1. INTRODUCTION

In part I, the systems of coupled nonlinear partial differential equations obtained in the context of the polyharmonic framework remain difficult to handle and numerically delicate to tackle, especially in a full 3D vectorial context. In many cases, it is enough to deal with weakly nonlinear regimes, aka the perturbative regime. In any case, we are going to take the infinite system Eqs. (18) or Eqs. (23), part I, as a starting point and see that quite naturally we are going to come across trivial cascading nonlinear problems that are much simpler to process digitally.

In the third section, we address the delicate issue of energy. We answer the question on how the energy of the incident field is distributed between oscillating fields at various frequencies and the energy dissipated by the Joule effect. Then comes the question of what is a lossless nonlinear material in the polyharmonic context. In this study, we take into account the dispersive nature of materials, which leads us to generalize Kleinman’s relations. We give a number of examples that show that energy transfers from one frequency to another, and the notion of a lossless material is far from being obvious and sometimes leads to surprising and even unexpected conclusions.

In the fourth and fifth sections, we address the problem of the tensorial nature of quantities such as ${\chi _{(2)}}$ and ${\chi _{(3)}}$. We will conclude that they are not, strictly speaking, tensors, but the qualities sought by a tensor are nevertheless preserved. Due to the application of the very general Neumann’s principle, this will allow us to considerably reduce the number of elements of the tensor, which is obviously fundamental in practice. We will take this opportunity to give a complete example of a material with an exotic symmetry. We will answer the question of what an isotropic material is in the context of second and third order nonlinearities.

2. PERTURBATIVE APPROACH OF SCATTERING EQUATION SYSTEMS

A. With a ${\chi _{(2)}}$ Term

For the sake of simplicity, only the second order is taken into account but the generalization to the third order is straightforward. We then seek a solution to Eqs. I.(18) [1] in the form of a power series expansion of a small parameter $\eta$ that appears in the source term. In other words, we have to replace in the set of Eqs. I.(18), the source term ${\textbf{J}}$ by simply $\eta {\textbf{J}}$. Each electric field ${{\textbf{E}}_p}$ can be expanded as per

For Eq. (1) to be a solution of Eqs. I.(18) for any value of the parameter $\eta$, we require that the terms in Eqs. I.(18) be proportional to $\eta$, ${\eta ^2}$, ${\eta ^3}$, etc., each satisfying the equation separately. The terms corresponding respectively to $\eta$, ${\eta ^2}$, ${\eta ^3}$ are

Given the importance of the notations introduced in part I and used above and the fact that they are not standard, let us recall the meaning of ${\textbf{M}}_p^{{\rm{lin}}}$ and the notations $\langle \langle {{\textbf{E}}_1},{{\textbf{E}}_2}\rangle \rangle$ [2], for instance,

From Eq. (2), we get

It then turns out that the terms ${\textbf{E}}_p^{(1)}$ vanish for every $|p| \ne 1$. In short, we have

The sum appearing in Eq. (3) is now very sparse:

Recalling that ${{\textbf{E}}_0}$ is null, this term vanishes if $|p| \ne 2$, and as a result, ${\textbf{E}}_p^{(2)} = 0$. If $p = 2$, we get

Table 1. Couples $(p,q)$ Giving Rise to Nonvanishing Terms Have the Symbol “✓”; Others Have the Symbol “0”

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The whole process is summarized in Fig. 1.

 

Fig. 1. Perturbative approach amounts to solving only four trivially nonlinear cascading problems.

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B. With ${\chi _{(2)}}$ and ${\chi _{(3)}}$ Terms

Our starting point is now Eqs. I.(23), whereas the expansion Eq. (1) remains valid. It turns out that Eqs. (2) and (3) are unchanged. After some elementary computations, Eqs. (10) and (11) must be slightly modified as per

3. ELECTROMAGNETIC ENERGY IN NONLINEAR MATERIALS

A. Goal

During a linear process, it is easy to make an energy balance even if the material is lossy. It is enough on one hand to consider the energy of the incident wave and on the other hand the energy of the diffracted wave and the losses by Joule effect if necessary. But in the nonlinear case, it is another matter. Even in the simplified case of the polychromatic approach to a monochromatic incident wave, it is more complicated. On one hand, it is a question of considering the energy of the incident wave and, on the other hand, the energy of the different parts of the diffracted wave oscillating at different frequencies and losses by the Joule effect corresponding to each of the frequencies at stake. A fundamental question then arises. Under what conditions on permittivity tensors, are the materials lossless? The reader probably knows the answer in the linear case. The linear tensor must have Hermitian symmetry in the general case, and in the particular case where the material is both linear and isotropic, the susceptibility must be real. For generalizing this result, we need to find a precise definition of electromagnetic energy in the polychromatic case which is the subject of this section.

