Abstract

We consider the problem of using the photoacoustic effect to image the optical properties of tissue. A region of tissue is assumed to be illuminated by frequency-modulated light that creates an ultrasonic wave of the same frequency. This wave is detected on a passive array of receiving transducers distributed over a circular or a cylindrical aperture. If the frequency is swept over a broad band (or, equivalently, if we illuminate with a pulse and Fourier transform the response), then a spatial map of a parameter that depends on the optical absorption coefficient of the tissue can be recovered. Analytical inversion formulas are derived in both two and three dimensions. The effects of band-limited data on image quality are also investigated.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Vo-Dinh, ed., Biomedical Photonics Handbook (CRC Press, Boca Raton, Fla., 2003).
  2. R. A. Kruger, P. Liu, “Photoacoustic ultrasound: theory,” in Laser–Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134A, 114–118 (1994).
  3. A. A. Oraevsky, R. O. Esenaliev, S. L. Jacques, F. K. Tittel, “Laser optic-acoustic tomography for medical diagnostics: principles,” in Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, T. Vo-Dinh, eds. Proc. SPIE2676, 22–31 (1996).
    [CrossRef]
  4. C. G. A. Hoelen, F. F. M. de Mul, R. Pongers, A. Dekker, “Three-dimensional photoacoustic imaging of blood vessels in tissue,” Opt. Lett. 23, 648–650 (1998).
    [CrossRef]
  5. F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
    [CrossRef]
  6. M. Kempe, M. Larionov, D. Zaslavsky, A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am. A 14, 1151–1158 (1997).
    [CrossRef]
  7. S. Leveque-Fort, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
    [CrossRef]
  8. J. Mobley, B. M. Cullum, T. Vo-Dinh, “Method for the simultaneous acquisition of photoacoustic and ultrasonic spectra,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems II, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE3911, (2000).
    [CrossRef]
  9. J. Mobley, B. M. Cullum, T. Vo-Dinh, “Ultrasonic diffraction in the design of photoacoustic probes,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems III, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4254, 151–163 (2001).
    [CrossRef]
  10. J. Mobley, T. Vo-Dinh, “Opto-ultrasonic system for generation of ultrasound and optical detection,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems IV, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4615, 173–179 (2002).
    [CrossRef]
  11. S. J. Norton, M. Linzer, “Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical and spherical apertures,” IEEE Trans. Biomed. Eng. BME-28, 202–220 (1981).
    [CrossRef]
  12. Y. Xu, M. Xu, L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography—II: cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829–833 (2002).
    [CrossRef] [PubMed]
  13. A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
    [CrossRef]
  14. In the 600–1000-nm near-infrared window, the intensity distribution I(r) is dominated primarily by multiple scattering within the tissue and to a lesser extent by optical absorption. A reasonable assumption is that the intensity distribution in the tissue, for the purpose of computing I(r), is dominated entirely by the multiple scattering. One would expect that a good first-order approximation should result from computing I(r) under the assumption of a homogeneous-tissue model with a constant (mean) scattering cross section. Then the spatial variations in the optical absorption coefficient α(r), which is what we wish to image, arise entirely from the first-order dependence of fν(r) on α(r), as defined in Eq. (5). Thus we neglect a small second-order contribution that may arise through a dependence on I(r). See Ref. 1 for comprehensive articles on the diffusion of light in tissue.
  15. P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 365, Eq. (7.3.15).
  16. Ref. 15, p. 365.
  17. G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), p. 429.

2002 (1)

Y. Xu, M. Xu, L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography—II: cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829–833 (2002).
[CrossRef] [PubMed]

2001 (1)

S. Leveque-Fort, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

1998 (1)

1997 (1)

1986 (1)

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
[CrossRef]

1981 (1)

S. J. Norton, M. Linzer, “Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical and spherical apertures,” IEEE Trans. Biomed. Eng. BME-28, 202–220 (1981).
[CrossRef]

Boccara, A. C.

S. Leveque-Fort, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

Brooksby, G. W.

F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Cullum, B. M.

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Ultrasonic diffraction in the design of photoacoustic probes,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems III, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4254, 151–163 (2001).
[CrossRef]

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Method for the simultaneous acquisition of photoacoustic and ultrasonic spectra,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems II, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE3911, (2000).
[CrossRef]

de Mul, F. F. M.

Dekker, A.

Esenaliev, R. O.

