Abstract

We study the behavior of Fabry-Perot micro-optical resonators based on cylindrical reflectors, optionally combined with cylindrical lenses. The core of the resonator architecture incorporates coating-free, all-silicon, Bragg reflectors of cylindrical shape. The combined effect of high reflectance and light confinement produced by the reflectors curvature allows substantial reduction of the energy loss. The proposed resonator uses curved Bragg reflectors consisting of a stack of silicon-air wall pairs constructed by micromachining. Quality factor Q ~1000 was achieved on rather large cavity length L = 210 microns, which is mainly intended to lab-on-chip analytical experiments, where enough space is required to introduce the analyte inside the resonator. We report on the behavioral analysis of such resonators through analytical modeling along with numerical simulations supported by experimental results. We demonstrate selective excitation of pure longitudinal modes, taking advantage of a proper control of mode matching involved in the process of coupling light from an optical fiber to the resonator. For the sake of comparison, insight on the behavior of Fabry-Perot cavity incorporating a Fiber-Rod-Lens is confirmed by similar numerical simulations.

© 2013 OSA

1. Introduction

Coating-free silicon mirrors and corresponding optical cavities are of special interest, as they offer excellently combined optical and thermomechanical behaviors. First, single crystal silicon has low thermoelastic loss [1] leading to mechanical quality factors as high as 109 [2]. Secondly, silicon has low absorption in the near infrared and, moreover, it offers the remarkable possibility of building monolithic, coating-free, high quality mirrors, as recently reported [3]. Primary needs for such high quality optical mirrors and cavities relate to fundamental research in several areas: gravitational wave detection [4], exploration of quantum limits through sub-Kelvin optical cooling [5,6], lasers and cavity quantum electrodynamics experiments under high confinement [7]. In these areas, the concern is not only about achieving high optical quality factors (Q) but also small modal volumes (V), leading to a figure of merit of the form (Q/V) [810].

On the other hand, optical cavities with other specific properties are also required in biology and environmental science, namely in instruments dedicated either to single biological entities or microscopic particles for optical trapping [11], manipulation [12] and characterization [13], as well as fluid mixtures for analysis by optical spectroscopy techniques [14,15]. When targeting such instruments in an ultra-compact lab-on-chip format, the main issue for optical cavities is to leave a distance (L), between the mirrors, which is long enough to allow both the introduction of a particle inside the cavity and a sufficiently long optical path for absorption spectroscopy. At the same time, light must remain confined inside the cavity despite the increased length of the latter for the purpose of achieving high finesse and quality factor. Most reports on high quality optical micro and nanoresonators relate to guided propagation of light inside rings, disks, or Fabry-Perot (FP) cavities with access waveguides [16,17]. They are also based on light confinement in a space in the order of 10 µm or less.

Practically, the larger is the cavity; the lower is the Q-factor of the optical resonator in the micro-scale. This trend relates to expansion of the Gaussian beam inside the cavity after several round trips; the beam eventually escapes from the cavity due to the short height of the reflectors. Therefore, the challenge of combining both high L and high Q can be evaluated by introducing Q.L as a figure of merit. Earlier work on FP microcavities involving light propagation in air limited L to few tens of microns [1620]. The highest recorded Q.L was around 106 μm for a silicon cavity of length L = 12 μm [17]. In this work, we achieved Q.L = 1.6 x105 μm for a much larger cavity of length L = 210 μm.

2. All-silicon resonator with cylindrical Bragg reflectors

2.1 Silicon resonator architecture

Here below, we study all-silicon optical resonators, without any additional coating material, whose cavity lengths are above 200 microns and whose mirrors have a cylindrical shape. Their high reflectivity is achieved thanks to monolithic silicon Bragg reflectors, which consist of a stack of silicon-air wall pairs constructed by micromachining of single-crystal silicon. In general, an important limitation of conventional micromachined FP cavities constructed with planar mirrors (Fig. 1(a) ) is due to their poor focusing capability: after some round trips, the Gaussian beam injected from an optical fiber diverges and escapes from the cavity, which increases the cavity loss; this, in turn, decreases the Q-factor of the cavity and the related resonance line width. Our effort was then directed towards setting a technique apt to focus the diverging beam and to confine the light inside the cavity.

 

Fig. 1 Schematic representation of Fabry-Perot architectures with different mirror shapes; (a) planar mirrors, (b) cylindrical mirrors providing 1D-confinment of light, (c) spherical mirror providing 2D-confinment of light and (d) cylindrical mirrors combined with a fiber-rod-lens, also providing 2D-confinment of light.

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The proposed cavity architecture (Fig. 1(b)) consists of two cylindrical Bragg reflectors with radii of curvature ρ of 140 μm. The Bragg reflectors are built from alternation of silicon and air layers whose thicknesses were chosen to be 3.67 μm for silicon and 3.48 μm for air, fulfilling the requirement of odd multiple of quarter of the wavelength λ (λ = 1550 nm in air). The cavity length L is 210 μm, corresponding to 3ρ/2. The device is realized in a single technology step using Deep Reactive Ion Etching (DRIE) [21] of a silicon wafer on a depth of 77 μm.

Scanning Electron Microscope (SEM) photos of two fabricated devices are shown in Fig. 2 . For the first device, the cavity mirrors are built from a single silicon layer (Fig. 2(a)). For the second device, the mirrors are built from multiple silicon layers with air layers in-between (Fig. 2(b)), with the aim of increasing the mirror reflectance and hence, the cavity Q-factor.

 

Fig. 2 SEM photos of a fabricated curved FP cavity with (a) single silicon layer, (b) multiple silicon layers (three in this case); the inset shows a magnified view on a silicon-air Bragg reflector.

