Abstract

In this tutorial, we present a general model linking the data provided by any optical diffraction microscope to the sample permittivity. Our analysis is applicable to essentially all microscope configurations, in transmission or reflection mode, using scanning or full-field illumination, with or without interferometric measurements. We include also a generalization of our analysis to vector fields.

© 2018 Optical Society of America

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References

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    [Crossref]
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  3. D. L. Marks, T. S. Ralston, S. A. Boppart, and P. Scott Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am. A 24, 1034–1041 (2007).
    [Crossref]
  4. Moving the sample yields a correlation while moving the source [in a translationally invariant microscope where G(r,i)=G(r−i)] yields a convolution. In most cases, the source is moved in the transverse plane while the sample is moved along the axial direction. For a more in-depth discussion of this difficulty, see D. S. Goodman, “Stationary optical projectors,” Ph.D. disseration (University of Arizona, 1979), http://hdl.handle.net/10150/298640 , Chap. 10.
  5. R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32, 3107–3109 (2007).
    [Crossref]
  6. C. J. R. Sheppard, S. S. Kou, and J. Lin, “The Green-function transform and wave propagation,” Front. Phys. 2, 1–10 (2014).
    [Crossref]
  7. D. Gabor, “A new microscope principle,” Nature 161, 777–778 (1948).
    [Crossref]
  8. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [Crossref]
  9. E. Beaurepaire, A. C. Bocarra, M. Lebec, L. Blanchot, and H. Saint-Jalmes, “Full-field optical coherence microscopy,” Opt. Lett. 23, 244–246 (1998).
    [Crossref]
  10. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
    [Crossref]
  11. O. Haeberlé, K. Belkebir, H. Giovannini, and A. Sentenac, “An introduction to diffractive tomographic microscopy,” J. Mod. Opt. 57, 686–699 (2010).
    [Crossref]
  12. J. Dyson, “An interferometer microscope,” Proc. R. Soc. A 204, 170–187 (1950).
    [Crossref]
  13. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [Crossref]
  14. S. B. Mehta and C. J. R. Sheppard, “Quantitative phase-gradient imaging at high resolution with asymmetric illumination-based differential phase contrast,” Opt. Lett. 34, 1924–1926 (2009).
    [Crossref]
  15. R. Barankov, J.-C. Baritaux, and J. Mertz, “High-resolution 3D phase imaging using a partitioned detection aperture: a wave-optic analysis,” J. Opt. Soc. Am. A 32, 2123–2135 (2015).
    [Crossref]
  16. L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

2015 (1)

2014 (1)

C. J. R. Sheppard, S. S. Kou, and J. Lin, “The Green-function transform and wave propagation,” Front. Phys. 2, 1–10 (2014).
[Crossref]

2010 (1)

O. Haeberlé, K. Belkebir, H. Giovannini, and A. Sentenac, “An introduction to diffractive tomographic microscopy,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

2009 (1)

2007 (2)

1998 (1)

1994 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

1986 (1)

C. J. R. Sheppard and T. Wilson, “On the equivalence of scanning and conventional microscopes,” Optik 73, 39–43 (1986).

1985 (1)

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref]

1950 (1)

J. Dyson, “An interferometer microscope,” Proc. R. Soc. A 204, 170–187 (1950).
[Crossref]

1948 (1)

D. Gabor, “A new microscope principle,” Nature 161, 777–778 (1948).
[Crossref]

Barankov, R.

Baritaux, J.-C.

Beaurepaire, E.

Belkebir, K.

O. Haeberlé, K. Belkebir, H. Giovannini, and A. Sentenac, “An introduction to diffractive tomographic microscopy,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

Blanchot, L.

Bocarra, A. C.

Boppart, S. A.

Carminati, R.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

de Rosny, J.

Dyson, J.

J. Dyson, “An interferometer microscope,” Proc. R. Soc. A 204, 170–187 (1950).
[Crossref]

Fink, M.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Fujimoto, J. G.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[Crossref]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Gabor, D.

D. Gabor, “A new microscope principle,” Nature 161, 777–778 (1948).
[Crossref]

Giovannini, H.

O. Haeberlé, K. Belkebir, H. Giovannini, and A. Sentenac, “An introduction to diffractive tomographic microscopy,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

Goodman, D. S.

