## Abstract

The fundamentals of a new high contrast technique for optical microscopy, named “Surface Enhanced Ellipsometric Contrast” (SEEC), are presented. The technique is based on the association of enhancing contrast surfaces as sample stages and microscope observation between cross polarizers. The surfaces are designed to become anti-reflecting when used in these conditions. They are defined by the simple equation *r _{p}* +

*r*= 0 between their two Fresnel coefficients. Most often, this equation can be met by covering a solid surface with a single

_{s}*λ*/4 layer with a well defined refractive index. A higher flexibility is obtained with multilayer stacks. Solutions with an arbitrary number of all-dielectric

*λ*/4 layers are derived.

© 2007 Optical Society of America

## 1. Introduction

Improving contrast in optical microscopy has been a constant challenge for centuries and is still an active field of research. A contrast C between a film and a surface is usually defined as the relative difference between the film intensity *I _{F}* and the bare surface intensity

*I*,

_{S}*C*= (

*I*-

_{F}*I*)/(

_{S}*I*+

_{F}*I*). It is optimal if the intensity of the substrate goes to zero. For that reason anti reflecting (AR) surfaces can be used in order to improve contrast for thin film detection or visualization using reflected light techniques [1]. In 1986, Ausserré and coworkers have proposed a sensitive thin film (∼10 nm) imaging method based on the existence of an ellipsometric contrast when working with a high aperture illumination between cross-polarized filters [2]. Recently, a drastic improvement of this contrast technique has been introduced [3] that increases the present sensitivity of optical microscopy by more than an order of magnitude. This new technique is well adapted for the imaging of molecular films or sub-molecular films at surfaces. With the help of a simple optical microscope, it allows to probe in real time important kinetic phenomena such as wetting-dewetting, phase transitions and adsorption. It is therefore expected to find important applications in the fields of surface chemistry, micro-fluidics and biotechnology. In particular, the SEEC technique has a high potential for label-free reading of biochips since it may combine high sensitivity, high resolution and immersion microscopy.

_{S}The technique is based on the association of enhancing contrast surfaces as sample stages and microscope observation under incoherent illumination between cross polarizers. The specificity of these surfaces is that they do not change the polarization state of light upon reflection. Therefore they become anti-reflecting surfaces for polarized light in extinction conditions (AR-X-Pol). This specificity is lost when a film is present on the surface. As a consequence, the reflected intensity is different from zero and the film is visualized on dark background. Because the change of the polarization state of the reflected light is the origin of film imaging, we name the technique “Surface Enhanced Ellipsometric Contrast” (SEEC).

A classical anti-reflective (AR) surface [3,4] is defined for non polarized light. The reflected light intensity *I _{S}* of the surface is linked to its Fresnel coefficients (

*r*, parallel and

_{p}*r*, orthogonal) by the relationship $\frac{{I}_{S}}{{I}_{1}}=\frac{1}{2}\left({\mid {r}_{p}\mid}^{2}+{\mid {r}_{s}\mid}^{2}\right)$ in which

_{s}*I*

_{1}is the incident light intensity. The antireflection condition

*I*= 0 requires that both

_{S}*r*and

_{p}*r*are null. These two conditions can only be met with normal incidence and can be satisfied with a solid bearing a single layer. In this case the optical thickness of the layer is λ/4 and its refractive index

_{s}*n*

_{1}must fulfill the relationship

*n*

^{2}

_{1}=

*n*

_{0}

*n*

_{2}[5], where

*n*

_{0}and

*n*

_{2}denote the ambient and solid refractive index.

In this paper, we will first establish in detail the anti-reflecting condition for a surface illuminated with incoherent convergent light and observed between cross polarizers. This condition generates a new family of solid supports that can be used as sample stages for high resolution and high sensitivity imaging. In the second part of this paper, we will describe part of this family. In the simplest cases, these substrates consist in a solid support covered with a single appropriate layer. More generally, they can be obtained by covering a solid support with a quarter wave multilayer stacking obeying a simple relationship between layer indices. Using these results, one will be able to design high contrast SEEC surfaces either for low aperture microscopy under a conventional polarizing microscope or for high resolution microscopy by adding a ring aperture diaphragm to the microscope.

