Toward perfect antireflection coatings: numerical ... - OSA Publishing

2 downloads 0 Views 253KB Size Report
Lockheed Way, Department 2551, Building 611, Plant 10, Palm- ... 1d .12. On the basis of past calculations it is well known that, for a satisfactory performance, ...
Toward perfect antireflection coatings: numerical investigation J. A. Dobrowolski, Daniel Poitras, Penghui Ma, Himanshu Vakil, and Michael Acree

A perfect antireflection 共AR兲 coating would remove completely the reflection from an interface between two media for all wavelengths, polarizations, and angles of incidence. The degree to which this can be achieved is investigated numerically. It is shown that wideband solutions can be found provided that layers can be deposited with refractive indices that are close to that of the low-index medium. Thus realistic solutions exist for interfaces between two solid media. Narrow-band high-angle AR solutions are also possible for polarized light and for unpolarized light in the vicinity of certain reststrahlen bands. © 2002 Optical Society of America OCIS codes: 310.1210, 310.1620, 310.6860.

1. Introduction

Ever since the seminal 1966 paper by Turner and Baumeister on contiguous quarter-wave stacks,1 optical thin-film scientists have been aware of the fact that it is possible to achieve a reflectance approaching unity over an extended spectral region and for all angles of incidence. The mathematical conditions to achieve such results are found in a more recent paper.2 In the past three or four years a flurry of papers were published dealing with the design and construction of so-called omnidirectional mirrors that, according to the definition adopted for this term, reflect all light of a narrow range of wavelengths incident at all angles of incidence.3 A number of application areas for such omnidirectional mirrors have been cited. Antireflection 共AR兲 coatings are used on optical surfaces primarily to prevent loss of light and to reduce stray light produced by multiple reflections between different surfaces of an optical system in its operating spectral region. Hundreds of papers and J. A. Dobrowolski 共[email protected]兲, D. Poitras, and P. Ma are with the Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6. H. Vakil is with the General Electric Company, Corporate Research and Development, 1 Research Circle, Niskayuna, New York 12309. M. Acree is with the Lockheed Martin Company, 1011 Lockheed Way, Department 2551, Building 611, Plant 10, Palmdale, California 93599-2551. Received 24 October 2001; revised manuscript received 17 December 2001. 0003-6935兾02兾163075-09$15.00兾0 © 2002 Optical Society of America

two books4,5 are devoted to this narrow but technologically important area of thin-film optics. Of these, a number deal with the design of AR coatings for unpolarized light and angles up to 30° or 60° from the normal.3,6 – 8 The design of AR coatings for angles within one or two degrees of 90° for one polarization has also been considered.9,10 The diametrically opposite concept of an omnidirectional mirror would be that of an omnidirectional AR coating. Such a coating, when deposited onto an interface between two media, would suppress for a narrow spectral region the reflection of s- and p-polarized light incident on the interface at all angles of incidence except 90°. An omnidirectional AR coating that is effective over an extended spectral region might be called a perfect AR coating. Are perfect AR coatings conceivable? In this paper we investigate the feasibility of the numerical design of perfect AR coatings according to the above definition. Of course, the utility of AR numerical coating designs found in this study will depend on the width of the spectral region over which they are effective. Existing normal-incidence AR coating designs typically operate over spectral ranges defined by lower wavelengths ␭L and upper wavelengths ␭U for which 0.85 ⬍ ␭U兾␭L ⬍ 5.0.11 We show in this paper that, because of the range of the optical constants of the available real coating materials, the design of perfect AR coatings is a more difficult problem than the design of perfect mirrors. In Section 2 we briefly review various types of AR coatings, with particular emphasis on inhomogeneous-layer AR coatings. This is followed, in Section 3, by an outline of a numerical approach 1 June 2002 兾 Vol. 41, No. 16 兾 APPLIED OPTICS

3075

Fig. 1. Structure and effective refractive-index profiles of various types of AR coating. 共a兲–共c兲 Homogeneous single-layer, digital, and multilayer AR coatings; 共d兲–共f 兲 inhomogeneous single-layer, structured, and complex AR coatings 共from Fig. 8, Ref. 11兲.

adopted for the design of broadband wide-angle AR coatings. Some numerical results are presented in Section 4. A discussion of the prospects for the experimental implementation of variants of the solutions presented here is given in Section 5. This is followed by some general conclusions 共Section 6兲. 2. Basic Types of Antireflection Coatings

AR coatings can be classified into two basic types: those based on homogeneous layers and those that consist of an inhomogeneous layer 共Fig. 1兲.11 A.