B. Stochastic Mean of Electric Power Density

1. Generality

The starting point is the so-called Poynting equality

It turns out that the derivative of the density of energy ${\partial _t}w$ is naturally split into two parts—one associated with the magnetic part ${w_m}$ and one associated with the electric part ${w_e}$:

For nonmagnetic material, ${\textbf{B}} = {\mu _0} {\textbf{H}}$, and we deduce

For the electric part,

2. Term of Order 0, ${\langle {\partial _t}w_e^{(0)}\rangle _\sharp}$: Vacuum

In our polyharmonic approach, i.e., with a single monochromatic incident field, the electric field ${\textbf{E}}$ is given by

To obtain ${\langle {\partial _t}w_e^{(0)}\rangle _\sharp}$, it is enough to notice that the mean process is linear and that ${\langle {e^{- i(p + q){\omega _I}t}}\rangle _\sharp} = {\delta _{p, - q}}$:

The sum vanishes, and we obtain an expected result: ${\langle {\partial _t}w_e^{(0)}\rangle _\sharp} = 0$. In other words, the stochastic mean of the electromagnetic energy variation is null as expected.

3. Term of Order 1, ${\langle {\partial _t}w_e^{(1)}\rangle _\sharp}$: Linear Response

To obtain the term ${\langle {\partial _t}w_e^{(1)}\rangle _\sharp}$, all we have to do is to replace ${{\textbf{E}}_q}$ by the expression $\langle \langle {{\textbf{E}}_q}\rangle \rangle$ in Eq. (14):

First notice that for any ${q_0}$, ${{\textbf{E}}_{{q_0}}} \cdot {\chi _{(1), - {q_0}}}{{\textbf{E}}_{- {q_0}}} = \overline {{{\textbf{E}}_{- {q_0}}} \cdot {\chi _{(1),{q_0}}}{{\textbf{E}}_{{q_0}}}}$. The mean value ${\langle {\partial _t}w_e^{(1)}\rangle _\sharp}$ is then real as expected, and second, we have

By making use of Hermitian symmetry, we get

4. Term of Order 2, ${\langle {\partial _t}w_e^{(2)}\rangle _\sharp}$: Quadratic Response

The task now is to resume the calculations already made in the linear framework. This time the part is more delicate because unlike in the linear case, there are interactions between the different frequencies that will have to be taken into account:

As a result,

In a more symmetric manner, by re-indexing the double sum ($r \to r + q$),

The trick now is to use the properties of the different components of the sum to reduce the set to which the summation indices $q$ and $r$ belong [4]. In this case, we can reduce to one-eighth of the set ${\mathbb Z^2}$, i.e., $q \gt 0$ and $q \le r$. For this purpose, we introduce the double sequence ${G_{r,q}}$, defined by

Provided that $G$ has the two aforementioned properties, we have the following equality:

We are now close to the goal. We remark indeed that the last three terms contain the fields ${{\textbf{E}}_r}$, ${{\textbf{E}}_q}$, and ${{\textbf{E}}_{- (r + q)}}$ but not in the same order. To rearrange them, it suffices to make use of the transposition operator on the tensor ${\chi _{(2)}}$ (see part III, Section 3.D, for further explanations). For instance, we have the following equality:

We are now able to recast the expression of ${\langle {\partial _t}w_e^{(2)}\rangle _\sharp}$:

The material is said to be lossless for the quadratic response if for any couples of integers $(r,q)$, the following conditions are fulfilled:

5. Term of Order 3, ${\langle {\partial _t}w_e^{(3)}\rangle _\sharp}$: Cubic Response

We can easily convince ourselves that we can obtain an expression of the type

6. Term of order $n$, ${\langle {\partial _t}w_e^{(n)}\rangle _\sharp}$: nth Order Response

In this section, we give the generalization of the previous results in an arbitrary order:

7. Kleinman’s Relations

When dealing with instantaneous media, the susceptibility tensor fields no longer depend on the frequency:

To put it in a nutshell, the tensor ${\chi _{0,(n)}}$ is said to be completely symmetric. This is the famous result given by Kleinman in [6].

