Abstract

The properties of surface modes present in the junction of two different one-dimensional photonic crystals placed in series are studied to assess the possibility of applying them in the design of narrow-bandpass filters. To design bandpass filters that are similar in many respects to multiple-cavity Fabry–Perot filters, we also consider the coupling conditions for these surface modes for multiple photonic crystals placed in series to form a multiple-junction system.

© 2004 Optical Society of America

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References

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    [CrossRef]
  7. J. Mizuguchi, Y. Tanaka, S. Tamura, and N. Notomi, "Focusing of light in a three dimensional cubic photonic crystal," Phys. Rev. B 67, 075109 (2003).
    [CrossRef]
  8. F. 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).
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  13. F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59 (1999) 15112�??15120.
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  14. Xianmin Yi, Pochi Yeh, and John Hong, "Nonexistence of well-confined surface waves on obliquely cut surfaces of one-dimensional photonic crystals," J. Opt. Soc. Am. B 18 (2001) 352�??357.
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  15. Francisco Villa and J. A. Gaspar-Armenta, "Electromagnetic surface waves: photonic crystal-photonic crystal interface," Opt. Commun. 223, 109�??115 (2003).
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  16. Jorge A. Gaspar Armenta and Francisco Villa, �??Photonic surface-wave excitation: photonic crystal-metal interface,�?? J. Opt. Soc. Am. B 20, 2349�??2354 (2003).
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  18. R. R. Austin, �??Narrow band interference light filter,�?? U.S. Patent 3,528,726 (Sept. 15, 1970).
  19. H. A. Macleod, Thin Film Optical Filters (McGraw Hill, New York, 1989).
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    [CrossRef] [PubMed]
  21. J. A. Gaspar-Armenta, Francisco Villa, and T. López-Ríos,"Surface waves in finite one-dimensional photonic crystals: mode coupling," Opt. Commun. 216 379�??394 (2003).
    [CrossRef]
  22. H. A. Macleod, "Challenges in the design and production of narrow band filters for optical fiber telecommunications," in Optical and Infrared Thin Films, Michael L. Fulton ed., Proc. SPIE 4094, 46�??57 (2000).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

W. M. Robertson and M. S. May, �??Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,�?? Appl. Phys. Lett. 74 1800�??1802 (1999).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

Francisco Villa and J. A. Gaspar-Armenta, "Electromagnetic surface waves: photonic crystal-photonic crystal interface," Opt. Commun. 223, 109�??115 (2003).
[CrossRef]

J. A. Gaspar-Armenta, Francisco Villa, and T. López-Ríos,"Surface waves in finite one-dimensional photonic crystals: mode coupling," Opt. Commun. 216 379�??394 (2003).
[CrossRef]

F. 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]

Opt. Lett. (1)

Phys. Rev. B (2)

J. Mizuguchi, Y. Tanaka, S. Tamura, and N. Notomi, "Focusing of light in a three dimensional cubic photonic crystal," Phys. Rev. B 67, 075109 (2003).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59 (1999) 15112�??15120.
[CrossRef]

Proc. SPIE (1)

H. A. Macleod, "Challenges in the design and production of narrow band filters for optical fiber telecommunications," in Optical and Infrared Thin Films, Michael L. Fulton ed., Proc. SPIE 4094, 46�??57 (2000).

Other (6)

R. R. Austin, �??Narrow band interference light filter,�?? U.S. Patent 3,528,726 (Sept. 15, 1970).

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

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).

S. Johnson and J. D. Joannopoulos, Photonic Crystals: the Road from Theory to Practice (Kluwer Academic Publishers, Boston 2002)

K. Sakoda, Optical Properties of Photonic Crystals (Springer Verlag, Heidelberg, 2001).

Pochi Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (15)

Fig. 1.
Fig. 1.

Periodic multilayer of symmetric periods. η 0 and ηs stand for the optical admittance of the incidence and the transmission media, respectively. In this multilayer the indicated symmetric period p/2q p/2 indicates that layers of a material with the refractive index np have the physical thickness dp /2.

Fig. 2.
Fig. 2.

Equivalent functions versus reduced frequency under normal incidence (β̄=0). The system considered is np =2.1, nq =1.444, and npdp =nqdq =λ 0/4, with λ 0=1516.2 nm. The imaginary part of the equivalent phase thickness given in the lower graph was multiplied by a factor of eight to make the function more clearly visible on the same scale as the real part of the same function. The real part of δe is given in units of π radians.

Fig. 3.
Fig. 3.

Imaginary part of equivalent functions of two C-PCs. The parameters are the same as in previous example.

Fig. 4.
Fig. 4.

Multiple values of the cosine function of the equivalent phase function.

Fig. 5.
Fig. 5.

Reflectance as a function of reduced frequency for single-junction system with a C-PC of six periods. The integer numbers in the ω̄ axis represent the order where the bandgaps are centered.

Fig. 6.
Fig. 6.

Transmittance as a function of wavelength. The peak indicated by a solid curve corresponds to 1DPCs of 10 periods each. The peak on the dotted curve illustrates a mode resulting from a system with C-PCs of 14 periods.

Fig. 7.
Fig. 7.

Band structure for the TE case. The nonshaded bands represent the regions of high reflectance where the surface modes can be present. Surface electromagnetic excitations are indicated by dotted curves. One example of a surface mode is shown in the inset, corresponding to the point indicated by a diamond for β̄=2.5, ω̄=1.717. If we consider an incident medium with refractive index n 0=2.0, then the mode appears at θ 0=46.7°. The dotted-dashed line represents the light line for vacuum.