A. A. Oraevsky, R. O. Esenaliev, S. L. Jacques, F. K. Tittel, “Laser optic-acoustic tomography for medical diagnostics: principles,” in Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, T. Vo-Dinh, eds. Proc. SPIE2676, 22–31 (1996).
[CrossRef]

Genack, A. Z.

Hoelen, C. G. A.

Ingard, K. U.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 365, Eq. (7.3.15).

Jacques, S. L.

A. A. Oraevsky, R. O. Esenaliev, S. L. Jacques, F. K. Tittel, “Laser optic-acoustic tomography for medical diagnostics: principles,” in Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, T. Vo-Dinh, eds. Proc. SPIE2676, 22–31 (1996).
[CrossRef]

Kempe, M.

Kruger, R. A.

R. A. Kruger, P. Liu, “Photoacoustic ultrasound: theory,” in Laser–Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134A, 114–118 (1994).

Larionov, M.

Leveque-Fort, S.

S. Leveque-Fort, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

Linzer, M.

S. J. Norton, M. Linzer, “Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical and spherical apertures,” IEEE Trans. Biomed. Eng. BME-28, 202–220 (1981).
[CrossRef]

Liu, P.

R. A. Kruger, P. Liu, “Photoacoustic ultrasound: theory,” in Laser–Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134A, 114–118 (1994).

Marks, F. A.

F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Mobley, J.

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Ultrasonic diffraction in the design of photoacoustic probes,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems III, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4254, 151–163 (2001).
[CrossRef]

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Method for the simultaneous acquisition of photoacoustic and ultrasonic spectra,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems II, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE3911, (2000).
[CrossRef]

J. Mobley, T. Vo-Dinh, “Opto-ultrasonic system for generation of ultrasound and optical detection,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems IV, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4615, 173–179 (2002).
[CrossRef]

Morse, P. M.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 365, Eq. (7.3.15).

Norton, S. J.

S. J. Norton, M. Linzer, “Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical and spherical apertures,” IEEE Trans. Biomed. Eng. BME-28, 202–220 (1981).
[CrossRef]

Oraevsky, A. A.

A. A. Oraevsky, R. O. Esenaliev, S. L. Jacques, F. K. Tittel, “Laser optic-acoustic tomography for medical diagnostics: principles,” in Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, T. Vo-Dinh, eds. Proc. SPIE2676, 22–31 (1996).
[CrossRef]

Pongers, R.

Pottier, L.

S. Leveque-Fort, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

Selb, J.

S. Leveque-Fort, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

Tam, A. C.

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
[CrossRef]

Tittel, F. K.

A. A. Oraevsky, R. O. Esenaliev, S. L. Jacques, F. K. Tittel, “Laser optic-acoustic tomography for medical diagnostics: principles,” in Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, T. Vo-Dinh, eds. Proc. SPIE2676, 22–31 (1996).
[CrossRef]

Tomlinson, H. W.

F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Vo-Dinh, T.

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Ultrasonic diffraction in the design of photoacoustic probes,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems III, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4254, 151–163 (2001).
[CrossRef]

J. Mobley, T. Vo-Dinh, “Opto-ultrasonic system for generation of ultrasound and optical detection,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems IV, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4615, 173–179 (2002).
[CrossRef]

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Method for the simultaneous acquisition of photoacoustic and ultrasonic spectra,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems II, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE3911, (2000).
[CrossRef]

Wang, L. V.

Y. Xu, M. Xu, L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography—II: cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829–833 (2002).
[CrossRef] [PubMed]

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), p. 429.

Xu, M.

Y. Xu, M. Xu, L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography—II: cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829–833 (2002).
[CrossRef] [PubMed]

Xu, Y.

Y. Xu, M. Xu, L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography—II: cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829–833 (2002).
[CrossRef] [PubMed]

Zaslavsky, D.

