## Through-focus phase retrieval and its connection to the spatial correlation for propagating fields |

Optics Express, Vol. 21, Issue 5, pp. 5550-5560 (2013)

http://dx.doi.org/10.1364/OE.21.005550

Acrobat PDF (1129 KB)

### Abstract

Through-focus phase retrieval methods aim to retrieve the phase of an optical field from its intensity distribution measured at different planes in the focal region. By using the concept of spatial correlation for propagating fields, for both the complex amplitude and the intensity of a field, we can infer which planes are suitable to retrieve the phase and which are not. Our analysis also reveals why all techniques based on measuring the intensity at two Fourier-conjugated planes usually lead to a good reconstruction of the phase. The findings presented in this work are important for aberration characterization of optical systems, adaptive optics and wavefront metrology.

© 2012 OSA

## 1. Introduction

12. B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. **28**, 801–803 (2003). [CrossRef] [PubMed]

## 2. Spatial correlation for propagated fields

*c*(0,

*z*) can be expressed in terms of the angular spectrum of the field at

*z*= 0 only as (Plancherel’s theorem) where the angular spectrum at

*z*= 0 is defined as and

*m*is In Eq. (3),

*p*and

*q*represent the spatial frequencies in Fourier space. The integration domain 𝔇, that appears in the integrals in Eq. (2), is a subset of the homogeneous domain only. This means that we are neglecting any evanescent wave in the spectrum. This assumption is surely applicable when one deals with fields that propagate over distances longer than the wavelength

*λ*. In case of a focused field, the domain 𝔇 will coincide with the domain {(

*p,q*) |

*p*

^{2}+

*q*

^{2}

*=*(NA/

*λ*)

^{2}}, with NA the numerical aperture of the optical system.

## 3. Through-focus phase retrieval: experimental implementation and reconstruction

### 3.1. Experimental setup

*λ*= 633

*nm*) is coupled into the single mode fiber and is collimated by a lens of focal length

*f*= 80 cm. The highly collimated beam is then focused by the test lens placed on two translation stages. A plane in the focal region of the lens under test is imaged by the imaging system consisting of the imaging objective, the tube lens and the CCD camera. The aberrations of the test lens influence the intensity distribution in the planes across the focal region. The test lens is then moved along the optical axis by a mechanical translation stage and several approximately equidistant intensity planes in the vicinity of the focal plane are recorded in the camera. The movement along the optical axis is recorded with a displacement interferometer for the precise location of the intensity planes. The test lens in the present setup is a microscope objective with a numerical aperture of NA = 0.4. The imaging objective is of higher numerical aperture (NA = 0.9) in order to capture the complete angular spectrum of the test objective. Intensity planes in the vicinity of the focal plane are recorded in the camera with the same acquisition time. The pixel width of the 8-bit CCD camera is 3.75

*μ*m with 1600 × 1200 pixels in x and y directions respectively. The magnification of the measurement system has been calibrated by moving the test lens in the direction perpendicular to the optical axis with a piezo stage. The piezo movement has been calibrated with a displacement interferometer before the measurements were performed. A linear fit to the recorded positions of the focal spots reveals the magnification of the imaging system. In the present configuration of the experimental setup, the magnification is 50.4. The incident wavefront has been characterized through a Shack Hartmann sensor for high degree of collimation with rms ≃

*λ*/100 on a beam size of 8 mm, which corresponds to the diameter of the test objective. The aberrations of the measurement system have been calibrated for precise measurements. More specifically, the aberrations of the imaging objective have been characterized with an interferometer while the aberrations of the tube lens could not be measured with the interferometer due to parasitical interference fringes of the internal surfaces of the tube lens. However, the aberrations of the tube lens and imaging objective resulted to have no significant impact on the final reconstruction on the phase. Dark frames have been recorded before the through-focus measurements by blocking the illumination beam before the lens. These dark frames account for stray light in the setup and have been subtracted from the intensity measurements in the focal region. Each intensity plane is the average of 50 intensity maps at the same location. These dark frames account for stray light in the setup and have been subtracted from the intensity measurements in the focal region. Each intensity plane is the average of 50 intensity maps at the same location. The range of measurements is few microns (≈ 10

*μ*m) in both directions with respect to the focal plane.

