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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 12, NO. 2, MARCH/APRIL 2006

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Femtosecond Optical Parametric Amplification With Dispersion Precompensation Peng Yuan, Liejia Qian, Hang Luo, Heyuan Zhu, and Shuangchun Wen

Abstract—In this paper, we study a midinfrared femtosecond optical parametric amplifier (OPA) that is severely affected by group velocity dispersion (GVD). Both theoretical and experimental results show that GVDs in nonlinear crystals will significantly degrade the performance of a femtosecond OPA. By introducing a prechirp to the pump pulse, the effect of GVD can be effectively compensated. A lithium-niobate-crystal-based femtosecond OPA demonstrates that the conversion efficiency with optimally prechirped pumping is nearly twice that of the nonchirp case, and the output pulses can be further compressed to nearly their Fourier-transform limit by prism pairs. Index Terms—Femtosecond pulses, group-velocity dispersion (GVD), optical parametric amplifier (OPA).

I. INTRODUCTION VER the last decade, there have been spectacular developments in ultrafast laser technology due to the introduction of broadband solid-state laser materials, and to Kerr lens mode-locking and amplification techniques. These advances, along with the new nonlinear optical crystals, have made optical parametric amplification (OPA) a routine technique for generating tunable femtosecond pulses covering visible through midinfrared spectral ranges. [1] Basically, femtosecond OPAs were developed along with the experience of long-pulse pumped OPAs, by imposing special requirements on nonlinear interactions of femtosecond pulses; i.e., the spectral acceptances of the crystal need to be sufficiently large compared with the pump pulse bandwidth. Equivalently, in the temporal domain, the overall group-velocity mismatches (GVMs) among the three waves in the crystal need to be small compared with the pump pulse duration. Thus, knowledge of long-pulse OPAs can be directly applied to the prediction of performance of femtosecond OPAs in the case when GVMs are negligible [2]. One crucial emphasis in a femtosecond OPA design is to search for an appropriate crystal with both a large effective nonlinear coefficient and small GVMs in the spectral range of interest. To date, β-barium borate (BBO) is a commonly used

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Manuscript received July 14, 2005; revised January 24, 2006. This work was supported in part by Science and Technology Commission of Shanghai Grant 05SG02 and Grant 05JC14005 and in part by Natural Science Foundation of China Grant 60538010 and Grant 10376009. P. Yuan, L. Qian, H. Luo, and H. Zhu are with the State Key Laboratory for Advanced Photonic Materials and Devices, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). S. Wen was with the State Key Laboratory for Advanced Photonic Materials and Devices, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China. He is now with the School of Computer and Communications, Hunan University, Changsha 410082, China (e-mail: [email protected]). Digital Object Identifier 10.1109/JSTQE.2006.872723

crystal for visible and near infrared OPAs [3]–[6]. Pulse durations of both the signal and idler are usually comparable to that of the pump pulses. By resorting to the advantage of perfect group-velocity matching in a noncollinear phase-matching configuration, BBO-based noncollinear OPA is one of the effective approaches to the generation of few-cycle optical pulses. [5]–[9] More recently, stabilizations of the carrier-envelope offset phase (CEP) in OPAs were demonstrated, which revealed the unique features of CEP in OPAs [10]–[12], Midinfrared (MIR) femtosecond OPAs, on the other hand, rely mainly on nonlinear crystals like LiNbO3 (LN), KTiOPO4 (KTP), and their isomorphs (KNB, RTA, etc.), and either seeded OPA configuration or difference-frequency generation was adopted to improve the efficiency. [13]–[15] These MIR OPA crystals show a common property that GVMs are much smaller than that of BBO in the visible and near infrared and, hence, the crystal lengths can be designed to be longer. Unlike a BBO-based OPA, groupvelocity dispersion (GVD) in a MIR OPA would be as important as GVM, and affects the performance of the device considerably. Consequently, the design criteria of a MIR femtosecond OPA should be somewhat different than the conventional OPAs. For example, the typical length of BBO crystal is only 1 mm (noncollinear) to 7 mm (collinear), limited by GVMs, and thus GVD is negligible in many common situations with a ∼100-fs pump pulse, while crystal lengths of ∼2–4 cm are generally adopted in MIR OPAs. Surprisingly, the fundamental issues concerning the impact of GVD on femtosecond OPAs have not been clearly discussed in the literature. In this paper, we study MIR OPA where GVD plays a significant role. We show that GVDs in crystals will degrade the performance of OPA. It is demonstrated both numerically and experimentally that introducing an appropriate prechirp to the pump pulse may provide an effective means to compensate the effect of GVD, and to achieve high conversion efficiency. The results obtained in this paper can be a useful guideline in designing and optimizing a femtosecond OPA in MIR. In Section II, we present a detailed theoretical study of femtosecond OPA with GVDs, which focuses on the optimization of OPA efficiency by dispersion precompensation. The experimental results of a lithium-niobate-crystal-based femtosecond OPA are presented in Section III, which demonstrate that prechirp of the pump pulse is a new degree of freedom for optimizing a femtosecond OPA in MIR. Finally, conclusions are given in Section IV. II. NUMERICAL SIMULATIONS OPA with femtosecond pulses can be treated using the nonlinear-coupled wave equations in the time domain.

