1. Introduction

Tunable bandpass filters are used in fiber communication [1], hyperspectral imaging [2], fluorescence microscopy [3], and radiometry/photometry [4] systems. Desirable properties include fast tuning over a wide range, high angular acceptance (wide field of view), large clear aperture, flat-top passband shape [2], high transmission in the passband, variable pass bandwidth, high out-of-band rejection [2,3], and wide blocking range. While no single technology has been able to satisfy all of these requirements, their relative importance is of course application-dependent. For example, some imaging systems require filters with high angular acceptance and aperture, but often with relaxed specifications on pass-band characteristics. Conversely, many non-imaging applications of tunable filters can be addressed by commercially available fiber-coupled instruments (see for example [5]).

Length-tuned fiber- or MEMS-based Fabry-Perot filters [1] and conventional angle-tuned thin-film filters can provide narrow to moderate tuning range [3,6–8]. When a wider range is required, liquid-crystal (LC) [9] and acousto-optic (AO) [2] filters are commonly employed. Both can provide tuning over several hundred nanometers in wavelength and can accommodate high angular field of view (several degrees), but they suffer from sub-optimal pass-band shape, relatively high loss (< 50% transmission in the passband for polarized light), and poor out-of-band blocking [3]. An alternative approach is the angle-tuned volume-Bragg-grating (VBG) filter [4], which can operate over several hundred nanometers in wavelength and provides high out-of-band blocking. The VBG filter has limited angular acceptance (~1 mrad), and thus has often been used in combination with a fiber-coupled, broadband supercontinuum laser source. However, it is worth noting that, through appropriate software correction to compensate angular shift of the passband, VBG filters can also be used for imaging spectroscopy [10].

While attracting theoretical interest, tunneling-based bandpass filters have historically been viewed as technologically impractical [11]. The reasons for this include their extreme angular sensitivity and polarization dependence, typically high insertion loss, and need for high-index coupling prisms and high incident angles when traditional solid tunneling layers are employed [11]. We recently demonstrated [12] a tunneling filter that addresses several of these shortcomings, in part by employing an air gap tunneling layer. Notably, we showed that, with the use of high-index-contrast admittance-matching stacks, the filter can simultaneously act as a broadband polarizer, eliminating the problem of polarization-dependent pass-bands associated with traditional tunneling filters. Furthermore, the air gap tunneling layer allows the use of standard glass or fused silica coupling prisms and operation at relatively convenient incident angles near 45 degrees. While the angular sensitivity of the pass-band is still very high (~40-50 nm/degree), it can result in an angular acceptance comparable to that of the commercial VBG filters (~0.5 mrad). Moreover, for applications that can accommodate such a restriction, high angular sensitivity confers potential for broadband and rapid tuning through the use of mechanical rotational stages. Here, we describe a strategy that combines angle tuning with piezo-based tuning of the air gap tunneling layer thickness. This enables the position of a flat-top pass-band, with fractional bandwidth Δλ/λ ~1%, to be varied over an extremely broad range (~1000 – 1800 nm wavelength), using a single optical prism assembly. Both a detailed theoretical treatment and an experimental verification are provided.

2. Admittance-matched tunneling - theory

As shown in Fig. 1, a representative structure comprises an air gap symmetrically bounded by thin-film-stack-coated prisms, with light incident at an angle subject to total internal reflection (TIR) at the air interfaces. Consider the central 3 layers comprising the air gap and the adjacent ‘phase matching’ layers. As discussed previously [12], a resonant tunneling passband can occur when the effective admittance (η_EFF) of this tri-layer is made real and equal to the effective admittance ‘looking into’ the periodic-multilayer-coated prisms (i.e. η_Q, labeled in Fig. 1). In [12], we provided a partial analytic theory for achieving such a condition. We also showed that admittance matching for TE-polarized light resulted in rejection of TM-polarized light over a broad wavelength range, and vice-versa. As mentioned above, this polarizing property conveys a significant advantage over conventional tunneling-based filters. In the following, we present a more complete theoretical treatment, which provides additional insight and allows a more rational design of the desired flat-top bandpass filter response. In the interest of brevity, we will restrict the discussion to the case of designing a TE-polarization passband; the TM-polarization design follows easily [12].

