Abstract

We show theoretically that the absolute frequency stability of a solid-state millimeter-scale whispering gallery mode resonator can reach one part per 1014 per 1 s integration time if proper crystalline material as well as proper stabilization technique is selected. Both the fluctuations of the resonator temperature and the fluctuations of the temperature in the mode volume can be measured with the sensitivity better than the fundamental thermodynamic limit and actively compensated.

© 2007 Optical Society of America

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2007 (4)

2006 (8)

T. J. Johnson, M. Borselli, and O. Painter, "Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator," Opt. Express 14, 817-831 (2006).
[CrossRef] [PubMed]

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
[CrossRef]

H. Stoehr, F. Mensing, J. Helmcke, and U. Sterr, "Diode laser with 1 Hz linewidth," Opt. Lett. 31, 736-738 (2006).
[CrossRef] [PubMed]

A. B. Matsko and V. S. Ilchenko, "Optical resonators with whispering gallery modes I: basics," IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, "Optical resonators with whispering gallery modes II: applications," IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006).
[CrossRef]

I. S. Grudinin, A. Savchenkov, A. B. Matsko, D. Strekalov, V. Ilchenko, and L. Maleki, "Ultra high Q crystalline microcavities," Opt. Commun. 265, 33-38 (2006).
[CrossRef]

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, and L. Maleki, "White-light whispering gallery mode resonators," Opt. Lett. 31, 92-94 (2006).
[CrossRef] [PubMed]

2005 (4)

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, "Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef] [PubMed]

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, 033901 (2005).
[CrossRef] [PubMed]

M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, "Simple and compact 1-Hz laser system via an improved mounting configuration of a reference cavity," Opt. Lett. 30, 1815-1817 (2005).
[CrossRef] [PubMed]

A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, "Nonstationary nonlinear effects in optical microspheres," J. Opt. Soc. Am. B 22, 459-465 (2005).
[CrossRef]

2004 (6)

M. L. Gorodetsky and I. S. Grudinin, "Fundamental thermal fluctuations in microspheres," J. Opt. Soc. Am. B 21, 697-705 (2004).
[CrossRef]

S. A. Webster, M. Oxborrow, and P. Gill, "Subhertz-linewidth Nd:YAG laser," Opt. Lett. 29, 1497-1499 (2004).
[CrossRef] [PubMed]

K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804R (2004).
[CrossRef]

2003 (3)

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

M. Eichenseer, A. Yu. Nevsky, Ch. Schwedes, J. von Zanthier, and H. Walther, "Towards an indium single-ion optical frequency standard," J. Phys. B 36, 553-559 (2003).
[CrossRef]

2002 (2)

2001 (2)

K. Shimamura, H. Sato, A. Bensalah, V. Sudesh, H. Machida, N. Sarukura, and T. Fukuda, "Crystal growth of fluorides for optical applications," Cryst. Res. Technol. 36, 801-813 (2001).
[CrossRef]

V. V. Datsyuk and I. A. Izmailov, "Optics of microdroplets," Usp. Fiz. Nauk 171, 1117-1129 (2001) V. V. Datsyuk and I. A. Izmailov,[Phys. Usp. 44, 1061-1073 (2001)].
[CrossRef]

2000 (2)

M. H. Fields, J. Popp, and R. K. Chang, "Nonlinear optics in microspheres," Prog. Opt. 41, 1-95 (2000).
[CrossRef]

S. M. Etzel, A. H. Rose, and C. M. Wang, "Dispersion of the temperature retardance in SiO2 and MgF2," Appl. Opt. 39, 5796-5800 (2000).
[CrossRef]

1999 (3)

1998 (2)

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

A. W. Sleight, "Isotropic negative thermal expansion," Annu. Rev. Mater. Sci. 28, 29-43 (1998).
[CrossRef]

1997 (2)

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering gallery mode resonances using a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
[CrossRef] [PubMed]

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1 Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

1995 (1)

F. L. Walls and J. R. Vig, "Fundamental limits on the frequency stabilities of crystal oscillators," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 576-589 (1995).
[CrossRef]

1994 (2)

1992 (3)

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-Hertz relative frequency stabilization of 2-diode laser-pumped Nd:YAG lasers locked to a Fabry-Perot interferometer," IEEE J. Quantum Electron. 28, 1106-1117 (1992).
[CrossRef]

J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
[CrossRef]

M. L. Gorodetsky and V. S. Ilchenko, "Thermal nonlinear effects in optical whispering-gallery microresonators," Laser Phys. 2, 1004-1009 (1992).

