Abstract

Applicability of inverse scattering based imaging procedures can be broadened by developing new approaches exploiting only amplitude data. As a matter of fact, this can open the way to simpler and less expensive measurement set-ups. In this respect, a two-step based procedure for solving electromagnetic nonlinear inverse scattering problems from only amplitude measurements of the total field has been recently proposed [1,2]. However, in these latter both amplitude and phase of the incident field are still required. In this contribution, we show the possibility of achieving this information from the measured amplitude distribution of the incident field on the observation domain. In particular, a three steps imaging technique which exploits only amplitude measurements of the total and incident fields has been developed. The proposed procedure has been tested against benchmark experimental data available in the literature. The obtained results fully confirm the possibility of achieving faithful reconstructions of unknown targets without performing any phase measurements and any approximation on the scattering equations involved in the inverse scattering problems.

© 2007 Optical Society of America

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References

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  1. L. Crocco, M. D’Urso, T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
    [CrossRef]
  2. O. M. Bucci, L. Crocco, M. D’Urso, T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
    [CrossRef]
  3. M. A. Fiddy, M. Testorf, "Inverse scattering method applied to the synthesis of strongly scattering structures," Opt. Express 14, 2037-2046 (2006).
    [CrossRef] [PubMed]
  4. P. C. Chaumet, K. Belkebir and A. Sentenac, "Superresolution of three-dimensional optical imaging by use of evanescent waves," Opt. Lett. 29, 2740-2742 (2004).
    [CrossRef] [PubMed]
  5. R. E. Kleinman and P. M. van den Berg, "A Contrast Source inversion method," Inverse Probl. 13, 1607-1620 (1997).
    [CrossRef]
  6. P. C. Chaumet, K. Belkebir, and R. Lencrerot, "Three-dimensional optical imaging in layered media,"Opt. Express 14, 3415-3426 (2006).
    [CrossRef] [PubMed]
  7. M. H. Maleki, A. J. Devaney, A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 10, 1356-1363 (1992).
    [CrossRef]
  8. M. H. Maleki, A. J. Devaney, "Phase retrieval and intensity-only reconstruction algorithms from optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
    [CrossRef]
  9. A. Litman and K. Belkebir, "Two-dimensional inverse profiling using phaseless data," J. Opt. Soc. Am. A 23, 2737-2746 (2006).
    [CrossRef]
  10. T. Takenaka, D. J. N. Wall, H. Harada, M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
    [CrossRef]
  11. S. Caorsi, A. Massa, M. Pastorino, A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
    [CrossRef]
  12. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, "Role of the support and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999)
    [CrossRef]
  13. O. M. Bucci, C. Gennarelli and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples," IEEE Trans. Antennas Propag 46, 351-359 (1998).
    [CrossRef]
  14. D. L. Misell, "A method for the solution of the phase problem in Electron Microscopy," J. Phys. D 6, L6-L9 (1973).
    [CrossRef]
  15. R. J. Fienup, "Reconstruction of a complex valued object from the modulus of its Fourier transform pairs from noisy data II. The non-linear problem of phase retrieval," J. Integr. Eq. 77-125 (1985).
  16. R. P. Millane, "Phase retrieval in crystallography and optics," J. Opt. Soc. Am. A 7, 394-411 (1990).
    [CrossRef]
  17. R. Harrison, "Phase retrieval in crystallography," J. Opt. Soc. Am. A 7, 1046-1055 (1990)
  18. K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, (2005).
  19. T. Isernia, L. Crocco and M. D’Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (2004).
    [CrossRef]
  20. L. Crocco, M. D’Urso and T. Isernia, "Testing the Contrast Source Extended Born method against real data: the TM case," Inverse Probl. 21, S33- S50, (2005)
    [CrossRef]
  21. T. M. Habashy, R. Groom, W. B. Spies, "Beyond the Born and Rytov Approximations: a Non-Linear Approach to Electromagnetic Scattering," J. Geophys. Research 98, 1795-1775 (1993)
    [CrossRef]
  22. P. Debye, "Das Verhalten von Lichtwellen in der Nahe eines Brenpunktes oder einer Brennlinie," Ann. P. Physik 4, 30- 57 (1909).
  23. O. M. Bucci, T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Science 32, 2123-2138 (1997).
    [CrossRef]
  24. O. M. Bucci, L. Crocco, T. Isernia, "Improving the reconstruction capabilities in inverse scattering problems by exploiting ‘near proximity’ set-ups," J. Opt. Soc. Am. A 16, 1788-1798 (1999).
    [CrossRef]
  25. T. Isernia, G. Leone and R. Pierri, "Phase retrieval of radiated fields," Inverse Probl. 11183-203 (1995).
    [CrossRef]
  26. R. Pierri, A. Cutolo, T. Isernia, I. Izzo, L. Zeni, "Transverse mode analysis of a laser beam by near-and far-field intensity measurements," Appl. Opt. 34, 7974-7978 (1995).
    [CrossRef] [PubMed]
  27. T. Isernia, G. Leone, and R. Pierri, "New technique for estimation of far field from near zone phaseless data," Electron. Lett. 27, 652-654 (1991).
    [CrossRef]
  28. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Inst. of Physics; Bristol & Philadelphia, UK (1998).
    [CrossRef]

