## A vibration-insensitive optical cavity and absolute determination of its ultrahigh stability

Optics Express, Vol. 17, Issue 11, pp. 8970-8982 (2009)

http://dx.doi.org/10.1364/OE.17.008970

Acrobat PDF (1297 KB)

### Abstract

We use the three-cornered-hat method to evaluate the absolute frequency stabilities of three different ultrastable reference cavities, one of which has a vibration-insensitive design that does not even require vibration isolation. An Nd:YAG laser and a diode laser are implemented as light sources. We observe ~ 1 Hz beat note linewidths between all three cavities. The measurement demonstrates that the vibration-insensitive cavity has a good frequency stability over the entire measurement time from 100 *μ*s to 200 s. An absolute, correlation-removed Allan deviation of 1.4 × 10^{−15} at 1 s of this cavity is obtained, giving a frequency uncertainty of only 0.44 Hz.

© 2009 Optical Society of America

## 1. Introduction

1. T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al^{+} and Hg^{+} single-ion optical clocks; metrology at the 17th decimal place,” Science **319**, 1808–1812 (2008). [CrossRef] [PubMed]

2. A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10^{−16} fractional uncertainty by remote optical evaluation with a Ca clock,” Science **319**, 1805–1808 (2008). [CrossRef] [PubMed]

3. T. Schneider, E. Peik, and C. Tamm, “Sub-Hertz optical frequency comparisons between two trapped ^{171}Yb^{+} ions,” Phys. Rev. Lett. **94**, 230801 (2005). [CrossRef] [PubMed]

4. Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, “Improved absolute frequency measurement of the ^{115}In^{+} 5*s*^{2}^{1}S_{0-5}*s*5_{p}^{3}P_{0} narrowline transition: progress towards an optical frequency standard,” Laser Phys. **17**, 1017–1024 (2007). [CrossRef]

5. 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]

6. C. Salomon, D. Hils, and J. L. Hall, “Laser stabilization at the millihertz level,” J. Opt. Soc. Am. B **5**, 1576–1587 (1988). [CrossRef]

7. B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. **82**, 3799–3802 (1999). [CrossRef]

8. A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, “Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1× 10^{−15},” Opt. Lett. **32**, 641–643 (2007). [CrossRef] [PubMed]

9. S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A **77**, 033847 (2008). [CrossRef]

8. A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, “Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1× 10^{−15},” Opt. Lett. **32**, 641–643 (2007). [CrossRef] [PubMed]

9. S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A **77**, 033847 (2008). [CrossRef]

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

12. J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-Pérot cavities,” Phys. Rev. A **77**, 053809 (2008). [CrossRef]

13. T. Liu, Y. N. Zhao, V. Elman, A. Stejskal, and L. J. Wang, “Characterization of the absolute frequency stability of an individual reference cavity,” Opt. Lett. **34**, 190–192 (2009). [CrossRef] [PubMed]

## 2. Experimental Setup

### 2.1. Reference cavities

14. L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, “Vibration-induced elastic deformation of Fabry-Perot cavities,” Phys. Rev. A **74**, 053801 (2006). [CrossRef]

15. A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, “A Nd:YAG Laser with short-term frequency stability at the Hertz-level,” Opt. Commun. **210**, 91–100 (2002). [CrossRef]

16. T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, “Narrow linewidth light source for an ultraviolet optical frequency standard,” Appl. Phys. B **87**, 227–232 (2007). [CrossRef]

17. 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]

*L*is the cavity length excluding the two mirrors, position

*D*is along the cavity axis. Square “cutouts” are made on the bottom of the cylindrical spacer and the cavity is supported at four points. The cutouts compensate for vertical forces, and vibration insensitivity is achieved through the special cavity shape and symmetrical mounting at the Airy points.

