## Rapid quantitative phase imaging for partially coherent light microscopy |

Optics Express, Vol. 22, Issue 11, pp. 13472-13483 (2014)

http://dx.doi.org/10.1364/OE.22.013472

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### Abstract

Partially coherent light provides promising advantages for imaging applications. In contrast to its completely coherent counterpart, it prevents image degradation due to speckle noise and decreases cross-talk among the imaged objects. These facts make attractive the partially coherent illumination for accurate quantitative imaging in microscopy. In this work, we present a non-interferometric technique and system for quantitative phase imaging with simultaneous determination of the spatial coherence properties of the sample illumination. Its performance is experimentally demonstrated in several examples underlining the benefits of partial coherence for practical imagining applications. The programmable optical setup comprises an electrically tunable lens and sCMOS camera that allows for high-speed measurement in the millisecond range.

© 2014 Optical Society of America

## 1. Introduction

1. F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. **31**, 178–180 (2006). [CrossRef] [PubMed]

5. Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat Photon **7**, 113–117 (2013). [CrossRef]

3. M. Kim, *Digital Holographic Microscopy: Principles, Techniques, and Applications*, Springer Series in Optical Sciences (Springer, 2011). [CrossRef]

5. Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat Photon **7**, 113–117 (2013). [CrossRef]

8. J. A. Rodrigo, T. Alieva, G. Cristóbal, and M. L. Calvo, “Wavefield imaging via iterative retrieval based on phase modulation diversity,” Opt. Express **19**, 18621–18635 (2011). [CrossRef] [PubMed]

9. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. **133**, 339–346 (1997). [CrossRef]

10. C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express **21**, 24060–24075 (2013). [CrossRef] [PubMed]

10. C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express **21**, 24060–24075 (2013). [CrossRef] [PubMed]

12. B. Rappaz, A. Barbul, Y. Emery, R. Korenstein, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Comparative study of human erythrocytes by digital holographic microscopy, confocal microscopy, and impedance volume analyzer,” Cytometry Part A **73A**, 895–903 (2008). [CrossRef]

13. B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat Photonics **6**, 355–359 (2012). [CrossRef] [PubMed]

14. M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. **38**, 3858–3861 (2013). [CrossRef] [PubMed]

15. B. Kemper, S. Sturwald, C. Remmersmann, P. Langehanenberg, and G. von Bally, “Characterisation of light emitting diodes (LEDs) for application in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Lasers Eng. **46**, 499–507 (2008). [CrossRef]

18. T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat Photonics **8**, 256–263 (2014). [CrossRef]

15. B. Kemper, S. Sturwald, C. Remmersmann, P. Langehanenberg, and G. von Bally, “Characterisation of light emitting diodes (LEDs) for application in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Lasers Eng. **46**, 499–507 (2008). [CrossRef]

16. P. Langehanenberg, G. v. Bally, and B. Kemper, “Application of partially coherent light in live cell imaging with digital holographic microscopy,” J. Mod. Opt. **57**, 709–717 (2010). [CrossRef]

*J*(

*τ*)Γ(

**r**

_{1},

**r**

_{2}) [19]. The temporal function

*J*(

*τ*) describes the coherence gating effect often exploited for axial optical sectioning [20

20. I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys-Berlin **524**, 787–804 (2012). [CrossRef]

*λ*≪

*λ*

_{0}) with nearly Gaussian spectrum, the coherence length is often approximated as

*λ*

_{0}is the central wavelength and Δ

*λ*is the full width at half maximum (FWHM) of the spectrum [19].

**r**

_{1},

**r**

_{2}), referred to as mutual intensity (MI), describes the image formation of a sample’s layer. In the scalar paraxial approximation the MI of a two-dimensional (2D) wavefield is described by a complex-valued 4D MI defined as Γ(

**r**

_{1},

**r**

_{2}) = 〈

*f*(

**r**

_{1})

*f*

^{*}(

**r**

_{2})〉, where:

**r**

_{1,2}= (

*x*,

*y*)

_{1,2}is a position vector in a plane transverse to the light propagation direction and 〈

*·*〉 stands for ensemble averaging. Completely coherent beams are characterized by Γ

*(*

_{c}**r**

_{1},

**r**

_{2}) =

*f*(

**r**

_{1})

