Abstract

We discuss experimental and theoretical aspects of a low-noise fiber-laser frequency comb, including the experimental configuration and the major contributions to the frequency noise and linewidth of the individual comb modes. Intracavity noise sources acting on the mode-locked laser determine the free-running comb linewidth and include environmental changes, pump noise, and amplified spontaneous emission (ASE). Extracavity noise sources acting outside of the laser typically determine the signal-to-noise ratio on the comb lines and include environmental effects, shot noise, and noise generated during supercontinuum generation. Feedback strongly suppresses these intracavity noise contributions, yielding a system that operates with comb linewidths and timing jitter below the quantum limit set by the intracavity ASE. Finally, we discuss correlations in the residual noise across a phase-locked comb.

© 2007 Optical Society of America

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2007 (1)

J. J. McFerran, W. C. Swann, B. R. Washburn, and N. R. Newbury, "Suppression of pump-induced frequency noise in fiber-laser frequency combs leading to sub-radian fceo phase excursion," Appl. Phys. B 86, 219-227 (2007).
[CrossRef]

2006 (6)

2005 (11)

P. Kubina, P. Adel, F. Adler, G. Grosche, T. W. Hänsch, R. Holzwarth, A. Leitenstorfer, B. Lipphardt, and H. Schnatz, "Long-term comparison of two fiber based frequency comb systems," Opt. Express 13, 904-909 (2005).
[CrossRef] [PubMed]

K. W. Holman, D. D. Hudson, J. Ye, and D. J. Jones, "Remote transfer of a high-stability and ultralow-jitter timing signal," Opt. Lett. 30, 1225-1227 (2005).
[CrossRef] [PubMed]

A. Schlatter, B. Rudin, S. C. Zeller, R. Paschotta, G. J. Spuhler, L. Krainer, N. Haverkamp, H. R. Telle, and U. Keller, "Nearly quantum-noise-limited timing jitter from miniature Er:Yb:glass lasers," Opt. Lett. 30, 1536-1538 (2005).
[CrossRef] [PubMed]

E. Benkler, H. R. Telle, A. Zach, and F. Tauser, "Circumvention of noise contributions in fiber laser based frequency combs," Opt. Express 13, 5662-5668 (2005).
[CrossRef] [PubMed]

I. Hartl, G. Imshev, M. E. Fermann, C. Langrock, and M. M. Fejer, "Integrated self-referenced frequency-comb laser based on a combination of fiber and waveguide technology," Opt. Express 13, 6490-6496 (2005).
[CrossRef] [PubMed]

D. D. Hudson, K. W. Holman, R. J. Jones, S. T. Cundiff, J. Ye, and D. J. Jones, "Mode-locked fiber laser frequency-controlled with an intracavity electro-optic modulator," Opt. Lett. 30, 2948-2950 (2005).
[CrossRef] [PubMed]

R. W. Fox, B. R. Washburn, N. R. Newbury, and L. Hollberg, "Wavelength references for interferometry in air," Appl. Opt. 44, 7793-7801 (2005).
[CrossRef] [PubMed]

B. R. Washburn, W. C. Swann, and N. R. Newbury, "Response dynamics of the frequency comb output from a femtosecond fiber laser," Opt. Express 13, 10622-10633 (2005).
[CrossRef] [PubMed]

N. R. Newbury and B. R. Washburn, "Theory of the frequency comb output from a femtosecond fiber laser," IEEE J. Quantum Electron. 41, 1388-1402 (2005).
[CrossRef]

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, "Low-noise synthesis of microwave signals from an optical source," Electron. Lett. 41, 650-651 (2005).
[CrossRef]

2004 (12)

L.-S. Ma, Z. Bi, A. Bartels, L. Robersson, M. Zucco, R. S. Windeler, G. Wilpers, C. Oates, L. Hollberg, and S. A. Diddams, "Optical frequency synthesis and comparison with uncertainty at the 10−19 level," Science 303, 1843-1845 (2004).
[CrossRef] [PubMed]

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, "Frequency stabilization of mode-locked erbium fiber lasers using pump power control," Appl. Phys. B 78, 321-324 (2004).
[CrossRef]

R. Paschotta, "Noise of mode-locked lasers (Part II): timing jitter and other fluctuations," Appl. Phys. B 79, 163-173 (2004).
[CrossRef]

