1. Introduction

The characterization or the detection of transparent living cells is challenging even with advanced optical technologies. To observe cells, people tend to use special optical microscopy [1,2] such as phase contrast (PC), differential interference contrast (DIC), polarization, fluorescence, or conformal microscopy to capture the clear pictures of the cells, or with spectroscopy or scattering [3-6] to extract the size or optical properties of the cells. Spectroscopy and scattering methods give no picture information of the cells, and the spectroscopy method cannot even determine intrinsic optical property data of the cells. What we want to investigate in this paper is whether one can use diffraction to characterize transparent living cells for both their optical property and sizes.

Diffraction is an optical phenomenon that light changes its directions when it encounters an aperture or an obstacle, or such a 1D or 2D periodical array, whose size is close to the optical wavelength. Diffraction limits optical instruments from observing sub-micro details and has rarely been utilized to observe cells. Recently researchers reported detecting the concentration of the cells by the diffraction of a pre-arranged matrix pattern formed by Escherichia coli bacteria when they grew [7-9]. King et al. used diffraction to enlarge images formed by an optical system in front of a photodetector and later corrected the enlarged but deformed image by software [10]. In this paper, we propose a new concept to detect transparent cells by diffraction of an aperture or an aperture array. Different from the reported diffraction applications for cell detections, in our method, the cells are in arbitrary positions, and the aperture forms the image instead of amplifying an existing image formed by a lens. Most of all, it is integrated in a robust and inexpensive MEMS chip.

Our MEMS chip consists of an optical aperture and a microfluidic chamber aligned with it. Cells can be injected and trapped in the fluidic chamber, and a laser beam is shed on the aperture to produce a far-field diffraction pattern of the transparent optical system, which comprises the aperture, the cells, the microfluidic chamber and the buffer. The diffraction pattern is very sensitive to any change in the cells, including the shape, position, optical absorption and refractive index of each cell, and can be used for cell characterizations. But the images of the cells are deformed in the diffraction pattern, in this paper, we develop a complete set of software to extract the optical information of the cells from their diffraction patterns when various aperture and chamber shapes are used, and the correctness of our simulation is verified by experiments.

This MEMS device can be fabricated into a biosensor chip to characterize living cells, micro-sized transparent polymer spheres, or insoluble droplets in an aqueous solution.

2. Device design

An example of our proposed diffraction system is shown in Fig. 1. The detection for this device is achieved by recording the far-field diffraction pattern of the aperture when the device’s backside is illuminated by a laser beam.

 

Fig. 1. Detection of cells by aperture diffraction (two cells with their nuclei are inside a pyramidal chamber). The circular aperture illuminated by laser generates a diffraction pattern on the detection plane which can be recorded by a CCD camera or a photodetector array. The buffer deflects the light beam, causing the light spot on P’ to actually appear on P”.

Download Full Size | PDF

In the MEMS fabrication and also the diffraction simulations, we include several kinds of aperture shapes as drawn in Fig. 2. The aperture can be circular or rectangular ones without or with their centers blocked as shown in Figs. 2(a)-(d), or an array combination of one of these kinds of apertures. We find out that the middle blocked apertures enhance the contrasts of the diffraction pattern, and the aperture array will magnify the optical intensity.

 

Fig. 2. Apertures that can be used for optical diffraction, with the shapes of (a) circular, (b) rectangular, (c) circular with middle blocked, (d) rectangular with middle blocked, (e) an aperture array formed by the ones in (a)-(d), and we suppose each element of (e) is the same.

Download Full Size | PDF

We design the chamber of our MEMS device in cylindrical, cuboidal, pyramidal or conical shapes. The first 3 kinds are planar-shaped that are not able to trap the cells as easy as the conical one, but they can be fabricated in several layers with size steps (analogous to a conical shape) to trap the cells, as depicted in Fig. 3. By slow injection, the cells can move slowly and stop at the area with a trap. The cell is not necessary to be stable, because the picture of the diffraction pattern is taken within a few seconds, and we even can video the diffraction patterns when a cell is moving and later analyze the recorded video frame by frame.

 

Fig. 3. Various chamber shapes can be fabricated in the MEMS device for detecting the cells with an optical aperture. The chambers are: (a) cylindrical, (b) pyramidal, (c) cuboidal, and (d) conical. It is possible to fabricate a planar chamber in several layers in order to trap the cells, so (a)-(c) show stacked chambers of up to 3 layers. In MEMS fabrication, each layer of the chamber might be misaligned from the center of the aperture by fabrication error, so the central line of the aperture is drawn to indicate the possible misalignment. Both side and top views are drawn for each kind of the chambers.

