1. Introduction

The second harmonic generation process has been widely investigated beginning with seminal contributions by authors [1–4] that derived the now well-known theory described in every nonlinear optics textbook. Typically, second harmonic generation processes are investigated and used at or near the phase-matching condition, so that the pump and generated frequencies travel inside the nonlinear material experiencing similar propagation velocities, and yielding as large a conversion efficiency as possible. Outside of the phase matching condition conversion efficiencies decrease rapidly [1].

Since the achievement of large conversion efficiency was the most sought-after outcome, the study of propagation phenomena under phase mismatched condition was seldom tackled. Investigations typically focused on the achievement of two main goals: a) compensation of material dispersion by using materials with alternate orientations of poling [5–7], and b) realization of new materials, either organic [8] or structured [9–10], in order to achieve a larger effective nonlinear coefficient.

Parametric processes were also investigated for signal processing purposes by observing that a small phase-mismatch between generating and generated beams can induce phase-modulation of the fundamental frequency by means of cascaded parametric generation [11] as well as pulse breaking [12–15]. However, even for large phase-mismatch Noordam et al [16] theoretically and experimentally demonstrated that two second harmonic pulses are generated. Theoretical aspects of such phenomena have also been discussed by M. Mlejnek et al [17] in a birefringent semiconductor and by Su et al [18], both experimentally and theoretically, in BBO crystals. In all cases one finds that the second harmonic signal breaks-up to form two distinct pulses.

It was not until recently, however, that the generation of two distinct components was understood as a phase locking mechanism that causes the pump to trap portions of the harmonics [19], an effect later experimentally confirmed even in absorbing materials [20]. In ref. [20] an investigation of harmonic generation was conducted at normal incidence in a GaAs substrate, at wavelengths below the band edge (~890nm). A 100-fs pump pulse tuned to 1300nm generated signals at 650nm (SH) and 433nm (TH). The harmonic signals then became trapped by the pump pulse and subsequently propagated through the 450-micron thick GaAs sample without being absorbed, notwithstanding the fact that typical absorption lengths in GaAs below the band edge are much smaller than 100nm. In contrast, both second and third harmonic homogenous pulses were immediately absorbed.

Although at first counter-intuitive, the suppression of absorption that portions of the harmonic signals experience can be understood within the frame-work of the phase locking mechanism. In refs.[19] it was shown theoretically that the real part of the effective index of refraction that the harmonics experience is equal to that of the pump. This may be done by performing a full spectral decomposition of the fields, and by calculating the effective index of refraction n_eff=c<k>/<ω> as the ratio of expectation values $<k>=\underset{k=-\infty }{\overset{k=\infty }{\int }}k{\mid 𝓗(\omega ,k)\mid }^2\mathrm{dk}\mathrm{and}<\omega >=\underset{\omega =-\infty }{\overset{\omega =\infty }{\int }}\omega {\mid 𝓗(\omega ,z)\mid }^2\mathrm{d\omega }$ [19]. The E field may also be used in the calculation. Once the real part of the index n_eff is mapped onto the pump index of refraction, the imaginary part of the index is obtained by performing a Kramers-Kronig reconstruction, which in turn yields an imaginary part for the second harmonic field that is identical to the index of refraction for the pump field. Therefore, it is sufficient to tune the pump to a region of transparency to guarantee suppression of absorption at the harmonic wavelengths.

In the present work we theoretically predict and experimentally demonstrate that under conditions of a phase mismatch and oblique incidence, away from absorption resonances, the phase locked pulses display the same phase and group velocities as the pump field. This determination is made by performing second harmonic generation at 400 nm in a lithium niobate crystal using femtosecond pulses, and by monitoring pulse trajectories and group velocities inside the sample.

2. Numerical simulations

To model oblique incidence we use a numerical model similar to that used in ref. [20], with the addition of a transverse coordinate to include diffraction and propagation on a two-dimensional plane. We use a Lorentz oscillator model to describe material dispersion, and for simplicity we assume a TM-polarized incident pump field. The fields may be written as a superposition of harmonics as follows:

where ℰ(z,y,t), 𝓗^ℓω(z,y,t) are generic, spatially- and temporally-dependent, complex envelope functions; k and ω are carrier wave vector and frequency, respectively, and ℓ is an integer. Eqs.(1) are a convenient representation of the fields, and no a priori assumptions are made about the field envelopes. We have also assumed that a TM-polarized incident field generates similarly polarized harmonics. The linear response of the medium is described by a Lorentz oscillator model: $\epsilon \left(\omega \right)=1-\frac{{\omega }_{𝒫}^2}{{\omega }^2+\mathrm{i\gamma \omega }-{\omega }_r^2}$, and μ(ω) = 1, where γ, ω_p, and ω_r are the damping coefficient, the plasma and resonance frequencies, respectively. The second order nonlinear polarization is: P_NL = χ ⁽²⁾ E ². Expanding the field into its components yields nonlinear polarization terms at the fundamental and second harmonic frequencies: 𝒫 (z,t)= 2χ ⁽²⁾ _ωℰ^* _ω and ℰ_2ω (z,t) = χ ⁽²⁾ _2ωℰ² _ω. Assuming that polarization and currents may be decomposed as in Eqs.(1), we obtain the following Maxwell-Lorentz system of equations for the ℓ^th field components, in the scaled two-dimensional space (ỹ,ξ) plus time (τ) coordinate system:

In Eqs.(2), the functions 𝒥, 𝒫, 𝒫_NL refer to linear electric currents, polarization, and nonlinear polarization, respectively. The coordinates are scaled so that ξ = z/λ ₀ , ỹ=y/λ ₀, τ = ct/λ ₀, ${\omega }_0=\frac{2\mathrm{\pi c}}{{\lambda }_0}$, where λ ₀=1μm is just reference wavelength;γ, β_ℓω= 2πℓω/ω ₀, β = 2π ω_r/ ω ₀, ω_p, are the scaled damping coefficient, wave-vector, resonance and electric plasma frequencies for the ℓ^th harmonic, respectively. θ_i is the angle of incidence of the pump field. The equations are solved using a split-step, fast Fourier transform-based pulse propagation algorithm [20] that advances the fields in time. The basis for understanding the generation of two distinct signals can be found in reference [2], and as mentioned earlier it is based on the mathematical solution of the homogenous and inhomogeneous wave equations at the second harmonic frequency. Continuity of the tangential components of all the fields at the boundary, a condition automatically and dynamically included in Eqs.(2), guarantees the generation of the two forward-propagating components that interfere in the vicinity of the entry surface and give rise to Maker fringes [21]. In Fig.1 we show the results of a simulation performed in a generic Lorentz medium 80microns thick, where a pump pulse approximately 100fs in duration generates two distinct second harmonic pulses that follow independent trajectories: one follows the pump (yellow arrow), the other refracts according to Snell’s law (red arrow), so that at the medium’s exit two spatially separate spots are clearly visible. From the figure it is easy to see that the SH pulse described by the yellow arrow spatially and temporally overlaps the pump pulse at all times. In contrast, the homogenous pulse, described by the red arrow, propagates much more slowly and exits the medium long after the phase locked pulse finds its way out.

 

Fig. 1. Numerical simulation of pulsed second-harmonic in a generic Lorentz medium, having γ=10^-8, ω_p=4, ω₀=4, under phase-mismatched conditions. The small γ keeps absorption at negligible levels. The yellow box delineates the medium. (a) pump pulse. (b)SH pulses. Two SH signals are discernable, one that tracks the pumps pulse (yellow arrow), the other that refracts according to material dispersion (red arrow). These two components travel with different group velocities and thus tend to separate as distance is gained inside the sample. The leading pulse is phase-locked and trapped by the pump; the second pulse propagates freely and at a group velocity approximately three times smaller compared to the pump.

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3. Experiment

The pulse locking was experimentally monitored by performing a SHG experiment using the set-up shown in fig.2. 110-fs light pulses from an amplified Ti-Sapphire laser system with energies of few ten’s of micro-joule are focused onto the XZ face of a congruent lithium niobate crystal (from a commercial z-cut wafer) and propagated along the Y direction. At the entrance face the beam waist was about 600 μm, with intensities of the order of a few GW/cm². A special prismatic shape of the sample ensured the output face to be 20° tilted from the propagation direction. Thus, this sample acts simultaneously as a generator and a spectrum analyser, as each wavelength refracts at different angle and according to Snell’s law.

 

Fig. 2. The experimental set-up. A prismatic lithium niobate crystal was used to generate the second harmonic 400 nm signal from a 800 nm pump pulse. The titled output face of the crystal forced the pulses to exit at different angles, according to their refractive indices. The locked arm was then refocused onto a thin BBO crystal at perfect phase-matching for ω+2ω=3ω interaction.

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4. Phase-velocity locking

The phase-velocity locking was initially monitored by recording the output angular distribution with a CCD camera in the reference position. Three cases were recorded according to the injected input beam.