 

Fig. 1 A schematic illustration of the tunneling filter concept is shown. Light is incident at an angle subject to total internal reflection at the air gap interfaces. The air gap is symmetrically bounded by prisms, each coated with a periodic Bragg reflector (2-period case shown) terminated by a ‘phase matching’ layer (n_PH = n_L here). The air gap and the phase matching layers can be replaced by an ‘equivalent’ layer with effective admittance η_EFF at a given input angle and wavelength. For resonant tunneling to occur, η_EFF and η_Q (the admittance ‘presented’ by the periodic multilayer/prism substrate combination) must be real and equal.

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The symmetric tri-layer comprising the air gap and the phase matching layers can be replaced by an equivalent layer with effective admittance η_EFF, where [13]:

Equation (1) allows the effective admittance to be calculated for a given set of (air gap and phase matching) layer thicknesses and input angle. We previously described [12] how to analytically predict the value of d_PH needed to produce a real value of η_EFF for a given combination of d_A and θ_IN. Once d_PH is determined accordingly, a matching stack of quarter-wave (i.e. for a given θ_IN) high- and low-index layers can be chosen to produce a real η_Q (see Fig. 1) as close as possible to the resultant η_EFF, thereby producing a resonant tunneling passband. However, that design procedure involves trial-and-error and iteration. For a given input medium (i.e. prism) and input angle θ_IN, a more direct synthesis results from first choosing a quarter-wave matching stack (QWS), which for assumed lossless materials results in an easily calculated and real value of η_Q. Then, the exercise is to determine combinations of d_PH and d_A which result in η_EFF = η_Q. Starting from Eq. (1), the following relationship between μ_A and δ_PH (i.e. for a given η_EFF = η_Q) can be derived using an analysis similar to that of van der Laan et al. [15]:

At this point, it is useful to consider a specific example. For comparison to our previously reported TE filter [12], we assume λ = 1550 nm, n_H = 3.7 (representing a-Si), n_L = n_PH = 1.46 (representing SiO₂) and n_IN = 1.44 (representing fused quartz prisms). Furthermore, we assume θ_IN = 48 degrees, and that the matching stack has the form (H·L)^Z, where H and L represent quarter-wave layers (d_H = 109.4 nm and d_L = 390.2 nm at this angle), and Z is the number of periods. For such a stack, η_Q = η_IN (η_L/η_H)^2·Z, resulting in η_Q << 1, as necessary to match the typical values of real η_EFF attainable for the air gap tunneling layer and TE-polarized light [12]. For TM-polarized light, η_EFF >> 1 is typical, and (H·L)^Z·H matching stacks are thus appropriate. Figure 2(a) is a plot of solutions to Eq. (2), in the range 0 < δ_PH < π/2 [15], for Z = 1, 2, and 3, corresponding to η_Q = 7.6x10⁻², 6.0x10⁻³, and 4.7x10⁻⁴ (in free-space units [14]), respectively. As is evident, an infinite number of combinations of d_A and d_PH can produce the required admittance match in each case. However, note that each curve is peaked, and that the location of the peak (versus d_PH) is relatively independent of η_Q (i.e. of Z).

 

Fig. 2 (a) Plots of d_A versus d_PH that produce an admittance match to 1, 2, and 3 period quarter-wave stacks, for λ = 1550 nm and θ_IN = 48 degrees. The circled data points are referenced in subsequent plots. (b) Plots of transmittance versus wavelength for the 2-period QWS case, and for the combinations of d_A and d_PH labeled as points 1 (blue solid line), 2 (blue dashed line) and 3 (blue dash-dotted line) in part (a). (c) Plots of transmittance versus wavelength for the combinations of d_A and d_PH corresponding to the peaks of the curves in part(a), labeled as points 1 (Z = 2, blue solid line), 4 (Z = 3, red dashed line), and 5 (Z = 1, green dotted line).

