Abstract

We report 3-D imaging of density in a supersonic expansion using beam-deflection optical tomography. Quantitative high-resolution images with absolute accuracy of 3%, dynamic range of 500:1, and spatial resolution to within a factor of 1.7 of the diffraction limit were produced with a He–Ne laser and simple apparatus. Theory shows that the spatial frequency content of beam-deflection measurements is well suited for tomographic reconstruction. The theory for the diffraction-limited resolution for tomography is presented.

© 1988 Optical Society of America

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References

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  1. G. W. Faris, R. L. Byer, “Beam-Deflection Optical Tomography,” Opt. Lett. 12, 72 (1987).
    [CrossRef] [PubMed]
  2. G. W. Faris, R. L. Byer, “Quantitative Three-Dimensional Optical Tomographic Imaging of Supersonic Flows,” Science 238, 1700 (1987).
    [CrossRef] [PubMed]
  3. G. W. Faris, R. L. Byer, “Beam-Deflection Optical Tomography of a Flame,” Opt. Lett. 12, 155 (1987).
    [CrossRef] [PubMed]
  4. G. W. Faris, R. L. Byer, “Quantitative Optical Tomographic Imaging of a Supersonic Jet,” Opt. Lett. 11, 413 (1986).
    [CrossRef] [PubMed]
  5. K. E. Bennett, G. W. Faris, R. L. Byer, “Experimental Optical Fan Beam Tomography,” Appl. Opt. 23, 2678 (1984).
    [CrossRef] [PubMed]
  6. J. Stricker, “Analysis of 3-D Phase Objects by Moiré Deflectometry,” Appl. Opt. 23, 3657 (1984).
    [CrossRef] [PubMed]
  7. C. Saekeang, P. L. Chu, T. W. Whitbread, “Nondestructive Measurement of Refractive-Index Profile and Cross-Sectional Geometry of Optical Fiber Preforms,” Appl. Opt. 19, 2025 (1980); T. Okoshi, M. Nishimura, “Measurement of Axially Nonsymmetrical Refractive-Index Distributions of Optical Fiber Preforms by a Triangular Mask Method,” Appl. Opt. 20, 2407 (1981); P. L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. QE-18, 524 (1982).
    [CrossRef] [PubMed]
  8. H. Hertz, “Experimental Determination of 2-D Flame Temperature Fields by Interferometric Tomography,” Opt. Commun. 54, 131 (1985).
    [CrossRef]
  9. R. D. Matulka, D. J. Collins, “Determination of Three-Dimensional Density Fields from Holographic Interferograms,” J. Appl. Phys. 42, 1109 (1971).
    [CrossRef]
  10. R. Snyder, L. Hesselink, “Measurement of Mixing Fluid Flows with Optical Tomography,” Opt. Lett. 13, 87 (1988).
    [CrossRef] [PubMed]
  11. T. Okoshi, M. Nishimura, “Measurement of Axially Non-symmetrical Refractive-Index Distribution of a Single-Mode Fiber by a Multidirectional Scattering-Pattern Method,” IEEE/OSA J. Lightwave Technol. LT-1, 9 (1983).
    [CrossRef]
  12. N. A. Massie, “Real-Time Digital Heterodyne Interferometry: A System,” Appl. Opt. 19, 154 (1980).
    [CrossRef] [PubMed]
  13. L. A. Vasil’ev, Schlieren Methods, translated by A. Baruch (Isreal Program for Scientific Translations, New York, 1971).
  14. R. W. Lewis, R. E. Teets, J. A. Sell, T. A. Seder, “Temperature Measurements in a Laser-Heated Gas by Quantitative Shadowgraphy,” Appl. Opt. 26, 3695 (1987).
    [CrossRef] [PubMed]
  15. W. J. Stewart, “Optical Fiber and Preform Profiling Technology,” IEEE J. Quantum Electron. QE-18, 1451 (1982).
    [CrossRef]
  16. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spetroscopy and Detection,” Appl. Opt. 20, 1333 (1981); W. Zapka, A. C. Tam, “Noncontact Optoacoustic Determination of Gas Flow Velocity and Temperature Simultaneously,” Appl. Phys. Lett. 40, 1015(1982); J. A. Sell, “Gas Velocity Measurements Using Photothermal Deflection Spectroscopy,” Appl. Opt. 24, 3725 (1985); Q. Kong, J. Fischer, F. Träger, “Surface Temperature and Thin Film Investigations by Pulsed Laser Photothermal Displacement Spectroscopy,” in Photoacoustic and Photothermal Phenomena, P. Hess, J. Pelzl, Eds. (Springer-Verlag, New York, 1988).
