Inverse propagation algorithm for angstrom accuracy interferometer
Max L. Krieg
*aand Joseph J. M. Braat
aa
Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ, Netherlands.
ABSTRACT
This paper will illustrate several approaches to retrieving the shape of aspherical reflective surfaces as used in EUV Lithography, from measurements from a previously reported angstrom-accuracy interferometer. First, the working principles of the interferometer will be reviewed, and typical measurement data expected from the instrument will be presented. Several methods will then be introduced for retrieving the reflector shape from such measurements. These methods will include approaches based on ray tracing, approximate diffraction calculations, and linearization of rigorous diffraction calculations which use a novel numerical scheme to reduce calculation time of the diffraction integral. The methods will be compared on the basis of accuracy, calculation time and extendibility.
Keywords: Inverse propagation, diffraction integral, ray tracing, interferometry. 1. INTRODUCTION
The accurate measurement of the low spatial frequencies of aspheric reflector substrates for extreme ultraviolet lithography (EUVL) requires the use of novel metrology techniques. An interferometer being developed at the Delft University of Technology circumvents the problematic issues associated with the need for accurate reference optics by instead using the wavefront from a single mode fiber as reference surface. The instrument measures the optical path difference (OPD) arising from interference between such a reference wavefront, and a wavefront aberrated by the substrate under test. This optical path difference can be used to reconstruct the wavefront reflected from the substrate at the measurement plane. Relating this wavefront back to the substrate’s shape is an exercise in inverse propagation. The target accuracy of the instrument, using a metrology wavelength of 633nm, is 0.1nm. Consequently, issues relating to diffraction cannot be ignored. We will start by reviewing the working principle of the interferometer to define the inverse propagation problem. We then proceed to illustrate a novel algorithm to retrieve the mirror shape from our measurements, based on ray-tracing. Subsequently, the influence of diffraction on this approach will be discussed and a number of alternative approaches will be outlined. Several such methods rely on the accurate calculation of the forward diffraction problem. For this purpose, novel numerical schemes allowing the calculation of diffraction from arbitrary, non-planar reflectors were developed, which will also be discussed here.
2. THE INTERFEROMETER
The details of the novel interferometer have previously been reported elsewhere1,4,5, but a brief review of the working principle will serve to explain the nature of the inverse propagation problem.
Figure 1 shows a schematic diagram of the interferometer. Two fibers serve as point sources, providing wavefronts that can be considered spherical within the measurement accuracy3,6. Light from the object fiber first falls onto the mirror substrate where it is reflected, passing through a focus and finally interferes with light from the reference fiber at the detection plane. To minimize the OPD gradients, and hence the fringe density, over the detection plane, the fiber tips should lie on a plane parallel to the detection plane and the focus should occur half way between the two fiber tips. This requires that the mirror substrate be placed approximately a distance R, equal to the substrate's radius of best fit, away from the fiber tips. The detector array should be placed at a distance away from the focus such that the resulting illumination just covers the array. For typical mirror substrates7, the fringe density can exceed the pixel density, and multiple wavelength interferometry is therefore used to retrieve the absolute OPD in the face of the significant under-sampling of the fringes8.
*
m.l.krieg@tnw.tudelft.nl; phone: +31 15 278 2455; fax: +31 15 278 8105
Mirror Detector Multiple Wavelength Light Source Object fiber Reference fiber
Figure 1: Schematic interferometer setup.
To recover the required range of spatial frequencies, the detector array must contain at least 25x25 sensors. The available lightsource- and detection- systems permit the use of either phase-shifting interferometry or heterodyne interferometry at much higher resolutions than this. For the remainder of this treatment, we will assume that an OPD measurement with an accuracy exceeding 0.1nm is available at the detection plane, with a resolution of at least 25x25 pixels over the projected area of the substrate.
3. RAYTRACING APPROACH
In discussing the raytracing approach, we will use the notation shown in Figure 2:
PS PP PO PF PD Mirror Detector rOS ~ PR PO –Location of point-source
PP –Reflection pt. of principal ray.
