Toward perfect antireflection coatings. 2. Theory Daniel Poitras and J. A. Dobrowolski
Recently we performed a numerical investigation of antireflection coatings that reduce significantly the reflection over a wide range of wavelengths and angles of incidence, and we proposed some experiments to demonstrate their feasibility. We provide a theoretical description of omnidirectional antireflection coatings that are effective over a wide range of wavelengths. © 2004 Optical Society of America OCIS codes: 310.1210, 310.6860.
1. Introduction
In 1880 Lord Rayleigh was one of the first to mathematically recognize the potential of gradual transition layers as antireflection 共AR兲 coatings.1 He stated, “No one would expect a ray of light to undergo reflection in passing through the earth’s atmosphere as a consequence of the gradual change of density with elevation.” Many decades later, his idea was implemented in devices for eliminating reflections of sound2 and light.3 However, a few years after Lord Rayleigh’s publication, Wood proved experimentally 共Fig. 1兲 that even graded atmospheric layers are, in fact, reflective at grazing angles of incidence.4,5 As magnificent as this type of reflection at grazing angles can appear in nature, it does limit seriously the performance of antireflection devices. At a time when papers on omnidirectional mirrors have become popular, we suggest a general method for enhancing the efficiency of existing AR designs at grazing angles of incidence, making them quasi-omnidirectional, with only minor effects on their properties at normal incidence. 2. Theory
We have shown numerically in a previous paper that it is possible, starting with an AR coating of the gradual transition-layer type, to arrive at a multilayer AR coating consisting of only a few layers and that is effective over a wide range of angles of incidence.6
The authors are with the Institute for Microstructural Sciences, National Research Council of Canada, 1200 Montreal Road, Ottawa, Ontario K1A 0R6, Canada. The e-mail address of D. Poitras is
[email protected]. Received 7 June 2003; revised manuscript received 23 October 2003; accepted 28 October 2003. 0003-6935兾04兾061286-10$15.00兾0 © 2004 Optical Society of America 1286
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Such solutions require layers with refractive indices that are close to that of the ambient medium. In this section we propose to modify the expression for an inhomogeneous transition layer to explain theoretically the refractive-index profiles with good highangle performances calculated in Ref. 6. We present here a set of inhomogeneous layer designs with low reflectance over a wide range of wavelengths and angles of incidence. We explore the properties of this type of coating and use them to deduce solutions that are relatively thin and consist of only a few layers. For easier correlation with the results of Ref. 6, the AR coatings that we consider in this paper correspond for the most part to an interface between two media of refractive indices 3.0 and 1.0. A. Theoretical Aspects of Optical Coatings at Oblique Angles
Procedures for the calculation of optical thin-film properties at oblique angles of incidence are well known and can be found in many textbooks.7,8 We will review and comment on the few simple rules that affect the properties of omnidirectional AR coatings. 共i兲 Snell–Descartes law. This rule, generally represented for an inhomogeneous medium by the expression d 关n共 z兲sin 共 z兲兴 ⫽ 0 , dz
(1)
indicates that large variations of n with z within the layer will induce large variations in the propagating angle of the beam. We will see later in Subsection 2.3C that large variations of 共z兲 must be avoided in omnidirectional AR coatings. 共ii兲 Effect of polarization. At oblique angles of incidence, the refractive index of each material is split
Fig. 1. Schematic drawings and pictures adapted from Wood’s publication in 18994 illustrating 共a兲 the physical cause leading to a mirage effect, 共b兲 an experimental setup for artificially creating mirage effects, and 共c兲 some pictures showing the effect obtained as the sand was heated 共from left to right兲.
into two effective refractive indices, or admittances 共Fig. 2兲: s ⫽ n共 z兲cos 共 z兲, p ⫽ n共 z兲兾cos 共 z兲.
