Abstract

Cut-off filters are usually operating at oblique incidence and exhibit polarization dependence properties. We propose a simple approach to design cut-off filters with low linear polarization sensitivity (LPS) based on dielectric-metal-dielectric (DMD) stacks. The designing method is derived from the theory of optical film characteristic matrix. The admittance loci of the film are adjusted to achieve similar spectral properties of s- and p-polarized light at oblique incidence. Different film structures are designed non-polarizing at different angles of incidence with the method. The results show that the designing method is efficient for designing non-polarizing cut-off filters, which are widely used in non-polarizing optical system.

© 2013 Optical Society of America

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]

LPS=|TmaxTmin|Tmax+Tmin=|TpTs|Tp+Ts.
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 [915]. 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 [1015]. 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
[BC]={j=1K[cosδjisinδj/ηjiηjsinδjcosδj]}[1ηsub],
whereδj=2πNjdjcosθj/λ, ηj is optical admittance of the jth layer, dj is physical thickness of the jth layer, and ηsub is optical admittance of substrate. As the incident angle θ0 is given, refractive angle θj can be get from Snell’s law, i.e.
N0sinθ0=Njsinθj,
where N0, Nj and Nsub 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
{ηrs=Nrcosθrfors-polarizedlightηrp=Nr/cosθrforp-polarizedlight.
The complex refractive index Nr = nr - ikr, can be simplified as Nr = nr for transparent dielectric or as Nr = - ikr 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
[cosδjisinδj/ηjiηjsinδjcosδj]
is called characteristic matrix of the jth layer. For transparent dielectric layer, parameters δj and ηj are real. For ideal metal layer,

cosθj=1sin2θj=1+n02sin2θ0/kj2,
{ηjs=Njcosθj=-ikj1+n02sin2θ0/kj2fors-polarizedlightηjp=Nj/cosθj=-ikj/1+n02sin2θ0/kj2forp-polarizedlight,
δj=2πNjdjcosθj/λ=-i2πkjdjcosθj/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=(1R)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=(η0Yη0+Y)(η0Yη0+Y)*=(1Y/η01+Y/η0)(1Y/η01+Y/η0)*.
Note that Y / η0 is a function given by
Y/η0=f(ηsub,η0,η1,,ηj,,ηK,δ1,,δj,,δK).
For incident medium, each layer or substrate, ηr is defined in Eq. (4). At small angle of incidence, from Taylor series expansion of ηr with sin2θr as independent variable, first order analytical expression of ηr is given by
{ηrsNr(112sin2θr)=Nr(112N02Nr2sin2θ0)fors-polarizedlightηrpNr(1+12sin2θr)=Nr(1+12N02Nr2sin2θ0)forp-polarizedlight.
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
{(Y/η0)sf(Nsub,N0,N1,,Nj,,NK,δ1,,δj,,δK)Δ(Y/η0)fors-polarizedlight(Y/η0)pf(Nsub,N0,N1,,Nj,,NK,δ1,,δj,,δK)+Δ(Y/η0)forp-polarizedlight.
Where, Δ(Y/η0) is the expression of linear variation induced by sin2θ0. If f (Nsub, N0, N1,, Nj,, NK, δ1,, δj,, δK) is designed to be 1, the reflectance R of different polarized light is nearly same, as shown in the following relations
{Rs|Δ(Y/η0)|2|2-Δ(Y/η0)|2|Δ(Y/η0)|24fors-polarizedlightRp|Δ(Y/η0)|2|2+Δ(Y/η0)|2|Δ(Y/η0)|24forp-polarizedlight.
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 di can be given by

dj=δjλ2πNjcosθj=tjcosθj.

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 n12/nsubnsubn32/n0n0, 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 (Nsub) 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 dj 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 Ns).

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.

Tables Icon

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

Tables Icon

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

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 tj 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 | t1D′1/t2M′/t1D″1 | Air at normal incidence.

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Fig. 6 Admittance locus of BK7 | (t1D′1/t2M′/t1D″1)x/t1D1/t2M/t3D3 | Air at normal incidence.

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The designed thickness of each layer (d1, d2, d1, d1, d2 and d3) changes with angle of incidence, following the rule of dj = tj/cosθj. Now non-polarizing cut-off filters with the structure BK7 | (d1D′1/d2M′/d1D″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.

