We discuss the effect of second harmonic generation via the Čerenkov-like process in nonlinear bulk media and waveguides. We show that in both schemes the Čerenkov harmonic emission represents in fact a nonlinear Bragg diffraction process. It is therefore possible, for the first time, to describe the bulk and waveguide Čerenkov emission uniformly by considering the spatial modulation of the second-order nonlinear polarization. This is also experimentally illustrated by studying the Čerenkov second harmonic generation at the boundary of a nonlinear quadratic medium via the total internal reflection inside the nonlinear crystal.
© 2013 Optical Society of America
Nonlinear Čerenkov Second Harmonic Generation (SHG) [1, 2] is a special type of frequency doubling process that attracts lots of attention due to its versatile applications in all-optical processing  and non-destructive visualization of ferroelectric domains [4, 5], to mention a few. For a fundamental wave propagating in a normally dispersive nonlinear medium with wave vector k1 = ω1n(ω1)/c (n is refractive index and c is speed of light), the emitted Čerenkov Second Harmonic (SH) propagates with the wave vector k2 = ω2n(ω2)/c (at the frequency ω2 = 2ω1) at an angle cosθC = n(ω1)/n(ω2). Graph in Fig. 1(a) illustrates emission of Čerenkov SH signal as a cone in as-grown strontium barium niobate crystal with fundamental beam propagating along its optical axis.
The common practical realizations of the Čerenkov frequency conversion involve either planar waveguides [6,7] or bulk crystals with a spatially modulated second-order nonlinear coefficient [8,9]. Accordingly, different theoretical approaches have been used to describe interaction in these two systems separately. The waveguide-based emission has been commonly analyzed using the coupled mode theory [7, 10–13]. In the case of bulk media the Čerenkov emission has been typically associated with periodic, random or even single spatially isolated modulations of the second-order nonlinearity such as in periodically or randomly poled ferroelectric crystals. Moreover, it has been shown that Čerenkov SH signal can be emitted even in homogeneous nonlinear medium provided the fundamental beam maintains strong spatial confinement in propagation .
The fact that the Čerenkov SH is generated efficiently in so diverse systems indicates that the physical nature of this process is of very fundamental nature. One can consider the nonlinear Čerenkov radiation in terms of the phase matching which expresses phase relation between interacting waves [see Fig. 1(b)]Eq.(1)]. As this appears to ensure an efficient energy flow from the fundamental to the SH wave, little attention to the role of the transverse phase-matching condition in the Čerenkov emission [Eq. (2)] has been paid so far. However, the requirement of the momentum conservation in the Čerenkov second harmonic generation requires that the full (vectorial) phase matching condition must be satisfied.
In this work we analyze both effects of transverse and longitudinal phase matching conditions on the efficiency of nonlinear Čerenkov radiation. We show that the nonlinear Čerenkov processes in both planar waveguide and bulk nonlinear photonic crystal represent a particular type of the so-called nonlinear Bragg diffraction and hence can be described in a uniform way by considering the spatial modulation of the nonlinear polarization. We also present experimental results that provide additional support of these conclusions.
2. Čerenkov second harmonic generation in bulk materials
The nonlinear Čerenkov radiation has been recently intensively studied in nonlinear photonic crystals with spatial modulation to their second-order nonlinear coefficient. It has been shown that when the fundamental frequency (FF) beam propagates transversely to the direction of the nonlinear modulation (FF along z-axis in our case), a strong emission of Čerenkov SH is observed in the direction θC = cos−1(2k1/k2) [see Fig. 2] [4,5,8,9,15–21]. In this case, owing to the particular configuration of the modulation of second-order nonlinearity, the longitudinal and transverse phase matching relations read as22].
However, there are two issues with this approach to Čerenkov emission. On one hand, the direction of the emitted second harmonic wave is fixed solely by the longitudinal phase matching condition and is independent of the spatial modulation period of nonlinearity. On the other hand, the fulfillment of transverse quasi-phase matching condition usually requires very high orders of RLVs to compensate for the missing momentum in the transverse direction [see Fig. 3(a)]. In this case, even assuming that the effective nonlinearity associated with high order process is sufficiently high, there may not even be any reciprocal lattice vectors (G) available to exactly compensate the phase mismatch in the transverse direction.
The issue becomes even murkier in light of the recent observations of nonlinear Čerenkov radiation induced by a tightly focused fundamental beam illuminating only a single ferroelectric domain wall, i.e. the interface separating two adjacent oppositely oriented ferroelectric domains [see Fig. 2(b)]. Both experimental and numerical studies showed that the Čerenkov signal was strong when the FF was exactly located at the domain wall while it became weaker and finally disappeared when shifting the beam from the wall into the nonlinear domain [4, 5, 19]. These results suggest that the Čerenkov emission originates from the localized modulation of the nonlinear polarization. This situation differs from the cases of periodically poled crystals as now the periodic variation of the sign of χ(2) is missing. It was suggested that the Čerenkov emission from the single domain wall may be originated from the nonlinearity enhancement and crystal symmetry transformation within the domain wall emphasized the role of domain wall [5, 9, 16, 20, 23, 24]. However, as we will show in the next section, all these experimental observations and related issues can be explained by considering the spatial modulation of the nonlinear polarization.
