## General symmetries and simplifications In [[Classical origin of optical nonlinearity]], we derived the second order nonlinear susceptibility with permutation method with similar approach in derivation of Lorentz model. In the derivation we did not (or say cannot) determine the symmetry property of the system. Consider the most general case, three distinct frequencies, $\omega_{1},\ \omega_{2},\ \omega_{3}$ and $\omega_{3}=\omega_{1}+\omega_{2}$. And subscripts $i,\ j,\ k$ are Cartesian components. From the [[Nonlinear susceptibility|definition of polarization]], we have $P_i(\omega_n + \omega_m) = \varepsilon_0 \sum_{jk} \sum_{(nm)} \chi_{ijk}^{(2)}(\omega_n + \omega_m, \omega_n, \omega_m) E_j(\omega_n) E_k(\omega_m)$ This leads to twelve tensor quantities in total, namely, $\chi_{ijk}^{(2)}(\omega_1, \omega_3, -\omega_2), \quad \chi_{ijk}^{(2)}(\omega_1, -\omega_2, \omega_3), \quad \chi_{ijk}^{(2)}(\omega_2, -\omega_1, \omega_3), $ $ \chi_{ijk}^{(2)}(\omega_3, \omega_1, \omega_2), \quad \chi_{ijk}^{(2)}(\omega_3, \omega_2, \omega_1),\quad \chi_{ijk}^{(2)}(\omega_2, \omega_3, -\omega_1)$ and those with negative frequencies >[!Note]- >One may count possible combinations by defining the output frequencies. For $\omega_{3}$, possible combinations could only be $\chi_{ijk}^{(2)}(\omega_3, \omega_1, \omega_2)$ and $\chi_{ijk}^{(2)}(\omega_3, \omega_2, \omega_1)$; For $\omega_{2}$, they are $\chi_{ijk}^{(2)}(\omega_2, -\omega_1, \omega_3)$ and $\chi_{ijk}^{(2)}(\omega_2, \omega_3, -\omega_1)$; For $\omega_{1}$, they are $\chi_{ijk}^{(2)}(\omega_1, \omega_3, -\omega_2)$ and $\chi_{ijk}^{(2)}(\omega_1, -\omega_2, \omega_3)$. >Then count those negative frequencies. Without further simplification, the permutation on Cartesian indices would give $3^{3}=27$ different terms, so the total number of combinations would be $27\times 12=324$. In this chapter, we will dig into how these tensor quantities look like and simplify them to something we can apply in actual experiments or applications. ### Reality of the field: simplification on negative frequencies The electric fields and our polarization response are physically, measurable quantities, therefore, they are real. For input field $\omega_{n}$ and $\omega_{m}$ and SFG, the polarization is $\tilde{P}_i(\mathbf{r}, t) = P_i(\omega_n + \omega_m) e^{-i(\omega_n + \omega_m)t} + P_i(-\omega_n - \omega_m) e^{i(\omega_n + \omega_m)t}$ And if this $\tilde{P}_i(\mathbf{r}, t)$ is real, then $P_i(\omega_n + \omega_m)$ and $P_i(-\omega_n - \omega_m)$ must satisfy $P_i(-\omega_n - \omega_m) = P_i(\omega_n + \omega_m)^*$ >[!Note]- >This could be a conclusion that people commonly apply, but here we show the proof: >Given a real-valued function $f(t)$, expressed as a sum of positive and negative frequency components: >$f(t) = A(\omega) e^{-i\omega t} + B(\omega) e^{i\omega t},$ >where $A(\omega)$ and $B(\omega)$ are complex-valued functions of frequency $\omega$. >**Claim**: If the function $f(t)$ is real, then the complex amplitudes $A(\omega)$ and $B(\omega)$ must satisfy the following relation: >$B(\omega) = A(\omega)^*,$ >where $A(\omega)^*$ denotes the complex conjugate