IEEE Trans. Biomed. Eng. (1)

S. J. Norton, M. Linzer, “Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical and spherical apertures,” IEEE Trans. Biomed. Eng. BME-28, 202–220 (1981).
[CrossRef]

IEEE Trans. Med. Imaging (1)

Y. Xu, M. Xu, L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography—II: cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829–833 (2002).
[CrossRef] [PubMed]

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

Opt. Commun. (1)

S. Leveque-Fort, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

Opt. Lett. (1)

Rev. Mod. Phys. (1)

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
[CrossRef]

Other (11)

In the 600–1000-nm near-infrared window, the intensity distribution I(r) is dominated primarily by multiple scattering within the tissue and to a lesser extent by optical absorption. A reasonable assumption is that the intensity distribution in the tissue, for the purpose of computing I(r), is dominated entirely by the multiple scattering. One would expect that a good first-order approximation should result from computing I(r) under the assumption of a homogeneous-tissue model with a constant (mean) scattering cross section. Then the spatial variations in the optical absorption coefficient α(r), which is what we wish to image, arise entirely from the first-order dependence of fν(r) on α(r), as defined in Eq. (5). Thus we neglect a small second-order contribution that may arise through a dependence on I(r). See Ref. 1 for comprehensive articles on the diffusion of light in tissue.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), p. 365, Eq. (7.3.15).

Ref. 15, p. 365.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), p. 429.

T. Vo-Dinh, ed., Biomedical Photonics Handbook (CRC Press, Boca Raton, Fla., 2003).

R. A. Kruger, P. Liu, “Photoacoustic ultrasound: theory,” in Laser–Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134A, 114–118 (1994).

A. A. Oraevsky, R. O. Esenaliev, S. L. Jacques, F. K. Tittel, “Laser optic-acoustic tomography for medical diagnostics: principles,” in Biomedical Sensing, Imaging, and Tracking Technologies I, R. A. Lieberman, H. Podbielska, T. Vo-Dinh, eds. Proc. SPIE2676, 22–31 (1996).
[CrossRef]

F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Method for the simultaneous acquisition of photoacoustic and ultrasonic spectra,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems II, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE3911, (2000).
[CrossRef]

J. Mobley, B. M. Cullum, T. Vo-Dinh, “Ultrasonic diffraction in the design of photoacoustic probes,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems III, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4254, 151–163 (2001).
[CrossRef]

J. Mobley, T. Vo-Dinh, “Opto-ultrasonic system for generation of ultrasound and optical detection,” in Biomedical Diagnostic, Guidance, and Surgical Assist Systems IV, T. Vo-Dinh, W. S. Grunfest, D. A. Benaron, eds., Proc. SPIE4615, 173–179 (2002).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Cylindrical array of ultrasonic receiving elements. Illumination is along the axis of the cylinder.

Fig. 2
Fig. 2

(a) Circular array of isotropic receiving elements. The object function, Sν(r, ϕ), is 2-D, that is, independent of the z-coordinate. (b) Circular array of receiving elements long in the vertical dimension, so the signals are integrated with respect to z. Here the physical object function, Sν(r, ϕ, z), is 3-D, but the z-averaged object, S¯ν(r, ϕ), given by Eq. (12), is reconstructed. The mathematical solutions to the problems depicted in (a) and (b) are identical.

Fig. 3
Fig. 3

Reconstruction of the ring anomaly defined mathematically by the ring function [Eq. (52)], which gives rise to the data represented by Eq. (54) for the three indicated bandwidths.

Fig. 4
Fig. 4

Reconstruction of two nearby points, with each point defined mathematically by Eq. (53), which gives rise to the data represented by Eq. (55).

Equations (56)

