## Two-dimensional local density of states in two-dimensional photonic crystals

Optics Express, Vol. 8, Issue 3, pp. 191-196 (2001)

http://dx.doi.org/10.1364/OE.8.000191

Acrobat PDF (4769 KB)

### 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.

© Optical Society of America

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 , **11**28–33 (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]

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]

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]

*ρ*(

**r**,

*ω*)) is calculated from the electric Green’s tensor

**G**

^{e}according to [10]

**G**

^{e}(

**r**,

**r**

_{s};

*ω*) is the 3×3 electric field Green’s tensor at the field position

**r**corresponding to a current source at

**r**

_{s}of frequency

*ω*. Each column of

**G**

^{e}represents the components of an electric field vector [

*u*=

*x,y,z*axes respectively. For a 2D in-plane problem, the polarizations decouple and the fields are specified by a single, longitudinal components:

*V*=

*E*

_{z}in the case of TM or

*E*

_{||}polarization, and by

*V*=

*H*

_{z}for TE of

*H*

_{‖}polarization. For TM polarization,

*V*

_{z}is determined directly by the the solution of

*n*(

**r**) denotes the refractive index. For TE polarization, the process is less direct and requires the solution of two scalar problems for

*H*

_{z}satisfying

**u**=

*x̂*,

*ŷ*. Here,

**G**

^{e}is characterized by a 2×2 tensor the elements of which are calculated according to (

*G*

_{xu},

*G*

_{yu})=-

*i*(

*ẑ*×∇

*kn*

^{2}).

*N*

_{c}parallel, non-overlapping dielectric cylinders of radii

*a*

_{l}, refractive indices

*n*

_{l}, centered at positions

*c*

_{l}in a medium with refractive index

*n*

_{b}=1. More details of the method are given in [11] and here we outline the approach, focusing on its attractive computational features. In the vicinity of the

*I*

^{th}cylinder, we expand fields in

*local coordinates*(

*r*

_{l},

*θ*

_{l}) with origin at

*c*

_{l}to facilitate the imposition of boundary conditions. For either polarization,

*V*is expressed in exterior

*r*

_{l}>

*al*and interior rl<al multipole expansions

**C**with

**B**and

**D**. Note that the terms with the

*D*coefficients are associated with possible interior sources. In vector notation, with

**B**

^{l}=[

**A**and

**D**), we may express the boundary conditions in terms of cylindrical harmonic reflection

*R*

^{l}=[

**T**

^{l}=[

**B**

^{l}are determined from the field identity (6)that expresses the regular part of the field (

**A**

^{l}) in terms of fields scattered by all other cylinders (

**B**

^{J},

*j*≠

*l*) and real sources associated with the source terms arising in the wave equations. The field identity, whose derivation is given elsewhere [11], reduces to the partitioned system

**S**

^{lj}characterize the multipole contributions due to sources associated with each scatterer (and follow from Graf’s addition theorem for Bessel functions), while the elements of

**Q**

^{l}and

**K**

^{l}are the multipole coefficients of the real, external and internal sources, respectively. The Green’s function is then formed from the exterior form and the interior (for cylinder

*l*) form of

*V*:

*V*

_{0}is the solution of the field problem in the absence of scatterers and, for TM polarization, is given by

*G*

_{0}=

*kn*|

**r**-

**r**

*s*|)/(4

*i*), the Green’s function for a homogeneous medium of refractive index

*n*=

*n*

_{b}and

*n*=

*n*

_{l}in the exterior and interior forms respectively. The

*X*

^{ext}and

*X*

^{int}are switch functions, with

*X*

^{ext}(

**r**,

**r**

_{s})=1 for a source exterior to all cylinders,

*X*

^{int}(

**r**

_{l},

**r**

_{s})=1 for a source interior to cylinder

*l*, and vanishing for all other source locations. The accuracy of the method is governed by the number of terms that are retained in the field expansions (4). All calculations below have relative accuracies better than 10

^{-4}, requiring 11 basis functions (i.e.

*m*∈[-5

5. S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news , **11**28–33 (2000). [CrossRef]

5. S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news , **11**28–33 (2000). [CrossRef]

*n*

_{b}) differs from unity, the solution may be obtained from (7) by rescaling the wave number

*k*→

*kri*

_{b}and the refractive indices of the cylinders

*nl*→

*nl/nb*.

