Abstract

Using 3.6- and 5.3-fs pulses, we demonstrated theoretically and experimentally that fringe-resolved autocorrelation (FRAC) traces are distorted by bandwidth limitations of the second-harmonic generation (SHG) in 10-µm-thick, type I β -BaB2O4 for pulses shorter than sub-5 fs. In addition, detailed numerical analysis of the SHG showed that the optimum crystal angle where the FRAC trace distortion becomes minimum is in disagreement not only with the phase-matching angle but also with the angle where the FRAC signal intensity becomes maximum. Furthermore, the apparent pulse duration measured at a nonoptimum angle was confirmed to become shorter than that of its transform-limited pulse, in excellent agreement with the calculated result.

©2004 Optical Society of America

1. Introduction

A fringe-resolved autocorrelation (FRAC) method [1] with second-harmonic (SH) generation is widely used for the measurement of ultrashort optical pulses even with durations shorter than 10 fs because of its simplicity. However, this method has several problems. First, it is difficult to measure pulses with spectra of more than one octave because a bandpass filter is needed to separate the SH signal from fundamental pulses to be measured and hence distorts its SH signal. Second, the method requires the assumption of a pulse shape, for example, Gaussian, sech2. This ambiguity causes a determinate error of the measured pulse duration. The former problem is difficult to solve in principle, but the latter may be avoided by comparing a measured FRAC trace with a calculated trace by use of its spectrum. Nevertheless, the measured pulse duration tends to be evaluated to be shorter [2, 3, 4, 5] than that obtained by other measurement methods, for example, frequency-resolved optical gating (FROG) [2, 4, 6] and spectral phase interferometry for direct electric-field reconstruction (SPIDER) [7, 8, 11]. In addition, the measured pulse duration is sometimes evaluated to be shorter than that of its Fourier–transform-limited (TL) pulses. However, the origin of these kinds of problems and quantitative limitation of the FRAC method have not, to our knowledge, been closely scrutinized, except in Refs. [9] and [10]. In the former [9], the effect of group-velocity mismatch on the intensity autocorrelation method was discussed. On the other hand, in the latter [10], the influence of the asymmetric Michelson interferometer in FRAC measurement for sub-10-fs pulses was identified.

In this paper, we investigate the influence of the frequency-dependent efficiency of the SH generation of few-cycle pulses to be measured by the FRAC method as the main origin. For ultrabroadband optical pulses, we cannot consider that the SH electric field generated even in a thin nonlinear crystal is proportional to the square of the incident temporal electric field because of the limitation of phase-matching bandwidth in the SH generation. We verify theoretically and experimentally such a bandwidth-limitation effect in FRAC measurement with type I, 10-µm thick β -BaB2O4 (BBO), using two optical pulses of 5.3 and 3.6 fs. Furthermore, we demonstrate that the boundary pulse duration between the validity and the limitation of the FRAC method is in the sub-5-fs range.

2. Equations for FRAC signals

FRAC signal is obtained by measuring the intensity of the SH wave generated from an incident pulse pair with a relative delay time τ as a function of its τ by use of a “slow” detector. The ideal signal S FRAC(τ) is expressed by the following well-known form [1]:

SFRAC(τ)=2G(0)+4G(τ)+4Re[F1(τ)]+2Re[F2(τ)],

where

G(τ)=E(t)2E(tτ)2dt,
F1(τ)=[E(t)2+E(tτ)2]E*(t)E*(tτ)dt,
F2(τ)=E2(t)E*2(tτ)dt,

and the linearly polarized electric field E(t) of the positive frequency ω component in a complex representation is expressed by the real electric field ε (t) including a carrier-wave term with a time-dependent phase φ(t), as follows:

E(t)=0ε˜(ω)exp[iωt]dω,ε˜(ω)=ε(t)exp[iωt]dt.

However, the SH generation is limited by a phase-matching bandwidth in a nonlinear crystal. Especially, for few-optical-cycle pulses, the investigation of the bandwidth limitation effect is indispensable. Here, for the sake of simplicity, let us consider the frequency-dependent efficiency of the SH generation from a thin nonlinear crystal in the frequency domain.