^{3}kg/m

^{3}, elastic modulus of 6.67 × 10

^{10}N/m

^{2}, and Poisson’s ratio of 0.17. The spacer has a length of approximately 10 cm and an outside diameter of approximately 10 cm. Since the vibration frequencies that significantly contribute to the cavity instability are less than 10 Hz, they can be considered as dc relative to the first structural resonance of the cavity, which is about 10 kHz. Therefore, we choose a static stress-strain model with a gravity-like force applied on the cavity in the vertical direction, as shown in Fig. 1(a). To reduce modeling computation time, we take advantage of the symmetry of the cavity and only calculate a quarter section of the cavity excluding the mount. As only symmetric solutions are considered, the planes of symmetry are constrained to move only along the plane surfaces. The support is modeled as a point constrain on the horizontal cutout surface in order to prevent the cavity spacer from translational or rotational movements.

*L*/d

*g*at different supporting points

*D*is shown in Fig. 2. Here d

*L*presents the change in cavity length caused by d

*g*, a small change in the acceleration

*g*. Both d

*g*= 0.0981 m/s

^{2}and d

*g*= 0.981 m/s

^{2}are calculated, and the results are nearly the same, showing good linearity. The results indicate that there exists an optimal supporting position where two cavity mirrors are parallel and insensitive to the vertical accelerations. At this point, the cavity deforms under the gravity, but the axial distance between the centers of the mirror inner surfaces remains unchanged. Furthermore, the axial displacement can be calculated as a function of support position and cut depth. We choose a cut depth of 5 mm at a vertical position of 4.5 mm below the mid-plane. The calculated result shows a d

*L*/d

*g*< 10

^{−14}s

^{2}acceleration sensitivity.

^{−8}mbar. The vacuum chamber is placed inside a temperature-stabilized aluminum box with a digital PID control. The temperature is stabilized at 27 °C with a root-mean-square (RMS) fluctuation of less than 1 mK. To reduce the influence of acoustic noise, the vacuum chamber is surrounded by plastic foams containing a layer of lead septum. The cavity and other optical components sit on top of a 10-cm-thick breadboard without any vibration control. The three platforms for cavities 1, 2 and 3 are located on top of an unfloated optical table inside an acoustically isolated cabin (lab 1).

### 2.2. Laser locking

4. Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, “Improved absolute frequency measurement of the ^{115}In^{+} 5*s*^{2}^{1}S_{0-5}*s*5_{p}^{3}P_{0} narrowline transition: progress towards an optical frequency standard,” Laser Phys. **17**, 1017–1024 (2007). [CrossRef]

16. T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, “Narrow linewidth light source for an ultraviolet optical frequency standard,” Appl. Phys. B **87**, 227–232 (2007). [CrossRef]

_{00}cavity mode. The laser beam then passes through an electro-optical modulator (EOM1), which generates two 10.5 MHz sidebands around the carrier frequency for PDH frequency locking. The temperature of EOM1 is stabilized to avoid thermal-induced fluctuations in the polarization of the laser light. Approximately 30

*μ*W power is directed into cavity 1, with a proper confocal lens for cavity mode-matching to obtain a maximum coupling efficiency. The reflected light from cavity 1 is detected by a photodiode (PD1). An additional AOM (AOM2) is used as an optical isolator to prevent standing waves on PD1. The signal from PD1, after demodulation, is used as an error signal to feedback control the frequency of the MISER. This feedback signal is applied to a piezo that is glued on top of the MISER crystal. It deforms the crystal, and changes the optical path length of the oscillator. To increase the dynamic range of the feedback control, an additional thermal controller is servo controlled to change the temperature of the laser crystal. This combined feedback system typically can maintain continuous frequency locking for a full day. Part of the main beam (output 1) is split off and reserved for the three-cornered-hat measurement.

_{00}modes of cavity 1 and 2, and is also used for PDH locking to cavity 2. Double-passing arrangement allows fast frequency corrections without causing beam deviation. The light after double-passing AOM3 is separated into two parts. A small part of light, approximately 30

*μ*W, is mode-matched into cavity 2. The servo bandwidth for AOM3 locking is ~ 15 kHz. The other part of the light (output 2) is used for the three-cornered-hat measurement.

