Abstract

We demonstrate that fiber-based frequency combs with multi-branch configurations can transfer both linewidth and frequency stability to another wavelength at the millihertz level. An intra-cavity electro-optic modulator is employed to obtain a broad servo bandwidth for repetition rate control. We investigate the relative linewidths between two combs using a stable continuous-wave laser as a common reference to stabilize the repetition rate frequencies in both combs. The achieved energy concentration to the carrier of the out-of-loop beat between the two combs was 99% and 30% at a bandwidth of 1 kHz and 7.6 mHz, respectively. The frequency instability of the comb was 3.7×10−16 for a 1 s averaging time, improving to 5-8×10−19 for 10000 s. We show that the frequency noise in the out-of-loop beat originates mainly from phase noise in branched optical fibers.

© 2010 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [PubMed]
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    [CrossRef] [PubMed]

2009

2008

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A 77(3), 033847 (2008).
[CrossRef]

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevicius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[CrossRef]

Y. Nakajima, H. Inaba, F. L. Hong, A. Onae, K. Minoshima, T. Kobayashi, M. Nakazawa, and H. Matsumoto, “Optimized amplification of femtosecond optical pulses by dispersion management for octave-spanning optical frequency comb generation,” Opt. Commun. 281(17), 4484–4487 (2008).
[CrossRef]

2007

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 f (ceo) phase excursions,” Appl. Phys. B 86(2), 219–227 (2007).
[CrossRef]

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

F. Adler, A. Sell, F. Sotier, R. Huber, and A. Leitenstorfer, “Attosecond relative timing jitter and 13 fs tunable pulses from a two-branch Er:fiber laser,” Opt. Lett. 32(24), 3504–3506 (2007).
[CrossRef] [PubMed]

2006

2005

2004

2003

2002

2000

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(5466), 635–639 (2000).
[CrossRef] [PubMed]

1999

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82(19), 3799–3802 (1999).
[CrossRef]

T. Okuno, M. Onishi, T. Kashiwada, S. Ishikawa, and M. Nishimura, “Silica-based functional fibers with enhanced nonlinearity and their applications,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1385–1391 (1999).
[CrossRef]

T. Udem, J. Reichert, R. Holzwarth, and T. W. Hansch, “Absolute optical frequency measurement of the cesium D-1 line with a mode-locked laser,” Phys. Rev. Lett. 82(18), 3568–3571 (1999).
[CrossRef]

1994

1993

1990

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57(7), 653–655 (1990).
[CrossRef]

1989

Y. Kimura, K. Suzuki, and M. Nakazawa, “46.5dB gain in Er3+-doped fiber amplifier pumped by 1.48 μm GaInAsP laser diodes,” Electron. Lett. 25(24), 1656–1657 (1989).
[CrossRef]

Adler, F.

Bartels, A.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 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(10), 1081–1083 (2004).
[CrossRef] [PubMed]

Baumann, E.

Bergquist, J. C.

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82(19), 3799–3802 (1999).
[CrossRef]

Bi, Z. Y.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[CrossRef] [PubMed]

Braje, D.

Coddington, I.

Coen, S.

Cruz, F. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82(19), 3799–3802 (1999).
[CrossRef]

Cundiff, S. T.

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(21), 2948–2950 (2005).
[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(5466), 635–639 (2000).
[CrossRef] [PubMed]

Daimon, Y.

Diddams, S. A.

S. A. Diddams, M. Kirchner, T. Fortier, D. Braje, A. M. Weiner, and L. Hollberg, “Improved signal-to-noise ratio of 10 GHz microwave signals generated with a mode-filtered femtosecond laser frequency comb,” Opt. Express 17(5), 3331–3340 (2009).
[CrossRef] [PubMed]

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[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(10), 1081–1083 (2004).
[CrossRef] [PubMed]

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[CrossRef] [PubMed]

E. N. Ivanov, S. A. Diddams, and L. Hollberg, “Experimental study of noise properties of a Ti: Sapphire femtosecond laser,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50(4), 355–360 (2003).
[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(5466), 635–639 (2000).
[CrossRef] [PubMed]

Dudley, J. M.

