Abstract

A method is presented for recovering the intensity depth profile, by confocal optical microscopy, in transparent and amorphous samples with low scattering. The response function of a confocal Raman microscope has been determined by using the second Rayleigh-Sommerfeld diffraction integral and scalar wave optics within paraxial approximation, taking into account the refractive index mismatch between the sample and the medium surrounding the objective lens. An iterative multi-fitting-scheme, based on the conjugate gradient method and Brent algorithm, allowed to fit several depth profile curves simultaneously and retrieve the beam waist, the signal amplitude and the position of the sample surface. The reliability and accuracy of the theoretical procedure has been investigated through comparison with experimental measurements of the Raman depth profiles for different pinhole diameters. The model is shown to provide accurate description of the effect of the mismatch of the refractive index and of the dependence of the Raman signal on the depth with discrepancies lower than 3%. This procedure constitutes a first step towards the development of a manageable theoretical framework, amenable to a relatively simple numerical implementation, for the solution of the ’inverse’ problem of finding the correct reconstruction of unknown profiles of chemical species within the sample, starting from experimental information gathered from micro-Raman depth profiling.

© 2016 Optical Society of America

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References

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  5. J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
    [Crossref]
  6. P. Eaton, P. Holmes, and J. Yarwood, “ATR/FT-IR and Raman microscopic investigation of diffusion and distribution of silane coupling agents in pvc films,” Appl. Spectrosc. 54, 508–516 (2000).
    [Crossref]
  7. C. Mura, “Raman microscopic studies of the distribution of the fungicide fluorfolpet in plasticised pvc films,” Polymer 41, 8659–8671 (2000).
    [Crossref]
  8. J. White, W. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105, 41–48 (1987).
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  9. S. Cha, P. C. Lin, L. Zhu, P.-C. Sun, and Y. Fainman, “Nontranslational three-dimensional profilometry by chromatic confocal microscopy with dynamically configurable micromirror scanning,” Appl. Opt. 39, 2605–2613 (2000).
    [Crossref]
  10. J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
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    [Crossref]
  24. C. Sourisseau and P. Maraval, “Confocal Raman microspectrometry: a vectorial electromagnetic treatment of the light focused and collected through a planar interface and its application to the study of a thin coating,” Appl. Spectrosc. 57, 1324–1332 (2003).
    [Crossref] [PubMed]
  25. L. Baia, K. Gigant, U. Posset, G. Schottner, W. Kiefer, and J. Popp, “Confocal micro-Raman spectroscopy: theory and application to a hybrid polymer coating,” Appl. Spectrosc. 56, 536–540 (2002).
    [Crossref]
  26. A. Gallardo, S. Spells, R. Navarro, and H. Reinecke, “Confocal Raman microscopy: how to correct depth profiles considering diffraction and refraction effects,” J. Raman Spectrosc. 38, 880–884 (2007).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  29. A. M. Macdonald and A. S. Vaughan, “Numerical simulations of confocal Raman spectroscopic depth profiles of materials: a photon scattering approach,” J. Raman Spectrosc. 38, 584–592 (2007).
    [Crossref]
  30. J. P. Tomba, M. de la Paz Miguel, and C. J. Perez, “Correction of optical distortions in dry depth profiling with confocal Raman microspectroscopy,” J. Raman Spectrosc. 42, 1330–1334 (2011).
    [Crossref]
  31. Y. Maruyama and W. Kanematsu, “Confocal volume in laser Raman microscopy depth profiling,” J. Appl. Phys. 110, 103107 (2011).
    [Crossref]
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    [Crossref]
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  36. J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering (Springer, 2006).
  37. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(John Wiley & Sons, Inc., 2001).
  38. A. Papoulis, Systems and Transforms With Applications in Optics (Robert Krieger Publishing Company, 1968).
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    [Crossref] [PubMed]
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2012 (1)

M. de la Paz Miguel and J. P. Tomba, “A comparison of different approaches for depth profiling of films and coatings by confocal Raman microscopy,” Prog. Org. Coat. 74, 43–49 (2012).
[Crossref]

2011 (2)

J. P. Tomba, M. de la Paz Miguel, and C. J. Perez, “Correction of optical distortions in dry depth profiling with confocal Raman microspectroscopy,” J. Raman Spectrosc. 42, 1330–1334 (2011).
[Crossref]

