## Abstract

We present a model describing the image formation in DIC (Differential Interference Contrast) mode microscopy, by including the actual refractive indexes and reflection coefficients of objects and substrates. We calculate the contrast of flat and level objects of nanometric thickness versus the bias retardation Γ and the numerical aperture NA. We show that high contrasts, of the edge and of the inner object, can be achieved in DIC mode with special anti-reflective substrates and large NA values. The calculations agree with contrast measurements on nanometric steps of silica and explain also the extreme ability to detect single molecules (stretched DNA molecules).

©2008 Optical Society of America

## 1. Introduction

Direct observation of nanometric objects with the naked eye or with a conventional optical microscope is becoming a reality. The Brewster microscopy, invented in the 1990’s, allows observing objects of nanometric thickness at the air/water interface [1]. More recently, the SEEC (Surface Enhanced Ellipsometric Contrast) technique, has opened the way to the visualization of objects of nanometric thickness deposited on solid substrates [2,3]. Both techniques are based on the observation, in reflection mode, of an object positioned at the interface between two media. The principle of these techniques is to work in conditions where the reflecting signal on the bare interface is null. The contrast of an object versus the dark background of the bare interface is therefore maximized. In the SEEC technique, the substrates, called Surfs in the following, are engineered to be anti reflective for polarized light. SEEC observation requires simply an upright episcopic microscope with crossed polarizers illumination. Preliminary experiments have shown that the observation in Differential Interference Contrast (DIC) mode microscopy can also improve the detection limits of the SEEC technique. The detection of isolated objects, like elongated DNA molecules, has even been possible [2].

The principle of DIC microscopy is based on the splitting up of a polarized illuminating beam into two beams that are coherent, have orthogonal polarizations and are deviated by a small angle. The final image results from the interference between two slightly shifted images of the object, which permits to reveal local variations in optical paths on the object, e.g. the edges [4]. Several theoretical models have already been proposed for understanding DIC image formation [4–9]. They have lead to an in depth understanding of the image formation in DIC mode versus the lateral shear, the bias retardation Γ and the numerical aperture NA. To our knowledge, all the models for reflection mode DIC consider only perfectly reflecting surfaces. In this context, the calculated images correspond to topographic images. In practice, DIC images depend also on the reflectivity of the specimen. Hence, a flat and level surface presenting regions with different reflection properties will show regions of different brightness in DIC microscopy. Our goal here is to develop a model for DIC imaging with non ideal reflecting surfaces. Such a model is specially relevant for the objects with nanoscopic thickness (h≥1 nm). Indeed, the topographic contribution to DIC image formation is very limited for ultra thin objects. We will demonstrate that the use of Surfs substrates permits to improve this limitation and opens new opportunities for DIC imaging. As Surfs can be designed for any given angle of illumination, Surfs designed for high NA values allow imaging in DIC mode with high NA values and high contrast.

In this paper, we first calculate the reflected intensity in DIC mode for any reflective substrate by taking into account the phase-shift between the two beams (bias retardation), the numerical aperture, and also the actual reflection parameters of the substrate. We focus on the case of a flat and level step object with sharp edges and we calculate the contrasts of the object edges and of the inner part of the object. We then apply these calculations to Surf and silicon wafer substrates and compare the results to experimental measurements on calibrated steps.

## 2. Image intensity

#### 2.1 Image intensity in crossed polarizers mode

Let us first introduce the calculation of the reflected intensity in crossed polarizers mode [3]. We consider a uniform monochromatic plane wave passing through a non-depolarizing optical system made of a polarizer, a sample and an analyzer. We first consider that all the beams converge with an incidence θ_{0} on the reflective surface. For each beam, the orientations of the polarizer and the analyzer around the beam axis are specified by the azimuth angles P and A measured from the direction of the axis -p (for parallel) of an arbitrary reference plane of incidence. The amplitude and phase of the reflected light, for one plane of incidence, can be calculated by use of Jones optical calculus [11]. Let A_{0} be the amplitude of the incident light. The components of the Jones vector emerging from the analyzer can be expressed as:

$$\left[-------\mathrm{ANALYZER}-----\right]\left[\mathrm{SAMPLE}\right]\left[--\mathrm{POLARIZER}--\right]$$

where R_{p} and R_{s} are the reflection coefficients of the sample for the polarization directions respectively parallel and perpendicular to the plane of incidence. If φ̟ is the azimuth of the plane of incidence relative to the direction of the polarizer (φ=P) and polarizers are crossed (A=π/2-φ), the evaluation of the matrix sequence for crossed polarizers leads to:

The total reflected intensity, for a cone of light with incidence θ_{0}, is obtained by integrating the contribution of each incident plane:

where Σ is a constant characteristic of the microscope geometry and E̅_{A} is the conjugate complex of E_{A}.

In order to normalize the intensity, we define a reference intensity I_{0} as the intensity that would be reflected with an incident beam of amplitude A_{0} on a perfectly reflecting surface (R_{s}=-R_{p}=1) with no polarizing components. With these conditions, I_{0} is equal to:

and the normalized intensity in crossed polarizers mode I_{NX} is given by:

#### 2.2 Image intensity in reflected DIC mode

In a DIC microscope, the linearly-polarized light waves entering the Nomarski prism are divided into two components of perpendicular polarizations and of equal amplitudes. The waves are spatially separated (angle deviation) and phase shifted (bias retardation Γ). After reflection upon the surface, the passage through the Nomarski prism recombines the wavefronts of each polarization, so that they are collinear again when passing the analyzer. The two beams being coherent, they add vectorially and the analyzer selects one polarization component to produce the DIC image.

Let us consider the simple case of a flat step with sharp edges of refractive index n and thickness h deposited on a substrate of refractive index n_{S} (Fig. 1) in an ambient medium of refractive index n_{0}. We note R_{p} and R_{s} the reflection coefficients corresponding to the surface “substrate+step”, and R′_{p} and R′_{s} the reflection coefficients corresponding to the substrate. The image of the edge of the step in DIC mode corresponds to the case where one beam of the DIC mode reflects on the bare substrate and the other beam of the DIC mode reflects on the step. For a given plane of incidence of azimuth φ, the output intensity can be calculated by the following Jones operations sequence:

$$\left[---\mathrm{ANALYZER}---\right]\phantom{\rule{.9em}{0ex}}\left[------------\mathrm{SAMPLE}\&\mathrm{NOMARSKI}----------\right]\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\left[---\mathrm{POLARIZER}---\right]$$

where Γ is the bias retardation introduced by the Nomarski prism between the two shifted beams, and β_{0} is the phase shift resulting from the topography between the two shifted beams. The Jones operation sequence of Eq. (6) is inspired from Holzwarth, *et al* [9]. First, the field components are calculated after the polarizer (matrix M_{1}) in the referential of the incident plane (matrix M_{2}). The two beams split by the Nomarski prism are then calculated via the top and the bottom arms of Eq. (6). The two orthogonal matrices M_{3} and M_{3}′ represent the splitting in two beams with equivalent intensity repartition (angle between the Nomarski prism and the polarizer axes is 45°) and orthogonal polarizations. The effect of the reflection on the substrate is calculated with matrix M_{4} for one beam, whereas the effect of the reflection on the sample combined with the phase bias of the Nomarski prism is calculated with matrix M_{4}′ for the other beam. After addition of the resultant fields, the final coordinates are calculated in the referential of the analyzer (matrix M_{5}) and the effect of the analyzer is applied (matrix M_{6}).

