Abstract

We study theoretically and numerically the second harmonic generation in a nonlinear crystal with random distribution of ferroelectric domains. We show that the specific features of disordered domain structure greatly affect the emission pattern of the generated harmonics. This phenomena can be used to characterize the degree of disorder in nonlinear photonic structures.

©2010 Optical Society of America

Introduction

Quasi-phase matching is one possible method of choice to compensate for the lack of exact phase synchronism among interacting waves in quadratic nonlinear media ensuring a high efficiency of frequency mixing [1,2]. It employs spatial periodic changes in the sign of nonlinearity. This is achieved by periodic poling of ferroelectric nonlinear crystals such as lithium niobate, lithium tantalate, etc [3]. However, the exact periodic poling ensures the fulfilment of the phase matching condition only for a particular choice of wavelengths of the interacting waves; thus restricting its practical applicability. To remove this constrain, quasi-periodic poled structures or sequences of periodic regions with different periods have been realized [4, 5]. Naturally grown ferroelectric crystals such as Strontium Barium Niobate (SBN) often exhibit an irregular multi-domain structure with rod-like domains [6–9]. Because of the random size and spatial distribution of these domains, the unpoled SBN crystal belongs to the class of disordered nonlinear quadratic crystals. Such media have attracted a considerable attention in recent years due to their potential benefits for nonlinear parametric processes [10]. Despite being usually considered as disadvantageous in typical nonlinear optical phenomena, the randomness of the nonlinear media offers potential applications in broadband frequency generation [11–18] and ultra-short pulse monitoring [19]. These applications utilize the fact that the crystal with disordered anti-parallel domains forms, in fact, a two-dimensional photonic nonlinear structure with an infinite number of χ (2) reciprocal vectors corresponding to various local nonlinear gratings. Consequently, such structure enables phase matching of various parametric processes over an ultra-broad spectral range and in different spatial directions, including a direction transverse to the beam propagation [15, 20, 21]. Figure 1(a) represents a schematic illustration of such type of random crystal placed in a typical experimental set-up. The inverted ferroelectric domains are represented by bright and dark colors, respectively.

However, recent experiments with random domain SBN crystals also demonstrated that the emission pattern of the generated waves may differ drastically from sample to sample [see Fig. 1(b), 1(c)], being either spatially homogeneous or exhibiting distinct intensity peaks [16, 22]. This behavior points towards the difference in the actual distributions of ferroelectric domains in different samples. In fact, observations of random domains ranging from tens of nanometers to few microns have been reported in the literature [6, 8, 9, 13]. Moreover, it has been even suggested that the domain-walls induced light scattering of the fundamental wave plays significant role in the SH generation [17].

In order to obtain a better understanding of the nonlinear interaction in random media, in this work we analyze theoretically and numerically the role of the domain distribution on the efficiency and transverse characteristics of the SH emission in a quadratic random crystal. The numerical simulations obtained are supported by our experimental evidence of the SH emission (see for example Fig. 1(b), 1(c) and Ref. [20]).

Modeling random ferroelectric domain pattern

In order to simulate the 2D ferroelectric domain distribution for our study we will consider a simplified model which includes the most significant aspects of the experimentally reported distributions in SBN crystals [9, 17]. To mimic the process of domain formation we adopt the following strategy. Firstly, we assume that the individual domains (our building blocks) have form of rods with a circular transverse profile. While the assumption about circular shape follows from the experimental observations [23] it is, in fact, not that critical because, as we see below, the macroscopic pattern of nonlinearity distribution consists of regions of opposite sign of the nonlinearity which may have very complicated (not necessarily circular) shape. As we consider light propagation and emission in the plane perpendicular to the longer dimension of the domains only their transverse structure is taken into account here. A normal (Gaussian) distribution of the domain sizes is chosen with a certain mean diameter (ρ 0) and variance (σ). Using this distribution a number of domains is randomly generated and randomly placed in a rectangular area representing the size of the sample. In all the cases discussed here at least two thousand domains were used to create a random domain pattern even if the domains number that take part in the process may depend on actual geometry of interaction. As a rule of thumb this number should be large compared to the actual size of the input optical beam [24]. In the next step each domain is randomly assigned a particular sign of the nonlinearity (positive/negative). Some particular realizations of the 2D domains pattern for different choice of statistical parameters are shown in the top row of Fig. 2(a)–2(c). The black and white colors denote the antiparallel oriented domains with opposite signs of the quadratic nonlinearity. These kind of generated patterns are directly used in the numerical simulations. A completely random assignment leads sometimes to large agglomeration of domains with the same sign of the χ (2) nonlinearity [Fig.2(c) top row]. To remedy this, and to create structures with very fine domain pattern the sign of each particular domain was assigned taking into account the signs of its closest neighborhood, such as in Figs. 2(a) and 2(b). The algorithm implemented here integrates the area close to the domain under study; if in this area there is a prevalence of positive χ (2) sign, a negative sign will be chosen and vice versa. The non occupied zones among the domains are considered to form the domain walls and they amount to around 10% of the whole area of the sample which is in agreement with the experimental observations in similar poled structures [25]. In the region of the domain walls the nonlinearity is assigned a null value. However, we allowed in this region for a small linear refractive index change of the order of 10-3 to account for the strong discontinuity of the vector of spontaneous polarization [25, 26]. This weak linear index variation will produce a linear scattering of light. We will talk about this effect later in the paper.

 

Fig. 1. (a) - schematic representation of the second harmonic generation in multi-domain disordered structure in ferroelectric crystal. (b–c) Experimentally recorded various emission patterns in SHG in three different samples of as-grown SBN crystal with random domain distribution. (b) planar (almost 360°) transverse emission. The red arrow indicates the direction of the fundamental beam; (c) examples of SH emission from two different samples of as-grown SBN crystal; top graph: narrow angle forward emission with two distinctive maxima; bottom graph: wide angle forward emission.

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The insets in Fig. 2(a) and 2(b) depict corresponding histograms of normalized number of domains versus domain size. Such histograms were previously used to characterize actual samples of as grown SBN crystals with disordered domain pattern [17] but do not contain all the needed information on the distribution. For example, it is not possible to retrieve the distribution of the sign of the nonlinearity. Moreover, the histograms can be ambiguous since it is possible to choose either the diameter or the area of the domains as discriminating parameter. Notice how, starting from the same statistical parameters, it is possible to draw two completely different random patterns by either avoiding [Fig. 2(b)] or allowing the agglomeration [Fig. 2(c)].

 

Fig. 2. Examples of particular realizations of the random 2D domain pattern. (a) ρ 0 = 0.3μm, σ = 0.05μm, (b) ρ 0 = 0.3μm, σ = 0.1μm, (c) ρ0 = 0.3μm, σ = 0.1μm with agglomerations of the domains. The insets depict histograms of the resulting domain diameters. Bottom row: corresponding spatial Fourier spectrum of the domain patterns. Note that the agglomeration of domains shown in (c) results drastically changes its spatial Fourier spectrum and hence will have a strong impact on SH emission.