Feder, K. S.

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, “Fiber-laser frequency combs with subhertz relative linewidths,” Opt. Lett. 31(20), 3046–3048 (2006).
[CrossRef] [PubMed]

Fejer, M. M.

Fermann, M. E.

Foreman, S. M.

Fortier, T.

Gill, P.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A 77(3), 033847 (2008).
[CrossRef]

Giorgetta, F. R.

Grosche, G.

Hall, J. L.

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(5466), 635–639 (2000).
[CrossRef] [PubMed]

L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places - accurate cancellation of phase noise introduced by an optical-fiber or other time-varying path,” Opt. Lett. 19(21), 1777–1779 (1994).
[CrossRef] [PubMed]

Hansch, T. W.

T. Udem, J. Reichert, R. Holzwarth, and T. W. Hansch, “Absolute optical frequency measurement of the cesium D-1 line with a mode-locked laser,” Phys. Rev. Lett. 82(18), 3568–3571 (1999).
[CrossRef]

Hartl, I.

Haus, H. A.

Hirai, A.

Hirano, M.

Hollberg, L.

S. A. Diddams, M. Kirchner, T. Fortier, D. Braje, A. M. Weiner, and L. Hollberg, “Improved signal-to-noise ratio of 10 GHz microwave signals generated with a mode-filtered femtosecond laser frequency comb,” Opt. Express 17(5), 3331–3340 (2009).
[CrossRef] [PubMed]

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 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(10), 1081–1083 (2004).
[CrossRef] [PubMed]

E. N. Ivanov, S. A. Diddams, and L. Hollberg, “Experimental study of noise properties of a Ti: Sapphire femtosecond laser,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50(4), 355–360 (2003).
[CrossRef] [PubMed]

Holman, K. W.

Holzwarth, R.

T. Udem, J. Reichert, R. Holzwarth, and T. W. Hansch, “Absolute optical frequency measurement of the cesium D-1 line with a mode-locked laser,” Phys. Rev. Lett. 82(18), 3568–3571 (1999).
[CrossRef]

Hong, F. L.

T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima, and F. L. Hong, “One-dimensional optical lattice clock with a fermionic Yb-171 isotope,” Appl. Phys. Express 2 (2009).
[CrossRef]

Y. Nakajima, H. Inaba, F. L. Hong, A. Onae, K. Minoshima, T. Kobayashi, M. Nakazawa, and H. Matsumoto, “Optimized amplification of femtosecond optical pulses by dispersion management for octave-spanning optical frequency comb generation,” Opt. Commun. 281(17), 4484–4487 (2008).
[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(12), 5223–5231 (2006).
[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(21), 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(17), 1516–1518 (2003).
[CrossRef] [PubMed]

K. Hosaka, H. Inaba, Y. Nakajima, M. Yasuda, T. Kohno, A. Onae, and F. L. Hong, “Evaluation of the clock laser for an Yb lattice clock using an optical fiber comb,” To be published in IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
[PubMed]

Hosaka, K.

T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima, and F. L. Hong, “One-dimensional optical lattice clock with a fermionic Yb-171 isotope,” Appl. Phys. Express 2 (2009).
[CrossRef]

K. Hosaka, H. Inaba, Y. Nakajima, M. Yasuda, T. Kohno, A. Onae, and F. L. Hong, “Evaluation of the clock laser for an Yb lattice clock using an optical fiber comb,” To be published in IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
[PubMed]

Huber, R.

Hudson, D. D.

Inaba, H.

