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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 11 — May. 28, 2007
  • pp: 6768–6773
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Optical resonators with ten million finesse

Anatoliy A. Savchenkov, Andrey B. Matsko, Vladimir S. Ilchenko, and Lute Maleki  »View Author Affiliations


Optics Express, Vol. 15, Issue 11, pp. 6768-6773 (2007)
http://dx.doi.org/10.1364/OE.15.006768


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Abstract

We report a significant improvement of the finesse and quality factor of a calcium fluoride whispering gallery mode optical resonator achieved with iterative thermal annealing. The initial and final values of the resonator finesse are i ≈ 5 × 105 and f ≥ 107 respectively. The measurements are conducted at room temperature in the laboratory atmosphere.

© 2007 Optical Society of America

1. Introduction

Resonators are usually characterized with two partially dependent parameters – finesse () and quality factor (Q). The finesse of an empty Fabry-Perot (FP) resonator is defined solely by the reflectivity of its mirrors and is calculated as = π√R/(1 - R). The maximum reported value of reflectivity R≃ 1 - 1.6 × 10-6 is achieved with dielectric mirrors [1

1. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363–365 (1992). [CrossRef] [PubMed]

,2

2. C. J. Hood, H. J. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001). [CrossRef]

]. A FP resonator made with the mirrors has finesse = 1.9 × 106. Further practical increase of the finesse of FP resonators is problematic because of the absorption and scattering of light in the mirror material [3

3. H. R. Bilger, P. V. Wells, and G. E. Stedman, “Origins of fundamental limits for reflection losses at multilayer dielectric mirrors,” Appl. Opt. 33, 7390–7396 (1994). [CrossRef] [PubMed]

, 4

4. H.-J. Cho, M.-J. Shin, and J.-C. Lee, “Effects of substrate and deposition method onto the mirror scattering,“, Appl. Opt. 45, 1440–1446 (2006). [CrossRef] [PubMed]

], though fundamental limit on the reflection losses given by the internal material losses and by thermodynamic density fluctuations is of the order of parts in 109 [3

3. H. R. Bilger, P. V. Wells, and G. E. Stedman, “Origins of fundamental limits for reflection losses at multilayer dielectric mirrors,” Appl. Opt. 33, 7390–7396 (1994). [CrossRef] [PubMed]

]. Quality factor of a resonator depends on both its finesse and its geometrical size. The one-dimensional FP resonator has Q = 2 L/λ, where L is the distance between the mirrors, and λ is the wavelength. It is easy to see that the quality factor of the resonator is essentially unlimited because L is unlimited.

and Q are typically equally important in the majority of applications. In some cases, though, finesse is technically more valuable than the quality factor. For instance, the buildup of optical power inside the resonator and the Purcell factor [5

5. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

] are proportional to finesse. Sometimes quality factor is more valuable. For example, the inverse threshold power of intracavity hyperparametric oscillation is proportional to Q 2 [6

6. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005). [CrossRef]

], and efficiency of parametric frequency mixing is proportional to Q 3 [7

7. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and Lute Maleki, “Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities,“ Phys. Rev. Lett. 92, 043903 (2004). [CrossRef] [PubMed]

]. Therefore, it is important to know both the maximally achievable finesse and quality factor values of a resonator.

2. Measurement technique

3. Annealing procedure

Fig. 1. Left: Ring-down signal after one annealing step. Right: Ring-down signal after three annealing steps. The exponential fit (red solid curve) is nonlinear in the logarithmic scale because the exponents have constant offsets. Keeping in mind that Q = ωτ/2, and ω = 2πc/λ, λ = 1.55 μm, we find that the values of the quality factors after the first and third annealing steps are Q = 1.2 × 1010 and Q = 6.7 × 1010 respectively. It is important to note that the initial value of the Q-factor corresponds to the earlier observations [10], while the final value is the apparent improvement at the given wavelength.