The SH electric field generated from a nonlinear crystal is derived from a nonlinear propagation equation [6]. The intensity I SH(Ω,τ) of the SH electric field at an SH frequency Ω and a delay time τ with respect to a uniaxial negative nonlinear crystal such as BBO is given by

ISH(Ω,τ)Ωne(Ω)ω(Ωω)exp[iV(Ω,ω)L]sinc[V(Ω,ω)L]
×[ne2(Ω)1][no2(Ωω)1](no2(ω)1)
×E˜(ω)E˜(Ωω)(1+exp[iωτ]){1+exp[i(Ωω)τ]}dω2
ISH,no_SFA(Ω,τ)

(SFA in the subscript is the shortened representation of spectral filter approximation, which is mentioned in the next paragraph) with a quantity concerning the type I phase-matching condition

V(ω,Ωω)=ko(ω)+ko(Ωω)ke(Ω)2,

where n o(ω) and n e(ω) are linear refractive indices for ordinary and extraordinary beams, respectively; k o(ω) and k e(ω) are wave numbers for ordinary and extraordinary beams, respectively; L is the crystal length; and (ω) is the Fourier transform of the incident pulse electric field E(t). We have assumed that the beam diameter depends on the frequency and is proportional to 1/√ω as Eq. (9) in Ref. [6] (It is experimentally confirmed that the beam diameter has such dependence for our hollow fiber, although it is thought that the beam waist has no frequency dependence for a hollow fiber as pointed out in Ref. [6]). In addition, we approximated the second order susceptibility χ <2>(Ω,ω,Ω-ω) by the product of the linear susceptibility χ <1>(ω)=χ(n 2(ω)-1), that is, χ <2>(Ω,ω,Ω-ω)∝χ(ne2 (Ω)-1)(no2 (ω)-1)(no2 (Ω-ω)-1).

Under approximations [ω(1-ω/Ω)]1/2≃√Ω/2 and V(ω,Ω-ω)≃V(Ω/2,Ω/2) (we expanded k o(ω) and k o(Ω-ω) into the Taylor series around Ω/2 up to the first order) in Eq. (4) [6], the intensity of the SH field can be conveniently decomposed to a product of a spectral filter function R(Ω) and an ideal SH intensity I SH,ideal(Ω,τ) without taking account of the influence of the bandwidth limitation resulting from the SH wave conversion, as follows [the I SH,no_SFA(Ω,τ) of Eq. (4) is the equation without those approximations]:

ISH,no_SFA(Ω,τ)R(Ω)ISH,ideal(Ω,τ)ISH,with_SFA(Ω,τ),

where

R(Ω)=Ω34ne(Ω)[(ne2(ω)1)(no2(Ω2)1)2sinc[V(Ω2,Ω2)L]]2,
ISH,ideal(Ω,τ)=E˜(ω)E˜(Ωω)(1+exp[iωτ])(1+exp[i(Ωω)τ])dω2.

We will call the decomposition “spectral filter approximation.”

Three different signals S SH,α (τ)s are obtained by integrating I SH,α(Ω,τ)s over the whole spectral range (α=no_SFA, with SFA and ideal):

SSH,α(τ)=ISH,α(Ω,τ)dΩ(α=no_SFA,with_SFAandideal).

In particular, it is easily confirmed that the FRAC signal S SH,ideal is equivalent to Eq. (1).

Let us introduce the convenient value εα,β for the evaluation of the difference between FRAC signals S SH,α(τ)s,

RMS (root means square) error ε α,β

εα,β(sα(τ)sβ(τ))2dτ(α,β=no_SFA,with_SFAandideal),

and the value rα,β for the evaluation of the difference between full widths at half-maximum (FWHMs) of their envelopes,

ratio of FWHMs of the envelope r α,β

rα,βΔtαΔtβ(α,β=no_SFA,with_SFAandideal),

where

sα(τ)8SSH,α(τ)SSH,α(0)(α=no_SFA,with_SFAandideal)

and Δtα is FWHM of the envelope of sα (τ). For Gaussian transform-limited (TL) pulses, we have the well-known value Δt ideal/tp,inTL=1.7, where tp,inTL is its pulse duration. The envelope function is calculated by absolute values of oscillating terms in sα (τ) (corresponding to F 1(τ) and F 2(τ) in Eq. [1)].

For example, in the case of α=ideal, the envelope function S env(τ) is exactly equivalent to the following equation in the temporal domain [1],

Senv(τ)=2G(0)+4G(τ)+4F1(τ)+2F2(τ).

3. Numerical analysis: deviation of practical FRAC signal from ideal FRAC signal

Figure 1 shows an example of the difference among three FRAC signals of s ideal(τ), s no_SFA(τ), and s with SFA(τ), where tp,inTL=4.01 fs, spectral FWHM Δν in=110 THz, the center wavelength λc=600 nm, the angle between the incident pulse-train beam and the optic axis of 10-µm-thick BBO θ=25° (hereafter called the θ crystal angle). The dispersion and related values of n o and n e of type I BBO are taken from Ref. [12]. This case indicates that the apparent pulse duration obtained from the FRAC signal s no_SFA(τ) that considers the bandwidth-limited SH-generation effect without spectral filter approximation is shorter than the correct pulse duration obtained from the FRAC signal s ideal(τ), which is not affected by its effect at all.

