We demonstrate theoretically how the time-dependent phase modulation induced by molecular alignment can be used to tune the frequency of few-cycle femtosecond laser pulses continuously. Using impulsively excited alignment in N2, the central wavelength of an initial 800 nm, 5 fs Gaussian pulse can be tuned from 324.6 to 4237.3 nm. The aligned N2 molecules are obtained by pretransmitting another 800 nm, 100 fs linearly polarized laser pulse of intensity 3.5×1013 Wcm-2. The number of optical cycles contained in the frequency-tuned pulse is almost unchanged after ideal chirp compensation.
©2009 Optical Society of America
Nowadays with the rapid development of laser physics the generation of few-cycle laser pulses brings great opportunities to science and technology. The pulses have been applied in various fields, for example for measurements with high time resolution of electron dynamics in atoms and molecules , wake-field particle acceleration , and the generation of single attosecond pulses . Moreover, numerous applications additionally demand laser sources with frequency tunability over a broad spectral range, and therefore methods to generate these laser pulses are desired. In most such studies, optical parametric amplification (OPA) is a well established technique used to produce frequency-tunable laser pulses with typical pulse widths in the 50-200 fs range . It is difficult to generate frequency-tunable few-cycle laser pulses because it is very hard to satisfy the phase matching requirement over a broad spectrum. Recently 12 fs laser pulses tunable in the visible spectrum and 13 fs mid-infrared pulses have been obtained respectively by four-wave mixing during filamentation in gases [5, 6]. However, few-cycle pulses tunable across a broad spectral range are still lacking.
Over the last few years, several approaches for the modulation of light based on molecular Raman effects have been demonstrated. Impulsively and adiabatically excited Raman active mediums were used for pulse compression, e.g. for the generation of ultrashort fs-pulse trains or single ultrashort pulses [7-11]. In addition to pulse compression, the modulated instantaneous susceptibility of the medium can also induce frequency conversion. Using solid hydrogen adiabatically excited by two narrow-linewidth laser fields, Kien et al. theoretically described the frequency shifting of probe pulses that were injected into the medium with certain delays . In experimental studies, Korn et al. mentioned a slight ~ 25 nm spectral shift of the time-delayed probe pulse using impulsively excited vibration in SF6 . In this letter, we demonstrate an approach using impulsively excited alignment in N2 filling in a hollow fiber to tune the frequency of pulses. This technique is more realistic for experimental implementation and is suitable for few-cycle pulses. Continuous frequency tunability from the UV to the mid-IR is obtained. Kien et al. found the number of oscillation conservation for the limit of negligible dispersion. In our calculation the dispersion can not be neglected but the number of optical cycles contained in the frequency-tuned pulse remains unchanged after ideal phase compensation.
Similar to the pump-probe approach, an intense, short, linearly polarized pulse pretransmits through a hollow fiber and aligns the gas molecules inside it . The molecular alignment induces a temporal modulation of the refractive index, n(t), that undergoes periodic full revivals at a period of Tr= h/2B , where h is Plank’s constant and B is the rotational constant of the molecules. In the case of N2, full revivals occur at Tr = 8.38 ps. When the time-delayed, identically polarized signal pulse is sent into the aligned medium, its spectrum is modulated by the modulation of n(t). As the signal pulse is much shorter than Tr it experiences just a part of the revival. Different time delays make the signal pulse face different modulations. Previous works have been reported on pulse compression by selecting an appropriate region of the revival that broadens extremely the spectrum of the pulse [13-15]. However, we adjust the time delays to correspond to the regions where the change of the refractive index can be assumed to be approximately linear, i.e. Δn=αt, in which α is the linear coefficient. The linear modulation can induce the same frequency shift across the entire pulse, which can be estimated using the formula Δω=-dΔϕ/dt. Hence, the carrier frequency will shift from the initial ω0 to ω0 =ω0+Δω. Then through propagation continuous frequency shift with broad tunability extending from the UV to the mid-IR is obtained. Different from the OPAs, the intense pulse does not participate in the course of frequency shift directly, and it can be optimized to obtain a perfect medium response decoupled from the details of the delayed signal pulse that is frequency shifted.
