## Understanding parametric nonlinear optical processes Consider lossless and dispersionless medium. Some common nonlinear effects may happen. The main focus is 2nd and 3rd order nonlinear effect. ### Second harmonic generation ![[Drawing 2024-09-04 20.51.51.excalidraw.svg]] Consider a laser beam with electric field strength $\tilde{E}(t) = E e^{-i \omega t} + E^* e^{i \omega t} = E e^{-i \omega t} + \text{c.c.}$ >[!Question] >There is a problem related to intuitions that, using the plus complex conjugate form is mathematically reasonably, but what's the intuitions to write the field in this way. We do not have negative frequencies and wavevectors, so how can one understand the latter part? >>[!Answer] >>Of course one can understand it as purely mathematical simplification, or as the expansion of frequencies, just like in Fourier transform. But is there some specific meanings related to actual physical processes? These frequencies played a role in nonlinear processes. Then from the definition, we have $\tilde{P}^{(2)}(t)=\varepsilon_{0}\chi^{(2)}\tilde{E}(t)^{2}$ $\tilde{P}^{(2)}(t) = 2 \varepsilon_0 \chi^{(2)} E E^* + \left( \varepsilon_0 \chi^{(2)} E^2 e^{-i 2 \omega t} + \text{c.c.} \right)$ The 2nd order polarization contains two terms, the first one is frequency independent (a static field) and does not generate electromagnetic radiation, and it causes [optical rectification](https://en.wikipedia.org/wiki/Optical_rectification); the second one gives a $2\omega$ wave, namely second harmonic generation. >[!Example] >SHG is the technique that commercial green lasers, [Nd:YAG laser](https://en.wikipedia.org/wiki/Nd:YAG_laser) applied. It turn 1064 nm wavelength infrared laser light into the green 532 nm light. The energy-level diagram is shown below. The solid line is ground states, and the dashed lines are virtual states (namely not actual energy eigenlevels). ![[Drawing 2024-09-04 21.02.19.excalidraw.svg]] ### Sum and difference frequency generation Similarly, for the field, $\tilde{E}(t) = E_1 e^{-i \omega_1 t} + E_2 e^{-i \omega_2 t} + \text{c.c.}$ Insert into $\tilde{P}^{(2)}(t)=\varepsilon_{0}\chi^{(2)}\tilde{E}(t)^{2}$ we have $\tilde{P}^{(2)}(t) = \varepsilon_0 \chi^{(2)} \left[ E_1^2 e^{-2i \omega_1 t} + E_2^2 e^{-2i \omega_2 t} + 2E_1 E_2 e^{-i (\omega_1 + \omega_2) t} + 2E_1 E_2^* e^{-i (\omega_1 - \omega_2) t} + \text{c.c.} \right] + 2 \varepsilon_0 \chi^{(2)} \left[ E_1 E_1^* + E_2 E_2^* \right]$ or write in a compact form using $\tilde{P}^{(2)}(t) = \sum_{n} P(\omega_n) e^{-i \omega_n t}$ with all positive and negative frequencies. They are given by $\begin{aligned} P(2\omega_1) &= \varepsilon_0 \chi^{(2)} E_1^2 \\ P(-2\omega_1) &= \varepsilon_0 \chi^{(2)} E_1^{*2} \quad (\text{SHG}), \\ P(2\omega_2) &= \varepsilon_0 \chi^{(2)} E_2^2 \\ P(-2\omega_2) &= \varepsilon_0 \chi^{(2)} E_2^{*2} \quad (\text{SHG}), \\ P(\omega_1 + \omega_2) &= 2 \varepsilon_0 \chi^{(2)} E_1 E_2 \\ P(-\omega_1 - \omega_2) &= 2 \varepsilon_0 \chi^{(2)} E_1^{*} E_2^{*} \quad (\text{SFG}), \\ P(\omega_1 - \omega_2) &= 2 \varepsilon_0 \chi^{(2)} E_1 E_2^* \\ P(\omega_2 - \omega_1) &= 2 \varepsilon_0 \chi^{(2)} E_1^* E_2 \quad (\text{DFG}), \\ P(0) &= 2\varepsilon_0 \chi^{(2)} (E_1 E_1^* + E_2 E_2^*) \quad (\text{OR}). \end{aligned}$ >[!Notice] >For optical rectification, i.e., $P(0) = 2\varepsilon_0 \chi^{(2)} (E_1 E_1^* + E_2 E_2^*)$, although in the book we have a factor 2 and having a $\chi^{(2)}$ for both field ([[nonlinear optics Robert.W.Boyd.pdf#page=20&selection=58,0,301,0|nonlinear