Abstract

A method for sizing nanoparticles using phase-stepping interferometry has been developed recently by Little et al. [Appl. Phys. Lett. 103, 161107 (2013) [CrossRef]  ]. 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

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References

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  1. B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
    [CrossRef]
  2. J. C. Wyant, C. L. Koliopoulos, B. Bhushan, D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986).
    [CrossRef]
  3. G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef]
  4. D. J. Little, D. M. Kane, “Measuring nanoparticle size using optical surface profilers,” Opt. Express 21, 15664–15675 (2013).
    [CrossRef]
  5. D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
    [CrossRef]
  6. “Guide to the expression of uncertainty in measurement,” 2nd ed., (International Organization for Standardization, Geneva, 1995).
  7. “Extension to any number of output quantities—Part 3,” International Organization for Standardization, Geneva, (2011).
  8. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  9. J. J. Lennon, “Red-shifts and red herrings in geographical ecology,” Ecography 23, 101–113 (2000).
    [CrossRef]
  10. J. M. Halley, W. E. Kunin, “Extinction risk and the 1/f family of noise models,” Theor. Popul. Biol. 56, 215–230 (1999).
    [CrossRef]

2013

D. J. Little, D. M. Kane, “Measuring nanoparticle size using optical surface profilers,” Opt. Express 21, 15664–15675 (2013).
[CrossRef]

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

2000

J. J. Lennon, “Red-shifts and red herrings in geographical ecology,” Ecography 23, 101–113 (2000).
[CrossRef]

1999

J. M. Halley, W. E. Kunin, “Extinction risk and the 1/f family of noise models,” Theor. Popul. Biol. 56, 215–230 (1999).
[CrossRef]

1990

1986

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986).
[CrossRef]

1985

1953

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Basila, D.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986).
[CrossRef]

Bhushan, B.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986).
[CrossRef]

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef]

Burgess, T.

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

Chim, S. S. C.

Gao, Q.

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

Halley, J. M.

J. M. Halley, W. E. Kunin, “Extinction risk and the 1/f family of noise models,” Theor. Popul. Biol. 56, 215–230 (1999).
[CrossRef]

Jagadish, C.

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

Joyce, A.

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

Kane, D. M.

D. J. Little, D. M. Kane, “Measuring nanoparticle size using optical surface profilers,” Opt. Express 21, 15664–15675 (2013).
[CrossRef]

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

Kino, G. S.

Koliopoulos, C. L.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986).
[CrossRef]

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef]

Kunin, W. E.

J. M. Halley, W. E. Kunin, “Extinction risk and the 1/f family of noise models,” Theor. Popul. Biol. 56, 215–230 (1999).
[CrossRef]

Kuruwita, R. L.

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

Lennon, J. J.

J. J. Lennon, “Red-shifts and red herrings in geographical ecology,” Ecography 23, 101–113 (2000).
[CrossRef]

Little, D. J.

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

D. J. Little, D. M. Kane, “Measuring nanoparticle size using optical surface profilers,” Opt. Express 21, 15664–15675 (2013).
[CrossRef]

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M. N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, A. H.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Wyant, J. C.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986).
[CrossRef]

B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, D. M. Kane, “Phase-stepping interferometry of GaAs nanowires: determining nanowire radius,” Appl. Phys. Lett. 103, 161107 (2013).
[CrossRef]

Ecography

J. J. Lennon, “Red-shifts and red herrings in geographical ecology,” Ecography 23, 101–113 (2000).
[CrossRef]

J. Chem. Phys.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Tribol.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, D. Basila, “Development of a three-dimensional, noncontact digital optical profiler,” J. Tribol. 108, 1–8 (1986).
[CrossRef]

Opt. Express

Theor. Popul. Biol.

J. M. Halley, W. E. Kunin, “Extinction risk and the 1/f family of noise models,” Theor. Popul. Biol. 56, 215–230 (1999).
[CrossRef]

Other

“Guide to the expression of uncertainty in measurement,” 2nd ed., (International Organization for Standardization, Geneva, 1995).

“Extension to any number of output quantities—Part 3,” International Organization for Standardization, Geneva, (2011).

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Figures (10)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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 .

Fig. 4.
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 .

Fig. 5.
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.

Fig. 6.
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. 7.
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.

Fig. 8.
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.

Fig. 9.
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.

Fig. 10.
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 two-dimensional power spectrum over the angular range 0 to 2 π . 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.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

ϕ image ( x , y ) ϕ ref = tan 1 ( j = 1 N sin ( ϕ j ) Q j τ d x d y j = 1 N cos ( ϕ j ) Q j τ d x d y ) .
ϕ j = 2 k ( z j z ref ) ,
tan 1 ( j = 1 N y i = 1 N x sin ( 2 k ( z i j z ref ) ) τ ( x i j x , y i j y ) Δ x Δ y j = 1 N y i = 1 N x cos ( 2 k ( z i j z ref ) ) τ ( x i j x , y i j y ) Δ x Δ y ) ,
τ ( x , y ) = J 1 ( NA × k x 2 + y 2 ) / x 2 + y 2 ,
z i j = z i j + Δ z i j .
SRF ( x , y ) = σ ϕ ( x , y ) / σ noise ,
Δ z i j = IFFT ( S i j e i θ i j ) ,
S i j = A ( k x i 2 + k y j 2 ) β / 2 ,
z ( x , y ) z ref = { R + R 2 x 2 y 2 x 2 + y 2 R 2 0 x 2 + y 2 > R 2 ,
z ( x , y ) z ref = { R + R 2 x 2 | x | R 0 | x | > R ,

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