ˇ Unified approach to Cerenkov second harmonic generation Vito Roppo,1,∗ Ksawery Kalinowski,2 Yan Sheng,2 Wieslaw Krolikowski,2 Crina Cojocaru,3 and Jose Trull3 1 Laboratoire
de Photonique et de Nanostructures CNRS UPR 20, Route de Nozay 91460 Marcoussis, France 2 Laser Physics Center, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia 3 Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Rambla Sant Nebridi, 08222 Terrassa, Barcelona, Spain ∗ [email protected]
Abstract: We discuss the effect of second harmonic generation via the ˇ Cerenkov-like process in nonlinear bulk media and waveguides. We show ˇ that in both schemes the Cerenkov harmonic emission represents in fact a nonlinear Bragg diffraction process. It is therefore possible, for the first ˇ time, to describe the bulk and waveguide Cerenkov emission uniformly by considering the spatial modulation of the second-order nonlinear polarˇ ization. This is also experimentally illustrated by studying the Cerenkov 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 OCIS codes: (190.2620) Harmonic generation and mixing; (190.4223) Nonlinear wave mixing; (190.7220) Upconversion.
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#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25715
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1.
Introduction
ˇ Nonlinear Cerenkov 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 vecˇ Second tor k1 = ω1 n(ω1 )/c (n is refractive index and c is speed of light), the emitted Cerenkov Harmonic (SH) propagates with the wave vector k2 = ω2 n(ω2 )/c (at the frequency ω2 = 2ω1 ) ˇ SH sigat an angle cos θC = n(ω1 )/n(ω2 ). Graph in Fig. 1(a) illustrates emission of Cerenkov nal as a cone in as-grown strontium barium niobate crystal with fundamental beam propagating #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25716
along its optical axis. ˇ The common practical realizations of the Cerenkov 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 ˇ using the coupled mode theory [7, 10–13]. In the case of bulk media the Cerenkov 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 ˇ crystals. Moreover, it has been shown that Cerenkov SH signal can be emitted even in homogeneous nonlinear medium provided the fundamental beam maintains strong spatial confinement in propagation [14]. ˇ The fact that the Cerenkov 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 ˇ Cerenkov radiation in terms of the phase matching which expresses phase relation between interacting waves [see Fig. 1(b)] k2z − 2k1z k2x − 2k1x
= =
0, Δkx ,
(1) (2)
which express the longitudinal and transverse projections of the general vectorial phase matchˇ ing relation, respectively. Hence, by definition, the nonlinear Cerenkov process is always longitudinally phase matched [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 ˇ condition in the Cerenkov emission [Eq.(2)] has been paid so far. However, the requirement of ˇ the momentum conservation in the Cerenkov second harmonic generation requires that the full (vectorial) phase matching condition must be satisfied.
ˇ Fig. 1. (a) Experimentally recorded Cerenkov 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 ˇ second harmonic emission. The Cerenkov SH emission is the special case when the emission angle is defined by the fulfillment of the longitudinal phase matching condition.
In this work we analyze both effects of transverse and longitudinal phase matching conditions ˇ ˇ on the efficiency of nonlinear Cerenkov radiation. We show that the nonlinear Cerenkov 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. #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25717
2.
ˇ Cerenkov second harmonic generation in bulk materials
ˇ The nonlinear Cerenkov 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 ˇ the nonlinear modulation (FF along z-axis in our case), a strong emission of Cerenkov SH is −1 observed in the direction θC = cos (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 k2 cos θ − 2k1 = 0, k2 sin θ − G = Δkx .
(3) (4)
Here the transverse phase matching condition is modified by the contribution from the reciprocal lattice vectors (RLV) G which are associated with Fourier components of the χ (2) modulation in the transverse direction. In the particular case of a periodic modulation, |G| = 2mπ /Λ, where Λ is the periodicity of the modulation and m is an integer representing the so called Raman-Nath nonlinear diffraction orders [22]. ˇ However, there are two issues with this approach to Cerenkov 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.