The alignment-induced modulation of the refractive index is given by n(t)=1+2πN[α⊥+Δα<cos2θ(t)>], where N is the molecular number density, Δα(ω)=α∥(ω)-α⊥(ω), α∥(ω) and α⊥(ω) are the components of polarizability, parallel and perpendicular to the molecular axis, and <cos2θ(t)> is the quantum and thermally averaged value that is usually used to characterize the extent of alignment. It is calculated by solving the Schrödinger equation 
where Ĵ is the angular momentum operator, I is the moment of inertia, and |Epre(t)| is the envelope of the intense pulse that aligns the molecules and is polarized along the x axis.
Eq. (1) describes the evolution of molecular states in a certain intense laser field. Since the intense pulse will change as it propagates, the solution for the propagation of the intense pulse is required in order to know precisely the modulation of the refractive index. Neglecting back reflection and for a refractive index n(t) close to unity, the intense pulse propagating along the z axis in the hollow fiber can be described by the so-called reduced Maxwell equation 
where c is the velocity of light in vacuum. For the intense pulse, the polarization Ppre of the medium is given by Ppre=Pθ+PK, where PK=χ(3)(N) |Epre|2Epre is due to the Kerr effect, and χ(3)(N) is the third-order susceptibility of the medium. The x component of the polarization Pθ in the molecules aligned at an angle θ to the x axis is Pθ(z, ω)=4πN[α⊥(ω)+Δα(ω)<cos2θ(z, t)>]Epre(z, ω).
The polarizabilities are normalized to their static values, α⊥,∥(ω)=α0⊥,∥f⊥,∥(ω), where f⊥,∥(ω) are given by the Sellmeier-type expressions, f(ω)= ∑ iSi/(1-ω2/ωi2). We set f⊥(ω)=f∥(ω)=f(ω) and the function f(ω) is taken from the frequency dependent linear refractive index n(ω) .
Eq. (2) is solved numerically by standard finite-difference method in the moving reference frame τ=t-z/vs with the fixed speed vs near that at which the intense pulse propagates. The molecular medium, N2 (α0∥=1.45×10-30m3, α0⊥=2.38×10-30m3, χ(3)=1.16×10-25m3/V2), is at temperature of 90 K and has a density of N=1.0748×1026 m-3. The initial 100 fs, 800 nm, intense pulse at intensity of 3.5×1013 Wcm-2 is described as E(z=0, τ)=E0exp(-τ2/τ02-iω0τ).
3. Propagation results of the intense pulse
As the intense pulse propagates through the gas medium, it experiences a higher refractive index at the pulse tail caused by the molecular alignment induced by itself. The pulse also undergoes the self-steepening process which is due to the third order nonlinearity. All the effects including dispersion lead to a modification of the intense pulse as shown in Fig. 1(a). Therefore the modulation of the refractive index also changes across the medium. The resulting modulation of the refractive index both at the input and at the 54 cm output of the fiber is shown in Fig. 1(b). Even the modulation decreases along the medium, it is still adequate to induce large frequency shifts.
4. Propagation results of the signal pulse
4.1 Frequency blue-shift
The propagation equation for the initial 5 fs, 800 nm signal pulse is also described by Eq. (2) with polarization Ps=Pθs only resulted from the induced alignment Pθs(z, ω)=4πN[α⊥(ω)+Δα(ω) <cos2θ(z, τ-τd)>]Es(z, ω), where τd is the time delay between the two pulses at the input of the hollow fiber. The equation is also solved in the moving reference frame τ=τ-z/vs with the same fixed speed vs used for the intense pulse. Using this moving frame we can calculate the influence of the group velocity walk-off between the two pulses. In our calculation, the intensity of the signal pulse is restricted to avoid any nonlinear effects and modulations to the induced alignment of the molecules.
Frequency blue-shift is obtained when τd corresponds to ~ 3780 fs in Fig. 1(b). The pulse spectrum continuously shifts to the blue as it propagates, as shown in Fig. 2(a). With this time delay the signal pulse undergoes an approximately linear decreasing refractive index, and it leads to same positive frequency shift over the entire pulse. The carrier frequency at the propagation length of 21 cm is about 2.4642 times of the initial ω0. Hence, the central wavelength shifts from an initial 800 nm to 324.6 nm. Fig. 2(b) shows the calculated 26.52 fs pulse and its 2.29 fs Fourier-transform-limited pulse at 21cm output. Assuming ideal linear modulation and no dispersion, chirp will not occur and thus the number of optical cycles contained in the signal pulse will not change. The ratios of the number of optical cycles of the Fourier-transform-limited pulse to that of the initial pulse at different propagation lengths are shown in the inset of Fig. 2(b). Note that as long as the signal pulse is confined to the linear region, the number of optical cycles contained in the signal pulse remains nearly constant after ideal chirp compensation.