Fig. 8.
Fig. 8.

Band structure for the TM case. As in Fig. 7, the surface modes are indicated by dashed curves. One example of surface mode under this polarization is shown in the inset corresponding to the point indicated by a diamond for β̄=4, ω̄=2.608. If we consider the incident medium with a refractive index n 0=2.0, then the mode appears at θ 0=50.07°. The points where the bandgaps narrow and close represent the Brester points (spheres).

Fig. 9.
Fig. 9.

Transmittance as a function of wavelength. The peak on the dotted curve illustrates coupled modes resulting from a system (H/2 L H/2)12 (L/2 H L/2)25(H/2 L H/2)12. In this case the peaks are almost overlapping. The double peak indicated by the solid curve corresponds to the system (H/2 L H/2)12 (L/2 H L/2)15(H/2 L H/2)12.

Fig. 10.
Fig. 10.

Electric field magnitude as a function of the optical thickness normalized to λ 0/4. This field corresponds to the system (H/2 L H/2)12 (L/2 H L/2)25(H/2 L H/2)12 with vacuum as incidence and transmission media.

Fig. 11.
Fig. 11.

Electric field magnitude as a function of the optical thickness normalized to λ 0/4. This field corresponds to the system (H/2 L H/2)12 (L/2 H L/2)15(H/2 L H/2)12 with vacuum as incidence and transmission media.

Fig. 12.
Fig. 12.

Separation of two coupled modes as a function of the number of periods of the middle 1DPC in the system.

Fig. 13.
Fig. 13.

Electric field magnitude for a triple-junction filter (H/2 L H/2)9 (L/2 H L/2)18(H/2 L H/2)18(L/2 H L/2)9. The materials are those used in previous examples, and the design wavelength was weakly modified to 1516.18 nm to center the peak at 1550 nm.

Fig. 14.
Fig. 14.

Transmittance versus wavelength of the triple-junction filter given in Fig. 13.

Fig. 15.
Fig. 15.

Group-delay dispersion as a function of wavelength of the system given in Fig. 13. The vertical axis is given in squared picoseconds.

Equations (45)

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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 ¯ zj d j
η j = { y k ¯ zj ω ¯ TE polarization yn j 2 ω ¯ k ¯ zj TM polarization
k ¯ zj = n j 2 ω ¯ 2 β ¯ 2
M ( η e , δ e ) = [ cos ( δ e ) i η e sin ( δ e ) i η e sin ( δ e ) cos ( δ e ) ] ,
M 11 = cos ( δ p ) cos ( δ q ) ρ + sin ( δ p ) sin ( δ q ) ,
M 22 = M 11 ,
M 21 = i η p [ sin ( δ p ) cos ( δ q ) + ρ + cos ( δ p ) sin ( δ q ) ρ sin ( δ q ) ] ,
M 12 = i η p [ 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 ) .
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 ) ] .
δ e = δ ˜ e .
cos ( δ ˜ e ) = cos ( δ p ) cos ( δ q ) ρ + sin ( δ p ) sin ( δ q ) .
δ e = δ ˜ e + 2 k π ,
η e η ˜ e = η p η q .
M ˜ 12 = i η p [ sin ( δ p ) cos ( δ q ) + ρ + cos ( δ p ) sin ( δ q ) ρ sin ( δ q ) ] .
M 21 η p = η q M ˜ 12 .
M 21 = i η e sin ( δ e ) ,
M ˜ 12 = i η ˜ e sin ( δ ˜ e ) ,
η 0 + η ˜ e + i ( η 0 η ˜ e η e + η e ) tan ( σ δ e ) = 0 ,
η e = η 0 ,
η e = η ˜ e .
cos ( δ ) = ( η p η q η p + η q ) 2 .
δ δ H = δ L = π 2 λ 0 λ ,
δ = π 2 ω ¯ .
λ 1 = λ 0 2 π cos 1 ( n H n L n H + n L ) 2 .
δ m = [ 2 m 1 + ( 1 ) m ] π 2 ( 1 ) m δ 1 ,
δ m = π 2 λ 0 λ m .
λ m = λ 0 2 m 1 + ( 1 ) m ( 1 ) m 2 π cos 1 ( n H n L n H + n L ) 2 .
[ B C ] = [ cos ( σ ˜ δ ˜ e ) i η e sin ( σ ˜ δ ˜ ) i η e sin ( σ ˜ δ ˜ ) cos ( σ ˜ δ ˜ ) ] [ cos ( σ δ e ) i η e sin ( σ δ e ) i η e sin ( σ δ e ) cos ( σ δ e ) ] [ 1 η s ] ,
η 0 B = C .
( η s 2 η e 2 ) [ tan ( σ ˜ δ e ) tan ( σ δ e ) ] = 0 .
2 η ˜ e + i ( η ˜ e 2 η e + η e ) tan ( σ δ e ) = 0 .
η e 4 + 2 η H η L a η e 2 + ( η H η L ) 2 = 0 ,
a = 1 + e σ Im ( δ e ) 1 e σ Im ( δ e )
η e ± = η H η L a ± a 2 1 .
η e ± η e ( ω ¯ 1 ) + a ( ω ¯ ± ω ¯ 1 ) ,
α i π 2 η H Δ ( 1 + Δ 2 ) 2 ( 1 + Δ ) ,
Δ = η H η L η H + η L .
e Im ( δ e ) b b 2 1 ,
b = Δ 4 ( 1 + ρ + ) ρ + .
a = 1 b b 2 1 1 + b + b 2 1 .
ω ¯ ± ω ¯ 1 ± η e + η e η e + + η e .

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