Quantitative SLM-based differential
interference contrast imaging

doi:10.1364/OE.18.014063

Quantitative SLM-based differential interference contrast imaging

Timothy J. McIntyre, Christian Maurer, Stephanie Fassl, Saranjam Khan, Stefan Bernet, and Monika Ritsch-Marte »View Author Affiliations

Optics Express, Vol. 18, Issue 13, pp. 14063-14078 (2010)
http://dx.doi.org/10.1364/OE.18.014063

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▸ Article Outline
– Abstract
1. Introduction
2. Experimental Approach
3. Image Analysis
4. Results and Discussion
5. Conclusion
– References and links

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Abstract

We describe the implementation of quantitative Differential Interference Contrast (DIC) Microscopy using a spatial light modulator (SLM) as a flexible Fourier filter in the optical path. The experimental arrangement allows for the all-electronic acquisition of multiple phase shifted DIC-images at video rates which are analyzed to yield the optical path length variation of the sample. The resolution of the technique is analyzed by retrieving the phase profiles of polystyrene spheres in immersion oil, and the method is then applied for quantitative imaging of biological samples. By reprogramming the diffractive structure displayed at the SLM it is possible to record the whole set of phase shifted DIC images simultaneously in different areas of the same camera chip. This allows for quantitative snap-shot imaging of a sample, which has applications for the investigation of dynamic processes.

1. Introduction

Differential Interference Contrast (DIC) imaging is traditionally performed using Wollaston prisms to create two slightly offset coherent images of a sample which are overlapped in the image plane [1

G. Nomarski, “Microinterferometrie differentiel a ondes polarisees,” J. Phys. Radium 16, 9S–11S (1955).

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004). [CrossRef] [PubMed]

]. The resultant interference pattern is related to the gradient of the optical path length of the sample which can, in principle, be extracted from a single image [3

C. Preza, D. L. Snyder, and J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999). [CrossRef]

]. However, variations in the intensity of the light source and in the transmissivity of the sample introduce large uncertainties in the final result. An improved approach is to record multiple DIC images while varying the phase difference between the two coherent beams or recording of images with orthogonal shearing directions [4–6

M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. 45, 460–469 (2006). [CrossRef] [PubMed]

]. Three or more such images typically provide sufficient information to extract the linear phase gradient imparted by the sample, from which the quantitative phase distribution of the sample can be calculated.

Quantitative phase information can also be recorded with holographic methods. In the so called “Digital Holographic Microscope” (DHM) the bright field image is interfered with an external reference wave. If the reference wave has a sufficiently large inclination with respect to the image wave, a single image is sufficient for the calculation of the phase topography [7–9

B. Kemper and G. von Bally, “Digital holographic microscopy for life cell applications and technical inspection,” Appl. Opt. 47, A52–A61 (2008). [CrossRef] [PubMed]

]. The digital holographic microscope has been improved by different methods. Multiple wavelength holography [10

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009). [CrossRef] [PubMed]

], dual interference channel microscopy [11

N. T. Shaked, M. T. Rinehart, and A. Wax, “Dual-interference-channel quantitative phase microscopy of live cell dynamics” Opt. Lett. 34, 767–769 (2009). [CrossRef] [PubMed]

], Hilbert phase microscopy [12

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]

] and instantaneous spatial light interference microscopy [13

H. Ding and G. Popescu, “Instantaneous spatial light interference microscopy,” Opt. Express 18, 1569–1575 (2010). [CrossRef] [PubMed]

] are some of the enhanced techniques for quantitative phase measurements. In all these approaches at least partial coherence of the illumination light is necessary. High spatial coherence may lead to speckle, which reduces the signal to noise ratio of the image.

Alternatively, Fourier microscopy may be used. One can, for example, shift the sample slightly out of focus and calculate the phase distribution from three diffraction patterns [14

A. Grjasnow, A. Wuttig, and R. Riesenberg, “Phase resolving microscopy by multi-plane diffraction detection,” J. Microsc. 231(1), 115–123 (2008). [CrossRef] [PubMed]

]. If a spatially coherent light source is used, the zero order Fourier component can be shifted with respect to higher Fourier components. As one example, spiral phase contrast can be used to calculate the phase [15

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006). [CrossRef] [PubMed]

, 16

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase fourier filter,” J. Opt. A, Pure Appl. Opt. 11(8), 094023 (2009). [CrossRef]

The advantage of Fourier microscopy is that spatially incoherent illumination can also be used, not giving rise to disturbing speckle. DIC microscopy has proved to be applicable for thick samples, because it enhances the contrast only in a small axial slice, where the thickness of the sharp region is determined by the condenser and imaging objective. A recent demonstration of DIC microscopy using a liquid crystal spatial light modulator provides one avenue for recording multiple sequentially recorded images to obtain quantitative optical path length measurements [17

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

]. In place of the Wollaston prisms, an SLM is used to generate and overlap two displaced images of the sample which yields the differential interference pattern. A similar principle has been also used in x-ray microscopy using specially fabricated diffractive optical elements [18

E. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein, “Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003). [CrossRef] [PubMed]

]. The use of an SLM to generate these images offers a powerful but simple approach to achieve this goal not offered by previous techniques. The shear, orientation and phase difference of the images are all computer controlled and can be varied at video rates. This allows for the fast recording of the necessary images followed by computer processing to obtain the final result. In this paper we demonstrate the technique by measuring the thickness and refractive index of polystyrene spheres placed in immersion oil. We also show applications to measurement of the optical path length of red blood cells and chromosomes. Finally we demonstrate a modification of the method, where the SLM is reprogrammed to generate six phase shifted DIC images with different orientations of the shear angle simultaneously. The images are diffracted by the SLM into six different directions, which correspond to six adjacent positions in the image plane. Thus a single camera exposure provides a set of images which is sufficient to quantitatively evaluate the phase distribution of the sample.

