## Harmonically matched grating-based full-field quantitative high-resolution phase microscope for observing dynamics of transparent biological samples

Optics Express, Vol. 15, Issue 26, pp. 18141-18155 (2007)

http://dx.doi.org/10.1364/OE.15.018141

Acrobat PDF (3302 KB)

### Abstract

We have developed a full-field high resolution quantitative phase imaging technique for observing dynamics of transparent biological samples. By using a harmonically matched diffraction grating pair (600 and 1200 lines/mm), we were able to obtain non-trivial phase difference (other than 0° or 180°) between the output ports of the gratings. Improving upon our previous design, our current system mitigates astigmatism artifacts and is capable of high resolution imaging. This system also employs an improved phase extraction algorithm. The system has a lateral resolution of 1.6 µm and a phase sensitivity of 62 mrad. We employed the system to acquire high resolution phase images of onion skin cells and a phase movie of amoeba proteus in motion.

© 2007 Optical Society of America

## 1. Introduction

1. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica **9**, 686–698 (1942). [CrossRef]

2. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica **9**, 974–986 (1942). [CrossRef]

3. R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. **69**, 193–221 (1969). [PubMed]

4. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. **26**, 349–393 (1988). [CrossRef]

13. D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. **23**, 783–785 (1998). [CrossRef]

4. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. **26**, 349–393 (1988). [CrossRef]

6. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature methods **4**, 717–719 (2007). [CrossRef] [PubMed]

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

9. J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express **15**, 7231–7242 (2007). [CrossRef] [PubMed]

10. T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. **30**, 1165–1167 (2005). [CrossRef] [PubMed]

11. G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Optical measurement of cell membrane tension,” Phys. Rev. Lett. **97**, 218101 (2006). [CrossRef] [PubMed]

12. M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. **31**, 1462–1464 (2006). [CrossRef] [PubMed]

13. D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. **23**, 783–785 (1998). [CrossRef]

## 2. G1G2 interferometery concept

*P*and

_{r}*P*are the reference power and sample power, respectively;

_{s}*ψ*and

_{r}*ψ*are the phase of the reference beam and sample beam before incident on the beamsplitter;

_{s}*θ*is the common phase shift. We note that the interference terms are 180° out of phase with each other. As such, it is generally impossible to extract phase information from this system without resorting to some form of phase encoding. An often cited reason to explain this trivial interference phase relationship is that the power influx and outflux from this optical system must be conserved – conservation requires the interference terms to be equal and opposite.

15. Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, “Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements,” Opt. Express **14**, 8127–8137 (2006). [CrossRef] [PubMed]

*m*is the diffracted order,

*x*

_{0}is the displacement of the grating and Λ is the grating period; sgn() is the sign function. This implies that the detected signals are given by:

_{1}, Λ

_{2}that satisfy Λ

_{1}=2Λ

_{2}. If the G1G2 grating is used as a beam splitter/combiner in an interferometer setup, as shown in Fig. 1(c), the additional phase of the diffracted beams ϕ

*(k=R, S represent the reference or sample beam; n=1, 2 is the port number) can be written as, according to Eq. (3),*

_{k,n}*is the additional phase shift of the*

_{m,Gn}*m*

^{th}diffracted order of the grating

*Gn*;

*x*

_{1},

*x*

_{2}are the displacements of G1, G2, respectively. Thus the phases of interference signals and their difference are

*ϕ*. The grating pair employed in our current experiment was fabricated by using three-beam interference on a holographic plate as detailed in Ref. [14

14. J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, “Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer,” Appl. Phys. Lett. **90**, 151123 (2007). [CrossRef]

## 3. Experiment method

*ϕ*to be equal to 92°±8°. More details about the grating pair can be found in Ref. 14.

14. J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, “Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer,” Appl. Phys. Lett. **90**, 151123 (2007). [CrossRef]

*x*

_{1}away from the center of object field will be mapped to a point

*x*

_{2}from the image field’s center where

*x*

_{2}is given by this expression:

*f*,

_{1}*f*are the focal length of the objective and the lens, respectively;

_{2}*λ*is the wavelength of the laser;

*d*is the grating period. Here we assume that both the objective and the lens are perfect paraxial lens. As

*x*

_{2}is not linearly dependent on

*x*

_{1}, the resulting image will appear distorted. When

*x*

_{1}is small, Eq. (9) can be simplified as

*x*

_{2}is proportional to

*x*

_{1}and there is no distortion. In our setup,

*f*

_{1}=16.5 mm,

*f*

_{2}=200 mm, λ=632.8 nm, d=1/600 mm. For the field of view with a maximum offset,

*x*

_{1}=0.16 mm, the difference between the two

*x*

_{2}calculated by (9) and (10) is 4.5 µm, which is smaller than the size of a pixel on the CCDs (pixel size =4.7 µm). Beyond this field of view, the distortion does cause a slight deterioration in resolution. This problem can be resolved by appropriate spatial rescaling of the raw data images.