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Second harmonic emission by disordered domain structure

In order to fully characterize the domain pattern as a quasi-phase matching nonlinear photonic structure we use the spatial Fourier spectrum of the nonlinearity distribution. Such spectrum represents the domain of reciprocal vectors G⃗ which will be employed to fulfil the phase matching condition k⃗ 2 − 2k⃗ 1G⃗ = 0 which determines the direction and efficiency of the quadratic process. Here k⃗ 2, k⃗ 1 are the wave-vectors of the second harmonic and fundamental waves, respectively, and G⃗ can be expressed as G⃗ = (π/Λ)n⃗ where Λ represents periodicity along the direction n⃗. In the bottom row in Fig. 2 we depict the modulus of the Fourier spectrum corresponding to the domain patterns shown in the top row. In these graphs the strength of the spectral components is directly represented by their brightness. It is clear that different domain distributions lead to drastically different structure of the reciprocal space which, as we will show later, will subsequently affect the second harmonic emission. In fact, this property has been utilized to control the shape of the generated SH beam in specially designed structures [27, 28]. As it is clear from the bottom row of the Fig. 2(a), 2(b), a real spatially random domain pattern leads to a circularly shaped Fourier spectrum. The radius of the circle will be directly related to the mean domain size and its width from the radius will be related with the standard deviation. In particular, domain distributions with narrow (broad) standard deviation lead to the correspondingly narrow (broad) width of the reciprocal distribution of the G⃗ vectors. And domain distributions with small (big) mean domain size lead to the correspondingly large (small) radius of the circular distribution. As comparison, the bottom row of Fig. 2(c) depicts the spatial Fourier spectrum of the distribution with large rate of domains agglomeration. Here the structure consists effectively of large area domains of the size of the agglomerated areas, and consequently its spatial spectrum shrinks toward very small components. In order to see how the particular nonlinear structures affect emission of the second harmonic we need to consider the phase matching condition. In Fig. 3 we plot the three particular scenarios associated with the random domain structures. The green circle represents possible directions of the wave-vector of the second harmonic while the yellow ring represents the more considerable region of the reciprocal vectors G⃗ provided by the randomness of the domain pattern. As the pattern is isotropic so is the distribution of the G⃗ vectors. The most efficient second harmonic emission is expected to occur in the directions determined by the intersection of the green circle and yellow rings where the phase matching conditions is fulfilled with the stronger coefficients of the reciprocal spectrum. It is clear that because of availability of isotropically oriented G vectors the generation of the second harmonic is expected to be non-collinear with a broad spatial distribution of the intensity. The two cases shown in Fig. 3(a), 3(b) correspond to identical domain pattern but they differ in the wavelength of the fundamental wave. Hence, in the first case (a) the SH is expected to be emitted predominantly in two angular directions. On the other hand the same structure used with shorter fundamental wave [as in Fig. 3(b)] should lead to mostly forward emission centered around the direction of the fundamental wave. Finally, the domain pattern with much smaller average size is expected to lead to strongest emission oriented at large angles as in Fig. 3(c). As the geometric regions of the strongest phase matching shown in these three examples differ significantly one may expect this property to be reflected in the SH emission pattern. However, the actual spatial intensity distribution of the generated SH wave will be determined not only by the phase matching condition but also by the strength of the nonlinearity for the particular emission direction which is strongly affected by the randomness of the system.

 

Fig. 3. Diagram illustrating few possible scenarios of the phase matching condition for the SH emission in medium with disordered ferroelectric domain structure. Dashed green arrows indicate possible directions of strong emission of SH. (a–b) same disordered domain distribution, different wavelengths of the fundamental wave - λF (λF is shorter in case (b)); (c) random domain distribution with the mean value significantly smaller than in cases (a–b)

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In order to explore this further we employ the approach of Dolino [29] and Le Grand [24]. In this approach the intensity of the SH wave generated in the medium consisting of randomly distributed antiparallel domains can be expressed as

I(2ω)I(ω)defff(q,ρ0,σ)
 

Fig. 4. Theoretically predicted angular distribution of the second harmonic emission in the quadratic crystal with random domain distribution [Eq. (1)]. (a) constant average domain diameter ρ 0 = 1μm, (b) constant dispersion σ = 1μm. In all simulations we assumed λF = 1μm. (c) The effect of varying of the wavelength of the fundamental wave on the emission angle of the SH. Here ρ 0 = 1μm and σ = 1μm. In all graphs the dashed line represents phase matching condition with a single reciprocal vector |G⃗mean| = π/ρ 0.

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where I (ω) is the intensity of the fundamental wave and deff is the effective χ (2) nonlinearity which depends on material parameters and geometry of the interaction. The function f(q, ρ 0, σ) [Eq. (2)] represents the effect of disorder in the domain distribution.

f(q,ρ0,σ)=4Lq2.1eq2σ21+eσ2q2+2cos(qρ0)eσ2q2/2.

where L is the propagation distance and q = |Δk| is the absolute value of the phase mismatch. The linear dependence of I (2ω) on propagation distance is the signature of disorder which makes the process of SH build-up to be incoherent [10, 30–32]. As the phase mismatch parameter q is in our geometry of interaction uniquely linked to the angle between propagation direction of the SH and fundamental waves, the relation (1) provides, in fact, angular distribution of the SH intensity. The contour plots in Fig. 4 illustrate the relation Eq. (1) as a function of emission angle and/or statistical parameters of the domain distribution. From Fig. 4(a) it is evident that for low domain dispersion the structure is almost perfectly periodic leading to very directed transverse emission determined by the phase matching condition which in this case is satisfied in two particular directions of the SH. As the domain dispersion increases the more G vectors contribute towards the phase matching which results in a forward emission of the SH. Plot in Fig. 4(b) illustrates the effect of the mean value of the domain diameter. For fine domain pattern (at given dispersion) many G vectors participate in the SH generation leading to very broad emission. As the average domain diameter increases the SH emission becomes less spread approaching the forward emission for large ρ 0. Finally, in Fig. 4(c) we demonstrate the effect of varying the wavelength of the fundamental wave on the SH generation. In this particular example ρ 0 = σ = 1μm. The dashed line in all graphs in Fig. 4 represents emission of the SH determined solely by the phase matching condition taking into account only one reciprocal vector |G⃗mean| = 2π/(2ρ 0) corresponding to the mean diameter of the domains. It is clear that while this simplified approach is sufficiently accurate for weak disorder it leads to erroneous results for high degree of randomness of the periodic structure. In addition it cannot be used to determine the spatial distribution of the emitted SH.