ˇ Fig. 2. (a) Scheme of Cerenkov SH emission in a 1D periodically poled LiNbO3 crystal using fundamental beam at λ =1300nm. The right inset shows experimentally recorded farfield SH image. The SH spots at small angles (θRN ) represent the Raman-Nath emission ˇ SH. The SH beams are generally while the spot at bigger angles θC are the Cerenkov 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 ˇ of Cerenkov 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. ˇ 2(a), here only the Cerenkov SH spots are visible. See Fig. 3(b) for the corresponding phase matching diagram.
ˇ The issue becomes even murkier in light of the recent observations of nonlinear Cerenkov 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 ˇ [see Fig. 2(b)]. Both experimental and numerical studies showed that the Cerenkov signal was #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25718
ˇ Fig. 3. Phase matching diagrams of the Cerenkov 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 illumiˇ signals nating χ (2) modulation at an oblique angle. In this last case, even if both Cerenkov 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 ).
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 ˇ results suggest that the Cerenkov emission originates from the localized modulation of the nonlinear polarization. This situation differs from the cases of periodically poled crystals as ˇ now the periodic variation of the sign of χ (2) is missing. It was suggested that the Cerenkov 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.
ˇ Cerenkov 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]:
∂ E1 ∂z ∂ E2 ∂z
= =
ω12 χ (2) (x) ∗ i ∂ 2 E1 − i E1 E2 ei(k2 −2k1 )z , 2k1 ∂ x2 k1 c2 ω22 χ (2) (x) 2 i(2k1 −k2 )z i ∂ 2 E2 − i E1 e . 2k2 ∂ x2 2k2 c2
(5)
These equations can be numerically solved using standard Fast-Fourier-transform method for various nonlinearity profiles. In a recent work [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 ˇ the nonlinear Cerenkov process in the transverse direction and consequently leading to efficient ˇ Cerenkov 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 ˇ components, and subsequently more efficient the nonlinear Cerenkov emission.
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25719
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 ˇ amplitude of the Cerenkov signal ESH ∝ z
∞ −∞
χ (2) (x)E12 (x, z)eikc x dx,
(6)
ˇ where kc denotes the transverse component of the wave-vector of the Cerenkov second harmonic. It is clear that the amplitude of the SH is given by just a Fourier integral of the nonlinear polarization. Obviously this integral acquires large value only when its kernel exhibits fast spatial modulation which can be achieved either by varying the strength of nonlinearity or by using strongly spatially localized fundamental beam in a homogeneous medium, or both. In fact ˇ the appearance of Cerenkov SH signal in [2] 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 ˇ emission of Cerenkov SH in homogeneous nonlinear crystal [14]. 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 ˇ larger than the phase mismatch of Cerenkov radiation in the transverse direction. This means that in this system, in principle, there is no reciprocal lattice vectors which would enable efˇ ˇ ficient Cerenkov emission. However, as we see the Cerenkov SH signal is still generated. At
.
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 = 0.2μ m in this case) ensures that the reciprocal lattice vectors corresponding to the periodic χ (2) modulation cannot fulfil the Bragg condition. ˇ However, the Cerenkov SH signal is still clearly emitted at kc ≈ ±5μ m−1 . Here we assumed the refractive index to be n1 (λ = 1.2μ m) =1.125 and n2 (λ = 0.6μ m)=1.179
ˇ this point it is then clear that the nonlinear Cerenkov 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 ˇ matching of the interaction process. In this context the Cerenkov SHG is actually a particular case of a well known nonlinear Bragg diffraction [22, 26–29] with wave vectors of the funda representing the spectrum of the spatial modulation of mental, second harmonic and vector G
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25720
nonlinearity, satisfying vectorially the following relation k2 − 2k1 = G.
