Abstract

The spectral separation of resonances within multiwavelength resonant grating filters is studied through the use of a grating layer with constant structural parameters in combination with single-layer, multimode waveguides and multilayer, multimode waveguides. The use of multilayer, multimode waveguides are shown to provide the ability to control resonance separations for multiwavelength filters from hundreds of nanometers to only a few nanometers.

© 2007 Optical Society of America

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References

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  1. R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
    [CrossRef]
  2. S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32, 2606-2613 (1993).
    [CrossRef] [PubMed]
  3. R. Magnusson and S. S. Wang, "Transmission bandpass guided-mode resonance filters," Appl. Opt. 34, 8106-8109 (1995).
    [CrossRef] [PubMed]
  4. S. Tibuleac and R. Magnusson, "Reflection and transmission guided-mode resonance filters," J. Opt. Soc. Am. A 14, 1617-1626 (1997).
    [CrossRef]
  5. D. K. Jacob, S. C. Dunn, and M. G. Moharam, "Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams," J. Opt. Soc. Am. A 18, 2109-2120 (2001).
    [CrossRef]
  6. A. Sentenac and A. L. Fehrembach, "Angular tolerant resonant grating filters under oblique incidence," J. Opt. Soc. Am. A 22, 475-480 (2005).
    [CrossRef]
  7. Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
    [CrossRef]
  8. S. Boonruang, A. Greenwell, and M. G. Moharam, "Multiline two-dimensional guided-mode resonant filters," Appl. Opt. 45, 5740-5747 (2006).
    [CrossRef] [PubMed]
  9. S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).
  10. L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993).
    [CrossRef]
  11. W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, "Root finding and nonlinear sets of equations," in Numerical Recipes in C++ (Cambridge U. Press, 2002).
  12. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  13. H. A. Macleod, Thin Film Optical Filters, 3rd ed. (Taylor & Francis, 2001).
    [CrossRef]

2006 (1)

2005 (1)

2002 (2)

Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
[CrossRef]

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, "Root finding and nonlinear sets of equations," in Numerical Recipes in C++ (Cambridge U. Press, 2002).

2001 (2)

1997 (1)

1995 (2)

1993 (2)

L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32, 2606-2613 (1993).
[CrossRef] [PubMed]

1992 (1)

R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
[CrossRef]

1956 (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Boonruang, S.

Dunn, S. C.

Fehrembach, A. L.

Flannery, B. P.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, "Root finding and nonlinear sets of equations," in Numerical Recipes in C++ (Cambridge U. Press, 2002).

Gaylord, T. K.

Grann, E. B.

Greenwell, A.

Jacob, D. K.

Li, L.

L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

Liu, Z. S.

Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters, 3rd ed. (Taylor & Francis, 2001).
[CrossRef]

Magnusson, R.

Moharam, M. G.

Pommet, D. A.

Press, W. H.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, "Root finding and nonlinear sets of equations," in Numerical Recipes in C++ (Cambridge U. Press, 2002).

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Sentenac, A.

Teukolsky, S. A.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, "Root finding and nonlinear sets of equations," in Numerical Recipes in C++ (Cambridge U. Press, 2002).

Tibuleac, S.

Vetterling, W. T.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, "Root finding and nonlinear sets of equations," in Numerical Recipes in C++ (Cambridge U. Press, 2002).

Wang, S. S.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
[CrossRef]

J. Mod. Opt. (1)

L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

J. Opt. Soc. Am. A (4)

Sov. Phys. JETP (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Other (2)

H. A. Macleod, Thin Film Optical Filters, 3rd ed. (Taylor & Francis, 2001).
[CrossRef]

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery, "Root finding and nonlinear sets of equations," in Numerical Recipes in C++ (Cambridge U. Press, 2002).

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Figures (3)

Fig. 1
Fig. 1

(Color online) (a) Index distribution and spatial mode profile for the first two modes supported by the single wave-guiding layer GMR structure. (b) Zeroth-order reflection response for the structure calculated using RCWA showing the spectral resonance response. (c) Real modal index distribution of the modes supported by the structure as well as the modal index of the ± 1 tangential diffracted order in free space, whose intersection indicates the location of a resonance.

Fig. 2
Fig. 2

(Color online) (a) Index distribution and spatial mode profile for the first two modes supported by the multilayer wave-guiding layer GMR structure. (b) Zeroth-order reflection response for the structures calculated using RCWA showing the spectral resonance response. (c) Real modal index distribution of the modes supported by the structures as well as the modal index of the ± 1 tangential diffracted order in free space, whose intersection indicates the location of a resonance.

Fig. 3
Fig. 3

(Color online) Sketch of a multilayer slab waveguide where the origin of the coordinate system is located at the substrate interface.

Equations (49)