Homogeneous Antireflection Coatings

A single homogeneous layer of refractive index n will reduce to zero the normal-incidence reflectance of an interface between a substrate of refractive index ns and a medium of refractive index nm for light of wavelength ␭ provided that its optical thickness is equal to ␭兾4 and the refractive indices satisfy the relation n ⫽ 共ns nm兲0.5 关Fig. 1共a兲兴. When nm ⫽ 1.0, this relationship can be satisfied only with dense films of existing coating materials provided that ns ⬎ 2.0. Most optical glasses have indices that are lower than this. A lower effective index n of the material is achieved when the layer is porous or when the layer is patterned 关Fig. 1共b兲兴. In the latter case the dimensions of the patterns must be less than the wavelength of light. Finally, through use of multiple thin layers, it is possible to obtain zero reflectance at one or more wavelengths even if the refractive-index relationship given above is not satisfied 关Fig. 1共c兲兴. B.

Inhomogeneous Antireflection Coatings

It is also possible for one to reduce the reflectance of the interface between a substrate and a medium by depos3076

APPLIED OPTICS 兾 Vol. 41, No. 16 兾 1 June 2002

iting onto the substrate an inhomogeneous layer with a refractive index that varies gradually from ns to nm 关Fig. 1共d兲兴.12 On the basis of past calculations it is well known that, for a satisfactory performance, the optical thickness of the inhomogeneous layer must exceed at least 1␭U or 2␭U.13 But the overall thickness of the layer is not critical: The primary reason why an inhomogeneous layer acts as an AR coating is that, in essence, in such a layer the large contribution to the reflection from abrupt interfaces is removed. However, inhomogeneous layers are more difficult to deposit with precise control. It is also well known that the spectral properties of an inhomogeneous layer can be approximated by use of a number of thin layers of equal optical thickness provided that ␭L is appreciably larger than the wavelength for which the optical thickness of the individual layers is a half-wave. The performance of the resulting multilayer will not be much worse than that of the original inhomogeneous layer, provided that the number of sublayers is large enough so that the refractive-index difference between adjacent layers is small compared with 共ns–nm兲. When this is so, the reflectances at the interfaces of the sublayers remain small. When nm ⫽ 1.0, it is possible to achieve zero reflectance with a structured coating of index ns, provided its thickness exceeds at least one or two waves and that the lateral dimensions of the patterns are less than the wavelength of light 关Fig. 1共e兲兴. When these conditions are satisfied the effective medium theory can be applied and the material–air structure can be represented by a series of thin films with refractive indices that vary gradually from unity 共air兲 to ns, the index of the substrate, as suggested in the above paragraph. For more details on basic types of antireflection coating, see Ref. 11. 3. Basic Design Approach

In this paper we are mainly concerned with the numerical design of AR coatings for unpolarized light incident upon a surface at angles of incidence ranging from 0° to 90°. We arbitrarily selected a substrate of refractive index 3.00 and the 5.0 – 8.0-␮m wavelength region. Figures 3 and 5 show calculated results that, unless otherwise stated, are based on nondispersive optical constants. The introduction of dispersion will not materially affect the results as long as the materials that are eventually chosen to implement such designs do not have absorption bands in the vicinity of the spectral range of interest. Although the calculations presented apply to a substrate of refractive index 3.00 and the spectral region extends from 5.0 to 8.0 ␮m, the general approach presented here should be valid for the design of similar AR coatings for substrates with other refractive indices and for other spectral regions. Figure 2 shows the angular reflectance for p- and s-polarized light of a single untreated 3.00–1.00 interface. Above 70° the reflectance rises sharply and there is appreciable polarization splitting. These two effects are the main reasons why it is difficult to design AR coatings that are effective over a wide range of angles.