PF –Focal Point
PD –Point of interest on detector
PS –Point of reflection on mirror surface
PR –Position of reference fiber.
n
(PS ) –Normal to mirror atPS (not shown)r
OS –Vector fromPO to PSFigure 2: Notation convention for raytracing approach.
In order to clarify the raytracing based inversion approach, we will first outline the forward problem of calculating the OPD at the detector using raytracing. As we will show shortly, the process of inversion turns out to be considerably less complicated than the forward problem, since extra information is available for the inversion.
The initial task consists of finding the points of reflection on the mirror surface (PS) for every pixel location (PD). We begin by choosing a set of 25x25 ray-directions which generously cover the complete mirror aperture. We then iterate the length of each ray, until its intersection with the mirror is found to within the desired accuracy. The normal to the surface of the mirror at that point is evaluated, and the direction of the reflected ray is calculated from the law of reflection. The intersection of this ray with the detection plane is then found analytically. The resulting ray intersections with the detection plane will not coincide with our pixel locations, and so we interpolate between the initial ray directions to find ray directions which should intersect more closely to our pixels. The process of tracing this new set of rays to the detection plane is then repeated. To achieve an accuracy better than 0.1nm for the location of the ray intersections, three to four iterations of this interpolation and raytracing process are required. The whole procedure takes a matter of seconds on a modern computer.
For every pixel location (PD), we now have the corresponding point of reflection on the mirror (PS), from which we can calculate the total optical path length (OPL) of the ray from the object fiber tip (PO) to the pixel (rOS+ rDS). The OPD is
found by subtracting the path from the reference fiber (PO) to the pixel:
D OS SD RD
OPD P r r r (1)
In this way we can get an OPD map for our detector. By way of example, we will use the case of an aspheric mirror with radius of curvature (ROC) 340mm, aperture radius of 80mm and a deviation from the best-fit sphere of approximately 3.5Pm, with a 30nm “bump” in one quadrant of the mirror (See Figure 3 for the mirror shape and Figure 4 for the resulting OPL and OPD maps).
x (mm)
y (mm)
Difference from best−fit sphere
−50 0 50 −80 −60 −40 −20 0 20 40 60 80 −2 −1.5 −1 −0.5 0 0.5 1 1.5 µm x (mm) y (mm)
Difference from ideal mirror shape
−50 0 50 −80 −60 −40 −20 0 20 40 60 80 0 5 10 15 20 25 nm
Figure 3: Asphericity and deviation from ideal mirror shape of example mirror
x (mm)
y (mm)
OPL map on detector
−3 −2 −1 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 715.45 715.5 715.55 715.6 715.65 715.7 715.75 715.8 715.85 715.9 715.95 mm
OPD map on detector
x (mm) y (mm) −3 −2 −1 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 680.4 680.45 680.5 680.55 680.6 680.65 mm
Figure 4: Optical path length (OPL) from object fiber to detector and optical path difference (OPD) at detector. Although we do not have access to the absolute OPD from our interferometer, we have a very sensitive measurement of the relative OPD. We can estimate the offset of the absolute OPD from the relative OPD to within fractions of a millimeter for a well-constructed interferometer, by using measurements of the geometry of the set-up.
We will first show how we can retrieve the shape of the mirror, given the exact positions of the key components of the interferometer and the true value of the OPD. The consequences of errors in our estimates of these quantities, and the steps required to correct for them will be discussed later.
We begin by calculating the OPL from our measurement of the OPD at the detector. From the OPL between the object fiber tip and the pixel position, we can conclude that the point of reflection (PS) lies somewhere on a unique prolate spheroid with PO and PDas foci (Figure 5).
PS PO PD Mirror Pixel Position Object fiber tip
Figure 5: Prolate spheroid traced out by fixed OPL around fiber tip and pixel location.
