HEW simulations and quantification of the microroughness requirements for X-ray telescopes by means of numerical and analytical methods

Transcription

1 HEW simulations and quantiication o the microroughness requirements or X-ray telescopes by means o numerical and analytical methods D. Spiga a*, G. Cusumano b, G. Pareschi a a INAF/ Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-387 Merate (LC), Italy b INAF/Istituto di Astroisica Spaziale e Fisica Cosmica, Via La Mala 153, I-9146 Palermo, Italy ABSTRACT Future X-ray telescopes like SIMBOL-X will operate in a wide band o the X-ray spectrum (rom.1 to 8 kev); these telescopes will extend the optical perormances o the existing sot X-ray telescopes to the hard X-ray band, and in particular they will be characterized by a angular resolution (conveniently expressed in terms o HEW, Hal-Energy- Width) less than arcsec. However, it is well known that the microroughness o the relecting suraces o the optics causes the scattering o X-rays. As a consequence, the imaging quality can be severely degraded. Moreover, the X-ray scattering can be the dominant problem in hard X-rays because its relevance is an increasing unction o the photon energy. In this work we consistently apply a numerical method and an analytical one to evaluate the X-ray scattering impact on the HEW o an X-ray optic, as a unction o the photon energy: both methods can also include the eects o igure errors in determining the inal HEW. A comparison o the results obtained with the two methods or the particular case o the SIMBOL-X X-ray telescope will be presented. Keywords: X-ray telescopes, Hal-Energy-Width, X-ray Scattering, Power Spectral Density 1. INTRODUCTION The launch o a number o X-ray observatories, in addition to those already operating (Chandra, XMM-Newton, SWIFT), is oreseen in the near uture. Some o them, like SIMBOL-X 1, Constellation-X/HXT, NeXT 3, XEUS/HXT 4, will extend the current techniques or X-ray ocusing (to date, adopted only below 1 kev) to the hard (E > 1 kev) X-ray band by adopting very shallow incidence angles (.1.4 deg) and graded multilayer coatings 5,6. For example, the SIMBOL-X sensitivity band is.5 8 kev, with a required angular resolution o 15 arcsec. Other X-ray observatories will instead take the advantage o a very large Field o View (1.5 deg diameter or the Wide Field Imager instrument aboard EDGE 7 ) in a more limited energy band (.3 6 kev), made possible by the adoption o optics constituted by mirrors with polynomial proiles 8,9. These mirror proiles enable, at the expense o a slight degradation o the on-axis angular resolution, a much better o-axis imaging quality with respect to the widespread Wolter-I mirrors,. An important point or all these instruments is the request o a good angular resolution, heretoore expressed in terms o HEW (Hal Energy Width). The optics, in particular, should be characterized by imaging perormances comparable to those o XMM-Newton (15 arcsec HEW) and SWIFT-XRT ( arcsec HEW). For this reason, the actors that could lead to a degradation o the HEW have to be careully analyzed and corrected. Departures o the mirror proile rom the nominal one ( igure errors ), and mirrors misalignments, are a requent actor o imaging degradation in X-ray optics. As ar as the optical path deviations introduced by these deects are much larger than the photon wavelength,, the ocal spot blurring can be treated with geometrical optics tools. Thereore, the impact o the deormations on the HEW - considered as independent o - can be evaluated along with ray-tracing codes. * daniele.spiga@brera.ina.it, phone , ax

2 Another breakthrough, however, can occur as a consequence o relecting suraces microroughness. When X-rays impinge at a grazing-incidence angle θ i on a rough surace with rms σ, its relectivity decays exponentially, ollowing the well-known Debye-Waller ormula 5 : 16π sin ϑ = iσ R σ R exp, where R σ is the measured relectivity in the direction specular to that o incidence and R is the relectivity as computed rom the Fresnel equations (i.e., or an ideally smooth mirror). Besides the relectivity reduction, the missing X-ray photons are scattered in the surrounding directions. X-ray Scattering (XRS) rom rough suraces is a well-known and studied eect 1,11 that can seriously degrade the angular resolution, because it is an increasing unction o the photon energy E. Thereore, the XRS contribution to the HEW is o increasing relevance as one moves to the hard band o the X-ray spectrum, where it can even dominate the igure error term. XRS is a surace diraction eect, taking place whenever the optical path dispersion due to the surace relie becomes comparable to. For this reason, the limit between igure errors and XRS is not ixed but depends on θ i and. A possible criterion (proposed by Aschenbach 1 ) considers the rms o each Fourier components o the surace. I this component ulills the smooth-surace limit, i.e., 4πσ sinθ i <, is dominated by roughness; otherwise, it should be mainly considered as geometric deormation. We plot in Fig. 1 the limiting σ values that mark the boundary between the two regimes, in the