# Modeling of Random Quasi-Phase-Matching in Birefringent ...

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Modeling of Random Quasi-Phase-Matching in Birefringent Disordered Media

Jolanda S. Muller, Andrea Morandi, Rachel Grange, and Romolo Savo∗

Optical Nanomaterial Group, Institute for Quantum Electronics,Department of Physics, ETH Zurich, Zurich, Switzerland

(Dated: November 23, 2020)

We provide a vectorial model to simulate second-harmonic generation (SHG) in birefringent,

transparent media with an arbitrary configuration of nonlinear (χ(2)) crystalline grains. We applythis model on disordered assemblies of LiNbO3 and BaTiO3 grains to identify the influence of thebirefringence on the random quasi-phase-matching process. We show that in monodispersed assem-blies, the birefringence relaxes the grain size dependence of the SHG efficiency. In polydispersedassemblies with sufficiently large grains, we find that the birefringence introduces an SHG efficiencyenhancement of up to 54% compared to isotropic reference crystals, which is grain size independent.This enhancement increases linearly with the grain size, if the birefringent grains can be phasematched. These two different scaling behaviours are used in Kurtz and Perry’s powder-technique toidentify the phase-matchability of a material. We show on the example of LiNbO3 and ADP thatthis technique cannot be applied when the grains get smaller than the coherence length, becausethe SHG scaling with the grain size becomes material specific.

Optical frequency conversion through phase match-ing in birefringent nonlinear (χ(2)) crystals has enableda variety of applications in laser technology [1, 2] andplays a key role in quantum source schemes [3]. Thebirefringence provides an orientation- and polarization-dependent refractive index, which is used to compen-sate for the phase lag accumulated by the mixing waves.This avoids the destructive interference of the generatedwaves, which starts to occur at lengths larger than the co-herence length in any not phase-matched configuration.Random quasi-phase-matching (RQPM) is an alternativeapproach for removing destructive interference in opticalfrequency conversion [4, 5]. It relies on the disordered dis-tribution of the χ(2)-domains of certain media, such aspolycrystalline materials [5–7], scattering powders [8–11]and bottom-up assembled photonic structures [12, 13].Under a pump excitation the domains, or grains, gener-ate nonlinear waves with random phases and amplitudes.This leads to the cancellation of the interference termsand to the linear accumulation of the generated powerwith the number of domains. While RQPM is gener-ally less efficient than phase matching, it provides numer-ous advantages, such as an ultra-broad acceptance band-width (tunability of the pump over hundreds of nanome-ters), the use of low-cost materials, and relaxed con-straints on the polarization and the angle of incidenceof the pump. These features make RQPM attractivefor a variety of applications, such as optical-parametric-oscillators [14], ultrafast mid-infrared lasers [15], and ul-trafast autocorrelators [16]. The interest on RQPM hasbeen largely focused on isotropic (i.e. non-birefringent)crystals such as ZnSe, both experimentally and theoret-ically [5, 17, 18]—one reason for this is that RQPM en-ables taking advantage of the large nonlinearity of theseisotropic crystals, even if they are non-phase-matchable.First modeling of RQPM relied on the scalar approxima-

tion of both the optical field and the χ(2)-susceptibiliy ofthe grains [17, 19]. Very recently, more comprehensivevectorial models have been developed to study the non-linear speckle statistics [18] and the supercontinuum gen-eration from isotropic disordered polycrystals [20]. SHGin birefringent disordered materials has only been mod-eled by averaging over the single grain generation, with-out considering the effects of the propagation through thebirefringent medium on the polarization states [21, 22].Up to now, no comprehensive model that explicitly ac-counts for the birefringence in RQPM has been presented.

Here, we introduce a vectorial model that considersthe full three dimensional rotation (i.e. full χ(2)-tensor)as well as the birefringence of each grain. The generationand propagation of the waves is calculated from grain-to-grain, considering the phases of the ordinary and extraor-dinary beam components separately. This is done withinthe approximation of no scattering at the grain-to-grainboundaries. We explore the effects of the birefringenceon RQPM by calculating the second-harmonic genera-tion (SHG) in disordered assemblies of lithium niobate(LiNbO3) and barium titanate (BaTiO3). We find thatin general the birefringence is beneficial for RQPM. Inmonodispersed assemblies, the birefringence relaxes thegrain size dependence of the SHG efficiency. In poly-dispersed assemblies, it introduces an efficiency enhance-ment of up to 54% compared to an isotropic referencematerial. Thanks to the birefringence, our model cansimulate grains in a phase-matchable regime, in whichthe SHG enhancement increases linearly with the grainsize. This allows us to draw a comparison to the powder-characterization technique of Kurtz and Perry [8, 22],which is widely applied to identify the phase-matchabilityof a crystalline material. Contrary to the established be-haviour at large grain sizes, we find that the SHG scal-ing with the grain size cannot unequivocally identify thephase-matchability of a crystal when the grain size issmaller than the coherence length.

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I. MODEL

The presented model considers a disordered threedimensional cuboid structure, consisting of quadratic(χ(2) 6= 0), birefringent, crystalline grains, as depicted inFig. 1a. Similarly to other models [17, 18], the three di-mensional problem is reduced to a one dimensional prob-lem by breaking down the cuboid into multiple paral-lel one dimensional sticks. The SHG of each individualstick is calculated separately. Fig. 1b shows one disor-der configuration of such a stick, in which the grainshave random sizes and orientations. The size Xn is ran-domly chosen from a Gaussian distribution with an av-erage grain size X and a polydispersity σ. The randomorientation is defined through the rotation between thelab frame (a,b,c) and the crystal frame of each grain(x,y,z), with the Euler angles (ϕ, ϑ, γ)n, as shown inFig. 1e. For this, the lab frame axes are defined asthe c-axis parallel to the propagation direction k andthe two orthogonal directions a and b, while the z axisis the optic axis of the crystal. For a uniform distri-bution of crystal orientations, we choose the rotationaround z as ϕ ∈ [0, 2π], the angle between k and z asϑ = arccos(u) with u ∈ [−1, 1] [18], and the rotationaround k as γ ∈ [0, 2π]. The refractive indices of theordinary (o) and extraordinary (e) axes are given by no

and ne(ϑ) =(

sin(ϑ)2/(ne)2 + cos(ϑ)2/n2

o

)−1/2, with ne

the extraordinary refractive index at ϑ = π/2 [1]. A visu-alization of the calculation process in one single grain isshown in Fig. 1d. The incoming beam (pump) is definedas the plane wave (red)

E(ω) = (eaeiφa cosβ + ebeiφb sinβ)Eωei(kc−ωt)

with amplitude Eω, frequency ω, wavevector k, the start-ing phases φa and φb, and a polarisation angle β in the labframe. The vectors ea and eb are the unit vectors alongthe a and b axes. At the beginning of the grain, the elec-tric fields in the lab frame Elab, are transformed into thereference frame of the respective crystal Ecry = R ·Elab