2. Experimental Approach

The optical arrangement is similar to previous qualitative imaging experiments described in [17

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

]. Figure 1 shows the experimental setup: The frequency doubled output from a continuous-wave Nd:YAG at a wavelength of 532 nm is expanded by a telescope and passes through a rotating diffuser plate and a subsequent condenser lens, and then homogeneously illuminates the sample. The rotating diffuser removes the effective coherence of the laser beam by time averaging [19

A. Y. M. Ng, C. W. See, and M. G. Somekh, “Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator,” J. Microsc. 214(3), 334–340 (2004). [CrossRef] [PubMed]

, 20

C. Maurer, S. Khan, S. Fassl, S. Bernet, and M. Ritsch-Marte, “Depth of field multiplexing in microscopy,” Opt. Express 3, 3023–3034 (2010). [CrossRef]

] - it was found that an averaging time of 1ms is sufficient to reduce the contrast of the disturbing speckle pattern to an acceptable level. The spot diameter determines the effective numerical aperture of the illumination. The spot diameter can be changed in our setup, where a diameter of 6mm uses the whole numerical aperture of the subsequent condenser. If the condenser and the imaging objective have the same numerical aperture, the spatial coherence of the illumination is minimal. For smaller spot diameters the coherence increases until the maximum coherence for this system is reached, for a diffraction limited spot on the diffuser. The light transmitted through the sample passes through a microscope objective and telescope relay optics (consisting of a 160mm tube lens suitable for Zeiss objectives and an achromatic lens with a focal length of 150 mm) to an SLM placed in a Fourier plane of the imaging setup. At an intermediate image plane between the two telescope lenses a slit aperture is placed. This aperture restricts the field of view to about 50µm, which is necessary to allow a separation of the diffraction orders of the SLM.

Fig. 1. Experimental setup. The beam path and all used lenses are shown in the image. For a detailed explanation see the text.

The high resolution reflective SLM is programmed to act as a diffractive optical element (DOE). For the sequentially recorded images, the mask consists of two phase gratings with slightly different line spacings combined randomly on a pixel-by-pixel basis. The first order diffraction from the mask displayed on the SLM is imaged onto a video camera (DVC-1412, DVC Co) with a 100mm lens. The pixel pitch of the camera is sufficient to record diffraction limited images with objective magnifications higher than 20×. The relative line spacings of the phase gratings controls the beam shear in the image plane while the grating orientation controls the shearing direction. As well, the relative phase of the gratings controls the phase difference between the two overlapped images. It is worth noting that control of these three parameters is fully electronic (set by the displayed mask on the SLM) and there are no mechanical changes required (such as piezo-electric driven stages) during acquisition. This means that the parameters can be accurately controlled, quickly changed and are highly repeatable. Details of the camera, SLM and laser are given in Table 1.

The masks were computer generated by a dedicated software program. The algorithm calculates the patterns for the two blazed diffraction gratings to generate the two slightly displaced beams and then creates a mask on a pixel-by-pixel basis. At each pixel, the pattern at the corresponding position of one of the two gratings is randomly chosen [21

J. A. Davis and D. M. Cottrell, “Random mask encoding of multiplexed phase-only and binary phase-only filters,” Opt. Lett. 19, 496–498 (1994). [CrossRef] [PubMed]

]. The mask, displayed as a grayscale image on the computer monitor, is converted to a phase image on the SLM appropriate for the wavelength used in the measurements. The software allows interactive selection of the mask settings or can be preprogrammed to display a selection of masks for timed intervals. A measurement consisted of selecting a given shear and then displaying masks for three phase differences (0, 2π/3, 4π/3) for horizontal and then vertical displacements. Each mask was displayed for approximately one half of a second while recording the image on a CCD video camera.

For single shot technique, a more complex mask was generated prior to the measurements using a separate algorithm. The mask was displayed on the SLM for the duration of the experiment.

3. Image Analysis

The minimum requirement when using our DIC technique for the reconstruction of the optical path length of a sample is the recording of three independent images with known phase differences between the two recorded beams in each image. Such images can be analyzed to extract the differential phase shift imparted by the sample independent of the spatial variation of the laser intensity and the transmissivity of the sample. A more accurate measurement can be achieved by recording two sets of images recorded with orthogonal shears which then defines the complete gradient field. A description of the steps involved to obtain the optical path length is given below.

3.1. Phase analysis

The equation for the electric field of a beam of light which has been passed through a sample can be written as

E (x, y) = A (x, y) \cos [k z - ω t + ϕ (x, y)],

(1)

where (x,y) define coordinates in the plane of the sample seen as horizontal and vertical displacement in the images, respectively, z is the direction normal to the sample, t is the time, A is the amplitude of the field, k its wave vector, ω its angular frequency, and ϕ (x,y) is the phase shift imparted on the beam by the sample. When two such coherent beams are overlapped and allowed to interfere, the resulting time averaged irradiance seen by the camera can be written as

I = {∣ E_{1} + E_{2} ∣}^{2} = \frac{A_{1}^{2} + A_{2}^{2}}{2} + A_{1} A_{2} \cos [ϕ_{1} - ϕ_{2}] + I_{INC},

(2)

where subscripts 1 and 2 refer to parameters for each of the beams and the dependence of the (x,y)-coordinates is inferred. The term I_INC = I_INC(x,y) refers to an incoherent background which may be present due to scattered laser light or from other external sources.

