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Applied Physics B

, 124:136 | Cite as

Spectral characterization of silicon photonic crystal slab using out-of-plane light coupling arrangement

  • K. V. Ummer
  • R. Vijaya
Article
  • 82 Downloads

Abstract

Fano resonance is analyzed by measuring the transmission and reflection from a photonic crystal slab containing a hexagonal symmetry of holes in silicon fabricated by electron beam lithography. The spectral features in the measured transmission and angle dependent reflection are in good agreement with those calculated using rigorous coupled wave analysis method (using DiffractMOD module of RSoft\(^{\mathrm {TM}}\)). This provides a convincing proof to this versatile and convenient out-of-plane measurement technique. In our photonic crystal, the quality factors measured at 1314, 1328, 1377, 1599, and 1630 nm are high and can be utilized for designing low-threshold micro-lasers and narrowband optical filters in the silicon slab platform.

1 Introduction

Photonic crystal slab (PCS) is an attractive platform for miniaturization of optical devices with exotic optical properties such as negative refraction [1, 2] and super-collimation [3, 4]. In designing such devices, the in-plane periodicity of the order of wavelength is introduced in a higher-index dielectric slab surrounded by a lower-index medium such as air, so that light can confine in the vertical direction by total internal reflection and within the plane by guided modes. Such guided modes are completely isolated from the background and are confined within the slab [5]. These modes can be utilized for controlling the flow of light within the slab plane for different applications which helps in large-scale on-chip integration of optical components. But, one cannot directly couple an external radiation source to the guided modes in vertical direction [6]. The light from outside the PCS can resonantly couple only to the modes known as guided resonance modes [7]. These slab modes are located in the band structure above the light line corresponding to the background medium and are known as radiating modes. The guided modes have an infinite lifetime within the slab, whereas the radiating modes have a finite lifetime and can lose their energy to the background and hence also known as leaky modes.

The light incident vertically on the PCS can experience the lateral periodicity of the PCS and can couple to leaky modes with a finite lifetime, and also introduce a phase shift to obtain complex resonance behavior in the reflection or transmission spectrum which is known as the Fano resonance [8, 9]. The Fano resonance is the phenomenon found in systems when discrete quantum states interfere with a continuum of states and has been observed from many systems such as nanostructures and meta-materials [10, 11, 12, 13, 14]. In PCS, the Fano resonance arises from the interaction of excited leaky modes of the slab with the incident external radiation [15]. Thus, the frequency at which Fano resonances occur provides the details of the leaky modes of the PCS which can be excited by the light incident in the vertical direction. Hence the leaky modes are the channels for the PCS to interact with the external medium in vertical direction and can be studied with the light coupled from outside the PCS. They are very convenient since specialized coupling tapers are not required during the fabrication process. In addition, these Fano resonances are of special interest while designing efficient optical devices such as low-threshold lasers [16, 17, 18, 19], light-emitting diodes [20, 21] and narrowband optical filters [5, 22, 23] in the PCS geometry, in view of their spectral selectivity.

To understand completely how the eigenmodes of an un-patterned slab change upon introducing the texturing, one requires a systematic study on the characterization of the PCS. Villeneuve et al. studied theoretically the 3-D confinement of photons from PCS with class III and V semiconductors and from one-dimensional ridge waveguides [6]. Pacradouni et al. subsequently modelled the band structure obtained from experimental data of reflection spectrum from the PCS with square lattice of air holes in AlGaAs background [24]. Paddon et al. and Cowan et al. [25, 26] studied the formalism of resonant light coupling to guided and radiation modes using Green’s function technique. Ochiai and Sakoda suggested the free photon approximation for the calculation of band structure of the PCS [27]. Tikhodeev et al. formulated numerical method based on scattering matrix for studying the optical response of guided resonance in transmission spectrum from PCS with square lattice symmetry [28]. S. Fan and Joannopoulos theoretically studied the guided resonance [29] and its spectral features where they analyzed that the diameter of the holes is crucial to control the vertical coupling of the light from PCS to the external medium. Gippius et al. have studied theoretically the optical response of PCS with asymmetric unit cell [30]. In the present work, we study the characterization of silicon (Si) PCS made on silicon-on-insulator (SOI) wafer with vertical coupling method.

