## A novel phase retrieval technique based on propagation diversity via a dielectric slab

Optics Express, Vol. 16, Issue 10, pp. 7418-7427 (2008)

http://dx.doi.org/10.1364/OE.16.007418

Acrobat PDF (131 KB)

### 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

## 1. Introduction

2. D. L. Misell, “A method for the solution of the phase retrieval problem in electronic microscopy,” J. Phys. D: Appl. Physics **6**, L6–L9 (1973). [CrossRef]

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**, 2332–3241, (2007). [CrossRef]

7. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “On the local minima in phase reconstruction algorithms,” *Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

*amplitude*of the measured and the reconstructed near-field [2

2. D. L. Misell, “A method for the solution of the phase retrieval problem in electronic microscopy,” J. Phys. D: Appl. Physics **6**, L6–L9 (1973). [CrossRef]

8. J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

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]

*square amplitude*of field as data of the problem. In this way, the phase retrieval problem is formulated as the inversion of a quadratic operator [7

7. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “On the local minima in phase reconstruction algorithms,” *Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

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]

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]

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]

7. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “On the local minima in phase reconstruction algorithms,” *Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

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]

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]

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]

*Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

**16**, 1845–1856 (1999). [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**, 3135–2141 (2005). [CrossRef]

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]

*single*scanning 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, a measurement is performed under free-space propagation and the other one by exploiting a dielectric slab with known properties (dielectric permittivity and thickness) that fills partly the space between the aperture and the measurement plane.

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.

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]

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.

## 2. The Formulation

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**, 3135–2141 (2005). [CrossRef]

*xy*plane and of extent 2

*a*and 2

*b*along the

*x*- and

*y*-axis respectively. The aperture field is assumed to have only the

*y*-directed transverse component.

*y*-component of the radiated field over the plane at

*z*=

*z*

_{1}can be expressed as

*u*=

*β*sin

*θ*cos

*φ*,

*ν*=

*β*sin

*θ*sin

*φ*,

*β*=2

*π*/

*λ*and λ is the wavelength.

*Ê*(

*u*,

*ν*) is the plane wave spectrum (PWS) of the aperture field and the time dependence exp(

*jωt*) has been assumed and omitted.

**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**, 3135–2141 (2005). [CrossRef]

*jw*(

*z*

_{2}–

*z*

_{1})) becomes more and more relevant to ensure diversity between the two data sets so that the information content increases.

*z*=

*z*

_{1}. In particular, it exploits as a first measurement the same of eq. (1). Differently, the second measurement is achieved by partially filling the space between the aperture plane and the measurement plane located at

*z*=

*z*

_{1}with a dielectric slab of relative dielectric permittivity

*ε*and thickness

_{r}*d*. By neglecting depolarization effects, the second near field measurement is given as

*τ*(

*u*,

*ν*)=[

*u*

^{2}

*T*(

_{0}*u*,

*ν*)+

*ν*

^{2}

*T*(

_{p}*u*,

*ν*)]/(

*u*

^{2}+

*ν*

^{2}) accounts for the transmission of the PWS through the slab (apart from the exp(-

*jw*) term),

_{d}d*T*(

_{p}*u*,

*ν*),

*T*

_{0}(

*u*,

*ν*) are the transmission coefficients in the perpendicular and horizontal polarization with respect to the incidence plane of each plane wave, respectively. It can be verified easily that in many circumstances the phase function of

*τ*(

*u*,

*ν*) is slowly varying so that in (3) it can be now appreciated how the term exp[(-

*j*(

*w*-

_{d}*w*)

*d*], where

*jw*(

*z*

_{2}–

*z*

_{1})] in (2).

*L*

_{1}=

*T*

_{1}and

*L*

_{2}=

*T*

_{2},

*L*

_{2}=

*T̂*

_{2}for the two-planes and the single plane/dielectric slab case, respectively. Thus the phase-retrieval problem at hand can be cast as the solution of the equation (4), which is searched for as the

*global minimum*of the functional

^{2}is the usual quadratic norm in the data space,

*M̃*

^{2}

_{1}and

*M̃*

^{2}

_{2}are the measurement errors and noise affected versions of the actual square amplitudes

*M*

^{2}

_{1}and

*M*

^{2}

_{2}, respectively.

