## Abstract

Frequency Domain Analysis of measurement results from coherence scanning interferometers provides an estimate of the topography of the surface scattering the light, with claims of nm levels of accuracy being achieved. In the following work we use simulations of the measurement result to show that the limited set of spatial frequencies passed by the instrument can lead to errors in excess of 200 nm for surfaces with curvature. In addition we present a method that takes the uncertainty in the amplitude and phase of each element of the transfer function and provides the upper and lower limit on the location of the surface provided by the Frequency Domain Analysis method. This provides an idea of the level of accuracy that the spatial frequency components must be known to in order to reproduce curved surfaces well.

© 2015 Optical Society of America

## 1. Introduction

Instruments such as Coherence Scanning Interferometers (CSI) have been developed in order to provide an estimate of the topography of a surface *via* optical measurements. The result of the measurement is not the surface, but instead it is the optical signal created by the light scattered by the object; one of several methods, such as detection of the peak visibility in the fringes [1], envelope detection [2, 3], centroid of the square of the signal [4, 5], wavelet techniques [6, 7], and Frequency Domain Analysis (FDA) [8–10], are subsequently used to estimate the surface that scattered the light. The successful recovery of the location of the interface is important if a comparison between the results from optical measurements and contact type measurements is to be attempted, a comparison that is important if the role of the surface being measured involves it being in contact with other objects. In addition if the function of the object is related to the precise form of the surface, i.e. for optical components, or when the absolute shape and size of an object is required, it is imperative that an accurate measurement can be made, or at least that the result of a measurement is understood.

In [11] the result of a measurement of a strongly scattering object *via* CSI was shown, subject to certain limitations such as the absence of multiple scattering, to be given by the convolution of the Point Spread Function (PSF) of the instrument with a function that only takes a value at the interface between the different materials. While this is only true in a limited set of cases, where it is true there is the possibility to examine the ability of a surface extraction method to recover the correct scattering object. In the following work we examine the ability of the Frequency Domain Analysis (FDA) method to extract the correct surface from a simulated measurement result. It is shown that when there is curvature to the surface being imaged the limited set of spatial frequencies passed by the instrument leads to errors in the determined topography. This method is chosen for the ease by which it can be addressed mathematically, and the errors that are shown are likely to occur with other surface extraction techniques in a similar fashion, hence this should not be seen specifically as a criticism of the FDA method.

Further errors in the topography gained arise due to the way in which an instrument modifies the spatial frequencies in the image, behaviour which is captured by the Transfer Function (TF) when Fourier optics is applicable. This paper demonstrates how an uncertainty in the elements of the TF can affect the topography found *via* FDA, providing upper and lower limits to the estimated surface, which in turn provides an indication of the accuracy to which the TF must be determined in order to provide a given measurement accuracy.

The structure of the paper is as follows; in section 2 the model that is used is introduced, and simulation results presented. In section 3 the determination of the surface *via* the FDA method is discussed, and the results from section 2 are used to examine the ability of the method to recover curved surfaces. Finally, in section 4 a given uncertainty associated with each element of the TF is propagated through a measurement model, allowing the uncertainty associated with the surface determined by the FDA method to be calculated, with an example using simulated measurement results being presented.

## 2. Simulation of the measurement of a strongly scattering object

An accurate simulation of the result of a measurement allows us to deal with a clean system, i.e. one where there is no noise or unintentional imperfections in the microscope. Some care must be taken to avoid numerical artefacts and to create a realistic model, however, if this is done it should be possible to examine the limits of the method, and how results are affected by errors that would occur in real systems.

There are two limits in which we can easily look at the scattering and construct a simulation; firstly for weakly scattering objects, where the first order Born approximation is sufficient [12], and secondly, within certain limits, for strongly scattering objects [11]. In the first case, for a plane wave illumination and at a large distance from the scattering object, the complex amplitude of the scattered field in any direction is related to a single term of the Fourier Transform (FT) of the scattering potential [12]. In the second case, if certain criteria are met such as the surface being smooth, slowly varying and in the absence of multiple scattering, then the only contribution to a measurement in the far field in each direction comes from regions of the surface whose surface normal corresponds to that direction [11]. It is this second case that will be considered in the rest of this paper. The limits to the cases where it is applicable will not be examined in detail here, and the interested reader should look at [11], suffice to say that the case where a perfectly conducting sphere, whose radius is greater than around 10 µm, is imaged by a lens of up to 0.7 NA lies within the applicable regime.