The Fano resonance has been analyzed numerically by many groups from passive PCS as well as from functional active PCS devices. Koshino has studied theoretically the Fano resonance from one-dimensional PCS [31]. Chen et al. has studied the optical filtering properties using Fano-resonance with a square lattice on SOI wafer [22]. Recently, Limonov et al. has studied the distinction between Fano-resonance and other resonances found in photonics [15]. Prodan et al. experimentally investigated the guided resonance from square lattice PCS made by laser interference lithography [32]. Paraire and Benachour have investigated the PCS band diagram experimentally using surface coupling technique of the incident beam with a silica protection layer on the top of the device layer [33], where they have measured the transmission and reflection spectrum using Fourier transform infrared spectrometer. Benjamin et al. proposed the mapping of the dispersion diagram from the spatiotemporal evolution of the fields in the polaritonic platform using THz-frequency PCS with square lattice symmetry [34]. Wierer et al. mapped the band structure of their active PCS by collecting angle-resolved emission of the InGaN quantum well from their LED device [21]. Blanchard et al. analyzed the coupling of incident light with the modes which are optical bound states in the radiation continuum (BICs) of the band structure [35]. The quality factors of the BICs are very high as their energy does not decay to the background and thus have attracted research recently [36, 37, 38, 39, 40]. Besides the theoretical studies on guided resonances from PCS, the reports on experimental characterization studies are very few. Hence, we have carried out this experimental characterization of guided resonance modes to evaluate their tolerance to fabrication limitations. In the present work, we have used electron beam lithography (EBL) and fabricated a PCS with a hexagonal lattice of air holes in a SOI substrate. The ratio (r / a) of radius of the hole to the lattice constant is 0.227 with the lattice parameter of a = 664 nm. The device layer thickness (Si, \(\varepsilon\) = 12) is 500 nm, the box layer thickness (SiO\(_2\), n = 1.44) is 3 \(\upmu\)m and handle layer (Si) thickness is 623 \(\upmu\)m in our SOI substrate.

We have measured the guided resonances with a very simple, specially-designed experimental set-up. We have also calculated the guided resonance frequencies and compared them to the measured results. In the following sections, the fabrication of PCS and theoretical modeling of the guided resonances has been discussed in Sect. 2. The transmission and reflection measurements are discussed respectively in Sect. 3 and Sect. 4 while in Sect. 5, the detailed discussion on guided resonance modes fit to the Fano resonances has been discussed and the quality factors of the resonances of the measured reflection spectrum have been compared with the calculated reflection spectrum and finally, conclusions are in Sect. 6.

2 Fabrication of PCS and modeling of guided resonance modes

Fig. 1

FESEM image of the fabricated Si PCS on SOI wafer with lattice parameter, a = 664 nm and diameter of holes = 301 nm. The scale bar is 200 nm. The inset figure shows a 30\(\,^{\circ }\) angular view

The hexagonal array of holes is made on SOI wafer by EBL. After RCA cleaning of the wafer, a photoresist (PMMA C3) of thickness 200 nm has been spin coated over SOI wafer at 6000 rpm for 45 s followed by baking at 180 \(^{\circ }\)C for 2 min by keeping on a hot plate. Raith e-LINE machine has been used for writing the lattice structure on the photoresist with a 187.5 dose, 20 kV and 10 \(\upmu\)m aperture size. The overall area of the structure is 500 \(\upmu\)m \(\times\) 500 \(\upmu\)m. The electron beam exposed wafer is dipped for 40 s in the developer which has MIBK: IPA concentration of 1:3. The reactive ion etching (RIE) of the developed wafer for 30 s is carried out using SF\(_6\) and C\(_4\)F\(_8\) gases. Finally PMMA photoresist has been removed by ashing for 2 min.