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fiber grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgensen, "Phase-locked erbium-fiber-laser-based frequency comb in the near infrared," Opt. Lett. 29, 250-252 (2004).
[CrossRef] [PubMed]

H. Hundertmark, D. Wandt, N. Haverkamp, and H. R. Telle, "Phase-locked carrier-envelope-offset frequency at 1560nm," Opt. Express 12, 770-775 (2004).
[CrossRef] [PubMed]

A. Bartels, C. W. Oates, L. Hollberg, and S. A. Diddams, "Stabilization of femtosecond laser frequency combs with subhertz residual linewidths," Opt. Lett. 29, 1081-1083 (2004).
[CrossRef] [PubMed]

B. R. Washburn and N. R. Newbury, "Phase, timing, and amplitude noise on supercontinua generated in microstructure fiber," Opt. Express 12, 2166-2175 (2004).
[CrossRef] [PubMed]

B. R. Washburn, R. Fox, N. R. Newbury, J. W. Nicholson, K. Feder, P. S. Westbrook, and C. G. Jørgensen, "Fiber-laser-based frequency comb with a tunable repetition rate," Opt. Express 12, 4999-5004 (2004).
[CrossRef] [PubMed]

J. W. Nicholson, P. S. Westbrook, K. S. Feder, and A. D. Yablon, "Supercontinuum generation in UV irradiated fibers," Opt. Lett. 29, 2363-2365 (2004).
[CrossRef] [PubMed]

T. R. Schibli, K. Minoshima, F.-L. Hong, H. Inaba, A. Onae, H. Matsumoto, I. Hartl, and M. E. Fermann, "Frequency metrology with a turnkey all-fiber system," Opt. Lett. 29, 2467-2469 (2004).
[CrossRef] [PubMed]

F. Adler, K. Moutzouris, A. Leitenstorfer, H. Schnatz, B. Lipphardt, G. Grosche, and F. Tauser, "Phase-locked two-branch erbium-doped fiber laser system for long-term precision measurements of optical frequencies," Opt. Express 12, 5872-5880 (2004).
[CrossRef] [PubMed]

2003 (5)

F. Tauser, A. Leitenstorfer, and W. Zinth, "Amplified femtosecond pulses from an Er:fiber system: nonlinear pulse shortening and self-referencing detection of the carrier-envelope phase evolution," Opt. Express 11, 594-600 (2003).
[CrossRef] [PubMed]

J. W. Nicholson, M. F. Yan, P. Wisk, J. Fleming, F. DiMarcello, E. Monberg, A. Yablon, C. G. Jørgensen, and T. Veng, "All-fiber, octave-spanning supercontinuum," Opt. Lett. 28, 643-645 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen,S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

F.-L. Hong, K. Minoshima, A. Onae, H. Inaba, H. Takada, A. Hirai, H. Matsumoto, T. Sugiura, and M. Yoshida, "Broad-spectrum frequency comb generation and carrier-envelope offset frequency measurement by second harmonic generation of a mode-locked fiber laser," Opt. Lett. 28, 1-3 (2003).
[CrossRef]

2002 (5)

2001 (2)

H. A. Haus and E. P. Ippen, "Group velocity of solitons," Opt. Lett. 26, 1654-1656 (2001).
[CrossRef]

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

2000 (3)

1999 (2)

H. Kubota, K. R. Tamura, and M. Nakazawa, "Analyses of coherence-maintained ultrashort optical pulse trains and supercontinuum generation in the presence of soliton-amplified spontaneous-emission interaction," J. Opt. Soc. Am. B 16, 2223-2232 (1999).
[CrossRef]

H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, "Carrier-envelope offset phase control: a novel concept for absolute optical frequency and ultrashort pulse generation," Appl. Phys. B 69, 327-332 (1999).
[CrossRef]

1998 (1)

M. Nakazawa, K. Tamura, H. Kubota, and E. Yoshida, "Coherence degradation in the process of supercontinuum generation in an optical fiber," Opt. Fiber Technol. 4, 215-223 (1998).
[CrossRef]

1997 (1)

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]

1996 (1)

S. Namiki, E. P. Ippen, H. A. Haus, and K. Tamura, "Relaxation oscillation behavior in polarization additive pulse mode-locked fiber ring lasers," Appl. Phys. Lett. 69, 3969-3971 (1996).
[CrossRef]

1993 (2)

1991 (2)

