Generation of single-cycle mid-infrared pulses via coherent synthesis

Fen Ma; Hongjun Liu; Nan Huang; Qibing Sun

doi:10.1364/OE.20.028455

1. Introduction

Coherent synthesis is the term applied to creating a single pulse from parent pulses so that the generated pulse is fully coherent across the entire bandwidth, and offers a route to produce isolated sub-femtosecond optical pulses [1–4]. In order to artificially synthesize two or more initial pulses, two fundamental prerequisites are that the pulse sequences share a common repetition frequency (f_rep) and a common carrier-envelope offset (CEO) frequency. Sufficient repetition-frequency stabilization and CEO-frequency control have been achieved between two distinct femtosecond lasers [3,5,6] as well as between a pump source and a synchronously-pumped optical parametric oscillator (OPO) [7–9]. With two independent lasers, not only their relative pulse timing is synchronized at the femtosecond level, but their carrier frequencies are also phase-locked. Under this scheme, the greatest challenge of synthesis is to obtain sufficient stabilization of the repetition frequencies, which allows coherent combination of the pulses over significant time periods. In contrast, coherent synthesis based on a synchronously-pumped OPO benefits from inherently passive synchronization between all the parametrically generated pulses and its pump source. In this case, the predominant issue is locking the CEOs of all the participating pulses to a common value. These problems as mentioned above have been circumvented by a certain control system, including an acousto-optic modulator (AOM) [3,10], phase-frequency detector (PFD) circuits [7,9], and an acousto-optic frequency shifter (AOFS) [11]. Furthermore, the synthesized pulses obtained to date merely lie in the near-infrared spectral range [3–5, 7–10].

However, the driving laser sources at longer wavelengths are suggested to obtain higher photon energy harmonic supercontinuum for the shorter attosecond pulses [12–14] and clearer separation of multiphoton and tunneling processes [15]. The mid-infrared lasers are beneficial for suppressing multiphoton ionization in favor of the tunneling ionization mechanism because the ponderomotive energy in a strong-field interaction is scaled with λ². Therefore, the phase-stabilized mid-IR ultrashort pulsed sources, at wavelengths longer than 3 μm with single-cycle duration, are greatly sought for the ultrafast and strong-ðeld laser physics [16].

In this paper, we present a new approach, based on coherent wavelength multiplexing of both idlers from difference frequency generation (DFG) processes, to generate single-cycle mid-infrared pulses. We numerically investigate the distinct DFG processes pumped by a single laser and seeded with different portions of the white-light continuum (WLC). Any optical waveform supported by the combined spectrum can be produced by the design of these parameters, including the spectra superposition, the difference of two CEPs, the relative timing and the chirp ratio between the idlers. Through the optimal design of the parameters, the single-cycle mid-infrared pulses with a center wavelength of 4233 nm are obtained with combined spectra spanning from 3000 nm to 7000 nm. This work represents a new and flexible approach to the synthesis of coherent light, providing a useful tool for investigating the strong-field physics.

2. Principles for direct coherent synthesis

The pulse train emitted from a mode-locked femtosecond laser consists of a comb of modes f_n obeying the equation $f_{n} = n f_{r e p} + f_{C E O}$ , where n is the integer harmonic number and f_CEO shifts the comb as a whole from the exact harmonics of f_rep. Owning to the discrepancy in the group- (v_g) and phase-velocity (v_p), there is a phase slip between the carrier phase and the envelope peak for each of the successive pulses derived from the laser, which can be expressed as Φ. Denoting this pulse-to-pulse phase slip by Δφ, it is mapped to coordinates $f_{C E O} = f_{r e p} Δ ϕ / 2 π$ in the frequency domain [2,3,17]. In the most general case, the phase and frequency offsets are arbitrary. If the parent pulses are from a single pump source, they inherit the identical repetition frequency of the pump. Under this condition, the equal CEO frequencies of two pulses are imperative for coherent synthesis.

Figure 1 exhibits the configuration of proposed direct synthesis approach, including a femtosecond pump laser, a photonic crystal fiber (PCF) which is used to generate a broadband WLC, two broadband DFGs seeded by distinct portions of the WLC and coherent combination of the two idlers in order to synthesize high energy single-cycle pulses.