2006 (4)

2005 (1)

L. Crocco, M. D’Urso and T. Isernia, "Testing the Contrast Source Extended Born method against real data: the TM case," Inverse Probl. 21, S33- S50, (2005)
[CrossRef]

2004 (3)

2003 (1)

S. Caorsi, A. Massa, M. Pastorino, A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

1999 (2)

1998 (1)

O. M. Bucci, C. Gennarelli and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples," IEEE Trans. Antennas Propag 46, 351-359 (1998).
[CrossRef]

1997 (3)

R. E. Kleinman and P. M. van den Berg, "A Contrast Source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

T. Takenaka, D. J. N. Wall, H. Harada, M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

O. M. Bucci, T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Science 32, 2123-2138 (1997).
[CrossRef]

1995 (2)

1993 (2)

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

T. M. Habashy, R. Groom, W. B. Spies, "Beyond the Born and Rytov Approximations: a Non-Linear Approach to Electromagnetic Scattering," J. Geophys. Research 98, 1795-1775 (1993)
[CrossRef]

1992 (1)

M. H. Maleki, A. J. Devaney, A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 10, 1356-1363 (1992).
[CrossRef]

1991 (1)

T. Isernia, G. Leone, and R. Pierri, "New technique for estimation of far field from near zone phaseless data," Electron. Lett. 27, 652-654 (1991).
[CrossRef]

1990 (2)

R. P. Millane, "Phase retrieval in crystallography and optics," J. Opt. Soc. Am. A 7, 394-411 (1990).
[CrossRef]

R. Harrison, "Phase retrieval in crystallography," J. Opt. Soc. Am. A 7, 1046-1055 (1990)

1973 (1)

D. L. Misell, "A method for the solution of the phase problem in Electron Microscopy," J. Phys. D 6, L6-L9 (1973).
[CrossRef]

1909 (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nahe eines Brenpunktes oder einer Brennlinie," Ann. P. Physik 4, 30- 57 (1909).

Belkebir, K.

Bucci, O. M.

O. M. Bucci, L. Crocco, M. D’Urso, T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, "Improving the reconstruction capabilities in inverse scattering problems by exploiting ‘near proximity’ set-ups," J. Opt. Soc. Am. A 16, 1788-1798 (1999).
[CrossRef]

O. M. Bucci, C. Gennarelli and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples," IEEE Trans. Antennas Propag 46, 351-359 (1998).
[CrossRef]

O. M. Bucci, T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Science 32, 2123-2138 (1997).
[CrossRef]

Caorsi, S.

S. Caorsi, A. Massa, M. Pastorino, A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Chaumet, P. C.

Crocco, L.

Cutolo, A.

D’Urso, M.

O. M. Bucci, L. Crocco, M. D’Urso, T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
[CrossRef]

L. Crocco, M. D’Urso and T. Isernia, "Testing the Contrast Source Extended Born method against real data: the TM case," Inverse Probl. 21, S33- S50, (2005)
[CrossRef]

T. Isernia, L. Crocco and M. D’Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (2004).
[CrossRef]

L. Crocco, M. D’Urso, T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
[CrossRef]

Debye, P.

P. Debye, "Das Verhalten von Lichtwellen in der Nahe eines Brenpunktes oder einer Brennlinie," Ann. P. Physik 4, 30- 57 (1909).

Devaney, A. J.

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

M. H. Maleki, A. J. Devaney, A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 10, 1356-1363 (1992).
[CrossRef]

Fiddy, M. A.

Fienup, R. J.

R. J. Fienup, "Reconstruction of a complex valued object from the modulus of its Fourier transform pairs from noisy data II. The non-linear problem of phase retrieval," J. Integr. Eq. 77-125 (1985).

