Abstract

We reduce the sidelobes that are present in the bandpass zones of rugate filters, using antireflection coatings to design dichroic rugate filters. Our designs are based on the formalism of equivalent systems that allow us to consider a rugate filter a simple equivalent layer with equivalent optical admittance and equivalent phase thickness functions. Rugate filters with apodization were considered.

© 2006 Optical Society of America

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References

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  1. W. H. Southwell, "Spectral response calculations of rugate filters using coupled-wave theory," J. Opt. Soc. Am. A 5, 1558-1564 (1988).
    [CrossRef]
  2. W. H. Southwell, "Using apodization functions to reduce sidelobes in rugate filters," Appl. Opt. 28, 5091-5094 (1989).
    [CrossRef] [PubMed]
  3. W. H. Southwell and R. L. Hall, "Rugate filter sidelobe suppression using quintic and rugated quintic matching layers," Appl. Opt. 28, 2949-2951 (1989).
    [CrossRef] [PubMed]
  4. A. Thelen, "Design of optical minus filters," J. Opt. Soc. Am. 61, 365-369 (1971).
    [CrossRef]
  5. F. Aguayo-Ríos, F. Villa-Villa, and J. A. Gaspar-Armenta, "One-dimensional photonic crystals of inhomogeneous thin films: band structure of rugate filters," Opt. Commun. 244, 259-267 (2005).
    [CrossRef]
  6. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, 1989).
  7. F. Villa-Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
    [CrossRef]
  8. J. A. Gaspar-Armenta and F. Villa-Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am B 21, 405-412 (2004).
    [CrossRef]
  9. R. Jacobson, "Light reflection from films of continuously varying refractive index," in Progress in Optics, E.Wolf, ed. (North-Holland, 1966), Vol. 5, pp. 247-286.
    [CrossRef]
  10. H. A. Abu-Sadia, A. I. Al-Sharif, and I. O. Abu Aljarayesh, "Rugate filter sidelobe suppression using half-apodization," Appl. Opt. 32, 4831-4835 (1993).
    [CrossRef]
  11. K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
    [CrossRef]

2005 (1)

F. Aguayo-Ríos, F. Villa-Villa, and J. A. Gaspar-Armenta, "One-dimensional photonic crystals of inhomogeneous thin films: band structure of rugate filters," Opt. Commun. 244, 259-267 (2005).
[CrossRef]

2004 (2)

J. A. Gaspar-Armenta and F. Villa-Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am B 21, 405-412 (2004).
[CrossRef]

K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
[CrossRef]

2003 (1)

F. Villa-Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

1993 (1)

1989 (2)

1988 (1)

1971 (1)

Abu-Sadia, H. A.

Aguayo-Ríos, F.

F. Aguayo-Ríos, F. Villa-Villa, and J. A. Gaspar-Armenta, "One-dimensional photonic crystals of inhomogeneous thin films: band structure of rugate filters," Opt. Commun. 244, 259-267 (2005).
[CrossRef]

Aljarayesh, I. O. Abu

Al-Sharif, A. I.

Gaspar-Armenta, J. A.

F. Aguayo-Ríos, F. Villa-Villa, and J. A. Gaspar-Armenta, "One-dimensional photonic crystals of inhomogeneous thin films: band structure of rugate filters," Opt. Commun. 244, 259-267 (2005).
[CrossRef]

J. A. Gaspar-Armenta and F. Villa-Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am B 21, 405-412 (2004).
[CrossRef]

F. Villa-Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

Hall, R. L.

Jacobson, R.

R. Jacobson, "Light reflection from films of continuously varying refractive index," in Progress in Optics, E.Wolf, ed. (North-Holland, 1966), Vol. 5, pp. 247-286.
[CrossRef]

Kaminska, K.

K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
[CrossRef]

Kimura, K.

K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, 1989).

Ramos-Mendieta, F.

F. Villa-Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

Robbie, K.

K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
[CrossRef]

Southwell, W. H.

Suzuki, M.

K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
[CrossRef]

Taga, Y.

K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
[CrossRef]

Thelen, A.

Villa-Villa, F.

