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Dual-band windows (DBW) transparent for fundamental and second harmonic generation frequencies
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Thin Solid Films 516 (2008) 5470 – 5473 www.elsevier.com/locate/tsf
Dual-band windows (DBW) transparent for fundamental and second harmonic generation frequencies Alexandre Ghiner a , Mikhail Skvortsov b , Gregory Surdutovich b,⁎ a
b
Physics Department, UFMA, CEP 65080-040, Sao Luis, MA, Brazil Laser Physics Institute of Siberian Branch of RAS, Novosibirsk 630090, Russia Available online 13 July 2007
Abstract Using the known “looking glass” transformation property (z → 2π − z, y → 2π − y) of the optical phase thickness z and y of matching layers of two-layer anti-reflection coating, together with the fact that optical characteristics of any film do not change after addition of a half-wavelength layer, we designed dual-band anti-reflection coatings transparent at any preset wavelengths λ1 and λ2. On the basis of this result common fractional anti-reflection coatings for second and higher harmonics generation using dispersionless coating materials are developed. Explicit analytical relationships between refractive indices of the layers and substrate are deduced. Since for second harmonic generation the dispersion of materials may be a factor we show how to compensate the known dispersion of the coating materials by special choice of dispersion of a suitable substrate. © 2007 Elsevier B.V. All rights reserved. Keywords: Anti-reflection coatings; Harmonics generation
1. Introduction A good anti-reflective coating is vital for solar cell performance, nonlinear optics and many other applications[1–3]. For a substrate with low refractive index, s, often it is very difficult or even impossible to find for manufacture of a single-layer antireflection pffifficoating an optical material with still lower refractive index, s. The conventional adoption of the double-layered coatings effectively reduces limitations on values of the refractive indices of a suitable pair of the coating materials. Two remaining free parameters of a double-layer coating, i.e., thicknesses of the layers, after selection of suitable materials, are always enough to attain null reflectivity of the double-layered interface. Moreover, we will show that under this generally accepted approach not all potentialities of such coatings are utilized and will demonstrate how to extend these potentialities. Limiting ourselves to the case of the normal light incidence one finds that any two-layer anti4ktdt n reflection coating with phase thickness z ¼ 4knd k and y ¼ k (dn and dt physical thickness of the first and second layers,
⁎ Corresponding author. Fax: +7 383 333 2067. E-mail address: [email protected] (G. Surdutovich). 0040-6090/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2007.07.026
correspondingly, λ is the wavelength), always allows the following “looking-glass” transformation: z+ → 2π − z−, y+ → 2π −y−. In each of such pairs of the solution (z−,y+) or (z+,y−) one of the layers is always thicker and another thinner than quarterwavelength thickness π [4–6]. An example of phase thickness dependences is shown in Fig. 1. Equations for lower (z−,y− ≤ π) and upper (z+,y+ ≥ π) branches have the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs 1Þðt 2 sÞ z ¼ 2 arctan n ðn2 sÞðt 2 n2 sÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ðs 1Þðn2 sÞ y ¼ 2 arctan t ðt 2 sÞðt 2 n2 sÞ yþ ¼ 2k y ; zþ ¼ 2k z ; where n, t and s are indices of the first, second layers and substrate, correspondingly. 2. Double-band anti-reflection coatings Duality of the solutions prompts a hypothesis that one may to yield zero reflectance at two different wavelengths λ1 and λ2 (assume λ1 N λ2). It means that passing from solution (z−(λ1), y+(λ1)) to (z+(λ2), y−(λ2)), under change of wavelength from λ1
A. Ghiner et al. / Thin Solid Films 516 (2008) 5470–5473
5471
thickness always may be realized under p = 0 condition. With decreasing λ on the road to the “looking-glass” solution z−(λ2), y+(λ2) the value z−(λ2) may surpass 2π, i.e. reach some zone with p ≥ 1, whereas the phase thickness y occurs to be larger π. Then the solution z−(λ2) + 2πp, y+(λ2) may be realized. The condition for such realization is yþ ðk2 Þ z ðk2 Þ þ 2kp k1 ¼ ¼ ¼ gN1 y ðk1 Þ zþ ðk1 Þ k2 Fig. 1. Upper (z+, y+ ≥ π) and lower (z−, y− ≤ π) branches of the phase thicknesses z (thin lines) and y (thick lines) of the first and second layers as function of the refractive index t of a second layer under fixed value of the first layer index n = 1.92 and substrate's index s = 1.515. The “branch-converging” pffiffi point y = z = π corresponds to t ¼ n s. This graphic relates to the values n ant t into domain I (see Fig. 3).