energy and incidence angles range o interest or X-ray telescopes. Considering the typical relations between incidence angles and photon energies in X-ray optics, the boundary on the rms or a single spatial requency lies between.5 and nm or the telescopes SIMBOL-X, EDGE and XEUS. (1) Fig. 1: separation o the mirror igure error and XRS to the HEW. Angles and energy ranges or the SIMBOL-X, EDGE and XEUS X-ray telescopes are also indicated. We did not highlight the ull energy ranges o the telescopes to improve the igure readability. In order to keep the HEW o X-ray optics or imaging telescopes at the required level or a scientiic relevance o the images, clear requirements on the mirror igure and microroughness have to be stated. In the ollowing we shall ocus on the problem o deining surace roughness tolerances, by means o the simulation o the expected HEW o an X-ray mirror module. To this end, the surace roughness will be conveniently expressed in terms o its Power Spectral Density 11 (PSD), because it is the physical quantity that directly determines the intensity and the distribution o the XRS. The PSD provides us with a detailed characterization o the surace roughness amplitude as a unction o the spatial requency (or, equivalently, the spatial wavelength l = 1/). The rms roughness σ in a given spectral band Δ can be obtained by integration o the PSD P(): σ = P ( ) d. Δ ()

3 The σ parameter will thereore depend on the particular spectral band o interest or sensitivity. A requent PSD modelization or an optically-polished surace is a power-law spectrum 11, 1 : K n P ( ) =, n where n is a real number taking on values in the interval (1: 3) and K n is a normalization constant. The roughness power spectrum o several substrates or X-ray mirrors well approximates this ormula 14 in a wide spatial requency range. X- ray mirrors, indeed, can deviate rom this model, especially i a relective coating has been deposited onto the substrate, owing to the roughness growth 15 that may occur in a variable extent in the high-requency range. It should be noted that the σ parameter is not suicient to describe the surace aspect or to return quantitative inormation o the HEW: suraces with the same rms can be very dierent (see Fig. ) and the same σ in dierent requency ranges can have a completely dierent impact on the imaging degradation. For instance, given a ixed σ value, a smoothly-decreasing PSD (n 1) will comprise more high-requency roughness than a steeply-decreasing one (n 3), and since high requencies are responsible or large-angle XRS, it will cause a aster increase o the HEW with the photon energy. This is a simple consequence o the grating ormula 1, that relates the scattering/incidence angles (θ s and θ i ) to : cosϑi cosϑs =. (4) The evaluation o the HEW rom a surace PSD has been perormed in the past years by several authors (e.g., De Korte et al. 16 ; Christensen et al. 17 ; Harvey et al. 18 ; O Dell et al. 19 ; Willingale ; Zhao and Van Speybroeck 1 ) using dierent methods to address the computation o the X-ray scattering distribution, to be convolved with the igure errors to return the mirrors PSF, and consequently the HEW. These methods, though very accurate, require the calculation o the whole PSF or a single photon energy, and thereby require a considerable computational eort. Hence, it would be advantageous to ind out a proper methodology to iner the HEW rom a direct calculation on the characteristic parameters o the surace roughness. In this work we attempt to provide with quantitative predictions o the HEW or the SIMBOL-X optical module, as a unction o the photon energy, starting rom given assumptions on the surace inishing level. The most innovative eature o the SIMBOL-X 1 X-ray telescope will be the unprecedented extension o imaging capabilities to the hard X-ray band up to 8 kev. Even i this extension will be mainly made possible by the adoption o graded multilayer coatings, the incidence angles will have to be very shallow (see Tab. 1) to ensure a good relectivity over the photon energy band o sensitivity. As the mirror module diameter is similar in size to XMM (7 mm), the ocal length will be very long ( m). With the present technology, such a ocal length cannot be managed with a single spacecrat, thereore the ormation-light coniguration has to be adopted. Since the energy band o SIMBOL-X is extended up to the hard X-ray energy band, the mirrors surace smoothness has to be much better than or the case o XMM-Newton (σ ~ 7 Å) to ulill the arcsec HEW requirement at 3 kev (see Tab. 1), which is needed to resolve astronomical targets like the hard X- ray background (XRB) in the peak region. For this reason, the quantiication o the HEW due to X-ray scattering is a very important issue in order to establish the surace smoothness tolerance to be required or the SIMBOL-X mirrors. (3) Å Å Fig. : two dierent simulated suraces with the same size (1 µm) and rms (1 Å), but PSD with dierent spectral indexes:.3 (let) and 1.4 (right). The larger high-requency content is apparent in the second case. Thereore, the imaging degradation will be more severe at high energies.