 

Fig. 5. Numerical simulations - generation of the second harmonic by a femtosecond laser pulse in a disordered structure. A fundamental pulse in initially placed in the air and propagates toward the sample in the right direction. When it reaches the first air/sample interface a portion is backward reflected and the remaining part keep propagating forward. Due to the presence of the nonlinearity a SH pulse is generated and propagates forward. Left column: Spatial intensity distribution of the fundamental (Media 1) (a) and second harmonics (b-d) for three different random samples in three different numerical simulations. Middle column - the structure of the interacting waves in the spatial Fourier space. Note differences in the emission angle. Right column - energy of the fundamental and second harmonics as a function of the time of propagation. The statistical parameters of the domain distribution are: (b) - ρ 0 = 3μm, σ = 0.3μm, λF = 1.064μm, here the sample is bigger to arrange an enough number of domains (Media 2); (c) ρ 0 = 0.9μm, σ = 0.1μm, λF = 1.064μm (Media 3); (d) ρ 0 = 0.3μm, σ = 0.05μm, λF = 0.800μm (Media 4).

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Numerical simulations

To test the above discussed analytical predictions we resorted to numerical simulations of second harmonic generation in random domain structures. We numerically solved the Maxwell’s equations assuming that the random quadratic medium (Fig. 2) is illuminated with a Gaussian fundamental pulse. We used the material parameters which correspond to those of as-grown SBN crystals [26]. It should be stressed here that Vidal and Martorell [33] were perhaps the first ones to investigate numerically second harmonic emission in multi-domain crystal. They considered only 1D structure and used the matrix transfer approach. Our simulations of full Maxwell equations were conducted using optical pulses and finite beams in two dimensional medium. Therefore we could describe SH emission at arbitrary angle and did not have to perform statistical averaging in order to obtain relevant physical quantities such as the energy of the generated waves. To simulate the light propagation through SBN crystal we use a numerical model similar to that discussed before in [34] with the addition of a transverse coordinate to include diffraction and propagation in a two-dimensional plane. We use a Lorentz oscillator model to describe material dispersion and, for simplicity, we assume a TM-polarized incident pump field. The electric and magnetic fields may be written as a superposition of the harmonics as follows:

H=x̂l=1(Hy(z,y,t)eil(kzωt)+c.c.)
E=ŷl=1(Ey(z,y,t)eil(kzωt)+c.c.)+ẑl=1(Ez(z,y,t)eil(kzωt)+c.c.)

where Hy(z,y,t), Ey(z,y,t) and Ez(z,y,t) are generic, spatially- and temporally-dependent, complex envelope functions; k and ω are carrier wave vector and frequency, respectively, and l is an integer. Equations (3), (4) are a convenient representation of the fields, and no a priori assumptions are made about the field envelopes. The second order nonlinear polarization is P NL = χ 2 E 2. Expanding the field into its components yields nonlinear polarization terms at the fundamental and harmonics frequencies (taking only the leading terms): Pω(z,t) = 2χ 2 ω E * ω E 2ωand P2ω(z,t) = 2χ 2 2ω E 2 2ω. The linear response of the medium is described by a Lorentz oscillator model: ε(ω) = 1 − ω 2 p/(ω)2 + iγωω 2 r) and μ(ω) = 1, where γ, ωp, and ωr are the damping coefficient, the plasma and resonance frequencies, respectively. Assuming that polarization and currents may be decomposed as in Eqs. (3), (4), we obtain the following Maxwell-Lorentz system of equations for the lth field components, in the scaled two-dimensional space (, ξ) plus time (τ) coordinate system:

Hxτ=l(Hx+Ezsinθi+Eycosθi)Ezy˜+Exξ
Eyτ=l(Ey+Hxcosθi)4π(JyiβlPy)+4βlPy,NLPy,NLτ+Hxξ
Ezτ=l(Ez+Hxsinθi)4π(JziβlPz)+4βlPz,NLPz,NLτHxy˜
Jyτ=(2iβlγ)Jy+(βl2+βlβr2)Py+πωp2ω02Ey
Jzτ=(2iβlγ)Jz+(βl2+βlβr2)Pz+πωp2ω02Ez
Jy=Pyτ,Jz=Pzτ

In Eqs. (5)–(10), the functions J,P,PNL refer to linear electric currents, polarization, and nonlinear polarization, respectively. The coordinates are scaled so that ξ = z/λ 0, = y/λ 0, τ = ct/λ 0, ω 0 = 2πc/λ 0, where λ 0 = 1μm is just reference wavelength; γ, β = 2πlω/ω 0, β = 2πωr/ω 0, ωp are the scaled damping coefficient, wave-vector, resonance and electric plasma frequencies for the lth harmonic, respectively. θi is the angle of incidence of the pump field. The equations are solved using a split-step, fast Fourier transform-based pulse propagation algorithm that advances the fields in time [35]. Contrary to the most common nonlinear numerical codes, the one implemented here is able to solve the complete 2D vectorial Maxwell’s equations in the time domain without slowly varying envelope approximation. For this reason all angles of propagation and/or diffraction are allowed. Furthermore, no undepleted pump approximation is used. A number of numerical simulations have been performed with different working conditions. The fundamental pulse length in the plane of the nonlinear domain was in the range 6–30 optical cycles. We varied the fundamental wavelength from 790nm up to 1550nm. The data for the index of refraction were obtained fitting the Sellemier’s equation with the coefficient as in [26] to a general Lorentz curve. Figure 5 shows the results of some of these simulations obtained assuming the λF = 800,1064nm. In this figure the plots in the left column depict the spatial intensity distribution of the fundamental and generated second harmonics for three different random samples (from the top to bottom). Middle column shows the transverse structure of the fundamental and second harmonics in the spatial Fourier space (or equivalently, in the far field) and the plots in the right column depicts the energy of the fundamental and second harmonics as a function of the time of propagation. In the Fig. 5(a) is depicted the typical propagation of the fundamental pulse. Initially located in the air, it impinges normally in the sample. Due to the index mismatch, part of the pulse is backward reflected and the remaining is forward propagating in the material. The k-spectrum of the fundamental field shows a very narrow peak with coordinates kx = 2π/λ and ky = 0 during the first part of the propagation in the air and kx = 2πn(λF)/λF and ky = 0 after entering in the material, where n(λF) is the index of refraction at the fundamental frequency. Since the efficiency of the nonlinear process is low, the energy of the fundamental is practically constant during the propagation in the SBN. The graphs in Fig. 5 demonstrate dramatic variations in the far field of the emitted SH depending on the characteristics of the domain pattern. Note the differences in the emission angles in cases (b), (c) and (d) which correspond to the domain distributions from large to small average domain sizes, namely 3,0.9 and 0.3μm. The angular distribution of the intensity of the generated SH fields reflects the spatial distributions of the domains. As expected, the energy of the generated SH clearly displays a linear-like growth. This is indication of the incoherent character of the build-up of the SH and is in full agreement with earlier experiments and theoretical prediction [10, 30, 31]. The difference in the maximum reached values of the SH fields and energies is due to the different input intensities of the fundamental as initial condition. For example, in the simulation in Fig. 5(b) the input intensity is three orders of magnitude higher than the simulation in Fig. 5(c). Comparing the numerical simulations with the analytical predictions [Eq. (1)] we find them in very good agreement. Therefore the simple statistical approach appears to be a powerful tool in predicting spatial distribution of the generated harmonic in media with disordered domain structure. This suggests that experimental observation of second harmonic pattern can be employed to obtain information about the degree of the disorder of the ensuing domain structure. This aspect is currently being experimentally investigated. To give an idea of the power of this method, as it follows from the results of this study we can evaluate that to have second harmonic light propagating at the transverse and backward directions the sample needs to possess nonlinear domains at least of the order of 10 – 50nm. This consideration is very important as far as the choice of the domain visualization technique with appropriate resolution is concerned [36].

