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We describe out-coupling of visible band light from a tapered hollow waveguide with TiO2/SiO2 Bragg cladding mirrors. The mirrors exhibit an omnidirectional band for TE-polarized modes in the ~490 to 570 nm wavelength range, resulting in near-vertical radiation at mode cutoff positions. Since cutoff is wavelength-dependent, white light is spatially dispersed by the taper. Resolution on the order of 2 nm is predicted and corroborated by experimental results. These tapers can potentially form the basis for compact micro-spectrometers in lab-on-a-chip and optofluidic micro-systems.
©2012 Optical Society of America
1. Introduction
There is currently much interest in the resonant delay and spatial dispersion of white light using tapered waveguides, which Tsakmakidis et al. termed the ‘trapped rainbow’ effect [1]. Most of this work is based on a slow-light confinement mechanism made possible by a negative Goos-Hӓnchen shift occurring in waveguides with a negative-index (NI) core [1] or claddings [2,3]. For TM-polarized modes, analogous effects are possible using negative-permittivity (NP, i.e. plasmonic) materials [4,5]. While it was reported that light could, in principle, be brought to a ‘complete standstill’ [1], practical realities limit the storage time and spectral resolution. First, there is an unavoidable mode-coupling that occurs at the critical stopping thickness [6], which results in loss of power by back-reflection and radiation [2,7]. Second, practical NI and NP materials employ metals, which results in loss of power by absorption. Because of this, relatively modest wavelength resolution (ex. ~40 nm for a metal-clad tapered waveguide [8]) has been reported for experimental NI and NP devices [3,8,9].
Photonic-crystal-based waveguides can also support slow-light modes, based either on negative refraction [7], coherent backscattering, or omnidirectional reflection by periodic claddings [10]. Moreover, a simple Bragg mirror (i.e. a 1-D photonic crystal) can provide omnidirectional reflection [11], provided it has sufficient index contrast and the light is incident from a lower index medium. If only TE-polarized light is considered, the conditions for achieving omnidirectional reflection are significantly relaxed [11]. In accordance with these facts, omnidirectional mode confinement [12] and slow-light modes [13] are inherent features of a Bragg waveguide with a low-index (ex. air) core.
In previous work [14,15], we showed that a tapered, air-core Bragg waveguide can be used to spatially disperse a polychromatic signal. Light within the omnidirectional band of the cladding mirrors is radiated out-of-plane as a given mode approaches its critical cutoff thickness, the position of which is wavelength dependent. For application to high-resolution spectrometry, we furthermore proposed that this radiated light should be collected by a low-numerical-aperture (low-NA) optic [15]. Those previous results (and similar results reported by Koyama et al. [16]) were restricted to the near-infrared range, around 1550 nm wavelength. Here, we report results for analogous waveguide tapers with TiO2/SiO2-based cladding mirrors, for operation at visible wavelengths. With resolution ~2 nm and an operational bandwidth ~80 nm, these devices could find application for fluorescence spectroscopy in lab-on-a-chip and optofluidic micro-systems [17].
2. Device concept and design
A schematic illustration of an air-core, slab Bragg waveguide taper is shown in Fig. 1(a) , including a ray-optics depiction of light guidance for wavelengths within the omnidirectional band of the cladding mirrors. As a guided mode propagates towards the narrow end of the taper, it is adiabatically transformed into a vertical cavity resonant mode at cutoff [12,14]. For integer mode ordering (m = 1,2,3…) and neglecting field penetration into the dielectric mirrors, cutoff occurs at core thickness dcm = mλ/2, where λ is free-space wavelength. Thus, polychromatic light is dispersed by the taper, and its spectrum can be extracted from the spatial distribution of the light radiated at cutoff. As discussed in detail elsewhere [15], the guided mode is subject to back-reflection, so that a standing wave radiation pattern is observed on the input side of the cutoff point. However, a high-resolution spectrometer can be implemented by placing a low-NA optic in front of the detector, in order to preferentially detect the near-vertically radiated light associated with the cutoff point.