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The shape of the curves in Fig. 2(a) can be understood from the fact that the physical basis for resonant tunneling is the excitation of coupled surface modes at the air interfaces [16–18]. In general, these two coupled surface states result in two transmission peaks, associated with a symmetric and an anti-symmetric ‘super-mode’ (see Fig. 4 of [17]). For a given curve, the maximum (e.g. points 1, 4 or 5 labeled in Fig. 2(a)) corresponds to a critical coupling condition at which the symmetric and anti-symmetric resonances merge, producing a single flat-top transmission feature [17]. On the left side of this maximum (e.g. point 2 labeled in Fig. 2(a)), every point on the curve represents a combination of d_A and d_PH that aligns the symmetric mode (i.e. lower energy) tunneling peak to the design wavelength. On the right side of the maximum (e.g. point 3 labeled in Fig. 2(a)), every point aligns the anti-symmetric mode (i.e. higher energy) tunneling peak to the design wavelength. This behavior is confirmed by the transfer-matrix [14] simulation results shown in Fig. 2(b), which are for the Z = 2 stacks and for three combinations of d_A and d_PH (labeled as points 1,2, and 3 in Fig. 2(a)) that lie on the admittance-matching curve.

Thus, the maximum of any particular curve of the type shown in Fig. 2(a) represents a particularly desirable condition for bandpass filtering applications. The location of this maximum can be determined analytically by differentiating Eq. (2), leading to a quadratic solution:

As the number of periods in the quarter-wave stacks is increased, the flat-top resonant tunneling passband becomes increasingly narrow. This can be traced to the higher phase dispersion and reflectance for increasing Z. The transmittance curves for flat-top passband conditions and Z = 1, 2, and 3 (corresponding to points labeled as 5, 1, and 4, respectively, in Fig. 2(a)) are plotted in Fig. 2(c). A flat-top passband condition is confirmed in each case, but with the FWHM dropping from ~140 nm to ~12 nm to ~0.8 nm, and the out-of-band rejection also increasing dramatically with increasing Z. Note that the secondary ‘peak’ near 1000 nm wavelength for the Z = 3 case is actually the edge of the ‘stop band’ associated with the matching stacks. Thus, the out-of-band rejection in these filters is attributable to both TIR at the air interface and coherent back-scattering within the periodic matching stacks. The narrow bandwidth and high out-of-band rejection predicted for the Z = 3 case is intriguing, but our numerical simulations indicate that this pass-band could only be observed with extremely well-collimated light and extremely high symmetry in the matching stacks, consistent with [17]. For example, if the thickness of the two phase matching layers (which is particularly critical) is mismatched by only ~0.2%, the predicted peak transmission is reduced by approximately half. The observation of such narrow passbands remains an interesting topic for future study, but might be impractical for most applications.

On the other hand, our previous work, which employed a conventional magnetron sputtering system [12], has already demonstrated the practicality of the Z = 2 case. The mismatch can be as high as 2-3% in this case, without significant degradation of the pass-band characteristics. Moreover, the angular sensitivity of the pass-band, while still high, is within a range that is compatible with off-the-shelf collimation optics (see Section 4). For these reasons, we restrict the remainder of the discussion to the Z = 2 case.

3. Tuning by varying the air gap and angle

We turn our attention to the prospect of achieving a tunable version of the filter described in Section 2. One option would be to implement a linearly varied version of the filter, where all layers (including the air gap) are proportionally tapered along one axis. However, this would be a rather challenging structure to fabricate. Moreover, linearly varied tunable filters (LVTFs) tend to suffer from slow tuning speeds, due to the need for linear translation of a large part, and passband broadening, due to spatial convolution effects [6]. Angular tuning of thin film filters is an alternative approach, which has been used extensively for narrow-range tuning in fiber systems and more recently for moderate-range tuning in fluorescence and Raman spectroscopy systems [7,8]. The latter work uses specialized thin-film filters (with high layer count) developed by Semrock [6], which can be tuned over a range equal to ~10% of their normal-incidence passband center wavelength. Angle-tuning is also the mechanism employed in most grating-based tunable filters including the VBG filters discussed above [4].