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FP cavities with planar mirrors are not optimal regarding loss since the diverging light rapidly escapes the cavity after some round-trips. Hence, they are referred to as unstable resonators [22]. Thus, FP cavities with planar mirrors mostly have short cavity lengths of no more than tens of microns. In the present design, we reduce the issue of loss along one transverse direction by introducing silicon-air Bragg reflectors of cylindrical shape, as schematically depicted in the inset of Fig. 2(b). The resulting cylindrical mirror might be considered as a concave lens whose optical axis, z, is in the plane of the silicon surface; it has a focusing capability in the x direction parallel to the plane of the silicon wafer. In the out-of-plane, y direction, the cavity behaves like an ordinary planar mirror and the beam therefore diverges. It is worth mentioning that mirrors of spherical shapes (Fig. 1(c)) would behave much better but they are difficult to realize using current microfabrication technologies. Confinement of light inside the cavity and the acquired spot size in the transverse plane are also shown in Fig. 5 along with both the excitation and the detection fibers. In an earlier report from our group [23], we introduced an alternative solution to the spherical mirrors configuration. In that report, a Fiber-Rod-Lens (FRL) was used to provide light confinement in the out-of-plane direction, which complements the in-plane confinement achieved thanks to the cylindrical mirrors (Fig. 1(d)). Though the recorded Q-factor (~9000) was much higher, the device under consideration in the present paper offers the advantages of being all-silicon and of having a much simpler geometrical layout, which makes it suitable for a detailed analysis. Analytical modeling is accessible for the purpose of comparing between experimental results and simulations. Indeed, such comparison appeared to be very useful for a better understanding of the depicted behavior of selective mode excitation as described hereafter. As mentioned here above, we expect 2D confinement for the light beam in the architecture with cylindrical Bragg reflectors, while 3D confinement of the light beam is expected for the architecture with FRL. This issue has been justified by referring to numerical simulations as demonstrated in Fig. 3 . HFSS-FEM simulations have been carried out on arbitrary design parameters elected for each architecture. Obviously, the cylindrical cavity (Fig. 3(a)) exhibits an elliptical spot size for the light beam bouncing inside its boundaries, as observed on the mid-plane monitor inserted in this cavity. For the FRL cavity instead (Fig. 3(b)), the spot size shrinks to a circle of much smaller size, which confirms our prior expectations about the 2D and the 3D confinements. This is also consistent with the experimental trend of the Q-factors of 1000 and 9000, obtained for 2D and 3D confinement architectures respectively. Knowing that FEM simulations are lengthy and require large computational resources, we made use of the design symmetry along XZ and YZ planes to overcome such limitation. Hence, only one quarter of the cavity volume has been simulated for both the simple and the FRL cavities.

 

Fig. 3 HFSS simulation results illustrating the issue of light confinement (a) 2D confinement inside FP cavity based on cylindrical Bragg mirrors. The light beam is confined only along XZ, at λ = 1480 nm, thanks to the cylindrical surfaces while it diverges along the Y-axis where the cavity behaves as a conventional FP cavity with planar reflectors. (b) 3D light confinement inside FP cavity based on cylindrical Bragg mirrors and a Fiber-Rod-Lens (FRL). The light beam exhibits 3D confinement, at λ = 1455 nm, thanks to the combination of cylindrical reflectors and the FRL. Geometrical parameters of the simulated cavities will be given in section 2.4

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2.2 Resonator analytical model

In an earlier report from our group [24], the cavity microfabrication details were given together with some basic notions about the cavity model; the latter are recalled herein for the sake of reminding. Moreover, we elaborate here on several new aspects of analytical modeling, not only for the derivation of resonant frequencies and mode shapes but also for the analytical estimation of the Q-factor. Furthermore, a mode de-convolution process is implemented for the purpose of analyzing experimental results, leading to a better illustration of the observed process of selective mode excitation.

From Fig. 4 , we can see that the light field injected by the input fiber undergoes three successive transformations with the corresponding power losses, when passing through the whole optical system before it propagates into the output fiber. The corresponding (power) transfer functions are used to build a complete model for studying the resonator behavior. As evident, it involves the multiplication of three different wavelength dependent transmission coefficients, T = Γ·Hcav·O such that:

 

Fig. 4 Schematic representation of the measurement setup with an inset related to the arrangement of the curved cavity with respect to positions of the fiber input and fiber output.

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  • (i) Input coupling efficiency Γ: this first term involves power coupling of the light field coming out from the input lensed fiber to the cavity entrance. The corresponding input coupling losses are described by the coefficient Γ, referred as the (input) coupling efficiency.
  • (ii) Cavity transmittance Hcav: this second term describes the cavity response at the different resonance modes. It involves an intra-cavity round-trip coupling efficiency γ, as detailed below.
  • (iii) Output coupling efficiency O: This third term concerns the coupling from the cavity to the output fiber. Since the chosen output fiber can accommodate spot sizes up to 56 µm, it has good collection efficiency and its output coupling efficiency O can be considered very close to unity: O ~1. Therefore the overall transfer function of the power transmittance T reduces to the product T = Γ·Hcav.

Before presenting the experimental results and details about theoretical analysis, we highlight the basic assumptions of this model:

  • (a) EFiber is the transverse component of the electromagnetic field injected from the lensed fiber and is considered as a pure Gaussian beam that is given by:
    EFiber(x,y,z=0)=2wxwye(x2wx2+y2wy2)ejk(x22Rx+y22Ry)
  • (b) Ψm,n denotes the electric field transverse components representing the cavity electromagnetic resonant modes. The cavity, case of study, is supposed to support Hermite-Gaussian transverse modes Ψm,n [22], which are the typical modes of resonators made from spherical mirrors and whose analytical expression was adapted here to the cylindrical shape of the mirrors. The degeneracy applies for order m only, along the x direction, while the fundamental (order n = 0) Gaussian mode profile is kept along the y direction. Hereafter, we therefore consider only the transverse part of the modes supported by the cylindrical cavity:
    ψm,0(x,y)=2wxwyHm(2xwx)e(x2wx2+y2wy2)ejk(x22Rx+y22Ry)

    where wx and wy denote the beam size in the x and the y directions; Rx and Ry are the beam radii of curvature in the x and y directions; Hmis the Hermite polynomial function of order m.

  • (c) As we are working in the limits of the paraxial approximation, we adopted the scalar notation for Ψm,0 as no longitudinal component is observed. Indeed, this is acceptable as the divergence angle of the studied Gaussian beam is less than 3.5° in our case and thus, longitudinal components of the electric field could be neglected [22].
  • (d) In addition, as we consider only the modes at the cavity entrance z = 0, then, the longitudinal (z) dependence of the cavity modes and the corresponding third mode order q do not appear in Eq. (1). Actually, there is a different set of transverse modes Ψm,0 for each longitudinal mode of order q. All these modes have their own resonance frequencies, written as follows:
    νm,0,q=c2L[q+(1+m)arccos(1L/ρ)π]
  • (e) All calculations that will follow, concerning the fiber-to-cavity coupling efficiency Γ and the intra-cavity round-trip coupling efficiency γ are done numerically based on the corresponding equations presented hereafter. The basic assumptions are the following: (i) the calculation domain extends from [-100 µm, 100 µm] in both X and Y directions; (ii) the Bragg mirrors are assumed to be thin and transparent in the infra-red range. Thus, the injected field crosses the cavity, keeping the initial values of the beam size and radii of curvature; (iii) the beam size differs in the X and Y directions as the curved mirror focuses the beam in X while it has no effect on it in the Y direction.

2.3 Resonator experimental characterization

The optical setup, schematized in Fig. 4, consists of an Agilent tunable laser source (TLS) module 81949A used to inject the laser light, and an Anritsu MS9710B optical spectrum analyzer (OSA) used as a powermeter to measure the transmitted light. A visible laser light at 635 nm was also used for alignment purposes. The visible light and the infrared light are coupled through a directional coupler and injected into the lensed fiber to the device under test (DUT). A similar lensed fiber is used to collect the light transmitted through the device and to couple it directly to the OSA. The lensed fibers from Corning, have a spot size of 18 μm and a working distance of 300 μm. Six-axis positioners were used to align each fiber in the input and output grooves. All elements were mounted on an optical table to reduce vibration effects.

