## Quantitative phase microscopy using defocusing by means of a spatial light modulator

Optics Express, Vol. 18, Issue 7, pp. 6755-6766 (2010)

http://dx.doi.org/10.1364/OE.18.006755

Acrobat PDF (1420 KB)

### Abstract

A new method for recovery the quantitative phase information of microscopic samples is presented. It is based on a spatial light modulator (SLM) and digital image processing as key elements to extract the sample’s phase distribution. By displaying a set of lenses with different focal power, the SLM produces a set of defocused images of the input sample at the CCD plane. Such recorded images are then numerically processed to retrieve phase information. This iterative process is based on the wave propagation equation and leads on a complex amplitude image containing information of both amplitude and phase distributions of the input sample diffracted wave front. The proposed configuration is a non-interferometric architecture (conventional transmission imaging mode) where no moving elements are included. Experimental results perfectly correlate with the results obtained by conventional digital holographic microscopy (DHM).

© 2010 OSA

## 1. Introduction

1. D. Gabor, “A new microscopic principle,” Nature **161**(4098), 777–778 (1948). [CrossRef] [PubMed]

2. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**(2), 179–181 (1994). [CrossRef] [PubMed]

6. V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Lett. **34**(10), 1492–1494 (2009). [CrossRef] [PubMed]

7. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and Ch. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. **30**(5), 468–470 (2005). [CrossRef] [PubMed]

11. V. Micó, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express **16**(23), 19260–19270 (2008). [CrossRef]

7. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and Ch. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. **30**(5), 468–470 (2005). [CrossRef] [PubMed]

12. H. Iwai, C. Fang-Yen, G. Popescu, A. Wax, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Quantitative phase imaging using actively stabilized phase-shifting low-coherence interferometry,” Opt. Lett. **29**(20), 2399–2401 (2004). [CrossRef] [PubMed]

14. Y. K. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Fresnel particle tracing in three dimensions using diffraction phase microscopy,” Opt. Lett. **32**(7), 811–813 (2007). [CrossRef] [PubMed]

2. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**(2), 179–181 (1994). [CrossRef] [PubMed]

17. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. **40**(34), 6177–6186 (2001). [CrossRef]

25. 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**(24), 3234–3241 (2003). [CrossRef] [PubMed]

*et al*[24

24. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. **23**(11), 817–819 (1998). [CrossRef]

20. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. **73**(11), 1434–1441 (1983). [CrossRef]

22. M. R. Teague, “Image formation in terms of transport equation,” J. Opt. Soc. Am. A **2**(11), 2019–2026 (1985). [CrossRef]

*et al*[26

26. G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. **30**(8), 833–835 (2005). [CrossRef] [PubMed]

28. P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a volume-speckle field,” Opt. Lett. **32**(7), 733–735 (2007). [CrossRef] [PubMed]

*et al*validated their method working without lenses and displacing axially the CCD to provide the different intensity measurement planes. After that, reconstruction algorithms based on iteration of the wave propagation equation allow the recovery of the object’s complex amplitude wavefront. This single-beam multiple-intensity reconstruction (SBMR) method has also been validated in shape, deformation and angular displacement measurements of three-dimensional (3D) objects [29

29. A. Anand, V. K. Chhaniwal, P. Almoro, G. Pedrini, and W. Osten, “Shape and deformation measurements of 3D objects using volume speckle field and phase retrieval,” Opt. Lett. **34**(10), 1522–1524 (2009). [CrossRef] [PubMed]

30. P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Angular displacement and deformation analyses using a speckle-based wavefront sensor,” Appl. Opt. **48**(5), 932–940 (2009). [CrossRef] [PubMed]

24. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. **23**(11), 817–819 (1998). [CrossRef]

26. G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. **30**(8), 833–835 (2005). [CrossRef] [PubMed]

30. P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Angular displacement and deformation analyses using a speckle-based wavefront sensor,” Appl. Opt. **48**(5), 932–940 (2009). [CrossRef] [PubMed]

24. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. **23**(11), 817–819 (1998). [CrossRef]

25. 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**(24), 3234–3241 (2003). [CrossRef] [PubMed]

30. P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Angular displacement and deformation analyses using a speckle-based wavefront sensor,” Appl. Opt. **48**(5), 932–940 (2009). [CrossRef] [PubMed]