(Fig. 1e), where R = Z1(ϕ)X2(ϑ)Z3(γ) is the Euler-transformation matrix for a rotation around the originalz-axis (Z1 = z) by ϕ, the new x-axis (X2 = o) by ϑ andfinal z-axis (Z3 = k) by γ (see Supplementaries II). Inthe crystal frame (Fig. 1d), the beams are decomposedinto their components along the o and e axes and eachpolarisation combination of the pump (oo, eo, oe, ee) gen-erates a second-harmonic field Eugen(2ω,Xn) (blue) along

u ∈ {o, e}, which is given at the end of the nth singlegrain according to:

Eugen(2ω,Xn) =∑

v,w

i(2ω)2

2ε0c2ku3Pu,vw

(ei∆k

u,vwXn − 1

i∆ku,vw

)eik

u3Xn

(1)With the phase mismatch ∆ku,vw = kv(ω) + kw(ω) −ku(2ω), the wave vector ku3 of the second-harmonic alongu, and Pu,vw = 〈eu,P vw〉 the second-harmonic po-larization along v, w ∈ {o, e} projected onto the o/e-

FIG. 1. (a) Sketch of a three dimensional disordered assemblyof second-order nonlinear grains, consisting of multiple par-allel one dimensional sticks. (b) Each stick contains grainsof varying size Xn and orientation (ϕ, ϑ, γ)n, represented bythe variation in colour. (c) Phasor (an) representation ofthe interference between the second-harmonic waves gener-ated by the grains within the stick. Each stick correspondsto a single random walk in the SHG complex plane, in whichthe step length is the amplitude of the second-harmonic field.(d) Schematics of the process to propagate, generate and in-terfere the beams. The input (left) is transformed into thecrystal frame, propagated, summed with the generated SHG,and transformed back into the lab frame. (e) Depiction of theangles between the crystal frame (x,y,z) and the lab frame(a,b,c) with the transformation matrix R. β is the polarisa-tion angle of the input beam in case of a linearly polarizedbeam (φa = φb). The ordinary/extraordinary axes are indi-cated with o and e.

axis (eu unit vector along u ∈ {o, e}), where P vwi =2ε0∑jk dijkE

vjE

wk . The second-order nonlinear tensor

dijk is used in its contracted matrix form to calculateP vwi (see Supplementaries II). The pump E(ω, 0), as wellas the total second-harmonic from all previous crystalsEpre(2ω, 0) are propagated through the grain. For bothbeams, the components along the o and e axis are prop-agated separately: Eu(ω,Xn) = Eu(ω, 0)eik

u(ω)Xn . Atthe end of the grain, the second-harmonic from the pre-

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vious grains is summed with the second-harmonic gener-ated in the current grain Eutot(2ω,Xn) = Eupre(2ω,Xn) +Eugen(2ω,Xn) (* in Fig. 1d). The amplitude, polarisationand phase of the fields is explicitly considered in everystep. This way, the model can account for interference ef-fects. The fields of the propagated pump Eu(ω,Xn) andthe total second-harmonic field Eutot(2ω,Xn) are trans-formed back into the lab frame, forming the outputof the nth grain. To simulate the propagation of thelight through the stick, we implemented a folding algo-rithm, which iteratively processes the sequence of grains.In each step, it takes the data of the current element(grain n) of the sequence and the result of the previousstep (grain n−1) as input. The information is processedto generate the output of the nth grain which then formsthe input of the (n+ 1)th grain. The SHG intensities ofeach stick Istick = cε0(|Eatot|2 + |Ebtot|2)/2 are averagedto yield the intensity of the cuboid. The model assumesan undepleted pump, no absorption within the grains, noreflections at the grain-to-grain interfaces, and the prop-agation only in the forward direction (no walk-off angle).

As an initial test of the algorithm, we computed theSHG from structures characterised by mono- and poly-dispersed grain sizes without randomisation of the grainorientation. The system behaves identical to a singlecrystal, periodically showing a totally constructive anddestructive SHG along the length of the system. As ex-pected, there is no dependence on the size distributionof the grains (see Supplementaries III). When introduc-ing disorder in grain orientation and grain size, the SHGintensity of the single stick depends strongly on the spe-cific disorder configuration. In this case, the SHG in-terference along the stick corresponds to the trajectoryof a single random walk in the complex plane, as de-picted in Fig. 1c. As such, the SHG intensity of a singlestick cannot be correlated with the number of grains N .Only upon averaging the intensities of multiple sticks (i.e.along the transverse dimensions of the cuboid), the totalSHG intensity grows linearly with N , providing evidenceof RQPM [17]. To quantitatively study the influence ofbirefringence, we introduce the isotropic analogue of abirefringent material, defined as the material that hasthe same χ(2)-tensor as the original material, but withne = no. Hence, the coherence length of the isotropicanalogue Liso

c is not angle dependent. This stands incontrast to the coherence length in a birefringent mate-rial Lbir

c (ϕ, ϑ, γ)n, which depends on the angle, as well asthe combination of beam polarizations. Thus, a randomdistribution of the grain orientations does not affect thecoherence length of isotropic grains, while it randomisesthe coherence length of birefringent grains.

II. RESULTS

Monodispersed Assemblies.— One noteworthy result ofthis study, which is not exclusively related to birefrin-gence, is that even monodispersed structures (without

FIG. 2. Scaling of the SHG intensities with the number ofgrains N in birefringent LiNbO3 and in its isotropic analogueat a pump wavelength of 930 nm. SHG intensities are aver-aged over 1000 sticks, leading to a linear growth. Its sloperepresents the SHG efficiency for a certain grain size, and theshaded area shows the possible efficiencies in monodispersedassemblies. The insets explicitly show the SHG intensityin monodispersed assemblies only as a function of the grainsize X for N = 100. In polydispersed assemblies (σ = 30%)with large grains (X > 3 Liso

c in the isotropic and X > 8 Lisoc

in the birefringent material) the efficiency has a stable valuecorresponding to the dotted line. (a) The isotropic analoguehas zero-efficiency minima when X is an even multiple of Liso

c ,and efficiency maxima when X is an odd multiple of Liso

c . (b)The birefringent assembly has its minimum at X = 9.25 Liso

c

and its maximum at X = 3.47 Lisoc .

size randomisation) sustain RQPM, as long as the grainsare randomly oriented. This is shown in Fig. 2a-b forLiNbO3. The SHG efficiency per grain is measured bythe slope of the linear growth, which depends on the grainsize X. Its variation is represented by the shaded areaand can be seen in the insets of Fig. 2a-b, which reportthe SHG intensity as a function of the grain size. Theisotropic analogue, Fig. 2a, shows a maximal efficiencywhen X is an odd multiple of Liso

c and a minimal (zero)efficiency when X is an even multiple of Liso

c . The lattercase corresponds to the totally destructive SHG inter-ference within the single grains. This is consistent withthe results of Vidal. et al. [17] at low polydispersity.The effect of the birefringence is shown in Fig. 2b. Themaximal as well as the minimal efficiency is increasedcompared to the isotropic analogue. In particular, thelower bound of the cone has a non-zero slope and theSHG never vanishes. Indeed, each birefringent grain hasan orientation-dependent coherence length Lbir

c (ϕ, ϑ, γ)n,prompting that there is no specific grain size at which to-tal destructive interference occurs for all grains. BaTiO3

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shows the same effect, but with a narrower cone of possi-ble efficiencies (see Supplementaries IV). The consideredcase of monodispersed assemblies highlights that polydis-persity is neither a sufficient nor a necessary condition toachieve RQPM. This stands in contrast to the randomorientation of grains, which plays a key role in reachingthe RQPM regime.