The goal of the measurements is to extract Δϕ = ϕ₁ − ϕ₂ and this can be achieved by recording three images labeled A, B and C with extra imposed phase shifts of 0, 2π/3, and 4π/3, respectively between the beams, specifically

I_{A} = \frac{A_{1}^{2} + A_{2}^{2}}{2} + A_{1} A_{2} \cos [Δ ϕ] + I_{INC}

(3)

I_{B} = \frac{A_{1}^{2} A_{2}^{2}}{2} + A_{1} A_{2} \cos [Δ ϕ - \frac{2 π}{3}] + I_{INC}

(4)

I_{C} = \frac{A_{1}^{2} A_{2}^{2}}{2} + A_{1} A_{2} \cos [Δ ϕ - \frac{4 π}{3}] + I_{INC} .

(5)

The phase difference between the two beams can then be extracted as

Δ ϕ (x, y) = \tan^{- 1} {\frac{I_{A} \sin (0) + I_{B} \sin (\frac{2 π}{3}) + I_{C} \sin (\frac{4 π}{3})}{I_{A} \cos (0) + I_{B} \cos (\frac{2 π}{3}) + I_{C} \cos (\frac{4 π}{3})}} .

(6)

Equation (6) represents the wrapped (modulo 2π) phase difference that the two beams experience as they pass through the sample. For samples where this difference exceeds 2π, it is necessary to unwrap the phase [22

D.C. Ghiglia and M.D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software, (New York: Wiley-Interscience, 1998).

Table 1. Experimental Parameters

Camera Scale (for 40× oil objective)	1 pixel = (0.122 ± 0.002) µm
SLM Size	1920 × 1080 pixels
SLM Resolution	8.0 µm / pixel
Laser Wavelength	532 nm

3.2. Optical path length determination

The phase shift imparted on a beam of light of wavelength λ by a sample of thickness h(x,y), and uniform refractive index n, placed in a surrounding medium of uniform refractive index n_s, is given by

ϕ (x, y) = 2 π (n - n_{s}) \frac{h (x, y)}{λ} = 2 π \frac{Λ (x, y)}{λ},

(7)

where Λ is the optical path length (OPL) of the sample relative to the surroundings.

The phase difference between two overlapped beams, where one is displaced from the other by a distance of Δx_r in the x-direction is

Δ ϕ_{x} (x, y) = \frac{2 π}{λ} [Λ (x, y) - Λ (x + Δ x_{r}, y)] .

(8)

Rearranging this equation, one can estimate the variation of the OPL with position as

\frac{\partial Λ (x, y)}{\partial x} \approx \frac{λ}{2 π} \frac{Δ ϕ_{x} (x, y)}{Δ x_{r}},

(9)

while for a shear in the y direction of magnitude Δy_r we have

\frac{\partial Λ (x, y)}{\partial y} \approx \frac{λ}{2 π} \frac{Δ ϕ_{y} (x, y)}{Δ y_{r}} .

(10)

Our technique allows the experimental determination of Δϕ_x, Δϕ_y, Δx_r, and Δy_r and hence, if we can reconstruct the optical path length difference from these gradient fields, then we have our final quantitative result. The reconstruction of a parameter from its gradient fields is a classical problem common in many areas of research including computer vision. Here we make use of the method of Agrawal et al. [23

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” International conference on computer vision (ICCV) (2005).

, 24

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” European conference on Computer Vision (ECCV) (2006).

] which is an algebraic approach that enforces integrability of the equations in the presence of noise.

3.3. Shear

Application of Eqs. (9) and (10) requires a knowledge of the shears (Δx_r and Δy_r) introduced to record the images [6

B. Heise and D. Stifter, “Quantitative phase reconstruction for orthogonal-scanning differential phase-contrast optical coherence tomography,” Opt. Lett. 34, 1306–1308 (2009). [CrossRef] [PubMed]

]. While it is possible to measure the displacement by close examination of characteristic features in the field of view, this gives limited accuracy. An alternative approach is to examine the displacement observed between zeroth and first order from the main grating frequency and use this, together with the known grating frequencies, to determine the displacement. This proceeds as follows.

Table 2. Measured and calculated values for the bead DIC measurements with a shear of 0.126 µm

Value	Symbol	Pixels	µm
Grating 1 spacing	a₁	5.000	40.00
Grating 2 spacing	a₂	4.988	39.90
First order displacement	X_0–1	412±2	50.3±0.3
Calculated shears	Δx_r,Δy_r	1.04	0.126±0.002

For a grating with a line spacing of a, illuminated with light at a wavelength λ, the angle of the first order diffraction is

θ = \sin^{- 1} {\frac{λ}{a}} .

(11)

The displacement in the back focal plane of the camera lens will be

x = L \tan (θ) = L \tan (\sin^{- 1} {\frac{λ}{a}}) \approx L \frac{λ}{a}

(12)

under the assumption of small diffraction angles (which is valid in our setup). Here L is the distance between the lens and the image plane. For two diffraction gratings of spacings a₁ and a₂, the difference in the displacements of the respective images is

Δ x_{r} = x_{2} - x_{1} = L (\frac{λ}{a_{2}} - \frac{λ}{a_{1}}) .

(13)

L can be determined from the experimental arrangement but is here calculated from the measured displacement X_0–1 between first order and zeroth order when displaying a grating with line spacing a₁ at the SLM by

L = \frac{a_{1} X_{0 - 1}}{λ} .

(14)

Combining these equations gives a final result of

Δ x_{r} = X_{0 - 1} (\frac{a_{1}}{a_{2}} - 1) .

(15)

A similar approach can be used to determine the shear in y-direction.

