Ultrafast waveform synthesis and characterization using coherent Raman sidebands in a reflection scheme

Kai Wang; Miaochan Zhi; Xia Hua; Alexei V. Sokolov

doi:10.1364/OE.22.021411

1. Introduction

Optical pulses with total duration equal to or less than one cycle are essential for ultrafast time-resolved spectroscopy and attosecond science. Currently, several types of sources are available for optical pulses of a few-femtosecond or sub-femtosecond duration [1]. High harmonic generation (HHG) is one of the most successful techniques in producing attosecond pulses in the extreme ultraviolet or the soft X-ray region, but not in the optical region [2]. The molecular modulation technique, which relies on broadband coherent Raman scattering to provide the required optical bandwidth, has been developed to produce such ultrashort optical pulses since the 1990s [1]. This technique complements HHG and may lead to the generation of the attosecond pulses in the optical range.

Experiments in molecular modulation have shown potential in producing ultrafast optical pulses. Through adiabatic excitation of the D₂ molecular transitions with nanosecond pulses, the synthesis of a pulse train with nearly single-cycle waveforms was demonstrated [3, 4]. Utilizing an impulsive excitation technique, Zhavoronkov and Korn have generated pulses with a period of 3.8 fs in a hollow fiber filled with SF₆ [5]. Coherence properties of a multioctave stimulated Raman frequency comb produced in a hydrogen-filled hollow-core photonic crystal fiber were studied by Wang et al. [6]. Yavuz et al. extended the molecular modulation technique using continuous-wave lasers in a gas-filled cavity for precision spectroscopy applications [7]. An octave-spanning Raman comb with carrier-envelope offset control was achieved by Suzuki et al. in gaseous parahydrogen [8]. Remarkably, an “Optical Function Generator” which will enable arbitrary waveform shaping was accomplished by using Raman sidebands with commensurate frequencies in Kung’s group [9]. Worthy of mention, there is another similar technique based on cascaded four wave mixing (CFWM) processes, which showed the potential for the generation of ultrafast optical pulses, especially in the UV range [10–12].

Besides exciting Raman transitions in a gaseous medium, researchers have extended molecular modulation technique to solids driven by femtosecond laser pulses [13,14]. Different from collinear sideband generation in gas due to the dispersion of the solid state, the sidebands of coherent Raman scattering are generated at different output angles. Different schemes have been implemented to synthesize ultrafast waveforms using these Raman sidebands. For example, with the assistance of a prism and pulse shapers based on an acousto-optic programmable dispersive filter (AOPDF), coherent Raman sidebands generated in diamond have been combined and 2-cycle optical pulses [15] were obtained. Moreover, the scheme in Ref. [15] also shows the potential to generate sub-cycle optical vortices [16, 17].

In addition to the generation of a broadband spectrum, the characterization of ultrafast waveforms synthesized using Raman sidebands is also important for ultrafast optical science. Both Matsubara et al. [14] and Pandiri et al. [18] used the spectral phase interferometry for direct electrical field reconstruction (SPIDER) technique to measure pulses in their work. Goda et al. [19] and Shverdin et al. [20] characterized their waveforms using a technique based on four wave mixing. In addition, the relative spectral phase can be characterized by the sum frequency generation among different sidebands [16, 21].

In the scheme using AOPDF pulse shapers [15], the energy of the waveform is limited by the damage threshold of the acousto-optic birefringent crystal. Meanwhile, the bandwidth of a single AOPDF is usually smaller than an octave. It would require several AOPDFs for subcycle optical pulse synthesis and it would be difficult to implement such an experiment in the lab. In this paper, we propose a novel reflection scheme for the characterization and synthesis of waveforms using coherent Raman sidebands produced from Raman active crystals. In this experiment, spherical mirrors were used to reflect and refocus the generated beams back to the crystal. The sidebands interact with each other on the crystal and an interferogram is created based on the Raman interaction. The relative spectral phases of the sidebands could be retrieved from the interferogram using theoretical simulations. Furthermore, in order to synthesize the ultrafast waveform, we inserted a pulse shaper based on a micro-electro-mechanical systems (MEMS) deformable mirror (DM) to adjust the relative spectral phases. Our scheme experimentally demonstrated the capability of synthesizing an ultrafast waveform using coherent Raman sidebands.