Polydispersed Assemblies.— In Fig. 3a the SHG in-tensity of LiNbO3 assemblies (red) and of their isotropicanalogue (blue) is shown for an increasing average grainsize 0 ≤ X ≤ 20 Liso

c , a polydispersity σ = 30%, anda fixed number of grains N = 100. Both curves followthe expected trend for RQPM [17]. In the small grainregime (X > Liso

c ) the SHG intensity grows with the grainsize without an appreciable difference between the bire-fringent and the isotropic case. Maximal intensities arereached when X = Liso

c , for the isotropic analogue andfor slightly larger grains in birefringent LiNbO3. Afteran initial peak, the SHG gets independent of the averagegrain size, at ≈ 3 Liso

c for the isotropic analogue and at≈ 8 Liso

c for the birefringent LiNbO3. In this stabilisedlarge-grain regime, the SHG intensity of the birefringentLiNbO3 is significantly larger than the stable value ofthe isotropic analogue. Specifically, we observe a peakenhancement of 54 ± 2% at 3 Liso

c . The effect is weakerin BaTiO3, where the birefringence leads to an SHG en-hancement of 9.25± 0.8% (see Supplementaries IV). Theenhancement can also be seen on the slope of the of dot-ted lines in Fig. 2a-b, which report the linear scalingof the SHG intensity for assemblies with polydispersedgrains at large grain sizes.

The origin of this overall enhancement lies in the dis-tribution of the SHG intensities generated by the indi-vidual grains Igen = cε0(|Eogen|2 + |Eegen|2)/2 within thecuboid, which are shown in Fig. 3b for both the bire-fringent (red) and the isotropic (blue) LiNbO3. Thesetwo SHG intensity distributions have been obtained byextracting Igen of the individual grains from the com-plete simulation depicted in Fig. 1d, thus considering thespecific polarization state of the pump (phase retarda-tion between the o and e components) that the propaga-tion through the previous birefringent grains introduced.The birefringence significantly widens the range of pos-sible SHG intensities in the individual grains, due to theincreased coherence lengths Lbir

c (ϕ, ϑ, γ)n > Lisoc for cer-

tain grain orientations. Consequently, the average SHGintensity of the individual grains of birefringent LiNbO3

is 54.1± 0.3% higher than the isotropic analogue. Simi-larly for BaTiO3, the enhancement is 9.24± 0.1%, whosesingle-grain SHG intensity distribution is shown in theSupplementaries IV. The enhancements of Igen are con-sistent with the enhancement of the assembly calculatedby considering the vectorial propagation of all compo-nents. We highlight that the birefringence is only bene-ficial in grains larger than the isotropic coherence lengthXn > Liso

c . In smaller grains, the grain size Xn, ratherthan the coherence length, is the length scale limitingtheir SHG intensity (see Supplementaries V). To under-

FIG. 3. (a) Scaling of the SHG intensity with the averagegrain size for disordered assemblies of isotropic LiNbO3, bire-fringent LiNbO3, and birefringent LiNbO3 in the single grain(SG) approximation (single grain average ×N). The assem-blies have a grain size polydispersity σ = 30%, a fixed numberof grains per stick N = 100, an averaging over 1000 sticks, andare simulated at a pump wavelength of 930 nm. At this wave-length, phase matching in LiNbO3 is not possible. (b) Proba-bility distributions of the SHG intensities generated by the in-dividual grains within the assembly (X = 3 Liso

c and σ = 30%)for isotropic LiNbO3, birefringent LiNbO3, and birefringentLiNbO3 in the single grain (SG) approximation (N = 1, lin-early polarized input beam). Probabilities are calculated asthe relative frequency of appearance of an SHG intensity valueover 107 grains. The SHG binning is 0.002. The SHG intensi-ties are multiplied by 100 to be directly comparable with theintensities of the 100-grain-assemblies in (a).

line the importance of modeling the phase propagationbetween the grains, we calculated the SHG scaling alsoin the single grain (SG) approximation, i.e. by consid-ering randomly oriented grains that are independentlyilluminated with a linearly polarized pump. Their aver-age SHG is then multiplied by N to get the SHG effi-ciency of the assembly ISG

tot = N × 〈ISGgen〉. As shown in

Fig. 3a (grey line), the SG approximation overestimatesthe SHG enhancement in the birefringent assembly ofLiNbO3 almost by a factor 2. This is because the SGapproximation ignores the retardation introduced in thepump and hence calculates an inaccurate distribution ofsingle grain intensities, as shown in Fig. 3b (grey area).Isotropic materials, on the other hand, can be accuratelysimulated with the SG approximation, since they do notmodify the polarization state of the pump.

Polydispersed Assemblies under Phase-matchable Con-ditions.— So far, we have performed our analysis in anon-phase-matchable regime. This means that the wave-

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lengths were chosen such that there is no crystal orienta-tion for which phase matching is possible in the studiedmaterials. Since our model considers birefringence, wecan numerically investigate the phase-matchable regimein RQPM. For this, we consider LiNbO3 at 1200 nm (seeSupplementaries I) and compare it to the non-phase-matchable regime by using the corresponding isotropicanalogue. We investigate how the SHG scaling with theaverage grain size X is affected by the phase-matchabilityof the grains, since this scaling behavior is consideredexperimentally to discriminate between phase-matchableand non-phase-matchable materials by using powdersamples. This method was first introduced in the seminalwork of Kurtz and Perry (KP) [8] and recently revisedby Aramburu et. al. [22]. The calculated SHG scalingswith increasing average grain size X for LiNbO3 and forits isotropic analogue are shown in Fig. 4a. To be com-parable with KP, the length of the cuboid is kept fixed,which reduces the total number of grains when increas-ing the grain size. This re-scales the SHG efficiency ofFig. 3 by 1/X, e.g. the constant SHG efficiency of theisotropic analogue from Fig. 3 scales with 1/X in Fig. 4.Another effect of the constraint on the total length of thecuboid, is the shift in the most efficient grain size fromX ∼ 1 Liso

c to X ∼ 0.7 Lisoc . For comparison with the

results reported by KP [8], we performed the same cal-culations on (Ammonium Dihydrogen Phosphate) ADP,which are also shown in Fig. 4a. Our model, is able toreproduce the results obtained experimentally by KP inthe large-grain regime (X > 5 Liso

c ). Namely, in the bire-fringent and phase-matchable case, the SHG intensity isindependent of X, while we observe a 1/X dependencefor its non-phase-matchable isotropic analogue for bothADP and LiNbO3. This implies that the enhancementof the SHG intensity, which is introduced by the birefrin-gence, grows linearly with the average grain size as soonas some of the grains can be phase matched. This lineardependence is clearly visible when considering the SHGscaling without constraint on the cuboid length and witha fixed number of grains (Supplementaries VI).

