The possibility of achieving large nonlinear phase shift has attracted renewed interest in the parametric interaction of a fundamental frequency (FF) beam with its second harmonic (SH). A large nonlinear phase shift is expected for small phase mismatch Δk=k(2ω)-2k(ω) between the SH and FF beams wave vector respectively during type 1 interaction. While this effect was predicted a long time ago [1], there has been an increasing interest in cascaded processes that take place in a material with second-order optical susceptibility. Such phase shift modifies the fundamental wave propagation [2] as well as the harmonic spectrum [3]. The recent attention to cascaded process is due to the possibility to achieve large, fast third order nonlinearities for device applications [4]. For bulk crystals, the cascaded nonlinear phase shift has been reported for a beam propagating in CDA and KTP crystal using a form of time resolved interferometry [5] or z-scan technique respectively [6]. This effect has also been evidenced measuring the self-diffraction of two picosecond and frequency-degenerated pulses interacting in a β-barium borate crystal (BBO) [7]. In a similar experiment, the efficiency of sum-frequency of two collinear but slightly frequency shifted picosecond pulses was recorded in a BBO crystal close to Δk=0 [8]. In these later experiments [7,8], a theoretical analysis considering small diffraction [7] or conversion efficiency [8] was performed. It was shown that to explain the phenomenon, one has to firstly consider both the real and imaginary part of the cascaded nonlinear index of refraction and secondly to introduce the third order susceptibility that cannot be neglected at large phase mismatch. However, such an analysis does not hold at Δk=0 where diffraction and conversion efficiency are very high. Therefore one may wonder about the evolution at Δk≈0, a condition where the perturbation analysis so far proposed does not hold. Cascaded phenomena have also been observed in optical parametric amplifier or oscillator [9,10].

In this letter, we address the cascaded effect produced during the second harmonic generation of femtosecond pulses in the frequency domain at Δk≈0. We mix two pulses at frequency ω-Δω and ω+Δω contained within a single femtosecond laser beam in a barium borate crystal (BBO) and we observe the FF and SH spectra of this beam around ω and 2ω. Higher order cascading phenomena are evidenced in both the FF and SH spectra. At Δk=0, our theoretical analysis derived from the solution given by Armstrong et al. [11] very agrees well with our experimental data. This allows a simple and direct measurement of the effective χ⁽²⁾. When Δk≠0, it is shown that to explain the measured spectral density of the FF and SH waves, both the second order (i.e.: cascaded χ⁽²⁾) as well as the intrinsic χ⁽³⁾ third order (self phase modulation) optical nonlinear phenomena have to be taken into account. Both simple analysis as well as numerical simulations well agrees with our experimental data. Depending on the sign of Δk, the χ⁽²⁾ and χ⁽³⁾ nonlinear contribution can either adds-up or cancel each other. This latter phenomenon allows one to evaluate the χ⁽³⁾ contribution against the χ⁽²⁾ contribution.