References and links

1. P. N. Slater, Remote Sensing: Optics and Optical Systems (Addison-Wesley, 1980).

2. P. Ma, F. Lin, and J. A. Dobrowolski, “Design and manufacture of metal/dielectric long-wavelength cutoff filters,” Appl. Opt. 50(9), C201–C209 (2011). [CrossRef]   [PubMed]  

3. P. H. Berning and A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. 47(3), 230–239 (1957). [CrossRef]  

4. L. G. Schulz, “The optical constants of silver, gold, copper, and aluminum. I. the absorption coefficient k,” J. Opt. Soc. Am. 44(5), 357–362 (1954). [CrossRef]  

5. P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta (Lond.) 8(2), 105–119 (1961). [CrossRef]  

6. W. L. Barnes, T. S. Pagano, and V. V. Salomonson, “Prelaunch characteristics of the moderate resolution imaging spectroradiometer (MODIS) on EOS-AM1,” IEEE Trans. Geosci. Rem. Sens. 36(4), 1088–1100 (1998). [CrossRef]  

7. G. Meister, E. J. Kwiatkowska, B. A. Franz, F. S. Patt, G. C. Feldman, and C. R. McClain, “Moderate-resolution imaging spectroradiometer ocean color polarization correction,” Appl. Opt. 44(26), 5524–5535 (2005). [CrossRef]   [PubMed]  

8. H. R. Gordon, T. Du, and T. Zhang, “Atmospheric correction of ocean color sensors: analysis of the effects of residual instrument polarization sensitivity,” Appl. Opt. 36(27), 6938–6948 (1997). [CrossRef]   [PubMed]  

9. V. R. Costich, “Reduction of polarization effects in interference coatings,” Appl. Opt. 9(4), 866–870 (1970). [CrossRef]   [PubMed]  

10. J. Ciosek, J. A. Dobrowolski, G. A. Clarke, and G. Laframboise, “Design and manufacture of all-dielectric nonpolarizing beam splitters,” Appl. Opt. 38(7), 1244–1250 (1999). [CrossRef]   [PubMed]  

11. A. Thelen, “Nonpolarizing interference films inside a glass cube,” Appl. Opt. 15(12), 2983–2985 (1976). [CrossRef]   [PubMed]  

12. A. Thelen, “Nonpolarizing edge filters,” J. Opt. Soc. Am. 71(3), 309–314 (1981). [CrossRef]  

13. A. Thelen, “Nonpolarizing edge filters: Part 2,” Appl. Opt. 23(20), 3541–3543 (1984). [CrossRef]   [PubMed]  

14. H. Qi, R. Hong, K. Yi, J. Shao, and Z. Fan, “Nonpolarizing and polarizing filter design,” Appl. Opt. 44(12), 2343–2348 (2005). [CrossRef]   [PubMed]  

15. X. Ma, D. Liu, F. Zhang, and Y. Yan, “Non-polarizing broadband dichroic mirror,” J. Opt. A, Pure Appl. Opt. 9(7), 573–576 (2007). [CrossRef]  

16. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, Bristol, UK, 2001).

17. K. Wu, C. C. Lee, and T. L. Ni, “Advanced broadband monitoring for thin film deposition through equivalent optical admittance loci observation,” Opt. Express 20(4), 3883–3889 (2012). [CrossRef]   [PubMed]  

18. Q. Y. Cai, Y. X. Zheng, D. X. Zhang, W. J. Lu, R. J. Zhang, W. Lin, H. B. Zhao, and L. Y. Chen, “Application of image spectrometer to in situ infrared broadband optical monitoring for thin film deposition,” Opt. Express 19(14), 12969–12977 (2011). [CrossRef]   [PubMed]  

19. C. C. Lee and Y. J. Chen, “Multilayer coatings monitoring using admittance diagram,” Opt. Express 16(9), 6119–6124 (2008). [CrossRef]   [PubMed]  

20. C. C. Lee, K. Wu, S. H. Chen, and S. J. Ma, “Optical monitoring and real time admittance loci calculation through polarization interferometer,” Opt. Express 15(26), 17536–17541 (2007). [CrossRef]   [PubMed]  

21. T. W. Allen and R. G. DeCorby, “Conditions for admittance-matched tunneling through symmetric metal-dielectric stacks,” Opt. Express 20(S5Suppl 5), A578–A588 (2012). [CrossRef]   [PubMed]  

