Abstract

Angularly diverse or partially coherent illumination is widely used for optical, x-ray, and electron microscopy. A long-standing challenge in developing new partially coherent approaches is that the nonlinear image formation model does not allow physical intuition into how the imaging and illumination pupils impact contrast and resolution. We report a phase-space model, the phase-space imaging kernel, for partially coherent systems that describes image formation in terms of a convolution and is analogous to the point spread function model for coherent imaging. We simulate phase-space imaging kernels for brightfield and differential interference contrast (DIC) microscopes to explain a seemingly paradoxical experimental result that the DIC image of a point depends on the coherence of the illumination. We discuss interpretation of the spatial and spatial-frequency marginals of the kernel. We expect this intuitive model and simulations to facilitate design of novel computational schemes for phase imaging and optical lithography.

© 2015 Optical Society of America

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References

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  1. S. B. Mehta and C. J. R. Sheppard, Opt. Lett. 35, 348 (2010).
    [Crossref]
  2. S. B. Mehta and C. J. R. Sheppard, J. Mod. Opt. 57, 718 (2010).
    [Crossref]
  3. H. H. Hopkins, Proc. R. Soc. London 217, 408 (1953).
    [Crossref]
  4. K. Yamazoe, J. Opt. Soc. Am. A 29, 2591 (2012).
    [Crossref]
  5. S. B. Mehta and C. J. R. Sheppard, Appl. Opt. 49, 2954 (2010).
    [Crossref]
  6. S. B. Mehta and R. Oldenbourg, Biomed. Opt. Express 5, 1822 (2014).
    [Crossref]
  7. S. B. Mehta and C. J. R. Sheppard, Opt. Express 16, 19462 (2008).
    [Crossref]
  8. J. Ville, Cables et Transmissions 2A, 61 (1948) [translated from French in 1958 by I. Selin of the RAND Corporation].
  9. C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, Proc. R. Soc. Lond. A 387, 171 (1983).
    [Crossref]
  10. D. K. Hamilton and C. J. R. Sheppard, J. Microsc. 133, 27 (1983).
    [Crossref]
  11. C. J. Cogswell and C. J. R. Sheppard, J. Microsc. 165, 81 (1992).
    [Crossref]

2014 (1)

2012 (1)

2010 (3)

2008 (1)

1992 (1)

C. J. Cogswell and C. J. R. Sheppard, J. Microsc. 165, 81 (1992).
[Crossref]

1983 (2)

C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, Proc. R. Soc. Lond. A 387, 171 (1983).
[Crossref]

D. K. Hamilton and C. J. R. Sheppard, J. Microsc. 133, 27 (1983).
[Crossref]

1953 (1)

H. H. Hopkins, Proc. R. Soc. London 217, 408 (1953).
[Crossref]

1948 (1)

J. Ville, Cables et Transmissions 2A, 61 (1948) [translated from French in 1958 by I. Selin of the RAND Corporation].

Cogswell, C. J.

C. J. Cogswell and C. J. R. Sheppard, J. Microsc. 165, 81 (1992).
[Crossref]

Cox, I. J.

C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, Proc. R. Soc. Lond. A 387, 171 (1983).
[Crossref]

Hamilton, D. K.

D. K. Hamilton and C. J. R. Sheppard, J. Microsc. 133, 27 (1983).
[Crossref]

C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, Proc. R. Soc. Lond. A 387, 171 (1983).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. London 217, 408 (1953).
[Crossref]

Mehta, S. B.

Oldenbourg, R.

Sheppard, C. J. R.

S. B. Mehta and C. J. R. Sheppard, Appl. Opt. 49, 2954 (2010).
[Crossref]

S. B. Mehta and C. J. R. Sheppard, J. Mod. Opt. 57, 718 (2010).
[Crossref]

S. B. Mehta and C. J. R. Sheppard, Opt. Lett. 35, 348 (2010).
[Crossref]

S. B. Mehta and C. J. R. Sheppard, Opt. Express 16, 19462 (2008).
[Crossref]

C. J. Cogswell and C. J. R. Sheppard, J. Microsc. 165, 81 (1992).
[Crossref]

C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, Proc. R. Soc. Lond. A 387, 171 (1983).
[Crossref]

D. K. Hamilton and C. J. R. Sheppard, J. Microsc. 133, 27 (1983).
[Crossref]

Ville, J.

