Neumann’s principle based eigenvector approach for deriving non-vanishing tensor elements for nonlinear optics

Abstract

Physical properties are commonly represented by tensors, such as optical susceptibilities. The conventional approach of deriving non-vanishing tensor elements of symmetric systems relies on the intuitive consideration of positive/negative sign flipping after symmetry operations, which could be tedious and prone to miscalculation. Here, we present a matrix-based approach that gives a physical picture centered on Neumann’s principle. The principle states that symmetries in geometric systems are adopted by their physical properties. We mathematically apply the principle to the tensor expressions and show a procedure with clear physical intuition to derive non-vanishing tensor elements based on eigensystems. The validity of the approach is demonstrated by examples of commonly known second and third-order nonlinear susceptibilities of chiral/achiral surfaces, together with complicated scenarios involving symmetries such as D₆ and O_h symmetries. We then further applied this method to higher-rank tensors that are useful for 2D and high-order spectroscopy. We also extended our approach to derive nonlinear tensor elements with magnetization, which is critical for measuring spin polarization on surfaces for quantum information technologies. A Mathematica code based on this generalized approach is included that can be applied to any symmetry and higher order nonlinear processes.

INTRODUCTION

When modeling a physical phenomenon, its properties can be represented by algebraic objects called tensors, for simplicity in mathematical calculations. The tensor representation can be categorized by its ranks. For example, density, volume, and mass can be described by zeroth-rank tensors, i.e., scalars; electric field, velocity, temperature gradient can be represented by first-rank tensors, i.e., vectors; strain, piezoelectricity, and hyperpolarizability can be expressed as higher-rank tensors.¹ In nonlinear optics, optical signals are also mathematically described by tensors.² The most commonly known are second-order susceptibilities (χ⁽²⁾) representing second harmonic generation (SHG), sum frequency generation (SFG) and different frequency generation (DFG) that are third-rank tensors, third-order susceptibilities (χ⁽³⁾) describing transient absorption, two-dimensional optical spectroscopy, and coherent antistoke Raman spectroscopy (CARS) that are fourth-rank tensors.^3–21

One convenience of representing physical properties with abstract tensors is that many physical systems have symmetry, and their associated physical properties also adopt the same symmetry, which can be derived using tensor-based linear algebra. For example, the methyl group belongs to the C_3v point group and has a dipole moment (vector, first-rank tensor) along the principal axis denoted as μ_z, while the cubic system does not have a dipole moment due to centrosymmetry.²² Yet, the derivation involving higher rank tensors could be complicated and tedious. In current literature, the non-vanishing tensor elements are derived in a conventional way, where a tensor element has to be zero if it changes sign after a symmetry operation, such as 2-fold rotation, inversion, or reflection.^23–27 However, such a method is difficult for deriving non-vanishing elements with complicated symmetry, for instance, D₆ symmetry and octahedral systems. Perhaps because of the tedious and non-intuitive derivations, most researchers rely on the now well-accepted tabulation of non-vanishing elements of mainly third- and fourth-rank tensors in textbooks.^2,28 However, historically, corrections were made to the initial derivations after some incorrect results were implemented in research for 20 years.^2,28–31 Moreover, only χ⁽²⁾ and χ⁽³⁾ have well-established results for non-vanishing elements of different symmetric systems.^5,32,33 With the advancement of laser technologies, higher order nonlinear signals (e.g., 2D and 3D spectroscopy) are popular now and have led to new physical insights into molecular and materials systems.^34–39 However, the derivation of their non-vanishing elements could be more confusing, for example, fifth-order susceptibility for 3D spectroscopy (a sixth-rank tensor) has a total of 729 tensor elements!⁴⁰