to λ2 while maintaining null reflectivity, one should guarantee implementation of the equality yþ ðk1 Þ z ðk1 Þ k2 ¼ ¼ : y ðk2 Þ zþ ðk2 Þ k1
ð2Þ
Since y+, z+ ≥ π and y−, z− ≤ π the first fraction always should be ≥ 1, while the second fraction should be ≤1. Therefore, at first sight it seems that unique solution is the trivial one: y = z = π, i.e. λ1 = λ2 (see the lower-upper “branchconverging” point in Fig. 1). It is possible, however, to overcome this obstacle if take into consideration such evident fact that optical properties of any layer do not change after of the quarter-wavelength increase of its phase thickness. Therefore, the optical characteristics of any film with phase thickness z are identical to characteristics of a film with z + 2πp, where p is an integer. Let us choose wavelengths with scaling ratio parameter g ¼ kk12 . If one starts from sufficiently long wavelength λ1 then the solution z+(λ1), y−(λ1), with lesser total double-layer
Fig. 2. The dependences of the refractive indices n and t on the scaling parameter γ. The solution z = 2y corresponds to the branch p = 1 and located in the interval 1 b γ b 3, whereas solution z = 3y lies in the interval 3 b γ b 5 and corresponds to branch p = 2. These values are acceptable for practical realizations. The points a and b under γ = 2 correspond to the values n and t given in Table 1 (lines 3 and 4).
ð3Þ
where y−(λ1) + y+(λ2) = 2π, z+(λ1) + z−(λ2) = 2π(p + 1). In terms of two auxiliary parameters α and β Eqs. (1),(3) acquire the form n t2 s ¼a t n2 s
nt
s1 ¼ b: t 2 n2 s
ð4Þ
These parameters depend on selected for realization scaling parameter γ and branch p. a ¼ tank
gp k =tan ; gþ1 gþ1
b ¼ tanp
gp k tan : gþ1 gþ1 ð5Þ
When p = 1 (the first branch) between these parameters springs up the relationship β = α + 2 [7]. Then parametric equations for n and t take the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 þ ðs 1Þ2 þ 4b2 s sðx þ aÞ 2 t ¼ ; n ¼ xt; where x ¼ : xð1 þ axÞ 2bs ð6Þ In Fig. 2 are shown the dependences of the refractive indices n and t on the scaling parameter γ for substrate with a given refractive index s. The value of the variable parameter x (as function γ) strongly depends on the permissible domain pffiffi in (n,t) plane (see Fig. 3). In the domain I the value xb1= s, in the pffiffi domain II 1= sbxb1 and in the domain III x N 1. In domains I and III exist only solutions z = 2y (branch p = 1) and z = 3y (p = 2), correspondingly. In the domain II there are both of these
Fig. 3. The allowable domains (I–III) of the refractive indices n, t of a pair of optical materials for a fixed substrate s. The most practically interesting for pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi pffiffiffi DBWs is the domain II, where sbtbn s and t= sbnb s.
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A. Ghiner et al. / Thin Solid Films 516 (2008) 5470–5473
Table 1 The examples of DBW for fundamental and second harmonic generation s
t0
n0
dt/λ1
dn/λ1
1.5 2.4 2.6 (a) 4 (b)
1.3101 1.7888 1.8857 2.49924
1.1449 1.3416 1.3787 1.60048
0.12721 0.09317 0.08838 0.06668
0.1456 0.1242 0.1208 0.1041
band two-layer coating for dispersionless materials. In Table 1 are given some examples (all into domain II in Fig. 3) of the dual-band anti-reflection for fundamental and second harmonic generation frequencies coatings deposited onto the different substrates. In Fig. 4 an example of DBW (last line in Table 1) for substrate with refractive index s = 4 is shown.
Here z−(λ1) = y−(λ1) = 2/3π and z+(λ1) = 2y−(λ1), z+(λ2) = 2z−(λ2) + 2π.
4. Dispersion-managed coatings
solutions. Therefore, suitably designed double-layer coatings with the integral multiple phase thickness of layers (ratio 1:1 + p) allow to create anti-reflection device at the preset wavelengths in the interval 1 b λ1/λ2 b 5.
