## Wavefield imaging via iterative retrieval based on phase modulation diversity |

Optics Express, Vol. 19, Issue 19, pp. 18621-18635 (2011)

http://dx.doi.org/10.1364/OE.19.018621

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### Abstract

We present a fast and robust non-interferomentric wavefield retrieval approach suitable for imaging of both amplitude and phase distributions of scalar coherent beams. It is based on the diversity of the intensity measurements obtained under controlled astigmatism and it can be easily implemented in standard imaging systems. Its application for imaging in microscopy is experimentally studied. Relevant examples illustrate the approach capabilities for image super-resolution, numerical refocusing, quantitative imaging and phase mapping.

© 2011 OSA

## 1. Introduction

1. U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A **11**, 2011–2015 (1994). [CrossRef]

5. V. Mico, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express **16**, 19260–19270 (2008). [CrossRef]

11. Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, “Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm,” Opt. Express **11**, 3234–3241 (2003). [CrossRef] [PubMed]

12. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. **22**, 1268–1270 (1997). [CrossRef] [PubMed]

13. I. Yamaguchi, J. ichi Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. **40**, 6177–6186 (2001). [CrossRef]

14. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett. **13**, 619–621 (1988). [CrossRef] [PubMed]

16. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A **75**, 043805 (2007). [CrossRef]

16. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A **75**, 043805 (2007). [CrossRef]

18. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. **85**, 4795–4797 (2004). [CrossRef]

17. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. **93**, 023903 (2004). [CrossRef] [PubMed]

18. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. **85**, 4795–4797 (2004). [CrossRef]

19. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. **91**, 203902 (2003). [CrossRef] [PubMed]

23. J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt. Express **19**, 6064–6077 (2011). [CrossRef] [PubMed]

23. J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt. Express **19**, 6064–6077 (2011). [CrossRef] [PubMed]

23. J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt. Express **19**, 6064–6077 (2011). [CrossRef] [PubMed]

24. L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration correction for an arbitrary linear imaging system,” Phys. Rev. Lett. **87**, 123902 (2001). [CrossRef] [PubMed]

26. T. C. Petersen and V. J. Keast, “Astigmatic intensity equation for electron microscopy based phase retrieval,” Ultramicroscopy **107**, 635–643 (2007). [CrossRef] [PubMed]

**19**, 6064–6077 (2011). [CrossRef] [PubMed]

*a priori*knowledge about the object, that allows its implementation in imaging systems. The proposed technique and its performance are briefly discussed in Section 2. An optical system for its experimental implementation, which can be easily inserted in other imaging devices, is described in Section 3. The Section 4 is devoted to the applications of the proposed approach for imaging in the context of transmission microscopy. In particular, we study the method’s capabilities for relevant imaging tasks such as resolution enhancement of the retrieved image, numerical refocusing and quantitative phase mapping. The problem of detection and localization of micron-sized particles is also studied by using the proposed approach, which experimental results are compared with the ones provided by a conventional in-line holography technique. Relevant parameters (e.g. shape and refractive index) of such miro-particles are determined form the retrieved signal and compared with the values quoted by the manufacturer. Concluding remarks are drawn in Section 5.

## 2. Algorithm for iterative wavefield retrieval

*f*(

*x*,

*y*) under study. The modulated beam is free-space propagated a fixed distance

*z*and it is expressed as

*f*(

_{out}**r**

*;*

_{o}*j*) = F

*[*

_{z}*f*(

**r**

*)*

_{in}*L*(

_{j}**r**

*)](*

_{in}**r**

*) : being*

_{o}**r**

_{in}_{,}

*= (*

_{o}*x*

_{in}_{,}

*,*

_{o}*y*

_{in}_{,}

*) the input and output spatial coordinates,*

_{o}*j*is an integer and

*λ*is the wavelength. Here

*L*(

_{j}**r**

*) stands for the transmittance function of the convergent cylindrical lens with variable focal length f*

_{in}*=*

_{j}*α*that controls the astigmatism of the constraint images |

_{j}z*f*(

_{out}**r**

*;*

_{o}*j*)| (square root of the measured intensity) and, therefore, provides the required information diversity.

