Eur. Phys. J. D (2015) 69: 273 DOI: 10.1140/epjd/e2015-60478-7
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Correlation of optical reflectivity with numerical calculations for a two-dimensional photonic crystal designed in Ge Marius Adrian Husanu1 , Dana Georgeta Popescu1,a , Constantin Paul Ganea1 , Iulia Anghel2,3 , and Camelia Florica1 1 2 3
National Institute of Materials Physics, Atomistilor 105b, 077125 Magurele-Ilfov, Romania National Institute for Laser, Plasma & Radiation Physics, Atomistilor 409, 077125 Magurele, Romania University of Bucharest, Faculty of Physics, Atomistilor 405, 077125 Magurele, Romania Received 20 August 2015 / Received in final form 5 October 2015 c EDP Sciences, Societ` Published online 10 December 2015 – a Italiana di Fisica, Springer-Verlag 2015 Abstract. A two dimensional photonic crystal (2DPhC) with triangular symmetry is investigated using optical reflectivity measurements and numerical calculations. The system has been obtained by direct laser writing, using a pulsed laser (λ = 775 nm), perforating an In-doped Ge wafer. A lattice of holes with welldefined symmetry has been designed. Analyzing the spectral signature of PBGs recorded experimentaly with finite difference time domain theoretical calculations one was able to prove the relation between the geometric parameters (hole format, lattice constant) of the system and its ability to trap and guide the radiation in specific energy range. It was shown that at low frequency and telecommunication ranges of transvelsal electric modes photonic band gap occur. This structure may have potential aplications in designing photonic devices with applications in energy storage and conversion as potential alternative to Si-based technology.
1 Introduction In the last decades, many interesting properties concerning the manipulation of light at wavelength scale in periodic dielectric structures have aroused [1–8] in all of the three dimensions. These materials, known as photonic crystals (PhCs), display forbidden frequency region where electromagnetic wave cannot be steered along any direction, called photonic band gaps (PBG). This may cause the appearance of certain peculiar physical phenomena. The presence of PBGs can change dramatically the properties of light for permitting the realization of ultracompact photonic integrated circuits, capable of photon localization and light diffraction in UV, visible and near infrared. Some of the potential applications consist in waveguides [4,9], sensors [10], PhC fibers [11], optical switches [12,13], optical filters [14], magnetic PhCs [15]. One can deduce the PBGs for a certain photonic crystal design by studying the dispersion diagrams, characterized by photonic mode energies depending on the wave vector. The optical response of these periodic structures can be conveniently tuned by modeling the lattice constant, refractive index and sample thickness. The advantage of the photonic crystals compared with their conventional counterparts are the very small feature sizes, long life period, better confinement of light in ultra-small spaces and high speed of operation. Even if three dimensional PhCs are ideal for controlling the emission and propagation of light [3], it is handier a
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to use two dimensional PhCs because they can be easily fabricated. Till now silicon has been established as the material of choice for industry. However, in photonics, it has some constrains due to its indirect bandgap and the limited degree of freedom in material design. On the other hand, germanium (Ge) had some limitation because of the absence of stable native oxide for gate insulation in the metaloxide semiconductor field-effect transistors (MOSFET). Nevertheless, as a result of the progress on the device miniaturization on high dielectric systems the door for replacements of Si as channel materials was opened. Thus Ge offers the advantage of direct CMOS compatibility [16], a higher refractive index (n ∼ 4.0) and low dispersion properties in a wide range of temperatures. It is also a valuable photodetector material for applications in on-chip data distribution [17,18] and for emitters [19], lasers [20] and solar cells applications [21]. In this paper we present a study regarding the feasibility of using germanium as a substrate for applications in photonics, i.e. 2D PhC. The structure is numerically and experimentally investigated by using the finite-difference time-domain (FDTD) calculations and Fourier transform infrared spectroscopy (FTIR). The present design was built by drilling the Ge surface with a ultra-short pulses laser at sub-micrometer resolution. To achieve the desired structure scanning electron microscopy (SEM) was used and the substrate was previously investigated by means of X-Ray Photoelectron Spectroscopy [22,23].
