Abstract
We analyze experimentally light scattering from ͑2͒ nonlinear gratings and observe two types of secondharmonic frequency-scattering processes. The first process is identified as Raman-Nath type nonlinear diffraction that is explained by applying only transverse phase-matching conditions. The angular position of this type of diffraction is defined by the ratio of the second-harmonic wavelength and the grating period. In contrast, the second type of nonlinear scattering process is explained by the longitudinal phase matching only, being insensitive to the nonlinear grating period. © 2009 Optical Society of America OCIS codes: 190.4420, 190.2620, 050.1940 Second-harmonic generation (SHG) is one of the most studied parametric processes in nonlinear optics. Often, to achieve efficient SHG, this process requires periodic modulation of the ͑2͒ components, the socalled quasi-phase-matching (QPM) [1] . Usually, QPM is realized by collinear propagation of the fundamental frequency (FF) and second-harmonic (SH) beams along the QPM grating vector. However, when the FF beam and the QPM grating vector are noncollinear, then the SH beam diffracts from the ͑2͒ grating in a fashion similar to a linear beam diffraction on an index grating.
In the linear optics it is well established that propagation of a light beam in periodic refractive index structures leads either to Bragg diffraction when the full vectorial phase-matching condition is satisfied or to Raman-Nath diffraction otherwise [2] . In the latter case since the full vectorial phase matching is not fulfilled, a multiorder diffraction takes place, similar to the case of light scattering by acoustic waves.
Analogously, nonlinear Bragg diffraction is realized at a specific angle between the beams [3], determined by the vectorial phase matching 2k 1 + G n = k 2 , where G n is the grating vector and k 1 , k 2 are the wave vectors of the FF and SH beams, respectively [ Fig. 1(a) ]. If this vectorial condition is not satisfied, multiorder nonlinear SH diffraction should be seen, in an analog to linear Raman-Nath diffraction [ Fig.  1(b) ].
Nonlinear Bragg diffraction has been observed in a number of structures, including naturally laminated crystals [3] [4] [5] , as well as artificially created nonlinear one-dimensional [6-9] and annular [10] ͑2͒ gratings. On the other hand, to the best of our knowledge, the nonlinear Raman-Nath diffraction has not been discussed so far, simply because QPM technology is mainly used for photonics components with collinear propagation. In fact, the first experimental indication of this phenomena has been reported in our recent work [10] , where the first-and second-order nonlinear diffraction in annular QPM structure was observed.
In this Letter we study the nonlinear diffraction of beams from a one-dimensional nonlinear grating, realized by periodic spatial alternation of the sign of the relevant components of the ͑2͒ tensor. We observe multiple-order nonlinear diffraction in the SHG processes, which represents the Raman-Nath contribution to the process. In addition, we analyze the diffracted SH beams that are governed only by longitudinal phase-matching conditions and can be classified as noninteger order nonlinear diffraction, having no analog in linear optics. 

Second-harmonic generation (SHG) is one of the most studied parametric processes in nonlinear optics. Often, to achieve efficient SHG, this process requires periodic modulation of the ͑2͒ components, the socalled quasi-phase-matching (QPM) [1] . Usually, QPM is realized by collinear propagation of the fundamental frequency (FF) and second-harmonic (SH) beams along the QPM grating vector. However, when the FF beam and the QPM grating vector are noncollinear, then the SH beam diffracts from the ͑2͒ grating in a fashion similar to a linear beam diffraction on an index grating.
In the linear optics it is well established that propagation of a light beam in periodic refractive index structures leads either to Bragg diffraction when the full vectorial phase-matching condition is satisfied or to Raman-Nath diffraction otherwise [2] . In the latter case since the full vectorial phase matching is not fulfilled, a multiorder diffraction takes place, similar to the case of light scattering by acoustic waves.
Analogously, nonlinear Bragg diffraction is realized at a specific angle between the beams [3] , determined by the vectorial phase matching 2k 1 + G n = k 2 , where G n is the grating vector and k 1 , k 2 are the wave vectors of the FF and SH beams, respectively [ Fig. 1(a) ]. If this vectorial condition is not satisfied, multiorder nonlinear SH diffraction should be seen, in an analog to linear Raman-Nath diffraction [ Fig.  1(b) ].
Nonlinear Bragg diffraction has been observed in a number of structures, including naturally laminated crystals [3] [4] [5] , as well as artificially created nonlinear one-dimensional [6] [7] [8] [9] and annular [10] ͑2͒ gratings. On the other hand, to the best of our knowledge, the nonlinear Raman-Nath diffraction has not been discussed so far, simply because QPM technology is mainly used for photonics components with collinear propagation. In fact, the first experimental indication of this phenomena has been reported in our recent work [10] , where the first-and second-order nonlinear diffraction in annular QPM structure was observed.
In this Letter we study the nonlinear diffraction of beams from a one-dimensional nonlinear grating, realized by periodic spatial alternation of the sign of the relevant components of the ͑2͒ tensor. We observe multiple-order nonlinear diffraction in the SHG processes, which represents the Raman-Nath contribution to the process. In addition, we analyze the diffracted SH beams that are governed only by longitudinal phase-matching conditions and can be classified as noninteger order nonlinear diffraction, having no analog in linear optics. In our experiments we use periodically poled lithium niobate (PPLN) and stoichiometric lithium tantalate (SLT) samples (typical thickness 0.5 mm) in the arrangement shown in Fig. 1(c) . The fundamental pump beam from a regenerative amplifier, delivering 10 ps, 1 mJ pulses at 1053 nm with repetition rate of 20 Hz, is directed at a small angle with respect to the Z axis. The laser beam is focused with a lens creating a spot size of Ϸ400 m in the plane of the crystal.