Fig. 1 Schematic of coherent synthesis for single-cycle mid-infrared pulses generation.

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Here the WLC is generated in a PCF with all-normal group velocity dispersion (GVD) in which solitons cannot form and spectral broadening is mainly via self phase modulation (SPM) [18–20]. When the pulse propagates in the PCF, the variation of the pulse amplitude can be analyzed based on the nonlinear Schrödinger (NLS) equation described as follows:

\frac{\partial A (z, t)}{\partial z} + \frac{i β_{2}}{2} \frac{\partial^{2} A (z, t)}{\partial t^{2}} - i γ {| A (z, t) |}^{2} A (z, t) = 0,

where A(z, t) is the slowly varying amplitude of the pulse envelope, t is the relative time in a frame of reference co-moving with the pulse at the group velocity, z is the spatial coordinate along the fiber length. Parameter β₂ is the second order dispersion coefficient. The coefficient γ is defined as

γ = n_{2} ω_{0} / c A_{e f f}

, where n₂ is the Kerr coefficient, ω₀ is the central angular frequency, c is the velocity of light in vacuum, and A_eff is the effective mode area. Since SPM is a purely intensity-dependent nonlinear optical process and preserves the coherence of input pulses, the WLC generated by SPM has the same, random, CEP equal to the pump beam [21–23] and produces high spectral coherence. As both pulses are derived from the single laser, the pump comb can be described as

f_{P} = n_{1} f_{r e p} + f_{C E O}

, whereas the signal comb is

f_{S} = n_{2} f_{r e p} + f_{C E O}

with n₁> n₂. Based on conservation law of energy, the generated idler comb through DFG is f_I with vanishing f_CEO as expressed in Eq. (2) [23]:

f_{I} = f_{P} - f_{S} = (n_{1} f_{r e p} + f_{C E O}) - (n_{2} f_{r e p} + f_{C E O}) = (n_{1} - n_{2}) f_{r e p},

consequently, an offset-free idler can be achieved by DFG in a nonlinear crystal. Owing to the unique feature of DFG, coherent synthesis of independent idlers can be simply achieved, which avoids the need for the complicated control system in either two separate lasers or a synchronously-pumped OPO.

3. Coherent synthesis and waveform optimization

In order to pave the way to optimize the experimental results, the value of each parameter is taken as consistent as possible with the off-the-shelf components. In the following simulations, considering the practical parameters of a general laser, the pulse duration, center wavelength and repetition rate of the pump are taken as 100 fs, 1064 nm and 50 MHz, respectively. The pump pulse is supposed to exhibit a Gaussian profile in the time domain. The output beam has an average power of 0.3 W and is split into three branches. One of the branches with 90 mW average power is focused into the 30 cm-long PCF [19]. Based on Eq. (1), the spectrum region of WLC can be calculated, covering 772–1637 nm (–40 dB) as shown in Fig. 2 . The WLC is then divided into two branches via a long-pass filter, which the reflected pulse is centered at 1350 nm with an average power of 24 mW and the transmitted pulse is centered at 1400 nm with an average power of 20 mW.

Fig. 2 Spectra of the input pulse and output pulse.

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Other two branches with 0.21 W average power are acted as the pumps in the subsequent DFGs. By quasi-phase matching (QPM) both pumps are down-converted in two gain media of MgO-doped periodically poled Lithium Niobate (PPLN) crystals. Under the perfect phase matching $Δ k = k_{P} - k_{S} - k_{I} - k_{Λ} = 0$ where k signifies the wave vector, the generated signal region depends on the different poling periods of PPLN with the stationary pump region as shown in Fig. 3 . For the reflected pulse the poling period is taken as 25.5 μm and the signal spans from 1309 to 1388 nm, while for the transmitted one the poling period is taken as 27.5 μm and the signal spans from 1374 to 1441 nm. To ensure tight synchronization, we assume that the pump and the signal propagate the same distance and the time delay is accurately controlled.