Y. Maruyama and W. Kanematsu, “Confocal volume in laser Raman microscopy depth profiling,” J. Appl. Phys. 110, 103107 (2011).
[Crossref]

2010 (1)

N. J. Everall, “Confocal Raman microscopy: common errors and artefacts,” Analyst 135, 2512–2522 (2010).
[Crossref] [PubMed]

2007 (4)

A. M. Macdonald and A. S. Vaughan, “Numerical simulations of confocal Raman spectroscopic depth profiles of materials: a photon scattering approach,” J. Raman Spectrosc. 38, 584–592 (2007).
[Crossref]

A. Gallardo, S. Spells, R. Navarro, and H. Reinecke, “Confocal Raman microscopy: how to correct depth profiles considering diffraction and refraction effects,” J. Raman Spectrosc. 38, 880–884 (2007).
[Crossref]

J. P. Tomba, L. M. Arzondo, and J. M. Pastor, “Depth profiling by confocal Raman microspectroscopy: semi-empirical modeling of the Raman response,” Appl. Spectrosc. 61, 177–185 (2007).
[Crossref] [PubMed]

N. Everall, J. Lapham, F. Adar, A. Whitley, E. Lee, and S. Mamedov, “Optimizing depth resolution in confocal Raman microscopy: a comparison of metallurgical, dry corrected, and oil immersion objectives,” Appl. Spectrosc. 61, 251–259 (2007).
[Crossref] [PubMed]

2004 (2)

N. Everall, “Depth profiling with confocal raman microscopy is one of the techniques of choice for investigating heterogeneous sys,” Spectroscopy 19, 16–24 (2004).

N. Everall, “Depth profiling with confocal Raman microscopy, part II,” Spectroscopy 19, 16 (2004).

2003 (2)

2002 (2)

L. Baia, K. Gigant, U. Posset, G. Schottner, W. Kiefer, and J. Popp, “Confocal micro-Raman spectroscopy: theory and application to a hybrid polymer coating,” Appl. Spectrosc. 56, 536–540 (2002).
[Crossref]

J. L. Bruneel, J. C. Lassègues, and C. Sourisseau, “In-depth analyses by confocal Raman microspectrometry: experimental features and modeling of the refraction effects,” J. Raman Spectrosc. 33, 815–828 (2002).
[Crossref]

2001 (1)

2000 (7)

N. J. Everall, “Modeling and measuring the effect of refraction on the depth resolution of confocal Raman microscopy,” Appl. Spectrosc. 54, 773–782 (2000).
[Crossref]

P. Eaton, P. Holmes, and J. Yarwood, “ATR/FT-IR and Raman microscopic investigation of diffusion and distribution of silane coupling agents in pvc films,” Appl. Spectrosc. 54, 508–516 (2000).
[Crossref]

S. Cha, P. C. Lin, L. Zhu, P.-C. Sun, and Y. Fainman, “Nontranslational three-dimensional profilometry by chromatic confocal microscopy with dynamically configurable micromirror scanning,” Appl. Opt. 39, 2605–2613 (2000).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
[Crossref]

C. Sammon, C. Mura, P. Eaton, and J. Yarwood, “Raman microscopic studies of polymer surfaces and interfaces,” Analusis 28, 30–33 (2000).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
[Crossref]

C. Mura, “Raman microscopic studies of the distribution of the fungicide fluorfolpet in plasticised pvc films,” Polymer 41, 8659–8671 (2000).
[Crossref]

1999 (1)

C. Sammon, S. Hajatdoost, P. Eaton, C. Mura, and J. Yarwood, “Materials analysis using confocal Raman microscopy,” Macromolecular 141, 247–262 (1999).

1998 (1)

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc.-Oxford 192, 90–98 (1998).
[Crossref]

1997 (1)

C. J. De Grauw, N. M. Sijtsema, C. Otto, and J. Greve, “Axial resolution of confocal Raman microscopes: Gaussian beam theory and practice,” J. Microsc. 188, 273–279 (1997).
[Crossref]

1996 (1)

1992 (1)

1991 (1)

C. J. R. Sheppard and C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

1988 (1)

1987 (1)

J. White, W. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105, 41–48 (1987).
[Crossref] [PubMed]

Adar, F.

Amos, W.

J. White, W. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105, 41–48 (1987).
[Crossref] [PubMed]

Arzondo, L. M.

Baia, L.

Baldwin, K. J.

Batchelder, D. N.