(2)
As a consequence, the absolute value of the Fresnel coefficient 共or complex-amplitude reflection coefficient兲 will increase significantly at grazing angles 共see Fig. 3兲, and this complicates the design of omnidirectional AR coatings. 共iii兲 Phase thickness. The beam entering a thin film at an oblique angle of incidence sees a different material admittance and a different film thickness, and, as a result, a new effective optical thickness is given by 关n共 z兲dz兴 ⫽ n共 z兲dz cos 共 z兲,
Fig. 3. Polarization effect at oblique incidence: Fresnel coefficients at interfaces with different refractive-index-step sizes.
One can see that the largest effect on the optical thickness will occur at grazing angles and for low refractive-index materials. 共iv兲 Total internal reflection. Consider the thinfilm system depicted in Fig. 5. Here the refractive indices of the substrate and of the ambient media are ns, na, and those of the layers on either side of the ith interface are ni⫺1, ni. An incident beam arriving at an ith interface at which ni⫺1兾ni ⬍ 1 will be totally reflected if the angle of incidence is larger than the critical angle for this interface, given by c ⫽ arcsin共n i⫺1兾n i 兲.
(4)
(3)
where 共z兲 is given by Eq. 共1兲. Figure 4 shows the quantity sec 共z兲 for six refractive indices n and for different angles of incidence on the surface of the film.
Fig. 2. Polarization effect at oblique incidence: splitting of admittance for materials with different refractive indices.
However, this critical angle must also correspond to a realistic angle of incidence within the ambient me-
Fig. 4. Reduction factor sec共兲 for the phase thickness at oblique angles of incidence, as a function of , the angle of propagation in materials of different refractive indices. 20 February 2004 兾 Vol. 43, No. 6 兾 APPLIED OPTICS
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mogeneous layer. The quintic profile given by Eq. 共6兲 has a continuously varying refractive index at both the substrate and the ambient medium interfaces with zero first- and second derivatives at these interfaces. Another efficient profile consists of a half-period of an exponential sine 共ES兲.10 By making use of the rugate filter theory developed by Bovard11 and of a rescaling method found by Larouche and described in Ref. 12, we can write
Fig. 5. Schematic representation of a multilayer system.
dium, and so the overall critical angle at the surface i of the coating will be Cinc ⫽ arcsin共n i⫺1兾n a兲.
(5)
As can be seen from Fig. 6, the critical angle will asymptotically approach 90° as ni⫺ 1兾na gets closer to unity. As a result, abrupt step-down interfaces are allowed in ideal omnidirectional AR coatings, providing that the refractive indices of the coating materials are not exceeded by the refractive index of the ambient medium. B.
Inhomogeneous Antireflection Layer
As noted in Section 1, graded transition layers are regarded by many as excellent AR devices for normal incidence of light. Some studies have been carried out to find the refractive-index profiles n共z兲 of inhomogeneous layers that yield the lowest reflectance. Southwell found that quintic profiles of the type represented by the expression9
冋 冉冊 冉冊
n Q共 z兲 ⫽ n max ⫺ 共n max ⫺ n min兲 10 ⫹6
冉 冊册 z d
z d
3
⫺ 15
z d
4
5
(6)
act as efficient AR coatings. Here z, the metric thickness, has its origin at the substrate–film interface and d is the total metric thickness of the inho-
(
n ES共 x兲 ⫽ max exp
1 2
ln共 max兾 min兲
再 冋冉 冊
⫻ sin
册
x ⫹ 兾2 ⫺ sin共兾2兲 x tot
冎)
, (7)
where x ⫽ 兰 zn共 z⬘兲dz⬘ and x tot⫽兰 dn共z⬘兲dz⬘ are the 0
0
optical distances from the substrate and total optical thickness, respectively. When this equation is expressed in the form of a Taylor series, a 20th-order polynomial is needed to reproduce adequately the above refractive-index profile. This means that Eq.共6兲 is not a polynomial approximation for Eq. 共7兲. One should also note that, although changing the sin共兾2兲 phase term in Eq.