Fig. 3 Signal wavelength region depends on the poling period of PPLN in the case of the perfect phase matching with the fixed pump wavelength. The signal covers 1309–1388 nm with the poling period of 25.5 μm (dashed line), whereas the signal covers 1374–1441 nm with the poling period of 27.5 μm (dot-dashed line).

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The theoretical description of DFG process is based on the coupled-wave equations for three-wave mixing process. Taking into account that the pump is a femtosecond pulse, the coupled-wave equations can be described as Eq. (3) [24], providing a basis for calculating the intensity variation of pump, signal and idler pulses.

\begin{array}{l} \frac{\partial E_{S}}{\partial z} + \frac{n_{S}}{c} \frac{\partial E_{S}}{\partial t} + α_{S} E_{S} = - i \frac{ω_{S} d_{e f f}}{n_{S} c \cos^{2} β_{S}} E_{I}^{*} E_{P} \exp (- i Δ k z) \\ \frac{\partial E_{I}}{\partial z} + \frac{n_{I}}{c} \frac{\partial E_{I}}{\partial t} + α_{I} E_{I} = - i \frac{ω_{I} d_{e f f}}{n_{I} c \cos^{2} β_{I}} E_{S}^{*} E_{P} \exp (- i Δ k z) \\ \frac{\partial E_{P}}{\partial z} + \frac{n_{P}}{c} \frac{\partial E_{P}}{\partial t} + α_{P} E_{P} = - i \frac{ω_{P} d_{e f f}}{n_{P} c \cos^{2} β_{P}} E_{S} E_{I} \exp (i Δ k z), \end{array}

where α_x and β_x (x = P, S, I) represent the absorption coefficient and the walk-off angle, respectively. Considering the damage threshold of the PPLN crystal [25], the intensities of pump and signal (centered at 1400 nm) in the DFG 1 are taken as 600 MW/cm² and 140 MW/cm², whereas the corresponding intensities in the DFG 2 are set to be 620 MW/cm² and 150 MW/cm², respectively. According to Eq. (3), the relationship between the crystal length and the power densities of the pump, signal and idler is shown in Fig. 4 . It is found that the power of idler increases to a maximum value along the crystal length until the power of pump is fully depleted. After that, the power begins to decrease because the energies of the signal and idler flow back to the pump. Obviously, the maximum powers of both idlers are obtained at the length of 2.1 mm. Here, two gain media are set to be 2 mm long in the following calculations.

Fig. 4 Relationship between the crystal length and the power density in the DFG 1 (a) and DFG 2 (b).

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Based on Eq. (3), the spectra of two idlers can be calculated as shown in Fig. 5(a) . Both idlers have a vanishing f_CEO and a common f_rep of the pump, therefore, a synthesized pulse is generated by coherently stitching the optical bandwidths together, providing a wider bandwidth than an individual laser output.

Fig. 5 (a) Spectra of two idlers from DFGs. The intensity of idler 1 is normalized to that of the idler 2. Inset: the corresponding linear values of both idlers intensities; (b) Spectrum of the synthesized pulse centered at 4600 nm. Inset: the temporal profile of synthesized pulses.

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In the synthesis, the parameters, including the spectra superposition, the relative timing (Δt), the chirp ratio (c₁/c₂) and the difference of CEPs ( $Δ Φ = Φ_{1} - Φ_{2}$ ) between the idlers, are adjustable. It is found that precise waveform shaping can be accomplished by the control of each parameter. For example, Fig. 5(b) depicts the spectrum intensity of the synthesized pulse without the CEPs difference as well as the relative timing between two idlers.

The impacts of these above parameters on the synthesized waveform are further calculated via second-order autocorrelation, offering guidelines for waveform optimization. The second-order autocorrelation includes intensity autocorrelation and interferometric autocorrelation, which are described as Eq. (4) and Eq. (5), respectively.

I (τ) = \int_{- \infty}^{+ \infty} I (t) I (t - τ) d t,

I (τ) = {\int_{- \infty}^{+ \infty} | {(E (t) + E (t - τ))}^{2} |}^{2} d t,

where E(t) is the electric-field of synthesized pulses and I(t) is the corresponding intensity.