2. Ultrafast waveform synthesis and characterization in a reflection scheme

The experimental setup is shown in Fig. 1. It consisted of a Ti:Sapphire amplifier which generated 40 fs pulses at a 1 kHz repetition rate and a central wavelength of 806 nm. The beam was divided into two by a beam splitter [60:40 (R:T)]. About 40% of the beam was used as a pump beam and the rest was used to pump an optical parametric amplifier (OPA). The second harmonic of the idler beam generated by the OPA with a central wavelength of 900 nm was then used as the Stokes beam (following coherent anti-Stokes Raman spectroscopy (CARS) convention, we denote the shorter 806 nm wavelength beam as the pump beam and the longer wavelength 900 nm as the Stokes beam). The power of the pump was around 10 mW and the power of Stokes was around 2 mW. After focusing using a 50 cm lens, the beam intensity of the pump was around 2 × 10¹¹W/cm², which was similar to that in Ref. [22]. The pump and Stokes beams were crossed in a 1-mm thick synthetic single-crystal diamond at an angle of 3.7 degrees to obtain coherent Raman sideband generation. The properties of the coherent Raman sidebands generated in diamond have been studied by Zhi et al. [22].

Fig. 1 Schematic of the experimental setup. Two concave spherical mirrors are used to reflect beams back to the crystal. To avoid the interplay of the beams between the incident path and the reflection path, the reflection spots were about 500 μm offset from the incident spot. The Raman sidebands in the second path are collected with another spherical mirror and recorded by the spectrometer to obtain the interferogram.

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We optimized the generation to suppress unwanted nonlinearities. The sidebands up to AS9 have been generated with a lower input energy after optimization. We then used 2-inch concave spherical mirrors with focal lengths of 10 cm to reflect the beams back to the crystal. We used two spherical mirrors. One mirror reflected the pump and Stokes beams and the second mirror reflected sidebands AS3, AS4, AS5, AS6, and AS7 (a single 2-inch mirror could reflect up-to five sidebands due to the aperture). The spherical mirrors were about twice the focal length away from the crystal. In this 2f-2f configuration, the sidebands reflected by one mirror were refocused to the same spot on the crystal. When we overlapped the two spots projected by two spherical mirrors spatially, the beams interacted with each other on the crystal with phase matching angles [23].

The second mirror that reflected Raman sidebands was put on a translation stage, which was motorized with a step size of 0.2 μm corresponding to a 1.2 fs time delay. We recorded the spectrum of AS3–AS7 as we scanned the time delay. This produced an interferogram with the intensity I(ω) of Raman sidebands as a function of time delay, as displayed in Fig. 2. The interferogram is equivalent to a cross-correlation frequency-resolved optical gating (XFROG) trace based on the Raman nonlinear interaction. When only one sideband (for example, AS3) is reflected from the second mirror and interacts with the pump and Stokes beams from the first mirror, Raman scattering occurs with a frequency shift equal to the Raman shift. If two sidebands, for example, AS3 and AS4, are reflected back to the crystal from the second mirror, the spectral intensities of AS3 and AS4 oscillate as a function of the time delay (Fig. 2(a)). The frequency of the oscillation is nearly equal to the central frequency difference of the two sidebands. The frequencies of the intensity oscillations of AS3 and AS4 are characterized through a Fourier transformation of the time delay scanning (Fig. 2(b)). When there are multiple sidebands, for example, three sidebands, AS3, AS4, and AS5, reflected back to the crystal from the second mirror, the spectral intensities of AS3, AS4, and AS5 oscillate at slightly different frequencies (Fig. 2(c)). If non-adjacent sidebands, for example, AS3 and AS5, are reflected back to the crystal, a new beam which has the same peak frequency as AS4 is generated with an intensity oscillation whose frequency is equal to the peak frequency difference of AS3 and AS5 (Fig. 2(d)).

Fig. 2 Spectrograms of Raman sidebands. (a) The spectrogram of AS3 and AS4. (b) The Fourier transform for the delay time scanning at different wavelengths of the spectrogram produced by AS3 and AS4. The blue(red) line is the average result of the Fourier transform of AS3(AS4) at different wavelengths. (c) The spectrogram of AS3, AS4, and AS5. (d) The spectrogram of AS3 and AS5. All the spectrograms have the same delay (zero).

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From Fig. 2, we also note that the central frequencies of the sidebands are shifted due to the cross phase modulation. Moreover, the oscillation period on one sideband changes slightly during the scanning (this can be seen by measuring the distance between adjacent intensity extrema). This is the result of several factors, such as the aberration of the concave spherical mirror, the chirp of the beam, etc..

