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New photonic microstructures are proposed for an efficient light trapping in low index media. Cylindrical hollow cavities formed by bending a photonic crystal membrane are designed. Using numerical simulations, strong confinement of photons is demonstrated for very open resonators. The resulting strong light matter interaction can be exploited in optical devices comprising an active material embedded in a low index matrix like polymer or even gaz.
© 2013 Optical Society of America
1. Introduction
The general approach to achieve strong confinement of photons consists in high index contrast structuring of space at the wavelength scale. Numerous configurations have been used which demonstrate the storage of light during a long time (high Q factor) into microcavities (small mode volume), leading to high Purcell effect [1]. Microdisks [2], nanopillars [3,4] or photonic crystal (PhC) [5] based resonators have been tremendously exploited to achieve strong light matter interaction in high index materials, e.g. semiconductors, thus resulting in the production of low threshold microlasers or efficient quantum devices. However, light confinement in low index material is required for applications where the active material is embedded in a gas, a liquid or, more generally, a dielectric with low permittivity. This is often the case for sensing applications or for sources based on nano-emitters like colloidal quantum dots or rare earth emitters. For that purpose, slot waveguides [6–8], “air” Bloch modes in PhC [9] or plasmon resonances at the surface of metals [10] have been proposed. However, in these structures, although strong light confinement can be reached, the air apertures are tiny, with dimensions from 10nm to a few 100 nm, and can be quite resilient to the introduction of an active material. In this paper a new microphotonic structure is proposed, called Photons Cage, where light is efficiently confined in a low index medium but which is sufficiently open to welcome (or to be impregnated by) additional material.
Schematically, an optical cavity can be seen as a certain volume of space enclosed by mirrors. For 1D cavity, efficient mirrors like Bragg reflectors are commonly used to trap photons in low index material. This has been extended to 2D cavities, or hollow core fibers, where the Bragg mirror stack is bent to get periodicity along the radius of a cylinder [11]. More recently, an advantageous alternative to Bragg mirrors has been proposed which consists in exploiting a monolayer diffracting structures, called Photonic Crystal Membrane (PCM) [12,13] or High Contrast Gratings [14,15] as broadband efficient reflectors. Highly selective 1D air vertical microcavities have been achieved by associating two suspended PCMs [16]. An intuitive way to confine light in more than one direction is to bend such a PCM in order to define a cylindrical 2D microcavity (Fig. 1), or 2D Photon Cage.
Fig. 1 Schematic views of (left) a PCM based vertical microcavity and (right) a cylindrical 2D Photon Cage. D is the pillars diameter and Λ is the period of the PCM and the length of the arc of circle between two neighboring pillars.
Its geometry is defined by an array of high refractive index pillars periodically arranged on a cylinder, embedded in a low refractive index environment. As illustrated on Fig. 1, this cavity can be seen as a cylindrical deformation of the resonator we proposed in [16]. This object aims at confining light inside the core of the cylinder and its optical properties are determined by the radius of the cavity and by the reflectivity properties of the arrangement of pillars. In the following, for sake of illustration, we are seeking for resonant TE polarized modes (electric field parallel to the pillars axis), with wavelength around 1.5µm, in a Photon Cage defined by silicon pillars in air. Such a structure can be fabricated, among different methods, using a top down approach including deep Reactive Ion Etching [17]. The designs which are presented below are of course scalable for different optical frequencies and similar results can be found for TM polarization and for different material couples, such as semiconductor pillars embedded in silica, as long as the index contrast between both materials remains large (typically >2).
The paper is organized as follows. A first section is devoted to the design of a planar broadband PCM mirror. Then, the main properties of the resonant modes of a 2D photons cage are studied by 2D Finite Difference Time Domain (FDTD) simulations and a high Q factor 2D cage is designed. In the last section, 3D FDTD calculations are performed to determine the confinement properties of a real cage with finite height. Routes to the production of efficient 3D photon cages are proposed in the conclusion.
2. Design of a PCM mirror
In a 2D Photon Cage, the shape and dimensions of the pillars, and the length of the arc of circle – i.e. the period Λ– between each pillar are the main controlling parameters of the reflectivity of the cylindrical mirror. We first assume that the reflectivity properties of the cylindrical pillars arrangement are closed to those of its planar counterpart. This assumption is partly justified by the specific properties of 1D PCMs. Indeed, we have shown in [12] that the reflectivity of such mirrors comes from the interaction between free space waves and resonant guided modes of the PhC membrane exhibiting a slow group velocity and, therefore, a flat dispersion curve. This property results in mirrors with a large angular acceptance [18].