The principle of the proposed tunable filter is illustrated schematically in Fig. 3(a). Assume that the prisms are uniformly coated with a thin film stack, designed to provide an admittance-matched tunneling band at a particular wavelength and incident angle, as described in Section 2. For illustration purposes, we will assume the values cited for the 2-period filter above (i.e. n_IN = 1.44, n_H = 3.7, n_L = n_PH = 1.46, d_H = 109.4 nm, d_L = 390.2 nm, and d_PH^# = 91 nm) representing a QWS at θ_IN = 48 degrees, terminated by a phase adjusting layer chosen to produce a flat-top bandpass response centered at 1550 nm wavelength. Now consider changing the incident angle to some new value θ_IN*. At this new angle, the matching stack (i.e. HLHL) is no longer a QWS, either at 1550 nm or at any other wavelength, and its input admittance is in general complex, η_Q = η_Q^’-i·η_Q^”, as easily calculated for example using transfer matrices. At any given wavelength, a real input admittance can be restored by adding an ‘extra’ low-index layer of phase thickness [14]:

Now, at the new input angle and using η_Q*(λ) determined above, Eqs. (2) and (3) can be solved to determine the values of phase-matching layers and air gap thicknesses, d_PH*(λ) and d_A*(λ), respectively, that are required to produce a flat-top tunneling passband. For a given angle, the spectral location of this passband is thus determined by the following condition:

 

Fig. 3 (a) A schematic illustration of the proposed tuning mechanism is shown. For a change in incident angle (θ_in to θ_in*), the effective position of the interface between the phase matching layer and the matching stack shifts by some amount Δd (which can be positive or negative). The flat-top tunneling passband occurs at a new wavelength for which η_Q* and η_EFF* are real and equal, and with the air gap thickness d_A* tuned to an appropriate value. (b) An example of the graphical determination of the flat-top tunneling wavelength for a particular incident angle (53 degrees) is shown. As described in the main text, for each incident angle there is a single wavelength at which d_PH* = (d_PH^# - Δd), which in turn makes η_Q* real while also enabling η_EFF* = η_Q* through appropriate choice of d_A*.

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It follows that the spectral position of the flat-top pass-band can be continuously varied over a wide range by simultaneously tuning the air gap thickness and the incident angle. Plots of the predicted resonant tunneling wavelength, and the corresponding required air gap thickness, are shown in Fig. 4. For the matching stacks assumed above, continuous tuning over a wavelength range from ~1000 nm to ~1800 nm is possible through angular tuning in the ~44 to 62 degrees range and corresponding air gap thicknesses in the ~7 to 1 μm range. Results with refractive index dispersion (calculated using previously described models for our sputtered SiO₂ and a-Si films [19]) taken into account are also shown.

 

Fig. 4 (a) Plot showing the predicted flat-top passband wavelength versus input angle, for the filter structure described in the main text. (b) Plot showing the predicted air gap thickness required to produce the flat-top passband at the corresponding wavelengths shown in (a). The solid (dashed) curves were calculated without (with) material dispersion taken into account.

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Representative passbands, calculated using transfer matrices, are shown in Fig. 5 for selected input angle and air gap combinations. Of course, unity transmission features are only possible in an idealized scenario neglecting loss and scattering. The impact of including absorption loss for the a-Si layers is shown for the two pass-bands near the opposite ends of the tuning range in Fig. 5(a). We used an extinction coefficient model from our previous work [19], which gives κ_aSi ~0.006 and ~0.0006 at λ = 1000 nm and 1800 nm, respectively. As shown, the predicted impact on peak transmittance is tolerable, and is consistent with our previous experimental results [12]. Moreover, a-Si films have potential for significantly lower loss, for example through optimized hydrogenation [20].

 

Fig. 5 (a) Transmittance versus wavelength for a series of input angle / air gap combinations corresponding to the flat-top passband conditions depicted in Fig. 4. A flat-top pass-band centered at the predicted wavelength is verified in each case. The FWHM linewidth varies from ~10 nm at the short wavelength end of the range to ~20 nm at the long wavelength end. For the two cases near the ends of the tuning range, the bold curves were calculated with the inclusion of absorptive loss for the a-Si layers (see main text). (b) The data from part (a) is shown on a logarithmic scale in order to reveal the predicted out-of-band rejection.