4 Tab. 1. Main eatures o the SIMBOL-X optical module, compared to those o XMM-Newton XMM-Newton SIMBOL-X Focal length 7.5 m m Min diameter 3 mm 3 mm Max diameter 7 mm 7 mm Min incidence angle.8 deg.11 deg Max incidence angle.67 deg.5 deg Number o shells 58 1 Energy band.1 1 kev.5 8 kev Eective area (1 kev) ~ 14 cm² ~ 14 cm² Eect. area (3 kev) - ~ 45 cm² Required HEW (1 kev) 15 arcsec 15 arcsec Required HEW (3 kev) - arcsec To this end, we shall either make use o a numerical routine or ollow an analytical approach. The numerical code (Sect. ), based on the treatment discussed by Green et al. 3, was written by Sacco et al. (1996) and integrated in a raytracing FORTRAN program to interpret the PSF o the Beppo-SAX X-ray telescope on the basis o simple assumptions or the surace roughness o the X-ray mirrors. The analytical approach (Sect. 3) has been developed by one o us (D. Spiga 4 ), and enables an immediate translation o a surace PSD into the expected XRS term o the HEW. In addition, it enables the reverse computation or a single mirror shell (rom the HEW trend to the PSD). In Sect. 4 we deal with a comparison o an application o the two methods to the SIMBOL-X optical module. The results are briely summarized in Sect. 5.. COMPUTATION OF THE SIMBOL-X HEW: MODIFIED RAY-TRACING ROUTINE The simulation o the imaging degradation caused by mirrors deormations can be oten perormed along with X-ray tracing routines, simulating a set o rays that ollow the geometrical optics laws. The same approach cannot be used to account or XRS, because the concept o ray becomes no longer applicable. However, the irst-order approximation XRS theory 1 can be applied to correct the enlargement o the HEW obtained rom usual X-ray routines as ollows: 1. given a proper PSD (or, equivalently, a sel-correlation unction) modelization, the scattering intensity or X- ray photons (impinging on the optic at the incidence angle θ i ) is calculated at each scattering angle θ s.. each photon used in the ray-tracing simulation is assigned a scattering likelihood proportional to the intensity o the X-ray scattering distribution. The adopted procedure 3, rather than a PSD, takes as input two characteristic parameters o the surace roughness, that can be derived rom the PSD. The rms roughness σ (Eq. ) and the surace correlation length τ, that can be deined as the average spatial wavelength o the surace micro-relie. The parameter τ in a spatial requency window Δ can be computed along with the ormula τ = πσ Δ m Δ Δ, (5) where m Δ is the rms slope, measured in radians. This parameter is an index o the steepness o the rough eatures o the surace, and can be calculated rom the PSD in the spectral band Δ: Δ = m (π ) P( ) d. Δ (6)

5 We can expect that the HEW, at a given photon energy, is an increasing unction o the σ parameter. Similarly, as a smaller τ indicates that the roughness is mainly concentrated at high requencies (that scatter X-rays at higher angles), the HEW is expected to increase as τ decreases. The numerical routine has been applied to 5 couples o σ and τ parameters (σ in the interval 3-4 Å, τ between 11 and µm) or the entire SIMBOL-X module. The HEW has been directly derived rom the simulated photon distribution on the ocal plane, assumed to have a 31 arcsec radius, similar to that o SIMBOL-X. The HEW trends, rom 1 to 6 kev, are plotted in Fig. 3. To account or possible mirror proile deormations, 15 arcsec igure error were added in quadrature to the calculated X-ray scattering terms. The calculated trends exhibit a clear increase o the HEW with the photon energy, as expected. At a ixed photon energy, the HEW is an increasing unction o the rms roughness, and is particularly sensitive to small variations o the correlation length. These results are in agreement with our qualitative discussion. It is worth noting that the applied program is currently being used also to quantiy the impact o the stray light in the SIMBOL-X optics, that also strongly depend on XRS caused by surace roughness. The evaluation o this eect, as well as the optimization o methods to reduce it, is also described in this volume 5. Fig. 3: results o the application o the numerical routine to the SIMBOL-X optical module, or dierent σ and τ values. 3. ANALYTICAL DERIVATION OF THE HEW FROM THE PSD Another possibility to predict the optical perormances o an X-ray optical module is the analytical translation o a surace PSD into the expected HEW 4. This method, based on the well-known theory o X-ray scattering rom rough suraces 1, can be applied to any PSD, and holds or an optical system with an arbitrary number N o relections at the same grazing incidence angle θ i. Moreover, it is reversible: that is, rom a given HEW() requirement one can derive a corresponding mirror surace PSD unction, that can be assumed as microroughness tolerance or the mirrors surace. We consider irstly a single mirror. The application o this method requires that: 1) X-ray relection and scattering occur in grazing incidence, so that the XRS chiely lies in the plane o incidence. ) The mirrors surace is smooth, so that the XRS theory 1 can be applied, and the scattering angles are always small when compared with the incidence angle. 3) The surace is coated with either a single layer coating, or a graded multilayer with a smoothly decreasing relectivity with the incidence angle and the photon energy. 4) The relective coating roughness is described by a single PSD (i.e., no roughness growth throughout the multilayer stack).

