A resonant diffraction grating comprising a mirror, a dielectric layer and a high index corrugation at the layer-air interface is shown to exhibit off-Littrow the record diffraction efficiency of 99% in the -1st reflected order at 1064 nm wavelength thanks to the excitation of a leaky mode of the layer. Such high figure is obtained by a grating 5 to 10 times shallower than in current attempts to realize high efficiency all-dielectric gratings.
©2005 Optical Society of America
Unlike their standard metallic counterparts, resonant all-dielectric gratings are capable of diffracting an incident free space wave with a diffraction efficiency of up to 100% theoretically [1–2]. They are composed of a dielectric film and of a surface corrugation or film index modulation. Since diffraction in resonant gratings involves the excitation of a waveguide mode, the efficiency possibly exhibits high polarization, angular and wavelength selectivity, which offers new functions and performances to the field of optical filtering.
There are mainly two types of resonant phenomena taking place in a dielectric slab associated with a periodical corrugation. The first type relies upon the excitation of a propagating mode of the dielectric layer used as a slab waveguide. Under the condition αD≫2π (D is the incident beam diameter, α is the radiation coefficient of the considered mode of the grating waveguide [3–4]), and in the neighbourhood of the waveguide mode synchronism condition, the corrugated waveguide exhibits close to 100% reflection of a finite beam in the direction of the zeroth order . This resonant effect , called anomalous reflection , or resonant reflection , can be understood phenomenologically as a destructive interference in the transmission medium between the wave crossing the waveguide grating without being diffracted (the zeroth transmitted order) and the wave trapped in the slab waveguide by +1st or −1st order coupling, and re-radiated out into the transmission medium.
The second type of resonant grating is the subject of the experimental result of the present paper. The mechanism is quite different and much less known in the scientific community although implicitly used sometimes as the result of numerical computing . No truly guided mode is involved. The field trapping is achieved here by the refractive excitation of a leaky mode of a mirror based layer [8–9]. The grating acts as a valve allowing the cancellation of the Fresnel reflection and thus imposing in principle 100% diffraction efficiency in the −1st order [2;10]. We are reporting here the record experimental value of more than 99% diffraction efficiency (a comparable result was recently reported  for a similar structure under the Littrow mounting which is known to always permit close to 100% diffraction efficiency ; the present result is obtained off-Littrow). A further achievement reported here is that the grating is made in the last high index layer of a multilayer stack which permits maximum efficiency to be obtained by 5 to 10 times shallower grating grooves.
2. Operation principle and structure design
Let an incident free space wave impinge under incidence angle θi onto a dielectric layer placed on a mirror. If the refraction angle θf in the layer of refractive index nf, the average layer thickness h and the wavelength λ satisfy the condition
the incident wave excites a leaky mode resonance of order m  in the mirror layer. The incident field is trapped in the layer with some rate of leakage into the cover medium, which can be controlled by the reflection coefficient at the film-cover interface. k0=2π/λ is the vacuum wave number, ϕm is the reflection phase shift at the mirror and ϕc is the reflection phase shift at the layer-cover interface in case of incidence from the layer. In the absence of grating there are two contributions to the Fresnel reflection : the wave firstly reflected at the layer surface, and the trapped leaky mode field re-radiating into the cover. The presence of a grating, as sketched in figure 1, modifies the balance between these two reflection contributions through its 0th order; furthermore, its −1st order represents an additional output port for the field trapped in the layer. The second contribution results from a resonance. Under the hypothesis that dispersion equation (1) is fulfilled, its phase is opposite to the phase of the first contribution. There is consequently the possibility of cancelling the overall reflection in the Fresnel reflection direction by rendering the modulus of the two contributions equal. If moreover the −1st diffraction order is the only one which can propagate, the light has nowhere else to propagate but to be diffracted with 100% efficiency (minus the metal loss if the mirror is metallic). This resonant diffraction effect is polarization selective since ϕm and ϕc are usually polarization dependent. It can be highly wavelength selective if the leaky mode resonance is sharp, i.e., if the incidence angle θi is large and if the mirror is essentially lossless.
