Reflective cavity couplers based on resonant waveguide gratings

doi:10.1364/OE.19.016466

Reflective cavity couplers based on resonant waveguide gratings

Stefanie Kroker, Thomas Käsebier, Frank Brückner, Frank Fuchs, Ernst-Bernhard Kley, and Andreas Tünnermann »View Author Affiliations

Optics Express, Vol. 19, Issue 17, pp. 16466-16479 (2011)
http://dx.doi.org/10.1364/OE.19.016466

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Abstract

We report on a novel concept for reflective diffractive cavity couplers based on resonant waveguide gratings instead of multilayer coatings. The diffracting or rather beam splitting properties are induced to the subwavelength structures by a periodic parameter modulation of the ridges. Since such a perturbation of the highly reflective system also enhances transmission stacks of two and three reflectors are considered to retrieve transmittivities as low as possible. Our calculations show that transmissions of less than 10⁻⁴ are possible for different configurations based on silicon and silica. The results of first technological tests for the realization of stacked T-shape structures are presented. With a total effective layer thickness not exceeding 1.1 μm the discussed approaches are expected to remarkably reduce the urgent problem of coating thermal noise of conventional components for high-precision metrology.

1. Introduction

Thermal noise is one of the most important limitations for experiments in the field of optical high-precision metrology such as the realization of frequency stabilized lasers or the detection of gravitational waves (GW). Usually these GW detectors are based on Michelson interferometers with additional arm cavity mirrors for high light-power built-ups [1

P. Fritschel, D. Shoemaker, and R. Weiss, “Demonstration of light recycling in a Michelson interferometer with Fabry-Perot cavities,” Appl. Opt. 31, 1412–1418 (1992). [CrossRef] [PubMed]

, 2

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rüdiger, W. Winkler, and K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational-wave detectors,” Phys. Lett. A 175, 273–276 (1993). [CrossRef]

]. During the last years large efforts have been made to improve their sensitivity. For example, all-reflective optical components as substitutes for transmissive beam splitters or cavity couplers were investigated [3

R. W. P. Drever, “Concepts for extending the ultimate sensitivity of interferometric gravitational gravitational-wave detectors using non-transmissive optics with diffractive or holographic coupling,” in Proceedings of the seventh Marcel Grossman meeting on general relativity , M. Keiser and R. T. Jantzen (eds.), World Scientific Singapore (1995).

–7

M. Britzger, D. Friedrich, S. Kroker, F. Brückner, O. Burmeister, E.-B. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractively coupled Fabry-Perot resonator with power-recycling,” Opt. Express 19, 14964–14975 (2011). [CrossRef] [PubMed]

] to avoid any effects of a potential optical substrate absorption, such as thermal lensing losses or thermo-refractive noise, as well [8

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational-wave detectors,” Phys. Lett. A 194, 124–132 (1994). [CrossRef]

, 9

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational-wave antennae,” Phys. Lett. A 271, 303–307 (2000). [CrossRef]

]. Here especially the coupling via a low-efficient three-port grating is promising. Another issue is that all conventional reflective optics based on multiple beam interference require stacks of about 40 dielectric layers making a large contribution to Brownian coating thermal noise, since this noise type scales linearly with the effective layer thickness [10

Y. Levin, “Internal thermal noise for LIGO test masses: a direct approach,” Phys. Rev. D 57, 659–663 (1998). [CrossRef]

, 11

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002). [CrossRef]

]. Therefore alternative concepts are necessary. One way to avoid the multilayer stacks in order to reduce the layer thickness up to one order of magnitude is to use the phenomenon of resonant light coupling in resonant waveguide gratings (RWG) [12

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985). [CrossRef]

, 13

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006). [CrossRef]

]. So far RWGs have only been used for transmissive cavity couplers and normally incident light.

In this paper we propose to combine both independent approaches of RWGs and reflective cavity couplers with prospects to an all-reflective as well as coating reduced interferometry. This has, to the best of our knowledge, never been considered before. Instead of a very recent work about reflective blazed gratings [14

Z. Peng, D. A. Fattal, A. Faraon, M. Fiorentino, J. Li, and R. G. Beausoleil, “Reflective silicon binary diffraction grating for visible wavelengths,” Opt. Lett. 36, 1515–1517 (2011). [CrossRef] [PubMed]

] based on resonant silicon structures for the visible range here it is particularly investigated how defined diffraction efficiencies can be realized with a total transmission kept significantly below 1%. The focus is on T-shape structures, because the undercut helps to realize a large index contrast between the involved materials being useful for huge angular and spectral bandwidths as well as a good feasibility of ultra-high reflectivities.

In the following, first we discuss some basic aspects to realize this combination. Afterwards we present restrictions for the possible grating parameters to alleviate the rigorous design process before we demonstrate concrete configurations for cavity couplers. For the setups also a discussion of the tolerance with respect to deviations of the grating parameters from the designed values is given. At the end we demonstrate a technology which is suitable for the fabrication of such structures. We consider the material combination (amorphous) silicon (n = 3.8) and silica (n = 1.48) on crystalline silicon. But, the discussion holds also for other material combinations and even for a monolithic implementation made of silicon. Due to its excellent mechanical properties at cryogenic temperatures it is a very promising material for future GW detectors which are planned to work at low temperatures [15

R. Nawrodt, A. Zimmer, T. Koettig, C. Schwarz, D. Heinert, M. Hudl, R. Neubert, M. Thürk, S. Nietzsche, W. Vodel, P. Seidel, and A. Tünnermann, “High mechanical Q-factor measurements on silicon bulk samples,” J. Phys. Conf. Ser. 122, 1–9 (2008). [CrossRef]

, 16

X. Liu, C. L. Spiel, R. D. Merithew, R. O. Pohl, B. P. Nelson, Q. Wang, and R. S. Crandall, “Internal friction of amorphous and nanocrystalline silicon at low temperatures,” Mat. Sci. Eng. A 442, 307–313 (2006) [CrossRef]

]. In the following the term beam splitter is used as a synonym for cavity coupler.

