Abstract

The Fourier transform method to design graded-index optical filters, that relates the desired reflection spectrum and the index profile through the use of a Q function, has two important drawbacks: (1) It relies on approximate Q functions, and (2) it does not account for the dispersion of the index of refraction. The former is usually addressed by an iterative correctionprocess. We propose to address the latter by scaling the wavelength in the Fourier transform by the optical thickness of the filter and to multiply the Q function by a wavelength-dependent correction factor. We demonstrate the high effectiveness of this approach by the performance of optical filters designed with such correction factors using the optical properties of SiO2/TiO2 mixtures.

© 2007 Optical Society of America

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References

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  1. B. G. Bovard, "Rugate filter theory: an overview," Appl. Opt. 32, 5427-5442 (1993).
    [CrossRef] [PubMed]
  2. D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
    [CrossRef]
  3. E. Delano, "Fourier synthesis of multilayer filters," J. Opt. Soc. Am. 57, 1529-1533 (1967).
    [CrossRef]
  4. L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 23, 229-237 (1974). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).
  5. J. A. Dobrowolski and D. Lowe, "Optical thin film synthesis program based on the use of Fourier transforms," Appl. Opt. 17, 3039-3050 (1978).
    [CrossRef] [PubMed]
  6. P. G. Verly and J. A. Dobrowolski, "Iterative correction process for optical thin film systhesis with the Fourier transform method," Appl. Opt. 29, 3672-3684 (1990).
    [CrossRef] [PubMed]
  7. H. Fabricius, "Gradient-index filters: Conversion into a two-index solution by taking into account dispersion," Appl. Opt. 31, 5216-5220 (1992).
    [CrossRef] [PubMed]
  8. D. Poitras, S. Larouche, and L. Martinu, "Design and plasma deposition of dispersion-corrected multiband rugate filters," Appl. Opt. 41, 5249-5255 (2002).
    [CrossRef] [PubMed]
  9. 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]
  10. L. Sossi, "On the theory of the synthesis of multilayer dielectric light filters," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 25, 171-176 (1976). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).
  11. P. G. Verly, J. A. Dobrowolski, W. J. Wild, and R. L. Burton, "Synthesis of high rejection filters with the Fourier transform method," Appl. Opt. 28, 2864-2875 (1989).
    [CrossRef] [PubMed]
  12. B. G. Bovard, "Derivation of a matrix describing a rugate dielectric thin film," Appl. Opt. 27, 1998-2005 (1988).
    [CrossRef] [PubMed]
  13. B. G. Bovard, "Fourier transform technique applied to quarterwave optical coatings," Appl. Opt. 27, 3062-3063 (1988).
    [CrossRef] [PubMed]
  14. R. Szipocs and A. Koházi-Kis, "Theory and design of chirped dielectric laser mirrors," Appl. Phys. B 65, 115-135 (1997).
    [CrossRef]
  15. H. Chang, S.-S. Lee, M.-R. Chol, and S. Lim, "Inhomogeneous optical filter design with the use of a Riccati equation," Microwave Opt. Technol. Lett. 22, 140-144 (1999).
    [CrossRef]
  16. S. Larouche and L. Martinu, "A simple way to include dispersion in the design of graded-index optical filters by the fourier transform method," in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), p. TuB6.

2004

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]

2002

1999

D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
[CrossRef]

H. Chang, S.-S. Lee, M.-R. Chol, and S. Lim, "Inhomogeneous optical filter design with the use of a Riccati equation," Microwave Opt. Technol. Lett. 22, 140-144 (1999).
[CrossRef]

1997

R. Szipocs and A. Koházi-Kis, "Theory and design of chirped dielectric laser mirrors," Appl. Phys. B 65, 115-135 (1997).
[CrossRef]

1993

1992

1990

1989

1988

1978

1976

L. Sossi, "On the theory of the synthesis of multilayer dielectric light filters," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 25, 171-176 (1976). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

1974

L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 23, 229-237 (1974). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

1967

Bovard, B. G.

Burton, R. L.

Chang, H.

H. Chang, S.-S. Lee, M.-R. Chol, and S. Lim, "Inhomogeneous optical filter design with the use of a Riccati equation," Microwave Opt. Technol. Lett. 22, 140-144 (1999).
[CrossRef]

Chol, M.-R.

H. Chang, S.-S. Lee, M.-R. Chol, and S. Lim, "Inhomogeneous optical filter design with the use of a Riccati equation," Microwave Opt. Technol. Lett. 22, 140-144 (1999).
[CrossRef]

Delano, E.

Dobrowolski, J. A.

Fabricius, H.