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Fig. 1. Schematic representation of the principle DLW.
The experimental setups and FDTD method used in the present work are depicted in Sections 2 and 3. The details of the experiment and numerical results are presented in Section 4. The potential applications and the conclusions for PhC system is summarized in Section 5.
2 Experimental details Direct laser writing (DLW) using ultrafast lasers became a popular technique for fabrication of two dimensions systems [24–26]. The periodic structure was realized by a femtosecond laser ablation setup (Fig. 1) in a p-type Ge (001) wafer using a a femtosecond regenerative amplifier (CPA-2101 system from Clark-MXR), which delivers pulses at 2 kHz repetition rate. The pulse duration was 200 fs, the central wavelength 775 nm, and maximum laser energy of about 0.6 mJ per pulse. A Mitutoyo microscope objective with high magnification (100×) and numerical aperture (NA = 0.5) was used to focus (spot diameter 2 μm) the laser beam on the sample. In order to reach the desired structure, the optimal laser intensity and the number of pulses were subsequently determined (N = 15). The optimum processing parameters were established by mapping equidistant points every 15 μm, using a 10 × 10 points with the lattice constant of 1.5 μm for different energies and number of pulses. The laser power was varied from 0.02 mW up to 0.1 mW and a 35 nJ energy was chosen for most favorable processing structure. To evaluate the diameters of the ablated holes and the morphology of the samples a Scanning Electron Microscope (SEM) were used. This process allows choosing the best ablation quality and structure in order to use it further in the experiments. A Fourier Transform Infrared Spectrometer (FTIR) – Perkin Elmer Spectrum 100 Spectrometer – was used for recording the reflectivity of the photonic crystals. The measurements were carried out at room temperature in the 7800 cm−1 –650 cm−1 range. The obtained parameters of the constituent structure allowed the computation of the photonic band gaps and reflectivity spectra.
3 Numerical details The theoretical computations performed with the MIT photonic bands (MPB) Package [27] and MIT
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electromagnetic equation propagation (MEEP) [28] can provide very accurate prediction on the properties of photons propagating in various structures and therefore it is very helpful as complementary investigation to experimental analysis. MPB is a freely available software package, based on finite difference frequency-domain (FDFD) simulations of arbitrary electromagnetic (EM) structures, which directly computes eigenstates and eigenvalues of the Maxwell equation (1) in the frequency domain [29]: 1 ω 2 ∇× ∇ × H (r) = H (r) , (1) ε(r) c where ε(r) is the dielectric function and cis the speed of light in vacuum, ( ωc )2 is the eigenvalue. Because of the periodicity of ε(r), Bloch’s theorem can be used to expand both magnetic field H (r) and ε(r) in terms of plane waves [30,31]. The MPB method takes a Fourier transform over an infinite repetition of the computational cell, into all directions in order to avoid any step discontinuities. The repetition of computational cells introduces a new period in vertical direction, hence MPB can only calculate the guided modes at high precision, since those are strong localized within the PhC slab. Still, the weak guided modes, called leaky modes, remain dependent on the period introduced in vertical direction. The strength of MPB is that it can calculate band structures and eigenstates very accurately, being able to acquire every eigenstate, and even modes with closely spaced frequencies appearing as individual modes, the results improving with every new computational iteration. MEEP is also an open source FDTD electrodynamics code developed to model electromagnetic systems, which is applied to computing the transmission or scattering spectra from some finite structure [28,29], and can be used to handle 2D and 3D photonic structures and to model photonic crystals, waveguides and other optical structures. The propagation of fields is calculated directly as a function of space and time, according to Maxwell’s equations, requiring however to specify, within a computational cell, light sources and flux planes, to generate light and follow its propagation through the structure. After the computational cell has been divided in a discreet spatial grid the electromagnetic fields are evolved in discreet time steps [28]. MEEP can analyze a myriad of electromagnetic situations allowing the direct investigation in time for given light sources of the resulting field patterns. Transmission and reflection spectra can be calculated by a single computation followed by a Fourier-transformation as the response to a short pulse emitted by the source. In our case, two simulation runs were performed: one with the empty cell in order to acquire the source spectrum, and another with the studied geometry. Thus the needed flux of the structure is the reflection spectrum resulted in the second run normalized with the source one. Unlike MPB, MEEP can have difficulties resolving modes that are closely spaced in frequencies, but it calculates all frequency modes at the same time, not consecutively.