8. Examples with Low Degree Materials Involving ${\chi _{(1)}}$ and ${\chi _{(2)}}$

To give an example, we will look in detail at the case where the order is two and for a ${\chi _{(2)}}$ material. We will see that energy transfer in the nonlinear domain is quite subtle and sometimes quite unexpected.

With ${d_1} = 2$ and a material involving ${\chi _{(1)}}$ and ${\chi _{(2)}}$. In this case, we have ${\partial _t}{w_e} = {\partial _t}w_e^{(0)} + {\partial _t}w_e^{(1)} + {\partial _t}w_e^{(2)}$, which leads to

We know how the linear term behaves. Among other things, it cancels itself out provided that the ${\chi _{(1)}}$ term respects Hermitian symmetry. We focus our attention on the quadratic term ${\langle {\partial _t}w_e^{(2)}\rangle _\sharp}$, which yields

In the case where ${d_1} = 2$, the only nonvanishing term is ${{\textbf{P}}_{(2),2}} = {\varepsilon _0}\langle \langle {{\textbf{E}}_1},{{\textbf{E}}_1}\rangle \rangle$ [see system Eq. I.(20)]. The term ${\langle {\partial _t}w_e^{(2)}\rangle _\sharp}$ is then reduced to a single term, namely,

In the perspective of considering materials without Joule leakage, it must be possible to cancel this term. But it seems pointless to use Kleinman-type relations since for this we would need at least two terms, and we have only one. But this does not take into account the interactions between the different processes. The second equation of the system Eq. I.(20) gives

This latest term corresponds to the density of energy of the electric field ${{\textbf{E}}_2}$ in the linear case. Among other things, if we are interested in the total energy ${{\cal E}_2}$ associated with the quadratic term, let us integrate ${\langle {\partial _t}w_e^{(2)}\rangle _\sharp}$ on the overall space:

After an integration by parts, it turns out that the term involving the $\nabla \times \nabla \times$ disappears:

Once again, if the tensor ${\chi _{(1)}}$ has Hermitian symmetry, the term ${\varepsilon _r}(2{\omega _I}) {{\textbf{E}}_2} \cdot \overline {{{\textbf{E}}_2}}$ is real-valued, and as a result, ${{\cal E}_2}$ vanishes. On the other hand, if the material at stake is isotropic and passive, it appears that ${{\cal E}_2}$ is negative as expected. But it is worth noting that the aforementioned expression does not depend upon the ${\chi _{(2)}}$ materials! Formulas dealing with energy transfer are indifferent to ${\chi _{(2)}}$ characteristics. They work in the same way, irrespective of whether the materials are of the Kleinman type.

With ${d_\infty} = 2$ and a material involving ${\chi _{(1)}}$ and ${\chi _{(2)}}$. The discussion may be resumed up to Eq. (20). But now we have two nonvanishing terms for ${{\textbf{P}}_{(2)}}$ [see Eqs. I. (21)]:

We then obtain the corresponding quadratic term by making use of Eq. (20):

The first term has to be rearranged as per

If the material is of Kleinman type for the frequencies at stake, we have

As a conclusion, if the material is of Kleinman type, the quadratic term vanishes.

9. Examples with Low Degrees with a Material Involving ${\chi _{(1)}}$, ${\chi _{(2)}}$, and ${\chi _{(3)}}$

With ${d_\infty} = 2$ and a material involving ${\chi _{(1)}}$, ${\chi _{(2)}}$, and ${\chi _{(3)}}$. The system we are interested in is reduced to the nonlinear Eq. I.(24). This time, we pay attention to the cubic term ${\langle {\partial _t}w_e^{(3)}\rangle _\sharp}$:

We wonder if this term can be eliminated. We can consider Eq. I.(24) as we did in a preceding section, but the presence of the source ${{\textbf{J}}_1}$ does not allow to conclude. We are going to change tactics and reduce the field of our investigations to isotropic media. In that case, the (0,4)-tensor ${\chi _{(3)}}$ can be written in the following manner:

The cubic term is then

As a conclusion, for cancelling the cubic term, it is enough to consider ${\chi _{(3)}}$ media with a real isotropic tensor.