We investigated in detail estimation values εα,β and rα,β as functions of temporalFWHM tp,inTL of Gaussian TL pulses to be measured (the corresponding spectral FWHM Δν in) and the crystal angle θ for different pulse center wavelengths λcs (Figs. 2 and 3). In the following numerical calculations, we assumed that the nonlinear crystal for the SH generation is type I 10-µm-thick BBO and limited the Gaussian spectrum to 400–2600 nm in consideration of the fact that the transparency range of BBO is approximately 200–2600 nm (we permitted the Gaussian pulses to have over-one-octave bandwidth). As a result, the following important findings were obtained (Figs. 25):

1. The relations ε no_SFA,with_SFAε no_SFA,ideal and |r no_SFA,with_SFA-1|≪|r no_SFA,ideal-1| are satisfied over the whole tp,inTL-θ region (441 fs≥t p,in ≥2.3 fs; 10°≤θ≤ 45°, λc=600, 700 and, 800 nm; see Fig. 2), and any critical difference between s no_SFA(τ) and s with_SFA(τ) is not found (for example, see Fig. 1). Therefore, it is valid that we interpret results by utilizing the filter function R(Ω).

2. Note the angle θ opt where the value of ε no_SFA,ideal is minimum (=εno_SFA,idealmin). For example, εno_SFA,idealmin/ε max=0.155 at θ opt=39.2°, tp,inTL=5.0 fs (FRAC ideal’s (=s ideal(τ)’s) FWHM Δt ideal=8.51 fs and Δν in=88 THz) and λc=600 nm, where ε max=7.083×10-15 indicated the greatest value of all the calculated εα,β s. This corresponds to the result that Δt no_SFA=8.99 fs is longer than Δt ideal=8.51 fs. For λc=600, 700, and 800 nm, ε no_SFA,ideals become the least (=εno_SFA,idealmin) around θ opt=39°, 31°, and 24° in the range of tp,inTL of 2.3-441 fs, respectively (see dotted lines in Fig. 2). Each angle nearly corresponds to the angle θ R where the wavelength at the maximum value of R(Ω) equals half of λc (θ R=38.5°, 30°, and 22.9° for λc=600, 700, and 800 nm, respectively (see Table. 1). It should be noted that these optimal angles θ opts do not agree with phase-matched angles θ pm of the crystal (θ pm=40.2°, 33.4°, and 29° for 600, 700, and 800 nm, respectively). In addition, their θ opts do not agree with the angles θ maxs where the strength of the FRAC signal becomes maximum (see Table. 1).

3. The longer λc is, the smaller ε no_SFA,ideal is. This is because θ R has the smaller value for pulses with longer λc, and the bandwidth of R(Ω) with the smaller θ R is the broader (see Fig. 4). That is, pulses with the longer center wavelength are less affected by the filter effect of the nonlinear crystal.

4. The duration ratio r no_SFA,ideal at the angle θ opt where ε no_SFA,ideal has the minimum value with respect to tp,inTL is shown in Fig. 5. The ratio r no_SFA,ideal is always larger than 1 and increases with the decrease of tp,inTL. That is, the apparent pulse duration at θθ opt should be evaluated to be slightly longer than the correct one.

5. Except for the θθ opt region, there is the region where r no_SFA,ideal is less than 1; that is, the apparent pulse duration should be shorter than the correct one [for example, see the rα,β <1 region in Figs. 3(A) and 3(B)].

 figure: Fig. 1.

Fig. 1. Calculated FRAC traces for Gaussian TL pulses (tp,inTL=4.01 fs, Δν in, 110 THz; center wavelength, 600 nm; θ=25°; ε no_SFA,ideal/ε max=0.078; ε no_SFA,with SFA/ε max=0.0053; r no_SFA,ideal=0.88) (a) with taking account of the bandwidth-limited SH-generation effect without filter approximation [s no_SFA(τ)], (b) after filter approximation [s with SFA(τ)] and (c) without the bandwidth limitation effect [s ideal(τ)].

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 figure: Fig. 2.

Fig. 2. Contour plots of εα,β max (ε max=7.083×10-15) as functions of the input pulse duration tp,inTL and the crystal angle θ. ε no_SFA,ideal/ε max at (A) λc=600, (B) 700, and (C) 800 nm. (D) ε with SFA,ideal/ε max and (E) ε no_SFA,with SFA/ε max at λc=600 nm.

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Tables Icon

Table 1. θpm, θR, θopt, θmax for TL Gaussian pulses with λc=600, 700, and 800 nm (see the text for notation).

 figure: Fig. 3.

Fig. 3. Contour plots of rα,β as functions of the input pulse duration tp,inTL and the crystal angle θ. r no_SFA,ideal at (A) λc=600, (B) 700, and (C) 800 nm. (D) r with_SFA,ideal and (E) r no_SFA,with SFA at λc=600 nm.

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 figure: Fig. 4.

Fig. 4. Plots of the SH frequency-dependent filter function R(Ω) of (A) normalized ones and (B) nonnormalized ones at various values of crystal angle θ.