 

Fig. 3. Experimental images of the fundamental (800 nm) and second harmonic (400 nm) signals exiting the tilted output surface. Each position corresponds to a different output angle according to the Snell’s law.

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In Figure 3(I) we describe the case of injecting a pump e-polarised beam (A), and generating two e-polarised second-harmonic pulses, thus achieving type-0 polarisation coupling ee-e (Δk_ee-e=2.45·10⁴ cm^-1). One of the beams (B) exits at the angle that corresponds to n_e(400nm), while the other (C) exits exactly overlapped with the pump beam, i.e. at the angle that corresponds to n_e(800nm). This result demonstrates that the generated second harmonic signal separates in two different parts, one freely propagating without further interaction with the pump pulse, the other locked with the pump, propagating exactly at the same phase-velocity (A-C), i.e. having exactly the same refractive index and following exactly the same trajectory as the pump. It should be mentioned here that an unexpected high conversion efficiency was observed for the locked second-harmonic beam, whose average power was of the order of 10^-3-10^-2 of the pump one.

In Figure 3(II) we show the case of an o-polarised fundamental frequency injected in the crystal (D), resulting in type I coupling oo-e (Δk_oo-e=1.20·10⁴ cm^-1). The generated SH signal is once again broken in two fragments, one freely propagating (E), and the other (F) once again phase-locked to the pump (D-F) that exits the lithium niobate crystal at the same angle as the fundamental frequency. Also in this case there is a transfer of refractive index from the generating to the generated wavelength: in fact the e-polarised locked pulse at 400nm experiences exactly the ordinary refractive index of 800nm pump pulses.

Finally in Figure 3(III) we display the case when both e- and o-polarisations are injected inside the sample (G, H), such that type-0, type-I and type-II (oe-e, Δk_oe-e=1.8·10⁴ cm^-1, oe-o, Δk_oe-o=3.5·10⁴ cm^-1) couplings are obtained simultaneously. The fundamental e-polarisation (G) generates two e-polarised second harmonic pulses (M, O) according to the type-0 coupling. The fundamental o-polarisation generates two e-polarised second harmonic pulses according to type-I coupling, (M, N). The temporal overlapping of the fundamental o- and e-polarisations generates a second harmonic signal (L) by type-II coupling, which is the only second harmonic o-polarisation. Indeed, due to group velocity mismatch of the two pumps, the phase-locked component cannot be sustained because the two cross-polarised pump pulses rapidly separate and the inhomogeneous (forced) term in the second harmonic field equation vanishes. Thus, only the pulse associated with the homogenous term is recorded at the output.

5. Group-velocity locking

The group velocity locking was monitored in the case of type 0 ee-e second-harmonic generation, the most efficient one. The CCD camera was removed from the reference position and, by using a spherical mirror, the arm containing the locked pulses was refocused inside a 100 μm thick BBO crystal. The BBO was cut in order to achieve perfect phase-matching for the interaction ω+2ω=3ω. This interaction is possible only if the ω and 2ω pulses arrive simultaneously onto the BBO crystal, and its detection would certainly demonstrate that group-velocity locking was taking place inside the LNB crystal. Indeed, using our setup a generated ultraviolet (3ω) pulse train was monitored by the PMT on the scope screen (Figure 4(I)).

 

Fig. 4. The generated 3ω signal as recorded by the photomultiplier tube (I), with a thin (II) or a thick (III) glass plate at the reference position of the experimental set-up in fig.2.

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To be sure that the 3ω signal was not generated inside the LNB crystal, a 100 μm thick glass plate was positioned at the reference position in order to remove any UV light present there. The PMT was still able to register a pulse train (figure 4(II)). In order to be sure of the pulse temporal overlapping as well as of the absence of any 3ω generated inside the BBO crystal, an 8 mm thick glass plate was positioned at the reference position. The glass plate separates the ω and 2ω pulses temporally, so that the two signals arrive at the BBO trailing each other. In that case no 3ω light was observed (figure 4(III)).

6. Conclusions

In conclusion we have experimentally demonstrated that pulsed second harmonic generation under phase and group velocity mismatch conditions gives rise to phase-locked fundamental and second harmonic pulses that propagate with similar effective dispersive properties and locked group-velocities. The locking occurs with type-0 and type-I polarisation couplings, while it is not present with type-II coupling. The absence of a phase locked signal in the latter case can be attributed to the lack of any sustained pump fields overlap. Thus, in the locked cases we have examined a complete imprinting of refractive index and group velocity occurs between the pump and the second harmonic pulse, with the locked pulse at 400 nm experiencing exactly the refractive index of its pump at 800 nm.