3. Čerenkov SHG as a nonlinear Bragg diffraction
The interaction of the fundamental and second harmonic waves in a stationary 2-dimensional case is described by the following system of wave equations :25] we have simulated SH emission with fundamental beam propagating along different types of nonlinear boundaries including step-like profiles such as abrupt change of the sign of nonlinearity or change of its strength across the boundary. Our simulations showed that any single localized χ(2)modulation serves as a source of continuous Fourier components (reciprocal wave vectors), enabling to phase match the nonlinear Čerenkov process in the transverse direction and consequently leading to efficient Čerenkov SHG. This has been further confirmed by varying the steepness of the non linearity jump. Calculations showed that the sharper is the χ(2) modulation, the stronger are the Fourier components, and subsequently more efficient the nonlinear Čerenkov emission.
This behavior is even clearer if one considers the frequency conversion assuming undepleted fundamental beam. Then the system of equation Eq. (5) yields the following formula for the amplitude of the Čerenkov signal2] was a result of assuming the sharp (rectangular) boundaries of the fundamental beam. Moreover, this also explain the recent experimental results where a fundamental wave in the form of nondiffracting Bessel beam leads to appreciable emission of Čerenkov SH in homogeneous nonlinear crystal .
In Fig. 4 we illustrate the SH emission in a medium with periodically modulated χ(2) nonlinearity (period Λ) in the regime when the smallest reciprocal lattice vector G = 2π/Λ is already larger than the phase mismatch of Čerenkov radiation in the transverse direction. This means that in this system, in principle, there is no reciprocal lattice vectors which would enable efficient Čerenkov emission. However, as we see the Čerenkov SH signal is still generated. At this point it is then clear that the nonlinear Čerenkov radiation does not necessarily require a specific modulation pattern of nonlinearity χ(2). Any variation of χ(2) in the transverse direction can provide non-zero Fourier coefficients of reciprocal wave vector to ensure the full phase matching of the interaction process. In this context the Čerenkov SHG is actually a particular case of a well known nonlinear Bragg diffraction [22, 26–29] with wave vectors of the fundamental, second harmonic and vector G⃗ representing the spectrum of the spatial modulation of nonlinearity, satisfying vectorially the following relation
While in general in the Čerenkov SHG the relevant reciprocal vector is always perpendicular to the nonlinear modulation, the same arguments hold also for general oblique incidence of fundamental beam. The corresponding phase matching diagram is sketched in Fig. 3(c). In this case the Čerenkov SH is generated again symmetrically with respect to the normal to the non-linearity modulation. However the oppositely emitted signals may differ in the intensity as they are generated in interactions involving different reciprocal vectors with different nonlinearity strengths .
4. Čerenkov SHG in planar waveguide
In this section we demonstrate how the approach of nonlinear polarization modulation describes the Čerenkov SH emission in the waveguide geometry (see Fig. 5). Let us consider the guided mode of the simple slab waveguide with refractive index n0 and width 2d. Refractive index of the surrounding medium is n1. The fundamental mode propagating along the z-axis has the following form
The guided mode will generate Čerenkov SH signal propagating towards substrate and the cladding. Let us treat, for now, the SH emission process as if it was taking place in homogeneous medium with spatial profile of nonlinearity and the form of fundamental wave determined by the guiding geometry. Then the amplitude of the Čerenkov signal will be given by the Fourier integral Eq. (6) with the transverse component of the wave-vector kc given by
In Fig. 6 we depict the normalized intensity of Čerenkov SH signal emitted in a waveguide geometry as a function of waveguide width (2d) for the case when nonlinearity is located in the guiding layer (red/solid line) and substrate (black/dashed line). The lines represent the simple formulas Eq. (15) and Eq. (17), while the points depict results of the exact numerical simulation of the SH emission process [Eqs. (5)]. The excellent agreement between the exact and simplified approach confirms the simple physical picture of the emission process based on spatial modulation of nonlinearity (or, more generally, the nonlinear polarization).
To complement the above discussion we demonstrate in Fig. 7 numerically simulated second harmonic generation in a waveguide geometry with the χ(2) nonlinearity being artificially uniform across the whole structure. The top (bottom) panel depicts near (far) field of the fundamental and second harmonic. The formation of Čerenkov SH wave with the wave vector kc ≈ ±5μm−1 is clearly visible (the refractive indices are the same as in Fig. 4). While in this particular case the χ(2) nonlinearity is constant the nonlinear polarization is still spatially modulated. Therefore the transverse phase matching necessary for Čerenkov generation is ensured solely by the wave-vectors originating from the waveguide-mediated spatial confinement of the fundamental beam, following the formula Eq. (6). The Čerenkov signal is emitted at kc = ±5μm−1 and grows monotonically with propagation distance.
