1. Introduction

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 [3] 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 cosθ_C = 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.

 

Fig. 1 (a) Experimentally recorded Čerenkov conical SH emission from an as-grown strontium barium niobate crystal illuminated by the fundamental beam at λ =800nm. (b) Phase matching diagram for SHG in normally dispersive medium. The circle represents wavevectors of the SH. We can distinguish the typical collinear and the general non-collinear second harmonic emission. The Čerenkov SH emission is the special case when the emission angle is defined by the fulfillment of the longitudinal phase matching condition.

Download Full Size | PDF

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 [14].

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)]

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 θ_C = 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 as

 

Fig. 2 (a) Scheme of Čerenkov SH emission in a 1D periodically poled LiNbO₃ crystal using fundamental beam at λ =1300nm. The right inset shows experimentally recorded far-field SH image. The SH spots at small angles (θ_RN) represent the Raman-Nath emission while the spot at bigger angles θ_C are the Čerenkov SH. The SH beams are generally emitted as doublets with orthogonal polarization because of birefringence of the crystal. See Fig. 3(a) for the corresponding phase-matching scheme. (b) Schematic representation of Čerenkov SH emission via spatially localized modulation of nonlinearity. The right inset shows experimentally recorded far-field SH image [4]. Unlike the case depicted in Fig. 2(a), here only the Čerenkov SH spots are visible. See Fig. 3(b) for the corresponding phase matching diagram.

Download Full Size | PDF

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.

 

Fig. 3 Phase matching diagrams of the Čerenkov SHG in: (a) 1D periodically modulated nonlinearity, as shown in Fig. 2(a). The phase-mismatched emissions at small angles given by the action of the single reciprocal lattice vector are the Raman-Nath SHs. (b) localized nonlinear modulation, as shown in Fig. 2(b). c) general case with fundamental beam illuminating χ⁽²⁾ modulation at an oblique angle. In this last case, even if both Čerenkov signals are still emitted symmetrically with respect to the normal to nonlinearity modulation, they differ in intensity as they are formed using different reciprocal vectors (G₁ and G₂).

Download Full Size | PDF

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]:

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 signal

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

 

Fig. 4 Numerically simulated fundamental (left) and generated second harmonic (rigt) in a nonlinear medium with a periodically modulated nonlinearity in transverse direction. The short period of the modulation (Λ = 0.2μm in this case) ensures that the reciprocal lattice vectors corresponding to the periodic χ⁽²⁾ modulation cannot fulfil the Bragg condition. However, the Čerenkov SH signal is still clearly emitted at k_c ≈ ±5μm⁻¹. Here we assumed the refractive index to be n₁(λ = 1.2μm) =1.125 and n₂(λ = 0.6μm)=1.179.

Download Full Size | PDF

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 [21].

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

 

Fig. 5 Illustrating of the concept of Čerenkov SH emission in a waveguide. FF and SH denote fundamental and second harmonic beams, respectively. Here fundamental beam propagates as guided mode while second harmonic is a radiation mode of the waveguiding structure. The graph depicts also the phase matching diagram corresponding to this physical situation with the reciprocal vector G⃗ provided by the jump of the nonlinearity (see sec.III–IV for details). The dashed line indicates that the SH cannot propagates into the air because of the total internal refection.

Download Full Size | PDF

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 k_c 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).

 

Fig. 6 Normalized strength of Čerenkov signal excited by the fundamental mode of the symmetric waveguide as a function of the waveguide width (2d). Line represents result obtained by using simple analytical approach based on the transverse modulation of the nonlinearity [Eqs. (15) and (17)]. Points depict results evaluated by numerically solving the propagation equations Eq. (5) in waveguide geometry. Solid and dashed lines correspond to the nonlinearity located in the guiding layer and substrate, respectively.

Download Full Size | PDF

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 k_c ≈ ±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 k_c = ±5μm⁻¹ and grows monotonically with propagation distance.

 

Fig. 7 Numerically simulated SHG in waveguide geometry (waveguide width 2d = 1μm) with uniform second order nonlinearity. Left (right) - evolution of the intensity of the fundamental (second harmonic) beam. The Čerenkov SH signal is emitted at k_c ≈ ±5μm⁻¹.

Download Full Size | PDF

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. [30] 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 E_n=1 μJ.

 

Fig. 8 SH emission via total internal reflection in nonlinear medium. (a) Experimental setup. Red line represents the fundamental frequency (FF) beam. Green lines represent the SH beams: VSH - virtual SH, CSH - Čerenkov SH, FSH - forward SH. α_S - FF incidence angle, α_D - detector position angle. P - polarizer, F - short pass filter, DET - detector. (b) The phase matching condition for the Čerenkov SHG. (c) The image of the observed SH generated at the boundary between nonlinear crystal and air. The red dashed line indicates the FF, green lines represent generated SH. The intense red diffusive spots are the observed SH beams as seen on the paper screen (no FF filter used here). (i) lithium niobate crystal, (ii)–((iv) CCD images (linear scale) of the second harmonic (FF beam filtered out): (ii) VSH, (iii) CSH and (iv) FSH.

Download Full Size | PDF

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) [33]. 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 α_FSH=α_FF.

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 θ_C = cos⁻¹ (n₁ cosα_FF/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 α_D as a function of the FF incidence angle α_S 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.

 

Fig. 9 Experimentally measured angular positions (detector angle α_D) of second harmonic signals as function of FF incident angle α_S (outside the crystal). Experimental data is shown as triangles while lines depict theory.

Download Full Size | PDF

6. Conclusions

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.