Figure 1 shows the field emission scanning electron microscope (FESEM) image of the fabricated PCS as a hexagonal lattice of air holes on SOI wafer by EBL. The shortest distance between the centers of the adjacent holes is a = 664 nm with a \(\pm \,\)4 nm deviation after the fabrication process. The inset figure shows the 30\(\,^{\circ }\) angular view of the holes which shows the neat, vertical side walls. The three dimensional projected band structure towards background medium (n\(_{\mathrm {eff}}\) = 1.44 corresponding to the box layer; since the hole is not piercing the box layer, the \(n_{\mathrm {eff}}\) is same as the index of box layer) which is calculated using the BandSOLVE module of RSoft\(^{\mathrm {TM}}\) is shown in Fig. 2a. The number of plane waves used is 32*32*32 with eigenvalue resolution of 10\(^{-6}\). The number of bands is 24 and the number of k-path difference is 120. A supercell containing an in-plane unit cell with out-of-plane periodicity chosen as four times the lattice period (4a) is also considered in the band calculation.
Fig. 2

a Three dimensional band structure for hexagonal photonic crystal slab for r / a = 0.227. The modes above the light line are called leaky modes and shown as thin blue curves. The modes below the light line are guided modes and are shown with thick blue curves. The green shaded region is in-plane band-gap. The inset figure shows the first Brillouin zone with the high symmetry directions of the lattice. The schematic for out-of-plane reflection measurement along \({\Gamma }\)K direction of the hexagonal lattice. The light incident from the background at an incident angle from the normal to the PCS plane provides the in-plane k vector for the leaky modes

The modes below the light line (\(n_{\mathrm {eff}}\) = 1.44) are the guided modes (thick blue curves) and these remain confined within the PCS. The modes above the light line are the radiating modes (thin blue curves). The inset figure shows the first Brillouin zone along with the high symmetry directions. The green shaded region is the in-plane stopband ranging in reduced frequency of \(a/\lambda\) = 0.4618–0.4441 and the corresponding wavelength range is 1437.85–1495 nm. The in-plane stopband can act as a micro-cavity for emitters embedded within the PCS. The in-plane stopband can be utilized for controlling the spontaneous emission of embedded emitters for low threshold lasing, similar to the studies using three-dimensional active PhCs [41, 42, 43].

The reflection spectrum has been measured at different angles from the normal to PCS along \({\Gamma }\)K and \({\Gamma }\)M directions which will be discussed in the next section. The reflection and transmission are also obtained by calculation using the DiffractMOD module of RSoft\(^{\mathrm {TM}}\). The light source used in DiffractMOD is polarized, either P-polarized (electric field within the plane of reflection) or S-polarized (electric field perpendicular to the plane of reflection). The harmonics along x and z axis are chosen respectively as 6 and 4 with an index resolution of \(\sim\) 18 nm. The wavelength resolution for the incident light source is chosen as 0.1 nm. The frequency dependent dielectric constant for Si slab (device layer and handle layer) has been considered in the simulations. The transmission plane of the domain for the calculation of reflection or transmission has been kept within the handle layer to reduce the time for simulation and to extract out the Fabry-Perot fringes since our focus is on the resonance features from PCS.

Figure 2b represents the schematic for coupling of the external light with the leaky modes of the photonic crystal slab along \({\Gamma }\)K which is one of the high-symmetry directions. The external radiation cannot couple to the guided modes since the in-plane k-vector of the guided modes does not match with the light incident in the vertical direction. But the leaky modes of the photonic crystal can be excited by the vertical incident light at different angles from the normal to the plane of the PCS.
Fig. 3