Obviously, with allowance for the dispersion can ruin all this approach. It is possible, however, to generalize this method by taking into account not only dispersion of the coating materials but also the compensating substrate's dispersion. For this two condition should be satisfied. The first one is the condition that the new point (n(λ2), t(λ2)) should remain at the universal curve of Eq. (8) (see Fig. 5). On the other hand, one should match the position of this new point with new substrate's index value s (λ2). In other words, it is necessary to maintain
3. Anti-reflection coatings for second harmonic generation In the case of the harmonics generation the scaling parameter g ¼ kk12 should be selected as an integer. For the second harmonic γ = 2 and for the first (p = 1) branch the parameters α and β Eq. (5) turn into α = 1, β = 3. As a result, now the refractive indices may be expressed in the explicit form in terms of the substrate refractive index s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s þ ðs 1Þ2 þ 36s t2 ¼ s ; 6 ð7Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 s 1 þ ðs 1Þ þ 36s s n2 ¼ n¼ : t 6 Excluding substrate's parameter s we come to the following universal (valid for any substrate) relationship between refractive indices of the layers t¼n
3n þ 1 : n2 þ 3 2
ð8Þ
AðntÞ As ¼ : Ak Ak
ð9Þ
Practically it means, that dispersion of one material may be arbitrary and then the second material's dispersion should keep new (n(λ2), t(λ2))-point at the curve (8). After that one need find substrate with the dispersion Δs = n2(λ2) − n2(λ1), as it is shown in Fig. 5. As it is follows from the form of curve (8) the derivative An At of this curve is usually a third of unity, i.e., the first layer is the most sensitive to presence of a dispersion, whereas the second layer's dispersion is of a lesser importance. 5. Conclusion
Further consideration strongly depends on values of the parameters n and t. In Fig. 3 are shown three such characteristic domains for fixed value of s. It is general solution of the dual-
To conclude, we have demonstrated the possibility to enhance the potentialities of a two-layer coating by a selfconsistent choice of four its variables (phase thicknesses and refractive indices) matched with the refractive index s of a substrate and have derived exact analytical expressions
Fig. 4. The reflectance of DBW with parameters shown in Table 1(line 4) under passage from fundamental, λ1, to second harmonic generation, λ2, wavelength.
Fig. 5. Compensation of the coating materials' dispersion under two-fold decreasing of the wavelength by a suitable substrate with varying dispersion sa − sb = nata − nbtb in the same spectral interval.
A. Ghiner et al. / Thin Solid Films 516 (2008) 5470–5473
connecting all these parameters. Using the well-known “looking glass” transformation property of the double film together with a fact that optical properties of any film remain invariable after addition of a half-wavelength layer we have designed dual-band anti-reflection coatings maintaining zero reflectance at any selected two wavelengths. In particular, for the fundamental and second harmonic generation frequencies common fractional anti-reflection coatings (DBW) with layers' phase thicknesses 2π/3 (see Table 1) in the domain II (see Fig. 3) may be realized. It is shown, that an accurate fitting of a second (inner) layer refractive index t is always more important than fitting of the exact valor n of the outer layer (Fig. 2). An important advantage of the developed approach consists in opportunity to avoid the negative influence dispersion of the materials on performance of these DBWs by a suitable choice and matching dispersions of materials and substrate (Fig. 5). Note, that enhancement of the potentialities of even one-layer coating may be achieved taking into account its uniaxial anisotropy, as was shown in [8] at the example of the Brewster-angle window (BAW) transparent to
5473
both polarizations. Furthermore, the problem of enhancement potentialities of a two-layer coating for cathode ray tubes was considered earlier numerically [9] in somewhat another context using the extinction coefficient of the interior layer as a fee adjusting parameter to suppress sample's reflectivity. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
A. Bideau-Mehu, et al., J. Phys., E J. Sci. Instrum. 17 (1984) 265. M. Cid, N. Stem, et al., Surf. Coat. Technol. 106 (1998) 117. B.S. Richards, Sol. Energy Mater. Sol. Cells 79 (2003) 369. H.A. Macleod, Thin-film Optical Filters, McGraw-Hill, New-York, NY, 1989. A. Thelen, Optical Interference Coatings, McGraw-Hill, 1989. J.A. Dobrovolski, in: M. Bass (Ed.), Handbook of Optics, McGraw-Hill, New York, NY, 1995. G.I. Surdutovich, R.Z. Vitlina, Optical Interference Coatings, Banff, Canada, 15–20 July, 2001, TuB4. G.I. Surdutovich, R.Z. Vitlina, V. Baranauskas, Thin Solid Films 355–356 (1999) 60. T. Oyama, et al., Thin Solid Films 351 (1999) 235.