Moving the sample yields a correlation while moving the source [in a translationally invariant microscope where G(r,i)=G(r−i)] yields a convolution. In most cases, the source is moved in the transverse plane while the sample is moved along the axial direction. For a more in-depth discussion of this difficulty, see D. S. Goodman, “Stationary optical projectors,” Ph.D. disseration (University of Arizona, 1979), http://hdl.handle.net/10150/298640 , Chap. 10.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Haeberlé, O.

O. Haeberlé, K. Belkebir, H. Giovannini, and A. Sentenac, “An introduction to diffractive tomographic microscopy,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

Hee, M. R.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[Crossref]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Izatt, J. A.

Kou, S. S.

C. J. R. Sheppard, S. S. Kou, and J. Lin, “The Green-function transform and wave propagation,” Front. Phys. 2, 1–10 (2014).
[Crossref]

Landau, L.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Lebec, M.

Lifshitz, E.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Lin, J.

C. J. R. Sheppard, S. S. Kou, and J. Lin, “The Green-function transform and wave propagation,” Front. Phys. 2, 1–10 (2014).
[Crossref]

Marks, D. L.

Mehta, S. B.

Mertz, J.

Owen, G. M.

Pierrat, R.

Pitaevskii, L.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Ralston, T. S.

Saint-Jalmes, H.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Scott Carney, P.

Sentenac, A.

O. Haeberlé, K. Belkebir, H. Giovannini, and A. Sentenac, “An introduction to diffractive tomographic microscopy,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, S. S. Kou, and J. Lin, “The Green-function transform and wave propagation,” Front. Phys. 2, 1–10 (2014).
[Crossref]

S. B. Mehta and C. J. R. Sheppard, “Quantitative phase-gradient imaging at high resolution with asymmetric illumination-based differential phase contrast,” Opt. Lett. 34, 1924–1926 (2009).
[Crossref]

C. J. R. Sheppard and T. Wilson, “On the equivalence of scanning and conventional microscopes,” Optik 73, 39–43 (1986).

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Streibl, N.

Swanson, E. A.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[Crossref]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Wilson, T.

C. J. R. Sheppard and T. Wilson, “On the equivalence of scanning and conventional microscopes,” Optik 73, 39–43 (1986).

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref]

Front. Phys. (1)

C. J. R. Sheppard, S. S. Kou, and J. Lin, “The Green-function transform and wave propagation,” Front. Phys. 2, 1–10 (2014).
[Crossref]

J. Mod. Opt. (1)

O. Haeberlé, K. Belkebir, H. Giovannini, and A. Sentenac, “An introduction to diffractive tomographic microscopy,” J. Mod. Opt. 57, 686–699 (2010).
[Crossref]

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

Nature (1)

D. Gabor, “A new microscope principle,” Nature 161, 777–778 (1948).
[Crossref]

Opt. Lett. (4)

Optik (1)

C. J. R. Sheppard and T. Wilson, “On the equivalence of scanning and conventional microscopes,” Optik 73, 39–43 (1986).

Proc. R. Soc. A (1)

J. Dyson, “An interferometer microscope,” Proc. R. Soc. A 204, 170–187 (1950).
[Crossref]

Science (2)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref]

Other (2)

Moving the sample yields a correlation while moving the source [in a translationally invariant microscope where G(r,i)=G(r−i)] yields a convolution. In most cases, the source is moved in the transverse plane while the sample is moved along the axial direction. For a more in-depth discussion of this difficulty, see D. S. Goodman, “Stationary optical projectors,” Ph.D. disseration (University of Arizona, 1979), http://hdl.handle.net/10150/298640 , Chap. 10.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

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Figures (4)