Thus, a systematic, general approach with a clear physical intuition is beneficial to derive nonlinear optical susceptibilities (and higher rank tensors in general) based on the symmetry of the systems. We here present a matrix-based approach to solve for the non-vanishing tensor elements of systems with any symmetry and also reveal the relationships between non-vanishing tensor elements. This approach is based on solving the eigenvectors of symmetry-based operators of a point group that follows the Neumann’s principle. This general approach can be applied to tensors of any rank and systems with complicated symmetries. We showed that this approach solves the same underlying linear algebra as Refs. 33 and 41. We noted that although symmetry-based nonlinear susceptibility has been derived before, the approaches were elusive to researchers new to this field. The present work provides a clear physical picture and straightforward mathematical derivation of the reasons why certain tensors survive and hold specific relationships with other tensors under the symmetry operations. For broader usage by the research community, detailed tabulations of second (SFG, DFG, and SHG), third (transient absorption,CARS, 2D IR, 2D electronic etc.), fourth (2D SFG) to fifth (2D Raman, 3D IR) order nonlinear optical susceptibilities of common point groups, and second order susceptibilities under magnetizations are obtained and provided in the supplementary material. We also included a Mathematica code to efficiently derive non-vanishing tensor elements of interest. In the result section, we showed selected examples of how to derive the non-vanishing second to fifth order nonlinear susceptibilities of common point groups based on the Mathematica code provided, and further discussed the physical implications of experiments.

METHOD

The central concept of this method is Neumann’s principle, stating that symmetries in geometric systems are also possessed by the physical properties of the system.^{29,30,42–44} Intuitively, an example is that if a molecule is unchanged by a $C_3^1$ rotation about its principal axis (z), so is its linear optical susceptibilities represented by first rank tensors. Thus, one can apply the $C_3^1$ rotation to μ, which is a vector (or first rank tensor), and it is straightforward to see that only μ_z remains unchanged and thereby survives the operation.

However, we can also use Neumann’s principle in an algebraic way to solve this problem. The mathematical expression of Neumann’s principle is that for a physical property , in a vectorized form (vectorized tensor V), it should obey MV = V, where M is the Kronecker product of a symmetry operation S.^45,46 That is, to solve for the non-vanishing tensor elements, one needs to find the eigenvectors of M whose corresponding eigenvalues are equal to 1. Certain conditions or relationships between tensor elements are then encoded in the eigenvectors. Mathematically, the equation can also be rearranged as $\left(M-I\right)V=0$, where I is an identity matrix with the same dimension as M. Therefore, solving for eigenvectors with eigenvalue 1 becomes solving for the nullspace of the matrix (M − I). To implement this idea back to the linear optical susceptibility with the C₃ rotation symmetry problem, it equals to solving the nullspace of $(C_3^1-I)$.

While it might seem to be unnecessary, this concept of Neumann’s Principle based EigenVector approach (NPEV) becomes handy when considering higher-rank tensors because there are 3ⁿ elements for an nth rank tensor, and we have to find the correlations between each element, which is 3ⁿ × 3ⁿ in size. For example, for a fifth-rank tensor, which contains 243 tensor elements, it could become difficult to follow using traditional methods. However, one can mathematically use the Kronecker products of the symmetry operator S to describe the linear map between the old and new tensors of any rank and automate the derivation process algebraically.⁴⁷ For a symmetry operation denoted by matrix S (3 × 3 in 3-dimensional coordinate system) applied to a vectorized n-th rank tensor, the mapping matrix M is the Kronecker product of the matrix S with itself n times, expressed as M = S ⊗ S ⊗ S ⊗ S … (S appears n times). In our Mathematica code, Kronecker products of most common symmetry operations of third to fifth-rank tensors are provided. For a fifth-rank tensor, $M=C_3^1\otimes C_3^1\otimes C_3^1\otimes C_3^1\otimes C_3^1$ is a 243 × 243 matrix, describing the mapping of old tensor to the new one after a $C_3^1$ rotation. Then, to solve the non-zero tensor elements and their relationships, essentially, one just needs to solve the eigenvectors of MV = IV. The eigenvectors thereby remain unchanged under the symmetry operation, and the original tensor elements that compose the eigenvectors should be non-zero and also satisfy the relationships between the eigenvectors. For example, to determine the surviving polarizability element α_ij (second-rank tensor with 3 × 3 elements) after $C_3^1$ rotation about z, we can first vectorize it into a 9 × 1 vector. Then the corresponding mapping matrix M is $\left(C_3^1\otimes C_3^1\right)$, a Kronecker product of $C_3^1$ with itself. By solving the nullspace of $\left(M-I\right)$, we can determine two independent elements α_3,3 and α_1,1 = α_2,2 will survive.

