1. Introduction

Planar high index contrast 2-D photonic crystals of subwavelength characteristic dimension exhibit interesting electromagnetic effects seen from the viewpoint of confined fields propagating in-plane for guiding, redirecting, reflecting, coupling and filtering slow waves trapped in this very plane [1]. This domain has made important progress thanks to the availability of the needed fabrication technologies, from the origination of the sub-wavelength structures by electron beam writing to the high aspect ratio micro-structuring in a semiconductor layer by reactive-ion etching (RIE) [2]. In contrast, the electromagnetic behavior of planar 1-D and 2-D micro-structures excited by free space waves has received less attention from the photonic crystal community. Pioneering work was made as in ref [3]. where a concept of “2.5 dimension photonic crystal” was introduced. Free-space wave interaction with planar periodic structures has actually been extensively explored for more than three decades by the community of integrated optics [4] in the objective of coupling an incoming free space wave to a waveguide mode [5]. Narrow band resonant reflection was first identified and experimentally demonstrated since the mid-eighties by Sychugov et al. [6] as due to the excitation of a waveguide resonance. Since then, much theoretical as well as experimental works have been reported on this effect [7]. With the advent of the photonic crystal era, the question has arisen of the existence of this effect of resonant reflection and of its features when the waveguide is of high index contrast and is strongly perturbed by deep trenches [8]. The occurrence of ultra-narrow-band resonant reflection in such structures under normal incidence was revealed numerically and explained phenomenologically in [9] and confirmed recently on a grating-mode basis [10]. Ultra-narrow-band was also anticipated in high index contrast gratings under slightly off-normal incidence [11]. The authors of the present paper have extended the classical waveguide grating rationale to such high contrast configurations and have succeeded in giving a physically intelligible procedure for setting the opto-geometrical conditions giving rise to ultra-narrow band resonant reflection under normal incidence [9, 12]. In [13] we have used silicon micro-machining to fabricate a test structure with high precision in the aim of demonstrating the expected effect at terahertz frequencies (THz). The experiments have evidenced the presence under normal incidence of a narrow transmission dip, but it has not been possible at that time to experimentally demonstrate that this close to zero transmission dip does actually correspond to a high reflection peak since under normal incidence it is difficult to measure a reflection spectrum with high accuracy. It was not possible either to conclude that, despite the absence of propagating diffraction orders of order larger than 0, a zero transmission necessarily implies a 100% reflection since the reflection mechanism is highly resonant and high-resistivity silicon may exhibit non-negligible absorption losses in the THz range for a guided mode.

The present paper demonstrates experimentally that zero transmission does correspond to close to 100% reflection over a narrow spectrum indeed. In addition, we bring here some early analytic material for an alternative phenomenological explanation of the resonant reflection effect on the basis of the true grating modes propagating up and down the walls and slits, instead of considering waveguide modes propagating along the plane of the pierced membrane as we did in Ref [12]. The waveguide mode representation is conceptually very satisfactory; it is however difficult to develop it further to an accurate quantitative model because it involves a number of phenomenological parameters which under normal incidence depend rather sharply on the optogeometrical parameters, and therefore are not suitable for a meaningful phenomenological model. It was recently shown by Chang-Hasnain et al. [14] that the representation based on the interplay of the true grating modes is well suited for accounting for the sharp resonant reflection effect in the considered high index contrast structures, and that the resonance effect is that of a supermode of the finite thickness segmented membrane as a linear combination of two grating modes of the infinitely thick set of alternate air gaps and high index walls. It must be reminded that this modal analysis of ultra-narrow resonances was made years ago, and on 2D gratings, by the recognized pioneers of this modal representation [15]. In what follows the operation of the resonant grating will first be described on the basis of the phenomenological in-plane waveguide mode representation [12]. After a short description of the fabrication technology, the experimental demonstration of ultra-narrow band reflection will be presented and the results compared with an exact numerical modeling. Finally, the grating-mode representation of the ultra-narrow band resonant reflection will be explained, as first outlined in Ref [16], and applied for the identification, the sorting out and classification of the supermode resonances.