(7)
As a result, any variation of χ (2) in the transverse direction already constitutes a sufficient condition for its emission. There is no restriction on nonlinearity modulation which can be of any type including periodic, quasi-periodic, or even spatially localized, with the only consequence being the difference in efficiency of the process. ˇ While in general in the Cerenkov 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 ˇ case the Cerenkov SH is generated again symmetrically with respect to the normal to the nonlinearity 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.
ˇ Cerenkov SHG in planar waveguide
In this section we demonstrate how the approach of nonlinear polarization modulation describes ˇ the Cerenkov 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 (8) E(x, z) = E(x)eiβ z , where β is propagation constant and E(x) is of the form: ⎧ 2 2 ⎪ x ≤ −d ⎨A0 /(1 + κ /γ ) exp (κ (x + d)) E(x) = A0 cos(γ x) |x| ≤ d ⎪ ⎩ A0 /(1 + κ 2 /γ 2 ) exp (−κ (x − d)) x ≥ d.
(9)
Here κ , γ and β satisfy the following relations:
κ2 γ2 κ γ
= β 2 − k02 n21
(10)
=
k02 n20 − β 2
(11)
=
tan(γ d),
(12)
and k0 = 2π /λ , with λ being the fundamental wavelength. ˇ The guided mode will generate Cerenkov 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 ˇ the guiding geometry. Then the amplitude of the Cerenkov signal will be given by the Fourier integral Eq. (6) with the transverse component of the wave-vector kc given by kc2 + 4β 2 = k22 . Below we will calculate ESH for two cases of nonlinearity modulation. 1. Nonlinearity in the guiding layer only (2) χ0 for |x| ≤ d (2) χ (x) = 0 otherwise.
(13)
(14)
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25721
ˇ Fig. 5. Illustrating of the concept of Cerenkov 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 provided by the jump of the nonlinearity (see sec.IIIsituation with the reciprocal vector G IV for details). The dashed line indicates that the SH cannot propagates into the air because of the total internal refection.
Then we get ESH ∝
d −d
(2) χ0 A20 cos2 (γ x)eikc x dx
(2) = A20 χ0
sin(kc d) sin(2γ + kc )d sin(kc − 2γ )d + + kc 2γ + kc kc − 2γ
(15)
2. Nonlinearity in the substrate (or cladding) only (2) χ0 for x ≤ −d (2) χ (x) = 0 otherwise This gives
−d (2) ESH ∝ χ0 A21 e2κ (x+d) eikc x dx = −∞
(16)
(2)
A20 χ0 e−ikc d 1 + κ 2 /γ 2 2κ + ikc
(17)
ˇ In Fig. 6 we depict the normalized intensity of Cerenkov 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 funˇ damental and second harmonic. The formation of Cerenkov SH wave with the wave vector −1 kc ≈ ±5μ m 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 ˇ modulated. Therefore the transverse phase matching necessary for Cerenkov generation is ensured solely by the wave-vectors originating from the waveguide-mediated spatial confinement #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25722
1.4
x 10
−6
SH intensity (a.u.)
1.2 1 0.8 0.6 0.4 0.2 0 2
2.5
3
3.5
4
4.5
5
d (micron)
ˇ Fig. 6. Normalized strength of Cerenkov 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.
ˇ of the fundamental beam, following the formula Eq. (6). The Cerenkov signal is emitted at kc = ±5μ m−1 and grows monotonically with propagation distance.
Fig. 7. Numerically simulated SHG in waveguide geometry (waveguide width 2d = 1μ m) with uniform second order nonlinearity. Left (right) - evolution of the intensity of the funˇ damental (second harmonic) beam. The Cerenkov SH signal is emitted at kc ≈ ±5μ m−1 .