*n*

_{c}, although we stress that neither restriction is necessary. The cylinders, with

*n*

_{c}=3, are arranged in a square lattice with period

*d*and normalized radius

*a*/

*d*=0.3, corresponding to an area fraction of 28.3 percent. The corresponding infinite structure for

*TM*polarization has a band gap for 2.986<λ/

*d*<3.771.

*x*

_{s}

*, y*

_{s})=(0, 7,3) radiating into a cluster of

*N*

_{c}=81 cylinders, while Fig. 2 shows a section of Fig. 1 along the line

*x*=0. For a wavelength in the gap (left panel, Fig. 1), the field decreases exponentially into the crystal, while in the pass band there is no such behavior. Note the divergence of the magnitude of the Green’s function (associated with the singularity of its real part) at the source point. Also note that in the pass band, the Green’s function has pronounced interference minima (Fig. 2), but in general oscillates about a trend line that is relatively flat.

*ρπc*

^{2}/2

*ω*=0.25. The contrast in the LDOS between points inside and outside of the cylinders is of order 5×10

^{4}. However, other wavelengths show enhanced emission from subsets of the cylinders (right panel). From the video clips (see Fig. 5), we see that as the band gap is entered from the short wavelength side, the LDOS initially decreases within the cylinders at the centre of the cluster, gradually evolving into the surrounding matrix. As the wavelength increases, the region of suppressed emission increases and approaches the edge of the cluster. At the long wavelength edge of the gap the high values of the LDOS emerge from inside the cylinders. For wavelengths greater than λ/

*d*=3.8, there is oscillatory character in the LDOS and, while the physics of this behavior is not yet clear, it is likely to be associated with Fabry-Perot interference effects between layers, or with resonances of, or between, individual cylinders. Some further investigation of this appears necessary.

*n*

_{l}=1) with radii

*a/d*=0.48 in a high index matrix (

*n*

_{b}=√13) for both pass band and gap wavelengths. For this geometry, the LDOS exhibits six-fold symmetry (rather than four-fold symmetry of previous examples), together with strong interference and diffraction effects. We note further that for TE polarization the LDOS is discontinuous at the cylinder surfaces, whereas for TM polarization it is continuous.

^{4}and, indeed, the video clips show complex and interesting behavior that merits further investigation.

## References and links

1. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. |

3. | K. Busch and S. John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett. |

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 |

5. | S. Fan and J.D. Joannopoulos, “Photonic crystals: towards large-scale integration of optical and optoelectronic circuits,” Optics & photonics news , |

6. | J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic Band Gap Guidance in Optical Fibers,” Science |

7. | R. Spirk, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. |

8. | S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Technology |

9. | A. Moroz, “Minima and maxima of the local density of states for one-dimensional periodic systems,” Europhys. Lett. |

10. | G.S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. Parts I–III.” Phys. Rev. A , |

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 |

13. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals”, Phys. Rev. B |

14. | B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals”, J. Opt. Soc. Am. A |

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

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Focus Issue: Photonic bandgap calculations

**History**

Original Manuscript: November 13, 2000

Published: January 29, 2001

**Citation**

Ara Asatryan, Sebastien Fabre, Kurt Busch, Ross McPhedran, Lindsay Botten, Martijn de Sterke, and Nicolae A. Nicorovici, "Two-dimensional local density of states in two-dimensional
photonic crystals," Opt. Express **8**, 191-196 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-191

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### References

- 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]
- 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]
- S. Fan, and J.D. Joannopoulos,"Photonic crystals: towards large-scale integration of optical and optoelectronic circuits," Optics & photonics news, 11 28-33 (2000). [CrossRef]
- 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]
- R. Spirk, B. A. van Tiggelen, A. Lagendijk, "Optical emission in periodic dielectrics," Europhys. Lett. 35, 265-270 (1996). [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. Moroz, "Minima and maxima of the local density of states for one-dimensional periodic systems," Europhys. Lett. 46, 419-424 (1999). [CrossRef]
- G. S. Agarwal,"Quantum electrodynamics in the presence of dielectrics and conductors. Parts I-III," Phys. Rev. A 11, 230-264 (1975). [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.
- 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]
- 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]
- B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A 17, 1012-1020 (2000). [CrossRef]
- J. D. Joanopoulos, R. D. Meade and J. N. Winn, "Photonic Crystals: Molding the Flow of Light," (Princeton University Press. Princeton, NJ, 1995).

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