Abstract

We calculate the quantum-limited shape of the comb lines from a mode-locked Ti:sapphire laser using experimentally-derived parameters for the linear response of the laser to perturbations. The free-running width of the comb lines is found across the laser spectrum. By modeling the effect of a simple feedback loop, we calculate the spectrum of the residual phase noise in terms of the quantum noise and the feedback parameters. Finally, we calculate the frequency uncertainty in an optical frequency measurement if the limiting factor is quantum noise in the detection of the optical heterodyne beat.

© 2008 Optical Society of America

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References

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  1. R. Loudon, Quantum Theory of Light (Clarendon, Oxford, UK, 2000).
  2. A. L. Schawlow and C. H. Townes, "Infrared and Optical Masers," Phys. Rev. 112, 1940-1949 (1958).
    [CrossRef]
  3. H. Haus and A. Mecozzi, "Noise of mode-locked lasers," IEEE J. Quantum Electron. 29, 983-996 (1993).
    [CrossRef]
  4. S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
    [CrossRef]
  5. F. X. ¨artner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-Cycle Pulses Directly from the Laser," in Few-cycle Laser Pulse Generation and its Applications, F. X. Kartner, ed., Topics in Applied Physics, (Springer, Berlin, Germany, 2004) Vol. 95.
    [CrossRef]
  6. R. Paschotta, "Noise of mode-locked lasers (Part I): numerical model," Appl. Phys. B 79, 153-162 (2004).
    [CrossRef]
  7. R. Paschotta, "Noise of mode-locked lasers (Part II): timing jitter and other fluctuations," Appl. Phys. B 79, 163-173 (2004).
    [CrossRef]
  8. N. Newbury and B. Washburn, "Theory of the frequency comb output from a femtosecond fiber laser," IEEE J. Quantum Electron. 41, 1388-1402 (2005).
    [CrossRef]
  9. 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]
  10. L. Matos, O. D. Mucke, J. Chen, and F. X. Kartner, "Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers," Opt. Express 14, 2497-2511 (2006).
    [CrossRef] [PubMed]
  11. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Noise-induced linewidth in frequency combs," Opt. Lett. 31, 1875-1877 (2006).
    [CrossRef] [PubMed]
  12. N. R. Newbury and W. C. Swann, "Low-noise fiber-laser frequency combs," J. Opt. Soc. Am. B 24, 1756-1770 (2007).
    [CrossRef]
  13. C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. R. Schibli, and S. T. Cundiff, "Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling," Opt. Express 15, 6677-6689 (2007).
    [CrossRef] [PubMed]
  14. J. K. Wahlstrand, J. T. Willits, T. R. Schibli, C. R. Menyuk, and S. T. Cundiff, "Quantitative measurement of timing and phase dynamics in a mode-locked laser," Opt. Lett. 32, 3426-3428 (2007).
    [CrossRef] [PubMed]
  15. NIST technical note 1337: Characterization of clocks and oscillators, D. B. Sullivan, D. W. Allan, D. A. Howe, and F. L. Walls, eds. (U.S. Government printing office, Washington, 1990).
  16. P. Wai and C. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technology 14, 148-157 (1996). See the Appendix.
    [CrossRef]
  17. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Carrier-envelope phase slip of ultrashort dispersion-managed solitons," Opt. Lett. 29, 1808-1810 (2004).
    [CrossRef] [PubMed]
  18. D. Kleppner, "Physics - a milestone in time-keeping," Science 319, 1768-1769 (2008).
    [CrossRef] [PubMed]
  19. E. Rubiola, "On the measurement of frequency and of its sample variance with high-resolution counters," Rev. Sci. Instrum. 76, 054703 (2005).
    [CrossRef]
  20. S. T. Dawkins, J. J. McFerran, and A. N. Luiten, "Considerations on the measurement of the stability of oscillators with frequency counters," IEEE Trans. Ultrason., Ferroelectr., Freq. Control 54, 918-925 (2007).
    [CrossRef]
  21. L.-S. Ma, Z. 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, 1843-1845 (2004).
    [CrossRef] [PubMed]

