Abstract

When tomography is performed with electromagnetic or acoustical radiation, refraction may cause sufficient bending of the probing rays that ordinary reconstruction algorithms, which are based on the assumption of straight rays, do not yield accurate results. The resulting problem of reconstructing the refractive-index distribution of an object from time of flight or optical path length data is nonlinear. Various approaches to solving this problem approximately have been proposed and subjected to modest numerical studies. These include iterative algorithms and techniques based on linearized inverse scattering theory. One exception is the case of axisymmetric objects for which an exact solution is known.

© 1985 Optical Society of America

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References

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  1. E. J. Farrell, “Tomographic Imaging of Attenuation with Simulation Correction for Refraction,” Ultrason. Imaging 3, 144 (1981).
    [CrossRef] [PubMed]
  2. K. M. Pan, C. N. Liu, “Tomographic Reconstruction of Ultrasonic Attenuation with Correction for Refractive Errors,” IBM J. Res. Dev. 25, 71 (1981).
    [CrossRef]
  3. J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
    [CrossRef]
  4. T. Itoh, J.-S. Choi, C. Ksai, M. Nakajima, “Correction for Refraction on Ultrasonic CT,” Jpn. J. Appl. Phys. 21, 149 (1981).
  5. J. W. Eberhard, “Quantitative Imaging in Nondestructive Evaluation (NDE) by Ultrasonic Time-of-Flight (TOF) Tomography,” Mater. Eval. 68 (1982).
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 6.
  7. K. E. Bullen, An Introduction to the Theory of Seismology (Cambridge U. P., New York, 1963).
  8. R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
    [CrossRef]
  9. C. M. Vest, “Interferometry of Strongly Refracting Axisymmetric Objects,” Appl. Opt. 14, 1601 (1975).
    [CrossRef] [PubMed]
  10. G. P. Montgomery, D. L. Reuss, “Effects of Refraction on Axisymmetric Flame Temperatures Measured by Holographic Interferometry,” Appl. Opt. 21, 1373 (1982).
    [CrossRef] [PubMed]
  11. S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
    [CrossRef] [PubMed]
  12. S. Cha, “Reconstruction of Strongly Refracting Asymmetric Fields from Interferometric Measurements,” Doctoral Dissertation, U. Michigan, Ann Arbor (1980).
  13. A. H. Andersen, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
    [CrossRef]
  14. J. F. Greenleaf, S. A. Johnson, A. H. Lent, “Measurement of Spatial Distribution of Refractive Index in Tissues by Ultrasonic Computer Assisted Tomography,” Ultrasound Med. Bio. 3, 327 (1978).
    [CrossRef]
  15. G. H. Glover, J. C. Sharp, “Reconstruction of Ultrasound Propagation Speed Distributions in Soft Tissue: Time-of-Flight Tomography,” IEEE Trans. Sonics Ultraon. SU-24, 229 (1977).
    [CrossRef]
  16. H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D. 11, L181 (1978).
    [CrossRef]
  17. G. C. McKinnon, X. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrason. Imaging 2, 48 (1980).
    [CrossRef] [PubMed]
  18. S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: A Perturbation Approach,” Ultrason. Imaging 4, 201 (1982).
    [CrossRef] [PubMed]
  19. K. Iwata, R. Nagata, “Calculation of Refractive Index Distribution from Interferograms Using the Born and Rytov’s Approximation,” Jpn. J. Appl. Phys. 14-1, 379 (1975).
  20. S. K. Kenue, J. F. Greenleaf, “Limited Angle Multifrequency Diffraction Tomography,” IEEE Trans. Sonics Ultrason. SU-29, 213 (1982).
    [CrossRef]
  21. A. J. Devaney, “Inverse Scattering as a Form of Computed Tomography,” Proc. Soc. Photo-Opt. Instrum. Eng. 358, 10 (1982).
  22. A. J. Devaney, “A Filtered Backpropagation Algorithm for Diffraction Tomography,” Ultrason. Imaging 4, 336 (1982).
    [CrossRef] [PubMed]
  23. A. J. Devaney, “Coherent Optical Tomography,” in Technical Digest, Topical Meeting on Industrial Applications of Computed Tomography and NMR Imaging (Optical Society of America, Washington, D.C., 1984), paper TuD3.
  24. R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
    [CrossRef]
  25. R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
    [CrossRef]

1982 (7)

J. W. Eberhard, “Quantitative Imaging in Nondestructive Evaluation (NDE) by Ultrasonic Time-of-Flight (TOF) Tomography,” Mater. Eval. 68 (1982).

