Abstract

This paper deals with a novel technique to determine the far field of an aperture starting from the knowledge of two near-field intensity data sets collected over the same measurement plane. The diversity between the two intensity data sets is achieved by ensuring different conditions of the near field propagation between the aperture and the measurement plane. In particular, one measurement is performed under free-space propagation condition while the second one is performed by exploiting a dielectric slab, with known properties, filling partly the space between the aperture and the measurement plane. A phase retrieval technique, that faces a non linear inverse problem, is solved by assuming as unknown the plane wave spectrum of the aperture field. The feasibility of the novel approach is presented also in comparison with the usual near field phase retrieval technique exploiting measurements of the near field intensity over two scanning planes.

© 2008 Optical Society of America

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  1. C. Giacovazzo, Direct Phasing in Cristallography (Oxford Press 1988).
  2. D. L. Misell, "A method for the solution of the phase retrieval problem in electronic microscopy," J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
    [CrossRef]
  3. B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST flight and test-bed telescope," in Space Telescopes and Instrumentation I: Optical Infrared and Miliimeter, J. C. Mather, H. A. McEwan and M. W. M de Graauw, eds. Proc. SPIE 6265 (2006).
  4. R. Pierri, G. D�??Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. 47, 792-802 (1999).
    [CrossRef]
  5. S. R. Razavi and Y. Rahmat-Samii, "A new look at phaseless planar near-field measurements: limitations, simulations, measurements, and a hybrid solution," IEEE Antennas Propag. Mag. 49, 170-178 (2007).
    [CrossRef]
  6. G. Hislop, G. C. James, and A. Hellicar, "Phase Retrieval of Scattered Fields," IEEE Trans. Antennas Propag. 55, 2332-2341 (2007).
    [CrossRef]
  7. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996).
    [CrossRef]
  8. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  9. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "Role of support information and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A. 16, 1845-1856 (1999).
    [CrossRef]
  10. G. R. Brady and J. R. Fienup, "Nonlinear optimization algorithm for retrieving the full complex pupil function," Opt. Express 14, 474-486 (2006).
    [CrossRef] [PubMed]
  11. S. Marchesini, "A unified evaluation of iterative projection algorithms for phase retrieval," Rev. Sci. Instrum. 78, 011301 1-10, (2007).
  12. G. Leone, R. Pierri, and F. Soldovieri, "Reconstruction of complex signals from intensities of Fourier-transform pairs," J. Opt. Soc. Am. A 13, 1546-1556 (1996).
    [CrossRef]
  13. T. Isernia, G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. Antennas Propag. 44, 701-710 (1996).
    [CrossRef]
  14. F. Soldovieri, A. Liseno, G. D�??Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. 53, 3135-3141 (2005).
    [CrossRef]
  15. J. Ala-Laurinaho, P. R. Foster, G. J. Junkin, T. Hirvonen, A. Letho, D. H. Martin, A. D. Olver, R. Padman, C. Parini, A. V. Raisanen, T. Sehm, J. Tuovinen, and R. J. Wylde, "Comparison of antenna measurement techniques for 200-1500 GHz," in Proc. 20th ESTEC Antenna Workshop Millimeter Wave Antenna Technol. Antenna Measurements, Noordwijk, (The Netherlands, June 1997), pp. 345-351.
  16. D. Luenberger, Linear and Nonlinear Programming ( Reading, MA: Addison-Wesley, 1987).
  17. H-E. Hwang and P. Han, "Signal reconstruction algorithm based on a single intensity in the Fresnel domain," Opt. Express 15, 3766-3776 (2007).
    [CrossRef] [PubMed]

2007 (3)

S. R. Razavi and Y. Rahmat-Samii, "A new look at phaseless planar near-field measurements: limitations, simulations, measurements, and a hybrid solution," IEEE Antennas Propag. Mag. 49, 170-178 (2007).
[CrossRef]

G. Hislop, G. C. James, and A. Hellicar, "Phase Retrieval of Scattered Fields," IEEE Trans. Antennas Propag. 55, 2332-2341 (2007).
[CrossRef]

H-E. Hwang and P. Han, "Signal reconstruction algorithm based on a single intensity in the Fresnel domain," Opt. Express 15, 3766-3776 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

F. Soldovieri, A. Liseno, G. D�??Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. 53, 3135-3141 (2005).
[CrossRef]

1999 (2)

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

R. Pierri, G. D�??Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. 47, 792-802 (1999).
[CrossRef]

1996 (3)

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. Antennas Propag. 44, 701-710 (1996).
[CrossRef]

G. Leone, R. Pierri, and F. Soldovieri, "Reconstruction of complex signals from intensities of Fourier-transform pairs," J. Opt. Soc. Am. A 13, 1546-1556 (1996).
[CrossRef]

1982 (1)

1973 (1)

D. L. Misell, "A method for the solution of the phase retrieval problem in electronic microscopy," J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
[CrossRef]

Brady, G. R.

