Second harmonic generation by micropowders: a revision of the Kurtz–Perry method and its practical application

Abstract

We theoretically study the second harmonic generation by powder crystal monolayers and by thick samples of crystalline powder with particle size in the range of microns. Contrary to usual treatments, the light scattering by the particles is explicitly introduced in the model. The cases of powder in air and in an index-matching liquid under the most common experimental geometries are considered. Special attention is paid to the possibility of determining the value of some nonlinear optical coefficients from the experiments. The limitations and shortcomings of the classical Kurtz and Perry method (Kurtz and Perry in J Appl Phys 39:3798, 1968) and the most common practical misuses of it are discussed. It is argued that many of the experimental works based on that method oversimplify the technique and contain important errors. In order to obtain reliable values of the nonlinear coefficients, an appropriate experimental configuration and analysis of the data are pointed out. The analysis is especially simple in the case of uniaxial phase-matchable materials for which simple analytical expressions are derived.

1 Introduction

The second-order nonlinear optical susceptibilities d _mn [1–6] are the most important parameters in the determination of the performance of nonlinear optical devices, such as frequency doublers, optical parametric oscillators and, in general, in many kinds of electrooptic processes at optical frequencies. The determination of those coefficients is usually achieved through measurements of the second harmonic generation (SHG) on single crystals [7, 8]. Maker fringe technique or any of its variants can yield accurate values of the individual d _mn provided that large single crystals free from defects are available. In these experiments, several crystals have to be oriented, cut and polished along prescribed directions to determine the full d tensor. The time and effort needed is considerable, so the technique is not appropriate for a material survey. In addition, on many occasions, the method simply cannot be applied because crystals are difficult to grow. In those cases, the powder method developed by Kurtz and Perry (KP) [9] is commonly used. The technique gives an estimate of the nonlinear optical capability of the material and permits to know whether the phase-matching (PM) condition can be achieved or not.

Other alternatives to the KP method have been proposed in the literature, such as the SHG by the evanescent wave under total reflection conditions [10], or the SHG by powder crystal monolayers [11]. It has been experimentally proven that the last method is able to give reliable and precise values of d _mn [11]. However, the KP method is still by far the most popular among researchers for evaluating new materials (more than 200 citations in 2012). Nevertheless, problems can arise with the method when more than information about the order of magnitude of the d _mn coefficients is required. In this respect, it is interesting to point out that it is not unusual to find errors of one order of magnitude for the d _mn values determined with this method [1, 12]. There are two kinds of reasons that explain these problems.

On the one hand, the KP theory itself has certain limitations. One important shortcoming is that the scattering by the powder particles of the sample is neglected. This is hardly justified. Even when the powder is mixed with a liquid with approximately the same refractive index as that of the crystal, the length of the sample is usually too long to ignore the effect of light scattering. Another important parameter not considered in the theory is the packing fraction of the powders. In this regard, it is evident that the magnitude of the SHG signal depends on the packing density and, in fact, care should be taken in the powder packing to obtain reproducible results. Finally, some other deficiencies derive from the fact that the KP analysis initially assumes a material with cubic symmetry, and the expressions for the effective values of the SHG efficiencies are calculated without considering the different wave vector mismatches for all the interactions involved in the SHG process.

There are also experimental issues to be commented. Though the original KP method requires the addition of an index-matching fluid to the powder to minimize the scattering, this is seldom used in practice. On the other hand, normally, there is no control on the powder packing fraction and, in the vast majority of cases, it is assumed that the SHG signal generated in a particular direction is proportional to the total integrated SHG flux. Depending on the conditions, it is understood that these practices can give misleading results, but they are usually accepted among researchers.

In this paper, we theoretically study the SHG by crystalline powders with grain size in the range of microns. The problem of SHG in nanopowders will not be treated here [13–15]. Special attention is devoted to account for the weak points of the KP technique referred above and, in particular, the effects of the light scattering are analyzed thoroughly. We will show that an appropriate analysis of the collected data can be used to obtain reliable values of the nonlinear susceptibilities. In the particular case of uniaxial phase-matchable materials, simple analytical expressions are derived. This paper is organized as follows. In Sect. 2, we deal with the SHG generated by powder crystal monolayers, completing and detailing the theory outlined in [11]. This model is used in Sect. 3 as a starting point to study the SHG of thick powder samples. The KP method is critically examined and the scattering is explicitly introduced. Finally, powders in air and in an index-matching fluid are studied under the most typical experimental conditions used in practice. The results of the calculations are compared with experimental data available in the literature. Some conclusions are drawn in Sect. 4.

2 SHG by monolayers of crystalline powder

We will study the SHG by a monolayer of crystalline powder randomly oriented [16, 17]. It will be assumed that the primary beam incident on the sample can be described by a plane wave. The powder is immersed in an index-matching liquid in order to reduce the light scattering and, therefore, simplify the analysis of the SHG output. Additionally, we will assume that the absorption of the material is negligible for the wavelengths involved in the SHG process. By use of sieves, all the particles in the sample can be achieved to have approximately the same average size r. Let us focus on one single particle of the layer. The primary beam will cross it along a certain direction that can be characterized by the angles θ (polar) and φ (azimuthal) with respect to the principal dielectric axes of the material [2]. As the powder particles are immersed in an index-matching liquid usually between two parallel microscope slides, we will assume that the SHG wave produced by the powder particle is similar to the one generated by the same primary beam propagating along the (θ, φ) direction through a slab of length r. Besides, since walk-off angles are small (typically a few degrees), we will neglect the spatial walk-off and assume, for all purposes, the collinear propagation of the waves. Finally, under usual experimental conditions, the field amplitude of the incident wave is undepleted [3] so that the field amplitude of the SHG wave at the end of the particle will be given by [4–6]:

$$E_{i}^{2\omega } (r) = - i\omega \varepsilon_{o} \sqrt {\frac{{\mu_{o} }}{{\varepsilon_{2i} }}}\sum\limits_{j,k} {d_{ijk} E_{j}^{\omega } E_{k}^{\omega } \left( {\frac{{{\text{e}}^{{i\varDelta k_{ijk} r}} - 1}}{{i\varDelta k_{ijk} }}} \right)} ,$$

(1)

where the subscripts i, j and k can take the values {X, Y}, which are chosen to be along the polarization direction of the eigenmodes of propagation in the crystal, i.e., {X, Y} = {o (“ordinary”), e (“extraordinary”)} in the case of uniaxial materials and {X, Y} = {s (“slow”), f (“fast”)} in the case of biaxial materials. ε _2i is the permittivity of the material for the SHG field $E_{i}^{2\omega }$ and $\varDelta k_{ijk}$ is the wave vector mismatch of the interacting waves: $\varDelta k_{ijk} = k_{i}^{2\omega } - (k_{j}^{\omega } + k_{k}^{\omega } )$. $d_{ijk}$ is the nonlinear optical coefficient [2, 18] for the interaction: $\left( {E_{j}^{\omega } ,E_{k}^{\omega } } \right) \to E_{i}^{2\omega }$. As $\varDelta k_{ijk}$ and $d_{ijk}$ are symmetrical with respect to permutations of the last two indices {j, k}, there will be only three different terms on the right side of Eq. (1). Usually the beam incident on the sample is generated by a laser and the light is linearly polarized. If α is the angle between the plane of polarization of the incident wave and the polarization direction of the eigenmode corresponding to the o wave in the case of a uniaxial material (or the s wave in a biaxial material), the field amplitudes of the fundamental wave to introduce in Eq. (1) will be:

$$\left\{ {E_{o}^{\omega } = E^{\omega } \cos \alpha,\quad E_{e}^{\omega } = E^{\omega } \sin \alpha } \right\}.$$

(2)

The intensity of the SHG wave that emerges from the particle will be given by:

$$I_{2} \left( {r,\theta ,\varphi ,\alpha } \right) = \frac{1}{2}c\varepsilon_{o} \left( {n_{2o} \left| {E_{o}^{2\omega } } \right|^{2} + n_{2e} \left| {E_{e}^{2\omega } } \right|^{2} } \right).$$

(3)

If the powder particles of the monolayer are randomly oriented and the laser spot size on the sample is large enough, the incident beam will cross particles oriented along all possible crystal directions. In consequence, the SHG intensity generated by the sample will be the result of averaging Eq. (3) over three angles $(\theta ,\varphi ,\alpha )$. We will carry out the averaging process in two steps: first on α, then on (θ, φ). After the first step, we obtain:

$$\begin{aligned} I_{2} \left( {r,\theta ,\varphi } \right) \equiv \left\langle {I_{2} \left( {r,\theta ,\varphi ,\alpha } \right)} \right\rangle_{\alpha } & = \frac{{2\omega^{2} }}{{\varepsilon_{o} c^{3} }} I_{1}^{2} \left[ {\frac{3}{8} \frac{{d_{ooo}^{2} }}{{n_{2o} n_{1o}^{2} }} \left| {\varDelta_{ooo} } \right|^{2} } \right. \\ & \quad + \frac{1}{4} \frac{{d_{ooo} d_{oee} }}{{n_{2o} n_{1o} n_{1e} }} \Re \left[ {\varDelta_{ooo} \varDelta_{oee}^{*} } \right] \\ & \quad + \frac{1}{2} \frac{{d_{ooe}^{2} }}{{n_{2o} n_{1o} n_{1e} }} \left| {\varDelta_{ooe} } \right|^{2} + \frac{3}{8} \frac{{d_{oee}^{2} }}{{n_{2o} n_{1e}^{2} }} \left| {\varDelta_{oee} } \right|^{2} \\ & \quad + \frac{3}{8} \frac{{d_{eoo}^{2} }}{{n_{2e} n_{1o}^{2} }} \left| {\varDelta_{eoo} } \right|^{2} + \frac{1}{4} \frac{{d_{eoo} d_{eee} }}{{n_{2e} n_{1o} n_{1e} }} \Re \left[ {\varDelta_{eoo} \varDelta_{eee}^{*} } \right] \\ & \left. {\quad + \frac{1}{2} \frac{{d_{eoe}^{2} }}{{n_{2e} n_{1o} n_{1e} }} \left| {\varDelta_{eoe} } \right|^{2} + \frac{3}{8} \frac{{d_{eee}^{2} }}{{n_{2e} n_{1e}^{2} }} \left| {\varDelta_{eee} } \right|^{2} } \right]. \\ \end{aligned}$$

(4)

Here, n _1o,e and n _2o,e represent the principal refractive indices for the fundamental and the second harmonic waves respectively, and $\varDelta_{ijk} \equiv \left( {\exp (i\varDelta k_{ijk} r) - 1} \right)/(i\varDelta k_{ijk} )$. I ₁ is the intensity of the fundamental wave within the sample and I ₂ the SHG intensity at the end of the sample. Note that, unlike what is usually considered [9], all types of wave interactions (oo → o, oe → o, ee → o, …) contribute, in general, to the SHG intensity, each of them with a different wave vector mismatch. Substituting {o → s, e → f} in Eqs. (2)–(4), the equivalent expressions for biaxial materials are obtained.

2.1 Averaging on (θ, φ)

In Eq. (4), the variables that depend on the angles (θ, φ) are {d _ijk (θ, φ), Δ _ijk (θ), n _e (θ)} in uniaxial materials, and {d _ijk (θ, φ), Δ _ijk (θ, φ), n _s (θ, φ), n _f (θ, φ)} in biaxial materials. We have carried out numerically the averaging of $I_{2} (r,\theta ,\varphi )$ over (θ, φ) for different compounds and different particle sizes.

2.1.1 Non-phase-matchable materials (NPMM)

We have studied the dependence between I ₂ and r in quartz powders (point group 32) excited by the same fundamental wavelength (λ ₁ = 0.694 μm) used in the experimental study of Ref. [17]. The refractive indices required for the calculations [n _o,e (λ ₁, λ ₂)] have been taken from [19]. The total SHG intensity emitted by the layer is shown in Fig. 1a. As can be observed, it oscillates with a period of about 15 μm, which approximately corresponds to twice the coherence length [4] averaged over all crystal directions and all types of wave interactions. If the variation of particle sizes in the sample is >15 μm, only the average value of the intensity in that range will be observed (dashed line). The SHG intensity will be therefore independent of r, in agreement with the experimental results [17]. In Fig. 1b, the contribution of different types of wave interactions to I ₂ is shown. It can be observed that although the oo → o interaction predominates, the rest of interactions are not negligible and they should be considered in the analysis of the SHG intensity produced by the layer. In fact, if we compare Fig. 1a, b, we can observe that the average value of the total intensity is about twice larger than that produced only by the oo → o interaction.