Thin Solid Films 516 (2008) 5470 – 5473 www.elsevier.com/locate/tsf
Dual-band windows (DBW) transparent for fundamental and second harmonic generation frequencies Alexandre Ghiner a , Mikhail Skvortsov b , Gregory Surdutovich b,⁎ a
b
Physics Department, UFMA, CEP 65080-040, Sao Luis, MA, Brazil Laser Physics Institute of Siberian Branch of RAS, Novosibirsk 630090, Russia Available online 13 July 2007
Abstract Using the known “looking glass” transformation property (z → 2π − z, y → 2π − y) of the optical phase thickness z and y of matching layers of two-layer anti-reflection coating, together with the fact that optical characteristics of any film do not change after addition of a half-wavelength layer, we designed dual-band anti-reflection coatings transparent at any preset wavelengths λ1 and λ2. On the basis of this result common fractional anti-reflection coatings for second and higher harmonics generation using dispersionless coating materials are developed. Explicit analytical relationships between refractive indices of the layers and substrate are deduced. Since for second harmonic generation the dispersion of materials may be a factor we show how to compensate the known dispersion of the coating materials by special choice of dispersion of a suitable substrate. © 2007 Elsevier B.V. All rights reserved. Keywords: Anti-reflection coatings; Harmonics generation
1. Introduction A good anti-reflective coating is vital for solar cell performance, nonlinear optics and many other applications[1–3]. For a substrate with low refractive index, s, often it is very difficult or even impossible to find for manufacture of a single-layer antireflection pffifficoating an optical material with still lower refractive index, s. The conventional adoption of the double-layered coatings effectively reduces limitations on values of the refractive indices of a suitable pair of the coating materials. Two remaining free parameters of a double-layer coating, i.e., thicknesses of the layers, after selection of suitable materials, are always enough to attain null reflectivity of the double-layered interface. Moreover, we will show that under this generally accepted approach not all potentialities of such coatings are utilized and will demonstrate how to extend these potentialities. Limiting ourselves to the case of the normal light incidence one finds that any two-layer anti4ktdt n reflection coating with phase thickness z ¼ 4knd k and y ¼ k (dn and dt physical thickness of the first and second layers,
⁎ Corresponding author. Fax: +7 383 333 2067. E-mail address: [email protected] (G. Surdutovich). 0040-6090/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2007.07.026
correspondingly, λ is the wavelength), always allows the following “looking-glass” transformation: z+ → 2π − z−, y+ → 2π −y−. In each of such pairs of the solution (z−,y+) or (z+,y−) one of the layers is always thicker and another thinner than quarterwavelength thickness π [4–6]. An example of phase thickness dependences is shown in Fig. 1. Equations for lower (z−,y− ≤ π) and upper (z+,y+ ≥ π) branches have the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs 1Þðt 2 sÞ z ¼ 2 arctan n ðn2 sÞðt 2 n2 sÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ðs 1Þðn2 sÞ y ¼ 2 arctan t ðt 2 sÞðt 2 n2 sÞ yþ ¼ 2k y ; zþ ¼ 2k z ; where n, t and s are indices of the first, second layers and substrate, correspondingly. 2. Double-band anti-reflection coatings Duality of the solutions prompts a hypothesis that one may to yield zero reflectance at two different wavelengths λ1 and λ2 (assume λ1 N λ2). It means that passing from solution (z−(λ1), y+(λ1)) to (z+(λ2), y−(λ2)), under change of wavelength from λ1
A. Ghiner et al. / Thin Solid Films 516 (2008) 5470–5473
5471
thickness always may be realized under p = 0 condition. With decreasing λ on the road to the “looking-glass” solution z−(λ2), y+(λ2) the value z−(λ2) may surpass 2π, i.e. reach some zone with p ≥ 1, whereas the phase thickness y occurs to be larger π. Then the solution z−(λ2) + 2πp, y+(λ2) may be realized. The condition for such realization is yþ ðk2 Þ z ðk2 Þ þ 2kp k1 ¼ ¼ ¼ gN1 y ðk1 Þ zþ ðk1 Þ k2 Fig. 1. Upper (z+, y+ ≥ π) and lower (z−, y− ≤ π) branches of the phase thicknesses z (thin lines) and y (thick lines) of the first and second layers as function of the refractive index t of a second layer under fixed value of the first layer index n = 1.92 and substrate's index s = 1.515. The “branch-converging” pffiffi point y = z = π corresponds to t ¼ n s. This graphic relates to the values n ant t into domain I (see Fig. 3).