**19**, 6064–6077 (2011). [CrossRef] [PubMed]

*j*= 1, 2, ... ,

*M*), which includes the intensity distribution of the object field

*f*(

**r**

*) whose direct acquisition is not always possible. While, in this work, we generalize this approach to retrieve both amplitude and phase distributions of the object field. In this case the diversity required for accurate signal recovery is provided by modulating the object field*

_{in}*f*(

**r**

*) with an astigmatic phase in two orthogonal directions. For instance, a cylindrical lens*

_{in}*L*(

_{j}**r**

*) oriented according to the*

_{in}*x*– and

*y*–axis can be used for odd and even

*j*, respectively. Notice that the orientation of the cylindrical lenses (e.g. along the

*x*– and

*y*–axis) is arbitrary if they remain orthogonally oriented. Nevertheless, because the constraint images are often registered in a square area of the CCD chip (as it is in our case), it could be useful rotating them by an angle

*π*/4 for optimizing the acquisition of such an astigmatic field. In this case, the lenses necessary for wavefield retrieval can be easily expressed as which corresponds to a cylindrical lens rotated at

*π*/4 and –

*π*/4 for odd and even

*j*, respectively.

*ω*

_{k=1,j=1}the pupil function P(

*x*,

*y*) of the lens is applied. The pupil function serves as a realistic estimation. In each iteration loop

*k*= 1, 2, ...,

*N*, all the measured constraints images

*E*= |F

_{j}*[*

_{z}*f*(

**r**

*)*

_{in}*L*(

_{j}**r**

*)]| are applied as shown in Fig. 1, where*

_{in}*j*= 1, 2, ...,

*M*(

*M*constraint images). Therefore after

*N*×

*M*iterations it is expected a successful wavefield retrieval:

*ω*

_{N,M}(

*x*,

*y*) =

*f*(

**r**

*). Notice that the entire wavefield is updated in each iteration step rather than just the phase distribution, which is done in the retrieval algorithm developed in Ref. [23*

_{in}**19**, 6064–6077 (2011). [CrossRef] [PubMed]

*r*

^{2}=

*x*

^{2}+

*y*

^{2},

*w*is the beam waist, and

*p*and azimuthal index

*l*. In particular, we study the input beam

*λ*= 532 nm and the propagation distance

*z*= 10 cm that correspond to our experimental setup described in Section 3. We remind that the LG beams are stable under free-space propagation, which means that its form is preserved except for scaling. Moreover a LG mode cannot be distinguished from its complex conjugate (twin-image) under propagation. These facts make difficult the wavefield retrieval of a LG mode and therefore it is well-suited for testing signal recovery algorithms.

*ℰ*

_{k,j}= |F

*[*

_{z}*ω*

_{k,j}·

*L*(

_{j}**r**

*)]|, corresponding to each iteration, with respect to the measured ones*

_{in}*E*as it follows The corresponding wavefields retrieved after

_{j}*N*×

*M*= 96 are shown in Fig. 2(b), while the corresponding evolution of the RMSE is shown in Fig. 2(c). These results demonstrate that the increase of the number

*M*of constraint images significantly speeds up the convergence of the RMSE. Convergence in the wavefield retrieval is reached at

*N*×

*M*= 96 iterations with a RMSE value of 0.04 (

*M*= 2), 0.01 (

*M*= 3), and 5 × 10

^{−3}(

*M*= 4). We conclude that at least three constraint images are necessary to obtain successful signal retrieval, using a low number of iterations. We remind that each constraint image

*E*is associated with a single value of the focal length given as f

_{j}*=*

_{j}*α*[see the lens function Eq. (2)], where the orientation of the cylindrical lens is alternatively varied according with the rotation angle (−1)

_{j}z

^{j}*π*/4 . In particular, we have chosen

*α*

_{1}= 0.35,

*α*

_{2}= 0.5,

*α*

_{3}= 0.75 and

*α*

_{4}= 1.

