Abstract

The Fourier transform technique developed for the design of variable refractive index coatings such as rugate filters is improved to achieve an accurate correspondence between optical properties and the refractive index profile. An application to the design of narrowband reflectors is presented.

© 1990 Optical Society of America

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References

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  1. E. Delano, “Fourier Synthesis of Multilayer Filters,” J. Opt. Soc. Am. 57, 1529–1533 (1967).
    [CrossRef]
  2. L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).
  3. L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).
  4. L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).
  5. J. A. Dobrowolski, D. Lowe, “Optical Thin Film Synthesis Program Based on the Use of Fourier Transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  6. B. G. Bovard, “Derivation of a Matrix Describing a Rugate Dielectric Thin Film,” Appl. Opt. 27, 1998–2005 (1988).
    [CrossRef] [PubMed]
  7. B. G. Bovard, “Fourier Transform Technique Applied to Quarterwave Optical Coatings,” Appl. Opt. 27, 3062–3063 (1988).
    [CrossRef] [PubMed]
  8. B. G. Bovard, “Quarterwave Optical Coatings and Fourier Transform Technique,” in Technical Digest, OSA Annual Meeting (Optical Society of America, Washington, DC, 1988), paper WAA8.
  9. H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986).
    [CrossRef]
  10. S. M. Bozic, Digital and Kalman Filtering. An Introduction to Discrete Time Filtering and Optimum Linear Estimation (Edward Arnold, London, 1979).

1988 (2)

1978 (1)

1976 (1)

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).

1974 (1)

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

1968 (1)

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

1967 (1)

Bovard, B. G.

B. G. Bovard, “Fourier Transform Technique Applied to Quarterwave Optical Coatings,” Appl. Opt. 27, 3062–3063 (1988).
[CrossRef] [PubMed]

B. G. Bovard, “Derivation of a Matrix Describing a Rugate Dielectric Thin Film,” Appl. Opt. 27, 1998–2005 (1988).
[CrossRef] [PubMed]

B. G. Bovard, “Quarterwave Optical Coatings and Fourier Transform Technique,” in Technical Digest, OSA Annual Meeting (Optical Society of America, Washington, DC, 1988), paper WAA8.

Bozic, S. M.

S. M. Bozic, Digital and Kalman Filtering. An Introduction to Discrete Time Filtering and Optimum Linear Estimation (Edward Arnold, London, 1979).

Delano, E.

Dobrowolski, J. A.

Kard, P.

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

Lowe, D.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986).
[CrossRef]

Sossi, L.

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

Appl. Opt. (3)

Eesti. NSV Tead. Akad. Toim. Fuus. Mat. (3)

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti. NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

J. Opt. Soc. Am. (1)

Other (3)

B. G. Bovard, “Quarterwave Optical Coatings and Fourier Transform Technique,” in Technical Digest, OSA Annual Meeting (Optical Society of America, Washington, DC, 1988), paper WAA8.

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986).
[CrossRef]

S. M. Bozic, Digital and Kalman Filtering. An Introduction to Discrete Time Filtering and Optimum Linear Estimation (Edward Arnold, London, 1979).

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Figures (13)

Fig. 1
Fig. 1

Reflectance computed by the Fourier transform technique (diamonds) compared with reflectance obtained by matrix multiplication (solid line). The reflectance is calculated at normal incidence and the parameters are nh = 1.55, nl = 1.45, and q = 50.

Fig. 2
Fig. 2

Phase factor of the Fourier transform of the logarithmic derivative of a quarterwave stack (HL)4. H stands for a quarterwave of a high index material and L for a quarterwave of low index material. The outer media have the same refractive index as the high index layer.

Fig. 3
Fig. 3

Refractive index profile obtained by Fourier transform of the reflectance of a quarterwave stack. The calculations are performed assuming a phase factor such as the one presented in Fig. 2. The original design is (HL)50 between two high index media. The original high refractive index is 1.55 and the original low refractive index is 1.45.