J. Ville, Cables et Transmissions 2A, 61 (1948) [translated from French in 1958 by I. Selin of the RAND Corporation].

Yamazoe, K.

Appl. Opt. (1)

Biomed. Opt. Express (1)

Cables et Transmissions (1)

J. Ville, Cables et Transmissions 2A, 61 (1948) [translated from French in 1958 by I. Selin of the RAND Corporation].

J. Microsc. (2)

D. K. Hamilton and C. J. R. Sheppard, J. Microsc. 133, 27 (1983).
[Crossref]

C. J. Cogswell and C. J. R. Sheppard, J. Microsc. 165, 81 (1992).
[Crossref]

J. Mod. Opt. (1)

S. B. Mehta and C. J. R. Sheppard, J. Mod. Opt. 57, 718 (2010).
[Crossref]

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

Opt. Express (1)

Opt. Lett. (1)

Proc. R. Soc. Lond. A (1)

C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, Proc. R. Soc. Lond. A 387, 171 (1983).
[Crossref]

Proc. R. Soc. London (1)

H. H. Hopkins, Proc. R. Soc. London 217, 408 (1953).
[Crossref]

Supplementary Material (3)

NameDescription
» Visualization 1: AVI (755 KB)      PSI-kernel and marginals for brightfield microscope at variable illumination aperture
» Visualization 2: AVI (885 KB)      PSI-kernel and marginals for DIC microscope with bias of 0° at variable illumination aperture
» Visualization 3: AVI (808 KB)      PSI-kernel and marginals for DIC microscope with bias of 90° at variable illumination aperture

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Figures (4)

Fig. 1.
Fig. 1. Frame at S=0.5 from Visualization 1. (a) The PSI-kernel of a brightfield microscope. The kernel is displayed using a nested coordinate system, the outer coordinates describing the space coordinates and the inner coordinates describing the spatial frequency. (b) The spatial marginal is the sum within inner tiles. (c) The frequency marginal is the sum of inner tiles.
Fig. 2.
Fig. 2. Experimental images of a point under brightfield and DIC microscopes with multiple illumination aperture sizes. These are images of a point defect in an aluminum coating on a coverslip, acquired with 100× 1.3 NA objective at a wavelength of λ=532nm. The shear of the DIC prism was measured [5] to be 2Δ=0.5λ/NA, and bias (2ϕ) was set to either 0° or 90°(180° bias provides brightfield-like contrast). Scale: 200 nm.
Fig. 3.
Fig. 3. Frame at S=0.5 from Visualization 2. (a) The simulated PSI-kernel of the DIC microscope with shear of 0.5λ/NA and bias of 0°, (b) the spatial marginal of the PSI-kernel, or the DIC image of a point, and (c) the frequency marginal of the PSI-kernel.
Fig. 4.
Fig. 4. Frame at S=0.5 from Visualization 3. (a) The simulated PSI-kernel of the DIC microscope with shear of 0.5λ/NA and bias of 90°, (b) the spatial marginal of the PSI-kernel, or the DIC image of a point, and (c) the frequency marginal of the PSI-kernel.

Equations (12)

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

m=12(m1+m2),m=m1m2;x=12(x1+x2),x=x1x2.
Ψ(m,x)=T(m+m2)T*(mm2)C(m,m)×exp(2πim·x)dm,
Ψ(m,x)=Fm1[M(m,m)C(m,m)].
Ψ(m,x)=W(m,x)xK(m,x),
Wi(m,x)=W(m,x)xWh(m,x),
CDIC(m,m)=C(m,m)sin[2π(mxmx2)Δϕ]×sin[2π(mx+mx2)Δϕ]=C(m,m)[cos(2πmxΔ)cos(4πmxΔ2ϕ)],
KDIC(m,x)=K(m,x)x[δ(xΔ)+δ(x+Δ)δ(x)cos(4πmxΔ2ϕ)].
Ψ(m,x)dx=Fm1[M(m,m)C(m,m)]dx.
Ψ(m,x)dx=M(m,0)C(m,0),
t(x)=a(x)exp[i2πm0(x)·x].
W(m,x)=a2(x)δ[mmo(x)],
I(x)=a2(x)C[mo(x),0].

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