Band structure along a \({\Gamma }\)K direction and b \({\Gamma }\)M direction for Si PCS. The light line for the background upper medium (air) and bottom (box layer) are shown respectively with brown open circle and closed green circle symbols. The \(k_\Vert\) vector component for the light incident at an angle for reduced frequency ranges \(a/\lambda\)= 0– 0.59 subtends straight line above the light line in the band structure. Such \(k_\Vert\) straight lines for incidence angles 35\(\,^{\circ }\) to 75\(\,^{\circ }\) are shown with black solid lines. The shaded regions in grey colors are the in-plane stopband along \({\Gamma }\)K and \({\Gamma }\)M directions

Figure 3a and b represent the band structure respectively along \({\Gamma }\)K and \({\Gamma }\)M directions. The light line for the background upper (air) medium and bottom (box layer) medium are shown respectively with grey hollow circle and filled green circle symbols. The modes above the light lines (thin curves) are leaky modes and the modes below the light line (thick curves) are guided modes well-confined within the slab from the background cladding. The modes between the light line for air and oxide layer cladding are guided with respect to air but are leaky modes with respect to the oxide layer. Hence these modes are more confined to the slab from air and less confined from the oxide layer. Therefore the modes below the light line for box layer (SiO\(_2\), n = 1.44) can be considered as purely guided modes of the PCS. The gray shaded regions represent the in-plane stopband. Along \({\Gamma }\)K direction (in Fig. 3a), the wavelength ranges for three in-plane stopbands are 1437.6–1495.5, 1504–1616.4, and 1675.9–1846.5 nm, whereas along the \({\Gamma }\)M direction (in Fig. 3b), these are 1682.7–1722.9, 1736.4–1807.8 nm, and 1809.8–2007.9 nm for the lattice constant used in the experimental sample. The \(k_\Vert\) wave vector for a plane wave incident at a wavelength of \(\lambda _{{\text {air}}}\) is given by
$$\begin{aligned} k_{\parallel }=k\sin \theta =\frac{2\pi }{\lambda _{{\text {air}}}}n\sin \theta \end{aligned}$$
(1)
where \(\theta\) is the angle of incident light with the normal to the plane of the PCS and n is the refractive index of the background.

For a fixed incident angle \(\theta\) and crystallographic symmetry direction, the \(k_\Vert\) vector component for the light incident at an angle for the reduced frequency range of \(a/\lambda\) = 0.0–0.59 subtends straight line above the light line in the band structure. Such \(k_\Vert\) straight lines for incidence angles of 35\(\,^{\circ }\) to 75\(\,^{\circ }\) are shown with black solid straight lines in Fig. 3a and 3b. Here these lines are drawn with respect to the oxide layer cladding with an effective index of 1.44. At the point of intersection of the \(k_\Vert\) line with the slab leaky modes, the matching condition for the in-plane k vector of the light from the background with the in-plane k-vector of the PCS leaky modes will be preserved and hence makes coupling possible. These intersection points are expected to provide the frequency of resonance peak observed in reflection or transmission spectrum. One may note that for asymmetric PCS, for an incident angle \(\theta\) in measurement (Fig. 2b), the \(k_\Vert\) line with respect to top and bottom cladding will be different. On the other hand, for a symmetric PCS the \(k_\Vert\) line at a particular angle will be unique with respect to the background medium. In this work, since the slab is asymmetric, the leaky modes excited by light incident at an angle correspond to two \(k_\Vert\) straight lines in the band structure with respect to the air and oxide layer backgrounds. The intersection of these lines with the leaky modes gives the frequency of the Fano resonance or guided resonance.

3 Experimental set-up for reflection and transmission measurements

Fig. 4

a Experimental set-up used for reflection and transmission measurements. b Transmission spectrum of PCS with r / a = 0.227 measured experimentally for un-patterned SOI (blue dashed curve) and from PCS (blue solid curve). The calculated transmission spectrum using DiffractMOD is shown with black dotted curve. The dip in transmission denoted by arrows represents the guided resonances which can couple resonantly with the external radiation

The experimental set-up for reflection and transmission measurements is shown in Fig. 4a. A super-luminescent laser diode emitting in the wavelength range of 1.25–1.75 \(\upmu\)m is used as the light source for this study. The output light from the source is not polarized and is incident on PCS through a bare optical fiber. The sample is kept on a rotational stage (black circle in the figure) and the reflected/transmitted beam is collected through a tapered optical fiber (D) which is kept on another rotational stage (shown by a green circle in the figure). The collected light is analyzed with an optical spectrum analyzer (OSA) which has a resolution of 0.065 nm in the wavelength range of 1.25–1.65 \(\upmu\)m.

