## Surface enhanced ellipsometric contrast (SEEC) basic theory and λ/4 multilayered solutions

Optics Express, Vol. 15, Issue 13, pp. 8329-8339 (2007)

http://dx.doi.org/10.1364/OE.15.008329

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### 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

*I*and the bare surface intensity

_{F}*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

_{S}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]

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]

*I*of the surface is linked to its Fresnel coefficients (

_{S}*r*, parallel and

_{p}*r*, orthogonal) by the relationship

_{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.

## 2. General

*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*and

_{P}*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}*r*and

_{p}*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}*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:

*β*=

*ϕ*-

*φ*±

*π*:

*φ*,

*φ*+

*dφ*] to the image intensity is:

*φ*while keeping a single incidence angle θ

_{0}leads to:

_{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}_{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:

*c*= cos

_{k}*θ*. We thus define:

_{k}*c*

^{2}

_{i-1}=

*c*

^{2}

*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.*

_{i}## 3. Single layer Solutions

_{p}and r

_{s}:

*r*are real quantities. Then the equation

_{p(0,1)}r_{s(0,1)}, r_{P(1,2)}and r_{s(1,2)}*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*e*

^{-2jβ1}= +1

*k*being an integer. It gives respectively:

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

*θ*

_{0}and we get:

*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}.

_{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.

*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

_{0n+1}=0 in the particular case where each layer has a quarter wave thickness given by

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

*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:

*r*or

_{p}*r*.

_{s}*r*

_{i-1,n+1}Fresnel coefficients, we define the sum and the product of

*r*and

_{p}*r*:

_{s}*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

*n*≥

*i*≥ 1):

*A*(

*i*-1,

*n*+1) and

*B*(

*i*-1,

*n*+1) as follows:

*ξ*

_{i-1,n+1}and then

*ξ*

_{i-2,n+1}with the help of Eq. (8), Eq. (11) and Eq. (25):

*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.

*i*= 1 and

*n*= 1 and specifying the Fresnel coefficients with the help of Eq. (8), we

*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:

*θ*

_{0}, we obtain the general low aperture relationship to be satisfied by the indices

*n*of the different layers:

_{i}## 5. Summary

## References and links

01. | T. Sandström, M. Stenberg, and H. Nygren, “Visual detection of organic monomolecular films by interference colors,” Appl. Opt. |

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. |

03. | D. Ausserré and M.-P. Valignat, “Wide field optical imaging of surface nanostructures,” Nano Lett. |

04. | A. Musset and A. Thelen, “Multilayer antireflection coating,” in |

05. | J. T. Cox and G. Hass “Antireflection coatings for optical and infrared materials,” in |

06. | R. M. A. Azzam and N. M. Bashara, |

07. | G. B. Airy, Phil. Mag., |

08. | L. G. Parratt, “Surface studies of solids by total reflection of X-Rays,” Phys. Rev. |

**OCIS Codes**

(180.0180) Microscopy : Microscopy

(240.0240) Optics at surfaces : Optics at surfaces

(260.0260) Physical optics : Physical optics

(310.0310) Thin films : Thin films

**ToC Category:**

Microscopy

**History**

Original Manuscript: December 14, 2006

Revised Manuscript: April 23, 2007

Manuscript Accepted: April 24, 2007

Published: June 18, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

D Ausserré and M.-P. Valignat, "Surface enhanced ellipsometric contrast (SEEC) basic theory and λ/4 multilayered solutions," Opt. Express **15**, 8329-8339 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-13-8329

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### References

- T. Sandström, M. Stenberg, and H. Nygren, "Visual detection of organic monomolecular films by interference colors," Appl. Opt. 24,472-479 (1985). [CrossRef] [PubMed]
- 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]
- D. Ausserré, and M.-P. Valignat, "Wide field optical imaging of surface nanostructures," Nano Lett. 6,1384-1388 (2006). [CrossRef] [PubMed]
- 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]
- 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.
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, (Elsevier, Amsterdam 1987).
- G. B. Airy, Phil. Mag. 2, 20 (1833).
- L. G. Parratt, "Surface studies of solids by total reflection of X-Rays," Phys. Rev. 95, 359-369 (1954). [CrossRef]

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