25. 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**(24), 3234–3241 (2003). [CrossRef] [PubMed]

27. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. **45**(34), 8596–8605 (2006). [CrossRef] [PubMed]

*et al*reported on a different method where the set of diffracted patterns is recorded not by displacing the CCD but by tuning the illumination wavelength [31

31. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. **33**(4), 309–311 (2008). [CrossRef] [PubMed]

**11**(24), 3234–3241 (2003). [CrossRef] [PubMed]

27. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. **45**(34), 8596–8605 (2006). [CrossRef] [PubMed]

32. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. **29**(21), 2503–2505 (2004). [CrossRef] [PubMed]

33. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express **13**(3), 689–694 (2005). [CrossRef] [PubMed]

34. Ch. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Phase contrast microscopy with full numerical aperture illumination,” Opt. Express **16**(24), 19821–19829 (2008). [CrossRef] [PubMed]

35. T. J. McIntyre, Ch. Maurer, S. Bernet, and M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. **34**(19), 2988–2990 (2009). [CrossRef] [PubMed]

**23**(11), 817–819 (1998). [CrossRef]

26. G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. **30**(8), 833–835 (2005). [CrossRef] [PubMed]

27. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. **45**(34), 8596–8605 (2006). [CrossRef] [PubMed]

## 2. Analysis of the proposed method and experimental calibration

**30**(8), 833–835 (2005). [CrossRef] [PubMed]

31. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. **33**(4), 309–311 (2008). [CrossRef] [PubMed]

*I*, where

_{N}*N*varies from 1 to 9. Obviously, each one of these 9 images is obtained by a different value of the lens displayed at the SLM and is connected to a different transversal section of the input object. In Fig. 3 we can see the whole set of images where case (e) corresponds to the SLM no-lens case. Since the final image magnification at infinity corrected imaging configuration depends on the ratio between the focal lengths of the tube lens and the microscope objective, an increase in the power of the tube lens implies a reduction in the overall image magnification. Thus, positive lenses at the SLM will produce a reduction in the image magnification [Fig. 3, from (a) to (d)] while negative lenses will increase the magnification of the image [Fig. 3, from (f) to (i)]. Both cases are compared with the no-lens SLM imaging case [Fig. 3(e)].

*I*inputs must be matched in magnification and transverse location. Otherwise, the numerical propagation will need a magnification control and will cause costly computational difficulties. Moreover the precise propagation distance between planes is also needed.

_{N}*β’*is just a quotient between the following focal lengths in the system taking into account the distance

*e*between the SLM and the tube lenswhere

*f*,

_{O}, f_{T}*f*are the focal lengths of the microscope lens, the tube lens and the lens encoded in the SLM, respectively. On the other hand, the axial location of the image is given by the back focal length

_{S}*D*defined as

*e*between the SLM and the tube lens has to be measured on the system. Owing to the system complexity there is a significant uncertainty in its measurement. For experimental simplicity the magnification and axial location of the images can be obtained from the recorded images, without the need for accurate physical measurements. This significantly simplifies the experiments, at the price of some parameters matching steps that can be automated.

11. V. Micó, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express **16**(23), 19260–19270 (2008). [CrossRef]

*RS(x,y;d) = FT*, where

^{−1}{FT{U(x,y)}⋅FT{h(x,y;d)}}*RS(x,y)*is the propagated wave field,

*U(x,y)*is the recorded hologram,

*h(x,y)*is the impulse response of free space propagation (the definition of

*h(u,v;d)*can be found in Ref [15], page 115, Eq. (3).73),

*(x,y)*are the spatial coordinates,

*FT*is the Fourier transform operation (realized with the FFT algorithm) and

*d*is the propagation distance.

*I’*) having both the same object size and the same lateral position is obtained. Figure 4 depicts this new set of

_{N}*I’*images. And on the other hand, the propagation distance between the different images is known in a precise way.

