1. Introduction

In a waveguide clad by omnidirectional reflectors, there is no critical angle for the guided modes. This enables unique functionality (not possible for waveguides based on total internal reflection (TIR) or anti-resonant reflection) such as the low-loss guiding of slow light [1–3]. Omnidirectionality is a property associated with both metallic mirrors and photonic crystal (PC) mirrors. At optical frequencies, however, metallic mirrors are absorptive and threedimensional PCs remain challenging to fabricate. Alternatively, a one-dimensional PC can exhibit bands of omnidirectional reflection for light incident from a lower index medium [4]. Bragg mirrors with this property, called omnidirectional dielectric reflectors (ODRs), have recently been used as claddings for hollow fibers [5] and integrated waveguides [6–9].

Regardless of the guiding mechanism, efficient coupling between external beams and planar waveguide modes is a long-standing challenge. When preparation of a cleaved end facet is not practical, coupling can be realized using a prism, a diffraction grating [10], or an angled micromirror [11]. An alternative is to use a waveguide with a tapered thickness core [12], where coupling is afforded by mode radiation at cutoff. This latter technique has not been widely used, in part because controlled fabrication of vertically tapered waveguides on a planar platform requires advanced processing techniques such as gray-scale lithography. Furthermore, for waveguides based on TIR, light is radiated nearly parallel to the plane of the substrate at cutoff [12]. For interfacing to external sources, detectors, and fibers, surfacenormal coupling is more convenient [10,11] and more compatible with a massively parallel input/output (I/O) coupling scheme [13,14].

In the case of a leaky waveguide, direct coupling (i.e. without a diffraction grating or prism) between a free space beam and a quasi-guided mode is possible, provided the mode has effective index less than unity. Based on this, Lederer, et al., [15] proposed a technique in which the reflectivity of a cladding mirror is locally reduced in order to facilitate I/O coupling at desired locations along a solid-core Bragg waveguide. It is worth noting that the air-guided modes of a hollow waveguide satisfy the sub-unity index requirement (otherwise the field is evanescent in the air layer) and, therefore, can always be directly coupled in this way.

The coupling mechanism described here (and in [16]) can be considered a hybrid of those based on the tapered core [12] and the partially transmitting cladding [15]. When an optical mode in a tapered ODR-clad waveguide approaches its effective cutoff thickness, rays associated with that mode approach normal incidence on the cladding mirrors [1–3], a standing wave forms, and the light leakage per unit length diverges. This enables localized coupling between guided modes and normally incident free-space beams. Furthermore, since the core thickness at cutoff is wavelength dependent, these tapers also provide a mechanism for spatially dispersing a polychromatic light signal. In the following, we provide a theoretical and experimental basis for this behavior and suggest possible applications.

2. Brief background and structural details

The waveguides discussed below have gold-terminated ODR cladding mirrors [17], and their fabrication (by a self-assembly buckling process) was described elsewhere [9]. Amongst other features, the process enabled the fabrication of low-defect, tapered hollow waveguide channels as shown in Fig. 1(a). The tapered channels (tapered in both height and width [8,18]) were realized by defining linearly tapered regions of low adhesion, as shown schematically in Fig. 1(b). For light-guiding experiments, samples were cleaved part way along the taper. This reduced the number of modes supported at the wide end, thereby simplifying the analysis to some extent.

 

Fig. 1. (a). Microscope photograph showing portions of 3 adjacent hollow waveguides, with base width tapered from 80 to 20 µm. (b). Schematic of the mask layout used to define the region of delamination for the tapers. For light-guiding experiments, the samples were cleaved part way along the tapers as indicated. (c) Schematic cross-section (end view) of a buckled hollow channel with upper (curved) and lower cladding mirrors. Layer details are provided in the main text. (d) Peak buckle height versus base width as determined from AFM scans at several points along a typical taper. The red line is a linear fit.

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From elastic buckling theory [18], the vertical profile of a straight-sided delamination buckle (also known as an Euler column) versus the lateral coordinate y is given by:

where b is the half-width of the delaminated strip. Furthermore, the peak height of the buckle is predicted to be [18]:

where h is the height of the film (or stack of films) subject to buckling and b_min is the minimum half-width for spontaneous buckle formation, for a given pre-buckle compressive stress resultant and a given set of elastic film properties. It follows that for b≫b_min, the peak buckle height is proportional to the buckle width. As discussed elsewhere [8], plastic folding [19] plays a role in the formation of our buckled waveguides, so that the elastic buckling theory is not strictly applicable. Nevertheless, AFM measurements revealed a nearly linear relationship between the peak height and the base width [see Fig. 1(d)]. Thus, the hollow channels [arising from the delamination patterns shown in Fig. 1(b)] exhibit an approximately linear taper in both width and height.