2. The Way to Cavity Couplers Based on Resonant Waveguide Gratings

For interferometric high-precision metrology highly reflective optical components are essential as the end mirrors of the interferometer arms. To replace transmissive cavity couplers (Fig. 1(a)) different types of reflection gratings have been extensively investigated. In doing so, highly and low efficient reflection gratings have been realized which are used as two-port and three-port cavity couplers, respectively (see Fig. 1(b) and 1(c)) [5

A. Bunkowski, O. Burmeister, T. Clausnitzer, E.-B. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Optical characterization of ultrahigh diffraction efficiency gratings,” Appl. Opt. 45, 5795–5799 (2006). [CrossRef] [PubMed]

–7

]. Because of their better technological feasibility the low-efficient gratings are in the focus of the present paper. They consist of a diffraction grating on top of a conventional highly-reflective multilayer coating, where the layer stack has to be designed such that the total transmission T = Σ _i T_i is kept as low as possible. We now want to discuss how the use of this stack can be avoided in order to reduce the effective layer thickness and with it coating thermal noise. Just recently, it has been demonstrated that the cavity mirrors or transmissive cavity couplers can not only be realized by multilayer based systems but also by resonant waveguide gratings which comprise a structured high-index film on top of a low index layer or substrate as depicted in Fig. 2(a). Also T-shaped RWGs (see Fig. 2(b)) where the effective index of the sublayer is controlled via its duty cycle are able to provide high reflectivities [17

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Monolithic dielectric surfaces as new low-loss light-matter interfaces,” Opt. Lett. 33, 264–266 (2008). [CrossRef] [PubMed]

, 18

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and Roman Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104, 163903 (2010). [CrossRef] [PubMed]

]. As a fundamental requirement for reflectivities close to unity the period of such a grating needs to be smaller than the wavelength of the incident light to suppress any additional diffraction order in reflection. These subwavelength structures which act as a mirror can be optimized for a certain angle of incidence as shown in Fig. 1(d) (solid and dashed line) but usually high reflectivities are obtained only in a quite narrow angular or wavelength band [19

D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. 18, 2109–21120 (2001). [CrossRef]

, 20

F. Brückner, S. Kroker, D. Friedrich, E.-B. Kley, and A. Tünnermann, “Widely tunable monolithic narrowband grating filter for near-infrared radiation,” Opt. Lett. 36, 436–438 (2011). [CrossRef] [PubMed]

]. To combine now the concept of RWGs with reflective cavity couplers for a coating reduced component at least one additional port (diffraction order) for the defined coupling into the cavity is necessary. Hence, we partially need to give up the subwavelength condition for the period. This can be done by a periodic modulation of the highly reflective waveguide grating structure. According to the grating equation for a three-port coupler the period of this perturbation in dependence of the illuminating wavelength (for normal incidence) is:

λ \leq p_{B S} < 2 λ,

(1)

to allow the propagation of exactly three diffraction orders and hence outgoing ports of the grating. Such a diffracting period can only be superposed as an integer multiple of the RWG period p, since a grating ridge can either be manipulated or not. Two possibilities to combine the double-T configuration with diffracting properties are to modulate the depth of the upper grating layer(s) or to induce the perturbation with periodically changing the duty cycle of the grating ridges as shown in Figs. 3(a) and 3(b). For the three-port grating the coupling finally is realized via the first diffraction orders in reflection R _±1 (see Fig. 1(c)) [6

A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. Tünnermann, “Low-loss grating for coupling to a high-finesse cavity,” Opt. Lett. 29, 2342–2344 (2004). [CrossRef] [PubMed]

]. The reflectivity of the component at normal incidence determines the finesse of the cavity, while the reflectivities off normal incidence will affect the total transmission of the beam splitter. Recently, we have shown that also with large index-contrast RWGs it is not possible to simultaneously realize reflectivities at normal incidence and at large angles [21

S. Kroker, F. Brückner, E.-B. Kley, and A. Tünnermann, “Enhanced angular tolerance of resonant waveguide grating reflectors,” Opt. Lett. 36, 537–539 (2010). [CrossRef]

] (see also solid curve in Fig. 1(d)). This implies that the multilayer stack of the conventional beam splitter gratings cannot simply be replaced by a single modulated waveguide grating, because the angular tolerance of the RWG is not large enough. However, one can significantly enhance the angular bandwidth by stacking two T-shaped RWGs on top of each other as proposed in [21

S. Kroker, F. Brückner, E.-B. Kley, and A. Tünnermann, “Enhanced angular tolerance of resonant waveguide grating reflectors,” Opt. Lett. 36, 537–539 (2010). [CrossRef]

] and plotted in Fig. 1(d) (dotted line). With this approach it should be possible to ensure a low total transmission of an all-reflective beam splitter grating.

Fig. 1 (Color online) (a) Sketch of a transmissively coupled cavity. Reflective cavity coupling by means of two-port (b) and a three-port grating coupler (c). The total transmission is the sum of all transmitted diffraction orders and should be minimized. (d) Example for the dependence of the reflectivity for a T-shaped RWG based on silicon and silica optimized for normal incidence and an angle of 45° and angular behavior of the reflectivity for a double T-shaped grating.

Fig. 2 (a) Conventional configuration for a resonant waveguide grating. (b) T-shaped grating.