5. Čerenkov SHG at the boundary of linear and nonlinear media
As discussed in the previous sections, the nonlinear Čerenkov process can be initiated by any sufficiently sharp changes of the quadratic nonlinear polarization. This means that even oblique illumination of the boundary between nonlinear media and linear should generate the Čerenkov signal, where the second-order nonlinear coefficient changes from its value to zero. In fact this effect was already implicitly pointed out in the early works of Armstrong et al.  and Bloembergen [31,32]. However, it was not not originally cast in terms of nonlinear Čerenkov radiation. The appearance of the SH signal was explained by invoking the requirement of the continuity of electromagnetic fields at the boundary between linear medium and nonlinear crystal. While this approach does predict the appearance of the SH beam at Čerenkov angle, it cannot reveal its propagation dynamics. Only solving the full Maxwell’s equations it is possible to infer the phase-matched nature of the process with the propagation distance.
To complete our discussions in the following section we will provide a clear experimental evidence of the role of the boundary of the nonlinear crystal in the formation of the Čerenkov beam. In order to resolve various emitted SH signals we will utilize the total internal reflection of the fundamental wave inside the lithium niobate crystal.
Figure 8(a) shows the experimental setup. The extraordinary polarized fundamental beam (solid read line) enters the 5% MgO doped homogeneous LiNbO3 crystal at point O. It is then totally internally reflected at the C1–C2 crystal facet at point A and exits crystal at point B. This situation is a variation of the scheme in Fig. 3(b), where the fundamental beam now propagates at an oblique angle with respect to the domain wall. In addition, the jump of χ(2) is accompanied by a change of the refractive index. The parameters of the incident laser beam were: fundamental beam wavelength λ = 1200 nm, pulse duration τ =200 fs, repetition rate f =1 kHz and pulse energy En=1 μJ.
We observed three SH beams exiting the C2–C3 surface of the crystal, as shown in the photo in Fig. 8. Images (ii–iv) in Fig. 8(c) show the individual SH beams recorded with a CCD camera (in a linear intensity scale). The first SH beam [left spot in the Fig. 8(c)] originates from the sum frequency mixing of the incident and totally internally reflected fundamental beams at point A. This is the SH generated from the so called virtual beam (VSH) . This SH beam propagates exactly along the C1–C2 surface of the crystal and is it due to the classical non-collinear interaction of one photon from each of the two symmetric FF beam (incident and reflected).
The second SH beam is a forward second harmonic (FSH) emitted collinearly with the fundamental wave. This signal comprises two different components. The first one is the so called inhomogeneous SH beam which is collinear and phase-locked with the fundamental beam [34, 35]. This is generated at both C1–C4 and C1–C2 surfaces. The second component of the FSH wave is a Čerenkov SH signal generated at the entrance facet (C1–C4) of the crystal (point O). One can show that in the geometry considered here both components overlap and hence they are indistinguishable when they leave the C2–C3 facet of the crystal. The FSH emission angle with respect to the C1–C2 crystal facet is αFSH=αFF.
The third SH beam denoted as CSH represents the Čerenkov SH wave generated at the C1–C2 facet of the crystal at the point A. This Čerenkov signal is stronger than the FSH because it is generated by both incident fundamental beam and its internally reflected replica. The resulting total intensity around point A is much stronger then the one at point O. The direction of propagation of such Čerenkov with respect to the C1–C2 crystal facet can be calculated based on the phase matching diagram shown in Fig. 8(b). The Čerenkov angle is then determined as θC = cos−1 (n1 cosαFF/n2).
To confirm the origin of the three SH signals we measured the SH emission angle as a function of the FF incident angle. Figure 9 shows the position of the detector αD as a function of the FF incidence angle αS outside the crystal. The red, green and blue solid lines represent the theoretical detector position calculated for the CSH, FSH and VSH respectively, using the phase-matching scheme in Fig. 8(a). The triangles represent experimental data. The agreement between theory and experiment further confirms the correct physical origin of all generated second harmonic waves and in particular the Čerenkov beam.
In conclusion we have discussed the physical nature of the nonlinear Čerenkov SH generation. We have shown that the reported earlier observations of this effect in bulk media and waveguide settings represent in fact different aspects of exactly the same physical phenomenon of the nonlinear Bragg diffraction. Consequently, all features of the Čerenkov SH emission can be explained by considering the effect of spatially modulated nonlinear polarization. As a consequence we have shown that the Čerenkov signal appears every time a strong fundamental beam crosses a nonlinear interface. We also demonstrated experimentally this process by utilizing a total internal reflection of the fundamental beam inside the nonlinear crystal.
This work was supported by the Australian Research Council and US Army Research Office (contracts W911NF-11-1-0287 and W911NF-12-1-0589).
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