In order to understand the basic idea of our experiment, let us consider two collinear, equally balanced and temporally synchronized pulses, whose central frequency are ω-δω and ω+δω respectively, entering a thin nonlinear crystal cut for type I second harmonic generation (SHG) phase matching at ω. The crystal is thin so that its spectral acceptance is large compared to δω. At Δk=0, during sum frequency generation (SFG) in the crystal, three peaks centered at 2ω, 2(ω+δω), 2(ω-δω) are generated and appear in the SFG spectrum. In turn, due to the coupling between the FF and SH pulses through cascaded χ⁽²⁾ phenomenon, the generated SF pulses modify the spectrum of the FF pulses, and new spectral components at ω+3δω and ω-3δω appear. These processes repeated back and forth, produces new spectral components at both 2(ω±n δω) and ω±(2n+1)δω, where n is a positive integer. It can be easily shown that the amplitude of the 2nδω and (2n+1)δω spectral components are linked to (χ⁽²⁾)^2n-1 and (χ⁽²⁾)²ⁿ processes respectively. Higher order cascading effects can therefore be easily evidenced. When Δk is large, the amplitudes of the spectral components centered at 2(ω+δω) as well as the spectral components ω±3δω decreases. In fact when Δk is large enough, the third order optical susceptibility χ⁽³⁾ also contribute. Indeed, the nonlinear polarization P⁽³⁾~χ⁽³⁾ |E²|E generated by cross phase modulation of the pulses at ω+δω and ω-δω also generate spectral components at the frequencies ω±3δω. Since to induce large ±Δk only small angle tilt ±Δθ around the phase matching angle θ are needed, one can consider that χ⁽³⁾(θ±Δθ)~χ⁽³⁾(θ), and therefore that χ⁽³⁾ contribution to the spectral components at ω±3δω remains constant. However, as one changes the sign of Δk from positive to negative, an another phenomenon occurs: the phase of the nonlinear polarization generated at ω±3δω by the cascaded second order nonlinear effect (P⁽²⁾(ω ±3δω)~(χ⁽²⁾)²/±Δk) changes of sign while the one due to third order nonlinear effect remains constant. Accordingly the fields generated by these two nonlinear polarizations can either interfere constructively or destructively. This constructive or destructive interference gives a way to monitor the change of the phase shift experienced by the fundamental pulse in the crystal. This makes possible to evaluate the χ⁽³⁾ contribution against the χ⁽²⁾ contribution.

 

Fig. 1. Experimental set-up

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Our experimental set-up, sketched on Fig. 1, used 120 fs Ti:Sapphire pulses amplified at 1-kHz repetition rate. The spectral density of FF pulse centered at 812 nm was 8 nm wide. The output energy of the regenerative system was initially about 0.5 mJ. To generate two synchronized and almost Fourier transformed pulses, we used an amplitude mask placed in the compressor of the regenerative amplifier. The mask was simply a combination of two slits cut in a black sheet of paper. Adjusting the width and the position of the slit allows us to select frequency and the spectral width of each pulse. For the experiments described hereafter, the pulse duration was deduced from the spectra considering the FF pulses are Fourier transform limited (τ=455 fs). The amplitude filter was placed slightly off the Fourier plane of the compressor so that no significant satellite pulses was produced. The insertion of this amplitude filter in the compressor reduced the output energy of the amplified system to ~50 µJ. The spatially gaussian outgoing beam was then collimated to a beam waist size (FWHM) w0 of ~1.1 mm with an afocal system. The SFG was produced in a 0.5 mm thick BBO crystal cut for type I phase matching at 800 nm (θ=28,3°, ϕ=30°). The spectral acceptance of this thin crystal is large compared to the spectrum of the pulses. Moreover, the small thickness of the crystals made it possible to discount all the phenomena related to the dispersion of the FF and SH pulses. To continuously control the FF pulse energy at the input of the crystal, a zero order λ/2 wave-plate followed by a Glan prism were placed in front of the afocal system. We checked that this system modified neither the spectrum nor the pulse duration of the fundamental pulse. In the crystal, the beam size was considered to be large enough for the divergence of the beam to be smaller than the angular acceptance of the crystals used. Hence the spatial walk-off and beam size variations between the SH and FF pulses were negligible. The central part of the spatial profile of both the FF and SH pulses at the exit of the crystal were imaged on the input slit of an imaging spectrometer. The thickness of the slit was equal to 10 micrometers.

The SH and FF spectra recorded at 406 nm in the center of the beam at the exit of the crystal when Δk=0 are shown on Fig. 2a and 2b respectively. As a comparison, typical spectra of the FF pulse yielded by the regenerative amplifier before and after we placed the amplitude mask are shown on Fig. 2c and 2d respectively

 

Fig. 2. a-b: Spectrum of the SH and FF pulses at the exit of the crystal for Δk=0. c-d: FF spectrum before and after the filtering of the laser pulses.