The cladding mirrors are multilayer Bragg reflectors comprising polyamide-imide (PAI) polymer and Ge₃₃As₁₂Se₅₅ (IG2) chalcogenide glass. As discussed extensively in our previous work [8,9,20,21], these mirrors can provide low-loss, omnidirectional reflection in the near infrared. Furthermore, we recently showed [17] that termination of the mirrors by a metal layer increases their angle-averaged reflectance and omnidirectional bandwidth, while reducing the number of required layers. For the waveguides studied here [see Fig. 1(c)], complete details on the material and film properties can be found elsewhere [9]. Briefly, the bottom cladding is a 5.5 period Bragg mirror with ~290 nm thick PAI layers (n~1.65) and ~135 nm thick IG2 layers (n~2.55). This mirror was deposited overtop a metal (Au/Cr) bilayer (~40 nm thick). The top cladding is a Au-terminated, 4 period Bragg mirror comprising ~290 nm thick PAI layers and ~140 nm thick Ag-doped IG2 (Ag:IG2) layers (n~2.95). The first Ag:IG2 layer was made thicker (~270 nm), in part to improve the reflectance for TM polarized light at glancing incidence [17] and in part to ensure reliable delamination of the upper mirror [9]. However, the thick layer also has a detrimental impact on the propagation loss of TE modes [9], and improved designs should be possible.

 

Fig. 2. (a). Angle-averaged reflectance versus wavelength, as predicted by a planar transfermatrix model, for the top and bottom cladding mirrors of the as-fabricated waveguides. (b) Reflectance versus incident angle at a wavelength of 1550 nm.

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The reflectance of these mirrors, for light incident from the air core side and neglecting the curvature of the upper mirror, was predicted using transfer matrices and the material dispersion relations described elsewhere [17,20]. The wavelength-dependent, complex refractive index of Au was taken into account, but loss in the glass and polymer layers was neglected [17]. The out-of-plane coupling mechanism discussed below is dependent on omnidirectional reflection from both the top and bottom cladding mirrors. One measure of omnidirectionality is the angle-averaged reflectance [17], which is plotted versus wavelength in Fig. 2(a). The mirrors exhibit a band of overlapping omnidirectionality, which for TM polarized light extends from ~1450 to ~1650 nm and for TE polarized light extends from ~1450 nm to ~1850 nm. To further illustrate, the predicted reflectance versus incidence angle at 1550 nm is plotted in Fig. 2(b). For both mirrors, a high reflectance (R>0.995) is predicted for all incident angles and for both polarization states. Note that the agreement between theoretical and experimental reflectance has been consistently excellent for mirrors based on these materials [8, 9, 17, 20, 21].

3. Tapered Bragg waveguides with omnidirectional claddings – theoretical analysis

We first present results from a planar (slab) model for the air-core waveguides. The slab model is a reasonable approximation to the actual waveguides, which have low height-towidth aspect ratio. Since leaky air-guided modes couple directly to external plane waves (see Section 1), modes of the slab waveguide can be solved in a straightforward way using transfer-matrices [22]. In other words, since the slab Bragg waveguide is formally equivalent to a Fabry-Perot cavity [23], the ray angles associated with low loss modes appear as dips in reflection (or peaks in transmission) versus incidence angle. We consider two different slab structures in the following discussion, as shown in Fig. 3. The first structure is symmetric and has Bragg mirrors tuned to be quarter-wavelength stacks (QWS) for normal incidence at 1600 nm. The second structure is representative of the as-fabricated and tested waveguides, described in detail in Section 2. We used the dispersion models described above, and again neglected loss in the glass and polymer layers.

 

Fig. 3. (a). Schematic of a quasi-symmetric air-core slab waveguide with quarter-wavelength Bragg cladding mirrors and high index layers adjacent to the core. In general, N period mirrors comprising IG2 glass (n~2.55) and PAI polymer (n~1.65) were assumed. Forward and backward propagating plane-waves in the external medium and the core (both air) are indicated. (b). Schematic of the slab model representing the as-fabricated, asymmetric waveguides. The 5.5 period bottom mirror has a low index layer adjacent to the core. In the 4 period top mirror, all the IG2 glass layers are Ag-doped (n~2.95) and the layer adjacent to the core is approximately twice as thick as the other Ag:IG2 layers. Both top and bottom mirrors were terminated by a gold layer. The ray bouncing at angle ϕ_m represents a low loss, leaky mode.