With ${d_1} = 2$ and a material involving ${\chi _{(1)}}$, ${\chi _{(2)}}$, and ${\chi _{(3)}}$. We refer to the system Eq. I.(25), which is the same as the one we have seen before with ${d_1} = 2$ and a material involving ${\chi _{(1)}}$ and ${\chi _{(2)}}$.

4. COVARIANT DESCRIPTION IN NONLINEAR MEDIA

A. General Comments

The ambition of this section is modest: it is to discover the tensorial nature of the characteristics of the materials already mentioned, namely, the $\chi$s. The fact that ${\chi _{(n)}}$ obviously have $n + 1$ legs (indices) does not necessarily make them tensors of rank $n + 1$. On the other hand, if these quantities were indeed tensors, it remains to be determined whether they are completely covariant or completely contravariant, or whether they are partially covariant and partially contravariant. It is obviously out of the question to recall even briefly the theory of tensor analysis. We refer the reader to the very numerous works on the subject such as, for example, [7].

To introduce the quantities we are going to need, it is apropos to introduce a tensor $\mathbb Q$ of order 2, skewsymmetric of dimension 4, our four-dimensional space. Expressed on an ad hoc basis, this tensor is written as a $4 \times 4$ matrix with six independent components. We are going to attach three of these six components to a vector that we will call ${\textbf{u}}$ and the three others to another vector that we will call [8] ${\textbf{v}}$. We have then

Conversely, vectors ${\textbf{u}}$ and ${\textbf{v}}$ can be “recovered” from the components of $\mathbb Q$. Assuming that this tensor is twice covariant, we note it then as ${\mathbb Q_{\cdot , \cdot}}$, and we will call ${\mathbb Q_{\nu ,\mu}}$ the components of $\mathbb Q$ with $\nu ,\mu \in \{{0,1,2,3} \}$ and ${\mathbb Q_{i,j}}$ the components of $\mathbb Q$ with $i,j \in \{{1,2,3} \}$ [9]. We then obtain

We are now in a position to introduce the electromagnetic tensor field ${\mathbb F_{\cdot , \cdot}}$, which is twice covariant:

The two Maxwell equations showing fields ${\textbf{E}}$ and ${\textbf{B}}$, which also happen to be the ones that do not show sources, are equivalent to the Gauss–Faraday law written in the covariant form

It is left to the reader to see what happens to Eq. (23) for $\beta = 2$ and $\beta = 3$. We obtain without too much surprise $\nabla \times {\textbf{E}} + {\partial _t}{\textbf{B}} = {\textbf{0}}$.

We introduce the magnetization-polarization tensor $\mathbb P$ and electric displacement tensor $\mathbb D$:

The three tensors $\mathbb D$, $\mathbb P$, and $\mathbb F$ [11] are related by

Finally, if we introduce the quadri-current $\mathbb J = (c\rho ,{\textbf{j}})$, it is possible to prove that the two Maxwell equations involving ${\textbf{D}}$, ${\textbf{H}}$ and the sources give

For the sake of simplicity, we suppose that the involved media are instantaneous. In that case, as explained before for the electric field, we expand the tensor ${\mathbb P^{\cdot , \cdot}}$ as a series:

Tensors $\mathbb F$ are skewsymmetric and can be inverted: tensors $\mathbb K$ are therefore not unique (see discussion on $\chi$’s symmetries in part I, for instance). On the other hand, there is only one fully symmetrical tensor $\mathbb K$, which is skewsymmetric by swapping two indices $2k + 1$ and $2k + 2$ (e.g., $\mu$ and $\nu$ or ${\alpha _k}\;{\beta _k}$) and symmetric by swapping four indices grouped two by two ($2k + 1$, $2k + 2$) and ($2l + 1$, $2l + 2$) [e.g., the couple ($\mu$, $\nu$) with the couple (${\alpha _k}$, ${\beta _k}$)]. In the following, the tensors $\mathbb K$ are supposed to be completely symmetric. We have, for instance,

In the following, we focus our attention on the first three terms.

