Optical/plasma wave interference to explain the formation of self-organized

Surface ripples with a period equal to the wavelength of incident laser radiation and that are likewise aligned in a direction perpendicular to the electric field of the light wave, have been observed in many experiments involving laser deposition [90] and laser ablation [95]. Such gratings are generated as a result of interference between the light field and the surface plasmon-polariton wave that is launched because of initial random surface inhomogeneities. However, until now there has been no observation of periodic structures being generated within the bulk of a material just by a single writing laser beam. We suggest that the self-organized nano-structures are produced by a pattern of interference between the incident laser radiation and a plasma wave generated within the sample.

Once a high free electron density is produced by multiphoton ionization, the material in the focal volume is a plasma. A characteristic peculiarity distinguishing a plasma from other media is the possibility of existence therein of longitudinal electrical oscillations (plasma or Langmuir waves), namely waves whose electric field component is parallel to the direction of propagation (Sec. A.2.1). When an electromagnetic wave is propa- gating through a plasma in the presence of density fluctuations, plasma waves can be generated [96].

We suggest that once generated, a Langmuir wave interferes with the electromagnetic wave creating a periodic modulation of the electron plasma concentration, which finally becomes frozen within the material. The electron plasma wave is efficiently generated only with wave vector kpl in the plane of light polarization and only in the direction

defined by conservation of the longitudinal component of the momentum (Fig. 6.7 (a)). Consequently, the interference pattern has to be perpendicular to the direction of the laser propagation (hence parallel to the polarization of the laser). The amplitude of the wave vector of the plasma wave is defined by this momentum conservation condition [54]:

kgr = 2_Λπ =

q

k2

pl−kw2, (6.1)

wherekw is given by the dispersion relation of an electromagnetic wave propagating in

plasma ω2 =ω_p2+ c 2 n2 bg k2_w, (6.2)

Chapter 6 Self-assembled periodic nanostructures by femtosecond direct writing 76

with ω_p the plasma frequency (A.19). Hence, given the material and the optical fre- quency,kw only depends on the plasma electron density. kpl in Equation 6.1 is given by

the dispersion equation for the plasma wave (A.27):

ω2_pl =ω_p2+3 2v

2

ek2pl,

where for the energy conservationωpl =ω.

Figure 6.7 (a) elucidates the interference mechanism between the electron plasma wave and the electromagnetic field giving rise to the grating with period Λ, offering an ex- planation of the appearance of the self-assembled nanostructures. Indeed, the presence of Λ2 = Λ/2, observed in Figure 6.4 can now be explained, considering the interference

between the two plasma waves satisfying (6.1) (Fig. 6.7 (b)). In analogy, also Λ3may be

explained as a result of interference between waves. In this case, though, it is necessary to invoke the interaction of all the three waves, namely the two plasma waves and the electromagnetic field as sketched in Figure 6.7 (c). If this is the case, it can be seen that Λ₃ =λ_w. Indeed, this is in good agreement with the experimental results on the value of Λ3 reported in Table 6.1. The wavelength of the electromagnetic wave, which in air

is 800nm, due to the dispersion becomes ∼ 550nm in glass. In presence of a plasma

λ_w increases according to (6.2), whereλ_w = 2π/k_w. Using (6.2) to calculate the plasma frequency wp, from Equation A.19 the maximum measured value ofλw = 580nm cor-

responds to a plasma density of∼1020_cm−3_{, which is in excellent agreement with the}

value found in [73].

Although this theory of optical/plasma wave interference is extremely interesting and would justify the presence of all the three periods observed experimentally, however it leaves space to some criticism. In particular, from Equation A.27 we can see that the amplitude of the plasma wave vector depends on the electron density (via ωp Equa-

tion A.19) and electron temperature (via ve), which are both spatially and temporally

varying. This makes difficult to understand the mechanisms which preserve the period- icity while the plasma is cooling and highlight the need for further investigations.

Chapter 6 Self-assembled periodic nanostructures by femtosecond direct writing 77 w

k

Λ

=

2π

k

k

k

k

k

k

k

=

Λ

=

2π

k

k

k

Figure 6.7: Diagram explaining the interference between the light wave and plasma waves, giving rise to the three periods observed in Figure 6.4. kpl is wave vector

of electron plasma wave,kw is wave vector of light. (a): interference diagram for the

grating of period Λ (b): Two waves interference between the two plasma waves satisfying (6.1) and originating in a grating with period Λ2= Λ/2. (c): Three-waves interference between two plasma waves and the light wave originating in a grating of period Λ3.

Chapter 6 Self-assembled periodic nanostructures by femtosecond direct writing 78