The above paragraph described the key idea of the NPEV approach, and to determine the non-zero tensor elements of a point group, one, in principle, just needs to repeat the same procedure over all symmetry operators, which could still be tedious. However, there are still a few tricks to be implemented. Below, we present a practical three-step to implement this NPEV approach for non-vanishing tensors [Fig. 1(a)]. The three steps are: 1. Identify generating operations for a specific point group; 2. Compose the mapping matrix M; and 3. Diagonalize the matrix M and inspect eigenvectors to reveal non-zero tensor elements and their relationships. As a prerequisite knowledge of the three-step approach, we show the basics of tensor vectorization and position index calculation to assist the inspection of eigenvectors to retrieve corresponding element correlations in Fig. 1(b). A number subscript is used as a generalized format to denote different coordinate systems. To use xyz or abc coordinates and subscripts (i.e., tensor elements shown as β_xyzy or β_abcb instead of β₁₂₃₂), users can uncomment corresponding command lines in the provided Mathematica code. One only needs to select from the common point groups already composed by us, and a tabulation of surviving tensor elements would be computed and displayed. The second to fifth order optical susceptibilities useful for 2D spectroscopy and high order nonlinear optics have been summarized in the supplementary material.

FIG. 1.

Basics of non-vanishing tensor element derivation. (a) Flowchart of the NPEV 3-step approach. First, finding the generating operations of a point group. Second, building a mapping matrix M based on the Kronecker product of the generating operations. Third, computing the eigenvectors of M whose eigenvalues are 1, i.e., nullspace of (M − I), and finding the non-vanishing elements and their relationships. (b) An example of tensor vectorization of a third-rank tensor β_i,j,k and the calculation of the position index of a tensor element.

Identifying generating operations

It is convenient to describe molecular systems using point groups, in each of which there exists a set of symmetry operations, such as n-fold rotation, reflection (mirror plane), and inversion.^30,42 For example, the C_3v point group has $C_3^1$, $C_3^2$, E, ${\sigma }_{\nu }^a$, ${\sigma }_{\nu }^b$, and ${\sigma }_{\nu }^c$ symmetry operations, denoting a 3-fold rotation, twice of a 3-fold rotation, identity, and three vertical mirror planes containing the C₃ axis, respectively. Yet, only a subset of operations is necessary to express all these symmetry operations. This subset is the generating operations.^30,42 For instance, in the C_3v point group, $C_3^1$ and ${\sigma }_{\nu }^a$ are the generating operations, while the other symmetry operations can be expressed as: $C_3^2=C_3^1\cdot C_3^1$, $E=C_3^1\cdot C_3^1.C_3^1$, ${\sigma }_{\nu }^b={\sigma }_{\nu }^a\cdot C_3^1$, ${\sigma }_{\nu }^c={\sigma }_{\nu }^a\cdot C_3^2$. Since the generating operations can fully characterize the spatial symmetries of a point group, these elemental operations are sufficient to derive all the non-vanishing tensor elements of a point group. The generating operations of seven crystal systems and 32 crystallographic point groups are summarized by Briss.⁴² In the case of the C_3v point group, the generating matrices are

$$C_3^1=\left(\begin{array}{ccc}\hfill -1/2\hfill & \hfill -\sqrt{3}/2\hfill & \hfill 0\hfill \\ \hfill \sqrt{3}/2\hfill & \hfill -1/2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d}{\sigma }_{\nu }^a=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1,\hfill \end{array}\right),$$

which are defined based on a right-handed coordinate system, where the z axis is the C₃ axis and zx plane is the ${\sigma }_{\nu }^a$ mirror plane, shown in Fig. 2.

FIG. 2.

Symmetry operations of the C_3v point group in a right-handed xyz coordinate system.

