Nonlinear Optics & Harmonic Generation
Nonlinear Optics & Harmonic Generation
At high intensities, polarization becomes nonlinear: $P=\epsilon_0(\chi^{(1)}E+\chi^{(2)}E^2+\chi^{(3)}E^3+\cdots)$. The $\chi^{(2)}$ term (present only in non-centrosymmetric crystals) drives second-harmonic generation (SHG, $\omega\to2\omega$) and sum/difference-frequency mixing.
Efficient conversion requires phase matching: $\Delta k=k_{2\omega}-2k_\omega=0$. Achieved by birefringent angle tuning or quasi-phase-matching (QPM) via periodic poling.
SHG Power Scaling
Under perfect phase matching ($\Delta k=0$): $$P_{2\omega}\propto(\chi^{(2)})^2\,P_\omega^2\,L^2.$$ Power grows quadratically with pump power and crystal length.
Phase-Matching Condition
$$\Delta k=k_3-k_1-k_2=\frac{n_3\omega_3-n_1\omega_1-n_2\omega_2}{c}=0.$$ Satisfied by birefringent angle or temperature tuning, or by QPM using a reciprocal lattice vector.
Example 1
A 1064-nm Nd:YAG laser pumps a KDP crystal. What wavelength does SHG produce?
Solution: $\lambda_\text{SHG}=1064/2=532\,\text{nm}$ (green).
Example 2
Crystal length $L$ is doubled under phase matching. How does SHG output change?
Solution: $P_{2\omega}\propto L^2$, so doubling $L$ quadruples the SHG output.
Practice
- Why does $\chi^{(2)}$ vanish in centrosymmetric media?
- What is quasi-phase-matching and how does periodic poling achieve it?
- Name two applications of optical parametric amplification.
- How does SHG intensity scale with pump intensity?
Show Answer Key
1. $\chi^{(2)}$ is a third-rank tensor. Under inversion symmetry ($\mathbf{r}\to-\mathbf{r}$): $P_i^{(2)} = \epsilon_0\chi_{ijk}^{(2)}E_jE_k$. Under inversion, $\mathbf{P}$ and $\mathbf{E}$ both change sign: $-P_i = \chi_{ijk}(-E_j)(-E_k) = \chi_{ijk}E_jE_k = P_i$, so $P_i = -P_i = 0$. Hence $\chi^{(2)} = 0$ in centrosymmetric media.
2. Phase matching requires $\Delta k = k_{2\omega} - 2k_\omega = 0$, but normal dispersion makes $n(2\omega) > n(\omega)$. Periodic poling (alternating the sign of $\chi^{(2)}$ with period $\Lambda = 2\pi/\Delta k$) provides a reciprocal lattice vector $G = 2\pi/\Lambda$ that compensates: $\Delta k - G = 0$. This is quasi-phase-matching — less efficient per period but works at any wavelength by tuning $\Lambda$.
3. (1) Tunable mid-IR/THz generation (OPO/OPA for spectroscopy, sensing). (2) Quantum optics — generating entangled photon pairs (spontaneous parametric down-conversion). Also: ultrafast pulse amplification (chirped-pulse OPA), wavelength conversion in telecom.
4. SHG: $P^{(2)} \propto \chi^{(2)}E^2$, so $I_{2\omega} \propto |P^{(2)}|^2 \propto I_\omega^2$ (quadratic in pump intensity). Doubling the pump intensity quadruples the SHG output. At high conversion efficiency (depleted pump regime), the scaling saturates.