L. B. Mercer, "1/f frequency noise effects on self-heterodyne linewidth measurements," J. Lightwave Technol. 9, 485-493 (1991).
[CrossRef]

H. A. Haus, "Quantum noise in a solitonlike repeater system," J. Opt. Soc. Am. B 8, 1122-1126 (1991).
[CrossRef]

1985 (2)

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. 21, 1806-1813 (1985).
[CrossRef]

S. R. Bramwell, D. M. Kane, and A. I. Ferguson, "Frequency offset locking of a synchronously pumped mode-locked dye laser," Opt. Commun. 56, 112-116 (1985).
[CrossRef]

1982 (1)

D. S. Elliott, R. Roy, and S. J. Smith, "Extracavity laser band-shape and bandwidth modification," Phys. Rev. A 26, 12-18 (1982).
[CrossRef]

1975 (1)

F. L. Walls and A. DeMarchi, "RF spectrum of a signal after frequency multiplication; measurement and comparison with a simple calculation," IEEE Trans. Instrum. Meas. IM-24, 210-217 (1975).
[CrossRef]

Adel, P.

Adler, F.

Agrawal, G. P.

Amy-Klein, A.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Bartels, A.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, "Low-noise synthesis of microwave signals from an optical source," Electron. Lett. 41, 650-651 (2005).
[CrossRef]

A. Bartels, C. W. Oates, L. Hollberg, and S. A. Diddams, "Stabilization of femtosecond laser frequency combs with subhertz residual linewidths," Opt. Lett. 29, 1081-1083 (2004).
[CrossRef] [PubMed]

L.-S. Ma, Z. Bi, A. Bartels, L. Robersson, M. Zucco, R. S. Windeler, G. Wilpers, C. Oates, L. Hollberg, and S. A. Diddams, "Optical frequency synthesis and comparison with uncertainty at the 10−19 level," Science 303, 1843-1845 (2004).
[CrossRef] [PubMed]

Benkler, E.

Bergquist, J. C.

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

Bi, Z.

L.-S. Ma, Z. Bi, A. Bartels, L. Robersson, M. Zucco, R. S. Windeler, G. Wilpers, C. Oates, L. Hollberg, and S. A. Diddams, "Optical frequency synthesis and comparison with uncertainty at the 10−19 level," Science 303, 1843-1845 (2004).
[CrossRef] [PubMed]

Bize, S.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Bramwell, S. R.

S. R. Bramwell, D. M. Kane, and A. I. Ferguson, "Frequency offset locking of a synchronously pumped mode-locked dye laser," Opt. Commun. 56, 112-116 (1985).
[CrossRef]

Brown, T.

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fiber grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

Chambon, D.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Chardonnet, C.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Cho, G. C.

I. Hartl, T. R. Schibili, G. Imbeshev, G. C. Cho, M. N. Fermann, K. Minoshima, A. Onae, F.-L. Hong, H. Matsumoto, J. W. Nicholson, and M. F. Yan, "Carrier envelope phase locking of an in-line, low-noise Er fiber system," in Conference on Lasers and Electro-Optics (Optical Society of America, 2004), p. 59.

Clairon, A.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Coddington, I.

Coen, S.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen,S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

Corwin, K. L.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen,S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2002).
[CrossRef]

Cundiff, S. T.

Curtis, E. A.

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

Daimon, Y.

Daussy, C.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

DeMarchi, A.

F. L. Walls and A. DeMarchi, "RF spectrum of a signal after frequency multiplication; measurement and comparison with a simple calculation," IEEE Trans. Instrum. Meas. IM-24, 210-217 (1975).
[CrossRef]

Desurvire, E.

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, 1994).

Diddams, S. A.

K. Kim, S. A. Diddams, P. S. Westbrook, J. W. Nicholson, and K. S. Feder, "Improved stabilization of a 1.3μm femtosecond optical frequency comb by use of a spectrally tailored continuum from a nonlinear fiber grating," Opt. Lett. 31, 277-279 (2006).
[CrossRef] [PubMed]

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, "Low-noise synthesis of microwave signals from an optical source," Electron. Lett. 41, 650-651 (2005).
[CrossRef]

B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgensen, "Phase-locked erbium-fiber-laser-based frequency comb in the near infrared," Opt. Lett. 29, 250-252 (2004).
[CrossRef] [PubMed]

