Abstract

Detection and evaluation of time- and space-domain fields scattered by one-dimensional objects is considered for applications in superresolved microscopy. Modal decomposition of scattered fields is used to estimate the information passed by band-limited diffraction from an object to the far field. The transinformation is shown to increase with the number of frequencies in the probe field.

© 1995 Optical Society of America

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References

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  4. P. C. Sun, E. N. Leith, “Superresolution by spatialtemporal encoding methods,” Appl. Opt. 31, 4857–4862 (1992).
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  5. J. C. Wyant, K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tools Manuf. 32, 5–10 (1992).
    [CrossRef]
  6. P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
    [CrossRef]
  7. Y. N. Ning, K. T. V. Grattan, A. W. Palmer, “Fibre-optic interferometric systems using low-coherence light sources,” Sensors Actuators A 30, 181–192 (1992).
    [CrossRef]
  8. H.-H. Liu, P.-H. Cheng, J. Wang, “Spatially coherent white-light interferometer based on a point fluorescent source,” Opt. Lett. 18, 678–680 (1993).
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  9. A. Bearden, M. P. O’Neill, L. C. Osborne, T. L. Wong, “Imaging and vibrational analysis with laser-feedback interferometry,” Opt. Lett. 18, 238–240 (1993).
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  10. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, New York, 1991).
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  11. J. Connes, “Recherches sur la spectroscopie par transformation de Fourier,” Rev. Opt. Theor. Instrum. 40, 45–48, 116–140, 171–190, 231–265 (1961).
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    [CrossRef]
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 6.1.
  14. R. J. McEliece, The Theory of Information and Coding (Addison-Wesley, Reading, Mass., 1977), Vol. 3.
  15. S. P. Luttrell, “A new method of sample optimization,” Opt. Acta 32, 255–257 (1985).
    [CrossRef]
  16. D. Blacknell, C. J. Oliver, “Information content of coherent images,” J. Phys. D 26, 1364–1370 (1993).
    [CrossRef]
  17. R. F. Harrington, Field Computation by Moment Methods (Institute of Electrical and Electronics Engineers, New York, 1987).
  18. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
    [CrossRef]
  19. J. E. Freund, R. E. Walpole, Mathematical Statistics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1980), Sec. 13.8.

1993 (4)

1992 (3)

Y. N. Ning, K. T. V. Grattan, A. W. Palmer, “Fibre-optic interferometric systems using low-coherence light sources,” Sensors Actuators A 30, 181–192 (1992).
[CrossRef]

P. C. Sun, E. N. Leith, “Superresolution by spatialtemporal encoding methods,” Appl. Opt. 31, 4857–4862 (1992).
[CrossRef] [PubMed]

J. C. Wyant, K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tools Manuf. 32, 5–10 (1992).
[CrossRef]

1986 (1)

1985 (1)

S. P. Luttrell, “A new method of sample optimization,” Opt. Acta 32, 255–257 (1985).
[CrossRef]

1977 (1)

J. W. Fleming, “Noise levels in broad band Fourier transform absorption spectrometry,” Infrared Phys. 17, 263–269 (1977).
[CrossRef]

1969 (1)

1967 (1)

1965 (1)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

1961 (1)

J. Connes, “Recherches sur la spectroscopie par transformation de Fourier,” Rev. Opt. Theor. Instrum. 40, 45–48, 116–140, 171–190, 231–265 (1961).

Bearden, A.

Blacknell, D.

D. Blacknell, C. J. Oliver, “Information content of coherent images,” J. Phys. D 26, 1364–1370 (1993).
[CrossRef]

Cheng, P.-H.

Connes, J.

J. Connes, “Recherches sur la spectroscopie par transformation de Fourier,” Rev. Opt. Theor. Instrum. 40, 45–48, 116–140, 171–190, 231–265 (1961).

Cox, I. J.

Creath, K.

J. C. Wyant, K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tools Manuf. 32, 5–10 (1992).
[CrossRef]

Di Francia, G. T.

Fleming, J. W.

J. W. Fleming, “Noise levels in broad band Fourier transform absorption spectrometry,” Infrared Phys. 17, 263–269 (1977).
[CrossRef]

Freund, J. E.

J. E. Freund, R. E. Walpole, Mathematical Statistics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1980), Sec. 13.8.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, New York, 1991).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 6.1.

