Open Access
Add to BibSonomyAbstract
We present a half-plane surface-integral equation (SIE) approach for modeling the optical phase response of a single nanowire under phase-stepping interferometric (PSI) microscopy. This approach calculates scattered fields exactly from the Helmholtz equation in this 2D problem, obviating the need for ray-optic approximations. It is demonstrated that refractive index metrology is enabled by this method, with precision as low as 7 × 10−5 possible for current state-of-the-art PSI microscopes. For nanowires of known refractive index, radii as small as 0.001λ are shown to yield a measurable phase signal and are therefore potentially measurable by this approach. Measurements are also demonstrated to be relatively insensitive to the spectral and coherence characteristics of the light source, the illumination conditions, and variations in nanowire cross-section shape. Prospects for measuring both the radius and refractive index simultaneously, and scope for generalizing this approach to arbitrary nanoparticle shapes are discussed.
© 2015 Optical Society of America
1. Introduction
Phase-stepping interferometric (PSI) microscopy is a widely used areal surface measurement technique, nominally capable of measuring surface height profiles with sub-nanometer precision [1, 2]. This technique uses a series of interferograms to establish the phase profile of the image field of the PSI instrument [3], which can then be interpreted as a height profile by equating phase changes with optical path-length differences. While PSI is a precise surface-profiling technique, it is often inaccurate because of errors that arise due to optical diffraction in the object volume, particularly for small or sharp surface features [4].
Previously, de Groot et al. demonstrated that phase measurements made using PSI microscopy can be used in conjunction with electromagnetic wave theory to characterize silicon gratings with lateral features much smaller than the Abbe diffraction resolution limit [5]. Little et al. performed a similar demonstration on GaAs nanowires [6], using linear-systems theory to model the response of the PSI microscope [7]. The concept of using observable optical quantities to quantify sub-diffraction-limited surface characteristics is generally termed Scatterfield microscopy [8]. Parallel, contact-free nano-characterization of multiple surface elements is possible using PSI microscopy, potentially enabling faster measurement than scanning-based methods such as atomic-force microscopy (AFM).
Here, we present a method that enables the characterization of nanoparticles with finite conductivity using PSI microscopy, where previously nanoparticles were assumed to be perfect conductors. This half-plane surface-integral equation (SIE) approach exactly models the optical field in the object volume by numerically solving the Helmholtz equation (Maxwell's equations in the general 3D case) [9], before transforming this object field using linear systems theory into an observable image field. This approach is more accurate than the method previously used by Little et al. [6], as it does not rely on ray-optic approximations to derive the object field for a given surface.
The primary motivation for developing this new approach is to enable refractive index metrology of single nanoparticles of known size and shape. Existing refractive index measuring techniques for nanoparticles generally require a colloidal suspension, and make no provision for the refractive index of the nanoparticles being complex [10, 11]. The SIE approach also opens up the prospect of using the amplitude and polarization of the scattered wave (in addition to its phase) to infer nanoparticle shape, in addition to size.
Details of the theoretical approach for calculating the PSI response of nanowires are presented in section 2 of this paper. Section 3 presents modeling results from applying this theory to nanowires of different sizes. Partially coherent illumination, changes in nanowire cross-section and changes in refractive index are also investigated. Results are discussed in section 4 in the context of nanoparticle characterization. The main conclusions of this work are presented in section 5.
2. The surface-integral equation (SIE) approach
Here, the SIE approach for calculating the object field of the PSI microscope is presented for cylindrical nanowires. The symmetry of these nanoparticles allowed calculation of this object field to be reduced to a 2D scalar problem, simplifying the calculation whilst retaining the salient features of this nanoparticle characterization technique. An analogous formulation for the 3D case can be found in [9].
Light scattered by cylindrical nanowires is modeled with the assumption that the length is large compared to the diameter, and that the nanowire axis is oriented parallel to the x-axis. The transverse electric field can then be written as E = (0, u(r), 0), where r is the spatial variable (x, z) in the upper half-plane z > 0. The scalar field u satisfies the 2D Helmholtz equation,
The case of a transverse magnetic field is similar and leads to equivalent equations. Here k is the wavenumber, given by;where n is the refractive index of the nanowire and D is the circular cross-section of the nanowire of radius R, bounded by the surface ∂D, whereA gap, δ, is required between the nanowire and the surface to avoid divergence of the surface integrals.The scalar field u can be written as u = ui + us, where ui is the incident wave, together with its reflection from the plane z = 0, thus
where k is the incident wavevector (kx, kz), and kr is the reflected wavevector (kx, -kz). The field ui satisfies Eq. (1) for r ∉ D and vanishes on the plane z = 0.The scattered field, us can be represented as
where ϕ and ψ are unknown surface potentials defined on ∂D, and S and K are the single- and double-layer potentials respectively. Here;andwhere n is the outward unit normal to the surface ∂D, and G is the half-plane Green's function;where the spatial variables r' and r” are given by (x', z') and (x', -z') respectively, and H0 is the 0th order Hankel function. G satisfies Eq. (1) for r ≠ r' ∈ D or ∉ D and vanishes on the plane z = 0. The unknown surface potentials, ϕ and ψ satisfy the system;where KT and T denote the normal derivatives of S and K respectively on ∂D, I is the identity operator and the subscripts ‘int’ and ‘ext’ denote the values of K, S and T using the interior and exterior values of k(r) respectively. Jump relations [12], were used in the derivation of Eq. (9).To solve Eq. (9), the system was discretized using a high-order Nyström scheme [13], which exhibits fast convergence when the surface, ∂D is smooth. The half-plane Green's function was implemented using image theory [14], which introduces a virtual image scatterer in the lower half-space.
After calculating u, the field at the image plane of the microscope, v, was calculated using linear-systems analysis as;
where H is the optical transfer function of the PSI microscope and * denotes convolution. The purpose of intersecting the object field a distance Δz from the object plane is to preserve the shift-invariance of the imaging system. The image field is independent of Δz if shift-invariance is preserved, as the exponent term in Eq. (10) undoes the defocus that arises from the distance Δz that separates u from the object plane.For calculations in this study, it was initially assumed that nanowires were illuminated with a monochromatic plane wave source, the optical system was aberration-free with a numerical aperture of 0.8, the nanowire and underlying surface were perfect conductors, and the gap between nanowire and surface, δ, was λ × 10−4. Several of these assumptions are relaxed over the course of the results presented in the next section.
3. Results
3.1 Image field phase as a function of nanowire radius
The central aim of this paper is to calculate the variation of the image field phase with nanoparticle size, as this enables phase values measured using PSI microscopy to be translated to nanowire radius (and nanoparticle size more generally). Figure 1 illustrates the steps by which this calculation is made. The object field, u, was calculated for nanowires of different radii (Fig. 1(a)), from which the image field, v, was calculated using Eq. (10). The phase of the image field was then calculated relative to the phase of the reflected wave in the reference arm of the PSI microscope, which was given by Arg(v)-kz (Fig. 1(b)). The maximum absolute phase difference, denoted ϕim (which occurred at the centre of the phase profile) was then calculated as a function of nanowire radius (Fig. 1(c)). Leading phases are taken as positive values to resemble outputs of commercial PSI systems.
Fig. 1 a) Real component of the scattered field calculated in the object volume (normalized to the incident amplitude) for a nanowire of radius 0.10λ. The x- and z-axes are normalized to the illumination wavelength. b) Calculated phase at the image plane (ϕim) for nanowires of radius 0.02λ, 0.04λ, 0.06λ, 0.08λ and 0.10λ, with larger nanowires yielding larger phase differences. Phase values are taken relative to the reference field in the PSI microscope, and positive phase values indicate a leading phase. c) Peak value of ϕim as a function of nanowire radius (normalized to the illumination wavelength) for different objective NAs.
Peak phase calculated as a function of normalized nanowire radius (radius divided by wavelength) is shown in Fig. 1 for NAs of 0.2, 0.4, 0.6 and 0.8. These are standard NAs for interferometric objectives available on PSI microscopes, and correspond to magnifications of 5 × , 20 × , 50 × and 115 × respectively. The curves shown in this figure can be used to measure nanowire radii from phase profiles measured using PSI. For example, a peak phase of 0.50 radians measured with a 0.8 NA microscope objective would correspond to a nanowire radius 0.067λ if the appropriate assumptions are satisfied. Nanowire radii measured in this way are not constrained by the Abbe diffraction limit, evidenced by the lack of a radius cut-off in Fig. 1©. Measurements are instead constrained by the phase precision of the PSI microscope, typically claimed to be (at best) between 0.25 and 2.5 mrad, depending on the system. If ϕim must exceed the measurement precision of the instrument (ϕmin) for a successful measurement, then radius measurements are limited to above 0.001λ-0.0035λ (for NA = 0.8). This corresponds to nanowire radii of around 0.50-1.75 nm in the visible, which is roughly 10 times less than previous lower bound estimates on nanoparticle size measurements using PSI [7].
Table 1 shows the modeled peak phase values compared to previous work, and also to experimental measurement using a Bruker-AXS NT-9800 surface profiler in PSI mode. The method presented in this paper yields peak phase values that are larger than from the modified linear-systems method in [6], by a factor of around 2.1 and 1.7 for 25 nm and 50 nm radius cylinders respectively. The discrepancy between the SIE approach and measurement is indicative of aberrations present in the optical system. If, for example, the optical transfer function is adjusted in accordance with the roll-off in the spatial frequency spectrum measured previously for a flat mirror [15], then the SIE approach models phases of 312 and 943 mrad for 25 and 50 nm radius nanowires respectively. This also emphasizes the importance of accurately evaluating the optical transfer function of the microscope system.
Table 1. Peak phase values for 25 nm and 50 nm radius nanowires acquired using different models for λ = 514 nm and NA = 0.8.
3.2 Partially spatially-coherent illumination
PSI microscopes use partially spatially-coherent illumination with a numerical aperture defined by the position of the aperture-stop in the Köhler illumination setup. The illumination NA defines the angular bandwidth of plane-wave components incident upon the object surface, and is distinct from the objective NA. Partially spatially-coherent illumination was modeled by dividing the incident field into 21 plane-wave components, with transverse wavevector components distributed evenly over the illumination bandwidth;
where NAill is the numerical aperture of the incident illumination. The field at the image plane due to each plane-wave component was calculated, then superimposed to get the total field at the image plane.Figure 2 shows that the phase difference between partially spatially-coherent illumination and plane-wave illumination is typically less than 1% for all but the smallest nanowires, with the change approaching 10% at the 0.001λ-0.0035λ detection threshold.
Fig. 2 Difference between calculated ϕim for partially spatially coherent illumination and plane-wave illumination (at normal incidence).
3.3 Partially temporally-coherent illumination
PSI microscopes also use illumination that is partially temporally coherent. The degree of temporal coherence is determined by the bandwidth of the illumination source, Δω, and is typically 2-20% of the central (carrier) frequency, ω0. Partially temporally-coherent illumination was modeled as 51 discrete frequency components between ω0 - Δω and ω0 + Δω, with an amplitude spectrum of the form;
An amplitude spectrum of this form yields the Gaussian interference envelope typically observed in PSI microscopes.Figure 3 indicates that the phase difference between partially temporally-coherent illumination and monochromatic illumination is less than 0.1% over the investigated range of nanowire radii. The effect of partially coherent illumination on the PSI phase measurement is therefore negligible. Calculations of phase versus nanowire radius can therefore proceed assuming monochromatic plane wave illumination with only a minor (~1%) discernible loss in accuracy.
Fig. 3 Difference between calculated ϕim for partially temporally-coherent illumination and monochromatic illumination.
3.4 Nanowires with hexagonal cross-sections
Scanning electron-microscopy of nanowires revealed that their cross-sections are not perfectly circular, but resemble hexagons with rounded points. It is therefore prudent to consider how this departure from the assumed nanowire shape affects phase measurements made using PSI microscopy. Nanowires with non-circular cross-section were modeled as supercircles [16], where the boundary of a rounded N-sided polygon centered at the origin is parameterized by polar coordinates (r, θ), with
where α = 4/N andHere p is an even natural number that parameterizes the amount of rounding. In particular, when p = 2 the parameterization (13)-(14) reduces to a circle, whilst the parameterization converges to an N-sided polygon as p → ∞.Changing the nanowire cross-section from a circle to a hexagon results in an increase in the phase value by around 4 - 6% depending on the normalized radius of the nanowire (Fig. 4). The cross-sectional shape of the nanowire therefore has a small, but significant influence on the measured phase at the image plane. More irregular cross-sectional shapes could impart a larger variation in the measured phase.
Fig. 4 Difference between calculated ϕim for nanowires with non-circular and circular cross-sections.
3.5 Avenues for refractive index metrology of nanowires
A key advantage of the SIE approach is the ability to analyze how PSI phase measurements vary with the refractive index of the nanowire. Here, complex refractive indices were considered, where the imaginary component of the refractive index, κ is related to the absorption, α via
If the radius of the nanowire is known via some other independent means, such as atomic-force microscopy or scanning-electron microscopy, then PSI microscopy can be used to determine the refractive index of the nanowire. A single phase measurement can only determine either the real or imaginary refractive index independently. A second phase measurement is required to fully characterize both real and imaginary refractive index components. The simplest way to obtain this second measurement is to use a second objective with a different NA. However, this reduces measurement sensitivity. An alternate approach is to use a second wavelength, which can be implemented in most PSI systems using a second color filter in conjunction with the standard white-light source, though consideration needs to be given to the effect of nanoparticle dispersion.The phase sensitivity to refractive index change, ∂ϕim/∂n (radians per refractive index unit), was calculated for three classes of materials; dielectrics, semiconductors and metals over (real) refractive index ranges of 1.3-1.7, 3.5-4.5 and 0.1-4.5 respectively, and characterized by imaginary refractive indices of 0, 0.3 and 1, 3 respectively. Minimum discernible real refractive index change, Δnmin was defined as;
where ϕmin is the phase precision of the PSI microscope. For brevity, Δnmin was calculated using the average value of ∂ϕim/∂n over the refractive index ranges mentioned previously. Figure 5 shows Δnmin as a function of nanowire radius for phase precisions of 0.25 and 2.5 mrads for dielectrics, semiconductors and metals.Fig. 5 Minimum discernible (real) refractive index change, Δnmin calculated as a function of nanowire radius, for a) dielectrics, c) semiconductors and e) metals (note different plot range), for ϕmin = 0.25 and 2.5 mrad; and ϕim versus real refractive index for nanowires of radii 0.07λ and 0.1λ for b) dielectrics, d) semiconductors and f) metals (for κ = 1).
Dielectrics exhibited the smoothest variation in Δnmin with nanowire radius because the scattered field was always dominated by the wave component transmitted through the nanowire (and reflected by the underlying surface). For metals, the variation in Δnmin exhibited a peak, which demarcated transmitted-wave dominated (small radii) and reflected-wave dominated (larger radii) scattered fields. Semiconductors exhibited a more complex interplay between transmitted and reflected waves.
For dielectrics Δnmin approached a minimum value of 7 × 10−5 for larger nanowires. For semiconductors, the minimum Δnmin was slightly larger, around 1.3 × 10−4, and occurred at normalized radii of around 0.06. For metals, the minimum Δnmin was 3.2 × 10−3 for κ = 3 and 1.0 × 10−3 for κ = 1. Values quoted are for a PSI phase measurement precision of 0.25 mrad, and can be multiplied by a factor of P/0.25 to obtain corresponding values for an arbitrary phase precision, P in mrad.
4. Discussion
A key finding of this study was that nanowires as small as 0.001λ could potentially be detected and measured using PSI microscopy. This corresponds to radii around 0.5 nm for visible illumination, and is nearly a factor of 10 smaller than previous lower-bound estimates on nanoparticle size measurable with this technique [7]. Aberrations in the PSI microscope, temperature fluctuations, mechanical vibrations, and other environmental effects can all serve to degrade this measurement limit to a higher value. Surface roughness on the nanowire and the underlying surface can also degrade measurement precision, depending on its power spectrum and its surface extent [15]. In contrast, the illumination conditions do not appear to affect the PSI phase measurement precision much at all. Factoring in the partial coherence of the illumination leads to a maximum phase divergence of around 5 mrad for larger (R > 0.1λ) nanowires.
Variation in the cross-section shape of the nanowire had a small but significant influence on the phase measurement; around 4 - 6% depending on the nanowire radius. This would impart a percentage uncertainty of at least 4 - 6% on radius measurements (more for smaller nanowire radii) if the nanowire cross-section is unknown. More irregular cross-section shapes could impart a larger uncertainty. Particle shape therefore needs to be determined in detail to maximize measurement accuracy. AFM or SEM can be used to achieve this, though doing so negates many of the advantages of using PSI microscopy in the first place. Ideally the shape of the particle should be determined optically; possibly using full amplitude and phase profiles, in conjunction with multiple wavelengths and objective NAs, and this is a topic for future work.
The second key finding in this study was that accurate refractive index metrology of single nanoparticles is possible if the size and shape of the nanoparticle is accurately known, through SEM for example. Measurement precisions between 10−2 and 10−4 are possible for the real refractive index component, and are comparable to other measurement techniques [10, 11], although it remains to be investigated whether these precisions hold for other particle shapes. Colloid-based techniques also appear to be better suited to smaller particles (< 0.05λ), where PSI microscopy favors larger particles.
One final point of investigation that remains is allowing for surfaces that have finite conductance, since it may not always be possible to choose the surface on which the nanoparticles are located. An extension of the modified Green's function approach that we use in (8) is applicable in the case that the surface is of dielectric material [17], but its use may present numerical difficulties [12], and is beyond the scope of this article. We plan to investigate the dielectric material case in a future work.
5. Conclusion
In this paper, a surface-integral equation (SIE) approach for measuring particle size using PSI microscopy was presented, improving accuracy of previous theoretical models and enabling refractive index metrology of nanoparticles. It was determined that the size of nanowire radii down to 0.001λ could be measured using PSI microscopy using instruments with the best currently-available phase-measurement precision (0.25 mrad). PSI phase measurements were found to be relatively insensitive to illumination conditions (variation < 5 mrad), and only moderately sensitive to the cross-section of the nanowire (variation of 5-10%).
Refractive index metrology of nanoparticles was also shown to be possible with particles of known size and shape. For nanowires on a perfectly conducting underlying surface, (real) refractive index sensitivities as low as 7 × 10−5 were demonstrated. Refractive index sensitivities depended on the size and magnitude of the imaginary component of the refractive index, and tended to be superior for materials with lower absorption.
Acknowledgment
This research was supported by an Australian Research Council (ARC) Linkage Infrastructure, Equipment, and Facilities Grant LE110100024; a Macquarie University Research Development Grant and ARC Discovery Project DP130102674.
References and links
1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]
2. B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24(10), 1489–1497 (1985). [CrossRef] [PubMed]
3. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]
4. F. Gao, R. K. Leach, J. Petzig, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008). [CrossRef]
5. P. De Groot, X. Colonna de Lega, J. Liesener, and M. Darwin, “Metrology of optically-unresolved features using interferometric surface profiling and RCWA modeling,” Opt. Express 16(6), 3970–3975 (2008). [CrossRef] [PubMed]
6. 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(16), 161107 (2013). [CrossRef]
7. D. J. Little and D. M. Kane, “Measuring nanoparticle size using Optical Surface Profilers,” Opt. Express 21(13), 15664–15675 (2013). [CrossRef] [PubMed]
8. R. M. Silver, B. M. Barnes, R. Attota, J. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46(20), 4248–4257 (2007). [CrossRef] [PubMed]
9. M. Ganesh, S. C. Hawkins, and D. Volkov, “An all-frequency weakly-singular surface integral equation for electromagnetism in dielectric media: Reformulation and well-posedness analysis,” J. Math. Anal. Appl. 412(1), 277–300 (2014). [CrossRef]
10. B. N. Khlebtsov, V. A. Khanadeev, and N. G. Khlebtsov, “Determination of the size, concentration, and refractive index of silica nanoparticles from turbidity spectra,” Langmuir 24(16), 8964–8970 (2008). [CrossRef] [PubMed]
11. Z. Wang, N. Wang, D. Wang, Z. Tang, K. Yu, and W. Wei, “Method for refractive index measurement of nanoparticles,” Opt. Lett. 39(14), 4251–4254 (2014). [CrossRef] [PubMed]
12. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed. (Springer, 2012), p. 40.
13. R. Kress, “Boundary integral equations in time-harmonic acoustic scattering,” Math. Comput. Model. 15(3-5), 229–243 (1991). [CrossRef]
14. J. C. Chao, F. J. Rizzo, I. Elshafiey, Y. J. Liu, L. Upda, and P. A. Martin, “General formulation for light scattering by a dielectric body near a perfectly conducting surface,” J. Opt. Soc. Am. A 13(2), 338–344 (1996). [CrossRef]
15. S. Onaka, “A simple equation giving shapes between a circle and regular N-sided polygon,” Philos. Mag. Lett. 85, 359–365 (2005). [CrossRef]
16. D. J. Little and D. M. Kane, “Measuring nanoparticle size using phase-stepping interferometry: quantifying measurement sensitivity to surface roughness,” Appl. Opt. 53(20), 4548–4554 (2014). [CrossRef] [PubMed]
17. W. Chew, Waves and Fields in Inhomogeneous Media, 1st edition. (Wiley, 1990), Chapt. 2.