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Fig. 1. Small-signal gain as functions of FTL position when GVM is large (solid square, L sp = L ip = 0.35 L) and small (solid triangle, L sp = L ip = 1.4 L). GVD effect is considered only at the pump (L 2p = 1.24 L). The nonlinear length L NL = 0.3 L, L is crystal length.

Third-order nonlinearity has only a minor influence on OPA performance, as shown in previous works, and thus is neglected here [2], [16]. To the first order in quadratic susceptibility χ(2) and neglecting all derivatives of the linear refractive index beyond the second order, the equations that govern the evolution of pump, signal, and idler pulses, respectively, are ∂As (z, t) LNL ∂As (z, t) LNL ∂ 2 As (z, t) + −i ∂z Lsp ∂t L2s ∂t2 = −i

λp Ap (z, t)A∗i (z, t)e−i∆k z λs

LNL ∂ 2 Ai (z, t) ∂Ai (z, t) LNL ∂Ai (z, t) + −i ∂z Lip ∂t L2i ∂t2 = −i

λp Ap (z, t)A∗s (z, t)e−i∆k z λi

LNL ∂ 2 Ap (z, t) ∂Ap (z, t) −i ∂z L2p ∂t2 = −iAs (z, t)Ai (z, t)ei∆k z where Aj (z, t) is the field envelope normalized to input pump field A0 , and j = p, s, i stand for the pump, signal, and idler waves, respectively. The time variable t is in the unit of the input pump pulse duration τ , and ∆k is the wave-vector mismatch among the three waves. We define the nonlinear length as LNL = nλp /(πχ(2) A0 ), which is a measure of the pump intensity. The dispersion (L2j ) and GVM (Ljk ) lengths are defined as usual. A Gaussian profile is assumed for the input pump field, Ap (0, t) = A0 exp(−t2 ). The nonlinear-coupled wave equations are solved numerically by the standard split-step approach. We first address the impact of GVD in crystal on femtosecond OPA (Fig. 1). The parameters used in our simulations correspond to LiNbO3 with λp = 800 nm, λs = 1054 nm, and λi = 3320 nm, and the GVDs at the pump, signal, and idler wavelengths are 295 fs2 /mm, 395 fs2 /mm, and −800 fs2 /mm, respectively. The transverse coordinate in Fig. 1 (also in Fig. 2) indicates