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Consider propagation of an air-guided ray, as shown in Fig. 3(b). For infinite period (unity reflectance) claddings, non-leaky modes exist and can be solved using the well-known phase consistency expression:

where Φ_T and Φ_B are the phase shifts on reflection from the top and bottom cladding mirrors, respectively, k_x is the transverse component of the propagation vector in the air core, and m is an integer (m=0,1,2,…). While (3) has often been applied to the analysis of leaky slab waveguides [3,23], it is only approximately valid in that case. For leaky modes, both k_x and k_z are complex numbers, so that, strictly speaking, a self-consistency condition for both the phase and amplitude is required [24,25]:

where r_T and r_B are the amplitude reflection coefficients of the top and bottom mirrors, as seen from the air core side for a given ray incident angle, and k_x is a complex-valued transverse propagation constant. From solutions of (4), complex modal propagation constants are obtained as follows:

where k₀ is the free-space wavenumber (applicable to the air core medium), β_m=n_eff k₀ determines the phase velocity of the leaky mode, and α_m is the intensity attenuation coefficient (due to sub-unity cladding reflectance) of the leaky mode. Furthermore, from a simple physical model that considers a ray bouncing off partially reflecting claddings, the attenuation coefficient of a leaky slab waveguide mode can be expressed as [24, 26]:

where n_eff=sinϕ_m, R_T and R_B are the reflectances from the top and bottom mirrors (for a ray incident at angle ϕ_m), and deff is the effective core thickness accounting for field penetration into each cladding mirror.

As discussed above, a straightforward transfer-matrix approach can be used to solve for modes in an air-core slab waveguide. As indicated in Fig. 3(a), Snell’s law requires that a plane wave incident from air will propagate with the same angle inside the core. Thus, the modes of the slab waveguide correspond to modes of the equivalent Fabry-Perot cavity. By plotting the plane wave transmission versus incident angle (i.e. versus β=k₀sinϕ), modal solutions (ϕ_m) correspond to peaks in the core transmittance parameter [see Fig. 3(a)]:

The peaks in T_C(β) are Lorentzian lineshape functions [22] centred at β=β_m. Furthermore, the FWHM (2Γ) of the line is equal to the radiation loss coefficient of the mode (i.e. α_m=2_Γ).

Modal analysis of the structures in Fig. 3 was carried out as follows. First, the plane wave response of the overall structure was calculated as a function of incident angle, using a relatively coarse step size. Mode positions were identified as peaks in T_C, and then the plane wave response versus incidence angle was recalculated using a much finer step size in the vicinity of these modes. The mode parameters (ϕ_m, etc.) were then extracted from T_C (β) as discussed above. Subsequently, r_T (ϕ_m) and r_B (ϕ_m) were determined using transfer matrices applied to each mirror separately, and (5) and (4) were used to verify the mode solutions. As a final check for consistency, we calculated the modal attenuation based on (6), as follows. Using ϕ_m determined in the previous steps, transfer matrices were used to obtain R_T(ϕ_m) and R_B(ϕ_m) and also to obtain the effective core thickness, given by [27]:

where L_T and L_B are the phase penetration depths [28] into the top and bottom mirrors, respectively. We found nearly perfect agreement between the attenuation calculated using (8) and (6) and that calculated from the FWHM of the lineshape functions.

It is illustrative to first consider the symmetric slab waveguide structure with QWS mirrors, shown in Fig. 3(a). In Fig. 4(a), T_C is plotted versus incident angle for several different core thicknesses and with 8 periods assumed for both the top and bottom mirrors. For clarity, core thickness was restricted to the single mode range (d<1.6 µm). For this structure, the bounce angle of the fundamental (m=0) mode approaches normal incidence on the cladding mirrors as the core thickness approaches d=d₀=λ₀ /2=0.8 µm. This limit represents mode cutoff, where both the phase velocity and attenuation of the mode diverge [see Fig. 4(b)]. Furthermore, as cutoff is approached the group velocity reduces [3] and, in the case of perfectly reflecting mirrors, the radiation pressure on the claddings (for fixed input power) diverges [2]. In general, the cutoff condition of the m^th order mode is given by d_m=(m+1)λ₀ /2=d₀+mλ₀ /2. In other words, the cutoff conditions are the mirror separations at which the equivalent Fabry-Perot cavity exhibits a normal-incidence resonance at the wavelength of interest. It is also interesting to note that the TE and TM modes become degenerate near cutoff [2, 3], implying the potential for the coupling mechanism to be polarization independent.