ˇ 5. Cerenkov SHG at the boundary of linear and nonlinear media ˇ As discussed in the previous sections, the nonlinear Cerenkov process can be initiated by any sufficiently sharp changes of the quadratic nonlinear polarization. This means that even oblique ˇ illumination of the boundary between nonlinear media and linear should generate the Cerenkov signal, where the second-order nonlinear coefficient changes from its value to zero. In fact this
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25723
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 ˇ SH beams: VSH - virtual SH, CSH - Cerenkov SH, FSH - forward SH. αS - FF incidence angle, αD - detector position angle. P - polarizer, F - short pass filter, DET - detector. (b) ˇ The phase matching condition for the Cerenkov 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.
effect was already implicitly pointed out in the early works of Armstrong et al. [30] and Bloemˇ bergen [31,32]. However, it was not not originally cast in terms of nonlinear Cerenkov 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 ˇ this approach does predict the appearance of the SH beam at Cerenkov 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 ˇ evidence of the role of the boundary of the nonlinear crystal in the formation of the Cerenkov 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 μ J. 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
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25724
120 Detector position αD (deg)
CSH 100 80
FSH
60 40
VSH
20 0 15
20 25 30 35 40 45 50 FF incidence angle αS (deg)
Fig. 9. Experimentally measured angular positions (detector angle αD ) of second harmonic signals as function of FF incident angle αS (outside the crystal). Experimental data is shown as triangles while lines depict theory.
propagates exactly along the C1-C2 surface of the crystal and is it due to the classical noncollinear 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 ˇ FSH wave is a Cerenkov 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 . ˇ The third SH beam denoted as CSH represents the Cerenkov SH wave generated at the C1ˇ C2 facet of the crystal at the point A. This Cerenkov 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 ˇ propagation of such Cerenkov with respect to the C1-C2 crystal facet can be calculated based ˇ on the phase matching diagram shown in Fig. 8(b). The Cerenkov angle is then determined as −1 θC = cos (n1 cos αFF /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 ˇ second harmonic waves and in particular the Cerenkov beam. 6.
Conclusions
ˇ In conclusion we have discussed the physical nature of the nonlinear Cerenkov 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
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25725
ˇ the nonlinear Bragg diffraction. Consequently, all features of the Cerenkov SH emission can be explained by considering the effect of spatially modulated nonlinear polarization. As a conseˇ quence we have shown that the Cerenkov 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. Acknowledgment This work was supported by the Australian Research Council and US Army Research Office (contracts W911NF-11-1-0287 and W911NF-12-1-0589).
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25726
de Photonique et de Nanostructures CNRS UPR 20, Route de Nozay 91460 Marcoussis, France 2 Laser Physics Center, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia 3 Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Rambla Sant Nebridi, 08222 Terrassa, Barcelona, Spain ∗ [email protected]
Abstract: We discuss the effect of second harmonic generation via the ˇ Cerenkov-like process in nonlinear bulk media and waveguides. We show ˇ that in both schemes the Cerenkov harmonic emission represents in fact a nonlinear Bragg diffraction process. It is therefore possible, for the first ˇ time, to describe the bulk and waveguide Cerenkov emission uniformly by considering the spatial modulation of the second-order nonlinear polarˇ ization. This is also experimentally illustrated by studying the Cerenkov 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 OCIS codes: (190.2620) Harmonic generation and mixing; (190.4223) Nonlinear wave mixing; (190.7220) Upconversion.