*M*) of constraint images has to be increased. It was previously demonstrated in the astigmatic phase retrieval approaches reported in Refs. [22

22. J. A. Rodrigo, H. Duadi, T. Alieva, and Z. Zalevsky, “Multi-stage phase retrieval algorithm based upon the gyrator transform,” Opt. Express **18**, 1510–1520 (2010). [CrossRef] [PubMed]

**19**, 6064–6077 (2011). [CrossRef] [PubMed]

20. C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express **17**, 11905–11915 (2009). [CrossRef] [PubMed]

19. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. **91**, 203902 (2003). [CrossRef] [PubMed]

27. L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E **63**, 037602 (2001). [CrossRef]

20. C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express **17**, 11905–11915 (2009). [CrossRef] [PubMed]

*M*= 3 constraint images the RMSE convergence is reached at 0.01 after 96 iterations, while using the Henderson’s algorithm about 576 iterations are required to reach convergence at the same RMSE value. Therefore the proposed technique provides an accurate and fast signal retrieval using tens instead of hundreds iterations. It is important to note that the same improvement ratio in the algorithm convergence is also obtained when

*M*= 2 and 4 constraint images are applied, which RMSE evolution is also shown in Fig. 2(d).

28. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. **44**, 407–414 (1997). [CrossRef]

*N*×

*M*= 96 iterations, where

*M*is the number of constraint images of 1024 × 1024 pixels. These facts make the described method a versatile tool for wavefield retrieval. Notice that almost real time calculation can be reached by using a highly parallel processor, as for example a graphic processing unit [29

29. T. Shimobaba, T. Ito, N. Masuda, Y. Abe, Y. Ichihashi, H. Nakayama, N. Takada, A. Shiraki, and T. Sugie, “Numerical calculation library for diffraction integrals using the graphic processing unit: the GPU-based wave optics library,” J. Opt. A, Pure Appl. Opt. **10**, 075308 (2008). [CrossRef]

## 3. Experimental implementation

*L*(

_{j}*x*,

*y*) necessary for wavefield retrieval [see Eq.(2)] are addressed into a programmable spatial light modulator (SLM) operating in phase-only modulation. Specifically, we use a reflective LCoS-SLM (Holoeye PLUTO, 8-bit gray-level, pixel pitch of 8

*μ*m and 1920 × 1080 pixels), which was calibrated for a 2

*π*phase shift at the wavelength

*λ*= 532 nm and corrected from static aberrations caused by the deviation of flatness of the SLM display, as reported in Ref. [23

**19**, 6064–6077 (2011). [CrossRef] [PubMed]

*λ*/15 RMS for a circular area of 4 mm in diameter of the SLM display, where the digital lenses are addressed, which guarantees for an accurate lens implementation. In this work we apply the same experimental setup described in Ref. [23

**19**, 6064–6077 (2011). [CrossRef] [PubMed]

*λ*= 532 nm, with output power of 300 mW) is spatially modulated by the SLM in order to generate such signals. Nevertheless, conventional glass cylindrical lenses can be used instead of such digital ones.

_{1}and RL

_{2}) are set as a 4-f system or telescope for spatial filtering to isolate the addressed signal from the unmodulated light as well as from unwanted diffraction terms caused by the discrete structure of the SLM display. Notice that this spatial filtering is achieved at the Fourier plane of the spherical relay lens RL

_{1}, whereas the SLM is located at its back focal plane. Therefore the encoded lens

*L*(

_{j}*x*,

*y*) is generated at the output plane (at

*z*

_{0}= 0) of the telescope, which corresponds to the focal plane of relay lens RL

_{2}, see Fig. 3. The cylindrical lens

*L*(

_{j}*x*,

*y*) addressed into the SLM was set with a focal length f

*=*

_{j}*α*/

_{j}z*s*

^{2}to deal with the telescope magnification

*s*. The intensity distribution of the outgoing beam is recorded by a CCD camera (Spiricon SP620U, 12-bit gray-level, pixel pitch of 4.4

*μ*m and 1620 × 1200 pixels), which is placed at distance

*z*for acquiring the constraint images.

