The reflectivity of single mirror surfaces at glancing (or grazing) angles greater than the critical angle is very small; by clicking here you can calculate the x-ray or extreme ultraviolet (EUV) reflectivity of a mirror material of your choice for a variety of wavelengths and angles of incidence. You will need to specify the mirror material, the angle of glancing incidence q, and the wavelength l (or photon energy E). The resident program will then use tabulated values for the complex refractive index ñ = 1 - d - i b to calculate the reflectivity (d is the decrement of the real part of the refractive index and b is the absorption index). As an example, you will find that copper has a normal incidence reflectivity about 10-3 for 15 nm radiation. But this calculation gives energy reflectivity I/I0. The magnitude of the amplitude reflectivity is (ñ-1)/(ñ+1) or the square root of the energy reflectivity, and for this example is about 1/30. Neglecting the effects of absorption for the moment, this means that if we can arrange to get the reflections from 30 copper-vacuum interfaces to add with the correct phase relationship, (because the amplitudes are complex) then we can achieve a reflectivity close to 1. Although materials with d or b = 0 do not exist, it is possible to choose pairs of real materials having satisfactory optical constants, and by depositing a stack of ultrathin films having alternately high and low values of d, as shown in the figure, to obtain usefully high reflectivities. Click here to see a diagram of a multilayer. (Note this figure is simplified because multiple reflections have been omitted, but it shows the principle).
The individual thicknesses of these film is adjusted so that the reflections from each interface add in phase at the top of the stack, in analogy to the multilayer mirrors used for the visible, infrared and UV regions. However, the films need to be much thinner than those used in these regions; visible multilayers require film thicknesses of hundreds of nanometers, but for x-rays the thickness of each film must be between 1 and 10 nanometers. The deposition of up to 100 or more ultrathin layers having constant and uniform thickness requires special technology. Multilayer coatings are most commonly made by a vacuum deposition process such as evaporation or sputtering. Both methods can yield good multilayer coatings.
The response of these artificial multilayers is strongly wavelength-selective. For incident x-rays of wavelength the reflectivity of the stack has peaks at the angles qm given by the Bragg equation:
ml=2dsin(qm)
where m, the order of the reflection is an integer 1,2,3... For x-ray imaging optics the multilayers are most commonly used in the first order, m = 1.
Because neither the high- nor the low-density material can have b = 0, x-rays will always be absorbed in the layers, reducing the reflectivity to a value less than 100%. However, with a careful choice of materials, multilayers can be made which have significant normal incidence reflectivity (from a few percent upwards) at wavelengths of 4.5 nm and longer. For shorter wavelengths we must depart from normal incidence; we see from equation (1) that, for normal incidence reflection at a wavelength 4.5 nm, d ~ 2 nm and the individual layers must be ~1 nm thick. It is very difficult to make high quality coatings that are thinner than this because each layer must be only 4 to 50 atoms thick, and thus it is difficult to achieve normal incidence reflectivity higher than a few percent at wavelengths shorter than about 4.5 nm. However, we see from equation (1) that if q ~ 1 deg, multilayer coatings with d ~ 2 nm will reflect 0.07 nm, or 17.7 keV x-rays. Hence multilayers can be used as coatings for glancing incidence optics up to energies of 10 or 20 keV or more. The benefits are larger angles of glancing incidence, and hence improved collecting area over optics coated with a single metal layer, and an energy selective effect - a reduction of the bandpass of the reflected radiation which is desireable in many experiments. Multilayers have been used to coat optical elements for instruments such as hard x-ray microprobes, spectrometers and monochromators at synchrotron radiation beamlines.
The reflecting power (reflectivity) of a multilayer coating depends on a number of factors, including not only the choice of materials but the degree to which they interact physically and chemically to wash out the contrast between the materials and the roughness of the interfaces. The roughness of the underlying substrate is also of prime importance - an rms roughness of the same order of magnitude as the layers thicknesses will spoil the performance of the most carefully deposited coatings. "Superpolished" substrates, with an roughness s ~ 0.1 nm are preferred. On such substrates the peak reflectivity of the coatings can approach 80% to 90% of what is theoretically predicted.