D???Elia, G.

F. Soldovieri, A. Liseno, G. D�??Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. 53, 3135-3141 (2005).
[CrossRef]

R. Pierri, G. D�??Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. 47, 792-802 (1999).
[CrossRef]

Fienup, J. R.

Han, P.

Hellicar, A.

G. Hislop, G. C. James, and A. Hellicar, "Phase Retrieval of Scattered Fields," IEEE Trans. Antennas Propag. 55, 2332-2341 (2007).
[CrossRef]

Hislop, G.

G. Hislop, G. C. James, and A. Hellicar, "Phase Retrieval of Scattered Fields," IEEE Trans. Antennas Propag. 55, 2332-2341 (2007).
[CrossRef]

Hwang, H-E.

Isernia, T.

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

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. Antennas Propag. 44, 701-710 (1996).
[CrossRef]

James, G. C.

G. Hislop, G. C. James, and A. Hellicar, "Phase Retrieval of Scattered Fields," IEEE Trans. Antennas Propag. 55, 2332-2341 (2007).
[CrossRef]

Leone, G.

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

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996).
[CrossRef]

G. Leone, R. Pierri, and F. Soldovieri, "Reconstruction of complex signals from intensities of Fourier-transform pairs," J. Opt. Soc. Am. A 13, 1546-1556 (1996).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. Antennas Propag. 44, 701-710 (1996).
[CrossRef]

Liseno, A.

F. Soldovieri, A. Liseno, G. D�??Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. 53, 3135-3141 (2005).
[CrossRef]

Misell, D. L.

D. L. Misell, "A method for the solution of the phase retrieval problem in electronic microscopy," J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
[CrossRef]

Pierri, R.

F. Soldovieri, A. Liseno, G. D�??Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. 53, 3135-3141 (2005).
[CrossRef]

R. Pierri, G. D�??Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. 47, 792-802 (1999).
[CrossRef]

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

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996).
[CrossRef]

G. Leone, R. Pierri, and F. Soldovieri, "Reconstruction of complex signals from intensities of Fourier-transform pairs," J. Opt. Soc. Am. A 13, 1546-1556 (1996).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. Antennas Propag. 44, 701-710 (1996).
[CrossRef]

Rahmat-Samii, Y.

S. R. Razavi and Y. Rahmat-Samii, "A new look at phaseless planar near-field measurements: limitations, simulations, measurements, and a hybrid solution," IEEE Antennas Propag. Mag. 49, 170-178 (2007).
[CrossRef]

Razavi, S. R.

S. R. Razavi and Y. Rahmat-Samii, "A new look at phaseless planar near-field measurements: limitations, simulations, measurements, and a hybrid solution," IEEE Antennas Propag. Mag. 49, 170-178 (2007).
[CrossRef]

Soldovieri, F.

F. Soldovieri, A. Liseno, G. D�??Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. 53, 3135-3141 (2005).
[CrossRef]

R. Pierri, G. D�??Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. 47, 792-802 (1999).
[CrossRef]

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

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996).
[CrossRef]

G. Leone, R. Pierri, and F. Soldovieri, "Reconstruction of complex signals from intensities of Fourier-transform pairs," J. Opt. Soc. Am. A 13, 1546-1556 (1996).
[CrossRef]

Appl. Opt. (1)

IEEE Antennas Propag. Mag. (1)

S. R. Razavi and Y. Rahmat-Samii, "A new look at phaseless planar near-field measurements: limitations, simulations, measurements, and a hybrid solution," IEEE Antennas Propag. Mag. 49, 170-178 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

G. Hislop, G. C. James, and A. Hellicar, "Phase Retrieval of Scattered Fields," IEEE Trans. Antennas Propag. 55, 2332-2341 (2007).
[CrossRef]

R. Pierri, G. D�??Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. 47, 792-802 (1999).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. Antennas Propag. 44, 701-710 (1996).
[CrossRef]

F. Soldovieri, A. Liseno, G. D�??Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. 53, 3135-3141 (2005).
[CrossRef]

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

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

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

J. Phys. D: Appl. Phys. (1)

D. L. Misell, "A method for the solution of the phase retrieval problem in electronic microscopy," J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
[CrossRef]

Opt. Express (2)

Radio Sci. (1)

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996).
[CrossRef]

Other (5)

S. Marchesini, "A unified evaluation of iterative projection algorithms for phase retrieval," Rev. Sci. Instrum. 78, 011301 1-10, (2007).