Composing mapping matrix

Once we identify the generating operations of the point group, we can create the mapping matrix. For each generating operation, we can calculate Kronecker product matrix M_i, and the Neumann’s principle equation M_iV^(N) = V^(N) must satisfy for vectorized Nth-rank tensors. By considering the generating operators, the number of symmetry operations for deriving non-zero tensors has already been reduced. However, we can take one more step to simplify the calculation. That is, if there are m generating operations, we can sum up all the Neumann’s principle equations and get $\frac{1}{m}(M_1+M_2\cdots +M_m)V^{(N)}=V^{(N)}$, which is equivalent to solve the nullspace of ($\frac{M_1+M_2\cdots +M_m}{m}-I$). In this way, we only need to diagonalize the matrix once to retrieve the surviving tensor elements.

Taking deriving the non-vanishing hyperpolarizability ${\beta }_{\mathit{ijk}}\left(\right.a$ third-rank tensor) elements for C_3v point group as the example again, with $C_3^1$ and ${\sigma }_{\nu }^a$ as the generating operations, the generating matrices are $M_1=C_3^1\otimes C_3^1\otimes C_3^1$ and $M_2={\sigma }_{\nu }^a\otimes {\sigma }_{\nu }^a\otimes {\sigma }_{\nu }^a$. The mapping matrix are then $M=\frac{M_1+M_2}{2}$, shown in Fig. 3.

FIG. 3.

Deriving non-vanishing third-rank tensor elements for the C_3v point group. (a) Following the 3-step NPEV approach, mapping matrix $M=\frac{M_1+M_2}{2}$ is created with two generating operations (3-fold rotation about principal axis z, reflection about zx plane) where $M_1=C_3^1\otimes C_3^1\otimes C_3^1$, and $M_2={\sigma }_{\nu }^a\otimes {\sigma }_{\nu }^a\otimes {\sigma }_{\nu }^a$. (b) Five eigenvectors (V₁ to V₅) of M are calculated with eigenvalues equal to 1 and a generalized form (a₁V₁ + a₂V₂…a₅V₅) of a vectorized 3rd-rank tensor that survives after $C_3^1$ rotation and ${\sigma }_{\nu }^a$ reflection is presented. The extracted non-vanishing elements (c) and the tabulated (d) surviving tensor representation. Numbering is for counting purposes only and is different from the vectorized tensor position indexing in Fig. 1.

Solving eigenvectors and extracting non-zero elements and their relationships

Now, we can solve MV^(N) = V^(N). The number of such eigenvectors tells us the number of independent non-vanishing tensor elements. If there exist k different eigenvectors (V₁, V₂, V₃, …, V_k), then the generalized form of a vectorized tensor that survives after the symmetry operation can be written as: a₁V₁ + a₂V₂ + a₃V₃, …, a_kV_k, where a₁, a₂, …, a_k are arbitrary coefficients for each eigenvector. A simple proof for the generalized solution can be carried out as: $M(a_1{V}_1+a_2{V}_2+a_3{V}_3+\cdots +a_k{V}_k)=M{a}_1{V}_1+M{a}_2{V}_2+M{a}_3{V}_3\dots M{a}_k{V}_k=a_{1}M{V}_1+a_{2}M{V}_2+a_{3}M{V}_3$$+\cdots +a_{k}M{V}_k=a_1{V}_1+a_2{V}_2+a_3{V}_3\dots a_k{V}_k$. It is readily satisfied that a₁V₁ + a₂V₂ + a₃V₃, …, a_kV_k is a solution.

Getting back to solving β_ijk for the C_3v point group, diagonalization shows only five eigenvectors [column of Fig. 3(b), V₁ to V₅] can have eigenvalues of 1, implying that there are only five independent non-vanishing β_ijk elements. These eigenvectors also encode the relationships between the original β_ijk elements. For example, taking the second eigenvector V₂ as an example [Fig. 3(b)], the eigenvector requires β_3,1,1 and β_3,2,2 to have the same amplitude. In other words, β_3,1,1 = β_3,2,2, though more complicated linear relationships can only be obtained by inspecting the generalized solution (see example in Figs. 4 and 5).

