Design of non-polarizing cut-off filters based on dielectric-metal-dielectric stacks

Qing-Yuan Cai; Hai-Han Luo; Yu-Xiang Zheng; Ding-Quan Liu

doi:10.1364/OE.21.019163

1. Introduction

Cut-off filters are widely used in optical system, especially in multichannel optical system [1]. To separate different wavebands, for instance, visible and infrared light, dielectric-metal-dielectric (DMD) stacks are often used in cut-off filters [2]. They are also used in band pass filters to get narrow pass band and wide cut-off region [3]. For the extinction coefficients of some metal materials, such as Ag, Au, Al, etc., are approximately proportional to wavelength in visible/infrared region [4], very thin metal layer induces bigger change of equivalent optical admittance of film at longer wavelength, which induces higher reflectance at longer wavelength. In many situations, optical films are operating at oblique incidence, which induces different transmission properties of s- and p-polarized light [5]. In some advanced optical systems, polarization state of the light contains key information, and very low linear polarization sensitivity (LPS) is required. LPS is defined as [6]

L P S = \frac{| T_{\max} - T_{\min} |}{T_{\max} + T_{\min}} = \frac{| T_{p} - T_{s} |}{T_{p} + T_{s}} .

Where, T is transmittance of different linear polarized light. It means all the optical elements must be designed and fabricated non-polarizing [7, 8]. A lot of works have been done on the subject of polarization of optical films [9–15]. On the subject of designing non-polarizing cut-off filters (edge filters, heat mirrors or dichroic beam-splitters), several approaches to the design of non-polarizing all-dielectric cut-off filters have been suggested in the references [10–15]. Few works have been done on the design of non-polarizing cut-off filters based on DMD stacks.

For the cut-off filters with the structure BK7 | 45nm ZnS/11nm Ag/45nm ZnS | Air, the polarization spectra at incident angle of 45° are showed in Fig. 1. In the transmission spectral region 400-800 nm, the polarization properties are: average LPS ≈2%, maximal LPS > 4%. The conventionally designed cut-off filters are not suitable for non-polarizing optical systems with the requirement of LPS < 2% in the visible region.

Fig. 1 Polarization spectra of 45° incident light on BK7 | 45nm ZnS/11nm Ag/45nm ZnS | Air.

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Admittance loci technology has been widely used in design [16], analysis [17, 18] and coating monitoring [19] of optical film. It can be applied at oblique incidence, and gives transmission characteristics of linear polarized incident light, which is s- or p-polarized light. It has been used to assist analyzing polarization state of light [20] and designing DMD stack [21]. To minimize LPS, the transmission characteristics of both s- and p-polarized light must be analyzed about their dependence on incident angle. Using the theory of optical film characteristic matrix and admittance loci technology, we do some analysis and mathematical derivation, and propose a simple approach to design non-polarizing cut-off filters based on DMD stacks.

2. Principle of design

The principle is derived from the theory of optical film characteristic matrix [16]. Optical film can be considered as a reference surface with equivalent optical admittance Y, which is defined as

Y = C / B,

where C and B are given by

[\begin{matrix} B \\ C \end{matrix}] = {\prod_{j = 1}^{K} [\begin{matrix} \cos δ_{j} & i \sin δ_{j} / η_{j} \\ i η_{j} \sin δ_{j} & \cos δ_{j} \end{matrix}]} [\begin{matrix} 1 \\ η_{s u b} \end{matrix}],

where

δ_{j} = 2 π N_{j} d_{j} \cos θ_{j} / λ

, η_j is optical admittance of the jth layer, d_j is physical thickness of the jth layer, and η_sub is optical admittance of substrate. As the incident angle θ₀ is given, refractive angle θ_j can be get from Snell’s law, i.e.

N_{0} \sin θ_{0} = N_{j} \sin θ_{j},

where N₀, N_j and N_sub represent the complex refractive indices of incident medium, the jth layer and substrate, respectively. The optical admittance of incident medium, each layer or substrate is given by

{\begin{cases} η_{r s} = N_{r} \cos θ_{r} for s-polarized light \\ η_{r p} = N_{r} / \cos θ_{r} for p-polarized light \end{cases} .