6 5) Mirror proile errors and microroughness act independently rom each other, so we can disentangle the mirror igure error HEW term rom the scattering term as ollows: HEW (") # H + H ("), (7) where H represents the HEW due to geometrical deormation, whereas the energy-dependent term H() denotes the HEW caused by the XRS. In the ollowing we shall see how the H() unction is analytically related to the surace PSD P(). 3.1 From the PSD to the HEW The H() unction or an optical system with N = 1,, identical relections can be computed rom the PSD P() along with the integral equation 4 / N P( ) d = ln, (8) 16π sin ϑ N 1 that allows one to calculate the lower integration limit. Mirror shells with Wolter-I proile all in the case N =. Then, or all N, H() is calculated along with the Eq. 4 in this approximate orm i i H( ) =. (9) sinϑ Application o these ormulae to all photon wavelength o interest returns the desired unction. The HEW trend or the mirror shell is then obtained by using the Eq. 7. We shall provide in Sect. 4 a detailed example o HEW computation rom a PSD, using the Eqs. 8 and 9. It is worth noticing that the Eq. 8 is derived supposing that the X-ray detector that collects the scattered photons is very large. This assumption justiies the upper integration limit cosθ i / /, that corresponds to a photon backscattering, even i the PSD at so large requencies usually is usually irrelevant or the calculation, because it has so small values that contributes negligibly to the integral in Eq. 8. However, in practice the detector size has always a inite angular radius r and scattered photons beyond r are lost, so they do not contribute to the imaging degradation (they cause, indeed, eective area loss). I the HEW is computed rom a EE unction being normalized to its maximum measured value, the upper integration limit in the Eq. 8 should be modiied as ollows: r sin ϑi r =. (1) The eect o this substitution is to diminish the HEW because the high-requency tail o the PSD is ruled out rom the integration, so one should integrate down to smaller to return the constant on right-hand side o Eq From the HEW to the PSD The Eqs. 8 and 9 can be inverted in order to derive the PSD rom a H() unction. We can immediately derive the requency at which the PSD is evaluated rom the Eq. 9: sinϑi ( ) = H ( ), (11) whilst the PSD at the requency at can be immediately computed 4 rom the derivative o the ratio H()/ : P( ) d H( ) 1 + d 4π sin 3 ϑ i N ln. (1) N 1 Thereore, i one wants to ind the surace PSD responsible or to a measured HEW trend in a deinite photon wavelength band, this equation can be very useul. Similarly, it can be used to translate a HEW requirement into a surace PSD, that can be assumed as roughness tolerance or the X-ray mirrors. For the Eq. 1 to be valid, we have to

7 suppose that the HEW has been computed rom the PSF over an angular range much larger than the HEW. For instance, i the X-rays range o interest is the interval [ M, m ] (with M > m ) and the corresponding H() values increase (monotonically) in the range [H m, H M ], with H M > H m, the PSD can be computed up to the maximum spatial requency or the validity o Eq. 1, it has to be r >> MAX. M H sin ϑ i = : AX M m It is important to note that, even i the Eq. 1 is able to provide with a PSD in a deinite spectral range (determined by the extent o the X-ray wavelength range and the HEW values), it does not return alone a complete surace characterization equivalent to the requested/measured HEW trend, because the PSD can be non-unique beyond MAX. Nevertheless, the Eq. 8 with = m and = MAX sets another constraint on the PSD at larger requencies than MAX : That relation is able to bound the PSD at high requencies, that inluences all the HEW trend, and in particular the one at higher photon energies. As beore, whenever the HEW becomes comparable to the detector size, the substitution / r should be done (Eq. 1). 