Abstract

Three-mode optoacoustic parametric amplifiers (OAPAs), in which a pair of photon modes are strongly coupled to an acoustic mode, provide a general platform for investigating self-cooling, parametric instability and very sensitive transducers. Their realization requires an optical cavity with tunable transverse modes and a high quality-factor mirror resonator. This paper presents the design of a table-top OAPA based on a near-self-imaging cavity design, using a silicon torsional microresonator. The design achieves a tuning coefficient for the optical mode spacing of 2.46MHz/mm. This allows tuning of the mode spacing between amplification and self-cooling regimes of the OAPA device. Based on demonstrated resonator parameters (frequencies 400kHz and quality-factors 7.5×105) we predict that the OAPA can achieve parametric instability with 1.6 μW of input power and mode cooling by a factor of 1.9×104 with 30 mW of input power.

© 2014 Optical Society of America

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2013 (3)

C. Blair, S. Susmithan, C. Zhao, F. Qi, L. Ju, and D. Blair, “Radiation pressure excitation of test mass ultrasonic modes via three mode opto-acoustic interactions in a suspended Fabry–Pérot cavity,” Phys. Lett. A 377, 1970–1973 (2013).
[CrossRef]

F. A. Torres, P. Meng, L. Ju, C. Zhao, D. G. Blair, K.-Y. Liu, S. Chao, M. Martyniuk, I. Roch-Jeune, R. Flaminio, and C. Michel, “High quality factor mg-scale silicon mechanical resonators for 3-mode optoacoustic parametric amplifiers,” J. Appl. Phys. 114, 014506 (2013).
[CrossRef]

R. Nawrodt, C. Schwarz, S. Kroker, I. W. Martin, R. Bassiri, F. Brückner, L. Cunningham, G. D. Hammond, D. Heinert, J. Hough, T. Käsebier, E.-B. Kley, R. Neubert, S. Reid, S. Rowan, P. Seidel, and A. Tünnermann, “Investigation of mechanical losses of thin silicon flexures at low temperatures,” Class. Quantum Grav. 30, 115008 (2013).
[CrossRef]

2012 (4)

S. P. Vyatchanin, “Parametric oscillatory instability in laser gravitational antennas,” Phys. Usp. 55, 302–305 (2012).

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[CrossRef]

G. Bahl, M. Tomes, F. Marquardt, and T. Carmon, “Observation of spontaneous Brillouin cooling,” Nat. Phys. 8, 203–207 (2012).
[CrossRef]

E. Serra, A. Borrielli, F. S. Cataliotti, F. Marin, F. Marino, A. Pontin, G. A. Prodi, and M. Bonaldi, “An ultra-low dissipation micro-oscillator for quantum opto-mechanics,” Phys. Rev. A 86, 051801 (2012).
[CrossRef]

2011 (2)

J. Miller, M. Evans, L. Barsotti, P. Fritschel, M. MacInnis, R. Mittleman, B. Shapiro, J. Soto, and C. Torrie, “Damping parametric instabilities in future gravitational wave detectors by means of electrostatic actuators,” Phys. Lett. A 375, 788–794 (2011).
[CrossRef]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).

2010 (7)

G. M. Harry, “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[CrossRef]

M. Evans, L. Barsotti, and P. Fritschel, “A general approach to optomechanical parametric instabilities,” Phys. Lett. A 374, 665–671 (2010).
[CrossRef]

J. P. Davis, D. Vick, D. C. Fortin, J. A. J. Burgess, W. K. Hiebert, and M. R. Freeman, “Nanotorsional resonator torque magnetometry,” Appl. Phys. Lett. 96, 072513 (2010).
[CrossRef]

J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett. 104, 033901 (2010).
[CrossRef]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[CrossRef]

J. B. Khurgin, “Viewpoint: phonon lasers gain a sound foundation,” Physics 3, 16 (2010).
[CrossRef]

B. Chalopin, A. Chiummo, C. Fabre, A. Matre, and N. Treps, “Frequency doubling of low power images using a self-imaging cavity,” Opt. Express 18, 8033–8042 (2010).
[CrossRef]