A. H. Andersen, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: A Perturbation Approach,” Ultrason. Imaging 4, 201 (1982).
[CrossRef] [PubMed]

S. K. Kenue, J. F. Greenleaf, “Limited Angle Multifrequency Diffraction Tomography,” IEEE Trans. Sonics Ultrason. SU-29, 213 (1982).
[CrossRef]

A. J. Devaney, “Inverse Scattering as a Form of Computed Tomography,” Proc. Soc. Photo-Opt. Instrum. Eng. 358, 10 (1982).

A. J. Devaney, “A Filtered Backpropagation Algorithm for Diffraction Tomography,” Ultrason. Imaging 4, 336 (1982).
[CrossRef] [PubMed]

G. P. Montgomery, D. L. Reuss, “Effects of Refraction on Axisymmetric Flame Temperatures Measured by Holographic Interferometry,” Appl. Opt. 21, 1373 (1982).
[CrossRef] [PubMed]

1981 (6)

S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
[CrossRef] [PubMed]

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

E. J. Farrell, “Tomographic Imaging of Attenuation with Simulation Correction for Refraction,” Ultrason. Imaging 3, 144 (1981).
[CrossRef] [PubMed]

K. M. Pan, C. N. Liu, “Tomographic Reconstruction of Ultrasonic Attenuation with Correction for Refractive Errors,” IBM J. Res. Dev. 25, 71 (1981).
[CrossRef]

J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
[CrossRef]

T. Itoh, J.-S. Choi, C. Ksai, M. Nakajima, “Correction for Refraction on Ultrasonic CT,” Jpn. J. Appl. Phys. 21, 149 (1981).

1980 (2)

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[CrossRef]

G. C. McKinnon, X. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrason. Imaging 2, 48 (1980).
[CrossRef] [PubMed]

1978 (2)

H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D. 11, L181 (1978).
[CrossRef]

J. F. Greenleaf, S. A. Johnson, A. H. Lent, “Measurement of Spatial Distribution of Refractive Index in Tissues by Ultrasonic Computer Assisted Tomography,” Ultrasound Med. Bio. 3, 327 (1978).
[CrossRef]

1977 (1)

G. H. Glover, J. C. Sharp, “Reconstruction of Ultrasound Propagation Speed Distributions in Soft Tissue: Time-of-Flight Tomography,” IEEE Trans. Sonics Ultraon. SU-24, 229 (1977).
[CrossRef]

1975 (2)

K. Iwata, R. Nagata, “Calculation of Refractive Index Distribution from Interferograms Using the Born and Rytov’s Approximation,” Jpn. J. Appl. Phys. 14-1, 379 (1975).

C. M. Vest, “Interferometry of Strongly Refracting Axisymmetric Objects,” Appl. Opt. 14, 1601 (1975).
[CrossRef] [PubMed]

1968 (1)

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

Andersen, A. H.

A. H. Andersen, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

Anderson, D. L.

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

Balanis, C. A.

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

Bates, X.

G. C. McKinnon, X. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrason. Imaging 2, 48 (1980).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 6.

Brandenburger, G. H.

J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
[CrossRef]

Bullen, K. E.

K. E. Bullen, An Introduction to the Theory of Seismology (Cambridge U. P., New York, 1963).

Cha, S.

S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
[CrossRef] [PubMed]

S. Cha, “Reconstruction of Strongly Refracting Asymmetric Fields from Interferometric Measurements,” Doctoral Dissertation, U. Michigan, Ann Arbor (1980).

Choi, J.-S.

T. Itoh, J.-S. Choi, C. Ksai, M. Nakajima, “Correction for Refraction on Ultrasonic CT,” Jpn. J. Appl. Phys. 21, 149 (1981).

Devaney, A. J.

A. J. Devaney, “Inverse Scattering as a Form of Computed Tomography,” Proc. Soc. Photo-Opt. Instrum. Eng. 358, 10 (1982).