Bhatia, A.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

Booth, M. J.

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc.-Oxford 192, 90–98 (1998).
[Crossref]

Born, M.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

Brent, R. P.

R. P. Brent, Algorithms for Minimization Without Derivatives, Prentice-Hall Series in Automatic Computation (Prentice-Hall, 1973).

Bruneel, J. L.

J. L. Bruneel, J. C. Lassègues, and C. Sourisseau, “In-depth analyses by confocal Raman microspectrometry: experimental features and modeling of the refraction effects,” J. Raman Spectrosc. 33, 815–828 (2002).
[Crossref]

Cha, S.

Cogswell, C. J.

C. J. R. Sheppard and C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

ColinJ, X. C. M. C.

X. C. M. C. ColinJ, R. Sheppard, and M. Roy, Signal Level in Confocal Microscopes in Handbook of Biological Confocal Microscopy(Springer, 2006).

De Grauw, C. J.

C. J. De Grauw, N. M. Sijtsema, C. Otto, and J. Greve, “Axial resolution of confocal Raman microscopes: Gaussian beam theory and practice,” J. Microsc. 188, 273–279 (1997).
[Crossref]

de la Paz Miguel, M.

M. de la Paz Miguel and J. P. Tomba, “A comparison of different approaches for depth profiling of films and coatings by confocal Raman microscopy,” Prog. Org. Coat. 74, 43–49 (2012).
[Crossref]

J. P. Tomba, M. de la Paz Miguel, and C. J. Perez, “Correction of optical distortions in dry depth profiling with confocal Raman microspectroscopy,” J. Raman Spectrosc. 42, 1330–1334 (2011).
[Crossref]

Eaton, P.

C. Sammon, C. Mura, P. Eaton, and J. Yarwood, “Raman microscopic studies of polymer surfaces and interfaces,” Analusis 28, 30–33 (2000).
[Crossref]

P. Eaton, P. Holmes, and J. Yarwood, “ATR/FT-IR and Raman microscopic investigation of diffusion and distribution of silane coupling agents in pvc films,” Appl. Spectrosc. 54, 508–516 (2000).
[Crossref]

C. Sammon, S. Hajatdoost, P. Eaton, C. Mura, and J. Yarwood, “Materials analysis using confocal Raman microscopy,” Macromolecular 141, 247–262 (1999).

Escobar, I.

I. Escobar, E. Sánchez-Ortiga, G. Saavedra, and M. Martínez-Corral, New analytical tools for evaluation of spherical aberration in optical microscopy(Springer, 2011).

Everall, N.

N. Everall, J. Lapham, F. Adar, A. Whitley, E. Lee, and S. Mamedov, “Optimizing depth resolution in confocal Raman microscopy: a comparison of metallurgical, dry corrected, and oil immersion objectives,” Appl. Spectrosc. 61, 251–259 (2007).
[Crossref] [PubMed]

N. Everall, “Depth profiling with confocal raman microscopy is one of the techniques of choice for investigating heterogeneous sys,” Spectroscopy 19, 16–24 (2004).

N. Everall, “Depth profiling with confocal Raman microscopy, part II,” Spectroscopy 19, 16 (2004).

Everall, N. J.

Fainman, Y.

Fordham, M.

J. White, W. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105, 41–48 (1987).
[Crossref] [PubMed]

Froud, C. A.

Gabor, D.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

Gallardo, A.

A. Gallardo, S. Spells, R. Navarro, and H. Reinecke, “Confocal Raman microscopy: how to correct depth profiles considering diffraction and refraction effects,” J. Raman Spectrosc. 38, 880–884 (2007).
[Crossref]

Gibson, S. F.

Gigant, K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier OpticsMcGraw-HillSeries in Electrical and Computer Engineering, No. 5 in McGraw-Hill Series in Electrical and Computer Engineering (McGraw-Hill, 1996).

Greve, J.

C. J. De Grauw, N. M. Sijtsema, C. Otto, and J. Greve, “Axial resolution of confocal Raman microscopes: Gaussian beam theory and practice,” J. Microsc. 188, 273–279 (1997).
[Crossref]

Gu, M.

M. Gu, Advanced Optical Imaging Theory, Springer Series in Optical Sciences (Springer, 2000).
[Crossref]

Hajatdoost, S.

C. Sammon, S. Hajatdoost, P. Eaton, C. Mura, and J. Yarwood, “Materials analysis using confocal Raman microscopy,” Macromolecular 141, 247–262 (1999).