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Fig. 2. SEM plan-view image of the tested 2D photonic crystal structure.
4 Results and discussion In order to investigate the properties of two-dimensional photonic crystals designed in a germanium matrix of permittivity ε = 16.2 (the imaginary component of the dielectric constant is assumed to be zero, because it does not affect the dispersion relation of a periodic structure) regarding the effects which appear at the surface, the reflectance spectra for the considered structure is experimentally determined and calculated. A plan-view image of the prepared 2D PhC structure obtained in SEM is depicted in Figure 2. Drilled germanium ensures the large in-plane refractive index contrast (>3) required for robust band gaps and guided mode design. The radii of the cylinders are R = 0.28a μm, where a = 1.5 μm represent the lattice constant. Using these parameters the PBG, the reflectivity and the magnetic and electric field distribution were calculated. Thus the 2D PhC structure under study comprises a triangular lattice of infinitely long air cylinders, with R radius arranged in Ge background in the xy plane which can be seen in the inset of Figure 3b. The dielectric constant a was set to examine the imaging properties of the 2D PhC structure at near-infrared frequency regime. The air-filling factor for our tested structure is: f=
2
πr2 π (0.28a) 2 = = (0.28) π ∼ = 0.246. 2 a a2
Figure 3 shows a comparison between the MPB and MEEP calculations and the FTIR data. The dispersion relations gather the computed TE (the solid lines) and TM modes (dashed lines) of the structure under study vs the special k-points labeled in the Brillouin zone. The frequencies are expressed in c/a units, with c the speed of light and a the lattice constant. The in-plane vector k|| goes along the edge of irreducible Brillouin zone (from Γ to M to K) as shown in the inset in Figure 3a. The component of spectral reflectivity ρ(θ, λ) is measured in infrared range by using FTIR spectrometer [32]: ρ (θ, λ) =
Iin (θ, λ) . Iref (θ, λ)
The spectral reflectivity of the system was attained by normalizing the collected intensity of the photonic crystal, Iin (θ, λ), to the reference one, Iref (θ, λ), previously measured. Due to the noise in the measurements the experimental reflectivity spectrum presents some rapid oscillations. For maximum refraction the photons with energies lying within the bandgaps do not propagate within the PhC, while photons with energies lying within gaps do propagate. Evaluation of the band diagrams reveals the presence of 2 bandgaps for TE modes and no gaps for TM ones. The simulations show the existence of a large TE gap with the central frequency at ω1 = 0.2 and the occurrence of a smaller gap centered in ω2 = 0.5 with Δω 1e = 0.051 and Δω 2e = 0.03 bandgap widths and a bandgap relative width η = Δω/ω of η1e = 25.5 %, η2e = 6%. There are also some partial gaps near the center of the Brillouin zone, marked with • symbols in Figures 3a and 3c such as the ones occurring between the ones from the 2th and the 3th and from the 5th and 6th bands for TE modes and the 1st and the 2nd band, the 4nd and the 5rd, the 6th and 7th and the 7th and the 8th bands for TM modes in the K direction. Careful inspection of the dispersing bands near the point (kx = ky = 0, i.e. at normal incidence), discloses a fair agreement with the experimental and calculated reflectivity spectra for drilled Ge sample as a function of wavelength, as can be seen in Figures 3b and 3c. The discrepancies between the frequencies at the edge of the gaps, such as the one encounter at ω ∼ 0.6, could arise from the finite size of the experimental sample, the optical response of PhCs being closely related to the quality of the system. Also, another reason for the variations and intensity loss in the regions at higher frequencies is the presence of leaky, radiative modes, localized within the radiation continuum, falling above the ω = ck light cone. Additionally, we must take into account the geometry of the experiment, performed with uniform-polarized light. The light is being scattered out of the normal direction, affecting the experimental spectra, yet allowing distinguishing the regions featured by bandgaps. Nevertheless, Hsu et al. [33,34] shown that even in this situation perfect cancellation between reflected and forward leakage may occur. In this