FIG. 4.

Deriving non-vanishing fourth-rank tensor elements for C_∞v point group. (a) Following the 3-step NPEV approach, mapping matrix $M=1/2\left(\frac{{\int }_0^{2\pi }{M}_1\mathrm{d}\theta }{{\int }_0^{2\pi }\mathrm{d}\theta }+\frac{{\int }_0^{\pi }{M}_2\mathrm{d}\phi }{{\int }_0^{\pi }\mathrm{d}\phi }\right)$ is created with two set of generating operations (C_∞ rotation about the z axis and infinite number of vertical mirrors σ_v containing the C_∞ axis) where M₁ = C_∞z ⊗ C_∞z ⊗ C_∞z ⊗ C_∞z and M₂ = σ_v ⊗ σ_v ⊗ σ_v ⊗ σ_v. (b) Ten eigenvectors (V₁ to V₁₀) of M are calculated with eigenvalues equal to 1 and a generalized form (a₁V₁ + a₂V₂…a₁₀V₁₀) of a vectorized fourth-rank tensor that survives after C_∞ rotation and infinite σ_v reflections is presented. (c) The tabulated surviving tensor representation. Numbering is for counting purposes only and different from the vectorized tensor position indexing in Fig. 1.

FIG. 5.

NPEV approach to derive non-vanishing SHG hyperpolarizability tensor elements of graphene monolayer with D₆ symmetry (a) Definition of lab coordinate and graphene monolayer coordinate. (b) without and with magnetization along the (c) x axis, (d) y axis, and (e) z axis.

A faster way to extract all relationships between non-vanishing elements is from the generalized solution of β_ijk in the C_3v point group. It is a₁V₁ + a₂V₂, …, a₅V₅, shown in Fig. 3(b) denoted as the column “Sum[a_kV_k].” Focusing on the non-vanishing elements extracted [Fig. 3(c)], at vector position 0 it is an arbitrary number −a₅, whereas at position 4, 10, and 12 the number are all a₅, indicating that the tensor elements at position 0 is equal to the negative of those at position 4, 10, and 12, simplified as: β_1,2,2 = β_2,1,2 = β_2,2,1 = −β_1,1,1. Similarly, we can obtain linear relationships of: β_1,1,3 = β_2,2,3, β_1,3,1 = β_2,32, and β_3,1,1 = β_3,2,2, for a total of five independent elements [Fig. 3(d)]. With all relationships encoded in one single generalized solution, we can automate the inspection process in Mathematica to readily display the surviving elements and their relationships, which could be more difficult to implement for higher order nonlinear optical signals, using the alternative approach described in the latter section.

Up until now, we have shown the complete procedure of using the NPEV approach to derive non-vanishing tensor elements of a symmetry point group, using the C_3v point group as an example. In the result and discussion, we will briefly show a few more cases of different ranks of tensors with various symmetries that are related to different common nonlinear optical techniques, most of whose non-vanishing tensors have not been reported. We finally discuss the relationship between our NPEV approach and one method in the literature.^33,41

RESULTS AND DISCUSSIONS

Third-order susceptibilities (fourth-rank tensors) of C_∞ and C_∞v symmetry

We start to show the results of third order susceptibility of isotropic achiral surfaces, which have a C_∞v symmetry. Surfaces and interfaces are of significant interest in many fields.^48–53 They are widely studied by SFG owing to its sensitivity to non-centrosymmetry that inherently exists at interfaces. 2DIR is also reported to study monolayers on interfaces, using attenuated total internal reflection geometry, yet a clear derivation and tabulation of surviving tensor elements are missing.^54–56