S. Hajatdoost and J. Yarwood, “Depth profiling of polg(methyl methacrylate), poly(vinyl alcohol) laminates by confocal Raman microspectroscopy,” Appl. Spectrosc. 50, 558–564 (1996).
[Crossref]

Hayward, I. P.

Holmes, P.

Kanematsu, W.

Y. Maruyama and W. Kanematsu, “Confocal volume in laser Raman microscopy depth profiling,” J. Appl. Phys. 110, 103107 (2011).
[Crossref]

Kiefer, W.

Lanni, F.

Lapham, J.

Lassègues, J. C.

J. L. Bruneel, J. C. Lassègues, and C. Sourisseau, “In-depth analyses by confocal Raman microspectrometry: experimental features and modeling of the refraction effects,” J. Raman Spectrosc. 33, 815–828 (2002).
[Crossref]

Laven, J.

Lee, E.

Lin, P. C.

Macdonald, A. M.

A. M. Macdonald and A. S. Vaughan, “Numerical simulations of confocal Raman spectroscopic depth profiles of materials: a photon scattering approach,” J. Raman Spectrosc. 38, 584–592 (2007).
[Crossref]

Mamedov, S.

Maraval, P.

Martínez-Corral, M.

I. Escobar, E. Sánchez-Ortiga, G. Saavedra, and M. Martínez-Corral, New analytical tools for evaluation of spherical aberration in optical microscopy(Springer, 2011).

Maruyama, Y.

Y. Maruyama and W. Kanematsu, “Confocal volume in laser Raman microscopy depth profiling,” J. Appl. Phys. 110, 103107 (2011).
[Crossref]

Mijangos, C.

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
[Crossref]

Muller, M.

M. Muller, Introduction to confocal fluorescence microscopy, Tutorial Text Series (Society of Photo Optical, 2006).

Mura, C.

C. Mura, “Raman microscopic studies of the distribution of the fungicide fluorfolpet in plasticised pvc films,” Polymer 41, 8659–8671 (2000).
[Crossref]

C. Sammon, C. Mura, P. Eaton, and J. Yarwood, “Raman microscopic studies of polymer surfaces and interfaces,” Analusis 28, 30–33 (2000).
[Crossref]

C. Sammon, S. Hajatdoost, P. Eaton, C. Mura, and J. Yarwood, “Materials analysis using confocal Raman microscopy,” Macromolecular 141, 247–262 (1999).

Navarro, R.

A. Gallardo, S. Spells, R. Navarro, and H. Reinecke, “Confocal Raman microscopy: how to correct depth profiles considering diffraction and refraction effects,” J. Raman Spectrosc. 38, 880–884 (2007).
[Crossref]

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc.-Oxford 192, 90–98 (1998).
[Crossref]

Nemoto, S.

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering (Springer, 2006).

Otto, C.

C. J. De Grauw, N. M. Sijtsema, C. Otto, and J. Greve, “Axial resolution of confocal Raman microscopes: Gaussian beam theory and practice,” J. Microsc. 188, 273–279 (1997).
[Crossref]

Papoulis, A.

A. Papoulis, Systems and Transforms With Applications in Optics (Robert Krieger Publishing Company, 1968).

Pastor, J. M.

Perez, C. J.

J. P. Tomba, M. de la Paz Miguel, and C. J. Perez, “Correction of optical distortions in dry depth profiling with confocal Raman microspectroscopy,” J. Raman Spectrosc. 42, 1330–1334 (2011).
[Crossref]

Popp, J.

Posset, U.

Reinecke, H.

A. Gallardo, S. Spells, R. Navarro, and H. Reinecke, “Confocal Raman microscopy: how to correct depth profiles considering diffraction and refraction effects,” J. Raman Spectrosc. 38, 880–884 (2007).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
[Crossref]

Roy, M.

X. C. M. C. ColinJ, R. Sheppard, and M. Roy, Signal Level in Confocal Microscopes in Handbook of Biological Confocal Microscopy(Springer, 2006).

Saavedra, G.

I. Escobar, E. Sánchez-Ortiga, G. Saavedra, and M. Martínez-Corral, New analytical tools for evaluation of spherical aberration in optical microscopy(Springer, 2011).

Sacristán, J.

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(John Wiley & Sons, Inc., 2001).