共7兲 would alter the refractive-index matching at the interface so that the coating would no longer be a transition layer, the coating will still have reflectance minima at wavelengths corresponding to 4xtot兾共2m ⫹ 1兲, for values of m ⫽ 1, 2, 3. . . .10 A minimum thickness of the inhomogeneous transition layer is required for achieving a low reflectance. Although the exact value of this minimum thickness depends on the profile of the inhomogeneous transition layer and on the reflectance required at a target wavelength 0, it is generally accepted that a minimum optical thickness of 0兾2 is required.7 In much thinner transition layers 共or at longer wavelengths兲 in which d ⬍⬍ , the electromagnetic field does not vary significantly inside the coating, and so an abrupt interface is perceived by the incident electromagnetic radiation.3 In the opposite case in which d ⬎⬎ , the refractive index n appears to vary slowly with the layer thickness when compared with the wavelength of the incident radiation 共the WKBJ approximation holds3兲, and for this reason internal reflections within the coating can be neglected. This permits one to establish an approximate lower limit on the reflectance R, R ⬃
冉 冊 2d
2m
,
(8)
which is valid when all derivatives up to the mth order are continuous at the boundaries of the graded layer.3 C. Novel Modified Inhomogeneous Transition-Layer Profiles Fig. 6. Variation of the critical angle of incidence as a function of the ratio ni兾na. 1288
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In Fig. 3 are shown the calculated Fresnel coefficients, plotted as a function of angle of incidence, of
Fig. 7. Original and modified quintic and ES refractive-index profiles of inhomogeneous transition layers 关column 共1兲兴; the performance of the coatings as a function of angle of incidence, for a wavelength ⫽ 0.01625d 关column 共2兲兴, and as a function of wavelength, for various angles of incidence 关column 共3兲兴.
an interface in which the media on either side of the interface differ in refractive index by 0.01, 0.1, and 1.0. This diagram suggests that eliminating abrupt interfaces in multilayer coatings can reduce polarization effects. For such coatings, the Fresnel coefficients are given by 1 d s n⬘ ⬘ s ⫽ ⫽ ⫺ tan , 2 s dz 2n 2 p ⫽
(9)
1 d p n⬘ ⬘ ⫽ ⫹ tan . 2 p dz 2n 2
These expressions show that not only abrupt interfaces but also large variations of the propagation angle in the coating must be avoided to reduce the effect of polarization. According to Eqs. 共9兲, the solution should have a smooth monotonic refractive-index profile. In addition to the splitting of the admittance owing to polarization, the cos 共z兲 factor 关Eq. 共3兲兴 for the effective optical thickness will significantly deform the refractive-index profile at grazing angles. To partly overcome this problem, we introduce new inhomogeneous transition-layer profiles generated by applying a transformation to the z axis describing the refractive-index profiles of quintic and ES coatings.
To calculate the new z axis, we apply the following equation to each sublayer thickness dz 共or optical thickness dx兲: dz new ⫽
冋
dz 2
nM 1⫺ sin共 0max兲 2 n共 z兲 2
册
1兾2
.
(10)
Here nM and n共z兲 are the refractive indices of the ambient medium and of the inhomogeneous transition layer at a position z within it, respectively, and 0max is the largest angle of incidence considered at the surface of the coating. It is important to emphasize that the transformation of the axis given in Eq. 共10兲 could be applied to any other inhomogeneous refractive-index profile, such as rugate filters, which would then become similar to the power-sine rugate filters found by Bovard,13 or to the refractive-index profiles found numerically by Verly.14 Thick quintic and ES profiles, given by Eqs. 共6兲 and 共7兲, respectively, and the new modified refractive-index profiles, obtained after Eq. 共10兲 and rescaling are applied, are shown in Fig. 7.15 In Figs. 7共a3兲 and 7共b3兲 we see that the average reflectance near normal incidence can reach lower values for the ES profile than for the quintic profile at specific wavelengths.10 However, the overall reflectance of the modified quintic is lower than that of the 20 February 2004 兾 Vol. 43, No. 6 兾 APPLIED OPTICS
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Fig. 8. Variation of the reflectance at an angle of incidence of 89°, as a function of the total metric thicknesses of the original and modified quintic profiles divided by the wavelength.