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4. Experiments

We carried out few-cycle-pulse experiments for verification of the above-mentioned filter effect. Our experimental setup was almost the same as in Ref. [8]. Optical pulses from a Ti:sapphire laser amplifier system (repetition rate, 1 kHz; center wavelength, 790 nm; pulse duration, <30 fs; pulse energy, ~200µJ/pulse) were broadened by self-phase modulation (SPM) in an Ar-filled hollow fiber and then guided into a feedback chirp compensation system [8] that consists of a modified-SPIDER (M-SPIDER) apparatus with a high sensitivity [7] and a 4-f phase compensator with a liquid-crystal spatial light modulator [2]. The bandwidth of the spectrum was limited to 530–980 nm by use of a slit in the 4-f phase compensator, to avoid the background-light noise resulting from fundamental pulses in FRAC measurements. We used a reflective dielectric filter with a nearly flat reflectance over approximately 250–500 nm (cutoff 500 nm) to separate the SH signal from fundamental pulses. It had been confirmed that a spectral sensitivity of a photodiode for detection of the SH signal does not give the practical change to FRAC traces by numerical calculation in our present case. The chirp of the fiber output pulses was almost completely compensated for by this feedback system, and then compensated pulses were independently measured by the FRAC apparatus with 10-µm-thick BBO (type I, θ=29°). To compare FRAC and M-SPIDER measurements precisely, we aligned those apparatuses to equalize optical path lengths to nonlinear crystals and used beams splitters with the same properties (thickness, 0.5 mm; substrate, fused-silica; 20% reflection and 80% transmission for s polarization from 400 to 1300 nm) for symmetric Michelson interferometers [10] in both apparatuses.

 figure: Fig. 5.

Fig. 5. Plots of r no_SFA,ideal at ε=εno_SFA,idealmin for the input pulse duration tp,inTLν in). Each plot for (a) λc, 600 nm; (b) λc, 700 nm; and (c) λc, 800 nm.

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To clarify the practical limitation of the FRAC method, experiments for two cases (cases M-1 and M-2) were performed.

• Case M-1: 5.3-fs (t p,in), 2.2-cycle pulses with a center wavelength of 725.1 nm (λc) and a TL pulse duration of 4.42 fs (tp,inTL) [Figs. 7(A) and 7(B)]. An Ar pressure of p=1.0 atm.

• Case M-2: 3.6-fs (t p,in), 1.7-cycle pulses with a wavelength of 617.5 nm (λc) and a TL pulse duration of 3.5 fs (tp,inTL) [Figs. 8(A) and 8(B)]. An Ar pressure of p=3.0 atm.

The center wavelength λc was calculated by an equation

2πcc=∫-∞ω|(ω)|2dω/∫-∞|(ω)|2dω.

4.1. Comparison between TL-pulse FRAC signals based on measured and corresponding Gaussian spectra

First of all, we calculated and compared FRAC traces of TL pulses using two measured spectral cases M-1 and M-2 and those using corresponding Gaussian spectral cases G-1 and G-2. We discussed them using the same pulse duration tp,inTL and center wavelength λc.

 figure: Fig. 6.

Fig. 6. (A) Measured spectra at the pressure p=1.0 and 3.0 atm. (B) Dependency of ε no_SFA,ideal on the crystal angle θ for (a) TL pulse at p=1.0 atm, (b) retrieved one by M-SPIDER measurement at p=1.0 atm, (c) TL pulse at p=3.0 atm, and (d) retrieved one by M-SPIDER measurement at p=3.0 atm. (C) Dependency of r no_SFA,ideal on the crystal angle θ for (e) TL pulse at p=1.0 atm, (f) retrieved one by M-SPIDER measurement at p=1.0 atm, (g) TL pulse at p=3.0 atm, and (h) retrieved one by M-SPIDER measurement at p=3.0 atm.

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It was confirmed that the θ opt between cases M-i and G-i (i=1 and 2) has almost the same value as well as θ max (see Table 2) and θ optθ R, but εno_SFA,idealmin and r no_SFA,ideal have different values in the different cases. However, for ~10-fs TL pulses θ opt, εno_SFA,idealmin, and r no_SFA,ideal for measured and Gaussian spectral cases were not different. In addition, the filter effect of the nonlinear crystal hardly caused evaluation errors for ~10-fs pulse durations.

Figures 6(B) and 6(C) show dependencies of the normalized ε no_SFA,ideal and the r no_SFA,ideal on the crystal angle θ for cases M-1 and M-2. Those figures indicate that the θ opts for cases M-1 and M-2 are 29.7° and 36.8°, where both measured apparent pulse durations become slightly longer than correct pulse durations. Around θ=34° [around S in Fig. 6(C)], r no_SFA,ideals are ~1, although ε no_SFA,ideals do not have the minimum values. However, we should not measure the FRAC trace around such region of θ~34°. This is because the FRAC signal is often distorted so remarkably, except at θ~θ opt, that the spacing of the fringe differs, especially in its both wings, from that at θ~θ opt, even though the apparent FRAC-envelope ratio r no_SFA,ideal becomes ~1.