For a wavelength λ, β_{0} is defined as:

By multiplying out the matrices in the top and the bottom arms of Eq. (6), then adding the resulting field, and finally multiplying by the analyzer matrix, one obtains the output field E⃗_{A}:

At this point, it is convenient to use the ellipsometric conventions in place of the reflection coefficients. The ellipsometric angles Ψ, Ψ′, Δ, Δ′, Δ_{s} and Δ′_{s} are defined by
$\frac{{R}_{p}}{{R}_{s}}=\mathrm{tan}\Psi \mathrm{exp}\left(i\Delta \right)$
,
$\frac{{R\prime}_{p}}{{R}_{s}}=\mathrm{tan}\Psi \prime \mathrm{exp}\left(i\Delta \prime \right)$
, R_{s}=|R_{s}|exp(iΔ_{s}), and R′_{s}=|R′_{s}|exp(iΔ′_{s}). Integration over φ leads to the normalized reflected intensity for the edge of the step I_{DIC-edge} as:

$$-\mid {R}_{s}\mid \mid R{\prime}_{s}\mid \left(\mathrm{tan}\Psi \mathrm{tan}\Psi \prime \mathrm{cos}\left(\Delta -\Delta \prime +{\Delta}_{s}-{\Delta \prime}_{s}+\Gamma +2{\beta}_{0}\right)+\mathrm{cos}\left({\Delta}_{s}-{\Delta}_{s}\prime +\Gamma +2{\beta}_{0}\right)\right)]$$

Note that Eq. (9) implicitly assumes that the edge is perpendicular to the shear direction. As I_{DIC-edge} is a function of the angle of incidence, via the reflection coefficients, the total reflected flux I*_{DIC-edge} in the convergent illumination conditions of a microscope is obtained by integration of Eq. (9) over the incident light cone as:

where θ_{max} is the half angle of the cone of specimen light accepted by the objective lens. θ_{max} depends on the numerical aperture.

Equation (9) can also be used to compute the average intensity I_{DIC-flat} of a flat surface. In this case, the two beams of the DIC mode are reflected by the same surface. Setting R_{p}=R′_{p}, R_{s}=R′_{s} and β_{0}=0 in Eq. (9), I_{DIC-flat} can be expressed as:

Equation (11) will be used to calculate the intensity of the bare substrate and of the inner part of a flat object.

## 3. Contrast

#### 3.1 Contrast in crossed polarizers and DIC mode

As introduced previously, we consider the case of a single step with sharp edges (Fig. 1). The contrast between the step and the substrate in crossed polarizers mode C_{x} is equal to:

where the superscript star sign * indicates that the entities have been integrated over the incident light cone. By the same token, the contrast C_{DIC-step} between the substrate and the inner part of the step in DIC mode is equal to:

One can notice that C_{DIC-step} is theoretically equivalent to C_{X} for Γ=0, i.e. the DIC mode is equivalent to crossed polarizers mode when no bias retardation is applied. Finally, since the edge of the step appears with a specific contrast in DIC mode, one can also define the contrast C_{DIC-step} of the edge of the step as compared to the substrate as:

#### 3.2 DIC mode contrast in the limit case of normal incidence

We first apply our contrast calculations to the limit case of normal incidence. We consider two model specimens. The first specimen consists of a flat step with sharp edges made of the same medium as the substrate [Fig. 2(a)]. If we consider that the surfaces are perfect mirrors (R_{s}=-R_{p}=R′_{s}=-R′_{p}=1) then Eq. (9) reduces to:

which corresponds to the expression of the intensity found in the literature for calculations that do not take into account the actual reflection coefficients of surfaces [9]. It is worth noticing that the reflected intensities are the same for the step and for the substrate (since they have the same refractive index), so that the contrast between the step and the substrate C_{DIC-step} is null. However, there is a contrast on the edge of the step which can be expressed, via Eq. (14), as:

The calculated values of C_{DIC-edge} versus the bias retardation Γ for a step of thickness h=41 nm are reported on Fig. 3. The contrast is not symmetric vs Γ. It presents two extrema of absolute value equal to one. The positive extremum corresponds to a perfect extinction of the substrate for a value of the bias retardation set to zero, while the negative extremum corresponds to the particular bias retardation value that cancels the phase shift arising from the topography, i.e. Γ=-2β_{0} [cf. Eq. (7)].