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cosh ( γ 1 d 1 ) cosh ( γ 2 d 2 ) 1 2 ( γ 1 γ 2 + γ 2 γ 1 ) sinh ( γ 1 d 1 ) × sinh ( γ 2 d 2 ) = 0 ,
( 2 π λ ) 2 μ p ε p = γ p 2 + β g r a t i n g 2 ,
γ p = γ p μ p  for   TE   waveguides , γ p = γ p ε p  for   TM   waveguides .
S ubstrate : x 0
E substrate = F 0 exp ( + γ 0 x ) exp ( j β z ) y ^ ,
H substrate = F 0 ( β ω μ x , 0 x ^ + j γ 0 ω μ z , 0 z ^ ) exp ( + γ s u b x ) × exp ( j β z ) ,
Cover : x i = 1 N d i
E cover = F N + 1 + exp ( γ N + 1 x ) exp ( j β z ) y ^ ,
H cover = F N + 1 + ( β ω μ x , N + 1 x ^ j γ N + 1 ω μ z , N + 1 z ^ ) exp ( γ N + 1 x ) × exp ( j β z ) ,
N   layers : i = 1 p 1 d i x i = 1 p d i ,
0 x p , ( f r o m _ s u b s t r a t e _ s i d e ) d p , ( f r o m _ s u b s t r a t e _ s i d e )
E f i l m , p = { F p exp [ γ p ( x d p ) ] + F p + exp ( γ p x ) } × exp ( j β z ) y ^ ,
H f i l m , p = ( β ω μ x { F p exp [ γ p ( x d p ) ] + F p + exp ( γ p x ) } x ^ + j γ p ω μ z { F p exp [ γ p ( x d p ) ] F p + exp ( γ p x ) } z ^ ) exp ( j β z ) .
F p + = F p s p X p , F p = F p + c p X p ,
X p = exp ( γ p d p ) , γ p = γ p / μ z , p .
E f i l m , p = F p { exp [ γ p ( x p d p ) ] + s p X p exp ( γ p x p ) } × exp ( j β z ) y ^ ,
H f i l m , p = F p ( β ω μ x { exp [ γ p ( x p d p ) ] + s p X p exp ( γ p x p ) } x ^ + j γ p ω { exp [ γ p ( x p d p ) ] s p X p exp ( γ p x p ) } z ^ ) × exp ( j β z ) ,
E f i l m , p = F p + { c p X p exp [ γ p ( x p d p ) ] + exp ( γ p x p ) } × exp ( j β z ) y ^ ,
H f i l m , p = F p + ( β ω μ x { c p X p exp [ γ p ( x p d p ) ] + exp ( γ p x p ) } x ^ + j γ p ω { c p X p exp [ γ p ( x p d p ) ] exp ( γ p x p ) } z ^ ) exp ( j β z ) .
S u b s t r a t e S i d e
F p 1 ( 1 + s p 1 X p 1 2 ) = F p X p ( 1 + s p ) ,
γ p 1 F p 1 ( 1 s p 1 X p 1 2 ) = γ p F p X p ( 1 s p ) ,
C o v e r S i d e
F p + 1 ( 1 + c p + 1 X p + 1 2 ) = F p X p ( 1 + c p ) ,
γ p + 1 F p + 1 ( 1 c p + 1 X p + 1 2 ) = γ p F p X p ( 1 c p ) .
S u b s t r a t e S i d e
F p 1 = F p X p ( 1 + s p ) ( 1 + s p 1 X p 1 2 ) ,
C o v e r S i d e
F p + 1 + = F p + X p ( 1 + c p ) ( 1 + c p + 1 X p + 1 2 ) .
s p γ p ( 1 + s p 1 X p 1 2 ) γ p 1 ( 1 s p 1 X p 1 2 ) γ p ( 1 + s p 1 X m 1 2 ) + γ m 1 ( 1 s p 1 X p 1 2 ) ,
c p γ p ( 1 + c p + 1 X p + 1 2 ) γ p + 1 ( 1 c p + 1 X p + 1 2 ) γ p ( 1 + c p + 1 X p + 1 2 ) + γ p + 1 ( 1 c p + 1 X p + 1 2 ) .
s p = γ p γ p 1 s γ p + γ p 1 s , s 0 = 0 ,
c p = γ p γ p + 1 c γ p + γ p + 1 c , c N + 1 = 0 ,
γ p 1 s = γ p 1 ( 1 s p 1 X p 1 2 ) ( 1 + s p 1 X p 1 2 ) ,
γ p + 1 c = γ p + 1 ( 1 c p + 1 X p + 1 2 ) ( 1 + c p + 1 X p + 1 2 ) .
S u b s t r a t e S i d e
F r e f 1 ( 1 + s r e f 1 X r e f 1 2 ) = F r e f X r e f + F r e f + ,
γ r e f 1 F r e f 1 ( 1 s r e f 1 X r e f 1 2 ) = j κ ( F r e f X r e f F r e f + ) ,
F r e f + = exp ( j κ d r e f ) s r e f F r e f ,
s r e f = κ r e f ( 1 + s r e f 1 X r e f 1 2 ) + j γ r e f 1 ( 1 s r e f 1 X r e f 1 2 ) κ r e f ( 1 + s r e f 1 X r e f 1 2 ) j γ r e f 1 ( 1 s r e f 1 X r e f 1 2 ) ,
s r e f = j κ γ r e f 1 s j κ + γ r e f 1 s = exp ( j 2 ϕ s ) ,
C o v e r S i d e
F r e f + 1 ( 1 + c r e f 1 X r e f + 1 2 ) = F r e f + X r e f + F r e f ,
γ r e f + 1 F r e f + 1 ( 1 c r e f + 1 X r e f + 1 2 ) = j κ ( F r e f + X r e f F r e f ) ,
F r e f = exp ( j κ d r e f ) c r e f F r e f + ,
c r e f = κ r e f ( 1 + c r e f + 1 X r e f + 1 2 ) + j γ r e f + 1 ( 1 c r e f + 1 X r e f + 1 2 ) κ r e f ( 1 + c r e f + 1 X r e f + 1 2 ) j γ r e f + 1 ( 1 c r e f + 1 X r e f + 1 2 ) ,
c r e f = j κ γ r e f + 1 c j κ + γ r e f + 1 c = exp ( j 2 ϕ c ) .
exp [ j 2 ( κ d r e f ϕ c ϕ s ) ] = 1 .
κ d r e f ϕ c ϕ s = m π .

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