B. Linear Term

The linear term is given by $\mathbb P_{(1)}^{\mu ,\nu}$, with

Let us expand the implicit sum! For that, once again, we have to study separately the cases $\mu= 0$ and $\mu= i$:

But ${M^k} = - \epsilon _{i,j}^k \mathbb P_{(1)}^{i,j}$. We then obtain the expression for the components of the vector ${\textbf{M}}$:

For nonchiral materials (see [12], for instance), the “crossed terms” vanish, namely, terms $\mathbb K_{(1)}^{0,j,k,l}$ and $\mathbb K_{(1)}^{i,j,0,l}$:

The first term has to be compared with the “common” constitutive material relation ${{\textbf{P}}^{(1)}} = {\epsilon _0} {\chi _e}{\textbf{E}}$. We then obtain the first relation:

The relation between relative permittivity ${\epsilon _r}$ and ${\chi _e}$ being simply ${\epsilon _r} = {\rm{Id}} + {\chi _{\rm{e}}}$, the components of the relative permittivity are of course

To obtain the second one, there is one more step. First compute the vector ${\textbf{H}}$ by the relation ${\textbf{H}} = \frac{1}{{{\mu _0}}}{\textbf{B}} - {\textbf{M}}$:

We now have to compare with ${\textbf{H}} = \frac{1}{{{\mu _0}}}\mu_r^{- 1}{\textbf{B}}$ to obtain the different components of the quantity $\mu_r^{- 1}$, namely,

C. Quadratic Term

The quadratic term is given by $\mathbb P_{(2)}^{\mu ,\nu}$, with

In this subsection, we restrict ourselves to the second order polarization vector $\mathbb P_{(2)}^{0,\nu}$. To obtain the magnetic counterpart, the reader will be able to draw inspiration from what has been done in the linear case:

The vector ${{\textbf{P}}_{(2)}}$ is simply given by

D. Cubic Term

It is left to the reader to calculate all the terms. It is obviously not difficult to find the term that interests us, if we are interested in only the fully electrical part. The vector ${{\textbf{P}}_{(3)}}$ is given by its components

E. Are Susceptibility Tensors Really Tensors?

Let ${\textbf{v}}$ be a vector and let ${{\textbf{v}}_{|{\cal B}}}$ (resp., ${{\textbf{v}}_{|{\cal B}^\prime}}$) be the column-vector associated with ${\textbf{v}}$ in the basis ${\cal B}$ (resp., ${\cal B}^\prime $) [13]. The link between the two aforementioned quantities is made through the matrix $p = \{p_\nu ^\mu {\} _{\nu ,\mu \in \{0,1,2,3\}}}$, namely,

More generally, if $\mathbb T$ is a $(p,q) $-tensor,

It is then clear that the equality

Coming back to Eq. (27), we obtain

Put another way:

In the same fashion,

5. NEUMANN’S PRINCIPLE AND SOME EXTRA SYMMETRIES FOR THE SUSCEPTIBILITIES

A. Generalities

We first recall Neumann’s principle. This principle, also called principle of symmetry, states that, if a crystal is invariant with respect to certain symmetry operations, any of its physical properties must also be invariant with respect to the same symmetry operations, or otherwise stated, the symmetry operations of any physical property of a crystal must include the symmetry operations of the point group of the crystal. In other words, if a crystal is invariant according to a symmetry $\rho$ characterized by a matrix $r_i^{{i^\prime}}$ in a certain basis, the different susceptibilities ${\chi _{(n)}}$ are invariant according to this symmetry. For such a symmetry, if Eq. (28) holds, Neumann’s principle simply reads

B. Centrosymmetric Materials

In crystallography, a point group that contains an inversion center as one of its symmetry elements is centrosymmetric. In such a point group, for every point (${x_1},{x_2},{x_3}$) in the unit cell, there is an indistinguishable point (${-}{x_1}, - {x_2}, - {x_3}$). Such point groups are also said to have inversion symmetry. Point reflection is a similar term used in geometry. Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect [15], or as we will see now, nonlinearities of even order. In practice, for such materials, applying the Neumann’s principle written in Eq. (28) for $r = - {\rm{I}}{{\rm{d}}_3}$ leads to $r_{{{i^\prime_1}}}^{{i_1}} = - \delta _{{{i^\prime_1}}}^{{i_1}}$. There are then two radically opposed possibilities depending on the parity of the order of the nonlinearities.

