## Abstract

We report a simple all-fiber design of an Er-doped laser system that is capable of generating widely tunable two-cycle pulses. In particular, 13-fs pulses at a wavelength of 1.7 *μ*m are produced. The mechanism of pulse shortening is identical to the higher-order soliton compression and is supported by modeling based on the slowly evolving wave approximation, which is well suited for down to single-cycle pulse propagation in nonlinear dispersion-shifted fibers.

© 2011 Optical Society of America

## 1. Introduction

Progress in laser science made in the recent decade has established a new field of ”extreme nonlinear optics,” which involves very short laser pulses comprising only a few optical cycles (see, e.g., [1]). Few-cycle pulses have found important applications, e.g., in high harmonic generation, frequency metrology, attosecond science, and THz generation as well. The main source for such extremely short pulses is solid-state mode-locked bulk lasers, in particular, Ti:sapphire lasers. Few-cycle pulses can be generated directly from a Ti:Sa oscillator using specifically designed dispersion compensation optics [2] which is still impossible with fiber oscillators. However, recently Er-doped fiber lasers have demonstrated their ability for delivering few-cycle optical pulses in the near and mid IR [3, 4]. This may have a strong impact on femtosecond science and technology mostly owing to their prominent advantages, such as compactness, ease of use, cost efficiency, and long-term stability.

In general, there are several main approaches to obtain few-cycle optical pulses [5]. The first one is the above-mentioned generation directly from a Kerr-lens mode-locked Ti:Sa oscillator. The second one is based on external nonlinear optical shaping of femtosecond pulses using different kinds of optical fibers. Input femtosecond pulses (converted in nonlinear fibers) can be obtained from different sources: solid lasers and fiber lasers as well. Baltuška *et al.* demonstrated 4.5-fs pulses resulting from the fiber-compressed output of a cavity-dumped Ti:sapphire laser [6]. Kieu *et al.* reported a fiber laser system based on a carbon nanotube saturable absorber that is capable of generating 14-fs pulses [7]. There were also some papers demonstrating generation of few-cycle optical pulses using soliton-effect self-compression in photonic crystal fibers and photonic nanowires [8, 9]. Other researchers employed more sophisticated schemes, such as optical field synthesis when the fields from different spectral parts of the pulse transmitted through nonlinear media are combined [10], as well as the technique based on phase manipulation by feedback adjustment with precise computer analysis [11], both providing the shortest durations attained at the present.

In this paper, we report a simple technique to generate routinely nJ-level few-cycle optical pulses with no post compression directly from an all-fiber Er-doped laser setup. Although the design we used is similar to those that have already proposed for the sub-two-cycle pulse generation [3, 8, 9], we, first, use standard telecommunication components and do not use photonic crystal fibers commonly employed for few-cycle pulse production in the near infrared [8, 9] and, second, as compared with Ref. [3], our system is an all-fiber one and produces pulses in the range of 1.6–2 *μ*m. The technique of second-harmonic generation frequency-resolved optical gating (SHG FROG) is also applied to measure as short as two-cycle pulses.

## 2. Experimental setup

The schematic diagram of the laser system is shown in Fig. 1. We start with a polarization mode-locked Er:fiber master oscillator delivering positively chirped femtosecond pulses with a bandwidth of 30 nm centered at a wavelength of 1.57 *μ*m and following with a repetition rate of 49 MHz. The generated pulses are coupled to a 2.5 m long single-mode fiber (SMF-28) of a length of 2.5 m where they are compressed due to anomalous dispersion and are stretched again before entering the Er-doped gain fiber 2.1 m long. The pulses are amplified in a forward and backward diode-pumped Er:fiber amplifier to about 110 mW. The scheme is based on non-polarization-maintaining (non-PM) fibers, and the polarization controller (PC) is used to set a linear polarization state. The next element of the laser system is a 25 cm piece of SMF-28 working as a solitonic pre-compressor. We use dispersion-managed amplification scheme to escape non-linear pulse distortion inside active fibers. The laser optimization was supported by numerical simulation based on the generalized nonlinear Schrödinger equation [12]. The laser source design is similar to that reported in [13]. We retrieve the pulse shape at the output by using SHG FROG, which relies on the measurement of spectrally resolved second-order noncollinear autocorrelation. The retrieved pulse shape is depicted in Fig. 2(a). The corresponding pulse duration is 52 fs. The pulse spectra measured and calculated from the reconstructed pulses are shown in Fig. 2(b).

