14. Reflecting Optics: Multilayers

14. RE FLECTlN G OPTICS: MULTILAYERS Eberhard Spiller Spiller X-Ray Optics Mt. Kisco, NewYork

14.1 Introduction Multilayer coatings of thin film are used to modify the optical properties of surfaces. Enhancement or reduction of reflectivity or transmission of mirrors and lenses are well-known examples in the visible region. Coatings are also used as spectral filters and as polarizers and phase retarders. The possible performance of a multilayer coating is limited by the optical constants of the materials available as thin films. The ideal situation, that absorption- and scatter-free films of different refractive index are available, permits practically unlimited optical performance for a coating. Coatings can be designed for nearly any specification; reflectivities can be 100% and one can design mirrors to produce any arbitrary reflectivity curve R(1). The absorption of materials is the most severe limitation for the performance of coatings in the vacuum ultraviolet (VUV) region. Absorption-free materials of “high index” are only available for II > 150 nm and multilayer mirrors with reflectivities R > 95% are still available down to this wavelength [l, 21. Single films of A1 and Be have good reflectivities close to 90% for photon energies lower than their plasma resonance (Fig. 1) [3,4]. However, this reflectiviy can only be obtained when oxidation of the surface is prevented requiring evaporating and using the film in ultra high vacuum without ever exposing it to oxygen. Films of LiF and MgF2 still have very low absorption for wavelengths 1 > 110 nm and can be used to overcoat aluminum to prevent oxidation. The thickness of the films can be adjusted such that the amplitude reflected from the top of this film adds in phase to that reflected from Al, thus enhancing the reflectivity. Multilayers of A1 and MgF2 can enhance the reflectivity even further UP to R = 96% [S]. There are no absorption-free thin-film materials for A < 110 nm. In the wavelength region II = 80-1 10 nm the absorption index p-the imaginary part of the complex index of refraction n“ = 1 - 6 $ - a f all stable materials is between 0.1 and 1. Overcoating an A1 or Be mirror with any other material does not produce a substantial enhancement because the absorption in the overlayer attenuates the amplitude from the metal more than the additional boundary adds. However, it is possible to use such overlayers to suppress undesired wavelengths

+

27 1 EXPERIMENTAL METHODS IN THE PHYSICAL SCIENCES Val 31 ISBN 0-12-475978-5

Copyright 0 1998 by Academic Press All rights of reproduction in any form reserved

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REFLECTING OPTICS: MULTILAYERS

0.8

0.2 0 40

60

80 100 120 WAVELENGTH (nm)

140

FIG. 1. Normal incidence reflectivity of some materials near their plasma frequency calculated with the Drude model with parameters from [3].

more than desired ones. This option has been used to image the 1 = 83.4 nm emission from O+ ions in the ionosphere while suppressing the much stronger hydrogen lines at I = 102.5 and 121.6 nm [6,7]. The alkali metals with plasma frequencies in the 3- to 5-eV range have low absorption [8,9] in the 1 = 100 nm region and could be suitable for multilayer coatings. They have up to now not been used in multilayer structures because of their reactivity. Theoretically, a peak reflectivity R 40% with a bandwidth around 60 nm can be obtained with a K-C multilayer at I = 100 nm, and a top layer of carbon might be sufficient to seal the alkali metal from the atmosphere. The absorption of the lighter elements decreases to values close to p = 0.01 around I = 70 nm. Radiation can propagate several wavelengths into materials allowing deeper boundaries to add their reflected amplitudes to that of the top surface, making it possible to enhance or modify the reflectivity of the best single-film materials with multilayer structures. Absorption of all materials decreases dramatically toward still shorter wavelengths, roughly proportional to ,I3 for I < 20 nm and away from absorption edges. Simultaneously the refractive index of all materials approaches 1 with 6 cc A’, while the reflectivity from a single boundary becomes very small with R cc I4 and a value around R = lo-’ for 1 = 6 nm. However, it is now possible to enhance this small reflectivity by adding the reflected amplitudes from a large number of boundaries in a multilayer coating. Theoretically, one can obtain usefbl reflectivities above R = 10% for any wavelength below 1 = 20 nm with values of R > 80% in some regions [lo]. The decrease in the reflectivity of a single boundary with decreasing wavelengths is compensated by increasing the number of layers in a coating. Reflectivity enhancements over that of a single boundary can be higher than a factor of 10,000.