optics Robert.W.Boyd, page 20]]), which is derived from direct calculation. In actual experiments, people might write different input field separately. >>[!Note] >>One should realize that the factor 2 before SFG and OR have different origin. For SFG, we have two distinct input field, therefore we have this factor. However, for OR, although we have only one field like SHG, but $P(0)=P(-0)$, with no negative frequency, the polarization combines $EE^{*}$ and $E^{*}E$ so we have this factor. ^83ffdb Since the positive and negative frequencies are just complex conjugate, the negative expression are typically not written explicitly. Among these polarization, we call $P(\omega_1 + \omega_2) = 2 \varepsilon_0 \chi^{(2)} E_1 E_2$ the sum frequency generation, ![[Drawing 2024-09-04 23.56.40.excalidraw.svg]] and $P(\omega_1 - \omega_2) = 2 \varepsilon_0 \chi^{(2)} E_1 E_2^*$ difference frequency generation (also their complex conjugate). ![[Drawing 2024-09-04 23.58.11.excalidraw.svg]] > [!Notice] > Although looks similar, SFG and DFG are different process. Consider the energy conservation condition. For SFG, one have to have two photons satisfy the phase matching condition, then convert to one photon as the output. But for the DFG, the input photon would be $\omega_{1}$, this higher frequency photon will get destroyed and $\omega_{2}$, also the input frequency is created to satisfy $\omega_{3}=\omega_{1}-\omega_{2}$. Therefore, the presence of $\omega_{2}$ field stimulated this two-photon emission process. The DFG process could be considered as an amplification process, called optical parametric amplification. > It is possible that DFG happens even with no presence of $\omega_{2}$ field, but the probability is much smaller. This is called parametric fluorescence, or spontaneous parametric down conversion, SPDC. This method is applied in many quantum optics experiments to get photon pair. > >[!Notice] > >When looking at the geometry of interaction plot, **DO NOT TREAT IT AS A QUANTUM-MECHANICAL PROCESS!** The plot is for electric fields and the light-matter interaction relies the present of field and matter. The inputs are **fields** with these frequencies, rather than photons. This is not a photon interaction. After all nonlinear optics and susceptibility is still under the wave description. > >If consider $\omega_{1}$, $\omega_{2}$ as photons, then energy conservation would be invalid for DFG. > >>[!Question] > >>I honestly do not understand how classical fields are represented in QFT. >[!Example] >By placing a material with proper $\chi^{(2)}$ into a [[Introduction to optical resonators|resonator]] and under phase matching condition, we could get a broadly tunable device to have (any) $\omega_{2}$ as output (by changing phase matching condition, like rotate the crystal). Such device is called the optical parametric oscillator. And we call $\omega_{1}$ as the pump frequency, the desired output $\omega_{2}$ as signal and unwanted frequency $\omega_{3}$ as idler frequency. >![[Drawing 2024-09-05 14.11.11.excalidraw.svg]] ### Third order nonlinear effects The third order polarization is $\tilde{P}^{(3)}(t) = \varepsilon_0 \chi^{(3)} \tilde{E}(t)^3$ Consider the general form, $\tilde{E}(t) = E_1 e^{-i \omega_1 t} + E_2 e^{-i \omega_2 t} + E_3 e^{-i \omega_3 t} + \text{c.c.