Grattan, K. T. V.

Y. N. Ning, K. T. V. Grattan, A. W. Palmer, “Fibre-optic interferometric systems using low-coherence light sources,” Sensors Actuators A 30, 181–192 (1992).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Institute of Electrical and Electronics Engineers, New York, 1987).

Leith, E. N.

Liu, H.-H.

Lukosz, W.

Luttrell, S. P.

S. P. Luttrell, “A new method of sample optimization,” Opt. Acta 32, 255–257 (1985).
[CrossRef]

McEliece, R. J.

R. J. McEliece, The Theory of Information and Coding (Addison-Wesley, Reading, Mass., 1977), Vol. 3.

Ning, Y. N.

Y. N. Ning, K. T. V. Grattan, A. W. Palmer, “Fibre-optic interferometric systems using low-coherence light sources,” Sensors Actuators A 30, 181–192 (1992).
[CrossRef]

O’Neill, M. P.

Oliver, C. J.

D. Blacknell, C. J. Oliver, “Information content of coherent images,” J. Phys. D 26, 1364–1370 (1993).
[CrossRef]

Osborne, L. C.

Palmer, A. W.

Y. N. Ning, K. T. V. Grattan, A. W. Palmer, “Fibre-optic interferometric systems using low-coherence light sources,” Sensors Actuators A 30, 181–192 (1992).
[CrossRef]

Richmond, J. H.

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

Sandoz, P.

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
[CrossRef]

Sheppard, C. J. R.

Sun, P. C.

Tribillon, G.

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
[CrossRef]

Walpole, R. E.

J. E. Freund, R. E. Walpole, Mathematical Statistics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1980), Sec. 13.8.

Wang, J.

Wong, T. L.

Wyant, J. C.

J. C. Wyant, K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tools Manuf. 32, 5–10 (1992).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

Infrared Phys. (1)

J. W. Fleming, “Noise levels in broad band Fourier transform absorption spectrometry,” Infrared Phys. 17, 263–269 (1977).
[CrossRef]

Int. J. Mach. Tools Manuf. (1)

J. C. Wyant, K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tools Manuf. 32, 5–10 (1992).
[CrossRef]

J. Mod. Opt. (1)

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. D (1)

D. Blacknell, C. J. Oliver, “Information content of coherent images,” J. Phys. D 26, 1364–1370 (1993).
[CrossRef]

Opt. Acta (1)

S. P. Luttrell, “A new method of sample optimization,” Opt. Acta 32, 255–257 (1985).
[CrossRef]

Opt. Lett. (2)

Rev. Opt. Theor. Instrum. (1)

J. Connes, “Recherches sur la spectroscopie par transformation de Fourier,” Rev. Opt. Theor. Instrum. 40, 45–48, 116–140, 171–190, 231–265 (1961).

Sensors Actuators A (1)

Y. N. Ning, K. T. V. Grattan, A. W. Palmer, “Fibre-optic interferometric systems using low-coherence light sources,” Sensors Actuators A 30, 181–192 (1992).
[CrossRef]

Other (5)

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, New York, 1991).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 6.1.

R. J. McEliece, The Theory of Information and Coding (Addison-Wesley, Reading, Mass., 1977), Vol. 3.

R. F. Harrington, Field Computation by Moment Methods (Institute of Electrical and Electronics Engineers, New York, 1987).

J. E. Freund, R. E. Walpole, Mathematical Statistics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1980), Sec. 13.8.

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Figures (11)

Fig. 1
Fig. 1

White-light microscope.

Fig. 2
Fig. 2

Object geometry.

Fig. 3
Fig. 3

Normalized singular values for the signal field, the signal with Gaussian noise added, and the noise by itself.

Fig. 4
Fig. 4

Transinformation versus noise level (relative to the highest singular value).

Fig. 5
Fig. 5

Transinformation versus number of strips for a fixed geometry.

Fig. 6
Fig. 6

Normalized singular values for two frequencies with varying frequency separations for a PEC object. f1 corresponds to the first frequency, and f2 corresponds to the lowest frequency. The corresponding ratios of the two frequencies are given in the legend.

Fig. 7
Fig. 7

Singular values for two frequencies with varying frequency separations for a dielectric object.