Fig. 1

a Theoretical SHG intensity produced by a monolayer of crystalline quartz powder as a function of particle size (continuous line) for λ ₁ = 0.694 μm. Dashed line represents the SHG signal averaged over all thicknesses. b Contribution of several types of wave interactions to the SHG intensity generated by the monolayer: $oo \to o$ (continuous line), $oe \to o$ (dashed line), and $oo \to e$ (dotted line)

2.1.2 Phase-matchable materials (PMM)

We have studied 3 uniaxial compounds with point group $\bar{4}2$ m [potassium dihydrogen phosphate (KDP), ammonium dihydrogen phosphate (ADP), rubidium dihydrogen phosphate (RDP)] and one biaxial compound with point group 2 [potassium tartrate (DKT)]. The fundamental wavelength (λ = 0.694 μm) was also the one used in the experimental study of Ref. [17]. For that wavelength, all the studied materials show a PM of type I [oo → e (uniaxial), ss → f (biaxial)]. The values of the variables required for the calculations have been taken from [2, 20–23]. The results obtained are shown in Fig. 2. It can be observed that for r ≥ 20 μm, the SHG intensity generated by the layer is essentially linear on the particle size, regardless of the uniaxial or biaxial character of the material. The relative values of the slopes of the straight lines are in good agreement with those experimentally determined [17]:

Fig. 2

SHG intensity produced by a monolayer of crystalline powder of KDP, ADP, RDP and DKT as a function of particle size for λ ₁ = 0.694 μm. The values of the refractive indices for the wavelengths involved have been taken from Kirby and DeShazer [20] (KDP, ADP, RDP) and Hobden [21] (DKT). The values of the nonlinear optical coefficients relative to d ₁₄(KDP) used in the numerical calculations are: ADP [22]: d ₁₄ = 1.1; RDP [2]: d ₁₄ = 0.92; DKT [23]: d ₂₁ = 0.28, d ₂₂ = 0.9, d ₂₃ = 0, d ₂₅ = 0.4. The inset shows the residual oscillations with respect to the linear behavior for RDP and KDP

	Experimental	Model
Slope(RDP)/slope(KDP)	2.3	2.7
Slope(ADP)/slope(KDP)	1	1.15
Slope(DKT)/slope(KDP)	0.7	0.7

In addition, for r large enough, the SHG intensity due to the PM mode prevails over the rest. Thus, the summation on the right-hand side of Eq. (4) is reduced to only one dominant term. The linear dependence between I ₂ and r seems surprising at first sight, since it is well known [4–6] that the SHG signal generated along the PM direction grows as r ². The reason for that dependence will be discussed in the next section.

2.2 Analytical expressions for uniaxial PMM

We will analyze the case of uniaxial PMM. Only the term corresponding to the type of wave interaction for which the PM occurs will be considered. If the PM angle is θ _c  ≠ 90° and the powder particle size is large enough, the SHG intensity emitted by the layer will be given by (see Appendix 1):

$$I_{2} (r) = \frac{{8\pi^{2} }}{{\varepsilon_{o} c\lambda_{1} }} I_{1}^{2} {\text{ PolFact }}n{\text{Fact}}\left\langle {d_{\text{eff}}^{2} } \right\rangle r,$$

(5)

where

• λ ₁ and I ₁ are the wavelength and the intensity of the fundamental wave, respectively;

• PolFact is the “polarization factor” that depends on the type of PM (see Table 1);

  Table 1 Polarization factor (PolFact) for each type of PM

• nFact is a function of the principal refractive indices of the material for the wavelengths involved {(n _1o , n _1e ), (n _2o , n _2e )}. It depends on the sign of the birefringence and the type of PM (see Table 2). The angle of PM (θ _c ) can be calculated for every type of PM using the expressions given in Table 3. The extraordinary refractive index along the PM direction [n _e (θ _c )] is given by [2]:

  Table 2 nFact for positive and negative uniaxial materials and both types of PM

  Table 3 Cos(θ _c ) for positive and negative uniaxial materials and both types of PM [2]

  $$n_{e} (\theta_{c} ) = \frac{{n_{o} n_{e} }}{{\sqrt {n_{o}^{2} + \left( {n_{e}^{2} - n_{o}^{2} } \right)\cos^{2} (\theta_{c} )} }};$$

  (6)

• $\left\langle {d_{\text{eff}}^{2} } \right\rangle$ is the average value along the PM direction θ _c of the square of the nonlinear coefficient for the type of PM considered (jk → i):

  $$\left\langle {d_{\text{eff}}^{2} } \right\rangle \equiv \left\langle {d_{ijk}^{2} \left( {\theta_{c} ,\varphi } \right)} \right\rangle_{\varphi } \equiv \frac{1}{2\pi} \int\limits_{0}^{2\pi } {d_{ijk}^{2} \left( {\theta_{c} ,\varphi } \right)} {\text{d}}\varphi .$$

  (7)

  It depends on the sign of the birefringence, the type of PM and the crystal class (see Table 4).

  Table 4 $\left\langle {d_{\text{eff}}^{2} } \right\rangle$ for all non-centrosymmetrical uniaxial crystal classes

Equation (5) clearly shows that the dependence of the SHG intensity emitted by the layer is linear with the powder particle size and not quadratic. The reason is that although the SHG signal generated along the PM direction grows as r ², the angular width Δθ around the PM direction for which the SHG is significantly emitted varies as r ⁻¹ [see Eq. (60) and Fig. 16 in Appendix 1]. Thus, when r increases, the number of particles correctly oriented for which the fundamental wave propagates around the PM direction decreases as r ⁻¹. As the SHG intensity generated by those particles increases as r ², the total SHG intensity emitted by the ensemble will depend linearly on the powder particle size. This reasoning also applies to biaxial PMM (see Appendix 1). Nevertheless, in that case, the PM direction loci are complicated [21, 24–26], the θ angle does not remain constant and no simple analytical expression like Eq. (5) can be derived. On the other hand, it should be noted that sometimes both types of PM could be present for the same wavelength. This is the case, for example, of KDP for λ ₁ = 1.064 μm [2, 8]. In a bulk crystal, only one type of PM can be selected by proper orientation of the sample. In the case of a random monolayer, that situation cannot be achieved. Therefore, both types of PM will coexist and the total SHG intensity generated by the layer will be described by the sum of Eq. (5) for both types of PM [11].

Some coefficients of the second-order nonlinear tensor d _mn can be easily determined in this kind of materials, as I ₂(r) has an analytical expression (Eqs. (5), (6) together with Tables 1, 2, 3, 4). Thus, from the slope of the straight line that best fits the experimental data I ₂ versus r, the value of one coefficient (materials belonging to crystal classes: 6, 4, 6mm, 4mm, $\bar{6}m2$, 32, $\bar{4}2m$, and sometimes 3m) or a combination of some of them (materials belonging to crystal classes: $\bar{6}$, $\bar{4}$, and sometimes 3 and 3m) can be obtained. Except for crystal classes 3 and 3m, the value of those coefficients can be determined even when both types of PM coexist. The values of coefficients d _mn obtained from experimental data using this procedure have proven to be reliable and precise [11].

2.3 The case of NPMM and biaxial PMM

As we have seen in Sect. 2.1.1, in the case of NPMM, the SHG intensity generated by the layer is, in practice, independent of the particle size: I ₂(r) = C. The value of some coefficients {d _mn } can be determined by equating C to the result obtained by numerically averaging Eq. (4) on $(\theta ,\varphi ,r)$. As there is only one experimental datum (C), the method will be especially useful for materials belonging to crystal classes with only one nonzero coefficient. Assuming the Kleinman symmetry [27], these classes are: $\bar{4}3m$, 23, 222, $\bar{6}m2$, 32, $\bar{4}2m$.

. In the case of biaxial PMM, the value of some coefficients d _mn can be determined by equating the slope of the straight line that best fits the experimental data I ₂(r) to the expression [see Eq. (64)]

$$I_{2} (r) = \frac{{8\pi^{2} }}{{\varepsilon_{o} c\lambda_{1} }}I_{1}^{2} {\text{PolFact}}\left\langle {d_{\text{eff}}^{2} } \right\rangle r.$$

(8)

The procedure is inevitably numerical because $\left\langle {d_{\text{eff}}^{2} } \right\rangle$ is now a complicated function [see Eqs. (64), (65) in Appendix 1]. The method can be especially useful for materials of the 222 crystal class, with only one independent nonlinear coefficient.

To end this section, we would like to comment that, in all types of materials, I ₂(r) is extremely sensitive to the value of the principal refractive indices of the material for the wavelengths involved. Therefore, a larger I ₂(r) (NPMM) or a larger slope of I ₂(r) versus r (PMM) does not necessarily imply higher nonlinear coefficients. In fact, it can be shown that if we want to determine d _mn accurately, the principal refractive indices for both wavelengths have to be known with an error <0.01.

2.4 Use of polarizers

Let us suppose that the sample is placed between two linear polarizers and that the angle between their polarizing axes is β. If α is the angle between the axis of the first polarizer and the polarization direction of the eigenmode corresponding to the o wave (or the s wave) of one powder particle of the layer, then the field amplitude of the SHG wave produced by that particle along the axis of the analyzer will be:

$$\begin{aligned} E_{{{\text{anal}} .}}^{2\omega } \left( {r,\theta ,\varphi ,\alpha ,\beta } \right) & = E_{o}^{2\omega } \left( {r,\theta ,\varphi ,\alpha } \right)\cos \left( {\alpha + \beta } \right) \\ & \quad + E_{e}^{2\omega } \left( {r,\theta ,\varphi ,\alpha } \right)\sin \left( {\alpha + \beta } \right) \\ \end{aligned}$$

(9)

where $E_{o}^{2\omega } (r,\theta ,\varphi ,\alpha )$ and $E_{e}^{2\omega } (r,\theta ,\varphi ,\alpha )$ are given by Eqs. (1) and (2). In consequence, the SHG intensity passing through the analyzer when the total ensemble of powder particles is considered will be:

$$I_{2} (r,\beta ) \equiv \left\langle {\frac{1}{2}c\varepsilon_{o} n_{2} \left| {E_{{{\text{anal}}.}}^{2\omega } \left( {r,\theta ,\varphi ,\alpha ,\beta } \right) } \right|^{2} } \right\rangle_{\alpha ,\theta ,\varphi } .$$

(10)

The result obtained depends on the angle between polarizers β. In the case of uniaxial PMM, it can be shown that Eq. (10) leads to Eq. (5), being PolFact given by the third column of Table 1 [11]. It can be observed that depending on the type of PM, the SHG output depends on β (type I) or not (type II). Hence, in those cases, in which both types of PM coexist, the use of polarizers will allow to separate their contribution to the total SHG intensity [11]. This result is particularly useful for materials belonging to crystal classes 3 and 3m as this method would allow the determination of the value of more coefficients d _mn . Similarly, in the case of NPMM or biaxial PMM, the use of polarizers would allow to obtain the value of more nonlinear optical coefficients but now that determination must be carried out numerically.

3 SHG by thick samples of crystalline powder

The first systematic experimental study of the SHG intensity generated by thick samples of crystalline powder ($L \gg r$, where L represents the sample thickness) was carried out by KP [9]. Their main results can be summarized as follows:

• a. Angular distribution of the SHG intensity:

  • Powder in air The sample behaves like an isotropic planar radiator obeying Lambert’s (cos) law. The SHG intensity emitted by the back surface of the sample is greater than or equal to that emitted by the front surface (see Fig. 3 of Ref. [9]). If the particle size decreases, the SHG intensity emitted by the sample in the backward direction increases while that emitted in the forward direction diminishes.

  • Powder in index-matching liquid The SHG intensity is emitted in a narrow cone in the forward direction. The total SHG intensity is much greater than that produced by powders in air.

• b. Dependence on sample thickness L:

  • The SHG intensity generated by a sample immersed in an index-matching liquid varies linearly with L.

• c. Dependence on powder particle size:

  • NPMM: The SHG intensity shows a maximum when r is close to the average coherence length ($\left\langle {l_{\text{c}} } \right\rangle$). For greater values of particle size, the intensity depends inversely on r. When the spread of the values of r in the sample is small enough, an oscillation of period 2 $\left\langle {l_{\text{c}} } \right\rangle$ superimposed to the r ⁻¹ dependence can be observed.

  • PMM in index-matching liquid: The SHG intensity increases with particle size up to r ≈ 150 μm. For greater values of r, it remains constant (see Fig. 6 in [9, 28]).

Some of these results, particularly (b) and (c), can be qualitatively explained using a simple model, as we will show below.