    [CrossRef] [PubMed]
  17. R. N. Bracewell, A. C. Riddle, “Inversion of Fan-Beam Scans in Radio Astronomy,” Astrophys. J 150, 427 (1967).
    [CrossRef]
  18. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 108–112.
  19. L. W. Casperson, “Gaussian Light Beams in Inhomogeneous Media,” Appl. Opt. 12, 2434 (1973).
    [CrossRef] [PubMed]
  20. Ref. 17, pp. 117–118.
  21. Ref. 17, pp. 61–62.
  22. S. W. Rowland, “Computer Implementation of Image Reconstruction Formulas,” in Image Reconstruction from Projections, Implementation and Applications, G. T. Herman, Ed. (Springer-Verlag, New York, 1979), pp. 9–79.
    [CrossRef]
  23. H. H. Barrett, W. Swindell, “Analog Reconstruction Methods for Transaxial Tomgraphy,” Proc. IEEE 65, 89 (1977).
    [CrossRef]
  24. A. Gersho, “Principles of Quantization,” IEEE Trans. Circuits Syst. CAS-25, 427 (1978).
    [CrossRef]
  25. L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (June1974).
  26. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), p. 22.
  27. P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
    [CrossRef] [PubMed]
  28. P. M. Joseph, R. A. Schulz, “View Sampling Requirements in Fan Beam Computed Tomography,” Med. Phys. 7, 692 (1980).
    [CrossRef] [PubMed]
  29. D. L. Snyder, J. R. Cox, “An Overview of Reconstructive Tomography and Limitations Imposed by a Finite Number of Projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. TerPogossian, M. E. Phelps, G. L. Brownell, J. R. Cox, D. O. Davis, R. G. Evans, Eds. (University Park Press, Baltimore, 1977), pp. 3–32.
  30. B. F. Logan, “The Uncertainty Principle in Reconstructing Functions from Projections,” Duke Math. J. 42, 661 (1975).
    [CrossRef]
  31. A. J. Duerinckx, A. Macovski, “Nonlinear Polychromatic and Noise Artifacts in X-Ray Computed Tomography Images,” J. Comput. Assist. Tomogr. 3, 519 (1979).
    [CrossRef] [PubMed]
  32. K. E. Bennett, R. L. Byer, “Fan-Beam-Tomography Noise Theory,” J. Opt. Soc. Am. A 3, 624 (1986).
    [CrossRef]
  33. A. Rose, “A Unified Approach to the Performance of Photographic Film, Television Pickup Tubes, and the Human Eye,” J. Soc. Motion Pict. Eng. 47, 273 (1946).
  34. D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording with Resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
    [CrossRef]
  35. A. J. Devaney, “A Computer Simulation Study of Diffraction Tomography,” IEEE Trans. Biomed. Eng. BME-30, 377 (1983).
    [CrossRef]
  36. S. Leeman, M. A. Fiddy, L. Zapalowski, “Born Versus Rytov: is the Debate Over?” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 11 (1985).
  37. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford,. 1970), p. 333.
  38. A. E. Siegman, Lasers (Oxford U.P., London, 1986), p. 665.
  39. E. Wolf, “Light Distribution near Focus in an Error-Free Diffraction Image,” Proc. R. Soc. London, Ser A 204, 533 (1951).
    [CrossRef]
  40. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA1974), p. 354.
  41. G. W. Faris, “Quantitative Optical Tomographic Imaging of Fluid Flows and Flames,” Ph.D. Dissertation, Stanford U., Stanford, CA (1986), p. 30.
  42. F. J. Weinberg, Optics of Flames (Butterworth, London, 1963), p. 24.
  43. P. A. Thompson, Compressible-Fluid Dynamics (McGraw-Hill, New York, 1972), p. 371.

1988

1987

1986

1985

S. Leeman, M. A. Fiddy, L. Zapalowski, “Born Versus Rytov: is the Debate Over?” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 11 (1985).

