Abstract

We propose to refine the refractive index of the layers composing optical filters while keeping their optical thicknesses constant. Using this technique, one can optimize filters made of quarter-wave layers using conventional optimization techniques, while preserving the possibility to use turning-point monitoring during their fabrication. Application of this method to the design of a dual narrowband filter and a tilted edge filter demonstrates its effectiveness.

© 2008 Optical Society of America

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References

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  1. H. A. Macleod, “Turning value monitoring of narrow-band all-dielectric thin-film optical filters,” Opt. Acta 19, 1-28 (1972).
    [CrossRef]
  2. A. V. Tikhonravov and M. K. Trubetskov, “Mathematical modeling of multilayer filters for telecommunication,” Computational Mathematics and Modeling 14, 74-84 (2003).
    [CrossRef]
  3. P. Baumeister, “Design of a coarse WDM bandpass filter using the Thelen bandpass design method,” Opt. Express 9, 652-657 (2001).
    [CrossRef] [PubMed]
  4. A. Thelen, “Equivalent layers in multilayer filters,” J. Opt. Soc. Am. 56, 1533-1538 (1966).
    [CrossRef]
  5. P. G. Verly, “Simple technique for the accurate design of square bandpass WDM interference filters,” Proc. SPIE 5250, 378-383.
  6. P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955-958 (1958).
    [CrossRef]
  7. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).
  8. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
    [CrossRef]
  9. C. J. van der Laan and H. J. Frankena, “Fast computation method for derivatives of multilayer stack reflectance,” Appl. Opt. 17, 538-541 (1978).
    [CrossRef]
  10. P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis for inhomogeneous optical coatings,” Appl. Opt. 36, 1487-1495 (1997).
    [CrossRef] [PubMed]
  11. S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219-C230 (2008).
    [CrossRef] [PubMed]
  12. Y. Soudagar, F. Bussières, J. M. Fernandez, and N. Godbout, “Creation of a two-photon 4-qubit square cluster state in optical fibres,” in Frontiers in Optics 2007/Laser Science XXIII/Organic Materials and Devices for Displays and Energy Conversion (Optical Society of America, 2007), paper JTuA6.
    [PubMed]
  13. D. Poitras, “Asymetrical dual-cavity filters and their application to thickness uniformity monitoring,” Opt. Express 11, 3393-3403 (2003).
    [CrossRef] [PubMed]
  14. S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
    [CrossRef]
  15. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427-5442 (1993).
    [CrossRef] [PubMed]
  16. S. Larouche and L. Martinu, “Optical filters with constant optical thickness and refined refractive indices,” in Optical Interference Coatings on CD-ROM (Optical Society of America, 2007), paper TuD8.

2008 (1)

2007 (2)

Y. Soudagar, F. Bussières, J. M. Fernandez, and N. Godbout, “Creation of a two-photon 4-qubit square cluster state in optical fibres,” in Frontiers in Optics 2007/Laser Science XXIII/Organic Materials and Devices for Displays and Energy Conversion (Optical Society of America, 2007), paper JTuA6.
[PubMed]

S. Larouche and L. Martinu, “Optical filters with constant optical thickness and refined refractive indices,” in Optical Interference Coatings on CD-ROM (Optical Society of America, 2007), paper TuD8.

2004 (1)

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

2003 (2)

D. Poitras, “Asymetrical dual-cavity filters and their application to thickness uniformity monitoring,” Opt. Express 11, 3393-3403 (2003).
[CrossRef] [PubMed]

A. V. Tikhonravov and M. K. Trubetskov, “Mathematical modeling of multilayer filters for telecommunication,” Computational Mathematics and Modeling 14, 74-84 (2003).
[CrossRef]

2001 (2)

1997 (1)

1993 (1)

1978 (1)

1972 (1)

H. A. Macleod, “Turning value monitoring of narrow-band all-dielectric thin-film optical filters,” Opt. Acta 19, 1-28 (1972).
[CrossRef]

1966 (1)

1958 (1)

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).

Baumeister, P.