modified ES at all angles 关see Fig. 7, rows 共c兲 and 共d兲兴. Figures 7共b2兲 and 7共b3兲 show the oscillatory nature of the reflectance of the ES profile, which acts as an absentee layer at several wavelengths. The distribution of the points used for numerically describing the modified refractive-index profiles can also affect the calculated spectral properties. After applying Eq. 共10兲 to an inhomogeneous transition layer, one obtains a curve with fewer points in the part of the new profile that corresponds to the lower values of the refractive index. The main effect of this is a large increase in the reflectance at the shorter wavelengths and a narrowing of the spectral region in which the coating has a good performance. This will be further discussed in Subsection 2.F on discretization. Another consequence of this is the deterioration of the performance of the AR at angles close to 89°. One can avoid these effects either by using a very large number of points 共more than 10,000兲 or by redistributing the points evenly within the new profile, through the use of an interpolation algorithm, such as a cubic spline.16 From a comparison of the performances of the quintic and modified quintic profiles at different angles of incidence shown in Figs. 7共a2兲 and 7共c2兲, it can be seen that the performance of the modified profiles at high angles is significantly improved, leading to almost perfect AR coatings. In Fig. 8 are shown plotted the average calculated reflectances at an angle of incidence of 89° of layers with quintic and modified quintic refractive-index profiles when plotted against the ratio d兾 共total metric thickness兾wavelength兲. Note that the higher this ratio, the lower the resulting average reflectance. In the following subsections, we will describe some other properties of the modified quintic profile. D. Understanding the Performance of the Modified Profiles
One can better understand the behavior of the modified quintic refractive-index profiles by looking at the 1290
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Fig. 9. Variation with thickness of the admittances for s- and p-polarized light of the modified quintic refractive-index profile of Fig. 7共c1兲, plotted for different angles of incidence.
effective profiles that are “seen” by the light at different angles of incidence. Figure 9 shows, for various oblique angles of incidence, the admittances 关from Eqs. 共2兲兴 as a function of the effective optical thickness 关given by Eq. 共3兲兴. At higher angles, the p-polarized light sees two adjacent graded layers, one of which gets thinner as the angle increases and tends toward a large abrupt interface at 89° 关see Fig. 9共l兲兴. The effect on s-polarized light is less apparent at first glance, but it is nevertheless still pronounced: As the admittance at the surface of the coating at the medium interface gets closer to zero, the value of the Fresnel coefficient at this point is substantially in-
Fig. 10. Variation of the Fresnel reflection coefficient for different departures of the admittance of the inhomogeneous layer from the medium admittance at the interface of the layer and the medium.