 

Fig. 4. Modal analysis results for the symmetric QWS slab waveguide [Fig. 3(a)], with 8 period top and bottom mirrors and at the resonant wavelength (1600 nm) of the mirrors. (a). The core transmittance parameter T_C versus incidence angle, for several core thicknesses (in µm) as indicated. The Lorentzian lines correspond to the m=0 mode in this case. (b) The predicted effective index and attenuation versus core thickness, for the 3 lowest order modes. The vertical dotted lines indicate the cutoff thicknesses of these modes.

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In keeping with the results of Fig. 4, a λ₀ /2 cutoff condition is often cited for the fundamental mode of a hollow Bragg waveguide [3]. However, this condition applies only when the normal incidence resonance of the slab structure is not affected by field penetration into the cladding mirrors. For Bragg mirrors, this is only true at the specific wavelength for which the layers (of both top and bottom mirrors) are tuned to have quarter-wave thickness at normal incidence [28], as was the case in [3]. A λ₀ /2 cutoff condition for the fundamental mode also applies in the idealized case of lossless metallic mirrors [2], where the penetration into the mirrors is zero. In all other cases, an effective core thickness (incorporating penetration into the mirrors) must be used to assess the normal incidence resonance condition.

From the analysis above, we can make some approximate predictions regarding the propagation of light inside a tapered-core hollow waveguide. As shown in Fig 5(a), the rays associated with a given mode approach normal incidence in the vicinity of the cutoff thickness. From (6) and assuming omnidirectional mirrors, the divergence in the loss can be attributed to the divergence in the number of ray bounces per unit length near cutoff. At cutoff, the mode is equivalent to a normal incidence resonance of the slab structure, and remaining power is radiated in a surface-normal direction. Of course, the ray picture neglects key features of the process. For example, since the mode propagation constant becomes increasingly complex near cutoff (and purely imaginary beyond cutoff), some mode power is also reflected in the backwards direction. Thus, a standing wave is expected to form, as is well known from the treatment of analogous tapered waveguides (with conducting metal walls) in the microwave regime [29]. Nevertheless, the gross details of the ray optics model are consistent with finite difference simulations of a similar structure by Miura, et al. [16].

Results from the slab model representing the as-fabricated waveguides [Fig. 3(b)] are plotted in Figs. 5(b)–5(d). The transverse intensity profiles of the 3 lowest order modes are plotted in Fig. 5(b), for a core thickness of 3 µm and a wavelength of 1600 nm. The evolution of the modal effective indices and radiation loss with decreasing core thickness are plotted in Figs. 5(c) and 5(d), for a wavelength of 1600 and 1520 nm, respectively. For the asymmetric waveguide, the fundamental ‘air-guided’ mode exists even for vanishing air-core thickness (i.e. there is no cutoff condition related to a reduction in the separation of the cladding mirrors). This is mainly due to the thicker Ag:IG2 layer in the top mirror, which plays the role of a cavity defect when the two mirrors are brought into contact. It should be emphasized that we are referring only to ‘air-guided’, leaky modes with n_eff<1 in the present analysis. The Bragg structure also supports index guided modes with n_eff≫1 (especially centered on the thick Ag:IG2 layer). Also, since absorption in the Ag:IG2 layers was neglected, the loss of the m=0 mode is underestimated for vanishing mirror separation.

 

Fig. 5. Results from an analysis of radiation at cutoff in a tapered, ODR-clad slab waveguide. (a) Schematic illustration showing a guided ray approaching normal incidence as the core is tapered to the cutoff thickness of the associated mode. The length of the arrows in the external medium indicates the increased leakage of the mode with decreasing core thickness. Radiation through the bottom mirror also occurs. (b)-(d) Results from the ray optics model applied to the asymmetric slab structure, representative of the as-fabricated waveguides: (b) Transverse intensity profiles (in the air core) of the 3 lowest order modes, for a core thickness of 3 µm and a wavelength of 1600 nm. (c) The predicted effective index and attenuation versus core thickness, for the 5 lowest order modes at 1600 nm. (d) As in part (c), but for the 6 lowest order modes at 1520 nm. The vertical dotted lines indicate the cutoff thicknesses in each case.