The spectral responses shown in Fig. 5 relate to cavities having one and two silicon layers per mirror and separations of 210 μm and 195.6 μm, respectively. Table 1 shows the experimentally observed resonance wavelengths in the explored range, together with the calculated values obtained from Eq. (3). For each resonance, the mode orders are also indicated. The wavelength spacing between two consecutive longitudinal resonant modes, referred as free spectral range (FSR), was found to be 5.375 nm for the single-silicon layer mirror cavity and 5.625 nm for the two-silicon layer mirror cavity. These values are somewhat close to the theoretical FSRs estimated to be 5.537 nm and 5.953 nm, respectively. We ascribed the observed mismatch to the cavity effective length Leff. Indeed, in our case, the cavity mirrors consist of concave Distributed Bragg Reflectors (DBR) that can be reduced to two single reflecting walls located inside their corresponding DBRs [25]. Thus, the effective length of the cavity becomes larger than the physical distance measured between the inner shells of the DBRs. In addition, one can notice from Fig. 5 that the cavity becomes eventually more selective as the number of silicon layers increases. For the two-layer design, the mirror reflectivity is expected theoretically to be 97.3%, compared to 72% for the single-layer design. A comparison between the results is shown in Table 2 .

 

Fig. 5 Measured spectral responses for a FP cavity with cylindrical silicon-air Bragg reflectors having one and two silicon layers per mirror, respectively, illustrating the improvement of resonator finesse with the number of layers.

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Tables Icon

Table 1. Theoretical and experimental resonance wavelengths for the different (m,n,q) cavity modes in wavelength range between 1528 and 1545 nm. The wavelength values observed experimentally are given in italic.

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Table 2. Summary of experimental results comparing single and double layers cavities of the same length

Another experiment was done to study the power coupling efficiency and related mode matching between the fiber and the cavity, targeting a selective excitation of the longitudinal

modes. For this purpose, the position zin of the input fiber with respect to the entrance mirror was modulated; three different positions were tested: zin = 150 μm, zin = 300 μm and zin = 460 μm, the detection fiber was replaced by another lensed fiber characterized by its better collection efficiency. The corresponding recorded spectral responses are superimposed as shown in Fig. 6 . Observing the cavity spectral responses, we notice a form of periodicity: the pattern being repeated includes not only the main peaks of the longitudinal modes, referred to as modes of type (0,0), but also other peaks corresponding to (2,0) transverse modes. These modes were identified by calculating the resonance wavelengths of the different cavity modes as shown in Table 1. It is worth to mention that the typical value of the optical insertion loss is about 28 dB at the peak of the resonance wavelength.

 

Fig. 6 Measured spectral responses of a FP cavity having a single silicon layer per mirror. The fiber-to-mirror coupling distance was varied zin = 150 µm, 300 µm and 460 µm. The most selective coupling to fundamental modes of type (0,0) is achieved when zin = 300 µm, also corresponding to twice the position of the fiber beam waist. As for the transverse mode (2,0), either effective coupling or noticeable extinction (of ratio of 7:1) is obtained, when zin = 150 µm and zin = 300 µm, respectively.

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2.4 Resonator numerical model

As pointed out earlier, HFSS-FEM was deployed for the numerical simulation of the investigated optical resonator. In general, the simulation conditions applied for the different cases were as follow:

  • i. All results pertain to a Gaussian beam excitation with TE polarization. The beam waist, propagating along the positive z-direction, is located at the cavity entrance, and its spot size will be specified for each case accordingly. Radiation boundary conditions have been applied for the studied geometries and the surrounding external media is the free space.
  • ii. If cavities with real dimensions are to be simulated, enormous calculations resources will be required. To overcome this problem; scaled down miniaturized versions of the cavities have been designed and simulated. Moreover, to render the simulation more efficient, we exploited the symmetry of the design along the XY and the YZ planes to simulate only one quarter the cavity volume.
  • iii. For further simplification and size reduction, cavities with single silicon Bragg layer per mirror have been designed simulated. Since single thin silicon layers have been explored (thickness equivalent to quarter the wavelength) for the mirrors, the meshing has been adjusted to assure a least two meshes within the silicon layer thickness.
  • iv. The transmission response is calculated as the ratio between the transmitted power and the input power at the different excitation wavelengths. These powers components are obtained by integrating the Poynting vectors over the external surfaces of the studied volume in the input and the output. For calculating the input power, only the incident field is considered while the calculation of the transmitted power is obtained by integrating the total field transmitted through the cavity.

Based on these conditions, we simulated two cases of study: The first concerns to the simple curved cavity while the second concerns the FRL cavity. The geometrical parameters specific to the first case are the radius of curvature taken as 3 μm, the thickness of the silicon layer taken equal to 111.4 nm and the physical length of the cavity taken equal to 3.98 μm. The spot size of the exciting Gaussian beam is 1.56 μm. The acquired spectral response is shown in Fig. 7(a) .

 

Fig. 7 Spectral response obtained by HFSS simulation; (a) for the simple curved cavity, the mode orders are mentioned beside each resonance peak, (b) Spectral response of the FRL cavity obtained by HFSS simulation.

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The second case of study pertains to the FRL cavity. The geometrical parameters of this design are the radius of curvature taken as 6 μm, the thickness of the silicon layer taken equal to 111.4 nm, the physical length of the cavity taken equal to 4.78 μm and the fiber diameter is equal to 1.9 μm. The spot size of the exciting Gaussian beam is 0.9 μm. The corresponding spectral response is shown in Fig. 7(b).

Further analysis of these simulation results justifies the issue of the Q-factor improvement upon the addition of the FRL as in the second design. In fact, the simple curved cavity exhibits a Q-factor of 48 at λ = 1490 nm while the FRL design exhibits a Q-factor of 144 at λ = 1455 nm that is three times improvement with respect to the simple curved cavity. These values for Q are just mentioned here to provide a qualitative comparison between both designs. Nevertheless, a quantitative comparison between the simulation and the experimental results is invalid in our case due to many factors. Among these is the difference between the dimensions of the simulated and the real cavity which has a strong impact on the value of the Q-factor.

Beside the Q-factor improvement, the numerical spectral response obtained for the simple curved cavity reveals the excitation of higher order modes and this is justified by reference to the experimental results shown in Fig. 6 and in Table 3 . In this table, a comparison is established between the values of the resonance wavelengths obtained by the analytical formula (Eq. (3) and the simulation result obtained by HFSS. Results are rather close and this confirms our earlier anticipations about the excitation of higher order mode. Indeed, the noticed discrepancy is mainly attributed to the variation of the effective cavity length [25] over the large spanned spectral range which extends up to 300 nm and this in turn, affects the positions of the resonance wavelengths. Another reason for this discrepancy which is more pronounced at the higher order modes is that these modes are not purely (2,0,q) or (4,0,q), but they involve a non-negligible contribution coming from the neighbor fundamental mode and this effect also, leads to the shift of their resonance wavelengths. It is to be noted that for the FRL design the spectral response includes also side peaks which are most probably due to the excitation of higher order modes as demonstrated by the simulation result presented in Fig. 8 where we observe multi-spots for the mid-plane cartography. Oppositely to the previous case, the FRL cavity providing the 3D confinement does not have a robust analytical model and so, a similar treatment cannot be conducted as in the previous study related to the simply curved cavities with 2D confinement. We therefore concentrate in what follows on the latter cavities to elaborate more on the analysis supported by theoretical modeling.