Let nd0 be the normal-incidence optical thicknesses of the N sublayers. Then nd␾ is the effective optical thickness of the sublayer of index n for an angle of incidence ␪ given by the expression nd ␾ ⫽ nd 0 cos(␾) The ratio of the normal-incidence to obliqueincidence optical thicknesses are given by the expression nd 0/nd ␾ ⫽ sec(␾).

Fig. 2. Reflectance as a function of angle of incidence for s- and p-polarized light of an interface between a substrate of index 3.00 and air.

Figure 3共a兲 represents the refractive-index profile of a conventional step-down broadband three-layer AR coating for a surface of refractive index 3.00 designed for normal incidence of light. The refractive indices and metric thicknesses of this and other layer systems described in this paper that consist of only a few layers are also given in Table 1. Figure 3共b兲 shows the angular variation of the average reflectance Rav of this coating for p- and s-polarized light at 6.5 ␮m. Above 70° the calculated average reflectance exceeds 0.10 and rises steeply for higher angles. Figure 3共c兲 depicts the average reflectance for unpolarized light of the same coating for light incident at angles of 30°, 50°, 60°, 70°, 80°, and 85° in the spectral region 5.0 ␮m ⬍ ␭ ⬍ 8.0 ␮m. 共The reflectance at 90° of all surfaces is unity.兲 Similar coatings can be manufactured for the visible and for those wavelengths in the near-infrared spectral region for which stable, approximately quarter-wave-thick layers of MgF2 can be produced. These data are presented here so that the calculated angular performance of the subsequent designs can be compared with that of a typical normal-incidence AR coating design. It has been observed earlier that AR coatings based on inhomogeneous layers are less sensitive to angle of incidence.14 For this reason we start our calculations with an inhomogeneous-layer model. We assume for simplicity that the refractive index of this inhomogeneous layer varies linearly between the refractive indices nm and ns, the indices of the medium, and of the substrate. Let us further assume that this linearly inhomogeneous layer can be adequately modeled by an N-layered multilayer structure in which the refractive-index difference ␦n between any two adjacent sublayers is 共ns ⫺ nm兲兾共N ⫹ 1兲. For an angle of incidence ␪ the angle of refraction ␾ within a thin sublayer of index n in this inhomogeneous layer is given by Snell’s law: n M sin(␪) ⫽ n sin(␾).

The variation of this ratio with angle of incidence ␪ for different sublayer refractive indices n is shown in Fig. 4. For sublayers of refractive indices n close to nm, the value of this ratio for high values of ␪ lies between 10 and 100, whereas for sublayers of refractive indices n close to the value of ns the effective thickness is not much different from that for normal incidence. It is obvious from Fig. 4 that the thickness of an inhomogeneous-layer AR coating designed for use at angles greater than 70° will need to be much larger than that for a coating intended for use at normal incidence only. In view of the above, the following numerical design approach was adopted for the design of the broadband wide-angle AR coatings. The starting point was a thick inhomogeneous AR coating approximated by a large number of homogeneous layers of equal optical thicknesses defined for an oblique angle of incidence. The difference ␦n between the refractive indices of two adjacent sublayers was small and constant. Unless otherwise stated, all media and layers were assumed to be nondispersive and nonabsorbing. To reduce the overall thickness of the final AR coating design and make it more practical, the thicknesses of the layers were then refined, thin and half-wave layers were removed, and adjacent layers with close indices were consolidated whenever these operations did not have too much of a detrimental effect on the AR performance, angular variation, or bandwidth of the coating. Nevertheless, these operations did result in somewhat smaller angular and wavelength ranges of the final design. 4. Numerical Results

