Second Harmonic Generation

In the class lectures, we focus on Linear, Isotropic and Homogeneous (LIH) dielectrics. Linearity is a good approximation for weak electric fields while isotropy would hold for non-crystalline (amorphous) materials. For crystalline materials, isotropy is clearly violated as all directions are not identical in a crystal. So for weak fields, the polarization $\mathbf{P}$ is related to the applied electric field $\mathbf{E}$ in the following way:

(1)
\begin{align} P_i = \epsilon_0\left(\chi^{(1)}_{ij}\ E_j \right)\ , \end{align}

where $\chi^{(1)}_{ij}$ is a rank two tensor and the repeated(or dummy) index ($j$ in this case) is summed over. The number of independent components is determined by the symmetries of the crystal. Isotropy implies that $\chi^{(1)}_{ij} = \chi \delta_{ij}$ thereby recovering the single function that was introduced in the context of LIH dielectrics. Homogeneity implies that the tensor is a constant independent of location.

For a non-linear dielectric, one defines the following nonlinear susceptibilities1:

(2)
\begin{align} P_i = \epsilon_0\left(\chi^{(1)}_{ij}\ E_j + \chi^{(2)}_{ijk}\ E_j E_k + \chi^{(3)}_{ijkl}\ E_j E_k E_l +\cdots\right) \end{align}

where the rank $(m+1)$ tensors $\chi^{(m)}$ for $m=2$ represents quadratic non-linearity and cubic nonlinearity for $m=3$. These nonlinearities become prominent in strong (possibly, time-independent) electric fields, say for instance, as the ones generated by light from a laser. The quadratic and cubic terms leads to the generation of light at frequencies that are double and triple the frequency of the incident light. These are called higher harmonics and more specifically, the second and third harmonics. For instance, one can convert light from a Ti-Sapphire laser with wavelength 800nm (infrared) to light with wavelength 400nm (violet) which is in the visible range due to a non-zero $\chi^{(2)}$ in a material. However, for $\chi^{(2)}$ to be non-zero, the crystal that should not possess inversion symmetry. (Why?)

## SHG at IITM

Below we show two examples of Second Harmonic Generation (SHG) from the Optics Lab of Prof. Prem Bisht2 (one of the teachers in this course) at the Department of Physics, IITM.

The above photo shows a second harmonic crystal BBO (Barium Borate) cut at 29.6 degrees from its optical axis, mounted on teflon which is fitted on a black, square shaped crystal mount from a company "PROMPT". The crystal is 2 mm thick with a diameter of 3 mm. The beam is from a femtosecond laser (of wavelength 800 nm — just outside the visible range) falls on the front side of this crystal with a focused beam with a beam waist of about 2 mm. On the far side , a blue streak (400 nm) with a whitish spot is seen on a paper screen kept at about 10 cm from the crystal . The whitish spot is actually blue but over-exposure of the photographic plate gives a whitish impression. Believe it or not, the red colour at the crystal is due to the laser pulse of 800 nm. This is due to requirement of 'large wavelength range' for generation of the short burst of femtosecond light. The Gaussian pulse has an edge (at lower wavelength) which is close to the visible region of our eye and it gives a wrong impression that the 'red light' is weaker than the 'blue'. Actually it is other way around with the high power in the invisible infrared region!

Similarly, in the photo above, we can also see green SHG from the fundamental of a Nd:YAG laser (1064 nm - infrared and hence not visible). The multiple images are due to diffraction.

These lasers are available at the Department of Physics and are being used for the fundamental research in optics, spectroscopy and photonics. You are welcome to visit and/or work in these labs!