We can derive the non-vanishing elements of third-order susceptibility denoted by β_i,j,k,l for interfaces represented by the C_∞v point group via the NPEV approach. C_∞v point group consists of C_∞ rotation about z axis and infinite number of vertical mirrors σ_v containing the C_∞ axis [Fig. 4(a)]. Mathematically, an arbitrary rotation about the z axis can be expressed by a rotation matrix $C_{\infty z}=\left(\begin{array}{ccc}\hfill \cos \theta \hfill & \hfill -\sin \theta \hfill & \hfill 0\hfill \\ \hfill \sin \theta \hfill & \hfill \cos \theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$. Similarly, we can describe a mirror containing a z axis by ${\sigma }_v=\mathrm{R}\mathrm{e}\mathrm{fl}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}\left[\cos \phi ,\sin \phi ,0\right]$, which is a Mathematica built-in function to describe reflection in a mirror passing through origin and perpendicular to vector $\left[\cos \phi ,\sin \phi ,0\right]$, a vector in the xy plane. We then compose the mapping matrix from two main generating operations, C_∞z and σ_v. To sum up all Neumann’s principle equation and get $\frac{1}{m}(M_1+M_2+\cdots +M_m)V^{(N)}=V^{(N)}$ as described earlier, we integrate the Kronecker product with respect to θ and φ, i.e., the mapping matrix $M=1/2\left(\frac{{\int }_0^{2\pi }{M}_{C_{\infty z}}\mathrm{d}\theta }{{\int }_0^{2\pi }\mathrm{d}\theta }+\frac{{\int }_0^{\pi }{M}_{{\sigma }_v}\mathrm{d}\phi }{{\int }_0^{\pi }\mathrm{d}\phi }\right)$ [Fig. 4(b)] because there are infinite number of symmetry operations. The eigensystem of M has ten eigenvectors with eigenvalues of 1, indicating that there are ten independent non-vanishing elements [Fig. 4(c)]. Noticeably, from the generalized solution, we can see that β_1,2,2,1 and β_2,1,1,2 are degenerate and are dependent on other elements via the linear relationship denoted as: β_1,2,2,1 = β_2,1,1,2 = β_1,1,1,1 − β_1,1,2,2 − β_1,2,1,2, which agrees with the relationships of 2D spectroscopy of various polarizations of isotropic systems. One interesting result is that β₃₃₃₃ is different from β_1,1,1,1 or β_2,2,2,2, which should be equal in a bulk isotropic system. Thus, controlling the absolute polarization (i.e., S vs P polarization) could allow one to probe different susceptibility elements of interfacial molecules and reveal molecular orientations.

More interestingly, we can also derive the non-vanishing elements for C_∞ point group, such as an isotropic chiral interface. Because of the lower symmetry than C_∞v, more elements survive, with 19 independent ones and 41 in total (see supplementary material). Some elements are β_1,2,2,2 = −β_2,1,1,1 = −β_2,1,2,2 − β_2,2,1,2 − β_2,2,2,1, which can be measured using PSSS polarization (from left to right are the polarizations of signal, and third to first pulses). The signals of such polarization are absent from the achiral system, similar to the bulk 2D spectroscopy.^57,58 This is very similar to the well-known example in SFG spectroscopy: chiral tensor elements (β_3,2,1 = −β_3,1,2, β_2,3,1 = −β_1,3,2, β_2,1,3 = −β_1,2,3) that will otherwise vanish in the C_∞v symmetry starts to survive in the C_∞ point group. As shown in the next section, when symmetry is broken by magnetization, it also induces new SHG/SFG signals.

Second order susceptibility (SHG) with magnetizations

Another way to break symmetry is by introducing a magnetic field. As a result, the symmetry of an object is lowered, with more non-vanishing tensor elements arising.²⁴ SHG has been developed to probe surface magnetization and it is of technical interest to know what elements will survive under magnetization, which is beneficial to the study of 2D materials under magnetic field.^24,59–64 In principle, by knowing the symmetry of the system with and without magnetization, the NPEV approach can compute the surviving elements.

To derive surviving tensor elements for SHG, one can add a permutation matrix when composing the mapping matrix to account for the last two interchangeable indices in SHG, i.e., β_ijk = β_ikj. In fact, here we are treating the permutation as a symmetry operation M_p, which again shows the power of summing up all of Neumann’s principle equations to compose the mapping matrix $M=\frac{1}{m}(M_1+M_2+\cdots +M_m)$. The 27 × 27 permutation matrix is included in the supplementary material and our Mathematica code.