3.3 The case o a ractal surace I the power-law model (Eq. 3) is suitable to extensively describe the surace PSD, it can be used to derive an analytical expression or the H() unction 4, by means o the Eq. 8. The most interesting result is that the HEW inherits the powerlaw trend rom the PSD, γ (13) sinϑi H( ), (14) and the spectral index γ is related to that o the PSD, n, along with the simple, algebraic relation 3 n γ =. (15) n 1 This result is interesting because the requirement 1 < n < 3 required by the theory o ractal suraces becomes equivalent to the statement γ >, indicating that the HEW is an increasing unction o the photon energy E 1/. Moreover, it highlights the relevant impact that the spectral index n has on the dependence. For example, i n 3, γ : the HEW becomes nearly energy-independent i the spectral index n goes close to its maximum allowed value. On the contrary, the case n 1 implies γ + and the increase o the HEW becomes very quick. Finally, a n = power-law index would cause a linear growth o the H() term (γ = 1). This discussion makes apparent the importance o a steeply-decreasing surace PSD in X-ray mirror design and abrication, especially in the hard (> 1 kev) X-ray band, where H() can dominate the igure error contribution. In general, the Eq. 14 is not valid or PSDs that are not power-laws. Instead, the most general relation between the local power-law index o the PSD, d(log P) n =, d(log ) and the corresponding local power-law index o the H() unction (the correspondence is established by the Eq. 9), is given by the more complex equation (16) d(log H) γ =, (17) d(log) 3 + γ dγ =, (18) 1+ γ (1 + γ ) d n that reduces to the Eq. 15 or constant γ (i.e. or a power-law PSD). However, i the HEW has only a inite number o power-law index changes, the derivative in Eq. 18 is zero almost everywhere. In other words, the Eq. 14 retains its validity also locally, except at requencies where power-law indexes change. The derivation o the Eq. 18 is postponed in appendix A.

8 4. COMPARISON OF THE TWO METHODS FOR THE SIMBOL-X OPTICAL MODULE A comparison o the results o the simulation described in Sect. and the results that can be obtained using the method described in Sect. 3 has been perormed. This will also allow us not only to cross-check the two dierent approaches, but also to return an estimation o the X-ray scattering impact on the HEW o the SIMBOL-X optical module. This will also enable the evaluation o tolerable surace roughness levels. The PSD taken in order to compute the HEW trend using the results o Sect. 3 is a simulation at low requency, aimed at returning a slowly-increasing HEW rom 15 arcsec at 1 kev up to arcsec at 3 kev. I we assume the HEW at 1 kev to be essentially due to igure errors, adopting a HEW interpolation o the two values and using Eqs. 8,13,14 (assuming N= and θ i =.18 deg, the incidence angle o an intermediate mirror shell o SIMBOL-X), we can derive a PSD rom the HEW trend. The PSD is plotted in Fig. 4 (marks) and covers only the low-requencies regime (spatial wavelengths l = 1/ > 48 µm). As mentioned in Sect. 3, the HEW requirement does not allow us to derive the PSD at higher requencies, unless we extend the required HEW trend at higher energies. However, in order to assign realistic values to the PSD, we adopt or the high-requency regime a measured PSD. The sample under test, assumed to be representative or the achievable smoothness levels in X-ray optics, is a Gold-coated X-ray mirror shell prototype sample, available at INAF/OAB. Metrological instrumentation operated at INAF/OAB (AFM, optical proilometers, XRS measurements at 8.5 kev) allowed us to characterize its roughness in terms o PSD at spatial wavelengths shorter than 6 µm. The merging o the simulated and measured PSDs (Fig. 4, solid line) returns a roughness description over a very wide spectral range. In addition, it represents a realistic surace inishing level at all spatial wavelengths considered here. The HEW scattering term was thereby derived rom the PSD in Fig. 4 via Eqs. 8 and 9. The computation was done rom 1 to 65 kev, or each SIMBOL-X mirror shell incidence angle. In order to make the results comparable with the simulation carried out in Sect., we accounted or the inite size o the detector (r = 31 arcsec) by modiying the upper integration limit (Eq. 1). Moreover, 15 arcsec were added in quadrature (Eq. 7) to the scattering term H() to allow or mirror igure errors: the same contribution was assumed in the simulations in Sect.. As an example, calculated HEW(E) or three SIMBOL-X mirror shells are plotted in Fig. 5. Notice that the HEW trends exhibit an initial saturation, ollowed by a steep divergence at the highest energies. This can be ascribed to the slope change o the PSD at ~ µm (see Fig. 4), that causes a sudden increase o the local spectral exponent o the HEW, γ (Eq. 18), when becomes suiciently small to set the requency in the smoother part o the PSD. Fig. 4: the PSD being utilized to simulate the HEW o the SIMBOL-X optical module. The PSD at low requencies (triangles) is derived rom the requirement that the HEW increases slowly rom 15 arcsec at 1 kev up to 3 kev. The remaining high-requency PSD is measured rom a Gold-coated X-ray mirror.

