Abstract

We calculate the two-dimensional local density of states (LDOS) for two-dimensional photonic crystals composed of a finite cluster of circular cylinders of infinite length. The LDOS determines the dynamics of radiation sources embedded in a photonic crystal. We show that the LDOS decreases exponentially inside the crystal for frequencies within a photonic band gap of the associated infinite array and demonstrate that there exist “hot” and “cold” spots inside the cluster even for wavelengths inside a gap, and also for wavelengths corresponding to pass bands. For long wavelengths the LDOS exhibits oscillatory behavior in which the local density of states can be more than 30 times higher than the vacuum level.

© 2001 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  3. K. Busch and S. John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
    [CrossRef]
  4. O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
    [CrossRef] [PubMed]
  5. S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news,  1128–33 (2000).
    [CrossRef]
  6. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476–1478 (1998)
    [CrossRef] [PubMed]
  7. R. Spirk, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
    [CrossRef]
  8. S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Technology 17, 1931–1943 (1999).
    [CrossRef]
  9. A. Moroz, “Minima and maxima of the local density of states for one-dimensional periodic systems,” Europhys. Lett. 46, 419–424 (1999).
    [CrossRef]
  10. G.S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. Parts I–III.” Phys. Rev. A,  11, 230–264 (1975).
    [CrossRef]
  11. A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.
  12. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  13. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
    [CrossRef]
  14. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals”, J. Opt. Soc. Am. A 17, 1012–1020 (2000).
    [CrossRef]
  15. J. D. Joanopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals: Molding the Flow of Light,” Princeton University Press. Princeton, NJ, 1995).

2000 (2)

S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news,  1128–33 (2000).
[CrossRef]

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals”, J. Opt. Soc. Am. A 17, 1012–1020 (2000).
[CrossRef]

1999 (4)

S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Technology 17, 1931–1943 (1999).
[CrossRef]

A. Moroz, “Minima and maxima of the local density of states for one-dimensional periodic systems,” Europhys. Lett. 46, 419–424 (1999).
[CrossRef]

K. Busch and S. John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

1998 (2)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476–1478 (1998)
[CrossRef] [PubMed]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

1996 (2)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

R. Spirk, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1975 (1)

G.S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. Parts I–III.” Phys. Rev. A,  11, 230–264 (1975).
[CrossRef]

Agarwal, G.S.

G.S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. Parts I–III.” Phys. Rev. A,  11, 230–264 (1975).
[CrossRef]

Asatryan, A. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476–1478 (1998)
[CrossRef] [PubMed]

Botten, L.C.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476–1478 (1998)
[CrossRef] [PubMed]

Busch, K.

K. Busch and S. John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Technology 17, 1931–1943 (1999).
[CrossRef]

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.

Dapkus, P.D.

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Enoch, S.

Fan, S.

S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news,  1128–33 (2000).
[CrossRef]

Gralak, B.

Joannopoulos, J.D.

S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news,  1128–33 (2000).
[CrossRef]

Joanopoulos, J. D.

J. D. Joanopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals: Molding the Flow of Light,” Princeton University Press. Princeton, NJ, 1995).

John, S.

K. Busch and S. John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Technology 17, 1931–1943 (1999).
[CrossRef]

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Kim, I.

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476–1478 (1998)
[CrossRef] [PubMed]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Lagendijk, A.

R. Spirk, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

Lee, R.K.

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Martijn de Sterke, C.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.

McPhedran, R. C.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.

Meade, R. D.

J. D. Joanopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals: Molding the Flow of Light,” Princeton University Press. Princeton, NJ, 1995).

Moroz, A.

A. Moroz, “Minima and maxima of the local density of states for one-dimensional periodic systems,” Europhys. Lett. 46, 419–424 (1999).
[CrossRef]

Nicorovici, N. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

O’Brien, J. D.

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Painter, O.