The measurement of a perfectly electrically conducting object *via* CSI was shown [11] to be equivalent to the convolution of the PSF of the system and a function that takes a value only at the interface between the two materials, ∆(**r**) = 4*πiδ*(**r**;*S*)*W*. Here *δ*(**r**;*S*) is equal to 1 where the position vector **r** lies on the surface, and is equal to zero elsewhere. *W* is a window function limiting the section of the surface to that where the surface normal lies within NA of the instrument. The definition of the surface is slightly different to the definition in [11], however this change has no significant effect on the ideas presented there, and really just shifts a couple of terms from the TF into the new ∆.

#### 2.1. The Transfer Function

The measurement in the spatial frequency domain is then given by [11],

where $\tilde{O}(\mathbf{k})$ is the FT of the output of the measurement, $\tilde{\mathrm{\Delta}}(\mathbf{k})$ is the FT of the relevant section of the surface, $\tilde{H}(\mathbf{k})$ is the TF of the instrument, and the wavevector**k**, where |

**k**| = 2

*π/λ*. The TF is a complex valued function that describes how an instrument modifies the relative amplitudes and phases of the spatial frequency elements in a measurement. The passband of the instrument corresponds to the non-zero elements of the TF, the location of which are given for a CSI by the construction in [13]. A 2-D slice through

*k*- space passing through the

*k*axis is illustrated in Fig. 1 with the spatial frequencies that are passed being marked in green. These passed spatial frequencies lie within a cone with the same half angle as the lens, Θ, are within a circle of radius 2

_{z}*k*, and are above two circles of radius

_{max}*k*centred on the points a distance

_{min}*k*from the apex of the cone along the edges. Here

_{min}*k*is the wavenumber corresponding to the shortest wavelength of the illuminating light,

_{max}*k*= 2

_{max}*π/λ*, and

_{min}*k*is the wavenumber corresponding to the longest wavelength of the illuminating light. This construction is symmetric throughout a rotation about the optical axis of the system, which aligns with the

_{min}*k*axis. This set of spatial frequencies is related to both the numerical aperture of the objective lens and the wavelengths of the light used. The exact nature of the TF depends upon such things as the (in)coherence of the light source, and a far fuller description can be found in [14].

_{z}The output of the measurement therefore corresponds to the spatial frequencies of the surface function, modified by the multiplication with the TF. For simplicity the TF will, initially, be equal to one within the pass band and be zero elsewhere. This corresponds to a system that passes all of the spatial frequencies within the passband without changing their relative phases or amplitudes. If there was no noise, and the true values of the TF were known, this situation could also be obtained after the image had been recorded by correcting all of the spatial frequencies to remove the errors introduced by the instrument. The case where the TF is not known perfectly is considered in section 4, and the work in that section may also be used to examine how the calculated topography is changed when a correction is not applied.

#### 2.2. The FT of the surface

The accuracy of the simulation depends upon the ability to obtain the correct FT of the surface being imaged. As such we will look at the case where the object being measured is a sphere, where the relevant section of the surface is the cap of a sphere that lies within a cone with a half angle equal to that of the objective lens of the microscope, the FT of which is given by [15],

*r*

_{0}is the radius of the sphere, set to 20 µm, and Θ is the half angle of the objective lens. Equation (2) is the convolution of the FT of an infinitesimally thin spherical shell centred on the origin, $4\pi {r}_{0}^{2}\text{sinc}\left({\mathrm{r}}_{0}\mathrm{k}\right)$, and the FT of a function that in real space is equal to one between

*z*=

*r*

_{0}

*cos*Θ and

*z*=

*r*

_{0}

*cos*Θ+2

*r*

_{0}, the region of space where the surface of the sphere is retained, and is zero elsewhere [15]. The FT of this second function, that of the sheet, is zero everywhere except on the

*k*axis. This means that the three dimensional convolution of the two functions is equivalent at all points to a one dimensional convolution along the line where the non-zero values of the second function are found.