Gennarelli, C.

O. M. Bucci, C. Gennarelli and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples," IEEE Trans. Antennas Propag 46, 351-359 (1998).
[CrossRef]

Groom, R.

T. M. Habashy, R. Groom, W. B. Spies, "Beyond the Born and Rytov Approximations: a Non-Linear Approach to Electromagnetic Scattering," J. Geophys. Research 98, 1795-1775 (1993)
[CrossRef]

Habashy, T. M.

T. M. Habashy, R. Groom, W. B. Spies, "Beyond the Born and Rytov Approximations: a Non-Linear Approach to Electromagnetic Scattering," J. Geophys. Research 98, 1795-1775 (1993)
[CrossRef]

Harada, H.

T. Takenaka, D. J. N. Wall, H. Harada, M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Harrison, R.

R. Harrison, "Phase retrieval in crystallography," J. Opt. Soc. Am. A 7, 1046-1055 (1990)

Isernia, T.

O. M. Bucci, L. Crocco, M. D’Urso, T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
[CrossRef]

L. Crocco, M. D’Urso and T. Isernia, "Testing the Contrast Source Extended Born method against real data: the TM case," Inverse Probl. 21, S33- S50, (2005)
[CrossRef]

L. Crocco, M. D’Urso, T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
[CrossRef]

T. Isernia, L. Crocco and M. D’Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (2004).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, "Role of the support and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999)
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, "Improving the reconstruction capabilities in inverse scattering problems by exploiting ‘near proximity’ set-ups," J. Opt. Soc. Am. A 16, 1788-1798 (1999).
[CrossRef]

O. M. Bucci, T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Science 32, 2123-2138 (1997).
[CrossRef]

R. Pierri, A. Cutolo, T. Isernia, I. Izzo, L. Zeni, "Transverse mode analysis of a laser beam by near-and far-field intensity measurements," Appl. Opt. 34, 7974-7978 (1995).
[CrossRef] [PubMed]

T. Isernia, G. Leone and R. Pierri, "Phase retrieval of radiated fields," Inverse Probl. 11183-203 (1995).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "New technique for estimation of far field from near zone phaseless data," Electron. Lett. 27, 652-654 (1991).
[CrossRef]

Izzo, I.

Kleinman, R. E.

R. E. Kleinman and P. M. van den Berg, "A Contrast Source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

Lencrerot, R.

Leone, G.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, "Role of the support and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999)
[CrossRef]

T. Isernia, G. Leone and R. Pierri, "Phase retrieval of radiated fields," Inverse Probl. 11183-203 (1995).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "New technique for estimation of far field from near zone phaseless data," Electron. Lett. 27, 652-654 (1991).
[CrossRef]

Litman, A.

Maleki, M. H.

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

M. H. Maleki, A. J. Devaney, A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 10, 1356-1363 (1992).
[CrossRef]

Massa, A.

S. Caorsi, A. Massa, M. Pastorino, A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Millane, R. P.

Misell, D. L.

D. L. Misell, "A method for the solution of the phase problem in Electron Microscopy," J. Phys. D 6, L6-L9 (1973).
[CrossRef]

Pastorino, M.

S. Caorsi, A. Massa, M. Pastorino, A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Pierri, R.

Randazzo, A.

S. Caorsi, A. Massa, M. Pastorino, A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

Saillard, M.

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, (2005).

Savarese, C.

O. M. Bucci, C. Gennarelli and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples," IEEE Trans. Antennas Propag 46, 351-359 (1998).
[CrossRef]

Schatzberg, A.

M. H. Maleki, A. J. Devaney, A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 10, 1356-1363 (1992).
[CrossRef]

Sentenac, A.

Soldovieri, F.

Spies, W. B.

T. M. Habashy, R. Groom, W. B. Spies, "Beyond the Born and Rytov Approximations: a Non-Linear Approach to Electromagnetic Scattering," J. Geophys. Research 98, 1795-1775 (1993)
[CrossRef]

Takenaka, T.

T. Takenaka, D. J. N. Wall, H. Harada, M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Tanaka, M.

T. Takenaka, D. J. N. Wall, H. Harada, M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Testorf, M.

van den Berg, P. M.

R. E. Kleinman and P. M. van den Berg, "A Contrast Source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

Wall, D. J. N.

T. Takenaka, D. J. N. Wall, H. Harada, M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Zeni, L.

Ann. P. Physik (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nahe eines Brenpunktes oder einer Brennlinie," Ann. P. Physik 4, 30- 57 (1909).