2009 (2)

S. A. Chandorkar, R. N. Candler, A. Duwel, R. Melamud, M. Agarwal, K. E. Goodson, and T. W. Kenny, “Multimode thermoelastic dissipation,” J. Appl. Phys. 105, 043505 (2009).
[CrossRef]

C. Zhao, L. Ju, H. Miao, S. Gras, Y. Fan, and D. G. Blair, “Three-mode optoacoustic parametric amplifier: a tool for macroscopic quantum experiments,” Phys. Rev. Lett. 102, 243902 (2009).
[CrossRef]

2008 (5)

H. Miao, C. Zhao, L. Ju, S. Gras, P. Barriga, Z. Zhang, and D. G. Blair, “Three-mode optoacoustic parametric interactions with a coupled cavity,” Phys. Rev. A 78, 063809 (2008).
[CrossRef]

R. Nawrodt, A. Zimmer, T. Koettig, C. Schwarz, D. Heinert, M. Hudl, R. Neubert, M. Thürk, S. Nietzsche, W. Vodel, P. Seidel, and A. Tünnermann, “High mechanical Q-factor measurements on silicon bulk samples,” J. Phys. 122, 012008 (2008).

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef]

C. Zhao, L. Ju, Y. Fan, S. Gras, B. J. J. Slagmolen, H. Miao, P. Barriga, D. G. Blair, D. J. Hosken, A. F. Brooks, P. J. Veitch, D. Mudge, and J. Munch, “Observation of three-mode parametric interactions in long optical cavities,” Phys. Rev. A 78, 023807 (2008).
[CrossRef]

K. Uhlig, “Dry dilution refrigerator with high cooling power,” AIP Conf. Proc. 985, 1287 (2008).
[CrossRef]

2007 (2)

T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, “An all-optical trap for a gram-scale mirror,” Phys. Rev. Lett. 98, 150802 (2007).
[CrossRef]

G. M. Harry, M. R. Abernathy, A. E. Becerra-Toledo, H. Armandula, E. Black, K. Dooley, M. Eichenfield, C. Nwabugwu, A. Villar, D. R. M. Crooks, G. Cagnoli, J. Hough, C. R. How, I. MacLaren, P. Murray, S. Reid, S. Rowan, P. H. Sneddon, M. M. Fejer, R. Route, S. D. Penn, P. Ganau, J.-M. Mackowski, C. Michel, L. Pinard, and A. Remillieux, “Titania-doped tantala/silica coatings for gravitational-wave detection,” Class. Quantum Grav. 24, 405–415 (2007).
[CrossRef]

2006 (7)

L. Ju, S. Gras, C. Zhao, J. Degallaix, and D. G. Blair, “Multiple modes contributions to parametric instabilities in advanced laser interferometer gravitational wave detectors,” Phys. Lett. A 354, 360–365 (2006).
[CrossRef]

Y. Mita, M. Kubota, T. Harada, F. Marty, B. Saadany, T. Bourouina, and T. Shibata, “Contour lithography methods for drie fabrication of nanometre–millimetre-scale coexisting microsystems,” J. Micromech. Microeng. 16, S135–S141 (2006).
[CrossRef]

S. Reid, G. Cagnoli, D. R. M. Crooks, J. Hough, P. Murray, S. Rowan, M. M. Fejer, R. Route, and S. Zappe, “Mechanical dissipation in silicon flexures,” Phys. Lett. A 351, 205–211 (2006).
[CrossRef]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

A. Schliesser, P. Delaye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor,” Phys. Rev. Lett. 97, 133601 (2006).
[CrossRef]

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

2005 (2)

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 33901 (2005).
[CrossRef]

C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, “Parametric instabilities and their control in advanced interferometer gravitational-wave detectors,” Phys. Rev. Lett. 94, 121102 (2005).
[CrossRef]

2004 (1)

U. Gysin, S. Rast, P. Ruff, E. Meyer, D. W. Lee, P. Vettiger, and C. Gerber, “Temperature dependence of the force sensitivity of silicon cantilevers,” Phys. Rev. B 69, 045403 (2004).
[CrossRef]

2002 (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Analysis of parametric oscillatory instability in power recycled LIGO interferometer,” Phys. Lett. A 305, 111–124 (2002).
[CrossRef]

2001 (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry–Perot interferometer,” Phys. Lett. A 287, 331–338 (2001).
[CrossRef]

1999 (1)

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

1989 (1)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1969 (1)

Abernathy, M. R.