4. Results and Discussion

4.1. Effect of illumination coherence

DIC imaging requires good fringe visibility to provide clear images. For a technique that produces two displaced images that interfere, this is best achieved by using illumination with a high level of coherence. A theoretical investigation of the coherence requirements has been performed for the case of x-ray DIC imaging with diffractive optical elements in [25

O. von Hofsten, M. Bertilson, and U. Vogt, “Theoretical development of a high-resolution differential-interference-contrast optic for x-ray microscopy,” Opt. Express 16, 1132–1141 (2008). [CrossRef] [PubMed]

]. The technique used here generates the images using a phase-only mask which not only generates the two images but also produces other artefacts due to the random structure present. To explore the importance of the level of spatial coherence in the illuminating light, a series of images was recorded in which the illumination coherence was varied by changing the spot diameter in the diffuser plane. The mask on the SLM was created consisting of random samplings of two blazed gratings with frequencies of 20.00 cycles per 100 pixels and 20.20 cycles per 100 pixels respectively which generated two images of the sample spaced approximately 0.5 µm in the object plane. A phase difference of π/2 was set between the gratings which allows the discrimination between increases and decreases in the optical path length of the sample.

Fig. 2. DIC images and intensity plots of a polystyrene bead for a π/2 phase difference illustrating the effect of illumination spatial coherence. (a) High level of coherence (NA 0.01); (b) Medium coherence (NA 0.38); (c) Low coherence (NA 0.75).

Figure 2 shows a selection of images of a polystyrene bead placed in immersion oil. The bead is imaged through a 40× oil immersion objective with a numerical aperture of 1.25. As a condenser, a 60× oil immersion objective (NA = 1.3) was used. The width of the illumination beam in the front focal plane of the condenser determines the effective numerical aperture of the illumination. As discussed earlier, the diameter of the illumination beam can be adjusted, such that the NA of the condenser is (0.01, 0.38, 0.75) for Figs. 2(a), 2(b), 2(c), respectively. An intensity cross-section, taken from top to bottom through the center of the bead is also shown in each case. For a high level of coherence, the expected increase and decrease of intensity is observed near the edges of the bead where the gradients are largest. However, the signal to noise ratio is poor and there are extra interference effects that result from the phase only nature of the mask. Figure 2(b) shows an image in which the spatial coherence of the light has been lowered. Under these conditions, time independent interference is now only observed between regions separated by roughly the shear imposed between the two beams. The signal is far less noisy as the artefacts present in the first image now tend to be smoothed by time averaging. The intensity pattern shows a relatively noise free classical DIC profile. For low coherence, below the shear between the two beams, the image no longer displays interference effects, reverting to simply the sum of the intensities in each beam. This is shown in Fig. 2(c). All further recordings were taken with an illumination coherence equivalent to that shown in Fig. 2(b).

4.2. Sequential image acquisition

To illustrate the quantitative capabilities of the technique, the optical path length of a polystyrene bead (6µm Polybead®Microspheres, Polysciences) was measured utilizing the capability of the SLM to quickly and easily produce images with displacements in two orthogonal directions in the object plane. A set of six images of the bead was recorded - three different phase offsets for each shear direction - and an analysis performed to extract the differential phase for each direction and then the optical path length. The shears for each direction were here both set by choosing a difference in the grating frequencies of 0.05 cycles per 100 pixels yielding a shear in the image plane of 0.126 µm. Measured and calculated parameters for the experiments are shown in Table 2.

Fig. 3. Raw DIC images at a shear of 0.126 µm. (a)–(c) Horizontal (x) shear with phase differences of 0, 2π/3 and 4π/3 respectively. (d)–(f) Vertical (y) shear with phase differences of 0, 2π/3 and 4π/3 respectively.

Figure 3 shows the raw DIC images for two orthogonal shearing directions and varying phase differences between the recording beams. The overall intensity of the image depends on the phase difference between the two beams - highest intensities are recorded for zero phase differences while lower intensities are seen when the beams are partly out of phase. The intensity variations seen within the bead are consistent with the gradients in the thickness of the bead.

The three phase images for each displacement were used to obtain the phase gradients due to the sample, Δϕ_x and Δϕ_y, by the application of Eq. (6). These are shown in Fig. 4. Both images have low noise levels, and appear as expected for a spherical bead. The surface reconstruction process of Agrawal et al. [23

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” International conference on computer vision (ICCV) (2005).

, 24

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” European conference on Computer Vision (ECCV) (2006).

] was then applied to the results of the measurements to obtain the optical path length for the bead. Shears in both directions are in principle not necessary, but they improve the quality of the recorded images, allowing to cross-check two data sets, and to average over the results. The two-dimensional distribution is shown in Fig. 4(c) while a cross-section through the center of the bead is shown in Fig. 4(d). The image shows that the analysis has successfully retrieved the expected distribution of the bead with a very low level of noise. The major sources of possible systematic errors in these measurements are the determination of the shear and the determination of the scale of the image. The combined contribution of these factors is estimated to be about 3%.

It is worth noting that the results presented in Fig. 4 represent measurements using the minimum number of images necessary to calculate the optical path length difference using the current technique. Several different approaches are possible for improving the signal to noise [26

C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990). [CrossRef]

]. The phase extraction can, for instance, be improved by recording more than three images with equally distributed phase differences. Alternatively, the optical path length calculation can be improved by recording images with various displacements differing in shear magnitude and direction.

Fig. 4. Measured phase gradients and optical path lengths for a polystyrene bead in immersion oil. (a) Phase gradient for a shear in the horizontal (x) direction; (b) Phase gradient for a shear in the vertical (y) direction; (c) Recovered two-dimensional optical path length difference; (d) Cross-sectional optical path length difference from left to right through the center of the bead shown in (c).

4.3. Recovery of thickness and refractive index

Without accurate knowledge of the refractive indices of the bead and the surrounding fluid, it is not possible to directly determine the axial thickness of the measured object. However, for the presented measurements we can make use of the fact that the bead is spherical in shape to determine both the radius of the bead and the refractive index difference between the bead and the fluid. The analysis proceeds as follows.

