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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 3, Iss. 1 — Jan. 29, 2008
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Harmonically matched grating-based full-field quantitative high-resolution phase microscope for observing dynamics of transparent biological samples

Jigang Wu, Zahid Yaqoob, Xin Heng, Xiquan Cui, and Changhuei Yang  »View Author Affiliations


Optics Express, Vol. 15, Issue 26, pp. 18141-18155 (2007)
http://dx.doi.org/10.1364/OE.15.018141


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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

Observing unstained transparent biological samples with sufficiently high resolution is important for a wide range of biomedical studies. Besides conventional qualitative techniques such as Zernike phase contrast [1

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

,2

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

] and Nomarski differential interference contrast (DIC) microscopy [3

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]

], various full-field quantitative phase imaging techniques [4

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

13

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]

] have been developed recently that are well suited for this purpose. Most of these methods involve the use of interferometry in one form or another. Some of the prominent techniques are: 1) Phase shifting interferometry schemes [4

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

6

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]

] – where two or more interferograms with different phase shifts are acquired sequentially and a phase image is generated from them. 2) Digital holography [7

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

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]

] or Hilbert phase microscopy [10

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

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]

] – where high frequency spatial fringes encoded in the interferogram are demodulated to generate the phase image. 3) Swept-source phase microscopy [12

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

] – where modulation in the interferogram generated by a wavelength sweep can be processed to create a phase image. 4) Polarization quadrature microscopy [13

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]

] – where phase images are generated by a polarization based quadrature interferometer.

Recently, our group demonstrated a full-field phase imaging technique based on the substitution of a beamsplitter with a harmonically matched diffraction grating pair (G1G2 grating) [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]

, 15

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]

] in an interferometer – we named the technique G1G2 interferometry. With this optical element substitution, we were able to create a non-trivial phase relationship between the different output ports of the interferometer. Conceptually, the operating principle shares many similarities to the multiport fiber coupler based interferometry method [16

16. 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]

, 17

17. Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006) [CrossRef] [PubMed]

]. As G1G2 interferometry involves a minimal change of optical elements – replacement of a beamsplitter, it can be potentially adapted into a wide range of interferometer systems. Phase imaging based on this technique involves simultaneously acquiring images from two or more of the output ports; the imaging speed of this technique is limited only by the cameras’ speed. This is an advantage over phase shifting interferometry methods, where the imaging speed is additionally limited by the speed of the phase shifting process. Another appealing aspect of this technique is that data processing is relatively simple. Yet another advantage of this technique is that, unlike some of the other phase techniques, its phase imaging capability does not require a tradeoff in field of view (see discussion in Ref. 5). Specifically, methods such as digital holography and Hilbert phase microscopy perform phase measurements by implementing a spatial sinusoidal interference – each resolvable point on the image is limited by the interference fringe and is by necessity several sensor pixels wide. In comparison, each resolvable point in G1G2 interferometry imaging can be as small as a single sensor pixel.

This paper reports on our recent progress. We have addressed the two abovementioned design issues and have developed a more robust G1G2 interferometer system. Using this system, we were able to acquire our first high resolution phase images of biological samples based on the G1G2 interferometry principle. The rest of this paper is structured as follows. In Section 2, we summarize the concept of G1G2 interferometry. In Section 3, we present our current experimental setup and discuss the means by which we reduced aberrations. In Section 4, we present our modified phase extraction algorithm that can accommodate the usage of reference and sample beams of comparable strength in the interferometer. In Section 5, we show phase images and videos acquired with our system – this illustrates the capability of the G1G2 interferometer method for biological studies. Finally, we summarize our work in Section 6.

2. G1G2 interferometery concept

This section briefly summarizes the G1G2 interferometry concept. Interested readers are invited to read Ref. 14 and 15 for more in depth explanations.

Fig. 1. (a). Simple beamsplitter based interferometer: 180o phase shift between output ports; (b) Single shallow grating based interferometer: still 180o phase shift between output ports; (c) G1G2 grating based interferometer: non-trivial phase shift can be obtained.