H. Hertz, “Experimental Determination of 2-D Flame Temperature Fields by Interferometric Tomography,” Opt. Commun. 54, 131 (1985).
[CrossRef]

1984

1983

A. J. Devaney, “A Computer Simulation Study of Diffraction Tomography,” IEEE Trans. Biomed. Eng. BME-30, 377 (1983).
[CrossRef]

T. Okoshi, M. Nishimura, “Measurement of Axially Non-symmetrical Refractive-Index Distribution of a Single-Mode Fiber by a Multidirectional Scattering-Pattern Method,” IEEE/OSA J. Lightwave Technol. LT-1, 9 (1983).
[CrossRef]

1982

W. J. Stewart, “Optical Fiber and Preform Profiling Technology,” IEEE J. Quantum Electron. QE-18, 1451 (1982).
[CrossRef]

1981

1980

1979

A. J. Duerinckx, A. Macovski, “Nonlinear Polychromatic and Noise Artifacts in X-Ray Computed Tomography Images,” J. Comput. Assist. Tomogr. 3, 519 (1979).
[CrossRef] [PubMed]

1978

A. Gersho, “Principles of Quantization,” IEEE Trans. Circuits Syst. CAS-25, 427 (1978).
[CrossRef]

1977

H. H. Barrett, W. Swindell, “Analog Reconstruction Methods for Transaxial Tomgraphy,” Proc. IEEE 65, 89 (1977).
[CrossRef]

1975

B. F. Logan, “The Uncertainty Principle in Reconstructing Functions from Projections,” Duke Math. J. 42, 661 (1975).
[CrossRef]

1974

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (June1974).

1973

1971

R. D. Matulka, D. J. Collins, “Determination of Three-Dimensional Density Fields from Holographic Interferograms,” J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

1967

R. N. Bracewell, A. C. Riddle, “Inversion of Fan-Beam Scans in Radio Astronomy,” Astrophys. J 150, 427 (1967).
[CrossRef]

1951

E. Wolf, “Light Distribution near Focus in an Error-Free Diffraction Image,” Proc. R. Soc. London, Ser A 204, 533 (1951).
[CrossRef]

1946

A. Rose, “A Unified Approach to the Performance of Photographic Film, Television Pickup Tubes, and the Human Eye,” J. Soc. Motion Pict. Eng. 47, 273 (1946).

Amer, N. M.

Barrett, H. H.

H. H. Barrett, W. Swindell, “Analog Reconstruction Methods for Transaxial Tomgraphy,” Proc. IEEE 65, 89 (1977).
[CrossRef]

Bennett, K. E.

Boccara, A. C.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford,. 1970), p. 333.

Bracewell, R. N.

R. N. Bracewell, A. C. Riddle, “Inversion of Fan-Beam Scans in Radio Astronomy,” Astrophys. J 150, 427 (1967).
[CrossRef]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 108–112.

Byer, R. L.

Casperson, L. W.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), p. 22.

Chu, P. L.

Collins, D. J.

R. D. Matulka, D. J. Collins, “Determination of Three-Dimensional Density Fields from Holographic Interferograms,” J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

Cox, J. R.

D. L. Snyder, J. R. Cox, “An Overview of Reconstructive Tomography and Limitations Imposed by a Finite Number of Projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. TerPogossian, M. E. Phelps, G. L. Brownell, J. R. Cox, D. O. Davis, R. G. Evans, Eds. (University Park Press, Baltimore, 1977), pp. 3–32.

Denk, W.

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording with Resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Devaney, A. J.

A. J. Devaney, “A Computer Simulation Study of Diffraction Tomography,” IEEE Trans. Biomed. Eng. BME-30, 377 (1983).
[CrossRef]

Duerinckx, A. J.

A. J. Duerinckx, A. Macovski, “Nonlinear Polychromatic and Noise Artifacts in X-Ray Computed Tomography Images,” J. Comput. Assist. Tomogr. 3, 519 (1979).
[CrossRef] [PubMed]

Faris, G. W.

Fiddy, M. A.

S. Leeman, M. A. Fiddy, L. Zapalowski, “Born Versus Rytov: is the Debate Over?” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 11 (1985).