Figure 5 shows an example of a graphene monolayer of D₆ symmetry (z axis is the principal axis) with magnetization along different directions. We define lab coordinate as x′y′z′ and graphene coordinate as xyz, with the two systems sharing z′(z) axis [Fig. 5(a)]. x′z′ is incident plane (S polarization along y′) and sample is in x′y′ plane. xy and x′y′ describes the same plane, but are rotated apart about z(z′) axis by angle ϕ. There are three generating operations for a D₆ point group: a 6-fold rotation about z, a 2-fold rotation about x, and a 2-fold rotation about y. Without magnetization [Fig. 5(b)], there are four surviving SHG hyperpolarizability elements. All these elements (β_2,1,3 = β_2,3,1 = −β_1,2,3 = −β_1,3,2) have a z component (i.e., P polarization required for measurement), meaning the SHG signal in SSS polarization (i.e., y′y′y′) will vanish.

When magnetization is along x axis [Fig. 5(c)], the C_6Z and C_2Y rotation symmetry are broken, and it only has a C_2X rotation symmetry left. There are now 13 non-vanishing SHG hyperpolarizability elements. Aside from the four elements that exist without magnetizations, which are also even to magnetization, i.e., β(M) = β(−M), the other nine new elements enabled by magnetizations are odd, i.e., −β(M) = β(−M). The SHG signal in SSS polarization will now show up, with odd elements β_1,1,1, β_1,2,2, β_2,1,2, and β_2,2,1 contributing to the signal. Similarly, with magnetization along the y axis [Fig. 5(d)], only a C_2Y rotation axis is left, leading to 13 surviving elements and nine to be odd. β_1,1,2, β_1,2,1, β_2,1,1, and β_2,2,2 will be contributing to the SHG signal in SSS polarization. In contrast, a magnetization along the z axis will keep the C_6Z rotation axis, bringing a total of 11 non-vanishing elements and seven odd ones. To differentiate a z-direction magnetization from the in-plane or no magnetization cases, we can measure the SHG signals of SSS and PPP. With no magnetization, only the so-called chiral SHG signal would appear, and there should be no PPP or SSS signals. However, when the z-direction magnetization is on, it should induce PPP signals but no SSS signals [Fig. 5(e)]. We note that all magnetizations are derived relative to the internal (molecular frame) because it is easier to define the molecular symmetry this way. To transfer the response back to the lab frame, an Euler rotation needs to be implemented between the molecular and lab frame.

One can also deduce the surviving SHG/SFG hyperpolarizability elements for an interface/surface with C_∞v symmetry with different magnetization directions. The results are summarized in the supplementary material and match well with the reported general form of nonlinear magneto-optical tensor χ⁽²⁾ for interface/surface.⁶⁵ One important consideration for deriving magnetization related tensors is that when a mirror plane is involved, sign flipping of magnetization might occur because it is a pseudovector.⁶⁶ Operations that change the sign of magnetization will break the symmetry and is not possessed by the system any more. For example, when magnetization is along the x axis, the C_∞ rotation symmetry (defined as the z axis) is broken, and only one yz mirror is left, because it is the only symmetry operation that does not flip the sign of magnetization. The xz mirror plane is broken because it would otherwise change the magnetization, which is a pseudovector, to the opposite direction, i.e., the cross product of MAG = e_y × e_z becomes −MAG = −e_y × e_z with xz mirror imaging (reflection of y).⁶⁶

Fifth order nonlinear susceptibility for 3D IR (sixth-rank tensor) of cubic centrosymmetry