MULTILAYER THEORY

273

14.2 Multilayer Theory To calculate the optical properties of a multilayer structure one needs Snell’s law to obtain the direction of propagation in each of the layers and the Fresnel formulas for the amplitude reflection and transmission coefficients r,, and t,, at the boundaries between the layers. One assumes that there is no scattering in the volume of a film; only at the boundaries is an incoming wave split into a transmitted and reflected wave. By introducing the parameters q1= (4n/L)fi1 cos iz for s polarization or q1 = (4n/Lri‘,)cos il for p polarization, we can express the reflected and transmitted amplitudes at the boundary between two materials as

The q-values for s polarization are proportional to the change in the momentum of a photon perpendicular to the boundary, often called the momentum transfer in x-ray optics, while the name “effective index” is used in the literature on optical coatings (usually defined without the factor 4nlL). The angle 0; is the propagation angle in each of the materials and is obtained from the angle io in the incident medium of index no with Snell’s law fi; sin ii= no sin q$, as cos ii= dl

(n0/fii)’ sin2 do.

(3) Figure 2 shows the field amplitudes ai and bi of the forward and backward running waves in each of the layers. The amplitude reflection in the incident -

FIG.2. Geometry of a multilayer strucure with forward running waves a, and backward running waves bi. The dashed lines represent the boundaries of the films in the optical theories or the atomic planes in x-ray diffraction.

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REFLECTING OPTICS: MULTILAYERS

medium is given by r = bo/aoand the transmitted amplitude by t = a,+ I/ao if n 1 represents the medium below the multilayer structure. For all calculations we assume that no radiation enters the structure from below: b n f l= 0. The field amplitudes in each layer in Fig. 2 are coupled to those of the adjacent layers by linear equations:

+

+ b, ei"tlo, b1 = al r12+ b2 eip2tll, al = aotol + bl e2irlr10, a2 = a l tI2eiC2 + b2e2i(02r21, bo = aorol

eiql

(4)

The r, and t, are the Fresnel reflection and transmission coefficients, respectively, for the i , j boundary and iiis the phase delay due to the propagation through layer i :

Numerous methods are available for solving the system of linear equations in Eq. (1) that are described in many textbooks and papers [ll-151. 14.2.1 Recursive Method [I61

By solving Eq. (1) first for a single film with two boundaries (layer 3 in Fig. 2 represents the substrate) and using the identity

+

t12t21 r?, = 1

(6)

for the Fresnel coefficients, one obtains for the reflected amplitude

where r, = rol and rb = r12 are the reflection coefficients of the top and bottom boundaries, respectively, of the film. Equation (7) remains valid if the thin film is deposited on top of a multilayer structure; for that case rb represents the amplitude reflectivity of the multilayer into the film. Therefore, we can calculate the reflectivity of any multilayer structure by repeated application of Eq. (7) starting from the bottom layer on the substrate and continuing until the boundary between the top layer and the incident medium is reached.

MULTILAYER THEORY

275

14.2.2 Matrix Methods [ll, 171 Equations (4) can be rearranged into matrix form. One can describe the transfer of the field amplitudes over a boundary and the propagation through a layer with 2 X 2 matrices to obtain

It follows that the amplitudes in the incident medium and the substrate are connected by

= 0 in the with Mi the matrix of each layer as defined in Eq. (8). Using matrix for the substrate gives the fields at the top of the substrate. The reflected and transmitted amplitudes from the multilayer structure can be obtained from the elements of the product matrix mii defined in Eq. (9):

r = bola0

= m21/m11,

t = an+l/ao=

n ti,i+llmll.

