Abstract

We address the performance of transmission geometry volume holograms as depth-selective imaging elements. We consider two simple implementations using holograms recorded with spherical and plane beams. We derive the point-spread function (PSF) of these systems using volume diffraction theory and use the PSF to estimate depth resolution. Furthermore, we show that appropriately designed objective optics can significantly improve the depth resolution or the working distance of plane-wave reference holographic imaging systems. These results are confirmed experimentally and demonstrated for objects with millimeter axial features, imaged from the 5- to 50-cm range.

© 2004 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2002 (4)

2001 (1)

G. Barbastathis, A. Sinha, “Information content of volume holographic imaging,” Trends Biotechnol. 19, 383–392 (2001).
[CrossRef] [PubMed]

1999 (2)

G. Barbastathis, D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

G. Barbastathis, M. Balberg, D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. 24, 811–813 (1999).
[CrossRef]

1998 (1)

1996 (2)

1995 (1)

D. Psaltis, F. Mok, “Holographic memories,” Sci. Am. 273, 70–76 (1995).
[CrossRef]

1994 (4)

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

M. Subbarao, S. Gopal, “Depth from defocus: a spatial domain approach,” Intl. J. Comput. Vision 13, 271–294 (1994).
[CrossRef]

D. Psaltis, F. Mok, H. Y.-S. Li, “Nonvolatile storage in photorefractive crystals,” Opt. Lett. 19, 210–212 (1994).
[CrossRef] [PubMed]

E. R. Dowski, W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1994).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1990 (1)

D. Brady, S. Lin, X. G. Gu, D. Psaltis, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[CrossRef]

1989 (1)

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

1987 (1)

H. Fisk-Johnson, “An improved method for computing a discrete Hankel transform,” Comput. Phys. Commun. 43, 181–202 (1987).
[CrossRef]

1986 (1)

G. Binnig, C. F. Quate, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

1983 (1)

G. Binnig, H. Rohrer, C. Gerber, E. Weibel, “7 × 7 reconstruction on Si(111) resolved in real space,” Phys. Rev. Lett. 50, 120–123 (1983).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1966 (1)

1963 (1)

1962 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–779 (1948).
[CrossRef]

Balberg, M.

Barbastathis, G.

Bashaw, M. C.

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

Binnig, G.

G. Binnig, C. F. Quate, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

G. Binnig, H. Rohrer, C. Gerber, E. Weibel, “7 × 7 reconstruction on Si(111) resolved in real space,” Phys. Rev. Lett. 50, 120–123 (1983).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, Cambridge, U.K., 1998).

Brady, D.

D. Brady, S. Lin, X. G. Gu, D. Psaltis, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[CrossRef]

Brady, D. J.

G. Barbastathis, D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

G. Barbastathis, M. Balberg, D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. 24, 811–813 (1999).
[CrossRef]

Cathey, W. T.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, San Diego, Calif., 1990), Chap. 4, pp. 143–169.

Dowski, E. R.

Fisk-Johnson, H.

H. Fisk-Johnson, “An improved method for computing a discrete Hankel transform,” Comput. Phys. Commun. 43, 181–202 (1987).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Furtak, T. E.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–779 (1948).
[CrossRef]

Gerber, C.

G. Binnig, C. F. Quate, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

G. Binnig, H. Rohrer, C. Gerber, E. Weibel, “7 × 7 reconstruction on Si(111) resolved in real space,” Phys. Rev. Lett. 50, 120–123 (1983).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gopal, S.

M. Subbarao, S. Gopal, “Depth from defocus: a spatial domain approach,” Intl. J. Comput. Vision 13, 271–294 (1994).
[CrossRef]

Grangeat, P.

P. Grangeat, “Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform,” in Lecture Notes in Mathematics, Vol. 1497 of Mathematical Methods in Tomography, G. T. Herman, A. K. Louis, F. Natterer, eds. (Springer-Verlag, Berlin, 1990).

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gu, X. G.

D. Brady, S. Lin, X. G. Gu, D. Psaltis, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[CrossRef]

Gu, X.-G.

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

Heanue, J. F.

