We describe a micro-spectrometer that exploits out-of-plane radiation at mode cutoff in a tapered leaky waveguide clad by omnidirectional Bragg reflectors. The device can be viewed as a side-coupled, tapered Fabry-Perot cavity. An effective-index transfer-matrix model reveals that optimal resolution is dependent on the reduction or mitigation of back-reflection and standing waves leading up to the cutoff point. We address this by insertion of low numerical aperture optics between the taper and the detector, and demonstrate an experimental resolution as small as ~1 nm and operating bandwidth >100 nm in the 1550 nm range, from a tapered waveguide with footprint ~50 μm x 500 μm. The device combines the small size of a Fabry-Perot instrument with the detector array compatibility and fixed optics of a grating-based instrument.
©2009 Optical Society of America
With applications in biochemistry, medical diagnostics, environmental monitoring, and industrial process control, the global market for spectrometers is predicted to reach $4 billion by 2012 [1,2]. Because of this, and due to the emergence of lab-on-chip (LOC) Microsystems [3–5], the integration and miniaturization of spectrometers is a subject of great interest. A key figure of merit is the resolving power, RP ≡λ0 /Δλ, where Δλ is the FWHM resolution at free-space wavelength λ0. RP ~10-100 is typical of most micro-spectrometers reported to date, but there is a need for devices with RP ~1000 or greater .
Chip-scale grating spectrometers typically operate in the Fresnel diffraction regime with RP less than ~20 for a lens-free design . Yee et al.  employed an external lens and achieved RP ~75 for a 0.6 cm grating-to-detector array spacing. More recently, Grabarnik et al. [7,8] employed flat grating elements for both dispersion and focusing and achieved Δλ ~3 nm in the 450-750 nm range, for a device with optical part dimensions ~3 mm x 3 mm x 11 mm. Slab waveguide devices with integrated gratings and focusing optics [5,6] have also been widely studied. For example, LIGA-fabricated spectrometers with optical part dimensions ~1 cm x 2 cm offer RP ~50-80 . Another approach makes use of a waveguide grating coupler and a free-space propagation section ; for example, Chaganti et al.  reported Δλ <4.6 nm in the 488-632 nm range for a device having optical part dimensions ~1.2 cm x 0.4 cm x 0.4 cm.
As an alternative to grating instruments, Fabry-Perot interferometers can provide superior throughput and resolution from a small optical part. However, it is challenging to fabricate etalons with sufficiently reflective, flat, and parallel mirrors, and the tradeoff between resolution and free spectral range can be a disadvantage [6,11]. Nevertheless, the Fabry-Perot is well suited to micro-spectrometry because it can be placed in close contact with the photodetector .
The conventional plane-mirror Fabry-Perot selects one spectral component per mode order from an incident beam. To obtain a spectrum, it is necessary to tune the resonant condition (i.e. the index or length of the cavity) or to broadly illuminate an ‘array’ of Fabry-Perot filters with spatially varying mirror spacing. In the former category are tunable MEMS devices , commercial versions of which provide Δλ ~1 nm over ~200 nm bandwidth in the near-infrared  and can capture a full spectrum in ~50 μs. These impressive specifications are offset by the presence of moving parts and (typically) the need for large electrostatic deflection voltages. In the latter category are filter array and wedge filter (linear variable filter) spectrometers. Recently, a filter array spectrometer with optical part dimensions 12 mm x 12 mm was reported  to provide Δλ <3.2 nm in the 722-880 nm range and a wedge filter spectrometer with largest optical part dimension ~10 mm was reported  to provide RP ~100 in the 400-700 nm range. However, the need for broad-area, uniform, and highly collimated illumination of these spectrometers is a disadvantage for many applications. They also suffer from low optical throughput, since only a fraction of the incident light (i.e. the resonant wavelength) reaches the underlying photodetector at a given location .