3.1 Simplified model

Let us consider a thick sample of crystalline powder immersed in an index-matching liquid. A sample of thickness L can be modeled by a sequence of N = L/r monolayers of crystalline powder as those considered in Sect. 2. Assuming that the fundamental incident beam is not altered when passing through the sample and that the SHG intensities generated by the layers add up incoherently then:

$$I_{{2,{\text{sample}}}} (r) = NI_{{2,{\text{layer}}}} (r) = \frac{L}{r}I_{{2,{\text{layer}}}} (r).$$

(11)

According to this equation, the SHG intensity generated by the sample will depend linearly on L. Besides, in the case of a NPMM, the oscillations of period 2$\left\langle {l_{\text{c}} } \right\rangle$ of the function I _2,layer(r) (see Sect. 2.1.1) will be damped by a factor r ⁻¹. If the error in the determination of r is >2$\left\langle {l_{\text{c}} } \right\rangle$ only a r ⁻¹ dependence will be observed. Finally, in the case of a PMM, $I_{{2,{\text{layer}}}} (r) \propto r$ [see Eq. (5)] and the SHG intensity generated by the sample will be independent of the particle size. Eq. (11) can, therefore, reproduce the main features of the L and r dependence of I _2,sample experimentally observed. Besides, in the case of a PMM immersed in an index-matching liquid, the value of $\left\langle {d_{\text{eff}}^{2} } \right\rangle$ can be easily obtained from the constant value of I _2,sample and Eqs. (5) and (11).

Now, some weak aspects of the KP theory will be pointed out. The first criticism has to do with the approach to calculate the SHG by the sample. In the model proposed by KP [9], the sample is described as a 1D arrangement of N = L/r particles randomly oriented along the sample thickness. Due to the arbitrary orientation of each particle, the average contribution per particle to the SHG intensity $\bar{I}_{{2,{\text{part}} }} (r)$ is calculated by averaging $I_{{2,{\text{part}}}} (r,\theta ,\varphi )$ over the angles (θ, φ): $\bar{I}_{{2,{\text{part}}}} (r) \equiv \left\langle {I_{{2,{\text{part}}}} (r,\theta ,\varphi )} \right\rangle_{\theta ,\varphi }$. The SHG intensity generated by the sample will be given by:

$$I_{{2,{\text{sample}}}} (r) = N\bar{I}_{{2,{\text{part}}}} (r) = \frac{L}{r}\bar{I}_{{2,{\text{part}}}} (r).$$

(12)

Nevertheless, this assumption is not always correct since the SHG contribution of the different particles along the sample thickness is coherent, especially for small N values. However, even in this case, the coherence does not manifest in practice since the illuminated area of the sample contains a large number of particles, and it is in fact the averaging over this ensemble which is equivalent to an incoherent contribution of all the particles in the sample. Since $I_{{2,{\text{layer}}}} (r)$ and $\bar{I}_{{2,{\text{part}}}} (r)$ are equal, Eqs. (11) and (12) are identical.

In any case, the equations proposed by KP for $\bar{I}_{{2,{\text{part}}}} (r)$ in [9] have some limitations:

• The analysis is restricted to materials with cubic symmetry.

• Only one type of wave interaction is considered. However, in the case of NPMM, it has been shown in Sect. 2.1.1 that many wave interactions can appreciably contribute to the SHG intensity. On the other hand, in the case of PMM, both types of PM (I and II) can be achieved for certain wavelengths simultaneously [11], and the equations are different for every type of PM (see Eq. (5) and Tables 1, 2, 3, 4).

In addition, the equations proposed by KP in [9] have other shortcomings:

• The effect of the polarization of the incident beam is not considered. Even for unpolarized light, the averaging on α gives rise to different factors depending on the type of wave interaction [see Eqs. (4), (5) and Polfact in Table 1).

• The equations for $\left\langle {d_{\text{eff}}^{2} } \right\rangle$ given in the Appendix of Ref. [9] and deduced in [29] are not valid for non-cubic crystals. The correct expressions are given in Table 4 of the present paper.

On the other hand, both models (“multilayer” and “1D”) have a common and very important shortcoming: They disregard the inevitable scattering by the powder particles of the sample. In consequence, two essential variables of the system, the volume fraction of the sample occupied by the particles (f) and the refractive index ratio (n/n _liq), do not appear in the equations. But a perfect index-matching is usually impossible to achieve. Sometimes it is not feasible to find a proper liquid because the refractive index of the powder is very high or there is a chemical reaction between liquid and powder. Even when the refractive index of the powder is nearly matched by the liquid, the sample is usually so thick that the scattering cannot be neglected. In addition, on many occasions, index matching is simply impossible even in principle, because of the existence of moderate birefringence in the material. In this respect, it is interesting to point out that the vast majority of experiments are carried out on dry powders and, however, the effects of scattering are systematically ignored.

In the case of PMM, there is also an intriguing disagreement between the r dependence of I _2,sample predicted by Eqs. (11) [or (12)] and the real dependence on r experimentally observed. As is shown in Appendix 1, the SHG intensity produced by a monolayer of a PMM becomes linearly dependent on r for $r \gg \pi /\left| {\beta_{ijk} } \right|$ [Eq. (61)]. Typical values of $\left( {\pi /\left| {\beta_{ijk} } \right|} \right)$ range from 1 to 20 μm. Numerical calculations carried out for different materials confirm this result and show that the relation between I ₂ and r already becomes linear for r ≥ 20 μm (see Fig. 2). Therefore, according to Eq. (11) I _2,sample(r) should become constant for $r_{c} \cong 20$ μm. Similarly, KP [9] deduce that I _2,sample would be independent on the powder particle size for:

$$r \gg \frac{{\varGamma_{\text{PM}} }}{{\sin (\theta_{c} )}} = \frac{{\left( {\pi \sin (\theta_{c} )/\beta } \right)}}{{\sin (\theta_{c} )}} = \frac{\pi }{\beta }.$$

(13)

This equation is equal to Eq. (61) so the previous discussion also applies here and $r_{c} \cong 20$ μm. These predictions are, however, in contradiction with experiments: it is observed that I_2,sample increases with particle size up to $r \approx 150$ μm, and only for greater values of r, it remains constant (see Fig. 6 in [9, 28]). Equations (11) and (12) are, therefore, unable to explain why the value of I _2,sample (r) does not saturate until the particle size is as large as $r \cong 7r_{c}$. The reason of that disagreement remains unexplained.

In the next section, a model that takes into account the scattering by the powder particles in the sample will be proposed and the origin of the peculiar r dependence of the SHG intensity generated by a PM material in index-matching liquid for $r \le 7r_{c}$ will be clarified.

3.2 Model with scattering

The rest of the present paper will be devoted to study the effects of the light scattering in the SHG process by micropowders. We will see that the consideration of the scattering influence is critical for the correct interpretation of experiments not only in the case of dry powders but also in the case of powders immersed in an index-matching liquid. In some cases, the proper analysis of scattering is indispensable even to obtain mere qualitative information of the nonlinear behavior of the materials. When the experimental conditions are properly controlled, we will show that the SHG intensities can be related with expressions already deduced for the case of monolayers. Only in these situations, quantitative data can be extracted for the d _mn coefficients.

3.2.1 Scattering of the primary beam

We will first introduce several concepts related to light scattering by particles. When a light wave is incident on a particle, it is partially scattered. The ability of the particle to scatter light is described by “the scattering cross section (σ _s)” [30–33]. It depends on the refractive index ratio $m \equiv n/n_{m}$ between the scatterer (n) and the surrounding medium (n _m ), and the ratio r/λ _m between the particle size (r) and the light wavelength in the medium (λ _m  = λ/n _m ). The scattering of a plane wave by one particle, even of spherical shape, is anisotropic [30–33]. The asymmetry of the angular distribution of scattering depends, among other factors, on the particle size r [30–33] and the refractive index ratio m. When the ratio (r/λ _m ) is small, the scattering is not very anisotropic (Rayleigh scattering). But as the particle size increases, more light tends to scatter in the forward direction (Mie scattering) (see Fig. 3). On the other hand, when the index mismatch between the scatterer and the surrounding medium decreases (m → 1), the light also tends to be scattered forward.

Fig. 3

Schematic representation of Rayleigh and Mie scattering on spherical particles. The arrows indicate the scattering intensity at particular directions. The light is incident from the left hand side

When the ratio (r/λ _m ) is large, as is usually the case in the experiments of SHG by micropowders, the real shape of the particles has a small relevance on the global scattering properties of the system and can be substituted by area-equivalent spheres [32]. The scattering cross section is then written as:

$$\sigma_{s} (R) \equiv \pi R^{2} Q_{s} (R),$$

(14)

where R is the radius of the area-equivalent sphere and Q _s (R) the “efficiency factor for scattering” that can be computed using the Mie theory for spheres [30–33]. The distribution of particle sizes usually has a dispersion high enough (larger than the wavelength) to wipe out Mie resonances [32]. When a light wave propagates through a thick powder sample, it will be multiply scattered. The average distance between two consecutive scattering events is called “scattering mean free path (l _s)” and, assuming the independent-scattering approximation, it is related to σ _s by [34]:

$$l_{s} = \frac{1}{{\rho \sigma_{\text{s}} }},$$

(15)

where ρ is the number of scatterers per unit volume in the sample. In terms of the volume fraction of the sample occupied by the scatterers f = ρ (4/3)πR ³:

$$l_{s} = \frac{1}{{\rho \sigma_{\text{s}} }} = \frac{{4\pi R^{3} }}{{3f\sigma_{\text{s}} }} = \frac{{4\pi R^{3} }}{{3f\pi R^{2} Q_{s} (R)}} = \frac{4R}{{3fQ_{s} (R)}}.$$

(16)

This equation can be used to calculate the dependence of l _s on the particle size (r = 2R). The result obtained for powder particles of ADP (n = 1.49) in air is shown in Fig. 4. The values of Q _s (R) have been computed using the Mie theory for spheres and f = 0.64, the maximum value for a random distribution of hard spheres [35, 36]. It can be observed that l _s depends linearly on the particle size. The reason is that for the particle sizes considered (2π/λ _m)R $\gg$ 1 and, since the particles are non-absorbing, it can be shown that $Q_{s} (R) \cong 2$ [30, 31]. Therefore, for large powder particles in air (r ≥ 20 μm), Eq. (16) can be reduced to:

$$l_{s} \cong \frac{r}{3f}.$$

(17)

Fig. 4

Scattering mean free path (l _s ) (triangles) and “effective mean free path” (l) as a function of particle size for powder particles of ADP in air. It has been assumed: n(ADP) = 1.49, f = 0.64, and λ ₁ = 1.064 μm. The values of l _s and l have been computed using the Mie theory for spheres. The effective mean free path has been calculated for different values Δγ of the half-angle of the cone around the forward direction: Δγ = 1° (circles), Δγ = 5° (squares), and Δγ = 10° (diamonds)

The effect of r and m on the anisotropic distribution of scattering can be observed in Fig. 5. Q _{s,“non-forward”} represents the efficiency factor for scattering out of a cone of half-angle Δγ = 1° around the forward direction, and Q _s the total efficiency factor for scattering. It can be observed that the efficiency for scattering out of the forward direction is greater for small particle sizes and high refractive index ratio.

Fig. 5

Q _{s,“non-forward”}/Q _s as a function of particle size for powder particles of ADP. The forward direction has been taken inside the cone of half-angle Δγ = 1°. Two cases have been considered: powders in air (refractive index ratio m = 1.49) and powders immersed in an index-matching liquid with m = 1.001. The values have been computed using the Mie theory for spheres. It has been assumed: n(ADP) = 1.49 and λ ₁ = 1.064 μm. Continuous lines are only eye guides

In the case of non-absorbing particles in air of very large size in comparison with the wavelength, we will show that (Q _{s,“non-forward”}/Q _s ) $\cong$ 1/2. If (2π/λ _m )R $\gg$ 1, the light can be considered to be scattered by three different processes: diffraction (Q _diff), reflection and refraction ($Q_{\text{reflec}} + Q_{\text{refrac}} \equiv Q_{r}$), i.e., Q _s  = Q _diff +Q _r [30, 31]. It can be demonstrated that the diffracted light is confined to a narrow set of angles about the forward direction and Q _diff = 1 [30, 31]. On the contrary, the light reflected or refracted gives rise to a less intense radiation into all directions with an angular distribution dependent on the refractive index of the particle. Thus $Q_{{s,\hbox{``}{\text{non - forward}}\hbox{''}}} \cong Q_{r}$. As the particles are very large Q <sub>s</sub>  → 2 [30, 31] so that Q <sub>r</sub>  = Q <sub>s</sub>  − Q <sub>diff</sub> → 1. Therefore, $\left( {Q_{{s,\hbox{``}{\text{non - forward}}\hbox{''}}} /Q_{s} } \right) \cong \left( {Q_{r} { / }Q_{s} } \right) \to 1 /2$ (see circles in Fig. 5). In the case of particles of very large size immersed in an index-matching liquid, when m → 1, all scattered light is confined to very small angles around the forward direction, and Q _{s,“non-forward”} → 0 (see squares in Fig. 5).