Finally, we would like to comment on the role of the light scattering on the SH emission. Due to the discontinuity in the direction of spontaneous polarization between domains a small jump of the linear index of refraction is present in the area of the domain walls. For instance, according to [26] this index modulation is Δn λ=852nm = 1.0×10-4 and Δn λ=404nm = 2.6×10-3. Obviously the strength of light scattering induced by the domain walls will be at least one order of magnitude weaker for infrared than visible light. Thus, the SH light will always experience stronger scattering than the fundamental wave while propagating in the same sample. It has been suggested before that the wide angle emission of the second harmonic in experiments with multi-domain SBN crystals could be attributed to the emission of the SH signal by scattered fundamental beam [17]. To test this hypothesis we included the effect of index variation in the domain walls in our numerical simulations. In fact all graphs depicted in Fig. 5 have been obtained assuming the index jumps at fundamental frequency of Δn = 0.01. No appreciable effect in both fundamental beam nor the second harmonic has been observed. This confirms that the transverse emission of the SH is caused solely by the spatial modulation of the nonlinearity resulting from the random domain pattern. However, it should be stressed that in the situation when the strong scattering of the second harmonic is present it may indeed affect its overall intensity distribution.

Conclusions

In conclusion, we have studied the effect of random domain distribution on the emission of second harmonic generation in a nonlinear crystal with disordered ferroelectric domains. We demonstrated that there is direct connection between the statistical properties of the domain structure and the far field of the second harmonic. This effect can be employed to characterize the degree of randomness of the quasi-phase matching domain pattern. We are currently working on experimental verification of these results in SBN crystals with disordered domain structure.

Acknowledgements

This work was supported by the Australian Research Council, Spanish Government (FIS2008-06024-C03-02), Catalan Government (2009 SGR 1168 and BE 2009) and the COST Action MP0702. V.R. thanks Cristian Ciracì for helpful discussions and suggestions with the images. The authors thank D. Neshev for collaboration.

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27. Y. Q. Qin, C. Zhang, and Y. Y. Zhu, et al. “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100, 063902 (2008). [CrossRef]   [PubMed]  

28. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009). [CrossRef]  

29. G. Dolino, “Effects of domain shapes on second harmonic scattering in Triglycine Sulfate,” Phys. Rev. B 6, 4025–4035 (1972). [CrossRef]  

30. E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001). [CrossRef]  

31. E. Yu. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Sov. J. Quantum Electron. 34, 227–232 (2004). [CrossRef]  

32. I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008). [CrossRef]  

33. X. Vidal and J. Martorell, “Generation of Light in Media with a Random Distribution of Nonlinear Domains,” Phys. Rev. Lett. 97, 013902 (2006). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

35. M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994). [CrossRef]  

36. E. Soergel, “Visualization of ferroelectric domains in bulk single crystals,” Appl. Phys. B 81, 729–752 (2005). [CrossRef]  

References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]
  2. P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 3523–39 (1963).
    [Crossref]
  3. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched 2nd harmonic-generation - tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [Crossref]
  4. S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
    [Crossref]
  5. U. K. Sapaev and G. Assanto, “Efficient high-harmonic generation in engineered quasi-phase matching gratings,” Opt. Express 16, 1–6 (2008).
    [Crossref] [PubMed]
  6. K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
    [Crossref]
  7. M. O. Ramirez, D. Jaque, L. Ivleva, and L. E. Bausa, “Evaluation of ytterbium doped strontium barium niobate as a potential tunable laser crystal in the visible,” J. Appl. Phys. 95, 6185–6191 (2004).
    [Crossref]
  8. V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, “Nanopolar structure in SrxBa1-xNb2O6 single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy,” Phys. Rev. B 77, 054105 (2008).
    [Crossref]
  9. R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
    [Crossref]
  10. M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
    [Crossref] [PubMed]
  11. M. Horowitz, A. Bekker, and B. Fischer, “Broad-band 2nd-harmonic generation in SrxBa1-xNb2O6 by spread-spectrum phase-matching with controllable domain gratings,” Appl. Phys. Lett. 62, 2619–2621 (1993).
    [Crossref]
  12. S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
    [Crossref]
  13. J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, “Simultaneous generation of coherent light in the three fundamental colors by quasicylindrical ferroelectric domains in Sr0.6Ba0.4(NbO3)2,” Appl. Phys. Lett 81, 4106–4108 (2002).
    [Crossref]
  14. J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
    [Crossref]
  15. A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90, 243901 (2003).
    [Crossref] [PubMed]
  16. R. Fischer, D. N. Neshev, S. M. Saltiel, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
    [Crossref]
  17. P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium Barium Niobate as a Multifunctional Two-Dimensional Nonlinear Photonic Glass,” Adv. Funct. Mater. 18, 709–715 (2008).
    [Crossref]
  18. W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Yu. Kivshar “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Opt. Express 17, 20117–20123 (2009).
    [Crossref] [PubMed]
  19. R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
    [Crossref]
  20. V. Roppo, D. Dumay, J. Trull, C. Cojocaru, S. M. Saltiel, K. Staliunas, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Planar second-harmonic generation with noncollinear pumps in disordered media,” Opt. Express 16, 14192–14199 (2008).
    [Crossref] [PubMed]
  21. S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, and Yu. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527–529 (2008).
    [Crossref] [PubMed]
  22. J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007).
    [Crossref] [PubMed]
  23. J. J. Romero, D. Jaque, and J. Garca Sol, “Diffuse multiself-frequency conversion processes in the blue and green by quasicylindrical ferroelectric domains in Nd3 + :Sr0.6Ba0.4(NbO3)2 laser crystal,” Appl. Phys. Lett. 78, 1961–1963 (2001).
    [Crossref]
  24. Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
    [Crossref]
  25. A. L. Aleksandrovskii, O. A. Gliko, I. I. Naumova, and V. I. Pryalkin, “Linear and nonlinear diffraction gratings in lithium niobate single crystals with a periodic domain structure,” Quantum Electron. 26, 641–643 (1996).
    [Crossref]
  26. Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
    [Crossref]
  27. Y. Q. Qin, C. Zhang, and Y. Y. Zhu, et al. “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100, 063902 (2008).
    [Crossref] [PubMed]
  28. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
    [Crossref]
  29. G. Dolino, “Effects of domain shapes on second harmonic scattering in Triglycine Sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
    [Crossref]
  30. E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001).
    [Crossref]
  31. E. Yu. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Sov. J. Quantum Electron. 34, 227–232 (2004).
    [Crossref]
  32. I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008).
    [Crossref]
  33. X. Vidal and J. Martorell, “Generation of Light in Media with a Random Distribution of Nonlinear Domains,” Phys. Rev. Lett. 97, 013902 (2006).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  35. M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
    [Crossref]
  36. E. Soergel, “Visualization of ferroelectric domains in bulk single crystals,” Appl. Phys. B 81, 729–752 (2005).
    [Crossref]