Here, we describe a micro-spectrometer that can be viewed as a side-coupled Fabry-Perot filter, based on our recent report  of wavelength-dependent, out-of-plane coupling in tapered hollow waveguides with omnidirectional claddings. Tapers have been widely employed in guided wave optics, for coupling, mode conversion, and field concentration. However, the presence of omnidirectional claddings creates the potential for new functionality. The proposed spectrometer combines desirable features of diffraction grating instruments, such as fixed optics and spatial separation of wavelengths, with desirable features of Fabry-Perot instruments, such as high RP and potential for close contact between the wavelength discrimination element and the detector. Furthermore, it addresses some of the aforementioned drawbacks of conventional Fabry-Perot spectrometers. In the following, a detailed theoretical analysis is corroborated by experimental results obtained using previously described waveguide tapers (details regarding the fabrication process were reported in reference ).
2. Micro-spectrometry using a tapered Bragg waveguide – analysis
In a tapered waveguide clad by omnidirectional reflectors (see Fig. 1(a) ), there is no critical angle for mode guidance. At cutoff, the waveguide mode effectively becomes a normal-incidence resonant mode of the equivalent Fabry-Perot structure. Similar behavior is well-known from early studies of microwave waveguides with metallic boundaries , and related work on tapered Bragg waveguides was reported by Koyama et al. [19,20].
These tapers can be viewed as side-coupled Fabry-Perot cavities  or perhaps more accurately as side-coupled wedge filters . Light launched into a guided mode at the wide end propagates until it reaches its cutoff point, where (assuming at least one of the cladding mirrors is not completely opaque or reflective) it is partially radiated in a nearly surface-normal direction. The cutoff position is wavelength-dependent, so that each spectral component of a polychromatic input signal is radiated at a unique position. Specifically, the longer wavelength components of the signal are radiated nearer the large end of the taper.
As shown schematically in Fig. 1(b), a compact spectrometer could be realized by combining the tapered waveguide with a photodetector array, possibly with an intervening angular filter to optimize resolution as explained below. Spectral content of the input signal is captured without need for moving parts or tuning of the interferometer. Furthermore, because the taper imparts spatial dispersion, the throughput can potentially be higher than is typical for conventional wedge spectrometers.
As discussed below, the resolving power and efficiency are affected by:
- (i) the effective size of the pixels in the detector array combined with the spatial dispersion of the taper (which is related to the taper slope);
- (ii) the reflectance of the cladding mirrors at normal incidence, which combined with the air core thickness places an upper limit on the RP (i.e. Q-factor) of the cavity;
- (iii) the formation of a standing wave radiation pattern due to back-reflection into the reverse propagating mode, and;
- (iv) the possible existence of multiple lateral modes in the actual (non-slab) waveguide .
As described in Section 3, preferential coupling to the fundamental lateral mode is possible by alignment of the input beam, allowing contribution (iv) to be neglected and a slab model to be employed for first-order analysis. In the following, we use analytical results and a simple numerical model to assess the impacts of (i) to (iii).
2.1. Vertical Fabry-Perot cavity model
For preliminary analysis we neglect back-reflection and standing wave effects (contribution (iii) above), and model the tapered slab waveguide near mode cutoff as a tapered vertical cavity. In that case, the FWHM resolution can be approximated as follows:16]. Thus, assuming a constant taper slope and neglecting the wavelength dependence of field penetration into the claddings, the dispersion can be expressed:Eq. (2) overestimates Δλ0/Δδ by a factor of ~2-4 depending on the exact details of the mirror .
Optical efficiency is another key parameter for a spectrometer. The light-gathering ability of bulk-optic spectrometers is commonly described in terms of optical etendue. A waveguide-coupled spectrometer [9,10,23] generally has relatively low efficiency when gathering light from an extended source, but is not strictly characterized by an etendue . Moreover, for many applications the optical throughput is a more relevant parameter. Throughput depends on the etendue of both the source and the spectrometer (i.e. on the coupling efficiency), and can be improved by appropriate input coupling optics . For the proposed spectrometer, the signal to be analyzed might even originate from inside the hollow waveguide (such as the fluorescence of an embedded analyte), and the presence of omnidirectional claddings can actually enhance spontaneous emission into a guided mode . For these reasons, we focus on optical throughput below.