*Ê*(

*u*,

*ν*)≈0 outside a circular domain Ω of radius

*ρ*comprising all the directions

*u*

^{2}+

*v*

^{2}≤

*ρ*

^{2}contained within the

*visible domain*, namely

*ρ*<

*β*. Due to the Fourier transform relationship between the PWS

*Ê*(

*u*,

*ν*) and the aperture field, that is in turn is defined on the rectangular domain of extent 2

*a*×2

*b*, the

*Ê*(

*u*,

*ν*) PWS function is amenable of a representation through a Shannon sampling series [4

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]

**44**, 701–710, (1996). [CrossRef]

**53**, 3135–2141 (2005). [CrossRef]

*Ê*=

_{nm}*Ê*(

*nπ*/

*a*,

*mπ*/

*b*) and

*sin c*(

*x*)=

*sin*(

*x*)/

*x*.

*N*and

*M*in (6) are chosen in order to satisfy the relation (

*Nπ*/

*a*)

^{2}+(

*Mπ*/

*b*)

^{2}≤

*ρ*

^{2}; this relation means that the spectrum samples searched for as actual unknowns are the ones belonging to the a priori known circular domain Ω of radius

*ρ*where the spectrum is assumed to be significantly different from zero. The square amplitude of the near field are represented through their samples at the uniform step of a quarter of a wavelength [4

**47**, 792–802 (1999). [CrossRef]

**44**, 701–710, (1996). [CrossRef]

**53**, 3135–2141 (2005). [CrossRef]

*Ê*. This operation, as well as the other ones of the iterative procedure, are performed in an efficient way thanks to the FFT technique [4

_{nn}**47**, 792–802 (1999). [CrossRef]

**44**, 701–710, (1996). [CrossRef]

*Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

**16**, 1845–1856 (1999). [CrossRef]

**13**, 1546–1556 (1996). [CrossRef]

## 3. The two planes implementation

*Ê*and thus the local minima problem arises. In fact, the large number of unknown parameters to be searched for such a problem makes it feasible only the adoption of deterministic minimization schemes in order to achieve the global minimum of the functional in (5). However, these schemes are able only to achieve a minimum closer to the starting point of the procedure and can be trapped into a local minimum that can be completely different from the global one.

_{nn}*Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

**16**, 1845–1856 (1999). [CrossRef]

**13**, 1546–1556 (1996). [CrossRef]

*ν*=

*0*has a more general validity, since, in virtue of the dependence of

*w*from the variable

*u*+

^{2}*ν*, this behaviour is similar to the one along all the cut-lines passing through the origin of the (

^{2}*u*,

*ν*) plane.

*2a*=

*2b*=

*14*λ. The square amplitude of the near field is measured over two planes at

*z*

_{1}=10

*λ*and

*z*

_{2}=6

*λ*at 128×128 measurement points equally spaced by λ/4.

*Ê*(

*u*,

*ν*) are considered [4

**47**, 792–802 (1999). [CrossRef]

*Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

**47**, 792–802 (1999). [CrossRef]

*Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

*Ê*(

*u*,

*ν*) are reliably estimated, the solution is improved by gradually increasing the number of unknowns until all the samples falling within the domain Ω with radius

*ρ*=0.8

*β*are considered. Finally, the result is improved by adopting the weighted formulation [7

*Radio Science* , **31**, 1887–1899 (1996). [CrossRef]

**16**, 1845–1856 (1999). [CrossRef]

**13**, 1546–1556 (1996). [CrossRef]

*z*

_{2}=6

*λ*(see Fig. 2).

## 4. The single plane/dielectric slab implementation

*ε*and thickness

_{r}*d*, when the two measurements of the intensity of the near field are performed over the same scanning plane at

*z*=

*z*

_{1}.