2009 (3)

W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Yu. Kivshar “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Opt. Express 17, 20117–20123 (2009).
[Crossref] [PubMed]

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

2008 (8)

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008).
[Crossref]

U. K. Sapaev and G. Assanto, “Efficient high-harmonic generation in engineered quasi-phase matching gratings,” Opt. Express 16, 1–6 (2008).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, “Nanopolar structure in SrxBa1-xNb2O6 single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy,” Phys. Rev. B 77, 054105 (2008).
[Crossref]

P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium Barium Niobate as a Multifunctional Two-Dimensional Nonlinear Photonic Glass,” Adv. Funct. Mater. 18, 709–715 (2008).
[Crossref]

V. Roppo, D. Dumay, J. Trull, C. Cojocaru, S. M. Saltiel, K. Staliunas, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Planar second-harmonic generation with noncollinear pumps in disordered media,” Opt. Express 16, 14192–14199 (2008).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, and Yu. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527–529 (2008).
[Crossref] [PubMed]

Y. Q. Qin, C. Zhang, and Y. Y. Zhu, et al. “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100, 063902 (2008).
[Crossref] [PubMed]

2007 (2)

J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
[Crossref]

2006 (2)

X. Vidal and J. Martorell, “Generation of Light in Media with a Random Distribution of Nonlinear Domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[Crossref]

2005 (1)

E. Soergel, “Visualization of ferroelectric domains in bulk single crystals,” Appl. Phys. B 81, 729–752 (2005).
[Crossref]

2004 (3)

M. O. Ramirez, D. Jaque, L. Ivleva, and L. E. Bausa, “Evaluation of ytterbium doped strontium barium niobate as a potential tunable laser crystal in the visible,” J. Appl. Phys. 95, 6185–6191 (2004).
[Crossref]

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
[Crossref] [PubMed]

E. Yu. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Sov. J. Quantum Electron. 34, 227–232 (2004).
[Crossref]

2003 (2)

J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
[Crossref]

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90, 243901 (2003).
[Crossref] [PubMed]

2002 (2)

J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, “Simultaneous generation of coherent light in the three fundamental colors by quasicylindrical ferroelectric domains in Sr0.6Ba0.4(NbO3)2,” Appl. Phys. Lett 81, 4106–4108 (2002).
[Crossref]

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

2001 (4)

Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
[Crossref]

J. J. Romero, D. Jaque, and J. Garca Sol, “Diffuse multiself-frequency conversion processes in the blue and green by quasicylindrical ferroelectric domains in Nd3 + :Sr0.6Ba0.4(NbO3)2 laser crystal,” Appl. Phys. Lett. 78, 1961–1963 (2001).
[Crossref]

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

1998 (1)

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[Crossref]

1997 (1)

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

1996 (1)

A. L. Aleksandrovskii, O. A. Gliko, I. I. Naumova, and V. I. Pryalkin, “Linear and nonlinear diffraction gratings in lithium niobate single crystals with a periodic domain structure,” Quantum Electron. 26, 641–643 (1996).
[Crossref]

1994 (1)

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[Crossref]

1993 (1)

M. Horowitz, A. Bekker, and B. Fischer, “Broad-band 2nd-harmonic generation in SrxBa1-xNb2O6 by spread-spectrum phase-matching with controllable domain gratings,” Appl. Phys. Lett. 62, 2619–2621 (1993).
[Crossref]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched 2nd harmonic-generation - tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

1972 (1)

G. Dolino, “Effects of domain shapes on second harmonic scattering in Triglycine Sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
[Crossref]

1963 (1)

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 3523–39 (1963).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Akozbek, N.

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).
[Crossref] [PubMed]

Aleksandrovskii, A. L.

A. L. Aleksandrovskii, O. A. Gliko, I. I. Naumova, and V. I. Pryalkin, “Linear and nonlinear diffraction gratings in lithium niobate single crystals with a periodic domain structure,” Quantum Electron. 26, 641–643 (1996).
[Crossref]

Aragó, C.

J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
[Crossref]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Assanto, G.

Aubry, R.

Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
[Crossref]

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
[Crossref] [PubMed]

Bausa, L. E.

P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium Barium Niobate as a Multifunctional Two-Dimensional Nonlinear Photonic Glass,” Adv. Funct. Mater. 18, 709–715 (2008).
[Crossref]

M. O. Ramirez, D. Jaque, L. Ivleva, and L. E. Bausa, “Evaluation of ytterbium doped strontium barium niobate as a potential tunable laser crystal in the visible,” J. Appl. Phys. 95, 6185–6191 (2004).
[Crossref]

Bekker, A.

M. Horowitz, A. Bekker, and B. Fischer, “Broad-band 2nd-harmonic generation in SrxBa1-xNb2O6 by spread-spectrum phase-matching with controllable domain gratings,” Appl. Phys. Lett. 62, 2619–2621 (1993).
[Crossref]

Betzler, K.

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90, 243901 (2003).
[Crossref] [PubMed]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Bloemer, M. J.

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).
[Crossref] [PubMed]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched 2nd harmonic-generation - tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Centini, M.

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).
[Crossref] [PubMed]

Chirkin, A. S.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008).
[Crossref]

E. Yu. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Sov. J. Quantum Electron. 34, 227–232 (2004).
[Crossref]

E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Cojocaru, C.

Crenshaw, M. E.

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[Crossref]

Dec, J.

V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, “Nanopolar structure in SrxBa1-xNb2O6 single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy,” Phys. Rev. B 77, 054105 (2008).
[Crossref]

DeMattei, R. C.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[Crossref]

Dolino, G.

G. Dolino, “Effects of domain shapes on second harmonic scattering in Triglycine Sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
[Crossref]

Dörfler, U.

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Dumay, D.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Fazio, E.

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).
[Crossref] [PubMed]

Feigelson, R. S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[Crossref]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched 2nd harmonic-generation - tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Fischer, B.

M. Horowitz, A. Bekker, and B. Fischer, “Broad-band 2nd-harmonic generation in SrxBa1-xNb2O6 by spread-spectrum phase-matching with controllable domain gratings,” Appl. Phys. Lett. 62, 2619–2621 (1993).
[Crossref]

Fischer, R.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, and Yu. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527–529 (2008).
[Crossref] [PubMed]

J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
[Crossref]

R. Fischer, D. N. Neshev, S. M. Saltiel, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[Crossref]

Foreman, J. V.