_{N}*U*coming from the square root of the first intensity image [

_{1}(x,y)*I’*or Fig. 4(a)] multiplied by an initial constant phase:

_{1}*U*, being

_{1}(x,y) = I’_{1}(x,y) exp(iφ_{0}(x,y))*φ*. This initial complex amplitude distribution

_{0}(x,y) = 0*U*is digitally propagated to the next measured plane, that is, to the image represented by

_{1}(x,y)*I’*[or Fig. 4(b)]. Once again, numerical computation of the R-S equation by convolution approximation is considered but now the recorded hologram is the complex amplitude distribution

_{2}*U*and the Fresnel approximation is used in the calculation of the Fourier transform of the impulse response

_{1}(x,y)*H(u,v;d) = FT{h(x,y;d)}*, where

*(u,v)*are the spatial-frequency coordinates and the definition of

*H(u,v;d)*can be found in Ref [15], page 117, Eq. (3).84. Then, the calculation of the propagated wave field from the first measured intensity to the second one separated by a distance of

*d*is simplified to

_{2}*RS*, where

_{2}(x,y;d_{2}) = FT^{−1}{T_{1}(u,v) H(u,v;d_{2})}*T*is the Fourier transform of the initial complex wave field.

_{1}(u,v) = FT{U_{1}(x,y)}*RS*. However, for subsequent propagations (N≠1), we retain the phase distribution

_{N + 1}(x,y;d_{N + 1}) = FT^{−1}{T_{N}(u,v) H(u,v;d_{N + 1})}*φ*incoming from the previous propagation and replace the obtained amplitude by the square root of the intensity measured at that plane:

_{N}(x,y)*T*. Let us name the iterative process from the first image (

_{N}(u,v) = FT{U_{N}(x,y)} = FT{(I’_{N}(x,y))^{1/2}exp(iφ_{N}(x,y))}*I*) to the last one (

_{1}*I*) step by step considering all the images as a

_{9}*cycle*. Thus, once one cycle is performed, we propagate from

*I*to

_{9}*I*and the iterative process starts again, that is, a second cycle is considered. This iterative process is repeated until the quality of the reconstructed image at the imaging plane [

_{1}*I*or Fig. 3(e)] will be smaller than some predefined threshold which is obtained by computing the root mean square error (rmse) between the real image

_{5}*I’*and the one being obtained from the last image of the set (

_{5}*I’*) propagated to plane number 5 after a given number of cycles. We found that the rmse stabilizes after 5-6 cycles.

_{9}*I’*[Fig. 4(i)] without applying the proposed approach, and case (c) represents the resulting image obtained after 6 cycles. Figure 5(d) plots the normalized variation of the rmse as the number of iterations of the whole cycle increases (from 1 to 25). We stop the iteration process at cycle number 6 when the rmse value equals the background rmse. We can see as no image reconstruction is possible when the proposed approach is not considered [case (b)] while a very good image quality is reconstructed by considering only 6 cycles in the iteration process.

_{9}## 3. Experimental validation

*l = exp(ik(x*being

^{2}+ y^{2}))*(x,y)*the spatial coordinates and

*k*the variable parameter that modifies the focal length of the lens. Since we vary

*k*in the range [-0.0002, 0.0002], the focal length is modified from 0 to ± 4 meters, approximately. Along this range, we linearly varied the focal length with fixed increments allowing the recording of the whole set of 9 images. No evidence of another type of spacing (non-linear) between images has been noticed. Although the SLM allows a higher focal length variation, we stopped at this value since lower focal lengths than 4 m imply the presence of additional lens diffraction orders in the recorded images. One can see one of those high diffraction orders produced by the SLM lens appearing as a small vertical white rectangle in the centre of the image depicted in Fig. 8(b). Once the whole set of 9 recorded images is stored in the computer’s memory, the images are shifted and rescaled according to the values provided by the USAF test case.

## 4. Conclusions and discussion

**30**(8), 833–835 (2005). [CrossRef] [PubMed]

**11**(24), 3234–3241 (2003). [CrossRef] [PubMed]

**48**(5), 932–940 (2009). [CrossRef] [PubMed]

**11**(24), 3234–3241 (2003). [CrossRef] [PubMed]

**30**(8), 833–835 (2005). [CrossRef] [PubMed]

31. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. **33**(4), 309–311 (2008). [CrossRef] [PubMed]

**45**(34), 8596–8605 (2006). [CrossRef] [PubMed]

**33**(4), 309–311 (2008). [CrossRef] [PubMed]

## Acknowledgements

## References and links

1. | D. Gabor, “A new microscopic principle,” Nature |

2. | U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

3. | U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A |

4. | L. P. Yaroslavsky, Digital Holography and Digital Image Processing: Principles, Methods, Algorithms (Kluwer, 2003). |