creased 共see Fig. 10兲. As a result, Fig. 9 demonstrates that the performance of the modified quintic refractive-index profile at high angles of incidence is, for both polarizations, mainly dominated by the contribution of the ambient-side portion of the coating to the reflectance. To reduce the overall reflectance at high angles, one has to extend significantly the thickness of that portion of the coating so that its refractive index approximates the ambient index as closely as possible. We noted earlier that an inhomogeneous transition layer starts to act as a good AR coating when its optical thickness is equal to or greater than half the target wavelength. Using this rule of thumb and assuming that the large increase in admittance seen in Fig. 9共l兲 is linear between 4 and 57.5, we found that this part of the coating only must have a metric thickness greater than 250 to obtain a low reflectance at 89°. Because the thicknesses of the nonflat portions of the inhomogeneous transition layers of the modified quintic and ES curves 关Figs. 7共c1兲 and 7共d1兲兴 are smaller than those in the corresponding original curves 关Figs. 7共a1兲 and 7共b1兲兴, one would expect the reflectances of the modified curves to be higher than those of the original curves at near-normal angles of incidence. An inspection of the second column of Fig. 7 shows that this is so. A surprising phenomenon is seen in Fig. 11, in which a large portion of the modified quintic profile is removed from the substrate side of the coating, and a new substrate material is introduced to match this new truncated profile: As shown in Fig. 11共b兲, the reflectance is almost unaffected by these changes. This has also been demonstrated numerically in our earlier paper.6 The reason for this is that the refractive-index profile continues to change gradually from that of the ambient medium to that of the new substrate, and this is true for all angles of incidence
Fig. 11. Comparison of the performances of an inhomogeneous AR coating with a modified quintic profile designed for an interface between media of refractive indices 3.0 and 1.0 and of an AR coating for a substrate of refractive index 1.5, which consists of a fraction of the same quintic profile: 共a兲 refractive-index profiles and 共b兲 average reflectance spectra for 0 – 80° and 89° angles of incidence.
and for both polarizations of the incident light. The main effect of the refractive-index truncation shown in Fig. 11共a兲 is the introduction of a discontinuity of the profile derivative at the substrate interface, and this leads to an increase of the Fresnel coefficient at this interface 关Eqs. 共9兲兴. As a result, an interference pattern becomes apparent at large angles of incidence, for which an abrupt profile is seen close to the surface 共Fig. 9兲, and the lower limit of the reflectance should increase 关expression 共8兲兴. A last comment concerning the modified quintic profile is that the refractive-index sensitivity of the ambient-side portion of the coating will not be reduced significantly in the case of a solid–solid interface, with an ambient index that is greater than 1. Although the observations made so far in this paper concerned AR coatings designed for use in air, they still hold for solid–solid inhomogeneous layer AR coatings, as also demonstrated in Ref. 6. E. Thickness Truncation of the Modified Quintic Profile at the Ambient Medium End
An obvious problem with the use of a modified quintic refractive-index profile to produce an omnidirectional AR in air is that the required low refractive index should be ideally equal to unity. This led us to the study of the effect of truncation of the thickness of the modified quintic refractive-index profile of Fig. 7共c1兲 on the resulting reflectance. Figures 9 and 10 suggest that the truncation of the profile at the ambient medium side will strongly affect the reflectance for grazing angles of incidence. Figure 12 shows the 20 February 2004 兾 Vol. 43, No. 6 兾 APPLIED OPTICS
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lower than 80° is only slightly affected even when half of the refractive-index profile is removed. Provided that the coating is graded and matches the refractive index of the emergent medium at the substrate interface, the main contribution to the reflectance will come from the interface with the ambient medium, and it will be given by R⬇ F.
Fig. 12. Effect of truncation of the overall thickness of an AR coating with a modified quintic refractive-index profile on the performance at angles of incidence of 0°, 80°, and 89°: 共a兲 refractiveindex profile and 共b兲 reflectance at ⫽ 0.01625d as a function of the amount of truncation.
effect on the calculated average reflectance 共at three angles of incidence兲 of truncating the modified quintic refractive-index profile at the ambient medium interface by different amounts. Truncation has the most significant effect on the reflectance of the coating at an angle of 89°. However, the performance at angles
冏
共d兲 ⫺ a 共d兲 ⫹ a
冏
2
.