Download Full Size | PDF

Fig. 4. gives a fabrication process for the MEMS device with above aperture and chamber designs by using polydimethylsiloxane (PDMS), a kind of transparent polymer. Soft lithography on PDMS is one of the methods to form the microfluidics we want, and in this way the device is cost-effective and disposable. Prior to PDMS stamping, a protruding mold for microfluidics will be fabricated on silicon with a specific shape of the chamber. The process for fabricating the silicon mold is shown in Figs. 4(a)-(b), which exemplify how a two-layer pyramidal chamber mold is fabricated. For a chamber with several layers to trap the cells, the layers are fabricated one by one with different lithography masks. The cylindrical or cuboidal shapes on the mold can be fabricated by deep reaction ion etching (DRIE) of the silicon, and the pyramidal and the conical shapes on the mold can respectively be achieved by silicon potassium hydroxide (KOH) wet etching and silicon tip fabrication method. The misalignments among the layers (shown in Fig. 3) affect the diffraction patterns and can be corrected in our simulations. In Fig. 4(c), the optical aperture is fabricated at the center of the device by depositing a layer of aluminum or chromium onto a photoresist layer patterned by lithography, followed by the metal lift-off in acetone. PDMS is then spin-coated on the glass substrate with optical apertures fabricated and stamped by the silicon mold to form the microfluidics, as shown in Fig. 4(d). Later in Fig. 4(e), the top PDMS layer is activated by plasma to bond with a top glass cover with microfluidic inlet/outlet pre-drilled.

 

Fig. 4. Fabrication process of the MEMS device with an optical aperture and a two-layer pyramidal chamber. The steps are: (a) etch a small Si tip for the pyramidal chamber; (b) etch a wider pyramidal chamber and a channel on Si mold; (c) fabricate the aperture by a metal layer; (d) spin-coat PDMS on the bottom glass and stamp the PDMS with Si mold to form the microfluidics; (e) bond the PDMS microfluidics with an inlet/outlet pre-drilled glass cover.

Download Full Size | PDF

3. Mathematical derivatives

The algorithms for calculating the diffraction patterns with all kinds of cells, apertures and chambers are presented in this section.

3.1 Diffraction calculation method for various apertures

We derive the diffraction pattern calculations with the Fresnel-Kirchhoff diffraction integral method [11]. In the optical setup shown in Fig. 1, we regard the aperture or aperture array, the chamber, the buffer and the cells as an optical diffraction system. When a ray emits from a point P(x,y) on the aperture plane X-Y and sheds onto a point P’(x’, y’) on the detection plane, the ray PP’ passes through a particular optical path. In our analysis, assuming the positions of these materials are known, the effective optical path for the ray is calculated by:

where n_k is the refractive index of the material that the ray encountered, L_k is the length that the ray travels inside the material, and the material can be the chamber, or the buffer, or the cells and their nuclei. Because some materials have optical absorption, for example, nuclei have higher optical density, Φ is assumed to be a complex, with its real part relates to the phase of the ray and imaginary part relates to the optical loss.

For a single aperture, the normalized amplitude of the light at a point P’(x’,y’) on the detection plane is the integration of e^{ik Φ} in the aperture area S, which can be written as:

where k=2π/λ is the wave number of the optical light, S is the opening area of the aperture. For instance, a circular aperture on the X-Y plane with its center (a, b) in the X-Y-Z coordinate system is expressed as (x - a)² + (y - b)² = r ², where r is the radius of the aperture.

The light intensity at the point P’(x’, y’) on the detection plane is the conjugate product of the amplitude in Eq. (2) and can be calculated by:

For the aperture array, the optical amplitude caused by each aperture at P’(x’, y’) on the detection plane is calculated individually by shifting the location of each aperture, and later adding up their amplitudes altogether. The normalized optical amplitude at P’ is:

where i represents each aperture, N_x, N_y are respectively the numbers of the apertures along the X and Y axes in the aperture array, S_i is the area of each aperture. Thus the light intensity at P’(x’, y’) is expressed as:

where I_ax, I_ay are respectively defined as $I_{\mathit{ax}}=\{\begin{array}{c}\frac{\left(N_x-1\right)}{2}\mathit{if}\mathrm{mod}\left(N_x-2\right)=0\\ \mathit{floor}\left(\frac{N_x}{2}\right)\mathit{otherwise}\end{array},I_{\mathit{ay}}=\{\begin{array}{c}\frac{\left(N_y-1\right)}{2}\mathit{if}\mathrm{mod}\left(N_y,2\right)=0\\ \mathit{floor}\left(\frac{N_y}{2}\right)\mathit{otherwise}\end{array}$, and we suppose each aperture in the array has a same shape and a same area of S.

According to Eqs. (1)-(5), the diffraction intensity at each point of the detection plane can be obtained, and the whole diffraction pattern comes out by calculating the points one by one.