Moreover, we find that when pulse durations are measured around θ≃40° for case M-1 and around θ≃20° for case M-2, both apparent durations become shorter than correct pulse durations, as previously reported [2, 3, 4, 5].

4.2. Experimental comparison between directly measured and M-SPIDER-retrieved FRAC signals

We compared measured FRAC traces with traces retrieved from M-SPIDER measurements [7, 8] for cases M-1 and M-2. Experimental results are shown in Fig. 7(C) for M-1 and Fig. 8(C) for M-2. Curves (a), (b), and (c) of (A) in those Figs. 7 and 8 show the intensity spectrum (∝|(ω)|2), the spectral phase before chirp compensation, and the spectral phase [ϕ(ω)] after compensation, respectively. Curves (d), (e), and (f) of (B) show the temporal intensity [I(t)], the temporal phase [φ(t)], and the TL pulse profile, respectively. Curves (i) of (C) show FRAC traces measured at the angle θ=29° (close to the θ opt for case M-1; see Table. 2) of 10-µm thick BBO, and curves (ii) and (iii) of (C) show FRAC traces retrieved from s no_SFA(τ) and s ideal(τ) in Eq. (12) using M-SPIDER results of ϕ(ω) and |(ω)|, respectively.

In case M-1 (t p,in=5.3 fs, 2.2 cycles, tp,inTL=4.42 fs, and λc=725.1 nm), we could not find explicit differences among curves (i)–(iii), as expected from calculated results of Figs. 6(B) and 6(C) [curves (b) and (f)]. On the other hand, in case M-2 (t p,in=3.6 fs, 1.7 cycles, tp,inTL=3.50 fs, and λc=617.5 nm), the measured FRAC trace [curve (i)] agrees with the retrieved trace [curve (ii)] with taking account of the bandwidth-limited SH-generation effect without spectral filter approximation, but there is the unignorable difference between curves (i) and (iii) (the retrieved trace without the spectral filter effect), as expected from calculated results of Figs. 6(B) and 6(C) [curves (d) and (h)]. Therefore, it was verified that the FRAC trace of pulses shorter than sub-5 fs was seriously distorted by the spectral filter effect because of the nonlinear crystal. In addition, in comparing curves (ii) and (iii) in Fig. 8(C), we found that Δt no_SFA is 11% smaller than Δt ideal. That is to say, the apparent pulse duration may be evaluated to be 3.16 fs. This implies that the apparent pulse duration evaluated by the FRAC measurement is shorter than that of the TL pulse. This result agrees with not only the calculated result of Fig. 6 but also previously reported results [2, 3, 4, 5].

When the deviation of r no_SFA,ideal from 1 is less than ~5% at θ~θ opt, we could not recognize any practical difference in the FRAC traces between s no_SFA(τ) and s ideal(τ) for all the FRAC traces calculated in this paper. Therefore, we can assume that such FRAC traces are not nearly distorted, and it is not necessary to take account of the bandwidth-limitation effect. In fact, in accumulative measurements of one FRAC trace of usual, repetitive pulses, a~5% margin of the error is assumed to be reasonable when one considers the actual FRAC trace noise, the fluctuation of the pulse intensity, and the delay time error.

5. Conclusion

The practical limitation of the FRAC method with 10-µm-thick BBO for characterization of few-optical-cycle pulses with center wavelengths of 600–800 nm was demonstrated experimentally and theoretically. For TL pulses longer than sub-5 fs, the FRAC measurement under the suitable crystal angle is consistently available. On the other hand, for TL pulses shorter than sub-5 fs, the FRAC measurement even under the suitable crystal angle does not provide the correct result, because of the spectral filter R(Ω) effect of the nonlinear crystal. However, numerical consideration and correction of this R(Ω) effect greatly reduce the evaluation errors of FRAC measurements and improve reliability of their measurement results.

Tables Icon

Table 2. θpm, θR, θopt, and θmax for Gaussian spectrum (G-i) and measured spectrum (Mi) (i=1, tp,inTL=4.42 fs λc=725.1 nm; i=2, tp,inTL=3.5 fs λc=617.5 nm (see text for notationa).

 figure: Fig. 7.

Fig. 7. Experimental results (Ar-gas pressure, 1.0 atm; fiber input, 150 µJ/pulse; fiber output, 16.5µJ/pulse). (A) (a) Intensity spectrum and spectral phases (b) before and (c) after feedback chirp compensation. (B) (d) Temporal intensity (pulse duration: 5.3 fs, center wavelength: 725.1 nm) and (e) temporal phase after chirp compensation. (f) Temporal intensity of the Fourier transform-limited pulse (pulse duration: 4.42 fs). (C) (i) Measured FRAC trace and retrieved FRAC traces (ii) with and (iii) without taking account of the filter effect by the nonlinear crystal.