Notably, the scaling behavior of the two materials isvery different in the small-grain regime (X < 5 Liso

c ), asshown in Fig. 4b. For ADP, one can see the monotonic in-crease in the phase-matchable case, while the non-phase-matchable case shows an SHG peak followed by a de-creasing trend, in agreement with the results of KP [8].Surprisingly, for LiNbO3, there is no significant differ-ence between the phase-matchable and the non-phase-matchable case for X < 5 Liso

c . Both show an initialpeak followed by a decreasing trend and they could notbe distinguished by a scaling experiment. This resultshows that for average grain sizes X smaller than 5 Liso

c

the SHG scaling discovered by KP is not universal, butrather becomes material specific. In LiNbO3, the sourceof the deviation from the phase-matchable scaling of KPcan be found in the largest tensor element d33 whichdetermines the Ez · Ez component of the SHG. This isa non-phase-matchable tensor element since the electric

FIG. 4. Scaling of the SHG intensity with the average grainsize X in disordered assemblies of LiNbO3 and ADP and theirisotropic analogues at a pump wavelength of λ = 1200 nm.At this wavelength, phase matching is possible in both ma-terials. The total length of the sticks is kept constant andthe grains have a polydispersity σ = 30%. (a) The totallength of the sticks is fixed at 1000 Liso

c . When changing X,the number of grains per stick varies between 1000 grains (atX = 1 Liso

c ) to 10 grains (at X = 100 Lisoc ). The phase-

matchable materials show a different scaling in comparisonto their not phasematchable isotropic analogues. Non-phase-matchable LiNbO3 can also be simulated by using a shorterpump wavelength which is shown in the Supplementaries VI.(b) Higher resolution simulation of the small grain size regime,performed at a reduced stick length of 50 Liso

c for computa-tional efficiency. The shift in the y-scale compared to (a) isdue to the different stick length. In this small grain regimethe scaling of the SHG intensity is not determined solely bythe phase-matchability, but becomes material specific.

field of the SHG is orthogonal to the z-axis for the ordi-nary beam component, and hence its contribution is zeroin the phase-matchable configuration (oo-e). However,for randomly oriented grains smaller than Liso

c , the con-tributions of the d33 tensor element dominate the totalSHG. For such small grains the maximum SHG inten-sity is limited by the grain size rather than the coherencelength. At larger grain sizes, the smaller—but phase-matchable—coefficient starts to contribute significantlyto the total SHG and leads to the KP behaviour. In ADP,all tensor components are the same, and therefore influ-ence the phase-matchable and the non-phase-matchableSHG in equal parts. We conclude that only when thephase-matchable tensor components are much smallerthan the non-phase-matchable ones, the disordered as-sembly behaves like the non-phase-matchable case of KPin the small grain regime.

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III. CONCLUSIONS

We have developed a vectorial model that is capableof calculating the second-harmonic generation throughRQPM in any transparent, birefringent, χ(2)-disorderedmedium. This gives the presented model a wide rangeof applicability: from single grains, to disordered as-semblies, to assemblies with specific size and orientationcorrelations between the grains. We employ the pre-sented model to explore the effects of the birefringenceon the RQPM process, by considering fully disorderedassemblies of LiNbO3, BaTiO3, and ADP in both non-phasematchable and phase-matchable conditions. Wepinpoint the role played by the birefringence througha comparison with isotropic analogue materials havingne = no. While no influence of the birefringence is appre-ciable for smaller grains, the birefringence starts to play anotable role once the grains in the system are larger thanthe isotropic coherence length Liso

c in three studied situ-ations (monodispersed, polydispersed, and polydispersedwith phase matching). We identify the random orienta-tion of the grains as the key feature to enter the RQPMregime, a property enabling RQPM even in monodis-persed polycrystalline assemblies. With a proper choiceof grain size, the monodispersed assembly can outper-form the SHG efficiency of a polydispersed assembly inthe large grain regime. This property reveals the poten-tial of the nonlinear generation from layered structures(similar to periodically poled ones) which would requireto control only the layer thickness, without restrictions inthe crystal orientation. In this monodispersed case, thebirefringence introduces a randomisation of the coherencelength that relaxes the grain size dependence of the SHGintensity. In polydispersed assemblies, the birefringenceof the material leads to a grain size independent efficiencyincrease of up to 54% in comparison to the correspond-ing isotropic analogue. This efficiency increase is ascribedto an increase in the average grain efficiency, stemming

from larger coherence lengths for certain grain orienta-tions. We show that the explicit grain-to-grain propaga-tion of the polarization components is necessary for theaccurate prediction of this SHG efficiency enhancementand in general for the accurate description of the three-wave mixing process in disordered birefringent media.We show that the Kurtz and Perry method to discrimi-nate between phase-matchable and non-phase-matchablematerials, cannot be applied when the grains are smallerthan the coherence length, e.g. nano-powders. In thissmall grain regime, the dependence of the SHG on the av-erage grain size is given by the specific χ(2) tensor of thegrains and thus becomes material specific. Our findingsshow how a larger set of materials, including birefringentcrystals, can serve for RQPM applications and clarify thelimitations of a widely-applied method for the character-ization of nonlinear optical materials. Due to the nonresonant nature of the SHG and the RQPM, we expectour results to be valid in a wide range of wavelengths(e.g. hundreds of nanometers for metal-oxides), as longas the absorption is negligible. The model could be ex-tended to include absorption within the grains, scatteringeffects at the interfaces, and electro-optic effects. More-over, it could be generalized to study other three-wavemixing processes such as sum- and difference-frequencygeneration.

IV. ACKNOWLEDGEMENTS

This research has received funding from the Euro-pean Union’s Horizon 2020 research and innovation pro-gram under the Marie Sk lodowska-Curie grant agreementno. 800487 (SECOONDO) and from the European Re-search Council under the grant agreement no. 714837(Chi2-nano-oxides). We thank the Swiss National Sci-ence Foundation (SNF) grant 150609.

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[13] R. Savo, A. Morandi, J. S. Muller, F. Kaufmann,F. Timpu, M. Reig Escale, M. Zanini, L. Isa, andR. Grange, Broadband mie driven random quasi-phase-matching, Nature Photonics 10.1038/s41566-020-00701-x(2020).

[14] Q. Ru, N. Lee, X. Chen, K. Zhong, G. Tsoy, M. Mirov,S. Vasilyev, S. B. Mirov, and K. L. Vodopyanov, Op-tical parametric oscillation in a random polycrystallinemedium, Optica 4, 617 (2017).