From a single-point measurement it is therefore impossible to uniquely determine the point of reflection. However, we may use the OPL of neighboring pixels to estimate the normal to the wavefront at the pixel, and hence the direction of the ray rSD. This provides us with a unique solution for the point of reflection, given by the intersection of a line from PD
and direction rSD, with the prolate spheroid. Mathematically, this is equivalent to solving a quadratic equation:
2 2 2 2 2 2 2 ' 2 DO A A B DS DS B A DO r d OPL d d r r OPL d d OPL r § · ¨ ¸ © ¹ (2) with: ' ,DS DO , ' A A B DS DO DO D r r d d d r r r O r (3)
Where
r
'
DSis the unit vector in the direction of the calculated wavefront normal.The mirror shape retrieved with this method exhibits a remarkable stability with respect to errors in the estimated ray direction, as a consequence of the identical local gradients of the prolate spheroid and the mirror surface at the point of reflection. Even the curvatures of the two surfaces are matched very closely, so that the estimate of the mirror shape is correct to better than first order for an error in the ray direction.
Nonetheless, care must be taken when calculating the wavefront normals from our sampled OPD function. The fact that our simulations automatically generate the correct ray directions, allows us to directly evaluate and compare the accuracy of schemes to perform this task. The best results were obtained by fitting a bivariate quadratic (including cross-terms) to six points surrounding the point of interest, in an arrangement similar to that shown in 25.3.27 of Abramowitz & Stegun9. This method has a limited amount of noise suppression, having six fitting parameters compared to the seven data points used in the fit.
Performing the inversion on ideal data shows some residual error, dominated by edge effects of the wavefront normal retrieval algorithm (Figure 6). We will subsequently subtract this residual error from retrieved surface shapes of non-ideal mirrors, giving a perfect retrieval for the non-ideal mirror shape by definition.
We now simulate the OPD values for our aberrated mirror shape with a Gaussian bump, and use these data as input for our inverse propagation algorithm. The error figure between the actual and retrieved shapes in Figure 6 shows that the algorithm performs within expected parameters for ideal data, having an rms error of merely 2.0pm.
We will now consider the effect of errors in our assumptions about the positions of the various interferometer components.
−60 −40 −20 0 20 40 −40 −20 0 20 40 −15 −10 −5 0 x (mm) Residual error for retrieval of ideal mirror (rms=1.3pm)
y (mm) Figure Error (pm) −60 −40 −20 0 20 40 −40 −20 0 20 40 −8 −6 −4 −2 0 2 x (mm) Residual error for retrieval of aberrated mirror (rms=2.0pm)
y (mm)
Figure Error (pm)
Figure 6: Residual error when directly inverting ideal, simulated data (rms=1.3pm), and when inverting data from a mirror aberrated by a Gaussian “bump” (rms=2.0pm) c.f. Figure 3.
An error in any of the listed parameters will cause a figure error to be introduced into our retrieved mirror shape. Piston, tilt and defocus errors are considered acceptable within certain limits by the manufacturers of such mirrors14 and have therefore been subtracted from Figures 7-10, which show the error in the retrieved mirror shape for a 1Pm error in the various component positions. It should be clear that the most severe figure errors are introduced by incorrect estimates in the horizontal positions of interferometer components, where they cause what appears to be shear-errors in the direction of displacement. The positioning accuracy of the fibers in the xy plane can be considered to be in the 1-5Pm range, while the z-positions can be determined with even better accuracy, although this is not required. The CCD horizontal position can also be determined to within about 1Pm, by using imaged reference markings on the mirror substrate (such as a central obstruction, for example). This is still not sufficient to guarantee a figure error of 0.1nm however.
To decrease the influence of these errors, we can try to optimize the input position parameters to our inverse algorithm such, that they give a best fit of the resulting retrieved mirror shape to the ideal mirror shape. This procedure is likely to result in an overly optimistic estimate of the error figure for our mirror, since any actual figure errors present on our mirror of the type shown in Figures 7 – 10 will be significantly attenuated by such a fitting procedure. Fortunately, the most significant of these figure errors are non-rotationally symmetric errors. Consequently, a simple rotation of the mirror about its axis can be used to confirm whether the detected error is actually present on the surface, or is due to an incorrect estimate is the position parameters.
