Abstract

The use of an optical waveguide to attain a numerical aperture of unity in computational coherent optical imaging applications is described. It is shown that for the case of a one-dimensional (slitlike) object radiating into an optical waveguide consisting of two plane-parallel mirrors the complex field amplitude across any cross section of the waveguide contains sufficient information to reconstruct the object’s transmittance function with a numerical aperture of unity. We include the derivation of an inversion algorithm for performing the object reconstruction as well as computer simulations of the procedure.

© 2005 Optical Society of America

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References

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  1. E. Abbé, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. f. Mikroskopiscke Anat. 9, 413–668 (1873).
    [CrossRef]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge UK, 1999).
    [CrossRef]
  4. E. G. Williams, J. D. Maynard, “Holographic imaging without the wavelength resolution limit,” Phys. Rev. Lett. 45, 554–557 (1980).
    [CrossRef]
  5. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  6. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
    [CrossRef]
  7. J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  8. A. J. Devaney, M. H. Maleki, “Phase retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
    [CrossRef]
  9. I. Yamaguchi, J. Kato, S. Ohta, J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6186 (2001).
    [CrossRef]
  10. P. Guo, A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
    [CrossRef] [PubMed]
  11. A. J. Devaney, P. Guo, “Digital holographic microscopy,” in Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics, T. P. Jannson, ed. SPIE press monograph, ISBN 0-8194-5441-9 (SPIE Press, Bellingham, Wash., 2004), pp. 179–200 .
  12. A. A. Egorov, “Theory of waveguide optical microscopy,” Laser Phys. 8, 536–540 (1998).
  13. J. C. Wendoloski, “The reconstruction of a spatially incoherent two-dimensional source in an acoustically rigid rectangular cavity,” J. Acoust. Soc. Am. 107, 51–69 (2000).
    [CrossRef] [PubMed]
  14. G. A. Athanassoulis, “Three-dimensional acoustic scattering from a penetrable layered cylindrical obstacle in a horizontally stratified ocean waveguide,” J. Acoust. Soc. Am. 107, 2406–2417 (2000).
    [CrossRef] [PubMed]
  15. P. Roux, M. Fink, “Time reversal in a waveguide: study of the temporal and spatial focusing,” J. Acoust. Soc. Am. 107, 2418–2429 (2000).
    [CrossRef] [PubMed]
  16. J. A. Fawcett, “A method of images for a penetrable acoustic waveguide,” J. Acoust. Soc. Am. 113, 194–204 (2003).
    [CrossRef] [PubMed]
  17. R. M. Shubair, Y. L. Chow, “A simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half spaces,” IEEE Trans. Antennas Propag. 41, 806–812 (1993)
    [CrossRef]
  18. A. D. Puckett, M. L. Peterson, “A time-reversal mirror in a solid circular waveguide using a single, time-reversal element,” ARLO 4, 31–36 (2003).
    [CrossRef]
  19. M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [CrossRef]

2004 (1)

2003 (2)

J. A. Fawcett, “A method of images for a penetrable acoustic waveguide,” J. Acoust. Soc. Am. 113, 194–204 (2003).
[CrossRef] [PubMed]

A. D. Puckett, M. L. Peterson, “A time-reversal mirror in a solid circular waveguide using a single, time-reversal element,” ARLO 4, 31–36 (2003).
[CrossRef]

2001 (1)

2000 (3)

J. C. Wendoloski, “The reconstruction of a spatially incoherent two-dimensional source in an acoustically rigid rectangular cavity,” J. Acoust. Soc. Am. 107, 51–69 (2000).
[CrossRef] [PubMed]

G. A. Athanassoulis, “Three-dimensional acoustic scattering from a penetrable layered cylindrical obstacle in a horizontally stratified ocean waveguide,” J. Acoust. Soc. Am. 107, 2406–2417 (2000).
[CrossRef] [PubMed]

P. Roux, M. Fink, “Time reversal in a waveguide: study of the temporal and spatial focusing,” J. Acoust. Soc. Am. 107, 2418–2429 (2000).
[CrossRef] [PubMed]

1998 (1)

A. A. Egorov, “Theory of waveguide optical microscopy,” Laser Phys. 8, 536–540 (1998).

1993 (2)

A. J. Devaney, M. H. Maleki, “Phase retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
[CrossRef]

R. M. Shubair, Y. L. Chow, “A simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half spaces,” IEEE Trans. Antennas Propag. 41, 806–812 (1993)
[CrossRef]

1992 (1)

1982 (1)

1980 (1)

E. G. Williams, J. D. Maynard, “Holographic imaging without the wavelength resolution limit,” Phys. Rev. Lett. 45, 554–557 (1980).
[CrossRef]

1976 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1873 (1)

E. Abbé, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. f. Mikroskopiscke Anat. 9, 413–668 (1873).
[CrossRef]

Abbé, E.

E. Abbé, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. f. Mikroskopiscke Anat. 9, 413–668 (1873).
[CrossRef]

Athanassoulis, G. A.