*f*

^{*}(

**r**

_{2}) while Schell-model partially coherent beams (SMBs) [21] are described by Γ(

**r**

_{1},

**r**

_{2}) =

*f*(

**r**

_{1})

*f*

^{*}(

**r**

_{2})

*γ*(

**r**

_{1}−

**r**

_{2}) = Γ

*(*

_{c}**r**

_{1},

**r**

_{2})

*γ*(Δ

**r**), where

*γ*(Δ

**r**) is an equal-time complex degree of spatial coherence (DSC). The Schell model is widely used in different imaging applications, including microscopy. For instance, in bright field microscopes under Köhler illumination with incoherent source, the SMB corresponds to the beam scattered by an object described by the complex modulation function

*f*(

**r**). In this case the DSC,

*γ*(Δ

**r**), at the sample plane is given by the Fourier transform of the intensity distribution of the illumination beam in the condenser lens aperture, according to the van-Cittert-Zernike theorem [19]. Therefore, the SMB is present in illumination schemes with different sources such as LEDs, halogen lamps and randomized laser beams, however, its experimental characterization is challenging. The DSC can be tuned in several ways, for example, by spatial filtering of the incoherent source or by using a moving diffuser in the case of a coherent source (e.g. laser) [22

22. C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, “Fast measurements of concentration profiles inside deformable objects in microflows with reduced spatial coherence digital holography,” Appl. Opt. **47**, 5305–5314 (2008). [CrossRef] [PubMed]

24. J. A. Rodrigo and T. Alieva, “Recovery of Schell-model partially coherent beams,” Opt. Lett. **39**, 1030–1033 (2014). [CrossRef] [PubMed]

*μ*m) is often used as a spatial filter for LED and halogen sources in order to obtain high spatial coherence but low temporal coherence [15

15. B. Kemper, S. Sturwald, C. Remmersmann, P. Langehanenberg, and G. von Bally, “Characterisation of light emitting diodes (LEDs) for application in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Lasers Eng. **46**, 499–507 (2008). [CrossRef]

18. T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat Photonics **8**, 256–263 (2014). [CrossRef]

*f*(

**r**) and the DSC of the illumination beam. Its performance is experimentally demonstrated in several examples which underline the advantages of partial coherent light for speckle-noise and cross-talk reduction. The reconstructed object wavefield can be numerically refocused for accurate topography analysis or inspection of the sample. The proposed experimental setup comprises a high-speed sCMOS camera and electrically tunable fluidic lens (ETL), which focal length is tuned in few milliseconds by applying current control. In contrast to the widely used liquid-crystal spatial light modulators, the ETLs have faster response. In addition, they are made from low dispersion and polarization preserving materials. Such ETLs have been used for high-speed volumetric imaging [25

25. F. O. Fahrbach, F. F. Voigt, B. Schmid, F. Helmchen, and J. Huisken, “Rapid 3D light-sheet microscopy with a tunable lens,” Opt. Express **21**, 21010–21026 (2013). [CrossRef] [PubMed]

10. C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express **21**, 24060–24075 (2013). [CrossRef] [PubMed]

## 2. Principle of the MI recovery technique

*d*from the ETL as sketched in Fig. 1(a). Therefore, the input beam is focused into the ETL and then imaged into the detector plane (e.g. sCMOS camera) for several values of the ETL’s focal length: f

*. This measurement setup can be easily attached to the microscope as sketched in Fig. 1(b). We underline that the tube lens and RL are identical.*

_{m}*γ̃*(

_{m}**r**

*) is the scaled Fourier transform (FT) of the input DSC:*

_{o}*γ̃*(

_{m}**r**

*) =*

_{o}*FT*[

*γ*(

**r**)](

**r**

*) with*

_{o}S_{m}*S*= f

_{m}*/ (*

_{m}*d*− f

*) being the scaling factor, see Appendix. Note that*

_{m}*FT*[·] indicates the FT optically performed by the RL. The expression Eq. (1) is the convolution between the

*coherent intensity*

*γ̃*(

_{m}**r**

*), where*

_{o}*ℱ*

^{(m)}describes the propagation of

*f*(

**r**) through the system (see Appendix). This makes possible the MI recovery of SMBs as it was experimentally demonstrated in [24