Tables Icon

Table 3. Theoretical and numerical resonance wavelengths for the different (m,n,q) cavity modes in wavelength range between 1250 and 1560 nm. The wavelength values observed experimentally are given in italic.

 

Fig. 8 HFSS simulation for the FRL cavity at λ = 1518.7 nm. Multi-spot are observed at the mid-plane and they reveal the excitation of higher order modes in the FRL cavity.

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3. Analysis of simple curved cavities

3.1 Fiber-to-cavity power coupling efficiency Γ

As the field injected from the lensed fiber (Gaussian beam) is not perfectly matched to the cavity modes (Hermite Gaussian modes), there is a need for estimating this mismatch, as it represents also a source of losses. There is also an interest of controlling this mismatch for the purpose of selective mode excitation. This source of loss was taken into account in our resonator model by the power coupling efficiency term Γ, which is also part of the resonator transfer function. It is obvious that Γ depends on the field distribution and therefore on the cavity mode order (m,n).

Numerical calculations were used to evaluate this power coupling efficiency between the field EFiber injected from a single mode optical fiber (fundamental Gaussian mode) and the field of the different transverse cavity modes Ψm,0 at the cavity entrance. The calculation is done according to a normalized overlap integral:

Γm,0=|(D)EFiber(x,y,0)ψm,0(x,y,0)dxdy|2(D)|EFiber(x,y,0)|2dxdy(D)|ψm,0(x,y,0)|2dxdy

The injected fundamental mode (EFiber) is purely Gaussian; it has a beam waist of w0 = 18 μm and waist position at 150 μm from the fiber output. The transverse cavity modes (Ψm,0) from m = 0 to m = 4 were examined. The calculated coupling efficiencies Γm,0 are shown in Table 4 below for a wavelength of 1530 nm and coupling distances zin of 150 μm, 300 μm and 460 μm (same values were chosen for the measurements shown in Fig. 6). Since the integral calculation is done numerically, the mentioned numbers involve a very small numerical error which makes the sum of the coupling efficiencies slightly less than 100%. Neglecting this numerical error, we can consider that there is almost no coupling to modes higher than order (4,0) except in the case where zin = 460 μm, where only 99,7% of the available power is coupled to the modes up to (4,0), leaving the remaining 0,3% of the injected power to higher order modes. The limits of integration were taken between [-100 μm, 100 μm] in both x axis and y axis, these limits being large enough for the considered beam size, whose waist is 18 µm. The simulation results show that only the modes with odd peaks can be excited since the coupling with modes of even peaks is null due to symmetry reasons. In addition, the coupling efficiency decreases significantly as the mode order m increases and, moreover, it varies as a function of the fiber input position (zin). A case of particular interest is when the coupling distance zin is equal to 300 μm, leading to very selective excitation of the longitudinal modes, that is, the fundamental modes (0,0), with a drastic extinction of all other transverse modes.

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Table 4. Power coupling efficiencies Γm,0 between the fiber mode (Gaussian) and the (m,0) resonator modes (Hermite-Gaussian); calculated values at λ = 1530 nm and values obtained by fitting to experimental results.

Next is the comparison between our theoretical and experimental results. The analysis is done by comparing the calculated coupling efficiencies shown in Table 4 and the experimental spectral responses shown in Fig. 6, where modes are identified through their resonance wavelengths. First, it appears from peak wavelengths in the measured spectra shown in Fig. 6 that the odd resonant modes (1,0) and (3,0) are not excited, as expected from our calculations of the corresponding coupling efficiencies shown in Table 4. Comparing the calculations presented in Table 4 and the experimental results, we can infer that the mode of type (2,0) becomes noticeable at zin = 150 μm and 460 μm, as its coupling efficiency is non-negligible at these positions. The mode (2,0) has the highest coupling efficiency at zin = 150 μm. Mode (4,0) has much reduced coupling efficiency and, as expected, it is not easily observable in the experiments.

On the other hand, it is worth mentioning that the fundamental resonant mode (0,0) is best coupled at zin = 300 μm (99.93%) while the modes (2,0) and (4,0) are weakly coupled as their resonance peaks are not very high. This can be explained by the fact that the beam size at the cavity edges is similar to the fiber mode at zin = 300 μm, which leads to optimal conditions for selective mode coupling with the cavity fundamental (longitudinal) modes together with noticeable extinction of the transverse modes.

Further analysis of the experimental results enabled the evaluation of Γ for the different modes at different positions zin. From the experimental results presented above, noticeable coupling is observed only on the modes TEM00 and TEM20 of the cavity. Based on this conclusion, we simulated a deconvolution process where we made a linear combination of these two modes with different coupling coefficients Γ00 and Γ20 and then, we fitted this combination to the experimental responses as shown in Fig. 9 .

 

Fig. 9 Fitting of measured data to a combination of only two contributing modes of type TEM00 and TEM20: a) for zin = 150 µm, Γ00 = 63% and Γ20 = 37%, b) for zin = 300 µm, Γ00 = 99% and Γ20 = 1% and c) for zin = 460 µm, Γ00 = 89% and Γ20 = 11%

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In this process, the resonance peaks were described by Airy functions given by Eq. (5) below, with effective mirror reflectance eff=γ=61% and cavity length L around 214 μm. Matching is achieved between the trial values of Γ00 and Γ20 corresponding to the best fit with experiments, and those obtained by numerical calculations using Eq. (4). While this matching is very good in the case of zin = 300 µm and zin = 460 µm, a quite large discrepancy is observed in the case of zin = 150 µm. This deficiency is not fully understood and might be attributed to angular misalignment of the input fiber that led to a strong coupling to mode (2,0) (37% instead of 3.14%). Another possible explanation might be related to the fact that the excitation spot size is smaller than the mode size, which happens when zin = 150 µm, also corresponding to the position of the optical fiber beam waist.

The mode addition detailed so far has also been proved after performing a numerical simulation on HFSS for the scaled down simple curved cavity discussed earlier. The simulation is carried out at λ = 1420.8 nm, associated nominally to mode (2,0,4). Accordingly, we were expecting four lobes inside the cavity under these conditions. Instead, the result, shown in Fig. 10 , reveals a combined effect of four lobes (higher part of the cavity) and five lobes (central part of the cavity) corresponding to modes (2,0,4) and (0,0,5). As noticed from the field map, the lobes amplitudes diminish due to the difference between the periodicity of both modes.