G. M. Harry, M. R. Abernathy, A. E. Becerra-Toledo, H. Armandula, E. Black, K. Dooley, M. Eichenfield, C. Nwabugwu, A. Villar, D. R. M. Crooks, G. Cagnoli, J. Hough, C. R. How, I. MacLaren, P. Murray, S. Reid, S. Rowan, P. H. Sneddon, M. M. Fejer, R. Route, S. D. Penn, P. Ganau, J.-M. Mackowski, C. Michel, L. Pinard, and A. Remillieux, “Titania-doped tantala/silica coatings for gravitational-wave detection,” Class. Quantum Grav. 24, 405–415 (2007).
[CrossRef]

Agarwal, M.

S. A. Chandorkar, R. N. Candler, A. Duwel, R. Melamud, M. Agarwal, K. E. Goodson, and T. W. Kenny, “Multimode thermoelastic dissipation,” J. Appl. Phys. 105, 043505 (2009).
[CrossRef]

Arcizet, O.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor,” Phys. Rev. Lett. 97, 133601 (2006).
[CrossRef]

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

Armandula, H.

G. M. Harry, M. R. Abernathy, A. E. Becerra-Toledo, H. Armandula, E. Black, K. Dooley, M. Eichenfield, C. Nwabugwu, A. Villar, D. R. M. Crooks, G. Cagnoli, J. Hough, C. R. How, I. MacLaren, P. Murray, S. Reid, S. Rowan, P. H. Sneddon, M. M. Fejer, R. Route, S. D. Penn, P. Ganau, J.-M. Mackowski, C. Michel, L. Pinard, and A. Remillieux, “Titania-doped tantala/silica coatings for gravitational-wave detection,” Class. Quantum Grav. 24, 405–415 (2007).
[CrossRef]

Arnaud, J. A.

Aspelmeyer, M.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Bahl, G.

G. Bahl, M. Tomes, F. Marquardt, and T. Carmon, “Observation of spontaneous Brillouin cooling,” Nat. Phys. 8, 203–207 (2012).
[CrossRef]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).

Barriga, P.

C. Zhao, L. Ju, Y. Fan, S. Gras, B. J. J. Slagmolen, H. Miao, P. Barriga, D. G. Blair, D. J. Hosken, A. F. Brooks, P. J. Veitch, D. Mudge, and J. Munch, “Observation of three-mode parametric interactions in long optical cavities,” Phys. Rev. A 78, 023807 (2008).
[CrossRef]

H. Miao, C. Zhao, L. Ju, S. Gras, P. Barriga, Z. Zhang, and D. G. Blair, “Three-mode optoacoustic parametric interactions with a coupled cavity,” Phys. Rev. A 78, 063809 (2008).
[CrossRef]

Barsotti, L.

J. Miller, M. Evans, L. Barsotti, P. Fritschel, M. MacInnis, R. Mittleman, B. Shapiro, J. Soto, and C. Torrie, “Damping parametric instabilities in future gravitational wave detectors by means of electrostatic actuators,” Phys. Lett. A 375, 788–794 (2011).
[CrossRef]

M. Evans, L. Barsotti, and P. Fritschel, “A general approach to optomechanical parametric instabilities,” Phys. Lett. A 374, 665–671 (2010).
[CrossRef]

Bassiri, R.

R. Nawrodt, C. Schwarz, S. Kroker, I. W. Martin, R. Bassiri, F. Brückner, L. Cunningham, G. D. Hammond, D. Heinert, J. Hough, T. Käsebier, E.-B. Kley, R. Neubert, S. Reid, S. Rowan, P. Seidel, and A. Tünnermann, “Investigation of mechanical losses of thin silicon flexures at low temperatures,” Class. Quantum Grav. 30, 115008 (2013).
[CrossRef]

Bäuerle, D.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Becerra-Toledo, A. E.