The second specimen consists of a flat step with sharp edges deposited on a flat substrate with a different refractive index [Fig. 2(B)]. We have used Eq. (14) to compute C_{DIC-edge} for a silica step of refractive index n_{SiO2}=1.46 and thickness h=41 nm deposited on a silicon substrate of refractive index n_{Si}=4-0.02i (Fig. 3). The maximum of positive contrast is obtained for Γ=0, like for the topographic specimen of case A. Indeed, this maximum results only from the extinction condition of the substrate. All other values differ noticeably between case A and case B. The absolute minimum of the contrast is shifted to lower values of the bias retardation Γ. The value of Γ needed in case B to cancel the phase shift induced by the topography of the specimen and the phase shift resulting from the presence of a layer standing on top of a “semi-infinite” medium is noticeably different from the one needed in case A to cancel the phase shift induced by the topography of the specimen only. As a general comment, Fig. 3 demonstrates clearly that our complete calculation improves substantially the description of DIC imaging.

#### 3.3 Contrast in DIC mode with convergent illumination

The novelty of our modeling of the contrast in DIC mode is not limited to taking into account actual refractive index and reflection coefficients of the objects in normal incidence. It also pays attention to the case of oblique incidence illumination and the effect of the numerical aperture of the microscope. Let consider the case of a step with sharp edges on a substrate with a different refractive index [Fig. 2(b)]. Figure 4 reports the calculated contrasts C_{DIC-edge} and C_{DIC-step} vs. the bias retardation Γ for three different numerical apertures (NA=0, 0.5 and 0.8) and two objects of different thicknesses (h=4 and 41 nm). The effect of NA on C_{DIC-edge} is almost negligible for thick objects (h=41 nm), but is on the contrary very important for thin objects (h=4 nm). Indeed, for thin objects, C_{DIC-edge} decreases from 1 at NA=0 to 0.25 at NA=0.8. Also, the calculations show that the maximum of contrast appears at higher values of bias retardation Γ when NA increases: the bias retardation Γ_{max} at which C_{DIC-edge} is maximum is 0°, 2° and 11° for respectively NA=0, 0.5 and 0.8. Indeed, at oblique incidence, the extinction conditions and consequently the maximum of the contrast depends on the reflective properties of the sample and the NA. The case of C_{DIC-step} (Fig. 4) is quite different. One can see that the NA has a very strong effect on C_{DIC-step} both for thin (h=4 nm) and for thick (h=41 nm) steps. The maximum of contrast is always at Γ=0 whatever the NA value and the contrast decreases monotonously for increasingly positive and negative values of Γ. However, it appears that the decrease of C_{DIC-step} vs Γ is much smoother when the NA is increased.

## 4. DIC and Surfs

#### 4.1 Wafer versus Surf

The Surfs are substrates specifically engineered to increase the contrast of objects in polarized microscopy. More precisely, they are designed to fulfill the condition R_{p}+R_{s}=0 for a given angle of incidence. As regard to Eq. (5), the reflected intensity on Surfs observed between crossed polarizers is null for the chosen angle of incidence. This optimized dark background implies an optimum of contrast for deposited objects. The question now is to determine how Surfs substrates, developed for crossed polarizers microscopy, can also present advantages for DIC microscopy. In the following calculations, the optical characteristics of the Surf substrates correspond to the ones of silicon substrates covered with 106 nm of silica. Such substrates are indeed good approximations of Surfs substrates for observation in air [2]. Figure 5 presents a comparison, between a silicon wafer and a Surf substrate, of the theoretical contrasts in crossed polarizers mode (or DIC mode with Γ=0) and DIC mode (or DIC mode with Γ≠0) versus the thickness of specimen. The specimen are flat steps of silica with sharp edges and the bias retardation is set to 5° for Fig. 5. On a silicon wafer, the contrast of the step versus the substrate is larger in crossed polarizers mode (C_{X}) than in DIC mode (C_{DIC-step}). However, both contrasts are rather low, which means that the detection of very thin objects is inefficient on a silicon wafer. On a Surf substrate, the two contrasts C_{X} and C_{DIC-step} are increased by a large factor. For 1 nm thick objects, the contrasts are respectively increased by a factor 74 and 104. This has two important consequences for experimental observations. Firstly, the performance of DIC mode on Surfs is now comparable to the performance of crossed polarizers mode. Secondly, Surfs make it possible to detect contrast variation induced by flat nanometric steps. To shed more light on the coupling between DIC mode microscopy and Surfs, we now compare the efficiency of this edge detection on a silicon wafer and a Surf substrate. The calculated edge contrast C_{DIC-step} for the sharp silica step on a silicon and on a Surf substrate are reported on Fig. 5. This contrast is clearly much larger on Surfs than on silicon wafers in the whole range of thicknesses investigated and especially for the smallest thickness range (h<10 nm). This explains the extreme detection capabilities of DIC mode microscopy on Surfs, which permit to image such tiny objects as isolated stretched DNA molecules [3].