## 2. General

We consider an optical microscope working in reflection mode. A direct laboratory frame (*x*⃗, *y*⃗, *z*⃗) is attached to the microscope. The optical axis of the microscope is along *z*⃗ and is oriented upwards. The directions of the two polarizing plates are *u⃗ _{P}* and

*u⃗*. Because of the radial symmetry of the instrument, and as long as

_{A}*u⃗*is fixed, one may choose

_{P}*u⃗*=

_{P}*x*⃗ without lack of generality. We name ϕ the oriented angle between

*u⃗*and

_{P}*u⃗*in the laboratory frame. We assume a quasi-monochromatic illumination with wavelength λ. The cone of light impinging the sample integrates beam contributions with incidence angle θ

_{A}_{0}ranging from θ

_{0min}to θ

_{0Max}and azimuth φ ranging from 0 to 2π. Let us first consider a single incidence θ

_{0}and a given azimuth φ. Following Azzam and Bashara sign conventions [6], the frame attached to the light beam is (

*p⃗*,

_{i}*s⃗*,

_{i}*k⃗*) before reflection and is (

_{i}*p⃗*

_{r}*s⃗*,

_{r}*k⃗*) after reflection, where

_{r}*p⃗*and

*s⃗*are electric field unit vectors respectively parallel and perpendicular to the plane of incidence, and where the wavevectors

*k⃗*and

_{i}*k⃗*are taken above the objective lens in order to confine the calculation in two dimensions. The two frames are more precisely defined by the drawing displayed in Fig. 1.

_{r}The surface reflection is characterized by the two Fresnel coefficients *r _{p}* and

*r*respectively parallel and perpendicular to the plane of incidence defined by the azimuth. Setting to E

_{s}_{i0}the amplitude of the non polarized initial beam, the output emerging field in a direct frame (

*u⃗*,

_{A}*v⃗*,

_{A}*k⃗*) attached to the second polarizer is given by [6]:

_{r}In this formula, the meaning of the different matrices, from right to left is the following: First the non polarized initial field is reduced in amplitude when passing through the first polarizer, then it is expressed in the (*p⃗ _{i}*,

*s⃗*,

_{i}*k⃗*) frame, then reflected on the sample surface – where it becomes automatically expressed in the (

_{i}*p⃗*

_{r}*s⃗*,

_{r}*k⃗*) frame due to sign conventions in the Fresnel coefficients-, then expressed in the analyzer frame (

_{r}*U⃗*,

_{A}*v⃗*,

_{A}*k⃗*) and then projected along the analyzer. Angle φ is the angle made by

_{r}*p⃗*and

_{i}*u⃗*=

_{P}*x⃗*in the (

*p⃗*,

_{i}*s⃗*,

_{i}*k⃗*) frame and angle β is the angle made by

_{i}*p⃗*and

_{r}*u⃗*in the (

_{A}*p⃗*,

_{r}*s⃗*,

_{r}*k⃗*) frame. Notice that the dependence in θ

_{r}_{0}is entirely contained in the reflection Jones matrix. Expanding the matrices sequence one gets:

Ellipsometry users are familiar with such an expression where everything is expressed with respect to a fixed plane of incidence. Since in optical microscopy there is no privileged azimuth, it is more convenient to write the result in terms of angles φ and ϕ using the relationship *β* = *ϕ* - *φ* ± *π*:

Assuming that the source is incoherent, two different beams reflected by the sample cannot interfere and the light intensity in the image is obtained by adding intensities of the reflected beams. The contribution of the elementary azimuth range [*φ*, *φ* + *dφ*] to the image intensity is: $\frac{2\pi}{{I}_{0}}\frac{\mathrm{dI}}{d\phi}=\frac{1}{8}\left\{{\mid {r}_{p}-{r}_{s}\mid}^{2}{\mathrm{cos}}^{2}\varphi +{\mid {r}_{p}+{r}_{s}\mid}^{2}{\mathrm{cos}}^{2}\left(\varphi -2\phi \right)+2\left({\mid {r}_{p}\mid}^{2}-{\mid {r}_{s}\mid}^{2}\right)\mathrm{cos}\varphi \mathrm{cos}\left(\varphi -2\phi \right)\right\}$ Summing over all azimuths *φ* while keeping a single incidence angle θ_{0} leads to:

In the two equations above, I_{0} is the intensity one would get using the same microscope with a perfectly reflecting sample (defined by *r _{p}* =-

*r*= 1) in the absence of any polarizer. The first term in the second member of Eq. (4) is half of the intensity one would get assuming no interference between

_{s}*r*and

_{p}*r*. The second term comes from interference between

_{s}*r*and

_{p}*r*. Indeed, although the beam is incoherent, the two amplitude components

_{s}*r*and

_{p}*r*have a well defined phase relationship and may interfere. Let us consider a thin sample film deposited on part of a flat solid support. The contrast between the film and the bare substrate is maximum (C=1) when the substrate intensity is zero. This is only possible when the two polarizers are crossed. Then Eq. (4) becomes:

_{s}and the support must satisfy the relationship

Equation (6) defines a non depolarizing surface.