The intensity reflectiviy becomes R = rr*, while the transmitted intensity is T = tt*(qn+i/qO), where the values for q o and q n c l are those for s polarization; they are proportional to the momentum of the photon perpendicular to the cos d j . Polarization is boundaries. For p polarization one defines qi = (4nll~7~) included in the multilayer program through the proper Fresnel coefficients, most conveniently by expressing them as a function of q. In the optics literature the variable q (without the factor 4n/l) is often called the effective refractive index. Another convenient matrix formalism due to AbelCs [12, 171 introduces a matrix for each film that contains only parameters of this film. However, it is more convenient to include the influence of boundary roughness as a reduction in the amplitude reflection coefficients rii in the matrix form given in Eq. (8). 14.2.3 Boundary Imperfections and Reflectivity Reduction

The atomic structure of matter makes it impossible to produce sharp boundaries between two materials. All real boundaries have a certain width due either to diffusion of the two materials or to roughness of the interface. The reflectivity of each boundary, represented by a 6 function at the interface for sharp interfaces, is spread over the width of the interface and is often represented as a Gaussian

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REFLECTING OPTICS: MULTILAYERS

where B is the width of the boundary and ro is the amplitude reflectivity of the ideal sharp interface. The total reflectivity is reduced because the amplitudes reflected from different depth within the transition layer now add with different phases. For the case that ro 4 1 and refraction can be neglected, one can obtain the reflectivity reduction as a function of wavelength or incidence angle by Fourier transform from coordinates z to momentum transfer q [18] of Eq. (1 1):

The second part of Eq. (12) is obtained by replacing q with the period A of a multilayer that has maximum reflectivity at order m using Eq. (21) below and neglecting refraction. It has been shown [18-221 that one can describe the reduction of the reflectivity, even for the case of very small grazing angles of incidence (do = 90"), where refraction and reflectivity are large, with a small modification of Eq. (12): r(ql, q d

= roe

-o.sq,

q2"2

,

(13)

where ql and q2 are defined in the two media far from the boundaries. The reduction factor for the intensities-the square of Eqs. (12) or (13)-is called the Debye-Waller factor and was originally derived to describe the reduction of the x-ray diffraction peaks by thermal motion of the atoms [23-251. While the q values are complex numbers for absorbing media, one sometimes uses only their real part to calculate the reflectivity reduction. One can also calculate the influence of the transition layer on the reflectivity by dividing it into very thin homogenous films with an index distribution that describes the transition. Inserting the reduced reflectivity values from Eqs. (12) or (13) into Eqs. (7) or (8) reduces the computation time. It is easy to write computer programs that calculate the performance of any multilayer structure, and personal computers are sufficiently fast to give results within seconds. Programming is especially convenient with modern high-level mathematics packages. Arithmetic with complex matrices is often included as a building block, making the matrix methods easy to program and fast to run. Programs using the recursive methods are slower because they require a loop from layer to layer. For good speed one should use a compiled program for the recursive method. 14.2.4 Boundary Roughness and Diffuse Scattering

A two-dimensional (2-D) Fourier transform of the deviation of the boundary heights z(?) from its mean gives the 2-D power spectral density (PSD) of the

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MULTILAYER THEORY

boundary roughness as a function of spatial frequency period A, = 1 , Ay = lYy:

vx

= (f x

,f y ) or spatial

Each spatial frequency in the roughness spectrum scatters radiation in a direction determined by

It,,,

=

It" + m d s ,

(15)

where is proportional to the photon momentum (lkl = 2nnnlA) and ds= 2nf is the momentum parallel to the surface that is transferred to the photon from this Fourier component of the roughness. Equation (15) is equivalent to the grating equation sin bout- sin @in

=

mAIA,.