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Unlike the fundamental mode, the higher-order air-guided modes do exhibit a cutoff thickness. Furthermore, their divergence in phase velocity and attenuation near cutoff is similar to the results shown in Fig. 4. As for the QWS case, the cutoff thicknesses of subsequent modes are spaced by λ₀ /2 (i.e. d_m=d _m-1+λ₀ /2). This is due to the fact that the normal-incidence field penetration into each mirror is invariant with respect to the mirror separation. The results for λ₀=1600 nm are plotted in Fig. 5(c), showing d₁ ~0.51 µm, d₂ ~1.31 µm, etc. The results for λ₀=1520 nm are plotted in Fig. 5(d), showing d₁ ~0.29 µm, d₂ ~1.05 µm, etc.

Since the real waveguides have a buckled shape and provide two-dimensional confinement, we used a commercial finite-difference mode solver (ModeSolutions 2.0, Lumerical Solutions, Inc.) to augment the ray optics results. This software provides mode field profiles, radiation loss, and dispersion for arbitrary (two-dimensional) leaky waveguide structures. We assumed a raised cosine profile (see (1)) for the buckle waveguides and used the same layer thicknesses and material properties as described above. All of the results shown were obtained for a wavelength of 1600 nm and for TE polarization. Figure 6 shows several mode field profiles predicted for a peak core height of 3.5 µm and base width 67 µm, representative of the large end of a cleaved taper. As expected from the slab analysis, this core size supports modes with multiple lobes in the vertical direction. Furthermore, the lateral confinement splits each vertical (transverse) mode into a family of lateral sub-modes. In the following, the modes are referred to by a TE_mn (TM_mn) labeling convention, where (m+1) indicates the number of lobes (i.e. anti-nodes inside the air core) in the transverse (x) direction and (n+1) indicates the number of lobes in the lateral (y) direction.

 

Fig. 6. Selected low-order mode field profiles predicted by a commercial, two-dimensional finite difference mode solver are shown (not to scale), for a buckled waveguide with 67 µm base width and 3.5 µm peak core height. The wavelength was set to 1600 nm. (a) TE₀₀, TE₀₁, and TE₀₂ modes. (b) TE₁₀, TE₁₁, and TE₁₂ modes. (c) TE₂₀, TE₂₁, and TE₂₂ modes.

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To support the conclusions regarding mode cutoff from the ray optics model, we obtained finite difference solutions for a range of core sizes. We used peak heights and base widths obtained from AFM measurements as input to the simulator. Some of these results are plotted in Fig. 7, showing that the variation in the effective index and radiation loss is in good agreement with the ray optics model [compare to Fig. 5(c)].

 

Fig. 7. Selected results from the finite difference mode solver are shown, for a wavelength of 1600 nm. In each figure, the dashed vertical lines are the cutoff thicknesses predicted by the ray optics model for an equivalent thickness slab waveguide. (a) Predicted modal effective indices versus peak core height. (b) Predicted modal attenuation versus peak core height.

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Because of the lateral confinement, each vertical mode from the ray optics model becomes a family of modes in the two-dimensionally confining structure. To illustrate this, the effective index and radiation loss curves for the lowest 3 horizontal sub-modes are plotted for m=0 and m=1 in Figs. 7(a) and 7(b). In fact, ~10 low-loss horizontal sub-modes are predicted for the first few vertical mode families. Furthermore, the horizontal sub-modes are closely spaced in effective index, as expected from the large width of the buckle waveguides relative to their peak height. Within a given vertical mode family (i.e. for fixed m), modes with higher lateral mode number exhibit higher loss and attain cutoff for a larger core thickness. This is again as expected, since at cutoff β=k_z=0 and k ² _x+k ² _y=k ² ₀. Higher mode number n implies a higher value of k_y, and therefore a smaller value for k_x at the cutoff point.

