Abstract

In this paper we discuss in detail the underlying theory of a novel method that allows the characterizing of ultrashort laser pulses to be achieved in an analytical way. MEFISTO, (measuring the electric field by interferometric spectral trace observation) is based on a Fourier analysis of the information contained in a spectrally resolved interferometric correlation and can be applied to both situations: the characterization of an unknown pulse (MEFISTO) or to the simultaneous characterization of two different unknowns pulses (Blind-MEFISTO). The theoretical development and experimental practical implications are discussed in both situations.

©2006 Optical Society of America

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References

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  1. J.-C. Diels, E. W. Van Stryland, and D. Gold, “Investigation of the parameters affecting subpicosecond pulse duration of passively mode-locked dye laser,” in Proceedings, First International Conference on Picosecond Phenomena, (Springer-Verlag, New York, 1978), pp. 117–120.
  2. J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. 25, 93–95 (1978).
    [Crossref]
  3. C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
    [Crossref]
  4. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
    [Crossref]
  5. D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
    [Crossref] [PubMed]
  6. J. L. A. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991).
    [Crossref] [PubMed]
  7. G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express 13, 2617–2626 (2005).
    [Crossref] [PubMed]
  8. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, E. J. Gualda, and D. Artigas, “Ultrashort pulse characterisation with SHG collinear-FROG,” Opt. Express 12, 1169–1178 (2004).
    [Crossref] [PubMed]
  9. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Starch-based second-harmonic-generated collinear frequency-resolved optical gating pulse characterization at the focal plane of a high-numerical-aperture lens,” Opt. Lett. 29, 2282–2284 (2004).
    [Crossref] [PubMed]
  10. I. Amat-Roldán, I. G. Cormack, P. Loza-Alvarez, and D. Artigas, “Measurement of electric field by interferometric spectral trace observation,” Opt. Lett. 30, 1063–1065 (2005).
    [Crossref] [PubMed]
  11. D. T. Reid, P. Loza-Alvarez, C. T. A. Brown, T. Beddard, and W. Sibbett, “Amplitude and phase measurement of mid-infrared femtosecond pulses by using cross-correlation frequency-resolved optical gating,” Opt. Lett. 25, 1478–1480 (2000).
    [Crossref]
  12. K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463–2466 (1995).
    [Crossref]
  13. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).
  14. B. Seifert, H. Stolz, and M. Tasche, “Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness,” J. Opt. Soc. Am. B 21, 1089–1097 (2004).
    [Crossref]

2005 (2)

2004 (3)

2000 (1)

1998 (2)

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[Crossref]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[Crossref]

1995 (1)

1993 (1)

1991 (1)

1978 (1)

J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. 25, 93–95 (1978).
[Crossref]

Amat-Roldán, I.

Artigas, D.

Beddard, T.

Benedict, G.

J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. 25, 93–95 (1978).
[Crossref]

Brown, C. T. A.

Chilla, J. L. A.

Cormack, I. G.

DeLong, K. W.

Diels, J.-C.

J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. 25, 93–95 (1978).
[Crossref]

J.-C. Diels, E. W. Van Stryland, and D. Gold, “Investigation of the parameters affecting subpicosecond pulse duration of passively mode-locked dye laser,” in Proceedings, First International Conference on Picosecond Phenomena, (Springer-Verlag, New York, 1978), pp. 117–120.

Gold, D.

J.-C. Diels, E. W. Van Stryland, and D. Gold, “Investigation of the parameters affecting subpicosecond pulse duration of passively mode-locked dye laser,” in Proceedings, First International Conference on Picosecond Phenomena, (Springer-Verlag, New York, 1978), pp. 117–120.

Gualda, E. J.

Iaconis, C.

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[Crossref]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[Crossref]

Kane, D. J.

Loza-Alvarez, P.

Martinez, O. E.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).

Reid, D. T.

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).

Seifert, B.

Sibbett, W.

Steinmeyer, G.

Stibenz, G.

Stolz, H.

Tasche, M.

Trebino, R.

Van Stryland, E. W.

J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. 25, 93–95 (1978).
[Crossref]

J.-C. Diels, E. W. Van Stryland, and D. Gold, “Investigation of the parameters affecting subpicosecond pulse duration of passively mode-locked dye laser,” in Proceedings, First International Conference on Picosecond Phenomena, (Springer-Verlag, New York, 1978), pp. 117–120.