In order to obtain the relevant contrast/intensity for visualization with an optical microscope, one must take into account all angles belonging to the incident light solid angle, which we suppose defined by two cutting angles θ_{min} and θ_{max}. Assuming a homogeneous source, the contribution of an elementary solid angle to the reference intensity is *dI*
_{0}(*θ*,*θ* + *dθ*) = α sin *θdθ*, where α is a constant factor, and the integrated reference intensity becomes: ${I}_{0}({\theta}_{\mathrm{inix}},{\theta}_{max})=\alpha \underset{{\theta}_{\mathrm{inix}}}{\overset{{\theta}_{max}}{\int}}\mathrm{sin}\theta d\theta $ The normalized intensity between cross polarizers is therefore:

This expression will be used in numerical examples, while analytical solutions will be derived for single incidence. In the following, we look for surfaces obeying Eq. (6).

At the interface between two semi-infinite media (index i-1 and i), the parallel and perpendicular Fresnel coefficients are:

In order to solve Eq. (6), it is convenient to introduce the sum and the product of the two Fresnel coefficients. We note *c _{k}* = cos

*θ*. We thus define:

_{k}and

It is easy to establish that

Equation (6) cannot be satisfied unless *c*
^{2}
_{i-1} = *c*
^{2}
* _{i}* which is only possible but always true for normal incidence. For oblique incidence, it is necessary to add at least one layer in between the two semi-infinite media in order to obtain a solution. We look for the properties of this layer.

## 3. Single layer Solutions

The Fresnel coefficients of the stacking made of media 0 (impinging, semi-infinite), 1 (intermediate layer) and 2 (emerging, semi-infinite) can be calculated using the same Drude formula [7] for r_{p} and r_{s}:

where ${\beta}_{1}=\frac{{2\pi n}_{1}{e}_{1}{\mathrm{cos}\theta}_{1}}{\lambda},{e}_{1}$ being the layer thickness. From there, the sum of the parallel and perpendicular coefficients can be written as

In the following, we restrict our search to dielectric media. Then *r _{p(0,1)}r_{s(0,1)}, r_{P(1,2)} and r_{s(1,2)}* are real quantities. Then the equation

*σ*= 0 has real solutions and the value of the phase factor

_{(0,2)}*e*

^{-2jβ1}is either -1 or +1.

*If e^{-2jβ1} = -1, we have*

and if *e*
^{-2jβ1} = +1

*k* being an integer. It gives respectively:

and

Here θ_{1} holds for the angle of incidence in medium 1. Solutions of Eq. (6) are given either by

or by

We are looking for the properties of the intermediate layer that is to insert in between two different semi-infinite media. The second equation has no solution and only Eq. (16) is considered. The corresponding cosine equation is:

Combining with the Snell law, it gives the angle dependant index relationship:

Or equivalently

Equation (18) is valid for any single incidence as would be selected by a ring aperture diaphragm set on a microscope.

At low aperture, the relationship between refractive indices is at first order independent of *θ*
_{0} and we get:

This equation shows that the absolute upper limit of *n*
_{1} is *n*
_{0}√2. For observations in air, the intermediate layer must have a very low refractive index. To make it easier to realize, a high index solid support must be used. A silicon wafer is well adapted. For visible light, the imaginary part of its refractive index is low so it can be treated as a dielectric material with a refractive index close to 4. Then the optimal refractive index of the intermediate layer is found to be about 1.37. This is close to the refractive index of MgF2, a material which is widely used for making classical AR layers on glass. For observations in water, the optimal layer on the same support has a refractive index 1. 75. It can be made for instance with Y_{2}0_{3}.

In order to be quantitative, we consider a sample made of a silicon wafer covered with a single layer with refractive index 1.36 which from Eq. (18) is optimal for a single incidence angle θ_{0}=15°. For sake of simplicity, we calculate the contrast of a film having the same refractive index as the intermediate layer. In other words, the sample film is just a step in the intermediate layer thickness. Figure 2 shows the theoretical variation of the step contrast in air as a function of intermediate layer thickness for three illumination geometries. The wavelength is 540 nm. In all cases, the step height value is fixed to 0.1 nm. Curve a) is obtained with single angle of incidence θ_{0} = 15°. Curve b) is obtained with a full cone of incidence with θ_{max} = 20°, as would be the case with a low aperture microscope illumination. Curve c) is obtained when the angle of incidence is limited between θ_{min} = 12° and θ_{max} = 17° as would be obtained by using an aperture ring. Notice in curve 2a that a perfect contrast (C=1) is obtained when the intermediate layer thickness satisfies Eq. (15). Since Eq. (15) is angle dependent, this contrast is smeared out (curves 2b and 2c) when the illumination aperture range increases. However, it remains good enough in all cases to detect the 0.1 nm layer with the eye.