T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima, and F. L. Hong, “One-dimensional optical lattice clock with a fermionic Yb-171 isotope,” Appl. Phys. Express 2 (2009).
[CrossRef]

Y. Nakajima, H. Inaba, F. L. Hong, A. Onae, K. Minoshima, T. Kobayashi, M. Nakazawa, and H. Matsumoto, “Optimized amplification of femtosecond optical pulses by dispersion management for octave-spanning optical frequency comb generation,” Opt. Commun. 281(17), 4484–4487 (2008).
[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(12), 5223–5231 (2006).
[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(21), 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(17), 1516–1518 (2003).
[CrossRef] [PubMed]

K. Hosaka, H. Inaba, Y. Nakajima, M. Yasuda, T. Kohno, A. Onae, and F. L. Hong, “Evaluation of the clock laser for an Yb lattice clock using an optical fiber comb,” To be published in IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
[PubMed]

Ippen, E. P.

Ishikawa, S.

T. Okuno, M. Onishi, T. Kashiwada, S. Ishikawa, and M. Nishimura, “Silica-based functional fibers with enhanced nonlinearity and their applications,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1385–1391 (1999).
[CrossRef]

Itano, W. M.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82(19), 3799–3802 (1999).
[CrossRef]

Ivanov, E. N.

E. N. Ivanov, S. A. Diddams, and L. Hollberg, “Experimental study of noise properties of a Ti: Sapphire femtosecond laser,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50(4), 355–360 (2003).
[CrossRef] [PubMed]

Jones, D. J.

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(21), 2948–2950 (2005).
[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(5466), 635–639 (2000).
[CrossRef] [PubMed]

Jones, R. J.

Jungner, P.

Kashiwada, T.

T. Okuno, M. Onishi, T. Kashiwada, S. Ishikawa, and M. Nishimura, “Silica-based functional fibers with enhanced nonlinearity and their applications,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1385–1391 (1999).
[CrossRef]

Kimura, Y.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57(7), 653–655 (1990).
[CrossRef]

Y. Kimura, K. Suzuki, and M. Nakazawa, “46.5dB gain in Er3+-doped fiber amplifier pumped by 1.48 μm GaInAsP laser diodes,” Electron. Lett. 25(24), 1656–1657 (1989).
[CrossRef]

Kirchner, M.

Kobayashi, T.

Y. Nakajima, H. Inaba, F. L. Hong, A. Onae, K. Minoshima, T. Kobayashi, M. Nakazawa, and H. Matsumoto, “Optimized amplification of femtosecond optical pulses by dispersion management for octave-spanning optical frequency comb generation,” Opt. Commun. 281(17), 4484–4487 (2008).
[CrossRef]

Kohno, T.

T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima, and F. L. Hong, “One-dimensional optical lattice clock with a fermionic Yb-171 isotope,” Appl. Phys. Express 2 (2009).
[CrossRef]

K. Hosaka, H. Inaba, Y. Nakajima, M. Yasuda, T. Kohno, A. Onae, and F. L. Hong, “Evaluation of the clock laser for an Yb lattice clock using an optical fiber comb,” To be published in IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
[PubMed]

Krauss, G.

Kubota, H.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57(7), 653–655 (1990).
[CrossRef]

Kurokawa, K.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57(7), 653–655 (1990).
[CrossRef]

Langrock, C.

Le Coq, Y.

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

Leitenstorfer, A.

Lipphardt, B.

Lorini, L.

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

Ma, L. S.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[CrossRef] [PubMed]

L. S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places - accurate cancellation of phase noise introduced by an optical-fiber or other time-varying path,” Opt. Lett. 19(21), 1777–1779 (1994).
[CrossRef] [PubMed]

Marcinkevicius, A.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevicius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[CrossRef]

Martin, M. J.

M. J. Martin, S. M. Foreman, T. R. Schibli, and J. Ye, “Testing ultrafast mode-locking at microhertz relative optical linewidth,” Opt. Express 17(2), 558–568 (2009).
[CrossRef] [PubMed]

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevicius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[CrossRef]

Matsumoto, H.

McFerran, J. J.

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 f (ceo) phase excursions,” Appl. Phys. B 86(2), 219–227 (2007).
[CrossRef]

W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, “Fiber-laser frequency combs with subhertz relative linewidths,” Opt. Lett. 31(20), 3046–3048 (2006).
[CrossRef] [PubMed]

Millo, J.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A 77(3), 033847 (2008).
[CrossRef]

Minoshima, K.