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We note on Fig. 2a the occurrence of 5 peaks in the spectral density of the SH pulses. While the peaks at the frequency 404, 406 and 407 nm are the obvious consequence of the SF phenomena, the new peaks at 403 and 409 nm evidence the high order cascading phenomena. Indeed, during their propagation in the crystal, the FF pulse generates the SF pulse (i.e. χ⁽²⁾ process), which alter the FF spectrum (i.e. χ⁽²⁾:χ⁽²⁾ process) and in turn modifies the SF spectrum (i.e. χ⁽²⁾:χ⁽²⁾:χ⁽²⁾ process). As we previously noted, this phenomenon where SF conversion efficiency is high and lot of new spectral components are generated, cannot be simply accounted by the analysis previously proposed [10].

 

Fig. 3. a: Theoretical (—) and experimental (∙) evolutions of the peak at 2ω of the SH spectrum versus the total FF pulse energy. b: Theoretical (—) and experimental evolution of the peak at ω-3δω (∘) and ω+3δω (∙) of the FF spectrum versus the total input FF pulse energy. c: Theoretical (—), numerical (----) and experimental (∙) evolution of the peak at ω0±3δω for 50 µJ of input FF pulse energy versus the phase mismatch Δk.

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On Fig. 3a and 3b, we have reported the evolution of the peak at 406 nm, versus the overall intensity in the BBO crystal at Δk=0. On Fig. 3b, we have also reported the evolution of peak at ~802 and ~820 nm (i.e. ω₀±3δω) at Δk=0 versus the overall intensity in the BBO crystal. On Fig. 2b it is important to note that these two peaks at ~802 and ~820 nm are, in this case, mainly due to the coupling of the FF and SF pulses in the crystal through the χ⁽²⁾ (i.e. χ⁽²⁾:χ⁽²⁾ process). When we recorded the evolution of the amplitude of these peaks versus Δk, we noted that for large Δk (|Δk|>1200 cm^-1) when the SF conversion efficiency is small, their amplitudes remains almost unchanged. Evolution of the slope of these peaks versus Δk is reported on fig. 3c. We note the gap of the signal between Δk>0 and Δk<0. It is related to the constructive or destructive interference of field due to χ⁽²⁾:χ⁽²⁾, that reverses its sign when Δk reverses its sign, with the field due to χ⁽³⁾, that remains constant when Δk changes.

The data presented on Fig. 2a–b are qualitatively in good agreement with the physical picture that we have drawn previously. These data can be analyzed more quantitatively considering that the electric field E(t) of the FF laser pulses are plane waves that have the following temporal profile at the input face of the BBO crystal: E(t)=E₀cos(δωt)exp[-(t/τ)²]exp(-iωt)=A₁₀(t) exp(-iωt), where τ is the duration of the pulses centered at ω+δω or ω-δω. According to the analytical solution proposed by Armstrong et al [11], the module ρ_1,2=(A_1,2 A*_1,2)½ of the slowly varying amplitude of FF and SH pulses at the point z in the crystal have the following expression:

where n, c and χ⁽²⁾ are respectively the index of refraction of the crystal for either the SH or FF pulse, the speed of light and the effective second order susceptibility of the crystal. Knowing the pulse energy, the beam size within the BBO crystal, the pulse duration τ of the FF pulses, we computed the field amplitude E₀ in the center of the beam (E₀~1.4 10⁸ V/m). The pulse duration was deduced from the spectra considering the FF pulses are Fourier transform limited (τ=455 fs). Then, we computed the FF and SH spectrum at the exit of the BBO crystal taking into account the effective susceptibility ${{\mathrm{\chi }}_{\text{eff}}}^{\left(2\right)}$=4.18 pm/V) and the indices at the phase matching condition (n=1.6603) of the BBO. These data are shown in full lines on Fig. 2a–b and Fig. 3a–b. Without any adjustable parameters, the computations very well agree with the experimental data and reproduce well the evolution of the data presented on Fig. 3a and 3b respectively.

