What is the essence of **Second Harmonic Generation** (SHG)?

Let's assume we have a laser beam at a given wavelength impinging on a non-linear crystal. If conditions are properly chosen (see below), we observe not only the original laser beam after the non-linear crystal, but a second laser beam having a different color. Closer inspection yields that the additional beam has exactly twice the frequency or half the wavelength of the original laser beam. Therefore, we talk about the second harmonic of the first one. But how does this come about?

Usually an applied electric field will produce a polarization in a material which is proportional to the electric field itself. If, however, the electric field gets larger - fields of the required magnitude can easily be generated by lasers, it can produce a contribution to the polarization which is proportional to the square of the electric field. Thus, the polarization reacts in a non-linear way to the electric field. The polarization in turn then produces electric field components at additional frequencies. The effect is comparable to a stereo system with the volume turned up. At some point the amplifiers will react in a non-linear fashion and additional frequencies occur as a result: the sound is distorted.

Assuming that laser radiation made up of two different frequencies is impinging on the crystal, one can show by use of the trigonometric addition theorems that due to the non-linear interaction different cases may occur as a result on the non-linear interaction: second harmonic generation of either frequency, sum frequency generation (SFG), difference frequency generation (DFG) and optical rectification (OR). The exact process depends on (1) the type of non-linear crystal (2) the geometry at which the crystal is cut (3) polarization of the input beams.

Two other conditions govern the process: (1) conservation of energy and (2) conservation of momentum which is sometimes called phase matching.

Conservation of energy can be understood in a particle picture quite easily. Two photons at the fundamental frequency are annihilated and one at the second harmonic is created. Thus, the overall energy of the process is conserved.

In the particle picture of SHG, momentum conservation demands that the two momenta of the annihilated photons add up to the momentum of the created photon at the second harmonic. In the wave picture momentum conservation is referred to as the phase matching condition. The phases of the second harmonic wave must be in phase throughout the crystal such that the different parts interfere constructively at the end of the crystal and a macroscopic wave at the second harmonic emerges form the crystal. In a collinear, type-I configuration (i.e. when the fundamental and second harmonic propagates collinearly through the crystal) the phase matching condition is identical to the condition that the indices of refraction of the fundamental and second harmonic wave are identical, which means that the phase velocities of the beams are identical.

Phase matching is only possible in birefringent materials, i.e. materials whose index of refraction depends on the polarization and direction of propagation of light. One distinguishes two types of phase matching: type-I and type-II, respectively. In type-I processes the two photons at the fundamental frequency have the same polarization, in a type-II process the two photons have orthogonal polarization.

Photograph of light at different wavelength/frequencies dispersed by the prism in the foreground. The green dot is originating from a green laser which is pumping a Ti:sapphire laser producing the red spot. The brighter blue spot is produced by second harmonic generation (SHG) of the fundamental red light; the faint blue spot to the left is the third harmonic of the red dot. It was produced by sum frequency generation (SFG) of the red and blue radiation.

The efficiency of SHG scales quadratically with the intensity the input beams and assuming interaction of Gaussian beams linearly with the length of the non-linear crystal. A potential problem is the so called walk-off between the different interacting beams. Walk-off is a feature of birefringent crystals and referrs to the fact that in general for the so-called extra-ordinary ray phase front and direction of energy transport are tilted with respect to each other.

- R.W. Boyd,
*Non-linear Optics, Academic Press*, (1992, San Diego) - Y.R. Shen,
*The Principles of Nonlinear Optics*, Wiley, (1984, New York) - A. Ashkin, G.D. Boyd, J.M. Dziedzic,
*Resonant optical second harmonic generation and mixing*, IEEE J. QE**2**(1966), p. 109-124 - G.D. Boyd and D.A. Kleinman,
*Parametric interaction of focussed Gaussian light beams*, J. Appl. Phys.**39**(1968) pp. 3597-3637 - R.L. Byer,
*Parametric Oscillators and non-linear Materials*in Nonlinear Optics - Proceedings of the 16th Scottish Universities Summer School in Physics, 1975, - P.G. Harper and B.S. Wherrett (eds), Academic Press (1977, New York), pp. 47-160
- SNLO - software written by A. Smith to calculate many aspects of SHG and more

Prof. Dr. Thomas Walther

Laser und Quantenoptik

Institut für Angewandte Physik

Fachbereich 05 - Physik

Technische Universität Darmstadt

Schlossgartenstr. 7

D-64289 Darmstadt

+49 6151 16-20831 (Sekretariat)

+49 6151 16-20834 (Fax)

Thomas.Walther@physik.tu-...

06.03.17