We remove the direction-of-time ambiguity by making a second SHG FROG trace after chirping the pulse with a thick piece of glass. The two resulting pulses are consistent with only one direction of time [14].

A key element is a short segment of highly nonlinear dispersion-shifted fiber (HNL-DSF) with dispersion profile as in the inset of Fig. 1, directly spliced to the SMF-28 pre-compression fiber (the splice loss is 10–15 %). As was shown previously [15, 16], such fibers are well suited for supercontinuum generation if specific supercontinuum propeties are needed. Here we show that by precisely defining the HNL-DSF parameters, such as the point of zero group velocity dispersion (GVD) wavelength and a dispersion profile as a function of wavelength, allow us to achieve optimal compression to the shortest duration. Compression of a 52-fs input pulse down to a 13-fs, comprising two optical cycles at a wavelength of 1.7 *μ*m, is experimentally realized. However, we first describe modeling of few-cycle pulse propagation in HNL-DSFs in order to carefully choose the fiber parameters needed.

## 3. Modeling

To design an all-fiber turnkey laser source for few-cycle pulse generation, precise modeling of such extremely short pulse propagation in nonlinear fibers should be available. Strictly speaking, for this modeling slowly varying envelope approximation is not valid any more, however dealing with few-cycle pulses one often uses the generalized nonlinear Schrodinger equation, particularly including higher-order dispersion terms [3, 12, 17]. The main reason is that, if the third harmonic and associated four-wave mixing are not effectively generated during pulse propagation, then such description gives suitable accuracy of modeling, taking less computer resources. However, the minimal pulse width should comprise at least several field oscillations, depending on particular examples [18].

To be sure that all important nonlinear processes are included and to make precise definition of the DSF parameters, we develop for few-cycle pulse propagation in a single-mode silica fiber a one way propagation equation (slowly evolving field approximation) for the Fourier transformed field *G(z,* *ω**)* similar to that given by [18, 19]:

*E(z, t)*is the real laser field,

*F̂*is the Fourier transform operator,

*β*(

*ω*) is a function describing actual fiber dispersion,

*z*is a distance along the fiber,

*ω*is the angular frequency, and

*c*is the speed of light in vacuum,

*n*

_{0}is an unperturbed refractive index at the central frequency

*ω*

_{0},

*P*is a nonlinear polarization which can be described for the linearly polarized laser by the following expression [20, 21]:

_{NL}The response function *a(t)* should include both the quasi-instantaneous Kerr as well as the retarded Raman contributions [21, 22]. We use the following analytic form of the Raman response function [22]:

The parameters *τ*_{1} and *τ*_{2} are two adjustable parameters and are chosen to provide a good fit to the actual Raman-gain spectrum in silica. Their appropriate values are *τ*_{1} = 12.2 fs and *τ*_{2} = 32 fs [22]. The third (nonlinear) term in Eq. (1) should be written as

*δ*(

*t*) is the Dirac delta function, the nonlinear parameter

*γ*is defined as

*γ*=

*n*

_{2}

*ω*

_{0}/

*cA*,

_{eff}*n*

_{2}is the nonlinear-index coefficient, and the parameter

*A*is known as the effective core area [12]. If one does not take into account Raman scattering,

_{eff}*f*= 0. But, actually, the fraction

_{R}*f*is estimated to be about 0.18 [22].

_{R}We employ the standard split-step Fourier method for numerical integration of Eq. (1). To get the shortest pulse width we have performed a parametric scan for 50-fs pulse propagation in a dispersion-shifted silica fiber in the range of zero GVD wavelength *λ _{ZD}* = 1.35 – 1.55

*μ*m. To estimate roughly the range of optimal fiber parameters, we take into account the dispersion as a linear function of frequency (

*β*

_{3}= 0.05 ps

^{3}/km ). Numerical results are depicted in Fig.3. The upper row demonstrates intensity distributions in the time domain but the lower one demonstrates them in the frequency domain. Each column corresponds to the zero GVD wavelength

*λ*written under it. The blue curves are obtained for the input pulse energy of 1.5 nJ, they illustrate higher-order soliton compression before the fission into separated spectral components. The fiber length

_{ZD}*L*provides generation of the shortest pulses. The red curves illustrate the possibility of few-cycle pulse generation after pulse fission. It is clearly seen that DSF with