1. Even Terms

This concerns the terms ${\chi _{(1)}}$, ${\chi _{(3)}}$, etc., since the order of the tensors is shifted by one. The term ${\chi _{(1)}}$ is represented by a pseudo-tensor of order 2, the term ${\chi _{(2)}}$ by a pseudo-tensor of order 3, etc. For these materials, the relationship Eq. (28) takes the following form:

2. Odd Terms

In practice, this mainly concerns the term ${\chi _{(2)}}$. In this case, we have

We are forced to face the facts: there are no centrosymmetric materials associated with even nonlinearities! However, we should not jump to conclusions. Indeed, let us imagine that we have a centrosymmetric material in a half-space ${x_3} \lt 0$ and a vacuum for the upper half-space ${x_3} \gt 0$. There may be a “${\chi ^{(2)}}$ phenomenon-like”, i.e., the existence of a response proportional to the square of the electric field, because in the vicinity of the interface, there is a break in the central symmetry: we then speak of surface nonlinearity (see [16], for instance).

C. Isotropic Materials

A material is said to be isotropic if Neumann’s principle holds for every symmetry $\rho$.

1. Linear Term

For the linear term ${\chi _{(1)}}$, Neumann’s principle takes the following form:

2. Quadratic Term

For material to be isotropic, it must be centrosymmetric. We have seen that there is no centrosymmetric ${\chi _{(2)}}$ material. The answer is indisputable: there is no such thing as an isotropic ${\chi _{(2)}}$ material. On the other hand, if we disregard the centrosymmetric character, then there may be isotropic materials for which the ${\chi _{(2)}}$ tensors are proportional to the Levi–Civita tensor [17].

3. Cubic Term

The demonstration can be expected to be more delicate than in the linear case. If not difficult, it is at least tedious. It can be shown that ${\chi _{(3)}}$ is an isotropic “tensor” if its components are linked by the following relationship:

This equality calls for a few comments. First of all, the number of independent components goes from 81 to three! This is obviously a considerable gain. At first sight, it seems that index 3 has disappeared in favor of indices 1 and 2. This is not the case: it is just an arbitrary choice. We can indeed show the equivalence with

It is of importance to notice that due to the intrinsic permutation symmetry [see Eq. I.(11)], this condition would lead to the nullity of ${\chi _{(3)}}$ when dealing with an instantaneous material.

D. Full Study of a Tetragonal Crystal

The intention of this section is to study the linear, quadratic, and cubic susceptibilities of a crystal that has the symmetries of an octo-trapezohedron, a geometrical object represented in Fig. 2. This figure is invariant according to two rotations: one around the axis $O{x_3}$ with an angle of $\pi /2$ [see Fig. 2(b)]; the other is a little more complex. It is invariant as suggested by Figs. 2(c) and 2(d) according to a composition of rotations, one along the axis $O{x_1}$ as in Fig. 2(c) (or $O{x_2}$) with an angle of $\pi$ and then one along the axis $O{x_3}$, with an angle of $\pi /2$ as in Fig. 2(d).

 

Fig. 2. Different rotations of an octo-trapezohedron. The faces have been colored for the sake of clarity and are not a geometric indication (red, blue, black, or white faces are equivalent). (a) Octo-trapezohedron “before” a rotation. The frame ($O;\;{x_1},\;{x_2},\;{x_3}$) is the laboratory’s frame, whereas the frame ($O;\;{{{x}}^\prime}_1,\;{{{x}}^\prime}_2,\;{x^\prime _3}$) is linked to the trapezohedron. In this subfigure, the two frames are identical. (b) Octo-trapezohedron “after” the rotation about the $O{x_3}$ axis (dotted-dashed line) with an angle of $\pi /{{2}}$. Note that the primed frame has rotated by $\pi /{{2}}$ around the $O{x_3}$ axis with the solid. (c) Octo-trapezohedron “after” the rotation about the $O{x_1}$ axis with respect to the first position in sub-figure (a) (dotted-dashed line) with an angle of $\pi$. Note that the primed frame has rotated by $\pi$ around the $O{x_1}$ axis with the trapezohedron. The solid is not superimposable on the initial solid. (d) Octo-trapezohedron “after” a second rotation about the $O{x_3}$ axis with respect to the third position in subfigure (c) (dotted-dashed line) with an angle of $\pi /{{4}}$. This time, the solid is superimposable on the initial solid depicted in sub-figure (a). (e) Octo-trapezohedron “after” a single rotation about the axis characterized by the vector ${\rm{u}} = {({{0}},\;{{1}}\;{ - }\surd {{2}},\;{{1}})^T}$ (dotted-dashed line) with respect to the initial position in subfigure (a) with an angle of $\pi$. This time, the solid is superimposable on the initial solid depicted in sub-figure (a). (f) Same octo-trapezohedron at the same position as in sub-figure (e) but with another point of view: the axis of symmetry points in the direction of the observer. The symmetry of this solid seems more intuitive in this way.