Fournier, D.

Gersho, A.

A. Gersho, “Principles of Quantization,” IEEE Trans. Circuits Syst. CAS-25, 427 (1978).
[CrossRef]

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA1974), p. 354.

Hertz, H.

H. Hertz, “Experimental Determination of 2-D Flame Temperature Fields by Interferometric Tomography,” Opt. Commun. 54, 131 (1985).
[CrossRef]

Hesselink, L.

Jackson, W. B.

Joseph, P. M.

P. M. Joseph, R. A. Schulz, “View Sampling Requirements in Fan Beam Computed Tomography,” Med. Phys. 7, 692 (1980).
[CrossRef] [PubMed]

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Lanz, M.

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording with Resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Leeman, S.

S. Leeman, M. A. Fiddy, L. Zapalowski, “Born Versus Rytov: is the Debate Over?” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 11 (1985).

Lewis, R. W.

Logan, B. F.

B. F. Logan, “The Uncertainty Principle in Reconstructing Functions from Projections,” Duke Math. J. 42, 661 (1975).
[CrossRef]

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (June1974).

Macovski, A.

A. J. Duerinckx, A. Macovski, “Nonlinear Polychromatic and Noise Artifacts in X-Ray Computed Tomography Images,” J. Comput. Assist. Tomogr. 3, 519 (1979).
[CrossRef] [PubMed]

Massie, N. A.

Matulka, R. D.

R. D. Matulka, D. J. Collins, “Determination of Three-Dimensional Density Fields from Holographic Interferograms,” J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

Nishimura, M.

T. Okoshi, M. Nishimura, “Measurement of Axially Non-symmetrical Refractive-Index Distribution of a Single-Mode Fiber by a Multidirectional Scattering-Pattern Method,” IEEE/OSA J. Lightwave Technol. LT-1, 9 (1983).
[CrossRef]

Okoshi, T.

T. Okoshi, M. Nishimura, “Measurement of Axially Non-symmetrical Refractive-Index Distribution of a Single-Mode Fiber by a Multidirectional Scattering-Pattern Method,” IEEE/OSA J. Lightwave Technol. LT-1, 9 (1983).
[CrossRef]

Pohl, D. W.

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording with Resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Riddle, A. C.

R. N. Bracewell, A. C. Riddle, “Inversion of Fan-Beam Scans in Radio Astronomy,” Astrophys. J 150, 427 (1967).
[CrossRef]

Rose, A.

A. Rose, “A Unified Approach to the Performance of Photographic Film, Television Pickup Tubes, and the Human Eye,” J. Soc. Motion Pict. Eng. 47, 273 (1946).

Rowland, S. W.

S. W. Rowland, “Computer Implementation of Image Reconstruction Formulas,” in Image Reconstruction from Projections, Implementation and Applications, G. T. Herman, Ed. (Springer-Verlag, New York, 1979), pp. 9–79.
[CrossRef]

Saekeang, C.

Schulz, R. A.

P. M. Joseph, R. A. Schulz, “View Sampling Requirements in Fan Beam Computed Tomography,” Med. Phys. 7, 692 (1980).
[CrossRef] [PubMed]

Seder, T. A.

Sell, J. A.

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (June1974).

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U.P., London, 1986), p. 665.

Snyder, D. L.

D. L. Snyder, J. R. Cox, “An Overview of Reconstructive Tomography and Limitations Imposed by a Finite Number of Projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. TerPogossian, M. E. Phelps, G. L. Brownell, J. R. Cox, D. O. Davis, R. G. Evans, Eds. (University Park Press, Baltimore, 1977), pp. 3–32.

Snyder, R.

Spital, R. D.

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Stewart, W. J.

W. J. Stewart, “Optical Fiber and Preform Profiling Technology,” IEEE J. Quantum Electron. QE-18, 1451 (1982).
[CrossRef]

Stockham, C. D.

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Stricker, J.

Swindell, W.

H. H. Barrett, W. Swindell, “Analog Reconstruction Methods for Transaxial Tomgraphy,” Proc. IEEE 65, 89 (1977).
[CrossRef]

Teets, R. E.

Thompson, P. A.

P. A. Thompson, Compressible-Fluid Dynamics (McGraw-Hill, New York, 1972), p. 371.

Vasil’ev, L. A.