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]

Koházi-Kis, A.

R. Szipocs and A. Koházi-Kis, "Theory and design of chirped dielectric laser mirrors," Appl. Phys. B 65, 115-135 (1997).
[CrossRef]

Larouche, S.

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]

D. Poitras, S. Larouche, and L. Martinu, "Design and plasma deposition of dispersion-corrected multiband rugate filters," Appl. Opt. 41, 5249-5255 (2002).
[CrossRef] [PubMed]

S. Larouche and L. Martinu, "A simple way to include dispersion in the design of graded-index optical filters by the fourier transform method," in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), p. TuB6.

Lee, S.-S.

H. Chang, S.-S. Lee, M.-R. Chol, and S. Lim, "Inhomogeneous optical filter design with the use of a Riccati equation," Microwave Opt. Technol. Lett. 22, 140-144 (1999).
[CrossRef]

Lim, S.

H. Chang, S.-S. Lee, M.-R. Chol, and S. Lim, "Inhomogeneous optical filter design with the use of a Riccati equation," Microwave Opt. Technol. Lett. 22, 140-144 (1999).
[CrossRef]

Lowe, D.

Martinu, L.

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]

D. Poitras, S. Larouche, and L. Martinu, "Design and plasma deposition of dispersion-corrected multiband rugate filters," Appl. Opt. 41, 5249-5255 (2002).
[CrossRef] [PubMed]

D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
[CrossRef]

S. Larouche and L. Martinu, "A simple way to include dispersion in the design of graded-index optical filters by the fourier transform method," in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), p. TuB6.

Poitras, D.

D. Poitras, S. Larouche, and L. Martinu, "Design and plasma deposition of dispersion-corrected multiband rugate filters," Appl. Opt. 41, 5249-5255 (2002).
[CrossRef] [PubMed]

D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
[CrossRef]

Rats, D.

D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
[CrossRef]

Soro, J. M.

D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
[CrossRef]

Sossi, L.

L. Sossi, "On the theory of the synthesis of multilayer dielectric light filters," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 25, 171-176 (1976). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 23, 229-237 (1974). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

Szipocs, R.

R. Szipocs and A. Koházi-Kis, "Theory and design of chirped dielectric laser mirrors," Appl. Phys. B 65, 115-135 (1997).
[CrossRef]

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]

Verly, P. G.

von Stebut, J.

D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
[CrossRef]

Wild, W. J.

Appl. Opt.

Appl. Phys. B

R. Szipocs and A. Koházi-Kis, "Theory and design of chirped dielectric laser mirrors," Appl. Phys. B 65, 115-135 (1997).
[CrossRef]

Eesti NSV Teaduste Akadeemia Toimetised Füüsika

L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 23, 229-237 (1974). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

L. Sossi, "On the theory of the synthesis of multilayer dielectric light filters," Eesti NSV Teaduste Akadeemia Toimetised Füüsika , Matemaatika 25, 171-176 (1976). English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

J. Opt. Soc. Am.

J. Vac. Sci. Technol. A

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]

Microwave Opt. Technol. Lett.

H. Chang, S.-S. Lee, M.-R. Chol, and S. Lim, "Inhomogeneous optical filter design with the use of a Riccati equation," Microwave Opt. Technol. Lett. 22, 140-144 (1999).
[CrossRef]

Surf. Coat. Technol.

D. Rats, D. Poitras, J. M. Soro, L. Martinu, and J. von Stebut, "Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings," Surf. Coat. Technol. 111, 220-228 (1999).
[CrossRef]

Other

S. Larouche and L. Martinu, "A simple way to include dispersion in the design of graded-index optical filters by the fourier transform method," in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), p. TuB6.

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Figures (6)

Fig. 1
Fig. 1

Index of refraction of SiO 2 / TiO 2 mixtures deposited by plasma-enhanced chemical vapor deposition, and characterized using spectroscopic ellipsometry and spectrophotometry [9].

Fig. 2
Fig. 2

Index profile of a model triple-band rugate filter made of SiO 2 / TiO 2 mixtures from Fig. 1 at the wavelengths of the three bands centered at 450   nm , 550   nm , and 750   nm . As the wavelength increases, the amplitude of the index profile and its total double optical thickness diminish.

Fig. 3
Fig. 3

Effect of dispersion on the transmission spectrum of a model triple-band rugate filter made of SiO 2 / TiO 2 (continuous curve) compared to that of a filter made of dispersionless material (dashed curve) for λ 0 = 550   nm .