_{z}When a numerical evaluation of Eq. (2) is carried out, the volume of space over which the integral is evaluated needs to be truncated due to the finite amount of memory and computation time. As the 3-D convolution is equivalent to the 1-D convolution along the line where the FT of the second function has non-zero values, a far greater volume of *k*- space can be incorporated than if a full three dimensional convolution was to be carried out. Therefore the results of a numerical evaluation of Eq. (2) may be more accurate than if both functions had non-zero values extending in all directions.

In the following work the limits to the integral in Eq. (2) are reduced to ±1000×(2*π/r*_{0}) = *±*3.142 × 10^{8} m^{−1} for the numerical calculations, with the two functions that are convolved being evaluated at points between these limits every 2*π/*(20*r*_{0}) m^{−1}. Truncating integrals such as this is appropriate as long as the section outside of the truncated region contributes a sufficiently small amount to full integral. Looking at the integral, there are two sinc functions, which have a magnitude that drops away from a central peak. The highest spatial frequency that *F*(**k**) needs to be evaluated for in Eq. (2) is the highest spatial frequency passed by the microscope, 2*π/k _{max}*, which for an illumination whose shortest wavelength is 400 nm is 1.571×10

^{7}m

^{−1}. Looking at the terms in the integral in Eq. (2), it can be seen that the term sinc(

*r*

_{0}.(

*k*−

_{z}*κ*)) has a peak that lies at

*k*and drops away from this point. As the maximum value of

_{z}*k*that the integral will be evaluated for is ≪ than the limit to the integration, the peak of this function will be retained. In addition the term $\text{sinc}({r}_{0}{[{\kappa}^{2}+{k}_{x}^{2}+{k}_{y}^{2}]}^{(1/2)})$ has a peak on the

_{z}*k*plane, with a magnitude dropping away along the

_{x,y}*z*- axis. As the magnitude of each of these terms drop away from the central peak, it may be hoped that the contribution to the integral from the region outside of the limits used is small compared to that within them, and thus sufficient accuracy in the FT is gained. In order to show that the effects seen later are due to the truncation of the

*k*- space by the TF, and not due to numerical errors in the evaluation of the FT, the results are compared to those obtained using a larger section of

*k*- space, where the success of the method in recovering the correct surface would imply that this is indeed the case.

#### 2.3. An example of the simulation

In a real measurement, the data will typically be recorded by the pixels on a CCD, and a set of images will be recorded with either the object or the reference mirror being shifted by a distance of the order of 80 nm between each image. To simplify matters the CCD is assumed to record the signal perfectly at the centre of each pixel, and the data is recorded at points in space 80 nm apart, meaning the data points are equally spaced in each direction. While in real instruments the separation of the data points may be greater than this, for example it is approximately 160 nm for one of our commercial instruments when a 50× magnification is used, assuming equal separation in each direction simplifies matters without significantly changing anything.

The simulation will therefore consist of a three dimensional block in physical space, 701 data points in each direction and spaced at intervals of 80 nm, giving a cube of space that is 56.08 µm along each side and is centred on the origin. In spatial frequency space this is represented by a set of 701 by 701 by 701 spatial frequencies, each separated by *dk* = 1.12 × 10^{5} m^{−1}. Thus −3.92 × 10^{7} m^{−1} *≤ k _{x,y,z} ≤* 3.92 × 10

^{7}m

^{−1}. The wavelength of the illuminating light will be from 400 nm to 600 nm, giving a maximum spatial frequency in the TF of 3.141×10

^{7}m

^{−1}.

A simulation of the result of the measurement of the cap of a sphere is shown in Fig. 2. The optical axis of the system is aligned with the *z*- axis. All of the images in spatial frequency space show the data that lies in the plane *k _{y}* = 0, while in real space

*y*= 0, though the results are symmetric throughout a rotation about the

*k*and

_{z}*z*axes respectively. Figure 2(a) shows log |FT| between −3.92 × 10

^{7}m

^{−1}

*< k*< 3.92 × 10

_{x}^{7}m

^{−1}and 0 m

^{−1}

*< k*< 3.12×10

_{z}^{7}m

^{−1}, while Fig. 2(b) shows the spatial frequencies that are passed by the instrument (shown in red) and those that are not (shown in blue) for an NA of 0.7 and light with wavelengths between 400 nm and 600 nm. In this case the red region corresponds to the TF having a value of one. Figure 2(c) shows log|FT| multiplied by the TF, and Fig. 2(d) shows a slice through the real part of the 3-D Inverse Fourier Transform (IFT) of the data represented in Fig. 2(c). Multiplying the result shown in Fig. 2(d) by two gives the fringes that would be recorded when the object is measured using a CSI.