2008 (1)

D. Kleppner, "Physics - a milestone in time-keeping," Science 319, 1768-1769 (2008).
[CrossRef] [PubMed]

2007 (4)

2006 (3)

2005 (2)

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

E. Rubiola, "On the measurement of frequency and of its sample variance with high-resolution counters," Rev. Sci. Instrum. 76, 054703 (2005).
[CrossRef]

2004 (4)

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Carrier-envelope phase slip of ultrashort dispersion-managed solitons," Opt. Lett. 29, 1808-1810 (2004).
[CrossRef] [PubMed]

R. Paschotta, "Noise of mode-locked lasers (Part I): numerical model," Appl. Phys. B 79, 153-162 (2004).
[CrossRef]

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

2003 (1)

S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

1996 (1)

P. Wai and C. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technology 14, 148-157 (1996). See the Appendix.
[CrossRef]

1993 (1)

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

1958 (1)

A. L. Schawlow and C. H. Townes, "Infrared and Optical Masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Ablowitz, M. J.

Bartels, A.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Bi, Z.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Chen, J.

Cundiff, S. T.

Dawkins, S. T.

S. T. Dawkins, J. J. McFerran, and A. N. Luiten, "Considerations on the measurement of the stability of oscillators with frequency counters," IEEE Trans. Ultrason., Ferroelectr., Freq. Control 54, 918-925 (2007).
[CrossRef]

Diddams, S. A.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Haus, H.

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

Hollberg, L.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Ilan, B.

Kartner, F. X.

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]

Kleppner, D.

D. Kleppner, "Physics - a milestone in time-keeping," Science 319, 1768-1769 (2008).
[CrossRef] [PubMed]

Luiten, A. N.

S. T. Dawkins, J. J. McFerran, and A. N. Luiten, "Considerations on the measurement of the stability of oscillators with frequency counters," IEEE Trans. Ultrason., Ferroelectr., Freq. Control 54, 918-925 (2007).
[CrossRef]

Ma, L.-S.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Matos, L.

McFerran, J. J.

S. T. Dawkins, J. J. McFerran, and A. N. Luiten, "Considerations on the measurement of the stability of oscillators with frequency counters," IEEE Trans. Ultrason., Ferroelectr., Freq. Control 54, 918-925 (2007).
[CrossRef]

Mecozzi, A.

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

Menyuk, C.

P. Wai and C. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technology 14, 148-157 (1996). See the Appendix.
[CrossRef]

Menyuk, C. R.

Mucke, O. D.

Newbury, N.

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

Newbury, N. R.

Oates, C.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Paschotta, R.

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]

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

R. Paschotta, "Noise of mode-locked lasers (Part I): numerical model," Appl. Phys. B 79, 153-162 (2004).
[CrossRef]

Robertsson, L.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Rubiola, E.

E. Rubiola, "On the measurement of frequency and of its sample variance with high-resolution counters," Rev. Sci. Instrum. 76, 054703 (2005).
[CrossRef]

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, "Infrared and Optical Masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Schibli, T. R.

Schlatter, A.

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]

Smith, R. P.

Swann, W. C.

Telle, H. R.

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]

Townes, C. H.

A. L. Schawlow and C. H. Townes, "Infrared and Optical Masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Wahlstrand, J. K.

Wai, P.

P. Wai and C. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technology 14, 148-157 (1996). See the Appendix.
[CrossRef]

Washburn, B.

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

Willits, J.

Willits, J. T.

Wilpers, G.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Windeler, R. S.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Ye, J.

S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

Zeller, S. C.

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]

Zucco, M.