Fig. 1 (a) Schematic illustration of ray guidance in a slowly tapered air-core Bragg waveguide. Light within the omnidirectional band of the mirrors is adiabatically transformed from a guided mode to a vertical Fabry-Perot mode at cutoff. (b) Predicted reflectance of the TiO2/SiO2 mirrors used in the present work (silicon substrate), for TE-polarized light at incidence angles of 0, 20, and 75 degrees. Layer thicknesses were assumed to be 61 nm and 90 nm for TiO2 and SiO2, respectively. The symbols are experimental data at 20 degrees.
Owing to the limited index contrast available from transparent materials in the visible range, it is a practical challenge to fabricate Bragg mirrors possessing a truly omnidirectional reflection band [18]. As mentioned, however, an omnidirectional band for TE-polarized light is more easily realized. Here, we employed 6.5-period TiO2/SiO2 Bragg mirrors (starting and ending with TiO2) deposited by electron beam evaporation onto glass and silicon substrates. For modeling purposes we used standard dispersion models for SiO2 and the dispersion relation from Kim [19] for TiO2. Optical constants extracted from variable angle spectroscopic ellipsometry (VASE) measurements confirmed the validity of these models. The refractive index at 550 nm wavelength was estimated to be ~1.46 and ~2.35 for SiO2 and TiO2, respectively. The target layer thicknesses were accordingly chosen to produce a normal-incidence stop-band centered at this same wavelength.
As shown in Fig. 1(b), these mirrors provide an omnidirectional reflection band for TE-polarized light, spanning the wavelength range from ~490 nm to ~570 nm (using a >98%-reflectance criterion). Thus, visible wavelengths in the cyan, green, and yellow ranges are expected (for TE-polarized modes) to be subject to the vertical out-coupling mechanism described above. Blue (i.e. shorter wavelength) light is well-guided at the wide end of the tapers, where guided rays are far from cut-off and have high incidence angles on the cladding mirrors. However, since the mirrors are not highly reflective for normal-incidence in this range, blue light will leak out more gradually and radiate less vertically as cutoff positions are approached. On the other hand, longer wavelengths outside the omnidirectional band (ex. in the orange and red range here) are not expected to be well-guided, nor significantly present in the mode cut-off radiation of interest.
If the standing-wave radiation pattern is mitigated as described above, then the wavelength resolution of the taper spectrometer can be approximated as [15]:
where m is the vertical mode order (m = 1,2,3 …), zp is the effective detector array pixel size (accounting for magnification), R is the normal-incidence reflectance of the cladding mirrors, and the spatial dispersion imparted by the taper is given by:where ST is the absolute value of the taper slope, and K is a phase-shift coefficient [20] accounting for field penetration into the cladding mirrors at normal incidence. Here, we are implicitly assuming that the reflectance and phase shift are approximately equal for the upper and lower mirrors. Typically, K is slightly less than 1, and for our mirrors K ~0.9 [20]. Finally, the free spectral range (FSR) in terms of wavelength can be expressed δλFSR ~λ/(m + 1). Thus, completely analogous to the well-known properties of a diffraction grating, higher-order modes provide higher spatial dispersion and resolution, but at the expense of reduced FSR. In any case, the omnidirectional bandwidth of the cladding mirrors ultimately limits the operating range. Thus, it is sufficient that the FSR exceeds the omnidirectional bandwidth. For the present tapers, the omnidirectional bandwidth is ~80 nm centered near 530 nm; the lowest 5 mode orders are expected to have FSR that exceeds this bandwidth.3. Fabrication
Waveguide tapers were assembled using a wafer-bonding technique, as illustrated schematically in Fig. 2(a) . The TiO2/SiO2 mirrors described above were deposited onto unheated Si wafers and glass slides, by electron beam evaporation at base pressure ~5x10−7 Torr. The TiO2 layers were deposited in an oxygen environment at pressure ~5x10−5 Torr, in order to ensure stoichiometry [19]. Photoresist (SU-8, 2010, Microchem) was diluted with solvent, spun-cast onto the mirror surface of the Si substrate, and patterned to leave rectangular posts (100 μm x 50 μm and ~5.9 μm in height) at defined locations. Then, both the glass and Si substrates were diced into ~3 to 6 mm long pieces. The Si substrate was diced along pre-defined scribe lines so that the pieces had a linear array of posts at one end only. Finally, a glass piece and a silicon piece were clamped together with their mirror coatings facing each other, and fixed in place using an UV-curable epoxy (NOA-61, Norland). Slab waveguide tapers formed in this way were analyzed by optical and scanning electron microscopy (Fig. 2(b)), and tapers with a linear profile and a nearly vanishing mirror separation (i.e. zero air-core height) at the narrow end were selected for optical experimentation. The process yield was greater than 50%, with sample failure attributed to debris on the mirrors or leakage of the epoxy into the space between mirrors.