(16)

For small roughness there is only scattering into the first order (m = _t 1) and we have a one-to-one relationship between each spatial period and the direction of the scattered radiation. The total roughness of a surface that is isotropic is given by

withf2 =L2+f,'. All experiments measure roughness only over a finite bandwidth and the value obtained is that obtained from Eq. (17) over that finite band. The band is limited by the resolution and image size in microscopy and by the range of angles in a scattering experiment. Measurements of scattered light have a theoretical limit for the smallest period of A12 for a scattering angle of 180"; the experimental limit is in most cases determined by the lowest detectable intensity and is around 50 nm for the boundaries in good multilayer mirrors. All spatial periods, even those smaller than A12 up to a largest period, determined by the acceptance angle of the detector, reduce the reflectivity. Large spatial periods diffract only into very small angles from the scattered beam and are not easily distinguished from the specular reflectivity in an experiment. The amount of light scattered from a surface with small roughness into a range of small angles from specular can be estimated from the bidirectional reflectivity distribution function (BRDF) [26-3 I]:

where q5i and bs are the incident and scattering angles, respectively, in the plane

278

REFLECTING OPTICS: MULTILAYERS

of incidence and R(q$) and R(&) are the specular reflectivities for these angles. The cos 4 factors describe the reduction in phase shift produced by a change in heights for off-normal incidence and the change in the widths of the scattered beam. Polarization effects have been omitted in Eq. (1 8); they can be neglected at grazing and near-normal incidence and only become important at intermediate angles. Calculating the amount of scattering from a multilayer structure is considerably more difficult than the calculation of the specular reflectivity and most authors have used approximations in their theories [32-351. The usual multilayer calculation including a value CT for the width of the boundaries is used to calculate all specular amplitudes ai and bi (see Fig. 2) within the multilayer structure. The amount of scattering of these amplitudes at the boundaries is obtained from the PSD of the boundaries and each scattered amplitude is propagated to the surface of the structure. For the addition of the scattered amplitudes from different layers it is important to know the degree of correlation between the roughness of different boundaries. For perfect correlation (i.e., all boundaries have the same shape, there is perfect replication of the roughness from layer to layer) the phase differences between the waves scattered from different boundaries are the same as those for a specular beam of the same direction. A scan through the scattered radiation (with a fixed input beam) shows similar interference structure as the specularly reflected beam at that angle. When there is no correlation between the contributions from different boundaries the scattered waves are added with random phases, which is equivalent to just adding the intensities. Practically no interference structure due to the multilayer is visible in the scattered field. The strength of the interference structure has been used to determine the degree of correlation between the roughness of different boundaries. Long spatial periods are usually replicated from layer to layer, producing strong correlation while small period roughness is uncorrelated. The degree of correlation between boundaries has been determined for many systems and the transition between uncorrelated and correlated roughness occurs at periods around 10 nm [36-411. Theoreticians often prefer to use autocorrelation functions instead of power spectra to characterize rough surfaces [32,42]. While that approach is in principle equivalent-the autocorrelation function and the power spectrum are Fourier transform pairs-the power spectrum is much more useful for experimental data [39,43,44]. Instrument resolution enters as a multiplication factor in the power spectrum and can easily be recognized and corrected for in a plot of the data; it is straightforward to combine partial power spectra obtained from different instruments. Scattering is directly given by the PSD at the spatial frequency that corresponds to the scattering angle. When using autocorrelation functions one usually has to extrapolate the experimental data with an analytical function to calculate the amount of scattering.

MULTILAYER DESIGN

279

14.3 Multilayer Design 14.3.1 High-Reflectivity Mirrors

The standard design for high reflectivity is the quarter-wave stack. One deposits two materials of different refractive index (high = H, low = L) on top of each other. The thickness of the layers is selected such that all boundaries add in phase to the reflected wave. This requires that the phase delay in propagating each layer [Eq. (5)] is 90" or 114, producing a round-trip propagation delay of 180"; because r12= - r 2 1 , a 360" phase shift occurs between the reflected amplitudes from adjacent boundaries. For normal incidence the optical thickness of each layer is nd = 114, Selecting two materials with a large difference in the refractive index produces a large reflected amplitude at each boundary, reduces the number of layers required for good reflectivity , and increases the spectral or angular bandwidth. For the case in which all layers are absorption free one can obtain a reflectivity very close to 100% by using a sufficiently large number of layers, even if the refractive index difference between layers is small. A first estimate for the number of layers required is obtained from NrI2= 1 and for the spectral resolution from 1lA1 = N. The quarter-wave stack is the design of choice for high-reflectivity mirrors in the visible region and is used in the UV region for wavelengths 1 > 150 nm. At shorter wavelengths good-quality, absorption-free high-index materials are no longer available, and for A< 110 nm no absorption-free material is available. Absorbing materials require a modified design. One can understand the basic design ideas by noting that the superposition of the forward and backward running waves (aiand b, in Fig. 2) produces a standing wave field inside the multilayer structure and that the absorption of a thin, strongly absorbing material can be very small if it is located at a node of this standing wave field. Alternating thin absorber films with spacer layers of low absorption located around the antinodes allow one to produce multilayer structures with minimized total absorption and maximum reflectivity. In these structures, all periods still add in phase, but the two boundaries within one period are not in phase. The optimum thickness of the two materials can easily be found numerically. One usually defines the ratio* y = dH/(dH+ dL),where dH and dL represent the thickness of the reflector (H) and spacer (L) layer. For a large number of layers the optimum ratio can be estimated from [45]

