Pulse shaping via Backward Second Harmonic Generation

: We numerically demonstrate pulse shaping by backward frequency doubling of femtosecond laser pulses in engineered quasi-phase-matched waveguides. We employ optimal control theory to access the regime of pump depletion. poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun. 268, 110-114 (2006).


Introduction
In a variety of applications, the precise synthesis and manipulation of the profile of optical pulses, i.e. optical pulse shaping, is required. Among others we like to mention spectroscopy, biological imaging, metrology and industrial tests [1][2] The demand for controlled temporal profiles equally applies to all timescales, from femtosecond to pico-and nano-second laser excitations and in various spectral ranges. Shapers should ideally provide high-quality pulses with the desired profiles and (central) wavelength, similar to electronic waveform generators but in distinct time and frequency scales.
To date, conventional picosecond-femtosecond pulse shaping has been implemented using linear effects, for example with free-space bulk optics including diffraction grating pairs and lenses [3]. The drawbacks associated with this approach relate to the need for high-quality optical elements of appreciable size, together with very strict alignment tolerances and the consequently limited integrability with waveguides. In such linear pulse shapers, the input spectrum is modulated (in amplitude and phase) in order to obtain the desired output profile in time. In most cases the input pulse is considered a given constraint and its bandwidth determines the maximum bandwidth of the output. Even though the latter drawback can be overcome using wide bandwidth inputs, this approach often limits the efficiency, inasmuch as desired waveforms of both narrow and large bandwidths can be required from the same input, e.g. in spectroscopy. For instance, the detection of a specific Raman transition might require the shaping of two laser pulses: the Raman pump field (e.g. at 750 nm) with a narrow bandwidth and ps duration; and the Raman probe with a broadband femtosecond pulse (e.g. from NIR to 750 nm) that stimulates the scattering of a particular vibrational mode [1]. Nowadays, since femtosecond laser sources are available in the wavelength range from 1.3 to 2.0 micrometers, an alternative approach to pulse synthesis in the visible or near-infrared consists in simultaneously frequency doubling and shaping a broadband fundamental frequency (FF) source.
Hereby we propose and numerically demonstrate a novel approach which, starting with a femtosecond near-infrared source, is able to produce and shape both the picosecond Raman pulse and a suitable broadband probe pulse at twice the input frequency. The method is based on Backward Second Harmonic Generation (BSHG) from the fundamental frequency source in an engineered Quasi-Phase-Matching (QPM) grating realized by periodically poling a ferroelectric lithium niobate waveguide.
In BSHG, since the fundamental and second harmonic (SH) waves propagate in opposite directions, the group velocity mismatch (GVM) plays an essential role in the process of second harmonic generation. In this context, GVM is usually considered a very detrimental effect for the conversion efficiency and, in contrast with the forward SHG case, does not stem from material properties but from the process geometry itself: FF and SH propagate in opposite directions and therefore, even in the absence of material dispersion, GVM is not zero but equal to the sum of the group velocities (V 1 and V 2 ) of the two waves coupled by the nonlinear process; hence, the temporal walk-off per unit length between FF and SH is δ = (1/V 2 + 1/V 1 ). Although GVM is indeed undesired when high doubling efficiencies are sought, in the framework of pulse shaping it can be conveniently exploited to yield and optimize the pulse shaping process. Shaping of second-harmonic pulses has been previously considered in the case of forward SHG [21][22][23][24][25][26] but never discussed with reference to a backward process, where a huge GVM may allow for shaping possibilities not accessible in forward configurations.
As we will demonstrate in the following, as a general rule of thumb one can think of a shaped output SH generated by a given FF according to two major constraints: (i) the duration T of the SH pulse is bounded by: T < δ L, being L the length of the structure: e.g. for a 1 mm long lithium niobate sample, the upper bound is of the order of 10 ps. This maximum duration determines the minimum bandwidth B SH of the shaped pulse. In femtosecond stimulated Raman spectroscopy, a small bandwidth is required for the Raman pump because it determines the instrument resolution; typical required values are of the order of 3 − 17 cm −1 , corresponding to transform limited pulses with a few ps duration (B SH T ≥ 0.44). Note that, for a Lithium Niobate sample, the temporal walk-off per unit length between FF and SH in the backward case is roughly 30 times larger than in the forward case; thus pulse shaping using backward interactions could lead to devices 30 times smaller. For example, if a temporal duration T of 30 ps is envisaged, then a pulse shaper based on forward-SHG would require an unpractical device 10 cm long, while a pulse shaper based on BSHG could be designed using a length of 0. 33