References

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  • |
  • |
  • |

  1. P. N. Slater, Remote Sensing: Optics and Optical Systems (Addison-Wesley, 1980).
  2. P. Ma, F. Lin, and J. A. Dobrowolski, “Design and manufacture of metal/dielectric long-wavelength cutoff filters,” Appl. Opt. 50(9), C201–C209 (2011).
    [Crossref] [PubMed]
  3. P. H. Berning and A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. 47(3), 230–239 (1957).
    [Crossref]
  4. L. G. Schulz, “The optical constants of silver, gold, copper, and aluminum. I. the absorption coefficient k,” J. Opt. Soc. Am. 44(5), 357–362 (1954).
    [Crossref]
  5. P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta (Lond.) 8(2), 105–119 (1961).
    [Crossref]
  6. W. L. Barnes, T. S. Pagano, and V. V. Salomonson, “Prelaunch characteristics of the moderate resolution imaging spectroradiometer (MODIS) on EOS-AM1,” IEEE Trans. Geosci. Rem. Sens. 36(4), 1088–1100 (1998).
    [Crossref]
  7. G. Meister, E. J. Kwiatkowska, B. A. Franz, F. S. Patt, G. C. Feldman, and C. R. McClain, “Moderate-resolution imaging spectroradiometer ocean color polarization correction,” Appl. Opt. 44(26), 5524–5535 (2005).
    [Crossref] [PubMed]
  8. H. R. Gordon, T. Du, and T. Zhang, “Atmospheric correction of ocean color sensors: analysis of the effects of residual instrument polarization sensitivity,” Appl. Opt. 36(27), 6938–6948 (1997).
    [Crossref] [PubMed]
  9. V. R. Costich, “Reduction of polarization effects in interference coatings,” Appl. Opt. 9(4), 866–870 (1970).
    [Crossref] [PubMed]
  10. J. Ciosek, J. A. Dobrowolski, G. A. Clarke, and G. Laframboise, “Design and manufacture of all-dielectric nonpolarizing beam splitters,” Appl. Opt. 38(7), 1244–1250 (1999).
    [Crossref] [PubMed]
  11. A. Thelen, “Nonpolarizing interference films inside a glass cube,” Appl. Opt. 15(12), 2983–2985 (1976).
    [Crossref] [PubMed]
  12. A. Thelen, “Nonpolarizing edge filters,” J. Opt. Soc. Am. 71(3), 309–314 (1981).
    [Crossref]
  13. A. Thelen, “Nonpolarizing edge filters: Part 2,” Appl. Opt. 23(20), 3541–3543 (1984).
    [Crossref] [PubMed]
  14. H. Qi, R. Hong, K. Yi, J. Shao, and Z. Fan, “Nonpolarizing and polarizing filter design,” Appl. Opt. 44(12), 2343–2348 (2005).
    [Crossref] [PubMed]
  15. X. Ma, D. Liu, F. Zhang, and Y. Yan, “Non-polarizing broadband dichroic mirror,” J. Opt. A, Pure Appl. Opt. 9(7), 573–576 (2007).
    [Crossref]
  16. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, Bristol, UK, 2001).
  17. K. Wu, C. C. Lee, and T. L. Ni, “Advanced broadband monitoring for thin film deposition through equivalent optical admittance loci observation,” Opt. Express 20(4), 3883–3889 (2012).
    [Crossref] [PubMed]
  18. Q. Y. Cai, Y. X. Zheng, D. X. Zhang, W. J. Lu, R. J. Zhang, W. Lin, H. B. Zhao, and L. Y. Chen, “Application of image spectrometer to in situ infrared broadband optical monitoring for thin film deposition,” Opt. Express 19(14), 12969–12977 (2011).
    [Crossref] [PubMed]
  19. C. C. Lee and Y. J. Chen, “Multilayer coatings monitoring using admittance diagram,” Opt. Express 16(9), 6119–6124 (2008).
    [Crossref] [PubMed]
  20. C. C. Lee, K. Wu, S. H. Chen, and S. J. Ma, “Optical monitoring and real time admittance loci calculation through polarization interferometer,” Opt. Express 15(26), 17536–17541 (2007).
    [Crossref] [PubMed]
  21. T. W. Allen and R. G. DeCorby, “Conditions for admittance-matched tunneling through symmetric metal-dielectric stacks,” Opt. Express 20(S5Suppl 5), A578–A588 (2012).
    [Crossref] [PubMed]