+

*The definition y = q H d H / ( q H d H qLdL)is more suitable for the discussion of the performance of a structure when one varies wavelength or incidence angle, and we will assume this definition when discussing multilayer performance.

280

REFLECTING OPTICS: MULTILAYERS

which for the case PLG PH can be approximated with

In general the optimum thickness ratio depends on the number of layers in the structure and on the location in the stack. The standing wave is most pronounced near the top of a good reflector where the ai and bi (see Fig. 2) are of the same magnitude. The absorber layers near the top will be thinner to take advantage of the low intensity near the nodes for reduced absorption; however, there is barely any modulation of the intensity by the standing wave near the bottom of the structure and film thickness is close to a quarter-wave there. Figure 3 shows an example for the change of y in such an optimized design. From the bottom to the top of the stack the thickness ratio changes from the quarter-wave stack ( y = 0.5) to a value close to y = 0.31, the value obtained from Eq. (19) for the optimum periodic design with a large number of layers. The reflectivity and the ratio y saturate when the absorption in the spacer layer prevents radiation from reaching the deeper layers. Therefore, one has to select the material with the lowest absorption at a specific wavelength if one wants to reach high reflectivities. Table I is a list of such best materials at their respective absorption edges where they have the smallest absorption. The table also gives the number of periods N,, that radiation can travel through each of the materials at normal incidence (see later discussion). In, practice the reflectivity curves R(A) for the design illustrated in Fig. 3 and the optimum periodic design are indistinguishable, because the bottom of the stack, where the two designs differ, contributes very little weight to the reflectivity. The differences between the two designs become significant at longer Mo-si, ~ = i 3 5 A

0

20 40 LAYER NUMBER

i

n

60

FIG.3. Calculated increase of the reflectivity (full curve) versus the number of layers for a Mo-Si multilayer mirror, where the thickness ratio is optimized in each period to give the largest increase in reflectivity [18].

28 1

MULTILAYER DESIGN

TABLE I. Absorption Index /3, Linear Absorption Coefficient 01 = 4np/l, and N,,, at Normal Incidence for Good Spacer Materials at Their Best Wavelength Near Absorption Edges

Mg-L Al-L Si-L Be-K Y-M B-K C-K Ti-L N-K sc-L v-L 0-K Mg-K AI-K Si-K Sic TIN Mg2Si Mg$i

25.1 17.1 12.3 11.1 8.0 6.6 4.37 3.14 3.1 3.19 2.43 2.33 0.99 0.795 0.674 0.674 3.15 25. I 0.99

1.74 2.7 2.33 1.85 4.47 2.34 2.26 4.5 1 1.o

3.0 6.1 1.o

1.74 2.7 2.33 3.2 5.22 1.94 1.94

7.5 x 4.2 x 1.6 x 1.0 x 3.5 x 4.1 x 1.9 x

IO-~ IO-~ 10-~ lo-’ 10-~ IO-~ IO-~ IO-~ IO-~ 10-~ IO-~ lo-’

4.9 x 4.4 x 2.9 x 3.4 x 2.2 x 6.6 X 6.5 X 4.2 X 6.2 X 4.9 x 1 0 - ~ 7.4 x lo-’ 6.8 X