A. Bartels, C. W. Oates, L. Hollberg, and S. A. Diddams, "Stabilization of femtosecond laser frequency combs with subhertz residual linewidths," Opt. Lett. 29, 1081-1083 (2004).
[CrossRef] [PubMed]

L.-S. Ma, Z. Bi, A. Bartels, L. Robersson, M. Zucco, R. S. Windeler, G. Wilpers, C. Oates, L. Hollberg, and S. A. Diddams, "Optical frequency synthesis and comparison with uncertainty at the 10−19 level," Science 303, 1843-1845 (2004).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen,S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

J. Ye, J. L. Hall, and S. A. Diddams, "Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum," Opt. Lett. 25, 1675-1677 (2000).
[CrossRef]

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]

DiMarcello, F.

Drullinger, R. E.

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

Dudley, J. M.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen,S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

Dunlop, A. E.

H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, "Carrier-envelope offset phase control: a novel concept for absolute optical frequency and ultrashort pulse generation," Appl. Phys. B 69, 327-332 (1999).
[CrossRef]

Elliott, D. S.

D. S. Elliott, R. Roy, and S. J. Smith, "Extracavity laser band-shape and bandwidth modification," Phys. Rev. A 26, 12-18 (1982).
[CrossRef]

Fallnich, C.

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, "Frequency stabilization of mode-locked erbium fiber lasers using pump power control," Appl. Phys. B 78, 321-324 (2004).
[CrossRef]

Feder, K.

B. R. Washburn, R. Fox, N. R. Newbury, J. W. Nicholson, K. Feder, P. S. Westbrook, and C. G. Jørgensen, "Fiber-laser-based frequency comb with a tunable repetition rate," Opt. Express 12, 4999-5004 (2004).
[CrossRef] [PubMed]

P. S. Westbrook, J. W. Nicholson, K. Feder, and A. D. Yablon, "UV processing of highly nonlinear fibers for enhanced supercontinuum generation," in Optical Fiber Conference (Optical Society of America, 2004).

Feder, K. S.

Fejer, M. M.

Ferguson, A. I.

S. R. Bramwell, D. M. Kane, and A. I. Ferguson, "Frequency offset locking of a synchronously pumped mode-locked dye laser," Opt. Commun. 56, 112-116 (1985).
[CrossRef]

Fermann, M. E.

Fermann, M. N.

I. Hartl, T. R. Schibili, G. Imbeshev, G. C. Cho, M. N. Fermann, K. Minoshima, A. Onae, F.-L. Hong, H. Matsumoto, J. W. Nicholson, and M. F. Yan, "Carrier envelope phase locking of an in-line, low-noise Er fiber system," in Conference on Lasers and Electro-Optics (Optical Society of America, 2004), p. 59.

Fleming, J.

Fortier, T. M.

Fox, R.

Fox, R. W.

Goncharov, A.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Grosche, G.

Guinet, M.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Hall, J. L.

J. Ye, J. L. Hall, and S. A. Diddams, "Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum," Opt. Lett. 25, 1675-1677 (2000).
[CrossRef]

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]

Hänsch, T. W.

Hartl, I.

Haus, H. A.

H. A. Haus and E. P. Ippen, "Group velocity of solitons," Opt. Lett. 26, 1654-1656 (2001).
[CrossRef]

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]

S. Namiki, E. P. Ippen, H. A. Haus, and K. Tamura, "Relaxation oscillation behavior in polarization additive pulse mode-locked fiber ring lasers," Appl. Phys. Lett. 69, 3969-3971 (1996).
[CrossRef]

H. A. Haus and A. Mecozzi, "Noise of mode-locked lasers," IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, "77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser," Opt. Lett. 18, 1080-1083 (1993).
[CrossRef] [PubMed]

H. A. Haus, "Quantum noise in a solitonlike repeater system," J. Opt. Soc. Am. B 8, 1122-1126 (1991).
[CrossRef]

Haverkamp, N.

Hirai, A.

F.-L. Hong, K. Minoshima, A. Onae, H. Inaba, H. Takada, A. Hirai, H. Matsumoto, T. Sugiura, and M. Yoshida, "Broad-spectrum frequency comb generation and carrier-envelope offset frequency measurement by second harmonic generation of a mode-locked fiber laser," Opt. Lett. 28, 1-3 (2003).
[CrossRef]

Hirano, M.

