Single interface reflectance and transmittance

Unlike in the visible range, where e.g. simple Ag-coated mirrors can be used for the broadband and effective (R(λ>400nm)>90%) guidance of light under nearly arbitrary angles of incidence, no comparable optics are available in the XU V range as will be in- vestigated in the following.

The single interface reflectivity is given by the Fresnel coeffi- cients. In the following the results from retrievals of both Snell’s law and the Fresnel coefficients from Maxwell’s equations [34] are summarized. The more detailed deduction of these can be found in appendix C. These formulas are prerequisites for the mathematical treatment of multilayer mirrors in the next section.

Figure 2.9 displays the model of two layers with the refractive indexes n˜1 and n˜2 forming an interface. Monochromatic light of

wavelength λ, incident on the interface under a propagation an- gle θi is assumed. As derived in appendix C claiming continuity

Figure 2.9: Schematic model of an interface between two materials 1 and 2.

of parallel electric field components at an interface at any time, directly yields Snell’s law:

θr = θi and n˜tsinθt = ˜nisinθi (2.25) The single interface reflectivity on the other hand can be deduced by utilizing continuity relations of the perpendicular field compo- nents (appendix C).

The resulting equations adopt a slightly different form, depend- ing on the polarization of the incoming light. But one can actually separate each beam of arbitrary polarization into two components: One component, where the electric field vector is parallel to the plane of incidence (p-polarization) and one where the electric field vector is perpendicular to the plane of incidence (s-polarization) including a phase delay φsp between the two components:

~

E = Es~1s +Ep~1peiφsp (2.26) Thereby the reflection of each component can be calculated sep- arately for s and p polarization and can be merged again by a superposition of the two reflected components following equation 2.26.

As deduced in appendix C one finally ends up with the polarization- dependent Fresnel coefficients, which generally describe the trans- mitted and the reflected field amplitude at an interface between linear, homogeneous, and non-magnetic media.

r1,2 = q1 −q2 q1+q2 = −r2,1; t1,2 = 2q1 q1+q2 (2.27)

The equations in 2.27 have the same appearance both for s and p

s-polarization qa (aε[1,2]) is the momentum transfer and twice the perpendicular component of the absolute value of the wave vector

q_as = 2ka⊥ = 4_λπn˜acosθa. The p-polarized notation can be simplified to: qp

a = cosθa/n˜a.

(a) s-polarization (b)p-polarisation

Figure 2.10: Simulated single interface reflectivity of anSi surface at vacuum dependent on both the incidence angle θ and the photon energy.

Figure 2.10 shows exemplarily the outcome of a simulation of a single Si surface for XU V light impinging under vacuum. The reflectivity R1,2 = |r1,2|2 is plotted in false color representation

against both the angle of incidence θ and of the energy _~ω. Each panel shows the result of one polarization.

Summarizing the most important features of this simulation, one finds:

• The single interface reflectivity of both for s and p polarized light drops with increasing energy as the optical constants approach 1.

• Only in the vicinity of an absorption edge (e.g. at 99 eV in the case of Si) where the optical constants exhibit disconti- nuities, upper tendency is broken and higher single interface reflectivity becomes possible.

• High reflectivity of nearly 100% is found at very small graz- ing incidence angles and total external reflection occurs if

• In the case of s polarization, the single interface reflectivity increases with increasing angle.

• In the case of p polarization, the reflectivity drops around

θ ≈ 45o _{due to Brewster angle reflection. θ}

B = arctan(n_n˜_˜2₁) ≈

arctan(1) = 45o_.

2.2.3.1 Grazing incidence optics

The broadband and high reflectivity of single interfaces at small grazing angles allow one to use bulk reflectors for the guidance of attosecond pulses. The small penetration depth at total external reflection angles ensures a flat phase reflection. For example a gold coated parabola is being used as focusing optic in the attosecond beam-line setup AS2 (appendix F, [35]). But beside their high reflectance and broadband beam guidance, these optics exhibit no possibility for as pulse-shaping.

2.2.3.2 Normal incidence

All presented attosecond experiments are performed at θ = 5o _or 45o, thus far-off grazing incidence angles. In this case, equation 2.27 can be approximated and written depending on the differential of the real (∆δ = δ1 −δ2) and the imaginary (∆β = β1 −β2) parts

of the index of refraction [33] of two materials 1 and 2:

R1,2 =

∆δ2_{+ ∆β}2

4cos4_(θ 0)

(2.28)

One finds that the single normal interface reflectivity is propor- tional to the distance of the optical constants in the complex plain (Pythagorean theorem). So both a large separation of the diffrac- tion part δ and in the absorption partβ results in higher reflectiv- ity. Please note, that the typical single interface reflectivity is very low (≈1% at 30 eV and≈0.1%/ 0.01% at 100eV/200eV). XU V

optics with a higher reflectivity can be implemented by multilayer interference coatings as introduced in the following section.