Click here to calculate the reflectance of a multilayer. You will be asked to fill in a form specifying the materials A and B of the layers, the thicknesses dA and dB, the number N of layer pairs, the rms roughness s of the interfaces, and the ranges of angles and wavelengths (energies) over which you want the calculations to be made. As a start, try the following examples:
| N | Material | Thickness (nm) | s(nm) | q(deg) | Energy (eV) | ||
| A | B | dA | dB | ||||
| 50 | Mo | Si | 2.1 | 4.8 | 0 | 90 | 80-120 |
| 70 | Mo | Be | 2.06 | 3.66 | 0.6 | 90 | 90-130 |
| 100 | W | C | 1.0 | 2.0 | 0 | 0 -10 | 8000 |
The first example illustrates the normal incidence reflectivity that can be achieved with molybdenum and silicon, at wavelengths just longer than the Si L3 edge at 12.4 nm (100 eV). While this calculation predicts about 70% reflectivity, in practice roughness limits the reflectivity to about 65%. To see the effect of roughness on the reflectivity, try the calculation again with various values of s. Mo/Si multilayer mirrors have been used to construct normal incidence optical systems that are analogs of mirror optics used for visible light. For example, x-ray telescopes have been used to photograph the hot outer atmosphere of the sun at a wavelength around 18 nm. Multilayer coated microscope objectives operating at a variety of energies from 50 to 200 eVhave been made for photoelectron microscopy . There is also an effort under way to develop multilayer coated optical systems for such applications as EUV lithography at a wavelength of 13.5 nm
The second example illustrates the highest reflectivity yet obtained with a normal incidence multilayer. This is about 70%, at wavelengths around 11.2 nm or 111 eV, using Mo/Be multilayers.
The third example is for a W/C multilayer that could be used as a mirror for hard x-rays. The calculation shows the reflection of several orders of Cu Ka x-rays (0.154 nm or 8.048 keV) at the angles qm. Repeating this calcualation with some added interface roughness will demonstrate the strong effect this roughness has on the higher orders.
The deposition of up to 100 or more ultrathin layers having constant and uniform thickness requires special vacuum deposition technology, most commonly vacuum evaporation or sputtering. Multilayer mirrors are made at CXRO by the sputtering process in the apparatus shown schematically in the figure. The substrate on which the deposition is made is mounted on a rotating table over a pair of magnetron sputtering sources, of the two component materials of the multilayer. The power to these sources is accurately controlled by stable precision power supplies, as is the sputtering gas pressure. The thickness of each layer is then determined by the distance of the substrate from the sources and the time it spends over each source. As the table rotates at a constant predetermined speed, the alternating layers are built up as the substrate passes over first one source and then the other. D.C. power is used to sputter metals and other conducting materials, and R.F. power is used for insulators. In this way multilayers composed of a wide variety of material pairs, of pure elements and of compounds, can be made.
A large variety of multilayers has been made and their reflectivity measured by various groups of experimenters over the years. A database of reported reflectivities has been assembled from surveys taken at the biennial Physics of X-ray Multilayer Structures conferences.
The calculational method used for the examples is described in: "Layered synthetic microstructures as Bragg diffractors for X rays and extreme ultraviolet: theory and predicted performance"; J. H. Underwood and T. W. Barbee, Jr. Applied Optics 20, 3027 (1981).
“Molybdenum/Beryllium Multilayer Mirrors for Normal Incidence in the Extreme Ultraviolet” ; K. M. Skulina, C. S. Alford, R. M. Bionta, D. M. Makowieki, E. M. Gullikson, R. Soufli, J. B. Kortright, and J. H. Underwood, Appl. Opt. 34, 3727 (1995).
General: "Soft X-ray Optics" Eberhard Spiller, SPIE,
Bellingham Washington (1994).