Figure 3共d兲 shows the refractive-index profile of a 200-layer simulation of an inhomogeneous-layer AR coating for a 3.00 –1.00 interface. The refractive indices of adjacent layers in this system differ from one another by 0.01. All the sublayers have one quarterwave optical thickness 共QWOT兲 at a wavelength of 5.5 ␮m and for an angle of incidence of 85°. The total thickness of the AR coating is 369.9 ␮m. Note that because the optical thicknesses were defined to be equal for an angle of incidence of 85°, a straight line can no longer approximate the refractive-index profile. First, the average reflectance for s- and p-polarized light of this system was calculated as a function of angle for light of a 6.5-␮m wavelength. The average reflectance remains less than 0.05 for all angles up to approximately 85° 关Fig. 3共e兲兴. Figure 1 June 2002 兾 Vol. 41, No. 16 兾 APPLIED OPTICS

3077

Fig. 3. Refractive-index profiles 共column 1兲, angular variation of the average reflectance for unpolarized light 共column 2兲, and spectral variation of the average reflectance for 30°, 50°, 60°, 70°, 80°, and 85° 共column 3兲 of AR coatings for different interfaces. 共a兲–共c兲 Conventional 3-layer AR coating for a 3.00 –1.00 interface; 共d兲–共f 兲 200-layer AR coating for a 3.00 –1.00 interface; 共g兲–共i兲 47-layer AR coating for a 1.48 –1.00 interface; 共j兲–共l兲 single-layer AR coating for a 3.00 –1.48 interface; 共m兲–共o兲 6-layer AR coating for a 3.00 –1.48 interface; 共p兲–共r兲 53-layer AR coating for a 3.00 –1.00 interface; 共s兲–共u兲 7-layer AR coating for a 3.00 –1.00 interface; 共v兲–共x兲 4-layer AR coating for a 3.00 –1.00 interface.

3共f 兲 shows that, for angles of incidence 30°, 50°, 60°, 70°, and 80°, the average reflectance is less than 0.01 across the whole 5.0 – 8.0-␮m spectral region. Note that the lowest refractive index in the design of Fig. 3共d兲 has a value of 1.01. Perhaps it is appropriate to mention here that Pohlack earlier recognized the potential of layers of refractive index close to unity for the single-layer AR coating of glass–air interfaces at oblique angles.7 Next, the profile of the refractive index is truncated so that it forms a 47-layer AR coating on a substrate of index 1.48 关Fig. 3共g兲兴. It can be seen from Figs. 3共h兲 and 3共i兲 that the average reflectance of the resulting system is quite comparable to that of Figs. 3共e兲 and 3共f 兲. Therefore step-down AR coatings of the same performance can be designed in this way for nonabsorbing substrates of any refractive index provided that the refractive indices of adjacent layers and the media differ from one another by 0.01. Additional calculations have shown that an increase and decrease in the value of the QWOT wavelength from the one cited above resulted in a deterioration of the performance. When a discrete3078

APPLIED OPTICS 兾 Vol. 41, No. 16 兾 1 June 2002

layer model replaces the inhomogeneous layer, interference effects take place at the abrupt interfaces between the individual layers. The deterioration becomes serious with longer QWOT wavelengths because the optical thickness will approach a half-wave and then the layers will become absentee layers. The performance deteriorates for shorter QWOT wavelengths because then the overall thickness of the system will no longer be large enough. The performance is degraded or improved when the refractiveindex difference between adjacent layers in the homogeneous-layer simulation is increased or decreased because this impacts on the accuracy of the approximation of the inhomogeneous layer. There are two main reasons why solutions of the type shown in Figs. 3共d兲 and 3共g兲 cannot be readily implemented at this time. One is that they call for solid films with refractive indices that are smaller than any that can be produced at this time, a problem that we address in Section 5. The other is the large overall thickness of these layer systems. We begin by addressing the second problem. It is not unreasonable to expect that, in a multi-

Fig. 3 (Continued).