We therefore start with a PCM consisting in a 1D periodic array of silicon pillars embedded in air.
Considering infinitely long pillars, we use 2D FDTD to derive the reflectivity spectra of the PCMs. Periodic boundary conditions and PML ones are used respectively for the x and z directions (see Fig. 2). The incident plane wave is TE polarized (electric field parallel to the pillars axis) and propagates in the z direction. The period of the array is Λ = 1µm. The filling factor (ff) of the PCM is defined as the ratio between the diameter (D) of the high index circular pillars and the period (Λ) of the array. Figure 2 shows the reflectivity map versus the wavelength and the filling factor, revealing high reflectivity zones (R>99%). At this stage, it should be kept in mind that, to create the photon cage, the 1D PCM will be bent along a circle. It means that it should be tolerant enough to support the deformation and that small changes in its dimensions should not have a too negative impact on the reflectivity. Looking at Fig. 2, the choice of the PCM is then straightforward as its dimensions should lie within a high reflectivity region which is spectrally broad and tolerant to ff variations. The chosen operating point is marked, with a dotted line, on the map of Fig. 2 and the corresponding reflectivity spectrum is plotted. Broadband mirrors with reflectivity higher than 99.8% are then obtained in a 1350-1550 nm range for a 270 nm pillar diameter. It is worth noting that a large fabrication tolerance of PCMs has been also observed by another group [19]: they have experimentally shown that a high reflectivity is maintained for a 20% variation of the PCM critical dimensions.
Fig. 2 Optical properties of a 1D PCM (see inset, Si cylindrical pillars in air, period Λ = 1µm): a) reflectivity map (the color scale is not linear) as a function of the wavelength and filling factor (see the text for definition); b) reflectivity spectrum for pillars with a diameter D = 0.27µm and a period Λ = 1µm.
3. Properties and design of a 2D photon cage software
A Photon Cage is designed, as described in Fig. 1, by bending the previously optimized mirrors in order to get a cylinder. The circular period of the array of pillars is chosen equal to the straight period of the PCM. Then, the microcavity is completely defined by the number of pillars. In this section, we consider infinitely high pillars in order to restrict our analysis to the 2D case. Now the cavity radius (or pillars number) must be optimized to obtain highly resonant and strongly confined modes.
For that purpose, 2D-FDTD calculations are carried out to characterize the resonant modes. In Fig. 3, calculated modes, for a 18 pillars cavity, are represented by points whose abscissas (resp. ordinates) are their wavelength (resp. Q factor). For comparison, the reflectivity of the previously selected PCM mirror is plotted (blue curve, right axis) as a function of the wavelength. Q factors as high as 8000 are obtained, indicating the robustness of the PCM mirror. For some modes, field intensity (Ez2) distributions are also shown. They are very similar to modes which are supported by a circular cavity enclosed by a perfect mirror, with a field distribution slightly affected by the periodic corrugation at the circular boundary. These modes can be sorted with 2 integers: a radial (n) and an azimuthal (m) orders which are the number of field nodes, inside the cylinder, in the corresponding directions.
Fig. 3 Left: left axis: Q factor as a function of the wavelength for the resonant modes of the cylindrical photon cage; right axis: reflectivity of the planar PCM mirror as a function of the wavelength. Right: electric field intensity maps for some of these modes. (Npillars = 18; Λ = 1µm; ff = 0.27)
Since it is aimed at confining the electromagnetic field in air, we will focus on modes with zero azimuthal order, such as the mode 1 at about 1.55 µm on Fig. 3. In this case the field is strongly confined in the center of the cage. Photon Cages with 10 to 30 pillars have been simulated. The number of pillars is then selected by finding which photon cage has the mode of null azimuthal order with the highest quality factor.
Figure 4 shows plots of the Q factor of each mode versus its resonant wavelength. It is worthwhile to notice the great consistency of the variation of the Q factor with the reflectivity of the 1D PCM, confirming that the optical properties of the 1D PC does not suffer significantly from the bending. For a cavity with a targeted operation wavelength close to 1.5µm, one can choose, for example, the mode with n = 3 at 1.560µm with Q~8300, which corresponds to a 18 pillars cavity. This high Q value can be further increased by applying a small modification on the pillar diameter: a decrease of only 4% (of the pillars diameter (0.26 instead of 0.27µm) leads to Q = 5x104.
