Abstract

We developed a new method aimed at designing short-pass filters, long-pass filters and filters blocking sidebands of Fabry-Perot bandpasses. The method is an automated version of a non-straightforward empirical approach invented as a result of many years’ experience in design and production of optical coatings. The method allows obtaining near-quarter-wave solutions in a few seconds. In many cases these solutions are more advantageous for deposition systems.

© 2015 Optical Society of America

1. Introduction

Due to tremendous progress in thin film technology and growing demands on coating performances, the development of new design approaches is still in a focus of optical coating engineers. Along with general purpose design techniques such as refinement, needle optimization, gradual evolution [1–5], special approaches aimed at different types of multilayer coatings are being actively developed. For example, specific methods for designing dispersive mirrors [6,7], minus and notch filters [8–11], WDM filters [12,13] have been implemented by different authors.

Very often, optical coating engineers in order to avoid complicated design solutions invent and use their own empirical approaches aimed at obtaining designs with some specific structures. Design preferences are often determined by technical requirements of deposition plants and monitoring systems in their disposal. In particular, for a wide range of filter design problems, quarter-wave stacks (QWS) or near-quarter-wave stacks (NQWS) solutions are more beneficial for the production.

NQWS do not contain thin layers. This represents an advantage in the case of deposition plants, that cannot produce layers thinner than 15 nm with a good uniformity because of a relatively low calotte rotation speed. For example, if the substrate holder rotates with the speed of about 30 rpm and the deposition rate is 0.3 nm/s, then the deposition of a 15 nm layer takes 50 sec or 25 rotations. Due to spatial non-uniformity of evaporated material flow and different positions of samples at the calotte, their layers may have different thicknesses. The deviation of layer thicknesses may reach 0.6 nm that is 4% of 15 nm layer. Therefore, in order to achieve good uniformity of about 1%, minimum 200 sec of deposition time or 100 rotations of the calotte are required.

In the case of the monochromatic optical monitoring, production yields of NQWS are higher, because large building blocks in such coatings can be monitored without changing the monitoring wavelength. Some real-life optical monitors (for example, Leybold OMS5000) use proven monitoring strategies that insure very good cut-off reliability for NQWS.

The new Sensitivity-Directed Refinement (SDR) method allows obtaining NQWS with excellent spectral performances. Using different settings of the method, it is possible to obtain not one but several design solutions. The method is highly appropriate for designing broadband blocking filters like short-pass, long-pass filters, or filters blocking sidebands of Fabry-Perot bandpasses. The robust version [14] of the SDR algorithm allows taking into account the requirement on stability of the design solutions to possible variations of the refractive indices as well as to errors in layer thicknesses.

In Section 2 the new SDR method is described in detail. In Section 3 the method is applied to three design problems. Final conclusions are presented in Section 4.

2. Description of the Sensitivity-Directed Refinement method

Denote the number of design layers as m, layer thicknesses as d1,...,dm. Let S^(λj) and S(d1,...,dm;λj) be target and actual spectral characteristics specified on the wavelength grid of interest {λj},j=1,...,L; and ΔSj – the tolerances. Deviations of actual spectral characteristics from the target ones are evaluated with the help of the merit function (MF):

MF(d1,,dm)=(1Lj=1L[S(d1,,dm;λj)S^(λj)ΔSj]2)1/2.

Let δH and δL be levels of relative errors in high and low index layers, respectively. Then, a set of m disturbed designs with layer thicknesses d1,,di(1+δH,L),,dm, i=1,,m is defined. For each disturbed design merit function variations (ΔMFi) are evaluated and layer sensitivities (LS) are calculated as:

ΔMFi=|MF(d1,...,di(1+δH,L),...,dm)MF(d1,...,dm)|,LSi=ΔMFimaxi=1,,mΔMFi100%.
With the help of Eq. (2), all layers can be ranged with respect to their LS values.

Step 0. A starting design is a combination of quarter-wave stacks with different wavelengths. The choice of a reasonable starting design can be usually done according to the positions of requested high reflection zones [1–3]. Designate starting layer thicknesses as dF,1,,dF,i,,dF,m. The letter “F” in the subscript denotes “Fixed” state of layer thicknesses.