We selected a fluorite WGM resonator with optical loss limited by the material attenuation, not by the surface scattering [18

18. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, Art. No. 051804R (2004). [CrossRef]

]. The resonator had 4.5 mm in diameter, 0.5 mm in thickness, and 32 μm in diameter of sidewall curvature. The resonator was placed into the center of a 3-feet long air-filled transparent tube made of annealed fused silica. The tube was installed into a 20 cm long horizontal tube furnace. The heated furnace core had approximately one inch in diameter and three inches in length. We increased the temperature of the furnace core from room temperature to 650 C during 3 hours, kept the temperature stabilized for one day, cooled the core back during 3 hours, and repolished the fluorite resonator. We repeated iterations three times keeping the same annealing duration but gradually decreasing the size of the grain of diamond slurry we used for polishing. We obtained a significant increase of the ringdown time at the end of the process.

4. Results

The measured ringdown spectrum did not change substantially after the first annealing stage. However, the ringdown time increased significantly after the third stage. The measured ring-down signals are shown in Fig. 1. A five-fold increase of the optical ringdown time is clearly observed. It is also worth to note that the measured quality factor is several times larger compared with the quality factor of calcium fluoride resonators observed previously at 1.55 μm (see [18

18. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, Art. No. 051804R (2004). [CrossRef]

, 10

10. I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,“ Phys. Rev. A 74, 063806 (2006). [CrossRef]

]).

The best ringdown signal measured with all the precautions is shown in Fig. 2(left). To ensure that the measured line is not a result of several consecutive excitations of the optical mode because of dithering of the laser carrier frequency we have evaluated the oscillation period of the beatnote (Fig. 2(right)). This shows that the carrier frequency indeed moves gradually from the resonance and the WGM has not been excited twice.

Fig. 2. Left: Ringdown signal after the third annealing step taken with low-power laser radiation (blue solid line). The theoretical fit of the signal is shown by the red dotted line. Right: Evaluated change of the period of the signal. The solid line stands for the linear fit of the time dependence of the period of the ringdown signal.

Let us discuss the fitting procedure to the theoretical curve in more detail. To evaluate the oscillation period we have determined the positions of the maxima and minima of the beatnote peaks as well as the zero crossings of the beatnote signal. We subtracted the time coordinate of each peak from the coordinate of the adjacent peak and obtained the averaged time for the two adjacent peaks. The same procedure has been repeated for the zero-crossing points. In this way we deduced the time dependence of the period of the beat note signal, shown in picture Fig. 2(right). The dependence is linear at the tail of the curve. The initial period has a different time dependence because the frequency of the WGM changes much faster immediately after the frequency of the pump laser is tuned away from the WGM. The change is determined by multiple nonlinear processes, e.g. WGM frequency shift due to after-interaction cooling of the resonator.

Using the linear approximation for the beatnote period we found the period of the waveform to fit the experimental data. We solved the equation ϕ(t + Period(t)/2) - ϕ(t) = π approximating it by a linear differential equation ϕ(t) = 2π/Period(t). using the experimental results we obtain Period(t) = ξ - ζt, where ξ =71.1 μs, and ζ = 0.11 (time t is measured in microseconds). As a result, we have the expression

ϕ(t)=Const(2πζ)ln(1ζtξ)
(1)

for the phase. The final oscillation waveform presented in Fig. 2(right) is given by cos[ϕ(t)].

Fig. 3. Change of the signal amplitude shown in Fig. 2(left) with time. Initial and final quality factors are Q = (7.9±0.5) × 1010 and Q = (3±1) × 1011 respectively. Solid line stands for the theoretical fit of the ringdown signal found using formalism presented in [19]. A small oscillation of the decaying signal may result from the residual scattering in the material becoming observable at the given value of quality factor.

5. Discussion

The annealing process discussed above improves the transparency of the material because an increased temperature results in the enhancement of the mobility of defects induced by the fabrication process, and also reduces any residual stress birefringence [20

20. B. Wang, R. R. Rockwell, and J. List, “Linear birefringence in CaF2 measured at deep ultraviolet and visible wavelengths”, J. Microlithography, Microfabrication, and Microsystems 3, 115–121 (2004). [CrossRef]

]. The increased mobility leads to the recombination of defects and their migration to the surface [16

16. V. B. Braginsky, V. S. Ilchenko, and K. S. Bagdassarov, “Experimental observation of fundamental microwave absorption in high quality dielectric crystalls,” Phys. Lett. A 120, 300–305 (1987). [CrossRef]

]. It is worth noting that our annealing technique is similar to the previously developed procedures [21

21. V. Deuster, M. Schick, Th. Kayser, H. Dabringhaus, H. Klapper, and K. Wandelt, “Studies of the facetting of the polished (100) face of CaF2,” J. Cryst. Growth 250, 313–323 (2003). [CrossRef]

, 22

22. Q. -Z. Zhao, J. -R. Qiu, X. -W. Jiang, C. -J. Zhao, and C. -S. Zhu, “Fabrication of internal diffraction gratings in calcium fluoride crystals by a focused femtosecond laser,” Opt. Express 12, 742–746 (2004). [CrossRef] [PubMed]

].