A. J. Devaney, “A Filtered Backpropagation Algorithm for Diffraction Tomography,” Ultrason. Imaging 4, 336 (1982).
[CrossRef] [PubMed]

A. J. Devaney, “Coherent Optical Tomography,” in Technical Digest, Topical Meeting on Industrial Applications of Computed Tomography and NMR Imaging (Optical Society of America, Washington, D.C., 1984), paper TuD3.

Dines, K. A.

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[CrossRef]

Eberhard, J. W.

J. W. Eberhard, “Quantitative Imaging in Nondestructive Evaluation (NDE) by Ultrasonic Time-of-Flight (TOF) Tomography,” Mater. Eval. 68 (1982).

Farrell, E. J.

E. J. Farrell, “Tomographic Imaging of Attenuation with Simulation Correction for Refraction,” Ultrason. Imaging 3, 144 (1981).
[CrossRef] [PubMed]

Glover, G. H.

G. H. Glover, J. C. Sharp, “Reconstruction of Ultrasound Propagation Speed Distributions in Soft Tissue: Time-of-Flight Tomography,” IEEE Trans. Sonics Ultraon. SU-24, 229 (1977).
[CrossRef]

Greenleaf, J. F.

S. K. Kenue, J. F. Greenleaf, “Limited Angle Multifrequency Diffraction Tomography,” IEEE Trans. Sonics Ultrason. SU-29, 213 (1982).
[CrossRef]

J. F. Greenleaf, S. A. Johnson, A. H. Lent, “Measurement of Spatial Distribution of Refractive Index in Tissues by Ultrasonic Computer Assisted Tomography,” Ultrasound Med. Bio. 3, 327 (1978).
[CrossRef]

Itoh, T.

T. Itoh, J.-S. Choi, C. Ksai, M. Nakajima, “Correction for Refraction on Ultrasonic CT,” Jpn. J. Appl. Phys. 21, 149 (1981).

Iwata, K.

K. Iwata, R. Nagata, “Calculation of Refractive Index Distribution from Interferograms Using the Born and Rytov’s Approximation,” Jpn. J. Appl. Phys. 14-1, 379 (1975).

Johnson, S. A.

J. F. Greenleaf, S. A. Johnson, A. H. Lent, “Measurement of Spatial Distribution of Refractive Index in Tissues by Ultrasonic Computer Assisted Tomography,” Ultrasound Med. Bio. 3, 327 (1978).
[CrossRef]

Kak, A. C.

A. H. Andersen, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

Kenue, S. K.

S. K. Kenue, J. F. Greenleaf, “Limited Angle Multifrequency Diffraction Tomography,” IEEE Trans. Sonics Ultrason. SU-29, 213 (1982).
[CrossRef]

Klepper, J. R.

J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
[CrossRef]

Ksai, C.

T. Itoh, J.-S. Choi, C. Ksai, M. Nakajima, “Correction for Refraction on Ultrasonic CT,” Jpn. J. Appl. Phys. 21, 149 (1981).

Lent, A. H.

J. F. Greenleaf, S. A. Johnson, A. H. Lent, “Measurement of Spatial Distribution of Refractive Index in Tissues by Ultrasonic Computer Assisted Tomography,” Ultrasound Med. Bio. 3, 327 (1978).
[CrossRef]

Linzer, M.

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: A Perturbation Approach,” Ultrason. Imaging 4, 201 (1982).
[CrossRef] [PubMed]

Liu, C. N.

K. M. Pan, C. N. Liu, “Tomographic Reconstruction of Ultrasonic Attenuation with Correction for Refractive Errors,” IBM J. Res. Dev. 25, 71 (1981).
[CrossRef]

Lytle, R. J.

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[CrossRef]

McKinnon, G. C.

G. C. McKinnon, X. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrason. Imaging 2, 48 (1980).
[CrossRef] [PubMed]

Miller, J. G.

J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
[CrossRef]

Mimbs, J. W.

J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
[CrossRef]

Montgomery, G. P.

Nagata, R.

K. Iwata, R. Nagata, “Calculation of Refractive Index Distribution from Interferograms Using the Born and Rytov’s Approximation,” Jpn. J. Appl. Phys. 14-1, 379 (1975).

