Signals in two-photon photoemission are proportional to the product of the intensities of the pump and probe pulses. To get a high efficiency in space the two beams must be well focused and have a good overlap. To get a high efficency in the time domain ultra-short pulses are used, which are generated by a femtosecond laser. The pulse width is usually in the order from 30 to 100 femtoseconds. In the following sections a short theoretical description of the femtosecond laser is given.
|| The two fundamental elements of a laser are the amplifier and a cavity which produces a positive feedback between the light and the amplifying medium. First we will have a look on the amplifier and the physics on which it is based
The physical principle of the amplification is the stimulated emission. A photon with an energy
traversing a two level system can either be absorbed or induce a de-excitation which generates another photon with the same properties. This is called stimulated emission. Another fundamental de-excitation is the spontaneous emission which is not dependent on the intensity of the incoming photons. The probability for the absorption of a light quanta is given by B12, for spontaneous emission by A21 and for stimulated emission by B21
In equilibrium the number of absorbed photons must be equal to the number of emitted photons.
Due to time-reversal symmetry the transtion probability for absorption equals the probability for spontaneous emission (B12=B21). The occupation of the two levels are according to Boltzmann's Law
Thus we get a spectral intensity of
which is equivalent to Planck's Law.
What condition must be given to get amplification by stimulated emission?
Looking on a system which is traversed by n photons per second the deviation with respect to the position z is
The constants a12, b12 and b21 are related to the Einstein coefficients. Neglecting the spontaneous emission the number of photons as a function of propagation is given by
Therefore a gain can just be obtained when there are more electrons in the excited state than in the ground state what is called population inversion.
It is obvious that a population inversion is not possible with a two level system. Therefore lasers are based on three or four level systems.
So far we have an amplifying medium. To get a positive feedback the light must traverse the amplifier several times with a phase difference of a multiple of . This is made possible by an optical cavity which is represented by two mirrors in which light is reflected. In an optimum cavity, light should make an infinte number of round trips. The geometry of light propagation in the cavity determines the transverse modes which are related to the electromagnetic field distribution. The longitudinal modes, which are important for femtosecond lasers, can be described by looking at the time-frequency property of the light. Taking two parallel mirrors with a distance L, a propagating wave interferes constructively with the reflected wave between the mirrors if
whereas m is an integer, c the speed of light and the period of the light. As gain is just possible for a certain bandwidth in the frequency domain, the number of active modes is finite.
So far we have a laser which emits continuously light. But how do we get short laser pulses? This can be explained by a process which is called mode locking. We will now have a closer look on this phenomenon.
The frequency spectra of most lasers are quite narrow and therefore it is impossible to get short pulses since these both quantities are related by Fourier transformation. But even a laser with a broad spectrum does not give ultra-short pulses necessariliy. Let us consider two modes which are polarized in the same direction and with the electromagnetic fields:
The detector measures the intensity I of the field and because its response time is much greater than the period of optical oscillations the mean of the intensity is detected:
The mode now depends on whether there is a relationship between
If these phase factors are random and independent, the integral term (beating term) averages to zero. Therefore the detector records a constant signal and the laser operates in a multimode regime.
In case of totally correlated phases, we get a sinusodial modulation with the frequency
In case of N modes we have multiple phase beating terms and the peaks become much narrower.
If the laser works in the free mode, the different modes will compete for the gain due to stimulated emission. This causes fluctuations in the phase.
But there is a way to regulate this competition:
If a nonlinear medium is inserted in the cavity, the fluctuation with the largest maximum is enhanced at the cost of the weaker ones while the light is propagating. At the optimum all the energy is concentrated in one maximum in the cavity. This corresponds to constant phase relation between the modes in the frequency domain (passive mode locking).
One can also give an explanation in the frequency domain. By an additional modulation close to the intermode frequency c/2L (L= cavity length) sidebands to each frequency mode are created which lie close to the neighboring frequency band and therefore compete with them for gain (active mode locking).
In a Ti:Sapphire laser the modes can lock without inserting a saturable absorber or any external modulation. This self locking is based on the Kerr Lens Effect. The Kerr effect states that the refractive index is a function of intensity
If n2is positiv a beam propagating through this medium, is more focused near the optical beam axis where its intensity is stronger. This is called self focusing.
The Ti:Sapphire crystal has this nonlinear property. The low intensity part of the beam can be removed by a pinhole. Thus high intensity maxima are much better amplified and the laser reaches the condition to give short pulses. A possible setup for such a laser is shown in the figure below