Theoretical analysis
In the process of (Type I) second harmonic generation in a non uniform quasi-phase-matching structure, the coupled nonlinear partial differential equations governing the propagation of the complex electric field amplitudes E m (z,t) (m = 1 fundamental frequency field FF, m = 2 second harmonic field SH) of two plane waves of central frequencies ω m (ω 2 = 2ω 1 ) and wave numbers k m travelling along the ±z axis read: with Δk = 2k 1 + k 2 being the wave-number mismatch, δ the sum of the inverse group velocities at FF and SH, β 1,2 the group velocity dispersions at FF and SH, respectively. The modulation of the second-order susceptibility is described by the grating function G(z), which represents a square wave with variable duty cycle and period. If the variations of period and duty cycle are slow compared to the spatial frequency of the grating, we can expand G(z) in the Fourier series G(z) = χ 0 Σ r g r (z) exp(ir f (z)) [χ 0 = ωd e f f /(n m c)] and, in the spirit of the rotating-wave approximation, retain only the terms ±r that are effective (i.e., resonant) to get [27]: where The coupled equations (2) can be numerically solved with various techniques. Since this is a two boundary problem (the FF input is given in z = 0 and the SH in z = L) we can employ a shooting approach. To solve eqs. (2) in the forward direction we used a fast Fourier transform method and a fourth-order Runge-Kutta routine [28,29]. First, we calculated BSHG profiles  Fig. 1. BSHG conversion efficiency versus z (1) and SH pulse profile (2) (in z = 0) for perfect QPM (Δk = 0), for various δ : black (δ =0), blue (δ =2), red (δ =5) and green (δ =10). BSHG conversion efficiency versus z (3) and SH pulse profile (4) (in z = 0) for δ = 10 and various Δk: black (Δk=0), blue (Δk=20), red (Δk=50) and green (Δk=100). In (4) the dashed profile is the input FF pulse. from transform-limited FF inputs in longitudinally uniform (perfectly periodic) QPM gratings. To this extent we set Eqs. 1 in dimensionless format and assumed a unity FF pulse duration (FWHM), L = 1, β 1,2 = 0 and G(z) = 1, respectively. Figure 1(1-2) show the resulting BSHG conversion efficiency vs z and the generated pulses (in z = 0) for Δk = 0 and various δ . Clearly, the role of GVM in the BSHG process is essential. The temporal profiles of the frequency doubled pulses broaden with δ and the conversion efficiency tends to a linear dependence on z, in agreement with Ref. [30]. In the case of a mismatched QPM (Δk = 0) and a fixed δ ( Fig.  1(3-4)), the results show that Δk severely affects the SH profiles, leading e.g. to splitting for Δk > 50 ( Fig. 1(4)). In the case of interest here, i. e. shaping by BSHG with a non-uniform QPM grating, the control of the output (z = 0) SH profile can be stated as follows: having fixed the FF input at z = 0, we need to determine the complex function χ(z) that minimizes the departure of the SH output [E 2 (z = 0,t)] from the target pulse shape [E 2,target (t)]. Once the complex χ(z) is obtained, we derive the non-uniform QPM modulation able to implement it. For real functions, to measure the "distance" between target and SH output we use: Extending the algorithm described in [21,31], the problem can be solved by finding the function χ(z) that minimizes the cost function J = J 1 + J 2 + c.c., [32,33] where J 1 takes into account the distance from the target and J 2 : imposes the fulfillment of eqns. (2); λ 1,2 (z,t) play the role of Lagrange multipliers. Setting to zero the functional derivatives of J with respect to λ 1 (z,t), λ 2 (z,t), E 1 (z,t), E 2 (z,t), E 1 (z = L,t), E 2 (z = 0,t), we then get six equations with the functional derivatives with respect to λ 1 and λ 2 providing the evolution equations for FF and SH, namely Eq. (2). The functional derivatives with respect to E 1 , E 2 are the evolution equations for the Lagrange multipliers: Finally, the functional derivatives with respect to E 1 (z = L,t), E 2 (z = 0,t) give the boundary conditions for Eq. (5): λ 1 (z = L,t) = 0 and λ 2 (z = 0,t) = −[E 2 (z = 0,t) − E 2,target (t)].
Using the functional derivative with respect to χ(z): we update the algorithm and determine the optimum χ(z) by the procedure: I: choose an initial guess for χ(z) and solve Eqs. (2); II: use the results obtained in the previous step to solve the evolution equations for the Lagrange multipliers (Eqs. (5)); III: update χ(z): χ(z) ← χ(z) + α δ J δ χ , with α a suitable constant; IV: if/when the obtained result is close enough to the target, the iterative procedure stops, otherwise it goes back to point 2.
Once the optimal nonlinear profile is found (i. e. we obtained the function χ(z) that minimizes the distance J 1 between target and SH output), the QPM can be implemented by assuming the use an r-order grating and considering only its r-th Fourier coefficient, i.e. g r (z) = 2 rπ sin(πrd c (z)) exp[−irπd c (z)], from which we get the duty cycle d c (z) and the period Λ of the square modulation:  Fig. 2, but for a chirped SH target pulse and a 10% conversion efficiency.
Since the dispersion length for the pump pulse is considerably larger than the sample length, we can neglect dispersion effects. As a first target, we choose a transform limited SH gaussian pulse with FWHM T FF = 5 ps and a conversion efficiency of 20%. Figure 2(a) displays the output SH intensity (circles) compared to the target (solid line): the agreement is perfect (after 100 iterations the relative error is < 10 −4 ). Fig. 2(b) graphs the output SH spectrum, transform limited with a bandwidth of 0.14 nm (2.9 cm −1 ). Figure 2(c) plots the normalized amplitude of the nonlinear coefficient χ(z). Since we sought a transform limited (i.e. real) target and neglected dispersion, the nonlinear coefficient is purely real, as visible also in Fig. 2(d).
Next, we consider a chirped gaussian SH pulse with bandwidth equal to a transform limited SH pulse of duration T SH = 40 f s and a 10% conversion efficiency. A long chirped SH target (rather than short and transform-limited) allows us to use the entire length of the crystal for frequency conversion. In such a way the maximum conversion efficiency, ultimately limited by the maximum value of the nonlinear coefficient, can be significantly enhanced. Figure  3(a) shows the agreement between SH output (circles) and target (solid line) pulses. Fig. 3(b) displays the output SH spectra, with a wide bandwidth of about 20 nm FWHM (i.e., 400 cm −1 ).
Let us now address the feasibility of the proposed devices. The current state of the art on periodic poling of Lithium Niobate allows for the realization of domain lengths as short as 200nm