Fig. 3 (a) Sketch of a depth modulated grating and (b) of a width modulated grating. The modulation also can be realized continuously with each ridge.

With an additional third layer of T-shaped RWGs it is possible to reduce the amount of transmitted light even more, because beside to the enhanced angular tolerance on the one hand one can e.g. decouple the perturbation of the upper layers from the rest of the grating. Already a double T-shaped structure exposes nine free parameters (four grating depths, four duty cycles, grating period). Therefore it is meaningful to reduce the range of relevant parameters by means of some theoretical considerations to alleviate the final design process.

3. Restriction of the Grating Parameter Ranges by Means of the Simplified Modal Theory

For the grating period p of a RWG as depicted in Fig. 2(a), the following condition has to be obeyed [13

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006). [CrossRef]

\frac{λ}{n_{h}} \leq p \leq \frac{λ}{n_{l}},

(2)

where n_h , n_l are the refractive indices of the high- and low-index regions, respectively. This condition guarantees that in the low-index layer only the zeroth diffraction order can propagate while in the high-index region also the first diffraction orders are permitted. If the geometrical grating parameters such as grating depth and duty cycle are chosen properly it is possible to retrieve constructive interference of the zeroth diffraction order into the back direction of the incoming light [22

A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. 14, 2985–2993 (1997). [CrossRef]

]. Basically, the low-index sublayer inhibits a coupling of the first evanescent diffraction orders into the substrate where their propagation might be possible. An intelligible explanation for the functionality of RWGs can also be given by the help of discrete modes propagating vertically up and down the grating. They were introduced by Botten et al. [23

I. C. Botten, M. S. Craig, R. C. McPhredan, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Act. 122, 413–428 (1981). [CrossRef]

] and can be understood as an analogon to modes in a slab waveguide. The effective indices n_eff of the grating modes (which correspond to the propagation constants in z-direction) can be calculated for TM-polarized (magnetic field vector parallel to y-axis) light with the following eigenvalue equation:

cos (β b) cos (γ c) - \frac{1}{2} (\frac{n_{2}^{2} β}{n_{1}^{2} γ} + \frac{n_{1}^{2} γ}{n_{2}^{2} β}) sin (β b) sin (γ c) = cos (\frac{2 π}{λ} sin φ p),

(3)

where

\begin{array}{l} β & = & \frac{2 π}{λ} \sqrt{n_{1}^{2} - n_{eff}^{2}} and, \\ γ & = & \frac{2 π}{λ} \sqrt{n_{2}^{2} - n_{eff}^{2}} \end{array}

(4)

are the x-components of the wavevectors in the ridge and groove, respectively. The geometrical grating and illumination parameters can be seen in Fig. 4(a). Those modes with

n_{e f f}^{2} > 0

are called propagating and those with

n_{eff}^{2} < 0

evanescent grating modes. For an optimum performance with large tolerances of the parameter design values it is necessary to excite exactly two grating modes being able to propagate [24

V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18, 16973–16988 (2010). [CrossRef] [PubMed]

]. In the case of normally incident plane wave this means the excitation of two symmetric ones. Figure 4(b) shows an example of the field distribution for two modes excited by TM-polarized light at normal incidence. Due to their different effective indices

n_{eff}^{0}

and

n_{eff}^{1}

they pick up a phase difference on their way to the bottom of the grating (see also Fig. 4(c)). The transmission of the zeroth diffraction order which is the only one supported by the silica sublayer can be completely suppressed, if the phase difference is an odd multiple of π. Basically, this fact holds also if we replace the film of SiO₂ by a grating as proposed in [17

]. But one has to take into account that with decreasing duty cycle f = b/p of the subgrating its effective properties will approach the ones of air (n_l → 1). Hence, according to Eq. (2) for a wavelength of 1550 nm the relevant grating periods would then range from 408 nm up to 1550 nm. Furthermore, it has to be assured, that in the silica grating only the fundamental mode is permitted in order that there is no additional phase difference between the modes in the subgrating. Hence, for the duty cycles of both gratings exists an upper and a lower limit. The upper boundary is reached if there are either two modes excited in the silica or three in the silicon grating. The lower one is set by the cut-off period of the second symmetric mode in the silicon grating. From Fig. 5 one can extract that due to the high index contrast between the materials for duty cycles f larger than 0.5 the limitation for the period is set by the excitation of the third symmetric grating mode in the silicon layer. We now consider a minimum duty cycle of 0.3 for the silica and a range of f = 0.5...0.7 for the silicon grating. This ensures a large contrast between the two gratings which provides a broadband reflection with respect to angle and wavelength [17

], and further it facilitates the technological realization. These duty cycles limit the period range to 435 nm< p <1246 nm. For f = 0.6 of the silicon grating this range reduces even more to 435 nm< p <1068 nm, which is about by a factor of two smaller than that marked by the dashed horizontal lines in Fig. 5. Furthermore, since close to the upper limit coupling of the incident light into evanescent grating modes will increase the optimal period is expected to be at the lower end of the given range.

Fig. 4 (Color online) (a) Geometrical grating parameters. (b) An example for the lateral field distribution of the magnetic field component H_y component for the first two grating modes excited by TM-polarized light at normal incidence. The grey bars represent the silicon grating ridges. (c) Illustration of the first two symmetric modes propagating through the grating. A phase difference of π between the modes at the bottom of the grating leads to minimum transmission of the zeroth diffraction order.

Fig. 5 (Color online) Lower (black) and upper (blue and red) limits for the range of relevant grating periods in dependence of the duty cycles of the involved silicon and silica grating. The horizontal (dotted) grey lines mark the periods for the effective indices 1 and 3.8, respectively. Note that for the T-shape gratings the duty cycle of the silica grating is chosen lower than that of the silicon grating.

