### Abstract

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- Abstract
- Acknowledgements
- References

All optical inscription of quasi phase matching structures in an x-cut LiNbO_{3} crystal is demonstrated. Quasi phase matching is obtained by periodically lowering the nonlinear refractive index of the crystal using focussed ultrashort pulses. The structures were used to frequency double 1.55 µm light. The converted signal could be enhanced by a factor of 70 with respect to the unmodified material. From these measurement it could be deduced that the nonlinearity has been periodically damped up to 20%.

Within the past years the aim to provide light in any wavelength regime has driven the field of nonlinear optics. Applications range from single frequency sources to broad band parametric frequency conversion. In most cases, second order nonlinearities are exploited for sum and difference frequency generation - with second harmonic generation (SHG) being the most prominent example.

Figure 1(a) displays as a function of propagation length in the nonlinear medium. All curves have been computed by numerically integrating the coupled-amplitude equation for a lossless medium [1]. For the case of perfect phase matching (), the dependence is quadratic (Fig. 1(a), black curve I), for a given mismatch (), back conversion starts to occur at the point where the phase mismatch amounts to π (Fig. 1(a), blue curve IV), thus at a coherence length of

- (1)

In a birefringent crystal, phase matching can be achieved for conversion of ordinary polarized light to extraordinary polarized light, by finding an axis of propagation for which their respective refractive indices are equal. Alternatively, back conversion can be diminished [2] (Fig. 1, curve III) or avoided (Fig. 1, curve II) with quasi phase matching (QPM). Here, the nonlinearity is altered with a period of . Ideally, the sign of the second order nonlinearity *d* is reversed (Fig. 1(b), green solid line).

Usually QPM structures are fabricated by means of electric field poling, where a high voltage is applied to the crystal through a patterned masking layer. This technique is restricted to the *d*_{33} component of the nonlinearity, consequently just *z*-cut crystals can be poled such that the individual domains range through the entire crystal. However, there are certain applications that require the use of *x*-cut crystals, e. g. electro-optic modulators. At the same time they would benefit from efficient frequency conversion mediated by QPM. While poling of *x*-cut crystals has been demonstrated with surface electrodes, the reversed domains extended only a few micrometers below the surface [3, 4].

Our aim is to realize a QPM structure that extends far into the volume of an x-cut LiNbO_{3} wafer. Instead of electric field poling, we use ultrashort laser pulses to structure the crystal. This approach has many benefits. Most importantly, it is more flexible since it does not require any micro structuring technology to lithographically define the respective poling patterns. Additionally, it can easily be combined with ultra-short pulse laser based direct writing of optical waveguides; thus enabling all-optical processing of integrated optical devices. Several functional elements could already be demonstrated with the direct write technique: modulators as well as frequency converters [5] and most recently Bragg reflectors [6]. QPM waveguides however were only demonstrated in crystals that were poled already [7-9]. Mailis et al. demonstrated that the field of a femtosecond UV-laser can induce a domain inversion [10]. Albeit, these modifications were limited to the surface of the *z*-cut sample.

In this paper, we aim not to switch the sign but to periodically lower the nonlinearity by as sketched in Figure 1(b), red dashed line. This approach has been sucessfully demonstrated on other platforms where domain poling is not possible, e.g. by reducing the nonlinearity of GaAs by ion-implantation [2, 11, 12], by Frozen-in field-induced second order nonlinearity in optical fibers [13] or more recently in the design of a meta-metarial with a QPM periodicity [14]. A Fourier transform of the second order nonlinearity profile yields

- (2)

for the effective second order nonlinearity *d*_{eff}. The efficiency of a QPM structure with lowered nonlinearity is well below that of a poled one, because back-conversion although reduced is allowed (Fig. 1(a), red dashed curve, III). Its conversion efficiency directly depends on how strongly the nonlinearity of the crystal is affected by the modification. Ultrashort pulses can locally alter the crystaline structure of LiNbO_{3} in two ways. They either raise the extraordinary refractive index (Type I) or destroy the crystaline structure, inducing stress fields (Type II) [15]. Two previous studies addressed how the nonlinearity is affected within fs written modifications. In both a Type I waveguide had been inscribed in a periodically poled LiNbO_{3} and the decrease of the second order nonlinearity was estimated by measuring the SHG generated within the waveguide and comparing it with the numerically calculated conversion efficiency [7, 8]. Lee et al. measured a substantially decreased normalized conversion efficiency of 0.0003 [7], while Osellame et al. achieved 6.5 and deduced that the second order nonlinearity can be conserved for a certain parameter regime within a Type I modification [8].

In the following we describe the results for the QPM structure that has been inscribed with 650 nJ pulses. Although this structures had high propagation losses ( dB/cm and dB/cm), a significant conversion could be obtained. Figure 3 shows the beam profiles of fundamental and second harmonic light after passage through the QPM structure. Note that both fundamental and second harmonic light are polarized parallel to the *z*-axis of the crystal, which clearly distiguishes quasi phase matching from polarization phase matching. While the beam width is conserved, the beam height of the second harmonic light is approximately a third of that of the fundamental. This indicates that the nonlinearity has not been modified homogenously over the cross-section. A possible explanation is that the modification of the non-linearity follows the same graded profile in the x-direction as the darkening (Fig. 2c). Thus, the beam size reduction by SHG is asymmetrically enhanced, because the conversion efficiency is higher in the central region of the structure.

Figure 4 displays the tuning curve at room temperature. QPM and best conversion is obtained at 1545 nm. For comparison, the SHG signal of the pristine crystal is also included (red curve). The conversion efficiency could be enhanced by a factor of up to 70 with respect to the bulk signal.

Figure 5 shows the power dependence at the phase matched wavelength. In the structure investigated, we did not observe saturation due to pump depletion. Thus, the effective second order nonlinearity can be determined from the quadratic power law

- (3)

where η denotes the overal efficiency of the QPM structure and

- (4)

is a proportionality constant, which depends on the length of the QPM structure , a loss term and the Boyd-Kleinmann focussing parameter (for the focussing conditions within the experiment) [17, 18]. By inserting the calculated *d*_{eff} and the measured bulk pm/V in Eq. (2), we estimated the contrast of the QPM structure to be . However, this is an upper limit, since our estimate assumes , thus we do not take systematic and statistical errors of fill factor and periodicity into account [19, 20].

QPM structures were also realized with lower and higher pulse energies, but yielded lower or no conversion. This has two reasons: For low pulse energies, the induced modification is primarily a positive index change, which seems to leave the nonlinearity almost untouched [8]. Higher pulse energies result in a partial darkening (Fig. 2(c)), which is not only accompanied by a significant decrease of the second order nonlinearity of up to 20 percent but also by significantly higher losses. Both factors have to be balanced to obtain a QPM structure with good efficiency. While the overall conversion efficiency is low compared to z-cut PPLN or surface poled x-cut PPLN waveguides [3, 4], our approach enables deeply embedding both modulator and frequency converter on a monolithic chip. In combination with current research on femtosecond induced second order nonlinearities in glass [21], QPM with damping domains might pave the way to truly three-dimensional nonlinear devices.