*λ*≥ 1.5

_{ZD}*μ*m is not fit for two-cycle pulse generation but DSF with a smaller

*λ*are suitable for this purpose in both regimes (shown in blue as well as in red). A large part of input energy flows fast to the normal dispersion range for

_{ZD}*λ*≥ 1.5

_{ZD}*μ*m. Although the Fourier-transform-limited pulses have a duration less than 6 fs (red spectra for

*λ*= 1.5

_{ZD}*μ*m and

*λ*= 1.55

_{ZD}*μ*m), separated components in the normal and anomalous dispersion ranges interfere and do not lead to few-cycle pulse production. It is clearly seen that for the input pulse energy of 3 nJ,

*λ*= 1.55

_{ZD}*μ*m as well as for the input pulse energy of 2.5 nJ,

*λ*= 1.5

_{ZD}*μ*m there are Raman-shifted solitons [12] (with central wavelength of 2.05

*μ*m and 1.95

*μ*m) respectively, but full intensity distributions are very complicated (for

*L*> 20 cm they separate in the time domain and their frequencies shift down). The red curves for

*λ*= 1.45

_{ZD}*μ*m demonstrate 17-fs Raman-shifted soliton with central wavelength of 2.0

*μ*m. The input pulse energy is taken to be 3.5 nJ (it is the maximal value for our Erbium setup limited by diode pump power). The efficiency of Raman soliton generation is higher for

*λ*= 1.45

_{ZD}*μ*m than for

*λ*≥ 1.5

_{ZD}*μ*m, so it has higher energy and shorter duration. The red curves for

*λ*= 1.4

_{ZD}*μ*m and

*λ*= 1.35

_{ZD}*μ*m show few-cycle pulses (not Raman solitons).

Further we study more carefully pulse propagation in the DSF with *λ _{ZD}*=1.4

*μ*m and realistic dispersion profile. Our encouraging factors were the following: (i) the high-order soliton compression effect extended to the few-cycle regime can provide the pulse compression down to single-cycle duration [23, 24] and (ii) two-cycle pulses were already produced by using photonic crystal fibers [8]. We believe that by carefully choosing the conventional dispersion shifted silica fibers few-cycle pulses can be also routinely produced just splicing a piece of HNL-DSF, thus providing a design of an all-fiber system. Based on these simulations some particular HNL-DSFs were chosen with the parameters providing optimal compression factor. An example of a such a fiber is shown in Fig. 4(c). Specifically for this fiber (

*λ*=1.4

_{ZD}*μ*m), taking into account actual dispersion in Eq. (1), the sech-shaped transform-limited pulse with a 49.4 fs duration and 2 nJ energy is launched in the slightly anomalous dispersion region (central wavelength of the pulse is

*λ*=1.57

_{C}*μ*m). The pulse evolution in the time and frequency domains is shown in Fig. 4(a), (b) respectively.

One can see typical spectrum broadening (supercontinuum generation in Fig. 4(b)) and pulse shortening (a), particularly to a shortest duration of 8.8 fs, which is, in fact, a sub-two-cycle pulse. The pulse shortening is due to the higher-order soliton compression with the soliton order of *N* = 4 at the input. The pulse reaches its minimum duration at the transient stage and after that it fissions into fundamental solitons. As shown in Ref. [23], these fundamental solitons can also have few-cycle durations. Specifically for the problem of interest, output pulses are wavelength shifted due to the intra-pulse Raman conversion effect [12], thus providing wide tunability in the 1.6–2.1 *μ*m range of silica transparency.

We also present intensity distributions of the output pulses in the time domain (Fig. 5) and in the frequency domain (Fig. 6) versus input pulse energy for different propagation lengths but the same input pulse envelope and DSF as in Fig. 4. As is clearly seen from Fig. 5, for a fixed fiber length there is an optimum for the input energy with respect to minimal width of the output pulse. For example, at 4.5 cm length to get the minimal pulse duration the input energy should be about 2 nJ. In Table 1 we summarized results on output pulse durations for different DSF lengths and pump energies. It should also be noted that with increasing input energies the output pulse spectrum, as depicted in Fig. 5, is broadened and shifted to lower wavelengths, even reaching the absorption border for fused silica about 2.1 *μ*m. For higher energies when the input pulse corresponds to a higher-order soliton, the output spectrum represents multi-peak distribution and actually corresponds to a number of solitons with different carrier wavelengths in the 1.6 – 2 *μ*m range.