Fig. 2 (a) Schematic illustration of a tapered slab Bragg waveguide fabricated by bonding a glass substrate to a patterned Si substrate. (b) SEM image showing the end facet (wide end) of a taper. For scale reference, the spacing between adjacent SU-8 posts is 150 μm.
The air-core height of the waveguides described below was typically tapered from ~5.9 μm to ~0 μm over a distance of ~3 - 6 mm, giving a taper slope ST ~1 - 2 μm/mm. Assuming light of wavelength ~530 nm, it follows that ~22 vertical slab mode orders are supported at the wide end of these tapers.
4. Experimental results
Various laser sources and a broadband super-continuum source (Koheras SuperK Red) were used to test the devices. Light was coupled into the wide end of the tapers, either as a collimated free-space beam or via an optical fiber. Light radiated from the top surface (i.e. through the glass substrate) was collected by an objective lens (5x, with NA = 0.12) and delivered to a silicon camera (Thorlabs model DCC1645C-HQ). As mentioned above, use of a low-NA objective lens suppresses the standing-wave radiation leading up to the cutoff point, thereby improving resolution.
To illustrate the spatial dispersion provided by the taper, Fig. 3 shows camera images captured with white light coupled into the taper. As evident from Fig. 3(a), and in agreement with the discussion above, ~20 ‘rainbow’ bands of radiation are observed. The position of particular mode orders was established using the green lasers described below. Near the left (i.e. wide end), the bands overlap due to the smaller FSR of the higher mode orders. Nearer the small end of the taper, the bands become clearly resolved, as shown for example in Fig. 3(b). Bands associated with the lowest ~5 mode orders are free of overlap from adjacent orders, and contain yellow (somewhat), green (especially), and cyan-blue light. These facts are entirely consistent with the theoretical description in Section 2. Note that the bright spots of scattered light in the images are due to inclusions in the glass slide substrate.
Fig. 3 (a) Microscope photograph showing a taper (~3 mm long) in its entirety, with white light coupled into the wide end of the taper, at the left side of the image. Approximately 20 partly overlapping rainbow bands are visible, each one associated with cut-off of a particular mode order (see text). The two lowest mode orders at the right appear dimmer because they are partly obscured by epoxy. (b) A higher magnification image of some ‘rainbows’ is shown. For sufficiently low mode orders on the right, the FSR is large enough such that the full range of colors (yellow, green, cyan-blue) within the cladding omnidirectional band is well resolved.
To assess the resolution of the tapers, we employed a pair of green lasers with peak wavelengths of ~532 and ~543 nm, respectively. These lasers were simultaneously launched using a free-space coupling method, with a slight angle between the beams. Figure 4(a) shows the cutoff radiation associated with ~20 mode orders along a taper ~6 mm in length. From Eq. (2), the spacing between the cutoff positions for the two lasers is expected to scale with the mode order. This was experimentally verified and is illustrated in the plot of Fig. 4(b). Fixing Δλ = 11 nm, this plot provides an estimate of the average taper slope, ST ~1.1 μm/mm, in good agreement with the value predicted from the taper dimensions.