to λ2 while maintaining null reflectivity, one should guarantee implementation of the equality yþ ðk1 Þ z ðk1 Þ k2 ¼ ¼ : y ðk2 Þ zþ ðk2 Þ k1
ð2Þ
Since y+, z+ ≥ π and y−, z− ≤ π the first fraction always should be ≥ 1, while the second fraction should be ≤1. Therefore, at first sight it seems that unique solution is the trivial one: y = z = π, i.e. λ1 = λ2 (see the lower-upper “branchconverging” point in Fig. 1). It is possible, however, to overcome this obstacle if take into consideration such evident fact that optical properties of any layer do not change after of the quarter-wavelength increase of its phase thickness. Therefore, the optical characteristics of any film with phase thickness z are identical to characteristics of a film with z + 2πp, where p is an integer. Let us choose wavelengths with scaling ratio parameter g ¼ kk12 . If one starts from sufficiently long wavelength λ1 then the solution z+(λ1), y−(λ1), with lesser total double-layer
Fig. 2. The dependences of the refractive indices n and t on the scaling parameter γ. The solution z = 2y corresponds to the branch p = 1 and located in the interval 1 b γ b 3, whereas solution z = 3y lies in the interval 3 b γ b 5 and corresponds to branch p = 2. These values are acceptable for practical realizations. The points a and b under γ = 2 correspond to the values n and t given in Table 1 (lines 3 and 4).
ð3Þ
where y−(λ1) + y+(λ2) = 2π, z+(λ1) + z−(λ2) = 2π(p + 1). In terms of two auxiliary parameters α and β Eqs. (1),(3) acquire the form n t2 s ¼a t n2 s
nt
s1 ¼ b: t 2 n2 s
ð4Þ
These parameters depend on selected for realization scaling parameter γ and branch p. a ¼ tank
gp k =tan ; gþ1 gþ1
b ¼ tanp
gp k tan : gþ1 gþ1 ð5Þ
When p = 1 (the first branch) between these parameters springs up the relationship β = α + 2 [7]. Then parametric equations for n and t take the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 þ ðs 1Þ2 þ 4b2 s sðx þ aÞ 2 t ¼ ; n ¼ xt; where x ¼ : xð1 þ axÞ 2bs ð6Þ In Fig. 2 are shown the dependences of the refractive indices n and t on the scaling parameter γ for substrate with a given refractive index s. The value of the variable parameter x (as function γ) strongly depends on the permissible domain pffiffi in (n,t) plane (see Fig. 3). In the domain I the value xb1= s, in the pffiffi domain II 1= sbxb1 and in the domain III x N 1. In domains I and III exist only solutions z = 2y (branch p = 1) and z = 3y (p = 2), correspondingly. In the domain II there are both of these
Fig. 3. The allowable domains (I–III) of the refractive indices n, t of a pair of optical materials for a fixed substrate s. The most practically interesting for pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi pffiffiffi DBWs is the domain II, where sbtbn s and t= sbnb s.
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A. Ghiner et al. / Thin Solid Films 516 (2008) 5470–5473
Table 1 The examples of DBW for fundamental and second harmonic generation s
t0
n0
dt/λ1
dn/λ1
1.5 2.4 2.6 (a) 4 (b)
1.3101 1.7888 1.8857 2.49924
1.1449 1.3416 1.3787 1.60048
0.12721 0.09317 0.08838 0.06668
0.1456 0.1242 0.1208 0.1041
band two-layer coating for dispersionless materials. In Table 1 are given some examples (all into domain II in Fig. 3) of the dual-band anti-reflection for fundamental and second harmonic generation frequencies coatings deposited onto the different substrates. In Fig. 4 an example of DBW (last line in Table 1) for substrate with refractive index s = 4 is shown.
Here z−(λ1) = y−(λ1) = 2/3π and z+(λ1) = 2y−(λ1), z+(λ2) = 2z−(λ2) + 2π.
4. Dispersion-managed coatings
solutions. Therefore, suitably designed double-layer coatings with the integral multiple phase thickness of layers (ratio 1:1 + p) allow to create anti-reflection device at the preset wavelengths in the interval 1 b λ1/λ2 b 5.