Since the aqueous buffer has a different refractive index compared to air, it causes a pattern shift or refractive effect, just as we notice an image shift in front of a piece of glass or a broken stick in water. When the chamber height is much smaller than the distance from the aperture plane to the detection plane, this refraction brings negligible phase shift, but changes the position where the ray hits the detection plane, i.e., the intensity we calculated for P’ on the detection plane in Fig. 1 will actually appear on point P”. By Snell’s law, the relationship between P’(x’, y’) and P”(x”, y”) is:

where n_b is the refractive index of the buffer, (x, y) is the coordinate of P on the aperture plane. So the amplitude of the light at point P”(x”, y”) is U(x’, y’) and should be calculated by $U\left(\frac{\mathit{x"}-x}{n_b}+x,\frac{\mathit{y"}-y}{n_b}+y\right)$, equivalently, the light amplitude and intensity at P’(x’, y’) are computed by $U\left(\frac{x\text{'}-x}{n_b}+x,\frac{y\text{'}-y}{n_b}+y\right)$ and $I\left(\frac{x\text{'}-x}{n_b}+x,\frac{y\text{'}-y}{n_b}+y\right).$

3.2 Cells and chambers in diffraction calculations

Because the intersected length of the ray in the cells, chambers and buffer can be calculated as long as the positions of these materials are expressed in the X-Y-Z coordinate system, this section further describes how to combine the influences of the cells and chambers into Eq. (1). As we have seen, once Eq. (1) is calculated, the diffraction pattern on the detection plane for all kinds of apertures can be obtained through Eqs. (2)-(6).

In our simulations, the cell is supposed to be transparent with a uniform refractive index allover its cytoplasm, and it includes a nucleus with a different refractive index and certain optical absorption. The cell and its nucleus are both spherical and can be expressed as:

where (d_x, d_y, t) is the center of the sphere in the X-Y-Z coordinate system, and R is the radius of the sphere.

When the microfluidic chamber is large or the diffraction detection area is small, the diffraction of the chamber won’t appear in the recorded diffraction pattern; otherwise, it becomes a background of the diffraction of the cells. For a one-layer chamber, whether it is misaligned to the aperture or not, its diffraction is calculated by setting its shape and position with respect to the aperture. For a multiple-layer misaligned chamber, each layer of the chamber should be calculated individually, in the same way as the one-layer chamber.

Assuming the misalignments of the chamber to the center of the aperture along the X and Y axes are respectively x_cm and y_cm, a cylindrical chamber can be depicted as:

where R_ch is the radius of the cylindrical chamber, h₁ and h₂ are respectively the distances from the bottom and the top of the chamber to the aperture.

Similarly, a cuboidal chamber is expressed as:

where a and b are the half-widths of the two sides of the rectangular viewed from the top, h₁ and h₂ are the distances from the bottom and the top of the chamber to the aperture.

A pyramidal chamber is calculated by the formula:

where a₁ and b₁ are the half-widths of the pyramid’s bottom rectangle, d is the distance from the bottom of the chamber to the aperture, γ is the slant angle of the pyramid as drawn in Fig. 3, and h is the height of the chamber.

with definitions of d, γ, and h the same as those in Eq. (10).

Our program checks whether the input positions of the cells and their nuclei are valid, for example, cells should be inside the chamber, a cell should not overlap to another cell or the chamber, and its nuclei should be inside it. If the original input positions of the cells are improper, a reminder will pop out and require adjustment of the cells’ locations.

We conducted all of our simulations for various apertures and chambers with both Labview and Mathcad, so as to cross check the correctness of our programs. Mathcad has higher mathematical accuracy, but it is slow in calculation and not as user-friendly as Labview. Our careful comparisons show that their accuracies are almost the same, so we used Labview in our diffraction analyses.

It is more important to extrapolate the optical parameters of the cells or nuclei than just to get the diffraction pattern. The solution to extrapolate a particular parameter is to fit the diffraction pattern by inputting a guess, and correcting this input datum till the simulated diffraction pattern well-fits the recorded one, i.e., within certain error or discrepancy. In detail, when a cell’s diffraction pattern is taken by the camera, we can find the minimum and maximum of the real picture and set them to be the same as the simulated pattern, so the contrast of these two patterns will be the same. After this process, we can add up the deviation of each pixel in these two patterns to indicate the error between them. Once this error reaches minimum, the parameter is found. This process is similar to numerically solving a nonlinear equation. Labview can automatically increase or decrease this input datum until a preset error is reached, at a reasonable speed and accuracy. Furthermore, Labview has the capability of communicating with instruments or controlling stages, it provides the potential that such a MEMS device might be built into a real-time and fully-automated cell detection system in the future.

4. Discussions

Our software is a complete set for various aperture and chamber combinations designed in this paper. In this section, we expand the functionality of our software and discuss the main issues in the aperture and chamber design.

In this section, we fix the laser wavelength at 0.532 μm. We regard each cell as a different kind which has a different size and refractive index, and its nucleus also has a different size and refractive index, as well as a specific optical absorption.

4.1 The diffraction patterns with various apertures

The aperture is the key to detect the cells in our designs, so we investigate its influence to the diffraction pattern, especially the detection limit caused by it.