Sammon, C.

C. Sammon, C. Mura, P. Eaton, and J. Yarwood, “Raman microscopic studies of polymer surfaces and interfaces,” Analusis 28, 30–33 (2000).
[Crossref]

C. Sammon, S. Hajatdoost, P. Eaton, C. Mura, and J. Yarwood, “Materials analysis using confocal Raman microscopy,” Macromolecular 141, 247–262 (1999).

Sánchez-Ortiga, E.

I. Escobar, E. Sánchez-Ortiga, G. Saavedra, and M. Martínez-Corral, New analytical tools for evaluation of spherical aberration in optical microscopy(Springer, 2011).

Schottner, G.

Sheppard, C. J. R.

C. J. R. Sheppard and C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

Sheppard, R.

X. C. M. C. ColinJ, R. Sheppard, and M. Roy, Signal Level in Confocal Microscopes in Handbook of Biological Confocal Microscopy(Springer, 2006).

Sijtsema, N. M.

C. J. De Grauw, N. M. Sijtsema, C. Otto, and J. Greve, “Axial resolution of confocal Raman microscopes: Gaussian beam theory and practice,” J. Microsc. 188, 273–279 (1997).
[Crossref]

Sourisseau, C.

C. Sourisseau and P. Maraval, “Confocal Raman microspectrometry: a vectorial electromagnetic treatment of the light focused and collected through a planar interface and its application to the study of a thin coating,” Appl. Spectrosc. 57, 1324–1332 (2003).
[Crossref] [PubMed]

J. L. Bruneel, J. C. Lassègues, and C. Sourisseau, “In-depth analyses by confocal Raman microspectrometry: experimental features and modeling of the refraction effects,” J. Raman Spectrosc. 33, 815–828 (2002).
[Crossref]

Spells, S.

A. Gallardo, S. Spells, R. Navarro, and H. Reinecke, “Confocal Raman microscopy: how to correct depth profiles considering diffraction and refraction effects,” J. Raman Spectrosc. 38, 880–884 (2007).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
[Crossref]

Stokes, A.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

Sun, P.-C.

Taylor, A.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(John Wiley & Sons, Inc., 2001).

Tomba, J. P.

M. de la Paz Miguel and J. P. Tomba, “A comparison of different approaches for depth profiling of films and coatings by confocal Raman microscopy,” Prog. Org. Coat. 74, 43–49 (2012).
[Crossref]

J. P. Tomba, M. de la Paz Miguel, and C. J. Perez, “Correction of optical distortions in dry depth profiling with confocal Raman microspectroscopy,” J. Raman Spectrosc. 42, 1330–1334 (2011).
[Crossref]

J. P. Tomba, L. M. Arzondo, and J. M. Pastor, “Depth profiling by confocal Raman microspectroscopy: semi-empirical modeling of the Raman response,” Appl. Spectrosc. 61, 177–185 (2007).
[Crossref] [PubMed]

Vaughan, A. S.

A. M. Macdonald and A. S. Vaughan, “Numerical simulations of confocal Raman spectroscopic depth profiles of materials: a photon scattering approach,” J. Raman Spectrosc. 38, 584–592 (2007).
[Crossref]

Wayman, P.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

White, J.

J. White, W. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105, 41–48 (1987).
[Crossref] [PubMed]

Whitley, A.

Wilcock, W.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

Wilson, T.

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc.-Oxford 192, 90–98 (1998).
[Crossref]

Wolf, E.

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering (Springer, 2006).

Yarwood, J.

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
[Crossref]

C. Sammon, C. Mura, P. Eaton, and J. Yarwood, “Raman microscopic studies of polymer surfaces and interfaces,” Analusis 28, 30–33 (2000).
[Crossref]

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
[Crossref]

P. Eaton, P. Holmes, and J. Yarwood, “ATR/FT-IR and Raman microscopic investigation of diffusion and distribution of silane coupling agents in pvc films,” Appl. Spectrosc. 54, 508–516 (2000).
[Crossref]

C. Sammon, S. Hajatdoost, P. Eaton, C. Mura, and J. Yarwood, “Materials analysis using confocal Raman microscopy,” Macromolecular 141, 247–262 (1999).

S. Hajatdoost and J. Yarwood, “Depth profiling of polg(methyl methacrylate), poly(vinyl alcohol) laminates by confocal Raman microspectroscopy,” Appl. Spectrosc. 50, 558–564 (1996).
[Crossref]

Zhu, L.