[15] S. Vasilyev, I. Moskalev, M. Mirov, S. Mirov, andV. Gapontsev, Mid-IR Kerr-lens mode-locked polycrys-talline Cr2+:ZnS lasers, in Ultrafast Bandgap Photonics,Vol. 9835, edited by M. K. Rafailov and E. Mazur, Inter-national Society for Optics and Photonics (SPIE, 2016)pp. 94 – 100.

[16] R. Fischer, D. N. Neshev, S. M. Saltiel, A. A. Sukho-rukov, W. Krolikowski, and Y. S. Kivshar, Monitoring ul-trashort pulses by transverse frequency doubling of coun-terpropagating pulses in random media, Applied Physics

Letters 91, 031104 (2007).[17] X. Vidal and J. Martorell, Generation of light in media

with a random distribution of nonlinear domains, Phys.Rev. Lett. 97, 013902 (2006).

[18] T. Kawamori, Q. Ru, and K. L. Vodopyanov, Compre-hensive model for randomly phase-matched frequencyconversion in zinc-blende polycrystals and experimentalresults for ZnSe, Phys. Rev. Applied 11, 054015 (2019).

[19] X. Chen and R. Gaume, Non-stoichiometric grain-growth

in znse ceramics for χ(2) interaction, Optical MaterialsExpress 9, 400 (2019).

[20] J. Gu, M. G. Hastings, and M. Kolesik, Simulation of har-monic and supercontinuum generation in polycrystallinemedia, JOSA B 37, 1510 (2020).

[21] J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Stal-iunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Kro-likowski, and Y. S. Kivshar, Second-harmonic parametricscattering in ferroelectric crystals with disordered nonlin-ear domain structures, Opt. Express 15, 15868 (2007).

[22] I. Aramburu, J. Ortega, C. Folcia, and J. Etxebarria, Sec-ond harmonic generation by micropowders: A revision ofthe kurtz-perry method and its practical application, Ap-plied Physics B 116 (2013).

SUPPLEMENTARY INFORMATIONModeling of Random Quasi-Phase-Matching in Birefringent Disordered Media

Jolanda S. Muller, Andrea Morandi, Rachel Grange, and Romolo Savo∗

Optical Nanomaterial Group, Institute for Quantum Electronics,Department of Physics, ETH Zurich, Zurich, Switzerland

(Dated: November 23, 2020)

CONTENTS

I. Material Parameters 2A. Nonlinear Optical Tensors and Refractive Indices 2

Lithium Niobate 2Barium Titanate 2Adenosine Diphosphate 2

B. Phase Mismatch and Coherence Length 2

II. Algorithm Details 4A. Reference Frame Transformations 4B. Non-linear Polarisation for Different Polarisation Directions 5C. Second Harmonic E-field at the End of One Grain (Derivation of Eq. (1)) 5D. Ensemble Averaging 6

III. Algorithm Verification 7A. Visualisation of the Wave Propagation and Summation 7B. The Single Crystal Limit and Quasi-Phase Matching 8

IV. SHG Intensity with Increasing Average Grain Size 9

V. SHG Intensity Distribution of Single Grains Smaller than the Coherence Length 10

VI. SHG Intensity Scaling for Phase-matchable Lithium Niobate 11

References 11

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I. MATERIAL PARAMETERS

A. Nonlinear Optical Tensors and Refractive Indices

Lithium Niobate

LiNbO3 is trigonal, point group 3m (C3v). The reduced χ(2) tensor of LiNbO3 is given in Eq. (1) [1]. The ordinaryand extraordinary refractive indices are calculated at room temperature (T = 20 °C) according to the Sellmeierequation [2]. At a wavelength of 930 nm the refractive indices are no = 2.2436 and ne = 2.1634.

dLNO =

0 0 0 0 d31 −d22

−d22 d22 0 d31 0 0d31 d31 d33 0 0 0

with

d22 = +2.1 pm/Vd31 = −4.2 pm/Vd33 = −27.0 pm/V

(1)

Barium Titanate

BaTiO3 is tetragonal, point group 4mm (C4v). The reduced χ(2) tensor of BaTiO3 is given in Eq. (2) [3].The ordinary and extraordinary refractive indices at different wavelengths are calculated according to the Sellmeierequation [4]. At a wavelength of 930 nm the refractive indices are no = 2.3502 and ne = 2.3079.

dBTO =

0 0 0 0 d15 00 0 0 d15 0 0d31 d31 d33 0 0 0

with

d15 = −17.0 pm/Vd31 = −15.7 pm/Vd33 = −6.8 pm/V

(2)

Adenosine Diphosphate

ADP is tetragonal, point group 42m (D2d). The reduced χ(2) tensor of ADP is given in Eq. (3) [5]. The ordinaryand extraordinary refractive indices at different wavelengths are calculated according to the Sellmeier equation [6].For example at a wavelength of 930 nm the refractive indices are no = 1.5114 and ne = 1.4708.

dADP =

0 0 0 d36 0 00 0 0 0 d36 00 0 0 0 0 d36

with d36 = 0.47 pm/V (3)

B. Phase Mismatch and Coherence Length

For isotropic materials, the phase mismatch between the funamental beam (ω) and the SHG (2ω) is independentof the beam polarizations and the orientation of the crystal and it is given as

∆k = 2|k(ω)| − |k(2ω)| = 2ω

c(nω − n2ω) (4)

with the wave vector k and the refractive indices nω and n2ω of the fundamental beam and the SHG, respectively.The coherence length is the length at which the relative phase shift of the generated waves reaches π

2 , and is defined

as Lisoc = π/∆k. In a birefringent crystal the refractive indices depend on the the beam polarization, and on the angle

between the optic axis of the crystal and the wave vector k. In this case, the general expression of the phase mismatchbetween the fundamental and the second harmonic ∆k takes into account the different polarisation combinations.This leads to ∆ku,vw = kv(ω) + kw(ω) − ku(2ω) with u, v, w ∈ {o, e} and is written as eight different values for ∆kdepending on the beam combinations: ∆ko,oo, ∆ko,ee, ∆ko,oe, ∆ko,eo, ∆ke,oo, ∆ke,ee, ∆ke,oe, ∆ke,eo. Consequently,the coherence length for each beam polarization combination is derived from the corresponding phase mismatchLbirc (ϑ) = π/∆ku,vw and it depends on ϑ as depicted in Fig. 1. We note that beam combinations of type e, oo

(solid black) have the largest coherence length, and the beam combination o, oo (solid blue) has an angle independentcoherence length. At 930 nm, LiNbO3 has a coherence length in the interval [1.14 µm, 9.40 µm], with the constant

3

FIG. 1. (a) Coherence lengths in LiNBO3 for different angles ϑ at 930 nm. At this wavelength all coherence lengths are finiteand phase matching is not possible. (b) Coherence lengths in LiNBO3 for different angles ϑ at 1100 nm. When the phase-matching condition is fulfilled, the coherence length gets infinitely large, as indicated with the red dashed line at θ (phm). (c)The phase-matching angle for different wavelengths in LiNbO3. The red shaded area indicates the wavelength range in whichthe phase matching condition does not have a solution.

o, oo-value of 1.88 µm and an average value of 2.314 µm. The coherence length of BaTiO3 lies within the interval[0.95 µm,1.87 µm], with a constant o, oo-value of 1.57 µm and an average value of 1.23 µm.