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

Hee, M. R.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hesselink, L.

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Katz, J.

J. Zhang, B. Tao, J. Katz, “Three-dimensional velocity measurements using hybrid HPIV,” in Developments in Laser Techniques and Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer, Berlin, Germany, 1997).

Klein, M. V.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kozma, A.

Lee, H.

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

Leith, E.

Leith, E. N.

Levene, M.

Li, H. Y.-S.

Li, W.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lin, S.

D. Brady, S. Lin, X. G. Gu, D. Psaltis, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[CrossRef]

Liu, W.

Marks, J.

Marr, D.

D. Marr, Vision (Freeman, New York, 1982).

Massey, N.

Milgram, J. H.

Minsky, M.

M. Minsky, “Microscopy apparatus,” U.S. patent3,013,467 (19December1961).

Mok, F.

Psaltis, D.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Quate, C. F.

G. Binnig, C. F. Quate, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

Rohrer, H.

G. Binnig, H. Rohrer, C. Gerber, E. Weibel, “7 × 7 reconstruction on Si(111) resolved in real space,” Phys. Rev. Lett. 50, 120–123 (1983).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, San Diego, Calif., 1990), Chap. 4, pp. 143–169.

Shih, T.

A. Sinha, W. Sun, T. Shih, G. Barbastathis, “N-ocular holographic 3d imaging,” in Proceedings of the OSA Annual Meeting (Optical Society of America, Washington, D.C., 2002), paper WD7.

Sinha, A.

A. Sinha, G. Barbastathis, “Volume holographic telescope,” Opt. Lett. 27, 1690–1692 (2002).
[CrossRef]

G. Barbastathis, A. Sinha, “Information content of volume holographic imaging,” Trends Biotechnol. 19, 383–392 (2001).
[CrossRef] [PubMed]

A. Sinha, W. Sun, T. Shih, G. Barbastathis, “N-ocular holographic 3d imaging,” in Proceedings of the OSA Annual Meeting (Optical Society of America, Washington, D.C., 2002), paper WD7.

Stein, A.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Subbarao, M.

M. Subbarao, S. Gopal, “Depth from defocus: a spatial domain approach,” Intl. J. Comput. Vision 13, 271–294 (1994).
[CrossRef]

Sun, W.

A. Sinha, W. Sun, T. Shih, G. Barbastathis, “N-ocular holographic 3d imaging,” in Proceedings of the OSA Annual Meeting (Optical Society of America, Washington, D.C., 2002), paper WD7.

Swanson, E. 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tao, B.

J. Zhang, B. Tao, J. Katz, “Three-dimensional velocity measurements using hybrid HPIV,” in Developments in Laser Techniques and Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer, Berlin, Germany, 1997).

Upatnieks, J.

van Heerden, P. J.

Weibel, E.

G. Binnig, H. Rohrer, C. Gerber, E. Weibel, “7 × 7 reconstruction on Si(111) resolved in real space,” Phys. Rev. Lett. 50, 120–123 (1983).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, Cambridge, U.K., 1998).

Yamaguchi, I.

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

Zhang, J.

J. Zhang, B. Tao, J. Katz, “Three-dimensional velocity measurements using hybrid HPIV,” in Developments in Laser Techniques and Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer, Berlin, Germany, 1997).

Zhang, T.

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Comput. Phys. Commun. (1)

H. Fisk-Johnson, “An improved method for computing a discrete Hankel transform,” Comput. Phys. Commun. 43, 181–202 (1987).
[CrossRef]

Intl. J. Comput. Vision (1)

M. Subbarao, S. Gopal, “Depth from defocus: a spatial domain approach,” Intl. J. Comput. Vision 13, 271–294 (1994).
[CrossRef]

J. Appl. Phys. (1)

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (London) (2)

D. Brady, S. Lin, X. G. Gu, D. Psaltis, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[CrossRef]

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–779 (1948).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. Lett. (2)

G. Binnig, C. F. Quate, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

G. Binnig, H. Rohrer, C. Gerber, E. Weibel, “7 × 7 reconstruction on Si(111) resolved in real space,” Phys. Rev. Lett. 50, 120–123 (1983).
[CrossRef]