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST flight and test-bed telescope," in Space Telescopes and Instrumentation I: Optical Infrared and Miliimeter, J. C. Mather, H. A. McEwan and M. W. M de Graauw, eds. Proc. SPIE 6265 (2006).

C. Giacovazzo, Direct Phasing in Cristallography (Oxford Press 1988).

J. Ala-Laurinaho, P. R. Foster, G. J. Junkin, T. Hirvonen, A. Letho, D. H. Martin, A. D. Olver, R. Padman, C. Parini, A. V. Raisanen, T. Sehm, J. Tuovinen, and R. J. Wylde, "Comparison of antenna measurement techniques for 200-1500 GHz," in Proc. 20th ESTEC Antenna Workshop Millimeter Wave Antenna Technol. Antenna Measurements, Noordwijk, (The Netherlands, June 1997), pp. 345-351.

D. Luenberger, Linear and Nonlinear Programming ( Reading, MA: Addison-Wesley, 1987).

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Figures (7)

Fig. 1.
Fig. 1.

Phase variation of exp[j(w(u,ν)-w(0,0))(z 1z 2)] for distances (z 1z 2) ranging from 1λ to 9λ with a step of 1λ.

Fig. 2.
Fig. 2.

Comparison between the ideal near field (blue line) and the retrieved near field (dashed red line) at the cut lines y=0 and x=0 of the plane at z 2=6λ.

Fig. 3
Fig. 3

Comparison between the phase of (exp{j[w(u,ν)-w(0,0)](z 1-z 2)} for z 1=10λ, z 2=6λ (blue line) and exp{j[wd (u,ν)-w(u,ν)-wd (0.0)+w(0,0)]d} for εr =6,d=6λ (red line).

Fig. 4.
Fig. 4.

Comparison between the ideal PWS (blue line) and the retrieved PWS (red line) at the cut line at constant v and passing through the point where the modulus of the PWS attains its maximum. Noise-free data.

Fig. 5.
Fig. 5.

Comparison between the ideal PWS (blue line) and the retrieved PWS (red line) at the cut line at constant u and passing through the point where the modulus of the PWS attains its maximum. Noise-free data.

Fig. 6.
Fig. 6.

Comparison between the ideal PWS (blue line) and the retrieved PWS (red line) at the cut line at constant v and passing through the point where the modulus of the PWS attains its maximum. Noisy data.

Fig. 7.
Fig. 7.

Comparison between the ideal PWS (blue line) and the retrieved PWS (red line) at the cut line at constant u and passing through the point where the modulus of the PWS attains its maximum. Noisy data.

Equations (10)

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

E ( x , y , z 1 ) = E ̂ ( u , v ) exp [ j ( u x + v y ) ] exp ( j w z 1 ) dudv = T 1 ( E ̂ )
E ( x , y , z 2 ) = E ̂ ( u , v ) exp [ j ( ux + vy ) ] exp ( jwz 2 ) dudv =
T 2 ( E ̂ ) = T 1 ( exp [ jw ( z 2 z 1 ) ] E ̂ ) = T 1 ( exp [ jw ( z 1 z 2 ) ] E ̂ )
E d ( x , y , z 1 ) = τ ( u , v ) E ̂ ( u , v ) exp [ j ( ux + vy ) ] exp [ j ( w d w ) d ] exp ( jwz 1 ) dudv
= T ̂ 2 ( E ̂ ) = T 1 ( exp [ j ( w d w ) d ] τ E ̂ )
( M 1 2 , M 2 2 ) = ( L 1 E ̂ 2 , L 2 E ̂ 2 )
Φ ( E ̂ ) = L 1 E ̂ 2 M ~ 1 2 2 + L 2 E ̂ 2 M ~ 2 2 2
E ̂ ( u , v ) = n = N N m = M M E ̂ nn sin c ( au n π ) sin c ( bv m π )
E a ( x , y ) = cos ( π x ( 2 a ) ) cos ( π y ( 2 b ) ) exp ( j β x sin ( 5 π 180 ) j 0.03 y 2 )
Ψ ( E ̂ ) = i , j ( L 1 E ̂ ij 2 M ~ 1 ij 2 ) 2 M ~ 1 ij 2 + ( L 2 E ̂ ij 2 M ~ 2 ij 2 ) 2 M ~ 2 ij 2

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