In principle, the synthesized pulse is varied with the distinct combined spectra region. To clarify the effect further, the impact of the spectra superposition between the idlers on the synthesized waveform is studied via the intensity autocorrelation as well as the interferometric autocorrelation. From the intensity autocorrelations (Figs. 6(a) –6(c)), it is found that the highest peak intensity is achieved when both idlers have identical center wavelength, and then the peak intensity is greatly reduced with the decrease in the spectra superposition between two idlers. A Gaussian pulse can be described as $E (t) = \exp (- t^{2} / 2)$ , and thus the intensity autocorrelation can be calculated based on Eq. (4), which is expressed as $I (τ) = \exp (- τ^{2} / 2)$ . It is concluded that the FWHM pulse duration ratio between the intensity autocorrelation and the initial pulse is $\sqrt{2}$ . Consequently, the calculated duration is 58 fs, 18 fs, and 52 fs, respectively. There is an optimal wavelength difference, within which a relative short duration and a high peak energy can be achieved simultaneously. However, in the case of the less spectra superposition, there are more oscillatory structures away from the central peak with higher energy as shown in Fig. 6(c), and in this case the synthesized pulse is not available for strong-field physics applications due to the temporal modulation of the synthesized waveform. The interferometric autocorrelations include the phase and profile information of the pulses as shown in the panels in Figs. 6(d)–6(f), which are mapped to the corresponding intensity autocorrelations as shown in the panels in Figs. 6(a)–6(c). As the spectra superposition is reduced, destructive interference or constructive interference in the synthesis process occurs, which is caused by the different intensities and phases of the idlers at the same frequency. Furthermore, the large center-wavelength difference between both idlers results in the large gap of the combined spectrum, as a result, the main peak energy reduces and the oscillatory structures appears.

Fig. 6 Second-order autocorrelation curves of the synthesized pulse varied with distinct parent pulses. Suppose that one centered at 4400 nm, the other is set to be 4400 nm in (a), 5000 nm in (b), and 5600 nm in (c), respectively. The intensity autocorrelation curves of synthesized pulses are shown in panels (a), (b) and (c), while panels (d), (e) and (f) exhibit the corresponding interferometric autocorrelations, respectively.

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The intensity autocorrelation of synthesized pulses with the CEP difference (ΔΦ) is numerically calculated for two conditions: (i) the parent pulses, coincident with our system, are centered at 4400 nm and 5000 nm, respectively (solid lines), and (ii) the parent pulses are centered at 4400 nm and 4500 nm, respectively (dashed lines). As shown in Fig. 7 , in the first condition, it is found that the peak energy is decreased when the CEP difference is varied from 0 to 0.8π, whereas exceeding 0.8π the peak energy is almost unchanged as well as the profile of synthesized pulses. In the second condition there are different quantitative results, but similar trends. The peak intensity is greatly reduced with the phase difference increased from 0 to 0.8π, while the profile of synthesized pulses is not changed clearly. Then as the phase difference exceeds 0.8π, the oscillatory structures are visible and the peak intensity continues to be reduced slightly. Apart from the effect of spectra superposition between parent pulses on the combined waveform, the trend appears because the π-phase difference causes destructive interference for the same frequency ingredients of parent pulses, which induces the pulse narrowed as well as the energy reduced. Moreover, the influence of CEP difference on the synthesized waveform will be weakened caused by the relative large center-wavelength difference between the parent pulses.

Fig. 7 The evolution of synthesized pulses for various phase difference as follows: 0 in (a), 0.8π in (b), 0.9π in (c), and π in (d) without relative timing as well as initial chirps. The intensity autocorrelations of synthesized pulses are shown between the parent pulses coincident with our system 4400 nm and 5000 nm, respectively (solid lines) and different from our system, centered at 4400 nm and 4500 nm, respectively (dashed lines).

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Figure 8 depicts the change of combined optical pulses with the relative timing Δt. With the increasing relative time, obviously, the peak intensity is reduced, which is transferred to the pedestal, and the duration is not changed until the relative timing exceeds 60 fs. After that, the synthesized pulse is stretched with the increasing relative time as shown in Figs. 8(c) and 8(d). The main reason is that destructive interference occurs between the two idlers, when the linear phase slip between the two idlers caused by the relative timing is accumulated to 0.8 π or more.