With the basic knowledge about grating modes it is also possible to set a limit on the range the thickness range of the silicon grating. As already mentioned, for high reflectivity, a phase difference of π between the two modes at the bottom of the grating is necessary, which corresponds to an optimal grating depth s:

s = \frac{λ}{2 (n_{eff}^{0} - n_{eff}^{1})} (2 m + 1), m \in ℕ .

(5)

The effective indices calculated from Eq. (3) are shown in Table 1. For the duty cycle of 0.7 the index contrast

n_{eff}^{0} - n_{eff}^{1}

covers a larger range than for 0.5. Thus, we can estimate the maximum range of the lowest relevant grating depths s (m = 0) to:

\frac{1550 nm}{2 (3.62 - 0)} < s < \frac{1550 nm}{2 (3.62 - 1.79)}

(6)

214 nm < s < 430 nm .

(7)

Table 1 Effective Indices for the Relevant Parameters of the Silicon Grating for TM-Polarization at Normal Incidence

Duty cycle	Lower limit	Upper limit
0.5	$n_{eff}^{0} = 2.57$	$n_{eff}^{0} = 3.6$
0.5	$n_{eff}^{1} = 0$	$n_{eff}^{1} = 1.43$

0.7	$n_{eff}^{0} = 2.95$	$n_{eff}^{0} = 3.62$
0.7	$n_{eff}^{1} = 0$	$n_{eff}^{1} = 1.79$

^* The lower limits of the periods for f = 0.5 and f = 0.7 are 519 nm and 435 nm. The upper limits are 1246 nm and 946 nm, respectively.

In order to achieve a good performance of the RWG in the entire angular spectrum also odd grating modes have to be taking into account. In this case the period range reduces to 519 nm < p < 756 nm (for f_Si = 0.5...0.7) and thickness range is 214 nm < s < 790 nm since with increasing angle of incidence the contrast between

n_{eff}^{0}

and

n_{eff}^{1}

becomes smaller. The final rigorous design of the structure is done by means of Rigorous Coupled Wave Analysis [25

M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave approach for planar diffraction gratings,” J. Opt. Soc. Am. 71, 811–818 (1981). [CrossRef]

]. For an exemplary single T-shaped waveguide grating we find the parameters shown in Fig. 6. Here, the thickness of the silica grating is determined as the minimum depth where the total transmission, which is the sum over all transmitted diffraction orders in the substrate, is in the range of 10⁻⁶. This design is our starting point for the optimization of the beam splitters based on RWGs.

Fig. 6 (Color online) Optimum geometric parameters for the grating illuminated at normal incidence. The silicon grating is made of amorphous silicon (n=3.8 at λ =1550 nm) whilst the substrate material is crystalline silicon (n=3.48).

4. Cavity Couplers Based on Resonant Waveguide Gratings

In the following we show how one can add diffracting properties to the highly reflective structures. For a three-port cavity coupler as depicted in Fig. 1(c) the period has to fulfill the condition λ < p_BS < 2λ, i.e. p_BS is larger than 1550 nm. This period has to be superposed onto the subwavelength T-structures. As already discussed the periodicity needs to be an integer multiple of the waveguide grating period. With the maximum relevant range for p from Sec. 3, we find p_BS = 3...7p. Generally, a certain diffraction efficiency can easier be realized with a small period than with larger ones if, e.g. a certain grating depth is assumed. That is why a variation of each third ridge is favorable.

We now first want to discuss the possibilities for beam splitters made by depth modulation. We aim for reflectivities R ₀ of 0.9995 and 0.9 to show that our approach applies for cavity couplers with finesses larger than 10⁴ but also for couplers with rather low finesses. For the design process we restrict ourselves to the optimization of the grating thickness and the grating period to show the possibilities of our approach. Of course, these disregarded additional degrees of freedom can lead to a further improvement of performance, however for the mechanical stability of the stacked structures it seems to be beneficial maintain the duty cycles of the primarily single-T design shown in Fig. 6.

4.1. Cavity Couplers by Depth Modulation

To realize a high finesse cavity coupler only a weak perturbation of the mirror is permitted. As an example we aim for a reflectivity R ₀ of 0.9995 and a total transmission T smaller than 10⁻⁴. By means of a nonlinear optimization algorithm [26

http://ab-initio.mit.edu/wiki/index.php/NLopt.

] we retrieve the grating parameters shown in Fig. 7(a), a R ₀ of the desired 0.9995 and T = 5 × 10⁻⁶. Only 0.0005 of the light in the cavity couples out into the first diffraction orders resulting in a rather high finesse of about 12600. For technological reasons the depth modulation was not chosen too small, although it is possible to obtain comparable diffraction efficiencies already with smaller perturbations. The grating parameters we found with the optimization are all in the ranges we estimated with the help of the modal model discussed in the previous section. The total effective thickness h_eff , which is the sum of all layer thicknesses multiplied by the corresponding duty cycle, of this modulated double-T configuration accounts for 725 nm. This is almost one order of magnitude smaller than the conventional HR-stack based grating couplers made of silica and tantala layers, which would have a thickness of about 5 μm for the design wavelength of 1550 nm.

Fig. 7 (Color online) (a) Optimized grating parameters for a high-finesse cavity coupler with R ₀ = 0.9995 and a total transmission of 5 × 10⁻⁶. (b) Optimized parameter set for R ₀ = 0.9 and T = 10⁻⁵. For both cases the duty cycles with respect to p of the design in Fig. 6 were maintained. The sketches are not true to scale.