 

Fig. 10 Field map of the numerical simulation carried for the simple curved cavity at λ = 1420.8 nm along the XZ plan. The combined effect of modes (2,0,4) and (0,0,5) is illustrated.

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3.2 Fiber-to-cavity power coupling efficiency Γ

The cavity (power) transmittance at the vicinity of a given resonant mode can be expressed by a proper Airy function:

Hcav=11+4γ(1γ)2sin2(2πLλ0)
where L, the cavity length, ℜ, the mirror reflectance, λo, the free space wavelength and the quantity γ = e–2αL defined as the power attenuation factor after a full round-trip inside the cavity, where α is the attenuation coefficient per unit length. In our case, the attenuation mechanism is ascribed to light beam divergence and γ is also referred as the intra-cavity round trip coupling efficiency. Our first objective was to combine both the characteristics of the high reflectivity mirrors (up to 99.99%) and the long cavity lengths, while keeping low loss. We adopted the quality factor Q, as expressed in Eq. (6) to quantify the cavity selectivity [26]:

Q=2πLλ0(γ1γ)

We notice from Eq. (6) that as the cavity length L increases the Q-factor increases accordingly for the case of a lossless cavity. On the contrary, if the cavity is lossy and, as γ = e–2αL, then Q increases first and then it decreases since the losses become important at large lengths where the exponential factor embedded in γ dominates. Therefore, one can potentially attain unequalled performances in terms of Q-factor on larger cavities, provided that the attenuation factor γ can be maintained close to unity, for instance, by implementing a focusing technique using curved mirrors, as it is considered in this work. For each mode order (m,n), γm,n relates to the corresponding cavity losses and it is mainly due to the light beam divergence inside the cavity. This parameter is estimated using the related cavity modes Ψm,n solely.

The intra-cavity round trip coupling efficiency γ = γm,0 of the (m,0) modes is estimated at different input positions (zin) of the excitation fiber, as illustrated in Fig. 4. γm,0 is obtained from the following overlap integral between the mode fields before and after full round trip along the cavity length, respectively:

γm,0=|(D)ψm,0(x,y,0)ψm,0*(x,y,2L)dxdy|2(D)|ψm,0(x,y,0)|2dxdy(D)|ψm,0(x,y,2L)|2dxdy
where (D) is the mirror domain. The obtained values of γ0,0 and the corresponding Q-factors derived from Eq. (6) are presented in Table 5 , where we considered a mirror reflectance ℜ = 72% corresponding to an ideal single layer Bragg reflectors.

Tables Icon

Table 5. Intra-cavity round-trip coupling efficiency γ0,0 and Q-factor for a cavity with single silicon layer, with expected mirror reflectance = 72%, calculated at λ = 1535 nm for mode (0,0) at three different distances zin.

It was found that the intra-cavity round trip coupling efficiency increases as zin increases due to the fact that the beam divergence becomes significant for values of z larger than the fiber working distance of 300 μm. We notice a discrepancy between the analytical and the experimental Q-factors. This issue is due to many reasons: (i) the mirror roughness which degrades the reflection coefficients to less than the expected value of 72%, (ii) the silicon intrinsic losses over the single layer thickness of 3.67 μm, (iii) the non-ideality of the optical alignment caused by the lensed fiber pointing error, (iv) for the case of zin = 460 μm, a portion of the light beam is passing over the filter. Calculation results show that γm,0 has negligible wavelength dependence (when considering different longitudinal mode orders q). Also, the values of γm,0 were found to be mainly related to the mode order n = 0 and keep unaffected by the values of mode order m. This can be understood as all Ψm,0 transverse modes expand similarly in the y direction, which is the major direction of energy loss. However, the most significant conclusion is that Q-factor is affected by γm,0, which, in turn, is a function of the fiber coupling distance zin; this is consistent with the wavefunction dependence on the spot size and the beam radii of curvature, as indicated in Eq. (2). As a final note, inspection of Fig. 6 reveals a good potential for the cylindrical FP cavity in Wavelength Division Multiplexing Systems either as an Optical Add-Drop Multiplexer (OADM), or as a Wavelength Selective Switch (WSS) or even as Mode Selective Filter (MSF).

4. Conclusions

To conclude, all-silicon Fabry-Perot cavities based on cylindrical Bragg mirrors have been studied. For the case of simple cylindrical cavity, an analytical model has been presented and it consists mainly of the Fiber-to-cavity coupling efficiency and the intra-cavity loss. The power coupling efficiency has been calculated analytically for different coupling distances zin. Moreover, a de-convolution process has been pursued on the measured results obtained for zin to evaluate the experimental coupling coefficients for modes TEM00 and TEM20. Results have been compared and some deficiencies, though limited, have been discussed as well. A series of HFSS-FEM numerical simulations have been conducted on both the simple and the FRL cavities to confirm the earlier analysis and predictions on the experimental results, namely: the spot size confinement in the cavity center, the excitation of higher order modes of type (2,0) and the mode mixing. In the last stage of this work, the mode intra-cavity losses due to beam expansion have been estimated numerically to evaluate its impact on the Q-factor. Here also, numerical and experimental results for the Q-factor have been compared. The reported behavior and performance predicts a strong potential for deploying the device in applications targeting OADM, WSS or MSF in optical communication systems.

References and links

1. C. Zener, “Internal friction in solids. Pt. II: general theory of thermoelastic internal friction,” Phys. Rev. 53(1), 90–99 (1938). [CrossRef]  

2. D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys. 30(5-6), 621–629 (1978). [CrossRef]  

3. F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104(16), 163903 (2010). [CrossRef]   [PubMed]  

4. 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 Gravity 19(5), 897–917 (2002). [CrossRef]  

5. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006). [CrossRef]   [PubMed]  

6. O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006). [CrossRef]   [PubMed]  

7. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002). [CrossRef]   [PubMed]  

8. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef]   [PubMed]  

9. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003). [CrossRef]   [PubMed]  

10. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23(4), 247–249 (1998). [CrossRef]   [PubMed]  

11. D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express 14(9), 4176–4182 (2006). [CrossRef]   [PubMed]  

12. D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco Jr, and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature 437(7056), 231–234 (2005). [CrossRef]   [PubMed]  

13. W. Z. Song, X. M. Zhang, A. Q. Liu, C. S. Lim, P. H. Yap, and H. M. M. Hosseini, “Refractive index measurement of single living cells using on-chip Fabry-Perot cavity,” Appl. Phys. Lett. 89(20), 203901 (2006). [CrossRef]  

14. S. Kassi, M. Chenevier, L. Gianfrani, A. Salhi, Y. Rouillard, A. Ouvrard, and D. Romanini, “Looking into the volcano with a mid-IR DFB diode laser and cavity enhanced absorption spectroscopy,” Opt. Express 14(23), 11442–11452 (2006). [CrossRef]   [PubMed]  