The transmission spectrum measured for the bare SOI wafer is shown (blue dashed curve) in Fig. 4b. The oscillatory trend (without any sharp features) is due to the uniform thickness of the slab. The experimentally measured transmittance from PCS, shown with blue solid curve in Fig. 4b, has multiple dips and these are denoted by arrows. Some of these dips are matching well with transmittance (black dotted curve) calculated using DiffractMOD, with p-polarized light source. To interpret these modes, we consider Fig. 3. The modes intersecting at the center of the Brillouin zone, \({\Gamma }\) point (which corresponds to the transmission direction) in Fig. 3 are shown in the first column of Table 1.
Table 1

The wavelength of resonance modes in transmittance

Modes intersecting at \({\Gamma }\) point (in Fig. 3)

Wavelength of resonance-measured (Exp) (nm)

Wavelength of resonance-calculated (DiffractMOD) (nm)

\(a/\lambda \,\,(a = 664\) nm)

\(\lambda\) (nm)

  

0.5306

1251.41

1263

1252.83

0.497

1336.02

1332.72

1349.34

0.4864

1365.05

1414.75

1389.41

0.4595

1445.05

1430

1448.02

0.4332

1532.78

1519

0.4209

1559.57

1558

0.4164

1594.62

1618

1608

It is clear from Table 1 that most of the measured (3rd column) and calculated (4th column—using DiffractMOD) resonances are matching well with each other. The deviations in wavelength between the measured and the calculated spectra are within the tolerance of slab thickness. It is observed from the calculation that the resonance peak shows a blue shift in wavelength as one reduces the slab (device layer) thickness. For a thickness variation of ±10 nm, the resonance peak shows a shift in wavelength of ±10 nm. The resonance modes—calculated and measured—are also in good agreement with the leaky modes (2nd column) of the PCS calculated from the band structure. This indicates that most of the slab leaky modes can be excited with simple out-of-plane experimental set up with an unpolarized light source.

4 Reflection measurements

Fig. 5

The measured reflection spectrum from patterned SOI (blue curves) and from bare SOI wafer (black dotted curve) at angles of 30\(\,^{\circ }\), 35\(\,^{\circ }\), 45\(\,^{\circ }\), 55\(\,^{\circ }\), 65\(\,^{\circ }\) and 75\(\,^{\circ }\) along a \({\Gamma }\)K direction and b along \({\Gamma }\)M direction. The resonance behavior can be observed from the reflection spectrum from patterned SOI as compared to the spectrum from reference background (bare SOI) wafer. An offset value is being added to the y axis for mapping the reflection spectrum at different angles

The reflection measurements at different angles from the normal are recorded for patterned and un-patterned (bare) SOI wafer using the set-up shown in Fig. 4a. The reflection spectrum from 30\(\,^{\circ }\) to 75\(\,^{\circ }\) has been measured. Figure 5a and b show the measured reflection spectrum respectively along \({\Gamma }\)K and \({\Gamma }\)M directions. The blue solid and black dotted curves represent the reflection spectrum respectively from patterned and un-patterned SOI wafer. The light from the super-luminescent laser diode is un-polarized. Hence in the reflection spectrum, the collective behavior of both the TE and TM mode-like features is found, depending on the polarized field component present in the incident light. One can observe the resonance features in the reflection spectrum from the patterned area which are absent in the reference SOI wafer.
Fig. 6