Fig. 4 Left: resonant wavelength (squares, dash lines) and quality factors (dots, full lines) for modes with null azimuthal order as a function of the number of pillars (each color corresponds to a radial order n from 1 to 7); the horizontal black line indicates, for the planar PCM, the wavelength range with R>99%. Right: electric field intensity distribution for the modes with radial orders n = 2 and 7.
The electric field intensity radial distribution is plotted on Fig. 5, showing a high concentration of the field at the center of the cage. More specifically, a mode surface of S = 0.7 λ2, defined as the inverse of the normalized electrical energy density maximum:
has been calculated. This value is extremely small compared to the physical cavity surface (around 11 λ2) and denotes the high capacity of this hollow structure to concentrate photons. It is striking that such a confinement is obtained for a cavity which is composed of more than 95% of air.Fig. 5 Left: electric field intensity along a diameter of the cage: the dashed lines indicate the boundaries of the cage. Right: 3D view of this distribution.
4. 3D photon cages
In this section, we consider the previously designed cage defined by 18 pillars with 0.26µm diameter. The diameter of the cage is then approximately 6µm. It is reminded that, from 2D calculations, the n = 3 resonant mode exhibits a wavelength of 1.56µm and a Q = 5x104.
3D FDTD calculations for photon cages with finite height h have been performed. For h = 10µm, the electric field intensity distribution of the mode, with zero vertical order, is depicted in Fig. 6. The photons not only remains laterally concentrated in the center of the photon cage but are also confined in the vertical direction. This vertical confinement is mainly due to the impedance mismatch, encountered by the photons escaping from the photon cage. However this confinement is limited, which results in a drastic reduction of the quality factor as compared to the 2D case. This is shown in Fig. 7(a), where the variation of the Q factor as a function of the cavity height is plotted: a 2500 Q-factor is obtained for a 10µm high cage. This indicates that vertical losses are strongly predominant in determining the life time of photons in the 3D finite height cavity.
Fig. 6 Mapping of the electric field intensity in a 10µm-high photon cage, cut along the pillars in the center of the cage. The cage extends vertically from −5 to 5 µm.
For completeness, we have considered the case of the same photon cage on top of a silica substrate (in place of air). It results to a Q factor of 1400, instead of 2500. These values indicate that, due to a higher index, the losses toward the silica substrate are 2 to 3 times larger than the losses towards the top air environment.
For a better insight into its physical properties, the photon cage can be considered as a vertical Fabry-Perot cavity where the mode described in section 3 is reflected at the top and bottom boundaries. In other terms, the impedance mismatch between the cage and the top or bottom medium can be characterized by a reflection coefficient RV. Following the analysis developed in [20], we estimate RV = 65% ± 10%.
To more accurately quantify the confinement properties of a microcavity, the Purcell factor, FP, is commonly used as a factor of merit. It can be written:
where V is the mode volume defined as the inverse of the normalized electrical energy density maximum:The evolution of FP as a function of the cavity height, h, is depicted on Fig. 7(b). The observed linear dependence of FP can be understood as follows. From Eq. (7) of reference [20], it is shown that the Q-factor is proportional to the height of the cage and to the group index of the mode propagating back and forth in the vertical direction with a propagation constant β. In our case β is close to zero where the group velocity vanishes and the group index has approximately a linear dependence with h. Therefore, as far as vertical losses are predominant, the Q-factor (resp. the Purcell factor) is a quadratic (resp. linear) function of h.
5. Conclusion
Hollow core photonic microstructures with a very large low index material filling factor (in excess of 90%), while showing very efficient confinement capability in the low index medium, have been demonstrated. These microstructures exhibit a Purcell factor, which is a figure of merit of the optical confinement, reaching several tenths for micrometer size photons cages. As we have shown in [17], they can be fabricated using a top down process.
The properties of the optical resonators presented in this paper result from the efficiency of PCMs to reflect photons even if they are bent or deformed. It must be noticed that a large variety of microstructures can be designed depending on the geometry of the cage and on the lattice which is drilled in the PCM. Following the pioneering work of Prinz [21], we have recently proposed [22, 23] to use the controlled mechanical relaxation of pre-stressed planar microstructures to engineer a variety of photons cages, e.g. spherical or helicoidal resonators. The properties of these “optical origamis” are under study.
Acknowledgments
The Region Rhône-Alpes is acknowledged for its financial support through the PhD grant of C. Sieutat.
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