## 3. Surface extraction using Frequency Domain Analysis

The calculation of the topography of a surface *via* FDA is a well documented technique [8–10], whereby the FT of the data recorded on each pixel is taken and, after unwrapping the phase, the relative location of the surface below it is determined from gradient of the phase w.r.t. wavenumber. This simple explanation of the technique assumes that there is no dispersion etc. due to the instrument, or dispersive behaviour in the object being imaged. If there is then, either corrections need to be made to take this into account, or errors will arise. In the work described here the microscope is assumed to be ideal, and so a calculation of the gradient should suffice.

That the location of the surface can be calculated from the gradient of the phase can be seen by considering the output of the instrument as the incoherent superposition of the sinusoidal interference patterns that would be found for each wavelength [10], with maxima occurring whenever the optical path length is equal to an integer multiple of *λ*. When the location of the interface is shifted by a distance *z*, the phase of each of the terms, *θ*, is shifted by *k _{z}z*, therefore

*dθ/dk*=

_{z}*z*.

The data on each pixel is considered independently to that found on the others, and the intensity values that are recorded as the object is scanned vertically are FT’d to obtain the phase in the range −*π* to *π* of each of the spatial frequency elements. A set of spatial frequencies around which the amplitude of the signal is greatest is selected [10], and should the phase change between the spatial frequency terms be sufficiently small the phase can be unwrapped and the gradient be found.

The process that is followed in this paper is illustrated in Fig. 3. The FT of the relevant cap of the sphere is calculated, and then multiplied by the TF of the instrument (a log plot of which is shown in Fig. 3(a)). The result of a measurement of the sphere is twice the real part of the 3-D IFT with the data on each (*x,y*) plane corresponding to an image recorded on the CCD, and different layers in the *z*- direction corresponding to different images, illustrated in Fig. 3(b). The data recorded on a single pixel corresponds to that on any of the lines of data in the *z*- direction, as illustrated by the blue dashed line in Fig. 3(b), and which is shown in Fig. 3(c). For each pixel a FT of this data is carried out and, in the region where there is a significant signal, the phase is unwrapped, and the gradient calculated. In Fig. 3(d) the red line shows the unwrapped phase of the term for each wavenumber, while the blue line shows a plot of the magnitude of the terms which has been scaled to match the phase plot; this has been superimposed just to show the wavenumbers where its magnitude is significant.

The 3-D IFT that relates the spatial frequencies in the image to the measured fringes indicate that the data on each pixel is still dependent upon all of the terms in the TF. It can be seen from the 3-D IFT which is given by

*f*(

**r**) is dependent upon all the non-zero terms in spatial frequency space.

To put this another way, when a FT is carried out on the data on each pixel, the value of the *k _{z}* term after the 1-D FT is related to all of the points in the TF that have the same component of

*k*. This link is explored more fully in the subsequent section, however, it can be seen that the fact that TF truncates the data in the

_{z}*k*,

_{x}*k*directions on each

_{y}*k*plane will alter the amplitude and phase of the

_{z}*k*term from that of the case where the full FT of the cap of the sphere is used.

_{z}While the most current literature that we are aware of states that the spatial frequencies that are used to calculate the gradient should be taken from the point where the amplitude of the signal is greatest, it may be the case that it is the point where the TF is broadest gives a better topography of the surface as more of the FT is retained. This will depend upon the specific FT of the surface, and how much truncating the data in this way modifies it.

It should be noted that the FT of an infinite plane whose normal in spherical polar coordinates is in the direction *θ* = *θ*_{0}, *ϕ* = *ϕ*_{0}, is given by

**r**

_{0}is a vector from the origin to the plane, and that is perpendicular to it. This FT only takes a value other than zero on the line

*θ*=

*θ*

_{0},

*ϕ*=

*ϕ*

_{0}, and as long as this lies within the passband of the TF, i.e. the angle of the normal w.r.t. the optical axis is less than the half angle of the lens, then the sheet should be reproduced well. This means that these instruments should be expected to work well when it is a flat surface that is being imaged, as the lateral truncation of the region in spatial frequency space does not remove any data from the FT. When more complex objects are imaged this will however have an effect.

In Figs. 4(a) and Figs. 4(b) a region of a slice through the |FT| of the cap of an infinitesimally thin spherical shell that lies within a cone with half angle of 30 degrees, i.e. the section that would be imaged by a microscope with an NA of 0.5, is shown. In Fig. 4(b) the data has been multiplied by the corresponding TF, while in Fig. 4(a) the FT is evaluated for the points where −3.92×10^{7} m^{−1} *< k _{x,y}* < 3.92×10

^{7}m

^{−1}and 0 m

^{−1}

*< k*< 3.12×10

_{z}^{7}m

^{−1}. An IFT is carried out on the two sets of data, followed by the one dimensional FT along the optical axis of the system and the location of the surface, shown by the red lines in Figs. 4(c) and Figs. 4(d), on each pixel is determined

*via*the FDA method. This is calculated using the gradient of the phase at 17 adjacent spatial frequencies centred on that marked by the red arrow in Figs. 4(a) and Figs. 4(b). The blue line in Figs. 4(c) and Figs. 4(d) corresponds to the ’ideal’ surface that was defined to generate the FT. In Figs. 4(e) and Figs. 4(f) the difference between the red and blue lines in Figs. 4(c) and Figs. 4(d), respectively, is shown. For the data in the left hand column a difference of about 5 nm is found between the two surfaces, while with the data on the right this has increase to about 100 nm, with a maximum peak to peak error of about 500 nm. The fact that the data shown in the left column reproduces the topography well indicates that it is not a numerical error that lead to the difference in the profiles that is seen in Fig. 4(f).

Figure 5 shows the same information as in Fig. 4, however this time it corresponds to an objective lens with a NA of 0.7. Once again the plots on the left correspond to a larger set of spatial frequency information than those on the right. The difference in the surfaces near the centre of the sphere is only a few nm. Once again, in the plots on the right the difference increases, though the difference is smaller than in the case for the 0.5 NA lens as would be expected due to the greater set of spatial frequencies in the passband.

## 4. Limits on the surface due to uncertainty in the TF

Up until this point the TF has been set to one for all of the spatial frequencies passed by the instrument, however this would be the ideal case and some modification to the amplitude and phase of each element is likely to be introduced by the microscope. The modifications to the amplitude and phase of each spatial frequency element that are due to the TF ideally need to be removed before the FDA analysis is performed, or need to be sufficiently small that they do not have a significant effect. If the TF is known then corrections can be made, however, as there may be an uncertainty in the values of the TF this leaves an uncertainty in the phase of the elements used to determine the location of the surface. The manner in which this uncertainty affects the determined topography may be examined by considering how the signal detected on each pixel is related to the points in 3-D *k*- space.

The phase of each of the spatial frequency elements recorded by each pixel throughout a measurement has so far been obtained by taking twice the real part of the 3-D IFT of the block of *k*- space that contains the non-zero product of the FT and the TF, in order to obtain the fringes, followed by a 1-D FT of the data that is recorded by each pixel, as was illustrated in Fig. 3. This process can, however, be simplified by just taking a 2-D IFT in the directions perpendicular to the optical axis of the system. While this approach ignores the fact that the measurement results in only the real part of the 3-D IFT being recorded, it can be shown that the phase of each of the terms is the same by either route. As it is only the phase that is used to determine the location of the surface, the computational effort can be saved.

Each of the points in *k*- space corresponds to a 3-D plane wave in real space, the phase and amplitude of which is given by its complex value, *A*** _{k}**. If a single point,

**k**, in

*k*- space is considered, the contribution in real space at the point

**r**is

By fixing the values of *x* and *y* to those of a pixel, the contribution to the signal due to a given 3-D plane wave, exp(−*i***r.k**), can be found by evaluating Eq. (5) at each of the *z* locations that images are taken at. Fixing the values of **k**, *x* and *y* in Eq. (5) and collecting all of the constant terms gives

*B*

_{k}*is the 2-D IFT of*

_{,x,y}*A*

**at the point (**

_{k}*x,y*). The complex value of

*B*

_{k}*gives the amplitude and phase of the wave exp(−*

_{,x,y}*ik*), which is measured on the pixel due to this 3-D plane wave.

_{z}zThe signal recorded on a pixel due to any point in *k*- space that has a component *k _{z}* is again a wave with spatial frequency

*k*. As it is easy to show that the summation of two waves with a given

_{z}*k*gives a further wave with the same spatial frequency, it can be seen that all of the plane waves with a given component

_{z}*k*contribute to the term with wavenumber

_{z}*k*in the 1-D FT. Therefore if the values of

_{z}*B*

_{k}*for each of the plane waves in the 3-D FT with a component*

_{,x,y}*k*that are passed by the microscope are found at the location of a pixel (

_{z}*x,y*), then their vector sum on the complex plane will give the magnitude and phase of the spatial frequency

*k*on that pixel.

_{z}*F*is the 1-D FT in the

_{z}*z*- direction. If an uncertainty in the magnitude and phase is included in each of these elements, then the method provides a route by which the uncertainty in the phase of the final vector can be found, and the possibility of finding a computationally fast estimate to the limits that the uncertainty in the values puts on the upper and lower bounds of the surface.

Each element of *B*_{k}* _{,x,y}* corresponds to a wave with magnitude |

*B*

_{k}*| and phase*

_{,x,y}*B*

_{k}*= atan2(*

_{,x,y}*Im*(

*B*

_{k}*)*

_{,x,y}*/Re*(

*B*

_{k}*)), where atan2 is the four quadrant inverse tangent. An uncertainty in the magnitude and phase of each of the vectors can be represented by a region on the complex plane where the vector may end, as shown by the green box in Fig. 6(a). The length of the vector is set in the range*

_{,x,y}*B*

_{k}*∆*

_{,x,y}±*B*

_{k}*, while the uncertainty in the angle is*

_{,x,y}*±*∆

*θ*

_{k}*.*

_{,x,y}A rapid estimate of the region can alternatively be gained by looking at the worst case. As can be seen from Fig. 6(a), the smallest circle about the end of the initial vector that encloses all of the points in the green box has a radius *U*_{0}, where

It can be seen that if this repeated for a second vector that can start at any point within this circle, and that itself has an uncertainty in magnitude and phase that leads to it ending within a circle of radius *U*_{1}, then the sum of these two vectors must lie within a circle of radius *U*_{0} +*U*_{1}, as shown in Fig. 6(b). This can be repeated for all of the vectors, leading to the sum providing a point (*x*_{0},*y*_{0}) where the vector ends, and a circle of radius

As the sum of the vectors must lie within a circle of radius *U _{max}*, centred on (

*x*

_{0},

*y*

_{0}), the maximum error in

*θ*is

*±*∆

*θ*where,

Figure 6(c) shows the sum of all of the components of the FT of the cap of the sphere imaged by a lens with NA = 0.7, when *k _{z}* = 2.23 × 10

^{7}m

^{−1}, with the uncertainty in each of the elements being set independently. The uncertainty in the length is randomly set between the limits of

*±*0.1%, and the uncertainty in the phase between ±0.01 degrees, with the value being selected via the use of the rand function in Matlab. The values for the limits of the uncertainty in magnitude and phase are selected just for illustration purposes. Varying these limits would demonstrate the accuracy that the

*k*- space elements in the FT of the object would need to be known in order to obtain a given limit to the location of the surface due to the uncertainty.

The green circle in Fig. 6(c) thus shows an example of the region in which the summation of the vector could end, and provides and upper and lower limit to the *θ* that may be found for the *k _{z}* element. By repeating this for each

*k*element, the situation illustrated in Fig. 6(d) is obtained, where the green crosses mark the maximum and minimum phase for each wavenumber when the uncertainty is included. By finding the maximum and minimum gradient that can be obtained for a least squares fit of a line to phase values between these limits, the upper and lower limit of the surface determined by the FDA method is found. Thus, if the TF was known sufficiently well to correct each term to this level of accuracy, then these limits provide an estimation of the error in the surface due to this uncertainty. This error is in addition to that demonstrated in the previous section. It should be noted that this calculation loses all meaning if ${\left({x}_{0}^{2}+{y}_{0}^{2}\right)}^{1/2}<|{U}_{\mathrm{max}}|$ as the origin lies within the circle, and the phase can take any value. That the FDA method demonstrably works in practice suggests that the changes in the magnitude and phase caused by a standard microscope are sufficiently small that this will not be a common problem, though as will be seen this may still lead to a significant error with curved surfaces such as the one considered here.

_{z}The surface without the uncertainty is found *via* a least squares fit of a line to the points marked in blue. In order to calculate the upper and lower location of the surface the maximum and minimum gradient of the line need to be found when the phase for each wavenumber is allowed to take any value from the lower limit to the upper limit. Fitting a least squares line of the form *y* = *β*_{1}*x* + *β*_{0} to *m* data points, where at point *n*, *x* = *w _{n}*, the wavenumber, and

*y*=

*p*+

_{n}*ε*, the phase, gives a gradient of

_{n}The change of the gradient, *β*_{1} due to a change of *ε _{n}* is

As *∂β*_{1}*/∂ε _{n}* is independent of

*ε*, all of the points where Eq. (12) is positive the uncertainty should be maximised to obtain the maximum value of

_{n}*β*

_{1}, and minimised when it is negative. The converse should be carried out to minimise

*β*

_{1}.

In Fig. 7 the blue line shows the locations of the surface without the uncertainty in the terms, while the magenta and green lines mark the upper and lower limits to where the surface can be found. The surface will be smooth, lying somewhere between these two limits.

There are still questions about how this result affects the topography, as if the errors are correlated then the topography of the surface may be unaffected, but shifted. If however the errors are slightly more random then the surface may be able to take a variety of forms between these two limits.

## 5. Conclusion

By carefully selecting cases for which the result of a measurement can accurately be modelled we have examined the ability of the Frequency Domain Analysis method to recover the correct interface of a scattering object. It has been shown that, due to the limited set of spatial frequencies that are passed by the instrument, errors of the order of a hundred nm may be found. These results are compared with those gained using spatial frequencies in a larger volume of *k*- space, a case where the surface is far more successfully recovered, in order to show that the effects found are not numerical artefacts.

If the Fourier transform of the surface of the object is not modified significantly by being truncated by the limited passband of the instrument then the surface will be recovered well, in the absence of multiple scattering. The FDA method uses only a small section of the passed spatial frequencies, and these are chosen as those where the signal is greatest, however, these may not give the most accurate surface. It has been shown that the phase of the terms in the 1-D FT of the measurement result on each pixel is related to all of the *k*- space terms that have the same *k _{z}* component. In may be the case that using points where the Transfer Function of the instrument is broadest may be better, though this will depend on the FT of the surface, a general rule would be hard to develop. In any case, only using a limited set of the spatial frequencies passed may mean that significant detail is missed. The limited lateral extent of the FT of a flat surface means that the truncation does not modify the data, so the instruments should recover these surfaces well.

Ideally the microscope will not modify the phase and amplitude of the spatial frequency elements in the FT of the surface, however in a real instrument this is unlikely to be the case, and these errors are compensated for in some way or deemed sufficiently small to be ignored. The TF describes how an instrument itself modifies the phase and amplitude of the spatial frequencies in the measurement and these modifications can be corrected for if the TF is known; this should be carried out before the FDA method is applied to the measurement results. We have demonstrated how an uncertainty in the TF, and therefore an uncertainty in the phase and amplitude of each of the terms translates into upper and lower limits to the surface recovered. It should be pointed out that the errors introduced are not correlated, and a uniform distribution was used. This does however inform us about how accurately the TF must be obtained in order to allow for a meaningful surface to be recovered.

## Acknowledgments

This work was funded by the NMS Engineering & Flow Metrology Programme 2011–2014 and the EMRP project Microparts. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. Many thanks go to Alistair Forbes (NPL) for several useful discussions.

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