Second harmonic generation

Second harmonic generation (SHG; also called frequency doubling) is a nonlinear optical process, in which photons interacting with a nonlinear material are effectively "combined" to form new photons with twice the energy, and therefore twice the frequency and half the wavelength of the initial photons.

Second harmonic generation was first demonstrated by P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich at the University of Michigan, Ann Arbor, in 1961. The demonstration was made possible by the invention of the laser, which created the required high intensity monochromatic light. They focused a ruby laser with a wavelength of 694 nm into a quartz sample. They sent the output light through a spectrometer, recording the spectrum on photographic paper, which indicated the production of light at 347 nm. Famously, when published in the journal Physical Review Letters (citation below), the copy-editor mistook the dim spot (at 347 nm) on the photographic paper as a speck of dirt and removed it from the publication.

In recent years, SHG has been extended to biological applications. Researchers Leslie Loew and Paul Campagnola at the University of Connecticut have applied SHG to imaging of molecules that are intrinsically second-harmonic-active in live cells, such as collagen, while Joshua Salafsky is pioneering the technique's use for studying biological molecules by labeling them with second-harmonic-active tags, in particular as a means to detect conformational change at any site and in real time. SH-active unnatural amino acids can also be used as probes.

Derivation of Second Harmonic Generation
The simplest case for analysis of second harmonic generation is a plane wave of amplitude $$E(\omega)$$ traveling in a nonlinear medium in the direction of its $$k$$ vector. A polarization is generated at the second harmonic frequency


 * $$ P(2\omega) = 2\epsilon_0d_{eff}(2\omega ;\omega,\omega)E^2(\omega), \,$$

where $$2d_{eff}=\chi^{(2)}$$.The wave equation at $$2\omega$$ (assuming negligible loss and asserting the slowly varying envelope approximation) is


 * $$\frac{\partial E(2\omega)}{\partial z}=-\frac{i\omega}{n_{2\omega}c}d_{eff}E^2(\omega)e^{i\Delta k z}$$

where $$\Delta k=k(2\omega)-2k(\omega)$$.

At low conversion efficiency ($$E(2\omega)<<E(\omega)$$ the amplitude $$E(\omega)$$ remains essentially constant over the interaction length, $$l$$. Then, with the boundary condition $$E(2\omega,z=0)=0$$ we get

$$E(2\omega,z=l)=-\frac{i\omega d_{eff}}{n_{2\omega}c}E^2(\omega)\int_0^l{e^{i\Delta k z}}=-\frac{i\omega d_{eff}}{n_{2\omega}c}E^2(\omega)l\frac{\sin{\Delta k l/2}}{\Delta k l/2}e^{i\Delta k l/2}$$

In terms of the optical intensity, $$I=n/2\sqrt{\epsilon_0/\mu_0}|E|^2$$, this is,

$$I(2\omega,l)=\frac{2\omega^2d^2_{eff}l^2}{n_{2\omega}n_{\omega}^2c^3\epsilon_0}(\frac{\sin{(\Delta k l/2)}}{\Delta k l/2})^2I^2(\omega)$$

This intensity is maximized for the phase matched condition $$\Delta k=0$$. If the process is not phase matched, the driving polarization at $$2\omega$$ goes in and out of phase with generated wave $$E(2\omega)$$ and conversion oscillates as $$\sin{(\Delta k l/2)}$$. The coherence length is defined as $$l_c=\frac{\pi}{\Delta k}$$. It does not pay to use a nonlinear crystal much longer than the coherence length. (Periodic poling and Quasi-phase-matching provide another approach to this problem.)

Second Harmonic Generation with Depletion
When the conversion to second harmonic becomes significant it becomes necessary to include depletion of the fundamental. One then has the coupled equations:

$$\frac{\partial E(2\omega)}{\partial z}=-\frac{i\omega}{n_{2\omega}c}d_{eff}E^2(\omega)e^{i\Delta k z}$$,

$$\frac{\partial E(\omega)}{\partial z}=-\frac{i\omega}{n_{\omega}c}d_{eff}^*E(2\omega)E^*(\omega)e^{-i\Delta k z}$$,

where $$*$$ denotes the complex conjugate. For simplicity, assume phase matched generation ($$\Delta k=0$$). Then, energy conservation requires that

$$n_{2\omega}[E^*(2\omega)\frac{\partial E(2\omega)}{\partial z}+c.c.]=-n_\omega[E(\omega)\frac{\partial E^*(\omega)}{\partial z}+c.c.]$$

where $$c.c.$$ is the complex conjugate of the other term, or

$$n_{2\omega}|E(2\omega)|^2+n_\omega|E(\omega)|^2=n_{2\omega}E_0^2$$.

Now we solve the equations with the premise

$$E(\omega)=|E(\omega)|e^{i\phi(\omega)}$$

$$E(2\omega)=|E(2\omega)|e^{i\phi(2\omega)}$$

We get

$$\frac{d|E(2\omega)|}{dz}=-\frac{i\omega d_{eff}}{n_\omega c}[E_0^2-|E(2\omega)|^2]e^{2i\phi(\omega)-i\phi(2\omega)}$$

$$\int_0^{|E(2\omega)|l}{\frac{d|E(2\omega)}{E_0^2-|E(2\omega)|^2}}=-\int_0^l{\frac{i\omega d_{eff}}{n_\omega c}dz}$$

Using

$$\int{\frac{dx}{a^2-x^2}}=\frac{1}{a}\tanh^-1{\frac{x}{a}}$$

we get

$$|E(2\omega)|_{z=l}=E_0\tanh{(\frac{-iE_0l\omega d_{eff}}{n_\omega c}e^{2i\phi(\omega)-i\phi(2\omega)})}$$

If we assume a real $$d_{eff}$$, the relative phases for real harmonic growth must be such that $$e^{2i\phi(\omega)-i\phi(2\omega)}=i$$. Then

$$I(2\omega,l)=I(\omega,0)\tanh^2(\frac{E_0\omega d_{eff}l}{n_\omega c})$$

or

$$I(2\omega,l)=i(\omega,0)tanh^2{(\Gamma l)}$$,

where $$\Gamma=\omega d_{eff}E_0/nc$$. From $$I(2\omega,l)+I(\omega,l)=I(\omega,0)$$, it also follows that

$$I(\omega,l)=I(\omega,0)sech^2{(\Gamma l)}$$.

Types of SHG
Second harmonic generation occurs in two types, denoted I and II. In Type I SHG two photons having ordinary polarization with respect to the crystal will combine to form one photon with double the frequency and extraordinary polarization. In Type II SHG, two photons having orthogonal polarization will combine to form one photon with double the frequency and extraordinary polarization. For a given crystal orientation, only one of these type of SHG occurs.

Common Uses
Second harmonic generation is used by the laser enthusiast industry to make green 532nm lasers from a 808nm source. The source is converted to 1064nm by a YAG crystal, then fed through a KDP second harmonic crystal. This is capped by an infrared filter to prevent leakage of any infrared that would be harmful to the human eyes