## Determination of absolute mode number using two mode-locked laser combs in optical frequency metrology

Optics Express, Vol. 15, Issue 8, pp. 4485-4492 (2007)

http://dx.doi.org/10.1364/OE.15.004485

Acrobat PDF (195 KB)

### Abstract

The mode number of the comb line involved in the optical frequency measurement was determined using two mode-locked, erbium-doped fiber laser combs operated at different repetition rates independently of the frequency fluctuation of the laser under measurement (LUM). A simple measurement process is presented to determine the difference in mode number between the two laser combs and the comb mode number, yielding the absolute frequency of the LUM.

© 2007 Optical Society of America

## 1. Introduction

1. Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency
metrology,” Nature **416**, 233–237
(2002). [CrossRef] [PubMed]

_{n}=nf

_{r}+f

_{o}, where n is mode number of the comb line; f

_{r}denotes the repetition rate of the pulse train, and f

_{o}denotes the CEO frequency.

2. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of
femtosecond mode-locked lasers and direct optical frequency
synthesis,” Science **288**, 635–639
(2000). [CrossRef] [PubMed]

_{om}and f

_{r}-f

_{om}that are smaller than the repetition frequency can be measured. The correct CEO frequency must be identified. Furthermore, when a beat frequency f

_{b}between the LUM and the comb line is measured, two lasers with a frequency f

_{b}higher and lower than the comb line frequency can generate the same beat frequency. Therefore, the frequency of the LUM can be given by f

_{L}= nf

_{r}+f

_{om}±f

_{b}or f

_{L}= (n+1)f

_{r}-f

_{om}±f

_{b}. The mode number and the signs of the coefficient of the CEO frequency and the beat frequency must be verified to determine the frequency of the LUM. An effective means of determining the mode number is based on approximate

*a priori*knowledge of the frequency of the LUM with an uncertainty of ±f

_{r}/4 [3

3. S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on
mode-locked lasers,” Rev. Sci. Instrum. **72**, 3749–3771
(2001). [CrossRef]

^{−7}is sufficiently accurate to meet the requirement of offering a priori knowledge of the frequency of the LUM. However, for lasers with low repeat rates, most frequency-stabilized fiber laser combs have repetition rates of no more than 100 MHz, and the commercial wavemeter is not accurate enough.

*et al*. presented a method for determining the comb mode number without using a wavemeter [4

4. L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S. Windeler, “A new method to determine the
absolute mode number of a mode-locked femtosecond laser comb used for
absolute optical frequency measurements,”
IEEE J. Sel. Top. Quantum Electron. **9**, 1066–1071
(2003). [CrossRef]

_{o2}-(±f

_{o1})+mf

_{r2}±f

_{b2}-(±f

_{b1})]/(f

_{r1}-f

_{r2}), where f

_{bi}is the beat frequency measured at a repetition rate of f

_{ri}; the CEO frequency is f

_{oi}; i is 1 or 2; m is difference between the mode numbers in the beat frequency measurement when the repetition rate is changed from f

_{r1}to f

_{r2}. Since they could not uniquely identify the signs of the CEO frequencies and the beat frequencies, they could determine the mode number only by comparing various groups of measurements, each of which involves numerous combinations. Also, the repetition rate had to be varied in several steps, depending on the frequency stability of the LUM. This work reports the determination of the mode number using two laser combs in optical frequency measurement with uniquely identified the CEO frequency and the beat frequency. From the corresponding beat frequency variation while changing the repetition rate and the measured CEO frequency, the beat frequency and the correct CEO frequency can be determined uniquely [5

5. H. Inaba, Y. Daimon, F.-L. Hong, A. Onae, K. Minoshima, T. R. Schibli, H. Matsumoto, M. Hirano, T. Okuno, M. Onishi, and M. Nakazawa, “Long-term measurement of optical
frequencies using a simple, robust and low-noise fiber based frequency
comb,” Opt. Express **14**, 5223–5231
(2006).
http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5223. [CrossRef] [PubMed]

## 2. Principles

_{b}is measured between the n

^{th}comb line and the LUM, lasers with a frequency f

_{b}higher and lower than the comb line frequency can generate the same beat frequency, as shown in Fig. 1(a). The right frequency of the LUM can be identified by varying the repetition rate. As the repetition rate is reduced, the comb frequency is also reduced. Figure 1(a) shows that the beat frequency increases for the LUM with a frequency of f

_{L1}and decreases for f

_{L2}. Therefore, observing the beat frequency variation while changing the repetition rate allows the frequency of the LUM relative to the beating comb line to be uniquely determined.