3.8 3.0 1.6 1.16 5.5 0.78 0.53 1.9 0.18 1.16 1.75 0.12 0.084 0.10 0.078 0.1 1 1.96 3.7 0.09

21 38 99 155 45 390 850 327 3,580 557 469 7,230 24,000 24,500 38,000 25,500 324 21 23,200

wavelengths (2 > 20 nm), where fewer layers are needed and absorption is stronger. The designs for high reflectivity differ in the thickness distribution within one period but fulfill the Bragg condition for the layer pair to ensure that all periods add in phase:

mi = 2neffAsin Oi,

(21)

where A is the period thickness, neff is the effective refractive index of the multilayer material, and Oiis the propagation angle from grazing in the structure. We can replace the angle 8, with the grazing angle of incidence in vacuum using Snell’s law [Eq. (3)] and obtain

mi

=

2AsinBo

J

26 1-sin2 o0 ’

for 6 G 1, p 4 6.

The value 6,ff in Eq. (22) is the weighted average index of the materials in the structure:

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REFLECTING OPTICS: MULTILAYERS

14.3.2 Multilayers as Filters Multilayer mirrors reflect only within a wavelength band around the Bragg peaks, and the reflectivity curve can be tuned in A by changing the angle of incidence. The width of the reflectivity curve is determined by the number of contributing boundaries and a large bandwidth can simply be obtained by using fewer layers. The smallest bandwidth or the largest possible number of contributing layers is determined either by absorption or by depletion of the incident beam due to reflection. Adding layers beyond that limit does not change the reflectivity curves because the bottom layers are not reached by radiation. The limit by depletion of the incident beam can be changed by changing the reflectivity of each period with an adjustment of the thickness of the reflector layer. In analogy to x-ray diffraction, the reflectivity of the reflector layer in one period is called the formfactor of the structure with the value

Rf = 4RI2sin2mny.

(24)

This equation is obtained from Eq. (7) with the assumption r,, rb < 1, using r, = -rb, and selecting the period A to fulfill the Bragg condition [Eq. (22)] for order m. Reduction of y to y < 0.5 for m = 1 produces smaller reflectivity per period, deeper penetration toward the bottom of the stack, and a corresponding reduction in bandwidth because more layers contribute. The limit on the penetration depth for this case is due to absorption in the spacer layer and defines the smallest bandwidth or highest resolution: N,,,

sin2 8 2np ’

= -.

-A_ - N m a x . AA

The largest bandwidth is obtained with a coating close to the quarter-wave stack ( y = 0.5) where all boundaries add in phase, and we need

layers to saturate the reflectivity. Selection of materials with the largest possible Fresnel coefficient produces the broadest reflectivity curves. The Fresnel reflection coefficients depend on the angle of incidence and on the polarization. For s polarization the reflectivity of a single boundary increases monotonously with the angle of incidence (or decreases with increasing grazing angle) to a value close to 100% at the critical angle. Therefore, a mirror designed to operate at small grazing angles can have high reflectivity and a large bandwidth even at a very short wavelength. With the angle of incidence as a free parameter we can produce mirrors with a spectral or angular resolution between at normal incidence (see Table I). 1 near the critical angle and the value of NmaX Very high resolution above lo4 is theoretically possible at x-ray wavelengths

MULTILAYER DESIGN

283

and is realized in the Bragg reflection from crystals. Due to roughness of the interfaces the reflectivity of deposited, amorphous, or polycrystalline multilayers becomes too small for most applications for periods A < 20 A. The resolution of multilayer mirrors is usually in the 50-100 range with a maximum around 250 [46,47]. Substantially higher resolution than is possible with a multilayer mirror can be obtained with multilayer gratings. The blazed grating in Fig. 4 can be described in two ways: It can be seen as a multilayer that contains steps, making it possible to reach deeper into the structure by eliminating the depletion of the incident beam in the deeper layers. It can also be seen as a blazed grating that is overcoated with a multilayer in such a way that all periods of grating and multilayer add in phase [48]. The resolution is given by the maximum phase difference between interfering beams and is equal to the number of steps times the step heights expressed as a multiple of A; this value is m = 5 for the example of Fig. 4. In the picture where the grating is the primary element, the resolution is the number of grating lines times m, and the multilayer just enhances the reflectivity. Chapter 18 gives a detailed discussion of multilayer gratings.