The height (in the z-direction) of a perfectly spherical bead of radius r, centred at (x₀,y₀) in the (x,y)-plane is

h (x, y) = 2 \sqrt{r^{2} - {(x - x_{0})}^{2} - {(y - y_{0})}^{2}},

(16)

where (x – x₀)² + (y – y₀)² < r². The phase shift imparted on light of wavelength λ, passing

Fig. 5. A plot of measured values and the fit corresponding to Eq. (18).

through the bead in a direction perpendicular to the (x,y)-plane is thus

ϕ (x, y) = \frac{4 π (n - n_{s}) \sqrt{r^{2} - {(x - x_{0})}^{2} - {(y - y_{0})}^{2}}}{λ} .

(17)

We rewrite this as

ϕ^{2} (x, y) = \frac{16 π^{2} {(n - n_{s})}^{2} r^{2}}{λ^{2}} - \frac{16 π^{2} {(n - n_{s})}^{2} [{(x - x_{0})}^{2} + {(y - y_{0})}^{2}]}{λ^{2}}

(18)

A plot of ϕ² against (x – x₀)² + (y – y₀)² yields a straight line from which the refractive index difference between the object and the surrounds can be determined from the slope of a best fit, and then the radius of the bead can be extracted from the y-intercept.

Before implementation, an exact determination of the position of the center of the sphere (x₀,y₀) is required. This can be obtained by noting that the phase gradient recorded with a shear in the x-direction must be symmetrical about the geometric center of the sphere in this direction, x₀. An automated routine was written which found this position using all the measured phase gradients in the image of the sphere. Similarly the image recorded with a shear in the y-direction can be used to determine y₀.

Figure 5 shows a plot of the measured phase and positions corresponding to Eq. (18). A point is plotted in the figure for each pixel in the image where the phase is greater than half the maximum phase. Phase values below this cut-off are ignored as the noise becomes larger for smaller phase shifts. The plot shows a clear linear distribution from which regression is used to determine the slope and intercept of the fitted line. The recovered refractive index difference between the sphere and the surrounding immersion oil equates to a value of (0.084 ± 0.003). Uncertainties are calculated from the straight line fit, from the uncertainty in the magnification of the optics and from the uncertainty of the shear. The quoted refractive index at 532 nm for the immersion oil used as the surrounding medium in the experiment is 1.518 and hence the determined refractive index of the polystyrene bead is (1.602 ± 0.003). This is consistent with the value quoted by the manufacturer (1.59–1.60 at 589 nm) and in good agreement with the literature value of Nikolov and Ivanov [27

I. D. Nikolov and C. D. Ivanov “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. 39, 2067–2070 (2000). [CrossRef]

] of 1.5986 at 532 nm although slightly higher than the value quoted by Ma et al. [28

X. Ma, J. Q. Lu, R. S. Scott, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003). [CrossRef]

] of 1.588.

We can now use the measured refractive index difference between the bead and the surrounding immersion oil to convert the measured optical path length shown in Fig. 4(c) into a measured thickness as shown in Fig. 6. The thickness has been used to generate a three-dimensional representation of the shape of the bead. All points above a horizontal plane through the center of the sphere represent a measured position of the bead’s surface based on the thickness. All points below this surface are mirrored from above. Also shown is a cross-section of the thickness which shows a close agreement between the thickness of the bead and a perfectly spherical shape. In general, the agreement is very good with the main departure being near the edges of the bead where the optical path length is shortest, the variation in the optical path length is the largest and noise becomes more significant.

Fig. 6. Recovered thickness for a polystyrene bead in immersion oil. (a) Three-dimensional representation. (b) A cross-section compared to the shape for a perfect sphere with the measured radius.

Table 3. Measured refractive index difference between the polystyrene bead and the immersion oil and bead radius obtained from the measurements with various shears

Shear (µm)	Refractive Index	Radius (µm)
0.13	0.084 ± 0.003	3.13 ± 0.09
0.25	0.090 ± 0.003	3.13 ± 0.09
0.51	0.080 ± 0.003	3.20 ± 0.09
0.76	0.076 ± 0.003	3.20 ± 0.09

The measurements presented above were repeated on the same bead for a further three different shears to investigate the effect of shear on the results. The outcome of this analysis is given in Table 3. The associated uncertainties quoted in the table result primarily from the uncertainties in the shear and the scale - the uncertainties in the linear fits of the type performed on the data shown in Fig. 5 are generally smaller and can be ignored. These results would seem to indicate that a larger shear leads to a larger determined radius and a smaller determined refractive index difference (although the second measured value in Table 3 is an exception in this). This was confirmed by a simulation of the process. It occurs because the images of the bead become broader in the direction of the shear resulting in a larger perceived radius and smaller refractive index difference. Hence as small a shear as possible is desired to achieve the best results. However, it is not possible to make the shear arbitrarily small. The finite size of the pixels on the SLM provides one limitation on the minimum shear - and this was found to be about equivalent to the smallest value here. Furthermore, a smaller shear leads to a smaller differential phase which is more sensitive to noise.

4.4. Biological samples - sequential imaging

The bead measurements showed that our SLM-DIC technique can successfully recover the thickness and shape of a spherical object. However, it would be of more interest to apply the technique to biological samples which provide more complex phase objects to investigate. Here we have attempted to image two types of samples - red blood cells and chromosomes.