9 Now, the HEW T o the entire optical module can be approximately calculated by averaging the HEW o each mirror shell, HEW k, over the respective eective areas A k (E), k =1 1: HEW ( E) T 1 HEW ( E) A k K = 1 = 1 K = 1 k k A ( E) The approximate validity o this assumption is also proven in another paper o this volume 6. Notice that the eect o the average over the eective areas is a partial compensation o the HEW divergence, because the mirror shells with the largest incidence angles (that enhance the XRS) have also the lowest energy cut-o in the relectivity. Thereore, the contribution to the HEW o the largest shells at high energies is relatively modest. It is worth noting that the calculated HEW is 15 arcsec at 1 kev and equals arcsec at 3 kev, close to the required arcsec or SIMBOL-X at that energy. ( E). (19) Fig. 5: the HEW as computed rom the PSD, or three mirror shells o SIMBOL-X at the incidence angles o:.11 deg (the smallest one).18 deg (the 55 th shell) and.5 deg (the largest one). The HEW increases with the energy and the increase is more marked at the largest angles. The oscillations in the HEW are caused by small luctuations o the PSD. Fig. 6: comparison o the HEW trend as derived via the modiied ray-tracing routine (marks) or several couples rms/correlation length, and the analytical computation rom the PSD. The agreement is satisactory i we assume σ = 3 Å (or less) and τ = µm as input parameters or the numerical simulation.

10 The result o the calculation is overplotted (see Fig. 6) to the results o the modiied ray-racing routine reerred to in Sect. : the HEW(E) unction just computed is consistent with the numerical simulation with the parameter values σ = 3 Å (or less) and τ = µm. Now, to complete the benchmarking o the two procedures, we have to veriy the consistency o these parameter values with the PSD we started rom. The mentioned consistency check can in principle be achieved by simply applying the Eqs. and 5. However, one should notice that the deinitions o these parameters depend strongly on the requency window Δ. Thus, we should ind out the spectral band that determines the HEW at each photon energy/incidence angle, beore computing the rms and the correlation length rom the PSD. To do this, notice that only a limited set o scattering angles [θ s m, θ s M ]contribute to the HEW 4. This range o scattering angles, or ixed values o and θ i, deines a spatial requency window [ m, M ], along with Eq. 4. More precisely, the spectral range o interest or each would in principle equal the integration range o Eq. 8: however, as we assumed a limited size o the detector over which the PSF is observed (31 arcsec radius), the upper integration limit has to be modiied according to Eq. 1. Thereore, we may adopt Δ H sin ϑi r sin ϑi =, as requency band o interest where we should calculate the parameters σ and τ. Unortunately, the evaluation is made more complicate by the dependence o the spectral band Δ on θ i and, whereas the numerical simulation was perormed or single-valued σ and τ, and or the entire set o angles o the SIMBOL-X optical module. We can, indeed, estimate the eective values or the parameters σ and τ, by calculating the average spectral band <Δ> over all, and over all the incidence angle set o SIMBOL-X mirror shells. The resulting spectral band turns out to be (. µm -1,.7 µm -1 ). Integrating the PSD (Eqs. and 5) in Fig. 4 within these limits, we can easily compute the eective parameters, σ and τ. The results are σ =.8 Å and τ = 9 µm, vs. 3. Å and µm which were required by the numerical simulation. Thereore, the two methods return similar values or the parameters characterizing the surace roughness, even i not exactly the same ones. It should be noted that the remaining discrepancy could be ascribed to the very approximate method we adopted or the comparison: a more detailed analysis would have required to simulate the photon distribution at each photon energy and or each mirror shell, using the actual σ - τ parameters, as computed in the respective spatial requencies band (Eq. ). 