Reparación de fisuras en estructuras

In 1965 the first experimental demonstration of an OPO was reported by Giordmane and Miller using Lithium Niobate [10] as the nonlinear crystal. Two beams with different frequencies travelling through a nonlinear crystal will generate a travelling polarisation wave at this different frequency. Provided that the polarisation wave travels with the same velocity as the freely propagation electromagnetic waves, cumulative growth will result. The two incident waves are termed pump and signal, having the frequency νp and νs, and the resulting third wave

is termed as the idler with frequency νi. Under proper conditions, the idler wave can

mix with the pump beam to produce a travelling polarisation wave at the signal frequency, phased such that the growth of the signal wave results. The process continues with the signal and the idler waves both growing and the pump wave decaying as a function of distance within the crystal. We mentioned before that, provided the polarisation wave travels at the same velocity as the freely propagating em wave, cumulative growth would result. The fundamental beam is labelled the pump beam, and the generated beams are labelled the signal and idler beams respectively, with the signal beam possessing the greater energy (shorter wavelength).

1 1 1

p s i

λ = λ +λ

[5.1]

where the subscripts p, s and i refer to the pump, signal and idler waves respectively. Within an OPO, the signal and idler outputs are generated by the interaction between the intense pump light and the weak quantum noise at the signal and idler frequencies. This interaction occurs via the second-order non-linearity of the crystal

147 medium, and results in the amplification of the signal and the idler frequencies at the expense of the pump. Energy conservation ensures that:

p s i

E =E +E

[5.2]

where Ep, Es and Ei are the energies of the pump, signal and idler photons respectively. For the interaction to occur, the three waves must be phase-matched within the crystal. As with frequency doubling (the exact opposite of optical parametric amplification at degeneracy – where the signal and idler photons have the same energy), phase matching can be achieved by temperature tuning or angle tuning the crystal however, in the case of an OPO, the signal and idler wavelengths change to follow the phase-matching conditions defined by the angle or temperature.

To understand the conversion process, it is necessary to re-write the equation in terms of the conservation of momentum:

I S

P k k

k = +

[5.3]

where k is known as the wave-vector:

( )

k = λω

[5.4]

where n(λ) is the wavelength dependent refractive index (dispersion).

The three refractive indices associated with the pump, signal and idler can be changed either in the propagation direction through the crystal (angle tuning) or by temperature variation (temperature tuning). When the three frequencies see refractive indices that satisfy equation 5.3 they are considered to be phase-matched.

Since the refractive index of a crystal is angularly (or temperature) dependent (depending on the polarisation direction within the crystal), the constraints of phase matching can typically require the non-linear crystal to be cut at a specific angle determined by dispersion, therefore, efficient frequency conversion can only be achieved within a narrow band of angles around the so-called “phase-matched angle”. Furthermore, once the angle is determined, phase matching can only be achieved for a narrow band of pump-frequencies [11]. Since the signal and idler intensities build

148 up from quantum noise and hence are initially very low, typically of the order of 10- 13 _{times lower than the pump intensity IP, they must be greatly amplified before any}

significant depletion of the pump occurs. The threshold condition for an optical parametric oscillators is exactly the same for that of a laser whereby oscillation will commence as soon as the parametric gain (in the case of the OPO) equals the intracavity loss (including loss due to the output coupler) of the OPO.

Provided therefore, that the phase-matching condition described by equation 5.3 is maintained, the signal and idler wavelengths will tune to satisfy equation 5.3 over a wide range of temperatures or angles. There are three different types of phase matching conditions within birefringent crystals (types 1, 2 and 3) whereby:

TYPE PUMP SIGNAL IDLER

1 e o o 2 e o e 3 e e o

Table 5.1 Showing the three phase-matching types within birefringent crystals.

In the table 5.1 o and e correspond to the ordinary and extraordinary refractive indices of the birefringent material in question (refractive indices in orthogonal directions to the direction of pump propagation). When comparing the more common type 1 and type 2 phase matching, we find that type 1 is more favourable when θmis near 900, whereas, type 2 leads to higher deff when θmlies near 450. For the purposes of this work we adopted type 1 phase-matching conditions in order to reduce the effect of Poynting vector walk-off. Poynting vector walk-off can occur in birefringent phase matching when the Poynting vector of an e-wave moves away from the k-vector when a wave propagates at an angle to the optic axis, whereas those of the o-wave remain collinear. This means that the interaction length is greatly reduced.

So within an OPO, under the proper conditions, the idler wave can mix with the pump beam to produce a travelling polarisation wave at the signal frequency, phased such that the growth of the signal wave results and vice versa for the idler wave.

149