Fig. 8 Intensity autocorrelations of synthesized pulses with altering the relative timing and keeping other two parameters unaltered. (a) Δt = 0, the duration is 18.4 fs; (b) Δt = 60 fs, the duration is 18.4 fs; (c) Δt = 65 fs, the duration is 19.8 fs; (d) Δt = 100 fs, the duration is 21.2 fs.

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Interferometric autocorrelations of the combined pulse with different chirp ratios between two idlers are illustrated in Fig. 9 . As the chirp ratio is increased, the synthesized pulse is narrowed and the main pulse energy is reduced. When the chirp ratio exceeds 5, more energy is transferred to the pedestal, and the temporal pedestal of the synthesized pulse is detrimental for some strong-field physics applications. Therefore, for achieving the shortest synthesized pulse, the chirp ratio cannot be increased infinitely and exists an optimal value of 5.

Fig. 9 Interferometric autocorrelations of combined pulses are illustrated with the varied chirp ratio between the idlers. The chirp ratio is assumed 0 in (a), 1 in (b), 3 in (c), 5 in (d), 6 in (e), and 7 in (f), respectively.

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The synthesized waveform is important for optimizing the high-harmonic generation (HHG) process [26], which is the only demonstrated technique for generating isolated attosecond pulses. The shortest duration of the synthesized pulse is a typical goal and will inform our optimization analysis. According to the desired investigations in the ultrafast and strong-ðeld laser physics, the optimization of the above parameters is made to pursue the trade-off between duration and pulse intensity. Through optimizing these distinct parameters, the shortest synthesized field can be obtained as shown in Fig. 10 , which lasts only a single cycle (amplitude FWHM) at a center wavelength of 4233 nm with the spectrum spanning from 3000 nm to 7000 nm. Owning to the less combined spectra region between two idlers, the oscillatory structures appear in the intensity autocorrelation curve of the synthesized pulse as shown in Fig. 10(a).

Fig. 10 Optimization of the synthesized pulse in our system. (a) The intensity autocorrelation of the optimal pulse and (b) the corresponding combined spectrum. Inset: The profile of optimal pulse in the temporal domain. The design of the correlative parameters is as follows: the relative timing Δt = 9 fs, the difference of CEPs ΔΦ = −0.82 rad, and the chirp ratio is 6. These parameters determine the waveform of synthesized pulses and shift the center wavelength of combined pulses.

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4. Conclusion

We have addressed a method for the generation of single-cycle pulses in the mid-infrared region, which is achieved based on coherent wavelength multiplexing of DFG processes. The impacts of the spectra superposition, the carrier-envelope phase (CEP) difference, the relative timing and the chirp ratio between the idlers on the waveform of combined pulses are analyzed in detail, which exhibit noteworthy conclusions and offer guidelines to the waveform optimization. The large center-wavelength difference of parent pulses results in a shorter duration of the combined pulse than the duration of combined pulse in the perfect overlap scheme. However, in the case of the small spectra superposition, the pulse is not available for strong-field physics because of the visible oscillatory structures. The profile of synthesized pulses is not affected by the CEP difference between both idlers until the value up to 0.8π or more, additionally, this influence is reduced by the decrease in the spectra superposition. The increased relative timing induces the energy transferred gradually to the pedestal, and the synthesized pulse is stretched as the relative timing is up to 60 fs or more. The duration of synthesized pulses is decreased with the increasing chirp ratio, and there is an optimal value of 5 to make the synthesized pulse shortest. Based on the optimal design of these parameters, we obtain a single cycle pulse in the mid-infrared region spanning from 3000 nm to 7000 nm without any compressed device. The single-cycle mid-infrared laser source provides the prototype of a novel optical tool for attosecond control of strong-ðeld physics experiments.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61078029 and 61178023.

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Generation of single-cycle mid-infrared pulses via coherent synthesis

Abstract

1. Introduction

2. Principles for direct coherent synthesis

3. Coherent synthesis and waveform optimization

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (5)

Optics Express