5. CONCLUSIONS AND FINAL REMARKS In the previous sections we have discussed some possible approaches to the problem o the HEW determination or an X- ray optic, in a photon energy band, rom the roughness characterization o the mirrors surace. To this aim, we utilized a modiied ray-tracing routine, allowing the simulation o the photons spread on the ocal plane due to mirror deormations and surace roughness rom two roughness parameters; this method takes the advantage o a simultaneous treatment o proile errors and roughness. We also utilized an alternative, analytical approach to directly derive the HEW scattering term rom the surace PSD, which can be applied to any PSD and takes the advantage o providing with analytical solutions to translate a PSD into a HEW trend, and vice versa. We perormed this way a HEW calculation or the SIMBOL-X optical module, and compared the results o the two methods, inding a quite satisactory agreement. The methodologies discussed above can be thereore applied to the deinition o microroughness tolerances or uture X-ray telescopes. An example has already been discussed in Sect. 4 or the case o the SIMBOL-X telescope, leading to the conclusion that the proposed PSD (or the proposed rms and the correlation length), assuming H = 15 arcsec, returns a HEW o arcsec at 3 kev, near the arcsec requirement. This small discrepancy can be explained as ollows: the required arcsec at 3 kev was used to derive the low-requency PSD via Eqs. 11 and 1. The PSD at higher requency than MAX = (48 µm) -1 could not be inerred, because our requested HEW was deined in a too limited range o energies to do that. Indeed, we are able to set a constraint on the integral o the PSD (Eq. 8) rom MAX (Eq. 13), up to the r (Eq. 1). In practice, always assuming or the incidence angles the average value θ i =.18 deg, we can derive or the integral o the mirrors PSD, between MAX = (48 µm) -1 and r = (7 µm) -1, the constraint r () P( ) d 5.6 A, (1) MAX

11 whereas the numerical integration o the PSD in the mentioned spectral range returns 6.3 Å. Thereore, the proposed PSD ulills the smoothness requirements only approximately. Detailed calculations o the XRS term o the HEW or the uture X-ray telescopes SIMBOL-X, EDGE, XEUS, and the consequent ormulation o microroughness PSD requirements, are carried out in another paper o the present proceedings 6. In that paper, it is also shown the relevant impact o high-requency microroughness in hard X-rays: in order to preserve the imaging quality in hard X-rays, uture developments o the X-ray optics have to concentrate on the microroughness damping, in particular at spatial wavelengths shorter than a ew hundreds micron. APPENDIX A. GENERAL RELATION BETWEEN PSD AND HEW POWER LAW INDEXES Using the Eq. 11, one can easily derive an equivalent orm o the Eq. 1: d P d The Eq. 17, in turn, can be developed as ollows: ) = 8π sin ( ϑ i whereas the Eq. 16 can be rewritten, using the Eq., as N ln. () N 1 dh 1 d( ) d γ = = = 1+, (3) H d d d d(log P) d d = = log log, d d d n note that all the constants that appear in the Eq. have been dropped out, because their logarithm is also a constant and would be canceled by the derivation in the Eq. 4. Furthermore, notice that d /d < (see Eq. 11) as long as the HEW is a decreasing unction o, so the second logarithm in the Eq. 4 has a positive argument. Now we have, rom Eq. 3, the substitution in the Eq. 4 yields d = (1 + d γ d n. [ log log log(1 + γ )] = d Carrying out the derivations and using the Eq. 5, we obtain d d log(1 + γ ) 3 + γ d log(1 + γ ) = + 1+ = + : d d 1+ γ d(log ) n (4) ) : (5) this ormula provides the relation between the two spectral indexes we were looking or, without hypotheses on the PSD shape. Ater some handling o the derivative and using again the Eq. 5, it takes the suggestive orm 3 + γ 1+ γ (1 + γ ) dγ d (6) (7) n =. (8) ACKNOWLEDGMENTS We thank O. Citterio, S. Basso, P. Conconi, F. Mazzoleni (INAF-OAB, Merate, LC, Italy), S. L. O Dell, B. Ramsey (NASA-MSFC, Huntsville, AL, USA), P. Gorenstein, S. Romaine (SAO- CA, Boston, MA, USA) or useul discussions. This research is unded by ASI (Italian Space Agency), MUR (the Italian Ministry or Universities), INAF (the National Institute or Astrophysics).