The overall throughput can be approximated as:
The last term in Eq. (3) is the radiation or out-coupling efficiency through the cladding of interest, accounting for radiation through the other cladding and absorption by the cladding layers. Using the vertical cavity model at cutoff, an estimate for out-coupling efficiency is given by the per-pass transmission of light as a fraction of the per-pass loss:
Lastly, we consider the potential operating range of the proposed spectrometer. As for all Fabry-Perot devices, the free spectral range (FSR) must be taken into account. However, the tapered waveguide device is expected to operate at the out-coupling point of a low vertical mode order. Thus, using a hard-mirror approximation, FSR~λ0/(m + 2), where λ0 is the resonant wavelength. For example, many of the experimental results below are for the m = 1 case, corresponding to FSR ~500 nm for a device operating near 1500 nm. In practice, the FSR is somewhat lower due to field penetration into the mirrors [22,26]. Nevertheless, the operating range is more likely to be limited by the omnidirectional bandwidth of the cladding mirrors. The omnidirectional bandwidth for the experimental devices below is ~1450-1650 nm for both polarizations or ~1450-1850 nm for TE polarized light . It is well known that omnidirectional bandwidth scales with index contrast and that various techniques, such as chirping or metal-termination of dielectric reflectors [17,27], can be employed to widen the omnidirectional stop-band.
2.2. Effective-index transfer matrix model
The analysis in the preceding section ignores important details such as back-reflection and standing wave formation in the vicinity of the cutoff point. For greater insight, we developed an effective-index transfer matrix model . A brief outline of the model is as follows. First, the complex refractive index (for a given mode and a discretized range of air-core height) was estimated by applying a numerical transfer matrix model to the sequence of layers in a slab waveguide model of the real waveguides . While the experimental devices are channel waveguides (see Fig. 2(a) ), they have a low height-to-width aspect ratio and the propagation of low order modes is well described by a slab model. For a given core height δi, this produces an effective index:Fig. 1(a)) and αi is the intensity attenuation factor.
Next, the tapered slab waveguide was replaced by a stack of effective index layers along the z direction as shown in Fig. 2(b), with the last layer K having its right boundary at the mode cutoff point (core thickness δC). Each layer was assigned an effective index according to its core height and its complex mode index solved in the first step. The input region was assumed to be air, with an effective index of 1. The output region was assigned a purely imaginary index, with a value equal to the imaginary part from section K. Note that the loss diverges rapidly near the cutoff point, so that the effective index in section K has a large imaginary component. Finally, transfer-matrices were used to determine the forward and reverse field amplitudes at all points along the waveguide axis . The step size along z was chosen sufficiently small to resolve standing wave features of interest.
The power dissipation in section i was calculated using the local effective index and the forward and backward propagating field components :
This model was applied to a layer structure representative of the as-fabricated channel waveguides, which are illustrated in Fig. 2(a) and were described in detail previously [16,17]. The mirrors are metal-terminated Bragg reflectors, and the power dissipation is mainly attributable to absorption in the outer gold layers. On the other hand, it is the radiated light that is experimentally measured. In qualitatively comparing the simulated and experimental results, we implicitly assume that the radiated power is a fixed proportion of the total power calculated by the transfer matrix model. This ignores the fact that the ratio is somewhat dependent on the ray incident angle. However, most of the power loss occurs in a short distance leading up to the cutoff point, over which the ray angle is at near-normal incidence on the cladding mirrors. Furthermore, finite difference simulations with a commercial software package (Lumerical FDTD Solutions) provided good corroboration of the transfer matrix model.