## 4. Improved processing algorithm

14. J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, “Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer,” Appl. Phys. Lett. **90**, 151123 (2007). [CrossRef]

*P*(

_{r1}*i, j*)and

*P*(

_{r2}*i, j*)are the reference powers at the pixels of CCD1 and CCD2, respectively;

*P*(

_{s1}*i, j*) and

*P*(

_{s2}*i, j*) are the sample powers at corresponding pixels of the CCDs;

*η*(

*i, j*)=

*P*(

_{s2}*i, j*)/

*P*(

_{s1}*i, j*) is the relative diffraction efficiency of the grating for the sample beam;

*A*is the interference factor determined by the coherence of the laser, ideally

*A*=2, in practice, the measured

*A*is 1.9; Δ

*ψ*(

*i, j*)=

*ψ*(

_{obj}*i, j*) +

*ψ*(

_{abe}*i, j*)+

*θ*, where

_{ran}*ψ*(

_{obj}*i, j*) is the optical phase change associated with presence of the sample,

*ψ*(

_{abe}*i, j*) is the phase aberration introduced by the grating as mentioned in the previous section,

*θ*is some random phase attributable to environment fluctuation and independent of pixel index (

_{ran}*i, j*); Δ

*ϕ*is the nontrivial phase shift between the interference signals caused by the G1G2 grating and is independent on the pixel index (

*i, j*).

*η*(

*i, j*).

*ψ*(

_{abe}*i, j*) and the nontrivial phase shift Δ

*ϕ*. This involves acquiring N (we use N=100) frame pairs of the interferograms in the absence of the sample. For pixel (i,j) (i=1…1024, j=1…768) of the CCD k (k=1,2), we can get a time series y

_{i,j,k}(n), n=1…N. For different n, there is a different random phase introduced by environmental disturbance. Since the time series for any of the two pixels have a phase difference between them, if we plot one time series versus another, we will get an elliptical profile. For the time series from the same CCD, we can let the first pixel (i=1,j=1) be the reference point and compute the phase difference for each CCD pixel with respect to the reference point by performing elliptic fitting [18] between y

_{i,j,k}(n) and y

_{1,1,k}(n). This computed phase difference is equal to the phase aberration

*ψ*. Similarly, by performing elliptic fitting between the corresponding pixels from the two CCDs, y

_{abe}_{i,j,1}(n) and y

_{i,j,2}(n), we can get the phase shift Δ

*ϕ*.

*P*(

_{r1}*i, j*) and

*P*(

_{r2}*i, j*).

*P*(

_{s1}*i, j*) and Δ

*ψ*(

*i, j*).

*i, j*) in the expression. This equation yields two solutions for

*P*. In order to find the right solution, the sample power

_{s1}*P*need to satisfy some condition. As shown in the appendix A, if the real

_{s1}*P*satisfies the following condition, we can always use the smaller solution of Eq. (13) as our solution:

_{s1}*ϕ*=90° and ideal coherence, A=2, the above condition becomes

*P*, we can then calculate the phase term Δ

_{s1}*ψ*by substituting

*P*in Eqs. (11) and (12). The phase of the sample plus some uniform random phase is then given by

_{s1}*ψ*+

_{abj}*θ*=Δ

_{ran}*ψ*-

*ψ*.

_{abe}*θ*tends to be constant over the entire image but it can vary in time. One approach to remove it from a time sequence of phase images is to look at the variation of

_{ran}*ψ*+

_{abj}*θ*at a location in the image where it is known that

_{ran}*ψ*is not varying. The time dependent variations can then be wholly attributed to

_{obj}*θ*. We can then subtract this value from each image at each time point. This is the approach we employed when we generated phase image movie sequences.