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Appl. Phys. B (3)

R. Paschotta, "Noise of mode-locked lasers (Part I): numerical model," Appl. Phys. B 79, 153-162 (2004).
[CrossRef]

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

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]

IEEE J. Quantum Electron. (2)

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

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

IEEE Trans. Ultrason., Ferroelectr., Freq. Control (1)

S. T. Dawkins, J. J. McFerran, and A. N. Luiten, "Considerations on the measurement of the stability of oscillators with frequency counters," IEEE Trans. Ultrason., Ferroelectr., Freq. Control 54, 918-925 (2007).
[CrossRef]

J. Lightwave Technology (1)

P. Wai and C. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technology 14, 148-157 (1996). See the Appendix.
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, "Infrared and Optical Masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Rev. Mod. Phys. (1)

S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

Rev. Sci. Instrum. (1)

E. Rubiola, "On the measurement of frequency and of its sample variance with high-resolution counters," Rev. Sci. Instrum. 76, 054703 (2005).
[CrossRef]

Science (2)

D. Kleppner, "Physics - a milestone in time-keeping," Science 319, 1768-1769 (2008).
[CrossRef] [PubMed]

L.-S. Ma, Z. 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, 1843-1845 (2004).
[CrossRef] [PubMed]

Other (3)

F. X. ¨artner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-Cycle Pulses Directly from the Laser," in Few-cycle Laser Pulse Generation and its Applications, F. X. Kartner, ed., Topics in Applied Physics, (Springer, Berlin, Germany, 2004) Vol. 95.
[CrossRef]

R. Loudon, Quantum Theory of Light (Clarendon, Oxford, UK, 2000).

NIST technical note 1337: Characterization of clocks and oscillators, D. B. Sullivan, D. W. Allan, D. A. Howe, and F. L. Walls, eds. (U.S. Government printing office, Washington, 1990).

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Figures (2)

Fig. 1.
Fig. 1.

The calculated free-running comb lineshape. (a) Power spectrum In(ω) for comb lines near the central frequency (black) and 200 THz from the central frequency (red). (b) Linewidth γ as a function of Ωn , the offset from ϖ eq. The red curve is the result with the full set of A coefficients measured in [13] and [14]. The black curve is the result when timing jitter caused by dispersion is neglected (by setting Aτϖ =0). The blue curve is the result when, in addition, timing and phase jitter driven by intensity fluctuations is neglected (by setting Aτw =0 and Aθw =0).

Fig. 2.
Fig. 2.

Calculated phase noise properties of the locked comb. Phase noise spectrum for comb lines near ϖ eq (black), 100 THz away (red), and 200 THz away (blue).

Equations (8)

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d v d T = A · v + S ,
d Δ ν i Δ ν j dT = k , l ( A ik + A jl ) Δ ν k Δ ν l + D ij ,
D ww = 2 w eq 2 1 τ ph h ¯ ϖ eq w eq , D wg = D gw = w eq g eq N ph , eq N 2 , eq 1 τ ph h ¯ ϖ eq w eq ,
D gg = 2 g eq 2 [ τ ph τ f N ph , eq N 2 , eq + ( N ph , eq N 2 , eq ) 2 ] 1 τ ph h ¯ ϖ eq w eq , D ϖ ϖ = t p 2 1 τ ph h ¯ ϖ eq w eq ,
D θθ = 2 F ( t p ) 1 τ ph h ¯ ϖ eq w eq , D ττ = t p 2 1 τ ph h ¯ ϖ eq w eq ,
Δ ν ˜ m Δ ν ˜ l * ( ω ) = ( i ω I A ) 1 · D · [ ( i ω I A ) 1 ] t ,
I n ( ω ) 0 d T cos ( ω T ) exp { 1 2 Ω n 2 [ Δ τ ( T ) ] 2 + Ω n Δ τ ( T ) Δ θ ( T ) 1 2 [ Δ θ ( T ) ] 2 } ,
S n ϕ ( ω ) = 1 2 Ω n 2 [ Δ τ ( ω ) ] 2 Ω n Δ τ ( ω ) Δ θ ( ω ) + 1 2 θ ( ω ) ] 2 .

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