Moutzouris, K.

Nakajima, Y.

T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima, and F. L. Hong, “One-dimensional optical lattice clock with a fermionic Yb-171 isotope,” Appl. Phys. Express 2 (2009).
[CrossRef]

Y. Nakajima, H. Inaba, F. L. Hong, A. Onae, K. Minoshima, T. Kobayashi, M. Nakazawa, and H. Matsumoto, “Optimized amplification of femtosecond optical pulses by dispersion management for octave-spanning optical frequency comb generation,” Opt. Commun. 281(17), 4484–4487 (2008).
[CrossRef]

K. Hosaka, H. Inaba, Y. Nakajima, M. Yasuda, T. Kohno, A. Onae, and F. L. Hong, “Evaluation of the clock laser for an Yb lattice clock using an optical fiber comb,” To be published in IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
[PubMed]

Nakazawa, M.

Y. Nakajima, H. Inaba, F. L. Hong, A. Onae, K. Minoshima, T. Kobayashi, M. Nakazawa, and H. Matsumoto, “Optimized amplification of femtosecond optical pulses by dispersion management for octave-spanning optical frequency comb generation,” Opt. Commun. 281(17), 4484–4487 (2008).
[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(12), 5223–5231 (2006).
[CrossRef] [PubMed]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57(7), 653–655 (1990).
[CrossRef]

Y. Kimura, K. Suzuki, and M. Nakazawa, “46.5dB gain in Er3+-doped fiber amplifier pumped by 1.48 μm GaInAsP laser diodes,” Electron. Lett. 25(24), 1656–1657 (1989).
[CrossRef]

Nelson, L. E.

Newbury, N. R.

E. Baumann, F. R. Giorgetta, J. W. Nicholson, W. C. Swann, I. Coddington, and N. R. Newbury, “High-performance, vibration-immune, fiber-laser frequency comb,” Opt. Lett. 34(5), 638–640 (2009).
[CrossRef] [PubMed]

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 f (ceo) phase excursions,” Appl. Phys. B 86(2), 219–227 (2007).
[CrossRef]

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, “Fiber-laser frequency combs with subhertz relative linewidths,” Opt. Lett. 31(20), 3046–3048 (2006).
[CrossRef] [PubMed]

B. R. Washburn and N. R. Newbury, “Phase, timing, and amplitude noise on supercontinua generated in microstructure fiber,” Opt. Express 12(10), 2166–2175 (2004).
[CrossRef] [PubMed]

Nicholson, J. W.

Nishimura, M.

T. Okuno, M. Onishi, T. Kashiwada, S. Ishikawa, and M. Nishimura, “Silica-based functional fibers with enhanced nonlinearity and their applications,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1385–1391 (1999).
[CrossRef]

Oates, C.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[CrossRef] [PubMed]

Oates, C. W.

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[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(10), 1081–1083 (2004).
[CrossRef] [PubMed]

Okuno, T.

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(12), 5223–5231 (2006).
[CrossRef] [PubMed]

T. Okuno, M. Onishi, T. Kashiwada, S. Ishikawa, and M. Nishimura, “Silica-based functional fibers with enhanced nonlinearity and their applications,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1385–1391 (1999).
[CrossRef]

Onae, A.

Onishi, M.

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(12), 5223–5231 (2006).
[CrossRef] [PubMed]

T. Okuno, M. Onishi, T. Kashiwada, S. Ishikawa, and M. Nishimura, “Silica-based functional fibers with enhanced nonlinearity and their applications,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1385–1391 (1999).
[CrossRef]

Oxborrow, M.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A 77(3), 033847 (2008).
[CrossRef]

Pugla, S.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A 77(3), 033847 (2008).
[CrossRef]

Quraishi, Q.

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

Ranka, J. K.

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(5466), 635–639 (2000).
[CrossRef] [PubMed]

Reichert, J.