The measured (black solid curve) and calculated (blue dotted and dot dashed curve) reflection spectrum from patterned SOI wafer along \({\Gamma }\)K direction at angles of a 45\(\,^{\circ }\) b 55\(\,^{\circ }\) and c 65\(\,^{\circ }\). The calculated reflection spectrum using RSoft\(^{\mathrm {TM}}\) DiffractMOD for p and s polarized incident light source are shown respectively with blue dotted and blue dot-dashed curves

The measured and calculated reflection spectrum along \({\Gamma }\)K direction at 45\(\,^{\circ }\), 55\(\,^{\circ }\) and 65\(\,^{\circ }\) are shown respectively in Fig. 6a, b and c. The measured spectrum using unpolarized light source is shown with black solid curve while the calculated spectrum is shown with blue solid (for p-polarized light) and blue dashed (for s-polarized light) curves. From Fig. 6, one can note that the measured reflection spectrum shows a collective response of both s and p polarization features. Some of the resonant features excited by measurements are matching well with those calculated with p- polarized light and some of them are overlapping with calculated spectrum using s-polarized light. Some resonances are found to be weakly excited. For example, the resonances at 1287, 1357, and 1408 nm in the 45\(\,^{\circ }\) reflection spectrum (in Fig. 6a), the resonances at 1290, 1310, 1382, 1470, 1541, and 1630 nm at 55\(\,^{\circ }\) (in Fig. 6b) and the resonances at 1301, 1314, 1354, 1443, 1472, 1603 and 1639 nm corresponding to 65\(\,^{\circ }\) (in Fig. 6c) are excited well. But the shape and width of some of the resonances measured are differing from the calculated ones. This might depend on the degree of the vertical side wall of the fabricated holes. Also, some deviations in wavelength between the measured and the calculated spectra are within the tolerance of slab thickness as discussed in Sect. 3. The experimental results show that the wavelength of the resonance features in reflection spectrum at different angles matched well with the point of intersection of the slab leaky modes with the constructed \(k_\Vert\) line drawn with respect to the background oxide layer (with n\(_{\mathrm {eff}}\) = 1.44). A few of the resonances are found to be matching with the \(k_\Vert\) line drawn with respect to air background (n\(_{\mathrm {eff}}\) = 1.00).
Fig. 7

The measured (black solid curve) and calculated (blue dotted and dot dashed curve) reflection spectrum from patterned SOI wafer along \({\Gamma }\)M direction at angles of a 45\(\,^{\circ }\) b 55\(\,^{\circ }\). The calculated reflection spectrum using RSoft\(^{\mathrm {TM}}\) DiffractMOD for p and s polarized incident light source are shown respectively with blue dotted and blue dot-dashed curves

Similarly, the measured and calculated reflection spectrum along \({\Gamma }\)M direction at incident angles of 45\(\,^{\circ }\) and 55\(\,^{\circ }\) are shown in Fig. 7. The reflection spectrum at 45\(\,^{\circ }\) (in Fig. 7a), is resembling with the reflection spectrum calculated using s-polarized light, while the reflection spectrum at 55\(\,^{\circ }\) (Fig. 7b) is better resembling with the calculated reflection spectrum using s-polarized light (in the longer wavelength side) and p-polarized light (in the lower wavelength side). The reduced frequencies of the guided resonance modes from the calculated and measured reflection spectrum, and the corresponding \(k_\Vert\) wave vectors (using Eq. 1) have been calculated. These mode frequencies and \(k_\Vert\) vectors are compared to those obtained from the band structure shown in Fig. 3. i.e., intersection of leaky modes with \(k_\Vert\) line (with n\(_{\mathrm {eff}}\) = 1.44 and n\(_{\mathrm {eff}}\) = 1.00) corresponding to different incident angles. These resonance frequencies and corresponding \(k_\Vert\) vectors are being used to map in the band structure (calculated using BandSOLVE) along \({\Gamma }\)K and \({\Gamma }\)M directions for the PCS which is shown respectively in Fig. 8a and b. The reduced frequency obtained from the experiment is shown with blue stars symbols and the modes obtained from the calculated reflection spectrum are shown with red circle symbols. The light line for box layer is shown with green dashed line. The radiating modes (thin blue curves) above the light line and guided modes (thick blue curves) below the light line are calculated using BandSOLVE. The \(k_\Vert\) line at different angles corresponding to oxide background is shown with black dotted straight lines. One can note that the measured and calculated resonance features are fairly coincident with each other. These resonance features are exactly falling on the point of intersection of leaky modes with the constructed \(k_\Vert\) line corresponding to different incident angles.
Fig. 8