14.3.3Supermirrors Mirrors with very large bandwidth beyond the limit given by Nmi,in Eq. (26) can be obtained by depositing multilayers with different periods, one for each

FIG. 4. A blazed grating overcoated with a matching multilayer structure can be described as a thick multilayer with steps in order to reach deeper layers and higher resolution. The drawing, where the step height is five multilayer periods, corresponds to a blazed grating used in the fifth order [48].

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REFLECTING OPTICS: MULTILAYERS

desired wavelength band, on top of each other. Absorption limits this procedure, or SIP is a quality measure for the design freedom. The and the ratio NmaxlNmin ratio is too small in the W V or XUV region for practical use. Where absorption-free materials are available, for example for visible light or cold neutrons, such mirrors are common, and the term supermirrors has been coined for them. The ratio SIP increases with higher photon energy and it becomes practical to produce such mirrors for x-rays for photon energies above 10 keV. The main application is the extension of the photon energy of grazing incidence optics beyond the critical angle. An example of such a mirror is a graded period W/Si mirror with 1200 layers and a reflectivity larger than 30% for photon energies between 20 and 70 keV at a grazing angle of 3 m a d [49, 501. 14.3.4 Multilayer Polarizers, Phase Retarders, and Beamsplitters

Any mirror used at nonnormal incidence has lower reflectivity for p polarization than for s polarization and can be used as polarizer. The minimum p polarization reflectivity at the Brewster angle occurs close to 45" in the VUV region, where the refractive index is close to one. The reflectivity for p polarization is zero at the Brewster angle for absorption-free materials, but increases with absorption. Therefore, the achievable degree of polarization is higher at the shorter wavelengths where absorption is smaller. Typical values for RJR, are 10 around 1= 30 nm and over 1000 around 1= 5 nm. However, the reflectivity of single boundaries is much too low (R = lop4for s polarization and 1= 5 nm) to make such polarizers useful. Multilayer structures change the theoretically possible degree of polarization very little, but can enhance the reflectivity to useful values [48]. Therefore, all multilayer x-ray mirrors are efficient polarizers when designed for high reflectivity at the Brewster angle. Phase retarders are more difficult to realize. The geometrical phase differences in a multilayer structure do not depend on polarization, and one cannot produce a reflector with high reflectivity for both polarizations and a 90" phase delay [51, 521. The situation is different in transmission. The p polarization passes the structure without reflection near the Brewster angle and is just attenuated by absorption. Some of the s polarization bounces back at each boundary. If we use the multilayer structure off-resonance, for instance, at one of the side minima in the reflectivity curve, we also have high transmission for s polarization, but due to the internal reflection, s-polarized radiation is delayed [53]. By selecting the number of layers and the materials one can find designs that produce 90" phase delays. Using the transmission maximum of a FabryPerot interference filter is another possibility. It is, of course, necessary to fabricate these structures on thin transmitting substrates or use them in a selfsupporting way. Multilayer phase retarders have been used as polarimeters in the 100-eV range [52,54-571.

MULTILAYER FABRICATION AND PERFORMANCE

285

Roughness of the boundaries reduces the reflectivity at each boundary and the phase retardation and has to be included when the performance of a design is calculated. Roughness makes it more difficult to reach large delays at higher photon energies. At 13. = 50 8, only a phase delay of 5" has been obtained up to

now [58].