Figure 3 compares the results of the transfer matrix model to experimental images of light radiation near cutoff. The microscope image in Fig. 3(a) shows the out-coupling of 4 vertical mode orders at a wavelength of 1600 nm, for a waveguide with width (2b) linearly tapered from ~66 to 10 μm over a distance of 4 mm (i.e. Δb/Δz is ~7 μm/mm). Near the large end of the taper, elastic buckling theory predicts that the peak buckle height would also exhibit a linear profile. This is supported by the approximately equal spacing of the cutoff positions for the m = 2 to 4 modes, since the difference in core height between adjacent orders at cutoff is ~λ0/2. However, buckle formation is influenced by both elastic and plastic deformation for the fabrication process employed . The larger spacing between the m = 2 and m = 1 spots indicates a lower slope at the small end of the taper. While the slope changes continuously near the small end, we found good agreement with experiment by using a dual-slope model as shown in Fig. 3(a). From profilometer measurements, the slope was estimated to be ~1.2 μm/mm and nearly constant for the portion of the taper encompassing the m = 2 to 5 cutoff points. Beyond z~2.2 mm, a constant slope ~0.9 μm/mm was assumed. Finally note that, due to the particular mirror design used , the fundamental (m = 0) vertical mode order does not exhibit cutoff.
Given these slope estimates, the lower plot in Fig. 3(a) shows the predicted radiation versus distance for the m = 1 to 4 vertical mode orders at 1600 nm. The qualitative features of the model are in good agreement with the experimentally captured image. Moreover, as shown in Fig. 3(b), the model nicely recreates the radiation pattern leading up to the cutoff point, including a predicted standing wave period ~8 μm that is in very good agreement with experimental observations.
For applications in spectroscopy or wavelength multiplexing, this radiation pattern presents some unique challenges. While the cutoff point is wavelength dependent, the standing wave could result in significant inter-channel cross-talk at a given detector location. This is illustrated by the effective-index transfer matrix results in Fig. 4(a) , which show the power radiated versus distance for two closely spaced wavelengths. While the main lobes are spatially dispersed and nicely resolved, it is evident that a detector pixel located at the main lobe for the 1600 nm light could also absorb significant power from the 1598 nm light (or from sources at any wavelength shorter than 1600 nm, within a certain range governed by the decay of the standing wave radiation streak).
This crosstalk problem is further illustrated by Fig. 4(b), which shows the power versus wavelength integrated over various lengths, but centered at the same point along the taper. This is meant to model the power collected by a detector pixel of finite size (zp in Eq. (1)), assuming the detector has a high NA and collects light radiated at all angles with equal efficiency. Note that the taper provides excellent out-of-band rejection on the long wavelength side, since longer wavelengths are cutoff prior to the detection point of interest. However, the out-of-band light at shorter wavelengths is significant and must be mitigated or reduced in order to exploit the device as a high resolution spectrometer.
There are several potential strategies for addressing the standing wave issue. First, it should be possible through appropriate taper design to reduce the size of the secondary peaks in the spectrum. For example, our simulations predict that both the taper slope and profile (linear versus non-linear) affect the standing wave pattern. Also, for higher reflectance cladding mirrors the radiation loss diverges more abruptly at the cutoff point and a larger fraction of the power is radiated in the main lobe. However, this must be weighed against an increased back-reflection towards the source. These design details are the subject of ongoing work. A more speculative strategy is to sample the standing wave pattern by a detector array with sufficiently small pixel dimensions, so as to resolve the interferogram produced by a polychromatic input beam. In principle, the instrument could be calibrated and the unknown spectrum extracted using an appropriate inverse transform . This approach is conceptually similar to the standing wave Fourier transform spectrometer described by Le Coarer et al. . However, their proposed device is predicated on the sampling of a standing wave pattern at the deep sub-micron scale and requires the development of appropriate nano-scale detectors. In the present case, the interferogram would be produced by slow-light standing waves formed near mode cutoff and the period of these standing waves (~5-10 μm) might enable sampling by conventional detector arrays. In this case, the taper might be designed to enhance rather than suppress the standing wave features and the excellent long wavelength suppression might benefit the spectral reconstruction .