The band structure is shown along with the reduced frequencies obtained from experiment (blue stars symbols) and from calculated (red circle symbols) reflection spectrum. The light line for box layer is shown with green dashed line and the in-plane stopband by green shaded regions. The blue thin curves (above light line) are radiating modes while guided modes shown by thick blue curves are calculated using BandSOLVE. The modes obtained from measurement are shown with blue star symbols while the modes calculated using DiffractMOD are shown with red circle symbols. The \(k_\Vert\) lines at different angles corresponding to background oxide layer are shown with black dotted straight lines while \(k_\Vert\) for air cladding is not drawn here

Since the PCS used here is asymmetric, a single incidence angle in experiments corresponds to two \(k_\Vert\) lines in the band structure—one due to air cladding and other due to oxide cladding layer (see Eq. 1). It is observed that a few of the resonant modes excited in measurement and calculation are found to lie at the point of intersection of radiating modes with the \(k_\Vert\) line corresponding to air background. These modes corresponding to 35\(\,^{\circ }\) to 75\(\,^{\circ }\) are also being mapped in Fig. 8, where \(k_\Vert\) straight lines are not drawn. Such a close coincidence between the calculated and measured data along with the calculated band structure provides a convincing proof for the vertical coupling experimental set-up.

5 Analysis of guided resonance

One can notice that the resonances observed in the reflection spectrum have different line shapes displaying maxima, minima or asymmetric behavior. These dispersive line shapes are arising from the interference between the light directly reflected from the PCS surface and the light which is coupled to leaky modes (with a finite life time) of the PCS and thus introducing a phase shift. The width (\(\gamma\)) and the center wavelength (\(\lambda _0\)) of the observed resonances are closely related to the quality factor (Q) of the leaky modes, which is given by (\(Q = \frac{\lambda _0}{\gamma }\)) the ratio of the center wavelength to the FWHM of the resonance. These resonances are related to the Fano resonance relation given by [8],
$$\begin{aligned} f(\lambda )=f_0\left[ 1+\frac{q^2-1+4q(\lambda -\lambda _0)/\gamma }{1+4q(\lambda -\lambda _0)^2/\gamma ^2}\right] \end{aligned}$$
(2)
where, \(f_0\) is the oscillator strength and q represents the asymmetry factor (degree of phase shift). For a value of q = 0, the incident light energy couples to the PCS leaky modes without having any phase shift and hence one can expect a dip in the reflection spectrum. The observed resonances in reflection spectrum are being fitted to the Fano relation given in Eq. 2 and the quality factors are extracted out. Figure 9 shows some of the resonances observed in the reflection spectrum along \({\Gamma }\)K and \({\Gamma }\)M direction of the hexagonal lattice of air holes. Appropriate offset value and \(f_0\) have been chosen to fit with the experimental data.
Fig. 9

The resonance features in the measured reflection spectrum (black solid curve) shown along with the fit to the Fano resonance (blue dotted) curve a at 55\(\,^{\circ }\) along \({\Gamma }\)K direction with \(\lambda _0\) = 1630.45 nm, \(\gamma\) = 3.4 nm and Q = 479.55, b at 65\(\,^{\circ }\) along \({\Gamma }\)K direction with \(\lambda _0\) = 1301.1 nm, \(\gamma\) = 3.4 nm and Q = 382.68, c at 65\(\,^{\circ }\) along \({\Gamma }\)K direction with \(\lambda _0\) = 1313.97 nm, \(\gamma\) = 2.3 nm and Q = 571.29 and d at 45\(\,^{\circ }\) along \({\Gamma }\)M direction with \(\lambda _0\) = 1376.9 nm, \(\gamma\) = 2.48 nm and Q = 555.2