The initial motivation for designing and fabricating such element was to replace one of the mirrors of an extended Sagnac laser cavity so as to filter a single longitudinal mode by exploiting the possibly strong grating dispersion, one narrow spectral line only satisfying the closed loop condition in the laser cavity of fixed geometry. An additional feature is the possibility of making the laser tunable by tilting the grating mirror. The light beam, of wavelength λ=1064 nm, is supposed to impinge onto the grating under the incidence angle of 73° and to diffract in the −1st order 70° off the normal with a grating period Λ of 560 nm. The requirement for such element to replace a laser mirror is obviously high diffraction efficiency. This was the first and necessary condition to satisfy and is consequently the subject of the present paper. Such a laser application is in contrast with the pioneering work of Perry et al.  on femtosecond pulse compression gratings and more recent works on similar high efficiency −1st order diffraction gratings  where the laser damage mechanism is related with electron discharges in the dielectric layers. This imposes the grating grooves to be etched into the last low index silica layer with the consequence that the grooves must be very deep: between 600 and 800 nm depth with a period of slightly more than 500 nm. Such gratings are very difficult to make. The damage mechanisms are more related with thermal effects in CW and in nano- and picosecond high power laser applications . In such applications, as well as in high pulse rate femtosecond laser machining applications, the grating can be etched into the high index layer with the consequence that a groove depth of less than 100 nm suffices.
The grating structure firstly comprises a multilayer mirror made of eighteen quarter wave layers for the prescribed incidence angle. The low index (nL) and high index (nH) layers are respectively made of SiO2 and Ta2O5 (nL=1.48 and nH=2.18 at λ=1064 nm in the case of ion plating layer deposition technology). The quarter wave layer thicknesses are and where θL and θH are the refracted angles which the incident wave makes in the low and high index quarter wave layers with respect to the normal. The multilayer mirror has a wide angular width; its role is primarily to act as the mirror for the leaky mode. It should however also reflect the −1st diffraction order of the grating propagating towards the substrate. The incident and diffracted beams being not far off angularly in the present case, a single periodic multilayer with wH=131 nm and wL=223 nm was used.
The leaky mode guiding corrugated layer lies on top of the multilayer mirror. It can be of low or of high index, or even of composite index provided the dispersion equation (1) or that given in ref.  is satisfied. The solution which was engineered in the present case was to combine a low index leaky mode guiding layer with a high index grating layer made of Ta2O5 as sketched in the inset of Fig. 1. This represents the very beneficial advantage of achieving a high efficiency grating with quite shallow grooves (hardly some tens of nanometers deep against several hundreds of nanometers if the leaky mode guiding layer was a unique silica layer). The reflection phase shift terms ϕs and ϕc in the leaky mode dispersion equation (1) applied to the leaky mode guiding layer for the TE polarization are in the present case ϕs=π, because the first layer of the mirror is a high index layer, and ϕc=0. Equation (1) for the fundamental mode (m=1) writes
which means that the leaky mode guiding layer must be a quarter wave layer. The fundamental leaky mode exhibits zero field amplitude at the multilayer mirror surface, but a field maximum at the air boundary where the corrugation is made.
However, the leaky mode guiding layer is here composed of two sub-layers: the low index layer (152 nm) and the average thickness of the high index corrugation. The latter consists of 50 nm deep quasi-rectangular grooves of close to 50/50 line/space ratio etched into a Ta2O5 layer of initial thickness of 70 nm; this is equivalent for the TE polarization to a uniform Ta2O5 layer of 20+25=45 nm thickness. The sum of the two contributions is equivalent to a single low index layer of 230 nm which is close to the thickness of a quarter wave layer at 1064 nm wavelength and under 73° incidence.