Bovard, B. G.

Bussières, F.

Y. Soudagar, F. Bussières, J. M. Fernandez, and N. Godbout, “Creation of a two-photon 4-qubit square cluster state in optical fibres,” in Frontiers in Optics 2007/Laser Science XXIII/Organic Materials and Devices for Displays and Energy Conversion (Optical Society of America, 2007), paper JTuA6.
[PubMed]

Fernandez, J. M.

Y. Soudagar, F. Bussières, J. M. Fernandez, and N. Godbout, “Creation of a two-photon 4-qubit square cluster state in optical fibres,” in Frontiers in Optics 2007/Laser Science XXIII/Organic Materials and Devices for Displays and Energy Conversion (Optical Society of America, 2007), paper JTuA6.
[PubMed]

Frankena, H. J.

Godbout, N.

Y. Soudagar, F. Bussières, J. M. Fernandez, and N. Godbout, “Creation of a two-photon 4-qubit square cluster state in optical fibres,” in Frontiers in Optics 2007/Laser Science XXIII/Organic Materials and Devices for Displays and Energy Conversion (Optical Society of America, 2007), paper JTuA6.
[PubMed]

Gujrathi, S. C.

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Klemberg-Sapieha, J. E.

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Larouche, S.

S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219-C230 (2008).
[CrossRef] [PubMed]

S. Larouche and L. Martinu, “Optical filters with constant optical thickness and refined refractive indices,” in Optical Interference Coatings on CD-ROM (Optical Society of America, 2007), paper TuD8.

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

H. A. Macleod, “Turning value monitoring of narrow-band all-dielectric thin-film optical filters,” Opt. Acta 19, 1-28 (1972).
[CrossRef]

Martinu, L.

S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219-C230 (2008).
[CrossRef] [PubMed]

S. Larouche and L. Martinu, “Optical filters with constant optical thickness and refined refractive indices,” in Optical Interference Coatings on CD-ROM (Optical Society of America, 2007), paper TuD8.

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Poitras, D.

Soudagar, Y.

Y. Soudagar, F. Bussières, J. M. Fernandez, and N. Godbout, “Creation of a two-photon 4-qubit square cluster state in optical fibres,” in Frontiers in Optics 2007/Laser Science XXIII/Organic Materials and Devices for Displays and Energy Conversion (Optical Society of America, 2007), paper JTuA6.
[PubMed]

Szymanowski, H.

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Thelen, A.

Tikhonravov, A. V.

A. V. Tikhonravov and M. K. Trubetskov, “Mathematical modeling of multilayer filters for telecommunication,” Computational Mathematics and Modeling 14, 74-84 (2003).
[CrossRef]

P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis for inhomogeneous optical coatings,” Appl. Opt. 36, 1487-1495 (1997).
[CrossRef] [PubMed]

Trubetskov, M. K.

A. V. Tikhonravov and M. K. Trubetskov, “Mathematical modeling of multilayer filters for telecommunication,” Computational Mathematics and Modeling 14, 74-84 (2003).
[CrossRef]

P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis for inhomogeneous optical coatings,” Appl. Opt. 36, 1487-1495 (1997).
[CrossRef] [PubMed]

van der Laan, C. J.

Verly, P. G.

P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis for inhomogeneous optical coatings,” Appl. Opt. 36, 1487-1495 (1997).
[CrossRef] [PubMed]

P. G. Verly, “Simple technique for the accurate design of square bandpass WDM interference filters,” Proc. SPIE 5250, 378-383.

Ann. Phys. (Paris) (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).

Appl. Opt. (4)

Computational Mathematics and Modeling (1)

A. V. Tikhonravov and M. K. Trubetskov, “Mathematical modeling of multilayer filters for telecommunication,” Computational Mathematics and Modeling 14, 74-84 (2003).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Vac. Sci. Technol. A (1)

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Opt. Acta (1)

H. A. Macleod, “Turning value monitoring of narrow-band all-dielectric thin-film optical filters,” Opt. Acta 19, 1-28 (1972).
[CrossRef]

Opt. Express (2)

Proc. SPIE (1)

P. G. Verly, “Simple technique for the accurate design of square bandpass WDM interference filters,” Proc. SPIE 5250, 378-383.