L. A. Vasil’ev, Schlieren Methods, translated by A. Baruch (Isreal Program for Scientific Translations, New York, 1971).

Weinberg, F. J.

F. J. Weinberg, Optics of Flames (Butterworth, London, 1963), p. 24.

Whitbread, T. W.

Wolf, E.

E. Wolf, “Light Distribution near Focus in an Error-Free Diffraction Image,” Proc. R. Soc. London, Ser A 204, 533 (1951).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford,. 1970), p. 333.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA1974), p. 354.

Zapalowski, L.

S. Leeman, M. A. Fiddy, L. Zapalowski, “Born Versus Rytov: is the Debate Over?” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 11 (1985).

Appl. Opt.

L. W. Casperson, “Gaussian Light Beams in Inhomogeneous Media,” Appl. Opt. 12, 2434 (1973).
[CrossRef] [PubMed]

N. A. Massie, “Real-Time Digital Heterodyne Interferometry: A System,” Appl. Opt. 19, 154 (1980).
[CrossRef] [PubMed]

C. Saekeang, P. L. Chu, T. W. Whitbread, “Nondestructive Measurement of Refractive-Index Profile and Cross-Sectional Geometry of Optical Fiber Preforms,” Appl. Opt. 19, 2025 (1980); T. Okoshi, M. Nishimura, “Measurement of Axially Nonsymmetrical Refractive-Index Distributions of Optical Fiber Preforms by a Triangular Mask Method,” Appl. Opt. 20, 2407 (1981); P. L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. QE-18, 524 (1982).
[CrossRef] [PubMed]

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spetroscopy and Detection,” Appl. Opt. 20, 1333 (1981); W. Zapka, A. C. Tam, “Noncontact Optoacoustic Determination of Gas Flow Velocity and Temperature Simultaneously,” Appl. Phys. Lett. 40, 1015(1982); J. A. Sell, “Gas Velocity Measurements Using Photothermal Deflection Spectroscopy,” Appl. Opt. 24, 3725 (1985); Q. Kong, J. Fischer, F. Träger, “Surface Temperature and Thin Film Investigations by Pulsed Laser Photothermal Displacement Spectroscopy,” in Photoacoustic and Photothermal Phenomena, P. Hess, J. Pelzl, Eds. (Springer-Verlag, New York, 1988).
[CrossRef] [PubMed]

K. E. Bennett, G. W. Faris, R. L. Byer, “Experimental Optical Fan Beam Tomography,” Appl. Opt. 23, 2678 (1984).
[CrossRef] [PubMed]

J. Stricker, “Analysis of 3-D Phase Objects by Moiré Deflectometry,” Appl. Opt. 23, 3657 (1984).
[CrossRef] [PubMed]

R. W. Lewis, R. E. Teets, J. A. Sell, T. A. Seder, “Temperature Measurements in a Laser-Heated Gas by Quantitative Shadowgraphy,” Appl. Opt. 26, 3695 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett.

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording with Resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Astrophys. J

R. N. Bracewell, A. C. Riddle, “Inversion of Fan-Beam Scans in Radio Astronomy,” Astrophys. J 150, 427 (1967).
[CrossRef]

Comput. Tomogr.

P. M. Joseph, R. D. Spital, C. D. Stockham, “The Effects of Sampling on CT Images,” Comput. Tomogr. 4, 189 (1980).
[CrossRef] [PubMed]

Duke Math. J.

B. F. Logan, “The Uncertainty Principle in Reconstructing Functions from Projections,” Duke Math. J. 42, 661 (1975).
[CrossRef]

IEEE J. Quantum Electron.

W. J. Stewart, “Optical Fiber and Preform Profiling Technology,” IEEE J. Quantum Electron. QE-18, 1451 (1982).
[CrossRef]

IEEE Trans. Biomed. Eng.

A. J. Devaney, “A Computer Simulation Study of Diffraction Tomography,” IEEE Trans. Biomed. Eng. BME-30, 377 (1983).
[CrossRef]

IEEE Trans. Circuits Syst.

A. Gersho, “Principles of Quantization,” IEEE Trans. Circuits Syst. CAS-25, 427 (1978).
[CrossRef]

IEEE Trans. Nucl. Sci.