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 figure: Fig. 8.

Fig. 8. Experimental results (Ar-gas pressure: 3.0 atm, fiber input: 140 µJ/pulse, fiber output: 18µJ/pulse). (A) (a) Intensity spectrum and spectral phases (b) before and (c) after feedback chirp compensation. (B) (d) Temporal intensity (3.6 fs, center wavelength: 617.5 nm) and (e) temporal phase after chirp compensation. (f) Temporal intensity of the Fourier transform-limited pulse (3.50 fs). (C) (i) Measured FRAC trace and retrieved FRAC traces (ii) with and (iii) without taking account of the filter effect by the nonlinear crystal.

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Acknowledgments

We are grateful to Akira Sugro (Citizen Co. Ltd.) for supplying a specially designed liquidcrystal SLM.

References and links

1. J.-C. M. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplietude and phase) with femtosecond accuracy,” Appl. Opt. 24, 1270 (1985). [CrossRef]   [PubMed]  

2. N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B 18, 1742 (2001). [CrossRef]  

3. A. Cheng, G. Tempea, T. Brabec, K. Ferencz, and F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI (Springer-Verlag, Berlin, 1998), p. 8. [CrossRef]  

4. Z. Cheng, A. Fürbach, S. Sartania, M. Lenzner, Ch. Spielmann, and F. Krausz, “Amplitude and chirp characterization of high-power laser pulses in the 5-fs regime,” Opt. Lett. 24, 247 (1999). [CrossRef]  

5. A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999). [CrossRef]  

6. A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459 (1999). [CrossRef]  

7. M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002). [CrossRef]  

8. K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro, “Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,” Opt. Lett. 28, 2258 (2003). [CrossRef]   [PubMed]  

9. A. M. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276 (1983). [CrossRef]  

10. Ch. Spielmann, L. Xu, and F. Krausz, “Measurement of interferometric autocorrelations: comment,” Appl. Opt. 36, 2523 (1997). [CrossRef]   [PubMed]  

11. R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002). [CrossRef]  

12. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

References

  • View by:

  1. J.-C. M. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplietude and phase) with femtosecond accuracy,” Appl. Opt. 24, 1270 (1985).
    [Crossref] [PubMed]
  2. N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B 18, 1742 (2001).
    [Crossref]
  3. A. Cheng, G. Tempea, T. Brabec, K. Ferencz, and F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI (Springer-Verlag, Berlin, 1998), p. 8.
    [Crossref]
  4. Z. Cheng, A. Fürbach, S. Sartania, M. Lenzner, Ch. Spielmann, and F. Krausz, “Amplitude and chirp characterization of high-power laser pulses in the 5-fs regime,” Opt. Lett. 24, 247 (1999).
    [Crossref]
  5. A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999).
    [Crossref]
  6. A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459 (1999).
    [Crossref]
  7. M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
    [Crossref]
  8. K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro, “Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,” Opt. Lett. 28, 2258 (2003).
    [Crossref] [PubMed]
  9. A. M. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276 (1983).
    [Crossref]
  10. Ch. Spielmann, L. Xu, and F. Krausz, “Measurement of interferometric autocorrelations: comment,” Appl. Opt. 36, 2523 (1997).
    [Crossref] [PubMed]
  11. R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
    [Crossref]
  12. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

2003 (1)

2002 (2)

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

2001 (1)

1999 (3)

Z. Cheng, A. Fürbach, S. Sartania, M. Lenzner, Ch. Spielmann, and F. Krausz, “Amplitude and chirp characterization of high-power laser pulses in the 5-fs regime,” Opt. Lett. 24, 247 (1999).
[Crossref]

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999).
[Crossref]

A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459 (1999).
[Crossref]

1997 (1)

1985 (1)

1983 (1)

A. M. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276 (1983).
[Crossref]

Baltuška, A.

A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459 (1999).
[Crossref]

Brabec, T.

A. Cheng, G. Tempea, T. Brabec, K. Ferencz, and F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI (Springer-Verlag, Berlin, 1998), p. 8.
[Crossref]

Cheng, A.

A. Cheng, G. Tempea, T. Brabec, K. Ferencz, and F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI (Springer-Verlag, Berlin, 1998), p. 8.
[Crossref]

Cheng, Z.

Diels, J.-C. M.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Ferencz, K.

A. Cheng, G. Tempea, T. Brabec, K. Ferencz, and F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI (Springer-Verlag, Berlin, 1998), p. 8.
[Crossref]

Fontaine, J. J.

Fürbach, A.

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Hirasawa, M.

