During my time I spent as student of physics in Austin, Texas, USA, I had the opportunity to conduct experiments with a laser system in the laboratory of Prof. Downer. In these experiments, I investigated nonlinear interaction of light and matter on crystalline surfaces of semiconductors. I want to sketch the basic ideas of the conducted experiments below in a comprehensive way. A more profound treatment and references can be found in my masters thesis and this article.

## Introduction: Higher Harmonics Generation at Crystalline Surfaces

The reflection of light at a shiny surface is mainly a linear process, i.e. the intensity of the reflected light depends linearly on the intensity of the incoming light. During the process of reflection, an electron captures a photon (=particle of light) and emits it after a very short periode in a certain direction.

At high light intensities, the electron may capture next to a first photon a second photon before emitting the first one. Instead of sending both photons away separately, the electron sends only one photon away which contains the sum of the energies of the two incoming photons. The doubled energy of the new photon leads to a double frequency of this photon. Thus such a generated photon is called "second harmonic".

The rate of generation of these second harmonics depends nonlinearly on the incident intensity of light. If one doubles the intensity of the incident beam, firstly the probability for the absorption of a photon by a certain electron doubles. Thus there are roughly twice as many electrons absorbing a photon. Secondely the probability of such a photon absorbing a second photon and consequently generation a second harmonic photon doubles as well. The rate of generation of second harmonics thus depend quadratically, and not linearly, on the incident intensity of light.

Sometimes it even happens that an electron does not only absorb two photons but three, four or more photons before emitting one photon with the sum energy of the absorbed photons. Photons generated in such a manner are called third, fourth or higher harmonics. The rate of generation of such photons raises with the third, fourth and higher powers, resp., of the light intensity of the incident light.

Theoretically, one only has to choose suitably high intensities of the incident light in order to reach high rates of generation of higher harmonics at crystalline surfaces. However, there is a fundamental practical limit: The incident light heats the crystal, which melts at high intensities. In this case the material is no longer crystalline.

This problem can be avoided using the following trick: Instead of a continuous beam of light, one may use a pulsed laser. Then it is possible to generate very short light pulses with extremely high intensities, while the average power of the beam is low and consequently the material does not melt. Due to the extremely high intensities of the incident light the conversion rate for higher harmonics is significantly amplified. At a constant average power of the incident light, the rate of generation of n-th harmonics is roughly proportional to one over the (n-1)-th power of the pulse duration.

In the experiments, I used a pulsed laser that generated 250000 pulses per second with a pulse duration of 200 fs, i.e. 0.0000000000002 seconds. In such a "beam" of light, the pulses are separated by 1.2 km and have have a length of just 0.06 mm. The pulse duration is shorted by a factor of 200 fs * 250000 Hz=0.00000005 in comparison to a continuous beam. Thus the rate of generation of, e.g., fourth harmonics is amplified by a gigantic factor of 1/(0.00000005)^3=8*10^21. Due to this effect it is possible to obtain a detectable signal of fourth harmonics from crystalline surfaces.

## Application: Analysis of Surfaces employing Higher Harmonics

Interestingly, the generation rate of higher harmonics does not only depend on the intensity of the incident light beam but also on the distribution of the electrons on the surface and in the bulk of the crystal. This fact can be understood, if one knows some more details about photons.

Each photon oscillates in a certain direction. The direction of oscillation of the generated photon corresponds to the direction of the oscillation of the electron involved, which in turn is determined by the directions of oscillation of the four incident Photons and the structure of the crystal embedding the electron.

Mathematically this connection can be described by a tensor of rank five. The tensor is similar to a five-dimensional matrix and connects the four directions of polarization of the incident photons to the direction of oscillation of the electron involved. This tensor contains 243(=3^5) entries. A lot of these entries are zero or are interdependent. One reason for this is that the incident photons cannot be distinguished and thus it does not make any sense that the tensor treats one of those photons in a different way as the others. A second reason is that the crystall used may have symmetries. If a crystal with a threefold rotational symmetry is turned by 120 degrees, the physical situation does not change. Therefore the generation rate of higher harmonics cannot change as well and the mathematical description of the generation by the tensor must reflect this. A deeper analysis of the impact of crystall symmetries on the tensor governing the generation of higher harmonics may be found in my masters thesis. It also contains routines to compute the dependencies of the entries of the tensors for an arbitrary spacial symmertry of the crystal using Mathematica 3.0 (for dipole and quadrupole radiation).

The advantage of n-th harmonics is that they may resolve a (n+1)-fold symmetry due to the use of an (n+1)-dimensional tensor. A threefold rotational symmetry, as can be found, e.g., on Si(111) surfaces, can be only be distinguished by second or higher harmonics.

Higher harmonics, e.g. fourth harmonics, may have a small penetration depth in the crystal to be analyzed. In such a case the obtained signal is generated mostly on the surface. If the symmetry of the surface changes, e.g. due to melting or due to the growth of a different layer of material in a growth chamber, the angular dependence of the signal may change dramatically. This may lead to an optical method to monitor the change of the surface of the crystall.

## Further Information

- My masters thesis and an article dealing with the generation of higher harmonics on crystalline surfaces of semiconductors.
- Mathematica 3.0 programs usable to analyse the symmetry properties of the susceptibility tensor governing the generation of higher harmonics by dipole radiation and by quadrupole radiation from my masters thesis.