Abstract

We present an experimental and numerical study of electromagnetic wave propagation in one-dimensional (1D) and two-dimensional (2D) systems composed of periodic arrays of dielectric scatterers. We demonstrate that there are regions of frequency for which the waves are exponentially attenuated for all propagation directions. These regions correspond to band gaps in the calculated band structure, and such systems are termed photonic band-gap (PBG) structures. Removal of a single scatterer from a PBG structure produces a highly localized defect mode, for which the energy density decays exponentially away from the defect origin. Energy-density measurements of defect modes are presented. The experiments were conducted at 6–20 GHz, but we suggest that they may be scaled to infrared frequencies. Analytic and numerical solutions for the band structure and the defect states in 1D structures are derived. Applications of 2D PBG structures are briefly discussed.

© 1993 Optical Society of America

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References

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  1. R. Dalichaouch, “Classical wave localization in one- and two-dimensional disordered structures at microwave frequencies,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1990).
  2. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
    [Crossref] [PubMed]
  3. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
    [Crossref] [PubMed]
  4. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [Crossref] [PubMed]
  5. J. Maynard, Phys. Rev. Lett. 62, 188 (1989).
  6. A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984).
  7. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” submitted to Phys. Rev. Lett.
  8. N. Kroll, D. Smith, and S. Schultz, “Photonic band-gap structures: a new approach to accelerator cavities,” presented at the Advanced Accelerator Concepts Workshop, Port Jefferson, New York.
  9. A. A. Maradudin, Department of Physics, University of California, Irvine, Irvine, Calif. 92717 (personal communication, 1992).
  10. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
    [Crossref]
  11. M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
    [Crossref]
  12. R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
    [Crossref]

1991 (5)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
[Crossref]

1989 (1)

J. Maynard, Phys. Rev. Lett. 62, 188 (1989).

1987 (1)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Armstrong, J. P.

R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
[Crossref]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” submitted to Phys. Rev. Lett.

Dalichaouch, R.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
[Crossref]

R. Dalichaouch, “Classical wave localization in one- and two-dimensional disordered structures at microwave frequencies,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1990).

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

Joannopoulos, J. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” submitted to Phys. Rev. Lett.

Kroll, N.

N. Kroll, D. Smith, and S. Schultz, “Photonic band-gap structures: a new approach to accelerator cavities,” presented at the Advanced Accelerator Concepts Workshop, Port Jefferson, New York.

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

Maradudin, A. A.

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

A. A. Maradudin, Department of Physics, University of California, Irvine, Irvine, Calif. 92717 (personal communication, 1992).

Maynard, J.

J. Maynard, Phys. Rev. Lett. 62, 188 (1989).

McCall, S. L.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
[Crossref]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” submitted to Phys. Rev. Lett.

Platzman, P. M.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
[Crossref]

Plihal, M.

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” submitted to Phys. Rev. Lett.

Schultz, S.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
[Crossref]

N. Kroll, D. Smith, and S. Schultz, “Photonic band-gap structures: a new approach to accelerator cavities,” presented at the Advanced Accelerator Concepts Workshop, Port Jefferson, New York.

Shambrook, A.

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Sheng, Ping

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Smith, D.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

N. Kroll, D. Smith, and S. Schultz, “Photonic band-gap structures: a new approach to accelerator cavities,” presented at the Advanced Accelerator Concepts Workshop, Port Jefferson, New York.

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984).

Nature (London) (1)

R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature (London) 354, 53 (1991).
[Crossref]

Opt. Commun. (1)

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Phys. Rev. B (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

Phys. Rev. Lett. (4)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

J. Maynard, Phys. Rev. Lett. 62, 188 (1989).

Other (5)

A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984).

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” submitted to Phys. Rev. Lett.

N. Kroll, D. Smith, and S. Schultz, “Photonic band-gap structures: a new approach to accelerator cavities,” presented at the Advanced Accelerator Concepts Workshop, Port Jefferson, New York.

