Re‡ectance Optimization for a Mo/Si Multilayer Structure

The simulation results in this thesis were all performed by using the language of technical computing design program MATLAB 7.1 (The MathWorks, Inc.), which is based on the characteristic matrix methodology [27] as discussed in Chapter 2. The characteristic matrix of the j th layer of the stack is given by :

is the phase thickness expressed in terms of the refractive index Nj and the physical thickness dj of the layers, wavelength , and the angle of internal refraction j. The optical admittances :

j = _jcos _j (TE pol.)

j = ^j

cos _j (TM pol.)

depend on the polarization of light at non-normal incidence. The assumptions inherent in these simulations are that the interfaces between the adjacent layers is perfect (i.e.

that there is no interfacial interdi¤usion and no scattering loss).

The optical constants of the various materials, namely the complex refractive index N = n ikare derived from atomic scattering factors by Henke et. al. and were obtained from http://henke.lbl.gov/optical_constants/asf server at Berkeley [36]. The values of n and k for the materials used in this thesis were downloaded as functions of wavelength from 10 nm to 42 nm and as such the wavelength dependence of n and k is implicit in all calculations. The values of n and k at the wavelength of particular technological interest are listed in Table 1. The main aim of this thesis is to investigate the performance enhancement of the re‡ectors.

In order to reduce the e¤ect on the re‡ectance response of the absorption losses, the thicknesses of the two components of the stack are adjusted slightly from the optical

path length of a quarter wavelength. The thickness of the high-absorption layer dh is set slightly lower than that of the low absorption layer dl. Thin …lm multilayer coatings greatly increase the re‡ection from surfaces in multi-element layer by making use of phase changes and the dependence of the re‡ectance on index of refraction. The idea behind multilayer coatings is that the creation of an interface by means of a thin …lm gives multiple re‡ected waves. The important property of a quarter-wave stacks is that a stack of quarter-wave layers of alternating refractive index high, low, high, low, . . . , etc., has the re‡ection from every interface in phase. This gives constructive interference between every re‡ection, all the re‡ections add together and a quarter-wave stack of enough layers acts as a mirror at the wavelength . To maintain the phenomenon of standing wave in conventional high re‡ective mirror (quarter-wave stack), using the concept of ideal Bragg crystal, to achieve the result of standing wave and reduce the absorption in order to enhance the re‡ectance.

A quarter-wave layer is one which

N d = =4,

where N = refractive index of thin …lm, d = thickness of thin …lm, and = wavelength of light used. Hence, we made the total thickness of multilayer equal to the quarter-wave layer and adjusted the thickness of Mo and Si layer, and the re‡ectance results are listed in Table 4.1. As seen in Table 4.1, it is appear that the re‡ectance at 13:5-nm wavelength is coparison of the quarter-wave stack and …ne-tuned Mo/Si multilayer structure which is used the way to adjust the layer thickness gradually. The …ne-tuned Mo/Si multilayer structure is a 40 pair Mo/Si system (dMo = 3:4nm and dSi = 3:6nm), yeilds a maximum re‡ectance around 71:25% of 13:5-nm wavelength, a peak re‡ectance of 72:34% at 13:60 nm, and the full-width-at-half-maximum (FWHM) is 0:758 nm.

Table 4.1: Re‡ectance versus individual thickness of Mo and Si layer.

d_{M o} (nm) d_Si (nm) Pair Number R _{= 13.5nm} (%) R_{p eak} (%) ( _peak)

quarter-wave stack 3.65 3.38 40 69.89 71.50 (13.62 nm)

…ne-tuned Mo/Si 4.0 3.0 40 69.55 69.55 (13.50 nm)

…ne-tuned Mo/Si 3.9 3.1 40 70.12 70.16 (13.52 nm)

…ne-tuned Mo/Si 3.8 3.2 40 70.56 70.71 (13.53 nm)

…ne-tuned Mo/Si 3.7 3.3 40 70.88 71.20 (13.55 nm)