^{−7}mbar) is maintained to decrease the cavity mode drift. The pre-stabilized light is then transferred through a 60-m-long single-mode fiber to lab 1. To further lock the diode laser to cavity 3, a similar locking scheme as used for cavity 2 is arranged, as shown in Fig. 4(d). Approximately 2 mW stabilized light (output 3) is reserved for the three-cornered-hat measurement.

### 2.3. Fiber noise compensation

18. B. C. Young, R. J. Rafac, J. A. Beall, F. C. Cruz, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Hg^{+} optical frequency standard: recent progress,” in *Laser Spectroscopy, proceedings of the XIV International Conference*,
R. Blatt, J. Eschner, D. Leibfried, and F. Schmidt-Kaler, eds. (World Scientific, Singapore, 1999), pp. 61–70.

*f*is the laser frequency at the fiber remote end,

_{remote}*f*

_{0}is the frequency of the original incoming light,

*f*is the center driving frequency of the AOM,

_{AOM}*f*is the noise frequency picked up by the fiber, and

_{noise}*f*is the modulation frequency imposed on the AOM by the phase-lock loop. The frequency of the retroreflected light after double-passing the AOM and the fiber,

_{mod}*f*, is

_{local}*f*+

_{AOM}*f*+

_{noise}*f*), is tightly phase-locked to a local reference frequency at 2

_{mod}*f*, such that

_{AOM}*f*+

_{noise}*f*remains zero. According to Eq. (1) and (2), it is clear that when the phase-lock loop is closed, both local and remote frequencies at the two ends of the fiber can retain their high spectral purity. The AOM is used locally, as shown in Fig. 5, so that the cables for the phase-lock loop can be as short as possible to avoid picking up extra noise.

_{mod}## 3. Experimental results

### 3.1. Classical three-cornered-hat measurement

*μ*s, we choose a 100 kHz sampling rate, and synchronize the three channels to one common external trigger. The Allan deviations of the three data sets are calculated, and are shown in Fig. 7. Linear frequency drifts of approximately 1 Hz/s are removed during data processing. We clearly observe a much better beat frequency stability for cavity 1–3 than those of cavity 1–2 and 2–3 over the entire time scale. The results agree with the linewidth measurement results. The last two beat frequency stabilities have more or less the same shape, mainly due to the relative poor performance of cavity 2. The beat frequency of cavity 2–3 is still slightly better than that of cavity 1–2 at all time scales, which suggests that the performance of cavity 3 is better than that of cavity 1.

*i*,

*j*, and

*k*refer to the three cavities,

*σ*(

_{ij}*τ*) denotes the relative frequency stability between cavity

*i*and

*j*,

*σ*(

_{i}*τ*) denotes the absolute frequency stability of cavity

*i*, and

*τ*is the averaging time, respectively. The absolute frequency stabilities

*σ*(

_{x}*τ*) (

*x*=

*i*,

*j*,

*k*) of the individual cavities are derived from Allan deviations of

*σ*(

_{ij}*τ*),

*σ*(

_{ik}*τ*), and

*σ*(

_{jk}*τ*) using Eq. (3), and are shown in Fig. 8(a). It is clear that cavity 3 has the best overall frequency stability. It is worth mentioning that even without AVI control, the mid-term stability of cavity 3 in the 1–200 s region has a mean value of only 2.8 × 10

^{−15}, almost two times better than that of cavity 1 (5.4 × 10

^{−15}), and four times better than that of cavity 2 (1.2 × 10

^{−14}). This is due to the optimized design of cavity 3. The best frequency stability of 1.3 × 10

^{−15}of cavity 3 is achieved at an averaging time of 0.4 s.