4.1.1 Single aperture comparisons

To show the diffraction difference caused by the apertures, in Fig. 5 we calculate the diffraction patterns of 3 single apertures when 3 cells are inside a cuboidal chamber. The 3 apertures in Fig. 5 are respectively a 1.2 μm × 0.8 μm rectangular aperture, a 0.6 μm in radius circular aperture, and a 0.6 μm in radius circular aperture blocked by a circle of 0.3 μm in radius. For each aperture, Fig. 5 shows its diffraction patterns of without and with 3 cells, and the diffraction difference obtained by deducting the diffraction pattern without cells from the one with 3 cells. Subtracting two diffraction patterns is an effective method to track the barely noticeable diffraction differences caused by cells, and can partially eliminate the background diffraction of the aperture and the chamber.

 

Fig. 5. The diffraction patterns without and with 3 cells, and their deducted diffraction patterns (obtained by deducting the diffraction pattern without cells from the one with cells), listed from left to right, for 3 single apertures in different shapes. The chamber is cuboidal with 16 μm × 14 μm in top view and 10 μm in height. (a)-(c) are calculated by a 1.2 μm × 0.8 μm rectangular aperture, (d)-(f) are by a 0.6 μm in radius circular aperture, and (g)-(i) are by an annular aperture with an outer radius of 0.6 μm and an inner radius of 0.3 μm. The detector is 30 cm × 30 cm and placed 10 cm away from the aperture. The refractive indices for the buffer and the chamber are respectively 1.33 and 1.46. The sizes and positions of 3 cells are set as following: the radii of the 3 cells and their nuclei are respectively (3, 3.5, 2.8) μm and (1, 0.9, 0.8) μm, the refractive indices of the cells and their nuclei are respectively (1.38, 1.4, 1.38) and (1.42, 1.45, 1.41), the nuclei have the optical absorptions of (1, 0.8, 1.2) μm-1, the X, Y, Z coordinates for the centers of the cells are respectively x = (2, 3, -4) μm, y = (4, -3, 2) μm and z = (5, 5.5, 5) μm, the X, Y, Z coordinates for the centers of the nuclei are respectively xn = (3, 3.5, -3) μm, yn = (4.3, -2.5, 1) μm and zn = (6, 7, 5.5) μm. In these diffraction patterns, X and Y represent the calculated points in the detection plane, and the color bar indicates the light intensity.

Download Full Size | PDF

Figure 5 shows that the aperture shape determines the basic shape of the diffraction pattern, that the rectangular aperture has an ellipsoidal diffraction pattern, while the circular and annular apertures have round diffraction patterns. Comparing the circular and the annular apertures, it is obvious that the middle blockage of the circular aperture greatly enhances the contrast of the diffraction patterns. In these diffraction patterns, the cuboidal chamber appears as a rectangular shadow, because the chamber is relatively small and the detector is large.

 

Fig. 6. The diffraction patterns of a cylindrical chamber without and with 3 cells, and their deducted diffraction patterns, from left to right, for various aperture arrays. Each array element is a circular aperture of 0.6 μm in radius, (a)-(c) are calculated for a single aperture, (d)-(f) are for a 2 × 2 array with aperture intervals of 2.4 μm, (g)-(i) are for a 3 × 3 array with aperture intervals of 2.4 μm, (j)-(l) are for a 3 × 1 array with aperture intervals of 7.2 μm. The detector is 15 cm × 15 cm and 10 cm away from the aperture. The refractive indices for the buffer and the chamber are respectively 1.33 and 1.46. In (a)-(i), the chamber is 8 μm in radius, the 3 cells are exactly the same as in Fig. 5; in (j)-(l), the chamber is 10 μm in radius, the cells are the same as in Fig. 5, except each cell is at the center of each aperture, i.e., the X, Y coordinates for the centers of the cells are x = (-7.2, 0, 7.2) μm, y = (0, 0, 0) μm. In diffraction patterns, X and Y represent the calculated points in the detection plane, the color bar shows the light intensity.

Download Full Size | PDF

4.1.2 Aperture array

Diffraction patterns are calculated for various aperture arrays formed by circular apertures of 0.6 μm in radius. Figures 6(a)-(i) present the calculated diffraction patterns of an 8 μm in radius cylindrical chamber without cells, with 3 cells, and the cells’ deducted diffraction patterns for a single aperture, a 2 × 2 aperture array, and a 3 × 3 aperture array. The interval between the centers of two adjacent apertures in these arrays is fixed at 2.4 μm. The detector is 15 cm × 15 cm and is placed 10 cm away from the aperture. The 3 cells are the same as those of Fig. 5. The single aperture in Figs. 6(a)-(c) is also the same as in Figs. 5(d)-(f), so their diffraction patterns are similar. But the detector size of Fig. 6 is only a quarter of that of Fig. 5, so the diffraction of the chamber cannot be seen in Fig. 6.