_{L1}in Fig. 1(a) is assumed to be the right frequency of the LUM. The right CEO frequency can be determined by observing the variation of the beat frequency while the CEO frequency is varied, as shown in Fig. 1(b). Let the two measured CEO frequencies be fom and f

_{r}-f

_{om}with f

_{om}<f

_{r}-f

_{om}. The CEO frequency can be shifted by changing the pump power. If f

_{om}increases as the pump power is varied, then f

_{r}-f

_{om}decreases. If f

_{om}is the correct CEO frequency, all of the comb frequencies will increase and the beat frequency f

_{b}should drop. However, if f

_{r}-f

_{om}is the right offset frequency, then all of the comb frequencies decrease and the beat frequency f

_{b}should increase. Therefore, after the location of the LUM is determined relative to the beating comb line, the right CEO frequency can be identified by observing the variation in the beat frequency with the CEO frequency.

_{b1}(f

_{b2}) be the beat frequency measured at a repetition rate of f

_{r1}(f

_{r2}) and a CEO frequency of f

_{o1}(f

_{o2}). Assume that the frequency of the LUM relative to the beating comb lines is determined according to the process described above, and is as plotted in Fig. 2.

_{r1}to f

_{r2}. From Eqs. (1) and (2), the mode number n can be derived as

*et al*. [4

4. L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S. Windeler, “A new method to determine the
absolute mode number of a mode-locked femtosecond laser comb used for
absolute optical frequency measurements,”
IEEE J. Sel. Top. Quantum Electron. **9**, 1066–1071
(2003). [CrossRef]

_{r}and f

_{o}are phase-locked to RF synthesizers, their values can be determined precisely. The main uncertainty in Eq. (3) arises from the uncertainty of f

_{b2}-f

_{b1}and f

_{r1}-f

_{r2}. The above method can be adopted to measure the frequency of the LUM using one laser comb or two laser combs. If only one comb is utilized, the two beat frequencies f

_{b1}and f

_{b2}cannot be measured simultaneously. Hence, the frequency fluctuation of the LUM cannot be subtracted in Eq. (3). Therefore, the measurement time must be increased to reduce the uncertainty or the repetition rate difference must be much larger than the frequency instability of the LUM. In contrast, when two laser combs are used, the frequency fluctuation of the LUM can be subtracted in Eq. (3) by measuring the two beat frequencies f

_{b1}and f

_{b2}simultaneously. Therefore, the determination of n is independent of the frequency fluctuation of the LUM.

_{r1}-f

_{r2})

^{2}/(f

_{r1}f

_{r2}) results in an error of only about 0.1 for n=10

^{7}, or a comb line with a wavelength of about 300 nm. Thus, the approximation in Eq. (4) is sufficiently accurate to determine m. The final term can be precisely measured and independent of the frequency fluctuation of the LUM when two laser combs are used. The uncertainty of the measured m should be much less than one to eliminate ambiguity. The uncertainty of the final term can be less than 10

^{−4}for a laser comb with f

_{r}and f

_{o}phase-locked to RF synthesizers with an uncertainty of 10

^{−12}. Therefore, a fraction uncertainty of much less than 1 from the first term is required to accurately determine m and 0.1 was chosen for more confidence, which corresponds to a relative uncertainty of 0.1/m. The quantity f

_{r1}/n dominates the uncertainty in the first term since f

_{r1}and f

_{r2}are phase-locked to the RF synthesizers; therefore, f

_{r1}/n with a relative uncertainty of 0.1/m is good enough to calculate m. The mode number n is of the order of 106 and m is less than 100 in this experiment.

_{r1}/n is determined as follows. When the repetition rate f

_{r1}is increased by f

_{r1}/n or decreased by f

_{r1}/(n+1), the mode numbers of the comb lines neighboring the LUM with a frequency of f

_{L1}are changed by one, as displayed in Figs. 3(a) and 3(c), respectively. Figures 3(b) and 3(d) show the corresponding movement of the beat signal in the RF spectrum; at final positions, one beat frequency is exactly the same as before and the other is shifted by an amount equals the change in repetition rate. The quantity f

_{r1}/(n+1) is the same as f

_{r1}/n with a relative error of only 1/n and is significantly accurate for calculating m using Eq. (3). Further adjusting the repetition rate by f

_{r1}/n

^{2}and -f

_{r1}/(n+1)2 in Figs. 3(a) and 3(c), respectively, can make the other beat signal the same as before. These quantities are significantly less than f

_{r1}/n and can be neglected in the estimation of m. Therefore, f

_{r1}/n can be determined by measuring the change in repetition rate that is required to shift the comb mode by one such that the two beat signals switch positions following the way as indicated in Figs. 3(b) or 3(d) until one beat signal is the same as before. The same method can be applied to the LUM with a frequency of f

_{L2}. The value of m equals the nearest integer calculated by Eq. (3).