14.4 Multilayer Fabrication and Performance Every thin-film deposition method can be used to fabricate multilayer x-ray mirrors. Thickness errors and boundary roughness are the most important parameters that have to be kept smaller than about N 1 0 for good performance. Thickness errors can be controlled as required in many systems. The deposition rate in sputtering systems [59] can be very well stabilized, and thickness control just by timing has produced multilayer structures with thickness errors below 0.1 8, per layer. Thermal deposition systems have larger variations; in these TABLE11. Multilayer Systems and Their Performance in the VUV

L > 150 nm 3, = 110-150 nm

1 < 110 nm 1= 80-11Onm

1 < 700 nm 1<20nm

A = 12.5-20 IUII

1 = 11.612.5 nm 1<10nm

A = 6.7-10

nm

1 = 4.5-6.7 nm a < 4.5

h
Absorption-free quarter-wave stack, R > 95%. A1-MgF2 mirrors and interference filters in transmission. Good absorption-free high index materials not available. MgF2, LiF still transparent. Mirrors: Al with MgF2 or LiF protection, R = 90%. Transmission filters: Al-MgF, FabryPerot. No absorption-free material available. Good reflection from unoxidized Al and Be; no protective overcoat available; p = 0.1-1 for all materials except alkali metals. Multilayers can not enhance reflectivity of clean A1 or Be, but can be used to suppress an undesired J. more than a desired 1. p < 0.01, decreases with decreasing 2 faster than 8. Multilayers can enhance reflectivity of best metals. Large enhancement of reflectivity with multilayers. For single surface R A4, but multilayer with large N can compensate for the loss in reflectivity. R = 67% with Mo-Si multilayers. R = 70% with Mo-Be multilayers. Boundary roughness affects the performance at normal incidence; high reflectivity can be maintained off-normal. R = 25% at normal incidence with B4C spacers. R = 10% at normal incidence with C spacers. R < 5% at normal incidence, higher off normal. 6//3 1. High-quality multilayers with little absorption possible, but roughness requires A > 2.5 nm and use at grazing incidence. R > 80% for 1 < 0.2 nm. Used as collimators and filters for synchrotron radiation and x-ray tubes.

*

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systems one can prevent the accumulation of thickness errors with in situ monitoring of the soft x-ray reflectivity during deposition [60]. A thickness error in one layer can be compensated for in the next layer and the accumulated error can be kept under 2 A, even after the deposition of hundreds of layers. Reduction of boundary roughness is a bigger challenge. The random arrival of atoms produces large roughness at high spatial frequencies and it is important to reduce that roughness as much as possible to get good reflectivity. Increased substrate temperatures, high kinetic energies of the evaporant, bombardment of the growing film with the sputter gas, or polishing the films with an ion beam are methods that have been used to move atoms away from the peaks of a surface and reduce roughness. Good multilayer mirrors have roughness values around 0 = 0.3 nm. We want to note that roughness values reported in the literature cannot be compared easily, because different authors or the same authors at different times use different methods to analyze their structures. A summary of the types of multilayer coatings used in the W V region and their performance is given in Table 11, and Fig. 5 is a plot of the peak normal incidence reflectivity achieved up to now. The drop in peak reflectivity toward shorter wavelengths for A < 10 nm is due to the increased influence of boundary roughness, whereas the drop at long wavelength (A > 13 nm) is due to the

0.6 x

2 L

0.4

I0 W

A

;0.2 LL

$ 0

40

100 WAVELENGTH

(A)

200

FIG.5. Achieved peak reflectivity near normal incidence of multilayer structures with spacer layers of C, 00; B4C, 00; Y , VV; Si, AA; Sc, 00; TiN, AA. (From a database at www-cxro.Ibl.gov/multilayer/survey.htmland [ 181.) The full curves are calculated; the two curves for Co-C are for 70 and 150 periods. The reflectivity of 10% at 3, = 3.14 nm (0) represents a 370-layer mirror of Cr-Sc with a period A = 16.3 p\ that was tuned to the region of anamalous dispersion of Sc by tilting it to 18" from normal [611.

REFERENCES

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increased absorption of the Si spacer layer. There is also a drop at the shortwavelength side of absorption edges (see Table I) where the best spacer layer material changes. One can maintain high reflectivities for shorter wavelengths than those in Fig. 5 by keeping the multilayer period fixed (around A > 3 nm) and tuning the Bragg peak toward shorter wavelengths by increasing the angle of incidence. At short wavelengths very high reflectivities are routinely being obtained at grazing incidence, for example, R > 80% for A = 1.54 A.

References 1. 2. 3. 4. 5. 6. 7.

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