of $A(\omega)$. >Proof: >1. Express the complex amplitudes in terms of real and imaginary parts: >Let $A(\omega) = a(\omega) + i b(\omega)$ and $B(\omega) = c(\omega) + i d(\omega)$, where $a(\omega), b(\omega), c(\omega), d(\omega)$ are real-valued functions of $\omega$. >2. Expand the expression for $f(t)$: >Using the Euler's formula for the exponentials: >$e^{-i\omega t} = \cos(\omega t) - i \sin(\omega t), \quad e^{i\omega t} = \cos(\omega t) + i \sin(\omega t),$ >substitute into $f(t)$: >$f(t) = (a(\omega) + i b(\omega)) (\cos(\omega t) - i \sin(\omega t)) + (c(\omega) + i d(\omega)) (\cos(\omega t) + i \sin(\omega t))$ >3. Simplify the expression: >Expanding both terms: >$f(t) = [a(\omega) \cos(\omega t) - b(\omega) \sin(\omega t)] - i [b(\omega) \cos(\omega t) + a(\omega) \sin(\omega t)]$ >$+ [c(\omega) \cos(\omega t) - d(\omega) \sin(\omega t)] + i [d(\omega) \cos(\omega t) + c(\omega) \sin(\omega t)]$ >Group the real and imaginary parts, >$f(t) = [a(\omega) + c(\omega)] \cos(\omega t) - [b(\omega) + d(\omega)] \sin(\omega t) + i \left\{[d(\omega) - b(\omega)] \cos(\omega t) + [c(\omega) - a(\omega)] \sin(\omega t)\right\}$ >4. Impose the condition that $f(t)$ is real: >For $f(t)$ to be real, the imaginary part must be zero. Therefore, we require: >$d(\omega) = b(\omega), \quad c(\omega) = a(\omega)$ >5. Conclude the relation between $A(\omega)$ and $B(\omega)$: >From the above, we have: >$B(\omega) = c(\omega) + i d(\omega) = a(\omega) + i b(\omega) = A(\omega)^*$ >Thus, $B(\omega) = A(\omega)^*$, as required. Similarly, for electric field, we have $E_j(-\omega_n) = E_j(\omega_n)^*, \quad E_k(-\omega_m) = E_k(\omega_m)^*$ If we substitute these results of negative frequencies into $P_i(\omega_n + \omega_m) = \varepsilon_0 \sum_{jk} \sum_{(nm)} \chi_{ijk}^{(2)}(\omega_n + \omega_m, \omega_n, \omega_m) E_j(\omega_n) E_k(\omega_m)$ we may have that for susceptibility with negative frequencies, there should be $\chi_{ijk}^{(2)}(-\omega_n - \omega_m, -\omega_n, -\omega_m) = \chi_{ijk}^{(2)}(\omega_n + \omega_m, \omega_n, \omega_m)^*$ Positive and negative frequencies are related. ### Intrinsic permutation symmetry: changing the order of input fields does not matter As presented in [[Classical origin of optical nonlinearity]] and [[Nonlinear susceptibility]], the intrinsic permutation symmetry indicates that the nonlinear susceptibility should not be different when simultaneously interchange frequency arguments and their Cartesian indices. For second order susceptibility, this means $\chi^{(2)}_{ijk}(\omega_n + \omega_m, \omega_n, \omega_m) = \chi^{(2)}_{ikj}(\omega_n + \omega_m, \omega_m, \omega_n)$ It is equivalent saying that the input field does not have an order. Intrinsic permutation symmetry is a fundamental property of the nonlinear susceptibilities, which arises from the principles of time invariance and causality, which applies universally ([[The elements of nonlinear optics|The elements of nonlinear optics, page 122]]). Note that this symmetry condition is purely out of convenience, alternative (unintuitive) definition could exist and does not violate actual physical phenomena. ### Reality