Nakajima, M.

T. Itoh, J.-S. Choi, C. Ksai, M. Nakajima, “Correction for Refraction on Ultrasonic CT,” Jpn. J. Appl. Phys. 21, 149 (1981).

Norton, S. J.

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: A Perturbation Approach,” Ultrason. Imaging 4, 201 (1982).
[CrossRef] [PubMed]

Pan, K. M.

K. M. Pan, C. N. Liu, “Tomographic Reconstruction of Ultrasonic Attenuation with Correction for Refractive Errors,” IBM J. Res. Dev. 25, 71 (1981).
[CrossRef]

Phinney, R. A.

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

Radcliff, R. D.

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

Reuss, D. L.

Schomberg, H.

H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D. 11, L181 (1978).
[CrossRef]

Sharp, J. C.

G. H. Glover, J. C. Sharp, “Reconstruction of Ultrasound Propagation Speed Distributions in Soft Tissue: Time-of-Flight Tomography,” IEEE Trans. Sonics Ultraon. SU-24, 229 (1977).
[CrossRef]

Sobel, B. E.

J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
[CrossRef]

Vest, C. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 6.

Appl. Opt. (3)

IBM J. Res. Dev. (1)

K. M. Pan, C. N. Liu, “Tomographic Reconstruction of Ultrasonic Attenuation with Correction for Refractive Errors,” IBM J. Res. Dev. 25, 71 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

J. R. Klepper, G. H. Brandenburger, J. W. Mimbs, B. E. Sobel, J. G. Miller, “Application of Phase-Insensitive Detection and Frequency-Dependent Measurements to Computed Ultrasonic Attenuation Tomography,” IEEE Trans. Biomed. Eng. BME-28, 186 (1981).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[CrossRef]

IEEE Trans. Sonics Ultraon. (1)

G. H. Glover, J. C. Sharp, “Reconstruction of Ultrasound Propagation Speed Distributions in Soft Tissue: Time-of-Flight Tomography,” IEEE Trans. Sonics Ultraon. SU-24, 229 (1977).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

S. K. Kenue, J. F. Greenleaf, “Limited Angle Multifrequency Diffraction Tomography,” IEEE Trans. Sonics Ultrason. SU-29, 213 (1982).
[CrossRef]

J. Acoust. Soc. Am. (1)

A. H. Andersen, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

J. Geophys. Res. Space Phys. (1)

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

J. Phys. D. (1)

H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D. 11, L181 (1978).
[CrossRef]

Jpn. J. Appl. Phys. (2)

T. Itoh, J.-S. Choi, C. Ksai, M. Nakajima, “Correction for Refraction on Ultrasonic CT,” Jpn. J. Appl. Phys. 21, 149 (1981).

K. Iwata, R. Nagata, “Calculation of Refractive Index Distribution from Interferograms Using the Born and Rytov’s Approximation,” Jpn. J. Appl. Phys. 14-1, 379 (1975).

Mater. Eval. (1)

J. W. Eberhard, “Quantitative Imaging in Nondestructive Evaluation (NDE) by Ultrasonic Time-of-Flight (TOF) Tomography,” Mater. Eval. 68 (1982).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

A. J. Devaney, “Inverse Scattering as a Form of Computed Tomography,” Proc. Soc. Photo-Opt. Instrum. Eng. 358, 10 (1982).

Ultrason. Imaging (4)

A. J. Devaney, “A Filtered Backpropagation Algorithm for Diffraction Tomography,” Ultrason. Imaging 4, 336 (1982).
[CrossRef] [PubMed]

G. C. McKinnon, X. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrason. Imaging 2, 48 (1980).
[CrossRef] [PubMed]

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: A Perturbation Approach,” Ultrason. Imaging 4, 201 (1982).
[CrossRef] [PubMed]

E. J. Farrell, “Tomographic Imaging of Attenuation with Simulation Correction for Refraction,” Ultrason. Imaging 3, 144 (1981).
[CrossRef] [PubMed]

Ultrasound Med. Bio. (1)

J. F. Greenleaf, S. A. Johnson, A. H. Lent, “Measurement of Spatial Distribution of Refractive Index in Tissues by Ultrasonic Computer Assisted Tomography,” Ultrasound Med. Bio. 3, 327 (1978).
[CrossRef]

Other (4)

S. Cha, “Reconstruction of Strongly Refracting Asymmetric Fields from Interferometric Measurements,” Doctoral Dissertation, U. Michigan, Ann Arbor (1980).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 6.