Ideally, a parameter optimization of this type would first be performed for a particularly well characterized surface, such as a spherical reflector, and the resulting calibrated interferometer parameters adopted for the retrieval of future mirror shapes. −60 −40 −20 0 20 40 −40 −20 0 20 40 −400 −200 0 200 400 x (mm) 1µm object fiber x−position error (rms= 0.1nm)
y (mm) Figure Error (pm) −60 −40 −20 0 20 40 −40 −20 0 20 40 −500 0 500 x (mm) 1µm object fiber y−position error (rms= 0.1nm)
y (mm) Figure Error (pm) −60 −40 −20 0 20 40 −40 −20 0 20 40 −4 −2 0 2 4 6 8 x (mm) 1µm object fiber z−position error (rms= 1.52pm)
y (mm)
Figure Error (pm)
Figure 7: Errors in the retrieved mirror shape for 1Pm x,y and z positioning errors of the object fiber. Piston, tilt and defocus have been removed.
−60 −40 −20 0 20 40 −40 −20 0 20 40 −1500 −1000 −500 0 500 1000 1500 x (mm) 1µm ref. fiber x−position error (rms= 0.32nm)
y (mm) Figure Error (pm) −60 −40 −20 0 20 40 −40 −20 0 20 40 −2000 −1000 0 1000 2000 x (mm) 1µm ref fiber y−position error (rms= 0.33nm)
y (mm) Figure Error (pm) −60 −40 −20 0 20 40 −40 −20 0 20 40 −140 −120 −100 −80 −60 −40 −20 0 20 40 x (mm) 1µm ref fiber z−position error (rms= 0.02nm)
y (mm)
Figure Error (pm)
Figure 8: Errors in the retrieved mirror shape for 1Pm x,y and z positioning errors of the reference fiber. Piston, tilt and defocus have been removed.
−60 −40 −20 0 20 40 −40 −20 0 20 40 −1000 −500 0 500 1000 x (mm) 1µm CCD x−position error (rms= 0.2nm) y (mm) Figure Error (pm) −60 −40 −20 0 20 40 −40 −20 0 20 40 −1500 −1000 −500 0 500 1000 1500 x (mm) 1µm CCD y−position error (rms= 0.22nm) y (mm) Figure Error (pm) −60 −40 −20 0 20 40 −40 −20 0 20 40 −50 0 50 100 150 x (mm) 1µm CCD z−position error (rms= 0.02nm) y (mm) Figure Error (pm)
Figure 9: Errors in the retrieved mirror shape for 1Pm x,y and z positioning errors of the detector (CCD). Piston, tilt and defocus have been removed.
−60 −40 −20 0 20 40 −40 −20 0 20 40 −20 0 20 40 60 x (mm) 1µm OPD error (rms= 0.01nm) y (mm) Figure Error (pm)
Figure 10: Errors in the retrieved mirror shape for a 1Pm OPD error. Piston, tilt and defocus have been removed.
Provided that the assumptions and approximations underpinning the raytracing approach are valid, this inversion technique gives an estimate of the figure of the reflector under test within ~2pm rms with our interferometer. The influence of positioning errors should be considered an effect of the interferometer type rather than of the inverse approach, but will most likely dominate the low-frequency error landscape.
Examples of reflectors for which the raytracing description is sufficient would be ones where the smooth reflective surface extends beyond the illuminated area, and contains no obstructions. For reflectors where the area of interest is close to the edges or contains obstructions, diffraction plays a significant role; distorting the optical phase from that calculated by geometrical optics, and hence affecting our OPD measurement.
4. EFFECT OF DIFFRACTION
To obtain an estimate of the extent to which diffraction can be expected to affect the above method, we turn to the paper by Sherman and Chew10, where the problem of focused fields encountering an aperture is treated using the Debye integral. The treatment is restricted to axially symmetric systems and hence not directly applicable to our interferometer, but the deviation from axial symmetry is considered sufficiently small for estimation purposes.