(11)
Discretization of the Modified Profile
The solutions presented so far consisted of inhomogeneous layers in which interference does not play a significant role. The inhomogeneous layer solutions have the advantage of being effective over a broad spectral region. However, achieving a good angular performance for a wavelength 0 requires a minimum thickness of approximately 500 共Fig. 8兲, which, in most cases, is inconvenient for thin-film solutions. A practical implementation of the continuous refractive-index profiles of Fig. 7 may be difficult with most fabrication techniques. For that reason, it is desirable to approximate the continuous profiles with discrete layers. Figure 13 共column 2兲 shows the calculated average reflectance spectra for light incident at angles of 0, 80°, and 89° obtained when an inhomogeneous modified quintic refractive-index profile is approximated by 1000, 10, and 5 layers with equal thicknesses 共column 1 in Fig. 13兲. Although the refractive-index profiles are similar, the reflectance spectra of the multilayer solutions are significantly different from that of an inhomogeneous coating, par-
Fig. 13. Effect of discretization of the refractive-index profile on the average spectral reflectance: 共row a兲 1000 sublayers, 共row b兲 10 layers, and 共row c兲 5 layers and 共column 1兲 refractive-index profiles, 共column 2兲 reflectance spectra, and 共columns 3 and 4兲 admittance diagrams at wavelengths of 15 and 5 m, respectively. The thickness of the five-layer solution was reduced so that its reflectance matched more closely those shown in 共a2兲 and 共b2兲. 1292
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ticularly in the short-wavelength part of the spectrum, in which the use of discrete layers results in a deterioration of the performance of the AR. In the long-wavelength region, however, one can see a surprising similarity between the spectra. One possible way to understand the effect of discretization is through a comparison of the admittance diagrams of the coatings. A few basic rules for understanding admittance diagrams are7,8 • The starting admittance is the admittance of the substrate, s. • The admittance is given by the expression
Yj ⫽
i j sin j ⫹ Y j⫺1 cos j , i cos j ⫹ Y j⫺1 sin j j
(12)
where j represents the admittance of the jth layer, Yj is the multilayer’s admittance after the jth layer, and j ⫽ 共2兾0兲 Re共nj兲dj cos j. • The reflectance is obtained from the surface admittance Y共d兲 and the ambient admittance 0, R ⫽ 兩关Y共d兲 ⫺ 0兴兾关Y共d兲 ⫹ 0兴兩2.
Columns 3 and 4 in Fig. 13 show the admittance diagrams corresponding to the different discrete designs of column 1, for two different wavelengths. The admittance loci consist of arcs of circles, each corresponding to a homogeneous layer. In the case of the inhomogeneous modified quintic refractiveindex profile 关approximated by 1000 layers in Figs. 13共a3兲 and 13共a4兲兴, the loci tend toward straight lines between the substrate and the ambient admittance values as the wavelength decreases, and we cannot distinguish the individual arcs. In such cases, interference effects still exist, since even small internal reflections can give rise to them, but they are greatly diminished and almost unaffected by the thickness values of the individual layers. On the other hand, the contributions to the admittances of individual layers are more clearly visible in the diagrams for the coarser approximations of the inhomogeneous layer, and the thicknesses of the individual layers also have a greater impact on the interference effect and on the resulting reflectance spectra. The reason for the deterioration of the AR properties of the multilayer coatings is that the optical thicknesses of the layers will be equal to a multiple of a half-wavelength at some short wavelengths and that there they will act as absentee layers 关see the circular admittance loci in Figs. 13共b4兲 and 13共c4兲兴. The AR effect is completely destroyed at those wavelengths at which the lowest-refractive-index material is an absentee layer, and they give rise to the reflectance maxima seen in Figs. 13共b2兲 and 13共c2兲. In view of the similarity of the reflectance spectra at the longer wavelengths and of the large impact of individual layer thicknesses on the interference effect in multilayer coatings 共columns 3 and 4, Fig. 13兲, it is
Fig. 14. Discrete five-layer profile reoptimized to reduce the reflectance in the 3.0 – 4.0-m spectral region, for angles of incidence as great as 89°: 共a兲 refractive index profile and 共b兲 reflectance spectra.