Walmsley, I. A.

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[Crossref]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[Crossref]

White, W. E.

Wong, V.

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

J.-C. Diels, E. W. Van Stryland, and G. Benedict, “Generation and measurement of 200 femtosecond optical pulses,” Opt. Commun. 25, 93–95 (1978).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Other (2)

J.-C. Diels, E. W. Van Stryland, and D. Gold, “Investigation of the parameters affecting subpicosecond pulse duration of passively mode-locked dye laser,” in Proceedings, First International Conference on Picosecond Phenomena, (Springer-Verlag, New York, 1978), pp. 117–120.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).

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Figures (8)

Fig. 1.
Fig. 1. Typical experimental set up needed to obtain a Frequency resolved interferometric correlation trace.

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Fig. 2.
Fig. 2. (a) Frequency resolved interferometric correlation trace and (b) its Fourier transform in the delay-frequency axis, showing the different spectral components.

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Fig. 3.
Fig. 3. Experimental data necessary to solve Eqs. (8) and (10). In (a) we show the amplitude and in (b) the phase corresponding to the two slices at κ=f 0 and κ=f 0+Δf obtained from the transformed trace in Fig. 2(b). We also show the spectra for the two fundamental pulses (c) and at the SHG frequency (d).

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Fig. 4.
Fig. 4. (a) Function Ω1(f,κ) and (b) Ω2(f,κ) at κ=f 0 and κ=f 0+Δf. Dashed lines shows the division between domains where the function cos-1[Ω(f,κ=f 0)] alternates the sign.

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Fig. 5.
Fig. 5. Retrieved pulses (a) E(f) and (b) G(f) using Eqs. (8) and (10) without alternate sign in when changing between the domains shown in Fig. 4.

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Fig. 6.
Fig. 6. Obtained pulses alternating the sign at the domains shown in Fig. 4. Result (a) E(f) and (b) G(f) using the sign combination (+,-) in Eqs. (8) and (10). The spurious solution is shown in (c) E(f) and (d) G(f) using the (-,+) sign combination. Here the discontinuity in the phase corresponds to a 2π change.

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Fig. 7.
Fig. 7. E(f) and G(f) calculated interferometric correlations (blue) and envelope of the interferometric correlation obtained from Fig. 2(a) as the time marginal of the interferometric trace (red). (a) Result using the solution in Figs. 6(a)–6(b). Here the red line coincides with the blue contour and is barely visible. (b) Result using the solution in Figs. 6(c)–6(d).

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Fig. 8.
Fig. 8. Retrieved phase for a pulse with (a) 20 dB and (b) 13 dB SNR. Solution in (b) has been obtained by low band pass filtering the functions Ω1(f,κ) and Ω2(f,κ) in the frequency axis.

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Equations (32)

