## Spatial symmetry of nonlinear susceptibility Now we focus on some more realistic aspects. These $\chi^{(n)}$s are no longer illusionary constants floating in absolute space, but actual material properties related to real crystals. This will possibly make these $\chi^{(n)}$ expressions even more simplified. With the help of [[Index (group theory)|group theory]], we know that all crystals can be classified into 32 crystal classes depending on their point group symmetries. ### Spatial symmetry of first order susceptibilities First check linear susceptibilities. Define $\chi^{(1)}_{ij}$ in the same manner, $i$ is the output direction, and $j$ is the input field. Then for different (7 in total) crystal families, we have, Triclinic: $\begin{bmatrix} xx & xy & xz \\ yx & yy & yz \\ zx & zy & zz \end{bmatrix}$ Monoclinic: $\begin{bmatrix} xx & 0 & xz \\ 0 & yy & 0 \\ zx & 0 & zz \end{bmatrix}$ Orthorhombic: $\begin{bmatrix} xx & 0 & 0 \\ 0 & yy & 0 \\ 0 & 0 & zz \end{bmatrix}$ Tetragonal, Trigonal, Hexagonal: $\begin{bmatrix} xx & 0 & 0 \\ 0 & xx & 0 \\ 0 & 0 & zz \end{bmatrix}$ Cubic, isotropic: $\begin{bmatrix} xx & 0 & 0 \\ 0 & xx & 0 \\ 0 & 0 & xx \end{bmatrix}$ Tetragonal, Trigonal, Hexagonal crystals, for having one particular direction $z$, are called uniaxial crystals and exhibit some symmetries. Besides cubic and isotropic (glasses, liquids, etc.), other crystal exhibits anisotropic in linear optical properties and will cause [[Birefringence|birefringence]]. >[!Note] >Names of Bravais lattices. Images are adapted from [Crystal system (wiki page)](https://en.wikipedia.org/wiki/Crystal_system). > > <table> <thead> <tr> <th rowspan="2">Crystal family</th> <th rowspan="2">Lattice system</th> <th rowspan="2">Point group (Schönflies notation)</th> <th colspan="4">14 Bravais lattices</th> </tr> <tr> <th>Primitive (P)</th> <th>Base-centered (S)</th> <th>Body-centered (I)</th> <th>Face-centered (F)</th> </tr> </thead> <tbody> <tr align="center"> <th colspan="2">Triclinic (a)</th> <td>C_i</td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Triclinic.svg" alt="aP"></td> <!-- Add your image here --> <td></td> <td></td> <td></td> </tr> <tr align="center"> <th colspan="2">Monoclinic (m)</th> <td>C_2h</td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Monoclinic.svg" alt="mP"></td> <!-- Add your image here --> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Base-centered_monoclinic.svg" alt="mS"></td> <!-- Add your image here --> <td></td> <td></td> </tr> <tr align="center"> <th colspan="2">Orthorhombic (o)</th> <td>D_2h</td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Orthorhombic.svg" alt="oP"></td> <!-- Add your image here --> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Base-centered_orthorhombic.svg" alt="oS"></td> <!-- Add your image here --> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Body-centered_orthorhombic.svg" alt="oI"></td> <!-- Add your image here --> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Face-centered_orthorhombic.svg" alt="oF"></td> <!-- Add your image here --> </tr> <tr align="center"> <th colspan="2">Tetragonal (t)</th> <td>D_4h</td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Tetragonal.svg" alt="tP"></td> <!-- Add your image here --> <td></td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Body-centered_tetragonal.svg" alt="tI"></td> <!-- Add your image here --> <td></td> </tr> <tr align="center"> <th rowspan="2">Hexagonal (h)</th> <th>Rhombohedral</th> <td>D_3d</td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Rhombohedral.svg" alt="hR"></td> <!-- Add your image here --> <td></td> <td></td> <td></td> </tr> <tr align="center"> <th>Hexagonal</th> <td>D_6h</td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Hexagonal_latticeFRONT.svg" alt="hP"></td> <!-- Add your image here --> <td></td> <td></td> <td></td> </tr> <tr align="center"> <th colspan="2">Cubic (c)</th> <td>O_h</td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Cubic.svg" alt="cP"></td> <!-- Add your image here --> <td></td> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Cubic-body-centered.svg" alt="cI"></td> <!-- Add your image here --> <td><img src="https://publish-01.obsidian.md/access/849815ab1c9f15832c15c41b08c52998/%40_Excalidraw/Cubic-face-centered.svg" alt="cF"></td> <!