*z*. Therefore, in order to reach a maximum NA, the distance

*z*was chosen as short as possible. Indeed, the shortest focal length that the SLM is able to implement at the considered wavelength is about f

*= 15 cm. Since the lens addressed in the SLM has a focal length given by f*

_{min}*=*

_{j}*α*/

_{j}z*s*

^{2}, where

*s*= 0.4 is the magnification of the telescope and

*α*= 0.35, the shortest propagation distance is

_{min}*z*= 10 cm. To reduce as much as possible the artifacts as well as ambiguities, caused by noisy experimental measurements and the limited dynamic range of the CCD camera, we use

*M*= 8 constraints images obtained for the transformation parameter values

*α*= 0.35, 0.42, 0.5, 0.62, 0.75, 0.87, 1, and 1.12. The analysis shows that at least four constraint images are necessary to obtain successful retrieval from experimental data.

*z*

_{0}= 0 where it is illuminated by the

*probe*beams corresponding to the lenses

*L*(

_{j}*x*,

*y*), where

*j*= 1, 2, ...,

*M*. While in the second setup scheme the light scattered by the object is directly imaged into SLM using a microscope objective. The main goal of these experiments is to demonstrate an accurate retrieval of the wavefield, from which the image of the specimen can be synthesized and analyzed.

## 4. Imaging in microscopy by means of wavefield retrieval

*i*) It is robust because the measurement does not require interferometry and the optical elements in the experimental setups are fixed,

*ii*) It provides an accurate wavefield recovery since ambiguities such as twin-image and defocus (typically found in conventional GS algorithms) are suppressed thanks to the constraint diversity,

*iii*) It is fast as in the acquisition of the required constraint images –the CCD camera and SLM can be synchronized to perform an automatic measurement at video rates (25 fps)– as well as in the retrieval algorithm convergence. We also underline that additional postprocessing tasks such as image resampling and digital alignment of the constraint images are not required in our approach.

### 4.1. Transmission microscopy scheme via multi-illumination probe beams

*z*

_{0}= 0, see Fig. 3) is sequentially illuminated by several

*probe*or

*scanning*beams P(

*x*,

*y*)

*L*(

_{j}*x*,

*y*), where P(

*x*,

*y*) is the circular pupil of the addressed lens. Specifically, such a pupil function is fixed with radius 0.8 mm and it corresponds to a circular area of 4 mm in diameter of the SLM display. Here the relay telescope magnification has been taken into account. The specimen image is obtained from the wave-field retrieved at

*z*

_{0}= 0, while the constraint images are acquired by the CCD camera as it is displayed in Fig. 4(a).

*z*= 10 cm in the central area of the CCD chip corresponding to 1024 × 1024 pixels. In order to enhance the resolution of the retrieved field, one can register the adjacent scattered field for every illumination beam by moving the CCD camera in the recording plane, for example as displayed in Fig. 4(b). It allows generating an extended synthetic aperture (SA) with an effective spatial cutoff frequency higher than the cutoff given by the limited CCD aperture. We underline that the extended SA in the Fourier domain is often applied for super-resolution imaging, see for example Refs. [5

5. V. Mico, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express **16**, 19260–19270 (2008). [CrossRef]

30. L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. **49**, 845–857 (2010). [CrossRef] [PubMed]

*superresolved*image is generated, by simple inverse Fourier transformation.