We therefore let our object fiber-tip coincide with the focal point, along the axis of our mirror, allowing us to use the results stated by Sherman and Chew:
Geometrically illuminated region:
^
`
^
`
(1) 0 sin (2) 0 sin , , sinsin sin sin
1
2 sin / 2
2 2 cos / 2
sin sin sin
1 2 sin / 2 2 2 cos / 2 m m ikr m m m ikr ik m m m m m ikr ik m m e D x y z A k r A k H k ik e erfc r A k H k ik e erfc r U T U T T T U T T S T T T T T U T T S T T T T | ª º u u ¬ ¼ ª º ¬ ¼ ª º u u ¬ ¼ ª º ¬ ¼ ikr ikr (4) Geometrical Shadow:
^
`
^
`
(1) 0 sin (2) 0 sinsin sin sin
1
, , 2 sin / 2
2 2 cos / 2
sin sin sin
1 2 sin / 2 2 2 cos / 2 m m m m m ikr ik m m m m m ikr ik m m A k H k ik D x y z e erfc ikr r A k H k ik e erfc i r U T U T T U T T S T T T T T U T T S T T T T ª º | u u ¬ ª º ¬ ¼ ª º u u ¬ ¼ ª º ¬ ¼ kr ¼ (5)
The notation used here is related to our notation as follows (zFD denoting the vertical distance of the detector to the focus): 2 2 1 ( , , ) sin O F D FD FD m x y P P x y z P z r r at geometrcal shadow U U T T T { § · ¨ ¸ © ¹
H0(1) and H0(1)denote the Hankel functions of the first and second kind. It is important to note that these results are cast
in a form that explicitly includes the geometrical optics contribution. We see that the diffraction contribution depends on the intensity of our incident wavefront only at the aperture boundary. This is consistent with the notion of a boundary diffracted wave, and allows us to attribute the effect of diffraction exclusively to the field at the aperture boundaries. That is to say that figure errors in the interior of the reflector should not alter the contribution due to diffraction, provided they introduce no caustics at the detector.
Figure 11 shows a cross-section of the field amplitude profile at a detector 35mm from focus, for a spherical mirror (Rm = 340mm) with a central obstruction (radius 25mm) illuminated with a uniform amplitude distribution. The figure also shows the difference in optical phase between the geometrical optics field and the diffracted field. The oscillations near the boundary exceed 0.2rad, which roughly translates into 20nm for the resulting figure error when using the raytracing approach. We see that the oscillations do not drop to zero very quickly, but increase in frequency.
The physical extent of our pixels will cause an averaging of the phase over the pixel area, so that the measured phase is actually the convolution of the actual phase with the pixel shape, sampled at the pixel locations, (see Figure 13). The phase fluctuations are soon under-sampled, but also attenuated by the measurement process. For our example geometry, we can say that diffraction introduces excessive figure errors over a rim 1.8mm wide, surrounding the central
obstruction at the detection plane. This translates to a rim 17mm wide on the mirror itself. We will refer to this region as the diffraction rim. In the diffraction rim, the OPD measurement deviates from that predicted by geometrical optics by more than 0.1nm, as a series of oscillations in the direction of the normal to the boundary of the obstruction.
2 2.5 3 3.5 4 4.5 5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 x at detector (mm) |E| (a.u.)
Diffraction near shadow boundary
Shadow Boundary
Figure 11: Amplitude of diffracted field near shadow boundary
2 2.5 3 3.5 4 4.5 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x at detector (mm)
Phase difference (rad)
Phase difference due to diffraction
Shadow boundary 3 3.5 4 4.5 5 5.5 6 −60 −40 −20 0 20 40 60 x at detector (mm)
Phase error (mrad)
Phase difference between geometrical and diffracted field
Figure 12: Phase-difference with geometrical field near shadow boundary, and magnified view further away.
3 3.5 4 4.5 5 5.5 6 −4 −3.5 −3 −2.5 −2 −1.5 −1 x at detector (mm)
Phase error (log
10
[rad])
Phase difference after convolution with pixel size
1mrad level = 0.1nm figure error 3 3.5 4 4.5 5 5.5 6 −30 −20 −10 0 10 20 30 x at detector (mm)
Phase error (mrad)
Phase difference as sampled by pixels
+/− 1mrad level = 0.1nm figure error
5. TEMPERED RAYTRACING APPROACH
The effects of diffraction on our raytracing method can be reduced in three ways: 1) Physical damping of diffraction effects.
2) Averaging out diffraction effects with variable obstructions. 3) Subtracting an estimate of the non-geometrical field.