4. Experimental results - outcoupling

In initial experiments, light (~1 mW launch power) from a tunable diode laser was passed through a fiber-based polarization controller and launched (either via a high NA fiber or an objective lens) into the large end of various tapers. The light radiated/scattered by the tapered waveguide was observed using an infrared camera attached to a microscope. All of the results discussed below are for TE polarized light, as verified by monitoring the light transmitted in the m=0 modes (which do not experience cutoff, as discussed above) at the output facet of the taper. Similar results were obtained for TM polarized light, and we verified that the coupling positions are approximately independent of polarization state. Furthermore, the TE and TM propagation losses near 1500 nm are similar (~6 and ~8 dB/cm, respectively, for guides with ~3 µm peak core height [9]) for the samples discussed here. Figure 8 is a series of images showing the light radiated from a typical taper, at several wavelengths in the 1520–1620 nm range. From AFM and microscope measurements, the large end of the taper was estimated to have a base width of ~67 µm and a peak core height of ~3.5 µm. The bright spot at the left of each image is the input coupling point. The bright, elongated streaks to the right of the input coupling point occur at wavelength-dependent positions along the length of the taper. From the discussion in Section 3 and below, these streaks can be attributed to out-of-plane radiation of light at mode cutoff. In fact, each streak corresponds to the radiation of a particular ‘family’ of modes with a shared transverse mode number m.

 

Fig. 8. Outcoupling results for a hollow Bragg waveguide with base width tapered from ~67 µm to ~10 µm. Light from a tunable laser was coupled into the large end of the taper, at the left of each image. Outcoupling streaks are attributed to radiation of 4 or 5 (depending on wavelength) vertical mode families, as indicated. Scale bar: 1 mm.

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In excellent agreement with the ray optics model (Fig. 5), 5 radiation streaks were observed at 1520 nm wavelength but only 4 streaks for wavelengths beyond 1600 nm. As expected, the coupling positions move towards the large end of the taper with increasing wavelength. For the taper shown, this shift corresponds to a spatial dispersion of ~500 nm/mm, which is 1–2 orders of magnitude higher than that provided by conventional wedge filters [23, 30].

We captured high magnification images of the individual out-coupling streaks shown in Fig. 8, by positioning a 60x microscope objective lens in close proximity to the surface of the tapered waveguides. This setup enabled the direct imaging of modal interference and standing wave patterns, as shown in Fig. 9. On close inspection, each elongated streak in Fig. 8 is actually an intricate standing wave pattern. Our interpretation of these patterns is as follows. Towards the left of a given streak (i.e. towards the large end of the taper), numerous individual modes within the associated vertical mode family are approaching cutoff. Each of these modes is subject to back-reflection and a diverging radiation loss. The interference of these forward and backward propagating mode components creates a pattern of bright radiation spots. Moving to the right along the same streak, the standing wave pattern becomes less complex as, one by one, the modes of higher lateral order reach their cutoff point. By making slight changes to the input coupling conditions and the wavelength, it was generally possible to isolate the cutoff point of the first 3 or 4 horizontal modes. For example, the cutoff points for the TE₃₁ and TE₃₀ modes are clearly identifiable in the image shown in Fig. 10(b). In every case, the right-most portion of the elongated streak is a simple (single-lobed) standing wave pattern, arising from the interference of the forward and backward propagating TE_m0 modes. A final spot, typically somewhat brighter than the preceding spots, is observed at the right-most position of each streak. We call this the ‘terminal’ cutoff point.

 

Fig. 9. High magnification images of the radiation streaks from Fig. 8 are shown, for a wavelength of 1600 nm. The images appear in order of position along the taper. (a) TE_4n radiation streak. (b) TE_3n radiation streak. (c) TE2n radiation streak. (d) TE_1n radiation streak.

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From the period of the standing waves (Λ_SW) in Fig. 9, it is possible to estimate the effective index of the TE_m0 modes at cutoff (i.e. λ₀=2n_effΛ_SW). We typically observed a ‘terminal’ standing wave period of ~10 µm (see Fig. 9(d)), corresponding to n_eff~0.08. In other words, the ray angle at cutoff is estimated to be approximately 5 degrees from normal. Light radiated prior to the terminal point emerges at slightly larger angles [see Fig. 5(a)], so that a distribution of angles will contribute to the far-field image of each radiation streak.

We conducted a preliminary analysis of the far-field radiation patterns, by collecting the radiated light with a multimode fiber (MMF). In this experiment, the fiber was attached to a micropositioner and aligned approximately normal to the surface of the samples. The fiber was scanned in a two-dimensional plane at fixed heights above the sample, and the collected power at each point was monitored by a photodetector attached to the other end of the fiber. The mode fields estimated in this way are a convolution of the fiber aperture and the actual emitted radiation pattern, so that the spot size is overestimated. Typical results are shown in Fig. 10. The radiation patterns were symmetric in the direction perpendicular to the waveguide axis (across the taper), which is expected from the symmetry of the waveguide. Furthermore, the radiation pattern typically exhibited a divergence angle of 5–10 degrees in the x-y plane.

