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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 4485–4492
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Determination of absolute mode number using two mode-locked laser combs in optical frequency metrology

Jin-Long Peng and Ren-Huei Shu  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 4485-4492 (2007)
http://dx.doi.org/10.1364/OE.15.004485


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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

Mode-locked (ML) femtosecond lasers have revolutionized optical frequency metrology over the past several years [1

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

]. In the time domain, an ML laser consists of periodic optical pulses, whereas in the frequency domain, these pulses consist of coherently superpositioned broadband optical frequency comb modes. Each comb line has a frequency that is an integer multiple of the repetition frequency plus a carrier-envelope offset (CEO) frequency, such that fn=nfr+fo, where n is mode number of the comb line; fr denotes the repetition rate of the pulse train, and fo denotes the CEO frequency.

Some issues must be dealt with when the ML laser comb is employed to measure the frequency of a laser under measurement (LUM). The repetition rate can be easily determined using a fast photodiode and a microwave frequency counter with certainty. However, the CEO frequency is generally detected using the self-referencing technique [2

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]

], and two possible CEO beat frequencies, fom and fr-fom that are smaller than the repetition frequency can be measured. The correct CEO frequency must be identified. Furthermore, when a beat frequency fb between the LUM and the comb line is measured, two lasers with a frequency fb higher and lower than the comb line frequency can generate the same beat frequency. Therefore, the frequency of the LUM can be given by fL= nfr+fom±fb or fL= (n+1)fr-fom±fb. 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 ±fr/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]

]. For a high repetition rate of the Ti:Sapphire laser in the order of GHz, a commercial wavemeter with an uncertainty of 2×10−7 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.

Ma 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]

]. They measured the beat frequencies at various repetition rates and determined the mode number by the formula, n=[±fo2-(±fo1)+mfr2±fb2-(±fb1)]/(fr1-fr2), where fbi is the beat frequency measured at a repetition rate of fri; the CEO frequency is foi; i is 1 or 2; m is difference between the mode numbers in the beat frequency measurement when the repetition rate is changed from fr1 to fr2. 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]

]. Two mode-locked, erbium-doped fiber laser combs, operated at different repetition rates, are utilized herein to determine the absolute mode number. The determination of the mode number is independent of the frequency fluctuation of the LUM. A simple measurement process was presented to determine the difference in mode number between the two laser combs and the absolute mode number involved in the optical frequency measurement. The following sections describe the principle of the technique and the experimental setup and results for the frequency measurement of an iodine-stabilized Nd:YAG laser.

2. Principles

When a beat frequency fb is measured between the nth comb line and the LUM, lasers with a frequency fb 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 fL1 and decreases for fL2. 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.

Fig. 1. (a) Determine the location of the LUM relative to the comb line by observing the beat frequency variation while reducing the repetition rate. (b) Identify the right CEO frequency by observing the beat frequency variation while changing the CEO frequency.

The comb mode number can be determined by measuring the beat frequencies at two repetition rates. Let fb1 (fb2) be the beat frequency measured at a repetition rate of fr1 (fr2) and a CEO frequency of fo1 (fo2). 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.

Fig. 2. The frequency of the LUM relative to the beating comb lines of two mode-locked lasers operated at different repetition rates.

Then, the frequency of the LUM can be expressed as

fL1=nfr1+fo1+fb1
(1)
fL1=(n+m)fr2+fo2+fb2
(2)

, where m is the shift in the mode number when the repetition rate is changed from fr1 to fr2. From Eqs. (1) and (2), the mode number n can be derived as

n=mfr2+fo2fo1+fb2fb1fr1fr2
(3)

This formula has the same form as that derived 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]

] except that the ambiguity of the sign of the offset frequency and the beat note is eliminated. The uncertainty of the measured n should be much less than unity to eliminate ambiguity. Accordingly, the relative uncertainties in the denominator and the numerator should be much less than 1/(2n). Since fr and fo are phase-locked to RF synthesizers, their values can be determined precisely. The main uncertainty in Eq. (3) arises from the uncertainty of fb2-fb1 and fr1-fr2. 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 fb1 and fb2 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 fb1 and fb2 simultaneously. Therefore, the determination of n is independent of the frequency fluctuation of the LUM.