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#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25715
12. G. J. M. Krijnen, W. Tormellas, G. I. Stegeman, H. J. W. M. Hoekstra, and P. V. Lambeck, “Optimization of second harmonic generation and nonlinear phase-shifts in the Cerenkov regime,” IEEE J. Quantum Electron. 32, 729–738 (1996). 13. R. Reinisch and G. Vitrant, “Phase matching in Cerenkov second-harmonic generation: A leaky-mode analysis,” Opt. Lett. 22, 760–762 (1997). 14. T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993). ˇ 15. Y. Sheng, W. J. Wang, R. Shiloh, V. Roppo, Y. F. Kong, A. Arie, and W. Krolikowski, “Cerenkov third-harmonic generation in χ (2) nonlinear photonic crystal,” Appl. Phys. Lett. 98, 241114 (2011). 16. X. W. Deng, H. J. Ren, H. Lao, X. F. Chen, “Noncollinear efficient continuous optical frequency doubling in periodically poled lithium niobate,” Appl. Phys. B 100, 755–758 (2010). 17. K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, “Wavelength and position tuning of ˇ Cerenkov second-harmonic generation in optical superlattice,” Appl. Phys. Lett. 99, 181128 (2011). ˇ 18. K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, “Enhanced Cerenkov second-harmonic emission in nonlinear photonic structures,” Opt. Lett. 37, 1832–1834 (2012). 19. Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Cerenkov-type second harmonic generation in periodically poled ferroelectric crystal,” J. Opt. Soc. Am. B 29, 312–318 (2012). 20. H. J. Ren, X. W. Deng, Y. L. Zheng, N. An, and X. F. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108, 223901 (2012). 21. W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Cerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. 35, 3790–3792 (2010). 22. S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling process,” Opt. Lett. 34, 848–850 (2009). 23. A. Fragemann, V. Pasiskevicius, and F. Laurell, “Second-order nonlinearities in the domain walls of periodically poled KTiOPO4 ,” App. Phys. Lett. 85, 375–377 (2004). 24. S. J. Holmgren, C. Canalias, and V. Pasiskevicius, “Ultrashort single-shot pulse characterization with high spatial resolution using localized nonlinearities in ferroelectric domain walls,” Opt. Lett. 32, 1545–1547 (2007). 25. Y. Sheng, V. Roppo, K. Kalinowski, and W. Krolikowski, “Role of a localized modulation of χ (2) in Cerenkov second-harmonic generation in nonlinear bulk medium,” Opt. Lett. 37, 3864–3866 (2012). 26. V. Berger, “Nonlinear Photonic Crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998). 27. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000). 28. S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of secondharmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008). 29. A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser and Photon. Rev. 4, 355–373 (2010). 30. J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). 31. N. Bloembergen, “Conservation laws in nonlinear optics,” J. Opt. Soc. Am. 70, 1429–1436 (1980). 32. N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. 181, 1261–1271 (1969). 33. S. M. Saltiel, D. N. Neshev, W. Krolikowski, N. Voloch-Bloch, A. Arie, O. Bang, and Yu. S. Kivshar, “Nonlinear diffraction from a virtual beam,” Phys. Rev. Lett. 104, 083902 (2010). 34. M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “ Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation,” Phys. Rev. Lett. 101, 113905 (2008). 35. E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “ Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
1.
Introduction
ˇ Nonlinear Cerenkov 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 vecˇ Second tor k1 = ω1 n(ω1 )/c (n is refractive index and c is speed of light), the emitted Cerenkov Harmonic (SH) propagates with the wave vector k2 = ω2 n(ω2 )/c (at the frequency ω2 = 2ω1 ) ˇ SH sigat an angle cos θC = n(ω1 )/n(ω2 ). Graph in Fig. 1(a) illustrates emission of Cerenkov nal as a cone in as-grown strontium barium niobate crystal with fundamental beam propagating #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25716
along its optical axis. ˇ The common practical realizations of the Cerenkov 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 ˇ using the coupled mode theory [7, 10–13]. In the case of bulk media the Cerenkov 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 ˇ crystals. Moreover, it has been shown that Cerenkov SH signal can be emitted even in homogeneous nonlinear medium provided the fundamental beam maintains strong spatial confinement in propagation [14]. ˇ The fact that the Cerenkov 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 ˇ Cerenkov radiation in terms of the phase matching which expresses phase relation between interacting waves [see Fig. 1(b)] k2z − 2k1z k2x − 2k1x
= =
0, Δkx ,
(1) (2)
which express the longitudinal and transverse projections of the general vectorial phase matchˇ ing relation, respectively. Hence, by definition, the nonlinear Cerenkov process is always longitudinally phase matched [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 ˇ condition in the Cerenkov emission [Eq.(2)] has been paid so far. However, the requirement of ˇ the momentum conservation in the Cerenkov second harmonic generation requires that the full (vectorial) phase matching condition must be satisfied.