Figure 7 shows a summary of measurements of a sample of red blood cells in a buffer solution. The imaging objective is a 40× water immersion objective with NA = 0.75. As before, a set of six sequential DIC images was recorded with three different phase differences at each of two different shear directions. The grating frequencies were set to give a shear of 0.5µm for each direction. Sample images for the horizontal shear and the corresponding recovered phase gradient are shown in the figure. One obvious problem observed in these images is that there is some motion of the blood cells as changes in shape are apparent in a number of the cells despite the fact that the images were recorded only over a time span of several seconds. Fortunately, for most cells this is not a problem and the technique can be used successfully. After analysis of the two phase gradient images (one for each shear direction), the optical path length difference between the cells and the surrounding solution can be determined. This is shown at the top right of the figure in a two-dimensional colour map representation, while a cross-section through one of the cells (as shown) is also given. The recovery of the quantitative phase information appears to be similar in quality to the measurements of the bead with only some loss of clarity in regions where cells have moved or changed shape between images.

A determination of the cell thickness (and hence volume) is not possible without a knowledge of the cell refractive index. Lacking measurements for our current experiments, we make use of the refractive index mentioned in the literature. In different experiments, all under physiological conditions, the refractive index was estimated to be close to 1.4 ± 0.1 depending on the haemoglobin concentration and wavelengths [29

N. Ghosh, P. Buddhiwant, A. Uppal, S. K. Majumder, H. S. Patel, and P. K. Gupta, “Simultaneous determination of size and refractive index of red blood cells by light scattering measurements,” Appl. Phys. Lett. 88, 084101 (2006). [CrossRef]

,30

Y. Park, M Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, and M. S. Feld, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105, 13730–13735 (2008). [CrossRef] [PubMed]

]. As the exact value for the refractive index for a wavelength of 532nm was not given, we used a refractive index of 1.40 for the red blood cells and 1.34 for the buffer solution (Δn = 0.06) to estimate cell thickness. Figure 8 shows a three-dimensional representation of the thickness of the cell and a cross-section through the cell center using this value. The cell is found to have a diameter of about 7.5µm and a peak thickness of around 2.1µm dropping to 0.8µm at the cell center. A determination of the cell volume is a simple exercise given these measurements.

The last application of the technique was the imaging of human chromosomes. The imaging objective was a 40× oil immersion objective with NA = 1.3. Six SLM-DIC images were recorded using the same SLM settings as for the red blood cells. A sample DIC image, the horizontal phase gradient and the final optical path length are shown in Fig. 9. The biggest challenge in these measurements is the small size of the chromosomes. For the shears possible here, these structures are close to the smallest size that can be comfortably detected. Nevertheless, the measurements allowed for the successful determination of the optical path length of the chromosomes as shown at the right of the figure. Improved imaging may be possible with a higher magnification and/or a finer SLM.

4.5. Single exposure measurement

One drawback of the above presented method is that multiple images must be recorded over a finite time interval - typically several seconds. For dynamic objects, any small amount of motion during the exposure will introduce uncertainties in the final result. Hence it is desirable to have a single shot image when motion is an issue. To demonstrate that single shot images can be obtained using this approach, we modified the experimental arrangement and generated a more complex mask for the recording.

Fig. 7. SLM-DIC imaging of blood cells in a buffer solution. (a)–(c) Sample raw images with a horizontal shear at a relative phase of 0, 2π/3 and 4π/3 respectively. (d) The recovered phase gradient from these images, Δϕ_x. (e) A colour map of the optical path length obtained from the x and y gradients. (f) A cross-section of the optical path length through one of the blood cells.

Fig. 8. Blood cell thickness determined using the current SLM-DIC measurements and a literature value for the refractive index difference between the cell and the buffer solution of 0.06. (a) Three-dimensional representation; (b) Cross-section through the center of the blood cell.

Fig. 9. SLM-DIC measurements of chromosomes. (a) A sample DIC image with a phase difference of 4π/3. (b) The recovered horizontal phase gradient Δϕ_x. (c) The optical path length difference obtained from the gradients.

Figure 10 shows a single shot image and the associated analysis applied to nominal 4.5µm diameter polystyrene beads in immersion oil. The beads were imaged through a 60× oil objective (NA=1.25) using an SLM to form the multiple images. The left panel in Fig. 10 shows six DIC images with different phase shifts and shearing directions simultaneously recorded in a single camera exposure. These images are formed from 12 individual beams (two beams per image) created by a mask generated using a Gerchberg-Saxton algorithm [31

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures”, Optik (Stuttg.) 35, 237–246 (1972).

]. In the left column of the figure, the shear direction is in the horizontal direction, with a displacement of the interfering images of 0.36µm, and phase differences of −2π/3,0,2π/3 from top to bottom. The shear direction in the right column is in the vertical direction and the phases are set to π/3, π, 5π/3 from top to bottom. Note that different phase values for the x- and y- shear were chosen in order to minimize the crosstalk between the -1 and 1 diffraction order of the gratings. The zero diffraction order of the SLM in the center of the image shows the bright field image. The different brightnesses of the images are, as before, due to the interference of the two superimposed images.

The analysis of the images proceeds as in the earlier measurements. The phase gradients are determined from the raw images and these are plotted in the upper row of the right side of Fig. 10. From the gradient field the final phase is calculated and converted to a thickness presuming a refractive index difference of 0.0800. As before, the approach gives a good bead profile although the result has more noise than the multiple shot images. The axial thickness of the beads is close to the nominal value of 4.5µm.

The method allows for quantitative imaging with a higher speed because only one frame is necessary for the imaging process. However, the exposure time must be increased for the same power laser to provide sufficient intensity in the image, and a lower field of view is obtained as all images are recorded simultaneously on the camera. The field of view for the used objective (60× oil immersion) in the object plane is 50µm × 67µm in the vertical and horizontal directions respectively. Note that the resolution of the images is not reduced because the mask displayed at the SLM uses the entire aperture. With the oil objective (NA = 1.3) the theoretical resolution limit is 320nm. The method could be further improved by the use of a lithographically produced DOE in place of the SLM which could improve both the efficiency of the diffraction and the quality of the images allowing measurements that are only restricted by the refresh time of the camera.