Step 1. Layer sensitivities of all starting design layers are calculated and a layer with LS=100% is to be found. Let k1 be the number of this layer. The merit function is minimized with respect to the thickness of this most sensitive layer:

MF(dF,1,...,dA,k1,...,dF,m)min

In Eq. (3) the letter “A” in the subscript denotes “Active” state of the chosen layer. As a result of the optimization, an intermediate design is obtained.

Step 2. Layer sensitivities of all layers of the intermediate design obtained at the Step 1 are evaluated, and a layer with the maximum layer sensitivity among all layers except for the layer numberk1 is found. Let k2 be the number of this layer. Note, that at this and next steps the found maximum LS value is not necessary 100% because some layers have been already activated at the previous steps and have been excluded from considerations. Then, the merit function Eq. (1) is optimized with respect to the layer thicknesses with the numbers k1 andk2:

MF(dF,1,,dA,k1,,dA,k2,,dF,m)min
After minimization of the merit function given by Eq. (4), a new intermediate design is found. Minimization of the merit function can be performed by various methods, for example, Quasi-Newton, Newton or Sequential Quadratic Programming (SQP) [15,16]. The only constraints on layer thicknesses are conditions di0, yet if necessary additional requirements dididi can be introduced (simple bounds constraints).

Further steps and termination. These iterations continue further, the number of active layers is increased at each step, and the value of MF is gradually decreased. The iterations are terminated when a desired spectral performance is achieved or all m layers become active.

For the sake of convenience, only the simplest version of the SDR has been presented above. In this version the number of layers activated at each step is p=1. In general, there are several algorithm parameters that can be controlled. First of all, at each step the method allows activating arbitrary number p1 of layers having maximum sensitivity values. Evidently, if all m layers are activated at the very beginning, then the SDR becomes a standard refinement procedure. Secondly, after several optimization steps, the insertion of new layers with the help of the needle optimization technique is possible as an additional option. At a needle step, only active design layers are considered as possible places for needle insertions. Thirdly, the levels of errors δH and δL can be also used as parameters of SDR.

3. Numerical examples

3.1 Example 1: designing a blocking filter

Let target transmittance be less than 1% in the spectral ranges from 400 nm to 480 nm and 680 nm to 806 nm and more than 99% in the range from 557 nm to 587 nm, see Fig. 1(a). The merit function is calculated on the wavelength grid containing 656 points. For this and next design problems, OptiLayer Software was used [17].

 figure: Fig. 1

Fig. 1 Target and actual transmittances of starting, SDR and refined designs (a); design structures: starting and SDR designs (b), starting and refined designs (c).

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In all examples, high and low index materials are Nb2O5 and SiO2, respectively. Their refractive indices are described by Cauchy formula (the wavelength is expressed in nanometers): n(λ)=A0+A1(1000/λ)2+A2(1000/λ)4 with A0=2.2185, A1=0.0218, A2=3.999103 for Nb2O5 and A0=1.4653, A1=0, A2=4.71104 for SiO2. The substrate is glass B270.

In order to explain the method clearly, the manual version of SDR is demonstrated first. A combination of two QWS is chosen as a starting design: the layers 1 through 26 have 1.75 QWOT (quarter-wave optical thickness) and the layers 27 through 46 have 1.0 QWOT; reference wavelength is 440 nm. Transmittance of the starting design is shown in Fig. 1(a), starting MF value is 11.6.

At the Step 1, LS for all design layers are calculated. The LS=100% corresponds to k1=14. After minimization of MF (Eq. (3)) with respect to d14, an intermediate design having MF=7.35 is obtained. At the Step 2, LS of all layers are calculated again and a layer with the maximumLSvalue among all layers with exception of the layer number 14 is searched for. It turns out, that the layer 24 is the most sensitive one at this step (k2=24). After minimization of MF Eq. (4) with respect to the thicknessesd14,d24, the second intermediate design is obtained and MF is decreased to the value 3.67. Further evolution of MF values and numbers of active layers are presented in Table 1. It is seen from Table 1 that only 9 steps of SDR are necessary to achieve zero MF value.