ˇ Fig. 1. (a) Experimentally recorded Cerenkov 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 ˇ second harmonic emission. The Cerenkov SH emission is the special case when the emission angle is defined by the fulfillment of the longitudinal phase matching condition.
In this work we analyze both effects of transverse and longitudinal phase matching conditions ˇ ˇ on the efficiency of nonlinear Cerenkov radiation. We show that the nonlinear Cerenkov 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. #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25717
2.
ˇ Cerenkov second harmonic generation in bulk materials
ˇ The nonlinear Cerenkov 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 ˇ the nonlinear modulation (FF along z-axis in our case), a strong emission of Cerenkov SH is −1 observed in the direction θC = cos (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 k2 cos θ − 2k1 = 0, k2 sin θ − G = Δkx .
(3) (4)
Here the transverse phase matching condition is modified by the contribution from the reciprocal lattice vectors (RLV) G which are associated with Fourier components of the χ (2) modulation in the transverse direction. In the particular case of a periodic modulation, |G| = 2mπ /Λ, where Λ is the periodicity of the modulation and m is an integer representing the so called Raman-Nath nonlinear diffraction orders [22]. ˇ However, there are two issues with this approach to Cerenkov 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.
ˇ Fig. 2. (a) Scheme of Cerenkov SH emission in a 1D periodically poled LiNbO3 crystal using fundamental beam at λ =1300nm. The right inset shows experimentally recorded farfield SH image. The SH spots at small angles (θRN ) represent the Raman-Nath emission ˇ SH. The SH beams are generally while the spot at bigger angles θC are the Cerenkov 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 ˇ of Cerenkov 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. ˇ 2(a), here only the Cerenkov SH spots are visible. See Fig. 3(b) for the corresponding phase matching diagram.
ˇ The issue becomes even murkier in light of the recent observations of nonlinear Cerenkov 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 ˇ [see Fig. 2(b)]. Both experimental and numerical studies showed that the Cerenkov signal was #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25718
ˇ Fig. 3. Phase matching diagrams of the Cerenkov 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 illumiˇ signals nating χ (2) modulation at an oblique angle. In this last case, even if both Cerenkov 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 ).
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 ˇ results suggest that the Cerenkov emission originates from the localized modulation of the nonlinear polarization. This situation differs from the cases of periodically poled crystals as ˇ now the periodic variation of the sign of χ (2) is missing. It was suggested that the Cerenkov 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.
ˇ Cerenkov 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]:
∂ E1 ∂z ∂ E2 ∂z
= =
ω12 χ (2) (x) ∗ i ∂ 2 E1 − i E1 E2 ei(k2 −2k1 )z , 2k1 ∂ x2 k1 c2 ω22 χ (2) (x) 2 i(2k1 −k2 )z i ∂ 2 E2 − i E1 e . 2k2 ∂ x2 2k2 c2
(5)
These equations can be numerically solved using standard Fast-Fourier-transform method for various nonlinearity profiles. In a recent work [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 ˇ the nonlinear Cerenkov process in the transverse direction and consequently leading to efficient ˇ Cerenkov 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 ˇ components, and subsequently more efficient the nonlinear Cerenkov emission.