2012 (2)

2011 (2)

2008 (1)

2007 (2)

2005 (2)

1999 (1)

1998 (1)

W. L. Barnes, T. S. Pagano, and V. V. Salomonson, “Prelaunch characteristics of the moderate resolution imaging spectroradiometer (MODIS) on EOS-AM1,” IEEE Trans. Geosci. Rem. Sens. 36(4), 1088–1100 (1998).
[Crossref]

1997 (1)

1984 (1)

1981 (1)

1976 (1)

1970 (1)

1961 (1)

P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta (Lond.) 8(2), 105–119 (1961).
[Crossref]

1957 (1)

1954 (1)

Allen, T. W.

Barnes, W. L.

W. L. Barnes, T. S. Pagano, and V. V. Salomonson, “Prelaunch characteristics of the moderate resolution imaging spectroradiometer (MODIS) on EOS-AM1,” IEEE Trans. Geosci. Rem. Sens. 36(4), 1088–1100 (1998).
[Crossref]

Baumeister, P.

P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta (Lond.) 8(2), 105–119 (1961).
[Crossref]

Berning, P. H.

Cai, Q. Y.

Chen, L. Y.

Chen, S. H.

Chen, Y. J.

Ciosek, J.

Clarke, G. A.

Costich, V. R.

DeCorby, R. G.

Dobrowolski, J. A.

Du, T.

Fan, Z.

Feldman, G. C.

Franz, B. A.

Gordon, H. R.

Hong, R.

Kwiatkowska, E. J.

Laframboise, G.

Lee, C. C.

Lin, F.

Lin, W.

Liu, D.

X. Ma, D. Liu, F. Zhang, and Y. Yan, “Non-polarizing broadband dichroic mirror,” J. Opt. A, Pure Appl. Opt. 9(7), 573–576 (2007).
[Crossref]

Lu, W. J.

Ma, P.

Ma, S. J.

Ma, X.

X. Ma, D. Liu, F. Zhang, and Y. Yan, “Non-polarizing broadband dichroic mirror,” J. Opt. A, Pure Appl. Opt. 9(7), 573–576 (2007).
[Crossref]

McClain, C. R.

Meister, G.

Ni, T. L.

Pagano, T. S.

W. L. Barnes, T. S. Pagano, and V. V. Salomonson, “Prelaunch characteristics of the moderate resolution imaging spectroradiometer (MODIS) on EOS-AM1,” IEEE Trans. Geosci. Rem. Sens. 36(4), 1088–1100 (1998).
[Crossref]

Patt, F. S.

Qi, H.

Salomonson, V. V.

W. L. Barnes, T. S. Pagano, and V. V. Salomonson, “Prelaunch characteristics of the moderate resolution imaging spectroradiometer (MODIS) on EOS-AM1,” IEEE Trans. Geosci. Rem. Sens. 36(4), 1088–1100 (1998).
[Crossref]

Schulz, L. G.

Shao, J.

Thelen, A.

Turner, A. F.

Wu, K.

Yan, Y.

X. Ma, D. Liu, F. Zhang, and Y. Yan, “Non-polarizing broadband dichroic mirror,” J. Opt. A, Pure Appl. Opt. 9(7), 573–576 (2007).
[Crossref]

Yi, K.

Zhang, D. X.

Zhang, F.

X. Ma, D. Liu, F. Zhang, and Y. Yan, “Non-polarizing broadband dichroic mirror,” J. Opt. A, Pure Appl. Opt. 9(7), 573–576 (2007).
[Crossref]

Zhang, R. J.

Zhang, T.

Zhao, H. B.

Zheng, Y. X.

Appl. Opt. (8)

IEEE Trans. Geosci. Rem. Sens. (1)

W. L. Barnes, T. S. Pagano, and V. V. Salomonson, “Prelaunch characteristics of the moderate resolution imaging spectroradiometer (MODIS) on EOS-AM1,” IEEE Trans. Geosci. Rem. Sens. 36(4), 1088–1100 (1998).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

X. Ma, D. Liu, F. Zhang, and Y. Yan, “Non-polarizing broadband dichroic mirror,” J. Opt. A, Pure Appl. Opt. 9(7), 573–576 (2007).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Acta (Lond.) (1)

P. Baumeister, “The transmission and degree of polarization of quarter-wave stacks at non-normal incidence,” Opt. Acta (Lond.) 8(2), 105–119 (1961).
[Crossref]

Opt. Express (5)

Other (2)

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, Bristol, UK, 2001).