layer in which interference effects are used, the many construction parameters 共thicknesses, refractive indices of all the layers兲 could be better exploited to achieve good AR coatings. If one were willing to design a coating for a somewhat narrower spectral region, one would expect a better solution for the same overall coating thickness or, preferably, a thinner solution of like performance than the one that can be obtained with an inhomogeneous-layer AR. Consider, for example, the design of an AR coating for a 3.00 –1.48 interface. A conventional singlelayer design 关Fig. 3共j兲兴 based on the relation n ⫽ 共ns nm兲0.5 has an average reflectance that is less than 0.01 for normal incidence across the spectral region of interest, but its performance at higher angles leaves a lot to be desired 关Figs. 3共k兲 and 3共l兲兴. However, if one or two layers with refractive indices close to 1.48 are used, it is rather easy to design a broadband wide-angle AR coating. The refractive-index profile of one such six-layer solution is shown in Fig. 3共m兲. Its angular and specular performance 关Figs. 3共n兲 and 3共o兲兴 compares with that of the system of Figs. 3共e兲 and 3共f 兲, yet the overall optical thickness of the system is only 17.9 ␮m. If this system is placed in series with the AR coating for the 1.48 –air interface of Fig. 3共g兲, a 53-layer system results 关Fig. 3共p兲兴. The calculated angular performance of this design

关Figs. 3共q兲 and 3共r兲兴 compares with that of the original 200-layer system shown in Figs. 3共d兲–3共f 兲, yet its overall optical thickness of approximately 139 ␮m is only a third of that of the original coating. There was little doubt that it should be possible to further reduce the overall thickness and the number of layers by further refinement. The solution of Fig. 3共p兲 served as a starting design. The merit function used in these calculations specified a target of zero average reflectance for s- and p-polarized light at wavelengths 5.0 ␮m ⬍ ␭ ⬍ 8.0 ␮m in steps of 0.1 ␮m and for angles of incidence 75°, 80°, and 85°. As stated above, during the thickness refinement, thin and half-wave layers were removed and layers with close refractive indices were consolidated whenever these operations did not have too much of a detrimental effect on the resulting performance. One solution was found in which the number of layers was reduced from 53 to 7 and in which the optical thickness was reduced from 120 ␮m to approximately 18.5 ␮m. The refractive-index profile of this system is shown in Fig. 3共s兲 and the spectral and angular performance in Figs. 3共t兲 and 3共u兲. The performance is comparable to that of the original inhomogeneouslayer system of Fig. 3共d兲, but the solution still calls for low refractive indices of 1.01 and 1.07. The four-layer solution depicted in Figs. 3共v兲–3共x兲 1 June 2002 兾 Vol. 41, No. 16 兾 APPLIED OPTICS