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25719
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 ˇ amplitude of the Cerenkov signal ESH ∝ z
∞ −∞
χ (2) (x)E12 (x, z)eikc x dx,
(6)
ˇ where kc denotes the transverse component of the wave-vector of the Cerenkov second harmonic. It is clear that the amplitude of the SH is given by just a Fourier integral of the nonlinear polarization. Obviously this integral acquires large value only when its kernel exhibits fast spatial modulation which can be achieved either by varying the strength of nonlinearity or by using strongly spatially localized fundamental beam in a homogeneous medium, or both. In fact ˇ the appearance of Cerenkov SH signal in [2] 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 ˇ emission of Cerenkov SH in homogeneous nonlinear crystal [14]. 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 ˇ larger than the phase mismatch of Cerenkov radiation in the transverse direction. This means that in this system, in principle, there is no reciprocal lattice vectors which would enable efˇ ˇ ficient Cerenkov emission. However, as we see the Cerenkov SH signal is still generated. At
.
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 = 0.2μ m in this case) ensures that the reciprocal lattice vectors corresponding to the periodic χ (2) modulation cannot fulfil the Bragg condition. ˇ However, the Cerenkov SH signal is still clearly emitted at kc ≈ ±5μ m−1 . Here we assumed the refractive index to be n1 (λ = 1.2μ m) =1.125 and n2 (λ = 0.6μ m)=1.179
ˇ this point it is then clear that the nonlinear Cerenkov 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 ˇ matching of the interaction process. In this context the Cerenkov SHG is actually a particular case of a well known nonlinear Bragg diffraction [22, 26–29] with wave vectors of the funda representing the spectrum of the spatial modulation of mental, second harmonic and vector G
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25720
nonlinearity, satisfying vectorially the following relation k2 − 2k1 = G.
(7)
As a result, any variation of χ (2) in the transverse direction already constitutes a sufficient condition for its emission. There is no restriction on nonlinearity modulation which can be of any type including periodic, quasi-periodic, or even spatially localized, with the only consequence being the difference in efficiency of the process. ˇ While in general in the Cerenkov 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 ˇ case the Cerenkov SH is generated again symmetrically with respect to the normal to the nonlinearity 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.
ˇ Cerenkov SHG in planar waveguide
In this section we demonstrate how the approach of nonlinear polarization modulation describes ˇ the Cerenkov 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 (8) E(x, z) = E(x)eiβ z , where β is propagation constant and E(x) is of the form: ⎧ 2 2 ⎪ x ≤ −d ⎨A0 /(1 + κ /γ ) exp (κ (x + d)) E(x) = A0 cos(γ x) |x| ≤ d ⎪ ⎩ A0 /(1 + κ 2 /γ 2 ) exp (−κ (x − d)) x ≥ d.
(9)
Here κ , γ and β satisfy the following relations:
κ2 γ2 κ γ
= β 2 − k02 n21
(10)
=
k02 n20 − β 2
(11)
=
tan(γ d),
(12)
and k0 = 2π /λ , with λ being the fundamental wavelength. ˇ The guided mode will generate Cerenkov 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 ˇ the guiding geometry. Then the amplitude of the Cerenkov signal will be given by the Fourier integral Eq. (6) with the transverse component of the wave-vector kc given by kc2 + 4β 2 = k22 . Below we will calculate ESH for two cases of nonlinearity modulation. 1. Nonlinearity in the guiding layer only (2) χ0 for |x| ≤ d (2) χ (x) = 0 otherwise.
(13)
(14)
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25721
ˇ Fig. 5. Illustrating of the concept of Cerenkov 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 provided by the jump of the nonlinearity (see sec.IIIsituation with the reciprocal vector G IV for details). The dashed line indicates that the SH cannot propagates into the air because of the total internal refection.
Then we get ESH ∝
d −d
(2) χ0 A20 cos2 (γ x)eikc x dx
(2) = A20 χ0
sin(kc d) sin(2γ + kc )d sin(kc − 2γ )d + + kc 2γ + kc kc − 2γ
(15)
2. Nonlinearity in the substrate (or cladding) only (2) χ0 for x ≤ −d (2) χ (x) = 0 otherwise This gives
−d (2) ESH ∝ χ0 A21 e2κ (x+d) eikc x dx = −∞
(16)
(2)
A20 χ0 e−ikc d 1 + κ 2 /γ 2 2κ + ikc
(17)
ˇ In Fig. 6 we depict the normalized intensity of Cerenkov 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 funˇ damental and second harmonic. The formation of Cerenkov SH wave with the wave vector −1 kc ≈ ±5μ m 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 ˇ modulated. Therefore the transverse phase matching necessary for Cerenkov generation is ensured solely by the wave-vectors originating from the waveguide-mediated spatial confinement #196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25722
1.4
x 10
−6
SH intensity (a.u.)