Fig. 4
Fig. 4

Transmission spectrum of the triple-band rugate filter made of SiO 2 / TiO 2 (continuous curve) compared to that of a filter made of dispersionless material (dashed curve) using the dispersion-corrected wavelength axis.

Fig. 5
Fig. 5

Reflection spectrum of a SiO 2 / TiO 2 filter designed using the correction factors (thin curve) is closer to the target (thick curve) than that of a filter designed without the correction factors (dashed curve). The index profile is defined at 650   nm .

Fig. 6
Fig. 6

Reflection spectrum of the filter presented in Fig. 5 after three iterations (thin curve) compared to the target (thick curve).

Equations (75)

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SiO 2 / TiO 2
SiO 2 / TiO 2
n ( x )
ln n ( x ) n m = 2 π 0 Q ( σ ) σ sin [ Ψ ( σ ) σ x ] d σ ,
σ = 2 π / λ
n m = n min n max
x = 2 0 z n ( u ) d u ,
Q ( σ ) = 1 2 ln 1 + R ( σ ) 1 R ( σ ) ,     Ψ ( σ ) = 0 ,
n ( z )
n ( z )
n ( x )
λ 0
λ 0
n ( x )
n ( x ) = n m i = 1 3 exp [ Q ¯ i sin ( 2 π x / λ i + ϕ i ) w ( x ) ] ,
Q ¯ i = 1 2 ln n m + Δ n i / 2 n m Δ n i / 2
Δ n i
λ i
ϕ i
w ( x )
SiO 2 / TiO 2
λ 0 = 550   nm
λ 0
1 2 ln n ( x ( λ ) , λ ) n m ( λ ) = 1 π 0 Q ( σ ) σ sin ( Ψ ( σ ) σ x ( λ ) ) d σ
λ 0
σ
Q ( σ )
1 2 ln n ( x ( λ 0 ) , λ 0 ) n m ( λ 0 ) = 1 π 0 Q ( σ ) σ sin ( Ψ ( σ ) σ x ( λ 0 ) ) d σ .
σ x ( λ ) = σ x ( λ 0 ) ,
Q ( σ ) ln [ n ( x ( λ ) , λ ) / n m ( λ ) ] = Q ( σ ) ln [ n ( x ( λ 0 ) , λ 0 ) / n m ( λ 0 ) ] .
σ = 1 S ( λ ) σ ,     λ = S ( λ ) λ ,
λ = 2 π / σ
S ( λ ) = x ( λ 0 ) x ( λ ) .
x ( λ 0 ) / x ( λ )
S ( λ ) = OT ( λ 0 ) OT ( λ ) ,
OT ( λ ) = d / 2 d / 2 n ( u , λ ) d u
OT ( λ )
OT ( λ )
n m ( λ )
S ( λ ) = n m ( λ 0 ) n m ( λ )
Q ( σ ) = K ( λ ) Q ( σ ) ,
K ( λ ) = ln [ n ( x ( λ 0 ) , λ 0 ) / n m ( λ 0 ) ] ln [ n ( x ( λ ) , λ ) / n m ( λ ) ] .
n ( x ( λ 0 ) , λ 0 ) n m ( λ 0 ) n max ( λ 0 ) n m ( λ 0 ) = n m ( λ 0 ) n min ( λ 0 ) ,
n ( x ( λ ) , λ ) n m ( λ ) n max ( λ ) n m ( λ ) = n m ( λ ) n min ( λ ) ,
K ( λ ) = ln n max ( λ 0 ) / n min ( λ 0 ) ln n max ( λ ) / n min ( λ ) .
n max / n min
K ( λ )
S ( λ )
30000   nm
SiO 2 / TiO 2
SiO 2 / TiO 2
n min
n max
λ 0
λ
d λ d λ > 0 .
d λ d λ = d S ( λ ) d λ λ + S ( λ ) d λ d λ ,
d S ( λ ) d λ > S ( λ ) λ .
S ( λ )
d d λ x ( λ 0 ) x ( λ ) > 1 λ x ( λ 0 ) x ( λ ) ,
1 x 2 ( λ ) d x ( λ ) d λ > 1 λ 1 x ( λ ) ,
d d λ 0 z n ( u , λ ) d u < 1 λ 0 z n ( u , λ ) d u .
0 z d n ( u , λ ) d λ d u < 1 λ 0 z n ( u , λ ) d u ,
d n ( u , λ ) / d λ
d n ( λ ) d λ < n ( λ ) λ .
SiO 2 / TiO 2
SiO 2 / TiO 2
450   nm
550   nm
750   nm
SiO 2 / TiO 2
λ 0 = 550   nm
SiO 2 / TiO 2
SiO 2 / TiO 2
650   nm

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