Fig. 10. Quantitative calculation of the phase with a single snapshot. (a) A single camera image showing the raw images with annotations giving the varying shear directions and relative phases. (b) Gradient field in the horizontal (x) direction. (c) Gradient field in the the vertical (y) direction. (d) Calculated phase distribution. (e) The profile along the line shown in (d).

5. Conclusion

Quantitative Differential Interference Contrast imaging has been demonstrated using a spatial light modulator to generate and overlap two slightly displaced images of a thin transparent sample. The flexibility offered by the SLM allows for the rapid recording of images with different displacements, orientations and phase offsets between the two interfering beams. The technique was successfully implemented to measure the thickness of polystyrene beads in immersion oil. The profiles extracted showed that the method accurately reproduced the expected thickness distribution of the bead. The method was also used to obtain quantitative images of blood cells in a buffer solution and of human chromosomes. A single shot imaging variation was also demonstrated that could be used to study objects for which motion presents difficulties during the normal recording time for images of several seconds.

The implementation presented here uses the minimum number of images required to calculate the sample thickness using the current approach. The quality of the final result could be improved by recording multiple images at the same phase difference and shear, or by recording sets of images with a range of values for the phase difference and the shear.

The technique opens up the possibility of performing quantitative DIC imaging using a simple laser illuminated microscope modified only to include an SLM (or DOE) in the back focal plane of the objective. Not only can such a set-up be used to perform quantitative DIC but other techniques such as dark field microscopy and phase contrast microscopy can be implemented with no change in geometry.

Acknowledgements

This experimental work was performed while TJM was on study leave at Innsbruck Medical University. TJM would like to pass on thanks to his co-authors for their generous support in hosting him during this time. The financial assistance of the University of Queensland is also acknowledged. This work was supported by the Austrian Science Foundation (FWF) Project No. P19582-N20. Christian Maurer was funded by the Christian Doppler Laboratory (CDL) CDL MS-MACH.

References and links

1.	G. Nomarski, “Microinterferometrie differentiel a ondes polarisees,” J. Phys. Radium 16, 9S–11S (1955).
2.	M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004). [CrossRef] [PubMed]
3.	C. Preza, D. L. Snyder, and J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999). [CrossRef]
4.	M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. 45, 460–469 (2006). [CrossRef] [PubMed]
5.	M. Shribak, J. LaFountain, D. Biggs, and S. Inoué, “Orientation-independent differential interference contrast microscopy and its combination with an orientation independent polarization system,” J. Biomed. Opt. 13, 014011 (2008). [CrossRef] [PubMed]
6.	B. Heise and D. Stifter, “Quantitative phase reconstruction for orthogonal-scanning differential phase-contrast optical coherence tomography,” Opt. Lett. 34, 1306–1308 (2009). [CrossRef] [PubMed]
7.	B. Kemper and G. von Bally, “Digital holographic microscopy for life cell applications and technical inspection,” Appl. Opt. 47, A52–A61 (2008). [CrossRef] [PubMed]
8.	F. Charriàre, N. Pavillon, T. Colomb, C. Depeursinge, T. J. Heger, E. A. D. Mitchell, P. Marquet, and B. Rappaz, “Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba,” Opt. Express 14(16), 7005–7013 (2006). [CrossRef]
9.	P. Ferraro, D. Alferi, S. D. Nicola, L. D. Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31(10), 1405–1407 (2006). [CrossRef] [PubMed]
10.	J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009). [CrossRef] [PubMed]
11.	N. T. Shaked, M. T. Rinehart, and A. Wax, “Dual-interference-channel quantitative phase microscopy of live cell dynamics” Opt. Lett. 34, 767–769 (2009). [CrossRef] [PubMed]
12.	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]
13.	H. Ding and G. Popescu, “Instantaneous spatial light interference microscopy,” Opt. Express 18, 1569–1575 (2010). [CrossRef] [PubMed]
14.	A. Grjasnow, A. Wuttig, and R. Riesenberg, “Phase resolving microscopy by multi-plane diffraction detection,” J. Microsc. 231(1), 115–123 (2008). [CrossRef] [PubMed]
15.	S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006). [CrossRef] [PubMed]
16.	C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase fourier filter,” J. Opt. A, Pure Appl. Opt. 11(8), 094023 (2009). [CrossRef]
17.	T. J. McIntyre, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. 34, 2988–2990 (2009). [CrossRef] [PubMed]
18.	E. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein, “Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003). [CrossRef] [PubMed]
19.	A. Y. M. Ng, C. W. See, and M. G. Somekh, “Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator,” J. Microsc. 214(3), 334–340 (2004). [CrossRef] [PubMed]
20.	C. Maurer, S. Khan, S. Fassl, S. Bernet, and M. Ritsch-Marte, “Depth of field multiplexing in microscopy,” Opt. Express 3, 3023–3034 (2010). [CrossRef]
21.	J. A. Davis and D. M. Cottrell, “Random mask encoding of multiplexed phase-only and binary phase-only filters,” Opt. Lett. 19, 496–498 (1994). [CrossRef] [PubMed]
22.	D.C. Ghiglia and M.D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software, (New York: Wiley-Interscience, 1998).
23.	A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” International conference on computer vision (ICCV) (2005).
24.	A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” European conference on Computer Vision (ECCV) (2006).
25.	O. von Hofsten, M. Bertilson, and U. Vogt, “Theoretical development of a high-resolution differential-interference-contrast optic for x-ray microscopy,” Opt. Express 16, 1132–1141 (2008). [CrossRef] [PubMed]
26.	C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990). [CrossRef]
27.	I. D. Nikolov and C. D. Ivanov “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. 39, 2067–2070 (2000). [CrossRef]
28.	X. Ma, J. Q. Lu, R. S. Scott, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003). [CrossRef]
29.	N. Ghosh, P. Buddhiwant, A. Uppal, S. K. Majumder, H. S. Patel, and P. K. Gupta, “Simultaneous determination of size and refractive index of red blood cells by light scattering measurements,” Appl. Phys. Lett. 88, 084101 (2006). [CrossRef]
30.	Y. Park, M Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, and M. S. Feld, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105, 13730–13735 (2008). [CrossRef] [PubMed]
31.	R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures”, Optik (Stuttg.) 35, 237–246 (1972).