24. J. A. Rodrigo and T. Alieva, “Recovery of Schell-model partially coherent beams,” Opt. Lett. **39**, 1030–1033 (2014). [CrossRef] [PubMed]

24. J. A. Rodrigo and T. Alieva, “Recovery of Schell-model partially coherent beams,” Opt. Lett. **39**, 1030–1033 (2014). [CrossRef] [PubMed]

*g*(

**r**

*) at the input of the ETL by using the measured intensities*

_{i}*f*(

**r**) =

*FT*[

*g*(

**r**

*)](*

_{i}**r**). Specifically, the iterative algorithm comprises two loops labeled with index

*m*= 1,...,

*M*and

*n*= 1,...,

*N*, where

*M*is the number of measurements and

*M*×

*N*is the total number of iterations. The wavefield at the detector plane is

*W*(

_{m,n}**r**

*) =*

_{o}*F*[

_{d}*g*(

_{n}**r**

*)*

_{i}*L*(

_{m}**r**

*)](*

_{i}**r**

*): corresponding to the Fresnel diffraction integral of the current wavefield estimate*

_{o}*g*(

_{n}**r**

*), where*

_{i}**21**, 24060–24075 (2013). [CrossRef] [PubMed]

25. F. O. Fahrbach, F. F. Voigt, B. Schmid, F. Helmchen, and J. Huisken, “Rapid 3D light-sheet microscopy with a tunable lens,” Opt. Express **21**, 21010–21026 (2013). [CrossRef] [PubMed]

**r**|/

*D*) (in our case with diameter

*D*= 5 mm) can be considered, thus it yields

*g*

_{1}(

**r**

*) =*

_{i}*FT*[circ(2|

**r**|/

*D*)](

**r**

*). The whole iterative process is described as it follows: (*

_{i}*i*) A new estimate of

*W*(

_{m,n}**r**

*) is obtained from the current version of*

_{o}*g*(

_{n}**r**

*), and then is replaced by the updated version where the current estimate of the intensity is*

_{i}*ii*) Then, the wavefield Eq. (3) is back-propagated to the ETL input plane obtaining a new estimate of the wavefield

*W*

_{m+1,n}(

**r**

*) is transformed into*

_{o}*W′*

_{m+1,n}(

**r**

*); (*

_{o}*iii*) the procedure described in (

*ii*) is performed using the rest of measured intensities, until

*m*=

*M*. Then

*W′*(

_{M,n}**r**

*) is inverted to obtain an updated estimate of the wavefield:*

_{o}*i*)–(

*iii*) is iterated over the index

*n*and stops (at

*n*=

*N*) when the estimated intensities

*γ̃*(

_{m}**r**

*) →*

_{o}*δ*(

**r**

*−*

_{o}**r**′

*), expression Eq. (3) reduces to the usual modulus constraint that replaces the amplitude by the measured one while retaining the phase.*

_{o}26. J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat Commun **3**, 993 (2012). [CrossRef] [PubMed]

**39**, 1030–1033 (2014). [CrossRef] [PubMed]

*i*) as it follows: with

*k*= 1,...,

*K*being the RLD iteration index, where

*γ̃*(

_{m}**r**

*) is a real-positive function. Applying this algorithm the object wavefield and the DSC can be successfully recovered from the experimental data [24*

_{o}**39**, 1030–1033 (2014). [CrossRef] [PubMed]

*ε*) given by:

*q*being the pixel index and

*Q*the number of pixels of the image. In the experiments described in the following Sections, the algorithm reached convergence with low RMS error (below 15 %) after few iterations (typically

*N*= 10 and

*K*= 2, for

*M*= 9).

3. M. Kim, *Digital Holographic Microscopy: Principles, Techniques, and Applications*, Springer Series in Optical Sciences (Springer, 2011). [CrossRef]

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

*Q*= 1024 × 1024 pixels). Nevertheless, to speed up the MI reconstruction, the computation was preformed using a Graphics Processing Unit (GPU, nVidia GeForce GTX550-Ti), thus reconstructing the object wavefield