Other (3)

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

S. Larouche and L. Martinu, “Optical filters with constant optical thickness and refined refractive indices,” in Optical Interference Coatings on CD-ROM (Optical Society of America, 2007), paper TuD8.

Y. Soudagar, F. Bussières, J. M. Fernandez, and N. Godbout, “Creation of a two-photon 4-qubit square cluster state in optical fibres,” in Frontiers in Optics 2007/Laser Science XXIII/Organic Materials and Devices for Displays and Energy Conversion (Optical Society of America, 2007), paper JTuA6.
[PubMed]

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Figures (2)

Fig. 1
Fig. 1

Design of a dual narrowband filter: index profile (left) and transmission spectrum (right) of the starting design (top), the filter obtained when the refractive index of all layers is allowed to vary (middle), and the filter obtained when only the refractive index of the cavities, the matching layer, and the coupling layers is allowed to vary (bottom).

Fig. 2
Fig. 2

Design of an s polarization edge filter working at 45 ° : index profile (left) and transmission spectrum (right) of the starting design (top) and the optimized filter (bottom).

Equations (36)

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MF = χ 2 = i = 1 m ( B i B ¯ i Δ B i ) 2 ,
M i = [ cos ϕ i ( i / η i ) sin ϕ i i η i sin ϕ i cos ϕ i ] ,
η i = { N i 2 α 2 s   polarization N i 2 / N i 2 α 2 p   polarization
ϕ i = 2 π λ N i 2 α 2 d i
α = N i sin θ i ,
M = [ m 11 m 12 m 21 m 22 ] = i = q 1 M i ,
d M d n j , 0 = i = q j + 1 M i d M j d n j , 0 i = j 1 1 M i .
d M j d n j , 0 | Re ϕ j , 0 = d M j d ϕ j d ϕ j d n j , 0 | Re ϕ j , 0 + d M j d η j d η j d N j d N j d n j , 0 ,
d M j d ϕ j = [ sin ϕ j ( i / η j ) cos ϕ j i η j cos ϕ j sin ϕ j , ] ,
d M j d η j = [ 0 ( i / η j 2 ) sin ϕ j i sin ϕ j 0 ] ,
d η j d N j = { N j N j 2 α 2 s     polarization N j N j 2 α 2 ( 2 N j 2 N j 2 α 2 ) p     polarization ,
d ϕ j d n j , 0 | Re ϕ j , 0 = d ϕ j d N j d N j d n j , 0 + d ϕ j d d j d d j d n j , 0 | Re ϕ j , 0 ,
d ϕ j d N j = 2 π λ d j N j N j 2 α 2 ,
d ϕ j d d j = 2 π λ N j 2 α 2 ,
d d j d n j , 0 | Re ϕ j , 0 = d j ( n j , 0 | N j , 0 | 2 Im N j , 0 Re N j , 0 k j , 0 | N j , 0 | 2 ) ,
N j , 0 = N j , 0 2 α 0 2 .
d d j d n j , 0 | Re ϕ j , 0 = d j n j , 0 | N j , 0 | 2 .
d d j d n j , 0 | Re ϕ j , 0 = d j n j , 0 ,
d R d n j , 0 = 2 r * d r d n j , 0 ,
r = η inc m 11 η ex m 22 + η inc η ex m 12 m 21 η inc m 11 + η ex m 22 + η inc η ex m 12 + m 21
d r d n j , 0 = Tr ( d M j d n j , 0 ψ r )
ψ r = t 2 η inc [ η inc ( 1 r ) ( 1 + r ) η inc η ex ( 1 r ) η ex ( 1 + r ) ] ;
t = 2 η inc η inc m 11 + η ex m 22 + η inc η ex m 12 + m 21
dMF d n j , 0 = 2 i = 1 m ( B i B ¯ i ( Δ B i ) 2 d B i d n j , 0 ) .
