## Abstract

Optical properties of overcoated microspheres are calculated and compared to those of planar multilayers, in regard to the sphere diameter. The classical criteria for in-situ optical monitoring is analyzed to control the growth of films on the spheres. Most coatings are multi-dielectric quarter-wave stacks used in the thin film community.

©2004 Optical Society of America

## 1. Introduction

Optical interference coatings have always been used to control light and waves, and the community of thin films has today reached a high level of expertise for all design, production and characterization techniques of these micro-components. However most of these filters are produced on plane substrates, and are hence designed for the control of specular light which is reflected and transmitted according to the Snell/Descartes relationships. For some applications of critical interest including cosmetics and paints (colour effects), furtivity (high reflectance dielectric particles) and biophotonics (particles with high radiation pressure), as well as optical communications (end coating of optical fibers)…, a demand exists to overcoat microspheres in order to reach specific properties similar or not to those of interferential multilayers. Such overcoated spheres can be used in an isolated form or in a powder form.

Within this framework we have built two specific deposition processes based on electron beam deposition [1,2] and ion beam sputtering, that allow overcoating of microspheres with diameters in the range 3 μm–300 μm. However a key problem lies in the in-situ monitoring process that one should use to control the growth of thin films on microspheres. In this paper we analyze the classical criteria used for quarter-wave stacks on planar substrates, which constitutes the basis of optical monitoring [3]. This criteria is investigated in regard to the sphere diameter. In a more general way we address the design problem of specific stacks to be deposited on the spheres, in order to reach spectral properties similar to planar multilayer mirrors, beam splitters and narrow-band filters…

## 2. The quarter-wave criteria

#### 2.1 Mie theory for overcoated spheres

The electromagnetic model that we use is the well-known Mie theory [4,5,6,7,8] that was extended to concentric multilayers deposited on microspheres. All constants were determined thanks to tangential field continuities at concentric radia. Materials can be dielectric or metallic. As usual the theory does not take account of interactions or multiple reflections between different microspheres. It allows to plot angular and wavelength variations of scattering from a unique overcoated sphere Notice that in the case of a 500 nm radius sphere, Mie theory is valid with an incident wavelength in the visible range. For smaller spheres radii, it was numerically verified that the scattered fields approach the Rayleigh scattering. An example is given in Fig. 1 where the initial radius of the silica substrate sphere is a_{0} = 500 nm, with an overcoating given by a 7 layer quarterwave mirror of design:

where H and B designate high and low index layers with optical thicknesses equal to a quarter-wavelength: n_{H.}e_{H} = n_{B.}e_{B} = λ_{0}/4, with λ_{0} = 633 nm, where n_{H} and n_{B} are respectively the high and low refractive indices and, e_{H} and e_{B} the mechanical thicknesses of layers. Thin film materials are Ta_{2}O_{5}/SiO_{2}. The angular scattering I(θ,λ,) is normalized to the incident flux, as we usually do for Angle Resolved Scattering (ARS) or BRDFcosθ curves to match scattering measurements [9,10,11].

We observe in Fig. 1 numerous oscillations characteristic of the large value of total radius (a_{t} = 1257.12 nm) of the overcoated sphere, that also depend on the mirror design. From this angular curve at a given wavelength λ, we are able to extract integrated curves by transmission (S_{T}) or reflection (S_{R}), that is, total scattering in the transmission (0°–90°) and reflection (90°–180°) ranges respectively:

Notice here that numerical integration is necessary since the analytical integrated Mie formulae are only valid for the sum S = S_{R} + S_{T}. Furthermore, I(θ,φ) is given by series expressions thanks to Mie theory [5]. The series should converge slowly when the radius of the sphere is larger than the wavelength. Convergence is systematically tested before each numerical result in particular by the Wiscombe criterium [12].

#### 2.2 The quarter-wave criteria

In a general way, the basic criteria to design, control or produce most planar optical coatings is that of quarterwave layers. It is well known that at a particular wavelength λ_{0}, specular reflection (R_{0}) or transmission (T_{0}) from a single transparent layer at normal illumination is a periodic function of the deposited optical thickness. The extrema are reached when the optical thickness (n.e) is a multiple of a quarter-wavelength, that is:

where k is an integer. In the same way, for a given dielectric thickness, variations of R and T versus wavelength λ reveal extrema at specific wavelengths λ_{q} with q integer given by:

This result originates from the fact that standard calculation of plane waves within a planar multilayer only involves trigonometric functions with a common period, which constitutes an initial step for most design problems. In the case of concentric multilayers, the problem is strongly different since the size effect is predominant and turns the calculation to be dependent on non periodic Bessel and Hankel functions with parameter p_{m} = 2π na_{m}/λ, where a_{m} designates the initial sphere radius or layer thicknesses.

This first criteria was studied and is plotted in Fig. 2 at wavelength λ_{0}= 633 nm, for a single high index (n_{H} = 2.25) thin film layer of Ta_{2}O_{5}. Each figure is calculated for a particular diameter of the initial sphere, and represents the variation of total scattering by reflection (S_{R}) or transmission (S_{T}) versus the thin film thickness.

When the sphere radius is rather large (a = 15 μm), the variation of S_{R} versus layer thickness (Fig. 2(a)) is quasi-periodic, with an average period given by: E_{H} = 71 nm. This value is quasi-identical to that of a layer deposited on a flat substrate, given by e_{H} = λ_{0}/(2nH) = 70.3 nm. Therefore the quarter-wave criteria is at least valid for high-index spheres of radia greater than 15 μm. When the sphere radius is reduced from 15 μm to 0.5 μm (see Figs. 2(b,c,d), the periodicity progressively vanishes but a pseudo-period can still be detected either on S_{R} or S_{T}.