15. J. M. Langridge, T. Laurila, R. S. Watt, R. L. Jones, C. F. Kaminski, and J. Hult, “Cavity enhanced absorption spectroscopy of multiple trace gas species using a supercontinuum radiation source,” Opt. Express 16(14), 10178–10188 (2008). [CrossRef]   [PubMed]  

16. M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett. 92(8), 081101 (2008). [CrossRef]  

17. M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Integrated waveguide Fabry-Perot microcavities with silicon/air Bragg mirrors,” Opt. Lett. 32(5), 533–535 (2007). [CrossRef]   [PubMed]  

18. B. Saadany, M. Malak, M. Kubota, F. Marty, Y. Mita, D. Khalil, and T. Bourouina, “Free-space tunable and drop optical filters using vertical Bragg mirrors on silicon,” J. Sel. Top. Quantum Electron. 12(6), 1480–1488 (2006). [CrossRef]  

19. R. St-Gelais, J. Masson, and Y.-A. Peter, “All-silicon integrated Fabry-Perot cavity for volume refractive index measurement in microfuidic systems,” Appl. Phys. Lett. 94(24), 243905 (2009). [CrossRef]  

20. A. Lipson and E. M. Yeatman, “A 1-D photonic band gap tunable optical filter in (110) silicon,” J. Microelectromech. Syst. 16(3), 521–527 (2007). [CrossRef]  

21. F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J. 36(7), 673–677 (2005). [CrossRef]  

22. A. Yariv, Quantum Electronics (Wiley, New York, USA 1989).

23. M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011). [CrossRef]  

24. M. Malak, F. Marty, N. Pavy, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250 μm-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst. 21(1), 171–180 (2012). [CrossRef]  

25. M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv 2(2), 022143 (2012). [CrossRef]  

26. T. Verdeyen, Laser Electronics (Prentice Hall, 1995).

References

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  • |

  1. C. Zener, “Internal friction in solids. Pt. II: general theory of thermoelastic internal friction,” Phys. Rev.53(1), 90–99 (1938).
    [CrossRef]
  2. D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys.30(5-6), 621–629 (1978).
    [CrossRef]
  3. F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
    [CrossRef] [PubMed]
  4. 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 Gravity19(5), 897–917 (2002).
    [CrossRef]
  5. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature444(7115), 75–78 (2006).
    [CrossRef] [PubMed]
  6. O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature444(7115), 71–74 (2006).
    [CrossRef] [PubMed]
  7. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science298(5597), 1372–1377 (2002).
    [CrossRef] [PubMed]
  8. K. J. Vahala, “Optical microcavities,” Nature424(6950), 839–846 (2003).
    [CrossRef] [PubMed]
  9. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(6926), 925–928 (2003).
    [CrossRef] [PubMed]
  10. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett.23(4), 247–249 (1998).
    [CrossRef] [PubMed]
  11. D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express14(9), 4176–4182 (2006).
    [CrossRef] [PubMed]
  12. D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature437(7056), 231–234 (2005).
    [CrossRef] [PubMed]
  13. W. Z. Song, X. M. Zhang, A. Q. Liu, C. S. Lim, P. H. Yap, and H. M. M. Hosseini, “Refractive index measurement of single living cells using on-chip Fabry-Perot cavity,” Appl. Phys. Lett.89(20), 203901 (2006).
    [CrossRef]
  14. S. Kassi, M. Chenevier, L. Gianfrani, A. Salhi, Y. Rouillard, A. Ouvrard, and D. Romanini, “Looking into the volcano with a mid-IR DFB diode laser and cavity enhanced absorption spectroscopy,” Opt. Express14(23), 11442–11452 (2006).
    [CrossRef] [PubMed]
  15. J. M. Langridge, T. Laurila, R. S. Watt, R. L. Jones, C. F. Kaminski, and J. Hult, “Cavity enhanced absorption spectroscopy of multiple trace gas species using a supercontinuum radiation source,” Opt. Express16(14), 10178–10188 (2008).
    [CrossRef] [PubMed]
  16. M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett.92(8), 081101 (2008).
    [CrossRef]
  17. M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Integrated waveguide Fabry-Perot microcavities with silicon/air Bragg mirrors,” Opt. Lett.32(5), 533–535 (2007).
    [CrossRef] [PubMed]
  18. B. Saadany, M. Malak, M. Kubota, F. Marty, Y. Mita, D. Khalil, and T. Bourouina, “Free-space tunable and drop optical filters using vertical Bragg mirrors on silicon,” J. Sel. Top. Quantum Electron.12(6), 1480–1488 (2006).
    [CrossRef]
  19. R. St-Gelais, J. Masson, and Y.-A. Peter, “All-silicon integrated Fabry-Perot cavity for volume refractive index measurement in microfuidic systems,” Appl. Phys. Lett.94(24), 243905 (2009).
    [CrossRef]
  20. A. Lipson and E. M. Yeatman, “A 1-D photonic band gap tunable optical filter in (110) silicon,” J. Microelectromech. Syst.16(3), 521–527 (2007).
    [CrossRef]
  21. F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J.36(7), 673–677 (2005).
    [CrossRef]
  22. A. Yariv, Quantum Electronics (Wiley, New York, USA 1989).
  23. M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
    [CrossRef]
  24. M. Malak, F. Marty, N. Pavy, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250 μm-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst.21(1), 171–180 (2012).
    [CrossRef]
  25. M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv2(2), 022143 (2012).
    [CrossRef]
  26. T. Verdeyen, Laser Electronics (Prentice Hall, 1995).

2012 (2)

M. Malak, F. Marty, N. Pavy, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250 μm-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst.21(1), 171–180 (2012).
[CrossRef]

M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv2(2), 022143 (2012).
[CrossRef]

2011 (1)

M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
[CrossRef]

2010 (1)

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
[CrossRef] [PubMed]

2009 (1)

R. St-Gelais, J. Masson, and Y.-A. Peter, “All-silicon integrated Fabry-Perot cavity for volume refractive index measurement in microfuidic systems,” Appl. Phys. Lett.94(24), 243905 (2009).
[CrossRef]

2008 (2)

J. M. Langridge, T. Laurila, R. S. Watt, R. L. Jones, C. F. Kaminski, and J. Hult, “Cavity enhanced absorption spectroscopy of multiple trace gas species using a supercontinuum radiation source,” Opt. Express16(14), 10178–10188 (2008).
[CrossRef] [PubMed]

M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett.92(8), 081101 (2008).
[CrossRef]

2007 (2)

M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Integrated waveguide Fabry-Perot microcavities with silicon/air Bragg mirrors,” Opt. Lett.32(5), 533–535 (2007).
[CrossRef] [PubMed]

A. Lipson and E. M. Yeatman, “A 1-D photonic band gap tunable optical filter in (110) silicon,” J. Microelectromech. Syst.16(3), 521–527 (2007).
[CrossRef]

2006 (6)