Proc. IEEE (1)

G. Barbastathis, D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098–2120 (1999).
[CrossRef]

Sci. Am. (1)

D. Psaltis, F. Mok, “Holographic memories,” Sci. Am. 273, 70–76 (1995).
[CrossRef]

Science (2)

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Trends Biotechnol. (1)

G. Barbastathis, A. Sinha, “Information content of volume holographic imaging,” Trends Biotechnol. 19, 383–392 (2001).
[CrossRef] [PubMed]

Other (12)

T. Wilson, ed., Confocal Microscopy (Academic, San Diego, Calif., 1990).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

P. Grangeat, “Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform,” in Lecture Notes in Mathematics, Vol. 1497 of Mathematical Methods in Tomography, G. T. Herman, A. K. Louis, F. Natterer, eds. (Springer-Verlag, Berlin, 1990).

D. Marr, Vision (Freeman, New York, 1982).

M. Minsky, “Microscopy apparatus,” U.S. patent3,013,467 (19December1961).

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, San Diego, Calif., 1990), Chap. 4, pp. 143–169.

J. Zhang, B. Tao, J. Katz, “Three-dimensional velocity measurements using hybrid HPIV,” in Developments in Laser Techniques and Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer, Berlin, Germany, 1997).

A. Sinha, W. Sun, T. Shih, G. Barbastathis, “N-ocular holographic 3d imaging,” in Proceedings of the OSA Annual Meeting (Optical Society of America, Washington, D.C., 2002), paper WD7.

H. Coufal, D. Psaltis, G. Sincerbox, eds., Holographic Data Storage (Springer, New York, 2000).
[CrossRef]

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, Cambridge, U.K., 1998).

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Figures (29)

Fig. 1
Fig. 1

(a) Two-dimensional imaging system cannot discriminate the distance between two objects; (b) a 3-D imaging system can recover depth information by scanning; (c) a 2½-D imaging system maps the height of a reflective surface to an intensity on the detector.

Fig. 2
Fig. 2

Simplified schematics of traditional holographic imaging: (a) recording and (b) readout by scanning. Schematics of VHI: (c) making VH lens and (d) readout.

Fig. 3
Fig. 3

Bragg selectivity of VHs: (a) recording, (b) Bragg-matched readout with a replica of the reference beam, (c) Bragg-mismatched readout results in a weak diffracted field, (d) a Bragg degenerate beam yields a strong diffracted beam. The Bragg degenerate beam is of a different wavelength and is incident at a different reference angle governed by Bragg selectivity.

Fig. 4
Fig. 4

Schematic of the active VHI system. A CCD camera monitors the diffracted beam to detect whether the light scattered by the object contains any Bragg-matched or Bragg degenerate components.

Fig. 5
Fig. 5

Spherical wave reference VHI: (a) recording and (b) readout (imaging).

Fig. 6
Fig. 6

Intensity pattern observed on the detector for spherical VHI for a (NA) of 0.07, θ s = 12° (0.21 rad), and δ = 6 mm. (a) Diffraction pattern caused by axial defocus, (b) crescent-shaped Bragg filter of a SR hologram, (c) diffracted pattern observed on the detector.

Fig. 7
Fig. 7

VHI is a shift-variant imaging system; the observed diffracted field changes with a change in the spatial location of the point source: (a) mutually incoherent point sources with no defocus, (b) mutually incoherent point sources with defocus δ.

Fig. 8
Fig. 8

VHI methods: (a) 3-D scanning with an integrating detector and focused illumination, (b) exploiting y degeneracy with a CCD and 2-D scanning with extended illumination.

Fig. 9
Fig. 9

Theoretical and experimental plots of longitudinal PSF for SR VHI.

Fig. 10
Fig. 10

Computed contour lines giving the fraction of the total diffracted power that falls within small circles centered at the Bragg-matched point versus selected values of the defocus term. Values corresponding to a defocus equal to the FWHM are indicated by the dotted-dashed lines.

Fig. 11
Fig. 11

Dependence of G SR on θ s .