We can characterize the ability of the particle to scatter the light incident on it out of the forward direction through a “non-forward scattering cross section” defined as:

$$\sigma_{{s,\hbox{``}{\text{non - forward}}\hbox{''}}} (R) \equiv \pi R^{2}Q_{{s,\hbox{''}{\text{non - forward}}\hbox{''}}} (R).$$

(18)

Assuming the independent-scattering approximation, the length:

$$l \equiv \frac{1}{{\rho \sigma_{{s,\hbox{``}{\text{non - forward}}\hbox{''}}} }}$$

(19)

represents the effective mean free path of light in the forward direction or the average distance the light travels before its direction of propagation is changed. For simplicity, we will refer to it as “the effective mean free path” of light in the sample. The dependence of l on the size of ADP powder particles in air is shown in Fig. 4. The different symbols in the figure account for several values of the half-angle of the cone considered in the definition of “forward direction.” It can be observed that l does not essentially depend on the precise value of Δγ, at least in the interval: 1° ≤ Δγ ≤ 10°. On the other hand, l depends linearly on the particle size and, in agreement with the previous discussion, is about twice l _s . Therefore, for large powder particles in air (r ≥ 20 μm), l can be approximated by the simple result:

$$l \cong 2l_{s} \cong \frac{2}{3f}r.$$

(20)

According to the experimental conditions used by KP in [9], we will assume that the sample can be modeled by a slab whose dimensions x and y are much larger than the sample thickness and that the primary beam incident along the z axis can be described by a plane wave. It will be also assumed that the beam diameter at the sample is much larger than the sample thickness. These conditions simplify the analysis of the problem that is essentially reduced to a 1D case in the z direction. The intensity of the primary beam that propagates through the sample along the z axis will decay exponentially with distance due to non-forward scattering as:

$$I_{1} (z) = I_{1,o} \exp ( - z/l_{1} ),$$

(21)

where l ₁ represents the effective mean free path of light defined in Eq. (19) for the wavelength of the primary beam. The light scattered out of the z axis will be diffused through the sample. As the SHG process is proportional to $I_{1}^{2} ,$ the contribution of the diffused intensity to the SHG is expected to be negligible, and the SHG radiation is essentially generated by I ₁(z). In the case, not considered here, of strongly scattering media constituted by nanoscatterers, the contribution of the diffused intensity to the SHG could be important [13, 14].

3.2.2 Average particle cross section for SHG

We will characterize the ability of a powder microparticle to generate SHG radiation by a “cross section for SHG.” To do this, we will use the expression obtained for the SHG radiation generated by a monolayer of powder particles.

As has been shown in Sect. 2, the SHG intensity generated by a monolayer of crystalline powders randomly oriented is:

$$I_{2} (r) \equiv \left\langle {I_{2} \left( {r,\theta ,\varphi } \right)} \right\rangle_{\theta ,\varphi } \equiv g(r)I_{1}^{2} ,$$

(22)

where $I_{2} (r,\theta ,\varphi )$ is given in Eq. (4). In the case of NPMM, g(r) is an oscillating function of r (see Sect. 2.1.1), while for PMM, it is a linear function of r [Eqs. (5), (8)]. The quantities I ₁ and I ₂ represent the intensities within the nonlinear optical material. If the particles are immersed in a medium of refractive index n _m , the value of the intensities in the medium ($I_{1,m}$, $I_{2,m}$) will be related to I ₁ and I ₂ by:

$$I_{1} = T_{1} I_{1,m} ,$$

(23)

$$I_{2,m} = T_{2} I_{2} ,$$

(24)

where T _i represents the transmittance between the medium and the material for the primary beam (i = 1) or the SHG beam (i = 2). For T _i , we will assume the expressions for normal incidence:

$$T_{i} \cong \frac{{4\bar{n}_{i} n_{m} }}{{\left( {\bar{n}_{i} + n_{m} } \right)^{2} }}.$$

(25)

Here $\bar{n}_{i} \equiv \left( {n_{io} + n_{ie} } \right)/2$ is the average refractive index of the material for the wavelength $\lambda_{i}$. In terms of $I_{1,m}$ and $I_{2,m}$ Eq. (22) can be rewritten as:

$$I_{2,m} (r) = T_{2} T_{1}^{2} g(r)I_{1,m}^{2} .$$

(26)

In Sect. 2, the occupation fraction of the layer has been assumed to be 100 %. If we also assume that, in average, the powder particles can be described as spheres of radius R = r/2, the number of particles in a region of the layer of area A will be:

$$N = \frac{A}{{\pi R^{2} }}.$$

(27)

We define the “average particle cross section for SHG” ($\bar{\sigma }_{sh}$) by requiring that the SHG power generated by $N = A/(\pi R^{2} )$ particles of cross section $\bar{\sigma }_{sh}$:

$$P_{2,m} = N\bar{\sigma }_{sh} I_{1,m}$$

(28)

to be equal to that generated by a region of the monolayer of area A:

$$P_{2,m} (r) = I_{2,m} (r)A.$$

(29)

Therefore:

$$\bar{\sigma }_{sh} = \pi R^{2} \frac{{I_{2,m} (r)}}{{I_{1,m} }} = \pi R^{2} T_{2} T_{1}^{2} g(r)I_{1,m} .$$

(30)

The effective particle cross section thus defined depends on the refractive index of the medium in which particles are immersed and the intensity of the primary beam, among other factors. The function g(r) has been deduced in Sect. 2 for a linearly polarized incident beam. However, as the light travels along the sample, it becomes progressively depolarized. Nevertheless, it can be demonstrated that the expressions for g(r) are also valid for unpolarized light. This can be understood since the sample has axial symmetry along the beam direction. Therefore, the cross section given by Eq. (30) will be valid for particles within a thick sample where the incident beam is essentially non-polarized. In this respect, it is irrelevant to use in the experiments linearly polarized or non-polarized fundamental light. Thus, the depolarization effects as the fundamental light travels through the sample are expected not to affect the measured SHG signal.

3.2.3 SHG by the sample

The primary beam that propagates through the sample will generate second harmonic radiation. The contribution of a thin slab at position z and thickness dz perpendicular to the beam direction can be expressed as ${\text{d}}I_{2}^{g} (z)$. Making use of Eq. (28), the intensity of second harmonic generated per unit length can be deduced to be:

$$\begin{aligned} \frac{{{\text{d}}I_{2}^{g} }}{{{\text{d}}z}}(z) = \rho \bar{\sigma }_{sh} I_{1} (z)& = \left( {\frac{f}{{(4/3)\pi R^{3} }}} \right) \left( {\pi R^{2} T_{2} T_{1}^{2} g(r)I_{1} (z)} \right)I_{1} \left( z \right) \\ &= f\frac{3}{4R}T_{2} T_{1}^{2} g(r)I_{1}^{2} (z) \\& = \frac{3}{2}fT_{2} T_{1}^{2} \left( {\frac{g(r)}{r}} \right)I_{1}^{2} (z), \\ \end{aligned}$$

(31)

where ρ represents the number of powder particles per unit volume, f is the volume fraction of the sample occupied by the powder particles, r is the particle size and I ₁(z) is the intensity of the primary beam that propagates through the sample along the z axis [Eq. (21)]. The SHG beam is collinear with the primary beam and, therefore, propagates along the z axis. This beam is subsequently scattered by the powder particles of the sample. Therefore, the intensity of the second harmonic beam that propagates along the z axis I ₂(z) will be described by the differential equation:

$$\frac{{{\text{d}}I_{2} (z)}}{{{\text{d}}z}} = \frac{{{\text{d}}I_{2}^{g} (z)}}{{{\text{d}}z}} - \frac{{I_{2} (z)}}{{l_{2} }},$$

(32)

where l ₂ represents the effective mean free path of light for the wavelength of the second harmonic. The part of the SHG beam that emerges through the sample face at z = L will be $I_{2} (z = L) \equiv I_{{2,{\text{beam}}}}$.

The second harmonic radiation scattered by the powder particles out of the z axis will be diffused throughout the sample and eventually will leave it through any of its faces (backward emission at z = 0, and forward emission at z = L). The intensity of second harmonic scattered out of the z axis per unit length will be given by:

$$\frac{{{\text{d}}I_{{2,{\text{scatt}}}} }}{{{\text{d}}z}}(z) = \frac{{I_{2} (z)}}{{l_{2} }}.$$

(33)

Therefore, the total intensity that emerges through both faces due to the second harmonic scattered throughout the sample will be given by:

$$I_{{2,{\text{scatt}}}} = \int\limits_{0}^{L} { \frac{{{\text{d}}I_{{2,{\text{scatt}}}} }}{{{\text{d}}z}}{\text{d}}z = \int\limits_{0}^{L} {\frac{{I_{2} (z)}}{{l_{2} }}dz} }.$$

(34)

The SHG process can be schematized as follows:

After the integration of the differential equation (32), assuming that $I_{2} (z = 0) = 0,$ the analytical expressions for I _2,beam and I _2,scatt can be obtained:

$$I_{{2,{\text{beam}}}} = q \left( {\frac{g(r)}{r}} \right) \frac{{l_{1} l_{2} }}{{\left( {2l_{2} - l_{1} } \right)}} \left( {\exp \left( { - \frac{L}{{l_{2} }}} \right) - \exp \left( { - \frac{2L}{{l_{1} }}} \right)} \right),$$

(35)

$$I_{{2,{\text{scatt}}}} = q\left( {\frac{g(r)}{r}} \right) \frac{{l_{1} l_{2} }}{{\left( {2l_{2} - l_{1} } \right)}} \left( {1 - \exp \left( { - \frac{L}{{l_{2} }}} \right) - \frac{{l_{1} }}{{2 l_{2} }}\left( {1 - \exp \left( { - \frac{2L}{{l_{1} }}} \right)} \right)} \right),$$

(36)

where

$$q \equiv \frac{3}{2}fT_{2} T_{1}^{2} I_{1,o}^{2} .$$

(37)

In the case of a quasi-perfect index-matching, the scattering of the second harmonic beam becomes negligible, and practically all second harmonic radiation is emitted through the sample face at z = L within a small solid angle around the z axis. The total intensity will be approximately given by I _2,beam. In contrast, when dry powders are considered, the scattering is strong, the second harmonic radiation is diffused throughout the sample and is emitted through both sample faces in all directions. The total intensity will be approximately given by I _2,scatt. In the general case, both terms, I _2,beam and I _2,scatt, will appreciably contribute to the intensity of second harmonic emitted by the sample, which will be given by

$$I_{{2,{\text{Tot}}}} = I_{{2,{\text{beam}}}} + I_{{2,{\text{scatt}}}} = q\left( {\frac{g(r)}{r}} \right)\frac{{l_{1} }}{2}\left( {1 - \exp \left( { - \frac{2L}{{l_{1} }}} \right)} \right).$$

(38)

As part of I _2,scatt is emitted through the sample face at z = 0, I _2,Tot has to be measured using an integrating sphere. I _2,Tot is independent on l ₂ because the scattering of the SHG beam is irrelevant in the evaluation of the total second harmonic radiation produced by the sample. In fact, I _2,Tot can also be obtained by integrating Eq. (31) over z.

In the next sections, we will use the previous expressions to analyze the SHG generated by thick samples of crystalline powder for different degrees of index matching. We will discuss separately the cases of PMM and NPMM.

3.3 Dry powder

In the case of dry powder, the effective mean free path l ₁ is usually small. If the sample is thick enough so $(l_{1} /2) \ll L,$ then the SHG signal is essentially generated in a region close to the front face of the sample (z = 0) and I _2,Tot becomes approximately independent of the sample thickness L:

$$I_{{2,{\text{Tot}}}} = q\left( {\frac{g(r)}{r}} \right)\frac{{l_{1} }}{2}\left( {1 - \exp \left( { - \frac{2L}{{l_{1} }}} \right)} \right) \cong q\left( {\frac{g(r)}{r}} \right)\frac{{l_{1} }}{2}.$$

(39)

On the other hand, as l ₁ depends linearly on the particle size [Eq. (20)], the dependence of I _2,Tot on r will be given only by the function g(r):

$$I_{{2,{\text{Tot}}}} \cong \frac{1}{2} T_{2} T_{1}^{2} I_{1,o}^{2} g(r).$$

(40)

3.3.1 PMM

In the case of PMM, g(r) is a linear function of r [Eqs. (22), (5), (8)] so, as long as $(l_{1} /2) \ll L,I_{{ 2,{\text{Tot}}}}$ will increase linearly with the particle size. We have calculated the dependence of I _2,Tot on r for samples of different thicknesses using Eq. (38). We have considered particles of ADP and f = 0.55. The results obtained are shown in Fig. 6. It can be observed that I _2,Tot increases linearly with the particle size and is essentially independent of L. Thus, if $(l_{1} /2) \ll L,$ the value of some coefficients of the second-order nonlinear optical tensor d _mn could be easily obtained from the slope of the straight line that best fits the experimental data I _2,Tot(r) and Eqs. (40), (22) and (5) or (8).

