## Laser frequency stabilization and control through offset sideband locking to optical cavities

Optics Express, Vol. 16, Issue 20, pp. 15980-15990 (2008)

http://dx.doi.org/10.1364/OE.16.015980

Acrobat PDF (204 KB)

### Abstract

We describe a class of techniques whereby a laser frequency can be stabilized to a fixed optical cavity resonance with an adjustable offset, providing a wide tuning range for the central frequency. These techniques require only minor modifications to the standard Pound-Drever-Hall locking techniques and have the advantage of not altering the intrinsic stability of the frequency reference. We discuss the expected performance and limitations of these techniques and present a laboratory investigation in which both the sideband techniques and the standard, non-tunable Pound-Drever-Hall technique reached the 100Hz/√Hz level.

© 2008 Optical Society of America

## 1. Introduction

_{0}, is one of the key figures of merit for a light source. In applications such as interferometry and spectroscopy, the linewidth limits the precision of the overall measurement. Various techniques are employed to reduce the linewidth of laser sources[1

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

2. J. Hall, L. Ma, M. Taubman, B. Tiemann, F. Hong, O. Pfister, and J. Ye, “Stabilization and frequency measurement of the *I*_{2}-stabilized Nd:YAG laser,” IEEE Trans. Instrum. Meas. **48**, 583–586 (1999). [CrossRef]

*Free Spectral Range*(FSR), which for a linear two-mirror cavity is given by

*c*is the speed of light and

*L*is the cavity length. For certain applications, such as generating an interference between two independent laser beams or probing spectroscopic features, it is advantageous to be able to adjust ν

_{0}with a resolution better than one FSR while still suppressing the free-running frequency noise.

3. L. Conti, M. D. Rosa, and F. Marin, “High-spectral-purity laser system for the Auriga detector optical readout,” JOSA B **20**, 462–468 (2003). [CrossRef]

4. H. Zhen, H. Ye, X. Liu, D. Zhu, H. Li, Y. Lu, and Q. Wang, “Widely tunable reflection-type Fabry-Perot interferometer based on relaxor ferroelectric poly(vinylidenefluoride-chlorotrifluoroethylene-trifluoroethylene),” Opt. Express , **16**, 9595–9600 (2008). [CrossRef] [PubMed]

5. F. Bondu, P. Fritschel, C. Man, and A. Brillet, “Ultrahigh-spectral-purity laser for the Virgo experiment,” Opt. Lett. **21**, 582–584 (1996). [CrossRef] [PubMed]

6. J. Ye and J. Hall, “Optical phase locking in the mircoradian domain: potential applications to NASA spaceborne optical measurements,” Opt. Lett. **24**, 1838–1840 (1999). [CrossRef]

## 2. Standard Pound-Drever-Hall (PDH) locking

*F*(

*ω*), where

*ω*≡2

*πν*. The amplitude of

*F*(

*ω*) goes to zero at the resonance frequencies (

*ω*≡2

_{n}*πn*·

*FSR*

*n*=1,2,3…) and approaches unity between them. The width of the resonance is characterized by the Finesse, ℱ, defined as

*is the full width at half minimum of |*

_{FWHM}*F*(

*ω*)|. It is the phase of

*F*(

*ω*) that contains the information about whether the light frequency is above or below the resonance. The phase of

*F*(

*ω*) begins at -

*π*rad far below resonance, increases monotonically to -

*π*/2 rad just below resonance, goes through a discontinuity of

*π*rad at resonance, and increases monotonically from

*π*/2rad to

*π*rad far above resonance. A measurement of the phase shift experienced by the reflected light can be used to generate an error signal for locking to the cavity resonance.

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

7. R. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. **17**, 490–505 (1946). [CrossRef] [PubMed]

*P*

_{0}is the power incident on the modulator,

*ω*is the angular frequency of the incoming light,

_{c}*β*is the modulation depth, and Ω is the angular frequency of the modulation. To first order in

*β*, the effect of phase modulation is to split the beam into three distinct frequency components: a carrier at

*ω*=

*ω*and two sidebands at

_{c}*ω*=

*ω*±Ω. For sufficiently large Ω, the sidebands are completely reflected when the carrier is near resonance, as shown in Fig. 1(a).