The first approach requires us to “vignette” the boundaries of the mirror. This could be achieved either with an external mask with a tapered transmission profile near the edges, or by applying an increasingly absorbing coating to the edges of the mirror, letting the reflectivity fall off smoothly over several hundred wavelengths (see Figure 14). While we may state that a wider vignetting rim will cause a smaller diffraction rim, we have to assume that the figure of the mirror is altered over the entire area where this vignetting takes place. The optimum thickness of such a border still needs to be determined by rigorous calculations.
Central obstruction Mirror
edge
Figure 14: Cross-section of mirror reflectivity before and after vignetting.
The fact that the diffraction oscillations appear to have a zero mean value can be exploited by placing a smaller aperture in front of the mirror, and either varying its position, orientation or both between repeated measurements, thus averaging out the diffraction effects. We will illustrate this process by averaging 25 phase profiles as in Figure 12, which have been shifted with respect to each other randomly in a range of 0-0.3mm. This is equivalent to an identical variation of the position of an added aperture. The resulting reduction of the diffraction rim to 0.8mm at the detector (equivalent to 7.8mm at the Mirror) can be seen in Figure 15.
2.5 3 3.5 4 4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 x at detector (mm)
Phase error (log
10
[rad])
Phase difference after using moving aperture technique
1mrad level = 0.1nm figure error
Figure 15: Reduction in phase error after using moving aperture technique (c.f. Figure 13)
If a good estimate of the expected diffraction pattern is available (taking into account the spatial filtering effect of our pixels), this can simply be subtracted from our measurements before proceeding with the raytracing approach. This requires a more flexible model than that presented in the previous section, which is restricted to axially symmetric systems. Such an algorithm will also be useful for the other approaches described in this paper, and will have to fulfill a number of requirements that rule our most asymptotic or approximate methods.
Our algorithm will have to be able to calculate the diffraction integral over an arbitrary three dimensional surface shape, not restricted to a plane or a spherical cap. This is due to the combination of large NAs and severe asphericities of our reflectors. The accuracy should be adjustable, to check that the algorithm converges to within our desired precision. Also, any assumptions or approximations made must influence the final phase by less than the 0.1mrad equivalent to our 0.1nm accuracy.
For these reasons alone, a direct numerical integration of the Rayleigh-Sommerfeld integral seems the most logical choice. Once results of such an approach are available, we are at liberty to use these to evaluate the applicability of approximate methods. It should be noted that the only assumption made for this approach is that the field vanishes to zero beyond the edge of the mirror.
6. NUMERICAL EVALUATION OF DIFFRACTION INTEGRAL FROM NON-PLANAR REFLECTORS
6.1 Brute Force approach
The brute force approach entails numerical integration of the Rayleigh-Sommerfeld diffraction integral.
PS PP PI PO PF PD Mirror Detector rID ~ PO –Location of point-source
PP –Reflection pt. of principal ray.
PF –Focal Point
PD –Point of interest on detector
PS –Point of stationary phase
PI –Arbitrary point, for integration.
n
(PI ) –Normal to mirror atPIr
ID –Vector fromPItoPDFigure 16: Notation convention for brute-force approach.
By making use of symmetries inherent in our situation, we can choose an adaptive type of quadrature which is scaleable, to allow us to trade off accuracy against computation time. For the purpose of this approach, we will use the notation shown in Figure 16. The point of stationary phase (PS) is approximated by the point of reflection of the geometric ray ending at PD.
As indicated, we will assume that the mirror is illuminated by a point-source. However, we allow for the possibility of a non-uniform amplitude distribution of this light source, Am(
r
OI), to model realistic fiber output. The electric field at PD is then given by:1 1 1 1 cos ( ), 1 OI ID I OI ikr ikr D ID ID OI ID ID Mirror Surface iC P I Mirror Surface Am r U P e e n P r dA ik r r ikr B P e dA ik O O § · ¨ ¸ © ¹
³³
³³
(6)We have separated out the real, slowly varying amplitude factor B(PI) and the quickly oscillating exponential, with argument C(PI). The contour-plot of C(PI) in Figure 17 for a typical mirror shows the apparent symmetry about the point of stationary phase. This symmetry is not perfect, especially if the mirror is aspheric. Nonetheless, by choosing a polar co-ordinate system in the plane normal to the mirror axis, centered on the point of stationary phase, we are able to choose a non-uniform spacing for the mantissa of the radial co-ordinateU, and a uniform spacing in the angular co-ordinateT, to take advantage of symmetry properties.