The complex refractive index N_r = n_r - ik_r, can be simplified as N_r = n_r for transparent dielectric or as N_r = - ik_r for ideal metal material. Here, an ideal metal is described as: for a lossless metal in which the refractive index, and hence the optical admittance, is purely imaginary, and given by −ik, the loci are a set of circles with centres on the real axis and passing through the points ik and −ik, which are on the imaginary axis [16]. The matrix

[\begin{matrix} \cos δ_{j} & i \sin δ_{j} / η_{j} \\ i η_{j} \sin δ_{j} & \cos δ_{j} \end{matrix}]

is called characteristic matrix of the jth layer. For transparent dielectric layer, parameters δ_j and η_j are real. For ideal metal layer,

\cos θ_{j} = \sqrt{1 - \sin^{2} θ_{j}} = \sqrt{1 + n_{0}^{2} \sin^{2} θ_{0} / k_{j}^{2}},

{\begin{cases} η_{j s} = N_{j} \cos θ_{j} = - i k_{j} \sqrt{1 + n_{0}^{2} \sin^{2} θ_{0} / k_{j}^{2}} for s-polarized light \\ η_{j p} = N_{j} / \cos θ_{j} = - i k_{j} / \sqrt{1 + n_{0}^{2} \sin^{2} θ_{0} / k_{j}^{2}} for p-polarized light \end{cases},

δ_{j} = 2 π N_{j} d_{j} \cos θ_{j} / λ = - i 2 π k_{j} d_{j} \cos θ_{j} / λ .

The potential transmittance of a layer or an assembly of layers is the ratio of the irradiance leaving by the rear, or exit, interface to that entering by the front interface [16]. The relation of transmittance T, reflectance R and potential transmittance ψ of optical film is given by

T = (1 - R) ψ .

The potential transmittance of a layer is dependent on admittance n-ik, phase thickness δ of the layer and the exit admittance X + iZ. When n = 0 or k = 0, the calculated ψ is constant 100% [16]. It means that the potential transmittance of ideal metal layer (n = 0) is constant 100%, whether at normal incidence or oblique incidence. For ideal-like metal layer with n ≈0, the potential transmittance difference of s- and p-polarized light is very small. For example, at wavelength 500nm, the potential transmittance difference of s- and p-polarized light on the structure BK7 | 10nm Ag | Air is only 0.5% at incident angle of 45° and 0.2% at incident angle of 30°, and the reflectance difference of s- and p-polarized light is 17.4% at incident angle of 45° and 8.8% at incident angle of 30°. The different transmission properties of s- and p-polarized light on DMD stacks are mainly determined by Y of s- and p-polarized light. To eliminate the transmittance difference of s- and p-polarized light, means to adjust reflectance of s- and p-polarized light, which is determined by Y, as shown in the expression

R = (\frac{η_{0} - Y}{η_{0} + Y}) {(\frac{η_{0} - Y}{η_{0} + Y})}^{*} = (\frac{1 - Y / η_{0}}{1 + Y / η_{0}}) {(\frac{1 - Y / η_{0}}{1 + Y / η_{0}})}^{*} .

Note that Y / η₀ is a function given by

Y / η_{0} = f (η_{s u b}, η_{0}, η_{1}, \dots, η_{j}, \dots, η_{K}, δ_{1}, \dots, δ_{j}, \dots, δ_{K}) .

For incident medium, each layer or substrate, η_r is defined in Eq. (4). At small angle of incidence, from Taylor series expansion of η_r with sin²θ_r as independent variable, first order analytical expression of η_r is given by

{\begin{cases} η_{r s} \approx N_{r} (1 - \frac{1}{2} \sin^{2} θ_{r}) = N_{r} (1 - \frac{1}{2} \frac{N_{0}^{2}}{N_{r}^{2}} \sin^{2} θ_{0}) for s-polarized light \\ η_{r p} \approx N_{r} (1 + \frac{1}{2} \sin^{2} θ_{r}) = N_{r} (1 + \frac{1}{2} \frac{N_{0}^{2}}{N_{r}^{2}} \sin^{2} θ_{0}) for p-polarized light \end{cases} .