### 4.2. Transmission microscopy scheme using a microscope objective

*z*

_{0}= 0 of the relay telescope, see Fig. 6. While the tube lens is digitally addressed into the SLM together with cylindrical lenses

*L*(

_{j}*x*,

*y*) necessary for the wavefield retrieval. Applying the algorithm described in Section 2 we are able to retrieve the wavefield at

*z*

_{0}= 0 from the measurement of the constraint images acquired by the CCD camera at the position z = 10 cm.

*n*, of the sample play an important role in the imaging processes because both affect to the transmitted wavefield. Therefore amplitude as well as phase distributions of the scattered field must be recovered for achieving quantitative analysis of the specimen. In particular, refractive index mapping, associated with phase distribution, is useful for biological applications because it reveals the cell activity, molecular dynamics, etc. We underline that in the last years tomographic techniques based on wavefield recovery have been developed for a 3D imaging of live cell, see for example [31

31. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express **17**, 266–277 (2009). [CrossRef] [PubMed]

*z*

_{0}= 0, see Fig. 7(a). The intensity [Fig. 7(b)] and phase [Fig. 7(c)] distributions of the light modulated by the specimen, projected into the conjugated plane

*z*

_{0}= 0, are simultaneously retrieved using the constraint images registered at the plane

*z*. Notice that the synthesized image Fig. 7(b) is slightly blurred with respect the image Fig. 7(a) due to the limited CCD aperture (cutoff frequency of 40 lp/mm) comprising the recording setup, as previously discussed. While the information content of the intensity images is almost the same, that in particular underlines the accuracy of the retrieval method, the phase map provides additional details about the cell structure. The nucleus and cell membrane are clearly distinguished in the unwrapped phase distribution as well as in its 3D representation displayed in Fig. 7(d) and 7(c), correspondingly. For accurate phase unwrapping we applied the algorithm reported in Ref. [32

32. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. **41**, 7437–7444 (2002). [CrossRef] [PubMed]

*z*

_{0}= 0 until all particles centers are found by locating the corresponding maxima in the diffracted field, see Fig. 9. Notice that the particles are labeled as they appear focused. Specifically, the spheres #5 – 7 are focused at

*z*= 11 ± 1

*μ*m, #8 – 10 at

*z*= 18 ± 1

*μ*m while the bead #11 is located at

*z*= 23 ± 1

*μ*m. Because the particles #1 and #2 are correspondingly attached to the ends of the #3 and #4 ones, their axial positions can not be accurately measured by applying this technique. Nevertheless, the light focused by the pairs of particles (#1 and #3) and (#2 and #4) is observed at

*z*= 5 ± 1

*μ*m and

*z*= 8 ± 1

*μ*m in the transversal position (plane

*XY*) indicated as bead #3 and #4, respectively.

3. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. **45**, 836–850 (2006). [CrossRef] [PubMed]

33. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express **18**, 13563–13573 (2010). [CrossRef] [PubMed]

*ϕ*(

*x*,

*y*) shown in Fig. 8(d), it is also possible to determine the bead’s radius (

*R*) and its refractive index

*n*. However, an accurate knowledge about the refractive index of the surrounding fluid (immersion oil,

_{b}*n*= 1.52 at 589 nm) is required. Because the particle is spherical in shape one can estimate both the radius of the bead and the refractive index difference

_{s}*n*–

_{b}*n*by linear fitting of the plot corresponding to the square of the phase

_{s}*ϕ*

^{2}(

*x*,

*y*) against

*r*

^{2}= (

*x*–

*x*)

_{o}^{2}+ (

*y*–

*y*)

_{o}^{2}. Notice that in the case of a spherical bead the square of the phase is theoretically given by the expression see Ref. [34

34. T. J. McIntyre, C. Maurer, S. Fassl, S. Khan, S. Bernet, and M. Ritsch-Marte, “Quantitative SLM-based differential interference contrast imaging,” Opt. Express **18**, 14063–14078 (2010). [CrossRef] [PubMed]