_{c}8. E. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. **69**, 79–87 (2001). [CrossRef]

*β*, by

*J*(

_{n}*x*) is the

*n*-order Bessel function of the first kind. When the carrier is near resonance, the bracketed term in (4) is purely imaginary and proportional to

_{th}*δν*, the frequency offset between the carrier and the cavity resonance. The proportionality constant is known as the frequency discriminant, which for a lossless cavity is given by

*P*

_{ref,Ω}using a photoreceiver and demodulating the output to recover the sin(Ω

*t*) component.

## 3. Sideband locking

### 3.1. Single Sideband (SSB) locking

*P*

_{ref,Ω}in (4) is still valid only one of the sidebands is resonant while the other sideband and the carrier are reflected. If we redefine the resonance frequencies as

*ω*≡2

_{n}*πn*·

*FSR*+(-)Ω for locking on the upper(lower) sideband, the sin(Ω

*t*) component is proportional to δν with a discriminant given by

*J*

_{2}(

*β*) term arises from interference between the resonant sideband and a second sideband at

*ω*±2Ω that appears when the expansion of (3) is taken to higher orders in

_{c}*β*. For small

*β*,

*D*is of opposite sign and a factor of two lower than

_{SSB}*D*. Once one of the sidebands is locked to the cavity resonance, the carrier frequency can be tuned by adjusting Ω.

_{PDH}### 3.2. Dual Sideband (DSB) locking

*P*

_{0}and angular frequency

*ω*that is phase-modulated with two sinusoidal signals of depth

_{c}*β*and angular frequency Ω

_{i}*(*

_{i}*i*=1,2). The electric field is given by,

*β*

_{1,2}, the result of the phase modulation is a carrier with angular frequency

*ω*, sidebands with angular frequencies

_{c}*ω*±Ω

_{c}_{1}, sidebands with angular frequencies

*ω*±Ω

_{c}_{2}, and sub-sidebands at

*ω*+Ω

_{c}_{1}±Ω

_{2}and

*ω*-Ω

_{c}_{1}±Ω

_{2}.

*ω*≥

*ω*is shown in Fig. 1(c) assuming Ω

_{c}_{1}>Ω

_{2}and

*β*

_{1}>

*β*

_{2}. Note that the spectral structure centered around

*ω*+Ω

_{c}_{1}with sidebands offset by ±Ω

_{2}is analogous to the PDH modulation spectrum in Fig. 1(a). In DSB locking, this structure (or the analogous one at

*ω*-Ω

_{c}_{1}) is used to generate an error signal by placing one of the

*ω*±Ω

_{c}_{1}sidebands on resonance and demodulating the reflected power with Ω

_{2}. The frequency discriminant is given by

_{1}.

_{2}. If the modulation depth of the first modulator is not large enough to sufficiently suppress the carrier, this error signal may even be larger than the desired DSB error signal. The situation becomes even more complex when additional resonances due to higher-order cavity spatial modes are present.

### 3.3. Electronic Sideband (ESB) locking

*ω*±Ω

_{c}_{2}sidebands. This can be accomplished by driving a single broadband phase-modulator with a phase-modulated drive signal. The drive signal has a carrier frequency of Ω

_{1}and is phase-modulated at Ω

_{2}with a depth of

*β*

_{2}. This signal is then used to drive the phase modulator with a depth of

*β*

_{1}. The electric field of the light exiting the modulator is of the form

*β*, the spectrum is identical to the DSB structure with the exception that the

_{i}*ω*±Ω

_{c}_{2}sidebands are removed. Figure 1(d) shows this spectrum for

*ω*≥

*ω*. The error signal is generated by placing one of the

_{c}*ω*±Ω

_{c}_{1}sidebands near resonance and demodulating with Ω

_{2}. As with DSB, the carrier is tuned by adjusting Ω

_{1}.

*ω*±Ω

_{c}_{1}sidebands and the

*ω*±Ω

_{c}_{1}±Ω

_{2}sub-sidebands is the same as for the DSB case. As a result, the frequency discriminant for ESB locking is identical to that given in (8) for DSB. The power in the carrier is increased to

*P*

_{0}

*J*

^{2}

_{0}(β1) for ESB versus

*P*

_{0}

*J*

^{2}

_{0}(

*β*

_{1})

*J*

^{2}

_{0}(

*β*

_{2}) for DSB.