T. Udem, J. Reichert, R. Holzwarth, and T. W. Hansch, “Absolute optical frequency measurement of the cesium D-1 line with a mode-locked laser,” Phys. Rev. Lett. 82(18), 3568–3571 (1999).
[CrossRef]

Robertsson, L.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[CrossRef] [PubMed]

Scheu, R.

Schibli, T. R.

Schnatz, H.

Sell, A.

Sotier, F.

Stentz, A.

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(5466), 635–639 (2000).
[CrossRef] [PubMed]

Sugiura, T.

Suzuki, K.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57(7), 653–655 (1990).
[CrossRef]

Y. Kimura, K. Suzuki, and M. Nakazawa, “46.5dB gain in Er3+-doped fiber amplifier pumped by 1.48 μm GaInAsP laser diodes,” Electron. Lett. 25(24), 1656–1657 (1989).
[CrossRef]

Swann, W. C.

E. Baumann, F. R. Giorgetta, J. W. Nicholson, W. C. Swann, I. Coddington, and N. R. Newbury, “High-performance, vibration-immune, fiber-laser frequency comb,” Opt. Lett. 34(5), 638–640 (2009).
[CrossRef] [PubMed]

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 f (ceo) phase excursions,” Appl. Phys. B 86(2), 219–227 (2007).
[CrossRef]

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, “Fiber-laser frequency combs with subhertz relative linewidths,” Opt. Lett. 31(20), 3046–3048 (2006).
[CrossRef] [PubMed]

Takada, H.

Tamura, K.

Tauser, F.

Udem, T.

T. Udem, J. Reichert, R. Holzwarth, and T. W. Hansch, “Absolute optical frequency measurement of the cesium D-1 line with a mode-locked laser,” Phys. Rev. Lett. 82(18), 3568–3571 (1999).
[CrossRef]

Washburn, B. R.

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 f (ceo) phase excursions,” Appl. Phys. B 86(2), 219–227 (2007).
[CrossRef]

B. R. Washburn and N. R. Newbury, “Phase, timing, and amplitude noise on supercontinua generated in microstructure fiber,” Opt. Express 12(10), 2166–2175 (2004).
[CrossRef] [PubMed]

Webster, S. A.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A 77(3), 033847 (2008).
[CrossRef]

Weiner, A. M.

Westbrook, P. S.

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, “Fiber-laser frequency combs with subhertz relative linewidths,” Opt. Lett. 31(20), 3046–3048 (2006).
[CrossRef] [PubMed]

Wilpers, G.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[CrossRef] [PubMed]

Windeler, R. S.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[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(5466), 635–639 (2000).
[CrossRef] [PubMed]

Yasuda, M.

T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima, and F. L. Hong, “One-dimensional optical lattice clock with a fermionic Yb-171 isotope,” Appl. Phys. Express 2 (2009).
[CrossRef]

K. Hosaka, H. Inaba, Y. Nakajima, M. Yasuda, T. Kohno, A. Onae, and F. L. Hong, “Evaluation of the clock laser for an Yb lattice clock using an optical fiber comb,” To be published in IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
[PubMed]

Ye, J.

Yoshida, M.

Yost, D. C.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevicius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[CrossRef]

Young, B. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82(19), 3799–3802 (1999).
[CrossRef]

Zinth, W.

Zucco, M.

L. S. Ma, Z. Y. Bi, A. Bartels, L. Robertsson, 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(5665), 1843–1845 (2004).
[CrossRef] [PubMed]

Appl. Phys. B

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 f (ceo) phase excursions,” Appl. Phys. B 86(2), 219–227 (2007).
[CrossRef]

Appl. Phys. Express

T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima, and F. L. Hong, “One-dimensional optical lattice clock with a fermionic Yb-171 isotope,” Appl. Phys. Express 2 (2009).
[CrossRef]

Appl. Phys. Lett.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57(7), 653–655 (1990).
[CrossRef]

Electron. Lett.