In Figs. 1(d) and 1(e) we show an example of the SH diffraction patterns obtained in one-dimensional ͑2͒ grating ͑⌳ = 14.6 m͒ in PPLN. The patterns consist of two types of spots: (i) central diffraction spots, grouped around the pump position, and (ii) peripheral diffraction spots, situated relatively far from the pump at both sides of the diffraction array [top and bottom pairs in Fig. 1(d) ]. The two neighboring peripheral spots at each side of the array are orthogonally polarized. The spots that are furthest away from the pump are polarized in the x -y plane, the other spots are polarized in the x -z plane. Since the pump is ordinary polarized, we identify the interactions that are responsible for these SH spots as O 2 [11] , however, in contrast to our experiments, they were due to second-order nonlinearities existing only in the domain walls.
The observed nonlinear diffraction patterns can be explained by employing the phase-matching conditions shown in Fig. 2 . The general vectorial phasematching condition 2k 1 + G n + ⌬k = k 2 is valid for both types of the SHG diffraction spots (the central and peripheral ones). Figure 2 shows one of the phasematching triangles (OCD) that governs the firstorder SH diffraction patterns, appearing close to the pump. It is constructed by the vectors k 2,o , 2k 1,o , BD= ⌬k TPM , and one of the grating vectors G o . The angular direction of the SH diffraction spot is defined only by the transverse phase-matching (TPM) conditions:
where 2 is the SH wavelength. This is a generic condition that holds for any periodically poled sample and does not depend on its refractive index. Our measurements of the first-order SH diffraction angles for four (PPLN and SLT) samples with different grating periods (7.5, 9.0, 14.6, 24.1 m) are shown in Fig.  3(a) . The experimentally measured angles are in excellent agreement with the predictions of Eq. (1). We note that Eq. (1) is known to describe normal incidence Raman-Nath diffraction in linear optics. For normal pump incidence, ␣ −m =−␣ +m . For the pump incident at the angle ␥ with respect to the Z axis toward the X axis (in the plane X-Z), again we have ␣ −m ͑␥͒ =−␣ +m ͑␥͒ but the angles are defined by sin ␣ m ͑␥͒ = ͑mG o / k 2,o ͒cos ␥, derived again using the TPM condition only. Calculating the external angles gives that ␤ m ͑␥͒ Ͼ ␤ m ͑␥ =0͒. The dependence ␤ m ͑␥͒ has been verified experimentally and the result is shown in Fig. 3(b) .
In summary, so far we have demonstrated that the observed central SH multiorder diffraction [ Fig. 1(e) ] is a nonlinear optical analog of Raman-Nath diffraction. In this process, the directions of the SH diffracted beams can be determined by applying the TPM conditions only.
Next, we describe the phase-matching conditions for the peripheral SH spots that appear far from the pump. Experiments with four different samples show that their angular positions, o and e [defined in Figs. 1(d) and 2] , do not depend on the poling period. This is in strong contrast to the direct angular dependence of the central diffraction spots on ⌳, as seen from Eq. (1). The measured angles can be explained only if we assume that the directions of these SH beams are defined uniquely by the longitudinal phase-matching (LPM) condition, seen as the triangle OAB in Fig. 2 , constructed by the vectors k 2,e , 2k 1,o , FA = ⌬k LPM , and m grating vectors, BF = mG o . Correspondingly, the phase-matching triangle for the ordinary SH wave is OBG.
For normal incidence, the measured angles o and e are found to be in accordance with those derived on the basis of LPM equations, 
͑2͒
These equations are also known to define the emission of Cherenkov SH generation [12] and noncollinear SHG in one-and two-dimensional random nonlinear gratings [13] [14] [15] . In the PPLN crystal n o Ͼ n e , so that For the pump propagating at an angle ␥ with respect to the Z axis [see inset in Fig. 4(b) ], the phase matching for the LPM diffraction spots is obtained from Eq. (2) by replacing 2k 1 with 2k 1 cos ␥. An increase of the angle ␥ thus leads to an increase in both ͉ o ͉ and ͉ e ͉. We should mention that the angular positions o and e are determined from the direction of the Z axis, so that increasing the angle ␥ leads to a reduction of the angle between the pump and one of the diffraction spots. This is illustrated by an additional experiment where the SH pattern is recorded as a function of the crystal rotation around the Y axis. The recorded images are shown in Fig. 4(a) . With the increase of the angle ␥ ext , the phasematching angles for both interactions increase. As seen from Fig. 4 , the angle o,e,ext − ␥ ext decreases initially with a growth of ␥ ext but, after reaching minimum value for ␥ ext Ϸ 40°, it starts to grow again. The theoretical dependencies of the external angles o,ext and e,ext on ␥ ext [ Fig. 4(b) ] are in very good agreement with the experimental results [ Fig. 4(a) ].
Although the period of the poled grating does not appear in Eqs. (2) , the LPM signals are not visible in experiments with single-domain crystals. On the other hand, if both TPM and LPM are satisfied simultaneously (the case of nonlinear Bragg diffraction [3] ), the respective diffraction order is much more intense than the neighboring SH diffraction spots.
In conclusion, we have observed second-harmonic patterns generated owing to multiorder nonlinear diffraction. We have found that the angular positions of the SH beams are defined only by the ratio of the wavelength to the grating period as in Raman-Nath diffraction in linear optics. The diffraction patterns include additional spots defined only by the longitudinal phase matching, not being observable without the nonlinear grating. For this reason we believe these peripheral SH spots can be identified as a noninteger order nonlinear diffraction. As recently shown [16] , similar type SH radiation can be used for single-shot reconstruction of femtosecond pulses. The observed phenomena can also find application in nonlinear microscopy and nondestructive evaluation of the domain structure in ferroelectric crystals. 

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