Re ϕ j , 0 = 2 π λ 0 Re N j , 0 2 α 0 2 d j ,
N j , 0 = N j , 0 2 α 0 2 .
d j = λ 0 2 π Re ϕ j , 0 Re N j , 0 .
d d j d n j , 0 | Re ϕ j , 0 = [ d d n j , 0 ( λ 0 Re ϕ j , 0 2 π Re N j , 0 ) ] Re ϕ j , 0 = λ 0 Re ϕ j , 0 2 π d d n j , 0 1 Re N j , 0 = λ 0 Re ϕ j , 0 2 π 1 Re 2 N j , 0 d Re N j , 0 d n j , 0 = d j Re N j , 0 d Re N j , 0 d n j , 0 .
N j , 0 = ( ( n j , 0 i k 0 ) 2 ( α 0 , r i α 0 , i ) 2 ) 1 / 2 = ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) + i ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) ) 1 / 2 .
a + i b = ± ( ( 1 2 ( a 2 + b 2 + a ) ) 1 / 2 ± i ( 1 2 ( a 2 + b 2 a ) ) 1 / 2 )
Re N j , 0 = ± ( 1 2 ( ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) ) 1 / 2 .
d Re N j , 0 d n j , 0 = 1 2 ± d d n j , 0 ( 1 2 ( ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) ) ± ( 1 2 ( ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) ) 1 / 2 = 1 4 1 2 d d n j , 0 ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + 2 n j , 0 ( 1 2 ( ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) ) 1 / 2 = 1 4 1 2 2 ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 n j , 0 + 2 ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) ( 2 k 0 ) ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + 2 n j , 0 ( 1 2 ( ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) ) 1 / 2 = 1 2 ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) n j , 0 ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) k 0 ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + n j , 0 ( 1 2 ( ( ( n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) 2 + ( 2 α 0 , r α 0 , i 2 n j , 0 k 0 ) 2 ) 1 / 2 + n j , 0 2 k 0 2 α 0 , r 2 + α 0 , i 2 ) ) 1 / 2 = 1 2 Re N j , 0 2 n j , 0 Im N j , 0 2 k 0 | N j , 0 2 | + n j , 0 Re N j , 0 = 1 2 ( Re N j , 0 2 n j , 0 Im N j , 0 2 k 0 | N j , 0 2 | Re N j , 0 + n j , 0 Re N j , 0 ) = 1 2 ( Re N j , 0 2 | N j , 0 2 | Re N j , 0 + 1 Re N j , 0 ) n j , 0 Im N j , 0 2 2 | N j , 0 2 | Re N j , 0 k 0 .
Re N j , 0 2 = Re 2 N j , 0 Im 2 N j , 0 , Im N j , 0 2 = 2 Re N j , 0 Im N j , 0 ,
| N j , 0 2 | = | N j , 0 | 2 = Re 2 N j , 0 + Im 2 N j , 0
d Re N j , 0 d n j , 0 = 1 2 ( Re 2 N j , 0 Im 2 N j , 0 | N j , 0 | 2 Re N j , 0 + 1 Re N j , 0 ) n j , 0 2 Re N j , 0 Im N j , 0 2 | N j , 0 | 2 Re N j , 0 k 0 = 1 2 Re 2 N j , 0 Im 2 N j , 0 + | N j , 0 | 2 | N j , 0 | 2 Re N j , 0 n j , 0 Im N j , 0 | N j , 0 | 2 k 0 = 1 2 Re 2 N j , 0 Im 2 N j , 0 + Re 2 N j , 0 + Im 2 N j , 0 | N j , 0 | 2 Re N j , 0 n j , 0 Im N j , 0 | N j , 0 | 2 k 0 = Re N j , 0 | N j , 0 | 2 n j , 0 Im N j , 0 | N j , 0 | 2 k 0 .
d d d n j , 0 | Re ϕ j , 0 = d ( n j , 0 | N j , 0 | 2 Im N j , 0 Re N j , 0 k 0 | N j , 0 | 2 ) .

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