2. The grating-coupled waveguide mode representation

A cross-section of the binary grating is shown in Fig. 1 . It represents a periodic grid of rectangular walls of high index n_g embedded in a uniform medium of index n_c, which is air in the present experimental model. In contrast to [9], where the grid is supported by a substrate of index n_s, the terahertz demonstration experiment involves a simple silicon grid in air, thus the embedding medium is uniform and of unity refractive index.

 

Fig. 1 Cross-section of the silicon/air grid showing the dimensions of the experimental element and the transverse electric field of the TE₀ guided mode.

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A systematic search for a structures and incidence conditions giving rise to ultra-narrow-band 100% reflection was initially made using the finite-difference time-domain method (FDTD) [9]. The incident field is essentially normal to the grid plane; it is laterally confined in the form of a Gaussian function. Under normal incidence the confined incident field only “sees” a restricted number of periods of the grid of unlimited extent. The FDTD method laboriously found a number of configurations where 100% reflection occurs, but ultra-narrow-band reflection only occurs under very specific conditions. These conditions were difficult to identify as the FDTD method is purely numerical and does not give much insight into the underlying resonance mechanism. That ultra-narrow-band reflection can be exhibited by a grating of which hardly 20 periods are illuminated looks astonishing, but is nevertheless true as verified by a completely different modeling method [17]. This was the motivation for the search of a phenomenological model accounting for such unusual effect. The model, as well as the design rule derived from it, were published in ref [12]. The rationale used in [12], based on waveguide- and multilayer optics with bridges to the concepts and terminology of photonic crystals, will by meaningfully applied in the following paragraphs to the extent needed for the understanding of the basic waveguide coupling mechanism.

First, the thickness of the silicon membrane is roughly set to provide high transmission for a normally incident beam outside resonance; the reflection pedestal of the future reflection peak will therefore be low as requested from a device point of view. This amounts to setting the condition for a Fabry-Perot resonator of half-wave thickness. The refractive index considered in this resonator is the effective index n_e0 of the fundamental grating mode propagating up and down the walls which is known to be inferior to n_g.

Secondly, the grid period Λ is adjusted to couple, by means of its ± 1st diffraction orders, the normally incident free-space wave to the fundamental guided mode propagating in-plane in the silicon membrane. This mode propagates along the segmented waveguide membrane normally to the grid grooves and re-radiates away normally into the incidence and transmission media. For high contrast interference in the incidence medium to take place between the Fresnel reflection of the incoming beam and the coupled and re-radiated wave, i.e. for a possible constructive reflection interference in the incident medium, the coupled guided wave must be laterally confined under the impact zone of the incident beam. If the waveguide grating is placed at the waist of a slightly focused incident beam, a substantial overlap of the reflected beam and the re-radiated wave cannot be achieved over a few grating periods only, unless the second order of the grating couples the right-propagating waveguide mode to its left-propagating counterpart, and conversely, so as to restrict the propagation length of the mode along the waveguide. Such a grating is designed by choosing a ratio between the air gap and silicon wall widths exhibiting a strong second order. This can be made by resorting to the results of the domain of multilayers if the coupled waveguide mode is sufficiently well confined vertically in the segmented silicon slab. One solution is to set the air gaps at a quarter wave width whereas the silicon walls are set to a three quarter wave width. Analytical expressions exist for the amplitude and phase of the reflection coefficient [12] and can readily be used to adjust the confinement of the coupled guided wave to a lateral extend which essentially corresponds to the diameter of the incident beam impact on the grid. These analytical expressions represent an approximate solution to the inverse problem of finding out the conditions for narrow band reflection of a slightly focused beam from a high contrast grating of which only a few periods are illuminated. Such solution corresponds to a 1D defectless 2nd order photonic crystal used at one of the band edges of the forbidden zone. A vivid physical interpretation is to state that the paradoxically narrow band reflection from a grating of a few periods only is a result of the intra-guide resonance or multi-roundtrip of the guided mode under the –2nd order reflection which is spectrally equivalent to a large number of periods “seen” under non-confined conditions. Figure 2 sketches the complete coupling mechanism whereby the first order of the grating couples a normally incident beam of small waist to the waveguide mode whereas the −2nd order intraguide coupling reflects the coupled mode and keeps the field narrowly confined under the impact zone of the incident beam.