A more straightforward approach is studied experimentally below. By using low NA optics to collect the light radiated from the taper, it is possible to suppress the secondary peaks prior to the detector. This can be understood with reference to Fig. 4(c), which shows the radiated power and effective ray angle versus distance as predicted by the effective-index model. Moving away from the main peak, the power in the secondary peaks is radiated at an increasingly large angle from normal (see Fig. 1(a)). Thus, suppression of off-normal light at the detector is equivalent to suppression of the out-of-band interference shown in Fig. 4(b). For the m = 1 case shown (representative of the structures studied below), the light associated with the main lobe corresponds to ray angles in the range ~0-6 degrees. This implies that a collection optic with NA~0.11 will primarily collect light from the terminal cutoff point while suppressing light in the secondary peaks. Also shown is the predicted power radiated to the right of a given point, enabling an estimate of ηprop from Eq. (3). For example, approximately 8% of the input power is dissipated within the main lobe for the case shown. If the detector receives power from this lobe only, it follows that ηprop ~0.08.
3. Experimental proof-of-concept
The experimental work employed hollow waveguide tapers fabricated using a previously described process . A high level of integration between a photodetector array and the tapered waveguide (see Fig. 1(b)) would be desirable for many applications, but is mainly an engineering challenge. To corroborate the theoretical predictions, we used either a fiber optic coupled to a single photodetector or an objective lens coupled to an infrared camera (see Fig. 5 ). In some cases, a polarizing beam splitter (PBS) or secondary lenses were placed in front of the camera. The fiber or objective lens was attached to a micro-positioner and could be stepped along the length of the taper. At the input facet of the tapered waveguide, light from either a tunable laser operating in the 1520-1620 nm range (Santec, model TSL-320) or from a broadband ASE source (MPB Communications, EFA-P18 C + L) was coupled via fiber-based polarization control optics and a tapered optical fiber (Oz Optics, focal spot size ~5 μm, working distance ~30 μm).
As described in detail elsewhere , these waveguides support a few vertical mode orders at the input end, each of which has several associated horizontal sub-modes. The modes can be labeled as TEmn or TMmn, where m is the vertical (transverse) mode number and n is the horizontal (in-plane or lateral) mode number. In practice, a particular vertical mode order (or several vertical mode orders) would be isolated by choosing the range of hollow core height spanning the adjacent photodector array. Furthermore, the height is controlled by the delamination regions defined in the self-assembly fabrication process. For the experimental devices discussed below, 4-5 vertical mode orders attain cutoff and the spectrum of a polychromatic input signal can be extracted from the radiation streak associated with each.
For a given vertical mode order, each horizontal sub-mode attains cutoff at a slightly different core thickness, with higher-order modes radiating nearer to the wide end of the taper. If input light is divided amongst the horizontal modes, a relatively complex standing wave pattern is observed near the cutoff position . On the other hand, a predominately single-lobed standing wave radiation pattern (see Fig. 3(b)) indicates minimal coupling to the higher order modes. Optimal resolution is expected when only the lowest order mode (i.e. the TEm0 and/or TMm0 mode) is excited, similar to other integrated optic spectrometers [31,32]. As evidenced below, even coarse alignment of the input fiber to the center of the hollow guides resulted in preferential excitation of these modes. This was aided by the higher propagation loss of higher-order modes, which provides some built-in mode filtering. In practice, a straight or curved waveguide section could be included prior to the taper (see Fig. 1(b)) to enhance this mode filtering effect. Moreover, it should be possible to fabricate single-mode tapers using a more conventional approach such as gray-scale lithography.