Fig. 4
Fig. 4

Refractive index profile obtained by modified Fourier transform of the reflectance of a quarterwave stack. The calculations are performed with Delano’s Q-function and assuming a phase factor such as the one presented in Fig. 2. The original design is (HL)50 between two high index media. The original high refractive index is 1.55 and the original low refractive index is 1.45.

Fig. 5
Fig. 5

Refractive index profile obtained by the modified Fourier transform technique (example 1). The total optical thickness is 19 μm. The desired reflectance is 90% and the desired halfwidth is 2.5%.

Fig. 6
Fig. 6

Reflectance curve corresponding to the index profile shown in Fig. 5 (example 1).

Fig. 7
Fig. 7

Refractive index of a reflector of period 8. The optical thickness of each layer is an eighthwave: 125 nm.

Fig. 8
Fig. 8

Reflectance curve of a reflector of period 8. The desired reflectance is 90% and the desired halfwidth is 2.5% (example 2).

Fig. 9
Fig. 9

Refractive index profile of the first 28-μm layer of example 3; no apodization.

Fig. 10
Fig. 10

Reflectance curve of the first 28-μm layer of example 3; no apodization. Desired reflectance 99.9%; obtained reflectance 99.72%.

Fig. 11
Fig. 11

Refractive index profile of the second 28-μm layer of example 3; Kaiser apodization.

Fig. 12
Fig. 12

Reflectance curve of the second 28-μm layer of example 3; Kaiser apodization. Desired reflectance 99.9%; obtained reflectance 91.68%.

Fig. 13
Fig. 13

Reflectance curve of the third 28-μm layer of example 3; no apodization. Desired reflectance 85%; obtained reflectance 91.24%.

Tables (1)

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Table I Use of the Modified Fourier Transform in the Design of Periodic Single Band Reflectors

Equations (23)

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r ( x ) = r · n = - p p - 1 [ δ ( x - 4 n + 3 4 σ 0 ) - δ ( x - 4 n + 1 4 σ 0 ) ] ,
r = 1 2 ln η h η l .
Q ( σ ) exp i Φ ( σ ) = - i r sin ( π σ λ 0 q ) cos ( π 2 σ λ 0 ) = - i r sin ( π q g ) cos ( π 2 g ) ,
R = tanh 2 Q Q = 1 2 ln 1 + R 1 - R
r ( x ) = - + Q ( σ ) exp [ i Φ ( σ ) + 2 i π σ x ] d σ ,
[ cos ( q γ ) i E - 1 sin ( q γ ) i E sin ( q γ ) cos ( q γ ) ] ,
sin ( γ 2 ) = cosh r · sin ( π 2 g ) ,
( E η h ) 2 = cos ( π g / 2 ) - tanh ( r ) cos ( π g / 2 ) + tanh ( r ) ,
R T = | sinh ( r ) sin ( q γ ) cos ( γ 2 ) | | r sin ( q r ) cos ( γ 2 ) | ,
tan ( Ψ ) tan ( Φ - π g q )
Φ π g q - γ q + π / 2 + m π ,
Φ π / 2 + m π .
v = 2 σ 0 π sin - 1 ( cosh r · sin π 2 g ) ,
Q ( v ) exp [ i Φ ( v ) ] = - + η ( x ) 2 η ( x ) exp ( - 2 i π v x ) d x ,
Q = R T .
tan ( Ψ ) tan ( Φ - 2 π v x 0 ) .
r = 1 2 ln ( 1 + sin π Δ g 0 2 1 - sin π Δ g 0 2 ) ,
v = σ 0 + 2 i π σ 0 cosh - 1 ( cosh r · sin π 2 g )
R ( v 0 ) = tanh 2 ( 2 q r ) Q 0 = 1 2 ln ( 1 + R ( v 0 ) 1 - R ( v 0 ) ) = 2 q r .
Q 0 = 1 2 ln ( 1 + R 0 1 - R 0 ) = 2 q r ,
cos ( π γ 2 Q 0 ) = cosh r · cos ( π 2 Δ g 0 ) ,
r = π Δ g 0 / 2 1 - π 2 8 Δ g 0 + π 2 8 Q 0 2 .
x 0 σ 0 = Q 0 4 r .

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