3079

Table 1. Construction Parameters of Some of the Systems

Figure 3共a兲 Layer Number

n

d 共␮m兲

Substrate 1 2 3 4 5 6 7 Medium

3.0000 2.3902 1.7321 1.3800

0.6799 0.9382 1.1775

1.0000

Figure 3共m兲 n

d 共␮m兲

3.0000 2.8000 2.5000 2.0000 1.6000 1.5000 1.4900

0.4706 0.4418 0.9778 2.1491 2.9897 3.7697

1.4800

Figure 5共a兲 Layer Number

n

d 共␮m兲

Substrate 1 2 3 4 5 6 Medium

3.0000 2.3000 1.3800 2.5000 1.3800 2.4000 1.3800 1.0000

0.1300 2.1094 1.0122 2.3670 2.4518 0.9809

n

d 共␮m兲

3.0000 2.8000 2.5000 2.0000 1.6000 1.3000 1.0700 1.0100 1.0000

0.3668 0.3841 0.7856 1.4003 1.7183 3.0972 7.0778

Figure 5共d兲

Figure 3共v兲 n

d 共␮m兲

3.0000 2.3000 1.4800 1.1000 1.0200

0.6596 1.1616 2.1789 5.4930

1.0000

Figure 5共g兲

Figure 5共j兲

n

d 共␮m兲

n

d 共␮m兲

n

d 共␮m兲

3.0000 1.3800 2.5000 1.3800 2.4000 1.3800

2.8080 0.8725 2.3624 0.7519 1.4141

3.0000 2.3000 1.4800 2.5000 1.3800 2.4000

0.6213 0.6594 0.7743 2.3454 0.5323

3.0000 2.5000 2.0000 1.6000 1.3800 SiO2

0.6025 1.0088 0.8828 1.4867 6.0220

1.0000

has a still smaller overall thickness 共approximately 11.2 ␮m兲 and is based on the low refractive indices 1.02 and 1.10, but, as a result, its performance is significantly worse. 5. Present and Future Possibilities of Experimental Implementation

In Section 4 it was shown that, in theory, it is almost possible to design perfect AR coatings consisting of relatively few layers and a small overall optical thickness. For example, it was shown that the high-angle spectral and angular performance of the conventional one-layer AR coating for a 3.00 –1.48 interface can be much improved by use of the six-layer AR coating shown in Figs. 3共m兲–3共o兲. The problem is that, for the special 共and most important兲 case when the incident medium is air, this type of a solution requires layers with refractive indices that are close to unity. For this reason the implementation of perfect AR coatings of the above type for substrate–air interfaces is much more difficult. Several different approaches can be considered. It is well known that with the Herpin equivalent index concept it is possible to design symmetrical layer combinations that, at one wavelength and for normal incidence of light, behave like a single layer with an equivalent index that lies outside the range of the actual materials.15–17 However, there are two problems with use of this approach to simulate a layer of low refractive index. The first is that the dispersion of such an equivalent refractive index is high. As a result any wide-angle AR coating designed on the basis of this approach is effective only over a narrow range of wavelengths and over a narrow range of angles. The second problem is that, for 3080

Figure 3共s兲

APPLIED OPTICS 兾 Vol. 41, No. 16 兾 1 June 2002

1.0000

1.0000

nonnormal angles of incidence, different layer combinations are required to achieve the same Herpin equivalent index for the two planes of polarization of the incident light. In Figs. 5共a兲–5共i兲 are shown the refractive-index profiles and calculated performances of three separate high-angle AR coatings designed for a wavelength of 6.5 ␮m and for the angular range of 75°– 85° for unpolarized as well as for s- and p-polarized light. However, it should be stated here that there exist other methods for the design of narrow-band, high-angle AR coatings that are not based on use of a step-down design as a starting point. The authors are grateful to one of the reviewers for contributing an 11-layer design for the 6.5-␮m wavelength and angles of incidence in the 75°– 85°

Fig. 4. Cosecant of the angle of refraction in layers of different refractive indices as a function of angle of incidence.

Fig. 5. Refractive-index profiles 共column 1兲, angular variation of the reflectance 共column 2兲, and spectral variation of the reflectance for indicated angles of incidence 共column 3兲 of AR coatings for 3.00 –1.00 interfaces. 共a兲–共c兲 Herpin equivalent index design for unpolarized light; 共d兲–共f 兲 Herpin equivalent index design for s-polarized light; 共g兲–共i兲 Herpin equivalent index design for p-polarized light; 共j兲–共l兲 AR coating for unpolarized light based on use of reststrahlen materials.

range with a performance that is quite superior to that of the coating depicted in Figs. 5共a兲–5共c兲. Another approach to the implementation of omnidirectional AR coatings could be based on use of reststrahlen materials. These are inorganic materials in which the dispersion of the optical constants is large in the neighborhood of wavelengths that excite lattice vibrations and give rise to sharp absorption bands. In particular, these materials also have narrow spectral regions in which the refractive index assumes values less than unity whereas the extinction coefficient of the material is still quite small. Many reststrahlen materials exist with absorption bands in the near- to far-infrared spectral regions.18 For example, Fig. 6 shows a plot of the optical constants of SiO2 as published in Palik’s book.19 At the wavelength of approximately 7 ␮m the refractive index of this material has a value of 1.00 and the extinction coefficient is still quite small. These dispersive optical constants were used to design a four-layer omnidirectional AR coating for a substrate of index 3.00 and for the 7.2-␮m wavelength. In Figs. 5共j兲–5共l兲 are shown the refractive-index profile and the angular and specular performances of the resulting coating. The refractive indices of the re-