Figure 3 shows the variation of the step contrast as a function of step thickness in the same three illumination geometries. The optimal thickness slightly depends on this geometry and has been numerically optimized in each case. It is 101.1 nm in case a), 102 nm in case b), and 101.4 nm in case c). Even in the less favourable case c), the slope at the origin is still as much as 10% per Angström, making easily visible any molecular layer.

When working at a molecular scale, the surface chemistry of the sample stage is decisive. Therefore it is important to build the intermediate layer with standard materials. Silica is certainly one of them. Equation (19) tells us that a silica layer is optimal when the refractive index of the substrate is 1.75 in water. Hence silica on sapphire is an interesting solution. However for observation in air, the upper limit for *n*
_{1} is *n*
_{0}√2 and there is no way to build a SEEC surface using a single silica layer. This example illustrates why it is important to look for multilayer solutions. In the line of what precedes, we will limit our search to quarter wave multilayers.

## 4. λ/4 layer stacking

In what follows, we will show that the quarterwave solutions of Eq. (6) are given by very simple cosine equations. Those equations, namely Eq. (30) and Eq. (31) are a generalization of the single layer Eq. (17). Then, in Eq. (32), we will reduce the low aperture solutions to a relationship between the refractive index of the stacking layers, namely Eq. (32). Here again, it will appear as a generalization of the single layer Eq. (19).

Let us consider a stacking of n layers between two semi-infinite media. We will note by subscript 0, 1, …i,… n+1 quantities which refer to the n+2 media. We are looking for the solution of σ_{0n+1}=0 in the particular case where each layer has a quarter wave thickness given by ${n}_{i}{e}_{i}\mathrm{cos}{\theta}_{i}=\frac{\lambda}{4}.$ According to the generalized Drude formula [8], the Fresnel coefficients *rp*
_{i-1,n+1} and *rs*
_{i-1n+1} of a stack of n-i+1 layers in between semi-infinite media i-1 and n+1 are linked to the Fresnel coefficients of the n-i lower layers (assuming material i semi-infinite) by a recursive relationship:

where r holds either for *r _{p}* or

*r*.

_{s}Generalizing Eq. (9) and Eq. (10) to *r*
_{i-1,n+1} Fresnel coefficients, we define the sum and the product of *r _{p}* and

*r*:

_{s}and

and we consider the ratio:

For λ/4 layers we can write the numerator and denominator of the last expression as

From Eq. (20), one could calculate a recursive relationship between *σ*
_{i-1,n+1} and *σ*
_{i,n+1}. However, this relationship is too complex and would be of little help for obtaining analytical
results. By contrast, one can check using Eq. (20) that the quantity $\xi =\frac{\sigma}{1+\pi}$ obeys the following simple propagating rule (*n* ≥ *i* ≥ 1):

We now define *A*(*i*-1,*n*+1) and *B*(*i*-1,*n*+1) as follows:

$$\left\{\begin{array}{c}A\left(i-1,n+1\right)={c}_{i-1}^{2}{c}_{n+1}^{2}\prod _{p=1}^{\frac{n-i}{2}}{c}_{2(p+i-1)}^{4}\\ B\left(i-1,n+1\right)=\prod _{p=1}^{\frac{n-i}{2}+1}{c}_{2\left(p+i-1\right)-1}^{4}\end{array}\right\}\phantom{\rule{.2em}{0ex}}\mathrm{when}\phantom{\rule{.2em}{0ex}}\left(n-i+1\right)\phantom{\rule{.2em}{0ex}}\mathrm{is}\phantom{\rule{.2em}{0ex}}\mathrm{odd}$$

First we demonstrate that if

then

Thus, we assume that Eq. (27) holds and we calculate *ξ*
_{i-1,n+1} and then *ξ*
_{i-2,n+1} with the help of Eq. (8), Eq. (11) and Eq. (25):

From Eq. (26) it is straightforward to show that, whatever the parity of n-i+1,

*A*(*i*-2,*n*+1) = *c*
^{2}
_{i}*c*
^{2}
_{i-2}
*A*(*i*,*n*+1) and *B*(*i*-2,*n*+1) = *c*
^{4}
_{i-1}
*B*(*i*,*n*+1), hence Eq. (28) is true.