The significant improvement of the Q-factor and finesse of a fluorite WGM resonator demonstrated here does not reach the fundamental limit. The straightforward annealing of a WGM resonator leads to Q > 1011 at 1.55 μm while our earlier theoretical prediction gives Q ≃ 1013 at this wavelength [18

18. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, Art. No. 051804R (2004). [CrossRef]

]. To improve the quality of the annealing we suggest using a larger furnace with much lower thermal gradients. Thermal gradients within a sample play the leading role in the defect redistribution [20

20. B. Wang, R. R. Rockwell, and J. List, “Linear birefringence in CaF2 measured at deep ultraviolet and visible wavelengths”, J. Microlithography, Microfabrication, and Microsystems 3, 115–121 (2004). [CrossRef]

]. The amplitude of the gradient can be reduced not only by increasing the volume of the oven but also by increasing thermal conductivity and optimizing shape of the container the resonator is placed into during the annealing procedure. A fluorite container is the best choice for annealing fluorite WGM resonators.

6. Conclusion

Acknowledgement

The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration and sponsorship from DARPA. A. Matsko appreciates useful discussions with D. Strekalov.

References and links

1.

G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363–365 (1992). [CrossRef] [PubMed]

2.

C. J. Hood, H. J. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A 64, 033804 (2001). [CrossRef]

3.

H. R. Bilger, P. V. Wells, and G. E. Stedman, “Origins of fundamental limits for reflection losses at multilayer dielectric mirrors,” Appl. Opt. 33, 7390–7396 (1994). [CrossRef] [PubMed]

4.

H.-J. Cho, M.-J. Shin, and J.-C. Lee, “Effects of substrate and deposition method onto the mirror scattering,“, Appl. Opt. 45, 1440–1446 (2006). [CrossRef] [PubMed]

5.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

6.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005). [CrossRef]

7.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and Lute Maleki, “Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities,“ Phys. Rev. Lett. 92, 043903 (2004). [CrossRef] [PubMed]

8.

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, 247–249 (1998). [CrossRef]

9.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip,” Appl. Phys. Lett. 85, 6113–6115 (2004). [CrossRef]

10.

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,“ Phys. Rev. A 74, 063806 (2006). [CrossRef]

11.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,“ Nature 415, 621–623 (2002). [CrossRef] [PubMed]

12.

I. S. Grudinin and L. Maleki, “Ultralow-threshold Raman lasing with CaF2 resonators,” Opt. Lett. 32, 166–168 (2007). [CrossRef]

13.

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]

14.

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]

15.

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]

16.

V. B. Braginsky, V. S. Ilchenko, and K. S. Bagdassarov, “Experimental observation of fundamental microwave absorption in high quality dielectric crystalls,” Phys. Lett. A 120, 300–305 (1987). [CrossRef]

17.

A. A. Savchenkov, A. B. Matsko, M. Mohageg, and L. Maleki, “Ringdown spectroscopy of stimulated Raman scattering in a whispering gallery mode resonator,” Opt. Lett. 32, 497–499 (2007). [CrossRef] [PubMed]

18.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, Art. No. 051804R (2004). [CrossRef]

19.

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Ring-down spectroscopy for studying properties of CW Raman lasers,“ Opt. Commun. 260, 662–665 (2006). [CrossRef]

20.

B. Wang, R. R. Rockwell, and J. List, “Linear birefringence in CaF2 measured at deep ultraviolet and visible wavelengths”, J. Microlithography, Microfabrication, and Microsystems 3, 115–121 (2004). [CrossRef]

21.

V. Deuster, M. Schick, Th. Kayser, H. Dabringhaus, H. Klapper, and K. Wandelt, “Studies of the facetting of the polished (100) face of CaF2,” J. Cryst. Growth 250, 313–323 (2003). [CrossRef]

22.

