1. INTRODUCTION

Writing a nonlinear electromagnetism course is obviously challenging, especially in the limited context of a few pages. I must confess publicly, limiting myself to half a page’s width has been a real headache for me, forcing me to employ treasures of ingenuity and feats of contracted notation. In addition, there are numerous authoritative monographs (e.g., [1–4]), and this is not intended to compete with them but rather to present general ideas on this field, with a few references to books or articles. This paper is therefore intended to be self-consistent. Indeed, since the emergence of lasers in the 1960s and consequently of sufficiently energetic sources, nonlinear optics has become a science per se, into which one can only enter with infinite caution. The difficulty of this field lies in the abundance of new effects linked to complex constitutive relations and to the extent that canonical mathematical framework is almost nonexistent. Indeed, it can be said that, with a few exceptions, the mathematical framework of linear phenomena is rather complete. By this latest statement, we mean that the framework in which the different mathematical quantities involved “live,” like fields and charge densities, is well established. This vector/tensor framework ensures that the problem is well posed, i.e., that it has a solution, that it is unique, and that, finally, this solution has a certain stability, i.e., that it is not too sensitive to the input of the problem. On the other hand, as soon as we enter nonlinear phenomena, we lose the principle of superposition and, consequently, many reassuring mathematical theorems and frameworks. Thus, in a nonlinear electromagnetism problem, if one does not take care, one can easily arrive at problems without a solution or with several solutions, which is an uncomfortable situation to say the least.

In this paper, I will therefore try to introduce the essential notions of nonlinear electromagnetism in a fairly rigorous and systematic way without falling into excessive mathematical rigour. In any case, as noted, the mathematical theory of this field of physics remains to be achieved. Be that as it may, this paper is rather addressed to those who have already studied Maxwell’s system in a linear context and who are interested in the field diffracted by an object in the resonance domain. In other words, one may not only wish to study nonlinearities from a qualitative but also from a quantitative point of view.

This modest work would not have been possible without the following two people. First of all, mon maître, the late Roger Petit [5], who introduced me to the magic of Maxwell’s equations 30 years ago and under whose supervision I was able to obtain my Ph.D. thesis. I like to imagine that this great scientific storyteller and this incorrigible chatterbox continues his conversation with Maxwell himself.

The second on the list is my office neighbor and friend André Nicolet, who introduced me to finite elements and differential geometry in return for useful hints in optics.

2. MAXWELL’S EQUATIONS AND THE ELECTROMAGNETIC CONSTITUTIVE RELATIONS

A. Microscopic and Mesoscopic Fields

The wavelengths of the visible spectrum are approximately between 400 and 800 nm. Throughout this tutorial, we will assume that the wavelengths used are at least greater than 300 nm. The size of an atom being of the order of an angstrom (${10^{- 10}}\; {\rm{m}}$), it seems impossible to detail the behavior of the light simultaneously at these two scales. We are therefore led to make the following hypotheses.

The waves described will evolve in homogenized environments. This implies that our model will not be able to describe the individual action of an atom on light or inversely of light on an atom. Obviously, it is illusory to know perfectly these homogenized environments by a study that is responsible for quantum statistical physics. Instead, we will start from Maxwell’s equations in matter by assuming that there are two additional vector fields, ${\textbf{D}}$ and ${\textbf{H}}$, which only make sense in a macroscopic way (say, a hundred nanometers), unlike ${\textbf{E}}$ and ${\textbf{B}}$, which make sense even at an atomic scale. For discussions on smoothed or homogenized values, we can refer to [6,7], for example. Within the framework of this approximation (which, it should be noted, includes all classical [8] electromagnetism), Maxwell’s equations are stated as follows:

Thus stated, it appears that Maxwell’s equations are subdivided into two subsets. The former two equations [Eq. (1)] do not reveal any sources and are relations between microscopic fields ${\textbf{B}}$ and ${\textbf{E}}$, while the latter two equations [Eq. (2)] link homogenized fields ${\textbf{H}}$ and ${\textbf{D}}$ and show the terms of sources ${\textbf{J}}$ and $\rho$, which are supposed to be known (see Fig. 1). The fields ${\textbf{E}}$ and ${\textbf{B}}$ are the so-called electric and magnetic fields. As for the fields ${\textbf{D}}$ and ${\textbf{H}}$, they are called displacement and magnetizing fields, even if they go by other names. We will see in detail the link between these four fundamental fields. It is now time we defined the problem of scattering in electromagnetism. Find the electromagnetic field, i.e., ${\textbf{B}}$ and ${\textbf{E}}$ on the one hand and ${\textbf{H}}$ and ${\textbf{D}}$ on the other. It appears that we have to add hypotheses to answer this problem, for, in each point ${\textbf{s}}$ of ${{\mathbb R}^3}$ and for any time $t$, we have 12 unknowns (the three components of the four vector fields ${\textbf{E}}({\textbf{s}},t)$, ${\textbf{B}}({\textbf{s}},t)$, ${\textbf{D}}({\textbf{s}},t)$, and ${\textbf{H}}({\textbf{s}},t)$); however, the Maxwell’s equations only give two vectorial and two scalar equalities. The set of Maxwell’s equations being independent of the medium in which the fields oscillate, we have to encode the (electromagnetic) characteristics of the medium, with two (vectorial) relations that give the inductive fields in function of the primitive fields.

 

Fig. 1. Schematic for electromagnetism. The left-hand side shows the microscopic quantities ${\textbf{E}}$ and ${\textbf{B}}$, whereas the right-hand side shows the derived mesoscopic quantities ${\textbf{J}}$, $\rho$, ${\textbf{D}}\rho$, and ${\textbf{H}}$. Moreover, Tonti’s diagram works as two flow diagrams (left-hand side and right-hand side). The value of a node ($\bullet$) equals the sum of the values associated with the incoming arrows ${\rightarrow\!\!\!-\!\!\!-\!\!\!-}$. The value associated with an arrow is given by the action of the operator associated with this arrow on the value of the node at the origin of the aforementioned arrow. Note that “spatial operators” ($\nabla$, $\nabla \cdot$, and $\nabla \times$) only act on vertical arrows, whereas the “temporal operator” (${\partial _t}$) only acts on horizontal arrows. We find in this way the relations (a) ${\textbf{E}} = \nabla V - {\partial _t}{\textbf{A}}$, (b) ${\textbf{0}} = \nabla \times {\textbf{E}} + {\partial _t}{\textbf{B}}$, (c) $0 = \nabla \cdot {\textbf{B}}$, and (d) ${\textbf{B}} = \nabla \times {\textbf{A}}$ for the left-hand side, and (e) $\nabla \cdot D = \rho$, (f) $0 = \nabla \cdot {\textbf{J}} + {\partial _t}\rho$, and (g) ${\textbf{J}} = \nabla \times {\textbf{H}} - {\partial _t}{\textbf{D}}$ for the right-hand side, where, of course, we have denoted by ${\textbf{A}}$ (resp. $V$) the vector (resp. scalar) potential. Additionally, the connection between the two sides (${--\!\!\!\gt\!--}$) of the diagram is given by the most classical constitutive relations such as the electric and magnetic constitutive relations and also (i) ${\textbf{D}} = \varepsilon {\textbf{E}}$, (ii) ${\textbf{B}} =\mu {\textbf{H}}$, (iii) ${\textbf{J}} = \sigma {\textbf{E}}$ (Ohm’s law), and (iv) ${\textbf{J}} = - \frac{1}{{{\Lambda ^2}}}{\textbf{A}}$ (London’s law). Note the use of the $\star$ symbol instead of the traditional space or $\times$ symbol in the constitutive relations. There are two reasons for this. First, to avoid confusion with the “vector multiplication” used for the curl operator $\nabla \times$; second, it is a nod to the Hodge operator [11,12].