reasonable to assume that the interference effect could be used to improve the performance of the AR coating in a restricted wavelength range through the proper choice of thicknesses of the individual layers. Figure 14 shows the results that were obtained when the thicknesses of the five-layer coating were refined. Of course, improvements of this kind have also been demonstrated in Ref. 6. The use of multilayer solutions of the type shown in Fig. 14 does not reduce the sensitivity of the coating to refractive-index values at the medium interface, as demonstrated in Figs. 10 and 12. If anything, the multilayer solutions are more sensitive to the refractive index of the layer at the medium interface: Its refractive index must match the ambient index as closely as possible to ensure a low Fresnel reflection coefficient at this interface 共Fig. 10兲. In addition, because thin-film interference is involved, the thicknesses of the layers must be more accurately controlled. In the special case of AR coatings for solid– solid interfaces, it is recommended that the coatings be deposited onto the lower-refractive-index substrate because then reoptimization of the remaining layers can be used to compensate for small errors in the fabrication of the most critical layers at the ambient side. This also avoids the need for an optical cement that precisely matches the refractive index of the ambient medium. G.
Effect of Dispersion and Absorption
Dispersion and absorption need to be considered when this type of AR coating is manufactured. Once again, because of the extreme sensitivity of the performance of AR coatings based on the modified quintic refractive-index profile to variations in the 20 February 2004 兾 Vol. 43, No. 6 兾 APPLIED OPTICS
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index, especially in the vicinity of the incidence medium, it is certain that dispersion will affect the performance of the coatings. A dramatic demonstration of this was the narrowing of the spectral width of the AR region in the wide-angle AR coating based on a reststrahlen material presented in Ref. 6. However, even in the visible and ultraviolet parts of the spectrum, dispersion causes the refractive index of coating materials to increase significantly at shorter wavelengths. This is also true for the critical low-refractive-index material at the ambient medium side. As a consequence, with real dispersive media, one should expect an increase in reflectance at grazing angles of incidence at shorter wavelengths, an effect that can only be partially compensated by refinement of the layer thicknesses in multilayer AR coatings. At normal incidence and short wavelengths, dispersion may also increase the reflectance of the inhomogeneous coating, since it will increase the ratio max兾min and will require a thicker transition layer at the substrate side. It has been shown numerically that absorption in the layer also results in deterioration of the performance of the AR coating, particularly if the substrate is transparent and if the transmittance should be maximized. However, more interesting is the case in which the substrate is absorbing and in which reflectance is the only parameter that needs to be minimized. We will consider here the particular case of an AR coating for amorphous silicon designed for wavelengths in the visible part of the spectrum. For such a case it has been shown numerically for normal incidence of light that a lower reflectance can be achieved when not only the refractive index 关Figs. 15 共column 1兲兴 but also the extinction coefficient 关Fig. 15 共column 2兲兴 is graded.17 Figures 15共c1兲 and 15共c2兲 provide an explanation for this observation: Owing to the complex admittance of the substrate, its matching with the ambient admittance cannot be done smoothly if k is not graded. As a result, the nonabsorbing quintic gives rise to interference and has a performance that is significantly worse than that of an absorbing quintic. An alternative approach is to use a phase-correction layer whose function is to match the substrate admittance with an admittance located on the real axis; a quintic profile can then be used to match this real admittance value to that of the ambient material. Figure 15 共column 3兲 illustrates the application of this technique and shows the resulting performance centered at a preselected wavelength 共600 nm兲. We observed that when the refractive index of the phase-correction layer is high, its thickness is small compared with the overall thickness of the coating and that the resulting performance does not change significantly with angle of incidence. 3. Conclusion
The theory of large-bandwidth quasi-omnidirectional inhomogeneous-layer AR coatings has been developed. It was shown that to obtain a wide-angle AR coating one has to distort the thickness axis of tradi1294
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Fig. 15. Effect of absorption on the performance of graded refractive-index AR coatings. 共Column 1兲 transparent inhomogeneous layer matches the real part of the substrate refractive index, 共column 2兲 layer with a graded refractive index and extinction coefficient, and 共column 3兲 graded transparent coating with a thin correction layer 关with n ⫽ Re共s兲 and d ⫽ 18.3 nm兴 at the substrate interface. Row 共b兲 shows the refractive-index profiles, and rows 共a兲 and 共c兲 show their respective reflectance spectra and admittance diagrams at the wavelength of 600 nm.