12 REFERENCES 1. G. Pareschi, P. Ferrando, The SIMBOL-X hard X-ray mission. Exp. Astron., (6). R. Petre, J. Lehan, S. L. O'Dell, et al., The Constellation-X Spectroscopy X-ray Telescope: Recent Technology Development., SPIE Proc. 666, 6661Q (6) 3. Y. Ogasaka, K. Tamura, R. Shybata, et al., Design and technology review o the X-ray Telescope System on board NeXT mission., SPIE Proc. 666, (6) 4. M. Turner, G. Hasinger et al., XEUS Physics o the Hot Evolving Universe. XEUS Cosmic Vision Proposal (7) 5. K. D. Joensen, P. Voutov, A. Szentgyorgyi, et al., Design o grazing-incidence multilayer supermirrors or hard X- ray relectors. Applied Optics 34(34), (1995) 6. Y. Tawara, K. Yamashita, H. Kunieda, et al., Development o a multilayer supermirror or hard x-ray telescopes SPIE Proc. 3444, (1998) 7. J. W. Van der Herder, L. Piro, T. Ohashi, L. Amati, et al., EDGE: Explorer or Diuse emission and Gamma-ray burst Explosions. SPIE Proc (present issue) 8. O Citterio, S. Campana, P. Conconi, M. Ghigo, F. Mazzoleni, H. W. Bräuninger, W. Burkert; A. Oppitz, X-ray optics or the WFXT telescope. SPIE Proc. 3766, (1999) 9. P. Conconi and S. Campana, Optimization o grazing incidence mirrors and its application to surveying X-ray telescopes. Astronomy & Astrophysics 37, (1) 1. E. L. Church and P. Z. Takács, Interpretation o glancing incidence scattering measurements. SPIE Proc. 64, (1986) 11. J. C. Stover, Optical Scattering: measurement and analysis, SPIE Optical Engineering Press (1995) 1. B. Aschenbach, Boundary between geometric and wave optical treatment o x-ray mirrors. SPIE Proc. 59, 59D (5) 13. E. L. Church, Fractal surace inish. Applied Optics, 7(8), (1988) 14. D. Vernani, G. Pareschi, F. Mazzoleni, D. Spiga, R. Valtolina, Surace characterization o superpolished substrates or X-ray and neutron multilayer optics. Materials Structure, 13(), (6) 15. R. Canestrari, D. Spiga, G. Pareschi, Analysis o microroughness evolution in X-ray astronomical multilayer mirrors by surace topography with the MPES program and by X-ray scattering. SPIE Proc. 666, (6) 16. P. A. J. de Korte, R. Giralt, J. N. Coste, C. Ernu, S. Frindel, J. Flamand., EXOSAT X-ray imaging optics. Applied Optics (6), pp (1981) 17. F. E. Christensen, A. Hornstrup, H. W. Schnopper Surace correlation unction analysis o high resolution scattering data rom mirrored suraces obtained using a triple-axis X-ray diractometer. Applied Optics, 7(8), (1988) 18. J. E. Harvey, E. C. Moran, W. P. Zmek, "Transer unction characterization o grazing incidence optical systems". Applied Optics 7(8), (1988) 19. S. L. O'Dell, R. F. Elsner, J. J. Kołodziejczak, M. C. Weisskop, J. P. Hughes, L. P. Van Speybroeck, X-ray evidence or particulate contamination on the AXAF VETA-1 mirrors. SPIE Proc. 174, (1993). R. Willingale, ROSAT wide ield camera mirrors. Applied Optics 7(8), (1988) 1. P. Zhao, L. P. Van Speybroeck, New method to model x-ray scattering rom random rough suraces. SPIE Proc. 4851, (3). O. La Marle, Simbol-X: extending the ield o hard X-ray astrophysics with the help o the ormation light concept. SPIE Proc (present issue) 3. J. Green, D. S. Finley, S. Bowyer, R. F. Malina, Surace evaluation o the grazing incidence mirrors or the extreme ultraviolet explorer. SPIE Proc. 733, D. Spiga, Analytical evaluation o the X-ray scattering contribution to imaging degradation in grazing-incidence X- ray telescopes. Astronomy & Astrophysics 468, (7) 5. G. Cusumano, M. Artale, V. Cotroneo, T. Mineo, G. Pareschi, SIMBOL-X: x-ray bale or stray-light reduction. SPIE Proc (present issue) 6. D. Spiga, Optical module HEW simulations or the X-ray telescopes SIMBOL-X, EDGE and XEUS. SPIE Proc (present issue)