This interferogram can be explained using the interaction between the pulse train and the coherent vibrational states driven by the pump and Stokes pulses. The pulse duration of a single sideband in our experiment is around 100 fs [24], which is much longer than the period of the coherence between the vibrational states (24 fs). When a single pulse interacts with a coherent state, it only produces Raman scattering (together with other nonlinearities) with a depletion of the energy on the sideband itself. However, when the multi-mutual coherent sidebands are reflected back and combined on the crystal, a pulse-train is obtained and the duration of a sub-pulse is smaller than the period of coherence. In this case, the interactions between the pulse and coherent states depend on the phases of the states. If the pulse train interacts with the phased state (ρ_ab > 0), the sideband with the longer wavelength is enhanced and the sideband with the shorter wavelength is reduced. When the pulse interacts with the antiphased state (ρ_ab < 0), the sideband with the shorter wavelength is enhanced and the longer wavelength sideband is reduced [25, 26]. During the phase scanning, the pulse train interacts with the phased state and the antiphased state alternatively and thereby, the intensity I(ω) oscillates [26].

In principle, the spectral phases could be retrieved from this interferogram based on Raman interaction. However, this is different from conventional XFROG trace retrieval [27]. In conventional frequency-resolved optical gating (FROG) or XFROG, the gated signal is distinguished from the source either in frequency or polarization. The amplitude of the signal is proportional to the amplitude of the source. Nevertheless, in the Raman interferogram, the signal is mixed with the original beam. As a result, in the theoretical model, the depletion of the beams should be included. Therefore, the conventional FROG retrieval algorithm does not directly work for the Raman interferogram and we use a different method to retrieve the spectral phase.

The interferogram based on the Raman interaction can be simulated with the propagation equations for the light in a Raman active medium. If the spectrum of one sideband is infinitely narrow, the analytical solution for the propagation of light in a Raman active medium can be obtained theoretically [28]. However, our experiments were done using femtosecond pulses, which requires extending the work of Ref. [28] to the broadband scenario.

The theory of coherent Raman scattering in Ref. [25, 28] is for a single frequency and the effective Hamiltonian has the following form :

\begin{array}{r} H_{eff} = \frac{\bar{h}}{2} [\begin{array}{l} Σ_{q} a_{q} {| E_{q} |}^{2} & Σ_{q} b_{q} E_{q} E_{q - 1}^{*} \\ Σ_{q} b_{q + 1}^{*} E_{q} E_{q + 1}^{*} & Σ_{q} d_{q} {| E_{q} |}^{2} - 2 Δ ω \end{array}] \\ = - \frac{\bar{h}}{2} [\begin{matrix} A & B \\ B^{*} & D - 2 Δ ω \end{matrix}] . \end{array}

We modify this equation for the broadband spectrum. We adopt the notation from Silberberg et al. [29] and write the nonlinear polarization as

P^{3} = P_{N R}^{3} (ω, τ) + P_{R}^{3} (ω, τ) = \int_{0}^{+ \infty} (χ_{N R}^{3} + χ_{R}^{3} (Ω)) E_{3} (ω - Ω, τ) S_{ps} (Ω) d Ω,

where

S_{ps} (Ω) \equiv \int_{0}^{+ \infty} E_{p} (ω^{'} + Ω) E_{s}^{*} (ω^{'}) d ω^{'},

here S_ps represents the coherence driven by the pump E_p and Stokes beam E_s,

P_{N R}^{3} (ω, τ)

and

P_{R}^{3} (ω, τ)

are the nonresonant and resonant polarization respectively, and

χ_{N R}^{3}

and

χ_{R}^{3} (Ω)

are the nonresonant and resonant nonlinear susceptibilities respectively. S_ps is replaced with ρ_ab(Ω) in the following equations. E₃ is the probe beam, and in the cascaded Raman process, it is E_q.

We take all sidebands as being far from the resonance as described in Ref. [3], assume a_q = b_q = a₀ and b₀ = b_q, and define the propagation constant β_q(ω) = ηh̄ω_qNa₀ for different wavelengths. Describing the interaction using a convolution of the broadband spectrum, the propagation equations can be written as

\frac{\partial E_{q} (ω, z)}{\partial z} = - j \frac{b_{0}}{a_{0}} β_{q} (\int_{0}^{+ \infty} χ (Ω) ρ_{a b} (Ω) E_{q - 1} (Ω - ω) d Ω + \int_{0}^{+ \infty} χ^{*} (Ω) ρ_{a b}^{*} (Ω) E_{q + 1} (Ω + ω) d Ω) .