manner, strong emission and localization can be attained by means of destructive interference, if the conditions of periodicity, inversion symmetry around a C2 axis and temporal invariance associated to non-magnetic systems are fulfilled simultaneously, as in the present case. However, the quality factor of the localized modes gets a finite lifetime when the local symmetry is locally broken. This leads to broader features in reflectance measurements, as can be seen at ω = 0.6. Figure 4 presents the analysis of the magnetic field patterns in the proximity of the high symmetry point Γ of the irreducible Brillouin zone. The study of these field distributions allows a better understanding of the formation of PBG for TE modes, and also
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Fig. 3. (a) Theoretical dispersion laws of a hexagonal PhC lattice that consists of air cylinders in air for TE (continuous line) and TM mode (dashed line). Inset: Ist Brillouin zone. (b) and (c) Experimental (blue) and theoretical (green) reflectivity spectra in the direction for the triangular PhC lattice of air cylinders in Ge matrix. Inset: calculated structure.
Fig. 4. Calculated magnetic field distribution of TE states for different bands in the proximity of Γ point of the PBG inside the hexagonal lattice of photonic structure. The colors indicate the amplitude of the magnetic field in the z direction.
the reflectivity spectra. They are characterized by two hexapolar modes (a, f) [32,34], a quadrupolar mode (b), a dipolar mode (c), a dodecapolar mode (d), and a decapolar mode (e). The present modes collect most of its magnetic fields energy in the high-dielectric regions in order to lower its frequency, explaining the large splitting between the first and second bands in the TE case. It was
shown [35] that for a configuration of holes in a dielectric medium, the air region support high frequency modes, while the dielectric hosts the low frequency range. The difference of resonance frequency between the upper and lower bands and the change of field pattern between Figures 4b and 4c, or Figures 4d and 4e give the PBGs widths. Due to the present difference in the field patterns, which
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emerge from the raise of the dielectric contrast in regular structures, the PBGs increase. Thus high index materials are crucial for producing photonic systems.
5 Conclusions A 2D photonic structure with hexagonal symmetry and a = 1.5 μm lattice constant was designed in a germanium wafer by femtosecond laser ablation. The current pattern was establish by enhancing the laser irradiation parameters, which lead to optimal conditions for designing the under study photonic system. The obtained configuration was further theoretically and experimentally investigated by means of finite difference time domain and finite difference frequency domain calcuations, SEM and FTIR measurements. The numerical calculations point out PBGs in the photonic crystal at telecommunication wavelengths range (1.33−1.53 μm) rising from the confinement of the leaky modes within the radiation continuum. Additionally, photonic band gaps at low energy range in the 2D photonic structure of air holes in Ge slab were spotted. Comparing the spectral signature of PBGs recorded experimentaly with theoretical calculations results one was able to prove the relation between the geometric parameters of the system and its ability to trap and guide the radiation in specific energy range. The analysis of photonic band gaps for the photonic structure in Ge wafer allows the understanding of optical properties in the regions described earlier. In conclusion, Fourier transform infrared measurements, scanning electron microscopy and theoretical calculations permit direct characterization of the band structure and modes representation important for designing photonic systems, which could represent an alternative to Si-based technology in microphotonic structure fabrication. Also laser ablation is a technique feasible for producing 2D PhCs, with the possibility of increasing the total processing surface and reducing the processing time by implementing rapid processing procedures such as pulse-shaping for multi-spot parallel processing. This work was financed by the Romanian UEFISCDI Agency under Contract PN 2-Partnerships No. 152/2011.
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