Fig. 6

Theoretical dependence of $I_{{2,{\text{Tot}}}}$ on r for samples of ADP powder in air of different thicknesses (in μm), λ ₁ = 1.064 μm and f = 0.55. The refractive indices required for the calculations have been taken from Kirby and DeShazer [20]. The function g(r) has been obtained by numerical averaging of Eq. (4). The values of l ₁(r) have been determined using the Mie theory for spheres and Eq. (19). $I_{{2,{\text{Tot}}}}$ has been calculated using Eq. (38). In the case of L = 500 μm, only particles with $r \le 240$ μm have been considered, so there are at least two particle layers in the sample

As the SHG light is emitted through both faces of the sample, I _2,Tot has to be measured using an integrating sphere. However, in the experiments, only the SHG light emitted in the backward direction is usually measured [37–49]. Now, we will explain some of those experimental results. Typically, the SHG light is collected by a parabolic mirror and sent to the detector [37–41, 44–49], or it is measured at different angles and added [42, 43]. This intensity corresponds to the part of I _2,scatt that is emitted through the sample face at z = 0.

From the theoretical point of view, the SHG intensity emitted in the backward direction can be determined by solving a last problem, which implies the analysis of the diffusion of I _2,scatt through the sample. The procedure is outlined in Appendix 2.

We have calculated the intensities of the diffuse SHG light emitted in the backward and forward directions, $I_{{2,{\text{diff}}}}^{\text{Back}}$ and $I_{{2,{\text{diff}}}}^{\text{Forw}},$ respectively, for different samples of ADP particles in air. In Fig. 7, the dependence of the ratio $I_{{2,{\text{diff}}}}^{\text{Back}} /I_{{2,{\text{diff}}}}^{\text{Forw}}$ on the particle size for samples of different thicknesses is shown. It can be observed that for small values of r, the SHG light is essentially emitted in the backward direction. As the particle size increases, the efficiency of light scattering in the forward direction by the powder particles increases (see Fig. 5) and, therefore, $I_{{2,{\text{diff}}}}^{\text{Forw}}$ increases too. Finally, when the particle size is large enough $I_{{2,{\text{dif}}f}}^{\text{Back}} \cong I_{{2,{\text{diff}}}}^{\text{Forw}}$. The behavior described in Fig. 7 accounts for the experimental results about the r-dependence of the backward and forward distribution of the SHG intensity. These results, which were already obtained by KP [9] had remained unexplained up to now. Concerning the effect of the sample thickness, if L decreases more diffuse intensity is emitted in the forward direction. Then, the ratio $I_{{2,{\text{diff}}}}^{\text{Back}} /I_{{2,{\text{diff}}}}^{\text{Forw}}$ decreases and the condition $I_{{2,{\text{dif}}f}}^{\text{Back}} \cong I_{{2,{\text{diff}}}}^{\text{Forw}}$ is attained for a smaller particle size. The dependence of $I_{{2,{\text{diff}}}}^{\text{Back}}$ on r for samples of different thicknesses is shown in Fig. 8a (f = 0.55), b (f = 0.30). It can be observed that $I_{{2,{\text{diff}}}}^{\text{Back}}$ increases with r and tends to saturate for large particle sizes. The saturation takes place at smaller r if L and/or f are decreased. Experimental data confirm these results [37–43, 46, 47]. Although those experimental curves resemble those obtained for PMM in index-matching liquid (Fig. 6 of Ref. [9]), the reason of the saturation is actually quite different: As the particle size is increased, an increasing fraction of the SHG light is emitted in the forward direction (see Fig. 7) and, therefore, $I_{{2,{\text{diff}}}}^{\text{Back}}$ tends to saturate. The determination of the value of some d _mn coefficients from the saturation value of $I_{{2,{\text{diff}}}}^{\text{Back}} (r)$ will be, therefore, quite difficult. Thus, many of the determinations of SHG efficiencies through measurements in the backward direction can be seriously questioned. Nevertheless, as the particle size dependence of $I_{{2,{\text{diff}}}}^{\text{Back}}$ is very different in NPMM (see next section), the conclusions about the NPMM or PMM character of the materials reported in [37–49] are still valid.

Fig. 7

Theoretical dependence of the ratio $I_{{2,{\text{diff}}}}^{\text{Back}}$/$I_{{ 2 , {\text{diff}}}}^{\text{Forw}}$ on r for samples of ADP powder in air of different thicknesses (in μm), using the same data as in Fig. 6. The values of l ₂(r) and l _tr,2(r) have been determined using the Mie theory for spheres and Eq. (19). r _i,2 has been computed according to the procedure given in [50]. In the case of L = 500 μm, only particles with r ≤ 240 μm have been considered, so there are at least two particle layers in the sample

Fig. 8

Theoretical dependence of $I_{{2,{\text{diff}}}}^{\text{Back}}$ on r for samples of ADP powder in air with: a f = 0.55, and b f = 0.30. Different sample thicknesses (in μm) have been considered. In the calculations, the same data and procedures as in Fig. 7 have been used. In the case of L = 500 μm, only particles with r ≤ 240 μm have been considered, so there are at least two particle layers in the sample

Measurements in the backward direction can, however, be used to determine the value of some nonlinear coefficients. If the sample is thick enough, practically no SHG radiation is emitted in the forward direction. For example, in the case of ADP powder with r ≤ 100 μm and f = 0.55, if L = 5,000 μm then $I_{{2,{\text{dif}}f}}^{\text{Back}} \ge 10 I_{{2,{\text{diff}}}}^{\text{Forw}}$ (see Fig. 7). Therefore, for these samples $I_{{2,{\text{diff}}}}^{\text{Back}} \cong I_{{2,{\text{Tot}}}}$ (see Figs. 6, 8a). Thus, as in the case of I _2,Tot analyzed before, the value of some coefficients d _mn can be obtained from the slope of the straight line that best fits the experimental data $I_{{2,{\text{diff}}}}^{\text{Back}} (r)$. No integrating sphere is needed in this method but it should be assured that the sample thickness is large enough so that conditions $I_{{2,{\text{diff}}}}^{\text{Forw}} \cong 0$ and $I_{{2,{\text{diff}}}}^{\text{Back}} \cong I_{{2,{\text{Tot}}}}$ are fulfilled for all particle sizes considered.

3.3.2 NPMM

In the case of NPMM, the function g(r) oscillates with a period $2\left\langle {l_{c} } \right\rangle$ (Sect. 2.1.1). Therefore, as long as $(l_{1} /2) \ll L,$ the function I _2,Tot(r) will also oscillate with the same period [Eq. (40)]. We have calculated the dependence of I _2,Tot on r for different samples of SiO₂ particles using Eq. (38). The results obtained are shown in Fig. 9. It can be observed that I _2,Tot(r) oscillates with a period of about 40 μm which approximately corresponds to twice the average coherence length of the material for the wavelength used (λ ₁ = 1.064 μm). If the dispersion in the particle sizes is >40 μm, only the average value of the intensity will be observed. In that case, I _2,Tot will be approximately independent of the particle size: $I_{{2,{\text{Tot}}}} (r) \cong C$. The value of C does not essentially depend on the sample thickness (see Fig. 9). Thus, if $(l_{1} /2) \ll L,$ the value of some coefficients d _mn could be obtained from the experimental determination of C and Eqs. (40), (22) and (4) (see Sect. 2.3).

Fig. 9

Theoretical dependence of $I_{{2,{\text{Tot}}}}$ on r for samples of SiO₂ powder in air of different thicknesses (in μm), λ ₁ = 1.064 μm, and f = 0.55. The refractive indices required for the calculations have been taken from Ghosh [19]. The function g(r) has been obtained by numerical averaging of Eq. (4). The values of l ₁(r) have been determined using the Mie theory for spheres and Eq. (19). $I_{{2,{\text{Tot}}}}$ has been calculated using Eq. (38). In the case of L = 500 μm, only particles with r ≤ 240 μm have been considered, so there are at least two particle layers in the sample. From the figure, it can be concluded that the sample thickness is irrelevant above a certain value no much larger than the particle size

The dependence of $I_{{2,{\text{diff}}}}^{\text{Back}}$ on r for L = 1,000 μm and two different values of f (f ₁ = 0.55, f ₂ = 0.30) are shown in Fig. 10. For small values of r, $I_{{ 2 , {\text{diff}}}}^{\text{Back}} (r) \cong I_{{2,{\text{Tot}}}} (r)$ (Fig. 7). However, when r is increased, the SHG radiation emitted in the forward direction increases and, since $I_{{2,{\text{Tot}}}} (r) \cong C$, $I_{{2,{\text{diff}}}}^{\text{Back}}$ decreases. In consequence, the function $I_{{2,{\text{diff}}}}^{\text{Back}} (r)$ shows a maximum at $r \cong \left\langle {l_{c} } \right\rangle$. This result is confirmed by the experimental data [37, 39, 44, 45, 48, 49]. Thus, in the case of a NPPM, the value of the average coherence length could be directly extracted from the experimental data $I_{{2,{\text{diff}}}}^{\text{Back}} (r)$ [37, 39, 44, 45, 48, 49]. Nevertheless, for $r \ge \left\langle {l_{c} } \right\rangle$, $I_{{2,{\text{diff}}}}^{\text{Back}} (r)$ will not decay with a $r^{ - 1}$ dependence, and the estimate of d _mn coefficients from $I_{{2,{\text{diff}}}}^{\text{Back}}$ data obtained in these kind of measurements is questionable.

Fig. 10

Theoretical dependence of $I_{{2,{\text{diff}}}}^{\text{Back}}$ on r for samples of SiO₂ powder in air with f ₁ = 0.55 (continuous line) and f ₂ = 0.30 (discontinuous line). The sample thickness is L = 1,000 μm. The same data as in Fig. 9 have been used in the calculations. The values of l ₂(r) and l _tr,2(r) have been determined using the Mie theory for spheres and Eq. (19). r _i,2 has been computed according to the procedure given in Zhu et al. [50]

However, as in the case of PMM, if the sample is thick enough (L = 5,000 μm, f = 0.55 and r ≤ 100 μm, for example) then $I_{{2,{\text{diff}}}}^{\text{Forw}} \cong 0$ and $I_{{2,{\text{diff}}}}^{\text{Back}} \cong I_{{2,{\text{Tot}}}}$. Thus $I_{{2,{\text{diff}}}}^{\text{Back}}$ will be approximately independent of the particle size $I_{{2,{\text{diff}}}}^{\text{Back}} \cong C$ and, as in the case of $I_{{2,{\text{Tot}}}}$ analyzed before, the value of some coefficients d _mn could be obtained from the experimental determination of C.

3.4 Powder in index-matching liquid

When the powders are immersed in an index-matching liquid, the primary beam is expected to be barely scattered and, therefore, l ₁ is large. If $(l_{1} /2) \gg L$ the intensity of the primary beam does not depend essentially on z and, according to Eq. (31), the second harmonic generated per unit length is the same throughout the sample thickness. Then $I_{{2,{\text{Tot}}}}$ becomes linearly dependent on L and f:

$$\begin{aligned} I_{{2,{\text{Tot}}}}& = q \left( {\frac{g(r)}{r}} \right)\frac{{l_{1} }}{2}\left( {1 - \exp \left( { - \frac{2L}{{l_{1} }}} \right)} \right) \cong q\left( {\frac{g(r)}{r}} \right)\frac{{l_{1} }}{2}\left( {1 - \left( {1 - \frac{2 L}{{l_{1} }}} \right)} \right) \\ & = q \left( {\frac{g(r)}{r}} \right)L = \frac{3}{2}fT_{2} T_{1}^{2} I_{1 ,o}^{2} \left( {\frac{g\left( r \right)}{r}} \right)L \equiv I_{{2 , {\text{Tot}}}}^{\lim } . \\ \end{aligned}$$

(41)

The dependence of the maximal SHG total intensity $I_{{2,{\text{Tot}}}}^{\lim }$ on the particle size will be described by the function $\left( {\frac{g(r)}{r}} \right).$

3.4.1 PMM

In the case of PMM, if the particle size is large enough g(r) is a linear function of r (typically r ≥ 20 μm, see Sect. 2.1.2, Appendix 1). Accordingly, the function $\left( {\frac{g(r)}{r}} \right)$ will increase with the particle size and, beyond a certain r, will remain essentially constant. We have calculated the dependence of ($I_{{2,{\text{Tot}}}}^{\lim } /q$) on r for samples of ADP with f = 0.30 and L = 1,000 μm. The result is shown in Fig. 11 (dashed line). It can be observed that $I_{{2,{\text{Tot}}}}^{\lim }$ increases with the particle size and for $r \ge r_{g} \cong 100$ μm stabilizes at a constant value. The value of r _g may seem surprisingly large as g(r) is already a linear function of r for r ≥ 20 μm. The reason is that for the wavelength considered (λ ₁ = 1.064 μm), g(r) has an appreciable y-intercept, $g(r) \cong ar + b.$. Hence, the function $\left( {\frac{g(r)}{r}} \right)$ only stabilizes when the particle size is such that $(b/r_{g} ) \ll a$. The profile of the curve for r < r _g depends on the sign of b. In the case of ADP, b < 0. The value of some d _mn coefficients can be easily obtained from the experimental determination of the constant value of $I_{{2,{\text{Tot}}}}$ at the “saturation” region (r ≥ r _g ) and Eqs. (41), (22), and (5) or (8).