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

Karasawa, N.

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B 18, 1742 (2001).
[Crossref]

Kobayashi, T.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999).
[Crossref]

Krausz, F.

Kusaka, S.

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

Lenzner, M.

Li, L.

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B 18, 1742 (2001).
[Crossref]

McMichael, I. C.

Morita, R.

K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro, “Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,” Opt. Lett. 28, 2258 (2003).
[Crossref] [PubMed]

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B 18, 1742 (2001).
[Crossref]

Nakagawa, N.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Oka, K.

Pshenichnikov, M. S.

A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459 (1999).
[Crossref]

Sakane, I.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999).
[Crossref]

Sartania, S.

Shigekawa, H.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B 18, 1742 (2001).
[Crossref]

Shirakawa, A.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999).
[Crossref]

Simoni, F.

Spielmann, Ch.

Suguro, A.

Takasaka, M.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999).
[Crossref]

Tempea, G.

A. Cheng, G. Tempea, T. Brabec, K. Ferencz, and F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI (Springer-Verlag, Berlin, 1998), p. 8.
[Crossref]

Weiner, A. M.

A. M. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276 (1983).
[Crossref]

Wiersma, D. A.

A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459 (1999).
[Crossref]

Xu, L.

Yamamoto, K.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

Yamane, K.

K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro, “Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,” Opt. Lett. 28, 2258 (2003).
[Crossref] [PubMed]

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

Yamashita, M.

K. Yamane, Z. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro, “Optical pulse compression to 3.4 fs in the monocycle region by feedback phase compensation,” Opt. Lett. 28, 2258 (2003).
[Crossref] [PubMed]

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B 18, 1742 (2001).
[Crossref]

Zhang, Z.

Appl. Opt. (2)

Appl. Phys. B (1)

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B 74, S225–229 (2002).
[Crossref]

Appl. Phys. Lett. (1)

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched non-collinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268 (1999).
[Crossref]

IEEE J. Quantum Electron. (2)

A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459 (1999).
[Crossref]

A. M. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276 (1983).
[Crossref]

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

Meas. Sci. Technol. (1)

R. Morita, M. Hirasawa, N. Karasawa, S. Kusaka, N. Nakagawa, K. Yamane, L. Li, A. Suguro, and M. Yamashita, “Sub-5 fs optical pulse characterization,” Meas. Sci. Technol. 13, 1710–1720 (2002).
[Crossref]

Opt. Lett. (2)

Other (2)

A. Cheng, G. Tempea, T. Brabec, K. Ferencz, and F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI (Springer-Verlag, Berlin, 1998), p. 8.
[Crossref]

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

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Figures (8)

Fig. 1.
Fig. 1. Calculated FRAC traces for Gaussian TL pulses (tp,inTL=4.01 fs, Δν in, 110 THz; center wavelength, 600 nm; θ=25°; ε no_SFA,ideal/ε max=0.078; ε no_SFA,with SFA/ε max=0.0053; r no_SFA,ideal=0.88) (a) with taking account of the bandwidth-limited SH-generation effect without filter approximation [s no_SFA(τ)], (b) after filter approximation [s with SFA(τ)] and (c) without the bandwidth limitation effect [s ideal(τ)].
Fig. 2.
Fig. 2. Contour plots of εα,β max (ε max=7.083×10-15) as functions of the input pulse duration tp,inTL and the crystal angle θ. ε no_SFA,ideal/ε max at (A) λc=600, (B) 700, and (C) 800 nm. (D) ε with SFA,ideal/ε max and (E) ε no_SFA,with SFA/ε max at λc=600 nm.
Fig. 3.
Fig. 3. Contour plots of rα,β as functions of the input pulse duration tp,inTL and the crystal angle θ. r no_SFA,ideal at (A) λc=600, (B) 700, and (C) 800 nm. (D) r with_SFA,ideal and (E) r no_SFA,with SFA at λc=600 nm.
Fig. 4.
Fig. 4. Plots of the SH frequency-dependent filter function R(Ω) of (A) normalized ones and (B) nonnormalized ones at various values of crystal angle θ.
Fig. 5.
Fig. 5. Plots of r no_SFA,ideal at ε=εno_SFA,idealmin for the input pulse duration tp,inTLν in). Each plot for (a) λc, 600 nm; (b) λc, 700 nm; and (c) λc, 800 nm.
Fig. 6.
Fig. 6. (A) Measured spectra at the pressure p=1.0 and 3.0 atm. (B) Dependency of ε no_SFA,ideal on the crystal angle θ for (a) TL pulse at p=1.0 atm, (b) retrieved one by M-SPIDER measurement at p=1.0 atm, (c) TL pulse at p=3.0 atm, and (d) retrieved one by M-SPIDER measurement at p=3.0 atm. (C) Dependency of r no_SFA,ideal on the crystal angle θ for (e) TL pulse at p=1.0 atm, (f) retrieved one by M-SPIDER measurement at p=1.0 atm, (g) TL pulse at p=3.0 atm, and (h) retrieved one by M-SPIDER measurement at p=3.0 atm.
Fig. 7.
Fig. 7. Experimental results (Ar-gas pressure, 1.0 atm; fiber input, 150 µJ/pulse; fiber output, 16.5µJ/pulse). (A) (a) Intensity spectrum and spectral phases (b) before and (c) after feedback chirp compensation. (B) (d) Temporal intensity (pulse duration: 5.3 fs, center wavelength: 725.1 nm) and (e) temporal phase after chirp compensation. (f) Temporal intensity of the Fourier transform-limited pulse (pulse duration: 4.42 fs). (C) (i) Measured FRAC trace and retrieved FRAC traces (ii) with and (iii) without taking account of the filter effect by the nonlinear crystal.
Fig. 8.
Fig. 8. Experimental results (Ar-gas pressure: 3.0 atm, fiber input: 140 µJ/pulse, fiber output: 18µJ/pulse). (A) (a) Intensity spectrum and spectral phases (b) before and (c) after feedback chirp compensation. (B) (d) Temporal intensity (3.6 fs, center wavelength: 617.5 nm) and (e) temporal phase after chirp compensation. (f) Temporal intensity of the Fourier transform-limited pulse (3.50 fs). (C) (i) Measured FRAC trace and retrieved FRAC traces (ii) with and (iii) without taking account of the filter effect by the nonlinear crystal.

