Abstract

Limits on the effective resolution of many optical near-field experiments are investigated. The results are applicable to variants of total-internal-reflection microscopy (TIRM), photon-scanning-tunneling microscopy (PSTM), and near-field-scanning-optical microscopy (NSOM) in which the sample is weakly scattering and the direction of illumination may be controlled. Analytical expressions for the variance of the estimate of the complex susceptibility of an unknown two-dimensional object as a function of spatial frequency are obtained for Gaussian and Poisson noise models, and a model-independent measure is examined. The results are used to explore the transition from near-zone to far-zone detection. It is demonstrated that the information content of the measurements made at a distance of even one wavelength away from the sample is already not much different from the information content of the far field.

© 2004 Optical Society of America

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  1. J.-J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
    [CrossRef]
  2. D. Courjon, K. Sarayeddine, M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  4. E. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Philos. Mag. 6, 356–362 (1928).
  5. E. Ash, G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
    [CrossRef]
  14. P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
    [CrossRef]
  15. P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A, Pure Appl. Opt. 4, S140–S144 (2002).
    [CrossRef]
  16. P. S. Carney, J. C. Schotland, “Theory of total-internal-reflection tomography,” J. Opt. Soc. Am. A 20, 542–547 (2003).
    [CrossRef]
  17. P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).
  18. D. G. Fischer, “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).
    [CrossRef]
  19. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1988), pp. 22–27.
  20. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  21. T. K. Moon, W. C. Sterling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, Upper Saddle River, N.J., 2000), pp. 140–141, 235–237.

2003

2002

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A, Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

2001

2000

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

D. G. Fischer, “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).
[CrossRef]

1997

J.-J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

1995

1992

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

1990

1989

D. Courjon, K. Sarayeddine, M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

1984

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

1981

1972

E. Ash, G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

1964

C. W. McCutchen, “Optical systems for observing surface to-pography by frustrated total internal reflection and interference,” Rev. Sci. Instrum. 35, 1340–1345 (1964).
[CrossRef]

1928

E. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Philos. Mag. 6, 356–362 (1928).

Ash, E.

E. Ash, G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Betzig, E.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Boltasseva, A.

P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Bozhevolnyi, S. I.

P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).

Carminati, R.

J.-J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Carney, P. S.

P. S. Carney, J. C. Schotland, “Theory of total-internal-reflection tomography,” J. Opt. Soc. Am. A 20, 542–547 (2003).
[CrossRef]

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A, Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).

Courjon, D.

Dong, C.

Fischer, D. G.

D. G. Fischer, “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).
[CrossRef]

Frazin, R. A.

P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).

Garcia, N.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1988), pp. 22–27.

Greffet, J.-J.

J.-J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Harootunian, A.

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Isaacson, M.

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Kwon, H.-S.

Leblanc, S.

Lewis, A.

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

McCutchen, C. W.

C. W. McCutchen, “Optical systems for observing surface to-pography by frustrated total internal reflection and interference,” Rev. Sci. Instrum. 35, 1340–1345 (1964).
[CrossRef]

Moon, T. K.

T. K. Moon, W. C. Sterling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, Upper Saddle River, N.J., 2000), pp. 140–141, 235–237.

Muray, A.

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Nicholls, G.

E. Ash, G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

Sarayeddine, K.

Schotland, J. C.

P. S. Carney, J. C. Schotland, “Theory of total-internal-reflection tomography,” J. Opt. Soc. Am. A 20, 542–547 (2003).
[CrossRef]

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A, Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).

So, P. T. C.

Spajer, M.

Sterling, W. C.

T. K. Moon, W. C. Sterling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, Upper Saddle River, N.J., 2000), pp. 140–141, 235–237.

Synge, E.

E. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Philos. Mag. 6, 356–362 (1928).

Temple, P. A.

Trautman, J. K.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Vigoureaux, J.-M.

Volkov, V. S.