Fig. 11

Theoretical dependence of ($I_{{2,{\text{Tot}}}} /q$) on r for ADP samples in an index-matching liquid with f = 0.30, L = 1,000 μm and λ ₁ = 1.064 μm. Different values of the index ratio m have been considered. The refractive indices required for the calculation have been taken from Kirby and DeShazer [20]. The function g(r) has been obtained by numerical averaging of Eq. (4). The values of l ₁(r) have been determined using the Mie theory for spheres and Eq. (19). ($I_{{2,{\text{Tot}}}} /q$) has been calculated using Eq. (38). The dashed line represents the function ($I_{{2,{\text{Tot}}}}^{\lim } /q$) calculated using Eq. (41)

However, as we will show below, the condition $(l_{1} /2) \gg L$ can be difficult to be satisfied in practice. In Fig. 11, the dependence of ($I_{{2,{\text{Tot}}}} /q$) on the particle size for different values of the index ratio m is shown. $I_{{2,{\text{Tot}}}}$ has been calculated using Eq. (38). It can be observed that the function $I_{{2,{\text{Tot}}}} (r)$ is strongly dependent on m. Besides, an extremely good index matching is needed ($m \cong 1.001$) for $I_{{2,{\text{Tot}}}} (r) \cong I_{{2,{\text{Tot}}}}^{\lim } (r)$. Even in this case, the scattering of the primary beam cannot be neglected for small particle sizes (see Fig. 5). This index matching cannot be usually achieved due to the material birefringence (Δn). In a birefringent medium, the best index matching can be estimated by

$$m_{\hbox{min} } \cong \frac{{\bar{n} + \left( {\varDelta n/2} \right)}}{{\bar{n}}} = 1 + \frac{\varDelta n}{{2\bar{n}}},$$

(42)

where $\bar{n} = (n_{o} + n_{e} )/2$. In the case of ADP for λ ₁ = 1.064 μm [20]: $m_{\hbox{min} } \cong 1.013$. It can be observed in Fig. 11 that for that index matching $I_{{2,{\text{Tot}}}} (r)$ is quite different from $I_{{2,{\text{Tot}}}}^{\lim } (r)$ and the saturation is not achieved. The unavoidable light scattering by each particle due to birefringence is large enough to prevent $(l_{1} /2) \gg L$. The scattering of the primary beam can be additionally reduced by decreasing the volume fraction occupied by the scatterers. But even for f = 0.05 (see Fig. 12), the saturation is not achieved. Thus, in the case of multiple scattering, condition $(l_{1} /2) \gg L$ can only be fulfilled by an extremely good index matching in materials with small birefringence. Only in those cases $I_{{2,{\text{Tot}}}} (r) \cong I_{{2,{\text{Tot}}}}^{\lim } (r).$

Fig. 12

Theoretical dependence of ($I_{{2,{\text{Tot}}}} /q$) on r for ADP samples in an index-matching liquid with m = 1.013, L = 1,000 μm and λ ₁ = 1.064 μm. Different values of f have been considered. In the calculations, the same data and procedures as in Fig. 11 have been used. The function ($I_{{2,{\text{Tot}}}}^{\lim } /q$) is also represented

It is interesting to point out that, for fixed f and L, a change from multiple to single scattering can be achieved by increasing the particle size. If A represents the cross-sectional area of the sample and N the total number of particles, then the ratio between the total area covered by the particles and A is:

$$\eta = \frac{{N\pi R^{2} }}{A} = \frac{{AL\rho \pi R^{2} }}{A} = L\left( {\frac{f}{{(4/3)\pi R^{3} }}} \right)\pi R^{2} = \frac{3fL}{2r},$$

(43)

where we have assumed that particles have, on average, spherical shape. If the values of L and f are kept fixed, an increase in the particle size will produce a decrease in η. If the particle size:

$$r \ge r_{s} = \frac{3}{2}fL,$$

(44)

then η ≤ 1, and the primary beam is only single scattered. In this case, the intensity of the primary beam incident on each particle is the same ($I_{1,o}$) and Eq. (38) is no longer valid.

The total intensity of the SHG emitted by the sample is [see Eq. (28), (30)]:

$$\begin{aligned} I_{{2,{\text{Tot}}}}^{\text{single}} & = \frac{{N\bar{\sigma }_{sh} I_{1,o} }}{A} = \frac{{\left( {AL\rho } \right)\bar{\sigma }_{sh} I_{1,o} }}{A} = L\left( {\frac{f}{{(4/3)\pi R^{3} }}} \right)\bar{\sigma }_{sh} I_{1,o} \\ & = L\left( {\frac{f}{{ (4 /3 )\pi R^{3} }}} \right)\left( {\pi R^{2} T_{2} T_{1}^{2} g(r)I_{1 ,o} } \right)I_{1 ,o} \\ & = \frac{3}{2}fT_{2} T_{1}^{2} I_{1 ,o}^{2} \left( {\frac{g(r)}{r}} \right)L. \\ \end{aligned}$$

(45)

It can be observed that $I_{{2,{\text{Tot}}}}^{\text{single}}$ is the function $I_{{2,{\text{Tot}}}}^{\lim } (r)$ analyzed before [Eq. (41)]. In consequence, when $r \ll r_{s}$, the system is in a multiple scattering regime and $I_{{2,{\text{Tot}}}}$ is given by Eq. (38), and when $r \gg r_{s}$ the system is in a single scattering regime and $I_{{2,{\text{Tot}}}}$ is given by Eq. (45). In Fig. 13, the calculated dependence of ($I_{{2,{\text{Tot}}}} /q$) on the particle size for ADP samples in an index-matching liquid (m = 1.013) is shown. Samples with f = 0.05 and L = 1,000 μm have been considered. Circles represent the values of $I_{{2,{\text{Tot}}}}$ in the multiple scattering regime and squares those in the single scattering regime. According to Eq. (44), the transition between regimes takes place at $r_{s} \cong 75$ μm. The continuous line represents the approximate dependence of $I_{{2,{\text{Tot}}}}$ on r that could be obtained in an experiment. This curve describes quite well the experimental results obtained by KP (Figure 6 of Ref. [9]).

Fig. 13

Theoretical dependence of ($I_{{2,{\text{Tot}}}} /q$) on r for ADP samples in an index-matching liquid with m = 1.013, f = 0.05, L = 1,000 μm and λ ₁ = 1.064 μm. Dots and dash-dotted line represent the values of ($I_{{2,{\text{Tot}}}} /q$) in a multiple scattering regime, and squares and short-dashed line the values of ($I_{{2,{\text{Tot}}}}^{\text{single}} /q$) in the single scattering regime. According to Eq. (44), the transition between regimes takes place at $r_{s} \cong 75$ μm. Solid line represents the approximate dependence of ($I_{{2,{\text{Tot}}}} /q$) on r that could be obtained in an experiment. In the calculations, the same data and procedures as in Fig. 11 have been used

As has been shown before, the values of some coefficients d _mn can be obtained from the experimental value of $I_{{2,{\text{Tot}}}}$ in the saturation region. In the case of materials of small birefringence immersed in a liquid with extremely good index matching, saturation is achieved for r > r _g. In all other cases, saturation can only be achieved inducing a transition to a regime of single scattering. In these cases, saturation is achieved for r larger than r _g and r _s . In this respect, it should be remarked that the values of r for which the saturation is experimentally observed are in good agreement with r _s . For example, in the experiments reported in [28], the diameter of the cross section of the sample is 8 mm, and approximately M = 7.5 mg of powder are used. Two materials are considered, ADP and l(+) glutamic acid HCl, with mass densities ρ _v  = 1.803 g cm⁻³ [2] and ρ _v  = 1.524 g cm⁻³ [51], respectively. Since $r_{s} = 3M/(2A\rho_{v} )$, then r _s  = 124 μm and r _s  = 147 μm, respectively, in good agreement with the experimental data (see Fig. 6 of Refs. [9, 28], respectively). In this regard, it is interesting to comment that the samples commonly used in this kind of experiments have unexpectedly small packing fractions. For example, in the two preceding cases $f = 2 r_{s} /(3L)$ results to be 0.08 and 0.10, respectively, if the sample thickness was L = 1,000 μm. Evidently, the size of f and L should be well known for any quantitative inference from the SHG intensities.

3.4.2 NPMM

In the case of NPMM, the function g(r) oscillates with a period $2\left\langle {l_{c} } \right\rangle$. Therefore, as long as $(l_{1} /2) \gg L,$ the function $\left( {\frac{g\left( r \right)}{r}} \right),$ and thus $I_{{2,{\text{Tot}}}}^{\lim } (r)$ [see Eq. (41)], will show oscillations of the same period as g(r) but damped by a factor r ⁻¹. The absolute maximum of $I_{{2,{\text{Tot}}}}^{\lim } (r)$ will be approximately located at $r_{\hbox{max} } \cong \left\langle {l_{c} } \right\rangle .$ If the dispersion in the particle size is >$2\left\langle {l_{c} } \right\rangle$ , only a maximum at r _max followed by a decay with a r ⁻¹ dependence will be observed. In consequence, when the logarithm of the experimental data $I_{{2,{\text{Tot}}}}^{\lim } (r)$ is plotted versus the logarithm of r, the slope s of the straight line that best fits the points should be: s = −1. From the y-intercept in that logarithmic plot, the average value of the function g(r) can be obtained: $\left\langle {g\left( r \right)} \right\rangle \equiv C.$. Using C and Eq. (4), the value of some coefficients d _mn can be determined. Besides, the approximate value of $\left\langle {l_{c} } \right\rangle$ can be directly obtained from the experimental value of r _max .

In order to analyze the effect of the index ratio on the SHG intensity, we have calculated, using Eq. (38), the dependence of ($I_{{2,{\text{Tot}}}} /q$) on r for samples of SiO₂ with different values of m. We have considered f = 0.05, L = 1,000 μm and λ ₁ = 1.064 μm. The average coherence length of the material for this wavelength is 20 μm. The results obtained are shown in Fig. 14. The values of ($I_{{2,{\text{Tot}}}}^{\lim } /q$) calculated using Eq. (41) have also been represented. The birefringence of SiO₂ is small and, therefore, a good index matching can be achieved. Using Eq. (42) and the values of the refractive indices of SiO₂ for λ ₁ [19]: $m_{\hbox{min} } \cong 1.003$. It can be observed in Fig. 14 that for this index ratio $I_{2,Tot} \left( r \right) \cong I_{2,Tot}^{\lim } \left( r \right).$ The curve $I_{{2,{\text{Tot}}}} (r)$ describes quite well the experimental data obtained by KP (Figure 5 of Ref. [9]). As $I_{{2,{\text{Tot}}}} (r) \cong I_{{2,{\text{Tot}}}}^{\lim } (r)$ the value of d ₁₁ and $\left\langle {l_{c} } \right\rangle$ of SiO₂ could be determined using the procedure described above. However, if the index matching is not so good, the scattering of the primary beam cannot be neglected. Thus, $I_{{2,{\text{Tot}}}} (r) < I_{{2,{\text{Tot}}}}^{\lim } (r)$ and, as the scattering is stronger for small particle sizes, the difference between $I_{{2,{\text{Tot}}}}$ and $I_{{2,{\text{Tot}}}}^{\lim }$ will be larger for small r values. Consequently, the oscillations of the function $I_{{2,{\text{Tot}}}} (r)$ will be damped by a factor r ^−s with s < 1. For example, for m = 1.01 (see Fig. 14) s = 0.63. In these cases, the determination of some d _mn coefficients from $I_{{2,{\text{Tot}}}} (r)$ will be, therefore, quite difficult. In a multiple scattering regime, the absolute maximum of $I_{{2,{\text{Tot}}}} (r)$ remains located at $r_{\hbox{max} } \cong \left\langle {l_{c} } \right\rangle$ (see Fig. 14). But, if a change to a single scattering regime is produced, then r _max can be different from $\left\langle {l_{c} } \right\rangle$.