K. E. Bullen, An Introduction to the Theory of Seismology (Cambridge U. P., New York, 1963).

A. J. Devaney, “Coherent Optical Tomography,” in Technical Digest, Topical Meeting on Industrial Applications of Computed Tomography and NMR Imaging (Optical Society of America, Washington, D.C., 1984), paper TuD3.

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Figures (4)

Fig. 1
Fig. 1

Optical ray passing through a strongly refracting axisymmetric object.

Fig. 2
Fig. 2

Formation of an interferogram of an asymmetric refracting object.

Fig. 3
Fig. 3

Ray passing through a mildly refracting object. It originates at transmitter T and impinges on receiver R.

Fig. 4
Fig. 4

Complex amplitude of a wave (initially plane) that has propagated through a refracting object.

Equations (34)

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

ϕ ( x ) = 2 x r 0 f ( r ) r d r ( r 2 x 2 ) 1 / 2 ,
f ( r ) = 1 π r r 0 ( d ϕ / d x ) d x ( x 2 r 2 ) 1 / 2 .
ϕ = n d s ,
d d s ( n d r d s ) = n ,
r n ( r ) sin ( i ) = p .
η = r n ( r )
ϕ ( p ) = 2 p η 0 ( η d ln r / d η ) η d η ( η 2 p 2 ) 1 / 2 .
η ( d ln r d η ) = 1 π η η 0 ( d ϕ / d p ) d p ( p 2 η 2 ) 1 / 2 .
r r 0 = exp [ 1 π η η 0 cosh 1 ( p π ) d ϕ d p d p p ] .
Δ ϕ ¯ ( p , θ ) = A B n ( r , ϕ ) d s + n 0 ( B C ¯ D E ¯ E F ¯ ) = P ˜ [ n ( r , ϕ ) ; n 0 ] .
D ( p , θ ) = Δ ϕ ˜ ( p , θ ) Δ ϕ ¯ ( p , θ )
Δ ϕ ¯ i ( p , θ ) = Δ ϕ ˜ ( p , θ ) D i ( p , θ ) .
n i ( r , θ ) n 0 = P ¯ 1 ( Δ ϕ ¯ i ) .
Δ ϕ ˜ i ( p , θ ) = P ˜ [ n i ( r , ϕ ) ; n 0 ] .
D i ( p , θ ) = Δ ϕ ˜ i Δ ϕ ¯ i .
n ( r ) = 1 + h ( r ) ,
T L ( R ) = L ( R ) n ( r ) d s ,
T L ( R ) = L ( R ) n ˆ ( R ) d s .
e ( R , n ˆ ) = T L ( R ) T L ( R ) .
y ( x ) = f ( x ) + O ( 2 )
n ( x , y ) = 1 + h 0 ( x ) + 2 f ( x ) h y ( x ) + ,
h 0 ( x ) = h ( x , 0 ) , h y ( x ) = ( h / y ) | y ¯ 0 .
e = 2 0 l ( f h y + 1 2 f 2 ) d x ,
d 2 f / d x 2 = h y ( x )
f ( 0 ) = f ( 1 ) = 0 .
w l a ,
U ( r ) = exp [ i K 0 ϕ ( r ) ]
2 U ( r ) + [ K 0 n ( r ) ] 2 U ( r ) = 0 .
i K 0 1 2 ϕ ( ϕ ) 2 + n 2 = 0 .
n = 1 + n 1 , ϕ = ϕ 0 + ϕ 1 ,
ϕ 0 = K · R .
ϕ b ¯ ϕ a ¯ = 0 ( b ¯ a ¯ ) n 1 ( x , y ) d s .
2 ϕ 1 + K 0 2 ϕ 1 = i 2 K 0 n 1 .
n ˆ ( U , V ) = ( u / 2 π K 0 ) exp [ i ( K 0 u ) X 0 ] ϕ ˆ 1 ( X 0 , υ , 0 ) ,

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