Fig. 12
Fig. 12

SR VHI systems (a) without objective optics and (b) with objective optics in between the hologram and the object.

Fig. 13
Fig. 13

Depth resolution Δz FWHM of a SR VHI system degrades quadratically with working distance (after Ref. 23).

Fig. 14
Fig. 14

(a) Theoretical and (b) experimental diffracted patterns on a CCD for SR VHI with a point-source displaced δ = 4 mm from the Bragg match.

Fig. 15
Fig. 15

Observed diffracted fields on a CCD for three mutually incoherent points: (a) points at the Bragg-matched plane and (b) points at a defocused plane δ = 4 mm.

Fig. 16
Fig. 16

SR VHI of the fabricated letters MIT. (a) The actual computer-aided design rendering of the object. (b) A volume holographic image of the object obtained by a complete lateral scan with the surface of the letter M placed at a Bragg-matched location, which consequently appears to be bright.

Fig. 17
Fig. 17

Plane-wave reference VHI schematic: (a) recording, (b) Bragg-matched readout, (c) Bragg-mismatched readout.

Fig. 18
Fig. 18

Intensity pattern observed on the detector for spherical VHI for a (NA) of 0.07, θ s = 12° (0.21 rad), and δ = 8 mm. (a) Diffraction pattern caused by axial defocus and the finite aperture of the collimating lens, (b) straight Bragg slit of a PR hologram, (c) diffracted pattern observed on the detector.

Fig. 19
Fig. 19

Longitudinal PSF for PR VHI.

Fig. 20
Fig. 20

VHI when the y degeneracy is exploited by use of a camera at the detector plane. (a) Line scanning of an object with small surface features, (b) line scanning of an object with large surface features.

Fig. 21
Fig. 21

Schematic for the design of an objective optical system to obtain high depth resolution at large working distances in VHI systems.

Fig. 22
Fig. 22

Appropriately designed objective optics can improve the Δz FWHM of a PR VHI system. (a) PR VHI schematic without objective optics, (b) PR VHI schematic with objective optics.

Fig. 23
Fig. 23

Appropriately designed telephoto system can improve working distance d without any degradation in the resolution. PSFs for a stand-alone PR VHI system (dashed curve) and a telephoto PR VHI system (solid curve) show that both systems have Δz FWHM ≈ 1 mm for d = 50 mm and d = 500 mm, respectively.

Fig. 24
Fig. 24

Plots of Δz FWHM versus object distance d for PR VHI (solid lines), SR VHI (dashed line), confocal systems (dotted line).

Fig. 25
Fig. 25

Calculation of the FOV for the telephoto PR VHI system; the chief ray makes an angle α with the optical axis.

Fig. 26
Fig. 26

(a) Theoretical (solid line) and experimental (asterisks) Δz FWHM versus a for fixed f and L, confirming the inversely proportional relationship to the (NA) in this case. (b) Theoretical (solid line) and experimental (asterisks) Δz FWHM versus f for fixed a and L, confirming the quadratic dependence on f.

Fig. 27
Fig. 27

(a) Theoretical and (b) experimental diffracted patterns on a CCD for PR VHI with a point-source displaced δ = 8 mm from a Bragg match.

Fig. 28
Fig. 28

PR VH images of the fabricated letters MIT placed 50.2 mm away from the entrance pupil of the system. (a) a PR VH image of the object obtained by a one-dimensional scan with the surface of the letter M placed at a Bragg-matched location, (b) an image of the object obtained by a one-dimensional scan with the surface of the letter I placed at a Bragg-matched location, (c) an image of the object obtained by a one-dimensional scan with the surface of the letter T placed at a Bragg-matched location.

Fig. 29
Fig. 29

PR VH images by collector optics of the fabricated letters MIT placed 500 mm away from the entrance pupil of the system. (a) A PR VHI image of the object obtained by a one-dimensional scan with the surface of the letter M placed at a Bragg-matched location, (b) an image of the object obtained by a one-dimensional scan with the surface of the letter I placed at a Bragg-matched location, (c) an image of the object obtained by a one-dimensional scan with the surface of the letter T placed at a Bragg-matched location.