For diffraction efficiencies in the range of several percent, the perturbation of the RWG would have to be so large that it would lead to an inacceptable increase of transmission. If such efficiencies are desired, a triple-T configuration can be used, instead. Here, one can enhance the depth modulation on the upper layer to reach higher diffraction efficiencies, which in the extreme case results in an incomplete upper silicon layer, as it is depicted in Fig. 7(b). In this case we optimized the grating depths and the period to R ₀ = 0.9 and a total transmission of 10⁻⁵, which corresponds to a efficiency of 0.05 into the first diffraction orders. The total effective layer thickness is with about 910 nm still far below that of conventional reflection gratings. By removing the upper silicon layer one can see that the functionality of the beam splitter indeed corresponds to our idea of inducing a perturbation into our highly reflective RWG: The calculated reflectivity of the remaining structure is unity at normal incidence. For some experimental setups eventually other diffraction angles might be more convenient. In this case, with the additional variation of the duty cycles, the optimum period and hence also the diffraction angles can be tuned. However, these supplementary free parameters do not provide the possibility for double-T depth modulated gratings with significantly higher diffraction efficiencies. For example with R ₀ = 0.9, i.e. diffraction efficiencies of 0.05, we do not find transmissions smaller than 0.004. From Figs. 7(a) and 7(b) we can extract that the thicknesses of the silicon gratings correspond pretty well to the range determined in Sec. 3. Since the incident wavefront is disturbed by the depth modulation in the upper layer we need to take into account the extended range of the thicknesses for the entire angular spectrum.

4.2. Cavity Couplers by Width Modulation

Another way to superpose the period of the beam splitter grating to the highly reflective RWG is to modulate the width of the grating ridges periodically. For an optimization, in addition to the variation of the grating period and the layer thicknesses also the duty cycles (and therefore the ridge widths) need to be adopted. With respect to a later technological feasibility we now restrict the duty cycles of the silicon gratings to f < 0.22 and those of the silica layers to f > 0.05 which are now defined with respect to p_BS . The first condition ensures a good possibility to etch large grating depths while the second should guarantee a sufficient mechanical stability of the structures. Since the silica grating will be etched isotropic the difference between both duty cycles f_Si – f_SiO ₂ must be kept constant for all ridges (see Sec. 6). Again we choose the period of the beam splitter p_BS three times the period of the RWG. For a good comparability we again optimized the structures to reflectivities R ₀ of 0.9995 and 0.9. The design profiles we obtained are illustrated in Figs. 8(a) and 8(b). For both diffraction efficiencies it is possible to find configurations with T < 10⁻⁴. For the high-finesse cavity couplers very small width modulations (f ₄ – f ₃)p_BS of only 15 nm are required while configuration with R ₀ 0.9 requires a difference between the ridge widths of about 100 nm because larger diffraction efficiencies have to be realized. The overview of the effective layer thicknesses of the width modulated beam splitters and the depth modulated ones are displayed in Table 2. It is obvious that with regard to the effective layer thicknesses, which scales linearly with coating thermal noise, for the high-finesse configuration the depth modulation is favorable. For the low-finesse cavity coupler both investigated approaches expose almost the same thickness. All presented designs still have an effective layer thickness which is far below that of a multilayer based beam splitter or cavity coupler grating. To finally evaluate the practicability of the structures it is important to figure out their tolerances with respect to fabricational deviations of the grating parameters from the designed values.

Fig. 8 (Color online) (a) Optimized grating parameters for a high-finesse cavity coupler with R ₀ = 0.9995 and a total transmission of 1 × 10⁻⁵. (b) Optimized parameter set for R ₀ = 0.9 and T = 7 × 10⁻⁵. Again, the sketches are not true to scale. A restriction for the duty cycles due to fabrication is that f ₃ – f ₁ = f ₄ – f ₁ (see also Sec. 6). The labeled duty cycles are in relation to the period of the beam splitter p_BS .

Table 2 Effective Layer Thicknesses h_eff for Beam Splitter Designs with Different Reflectivities R ₀

number T-shape layers modulation	2 depth	2 width	3 depth
R ₀ = 0.9995	h_eff = 725nm	h_eff = 1086nm	-
R ₀ = 0.9	-	h_eff = 892nm	h_eff = 910nm

5. Some Remarks on Parameter Tolerances

The structures discussed in the previous section expose a large number of adjustable parameters that can be changed. Therefore, it is hardly possible to set up a complete error budget containing all dimensions of the grating that can deviate from the design parameters. To ensure a good comparability we discuss the influence of deviations for the high-finesse configurations with depth and width modulation. As limits we consider the tolerances for R ₀ > 0.999 and a maximum total transmission of 5 × 10⁻⁴. These values still allow for finesses of 6300 and it is assured, that the intensity in the reflected first diffraction orders is not smaller than the total amount of transmitted light. Generally, it is more critical to meet the parameters of the two upper gratings than those of the lower ones, because for all considered configurations the upper waveguide gratings are mainly responsible for the high reflection of the device. For the depth of the upper silicon layers we find a range of 4 nm for the width modulated configuration and 20 nm for initial upper silicon layer of the beam splitter with depth modulation. In Figs. 9(a) and 9(b) the influence of the deviations from the designed duty cycle on the performance of the depth and width modulated beam splitters are illustrated. For the calculation the same deviation was added to all duty cycles of the design. It is obvious that compared to the depth modulated coupler the width modulated one facilitates almost double the tolerances to deviations of the duty cycles. In terms of ridge widths this corresponds to tolerances of 50 nm and 80 nm, respectively. Thus, for the discussed gratings the width modulated configurations should be favorable if the layer thicknesses are adjusted properly. However, it cannot be excluded that by means of further optimization also larger duty cycle tolerances for the depth modulated couplers can be found. The triple-T configuration (see Fig. 7(b)) exposes width deviations of about 200 nm which are still acceptable to provide transmissions lower than 5 × 10⁻⁴. This is by far more than the approximate 20 nm for the double-T gratings we discussed before. It can be understood by the fact that the lower gratings are almost perfectly decoupled from the perturbation. At normal incidence the grating stack without the very upper silicon layer has a reflectivity R ₀ of 100% with a large angular tolerance of more than 90% reflectivity up to an angle of incidence of 69° which is why it is able to efficiently suppress transmission. For all discussed configurations the exact depths of the silica layers are not a critical value because only a certain threshold has to be exceeded to suppress the transmission of the higher diffraction orders into the substrate.