of susceptibilities: additional properties for lossless medium Similarly to the linear susceptibilities, for lossless medium the susceptibilities are purely real. This could be derived by quantum mechanical expression. This means $\chi_{ijk}^{(2)}(-\omega_n - \omega_m, -\omega_n, -\omega_m) = \chi_{ijk}^{(2)}(\omega_n + \omega_m, \omega_n, \omega_m)^*=\chi_{ijk}^{(2)}(\omega_n + \omega_m, \omega_n, \omega_m)$ ### Full permutation symmetry: time reversal symmetry for lossless medium Because for lossless medium, time reversal symmetry is satisfied, we may even interchange between input and output fields. This means for second order susceptibilities, we have $\chi^{(2)}_{ijk}(\omega_n + \omega_m, \omega_n, \omega_m) = \chi^{(2)}_{jik}(-\omega_n, -(\omega_n + \omega_m), \omega_m)$ and from the reality of $\chi^{(2)}$ for lossless medium, changing frequencies does not change the value, $\chi^{(2)}_{ijk}(\omega_n + \omega_m, \omega_n, \omega_m) = \chi^{(2)}_{jik}(\omega_n, \omega_n + \omega_m, -\omega_m)=\chi^{(2)}_{kij}(\omega_m, \omega_n + \omega_m, -\omega_n)$ This can be derived from quantum mechanical expression of susceptibilities, or use field energy density. Check the book [[Nonlinear Optics]] for the derivation. ### Kleinman's symmetry: simplification under low frequency dependence In many cases, the input frequencies are small and much smaller than the lowest resonance frequency of the system. Under such condition, we can consider the nonlinear susceptibilities being independent with frequency, or say its dispersionless. Also, the medium is necessarily lossless when applied field frequencies are significantly smaller than $\omega_{0}$. Therefore, we have full permutation system simultaneously. This means we we can permutate **frequencies and indices** as we want. >[!Notice] >Although in previous and the following expressions, $i,\ j,\ k$ can be assigned to any Cartesian components. But in the expressions, they are **preassigned**. For example, if one assign $i=x,\ j=y,\ k=z$. Then in the expression like $\chi^{(2)}_{ijk}(\omega_n + \omega_m, \omega_n, \omega_m) =\chi^{(2)}_{kij}(\omega_m, \omega_n + \omega_m, -\omega_n)$, the indices keep being the same. This equality only establish for $\chi^{(2)}_{xyz}$ and $\chi^{(2)}_{zxy}$, and does not involve $\chi^{(2)}_{xxy}$ or $\chi^{(2)}_{yyz}$. >In short, $xyz$ and $xxz$ correspond to different values. Though under Kleinman symmetry, $xyz$ and $zxy$ are the same. For example, we have $\chi^{(2)}_{ijk}(\omega_n + \omega_m, \omega_n, \omega_m)=\chi^{(2)}_{ikj}(\omega_n + \omega_m, \omega_n, \omega_m)=\chi^{(2)}_{jik}(\omega_n + \omega_m, \omega_n, \omega_m)=\chi^{(2)}_{jki}(\omega_n + \omega_m, \omega_n, \omega_m)=\chi^{(2)}_{kij}(\omega_n + \omega_m, \omega_n, \omega_m)=\chi^{(2)}_{kji}(\omega_n + \omega_m, \omega_n, \omega_m)$ ### Contracted notation: $3\times 6$ matrix for 2nd order susceptibility >[!Notice] >Use of contracted notation does not require the medium being dispersionless, and does not need Kleinman's symmetry. It is perfectly fine for an lossy medium with dispersion. Just under this case, independent variables are 18 (only allow