P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Appl. Opt.

Appl. Phys. Lett.

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

J. Mod. Opt.

D. G. Fischer, “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A, Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Nature

E. Ash, G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Opt. Commun.

D. Courjon, K. Sarayeddine, M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Opt. Lett.

Philos. Mag.

E. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Philos. Mag. 6, 356–362 (1928).

Prog. Surf. Sci.

J.-J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Rev. Sci. Instrum.

C. W. McCutchen, “Optical systems for observing surface to-pography by frustrated total internal reflection and interference,” Rev. Sci. Instrum. 35, 1340–1345 (1964).
[CrossRef]

Science

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Ultramicroscopy

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

P. S. Carney, R. A. Frazin, S. I. Bozhevolnyi, V. S. Volkov, A. Boltasseva, J. C. Schotland, “A computational lens for the near-field,” Phys. Rev. Lett. (to be published).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1988), pp. 22–27.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

T. K. Moon, W. C. Sterling, Mathematical Methods and Algorithms for Signal Processing (Prentice Hall, Upper Saddle River, N.J., 2000), pp. 140–141, 235–237.

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Figures (3)

Fig. 1
Fig. 1

Base-10 logarithm of normalized, PSTM, susceptibility, Fourier space variance for detection heights of (0.03, 0.1, 0.3, 1.0, 10.0)λ as labeled above each panel. The coordinates are spatial frequency in units of k0=2π/λ. The data are taken to consist of two scans with incident wave vectors of ±2k0xˆ, where is a unit vector. Results of illumination by TM modes are shown on the left and by TE on the right. The plots are normalized so that the minimum variance of any point is 1.0. The linear gray scale runs from 0 to 4 and is shown at the bottom of the figure. Note that the range varies from panel to panel. Values of the normalized variance greater than 104 were set equal to 104 for clarity in display. It can be seen that at zd=1.0λ, the effective information content is already similar to that of the far-field limit, in which only the homogeneous modes are detected.

Fig. 2
Fig. 2

Same as Fig. 1, except that the data are assumed to consist of two scans with incident wave vectors of -2k0xˆ and 2k0yˆ, where ŷ is a unit vector.

Fig. 3
Fig. 3

Logarithm of the squared l2 norm of the system matrix inverse for the cases of measurements made in the two closer planes. The top four plots show results for the case of counterpropagating incident evanescent waves, and the bottom four plots show results for the case of the same orthogonal incident wave vectors discussed in the caption for Fig. 2. Coordinates are again spatial frequency in units of k0. The linear gray scale running from 0 to 4 is shown at the bottom of the figure.

Equations (50)