First, let us consider a simple interferometer, such as the one shown in Fig. 1(a). The detected signals at the output ports can be expressed as:

Port1:P1=Pr2+Ps2+PrPscos(ψsψr+θ)
(1)
Port2:P2=Pr2+Ps2PrPscos(ψsψr+θ)
(2)

where Pr and Ps are the reference power and sample power, respectively; ψr and ψs are the phase of the reference beam and sample beam before incident on the beamsplitter; 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.

However, power conservation does not intrinsically limit the interference terms to be trivially related. Multiport optical fiber coupler based interferometry methods [16

16. 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]

, 17

17. Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006) [CrossRef] [PubMed]

] clearly illustrate that the existence of multiple output ports allows non-trivial interference phase relationship to exist between output ports without violating power conservation. These methods indicate the possibility of using diffraction gratings in place of beamsplitters as a means for achieving non-trivial interference phase relationship in a full-field interferometer, as a diffraction grating can diffract light in more than two directions.

Now let us consider the shallow diffraction grating based interferometer scheme in Fig. 1(b). As is well known, a grating can confer additional phase shifts to the diffracted beams. For a shallow grating, the phase shifts can be written as [15

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]

]

ϕ(x0)=m[sgn(m)2πx0Λ+π2]
(3)

where 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:

Port1:P1=Pr1+Ps1+2Pr1Ps1cos(2πx0Λ+π2+θ)
(4)
Port2:P2=Pr2+Ps2+2Pr2Ps2cos(2πx0Λπ2+θ)
(5)

ϕR,1=ϕ0,G1G2=0,ϕS,1=ϕ+1,G1=2πx1Λ1+π2,
ϕR,2=ϕ1,G2=2πx2Λ2+π2,ϕS,2=ϕ1,G1=2πx1Λ1+π2
(6)

where ϕm,Gn is the additional phase shift of the 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

Δϕ1=ϕS,1ϕR,1=2πx1Λ1+π2,Δϕ2=ϕS,2ϕR,2=2πx1Λ1+2πx2Λ2
(7)
Δϕ=Δϕ2Δϕ1=4π(x2x1)Λ1π2
(8)

As long as 4π(x2x1)Λ1(n+12)π;nZ, we will have a non-trivial phase shift between the output ports of the interferometer. By adjusting the relative displacement between G1 and G2 grating during fabrication, we can set the phase shift Δϕ. 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

Our improved experimental setup is shown in Fig. 2. A HeNe laser (Thorlabs HRP120, wavelength of 632.8 nm) was split into reference beam and sample beam. In the reference arm, the laser was spatial filtered and expanded by objective 1 (Newport M-10X), a pinhole (diameter of 25µm) and lens 1 (focal length of 200 mm). The transmitted and diffracted reference beams were then collimated by lens 3 and 4, respectively. The focal lengths of lenses 2, 3, 4 were all equal to 200 mm. In the sample arm, objective 2 (Olympus UPlanFl 10X) and lens 3 (lens 4) made up the microscope system that imaged the sample onto the CCDs (The Imaging Source DMK 31BF03, 1024×768 pixels). The harmonically matched grating pair (G1G2 grating) served as the beam splitter/combiner. The G1G2 grating pair contains gratings of density 600 lines/mm and 1200 lines/mm. We measured Δϕ to be equal to 92°±8°. More details about the grating pair can be found in Ref. 14.

Fig. 2. Experimental setup for phase imaging. BS: beam splitter; O1 and O2: objective lenses 1 and 2; P: pinhole; L1-4: lens 1-4; S: sample; G1G2: the harmonically matched grating pair (G1G2 grating) on a holographic plate.

To correct this astigmatic aberration, we employed the imaging setup shown in Fig. 3(c), where the grating was put between the objective and the lens. In this case, the incident light associated with each point on the object plane was transformed into a collimated beam at the grating. The collimated beams would diffract from the grating as collimated beams. These would then transform back to point objects in the image plane by passage through the lens. Our Zemax simulations confirmed this fact [see Fig. 3(d)]. Figure 3(f) shows the image of the same letter “C” acquired using this setup, with which we can easily see that the obvious astigmatism aberration had been removed.