Let us now analyze the data presented on Fig. 3c. As Δk increases we expected to observe a decrease of the cascading phenomena. Therefore the amplitude of the peak at either 802 or 820 nm is expected to rapidly decrease and vanish as |Δk| increases. At variance as shown on fig. 3c, when |Δk| increases we noticed that the SH intensity rapidly decreases while the amplitude of the peaks remain almost constant. Note the pronounced difference between the amplitude of these peaks when Δk>0 and when Δk<0. In fact, as we have previously shown, when Δk is large, these peaks partly result from the cross-phase modulation experienced by the two FF pulses during their propagation in the crystal. If one consider the cross-phase modulation of the FF pulse, the equations for A₁ and A₂ are given by:

where γ=ω/(n.c). Armstrong et al. [11] have given a solution of the type SHG coupled wave equations when Δk=0 and χ⁽³⁾=0. This solution does not hold when χ⁽³⁾≠0. In this latter case, analytical solutions do not exist. To analytically solve these equations, we neglected the self-phase and cross-phase modulation experienced by the SF and FF pulses respectively and we used a perturbation method [12]. Neglecting the χ⁽³⁾ contribution, a similar method has been used to describe the self-diffraction process [13]. We developed A_1,2 with respect to the right hand terms of Eq. (3) and write A_1,2(z,t)=${\displaystyle \sum _{\mathrm{i}=0}^{\infty }}$${\mathrm{A}}_{1,2}^{\left(\mathrm{i}\right)}$(z,t) where ${{\mathrm{A}}_{1,2}}^{\left(\mathrm{i}\right)}$≫${{\mathrm{A}}_{1,2}}^{(\mathrm{i}+1)}$, ${\mathrm{A}}_1^{\left(0\right)}$(z,t)=A₁₀(t)=a₁₀(t)cos(δωt) and ${\mathrm{A}}_2^{\left(0\right)}$(z,t)=0. Inserting these expressions in Eq. (3), it can be easily shown that to the first order of perturbation yields:

where

and

Note that Eq. (4) doesn’t hold when Δk≈0. However, since when Δk≈0, one could neglect the third order contribution and therefore one could use the solution previously discussed [11]. Eq. (4) indicates that the amplitude of the spectral component at 3δω depends on ${{\mathrm{a}}_{10}}^3$. It is also interesting to note that the amplitude of these peaks is minimum when $\Delta k\prime ~\frac{{\omega }_0{\chi }_{\mathrm{eff}}^{\left(2\right)}{\chi }_{\mathrm{eff}}^{\left(2\right)}}{4\mathrm{nc}{\chi }^{\left(3\right)}}$. For such a phase mismatch, the contribution due to second order cascaded nonlinear effect compensates for the phase mismatch due to third order nonlinear phenomena. If |Δk|<<Δk’, the χ⁽³⁾ contribution to the peaks at 3δω is negligible compared to χ⁽²⁾:χ⁽²⁾ contribution. Therefore, the knowledge of Δk’ is important for practical application. However, the evaluation of Δk’ require one’s to have a realistic estimation of χ⁽³⁾. We estimated the χ⁽³⁾ amplitude by fitting the data as presented on Fig. 3 in solid line. The agreement between the fitting and experimental data is very good. The only adjustable fitting parameter in this figure is the amplitude of χ⁽³⁾ and we have found χ⁽³⁾=2.10^-21 m²/V². To further check the validity of our analysis, we have also numerically solved the Eq. (3). The result of this numerical resolution is shown in dashed line in Fig. 3. The agreement between the analytical and numerical solution is very good. This clearly confirms the validity of the approximations used to establish Eq. (4).

In conclusion we have proposed a simple method which, using femtosecond pulses, allows to evidence and evaluate in the spectral domain the cascading occurring during type I second harmonic generation. For small phase mismatch high order cascading phenomena have been observed and well agree with the analytical model previously proposed ¹³. Without any adjustable parameter, we where able to describe the spectra recorded in a BBO crystal when Δk~0. At large phase mismatch the competition between second order cascading and third order nonlinear optical phenomena is clearly evidenced. The proposed perturbation analysis well describes the observed phenomena and allows evaluating the χ⁽³⁾ contribution.