### 3.2. Correlation removed results

20. A. Premoli and P. Tavella, “A revisited three-cornered hat method for estimating frequency standard instability,” IEEE Trans. Instrum. Meas. **42**, 7–13 (1993). [CrossRef]

*S*,

*s*

_{11}=σ

_{13}

^{2},

*s*

_{22}=σ

_{23}

^{2}, and

*s*

_{12}=

*s*

_{21}=σ

_{13}σ

_{23}. From the covariance matrix

*S*, and using the relationship

*r*

_{11},

*r*

_{22},

*r*

_{33}are Allan variances of the individual cavities with correlation effect removed, and

*r*

_{12},

*r*

_{13},

*r*

_{23}are correlations between the cavity pairs, we can obtain the following expressions:

*r*

_{13},

*r*

_{23}, and

*r*

_{33}, an appropriate criterion ought to be formulated. One possible choice is to require the covariance matrices

*S*and

*R*to be positive definite, where

*R*is defined as

*r*

_{13},

*r*

_{23}, and

*r*

_{33}must always fulfill the positive definiteness of

*R*. This can be done by minimizing the “global correlation” among cavities. The detailed derivations are rather complicated and can be found in [20

20. A. Premoli and P. Tavella, “A revisited three-cornered hat method for estimating frequency standard instability,” IEEE Trans. Instrum. Meas. **42**, 7–13 (1993). [CrossRef]

*r*

_{13},

*r*

_{23}, and

*r*

_{33}. We define

*f*. We then choose the unique minimum positive root

*f*if it exists; otherwise, choose the

_{min}*f*= 0. With

_{min}*f*=

*f*, we can calculate

_{min}*r*

_{13},

*r*

_{23},

*r*

_{33}are determined to be

*r*

_{11},

*r*

_{12}, and

*r*

_{22}can be calculated from Eq. (6). The Allan deviation of a cavity

*x*(

*x*= 1,2,3) can then be calculated by using the diagonal matrix elements of

*R*,

^{−15}at 1 s. The correlations of the three cavity pairs are also shown in Fig. 9. At time scale shorter than 10 s, the

*R*matrix is diagonal with zero estimated correlations between cavities. Correlations appear after 10 s, most probably due to common temperature fluctuations and other environmental changes in lab 1.

### 3.3. Gaussian process criteria

*ξ*

_{ij}^{4}of any beat signals is equal to three times the respective Allan variance squared.

## 4. Conclusion

*μ*s to 200 s. A frequency stability of 1.3 × 10

^{−15}at 0.4 s is observed for this cavity, even without vibration isolation. We further investigate correlations between the reference cavities at different time scales, and observe that correlations between cavities start to appear from 10 s to 200 s according to calculated covariance matrix. The correlations are mainly caused by the same laboratory surroundings of the three cavities. An absolute, correlation-removed Allan deviation of 1.4 × 10

^{−15}at 1 s of the newly designed cavity is obtained. This gives a frequency uncertainty of only 0.44 Hz.

## Acknowledgments

## References and links

1. | T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al |

2. | A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10 |

3. | T. Schneider, E. Peik, and C. Tamm, “Sub-Hertz optical frequency comparisons between two trapped |

4. | Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, “Improved absolute frequency measurement of the |

5. | 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 |

6. | C. Salomon, D. Hils, and J. L. Hall, “Laser stabilization at the millihertz level,” J. Opt. Soc. Am. B |

7. | B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. |

8. | A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, “Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1× 10 |

9. | S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A |

10. | J. Millo, S. Dawkins, R. Chicireanu, D. Varela Magalhães, C. Mandache, D. Holleville, M. Lours, S. Bize, P. Lemonde, and G. Santarelli, “Ultra-stable optical cavities: design and experiments at LNE-SYRTE,” Proc. 2008 IEEE IFCS , 110114 (2008). |

11. | S. A. Webster, M. Oxborrow, and P. Gill, “Subhertz-linewidth Nd:YAG laser,” Opt. Lett. |

12. | J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-Pérot cavities,” Phys. Rev. A |