the position where the pump pulse reaches its Fourier transform limit (FTL). This FTL position can be experimentally adjusted inside the crystal by introducing a variable prechirp onto the pump pulse before reaching the OPA. Thus, the FTL position can be regarded as a measure of the amount of prechirp imposed on the pump pulse. For comparison, the intensity of the pump pulse at its corresponding FTL position for each case was kept identical, and only GVD at the pump wavelength was considered here. As expected, GVD in the crystal has little influence on OPA in the presence of large GVM (GVM length is much shorter than the crystal length), and the gain is almost independent of the prechirp of the pump pulse. In the case with negligible GVM, however, GVD will play an important role and the effect of pump pulse prechirping is prominent. Obviously, a Fourier-transformlimited pump pulse is not the right choice for the highest gain in this case. The results shown in Fig. 1 suggest that dispersion precompensation of femtosecond MIR OPA is necessary and can be accomplished by introducing a prechirp to the pump pulse. Detailed simulation results are summarized in Fig. 2. In the regime of small-signal amplification (Fig. 2(a)), the OPA gain can be maximized by prechirp of the pump pulse, which means that GVD at the pump in the crystal can be effectively compensated by a prechirp. In the case that GVMs are negligible and only GVD at the pump wavelength is considered, the simulation results are particularly interesting: the highest small-signal gain occurs when the FTL position is located exactly at the middle of the crystal. Either GVMs or GVDs at the signal and idler wavelengths will alter the FTL position needed for the maximized gain and reduce the gain as well, which eventually renders the technique of dispersion precompensation invalid if GVMs or GVDs at the signal and idler are sufficiently large. Noting the analogy between space and time, these simulation results reveal similar behaviors as that of second-harmonic generation with focused beams. [17] In the regime of saturated amplification (Fig. 2(b)), the qualitatively same features are obtained. However, in the case that GVD at the pump wavelength is dominant, the prechirp of the pump pulse for maximum conversion efficiency needs to be set so that its corresponding FTL position deviates from the center to the rear surface of the crystal. The required amount of the FTL position adjustment, or equivalently, the prechirp of the pump pulse, depends on the pump intensity. This phenomenon can be attributed to the effect of pump depletion. The pump pulse energy decreases slowly along the crystal due to pump depletion. Overall intensity inside the whole crystal will be highest by shifting the FTL position close to the end surface, and thus the conversion efficiency will be maximized in this situation. It is important to study the characteristics of the output pulses from the dispersion precompensated OPA. For the sake of simplicity, only the signal pulse was characterized, and similar properties of the idler can be anticipated. Fig. 3 shows the calculated signal pulses and their corresponding spectra for various conditions. It can be verified that the output pulse is nearly FTL as long as no significant GVD exists at the signal wavelength, which is also true even if there is a certain degree of GVM. With the presence of GVD at the signal,

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Fig. 2. (a) Small-signal gain (with A s (0) = 10−5 ), and (b) conversion efficiency (with A s (0) = 10−2 ) as functions of FTL position of pump pulse. The nonlinear length L NL = 0.3L. Solid square: L 2p = 1.24 L, L sp = L ip = ∞, L 2s = L 2i = ∞. Open circle: L 2p = 1.24 L, L sp = L ip = 0.7 L, L 2s = L 2i = ∞. Solid triangle: L sp = L ip = ∞, L 2p = 1.24 L, L 2s = 1.66 L and L 2i = −0.408 L.

Fig. 3. Computer simulation results of temporal (a) and spectral (b) characteristics of output signal pulse by pumping with optimally chirped pulse. The solid squares, open circles, and solid triangles correspond to the conditions given in Fig. 2 in sequence. The open triangle represents the compressed signal pulse of solid triangles.

however, the output pulse is chirped. While the pulse is relatively broader than without GVD, this chirped pulse can be compressed further to near its FTL by introducing an additional GVD with the same magnitude but the opposite sign of the crystal dispersion. In the dispersion precompensated OPA corresponding to the maximum conversion efficiency, the presence of GVMs will result in a small shift of the signal pulse spectrum and a reduced bandwidth as well. This phenomenon can be illustrated clearly by a comparison between the OPAs with FTL and optimally chirped pulses pumping, as shown in Fig. 4, which may be understood easily by viewing amplification of the signal pulse as a process of difference frequency generation between the pump and idler. A chirped pump pulse is spectrally decomposed in time domain, and also its shape will change with its propagation within the crystal due to the GVD effect. The overlap between the pump and idler pulses, hence, the frequency difference, varies along the crystal due to the GVM between them, as well as the shape change of the pump pulse in a complicated manner. As a result, a shifted signal spectrum with reduced bandwidth will be obtained. Meanwhile, the signal pulse

shape is gradually changed during its amplification along the crystal.