We first explored the polarization dependence of the out-coupling mechanism. For a slab waveguide clad by omnidirectional reflectors, the TE and TM modes are degenerate at cutoff [16,21]. For the buckled waveguides, we confirmed that the TEm0 and TMm0 modes nearly retain this property. As an example, the inset of Fig. 5 shows out-coupling radiation streaks corresponding to the TE30/TM30 modes captured at a fixed location and wavelength. Aside from a difference in brightness due to polarization dependent loss (PDL)  (which was compensated by adjusting the input power), the radiation patterns are nearly identical, and we verified this property for each of the vertical mode orders. With proper cladding design to minimize PDL, there is potential for polarization-independent operation. For many applications, this could represent a significant advantage over other integrated optic spectrometers [10,31,32].
Figure 6(a) shows the shift in the cutoff position of the m = 1 mode for a few representative tapers. Also shown is the shift predicted by the effective-index transfer matrix model, using the taper slope assumptions described in Section 2. The agreement between theory and experiment is good, with small discrepancies attributable to slight differences in slope between tapers and to local variations in slope for a given taper. Moreover, these variations likely arise from imperfect lithography steps in the self-assembly fabrication process. As shown below, plots such as these can be used to perform a mapping between the spatial and wavelength axes. From the slope of the theoretical curve, the spatial dispersion (DT) is predicted to vary between ~2.6 μm/nm at λ0 = 1520 nm and ~4.1 μm/nm at λ0 = 1620 nm. This is higher than the hard-mirror prediction (~1.1 μm/nm) of Eq. (2), due to significant field penetration into the claddings . Nevertheless, a spectrum in the 1520-1620 nm operating range could be resolved using a taper section of length <500 μm.
The impact of varying the effective NA of the light collecting optics is illustrated by Fig. 6(b), which shows a particular radiation streak captured with three different objective lenses: 20x (NA = 0.40), 10x (NA = 0.25), and 5x (NA = 0.12). In each case, the input power was adjusted so that the main out-coupling spot produced maximum pixel amplitude without saturating the camera. In this way, the images provide a qualitative comparison of the suppression of secondary spots relative to the main spot. In agreement with the discussion in Section 2, Fig. 6(b) shows that secondary peaks can be suppressed by the use of collection optics with sufficiently low NA, making it possible to resolve closely spaced wavelengths.
To quantify the effect of NA in the wavelength domain, we replaced the objective lens and camera by a cleaved single-mode fiber (Corning SMF-28) connected to a calibrated photodetector. The fiber has an effective NA~0.13, which is very close to the optimal value of 0.11 suggested in Section 2. With the fiber aligned at a series of fixed positions and within ~10 μm of the taper, power was collected as a function of input wavelength. As shown in Figs. 7(a) and 7(b), a single peak dominates the spectrum in each case, while the secondary peaks are suppressed by ~10-15 dB.
The normal-incidence reflectance of the cladding mirrors was estimated previously  as RT~0.996 and RB~0.998, which gives a mean value R~0.997. For a fiber collection optic, Eq. (1) should be modified per Δλ = (ΔλD2 + ΔλFP2)1/2, where ΔλD = zH/DT and zH is the FWHM of the fiber mode field intensity. For SMF-28 fiber at 1550 nm, the effective pixel size is zH~6 μm. For the hollow taper used (taper 3 from Fig. 6(a)), the dispersion is nearly constant in the 1550-1600 nm range, (1/DT) ~300 nm/mm. Using these values and the mirror reflectance mentioned, then ΔλD~1.8 nm, ΔλFP~0.8 nm and the predicted resolution is Δλ~2 nm, in good agreement with the experimental linewidth measured in this wavelength range. Near 1540 nm, the dispersion for this taper is much lower, (1/DT) ~900 nm/mm, resulting in a predicted linewidth Δλ~5.5 nm, also verified by the experimental data.