Compared with the single aperture, Figs. 6(d)-(i) show that the arrays diffract the light stronger than the single aperture, because the diffracted light beams from each aperture also interfere each other, cause a lot of deformations in the final diffraction pattern of the array. The existence of the cells will enhance such interference so the diffraction pattern of the one with 3 cells is totally different from the one without cells. This interference is also subjected to the interval of adjacent apertures, for instance, the interference in Fig. 6(j) is stronger than those in Figs. 6(d) and 6(g), because the array interval of Fig. 6(j) is 3 times larger.

Though it is not shown in Figs. 6(a)-(i), supposing the laser light intensity is 1 W/m², without 3 cells, the light powers detected on the detector for the single aperture, 2 × 2 and 3 × 3 aperture arrays are 1.08, 4.24 and 10.3 μW, respectively; with 3 cells, their light powers are 1.48, 4.46 and 9.08 μW, respectively. Obviously the light power is greatly improved by the aperture array. Without cells, this amplification factor is around N_x × N_y, where N_x and N_y are the aperture numbers along the X and Y axes in the aperture plane, respectively. With 3 cells, this amplification factor is slightly decreased.

People might wonder whether placing the cells in front of each aperture would give discrete diffraction patterns on the detector for each cell, under the condition that the interval of two adjacent apertures is large enough. In realty it is impossible, because no matter how large the apertures’ intervals are, the far-field diffraction pattern of an aperture array will only have one maximum at its center. We calculated such a case for a 3 × 1 aperture array in Figs. 6(j)-(l), when 3 cells (same as the ones in Figs. 6(a)-(i)) are each placed at the center of one of these 3 apertures. Figures 6(j)-(l) manifest that the diffraction patterns of the 3 apertures without cells are mixed and exhibit as one diffraction pattern at the center with fringes, and the diffraction patterns of the array with 3 cells are also mingled.

4.1.3 Detection limits caused by different apertures

The detection limit is an important factor because it indicates the extreme that people can expect from a detection system. For this device, the way to find the detection limit of the cell’s size or refractive index is to reduce the radius of a cell or the difference of the refractive indices between the cell and the buffer, till the diffraction pattern with the cell is very close to the one without the cell.

The detection limit for detecting a cell is mainly decided by the aperture, and subjected to the shape and size of the chamber and the detector. In order to discern the minimum diffraction variation, the cell’s diffraction pattern has to be deducted from the one without cell. The contrast in each diffraction pattern is calculated as the maximum pixel power minus the minimum pixel power on the detector, and the normalized difference contrast is defined as the contrast of the deducted diffraction pattern of the cell divided by the contrast of the diffraction pattern without cell. We can preset a value of normalized difference contrast as a threshold to identify the diffraction limit. Some detection limits of a cell’s radius and refractive index for various apertures are calculated in Table 1. In Table 1, we suppose the chamber is large and doesn’t influence the diffraction pattern. For single aperture cases, we can see that a smaller or a middle blocked aperture can detect smaller cells. But their minimum refractive index detectability is not much different, which is around 1.340 - 1.33 = 0.01. The aperture arrays can increase the detectability of the refractive index up to 1.333 - 1.33 = 0.003, about 3 times more sensitive than the single apertures. But the aperture arrays are not suitable for the detection of small cells, and their detectability for the cell’s size decreases as the number of apertures in the array increases. For example, as shown in Table 1, a 2 × 2 0.6 μm in radius aperture array can detect a cell of 0.4 μm in radius, while a 3 × 3 0.6 μm in radius aperture array only can detect a cell of 0.7 μm in radius.

Table 1. Detection limits of the radius and refractive index of a cell by various apertures. We suppose there is only one cell in a cylindrical chamber of 8 μm in radius and 10 μm in height. The cell has no nucleus, and we fixed its refractive index at 1.38 when calculating the detection limit of its size and fixed its radius at 3 μm when calculating the detection limit of its refractive index. The cell is at the center of the aperture (or aperture array), and 8 μm away from the aperture plane. The refractive indices for the buffer and chamber are 1.33 and 1.43, respectively. The detector is 20 cm × 20 cm and is placed 10 cm away from the aperture. We set the threshold of the normalized difference contrast to be 0.05.

View Table

4.2 The diffraction patterns with various chambers

4.2.1 The influence of basic chamber shapes

Figure 7 shows the diffraction patterns of a 0.6 μm in radius aperture for a one-layer chamber, when the chamber shapes are respectively cuboidal, cylindrical, pyramidal and conical. The chambers are roughly in the same size, and the detector is relatively large, so the diffraction patterns of the chambers are shown in the detector. Because the cells and their locations are the same for these 4 kinds of chambers, their diffraction patterns are similar when 3 cells exist. In Fig. 7, the diffractions of the chambers are the background of the cells’ diffraction patterns, and the deducted diffraction patterns cannot totally cancel out the influence of the chambers, that’s why for each kind of chamber, we have to analyze the diffraction patterns according to the particular size, shape and position of the chamber.