 

Fig. 2 Symbolic representation of the integrated ± 1st order free space wave coupling and ± 2nd order intraguide coupling for lateral field confinement.

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This single grating integrated mechanism is in sharp contrast with the recent attempt to achieve the field confinement by means of two lateral gratings of half the period of the central coupling grating in a distributed Bragg reflector (DBR) scheme [18]. What the waveguide grating resonance representation remarkably succeeds in accounting for – which the grating mode representation does not yet – is the very important feature of ultra-narrow-band reflection to be wide-band angularly. The 2nd grating order confinement of the guided mode over a restricted number of periods implies that its Fourier transform corresponds to a wide angular aperture beam. This property, associated with ultra-narrow spectral width, is an exclusive strong asset of this element seen as a filtering device. This will be shown vividly by the pulsed THz experiment in Section 3.

3. Silicon microsystem technology for the grid fabrication

An operation point of the resonant grating was defined approximately using the above described guidelines and submitted to an exact code based on the true-mode method [19]. Figure 1 represents the scaled cross-section of the segmented membrane calculated with the refractive index 3.4 for silicon in the THz domain under normal TE incidence. It has a period of 385 µm, a thickness of 208 µm, a silicon wall width of 174 µm, thus an air groove width of 211 µm and exhibits ultra-narrow reflection at 585 µm wavelength (i.e. at 0.513 THz). As anticipated, the exact code finds an ultra-narrow resonance with a grating silicon/air optical width ratio of about 3/1 showing large second order intra-guide coupling. Figure 3 shows a number of wavelength spectra.

 

Fig. 3 TE reflection spectra from the grating of Fig. 1. (a) Ultra-narrow-band reflection; (b) Effect of silicon ridge narrowing; (c) Effect of thicker silicon membrane; (d) Effect of silicon refractive index increase.

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Curve (a), with a reflection peak at 585 µm and a spectral width of 100 nanometers in wavelength units, is the nominal model to be fabricated. Against expectation, the segmented membrane thickness does not correspond to close to zero reflection outside resonance. This is discussed in Section 4. Spectrum (b) shows the effect of a 4% narrower silicon wall for the same period: the resonance shifts to a shorter wavelength since the effective index of the coupled mode decreases, and the resonance width increases dramatically to 2 µm. A 3% thicker silicon membrane causes a resonance shift to larger wavelength since the effective index increases, and the resonance width increases to about 2 µm too (curve (c)). A refractive index increase to 3.5 (curve (d)) also increases the effective index but does not affect much the spectral width which is about 300 nm; this will be discussed from a grating-mode point of view in Section 4. It must be pointed out that the above modeling results concern plane wave incidence.

The fabrication of the segmented waveguide is made from a 525-µm thick high resistivity (8 kΩ⋅cm), double-side polished silicon wafer with a 2 µm oxide layer on each side. The wafer is then covered by a photoresist film in which the grating printing (Λ = 385 µm) is made with a direct laser-writer system, then the front oxide mask is wet-etched. The second fabrication step consists in thinning down to 210 µm the silicon wafer with potassium hydroxide (KOH), the front side being protected. The third step is the deposition of an etch-stop layer at the bottom of the silicon membrane; this layer also acts as a “landing layer” by permitting the confinement of the cooling helium under the wafer up to the end of the silicon groove etching. Finally, deep anisotropic silicon etching is performed following the Bosch process [20]. Figure 4 represents the technological procedure used to fabricate the 1D silicon grating suspended in air, including intermediate fabrication steps.