Where

ρ_{a b} = \frac{1}{2} sin (θ) exp (j φ)

and B = |B|exp(jφ),

tan θ = \frac{2 | B |}{2 Δ ω - D + A}

. The line profile is described by

χ = \frac{α}{Ω - (ω_{p} - ω_{s}) - i γ}

. is a constant related to the spontaneous Raman cross-section. ω_s and ω_p are the angular frequencies of the pump and Stokes beams. γ is the half-width of Raman line. We set the boundary conditions following the experimental conditions. For example, in the experiment of Fig. 2, the boundary conditions are as follows: all the sidebands give zero contribution at z = 0, except E₀, E₋₁, E₃, E₄, E₅, E₆, and E₇. E_q = A_q * exp(j(φ_q)) and I_q = |A_q|². Here, A_q is the amplitude. At z = 0, E₀, E₋₁, E₃, E₄, E₅, E₆, and E₇ can be obtained from the measurements of the spectrometer. φ_q is the parameters that we need to retrieve using the simulation. We solve these partial differential equations numerically using the Runga-Kutta method.

The phase distortion of the coherent Raman sidebands can be divided into two parts, “global” phase distortion, which corresponds to the relative spectral phases among different Raman sidebands, and “local” phase distortion, which is the phase distortion of a single sideband induced by the dispersion [15]. In order to compare experimental results with the theoretical results, we normalize and smoothen the experimental results at first to eliminate the fluctuation and the noise of the experimental measurement. In our theoretical simulation, we minimize the difference between the theoretical and experimental [(I(ω, τ)_theory − I(ω, τ)_experiment)/I(ω, τ)_theory] with the “global” phase and “local” phase as the variables. We assume the time delay between different sidebands accounts for the “global” phase and the phase distortion accumulated during the propagation in the medium as the reason for the “local” phase distortion. The simulation results for the spectrogram of AS3–AS7 are shown in Fig. 3. The difference between the experimental result (Fig. 3(a)) and the theoretical simulation (Fig. 3(b)), [(I(ω, τ)_theory − I(ω, τ)_experiment)/I(ω, τ)_theory], is around 0.1 on average. This error will affect the reliability of our retrieval result (for example, if we use this technique to measure the time duration of the pulse, the result will have an error of of around 10%). The spectral phases over the spectrum for our simulation are shown in the inset of Fig. 3(c). The phases and periods of the intensity oscillations from the experiment (Fig. 3(a)) and the theoretical simulation (Fig. 3(b)) are in good agreement with each other but there is still some local difference. Generally, the phase distortion is due to dispersion in the crystal, cross-phase modulation, and some other parasitic nonlinear effects [13]. In our pulse retrieval algorithm, the spectral phases over the spectrum are the fitting parameters. During the simulation, we find that the best fit is achieved when the phase distortion for each individual sideband is equivalent to that produced by dispersion in a 200 μm thick crystal (even though in the experiment the crystal is 1 mm thick). Another way to look at this is to say that our spectral phases, which are shown in the inset of Fig. 3(c), can be understood as already containing the contributions from dispersion, cross-phase modulation, possibly mirror aberrations and other parasitic effects, without specifying them separately. Our result allows retrieving the pulse combined from the coherent Raman sidebands. Using the theoretical result, we calculated the shape of the pulse produced by the sidebands at the focal point of the spherical mirror in the time domain (Fig. 3(c)) and the electric field of the ultrafast waveform (Fig. 3(d)). These results have shown that, using the Raman interaction, it is possible to retrieve the relative spectral phases.

Fig. 3 The experimental interferogram of AS3–AS7 (a) and the theoretical interferogram (b). The theoretical simulation is in agreement with the experimental results within a 10% average error. In (c), we show the ultrafast waveform retrieved from the experimental interferogram. The inset is the spectral phase over 400–700nm for the simulation. In (d), we show the electric field of the ultrafast waveform calculated according to the theoretical result.

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3. Ultrafast waveform synthesis using a deformable mirror

The ultrafast waveform we produced, as described in the last section, was not a Fourier-transform limited waveform. In order to synthesize a Fourier-transform limited ultrafast waveform, we will need to adjust the relative spectral phases of the Raman sidebands. In principle, this can be done with the assistance of pulse shapers, for example, the pulse shaper based on AOPDFs [15] or a DM [30]. The scheme in Ref. [15] has been demonstrated as being able to synthesize and characterize ultrafast waveforms using the Raman sidebands generated in diamond. Nevertheless, due to the damage threshold of the pulse shapers (AOPDF), a high energy, ultrafast waveform may not be produced in that scheme.

In order to synthesize an ultrafast waveform using coherent Raman sidebands, we propose to use a pulse shaper based on the MEMS DM to assist the phase adjustment. The DM (1X140 Linear Array-DM 2.0 μm, Boston Micromachines Corporation) is a phase sensitive device whose surface could be deflected by the actuator. Due to the influence of neighboring actuators, the effective maximum width of the incident beam onto the continuous facesheet device we used is 600 μm [31], which limits the beam width of the sideband on the DM. We experimentally demonstrate that with the assistance of a DM, the synthesis of an ultrafast waveform using coherent Raman sidebands generated in a solid medium is possible.

