## Measuring nanoparticle size using optical surface profilers |

Optics Express, Vol. 21, Issue 13, pp. 15664-15675 (2013)

http://dx.doi.org/10.1364/OE.21.015664

Acrobat PDF (983 KB)

### Abstract

Optical surface profilers are state-of-the-art instruments for measuring surface height profiles. They are not conventionally applied to nanoparticle measurements due to the presence of diffraction artifacts. Here we use a theoretical model based on wave-optics to account for diffraction-based artifacts in optical surface profilers. This then enables accurate measurement of nanoparticles size of a known geometry. The model is developed for both phase shifting interferometry and vertical scanning interferometry modes of operation. It is demonstrated that nanosphere radii as small as 12 nm, and nano-cylinder radii as small as 10-15 nm can be measured from a standard profile measurement using phase shifted interferometry interpreted using the wave-optics approach.

© 2013 OSA

## 1. Introduction

1. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. **19**(11), 780–782 (1994). [CrossRef] [PubMed]

4. C. Urban and P. Schurtenberger, “Characterization of turbid colloidal suspensions using light scattering techniques combined with cross-correlation methods,” J. Colloid Interface Sci. **207**(1), 150–158 (1998). [CrossRef] [PubMed]

5. C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. **124**(Pt 2), 107–117 (1981). [CrossRef] [PubMed]

7. D. A. Agard and J. W. Sedat, “Three-dimensional architecture of a polytene nucleus,” Nature **302**(5910), 676–681 (1983). [CrossRef] [PubMed]

8. S. W. Hell, E. H. K. Stelzer, S. Lindek, and C. Cremer, “Confocal microscopy with an increased detection aperture: type-B 4Pi confocal microscopy,” Opt. Lett. **19**(3), 222–224 (1994). [CrossRef] [PubMed]

10. 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]

1. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. **19**(11), 780–782 (1994). [CrossRef] [PubMed]

2. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science **313**(5793), 1642–1645 (2006). [CrossRef] [PubMed]

11. S. W. Hell, “Far-Field Optical Nanoscopy,” Science **316**(5828), 1153–1158 (2007). [CrossRef] [PubMed]

12. F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. **19**(1), 015303 (2008). [CrossRef]

## 2. Method

10. 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]

14. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. **29**(26), 3784–3788 (1990). [CrossRef] [PubMed]

15. A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. **39**(13), 2107–2115 (2000). [CrossRef] [PubMed]

*z*position where coherence in the superposition is maximum. Optical surface profilers use a standard optical microscope to image the object surface with an interferometric objective to generate the required interference fringes and a CCD camera to record irradiance profiles [16

16. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. **29**(26), 3775–3783 (1990). [CrossRef] [PubMed]

### 2.1 Ray-optic model

*α*is the fraction of the illumination field incident on the object surface,

*ϕ*(

*x*,

*y*) is the phase profile at the object plane and

*ϕ*

_{ref}is the phase of the reference field. Constant terms are omitted for brevity. The first term in Eq. (1) arises from the object surface, while the second term arises from the flat reference surface and has constant phase.

#### 2.1.1 Ray-optic model for PSI

*δ*in steps of

*π*/2, the phase term in Eq. (2) can be retrieved using the following algorithmwhere subscripts denote the value of

*δ*corresponding to each irradiance profile. Equation (3) is one of many PSI algorithms that can potentially be used [17

17. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. **34**(22), 4723–4730 (1995). [CrossRef] [PubMed]

*π*(corresponding to a height of half of one wavelength) can be unambiguously determined.

#### 2.1.2 Ray-optic model for VSI

*z*. The field at the image plane under white light illumination iswhere

*Δz*is the translation of the object surface and

*ψ*is the amplitude of the illumination field. The corresponding irradiance isApplying Eq. (4) to Eq. (6) yieldswhere

_{0}*C*is a constant. The first term of Eq. (7) is recognised as the real part of the Fourier transform of the illumination field spectrum. Evaluating this Fourier transform yieldswhere

*ε*is the Fourier transform of |

*ψ(ω-ω*|

_{0})^{2}with respect to the conjugate variable

18. I. Gurov, E. Ermolaeva, and A. Zakharov, “Analysis of low-coherence interference fringes by the Kalman filtering method,” J. Opt. Soc. Am. A **21**(2), 242–251 (2004). [CrossRef] [PubMed]

*ε*has a well-defined peak at

*ξ*=

*ξ*with a corresponding envelope peak in Δ

_{0}*z*at Δ

*z*

_{0}then,That is, the position of the envelope peak depends linearly on the object surface height,

*z*. The surface profile is measured by detecting the peak in modulation as a function of position (denoted Δ

*z*) and equating it to the surface height. The constant terms in Eq. (9) are usually omitted since the absolute

_{0}*z*-position of the object surface is arbitrary.

### 2.2 Wave-optic model

#### 2.2.1 Wave-optic model for PSI

*τ*and Arg denotes the complex argument. If

*τ*is real, then Eq. (12) becomesSection 3 discusses how Eq. (12) and Eq. (13) can be used to determine nanoparticle size.

#### 2.2.2 Wave-optic model for VSI

*τ*is real thenExpanding

*τ*as a Taylor series about

*ω*gives

_{0}*x*and

*y*is the peak of the modulation envelope of Eq. (18) in Δ

*z*. Higher order approximations can be made by computing higher order terms of the Taylor series in Eq. (17).

## 3. Results and Discussion

### 3.1 Characterizing upright nano-cylinders using PSI

*R*and

*h*are the radius and height of the cylinder, respectively. If the effect of defocus is neglected, Eq. (4) can be used to obtain the phase profile in the object plane,Neglecting defocus is a reasonable approximation as the size of the nanoparticles is much less than the Rayleigh range of the objective. The phase profile measured using PSI can be found by substituting Eq. (20) into Eq. (13) and breaking up the convolution integral into regions of constant height

*τ*is a function of

*x-x’*and

*y-y’*. Equation (21) shows that for an upright cylinder, the convolution integrals reduce to integrals of the impulse response over the two regions of constant height (i.e. the top of the cylinder and the underlying surface), weighted by corresponding trigonometric terms. To evaluate Eq. (21) numerically, the coherent impulse response is assumed to be that of an aberration-free imaging system with a finite pupil diameter, namelywhere

*J*denotes a first-order Bessel function of the first kind and

_{1}*R*as

_{N}*ϕ*

_{image}

*(0, 0)*calculated as a function of

*R*using Eq. (21) for several different values of known height.

_{N}*R*can be determined from

_{N}*ϕ*

_{image}

*(0, 0)*, measured using optical surface profilometry.

*R*can then be determined using Eq. (23) without being constrained by the Abbe diffraction limit.

*R*is instead limited by the precision to which

*ϕ*

_{image}

*(0, 0)*is known. A precision of 2.5 mrad is typical for commercial OSPs. This limits measured normalized cylinder radii down to 0.10-0.15 (depending on cylinder height) if it is assumed that reliable measurement can only be achieved if

*ϕ*

_{image}

*(0, 0)*is greater than the precision of the OSP instrument. This corresponds to real cylinder radii of 10-15 nm for a NA of 0.85 and a wavelength of 514 nm. Values of

*ϕ*

_{image}

*(0, 0)*that correspond to several different radii can be distinguished by making multiple measurements with objectives of different NA, or by modelling and measuring multiple coordinates in the phase profile.

### 3.2 Characterizing spherical nanoparticles using PSI

*N*regions, labelled

*Q*each with a discrete height

_{j}*z*, where

_{j}*j*= 1, 2, ...,

*N*. Here, Eq. (13) becomes

*N*approaches infinity, in which case the summation terms in Eq. (24) become Lebesgue integrals;where the measure,

*μ*is given bywhere the integration region,

*Q*, varies as a function of

*z*. For a spherical nanoparticle the height profile can be parameterized aswhere

*ρ*is the radius of the sphere. The normalized sphere radius is defined asFigure 2 shows

*ϕ*

_{image}

*(0, 0)*calculated as a function of

*ρ*

_{N}using Eq. (24) for

*N*= 1, 5, 20, 50 and 200. Values of

*ρ*

_{N}down to 0.12 can be measured, corresponding to

*ρ*as small as 12 nm for a NA of 0.85 and a wavelength of 514 nm. The function shown in Fig. 2 exhibits three distinct regions; (i) a monotonically increasing region up until

*ρ*= 0.6, (ii) a dip region between

_{N}*ρ*= 0.6 and

_{N}*ρ*= 1.2 where

_{N}*ϕ*

_{image}

*(0, 0)*decreases with

*ρ*(even becoming negative), and (iii) a region from

_{N}*ρ*= 1.2 onward which is monotonically increasing. As

_{N}*ρ*becomes large

_{N}*ϕ*

_{image}

*(0, 0)*converges to a value of 4

*kρ,*the expected value if diffraction effects are ignored. This convergence is quite slow; for a NA of 0.85 and

*ρ*= 5.2,

_{N}*ϕ(0, 0)*is only 70% of 4

*kρ*. The dip region arises because of sign changes in the numerator and denominator of Eq. (24). The existence of this dip region can manifest as PSI artifacts in optical surface profiler systems where elevations appear as depressions and vice-versa.

### 3.3 Characterizing upright nano-cylinders using VSI

*k*is the central wavenumber of the illumination field spectrum. The modulation envelope (denoted

_{0}*I*) is approximated from Eq. (18) to beFinally, we take the irradiance spectrum of the white-light illumination to be a Gaussian function with central frequency

_{env}*ω*

_{0}and 1/

*e*

^{2}width Δ

*ω*. The Fourier transform is thenTaking the height profile of an upright nano-cylinder (Eq. (19)) and applying it to Eq. (29) yieldsIn the ray-optic model, there is only one modulation envelope for each point in

*x*and

*y*. In the wave-optic model by contrast, each point in

*x*and

*y*will consist of multiple modulation envelopes; one for each height present in the height profile, with an amplitude that depends on the integral of

*τ*over that corresponding height region. Since there are multiple modulation envelopes, the total envelope possesses multiple peaks, in which case identifying the peak can become ambiguous.

*h*is small, before splitting off into two envelopes, one due to light reflecting from the top of the cylinder and the other due to light reflecting from the underlying surface. Altering the cylinder radius changes the relative magnitude of each envelope, which alters how the position of the peak varies with height, as shown in Fig. 4. Two distinct regimes are evident in Fig. 4. The first regime occurs for larger

*R*(Eq. (23) where the second envelope (due to light reflected from the top of the cylinder) dominates the first, resulting in a continuous transition as the peak splits. The second regime occurs for smaller

_{N}*R*where the first envelope (due to light reflecting from the underlying surface) dominates, resulting in a discontinuous transition as the peak splits. For the curve corresponding to the first regime (

_{N}*R*= 1.56) the position of the highest envelope is monotonically increasing, while for the curve corresponding to the second regime (

_{N}*R*= 1.30) it is not.

_{N}### 3.4 Characterizing spherical nanoparticles using VSI

*N*regions, labelled

*Q*, of height

_{j}*z*where

_{j}*j*= 1, 2, ...,

*N*. Equation (31) can then be generalized asFor the limit where

*N*approaches infinity (i.e. for a continuous height profile), Eq. (32) becomeswhere

*ξ*

_{max}and

*ξ*

_{min}are the values of

*ξ*corresponding to the maximum and minimum in

*z(x, y)*respectively. Figure 5 shows the modulation envelopes calculated for a spherical nanoparticle with a height profile given by Eq. (27) at the origin.

*z*-coordinate asThe normalized peak position is defined as the peak position in these normalized coordinates. Figure 6 shows how Δ

*z*

_{N}varies with

*ρ*

_{N}_{.}Two distinct regions are evident. The first region, bounded by 0 ≤

*ρ*≤ 1.64 exhibits a single peak comprised of modulation envelopes from both surface and nanosphere regions and so is termed a hybrid peak. In this region the peak position of the modulation envelope exhibits a slow, monotonic increase with increasing

_{N}*ρ*. Here, the difference between the calculated value and the true normalized height (2

_{N}*ρ*) is substantial, with a difference exceeding two orders of magnitude for some values of

_{N}*ρ*. In the second region where

_{N}*ρ*> 1.64, the modulation envelope becomes dominated by modulations envelopes from the nanosphere regions and so is termed the sphere peak. In this region the measured peak position is much closer to the true sphere height, however there is a constant residual error which exists because the modulation envelope from the topological centre of the nanosphere cannot be truly isolated; modulation envelopes from other nanosphere regions will always place a downward bias on the peak position of the total modulation envelope. For

_{N}*ρ*> 2.4 there exists a distinct peak due to the modulation envelope from the surface region and is included only for completeness.

_{N}*ρ*as small as 0.19 can potentially be measured using VSI. This corresponds to

_{N}*ρ*as small as 19 nm for an NA of 0.85 and a central illumination wavelength of 514 nm.

## 4. Discussion

*ϕ*

_{image}precision is to real surface roughness. Close inspection of Eq. (21) and Eq. (24) reveal

*ϕ*

_{image}to essentially depend on a weighted average. The effect of random height variations are thus expected to cancel on average and so degradation in measurement precision of

*ϕ*

_{image}is expected to be minor for typical amounts of surface roughness (and indeed, the method has been demonstrated experimentally - see footnote on p.2). We are currently in the process of developing ways to quantify the sensitivity of

*ϕ*

_{image}precision to surface roughness.

## 5. Conclusion

## Acknowledgment

## References and links

1. | S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. |

2. | E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science |

3. | Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics |

4. | C. Urban and P. Schurtenberger, “Characterization of turbid colloidal suspensions using light scattering techniques combined with cross-correlation methods,” J. Colloid Interface Sci. |

5. | C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. |

6. | M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. |

7. | D. A. Agard and J. W. Sedat, “Three-dimensional architecture of a polytene nucleus,” Nature |

8. | S. W. Hell, E. H. K. Stelzer, S. Lindek, and C. Cremer, “Confocal microscopy with an increased detection aperture: type-B 4Pi confocal microscopy,” Opt. Lett. |

9. | T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A |

10. | B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. |

11. | S. W. Hell, “Far-Field Optical Nanoscopy,” Science |

12. | F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. |

13. | D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, and D. M. Kane, “Nanoparticle measurement in the optical far-field.” presented at the |

14. | B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. |

15. | A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. |

16. | G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. |

17. | P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. |

18. | I. Gurov, E. Ermolaeva, and A. Zakharov, “Analysis of low-coherence interference fringes by the Kalman filtering method,” J. Opt. Soc. Am. A |

19. | E. Hecht, |

20. | E. L. Church and P. Z. Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE |

21. | A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A |

22. | J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. |

23. | A. Harasaki and J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. |

**OCIS Codes**

(110.4850) Imaging systems : Optical transfer functions

(120.3940) Instrumentation, measurement, and metrology : Metrology

(180.3170) Microscopy : Interference microscopy

(110.3175) Imaging systems : Interferometric imaging

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 5, 2013

Manuscript Accepted: June 10, 2013

Published: June 21, 2013

**Citation**

Douglas J. Little and Deb M. Kane, "Measuring nanoparticle size using optical surface profilers," Opt. Express **21**, 15664-15675 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15664

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### References

- S. W. Hell, J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994). [CrossRef] [PubMed]
- E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006). [CrossRef] [PubMed]
- Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013). [CrossRef]
- C. Urban, P. Schurtenberger, “Characterization of turbid colloidal suspensions using light scattering techniques combined with cross-correlation methods,” J. Colloid Interface Sci. 207(1), 150–158 (1998). [CrossRef] [PubMed]
- C. J. R. Sheppard, T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124(Pt 2), 107–117 (1981). [CrossRef] [PubMed]
- M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000). [CrossRef] [PubMed]
- D. A. Agard, J. W. Sedat, “Three-dimensional architecture of a polytene nucleus,” Nature 302(5910), 676–681 (1983). [CrossRef] [PubMed]
- S. W. Hell, E. H. K. Stelzer, S. Lindek, C. Cremer, “Confocal microscopy with an increased detection aperture: type-B 4Pi confocal microscopy,” Opt. Lett. 19(3), 222–224 (1994). [CrossRef] [PubMed]
- T. E. Gureyev, A. Roberts, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12(9), 1932–1941 (1995). [CrossRef]
- B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24(10), 1489–1497 (1985). [CrossRef] [PubMed]
- S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316(5828), 1153–1158 (2007). [CrossRef] [PubMed]
- F. Gao, R. K. Leach, J. Petzing, J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19(1), 015303 (2008). [CrossRef]
- D. J. Little, R. L. Kuruwita, A. Joyce, Q. Gao, T. Burgess, C. Jagadish, and D. M. Kane, “Nanoparticle measurement in the optical far-field.” presented at the European Conference for Lasers and Electro-Optics (ECLEO), Munich, Germany, 12–16 May 2013, paper PD-B.9.
- B. S. Lee, T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29(26), 3784–3788 (1990). [CrossRef] [PubMed]
- A. Harasaki, J. Schmit, J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 39(13), 2107–2115 (2000). [CrossRef] [PubMed]
- G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29(26), 3775–3783 (1990). [CrossRef] [PubMed]
- P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef] [PubMed]
- I. Gurov, E. Ermolaeva, A. Zakharov, “Analysis of low-coherence interference fringes by the Kalman filtering method,” J. Opt. Soc. Am. A 21(2), 242–251 (2004). [CrossRef] [PubMed]
- E. Hecht, Optics (Addison Wesley Longman, 1998), Chap. 11.
- E. L. Church, P. Z. Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989). [CrossRef]
- A. Krywonos, J. E. Harvey, N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28(6), 1121–1138 (2011). [CrossRef] [PubMed]
- J. M. Coupland, J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008). [CrossRef]
- A. Harasaki, J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39(13), 2101–2106 (2000). [CrossRef] [PubMed]

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