Fig. 14

Theoretical dependence of ($I_{{2,{\text{Tot}}}} /q$) on r for SiO₂ samples in an index-matching liquid with f = 0.05, L = 1,000 μm and λ ₁ = 1.064 μm. Different values of the index ratio m have been considered. The refractive indices required for the calculation have been taken from Ghosh [19]. The function g(r) has been obtained by numerical averaging of Eq. (4). The values of l ₁(r) have been determined using the Mie theory for spheres and Eq. (19). ($I_{{2,{\text{Tot}}}} /q$) has been calculated using Eq. (38). The function ($I_{{2,{\text{Tot}}}}^{\lim } /q$), calculated using Eq. (41), is also represented

We finish this section with a comment about the measuring geometry and possible systematic errors in the determination of $\left\langle {l_{c} } \right\rangle$. Evidently, the total SHG intensity generated by the sample ($I_{{2,{\text{Tot}}}}$) has to be measured using an integrating sphere. Nevertheless, as the particles are immersed in an index-matching liquid, it is usually assumed that the scattering is negligible and all intensity is emitted in the forward direction. Consequently, only the SHG intensity emitted in the forward direction $I_{{2,{\text{beam}}}}$ is measured [28], and it is assumed that $I_{{2,{\text{Tot}}}} (r) \cong I_{{2,{\text{beam}}}} (r)$. However, even when the material birefringence is small, the dispersion of the refractive indices makes it impossible to achieve a good index matching at λ ₁ and λ ₂ simultaneously. Thus, the scattering of the SHG light out of the z axis cannot be negligible, especially for small particle sizes, and it is expected that $I_{{2,{\text{beam}}}}$ be smaller than $I_{{2,{\text{Tot}}}}$ for small r values. As $\left\langle {l_{c} } \right\rangle$ is usually small, the first oscillation of $I_{{2,{\text{Tot}}}} (r)$ will be weakened and the absolute maximum of $I_{{2,{\text{beam}}}} (r)$ will be located at $r_{{\hbox{max} ,{\text{beam}}}} > \left\langle {l_{c} } \right\rangle$. In Fig. 15, the dependence of ($I_{{2,{\text{beam}}}} /q$) on r for SiO₂ particles immersed in an index-matching liquid [m(λ ₁) = m(λ ₂) = 1.01) is shown. We have considered f = 0.05, L = 1,000 μm and λ ₁ = 1.064 μm. It can be observed that, although the absolute maximum of $I_{{2,{\text{Tot}}}}$ is located at $r_{\hbox{max} } \cong \left\langle {l_{c} } \right\rangle = 20$ μm (see Fig. 14), the absolute maximum of $I_{{2,{\text{beam}}}}$ is located at $r_{{\hbox{max} ,{\text{beam}}}} \cong 3\left\langle {l_{c} } \right\rangle = 60$ μm. The experimental data confirm this result (see Fig. 6 of Ref. [28]).

Fig. 15

Theoretical dependence of ($I_{{2,{\text{beam}}}} /q$) on r for SiO₂ samples in an index-matching liquid with m(λ ₁) = m(λ ₂) = 1.01, f = 0.05, L = 1,000 μm and λ ₁ = 1.064 μm. The refractive indices required for the calculations have been taken from Ghosh [19]. The function g(r) has been obtained by numerical averaging of Eq. (4). The values of l ₁(r) and l ₂(r) have been determined using the Mie theory for spheres and Eq. (19). ($I_{{2,{\text{beam}}}} /q$) has been calculated using Eq. (35)

4 Conclusions

We have presented a critical revision of the KP method and its practical application. The effects of light scattering have been explicitly included in the analysis of the SHG process. It has been found that some of the conclusions commonly accepted in the literature derived from the SHG curves are not always correct, precisely due to the scattering effects. The main goal of the present study is to establish what information can be extracted from the measurements depending on the experimental geometry. A summary of our main results is outlined in Table 5. In this table, the qualitative profiles of the SHG intensity versus r for different detection geometries and types of samples are presented for both PMM and NPMM. In addition, it is indicated whether the d _mn coefficients and coherence lengths can be determined or not, and the possibility to discriminate between the PMM or NPMM character depending on the experimental setup. Several important points must be highlighted:

Table 5 Qualitative profiles of the SHG intensity versus r for different detection geometries and types of samples for PMM and NPMM

• 1. In the case of dry powders (by far the most extended kind of samples in these experiments), no information about the d _mn coefficients can be obtained from the analysis of the SHG curves, unless an integrating sphere is included in the experimental setup to measure the whole SHG flux. In this case, Eq. (38) must be used, which implies that factors such as the packing fraction f, the sample thickness L or nFact must be known. This last term, which depends critically on the refractive indices involved in the process, implies that the ratio of SHG intensities from different materials is not equivalent to the ratio of the squares of their d _eff’s even in PMM. In the NPMM case, the corresponding equations are much more complicated. An interesting variant of the use of integrating spheres is the measurement of the backward SHG emission in very thick samples. In the limit of large L, all emissions are backward and the SHG signal in a particular direction is proportional to the total integrated flux. For packing fractions f = 0.5, a good approximation is obtained with thicknesses about 50 times the maximum grain size.

• 2. The scattering effects cannot be neglected even if powders in index-matching liquid are used, since a perfect matching is impossible in most cases. The detection of the forward SHG emission can certainly give reliable results for the d _mn coefficients. For PMM, d _mn can be obtained if the single scattering regime is achieved and, therefore, the SHG curve attains saturation. In this situation, the measurement is equivalent to that in monolayer samples [11]. Typically, the single scattering regime is reached beyond a certain particle size r _s , which depends on f and L [see Eq. (44)]. Thus, contrary to what is usually assumed, the SHG saturation is not related to any coherence length, but it has to do with a mere geometric factor. The d _mn coefficients can be deduced from Eq. (45).

• 3. The KP technique is often used to discriminate the materials between PMM and NPMM. A very extended criterion to carry out this classification is to check whether the SHG profile attains saturation or not. However, this strategy should be applied with caution. For example, in the case of dry powders, if the forward emission is detected, saturation trends may be observed in the SHG curves for NPMM materials. In general, for dry powders, the origin of the saturation is due to a redistribution of the SHG intensity in the back and forward directions as r increases, due to the modification of the scattering conditions. In this respect, the KP interpretation about the reason for the saturation cannot be applied either.

To end this paper, we would like to point out that a high percentage of the works that make use of the KP technique should be revised. In most cases, no information is given about some parameters influencing more or less critically the SHG signal, such as f, L or the refractive indices. Sometimes the kind of detection (backward or forward) is not specified either. Virtually, nobody uses integrating spheres or index-matching liquids and, on some occasions, the measurements are just carried out at a single r point. Under these circumstances, it is easy to understand that the information that can be deduced from those experiments is of scarce or null value.

Nevertheless, as we have shown, SHG measurements in micropowders can actually be used to deduce quantitative data of the nonlinear optical properties of solids. We have pointed out several experimental possibilities to achieve this goal (use of integrating spheres in the setup, measurement of the backward emission in thick dry powders or measurement of the forward emission in samples with index-matching liquids). The KP method can still be used successfully if it is not oversimplified and some elementary precautions are taken.

Appendices

Appendix 1: Derivation of expressions for I ₂(r) generated by monolayers of PMM

Let us first consider a uniaxial PMM that shows PM for a wave interaction jk → i. If the particle size is greater than the average coherence length for the rest of wave interactions, the SHG intensity generated by the wave interaction jk → i will prevail over the rest. In this case, Eq. (4) is reduced to:

$$I_{2} \left( {r,\theta ,\varphi } \right) = \frac{{2 \omega^{2} }}{{\varepsilon_{o} c^{3} }}I_{1}^{2} {\text{PolFact}}\frac{{d_{ijk}^{2} \left( {\theta ,\varphi } \right)}}{{n_{2i} (\theta ) n_{1j} (\theta ) n_{1k} (\theta )}} \left| {\varDelta_{ijk} \left( {r,\theta } \right)} \right|^{2} .$$

(46)

The meaning of all quantities that appear in Eq. (46) is given in Sect. 2 and Table 1. If the powder particles of the monolayer are randomly oriented and the laser spot size on the sample is large enough, the SHG intensity generated by the monolayer will be given by:

$$\begin{aligned} I_{2} (r) &= \left\langle {I_{2} \left( {r,\theta ,\varphi } \right)} \right\rangle_{\theta ,\varphi } = \frac{1}{4\pi } \int\limits_{0}^{2\pi } { \int\limits_{0}^{\pi } {I_{2} \left( {r,\theta ,\varphi } \right)\sin \theta {\text{d}}\theta {\text{d}}\varphi } } \\ & = \frac{1}{2\pi } \int\limits_{0}^{2\pi } { \int\limits_{0}^{\pi /2} {I_{2} \left( {r,\theta ,\varphi } \right)\sin \theta {\text{d}}\theta {\text{d}}\varphi } } \\ & = \frac{b}{2\pi } \int\limits_{0}^{2\pi } { \int\limits_{0}^{\pi /2} {B_{ijk} (\theta )d_{ijk}^{2} (\theta ,\varphi )f_{ijk} (r,\theta ){\text{d}}\theta {\text{d}}\varphi ,} } \\ \end{aligned}$$

(47)

where

$$b \equiv \frac{{2\omega^{2} }}{{\varepsilon_{o} c^{3} }}I_{1}^{2} {\text{PolFact}} = \frac{{8\pi^{2} }}{{\varepsilon_{o} c\lambda_{1}^{2} }}I_{1}^{2} {\text{PolFact}}$$

(48)

$$B_{ijk} (\theta ) \equiv \frac{\sin \theta }{{n_{2i} (\theta )n_{1j} (\theta )n_{1k} (\theta )}}$$

(49)

$$\begin{aligned} f_{ijk} (r,\theta ) \equiv \left| {\varDelta_{ijk} (r,\theta )} \right|^{2} & = \frac{{\sin^{2} \left( {\frac{{\varDelta k_{ijk} \left( \theta \right)}}{2} r} \right)}}{{\left( {\frac{{\varDelta k_{ijk} \left( \theta \right)}}{2}} \right)^{2} }} \\ & = r^{2} \text{sinc}^{2} \left( {\frac{{\varDelta k_{ijk} \left( \theta \right)}}{2} r} \right). \\ \end{aligned}$$

(50)

The function f _ijk (r, θ) has a large peak at the PM angle θ _c [Δ k _ijk (θ _c ) = 0] and successive secondary maxima of decreasing values (see Fig. 16). The peak value of the central maximum is r ² and its angular half width Δθ is given by: Δk _ijk (θ _c + Δθ) = 2π/r. Therefore, if the particle size r increases, the central peak grows sharply and narrows (see Fig. 16). For a particle size large enough, the function f _ijk (r, θ) will only show significant values for θ close to θ _c . Within that narrow angular interval:

Fig. 16

Theoretical SHG intensity generated by a single particle of KDP around the PM direction (θ _c  = 0.88 rad.) for different particle sizes: r = 20 μm (dotted line), r = 50 μm (continuous line) and r = 100 μm (dashed line). The intensities (proportional to r ²) have been normalized to make them comparable. λ ₁ = 0.694 μm and the PM is of type I ($oo \to e$)

• a. The functions B _ijk (θ) and d _ijk (θ, φ) do not appreciably change, so they can be substituted in (47) by their values at θ _c :

  $$\begin{aligned} I_{2} (r) & \cong \frac{b}{2\pi } \int\limits_{0}^{2\pi } { B_{ijk} (\theta_{c} )d_{ijk}^{2} (\theta_{c} ,\varphi )\int\limits_{0}^{\pi /2} { f_{ijk} (r,\theta ){\text{d}}\theta {\text{d}}\varphi } } \\ & = bB_{ijk} (\theta_{c} )\left[ {\frac{1}{2\pi }\int\limits_{0}^{2\pi } {d_{ijk}^{2} \left( {\theta_{c} ,\varphi } \right){\text{d}}\varphi } } \right]\int\limits_{0}^{\pi /2} {f_{ijk} \left( {r,\theta } \right){\text{d}}\theta = bB_{ijk} (\theta_{c} )\left\langle {d_{\text{eff}}^{2} } \right\rangle \int\limits_{0}^{\pi /2} {f_{ijk} (r,\theta ){\text{d}}\theta } ,} \\ \end{aligned}$$

  (51)

  where

  $$\left\langle {d_{\text{eff}}^{2} } \right\rangle \equiv \frac{1}{2\pi }\int\limits_{0}^{2\pi } {d_{ijk}^{2} (\theta_{c} ,\varphi ){\text{d}}\varphi .}$$

  (52)

