Abstract

We derive a simple equation to predict the center-wavelength shift of a Fabry-Perot type narrow-bandpass filter by using the conventional characteristic matrix method and the elastic strain model as the temperature varies. We determine the thermal expansion coefficient of substrate from the zero-shift condition of the center wavelength of the filter. The calculated shifts are in a good agreement with the experimental ones, in which the narrow-bandpass filters are prepared by plasma ion-assisted deposition on four substrates with different thermal expansion coefficients.

© 2004 Optical Society of America

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References

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  1. M. Gilo, N. Croitoru, “Properties of TiO2 films prepared by ion-assisted deposition using a gridless end-Hall ion source,” Thin Solid Films 283, 84–89 (1996).
    [CrossRef]
  2. R.Y. Tsai, C.S. Chang, C.W. Chu, T. Chen, F. Dai, D. Lin, S. Yan, A. Chang, “Thermally stable narrow-bandpass filter prepared by reactive ion-assisted sputtering,” Appl. Opt. 40, 1593–1598 (2001).
    [CrossRef]
  3. A. Zöller, R. Götzelmann, K. Matl, D. Cushing, “Temperature-stable bandpass filters deposited with plasma ion-assisted deposition,” Appl. Opt. 35, 5609–5612 (1996).
    [CrossRef] [PubMed]
  4. H. Takashashi, “Temperature stability of thin-film narrow-bandpass filters produced by ion-assisted deposition,” Appl. Opt. 34, 667–675 (1995).
    [CrossRef] [PubMed]
  5. H.A. Macleod, Thin-Film Optical Filters, 3rd ed. (IoP, Bristol, 2001).
    [CrossRef]
  6. Essential Macleod is commercial thin film software from Thin Film Center Inc, USA.

2001

1996

A. Zöller, R. Götzelmann, K. Matl, D. Cushing, “Temperature-stable bandpass filters deposited with plasma ion-assisted deposition,” Appl. Opt. 35, 5609–5612 (1996).
[CrossRef] [PubMed]

M. Gilo, N. Croitoru, “Properties of TiO2 films prepared by ion-assisted deposition using a gridless end-Hall ion source,” Thin Solid Films 283, 84–89 (1996).
[CrossRef]

1995

Chang, A.

Chang, C.S.

Chen, T.

Chu, C.W.

Croitoru, N.

M. Gilo, N. Croitoru, “Properties of TiO2 films prepared by ion-assisted deposition using a gridless end-Hall ion source,” Thin Solid Films 283, 84–89 (1996).
[CrossRef]

Cushing, D.

Dai, F.

Gilo, M.

M. Gilo, N. Croitoru, “Properties of TiO2 films prepared by ion-assisted deposition using a gridless end-Hall ion source,” Thin Solid Films 283, 84–89 (1996).
[CrossRef]

Götzelmann, R.

Lin, D.

Macleod, H.A.

H.A. Macleod, Thin-Film Optical Filters, 3rd ed. (IoP, Bristol, 2001).
[CrossRef]

Matl, K.

Takashashi, H.

Tsai, R.Y.

Yan, S.

Zöller, A.

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Figures (3)

Fig. 1.
Fig. 1.

The simulated transmission spectrum of the Fabry-Perot type narrow-bandpass filters on a fused silica substrate as a function of wavelength and temperature deviation, where the structure of basic stack is [air | (LH)10 (2L)4 (HL)10 | substrate] (L=SiO2, H=Ta2O5, λ 0=1550 nm).

Fig. 2.
Fig. 2.

The simulated thermal expansion coefficient (α S ) of the substrate for zero-shift of the center wavelength as a function of the order of cavity (q) for low- and high-index cavity filters (Low-index: SiO2, High-index: Ta2O5).

Fig. 3.
Fig. 3.

The measured center-wavelength shift of the filters against the temperature deviation.

Tables (3)

Tables Icon

Table 1. Δλ/ΔT from Takashashi’s experimental results [4] and Eq. (18) for various substrates.

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Table 2. Optical and thermal properties of glasses and thin films in this study.

Tables Icon

Table 3. Δλ/ΔT from the measured and the calculated results for various substrates

Equations (18)

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n f T = n f 0 + [ ( d n d T ) f Δ T + ( 1 n f 0 ( d n d T ) f Δ T ) ( A f Δ T 1 + ( 3 α f + A f ) Δ T ) ]
d f T = d f 0 { 1 + ( α f B f ) Δ T } ,
A f = 2 ( 1 2 v f ) ( 1 v f ) ( α s α f ) and B f = 2 v f ( 1 v f ) ( α s α f ) ,
[ M f ] = [ cos δ f ( i sin δ f ) n f i n f sin δ f cos δ f ]
δ f = 2 π λ n f d f .
δ f T = δ f 0 + ε f ,
[ M ] T = [ M 11 M 12 M 21 M 22 ] = ( [ M C ] [ M D ] ) p ( 2 [ M C ] ) q ( [ M D ] [ M C ] ) p ,
Y T = C T B T ,
[ B C ] T = [ M ] T [ 1 n S T ] ,
R T = n 0 T Y T n 0 T + Y T 2 ,
R min T = n 0 T n s T n 0 T + n s T 2 ,
Y T = n S T .
P ε H + Q ε L = 0 ,
P = n H T n L T and Q = ( n L T ) 2 + { ( n H T ) 2 ( n L T ) 2 } q
P = ( n L T ) 2 + { ( n H T ) 2 ( n L T ) 2 } q and Q = n H T n L T
ε f = 2 π λ T n f T d f T π 2
λ T = 4 [ P ( n H T d H T ) + Q ( n L T d L T ) P + Q ]
Δ λ λ 0 = λ T λ 0 λ 0 = 4 λ 0 [ P ( n H T d H T ) + Q ( n L T d L T ) P + Q ] 1

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