We experimentally determined the resolution of this imaging system to be equal to 1.6 µm (Sparrow’s criterion) by measuring the image profile of an effective point source (a hole of diameter 150 nm). The measured resolution agreed well with the theoretically calculated value of 1.2 µm based on the imaging system parameters.

This updated setup confers aberrations onto the input collimated light that is used to illuminate the sample [see Fig. 3(g)]. This is because the collimated light will be focused by the objective and will diverge when it enters the grating. The aberration of background beam will introduce unwanted pattern in the interferograms and unwanted phase aberration in the final phase image. Fortunately, we can measure and characterize this phase aberration during initial system calibration by removing the sample. This systematic error can then be removed from actual sample image measurements during data processing.

Fig. 3. Geometric aberration induced by the grating. (a) Previous imaging setup; (b) Astigmatism of the focal spots of previous setup; (c) Current imaging setup; (d) No aberration in the focal spot of current setup; (e) Image of letter “C” acquired by the previous setup; (f) Image of letter “C” acquired by the current setup; (g) Aberration of illumination beam caused by the grating diffraction in current setup.

One final aberration issue that we need to be concerned about is the spatial “stretch” and “compression” distortion on the off-axis diffracted raw data image due to the diffraction process. As shown in Fig. 4, the image will be stretched in one direction and compressed in the other direction, and this effect is opposite for +1 and -1 order diffraction of the grating. This distortion in principle will affect the matching of the two CCD images. However, the following calculation shows that the effect is small enough and can be neglected. The distortion is due to the fact that the angle of diffraction associated with the grating does not linearly depend on the incident angle. Mathematically, a point that is 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:

x2=f2tan[sin1(λdsin(tan1x1f1))sin1λd]
(9)

where f1, f2 are the focal length of the objective and the lens, respectively; 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

x2=f2f11λ2d2x1
(10)

where 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.

Fig. 4. “Stretch” and “compression” distortion caused by the diffraction of the grating in the imaging system.

4. Improved processing algorithm

In the current experiment, we removed this restriction and performed imaging processing without simplifying the involved equations. This section details the processing involved.

The detected signal in the corresponding pixels with pixel index (i,j) of the CCDs can be written as

P1(i,j)=Pr1(i,j)+Ps1(i,j)+APr1(i,j)Ps1(i,j)cos(Δψ(i,j))
(11)
P2(i,j)=Pr2(i,j)+η(i,j)Ps1(i,j)
+APr2(i,j)η(i,j)Ps1(i,j)cos(Δψ(i,j)+Δϕ)
(12)

The experiment procedure for phase extraction can be summarized as follows:

1) Determine the relative diffraction efficiency η(i, j).

2) Determine the phase aberration ψabe(i, j) and the nontrivial phase shift Δϕ. 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 yi,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

18. 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.

] between yi,j,k(n) and y1,1,k(n). This computed phase difference is equal to the phase aberration ψabe. Similarly, by performing elliptic fitting between the corresponding pixels from the two CCDs, yi,j,1(n) and yi,j,2(n), we can get the phase shift Δϕ.

3) Determine the reference power Pr1(i, j) and Pr2(i, j).

4) Acquire phase image of the sample. This involves placing the sample into the system and acquiring a frame pair from the two CCDs. The detected signals of corresponding pixels must satisfy Eqs. (11) and (12). The only remaining unknowns are Ps1(i, j) and Δψ(i, j).

By canceling Δψ(i, j) from Eqs. (11) and (12), we can obtain a quadratic equation for Ps1:

[1Pr1+ηPr22cos(Δϕ)ηPr1Pr2]Ps12[2P1Pr1Pr1+2P2Pr2Pr2
2cos(Δϕ)(P1Pr1)η+(P2Pr2)ηPr1Pr2+A2sin2(Δϕ)]Ps1
+[(P1Pr1)2Pr1+(P2Pr2)2Pr2η2cos(Δϕ)(P1Pr1)(P2Pr2)ηPr1Pr2]=0
(13)

For clarity, we omitted the functional dependency on (i, j) in the expression. This equation yields two solutions for Ps1. In order to find the right solution, the sample power Ps1 need to satisfy some condition. As shown in the appendix A, if the real Ps1 satisfies the following condition, we can always use the smaller solution of Eq. (13) as our solution:

Ps1Pr1+Ps2Pr22Ps1Pr1Ps2Pr2cos(Δϕ)<A24sin2(Δϕ)
(14)

For the ideal quadrature phase shift, Δϕ=90° and ideal coherence, A=2, the above condition becomes

Ps1Pr1+Ps2Pr2<1
(15)

The term θran 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 ψabj+θran at a location in the image where it is known that ψobj is not varying. The time dependent variations can then be wholly attributed to θran. 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.