Ho, P. T.

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. 21, 1806-1813 (1985).
[CrossRef]

Hollberg, L.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, "Low-noise synthesis of microwave signals from an optical source," Electron. Lett. 41, 650-651 (2005).
[CrossRef]

R. W. Fox, B. R. Washburn, N. R. Newbury, and L. Hollberg, "Wavelength references for interferometry in air," Appl. Opt. 44, 7793-7801 (2005).
[CrossRef] [PubMed]

L.-S. Ma, Z. Bi, A. Bartels, L. Robersson, M. Zucco, R. S. Windeler, G. Wilpers, C. Oates, L. Hollberg, and S. A. Diddams, "Optical frequency synthesis and comparison with uncertainty at the 10−19 level," Science 303, 1843-1845 (2004).
[CrossRef] [PubMed]

A. Bartels, C. W. Oates, L. Hollberg, and S. A. Diddams, "Stabilization of femtosecond laser frequency combs with subhertz residual linewidths," Opt. Lett. 29, 1081-1083 (2004).
[CrossRef] [PubMed]

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

Holman, K. W.

Holzwarth, R.

Hong, F. L.

Hong, F.-L.

T. R. Schibli, K. Minoshima, F.-L. Hong, H. Inaba, A. Onae, H. Matsumoto, I. Hartl, and M. E. Fermann, "Frequency metrology with a turnkey all-fiber system," Opt. Lett. 29, 2467-2469 (2004).
[CrossRef] [PubMed]

F.-L. Hong, K. Minoshima, A. Onae, H. Inaba, H. Takada, A. Hirai, H. Matsumoto, T. Sugiura, and M. Yoshida, "Broad-spectrum frequency comb generation and carrier-envelope offset frequency measurement by second harmonic generation of a mode-locked fiber laser," Opt. Lett. 28, 1-3 (2003).
[CrossRef]

I. Hartl, T. R. Schibili, G. Imbeshev, G. C. Cho, M. N. Fermann, K. Minoshima, A. Onae, F.-L. Hong, H. Matsumoto, J. W. Nicholson, and M. F. Yan, "Carrier envelope phase locking of an in-line, low-noise Er fiber system," in Conference on Lasers and Electro-Optics (Optical Society of America, 2004), p. 59.

Hudson, D. D.

Hundertmark, H.

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, "Frequency stabilization of mode-locked erbium fiber lasers using pump power control," Appl. Phys. B 78, 321-324 (2004).
[CrossRef]

H. Hundertmark, D. Wandt, N. Haverkamp, and H. R. Telle, "Phase-locked carrier-envelope-offset frequency at 1560nm," Opt. Express 12, 770-775 (2004).
[CrossRef] [PubMed]

Imbeshev, G.

I. Hartl, T. R. Schibili, G. Imbeshev, G. C. Cho, M. N. Fermann, K. Minoshima, A. Onae, F.-L. Hong, H. Matsumoto, J. W. Nicholson, and M. F. Yan, "Carrier envelope phase locking of an in-line, low-noise Er fiber system," in Conference on Lasers and Electro-Optics (Optical Society of America, 2004), p. 59.

Imshev, G.

Inaba, H.

Ippen, E. P.

H. A. Haus and E. P. Ippen, "Group velocity of solitons," Opt. Lett. 26, 1654-1656 (2001).
[CrossRef]

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]

S. Namiki, E. P. Ippen, H. A. Haus, and K. Tamura, "Relaxation oscillation behavior in polarization additive pulse mode-locked fiber ring lasers," Appl. Phys. Lett. 69, 3969-3971 (1996).
[CrossRef]

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, "77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser," Opt. Lett. 18, 1080-1083 (1993).
[CrossRef] [PubMed]

Itano, W. M.

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

Ivanov, E. N.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, "Low-noise synthesis of microwave signals from an optical source," Electron. Lett. 41, 650-651 (2005).
[CrossRef]

Jones, D. J.

Jones, R. J.

Jørgensen, C. G.

Kane, D. M.