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F. A. Torres, P. Meng, L. Ju, C. Zhao, D. G. Blair, K.-Y. Liu, S. Chao, M. Martyniuk, I. Roch-Jeune, R. Flaminio, and C. Michel, “High quality factor mg-scale silicon mechanical resonators for 3-mode optoacoustic parametric amplifiers,” J. Appl. Phys. 114, 014506 (2013).
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J. P. Davis, D. Vick, D. C. Fortin, J. A. J. Burgess, W. K. Hiebert, and M. R. Freeman, “Nanotorsional resonator torque magnetometry,” Appl. Phys. Lett. 96, 072513 (2010).
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J. P. Davis, D. Vick, D. C. Fortin, J. A. J. Burgess, W. K. Hiebert, and M. R. Freeman, “Nanotorsional resonator torque magnetometry,” Appl. Phys. Lett. 96, 072513 (2010).
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Figures (5)

Fig. 1.
Fig. 1.

Micromechanical silicon resonator with optical coating. (a) Dimensions: 20 mm by 20 mm frame, of thickness 500 μm, and three paddles (1 mm wide). The central paddle contains a 0.8 mm square 14-layer tantala/silica optical coating (total thickness 5.0μm). (b) Photograph showing paddles and central coating. (c) Finite element modeling (FEM) projection of the torsion mode of interest in which the central paddle has maximum amplitude, shown by the contour plot of deformation.

Fig. 2.
Fig. 2.

NSI optical-cavity concept and key parameters. A torsional microresonator (M1) interacts with the TEM00 mode and the TEM01 mode. We can continuously change the mode gap between the two optical modes by tuning the position of the lens LT. W1 and W2 are the waist sizes before and after the lens LT, respectively. In a practical cavity, the tuning lens LT is replaced with a tuning mirror MT (shown later in Fig. 4).

Fig. 3.
Fig. 3.

Mode gap as a function of the position of the micromechanical resonator relative to the lens, with fixed parameters of the radius of curvature and focal length, R0=50mm and f=50mm. The tuning gap is 325 μm and the tuning coefficient is 2.46MHz/mm. The tuning is almost linear and symmetric around the self-imaging point.

Fig. 4.
Fig. 4.

Experimental setup of the NSI cavity for three-mode interactions. The laser light passes through a PMC to clean up high-order modes, and is then phase modulated by an electro-optic modulator (EOM) for PDH locking of the laser frequency to the NSI cavity. A 400 kHz torsional microresonator acts as the end mirror of the NSI cavity, which interacts with the cavity TEM00 and TEM01 modes. The sum signal from the quadrant photodetector is used for PDH locking while its differential signal is used to monitor the resonator amplitude. The mode gap between the two optical modes is adjusted by tuning the position of MT.

Fig. 5.
Fig. 5.

Expected parametric gain for given laser input power values. Parametric instability (gain R=1) is achieved with 1.6 μW (left vertical–dotted line). The red curve is the achievable acoustic mode amplification and instability (for R1), and the blue curve is mode cooling. Our system allows for 30 mW of input power before the damage threshold of optical coatings is reached. With 30 mW of input power, a cooling factor 1.9×104 is reached (right vertical–dotted line).

Equations (16)

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R=8IinQ0Q1QmΛmL2ω0ωm2,
ν=c2L[p+απ(m+n+1)],
L1+L2=f+R0,
L3=f+f2/R0.
α=ϕ1+ϕ2+ϕ3.
ϕ=arctan(z/zR),
ϕ3=arctan(L3zR),
q=fqfq,
L2=f+(L3f)f2(L3f)2+zR2,
W1=f2W22(L3f)2+zR2.
ϕ2=arctan(L2zR),
zR=πW12λ.
R0=L1(1+(πW12λL1)2).
ϕ1=arctan(L1zR).
Δν=c2π(L1+L2+L3)(ϕ1+ϕ2+ϕ3kπ).
Teff=T01R,

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