The same study was performed for a low-index (n_{B} = 1.33) YF_{3} material, as shown in Figs. 3(a,b,c,d). In this case the average pseudo-period can be roughly seen, but for large diameters it is found to be approximately: E_{B} = 119 nm. This value is close to that of the classical case given by: e_{B} = λ_{0}/(2n_{B}) = 118.9 nm. When the radius is decreased, the periodicity is lost as expected.

So we conclude in this section that similar criteria can be obtained for the in-situ monitoring of multilayers deposited on flat or concentric substrates. The quarter-wave criteria remains valid for micro-spheres overcoated with high index layers, and for diameters greater than 1μm. The case of low index material deposition appears to be more complex.

## 3. The case of specific optical functions

#### 3.1 Case of a multilayer mirror

This section is now devoted to wavelength properties of the overcoated microspheres. The case of a multidielectric quarterwave mirror is plotted in Fig. 4 for different sphere radia in the range (15 μm–0.5 μm). Materials are those of the preceding section, that are Ta_{2}O_{5} and YF_{3}. The coating design is again: M7 = Air/HBHBHBH/Glass, with λ_{0} = 633 nm.

All scattering curves S_{R} by reflection are normalized and compared to specular reflection R of the planar multilayer. We observe in Figs. 4(a,b,c) that the spectral ranges of high reflectance can be easily observed, and rather similar to those of planar multilayers, in spite of a wavelength shift and a lower rejection rate. So we conclude that the spectral profiles are similar for concentric and planar multilayer mirrors, provided that the initial sphere radius is greater than a_{0} = 2.5 μm. When this radius is reduced, resonances occur in the reflectance bandpass, and become predominant for small radia.

#### 3.2 Case of a beam splitter

Another kind of design was studied and is plotted in Fig. 5 (planar multilayers) and Fig. 6 (concentric multilayers) for a specific non quarterwave beam-splitter (BS) of design:

$$\phantom{\rule{1.0em}{0ex}}/{e}_{H}=\mathrm{45,57}\mathrm{nm}/{e}_{B}=\mathrm{43,23}\mathrm{nm}/{e}_{H}=\mathrm{10,56}\mathrm{nm}\phantom{\rule{.2em}{0ex}}/\mathrm{Glass}\phantom{\rule{10.2em}{0ex}}$$

with e_{H} and e_{B} the mechanical thicknesses of high and low index materials, respectively. These thicknesses were determined by the needle method [13]. We observe than similar optical properties are emphasized for planar and concentric stacks, provided that the sphere radius is greater than 15 μm. For lower radia, angular scattering by reflection (S_{R}) and transmission (S_{T}) are dissociated.

#### 3.3 Case of a Fabry-Perot filters

The last design concerns a Fabry-Perot filter (FP) of design:

Results are given in Fig. 7 and show strong alteration of the optical properties, even for large sphere radia. This result can be correlated to the high sensitivity of Fabry-Perot filters.

## 4. Conclusion

Numerical calculation shows that the classical quarter-wave criteria of in situ optical monitoring can still be used to control the growth of films on microspheres. This result is mainly valid for high index thin films deposited on substrate spheres with diameters greater than 1 μm. The case of low-index materials appears more complex, except when the sphere diameter is greater than 30 μm. Notice that such monitoring should require an integrated sphere or a large receiver solid angle, since all calculation is performed for integrated values of scattering by reflection or transmission. Moreover, the in-situ scattering should be measured under the assumption that multiple reflections can be neglected between microspheres.

We have also given a range of sphere diameters where similar free space optical properties can be reached for quarter-wave stacks deposited on planar and spherical substrates. In the case of mirrors, the analogy remains valid for sphere diameters down to 1 μm. For beam splitters the sphere diameters should be greater than 30 μm. The analogy disappears for Fabry-Perot filters.

## References and links

**1. **C. Deumié, P. Voarino, and C. Amra, “Overcoated microspheres for the spectral control of diffuse light,” Appl. Opt. **41**, 3299–3305 (2002). [CrossRef] [PubMed]

**2. **C. Deumié, P. Voarino, and C. Amra, “Interferential powders for the spectral control of light scattering,” Advances in Optical Thin Films 2003, Proceedings of SPIE **5250**.

**3. **H.A. Macleod, *Thin Film Optical Filters*, (London Adam Hilger, 1986).

**4. **H.C. van de Hulst, *Light Scattering by Small Particles*, (Dover Publications, Inc. New York, 1981).

**5. **J.A. Stratton, *Electromagnetic theory*,( Mc Graw-Hill Book Company, New York, 1941).

**6. **C.F. Bohren and D.R. Hufman, *Absoption and scattering of light by small particules*, (Wiley-Interscience, New York, 1983).

**7. **G. Burlak, S. Koshevaya, J. Sanchez- Mondragon, and V. Grimalsky, “E/lectromagnetic oscillations in a multilayer spherical stack,” Opt. Commun. **180**, 49–58 (2000). [CrossRef]

**8. **F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. **34**, 7113–7124 (1995). [CrossRef] [PubMed]

**9. **C. Amra, “From light scattering to the microstructure of thin film multilayer,” Appl. Opt. **32**, 5481–5491 (1993). [CrossRef] [PubMed]

**10. **C. Amra, “Light scattering from multilayer optics. Part A: investigation tools,” J. Opt. Soc. Am. **11**, 197–210 (1994). [CrossRef]

**11. **C. Amra, “Light scattering from multilayer optics. Part B: application to experiment,” J. Opt. Soc. Am. **11**, 211–226 (1994). [CrossRef]

**12. **W.J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. **19**, 1505–1509 (1980). [CrossRef] [PubMed]

**13. **A.V. Tikhonarov, M.K. Trubetsov, and G.W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. **35**, 5493–5508 (1996). [CrossRef]