5. | J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. |

6. | V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Lett. |

7. | P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and Ch. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. |

8. | F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and Ch. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. |

9. | G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. |

10. | B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. |

11. | V. Micó, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express |

12. | H. Iwai, C. Fang-Yen, G. Popescu, A. Wax, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Quantitative phase imaging using actively stabilized phase-shifting low-coherence interferometry,” Opt. Lett. |

13. | S. Reichelt and H. Zappe, “Combined Twyman-Green and Mach-Zehnder interferometer for microlens testing,” Appl. Opt. |

14. | Y. K. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Fresnel particle tracing in three dimensions using diffraction phase microscopy,” Opt. Lett. |

15. | T. Kreis, Handbook of holographic interferometry: optical and digital methods (Wiley-VCH, 2005). |

16. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

17. | I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. |

18. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) |

19. | J. R. Fienup, “Phase retrieval algorithms: a comparision,” Appl. Opt. |

20. | M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. |

21. | N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. |

22. | M. R. Teague, “Image formation in terms of transport equation,” J. Opt. Soc. Am. A |

23. | G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. |

24. | A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. |

25. | 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 |

26. | G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. |

27. | P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. |

28. | P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a volume-speckle field,” Opt. Lett. |

29. | A. Anand, V. K. Chhaniwal, P. Almoro, G. Pedrini, and W. Osten, “Shape and deformation measurements of 3D objects using volume speckle field and phase retrieval,” Opt. Lett. |

30. | P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Angular displacement and deformation analyses using a speckle-based wavefront sensor,” Appl. Opt. |

31. | P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. |

32. | G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. |

33. | S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express |

34. | Ch. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Phase contrast microscopy with full numerical aperture illumination,” Opt. Express |

35. | T. J. McIntyre, Ch. Maurer, S. Bernet, and M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. |

**OCIS Codes**

(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: December 22, 2009

Revised Manuscript: February 27, 2010

Manuscript Accepted: March 1, 2010

Published: March 17, 2010

**Virtual Issues**

Vol. 5, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Luis Camacho, Vicente Micó, Zeev Zalevsky, and Javier García, "Quantitative phase microscopy using defocusing by means of a spatial light modulator," Opt. Express **18**, 6755-6766 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-7-6755

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### References

- D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef] [PubMed]
- U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33(2), 179–181 (1994). [CrossRef] [PubMed]
- U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11(7), 2011–2015 (1994). [CrossRef]
- L. P. Yaroslavsky, Digital Holography and Digital Image Processing: Principles, Methods, Algorithms (Kluwer, 2003).
- 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(5), 836–850 (2006). [CrossRef] [PubMed]
- V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Lett. 34(10), 1492–1494 (2009). [CrossRef] [PubMed]
- P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and Ch. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30(5), 468–470 (2005). [CrossRef] [PubMed]
- F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and Ch. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31(2), 178–180 (2006). [CrossRef] [PubMed]
- G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006). [CrossRef] [PubMed]
- B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47(4), A52–A61 (2008). [CrossRef] [PubMed]
- V. Micó, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express 16(23), 19260–19270 (2008). [CrossRef]
- H. Iwai, C. Fang-Yen, G. Popescu, A. Wax, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Quantitative phase imaging using actively stabilized phase-shifting low-coherence interferometry,” Opt. Lett. 29(20), 2399–2401 (2004). [CrossRef] [PubMed]
- S. Reichelt and H. Zappe, “Combined Twyman-Green and Mach-Zehnder interferometer for microlens testing,” Appl. Opt. 44(27), 5786–5792 (2005). [CrossRef] [PubMed]
- Y. K. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Fresnel particle tracing in three dimensions using diffraction phase microscopy,” Opt. Lett. 32(7), 811–813 (2007). [CrossRef] [PubMed]
- T. Kreis, Handbook of holographic interferometry: optical and digital methods (Wiley-VCH, 2005).
- I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]
- I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40(34), 6177–6186 (2001). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1978).
- J. R. Fienup, “Phase retrieval algorithms: a comparision,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]
- M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]
- N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]
- M. R. Teague, “Image formation in terms of transport equation,” J. Opt. Soc. Am. A 2(11), 2019–2026 (1985). [CrossRef]
- G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33(2), 209–218 (1994). [CrossRef] [PubMed]
- A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998). [CrossRef]
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