Substituting Eq. (11) into Eq. (10), and using first order analytical expression from Taylor series expansion of Y/η₀ with sin²θ₀ as independent variable, one obtains

{\begin{cases} {(Y / η_{0})}_{s} \approx f (N_{s u b}, N_{0}, N_{1}, \dots, N_{j}, \dots, N_{K}, δ_{1}, \dots, δ_{j}, \dots, δ_{K}) - Δ (Y / η_{0}) for s-polarized light \\ {(Y / η_{0})}_{p} \approx f (N_{s u b}, N_{0}, N_{1}, \dots, N_{j}, \dots, N_{K}, δ_{1}, \dots, δ_{j}, \dots, δ_{K}) + Δ (Y / η_{0}) for p-polarized light \end{cases} .

Where, Δ(Y/η₀) is the expression of linear variation induced by sin²θ₀. If f (N_sub_, N₀, N₁,…, N_j,…, N_K, δ₁,…, δ_j,…, δ_K) is designed to be 1, the reflectance R of different polarized light is nearly same, as shown in the following relations

{\begin{cases} R_{s} \approx \frac{{| Δ (Y / η_{0}) |}^{2}}{{| 2- Δ (Y / η_{0}) |}^{2}} \approx \frac{{| Δ (Y / η_{0}) |}^{2}}{4} for s-polarized light \\ R_{p} \approx \frac{{| Δ (Y / η_{0}) |}^{2}}{{| 2+ Δ (Y / η_{0}) |}^{2}} \approx \frac{{| Δ (Y / η_{0}) |}^{2}}{4} for p-polarized light \end{cases} .

So far, there is little difference between the transmittance of s- and p-polarized light, and an optical film with very low transmittance LPS can be designed. In the situation of oblique incidence, the designed physical thickness d_i can be given by

d_{j} = \frac{δ_{j} λ}{2 π N_{j} \cos θ_{j}} = \frac{t_{j}}{\cos θ_{j}} .

3. Designing procedure

The considered structure of non-polarizing cut-off filters is Sub | (D′₁/M′/D″₁)^x/D₁/M/D₃ | Air, as showed in Fig. 2. DMD is a basic stack of metal induced films. Here, the dielectric layers are of two different materials, for instance, Ta₂O₅ for D′₁, D″₁ and D₁, and SiO₂ for D₃. N₁ = n₁ and N₃ = n₃ are the refractive indices of the two dielectric materials. The design of D₁MD₃ stack is the key of designing non-polarizing cut-off filters. It is preferred that $n_{1}^{2} / n_{s u b} - n_{s u b} \approx n_{3}^{2} / n_{0} - n_{0}$ , to ensure that admittance loci of other wavelengths are similar with reference wavelength and end around the point of Y = n₀.

Fig. 2 The structure of non-polarizing cut-off filters based on DMD stacks.

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The designing procedure of D₁MD₃ stack includes the following steps using admittance analysis tool of optical film designing software, such as Essential Macleod:

1) Draw the expected admittance loci of first layer D₁ (circle SA′AA″) and the third layer D₃ (circle B″BB′O), which go through point S (N_sub) and point O (N₀) respectively, as shown in Fig. 3;
2) Find point A and point B, where the tangent through A of circle SA′AA″ is approximately parallel with the tangent through B of circle OB″BB′, and arc AB is admittance locus of metal layer M, such as Ag (It is an empirical way to ensure other wavelengths around reference wavelength to have similar admittance loci.);
3) Adjust the thickness of layer D₁ and M to ensure the admittance locus of the designed film Sub | D₁/M/D₃ | Air at normal incidence, is close to the expected admittance locus SABO from step 1) and 2);
4) Adjust the thickness of layer D₃ to ensure the simulated admittance locus end at point O;
5) The physical thickness of the three layers from steps 3) and 4) are represented by t₁, t₂, and t₃. At oblique incidence, the designed thickness of each layer should be d₁, d₂, and d₃ acquired in Eq. (14). When d_j is applied in the film, the polarization spectra around reference wavelength will be non-polarizing at oblique incidence;
6) In some cases, non-polarizing spectra in broader waveband are required. Further optimization of thickness of each layer should be performed with computer.