These results illustrate the impact of pixel size and taper dispersion on the potential linewidth of the proposed device. In an attempt to assess the ultimate resolution, we replaced the SMF by a tapered (lensed) fiber (Oz Optics) with a focal spot size ~5 μm and a working distance ~30 μm. Using a Gaussian beam approximation, this fiber has effective NA~0.17 and will collect light in the range ~0-10 degrees, implying (see Fig. 4(c)) that the output spectra will comprise the main peak and 2-3 secondary peaks. Furthermore, the effective pixel size is zH~2.9 μm. As above, spectra were obtained by scanning the wavelength of the input laser while keeping the tapered fiber at a series of fixed positions. Two representative scans are shown in Fig. 8(a) , one for the m = 2 mode and one for the m = 3 mode. Using (1/DT)~240 nm/mm for the m = 2 case (determined separately), the equation from above predicts ΔλD ~0.7 nm, ΔλFP ~0.5 nm, and Δλ ~0.75 nm. Both in terms of the linewidth and the presence of 2 significant satellite lobes, the data in Figs. 8(a) and 8(b) exhibit good agreement with theoretical predictions. As expected from the discussion in Section 2, the main lobe for the m = 3 case exhibits even narrower linewidth (~0.6 nm).
Note that for most MEMS-based cavities, Finesse is limited by defects rather than the theoretical mirror reflectance [6,11]. The present results are an exception, which we attribute in part to the smoothly curved structures that arise from buckling self-assembly . These results indicate a potential for sub-nanometer resolution over a 100 nm operational range near 1550 nm (RP >1500), provided the light is collected by a low-NA optical system and delivered to a detector array with sufficiently small effective pixel size. Furthermore, it should be possible to use an integrated angular filter [33,34] to restrict the effective NA.
For the tapers employed, the relative opacity of the outer Au layer (~50 nm thick) limits the out-coupling efficiency. From transfer-matrix simulations TT ~3x10−4, RT ~0.996, RB ~0.998, and thus Eq. (4) predicts ηrad ~0.05. From the effective index model in Section 2, ηprop ~0.08 for the fundamental m = 1 mode and assuming low NA collection optics. Ignoring all other sources of loss, the best-case throughput for m = 1 mode in these tapers can thus be estimated from Eq. (3) as ηT ~0.004. The efficiency is expected to be further reduced by input and output coupling losses for the mode of interest. The experimental results were consistent with these estimates. For input power ~1 mW and depending on the out-coupling point and collection fiber used, the peak power at the detector was in the range ~0.1-1 μW. The variation in throughput with position for a given mode (see Fig. 7 for example) was typically <3 dB. As discussed in Section 2, greatly improved throughput can be expected for an all-dielectric cladding design. In any case, the best value obtained is similar to efficiencies reported for some other integrated optic spectrometers, such as ηT ~0.001  and ηT ~0.0094  for grating coupler devices.
In the final experiment described, un-polarized light from the broadband EDFA source was coupled into a waveguide taper. As shown in Fig. 9(a) , this produces a series of spatially dispersed radiation streaks, wherein standing wave effects are obscured by interference between spectral components. As mentioned, each of the radiation streaks encodes the spectrum of the input source. To extract the spectrum, streaks were collected by a 5x objective lens (NA = 0.12), chosen for its near-optimal NA, and delivered to the InGaAs detector array (40 μm x 40 μm pixel size) of an infrared camera (Sensors Unlimited). Accounting for overall magnification, the effective pixel size was estimated as zp~6 μm.
Figure 9(b) shows an m = 3 radiation streak captured with this setup, along with the intensity distribution for a single row of pixels centered on the streak. Note that column-wise summation of pixels produced a plot with very similar shape. In spite of the relatively low throughput for these samples, we found it necessary to significantly attenuate the input power to avoid saturating the camera response. The taper dispersion for this mode was measured as above (see Fig. 6(a)), and is between ~6 and ~15 μm/nm in the 1520-1620 nm range. The resolution predicted by Eq. (1) is thus Δλ ~1.5 nm or less, with the worst-case resolution expected in the 1520-1540 nm range where dispersion is lowest. The actual resolution is undoubtedly somewhat worse due to non-idealities such as pixel crosstalk.
