative contribution of the scattering through the L3 fundamental mode and the scattering by the photonic crystal pattern. This can be controlled through the polarization of the incident radiation, which controls the effective phase difference between the two scattering channels. Figure 4 shows the real part of the *E _{x}* and

*E*components for the cases

_{y}*φ*= 0°, 49°, 55° and 90° at the reflectivity maxima of Fig. 3. There is a clear phase change of the photonic mode produced by the polarization angle of the incident field. The cases

*φ*= 49° and

*φ*= 55°, which have associated reversed lineshapes (see Fig. 3), evidence opposite phases in their electric field profile distributions. Such phase change remains up to the symmetric lineshape at 90°, generating an opposite phase between the symmetrical cases at 0° and 90°.

Since not only the field of the photonic mode contributes to the Fano interference, we examine also the field outside the crystal, which contains the continuum contribution. Figure 5 shows the electric field profiles Re{*E _{x}*}, Re{

*E*} and Re{

_{y}*E*} in the transversal planes

_{z}*y*= 0 and

*x*= 0 for the same 49° and 55° cases of Fig. 4. The excitation field is incident from bottom to top. Outside the crystal there is not a significative phase change of the field; the relative maxima and minima remain the same while the polarization angle produces the reversal of the reflectivity lineshape. In this way, the polarization angle of the incident wave changes significantly the phase of the mode only, and consequently the relative phase between the discrete and continuum contributions.

Another interesting phenomenon, first reported in the experimental work performed by Galli *et al.* [26], is the fact that when the radius of the excitation spot is increased the lineshape of the Fano resonance becomes symmetrical. This can be fully reproduced by our SM implementation. Our results are shown in Fig. 6 for the polarization angles *φ* = 55° and *φ* = 60°. The lineshapes become symmetrical when the spot radius *r _{s}* increases, up to the maximum value

*r*= 6

_{s}*a*allowed by the supercell size. Symmetrical lineshapes are also obtained for all other polarization angles when

*r*= 6

_{s}*a*, producing absolute values of the asymmetry factor, |

*q*|, larger than 39, after fitting the spectrum with the Fano formula [1]. The symmetrical lineshapes for large excitation spots can be understood as follows. When the size of the spot increases, the contribution to the reflected wave of the area away from the cavity region also increases. Thus, it is a good approximation to consider that the radiation reflected from the photonic crystal pattern (continuum contribution) has the same polarization of the incident wave for all frequencies. The cross-polarized collection then filters the continuum contribution and only radiation from the cavity is seen.

## 5. Conclusion

The scattering matrix method, with our strategic modifications to describe arbitrary geometrical excitation spots, showed its great generality to characterize processes in photonic crystals involving scattering states, which is a requirement to study an universal phenomenon such as Fano resonances. The results of this work give a clear insight about the Fano interference process in the L3 photonic crystal slab cavity. First of all, the polarization of the excitation field incident on the crystal is an important parameter to take into account in the reflectivity calculations for this kind of cavities, whose modes have a well defined polarization. And secondly, the interference phenomenon depends on the relative contribution of the scattering through the discrete state and the scattering to the continuum. Since in the photonic crystal this phenomenon is produced by the electromagnetic interference between these two channels, the relative contribution can be controlled through the effective phase difference between them. Therefore, the polarization of the incident radiation can be used to control the effective phase difference between the scattering through the L3 fundamental mode and the scattering through the photonic crystal pattern. The SM implementation here allowed us to reproduce the experimental results reported by Galli *et al.* [26], in which the spot size can be used to enhance the continuum contribution in the electromagnetic Fano interference.

## Acknowledgments

We acknowledge financial support from CNPq, FAPEMIG and CAPES (Brazil), and Colciencias (Colombia) within the project code 110156933525, contract number 0026-2013 and HER-MES code 17432.

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