*x*,

_{o}*y*) of the sphere was obtained from the numerical refocusing results previously discussed. It allows for easy determination of the phase profile of the bead needed for accurate fitting. In particular, we consider the bead #7 for which the phase profile determined by averaging in four directions is shown in Fig. 11(a). The corresponding best linear fitting is displayed in Fig. 11(b). A refractive index difference

_{o}*n*–

_{b}*n*= 0.063 ± 0.004 is obtained from the slope of the fitted line according with Eq. (5). Thus, the quoted refractive index at 532 nm is

_{s}*n*= 1.583 ± 0.004 whereas a bead’s radius value of

_{b}*R*= (5.0 ± 0.2)

*μ*m is found from the intercept of the fitted line. These values are consistent with the ones quoted by the manufacturer,

*R*= (5.0 ± 0.3)

*μ*m and

*n*= 1.59 – 1.60 at 589 nm. The refractive index determination is in good agreement with the value reported by Ma et al. [35

_{b}35. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. **48**, 4165 (2003). [CrossRef]

*n*= 1.588 at 532 nm), although slightly smaller than the one quoted in Ref. [36

_{b}36. I. D. Nikolov and C. D. Ivanov, “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. **39**, 2067–2070 (2000). [CrossRef]

*n*= 1.5986) and Ref. [34

_{b}34. T. J. McIntyre, C. Maurer, S. Fassl, S. Khan, S. Bernet, and M. Ritsch-Marte, “Quantitative SLM-based differential interference contrast imaging,” Opt. Express **18**, 14063–14078 (2010). [CrossRef] [PubMed]

*n*= 1.602). These results demonstrate that proposed approach can be used for determination of quantitative information about the object. Moreover, the considered examples illustrate the potential applications of the proposed wavefield retrieval approach for advanced microscopic imaging which includes enhanced resolution, numerical refocusing, quantitative imaging and phase mapping, etc.

_{b}## 5. Conclusions

## Acknowledgments

## References

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21. | T. Alieva and J. A. Rodrigo, “Iterative phase retrieval from Wigner distribution projections,” in |

22. | J. A. Rodrigo, H. Duadi, T. Alieva, and Z. Zalevsky, “Multi-stage phase retrieval algorithm based upon the gyrator transform,” Opt. Express |

23. | J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt. Express |

24. | L. J. Allen, M. P. Oxley, and D. Paganin, “Computational aberration correction for an arbitrary linear imaging system,” Phys. Rev. Lett. |

25. | W. McBride, N. L. O’Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr., Sect. A: Found. Crystallogr. |

26. | T. C. Petersen and V. J. Keast, “Astigmatic intensity equation for electron microscopy based phase retrieval,” Ultramicroscopy |

27. | L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E |

28. | D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. |

29. | T. Shimobaba, T. Ito, N. Masuda, Y. Abe, Y. Ichihashi, H. Nakayama, N. Takada, A. Shiraki, and T. Sugie, “Numerical calculation library for diffraction integrals using the graphic processing unit: the GPU-based wave optics library,” J. Opt. A, Pure Appl. Opt. |

30. | L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. |

31. | Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express |

32. | M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. |

33. | F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express |

34. | T. J. McIntyre, C. Maurer, S. Fassl, S. Khan, S. Bernet, and M. Ritsch-Marte, “Quantitative SLM-based differential interference contrast imaging,” Opt. Express |

35. | X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. |

36. | I. D. Nikolov and C. D. Ivanov, “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(090.1760) Holography : Computer holography

(100.2000) Image processing : Digital image processing

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(180.0180) Microscopy : Microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: July 6, 2011

Revised Manuscript: August 1, 2011

Manuscript Accepted: August 6, 2011

Published: September 8, 2011

**Virtual Issues**

Vol. 6, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

José A. Rodrigo, Tatiana Alieva, Gabriel Cristóbal, and María L. Calvo, "Wavefield imaging via iterative retrieval based on phase modulation diversity," Opt. Express **19**, 18621-18635 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18621

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