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As mentioned above, the high angular sensitivity of tunneling filters has traditionally been viewed as a major drawback, especially due to the existence of polarization-dependent pass-band splitting [11]. While the polarizing nature of the air gap tunneling filter described here significantly mitigates these issues, the high angular sensitivity of the polarized pass-band still limits the maximum acceptance angle. On the other hand, a high angular sensitivity imparts advantages, including the potential for wide tuning range and relatively fast tuning speeds [6]. The representative filter from above exhibits a shift in passband center wavelength on the order of ~40 nm / degree and pass-band bandwidths on the order of ~10 nm. For BPFs, the field of view (FOV) can be defined as the full angular width that results in a shift in center wavelength equal to half the passband width [21]. For the present filters, this implies that the incident light needs to be collimated with a half-angle less than ~0.5 mrad in order to avoid significant pass-band broadening. This is well within reach using commercial broadband collimators [22], as supported by our earlier experimental results [10] and confirmed below. It is also interesting to note that commercial VBG-based tunable filters have similar angular sensitivity and beam divergence restrictions [23]. Finally, it should be noted that it is always possible to limit angular acceptance through optical system design, to avoid passband broadening at the expense of throughput for sources with higher angular range. Similar tradeoffs are inherent in most diffraction grating instruments.

4. Experimental results

As an initial proof-of-concept, we assembled the system depicted in Fig. 6(a), by using nearly hemi-cylindrical lenses (uncoated N-BK7 glass, Edmund Optics, stock #35-020) for the coupling prisms. They were first mounted in a custom holder to facilitate sputtering deposition of Si/SiO₂ matching stacks onto their flat faces. In order to improve film adhesion between the matching stacks and the N-BK7 prisms, a thin (10-20 nm) layer of SiO₂ was first deposited, followed by two-period (i.e. Z = 2) stacks with layer thicknesses nominally as described in Sections 2 and 3. Next, pairs of lenses with well-matched coatings, as verified by photospectrometer measurements, were clamped together, and piezo-electric stack actuators (Thorlabs part no. PK2JA2P2) were glued to the lenses at each end (see Fig. 6(b)). In the following, we will refer to the piezo-bonded pair of hemi-cylindrical lenses as simply the ‘prism assembly’. Assuming the coated lens faces are flat and clamped in intimate contact, the ‘resting’ value of the air gap is approximately zero. The chosen piezo-actuators allow this gap to be increased up to a maximum value of ~8 μm.

 

Fig. 6 (a) Schematic depiction of the tunable filter assembly. Tuning is achieved by simultaneously controlling the incident angle (using a motorized rotational stage) and the thickness of the air gap between the matched hemi-cylindrical lenses. (b) Overhead photograph of the prism assembly mounted in an optical rail system (scale bar: 1 cm).

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The choice of BK-7 coupling prisms, in place of the fused silica prisms used previously [12] and assumed for the theoretical treatments above, was motivated by the availability of suitable off-the-shelf lenses. This change has minimal effect on the operation of the filter, other than to shift the angular position of the resonant passbands to slightly smaller values, in keeping with Snell’s law. For example, the 1550 nm flat-top passband predicted at ~48 degrees in Figs. 4 and 5 for fused silica prisms (n ~1.44) instead occurs at ~46 degrees for BK-7 prisms (n ~1.5).

The prism assembly was mounted on a rotational stage and aligned between fiber collimators (achromatic fiber ports, Thorlabs part no. PAF2-A4C). The use of hemi-cylindrical (or, as discussed below, hemi-spherical) input/output coupling prisms allows the angle of incidence to be varied by rotation of the prism assembly, without significant deflection of the beam path for a well-centered system. Diverging cylindrical lenses (Thorlabs part no. LK1363L2) were positioned between the collimators and the prism assembly, to cancel the focusing effects imparted by the curved surfaces of the coupling prisms. The lens focal lengths and positions were optimized through ray-tracing simulations (Zemax). Light from a broadband source, either a multiple-SLED-based instrument (LuxMux BeST-SLED) or a laser supercontinuum source (NKT SuperK Compact), was coupled via a polarization maintaining single mode fiber (SMF). The results shown below correspond to TE polarized input light. We verified (not shown) that the filters provide strong rejection of TM-polarized light (~OD4) across the entire wavelength range considered [12].