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

2p(r, t)-1v022p(r, t)t2=-αβCpI(r, t)t,
I(r, t)=I(r)exp(-iωt).
p(r, t)=p(r, ω)exp(-iωt),
2p(r, ω)+ωv02p(r, ω)=iωαβCp I(r).
fν(r)α(r)β(r)v0Cp(r).
Sν(r)fν(r)I(r).
2p(r, k)+k2p(r, k)=ikSν(r).
p(r0, k)=-ikd3rSν(r)gk(r|r0),
gk(r|r0)=exp(ik|r-r0|)4π|r-r0|.
p(k, ϕ0)=-ik0rdr02πdϕSν(r, ϕ)-dzgk(r|r0).
p¯(k, ϕ0)-p(R, ϕ0, z0, k)dz0.
S¯ν(r, ϕ)-Sν(r, ϕ, z)dz.
p¯(k, ϕ0)=-ik0rdr02πdϕS¯ν(r, ϕ)-dz0 gk(r|r0),
|r-r0|=[ρ2+(z-Z)2]1/2,
ρ2=r2+R2-2rR cos(ϕ-ϕ0).
-dzgk(r0|r)=i4H0(1)(kρ),
p(k, ϕ0)=k4 0rdr02πdϕSν(r, ϕ)H0(1)(kρ).
H0(1)(kρ)=m=-exp[im(ϕ0-ϕ)]Jm(kr)Hm(1)(kR),
p(k, ϕ0)=k4m=-0rdr02πdϕSν(r, ϕ)exp[im(ϕ0-ϕ)]Jm(kr)Hm(1)(kR).
Sν(r, ϕ)=n=-sn(r)exp(inϕ),
sn(r)=12π02πSν(r, ϕ)exp(-inϕ)dϕ.
p(k, ϕ0)=n=-pn(k)exp(inϕ0),
pn(k)=12π02πp(k, ϕ0)exp(-inϕ0)dϕ0.
pn(k)=πk20rdrsn(r)Jn(kr)Hn(1)(kR).
Pn(k)2pn(k)πkHn(1)(kR),
Pn(k)=0rdrsn(r)Jn(kr),
sn(r)=0kdkPn(k)Jn(kr).
Sν(r, ϕ)=02πdϕ00dkp(k, ϕ0)K(k, ϕ0|r, ϕ),
K(k, ϕ0|r, ϕ)1π2n=-Jn(kr)Hn(1)(kR)exp[in(ϕ-ϕ0)].
gk(r|r0)=i4πm=-exp[im(ϕ-ϕ0)]×0Jm(μr)Jm(μR) exp(iσ|z-z0|)σ μdμ,
σ=(k2-μ2)1/2if 0μ<ki(μ2-k2)1/2if 0k<μ.
Sν(r, ϕ, z)=n=-sn(r, z)exp(inϕ)
p(k, ϕ0, z0)=n=-pn(k, z0)exp(inϕ0),
sn(r, z)=12π02πSν(r, ϕ, z)exp(-inϕ)dϕ,
pn(k, z0)=12π02πp(k, ϕ0, z0)exp(-inϕ0)dϕ0.
pn(k, z0)=k20rdr-dzsn(r, z)×0μdμJn(μr)Jn(μR) exp(iσ|z-z0|)σ.
exp(iσ|z-z0|)σ=1iπ-exp(ikz(z-z0))kz2+μ2-k2dkz,
pn(k, kz)=-pn(k, z0)exp(ikzz0)dz0,
sn(r, kz)=-sn(r, z)exp(ikzz)dz,
pn(k, kz)=ki0rdrsn(r, kz)×0μdμ Jn(μr)Jn(μR)kz2+μ2-k2.
0μdμJn(μr)Jn(μR)μ2-c2=πiJn(Rc)Hn(1)(rc)/2,R<rπiJn(rc)Hn(1)(Rc)/2,r<R
0μdμ Jn(μr)Jn(μR)μ2-(k2-kz2)=πi2 Jn[r(k2-kz2)1/2]×Hn(1)[R(k2-kz2)1/2],
pn(k, kz)=πk20rdrsn(r, kz)Jn[r(k2-kz2)1/2]×Hn(1)[R(k2-kz2)1/2].
Pn(k, kz)2pn(k, kz)πHn(1)[R(k2-kz2)1/2],
Pn(k, kz)=k0rdrsn(r, kz)Jn[r(k2-kz2)1/2].
kzdkPn(k, kz)Jn[r(k2-kz2)1/2]=0rdrsn(r, kz)kzkdkJn[r(k2-kz2)1/2]×Jn[r(k2-kz2)1/2].
kzkdkJn[r(k2-kz2)1/2]Jn[r(k2-kz2)1/2]=0uduJn(ru)Jn(ru)=1r δ(r-r).
sn(r, kz)=kzPn(k, kz)Jn[r(k2-kz2)1/2]dk.
sn(r, kz)=2πkzpn(k, kz) Jn[r(k2-kz2)1/2]Hn(1)[R(k2-kz2)1/2]dk.
pn(k)=1Nm=0N-1p(k, ϕm)exp(-i2πnm/N),
Sν(rp, ϕq)=n=0N-1sn(rp)exp(i2πnq/N),
sn(r)=k1k2kdkPn(k)Jn(kr),n=0, 1,, N/2,
Sν(r, ϕ)=1r δ(r-r1),
Sν(r, ϕ)=1r δ(r-r1)δ(ϕ-ϕ1).
pring(k, ϕm)=πk2 J0(kr1)H0(1)(kR).
ppnt(k, ϕm)=k4 H0(1){k[r12+R2-2r1R×cos(ϕ1-ϕm)]1/2}.

Metrics