maining three layers were assumed to be nondispersive. The calculations indicate that, at that particular wavelength, the calculated reflectance for unpolarized light is less than 0.05 for all angles lower than 85°. This is a remarkable performance. In addition to the fact that such AR coatings are effective over only a narrow wavelength range, the other problem with this approach is that there is a limited

Fig. 6. Optical constants of SiO2 in the reststrahlen region. 1 June 2002 兾 Vol. 41, No. 16 兾 APPLIED OPTICS

3081

number of wavelengths in the infrared for which suitable reststrahlen materials exist. However, it may be possible that a limited tuning of the AR wavelength could be achieved through the deposition of layers that are mixtures of two or more materials. It may be that in the future some of the options depicted in Fig. 1 can be applied to the simulation of layers with low refractive indices. Homogeneous porous SiO2 and MgF2 layers of low refractive index for AR coatings have been produced in the past with, for example, deposition from solgel solutions or foams or by evaporation at oblique angles. For an introduction to this topic, see Fig. 8 and the accompanying text and citations in Ref. 11. Patterned homogeneous layers of the type shown in Fig. 1共b兲 can be produced either through photolithographic or laser etching techniques. However, in both cases it will be difficult to achieve stable and robust low-density layers with a metric thickness that is comparable to that of the remaining layers of the AR coating and with the lateral dimensions required for AR coatings of the type described in this paper. Inhomogeneous porous layers 关Fig. 1共d兲兴 for AR coatings have been produced in the past, for example, by various additive, subtractive, additive– subtractive, and replication methods. Patterned inhomogeneous layers 关Fig. 1共e兲兴 have been formed by the deposition of microspheres of transparent materials, by replication, by photomechanical means, or by evaporation at oblique angles. This topic is discussed in Fig. 19 and the corresponding text of Ref. 11. It may well be easier to produce inhomogeneous layers than to produce or simulate homogeneous layers of low index. If this were to be the case, then a logical solution to the problem would be to design a hybrid AR coating composed of an inhomogeneous layer combined with homogeneous films. 6. Conclusion

It has been shown that, with an inhomogeneous-layer AR coating model used as a starting point, it is almost possible to design perfect AR coatings provided that layer materials are available with refractive indices that are close to those of the incident medium. Numerical examples have been presented for 3.00 –1.00, 1.48 –1.00, and 3.00 –1.48 interfaces and for the 5.0 – 8.0-␮m spectral region, but it is obvious that similar results could also be obtained for any other interface. In this paper the design goal was to obtain solutions in which the reflectance for unpolarized light in the 5.0 – 8.0-␮m wavelength range stays below 0.02 and 0.05 for angles of incidence of up to 85° and 89°, respectively. In fact, in the solutions shown in this paper, the reflectances stay only below 0.05 for angles up to 85°. This is because the solutions consist of only a few layers. It will be shown in a future paper that better numerical results for glancing angles are possible if smaller index increments and more layers are used. Similar AR coatings to the ones shown in this paper can, of course, be designed for other spectral regions. 3082

