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.

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References

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2013 (1)

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]

2011 (1)

2008 (2)

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]

J. M.  Coupland, J.  Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[CrossRef]

2007 (1)

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

2006 (1)

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]

2004 (1)

2000 (3)

1998 (1)

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]

1995 (2)

1994 (2)

1990 (2)

1989 (1)

E. L.  Church, P. Z.  Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).
[CrossRef]

1985 (1)

1983 (1)

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

1981 (1)

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

Agard, D. A.

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

Betzig, E.

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]

Bhushan, B.

Bonifacino, J. S.

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]

Boss, D.

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]

Chim, S. S. C.

Choi, N.

Church, E. L.

E. L.  Church, P. Z.  Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).
[CrossRef]

Cotte, Y.

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]

Coupland, J. M.

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]

J. M.  Coupland, J.  Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[CrossRef]

Cremer, C.

Davidson, M. W.

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]

Depeursinge, C.

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]

Ermolaeva, E.

Gao, F.

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]

Groot, P.

Gureyev, T. E.

Gurov, I.

Gustafsson, M. G. L.

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]

Harasaki, A.

Harvey, J. E.

Hell, S. W.

Hess, H. F.

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]

Jourdain, P.

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]

Kino, G. S.

Koliopoulos, C. L.

Krywonos, A.

Leach, R. K.

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]

Lee, B. S.

Lindek, S.

Lindwasser, O. W.

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]

Lippincott-Schwartz, J.

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]

Lobera, J.

J. M.  Coupland, J.  Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[CrossRef]

Magistretti, P.

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]

Marquet, P.

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]

Nugent, K. A.

Olenych, S.

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]

Patterson, G. H.

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]

Pavillon, N.

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]

Petzing, J.

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]

Roberts, A.

Schmit, J.

Schurtenberger, P.

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]

Sedat, J. W.

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

Sheppard, C. J. R.

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

Sougrat, R.

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]

Stelzer, E. H. K.

Strand, T. C.

Takacs, P. Z.

E. L.  Church, P. Z.  Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).
[CrossRef]

Toy, F.

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]

Urban, C.

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]

Wichmann, J.

Wilson, T.

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

Wyant, J. C.

Zakharov, A.

Appl. Opt. (6)

J. Colloid Interface Sci. (1)

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]

J. Microsc. (2)

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]

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (2)

J. M.  Coupland, J.  Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[CrossRef]

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]

Nat. Photonics (1)

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]

Nature (1)

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

Opt. Lett. (2)

Proc. SPIE (1)

E. L.  Church, P. Z.  Takacs, “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE 1164, 46–59 (1989).
[CrossRef]

Science (2)

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

Other (2)

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.

E. Hecht, Optics (Addison Wesley Longman, 1998), Chap. 11.

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Figures (6)

Fig. 1
Fig. 1

Calculated ϕ(0, 0) as a function of normalised cylinder radius for heights of 20 nm, 50 nm and 100 nm. The discontinuity occurs because the range of the inverse tangent in Eq. (26) is limited to returning values between ± π.

Fig. 2
Fig. 2

Calculated ϕ(0, 0) as a function of ρN for an approximated height profile with N = 1, 5, 20, 50 and 200. The curve corresponding to N = 50 and N = 200 almost completely match, indicating good convergence is achieved for around 50 partitions. The inset shows the same curve over a smaller range for ρN up to 0.3.

Fig. 3
Fig. 3

Calculated Imod vs Δz for different cylinder heights for Δω = 1.29 × 1015 and ω0 = 3.67 × 1015.

Fig. 4
Fig. 4

Calculated Δz as a function of cylinder height. Solid and dashed lines indicate the position of the first and second modulation envelope peaks respectively. The dotted curve indicates the ideal measurement where the peak position is equal to the cylinder height.

Fig. 5
Fig. 5

Calculated Imod vs Δz for ρN = 2.08 and N = 5 at the origin, showing the modulation envelopes from each of the nanosphere height regions, labeled in order of ascending height Q2 - Q6 (colored curves), the modulation envelope from the underlying surface, Q1 (black curve) and the total envelope (dashed curve).

Fig. 6
Fig. 6

Calculated normalized peak position of the modulation envelope as a function of ρN. Sphere and surface peaks are peaks dominated by sphere and surface terms respectively, while hybrid peaks are those where sphere and surface terms have significant contributions.

Equations (34)

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ψ( x,y )=α e iϕ( x,y ) +( 1α ) e i ϕ ref ,
I( x,y )= α 2 + ( 1α ) 2 +2α( 1α )cos( ϕ( x,y ) ϕ ref ).
ϕ( x,y ) ϕ 0 = tan 1 ( I π/2 ( x,y ) I 3π /2 ( x,y ) I 0 ( x,y ) I π ( x,y ) ),
z( x,y ) z 0 = 1 2k ( ϕ( x,y ) ϕ 0 )
ψ( x,y,Δz )= ψ 0 ( ω ) e iωt ( α e iϕ( x,y,Δz,ω ) +( 1α ) e i ϕ ref ( ω ) )dω ,
I( x,y,Δz )= | ψ 0 ( ω ) | 2 ( α 2 + ( 1α ) 2 +2α( 1α )cos( ϕ( x,y,Δz,ω ) ϕ ref ( ω ) ) )dω .
I( x,y,Δz )= | ψ 0 ( ω ) | 2 cos( 2ω c ( z( x,y )Δz z ref ) )dω +C.
I( x,y,Δz )=ε( ξ )cos( ω 0 ξ )+C.
z( x,y ) z ref =Δ z 0 ( x,y )+ c ξ 0 /2 .
ψ image ( x,y )= ψ object ( x,y )τ( x,y ),
I( x,y )= | ( α e iϕ( x,y ) +( 1α ) e i ϕ ref )τ( x,y ) | 2 .
ϕ image ( x,y ) ϕ 0 =Arg( ( e iϕ( x,y ) τ( x,y ) )( e i ϕ 0 τ ¯ ( x,y ) ) ),
ϕ image ( x,y ) ϕ 0 = tan 1 ( sin( ϕ( x,y ) ϕ 0 )τ( x,y ) cos( ϕ( x,y ) ϕ 0 )τ( x,y ) ).
ψ image ( x,y,Δz )= ψ 0 ( ω ) e iωt ( α e iϕ( x,y,Δz,ω ) +( 1α ) e i ϕ ref ( ω ) )τ( x,y,ω )dω .
I( x,y,Δz )= | ψ 0 ( ω ) | 2 ( | ( 1α )τ( x,y ) | 2 + | α e iϕ( x,y,Δz,ω )+i ϕ ref ( ω ) τ( x,y ) | 2 +2Re( [ α e iϕ( x,y,Δz,ω )+i ϕ ref ( ω ) τ( x,y ) ][ ( 1α ) τ ¯ ( x,y ) ] )dω
I( x,y,Δz )= | ψ 0 ( ω ) | 2 ( cos( ϕ( x,y,Δz,ω ) ϕ ref ( ω ) )τ( x,y,ω ) ) dω +C.
I( x,y,Δz )= | ψ 0 ( ω ) | 2 ( cos( ϕ( x,y,Δz,ω ) ϕ ref ( ω ) )τ( x,y, ω 0 ) ) dω + | ψ 0 ( ω ) | 2 ( cos( ϕ( x,y,Δz,ω ) ϕ ref ( ω ) )( ω ω 0 ) τ( x,y,ω ) ω | ω= ω 0 ) dω ++C
I( x,y,Δz )=[ ε( ξ )cos( ω 0 ξ )+C ]τ( x,y, ω 0 )
z( x,y ) z ref ={ h x 2 + y 2 R 0 x 2 + y 2 >R .
ϕ( x,y ) ϕ 0 ={ 2kh x 2 + y 2 R 0 x 2 + y 2 >R .
ϕ image ( x,y ) ϕ 0 = tan 1 ( sin( 2kh ) x 2 + y 2 R τ d x d y +sin( 2k0 ) x 2 + y 2 >R τ d x d y cos( 2kh ) x 2 + y 2 R τ d x d y +cos( 2k0 ) x 2 + y 2 >R τ d x d y ),
τ ideal ( x,y )= J 1 ( r ) /r ,
R N =NA×kR.
ϕ image ( x,y ) ϕ 0 = tan 1 ( j=1 N sin( 2k( z j z ref ) ) Q j τ( x x, y y )d x d y j=1 N cos( 2k( z j z ref ) ) Q j τ( x x, y y )d x d y ),
ϕ image ( x,y ) ϕ 0 = tan 1 ( z min z max sin( 2k( z j z ref ) )dμ / z min z max cos( 2k( z j z ref ) )dμ )
μ( x,y,z )= Q( z ) τ( x x, y y )d x d y ,
z( x,y ) z ref ={ ρ+ ρ 2 x 2 y 2 x 2 + y 2 ρ 0 x 2 + y 2 >ρ
ρ N =NA×kρ.
I mod ( x,y,Δz )=ε( ξ )τ( x,y, ω 0 ).
ε( ξ )= e 4 ( Δω ) 2 ( z( x,y ) z ref Δz ) / 2 c 2 .
I mod ( x,y,Δz )=ε( 2( hΔz ) /c ) x 2 + y 2 R τ( x x, y y )d x d y +ε( 2( 0Δz ) /c ) x 2 + y 2 >R τ( x x, y y )d x d y
I mod ( x,y,Δz )= j=1 N ε( 2( z j Δz ) /c ) Q j τ( x x, y y )d x d y .
I mod ( x,y,Δz )= ξ min ξ max ε( ξ )dμ ,
Δ z N =NA×kΔz.

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