III. EXPERIMENTAL RESULTS The experimental setup is schematically shown in Fig. 5. The OPA configuration is basically similar to that adopted in [14] except that OPG instead of external seeding was applied. A commercial Ti:sapphire laser oscillator-regenerative amplifier (Spitfire, Spectra-Physics) with 500-µJ 50-fs output pulses at a 1-kHz repetition rate operating at 800 nm (with pulse-to-pulse stability better than 2%) was used as a pump source. Two 8 mmthick and 6 mm-thick uncoated MgO:LiNbO3 crystals (supplied by CASTECH), both cut at ϑ = 46.5◦ for type I o + o → e phase matching, were used in the OPG and OPA stages, respectively. The pump beam was divided by a beam splitter (M1) with a reflectance of ∼40%, and the two beams were telescoped down (T1, T2) to obtain the pump intensities of ∼70 GW/cm2 and ∼50 GW/cm2 for the two stages, respectively. As shown in Fig. 5, the OPG stage was double-pass in a collinear configuration,

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Fig. 4. Comparisons of temporal and spectral (inset) characteristics of the signal pulse between schemes of FTL (dashed line) and optimally chirped (solid line) pulses pumping.

Fig. 5. Schematic diagram of the experimental setup. M1, beam splitter; M2– M4, dichroic mirrors for 800 nm and 1 µm; M5–M8, silver mirrors; T1–T2, telescopes; C1–C2, lithium niobate crystals.

with the dichroic mirror (M4) reflecting the pump pulse while transmitting the signal pulse. Different from conventional OPAs, our dispersion precompensated femtosecond OPA introduces a prechirp to the pump pulse and final prism pairs (two in sequence) used for the signal pulse compression. The grating compressor in the regenerative Ti:sapphire amplifier provides a delicate means to create an appropriate prechirp on the pump pulse. Our OPA system has a capability of wavelength tuning. In order to demonstrate the idea of dispersion precompensation more explicitly, orientations of the crystals in the OPG/OPA were fixed in the experiments to generate signal at ∼1020 nm and the corresponding idler at ∼3.7 µm, where the gain of the OPA is maximum. The conversion efficiency from pump to signal was optimized by carefully adjusting the prechirp of the pump pulse. The measured duration of the pump pulse was used to estimate the prechirp. Fig. 6 shows the measured conversion efficiencies. The maximum efficiency obtainable with FTL pulse pumping was ∼5%. Imposing an anomalous dispersion to the pump pulse significantly improved the conversion efficiency,

Fig. 6. Conversion efficiency versus pump pulse energy. Solid square: FTL pulse pumping; solid circle: optimally chirped pulse pumping.

and ∼8% of the highest efficiency was obtained, which corresponds to ∼60% increase compared with the case of FTL pulse pumping. In the numerical simulations [Fig. 2(b)], up to ∼80% increase in conversion efficiency was predicted. However, the experimental conversion efficiencies were lower than the calculated results, since the transverse effects were neglected in the simulations. The duration of the pump pulse with the optimal prechirp was measured to be ∼120 fs against ∼50 fs for its corresponding FTL pulse (Fig. 7). The effective dispersion of ∼2000 fs2 was estimated based on the pulse duration measurements, assuming a linear chirp was imposed on the pump pulse. Such a chirp is comparable to the amount of overall GVD in the OPA system, but with opposite sign. These experimental results agree well with the numerical simulation results given in Section II. In addition, the system was stable with pulse-to-pulse fluctuation of only ±4%, and quite insensitive to the environment. Therefore, it has already offered practical uses in our laboratory. Only signal pulses were monitored in the experiment. In the case of FTL pulse pumping, the output signal wavelength was centered at 1020 nm with a bandwidth of 34 nm (inset, Fig. 8). The pulse duration was measured to be ∼250 fs (curve (a) in Fig. 8), and could be compressed to 120 fs by prism pairs (curve (b) in Fig. 8). By employing optimal prechirp, the duration of the output pulse was ∼230 fs (curve (a) in Fig. 9) directly from OPA, and was compressed further to ∼140 fs (curve (b) in Fig. 9). This results in a time-bandwidth product of 0.62, which corresponds to ∼1.4 times the Fourier transform limit assuming a Gaussian pulse shape. Remarkably, the signal wavelength was found to shift to 1045 nm with a reduced bandwidth of ∼16 nm by pumping with the optimally prechirped pulse, which is attributed to the combined effect of GVM and prechirp as discussed in Section II. Qualitatively, with optimally prechirped pulse pumping, both the numerical simulations and the experimental results give the same trend of wavelength shifting towards longer wavelength. However, the magnitude of the observed wavelength shift is larger than the calculated result (Fig. 4), which implies that the actual GVMs of the crystal might be larger than that used in those simulations.