Obviously, with allowance for the dispersion can ruin all this approach. It is possible, however, to generalize this method by taking into account not only dispersion of the coating materials but also the compensating substrate's dispersion. For this two condition should be satisfied. The first one is the condition that the new point (n(λ2), t(λ2)) should remain at the universal curve of Eq. (8) (see Fig. 5). On the other hand, one should match the position of this new point with new substrate's index value s (λ2). In other words, it is necessary to maintain
3. Anti-reflection coatings for second harmonic generation In the case of the harmonics generation the scaling parameter g ¼ kk12 should be selected as an integer. For the second harmonic γ = 2 and for the first (p = 1) branch the parameters α and β Eq. (5) turn into α = 1, β = 3. As a result, now the refractive indices may be expressed in the explicit form in terms of the substrate refractive index s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s þ ðs 1Þ2 þ 36s t2 ¼ s ; 6 ð7Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 s 1 þ ðs 1Þ þ 36s s n2 ¼ n¼ : t 6 Excluding substrate's parameter s we come to the following universal (valid for any substrate) relationship between refractive indices of the layers t¼n
3n þ 1 : n2 þ 3 2
ð8Þ
AðntÞ As ¼ : Ak Ak
ð9Þ
Practically it means, that dispersion of one material may be arbitrary and then the second material's dispersion should keep new (n(λ2), t(λ2))-point at the curve (8). After that one need find substrate with the dispersion Δs = n2(λ2) − n2(λ1), as it is shown in Fig. 5. As it is follows from the form of curve (8) the derivative An At of this curve is usually a third of unity, i.e., the first layer is the most sensitive to presence of a dispersion, whereas the second layer's dispersion is of a lesser importance. 5. Conclusion
Further consideration strongly depends on values of the parameters n and t. In Fig. 3 are shown three such characteristic domains for fixed value of s. It is general solution of the dual-
To conclude, we have demonstrated the possibility to enhance the potentialities of a two-layer coating by a selfconsistent choice of four its variables (phase thicknesses and refractive indices) matched with the refractive index s of a substrate and have derived exact analytical expressions
Fig. 4. The reflectance of DBW with parameters shown in Table 1(line 4) under passage from fundamental, λ1, to second harmonic generation, λ2, wavelength.
Fig. 5. Compensation of the coating materials' dispersion under two-fold decreasing of the wavelength by a suitable substrate with varying dispersion sa − sb = nata − nbtb in the same spectral interval.
A. Ghiner et al. / Thin Solid Films 516 (2008) 5470–5473
connecting all these parameters. Using the well-known “looking glass” transformation property of the double film together with a fact that optical properties of any film remain invariable after addition of a half-wavelength layer we have designed dual-band anti-reflection coatings maintaining zero reflectance at any selected two wavelengths. In particular, for the fundamental and second harmonic generation frequencies common fractional anti-reflection coatings (DBW) with layers' phase thicknesses 2π/3 (see Table 1) in the domain II (see Fig. 3) may be realized. It is shown, that an accurate fitting of a second (inner) layer refractive index t is always more important than fitting of the exact valor n of the outer layer (Fig. 2). An important advantage of the developed approach consists in opportunity to avoid the negative influence dispersion of the materials on performance of these DBWs by a suitable choice and matching dispersions of materials and substrate (Fig. 5). Note, that enhancement of the potentialities of even one-layer coating may be achieved taking into account its uniaxial anisotropy, as was shown in [8] at the example of the Brewster-angle window (BAW) transparent to
5473
both polarizations. Furthermore, the problem of enhancement potentialities of a two-layer coating for cathode ray tubes was considered earlier numerically [9] in somewhat another context using the extinction coefficient of the interior layer as a fee adjusting parameter to suppress sample's reflectivity. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
A. Bideau-Mehu, et al., J. Phys., E J. Sci. Instrum. 17 (1984) 265. M. Cid, N. Stem, et al., Surf. Coat. Technol. 106 (1998) 117. B.S. Richards, Sol. Energy Mater. Sol. Cells 79 (2003) 369. H.A. Macleod, Thin-film Optical Filters, McGraw-Hill, New-York, NY, 1989. A. Thelen, Optical Interference Coatings, McGraw-Hill, 1989. J.A. Dobrovolski, in: M. Bass (Ed.), Handbook of Optics, McGraw-Hill, New York, NY, 1995. G.I. Surdutovich, R.Z. Vitlina, Optical Interference Coatings, Banff, Canada, 15–20 July, 2001, TuB4. G.I. Surdutovich, R.Z. Vitlina, V. Baranauskas, Thin Solid Films 355–356 (1999) 60. T. Oyama, et al., Thin Solid Films 351 (1999) 235.