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).
[Crossref] [PubMed]

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P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 3523–39 (1963).
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Gainutdinov, R. V.

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

García Solé, J.

J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
[Crossref]

J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, “Simultaneous generation of coherent light in the three fundamental colors by quasicylindrical ferroelectric domains in Sr0.6Ba0.4(NbO3)2,” Appl. Phys. Lett 81, 4106–4108 (2002).
[Crossref]

Gliko, O. A.

A. L. Aleksandrovskii, O. A. Gliko, I. I. Naumova, and V. I. Pryalkin, “Linear and nonlinear diffraction gratings in lithium niobate single crystals with a periodic domain structure,” Quantum Electron. 26, 641–643 (1996).
[Crossref]

Gonzalo, J. A.

J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
[Crossref]

Gotoh, Y.

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

Granzow, T.

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

Gruverman, A.

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

Haidar, R.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
[Crossref] [PubMed]

Haus, J. W.

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).
[Crossref] [PubMed]

Herrero, R.

Higuchi, S.

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

Horowitz, M.

M. Horowitz, A. Bekker, and B. Fischer, “Broad-band 2nd-harmonic generation in SrxBa1-xNb2O6 by spread-spectrum phase-matching with controllable domain gratings,” Appl. Phys. Lett. 62, 2619–2621 (1993).
[Crossref]

Ivleva, L.

M. O. Ramirez, D. Jaque, L. Ivleva, and L. E. Bausa, “Evaluation of ytterbium doped strontium barium niobate as a potential tunable laser crystal in the visible,” J. Appl. Phys. 95, 6185–6191 (2004).
[Crossref]

Ivleva, L. I.

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

Jaque, D.

M. O. Ramirez, D. Jaque, L. Ivleva, and L. E. Bausa, “Evaluation of ytterbium doped strontium barium niobate as a potential tunable laser crystal in the visible,” J. Appl. Phys. 95, 6185–6191 (2004).
[Crossref]

J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
[Crossref]

J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, “Simultaneous generation of coherent light in the three fundamental colors by quasicylindrical ferroelectric domains in Sr0.6Ba0.4(NbO3)2,” Appl. Phys. Lett 81, 4106–4108 (2002).
[Crossref]

J. J. Romero, D. Jaque, and J. Garca Sol, “Diffuse multiself-frequency conversion processes in the blue and green by quasicylindrical ferroelectric domains in Nd3 + :Sr0.6Ba0.4(NbO3)2 laser crystal,” Appl. Phys. Lett. 78, 1961–1963 (2001).
[Crossref]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched 2nd harmonic-generation - tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Kalinowski, K.

Kaminskii, A. A.

J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, “Simultaneous generation of coherent light in the three fundamental colors by quasicylindrical ferroelectric domains in Sr0.6Ba0.4(NbO3)2,” Appl. Phys. Lett 81, 4106–4108 (2002).
[Crossref]

E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Kawai, S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[Crossref]

Kitamura, K.

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

Kivshar, Yu.

Kivshar, Yu. S.

Kleemann, W.

V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, “Nanopolar structure in SrxBa1-xNb2O6 single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy,” Phys. Rev. B 77, 054105 (2008).
[Crossref]

Kong, Y.

Krolikowski, W.

W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Yu. Kivshar “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Opt. Express 17, 20117–20123 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, and Yu. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527–529 (2008).
[Crossref] [PubMed]

V. Roppo, D. Dumay, J. Trull, C. Cojocaru, S. M. Saltiel, K. Staliunas, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Planar second-harmonic generation with noncollinear pumps in disordered media,” Opt. Express 16, 14192–14199 (2008).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
[Crossref]

J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[Crossref]

Kupecek, Ph.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
[Crossref] [PubMed]

Le Grand, Y.

Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
[Crossref]

Lee, H. S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[Crossref]

Lemasson, Ph.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
[Crossref] [PubMed]

Lukasiewicz, T.

V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, “Nanopolar structure in SrxBa1-xNb2O6 single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy,” Phys. Rev. B 77, 054105 (2008).
[Crossref]

Lysova, O. A.

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched 2nd harmonic-generation - tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Martorell, J.

X. Vidal and J. Martorell, “Generation of Light in Media with a Random Distribution of Nonlinear Domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

Mattauch, S.

Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
[Crossref]

Ming, N. B.

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Molina, P.

P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium Barium Niobate as a Multifunctional Two-Dimensional Nonlinear Photonic Glass,” Adv. Funct. Mater. 18, 709–715 (2008).
[Crossref]

Morozov, E. Yu.

E. Yu. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Sov. J. Quantum Electron. 34, 227–232 (2004).
[Crossref]

E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Nakamura, M.

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

Naumova, I. I.

A. L. Aleksandrovskii, O. A. Gliko, I. I. Naumova, and V. I. Pryalkin, “Linear and nonlinear diffraction gratings in lithium niobate single crystals with a periodic domain structure,” Quantum Electron. 26, 641–643 (1996).
[Crossref]

Neshev, D. N.

W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Yu. Kivshar “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Opt. Express 17, 20117–20123 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, and Yu. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527–529 (2008).
[Crossref] [PubMed]

V. Roppo, D. Dumay, J. Trull, C. Cojocaru, S. M. Saltiel, K. Staliunas, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Planar second-harmonic generation with noncollinear pumps in disordered media,” Opt. Express 16, 14192–14199 (2008).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
[Crossref]

J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[Crossref]

Odiu, C.

Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
[Crossref]

Ogawa, T.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[Crossref]

Ozheredov, I. A.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008).
[Crossref]

Pankrath, R.

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Pettazzi, F.

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).
[Crossref] [PubMed]

Poetsch, Ch.

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

Pryalkin, V. I.

A. L. Aleksandrovskii, O. A. Gliko, I. I. Naumova, and V. I. Pryalkin, “Linear and nonlinear diffraction gratings in lithium niobate single crystals with a periodic domain structure,” Quantum Electron. 26, 641–643 (1996).
[Crossref]

Qin, Y. Q.

Y. Q. Qin, C. Zhang, and Y. Y. Zhu, et al. “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100, 063902 (2008).
[Crossref] [PubMed]

Ramirez, M. O.

P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium Barium Niobate as a Multifunctional Two-Dimensional Nonlinear Photonic Glass,” Adv. Funct. Mater. 18, 709–715 (2008).
[Crossref]

M. O. Ramirez, D. Jaque, L. Ivleva, and L. E. Bausa, “Evaluation of ytterbium doped strontium barium niobate as a potential tunable laser crystal in the visible,” J. Appl. Phys. 95, 6185–6191 (2004).
[Crossref]

Razgonov, I. I.

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

Romero, J. J.