Analusis (1)

C. Sammon, C. Mura, P. Eaton, and J. Yarwood, “Raman microscopic studies of polymer surfaces and interfaces,” Analusis 28, 30–33 (2000).
[Crossref]

Analyst (1)

N. J. Everall, “Confocal Raman microscopy: common errors and artefacts,” Analyst 135, 2512–2522 (2010).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Spectrosc. (9)

C. Sourisseau and P. Maraval, “Confocal Raman microspectrometry: a vectorial electromagnetic treatment of the light focused and collected through a planar interface and its application to the study of a thin coating,” Appl. Spectrosc. 57, 1324–1332 (2003).
[Crossref] [PubMed]

C. A. Froud, I. P. Hayward, and J. Laven, “Advances in the raman depth profiling of polymer laminates,” Appl. Spectrosc. 57, 1468–1474 (2003).
[Crossref] [PubMed]

L. Baia, K. Gigant, U. Posset, G. Schottner, W. Kiefer, and J. Popp, “Confocal micro-Raman spectroscopy: theory and application to a hybrid polymer coating,” Appl. Spectrosc. 56, 536–540 (2002).
[Crossref]

K. J. Baldwin and D. N. Batchelder, “Confocal Raman microspectroscopy through a planar interface,” Appl. Spectrosc. 55, 517–524 (2001).
[Crossref]

N. J. Everall, “Modeling and measuring the effect of refraction on the depth resolution of confocal Raman microscopy,” Appl. Spectrosc. 54, 773–782 (2000).
[Crossref]

P. Eaton, P. Holmes, and J. Yarwood, “ATR/FT-IR and Raman microscopic investigation of diffusion and distribution of silane coupling agents in pvc films,” Appl. Spectrosc. 54, 508–516 (2000).
[Crossref]

S. Hajatdoost and J. Yarwood, “Depth profiling of polg(methyl methacrylate), poly(vinyl alcohol) laminates by confocal Raman microspectroscopy,” Appl. Spectrosc. 50, 558–564 (1996).
[Crossref]

J. P. Tomba, L. M. Arzondo, and J. M. Pastor, “Depth profiling by confocal Raman microspectroscopy: semi-empirical modeling of the Raman response,” Appl. Spectrosc. 61, 177–185 (2007).
[Crossref] [PubMed]

N. Everall, J. Lapham, F. Adar, A. Whitley, E. Lee, and S. Mamedov, “Optimizing depth resolution in confocal Raman microscopy: a comparison of metallurgical, dry corrected, and oil immersion objectives,” Appl. Spectrosc. 61, 251–259 (2007).
[Crossref] [PubMed]

J. Appl. Phys. (1)

Y. Maruyama and W. Kanematsu, “Confocal volume in laser Raman microscopy depth profiling,” J. Appl. Phys. 110, 103107 (2011).
[Crossref]

J. Cell Biol. (1)

J. White, W. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105, 41–48 (1987).
[Crossref] [PubMed]

J. Microsc. (1)

C. J. De Grauw, N. M. Sijtsema, C. Otto, and J. Greve, “Axial resolution of confocal Raman microscopes: Gaussian beam theory and practice,” J. Microsc. 188, 273–279 (1997).
[Crossref]

J. Microsc.-Oxford (1)

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc.-Oxford 192, 90–98 (1998).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Raman Spectrosc. (4)

A. M. Macdonald and A. S. Vaughan, “Numerical simulations of confocal Raman spectroscopic depth profiles of materials: a photon scattering approach,” J. Raman Spectrosc. 38, 584–592 (2007).
[Crossref]

J. P. Tomba, M. de la Paz Miguel, and C. J. Perez, “Correction of optical distortions in dry depth profiling with confocal Raman microspectroscopy,” J. Raman Spectrosc. 42, 1330–1334 (2011).
[Crossref]

J. L. Bruneel, J. C. Lassègues, and C. Sourisseau, “In-depth analyses by confocal Raman microspectrometry: experimental features and modeling of the refraction effects,” J. Raman Spectrosc. 33, 815–828 (2002).
[Crossref]

A. Gallardo, S. Spells, R. Navarro, and H. Reinecke, “Confocal Raman microscopy: how to correct depth profiles considering diffraction and refraction effects,” J. Raman Spectrosc. 38, 880–884 (2007).
[Crossref]

Macromol. Rapid Commun. (1)

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Depth profiling of modified pvc surfaces using confocal Raman microspectroscopy,” Macromol. Rapid Commun. 21, 894–896 (2000).
[Crossref]

Macromolecular (1)

C. Sammon, S. Hajatdoost, P. Eaton, C. Mura, and J. Yarwood, “Materials analysis using confocal Raman microscopy,” Macromolecular 141, 247–262 (1999).