Download Full Size | PDF

B. From General Principles to Nonlinear Susceptibility

1. First Step with Linear Media

When the polarization vector is proportional to the electric field, we are in the context of linear electromagnetism. Further, the mathematical framework can be made precise. One can consult the works of Cessenat [9] or Tonti [10], for example. We have reproduced here a Tonti-like diagram (see Fig. 1), summing up the Maxwell’s equations together with the main constitutive relations.

2. How to Link the Mesoscopic Vector Fields to the Microscopic Ones

Our aim is now to link the mesoscopic vector fields ${\textbf{D}}$ and ${\textbf{H}}$ to the microscopic ones ${\textbf{E}}$ and ${\textbf{B}}$ in a quite general context. For that, it is convenient to divide ${\textbf{D}}$ (resp. ${\textbf{H}}$) into two different parts:

The former appears in a vacuum, and the latter can be seen as a perturbation due to the different media. The vector field ${\textbf{P}}$ is the so-called polarization vector (${\textbf{P}}_{{\rm{per}}}^{\rm{e}}$ for the electric part and ${\textbf{P}}_{{\rm{per}}}^{\rm{m}}$ for the magnetic part) and measures the deviation of the response of the medium in relation to the vacuum. From a microscopic point of view, it corresponds to the mean density electric dipole momentum. This response may vary according to $t$ (respectively ${\textbf{s}}$)—if the material is nonhomogeneous in time (respectively spatially)—but also as a function of times other than $t$ (respectively places other than ${\textbf{s}}$), if the material is nonlocal in time (respectively spatially). We are restricted here to spatially local materials; if this approximation cannot be used, it is necessary to do to space what we will do to time (refer to [7], for example). The constitutive relations in a vacuum are reduced to

From now on, it is assumed that the materials involved are amagnetic, which is a reasonable assumption in the optical range. That is to say, the magnetic polarization vector can be considered null. In the sequel, only the electric polarization vector is present: The exponent as well to the index “per” can therefore be dropped. Thus, we simply denote ${\textbf{P}}$ instead of ${\textbf{P}}_{{\rm{per}}}^{{\rm e}}$. It remains to rely the polarization vector to the vector fields ${\textbf{E}}$ and ${\textbf{B}}$, which is, by far, the most difficult part. In a general context, the link between the polarization vectors and the microscopic quantities can be read as per

1. Causality: The causality of the medium implies that the set $T$ defined above is $T(t) =] - \infty ,t]$. This means that, to evaluate the polarization vector at a given time $t$, it is not necessary to know the microscopic fields at times after time $t$. This principle is general and a priori inviolable.

2. Locality in space: The locality in space of the medium implies that the set $S({\textbf{s}})$ is reduced to the single point ${\textbf{s}}$: $S({\textbf{s}}) = \{{\textbf{s}}\}$. That means that the properties of ${\textbf{P}}$ at a given point ${\textbf{s}}$ only depend on the microscopic fields at the same point. Unlike causality, some materials are spatially nonlocal.

3. Nonbianisotropy: A medium is said to be nonanisotropic if does not explicitly depend on the magnetic field. In other words, the electric polarization vector does not explicitly depend upon the magnetic field. Chiral materials are then not addressed.

4. Stationarity: A medium is said to be stationary in time (resp. in space) if the functional $\mathfrak{P}$ does not depend explicitly on time (resp. on space). It is of importance to stress that this hypothesis is debatable when dealing with nonlinear optics. Some effects such as saturation and damage appear, and the hypothesis on the stationarity in time must be questioned.