## 4. Technique comparison

### 4.1. Fundamental noise limits

*h*is Planck’s constant,

*λ*is the vacuum wavelength of the light, and

*P*

*is the reflected light power on resonance. For a perfectly coupled cavity,*

_{ref}*P*

*will be equal to*

_{ref}*P*

_{0}, the total power incident on the modulator(s), less the power of the resonant spectral component.

8. E. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. **69**, 79–87 (2001). [CrossRef]

*S*

_{shot,P}by the frequency discriminant,

*D*. Table 1 lists

*P*,

_{ref}*D*,

*β*, the optimal modulation depth for shot noise limited operation, and

_{opt}*S*

_{shot,ν}, the shot-noise limited frequency noise floor for

*β*=

*β*

*. The shot noise floors for the sideband systems in table 1 are five to six times larger than the floor for the traditional PDH system,*

_{opt}9. K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. **93** (2004). [CrossRef]

^{†}This is a theoretical optimum for perfect contrast and no technical noise. When the effects of finite contrast and technical noise are included, the optimum modulation depth will increase.

### 4.2. Technical noise

10. J. Alins, 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]

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

*t*) component in

*P*

_{ref,Ω}does not vanish but instead remains with a magnitude of 2

*P*

_{0}

*J*

_{0}(

*β*)

*J*

_{1}(

*β*). Since this offset appears in the opposite quadrature as the error signal, it does not couple completely but is instead reduced by a factor of ≈1-

*δθ*

^{2}, where

*δθ*is the error in the demodulation phase. DSB and ESB locking retain the symmetry of PDH locking and avoid this effect. There will, however, be certain values of the modulation frequencies, for example Ω

_{1}=2Ω

_{2}for DSB, where interference between the sidebands and sub-sidebands can generate RFAM at the demodulation frequency. This type of interference can be mitigated by maintaining Ω

_{1}≫Ω

_{2}.

_{1}±Ω

_{2}sub-sidebands) are introduced onto the light at higher frequencies than for DSB or PDH locking. To first order in

*β*

_{2}, passband ripple will lead to RFAM with an amplitude of

*G*′ (Ω

_{1}) is the derivative with respect to modulation frequency of the modulator response around the frequency Ω

_{1}. The amplitude can be reduced by reducing the modulation frequency, Ω

_{2}or, if necessary, measuring the modulator response and constructing a compensation filter for the RF drive that will reduce

*G*′ (Ω

_{1}).

### 4.3. Tuning range and bandwidth

12. B. Sheard, M. Gray, D. McClelland, and D. Shaddock, “Laser frequency stabilization by locking to a LISA arm,” Phys. Lett. A **320**, 9–21 (2003). [CrossRef]

## 5. Laboratory demonstration

### 5.1. Experimental configuration

### 5.2. Results & discussion

*µ*Hz was added to the EOM drive. The spectrum with the modulation tone was estimated using a periodograms with linear frequency bin spacing[14

14. P. Welch, “The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. **AU-15**, 70–73 (1967). [CrossRef]

15. M. Tröbs and G. Heinzel, “Improved spectrum estimation from digitized time series on a logarithmic frequency axis,” Measurement **39**, 120–129 (2005). [CrossRef]

^{4}–10

^{5}.

*f*≤40

*µ*Hz, the frequency noise is limited by thermal expansion of the cavities driven by variations in the room temperature. The thermal filter provided by the shields and vacuum tank surrounding the cavity result in the steep ~

*f*

^{-7}slope of the noise floor in this frequency band. Above 40

*µ*Hz, the limiting noise sources are less certain. Likely possibilities include pointing noise, vibration noise, RFAM, and intensity noise coupling into cavity length changes through absorption and heating.