A. A. Maradudin, Department of Physics, University of California, Irvine, Irvine, Calif. 92717 (personal communication, 1992).

R. Dalichaouch, “Classical wave localization in one- and two-dimensional disordered structures at microwave frequencies,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1990).

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Figures (7)

Fig. 1
Fig. 1

Top, calculated first five band gaps (the frequencies of the band-edge states) for a 1D lattice of slabs as a function of dielectric constant. The slabs have a lattice constant of d = 2 cm with a filling factor of f = 0.5 (slab width a = 1 cm). The dashed curves indicate those frequencies at which the symmetric and the antisymmetric standing modes are shifted in frequency by the same amount from the free-space unperturbed value as a function of dielectric constant. Hence the dashed curves intersect the band-edge states whenever the band gaps go to zero. These curves prove useful for the understanding of the band-edge and defect symmetries of Fig. 2. Bottom, depiction of one section of an infinite periodic lattice. In each region the solution to Maxwell’s equation yields left- and right-traveling waves.

Fig. 2
Fig. 2

Top, calculated defect modes in the first five band gaps of the 1D periodic lattice. As in Fig. 1, the lattice constant is d = 2 cm and f = 0.5. The symmetry of the first three defect modes is indicated: note the change in symmetry of the third mode on either side of the gap closing. The defect modes correspond to the removal of a single slab as depicted (bottom). Defect states will be either symmetric or antisymmetric about the origin of the defect (x = 0). The defect-mode frequencies are found by matching the standing-wave solutions in the defect region to the evanescent solutions in the semi-infinite lattice region.

Fig. 3
Fig. 3

Numerical calculation of the defect modes for d = 2 cm, f = 0.5 for 4 bands, 18 k vectors/band, and 37 reciprocal lattice vectors s = z for the slabs. Band-gap widths, positions, and defect frequencies all correspond to those found from the analytical calculation (Fig. 2). The three circled defect modes are those that have frequencies within the band gaps.

Fig. 4
Fig. 4

Computed band structure for the 2D triangular lattice along the symmetry directions indicated. Only the first six bands are plotted, showing two PBG’s; cylinder = 9, a/d = 0.375; 107 reciprocal lattice vectors are used.

Fig. 5
Fig. 5

Schematic representation of the scattering system (not to scale). 1, Bottom and sides of scattering chamber (made as one piece); 2, inlet and outlet waveguide ports; 3, dielectric cylinders placed on a square lattice. A missing cylinder is the defect. 4, Microwave absorber; 5, cover plate with holes for sampling electric energy density; 6, a tuned microwave probe, which is placed in the sampling holes in the coverplate and whose output is measured by a homodyne detector at two phases 90° apart.

Fig. 6
Fig. 6

Transmitted power through the triangular lattice versus frequency. (a) The incident power is along the [1, 0] direction of the lattice. (b) The incident power is along the ⌈0, 1⌉ direction. Both (a) and (b) show attenuation in the region that corresponds to the second frequency band gap. (c) Power once again is directed along the [1, 0] direction, but a cylinder has been removed from the lattice. Note the appearance of a defect mode, which has moved into the center of the gap.

Fig. 7
Fig. 7

Two mappings of the defect mode along the directions indicated in the insets. These data demonstrate that this defect mode is localized to within a few lattice spacings.

Equations (36)