G. A. Athanassoulis, “Three-dimensional acoustic scattering from a penetrable layered cylindrical obstacle in a horizontally stratified ocean waveguide,” J. Acoust. Soc. Am. 107, 2406–2417 (2000).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge UK, 1999).
[CrossRef]

Chow, Y. L.

R. M. Shubair, Y. L. Chow, “A simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half spaces,” IEEE Trans. Antennas Propag. 41, 806–812 (1993)
[CrossRef]

Devaney, A. J.

Egorov, A. A.

A. A. Egorov, “Theory of waveguide optical microscopy,” Laser Phys. 8, 536–540 (1998).

Fawcett, J. A.

J. A. Fawcett, “A method of images for a penetrable acoustic waveguide,” J. Acoust. Soc. Am. 113, 194–204 (2003).
[CrossRef] [PubMed]

Fienup, J. R.

Fink, M.

P. Roux, M. Fink, “Time reversal in a waveguide: study of the temporal and spatial focusing,” J. Acoust. Soc. Am. 107, 2418–2429 (2000).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Guo, P.

P. Guo, A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
[CrossRef] [PubMed]

A. J. Devaney, P. Guo, “Digital holographic microscopy,” in Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics, T. P. Jannson, ed. SPIE press monograph, ISBN 0-8194-5441-9 (SPIE Press, Bellingham, Wash., 2004), pp. 179–200 .

Kato, J.

Maleki, M. H.

Maynard, J. D.

E. G. Williams, J. D. Maynard, “Holographic imaging without the wavelength resolution limit,” Phys. Rev. Lett. 45, 554–557 (1980).
[CrossRef]

Mizuno, J.

Ohta, S.

Peterson, M. L.

A. D. Puckett, M. L. Peterson, “A time-reversal mirror in a solid circular waveguide using a single, time-reversal element,” ARLO 4, 31–36 (2003).
[CrossRef]

Puckett, A. D.

A. D. Puckett, M. L. Peterson, “A time-reversal mirror in a solid circular waveguide using a single, time-reversal element,” ARLO 4, 31–36 (2003).
[CrossRef]

Roux, P.

P. Roux, M. Fink, “Time reversal in a waveguide: study of the temporal and spatial focusing,” J. Acoust. Soc. Am. 107, 2418–2429 (2000).
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Schatzberg, A.

Shubair, R. M.

R. M. Shubair, Y. L. Chow, “A simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half spaces,” IEEE Trans. Antennas Propag. 41, 806–812 (1993)
[CrossRef]

Wendoloski, J. C.

J. C. Wendoloski, “The reconstruction of a spatially incoherent two-dimensional source in an acoustically rigid rectangular cavity,” J. Acoust. Soc. Am. 107, 51–69 (2000).
[CrossRef] [PubMed]

Williams, E. G.

E. G. Williams, J. D. Maynard, “Holographic imaging without the wavelength resolution limit,” Phys. Rev. Lett. 45, 554–557 (1980).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge UK, 1999).
[CrossRef]

Yamaguchi, I.

Appl. Opt. (2)

Arch. f. Mikroskopiscke Anat. (1)

E. Abbé, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. f. Mikroskopiscke Anat. 9, 413–668 (1873).
[CrossRef]

ARLO (1)

A. D. Puckett, M. L. Peterson, “A time-reversal mirror in a solid circular waveguide using a single, time-reversal element,” ARLO 4, 31–36 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. M. Shubair, Y. L. Chow, “A simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half spaces,” IEEE Trans. Antennas Propag. 41, 806–812 (1993)
[CrossRef]

J. Acoust. Soc. Am. (4)

J. C. Wendoloski, “The reconstruction of a spatially incoherent two-dimensional source in an acoustically rigid rectangular cavity,” J. Acoust. Soc. Am. 107, 51–69 (2000).
[CrossRef] [PubMed]

G. A. Athanassoulis, “Three-dimensional acoustic scattering from a penetrable layered cylindrical obstacle in a horizontally stratified ocean waveguide,” J. Acoust. Soc. Am. 107, 2406–2417 (2000).
[CrossRef] [PubMed]

P. Roux, M. Fink, “Time reversal in a waveguide: study of the temporal and spatial focusing,” J. Acoust. Soc. Am. 107, 2418–2429 (2000).
[CrossRef] [PubMed]

J. A. Fawcett, “A method of images for a penetrable acoustic waveguide,” J. Acoust. Soc. Am. 113, 194–204 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Laser Phys. (1)

A. A. Egorov, “Theory of waveguide optical microscopy,” Laser Phys. 8, 536–540 (1998).

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Phys. Rev. Lett. (1)

E. G. Williams, J. D. Maynard, “Holographic imaging without the wavelength resolution limit,” Phys. Rev. Lett. 45, 554–557 (1980).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge UK, 1999).
[CrossRef]

A. J. Devaney, P. Guo, “Digital holographic microscopy,” in Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics, T. P. Jannson, ed. SPIE press monograph, ISBN 0-8194-5441-9 (SPIE Press, Bellingham, Wash., 2004), pp. 179–200 .

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Figures (11)

Fig. 1
Fig. 1

Abbé’s theory of image formation in an optical microscope.