Light transmitted through the filter assembly was collected into a SMF or MMF and delivered to an optical spectrum analyzer (Yokogawa AQ6370B). All data shown below were obtained using the SMF output; the MMF output was used to more accurately assess the optical throughput of the prism assembly. Typical fiber-to-fiber insertion loss at the peak of the passband was ~10 dB for the SMF output and ~5 dB for the MMF output. Note that the tunneling filter, on its own, can exhibit very low insertion loss for well-collimated light, as per the results from Section 3 and as verified for a fixed filter previously [12]. Most of the extra insertion loss observed here can be attributed to the multiple uncoated surfaces in the optical path, and to imperfect compensation of beam divergence by the cylindrical surfaces. We expect that lower insertion loss can be achieved through further optimization, in particular if the cylindrical coupling prisms are replaced by spherical coupling prisms as discussed below.

For these studies, SMF-to-SMF coupling was optimized at a single wavelength and the coupling and alignment optics were then fixed as the passband wavelength was tuned. All experimental curves were spectrally calibrated against a base scan for the broadband source, captured without the prism assembly and diverging lenses in the optical path, and were normalized to their peak value to facilitate direct comparison with theory. Notably, the insertion loss was nearly constant (within ~2 dB) across the entire tuning range, verifying that small changes in angle and air gap do not significantly alter the beam path.

Figure 7(a) shows transmission curves for a typical passband centered near 1550 nm and for several different air gap thicknesses. Here, the incident angle was fixed and the piezo voltage was stepped in increments of 4 V, corresponding to increments in the air gap of ~0.14 μm. In keeping with the theoretical predictions, two separate peaks are observed for low values of the air gap thickness, and these peaks merge to produce a single passband feature when the air gap thickness is increased to an appropriate value.

 

Fig. 7 (a) Experimental passband centered near 1550 nm wavelength for an incident angle ~46 degrees and various values of the air gap thickness. The legend indicates the estimated gap in μm (see main text). (b) A typical passband feature (at ~1370 nm wavelength) is shown on a logarithmic scale, as theoretically predicted (blue dashed curve) and measured (red solid curve).

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Figure 7(b) shows a typical flat-top passband plotted on a logarithmic scale, revealing good agreement between theory and experiment. Some small notches were typically observed in the experimental data, and are likely artifacts arising from imperfect calibration. Notably, they were observed only in the case of SMF-to-SMF coupling, while different artifacts (a periodic ripple in the passband) was observed for SMF-to-MMF coupling. The measured out-of-band-rejection of ~OD4 at +/− 25 nm and ~OD6 at +/− 100 nm is in very good agreement with theoretical predictions. Theory also predicts higher rejection outside this interval, but the limited dynamic range of our experiment prevented direct corroboration.

We subsequently verified that we could tune the passband over a wide range of wavelength by simultaneously adjusting the incident angle and the air gap thickness. Figure 8(a) shows representative passbands spanning the ~1250-1700 nm wavelength range. From normal-incidence measurements, the prism assembly was determined to have a resting-point air gap of ~1.8 μm. This is likely attributable to the manual nature of the assembly, and prevented us from demonstrating passbands at shorter wavelengths. On the long wavelength end, we were limited by the operational range of our sources and the available OSA. The measured flat-top passbands exhibit wider FWHM linewidths (by ~40-50%) compared to the theoretical predictions in Fig. 5. This might be attributed to imperfect collimation of the input beam (angular convolution effects) combined with non-uniformities in the air gap and thin film thicknesses (spatial convolution effects), but is the subject of ongoing study. Nevertheless, the global behavior of the tunable filter is in excellent agreement with predictions.