We note that step 1 through 3 need only be done once during calibration.

Fig. 5. Compare unwrap algothms. (a) Wrapped image; (b) Unwrapped image by simple unwrap algorithm; (c) Unwrapped image by Flynn’s algorithm.

As phase images are intrinsically wrapped beyond the phase range of [0, 2π], we generally need to unwrap the acquired images when dealing with samples beyond a certain thickness. Due to the presence of noise in the phase image, simple unwrap algorithm generally does not work well. For our experiment, we instead chose to use the Flynn’s minimum discontinuity algorithm [19

19. D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.

] to unwrap the image. Flynn’s algorithm operates by identifying the lines of discontinuities and joining them into loops. Appropriate multiples of 2π are then added to each pixel enclosed by the loops to remove the phase wrap discontinuities. This unwrap algorithm worked well for our data. Figure 5 shows an example of the application of unwrap algorithms to our data. Figure 5(a) shows the wrapped phase image acquired by our system. Figure 5(b) shows an unwrapped phase image as generated by the simple unwrap algorithm. Figure 5(c) shows that the Flynn’s algorithm is capable of better phase unwrap performance.

To characterize the phase stability of our system, we used a cover glass as our sample and measured the phase difference between two different spots on the cover glass. The phase of the spots is the average of the pixels in the spots with corresponding object size of 1.2 µm×1.2 µm, matching the diffraction limit of the objective lens. The sample power incident on each spot was 1.6 nW (port 1) and 1.5 nW (port 2). The reference power incident on each spot was 7.6 nW (port 1) and 3.1 nW (port 2). The exposure time per image frame was 100 µs. The experiment results are shown in Fig. 6. The phase image is shown in Fig. 6(a), and the fluctuation of the phase difference versus time is shown in Fig. 6(b). The phase stability of our system is characterized by the standard deviation of the fluctuation which equals to 62 mrad (corresponding to 6.24 nm optical path length).

We can also estimate the shot noise limited phase noise of our system theoretically, as shown in appendix B. The estimated shot noise limit is 2 mrad, which is much smaller than the experimental measurement. The experimentally measured phase error can be largely attributed to the spatially uncorrelated power fluctuation observed in the experiment. We measured this power fluctuation to be ~3% for each pixel. Substituting this noise factor into our calculation yields a phase noise of ~80 mrad, which is comparable to our experimentally measured phase error.

In addition to the phase stability characterization, this set of experiments revealed another aspect of this experimental scheme – we can see some dim fringes in the phase image in Fig. 6(a). These fringes were caused by the relative larger phase noise near some special phase locations. For a simple explanation, consider the quadratic Eq. (13), aP 2 s1+bPs1+c=0, where a, b, c are the coefficients shown in the equation, the solutions are

Ps1=b±Δ2a,whereΔ=b24ac
(16)

The error of the solutions can be written as the function of the measured error of P1, P2:

δPs1=Ps1P1δP1+Ps1P2δP2
=bP1±(ΔP1)2Δ2aδP1+bP2±(ΔP2)2Δ2aδP2
(17)

Thus, when Δ is small, the error of the solution will be large. The fringes in Fig. 6(a) correspond to locations where this effect occurred. Fortunately, in our experiments, this additional phase noise was relatively small (~0.2 rad) and showed up in only a small region of the image.

Fig. 6. Measurement of the temporal phase stability. (a) Phase image of a cover glass, the two spots that are used to measure the phase stability are indicated; (b) Fluctuation of the phase of spot 2 with respect to spot 1 versus time, the standard deviation is 62 mrad.