 

Fig. 4 Fabrication process stages of a 1D silicon grating.

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Figure 5(a) shows a picture of the 4” silicon wafer in which 3 gratings were processed. The one used in this work is 1.5 × 7 cm² wide. Figure 5(b) presents an enlarged view of the rectangular silicon bars. Microscope inspection reveals that the silicon bars are slightly trapezoidal: their width is 174 nm at the KOH-etched side and 164 nm at the patterned resist side.

 

Fig. 5 Top view of the silicon gratings: (a) overall wafer with three grating patches; (b) optical microscope top-view.

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4. THz time-domain experimental demonstration of narrow band reflection

The experimental measurement of the THz electromagnetic response of the element is performed using THz time-domain spectroscopy (THz-TDS) [21]. The emitting and receiving antennas are low temperature grown GaAs photoswitches excited with 60 fs duration pulses at 800 nm wavelength delivered by a 82 MHz repetition rate Ti:Sapphire mode-locked laser. The setup radiates picosecond electromagnetic bursts whose spectrum ranges from 0.1 up to 5 THz. The quasi-optics THz system includes hemispheric silicon lenses attached to the photoswitches, and parabolic mirrors (PM) to collimate and focus the THz beam. For the measurement under normal incidence of both transmission [22] (Fig. 6(a) ) and reflection [23] (Fig. 6(b)), the sample is located at the focal point of a parabolic mirror whose illuminated area has a diameter of ~50 mm; at the focus position the THz beam waist is about 3 mm at 600 µm wavelength corresponding to a convergent beam of about 2 x 8°. Reference signals are measured without sample and using a copper mirror (close to 100% reflection) at the sample position respectively for transmission and reflection measurements. In the set up dedicated to reflectivity measurement (Fig. 6(b)), the reflected THz beam is directed to the receiver using a THz beam splitter made of a 4 mm thick silicon plate to prevent any etalon effect inside. In that case the beam incident onto the sample is rigorously the same in both transmission and reflection configurations. As the signals are recorded over a 120 ps time-window, the frequency resolution is limited to ~8 GHz, corresponding to ~8.7 µm at the resonance wavelength of 500-600 µm.

 

Fig. 6 THz-TDS measurement set up. PM stands for parabolic metal mirrors, the dashed element is the grating sample. (a) transmission measurement; (b) reflection measurement.

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The experimental transmitted and reflected relative intensities are represented in wavelength units in the graph of Fig. 7 . The blue solid line corresponds to the measured transmission. It exhibits the transmission dip at 575 µm close to the theoretical 585 µm spectral position of the reflection peak of Fig. 3, curve (a) with a spectral width of about 10 µm. It also exhibits a second reflection dip at 445 µm wavelength; this dip is very narrow as well although the structure was optimized for the ultra-narrow reflection peak at 585 nm wavelength. As the analysis in section 5 will illustrate clearly, the occurrence of an ultra-narrow resonance at some spectral position does not imply that the same structure exhibits another ultra-narrow resonance at another wavelength. The red dashed line in Fig. 7 is the complement to 1 of the transmission spectrum. The ordinate axis is limited to slightly above 1 as some measured intensities, which are relative to a reference measurement without the element, slightly exceed 1 as the alignment conditions for the reference and characterization measurements are not strictly identical. The red solid line represents the actually measured reflection spectrum. Although the number of measurable points within the narrow spectral width of the 575 µm peak is very small because of the limited frequency resolution of the TDS experiment, the presence of one point at more than 70% reflection, and the similarity between the reflection spectrum and (1-T) permit to state without ambiguity that the absence of transmission does correspond to resonant reflection.