## 6. Conclusion

## Acknowledgments

## References and links

1. | R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B |

2. | J. Hall, L. Ma, M. Taubman, B. Tiemann, F. Hong, O. Pfister, and J. Ye, “Stabilization and frequency measurement of the |

3. | L. Conti, M. D. Rosa, and F. Marin, “High-spectral-purity laser system for the Auriga detector optical readout,” JOSA B |

4. | H. Zhen, H. Ye, X. Liu, D. Zhu, H. Li, Y. Lu, and Q. Wang, “Widely tunable reflection-type Fabry-Perot interferometer based on relaxor ferroelectric poly(vinylidenefluoride-chlorotrifluoroethylene-trifluoroethylene),” Opt. Express , |

5. | F. Bondu, P. Fritschel, C. Man, and A. Brillet, “Ultrahigh-spectral-purity laser for the Virgo experiment,” Opt. Lett. |

6. | J. Ye and J. Hall, “Optical phase locking in the mircoradian domain: potential applications to NASA spaceborne optical measurements,” Opt. Lett. |

7. | R. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. |

8. | E. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. |

9. | K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. |

10. | J. Alins, 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 |

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

12. | B. Sheard, M. Gray, D. McClelland, and D. Shaddock, “Laser frequency stabilization by locking to a LISA arm,” Phys. Lett. A |

13. | P. Bender and K. Danzmann, and the LISA Study Team, “Laser interferometer space antenna for the detection of graviational waves, pre-Phase A report,” Tech. Rep. MPQ233, Max-Planck-Institut für Quantenoptik, Gärching (1998). 2nd ed. |

14. | P. Welch, “The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. |

15. | M. Tröbs and G. Heinzel, “Improved spectrum estimation from digitized time series on a logarithmic frequency axis,” Measurement |

**OCIS Codes**

(120.2230) Instrumentation, measurement, and metrology : Fabry-Perot

(140.3425) Lasers and laser optics : Laser stabilization

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 8, 2008

Revised Manuscript: August 15, 2008

Manuscript Accepted: August 18, 2008

Published: September 24, 2008

**Citation**

J. I. Thorpe, K. Numata, and J. Livas, "Laser frequency stabilization and control through offset sideband
locking to optical cavities," Opt. Express **16**, 15980-15990 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15980

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### References

- R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983). [CrossRef]
- J. Hall, L. Ma, M. Taubman, B. Tiemann, F. Hong, O. Pfister, and J. Ye, "Stabilization and frequency measurement of the I2-stabilized Nd:YAG laser," IEEE Trans. Instrum. Meas. 48, 583-586 (1999). [CrossRef]
- L. Conti, M. D. Rosa, and F. Marin, "High-spectral-purity laser system for the Auriga detector optical readout," J. Opt. Soc. B 20, 462-468 (2003). [CrossRef]
- H. Zhen, H. Ye, X. Liu, D. Zhu, H. Li, Y. Lu, and Q. Wang, "Widely tunable reflection-type Fabry-Perot interferometer based on relaxor ferroelectric poly(vinylidenefluoride-chlorotrifluoroethylene-trifluoroethylene)," Opt. Express, 16, 9595-9600 (2008). [CrossRef] [PubMed]
- F. Bondu, P. Fritschel, C. Man, and A. Brillet, "Ultrahigh-spectral-purity laser for the Virgo experiment," Opt. Lett. 21, 582-584 (1996). [CrossRef] [PubMed]
- J. Ye and J. Hall, "Optical phase locking in the mircoradian domain: potential applications to NASA spaceborne optical measurements," Opt. Lett. 24, 1838-1840 (1999). [CrossRef]
- R. Pound, "Electronic frequency stabilization of microwave oscillators," Rev. Sci. Instrum. 17, 490-505 (1946). [CrossRef] [PubMed]
- E. Black, "An introduction to Pound-Drever-Hall laser frequency stabilization," Am. J. Phys. 69, 79-87 (2001). [CrossRef]
- K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93 (2004). [CrossRef]
- J. Alins, 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]
- S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77, 033847 (2008). [CrossRef]
- B. Sheard, M. Gray, D. McClelland, and D. Shaddock, "Laser frequency stabilization by locking to a LISA arm," Phys. Lett. A 320, 9-21 (2003). [CrossRef]
- P. Bender and K. Danzmann, and the LISA Study Team, "Laser interferometer space antenna for the detection of graviational waves, pre-Phase A report," Tech. Rep. MPQ233, Max-Planck-Institut f¨ur Quantenoptik, Gärching (1998). 2nd ed.
- P. Welch, "The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms," IEEE Trans. Audio Electroacoust. AU-15, 70-73 (1967). [CrossRef]
- M. Tröbs and G. Heinzel, "Improved spectrum estimation from digitized time series on a logarithmic frequency axis," Measurement 39, 120-129 (2005). [CrossRef]

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