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I S H G ( f , τ ) = χ · ∣ F t { ( E ( t ) exp [ i 2 π f 1 t ] + G ( t − τ ) exp [ i 2 π f 2 ( t − τ ) ] ) 2 } ∣ 2
Y D C S H G ( f , κ ) = χ · ( ∣ E S H G ( f ) ∣ 2 + ∣ G S H G ( f ) ∣ 2 ) δ ( κ )
+ 4 χ ∫ − ∞ ∞ d f ′ E ( f ′ ) G ( f − f ′ ) E * ( f ′ − κ ) G * ( f − f ′ + κ )
Y κ ≈ 2 f 0 S H G ( f , κ ) = χ · E S H G ( f ) G S H G * ( f ) δ ( f − κ + 2 f 0 )
E ( t ) = F f − 1 { Y S H G ( f , κ = f + 2 f 0 ) χ · F t { ( G ( t ) ) 2 } }
Y κ ≈ f 0 S H G ( f , κ ) = 2 χ · E S H G ( f ) E * ( f + f 0 − κ ) G * ( κ − f 0 ) +
2 χ · G S H G * ( f ) G ( f + f 0 − κ ) E ( κ − f 0 )
R ( f , κ ) = 2 χ 1 U S H G ( f ) U ( f + f 0 − κ ) V ( κ − f 0 ) ×
exp [ ϕ S H G ( f ) − ϕ ( f + f 0 − κ ) − γ ( κ − f 0 ) − θ ( f , κ ) ] +
2 χ 2 · V S H G ( f ) V ( f + f 0 − κ ) U ( κ − f 0 ) ×
exp [ − γ S H G ( f ) + γ ( f + f 0 − κ ) + ϕ ( κ − f 0 ) − θ ( f , κ ) ]
ϕ S H G ( f ) − ϕ ( f ) − γ ( 0 ) = ± cos − 1 [ Ω 1 ( f , κ = f 0 ) ] + θ ( f , κ = f 0 )
γ S H G ( f ) − γ ( f ) − ϕ ( 0 ) = ± cos − 1 [ Ω 2 ( f , κ = f 0 ) ] − θ ( f , κ = f 0 )
ϕ S H G ( f ) − ϕ ( f + Δ f ) − γ ( − Δ f ) = ± cos − 1 [ Ω 1 ( f , κ = f 0 − Δ f ) ] + θ ( f , κ = f 0 − Δ f )
γ S H G ( f ) − γ ( f + Δ f ) − ϕ ( − Δ f ) = ± cos − 1 [ Ω 2 ( f , κ = f 0 − Δ f ) ] − θ ( f , κ = f 0 − Δ f )
Δ ϕ κ ( f ) = ϕ ( f + Δ f ) − ϕ ( f ) = ± cos − 1 [ Ω 1 ( f , κ = f 0 ) ] ∓ cos − 1 [ Ω 1 ( f , κ = f 0 − Δ f ) ] +
θ ( f , κ = f 0 ) − θ ( f , κ = f 0 − Δ f ) + γ ( 0 ) − γ ( − Δ f )
ϕ ( f ) = ϕ ( 0 ) + ∑ f ′ = Δ f f Δ ϕ κ ( f ′ ) .
Δ γ κ ( f ) = γ ( f + Δ f ) − γ ( f ) = ± cos − 1 [ Ω 2 ( f , κ = f 0 ) ] ∓ cos − 1 [ Ω 2 ( f , κ = f 0 − Δ f ) ] −
θ ( f , κ = f 0 ) + θ ( f , κ = f 0 − Δ f ) + ϕ ( 0 ) − ϕ ( − Δ f )
γ ( f ) = γ ( 0 ) + ∑ f ′ = Δ f f Δ γ κ ( f ′ ) .
Y δ S H G ( f , κ = 0 ) = χ · ( ( E 0 S H G ) 2 U S H G 2 ( f ) + ( G 0 S H G ) 2 V S H G 2 ( f ) )
Y CFROG SHG ( 0 , 0 ) = 4 χ E 0 2 G 0 2 P
∣ Y κ ≈ 2 f 0 S H G ( f , k = 2 f 0 + f ) ∣ = χ · E 0 S H G G 0 S H G U S H G ( f ) V S H G ( f )
χ 1 2 U S H G 2 ( f ) = Y CFRO SHG ( 0 , 0 ) 8 P [ Y δ S H G ( f , 0 ) ± ( Y δ S H G 2 ( f , 0 ) − 4 ( Y κ = 2 f 0 S H G ( f , 2 f 0 + f ) ]
χ 2 2 U S H G 2 ( f ) = Y CFRO SHG ( 0 , 0 ) 8 P [ Y δ S H G ( f , 0 ) ∓ ( Y δ S H G 2 ( f , 0 ) − 4 ( Y κ = 2 f 0 S H G ( f , 2 f 0 + f ) ]
Y FROG SHG ( f , 0 ) = 4 E 0 2 G 0 2 χ I E ( f ) ⊗ I G ( f )
Δ ϕ κ ( f ) = ϕ ( f + Δ κ ) − ϕ ( f ) = ± cos − 1 [ Ω ( f , κ = f 0 ) ] ∓ cos − 1 [ Ω ( f , κ = f 0 − Δ κ ) ] +
ϕ ( 0 ) − ϕ ( − Δ κ )
χ · E 0 SHG U SHG ( f ) = Y κ ≈ 2 f 0 SHG ( f , k = 2 f 0 + f )
Y FROG SHG ( f , κ = 0 ) = 4 χ E 0 4 ∫ − ∞ ∞ d f ′ U 2 ( f ′ ) U 2 ( f − f ′ ) = 4 χ E 0 4 · U 2 ( f ) ⊗ U 2 ( f ) .
χ E 0 2 U 2 ( f ) = 1 2 F t { F f − 1 { Y FROG SHG ( F , 0 ) } }

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