## Analysis of Fabry-Perot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors |

Optics Express, Vol. 21, Issue 2, pp. 2378-2392 (2013)

http://dx.doi.org/10.1364/OE.21.002378

Acrobat PDF (2790 KB)

### 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

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

^{9}[2

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]

*Q*) but also small modal volumes (

*V*), leading to a figure of merit of the form (

*Q/V*) [8

8. K. J. Vahala, “Optical microcavities,” Nature **424**(6950), 839–846 (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]

*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 [16

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]

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]

*Q.L*was around 10

^{6}μm for a silicon cavity of length

*L*= 12 μm [17

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]

*Q.L*= 1.6 x10

^{5}μm for a much larger cavity of length

*L*= 210 μm.

## 2. All-silicon resonator with cylindrical Bragg reflectors

### 2.1 Silicon resonator architecture

*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 3

*ρ*/2. The device is realized in a single technology step using Deep Reactive Ion Etching (DRIE) [21

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]

*Q*-factor.

*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

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]

### 2.2 Resonator analytical model

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 *μ*m-gap silicon Fabry–Perot cavities,” J. Microelectromech. Syst. **21**(1), 171–180 (2012). [CrossRef]

- (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 H*: this second term describes the cavity response at the different resonance modes. It involves an intra-cavity round-trip coupling efficiency_{cav}*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*·*H*._{cav}

*(b) Ψ*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_{m,n}*Ψ*[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,n}*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:where*w*and_{x}*w*denote the beam size in the_{y}*x*and the*y*directions;*R*and_{x}*R*are the beam radii of curvature in the_{y}*x*and*y*directions;*H*is the Hermite polynomial function of order_{m}*m*.*(c)*As we are working in the limits of the paraxial approximation, we adopted the scalar notation for*Ψ*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]._{m,0}*(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*Ψ*for each longitudinal mode of order_{m,0}*q*. All these modes have their own resonance frequencies, written as follows:*(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

*L*. 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

_{eff}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]

*z*of the input fiber with respect to the entrance mirror was modulated; three different positions were tested:

_{in}*z*= 150 μm,

_{in}*z*= 300 μm and

_{in}*z*= 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.

_{in}### 2.4 Resonator numerical model

- 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.

*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.

## 3. Analysis of simple curved cavities

### 3.1 Fiber-to-cavity power coupling efficiency Γ

*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).

*E*injected from a single mode optical fiber (fundamental Gaussian mode) and the field of the different transverse cavity modes

_{Fiber}*Ψ*at the cavity entrance. The calculation is done according to a normalized overlap integral:

_{m,0}*E*) is purely Gaussian; it has a beam waist of

_{Fiber}*w*= 18 μm and waist position at 150 μm from the fiber output. The transverse cavity modes (

_{0}*Ψ*) from

_{m,0}*m = 0*to

*m = 4*were examined. The calculated coupling efficiencies

*Γ*are shown in Table 4 below for a wavelength of 1530 nm and coupling distances

_{m,0}*z*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

_{in}*z*= 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

_{in}*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 (

*z*). A case of particular interest is when the coupling distance

_{in}*z*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.

_{in}*z*= 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

_{in}*z*= 150 μm. Mode (4,0) has much reduced coupling efficiency and, as expected, it is not easily observable in the experiments.

_{in}*z*= 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

_{in}*z*= 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.

_{in}*Γ*for the different modes at different positions

*z*. From the experimental results presented above, noticeable coupling is observed only on the modes TEM

_{in}_{00}and TEM

_{20}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

*Γ*and

_{00}*Γ*and then, we fitted this combination to the experimental responses as shown in Fig. 9 .

_{20}*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.

### 3.2 Fiber-to-cavity power coupling efficiency Γ

*L*, the cavity length, ℜ, the mirror reflectance,

*λ*, the free space wavelength and the quantity

_{o}*γ*=

*e*

^{–2}

*defined as the power attenuation factor after a full round-trip inside the cavity, where*

^{αL}*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]:

*γ*=

*γ*

_{m}_{,0}of the

*(m,0)*modes is estimated at different input positions (

*z*) of the excitation fiber, as illustrated in Fig. 4.

_{in}*γ*is obtained from the following overlap integral between the mode fields before and after full round trip along the cavity length, respectively:where (D) is the mirror domain. The obtained values of

_{m,0}*γ*and the corresponding

_{0,0}*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.

## 4. Conclusions

*z*. Moreover, a de-convolution process has been pursued on the measured results obtained for

_{in}*z*to evaluate the experimental coupling coefficients for modes TEM

_{in}_{00}and TEM

_{20}. 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. |

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. |

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. |

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 |

5. | D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature |

6. | O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature |

7. | H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science |

8. | K. J. Vahala, “Optical microcavities,” Nature |

9. | D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high- |

10. | D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High- |

11. | D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express |

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 |

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. |

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 |

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. | M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “In-plane microelectromechanical resonator with integrated Fabry–Perot cavity,” Appl. Phys. Lett. |

17. | M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Integrated waveguide Fabry-Perot microcavities with silicon/air Bragg mirrors,” Opt. Lett. |

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. |

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. |

20. | A. Lipson and E. M. Yeatman, “A 1-D photonic band gap tunable optical filter in (110) silicon,” J. Microelectromech. Syst. |

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. |

22. | A. Yariv, |

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. |

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 |

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 |

26. | T. Verdeyen, |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(230.1480) Optical devices : Bragg reflectors

(230.3990) Optical devices : Micro-optical devices

(140.3948) Lasers and laser optics : Microcavity devices

(080.4228) Geometric optics : Nonspherical mirror surfaces

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 28, 2012

Revised Manuscript: December 13, 2012

Manuscript Accepted: December 20, 2012

Published: January 24, 2013

**Virtual Issues**

Vol. 8, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Maurine Malak, Noha Gaber, Frédéric Marty, Nicolas Pavy, Elodie Richalot, and Tarik Bourouina, "Analysis of Fabry-Perot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors," Opt. Express **21**, 2378-2392 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-2378

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### References

- C. Zener, “Internal friction in solids. Pt. II: general theory of thermoelastic internal friction,” Phys. Rev.53(1), 90–99 (1938). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science298(5597), 1372–1377 (2002). [CrossRef] [PubMed]
- 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]
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