Y. Kimura, K. Suzuki, and M. Nakazawa, “46.5dB gain in Er3+-doped fiber amplifier pumped by 1.48 μm GaInAsP laser diodes,” Electron. Lett. 25(24), 1656–1657 (1989).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

T. Okuno, M. Onishi, T. Kashiwada, S. Ishikawa, and M. Nishimura, “Silica-based functional fibers with enhanced nonlinearity and their applications,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1385–1391 (1999).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

E. N. Ivanov, S. A. Diddams, and L. Hollberg, “Experimental study of noise properties of a Ti: Sapphire femtosecond laser,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50(4), 355–360 (2003).
[CrossRef] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control.

K. Hosaka, H. Inaba, Y. Nakajima, M. Yasuda, T. Kohno, A. Onae, and F. L. Hong, “Evaluation of the clock laser for an Yb lattice clock using an optical fiber comb,” To be published in IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
[PubMed]

Nat. Photonics

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevicius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008).
[CrossRef]

I. Coddington, W. C. Swann, L. Lorini, J. C. Bergquist, Y. Le Coq, C. W. Oates, Q. Quraishi, K. S. Feder, J. W. Nicholson, P. S. Westbrook, S. A. Diddams, and N. R. Newbury, “Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter,” Nat. Photonics 1(5), 283–287 (2007).
[CrossRef]

Opt. Commun.

Y. Nakajima, H. Inaba, F. L. Hong, A. Onae, K. Minoshima, T. Kobayashi, M. Nakazawa, and H. Matsumoto, “Optimized amplification of femtosecond optical pulses by dispersion management for octave-spanning optical frequency comb generation,” Opt. Commun. 281(17), 4484–4487 (2008).
[CrossRef]

Opt. Express

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(12), 5223–5231 (2006).
[CrossRef] [PubMed]

M. J. Martin, S. M. Foreman, T. R. Schibli, and J. Ye, “Testing ultrafast mode-locking at microhertz relative optical linewidth,” Opt. Express 17(2), 558–568 (2009).
[CrossRef] [PubMed]

A. Sell, G. Krauss, R. Scheu, R. Huber, and A. Leitenstorfer, “8-fs pulses from a compact Er:fiber system: quantitative modeling and experimental implementation,” Opt. Express 17(2), 1070–1077 (2009).
[CrossRef] [PubMed]

S. A. Diddams, M. Kirchner, T. Fortier, D. Braje, A. M. Weiner, and L. Hollberg, “Improved signal-to-noise ratio of 10 GHz microwave signals generated with a mode-filtered femtosecond laser frequency comb,” Opt. Express 17(5), 3331–3340 (2009).
[CrossRef] [PubMed]

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(6), 594–600 (2003).
[CrossRef] [PubMed]

B. R. Washburn and N. R. Newbury, “Phase, timing, and amplitude noise on supercontinua generated in microstructure fiber,” Opt. Express 12(10), 2166–2175 (2004).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Setup for a comb with an intra-cavity EOM and multi-branch configuration. Thick solid lines and curves represent optical fiber; thin solid lines represent beams in space, and broken lines represent electric wire. PD, photo detector; ISO, optical isolator; PBS, polarization beam splitter; Q, quarter wave plate; H, half wave plate; TEC, thermo-electric cooler; BPF, optical bandpass filter.

Fig. 2
Fig. 2

Experimental setup used to observe an out-of-loop beat between two combs referring a common CW laser. The CEO frequency in each comb is narrowed by fast phase locking. A narrow linewidth CW laser is used for repetition rate locking (578 nm) and a stabilized CW slave laser used for out-of-loop phase characterization (1064 nm). The slave laser is phase-locked to the 578 nm laser using a self-referenced fiber comb.

Fig. 3
Fig. 3

(a) Phase-locked CEO beat (in-loop beat) note of comb #1. Inset: High-resolution spectrum of the beat note. (b) Phase-locked heterodyne beat (in-loop beat) note between comb #1 and the 578 nm reference laser. We were able to obtain very clean in-loop beat spectra. These linewidths were 30 mHz, which was the resolution limit of the FFT analyzer. RBW, resolution bandwidth; VBW, video bandwidth.