4.2.2 The multi-layer and misaligned chambers

A multi-layer chamber can trap the cells better, but it will also affect the diffraction pattern. Figure 8 demonstrates the diffraction patterns of a 3-layer cuboidal chamber without and with 3 cells and their deducted patterns, both for well-aligned and misaligned chamber layers. The single aperture is 0.6 μm in radius as that of Fig. 7, the refractive indices of the buffer and the chamber, as well as the cells are the same as those of Fig. 7. So the diffraction patterns of the cells in Fig. 7 and Fig. 8 should be similar except the differences in the chambers in terms of their shapes, sizes, layers and misalignments.

According to Fig. 8, the multiple-layers or their misalignments seriously deform the diffraction patterns when without or with cells. The correction of the multiple-layer and the misalignments is important and compulsory for proper analyses of the cells, because some misalignments are unavoidable in MEMS fabrication, we have included the misaligned multilayer chamber simulation in our analysis software.

 

Fig. 7. The diffraction patterns of various shapes of one-layer chamber without and with 3 cells, and their deducted diffraction patterns, listed from left to right, with a single circular aperture of 0.6 μm in radius. All chambers are 15 μm high, the distance from the bottom of the chambers to the aperture is 3 μm. (a)-(c) are calculated with a cuboidal chamber with 16 μm × 14 μm in top view, (d)-(f) are with a cylindrical chamber in a radius of 8 μm, and (g)-(i) are with a pyramidal chamber whose bottom surface is 2 μm × 1.6 μm, and its slant angle (defined in Fig. 3) is 54.71°. (j)-(l) are with a conical chamber with a slant angle of 54.71°. The detector is 20 cm × 20 cm and 10 cm away from the aperture. The refractive indices for the buffer and the chamber are respectively 1.33 and 1.43. The sizes and positions of the 3 cells are set as following: the radii of the cells and their nuclei are respectively (3, 3.5, 2.8) μm and (1, 0.9, 0.8) μm, the refractive indices of the cells and their nuclei are respectively (1.38, 1.4, 1.38) and (1.42, 1.45, 1.41), the nuclei have optical absorptions of (1, 0.8, 1.2) μm-1, the X, Y, Z coordinates for the centers of the cells are respectively x = (0.5, 2, 3.5) μm, y = (1, 3, -3) μm and z = (8, 14, 11) μm, the X, Y, Z coordinates for the centers of the nuclei are respectively xn = (1, 2.5, 3) μm, yn = (1.3, 3.5, -3.2) μm and zn = (9, 15, 11.5) μm. For each diffraction pattern, X and Y represent the calculated points in the detection plane, and the color bar shows the light intensity.

Download Full Size | PDF

 

Fig. 8. The diffraction patterns for a 3-layer cuboidal chamber without and with 3 cells, and their deducted diffraction patterns, listed from left to right, with a single circular aperture of 0.6 μm in radius. The distance from the bottom of the chambers to the aperture is 3 μm. The refractive indices for the buffer and the chamber, as well as the parameters of the 3 cells are the same as those of Fig. 7. (a)-(c) are calculated with a 3-layer cuboidal chamber without any misalignment, the sizes of the top viewed rectangles for these layers (from bottom to top) are respectively 4 μm × 3.5 μm, 16 μm × 14 μm, and 20 μm × 17.5 μm, and the height of the 3- layers are respectively 4 μm, 4 μm, and 7 μm. (d)-(f) are with a 3-layer cuboidal chamber with each layer’s size the same as that of (a)-(c), but with some misalignments to the aperture. The misalignments of the centers of 3 chamber layers (bottom to top) to the center of the aperture along the X-axis are (2, 1.5, 0.5) μm, along the Y-axis are (1.5, 1, 0.5) μm. In these diffraction patterns, X and Y represent the calculated points in the detection plane, and the color bar indicates the light intensity.

Download Full Size | PDF

4.3 Movement of a cell

The diffraction pattern also can be used to detect the movement of a cell. Figure 9 illustrates when a cell is moving along the X-axis in a cylindrical camber, how its diffraction pattern changes. In the 6 diffraction patterns of Fig. 9, everything is the same except the position of the cell is varying from left to right at a step of 1.5 μm. The 6 patterns vary continuously, and the exact location of the cell in each pattern can be extrapolated through the data fitting in the analysis software. If the diffraction patterns are correlated with time, the speed of the cell can be detected.

4.4 Detect the diffraction with a photodetector array

Compared with the size of the aperture which is a few micrometers, a distance of 500 μm or 1 mm away from the aperture already can be regarded as far-field. Instead of using a CCD or video camera to record the diffraction patterns, the diffraction also can be detected by a small photodetector array placed or fabricated at this distance. A detector array with each pixel in a size of 80 μm × 80 μm and placed 1 mm away from the aperture is able to detect the diffraction patterns and is quite possible to be integrated into the MEMS chip. Later the current of each pixel in the photodector array, which is correlated with the optical power on that pixel, can be used to analyze the parameters of the cells. No doubt detecting the cells with a photodetector array will increase the functionalities and the integration of this MEMS device, and further reduce the cost of this diffraction detection method. Although the photodetector array is not as sensitive as the CCD camera, it might be enough for some simple applications.