As discussed above, the modulation of the refractive index n(t) used to tune the signal pulse decreases across the medium. Hence, at long propagation lengths the tuning efficiency will be low. However, before that can happen, the signal pulse will already shift in time into the nonlinear region because of group velocity walk-off. The signal pulse is finely confined to the approximately linear decreasing region before propagating to 21 cm. In the subsequent propagation the carrier frequency can continue to shift a little but the nonlinear effects become important, distorting the spectrum, as shown by the 27 cm spectrum in Fig. 2(a). To obtain larger tunability, the tuned pulse can be first compressed and then injected into a second gas-filled hollow fiber where the gas molecules are aligned by another intense pulse. This configuration can be referred simply as hollow-fiber cascading . If we assume ideal compressions and the same initial time delays between adjacent fibers, the pulse frequency continuously shifts upon propagation from one fiber to the next, thus enhancing the achievable frequency tunability. With this cascading system, the maximum tunability of the signal pulse would be limited only by the frequency-dependent absorption in the gas medium.
4.2 Frequency red-shift
Frequency red-shift is obtained when τd corresponds to ~ 7990 fs in Fig. 1(b). With this time delay the signal pulse experiences an approximately linear increasing refractive index and obtains the same negative frequency shift over the entire pulse. The calculated continuously red-shifted pulse spectrum is shown with respect to propagation length in Fig. 3(a). The carrier frequency at the propagation length of 54 cm corresponds to a wavelength of 4237.3 nm. The number of optical cycles contained in the signal pulse is also nearly constant after ideal chirp compensation, as shown in the inset of Fig. 3(b).
In contrast to the blue-shift, at this time delay the spectra of both the signal pulse and the intense pulse shift to the red, so the group velocity walk-off between the two pulses is very small, insuring a longer effective propagation length. In addition, with propagation the spectrum of the signal pulse becomes narrower and the effects of dispersion become smaller. Thus at 54 cm, the calculated 33.59 fs pulse is very close to the 28.28 fs Fourier-transform-limited pulse, as shown in Fig. 3(b). Therefore, we can obtain few-cycle pulses directly, even without any compression.
4.3 Optimization of input pulse for precompensation
Since the signal pulse, which is frequency shifted, is decoupled from the medium response, it can then be preshaped at the input to precompensate for dispersion and the slight nonlinear effects. For example, if the input signal pulse is optimized as shown in Fig. 4(a) and (b), then through 21 cm fiber a 2.26 fs, 323.8 nm pulse that is nearly Fourier-transform-limited is obtained, as shown in Fig. 4(c). Therefore, frequency shifted few-cycle pulses can be obtained by manipulating the input spectrum before the nonlinear medium rather than by compensating for the chirp of the generated spectrum.
In conclusion, we have demonstrated a new method to tune the frequency of few-cycle laser pulses from the UV to the mid-IR using ultrafast phase modulation in a hollow fiber filled with impulsively excited N2 molecules. There are three major factors restricting the tunability: (i) the decreasing modulation of the refractive index n(t) with propagation, (ii) group velocity walk-off between the intense pulse and the signal pulse that causes the signal pulse to shift in time out of the linear region, and (iii) absorption in the gas medium. By using a molecule with a larger difference in polarizabilities, enhancing the intensity of the intense pulse, and shaping the intense pulse to increase the alignment, larger tunability can be achieved. This approach is also suitable for longer pulses as long as pulse lengths do not exceed the linear region. Different molecules with smaller rotational constants can be chosen to optimize for different longer input pulse durations.
We gratefully acknowledge the support of the National Basic Research Program under Grants No. 2006CB806007 and 2006CB921601, and the National Science Foundation of China under Grants No. 10574006, 10634020 and 10521002.
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