1991 (1)

S. Shiller and R. L. Byer, "High-resolution spectroscopy of whispering gallery modes in large dielectric spheres," Opt. Lett. 16, 130-132 (1991).

1989 (1)

S. Biernacki and M. Scheffler, "Negative thermal expansion of diamond and zinc-blende semiconductors," Phys. Rev. Lett. 63, 290-293 (1989).
[CrossRef] [PubMed]

1988 (2)

D. T. Morelli and J. Heremans, "Thermal conductivity of single-crystal barium fluoride," J. Appl. Phys. 63, 573-574 (1988).
[CrossRef]

Ch. Salomon, D. Hils, and J. L. Hall, "Laser stabilization at the millihertz level," J. Opt. Soc. Am. B 5, 1576-1587 (1988).
[CrossRef]

1987 (1)

S. Andersson and G. Backstron, "Thermal conductivity and heat capacity of single-crystal LiF and CaF2 under hydrostatic pressure," J. Phys. C 20, 5951-5962 (1987).
[CrossRef]

1985 (1)

M. O. Manasreh and D. O. Pederson, "Elastic constants of barium fluoride from 300to1250 K," Phys. Rev. B 31, 3960-3964 (1985).
[CrossRef]

1984 (1)

1983 (2)

T. Toyoda and M. Yabe, "The temperature dependence of the refractive indices of fused silica and crystal quartz," J. Phys. D 16, L97-L100 (1983).
[CrossRef]

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]

1982 (2)

T. H. K. Barron, J. F. Collins, T. W. Smith, and G. K. White, "Thermal expansion, Gruneisen functions and static lattice properties of quartz," J. Phys. C 15, 4311-4326 (1982).
[CrossRef]

D. A. Ditmars, S. A. Ishihara, S. S. Chang, G. Bernstein, and E. D. West, "Enthalpy and heat-capacity standard reference material—synthetic sapphire (α-Al2O3) from 10to2250 K," J. Res. Natl. Bur. Stand. 87, 159-163 (1982).

1978 (1)

V. B. Braginsky and S. P. Vyatchanin, "Gravitational waves and limiting stability of self-excited oscillators," Sov. Phys. JETP 74, 828-832 (1978).

1974 (1)

W. M. Yim and R. J. Paff, "Thermal expansion of AIN, sapphire, and silicon," J. Appl. Phys. 45, 1456-1457 (1974).
[CrossRef]

1962 (1)

1961 (1)

G. A. Slack, "Thermal conductivity of CaF2, MnF2, CoF2, and ZnF2 crystals," Phys. Rev. 122, 1451-1464 (1961).
[CrossRef]

1958 (2)

R. Srinivasan, "Elastic constants of calcium fluoride," Proc. Phys. Soc. London 72, 566-575 (1958).
[CrossRef]

A. Duncanson and R. W. H. Stevenson, "Some properties of magnesium fluoride crystallized from the melt," Proc. Phys. Soc. London 72, 1001-1006 (1958).
[CrossRef]

1955 (1)

A. Smakula and V. Sils, "Precision density determination of large single crystals by hydrostatic weighing," Phys. Rev. 99, 1744-1746 (1955).
[CrossRef]

1949 (1)

S. S. Todd, "Heat capacities at low temperatures and entropies of magnesium and calcium fluorides," J. Am. Chem. Soc. 71, 4115-4116 (1949).
[CrossRef]

1938 (1)

K. S. Pitzer, W. V. Smith, and W. M. Latimer, "The heat capacity and entropy of barium fluoride, cesium perchlorate and lead phosphate," J. Am. Chem. Soc. 60, 1826-1828 (1938).
[CrossRef]

Annu. Rev. Mater. Sci. (1)

A. W. Sleight, "Isotropic negative thermal expansion," Annu. Rev. Mater. Sci. 28, 29-43 (1998).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. B (2)

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]

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
[CrossRef]

Cryst. Res. Technol. (1)

K. Shimamura, H. Sato, A. Bensalah, V. Sudesh, H. Machida, N. Sarukura, and T. Fukuda, "Crystal growth of fluorides for optical applications," Cryst. Res. Technol. 36, 801-813 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

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Figures (7)

Fig. 1
Fig. 1

Thermorefractive Allan variance of the frequency of a mode of a cylindrical calcium fluoride WGM resonator with R = 0.3 cm and L = 0.01 cm .