…ne-tuned Mo/Si 3.6 3.4 40 71.10 71.64 (13.57 nm)

…ne-tuned Mo/Si 3.5 3.5 40 71.22 72.02 (13.58 nm)

…ne-tuned Mo/Si 3.4 3.6 40 71.25 72.34 (13.60 nm)

…ne-tuned Mo/Si 3.3 3.7 40 71.19 72.61 (13.61 nm)

…ne-tuned Mo/Si 3.2 3.8 40 71.03 72.83 (13.62 nm)

…ne-tuned Mo/Si 3.1 3.9 40 70.77 73.00 (13.64 nm)

…ne-tuned Mo/Si 3.0 4.0 40 70.40 73.10 (13.65 nm)

…ne-tuned Mo/Si 2.9 4.1 40 69.92 73.12 (13.67 nm)

…ne-tuned Mo/Si 2.8 4.2 40 69.31 73.15 (13.68 nm)

…ne-tuned Mo/Si 2.7 4.3 40 68.54 73.08 (13.69 nm)

…ne-tuned Mo/Si 2.6 4.4 40 67.61 72.94 (13.70 nm)

…ne-tuned Mo/Si 2.5 4.5 40 66.46 72.73 (13.71 nm)

…ne-tuned Mo/Si 2.4 4.6 40 65.06 72.44 (13.72 nm)

…ne-tuned Mo/Si 2.3 4.7 40 63.36 72.05 (13.73 nm)

…ne-tuned Mo/Si 2.2 4.8 40 61.28 71.57 (13.74 nm)

…ne-tuned Mo/Si 2.1 4.9 40 58.73 70.98 (13.75 nm)

…ne-tuned Mo/Si 2.0 5.0 40 55.60 70.26 (13.76 nm)

The comparison spectral responses are shown in Figure 4-3. For the …ne-tuned multi-layer structure, the re‡ectance at 13:5-nm wavelength is increased from 69:89% to 71:25%;

the peak re‡ectance is increased from 71:50% at 13.62 nm to 72:34% at 13.60 nm, too. In order to …nd the maximum re‡ectance, we also calculated the re‡ectance as the function of pair number in the multilayer and the result is shown in Figure 4-4. As a conse-quence, we could …nd the maximun re‡ectance with 54 pair …ne-tuned Mo/Si multilayer structure (called Mo/Si ML structure) and still keeping the same value even though with more pairs. The spectral response for 54 pairs is then shown in Figure 4-5, which shows that the Mo/Si multilayer structure’s re‡ectance at 13:5-nm wavelength is 71:65% and the peak re‡ectance is 72:91% at 13.61 nm and the FWHM is 0:710 nm. The detailed parameters of the Figure 4-4 are listed in Tables 4.2 and 4.3.

12.0 12.5 13.0 13.5 14.0 14.5 15.0

Figure 4-3: The spectral response for quarter-wave stack and …ne-tuned Mo/Si ML struc-ture.

Table 4.2: The maximum re‡ectance at 13.5-nm wavelength of …ne-tuned Mo/Si ML structure with an increasing pair number.

Pair Number Rm ax (%) Pair Number Rm ax (%)

Table 4.3: Parameters of three kinds of Si-based ML structures.

d_{M o} (nm) d_Si (nm) Pair Number R _{= 13.5 nm} (%) R_{p eak} (%) FWHM (nm)

quarter-wave stack 3.65 3.38 40 69.89 71.50 0.753

…ne-tuned Mo/Si ML 3.40 3.60 40 71.25 72.34 0.758

0 10 20 30 40 50 60 70 80 90 100

Figure 4-4: The re‡ectance versus the pair number for the …ne-tuned Mo/Si ML structure.

12.0 12.5 13.0 13.5 14.0 14.5 15.0

0

Figure 4-5: The spectral re‡ectance comparison of the quarter-wave stack, …ne-tuned Mo/Si ML structure and Mo/Si ML structure.