}$ $\tilde{E}(t)^3$ would generate the following frequencies, only positive frequencies are shown here but do not forget their $\rm c.c$.. Intensity dependent refractive index, or self/cross phase modulation: $\omega_1, \omega_2, \omega_3$ Third harmonic generation: $3\omega_1, 3\omega_2, 3\omega_3$ And other frequencies (non-degenerate and degenerate four wave mixing): $(\omega_1 + \omega_2 + \omega_3), (\omega_1 + \omega_2 - \omega_3),(\omega_1 + \omega_3 - \omega_2), (\omega_2 + \omega_3 - \omega_1)$ $(2\omega_1 \pm \omega_2), (2\omega_1 \pm \omega_3),(2\omega_2 \pm \omega_1), (2\omega_2 \pm \omega_3), (2\omega_3 \pm \omega_1), (2\omega_3 \pm \omega_2)$ Write in similar format, we have $\tilde{P}^{(3)}(t) = \sum_{n} P(\omega_n) e^{-i \omega_n t}$ For $\omega_1, \omega_2, \omega_3$, namely SPM and CPM $\begin{aligned} P(\omega_1) &= \varepsilon_0 \chi^{(3)} (3E_1 E_1^* + 6E_2 E_2^* + 6E_3 E_3^*) E_1, \\ P(\omega_2) &= \varepsilon_0 \chi^{(3)} (6E_1 E_1^* + 3E_2 E_2^* + 6E_3 E_3^*) E_2, \\ P(\omega_3) &= \varepsilon_0 \chi^{(3)} (6E_1 E_1^* + 6E_2 E_2^* + 3E_3 E_3^*) E_3, \end{aligned}$For THG, $\begin{aligned} P(3\omega_1) &= \varepsilon_0 \chi^{(3)} E_1^3, \\ P(3\omega_2) &= \varepsilon_0 \chi^{(3)} E_2^3, \\ P(3\omega_3) &= \varepsilon_0 \chi^{(3)} E_3^3, \end{aligned}$ For non-degenerate FWM, $\begin{aligned} P(\omega_1 + \omega_2 + \omega_3) &= 6\varepsilon_0 \chi^{(3)} E_1 E_2 E_3, \\ P(\omega_1 + \omega_2 - \omega_3) &= 6\varepsilon_0 \chi^{(3)} E_1 E_2 E_3^*, \\ P(\omega_1 + \omega_3 - \omega_2) &= 6\varepsilon_0 \chi^{(3)} E_1 E_3 E_2^*, \\ P(\omega_2 + \omega_3 - \omega_1) &= 6\varepsilon_0 \chi^{(3)} E_2 E_3 E_1^*, \end{aligned}$ For degenerate FWM, $\begin{aligned} P(2\omega_1 + \omega_2) &= 3\varepsilon_0 \chi^{(3)} E_1^2 E_2, \\ P(2\omega_1 + \omega_3) &= 3\varepsilon_0 \chi^{(3)} E_1^2 E_3, \\ P(2\omega_2 + \omega_1) &= 3\varepsilon_0 \chi^{(3)} E_2^2 E_1, \\ P(2\omega_2 + \omega_3) &= 3\varepsilon_0 \chi^{(3)} E_2^2 E_3, \\ P(2\omega_3 + \omega_1) &= 3\varepsilon_0 \chi^{(3)} E_3^2 E_1, \\ P(2\omega_3 + \omega_2) &= 3\varepsilon_0 \chi^{(3)} E_3^2 E_2, \\ P(2\omega_1 - \omega_2) &= 3\varepsilon_0 \chi^{(3)} E_1^2 E_2^*, \\ P(2\omega_1 - \omega_3) &= 3\varepsilon_0 \chi^{(3)} E_1^2 E_3^*, \\ P(2\omega_2 - \omega_1) &= 3\varepsilon_0 \chi^{(3)} E_2^2 E_1^*, \\ P(2\omega_2 - \omega_3) &= 3\varepsilon_0 \chi^{(3)} E_2^2 E_3^*, \\ P(2\omega_3 - \omega_1) &= 3\varepsilon_0 \chi^{(3)} E_3^2 E_1^*, \\ P(2\omega_3 - \omega_2) &= 3\varepsilon_0 \chi^{(3)} E_3^2 E_2^*\end{aligned}$ >[!Notice] >The number before, ($1,\ 3, \ 6$ ) is the number of distinct permutations of the field frequencies that contribute to the term. For THG, the geometry and energy level description would be ![[Drawing 2024-09-05 18.13.02.excalidraw.svg]] In this case one could imagine there is only one input optical field. the $\omega$ output is essential self- (and cross-) phase modulation. But in this context, it is the intensity dependent refractive index. And the refractive index would be $n=n_{0}+n_{2} I$ this $n_{0}$ is the refractive index we typically talk about under low intensity field, while $n_{2}$ is the intensity dependent term, $I$ is the intensity of the incident wave, $n_2 = \frac{3}{4 n_0^2 \varepsilon_0 c} \chi^{(3)}$ Refractive dependent refractive index may cause self-focusing, self-trapping and beam breakup. This is talked in chapter 7 of the book. For general case (four wave mixing), the geometry would be ![[Drawing 2024-09-05 18.22.55.excalidraw.svg]] If we have sum frequency generations, or ![[Drawing 2024-09-05 18.24.56.excalidraw.svg]] If we have difference frequency generations. >[!Info] >For non-parametric process, see [[Parametric and nonparametric process]].