Fig. 4
Fig. 4

Schematic of the experimental setup in the frequency domain. Each comb is stabilized to a reference laser by phase locking f 0, 1, f 0, 2, f 1156, 1, and f 1156, 2 to microwave references. The slave laser is stabilized to comb #1 by phase locking f 1064, 1. f 1064, 2 is “out-of-loop”. Each comb component at 1156 nm is frequency-doubled to detect a beat note with the 578 nm reference laser.

Fig. 5
Fig. 5

Evaluation of the out-of-loop beat. (a) The RF spectrum of an out-of-loop beat observed with a spectrum analyzer. The spectrum is clean and its energy concentration to the coherent carrier is more than 99% at a bandwidth of 1 kHz. (b) The RF spectrum of an out-of-loop beat observed with a high-resolution FFT analyzer. The linewidth is less than the resolution limit of the FFT analyzer (7.6 mHz). The energy concentration to the coherent carrier is approximately 30% at a bandwidth of 7.6 mHz. (c) Recorded frequency of the out-of-loop beat for a measurement period of more than 16 hours. The averaging time per point is 0.1 s. (d) Allan deviations of the recorded frequency of the out-of-loop beat and all in-loop beats.

Fig. 6
Fig. 6

Relative frequency stability (Allan deviation) of the combs. The black circles represent the best stability of the out-of-loop beat of the combs. The red circles represent the stability when the fiber covers in the comb systems are partly removed, which represents the effect of fiber noise. The blue circles show the frequency stability when the pump power for the EDFA of the branch used to detect the CEO beat in comb #1 is decreased from 290 to 180 mW, which represents the effect of the ASE from an EDFA. These (black, red, and blue circles) are translated to the stability of one comb (divided by square root of 2). The green circles indicate the stability of the beat note between the reference laser and an identical comb in another branch, which shows whether or not the wavelength conversion in each comb is the dominant noise source.

Equations (7)

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ν n = f 0 + n f r + ε n, k ,
f 1156, 1 = ν 1156 ν m, 1 = ν 1156 f 0, 1 m f r, 1 ε m, 1 ,
ν n, 1 = f 0, 1 + n f r, 1 + ε n, 2 = ( 1 α ) f 0 , 1 + α f 0 , 1 + α m f r, 1 + α ε m, 1 α ε m, 1 + ε n, 2 = ( 1 α ) f 0 , 1 + α ν 1156 + α f 1156 , 1 + ε n, 2 α ε m, 1 ,
ν p, 2 = f 0, 2 + p f r, 2 + ε q, 4 = ( 1 β ) f 0 , 2 + β f 0 , 2 + β p f r, 2 + β ε p, 3 β ε p, 3 + ε q, 4 = ( 1 β ) f 0 , 2 + β ν 1156 + β f 1156 , 2 + ε q, 4 β ε p, 3 ,
ν 1064 = ν n , 1 + f 1064 , 1 = ν p , 2 + f 1064, 2 .
f out of loop = f 1064, 1 = ν p, 2 ν n , 1 + f 1064, 2 = ( 1 β ) f 0 , 2 ( 1 α ) f 0 , 1 + β f 1156 , 2 α f 1156 , 1 + f 1064 , 2 + ( β α ) ν 1156 + ε q, 4 ε n, 2 + α ε m, 1 β ε p, 3 .
< ( δ f out of loop ) 2 > = ( 1 α ) 2 < ( δ f 0 , 1 ) 2 > + ( 1 β ) 2 < ( δ f 0 , 2 ) 2 > + α 2 < ( δ f 1156 , 1 ) 2 > + β 2 < ( δ f 1156 , 2 ) 2 > + < ( δ f 1064 , 2 ) 2 > + ( β α ) 2 < ( δ ν 1156 ) 2 > + ( 2 + α 2 + β 2 ) < ε 2 > .

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