Abstract

Gradient-index filters from Lys & Optik were developed on the basis of an inverse Fourier transformation of the desired spectral characteristics. Filter designs with steep skirts and high reflection are possible because of a new correction method where analytical calculations for rugate structures are used as a basis for the initial corrections of the input characteristics. Overlaying of the quintic matching layers and successive approximation methods are used for the suppression of ripples on the transmission curve resulting from misadaptation between the gradient-index thin film and the surrounding media.

© 1992 Optical Society of America

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References

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  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). An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario KlA OS2, Canada.
  2. L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974). An English translation is available (see Ref. 1).
  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). An English translation is available (see Ref. 1).
  4. 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]
  5. B. G. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. 27, 1998–2005 (1988).
    [CrossRef] [PubMed]
  6. H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1986), Chap. 2, pp. 11–48.
  7. W. H. Southwell, “Spectral response calculations of rugate filters using coupled-wave theory,” J. Opt. Soc. Am. A 5, 1558–1564 (1988).
    [CrossRef]
  8. W. H. Southwell, R. L. Hall, “Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,” Appl. Opt. 28, 2949–2951 (1989).
    [CrossRef] [PubMed]

1989 (1)

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). An English translation is available (see Ref. 1).

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). An English translation is available (see Ref. 1).

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). An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario KlA OS2, Canada.

Bovard, B. G.

Dobrowolski, J. A.

Hall, R. L.

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). An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario KlA OS2, Canada.

Lowe, D.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1986), Chap. 2, pp. 11–48.

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). An English translation is available (see Ref. 1).

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974). An English translation is available (see Ref. 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). An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario KlA OS2, Canada.

Southwell, W. H.

Appl. Opt. (3)

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

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). An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario KlA OS2, Canada.

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974). An English translation is available (see Ref. 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). An English translation is available (see Ref. 1).

J. Opt. Soc. Am. A (1)

Other (1)

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1986), Chap. 2, pp. 11–48.

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Figures (10)

Fig. 1
Fig. 1

Spectral characteristic for a filter with steep skirts, high reflection, and a complex function.

Fig. 2
Fig. 2

Refractive-index profile obtained from calculations of the curve in Fig. 1. The total optical thickness is 10 μm, and the acceptable limits of the refractive index have been chosen to be 2.35 and 1.3.

Fig. 3
Fig. 3

Dashed curve shows the transmission corresponding to the refractive-index profile in Fig. 2. The solid curve shows the transmission curve that is obtained after there successive approximations. It is obvious that the transmission curve differs significantly from the desired characteristics in the transition regions.

Fig. 4
Fig. 4

Three successive approximations that have caused the refractive-index profile to drop partially outside the allowed index range.

Fig. 5
Fig. 5

Corrected version of the transmission curve in Fig. 1. By comparison it is clearly seen that the steep skirts and high reflection require correction.

Fig. 6
Fig. 6

Transmission curve obtained before (dashed curve) and after (solid curve) three successive approximations that correspond to Fig. 5. In comparison with Fig. 3 it is evident that the result obtained has been improved remarkably as a result of the developed initial correction method. In general the ripple is lower, and the skirts are steeper and placed more precisely.

Fig. 7
Fig. 7

Refractive-index profile that corresponds to the solid curve in Fig. 6. In Fig. 4 the refractive-index profile drops outside the allowed range. This case it remains within it, which shows clearly that the new correction methods make designing easier in more than one way.

Fig. 8
Fig. 8

Choosing surrounding media that have refractive indices in the range of, e.g., 1.5 introduces ripples caused by misadaptation at the boundaries on the transmission curve.

Fig. 9
Fig. 9

Refractive-index profile when quintic matching layers are overlayed at the final 0.5 μm at each end.

Fig. 10
Fig. 10

Final transmission curve corresponding to the refractive-index profile in Fig. 9. It shows how it is possible to reduce the ripple by overlaying the quintic matching layers. The transmission curve should be compared with the solid curve in Fig. 6.

Equations (22)

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- d n d x 1 2 n exp ( i k x ) d x = Q ( k ) exp [ i Φ ( k ) ] ,
k = 2 π / λ ,
x = 2 0 z n ( u ) d u ,
Q ( k ) = { 1 / 2 [ 1 / T ( k ) - T ( k ) ] } 1 / 2 .
Q ( k ) = { - ln [ T ( k ) ] } 1 / 2 .
Φ ( k ) = π k k min + k max - π 2 sin ( N π k - k min k max - k min ) ,
n ( x ) = exp { 2 π 0 Q ( k ) k sin [ Φ ( k ) - k x ] d k } .
Q ( k ) j = Q ( k ) j - 1 + [ Q o ( k ) - Q a ( k ) ] ,
if Q ( k ) j < 0 ,             then             Q ( k ) j = 0.
n i = n a + n p 2 sin ( k i x - Φ i ) ,
k i = 2 π × n a / λ i .
n p ( λ 1 ) = 2 n a λ 1 π O T arctanh { [ 1 - T ( λ 1 ) ] 1 / 2 } ,
n a = ( n H · n L ) 1 / 2 ,
R ( λ ) = β 2 sinh 2 ( s L ) s 2 cosh 2 ( s L ) + ( α / 2 ) 2 sinh 2 ( s L ) ,
β = π n p / ( 2 λ ) ,
α = 4 π n a ( 1 / λ - 1 / λ 1 ) ,
L = O T / n a ,
s = [ β 2 - ( α / 2 ) 2 ] 1 / 2 ,
d λ / λ = n p / ( 2 n a ) .
n ( x ) = n s + [ n ( x ) - n s ] ( 6 t 5 - 15 t 4 + 10 t 3 ) ,
t = ( x - x min ) / T             for             x - x min T ,
t = ( x max - x ) / T             for             x max - x T ,

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