PS PP PI PI rm Mirror max A Mirror
Figure 17: Contours of C(PI) over mirror surface, and polar co-ordinate system centered at PSused for numerical integration.
We re-write (1) to reflect this change in co-ordinate system as follows:
1
,
iC , D Mirror Surface U P B e dA ik U T U T O
³³
(7)The radial mantissa is chosen with increasing density, to account for the growing number of oscillations (Figure 18). As with the method of stationary phase, we see that the primary contribution to our integral will come from the stationary point and from an area around the boundary of the mirror, near the stationary point, where the contours are truncated. The region anterior to the stationary point with respect to the mirror axis contributes less to the integral due to the quick averaging that occurs by the large number of oscillations.
The results of a brute-force calculation equivalent to that shown in Figure 11 are shown in Figure 19, where they are also compared to the results of a boundary-diffracted wave (BDW) approach11. The accuracy of the numerical result which sample each oscillation with 100 points is better than 1%. The difference between the results obtained by using the Sherman formalism and the other two methods is explained by the different choice in boundary-conditions. The Sherman formalism requires a sharp drop-off of the angular spectrum, while the Rayleigh-Sommerfeld diffraction integral and the BDW approach of Born and Wolf assume the Kirchhoff boundary conditions – that is, that the field drops to zero at the boundaries in the spatial domain
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ρ −−> real ( e iC( ρ , θ ) )
Increasing oscillations with increasing ρ
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 θ (rad) real ( e iC( ρ , θ ) )
Relatively slow oscillations with θ
Figure 18: Behavior of integrand withU and T.
It should be noted here that because our mirror is a three-dimensional surface there are a number of implications for the evaluation of the integral. First, the obliquity factor must be evaluated with respect to the local surface normal.
Secondly, since (7) is an integral over 2 variables in the XY plane, the values of B and C are those of the points on the mirror which project to the corresponding point on the XY plane, and the area element, dA is actually the three-dimensional surface element on the mirror rather than the area element in the plane.
2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Comparing diffraction calculation approaches
x−position at detector (mm)
|E| (a.u.)
Brute−force 100pts/oscillation Brute−force 10pts/oscillation Bound. diffracted wave method Sherman solution to Debye integral
Figure 19: Comparison between brute-force approach, Boundary diffracted wave approach, and Sherman formalism.
This method is obviously very time consuming – a single point of Figure 19 taking more than 10 minutes to compute on a modern PC. An exhaustive comparison with various approximate schemes has not yet been completed, but it is hoped that such schemes exist which will permit a considerable saving of computational time. Figure 19 shows that good preliminary results were obtained with the BDW algorithm, which only takes seconds to calculate each point.
7. LINEARIZED ABERRATION ANALYSIS
As an alternative to the tempered raytracing approach, we can make use of our ability to solve the forward diffraction problem, to deduce aberrations present in the reflector substrate under test. This method requires us to assume that the actual mirror shape deviates from the nominal mirror shape by only a small amount.