The value m can be controlled and known by smoothly changing the repetition rate. Smoothly varying the repetition rate takes time if m is not small. A simple approach for determining m is described herein. From Eq. (3), m can be derived as

m=fr1fr2fr2nfo2fo1+fb2fb1fr2
=fr1fr2fr1n+n(fr1fr2)2fr1fr2fo2fo1+fb2fb1fr2
fr1fr2fr1nfo2fo1+fb2fb1fr2ifn(fr1fr2)2fr1fr21
(4)

For a repetition rate of 100 MHz and a repetition rate difference of 10 kHz, the term n(fr1-fr2)2/(fr1fr2) results in an error of only about 0.1 for n=107, 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 fr and fo 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 fr1/n dominates the uncertainty in the first term since fr1 and fr2 are phase-locked to the RF synthesizers; therefore, fr1/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.

The quantity fr1/n is determined as follows. When the repetition rate fr1 is increased by fr1/n or decreased by fr1/(n+1), the mode numbers of the comb lines neighboring the LUM with a frequency of fL1 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 fr1/(n+1) is the same as fr1/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 fr1/n2 and -fr1/(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 fr1/n and can be neglected in the estimation of m. Therefore, fr1/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 fL2. The value of m equals the nearest integer calculated by Eq. (3).

Fig. 3. Parts (a) and (c) show the shift of the comb mode neighboring the LUM at a frequency of fL1 for the repetition rate being increased by fr1/n and decreased by fr1/(n+1), respectively. Parts (b) and (d) show the corresponding movement of the beat signals. The beat signals cross at half of the repetition frequency, pass 0 and fr1, respectively one time as indicated. One beat frequency is exactly the same as before and the other one is changed by an amount equals the change in repetition rate.

3. Experimental setup and results

Figure 4 schematically depicts the frequency measurement of an iodine-stabilized Nd:YAG laser using two fiber laser combs. The two ML Er:fiber lasers were home-made ring lasers based on polarization additive pulse mode-locking (P-APM) [6

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]

]. One laser oscillator had two 10% output couplers, and therefore two output ports. The other had a single output coupler and the output was split into two branches using a 3 dB coupler. Each branch of the two ML lasers was amplified using an erbium-doped fiber amplifier, compressed with a section of a standard single mode fiber (SMF) and then spectrally broadened in a highly non-linear fiber (HNLF) to cover an octave-spanning supercontinuum from 1050 nm to 2100 nm. Each branch of supercontinuum has a total power of more than 100 mW. A single beam f-2f (SBF2F) self-referencing interferometer was utilized in one branch to detect the CEO frequency. Both the repetition rate and the offset frequency were stabilized to synthesizers with the time base referenced to a 10 MHz low-noise oven-controlled quartz oscillator, which was phase-locked to a global positioning system receiver disciplined Rb clock. The 10 MHz reference signal had an instability of less than 2×10−12 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]

].

The second harmonic generation of the Nd:YAG laser was locked to the 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.

Fig. 4. Schematic diagram of the optical frequency measurement using two mode-locked Er:fiber combs. Each laser has two branches of octave-spanning supercontinuum. One branch is for the frequency stabilization of the repetition rate and the CEO frequency, the other branch is for beating with the Nd:YAG laser. PC: polarization controller, SMF: single mode fiber

The repetition rates of the two laser combs were 100 MHz and 99.999 MHz, respectively. The self-referencing beat signals of the two ML lasers were stabilized to 167 MHz. The frequency of the Nd:YAG laser relative to the beating comb line and the right CEO frequencies were determined using the method described above. The CEO frequencies were constrained to be less than the repetition frequency and were both verified to be about 67 MHz. Notably, fo varies with the repetition rate, because fr+fo, and not fo, was controlled.