APPLIED OPTICS 兾 Vol. 41, No. 16 兾 1 June 2002

With present-day technology it should be possible now to manufacture perfect AR coatings of the type depicted in Fig. 3共m兲 for solid–solid interfaces. The broadband wide-angle designs for substrate–air interfaces cannot be implemented at the present time because there are no conventional coating materials that would yield solid films with sufficiently low refractive indices when deposited by conventional deposition methods. It should also be possible now to implement AR systems for substrate–air interfaces that are based on the Herpin equivalent index concept 关Figs. 5共d兲 and 5共g兲兴 and on reststrahlen materials 关Fig. 5共j兲兴. However, the resulting AR coatings will be effective over only a narrow range of wavelengths. In a subsequent paper a more rigorous theoretical treatment of the properties and the design of broadband wide-angle AR coatings will be presented. In that paper we will also describe the manufacture of prototype AR coatings for solid–solid interfaces and of a reststrahlen-material-based AR coating for a substrate–air interface. This research was first presented at the Optical Society of America’s Eighth Topical Meeting on Optical Interference Coatings held in Banff, Canada, 15–20 July 2001.20 References 1. A. F. Turner and P. W. Baumeister, “Multilayer mirrors with high reflectance over an extended spectral region,” Appl. Opt. 5, 69 –76 共1966兲. 2. K. V. Popov, J. A. Dobrowolski, A. V. Tikhonravov, and B. T. Sullivan, “Broadband high-reflection multilayer coatings at oblique angles of incidence,” Appl. Opt. 36, 2139 –2151 共1997兲. 3. Y. Fink, J. N. Winn, S. Fan, S. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679 –1682 共1998兲. 4. I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Prosvetlenie Optiki 共Antireflection Coating of Optical Surfaces兲 共State Publishers of Technical and Theoretical Literature, Moscow, 1946兲. 5. T. Sawaki, “Studies on anti-reflection films,” Research rep. 315 共Osaka Industrial Research Institute, Osaka, Japan, 1960兲. 6. P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta 8, 105–119 共1961兲. 7. H. Pohlack, “Zum Problem der Reflexionsverminderung optischer Gla¨ ser bei nichtsenkrechtem Lichteinfall,” in Jenaer Jahrbuch 1952, P. Go¨ rlich, ed. 共Fischer, Jena, Germany, 1952兲, pp. 103–118. 8. J. A. Dobrowolski and S. H. C. Piotrowski, “Refractive index as a variable in the numerical design of optical thin films,” Appl. Opt. 21, 1502–1510 共1982兲. 9. J. C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 共1989兲. 10. J. C. Monga, “Double-layer broadband antireflection coatings for grazing incidence angles,” Appl. Opt. 31, 546 –553 共1992兲. 11. J. A. Dobrowolski, “Optical properties of films and coatings,” in

12.

13. 14. 15. 16.

Handbook of Optics, M. Bass, ed. 共McGraw-Hill, New York, 1995兲, pp. 42.19 – 42.34. J. W. S. Rayleigh, “On reflections of vibrations at the confines of two media between which the transition is gradual,” Proc. London Math. Soc. 11, 51–56 共1880兲. H. A. Macleod, Thin Film Optical Filters 共Institute of Physics, Bristol, UK, 2001兲, p. 154. M. J. Minot, “The angular reflectance of single-layer gradient refractive-index films,” J. Opt. Soc. Am. 67, 1046 –1050 共1977兲. L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42, 806 – 810 共1952兲. M. C. Ohmer, “Design of three-layer equivalent films,” J. Opt. Soc. Am. 68, 137–139 共1978兲.

17. A. Thelen, Design of Optical Interference Coatings 共McGrawHill, New York, 1988兲, pp. 41–55. 18. J. A. Dobrowolski, “Coatings and filters,” in Handbook of Optics, W. G. Driscoll and W. Vaughan, eds. 共McGraw-Hill, New York, 1978兲, pp. 8.95– 8.102. 19. H. R. Philipp, “Silicon dioxide 共SiO2兲 共glass兲,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. 共Academic, Orlando, Fla., 1985兲, pp. 749 –764. 20. J. A. Dobrowolski, D. Poitras, P. Ma, M. Acree, and H. Vakil, “Towards perfect antireflection coatings,” in Optical Interference Coatings, Vol. 63 of OSA Trends in Optics and Photonics Postconference Digest 共Optical Society of America, Washington, D.C., 2001兲, paper TuA2–1-2.

1 June 2002 兾 Vol. 41, No. 16 兾 APPLIED OPTICS

3083