Appl. Opt. (1)

Electron. Lett. (1)

T. Isernia, G. Leone, and R. Pierri, "New technique for estimation of far field from near zone phaseless data," Electron. Lett. 27, 652-654 (1991).
[CrossRef]

IEEE Trans. Antennas Propag (1)

O. M. Bucci, C. Gennarelli and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples," IEEE Trans. Antennas Propag 46, 351-359 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

S. Caorsi, A. Massa, M. Pastorino, A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic Algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2752 (2003).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. Lett. (1)

T. Isernia, L. Crocco and M. D’Urso, "New tools and series for forward and inverse scattering problems in lossy media," IEEE Trans. Geosci. Remote Sens. Lett. 1, 331-337 (2004).
[CrossRef]

Inverse Probl. (3)

L. Crocco, M. D’Urso and T. Isernia, "Testing the Contrast Source Extended Born method against real data: the TM case," Inverse Probl. 21, S33- S50, (2005)
[CrossRef]

R. E. Kleinman and P. M. van den Berg, "A Contrast Source inversion method," Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

T. Isernia, G. Leone and R. Pierri, "Phase retrieval of radiated fields," Inverse Probl. 11183-203 (1995).
[CrossRef]

J. Geophys. Research (1)

T. M. Habashy, R. Groom, W. B. Spies, "Beyond the Born and Rytov Approximations: a Non-Linear Approach to Electromagnetic Scattering," J. Geophys. Research 98, 1795-1775 (1993)
[CrossRef]

J. Opt. Soc. Am. A (9)

J. Phys. D (1)

D. L. Misell, "A method for the solution of the phase problem in Electron Microscopy," J. Phys. D 6, L6-L9 (1973).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

T. Takenaka, D. J. N. Wall, H. Harada, M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Radio Science (1)

O. M. Bucci, T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Science 32, 2123-2138 (1997).
[CrossRef]

Other (3)

K. Belkebir and M. Saillard, "Special section: Testing inversion algorithms against real data: inhomogeneous targets," Inverse Probl. 21, (2005).

R. J. Fienup, "Reconstruction of a complex valued object from the modulus of its Fourier transform pairs from noisy data II. The non-linear problem of phase retrieval," J. Integr. Eq. 77-125 (1985).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Inst. of Physics; Bristol & Philadelphia, UK (1998).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Reference geometry.

Fig. 2.
Fig. 2.

Real (a) and imaginary (b) part of the measured incident field (red-dotted line), the estimated one (green-solid line) and the starting guess (blue-dotted line).

Fig. 3.
Fig. 3.

Amplitude distribution of the measured scattered field (a) and of the estimated one (b)

Fig. 4.
Fig. 4.

Phase distribution of the measured scattered field (a) and of the estimated one (b)

Fig. 5.
Fig. 5.

Real part of the estimated contrast function at f=4GHz (a), and f=8GHz (b)

Equations (11)

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J ( r , θ i ) p ( r ) E inc , i ( r , θ i ) = p ( r ) [ D g ( r r ´ ) J ( r ´ , θ i ) d r ´ J ( r , θ i ) D g ( r r ´ ) d r ´ ]
= p ( r ) A i mod [ J ( θ i ) ] r = ( r , θ ) D
E tot ( θ 0 , θ i ) E inc , e ( θ 0 , θ i ) = E s ( θ 0 , θ i )
= D g ( R o r ´ ) J ( r ´ , θ i ) d r ´ = A e [ J ( θ i ) ] R o = ( R o , θ 0 ) Γ o
p ( r ) = χ ( r ) [ 1 χ ( r ) D g ( r r ´ ) d r ´ ] 1 .
E inc ( r ´ , θ ´ ) = n = P P a n H n ( 2 ) ( β r ´ ) exp ( j n θ ´ ) ,
P = 2 β c + Δ N ,
M 2 ( r , θ ) n = P P a n H n ( 2 ) ( β r ) exp ( j n θ ) 2 2 ( r , θ ) Γ o .
B ( E s ) = Δ E s 2 + 2ℜ e [ E s ( E inc , e ) * ] ,
B ( E s ) = E tot ( θ o , θ i ) 2 E inc , e ( θ o , θ i ) 2 .
Φ ( p , J v ) = v = 1 N v J v pE inc v p A i mod [ J v ] 2 E inc v 2 + v = 1 N v E s v A e [ J v ] 2 E s v 2 + τ p 2 ,

Metrics