Scans along the z-axis [see Fig. 10(b)] revealed a slightly asymmetric beam, whose direction (defined by mapping the peak of the beam versus height above the waveguide) was typically 5–10 degrees off of normal to the sample surface, towards the direction of the output facet (i.e. towards the small end of the taper). The asymmetry and the slightly off-normal direction of the beam are both consistent with the simple model put forward in Fig. 5(a). Guided light radiates quite efficiently in the section of waveguide leading up to the nominal cutoff point, so that the rays producing much of the radiated beam are approaching but not exactly aligned with the surface normal. Encouragingly, the effective beam angle is in good agreement with the angle estimated from the standing wave patterns in Fig. 9. Given the standing waves observed, both forward and backward propagating light will contribute to the far-field radiation pattern. However, forward propagating light clearly makes the dominant contribution.

 

Fig. 10. Typical far-field profiles of the light radiated at cutoff, as picked up by a MMF probe scanned at various fixed heights above the sample surface. The case shown is for the TE_2n radiation streak of an 80–10 µm taper. (a) Intensity profile along the y -axis (normal to the axis of the waveguide). (b) Intensity profile along the z-axis (parallel to the axis of the waveguide).

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It is somewhat difficult to assess the efficiency of the coupling mechanism for the present samples, since the 1 mW launch power is split between 5–6 vertical mode families. As shown in Fig. 10, the peak power collected by the MMF probe was ~1 µW, or <0.1 % of the total input power. However, the tapers used here are far from optimized for coupling purposes. The outer metallic layers of the cladding mirrors were designed to reduce radiation loss, and are thicker than the skin depth of gold in the near infrared (~25 nm). Thus, a significant amount of light is lost in transmission through these layers. There is a possibility for resonant tunneling effects, and a full understanding would require a 3-dimensional numerical simulation. It should also be noted that the experiment only collects light emitted through the top mirror, while a similar amount of power is radiated through the bottom mirror. In practice, a completely reflective (or opaque) mirror could be used to suppress radiation in one direction. Furthermore, the reflectance/opacity of the other mirror could be reduced at locations where efficient radiation is expected and desired, such as by a patterned removal of the Au termination layer. For the all-dielectric structures simulated by Miura, et al. [16], radiation efficiency of 70% was reported. Clearly the geometry of the taper could be customized to ensure that only the lower order vertical modes are supported. In practice, it would often be desirable to couple the fundamental vertical mode family. A cutoff condition for these modes depends on proper design of the cladding mirrors, as discussed in Section 3.

For wavelength separation, the presence of multiple horizontal modes within each vertical mode family will have a negative impact on resolution, since each mode radiates at a slightly different position. For some applications, this could be mitigated by careful excitation of only select modes at the wide end of the taper [23]. The geometrical restrictions associated with buckle formation [8,9] make it challenging to realize purely single mode taper guides. However, such guides should be feasible using more conventional fabrication techniques.

It is expected that the spot size, the divergence angle, and the wavelength dispersion can all be customized to some extent through judicious choice of the mirror reflectance and the slope of the taper [16]. A full 3-dimensional simulation of these structures would be a useful improvement to the simple models presented here, but is left for future work.

5. Experimental results - incoupling

By reciprocity, the out-coupling of radiation discussed above suggests the possibility of coupling a free space beam (at nearly normal incidence) into a tapered ODR-clad waveguide. In initial experiments, we attached a MMF (50 µm graded index core) to a micropositioner and aligned it overtop a tapered waveguide, approximately normal to the surface. The separation between the cleaved facet of the fiber and the sample surface was ~50 µm, so that the spot size at the waveguide surface was significantly larger than the width of the taper. Light from a tunable laser was launched into the other end of the fiber, and ~5 mW power was delivered to the sample surface. An objective lens was used to collect light from the large cleaved end of the tapered waveguide, and this light was delivered to a photodetector. The power collected from the tapered waveguide was monitored as the MMF was scanned along the length of the taper, and a typical result is shown in Fig. 11.

 

Fig. 11. Plot of power measured at the input facet (large end) of a tapered (80–20 µm) hollow waveguide, versus position of a MMF used to normally illuminate the surface of the taper. The wavelength was fixed to 1560 nm for the case shown.