## 3. Experimental setup and results

6. L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring
lasers,” Appl. Phys. B **65**, 277–294
(1997). [CrossRef]

^{−12}for an integration time of over 1 s and an relative uncertainty of 10

^{−12}. The stabilized repetition frequency had an out-of-loop tracking stability of 2×10

^{−13}@1s and the fluctuation of the CEO frequency was of the order of mHz. Details of the frequency stabilization of the repetition rate and the CEO frequency have been presented elsewhere [7

7. J.-L. Peng, H. Ahn, R.-H. Shu, H.-C. Chui, and J. W. Nicholson, “Highly stable, frequency-controlled
mode-locked erbium fiber laser comb,”
Appl. Phys. B **86**, 49–53
(2007). [CrossRef]

*a*

_{10}component of the R(56)32-0 transition in the iodine molecule. The Nd:YAG laser at 1064 nm was coupled into a single mode fiber and split into two beams using a 3 dB coupler. The two beams were then combined with the two fiber laser frequency combs using two 3 dB couplers. Polarization controllers were used to ensure that the polarization of the Nd:YAG laser matched that of the fiber laser comb. After they had been filtered through a 10 nm bandpass filters with a center wavelength of 1064 nm, the beat signals were detected using InGaAS photodiodes. The detected beat signals had a signal-to-noise ratio of 28 dB in a 100 kHz resolution bandwidth.

_{o}varies with the repetition rate, because f

_{r}+f

_{o}, and not f

_{o}, was controlled.

_{b1}and f

_{b2}with f

_{r1}=100 MHz and f

_{r2}=99.999 MHz. The gate times of the counters were 1 s in all measurements. Both beat frequencies were generated by the Nd:YAG laser, beating with the neighboring comb lines with higher frequencies. The two beat frequencies were mixed in a double-balanced mixer and the difference frequency was counted using a counter, as shown in Fig. 5(c). The fact that the fluctuations are smaller than the individual beat frequency indicates that the frequency variations of the Nd:YAG laser were subtracted in the difference of the beat frequencies.

_{r1}was decreased from 100 MHz in step of 0.1 Hz until the two beat signals switching positions was observed in an RF spectrum analyzer following the way as indicated in Fig 3(b) or 3(d). The measured f

_{r1}/n was 35.5 Hz. The difference between the mode numbers calculated using Eq. (4) was 28.006. Therefore, m was determined to be 28. This number was also verified by evaluating the mode number shift as the repetition rate was varied smoothly from 100 MHz to 99.999 MHz.

_{r1}-f

_{r2}was obtained by mixing the two repetition rates using a frequency mixer and counted by the counter with a gate time of 1 s. The measured frequency of f

_{r1}-f

_{r2}was 1 kHz±36 μHz. Figure 5(c) plots the calculated mode number n

_{exp}given by Eq. (3). The mode number of the n

^{th}beating comb line, which generates the beat frequency f

_{b1}in Fig. 5(a), was determined to be 2816301. The frequencies measured using the two laser combs were 281630091746.299(1.3) kHz and 281630091746.293(1.3) kHz, which were completely agreed with those obtained when a historical

*a priori*knowledge of the frequency is available [8

8. H. Ahn, R.-H. Shu, R. S. Windeler, and J.-L. Peng, “Building a frequency-stabilized
mode-locked femtosecond laser for optical frequency
metrology,” IEEE Trans. Instrum. Meas. **54**, 767–770
(2005). [CrossRef]

_{exp}are similar mainly because the same measured f

_{r1}-f

_{r2}was used in the mode number calculation, which is the main source of the instability. Clearly, the determination of mode number is independent of the frequency fluctuation of the Nd:YAG laser.

_{b2}-f

_{b1}and f

_{r1}-f

_{r2}. The measured frequency fluctuation of f

_{b2}-f

_{b1}is around 50 Hz. Therefore, f

_{r1}-f

_{r2}must be much larger than 50 Hz such that the uncertainty of n can be much less than one. Further, the relative uncertainty of f

_{r1}-f

_{r2}should be much less than 1/(2n) as mentioned in section 2. Since f

_{r1}and f

_{r2}are phase-locked to the same RF source, the uncertainty of f

_{r1}-f

_{r2}is approximated by √2Δf

_{r}, where Δf

_{r}is the tracking uncertainty of the repetition frequency relative to the RF source and √2 arises from the combination of two uncertainties. In this experiment, Δ

_{fr}=2×10

^{−13}×100 MHz=20 μHz and n≅3×10

^{6}; The uncertainty of f

_{r1}-f

_{r2}is estimated to be 28 μHz, which is close to the measured value of 36 μHz. Therefore, f

_{r1}-f

_{r2}should be larger than 216 Hz according to the measured value, which corresponds to the shift of six comb modes. Although 1 kHz variation of the repetition rate was used, which corresponded to the shift of 28 comb modes, the technique of measuring m using Eq. (4) could determine m quickly. But, this technique relies on the judgment of the beat note position. Since f

_{r1}/n should be accurate to a relative uncertainty of 0.1/m, the beat note position should be accurate to n×(f

_{r1}/n)×(0.1/m), which equals to 0.33 MHz for f

_{r}=100 MHz and m=30; therefore the frequency of the LUM can not drift over this value during the shifting of one comb mode; otherwise, smoothly varying the repetition rate and counting the comb mode shift need to be used. No matter which method is used to determine m, the minimum required variation of the repetition rate does not depend on the frequency fluctuation of the LUM when two laser combs are used to measure the frequency.