Macromolecules (1)

J. Sacristán, C. Mijangos, H. Reinecke, S. Spells, and J. Yarwood, “Selective surface modification of pvc films as revealed by confocal Raman microspectroscopy,” Macromolecules 33, 6134–6139 (2000).
[Crossref]

Optik (1)

C. J. R. Sheppard and C. J. Cogswell, “Effects of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

Polymer (1)

C. Mura, “Raman microscopic studies of the distribution of the fungicide fluorfolpet in plasticised pvc films,” Polymer 41, 8659–8671 (2000).
[Crossref]

Prog. Org. Coat. (1)

M. de la Paz Miguel and J. P. Tomba, “A comparison of different approaches for depth profiling of films and coatings by confocal Raman microscopy,” Prog. Org. Coat. 74, 43–49 (2012).
[Crossref]

Spectroscopy (2)

N. Everall, “Depth profiling with confocal raman microscopy is one of the techniques of choice for investigating heterogeneous sys,” Spectroscopy 19, 16–24 (2004).

N. Everall, “Depth profiling with confocal Raman microscopy, part II,” Spectroscopy 19, 16 (2004).

Other (10)

M. Muller, Introduction to confocal fluorescence microscopy, Tutorial Text Series (Society of Photo Optical, 2006).

I. Escobar, E. Sánchez-Ortiga, G. Saavedra, and M. Martínez-Corral, New analytical tools for evaluation of spherical aberration in optical microscopy(Springer, 2011).

J. W. Goodman, Introduction to Fourier OpticsMcGraw-HillSeries in Electrical and Computer Engineering, No. 5 in McGraw-Hill Series in Electrical and Computer Engineering (McGraw-Hill, 1996).

M. Born, E. Wolf, A. Bhatia, D. Gabor, A. Stokes, A. Taylor, P. Wayman, and W. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 2000).

M. Gu, Advanced Optical Imaging Theory, Springer Series in Optical Sciences (Springer, 2000).
[Crossref]

R. P. Brent, Algorithms for Minimization Without Derivatives, Prentice-Hall Series in Automatic Computation (Prentice-Hall, 1973).

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering (Springer, 2006).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(John Wiley & Sons, Inc., 2001).

A. Papoulis, Systems and Transforms With Applications in Optics (Robert Krieger Publishing Company, 1968).

X. C. M. C. ColinJ, R. Sheppard, and M. Roy, Signal Level in Confocal Microscopes in Handbook of Biological Confocal Microscopy(Springer, 2006).

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Figures (5)

Fig. 1
Fig. 1 Schematic of a confocal Raman microscope from the point of view of a point-like emitter P located at a depth Δ inside a sample with refractive index n2. Σ is the surface of separation between the sample and the surrounding medium of refractive index n1.
Fig. 2
Fig. 2 Schematic representation of the transmitted field across the interface Σ. An optical ray (red dotted line) emitted by P moves towards the sample surface. The ray intercepts the interface at P1 and changes its direction (red solid line) due to the refractive index mismatch between the sample and the surrounding medium. The refracted ray hits the lens surface L0 at P′ that is distant d from the interface Σ and is distant r′ from P1. The unit vector n is the surface normal.
Fig. 3
Fig. 3 Schematic of a confocal Raman microscope from the point of view of the input source. The focal length of the lens L0 is fm. The nominal focal position is z = fmd and the actual focus position compared to nominal focal position is p = (n2/n1)z.
Fig. 4
Fig. 4 The normalized on-axis gaussian laser beam intensity Iill/I0 vs the depth Δ−(n2/n1)z, that is the distance of the point emitter from the actual focus position.
Fig. 5
Fig. 5 (a) Normalized confocal Raman depth profiles of a polystyrene thick sheet for pinholes sizes, D = 200 μm and D = 400 μm. Black points refer to the experimental data (intensity of the 997cm−1 Raman peak); (b) Normalized confocal Raman depth profiles of of a transparent homogeneous plastic plate made of allyl-diglycol-carbonate (NT43-927, Edmund Optics, Inc.) for three pinholes size, D = 400μm, 700μm, 1000μm. Black points refer to the experimental data reported in [31] (intensity of the 2955cm−1 Raman peak). Solid red line represent the theoretical model predictions.