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (June1974).

IEEE/OSA J. Lightwave Technol.

T. Okoshi, M. Nishimura, “Measurement of Axially Non-symmetrical Refractive-Index Distribution of a Single-Mode Fiber by a Multidirectional Scattering-Pattern Method,” IEEE/OSA J. Lightwave Technol. LT-1, 9 (1983).
[CrossRef]

J. Appl. Phys.

R. D. Matulka, D. J. Collins, “Determination of Three-Dimensional Density Fields from Holographic Interferograms,” J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

J. Comput. Assist. Tomogr.

A. J. Duerinckx, A. Macovski, “Nonlinear Polychromatic and Noise Artifacts in X-Ray Computed Tomography Images,” J. Comput. Assist. Tomogr. 3, 519 (1979).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Soc. Motion Pict. Eng.

A. Rose, “A Unified Approach to the Performance of Photographic Film, Television Pickup Tubes, and the Human Eye,” J. Soc. Motion Pict. Eng. 47, 273 (1946).

Med. Phys.

P. M. Joseph, R. A. Schulz, “View Sampling Requirements in Fan Beam Computed Tomography,” Med. Phys. 7, 692 (1980).
[CrossRef] [PubMed]

Opt. Commun.

H. Hertz, “Experimental Determination of 2-D Flame Temperature Fields by Interferometric Tomography,” Opt. Commun. 54, 131 (1985).
[CrossRef]

Opt. Lett.

Proc. IEEE

H. H. Barrett, W. Swindell, “Analog Reconstruction Methods for Transaxial Tomgraphy,” Proc. IEEE 65, 89 (1977).
[CrossRef]

Proc. R. Soc. London, Ser A

E. Wolf, “Light Distribution near Focus in an Error-Free Diffraction Image,” Proc. R. Soc. London, Ser A 204, 533 (1951).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

S. Leeman, M. A. Fiddy, L. Zapalowski, “Born Versus Rytov: is the Debate Over?” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 11 (1985).

Science

G. W. Faris, R. L. Byer, “Quantitative Three-Dimensional Optical Tomographic Imaging of Supersonic Flows,” Science 238, 1700 (1987).
[CrossRef] [PubMed]

Other

L. A. Vasil’ev, Schlieren Methods, translated by A. Baruch (Isreal Program for Scientific Translations, New York, 1971).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 108–112.

Ref. 17, pp. 117–118.

Ref. 17, pp. 61–62.

S. W. Rowland, “Computer Implementation of Image Reconstruction Formulas,” in Image Reconstruction from Projections, Implementation and Applications, G. T. Herman, Ed. (Springer-Verlag, New York, 1979), pp. 9–79.
[CrossRef]

D. L. Snyder, J. R. Cox, “An Overview of Reconstructive Tomography and Limitations Imposed by a Finite Number of Projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. TerPogossian, M. E. Phelps, G. L. Brownell, J. R. Cox, D. O. Davis, R. G. Evans, Eds. (University Park Press, Baltimore, 1977), pp. 3–32.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), p. 22.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford,. 1970), p. 333.

A. E. Siegman, Lasers (Oxford U.P., London, 1986), p. 665.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, MA1974), p. 354.

G. W. Faris, “Quantitative Optical Tomographic Imaging of Fluid Flows and Flames,” Ph.D. Dissertation, Stanford U., Stanford, CA (1986), p. 30.

F. J. Weinberg, Optics of Flames (Butterworth, London, 1963), p. 24.

P. A. Thompson, Compressible-Fluid Dynamics (McGraw-Hill, New York, 1972), p. 371.

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Figures (12)

Fig. 1
Fig. 1

Projection for tomographic reconstruction.

Fig. 2
Fig. 2

(a) Filter function for convolution backprojection algorithm in frequency domain; (b) filter function for convolution backprojection reconstruction from beam-deflection measurements in frequency domain.

Fig. 3
Fig. 3

Geometry for beam-deflection tomographic reconstructions.

Fig. 4
Fig. 4

(a) Scanned beam geometry for optical tomography; (b) imaging geometry for optical tomography.

Fig. 5
Fig. 5

Contour line for the normalized radius v of a circle transmitting 50% of the light intensity near a focus as a function of the normalized distance from focus u.