S. R. Bramwell, D. M. Kane, and A. I. Ferguson, "Frequency offset locking of a synchronously pumped mode-locked dye laser," Opt. Commun. 56, 112-116 (1985).
[CrossRef]

Keller, U.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, "Optical phase noise and carrier-envelope offset noise of mode-locked lasers," Appl. Phys. B 82, 265-273 (2006).
[CrossRef]

A. Schlatter, B. Rudin, S. C. Zeller, R. Paschotta, G. J. Spuhler, L. Krainer, N. Haverkamp, H. R. Telle, and U. Keller, "Nearly quantum-noise-limited timing jitter from miniature Er:Yb:glass lasers," Opt. Lett. 30, 1536-1538 (2005).
[CrossRef] [PubMed]

H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, "Carrier-envelope offset phase control: a novel concept for absolute optical frequency and ultrashort pulse generation," Appl. Phys. B 69, 327-332 (1999).
[CrossRef]

Kim, K.

Krainer, L.

Kubina, P.

Kubota, H.

Langrock, C.

Lee, W. D.

S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, "An optical clock based on a single trapped Hg+199 ion," Science 293, 825-828 (2001).
[CrossRef] [PubMed]

Leitenstorfer, A.

Li, Y.

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fiber grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

Lipphardt, B.

P. Kubina, P. Adel, F. Adler, G. Grosche, T. W. Hänsch, R. Holzwarth, A. Leitenstorfer, B. Lipphardt, and H. Schnatz, "Long-term comparison of two fiber based frequency comb systems," Opt. Express 13, 904-909 (2005).
[CrossRef] [PubMed]

F. Adler, K. Moutzouris, A. Leitenstorfer, H. Schnatz, B. Lipphardt, G. Grosche, and F. Tauser, "Phase-locked two-branch erbium-doped fiber laser system for long-term precision measurements of optical frequencies," Opt. Express 12, 5872-5880 (2004).
[CrossRef] [PubMed]

H. R. Telle, B. Lipphardt, and J. Stenger, "Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements," Appl. Phys. B 74, 1-6 (2002).
[CrossRef]

H. Schnatz, B. Lipphardt, and G. Grosche, "Frequency metrology using fiber-based fs-frequency combs," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO) (Optical Society of America, 2006), Paper CTuHl.

Lopez, O.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Lours, M.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
[CrossRef] [PubMed]

Luiten, A. N.

C. Daussy, O. Lopez, A. Amy-Klein, A. Goncharov, M. Guinet, C. Chardonnet, F. Narbonneau, M. Lours, D. Chambon, S. Bize, A. Clairon, G. Santarelli, M. E. Tobar, and A. N. Luiten, "Long-distance frequency dissemination with a resolution of 10(−17)," Phys. Rev. Lett. 94, 203904 (2005).
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K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2002).
[CrossRef]

Weber, K.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen,S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
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Wilpers, G.

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[CrossRef] [PubMed]

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L.-S. Ma, Z. Bi, A. Bartels, L. Robersson, M. Zucco, R. S. Windeler, G. Wilpers, C. Oates, L. Hollberg, and S. A. Diddams, "Optical frequency synthesis and comparison with uncertainty at the 10−19 level," Science 303, 1843-1845 (2004).
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K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, "Fundamental amplitude noise limitations to supercontinuum spectra generated in microstructure fiber," Appl. Phys. B 77, 269-277 (2003).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen,S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
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N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2002).
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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).
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B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgensen, "Phase-locked erbium-fiber-laser-based frequency comb in the near infrared," Opt. Lett. 29, 250-252 (2004).
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Figures (10)

Fig. 1
Fig. 1

Simplified schematic of the frequency comb output of a fiber-laser frequency comb. The comb teeth are simply the modes of the mode-locked laser. Supercontinuum generation after the laser extends this mode structure over a much wider spectral range (typically covering an octave of bandwidth) but the basic comb structure is preserved. The offset frequency, f 0 , is detected by self-referenced detection. The comb can then be locked to (a) a microwave reference by locking f r and f 0 to the reference or (b) an optical reference by locking one comb tooth at v n r e f to the optical reference and f 0 to a fixed fraction of the derived repetition rate.

Fig. 2
Fig. 2

Schematic of a fiber-laser frequency comb, which consists of a mode-locked fiber laser, Er-fiber amplifier, HNLF for supercontinuum generation, and detection of the offset frequency f 0 , repetition frequency f r , and frequency of an optical comb line ν n r e f (through heterodyne detetection with a cw reference laser). The comb is stabilized through electronic feedback to the cavity length and pump power (shown as a dashed line). Noise sources are shown in gray. Intracavity noise acting directly on the laser includes environmental perturbations, ASE-induced quantum noise, and pump fluctuations. Extracavity noise acting after the laser includes environmental perturbations, shot noise, and excess phase noise generated during supercontinuum generation.