#### 4.2 Experimental contrast

We have performed direct comparisons between our calculations and measurements on model specimens. One model specimen is made of nanometric silica steps deposited onto a Surf substrate (Fig. 6) and another model specimen is made of nanometric resin steps deposited onto a silicon wafer (Fig. 8). The thicknesses of the silica steps on Surf, calibrated by ellipsometry, are 10, 18, 26, 32, 37 and 41 nm. The thicknesses of the resin steps on silicon wafer, calibrated by AFM, are 27, 32, 45, 55, 66, 77 and 104 nm. The images were taken with an upright microscope (LEICA DMLM) and a color CCD camera (SONY 3CCD DSP). Figure 6 presents the image of the model specimen on Surf with a mercury source illumination in crossed polarizers mode and DIC mode. It is remarkable that objects as thin as 10 nm standing on Surf substrates are clearly visible with high contrasts in both modes. A rough comparison between Surf and silicon wafer can be done on Fig. 8 between the thinnest resin step on silicon (27 nm) and the third thinnest silica step on Surf (26 nm). The background is much darker and the contrast is noticeably higher on Surf than on silicon wafer. This confirms qualitatively the predictions of Fig. 5, i.e. that the contrast on a silicon wafer are poorer than on a Surf. Also, as predicted by our model, the contrasts on Surf are of the same order of magnitude in crossed polarizer mode and DIC mode. It also appears clearly on Fig. 6 that DIC mode permits to enhance the detection of the edges of the steps and that the edge and step contrasts increase with the thickness of the steps.

More quantitatively, it was possible to use the model specimen to probe our calculations of DIC mode imaging versus bias retardation Γ, nature of the substrate, and thickness of sample. The measurements have been taken with a monochromatic illumination (*λ*=540 nm) and constant camera settings (white balance, gain) for each data series.

On Fig. 7(a), the experimental edge contrasts C_{DIC-edge} for a step of height h=41 nm are reported versus Γ. It is remarkable that the general dependence of the contrast versus Γ is very similar to the theoretical prediction of Fig. 4(a). One difference is that the theoretical calculations predict a maximum contrast of 1, whereas the experimental maximum contrast is around 0.8. The ideal contrast value of 1 is only possible when optical components imperfections (polarizers, Nomarski prism, lenses) and CCD imaging noise sources (photon noise, dark noise, and read noise) are neglected. However, noise sources effects play an important role on experimental images. This can be evidenced, for instance, by comparing the contrast of a given object in crossed polarizers mode and in DIC mode. These two modes are theoretically equivalent, but in practice the contrast in DIC mode with Γ=0 appeared significantly lower than the contrast in crossed polarizers mode (data not shown here). This proves that the background noise induced by adding the Nomarski prism is sufficient to alter the experimental contrast. In order to take into account all different noise sources in our model, we decided, as a first approximation, to add a constant background intensity to the theoretical intensities. Using Eq. (10) and Eq. (14) with exact experimental parameters (i.e. n=1.46, n_{s}=4-0.02i, h=41 nm, NA=0.15, λ=540 nm) and adding a background intensity as the only fitting parameter, the experimental contrasts can be adjusted satisfactorily [Fig. 7(a)]. The optimum setting of the bias retardation Γ for a good contrast of the edges of the step appears finally to be a slightly positive value of Γ.