B. Saadany, M. Malak, M. Kubota, F. Marty, Y. Mita, D. Khalil, and T. Bourouina, “Free-space tunable and drop optical filters using vertical Bragg mirrors on silicon,” J. Sel. Top. Quantum Electron.12(6), 1480–1488 (2006).
[CrossRef]

D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express14(9), 4176–4182 (2006).
[CrossRef] [PubMed]

W. Z. Song, X. M. Zhang, A. Q. Liu, C. S. Lim, P. H. Yap, and H. M. M. Hosseini, “Refractive index measurement of single living cells using on-chip Fabry-Perot cavity,” Appl. Phys. Lett.89(20), 203901 (2006).
[CrossRef]

S. Kassi, M. Chenevier, L. Gianfrani, A. Salhi, Y. Rouillard, A. Ouvrard, and D. Romanini, “Looking into the volcano with a mid-IR DFB diode laser and cavity enhanced absorption spectroscopy,” Opt. Express14(23), 11442–11452 (2006).
[CrossRef] [PubMed]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature444(7115), 75–78 (2006).
[CrossRef] [PubMed]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature444(7115), 71–74 (2006).
[CrossRef] [PubMed]

2005 (2)

D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature437(7056), 231–234 (2005).
[CrossRef] [PubMed]

F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J.36(7), 673–677 (2005).
[CrossRef]

2003 (2)

K. J. Vahala, “Optical microcavities,” Nature424(6950), 839–846 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(6926), 925–928 (2003).
[CrossRef] [PubMed]

2002 (2)

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science298(5597), 1372–1377 (2002).
[CrossRef] [PubMed]

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 Gravity19(5), 897–917 (2002).
[CrossRef]

1998 (1)

1978 (1)

D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys.30(5-6), 621–629 (1978).
[CrossRef]

1938 (1)

C. Zener, “Internal friction in solids. Pt. II: general theory of thermoelastic internal friction,” Phys. Rev.53(1), 90–99 (1938).
[CrossRef]

Arcizet, O.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature444(7115), 71–74 (2006).
[CrossRef] [PubMed]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Basset, P.

M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv2(2), 022143 (2012).
[CrossRef]

Bourouina, T.

M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv2(2), 022143 (2012).
[CrossRef]

M. Malak, F. Marty, N. Pavy, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250 μm-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst.21(1), 171–180 (2012).
[CrossRef]

M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
[CrossRef]

B. Saadany, M. Malak, M. Kubota, F. Marty, Y. Mita, D. Khalil, and T. Bourouina, “Free-space tunable and drop optical filters using vertical Bragg mirrors on silicon,” J. Sel. Top. Quantum Electron.12(6), 1480–1488 (2006).
[CrossRef]

F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J.36(7), 673–677 (2005).
[CrossRef]

Bouwmeester, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature444(7115), 75–78 (2006).
[CrossRef] [PubMed]

Briant, T.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature444(7115), 71–74 (2006).
[CrossRef] [PubMed]

Britzger, M.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
[CrossRef] [PubMed]

Brückner, F.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
[CrossRef] [PubMed]

Burmeister, O.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
[CrossRef] [PubMed]

Burnham, D. R.

Bustamante, C.

D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature437(7056), 231–234 (2005).
[CrossRef] [PubMed]

Cagnoli, G.

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 Gravity19(5), 897–917 (2002).
[CrossRef]

Chenevier, M.

Clausnitzer, T.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
[CrossRef] [PubMed]

Cohadon, P.-F.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature444(7115), 71–74 (2006).
[CrossRef] [PubMed]

Collin, D.

D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature437(7056), 231–234 (2005).
[CrossRef] [PubMed]

Crooks, D. R. M.

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 Gravity19(5), 897–917 (2002).
[CrossRef]

Danzmann, K.

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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 Gravity19(5), 897–917 (2002).
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M. Malak, F. Marty, N. Pavy, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250 μm-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst.21(1), 171–180 (2012).
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M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
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M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
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M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv2(2), 022143 (2012).
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M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
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B. Saadany, M. Malak, M. Kubota, F. Marty, Y. Mita, D. Khalil, and T. Bourouina, “Free-space tunable and drop optical filters using vertical Bragg mirrors on silicon,” J. Sel. Top. Quantum Electron.12(6), 1480–1488 (2006).
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F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J.36(7), 673–677 (2005).
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R. St-Gelais, J. Masson, and Y.-A. Peter, “All-silicon integrated Fabry-Perot cavity for volume refractive index measurement in microfuidic systems,” Appl. Phys. Lett.94(24), 243905 (2009).
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D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys.30(5-6), 621–629 (1978).
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F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J.36(7), 673–677 (2005).
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B. Saadany, M. Malak, M. Kubota, F. Marty, Y. Mita, D. Khalil, and T. Bourouina, “Free-space tunable and drop optical filters using vertical Bragg mirrors on silicon,” J. Sel. Top. Quantum Electron.12(6), 1480–1488 (2006).
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M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv2(2), 022143 (2012).
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M. Malak, F. Marty, N. Pavy, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250 μm-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst.21(1), 171–180 (2012).
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M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
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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 Gravity19(5), 897–917 (2002).
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M. Malak, F. Marty, N. Pavy, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250 μm-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst.21(1), 171–180 (2012).
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M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
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R. St-Gelais, J. Masson, and Y.-A. Peter, “All-silicon integrated Fabry-Perot cavity for volume refractive index measurement in microfuidic systems,” Appl. Phys. Lett.94(24), 243905 (2009).
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O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature444(7115), 71–74 (2006).
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M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett.92(8), 081101 (2008).
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M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett.92(8), 081101 (2008).
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M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Integrated waveguide Fabry-Perot microcavities with silicon/air Bragg mirrors,” Opt. Lett.32(5), 533–535 (2007).
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F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J.36(7), 673–677 (2005).
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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 Gravity19(5), 897–917 (2002).
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B. Saadany, M. Malak, M. Kubota, F. Marty, Y. Mita, D. Khalil, and T. Bourouina, “Free-space tunable and drop optical filters using vertical Bragg mirrors on silicon,” J. Sel. Top. Quantum Electron.12(6), 1480–1488 (2006).
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F. Marty, L. Rousseau, B. Saadany, B. Mercier, O. Français, Y. Mita, and T. Bourouina, “Advanced etching of silicon based on deep reactive ion etching for silicon high aspect ratio microstructures and three dimensional micro- and nanostructures,” Microelectron. J.36(7), 673–677 (2005).
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F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
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D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature437(7056), 231–234 (2005).
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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(6926), 925–928 (2003).
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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 Gravity19(5), 897–917 (2002).
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R. St-Gelais, J. Masson, and Y.-A. Peter, “All-silicon integrated Fabry-Perot cavity for volume refractive index measurement in microfuidic systems,” Appl. Phys. Lett.94(24), 243905 (2009).
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M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett.92(8), 081101 (2008).
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M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Integrated waveguide Fabry-Perot microcavities with silicon/air Bragg mirrors,” Opt. Lett.32(5), 533–535 (2007).
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F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett.104(16), 163903 (2010).
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AIP Adv (1)