Figure 9(c) shows the data from Fig. 9(b) plotted versus wavelength, based on a pixel-to-wavelength mapping obtained using the tunable laser. Also plotted is the EDFA spectrum as measured using a high resolution (~0.05 nm) commercial optical spectrum analyzer (Anritsu model MS9710C). The agreement between the extracted and reference data is quite reasonable, except for the notch near 1545 nm. Using the laser, we verified that this correlates with a region of reduced throughput. Similar throughput notches, at seemingly random spectral locations, were observed for the spectra extracted from other radiation streaks. These variations might be due in part to small defects on the tapers. However, by varying the position of the input fiber, we found that the primary cause is the spectral dependence of the coupling efficiency. In nearly all cases, the orientation or presence of a PBS prior to the camera was found to have a negligible impact on the extracted spectrum, verifying the polarization independence of the device.
Aside from the spatial-to-spectral mapping used, the extracted spectrum is essentially raw data. Given a sufficiently stable and intimate coupling between the source fiber, the taper, and the detector array, throughput variations could be compensated through pre-calibration of the instrument. Moreover, resolving power and sensitivity might be enhanced by use of digital signal processing techniques with various fitting algorithms . For the proposed spectrometer, integration of the input coupling optics, the hollow waveguide taper, the angular filter, the photodetector array, and the digital processing circuitry is conceivable. We hope to explore some of these options in future work.
4. Discussion and conclusions
The micro-spectrometer described offers high resolution (Δλ ~1 nm) and moderate operating bandwidth (at least 100 nm) from an optical part with largest dimension ~1 mm. Operation in the 1550 nm wavelength region was demonstrated, in keeping with the omnidirectional band of the cladding mirrors employed. Operation at shorter or longer wavelengths should be feasible, given suitable low-loss cladding materials with sufficient index contrast and a compatible process for the fabrication of tapers. While the buckling-based fabrication method produces low-defect waveguides, the required tapers could also be fabricated using conventional approaches such as gray-scale lithography and sacrificial etching, enabling greater control over the modal properties of the guides. The resolution of the device is related to the normal-incidence reflectance of the cladding mirrors, and could be improved by appropriate cladding design. Furthermore, reasonably high throughput (ηT >0.05) should be possible through the use of low-loss, all-dielectric cladding mirrors. For the ultimate implementation suggested in Fig. 1(b), an outstanding challenge is the integration of an angular filter (to limit the effective NA) between the tapered waveguide and the detector array. We note that integrated structures for this purpose have previously been studied in the context of imaging applications  and wedge filter spectrometry .
Several chip-scale spectrometers from the literature share one or more features in common with the proposed device, and thus offer interesting comparisons. The device proposed by Le Coarer et al.  is similar in size, but ideally needs integrated nano-detectors that can probe a standing wave with sub-wavelength resolution. For proof-of-concept, they employed a grid of scattering nano-wires and an infrared camera, and reported Δλ ~4 nm over an operating bandwidth of 100 nm in the 1550 nm range. The device reported by Pezeshki et al.  relied on wavelength-selective coupling between a conventional waveguide and a leaky tapered Bragg waveguide formed in a relatively complex multilayer structure on GaAs. They demonstrated Δλ ~1 nm in the 730-750 nm range, for a device length ~1.5 cm. Devices based on mode cutoff in tapered two-dimensional photonic crystal waveguides have also been proposed , but only for applications requiring relatively coarse resolution (Δλ >10 nm).
It is worth reiterating that many of the compelling applications for chip-scale spectrometers are in microfluidic and LOC systems. The spectrometer described is based on a hollow waveguide that might be filled by a gas or liquid analyte , offering unique potential for integration within these systems. Furthermore, the out-of-plane coupling geometry could simplify monolithic integration with external optics (filters, etc.) and detector arrays.
The work was supported by the Natural Sciences and Engineering Research Council of Canada, the Microsystems Technology Research Initiative (MSTRI), and by TRLabs. Devices were fabricated at the Nanofab of the University of Alberta.
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