Fig. 9 (Color online) Influence of the grating duty cycle variations on the total transmission T (a) and the reflectivity R ₀ (b). The same deviations were added to all duty cycles of the design. Negative deviations represent duty cycles that are larger than the design values, positive deviations smaller ones.

6. Technological Realization of Stacked T-Shape Gratings

We illustrate the realization of a stacked RWG with the example of a double-T configuration. Basically, the process applies also to triple-Ts. For the fabrication of the structure, a silicon (100) wafer was first coated with a silicon-silica-silicon-silica layer system (see Fig. 10(a)). The thicknesses of the deposited thin films are 300 nm, 530 nm, 370 nm, and 520 nm being well within the range of the thicknesses needed for the actual cavity couplers. Afterwards the sample is prepared for structuring with an additional chromium and resist coating. The grating with a period of 640 nm and a duty cycle of 0.58 is defined by the use of electron beam lithography for an area of 9 × 9mm². These parameters are chosen for the first tests as intermediate values in the ranges we expect for the cavity couplers.

Fig. 10 (Color online) (a)–(d) Illustration of the fabrication process. (e) Scanning electron microscope (SEM) image of the fabricated double-T structures.

For the structurization after the development of the samples Inductively-Coupled-Plasma (ICP) etching processes are performed to completely etch through the thin film system consisting of the chromium, silicon and silica layers as depicted in Figs. 10(b) and 10(c). Afterwards wet etching by means of buffered hydrofluoric acid (HF) is done to selectively decrease the duty cycle of the silica subgratings against the silicon gratings (Fig. 10(d)). This isotropic etching process implies the need to keep the difference between the duty cycles of the silicon and silica grating constant for the width modulated beam splitter (see Sec. 4.2). From Fig. 10(e) it can be extracted that the widths of the ridges in the upper two layers are smaller than in the lower ones. This is due to a stronger exposure of this upper region to the etching plasma and can be taken into account for the design of the gratings. These differences and also the deviations from the rectangular grating profiles can still be minimized within an optimization process. The increase of the duty cycle of the silica grating at the bottom of the structure is caused by a limited material exchange during wet etching. It can be avoided by diluting the hydrofluoric acid and therewith retarding the etching process. The patterning was performed as a proof of principle before having the final design parameters. Hence, high reflectivities were not expected. Even under these conditions we measured a maximal reflectivity of 0.89 at a wavelength of 1570 nm and normal incidence. With larger duty cycles and slightly larger film thicknesses it is possible to drastically improve the performance of the reflectors at the design wavelength of 1550 nm. To the best of our knowledge, this is the first time the fabrication of such a waveguide grating configuration based on silicon and silica is presented. The depth modulation for the cavity couplers can be implemented by a pre-structurization of the surface by means of e-beam lithography and ICP etching. Afterwards a new layer of resist has to be deposited on the sample and the structurization of the stacked T-shaped gratings can be performed. For the width modulated gratings, no additional fabrication step is needed since the modulation can just be implemented within the electron beam patterning. Apart from that, the technological procedure remains the same.

7. Conclusion

We presented concepts for diffractive all-reflective beam splitters based on resonant waveguide gratings. For their realization a periodic modulation of the RWG is necessary to provide a grating with several ports. This modulation can be superposed via a variation of the depth of the upper grating layer or of the duty cycle. Both strategies give the possibility to realize total transmissions smaller than 10⁻⁴ for given reflectivities of 0.9995 or 0.9. We found that for reflectivities close to unity the depth modulated configuration is favorable with respect to coating thermal noise, because its thickness is about 300 nm less than its width modulated equivalent. For larger diffraction efficiencies R _±1 the triple-T structure has a thickness being slightly thicker than that for the width modulated structure which only needs a stack of two T-shaped gratings. However, the perturbation, represented by the incomplete upper silicon grating layer, of the triple-T grating is almost perfectly decoupled from the rest of the grating resulting in significantly larger parameter tolerances. We have shown that it is possible to fabricate stacked T-shaped grating structures which are a promising basis for future RWG based all-reflective diffractive beam splitters with reduced coating thermal noise. Since Brownian coating thermal noise scales linearly with the effective coating thickness the layer thickness reduction of minimally a factor of about four compared to conventionally silica-tantala based coatings should also lead to a significant reduction of thermal noise. Using the results of Harry et al. [11

] as well as the parameters of Liu et al. [16

] for the amorphous silicon layers a reduction of the power spectral density of a factor of about eight for the triple-T configuration can be expected. For smaller coating thicknesses the reduction is even larger. It has to be emphasized that this estimation does not take into account the influence of the surface structurization on coating thermal noise. These surface effects are under current investigation [27

R. Nawrodt, C. Schwarz, S. Kroker, I. W. Martin, F. Brückner, L. Cunningham, V. Große, A. Grib, D. Heinert, J. Hough, T. Käsebier, E.-B. Kley, R. Neubert, S. Reid, S. Rowan, P. Seidel, M. Thürk, and A. Tünnermann, “Investigation of mechanical losses of thin silicon flexures at low temperatures,” arXiv:1003.2893v1.