*f*(

**r**) and DSC in few seconds (typically 20 s).

## 3. Experimental setup and testing results

*λ*= 640±1 nm, coherence length ∼ 5 mm) and LED (Kingbright, GaAlAs,

_{laser}*λ*

_{0}= 648 ± 1 nm, Δ

*λ*= 12±2 nm corresponding to coherence length ∼ 17

*μ*m in water), which provide fully and partially temporal-spatial coherent illumination over the sample, correspondingly. In the transmission inverted microscope sketched in Fig. 1(b), the sample is enclosed between two glass coverslips (attached with a double-sided tape ∼ 90

*μ*m thick, glass thickness 0.17 mm) and imaged under Köhler illumination by an oil immersion objective (Olympus UPLSAPO, 1.4 NA, 100×, oil Cargille Type B). As a condenser we used a 10× objective (Nikon Plan Achromat, 0.25 NA).

*μ*m). The shift of the imaged plane is given as a function of the focal length (f) of the ETL, the effective magnification of the objective-tube lenses (

*M*= 83) and focal length f

*of the relay lens as it follows:*

_{RL}*n*= 1.33 is the refractive index of the immersion medium (water). The focal range of the ETL is 84 – 208 mm and therefore the available focusing scan range is about 30

_{s}*μ*m. Nevertheless, the measurement of the constraint images is performed in a shorter focusing scan range (15

*μ*m) according to the coherence gating of the LED illumination. As previously discussed, the proposed MI recovery technique reconstructs the object wavefield only within the coherence gate defined by the longitudinal coherence length of the LED. To avoid distortions in the reconstructed information (e.g. object’s phase and DSC) caused by the Brownian motion in the sample, the acquisition of the constraint images (1024 × 1024 pixels) was performed in about 125 ms at 144 frames per second (fps). This was achieved by using a periodic current driving signal of 2 Hz (triangular, current range 0 – 293 mA) applied to the ETL. Although 18 images are acquired in 125 ms, we considered only the nine ones belonging to the coherence gate. The corresponding focal lengths are: 122.4, 115.7, 109.9, 104.6, 100.0, 95.8, 91.8, 87.9 and 84.3 mm (range 140 – 293 mA, see Appendix). Note that the ETL can be set up to 10 Hz for faster measurement that requires high frame rate acquisition above 200 fps. In our case an electrical lens driver controller with 12-bit precision for current and frequency control has been used. Both sCMOS camera and ETL were controlled with a Labview program developed by us.

*μ*m bead diameter, Spherotech Lot. AD01) using laser (

*n*(640nm) = 1.587) and LED (

_{o}*n*(648nm) = 1.586) illumination. From the retrieved phase

_{o}*ϕ*(

**r**) of the object wavefield, the particle thickness

*t*(

**r**) is determined as it follows:

*t*(

**r**) =

*ϕ*(

**r**)

*λ*

_{0}/2

*π*Δ

*n*, where Δ

*n*=

*n*−

_{o}*n*is the refractive index difference between the bead (

_{s}*n*) and its surrounding medium (water,

_{o}*n*= 1.33) [3

_{s}3. M. Kim, *Digital Holographic Microscopy: Principles, Techniques, and Applications*, Springer Series in Optical Sciences (Springer, 2011). [CrossRef]

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

*μ*m. The reconstruction of the object wavefield completely fails [see thickness profile plotted by a cyan dotted-line in Fig. 2(c)] when the LED illumination is incorrectly assumed as fully spatial coherent. These facts prove that for accurate quantitative imaging the spatial coherence of the illumination has to be taken into account.

**46**, 499–507 (2008). [CrossRef]

16. P. Langehanenberg, G. v. Bally, and B. Kemper, “Application of partially coherent light in live cell imaging with digital holographic microscopy,” J. Mod. Opt. **57**, 709–717 (2010). [CrossRef]

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

**39**, 1030–1033 (2014). [CrossRef] [PubMed]

*μ*m, which was estimated from the measured power spectrum of the LED displayed in Fig. 3. In spite of the difference in the temporal coherence, the images of the particles at the chamber’s top and bottom are similar in both cases [see second row of Fig. 3(b) and 3(c)], when acquired by a conventional CMOS camera (exposure time of 60 ms, Thorlabs-DCC1240C). This demonstrates that low spatial coherence allows for speckle noise suppression. Nevertheless, in high-speed measurement the image [see third row of Fig. 3(b) and 3(c)] is completely free of speckle noise only under the LED illumination. A small amount of speckle noise persists in Fig. 3(b) due to the limited refresh rate of the time-varying diffuser with respect to the short acquisition rate of the sCMOS camera (10 ms in this case). Therefore, LED illumination is preferable for coherent noise suppression in high-speed measurements.