interchange of input fields, without intrinsic permutation, this number would be 27). If assume Kleinman's symmetry, the total independent variables are 10. Define a tensor whose components are $d$ and $d_{ijk} = \frac{1}{2} \chi_{ijk}^{(2)}$ Then we have the factor two before susceptibilities, $P_i(\omega_n + \omega_m) = \varepsilon_0 \sum_{jk} \sum_{(nm)} 2 d_{ijk} E_j(\omega_n) E_k(\omega_m)$ Then assign combinations of $jk$ to different numbers $l$. Here $1,\ 2,\ 3$ correspond to $x,\ y,\ z$ (and this is just how this notation works). $\begin{array}{c c c c c c c} jk: & 11 & 22 & 33 & 23, 32 & 31, 13 & 12, 21 \\ l: & 1 & 2 & 3 & 4 & 5 & 6 \end{array}$ We get the $d_{il}$ as $d_{il} = \begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{bmatrix}$ > [!Note] > If Kleinman symmetry holds, we can permutate the indices without worrying about frequencies. This means > $d_{12}=d(122)=d(212,221)=d_{26}$ > similarly, > $d_{14}=d(123,132)=d(231,213)=d_{25}=d(312,321)=d_{36}$ > $d_{24}=d(223,232)=d(322)=d_{32}$ > $d_{13}=d(133)=d(313,331)=d_{35}$ > $\cdots$ > After removing repeated variables, $d_{il}$ matrix now becomes > $d_{il} = \begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{16} & d_{22} & d_{23} & d_{24} & d_{14} & d_{12} \\ d_{15} & d_{24} & d_{33} & d_{23} & d_{13} & d_{14} \end{bmatrix}$ For SHG, the polarization components are $\begin{bmatrix} P_x(2\omega) \\ P_y(2\omega) \\ P_z(2\omega) \end{bmatrix} = 2 \varepsilon_0 \begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{bmatrix} \begin{bmatrix} E_x(\omega)^2 \\ E_y(\omega)^2 \\ E_z(\omega)^2 \\ 2E_y(\omega)E_z(\omega) \\ 2E_x(\omega)E_z(\omega) \\ 2E_x(\omega)E_y(\omega) \end{bmatrix}$ For SFG, we have an additional factor 2, so $\begin{bmatrix} P_x(\omega_3) \\ P_y(\omega_3) \\ P_z(\omega_3) \end{bmatrix} = 4 \varepsilon_0 \begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{bmatrix} \times \begin{bmatrix} E_x(\omega_1)E_x(\omega_2) \\ E_y(\omega_1)E_y(\omega_2) \\ E_z(\omega_1)E_z(\omega_2) \\ E_y(\omega_1)E_z(\omega_2) + E_z(\omega_1)E_y(\omega_2) \\ E_x(\omega_1)E_z(\omega_2) + E_z(\omega_1)E_x(\omega_2) \\ E_x(\omega_1)E_y(\omega_2) + E_y(\omega_1)E_x(\omega_2) \end{bmatrix}$ ### Effective value of $d$: turning a matrix to one single value If the geometry is fixed (i.e., fixed propagation direction and polarization), one could express the nonlinear polarization with effective $d_{eff}$ with scalar relation, like for SFG, $P(\omega_3) = 4 \varepsilon_0 d_{\text{eff}} E(\omega_1) E(\omega_2)$ and for SHG, $P(2\omega) = 2 \varepsilon_0 d_{\text{eff}} E(\omega)^2$ where $E(\omega) = |\mathbf{E}(\omega)|$ and $P(\omega) = |\mathbf{P}(\omega)|$. Actual calculation requires knowledge on the crystals systems and phase matching conditions. This will be discussed in [[Spatial symmetry of nonlinear susceptibility]] and [[Phase matching]]. >[!Example] >For a negative uniaxial crystal of crystal class $3m$, type I conditions, >$d_{\text{eff}} = d_{31} \sin \theta - d_{22} \cos \theta \sin 3\phi$ >and type II conditions, >$d_{\text{eff}} = d_{22} \cos^2 \theta \cos 3\phi$ >This example is given in page 40 in [[Nonlinear Optics]].