J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
[Crossref]

J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, “Simultaneous generation of coherent light in the three fundamental colors by quasicylindrical ferroelectric domains in Sr0.6Ba0.4(NbO3)2,” Appl. Phys. Lett 81, 4106–4108 (2002).
[Crossref]

J. J. Romero, D. Jaque, and J. Garca Sol, “Diffuse multiself-frequency conversion processes in the blue and green by quasicylindrical ferroelectric domains in Nd3 + :Sr0.6Ba0.4(NbO3)2 laser crystal,” Appl. Phys. Lett. 78, 1961–1963 (2001).
[Crossref]

Roppo, V.

Rosencher, E.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
[Crossref] [PubMed]

Rouede, D.

Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
[Crossref]

Saltiel, S. M.

W. Wang, V. Roppo, K. Kalinowski, Y. Kong, D. N. Neshev, C. Cojocaru, J. Trull, R. Vilaseca, K. Staliunas, W. Krolikowski, S. M. Saltiel, and Yu. Kivshar “Third-harmonic generation via broadband cascading in disordered quadratic nonlinear media,” Opt. Express 17, 20117–20123 (2009).
[Crossref] [PubMed]

V. Roppo, D. Dumay, J. Trull, C. Cojocaru, S. M. Saltiel, K. Staliunas, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Planar second-harmonic generation with noncollinear pumps in disordered media,” Opt. Express 16, 14192–14199 (2008).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, and Yu. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527–529 (2008).
[Crossref] [PubMed]

J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007).
[Crossref] [PubMed]

R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
[Crossref]

R. Fischer, D. N. Neshev, S. M. Saltiel, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[Crossref]

Sapaev, U. K.

Scalora, M.

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).
[Crossref] [PubMed]

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[Crossref]

Shumitski, A. V.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008).
[Crossref]

Shutov, I. V.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008).
[Crossref]

Shvartsman, V. V.

V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, “Nanopolar structure in SrxBa1-xNb2O6 single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy,” Phys. Rev. B 77, 054105 (2008).
[Crossref]

Sibilia, C.

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).
[Crossref] [PubMed]

Soergel, E.

E. Soergel, “Visualization of ferroelectric domains in bulk single crystals,” Appl. Phys. B 81, 729–752 (2005).
[Crossref]

Sol, J. Garca

J. J. Romero, D. Jaque, and J. Garca Sol, “Diffuse multiself-frequency conversion processes in the blue and green by quasicylindrical ferroelectric domains in Nd3 + :Sr0.6Ba0.4(NbO3)2 laser crystal,” Appl. Phys. Lett. 78, 1961–1963 (2001).
[Crossref]

Staliunas, K.

Sukhorukov, A. A.

R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
[Crossref]

Takekawa, S.

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

Terabe, K.

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

Tolstikhina, A. L.

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

Trull, J.

Tunyagi, A. R.

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90, 243901 (2003).
[Crossref] [PubMed]

Ulex, M.

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90, 243901 (2003).
[Crossref] [PubMed]

Vidal, X.

X. Vidal and J. Martorell, “Generation of Light in Media with a Random Distribution of Nonlinear Domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

Vilaseca, R.

Volk, T. R.

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Wang, W.

Ward, J. F.

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 3523–39 (1963).
[Crossref]

Wohlecke, M.

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

Woike, Th.

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

Yusupov, D. B.

E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Zhang, C.

Y. Q. Qin, C. Zhang, and Y. Y. Zhu, et al. “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100, 063902 (2008).
[Crossref] [PubMed]

Zhu, S.

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Zhu, Y. Y.

Y. Q. Qin, C. Zhang, and Y. Y. Zhu, et al. “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100, 063902 (2008).
[Crossref] [PubMed]

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Adv. Funct. Mater. (1)

P. Molina, M. O. Ramirez, and L. E. Bausa, “Strontium Barium Niobate as a Multifunctional Two-Dimensional Nonlinear Photonic Glass,” Adv. Funct. Mater. 18, 709–715 (2008).
[Crossref]

Appl. Phys. B (2)

R. V. Gainutdinov, T. R. Volk, O. A. Lysova, I. I. Razgonov, A. L. Tolstikhina, and L. I. Ivleva, “Recording of domains and regular domain patterns in strontium barium niobate crystals in the field of atomic force microscope,” Appl. Phys. B 95, 505–512 (2009).
[Crossref]

E. Soergel, “Visualization of ferroelectric domains in bulk single crystals,” Appl. Phys. B 81, 729–752 (2005).
[Crossref]

Appl. Phys. Lett (1)

J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, “Simultaneous generation of coherent light in the three fundamental colors by quasicylindrical ferroelectric domains in Sr0.6Ba0.4(NbO3)2,” Appl. Phys. Lett 81, 4106–4108 (2002).
[Crossref]

Appl. Phys. Lett. (6)

M. Horowitz, A. Bekker, and B. Fischer, “Broad-band 2nd-harmonic generation in SrxBa1-xNb2O6 by spread-spectrum phase-matching with controllable domain gratings,” Appl. Phys. Lett. 62, 2619–2621 (1993).
[Crossref]

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nb2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[Crossref]

R. Fischer, D. N. Neshev, S. M. Saltiel, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[Crossref]

K. Terabe, S. Takekawa, M. Nakamura, K. Kitamura, S. Higuchi, Y. Gotoh, and A. Gruverman, “Imaging and engineering the nanoscale-domain structure of a Sr0.61Ba0.39Nb2O6 crystal using a scanning force microscope,” Appl. Phys. Lett. 81, 2044–2046 (2002).
[Crossref]

R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Monitoring ultrashort pulses by the transverse frequency doubling of counterpropagating pulses in random media,” Appl. Phys. Lett. 91, 031104 (2007).
[Crossref]

J. J. Romero, D. Jaque, and J. Garca Sol, “Diffuse multiself-frequency conversion processes in the blue and green by quasicylindrical ferroelectric domains in Nd3 + :Sr0.6Ba0.4(NbO3)2 laser crystal,” Appl. Phys. Lett. 78, 1961–1963 (2001).
[Crossref]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched 2nd harmonic-generation - tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

J. Appl. Phys. (2)

M. O. Ramirez, D. Jaque, L. Ivleva, and L. E. Bausa, “Evaluation of ytterbium doped strontium barium niobate as a potential tunable laser crystal in the visible,” J. Appl. Phys. 95, 6185–6191 (2004).
[Crossref]

J. J. Romero, C. Aragó, J. A. Gonzalo, D. Jaque, and J. García Solé, “Spectral and thermal properties of quasiphase- matching second-harmonic generation in Nd3+:Sr0.6Ba0.4(NbO3)2 multi-self-frequency-converter nonlinear crystals,” J. Appl. Phys. 93, 3111–3113 (2003).
[Crossref]

JETP Lett. (1)

E. Yu. Morozov, A. A. Kaminskii, A. S. Chirkin, and D. B. Yusupov, “Second Optical Harmonic Generation in Nonlinear Crystals with a Disordered Domain Structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Nature (1)