5. Expandable media: We now tackle one of the most important hypothesis of nonlinear electromagnetism.

(a) Generality: At this stage, the aforementioned general assumptions are not enough. For reasons that we will clarify (in part) in Subsection 2.B, we will expand the polarization vector into integer powers of the electric field, the coefficients of this “polynomial” being denoted $Q$ in the sequel and “storing” the electromagnetic response of the matter. However, since the electric field is a vector field, we will have to take precautions. For defining this notion, it is apropos to introduce the $n$th order polarization vectors ${{\textbf{P}}_{(n)}}$, which are responses of the material in question proportional to the “$n$th power” of the electric field. However, the very notion of power of a vector has to be clarified. For this, we first introduce a vector quantity ${\tilde{\textbf{Q}}}_{(n)}^{\textbf{E}}$ defined as follows:

Now we are in a position to define the electric polarization vector field ${{\textbf{P}}_{(n)}}$:

You can also see another notation but which is strictly equivalent to the last one:

In other words, the $i$th component of  ${{\textbf{P}}_{(n)}}({\textbf{s}},t)$ reads

Remark: The writing of ${{\textbf{P}}_{(n)}}$ in Eq. (5) must be understood and especially the quantity ${\textbf{E}}({\textbf{s}},{t_1}) \cdots {\textbf{E}}({\textbf{s}},{t_n})$ within the integral, which appears as a strange and suspicious product of $n$ vectors ${\textbf{E}}({\textbf{s}},{t_1}), \cdots ,{\textbf{E}}({\textbf{s}},{t_n})$ (see Section 3.C for further explanations). Finally, it must be understood that all secrets of matter (dispersion, absorption, nonlinearity, etc.) are somehow encapsulated in its $Q$ functions. How to access such information even from an experimental point of view is obviously a challenge and even in the linear case (see [13], for example).

Then, a material is said to be expandable if the polarization vector ${\textbf{P}}$ can be expanded as per

Remark: The unit of ${Q_{(n)}}$ is ${m^{n - 1}} {V^{1 - n}} {s^{- n}}$, which depends on the order of the nonlinearity, $n$; thus, it is meaningless to compare the ${Q_{(n)}}$ for different $n$.

(b) Expandable media and beyond: In physics, it is necessary to determine the differences between general laws, which apply universally [here, Maxwell’s equations written in Eqs. (1) and (2)] and constitutive relations [here, the relationship among ${\textbf{P}}$ and ${\textbf{E}}$ and ${\textbf{B}}$ written in Eq. (3)], which, in turn, are the responsibility of the individual and reveal the details of a given material. Let’s take, for example, the simple example of a mass-spring system. From a macroscopic point of view, all physics are reduced to the stiffness of the spring linking the return force $F$ to the elongation $x$ of the type $F(x)$. Hooke’s law is in fact a power series limited to order 1 of $F$:

However, the facts impose that $F(0) = 0$ on one hand and that $F^\prime (0)$ is negative on the other (it is a restoring force). Thus, we have

Thus, we owe it to Hooke to neglect $o(x)$. More seriously, however, if we extend the spring sufficiently, we know that Hooke’s law needs to be reviewed. It even happens that, before breaking the restoring force no longer depends on the elongation, there is a phenomenon of saturation. One could therefore consider a law of the type

The question is then why we do not consider more general laws for the polarization vector and limit ourselves to the polynomial developments as suggested in Eq. (4). To answer this question, two different arguments must be put forward. The first is of a mathematical nature: the quantity ${{\tilde{\textbf{Q}}}_{(n)}}$ must be independent of the bases in which the tensors are expressed, and this considerably limits the choice of the mathematical quantities that can be used. The second argument is of a physical nature. It happens that polynomial development explains simply the appearance of the different harmonics.

3. Some Properties of Expandable Media

Of more fundamental nature, let us suppose that the medium is stationary; this means that, first, the spontaneous polarization does not depend explicitly on time, and, second, that for any point ${\textbf{s}}$ and for any duration time $T$, the effect of the electric vector field evaluated at the point ${\textbf{s}}$ and at the times ${\textbf{t}} = {t_1}, \cdots ,{t_n}$ on the polarization vector field evaluated at the point ${\textbf{s}}$ and at the time $t$ is the same than the effect of the electric vector field evaluated at the point ${\textbf{s}}$ and at the times ${t_1} - T, \cdots ,{t_n} - T$ on the polarization vector field evaluated at the point ${\textbf{s}}$ and at the time $t - T$. It is now time we introduce a new shorthand notation, which allows us to simplify the equations in the following. We define the “multione notation” ${\textbf{1}}$:

We can recast the expression $({t_1} - T, \cdots ,{t_n} - T)$ in a compact manner:

With this notation, the stationarity is expressed as

For $T = t$, the previous equation becomes

Thus, operators ${Q_{(n)}}$ depend, in addition to the space variable ${\textbf{s}}$, only on the duration of ${t_i} - t,i \in \{{1, \cdots ,n} \}$ and not on moments $t,{t_1}, \cdots ,{t_n}$ themselves. It is therefore legitimate to restrict the study to the polarization vector at time $t = 0$. When considering expandable media, a usual method is to define

Function ${R_{(n)}}$ is called the “response function of the $n$th order” because it allows us to write the relation as a convolution on ${{\mathbb R}^n}$ between ${R_{(n)}}$ and ${{\textbf{E}}_{(n)}}$ defined as follows:

By making use of Eq. (6), we have

With this notation, Eq. (5) becomes

By writing ${\hat{\textbf{E}}}({\textbf{s}},{\omega _k})$, the Fourier transform,

Plugging this last relation into Eq. (7), we obtain by letting ${\boldsymbol \omega} : = ({\omega _1}, \cdots ,{\omega _n})$

We recognize up to a multiplicative constant a Fourier transform

Remark: Because of lack of space, the fact that ${\tilde \chi _{(n)}}$ is a tensor field depending on both the space variable ${\textbf{s}}$ and the frequencies (${\omega _1},{\omega _2}, \cdots$) is sometimes dropped. Thus, we will note ${\tilde \chi _{(n)}}({\boldsymbol \omega})$ instead of ${\tilde \chi _{(n)}}({\textbf{s}},{\boldsymbol \omega})$. Moreover, for reasons that will be elucidated later, it will happen that one wants to emphasize the sum of the frequencies ${\omega _1} + {\omega _2} + \ldots + {\omega _n}$. This sum can be written thanks to the multione notation ${\omega _1} + {\omega _2} + \ldots + {\omega _n} = {\textbf{1}} \cdot {\boldsymbol \omega}$; further, one will note ${\tilde \chi _{(n)}}({\textbf{1}} \cdot {\boldsymbol \omega} ;{\boldsymbol \omega})$ despite the obvious redundancy rather than ${\tilde \chi _{(n)}}({\boldsymbol \omega})$.

C. Symmetries of the Susceptibility Tensors

The secret of the different media are encapsulated into the so-called susceptibility tensors. First, we have to bear in mind that these tensors are dreadful beasts. Indeed, in a given basis, they are represented by ${3^{(n + 1)}}$ different complex functions of $n + 3$ real variables $\tilde \chi _{(n)}^{i,{i_1} \cdots {i_n}}({\textbf{s}},{\omega _1}, \cdots ,{\omega _n})$ with $i,{i_1}, \cdots ,{i_n} \in \{{1,2,3} \}$. For instance, even for low $n$, say, three, for instance, we have to consider 81 complex functions of three space variables and three frequencies: ${\omega _1}$, ${\omega _2}$, and ${\omega _3}$. Second, as will be shown, these tensors are not defined in a unique way. The question is then to determine if these different components are independent; in other words, do these tensors exhibit some symmetries.

1. Intrinsic Permutation Symmetry

For the sake of simplicity, let us consider the $i$th component of the quadratic polarization vector ${{\textbf{P}}_{(2)}}$, which can be read as follows [14]:

Do not forget that the right-hand member has to be expanded with a summation over the dummy arguments, namely, ${i_1}$ and ${i_2}$ [15]. Let us suppose now that the susceptibilities involving the index $3$ vanish:

With variables ${\omega _1}$ and ${\omega _2}$ being dummy variables, it is possible to invert them, and the third component $P_{(2)}^{i,2,1}(t)$ becomes

It then exists an infinity of tensors ${\tilde \chi _{(2)}}$ giving the same polarization vectors ${{\textbf{P}}_{(2)}}$. Among all these compatible susceptibility tensors, we are able to build the symmetric tensor ${\chi _{(2)}}$ in the following way:

We are now in a position to generalize the previous result to the symmetries on the vector field ${{\textbf{P}}_{(n)}}$. For this purpose, let us develop the $i$th component of ${{\textbf{P}}_{(n)}}$:

It is then suitable to introduce the symmetrized susceptibility tensor of rank $n + 1$, namely, ${\chi _{(n)}}$:

2. Hermitian Symmetry

The fields at stake being real $({\textbf{E}},{\textbf{B}},{\textbf{P}}, \cdots)$, the harmonic counterparts satisfy the so-called Hermitian symmetry: ${\hat{\textbf{E}}}({\textbf{s}},{\omega}) = \overline {{\hat{\textbf{E}}}} ({\textbf{s}}, - \omega)$ for all ${\textbf{s}} \in {{\mathbb R}^3}$ and all real frequencies $\omega$, which implies the multidimensional Hermitian symmetry for the susceptibility tensor fields:

This last equation allows us to consider only positive frequencies.

3. SCATTERING EQUATION SYSTEMS

A. General Scattering Equation Systems

From Maxwell’s equations in Eqs. (1) and (2) and the general constitutive relations under the hypotheses stated in Subsection 2.B.2, and, after performing a Fourier transform, we obtain the scattering (or propagation) equation:

Finally, we will use the notation corresponding to a multilinear application (see Paragraph III.3.C for the properties of this form) [16]:

We then have

Note that, for $n = 0$ and $n = 1$, the factors ${{\hat{\textbf{P}}}_{(n)}}$ can be minimized:

As for ${{\hat{\textbf{P}}}_{(1)}}$,

The scattering equation system is therefore a nonlinear integro-differential equation [17]:

B. Polyharmonic Assumption

Equation (15) is still general. It is possible to consider transient mode, for instance, but this intregro-differential equation is very difficult to handle. It is the reason why, from now on, we focus on the case where the incident vector field is monochromatic at the frequency ${\omega _I}$: ${{\textbf{E}}^i}({\textbf{s}},t) = {\textbf{E}}_1^i({\textbf{s}}){e^{{\rm{i}}{\omega _I}t}} + {\textbf{E}}_{- 1}^i({\textbf{s}}){e^{- {\rm{i}}{\omega _I}t}}$ (we note ${\textbf{E}}_p^i({\textbf{s}})$ for ${{\hat{\textbf{E}}}^i}({\textbf{s}},p{\omega _I}))$ or equivalently in the case where the density of current ${\textbf{J}}(t)$ can be written as

In that case, the total electric field is monochromatic as the incident field

And, from Eq. (14), the polarization vector ${{\hat{\textbf{P}}}_{(2)}}$ becomes

This expansion allows us to integrate the polarization vectors:

This expansion can be re-indexed and is now compatible with Eq. (15). Nevertheless, the system in Eq. (15) is still too complicated to be solved, i.e., it still contains an infinite number of coupled partial nonlinear differential equations. To this aim, we introduce the notion of the degree, whose effect will be to obtain propagation equation systems described by a finite set of PDEs. In order to reduce the number of equations to a minimum for solving them numerically, we must use physics and what is generally observed in experiments that reveal nonlinear phenomena. It has been observed that tensors ${\chi _{(n)}}({p_1}{\omega _I}, \cdots ,{p_n}{\omega _I})$ vanish for sufficiently large integers ${p_1}, \cdots ,{p_n}$. Once again, we are doomed to make choices that seem arbitrary but not that much, as will later be shown, particularly with regard to energy balance criteria. To illustrate the choice we have to settle on, we give two definitions and will argue that these two definitions present interesting but different points of view.