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Peaks are evident, corresponding to positions of high input coupling efficiency. By comparing with the experiments described in Section 4, we verified that the in-coupling and out-coupling points are coincident for a given taper and wavelength. Note that the large spot size produced by the MMF results in the simultaneous excitation of numerous sub-modes within a particular vertical mode family. The overall insertion loss exceeds 30 dB, even at the peaks. This is partly due to the semi-opaque Au layer on the top surface of the tapers, as discussed in Section 4. Furthermore, the experiment is expected to be far from optimal from a mode overlap perspective, given the large size and diverging nature of the incident beam and the complex standing waves formed in the waveguide near mode cutoff.

To gain further insight, we replaced the MMF with a SMF and aligned the cleaved end of the SMF approximately 20 µm above the surface of a taper. In this case, the taper is illuminated by a single-lobed, approximately Gaussian beam with diameter ~10–15 µm. Figure 12 shows results for the case in which the fiber position was fixed, but the wavelength of the input laser was varied. In Fig. 12(a), the fiber was approximately centered (with respect to the lateral coordinate y) overtop a particular taper. In this case, a single peak dominates the wavelength scan, which we attribute to the resonant coupling of the TE₄₀ mode at this particular position along the taper.

In Fig. 12(b), the input SMF was intentionally misaligned relative to the central axis of the waveguide. In this case, the wavelength scan reveals a series of peaks with similar height. We attribute each of these to the resonant coupling of a particular horizontal sub-mode, all belonging to the same vertical mode family. This can be understood by considering the images in Fig. 9. Within a given TE_mn vertical mode family, the cutoff point moves slightly towards the large end of the taper for increasing horizontal mode order n. On the other hand, as the wavelength is decreased the cutoff point of each mode moves towards the small end of the taper. In the experiment described, the input coupling point is fixed and the launch fiber is offset so that its spatial mode overlaps to some degree with many of the guided modes. As the wavelength is varied, each sub-mode becomes resonant for a particular wavelength.

 

Fig. 12. Plot of power at the input facet versus wavelength, for normal incidence illumination by a SMF at a fixed position overtop a taper (80–10). (a) Typical result when the SMF is centered with respect to the taper axis, near the TE_4n cutoff point. (b) Typical result when the SMF was offset relative to the taper axis, near the TE_2n cutoff point. Note that the position of the SMF along the taper is different in the two cases, so that the n=0 peaks are not coincident in wavelength.

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6. Discussion and Conclusions

We have demonstrated an out-of-plane coupling mechanism in tapered, leaky waveguides clad by omnidirectional reflectors. Power radiates in a nearly surface-normal direction as the core thickness approaches the cutoff condition of a guided mode. While the coupling efficiency for the present devices was low (<0.1%), we believe that there is significant scope for improvement through design of the cladding layers and refinement of the taper geometry. Furthermore, the radiation spot size and the spatial dispersion of the coupling can be adjusted by modifying the slope, etc., of the taper.

In some ways, ODR-clad optical waveguides are analogous to metal-clad microwave waveguides [31]. This similarity could make it possible to translate waveguide-based elements from the microwave domain into the optical domain. For example, microwave delay line equalizers based on tapered hollow waveguides were developed almost 50 years ago [29]. With refinement, such as an increase in the mirror reflectance to suppress radiation loss, the tapered waveguides studied here could form the basis of an analogous class of optical filters. Furthermore, we note that tapered ODR-clad waveguides have been proposed as a novel geometry for realization of opto-mechanical coupling and radiation pressure effects [2].

The ability to spatially disperse a polychromatic signal suggests potential applications in on-chip spectroscopy [23,30]. In this scenario, a tapered hollow waveguide would be aligned overtop a linear detector array, forming an extremely compact and monolithic spectrometer. This design would address some traditional drawbacks associated with wedge-filter spectrometers, such as the need for wide area illumination [30]. The omnidirectional band of the cladding mirrors would limit the wavelength range of the spectrometer, but various techniques for broadening the omnidirectional bandwidth have been reported. Furthermore, since a hollow waveguide can be infiltrated with a gas or liquid analyte, such an on-chip spectrometer could be employed as a sensing system with applications in chemical and biological detection. The taper couplers could also be useful in the context of 3-dimensional, optical or optofluidic integration. For example, coupling between hollow waveguides on multiple levels of a chip could be achieved using opposing tapers on two different levels, possibly with a focusing optic integrated between them. The tapers could also play the role of wavelength multiplexers in this scenario, enabling multi-channel optical communication on a chip [13]. Finally, they could be used to illuminate or collect light at targeted locations along a microfluidic channel integrated on a separate plane. We hope to explore some of these applications in future work.