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

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    [CrossRef]
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    [CrossRef]
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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.

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]

Didelon, S.

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]

Doherty, A. C.

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

Douglass, D. H.

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]

Fejer, M. 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]

Français, O.

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]

Friedrich, D.

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]

Gianfrani, L.

Gram, R. Q.

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]

Gretarsson, A. 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]

Gutche, H. W.

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]

Harry, G. 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]

Heidmann, A.

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]

Hoffman, A. W.

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]

Hosseini, H. M. M.

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]

Hough, J.

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]

Hult, J.

Ilchenko, V. S.

Jarzynski, 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]

Jones, R. L.

Kaminski, C. F.

Kassi, S.

Khalil, D.

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]

Kimble, H. J.

Kippenberg, T. J.

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]

Kittelberger, S. E.

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]

Kleckner, D.

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

Kley, E.-B.

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]

Kubota, M.

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]

Lam, C. C.

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]

Langridge, J. M.

Laurila, T.

Lim, C. S.

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]

Lipson, A.

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]

Liu, A. Q.

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]

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]

Mabuchi, H.

Malak, M.

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]

Marty, F.

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]

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]

Masson, J.

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]

McGloin, D.

McGuigan, D. F.

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]

Mercier, B.

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]

Mita, Y.

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]

Nakagawa, N.

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]

Obaton, A.-F.

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]

Ouvrard, A.

Pavy, N.

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]

Penn, S. D.

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]

Peter, Y.-A.

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]

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]

Pinard, M.

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]

Pruessner, M. W.

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]

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]

Rabinovich, W. S.

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]

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]

Ritort, F.

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]

Romanini, D.

Rouillard, Y.

Rousseau, L.

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]

Rowan, S.

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]

Saadany, B.

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]

Salhi, A.

Saulson, P. R.

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]

Schnabel, R.

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]

Smith, S. B.

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]

Song, W. Z.

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]

Spillane, S. M.

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]

Startin, W. J.

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]

St-Gelais, R.

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]

Stievater, T. H.

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]

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]

Streed, E. W.

Tinoco, I.

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]

Tünnermann, A.

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]

Vahala, K. J.

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]

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

Vernooy, D. W.

Watt, R. S.

Yap, P. H.

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]

Yeatman, E. M.

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]

Zener, C.

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

Zhang, X. M.

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]

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).
[CrossRef]

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)

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]

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]

J. Sel. Top. Quantum Electron. (1)

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]

Microelectron. J. (1)

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]

Nature (5)

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]

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]

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]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. (1)

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

Phys. Rev. Lett. (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]

Science (1)

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

Other (2)

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

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

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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.

Tables Icon

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

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.

Tables Icon

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.

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.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

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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