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1. Linear Term

Two steps are necessary to find the symmetries of the linear susceptibility. First apply Eq. (31) for a rotation around the $O{x_3}$ axis with an angle $\pi /2$, i.e., with

We then compute the pseudo-tensor ${\chi ^\prime _{(1)}}$ defined as $\chi _{(1)}^{\prime {i_1},{i_2}}: = r_{{{i^\prime_1}}}^{{i_1}}r_{{{i^\prime_2}}}^{{i_2}}\chi _{(1)}^{{{i^\prime_1}},{{i^\prime_2}}}$:

By applying Neumann’s principle, we must make both terms equal: ${\chi ^\prime _{(1)}} = {\chi _{(1)}}$, i.e.,

We apply once again Neumann’s principle with the composition of rotation $r^\prime $ [18]:

We compute $\chi _{(1)}^{{\prime \prime} {i_1},{i_2}}: = r_{{\prime {i}_1}}^{\prime{i_1}}r _{{{i}_2}}^{\prime{i_2}}\chi _{(1)}^{{{i_1^\prime}},{{i_2^\prime}}}$, with ${\chi _{(1)}}$ written above:

We then deduce the final form of ${\chi _{(1)}}$:

The polarization vector is therefore not colinear to the electric field except if ${\chi _0} = {\chi ^\prime _0}$, which is one of the features of isotropic media.

2. Quadratic Term

The program given in part III, Section 4, provides the following result: the tensor ${\chi _{(2)}}$ is fully characterized by only one complex number named ${\chi _s}$:

In such a case, it remains to compute the forms $\langle \langle {{\textbf{E}}_p},{{\textbf{E}}_q}\rangle \rangle$, and due to the properties of the Levi–Civita symbol, the result is amazingly simple:

It turns then out that for this kind of ${\chi _{(2)}}$ material, all the contributions involving $\langle \langle {{\textbf{E}}_p},{{\textbf{E}}_q}\rangle \rangle$ disappear, almost magically, in Eqs. I.(22), I.(23), I.(25), I.(26), and I.(27), for instance.

Table 2. Eleven Elements Selected from the 29 Nonzero Elements of the ${\chi _{(3)}}$ Tensor^a

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3. Cubic Term

The cubic term does not yield as spectacular a result as the quadratic term, and the result remains extremely cumbersome to achieve without the use of a program. But since we have a program, why deprive ourselves of it? The program provides the information given in Table 2 and all the different elements of the “tensor.” Of course, it is difficult to display a fourth order pseudo-tensor. We have decided to represent it as a $3 \times 3$ matrix of $3 \times 3$ matrices. The “large matrix” gives the first two indices (the row index gives the first index and the column index gives the second) and the small ones give the next two (the row index gives the third index and the column index gives the fourth). But again, due to lack of space, we give the large matrix as three columns. Finally, we illustrate what has just been said with a few examples given directly on the following matrices, and the 11 generating elements given in Table 2 have been underlined. To be more precise, these elements are underlined if they are in their “own place.” For instance, in the following example, in part $\chi _{(3)}^{\cdot ,1, \cdot , \cdot}$, the element preceded by the symbol ($\star$) has not been underlined, for this element is in fact the element $\chi _{(3)}^{1,1,2,2}$ and has been replaced by a generating element given in Table 2: we have indeed $\chi _{(3)}^{1,1,2,2} = \chi _{(3)}^{1,1,3,3}$. In part $\chi _{(2)}^{\cdot ,2, \cdot , \cdot}$, we have, for instance, the term preceded by the symbol ($\bullet$): $\chi _{(3)}^{3,2,2,3} = - \chi _{(3)}^{3,2,3,2} - \chi _{(3)}^{3,3,2,2} + \chi _{(3)}^{3,3,3,3}$: