Measuring nanoparticle size using phase-stepping interferometry: quantifying measurement sensitivity to surface roughness Douglas J. Little* and Deb M. Kane MQ Photonics Research Centre, Department of Physics and Astronomy, Macquarie University, North Ryde, Sydney NSW 2109, Australia *Corresponding author: [email protected] Received 19 November 2013; revised 19 May 2014; accepted 24 May 2014; posted 29 May 2014 (Doc. ID 201657); published 9 July 2014
A method for sizing nanoparticles using phase-stepping interferometry has been developed recently by Little et al. [Appl. Phys. Lett. 103, 161107 (2013)]. We present an analytical procedure to quantify how sensitive measurement precision is to surface roughness. This procedure computes the standard deviation in the measured phase as a function of the surface roughness power spectrum. It is applied to nanospheres and nanowires on a flat plane and also a flat plane in isolation. Calculated sensitivity levels demonstrate that surface roughness is unlikely to be the limiting factor on measurement precision when measuring nanoparticle size using this phase-shifting-interferometry-based technique. The need to use an underlying surface that is very smooth when measuring nanoparticles is highlighted by the analysis. © 2014 Optical Society of America OCIS codes: (180.3170) Interference microscopy; (100.3175) Interferometric imaging; (120.3940) Metrology; (160.4236) Nanomaterials. http://dx.doi.org/10.1364/AO.53.004548
1. Introduction
Phase-stepping interferometry (PSI) is a widely used technique for remotely measuring three-dimensional surface height profiles [1–3]. Commercial PSI instruments are able to measure height profiles with subnanometer axial resolution. However, the lateral resolution is usually limited by diffraction, as with any conventional microscope system. This limited lateral resolution has traditionally meant that PSI can only be applied to nanometrology in a limited capacity. Recently, it was demonstrated that PSI can be used to measure the size of nanoscale objects of known geometry (sphere, cylinder, etc.) [4,5], by using a waveoptic model to compensate for the effect of diffraction in the PSI microscope. With this model, measured phase could be calculated as a function of nano-object 1559-128X/14/204548-07$15.00/0 © 2014 Optical Society of America 4548
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size, potentially enabling measurement of particles as small as 10 nm, given the precision of current state-ofthe-art PSI instruments [4]. Measurements of GaAs nanowire radii were demonstrated with an accuracy of 4% using this technique, following a modification of the coherent impulse response to account for the nonflatness of the object surface [5]. One practical complication is the presence of roughness on surfaces that the model assumes to be ideally smooth. To determine the precision of this PSI technique, the effect of surface roughness on phase measurements must therefore be carefully appraised. This essentially involves propagating the uncertainty in the object height profile (surface roughness) through the aforementioned wave-optic model to calculate the uncertainty in measured phase. The analytical procedure for this type of uncertainty analysis is documented in the International Organization for Standardization Guide to the Expression of Uncertainty in Measurement [6]. An alternate procedure is to calculate uncertainties numerically using
Monte Carlo simulations [7,8]. Monte Carlo simulations are often preferable to direct calculation due to the speed and relative simplicity of the approach. Here, the precision of PSI phase measurements is measured using Monte Carlo simulations of surface roughness within the original wave-optic model used in [4]. The sensitivity of phase precision to surface roughness amplitude is quantified as the sensitivity to roughness factor (SRF), defined as the phase precision per nanometer standard deviation of surface roughness. In Section 2, the procedure for calculating SRF is presented. In Section 3, SRFs are calculated for a flat plane, nanospheres on a flat plane, and nanowires lying prone on a flat plane, for f β roughness (surface noise) power spectra [9,10]. Surfaces where roughness was located only on a limited area were also investigated. A formula for estimating the SRF for various power spectra, surface geometries, and roughness area coverages is also presented. In Section 4, these results are discussed in the context of how they impact nanoparticle size measurements using phase measurement by an optical surface profilometer instrument. The impact surface roughness has on measurement precision is found to be negligible in most of the simulations carried out. This result is likely a more general one, as presented in the conclusion and outlook in Section 5. 2. Method A.
Calculating SRF
In the wave-optic model for calculating the phase in the image plane presented by Little and Kane [4], ϕimage of an optical surface profiler (OSP) from a specified phase in the object plane leads to Eq. (1) {see Eq. (24) in [5])}: 0P 1 RR N 0 0 sinϕ τdx dy j Q j1 A: RR j ϕimage x; y − ϕref tan−1 @PN 0 0 cosϕ j Qj τdx dy j1 (1)
to discretize the height profile, zx0 ; y0 , in x0 and y0 (Here, primes are used to denote coordinates in the object plane). Here, each discrete element of zx0 ; y0 can be labeled as a separate region. For a height profile discretized in a rectangular grid, the RHS of Eq. (1) can be expressed as tan−1 0P P 1 Ny Nx sin2kzij − zref τx0ij − x; y0ij − yΔx0 Δy0 j1 i1 @P P A; Ny Nx 0 − x; y0 − yΔx0 Δy0 cos2kz − z τx ij ref ij ij j1 i1 (3) where N x and N y are the total number of discrete elements in x and y. Δx0 , Δy0 , x0ij , and y0ij are the x and y intervals and the x and y coordinates, respectively, of the discrete elements, zij . The impulse response, τ, is taken to be the ideal impulse response for an imaging system with a finite numerical aperture q q τx; y J 1 NA × k x2 y2 ∕ x2 y2 ;
(4)
where k is the wavenumber and NA is the numerical aperture of the imaging system. In this study, a wavelength of 514 nm (giving a wavenumber of 12.224 μm−1 ), and an NA of 0.8 were assumed, reflecting real values used by commercial OSP instruments. Note that the problem is invariant with the quantity NA × k · x2 y2 1∕2. That is, halving NA × k and doubling the scale of the surface has no effect on the resultant measured phase. To introduce surface roughness, noise in the form of height perturbations, labeled Δz, is added to each discrete element in the surface height profile to obtain the “roughened” surface height profile, z0ij , given by z0ij zij Δzij :
(5)
Typical values of ϕimage range from radians for 100 nm sized particles to milliradians for 5 nm sized particles. Equation (1) arises as a result of approximating a surface with continuously varying height as N regions (labeled Qj ) of equal height. The left-hand side of Eq. (1) is the phase profile measured at the OSP’s image plane using PSI, relative to an arbitrary phase ϕref . On the right-hand side (RHS) of Eq. (1), τ denotes the coherent impulse response of the OSP’s imaging system and is a function of x0 –x and y0 –y. It is given by
The procedure for generating elements of Δzij is detailed in the next section. The addition of surface roughness to the surface height profile perturbs the measured PSI phase profile ϕimage x; y. This perturbation is random over a normal distribution, with standard deviation σ ϕ x; y. This standard deviation in phase represents the precision with which the phase from the (noise-free) surface zij can be ascertained. The aim of this study is to calculate σ ϕ x; y under various conditions. This is done by evaluating ϕimage x; y over multiple sets of Δzij . A SRF is defined as
ϕj 2kzj − zref ;
SRFx; y σ ϕ x; y∕σ noise ;
(2)
where zj is the constant height of region Qj , and zref is an arbitrary reference height. In order to numerically introduce surface roughness onto an ideally smooth surface, it is necessary
(6)
where σ noise is the standard deviation of Δzij. The linearity of Eq. (6) was verified up to a value of λ∕25 (around 20 nm for λ 514 nm). SRF was found by calculating σ ϕ x; y for different levels of surface 10 July 2014 / Vol. 53, No. 20 / APPLIED OPTICS
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roughness and then applying a linear regression. To determine SRF with a precision better than 5%, σ noise was increased from 0 to 5.0 nm in 0.5 nm increments, and 100 sets of height perturbations were used to calculate σ ϕ x; y for each value of σ noise. A natural unit for typical values of SRF is milliradians per nanometer. B.
Generating Surface Noise Elements
Noise can be characterized by three parameters; the shape of the power spectrum; the distribution; and the standard deviation, σ noise . Gaussian noise distributions have been considered here. To generate noise elements, we use the procedure outlined by Lennon [9], where noise elements are generated from elements of a predefined power spectrum Sij , using the following: Δzij IFFT
q Sij e−iθij ;
(7)
where θij is randomly generated with a uniform distribution over the interval [0, 2π) and IFFT denotes an inverse fast Fourier transform. In this study, we consider surface roughness (surface noise) with power spectra of the form Sij Ak2xi k2yj β∕2 ;
(8)
where kxi and kyj are discrete x and y spatialfrequency components, and the parameter β defines the color of the noise; β 0 conventionally denotes white noise, β −1 pink noise, and β −2 red (or Brownian) noise [6,7]. A was chosen so the standard deviation of the generated noise was equal to σ noise . An example of generated white, pink, and red surface roughness sets are shown in Figs. 1(a)–1(c). Figure 1(d) shows the spatial extent of a 50 nm radius sphere to link the spatial scale of the surface roughness features to that of a nanoparticle. In the simulations that follow; the effect of white, pink, and red surface roughness on the measurement of the phase profile of a 50 nm radius sphere on a flat plane is calculated. Unless stated otherwise, the surface roughness extends over both the sphere and the underlying flat surface.
Fig. 1. Sample surface roughness sets over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm for (a) white noise, (b) pink noise, and (c) red noise. The standard deviation of each noise set is the same. (a)–(c) are plotted on the same height grey scale. (d) Spatial extent of a nanosphere of radius 50 nm for comparison.
height profile of a sphere (upper surface) on a flat plane is defined as zx; y − zref
R
p R2 − x2 − y2 0
x2 y2 ≤ R2 ; x2 y2 > R2 (9)
where R is taken here to be 50 nm. Figure 2 shows SRF calculated as a function of N x · N y −1∕2 , which exhibits the expected linear dependence. The linear fit obtained via linear regression intersects the origin within statistical
3. Results A.
SRF for White Noise
For white noise, Δzij are statistically uncorrelated. This has some rather interesting consequences when attempting to calculate σ ϕ x; y. Close inspection of Eq. (3) reveals it to be a type of weighted average, and so the quantity σ ϕ x; y is a residual which is inversely proportional to N x · N y −1∕2 (i.e., the total number of discrete elements). Thus, in the continuous limit, σ ϕ x; y and the SRF are expected to converge precisely to zero. To demonstrate this, surface roughness sets are applied to a nanosphere lying on a flat plane. The 4550
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Fig. 2. SRF calculated for a sphere of radius 50 nm sitting on a flat surface with white surface roughness (noise) (β 0) as a function of the number of discrete elements raised to the power of −1∕2 over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm (circles). Line indicates a linear fit.
uncertainty, which is consistent with SRF vanishing as the number of discrete elements is made arbitrarily large. B.
SRF for Pink (β −1) and Red (β −2) Noise
White noise is a flawed model for surface roughness as real surfaces exhibit some degree of spatial height correlation. A better model for surface roughness is one where low-spatial-frequency noise dominates over high-spatial-frequency noise, which yields a more realistic spatial height correlation. One such type of noise is termed pink noise and is characterized by a β value of −1 and sometimes referred to as 1∕f noise or fractal noise. Figure 3 shows an example calculation of SRF versus N x · N y −1∕2 for the same surface as in the previous section, but with pink noise instead of white noise. The main difference here is that the residuals described by Eq. (3) converge to a finite nonzero value, calculated by performing a linear regression and taking the y-intercept. Residuals converge to a finite value because the surface roughness exhibits a degree of spatial correlation, preventing residuals from vanishing completely when an arbitrarily large number are included in the summation. Another type of noise where low-spatial-frequency components dominate even more than in pink noise is called red noise, which is characterized by a β value of −2. Red noise is also referred to as Brownian (or Brown) noise as its power spectrum is the integral of a white noise power spectrum and so can be modeled as a summation of random “kicks” as in a random walk, used to describe Brownian motion. Figure 4 shows an example calculation of SRF versus N x · N y −1∕2 for red noise, using the same surface as previously. Both a linear and quadratic fit were performed. In both cases, the residual converges to a higher value than for pink noise, reflecting the
Fig. 4. SRF calculated for a sphere of radius 50 nm sitting on a flat surface with red surface roughness (noise) (β −2), as a function of the number of discrete elements raised to the power of −1∕2 over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm (circles); a linear fit (solid line) with a y intercept of 11.5 mrad∕nm with a 95% confidence interval of 0.65 mrad∕nm, and a quadratic fit (dashed line) with a y intercept of 11.0 mrad∕nm with a 95% confidence interval of 0.30 mrad∕nm.
greater degree of spatial correlation present within the surface roughness. Linear fits are used in all following sections, as they yield the most conservative estimate of SRF, using grid densities ranging from 201 × 201 points N x · N y −1∕2 5 × 10−3 to 2001 × 2001 points N x · N y −1∕2 5 × 10−4 . C.
D.
Fig. 3. SRF calculated for a sphere of radius 50 nm sitting on a flat surface with pink surface roughness (noise) (β −1) as a function of the number of discrete elements raised to the power of −1∕2 over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm (circles). A linear fit (line) gives a y intercept of 0.80 mrad∕nm with a 95% confidence interval of 0.24 mrad∕nm.
SRF for Other Colored Noise (2 < β < 0)
A more complete way of characterizing the SRF for a given surface is the evaluation of the SRF for a range of β. The procedure described above for pink and red noise can be used for arbitrary β between 0 and −2. The resultant function, SRFβ, defines the sensitivity characteristics for a given surface in the presence of f β noise. Figure 5 shows SRFβ calculated for a sphere of 50 nm radius sitting on a flat surface, with 95% confidence intervals shown by shading. Variation in SRF with Surface Geometry
Previously, SRFs were calculated for 50 nm radius nanospheres on a flat surface only. Here, variations in surface geometry and the resultant effect on SRF are considered. First, consider the effect of nanosphere size on SRF. Figure 6 shows SRFβ calculated for spheres of radius 5 and 100 nm. It can be seen from Fig. 6(c) that SRFβ for both nanosphere radii is equal within statistical uncertainty. This indicates that SRFβ is most probably invariant with nanosphere size. Therefore, σ ϕ x; y mostly arises due to surface roughness on the flat plane rather than roughness on the nanosphere itself. Next, consider the effect of nonspherical surface geometries on SRF. Here, a nanocylinder (or nanowire) lying prone on a flat plane has been considered. The height profile of a nanocylinder (upper surface 10 July 2014 / Vol. 53, No. 20 / APPLIED OPTICS
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Fig. 5. SRF calculated for a sphere of radius 50 nm sitting on a flat surface as a function of β over the region −2 < β < 0 in steps of 0.2. Shaded region indicates the 95% confidence interval.
only) on a flat surface, with its axis oriented in the x direction, is defined as zx; y − zref
R
p R2 − x2 0
jxj ≤ R ; jxj > R
(10)
Fig. 7. SRF calculated for (a) 2 μm long cylinder with a radius of 50 nm sitting on a flat surface, (b) a flat surface only, and (c) the difference between SRF values calculated in (a) and (b). Filled circles indicate calculated values; shaded region indicates the 95% confidence interval.
geometry on SRF is negligible when the roughness extends over the whole surface. E. Variation in SRF with Surface Roughness Area
where R is the radius of the nanocylinder. It can be seen from Fig. 7(a) that SRFβ for nanocylinders lying prone on a flat surface is equal to that of the spheres on a flat plane within statistical uncertainty. This demonstrates that SRF is insensitive to the initial height profile, zij , when the roughness extends over the entire surface and only depends on the characteristics of the surface roughness noise elements Δzij . This was confirmed in Fig. 7(b), where a flat surface only exhibits the same SRF as other surface geometries, demonstrating that the effect of surface
It is realistic to suppose that for nanoparticles on a flat plane, the roughness characteristics of the nanoparticles will differ from that of the underlying surface. Figure 8 shows SRF calculated for a sphere with a radius of 50 nm sitting on a flat surface where the surface roughness extends over the nanosphere area only and over the flat plane area only. The relative contributions to SRFβ are clearly resolved, with the contribution from the underlying flat plane around 30 times greater than the contribution from the 50 nm radius nanosphere. Figure 9 shows SRF calculated for the same nanosphere compared to a flat plane where the surface roughness is only present over the same area extent as the nanosphere (i.e., a circle of radius 50 nm). That
Fig. 6. SRF calculated for (a) 5 nm radius sphere, (b) 100 nm radius sphere sitting on a flat surface, and (c) the difference between SRF values calculated in (a) and (b). Filled circles indicate calculated values; shaded region indicates the 95% confidence interval.
Fig. 8. SRF calculated for a sphere of radius 50 nm sitting on a flat surface for surface roughness on the sphere only (open circles) and on the flat surface only (filled circles). Shaded region indicates the 95% confidence interval.
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Fig. 9. SRF calculated for a sphere of radius 50 nm sitting on a flat surface for surface roughness on the sphere only [(a), dark circles] and a flat surface with roughness localized over a circle with a radius of 50 nm [(b), light circles]. Shaded regions indicate 95% confidence intervals.
is, the projected area over which the roughness is present is kept constant while the surface geometry is being changed. Here, the effect of surface geometry on SRFβ is clearly resolved. This demonstrates that the geometry is no longer a negligible factor in calculating SRFβ when the surface roughness is localized on the nanoparticle. 4. Relevance of SRF to PSI Phase Measurements
When measuring nanoparticles using OSPs, the question this paper seeks to answer is, “how much surface roughness is tolerable?” From the results in Section 3, it can be seen that the answer mainly depends on the power spectrum of the surface roughness and the area over which the roughness is present. For surfaces where the surface roughness is localized on the nanoparticle only, the surface geometry is also a factor. Here, we discuss results in Section 3 relative to quoted phase precisions for commercial OSP instruments, which typically vary from 2.5 to 0.25 mrad. In absolute terms, under conditions where SRF is highest (flat plane surface, red noise, surface roughness over the full area), the SRF is around 13 mrad∕nm. That is, the phase measurement made by the OSP is expected to deviate with a standard deviation of 13 mrad per nm of standard deviation in surface roughness. For such flat plane surfaces, roughness will be the limiting factor in measurement precision for surface roughness values above 0.02–0.2 nm (i.e., all but the smoothest of surfaces). A flat plane surface with roughness of 1 nm will result in a phase precision of 13 mrad, which equates to around 1% uncertainty in the measurement of a cylinder with radius 50 nm, and 16% uncertainty in the measurement of a sphere with radius 50 nm, for NA 0.8 and a wavelength of 514 nm (recalling that SRF is largely invariant with surface geometry when
the surface roughness extends over the full surface area). For pink noise power spectra, SRF is around 1 mrad∕nm, and so for flat plane surfaces with roughness under 0.25–2.5 nm (depending on the instrument), surface roughness will likely not be the limiting factor in the precision of the nanoparticle measurement. If the roughness is constrained to the nanoparticle only, then the SRF drops to 0.38 mrad∕nm for red noise power spectra and around 0.01 mrad∕nm for pink noise power spectra (for a 50 nm sphere). In these instances, roughness on the particle will only be a limiting factor for the most extreme cases of roughness. Good practice therefore demands that the flattest possible underlying surface is used in order to obtain the most precise measurements when measuring nanoparticles using PSI. Measuring surface roughness power spectra is possible using techniques such as PSI microscopy and atomic-force microscopy. Such measurements will always be bandwidth-limited due to instrument resolution. A degree of extrapolation is required to model surface roughness power spectra over the full spatial-frequency spectrum. An example of a surface roughness power spectrum measured using PSI microscopy is shown in Fig. 10. The surface measured was a polished mirror with σ noise 0.13 nm and a 750 μm × 750 μm field of view. For this surface, β was calculated to be −1.42 0.03. The roll-off at the upper end of the spatial frequency scale in Fig. 10 occurs due to attenuation of high-spatial-frequency components that occurs as a result of the finite spatial bandwidth of the PSI instrument.
Fig. 10. Power spectrum for a polished mirror surface (inset) with a 750 μm × 750 μm field of view and σ noise 0.13 nm. This power spectrum was obtained by integrating the isotropic twodimensional power spectrum over the angular range 0 to 2π. The gray scale in the inset corresponds to a height difference of 1.17 nm. β was calculated to be −1.42 0.03 from the gradient of the log–log fit shown (black line). Data from the “roll-off” present at the upper spatial frequency end of the scale was excluded from the log–log fit. 10 July 2014 / Vol. 53, No. 20 / APPLIED OPTICS
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5. Conclusion
In this paper, the effect of surface roughness (noise) on nanoparticle measurements using PSI has been quantified as a SRF, defined as the phase measurement precision per nanometer of standard deviation in the added surface noise. It was found that spectra dominated by low-spatial-frequency components (red noise) possess the highest SRF for flat planar or nearflat-planar surfaces: up to 13 mrad per nanometer of standard deviation of surface roughness. The projected area of the surface roughness was also found to be a major factor. Smaller areas yielded smaller SRFs by as much as a factor of 20 or more. This is significant in the context of nanoparticle measurement, as it is often the case that surface roughness is mostly localized on the nanoparticle. This also suggests that the underlying surface should be made as flat as possible to minimize SRF, and ultimately, precision of the PSI phase measurement. Surface geometry is also found to influence SRF, but to a lesser extent. A factor of 2–3 difference in SRF was calculated between nanospheres, nanowires, and flat planes. This difference was only discernible in cases where the surface roughness was localized on the nanoparticle. These results suggest that when measuring nanoparticles using PSI, the magnitude with which surface roughness affects measurement precision is comparable to, and in many cases less than, the precision of the instrument specified by the manufacturer. Only for surfaces where roughness possesses a red-noise-like power spectrum is it predicted that surface roughness will be a major limiting factor in measurement precision. Based on these results, it is recommended that when measuring nanoparticles using OSPs, the SRF should be calculated using the procedure described herein as part of the analytical procedure to test the feasibility of the measurement and to
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determine the contribution to the measurement uncertainty that the surface roughness will induce. The methodology reported here-in for simulating the impact of surface roughness on phase measurement in PSI is also useful to the broader range of measurement applications of the PSI technique. This research was supported by Australian Research Council Linkage Infrastructure, Equipment and Facilities Grant LE110100024; a Macquarie University Research Development Grant, and Australian Research Council Discovery Project Grant DP130102674. References 1. B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985). 2. J. C. Wyant, C. L. Koliopoulos, B. Bhushan, and D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986). 3. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990). 4. D. J. Little and D. M. Kane, “Measuring nanoparticle size using optical surface profilers,” Opt. Express 21, 15664–15675 (2013). 5. D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, and D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013). 6. “Guide to the expression of uncertainty in measurement,” 2nd ed., ISO 98:1995 (International Organization for Standardization, Geneva, 1995). 7. “Extension to any number of output quantities—Part 3,” International Organization for Standardization, Geneva, ISO/IEC 98-3:2008/Suppl 2:2011 (2011). 8. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953). 9. J. J. Lennon, “Red-shifts and red herrings in geographical ecology,” Ecography 23, 101–113 (2000). 10. J. M. Halley and W. E. Kunin, “Extinction risk and the 1/f family of noise models,” Theor. Popul. Biol. 56, 215–230 (1999).
A method for sizing nanoparticles using phase-stepping interferometry has been developed recently by Little et al. [Appl. Phys. Lett. 103, 161107 (2013)]. We present an analytical procedure to quantify how sensitive measurement precision is to surface roughness. This procedure computes the standard deviation in the measured phase as a function of the surface roughness power spectrum. It is applied to nanospheres and nanowires on a flat plane and also a flat plane in isolation. Calculated sensitivity levels demonstrate that surface roughness is unlikely to be the limiting factor on measurement precision when measuring nanoparticle size using this phase-shifting-interferometry-based technique. The need to use an underlying surface that is very smooth when measuring nanoparticles is highlighted by the analysis. © 2014 Optical Society of America OCIS codes: (180.3170) Interference microscopy; (100.3175) Interferometric imaging; (120.3940) Metrology; (160.4236) Nanomaterials. http://dx.doi.org/10.1364/AO.53.004548
1. Introduction
Phase-stepping interferometry (PSI) is a widely used technique for remotely measuring three-dimensional surface height profiles [1–3]. Commercial PSI instruments are able to measure height profiles with subnanometer axial resolution. However, the lateral resolution is usually limited by diffraction, as with any conventional microscope system. This limited lateral resolution has traditionally meant that PSI can only be applied to nanometrology in a limited capacity. Recently, it was demonstrated that PSI can be used to measure the size of nanoscale objects of known geometry (sphere, cylinder, etc.) [4,5], by using a waveoptic model to compensate for the effect of diffraction in the PSI microscope. With this model, measured phase could be calculated as a function of nano-object 1559-128X/14/204548-07$15.00/0 © 2014 Optical Society of America 4548
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size, potentially enabling measurement of particles as small as 10 nm, given the precision of current state-ofthe-art PSI instruments [4]. Measurements of GaAs nanowire radii were demonstrated with an accuracy of 4% using this technique, following a modification of the coherent impulse response to account for the nonflatness of the object surface [5]. One practical complication is the presence of roughness on surfaces that the model assumes to be ideally smooth. To determine the precision of this PSI technique, the effect of surface roughness on phase measurements must therefore be carefully appraised. This essentially involves propagating the uncertainty in the object height profile (surface roughness) through the aforementioned wave-optic model to calculate the uncertainty in measured phase. The analytical procedure for this type of uncertainty analysis is documented in the International Organization for Standardization Guide to the Expression of Uncertainty in Measurement [6]. An alternate procedure is to calculate uncertainties numerically using
Monte Carlo simulations [7,8]. Monte Carlo simulations are often preferable to direct calculation due to the speed and relative simplicity of the approach. Here, the precision of PSI phase measurements is measured using Monte Carlo simulations of surface roughness within the original wave-optic model used in [4]. The sensitivity of phase precision to surface roughness amplitude is quantified as the sensitivity to roughness factor (SRF), defined as the phase precision per nanometer standard deviation of surface roughness. In Section 2, the procedure for calculating SRF is presented. In Section 3, SRFs are calculated for a flat plane, nanospheres on a flat plane, and nanowires lying prone on a flat plane, for f β roughness (surface noise) power spectra [9,10]. Surfaces where roughness was located only on a limited area were also investigated. A formula for estimating the SRF for various power spectra, surface geometries, and roughness area coverages is also presented. In Section 4, these results are discussed in the context of how they impact nanoparticle size measurements using phase measurement by an optical surface profilometer instrument. The impact surface roughness has on measurement precision is found to be negligible in most of the simulations carried out. This result is likely a more general one, as presented in the conclusion and outlook in Section 5. 2. Method A.
Calculating SRF
In the wave-optic model for calculating the phase in the image plane presented by Little and Kane [4], ϕimage of an optical surface profiler (OSP) from a specified phase in the object plane leads to Eq. (1) {see Eq. (24) in [5])}: 0P 1 RR N 0 0 sinϕ τdx dy j Q j1 A: RR j ϕimage x; y − ϕref tan−1 @PN 0 0 cosϕ j Qj τdx dy j1 (1)
to discretize the height profile, zx0 ; y0 , in x0 and y0 (Here, primes are used to denote coordinates in the object plane). Here, each discrete element of zx0 ; y0 can be labeled as a separate region. For a height profile discretized in a rectangular grid, the RHS of Eq. (1) can be expressed as tan−1 0P P 1 Ny Nx sin2kzij − zref τx0ij − x; y0ij − yΔx0 Δy0 j1 i1 @P P A; Ny Nx 0 − x; y0 − yΔx0 Δy0 cos2kz − z τx ij ref ij ij j1 i1 (3) where N x and N y are the total number of discrete elements in x and y. Δx0 , Δy0 , x0ij , and y0ij are the x and y intervals and the x and y coordinates, respectively, of the discrete elements, zij . The impulse response, τ, is taken to be the ideal impulse response for an imaging system with a finite numerical aperture q q τx; y J 1 NA × k x2 y2 ∕ x2 y2 ;
(4)
where k is the wavenumber and NA is the numerical aperture of the imaging system. In this study, a wavelength of 514 nm (giving a wavenumber of 12.224 μm−1 ), and an NA of 0.8 were assumed, reflecting real values used by commercial OSP instruments. Note that the problem is invariant with the quantity NA × k · x2 y2 1∕2. That is, halving NA × k and doubling the scale of the surface has no effect on the resultant measured phase. To introduce surface roughness, noise in the form of height perturbations, labeled Δz, is added to each discrete element in the surface height profile to obtain the “roughened” surface height profile, z0ij , given by z0ij zij Δzij :
(5)
Typical values of ϕimage range from radians for 100 nm sized particles to milliradians for 5 nm sized particles. Equation (1) arises as a result of approximating a surface with continuously varying height as N regions (labeled Qj ) of equal height. The left-hand side of Eq. (1) is the phase profile measured at the OSP’s image plane using PSI, relative to an arbitrary phase ϕref . On the right-hand side (RHS) of Eq. (1), τ denotes the coherent impulse response of the OSP’s imaging system and is a function of x0 –x and y0 –y. It is given by
The procedure for generating elements of Δzij is detailed in the next section. The addition of surface roughness to the surface height profile perturbs the measured PSI phase profile ϕimage x; y. This perturbation is random over a normal distribution, with standard deviation σ ϕ x; y. This standard deviation in phase represents the precision with which the phase from the (noise-free) surface zij can be ascertained. The aim of this study is to calculate σ ϕ x; y under various conditions. This is done by evaluating ϕimage x; y over multiple sets of Δzij . A SRF is defined as
ϕj 2kzj − zref ;
SRFx; y σ ϕ x; y∕σ noise ;
(2)
where zj is the constant height of region Qj , and zref is an arbitrary reference height. In order to numerically introduce surface roughness onto an ideally smooth surface, it is necessary
(6)
where σ noise is the standard deviation of Δzij. The linearity of Eq. (6) was verified up to a value of λ∕25 (around 20 nm for λ 514 nm). SRF was found by calculating σ ϕ x; y for different levels of surface 10 July 2014 / Vol. 53, No. 20 / APPLIED OPTICS
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roughness and then applying a linear regression. To determine SRF with a precision better than 5%, σ noise was increased from 0 to 5.0 nm in 0.5 nm increments, and 100 sets of height perturbations were used to calculate σ ϕ x; y for each value of σ noise. A natural unit for typical values of SRF is milliradians per nanometer. B.
Generating Surface Noise Elements
Noise can be characterized by three parameters; the shape of the power spectrum; the distribution; and the standard deviation, σ noise . Gaussian noise distributions have been considered here. To generate noise elements, we use the procedure outlined by Lennon [9], where noise elements are generated from elements of a predefined power spectrum Sij , using the following: Δzij IFFT
q Sij e−iθij ;
(7)
where θij is randomly generated with a uniform distribution over the interval [0, 2π) and IFFT denotes an inverse fast Fourier transform. In this study, we consider surface roughness (surface noise) with power spectra of the form Sij Ak2xi k2yj β∕2 ;
(8)
where kxi and kyj are discrete x and y spatialfrequency components, and the parameter β defines the color of the noise; β 0 conventionally denotes white noise, β −1 pink noise, and β −2 red (or Brownian) noise [6,7]. A was chosen so the standard deviation of the generated noise was equal to σ noise . An example of generated white, pink, and red surface roughness sets are shown in Figs. 1(a)–1(c). Figure 1(d) shows the spatial extent of a 50 nm radius sphere to link the spatial scale of the surface roughness features to that of a nanoparticle. In the simulations that follow; the effect of white, pink, and red surface roughness on the measurement of the phase profile of a 50 nm radius sphere on a flat plane is calculated. Unless stated otherwise, the surface roughness extends over both the sphere and the underlying flat surface.
Fig. 1. Sample surface roughness sets over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm for (a) white noise, (b) pink noise, and (c) red noise. The standard deviation of each noise set is the same. (a)–(c) are plotted on the same height grey scale. (d) Spatial extent of a nanosphere of radius 50 nm for comparison.
height profile of a sphere (upper surface) on a flat plane is defined as zx; y − zref
R
p R2 − x2 − y2 0
x2 y2 ≤ R2 ; x2 y2 > R2 (9)
where R is taken here to be 50 nm. Figure 2 shows SRF calculated as a function of N x · N y −1∕2 , which exhibits the expected linear dependence. The linear fit obtained via linear regression intersects the origin within statistical
3. Results A.
SRF for White Noise
For white noise, Δzij are statistically uncorrelated. This has some rather interesting consequences when attempting to calculate σ ϕ x; y. Close inspection of Eq. (3) reveals it to be a type of weighted average, and so the quantity σ ϕ x; y is a residual which is inversely proportional to N x · N y −1∕2 (i.e., the total number of discrete elements). Thus, in the continuous limit, σ ϕ x; y and the SRF are expected to converge precisely to zero. To demonstrate this, surface roughness sets are applied to a nanosphere lying on a flat plane. The 4550
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Fig. 2. SRF calculated for a sphere of radius 50 nm sitting on a flat surface with white surface roughness (noise) (β 0) as a function of the number of discrete elements raised to the power of −1∕2 over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm (circles). Line indicates a linear fit.
uncertainty, which is consistent with SRF vanishing as the number of discrete elements is made arbitrarily large. B.
SRF for Pink (β −1) and Red (β −2) Noise
White noise is a flawed model for surface roughness as real surfaces exhibit some degree of spatial height correlation. A better model for surface roughness is one where low-spatial-frequency noise dominates over high-spatial-frequency noise, which yields a more realistic spatial height correlation. One such type of noise is termed pink noise and is characterized by a β value of −1 and sometimes referred to as 1∕f noise or fractal noise. Figure 3 shows an example calculation of SRF versus N x · N y −1∕2 for the same surface as in the previous section, but with pink noise instead of white noise. The main difference here is that the residuals described by Eq. (3) converge to a finite nonzero value, calculated by performing a linear regression and taking the y-intercept. Residuals converge to a finite value because the surface roughness exhibits a degree of spatial correlation, preventing residuals from vanishing completely when an arbitrarily large number are included in the summation. Another type of noise where low-spatial-frequency components dominate even more than in pink noise is called red noise, which is characterized by a β value of −2. Red noise is also referred to as Brownian (or Brown) noise as its power spectrum is the integral of a white noise power spectrum and so can be modeled as a summation of random “kicks” as in a random walk, used to describe Brownian motion. Figure 4 shows an example calculation of SRF versus N x · N y −1∕2 for red noise, using the same surface as previously. Both a linear and quadratic fit were performed. In both cases, the residual converges to a higher value than for pink noise, reflecting the
Fig. 4. SRF calculated for a sphere of radius 50 nm sitting on a flat surface with red surface roughness (noise) (β −2), as a function of the number of discrete elements raised to the power of −1∕2 over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm (circles); a linear fit (solid line) with a y intercept of 11.5 mrad∕nm with a 95% confidence interval of 0.65 mrad∕nm, and a quadratic fit (dashed line) with a y intercept of 11.0 mrad∕nm with a 95% confidence interval of 0.30 mrad∕nm.
greater degree of spatial correlation present within the surface roughness. Linear fits are used in all following sections, as they yield the most conservative estimate of SRF, using grid densities ranging from 201 × 201 points N x · N y −1∕2 5 × 10−3 to 2001 × 2001 points N x · N y −1∕2 5 × 10−4 . C.
D.
Fig. 3. SRF calculated for a sphere of radius 50 nm sitting on a flat surface with pink surface roughness (noise) (β −1) as a function of the number of discrete elements raised to the power of −1∕2 over the region −1 μm < x < 1 μm and −1 μm < y < 1 μm (circles). A linear fit (line) gives a y intercept of 0.80 mrad∕nm with a 95% confidence interval of 0.24 mrad∕nm.
SRF for Other Colored Noise (2 < β < 0)
A more complete way of characterizing the SRF for a given surface is the evaluation of the SRF for a range of β. The procedure described above for pink and red noise can be used for arbitrary β between 0 and −2. The resultant function, SRFβ, defines the sensitivity characteristics for a given surface in the presence of f β noise. Figure 5 shows SRFβ calculated for a sphere of 50 nm radius sitting on a flat surface, with 95% confidence intervals shown by shading. Variation in SRF with Surface Geometry
Previously, SRFs were calculated for 50 nm radius nanospheres on a flat surface only. Here, variations in surface geometry and the resultant effect on SRF are considered. First, consider the effect of nanosphere size on SRF. Figure 6 shows SRFβ calculated for spheres of radius 5 and 100 nm. It can be seen from Fig. 6(c) that SRFβ for both nanosphere radii is equal within statistical uncertainty. This indicates that SRFβ is most probably invariant with nanosphere size. Therefore, σ ϕ x; y mostly arises due to surface roughness on the flat plane rather than roughness on the nanosphere itself. Next, consider the effect of nonspherical surface geometries on SRF. Here, a nanocylinder (or nanowire) lying prone on a flat plane has been considered. The height profile of a nanocylinder (upper surface 10 July 2014 / Vol. 53, No. 20 / APPLIED OPTICS
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Fig. 5. SRF calculated for a sphere of radius 50 nm sitting on a flat surface as a function of β over the region −2 < β < 0 in steps of 0.2. Shaded region indicates the 95% confidence interval.
only) on a flat surface, with its axis oriented in the x direction, is defined as zx; y − zref
R
p R2 − x2 0
jxj ≤ R ; jxj > R
(10)
Fig. 7. SRF calculated for (a) 2 μm long cylinder with a radius of 50 nm sitting on a flat surface, (b) a flat surface only, and (c) the difference between SRF values calculated in (a) and (b). Filled circles indicate calculated values; shaded region indicates the 95% confidence interval.
geometry on SRF is negligible when the roughness extends over the whole surface. E. Variation in SRF with Surface Roughness Area
where R is the radius of the nanocylinder. It can be seen from Fig. 7(a) that SRFβ for nanocylinders lying prone on a flat surface is equal to that of the spheres on a flat plane within statistical uncertainty. This demonstrates that SRF is insensitive to the initial height profile, zij , when the roughness extends over the entire surface and only depends on the characteristics of the surface roughness noise elements Δzij . This was confirmed in Fig. 7(b), where a flat surface only exhibits the same SRF as other surface geometries, demonstrating that the effect of surface
It is realistic to suppose that for nanoparticles on a flat plane, the roughness characteristics of the nanoparticles will differ from that of the underlying surface. Figure 8 shows SRF calculated for a sphere with a radius of 50 nm sitting on a flat surface where the surface roughness extends over the nanosphere area only and over the flat plane area only. The relative contributions to SRFβ are clearly resolved, with the contribution from the underlying flat plane around 30 times greater than the contribution from the 50 nm radius nanosphere. Figure 9 shows SRF calculated for the same nanosphere compared to a flat plane where the surface roughness is only present over the same area extent as the nanosphere (i.e., a circle of radius 50 nm). That
Fig. 6. SRF calculated for (a) 5 nm radius sphere, (b) 100 nm radius sphere sitting on a flat surface, and (c) the difference between SRF values calculated in (a) and (b). Filled circles indicate calculated values; shaded region indicates the 95% confidence interval.
Fig. 8. SRF calculated for a sphere of radius 50 nm sitting on a flat surface for surface roughness on the sphere only (open circles) and on the flat surface only (filled circles). Shaded region indicates the 95% confidence interval.
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Fig. 9. SRF calculated for a sphere of radius 50 nm sitting on a flat surface for surface roughness on the sphere only [(a), dark circles] and a flat surface with roughness localized over a circle with a radius of 50 nm [(b), light circles]. Shaded regions indicate 95% confidence intervals.
is, the projected area over which the roughness is present is kept constant while the surface geometry is being changed. Here, the effect of surface geometry on SRFβ is clearly resolved. This demonstrates that the geometry is no longer a negligible factor in calculating SRFβ when the surface roughness is localized on the nanoparticle. 4. Relevance of SRF to PSI Phase Measurements
When measuring nanoparticles using OSPs, the question this paper seeks to answer is, “how much surface roughness is tolerable?” From the results in Section 3, it can be seen that the answer mainly depends on the power spectrum of the surface roughness and the area over which the roughness is present. For surfaces where the surface roughness is localized on the nanoparticle only, the surface geometry is also a factor. Here, we discuss results in Section 3 relative to quoted phase precisions for commercial OSP instruments, which typically vary from 2.5 to 0.25 mrad. In absolute terms, under conditions where SRF is highest (flat plane surface, red noise, surface roughness over the full area), the SRF is around 13 mrad∕nm. That is, the phase measurement made by the OSP is expected to deviate with a standard deviation of 13 mrad per nm of standard deviation in surface roughness. For such flat plane surfaces, roughness will be the limiting factor in measurement precision for surface roughness values above 0.02–0.2 nm (i.e., all but the smoothest of surfaces). A flat plane surface with roughness of 1 nm will result in a phase precision of 13 mrad, which equates to around 1% uncertainty in the measurement of a cylinder with radius 50 nm, and 16% uncertainty in the measurement of a sphere with radius 50 nm, for NA 0.8 and a wavelength of 514 nm (recalling that SRF is largely invariant with surface geometry when
the surface roughness extends over the full surface area). For pink noise power spectra, SRF is around 1 mrad∕nm, and so for flat plane surfaces with roughness under 0.25–2.5 nm (depending on the instrument), surface roughness will likely not be the limiting factor in the precision of the nanoparticle measurement. If the roughness is constrained to the nanoparticle only, then the SRF drops to 0.38 mrad∕nm for red noise power spectra and around 0.01 mrad∕nm for pink noise power spectra (for a 50 nm sphere). In these instances, roughness on the particle will only be a limiting factor for the most extreme cases of roughness. Good practice therefore demands that the flattest possible underlying surface is used in order to obtain the most precise measurements when measuring nanoparticles using PSI. Measuring surface roughness power spectra is possible using techniques such as PSI microscopy and atomic-force microscopy. Such measurements will always be bandwidth-limited due to instrument resolution. A degree of extrapolation is required to model surface roughness power spectra over the full spatial-frequency spectrum. An example of a surface roughness power spectrum measured using PSI microscopy is shown in Fig. 10. The surface measured was a polished mirror with σ noise 0.13 nm and a 750 μm × 750 μm field of view. For this surface, β was calculated to be −1.42 0.03. The roll-off at the upper end of the spatial frequency scale in Fig. 10 occurs due to attenuation of high-spatial-frequency components that occurs as a result of the finite spatial bandwidth of the PSI instrument.
Fig. 10. Power spectrum for a polished mirror surface (inset) with a 750 μm × 750 μm field of view and σ noise 0.13 nm. This power spectrum was obtained by integrating the isotropic twodimensional power spectrum over the angular range 0 to 2π. The gray scale in the inset corresponds to a height difference of 1.17 nm. β was calculated to be −1.42 0.03 from the gradient of the log–log fit shown (black line). Data from the “roll-off” present at the upper spatial frequency end of the scale was excluded from the log–log fit. 10 July 2014 / Vol. 53, No. 20 / APPLIED OPTICS
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5. Conclusion
In this paper, the effect of surface roughness (noise) on nanoparticle measurements using PSI has been quantified as a SRF, defined as the phase measurement precision per nanometer of standard deviation in the added surface noise. It was found that spectra dominated by low-spatial-frequency components (red noise) possess the highest SRF for flat planar or nearflat-planar surfaces: up to 13 mrad per nanometer of standard deviation of surface roughness. The projected area of the surface roughness was also found to be a major factor. Smaller areas yielded smaller SRFs by as much as a factor of 20 or more. This is significant in the context of nanoparticle measurement, as it is often the case that surface roughness is mostly localized on the nanoparticle. This also suggests that the underlying surface should be made as flat as possible to minimize SRF, and ultimately, precision of the PSI phase measurement. Surface geometry is also found to influence SRF, but to a lesser extent. A factor of 2–3 difference in SRF was calculated between nanospheres, nanowires, and flat planes. This difference was only discernible in cases where the surface roughness was localized on the nanoparticle. These results suggest that when measuring nanoparticles using PSI, the magnitude with which surface roughness affects measurement precision is comparable to, and in many cases less than, the precision of the instrument specified by the manufacturer. Only for surfaces where roughness possesses a red-noise-like power spectrum is it predicted that surface roughness will be a major limiting factor in measurement precision. Based on these results, it is recommended that when measuring nanoparticles using OSPs, the SRF should be calculated using the procedure described herein as part of the analytical procedure to test the feasibility of the measurement and to
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determine the contribution to the measurement uncertainty that the surface roughness will induce. The methodology reported here-in for simulating the impact of surface roughness on phase measurement in PSI is also useful to the broader range of measurement applications of the PSI technique. This research was supported by Australian Research Council Linkage Infrastructure, Equipment and Facilities Grant LE110100024; a Macquarie University Research Development Grant, and Australian Research Council Discovery Project Grant DP130102674. References 1. B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985). 2. J. C. Wyant, C. L. Koliopoulos, B. Bhushan, and D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986). 3. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990). 4. D. J. Little and D. M. Kane, “Measuring nanoparticle size using optical surface profilers,” Opt. Express 21, 15664–15675 (2013). 5. D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, and D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013). 6. “Guide to the expression of uncertainty in measurement,” 2nd ed., ISO 98:1995 (International Organization for Standardization, Geneva, 1995). 7. “Extension to any number of output quantities—Part 3,” International Organization for Standardization, Geneva, ISO/IEC 98-3:2008/Suppl 2:2011 (2011). 8. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953). 9. J. J. Lennon, “Red-shifts and red herrings in geographical ecology,” Ecography 23, 101–113 (2000). 10. J. M. Halley and W. E. Kunin, “Extinction risk and the 1/f family of noise models,” Theor. Popul. Biol. 56, 215–230 (1999).
