Abstract

The temporal coherence function of the femtosecond pulse train from femtosecond optical frequency comb (FOFC) has been studied. The theoretical derivation, which is based on the electric field equations of a pulse train, has been used to model the temporal coherence function of the FOFC and shows good agreement with experimental measurements which are taken with a modified Michelson interferometer. The theoretical and experimental points of view provide useful information for applications of FOFC in imaging and metrology.

© 2009 Optical Society of America

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References

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  1. J. Ye and S. T. Cundiff, Femtosecond optical frequency comb : principle, operation, and applications (Springer, New York, NY, 2005).
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    [CrossRef]
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  6. T. Yasui, K. Minoshima, and H. Matsumoto, "Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry," IEEE J. Quantum Electron. 37, 12-19 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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2008

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, "Experimental demonstration of distance measurement with a femtosecond frequency comb laser," J. Europ. Opt. Soc. Rap. Public.08003 Vol  3 (2008).

2005

2004

2002

2001

T. Yasui, K. Minoshima, and H. Matsumoto, "Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry," IEEE J. Quantum Electron. 37, 12-19 (2001).
[CrossRef]

2000

1998

1996

Berg, S. A.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, "Experimental demonstration of distance measurement with a femtosecond frequency comb laser," J. Europ. Opt. Soc. Rap. Public.08003 Vol  3 (2008).

Bhattacharya, N.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, "Experimental demonstration of distance measurement with a femtosecond frequency comb laser," J. Europ. Opt. Soc. Rap. Public.08003 Vol  3 (2008).

Brabec, T.

Chekhovsky, A. M.

Cui, M.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, "Experimental demonstration of distance measurement with a femtosecond frequency comb laser," J. Europ. Opt. Soc. Rap. Public.08003 Vol  3 (2008).

Golubev, A. N.

Gorbunkov, M. V.

Hansch, T. W.

Kim, S.-W.

Krausz, F.

Matsumoto, H.

Minoshima, K.

Oh, J. S.

Poppe, A.

Schouten, R. N.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, "Experimental demonstration of distance measurement with a femtosecond frequency comb laser," J. Europ. Opt. Soc. Rap. Public.08003 Vol  3 (2008).

Spielmann, C.

Xu, L.

Yamaoka, Y.

Yasui, T.

T. Yasui, K. Minoshima, and H. Matsumoto, "Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry," IEEE J. Quantum Electron. 37, 12-19 (2001).
[CrossRef]

Ye, J.

Appl. Opt.

IEEE J. Quantum Electron.

T. Yasui, K. Minoshima, and H. Matsumoto, "Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry," IEEE J. Quantum Electron. 37, 12-19 (2001).
[CrossRef]

J. Europ. Opt. Soc. Rap. Public.

M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, "Experimental demonstration of distance measurement with a femtosecond frequency comb laser," J. Europ. Opt. Soc. Rap. Public.08003 Vol  3 (2008).

Opt. Lett.

Other

J. Ye and S. T. Cundiff, Femtosecond optical frequency comb : principle, operation, and applications (Springer, New York, NY, 2005).

P. A. Atanasov, 14th International School on Quantum Electronics : laser physics and applications : 18-22 September, 2006, Sunny Beach, Bulgaria (SPIE, Bellingham, Wash., 2007), Chap. 2.

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Figures (6)

Fig. 1.
Fig. 1.

FOFC and its temporal coherence function. (a) Pulse train in the time domain. (b) Comb lines in the frequency domain. (c) TCF of FOFC.

Fig. 2.
Fig. 2.

Relative delay between pulse trains formed by an unbalanced Michelson interferometer (a) Simplified optical layout for interferometry. (b) Relative position between two pulse trains (integer h=1).

Fig. 3.
Fig. 3.

Data processing procedure for determination of ∣γ(τ)∣. (a) Interference fringes variation with the scan step of optical path length. (b) Band-pass filtering of Fourier spectra. (c) Determination of ∣γ(τ)∣ by inverse Fourier transforms.

Fig. 4.
Fig. 4.

Optical schematic. FOFC: femtosecond optical frequency comb, C: collimator, L: Lens, BS: beam splitter, PD: photo detector, SM: ultrasonic stepping motor, HM1–2: half mirror, S1–2: shutter, M1–2: mirror, PC: computer.

Fig. 5.
Fig. 5.

Interference fringes at three different situations. (a) Shutter S1 is closed. (b) Shutter S1 is opened and S2 is closed. (c) Shutter S1 and S2 are opened.

Fig. 6.
Fig. 6.

Reconstructed ∣γ(τ)∣ with different relative delay times.

Equations (18)

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E train ( t ) = A ( t ) exp ( i ω c t + i ( φ 0 + Δ φ ce t ) ) m = + δ ( t m T R ) ,
E ˜ train ( f ) = A ˜ ( f f c ) × m = + δ ( f m f rep f CEO ) ,
S ( f ) E ˜ train ( f ) 2 = A ˜ ( f f c ) 2 × m = + δ ( f m f rep f CEO ) ,
Γ ( τ ) = F 1 [ S ( f ) ]
F 1 [ A ˜ ( f f c ) 2 ] m = + δ ( τ m T R ) .
m = + δ ( f m f rep f CEO ) 2 = m = + δ ( f m f rep f CEO ) ,
F [ m = + δ ( t− m T R ) ] = m = + δ ( f m f rep f CEO ) .
Γ ( 0 ) = < E * ( 0 ) E ( 0 ) > = < E ( 0 ) 2 > = < I > .
γ ( τ ) = Γ ( τ ) Γ ( 0 ) F 1 [ A ˜ ( f f c ) 2 ] m = + δ ( τ m T R ) .
I ( t ) i = 1 N E ( t ) 2
E ( ( τ ) ) 2 = 1 2 E train 1 ( t ) + 1 2 E train 2 ( t + τ ) 2
1 2 E train 1 ( t ) 2 γ ( τ ) cos [ mod ( h × Δ φ ce , 2 π ) ]
I ( τ ) γ ( τ ) i = 1 N cos ( mod ( h × Δ φ ce , 2 π ) )
G ( f ) = F [ I ( τ ) ]
F [ γ ( τ ) cos ( mod ( n × Δ φ ce , 2 π ) ) ]
γ ˜ ( f ) [ 1 2 δ ( f + f CEO ) + 1 2 δ ( f f CEO ) ]
g ( t ) = F 1 [ γ ˜ ( f ) 1 2 δ ( f f CEO ) ] = 1 2 γ ( t ) exp ( ( t ) )
γ ( t ) = 2 × abs [ g ( t ) ]

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