The coherence length becomes infinitely large Lc → ∞, when the phase matching condition ∆k = 0 is satisfied,which means that both waves travel through the medium at the same speed. To allow for phase matching the refractiveindex of the fundamental wave (nω) must be equal to the refractive index of the SHG (n2ω). Considering the angledependent refractive indices of both frequencies the phase-matching condition becomes [7]:

sin2 ϑ

ne(2ω)2+

cos2 ϑ

no(2ω)2=

1

no(ω)2(5)

While the solution of this equation provides the wavelength dependent phasematching angle, it is not guaranteedthat a solution for every wavelength exists. For LiNbO3 a solution only exists between 1065 nm to 3732 nm. Thecorresponding phase-matching angles are reported in Fig. 1b, which shows the phase-matching angles from 500 nmto 4500 nm. It is possible that the birefringence of a material is not sufficient to allow for the solution of Eq. 5 at anywavelength. This is the case of BaTiO3 which is therefore never phase matchable.

4

II. ALGORITHM DETAILS

The propagation through the stick of N grains is performed with a fold function, which takes an input value (inputbeam), and a data structure (array of N grains) and reduces them to a single output value (output beam) throughrecursive processing with a function f (performing the single grain operations), by using the output of step n as input

of step (n + 1): ~En+1in (ω) = ~Enout(ω) for both ω and 2ω. The output of each grain is composed of the propagated

fundamental field, and the total SHG. The propagation is computed separately for the ordinary and extraordinarycomponents u ∈ {o, e} and for both frequencies ω and 2ω: Euprop(ω,Xn) = Eu(ω, 0) · eiXnk

u(ω). The total SHG isgiven by the sum of the propagated SHG input form the previous grain and the SHG generated within the currentgrain: Eutotal(2ω) = Eugen(2ω,Xn) + Euprop(2ω,Xn). The computational procedure is shown in simplified pseudo-codein Fig. 2.

Grains = [G1, G2, ..., GN]

Beams = InputBeams

for i = 1 to N

Beams = f(Beams, Grains(i))

TotalBeams(N) = Beams

end

OutputBeams = Beams

Single Grain Function f:

functions [BeamsLabOut] = f(BeamsLabIn, Grain)

BeamCrystalIn = transform2Crystal(BeamsLabIn, Grain)

BeamsCrystalOut.Fund = propagagte(BeamsCrystalIn.Fund, Grain)

Eprop = propagagte(BeamsCrystalIn.SHG, Grain)

Egen = generateSHG(BeamsCrystalIn.Fund, Grain)

BeamsCrystalOut.SHG = Eprop + Egen

BeamsLabOut = transform2Lab(BeamsOutCrystal, Grain)

end

FIG. 2. Pseudocode and schematic of the main functions in the algorithm.

A. Reference Frame Transformations

The transformation matrix R (c ≡ cos and s ≡ sin) is given by:

R =

c(ϕ)c(γ)− s(ϕ)c(ϑ)s(γ) −c(ϕ)s(γ)− s(ϕ)c(ϑ)c(γ) s(ϑ)s(ϕ)s(ϕ)c(γ) + c(ϕ)c(ϑ)s(γ) −s(ϕ)s(γ) + c(ϕ)c(ϑ)c(γ) −s(ϑ)c(ϕ)

s(ϑ)s(γ) s(ϑ)c(γ) c(ϑ)

(6)

After the transformation into the crystal frame, the E-fields are then projected onto the ordinary and extraordinaryaxis (o, e) for separate propagation. The ordinary axis is orthogonal to the plane defined by the propagation vector

kcry and the optic axis z and its unit vector is given by eo =z×kcry

|z×kcry| . The extraordinary axis is orthogonal to kcry

and to the optic axis z with the unit vector ee =kcry×eo

|kcry×eo| .

5

B. Non-linear Polarisation for Different Polarisation Directions

The non linear polarisation generated by a discrete set of incoming electric fields ~E(ωn) is given by Boyd [7]:

Pi(ωn + ωm) = 2ε0∑

jk

∑

nm

dijkEj(ωn)Ek(ωm) (7)

where dijk is the material specific χ(2) tensor. For the second harmonic generation there is just one fundamentalfrequency and therefore ωn = ωm = ω. Thus the sum over m and n is ignored. The E-fields Ej are decomposed intheir ordinary and extraordinary contribution Ej(ω) = Eoj (ω) +Eej (ω). For simplicity (ω is not written below). Thuswe can consider the contributions to the polarisation by the four combinations of the ordinary and extraordinary

complex fundamental ~E-fields (including the phase):

Pi(2ω) = 2ε0∑

jk

dijk(EojEok + EejE

ek + EojE

ek + EejE

ok)

= P ooi + P eei + P oei + P eoi where P vwi = 2ε0∑

jk

dijkEvjE

wk

(8)

Due to the Kleinmann Symmetry the tensor dijk is symmetric under exchange of j and k and is written in itscontracted matrix form dil [7] for the material coefficients, which we use with the matrix equation:

~P vw =

P vwx (2ω)P vwy (2ω)P vwz (2ω)

= 2ε0

d11 d12 d13 d14 d15 d16

d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

EvxEwx

EvyEwy

EvzEwz

EvyEwz + EvzE

wy

EvxEwz + EvzE

wx

EvxEwy + EvyE

wx

(9)

The resulting polarisations are generally not polarised along just one axis (o/e). Therefore, each ~P vw is decomposedalong the o/e-direction within the current crystal by taking the inner product [8] with the unit vector pointing in the

(extra-)ordinary direction: Pu,vw =< eu, ~P vw >. From this Eugen is calculated according to Eq. (1) in the main text,of which we show the derivation in the next section.