Definition 3.1: Degree ${d_1}$

The degree ${d_1}$ for the tensor ${\chi _{(n)}}$:

Definition 3.2: Degree ${d_\infty}$

The degree ${d_\infty}$ for the tensor ${\chi _{(n)}}$:

In other words, the terms $\langle \langle {{\textbf{E}}_{{p_1}}}, \cdots ,{{\textbf{E}}_{{p_n}}}\rangle \rangle$ vanish provided that $|{p_1}| + \cdots + |{p_n}| \gt d$ (resp. $\max (|{p_1}|, \cdots ,|{p_n}|) \gt d$). In this case, and disregarding from now on the static component ${{\textbf{E}}_0}({\textbf{s}})$, we will agree that the total field has the form

In other words, since just a finite number of frequencies have been considered, it is legitimate to consider that the different tensors $\chi$ do vanish for high frequencies. In addition, there is an infinite number of ways to pick the frequencies that are involved in the experiment. Needless to say, physics must have the last word [19]. In a process where a frequency that is a hundred times that of the incident frequency appears, this way of selecting the frequencies involved must be reviewed. But a priori, it remains that the polyharmonic approach remains valid, and only the notion of degree must be modified.

C. Short Digression About the Polyharmonic Assumption

It must be understood that the assumption of the harmonicity of the source has been made for want of anything better. A priori in Eq. (15) allows us to solve transient regime problems, for example, during extremely short and energetic waves as, for example, in the case of ultrafast pulse lasers. However, to the author’s knowledge, nonlinear integro-differential equations of this type are out of reach—not only for algorithmic and computational power reasons but also because the data of the susceptibility tensor for a wide frequency range remains an open question. A case of intermediate difficulty could have been considered, for example, by considering a bi-harmonic source, with two frequencies ${\omega _I}$ and ${\omega ^\prime _I}$. In this case, the source is described by

We will not discuss this case, even if the transition from the harmonic case to the bi-harmonic case is not insurmountable. However, bis repetita placent, the richness of the nonlinear world with harmonic source is sufficient for an introductory course. Pierre Godard’s thesis can be consulted for further readings (see [20]).

D. Nonlinearity of the Second Order

We will thus suppose from now on that no nonlinearity higher than the quadratic one is present. We will treat this order of nonlinearity with some detail; the other ones being direct but increasingly sophisticated generalizations of this case. Also, though its presence does not lead to high difficulties, we will neglect the spontaneous polarization. The propagation equation system in Eq. (15) remains difficult to be solved directly. We have

Finally, we will make use of the notation corresponding to bilinear application $\langle \langle {\hat{\textbf{E}}}({\omega _1}),{\hat{\textbf{E}}}({\omega _2})\rangle \rangle$. The propagation equation can be then written as per

1. Polyharmonic Assumption and the Notion of Degree for Nonlinearities of the Second Order

Under the hypotheses (polyharmonic assumption with ${{\textbf{E}}_0} = {\textbf{0}}$) described in the preceding section, the set of PDEs is

Nevertheless, the system in Eq. (18) is still too complicated to be solved even numerically, i.e., it still contains an infinite number of coupled partial differential equations. To this aim, we are going to use the notion of the degree described in quite a general context in the previous section, whose effect is to obtain propagation equation systems described by a finite set of PDEs.

In the Degree 1 (Linear Case)

The cases ${d_1} = 1$ and ${d_\infty} = 1$ give

In both cases, indeed, the total electric field is reduced to

The nonlinear contributions are null as expected. This means that, at this lowest order of nonlinearity ($n = 2$), both degrees lead, at their lowest order (${d_1} = 1$ or ${d_\infty} = 1$), to the linear approximation of nonlinear optics.

In the Degree 2 (Second Harmonic Generation)

With this degree, the total field is

 

Fig. 2. Diagrammatic representation of the degree 2 with the definition ${d_1}$ and ${d_\infty}$ corresponding to a ${\chi _{(2)}}$ material. (a) ${d_1} = {{2}}$. There is only one process of exchange of energy between the two frequencies. The pump is supposed to be an infinite tank. (b) ${d_\infty} = {{2}}$. The field oscillating at the frequency of the pump ${\omega _I}$ is changed by the one that oscillates at ${{2}}{\omega _I}$: the pump is said to be depleted.