We first calculate the diffraction pattern (B0) for the nominal mirror shape (A0), and then repeat this calculation for the
nominal mirror shape, aberrated by an adequate number (M) of mutually orthogonal aberration functions ('Aj) such as
the Zernike polynomials, giving us a set of detector-surface aberration functions, measured at N pixels. It should be noted that the addition of an aberration means that we are integrating over a different surface than the nominal surface:
0 0,1 0,2 0, 0 1 0,1 1,1 0,2 1,2 0, 1, 0 0,1 ,1 0,2 ,2 0, , , ,..., , ,..., , ,..., N N N M M M N A B B B A A B B B B B B A A B B B B B B o ' o ' ' ' ' o ' ' ' ! M N 2 B (8)
Provided that the amplitudes of the mirror-surface aberration functions were chosen to be sufficiently small, we may assume that the diffraction process is linear with respect to these aberrations:
1 1 2 0 1 2 M 0 1 2 M M M A B A A A a a a B B a a a A B ' ' § · § ¨' ¸ ¨' ·¸ ¨ ¸ ¨ | ¨ ¸ ¨ ¨ ¸ ¨ ¸ ¸ ¸ ' ' © ¹ © ! ! # # ¹ (9)
This is because a small local distortion of the nominal mirror shape may be regarded as equivalent to a proportional local retardation of the incident field on the nominal mirror surface. The linearity of the diffraction integral with respect to the incident field then guarantees the linearity of the diffracted field.
While the process may be linear, we cannot assume that the detector-surface aberration functions are mutually orthogonal. We therefore need to proceed with a regularized inversion scheme to decompose our actual measured diffraction pattern into the calculated detector-surface aberration functions. The resulting coefficients will then allow us
to infer the aberrated mirror shape which gave rise to the measured diffraction pattern. One such scheme is the truncated singular value decomposition (TSVD).
This approach can also be used with the raytracing formalism as the forward algorithm instead of the full diffraction treatment. In such a case, the forward mechanism would not include any diffraction effects, and the measured diffraction effects will simply be de-composed into contributions from our aberration functions. If no aberration functions resembling the diffraction pattern exist, it will be discarded as noise, creating a natural immunity against diffraction.
The drawback of this method is that it will work only for small aberrations from the nominal surface shape, and will only return aberrations of the type used in the decomposition rather than giving a full mirror surface-map as is the case for the raytracing inversion. Furthermore, if the brute-force method for calculating the forward problem is chosen, the computation time for the matrix mapping the mirror aberrations to changes in the diffraction pattern, will be NxM times longer than the calculation time for a single point – an unrealistic prospect with the currently available computation power. The use of faster, approximate forward algorithms will make this approach more feasible.
8. BACK-PROPAGATION
One last potential method for determining the shape of the reflector from our measurements of the OPL is the propagation of our measured wavefront back to the nominal mirror position. We cannot use this method to determine the mirror shape directly. Instead we will be deriving a retardation to the ideal field expected at the nominal surface, as described in the previous section, which can then be used to deduce the surface shape which would be equivalent to this a retardation. (See Figure 20)
Deformed Mirror Deformed Incidentwavefront
Figure 20: Equivalence of mirror form aberrations and deformations in the incident wavefront.
While being the most intuitively appealing, this method unfortunately suffers from the worst problems of both previously mentioned approaches. In terms of computation time, back-propagation would still be prohibitive in the absence of a more efficient forward algorithm. While the method would give the correct mirror shape if the exact diffraction pattern at the detector were available to us, the under-sampling and spatial filtering inherent in our measurement process will doubtlessly introduce a new class of distortions in the retrieved mirror shape.
Nonetheless, progress could be made along these lines if a similar tempering scheme was used as outlined in section 5. We would then only propagate a detector-surface OPL function where diffraction effects due to obstructions have already been removed. Propagating these back will result in mirror shape estimates where the obstructions are not retrieved correctly. This in itself is not an issue, as it is the mirror shape, not the shape of the obstruction which is of interest. However, the extent to which the various ways of “tempering” the measured diffraction pattern will affect the over-all mirror shape is not yet certain.
9. CONCLUSION
We have presented several potential approaches to retrieving the shape of reflectors under test in our novel interferometer. For reflectors where diffraction effects are negligible, a geometrical optics approach yields fast and accurate results. The geometrical optics approach in the presence of obstructions has been determined to be valid except within a so-called “diffraction rim” around the projected obstructions. Various ways to reduce the detrimental effects of diffraction were discussed. Alternative approaches not relying on geometrical optics were also outlined, and a numerical scheme to implement these was presented.
ACKNOWLEDGEMENTS
This work was supported by the Dutch Technology Foundation STW, by ASM Lithography, Veldhoven, The Netherlands, and by Carl Zeiss, Oberkochen, Germany.
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