_{ran}*aP*

^{2}

*+*

_{s1}*bP*+

_{s1}*c*=0, where

*a*,

*b*,

*c*are the coefficients shown in the equation, the solutions are

*P*,

_{1}*P*:

_{2}## 5. Imaging results

## 6. Summary

## Appendix A: derivation of equation (14)

*s*

_{0}and the other solution of Eq. (13) is

*s*

_{1}, then we have

*s*

_{1}>

*s*

_{0}, then we can choose the smaller solution of the equation and get

*s*

_{0}that we want. So we should have

*ϕ*)>0, thus

*ψ*, the above equation is satisfied, we must have

*ϕ*=-90°. Now the power detected in the two output ports can be written as

*P*and

_{1}*P*contain additive Gaussian white-noise terms

_{2}*x*

_{1},

*x*

_{2}with zero mean, respectively. For shot noise, the standard deviation of

*x*

_{1},

*x*

_{2}should be

^{20}

*h*is the Planck’s constant,

*ν*is the light frequency,

*η*is the quantum efficiency of the CCD, and

*τ*is the exposure time. Thus as shown in Fig. B1, the phase noise can be approximately expressed as:

*P*=5 nW,

_{r}*P*=1.5 nW,

_{s}*η*=0.9,

*τ*=100 µs, we have δ

*ψ*≤2 mrad.

*ψ*≤80 mrad. This corresponds well with the phase error measured in the experiment.

## Acknowledgments

## References and links

1. | F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica |

2. | F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica |

3. | R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. |

4. | K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. |

5. | K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. |

6. | W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature methods |

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

8. | B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express |

9. | J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express |

10. | T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. |

11. | G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Optical measurement of cell membrane tension,” Phys. Rev. Lett. |

12. | M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. |

13. | D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. |

14. | J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, “Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer,” Appl. Phys. Lett. |

15. | Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, “Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements,” Opt. Express |

16. | M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers,” Opt. Lett. |

17. | Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. |

18. | M. Pilu, A. W. Fitzgibbon, and R. B. Fisher, “Ellipse-specific direct least-square fitting,” in |

19. | D. G. Ghiglia and M. D. Pritt, |

20. | C. Yang, “Molecular Contrast Optical Coherence Tomography: A Review,” Photochemistry and Photobiology |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: September 6, 2007

Revised Manuscript: December 4, 2007

Manuscript Accepted: December 14, 2007

Published: December 19, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Jigang Wu, Zahid Yaqoob, Xin Heng, Xiquan Cui, and Changhuei Yang, "Harmonically matched grating-based full-field quantitative high-resolution phase microscope for observing dynamics of transparent biological samples," Opt. Express **15**, 18141-18155 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-26-18141

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### References

- F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942). [CrossRef]
- F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942). [CrossRef]
- R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted light microscopy," Z. wiss.Mikr. 69, 193-221 (1969). [PubMed]
- K. Creath, "Phase-measurement interferometry techniques," Prog. Opt. 26, 349-393 (1988). [CrossRef]
- K. J. Chalut, W. J. Brown, and A. Wax, "Quantitative phase microscopy with asynchronous digital holography," Opt. Lett. 15, 3047-3052 (2007).
- W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007). [CrossRef] [PubMed]
- P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005). [CrossRef] [PubMed]
- B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, "Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy," Opt. Express 13, 9361-9373 (2005). [CrossRef] [PubMed]
- J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, "Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition," Opt. Express 15, 7231-7242 (2007). [CrossRef] [PubMed]
- T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30,1165-1167 (2005). [CrossRef] [PubMed]
- G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006). [CrossRef] [PubMed]
- M. V. Sarunic, S. Weinberg, and J. A. Izatt, "Full-field swept-source phase microscopy," Opt. Lett. 31, 1462-1464 (2006). [CrossRef] [PubMed]
- D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, "Three-dimensional images generated by quadrature interferometry," Opt. Lett. 23, 783-785 (1998). [CrossRef]
- J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, 151123 (2007). [CrossRef]
- Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, "Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements," Opt. Express 14, 8127-8137 (2006). [CrossRef] [PubMed]
- M. A. Choma, C. Yang, and J. A. Izatt, "Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers," Opt. Lett. 28, 2162-2164 (2003). [CrossRef] [PubMed]
- Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, "Homodyne en face optical coherence tomography," Opt. Lett. 31, 1815-1817 (2006) [CrossRef] [PubMed]
- M. Pilu, A. W. Fitzgibbon, and R. B. Fisher, "Ellipse-specific direct least-square fitting," in Proceedings of IEEE International Conference on Image Processing (Lausanne, 1998), vol. 3, pp. 599-602.
- D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.
- C. Yang, "Molecular Contrast Optical Coherence Tomography: A Review," Photochemistry and Photobiology 81, 215-237 (2005) [CrossRef]

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