Second we will check that Eq. (27) is true for one and two layers. This will demonstrate that it is correct for an arbitrary number of layers.

From Eq. (24), we get:

Setting *i* = 1 and *n* = 1 and specifying the Fresnel coefficients with the help of Eq. (8), we ${\xi}_{\mathrm{0,2}}=\frac{{c}_{0}^{2}{c}_{2}^{2}-{c}_{1}^{4}}{{c}_{0}^{2}{c}_{2}^{2}+{c}_{1}^{4}}.$ Then using Eq. (25) and Eq. (8), we get ${\xi}_{\mathrm{0,3}}=\frac{{c}_{0}^{2}{c}_{2}^{4}-{c}_{1}^{4}{c}_{3}^{2}}{{c}_{0}^{2}{c}_{2}^{4}+{c}_{1}^{4}{c}_{3}^{2}}.$ Thus Eq. (27) has been iteratively demonstrated.

Since *ξ*
_{0,n+1} and *σ*
_{0,n+1} have same zeros, the general multi-(λ/4-layer) solution for dielectric materials is given by the following generalized cosine equations:

and

Like Eq. (18), Eq. (30), and (31) are valid for any single incidence angle.

Expanding Eq. (30) and Eq. (31) to second order in *θ*
_{0}, we obtain the general low aperture relationship to be satisfied by the indices *n _{i}* of the different layers:

The two equations finally merge into a single one:

This formula is useful to solve problems with no single layer solution. Most of the time, two layers are enough to find out a realistic stacking. Then Eq. (32) becomes:

As explained previously, working on a silica surface in water immersion is particularly interesting. Here a two layers solution is obtained with the sequence water, silica, high index, silicon, the high index value being close to 2.15. Figure 4(a) figures this stacking sequence and Fig. 4(b) shows the evolution of the computed contrast of a layer with the bare substrate as a function of its thickness. When working in air, we may also notice the sequence air, silica, high index, silica, with a high index value of 2.25.

Depending on the constraints to respect, it may be necessary to go to a higher number of layers. For instance, one can check from Eq. (33) that again there is no quarter-quarter solution when imposing air as the ambient medium, silicon as the supporting material and silica as the surface material. However, solutions are obtained when adding one more layer. As an example, an interesting low aperture solution is: air, silica, high index material, silica, silicon. Taking 1.47 as the refractive index of silica, we find the high index value to be close to 2.10.

## 5. Summary

In this paper, we have given the basic theory of a low cost imaging technique (SEEC) that allows detection of molecular and sub-molecular layers. The only instrument required is a standard polarization microscope. The technique imposes the use of surfaces with specific optical properties as sample stages. We have given the rules that allow to readily obtain SEEC surfaces by covering a solid with a λ/4 layer and the rules that permit to design SEEC surfaces with a high flexibility by covering the solid with a dielectric λ/4 multilayer stack. Then it is possible to combine the special optical properties of the SEEC surfaces with their desired physical and chemical properties.

## References and links

**01. **T. Sandström, M. Stenberg, and H. Nygren, “Visual detection of organic monomolecular films by interference colors,” Appl. Opt. **24**, 472–479 (1985). [CrossRef] [PubMed]

**02. **D. Ausserré, A.-M. Picart, and L. Léger, “Existence and role of the precursor film in the spreading of polymer liquids,” Phys. Rev. Lett. **57**, 2671–2674 (1986). [CrossRef] [PubMed]

**03. **D. Ausserré and M.-P. Valignat, “Wide field optical imaging of surface nanostructures,” Nano Lett. **6**, 1384–1388 (2006). [CrossRef] [PubMed]

**04. **A. Musset and A. Thelen, “Multilayer antireflection coating,” in *Progress in Optics*,
E. Wolf, ed., (North Holland Publ. Co., Amsterdam, 1970) Vol. 8 p. 201–237. [CrossRef]

**05. **J. T. Cox and G. Hass “Antireflection coatings for optical and infrared materials,” in *Physics of Thin Films*,
G. Hass and R.E. Thun, eds., (Academic Press, New York, 1968), Vol. 2 p. 239.

**06. **R. M. A. Azzam and N. M. Bashara, *Ellipsometry and Polarized Light*, (Elsevier, Amsterdam 1987).

**07. **G. B. Airy, Phil. Mag., **2**, 20– (1833).

**08. **L. G. Parratt, “Surface studies of solids by total reflection of X-Rays,” Phys. Rev. **95**, 359–369 (1954). [CrossRef]