_{b1}-f

_{b2}will be 22√2 kHz. The variation of the repetition rate must be larger than 32 kHz to get a fraction uncertainty of less than one, which corresponds to shifting more than 900 comb modes when 100 MHz fiber laser comb was used. In such case, smoothly varying the repetition rate and counting the mode number is not practical. Increasing the gate time by a factor of 22

^{2}can average down the frequency fluctuation from 22 kHz to 1 kHz, but it is time consuming and more integration time is needed for LUMs with higher frequency fluctuation. Using higher repetition rate of Ti: sapphire laser comb can reduce the mode number being shifted as shown by Ma

*et. al*. [4

4. L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S. Windeler, “A new method to determine the
absolute mode number of a mode-locked femtosecond laser comb used for
absolute optical frequency measurements,”
IEEE J. Sel. Top. Quantum Electron. **9**, 1066–1071
(2003). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency
metrology,” Nature |

2. | D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of
femtosecond mode-locked lasers and direct optical frequency
synthesis,” Science |

3. | S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on
mode-locked lasers,” Rev. Sci. Instrum. |

4. | L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S. Windeler, “A new method to determine the
absolute mode number of a mode-locked femtosecond laser comb used for
absolute optical frequency measurements,”
IEEE J. Sel. Top. Quantum Electron. |

5. | H. Inaba, Y. Daimon, F.-L. Hong, A. Onae, K. Minoshima, T. R. Schibli, H. Matsumoto, M. Hirano, T. Okuno, M. Onishi, and M. Nakazawa, “Long-term measurement of optical
frequencies using a simple, robust and low-noise fiber based frequency
comb,” Opt. Express |

6. | L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring
lasers,” Appl. Phys. B |

7. | J.-L. Peng, H. Ahn, R.-H. Shu, H.-C. Chui, and J. W. Nicholson, “Highly stable, frequency-controlled
mode-locked erbium fiber laser comb,”
Appl. Phys. B |

8. | H. Ahn, R.-H. Shu, R. S. Windeler, and J.-L. Peng, “Building a frequency-stabilized
mode-locked femtosecond laser for optical frequency
metrology,” IEEE Trans. Instrum. Meas. |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(140.4050) Lasers and laser optics : Mode-locked lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 5, 2007

Revised Manuscript: March 15, 2007

Manuscript Accepted: March 15, 2007

Published: April 3, 2007

**Citation**

Jin-Long Peng and Ren-Huei Shu, "Determination of absolute mode number using two mode-locked laser combs in optical frequency metrology," Opt. Express **15**, 4485-4492 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4485

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### References

- Th. Udem, R. Holzwarth, and T. W. Hänsch, "Optical frequency metrology," Nature 416, 233-237 (2002). [CrossRef] [PubMed]
- D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000). [CrossRef] [PubMed]
- S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Instrum. 72, 3749-3771 (2001). [CrossRef]
- L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S. Windeler, "A new method to determine the absolute mode number of a mode-locked femtosecond laser comb used for absolute optical frequency measurements," IEEE J. Sel. Top. Quantum Electron. 9, 1066-1071 (2003). [CrossRef]
- H. Inaba, Y. Daimon, F.-L. Hong, A. Onae, K. Minoshima, T. R. Schibli, H. Matsumoto, M. Hirano, T. Okuno, M. Onishi, and M. Nakazawa, "Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb," Opt. Express 14, 5223-5231 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5223. [CrossRef] [PubMed]
- L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, "Ultrashort-pulse fiber ring lasers," Appl. Phys. B 65, 277-294 (1997). [CrossRef]
- J.-L. Peng, H. Ahn, R.-H. Shu, H.-C. Chui, and J. W. Nicholson, "Highly stable, frequency-controlled mode-locked erbium fiber laser comb," Appl. Phys. B 86, 49-53 (2007). [CrossRef]
- H. Ahn, R.-H. Shu, R. S. Windeler, and J.-L. Peng, "Building a frequency-stabilized mode-locked femtosecond laser for optical frequency metrology," IEEE Trans. Instrum. Meas. 54, 767-770 (2005). [CrossRef]

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