1.2 1 0.8 0.6 0.4 0.2 0 2
2.5
3
3.5
4
4.5
5
d (micron)
ˇ Fig. 6. Normalized strength of Cerenkov 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.
ˇ of the fundamental beam, following the formula Eq. (6). The Cerenkov signal is emitted at kc = ±5μ m−1 and grows monotonically with propagation distance.
Fig. 7. Numerically simulated SHG in waveguide geometry (waveguide width 2d = 1μ m) with uniform second order nonlinearity. Left (right) - evolution of the intensity of the funˇ damental (second harmonic) beam. The Cerenkov SH signal is emitted at kc ≈ ±5μ m−1 .
ˇ 5. Cerenkov SHG at the boundary of linear and nonlinear media ˇ As discussed in the previous sections, the nonlinear Cerenkov process can be initiated by any sufficiently sharp changes of the quadratic nonlinear polarization. This means that even oblique ˇ illumination of the boundary between nonlinear media and linear should generate the Cerenkov signal, where the second-order nonlinear coefficient changes from its value to zero. In fact this
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25723
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 ˇ SH beams: VSH - virtual SH, CSH - Cerenkov SH, FSH - forward SH. αS - FF incidence angle, αD - detector position angle. P - polarizer, F - short pass filter, DET - detector. (b) ˇ The phase matching condition for the Cerenkov 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.
effect was already implicitly pointed out in the early works of Armstrong et al. [30] and Bloemˇ bergen [31,32]. However, it was not not originally cast in terms of nonlinear Cerenkov 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 ˇ this approach does predict the appearance of the SH beam at Cerenkov 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 ˇ evidence of the role of the boundary of the nonlinear crystal in the formation of the Cerenkov 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 μ J. 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
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25724
120 Detector position αD (deg)
CSH 100 80
FSH
60 40
VSH
20 0 15
20 25 30 35 40 45 50 FF incidence angle αS (deg)
Fig. 9. Experimentally measured angular positions (detector angle αD ) of second harmonic signals as function of FF incident angle αS (outside the crystal). Experimental data is shown as triangles while lines depict theory.
propagates exactly along the C1-C2 surface of the crystal and is it due to the classical noncollinear 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 ˇ FSH wave is a Cerenkov 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 . ˇ The third SH beam denoted as CSH represents the Cerenkov SH wave generated at the C1ˇ C2 facet of the crystal at the point A. This Cerenkov 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 ˇ propagation of such Cerenkov with respect to the C1-C2 crystal facet can be calculated based ˇ on the phase matching diagram shown in Fig. 8(b). The Cerenkov angle is then determined as −1 θC = cos (n1 cos αFF /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 ˇ second harmonic waves and in particular the Cerenkov beam. 6.
Conclusions
ˇ In conclusion we have discussed the physical nature of the nonlinear Cerenkov 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
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25725
ˇ the nonlinear Bragg diffraction. Consequently, all features of the Cerenkov SH emission can be explained by considering the effect of spatially modulated nonlinear polarization. As a conseˇ quence we have shown that the Cerenkov 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. Acknowledgment This work was supported by the Australian Research Council and US Army Research Office (contracts W911NF-11-1-0287 and W911NF-12-1-0589).
#196061 - $15.00 USD Received 19 Aug 2013; revised 23 Sep 2013; accepted 24 Sep 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025715 | OPTICS EXPRESS 25726