tional inhomogeneous transition-layer profiles. To obtain a coating that has a good performance for all angles of incidence as great as 89°, one would need the optical thickness of the coating to be of the order of 500 or more. During fabrication, the performance of such a coating would be especially sensitive to fabrication errors in the refractive index at the ambient side of the coating. If the inhomogeneous layer is approximated by a multilayer system and if interference effects caused by the finite Fresnel coefficients at the layer interfaces are exploited, we have shown that the number of layers, and the overall thickness of the AR coating, can be significantly reduced, providing that one is willing to reduce the effective bandwidth of the resulting AR coating and accept some deterioration in its performance at angles close to 89°. The applicability of the multilayer AR coatings in cases of solid–solid and air–solid interfaces has been numerically studied in Ref. 6. In the case of an air–
solid interface, it was shown that it should be possible to fabricate a quasi-omnidirectional AR coating efficient at a specific wavelength, by use of the optical properties of materials around their reststrahlen bands in the mid-IR. A future paper will describe the experimental manufacture of such coatings and their performances. Other approaches to produce such coatings could involve the use of structured surfaces.
9. 10.
11. 12.
References and Notes 1. Lord Rayleigh, “On reflection of vibrations at the confines of two media between which the transition is gradual,” Proc. London Math. Soc. 11, 51–56 共1880兲. 2. R. N. Gupta, “Reflection of sound waves from transition layers,” J. Acoust. Soc. Am. 39, 255–260 共1965兲. 3. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. 共NorthHolland, Amsterdam, 1966兲, Vol. 5, pp. 247–286. 4. R. W. Wood, “Some experiments on artificial mirages and tornadoes,” Philos. Mag. 47, 349 –353 共1899兲. 5. R. W. Wood, Physical Optics 共Macmillian, New York, 1934兲, p. 88. 6. J. A. Dobrowolski, D. Poitras, P. Ma, M. Acree, and H. Vakil, “Toward perfect antireflection coatings: numerical investigation,” Appl. Opt. 41, 3075–3083 共2002兲. 7. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. 共Institute of Physics, Bristol, UK, 2001兲. 8. Sh. A. Furman and A. V. Tikhonravov, Basics of Optics of
13.
14.
15.
16.
17.
Multilayer Systems 共Editions Frontie`res, Gif-sur-Yvette, France, 1992兲. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584 –586 共1983兲. D. Poitras, “Admittance diagrams of accidental and premeditated optical inhomogeneities in coatings,” Appl. Opt. 41, 4671– 4679 共2002兲. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 共1993兲. D. Poitras, S. Larouche, and L. Martinu, “Design and plasma deposition of dispersion-corrected multiband rugate filters,” Appl. Opt. 41, 5249 –5255 共2002兲. B. G. Bovard, “Graded index rugate filters: power-sine rugate structures,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 109 –125 共1993兲. P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 共1998兲. It is worth noting that when the ratio max兾min under the exponent in Eq. 共7兲 is increased, the profile of the ES transition layer becomes similar to the modified profile 关Fig. 7共d1兲兴. It may be preferable to use an uneven distribution of points, since 共i兲 its effect on the performance at short wavelengths and large angles of incidence is not critical and 共ii兲 because many layers with close indices would be hard to achieve in practice. P. V. Adamson, “Antireflecting surface coatings with continuously varying complex refractive index,” Tech. Phys. Lett. 26, 1003–1006 共2000兲.
20 February 2004 兾 Vol. 43, No. 6 兾 APPLIED OPTICS
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