Download Full Size | PDF

 

Fig. 3. Diagrammatic representation of the degree 3 with the definition ${d_\infty}$ corresponding to a ${\chi _{(2)}}$ material. See Eq. (22). Note that the intrinsic permutation symmetry has been used.

Download Full Size | PDF

The systems in Eqs. (20) and (21) describe one of the most famous effects of nonlinear optics: the second harmonic generation. Though the sources (the current or the incident field) oscillate at a single frequency ${\omega _I}$, the total electric field oscillates at ${\omega _I}$ and at $2{\omega _I}$. Astonishingly enough, we note that the two degrees lead to different systems: If we choose ${d_1} = 2$, the component ${{\textbf{E}}_1}$ is independent of ${{\textbf{E}}_2}$; on the contrary, if we choose ${d_\infty} = 2$, there is a counteraction of ${{\textbf{E}}_2}$ onto ${{\textbf{E}}_1}$.

In the Degree 3 (Third Harmonic Generation by Cascade Effect)

The reader may be surprised to see that frequency tripling effects appear with materials showing only nonlinearities of order 2 and no nonlinearities of order 3; however, one must be persuaded that a priori all harmonics are likely to be present with only ${\chi _{(2)}}$ materials. We will see in the following paragraph that the terms ${\chi _{(2)}}$ and ${\chi _{(3)}}$ intermingle in a rather subtle way and that, for example, the terms ${\chi _{(3)}}$ appear in effects associated with frequency doubling (see Fig. 3). In short, we will have to be vigilant.

E. Nonlinearity of the Third Order

The system in Eq. (18) was restricted to nonlinearities of the second order. Under the same hypotheses, but this time, with nonlinearities of second and third orders, the set of PDEs is

In the Degree 1 (Linear Case and Optical Kerr Effect)

In the Degree 2 (Second Harmonic Generation and Optical Kerr Effect)

In the Degree 3

4. VOCABULARY ABOUT PDEs

When obtaining the equations (see Section 3 and Section 2.B), one fundamental theoretical question is whether these equations are well-posed. The French mathematician Jacques Hadamard coined the notion of well-posedness. According to the definition, a problem is said to be well-posed if it satisfies the following criteria:

To the author’s best knowledge of these lines, the question of the well-posedness of the PDEs appearing in nonlinear framework remains an unresolved mathematical issue especially for unbounded structures.

A. Classification

The purpose of this section is modest, i.e., to classify the PDEs that we encounter in this tutorial and more generally in physics. We will note that it seems that the vocabulary concerning nonlinear PDEs is not fixed, contrary to that of linear PDEs, which are much more studied, as stated. The following books may be consulted (mainly devoted to scalar cases) [22].

If ${\textbf{u}}$ is a vector application of $n$ variables ${x_1},{x_2}, \cdots ,{x_n}$ and ${\textbf{F}}$ a vector (possibly scalar) application of $p$ variables, the general form of PDE for the function ${\textbf{u}}$ is

B. Extra Conditions

The PDEs per se cannot ensure the uniqueness of the solution. In Eq. (20), for instance, we can add any homogeneous solution of 20 (i.e., nonvanishing fields for which ${\textbf{M}}_1^{{\rm{lin}}}{{\textbf{E}}_1} = {\textbf{0}}$ and ${\textbf{M}}_2^{{\rm{lin}}}{{\textbf{E}}_2} = {\textbf{0}}$) to the expected result to yield another solution. This is why it is necessary to add extra conditions that depend on the nature of the problem. In a bounded problem, for instance, when dealing with cavities, these conditions are called the “boundary conditions” (${{\textbf{E}}_{|\partial \Omega}} \times {\textbf{n}} = {\textbf{0}}$, for example). In an unbounded problem, these conditions, called “Silver–Müller conditions” (or, simply, outgoing waves conditions), ensure the uniqueness of the problem and mean that the diffracted field has an appropriate decreasing far from the target and that it propagates in the correct direction, i.e., outgoingly.