Fig. 1.
Fig. 1. General configuration of a microscope. The microscope optical elements are depicted in gray while the sample is in yellow. W is the domain over which the permittivity variations resulting from the sample, δε, are non-zero. i is a point source in the illumination domain Ω and o a detection point of the observation domain Γ.
Fig. 2.
Fig. 2. Illustration of the key assumptions used for deriving the point spread function of a microscope with extended detectors on the transmission, ΓT, or reflection, ΓR, side: (1) The microscope without the sample has no absorbing components; (2) (left) the field in the absence of the sample created by a point source located on the reflection (transmission) detector at any point inside the sample domain, G(r,o), is assumed to be a sum of plane waves propagating exclusively in the positive (negative) z directions; (3) (right) the field G(r,i) created by the illumination source i at any point inside the sample domain is assumed to be a sum of plane waves propagating exclusively in the positive z direction. As a consequence, G*(r,i) is a sum of plane waves propagating toward negative z.
Fig. 3.
Fig. 3. Illustration of G(v,u) in a standard microscope mounted in a 4f configuration. The 3D Fourier transform of G, FT3D[G], is rotationally invariant about the kz axis.
Fig. 4.
Fig. 4. Support of the OTF of various microscopes. (a) Monochromatic transmission and reflection holography using plane-wave illumination. (b) Same as (a) but with pulsed or white light (corresponding to full-field coherent OCT in reflection). (c) Monochromatic reflection holography using focused or incoherent illumination (scanning OCT, full-field incoherent OCT, confocal reflectance microscopy). (d) Same as (c) but in transmission (corresponding to classical bright-field transmission microscopy). All the OTFs are rotationally invariant about the kz axis.

Equations (36)

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E(o,i)=G(o,i)+k02WG(o,r)δε(r)E(r,i)d3r,
E(o,i)=G(o,i)+δE(o,i),δE(o,i)k02WG(o,r)δε(r)G(r,i)d3r.
|E(o,i)|2=|G(o,i)|2+|δE(o,i)|2+2R[G*(o,i)δE(o,i)].
D=ΓG*(o,i)δE(o,i)d2o,
D(x)=H(r)δε(x+r)d3r,H(r)=k02G(r,i)ΓG*(o,i)G(o,r)d2o.
Hpoint(r)k02G(r,i)G(o,r).
Hconfocal(r)k02G2(r,i).
ΓR+ΓTG*(o,i)G(o,r)d2o=12ik0[G(r,i)G*(r,i)],
ΓRG*(o,i)G(o,r)d2o=12ik0G(r,i),ΓTG*(o,i)G(o,r)d2o=12ik0G*(r,i).
HR(r)=k02iG2(r,i),
HT(r)=k02i|G|2(r,i).
G(u,v)=i8π2pvu(k)γexp[ik·(uv)]d2k,
|δE(o,i)|2=k04W×WF(o,r1,r2)δε(r1)δε*(r2)F(i,r1,r2)dr13dr23,
2E(r,i)+k02εm(r)E(r,i)=δ(ri)k02δε(r)E(r,i),
r2G(r,r)+k02εm(r)G(r,r)=δ(rr),
E(o,i)=G(o,i)+k02WG(o,r)δε(r)E(r,i)d3r.
VolU2  VV2Ud3r=S(UVVU)·nd2r,
E(o,i)=E(i,o).
SG*(r,i)G(r,r)d2r=12ik0[G(r,i)G*(r,i)].
××E(r,i)k02ε¯m(r)E(r,i)=piδ(ri)+k02δε¯(r)E(r,i),
××G¯(r,r)k02ε¯m(r)G¯(r,r)=δ(rr)I¯,
E(o,i)=G¯(o,i)pi+k02WG¯(o,i)δε¯(r)E(r,i)d3r,
δE(o,i)k02WG¯(o,r)δε¯(r)G¯(r,i)pid3r.
VolU·××VV·××Ud3r,
=S(V××UU××V)·nd2r,
M¯V·U=M¯tU·V
E(o,i)·po=E(i,o)·pi.
SG¯(r,r)G¯*(r,i)pi*d2r=12ik0[G¯(r,i)pi*G¯*(r,i)pi*].
D=ΓG¯*(o,i)pi*·k02WG¯(o,r)δε¯(r)G¯(r,i)pid3rd2o,
D=k02Wδε¯(r)G¯(r,i)pi·ΓG¯(r,o)G¯*(o,i)pi*d2od3r.
DT=k02iWδε¯(r)G¯(r,i)pi·G¯*(r,i)pi*d3r,
DR=k02iWδε¯(r)G¯(r,i)pi·G¯(r,i)pi*d3r.
HT(r)=k02i|G¯(r,i)pi|2,
HR(r)=k02iG¯(r,i)pi*·G¯(r,i)pi.
Dpoint=k02G¯*(o,i)pi*·WG¯(o,r)δε¯(r)G¯(r,i)pid3r.
Hconfocalk02G¯(r,i)pi*·G¯(r,i)pi.

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