F. Aguayo-Ríos, F. Villa-Villa, and J. A. Gaspar-Armenta, "One-dimensional photonic crystals of inhomogeneous thin films: band structure of rugate filters," Opt. Commun. 244, 259-267 (2005).
[CrossRef]

J. A. Gaspar-Armenta and F. Villa-Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am B 21, 405-412 (2004).
[CrossRef]

F. Villa-Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

Appl. Opt. (3)

J. Appl. Phys. (1)

K. Kaminska, M. Suzuki, K. Kimura, Y. Taga, and K. Robbie, "Simulating structure and optical response of vacuum evaporated porous rugate filters," J. Appl. Phys. 95, 3055-3062 (2004).
[CrossRef]

J. Opt. Soc. Am B (1)

J. A. Gaspar-Armenta and F. Villa-Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am B 21, 405-412 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

F. Villa-Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

F. Aguayo-Ríos, F. Villa-Villa, and J. A. Gaspar-Armenta, "One-dimensional photonic crystals of inhomogeneous thin films: band structure of rugate filters," Opt. Commun. 244, 259-267 (2005).
[CrossRef]

Other (2)

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, 1989).

R. Jacobson, "Light reflection from films of continuously varying refractive index," in Progress in Optics, E.Wolf, ed. (North-Holland, 1966), Vol. 5, pp. 247-286.
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Periodically inhomogeneous thin film (rugate filter). represents the mean refractive index and 2Δn the peak-to-peak amplitude. ni and ns stand for the refractive indices of the incident and transmitting media, respectively.

Fig. 2
Fig. 2

Theorem of equivalence expressed graphically.

Fig. 3
Fig. 3

Rugate with symmetric period. The symmetry is along the vertical axis (dotted line), which can be outside the origin.

Fig. 4
Fig. 4

Equivalent optical admittance η e (normalized to the admittance of vacuum y) and phase thickness δ e of the symmetric rugate given in Fig. 2 under normal incidence ( β ¯ = 0 ) . Six hundred sublayers were considered for one period.

Fig. 5
Fig. 5

Index profile for a 50-cycle rugate filter upon a glass substrate (ns = 1.52) and with vacuum as the incidence medium (ni = 1); mean refractive index = 1.8 and amplitude Δn = 0.35 were used.

Fig. 6
Fig. 6

Reflectance of the rugate filter given in Fig. 5.

Fig. 7
Fig. 7

Schematic showing AR coating (layers with optical admittances η1 and η2 between the rugate and vacuum, and η3 and η4 between the rugate and the glass) on both sides of the inhomogeneous film.

Fig. 8
Fig. 8

Matching double-AR system.

Fig. 9
Fig. 9

Fifty-cycle rugate filter on a glass substrate and vacuum as the incidence medium with double-AR coating. The materials employed for the AR coatings were TiO2 and MgF2. They produce zero reflectance at ω ¯ = 1.5 .

Fig. 10
Fig. 10

Fifty-cycle rugate filter with Gaussian apodization.

Fig. 11
Fig. 11

Equivalent optical admittance η e and phase thickness δ e of the apodized system given by Eq. (14). We have normalized the equivalent admittance by dividing the admittance of vacuum by y.

Fig. 12
Fig. 12

Reflectance of the rugate filter with Gaussian apodization.

Fig. 13
Fig. 13

Reflectance of the apodized rugate filter with double-AR matching layers. The AR coatings produce zero reflectance at ω ¯ = 1.93 according to our design.

Equations (14)

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m ( η j , δ j ) = [ cos ( δ j ) i η j sin ( δ j ) i η j sin ( δ j ) cos ( δ j ) ] ,
δ j = 2 π Λ k ¯ Z j d j
η j = { y k ¯ zj / ω ¯ TE   polarization y n j     2 ω ¯ / k ¯ zj TM   polarization
k ¯ Z j = ( η j     2 ω ¯ 2 β ¯ 2 ) 1 / 2 ,
M ( η e , δ e ) = [ cos ( δ e ) i η e sin ( δ e ) i η e sin ( δ e ) cos ( δ e ) ] .
cos ( δ e ) = cos ( δ p ) cos ( δ q ) ρ + sin ( δ p ) sin ( δ q ) ,
η e = η p sin ( δ e ) [ sin ( δ p ) cos ( δ q ) + ρ + cos ( δ p ) sin ( δ q ) ρ sin ( δ q ) ] ,
ρ + = 1 2 ( η p η q + η q η p ) ,
ρ = 1 2 ( η p η q η q η p ) .
n ( z ) = n ¯ + Δ n cos ( 4 π Λ z ) ,
tan 2 δ a = ( η t η 0 ) ( η b     2 η 0 η t ) η a     2 ( η a     2 η t η b     2 η 0 ) ( η 0 η t η a     2 ) ,
tan 2 δ b = ( η t η 0 ) ( η 0 η t η a     2 ) η b     2 ( η a     2 η t η b     2 η 0 ) ( η b     2 η 0 η t ) ,
δ j = 2 π ω ¯ Λ n j d j ,
n ( z ) = n ¯ + Δ n cos ( 4 π Λ z ) exp [ ( z 3 ) 2 ] .

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