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2 E ( x , t ) x 2 = ( x ) c 2 2 E ( x , t ) t 2 ,
E ( x , t ) = E ( x ) exp ( - i ω t ) ,
- 2 E ( x ) x 2 - ( x ) - 1 c 2 ω 2 E ( x ) = ω 2 c 2 E ( x ) .
E ( x + d ) = exp ( i μ d ) E ( x ) ,
Region I E I ( x ) = A exp ( i k x ) + B exp ( - i k x ) , Region II E II ( x ) = f exp ( i q x ) + g exp ( - i q x ) , Region III E III ( x ) = C exp ( i k x ) + D exp ( - i k x ) ,
E I ( x ) = E II ( x ) x = - a / 2 , E I ( x ) = E II ( x ) x = - a / 2 , E II ( x ) = E III ( x ) x = + a / 2 , E II ( x ) = E II ( x ) x = + a / 2 .
[ C D ] = M [ A B ] ,
M = [ exp ( - i k a ) [ cos ( q a ) + i ( k q + q k ) sin ( q a ) 2 ] i ( q k - k q ) sin ( q a ) 2 - i ( q k - k q ) sin ( q a ) 2 exp ( i k a ) [ cos ( q a ) - i ( k q + q k ) sin ( q a ) 2 ] ] .
[ C exp ( i k d ) D exp ( - i k d ) ] = exp ( i μ d ) [ A B ] .
[ C exp ( i k d ) D exp ( - i k d ) ] = [ exp ( i k d ) 0 0 exp ( - i k d ) ] [ C D ] = T [ C D ] .
( T M - exp ( i μ d ) I ) [ A B ] = 0.
cos ( μ d ) = cos [ k ( d - a ) ] cos ( q a ) - 1 2 ( k q + q k ) sin [ k ( d - a ) ] sin ( q a ) .
σ cos h ( γ d ) = cos [ k ( d - a ) ] cos ( q a ) - 1 2 ( k q + q k ) sin [ k ( d - a ) ] sin ( q a ) .
E II ( x ) cos ( k x ) ,             symmetric states ,
E II ( x ) sin ( k x ) ,             antisymmetric states .
M 11 - σ exp ( γ d ) exp ( - i k d ) = M 12 , M 22 - σ exp ( γ d ) exp ( + i k d ) = M 21 ,
tan [ k ( d - a ) ] = - tan ( q a ) 2 { ( + 1 ) ± ( - 1 ) cos ( k d ) cos [ k ( d - a ) ] } .
E ( x ) = E μ ( x ) exp ( i μ x ) ,
E μ ( x + d ) = E μ ( x ) .
E μ ( x ) = G E G μ exp ( i G x ) ,
( x ) = 1 + G G exp ( i G x ) .
c 2 ω μ , G 2 μ + G 2 E G μ = G [ δ G G + G - G ] E G μ .
G - G = a d ( s - 1 ) sin ( G - G a / 2 ) G - G a / 2 .
( x ) = per ( x ) + def ( x ) ,
2 E ( x ) x 2 + ω 2 c 2 [ per ( x ) - 1 ] E ( x ) + ω 2 c 2 E ( x ) = - ω 2 c 2 def ( x ) E ( x ) .
E ( x ) = μ , n f μ , n E μ , n ( x ) ,
E μ , n * ( x ) per ( x ) E μ , n ( x ) d V = C μ , n δ μ μ δ n n ,
μ , n [ defect volume d V E μ , n * ( x ) def ( x ) E μ , n ] f μ , n = ( ω μ , n 2 - ω 2 ω 2 ) f μ , n ,
G - G = 2 π ( a d ) 2 J 1 ( G - G a ) G - G a ,
cos ( μ d ) = cos ( k d ) - U ω 2 c sin ( k d ) .
G - G = ( s - 1 ) ( a / d ) ,
( μ + G 2 - ω 2 c 2 ) E G = ( s - 1 ) a d ω 2 c 2 G E G .
1 = ( s - 1 ) a d n ( ω d / 2 π c ) 2 | μ d 2 π + n | 2 - ( ω d 2 π c ) 2 ,
1 = ( s - 1 ) a d ω d 4 π c n 1 [ ω d / ( 2 π c ) + μ d / ( 2 ) ] + n + 1 [ ω d / ( 2 π c ) - μ d / ( 2 π ) - n
cot ( π x ) = 1 π Lim N n = - N N 1 x - n ,
1 = - ( s - 1 ) a ω 4 c [ cot ( ω + μ ) d 2 + cot ( ω - μ ) d 2 c ] .

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