## 4. Quantitative phase imaging of biological specimens with partially coherent light

*In vivo*analysis of blood smears reveals important information for clinical studies. Quantitative phase imaging is a powerful tool for maker-free microscopy that has been extensively applied for analysis of red blood cells (RBCs) immersed in different environments [11

11. Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by plasmodium falciparum,” PNAS **105**, 13730–13735 (2008). [CrossRef] [PubMed]

12. B. Rappaz, A. Barbul, Y. Emery, R. Korenstein, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Comparative study of human erythrocytes by digital holographic microscopy, confocal microscopy, and impedance volume analyzer,” Cytometry Part A **73A**, 895–903 (2008). [CrossRef]

*z*= 0. We underline that the reconstructed object wavefield can be numerically refocused within the coherence gate, which is crucial in the analysis of the sample. Indeed, both RBCs and white blood cells (WBCs) can be accurately focused by numerical propagation of the object wavefield, as observed in Fig. 4(c) and 4(d) for z = 2.82

*μ*m. In contrast to the intensity image, these cells are clearly distinguished in the phase image Fig. 4(d). Moreover, small cell structures and platelets (2 – 3

*μ*m in diameter) are observed as indicated in the zoom inset Fig. 4(e). These structures, however, are washed out in the phase image obtained with the coherent laser illumination Fig. 4(f). Indeed, the overall image quality is significantly degraded and only RBCs are distinguished.

*γ*(Δ

**r**)| = exp(−|Δ

**r**|

^{2}/2

*σ*

^{2}) with

*σ*= 4.8

*μ*m, which is often referred to as lateral coherent length or spatial coherence radius.

20. I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys-Berlin **524**, 787–804 (2012). [CrossRef]

## 5. Discussion

*in-situ*, both the object wavefield and degree of spatial coherence of the illumination beam. Its optical implementation is straightforward in conventional microscopes at low cost, without altering the microscope design. In general, it can be combined with digital image processing methods which, for example, take into account the point spread function of the microscope.

13. B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat Photonics **6**, 355–359 (2012). [CrossRef] [PubMed]

14. M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. **38**, 3858–3861 (2013). [CrossRef] [PubMed]

**39**, 1030–1033 (2014). [CrossRef] [PubMed]

29. A. V. Martin, F. R. Che, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy **106**, 914–924 (2006). [CrossRef] [PubMed]

## Appendix

## A1. Intensity distribution of a partially coherent SMB at the output of the ABCD system

*ℱ*

^{(m)}. This transformation can be easily calculated by using the ray transfer matrix [30] of the setup comprising the RL (fixed focal length f

*) and ETL (variable focal length f*

_{RL}*). Specifically, it is given by where the kernel is with:*

_{m}*σ*= 1/

*λ*f

*,*

_{RL}*β*

_{1}= f

*/ (*

_{m}*d*− f

*),*

_{m}*β*

_{2}= −

*β*

_{1}f

*/f*

_{RL}*and*

_{m}*β*

_{3}= −

*β*

_{1}

*d*/f

*.*

_{m}**r**

_{1},

**r**

_{2}) at the input plane of the system, the intensity distribution at its output plane is expressed as it follows: Taking into account that the beam is described by the Schell model and introducing new variables, Δ

**r**=

**r**

_{1}−

**r**

_{2}and

**R**= (

**r**

_{1}+

**r**

_{2}) /2, the Eq. (6) is expressed as Since for the coherent case the intensity

*γ*(Δ

**r**) = 1, one can rewrite Eq. (7) as a convolution: where

*γ̃*(

_{m}**r**

*) is the scaled FT of the input DSC:*

_{o}*γ̃*(

_{m}**r**

*) =*

_{o}*FT*[

*γ*(Δ

**r**)](

**r**

_{o}β_{1}). This fact has been found for other ABCD systems in [19, 24

**39**, 1030–1033 (2014). [CrossRef] [PubMed]

26. J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat Commun **3**, 993 (2012). [CrossRef] [PubMed]

*FT*[·] is optically performed by RL.