13. | T. Liu, Y. N. Zhao, V. Elman, A. Stejskal, and L. J. Wang, “Characterization of the absolute frequency stability of an individual reference cavity,” Opt. Lett. |

14. | L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, “Vibration-induced elastic deformation of Fabry-Perot cavities,” Phys. Rev. A |

15. | A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, “A Nd:YAG Laser with short-term frequency stability at the Hertz-level,” Opt. Commun. |

16. | T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, “Narrow linewidth light source for an ultraviolet optical frequency standard,” Appl. Phys. B |

17. | T. Nazarova, F. Riehle, and U. Sterr, “Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser,” Appl. Phys. B |

18. | B. C. Young, R. J. Rafac, J. A. Beall, F. C. Cruz, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Hg |

19. | J. E. Gray and D. W. Allan, “A method for estimating the frequency stability of an individual oscillator,” in |

20. | A. Premoli and P. Tavella, “A revisited three-cornered hat method for estimating frequency standard instability,” IEEE Trans. Instrum. Meas. |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing

(140.2020) Lasers and laser optics : Diode lasers

(140.3580) Lasers and laser optics : Lasers, solid-state

(140.3425) Lasers and laser optics : Laser stabilization

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 3, 2009

Revised Manuscript: May 10, 2009

Manuscript Accepted: May 10, 2009

Published: May 13, 2009

**Citation**

Y. N. Zhao, J. Zhang, A. Stejskal, T. Liu, V. Elman, Z. H. Lu, and L. J. Wang, "A vibration-insensitive optical cavity and absolute determination of its ultrahigh stability," Opt. Express **17**, 8970-8982 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-11-8970

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### References

- T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008). [CrossRef] [PubMed]
- A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008). [CrossRef] [PubMed]
- T. Schneider, E. Peik, and C. Tamm, "Sub-Hertz optical frequency comparisons between two trapped 171Yb+ ions," Phys. Rev. Lett. 94,230801 (2005). [CrossRef] [PubMed]
- Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007). [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]
- C. Salomon, D. Hils, and J. L. Hall, "Laser stabilization at the millihertz level," J. Opt. Soc. Am. B 5,1576-1587 (1988). [CrossRef]
- B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999). [CrossRef]
- A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, "Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10?15," Opt. Lett. 32,641-643 (2007). [CrossRef] [PubMed]
- S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008). [CrossRef]
- J. Millo, S. Dawkins, R. Chicireanu, D. Varela Magalhães, C. Mandache, D. Holleville, M. Lours, S. Bize, P. Lemonde, and G. Santarelli, "Ultra-stable optical cavities: design and experiments at LNE-SYRTE," Proc. 2008 IEEE IFCS, 110-114 (2008).
- S. A. Webster, M. Oxborrow, and P. Gill, "Subhertz-linewidth Nd:YAG laser," Opt. Lett. 29,1497-1499 (2004). [CrossRef] [PubMed]
- J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008). [CrossRef]
- T. Liu, Y. N. Zhao, V. Elman, A. Stejskal, and L. J. Wang, "Characterization of the absolute frequency stability of an individual reference cavity," Opt. Lett. 34,190-192 (2009). [CrossRef] [PubMed]
- L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006). [CrossRef]
- A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, "A Nd:YAG Laser with short-term frequency stability at the Hertz-level," Opt. Commun. 210,91-100 (2002). [CrossRef]
- T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007). [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]
- B. C. Young, R. J. Rafac, J. A. Beall, F. C. Cruz, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Hg+ optical frequency standard: recent progress," in Laser Spectroscopy, proceedings of the XIV International Conference, R. Blatt, J. Eschner, D. Leibfried, and F. Schmidt-Kaler, eds. (World Scientific, Singapore, 1999), pp. 61-70.
- J. E. Gray and D. W. Allan, "A method for estimating the frequency stability of an individual oscillator," in Proc. 28th Frequency Control Symposium, (1974), pp. 243-246.
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