 

Fig. 8 (a) Experimental passbands centered at various wavelengths in the ~1250 – 1700 nm wavelength range, corresponding to selected combinations of angle and air gap indicated by the symbols in part b. FWHM linewidths vary from ~15 nm to ~30 nm, approximately 50% larger than the theoretical predictions shown in Fig. 5. (b) Comparison between theoretical and experimental combinations of air gap thickness (dashed curve and square symbols) and incident angle (solid curve and diamond symbols) required to observe a flat-top passband versus wavelength.

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Figure 8(b) shows a comparison of the theoretically predicted and experimentally observed combinations of incident angle and air gap thickness needed to observe the flat-top passband feature versus wavelength. The theoretical curves were based on the analysis in Sections 2 and 3, but with N-BK7 coupling prisms and a phase matching layer thickness of 88 nm assumed (as estimated from photospectrometer measurements). The experimental air gaps were estimated based on the manufacturer-supplied voltage-extension curve for the piezo actuators, under ‘low stiffness’ loading conditions [24]. The experimental angles were estimated from overhead images of the prism assembly captured by a camera (see Fig. 6(b) for example) at one particular setting, and then using the incremental angle readout supplied by the motorized rotational stage. Error bars for the air gap represent the range of voltage-extension variation (+/− 15%) specified by the manufacturer, and the error bars for the angle represent the estimated accuracy (+/− 0.02 rad) for the angle extracted from the camera images.

5. Discussion and conclusions

We have described and demonstrated a new type of bandpass filter, with potential to be tuned across a large portion of the near infrared spectrum. The physical basis of the filter is admittance-matched tunneling of light through an air gap, where the matching is achieved using a simple (4-layer) thin film stack. The spectral position of the resonant passband is tuned through simultaneous adjustments in the incident angle and the air gap thickness. We fabricated and tested a proof-of-principle system using hemi-cylindrical input/output prisms with an intervening air gap controlled by piezo actuators. Experimental results provided good corroboration of the theoretical predictions. Even simpler systems can be envisioned using hemi-spherical prisms, which might act as both the coupling and collimating/focusing elements and thereby eliminate the need for additional lenses. Moreover, systems operating in different wavelength regions (e.g. the visible band) could be realized through the use of appropriately transparent thin-film materials. It is also worth noting that resonant tunneling filters have recently been implemented in an on-chip MEMS platform [25]; this might be an interesting avenue to explore for miniaturization and integration of the present filter.

A further advantage is that operation of the tunable filter is not particularly sensitive to the exact thicknesses of the thin film matching layers, as long as the stacks are well-matched on each side of the air gap [12,17]. Thickness errors simply result in slight changes in the combinations of angle and air gap needed to observe the flat-top passband at a given wavelength. The ability to adjust the tunneling layer thickness during operation thus makes the filter quite robust to variations in the fabrication process. Moreover, it might be interesting to explore strategies for post-fabrication compensation or fine-tuning (e.g. through thermal tuning of the refractive index) of the matching stacks.

Finally, it is worth reiterating that resonant optical tunneling filters, after a few decades of study, were essentially deemed as impractical in the late 1960s [11], with Baumeister concluding: “It is doubtful if the single tunnel layer filter will ever be produced as a practical bandpass filter, owing to its disadvantages of a large angle shift and its strong polarization.” To our knowledge, this viewpoint has not changed significantly in the past 50 years, with resonant tunneling filters attracting only sporadic interest and mainly as a theoretical curiosity [14,25]. The filter described here in large part addresses the drawbacks that contributed to such a gloomy perspective. The use of an air gap tunneling layer enables convenient substrates and incident angles [26], and the use of high-index-contrast matching stacks can result in a strong rejection of one polarization state – i.e. the filter simultaneously plays the role of a polarizer, rather than exhibiting polarization-dependent passbands. By introducing an adjustable tunneling layer, the need for extreme control over thin film thicknesses is removed. Moreover, the tunable air gap allows us to turn the angular dependence into an advantage, potentially enabling passband tuning over a very large wavelength range. Combined with the modern ability to deposit thin films having vanishingly low absorption and scattering losses, our approach could spur renewed interest in the practical application of tunneling-based filters.