## A2. Calibration data of the ETL and its paraxial approximation

*n*= 1.299. It has a plano-convex shape that is accurately controlled in the focal tuning range 84 – 208 mm. We assume that the ETL can be described by the transmission function where its parabolic shape is given by with

_{lens}*ρ*

_{0}being the central deflection of the lens,

*a*= 5.5 mm is the semi-diameter of the lens and

*r*

^{2}=

*x*

^{2}+

*y*

^{2}. The parameter

*ρ*

_{0}is a function of the applied current

*I*(range 0 – 293 mA). Expression Eq. (10) and a Zemax model of the ETL were provided by the manufacturer.

_{ETL}*ρ*

_{0}and thus of the applied current: The term exp(

*i*2

*πρ*

_{0}

*n*) in Eq. (11) yields a variable phase shift with

_{lens}/λ*ρ*

_{0}=

*ρ*

_{0}(

*I*).

_{ETL}*ρ*

_{0}(

*I*) has to be determined. The back focal length of the ETL (

_{ETL}*f*) is given as a function of

_{ETL}*I*as displayed in Fig. 6(a), according to the manufacturer’s data. Note that the Zemax model of the ETL allows for estimation of both

_{ETL}*f*and the corresponding radius of curvature (

_{ETL}*R*). Since the central deflection of the lens is

*ρ*

_{0}(

*I*) can be estimated from

_{ETL}*R*(

*f*) taking into account the calibration

_{ETL}*f*(

_{ETL}*I*), see Fig. 6(b). In our case,

_{ETL}*ρ*

_{0}∈ [0.25, 0.55] mm and the ETL is well approximated by the thin tunable lens, Eq. (11), when using a

*correction function*of the current for the expression Eq. (12). This correction takes into account the effects of variable thickness of the real ETL in such a way that the thin lens behaves as a calibrated virtual ETL in the iterative algorithm. In Fig. 6(c) this correction function (

*I′*= 0.73

*I*+ 19.1 mA) is plotted as a function of the applied current

_{ETL}*I*. It has been estimated by linear fitting of the focal length shift of the uncorrected thin lens with respect the measured values

_{ETL}*f*. The calibrated focal length values used in the measurement (see Section 3) correspond to the current values: 140, 160, 180, 200, 220, 240, 260, 280 and 293 mA.

_{ETL}*λ*RMS, in the 80 % of ETL’s central aperture) can be included in this lens model to prevent distortions in the reconstructed image. Nevertheless, in the proposed setup [Fig. 1(b)] the object wavefield is focused into the ETL and thus illuminates the central region (about 40 % of the aperture) where such a phase aberration can be neglected.

## Acknowledgments

*Ministerio de Economía y Competitividad*is acknowledged for the project TEC2011-23629. We also thank Optotune AG (Switzerland) for technical assistance.

## References

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

3. | M. Kim, |

4. | P. Ferraro, A. Wax, and Z. Zalevsky, |

5. | Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat Photon |

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

7. | L. Camacho, V. Micó, Z. Zalevsky, and J. García, “Quantitative phase microscopy using defocusing by means of a spatial light modulator,” Opt. Express |

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

9. | T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. |

10. | C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express |

11. | Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by plasmodium falciparum,” PNAS |

12. | B. Rappaz, A. Barbul, Y. Emery, R. Korenstein, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Comparative study of human erythrocytes by digital holographic microscopy, confocal microscopy, and impedance volume analyzer,” Cytometry Part A |

13. | B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat Photonics |

14. | M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. |

15. | B. Kemper, S. Sturwald, C. Remmersmann, P. Langehanenberg, and G. von Bally, “Characterisation of light emitting diodes (LEDs) for application in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Lasers Eng. |

16. | P. Langehanenberg, G. v. Bally, and B. Kemper, “Application of partially coherent light in live cell imaging with digital holographic microscopy,” J. Mod. Opt. |

17. | S. O. Isikman, W. Bishara, and A. Ozcan, “Partially coherent lensfree tomographic microscopy [Invited],” Appl. Opt. |

18. | T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat Photonics |

19. | J. W. Goodman, |

20. | I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys-Berlin |

21. | A. C. Schell, “The multiple plate antenna,” Ph.D. thesis, Massachusetts Institute of Technology (1961). |