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Fig. 7. Typical intensity autocorrelation traces of (a) FTL and (b) optimally chirped pump pulses. Corresponding spectrum, same in both cases, is given in inset of Fig. 7(a).

Fig. 8. Typical intensity autocorrelation traces of (a) output signal pulse and (b) compressed signal pulse in configuration of FTL pulse pumping. Corresponding spectrum is given in inset.

Fig. 9. Typical intensity autocorrelation traces of (a) output signal pulse and (b) compressed signal pulse in configuration of optimally chirped pulse pumping. Corresponding spectrum is given in inset.

IV. CONCLUSION Femtosecond OPA in MIR suffers severe effects from GVD as well as GVM, which is very different from the usual OPAs in the visible and near infrared, where GVM plays a major role while GVD effect is negligible. Both theoretical and experi-

mental results have shown that GVDs in nonlinear crystals will significantly degrade the conversion efficiency of a femtosecond OPA. We have proposed and demonstrated a dispersion precompensated OPA in which the effect of GVD at the pump in crystals can be offset to some extent by introducing an optimal

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prechirp to the pump pulse. The behaviors of dispersion controlled OPA have been studied in detail. Due to the combined effect of GVM and prechirp, the output pulse spectrum will be shifted, and pulse shape will change during the amplification along the crystal. A lithium niobate crystal-based femtosecond OPA has shown that the conversion efficiency by pumping with an optimally prechirped pulse is nearly twice that of FTL pulse pumping. Output pulses can be compressed to nearly their Fourier-transform limit by prism pairs. We have demonstrated that prechirp of the pump pulse is a new degree of freedom for optimizing femtosecond OPA in MIR. REFERENCES [1] G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum., vol. 74, pp. 1–18, 2003. [2] J. C. Wang and J. K. Wang, “Experimental and theoretical analysis of white-light seeded, collinear phase-matching, femtosecond optical parametric amplification,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 21, pp. 45–56, 2004. [3] G. M. Gale, F. Hache, and M. Cavallari, “Broad-bandwidth parametric amplification in the visible: Femtosecond experiments and simulations,” IEEE J. Sel. Topics Quantum Electron., vol. 4, no. 2, pp. 224–229, Mar.– Apr. 1998. [4] S. R. Greenfield and M. R. Wasielewski, “Near transform-limited visible and near-IR femtosecond pulses from optical parametric amplifications using type II β-barium borate,” Opt. Lett., vol. 20, pp. 1394–1396, 1995. [5] G. Cerolluo, M. Nisoli, and S. DeSilvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett., vol. 71, pp. 3616–3618, 1997. [6] A. Shirakawa and T. Kobayashi, “Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000 cm-1 bandwidth,” Appl. Phys. Lett., vol. 72, pp. 147–149, 1998. [7] G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82 MHz optical parametric oscillator,” Opt. Lett., vol. 20, pp. 1562–1564, 1995. [8] T. Kobayashi, A. Shirakawa, and T. Fuji, “Sub-5-fs transform-limited visible pulse source and its application to real-time spectroscopy,” IEEE J. Sel. Topics Quantum Electron., vol. 7, no. 4, pp. 525–538, Jul.–Sep. 2001. [9] A. Baltuska, T. Fuji, and T. Kobayashi, “Visible pulse compression to 4 fs by optical parametric amplification and programmable dispersion control,” Opt. Lett., vol. 27, pp. 306–308, 2002. [10] C. Manzoni, G. Cerullo, and S. DeSilvestri, “Ultrabroadband self-phasestabilized pulses by difference-frequency generation,” Opt. Lett., vol. 29, pp. 2668–2670, 2004. [11] R. Th. Zinkstok, S. Witte, W. Hogervorst, and K. S. Eikema, “High-power parametric amplification of 11.8-fs laser pulses with carrier-envelope phase control,” Opt. Lett., vol. 30, pp. 78–80, 2005. [12] A. Baltuska, T. Fuji, and T. Kobayashi, “Controlling the carrier-envelope phase of ultrashort light pulses with optical parametric amplifiers,” Phys. Rev. Lett., vol. 88, p. 133901, 2002. [13] F. Rotermund, V. Petrov, F. Noack, M. Wittmann, and G. Korn, “Laser-diode-seeded operation of a femtosecond optical parametric amplifier with MgO:LiNbO3 and generation of 5-cycle pulses near 3 mm,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 16, pp. 1539–1545, 1999. [14] L. J. Qian, H. Y. Zhu, H. Luo, P. Yuan, T. Wang, G. Xu, and X. Q. Fu, “Quadratic nonlinear technologies for femtosecond lasers,” J. Korean Phys. Soc., vol. 46, pp. S233–S241, 2005. [15] G. M. Gale, G. Gallot, F. Hache, and R. Sander, “Generation of intense highly coherent femtosecond pulses in the mid infrared,” Opt. Lett., vol. 22, pp. 1253–1255, 1997. [16] G. M. Gale, M. Cavallari, and F. Hache, “Femtosecond visible optical parametric oscillator,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 15, pp. 702– 714, 1998. [17] G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys., vol. 39, pp. 3597–3639, 1968.

