Abstract

During supercontinuum formation in nonlinear fiber the presence of a noise seed on the input laser pulse can lead to significant excess noise on the generated output supercontinuum electric field. We relate pulse-averaged moments of this electric-field noise to the measured RF spectrum of the frequency comb formed by the supercontinuum. We present quantitative numerical results for the fundamental phase, timing, and amplitude noise on the frequency comb resulting from input quantum noise, including the scaling of the noise with different experimental parameters. This fundamental noise provides a lower limit to the phase stability of frequency combs that originate from microstructure fiber.

© 2004 Optical Society of America

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References

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Appl. Phys. B (3)

D. von der Linde, �??Characterization of noise in continuously operating mode-locked lasers,�?? Appl. Phys. B 39, 201-207 (1986).
[CrossRef]

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]

J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, and S. T. Cundiff, �??Excess noise generation during spectral broadening in a microstructured fiber,�?? Appl. Phys. B 77, 279-284 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus and A. Mecozzi, �??Noise of Mode-Locked Lasers,�?? IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

J. Sel. Topics in Quant. Electron. (2)

A. Baltuska, M. Uiberacker, E. Goulielmakis, R. Kienberger, V. S. Yakovlev, T. Udem, T. W. Hänsch, and F. Krausz, �??Phase-controlled amplification of few-cycle laser pulses,�?? J. Sel. Topics in Quant. Electron. 9, 972-989 (2003)
[CrossRef]

F. W. Helbing, G. Steinmeyer, and U. Keller, �??Carrier-envelope offset phase-locking with attosecond timing jitter,�?? J. Sel. Topics in Quant. Electron. 9, 1030-1040 (2003).
[CrossRef]

Nature (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, �??Optical Frequency Metrology,�?? Nature 416, 233 (2002).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (7)

Phys. Rev. Lett. (1)

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]

Science (2)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stenz, 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 (2000).
[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]

Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1.

Schematic showing the basic experimental setup for a heterodyne experiment. Two noisy pulse trains, exhibiting amplitude (an ), phase (φn ) and timing (δtn ) jitter, are optically filtered by a tunable bandpass filter (TBPF) and mixed on a photodetector of response h(t). The resulting voltage is recorded with a RF spectrum analyzer. The contributions of the self heterodyne (red) and heterodyne (blue) parts of the total RF spectrum (black) are shown.

Fig. 2.
Fig. 2.

(2.44 MB) A movie demonstrating the separate effect of the four jitter terms on a pulse train, compared to a noiseless pulse train. Amplitude jitter causes the peak power to vary per pulse with respect to the average. Phase jitter alters the arrival of the carrier oscillations with respect to time. The real component of the timing jitter alters the arrival time while the imaginary component is equivalent to a frequency jitter in the carrier oscillation.

Fig. 3.
Fig. 3.

(a) Supercontinuum spectrum, after filtered by a 8 nm Gaussian bandpass filter, generated by launching a 0.5 nJ laser pulse with a chirp of -282 fs2 into 8 cm of fiber. (b) The amplitude and phase jitter. (c) The complex timing jitter. (d) The CEO phase jitter. (e) The cross-correlation between the real timing jitter and the phase jitter. (f) The cross correlation between the imaginary timing jitter (frequency jitter) and the amplitude jitter. The jitter, defined as the square root of the total noise variance, is not calculated for wavelengths when the filtered intensity drops below -15 dB of the peak.

Fig. 4.
Fig. 4.

(a) Median spectral width as a function of initial chirp for an input pulse energy of 0.5 nJ and a fiber length of 8 cm. (b) The median amplitude and phase jitter. (c) The median complex timing jitter. (d) The median CEO phase jitter. For an input pulse bandwidth of 45 nm, the input pulse duration changes from 15 to 90 fs as the chirp changes from 0 to ±600 fs2

Fig. 5.
Fig. 5.

(a) Median spectral width as a function of pulse energy for chirp of -282 fs2 and a fiber length of 15 cm. (b) The corresponding median amplitude and timing jitter. (c) The corresponding median complex timing jitter. (d) Median spectral width as a function of fiber length for chirp of -282 fs2 and an input energy of 0.5 nJ. (e) The corresponding median amplitude and timing jitter. (f) The corresponding median complex timing jitter.

Equations (12)

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E n ( t + n T r ) ( 1 + r n ( t ) + i ϑ n ( t ) ) E 0 ( t ) e in ϕ CEO ,
S n ( t ) h ( t t n ) E ( i ) n ( t ) E ( j ) n * ( t ) dt t h ( t t n ) ( t t n ) E ( i ) n ( t ) E ( j ) n * ( t ) dt
t n t E ( i ) n ( t ) E ( j ) n * ( t ) dt E ( i ) n ( t ) E ( j ) n * ( t ) dt
a n A 1 r n ( t ) I 0 ( t ) dt ,
δ t R , n 2 A 1 r n ( t ) t I 0 ( t ) dt ,
S self ( f ) = H ( f ) 2 A 2 f r 2 { n δ ( f n f r ) + 4 S a ( f n f r ) + 4 π 2 f 2 S δt R ( f n f r ) } ,
φ n A 1 ϑ n ( t ) I 0 ( t ) dt ,
δt I , n 2 A 1 ϑ n ( t ) t I 0 ( t ) dt .
S cross ( f ) = H 2 ( f ) A 2 f r 2 n δ ( f n f r Δ f CEO ) + 2 S a ( f n f r Δ f CEO )
+ 2 S φ ( f n f r Δ f CEO ) + 2 π 2 f 2 S δt R ( f n f r Δ f CEO ) + 2 π 2 f 2 S δt I ( f n f r Δ f CEO ) ,
E n ( t + n T r ) ( 1 + a n ) e i φ n + i 2 πδ ν n t I 0 ( t δt R , n ) e i θ 0 ( t ) e in ϕ CEO ,
δ ϕ n , CEO θ n ( t ) n θ 0 ( t ) 0 φ n + 2 π ν 0 δt n , R .

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