Q. -Z. Zhao, J. -R. Qiu, X. -W. Jiang, C. -J. Zhao, and C. -S. Zhu, “Fabrication of internal diffraction gratings in calcium fluoride crystals by a focused femtosecond laser,” Opt. Express 12, 742–746 (2004). [CrossRef] [PubMed]

OCIS Codes
(230.5750) Optical devices : Resonators

ToC Category:
Optical Devices

History
Original Manuscript: April 12, 2007
Revised Manuscript: May 11, 2007
Manuscript Accepted: May 13, 2007
Published: May 17, 2007

Citation
Anatoliy A. Savchenkov, Andrey B. Matsko, Vladimir S. Ilchenko, and Lute Maleki, "Optical resonators with ten million finesse," Opt. Express 15, 6768-6773 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-11-6768


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References

  1. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari," Measurement of ultralow losses in an optical interferometer," Opt. Lett. 17, 363-365 (1992). [CrossRef] [PubMed]
  2. C. J. Hood, H. J. Kimble, and J. Ye, "Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity," Phys. Rev. A 64, 033804 (2001). [CrossRef]
  3. H. R. Bilger, P. V. Wells, and G. E. Stedman, "Origins of fundamental limits for reflection losses at multilayer dielectric mirrors," Appl. Opt. 33, 7390-7396 (1994). [CrossRef] [PubMed]
  4. H.-J. Cho, M.-J. Shin, and J.-C. Lee, "Effects of substrate and deposition method onto the mirror scattering," Appl. Opt. 45, 1440-1446 (2006). [CrossRef] [PubMed]
  5. E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681-681 (1946).
  6. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, "Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion," Phys. Rev. A 71, 033804 (2005). [CrossRef]
  7. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and Lute Maleki, "Nonlinear Optics and Crystalline Whispering Gallery Mode Cavities," Phys. Rev. Lett. 92, 043903 (2004). [CrossRef] [PubMed]
  8. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, "High-Q measurements of fusedsilica microspheres in the near infrared," Opt. Lett. 23, 247-249 (1998). [CrossRef]
  9. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004). [CrossRef]
  10. I. S. Grudinin, V. S. Ilchenko, and L. Maleki, "Ultrahigh optical Q factors of crystalline resonators in the linear regime," Phys. Rev. A 74, 063806 (2006). [CrossRef]
  11. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-623 (2002). [CrossRef] [PubMed]
  12. I. S. Grudinin and L. Maleki, " Ultralow-threshold Raman lasing with CaF2 resonators," Opt. Lett. 32, 166-168 (2007). [CrossRef]
  13. 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]
  14. 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]
  15. 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]
  16. V. B. Braginsky, V. S. Ilchenko, and K. S. Bagdassarov, "Experimental observation of fundamental microwave absorption in high quality dielectric crystalls," Phys. Lett. A 120, 300-305 (1987). [CrossRef]
  17. A. A. Savchenkov, A. B. Matsko, M. Mohageg, and L. Maleki, "Ringdown spectroscopy of stimulated Raman scattering in a whispering gallery mode resonator," Opt. Lett. 32, 497-499 (2007). [CrossRef] [PubMed]
  18. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, Art. No. 051804R (2004). [CrossRef]
  19. A. B. Matsko, A. A. Savchenkov, and L. Maleki, "Ring-down spectroscopy for studying properties of CWRaman lasers," Opt. Commun. 260, 662-665 (2006). [CrossRef]
  20. B. Wang, R. R. Rockwell, and J. List, "Linear birefringence in CaF2 measured at deep ultraviolet and visible wavelengths," J. Microlithogr., Microfabr., Microsyst. 3, 115-121 (2004). [CrossRef]
  21. V. Deuster, M. Schick, Th. Kayser, H. Dabringhaus, H. Klapper and K. Wandelt, "Studies of the facetting of the polished (100) face of CaF2," J. Cryst. Growth 250, 313-323 (2003). [CrossRef]
  22. Q. -Z. Zhao, J. -R. Qiu, X. -W. Jiang, C. -J. Zhao, and C. -S. Zhu, "Fabrication of internal diffraction gratings in calcium fluoride crystals by a focused femtosecond laser," Opt. Express 12, 742-746 (2004). [CrossRef] [PubMed]

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