22. | C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, “Fast measurements of concentration profiles inside deformable objects in microflows with reduced spatial coherence digital holography,” Appl. Opt. |

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

24. | J. A. Rodrigo and T. Alieva, “Recovery of Schell-model partially coherent beams,” Opt. Lett. |

25. | F. O. Fahrbach, F. F. Voigt, B. Schmid, F. Helmchen, and J. Huisken, “Rapid 3D light-sheet microscopy with a tunable lens,” Opt. Express |

26. | J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat Commun |

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

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

29. | A. V. Martin, F. R. Che, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy |

30. | J. W. Goodman, |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(110.0180) Imaging systems : Microscopy

(120.4630) Instrumentation, measurement, and metrology : Optical inspection

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

**ToC Category:**

Microscopy

**History**

Original Manuscript: March 28, 2014

Revised Manuscript: May 12, 2014

Manuscript Accepted: May 16, 2014

Published: May 27, 2014

**Virtual Issues**

Vol. 9, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

José A. Rodrigo and Tatiana Alieva, "Rapid quantitative phase imaging for partially coherent light microscopy," Opt. Express **22**, 13472-13483 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13472

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### References

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- R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- L. Camacho, V. Micó, Z. Zalevsky, J. García, “Quantitative phase microscopy using defocusing by means of a spatial light modulator,” Opt. Express 18, 6755–6766 (2010). [CrossRef] [PubMed]
- J. A. Rodrigo, T. Alieva, G. Cristóbal, M. L. Calvo, “Wavefield imaging via iterative retrieval based on phase modulation diversity,” Opt. Express 19, 18621–18635 (2011). [CrossRef] [PubMed]
- T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997). [CrossRef]
- C. Zuo, Q. Chen, W. Qu, A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21, 24060–24075 (2013). [CrossRef] [PubMed]
- Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by plasmodium falciparum,” PNAS 105, 13730–13735 (2008). [CrossRef] [PubMed]
- B. Rappaz, A. Barbul, Y. Emery, R. Korenstein, C. Depeursinge, P. J. Magistretti, P. Marquet, “Comparative study of human erythrocytes by digital holographic microscopy, confocal microscopy, and impedance volume analyzer,” Cytometry Part A 73A, 895–903 (2008). [CrossRef]
- B. Redding, M. A. Choma, H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat Photonics 6, 355–359 (2012). [CrossRef] [PubMed]
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- B. Kemper, S. Sturwald, C. Remmersmann, P. Langehanenberg, G. von Bally, “Characterisation of light emitting diodes (LEDs) for application in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Lasers Eng. 46, 499–507 (2008). [CrossRef]
- P. Langehanenberg, G. v. Bally, B. Kemper, “Application of partially coherent light in live cell imaging with digital holographic microscopy,” J. Mod. Opt. 57, 709–717 (2010). [CrossRef]
- S. O. Isikman, W. Bishara, A. Ozcan, “Partially coherent lensfree tomographic microscopy [Invited],” Appl. Opt. 50, H253–H264 (2011). [CrossRef] [PubMed]
- T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat Photonics 8, 256–263 (2014). [CrossRef]
- J. W. Goodman, Statistical Optics (Wiley&Sons, 2000).
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- A. C. Schell, “The multiple plate antenna,” Ph.D. thesis, Massachusetts Institute of Technology (1961).
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- J. A. Rodrigo, T. Alieva, “Recovery of Schell-model partially coherent beams,” Opt. Lett. 39, 1030–1033 (2014). [CrossRef] [PubMed]
- F. O. Fahrbach, F. F. Voigt, B. Schmid, F. Helmchen, J. Huisken, “Rapid 3D light-sheet microscopy with a tunable lens,” Opt. Express 21, 21010–21026 (2013). [CrossRef] [PubMed]
- J. Clark, X. Huang, R. Harder, I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat Commun 3, 993 (2012). [CrossRef] [PubMed]
- D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Optic 44, 407–414 (1997). [CrossRef]
- M. A. Herráez, D. R. Burton, M. J. Lalor, M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002). [CrossRef] [PubMed]
- A. V. Martin, F. R. Che, W. K. Hsieh, J. J. Kai, S. D. Findlay, L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, (Roberts&Company, 2005).

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