Peng Yuan was born in Shandong, China, on October 22, 1981. He received the B.S. degree in physics from Shandong University, Jinan, China, in 1999. He is currently working toward the Ph.D. degree at the State Key Laboratory for Advanced Photonic Materials and Devices, Department of Optical Science and Engineering, Fudan University, Shanghai, China. His research interests include nonlinear optics, ultrafast optics, and high-power lasers.

Liejia Qian was born in 1965. He received the Ph.D. degree in optics from Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China, in 1989. From 1991 to 1993, he was a Postdoctoral Researcher at the Politecnico de Milano, Milan, Italy, researching high-power solid-state lasers and intracavity frequency doubling. In 1993, he joined the Shanghai Institute of Optics and Fine Mechanics, Shanghai, China, as a Senior Researcher. He spent the academic years of 1997–1999 on leave at the Department of Applied Physics, Cornell University, Ithaca, NY. Since 2001, he has been a Professor in the Department of Optical Science and Engineering, Fudan University, Shanghai, China. He has published over 100 papers in international journals and conferences. His research interests include ultrafast science and technology, and quadratic nonlinear optics.

Hang Luo was born in China in 1981. He received the B.S. degree in physics from Fudan University, Shanghai, China, in 2003, where he is currently working toward the Ph.D. degree in optics. His research interests include nonlinear optics, ultrafast optics, and high-power lasers.

Heyuan Zhu received the B.S., M.S. and Ph.D. degrees in physics from Fudan University, Shanghai, China, in 1988, 1991 and 1994, respectively. He joined the Laboratory of Laser Physics and Optics at Fudan University in 1994. From 1996 to 1998 and in 2002, he was a Visiting Scholar at the University of California at Berkeley and Cornell University, Ithaca, NY, respectively. He joined the Department of Optical Science and Engineering at Fudan University in 2000 and is currently a Professor. His current research interests include the generation, to amplification, and diagnosis of ultrashort optical pulses as well as ultrafast phenomena. He has authored or coauthored over 50 journal papers.

Shuangchun Wen was born in 1966. He received the Ph.D. degree in optics from Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China, in 2002. He is currently a Professor at Hunan University, Changsha, China. He has published over 30 papers in international indexed journals and conferences. His current research involves optical communication, nonlinear optics, ultrafast phenomena, etc.