Fig. 2
Fig. 2

Basic diffraction experiment. In this illustration the transparency function τ ( x ) is a constant corresponding to an infinite slit with width 2 a .

Fig. 3
Fig. 3

Equivalent basic diffraction experiment. The aperture with transparency function τ ( x ) is extended periodically along the x axis with period L. If the period is sufficiently large, the spillover in the diffraction field from adjacent periods will be small and the results of this experiment will be close to those obtained in the basic experiment illustrated in Fig. 2.

Fig. 4
Fig. 4

Waveguide experiment. A waveguide consisting of two parallel mirrors located at x = ± a 0 is employed to capture all homogeneous plane waves diffracted by the transparency function τ ( x ) . The two mirrors require that the field vanish z at x = ± a 0 .

Fig. 5
Fig. 5

Equivalent waveguide experiment. The vanishing of the field at the mirror locations in the experiment illustrated in Fig. 4 results in the mathematically equivalent free-space diffraction experiment illustrated here. In this experiment the transparency function τ ( x ) is surrounded by its mirror image functions τ ( ± 2 a 0 x ) and the triplet is periodically extended along the entire x axis. We have included the mirrors at x = ± a 0 in the figure for clarity. They are not present in the actual equivalent experiment.

Fig. 6
Fig. 6

Magnitude of the diffracted field of a 50-μm slit in free space at a measurement distance of z = 45 mm . The thick horizontal line denotes the extent ( 2 a 0 = 5.2 mm ) of the detector array.

Fig. 7
Fig. 7

Reconstruction of the transparency function of the slit from the limited aperture ( 2 a 0 = 5.2 mm ) measurement of the diffracted field in free space.

Fig. 8
Fig. 8

Magnitude of the diffracted field of a 50-μm slit in a waveguide consisting of two plane-parallel mirrors separated by 2 a 0 = 5.2 mm . The field distribution is for a measurement distance of z = 45 mm , and the horizontal ( x ) axis covers the full extent of the finite detector array assumed to be 5.2 mm in extent.

Fig. 9
Fig. 9

Magnitude of the diffracted field in free space resulting from the transparency function whose amplitude and phase are shown by the solid curves in Fig. 10. The thick horizontal line denotes the extent of the detector array.

Fig. 10
Fig. 10

Reconstruction of the magnitude and phase of the transparency function from the limited aperture ( 2 a 0 = 5.2 mm ) diffracted-field data in free space. The solid curves represent the exact transparency function, and the dotted curves represent the reconstruction.

Fig. 11
Fig. 11

Magnitude of the diffracted field within the waveguide at z = 45 mm .

Equations (23)

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ψ ( r ) = 1 ( 2 π ) 2 d 2 K ρ ψ ̃ 0 ( K ρ ) exp [ i ( K ρ ρ + γ z ) ] ,
ψ ̃ 0 ( K ρ ) = d 2 ρ ψ 0 ( ρ ) exp ( i K ρ ρ )
γ = { k 2 K ρ 2 if K ρ k i K ρ 2 k 2 if K ρ > k } ,
ψ ̂ 0 ( ρ ) = 1 ( 2 π ) 2 K ρ < k sin θ d 2 K ρ ψ ̃ 0 ( K ρ ) exp ( i K ρ ρ ) ,
ψ ( x , z ) = n = A n exp [ i ( K n x + γ n z ) ] ,
γ n = { k 2 K n 2 if K n k i K n 2 k 2 if K n > k } ,
A n = 1 L L 2 L 2 d x τ ( x ) exp ( i K n x ) .
z f = ( 4 π a 4 λ ) 1 3 .
τ ̂ ( x ) = n = N h N h A n exp ( i K n x ) ,
K n max k a 0 z n max L a 0 λ z = a 0 z N h ,
τ ̂ ( x ) = n = ( a 0 z ) N h ( a 0 z ) N h A n exp ( i K n x ) .
τ ̂ ( x ) = τ ( x ) τ ( x 2 a 0 ) τ ( x + 2 a 0 ) , 2 a 0 < x < 2 a 0 .
ψ ̂ ( x , z ) = n = N h N h A ̂ n exp [ i ( K n x + γ n z ) ] ,
A ̂ n = 1 L L 2 L 2 d x τ ̂ ( x ) exp ( i K n x ) .
ψ ( x , z ) = ψ ̂ ( x , z ) , a 0 < x < a 0 .
K n x = 2 π n L m L N = 2 π N n m ,
ψ ̂ m ( z ) = ψ ̂ ( m δ x , z ) = n = N h N h A ̂ n exp ( i γ n z ) exp ( i 2 π N n m ) ,
N h < m < N h .
A ̂ n = exp ( i γ n z ) 1 N m = N h N h ψ ̂ ( m δ x , z ) exp ( i 2 π N n m ) ,
N h < n < N h .
ψ ̂ ( x , z ) = ψ ( x , z ) ψ ( x 2 a 0 , z ) ψ ( x + 2 a 0 , z ) ,
2 a 0 < x < 2 a 0 .
τ ̂ ( x ) = n = N h N h A ̂ n exp ( i 2 π n L x ) .

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