Fig. 2
Fig. 2

Temperature dependence of the thermorefractive deviation of the ordinary ( n 0 = Δ n + 1.37191 ) and extraordinary ( n e = Δ n + 1.38341 ) indexes of refraction of magnesium fluoride at 1.55 μ m . The dependencies are obtained using expressions n e T [ 0.04183 5.63233 × 10 4 T ] × 10 5 and n o T [ 0.09797 5.57293 × 10 4 T ] × 10 5 found from measurement data [54].

Fig. 3
Fig. 3

Power spectral density and Allan variance of the thermal expansion defined frequency fluctuations of a mode of a cylindrical magnesium fluoride WGM resonator with R = 0.3 cm and L = 0.01 cm .

Fig. 4
Fig. 4

Allan variance of the thermoelastic frequency fluctuations of a mode of a cylindrical magnesium fluoride WGM resonator with R = 0.3 cm and L = 0.01 cm .

Fig. 5
Fig. 5

(a) Thermal compensator for an optical WGM resonator. The compensator consists of (1) a rigid metal frame with thermal expansion coefficient α 1 ; (2) a metal or a glass wedge-shaped spacer with thermal expansion coefficient α 2 ; and (3) a WGM resonator sandwiched between rigid spacers on its top and bottom. (b) Top view of the wedge part of the thermal compensator. The dashed zone is an effective overlap cross section A 2 of spacer (2) and resonator (3). The cross-section area can be tuned continuously with high precision by shifting the spacer up and down with respect to spacer position shown in the picture.

Fig. 6
Fig. 6

Nonlinear thermal compensator for an optical WGM resonator. An enhanced temperature compensation compared with the compensation realized in the linear device (Fig. 5) is achieved by shifting of the working point towards both the first- and the second-order compensation temperature region. The temperature tuning is realized with heater (5). The nonlinearity is introduced by nonlinear element (4).

Fig. 7
Fig. 7

(a) Temperature compensated WGM resonator. The overall temperature of the resonator is kept higher than the room temperature. The output signal from a temperature sensor with an internal temperature reference is used to feed forward to a pressure actuator. The output signal can also be transferred to a voltage signal that changes the WGM frequency (if the resonator is made of quartz or other electro-optic material). (b) Oven controlled WGM resonator. The WGM resonator and other temperature sensitive components are in a temperature stabilized oven that is kept at a temperature where the WGM frequency has zero slope versus the oven temperature d ω d T = 0 (temperature feedback). (c) Microprocessor compensated WGM resonator. Temperature of the WGM resonator is measured and inserted into a smart optical element (low-Q resonator) that compensates for the frequency shift of the WGM. The optical element should memorize the dependence of the WGM frequency on the temperature.

Tables (2)

Tables Icon

Table 1 Linear and Nonlinear Thermorefractive Coefficients of the Ca, Ba, Mg F 2 , Sapphire, and Crystalline Quartz at T 300 K and λ = 1.5 μ m a

Tables Icon

Table 2 Thermorefractive, Thermal Expansion, and Thermoelastic of WGM Frequency Stability at Room Temperature a

Equations (13)

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( Δ ω TR ) 2 ω 2 = α n 2 k B T 2 C V m ρ ,
( Δ ω TE 1 ) 2 ω 2 = α l 2 k B T 2 C V r ρ ,
( Δ ω TE 2 ) 2 ω 2 = k B T β T 9 V r ,
S δ ω ω ( Ω ) k B α n 2 T 2 ρ C V m R 2 12 D [ 1 + ( R 2 D Ω 9 3 ) 3 2 + 1 6 ( R 2 D Ω 8 ν 1 3 ) 2 ] 1 .
σ 2 ( τ ) = 2 π 0 S δ ω ω ( Ω ) sin 4 ( Ω τ 2 ) ( Ω τ 2 ) 2 d Ω ,
S δ ω ω = ( Δ ω TE 1 ) 2 ω 2 2 R 2 π 2 D 1 + ( Ω R 2 D π 2 ) 2 .
Δ ω ̃ RF 2 ω ( α l o α l e ) Δ T R ,
Δ ω TM ω α l o α l o α l e Δ ω ̃ RF 2 ω .
Δ ω = [ ( α n + α l ) ω + d ω d F A 2 E 2 ( α 2 α 1 ) ] Δ T .
Δ ω = { ( α n + α l ) ω + d ω d F [ A 2 E 2 ( α 2 α 1 ) + A Δ T ] } Δ T .
Δ ω RF 1 = ω ( α n o α n e ) Δ T m ,
Δ ω RF 2 ω RF α l Δ T R .
Δ ω ̃ RF 1 = 2 ω ( α n ( ω ) α n ( 2 ω ) ) Δ T m ,

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