Tables (2)

Tables Icon

Table 1. θ pm, θ R, θ opt, θ max for TL Gaussian pulses with λc=600, 700, and 800 nm (see the text for notation).

Tables Icon

Table 2. θ pm, θ R, θ opt, and θ max for Gaussian spectrum (G-i) and measured spectrum (Mi) (i=1, tp,inTL=4.42 fs λc=725.1 nm; i=2, tp,inTL=3.5 fs λc=617.5 nm (see text for notation a ).

Equations (18)

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S FRAC ( τ ) = 2 G ( 0 ) + 4 G ( τ ) + 4 Re [ F 1 ( τ ) ] + 2 Re [ F 2 ( τ ) ] ,
G ( τ ) = E ( t ) 2 E ( t τ ) 2 d t ,
F 1 ( τ ) = [ E ( t ) 2 + E ( t τ ) 2 ] E * ( t ) E * ( t τ ) d t ,
F 2 ( τ ) = E 2 ( t ) E * 2 ( t τ ) d t ,
E ( t ) = 0 ε ˜ ( ω ) exp [ i ω t ] d ω , ε ˜ ( ω ) = ε ( t ) exp [ i ω t ] d t .
I SH ( Ω , τ ) Ω n e ( Ω ) ω ( Ω ω ) exp [ i V ( Ω , ω ) L ] sinc [ V ( Ω , ω ) L ]
× [ n e 2 ( Ω ) 1 ] [ n o 2 ( Ω ω ) 1 ] ( n o 2 ( ω ) 1 )
× E ˜ ( ω ) E ˜ ( Ω ω ) ( 1 + exp [ i ω τ ] ) { 1 + exp [ i ( Ω ω ) τ ] } d ω 2
I SH , no _ SFA ( Ω , τ )
V ( ω , Ω ω ) = k o ( ω ) + k o ( Ω ω ) k e ( Ω ) 2 ,
I SH , no _ SFA ( Ω , τ ) R ( Ω ) I SH , ideal ( Ω , τ ) I SH , with _ SFA ( Ω , τ ) ,
R ( Ω ) = Ω 3 4 n e ( Ω ) [ ( n e 2 ( ω ) 1 ) ( n o 2 ( Ω 2 ) 1 ) 2 sinc [ V ( Ω 2 , Ω 2 ) L ] ] 2 ,
I SH , ideal ( Ω , τ ) = E ˜ ( ω ) E ˜ ( Ω ω ) ( 1 + exp [ i ω τ ] ) ( 1 + exp [ i ( Ω ω ) τ ] ) d ω 2 .
S SH , α ( τ ) = I SH , α ( Ω , τ ) d Ω ( α = no _ SFA , with _ SFA and ideal ) .
ε α , β ( s α ( τ ) s β ( τ ) ) 2 d τ ( α , β = no _ SFA , with _ SFA and ideal ) ,
r α , β Δ t α Δ t β ( α , β = no _ SFA , with _ SFA and ideal ) ,
s α ( τ ) 8 S SH , α ( τ ) S SH , α ( 0 ) ( α= no _ SFA , with _ SFA and ideal )
S env ( τ ) = 2 G ( 0 ) + 4 G ( τ ) + 4 F 1 ( τ ) + 2 F 2 ( τ ) .

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