In this experiment, we obtained coherent Raman sidebands from the Raman active crystal PbWO₄. The Stokes beam was centered at 870 nm for the excitation of the Raman mode at 901 cm⁻¹ of PbWO₄. The pump and Stokes beams were crossed on the crystal at an angle of 3.7 degrees, and when the intensities were of the same order as described in the previous section, sidebands were generated. Before using the spherical mirror to reflect the beams back to the crystal, we inserted a DM into the region where the beams were still separate, as shown in Fig. 4. In this proof-of-principle experiment, the second mirror reflected three Raman sidebands, AS7, AS8, and AS9, and the first mirror reflected the pump and the Stokes beams. We used the DM to adjust the relative spectral phase of AS8 to generate different ultrafast waveforms. First, the DM applied a zero phase on AS8 and we produced an interferogram (Fig. 5(a)), which is used to retrieve the ultrafast waveforms. Next, we used the DM to apply a different spectral phase on AS8 which was the right amount to shift the intensity oscillation on AS8 by π in the interferogram. To show this π shift more clearly, in Fig. 5(c), we show the intensities at 550 nm wavelength (AS8) in these two situations. A π shift of the oscillation could be seen from this cross-section of the interferogram. Using the theoretical simulation, we retrieved the two ultrafast waveforms produced by the three coherent Raman sidebands (Figs. 5(e) and (f)). It is obvious that the different spectral phases of AS8 result in two different ultrafast waveforms in the experiments. Fig. 5(d) shows how the intensity oscillations vary at 550 nm as we adjust the spectral phase of AS8. These results demonstrate that with the assistance of the DM, we are able to control the synthesized ultrafast waveform by using coherent Raman sidebands.

Fig. 4 Schematics of the experimental setup with DM (1X140 Linear Array-DM 2.0 μm, Boston Micromachines Corporation). The DM is inserted before the spherical mirror to adjust the spectral phase. The inset displays the schematics of the actuator electrode and DM [31]. The active mirror area is supported by an array of 5X140 electrostatic actuators. Each actuator can be controlled to drive the DM surface to the desired shape.

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Fig. 5 The synthesis of ultrafast waveforms with a DM. (a) The spectrogram of AS7, AS8, and AS9 by applying zero phase on AS8 using the DM. (b) The spectrogram of AS7, AS8, and AS9 by applying a phase on AS8 such that the intensity oscillations of AS8 are out of phase from (a). (c) The cross section at 550nm from the previous two spectrograms for two different AS8 phases. The blue(red) line is from the spectrogram (a)((b)). (d) The intensity oscillations at 550nm in the interferogram change when we adjust the phase of AS8 continuously. The y-axis is the displacement of the surface of the DM. The displacement is estimated according to the manual. (e) and (f) show the pulses retrieved from the theoretical simulation of (a) and (b) separately.

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4. Conclusion

In this paper, we described a reflection scheme using spherical mirrors to assist ultrafast waveform synthesis and characterization using coherent Raman sidebands generated in a Raman active crystal. In this reflection scheme, the generated Raman sidebands are reflected back to the Raman crystal. We characterized the relative spectral phases of the sidebands by using the nonlinear Raman interaction. Furthermore, in order to synthesize the ultrafast waveforms, we inserted a MEMS DM in the beam paths to modify their spectral phases. Using Raman sidebands generated in Raman active crystal (PbWO₄), we demonstrated the feasibility of synthesizing an ultrafast waveform using the coherent Raman sidebands with the assistance of a DM. Moreover, to extend this reflection scheme to the study of ultrafast physics and chemistry, the beams were focused on the sample non-collinearly, which is different from other schemes for ultrafast pulse generation. The essence is that the ultrafast waveform is only produced near the focal point of the spherical mirror. Though the ultrafast waveform only maintains the profile for a short distance, this is already sufficient for the studies of ultrafast nonlinear phenomena and high-field physics.

Acknowledgments

This work is supported by the National Science Foundation (grant No. PHY-1307153) and the Welch Foundation (grant No. A1547). We thank Alexandra Zhdanova for valuable help.

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Ultrafast waveform synthesis and characterization using coherent Raman sidebands in a reflection scheme

Abstract

1. Introduction

2. Ultrafast waveform synthesis and characterization in a reflection scheme

3. Ultrafast waveform synthesis using a deformable mirror

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (4)

Optics Express