Recently, it has been demonstrated that the influence of the local grating period or duty cycle of the resonant reflector on the phase of the reflected light can be exploited to realize flat focusing reflectors based on resonant light coupling [28

F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18, 12606–12614 (2010). [CrossRef] [PubMed]

, 29

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010). [CrossRef]

]. Together with the results presented in this paper this gives the possibility to combine, e.g., a flat focusing broadband mirror with defined diffracting properties and an enhanced reflectivity or bandwidth. This integration of several optical functions into one flat compact optical component cannot be realized with multilayer stacks and might have an impact also for laser cavities beyond high-precision metrology, reflective phase masks or filter elements.

Acknowledgments

This research is supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich TR7.

References and links

1.	P. Fritschel, D. Shoemaker, and R. Weiss, “Demonstration of light recycling in a Michelson interferometer with Fabry-Perot cavities,” Appl. Opt. 31, 1412–1418 (1992). [CrossRef] [PubMed]
2.	J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rüdiger, W. Winkler, and K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational-wave detectors,” Phys. Lett. A 175, 273–276 (1993). [CrossRef]
3.	R. W. P. Drever, “Concepts for extending the ultimate sensitivity of interferometric gravitational gravitational-wave detectors using non-transmissive optics with diffractive or holographic coupling,” in Proceedings of the seventh Marcel Grossman meeting on general relativity , M. Keiser and R. T. Jantzen (eds.), World Scientific Singapore (1995).
4.	K.-X. Sun and R. L. Byer, “All-reflective Michelson, Sagnac, and Fabry-Perot interferometers based on grating beams splitters,” Opt. Lett. 23, 567–569 (1997). [CrossRef]
5.	A. Bunkowski, O. Burmeister, T. Clausnitzer, E.-B. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Optical characterization of ultrahigh diffraction efficiency gratings,” Appl. Opt. 45, 5795–5799 (2006). [CrossRef] [PubMed]
6.	A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. Tünnermann, “Low-loss grating for coupling to a high-finesse cavity,” Opt. Lett. 29, 2342–2344 (2004). [CrossRef] [PubMed]
7.	M. Britzger, D. Friedrich, S. Kroker, F. Brückner, O. Burmeister, E.-B. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractively coupled Fabry-Perot resonator with power-recycling,” Opt. Express 19, 14964–14975 (2011). [CrossRef] [PubMed]
8.	K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational-wave detectors,” Phys. Lett. A 194, 124–132 (1994). [CrossRef]
9.	V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational-wave antennae,” Phys. Lett. A 271, 303–307 (2000). [CrossRef]
10.	Y. Levin, “Internal thermal noise for LIGO test masses: a direct approach,” Phys. Rev. D 57, 659–663 (1998). [CrossRef]
11.	G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002). [CrossRef]
12.	L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985). [CrossRef]
13.	A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006). [CrossRef]
14.	Z. Peng, D. A. Fattal, A. Faraon, M. Fiorentino, J. Li, and R. G. Beausoleil, “Reflective silicon binary diffraction grating for visible wavelengths,” Opt. Lett. 36, 1515–1517 (2011). [CrossRef] [PubMed]
15.	R. Nawrodt, A. Zimmer, T. Koettig, C. Schwarz, D. Heinert, M. Hudl, R. Neubert, M. Thürk, S. Nietzsche, W. Vodel, P. Seidel, and A. Tünnermann, “High mechanical Q-factor measurements on silicon bulk samples,” J. Phys. Conf. Ser. 122, 1–9 (2008). [CrossRef]
16.	X. Liu, C. L. Spiel, R. D. Merithew, R. O. Pohl, B. P. Nelson, Q. Wang, and R. S. Crandall, “Internal friction of amorphous and nanocrystalline silicon at low temperatures,” Mat. Sci. Eng. A 442, 307–313 (2006) [CrossRef]
17.	F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Monolithic dielectric surfaces as new low-loss light-matter interfaces,” Opt. Lett. 33, 264–266 (2008). [CrossRef] [PubMed]
18.	F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and Roman Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104, 163903 (2010). [CrossRef] [PubMed]
19.	D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. 18, 2109–21120 (2001). [CrossRef]
20.	F. Brückner, S. Kroker, D. Friedrich, E.-B. Kley, and A. Tünnermann, “Widely tunable monolithic narrowband grating filter for near-infrared radiation,” Opt. Lett. 36, 436–438 (2011). [CrossRef] [PubMed]
21.	S. Kroker, F. Brückner, E.-B. Kley, and A. Tünnermann, “Enhanced angular tolerance of resonant waveguide grating reflectors,” Opt. Lett. 36, 537–539 (2010). [CrossRef]
22.	A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. 14, 2985–2993 (1997). [CrossRef]
23.	I. C. Botten, M. S. Craig, R. C. McPhredan, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Act. 122, 413–428 (1981). [CrossRef]
24.	V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18, 16973–16988 (2010). [CrossRef] [PubMed]
25.	M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave approach for planar diffraction gratings,” J. Opt. Soc. Am. 71, 811–818 (1981). [CrossRef]
26.	http://ab-initio.mit.edu/wiki/index.php/NLopt.
27.	R. Nawrodt, C. Schwarz, S. Kroker, I. W. Martin, F. Brückner, L. Cunningham, V. Große, A. Grib, D. Heinert, J. Hough, T. Käsebier, E.-B. Kley, R. Neubert, S. Reid, S. Rowan, P. Seidel, M. Thürk, and A. Tünnermann, “Investigation of mechanical losses of thin silicon flexures at low temperatures,” arXiv:1003.2893v1.
28.	F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18, 12606–12614 (2010). [CrossRef] [PubMed]
29.	D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.2770) Diffraction and gratings : Gratings
(230.1360) Optical devices : Beam splitters
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: June 22, 2011
Revised Manuscript: July 15, 2011
Manuscript Accepted: July 15, 2011
Published: August 11, 2011