*z*<

*z*. This can be also understood by the considerations below, when we first neglect the effect of the transmission coefficient

_{1}*τ*(

*u*,

*ν*).

*j*(

*w*-

_{d}*w*)

*d*] playing the same role as the above exp[

*jw*(

*z*

_{1}-

*z*

_{2})] term for the two-planes implementation. This consideration drives the choice of the relative dielectric permittivity and the thickness of the so that the phase of the exp[

*jw*(

*z*

_{1}-

*z*

_{2})] and exp[-

*j*(

*w*-

_{d}*w*)

*d*] terms behave in a very similar way. Once the value of the thickness of the slab

*d*=6

*λ*is assumed, the similarity between the two phase terms is achieved by choosing the relative dielectric permittivity

*ε*=6 (see Fig. 3).

_{r}*L*

_{2}(

*Ê*)=

*T*

_{1}(exp[-

*j*(

*w*-

_{d}*w*)

*d*]

*Ê*,

*ε*=6 and

_{r}*d*=6λ the global minimum is achieved. For sake of brevity, we do not report any figure of reconstruction relative to this test case.

*τ*(

*u*,

*ν*) in the operator

*T̂*

_{2}and, for the minimization, we follow the same two-step minimization strategy as the above Section with a progressive increase in the number of the searched unknowns followed by the exploitation of the weighted formulation in (8). This strategy allows us to achieve the actual solution as depicted by Figs. 4 and 5, where an excellent comparison is shown between the actual and the retrieved PWS along the cut lines at

*u*and

*ν*constant passing through the point of maximum PWS modulus.

## 5. Conclusions

*different conditions*under which the propagation of the near field from the aperture plane and the measurement plane occurs. In particular, one condition is concerned with the free-space propagation, while the other one is concerned with a propagation where the space between the aperture and measurement planes is partly filled with a dielectric slab of known properties.

**47**, 792–802 (1999). [CrossRef]

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]

## References and links

1. | C. Giacovazzo, |

2. | D. L. Misell, “A method for the solution of the phase retrieval problem in electronic microscopy,” J. Phys. D: Appl. Physics |

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 |

4. | R. Pierri, G. D’Elia, and F. Soldovieri, “A two probes scanning phaseless near-field far-field transformation technique,” IEEE Trans. Antennas Propag. |

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. , |

6. | G. Hislop, G.C. James, and A. Hellicar, “Phase Retrieval of Scattered Fields,” IEEE Trans. Antennas Propag. |

7. | T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “On the local minima in phase reconstruction algorithms,” |

8. | J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

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. , |

10. | G.R. Brady and J.R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express |

11. | S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Review of Scientific Instruments |

12. | G. Leone, R. Pierri, and F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A |

13. | T. Isernia, G. Leone, and R. Pierri, “Radiation pattern evaluation from near-field intensities on planes,” IEEE Trans. Antennas Propag. |

14. | F. Soldovieri, A. Liseno, G. D’Elia, and R. Pierri, “Global convergence of phase retrieval by quadratic approach,” IEEE Trans. Antennas Propag. |

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 |

16. | D. Luenberger, |

17. | H-E. Hwang and P. Han, “Signal reconstruction algorithm based on a single intensity in the Fresnel domain,” Opt. Express |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 25, 2007

Revised Manuscript: November 20, 2007

Manuscript Accepted: November 20, 2007

Published: May 7, 2008

**Citation**

Francesco Soldovieri, Giovanni Leone, and Rocco Pierri, "A novel phase retrieval technique based on propagation diversity via a dielectric slab," Opt. Express **16**, 7418-7427 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7418

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### References

- C. Giacovazzo, Direct Phasing in Cristallography (Oxford Press 1988).
- 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]
- 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).
- 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]
- 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]
- T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Sci. 31, 1887-1899 (1996). [CrossRef]
- J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- 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]
- 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]
- S. Marchesini, "A unified evaluation of iterative projection algorithms for phase retrieval," Rev. Sci. Instrum. 78, 011301 1-10, (2007).
- 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]
- 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. 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).
- 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]

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