Fig. 3 The expected admittance loci of each layer of D₁MD₃ stack [16].

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To get a sharper edge of transmission spectra of cut-off filters, it’s better to add more DMD stacks in the film. The added DMD stack is approximately symmetrical, and inserted between the D₁MD₃ stack and substrate, as Sub | (D′₁/M′/D″₁)^x/D₁/M/D₃ | Air. The designing procedure of D′₁M′D″₁ is similar to that of D₁MD₃ except that points O and S are same (N₀ is set as N_s).

4. Design of Ag-based non-polarizing cut-off filters

Ag is an ideal-like material in visible-infrared region for its k >> 1 and n ≈0. It is the metal used most frequently in metal composite film in visible-infrared region, and is also the best choice to design non-polarizing cut-off filters to confirm the validity of the proposed designing method.

In this case, layers M′ and M are Ag, layers D′₁, D″₁ and D₁ are Ta₂O₅, layer D₃ is SiO₂, and substrate is BK7. The reference wavelength is 500 nm. The choice of reference wavelength is dependent on average frequency of transmission spectra. Ta₂O₅-Ag-SiO₂, (where, an ultra-thin Al₂O₃ layer is inserted between Ag and SiO₂ to improve bonding strength in deposition process), is chosen as stack D₁MD₃. According to the designing steps 1) ~5), a designing table is made as Table 1. Stack D′₁M′D″₁ is Ta₂O₅-Ag-Ta₂O₅. The designing procedure of stack D′₁M′D″₁ is similar, and the designing table is Table 2.

Table 1. Designing Table for Ta₂O₅-Ag-SiO₂ sStack

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Table 2. Designing Table for Ta₂O₅-Ag-Ta₂O₅ Stack

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Before the process of film designing, structure BK7 | (t′₁D′₁/t′₂M′/t″₁D″₁)^x/t₁D₁/t₂M/t₃D₃ | Air is provided for admittance loci analysis. At the reference wavelength 500nm, the admittance locus of BK7 | t₁D₁/t₂M/t₃D₃ | air is showed in Fig. 4. The admittance locus of BK7 | t′₁D′₁/t′₂M′/t″₁D″₁ | Air is showed in Fig. 5, which starts from and ends at the same admittance point. For BK7 | (t′₁D′₁/t′₂M′/t″₁D″₁)^x/t₁D₁/t₂M/t₃D₃ | Air, the admittance locus is showed in Fig. 6. Parameters of thickness t_j are read directly from commercial optical film designing software Essential Macleod (Version 9.5.390, Thin Film Center, Inc., Tucson, USA).

Fig. 4 Admittance locus of BK7 | t₁D₁/t₂M/t₃D₃ | Air at normal incidence.

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Fig. 5 Admittance locus of BK7 | t′₁D′₁/t′₂M′/t″₁D″₁ | Air at normal incidence.

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Fig. 6 Admittance locus of BK7 | (t′₁D′₁/t′₂M′/t″₁D″₁)^x/t₁D₁/t₂M/t₃D₃ | Air at normal incidence.

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The designed thickness of each layer (d′₁, d′₂, d″₁, d₁, d₂ and d₃) changes with angle of incidence, following the rule of d_j = t_j/cosθ_j. Now non-polarizing cut-off filters with the structure BK7 | (d′₁D′₁/d′₂M′/d″₁D″₁)^x/d₁D₁/d₂M/d₃D₃ | Air is designed. With x set as different integers, different polarization spectra at incident angle of 30° are given, as shown in Fig. 7. At incident angles of 45° and 60°, x = 2 is set, and the designing results of polarization spectra are shown in Fig. 8 and Fig. 9. At incident angle of 45°, the average LPS is smaller than 0.5% in the range of 400-800 nm, which means it can be used in non-polarizing optical systems, where LPS < 2% is required.