 

Fig. 9. The diffraction patterns of a cell moving along the X-axis in an 8 μm radius cylindrical chamber. The centers of the cell and its nucleus are respectively with (a) x = -3 μm, xn = -2.5 μm; (b) x = -1.5 μm, xn = -1 μm; (c) x = 0, xn = 0.5 μm; (d) x = 1.5 μm, xn = 2 μm; (e) x = 3 μm, xn = 3.5 μm; (f) x = 4.5 μm, xn = 5 μm. The chamber is 15 μm high, and the distance from its bottom to the aperture plane is 3 μm. The detector is 30 cm × 30 cm and 10 cm away from the aperture. The refractive indices for the buffer and the chamber are respectively 1.33 and 1.43. The radii of the cell and its nucleus are respectively 3 μm and 1 μm, the refractive indices of the cell and its nucleus are respectively 1.38 and 1.42, the nucleus has an optical absorption of 1 μm-1. The Y and Z coordinates for the center of the cell are respectively y = 0 and z = 8 μm, the Y and Z coordinates for the center of the nucleus are yn = 0.3 μm and zn = 1 μm. In the diffraction patterns, X and Y represent the calculated points in the detection plane, and the color bar indicates the light intensity, which is fixed in (a)-(f) to be 0 ∼ 1 (from purple to red).

Download Full Size | PDF

 

Fig. 10. The detected powers for the pixels in a one-dimensional photodetector array placed 1 mm away from the aperture, when a cell is moving with the conditions in (a)-(f) be the same as in Figs. 9(a)-(f). The photodetector array consists of ten 80 μm × 80 μm pixels, the flaw between two adjacent pixels is 20 μm. Starting from the origin, it is placed in a line in the first quarter of the X’-Y’ coordinate system (refer to Fig. 1). In these patterns, the horizontal axis shows the number of the pixel, 0 is the one closest to the center. The vertical axis shows the detected optical power on each pixel, when supposing a laser light intensity of 1 W/m². The range of the pixel power in the vertical axis is fixed at 0 ∼ 0.016 μW.

Download Full Size | PDF

Because the diffraction pattern is calculated by data matrices, i.e., by calculating certain points on the detection plane, once we have calculated the diffraction pattern on the detection plane, the optical power on each pixel of a preset photodetector array is obtained by finding the calculated points that fall in the pixel, and adding up the power of these points.

As an example, Fig. 10 demonstrates that the movement of a cell in Fig. 9 can also be detected by a one-dimensional (1D) detector array, which consists of ten 80 μm × 80 µm detection pixels and is placed 1 mm away from the aperture plane. The array is arranged in a line along the X’-axis in the first quarter of the X’-Y’ coordinate system (on the detection plane as shown in Fig. 1). In each pattern of Figs. 10(a)-(f), the power detected by each pixel is indicated by a vertical bar, and the curve is obtained by connecting the power of each pixel. Figures 10(a)-(f) correspond to the diffraction patterns of Figs. 9(a)-(f). For example, each pixel power in Fig. 10(a) is the optical power that can be detected by that pixel (with a particular size and position) when the diffraction pattern is as Fig. 9(a), and a power combination of these 10 pixels in Fig. 10(a) uniquely relates to the specific location of the cell. Same explanation applies to Fig. 10(b)-(f), thus Fig. 10 can be used for analyzing the location (or movement speed) of the cell.

5. Experiments

Experiments were conducted to verify the validity of our simulations. We fabricated 3 kinds of apertures as shown in Fig. 11, including an annulus aperture, a 1 × 3 aperture array, and a 3 × 3 aperture array. The apertures were fabricated by depositing aluminum on a photoresist layer patterned by photolithography and later lifting-off the metal in acetone.

 

Fig. 11. The photos of the fabricated aperture and aperture arrays (left) and their diffraction patterns (without microparticles) obtained by experiments (middle) and simulations (right). From top to bottom: the annulus aperture with an inner diameter of 3 μm and an outer diameter of 7 μm; 1 × 3 aperture array with each element of 5 μm in diameter and 8 μm in interval; 3 × 3 aperture array with each element of 7 μm in diameter and 10 μm in interval. The scale bar of the microscope pictures is 20 μm.

Download Full Size | PDF

A Sony HDR-UX5E digital camera was employed to record the diffraction patterns of these apertures, when a laser beam at the wavelength of 0.633 μm was shed on the apertures. Figure 11 shows the optical apertures, the recorded and simulated diffraction patterns when no particles exist on the apertures. The experimental results match our calculations very well.

