Wai-Sze T. Lam
College of Optical Sciences
The University of Arizona
Tucson, Arizona 85721
Contact: waisze@email.arizona.edu
Abstract
The design and construction of optical coatings is an active branch of optical engineering. This paper explores the fundamental phenomenon of optical interference and the theory of optical interference coatings. Some applications design principles will also be introduced.
1. Introduction
Optical coatings improve the performance of a systems. It is difficult to find any modern optical system that does not have components with single or multiple layers of coatings. Scientists and engineers use the interference property of electromagnetic wave to vary the output light intensity as desire.
2. Theory
When two or more light waves
interact, they superimpose and form a resultant wave whose amplitude
depends on the phase difference of the incoming waves. If the waves interfere
constructively with a phase difference of
, where m an integer,
the resultant wave has a maximum intensity. If the waves interfere
destructively with a phase difference of
,
the resultant wave has a minimum intensity. Optical coatings manipulate this
interference characteristic to control the reflected wave intensity [1].
Consider figure 1, an incident light reaches an air-film interface. Some of it reflects back to the air and some of it transmits into the film. The transmitted part refracts in the film and reaches a film-substrate interface. Some of it reflects back to the air and some of it transmits into the substrate. The two reflected waves which are generated at the film-air interface and the substrate-film interface interact and produce a resultant reflected light waves.
Figure 1
The intensity of the net reflected light depends on the optical thickness of the film [2]. Optical thickness t of a film is given by,
where n is the
refractive index and d is the physical thickness of the film. If the
optical thickness of the film is a quarter wave, the phase difference of the two
reflected waves is
.
They become out of phase and interfere destructively. Then, the net
intensity
of the reflected wave is at minimum. If the optical thickness of the film is a half wave, the
phase difference of the two reflected waves is
. They become in-phase and interfere constructively.
Then, the net intensity of
the reflected wave is at maximum [3].
By conservation of energy, the transmittance T and reflectance R of a dielectric lossless film material is given by,
In this equation, film absorption is neglected because of its small thickness. More than one layer of coatings is used in most of the optical components to get a better performance, such as some infrared filter which has more than a hundred layers [4]. As the number of layers increases, the optical thickness created by the films is also increased. Then, the approximation of equation (2) becomes less and less accurate [2].
3. Early Development
A single layer coating is effective and inexpensive to produce. This coating with a quarter wave thickness is used in camera lenses and range finders to reduce their reflectance in some specific wavelength. The reflectance R of a coated surface is given by,
where ns and nf are the refractive index of the substrate and the coating. However, single interference film always yields insufficient results for transmission magnitude, transition sharpness and band width. Other types of multi-layer coatings which combine films of different refractive indexes and thicknesses were invented. Some of these coatings are specifically used in laser optical systems and infrared optical components which require higher transmittance and a much lower reflectance in visible range. The Rm reflectance of an m-layer lossless optical coating is given by,
where no and ns are the refractive index of the entrance medium (air) and the exit medium (substrate). M is the characteristic matrix given by,
where
nv is the refractive index of the vth
layer, 
is the phase thickness of the vth
layer. The phase thickness is given by,
where
is the wavelength of the vth layer, dv is the physical thickness of the vth
layer, and 
is the light angle in the vth
layer [4].
A dielectric mirror consists of a stacks of dielectric optical interference films. A typical dielectric mirror has eight to ten layers alternate between high and low reflective index, nH and nL, as shown in Fig. 2 [5].
Figure 2
They all have the same optical thickness of a specific quarter wavelength. At that specific wavelength, maximum reflectance is obtained. This maximum reflectance is bracketed by a region called “stopband”. (Fig.3) This stopband is created when light propagates through several pair of quarter wave stacks. Conventional metal mirrors have a constant 90% reflectance over the spectrum while dielectric mirrors can produce nearly 100% reflectance on a particular wavelength by increasing the number of layers [6].
Figure 3
4. Applications
Motion-picture projector employs optical coatings as a cold mirror and a heat reflector. A cold mirror reflects visible light and transmits thermal energy . On the other hand, a hot mirror transmits visible light and reflects thermal energy. To protect the movie film from the lamp heat, a hot mirror is placed in between the film gate and the lamp. As shown in figure 4, visible light is transmitted through and heat is reflected away from the film. To increase the visible light input to the film and let the heat from the lamp escape, the cold mirror is placed behind the lamp [2].
Figure 4
5. Conclusions
The applications of optical coatings are involved in everyday lives from antireflection coatings on eyeglasses to security coatings on bank notes [7]. Modern optical systems have more functions and require more different coatings designs. The technology of optical interference coatings has been expanded rapidly in the past fifty years and is expected it will continue to do so in the future.
References
[1] Nave, C.R., Hyper Physics, <http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/antiref.html>,
Georgia State University, 2005. (Visited 1st November, 2005)
[2] Baumeister, Philip and Pincus, Gerald, "Optical Interference Coating", Scientific American, Vol. 223, pp59, December 1970.
[3] Flory, François R., Thin Films For Optical Systems, New York: Marcel Dekker, Inc., 1995.
[4] Bach, Hans and Krause, Dieter, Thin Films on Glass, Berlin: Springer, 1997.
[5] Abelès, F. and Baumeister, P., “Multilayer reflectors with minimal dispersion of differential phase shift upon reflection”, Optical Communications, Vol. 93, pp1, September 1992.
[6] Savage, J. A., Infrared Optical Materials and their Antireflection Coatings, Bristol: Adam Hilger Ltd., 1985.
[7] Macleod, Augus, “The early days of optical coatings”, Journal of Optics A: Pure and Applied Optics, Vol. 1, pp779, September 1999.