 

Fig. 7 Measured relative transmission and reflection spectra (solid lines) of the fabricated silicon grating, illuminated by a 2x8° focused THz beam. The dashed line is the calculated complement to 1 of the measured transmission spectrum.

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The spectral width of these two resonances (between 5 and 10 µm) is not especially narrow if one compares it with the theoretical resonance widths of a plane wave as shown in Fig. 3 where the resonance width is notably smaller than 1 µm at its minimum of curve (a). However, the angular aperture of the incident beam in the experiment is not that of a quasi-plane wave: it is as much as 16° degrees, which implies that the measured spectral width should be as much as 100 µm if one applies the waveguide grating mode coupling phase-matching condition under oblique plane wave incidence. This is quickly shown by letting λ_θ be the wavelength at which resonant reflection occurs under the plane wave incidence angle θ: λ_θ = λ□ + Λsinθ where λ is the coupling wavelength under normal incidence (one assumes that the effective index of the mode of the segmented silicon waveguide does not vary notably with the wavelength over the resulting wavelength range). Consequently, the narrow spectral width of the resonance under focused beam incidence and the high contrast between the transmission dip and its pedestal bring the experimental evidence that the observed transmission dip does correspond to a sharp resonant reflection. The experiment also demonstrates that this resonant reflection exhibits a remarkably large angular tolerance as was anticipated from the phenomenological model summarized in section 2.

An experimental comparison was made on the same sample with a normally incident collimated beam whose total angular aperture is smaller than 0.4 degree. Figure 8 shows the transmission spectrum versus the frequency in THz over one decade. The low frequency components only “see” an effective high index medium whereas the first diffraction orders start propagating beyond about 0.8 THz. The frequency domain of interest extends between the frequency cutoff of the TE₂ grating-mode (see Section 5 hereunder) and the cutoff of the free-space first diffraction orders. The continuous blue line results from the exact modeling of the structure, the red dots stand for the experimental sampling points. As expected from Fig. 3 curve (a), there is a very sharp transmission dip associated with a sharp peak close to 0.5 THz as optimized. A zoom in a narrower frequency range around 0.5 THz permitted to state that the transmission dip of the 16° aperture beam is only twice as wide as that of the collimated beam.

 

Fig. 8 Transmission spectrum of the silicon grating under collimated normal incidence. Blue solid line: modeling of the fabricated structure; red dots: measured points.

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5. The grating-mode representation of ultra-narrow band reflection

Although the design approach based on the coupled propagating modes of the segmented waveguide gives a meaningful insight as to where the resonance conditions can be found, and permits to explain the wide angular spectrum of the reflection, this interpretation only leads to a rough quantitative design. An electromagnetically more exact and nonetheless instructive representation is found by analyzing the structure on the basis of the natural grating-modes propagating up and down the air slits and silicon walls, and coupling to each other and to the diffraction orders in the adjacent incidence and transmission free-spaces at the grating sides [14, 24]. The analysis of the structure exhibiting the ultra-narrow spectrum of Fig. 3(a) was made on the basis of the grating-mode representation with the help of the true-mode code [19]. This analysis provides the effective index n_e0 and n_e2 of the TE₀ and the TE₂ modes excited under normal incidence and their reflection and cross-coupling coefficients at both grating interfaces. This analysis reveals that all these parameters vary smoothly across the resonance, and that there is no significant accumulation of each TE₀ and TE₂ modal field at resonance which could account for the ultra-narrowness of the resonance. Thus, although the grating represents a Fabry-Perot resonator for each of the two TE₀ and TE₂ modes, there is nothing in the behavior of these modes considered one by one which can explain the narrow band reflection effect. It was recently shown [14] that the ultra-narrow band resonant reflection is that of a supermode which is a linear combination of both TE₀ and TE₂ grating-modes of the infinitely thick grating. This supermode is a mode of the finite thickness grating. At resonance, its transmission Poynting vector cancels out and its field within the grating reaches an arbitrarily large amplitude. Interestingly, this supermode resonance condition is very close – but not identical – to the Fabry-Perot resonance condition of the TE₀ and the TE₂ grating-modes when the latter coincide [16]. This is represented in the chart of Fig. 9 which is a mapping of the reflection coefficient in false colors versus the grating period and height (membrane thickness) with a Si-line/period ratio of 0.45; the wavelength is that of the short wavelength peak of Fig. 7 (at 445 µm wavelength) where the mode density is larger and makes the chart more intelligible.