Figure 12 presents the diffraction patterns similar to Fig. 11, when transparent polystyrene microparticles of 5 μm in diameter were blocking the apertures. As can be seen from the left column of Fig. 12, when the microparticles were placed on the aperture or aperture array, usually there was more than one particle on it. To accurately simulate the diffraction patterns, we observed the chip under a microscope and recorded the locations of these microparticles before simulating with our software. For the annulus aperture, there happened to have one particle at a side only, while the 1 × 3 and 3 × 3 aperture arrays both had 9 particles blocking the apertures. In Fig. 12, the simulated diffraction patterns of these particles well coincide with the recorded diffraction patterns, in terms of the positions and numbers of the bright strips or circles. Slight discrepancies between the simulations and experiments might result from the imperfectness in aperture fabrications, for instances, the aperture is a little rough at the side or deformed, or unexpected needle holes exist around the aperture.

 

Fig. 12. The photos of the fabricated aperture and aperture arrays with polystyrene microparticles (left) and their diffraction patterns obtained by experiments (middle) and simulations (right). From top to bottom: the annulus aperture with an inner diameter of 3 μm and an outer diameter of 7 μm, when one particle exists; 1 × 3 aperture array with each element of 5 μm in diameter and 8 μm interval, when 9 particles exist; 3 × 3 aperture array with each element of 7 μm in diameter and 10 μm interval, when 9 particles exist. The microparticles are with a diameter of 5 μm and a refractive index of 1.59, and their positions are adumbrated on the left of the microscopic pictures.

Download Full Size | PDF

Our experiments prove that our algorithm can accurately simulate the diffraction patterns. In applications, we can simplify the multi-cell situation by positioning only one microparticle or one cell in the trap (by conical or multi-layer chamber) to analyze its size or shape.

Figure 12 shows some complexity in the diffraction patterns for aperture arrays, this is because the microparticles were not well positioned that many of them fell in the array area and the diffraction pattern was a combination of all these microparticles. If there is only one microparticle in front of the array by microfluidic control, the diffraction pattern will not be complicated. When a single aperture is employed, if the aperture is small, its illumination area is limited and the diffraction system requires high accuracy in positioning the microparticles; if we simply enlarge the aperture, the resolution and detectability will decrease. Because an aperture array occupies a much larger area of the substrate, the microparticles can easily be captured even the positioning of the microparticles is inaccurate. Besides, the aperture array increases the optical intensity of the diffraction pattern. On the other hand, according to Table 1, a single aperture renders higher sensitivity in detecting the size of the microparticles, and an aperture array renders higher sensitivity in detecting the refractive index of the microparticles. So both a single aperture and an aperture array are practicable in real applications, and the selection of an appropriate aperture depends on the MEMS fabrication technology, the physical properties of the cells or microparticles, and the parameter under test.

Because the diffraction system can extrapolate one parameter each time, suppose we have a group of cells (in a same kind) without knowing the refractive index or the sizes, we can use a microscope to record the positions of the cells first, then record the cells’ diffraction pattern to get their refractive index by simulation, later we can use this refractive index value to characterize the size of each cell, when the cells slowly pass through the aperture in microfluidics one by one. In the size characterization stage, we only need a camera or a photodetector array.

6. Conclusions

This paper for the first time proposed using the diffraction of an optical aperture or an aperture array to characterize the size, or the refractive index, or the movement of transparent living cells. In this system, no optical lens is employed: a CCD camera records the diffraction patterns, and our software uses the deformed diffraction patterns to extract the information of the cells. In this paper, we developed a whole set of Labview programs for the diffraction analyses of the cells with various optical apertures and microfluidic chamber designs. The diffraction experiments conducted for different apertures with and without microparticles verified the correctness of our simulations.

According to our investigations, the circular or rectangular shape of the aperture does not influence the detectability of the cells, but an aperture with middle blockage enhances the edge contrast of the diffraction patterns, and an aperture array increases the optical power on the detection plane. A single and smaller aperture is more sensitive for detecting the size of the cells, and an aperture array is more sensitive for detecting the refractive index of the cells. For example, a 0.6 μm radius single aperture can detect a cell of 0.35 μm in radius or a refractive index change of 0.012, and a 3 × 3 0.6 μm radius aperture array can detect a cell of 0.7 μm in radius or a refractive index change of 0.004, when the detectable normalized difference contrast is set as 0.05 in both cases. A chamber might cause a background to the cells’ diffraction pattern, depending on the shape, size, misalignment, layers of the chamber, as well as the size of the detector, so the shape of the chamber must be considered in diffraction simulations. A multi-layer or conical chamber in MEMS design can trap the cells better.

The MEMS chip proposed in this paper is cost-effective and easy to fabricate. Once the chip is fabricated and the cells are injected, continuous cell characterizations can be realized. The CCD camera used to record the diffraction patterns also can be replaced by a 1D photodetector, which can be integrated into the chip to increase the robustness of the device and further reduce the cost of the detection.