P. N. Slater, Remote Sensing: Optics and Optical Systems (Addison-Wesley, 1980).

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Figures (9)

Fig. 1
Fig. 1 Polarization spectra of 45° incident light on BK7 | 45nm ZnS/11nm Ag/45nm ZnS | Air.
Fig. 2
Fig. 2 The structure of non-polarizing cut-off filters based on DMD stacks.
Fig. 3
Fig. 3 The expected admittance loci of each layer of D1MD3 stack [16].
Fig. 4
Fig. 4 Admittance locus of BK7 | t1D1/t2M/t3D3 | Air at normal incidence.
Fig. 5
Fig. 5 Admittance locus of BK7 | t1D′1/t2M′/t1D″1 | Air at normal incidence.
Fig. 6
Fig. 6 Admittance locus of BK7 | (t1D′1/t2M′/t1D″1)x/t1D1/t2M/t3D3 | Air at normal incidence.
Fig. 7
Fig. 7 Polarization spectra at incident angle of 30° with x = 0, 1, 2.
Fig. 8
Fig. 8 Polarization spectra at incident angle of 45° with x = 2.
Fig. 9
Fig. 9 Polarization spectra at incident angle of 60° with x = 2.

Tables (2)

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Table 1 Designing Table for Ta2O5-Ag-SiO2 sStack

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Table 2 Designing Table for Ta2O5-Ag-Ta2O5 Stack

Equations (16)

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LPS= | T max T min | T max + T min = | T p T s | T p + T s .
Y=C/B,
[ B C ]={ j=1 K [ cos δ j isin δ j / η j i η j sin δ j cos δ j ] }[ 1 η sub ],
N 0 sin θ 0 = N j sin θ j ,
{ η rs = N r cos θ r fors-polarized light η rp = N r /cos θ r forp-polarized light .
[ cos δ j isin δ j / η j i η j sin δ j cos δ j ]
cos θ j = 1 sin 2 θ j = 1+ n 0 2 sin 2 θ 0 / k j 2 ,
{ η js = N j cos θ j =-i k j 1+ n 0 2 sin 2 θ 0 / k j 2 fors-polarized light η jp = N j /cos θ j =-i k j / 1+ n 0 2 sin 2 θ 0 / k j 2 forp-polarized light ,
δ j =2π N j d j cos θ j /λ=-i2π k j d j cos θ j /λ.
T=( 1R )ψ.
R=( η 0 Y η 0 +Y ) ( η 0 Y η 0 +Y ) * =( 1Y/ η 0 1+Y/ η 0 ) ( 1Y/ η 0 1+Y/ η 0 ) * .
Y/ η 0 =f( η sub , η 0 , η 1 ,, η j ,, η K , δ 1 ,, δ j ,, δ K ).
{ η rs N r ( 1 1 2 sin 2 θ r )= N r ( 1 1 2 N 0 2 N r 2 sin 2 θ 0 )fors-polarized light η rp N r ( 1+ 1 2 sin 2 θ r )= N r ( 1+ 1 2 N 0 2 N r 2 sin 2 θ 0 )forp-polarized light .
{ ( Y/ η 0 ) s f( N sub , N 0 , N 1 ,, N j ,, N K , δ 1 ,, δ j ,, δ K )Δ( Y/ η 0 )fors-polarized light ( Y/ η 0 ) p f( N sub , N 0 , N 1 ,, N j ,, N K , δ 1 ,, δ j ,, δ K )+Δ( Y/ η 0 )forp-polarized light .
{ R s | Δ( Y/ η 0 ) | 2 | 2-Δ( Y/ η 0 ) | 2 | Δ( Y/ η 0 ) | 2 4 fors-polarized light R p | Δ( Y/ η 0 ) | 2 | 2+Δ( Y/ η 0 ) | 2 | Δ( Y/ η 0 ) | 2 4 forp-polarized light .
d j = δ j λ 2π N j cos θ j = t j cos θ j .

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