Fig. 3
Fig. 3

Example data illustrating the measurements of the fixed point for pump power and cavity length variations. (a) Counted change in f r and f 0 in a 0.1 s counter gate time for a 0.1 Hz square wave in pump power. The sign of Δ f 0 is ambiguous and must be determined by a separate reference laser. From these data, the fixed point is ν fix pump = ( 3.3 MHz 0.80 Hz ) × 50 MHz = 206 THz or 1460 nm . Also evident are a slow drift of the repetition rate and the thermal effects of changing the pump power. (b) Counted change in f r and f 0 in a 0.1 s counter gate time for a 0.1 Hz square wave modulation of the cavity length. The fixed point is ν fix length = ( 1.1 MHz 53 Hz ) × 50 MHz = 1.04 THz , as expected given the group and phase velocities of the optical fiber. Note that if an intracavity air path were used, the fixed point would be 0 Hz (see Subsection 4A). (c) Schematic illustrating the effect of cavity length and pump-power variations given the measured fixed points. Solid lines indicate comb modes with optical power, dashed lines indicate projected comb modes with no optical power.

Fig. 4
Fig. 4

Schematic showing the effect of various noise terms on the comb. Solid lines indicate comb modes with optical power, dashed lines indicate projected comb modes with no optical power. Intracavity noise sources (environmental length and loss fluctuations, pump noise and ASE-induced noise) cause frequency jitter, which is well described by a breathing motion of the comb about a fixed frequency. The jitter increases quadratically as one moves away from the fixed point. Depending on the noise, the jitter will decrease at higher modulation frequencies (which is not captured in this figure). The extracavity noise causes a strongly varying phase noise floor and is more difficult to quantify. Note the fixed point for the length fluctuations differs from Fig. 3 since here we assume all the fiber in the cavity is equally stretched whereas for the fiber stretcher only the properties of the fiber being stretched were important.(Also see Fig. 4 of Ref. [38]).

Fig. 5
Fig. 5

Calculated frequency noise PSD for a free-running comb for the mode at (a) f 0 , (b) 1535 nm , (c) 1126 nm . The total contribution (thick, black curve) comprises the three intracavity contributions: pump noise, ASE-induced quantum noise, and environmental fluctuations (on both cavity length and loss); and two extracavity contributions: environmental fluctuations and the combined effect of shot noise and excess noise generated during supercontinuum formation. This last term is directly connected with the SNR ratio of the measured signal and thus is labeled as “SNR limit” in the graphs. The SNR ranges from 43, 35, and 35 dB in a 300 kHz bandwidth for plots (a)–(c) (chosen to agree with data presented later). Other values used in the plots are listed in Table 1.

Fig. 6
Fig. 6

Linewidth across the comb. The total linewidth (solid black curve) has a dominant contribution from the pump noise (red, dotted curve) with additional contributions from the quantum limit (green, dashed-dotted curve), and environmental 1 f noise from cavity length fluctuations (blue, long dashed curve) and cavity loss fluctuations or 1 f pump fluctuations (brown, short dashed curve). The values and color coding used in these graphs are identical to those in Fig. 5. The contribution of the 1 f noise is calculated over an (arbitrary) time window of 1 ms . The total linewidth is approximated as the sum in quadrature of the individual components. The extracavity noise terms do not contribute significantly to the linewidth. The linewidth of the measured offset frequency beat is simply the value at 0 THz .

Fig. 7
Fig. 7

(a) Schematic of a low-noise fiber-laser frequency comb system. The comb is phase-locked to a cw reference oscillator and used to measure the frequency of a laser under test. For the optical heterodyne measurements, balanced detection is used to enhance the SNR. The laser design uses an air gap as shown in Ref. [12] but all-fiber designs are possible as well. (b) Supercontinuum after periodically poled lithium niobate. Note the enhanced signal at 1037 nm , which is the doubled light from 2074 nm . The enhanced signal at 1126 nm is from a fiber Bragg grating incorporated into the HNLF (Ref. [60]). (c) The unlocked f 0 beat signal from heterodyning the fundamental and doubled 2074 nm light, which has a 43 dB SNR in 300 kHz resolution bandwidth. The uptake on the edges of the beat is a result of the repetition rate beat and another f 0 beat. (There are two such beats in every f r = 50 MHz increment.)