With the assistance of the Mathematica program, we can automate the derivation process and obtain the non-vanishing elements of higher-rank tensors with complicated symmetry. The non-vanishing elements for fifth-harmonic generation in cubic centrosymmetric crystals (such as β_1,1,1,1,1,1, β_2,2,2,2,2,2, β_3,3,3,3,3,3) have been conventionally derived by considering positive/negative sign flipping after certain symmetry operations.⁴⁰ Here, we can derive all the non-vanishing elements easily with our NPEV approach by defining the mapping matrix as $M=\frac{M_1+M_2+M_3}{3}$, where M₁, M₂, and M₃ are the generating matrices and are related to inversion, 4-fold rotation about z axis, and 3-fold rotation about the cubic body diagonal, respectively. For an incident field without a z component, we can extract the non-vanishing elements, as demonstrated in Fig. 6, which matches well with literature.⁴⁰ A full list of non-vanishing sixth-rank tensor elements (for fifth-harmonic generation, 2D Raman and 3D IR) without restriction in incident field can be found in our Mathematica code (section χ⁽⁵⁾) and supplementary material.^36–38 In addition, non-vanishing susceptibilities of SFG, 2D IR, and 2D SFG of common symmetry are also summarized in the supplementary material.^35,67–69

FIG. 6.

Deriving non-vanishing sixth-rank tensor elements for fifth-harmonic generation in a cubic system (O_h point group) with no z component in incident fields. (a) Following the 3-step NPEV approach, mapping matrix $M=\frac{M_1+M_2+M_3}{3}$ is created with three generating operations (inversion, 4-fold rotation about principal axis z, 3-fold rotation about cubic body-diagonal), where M₁ = Inv ⊗ Inv ⊗ Inv ⊗ Inv ⊗ Inv ⊗ Inv, $M_2=C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1$, and $M_3=C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1$. 122 eigenvectors (V₁ to V₁₁₂) of M are calculated with eigenvalues equal to 1 and the simplified surviving tensor representation without the z component is shown (b). The full list of eigenvectors with the z component being considered can be found in our Mathematica code (section χ⁽⁵⁾) and supple. Numbering is for counting purposes only and is different from the vectorized tensor position indexing in Fig. 1.

Comparison to an alternative approach

In the aforementioned NPEV approach, we mathematically express Neumann’s principle as an eigensystem and look for the generalized expression of a vectorized tensor that survives after symmetry operations, which encodes the surviving elements and their linear relationships. Another approach is to directly apply the mapping matrix to a vectorized nth-rank tensor T⁽ⁿ⁾ and solve 3ⁿ linear equations in MT = T based on Neumann’s principle, similar to the reported ones.^33,41 The Hirose and Hingel teams implemented the alternative approach and rationalized the $M=\frac{M_1+M_2}{2}$ as averaging multiple indistinguishable configurations when rotating the plane of incidence about the C_n principal axis at a 2π/n interval.³³ Here, we seek to provide a clearer mathematical picture: The reason for $M=\frac{M_1+M_2}{2}$ is that by averaging all the mapping operators, the eigenvectors of the M that satisfy MT = T should also be eigenvectors of individual M_i. In the NPEV approach, the number of independent elements is the number of eigenvectors with eigenvalues of 1. The number is not readily achieved in the alternative approach. One needs to either inspect all the output (Fig. S1) or enumerate all possible combinations to find the ones that belong to the total symmetric representations, based on character table.

CONCLUSION

We demonstrate a 3-step NPEV approach to derive non-vanishing tensor elements of symmetric systems using tensor-based linear algebra. Although we specifically discussed its utility in nonlinear optics, in principle, such an approach can be applied to deriving tensor element relationships of any physical property, such as conductivity, piezoelectricity, or elasticity. By describing the derivation process with a clear physical picture and rigorous mathematical calculation, the present approach will help further develope the theoretical framework and understand signals from higher order nonlinear optics, materials under magnetizations, and other chemical physical phenomena.

SUPPLEMENTARY MATERIAL

See the supplementary material for tabulations of non-vanishing elements and their linear relationships in various point groups for SFG, 2DIR, 2D-SFG (fifth-rank tensor, χ⁽⁴⁾, fourth-order susceptibilities), and 3D-IR (sixth-rank tensor, χ⁽⁵⁾, fifth-order susceptibilities) applications.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Zishan Wu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Wei Xiong: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.