-- Add your image here --> </tr> </tbody> </table> ^b95a01 ### Inversion symmetry and second order nonlinear susceptibilities As shown in [[Classical origin of optical nonlinearity#Centrosymmetric media#Second order]], the second order nonlinear susceptibility is zero. In the book [[Nonlinear Optics]], the author presented a more intuitive way by showing the polarization under applied optical fields assuming instantaneous response. (But if we consider [[Time response of nonlinear optical processes]], we will have non-instantaneous response.) Take SHG as an example, assume input field and polarization direction is determined, we may write $-{\tilde{P}}(t) = \varepsilon_0 \chi^{(2)} \left[ - {\tilde{E}}(t) \right]^2=\varepsilon_0 \chi^{(2)} \left[ {\tilde{E}}(t) \right]^2 = {\tilde{P}}(t)$ since we have inversion symmetry for centrosymmetric medium, namely changing the sign of input field will automatically change the sign of polarization. This means the only possibility is $\chi^{(2)}=0$ for this input field and polarization. With similarly manner other directions could be processed. ![[Drawing 2024-09-13 01.56.00.excalidraw.svg]] In the plot we write $\chi^{(2)} = 0$ or $\chi^{(2)} \neq 0$, actually this should be valid for any even order nonlinear response. Just from waveform and inversion symmetry, we can already see, that linear response should give same (type of) waveform for ${\tilde{P}}(t)=\varepsilon_0 \chi^{(1)} {\tilde{E}}(t)$. And for $\chi^{{(2)}}$, if the medium allows inversion symmetry, or say, it's centrosymmetric, then the response can only be like the third plot because ${\tilde{P}}(t)=\varepsilon_0 \chi^{(2)} \left[ {\tilde{E}}(t) \right]^2$. Only for non-centrosymmetric medium, we may have second order effect and the response looks like the last plot. For the centrosymmetric medium, the time-averaged response is zero, and for non-centrosymmetric ones it's not zero for the medium responds differently for upward and downward direction (+ and -). ### Spatial symmetry, contracted notation $d$ and tables for second order susceptibilities Preorganized table for crystal classes and nonvanishing terms in second order susceptibilities are provided in [[Nonlinear Optics#^02fa5e|Table 1.5.2, page 46]]. Some conventional plots for some crystal classes showing zeros and nonvanishing terms under contracted notation are provided in [[Nonlinear Optics#^02fa5e|Figure 1.5.3, page 47-48]]. Some $d_{il}$ values are given in [[Nonlinear Optics#^c5c8a4|Table 1.5.3, page 49]]. Retrospect our simplification, we have the number of independent variable reduced from **324**, the original value, to **81** with reality of fields and intrinsic permutation applied. If we have lossless medium, we will have full permutation and the number will be further reduced to **27**. If we assign the output direction and use the contracted notation, we will have **18** independent variables. If we applied Kleinman's symmetry, the number becomes even smaller to **10**. And finally, with spatial symmetry of the crystals, we could possibly further reduce the number, from no reduction at all (triclinic), to all elements are vanished ($m3$ cubic). >[!Example] >This is a very good example mentioned in the book and differentiating birefringence and second order susceptibilities. >Birefringence is from **isotropic**, not non-centrosymmetric. >Second order susceptibilities are from **non-centrosymmetric**. >Being in cubic system does not necessarily mean that the medium is centrosymmetric. One example is zinc-blende structure, like $\rm GaAs$. (we had this in [[Semiconductor crystals]]) >![[Drawing 2024-09-13 02.21.28.excalidraw.svg|300]] >It is non-centrosymmetric. (while diamond is centrosymmetric) >But cubic system is **isotropic**, this means it does not have birefringence. So we lose some phase matching methods. >Crystals other than cubic are anisotropic, so birefringence could be noticed. But always keep in mind that no birefringence does not mean no second order susceptibilities. >[!Notice] >All ferroelectric crystals are non-centrosymmetric, but non-centrosymmetric itself is not adequate providing permanent, spontaneous electric dipole. $\rm GaAs$ could also be an example, it does not have permanent dipole. >We might talk about ferroelectric or multiferroic later in the future. ### Symmetry of third order susceptibilities Check [[Nonlinear Optics#^00906a|Table 1.5.4]].