Equations (20)

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U P ( P 1 ) = U 0 exp ( 1 n 2 k e m s ) s k e m = 2 π λ e m
U ( x , y , d | 0 , 0 , Δ ) = 1 2 π U P ( P 1 ) n exp ( 1 n 1 k e m r ) r d w i t h r = ( x x ) 2 + ( y y ) 2 + d 2 .
U ( x , y , d | 0 , 0 , Δ ) = Δ U 0 0 e ( 1 n 2 k e m s ) s exp ( 1 n 1 k e m r ) r ( 1 n 2 k e m s + 1 s 2 ) r d r w i t h s = r 2 + Δ 2
U ( x , y , d | 0 , 0 , Δ ) = = Δ U 0 d e ( 1 n 1 k e m d ) 0 e ( 1 n 2 k e m s ) s ( 1 n 2 k e m s + 1 s 2 ) exp [ 1 n 1 k e m ( x x ) 2 + ( y y ) 2 2 d ] r d r .
U ( x , y , d + 1 | Δ ) = e i k e m l 1 U ( x , y , d | Δ ) × e 1 n 1 k e m x 2 + y 2 2 f m e i 1 n 1 k e m ( x x ) 2 + ( y y ) 2 21 d x d y .
U ( x , y , L | Δ ) = e 1 k e m f t f t U ( x , y , d + 1 | Δ ) × e 1 k e m x 2 + y 2 2 f t e 1 k e m ( x x ) 2 + ( y y ) 2 2 f t d x d y
U ( r , L | Δ ) Δ 0 e ( 1 n 2 k e m s ) s ( 1 n 2 k e m s + 1 s 2 ) × e 1 f r 2 J 0 ( w r ) r d r
I p ( r , L | Δ ) = | U ( r , L | Δ ) | 2 2 π 0 r | U ( r , L | Δ ) | 2 d r .
I ( r , θ | ρ p , θ p , Δ ) = 1 | M | 2 I p ( r 2 + M 2 ρ p 2 2 M r ρ p cos ( θ θ p ) , L | Δ ) ,
I i l l ( ρ p , Δ , z ) = I 0 [ w 0 w ( Δ , z ) ] 2 e 2 ρ p 2 w 2 ( Δ , z ) ,
w ( Δ , z ) = w 0 1 + ( Δ ( n 2 / n 1 ) z n 2 z 0 ) 2 ,
I i l l ( 0 , Δ , z ) = I 0 1 + [ Δ ( n 2 / n 1 ) z z 0 ] 2 .
P ( r , θ , w 0 , z ) = ρ p I i l l ( ρ p , Δ , w 0 , z ) × I p ( r , θ | ρ p , θ p , Δ ) d ρ p d θ p d Δ ,
P t o t ( w 0 , z ) = 2 π 0 D / 2 0 2 π r P ( r , θ , w 0 , z ) d r d θ .
P t o t ( w 0 , z ) = 2 π ρ p I i l l ( ρ p , Δ , w 0 , z ) P p ( ρ p , Δ ) d ρ p d Δ .
P p ( ρ p , Δ ) = 0 D / 2 0 2 π I p ( r , θ | ρ p , θ p , Δ ) r d r d θ .
P ( A , w 0 , z 0 , z ) = A × P t o t ( w 0 , z z 0 ) .
χ t o t 2 ( w 0 ) = 1 N i = 1 N χ i 2 ( w 0 )
χ i 2 ( w 0 ) = 1 M i m = 1 M ( E m , i P ( A i , w 0 , z 0 , z m , i ) ) 2 σ m , i 2 .
w 0 , m i n = w 0 1 2 ( w 0 w a ) 2 [ χ t o t 2 ( w 0 ) χ t o t 2 ( w b ) ] ( w 0 w b ) 2 [ χ t o t 2 ( w 0 ) χ t o t 2 ( w a ) ] ( w 0 w a ) [ χ t o t 2 ( w 0 ) χ t o t 2 ( w b ) ] ( w 0 w b ) [ χ t o t 2 ( w 0 ) χ t o t 2 ( w a ) ] ,

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