3. Grating technology and experimental results
The multilayer was deposited by ion plating (Tafelmaier, Rosenheim, Germany) onto a 6.35 mm thick quartz wafer of 50 mm diameter. The grating was firstly created by exposing the resist coated multilayer mirror to an interferogramme at 442 nm wavelength, then transferred into the last high index layer by ion beam etching. As illustrated in Fig. 2 an AFM scan of the resulting corrugation shows a quasi-rectangular profile with a line/space ratio of 0.85 and a groove depth of 52 nm meaning that a 18 nm thick Ta2O5 layer remains under the grating. The theoretical diffraction efficiency in the −1st order of such a grating structure calculated at 1064 nm wavelength under an incidence angle of 73 ° is more than 99.7 %. Under these incidence and diffraction conditions the only propagating non-zero diffraction order is the −1st order. A microchip Nd:YAG laser was used to measure the angular spectrum of the −1st order diffraction of the device at 1064 nm wavelength.
Figure 3 compares the diffraction efficiencies measured and calculated in the −1st order at 1064 nm wavelength for the TE polarization upon a scan of the incidence angle θi in air at either side of the Littrow angle. The theoretical values were calculated by taking into account a groove depth of 52 nm and a line space ratio of 0.85. The experimental values were obtained at a location on the wafer showing the highest diffraction efficiency of more than 99 %. The measurement set up comprises two crossed translation stages for scanning the whole 50 mm diameter grating area, and a rotation stage. The latter controls the incidence angle of the collimated beam to within 0.02°. The incident power is 105 mW. The detector (Oriel 70263) follows the angular shift of the diffracted beam as a function of the incidence angle and measures the power to within 0.05 mW. The diffraction efficiency is defined as the ratio between the diffracted power and the incident power. The measurement were made without simultaneous referencing; however, the stability of the laser was tested to be within 1% during the experiment. The shape of the experimental curve is very similar to that of the theoretical curve, except a light shift in wavelength visible around 66° and under grazing incidence, and a slight sagging in the vicinity of the Littrow angle of 71.8 degrees (the Littrow condition could not be measured in this preliminary experiment). The diffraction efficiency is no more than 97% at the incidence angle of 73.4° (the first one measured past the Littrow angle). Nevertheless, several points located between 66.2° and 68.2° incidence reach values greater than 99%. The maximum diffraction efficiency is obtained under the incidence angle of 66.8 degrees i.e. a diffraction angle of 78.8 degrees. These small differences with the theoretical results observed across the grating wafer are due to small variations in the depth and/or the line space ratio of the grating. The influence of these geometrical variations on the diffraction efficiency is due to the alteration of the leaky mode resonance condition and the grating strength : a variation of depth and of line/space ratio changes the equivalent thickness h of the leaky mode propagating layer, leading to a non optimum field trapping. These variations also lead to a variation of the grating strength. The diffraction efficiency varies between more than 99% and 92 % efficiency all over the 50 mm diameter grating area. The relationship between the grating profile features across the grating wafer and the related diffraction efficiency distribution will be discussed in an extended paper.
The curves of Fig. 4 illustrate the diffraction efficiency of the TM polarization; as expected, the efficiency is practically zero because the TM leaky mode excitation condition is not fulfilled. Moreover, the multilayer is no perfect reflector for the TM orders and part of the incident power is transmitted in the 0th and −1st orders of the substrate. This explains why the sum of the two theoretical curves is not unity.
It was shown that an all-dielectric reflection grating can reach diffraction efficiencies above 99%, i.e., above the reflection coefficient of the best metal mirrors, and close to that of multilayer dielectric mirrors. This means that such element can now be envisaged for the low loss performance of a complex optical function usually requiring at least one standard mirror and one or more discrete optical elements such as a Lyot filter.
The phenomenological understanding proposed for this type of resonant grating has been explained and was shown to easily lead to a design, which is naturally capable of best satisfying the specifications under given constraints prior to any numerical modelling effort. It was shown in particular that such high efficiency can be obtained by the moderate technological effort of etching a few tens of nanometers deep grooves, i.e., between 5 and 10 times less than in the best known endeavours. The applicability of such resonant gratings goes well beyond the present application, still to be demonstrated, as a highly dispersive mirror. Systematic results concerning the grating uniformity, attempts of applying the technique of scatterometry for contactless, high resolution testing, as well as laser damage threshold investigation on this second type of resonant gratings will be reported in an extended paper.
The R&D as well as the applicative endeavours on resonant gratings are now performed within the Network of Excellence NEMO.
References and links
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