Tables Icon

Table 1. Evolution of the SDR in the case of blocking filter

The final design structure is plotted in Fig. 1(b). Transmittance of the final design is shown in Fig. 1(a). As mentioned above, only nine layers of the starting design were modified by SDR, thicknesses of other 37 layers were kept unchanged.

For comparison, the same design problem was solved with the help of a standard refinement (SQP method). The structure of the obtained design is presented in Fig. 1(c). The refined design also provides zero MF value and exhibits an excellent spectral performance Fig. 1(a). However, now not nine as before but all layer thicknesses have been varied that makes more difficult a choice of monochromatic monitoring strategy with a small number of different monitoring wavelengths.

3.2 Example 2: designing a short-pass filter

In this case target transmittance is more than 97% in the range from 400 nm to 480 nm and less than 1% in the range from 520 nm to 900 nm, see Fig. 2(a). Reference wavelength is 600 nm and 60-layer design consisting of two QWS is taken as a starting design. The layers 1 through 30 have 1.35 QWOT and the layers 31 through 60 have 1 QWOT. Transmittance of the starting design is shown in Fig. 2(a).

 figure: Fig. 2

Fig. 2 Target and actual transmittances of starting, SPF1 and SPF2 designs (a); design structures: starting and SPF1 designs (b), starting and SPF2 designs (c).

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After application of the two versions of the SDR withp=2, δH=1, δL=2 and with p=3, δH=1, δL=1, two different 60-layer NQWS, SPF1 and SPF2, are obtained. Both designs exhibit excellent spectral performance, see Fig. 2(a). Their structures in QWOT are shown in Fig. 2(b) and Fig. 2(c). In the course of SDR, 26 layers were activated in SPF1 design and 27 layers in SPF2 design.

3.3 Example 3: Designing a long-pass filter

Target transmittance is less than 1% in the spectral range from 425 nm to 670 nm and more than 97% in the range from 725 nm to 1200 nm, see Fig. 3(a). The merit function is calculated on the wavelength grid containing 309 points.

 figure: Fig. 3

Fig. 3 Target and actual transmittances of starting, LPF1 and LPF2 designs (a); design structures: starting and LPF1 design (b), starting and LPF2 design (c).

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Reference wavelength is 475 nm and a 51-layer design consisting of two QWS is taken as a starting design. The layers 1 through 26 have 1.28 QWOT and the layers 27 through 51 have 1 QWOT. Transmittance of the starting design is shown in Fig. 3(a).

After application of the two versions of the SDR with p=2, δH=1, δL=2 and with p=3,δH=2, δL=1, two different NQWS, LPF1 (51 layers) and LPF2 (49 layers), are obtained. It is seen from Fig. 3(a) that the both designs exhibit excellent spectral performance. Their structures in QWOT are shown in Fig. 3(b) and Fig. 3(c). In the course of two SDRs, 12 and 11 layers were activated.

Designs obtained by SDR algorithm can be practically monitored using the strategies presented in Table 2. We performed a series of simulations assuming monochromatic monitoring with the strategies specified in Table 2 and simulation parameters close to those considered in [18]. The SDR designs demonstrated high production yield during the simulations.

Tables Icon

Table 2. Practical monitoring strategies for the considered designs

4. Summary

The proposed SDR method takes into account important practical requirement of closeness of design layer thicknesses to layer thicknesses of quarter-wave stacks. The idea of the method is based on many years experience in the design and production of optical coatings. The automatic version of SDR provides a possibility to obtain a series of near-quarter-wave solutions for complicated design problems in a short time. The obtained solutions exhibit excellent spectral performances. SDR method demonstrates outstanding results in designing edge filters and blocking filters.

Acknowledgments

This work was supported by RFBR, research project No. 13-07-00846 А (“Mathematical modeling and simulation of optical nano-coatings deposition”) and by the Cluster of Excellence “Munich-Centre for Advanced Photonics” (http://www.munich-photonics.de/) of the German Research Foundation (DFG).