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London)  432, 374–376 (2004).
[Crossref] [PubMed]

Nature Photonics (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Opt. Commun. (2)

Y. Le Grand, D. Rouede, C. Odiu, R. Aubry, and S. Mattauch, “Second-harmonic scattering by domains in RbH2PO4 ferroelectric,” Opt. Commun. 200, 249–260 (2001).
[Crossref]

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Opt. Spectroscopy (1)

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulses in the Laue scheme,” Opt. Spectroscopy 105, 79–84 (2008).
[Crossref]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Phys. Rev. B (2)

V. V. Shvartsman, W. Kleemann, T. Lukasiewicz, and J. Dec, “Nanopolar structure in SrxBa1-xNb2O6 single crystals tuned by Sr/Ba ratio and investigated by piezoelectric force microscopy,” Phys. Rev. B 77, 054105 (2008).
[Crossref]

G. Dolino, “Effects of domain shapes on second harmonic scattering in Triglycine Sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
[Crossref]

Phys. Rev. Lett. (4)

X. Vidal and J. Martorell, “Generation of Light in Media with a Random Distribution of Nonlinear Domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Y. Q. Qin, C. Zhang, and Y. Y. Zhu, et al. “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100, 063902 (2008).
[Crossref] [PubMed]

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90, 243901 (2003).
[Crossref] [PubMed]

Phys. Stat. Solidi A (1)

Th. Woike, T. Granzow, U. Dörfler, Ch. Poetsch, M. Wohlecke, and R. Pankrath, ”Refractive Indices of Congruently Melting Sr0:61Ba0:39Nb2O6,” Phys. Stat. Solidi A 186, R13–R15 (2001).
[Crossref]

Quantum Electron. (1)

A. L. Aleksandrovskii, O. A. Gliko, I. I. Naumova, and V. I. Pryalkin, “Linear and nonlinear diffraction gratings in lithium niobate single crystals with a periodic domain structure,” Quantum Electron. 26, 641–643 (1996).
[Crossref]

Rev. Mod. Phys. (1)

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 3523–39 (1963).
[Crossref]

Science (1)

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Sov. J. Quantum Electron. (1)

E. Yu. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Sov. J. Quantum Electron. 34, 227–232 (2004).
[Crossref]

Supplementary Material (4)

» Media 1: MOV (106 KB)     
» Media 2: MOV (93 KB)     
» Media 3: MOV (136 KB)     
» Media 4: MOV (178 KB)     

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Figures (5)

Fig. 1.
Fig. 1. (a) - schematic representation of the second harmonic generation in multi-domain disordered structure in ferroelectric crystal. (b–c) Experimentally recorded various emission patterns in SHG in three different samples of as-grown SBN crystal with random domain distribution. (b) planar (almost 360°) transverse emission. The red arrow indicates the direction of the fundamental beam; (c) examples of SH emission from two different samples of as-grown SBN crystal; top graph: narrow angle forward emission with two distinctive maxima; bottom graph: wide angle forward emission.
Fig. 2.
Fig. 2. Examples of particular realizations of the random 2D domain pattern. (a) ρ 0 = 0.3μm, σ = 0.05μm, (b) ρ 0 = 0.3μm, σ = 0.1μm, (c) ρ0 = 0.3μm, σ = 0.1μm with agglomerations of the domains. The insets depict histograms of the resulting domain diameters. Bottom row: corresponding spatial Fourier spectrum of the domain patterns. Note that the agglomeration of domains shown in (c) results drastically changes its spatial Fourier spectrum and hence will have a strong impact on SH emission.
Fig. 3.
Fig. 3. Diagram illustrating few possible scenarios of the phase matching condition for the SH emission in medium with disordered ferroelectric domain structure. Dashed green arrows indicate possible directions of strong emission of SH. (a–b) same disordered domain distribution, different wavelengths of the fundamental wave - λF (λF is shorter in case (b)); (c) random domain distribution with the mean value significantly smaller than in cases (a–b)
Fig. 4.
Fig. 4. Theoretically predicted angular distribution of the second harmonic emission in the quadratic crystal with random domain distribution [Eq. (1)]. (a) constant average domain diameter ρ 0 = 1μm, (b) constant dispersion σ = 1μm. In all simulations we assumed λF = 1μm. (c) The effect of varying of the wavelength of the fundamental wave on the emission angle of the SH. Here ρ 0 = 1μm and σ = 1μm. In all graphs the dashed line represents phase matching condition with a single reciprocal vector |G⃗mean | = π/ρ 0.
Fig. 5.
Fig. 5. Numerical simulations - generation of the second harmonic by a femtosecond laser pulse in a disordered structure. A fundamental pulse in initially placed in the air and propagates toward the sample in the right direction. When it reaches the first air/sample interface a portion is backward reflected and the remaining part keep propagating forward. Due to the presence of the nonlinearity a SH pulse is generated and propagates forward. Left column: Spatial intensity distribution of the fundamental (Media 1) (a) and second harmonics (b-d) for three different random samples in three different numerical simulations. Middle column - the structure of the interacting waves in the spatial Fourier space. Note differences in the emission angle. Right column - energy of the fundamental and second harmonics as a function of the time of propagation. The statistical parameters of the domain distribution are: (b) - ρ 0 = 3μm, σ = 0.3μm, λF = 1.064μm, here the sample is bigger to arrange an enough number of domains (Media 2); (c) ρ 0 = 0.9μm, σ = 0.1μm, λF = 1.064μm (Media 3); (d) ρ 0 = 0.3μm, σ = 0.05μm, λF = 0.800μm (Media 4).

Equations (10)

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I ( 2 ω ) I ( ω ) d e f f f ( q , ρ 0 , σ )
f ( q , ρ 0 , σ ) = 4 L q 2 . 1 e q 2 σ 2 1 + e σ 2 q 2 + 2 cos ( q ρ 0 ) e σ 2 q 2 / 2 .
H = x ̂ l = 1 ( H y ( z , y , t ) e il ( kz ωt ) + c . c . )
E = y ̂ l = 1 ( E y ( z , y , t ) e il ( kz ωt ) + c . c . ) + z ̂ l = 1 ( E z ( z , y , t ) e il ( kz ωt ) + c . c . )
H x τ = l ( H x + E z sin θ i + E y cos θ i ) E z y ˜ + E x ξ
E y τ = l ( E y + H x cos θ i ) 4 π ( J y i β l P y ) + 4 β l P y , NL P y , NL τ + H x ξ
E z τ = l ( E z + H x sin θ i ) 4 π ( J z i β l P z ) + 4 β l P z , NL P z , NL τ H x y ˜
J y τ = ( 2 i β l γ ) J y + ( β l 2 + β l β r 2 ) P y + π ω p 2 ω 0 2 E y
J z τ = ( 2 i β l γ ) J z + ( β l 2 + β l β r 2 ) P z + π ω p 2 ω 0 2 E z
J y = P y τ , J z = P z τ

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