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Russel, P. St. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476–1478 (1998)
[CrossRef] [PubMed]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Scalora, M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Scherer, A.

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Spirk, R.

R. Spirk, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Tayeb, G.

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

van Tiggelen, B. A.

R. Spirk, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

Winn, J. N.

J. D. Joanopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals: Molding the Flow of Light,” Princeton University Press. Princeton, NJ, 1995).

Yablonovitch, E.

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Europhys. Lett. (2)

R. Spirk, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

A. Moroz, “Minima and maxima of the local density of states for one-dimensional periodic systems,” Europhys. Lett. 46, 419–424 (1999).
[CrossRef]

J. Lightwave Technology (1)

S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Technology 17, 1931–1943 (1999).
[CrossRef]

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

Optics & photonics news (1)

S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news,  1128–33 (2000).
[CrossRef]

Phys. Rev. A (1)

G.S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. Parts I–III.” Phys. Rev. A,  11, 230–264 (1975).
[CrossRef]

Phys. Rev. B (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

K. Busch and S. John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

Science (2)

O. Painter, R.K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P.D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science 282, 1476–1478 (1998)
[CrossRef] [PubMed]

Other (2)

A. A. Asatryan, K. Busch, R. C. McPhedran, L.C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length”, Phys. Rev. E submitted.

J. D. Joanopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals: Molding the Flow of Light,” Princeton University Press. Princeton, NJ, 1995).

Supplementary Material (3)

» Media 1: MOV (2489 KB)     
» Media 2: MOV (2187 KB)     
» Media 3: MOV (2409 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Plot of log10 |G| for a wavelength in the gap λ/d=3.5 (left panel) and in the pass band λ/d=2.5 (right panel). The black dotes indicate the positions of the source which have coordinates (0, 7.3) and the black circles indicate the cylinders.

Fig. 2.
Fig. 2.

Sections through Fig. 1 at x=0: green line for λ/d=3.5, blue line for λ/d=2.5. The red line is the Green function for a line source without scatterers for λ/d=3.5. The position of the source is indicated as x.

Fig. 3.
Fig. 3.

log10(ρπc 2=2ω) for TM polarization for λ=d=3.5 (left panel) and for λ=d=2.5 (right panel).

Fig. 4.
Fig. 4.

log10(ρπc 2=2ω) for TM polarization for λ=d=3.16 with enhanced emission inside the cylinders (left panel), and for λ=d=4.11 with enhanced emission in some of the cylinders.

Fig. 5.
Fig. 5.

Quicktime movies of LDOS for TM polarization. Left panel: top.mov (2.49MB) directly from above. Middle panel: above.mov (2.2 MB) from above. Right panel: below.mov (2.4MB) from below.

Fig. 6.
Fig. 6.

log10(ρπc 2/2ωnb2) for TE polarization for λ=d=2.25 in a band gap(left panel) and λ/d=3.0 in the pass band(right panel).

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

ρ ( r ; ω ) = 2 ω π c 2 Im Tr [ G e ( r , r ; ω ) ] .
( 2 + k 2 n 2 ( r ) ) V z ( r ) = δ ( r r s ) ,
( 2 + k 2 n 2 ( r ) ) V z u ( r ) = i z ̂ · [ × u δ ( r r s ) ] k ,
V ( r l ) = { m = [ A m l J m ( k r l ) + B m l H m ( 1 ) ( k r l ) ] e i m θ l , m = [ C m l J m ( k n l r l ) + D m l H m ( 1 ) ( k n l r l ) ] e i m θ l ,
B l = R l A l + T l D l .
B 1 R l j l S lj B j = R l Q l + T l K l
V ( r , r s ) = { χ ext ( r , r s ) V 0 ( r , r s ) + l = 1 N c m = B m l H m ( 1 ) ( k r l ) e i m θ l , χ int ( r l , r s ) V 0 ( r , r s ) + m = C m l J m ( k n l r l ) e i m θ l .

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