References and links

1. H. A. Macleod, Thin-Film Optical Filters, 4th ed. (Taylor & Francis, 2010).

2. P. Baumeister, Optical Coating Technology (SPIE Optical Engineering, 2004).

3. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

4. A. V. Tikhonravov, M. K. Trubetskov, and G. W. Debell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35(28), 5493–5508 (1996). [CrossRef]   [PubMed]  

5. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46(5), 704–710 (2007). [CrossRef]   [PubMed]  

6. N. Matuschek, F. X. Kartner, and U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35(2), 129–137 (1999). [CrossRef]  

7. M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Phase optimization of dispersive mirrors based on floating constants,” Opt. Express 18(26), 27613–27618 (2010). [CrossRef]   [PubMed]  

8. A. Thelen, “Design of optical minus filters,” J. Opt. Soc. Am. 61(3), 365 (1971). [CrossRef]  

9. K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008). [CrossRef]  

10. O. Lyngnes and J. Kraus, “Design of optical notch filters using apodized thickness modulation,” Appl. Opt. 53(4), A21–A26 (2014). [CrossRef]   [PubMed]  

11. U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010). [CrossRef]  

12. P. Baumeister, “Design of a coarse WDM bandpass filter using the Thelen bandpass design method,” Opt. Express 9(12), 652–657 (2001). [CrossRef]   [PubMed]  

13. A. V. Tikhonravov and M. K. Trubetskov, “Automated design and sensitivity analysis of wavelengh-division multiplexing filters,” Appl. Opt. 41(16), 3176–3182 (2002). [CrossRef]   [PubMed]  

14. M. K. Trubetskov, T. Amotchkina, and A. V. Tikhonravov, “Design of multilayer optical coatings with high stability to refractive index variations,” in Optical Interference Coatings, OSA Technical Digest Series (Optical Society of America, 2013), paper TD3.

15. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

16. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

17. A. V. Tikhonravov and M. K. Trubetskov, “OptiLayer software,” http://www.optilayer.com.

18. A. Zöller, M. Boos, H. Hagedorn, and B. Romanov, “Computer simulation of coating processes with monochromatic monitoring,” SPIE Proc. 7101, 71010G (2008).

References

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  1. H. A. Macleod, Thin-Film Optical Filters, 4th ed. (Taylor & Francis, 2010).
  2. P. Baumeister, Optical Coating Technology (SPIE Optical Engineering, 2004).
  3. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).
  4. A. V. Tikhonravov, M. K. Trubetskov, and G. W. Debell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35(28), 5493–5508 (1996).
    [Crossref] [PubMed]
  5. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46(5), 704–710 (2007).
    [Crossref] [PubMed]
  6. N. Matuschek, F. X. Kartner, and U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35(2), 129–137 (1999).
    [Crossref]
  7. M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Phase optimization of dispersive mirrors based on floating constants,” Opt. Express 18(26), 27613–27618 (2010).
    [Crossref] [PubMed]
  8. A. Thelen, “Design of optical minus filters,” J. Opt. Soc. Am. 61(3), 365 (1971).
    [Crossref]
  9. K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008).
    [Crossref]
  10. O. Lyngnes and J. Kraus, “Design of optical notch filters using apodized thickness modulation,” Appl. Opt. 53(4), A21–A26 (2014).
    [Crossref] [PubMed]
  11. U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010).
    [Crossref]
  12. P. Baumeister, “Design of a coarse WDM bandpass filter using the Thelen bandpass design method,” Opt. Express 9(12), 652–657 (2001).
    [Crossref] [PubMed]
  13. A. V. Tikhonravov and M. K. Trubetskov, “Automated design and sensitivity analysis of wavelengh-division multiplexing filters,” Appl. Opt. 41(16), 3176–3182 (2002).
    [Crossref] [PubMed]
  14. M. K. Trubetskov, T. Amotchkina, and A. V. Tikhonravov, “Design of multilayer optical coatings with high stability to refractive index variations,” in Optical Interference Coatings, OSA Technical Digest Series (Optical Society of America, 2013), paper TD3.
  15. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).
  16. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).
  17. A. V. Tikhonravov and M. K. Trubetskov, “OptiLayer software,” http://www.optilayer.com .
  18. A. Zöller, M. Boos, H. Hagedorn, and B. Romanov, “Computer simulation of coating processes with monochromatic monitoring,” SPIE Proc. 7101, 71010G (2008).