Fig. 4 (a) Camera image of the light radiated from the taper with simultaneous input (at the left) of 532 nm and 543 nm wavelength laser light. Each pair of lines corresponds to the cutoff position of a particular mode order; mode order decreases towards the right. (b) Plot of the spacing (Δz) between cutoff positions of the two lasers, versus the mode order.
Figure 5 shows representative plots of pixel intensity versus distance, extracted from images similar to that shown in Fig. 4(a). Column-wise averaging of pixels was used to reduce noise. The camera images were scaled using photolithographic features of known dimension, enabling a mapping between pixel number and spatial coordinate z. The experimental resolution can be approximated from the FWHM of the individual laser lines (see the labels in the plot of Fig. 5(a)), as dλE ~dz / (Δz/Δλ), where Δλ = 11 nm is the known wavelength difference of the two lasers. For the plots in Fig. 5, we estimate dλE ~2.1 nm and ~2.3 nm for modes 6 and 7, respectively. Similar resolution (typically within +/− 0.5 nm) was estimated for most of the mode orders. Note that a complete spectrum spanning the entire omnidirectional band is available from the cut-off radiation for a single low-order mode. Consistent with Eq. (2), the results show that spectral content can be extracted, with ~2 nm resolution over ~80 nm operating bandwidth, by imaging a section of the waveguide only ~100 μm in length (ex. for cut-off of the m = 6 mode).
Fig. 5 (a) Column-wise averaged pixel intensity versus z, for the m = 6 cutoff lines of the simultaneously launched 543 nm and 532 nm lasers (b) As in part (a), but for the m = 7 lines.
For the mirrors used here (see Fig. 1), the predicted normal-incidence reflectance is R ~0.992 and ~0.996 for the Si and glass substrates, respectively. The effective pixel size for our detection system is zp ~1.2 μm (the actual pixel size is 3.6 μm, and the collection optics have net magnification ~3x). Using an average R ~0.994 and ST ~1 μm/nm, Eq. (1) predicts dλ ~2 nm for m = 1 (in good agreement with the experimental values), but also predicts that resolution increases with mode order. For example, dλ < 1 nm is predicted for m = 6.
Several factors likely contribute to these discrepancies between theory and experiment. First, it should be noted that Eq. (2) is an approximate model only, and does not account for standing-wave effects that effectively broaden the radiation region near cutoff [15]. This is reflected by the asymmetry in the line-shapes from Fig. 5. Second, it is entirely probable that the mirrors have lower-than-expected reflectance, due to absorption and scattering. Third, camera non-idealities such as pixel cross-talk might also play a role in reducing the experimental resolution. Finally, the relative insensitivity of the experimental resolution with respect to mode order suggests that the collection optics play a limiting role. Notably, the relatively thick glass substrates introduce scattering and distortion, making it difficult to focus reliably on light radiating from the air core of the waveguides. We expect the results could be improved by the use of higher quality quartz substrates, or by implementation of channel tapers using buckling self-assembly [15].
Due to the highly multimodal nature of the tapers, we did not attempt to experimentally assess the light throughput efficiency. However, based on our previously reported analyses [15], we anticipate throughput as high as ~0.05 is possible with optimal input coupling.
5. Discussion and conclusions
Compared to the ‘trapped rainbow’ proposals [1–9], the spatial dispersion in the tapers described above is based on relatively conventional physics (i.e. omnidirectional mode confinement and cut-off). Furthermore, these devices can be fabricated using low-loss materials and thus have potential to provide high resolution. One limitation, in particular for visible-range operation, is that the operational bandwidth is limited to the omnidirectional range of the Bragg mirrors. However, this could be addressed by combining multiple tapers with offset reflection bands into a single spectrometer platform. Also, simulations show that a single taper based on GaP and SiO2 could provide a high-resolution operational band spanning nearly the entire visible range. Recent progress with respect to deposition of GaP thin films [21] makes this an interesting topic for future research.
Acknowledgment
The work was supported by the National Sciences and Engineering Research Council of Canada. B. D. and A. M. contributed equally to this work.
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