Thus, in the presented modeling few-cycle pulses can be easily produced in 3–5 cm long silica dispersion-shifted fibers when 1–3.5 nJ input pulses are used. It should also be noted that pulse shortening is accompanied by dispersive wave radiation (left green curve near 1 *μ*m in Fig. 6), which by properly choosing DSF can provide an efficient source of short femtosecond pulses in the one micron range [3], particularly as a seeding source for a high power Yb:Fiber amplifier [13, 25].

## 4. Experimental results

A segment of HNL-DSF was directly spliced to the Er:fiber system (see Fig. 1). HNL-DSF is fabricated by the standard technology [12]. Dispersion profile calculation is based on information about refractive-index profile of a cylindrical preform (not a step index) and diameter of drawn fiber. Eigenvalue equation is solved numerically for a fundamental mode. The nonlinearity of the DSF is higher than that one of SMF because of smaller mode diameter and estimated to be *γ*≃4 W^{−1} km^{−1}. The input pulse with a duration of 52 fs corresponds to the soliton order of *N* ≈ 4. To reconstruct the pulse intensity profile we use SHG FROG technique suitable for few-cycle or even single-cycle pulse measurements [6, 14]. We built scheme based on the Michelson interferometer containing reflective optics (golden mirrors) and broadband dielectric-coated beam splitter. The second harmonic spectra at different time delays are recorded with commercially available grating spectrometer.

We experimentally studied the pulse propagation along the HNL-DSF by simply cutting the corresponding pieces of the fiber. We also simulate the same situation assuming the input pulse to be the one shown in Fig. 2 (unlike the simulations depicted in Figs. 3–6 and Table 1 for input transform-limited sech-shape input pulse). Pulse evolution in the time and frequency domain is shown in Fig. 7 . One can see optical pulse before the experimental point of maximal compression, at the optimal point and after it.

The shortest experimentally obtained pulse has duration of 13 fs. Corresponding FROG traces of this pulse are shown in Fig. 8. We think that we did not get a shorter one as predicted by modeling because the cutting steps were not small enough.

Thus, we generate 13 fs pulse at a central wavelength of 1.7 *μ*m. Reconstructed intensity is shown in Fig. 9(a). The 13-fs pulse has a pedestal and carries about 70% of its energy in the form of narrow peak. Estimated peak power is 100 kW. Such situation of a narrow peak and broad pedestal is common for soliton compression mechanism. Our experimental results are in a good agreement with theoretical modeling. This is confirmed by comparing the measured, reconstructed, simulated spectra as well as reconstructed and simulated spectral phase as depicted in Fig. 9(b). We estimate that the Fourier-transform-limited pulses with such spectra have durations of about 10 fs.

## 5. Conclusion

We have shown a simple design of an all-fiber Er:doped laser system that can routinely generate few-cycle pulses tunable in the range of 1.6–2 *μ*m. The key element is a conventional dispersion-shifted silica fiber. Precisely choosing DSF parameters such as dispersion profile, nonlinearity coefficient and fiber length optical pulses as short as two-cycle duration can be produced in a turnkey system based on telecommunication components.

## Acknowledgments

This work was partly supported by the Russian Foundation for Basic Research through Grant No. 10-02-01241. E.A. Anashkina and A.V. Andrianov also acknowledge partial support from the RF President grant MK-4902.2011.2.

## References and links

**1. **M. Wegener, *Extreme Nonlinear Optics* (Springer-Verlag, Berlin, 2005).

**2. **S. Rausch, T. Binhammer, A. Harth, J. Kim, R. Ell, F.X. Kartner, and U. Morgner, “Controlled waveforms on the single-cycle scale from a femtosecond oscillator,” Opt. Express16, 9739–9745 (2008). [CrossRef] [PubMed]

**3. **A. Sell, G. Krauss, R. Scheu, R. Huber, and A. Leitenstorfer, “8-fs pulses from a compact Er:fiber system: quantitative modeling and experimental implementation,” Opt. Express17, 1070–1077 (2009). [CrossRef] [PubMed]

**4. **A. Andrianov, A. Kim, S. Muraviov, and A. Sysoliatin, “Wavelength-tunable few-cycle optical pulses directly from an all-fiber Er-doped laser setup,” Opt. Lett.15, 3193–3195 (2009). [CrossRef]