2014 (1)

2010 (2)

U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010).
[Crossref]

M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Phase optimization of dispersive mirrors based on floating constants,” Opt. Express 18(26), 27613–27618 (2010).
[Crossref] [PubMed]

2008 (1)

K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008).
[Crossref]

2007 (1)

2002 (1)

2001 (1)

1999 (1)

N. Matuschek, F. X. Kartner, and U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35(2), 129–137 (1999).
[Crossref]

1996 (1)

1971 (1)

Baumeister, P.

DeBell, G. W.

Hendrix, K. D.

K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008).
[Crossref]

Hulse, C. A.

K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008).
[Crossref]

Jakobs, S.

U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010).
[Crossref]

Kartner, F. X.

N. Matuschek, F. X. Kartner, and U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35(2), 129–137 (1999).
[Crossref]

Keller, U.

N. Matuschek, F. X. Kartner, and U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35(2), 129–137 (1999).
[Crossref]

Kraus, J.

Lappschies, M.

U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010).
[Crossref]

Lyngnes, O.

Matuschek, N.

N. Matuschek, F. X. Kartner, and U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35(2), 129–137 (1999).
[Crossref]

Ockenfuss, G. J.

K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008).
[Crossref]

Pervak, V.

Ploss, B.

U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010).
[Crossref]

Sargent, R. B.

K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008).
[Crossref]

Schallenberg, U.

U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010).
[Crossref]

Thelen, A.

Tikhonravov, A. V.

Trubetskov, M. K.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

N. Matuschek, F. X. Kartner, and U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35(2), 129–137 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Express (2)

Proc. SPIE (2)

U. Schallenberg, B. Ploss, M. Lappschies, and S. Jakobs, “Design and manufacturing of high-performance notch filters,” Proc. SPIE 7739, 77391X (2010).
[Crossref]

K. D. Hendrix, C. A. Hulse, G. J. Ockenfuss, and R. B. Sargent, “Demonstration of narrowband notch and multi-notch filters,” Proc. SPIE 7067, 706702 (2008).
[Crossref]

Other (8)

H. A. Macleod, Thin-Film Optical Filters, 4th ed. (Taylor & Francis, 2010).

P. Baumeister, Optical Coating Technology (SPIE Optical Engineering, 2004).

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

M. K. Trubetskov, T. Amotchkina, and A. V. Tikhonravov, “Design of multilayer optical coatings with high stability to refractive index variations,” in Optical Interference Coatings, OSA Technical Digest Series (Optical Society of America, 2013), paper TD3.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

A. V. Tikhonravov and M. K. Trubetskov, “OptiLayer software,” http://www.optilayer.com .

A. Zöller, M. Boos, H. Hagedorn, and B. Romanov, “Computer simulation of coating processes with monochromatic monitoring,” SPIE Proc. 7101, 71010G (2008).

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Figures (3)

Fig. 1
Fig. 1 Target and actual transmittances of starting, SDR and refined designs (a); design structures: starting and SDR designs (b), starting and refined designs (c).
Fig. 2
Fig. 2 Target and actual transmittances of starting, SPF1 and SPF2 designs (a); design structures: starting and SPF1 designs (b), starting and SPF2 designs (c).
Fig. 3
Fig. 3 Target and actual transmittances of starting, LPF1 and LPF2 designs (a); design structures: starting and LPF1 design (b), starting and LPF2 design (c).

Tables (2)

Tables Icon

Table 1 Evolution of the SDR in the case of blocking filter

Tables Icon

Table 2 Practical monitoring strategies for the considered designs

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

MF( d 1 ,, d m )= ( 1 L j=1 L [ S( d 1 ,, d m ; λ j ) S ^ ( λ j ) Δ S j ] 2 ) 1/2 .
ΔM F i =| MF( d 1 ,..., d i ( 1+ δ H,L ),..., d m )MF( d 1 ,..., d m ) |,L S i = ΔM F i max i=1,,m ΔM F i 100%.
MF( d F,1 ,..., d A, k 1 ,..., d F,m )min
MF( d F,1 ,, d A, k 1 ,, d A, k 2 ,, d F,m )min

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