Tables (4)

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Table 1 Measured Intensity Values for the Stand-Alone PR VHI (a.u.)

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Table 2 Ratios of Intensity Values Calculated from Table 1

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Table 3 Measured Intensity Values for PR VHI with the Telephoto System (a.u.)

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Table 4 Ratios of Intensity Values Calculated from Table 3

Equations (58)

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Efr=expi2π z-zfλ+iπ x-xf2+y-yf2λz-zf.
Esr=expi2π1-θs22zλ+i2πθsxλ.
Δr  Ef*rEsr,
Edr=η  EprΔrGr-rd3r,
Gr-r=expik|r-r||r-r|expi2π z-zλ+iπ x-x2+y-y2λz-z,
Epr=expi2π z-zpλ+iπ x-xp2+y-yp2λz-zp.
Edx, y=2πR2η-L/2L/2expiπCz×2πAzR2, 2πBzRdz.
Az=1λz-zf-1λz-zp,
Bxz=-xpλz-zp+xfλz-zf-xλF+θsλ,
Byz=-ypλz-zp+yfλz-zf-yλF,
Bz=Bxz2+Byz21/2,
Cz=xp2+yp2λz-zp-xf2+yf2λz-zf+x2+y2λF2-θs2λz.
u, v=01exp-i2 uρ2J0vρρdρ
|Edx, y; xp, yp, zp|2,
Ix, y; δIbθsF, 0; δ=2πR2δλzf2, 2πRλFx-θF2+y21/2sincx2+y2-θF2L2λF22.
NA=Rzf.
Δr=2λFLθs.
Ix, yIb=2πR2δλzf2, 2πRλFx-θsF2+y-Fδyzf21/22 ×sincx2+y2-F2θs2+δy2/zf2L2λF22.
Idδ=- Ix, ydxdy.
ΔzFWHMSR=GSRλzf2R2L,
ΔzFWHMSR=GSRλNA2L.
GSR=18.2θs.
ΔzFWHMSR=GSRλd2R2L.
δ=m12δ.
NAobj=ad.
NA=NAobjm1=Rzf.
ΔzFWHMSR=GSRλNAobj2L=GSRλd2a2L.
R=|m1|a
ΔzFWHMCF=GCFλr2d2.
PIPM=0.23±0.22,PTPM=0.13±0.07
Efr=expi2π zλ.
Esr=expi2π1-θs22zλ+i2πθsxλ.
Δr=expi2πλxθs-z θs22.
zp=ff-δδf2δ
Ãkp, kd=S ν Δr×expikp-kd · rd3r
E˜dkd= E˜pkpÃkp, kddkpxdkpy,
E˜pkpx, kpy=expizpkpx+kpy22|k|.
Edx, y=exp-iπ zpλxF-θs2+yF2sincL sin θsλxF-θs.
Ix, yIb=circx-θsF2+y2Faδ/f21/2×sinc2L sin θsλxF-θs,
Ix, yIb=circx-θsF2+y-Fδy/f21/2Faδ/f2×sinc2L sin θsλxF-θs.
IdI0=1π02πdϕ 01dρρ sinc2aL sin θsδλf2 ρ sin ϕ.
IdI0=1π02πdϕ 01dρρ sinc2aL sin θsδnλf2 ρ sin ϕ.
ΔzFWHM=GPRλf2aL,
GPR=5.34θs.
NAobj=ad,
ΔzFWHMPR=GPREFL2rL.
ΔzFWHMPRrNAobj2L.
ΔzFWHMPR=GPRrd2a2L.
rmin=cλL,
ΔzFWHMPRoptGPRrmind2a2L.
M=M11M12M21M22.
M12=-1EFL,
M11=FFLEFL.
f1=M11tM11-1-M12t,
f2=t1-M11.
ΔzFWHMPRopt=1.5×10-6d2,ΔzFWHMSR=20.45×10-6d2,ΔzFWHMCF=3.50×10-6d2.
FOV=2rEFLb+L.
ΔzFWHM=GPRλf2aL,

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