Citation
Stefanie Kroker, Thomas Käsebier, Frank Brückner, Frank Fuchs, Ernst-Bernhard Kley, and Andreas Tünnermann, "Reflective cavity couplers based on resonant waveguide gratings," Opt. Express 19, 16466-16479 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16466

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References

P. Fritschel, D. Shoemaker, and R. Weiss, “Demonstration of light recycling in a Michelson interferometer with Fabry-Perot cavities,” Appl. Opt. 31, 1412–1418 (1992). [CrossRef] [PubMed]
J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rüdiger, W. Winkler, and K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational-wave detectors,” Phys. Lett. A 175, 273–276 (1993). [CrossRef]
R. W. P. Drever, “Concepts for extending the ultimate sensitivity of interferometric gravitational gravitational-wave detectors using non-transmissive optics with diffractive or holographic coupling,” in Proceedings of the seventh Marcel Grossman meeting on general relativity , M. Keiser and R. T. Jantzen (eds.), World ScientificSingapore (1995).
K.-X. Sun and R. L. Byer, “All-reflective Michelson, Sagnac, and Fabry-Perot interferometers based on grating beams splitters,” Opt. Lett. 23, 567–569 (1997). [CrossRef]
A. Bunkowski, O. Burmeister, T. Clausnitzer, E.-B. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Optical characterization of ultrahigh diffraction efficiency gratings,” Appl. Opt. 45, 5795–5799 (2006). [CrossRef] [PubMed]
A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. Tünnermann, “Low-loss grating for coupling to a high-finesse cavity,” Opt. Lett. 29, 2342–2344 (2004). [CrossRef] [PubMed]
M. Britzger, D. Friedrich, S. Kroker, F. Brückner, O. Burmeister, E.-B. Kley, A. Tünnermann, K. Danzmann, and R. Schnabel, “Diffractively coupled Fabry-Perot resonator with power-recycling,” Opt. Express 19, 14964–14975 (2011). [CrossRef] [PubMed]
K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, R. Schilling, and W. Winkler, “Thermal lensing in recycling interferometric gravitational-wave detectors,” Phys. Lett. A 194, 124–132 (1994). [CrossRef]
V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational-wave antennae,” Phys. Lett. A 271, 303–307 (2000). [CrossRef]
Y. Levin, “Internal thermal noise for LIGO test masses: a direct approach,” Phys. Rev. D 57, 659–663 (1998). [CrossRef]
G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002). [CrossRef]
L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985). [CrossRef]
A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006). [CrossRef]
Z. Peng, D. A. Fattal, A. Faraon, M. Fiorentino, J. Li, and R. G. Beausoleil, “Reflective silicon binary diffraction grating for visible wavelengths,” Opt. Lett. 36, 1515–1517 (2011). [CrossRef] [PubMed]
R. Nawrodt, A. Zimmer, T. Koettig, C. Schwarz, D. Heinert, M. Hudl, R. Neubert, M. Thürk, S. Nietzsche, W. Vodel, P. Seidel, and A. Tünnermann, “High mechanical Q-factor measurements on silicon bulk samples,” J. Phys. Conf. Ser. 122, 1–9 (2008). [CrossRef]
X. Liu, C. L. Spiel, R. D. Merithew, R. O. Pohl, B. P. Nelson, Q. Wang, and R. S. Crandall, “Internal friction of amorphous and nanocrystalline silicon at low temperatures,” Mat. Sci. Eng. A 442, 307–313 (2006) [CrossRef]
F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Monolithic dielectric surfaces as new low-loss light-matter interfaces,” Opt. Lett. 33, 264–266 (2008). [CrossRef] [PubMed]
F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and Roman Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104, 163903 (2010). [CrossRef] [PubMed]
D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. 18, 2109–21120 (2001). [CrossRef]
F. Brückner, S. Kroker, D. Friedrich, E.-B. Kley, and A. Tünnermann, “Widely tunable monolithic narrowband grating filter for near-infrared radiation,” Opt. Lett. 36, 436–438 (2011). [CrossRef] [PubMed]
S. Kroker, F. Brückner, E.-B. Kley, and A. Tünnermann, “Enhanced angular tolerance of resonant waveguide grating reflectors,” Opt. Lett. 36, 537–539 (2010). [CrossRef]
A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. 14, 2985–2993 (1997). [CrossRef]
I. C. Botten, M. S. Craig, R. C. McPhredan, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Act. 122, 413–428 (1981). [CrossRef]
V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18, 16973–16988 (2010). [CrossRef] [PubMed]
M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave approach for planar diffraction gratings,” J. Opt. Soc. Am. 71, 811–818 (1981). [CrossRef]
http://ab-initio.mit.edu/wiki/index.php/NLopt .
R. Nawrodt, C. Schwarz, S. Kroker, I. W. Martin, F. Brückner, L. Cunningham, V. Große, A. Grib, D. Heinert, J. Hough, T. Käsebier, E.-B. Kley, R. Neubert, S. Reid, S. Rowan, P. Seidel, M. Thürk, and A. Tünnermann, “Investigation of mechanical losses of thin silicon flexures at low temperatures,” arXiv:1003.2893v1.
F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18, 12606–12614 (2010). [CrossRef] [PubMed]
D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010). [CrossRef]

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