5. Imaging results

Fig. 7. Images of “CIT” logo by our iamging system. (a) Intensity image; (b) Phase image; (c) 3D reconstruction of the phase image; (d) Step-height measurement.

Fig. 8. Images of onion skin cells. (a) Intensity image; (b) Phase image; (c) 3D reconstruction of the phase image.

To demonstrate the capability of our system to study biological movements and dynamics, we applied our system to observe the movement of an amoeba proteus. In the phase movie [Fig. 9(a)], we can clearly see the nucleus and contractile vacuole of the amoeba. The frame rate of the movie is 10 frames/second. The movement of the food vacuoles can be seen in the movie. Figure 9(b) shows the corresponding intensity movie, which had much poorer contrast.

Fig. 9. (a). (360 KB) MOVIE: phase movie of movement of amoeba proteus. The nucleus and contractile vacuole can be clearly seen. In the movie, the food vacuoles are moving inside the amoeba. The size of one frame is 147 µm (width)×123 µm (height) (372×312 pixels) [Media 1] (b) (1.66 MB) Movie: intensity movie of movement of amoeba proteus [Media 2].

6. Summary

We have developed an improved full-field quantitative phase imaging microscope system based on the use of a harmonically matched grating pair (G1G2 grating). This new system design significantly corrected the astigmatic aberration which had restricted the original interferometer design. The system also employed a new phase image processing algorithm which enabled a wide range of sample power to be used in the interferometer.

With these improvements, we were able to demonstrate high resolution phase imaging with a measured resolution of 1.6 µm and a phase sensitivity of 62 mrad (or an equivalent optical path length difference of 6.24 nm). We demonstrate the utility of this interferometer by imaging onion samples and rendering a movie of a moving amoeba proteus.

This phase imaging method is applicable for observing fast dynamics in biological samples as the image acquisition speed is only limited by the frame rate of the cameras. Another appealing aspect of this system is that the heart of the interferometer – the G1G2 grating is a planar device that can be easily designed and fabricated. The feasibility of creating G1G2 grating lithographically or by e-beam etching, also allows for more complicated G1G2 grating designs. For example, it may be interesting to combine Fresnel zone plate and G1G2 grating designs to implement flat phase imaging schemes. In addition, the concept of G1G2 interferometry can be applied to imaging at other wavelengths, such as X-ray or terahertz.

Appendix A: derivation of equation (14)

The derivation for the condition where we can choose one solution out of the two solutions that we get from the quadratic Eq. (13) is as follows. To begin, assume the sample power solution is s 0 and the other solution of Eq. (13) is s 1, then we have

s0+s1=2P1Pr1Pr1+2P2Pr2Pr22cos(Δϕ)(P1Pr1)η+(P2Pr2)ηPr1Pr2+A2sin2(Δϕ)1Pr1+ηPr22cos(Δϕ)ηPr1Pr2
(A.1)

If we always 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

2s0<2P1Pr1Pr1+2P2Pr2Pr22cos(Δϕ)(P1Pr1)η+(P2Pr2)ηPr1Pr2+A2sin2(Δϕ)1Pr1+ηPr22cos(Δϕ)ηPr1Pr2
(A.2)

Substitute P1 and P2 from Eqs. (11), (12), where Ps1=s 0, we get

s0(1Pr1+ηPr22cos(Δϕ)ηPr1Pr2)
<P1Pr1Pr1+P2Pr2Pr2cos(Δϕ)(P1Pr1)η+(P2Pr2)ηPr1Pr2+A22sin2(Δϕ)
=s0+APr1s0cos(Δψ)Pr1+ηs0+APr2ηs0cos(Δψ+Δϕ)Pr2
cosΔϕ[s0+APr1s0cos(Δψ)]η+[ηs0+APr2ηs0cos(Δψ+Δϕ)]ηPr1Pr2+A22sin2(Δϕ)
As0Pr1sin(Δψ)sin(Δψ+Δϕ)Aηs0Pr2sinΔϕsin(Δψ)+A22sin2(Δϕ)>0
(A.3)