C. Second Harmonic E-field at the End of One Grain (Derivation of Eq. (1))

We start from the coupled-amplitude equation provided by Boyd [7], which describes how the amplitude of thenon-linear wave (ω3) depends on the amplitudes of the fundamental waves (ω1 and ω2). In our case, ω1 = ω2 = ωand ω3 = 2ω.

dA3

dz=

iω3

2ε0n3cp3e

i∆kz (10)

A3 is the amplitude of the non-linear wave (envelope function). The non-linear polarisation is given by ~P3(z, t) =p3e

i(k1+k2)z + c.c.. We note that any starting phases of the fundamental E-fields (ϕ1 and ϕ2) given at z = 0, p = 0must be contained in p3, making it a complex quantity. Analogous to Boyd’s calculations [7], we derive the amplitudeof the sum-frequency field A3 at the end of a grain of length Xn, by integrating Eq. 10 from z = 0 to z = Xn andsubstituting n3 = c|k3|/ω3:

A3(Xn) =iω3

2ε0n3cp3

∫ Xn

0

ei∆kzdz =iω3

2ε0n3cp3

(ei∆kXn − 1

i∆k

)=

iω23

2ε0c2k3p3

(ei∆kXn − 1

i∆k

)(11)

In the birefringent case, the phase-mismatch ∆k becomes ∆ku,vw, dependent on the polarisation directions of thefundamental beam components v, w ∈ {o, e}, as well as the polarisation direction of the generated wave u ∈ {o, e}as established in section I B. Similarly, the non-linear polarisation p3 is given by the complex quantity ~Pu,vw with

6

u, v, w ∈ {o, e} as derived in the previous section. It contains the starting phase ϕ1 and ϕ2 of the to fundamentalbeam components at the beginning of the crystal. The amplitude of the non-linear field for one set of polarisationcombinations is given by:

Au,vw3 (Xn) =iω2

3

2ε0c2ku3· Pu,vw ·

(ei∆k

u,vwXn − 1

i∆ku,vw

)(12)

We retrieve the non-linear complex E-field at the end of the crystal along one polarisation direction throughmultiplication with the propagation term eik

uXn and summation over the different input polarisation combinations.Lastly, we substitute ω3 with 2ω, yielding eq. (1) presented in the main text:

Eugen(2ω,Xn) =∑

v,w

i(2ω)2

2ε0c2ku3· Pu,vw ·

(ei∆k

u,vwXn − 1

i∆ku,vw

)· eiku3Xn (13)

D. Ensemble Averaging

In Fig. 3, we demonstrate the difference between the simulation for one single stick, and the ensemble average overmultiple sticks discussed in the main manuscript. The SHG scaling with the number of grains for two sticks withindependent configurations of disorder, does not show a correlation with the number of grains. The SHG of the twosticks corresponds to two different, independent random walks in the SHG complex plane. Upon taking the ensembleaverage over 200 sticks, the SHG grows linearly with the number of grains. Taking the ensemble average is physicallyjustified by the fact, that the laser beam illuminates multiple parallel sticks at onces [9].

FIG. 3. SHG intensity for two single sticks with independent disorder configurations (green dotted, blue dashed) and for theensemble average over 200 sticks. After the ensemble average the SHG intensity depends linearly on the number of grains

7

III. ALGORITHM VERIFICATION

A. Visualisation of the Wave Propagation and Summation

Fig. 4a-d visualizes how the waves are propagated and summed by the algorithm within the first three grains forfour different configurations. For each configuration the top part of the image shows the Egen and Epre separatelyin each grain. The offsets from zero were added afterwards for visual clarity. The bottom part shows the sum of allthe contributions Etot, which is equivalent to the total SH-field along the assembly. An isotropic system with a fixedorientation is shown in Fig. 4a. As expected, the generated waves are generally not in phase with the propagatedwaves from the previous crystals. This constant phase-mismatch leads to constructive and destructive interference inperiodic intervals, leading to a | sin(x)| envelope function modulating the second harmonic E-filed along the structure.The maxima match odd and the minima even multiples of the coherence length. An isotropic system, with randomisedgrain orientations is shown in Fig. 4b. The amplitudes of the generated waves within each grain are randomised andthe phases have arbitrary relations. Therefore, the total E-field amplitude will grow and decline in a random fashion.However, on average (considering multiple randomised sticks) the field amplitude grows with the square root of thenumber of domains. This ultimately leads to a linear dependence of the intensity on the number of domains, which isknown as the RQPM. In a birefringent system with an not phase-matched fixed orientation for each grain, depictedin Fig. 4c, the fields are plotted in the lab-frame. Different colours are used for the components along the a and the baxis (dark/light blue). The SHG has different amplitudes and coherence lengths along the two different polarisationdirections. In the propagated waves, the amplitudes of the two orthogonal beam components changes within onedomain, because the birefringence introduces retardation, leading to an elliptically polarised beam which changesits eccentricity as well as its inclination during the propagation. Lastly, Fig. 4d shows a birefringent system in itsphase-matched orientation, with ϑ = ϑphm, ϕ = 0, γ = 0, and the polarisation of the incoming beam is set to β = 0.This corresponds to pure Type-II (oo-e) phase matching. In this orientation, the generated and propagated wavesare always in phase, which is correctly simulated by our algorithm. As a result, the amplitude of the total E-fieldincreases linearly, which ultimately leads to the quadratic dependence of the intensity with the sample length.

FIG. 4. For each system, the first three grains are shown, separated by a grey line. The generated and propagated wavesare shown individually in the top half and their resulting wave in the bottom half of a-d. (a) Isotropic system with fixedorientation. The coherence length is clearly visible in the resulting wave. (b) Birefringent system with fixed orientation. Thecontributions along a and b in the lab frame are shown in bark/light blue. (d) Birefringent system with fixed, phasematchedorientation. The generated and the propagated waves are always in phase. (d) Isotropic system with randomised orientation.The amplitude of the generated wave in each grain is random, leading to an overall randomised pattern. The phase shifts areidentical to the ones in the system with fixed orientations, because the refractive indices are isotropic.

8

B. The Single Crystal Limit and Quasi-Phase Matching

In order to verify that the amplitudes and phases are correctly propagated between the grains and that the depen-dence of the generated wave on the phase of the fundamental wave is implemented accurately, an array of grains withidentical parameters is compared to the reference simulation of a single grain (Fig. 5). The behaviour inside the singlegrain is directly derived from equation (1) in the main text as a function of the length of the crystal, which servesas the reference of the physically expected behaviour since no phase considerations at grain interfaces are necessaryfor the calculation. This single grain reference is shown as a gray continuous line. The single grain is compared toan assembly of multiple grains adding up to the same total length, which all have the same orientation as the singlecrystal. The output received from the simulation at the end of each grain is depicted with blue dots for monodispersedomains and with red crosses for polydisperse domains, as illustrated in Fig. 5a. This sanity check is performed fordifferent systems with a well-known physical behaviour: Fig. 5b,c,e show systems with a fixed crystal orientationfor an isotropic crystal, a birefringent crystal and a phase-matched birefringent crystal. All cases match the singlegrain behaviour perfectly, confirming the correct grain to grain transmission and propagation of all involved waves.Interestingly, in the birefringent case, one can see the superposition of the different beam components with dissimilarcoherence lengths. Lastly, we demonstrate that we can also simulate a quasi-phase matched system in Fig. 5d. Aquasi-phase-matched system consists of periodically poled layers with a thickness of one Liso

c each [10]. In our model,the poling is simulated by flipping the orientation of each grain with respect to the previous grain through a rotationof π around the angle γ. The intensity plateaus at the end of the grains. Before it starts decreasing, the next grainbegins (with a flipped orientation) and bringing the waves back in phase, thanks to a phase flip by π.