Figures 5(a) and 5(b) plot the measured beat frequencies fb1 and fb2 with fr1=100 MHz and fr2=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.

The difference in mode number between the beating comb lines of the two laser combs is determined by measuring the repetition frequency difference that is required to shift one comb mode near the frequency of the LUM as described in section 2. The repetition rate fr1 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 fr1/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.

The difference between the repetition frequencies fr1 -fr2 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 fr1-fr2 was 1 kHz±36 μHz. Figure 5(c) plots the calculated mode number nexp given by Eq. (3). The mode number of the nth beating comb line, which generates the beat frequency fb1 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]

].

Fig. 5. Parts (a) and (b) ( (e) and (f) ) are measured beat frequencies without (with) modulating the frequency of the Nd:YAG laser. Parts (c) and (g) are the measured beat frequency difference. Parts (d) and (h) are the calculated mode number.

The frequency fluctuation of the Nd:YAG laser was increased by intentionally varying the frequency-locking point with a 0.5 Hz modulation rate to modulate its frequency, and thus demonstrate that this technique can be applied to a large frequency fluctuation with a small repetition rate shift. Figures 5(e) and 5(f) show the measured beat frequencies with modulated frequency fluctuation of 25 kHz, which significantly exceeds the 1 kHz repetition rate shift. The two beat frequencies were mixed in a double-balanced mixer and the measured difference frequency is shown in Fig. 5(g), which shows no modulation that appears in the individual beat frequency. Figure 5(h) plots the measured mode number, which equals that measured without modulation. The fluctuations in the calculated nexp are similar mainly because the same measured fr1 -fr2 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.

The minimum required variation of the repetition frequency can be estimated from the following approximations. As mentioned before, the main uncertainty in Eq. (3) arises from that of fb2-fb1 and fr1-fr2. The measured frequency fluctuation of fb2-fb1 is around 50 Hz. Therefore, fr1 -fr2 must be much larger than 50 Hz such that the uncertainty of n can be much less than one. Further, the relative uncertainty of fr1-fr2 should be much less than 1/(2n) as mentioned in section 2. Since fr1 and fr2 are phase-locked to the same RF source, the uncertainty of fr1 -fr2 is approximated by √2Δfr, where Δfr 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, Δfr=2×10−13×100 MHz=20 μHz and n≅3×106; The uncertainty of fr1-fr2 is estimated to be 28 μHz, which is close to the measured value of 36 μHz. Therefore, fr1 -fr2 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 fr1/n should be accurate to a relative uncertainty of 0.1/m, the beat note position should be accurate to n×(fr1/n)×(0.1/m), which equals to 0.33 MHz for fr=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.

However, if only one laser comb with variation of the repetition rate was used to measure the frequency of the Nd:YAG laser with 22 kHz of random frequency fluctuation, the fluctuation of the beat frequency difference fb1-fb2 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 222 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]

], but longer integration time is still required for noisy LUMs. Furthermore, Ti:sapphire laser is much more expansive than fiber laser.

4. Conclusions

A simple measurement process was presented to determine the difference in mode number between the two laser combs and the absolute mode number involved in the optical frequency measurement. The determination of the absolute mode number involved in the optical frequency measurement using two erbium-doped fiber laser combs at different repetition rates was demonstrated to measure the frequency of an iodine-stabilized Nd:YAG laser. The determination of mode number was independent of the frequency fluctuation of the laser to be measured. A benchtop optical frequency counter with ocatve-spanning counting capability, based on this technique, is feasible.

Acknowledgments

We gratefully acknowledge J. W. Nicholson for providing the highly nonlinear fiber, as well as Y.-C. Cheng for her contributions. This work is supported by the Bureau of Standards, Metrology and Inspection of the Republic of China, Taiwan.

References and links

1.

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

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]

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]

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]

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]

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]

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]

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]

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

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

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