Quantronix Integra-C

The third laser system used in this thesis is Quantronix Integra-C system, which provides 130 fs long pulses at 800 nm central wave-length with 1 kHz repetition rate. Unlike in CDP TISSA-50 oscil-lator, in this system seed pulses for amplifier are generated using a fiber-oscillator [72, 73]. After frequency doubling, the oscillator provides 110 fs pulses at the central wavelength of 790 nm with 30-40 MHz repetition rate. The oscillator is passively mode-locked by a saturable absorber and the gain medium of the laser is a high gain Er-doped fiber.

Seed pulses in this system are amplified in two stages with two different amplifiers. The first amplifier is a regenerative ampli-fier [74, 75] and second multipass ampliampli-fier [76], like in Sect. 3.2.1.

Before amplification, the pulses are temporally stretched using the grating pair based stretcher and then amplified, whereafter they

Ultrashort laser pulse ablation

are compressed with another grating pair. The main difference be-tween this amplifier and the one described in Sect. 3.2.1 is that here there are two separate Ti:sapphire crystals, which makes possible the amplification of pulse energy up to 3.5 mJ.

4 Diffractive optics

In this Chapter the basics of the interference of electromagnetic waves, diffraction gratings, and more general diffractive optical el-ements (DOEs) are presented [77, 78]. All of these are used in the experimental part of this thesis. Their applications in the case of ultrashort pulses are discussed in Chapter 5.

4.1 INTERFERENCE

Interference of waves concerns phenomena that take place when two or more waves overlap each other. These waves may be mu-tually coherent meaning that their phase difference is constant in time. Additionally, the waves may have almost the same frequency and state of polarization. If all these conditions are true, the beams interfere strongly with each; in general, the interference pattern must be determined using the coherence theory of electromagnetic fields [79]. For example, if the polarization vectors of two linearly polarized plane waves are orthogonal, the waves do not interfere with each other at all.

The most classical example of interference is Thomas Young’s double slit experiment or simply Young’s experiment. In this ex-periment a wave passes through two slits near each other and gen-erates two separate waves behind the slits, which interfere with each other after propagation. The interference field oscillates quasi-periodically, varying regularly from minima to maxima. The pe-riod of the interference pattern depends on the distance between the Young pinholes and on the distance between the Young screen and observation plane. The visibility of the pattern depends on the light intensities at the two pinholes and on the degree of spatial coherence of the incident light, and can be used to determine the latter.

In this work we consider light fields originating from a single

laser, and therefore we may consider it as spatially fully coherent at each frequency. Since mode-locked lasers are employed, the light is also nearly coherent in spectral and temporal sense. Hence the only significant factors affecting the visibility of interference fringes are polarization and temporal overlap of the pulses.

Monochromatic two-beam interference fields always have sinu-soidal intensity distributions, while interference of three or four plane waves produces pattern consisting of elliptic or round spikes etc.. Let us first consider a plane electromagnetic wave E_i(r,ω) os-cillating at frequencyω: is the complex amplitude of the electric field vector and

ki =kxixˆ+kyiyˆ+kzizˆ (4.3) is the wave vector. Considering the interference of N plane waves, the time-averaged energy densityh^we(r¯)ican be expressed as

h^we(r¯)i= ^ε0n

Figure 4.1 illustrates the case of four-wave interference, where all waves propagate at the same angle θ with respect to the optical axis.

For example, let us take four waves that propagate in following directions:

k1(ω) =kxyxˆ+kzz,ˆ (4.5) k2(ω) =kxyyˆ+kzz,ˆ (4.6) k3(ω) =−^kxyxˆ+kzz,ˆ (4.7) k4(ω) =−^kxyyˆ+kzzˆ (4.8)

Diffractive optics

θ θ

kx

ky

kz

Ex1

Ey1

Ez1

Ex2

Ey2

Ez2

Ex3

Ey3

Ez3

Ex4

Ey4 Ez4

Figure 4.1: Interference of four waves propagating so that the angle between the optical axis and the wave vector of each wave is always angleθ.

where kxy = k0sinθ, kz = k0cosθ andk0 = ω/c. This case is the same as in Fig. 4.1, but let us now assume that all waves are linearly polarized and in same phase. Then the polarization and vectors can be represented as

E01(ω) = (Exxˆ−^Ezzˆ)/4, (4.9) E02(ω) = (E0xˆ)/4, (4.10) E03(ω) = (Exxˆ+Ezzˆ)/4, (4.11) E04(ω) = (E0xˆ)/4, (4.12) where Ex = E0cosθ and Ez = E0sinθ. Now we can calculate the

[µm]

Figure 4.2: Interference pattern of four linearly polarized waves with interference angle θ=30^oandλ=800nm.

Eventually the time-averaged energy density h^we(r¯)i of this four-wave interference pattern can be calculated:

h^we(r¯)i= ^ε0n This is periodic at ±^45o directions in the xy coordinate system. In Fig. 4.2 a few periods of time-averaged energy density (4.14) are plotted, when θ = 30^o andλ = 800 nm. From Equation (4.14) the spatial period of the four-wave time-average energy density can be

Diffractive optics wherenis the refractive index of the medium.

The visibility of the interference pattern can be defined as fol-lows:

V= ^Imax−^Imin

Imax+Imin, (4.16)

where Imax is maximum and respectively Imin minimum intensity value of the interference fields oscillation. As Equation (4.16) shows, the visibility of the interference pattern can vary between the values 0 ≤ ^V ≤ 1. In (4.14) the visibility is perfectV = 1. The spatial period (4.15) is valid for four-wave interference intensity distribu-tion. Respectively, it can be shown, that in case of the two-wave interference, interfering with angle of 2θ, the spatial period of the time-average energy density is

d= ^λ

2nsinθ. (4.17)