OCIS Codes
(110.0180) Imaging systems : Microscopy
(180.3170) Microscopy : Interference microscopy

ToC Category:
Microscopy

History
Original Manuscript: April 16, 2010
Revised Manuscript: May 28, 2010
Manuscript Accepted: June 6, 2010
Published: June 15, 2010

Virtual Issues
Vol. 5, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Timothy J. McIntyre, Christian Maurer, Stephanie Fassl, Saranjam Khan, Stefan Bernet, and Monika Ritsch-Marte, "Quantitative SLM-based differential interference contrast imaging," Opt. Express 18, 14063-14078 (2010)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-13-14063

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References

G. Nomarski, “Microinterferometrie differentiel a ondes polarisees,” J. Phys. Radium 16, 9S–11S (1955).
M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004). [CrossRef] [PubMed]
C. Preza, D. L. Snyder, and J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999). [CrossRef]
M. Shribak, and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. 45, 460–469 (2006). [CrossRef] [PubMed]
M. Shribak, J. LaFountain, D. Biggs, and S. Inoué, “Orientation-independent differential interference contrast microscopy and its combination with an orientation independent polarization system,” J. Biomed. Opt. 13, 014011 (2008). [CrossRef] [PubMed]
B. Heise, and D. Stifter, “Quantitative phase reconstruction for orthogonal-scanning differential phase-contrast optical coherence tomography,” Opt. Lett. 34, 1306–1308 (2009). [CrossRef] [PubMed]
B. Kemper, and G. von Bally, “Digital holographic microscopy for life cell applications and technical inspection,” Appl. Opt. 47, A52–A61 (2008). [CrossRef] [PubMed]
F. Charrière, N. Pavillon, T. Colomb, C. Depeursinge, T. J. Heger, E. A. D. Mitchell, P. Marquet, and B. Rappaz, “Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba,” Opt. Express 14(16), 7005–7013 (2006). [CrossRef]
P. Ferraro, D. Alferi, S. D. Nicola, L. D. Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31(10), 1405–1407 (2006). [CrossRef] [PubMed]
J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009). [CrossRef] [PubMed]
N. T. Shaked, M. T. Rinehart, and A. Wax, “Dual-interference-channel quantitative phase microscopy of live cell dynamics,” Opt. Lett. 34, 767–769 (2009). [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]
H. Ding, and G. Popescu, “Instantaneous spatial light interference microscopy,” Opt. Express 18, 1569–1575 (2010). [CrossRef] [PubMed]
A. Grjasnow, A. Wuttig, and R. Riesenberg, “Phase resolving microscopy by multi-plane diffraction detection,” J. Microsc. 231(1), 115–123 (2008). [CrossRef] [PubMed]
S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006). [CrossRef] [PubMed]
C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase Fourier filter,” J. Opt. A, Pure Appl. Opt. 11(8), 094023 (2009). [CrossRef]
T. J. McIntyre, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Differential interference contrast imaging using a spatial light modulator,” Opt. Lett. 34, 2988–2990 (2009). [CrossRef] [PubMed]
E. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein, “Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express 11, 2278–2288 (2003). [CrossRef] [PubMed]
A. Y. M. Ng, C. W. See, and M. G. Somekh, “Quantitative optical microscope with enhanced resolution using a pixilated liquid crystal spatial light modulator,” J. Microsc. 214(3), 334–340 (2004). [CrossRef] [PubMed]
C. Maurer, S. Khan, S. Fassl, S. Bernet, and M. Ritsch-Marte, “Depth of field multiplexing in microscopy,” Opt. Express 3, 3023–3034 (2010). [CrossRef]
J. A. Davis, and D. M. Cottrell, “Random mask encoding of multiplexed phase-only and binary phase-only filters,” Opt. Lett. 19, 496–498 (1994). [CrossRef] [PubMed]
D. C. Ghiglia, and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software, (New York: Wiley-Interscience, 1998).
A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” International conference on computer vision (ICCV) (2005).
A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” European conference on Computer Vision (ECCV) (2006).
O. von Hofsten, M. Bertilson, and U. Vogt, “Theoretical development of a high-resolution differential-interference contrast optic for x-ray microscopy,” Opt. Express 16, 1132–1141 (2008). [CrossRef] [PubMed]
C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990). [CrossRef]
I. D. Nikolov, and C. D. Ivanov, “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. 39, 2067–2070 (2000). [CrossRef]
X. Ma, J. Q. Lu, R. S. Scott, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165–4172 (2003). [CrossRef]
N. Ghosh, P. Buddhiwant, A. Uppal, S. K. Majumder, H. S. Patel, and P. K. Gupta, “Simultaneous determination of size and refractive index of red blood cells by light scattering measurements,” Appl. Phys. Lett. 88, 084101 (2006). [CrossRef]
Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, and M. S. Feld, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105, 13730–13735 (2008). [CrossRef] [PubMed]
R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

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