 

Fig. 9 Reflection coefficient mapping at 445 µm wavelength with 0.45 duty cycle. The underlined TE₀ and TE₂ Fabry-Perot resonance conditions of order 4 and 2 respectively cross at 360 µm period and 210 µm height.

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In the period domain below 263 µm the grating only supports the fundamental TE₀ grating-mode, therefore no interference between modes can take place at the interface between the grating and the adjacent media, thus the reflection coefficient cannot reach 100%. In the period domain above 445 µm, grating-modes of even order higher than 2 have a propagating character which renders the mode interplay too complex and gives rise to higher diffraction orders in the adjacent media. The period range between 263 and 445 µm is the one of concern here where the TE₀ and TE₂ grating modes only propagate and the 0th diffraction order only is non-evanescent in the adjacent semi-infinite media. The height-period plane (h,Λ) is marked out by two sets of curves of different slope: the set of curves corresponding to the TE₂ Fabry-Perot resonant transmission through the grating is the steeper since the effective index varies faster with the grating period, and the distance between curves is the larger since the effective index is the smaller. The denser and smaller slope set of curves corresponds to the TE₀ Fabry-Perot resonance; the effective index of the cutoffless TE₀ mode is large and varies slowly with the grating period. The order m of the Fabry-Perot resonance curves TE₀FP_m and TE₂FP_m increases with the silicon wall height. The two sets of curves intersect at two types of crossing points: those where the 100% reflection covers a large area and those where the 100% reflection is very narrow. A wavelength shift upwards amounts to a scan towards the bottom left in the (h,Λ) diagram, thus the ultra-narrow supermode resonance is in the neighborhood of the crossing points of small 100% reflection area. In other words, every crossing point between Fabry-Perot curves of orders differing by 2 has an ultra-narrow-band characteristic. In Fig. 9 the TE₀FP₄ and TE₂FP₂ curves cross at 360 µm period and 210 µm height, which corresponds approximately to the technologically realized grating parameters and thus represents the observed narrow-band reflection at 445 µm wavelength.

Figure 10 represents the reflection coefficient mapping in the height-wavelength plane (h, λ) with the period 385 µm and the 0.45 line/period ratio of the experimental element. The wavelength domain of interest is limited at the short wavelength side by the propagation of the free space 1st orders at 385 µm wavelength and by the propagation of the TE₄ mode below 397 µm, and at the right hand side by the cutoff of the TE₂ mode at 680 µm. In analogy to Fig. 9, two sets of thin black lines of steeper (smaller) positive slope are depicted, representing the TE₂ (TE₀) Fabry-Perot resonance condition. This chart permits to identify the waveguide and Fabry-Perot mode orders which the two supermode resonances are made of, and to locate approximately where they occur in the chart: following the labeling of Fig. 9, the 585 µm resonance occurs at the intersection of the TE₀FP₃ and TE₂FP₁ curves.

 

Fig. 10 Reflection coefficient mapping versus grating height and wavelength with 385 µm period and 0.45 duty cycle, and identification of the two resonances observed experimentally.