Without loss of generality, we can let sin(Δϕ)>0, thus

As0Pr1sin(Δψ+Δϕ)Aηs0Pr2sin(Δψ)+A22sin(Δϕ)>0
[As0Pr1cos(Δϕ)Aηs0Pr2]sin(Δψ)+As0Pr1sin(Δϕ)cos(Δψ)+A22sin(Δϕ)>0
(A.4)

If for all Δψ, the above equation is satisfied, we must have

[As0Pr1cos(Δϕ)Aηs0Pr2]2+[As0Pr1sin(Δϕ)]2+A22sin(Δϕ)>0
Ps1Pr1+Ps2Pr22Ps1Pr1Ps2Pr2cos(Δϕ)<A24sin2(Δϕ)
(A.5)

where Ps1=s 0 and Ps2s 0. This is the Eq. (14).

Appendix B: noise assessment based on system’s shot noise

In this appendix we estimate the phase noise associated with the system in the situation where shot noise is the only major noise source. To simplify the problem, we assume that the reference powers and sample powers are the same for the two output ports. We also assume the interference factor A=2 and the nontrivial phase shift Δϕ=-90°. Now the power detected in the two output ports can be written as

P1=Pr+Ps+2PrPscosΔψ
(B.1)
P2=Pr+Ps+2PrPssinΔψ
(B.2)

So

2PrPscosΔψ=P1PrPs
(B.3)
2PrPssinΔψ=P2PrPs
(B.4)

We shall assume the detected power P1 and P2 contain additive Gaussian white-noise terms x 1, x 2 with zero mean, respectively. For shot noise, the standard deviation of x 1, x 2 should be20

σx1=hνητP1,σx2=hνητP2
(B.5)

where 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:

δψσx12+σx222PrPs=12hνητP1+P2PrPs=22hνητPr+Ps+PrPs(cosΔψ+sinΔψ)PrPs
22hνητPr+Ps+2PrPsPrPs
(B.6)

For a typical set of numbers for our system, Pr=5 nW, Ps=1.5 nW, η=0.9, τ=100 µs, we have δψ≤2 mrad.

We note that the phase noise in shot noise limited detection is more than an order of magnitude smaller than the phase noise (62 mrad) observed in the experiment. This observation indicates that phase noise in our system is dominated by other sources. One probable source is the amount of spatially uncorrelated laser power fluctuations that are observed in our system. We measured the power fluctuation to be approximately equal 3% of the mean. If we substitute

σx1=0.03P1,σx2=0.03P2

to model this power fluctuation, we see that the phase noise can be expressed as:

δψσx12+σx222PrPs=0.015P21+P22PrPs
=0.0152(Pr+Ps)2+4PrP+4(Pr+Ps)PrPs(cosΔψ+sinΔψ)PrPs
0.0152(Pr+Ps)2+4PrP+4(Pr+Ps)2PrPsPrPs
(B.7)

Substituting the typical powers employed in the experiment, we find that δψ≤80 mrad. This corresponds well with the phase error measured in the experiment.

Fig.B1. 1. Schematic of the phase noise assessment.

Acknowledgments

This work is supported by NSF career award BES-0547657.

References and links

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]

5.

K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. 15, 3047–3052 (2007).

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]

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 13, 9361–9373 (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]

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]

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]

16.

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]

17.

Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006) [CrossRef] [PubMed]

18.

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.

19.

D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.

20.

C. Yang, “Molecular Contrast Optical Coherence Tomography: A Review,” Photochemistry and Photobiology 81, 215–237 (2005) [CrossRef]

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

  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]
  5. K. J. Chalut, W. J. Brown, and A. Wax, "Quantitative phase microscopy with asynchronous digital holography," Opt. Lett. 15, 3047-3052 (2007).
  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]
  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 13, 9361-9373 (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]
  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]
  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]
  16. 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]
  17. Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, "Homodyne en face optical coherence tomography," Opt. Lett. 31, 1815-1817 (2006) [CrossRef] [PubMed]
  18. 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.
  19. D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.
  20. C. Yang, "Molecular Contrast Optical Coherence Tomography: A Review," Photochemistry and Photobiology 81, 215-237 (2005) [CrossRef]

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