Fig. 6
Fig. 6

Peripheral resolution dp and central resolution d0 vs the f/No. for a focused plane wave (solid line, f/No. = L/D) and a Gaussian beam (dashed line, f/No. = L/2.65wff).

Fig. 7
Fig. 7

Diagram of supersonic gas flow, optical imaging, and data acquisition systems for beam-deflection optical tomography of a supersonic jet.

Fig. 8
Fig. 8

(a) Vertical section through a 3-D reconstructed image of nitrogen density in a supersonic jet; (b) logarithm of image in Fig. 8(a).

Fig. 9
Fig. 9

Horizontal sections through a 3-D reconstructed image of nitrogen in a supersonic jet.

Fig. 10
Fig. 10

Plot of nitrogen gas density vs radius for a section through the image shown in Fig. 9(b). Also shown is a no-free-parameter theoretical curve for the density.

Fig. 11
Fig. 11

Near-diffraction-limited reconstruction of supersonic jet at a height of 2.52 mm; (b) section through Fig. 10(a).

Fig. 12
Fig. 12

Effective point spread functions: (a) diffraction-limited probe beam width; (b) point spread function for reconstruction; (c) point spread function convolved with shock width; (d) diffraction-limited probe beam convolved with shock width.

Equations (40)

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p ( y , θ ) = - f ( x , y ) d x .
P ( Y , θ ) = - p ( y , θ ) exp ( - i 2 π y Y ) d y .
f ( x , y ) = 0 π - P ( Y , θ ) exp [ i 2 π Y ( - x sin θ + y cos θ ) ] Y d Y d θ .
f ( x , y ) = 0 π [ p ( y , θ ) * k ( y ) ] y = - x sin θ + y cos θ d θ ,
K ( Y ) = { Y Y < 1 2 Δ y , 0 Y > 1 2 Δ y ,
n ( r ) = n 0 [ 1 + n ^ ( r ) ] ,
d d x [ n 0 d r d x ] = n 0 ¯ n ^ ( r ) ,
d r d x = ¯ n ^ ( r ) d x .
α tan α = d y d x = d n ^ ( x , y ) d y d x .
α ( y , θ ) = d n ^ ( x , y ) d y d x ,
α ( y , θ ) d d y n ^ ( x , y ) d x .
p ( y , θ ) = n ^ ( x , y ) d x
A ( Y , θ ) = i 2 π Y P ( Y , θ ) .
n ^ ( y , θ ) = 0 π - P ( Y , θ ) exp [ i 2 π Y ( - x sin θ + y cos θ ) ] Y d Y d θ .
n ^ ( x , y ) = - i 2 π 0 π - A ( Y , θ ) × exp [ i 2 π Y ( - x sin θ + y cos θ ) ] sgn ( Y ) d Y d θ ,
sgn ( Y ) = { - 1 Y < 0 , 1 Y > 0.
n ^ ( x , y ) = 0 π [ α ( y , θ ) * k ( y ) ] y = - x sin θ + y cos θ d θ ,
K ( Y ) = { - i 2 π sgn ( Y ) Y < 1 2 Δ y , 0 Y > 1 2 Δ y ,
κ ( r ) = 1 π 2 r sin 2 [ π r 2 Δ y ]
κ ( m Δ y ) = { 0 m = 0 1 π 2 m Δ y m odd 0 m even .
w ( z ) = w 0 1 + ( λ z π w 0 2 ) 2 ,
d = 2 ln 2 w ,
d 0 = 2 ln 2 w 0 ,
d p = w 0 2 ln 2 [ 1 + ( λ l 2 π w 0 2 ) 2 ] .
w 0 = λ l 2 π .
d 0 * = 0.470 λ l
d p * = 0.664 λ l .
u = π D 2 2 λ L 2 z ,
v = π D λ L r .
d = 2 r = 2 λ L π D v ,
d 0 = 1.07 λ L D ,
d 0 = 1.22 λ L D .
u = π l D 2 4 λ L 2 .
d p = λ l π u v ,
L D = π l 4 λ u ,
f / No . = L D = 0.473 l λ .
d 0 * = 0.506 λ l ,
d p * = 0.628 λ l ,
w f f = λ z π w 0 .
ρ = k ( n - 1 ) ,

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