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Referring to the two peaks at 445 and 580 µm wavelength of Fig. 7 and identified by the red circles in Fig. 10, it is clear now that in a structure optimized for ultra-narrow resonance at 580 µm wavelength, the same structure will be off the ultra-narrow resonance condition at 445 µm wavelength as the optimal groove depths slightly differ; this is what the transmission spectrum of Fig. 8 confirms. As stated above, the supermode resonances responsible for the ultra-narrow-band reflection are not exactly at the crossing points of the Fabry-Perot curves [16]; they are at their close neighborhood as indicated by the red dots within the circles of the wavelength-height chart. The dispersion equation giving the exact position of the supermode resonances was derived algebraically. The equation itself and its derivation are too heavy to be reported in the present conceptual and experimental paper. The supermodes SM_0j2l can now be labeled with j representing the Fabry-Perot order of the TE₀ grating-mode, and l that of the TE₂ mode.

In contrast to the segmented waveguide mode representation, the grating-mode description permits to account for the background reflection off-resonance, here about 20% as observed in Fig. 3. The off-resonance reflection is essentially given by the direct transmission through the two TE₀ and TE₂ mode channels since, without resonant enhancement, the contribution of crosscoupling between modes and multiple back-reflections is negligible. In the present structure the modal method reveals that the transmitted amplitudes via the TE₀ and TE₂ modes are 0.46 and 0.43 respectively just outside resonance with a phase difference of 2π. Thus this constructive interference leads to a transmitted intensity of 0.89² = 79%, confirming the measured 20% off-resonance reflection. This background reflection can be tuned to zero by adjusting the line/space ratio, which will result in a Lorentzian resonance shape varying between 0% and 100% reflection.

6. Conclusion

It was shown experimentally in the THz frequency domain that the ultra-narrow-band cancellation of the transmission of a spatially limited beam through a 1D segmented high index membrane does correspond to a close to 100% reflection. The design of the functional structure was initially made on the basis of a segmented waveguide mode representation explaining phenomenologically the ultra-narrow-band character as well as the wide angular spectrum of the resonance. The bridge was then made with a representation based on the grating-modes which reveals meaningful features of the supermode resonances such as their background reflection, their bandwidth character depending on their order, their approximate location in the space of the optogeometrical parameters, and permits to label them. It is shown that this narrow-band effect is closely related to the excitation of the two first even grating modes approximately satisfying the Fabry-Perot resonance condition for each grating-mode. The experimental results are obtained in the time domain which does not permit a high resolution in the frequency domain. However, the reflection peaks are clearly evidenced and are at the spectral position where they were theoretically expected. Their spectral width is between 5 and 10 µm, i.e., a relative width of 1% which cannot be qualified as “ultra-narrow”; however, considering that such spectral width is obtained with an incident beam of about 16 degree total angular aperture, this bandwidth is definitely very narrow, i.e., a factor of about 15 smaller than the spectral width expected from simple phase matched grating mode coupling. The bridging which the present paper makes between the waveguide grating coupling and the grating-mode representations is not complete yet: whereas the grating mode vision accounts for the ultra-narrow resonance, it does not give the clue for its large angular width. Further theoretical work is needed to explore the grating mode interplay under oblique incidence with the excitation of the first odd mode and its possible role in energy storage [11].

In order to benefit from the silicon microsystem fabrication infrastructure, the demonstration of the narrow band reflection effect with large angular width was made in the THz range. In this domain the effect is however interesting for itself as an ultra-narrow band filter with large angular acceptance, since the THz radiation is most often very wide band and emitted with large divergence by quasi-point sources. For the same reasons of wide angular and narrow spectral width this effect is also highly interesting beyond the THz domain, e.g. in the far infrared where many molecules have their fundamental vibration mode. It is also applicable to the near infrared where, for instance, its angular acceptance comparable with the NA of single-mode optical fibers permits to envisage its use as an ultra-narrow band reflection filter monolithically integrated at the fiber tip; this can give rise to a new class of fiber point sensors as well as to fiber laser mirrors of line-width comparable with long fiber Bragg gratings. Further on the horizon, the theoretically unlimited field concentration associated with the ultra-narrow band reflection is worth considering from the viewpoint of nonlinear effects.