The experimental step contrasts C_{DIC-step} of the silica steps on Surf versus the bias retardation Γ are reported on Fig. 7(b). The general dependence of the step contrast vs Γ is similar for all steps. However, this dependence is very different from the theoretical prediction of Fig. 4(b). The experimental contrast shows a minimum vs Γ at Γ=0, whereas the theoretical contrast shows a maximum. This apparent discrepancy disappears when a constant background intensity is added to the theoretical calculations to take into account for stray scattering and noise effects [Fig. 7(b)]. Note that the same background value of 0.05*I_{0} permits to adjust all the data of Figs. 7(a) and 7(b). The results of Fig. 7(b) makes it clear that, in order to have a good contrast between two adjacent thin and flat surfaces, the bias retardation Γ has to be set to a value different from zero, the larger being the better. In the other hand, with large settings of Γ, the edge contrast is very small and all advantages of working in DIC mode are lost. In practice, ideal DIC settings will correspond to slightly positive Γ value to insure a high contrast of the inner part and of the edge of the step simultaneously.

Figure 8 presents the experimental and theoretical data on silicon wafer and Surf of step contrast variations versus the sample thickness.

All main features predicted by the calculations versus the object thickness for experimental conditions of Fig. 8 can be observed on the experimental pictures on silicon wafer, the contrast in crossed polarizers is positive for small thicknesses, passes by a maximum around 50 nm and becomes negative around 70 nm. On silicon wafer in DIC mode, the contrast is always negative on silicon wafer and decreases monotonically versus thickness. On Surf, the contrasts in crossed polarizers and DIC mode are always positive and are noticeably higher on Surf than on silicon wafers for the thinnest samples.

More quantitatively, the experimental contrasts can be adjusted by the theoretical model with additional noise as the only parameter. The noise added for all the adjustments of Fig. 8(c) is of 0.01*I_{0} for the crossed polarizer mode and of 0.02*I_{0} for the DIC mode. The slight noise difference in the two modes is consistent with a depolarizing effect of the Nomarki prism. The effect of noise addition is expected to be all the more important as the reflected intensity of the sample is low. Indeed, we checked that the additional noise i) hardly affects the calculations on silicon wafer, ii) lowers only slightly the contrast in DIC mode on Surf, and iii) lowers noticeably the contrasts on Surf in crossed polarizers mode. In the end, the contrast on Surf in crossed polarizers mode is very close to, nevertheless higher than, the contrast on Surf in DIC mode.

## 5. Conclusion

We have performed new calculations to describe the image formation in DIC (Differential Interference Contrast) mode microscopy. These calculations take into account the actual refractive indexes and reflection coefficients of objects and substrates. They have permitted to investigate the contrast of flat and level objects with sharp edges deposited on a substrate. We have calculated the contrast of the edges of the object and of the inner part of the object versus the bias retardation Γ and the numerical aperture NA. We have demonstrated the interest of coupling DIC microscopy with the SEEC (Surface Enhanced Ellipsometric Contrast) technique, as high contrasts and high resolution can be achieved in DIC mode on SEEC substrates. Finally, we have performed a quantitative analysis of experimental contrasts versus the bias retardation Γ, the nature of the substrate (silicon wafer or Surf) and the thickness of the sample. We have shown that the noise component of the reflected intensity can not be neglected with anti-reflective SEEC substrates. The technique could be even further improved by increasing the signal to noise ratio, e.g. by introducing a modulation of the polarization and performing video image processing [9,12].

## Acknowledgments

We are grateful to Nanolane company (Le Mans, France) for financial support and for providing us with the Surf surfaces and the silica steps on Surfs. We also want to thank Rhodia company for financial support and lending of their microfabrication facilities.

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