Fig. 7 Polarization spectra at incident angle of 30° with x = 0, 1, 2.

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Fig. 8 Polarization spectra at incident angle of 45° with x = 2.

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Fig. 9 Polarization spectra at incident angle of 60° with x = 2.

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5. Conclusion

Film structure Sub | (D′₁/M′/D″₁)^x/D₁/M/D₃ | Air based on DMD stacks is proved to be good for designing non-polarizing cut-off filters. Designing principle and procedure are derived from the theory of optical film characteristic matrix and admittance loci technology. Ag-based non-polarizing cut-off filters with high transmittance in visible region are designed via adjusting admittance loci of the film. The designed cut-off filters have excellent optical properties: high transmittance (> 90%) in visible region, high reflectance (> 90%) in infrared region, low transmittance LPS (< 0.5%) in visible region, and fulfills the need of non-polarizing optical system. The new method can also be used for designing non-polarizing cut-off filters using other metals.

Acknowledgments

This work has been financially supported by the Innovation Program of Shanghai Institute of Technical Physics of the Chinese Academy of Sciences (No. Q-DX-24), the National Natural Science Foundation of China (No. 61275160, 60938004), the STCSM project of China with the Grant No. 12XD1420600, 11DZ1121900.

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	D₁, Ta₂O₅		M, Ag		D₃, SiO₂
	N₁ = 2.15, t₁ = 25.0 nm		N₂ = 0.05 - 2.87i, t₂ = 7.1 nm		N₃ = 1.46, t₃ = 69.5 nm
θ₀	cosθ₁	d₁ (nm)	cosθ₂	d₂ (nm)	cosθ₃	d₃ (nm)
30°	0.973	25.7	~1.015	~7.0	0.940	74.0
45°	0.944	26.5	~1.030	~6.9	0.875	79.4
60°	0.915	27.3	~1.045	~6.8	0.805	86.3

	D′₁, Ta₂O₅		M′, Ag		D″₁, Ta₂O₅
	N₁ = 2.15, t′₁ = 34.6 nm		N₂ = 0.05 - 2.87i, t′₂ = 8.9 nm		N₁ = 2.15, t″₁ = 35.4 nm
θ₀	cosθ′₁	d′₁ (nm)	cosθ′₂	d′₂ (nm)	cosθ″₁	d″₁ (nm)
30°	0.973	35.6	~1.015	~8.8	0.973	36.4
45°	0.944	36.6	~1.030	~8.6	0.944	37.5
60°	0.915	37.8	~1.045	~8.5	0.915	38.7

	D₁, Ta₂O₅		M, Ag		D₃, SiO₂
	N₁ = 2.15, t₁ = 25.0 nm		N₂ = 0.05 - 2.87i, t₂ = 7.1 nm		N₃ = 1.46, t₃ = 69.5 nm
θ₀	cosθ₁	d₁ (nm)	cosθ₂	d₂ (nm)	cosθ₃	d₃ (nm)
30°	0.973	25.7	~1.015	~7.0	0.940	74.0
45°	0.944	26.5	~1.030	~6.9	0.875	79.4
60°	0.915	27.3	~1.045	~6.8	0.805	86.3

	D′₁, Ta₂O₅		M′, Ag		D″₁, Ta₂O₅
	N₁ = 2.15, t′₁ = 34.6 nm		N₂ = 0.05 - 2.87i, t′₂ = 8.9 nm		N₁ = 2.15, t″₁ = 35.4 nm
θ₀	cosθ′₁	d′₁ (nm)	cosθ′₂	d′₂ (nm)	cosθ″₁	d″₁ (nm)
30°	0.973	35.6	~1.015	~8.8	0.973	36.4
45°	0.944	36.6	~1.030	~8.6	0.944	37.5
60°	0.915	37.8	~1.045	~8.5	0.915	38.7

Design of non-polarizing cut-off filters based on dielectric-metal-dielectric stacks

Abstract

1. Introduction

2. Principle of design

3. Designing procedure

4. Design of Ag-based non-polarizing cut-off filters

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (2)

Equations (16)

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