Fig. 8
Fig. 8

Singular values for three frequencies for a PEC object.

Fig. 9
Fig. 9

Singular values for five frequencies for a PEC object.

Fig. 10
Fig. 10

Singular values for six frequencies for a dielectric object.

Fig. 11
Fig. 11

Singular values for six frequencies for a 50.0-μm-long PEC object illuminated by a field with a Gaussian envelope.

Tables (2)

Tables Icon

Table 1 Comparison of Normalized Transinformation for Two and Three Frequencies with Various Frequency Separations

Tables Icon

Table 2 Normalized Transinformation for Several Frequencies

Equations (30)

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I ( ρ , τ ) = | ψ ( t τ ) + ψ ( t ) * f ( ρ , t ) | 2 ,
I ( ρ , ω ) = i [ | ψ ( t ) | 2 i + | ψ ( t ) * f ( ρ , t ) | 2 i + 2 ψ ( t τ i ) ψ ( t ) * f ( ρ , t ) ] exp ( j ω τ i ) = i [ | ψ ( t ) | 2 i + | ψ ( t ) * f ( ρ i j , t ) | 2 i ] exp ( j ω τ i ) + 2 | Ψ ( ω ) | 2 F ( ρ , ω ) .
SNR s = 2 π τ β SNR r N s ,
H ( F ) = p ( f ) ln [ p ( f ) ] d f .
s = f + n .
I ( S ; F ) = H ( S ) H ( S | F ) = { p ( s ) ln [ p ( s ) ] d s } { p ( s , f ) ln [ p ( f | s ) ] d f d s } .
I ( S ; F ) = H ( S ) H ( N ) .
p ( f ) = 1 det ( π F c ) exp ( s F c 1 s ) ,
p ( n ) = 1 det ( π N c ) exp ( n N c 1 n ) ,
p ( s ) = 1 det [ π ( F c + N c ) ] exp [ s ( F c + N c ) 1 s ] .
H ( S ) = ln { det [ π ( F c + N c ) ] } + m ,
H ( N ) = ln [ det ( π N c ) ] + m ,
I ( S ; D ) = ln [ det ( F c + N c ) det ( N c ) ] .
F = [ f 1 , f 2 , , f n ] .
F c = F F n 1 .
I ( S ; D ) = ln [ j = 1 m ( σ f j 2 + σ n 2 ) ( σ n 2 ) m ] .
I ( S ; D ) ln [ j = 1 r σ f j 2 ( σ n 2 ) r ] = 2 j = 1 r ln ( σ f j / σ n ) .
E z inc ( ρ ) = exp ( j k l x ) ,
E z inc ( ρ ) η 0 k l 4 S c J z c ( ρ ) H 0 ( 2 ) ( k l | ρ ρ | ) d S = 0 ρ on S c ,
J z c ( ρ ) = j = 1 N c J j p j c ( ρ ) ,
p j c ( ρ ) = { 1 ρ in s j c 0 elsewhere .
V = ZJ ,
V i = E z inc ( ρ i c ) ,
Z i j = { η 0 k l 4 w j H 0 ( 2 ) [ k l ( | ρ i c ρ j c | ) 1 / 2 ] i j η 0 k l 4 w j [ 1 j 2 π ln ( γ k l w j 4 e ) ] i = j ,
E z inc ( ρ ) = j ω ɛ 0 [ ɛ r ( ρ ) 1 ] J z p ( ρ ) η 0 k l 4 × S p J z p ( ρ ) H 0 ( 2 ) ( k l | ρ ρ | ) d S ρ on S p ,
J z p ( ρ ) = j = 1 N J j p j p ( ρ ) ,
p j p ( ρ ) = { 1 ρ in s j p 0 elsewhere .
V i = E z inc ( ρ i p ) ,
Z ij = { η 0 ( π a j ) 1 / 2 4 J 1 ( a j π ) 1 / 2 H 0 ( 2 ) [ k l ( | ρ i p ρ j p | ) 1 / 2 ] i j η 0 ( π a i ) 1 / 2 4 H 1 ( 2 ) [ k l ( a i π ) 1 / 2 ] i = j ,
F ( ω l , ρ k s ) = η 0 k l 4 S q J z q ( ρ ) H 0 ( 2 ) ( k | ρ k r ρ | ) d S q , k = 1 , , n s ,

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