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

××E(r)-n2(z)k02E(r)=4πk02η(r)E(r),
η(r)=η(ρ),for0z<Δz,
=0,forzΔz.
Ei(r)=eiexp[ik(qi)r],
k(q)=[q, kz(q)],
kz(q)=(k02-|q|2)1/2.
Eαs(r)=k02d3rGαβ(r, r)eβiexp[iqi  ρ+ikz(qi)z]η(r),
Er(r)=erexp[ik(qr)r].
IT(ρ, zd)=EαrEαr*+EαsEαs*+EαsEαr*+EαrEαs*,
I(ρ, zd)EαsEαr*+EαrEαs*.
I˜(Q, zd)12πd2ρI(ρ, zd)exp[iQ  ρ].
I˜(Q, zd)=H(Q, zd)η˜(Q+qi-qr)+H*(-Q, zd)η˜*(-Q+qi-qr),
H(Q, zd)=ik022π eβr*eαiΓαβ(qr-Q)kz(qr-Q)×exp{i[kz(qr-Q)-kz*(qr)]zd}.
βn=2πL n,0nM-12=2πL [n-M],M-12+1nM-1.
(c˜1)mn(c˜1)¯mn=WH1(Qmn)ηˆ(Qmn+qi1-qr1)η˜(Qmn+qi1-qr1)+WH1*(QM-m,M-n)×η^*(QM-m,M-n+qi1-qr1)η˜*(QM-m,M-n+qi1-qr1),
(c˜2)p+m,q+n(c˜2)¯p+m,q+n=WH2(Qp+m,q+n)×ηˆ(Qp+m,q+n+qi2-qr2)η˜(Qp+m,q+n+qi2-qr2)+WH2*(Qp+M-m,q+M-n)×η^*(Qp+M-m,q+M-n+qi2-qr2)η˜*(Qp+M-m,q+M-n+qi2-qr2),
c1 c1¯=WH1+η^+η˜++WH1-*η^-*η˜-*,
c2 c2¯=WH2+η^+η˜++WH2-*η^-*η˜-*.
PB(b)=1(2πσ2)M2exp-12σ2n=0M-1m=0M-1bmn2.
bmn=k=0M-1l=0M-1b˜klexp-i2πkmM+lnM.
m=0M-1exp-i2π mM (k+p)=Mδ[k-(M-p)],
ln PB˜(b˜)=-M22σ2n=0M-1m=0M-1b˜mnb˜M-m,M-n+const.
ln PB˜(b˜)=-M22σ2b˜00b˜00*+2m=1(M-1)/2(b˜0mb˜0m*+b˜m0b˜m0*)+2n=1M-1m=1M-1/2b˜mnb˜mn*+const.
PBB(b1, b2)=PB(b1)PB(b2),
PB˜B˜(b˜1, b˜2)=PB˜(b˜1)PB˜(b˜2).
ln Pmn(η^-, η^+*, η^-*, η^+)
=-W2M2σ2 [|H1+(η^+-η˜+)+H1-*(η^-*-η˜-*)|2
+|H2+(η^+-η˜+)+H2-*(η^-*-η˜-*)|2]+const.
 
(Cov)-1=W2M2σ200ϱ00*ςϱ*00ς00,
(Cov)=σ2M2W2(ϱς-*)00ς-00-*ϱς-*00-ϱ00.
|η^+-η˜+|2¯=ϱσ2W2M2(ϱς-*).
c1¯c1*¯-c1¯c1*¯c1¯c2*¯-c1¯c2*¯c2¯c1*¯-c2¯c1*¯c2¯c2*¯-c2¯c2*¯
=M-4m=0M-1n=0M-1N1,mn¯00M-4m=0M-1n=0M-1N2,mn¯.
η^+η^+*¯-η˜+η˜+*=1W2(ϱς-*)|H2-|2M4m=0M-1n=0M-1N1,mn¯+|H1-|2M4m=0M-1n=0M-1N2,mn¯.
η^+η^+*¯-η˜+η˜+*=ρN0W2M2(ϱς-*).
A=H1+H1-*H2+H2-*,
ηˆ=η^+-η˜+η^-*-η˜-*,
ΔηˆA-1Δb˜.
Γ(q)
=1|q|2qx2hxx+qy2hyyqxqy(hxx-hyy)|q|qxhxzqxqy(hxx-hyy)qy2hxx+qx2hyy|q|qyhxz|q|qxhzx|q|qyhzx|q|2hzz,
hxx=kz2(q)k02 (θ1+R2θ2),
hxz=-|q|kz(q)k02 (θ1-R2θ2),
hyy=(θ1+R1θ2),
hzx=-|q|kz(q)k02 (θ1+R2θ2),
hzz=|q|2k02 (θ1-R2θ2),
R1(q)=kz(q)-kz(q)kz(q)+kz(q),
R2(q)=kz(q)-nkz(q)kz(q)+nkz(q),
θ1(q)=i 1-exp{i[kz(qi)-kz(q)]Δz}kz(qi)-kz(q),
θ2(q)=i 1-exp{i[kz(qi)+kz(q)]Δz}kz(qi)+kz(q).

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