### 3.2. Phase reconstruction

*c*(0,

_{int}*z*). In Fig. 4 (panel a) we show

*c*(0,

_{int}*z*) as obtained by using the measured through focus intensity distributions. As it can be easily seen, the curve is not symmetric, which is a manifestation of the presence of aberrations. In the same figure, panel b, we also plot the ideal degree of spatial correlation for both the intensity and for the field, in case of an ideal aberration-free system, uniformly illuminated. One can see that the position of the planes with minimal correlation does not change considerably from the aberrated to the aberration-free case. From the correlation for the intensities, we select the two measured distributions shown in Fig. 4, panel c and d, respectively. The first one corresponds to the field in focus, while the other is located 7.83

*μ*m far from the best focus. The measured correlation between the intensities at those two planes resulted to be

*c*(0, 7.83

_{int,meas}*μ*m) = 0.09, while the ideal one (no aberrations) for the fields and the intensities give |

*c*(0, 7.83

*μ*m)| = 0.03 and

*c*(0, 7.83

_{int}*μ*m) = 0.12, respectively. The two chosen intensity measurements were then used in a standard iterative phase retrieval algorithm, that cycled continuously between the planes

*z*= 0 and

*z*= 7.83

*μ*m. In each cycle of the algorithm, the intensity measured at each plane was used as constraint for the complex field that was propagated, back and forth, between the two planes. After few iterations (typically less than 50–60 already gave good results), the phase of the fields at both planes was obtained. In Fig. 5 we show such retrieved phases. In every phase retrieval method it is important to check whether the retrieved phase is correct or not. We have checked the correctness of our results in the following way. We have taken the complex field in focus (

*z*= 0), consisting of the measured amplitude and the retrieved phase (shown in Fig. 5, first column), and we have propagated it along

*z*. If the phase is the right one, then the intensity distributions obtained in this way should coincide with the intensities that have been measured. In Fig. 6 some of the measured and simulated field intensities are plotted where, the top row contains the measurements at different distances from the focal plane and the bottom row contains the simulations. As it is evident from the plots, the agreement between simulations and measurements is excellent, which confirms that the phase has been successfully retrieved. It is worth noting that even when the signal to noise ratio is low (such at

*z*= 15.17

*μ*m distance from the best focus) the calculated intensity shows a strong similarity with the measured one. In order to be more quantitative, we have computed the root mean square error (

*RMSE*) between each simulated and measured intensity pattern shown in Fig. 6. We have found that at

*z*= 4.04

*μ*m the

*RMS*deviation is 0.68%, at

*z*= 7.83

*μ*m is 0.38%, at

*z*= 11

*μm*is 0.47% and finally at

*z*= 15.17

*μ*m the deviation is 0.39%.

### 3.3. Role of SNR and a priori information

*a priori*information has been used. More specifically, the nominal numerical aperture of the microscope objective NA = 0.4 has been included in the iterative algorithm to retrieve the phase of the field. Considered that a microscope objective in air can, at most, have NA = 1, this has allowed us to throw away a lot of data (actually more than half of the field in the exit pupil) that contained only noise but no useful signal. This makes the reconstruction much more robust and less sensitive to eventually high correlations between the two intensity patterns used in the phase retrieval algorithm. This means that, in this specific case, even choosing the field in focus (at the plane

*z*= 0) and another field (at position

*z*) with a high value of the correlation

*c*(0,

_{int}*z*) could actually lead to an excellent reconstruction for the phase. In fact, while the increase in correlation is compensated by the said

*a priori*information, there is the additional advantage that close to the focal plane also the

*SNR*is generally high, which has beneficial effects on the phase reconstruction. Paradoxically, for extremely low (and known) values of NA the knowledge of only one measured intensity pattern would suffice to solve the phase problem. This would make unnecessary any further study based on the concept of spatial correlation introduced in the present work. On the other hand, for either high NA systems or for systems whose NA is unknown, making use of the concept of spatial correlation, as shown here, can be decisive. In order to clarify this aspect, let us perform the phase reconstruction by using the following sets of data:

- case 1: Intensity of the field in focus (
*z*_{1}= 0) and intensity of the field on*z*_{2}= 4.04*μm*, with*c*(0,_{int}*z*_{2}) = 0.82. - case 2: Intensity of the field in focus (
*z*_{1}= 0) and intensity of the field on*z*_{3}= 7.83*μm*, with*c*(0,_{int}*z*_{3}) = 0.09 (first minimum of the correlation curve). - case 3: Intensity of the field in focus (
*z*_{1}= 0) and intensity of the field on*z*_{4}= 11*μm*, with*c*(0,_{int}*z*_{4}) = 0.29 (first relative maximum of the correlation curve). - case 4: Intensity of the field in focus (
*z*_{1}= 0) and intensity of the field on*z*_{5}= 15.17*μm*, with*c*(0,_{int}*z*_{5}) = 0.03 (second minimum of the correlation curve).

*SNR*values. However, we can still draw some interesting conclusion. After running a phase retrieval algorithm in all these cases, assuming to know the NA, we have computed the root mean square error (

*RMSE*) between the simulated intensity and the measured one at the position

*z*= −4.62

*μ*m. We have obtained (NA = 0.4 known)

*RMSE*is the smallest. This is due to the fact that the field intensity at

*z*

_{2}= 4.04

*μ*m is characterized by the highest

*SNR*among the cases considered above. In order to be more quantitative, we have computed the

*SNR*for the measured intensity distributions (cases 1–4).

*SNR*has been defined as the ratio between the average intensity and the relative standard deviation of a set of 50 samples of the same intensity distribution, at a given plane. For the reader’s commodity, in Table 1 we report the values for the intensity correlations along with the

*SNR*,

*RMSE*for the planes listed above. The role of the a priori information can be pointed out by performing, once again, the reconstruction but assuming that the NA of the system is not known. For instance, for the case 1 and 2, under this assumption (NA unknown), we obtain

*a priori*information (NA is supposed to be unknown now). However, it is also evident that the situation is completely reversed now. In fact, since there is no

*a priori*information available that can compensate for eventual high correlation values, the rms error of case 1 becomes much larger than case 2, while with the said

*a priori*information available they showed very similar

*RMSE*values (with case 1 actually slightly better, as we have seen already). All this confirms our theoretical predictions. We expect that for high-NA systems choosing the proper plane for a phase reconstruction is a central element of the whole phase problem, considered that the even knowledge of NA does not help in that case.

## 4. Conclusions

## Acknowledgment

## References and links

1. | R. Loudon, |

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

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

4. | H. M. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. |

5. | H. M. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy |

6. | J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature |

7. | J. A. Rodrigo, T. Alieva, G. Cristbal, and M. L. Calvo, “Wavefield imaging via iterative retrieval based on phase modulation diversity,” Opt. Express |

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

9. | G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express |

10. | R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. |

11. | J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A |

12. | B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. |

13. | J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

14. | M. Born and E. Wolf, |

**OCIS Codes**

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(100.5070) Image processing : Phase retrieval

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 1, 2012

Revised Manuscript: December 12, 2012

Manuscript Accepted: December 13, 2012

Published: February 27, 2013

**Citation**

O. El Gawhary, A. Wiegmann, N. Kumar, S. F. Pereira, and H. P. Urbach, "Through-focus phase retrieval and its connection to the spatial correlation for propagating fields," Opt. Express **21**, 5550-5560 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5550

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### References

- R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart)35, 237–246 (1972).
- J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21, 2758–2769 (1982). [CrossRef] [PubMed]
- H. M. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett.93, 023903 (2004). [CrossRef] [PubMed]
- H. M. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy103153–164 (2005). [CrossRef] [PubMed]
- J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400342–344 (1999). [CrossRef]
- J. A. Rodrigo, T. Alieva, G. Cristbal, and M. L. Calvo, “Wavefield imaging via iterative retrieval based on phase modulation diversity,” Opt. Express1918621–18635 (2011). [CrossRef] [PubMed]
- M. R. Teague, “Image formation in terms of the transport equation,” J. Opt. Soc. Am. A2, 2019–2026 (1985). [CrossRef]
- G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express17624 – 639 (2009). [CrossRef] [PubMed]
- R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982). [CrossRef]
- J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998). [CrossRef]
- B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett.28, 801–803 (2003). [CrossRef] [PubMed]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2007), 883–891.

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