**5. **F.X. Kärtner, *Few-Cycle Laser Pulse Generation and Its Applications* (New York: Springer, 2004).

**6. **A. Baltuška, M.S. Pshenichnikov, and D.A. Wiersma, “Amplitude and phase characterization of 4.5-fs pulses by frequency-resolved optical gating,” Opt. Lett.23, 1474–1476 (1998). [CrossRef]

**7. **K. Kieu, R. Jason Jones, and N. Peyghambarian, “Generation of few-cycle pulses from an amplified carbon nanotube mode-locked fiber laser system,” *IEEE Photon. Technol. Lett.*22, 1521–1523 (2010). [CrossRef]

**8. **A.A. Amorim, M.V. Tognetti, P. Oliveira, J.L. Silva, L.M. Bernardo, F.X. Kärtner, and H.M. Crespo, “Sub-two-cycle pulses by soliton selfcompression in highly nonlinear photonic crystal fibers,” Opt. Lett.34, 3851–3853 (2009). [CrossRef] [PubMed]

**9. **M.A. Foster, A.L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express13, 6848–6855 (2005). [CrossRef] [PubMed]

**10. **G. Krauss1, S. Lohss, T. Hanke, A. Sell, S. Eggert, R. Huber, and A. Leitenstorfer, “Synthesis of a single cycle of light with compact erbium-doped fibre technology,” Nat. Photon. **4**, 33–36 (2010). [CrossRef]

**11. **M. Yamashita, K. Yamane, and R. Morita, “Quasi-Automatic Phase-Control Technique for Chirp Compensation of Pulses With Over-One-Octave BandwidthGeneration of Few- to Mono-Cycle Optical Pulses,” IEEE J. Selec. Top. Quantum. Electron. **12**213–222 (2006). [CrossRef]

**12. **G.P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

**13. **A.V. Andrianov, E.A. Anashkina, S.V. Muraviov, and A.V. Kim, “All-fiber design of hybrid Er-doped laser/Yb-doped amplifier system for high power ultrashort pulse generation,” Opt. Lett.35, 3805–3807 (2010). [CrossRef] [PubMed]

**14. **G. Taft, A. Rundquist, M.M. Murnane, I.P. Christov, H.C. Kapteyn, K.W. DeLong, D.N. Fittinghoff, M.A. Krumbügel, J.N. Sweetser, and R. Trebino, “Measurement of 10-fs laser pulses,” IEEE J. Sel. Top. Quantum Electron.2, 575–585 (1996). [CrossRef]

**15. **T. Hori, N. Nishizawa, T. Goto, and M. Yoshida, “Experimental and numerical analysis of widely broadened supercontinuum generation in highly nonlinear dispersion-shifted fiber with a femtosecond pulse,” J. Opt. Soc. Am. B **21**, 1969–1980 (2004). [CrossRef]

**16. **F. Tauser, F. Adler, and A. Leitenstorfer, “Widely tunable sub-30-fs pulses from a compact erbium-doped fiber source,” Opt. Lett. **29**, 516–518 (2004). [CrossRef] [PubMed]

**17. **Guoqing Chang, Li-Jin Chen, and Franz X. Krtner, “Fiber-optic Cherenkov radiation in the few-cycle regime,” Opt. Express **19**, 6635–6647 (2011). [CrossRef] [PubMed]

**18. **T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. **78**, 3282–3285 (1997). [CrossRef]

**19. **A.V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901–203904 (2001). [CrossRef] [PubMed]

**20. **R.W. Helwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl_{4},” Phys. Rev. A4, 2342–2347 (1971). [CrossRef]

**21. **R.W. Helwarth, J. Cherlow, and T.T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B11, 964–967 (1975). [CrossRef]

**22. **K.J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron.25, 2665–2673 (1989). [CrossRef]

**23. **S.A. Skobelev, D.V. Kartashov, and A.V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett.99, 203902–203905 (2007). [CrossRef]

**24. **A.V. Kim, S.A. Skobelev, D. Anderson, T. Hansson, and M. Lisak, “Extreme nonlinear optics in a Kerr medium: Exact soliton solutions for a few cycles,” *Phys. Rev. A*77, 043823 (2008). [CrossRef]

**25. **K. Kieu, R.J. Jones, and N. Peyghambarian, “High power femtosecond source near 1 micron based on an all-fiber Er-doped mode-locked laser,” Opt. Express18, 21350–21355 (2010). [CrossRef] [PubMed]