Fig. 8
Fig. 8

Free-running frequency noise PSD for the comb mode at (a) f 0 (near 112 MHz ), (b) 1535 nm , and (c) 1126 nm as measured by beating the comb against a narrow linewidth cw laser. The solid curve is the measured data and the dashed curve is the fitted data using the model in Section 3 (solid black curve in Fig. 5). The deviations in (b) in the middle frequencies are a result of the complex vibrational spectrum that we have not attempted to model.

Fig. 9
Fig. 9

Frequency noise PSD for the comb tooth at 1126 nm for an unlocked comb (solid black curve), a comb with just the offset frequency locked (solid gray curve), and after locking the comb tooth at 1126 nm to the cw laser (solid blue curve). The effect of locking only the offset frequency is the removal of the pump-induced noise, which otherwise dominates for Fourier frequencies from 1 to 10 kHz . The locked and unlocked noise PSD cross at an effective feedback bandwidth of 25 kHz . (b) An example in-loop, phase-locked heterodyne beat signal between the comb tooth near 1126 nm and the laser at 1126 nm for a resolution bandwidth of 3 kHz , corresponding to the blue frequency noise PSD in part (a). The central spike is a coherent peak with an instrument-limited linewidth (below 0.1 Hz ).

Fig. 10
Fig. 10

Counted optical beat signal, shifted to 1 mHz (solid curve) and the counted offset frequency beat signal, shifted to + 1 mHz , (dashed curve) for a 10 s gate time. The 150 μ Hz standard deviation in the counts is due to the frequency counter.

Tables (1)

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Table 1 Fixed Point, Frequency Dependence, and Magnitude of the Various Contributions to the Frequency Noise on the Comb Lines

Equations (19)

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S ν n X = ( ν n ν fix X ) 2 S r X ( f ) ,
( d f r d P ) 0 = f r 2 [ β 2 d ω c d P + ω r m s β 3 d ω r m s d P + Ω g 1 d g d P + δ ω c d A 2 d P ] ,
ν fix pump = ν c + f r ( d φ 2 π d P ) ( d f r f r d P ) 0 ν c ,
S r pump ( f ) = B 1 1 + ( f f 3 dB ) 2 S R I N pump [ 1 Hz ] ,
ν fix length = ν c ( 1 v group L v phase L ) = 0 to 3 THz ,
S r length ( f ) = ( f r v group L ) 2 S length ( f ) [ 1 Hz ] ,
ν f i x loss ν c .
S r loss ( f ) = B 1 1 + ( f + f 3 dB ) 2 S loss ( f ) [ 1 Hz ] ,
ν f i x A S E , timing = ν c ,
S r A S E , timing ( f ) = 2 f r 2 ( ( 1 + n s p ) h ν G P c i r c ) [ t r m s 2 + ( β 2 4 D g ω r m s ) ] [ 1 Hz ] ,
S ν n A S E , S T = 2 f r 2 ( 2 π ) 2 ( ( 1 + n s p ) h ν G P c i r c ) [ Hz 2 Hz ] ,
S ν n length , e x t ( f ) ( 2 π f ν n v p h ) 2 S length ( f ) [ Hz 2 Hz ] .
S ν n shot noise ( f ) + S ν n supercontinuum ( f ) = A f 2 [ Hz 2 Hz ] .
E ( t ) = A ̃ n e i ( 2 π ν n + φ n ) ,
Δ ν n = π S ν n ( 0 ) for S ν n ( f ) = constant ( white frequency noise ) ,
Δ ν n = π S ν n ( 0 ) f 3 dB for S ν n ( f ) = S ν n ( 0 ) 1 + ( f f 3 dB ) 2 , Δ ν n f 3 dB ,
Δ ν n = f i n s t for S φ n ( f ) = constant < ln ( 2 ) f N y q ,
Δ ν n = f i n s t for σ φ n 2 ( f i n s t 2 , f N y q ) < ln ( 2 ) , f i n s t = instrument - limited bandwidth ,
Δ ν n = 4 ln ( 2 ) K [ 4.3 + ln ( 4.3 × 4 π 2 K τ c 2.1 ) ] for S ν n ( f ) = K f , if ( 2 π ) 2 K τ c 1 ,

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