The shape of the resonances observed in reflection spectrum have been fit to the Fano relation and the Q factor has been measured. The different values of q factor correspond to different line shapes of the resonance peak. In Fig. 9a and d, the value of q is \(\sim\) 1 while in Fig. 9b q is chosen as 6 and in Fig. 9c q is zero. The measured Q factors from the measured experimental reflection data have been compared with the Q factors of the calculated reflection spectra using DiffractMOD. Table 2 shows the Q factors of the Fano resonances in the measured and calculated reflection spectrum at different angles. The centre wavelength of the resonance peaks in the measured reflection spectrum is matched well with the calculated reflection spectrum. The Q factor of the measured spectra being lower as compared to the calculated spectra might be due to the roughness of the side walls of the air holes in the samples, thus leading to additional energy loss of the coupled resonant modes.
Table 2

The measured and calculated quality factors of the Fano resonances observed in the reflection spectrum

Measured (experiment)

Calculated (DiffractMOD)

\(\lambda\) (nm)

Quality factor, Q

\(\lambda\) (nm)

Quality factor, Q

1289.91

248.54

1294.47

213.1

1313.97

571.29

1311.43

1457.2

1328.46

800.30

1324.80

2453

1376.9

555.2

1386.45

1205

1418.0

373.16

1413.85

537.6

1470.6

340.42

1480.3

251

1542.8

342.8

1542.3

1296.1

1553.47

319

1552.32

535.28

1599.4

869.24

1584.06

1085

1630.45

479.55

1638.2

1005

The width of the Fano resonances is crucial for the applications. For PCS applications such as miniaturized on-chip low-threshold lasers, one requires very narrow resonances (high quality factor) and this can be obtained by reducing the diameter of the holes [29]. For LEDs, the Fano resonance line width should match with the emission linewidth of the emitter in order to get maximum outward coupling of light generated within PCS [20, 44]. A higher quality factor at different wavelengths enables the wavelength selectivity of lasing with low threshold powers. We have achieved experimentally the highest quality factor of 1096 at 1490 nm which is useful for PCS applications in designing narrowband optical filters.

6 Conclusions

In conclusion, we have analyzed the guided resonances from the PCS fabricated by EBL. The fabricated PCS has been characterized optically by measuring the reflection spectrum at different angles by using a vertical coupling experimental set-up. The experimental reflection and transmission spectrum results are very well matched with the simulation results. This is clear from the transmission measurements as well as from the reflection measurements at different angles. From the sharp peaks in reflection corresponding to the Fano resonances, the quality factor of the modes can be obtained. Quality factors of 571 at 1314 nm, 800 at 1328 nm, 555 at 1377 nm, 869 at 1599 nm, and 479 at 1630 nm have been obtained. The guided resonances from these structures can be utilized for optical devices such as light emitting diodes, vertical emitting low-threshold lasers and narrowband filters. A high quality factor at different wavelength finds applications such as Si-based low-threshold microlasers with the freedom of wavelength tuning and narrowband optical filters.

Notes

Acknowledgements

K.V. Ummer thanks Ms. Suchita Yadav and Mr. Govind Kumar for their help during the experimental set-up. K.V. Ummer also thanks Dipak Rout for the technical help during the preparation of the manuscript. The fabrication work was carried out at the CeNSE, IISc Bangalore under INUP which is sponsored by DIT, MCIT, Government of India. The work was partially supported by (1) IRDE, Dehradun, India under the DRDO Nanophotonics program (ST-12/IRD-124), (2) DST, India under the India-Taiwan S&T co-operation project (GITA/DST/TWN/P-61/2014) and (3) SERB (EMR/2015/001450).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsIndian Institute of Technology KanpurKanpurIndia
  2. 2.Centre for Lasers and PhotonicsIndian Institute of Technology KanpurKanpurIndia

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