M. Malak, A.-F. Obaton, F. Marty, N. Pavy, S. Didelon, P. Basset, and T. Bourouina, “Analysis of micromachined Fabry-Perot cavities using phase-sensitive optical low coherence interferometry: insight on dimensional measurements of dielectric layers,” AIP Adv2(2), 022143 (2012).
[CrossRef]

Appl. Phys. Lett. (4)

M. Malak, N. Pavy, F. Marty, Y.-A. Peter, A. Q. Liu, and T. Bourouina, “Micromachined Fabry–Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett.98(21), 211113 (2011).
[CrossRef]

R. St-Gelais, J. Masson, and Y.-A. Peter, “All-silicon integrated Fabry-Perot cavity for volume refractive index measurement in microfuidic systems,” Appl. Phys. Lett.94(24), 243905 (2009).
[CrossRef]

W. Z. Song, X. M. Zhang, A. Q. Liu, C. S. Lim, P. H. Yap, and H. M. M. Hosseini, “Refractive index measurement of single living cells using on-chip Fabry-Perot cavity,” Appl. Phys. Lett.89(20), 203901 (2006).
[CrossRef]

M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett.92(8), 081101 (2008).
[CrossRef]

Class. Quantum Gravity (1)

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 Gravity19(5), 897–917 (2002).
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J. Low Temp. Phys. (1)

D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys.30(5-6), 621–629 (1978).
[CrossRef]

J. Microelectromech. Syst. (2)

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of Fabry-Perot architectures with different mirror shapes; (a) planar mirrors, (b) cylindrical mirrors providing 1D-confinment of light, (c) spherical mirror providing 2D-confinment of light and (d) cylindrical mirrors combined with a fiber-rod-lens, also providing 2D-confinment of light.

Fig. 2
Fig. 2

SEM photos of a fabricated curved FP cavity with (a) single silicon layer, (b) multiple silicon layers (three in this case); the inset shows a magnified view on a silicon-air Bragg reflector.

Fig. 3
Fig. 3

HFSS simulation results illustrating the issue of light confinement (a) 2D confinement inside FP cavity based on cylindrical Bragg mirrors. The light beam is confined only along XZ, at λ = 1480 nm, thanks to the cylindrical surfaces while it diverges along the Y-axis where the cavity behaves as a conventional FP cavity with planar reflectors. (b) 3D light confinement inside FP cavity based on cylindrical Bragg mirrors and a Fiber-Rod-Lens (FRL). The light beam exhibits 3D confinement, at λ = 1455 nm, thanks to the combination of cylindrical reflectors and the FRL. Geometrical parameters of the simulated cavities will be given in section 2.4

Fig. 4
Fig. 4

Schematic representation of the measurement setup with an inset related to the arrangement of the curved cavity with respect to positions of the fiber input and fiber output.

Fig. 5
Fig. 5

Measured spectral responses for a FP cavity with cylindrical silicon-air Bragg reflectors having one and two silicon layers per mirror, respectively, illustrating the improvement of resonator finesse with the number of layers.

Fig. 6
Fig. 6

Measured spectral responses of a FP cavity having a single silicon layer per mirror. The fiber-to-mirror coupling distance was varied zin = 150 µm, 300 µm and 460 µm. The most selective coupling to fundamental modes of type (0,0) is achieved when zin = 300 µm, also corresponding to twice the position of the fiber beam waist. As for the transverse mode (2,0), either effective coupling or noticeable extinction (of ratio of 7:1) is obtained, when zin = 150 µm and zin = 300 µm, respectively.

Fig. 7
Fig. 7

Spectral response obtained by HFSS simulation; (a) for the simple curved cavity, the mode orders are mentioned beside each resonance peak, (b) Spectral response of the FRL cavity obtained by HFSS simulation.

Fig. 8
Fig. 8

HFSS simulation for the FRL cavity at λ = 1518.7 nm. Multi-spot are observed at the mid-plane and they reveal the excitation of higher order modes in the FRL cavity.

Fig. 9
Fig. 9

Fitting of measured data to a combination of only two contributing modes of type TEM00 and TEM20: a) for zin = 150 µm, Γ00 = 63% and Γ20 = 37%, b) for zin = 300 µm, Γ00 = 99% and Γ20 = 1% and c) for zin = 460 µm, Γ00 = 89% and Γ20 = 11%

Fig. 10
Fig. 10

Field map of the numerical simulation carried for the simple curved cavity at λ = 1420.8 nm along the XZ plan. The combined effect of modes (2,0,4) and (0,0,5) is illustrated.

Tables (5)

Tables Icon

Table 1 Theoretical and experimental resonance wavelengths for the different (m,n,q) cavity modes in wavelength range between 1528 and 1545 nm. The wavelength values observed experimentally are given in italic.

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Table 2 Summary of experimental results comparing single and double layers cavities of the same length

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Table 3 Theoretical and numerical resonance wavelengths for the different (m,n,q) cavity modes in wavelength range between 1250 and 1560 nm. The wavelength values observed experimentally are given in italic.

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Table 4 Power coupling efficiencies Γm,0 between the fiber mode (Gaussian) and the (m,0) resonator modes (Hermite-Gaussian); calculated values at λ = 1530 nm and values obtained by fitting to experimental results.

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Table 5 Intra-cavity round-trip coupling efficiency γ0,0 and Q-factor for a cavity with single silicon layer, with expected mirror reflectance = 72%, calculated at λ = 1535 nm for mode (0,0) at three different distances zin.

Equations (7)

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E Fiber ( x,y,z=0 )= 2 w x w y e ( x 2 w x 2 + y 2 w y 2 ) e jk( x 2 2 R x + y 2 2 R y )
ψ m,0 ( x,y )= 2 w x w y H m ( 2 x w x ) e ( x 2 w x 2 + y 2 w y 2 ) e jk( x 2 2 R x + y 2 2 R y )
ν m,0,q = c 2L [ q+( 1+m ) arc cos( 1L/ρ ) π ]
Γ m,0 = | ( D ) E Fiber ( x,y,0 ) ψ m,0 ( x,y,0 )dx dy | 2 ( D ) | E Fiber ( x,y,0 ) | 2 dx dy ( D ) | ψ m,0 ( x,y,0 ) | 2 dx dy
H cav = 1 1+ 4 γ ( 1 γ ) 2 sin 2 ( 2πL λ 0 )
Q= 2πL λ 0 ( γ 1 γ )
γ m,0 = | ( D ) ψ m,0 ( x,y,0 ) ψ m,0 * ( x,y,2L )dx dy | 2 ( D ) | ψ m,0 ( x,y,0 ) | 2 dx dy ( D ) | ψ m,0 ( x,y,2L ) | 2 dx dy

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