FIG. 5. (a) Schematic representation a single grain split up in multiple grains of the same orientation either into monodisperseor polydisperse domains. In (b)-(e) the SHG intensity is normalised with the single grain peak intensity at Liso

c = 1.2 µm. (b)An isotropic assembly of 30 grains (size = 5

3Liso

c ) with a fixed orientation ( ϕ = 0, ϑ = π/2, γ = π/2) for every grain. (c) A

birefringent assembly with 30 grains (size = 53Liso

c ) with a fixed orientation (ϕ = 0, ϑ = π/2, γ = π/5) for every grain. Inb and c the intensity after each grain in the monodisperse assembly (blue dots) and in the assembly with 30% polydispersity(red crosses) follows the behaviour in a single crystal of the same orientation (grey). (e) Simulation of the QPH in an isotropicmedium, by periodic poling of the domains with a thickness of one coherence length. (f) Phase matching in a single crystal,polydisperse or monodisperse domains.

9

IV. SHG INTENSITY WITH INCREASING AVERAGE GRAIN SIZE

Fig. 6a-b shows the SHG intensity scaling with increasing average grain size 0 ≤ X ≤ 20Lisoc in monodispersed

assemblies, both, isotropic and birefringent. In both materials, the minimal and maximal efficiency in the oscillationare increased due to the birefringence. In LiNbO3 (Fig. 6a), the efficiency increase is more pronounced, while inBaTiO3 (Fig. 6b) the extreme of the oscillation are narrowed after the initial peak at one Liso

c . Interestingly, the localmaxima and minima in the oscillation in both birefringent materials appear periodically. In LiNbO3 this periodicityis larger than the Liso

c , while in BaTiO3 it is shorter than Lisoc . In the insets of 6a-b, the linear increase of the SHG

intensity with the number of grains is shown, for the grain sizes with the maximal and minimal efficiency (SHGintensity per grain). The shaded area represents all the possible values for different grain sizes.

Fig. 6c-d shows the SHG intensity with increasing average grain size 0 ≤ X ≤ 20 Lisoc in polydispersed assemblies.

The results for LiNbO3 (Fig. 6c) are presented and discussed in the main text. The similar behaviour of BaTiO3 isshown in Fig. 6d. In contrast to LiNbO3, BaTiO3 does not show a second peak around 3 Liso

c , but instead approachesdirectly a stable value for X > 3 Liso

c .

FIG. 6. The SHG intensity of an assembly with 100 grains as a function of the average grain size X, averaged over 1000 parallelsticks. (a),(b) In isotropic LiNbO3 and BaTiO3 (blue dotted) with monodispersed grain sizes, the SHG intensity shows maximaat odd and minima at even multiples of the coherence length. Including birefringence (solid red) the efficiency of the oscillationis offset from zero and the range of possible efficiencies is reduced. The insets of (a),(b) show the linear increase of the SHGintensity with the number of grains for different grain sizes. In LiNbO3 the birefringent maximum (max) is at X = 3.47 Liso

c

and the minimum (min) at X = 9.25 Lisoc . In BaTiO3 the birefringent maximum is at one Liso

c and the minimum at X = 9.2Liso

c . (c),(d) With polydispersed grain sizes, the dependence on the average grain size disappears completely above 3 Lisoc in

isotropic materials. In the corresponding birefringent materials, the stable value is only reached for larger average grain sizes,at ∼ 8 Liso

c in LiNbO3and at ∼ 4 Lisoc in BaTiO3. The insets of (c),(d) show the distribution of the SHG intensity generated by

the individual grains within the stick for isotropic and birefringent grains. The widening of the distribution leads to an increaseof the average grain intensity.

10

V. SHG INTENSITY DISTRIBUTION OF SINGLE GRAINS SMALLER THAN THE COHERENCELENGTH

The increase of the mean SHG intensity of the individual grains inside the simulated stick due to the birefringenceof the grains is responsible for the efficiency increase of the birefringent assembly in comparison to its isotropicanalogue. However, this only applies for grains larger than the isotropic coherence length Liso

c , where the increase ofthe orientation specific coherence length Lbir

c (ϕ, ϑ, γ)n in the birefringent medium provides an advantage. In Fig. 7,one can in fact see, that for grains smaller than Liso

c (average grain size X = 0.1 Lisoc ) the SHG distributions of

the individual grains in the stick is almost identical for both, isotropic and birefringent grains, independent of thematerial. In this small grain size regime, the maximal efficiency is limited by the size of the grain Xn, rather than itscoherence length Lbir

c (ϕ, ϑ, γ)n. We note that there is still a slight shift in the distribution favouring the birefringentmaterial. Interestingly, we also observe that the distribution of the single grain intensities differs significantly for thetwo materials.

FIG. 7. Distribution of the SHG intensities for 105 randomly oriented single grains, with an average grain size X = 0.1 Lisoc and

with σ = 30% polydispersity for LiNbO3 (a) and BaTiO3 (b). The mean intensities of all grains (vertical lines) are identicalfor the isotropic and the birefringent case for both materials.

11

VI. SHG INTENSITY SCALING FOR PHASE-MATCHABLE LITHIUM NIOBATE

The scaling of the SHG intensity of an assembly depends on the phase-matchability of the grains. In the mainmanuscript, we used the isotropic analogue as the not phasematchable reference. However, instead of using theisotropic analogue, not phase-matchable LiNbO3 can be obtained at a wavelength of 800 nm, as shown in sectionI B. For this case, Fig. 8a shows the scaling of the SHG intensity with the average grain size. While the LiNbO3 at1200 nm, which is phasematchable, shows a constant SHG scaling. Both the isotropic analogue at 1200 nm and theLiNbO3 at 800 nm show the same scaling behaviour. The only difference is a shift in the overall intensity due to thedifferent wavelength.

In the main manuscript we showed the scaling of the SHG intensity in a phase-matchable situation for sampleswith a constant thickness (constant stick length). In Fig. 8b, we show the scaling of the SHG intensity in birefringentgrains (phase-matchable) and isotropic grains with a constant number of grains, instead of a constant sample thickness.The birefringence introduces an enhancement of the SHG intensity, which grows linearly with the average grain sizebringing the birefringent phasematchable grains into a new scaling regime. This linear scaling is consistent with theresults found by Aramburu et al. [11].

FIG. 8. (a) The SHG intensity scaling with the average grain size for an assembly with a constant thickness. The LiNbO3

at 1200 nm is phase-matchable (solid line) and displays a constant SHG intensity for large average grain sizes. The isotropicLiNbO3 at 1200 nm as well as the birefringent LiNbO3 at 800 nm are not phase-matchable and show the same 1/X scaling withthe average grain size. (b) SHG intensity scaling with the average grain size for a LiNbO3 assembly with a constant number ofgrains at a phase-matchable wavelength of 1200 nm.

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