• b. If θ _c  ≠ π/2, the function Δk _ijk (θ) can be approximated by:

  $$\varDelta k_{ijk} \left( {\theta_{c} + \xi } \right) \cong \varDelta k_{ijk} \left( {\theta_{c} } \right) + \left. {\frac{{\text{d}\left( {\varDelta k_{ijk} \left( \theta \right)} \right)}}{{{\text{d}}\theta }}} \right|_{{\theta_{c} }} \xi = 0 + 2 \beta_{ijk} \xi ,$$

  (53)

  where β _ijk represents the angular sensitivity around the PM direction:

  $$\beta_{ijk} \equiv \frac{1}{2}\left. {\frac{{\text{d}\left( {\varDelta k_{ijk} \left( \theta \right)} \right)}}{{{\text{d}}\theta }}} \right|_{{\theta_{c} }} = \frac{\pi }{{\lambda_{1} }} \left. {\frac{{\text{d}\left[ {2 n_{2i} \left( \theta \right) - n_{1j} \left( \theta \right) - n_{1k} \left( \theta \right)} \right]}}{{{\text{d}}\theta }}} \right|_{{\theta_{c} }} .$$

  (54)

  Therefore:

  $$\int\limits_{0}^{\pi /2} {f_{ijk} (r,\theta ){\text{d}}\theta \approx r^{2} \int\limits_{{ - \theta_{c} }}^{{\pi /2 - \theta_{c} }} {\text{sinc}^{2} \left( {\beta_{ijk} r\xi } \right){\text{d}}\xi .} }$$

  (55)

  In addition, r is so large that the function $\text{sinc}^{2} \left( {\beta_{ijk} r\xi } \right)$ only has significant values for ξ close to 0, and the integration limits can be moved to ±∞ with no appreciable change in the value of the integral:

  $$\int\limits_{0}^{\pi /2} {f_{ijk} (r,\theta ){\text{d}}\theta\approx r^{2} \int\limits_{ - \infty }^{ + \infty } \text{sinc}^{2}\left( {\beta_{ijk} r \xi } \right){\text{d}}\xi = r^{2} \left(\frac{1}{{\left| {\beta_{ijk} } \right| r}} \int\limits_{ -\infty }^{ + \infty } \text{sinc}^{2} x{\text{d}}x\right) = r^{2} \left({\frac{\pi }{{\left| {\beta_{ijk} } \right| r}}} \right) =\frac{\pi }{{\left| {\beta_{ijk} } \right|}}r}.$$

  (56)

The analytical expression for I ₂(r) is finally obtained from Eqs. (51) and (56):

$$I_{2} (r) \approx bB_{ijk} \left( {\theta_{c} } \right)\left\langle {d_{\text{eff}}^{2} } \right\rangle \frac{\pi }{{\left| {\beta_{ijk} } \right|}}r = \frac{{8\pi^{2} }}{{\varepsilon_{o} c\lambda_{1} }}I_{1}^{2} {\text{ PolFact }}n{\text{Fact}}\left\langle {d_{\text{eff}}^{2} } \right\rangle r,$$

(57)

where

$$\begin{aligned} n{\text{Fact}} & \equiv \frac{1}{{\lambda_{1} }} B_{ijk} \left( {\theta_{c} } \right) \frac{\pi }{{\left| {\beta_{ijk} } \right|}} \\ & = \frac{{\sin \theta_{c} }}{{n_{2i} \left( {\theta_{c} } \right) n_{1j} \left( {\theta_{c} } \right) n_{1k} \left( {\theta_{c} } \right)}} \frac{1}{{\left| {\left. {\frac{{d\left[ {2 n_{2i} \left( \theta \right) - n_{1j} \left( \theta \right) - n_{1k} \left( \theta \right)} \right]}}{d\theta }} \right|_{{\theta_{c} }} } \right|}}, \\ \end{aligned}$$

(58)

In the derivation of Eq. (57), it has been assumed that r is large enough so that the function f _ijk (r, θ) only has significant values for θ close to θ _c. If Δθ represents the angular half width of the central peak of f _ijk (r, θ) around θ _c, this condition can be expressed as:

$$\varDelta \theta \ll 1.$$

(59)

Using Eq. (53) and taking into account that $\varDelta k_{ijk} (\theta_{c} + \varDelta \theta ) = 2\pi /r,$ we obtain

$$\varDelta \theta = \frac{\pi }{{\left| {\beta_{ijk} } \right|r}},$$

(60)

so condition (59) can be rewritten as:

$$r \gg \frac{\pi }{{\left| {\beta_{ijk} } \right|}}.$$

(61)

This is the same condition to be met by the thickness of a single crystal so that the PM direction is well defined [52]. Typical values of $\pi /\left| {\beta_{ijk} } \right|$ are in the range 1–20 μm.

If θ _c = π/2 then β _ijk  = 0 and the second derivative of Δk _ijk (θ) has to be considered in Eq. (53). Proceeding as before, a dependence $I_{2} (r) \propto r^{n}$ with n = {−1/2, 1/2, 3/2} is obtained depending on the type of uniaxial material (positive or negative), the type of PM and the crystal class.

In the case of biaxial PMM, the refractive indices depend on both θ and φ, so Eq. (47) can be written:

$$I_{2} (r) = \frac{b}{2\pi } \int\limits_{0}^{2\pi } { \int\limits_{0}^{\pi /2} { B_{ijk} \left( {\theta ,\varphi } \right)d_{ijk}^{2} \left( {\theta ,\varphi } \right)f_{ijk} \left( {r,\theta ,\varphi } \right){\text{d}}\theta {\text{d}}\varphi .} }$$

(62)

In this case, the integral is difficult to be worked out because the topology of the PM direction loci is complex [21, 24–26]. The function θ _c (φ) can be multivalued for certain azimuths φ or even not be defined for others. Proceeding as in the case of uniaxial materials:

$$\begin{aligned} I_{2} (r) & = \frac{b}{2\pi } \int\limits_{{{\text{PM }}\;{\text{loci}}}} {B_{ijk} \left( {\theta_{c} \left( \varphi \right),\varphi } \right)d_{ijk}^{2} \left( {\theta_{c} \left( \varphi \right),\varphi } \right)\left[ {\int\limits_{0}^{\pi /2} {f_{ijk} \left( {r,\theta ,\varphi } \right){\text{d}}\theta } } \right]{\text{d}}\varphi } \\ & \cong \frac{b}{2\pi } \int\limits_{{{\text{PM }}\;{\text{loci}}}} {B_{ijk} \left( {\theta_{c} \left( \varphi \right),\varphi } \right)d_{ijk}^{2} \left( {\theta_{c} \left( \varphi \right),\varphi } \right)\left[ {\frac{\pi }{{\left| {\beta_{ijk} \left( {\theta_{c} \left( \varphi \right),\varphi } \right)} \right|}}r} \right]{\text{d}}\varphi } \\ & = \frac{{8\pi^{2} }}{{\varepsilon_{o} c\lambda_{1} }} I_{1}^{2} {\text{ PolFact }}r \left[ {\frac{1}{2\pi }\int\limits_{{{\text{PM }}\;{\text{loci}}}} {\frac{\pi }{{\lambda_{1} }} \frac{{B_{ijk} \left( {\theta_{c} \left( \varphi \right),\varphi } \right)}}{{\left| {\beta_{ijk} \left( {\theta_{c} \left( \varphi \right),\varphi } \right)} \right|}}d_{ijk}^{2} \left( {\theta_{c} \left( \varphi \right),\varphi } \right){\text{d}}\varphi } } \right]. \\ \end{aligned}$$

(63)

Therefore, if the particle size is large enough, the SHG intensity generated by a monolayer of a biaxial PMM also increases linearly with r. The expression for I ₂(r) is formally similar to the uniaxial case (57)

$$I_{2} (r) = \frac{{8\pi^{2} }}{{\varepsilon_{o} c\lambda_{1} }}I_{1}^{2} {\text{PolFact}}\left\langle {d_{\text{eff}}^{2} } \right\rangle r,$$

(64)

but now

$$\begin{aligned} \left\langle {d_{\text{eff}}^{2} } \right\rangle & \equiv \frac{1}{2 \pi } \int\limits_{\begin{subarray}{l} {\text{PM }}\;{\text{loci:}} \\ \theta_{c} (\varphi )\end{subarray} } {\frac{\pi }{{\lambda_{ 1} }}\frac{{d_{ijk}^{2} (\theta ,\varphi )}}{{n_{2i} (\theta ,\varphi ) n_{1j} (\theta ,\varphi ) n_{1k} (\theta ,\varphi )}}\frac{\sin \theta }{{\left| { \beta_{ijk} (\theta ,\varphi ) } \right|}}{\text{d}}\varphi } \\ & \equiv \int\limits_{\begin{subarray}{l} \text{PM}\;{\text{ loci:}} \\ \theta_{c} (\varphi )\end{subarray} } {W (\theta ,\varphi )d_{ijk}^{2} (\theta ,\varphi ){\text{d}}\varphi.} \end{aligned}$$

(65)

Unlike Eq. (57), the term “nFact” does not explicitly appears in Eq. (64). The factors that give rise to it in uniaxial materials are now included in the definition of $\left\langle {d_{\text{eff}}^{2} } \right\rangle$ as they cannot be removed from the integral. Although a “$\left\langle {d_{\text{eff}}^{2} } \right\rangle$” can be defined in this case its meaning is questionable: It represents the average of $d_{ijk}^{2} (\theta ,\varphi )$ along the PM loci but with a rather complex weighting function W(θ, φ). In consequence, it will be quite difficult to get information of the d _mn coefficients directly from the value of this “$\left\langle {d_{\text{eff}}^{2} } \right\rangle$.” In contrast, in the case of uniaxial PMM, a meaningful definition of $\left\langle {d_{\text{eff}}^{2} } \right\rangle$ can be made [Eq. (52)].

Appendix 2: Calculation of the diffuse SHG intensities emitted in the backward and forward directions

The stationary distribution of the energy density of the scattered SHG light U ₂(z) will satisfy the diffusion equation [34, 53–57]:

$$\frac{{{\text{d}}^{2} U_{2} (z)}}{{{\text{d}}z^{2} }} = - \frac{p(z)}{{D_{2} }},$$

(66)

where p(z) is the source of diffusive light at the SHG frequency:

$$p(z) = \frac{{{\text{d}}I_{{2,{\text{scatt}}.}} (z)}}{{{\text{d}}z}} = \frac{{I_{2} (z)}}{{l_{2} }}$$

(67)

[see Eq. (33)], and D ₂ the diffusion constant at the SHG frequency:

$$D_{2} = \frac{{v_{2} l_{tr,2} }}{3}.$$

(68)

In Eq. (68), v ₂ and l _tr,2 represent the energy velocity and the transport mean free path [53, 54, 57–59] of the SHG light in the medium. In the independent-scattering approximation $l_{tr,2} = (\rho \sigma_{tr,2} )^{ - 1} ,$ where the transport scattering cross section $(\sigma_{tr,2} )$ (or radiation pressure cross section) can be computed using the Mie theory for spheres [30–34, 53]. In the experimental studies of SHG by micropowders in air the conditions: $L \gg l_{tr,2} \gg \lambda_{m}$ are usually fulfilled supporting the validity of the diffusion approximation. The energy density U ₂(z) must satisfy the boundary conditions [50]:

$$U_{2} (z = 0) - l_{{{\text{extra}},2}} \left. {\frac{{{\text{d}}U_{2} }}{{{\text{d}}z}}} \right|_{z = 0} = 0,$$

(69)

$$U_{2} \left( {z = L} \right) + l_{{{\text{extra}},2}} \left. {\frac{{{\text{d}}U_{2} }}{{{\text{d}}z}}} \right|_{z = L} = 0,$$

(70)

where l _extra,2 is the extrapolation length of the SHG light at the sample boundaries:

$$l_{{{\text{extra}},2}} \equiv \frac{2}{3} l_{tr,2} \left( {\frac{{1 + r_{i,2} }}{{1 - r_{i,2} }}} \right),$$

(71)

and r _i,2 the internal reflectivity at the boundaries for the diffuse SHG light (these parameters are calculated from f and the refractive indices as indicated in [50]). Once the differential equation (66) with conditions (69) and (70) is solved, the intensity of the diffuse SHG light that emerges through each face of the sample can be determined:

$$I_{{2,{\text{diff}}}}^{\text{Back}} = D_{2} \left. {\frac{{{\text{d}}U_{2} (z)}}{{{\text{d}}z}}} \right|_{z = 0} ,$$

(72)

$$I_{{2,{\text{diff}}}}^{\text{Forw}} = - D_{2} \left. {\frac{{{\text{d}}U_{2} (z)}}{{{\text{d}}z}}} \right|_{z = L} ,$$

(73)

and

$$I_{{2,{\text{diff}}}}^{\text{Back}} + I_{{2,{\text{diff}}}}^{\text{Forw}} = I_{{2,{\text{scatt}}}}.$$

(74)